Modelagem-de-Filas-de-Espera-Analise-e-Otimizacao-com-Winston-Albright

Fourth Edition Practical Management Science Winston Albright 12 A WAITINGLINE EXAMPLE As indicated earlier a mathematical model is a set of mathematical relationships that represent or approximate a real situation Models that simply describe a situation are called descriptive models Other models that suggest a desirable course of action are called optimization models To get started consider the following simple example of a mathematical model It begins as a descriptive model but then becomes an optimization model Consider a convenience store with a single cash register The manager of the store suspects that customers are waiting too long in line at the checkout register and that these excessive waiting times are hurting business Customers who have to wait a long time might not come back and potential customers who see a long line might not enter the store at all Therefore the manager builds a mathematical model to help understand the problem The manager wants the model to reflect the current situation at the store but it should also suggest improvements to the current situation A Descriptive Model This example is a typical waiting line or queueing problem Such problems are studied in detail in Chapter 13 The manager first wants to build a model that reflects the current situation at the store Later he will alter the model to predict what might make the situation better To describe the current situation the manager realizes that there are two important inputs to the problem 1 the arrival rate of potential customers to the store and 2 the rate at which customers can be served by the single cashier Clearly as the arrival rate increases andor the service rate decreases the waiting line will tend to increase and each customer will tend to wait longer in line In addition more potential customers will probably decide not to enter at all These latter quantities length of waiting line time in line per customer fraction of customers who dont enter are commonly referred to as outputs The manager believes he has some understanding of the relationship between the inputs and the outputs but he is not at all sure how to quantify this relationship This is where a mathematical model is useful By making several simplifying assumptions about the nature of the arrival and service process at the store as discussed in Chapter 13 the inputs can be related to the outputs In some cases when the model is sufficiently simple it is possible to write an equation for an output in terms of the inputs For example in one of the simplest queueing models if A is the arrival rate of customers per minute S is the service rate of customers per minute and W is the average time a typical customer waits in line assuming that all potential customers enter the store the following relationship can be derived mathematically W A SS A 11 This relationship is intuitive in one sense It correctly predicts that as the service rate S increases the average waiting time W decreases It also predicts that as the arrival rate A increases the average waiting time W increases Finally if the arrival rate is just barely less than the service rate that is the difference S A is positive but very small the average waiting time becomes quite large This model requires that the arrival rate be less than the service rate otherwise Equation 11 makes no sense In many other models there is no such equation that relates outputs to inputs or if there is it is too complex for the level of this book Nevertheless there may still be a mathematical procedure for calculating outputs from inputs and it may be possible to implement this procedure in Excel This is the case for the convenience store problem Again by making certain simplifying assumptions including the assumption that potential Overview of Applications by Discipline ECONOMICS Estimating sensitivity of demand to price 4554 Pricing problems 361377 431437 Assessing a utility function 526528 Estimating demand for products 904 Subway token hoarding 772 FINANCEAND ACCOUNTING Cost projections 2731 Finding a breakeven point 3139 Calculating NPV for production capacity decision 5560 Portfolio management 177182 353354 401406 452455 660662 Pension fund management 182186 Financial planning 215219 647652 Arbitrage opportunities in oil pricing 220224 Currency trading 225 Capital budgeting 299305 Estimating stock betas 407412 Stock hedging 419420 New product development 475476 547 643647 692699 Bidding for a government contract 485491 623627 653657 Investing with risk aversion 527531 Land purchasing decision 548 Liquidity risk management 621623 Estimating warranty costs 627632 Retirement planning 652657 Modeling stock prices 657658 Pricing options 658664 Investing for college 710 Bond investment 711 Revenue management at casinos 841842 Estimating costs 853860 862870 Forecasting overhead costs 905 HUMAN RESOURCES AND HEALTH CARE Fighting HIVAIDS 2122 DEA in the hospital industry 188194 Assigning MBA students to teams 474 Selecting a job 16221631 Drug testing for athletes 506513 MARKETING Determining an advertising schedule 135144 383387 1631610 16181621 Estimating an advertising response function 379383 Retail pricing 431437 Estimating a sales response function 448451 Cluster analysis of large cities 456460 Classifying subscribers of the WSJ 461464 New product marketing 514523 531532 Valuing a customer 667671 Reducing churn 671676 Marketing and selling condos 676680 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it OPERATIONS MANAGEMENT Queueing problems 47 788806 811814 817830 Ordering problems newsvendor 2327 573610 619 736739 Ordering with quantity discounts 4044 Manufacturing operations 6768 329334 Product mix problems 7094 100107 171176 306318 Production scheduling 108117 152162 442447 620 632637 Production inventory management 133134 16371639 Scheduling workers 145151 Aggregate planning 152162 Gasoline oil blending 163170 212214 Logistics problems 227228 229240 248256 Assigning workers to jobs 241243 Assigning school buses to routes 243247 Finding a shortest route 257261 Equipment replacement 261266 Airline crew scheduling 267272 Airline flight scheduling 273280 Global manufacturing and distribution 288289 Motor carrier selection 290292 Airline hub location 319324 Locating plants and warehouses 324328 388392 Cutting stock problems 335339 Plant expansion and retooling 350351 Selecting telecommunication carriers 352 Telephone call processing 836837 Railway planning 421422 Loading a gas truck 438442 Traveling salesperson problem 464467 Determining tradeoff between profit and pollution 16151618 Airline boarding strategies 551552 Demings funnel experiment 637641 Global supply chain decisions 713714 Economic order quantity models 718734 Ordering decisions with demand uncertainty 740754 Production planning in fashion industry 755760 Managing supply chain inventory 760764 Reducing work in progress 773774 Operations at banks 838839 Scheduling multiple projects 151152 15251528 Project scheduling with CPM 1581524 15301534 Forecasting problems 848853 879883 885895 906 SPORTS AND GAMES Rating NFL teams 393397 Playing craps 682685 NCAA basketball tournament 685689 MISCELLANEOUS Investment in US Space Systems 293294 Prioritizing projects at NASA 161162 Biotechnical engineering 549550 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing previous editions changes to current editions and alternate formats please visit wwwcengagecomhighered to search by ISBN author title or keyword for materials in your areas of interest Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Practical Management Science Wayne L Winston Kelley School of Business Indiana University S Christian Albright Kelley School of Business Indiana University With Cases by Mark Broadie Graduate School of Business Columbia University Lawrence L Lapin San Jose State University William D Whisler California State University Hayward 4TH EDITION Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Practical Management Science Fourth Edition Wayne L Winston S Christian Albright Vice President of Editorial Business Jack W Calhoun EditorinChief Joe Sabatino Senior Acquisitions Editor Charles McCormick Jr Senior Developmental Editor Laura Bofinger Ansara Editorial Assistants Nora Heink Courtney Bavaro Marketing Communications Manager Libby Shipp Marketing Manager Adam Marsh Senior Content Project Manager Holly Henjum Media Editor Chris Valentine Senior Buyer Manufacturing Miranda Klapper Production Service MPS Limited a Macmillan Company Art Director Stacy Jenkins Shirley Cover Designer Lou Ann Thesing Cover Image iStock Photo Rights Acquisitions Specialist John Hill 2012 2009 SouthWestern Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced transmitted stored or used in any form or by any means graphic electronic or mechanical including but not limited to photocopying recording scanning digitizing taping web distribution information networks or information storage and retrieval systems except as permitted under Section 107 or 108 of the 1976 United States Copyright Act without the prior written permission of the publisher ExamView is a registered trademark of eInstruction Corp Windows is a registered trademark of the Microsoft Corporation used herein under license Macintosh and Power Macintosh are registered trademarks of Apple Computer Inc used herein under license Library of Congress Control Number 2011922240 Student Edition PKG ISBN13 9781111531317 Student Edition PKG ISBN10 1111531315 Student Edition ISBN13 9781111531270 Student Edition ISBN10 1111531277 SouthWestern 5191 Natorp Boulevard Mason OH 45040 USA Cengage Learning products are represented in Canada by Nelson Education Ltd For your course and learning solutions visit wwwcengagecom Purchase any of our products at your local college store or at our preferred online store wwwcengagebraincom For product information and technology assistance contact us at Cengage Learning Customer Sales Support 18003549706 For permission to use material from this text or product submit all requests online at wwwcengagecompermissions Further permissions questions can be emailed to permissionrequestcengagecom Printed in the United States of America 1 2 3 4 5 6 7 15 14 13 12 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it To Mary my wonderful wife best friend and constant companion And to our Welsh Corgi Bryn who still just wants to play ball SCA To my wonderful family Vivian Jennifer and Gregory WLW Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it S Christian Albright got his BS degree in Mathematics from Stanford in 1968 and his PhD degree in Operations Research from Stanford in 1972 Since then he has been teaching in the Operations Decision Technologies Department in the Kelley School of Business at Indiana University He has taught courses in management science computer simulation and statistics to all levels of business students undergraduates MBAs and doctoral students He has also taught courses on database analysis to the US Army He has published over 20 articles in leading operations research journals in the area of applied probability and he has authored several books including Practical Management Science Data Analysis and Decision Making Data Analysis for Managers Spreadsheet Modeling and Applications and VBA for Modelers He jointly developed StatTools a statistical addin for Excel with the Palisade Corporation His current interests are in spreadsheet modeling and the development of VBA applications in Excel as well as Web programming with Microsofts NET technology On the personal side Chris has been married to his wonderful wife Mary for 40 years They have one son Sam who is currently finishing a law degree at Penn Law School Chris has many interests outside the academic area They include activities with his family especially traveling with Mary going to cultural events at Indiana University playing golf and tennis power walking and reading And although he earns his livelihood from statistics and management science his real passion is for playing classical music on the piano Wayne L Winston is Professor of Operations Decision Technologies in the Kelley School of Business at Indiana University where he has taught since 1975 Wayne received his BS degree in Mathematics from MIT and his PhD degree in Operations Research from Yale He has written the successful textbooks Operations Research Applications and Algorithms Mathematical Programming Applications and Algorithms Simulation Modeling Using RISK Data Analysis and Decision Making and Financial Models Using Simulation and Optimization Wayne has published over 20 articles in leading journals and has won many teaching awards including the schoolwide MBA award four times He has taught classes at Microsoft GM Ford Eli Lilly BristolMyers Squibb Arthur Andersen Roche PriceWaterhouseCoopers and NCR His current interest is in showing how spreadsheet models can be used to solve business problems in all disciplines particularly in finance and marketing Wayne enjoys swimming and basketball and his passion for trivia won him an appearance several years ago on the television game show Jeopardy where he won two games He is married to the lovely and talented Vivian They have two children Gregory and Jennifer About the Authors Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it v Preface xi 1 Introduction to Modeling 1 2 Introduction to Spreadsheet Modeling 21 3 Introduction to Optimization Modeling 67 4 Linear Programming Models 133 5 Network Models 227 6 Optimization Models with Integer Variables 293 7 Nonlinear Optimization Models 353 8 Evolutionary Solver An Alternative Optimization Procedure 421 9 Decision Making Under Uncertainty 475 10 Introduction to Simulation Modeling 551 11 Simulation Models 621 12 Inventory Models 713 13 Queueing Models 773 14 Regression and Forecasting Models 841 References 907 Index 913 Online Chapters 15 Project Management 151 16 Multiobjective Decision Making 161 Brief Contents Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it vii Preface xi CHAPTER 1 Introduction to Modeling 1 11 Introduction 3 12 A WaitingLine Example 4 13 Modeling Versus Models 7 14 The SevenStep Modeling Process 8 15 A Great Source for Management Science Applications Interfaces 14 16 Why Study Management Science 14 17 Software Included with This Book 16 18 Conclusion 18 CHAPTER 2 Introduction to Spreadsheet Modeling 21 21 Introduction 22 22 Basic Spreadsheet Modeling Concepts and Best Practices 22 23 Cost Projections 27 24 Breakeven Analysis 31 25 Ordering with Quantity Discounts and Demand Uncertainty 40 26 Estimating the Relationship between Price and Demand 45 27 Decisions Involving the Time Value of Money 55 28 Conclusion 61 Appendix Tips for Editing and Documenting Spreadsheets 65 CHAPTER 3 Introduction to Optimization Modeling 67 31 Introduction 68 32 Introduction to Optimization 69 33 A TwoVariable Product Mix Model 70 34 Sensitivity Analysis 83 35 Properties of Linear Models 94 Contents 36 Infeasibility and Unboundedness 97 37 A Larger Product Mix Model 100 38 A Multiperiod Production Model 108 39 A Comparison of Algebraic and Spreadsheet Models 118 310 A Decision Support System 118 311 Conclusion 121 Appendix Information on Solvers 128 CASE 31 Shelby Shelving 129 CASE 32 Sonoma Valley Wines 131 CHAPTER 4 Linear Programming Models 133 41 Introduction 134 42 Advertising Models 135 43 Worker Scheduling Models 145 44 Aggregate Planning Models 152 45 Blending Models 163 46 Production Process Models 171 47 Financial Models 177 48 Data Envelopment Analysis DEA 188 49 Conclusion 195 CASE 41 Amarco Inc 212 CASE 42 American Office Systems Inc 215 CASE 43 Lakefield Corporations Oil Trading Desk 220 CASE 44 Foreign Currency Trading 225 CHAPTER 5 Network Models 227 51 Introduction 228 52 Transportation Models 229 53 Assignment Models 241 54 Other Logistics Models 248 55 Shortest Path Models 257 56 Network Models in the Airline Industry 267 57 Conclusion 281 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 88 Cluster Analysis 455 89 Discriminant Analysis 461 810 The Traveling Salesperson Problem 464 811 Conclusion 469 CASE 81 Assigning MBA Students to Teams 474 CHAPTER 9 Decision Making under Uncertainty 475 91 Introduction 476 92 Elements of Decision Analysis 478 93 The PrecisionTree AddIn 492 94 Bayes Rule 505 95 Multistage Decision Problems 509 96 Incorporating Attitudes toward Risk 525 97 Conclusion 533 CASE 91 Jogger Shoe Company 547 CASE 92 Westhouser Paper Company 548 CASE 93 Biotechnical Engineering 549 CHAPTER 10 Introduction to Simulation Modeling 551 101 Introduction 552 102 Probability Distributions for Input Variables 554 103 Simulation and the Flaw of Averages 573 104 Simulation with BuiltIn Excel Tools 576 105 Introduction to RISK 587 106 The Effects of Input Distributions on Results 603 107 Conclusion 612 CASE 101 Ski Jacket Production 619 CASE 102 Ebony Bath Soap 620 CHAPTER 11 Simulation Models 621 111 Introduction 623 112 Operations Models 623 113 Financial Models 642 114 Marketing Models 667 115 Simulating Games of Chance 682 116 An Automated Template for RISK Models 690 viii Contents CASE 51 International Textile Company Ltd 288 CASE 52 Optimized Motor Carrier Selection at Westvaco 290 CHAPTER 6 Optimization Models with Integer Variables 293 61 Introduction 294 62 Overview of Optimization with Integer Variables 295 63 Capital Budgeting Models 299 64 FixedCost Models 306 65 SetCovering and LocationAssignment Models 319 66 Cutting Stock Models 335 67 Conclusion 340 CASE 61 Giant Motor Company 350 CASE 62 Selecting Telecommunication Carriers to Obtain Volume Discounts 352 CHAPTER 7 Nonlinear Optimization Models 353 71 Introduction 354 72 Basic Ideas of Nonlinear Optimization 355 73 Pricing Models 361 74 Advertising Response and Selection Models 379 75 Facility Location Models 388 76 Models for Rating Sports Teams 393 77 Portfolio Optimization Models 398 78 Estimating the Beta of a Stock 407 79 Conclusion 412 CASE 71 GMS Stock Hedging 419 CHAPTER 8 Evolutionary Solver An Alternative Optimization Procedure 421 81 Introduction 422 82 Introduction to Genetic Algorithms 425 83 Introduction to Evolutionary Solver 426 84 Nonlinear Pricing Models 431 85 Combinatorial Models 438 86 Fitting an SShaped Curve 448 87 Portfolio Optimization 452 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents ix 117 Using TopRank with RISK for Powerful Modeling 691 118 Conclusion 699 CASE 111 College Fund Investment 710 CASE 112 Bond Investment Strategy 711 CHAPTER 12 Inventory Models 713 121 Introduction 714 122 Categories of Inventory Models 715 123 Types of Costs in Inventory Models 717 124 Economic Order Quantity EOQ Models 718 125 Probabilistic Inventory Models 736 126 Ordering Simulation Models 749 127 Supply Chain Models 754 128 Conclusion 765 CASE 121 Subway Token Hoarding 772 CHAPTER 13 Queueing Models 773 131 Introduction 774 132 Elements of Queueing Models 776 133 The Exponential Distribution 779 134 Important Queueing Relationships 784 135 Analytical SteadyState Queueing Models 787 136 Approximating ShortRun Behavior Analytically 809 137 Queueing Simulation Models 815 138 Conclusion 831 CASE 131 Catalog Company Phone Orders 836 CASE 132 Pacific National Bank 838 CHAPTER 14 Regression and Forecasting Models 841 141 Introduction 843 142 Overview of Regression Models 844 143 Simple Regression Models 848 144 Multiple Regression Models 861 145 Overview of Time Series Models 874 146 Moving Averages Models 878 147 Exponential Smoothing Models 884 148 Conclusion 897 CASE 141 Demand for French Bread at Howies Bakery 904 CASE 142 Forecasting Overhead at Wagner Printers 905 CASE 143 Arrivals at the Credit Union 906 Online Chapters CHAPTER 15 Project Management 151 151 Introduction 152 152 The Basic CPM Model 154 153 Modeling Allocation of Resources 1514 154 Models with Uncertain Activity Times 1530 155 A Brief Look at Microsoft Project 1535 156 Conclusion 1538 CHAPTER 16 Multiobjective Decision Making 161 161 Introduction 162 162 Goal Programming 163 163 Pareto Optimality and TradeOff Curves 1613 164 The Analytic Hierarchy Process AHP 1622 165 Conclusion 1632 CASE 161 Play Time Toy Company 1637 References 907 Index 913 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it xi Practical Management Science provides a spreadsheet based exampledriven approach to management science Our initial objective in writing the book was to reverse negative attitudes about the course by making the subject relevant to students We intended to do this by imparting valuable modeling skills that students can appreciate and take with them into their careers We are very gratified by the success of the first three editions The book has exceeded our initial objectives We are especially pleased to hear about the success of the book at many other colleges and universities around the world The acceptance and excitement that has been generated has motivated us to revise the book and make the fourth edition even better When we wrote the first edition management science courses were regarded as irrelevant or uninteresting to many business students and the use of spreadsheets in management science was in its early stages of development Much has changed since the first edition was published in 1996 and we believe that these changes are for the better We have learned a lot about the best practices of spreadsheet modeling for clarity and communication We have also developed better ways of teaching the materials and we understand more about where students tend to have difficulty with the concepts Finally we have had the opportunity to teach this material at several Fortune 500 companies including Eli Lilly Price Waterhouse Coopers General Motors Tomkins Microsoft and Intel These companies through their enthusiastic support have further enhanced the realism of the examples included in this book Our objective in writing the first edition was very simplewe wanted to make management science relevant and practical to students and professionals This book continues to distinguish itself in the market in four fundamental ways Teach by Example The best way to learn modeling concepts is by working through examples and solving an abundance of problems This active learning approach is not new but our text has more fully developed this approach than any book in the field The feedback we have received from many of you has confirmed the success of this pedagogical approach for management science Integrate Modeling with Finance Marketing and Operations Management We integrate modeling into all functional areas of business This is an important feature because the majority of business students major in finance and marketing Almost all competing textbooks emphasize operations managementrelated examples Although these examples are important and many are included in the book the application of modeling to problems in finance and marketing is too important to ignore Throughout the book we use real examples from all functional areas of business to illustrate the power of spreadsheet modeling to all of these areas At Indiana University this has led to the development of two advanced MBA electives in finance and marketing that build upon the content in this book The inside front cover of the book illustrates the integrative applications contained in the book Teach Modeling Not Just Models Poor attitudes among students in past management science courses can be attributed to the way in which they were taught emphasis on algebraic formulations and memorization of models Students gain more insight into the power of management science by developing skills in modeling Throughout the book we stress the logic associated with model development and we discuss solutions in this context Because real problems and real models often include limitations or alternatives we include many Modeling Issues sections to discuss these important matters Finally we include Modeling Problems in most chapters to help develop these skills Provide Numerous Problems and Cases Whereas all textbooks contain problem sets for students to practice we have carefully and judiciously crafted the problems and cases contained in this book Each chapter contains four types of problems SkillBuilding Problems SkillExtending Problems Modeling Problems and Cases Most of the problems following sections of chapters ask students to extend the examples in the preceding section The endof chapter problems then ask students to explore Preface Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it xii Preface new models Selected solutions are available to students who purchase the Student Solution Files online and are denoted by the secondcolor numbering of the problem Solutions for all of the problems and cases are provided to adopting instructors In addition shell files templates are available for most of the problems for adopting instructors The shell files contain the basic structure of the problem with the relevant formulas omitted By adding or omitting hints in individual solutions instructors can tailor these shell files to best meet the individualspecific needs of their students New to the Fourth Edition The main reason for the fourth edition was the introduction of Excel 2010 Admittedly this is not nearly as much a game changer as Excel 2007 but it does provide new features that ought to be addressed In addition once we were motivated by Excel 2010 to revise the book we saw the possibility for a lot of other changes that will hopefully improve the book Important changes to the fourth edition include the following Undoubtedly the biggest change in Excel 2010 the one that affects our book the most is the new Solver addin Frontline Systems the develop of Solver swapped the previous version of Solver for what we used to provide separately Premium Solver Now all Excel 2010 users have essentially the old Premium Solver Therefore we no longer provide an academic version of Premium Solver with the book As discussed in detail in the optimization chapters the new Solver provides several important enhancements 1 It has a nicer more compact user interface 2 it appears to work better giving many fewer conditions for linear model not satisfied messages for models that are indeed linear and 3 it includes Evolutionary Solver which we continue to use for difficult nonsmooth models in Chapter 8 To make the book somewhat shorter we moved the old chapters 9 Multiobjective Decision Making and 15 Project Management online renumbering the former as 16 Based on user reports these are two of the lesscovered chapters in the book but they are still available online if you want to use them Of course the remaining chapters have been renumbered accordingly Both chapters are found on the Instructor web site and the students Essential Resource Web site Instructions for access to these sites are described later in this preface In the first optimization chapter Chapter 3 we replaced the introductory twovariable diet model with a simpler twovariable product mix model Then we follow it up with a larger version of the same basic product mix model We believe this should make the introduction to optimization easier for instructors to teach and for students to follow In the regression and forecasting chapter now numbered Chapter 14 we discontinued the use of the Analysis Toolpak and jumped directly into the Palisade StatTools addin We believe that StatTools is vastly superior to Analysis Toolpak so we decided to take full advantage of it One of the main strengths of this book has always been its numerous problems However some of these had been around for over a decade and were either totally out of date or required better data Therefore we deleted some problems added some brand new ones and changed the input data for many others We have included a file for instructors PMS4e Problem Databasexlsx that lists all of the changes One last change didnt make it into the book but we are offering it on a limited trial basis to instructors Specifically we have added several large optimization models more changing cells than Solver can handle to the instructor materials They are under Extra subfolders in the Example Files folders The motivation for these additions is to let students experience what it is like for managers who do not have access to optimization software What kinds of heuristics might they use Will these heuristics get anywhere near optimality For comparison we have provided optimal solutions If nothing else we believe these examples might make students appreciate the true power of optimization software such as Solver The Essential Resource Web Site for Students The tools offered with the fourth edition of Practical Management Science extend beyond the textbook Students purchasing a new textbook receive access to the Essential Resource Web site that accompanies this book For students who do not purchase a new textbook there are other access and product options available at CengageBraincom Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Preface xiii Software We continue to be very excited about offering the most comprehensive suite of software ever available with a management science textbook The commercial value of the software available with this text exceeds 1000 if purchased directly This software is available free with new copies of the fourth edition The following Palisade software is available from a link that is provided on the Essential Resource Web site Palisades DecisionTools Suite including the awardwinning RISK PrecisionTree StatTools TopRank RISKOptimizer NeuralTools and Evolver This software is not available with any competing textbook and comes in an educational version that is only slightly scaled down from the expensive commercial version StatTools replaces Albrights StatPro addin that came with the second edition If you are interested StatPro is still freely available from httpwwwkelleyiuedualbrightbooks although it will not be updated for Excel 2007 or 2010 For more information about the Palisade Corporation and the DecisionTools Suite visit Palisades Web site at httpwwwpalisadecom To make sensitivity analysis useful and intuitive we continue to provide Albrights SolverTable addin which is also freely available from httpwwwkelleyiuedualbrightbooks SolverTable provides data tablelike sensitivity output for optimization models that is easy to interpret Example Files Data Sets Problem Files and Cases Also on the Essential Resource Web site are the Excel files for all of the examples in the book as well as many data files required for problems and cases As in previous editions there are two versions of the example files a completed version and a template to get students started Because this book is so example and problem oriented these files are absolutely essential For instructors there is a third annotated version of each example file that provides even more insights into the model How to Access the Essential Resource Web Site Student Access Students are given access instructions to the Essential Resource Web site via the bindin card in new editions of their book Go to http logincengagebraincom click on Create an Account and then in the space provided enter the unique access code found on the access card bound in your new book Students who do not buy a new printed textbook may search CengageBraincom for other purchase options such as CourseMate which offers an eBook format of the book with access to the Essential Resource Web site Instructor Access Go to httplogincengagecom Use your current user account to sign in If you do not have an account follow the screen instructions to create one Verification of instructor status takes 24 to 48 hours for new accounts Once you are logged in type this textbooks ISBN number in the search box The ISBN is found on the back of your textbook You are then presented with selection options to add to your Bookshelf such as the Instructor Web site Student Essentials Resource Web site and CourseMate if applicable to your class Your selections will show up on your account home page for access to instructor and student materials Ancillaries Instructor Materials Adopting instructors can obtain the Instructors Resource CD IRCD from your regional Cengage Learning sales representative The IRCD includes PMS4e Problem Databasexlsx file which contains information about all problems in the book and the correspondence between them and those in the previous edition Solution files in Excel format for all of the problems and cases in the book and solution shells templates for selected problems in the modeling chapters PowerPoint presentation files Test Bank in Word format and now also in ExamView Testing Software Instructor ancillaries are also posted on the Instructor Web site Access instructions are described in the previous section Albright also maintains his own Web site at httpwwwkelleyiuedualbrightbooks Among other things this site includes errata for each edition Student Solutions Student Solutions for many of the oddnumbered problems indicated in the text with a colored box on the problem number are available in Excel format Students can purchase access to Student Solutions Files on CengageBraincom In the search window of Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it xiv Preface this Web site type in this books ISBN number found on the back cover of your book and hit enter A product page will show you Related Products you can purchase including the Student Solutions Companion VBA Book Soon after the first edition appeared we began using Visual Basic for Applications VBA the program ming language for Excel in some of our management science courses VBA allows you to develop decision support systems around the spreadsheet models An example appears at the end of Chapter 3 This use of VBA has been popular with our students and many instructors have expressed interest in learning how to do it For additional support on this topic a companion book VBA for Modelers 3e ISBN 1439079846 is available It assumes no prior experience in computer programming but it progresses rather quickly to the development of interesting and nontrivial applications The fourth edition of Practical Management Science depends in no way on this companion VBA book but we encourage instructors to incorporate some VBA into their management science courses This is not only fun but students quickly learn to appreciate its power If you are interested in adopting VBA for Modelers contact your local Cengage Learning representative Acknowledgments This book has gone through several stages of reviews and it is a much better product because of them The majority of the reviewers suggestions were very good ones and we have attempted to incorporate them We would like to extend our appreciation to Sue Abdinnour Wichita State University Robert Aboolian California State UniversitySan Marcos Mohammad Ahmadi University of Tennessee at Chattanooga Kelly Alvey Old Dominion University Jerry Bilbrey Anderson University Fattaneh Cauley Pepperdine University Gordon Corzine University of MassachusettsBoston Parthasarati Dileepan University of Tennessee at Chattanooga Ehsan Elahi University of MassachusettsBoston Kathryn Ernstberger Indiana University Southeast Levon R Hayrapetyan Houston Baptist University Max Peter Hoefer Pace University Harvey Iglarsh Georgetown University D K Kim Dalton State College Mary Kurz Clemson University Larry J LeBlanc Vanderbilt University Stephen Mahar University of North CarolinaWilmington James Morris University of WisconsinMadison Khosrow Moshirvaziri Caliornia State UniversityLong Beach Ozgur Ozluk San Francisco State University Susan Palocsay James Madison University Prakash P Shenoy University of Kansas Ekundayo Shittu Tulane University Steven Slezak California Polytechnic State UniversitySan Luis Obispo Christine Spencer University of Baltimore Robert Stoll Cleveland State University Charles Watts John Carroll University Yuri Yatsenko Houston Baptist University We would also like to thank two special people First we want to thank our previous editor Curt Hinrichs Although Curt has moved from this position and is no longer our editor his vision in the early years was largely responsible for the success of the first and second editions of Practical Management Science Second we were lucky to move from one great editor to another in Charles McCormick Jr Charles is a consummate professional he is both patient and thorough and his experience in the publishing business ensures that the tradition Curt started will be carried on In addition we would like to thank Marketing Manager Adam Marsh Senior Developmental Editor Laura Ansara Content Project Manager Holly Henjum Art Director Stacy Shirley Editorial Assistants Nora Heink and Courtney Bavaro and Project Manager at MPS Gunjan Chandola We would also enjoy hearing from youwe can be reached by email And please visit either of the following Web sites for more information and occasional updates httpwwwkelleyiuedualbrightbooks CengageBraincom Wayne L Winston winstonindianaedu S Christian Albright albrightindianaedu Bloomington Indiana January 2011 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 Introduction to Modeling C H A P T E R COMPLEX ALGORITHMS AND T HE SOFTO R APPR OACH SOLVE REALW ORLD PR OBLEMS A s you embark on your study of management science you might question the usefulness of quantitative methods to the real world A frontpage article in the December 311997edition of USA Today entitled Higher Math Delivers Formula for Success provides some convincing evidence of the applica bility of the methods you will be learning More recent evidence that math skills continue to be valuable can be found in the January 232006Business Week cover story Math Will Rock Your World You can find this article by doing a Web search for the title The subheading of the articleBusinesses turn to algo rithms to solve complex problems says it all Todays business problems tend to be very complexIn the pastmany managers and executives used a seat of the pants approach to solve problemsthat isthey used their business experience their intuitionand some thoughtful guesswork to solve problemsBut common sense and intuition go only so far in the solution of the complex problems busi nesses now face This is where management science modelsand the algo rithms mentioned in the title of the articleare so usefulWhen the methods in this book are implemented in userfriendly computer software packages that are applied to complex problemsthe results can be amazingRobert Crosswhose companyDFI Aeronomicssells algorithmbased systems to airlinesstates it suc cinctlyIts like taking raw information and spinning money out of it 1 Monkey Business Images2010Used under license from Shutterstockcom Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The methods in this book are powerful because they apply to so many problems and environments The article mentions the following success stories in which manage ment science has been applied others will be discussed throughout this book United Airlines installed one of DFIs systemswhich cost between 10 million and 20 millionUnited expected the system to add 50 million to 100 million annually to its revenues The Gap clothing chain uses management science to determine exactly how many employees should staff each store during the holiday rush Management science has helped medical researchers test potentially dangerous drugs on fewer people with better results IBM obtained a 93 million contract to build a computer system for the US Department of Energy that would do a onceimpossible task make exact realtime models of atomic blasts It won the contractand convinced the DOE that its sys tem was costeffectiveonly by developing management science models that would cut the processing time by half Hotelsairlinesand television broadcasters all use management science to implement a method called yield managementIn this methoddifferent prices are charged to different customersdepending on their willingness to payThe effect is that more customers are attracted and revenues increase Although most of this book describes how quantitative methods can be used to solve business problems solutions do not always need to be quantitatively based In a recent article Kimbrough and Murphy 2005 two academics located in Philadelphia describe how they were commissioned by the city to study the knowledge economy of the region and make recommendations on ways to improve its rate of growth Unlike most of the success stories chronicled in the Interfaces journal which is described in section 15 the authors state right away that they used no quantitative methods or mathematical models to develop recommendations for the city Instead they used a soft OR approach1 By this they imply that they used a systematic approach to formulate and solve their clients problem even though the approach does not employ quantitative methods Specifically Kimbrough and Murphy used two interrelated approaches in their study First using ideas in the management science and economics literature they developed a comprehensive framework for thinking about regional economic development This allowed them to identify the many factors that influence a regions economic vitality or lack thereof Second they interviewed a wide range of people from the region including researchers in science and technology business people government officials and acade mics Instead of asking these people what ought to be done they asked them to propose ideas or policy initiatives that might improve the regions economy As they state The results were gratifying The framework we developed focuses peoples thinking on prob lems bottlenecks and leverage points in the knowledge economy Asking for specific ideas produced a rich and constructive list of more than 50 promising realistic and detailed policy initiatives However the study went beyond brainstorming After conducting the interviews and analyzing the responses the authors made specific recommendations to their client 2 Chapter 1 Introduction to Modeling 1OR is an abbreviation for operations research another term for management science Over the years the two terms have become practically synonymous although some people in the field still prefer to be called manage ment scientists whereas others prefer to be called operations researchers Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it on initiatives that might be implemented to improve the knowledge economy Based on these recommendations the board of directors of Greater Philadelphia First adopted Six for Success a strategy that commits leaders to 1 attract more research dollars and expertise 2 implement strategies to accelerate science and technology 3 promote an entrepreneurial climate 4 launch a business marketing plan 5 leverage qualityoflife infrastructure and amenities and 6 streamline and rationalize businessoriented non profits Granted these ideas are not necessarily groundbreaking but they made sense to leaders in Philadelphia The important point is that they were developed through a systematic approach to solving a problemeven if it wasnt the quantitative approach discussed through most of this book 11 Introduction 3 11 INTRODUCTION The purpose of this book is to expose you to a variety of problems that have been solved successfully with management science methods and to give you experience in modeling these problems in the Excel spreadsheet package The subject of management science has evolved for more than 60 years and is now a mature field within the broad category of applied mathematics This book will emphasize both the applied and mathematical aspects of management science Beginning in this chapter and continuing throughout the rest of the book we discuss many successful management science applications where teams of highly trained people have implemented solutions to the problems faced by major compa nies and have saved these companies millions of dollars Many airlines and oil companies for example could hardly operate as they do today without the support of management sci ence In this book we will lead you through the solution procedure of many interesting and realistic problems and you will experience firsthand what is required to solve these problems successfully Because we recognize that most of you are not highly trained in mathematics we use Excel spreadsheets to solve problems which makes the quantitative analysis much more understandable and intuitive The key to virtually every management science application is a mathematical model In simple terms a mathematical model is a quantitative representation or ideal ization of a real problem This representation might be phrased in terms of mathemati cal expressions equations and inequalities or as a series of interrelated cells in a spreadsheet We prefer the latter especially for teaching purposes and we concentrate primarily on spreadsheet models in this book However in either case the purpose of a mathematical model is to represent the essence of a problem in a concise form This has several advantages First it enables managers to understand the problem better In par ticular the model helps to define the scope of the problem the possible solutions and the data requirements Second it allows analysts to employ a variety of the mathemati cal solution procedures that have been developed over the past half century These solu tion procedures are often computer intensive but with todays cheap and abundant computing power they are usually feasible Finally the modeling process itself if done correctly often helps to sell the solution to the people who must work with the system that is eventually implemented In this introductory chapter we begin by discussing a relatively simple example of a mathematical model Then we discuss the distinction between modeling and a collection of models Next we discuss a sevenstep modeling process that is used in essence if not in strict conformance in most successful management science applications Finally we discuss why the study of management science is valuable not only to large corporations but also to students like you who are about to enter the business world Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 4 Chapter 1 Introduction to Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove content at any time Figure 11 Descriptive Queueing Model for Convenience Store 1 Descriptive queueing model for convenience store 3 Inputs 4 Arrival rate customers per minute 05 5 Service rate customers per minute 04 6 Maximum customers before others go elsewhere 5 8 Outputs 9 Average number in line 222 10 Average time minutes spent in line 609 11 Percentage of potential arrivals who dont enter 271 Figure 12 Queueing Model with a Faster Service Rate 1 Descriptive queueing model for convenience store 3 Inputs 4 Arrival rate customers per minute 05 5 Service rate customers per minute 0556 6 Maximum customers before others go elsewhere 5 8 Outputs 9 Average number in line 141 10 Average time minutes spent in line 322 11 Percentage of potential arrivals who dont enter 126 Figure 13 Queueing Model with an Even Faster Service Rate 1 Descriptive queueing model for convenience store 3 Inputs 4 Arrival rate customers per minute 05 5 Service rate customers per minute 08 6 Maximum customers before others go elsewhere 5 8 Outputs 9 Average number in line 069 10 Average time minutes spent in line 142 11 Percentage of potential arrivals who dont enter 38 Figure 14 Queueing Model with Alternative Decisions 1 Decision queueing model for convenience store 3 Inputs Decision 1 Decision 2 Decision 3 4 Arrival rate customers per minute 05 05 05 5 Service rate customers per minute 04 0556 08 6 Maximum customers before others go elsewhere 5 5 5 8 Cost of extra person per hour 0 8 0 9 Cost of leasing new cash register per hour 0 0 11 10 Cost per customer per hour waiting in line 13 13 13 11 Cost per customer who doesnt enter the store 25 25 25 13 Outputs 14 Average number in line 222 141 069 15 Average time minutes spent in line 609 322 142 16 Percentage of potential arrivals who dont enter 271 126 38 18 Cost information 19 Cost of extra person per hour 0 8 0 20 Cost of leasing new cash register per hour 0 0 11 21 Cost per hour of waiting time 2887 1831 891 22 Cost per hour of lost customers 20329 9452 2852 24 Total cost per hour 23216 12082 4843 No tabular text present in this image for extraction model can be used at least not without modification to solve a companys real problem Unfortunately management science students have gotten the impression that all problems must be shoehorned into one of the textbook models The good news is that this emphasis on specific models has been changing in the past decade or two and our goal in this book is to continue that change Specifically this book stresses modeling not models The distinction between modeling and models will become clear as you proceed through the book Learning specific models is essentially a memoriza tion processmemorizing the details of a particular model such as the transportation model and possibly learning how to trick other problems into looking like a transportation model Modeling on the other hand is a process where you abstract the essence of a real problem into a model spreadsheet or otherwise Although many problems fall naturally into several categories successful modelers do not try to shoehorn each problem into one of a small number of wellstudied models Instead they treat each problem on its own merits and model it appropriately using all of the logical analytical or spreadsheet skills they have at their disposaland of course using their experience with previous models they have developed This way if they come across a problem that does not look exactly like anything they have ever seen they still have the skills and flexibility to model it successfully This doesnt mean you wont learn some classical models from management sci ence in this book in fact we will discuss the transportation model in linear programming the MM1 model in queueing the EOQ model in inventory and other classics These are important models that should not be ignored however you certainly do not have to mem orize these specific models They are simply a few of the many models you will learn how to develop The real emphasis throughout is on the modeling processhow a realworld problem is abstracted into a spreadsheet model of that problem We discuss this modeling process in more detail in the following section 14 THE SEVENSTEP MODELING PROCESS The discussion of the queueing model in section 12 presented some of the basic principles of management science modeling This section further expands on these ideas by charac terizing modeling as the following sevenstep process Step 1 Problem Definition The analyst first defines the organizations problem Defining the problem includes specify ing the organizations objectives and the parts of the organization that must be studied before the problem can be solved In the simple queueing model the organizations problem is how to minimize the expected net cost associated with serving customers Step 2 Data Collection After defining the problem the analyst collects data to estimate the value of parameters that affect the organizations problem These estimates are used to develop a mathematical model step 3 of the organizations problem and predict solutions step 4 In the conve nience store queueing example the manager needs to observe the arrivals and the checkout process to estimate the arrival rate A the service rate S and possibly other inputs Step 3 Model Development In the third step the analyst develops a model of the problem In this book we present many methods that can be used to model systems3 Models such as the equation for W 8 Chapter 1 Introduction to Modeling 3All of these models can generically be called mathematical models However because we implement them in spreadsheets we generally refer to them as spreadsheet models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it where an equation is used to relate inputs such as A and S to outputs such as W are called analytical models Most realistic applications are so complex however that an analytical model does not exist or is too complex to work with For example if the convenience store has more than one cash register and customers are allowed to join any line or jump from one line to another there is no tractable analytical modelno equation or system of equationsthat can be used to determine W from knowledge of A S and the number of lines When no tractable analytical model exists the analyst must often rely instead on a simulation model which approximates the behavior of the actual system Simulation models are covered in Chapters 10 and 11 Step 4 Model Verification The analyst now tries to determine whether the model developed in the previous step is an accurate representation of reality A first step in determining how well the model fits reality is to check whether the model is valid for the current situation As discussed previously to validate the equation for the waiting time W the manager might observe actual customer waiting times for several hours As we saw the equation for W predicts that when A 05 and S 04 the average customer spends 609 minutes in line Now suppose the manager observes that 120 customers spend a total of 750 minutes in line This indicates an average of 750120 625 minutes in line per customer Because 625 is reasonably close to 609 the managers observations lend credibility to the model In contrast if the 120 customers had spent 1200 minutes total in line for an average of 10 minutes per customer this would not agree very well with the models prediction of 609 minutes and it would cast doubt on the validity of the model Step 5 Optimization and Decision Making Given a model and a set of possible decisions the analyst must now choose the decision or strategy that best meets the organizations objectives We briefly discussed an optimization model for the convenience store example and we will discuss many other optimization models throughout the book Step 6 Model Communication to Management The analyst presents the model and the recommendations from the previous steps to the organization In some situations the analyst might present several alternatives and let the organization choose the best one Step 7 Model Implementation If the organization has accepted the validity and usefulness of the study the analyst then helps to implement its recommendations The implemented system must be monitored constantly and updated dynamically as the environment changes to ensure that the model enables the organization to meet its objectives Flowchart of Procedure and Discussion of Steps Figure 15 illustrates this sevenstep process As the arrows pointing down and to the left indicate there is certainly room for feedback in the process For example at various steps the analyst might realize that the current model is not capturing some key aspects of the real problem In this case the problem definition can be changed or a new model can be developed 14 The SevenStep Modeling Process 9 The following discussion explores these seven steps in more detail Step 1 Problem Definition Typically a management science model is initiated when an organization believes it has a problem Perhaps the company is losing money perhaps its market share is declining per haps its customers are waiting too long for serviceany number of problems might be evi dent The organization which we refer to as the client calls in a management scientist the analyst to help solve this problem4 In such cases the problem has probably already been defined by the client and the client hires the analyst to solve this particular problem As Miser 1993 and Volkema 1995 point out however the analyst should do some investigating before accepting the clients claim that the problem has been properly defined Failure to do so could mean solving the wrong problem and wasting valuable time money and energy For example Miser cites the experience of an analyst who was hired by the military to investigate overly long turnaround times between fighter planes landing and taking off again to rejoin the battle The military the client was convinced that the problem was caused by inefficient ground crewsif they worked faster turnaround times would cer tainly decrease The analyst nearly accepted this statement of the problem and was about to do classical timeandmotion studies on the ground crew to pinpoint the sources of their inefficiency However by snooping around he found that the problem lay elsewhere Specifically he learned that the trucks that refueled the planes were frequently late which in turn was due to the inefficient way they were refilled from storage tanks at another loca tion After this latter problem was solvedand its solution was embarrassingly simple the turnaround times decreased to an acceptable level without any changes on the part of the ground crews If the analyst had accepted the clients statement of the problem the real problem would never have been located or solved The moral of this story is clear If an analyst defines a problem incorrectly or too narrowly the solution to the real problem might never emerge In his article Volkema 1995 advocates spending as much time thinking about the problem and defining it properly as modeling and solving it This is undoubtedly good advice especially in realworld appli cations where problem boundaries are often difficult to define Step 2 Data Collection This crucial step in the modeling process is often the most tedious All organizations keep track of various data on their operations but the data are often not in the form the analyst requires In addition data are often stored in different places throughout the organization and in different formats Therefore one of the analysts first jobs is to gather exactly the right data and put the data into an appropriate and consistent format for use in the model 10 Chapter 1 Introduction to Modeling Problem definition Data collection Model development Model verification Possible feedback loops Optimization and decision making Model communication to management Model implementation Figure 15 Flowchart for the SevenStep Process 4Most organizations hire outside consultants sometimes academics to help solve problems However a number of large organizations employ a staff of management scientists who function as inside consultants It is important to solve the correct problem and defining that problem is not always easy The data collection step often takes the most time Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This typically requires asking questions of key people such as the cost accountants throughout the organization studying existing organizational databases and performing timeconsuming observational studies of the organizations processes In short it typically entails a lot of legwork In this book as in most management science textbooks we shield you from this datacollection process by supplying the appropriate data to develop and solve a model Although this makes the overall modeling process seem easier than it really is it is impractical in most class setting to have you go to companies and gather data In many cases it would not even be allowed for proprietary reasons Nevertheless we provide some insights with Where Do the Numbers Come From sections If nothing else these sections remind you that in real applications someone has to gather the necessary data Step 3 Model Development This step along with step 5 is where the analyst brings his or her quantitative skills into play After defining the clients problem and gathering the necessary data the analyst must develop a model of the problem Several properties are desirable for a good model First it should represent the clients real problem accurately If it uses a linear straightline func tion for costs when the real cost function is highly nonlinear curved the recommenda tions of the model can be very misleading Similarly if the model ignores an important constraint such as an upper bound on capacity its recommendations might not be possible to implement On the other hand the model should be as simple as possible Most good models where good really means useful capture the essence of the problem without getting bogged down in less important details They should be approximations of the real world not mirror images of every last detail Overly complex models are often of little practical use First overly complex models are sometimes too difficult to solve with the solution algorithms available Second complex models tend to be incomprehensible to clients If a client cannot understand a model the chances are that the models recommendations will never be implemented Therefore a good model should achieve the right balance between being too simple and too complex This is often much easier said than done Step 4 Model Verification This step is particularly important in real management science applications A client is much more likely to accept an analysts model if the analyst can provide some type of veri fication This verification can take several forms For example the analyst can use the model with the companys current values of the inputs If the models outputs are then in line with the outputs currently observed by the client the analyst has at least shown that the model can duplicate the current situation A second way to verify a model is to enter several sets of input values even if they are not the companys current input values and see whether the outputs from the model are reasonable One common approach is to use extreme values of the inputs to determine whether the outputs behave as they should For example for the convenience store queue ing model you could enter an extremely large service rate or a service rate just barely above the arrival rate in the equation for W In the first case you would expect the average waiting time to approach 0 whereas in the latter case you would expect it to become very large You can use equation 11 for W to verify that this is exactly what happens This provides another piece of evidence that the model is reasonable If the models outputs for certain inputs are not as expected there are two possible causes First the model could be a poor approximation of the actual situation In this case the analyst must refine the model until it lines up more accurately with reality Second the model might be fine but the analysts intuition might not be very good That is when 14 The SevenStep Modeling Process 11 Steps 3 and 5 developing and optimizing models are the steps emphasized most heavily in this book Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it asked what he or she thinks would happen if certain input values are used the analyst might provide totally wrong predictions In this case the fault lies with the analyst not the model Sometimes good models prove that peoples ability to predict outcomes in complex environments is lacking In such cases the verification step becomes harder because of political reasons office politics Step 5 Optimization and Decision Making After the problem has been defined the data has been collected and the model has been developed and verified it is time to use the model to recommend decisions or strategies In the majority of management science models this requires the optimization of an objective such as maximizing profit or minimizing cost The optimization phase is typically the most difficult phase from a mathematical standpoint Indeed much of the management science literature mostly from academics has focused on complex solution algorithms for various classes of models Fortunately this research has led to a number of solution algorithmsand computer packages that imple ment these algorithmsthat can be used to solve real problems The most famous of these is the simplex algorithm This algorithm which has been implemented by many commer cial software packages including Excels Solver is used on a daily basis to solve linear optimization models for many companies We take advantage of the simplex method in Chapters 3 through 5 Not all solution procedures find the optimal solution to a problem Many models are either too large or too complex to be solved exactly Therefore many complex problems use heuristic methods to locate good solutions A heuristic is a solution method that is guided by common sense intuition and trial and error to achieve a good but probably not optimal solution Some heuristics are quick and dirty whereas others are quite sophisti cated As models become larger and more complex good heuristics are sometimes the best that can be achievedand they are often perfectly adequate Step 6 Model Communication to Management The analyst must eventually communicate a model and its recommendations to the client To appreciate this step you must understand the large gap that typically exists between management science analysts and the managers of organizations Managers know their business but they often dont understand much about mathematics or mathematical modelseven spreadsheet implementations of these models The burden is therefore on the analyst to present the model in terms that nonmathematical people can understand other wise a perfectly good model might never see the light of day The best strategy for successful presentation is to involve key people in the organiza tion including top executives in the project from the beginning If these people have been working with the analyst helping to supply appropriate data and helping the analyst to understand the way the organization really works they are much more likely to accept the eventual model Step 6 therefore should really occur throughout the modeling process not just toward the end The analyst should also try to make the model as intuitive and userfriendly as possi ble Clients appreciate menudriven systems with plenty of graphics They also appreciate the ability to ask whatif questions and get answers quickly in a form that is easy to under stand This is one reason for developing spreadsheet models Although not all models can be developed in spreadsheets due to size andor complexity the spreadsheet approach in this book is an excellent choice whenever possible because most business people are com fortable with spreadsheets Spreadsheet packages support the use of graphics customized menus and toolbars data tables and other tools for whatif analyses and even macros that can be made transparent to users for running complex programs 12 Chapter 1 Introduction to Modeling A heuristic is a relatively simple solution method that often provides good but not necessarily optimal solutions Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Step 7 Model Implementation A real management science application is not complete until it has been implemented A successful implementation can occur only when step 6 has been accomplished That is the analyst must demonstrate the model to the client and the client must be convinced that the model adds real value and can be used by the people who need to use it For this reason the analysts job is not really complete until the system is up and running on a daily basis To achieve a successful implementation it isnt enough for management to accept the model the people who will run it every day must also be thoroughly trained to use it At the very least they should understand how to enter appropriate inputs run whatif analy ses and interpret the models outputs correctly If they conclude that the model is more trouble than its worth they might simply refuse to use it and the whole exercise will have been a waste of time An interesting trend as evidenced in many of the Interfaces articles discussed shortly is for analysts to build a userfriendly Excel front end for their clients even if the actual number crunching is performed behind the scenes in some nonExcel package Because many employees understand at least the basics of Excel a userfriendly front end makes the system much more attractive for daily use Many successful management science applications take on a life of their own after the initial implementation After an organization sees the benefits of a useful modeland of management science in generalit is likely to expand the model or create new models for uses beyond those originally intended Knowing that this is often the case the best analysts design models that can be expanded They try to anticipate problems the organization might face besides the current problem They also stay in contact with the organization after the initial implementation just in case the organization needs guidance in expanding the scope of the model or in developing new models This discussion of the sevenstep modeling process has taken an optimistic point of view by assuming that a successful study employs these seven steps in approximately this chronological order and that everything goes smoothly It does not always work out this way Numerous potential applications are never implemented even though the technical aspects of the models are perfectly correct The most frequent cause is a failure to commu nicate The analyst builds a complex mathematical model but the people in the organiza tion dont understand how it works and are reluctant to use it Also company politics can be a models downfall especially if the model recommends a course of action that top management simply does not want to followfor whatever reasons Even for applications that are eventually implemented the analyst doesnt always pro ceed through the seven steps exactly as described in this section He or she might backtrack considerably throughout the process For example based on a tentative definition of the problem a model is built and demonstrated to management Management says that the model is impressive but it doesnt really solve the companys problem Therefore the ana lyst returns to step 1 redefines the problem and builds a new model or modifies the orig inal model In this way the analyst generates several iterations of some or all of the seven steps before the project is considered complete The Model as a Beginning Not an End This book places heavy emphasis on developing spreadsheet models which is step 3 of the sevenstep modeling process We lead you stepbystep through the model development process for many examples and we ask you to do this on your own in numerous problems Given this emphasis it is easy to think of the completed model as the end of the process you complete the model and then proceed to the next model However a completed model is really a starting point After you have a working model of the problem you canand 14 The SevenStep Modeling Process 13 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it you shoulduse it as a tool for gaining insights For most models many whatif questions can be asked If the model has been developed correctly it should be capable of answering such whatif questions fairly easily In other words it should be relatively easy to perform sensitivity analysis on the model This is in fact how management science models are used in the business world They are typically developed to solve a particular problem but they are then used as a tool to analyze a number of variations of the basic problem For most of the examples in the book we not only show you how to develop a model to obtain an answer but we often include a section called Discussion of the Solution or a similar title and a section called Sensitivity Analysis The first of these asks you to step back and look at the solution Does it make sense Does it provide any insights espe cially surprising ones The second section indicates how the model can be expanded in one or more natural ways What happens if there is more or less of some scarce resource What happens if a new constraint is added The point is that before moving to the next model you should spend some time taking a close look at the model you just developed This is not just for pedagogical purposes it is exactly the way real management science projects proceed 15 A GREAT SOURCE FOR MANAGEMENT SCIENCE APPLICATIONS INTERFACES Many of the chapter openers in this book are based on successful management science applications that have been reported in the Interfaces journal This is a highly respected bimonthly journal that chronicles real applications of management science that have gen erated proven benefits often in the millions or even hundreds of millions of dollars The applications are in a wide range of industries are global and employ a variety of manage ment science techniques Of special interest are the JanuaryFebruary and since 1999 the SeptemberOctober issues Each JanuaryFebruary issue contains the winner and finalists for that years Franz Edelman Award for Achievement in Operations Research and the Management Sciences This is the professions most honored prize for the practice of management science The prize is awarded for implemented work that has had significant verifiable and preferably quantifiable impact Similarly each SeptemberOctober issue contains the winner and runnersup for that years Daniel H Wagner Prize for Excellence in Operations Research Practice Each prize is named after a pioneer in the field of operations research and man agement science and the winning papers honor them by documenting the practice of man agement science at its best The journal is probably available from your schools library either in paper or elec tronic format Check with your librarian about gaining access to Interfaces Its articles will confirm what we have been saying Management science makes a huge difference to both large and small organizations all over the world 16 WHY STUDY MANAGEMENT SCIENCE We hope that you are convinced by now that management science is an important area and that highly trained analysts are needed to solve the large and complex problems faced by the business world However unless you are one of the relatively few students who intends to become a professional management scientist you are probably wondering why you need to study management science This is a legitimate concern For many years those in the field of management science education received criticism from students and 14 Chapter 1 Introduction to Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it educators that management science courses were irrelevant for the majority of students who were required to take them Looking back it is difficult to argue with these critics Typical management courses were centered primarily around a collection of very specific models and worse a collection of mindnumbing mathematical solution techniques techniques that students were often required to implement by hand Some courses are probably still taught this way but we hope the number is decreasing rapidly Two forces have helped to change this tendency toward irrelevance First the many vocal critics motivated many of us to examine our course materials and teaching methods Certain topics have been eliminated and replaced by material that is more relevant and interesting to students We have certainly attempted to do so here The second force is the emergence of powerful computers and the accompanying easytouse software especially spreadsheet software With the availability of computers to do the number crunching there is no needexcept in advanced coursesto delve into the mathematical details of the solution techniques This task can be delegated to machines that are far better at it than humans The time formerly spent on such details can now be used to develop valuable modeling skills The intent in this book is not just to cover specific models and specific approaches to these models but to teach a more general approach to the modelbuilding process We believe that the spreadsheet approach is the best way to do this because it appeals to the largest audience We have been teaching our own courses with this spreadsheetmodeling approach for nearly two decadesto a wide range of business studentsand have received very few complaints about irrelevance In fact many students have stated that this is the most valuable business course they have taken The following are some of the rea sons for this newfound relevance The modeling approach emphasized throughout this book is an important way to think about problems in general not just the specific problems we discuss This approach forces you to think logically You must discover how given data can be used or which data are necessary you must determine the elements of the problem that you can control the decision variables and you must determine how the ele ments of the problem are logically related These logical thinking skills are valuable for your career regardless of the specific field you enter Management science is admittedly built around quantitative skillsit deals primarily with numbers and relationships between numbers Some critics object that not everything in the real world can be reduced to numbers but as one of our reviewers correctly points out a great deal that is of importance can As you work through the many models in this book your quantitative skills will be sharpened immensely In a business world driven increasingly by numbers quantitative skills are an important asset No matter what your spreadsheet abilities are when you enter this course by the time you are finished you will be a proficient spreadsheet user We deliberately chose the spreadsheet package Excel which is arguably the most widely used package other than wordprocessing packages in the business world today Many of our students state that the facility they gain in Excel is the most valuable part of the course That doesnt mean this is a course in spreadsheet fundamentals and neat tricks although you will undoubtedly pick up many useful tricks along the way A great spreadsheet packageand we strongly believe that Excel is the greatest spreadsheet package written to dategives you complete control over your model You can apply spread sheets to an endless variety of problems Excel gives you the flexibility to work in a way that suits your style best and it enables you to present results and often catch errors almost immediately As you succeed with relatively easy problems your con fidence will build and before long you will be able to tackle more difficult problems 16 Why Study Management Science 15 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it successfully In short spreadsheets enable everyone not just technical people to develop and use their quantitative skills Management science modeling helps you develop your intuition and it also indicates where intuition alone sometimes fails When you confront a problem you often make an educated or maybe not so educated guess at the solution If the problem is suffi ciently complex as many of the problems in this book are this guess will be fre quently wide of the mark In this sense the study of management science can be a humbling experienceyou find that your unaided intuition is often not very good But by studying many models and examining their solutions you can sharpen your intuition considerably This is sometimes called the Aha effect All of a sudden you see why a certain solution is so good The chances are that when you originally thought about the problem you forgot to consider an important constraint or a key relationship and this caused your poor initial guess Presumably the more problems you analyze the better you will become at recognizing the critical elements of new problems Experienced management scientists tend to have excellent intuition the ability to see through to the essence of a problem almost immediately However they are not born with this talent it comes through the kind of analysis you will be per forming as you work through this book 17 SOFTWARE INCLUDED WITH THIS BOOK Very few business problems are small enough to be solved with pencil and paper They require powerful software The software included in this book together with Microsoft Excel provides you with a powerful software combination that you will use for this course and beyond This software is being usedand will continue to be usedby leading com panies all over the world to solve large complex problems The experience you obtain with this software through working the examples and problems in this book will give you a key competitive advantage in the marketplace It all begins with Excel All the quantitative methods that we discuss are implemented in Excel Specifically in this edition we use Excel 20105 Although it is impossible to forecast the state of computer software into the longterm or even mediumterm future as we are writing this book Excel is the most heavily used spreadsheet package on the mar ket and there is every reason to believe that this state will persist for quite awhile Most companies use Excel most employees and most students have been trained in Excel and Excel is a very powerful flexible and easytouse package Although Excel has a huge set of tools for performing quantitative analysis we have included several addins with this book that make Excel even more powerful Access to addins are available on the Essential Resource Web site See the preface for details We discuss these briefly here and in much more depth in the specific chapters where they apply Together with Excel and the addins included in this book you have a wealth of software at your disposal The examples and stepbystep instructions throughout the book will help you to become a power user of this software This takes plenty of practice and a willingness to experiment but its certainly within your grasp When you are finished dont be surprised if you rate improved software skills as one of the most valuable things you have learned from the book 16 Chapter 1 Introduction to Modeling 5Excel 2007 was a big change from Excel 2003 and earlier versions The changes in Excel 2010 are much more minor So if you have been using Excel 2007 you will see very few changes here Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Builtin Excel Features Virtually everyone in the business world knows the basic features of Excel but relatively few know some of its more powerful features In short relatively few people are the power users we expect you to become by working through this book To get you started the file Excel Tutorialxlsx explains some of the intermediate features of Excelfeatures that we expect you to be able to use access this file on the textbooks Web site that accompanies new copies of this book These include the SUMPRODUCT VLOOKUP IF NPV and COUNTIF functions They also include range names data tables the Paste Special option the Goal Seek tool and many others Finally although we assume you can perform routine spreadsheet tasks such as copying and pasting the tutorial includes many tips to help you perform these tasks more efficiently Solver Addin In Chapters 38 and 16 we make heavy use of Excels Solver addin This addin devel oped by Frontline Systems not Microsoft uses powerful algorithmsall behind the scenesto perform spreadsheet optimization Before this type of spreadsheet optimization addin was available specialized nonspreadsheet software was required to solve opti mization problems Now you can do it all within a familiar spreadsheet environment SolverTable Addin An important theme throughout this book is sensitivity analysis How do outputs change when inputs change Typically these changes are made in spreadsheets with a data table a builtin Excel tool However data tables dont work in optimization models where we want to see how the optimal solution changes when certain inputs change Therefore we include an Excel addin called SolverTable which works almost exactly like Excels data tables This addin was developed by Albright In Chapters 38 and 16 we illustrate the use of SolverTable Palisade Decision Tools Suite In addition to SolverTable and builtin Excel addins we also have included on this text books essential resource Web site an educational version of Palisade Corporations pow erful Decision Tools suite All of the programs in this suite are Excel addins so the learning curve isnt very steep There are seven separate addins in this suite RISK StatTools PrecisionTree TopRank RISKOptimizer NeuralTools and Evolver6 We will use the first three most heavily in this book but all are useful for certain tasks and are described briefly below RISK The simulation addin RISK enables you to run as many replications of a spreadsheet sim ulation as you like As the simulation runs RISK automatically keeps track of the outputs you select and it then displays the results in a number of tabular and graphical forms RISK also enables you to perform a sensitivity analysis so that you can see which inputs have the most effect on the outputs Finally RISK provides a number of spreadsheet func tions that enable you to generate random numbers from a variety of probability distributions StatTools Palisade has also developed a statistics addin called StatTools which enhances the statisti cal capabilities of Excel Excels builtin statistical tools are rather limited It has several 17 Software Included in This Book 17 6The Palisade suite has traditionally included two standalone programs BestFit and RISKview The functional ity of both of these is now included in RISK so they are not included in the suite Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it functions such as AVERAGE and STDEV for summarizing data and it includes the Analysis ToolPak an addin that was developed by a third party However these tools are not suffi ciently powerful or flexible for the heavyduty statistical analysis that is sometimes required StatTools provides a collection of tools that help fill this gap Admittedly this is not a statistics book but StatTools will come in particularly handy in Chapter 14 when you study regression analysis and forecasting PrecisionTree The PrecisionTree addin is used in Chapter 9 to analyze decision problems with uncer tainty The primary method for performing this type of analysis is to draw a decision tree Decision trees are inherently graphical and they have always been difficult to implement in spreadsheets which are based on rows and columns However PrecisionTree does this in a very clever and intuitive way Equally important once the basic decision tree has been built it is easy to use PrecisionTree to perform a sensitivity analysis on the models inputs TopRank TopRank is a whatif addin used for sensitivity analysis It starts with any spreadsheet model where a set of inputs along with a number of spreadsheet formulas leads to one or more outputs TopRank then performs a sensitivity analysis to see which inputs have the largest effect on a given output For example it might indicate which input affects aftertax profit the most the tax rate the riskfree rate for investing the inflation rate or the price charged by a competitor Unlike RISK TopRank is used when uncertainty is not explicitly built into a spreadsheet model However it considers uncertainty implicitly by performing sensitivity analysis on the important model inputs RISKOptimizer RISKOptimizer combines optimization with simulation There are often times when you want to use simulation to model some business problem but you also want to optimize a summary measure such as a mean of an output distribution This optimization can be performed in a trialanderror fashion where you try a few values of the decision vari ables and see which provides the best solution However RISKOptimizer provides a more automatic and timeintensive optimization procedure NeuralTools In Chapter 14 we show how regression can be used to find a linear equation that quantifies the relationship between a dependent variable and one or more explanatory variables Although linear regression is a powerful tool it is not capable of quantifying all possible relationships The NeuralTools addin mimics the working of the human brain to find neural networks that quantify complex nonlinear relationships Evolver In Chapter 8 we show how Solver 2010s Evolutionary algorithm can be used to solve some nonsmooth nonlinear models that Solvers other algorithms cannot handle Evolutionary Solver uses genetic algorithms to solve these difficult problems Although we will not use it in this book Palisades Evolver addin is an alternative implementation of genetic algorithms 18 CONCLUSION In this chapter we have introduced the field of management science and the process of mathematical modeling To provide a more concrete understanding of these concepts we reviewed a simple queueing model We also explored a sevenstep modelbuilding process 18 Chapter 1 Introduction to Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it that begins with problem definition and proceeds through final implementation Finally we discussed why the study of management science is a valuable experience even if you do not intend to pursue a professional career in this field Dont worry if you dont understand some of the terms such as linear programming that were used in this chapter Although the sevenstep process is not too difficult to com prehend especially when discussed in the context of real applications it typically entails some rather complex logical relationships and mathematical concepts These ideas are pre sented in much greater detail in the rest of this book Specifically you will learn how to build spreadsheet models in Excel how to use them to answer whatif questions and how to find optimal solutions with the help of Excels Solver addin For practical reasons most of your work will take place in the classroom or in front of your own PC as you work through the examples and problems The primary emphasis of this book therefore is on steps 3 through 6 that is developing the model testing the model with different inputs optimizing the model and presenting and interpreting the results to a clientprobably your instructor Keep in mind however that with real problems you must take crucial steps before and after the procedures you will be practicing in this book Because real problems dont come as nicely packaged as those we discuss and because the necessary data are seldom given to you on a platter you will have to wrestle with the problems scope and precise data requirements when you solve problems in a real setting We have included modeling problems at the ends of most chapters These problems are not as well structured as the skill problems so the burden is on you to determine an appropriate structure and decide the necessary data Also because a mathematically accurate model doesnt necessarily result in a successful implementation your work is not finished just because the numbers check out To gain acceptance for a model an analyst must have the right combination of technical skills and people skills Try to keep this in mind as you write up your solutions to the problems in this book Dont just hand in a mass of numbers with little or no explana tion Sell your solution 18 Conclusion 19 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 21 Introduction to Spreadsheet Modeling C H A P T E R ANAL YSIS OF HIVAIDS M any of management sciences most successful applications are traditional functional areas of business including operations management logistics finance and marketing Indeed many such applications are analyzed in this book However another area where management science has had a strong influence over the past two decades has been the analysis of the worldwide HIVAIDS epidemic Not only have theoretical models been developed but even more important they have also been applied to help understand the epidemic and reduce its spread To highlight the importance of management science modeling in this areaan entire special issue MayJune 1998 of Interfacesthe journal that reports successful management science applications was devoted to HIVAIDS modelsSome of the highlights are discussed here to give you an idea of what management science has to offer in this important area Kahn et al1998 provides an overview of the problem They discuss how governmentspublichealth agenciesand healthcare providers must deter mine how best to allocate scarce resources for HIV treatment and prevention among different programs and populations They discuss in some depth how management science models have influencedand will continue to influence AIDS policy decisionsOther articles in the issue discuss more specific prob lemsCaulkins et al1998 analyze whether the distribution of difficultto reuse syringes would reduce the spread of HIV among injection drug users Based on their modelthey conclude that the extra expense of these types of syringes would not be worth the marginal benefit they might provide 2 Lise Gagneistockphoto Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Paltiel and Freedberg 1998 investigate the costs and benefits of developing and administering treatments for cytomegalovirus CMV an infection to which HIV carriers are increasingly exposed Retinitis CMVs most common manifestation is associated with blindness and sometimes death Their model suggests that the costs compare unfavorably with alternative uses of scarce resources Owens et al 1998 analyze the effect of womens relapse to highrisk sexual and needlesharing behavior on the costs and benefits of a voluntary program to screen women of childbearing age for HIV They find for example that the effect of relapse to highrisk behaviors on screening program costs and benefits can be substantial suggesting that behavioral interventions that pro duce sustained reductions in risk behavior even if expensive could be costsaving The important point is that these articles and others not mentioned here base their results on rigorous management science models of the HIVAIDS phenomenon In addi tion they are backed up with real data They are not simply opinions of the authors 22 Chapter 2 Introduction to Spreadsheet Modeling 21 INTRODUCTION This book is all about spreadsheet modeling By the time you are finished you will have seen some reasonably complexand realisticmodels Many of you will also be trans formed into Excel power users However we dont want to start too quickly or assume too much background on your part For practice in getting up to speed with basic Excel features we have included an Excel tutorial on this textbooks essential resource Web site See the Excel Tutorialxlsx file You can work through this tutorial at your own speed and cover the topics you need help with Even if you have used Excel extensively give this tutorial a look You might be surprised how some of the tips can improve your productivity In addition this chapter provides an introduction to Excel modeling and illustrates some interesting and relatively simple models The chapter also covers the modeling process and includes some of the less well known but particularly helpful Excel tools that are available These tools include data tables Goal Seek lookup tables and auditing commands Keep in mind however that our objective is not the same as that of the many howto Excel books on the market We are not teaching Excel just for its many inter esting features Rather we plan to use these features to provide insights into real busi ness problems In short Excel is a problemsolving tool not an end in itself in this book 22 BASIC SPREADSHEET MODELING CONCEPTS AND BEST PRACTICES Most mathematical models including spreadsheet models involve inputs decision vari ables and outputs The inputs have given fixed values at least for the purposes of the model The decision variables are those a decision maker controls The outputs are the ultimate values of interest they are determined by the inputs and the decision variables For example suppose a manager must place an order for a certain seasonal product This product will go out of date fairly soon so this is the only order that will be made for the product The inputs are the fixed cost of the order the unit variable cost of each item ordered the price charged for each item sold the salvage value for each item if any left in inventory after the product has gone out of date and the demand for the product The deci sion variable is the number of items to order Finally the key output is the profit or loss from the product This output can also be broken down into the outputs that contribute to Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 22 Basic Spreadsheet Modeling Concepts and Best Practices 23 it the total ordering cost the revenue from sales and the salvage value from leftover items These outputs must be calculated to obtain profit Spreadsheet modeling is the process of entering the inputs and decision variables into a spreadsheet and then relating them appropriately by means of formulas to obtain the outputs After you have done this you can then proceed in several directions You might want to perform a sensitivity analysis to see how one or more outputs change as selected inputs or decision variables change You might want to find the values of the decision vari ables that minimize or maximize a particular output possibly subject to certain con straints You might also want to create charts that show graphically how certain parameters of the model are related These operations are illustrated with several examples in this chapter Getting all the spreadsheet logic correct and producing useful results is a big part of the battle however we go farther by stressing good spreadsheet modeling practices You probaby wont be developing spreadsheet models for your sole use instead you will be sharing them with colleagues or even a boss or an instructor The point is that other people will be reading and trying to make sense out of your spreadsheet models Therefore you should construct your spreadsheet models with readability in mind Features that can improve readability include the following A clear logical layout to the overall model Separation of different parts of a model possibly across multiple worksheets Clear headings for different sections of the model and for all inputs decision vari ables and outputs Use of range names Use of boldface italics larger font size coloring indentation and other formatting features Use of cell comments Use of text boxes for assumptions and explanations Obviously the formulas and logic in any spreadsheet model must be correct however correctness will not take you very far if no one can understand what you have done Much of the power of spreadsheets derives from their flexibility A blank spreadsheet is like a big blank canvas waiting for you to insert useful data and formulas Almost anything is allowed However you can abuse this power if you dont have an overall plan for what should go where Plan ahead before diving in and if your plan doesnt look good after you start filling in the spreadsheet revise your plan The following example illustrates the process of building a spreadsheet model according to these guidelines We build this model in stages In the first stage we build a model that is correct but not very readable At each subsequent stage we modify the model to make it more readable You do not need to go through each of these stages explicitly when you build your own models You can often strive for the final stage right away at least after you get accustomed to the modeling process The various stages are shown here simply for contrast E X A M P L E 21 ORDERING NCAA TSHIRTS I t is March and the annual NCAA Basketball Tournament is down to the final four teams Randy Kitchell is a Tshirt vendor who plans to order Tshirts with the names of the final four teams from a manufacturer and then sell them to the fans The fixed cost of any order is 750 the variable cost per Tshirt to Randy is 8 and Randys selling price is 18 However this price will be charged only until a week after the tournament After that time Randy figures that interest in the Tshirts will be low so he plans to sell all remaining Some inputs such as demand in this example contain a considerable degree of uncertainty In some cases as in Example 24 later in this chapter this uncertainty is modeled explicitly Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Tshirts if any at 6 each His best guess is that demand for the Tshirts during the full price period will be 1500 He is thinking about ordering 1450 Tshirts but he wants to build a spreadsheet model that will let him experiment with the uncertain demand and his order quantity How should he proceed Objective To build a spreadsheet model in a series of stages all stages being correct but each stage being more readable and flexible than the previous stages Solution The logic behind the model is fairly simple but the model is built for generality Specifically the formulas used allow for the order quantity to be less than equal to or greater than demand If demand is greater than the order quantity Randy will sell all the T shirts ordered for 18 each However if demand is less than the order quantity Randy will sell as many Tshirts as are demanded at the 18 price and all leftovers at the 6 price You can implement this logic in Excel with an IF function A first attempt at a spreadsheet model appears in Figure 21 See the file TShirt Sales Finishedxlsx where each stage appears on a separate worksheet You enter a possible demand in cell B3 a possible order quantity in cell B4 and then calculate the profit in cell B5 with the formula 7508B4IFB3B418B418B36B4B3 This formula subtracts the fixed and variable costs and then adds the revenue accord ing to the logic just described 24 Chapter 2 Introduction to Spreadsheet Modeling 1 2 3 4 5 A B NCAA tshirt sales Demand Order Profit 1500 1450 13750 Figure 21 Base Model Excel Function IF Excels IF function is probably already familiar to you b ut it is too important not to dis cuss It has the syntax IFconditionresultIf TrueresultIfFalse The condition is any expression that is either true or false The two e xpressions resultIf True and resultIfFalse can be any expressions you would enter in a cell number s text or other Excel functions including other IF functions Note that if either e xpression is text it must be enclosed in double quotes such as IFScore90AB Finally condition can be complex combinations of conditions using the keywords AND or OR Then the syntax is for example IFANDScore160Score260FailPass This model in Figure 21 is entirely correct but it isnt very readable or flexible because it breaks a rule that you should strive never to break It hard codes input values into the profit formula A spreadsheet model should never include input numbers in formulas Instead the spreadsheet model should store input values in separate cells and then include cell references to these inputs in its formulas A remedy appears in Figure 22 Here the inputs have been entered in the range B3B6 and the profit formula in cell B10 has been changed to B3B4B9IFB8B9B5B910B8B6B9B8 Never hard code numbers into Excel formulas Use cell references instead Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 22 Basic Spreadsheet Modeling Concepts and Best Practices 25 This is exactly the same formula as before but it is now more flexible If an input changes the profit recalculates automatically Most important the inputs are no longer buried in the formula1 Still the profit formula is not very readable as it stands You can make it more read able by using range names The mechanics of range names are covered in detail later in this chapter For now the results of using range names for cells B3 through B6 B8 and B9 are shown in Figure 23 This model looks exactly like the previous model but the formula in cell B10 is now FixedordercostVariablecostOrderIFDemandOrder SellingpriceOrderSellingpriceDemandDiscountPriceOrderDemand This formula is admittedly more longwinded but it is certainly easier to read 1 2 3 4 5 6 7 8 9 10 A B NCAA tshirt sales Fixed order cost Variable cost Selling price Discount price Demand Order Profit 750 8 18 6 1500 1450 13750 Figure 22 Model with Input Cells 1 2 3 4 5 6 7 8 9 10 A B C D E F NCAA tshirt sales Fixed order cost Range names used Variable cost Selling price Discount price Order Demand Order Profit 750 8 18 6 1500 1450 13750 Demand Discountprice Fixedordercost Sellingprice Variablecost Model 3B8 Model 3B6 Model 3B3 Model 3B9 Model 3B5 Model 3B4 Figure 23 Model with Range Names in Profit Formula 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A B C D E F NCAA tshirt sales Fixed order cost 750 Range names used Variable cost 8 Demand Model 4B8 Selling price 18 Discountprice Model 4B6 Discount price 6 Fixedordercost Model 4B3 Order Model 4B9 Mod le gp cir e nille S 1500 Demand 4B5 Mod le elb co ts Va air 1450 Order 4B4 Costs Fixed cost 750 Variable costs 11600 Revenues Fullprice shirts 26100 Discountprice shirts 0 13750 ro tif P Figure 24 Model with Intermediate Outputs Randy might like to have profit broken down into various costs and revenues Figure 24 rather than one single profit cell The formulas in cells B12 B13 B15 and B16 are straightforward so they are not repeated here You can then accumulate these to get profit in cell B17 with the formula B12B13B15B16 1Some people refer to such numbers buried in formulas as magic numbers because they just seem to appear out of nowhere Avoid magic numbers Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Of course range names could be used for these intermediate output cells but this is prob ably more work than its worth You should always use some judgment when deciding how many range names to use If Randys assistant is presented with this model how does she know at a glance which cells contain inputs or decision variables or outputs Labels andor color coding can help to distinguish these types A blueredgray colorcoding style has been applied in Figure 25 along with descriptive labels in boldface The blue cells at the top are input cells the red cell in the middle is a decision variable and the gray cell at the bottom is the key output2 There is nothing sacred about this particular convention Feel free to adopt your own convention and style but be sure to use it consistently The model in Figure 25 is still not the last word on this example As shown in later examples you can create data tables to see how sensitive profit is to the inputs the demand and the order quantity You can also create charts to show any numerical results graphically But this is enough for now You can see that the model in Figure 25 is now much more readable and flexible than the orig inal model in Figure 21 Because good spreadsheet style is so important the appendix to this chapter discusses a few tools for editing and documenting your spreadsheet models Use these tools right away and as you progress through the book In the rest of this chapter we discuss a number of interesting examples and introduce important modeling concepts such as sensitivity analysis important Excel features such as data tables and even some important business concepts such as 26 Chapter 2 Introduction to Spreadsheet Modeling 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 A B C D E F NCAA tshirt sales Input Range elb s va air names used Fixed order cost 750 Demand Model 5B10 Variable cost 8 Discountprice Model 5B7 Selling price 18 Fixedordercost Model 5B4 Discount price 6 Order Model 5B13 Sellingprice Model 5B6 Uncertain variable Variablecost Model 5B5 1500 emand D Decision variable 1450 rder O Output variables Costs Fixed cost 750 Variable costs 11600 Revenues Fullprice shirts 26100 Discountprice shirts 0 13750 ro tif P Figure 25 Model with Category Labels and Color Coding 2This color convention shows up clearly in the Excel files that accompany the book However in this twocolor book shades of gray and blue it is difficult to see the colorcoding scheme We recommend that you look not only at the figures in the book but at the actual Excel files Spreadsheet Layout and Documentation If you want y our spreadsheets to be used and y ou want your value in your company to risegive a lot of thought to y our spreadsheet layout and then docu ment y our w ork car efully For la yout think about whether cer tain data ar e best oriented in r ows or columns whether y our w ork is better placed in a single sheet or in multiple sheets and so on For doc umentation use descriptive labels and headingscolor coding cell comments and text boxes to make your spreadsheets more readable It takes time and careful planning to design and then document y our spread sheet models but the time is w ell spent And if you come back in a few days to a spreadsheet model you developed and y ou cant make heads or tails of it dont be afraid to r edesign your work completely from the ground up FUNDAMENTAL INSIGHT Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it net present value To get the most from these examples follow along at your own PC starting with a blank spreadsheet It is one thing to read about spreadsheet modeling it is quite another to actually do it 23 COST PROJECTIONS In this next example a company wants to project its costs of producing products given that material and labor costs are likely to increase through time We build a simple model and then use Excels charting capabilities to obtain a graphical image of projected costs 23 Cost Projections 27 E X A M P L E 22 PROJECTING THE COSTS OF BOOKSHELVES AT WOODWORKS T he Woodworks Company produces a variety of customdesigned wood furniture for its customers One favorite item is a bookshelf made from either cherry or oak The com pany knows that wood prices and labor costs are likely to increase in the future Table 21 shows the number of boardfeet and labor hours required for a bookshelf the current costs per boardfoot and labor hour and the anticipated annual increases in these costs The top row indicates that either type of bookshelf requires 30 boardfeet of wood and 16 hours of labor Build a spreadsheet model that enables the company to experiment with the growth rates in wood and labor costs so that a manager can see both numerically and graphically how the costs of the bookshelves vary in the next few years Table 21 Input Data for Manufacturing a Bookshelf Resource Cherry Oak Labor Required per bookshelf 30 30 16 Current unit cost 550 430 1850 Anticipated annual cost increase 24 17 15 Business Objectives3 To build a model that allows Woodworks to see numerically and graphically how its costs of manufacturing bookshelves increase in the future and to allow the company to answer whatif questions with this model Excel Objectives To learn good spreadsheet practices to enable copying formulas with the careful use of relative and absolute addresses and to create line charts from multiple series of data Solution Listing the key variables in a table before developing the actual spreadsheet model is use ful so we will continue to do this in many later examples see Table 22 This practice forces you to examine the roles of the variableswhich are inputs which are decision variables and which are outputs Although the variables and their roles are fairly clear for this example later examples will require more thought 3In later chapters we simply list the Objective of each example as we did in Example 21 However because this chapter has been written to enhance basic spreadsheet skills we separate the business objectives from the Excel objectives Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Table 22 Key Variables for the Bookshelf Manufacturing Example Input variables Wood and labor requirements per bookshelf current unit costs of wood and labor anticipated annual increases in unit costs Output variables Projected unit costs of wood and labor projected total bookshelf costs The reasoning behind the model is straightforward You first project the unit costs for wood and labor into the future Then for any year you multiply the unit costs by the required numbers of boardfeet and labor hours per bookshelf Finally you add the wood and labor costs to obtain the total cost of a bookshelf Developing the Spreadsheet Model The completed spreadsheet model appears in Figure 26 and in the file Bookshelf Costsxlsx4 You can develop it with the following steps Figure 26 Bookshelf Cost Model A B C D E F G H I J K 1 Projecting bookshelf costs at Woodworks 2 3 Inputs 4 Requirements per bookshelf Cherry Oak 5 Boardfeet required 30 30 6 Labor hours required 16 16 7 8 Costs of wood Cherry Oak 9 Current cost per boardfoot 550 430 10 Projected annual increase 24 17 11 12 Cost of labor 13 Current cost per labor hour 1850 14 Projected annual increase 15 15 16 Projected costs 17 Cost per boardfoot Cost per hour Cost per bookshelf 18 Years from now Cherry Oak Labor Cherry Oak 19 0 550 430 1850 46100 42500 20 1 563 437 1878 46940 43163 21 2 577 445 1906 47796 43837 22 3 591 452 1935 48669 44521 23 4 605 460 1964 49558 45216 24 5 619 468 1993 50465 45922 25 6 634 476 2023 51389 46639 Always enter input values in input cells and then refer to them in Excel formulas Do not bury input values in formulas 1 Inputs You should usually enter the inputs for a model in the upperleft corner of a worksheet as you can see in the shaded ranges in Figure 26 using the data from Table 21 We have used our standard convention of coloring inputsthe numbers from the statement of the problemblue You can develop your own convention but the input cells should be distinguished in some way Note that the inputs are grouped logically and are explained with appropriate labels You should always document your spreadsheet model with informational labels Also note that by entering inputs explicitly in input cells you can refer to them later in Excel formulas 2 Design output table Plan ahead for how you want to structure your outputs We created a table where there is a row for every year in the future year 0 corresponds to the current year there are three columns for projected unit costs columns BD and there are two columns for projected total bookshelf costs columns EF The headings reflect this design Of course this isnt the only possible design but it works well The important point is that you should have some logical design in mind before diving in 4This textbooks essential resource Web site includes templates and completed files for all examples in the book where all of the latter have Finished appended to their file names However especially in this chapter we suggest that you start with a blank spreadsheet and follow the stepbystep instructions on your own 28 Chapter 2 Introduction to Spreadsheet Modeling 3 Projected unit costs of wood The dollar values in the range B19F25 are all calcu lated from Excel formulas Although the logic in this example is straightforward it is still important to have a strategy in mind before you enter formulas In particular you should always try to design your spreadsheet so that you can enter a single formula and then copy it This saves work and avoids errors For the costs per boardfoot in columns B and C enter the formula B9 in cell B19 and copy it to cell C19 Then enter the general formula B191B10 in cell B20 and copy it to the range B20C25 We assume you know the rules for absolute and relative addresses dollar sign for absolute no dollar sign for relative but it takes some planning to use these so that copying is possible Make sure you understand why we made row 10 absolute but column B relative Excel Tip Relative and Absolute Addresses in Formulas Relative and absolute addresses are used in Excel formulas to facilitate copying A dollar sign next to a column or r ow address indicates that the addr ess is absolute and will not change when copying The lack of a dollar sign indicates that the addr ess is relative and will change when copying After you select a cell in a formula you can pr ess the F4 k ey repeatedly to cycle through the relativeabsolute possibilities for example B4 both col umn and row relative B4 both column and r ow absolute B4 column r elative row absolute and B4 column absolute row relative 4 Projected unit labor costs To calculate projected hourly labor costs enter the formula B13 in cell D19 Then enter the formula D191B14 in cell D20 and copy it down column D 5 Projected bookshelf costs Each bookshelf cost is the sum of its wood and labor costs By a careful use of absolute and relative addresses you can enter a single formula for these costsfor all years and for both types of wood To do this enter the formula B5B19B6D19 in cell E19 and copy it to the range E19F25 The idea here is that the units of wood and labor per bookshelf are always in rows 5 and 6 and the projected unit labor cost is always in column D but all other references are relative to allow copying 6 Chart A chart is a valuable addition to any table of data especially in the business world so charting in Excel is a skill worth mastering Although not everyone agrees the many changes Microsoft made regarding charts in Excel 2007 and 2010 help you create charts more efficiently and effectively We illustrate some of the possibilities here but we urge you to experiment with other possibilities on your own Start by selecting the range E18F25yes including the labels in row 18 Next click on the Line dropdown list on the Insert ribbon and select the Line with Markers type You instantly get the basic line chart you want with one series for Cherry and another for Oak Also when the chart is selected that is it has a border around it you see three Chart Tools ribbons Design Layout and Format The most important button on any of these ribbons is the Select Data button on the Design ribbon It lets you choose the ranges of the data for charting in case 23 Cost Projections 29 Always try to organize your spreadsheet model so that you can copy formulas across multiple cells Typing dollar signs in formulas for absolute references is inefficient Press the F4 key instead Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 30 Chapter 2 Introduction to Spreadsheet Modeling Excels default choices which are based on the selected range when you create the chart are wrong Click on Select Data now to obtain the dialog box in Figure 27 On the left you control the series one series or multiple series being charted on the right you con trol the data used for the horizontal axis By selecting E18F25 you have the series on the left correct including the names of these series Cherry and Oak but if you didnt you could select one of the series and click on Edit to change it The data on the horizontal axis is currently the default 1 2 and so on To make it the data in column A click on the Edit button on the right and select the range A19A25 See Figure 28 Your chart is now correctly labeled and charts the correct data Beyond this you can experiment with vari ous formatting options to make the chart even better For example we rescaled the verti cal axis to start at 300 rather than 0 rightclick on the numbers on the vertical axis and select Format Axis or look at the many options on the Axes dropdown list on the Layout ribbon and we added a chart title at the top and a title for the horizontal axis at the bot tom see buttons on the Labels group on the Layout ribbon You can spend a lot of time finetuning chartsmaybe even too much timebut professionallooking charts are defi nitely appreciated Figure 27 Select Data Dialog Box Figure 28 Dialog Box for Changing Horizontal Axis Labels The many chart options are easily accessible from the three Chart Tools ribbons in Excel 2007 and 2010 Dont be afraid to experiment with them to produce professionallooking charts Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Fundamental Insight The Power of Chats A chart is typically much more informative to a business manager than the table of numbers it is based on Dont underestimate the power of Excel charts for getting your points across and include them in your spreadsheet models whenever possible However be prepared to do some investigating on your own Excel offers an abundance of chart types and chart options to choose from and they are not all equally suited to telling your story Using the Model for WhatIf Questions The model in Figure 26 can now be used to answer many whatif questions In fact many models are built for the purpose of permitting experimentation with various scenarios The important point is that the model has been built in such a way that a manager can enter any desired values in the input cells and all of the outputs including the chart will update automatically As a simple example if the annual percentage increases for wood costs are twice as high as Woodworks anticipated you can enter these higher values in row 10 and immediately see the effect as shown in Figure 29 By comparing bookshelf costs in this scenario to those in the original scenario the projected cost in year 6 for cherry bookshelves for example increases by about 55 from 51389 to 54226 Figure 29 Effect of Higher Increases in Wood Costs A B C D E F G H I J K 1 Projecting bookshelf costs at Woodworks 2 3 Inputs 4 Requirements per bookshelf Cherry Oak 5 Boardfeet required 30 30 6 Labor hours required 16 16 7 8 Costs of wood Cherry Oak 9 Current cost per boardfoot 550 430 10 Projected annual increase 48 34 11 12 Cost of labor 13 Current cost per labor hour 1850 14 Projected annual increase 15 15 16 Projected costs 17 Cost per boardfoot Cost per hour Cost per bookshelf 18 Years from now Cherry Oak Labor Cherry Oak 19 0 550 430 1850 46100 42500 20 1 576 445 1878 47336 43383 21 2 604 460 1906 48617 44287 22 3 633 475 1935 49944 45213 23 4 663 492 1964 51320 46162 24 5 695 508 1993 52746 47135 25 6 729 526 2023 54226 48132 A carefully constructed modelwith no input numbers buried in formulasallows a manager to answer many whatif questions with a few keystrokes You should appreciate by now why burying input numbers inside Excel formulas is such a bad practice For example if you had buried the annual increases of wood costs from row 10 in the formulas in columns B and C imagine how difficult it would be to answer the whatif question in the previous paragraph You would first have to find and then change all the numbers in the formulas which is a lot of work Even worse it is likely to lead to errors 24 BREAKEVEN ANALYSIS Many business problems require you to find the appropriate level of some activity This might be the level that maximizes profit or minimizes cost or it might be the level that allows a company to break evenno profit no loss We discuss a typical breakeven analysis in the following example 32 Chapter 2 Introduction to Spreadsheet Modeling E X A M P L E 23 BREAKEVEN ANALYSIS AT QUALITY SWEATERS T he Quality Sweaters Company sells handknitted sweaters The company is planning to print a catalog of its products and undertake a direct mail campaign The cost of printing the catalog is 20000 plus 010 per catalog The cost of mailing each catalog including postage order forms and buying names from a mailorder database is 015 In addition the company plans to include direct reply envelopes in its mailings and incurs 020 in extra costs for each direct mail envelope used by a respondent The average size of a customer order is 40 and the companys variable cost per order due primarily to labor and material costs averages about 80 of the orders valuethat is 32 The company plans to mail 100000 catalogs It wants to develop a spreadsheet model to answer the following questions 1 How does a change in the response rate affect profit 2 For what response rate does the company break even 3 If the company estimates a response rate of 3 should it proceed with the mailing 4 How does the presence of uncertainty affect the usefulness of the model Business Objectives To create a model to determine the companys profit and to see how sensitive the profit is to the response rate from the mailing Excel Objectives To learn how to work with range names to learn how to answer what if questions with oneway data tables to introduce Excels Goal Seek tool and to learn how to document and audit Excel models with cell comments and the auditing toolbar Solution The key variables appear in Table 23 Note that we have designated all variables as input variables decision variables or output variables Furthermore there is typically a key out put variable in this case profit that is of most concern In the next few chapters we refer to it as the objective variable Therefore we distinguish this key output variable from the other output variables that we calculate along the way Table 23 Key Variables in Quality Sweaters Problem Input variables Various unit costs average order size response rate Decision variable Number mailed Key output variable Profit Other output variables Number of responses revenue and cost totals Adopt some layout and formatting conventions even if they differ from ours to make your spreadsheets readable and easy to follow The logic for converting inputs and decision variable into outputs is straightforward After you do this you can investigate how the response rate affects the profit with a sensi tivity analysis The completed spreadsheet model appears in Figure 210 See the file Breakeven Analysisxlsx First note the clear layout of the model The input cells are colored blue they are separated from the outputs headings are boldfaced several headings are indented numbers are formatted appropriately and a list to the right spells out all range names we have used See the next Excel Tip on how to create this list Also following the conven tion we use throughout the book the decision variable number mailed is colored red and the bottomline output profit is colored gray Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 210 Quality Sweaters Model 1 Quality Sweaters direct mail model Range names used 2 Averageorder ModelB11 3 Catalog inputs Model of responses Fixedcostofprinting ModelB4 4 Fixed cost of printing 20000 Response rate 8 Numbermailed ModelB8 5 Variable cost of printing mailing 025 Number of responses 8000 Numberofresponses ModelE5 6 Profit ModelE13 7 Decision variable Model of revenue costs and profit Responserate ModelE4 8 Number mailed 100000 Total Revenue 320000 Totalcost ModelE12 9 Fixed cost of printing 20000 TotalRevenue ModelE8 10 Order inputs Total variable cost of printing mailing 25000 Variablecostofprintingmailing ModelB5 11 Average order 40 Total variable cost of orders 257600 Variablecostperorder ModelB12 12 Variable cost per order 3220 Total cost 302600 13 Profit 17400 We refer to this as the Create from Selection shortcut If you like it you can get the dialog box in Figure 211 even more quickly press CtrlShiftF3 Excel Tip Creating Range Names To create a range name for a range of cells which could be a single cell highlight the cells click in the Name Box just to the left of the Formula Bar and type a range name Alternatively if a column or row of labels appears next to the cells to be rangenamed you can use these labels as the range names To do this highlight the labels and the cells to be named for example A4B5 in Figure 210 select Create from Selection on the Formulas ribbon and make sure the appropriate box in the resulting dialog box see Figure 211 is checked The labels in our example are to the left of the cells to be named so the Left column box should be checked This is a very quick way to create range names and we did it for all range names in the example In fact by keeping your finger on the Ctrl key you can select multiple ranges5 After all your ranges are selected you can sometimes create all your range names in one step Note that if a label contains any illegal rangename characters such as a space the illegal characters are converted to underscores Figure 211 Range Name Create Dialog Box If you like this tip you can perform it even faster press the F3 key to bring up the Paste Name dialog box This works only if there is at least one range name in the workbook Excel Tip Pasting Range Names Including a list of the range names in your spreadsheet is useful for documentation purposes To do this select a cell such as cell G4 in Figure 210 select the Use in Formula dropdown list from the Formulas ribbon and then click on the Paste List option You get a list of all range names and their cell addresses However if you change any of these range names delete one for example the paste list does not update automatically you have to create it again 5Many users apparently believe range names are more work than they are worth This shortcut for creating range names helps to remedy this problem 24 Breakeven Analysis 33 DEVELOPING THE SPREADSHEET MODEL To create this model you can proceed through the following steps 1 Headings and range names We have named a lot of cells more than you might want to name but you will see their value when you create formulas In general we strongly support range names but it is possible to go overboard You can waste time naming ranges that do not really need to be named Of course you can use the Create from Selection shortcut described previously to speed up the process6 2 Values of input variables and the decision variable Enter these values and format them appropriately As usual we have used our blueredgray colorcoding scheme Note that the number mailed has been designated as a decision variable not as an input variable and it is colored red not blue This is because the company gets to choose the value of this variable Finally note that some of the values have been combined in the statement of the problem For example the 3220 in cell B12 is really 80 of the 40 average order size plus the 020 per return envelope To document this process comments appear in a few cells as shown in Figure 212 Figure 212 Cell Comments in Model 1 Great Threads direct mail model Range names used 2 3 Catalog inputs Model of responses Trial value will do sensitivity analysis on printing ModelB11 ModelB4 4 Fixed cost of printing 20000 Includes 010 for printing and 015 for mailing each catalog 8 Numbermailed ModelB8 5 Variable cost of printing mailing 025 Number of responses 8000 Numberofresponses ModelE5 6 Profit ModelE13 7 Decision variable Model of revenue costs and profit Responserate ModelE4 8 Number mailed 100000 Total Revenue 320000 Totalcost ModelE12 9 Fixed cost of printing 20000 ModelE8 10 Order inputs Includes 80 of the average 40 order size plus 020 per return envelope 11 Average order 40 Total variable cost of printing mailing 25000 Variablecostofprintingmailing ModelB5 12 Variable cost per order 3220 Total variable cost of orders 257600 Variablecostperorder ModelB12 13 Total cost 302600 Profit 17400 Excel Tip Inserting Cell Comments Inserting comments in cells is a great way to document your spreadsheet models without introducing excessive clutter To enter a comment in a cell rightclick on the cell select the Insert Comment item and type your comment This creates a little red mark in the cell indicating a comment and you can see the comment by resting the cursor over the cell When a cell contains a comment you can edit or delete the comment by rightclicking on the cell and selecting the appropriate item If you want all the cell comments to be visible for example in a printout as in Figure 212 click on the File tab or Office button in Excel 2007 then on Options Excel Options in Excel 2007 then on the Advanced link and select the Comment Indicator option from the Display group Note that the Indicator Only option is the default CHANGES IN EXCEL 2010 After Microsoft got all of us used to the Office button in the upper left corner of all Office 2007 applications it switched to a File tab in Office 2010 The menu structure under this File tab is slightly different from the structure under the Office button but the functionality is basically the same In particular this is where you go to change most of the Excel options 6 We have heard of one company that does not allow any formulas in its corporate spreadsheets to include cell references they must all reference range names This is probably too extreme but that companys formulas are certainly easy to read 34 Chapter 2 Introduction to Spreadsheet Modeling 3 Model the responses You have not yet specified the response rate to the mailing so enter any reasonable value such as 8 in the Responserate cell You will perform sensitivity on this value later on Then enter the formula NumbermailedResponserate in cell E5 Are you starting to see the advantage of range names 4 Model the revenue costs and profits Enter the formula NumberofresponsesAverageorder in cell E8 enter the formulas Fixedcostofprinting VariablecostofprintingmailingNumbermailed and NumberofresponsesVariablecostperorder in cells E9 E10 and E11 enter the formula SUME9E11 in cell E12 and enter the formula TotalrevenueTotalcost in cell E13 These formulas should all be selfexplanatory especially because of the range names used Excel Tip Entering Formulas with Range Names To enter a formula that contains range names you do not have to type the full range names You actually have two convenient options One you can point to the cells and range names will appear in your formulas Or two you can start typing the range name in the formula and after a few letters Excel will show you a list you can choose from Forming a OneWay Data Table Now that a basic model has been created the questions posed by the company can be answered For question 1 you can form a oneway data table to show how profit varies with the response rate as shown in Figure 213 Data tables are used often in this book so make sure you understand how to create them We will walk you through the procedure once or twice but from then on you are on your own First enter a sequence of trial values of the response rate in column A and enter a link to profit in cell B17 with the formula Profit This cell is shaded for emphasis but this isnt necessary In general other outputs could be part of the table and they would be placed in columns C D and so on There would be a link to each output in row 17 Finally highlight the entire table range A17B27 and select Data Table from the WhatIf Analysis dropdown list on the Data ribbon to bring up the Figure 213 Data Table for Profit 15 Question 1 sensitivity of profit to response rate Profit versus Response Rate 16 Response rate Profit 17 17400 18 1 37200 19 2 29400 20 3 21600 21 4 13800 22 5 6000 23 6 1800 24 7 9600 25 8 17400 26 9 25200 27 10 33000 24 Breakeven Analysis 35 Data tables are also called whatif tables They let you see what happens to selected outputs as selected inputs change Figure 214 Data Table Dialog Box dialog box in Figure 214 Fill it in as shown to indicate that the only input Responserate is listed along a column You can enter either a range name or a cell address in this dialog box The easiest way is to point to the cell When you click on OK Excel substitutes each response rate value in the table into the Responserate cell recalculates profit and reports it in the table For a final touch you can create a chart of the values in the data table To do this highlight the A18B27 range and select the type of chart you want from the Insert ribbon Then you can fix it up by adding titles removing the legend and making other modifications to suit your taste Excel Tool OneWay Data Table A oneway data table allows you to see how one or more output variables vary as a single input variable varies over a selected range of values These input values can be arranged vertically in a column or horizontally in a row We will explain only the vertical arrangement because it is the most common To create the table enter the input values in a column range such as A18A27 of Figure 213 and enter links to one or more output cells in columns to the right and one row above the inputs as in cell B17 of Figure 213 Then highlight the entire table beginning with the upperleft blank cell A17 in the figure select Data Table from the WhatIf Analysis dropdown list on the Data ribbon and fill in the resulting dialog box as in Figure 214 Leave the Row Input cell blank and use the cell where the original value of the input variable lives as the Column Input cell When you click on OK each value in the left column of the table is substituted into the column input cell the spreadsheet recalculates and the resulting value of the output is placed in the table Also if you click anywhere in the body of the table B18B27 in the figure you will see that Excel has entered the TABLE function to remind you that a data table lives here Note that the column input cell must be on the same worksheet as the table itself otherwise Excel issues an error As the chart indicates profit increases in a linear manner as the response rate varies More specifically each percentage point increase in the response rate increases profit by 7800 Here is the reasoning Each percentage point increase in response rate results in 100000001 1000 more orders Each order yields a revenue of 40 on average but incurs a variable cost of 3220 The net gain in profit is 780 per order or 7800 for 1000 orders Using Goal Seek From the data table you can see that profit changes from negative to positive when the response rate is somewhere between 5 and 6 Question 2 asks for the exact breakeven point You could find this by trial and error but it is easier to use Excels Goal Seek tool Essentially Goal Seek is used to solve a single equation in a single unknown Here the equation is Profit0 and the unknown is the response rate In Excel terminology the unknown is called the changing cell because you can change it to make the equation true The purpose of the Goal Seek tool is to solve one equation in one unknown It is used here to find the response rate that makes profit equal to 0 24 Breakeven Analysis 37 To implement Goal Seek select Goal Seek from the WhatIf Analysis dropdown list on the Data ribbon and fill in the resulting dialog box as shown in Figure 215 Range names or cell addresses can be used in the top and bottom boxes but a number must be entered in the middle box After you click on OK the Responserate and Profit cells have values 577 and 0 In words if the response rate is 577 Great Threads breaks even If the response rate is greater than 577 the company makes money if the rate is less than 577 the company loses money Of course this assumes that the company mails 100000 catalogs If it sends more or fewer catalogs the breakeven response rate will change Excel Tool Goal Seek The purpose of the Goal Seek tool is to solve one equation in one unknown Specifically Goal Seek allows you to vary a single input cell to force a single output cell to a selected value To use it select Goal Seek from the WhatIf Analysis dropdown list on the Data ribbon and fill in the resulting dialog box in Figure 215 Enter a reference to the output cell in the Set cell box enter the numeric value you want the output cell to equal in the To value box and enter a reference to the input cell in the By changing cell box Note that Goal Seek sometimes stops when the Set cell is close but not exactly equal to the desired value To improve Goal Seeks accuracy click on the File tab the Office button in Excel 2007 then Options Excel Options in Excel 2007 and then the Formulas link Then check the Enable iterative calculation box and reduce Maximum Change to any desired level of precision We chose a precision level of 0000001 For this level of precision Goal Seek searches until profit is within 0000001 of the desired value 0 Limitations of the Model Question 3 asks whether the company should proceed with the mailing if the response rate is only 3 From the data table see Figure 213 the apparent answer is no because profit is negative However like many companies we are taking a shortterm view with this rea soning The model does not include the fact that many customers who respond to direct mail will reorder in the future The company nets 780 per order If each of the respon dents ordered two more times say the company would earn 30007802 46800 more than appears in the model and profit would then be positive The moral is that man agers must look at the longterm impact of their decisions However if you want to incor porate the long term explicitly into the model you must build a more complex model Finally question 4 asks about the impact of uncertainty in the model Obviously not all model inputs are known with certainty For example the size of an order is not always 40it might range say from 10 to 100 When there is a high degree of uncertainty about model inputs it makes little sense to talk about the profit level or the breakeven response rate It makes more sense to talk about the probability that profit will have a certain value or the probability that the company will break even You will see how this can be done in the following example and in many more such examples in Chapters 10 through 12 Using the Formula Auditing Tool The model in this example is fairly small and simple Even so you can use a handy Excel feature to see how all the parts fit together This is the Formula Auditing tool which is available on the Formulas ribbon See Figure 216 Figure 215 Goal Seek Dialog Box Later chapters especially Chapters 10 through 12 deal explicitly with uncertainty Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 216 Formula Auditing Toolbar The Formula Auditing tool is indispensable for untangling the logic in a spreadsheet especially if someone else developed it The Trace Precedents and Trace Dependents buttons are probably the most useful buttons in this group To see which formulas have direct links to the Numberofresponses cell select this cell and click on the Trace Dependents button Arrows are drawn to each cell that directly depends on the number of responses as shown in Figure 217 Alternatively to see which cells are used to create the formula in the Totalrevenue cell select this cell and click on the Trace Precedents button Now you see that the Averageorder and Numberofresponses cells are used directly to calculate revenue as shown in Figure 218 Using these two buttons you can trace your logic or someone elses logic as far backward or forward as you like When you are finished just click on the Remove Arrows button Figure 217 Dependents of Numberofresponses Cell Figure 218 Precedents of Totalrevenue Cell Excel Tool Formula Auditing Toolbar The formula auditing toolbar allows you to see dependents of a selected cell which cells have formulas that reference this cell or precedents of a given cell which cells are referenced in this cells formula In fact you can even see dependents or precedents that reside on a different worksheet In this case the auditing arrows appear as dashed lines and point to a small spreadsheet icon By doubleclicking on the dashed line you can see a list of dependents or precedents on other worksheets These tools are especially 24 Breakeven Analysis 39 useful for understanding how someone elses spreadsheet works Unlike in pre2007 versions of Excel the Formula Auditing tools in Excel 2007 and 2010 are clearly visible on the Formulas ribbon You can place charts on the same worksheet as the underlying data or on separate chart sheetsThe choice is a matter of personal preference P R O B L E M S Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 The sensitivity analysis in the Quality Sweaters exam ple was on the response rate Suppose now that the response rate is known to be 8 and the company wants to perform a sensitivity analysis on the number mailed After all this is a variable under direct control of the company Create a oneway data table and a cor responding line chart of profit versus the number mailed where the number mailed varies from 80000 to 150000 in increments of 10000 Does it appear from the results you see here that there is an optimal number to mail from all possible values that maxi mizes profit Write a concise memo to management about your results 2 Continuing the previous problem use Goal Seek for each value of number mailed once for 80000 once for 90000 and so on For each find the response rate that allows the company to break even Then chart these values where the number mailed is on the hori zontal axis and the breakeven response rate is on the vertical axis Explain the behavior in this chart in a brief memo to management 3 In the Quality Sweaters model the range E9E11 does not have a range name Open your completed Excel file and name this range Costs Then look at the for mula in cell E12 It does not automatically use the new range name Modify the formula so that it does Then click on cell G4 and paste the new list of range names over the previous list SkillExtending Problem 4 As the Quality Sweaters problem is now modeled if all inputs remain fixed except for the number mailed profit will increase indefinitely as the number mailed increases This hardly seems realisticthe company could become infinitely rich Discuss realistic ways to modify the model so that this unrealistic behavior is eliminated Is the spreadsheet layout in Figure 212 the best possible layout This question is not too crucial because this model is so small However we have put all the inputs together usu ally a good practice and we have put all the outputs together in a logical order You might want to put the answers to questions 1 and 2 on separate worksheets but with such a small model it is arguably better to keep everything on a single worksheet We generally avoid separate worksheets unless things start getting bigger and more complex One other issue is the placement of the chart From the Chart Tools Design ribbon you can click on the Move Chart button to select whether you want to place the chart on the worksheet floating above the cells or on a separate chart sheet that has no rows or columns This choice depends on your personal preferenceneither choice is necessarily better than the otherbut for this small model we favor keeping everything on a single worksheet Finally we could have chosen the number mailed rather than the response rate as the basis for a sensitivity analysis A sensitivity analysis is typically based on an uncertain input variable such as the response rate or a decision variable that the decision maker con trols Fortunately there is no limit to the number of data tables you can create for a partic ular model MODELING ISSUES Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 25 ORDERING WITH QUANTITY DISCOUNTS AND DEMAND UNCERTAINTY In the following example we again attempt to find the appropriate level of some activity how much of a product to order when customer demand for the product is uncertain Two important features of this example are the presence of quantity discounts and the explicit use of probabilities to model uncertain demand Except for these features the problem is very similar to the one discussed in Example 21 EXAMPLE 24 ORDERING WITH QUANTITY DISCOUNTS AT SAMS BOOKSTORE Sams Bookstore with many locations across the United States places orders for all of the latest books and then distributes them to its individual bookstores Sams needs a model to help it order the appropriate number of any title For example Sams plans to order a popular new hardback novel which it will sell for 30 It can purchase any number of this book from the publisher but due to quantity discounts the unit cost for all books it orders depends on the number ordered Specifically if the number ordered is less than 1000 the unit cost is 24 After each 1000 the unit cost drops to 23 for at least 1000 copies to 2225 for at least 2000 to 2175 for at least 3000 and to 2130 the lowest possible unit cost for at least 4000 For example if Sams orders 2500 books its total cost is 22252500 55625 Sams is very uncertain about the demand for this bookit estimates that demand could be anywhere from 500 to 4500 Also as with most hardback novels this one will eventually come out in paperback Therefore if Sams has any hardbacks left when the paperback comes out it will put them on sale for 10 at which price it believes all leftovers will be sold How many copies of this hardback novel should Sams order from the publisher Business Objectives To create a model to determine the companys profit given fixed values of demand and the order quantity and then to model the demand uncertainty explicitly and to choose the expected profitmaximizing order quantity Excel Objectives To learn how to build in complex logic with IF formulas to get online help about Excel functions with the fx button to learn how to use lookup functions to see how twoway data tables allow you to answer more extensive whatif questions and to learn about Excels SUMPRODUCT function Solution The key variables for this model appear in Table 24 The primary modeling tasks are 1 to show how any combination of demand and order quantity determines the number of units sold both at the regular price and at the leftover sale price and 2 to calculate the total ordering cost for any order quantity After you accomplish these tasks you can model the uncertainty of demand explicitly and then find the optimal order quantity Table 24 Key Variables for Sams Bookstore Problem Input variables Unit prices table of unit costs specifying quantity discount structure Uncertain variable Demand Decision variable Order quantity Key output variable Profit Other output variables Units sold at each price revenue and cost totals The first step is to develop a spreadsheet model to calculate Sams profit for any order quantity and any possible demand Then you can perform a sensitivity analysis to see how profit depends on these two quantities Finally you can decide how Sams might choose the optimal order quantity DEVELOPING THE SPREADSHEET MODEL The profit model appears in Figure 219 See the file Quantity Discountsxlsx Note that the order quantity and demand in the Orderquantity and Demand cells are trial values Comments in these cells are a reminder of this You can put any values in these cells just to test the logic of the model The Orderquantity cell is colored red because the company can choose its value In contrast the Demand cell is colored green here and in later chapters to indicate that this input value is uncertain and is being treated explicitly as such Also note that a table is used to indicate the quantity discounts cost structure You can use the following steps to build the model Figure 219 Sams Profit Model Whenever the term trial value is used for an input or a decision variable you can be sure that we will follow up with a data table or in later chapters by running Solver to optimize 1 Inputs and range names Enter all inputs and name the ranges as indicated Note that the Create from Selection shortcut was used to name all ranges except for CostLookup and Probabilities For these latter two you can highlight the ranges and enter the names in the Name Boxthe manual method Why the difference To use the Create from Selection shortcut you must have appropriate labels in adjacent cells Sometimes this is simply not convenient 2 Revenues The company can sell only what it has and it sells any leftovers at the discounted sale price Therefore enter the formulas MINOrderquantityDemand IFOrderquantityDemand OrderquantityDemand0 and UnitssoldatregularpriceRegularprice UnitssoldatleftoverpriceLeftoverprice in cells B15 B16 and B17 The logic in the first two of these cells is necessary to account correctly for the cases when the order quantity is greater than demand and when it is less than or equal to demand Note that you could use the following equivalent alternative to the IF function in cell B16 MAXOrderquantityDemand0 42 Chapter 2 Introduction to Spreadsheet Modeling Excel Tool fx Button and Function Library Group If you want to learn more about how an Excel function operates click on the fx button next to the Formula bar This is called the Insert Function button although some people call it the Function Wizard If there is already a function such as an IF function in a cell and you then click on the fx button you will get help on this function If you select an empty cell and then click on the f x button you can c hoose a function to g et help on The same help is available from the Function Library group on the Formulas ribbon 3 Total ordering cost Depending on the order quantity you can find the appropriate unit cost from the unit cost table and multiply it by the order quantity to obtain the total ordering cost This can be accomplished with a complex nested IF formula but a much better way is to use the VLOOKUP function Specifically enter the formula VLOOKUPOrderquantityCostLookup2Orderquantity in cell B18 The VLOOKUP part of this formula says to compare the order quantity to the first leftmost column of the table in the CostLookup range and return the corresponding value in the second column because the last argument is 2 Excel Function VLOOKUP The VLOOKUP function acts like a tax table where you look up the tax corr esponding to your adjusted gross income from a table of incomes and taxes To use it first create a ver tical lookup table with values to use for comparison listed in the left column of the table and corresponding output values in as many columns to the right as you lik e See the CostLookup r ange in F igure 219 for an e xample Then the VLOOKUP function tak es three or four arguments 1 the value you want to compare to the values in the left column 2 the lookup table range 3 the index of the column you want the returned value to come from where the index of the left column is 1 the inde x of the next column is 2 and so on and optionally 4 TRUE for an approximate match the default or FALSE for an e xact match If you omit the last ar gument the values in the left column of the table must be entered in ascending order See online help for more details If the last argument is TRUE or is omitted Excel scans down the leftmost column of the table and finds the last entry less than or equal to the first argument In this sense it finds an approximate match There is also an HLOOKUP function that works exactly the same way except that the lookup table is arranged in rows not columns 4 Profit Calculate the profit with the formula RevenueCost TwoWay Data Table The next step is to create a twoway data table for profit as a function of the order quantity and demand see Figure 220 To create this table first enter a link to the profit with the formula Profit in cell A22 and enter possible order quantities and possible demands in column A and row 22 respectively We used the same values for both order quantity and demand from 500 to 4500 in increments of 500 This is not necessarythe demand could change in increments of 100 or even 1but it is reason able Perhaps Sams is required by the publisher to order in multiples of 500 Then select Data Table from the WhatIf Analysis dropdown list on the Data ribbon and enter the Demand cell as the Row Input cell and the Orderquantity cell as the Column Input cell see Figure 221 A twoway data table allows you to see how a single output varies as two inputs vary simultaneously Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 220 Profit as a Function of Order Quantity and Demand This is actually a preview of decision making under uncertainty To calculate an expected profit you multiply each profit by its probability and add the products This topic is covered in depth in Chapter 9 Figure 221 Dialog Box for TwoWay Data Table Excel Tool TwoWay Data Table A twoway data table allows you to see how a single output cell varies as you vary two input cells Unlike a oneway data table only a single output cell can be chosen To create this type of table enter a reference to the output cell in the topleft corner of the table enter possible values of the two inputs below and to the right of this corner cell and highlight the entire table Then select Data Table from the WhatIf Analysis dr opdown on the Data ribbon and enter references to the cells where the original two input variables live The Row Input cell corr esponds to the values along the top r ow of the table and the Column Input cell corresponds to the values along the leftmost column of the table When you clic k on OK Excel substitutes eac h pair of input values into these two input cells recalculates the spr eadsheet and enter s the corr esponding output value in the table By clicking on any cell in the body of the table you can see that Excel also enters the function TABLE as a reminder that the cell is part of a data table The resulting data table shows that profit depends heavily on both order quantity and demand and by scanning across rows how higher demands lead to larger profits But which order quantity Sams should select is still unclear Remember that Sams has complete control over the order quantity it can choose the row of the data table but it has no direct control over demand it cannot choose the column The ordering decision depends not only on which demands are possible but on which demands are likely to occur The usual way to express this information is with a set of probabilities that sum to 1 Suppose Sams estimates these as the values in row 35 of Figure 222 These estimates are probably based on other similar books it has sold in the past The most likely demands are 2000 and 2500 with other values on both sides less likely You can use these probabilities to find an expected profit for each order quantity This expected profit is a weighted average of the profits in any row in the data table using the probabilities as the weights The easiest way to do this is to enter the formula SUMPRODUCTB23J23Probabilities Figure 222 Comparison of Expected Profits in cell B38 and copy it down to cell B46 You can also create a bar chart of these expected profits as shown in Figure 222 Excel refers to these as column charts The height of each bar is the expected profit for that particular order quantity Excel Function SUMPRODUCT The SUMPRODUCT function takes two range arguments which must be exactly the same size and shape and it sums the products of the corresponding values in these two r anges For example the formula SUMPRODUCTA10B11E12F13 is a shortcut for a formula involving the sum of 4 pr oducts A10E12A11E13B10F12B11F13 This is an extremely useful function especially when the r anges involved are large and it is used repeatedly throughout this book Actually the SUMPR ODUCT function can have mor e than two r ange arguments all of the same size and shape but the most common use of SUMPRODUCT is when only two ranges are involved The largest of the expected profits 12250 corresponds to an order quantity of 2000 so we would recommend that Sams order 2000 copies of the book This does not guarantee that Sams will make a profit of 12250the actual profit depends on the eventual demandbut it represents a reasonable way to proceed in the face of uncertain demand You will learn much more about making decisions under uncertainty and the expected value criterion in Chapter 9 PROBLEMS SkillBuilding Problems 5 In some ordering problems like the one for Sams Bookstore whenever demand exceeds existing inventory the excess demand is not lost but is filled by expedited ordersat a premium cost to the company Change Sams model to reflect this behavior Assume that the unit cost of expediting is 40 well above the highest regular unit cost 6 The spreadsheet model for Sams Bookstore contains a twoway data table for profit versus order quantity and demand Experiment with Excels chart types to create a chart that shows this information graphically in an intuitive format Choose the format you would choose to give a presentation to your boss 7 In the Sams Bookstore problem the quantity discount structure is such that all the units ordered have the same unit cost For example if the order quantity is 2500 then each unit costs 2225 Sometimes the quantity discount structure is such that the unit cost for the first so many items is one value the unit cost for the next so many units is a slightly lower value and so on Modify the model so that Sams pays 24 for units 1 to 1500 23 for units 1501 to 2500 and 22 for units 2501 and above For example the total cost for an order quantity of 2750 is 150024 100023 25022 Hint Use IF functions not VLOOKUP SkillExtending Problems 8 The current spreadsheet model essentially finds the expected profit in several steps It first finds the profit in cell B19 for a fixed value of demand Then it uses a data table to find the profit for each of several demands and finally it uses SUMPRODUCT to find the expected profit Modify the model so that expected profit is found directly without a data table To do this change row 11 so that instead of a single demand there is a list of possible demands those currently in row 34 Then insert a new row below row 11 that lists the probabilities of these demands Next in the rows below the Profit Model label calculate the units sold revenue cost and profit for each demand For example the quantities in column C will be for the second possible demand Finally use SUMPRODUCT to calculate expected profit below the Profit row 9 Continuing Problem 5 create a twoway data table for expected profit with order quantity along the side and unit expediting cost along the top Allow the order quantity to vary from 500 to 4500 in increments of 500 and allow the unit expediting cost to vary from 36 to 45 in increments of 1 Each column of this table will allow you to choose an optimal order quantity for a given unit expediting cost How does this best order quantity change as the unit expediting cost increases Write up your results in a concise memo to management Hint You will have to modify the existing spreadsheet model so that there is a cell for expected profit that changes automatically when you change either the order quantity or the unit expediting cost See Problem 8 for guidelines 46 Chapter 2 Introduction to Spreadsheet Modeling Business Objectives To estimate the relationship between demand and price and to use this relationship to find the optimal price to charge Excel Objecti ves To illustrate Excels Trendline tool and to illustrate conditional formatting Solution This example is divided into two parts estimating the relationship between price and demand and creating the profit model Estimating the Relationship Between Price and Demand A scatterplot of demand versus price appears in Figure 224 This can be created in the usual way with Excels Scatter chart Obviously demand decreases as price increases but the goal is to quantify this relationship Therefore after creating this chart rightclick on any point on the chart to bring up the dialog box in Figure 225 This allows you to super impose several different curves including a straight line on the scatterplot We consider 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A B C Demand for golf clubs Month Price Demand 1 450 45 2 300 103 3 440 49 4 360 86 5 290 125 6 450 52 7 340 87 8 370 68 9 500 45 10 490 44 11 430 58 12 390 68 Figure 223 Demand and Price Data for Golf Clubs 90 100 110 120 130 40 50 60 70 80 90 280 320 360 400 440 480 520 Figure 224 Scatterplot of Demand Versus Price Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it three possibilities the linear power and exponential curves defined by the following general equations where y and x a general output and a general input correspond to demand and price for this example n Linear y a bx n Power y axb n Exponential y aebx Before proceeding we describe some general properties of these three functions because of their widespread applicability The linear function is the easiest Its graph is a straight line When x changes by 1 unit y changes by b units The constant a is called the intercept and b is called the slope The power function is a curve except in the special case where the exponent b is 1 Then it is a straight line Assuming that a is positive the shape of this curve depends primarily on the exponent b If b 1 y increases at an increasing rate as x increases If 0 b 1 y increases but at a decreasing rate as x increases Finally if b 0 y decreases as x increases An important property of the power curve is that when x changes by 1 y changes by a constant percentage and this percentage is approximately equal to b For example if y 100x235 then every 1 increase in x leads to an approximate 235 decrease in y The exponential function also represents a curve whose shape depends on the constant b in the exponent Again assume that a is positive Then if b 0 y increases as x increases if b 0 y decreases as x increases An important property of the exponential function is that if x changes by 1 unit y changes by a constant percentage and this percentage is approximately equal to 100 b For example if y 100e0014x then whenever x increases by 1 unit y decreases by approximately 14 Here e is the special number 27182 and e to any power can be calculated in Excel with the EXP function For example you can calculate e0014 with the formula EXP0014 48 Chapter 2 Introduction to Spreadsheet Modeling y 03546x 2113147 100 110 120 130 Linear Fit 40 50 60 70 80 90 100 280 320 360 400 440 480 520 Figure 226 BestFitting Straight Line y 58710642031x19082 100 110 120 130 Power Fit 40 50 60 70 80 90 100 280 320 360 400 440 480 520 Figure 227 BestFitting Power Curve Returning to the example if you superimpose any of these curves on the scatterplot of demand versus price Excel chooses the bestfitting curve of that type Better yet if you check the Display Equation on Chart option you see the equation of this bestfitting curve Doing this for each type of curve gives the results in Figures 226 227 and 228 The equations might not appear exactly as in the figures However they can be resized and reformatted to appear as shown Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 228 BestFitting Exponential Curve Exponential Fit Each of these curves provides the bestfitting member of its family to the demandprice data but which of these three is best overall You can answer this question by finding the mean absolute percentage error MAPE for each of the three curves To do so for any price in the data set and any of the three curves first predict demand by substituting the given price into the equation for the curve The predicted demand is typically not the same as the observed demand so you can calculate the absolute percentage error APE with the general formula APE Observed demand Predicted demand Observed demand 21 Then for any curve MAPE is the average of these APE values The curve with the smallest MAPE is the best fit overall The calculations appear in Figure 229 After manually entering the parameters of the equations from the scatterplots into column B you can proceed as follows 1 Predicted demands Substitute observed prices into the linear power and exponential functions to obtain the predicted demands in columns E F and G Specifically enter the formulas B19B20B4 B22B4B23 and B25EXPB26B4 in cells E19 F19 and G19 and copy them down their respective columns Figure 229 Finding the BestFitting Curve Overall A B C D E F G H I J 17 Parameters of bestfitting curves Prediction Absolute percentage error 18 Linear Linear Power Exponential Linear Power Exponential 19 Intercept 21131 5174 5080 5120 1498 1289 1378 20 Slope 03546 10493 11012 10694 187 691 383 21 Power 5529 5302 5378 1283 821 975 22 Constant 5871064 8365 7776 7965 273 958 738 23 Exponent 19082 10848 11748 11232 1322 601 1014 24 Exponential 5174 5080 5120 050 231 153 25 Constant 46651 9075 8673 8787 431 032 100 26 Exponent 000491 8011 7380 7584 1781 853 1152 27 3401 4155 4006 2442 767 1099 28 3756 4318 4207 1465 186 438 29 5883 5540 5649 143 448 261 30 7302 6675 6874 738 184 109 31 32 MAPE 968 588 650 2 Average percentage errors Apply Equation 21 to calculate APEs in columns H I and J Specifically enter the general formula ABSC4E19C4 in cell H19 and copy it to the range H19J30 Do you see why column C is made absolute Remember that this is where the observed demands are stored 3 MAPE Average the APEs in each column with the AVERAGE function to obtain the MAPEs in row 32 Evidently the power curve provides the best fit with a MAPE of 588 In other words its predictions are off on average by 588 This power curve predicts that each 1 increase in price leads to an approximate 19 decrease in demand Economists call this relationship elasticdemand is quite sensitive to price DEVELOPING THE PROFIT MODEL Now we move to the profit model using the bestfitting power curve to predict demand from price The key variables appear in Table 25 Note there is now one input variable unit variable cost and one decision variable unit price The red background for the decision variable distinguishes it as such The profit model is straightforward to develop using the following steps see Figure 230 Table 25 Key Variables for Golf Club Problem Input variable Unit cost to produce Decision variable Unit price Key output variable Profit Other output variables Predicted demand total revenue total cost 70 Figure 230 Profit Model A B C D E 1 Profit model using best fitting power curve for estimating demand 2 3 Parameters of bestfitting power curve from Estimation sheet 4 Constant 5871064 5 Exponent 19082 6 7 Monetary inputs 8 Unit cost to produce 250 9 10 Decision variable 11 Unit price trial value 400 12 13 Profit model 14 Predicted demand 63601 15 Total revenue 25441 16 Total cost 15900 17 Profit 9540 1 Predicted demand Calculate the predicted demand in cell B14 with the formula B4B11B5 This uses the power function that was estimated earlier 2 Revenue cost profit Enter the following formulas in cells B15 B16 and B17 B11B14 B8B14 and B15B16 The assumption here is that the company produces exactly enough sets of clubs to meet customer demand Maximizing Profit To see which price maximizes profit you can build the data table shown in Figure 231 Here the column input cell is B11 and the linking formula in cell B25 is B17 The corresponding scatter chart shows that profit first increases and then decreases You can find the maximum profit and corresponding price in at least three ways First you can attempt to read them from the chart Second you can scan down the data table for the maximum profit which is shown in the figure The following Excel Tip describes a third method that uses some of Excels more powerful features Excel Tip Conditional Formatting Cell B53 in Figure 231 is colored because it corresponds to the maximum profit in the column but Excels Conditional Formatting tool can do this for youautomatically To color the maximum profit select the range of profits B26B75 click on the Conditional Formatting dropdown arrow then TopBottom Rules and then Top 10 Items to bring up the dialog box in Figure 232 By asking for the top 1 item the maximum value in the range is colored You can experiment with the many other Conditional Formatting options This is a great tool 7 The value in cell B52 also appears to be the maximum but to two decimals it is slightly lower 26 Estimating the Relationship Between Price and Demand 51 71 Figure 231 Profit as a Function of Price A B C D E F G H I 19 Maximum profit from data table below with corresponding best unit price 20 Maximum profit 10409 21 Best price 530 22 23 Data table for Profit as a function of Unit price 24 Unit price Profit 25 9540 26 260 1447 27 270 2693 28 280 3769 29 290 4699 30 300 5506 31 310 6207 32 320 6815 33 330 7345 34 340 7805 35 350 8206 36 360 8554 37 370 8856 38 380 9118 39 390 9345 40 400 9540 41 410 9708 42 420 9851 43 430 9973 44 440 10075 45 450 10160 46 460 10230 47 470 10286 48 480 10330 49 490 10363 50 500 10387 51 510 10402 52 520 10409 53 530 10409 Maximum profit 54 540 10403 55 550 10391 56 560 10375 57 570 10354 58 580 10329 59 590 10300 60 600 10269 Profit versus Price 12000 10000 8000 Profit 6000 4000 2000 0 200 300 340 380 420 460 500 540 Price 52 Chapter 2 Introduction to Spreadsheet Modeling Figure 232 Conditional Formatting Dialog Box What about the corresponding best price shown in cell B21 of Figure 231 You could enter this manually but wouldnt it be nice if you could get Excel to find the maximum profit in the data table determine the price in the cell to its left and report it in cell B21 all automatically This is indeed possible Just enter the formula INDEXA26A75MATCHB20B26B7501 in cell B21 and the best price appears This formula uses two Excel functions MATCH and INDEX MATCH compares the first argument the maximum profit in cell B20 to the range specified in the second argument the range of profits and returns the index of the cell where a match appears The third argument 0 specifies that you want an exact match In this case the MATCH function returns 28 because the maximum profit is in the 28th cell of the profits range Then the INDEX function is called effectively as INDEXA26A75281 The first argument is the range of prices the second is a row index and the third is a column index Very simply this function says to return the value in the 28th row and first column of the prices range To learn more about these functions you can click on the fx button and examine the functions in the Lookup Reference category After experimenting you can see that the INDEX and MATCH combination solves the problem You dont have to memorize these functions although this combination really does come in handy Rather you can often solve a problem by investigating some of Excels less wellknown features You dont even need a manualeverything is in online help Sensitivity to Variable Cost We now return to question 2 in the example How does the best price change as the unit variable cost changes You can answer this question with a twoway data table Remember that this is a data table with two inputsone along the left side and the other across the top rowand a single output The two inputs for this problem are unit variable cost and unit price and the single output is profit The corresponding data table is in the range A83F168 the top part of which appears in Figure 233 To develop this table enter desired inputs in column A and row 83 enter the linking formula B17 in cell A83 it always goes in the topleft corner of a twoway data table highlight the entire table select Data Table from the WhatIf Analysis dropdown list and enter B8 as the Row Input cell and B11 as the Column Input cell Figure 233 Profit as a Function of Unit Cost and Unit Price As before you can scan the columns of the data table for the maximum profits and enter them manually in rows 79 and 80 Alternatively you can use the Excel features described in the previous Excel Tip to accomplish these tasks Take a look at the finished version of the file for details This file also explains how conditional formatting is used to color the maximum profit in each column of the table Then you can create a chart of maximum profit or best price versus unit cost The chart in Figure 233 shows that the maximum profit decreases but at a decreasing rate as the unit cost increases Limitations of the Model Question 3 asks you to step back from all these details and evaluate whether the model is realistic First there is no real reason to restrict golf club prices to multiples of 10 This was only required so that a data table could be used to find the profitmaximizing price Ideally you should search over all possible prices to find the profitmaximizing price Fortunately Excels builtin Solver tool enables you to accomplish this task fairly easily The problem of finding a profitmaximizing price is an example of an optimization model In optimization models you try to maximize or minimize a specified output cell by changing the values of the decision variable cells Chapters 38 and 16 contain a detailed discussion of optimization models A second possible limitation of the model is the implicit assumption that price is the only factor that influences demand In reality other factors such as advertising the state of the economy competitors prices strength of competition and promotional expenses also influence demand In Chapter 14 you will learn how to use multiple regression to analyze the dependence of one variable on two or more other variables This technique allows you to incorporate other factors into the model for profit A final limitation of the model is that demand might not equal sales For example if actual demand for golf clubs during a year is 70000 but the companys annual capacity is only 50000 the company will observe sales of only 50000 This will cause it to underestimate actual demand and the curvefitting method will produce biased predictions Can you guess the probable effect on pricing decisions Other Modeling Issues The layout of the Golf Club Demandxlsx file is fairly straightforward However note that instead of a single worksheet there are two worksheets partly for logical purposes and partly to reduce clutter There is one worksheet for estimation of the demand function and the various scatterplots and there is another for the profit model One last issue is the placement of the data tables for the sensitivity analysis You might be inclined to put these on a separate Sensitivity worksheet However Excel does not allow you to build a data table on one worksheet that uses a row or column input cell from another worksheet Therefore you are forced to put the data tables on the same worksheet as the profit model PROBLEMS SkillBuilding Problems 10 Suppose you have an extra six months of data on demands and prices in addition to the data in the example These extra data points are 35084 38572 41067 40062 33092 and 48053 The price is shown first and then the demand at that price After adding these points to the original data use Excels Trendline tool to find the bestfitting linear power and exponential trend lines Finally calculate the MAPE for each of these based on all 18 months of data Does the power curve still have the smallest MAPE 11 Consider the power curve y 10000x235 Calculate y when x 5 when x 10 and when x 20 For each of these values of x find the percentage change in y when x increases by 1 That is find the percentage change in y when x increases from 5 to 505 when it increases from 10 to 101 and when it increases from 20 to 202 Is this percentage change constant What number is it very close to Write a brief memo on what you have learned about power curves from these calculations 12 Consider the exponential curve y 1000e0014x Calculate y when x 5 when x 10 and when x 20 For each of these values of x find the percentage change in y when x increases by one unit That is find the percentage change in y when x increases from 5 to 6 when it increases from 10 to 11 and when it increases from 20 to 21 Is this percentage change constant When expressed as a decimal what number is it very close to Write a brief memo on what you have learned about exponential curves from these calculations SkillExtending Problem 13 In the profit model in this section we used the power curve to relate demand and price because it has the lowest MAPE However the exponential curve was not far behind Rework the profit model using the exponential curve to relate demand to price Write a brief memo indicating whether you get basically the same results as with the power curve or you get substantially different results 27 DECISIONS INVOLVING THE TIME VALUE OF MONEY In many business situations cash flows are received at different points in time and a company must determine a course of action that maximizes the value of cash flows Here are some examples Should a company buy a more expensive machine that lasts for 10 years or a less expensive machine that lasts for 5 years What level of plant capacity is best for the next 20 years A company must market one of several midsize cars Which car should it market To make decisions when cash flows are received at different points in time the key concept is that the later a dollar is received the less valuable the dollar is For example suppose you can invest money at a 5 annual interest rate Then 100 received now is essentially equivalent to 105 a year from now The reason is that if you have 100 now you can invest it and gain 005 in interest in one year If r 005 is the interest rate expressed as a decimal we can write this as 100 now 105 a year from now 1001 r 22 Dividing both sides of Equation 22 by 1 r we can rewrite it as 100 11 r now 100 a year from now 23 The value 11 r in Equation 23 is called the discount factor and it is always less than 1 The quantity on the left which evaluates to 0952 for r 005 is called the present value of 100 received a year from now The idea is that if you had 0952 now you could invest it at 5 and have it grow to 100 in a year In general if money can be invested at annual rate r compounded each year then 1 received t years from now has the same value as 11 rt dollars received todaythat is the 1 is discounted by the discount factor raised to the t power If you multiply a cash flow received t years from now by 11 rt to obtain its present value then the total of these present values over all years is called the net present value NPV of the cash flows Basic financial theory states that projects with positive NPVs increase the value of the company whereas projects with negative NPVs decrease the value of the company The rate r usually called the discount rate used by major corporations generally comes from some version of the capital asset pricing model The value of r used to evaluate any particular project depends on a number of things and can vary from project to project Because this is the focus of finance courses we will not pursue it here But given a suitable value of r the following example illustrates how spreadsheet models and the time value of money can be used to make complex business decisions The discount factor is 1 divided by 1 plus the discount rate To discount a cash flow that occurs t years from now multiply it by the discount factor raised to the t power The NPV is the sum of all discounted cash flows 56 Chapter 2 Introduction to Spreadsheet Modeling FUNDAMENTAL INSIGHT TheTimeValue of Money Money earned in the future is less valuable than money earned todayfor the simple reason that money earned today can be in vested to earn inter est Similarly costs incurred in the futur e ar e less costly than costs incurred today which is why you dont simply sum up revenues and costs in a multiperiod modelYou instead discount future revenues and costs f or a fair compari son with r evenues and costs incur red toda y The resulting sum of discounted cash flo ws is the net pr e sent value NPV and it f orms the cornerstone of much of financial theory and applications E X A M P L E 26 CALCULATING NPV AT ACRON A cron is a large drug company At the current time the beginning of year 0 Acron is trying to decide whether one of its new drugs Niagra is worth pursuing Niagra is in the final stages of development and will be ready to enter the market one year from now The final cost of development to be incurred at the beginning of year 1 is 93 million Acron estimates that the demand for Niagra will gradually grow and then decline over its useful lifetime of 20 years Specifically the company expects its gross margin revenue minus cost to be 12 million in year 1 then to increase at an annual rate of 10 through year 8 and finally to decrease at an annual rate of 5 through year 20 Acron wants to develop a spreadsheet model of its 20year cash flows assuming its cash flows other than the initial development cost are incurred at the ends of the respective years8 Using an annual discount rate of 12 for the purpose of calculating NPV the drug company wants to answer the following questions 1 Is the drug worth pursuing or should Acron abandon it now and not incur the 93 million development cost 2 How do changes in the model inputs change the answer to question 1 3 How realistic is the model Business Objectives To develop a model that calculates the NPV of Acrons cash flows to use this model to determine whether the drug should be developed further and then mar keted and to see how sensitive the answer to this question is to model parameters Excel Objecti ves To illustrate efficient selection and copying of large ranges and to learn Excels NPV function Solution The key variables in Acrons problem appear in Table 26 The first two rows contain the inputs stated in the problem We have made a judgment call as to which of these are known with some certainty and which are uncertain Although we wont do so in this chapter a thorough study of Acrons problem would treat this uncertainty explicitly probably with simulation For now you can accept the values given in the statement of the problem and leave the simulation for a later chapter 8To simplify the model taxes are ignored Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Table 26 Key Variables for Acrons Problem Input variables Development cost first year gross margin rate of increase during early years years of growth rate of decrease in later years discount rate Key output variable NPV Other calculated variables Yearly gross margins Figure 234 Acrons Model of 20Year NPV The model of Acrons cash flows appears in Figure 234 As with many financial spreadsheet models that extend over a multiyear period you enter typical formulas in the first year or two and then copy this logic down to all years In a previous edition we made the years go across not down In that case splitting the screen is useful so that you can see the first and last years of data Splitting the screen is explained in the following Excel Tip The main reason we modified the model to have the years go down not across is that it now fits easily on a screen without needing to split the screen 1 Calculating NPV at Acron Range names used Developmentcost ModelB4 Discountrate ModelB9 Grossmarginyear1 ModelB5 Grossmargin ModelB13B32 Increasethroughyear ModelB7 Rateofdecrease ModelB8 Rateofincrease ModelB6 Inputs Development cost 93 Gross margin year 1 12 Rate of increase 10 Increase through year 8 Rate of decrease 5 Discount rate 12 Cash flows End of year Gross margin 1 12000 2 13200 3 14520 4 15972 5 17569 6 19326 7 21259 8 23385 9 22215 10 21105 11 20049 12 19047 13 18095 14 17190 15 16330 16 15514 17 14738 18 14001 19 13301 20 12636 NPV 33003 Excel Tip Splitting the Screen To split the screen horizontally drag the separator just to the right of the bottom scrollbar to the left To split the screen vertically drag the separator just above the right scrollbar downward Drag either separator back to its original position to remove the split Developing the Spreadsheet Model To create the model complete the following steps See the file Calculating NPVxlsx 1 Inputs and range names Enter the given input data in the blue cells and name the ranges as shown As usual note that the range names for cells B4 through B9 can be created all at once with the Create from Selection shortcut as can the range name for the gross margins in column B In the latter case highlight the whole range B12B32 and then use the Create from Selection shortcut 2 Cash flows Start by entering the formula Grossmarginyear1 in cell B13 for the year 1 gross margin Then enter the general formula IFA14IncreasethroughyearB131Rateofincrease B131Rateofdecrease in cell B14 and copy it down to cell B32 to calculate the other yearly gross margins Note how this IF function checks the year index in column A to see whether sales are still increasing or have started to decrease Of course by using the rangenamed input cells in this formula you can change any of these inputs in cells B6 through B8 and the calculated cells will automatically update This is a much better practice than embedding the numbers in the formula itself Excel Tip Efficient Selection An easy way to select a large range assuming that the first and last cells of the range are visible is to select the first cell and then with your finger on the Shift key select the last cell Dont forget that you can split the screen horizontally andor vertically to make these first and last cells visible when the range is large This selects the entire range and is easier than scrolling9 Excel Tip Efficient Copying with CtrlEnter An easy way to enter the same formula in a range all at once is to select the range as in the preceding Excel Tip type the formula and press CtrlEnter both keys at once After you get used to this shortcut you will probably use it all the time 3 Net present value The NPV is based on the sequence of cash flows in column B From the general discussion of NPV to discount everything back to the beginning of year 1 the value in cell B13 should be multiplied by 11 r1 the value in cell B14 should be multiplied by 11 r2 and so on and these quantities should be summed to obtain the NPV Here r 012 is the discount rate Fortunately however Excel has a builtin NPV function to accomplish this calculation To use it enter the formula DevelopmentcostNPVDiscountrateGrossmargin in cell B34 The NPV function takes two arguments the discount rate and a range of cash flows Furthermore it assumes that the first cell in this range is the cash flow at the end of year 1 the second cell is the cash flow at the end of year 2 and so on This explains why the development cost is subtracted outside of the NPV functionit is incurred at the beginning of year 1 In general any cash flow incurred at the beginning of year 1 must be placed outside the NPV function To get some understanding of NPV note that the sum of the cash flows in column B is slightly more than 3414 million but the NPV aside from the development cost is only about 1260 million This is because values further into the future are discounted so heavily At the extreme the 12636 million cash flow in year 20 is equivalent to only 1263611 01220 0131 million now Excel Function NPV The NPV function takes two arguments the discount rate entered as a decimal such as 012 for 12 and a stream of cash flows These cash flows are assumed to occur in consecutive years starting at the end of year 1 If there is an initial cash flow at the beginning of year 1 such as an initial investment it should be entered outside the NPV function There is also an XNPV function that has three arguments a discount rate a series of cash flows and a series of dates when the cash flows occur Because these dates do not have to be equally spaced Use the CtrlEnter shortcut to enter a formula in a range all at once It is equivalent to copying The stream of cash flows in the NPV function must occur at the ends of year 1 year 2 and so on If the timing is irregular you can discount manually or you can use Excels XNPV function through time this function is considerably more flexible than the NPV function We will not use the XNPV function in this book but you can learn more about it in Excels online help Deciding Whether to Continue with the Drug NPV calculations are typically used to see whether a certain project should be undertaken If the NPV is positive the project is worth pursuing If the NPV is negative the company should look for other places to invest its money Figure 234 shows that the NPV for this drug is positive over 3 million10 Therefore if Acron is comfortable with its predictions of future cash flows it should continue with the development and marketing of the drug However Acron might first want to see how sensitive the NPV is to changes in the sales predictions After all these predictions are intelligent guesses at best One possible sensitivity analysis appears in Figure 235 Here you can build a oneway data table to see how the NPV changes when the number of years of increase the input in cell B7 changes Again the important question is whether the NPV stays positive It certainly does when the input variable is greater than its current value of 8 However if sales start decreasing soon enoughthat is if the value in B7 is 3 or lessthe NPV turns negative This should probably not concern Acron because its best guess for the years of increase is considerably greater than 3 Figure 235 Sensitivity of NPV to Years of Sales Increase Sensitivity to years of increase cell B7 33003 3 07190 4 01374 5 09687 6 17739 7 25516 8 33003 9 40181 10 47027 Another possibility is to see how long and how good the good years are To do this you can create the twoway data table shown in Figure 236 where cell B6 is the row input cell and cell B7 is the column input cell Now you can see that if sales increase through year 6 all reasonable yearly increases result in a positive NPV However if sales increase only through year 5 then a low enough yearly increase can produce a negative NPV Acron might want to step back and estimate how likely these bad scenarios are before proceeding with the drug Figure 236 Sensitivity of NPV to Years of Increase and Yearly Increase Sensitivity to rate of increase in early years cell B6 and years of increase cell B7 33003 5 6 7 8 9 10 3 13405 12184 10951 09708 08454 07190 4 08203 06352 04469 02554 00606 01374 5 03383 00897 01652 04265 06943 09687 6 01074 04195 07419 10750 14189 17739 7 05182 08934 12838 16899 21123 25516 8 08958 13330 17912 22711 27738 33003 9 12413 17392 22643 28182 34023 40181 10 15559 21125 27033 33306 39963 47027 10You might wonder why we didnt discount back to the beginning of the current year year 0 instead of year 1 This is a fairly arbitrary decision on our part To discount back to year 0 you would simply divide the current NPV by 112 The important point however is that this would have no bearing on Acrons decision A positive NPV would stay positive and a negative NPV would stay negative 27 Decisions Involving the Time Value of Money 59 Limitations of the Model Probably the major flaw in this model is that it ignores uncertainty and future cash flows are highly uncertain due mainly to uncertain demand for the drug Incorporating uncertainty into this type of model will be covered when we discuss simulation in Chapters 10 and 11 Aside from this uncertainty there are almost always ways to make any model more realisticat the cost of increased complexity For example you could model the impact of competition on Niagras profitability Alternatively you could allow Acron to treat its prices as decision variables However this might influence the likelihood of competition entering the market which would certainly complicate the model The point is that this model is only a start When millions of dollars are at stake a more thorough analysis is certainly warranted PROBLEMS SkillBuilding Problems 14 Modify Acrons model so that development lasts for an extra year Specifically assume that development costs of 72 million and 21 million are incurred at the beginnings of years 1 and 2 and then the sales in the current model occur one year later that is from year 2 until year 21 Again calculate the NPV discounted back to the beginning of year 1 and perform the same sensitivity analyses Comment on the effects of this change in timing 15 Modify Acrons model so that sales increase then stay steady and finally decrease Specifically assume that the gross margin is 12 million in year 1 then increases by 10 annually through year 6 then stays constant through year 10 and finally decreases by 5 annually through year 20 Perform a sensitivity analysis with a twoway data table to see how NPV varies with the length of the increase period currently 6 years and the length of the constant period currently 4 years Comment on whether Acron should pursue the drug given your results 16 Create a oneway data table in the Acron model to see how the NPV varies with discount rate which is allowed to vary from 8 to 18 in increments of 05 Explain intuitively why the results go in the direction they gothat is the NPV decreases as the discount rate increases Should Acron pursue the drug for all of these discount rates SkillExtending Problems 17 The NPV function automatically discounts each of the cash flows and sums the discounted values Verify that it does this correctly for Acrons model by calculating the NPV the long way That is discount each cash flow and then sum these discounted values Use Excel formulas to do this but dont use the NPV function Hint Remember that the discounted value of 1 received t years from now is 11 rt dollars today 18 In a situation such as Acrons where a onetime cost is followed by a sequence of cash flows the internal rate of return IRR is the discount rate that makes the NPV equal to 0 The idea is that if the discount rate is greater than the IRR the company will not pursue the project but if the discount rate is less than the IRR the project is financially attractive a Use Excels Goal Seek tool to find the IRR for the Acron model b Excel also has an IRR function Look it up in online help to see how it works and then use it on Acrons model Of course you should get the same IRR as in part a c Verify that the NPV is negative when the discount rate is slightly greater than the IRR and that it is positive when the discount rate is slightly less than the IRR 19 The XNPV function can calculate NPV for any possibly irregular series of cash flows Look this function up in Excels online help Then use it to develop a spreadsheet model that finds the NPV of the following series a payment of 25000 today assumed to be June 15 2010 and cash inflows of 10000 on March 1 2011 15000 on September 15 2011 8000 on January 20 2012 20000 on April 1 2012 and 10000 on May 15 2012 Discount these back to today using a discount rate of 12 82 c Graph profit as a function of the number of copiers for a daily demand of 500 copies for a daily demand of 2000 copies Interpret your graphs 23 Georgia McBeal is trying to save for her retirement She believes she can earn 10 on average each year on her retirement fund Assume that at the beginning of each of the next 40 years Georgia will allocate x dollars to her retirement fund If at the beginning of a year Georgia has y dollars in her fund by the end of the year it will grow to 11y dollars How much should Georgia allocate to her retirement fund each year to ensure that she will have 1 million at the end of 40 years What key factors are being ignored in this analysis of the amount saved for retirement 24 A European call option on a stock earns the owner an amount equal to the price at expiration minus the exercise price if the price of the stock on which the call is written exceeds the exercise price Otherwise the call pays nothing A European put option earns the owner an amount equal to the exercise price minus the price at expiration if the price at expiration is less than the exercise price Otherwise the put pays nothing The file P0224xlsx contains a template that finds based on the wellknown BlackScholes formula the price of a European call and put based on the following inputs todays stock price the duration of the option in years the options exercise price the riskfree rate of interest per year and the annual volatility in stock price For example a 40 volatility means approximately that the standard deviation of annual percentage changes in the stock price is 40 a Consider a sixmonth European call option with exercise price 40 Assume a current stock price of 35 a riskfree rate of 5 and an annual volatility of 40 Determine the price of the call option b Use a data table to show how a change in volatility changes the value of the option Give an intuitive explanation for your results c Use a data table to show how a change in todays stock price changes the options value Give an intuitive explanation for your results d Use a data table to show how a change in the options duration changes the options value Give an intuitive explanation for your results 25 Repeat parts ad of the previous problem for a sixmonth European put option with exercise price 40 Again assume a current stock price of 35 a riskfree rate of 5 and an annual volatility of 40 26 The file P0226xlsx lists sales in millions of dollars of Dell Computer during the period 19871997 where year 1 corresponds to 1987 a Fit a power and an exponential trend curve to these data Which fits the data better b Use your part a answer to predict 1999 sales for Dell c Use your part a answer to describe how the sales of Dell have grown from year to year d Search the Web for more recent Dell sales data Then repeat the preceding parts using all of the data 27 Dataware is trying to determine whether to give a 10 rebate cut the price 6 or have no price change on a software product Currently 40000 units of the product are sold each week for 45 apiece The variable cost of the product is 5 The most likely case appears to be that a 10 rebate will increase sales 30 and half of all people will claim the rebate For the price cut the most likely case is that sales will increase 20 a Given all other assumptions what increase in sales from the rebate would make the rebate and price cut equally desirable b Dataware does not really know the increase in sales that will result from a rebate or price cut However the company is sure that the rebate will increase sales by between 15 and 40 and that the price cut will increase sales by between 10 and 30 Perform a sensitivity analysis that could be used to help determine Datawares best decision 28 The file P0228xlsx gives the annual sales for Microsoft in millions of dollars for the years 19841993 where 1984 year 1 a Fit an exponential curve to these data b Assuming you are back in 1993 by what percentage do you estimate that Microsoft has grown each year based on this historical data c Why cant a high rate of exponential growth continue for a long time d Rather than an exponential curve what curve might better represent the growth of a new technology e Search the Web for more recent Microsoft sales data Then repeat the preceding parts using all the data 29 Assume that the number of units sold of a product is given by 100 05P 26A where P is the price in dollars charged for the product and A is the amount spent on advertising in thousands of dollars Each unit of the product costs 5 to produce Use a data table to find the combination of price and advertising that maximizes profit 30 A company manufacturers a product in the US and sells it in England The unit cost of manufacturing is 50 The current exchange rate dollars per pound is 151 The demand function which indicates how many units the company can sell in England as a function of price in pounds is of the power type with constant 27556759 and exponent 24 a Develop a model for the companys profit in dollars as a function of the price it charges in pounds Then use a data table to find the profitmaximizing price to the nearest pound 83 b If the exchange rate varies from its current value does the profitmaximizing price increase or decrease Does the maximum profit increase or decrease 31 The yield of a chemical reaction is defined as the ratio expressed as a percentage of usable output to the amount of raw material input Suppose the yield of a chemical reaction depends on the length of time the process is run and the temperature at which the process is run The yield can be expressed as follows Yield 9079 1095x1 1045x2 278112 2524x22 0775x1x2 Here x1 Temperature 12510 and x2 Time 30030 where temperature is measured in degrees Fahrenheit and time is measured in seconds Use a data table to find the temperature and time settings that maximize the yield of this process 32 A bond is currently selling for 1040 It pays the amounts listed in the file P0232xlsx at the ends of the next six years The yield of the bond is the interest rate that would make the NPV of the bonds payments equal to the bonds price Use Excels Goal Seek tool to find the yield of the bond 33 Assume the demand for a companys drug Wozac during the current year is 50000 and assume demand will grow at 5 a year If the company builds a plant that can produce x units of Wozac per year it will cost 16x Each unit of Wozac is sold for 3 Each unit of Wozac produced incurs a variable production cost of 020 It costs 040 per year to operate a unit of capacity Determine how large a Wozac plant the company should build to maximize its expected profit over the next 10 years 34 Consider a project with the following cash flows year 1 400 year 2 200 year 3 600 year 4 900 year 5 1000 year 6 250 year 7 230 Assume a discount rate of 15 per year a Find the projects NPV if cash flows occur at the ends of the respective years b Find the projects NPV if cash flows occur at the beginnings of the respective years c Find the projects NPV if cash flows occur at the middles of the respective years 35 A software company is considering translating its program into French Each unit of the program sells for 50 and incurs a variable cost of 10 to produce Currently the size of the market for the product is 300000 units per year and the English version of the software has a 30 share of the market The company estimates that the market size will grow by 10 a year for the next five years and at 5 per year after that It will cost the company 6 million to create a French version of the program The translation will increase its market share to 40 Given a 10year planning horizon for what discount rates is it profitable to create the French version of the software 36 The payback of a project is the number of years it takes before the projects total cash flow is positive Payback ignores the time value of money It is interesting however to see how differing assumptions on project growth impact payback Suppose for example that a project requires a 300 million investment at year 0 right now The project yields cash flows for 10 years and the year 1 cash flow will be between 30 million and 100 million The annual cash flow growth will be between 5 and 25 per year Assume that this growth is the same each year Use a data table to see how the project payback depends on the year 1 cash flow and the cash flow growth rate SkillExtending Problems 37 You are entering the widget business It costs 500000 payable in year 1 to develop a prototype This cost can be depreciated on a straightline basis during years 15 Each widget sells for 40 and incurs a variable cost of 20 During year 1 the market size is 100000 and the market is growing at 10 per year You believe you will attain a 30 market share Profits are taxed at 40 but there are no taxes on negative profits a Given your other assumptions what market share is needed to ensure a total free cash flow FCF of 0 over years 1 to 5 Note FCF during a year equals aftertax profits plus depreciation minus fixed costs if any b Explain how an increase in market share changes profit c Explain how an increase in market size growth changes profit d Use Excels auditing tool to show how the market growth assumption influences your spreadsheet 38 Suppose you are borrowing 25000 and making monthly payments with 1 interest Show that the monthly payments should equal 55611 The key relationships are that for any month t Ending month t balance Ending month t 1 balance Monthly payment Month t interest Month t interest Beginning month t balance Monthly interest rate Of course the ending month 60 balance must equal 0 39 You are thinking of starting Peaco which will produce Peakbabies a product that competes with Tys Beanie Babies In year 0 right now you will incur costs of 4 million to build a plant In year 1 you expect to sell 80000 Peakbabies for a unit price of 25 The price of 25 will remain unchanged through years 1 to 5 Unit sales are expected to grow by the same percentage g each year During years 1 to 5 Peaco incurs two types of costs variable costs and SGA selling general and administrative costs Each year variable costs equal 28 Conclusion 61 28 CONCLUSION The examples in this chapter provide a glimpse of things to come in later chapters You have seen the spreadsheet modeling approach to realistic business problems learned how to design spreadsheet models for readability and explored some of Excels powerful tools par ticularly data tables In addition at least three important themes have emerged from these examples relating inputs and decision variables to outputs by means of appropriate formu las optimization for example finding a best order quantity and the role of uncertainty uncertain response rate or demand Although you have not yet learned the tools to explore these themes fully you will have plenty of opportunities to do so in the rest of this book Summary of Key Management Science Terms Term Explanation Page Model inputs The numeric values that are given in any 22 problem statement Decision variables The variables a decision maker has control over 22 to obtain better solutions Model outputs The numeric values that result from combinations 22 of inputs and decision variables through the use of logical formulas Net present value NPV The current worth of a stream of cash flows that 55 occur in the future Discount rate Interest rate used for discounting future cash flows 55 to get the net present value Summary of Key Excel Terms Term Explanation Excel Page IF function Useful for implementing logic IFconditionresultIfTrue 24 resultIfFalse Relative absolute Useful for copying formulas A1 relative A1 or A1 mixed 29 cell addresses absolute row or column stays fixed A1 absolute press F4 to relative row or column moves cycle through possibilities Range names Useful for making formulas more Type name in Name box or use 33 meaningful Create from Selection shortcut CtrlShiftF3 Pasting range names Provides a list of all range names in Use Paste List from Use 33 the current workbook in Formula dropdown list F3 Cell comments Useful for documenting contents Rightclick on cell select Insert 34 of the cell Comment menu item Oneway data table Shows how one or more outputs Use Data Table from WhatIf 36 vary as a single input varies Analysis dropdown list Goal Seek Solves one equation in one unknown Use Goal Seek from WhatIf 37 Analysis dropdown list Formula Auditing Useful for checking which cells are Use Formula Auditing buttons 38 toolbar related to other cells through formulas on Formulas ribbon fx button Useful for getting help on On Formula Bar 42 Excel functions continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 62 Chapter 2 Introduction to Spreadsheet Modeling Summary of Key Excel Terms Continued Term Explanation Excel Page VLOOKUP function Useful for finding a particular value VLOOKUPvalueToCompare 42 based on a comparison lookupTable columnToReturn Twoway data table Shows how a single output varies Use Data Table from WhatIf 43 as two inputs vary Analysis dropdown list SUMPRODUCT Calculates the sum of products of SUMPRODUCTrange1range2 44 function values in two or more similar sized ranges Trendline tool Superimposes the bestfitting line With chart selected rightclick on 47 or curve of a particular type on any point and select Add Trendline a scatter chart or time series graph Conditional Formats cells depending on whether Use Conditional Formatting 51 formatting specified conditions hold on Home ribbon Splitting screen Useful for separating the screen Use screen splitters at top and right 57 horizontally andor vertically of scrollbars Efficient selection Useful for selecting a large While pressing the Shift key click 58 rectangular range on upperleft and bottomright cells of range Efficient copying Shortcut for copying a formula Select the range enter the formula 58 to a range and press CtrlEnter NPV function Calculates NPV of a stream of cash NPVdiscountRatecashFlows 58 flows at the ends of consecutive years starting in year 1 P R O B L E M S SkillBuilding Problems 20 Julie James is opening a lemonade stand She believes the fixed cost per week of running the stand is 5000 Her best guess is that she can sell 300 cups per week at 050 per cup The variable cost of producing a cup of lemonade is 020 a Given her other assumptions what level of sales volume will enable Julie to break even b Given her other assumptions discuss how a change in sales volume affects profit c Given her other assumptions discuss how a change in sales volume and variable cost jointly affect profit d Use Excels Formula Auditing tools to show which cells in your spreadsheet affect profit directly 21 You are thinking of opening a Broadway play I Love You Youre Mediocre Now Get Better It will cost 5 million to develop the show There are 8 shows per week and you project the show will run for 100 weeks It costs 1000 to open the theater each night Tickets sell for 5000 and you earn an average of 150 profit per ticket holder from concessions The theater holds 800 and you expect 80 of the seats to be full a Given your other assumptions how many weeks will the play have to run for you to earn a 100 return on the plays development cost b Given your other assumptions how does an increase in the percentage of seats full affect profit c Given your other assumptions determine how a joint change in the average ticket price and number of weeks the play runs influence profit d Use Excels Formula Auditing tools to show which cells in the spreadsheet are directly affected by the percentage of seats full 22 You are thinking of opening a small copy shop It costs 5000 to rent a copier for a year and it costs 003 per copy to operate the copier Other fixed costs of running the store will amount to 400 per month You plan to charge an average of 010 per copy and the store will be open 365 days per year Each copier can make up to 100000 copies per year a For one to five copiers rented and daily demands of 500 1000 1500 and 2000 copies per day find annual profit That is find annual profit for each of these combinations of copiers rented and daily demand b If you rent three copiers what daily demand for copies will allow you to break even Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it half of revenue During year 1 SGA costs equal 40 of revenue This percentage is assumed to drop 2 per year so during year 2 SGA costs will equal 38 of revenue and so on Peacos goal is to have profits for years 0 to 5 sum to 0 ignoring the time value of money This will ensure that the 4 million investment in year 0 is paid back by the end of year 5 What annual percentage growth rate g does Peaco require to pay back the plant cost by the end of year 5 40 Suppose the demand in thousands for a toaster is given by 100p² where p is the price in dollars charged for the toaster a If the variable cost of producing a toaster is 10 what price maximizes profit b The elasticity of demand is defined as the percentage change in demand created by a 1 change in price Using a data table show that the demand for toasters has constant elasticity that is the elasticity doesnt depend on the price Would this be true if the demand for toasters were linear in price 41 The file P0241xlsx contains the cumulative number of bits in trillions of DRAM a type of computer memory produced and the price per bit in thousandths of a cent a Fit a power curve that can be used to show how price per bit drops with increased production This relationship is known as the learning curve b Suppose the cumulative number of bits doubles Create a prediction for the price per bit Does the change in the price per bit depend on the current price 42 A large US drug company Pharmco has 100 million yen coming due in one year Currently the yen is worth 001 Because the value of the yen in US dollars in one year is unknown the value of this 100 million yen in US dollars is highly uncertain To hedge its risk Pharmco is thinking of buying oneyear put options on the yen with an exercise price of 0008 For example if the yen falls in value a year from now to 0007 the owner of the put receives 0001 The price of such a put is 000007 Show how the dollar value of Pharmcos receipts and hedging expenses depends on the number of puts purchased and the final yen exchange rate Assume final exchange rates between 0006 yen and 0015 yen are possible 43 The file P0243xlsx contains a template for a car loan Specifically once values are entered in the blue cells you need to enter formulas in the gray cells to calculate the amount financed the monthly payment assuming that monthly payments stay the same throughout the term of the loan the total interest paid and an amortization schedule For the latter fill in the entire gray area with formulas but use IF functions so that blanks appear past the term of the loan 44 The IRR is the discount rate r that makes a project have an NPV of 0 You can find IRR in Excel with the builtin IRR function using the syntax IRRrange of cash flows However it can be tricky In fact if the IRR is not near 10 this function might not find an answer and you would get an error message Then you must try the syntax IRRrange of cash flows guess where guess is your best guess for the IRR It is best to try a range of guesses say 90 to 100 Find the IRR of the project described in Problem 34 45 A project does not necessarily have a unique IRR Refer to the previous problem for more information on IRR Show that a project with the following cash flows has two IRRs year 1 20 year 2 82 year 3 60 year 4 2 Note It can be shown that if the cash flow of a project changes sign only once the project is guaranteed to have a unique IRR 46 The file P0246xlsx contains data on prices of products for several of a chain stores locations a discount schedule offered to customers depending on how much they spend and commission rates of the salespeople at the various stores Your job is to develop an invoice form Specifically you should enter formulas in the gray cells so that whenever data are entered in the blue cells the formulas in the gray cells calculate automatically As an extra use data validation in cell B23 so that the user can choose a city from a list of cities where the chain has its stores APPENDIX TIPS FOR EDITING AND DOCUMENTING SPREADSHEETS Editing and documenting your spreadsheet models is crucial and the following tips make these tasks much easier Format Appropriately Appropriate formatting can make a spreadsheet model much easier to read To boldface for example select one or more cells and click on the B button on the Home ribbon or press CtrlB Similarly to italicize indent increase or decrease the number of decimal places rightjustify or perform other common formatting tasks use the buttons on the Home ribbon or shortcut keys 66 Chapter 2 Introduction to Spreadsheet Modeling Use Range Names Naming ranges takes time but makes formulas much easier to read and understand To enter a range name highlight any cell or range of cells and enter a name for the range in the Name box just to the left of the Formula Bar If you want to edit or delete range names select Name Manager on the Formulas ribbon Here are some other options you have from the Defined Names group on the Formulas ribbon After you have named some ranges you can get a list of them in your spreadsheet by placing the cursor at the top of the range where you want the list to be placed selecting the Use in Formula dropdown list on the Formulas ribbon and clicking on the Paste List option Alternatively you can press the F3 button Suppose you have labels such as Fixed Cost Variable Cost Revenue and Profit in the range A3A6 with their values next to them in column B If you want to name the cells in column B with the labels in column A highlight the range A3B6 select Create from Selection on the Formulas ribbon or press CtrlShiftF3 and make sure the Left Column box is checked This creates the range names you want A similar trick works if you have descriptive labels above columns of data you want to name If you have a formula such as SUMA10A20 and then you name the range A10A20 Costs say the formula does not change automatically to SUMCosts However you can make it adapt to your new range name by selecting Apply Names from the Define Name dropdown list on the Formulas ribbon Sometimes you might want to use the same range name such as Totalcost on multiple worksheets of a workbook For example you might want Totalcost to refer to cell B26 in Sheet1 and to cell C59 in Sheet2 The trick is to use a sheetlevel name rather than a workbooklevel name for one or both versions of Totalcost This is easy to do from the Name Manager When you define a new name just select a worksheet as the Scope of the name Use Text Boxes Text boxes are very useful for documenting your work To enter an explanation or any other text into a text box click on the Text Box button on the Insert ribbon drag a box and start typing This technique is much better than typing explanations into cells because text boxes have word wrap Therefore text in text boxes is much easier to edit than text in cells Use Cell Comments Cell comments provide another good way to document your work To enter a comment in a cell select the cell and rightclick This brings up a dialog box which is also useful for other tasks such as formatting Click on the Insert Comment item to enter a comment If a comment is already in the cell this menu will contain Edit Comment and Delete Comment items The cells with comments should have small red triangles in their corners When you hover the cursor over the cell the comment appears Other Tips Finally we urge you once again to open the Excel Tutorialxlsx file on the Essential Resource Web site and work through it The file includes a number of techniques that will make you a better and more efficient Excel user Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 67 Introduction to Optimization Modeling C H A P T E R OPTIMIZING MANUFACTURING OPERATIONS AT GE PLASTICS T he General Electric Company GE is a global organization that must deliver products to its customers anywhere in the world in the right quantity at the right time and at a reasonable cost One arm of GE is GE Plastics GEP a 5 billion business that supplies plastics and raw materials to such industries as automotive appliance computer and medical equipment GEP has now been reorganized into GE Advanced Materials GEAM As described in Tyagi et al 2004 GEP practiced a polecentric manufacturing approach making each product in the geographic area Americas Europe or Pacific where it was to be delivered However it became apparent in the early 2000s that this approach was leading to higher distribution costs and mismat ches in capacity as more of GEPs demand was originating in the Pacific region Therefore the authors of the article were asked to develop a global optimi zation model to aid GEPs manufacturing planning Actually GEP consists of seven major divisions distinguished primarily by the capability of their products to withstand heatThe fastest growing of these divisions the high performance polymer HPP division was chosen as the pilot for the new global approach 3 Keith DannemillerCorbis Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it All GEP divisions operate as twoechelon manufacturing systemsThe first echelon consists of resin plantswhich convert raw material stocks into resins and ship them to the second echelonthe finishing plants These latter plants combine the resins with additives to produce various grades of the end productsEach physical plant consists of several plant lines that operate independentlyand each of these plant lines is capable of producing multiple products All end products are then shipped to GE Polymerland warehouses throughout the worldGE Polymerland is a wholly owned subsidiary that acts as the commercial front for GEPIt handles all customer sales and deliveries from its network of distribution centers and warehouses in more than 20 countriesBecause of its experience with customersGE Polymerland is able to aid the GEP divisions in their planning processes by supplying forecasts of demands and prices for the various products in the various global marketsThese forecasts are key inputs to the optimization model The optimization model itself attempts to maximize the total contribution margin over a planning horizon where the contribution margin equals revenues minus the sum of manufacturing material and distribution costsThere are demand constraints manufacturing capacity constraints and network flow constraints The decision variables include 1 the amount of resin produced at each resin plant line that will be used at each finishing plant line and 2 the amount of each end product produced at each finishing plant line that will be shipped to each geographic regionThe completed model has approximately 3100 decision variables and 1100 constraints and is completely linear It was developed and solved in Excel using LINGO a commercial optimization solver not Excels Solver addin and execution time is very fastabout 10 seconds The demand constraints are handled in an interesting way The authors of the study constrain manufacturing to produce no more than the forecasted demands but they do not force manufacturing to meet these demands Ideally manufacturing would meet demands exactly However because of its rapid growth capacity at HPP in 2002 appeared at the time of the study to be insufficient to meet the demand in 2005 and later years The authors faced this challenge in two ways First in cases where demand exceeds capacity they let their model of maximizing total contribution margin determine which demands to satisfyThe least profitable demands are simply not met Second the authors added a new resin plant to their model that would come on line in the year 2005 and provide much needed capacityThey ran the model several times for the year 2005 and later years experimenting with the location of the new plantAlthough some of the details are withheld in the article for confidentiality reasons the authors indicate that senior management approved the investment of a Europebased plant that would cost more than 200 million in plant and equipmentThis plant was planned to begin operations in 2005 and ramp up to full production capacity by 2007 The decision support system developed in the study has been a success at the HPP division since its introduction in 2002 Although the article provides no specific dollar gains from the use of the model it is noteworthy that the other GEP divisions are adopting similar models for their production planning 68 Chapter 3 Introduction to Optimization Modeling 31 INTRODUCTION In this chapter we introduce spreadsheet optimization one of the most powerful and flexible methods of quantitative analysis The specific type of optimization we will discuss here is linear programming LP LP is used in all types of organizations often on a daily basis to solve a wide variety of problems These include problems in labor scheduling inventory management selection of advertising media bond trading management of cash flows oper ation of an electrical utilitys hydroelectric system routing of delivery vehicles blending in oil refineries hospital staffing and many others The goal of this chapter is to introduce the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it basic elements of LP the types of problems it can solve how LP problems can be modeled in Excel and how Excels powerful Solver addin can be used to find optimal solutions Then in the next few chapters we will examine a variety of LP applications and we will also look at applications of integer and nonlinear programming two important extensions of LP 32 INTRODUCTION TO OPTIMIZATION Before we discuss the details of LP modeling it is useful to discuss optimization in general All optimization problems have several common elements They all have decision variables the variables whose values the decision maker is allowed to choose Either directly or indirectly the values of these variables determine such outputs as total cost revenue and profit Essentially they are the variables a company or organization must know to function properly they deter mine everything else All optimization problems have an objective function objective for short to be optimizedmaximized or minimized1 Finally most optimization problems have constraints that must be satisfied These are usually physical logical or economic restrictions depending on the nature of the problem In searching for the values of the decision variables that optimize the objective only those values that satisfy all of the constraints are allowed Excel uses its own terminology for optimization and we will use it as well Excel refers to the decision variables as the changing cells These cells must contain numbers that are allowed to change freely they are not allowed to contain formulas Excel refers to the objec tive as the objective cell There can be only one objective cell which could contain profit total cost total distance traveled or others and it must be related through formulas to the changing cells When the changing cells change the objective cell should change accordingly 32 Introduction to Optimization 69 The changing cells contain the values of the decision variables The objective cell contains the objective to be minimized or maximized The constraints impose restrictions on the values in the changing cells Finally there must be appropriate cells and cell formulas that operationalize the con straints For example one constraint might indicate that the amount of labor used can be no more than the amount of labor available In this case there must be cells for each of these two quantities and typically at least one of them probably the amount of labor used will be related through formulas to the changing cells Constraints can come in a variety of forms One very common form is nonnegativity This type of constraint states that changing cells must have nonnegative zero or positive values Nonnegativity constraints are usually included for physi cal reasons For example it is impossible to produce a negative number of automobiles There are basically two steps in solving an optimization problem The first step is the model development step Here you decide what the decision variables are what the objec tive is which constraints are required and how everything fits together If you are devel oping an algebraic model you must derive the correct algebraic expressions If you are developing a spreadsheet model the main focus of this book you must relate all variables with appropriate cell formulas In particular you must ensure that your model contains for mulas that relate the changing cells to the objective cell and formulas that operationalize the constraints This model development step is where most of your effort goes Nonnegativity constraints imply that changing cells must contain nonnegative values Typically most of your effort goes into the model development step 1Actually some optimization models are multicriteria models that try to optimize several objectives simultane ously However we will not discuss multicriteria models in this book Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The second step in any optimization model is to optimize This means that you must sys tematically choose the values of the decision variables that make the objective as large for maximization or small for minimization as possible and cause all of the constraints to be satisfied Some terminology is useful here Any set of values of the decision variables that sat isfies all of the constraints is called a feasible solution The set of all feasible solutions is called the feasible region In contrast an infeasible solution is a solution that violates at least one constraint Infeasible solutions are disallowed The desired feasible solution is the one that provides the best valueminimum for a minimization problem maximum for a maximiza tion problemfor the objective This solution is called the optimal solution Although most of your effort typically goes into the model development step much of the published research in optimization has been about the optimization step Algorithms have been devised for searching through the feasible region to find the optimal solution One such algorithm is called the simplex method It is used for linear models There are other more complex algorithms used for other types of models those with integer decision variables andor nonlinearities We will not discuss the details of these algorithms They have been programmed into the Excels Solver addin All you need to do is develop the model and then tell Solver what the objective cell is what the changing cells are what the constraints are and what type of model linear integer or nonlinear you have Solver then goes to work finding the best feasible solution with the appropriate algorithm You should appreciate that if you used a trialanderror procedure even a clever and fast one it could take hours weeks or even years to complete However by using the appropriate algorithm Solver typically finds the optimal solution in a matter of seconds Before concluding this discussion we mention that there is really a third step in the optimization process sensitivity analysis You typically choose the most likely values of input variables such as unit costs forecasted demands and resource availabilities and then find the optimal solution for these particular input values This provides a single answer However in any realistic situation it is wishful thinking to believe that all of the input values you use are exactly correct Therefore it is usefulindeed mandatory in most applied studiesto follow up the optimization step with whatif questions What if the unit costs increased by 5 What if forecasted demands were 10 lower What if resource availabilities could be increased by 20 What effects would such changes have on the optimal solution This type of sensitivity analysis can be done in an informal man ner or it can be highly structured Fortunately as with the optimization step itself good software allows you to obtain answers to various whatif questions quickly and easily 33 A TWOVARIABLE PRODUCT MIX MODEL We begin with a very simple twovariable example of a product mix problem This is a type of problem frequently encountered in business where a company must decide its product mixhow much of each of its potential products to produceto maximize its net profit You will see how to model this problem algebraically and then how to model it in Excel You will also see how to find its optimal solution with Solver Next because it contains 70 Chapter 3 Introduction to Optimization Modeling A feasible solution is a solution that satisfies all of the constraints The feasible region is the set of all feasible solutions An infeasible solution violates at least one of the constraints The optimal solution is the feasible solution that optimizes the objective An algorithm is basically a plan of attack It is a prescription for carrying out the steps required to achieve some goal such as finding an optimal solutionAn algorithm is typically translated into a computer program that does the work Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it T he PC Tech company assembles and then tests two models of computers Basic and XP For the coming month the company wants to decide how many of each model to assembly and then test No computers are in inventory from the previous month and because these models are going to be changed after this month the company doesnt want to hold any inventory after this month It believes the most it can sell this month are 600 Basics and 1200 XPs Each Basic sells for 300 and each XP sells for 450 The cost of component parts for a Basic is 150 for an XP it is 225 Labor is required for assembly and testing There are at most 10000 assembly hours and 3000 testing hours available Each labor hour for assembling costs 11 and each labor hour for testing costs 15 Each Basic requires five hours for assembling and one hour for testing and each XP requires six hours for assembling and two hours for testing PC Tech wants to know how many of each model it should produce assemble and test to maximize its net profit but it cannot use more labor hours than are available and it does not want to produce more than it can sell Objective To use LP to find the best mix of computer models that stays within the com panys labor availability and maximum sales constraints Solution In all optimization models you are given a variety of numbersthe inputsand you are asked to make some decisions that optimize an objective while satisfying all constraints We summarize this information in a table such as Table 31 We believe it is a good idea to create such a table before diving into the modeling details In particular you always need to identify the appropriate decision variables the appropriate objective and the con straints and you should always think about the relationships between them Without a clear idea of these elements it is almost impossible to develop a correct algebraic or spreadsheet model 33 A TwoVariable Product Mix Model 71 E X A M P L E 31 ASSEMBLING AND TESTING COMPUTERS Tables such as this one serve as a bridge between the problem statement and the ultimate spreadsheet or algebraic model Table 31 Variables and Constraints for TwoVariable Product Mix Model Input variables Hourly labor costs labor availabilities labor required for each computer costs of component parts unit selling prices and maximum sales Decision variables changing cells Number of each computer model to produce assemble and test Objective cell Total net profit Other calculated variables Labor of each type used Constraints Labor used Labor available Number produced Maximum sales The decision variables in this product mix model are fairly obvious The company must decide two numbers how many Basics to produce and how many XPs to produce Once these are known they can be used along with the problem inputs to calculate the only two decision variables you will see how it can be solved graphically Although this graphical solution is not practical for most realistic problems it provides useful insights into general LP models The final step is then to ask a number of whatif questions about the completed model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it number of computers sold the labor used and the revenue and cost However as you will see with other models in this chapter and the next few chapters determining the decision variables is not always this obvious An Algebraic Model In the traditional algebraic solution method you first identify the decision variables In this small problem they are the numbers of computers to produce We label these x₁ and x₂ although any other labels would do The next step is to write expressions for the total net profit and the constraints in terms of the xs Finally because only nonnegative amounts can be produced explicit constraints are added to ensure that the xs are nonnegative The resulting algebraic model is Maximize 80x₁ 129x₂ subject to 5x₁ 6x₂ 10000 x₁ 2x₂ 3000 x₁ 600 x₂ 1200 x₁ x₂ 0 To understand this model consider the objective first Each Basic produced sells for 300 and the total cost of producing it including component parts and labor is 150 511 115 220 so the profit margin is 80 Similarly the profit margin for an XP is 129 Each profit margin is multiplied by the number of computers produced and these products are then summed over the two computer models to obtain the total net profit The first two constraints are similar For example each Basic requires five hours for assembling and each XP requires six hours for assembling so the first constraint says that the total hours required for assembling is no more than the number available 10000 The third and fourth constraints are the maximum sales constraints for Basics and XPs Finally negative amounts cannot be produced so nonnegativity constraints on x₁ and x₂ are included For many years all LP problems were modeled this way in textbooks In fact many commercial LP computer packages are still written to accept LP problems in essentially this format Since around 1990 however a more intuitive method of expressing LP problems has emerged This method takes advantage of the power and flexibility of spreadsheets Actually LP problems could always be modeled in spreadsheets but now with the addition of Solver spreadsheets have the ability to solvethat is optimizeLP problems as well We use Excels Solver for all examples in this book A Graphical Solution When there are only two decision variables in an LP model as there are in this product mix model you can solve the problem graphically Although this graphical solution approach is not practical in most realistic optimization modelswhere there are many more than two decision variablesthe graphical procedure illustrated here still yields important insights for general LP models Recall from algebra that any line of the form ax₁ bx₂ c has slope ab This is because it can be put into the slopeintercept form x₂ cb abx₁ versions of polygons That is they are bounded by straight lines actually hyperplanes that intersect at several corner points There are five corner points in Figure 31 three of which are on the axes One of them is 00 When the dotted objective line is moved as far as pos sible toward better values the last feasible point it touches is one of the corner points The actual corner point it last touches is determined by the slopes of the objective and constraint lines Because there are only a finite number of corner points it suffices to search among this finite set not the infinite number of points in the entire feasible region4 This insight is largely responsible for the efficiency of the simplex method for solving LP problems A Spreadsheet Model We now turn our focus to spreadsheet modeling There are many ways to develop an LP spreadsheet model Everyone has his or her own preferences for arranging the data in the various cells We do not provide exact prescriptions but we do present enough examples to help you develop good habits The common elements in all LP spreadsheet models are the inputs changing cells objective cell and constraints Inputs All numerical inputsthat is all numeric data given in the statement of the problemshould appear somewhere in the spreadsheet Our convention is to color all of the input cells blue We also try to put most of the inputs in the upper left sec tion of the spreadsheet However we sometimes violate this latter convention when certain inputs fit more naturally somewhere else Changing cells Instead of using variable names such as xs spreadsheet models use a set of designated cells for the decision variables The values in these changing cells can be changed to optimize the objective The values in these cells must be allowed to vary freely so there should not be any formulas in the changing cells To designate them clearly our convention is to color them red Objective cell One cell called the objective cell contains the value of the objective Solver systematically varies the values in the changing cells to optimize the value in the objective cell This cell must be linked either directly or indirectly to the chang ing cells by formulas Our convention is to color the objective cell gray5 74 Chapter 3 Introduction to Optimization Modeling 4This is not entirely true If the objective line is exactly parallel to one of the constraint lines there can be multi ple optimal solutionsa whole line segment of optimal solutions Even in this case however at least one of the optimal solutions is a corner point 5Our blueredgray color scheme shows up very effectively on a color monitor For users of previous editions who are used to colored borders we find that it is easier in Excel 2007 and Excel 2010 to color the cells rather than put borders around them FUNDAMENTAL INSIGHT Geometry of LP Models and the Simplex Method The feasible region in any LP model is al ways a multi dimensional version of a pol ygon and the objectiv e is always a hyperplane the multidimensional version of a straight lineThe objective should always be moved as far as possible in the maximizing or minimizing dir ec tion until it just touches the edge of the feasible region Because of this geometrythe optimal solution is always a corner point of the polygonThe simplex method for LP works so w ell because it can sear ch through the finite number of corner points extremely efficiently and recognize when it has found the best corner pointThis rather simple insight plus its clever implementation in software packages has sa ved companies man y many millions of dollars in the past 50 years Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Our coloring conventions Color all input cells blue appears light blue on the printed page Color all of the changing cells red appears deep blue on the printed page Color the objective cell gray Constraints Excel does not show the constraints directly on the spreadsheet Instead they are specified in a Solver dialog box to be discussed shortly For example a set of related constraints might be specified by B16C16B18C18 This implies two separate constraints The value in B16 must be less than or equal to the value in B18 and the value in C16 must be less than or equal to the value in C18 We will always assign range names to the ranges that appear in the constraints Then a typical constraint might be specified as NumbertoproduceMaximumsales This is much easier to read and understand If you find that range names take too long to create you certainly do not have to use them Solver models work fine with cell addresses only Nonnegativity Normally the decision variables that is the values in the changing cells must be nonnegative These constraints do not need to be written explicitly you simply check an option in the Solver dialog box to indicate that the changing cells should be nonnegative Note however that if you want to constrain any other cells to be nonnegative you must specify these constraints explicitly Overview of the Solution Process As mentioned previously the complete solution of a problem involves three stages In the model development stage you enter all of the inputs trial values for the changing cells and formulas relating these in a spreadsheet This stage is the most crucial because it is here that all of the ingredients of the model are included and related appropriately In particular the spreadsheet must include a formula that relates the objective to the changing cells either directly or indirectly so that if the values in the changing cells vary the objective value varies accordingly Similarly the spreadsheet must include formulas for the various constraints usually their left sides that are related directly or indirectly to the changing cells After the model is developed you can proceed to the second stage invoking Solver At this point you formally designate the objective cell the changing cells the constraints and selected options and you tell Solver to find the optimal solution If the first stage has been done correctly the second stage is usually very straightforward The third stage is sensitivity analysis Here you see how the optimal solution changes if at all as selected inputs are varied This often provides important insights about the behavior of the model We now illustrate this procedure for the product mix problem in Example 31 WHERE DO THE NUMBERS COME FROM Textbooks typically state a problem including a number of input values and proceed directly to a solution without saying where these input values might come from However finding the correct input values can sometimes be the most difficult step in a realworld situation Recall that finding the necessary data is step 2 of the overall modeling process as discussed in Chapter 1 There are a variety of inputs in PC Techs problem some easy to find and others more difficult Here are some ideas on how they might be obtained The unit costs in rows 3 4 and 10 should be easy to obtain See Figure 32 These are the going rates for labor and the component parts Note however that the labor costs are probably regulartime rates If the company wants to consider overtime hours then the overtime rate and labor hours availability during overtime would be necessary and the model would need to be modified Figure 32 TwoVariable Product Mix Model with an Infeasible Solution A B C D E F G 1 Assembling and testing computers Range names used 2 Hoursavailable ModelD21D22 3 Cost per labor hour assembling 11 Hoursused ModelB21B22 4 Cost per labor hour testing 15 Maximumsales ModelB18C18 5 Numbertoproduce ModelB16C16 6 Inputs for assembling and testing a computer Totalprofit ModelD25 7 Basic XP 8 Labor hours for assembly 5 6 9 Labor hours for testing 1 2 10 Cost of component parts 150 225 11 Selling price 300 450 12 Unit margin 80 129 14 Assembling testing plan of computers 15 Basic XP 16 Number to produce 600 1200 17 18 Maximum sales 600 1200 20 Constraints hours per month Hours used Hours available 21 Labor availability for assembling 10200 10000 22 Labor availability for testing 3000 3000 23 24 Net profit this month Basic XP Total 25 48000 154800 202800 The resource usages in rows 8 and 9 often called technological coefficients should be available from the production department These people know how much labor it takes to assemble and test these computer models The unit selling prices in row 11 have actually been chosen by PC Techs management probably in response to market pressures and the companys own costs The maximum sales values in row 18 are probably forecasts from the marketing and sales department These people have some sense of how much they can sell based on current outstanding orders historical data and the prices they plan to charge The labor hour availabilities in rows 21 and 22 are probably based on the current workforce size and possibly on new workers who could be hired in the short run Again if these are regulartime hours and overtime is possible the model would have to be modified to include overtime DEVELOPING THE SPREADSHEET MODEL The spreadsheet model appears in Figure 32 See the file Product Mix 1xlsx To develop this model use the following steps 1 Inputs Enter all of the inputs from the statement of the problem in the shaded cells as shown 2 Range names Create the range names shown in columns E and F Our convention is to enter enough range names but not to go overboard Specifically we enter enough range names so that the setup in the Solver dialog box to be explained shortly is entirely in terms of range names Of course you can add more range names if you like or you can omit them altogether The following tip indicates a quick way to create range names Excel Tip Shortcut for Creating Range Names Select a range such as A16C16 that includes nice labels in column A and the r ange you want to name in columns B and C Then fr om the F ormulas ribbon select Cr eate fr om Selection and accept the default You automatically get the labels in cells A16 as the range name for the range B16C16 This shortcut illustrates the usefulness of adding concise but informative labels next to ranges you want to name 3 Unit margins Enter the formula B11B8B3B9B4B10 in cell B12 and copy it to cell C12 to calculate the unit profit margins for the two models Enter relativeabsolute addresses that allow you to copy whenever possible 4 Changing cells Enter any two values for the changing cells in the Numbertoproduce range Any trial values can be used initially Solver eventually finds the optimal values Note that the two values shown in Figure 32 cannot be optimal because they use more assembling hours than are available However you do not need to worry about satisfying constraints at this point Solver takes care of this later on 5 Labor hours used To operationalize the labor availability constraints you must calculate the amounts used by the production plan To do this enter the formula SUMPRODUCTB8C8Numbertoproduce in cell B21 for assembling and copy it to cell B22 for testing This formula is a shortcut for the following fully written out formula B8B16C8C16 The SUMPRODUCT function is very useful in spreadsheet models especially LP models and you will see it often Here it multiplies the number of hours per computer by the number of computers for each model and then sums these products over the two models When there are only two products in the sum as in this example the SUMPRODUCT formula is not really any simpler than the writtenout formula However imagine that there are 50 models Then the SUMPRODUCT formula is much simpler to enter and read For this reason use it whenever possible Note that each range in this function B8C8 and Numbertoproduce is a onerow twocolumn range It is important in the SUMPRODUCT function that the two ranges be exactly the same size and shape 6 Net profits Enter the formula B12B16 in cell B25 copy it to cell C25 and sum these to get the total net profit in cell D25 This latter cell is the objective to maximize Note that if you didnt care about the net profits for the two individual models you could calculate the total net profit with the formula SUMPRODUCTB12C12Numbertoproduce As you see the SUMPRODUCT function appears once again It and the SUM function are the most used functions in LP models Experimenting with Possible Solutions The next step is to specify the changing cells the objective cell and the constraints in a Solver dialog box and then instruct Solver to find the optimal solution However before you do this it is instructive to try a few guesses in the changing cells There are two reasons for doing so First by entering different sets of values in the changing cells you can confirm that the formulas in the other cells are working correctly Second this experimentation can help you to develop a better understanding of the model For example the profit margin for XPs is much larger than for Basics so you might suspect that the company will produce only XPs The most it can produce is 1200 maximum sales and this uses fewer labor hours than are available This solution appears in Figure 33 However you can probably guess that it is far from optimal There are still many labor hours available so the company could use them to produce some Basics and make more profit You can continue to try different values in the changing cells attempting to get as large a total net profit as possible while staying within the constraints Even for this small model with only two changing cells the optimal solution is not totally obvious You can only imagine how much more difficult it is when there are hundreds or even thousands of changing cells and many constraints This is why software such as Excels Solver is required Solver uses a quick and efficient algorithm to search through all feasible solutions or more specifically all corner points and eventually find the optimal solution Fortunately it is quite easy to use as we now explain Figure 33 TwoVariable Product Mix Model with a Suboptimal Solution A B C D E F G 1 Assembling and testing computers Range names used 2 Hoursavailable ModelD21D22 3 Cost per labor hour assembling 11 Hoursused ModelB21B22 4 Cost per labor hour testing 15 Maximumsales ModelB18C18 5 Numbertoproduce ModelB16C16 6 Inputs for assembling and testing a computer Totalprofit ModelD25 7 Basic XP 8 Labor hours for assembly 5 6 9 Labor hours for testing 1 2 10 Cost of component parts 150 225 11 Selling price 300 450 12 Unit margin 80 129 13 14 Assembling testing plan of computers 15 Basic XP 16 Number to produce 0 1200 17 18 Maximum sales 600 1200 19 20 Constraints hours per month Hours used Hours available 21 Labor availability for assembling 7200 10000 22 Labor availability for testing 2400 3000 23 24 Net profit this month Basic XP Total 25 0 154800 154800 78 Chapter 3 Introduction to Optimization Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove designated content at any time if subsequent rights restrictions require it USING SOLVER To invoke Excels Solver select Solver from the Data ribbon If there is no such item on your PC you need to load Solver To do so click on the Office button then Excel Options then AddIns and then Go at the bottom of the dialog box This shows you the list of avail able addins If there is a Solver Addin item in the list check it to load Solver If there is no such item you need to rerun the Microsoft Office installer and elect to install Solver It should be included in a typical install but some people elect not to install it the first time around The dialog box in Figure 34 appears6 It has three important sections that you must fill in the objective cell the changing cells and the constraints For the product mix prob lem you can fill these in by typing cell references or you can point click and drag the appropriate ranges in the usual way Better yet if there are any named ranges these range names appear instead of cell addresses when you drag the ranges In fact for reasons of readability our convention is to use only range names not cell addresses in this dialog box 33 A TwoVariable Product Mix Model 79 Figure 34 Solver Dialog Box in Excel 2010 6This is the new Solver dialog box for Excel 2010 It is more convenient than similar dialog boxes in previous versions because the typical settings now all appear in a single dialog box In previous versions you have to click on Options to complete the typical settings Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Excel Tip Range Names in Solver Dialog Box Our usual procedure is to use the mouse to select the relevant ranges for the Solver dialog box Fortunately if these ranges have already been named then the range names will automatically replace the cell addresses 1 Objective Select the Totalprofit cell as the objective cell and click on the Max option Actually the default option is Max 2 Changing cells Select the Numbertoproduce range as the changing cells 3 Constraints Click on the Add button to bring up the dialog box in Figure 35 Here you specify a typical constraint by entering a cell reference or range name on the left the type of constraint from the dropdown list in the middle and a cell reference range name or numeric value on the right Use this dialog box to enter the constraint NumbertoproduceMaximumsales Note You can type these range names into the dialog box or you can drag them in the usual way If you drag them the cell addresses shown in the figure eventually change into range names if range names exist Then click on the Add button and enter the constraint HoursusedHoursavailable Then click on OK to get back to the Solver dialog box The first constraint says to produce no more than can be sold The second constraint says to use no more labor hours than are available Figure 35 Add Constraint Dialog Box Excel Tip Inequality and Equality Labels in Spreadsheet Models The signs in cells B17C17 and C21C22 see Figure 32 or Figure 33 are not a necessary part of the Excel model They are entered simply as labels in the spreadsheet and do not substitute for entering the constraints in the Add Constraint dialog box However they help to document the model so we include them in all of the examples In fact you should try to plan your spreadsheet models so that the two sides of a constraint are in nearby cells with gutter cells in between where you can attach labels like or This convention tends to make the resulting spreadsheet models much more readable Solver Tip Entering Constraints in Groups Constraints typically come in groups Beginners often enter these one at a time such as B16B18 and C16C18 in the Solver dialog box This can lead to a long list of constraints and it is timeconsuming work It is better to enter them as a group as in B16C16B18C18 This is not only quicker but it also takes advantage of range names you have created For example this group ends up as Numbertoproduce MaximumSales 80 Chapter 3 Introduction to Optimization Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove designated content at any time if subsequent rights restrictions require it 4 Nonnegativity Because negative production quantities make no sense you must tell Solver explicitly to make the changing cells nonnegative To do this check the Make Unconstrained Variables NonNegative option shown in Figure 34 This automatically ensures that all changing cells are nonnegative In previous versions of Solver you have to click on the Options button and then check the Assume NonNegative option in the result ing dialog box 5 Linear model There is one last step before clicking on the Solve button As stated previously Solver uses one of several numerical algorithms to solve various types of mod els The models discussed in this chapter are all linear models We will discuss the prop erties that distinguish linear models shortly Linear models can be solved most efficiently with the simplex method To instruct Solver to use this method make sure Simplex LP is selected in the Select a Solving Method dropdown list in Figure 34 In previous versions of Solver you have to click on the Options button and then check the Assume Linear Model option in the resulting dialog box In fact from now on if you are using a pre2010 version of Excel and we instruct you to use the simplex method you should check the Assume Linear Model option In contrast if we instruct you to use a nonlinear algorithm you should uncheck the Assume Linear Model option 6 Optimize Click on the Solve button in the dialog box in Figure 34 At this point Solver does its work It searches through a number of possible solutions until it finds the optimal solution You can watch the progress on the lower left of the screen although for small models the process is virtually instantaneous When it finishes it displays the mes sage shown in Figure 36 You can then instruct it to return the values in the changing cells to their original probably nonoptimal values or retain the optimal values found by Solver In most cases you should choose the latter For now click on the OK button to keep the Solver solution You should see the solution shown in Figure 37 33 A TwoVariable Product Mix Model 81 Checking the Non Negative option ensures only that the changing cells not any other cells will be nonnegative Figure 36 Solver Results Message Solver Tip Messages from Solver Actually the message in Figure 36 is the one you hope for However in some cases Solver is not able to f ind an optimal solution in whic h case one of se veral other messa ges appears We discuss some of these later in the chapter Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 37 TwoVariable Product Mix Model with the Optimal Solution A B C D E F G 1 Assembling and testing computers Range names used 2 3 Cost per labor hour assembling 11 Hoursused ModelB21B22 4 Cost per labor hour testing 15 Maximumsales ModelB18C18 5 Numbertoproduce ModelB16C16 6 Inputs for assembling and testing a computer Totalprofit ModelD25 7 BasicXP 8 Labor hours for assembly 5 6 9 Labor hours for testing 1 2 10 Cost of component parts 150 225 11 Selling price 300 450 12 Unit margin 80 129 13 14 Assembling testing plan of computers 15 Basic XP 16 Number to produce 560 1200 17 18 Maximum sales 600 1200 19 20 Constraints hours per month Hours used Hours available 21 Labor availability for assembling 10000 10000 22 Labor availability for testing 2960 3000 23 24 Net profit this month Basic XP Total 25 44800 154800 199600 Discussion of the Solution This solution says that PC Tech should produce 560 Basics and 1200 XPs This plan uses all available labor hours for assembling has a few leftover labor hours for testing produces as many XPs as can be sold and produces a few less Basics than could be sold No plan can provide a net profit larger than this onethat is without violating at least one of the constraints The solution in Figure 37 is typical of solutions to optimization models in the following sense Of all the inequality constraints some are satisfied exactly and others are not In this solution the XP maximum sales and assembling labor constraints are met exactly We say that they are binding However the Basic maximum sales and testing labor constraints are nonbinding For these nonbinding constraints the differences between the two sides of the inequalities are called slack7 You can think of the binding constraints as bottlenecks They are the constraints that prevent the objective from being improved If it were not for the binding constraints on maximum sales and labor PC Tech could obtain an even larger net profit An inequality constraint is binding if the solution makes it an equality Otherwise it is nonbinding and the positive difference between the two sides of the constraint is called the slack 7 Some analysts use the term slack only for constraints and the term surplus for constraints We refer to both of these as slackthe absolute difference between the two sides of the constraint 82 Chapter 3 Introduction to Optimization Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove designated content at any time if subsequent rights restrictions require it In a typical optimal solution you should usually pay particular attention to two aspects of the solution First you should check which of the changing cells are positive as opposed to 0 Generically these are the activities that are done at a positive level In a product mix model they are the products included in the optimal mix Second you should check which of the constraints are binding Again these represent the bottlenecks that keep the objective from improving Binding and Nonbinding Constraints Most optimization models contain constraints expressed as inequalities In an optimal solution each such constraint is either binding holds as an equality or nonbinding It is extremely important to identify the binding constraints because the y are the constraints that prevent the objective from improving A typical constraint is on the a vailability of a resource If such a constraint is binding the objective could typicall y impr ove b y ha ving mor e of that resource But if such a r esource constraint is nonbinding more of that r esource w ould not impr ove the objective at all 34 SENSITIVITY ANALYSIS Having found the optimal solution it might appear that the analysis is complete But in real LP applications the solution to a single model is hardly ever the end of the analysis It is almost always useful to perform a sensitivity analysis to see how or if the optimal solution changes as one or more inputs vary We illustrate systematic ways of doing so in this section Actually we discuss two approaches The first uses an optional sensitivity report that Solver offers The second uses an addin called SolverTable that one of the authors Albright developed 341 Solvers Sensitivity Report When you run Solver the dialog box in Figure 36 offers you the option to obtain a sensitivity report8 This report is based on a wellestablished theory of sensitivity analysis in optimization models especially LP models This theory was developed around algebraic models that are arranged in a standardized format Essentially all such algebraic models look alike so the same type of sensitivity report applies to all of them Specifically they have an objective function of the form c1x1 cnxn where n is the number of decision variables the cs are constants and the xs are the decision variables and each constraint can be expressed as a1x1 anxn b a1x1 anxn b or a1x1 anxn b where the as and bs are constants Solvers sensitivity report performs two types of sensitivity analysis 1 on the coefficients of the objective the cs and 2 on the right sides of the constraints the bs 8It also offers Answer and Limits reports We dont find these particularly useful so we will not discuss them here 34 Sensitivity Analysis 83 We illustrate the typical analysis by looking at the sensitivity report for PC Techs product mix model in Example 31 For convenience the algebraic model is repeated here Maximize 80x1 129x2 subject to 5x1 6x2 10000 x1 2x2 3000 x1 600 x2 1200 x1x2 0 On this Solver run a sensitivity report is requested in Solvers final dialog box See Figure 36 The sensitivity report appears on a new worksheet as shown in Figure 389 It contains two sections The top section is for sensitivity to changes in the two coefficients 80 and 129 of the decision variables in the objective Each row in this section indicates how the optimal solution changes if one of these coefficients changes The bottom section is for the sensitivity to changes in the right sides 10000 and 3000 of the labor constraints Each row of this section indicates how the optimal solution changes if one of these availabilities changes The maximum sales constraints represent a special kind of constraintupper bounds on the changing cells Upper bound constraints are handled in a special way in the Solver sensitivity report as described shortly Figure 38 Solver Sensitivity Results A B C D E F G H 6 Variable Cells 7 Final Reduced Objective Allowable Allowable 8 Cell Name Value Cost Coefficient Increase Decrease 9 B16 Number to produce Basic 560 0 80 275 80 10 C16 Number to produce XP 1200 33 129 1E30 33 11 12 Constraints 13 Final Shadow Constraint Allowable Allowable 14 Cell Name Value Price RH Side Increase Decrease 15 B21 Labor availability for assembling Used 10000 16 10000 200 2800 16 B22 Labor availability for testing Used 2960 0 3000 1E30 40 Now lets look at the specific numbers and their interpretation In the first row of the top section the allowable increase and allowable decrease indicate how much the coefficient of profit margin for Basics in the objective currently 80 could change before the optimal product mix would change If the coefficient of Basics stays within this allowable range from 0 decrease of 80 to 1075 increase of 275 the optimal product mixthe set of values in the changing cellsdoes not change at all However outside of these limits the optimal mix between Basics and XPs might change 9If your table looks different from ours make sure you chose the Simplex LP method or checked Assume Linear Model in pre2010 versions of Solver Otherwise Solver uses a nonlinear algorithm and produces a different type of sensitivity report 84 Chapter 3 Introduction to Optimization Modeling To see what this implies change the selling price in cell B11 from 300 to 299 so that the profit margin for Basics decreases to 79 This change is well within the allowable decrease of 80 If you rerun Solver you will obtain the same values in the changing cells although the objective value will decrease Next change the value in cell B11 to 330 This time the profit margin for Basics increases by 30 from its original value of 300 This change is outside the allowable increase so the solution might change If you rerun Solver you will indeed see a changethe company now produces 600 Basics and fewer than 1200 XPs The reduced costs in the second column indicate in general how much the objective coefficient of a decision variable that is currently 0 or at its upper bound must change before that variable changes becomes positive or decreases from its upper bound The interesting variable in this case is the number of XPs currently at its upper bound of 1200 The reduced cost for this variable is 33 meaning that the number of XPs will stay at 1200 unless the profit margin for XPs decreases by at least 33 Try it Starting with the original inputs change the selling price for XPs to 420 a change of less than 33 If you rerun Solver you will find that the optimal plan still calls for 1200 XPs Then change the selling price to 410 a change of more than 33 from the original value After rerunning Solver you will find that fewer than 1200 XPs are in the optimal mix 34 Sensitivity Analysis 85 The reduced cost for any decision variable with value 0 in the optimal solution indicates how much better that coefficient must be before that variable enters at a positive level The reduced cost for any decision variable at its upper bound in the optimal solution indicates how much worse its coefficient must be before it will decrease from its upper bound The reduced cost for any variable between 0 and its upper bound in the optimal solution is irrelevant Now turn to the bottom section of the report in Figure 38 Each row in this section corresponds to a constraint although upper bound constraints on changing cells are omit ted in this section To have this part of the report make economic sense the model should be developed as has been done here where the right side of each constraint is a numeric constant not a formula Then the report indicates how much these rightside constants can change before the optimal solution changes To understand this more fully the concept of a shadow price is required A shadow price indicates the change in the objective when a rightside constant changes The term shadow price is an economic term It indicates the change in the optimal value of the objective when the right side of some constraint changes by one unit A shadow price is reported for each constraint For example the shadow price for the assembling labor constraint is 16 This means that if the right side of this constraint increases by one hour from 10000 to 10001 the optimal value of the objective will increase by 16 It works in the other direction as well If the right side of this constraint decreases by one hour from 10000 to 9999 the optimal value of the objective will decrease by 16 However as the right side continues to increase or decrease this 16 change in the objective might not continue This is where the reported allowable increase and allowable decrease are relevant As long as the right side increases or decreases within its allowable limits the same shadow price of 16 still applies Beyond these limits how ever a different shadow price might apply Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it You can prove this for yourself First increase the right side of the assembling labor constraint by 200 exactly the allowable increase from 10000 to 10200 and rerun Solver Dont forget to reset other inputs to their original values You will see that the objective indeed increases by 162003200 from 199600 to 202800 Now increase this right side by one more hour from 10200 to 10201 and rerun Solver You will observe that the objective doesnt increase at all This means that the shadow price beyond 10200 is less than 16 in fact it is zero This is typical When a right side increases beyond its allowable increase the new shadow price is typically less than the original shadow price although it doesnt always fall to zero as in this example Resource Availability and Shadow Prices If a r esource constraint is binding in the optimal solution the company is willing to pa y up to some amount the shadow price to obtain mor e of the resource This is because the objective improves by having more of the resource However there is typically a decreasing marginal eff ect As the compan y owns more and more of the r esource the shadow price tends to decr ease This is usuall y because other constraints become binding which causes extra units of this resource to be less useful or not useful at all The idea is that a constraint costs the company by keeping the objective from being better than it would be A shadow price indicates how much the company would be willing to pay in units of the objective to relax a constraint In this example the company would be willing to pay 16 for each extra assembling hour This is because such a change would increase the net profit by 16 But beyond a certain point200 hours in this examplefurther relaxation of the constraint does no good and the company is not willing to pay for any further increases The constraint on testing hours is slightly different It has a shadow price of zero In fact the shadow price for a nonbinding constraint is always zero which makes sense If the right side of this constraint is changed from 3000 to 3001 nothing at all happens to the optimal product mix or the objective value there is just one more unneeded testing hour However the allowable decrease of 40 indicates that something does change when the right side reaches 2960 At this point the constraint becomes bindingthe testing hours used equal the testing hours availableand beyond this the optimal product mix starts to change By the way the allowable increase for this constraint shown as 1E30 means that it is essentially infinite The right side of this constraint can be increased above 3000 indefinitely and absolutely nothing will change in the optimal solution The Effect of Constraints on the Objective If a constraint is ad ded or an existing constraint becomes mor e constraining f or example less of some resource is available the objective can only get worse it can ne ver impr ove The easiest wa y to understand this is to think of the f easible r egion When a constraint is added or an existing constraint becomes mor e constraining the f easible r egion shrinks so some solutions that w ere feasible before maybe even the optimal solution are no longer feasible The opposite is true if a constraint is deleted or an existing constraint becomes less constraining In this case the objective can only improve it can never get worse Again the idea is that when a constraint is deleted or an existing constraint becomes less constraining the f easible r egion expands In this case all solutions that w ere feasible before are still feasible and there are some ad ditional feasible solutions available 86 Chapter 3 Introduction to Optimization Modeling 342 SolverTable AddIn The reason Solvers sensitivity report makes sense for the product mix model is that the spreadsheet model is virtually a direct translation of a standard algebraic model Unfortunately given the flexibility of spreadsheets this is not always the case We have seen many perfectly good spreadsheet modelsand have developed many ourselvesthat are structured quite differently from their standard algebraicmodel counterparts In these cases we have found Solvers sensitivity report to be more confusing than useful Therefore Albright developed an Excel addin called SolverTable SolverTable allows you to ask sensitivity questions about any of the input variables not just coefficients of the objective and right sides of constraints and it provides straightforward answers The SolverTable addin is on this books essential resource Web site10 To install it simply copy the SolverTable files to a folder on your hard drive These files include the addin itself the xlam file and the online help files To load SolverTable you can proceed in one of two ways 1 Open the SolverTablexlam file just as you open any other Excel file 2 Go to the addins list in Excel click on the Office button then Excel Options then AddIns then Go and check the SolverTable item If it isnt in the list Browse for the SolverTablexlam file The advantage of the second option is that if SolverTable is checked in the addins list it will automatically open every time you open Excel at least until you uncheck its item in the list The SolverTable addin was developed to mimic Excels builtin data table tool Recall that data tables allow you to vary one or two inputs in a spreadsheet model and see instantaneously how selected outputs change SolverTable is similar except that it runs Solver for every new input or pair of inputs and the newest version also provides auto matic charts of the results There are two ways it can be used 1 Oneway table A oneway table means that there is a single input cell and any num ber of output cells That is there can be a single output cell or multiple output cells 2 Twoway table A twoway table means that there are two input cells and one or more output cells You might recall that an Excel twoway data table allows only one output SolverTable allows more than one It creates a separate table for each output as a function of the two inputs We illustrate some of the possibilities for the product mix example Specifically we check how sensitive the optimal production plan and net profit are to 1 changes in the selling price of XPs 2 the number of labor hours of both types available and 3 the maximum sales of the two models We assume that the model has been formulated and optimized as shown in Figure 37 and that the SolverTable addin has been loaded To run SolverTable click on the Run SolverTable button on the SolverTable ribbon You will be asked whether there is a Solver model on the active sheet Note that the active sheet at this point should be the sheet con taining the model If it isnt click on Cancel and then activate this sheet You are then given the choice between a oneway or a twoway table For the first sensitivity question choose the oneway option You will see the dialog box in Figure 39 For the sensitivity analysis on the XP selling price fill it in as shown Note that ranges can be entered as cell addresses or range names Also multiple ranges in the Outputs box should be separated by commas 34 Sensitivity Analysis 87 Solvers sensitivity report is almost impossible to unravel for some models In these cases Solver Table is preferable because of its easily interpreted results We chose the input range from 350 to 550 in increments of 25 fairly arbitrarily You can choose any desired range of input values 10It is also available from the Free Downloads link on the authors Web site at wwwkelleyiuedualbrightbooks Actually there are several versions of SolverTable available each for a particular version of Solver The one described in the text is for Solver in Excel 2007 or 2010 This Web site contains more information about these versions as well as possible updates to SolverTable Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 88 Chapter 3 Introduction to Optimization Modeling Figure 39 SolverTable One Way Dialog Box 1 2 3 4 5 6 7 8 9 10 11 12 13 A B C D E F G Oneway analysis for Solver model in Model worksheet Selling Price XP cell C11 values along side output cells along top Numbertoproduce1 Numbertoproduce2 Totalprofit 0 7 0 0 0 350 600 1166667 81833 375 600 1166667 11100 400 600 1166667 14016 425 560 1200 16960 450 560 1200 19960 475 560 1200 22960 500 560 1200 259600 0 525 560 1200 28960 550 560 1200 319600 Figure 310 SolverTable Results for Varying XP Price Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Excel Tip Selecting Multiple Ranges If you need to select multiple output ranges the trick is to keep your finger on the Ctrl key as you drag the ranges This automatically enters the separating commas for you Actually the same trick works for selecting multiple changing cell ranges in Solvers dialog box When you click on OK Solver solves a separate optimization problem for each of the nine rows of the table and then reports the requested outputs number produced and net profit in the table as shown in Figure 310 It can take a while depending on the speed of your computer and the complexity of the model but everything is automatic However if you want to update this tableby using different XP selling prices in column A for exampleyou must repeat the procedure Note that if the requested outputs are included in named ranges the range names are used in the SolverTable headings For example the label Numbertoproduce1 indicates that this output is the first cell in the Numbertoproduce range The label Totalprofit indicates that this output is the only cell in the Totalprofit range If a requested output is not part of a named range its cell address is used as the label in the SolverTable results Figure 311 Associated SolverTable Chart for Net Profit K L M N O P Q R 3 Data for chart When you select an output address from the dropdown list in cell K4 the chart will adapt to that output 4 Totalprofit 5 8183333 6 111000 7 1401667 8 169600 9 199600 10 229600 11 259600 12 289600 13 319600 14 15 16 17 Sensitivity of Totalprofit to Selling Price XP 350000 18 300000 19 250000 20 21 22 150000 23 24 25 26 27 350 375 400 425 450 475 500 525 550 Selling Price XP C11 28 29 34 Sensitivity Analysis 89 The outputs in this table show that when the selling price of XPs is relatively low the company should make as many Basics as it can sell and a few less XPs but when the sell ing price is relatively high the company should do the opposite Also the net profit increases steadily through this range You can calculate these changes which are not part of the SolverTable output in column E The increase in net profit per every extra 25 in XP selling price is close to but not always exactly equal to 30000 SolverTable also produces the chart in Figure 311 There is a dropdown list in cell K4 where you can choose any of the SolverTable outputs We selected the net profit cell D25 The chart then shows the data for that column from the table in Figure 310 Here there is a steady increase slope about 30000 in net profit as the XP selling price increases The second sensitivity question asks you to vary two inputs the two labor availabili ties simultaneously This requires a twoway SolverTable so fill in the SolverTable dialog box as shown in Figure 312 Here two inputs and two input ranges are specified and mul tiple output cells are again allowed An output table is generated for each of the output cells as shown in Figure 313 For example the top table shows how the optimal number of Basics varies as the two labor availabilities vary Comparing the columns of this top table it is apparent that the optimal number of Basics becomes increasingly sensitive to the available assembling hours as the number of available testing hours increases The SolverTable output also includes two charts not shown here that let you graph any row or any column of any of these tables The third sensitivity question involving maximum sales of the two models reveals the flexibility of SolverTable Instead of letting these two inputs vary independently in a two way SolverTable it is possible to let both of them vary according to a single percentage change For example if this percentage change is 10 both maximum sales increase by 90 Chapter 3 Introduction to Optimization Modeling Figure 312 SolverTable Two Way Dialog Box Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 313 TwoWay SolverTable Results A B C D E F G H I 3 Assembling hours cell D21 values along side Testing hours cell D22 values along top output cell in corner 4 Numbertoproduce1 2000 2500 3000 3500 4000 4500 5000 5 8000 600 250 160 160 160 160 160 6 8500 600 500 260 260 260 260 260 7 9000 600 600 360 360 360 360 360 8 9500 600 600 460 460 460 460 460 9 10000 600 600 560 560 560 560 560 10 10500 600 600 600 600 600 600 600 11 11000 600 600 600 600 600 600 600 12 11500 600 600 600 600 600 600 600 13 12000 600 600 600 600 600 600 600 14 15 Numbertoproduce2 2000 2500 3000 3500 4000 4500 5000 16 8000 700 1125 1200 1200 1200 1200 1200 17 8500 700 1000 1200 1200 1200 1200 1200 18 9000 700 950 1200 1200 1200 1200 1200 19 9500 700 950 1200 1200 1200 1200 1200 20 10000 700 950 1200 1200 1200 1200 1200 21 10500 700 950 1200 1200 1200 1200 1200 22 11000 700 950 1200 1200 1200 1200 1200 23 11500 700 950 1200 1200 1200 1200 1200 24 12000 700 950 1200 1200 1200 1200 1200 25 26 Totalprofit 2000 2500 3000 3500 4000 4500 5000 27 8000 138300 165125 167600 167600 167600 167600 167600 28 8500 138300 169000 175600 175600 175600 175600 175600 29 9000 138300 170550 183600 183600 183600 183600 183600 30 9500 138300 170550 191600 191600 191600 191600 191600 31 10000 138300 170550 199600 199600 199600 199600 199600 32 10500 138300 170550 202800 202800 202800 202800 202800 33 11000 138300 170550 202800 202800 202800 202800 202800 34 11500 138300 170550 202800 202800 202800 202800 202800 35 12000 138300 170550 202800 202800 202800 202800 202800 10 The trick is to modify the model so that one percentagechange cell drives changes in both maximum sales The modified model appears in Figure 314 Starting with the original model enter the original values 600 and 1200 in new cells E18 and F18 Do not copy the range B18C18 to E18F18 This would make the right side of the constraint 34 Sensitivity Analysis 91 Figure 314 Modified Model for Simultaneous Changes A B C D E F G H 1 Assembling and testing computers 2 3 Cost per labor hour assembling 11 4 Cost per labor hour testing 15 5 6 Inputs for assembling and testing a computer 7 Basic XP 8 Labor hours for assembly 5 6 9 Labor hours for testing 1 2 10 Cost of component parts 150 225 11 Selling price 300 450 12 Unit margin 80 129 13 14 Assembling testing plan of computers 15 Basic XP 16 Number to produce 560 1200 17 Original values change in both 18 Maximum sales 600 1200 600 1200 0 19 20 Constraints hours per month Hours used Hours available 21 Labor availability for assembling 10000 10000 22 Labor availability for testing 2960 3000 23 24 Net profit this month Basic XP Total 25 44800 154800 199600 E18F18 which is not the desired behavior Then enter any percentage change in cell G18 Finally enter the formula E181G18 in cell B18 and copy it to cell C18 Now a oneway SolverTable can be used with the percentage change in cell G18 to drive two different inputs simultaneously Specifically the SolverTable dialog box should be set up as in Figure 315 with the corresponding results in Figure 316 You should always scan these sensitivity results to see if they make sense For example if the company can sell 20 or 30 more of both models it makes no more profit than if it can sell only 10 more The reason is labor availability By this point there isnt enough labor to produce the increased demand It is always possible to run a sensitivity analysis by changing inputs manually in the spreadsheet model and rerunning Solver The advantages of SolverTable however are that it enables you to perform a systematic sensitivity analysis for any selected inputs and outputs and it keeps track of the results in a table and associated charts You will see other applications of this useful addin later in this chapter and in the next few chapters 343 Comparison of Solvers Sensitivity Report and SolverTable Sensitivity analysis in optimization models is extremely important so it is important that you understand the pros and cons of the two tools in this section Here are some points to keep in mind 92 Chapter 3 Introduction to Optimization Modeling 34 Sensitivity Analysis 93 Figure 315 SolverTable One Way Dialog Box 3 4 5 6 7 8 9 10 11 A B C D E F G change in max sales cell G18 values along side output cells along top Numbertoproduce1 Numbertoproduce2 Totalprofit B12 30 420 840 141960 80 20 480 960 162240 80 10 540 1080 182520 80 0 560 1200 199600 80 10 500 1250 201250 80 20 500 1250 201250 80 30 500 1250 201250 80 Figure 316 Sensitivity to Percentage Change in Maximum Sales Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Solvers sensitivity report focuses only on the coefficients of the objective and the right sides of the constraints SolverTable allows you to vary any of the inputs Solvers sensitivity report provides very useful information through its reduced costs shadow prices and allowable increases and decreases This same information can be obtained with SolverTable but it requires a bit more work and some experimentation with the appropriate input ranges Solvers sensitivity report is based on changing only one objective coefficient or one right side at a time This oneatatime restriction prevents you from answering certain questions directly SolverTable is much more flexible in this respect Solvers sensitivity report is based on a wellestablished mathematical theory of sensitivity analysis in linear programming If you lack this mathematical backgroundas many users dothe outputs can be difficult to understand especially for somewhat nonstandard spreadsheet formulations In contrast SolverTables outputs are straightforward You can vary one or two inputs and see directly how the optimal solution changes Solvers sensitivity report is not even available for integerconstrained models and its interpretation for nonlinear models is more difficult than for linear models SolverTables outputs have the same interpretation for any type of optimization model Solvers sensitivity report comes with Excel SolverTable is a separate addin that is not included with Excelbut it is included with this book and is freely available from the Free Downloads link at the authors Web site wwwkelleyiuedualbrightbooks Because the SolverTable software essentially automates Solver which has a number of its own idiosyncrasies some users have had problems with SolverTable on their PCs We have tried to document these on our Web site and we are hoping that the revised Solver in Excel 2010 helps to alleviate these problems In summary each of these tools can be used to answer certain questions We tend to favor SolverTable because of its flexibility but in the optimization examples in this chapter and the next few chapters we will illustrate both tools to show how each can provide useful information 35 PROPERTIES OF LINEAR MODELS Linear programming is an important subset of a larger class of models called mathemati cal programming models11 All such models select the levels of various activities that can be performed subject to a set of constraints to maximize or minimize an objective such as total profit or total cost In PC Techs product mix example the activities are the numbers of PCs to produce and the purpose of the model is to find the levels of these activities that maximize the total net profit subject to specified constraints In terms of this general setupselecting the optimal levels of activitiesthere are three important properties that LP models possess that distinguish them from general mathematical programming models proportionality additivity and divisibility We dis cuss these properties briefly in this section 94 Chapter 3 Introduction to Optimization Modeling 11The word programming in linear programming or mathematical programming has nothing to do with com puter programming It originated with the British term programme which is essentially a plan or a schedule of operations Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 351 Proportionality Proportionality means that if the level of any activity is multiplied by a constant factor the contribution of this activity to the objective or to any of the constraints in which the activity is involved is multiplied by the same factor For example suppose that the production of Basics is cut from its optimal value of 560 to 280that is it is multiplied by 05 Then the amounts of labor hours from assembling and from testing Basics are both cut in half and the net profit contributed by Basics is also cut in half Proportionality is a perfectly valid assumption in the product mix model but it is often violated in certain types of models For example in various blending models used by petroleum companies chemical outputs vary in a nonlinear manner as chemical inputs are varied If a chemical input is doubled say the resulting chemical output is not necessarily doubled This type of behavior violates the proportionality property and it takes us into the realm of nonlinear optimization which we discuss in Chapters 7 and 8 352 Additivity The additivity property implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint For example if the two PC models use respectively 560 and 2400 testing hours as in Figure 37 then the total number used in the plan is the sum of these amounts 2960 hours Similarly the additivity property applies to the objective That is the value of the objective is the sum of the contributions from the various activities In the product mix model the net profits from the two PC models add up to the total net profit The additivity property implies that the contribution of any decision variable to the objective or to any constraint is independent of the levels of the other decision variables 353 Divisibility The divisibility property simply means that both integer and noninteger levels of the activities are allowed In the product mix model we got integer values in the optimal solution 560 and 1200 just by luck For slightly different inputs they could easily have been fractional values In general if you want the levels of some activities to be integer values there are two possible approaches 1 You can solve the LP model without integer constraints and if the solution turns out to have fractional values you can attempt to round them to integer values or 2 you can explicitly constrain certain changing cells to contain integer values The latter approach however takes you into the realm of integer programming which we study in Chapter 6 At this point we simply state that integer problems are much more difficult to solve than problems without integer constraints 354 Discussion of Linear Properties The previous discussion of these three properties especially proportionality and additivity is fairly abstract How can you recognize whether a model satisfies proportionality and additivity This is easy if the model is described algebraically In this case the objective must be of the form a1 x1 a2 x2 an xn where n is the number of decision variables the as are constants and the xs are decision variables This expression is called a linear combination of the xs Also each constraint must be equivalent to a form where the left side is a linear combination of the xs and the right side is a constant For example the following is a typical linear constraint 3x1 7x2 2x3 50 It is not quite so easy to recognize proportionality and additivityor the lack of themin a spreadsheet model because the logic of the model is typically embedded in a series of cell formulas However the ideas are the same First the objective cell must ultimately possibly through a series of formulas in intervening cells be a sum of products of con stants and changing cells where a constant means that it does not depend on changing cells Second each side of each constraint must ultimately be either a constant or a sum of products of constants and changing cells This explains why linear models contain so many SUM and SUMPRODUCT functions It is usually easier to recognize when a model is not linear Two particular situations that lead to nonlinear models are when 1 there are products or quotients of expressions involving changing cells or 2 there are nonlinear functions such as squares square roots or logarithms that involve changing cells These are typically easy to spot and they guar antee that the model is nonlinear Whenever you model a real problem you usually make some simplifying assumptions This is certainly the case with LP models The world is frequently not linear which means that an entirely realistic model typically violates some or all of the three properties in this section However numerous successful applications of LP have demonstrated the useful ness of linear models even if they are only approximations of reality If you suspect that the violations are serious enough to invalidate a linear model you should use an integer or non linear model as we illustrate in Chapters 68 In terms of Excels Solver if the model is linearthat is if it satisfies the propor tionality additivity and divisibility propertiesyou should check the Simplex option or the Assume Linear Model option in pre2010 versions of Excel Then Solver uses the simplex method a very efficient method for a linear model to solve the problem Actually you can check the Simplex option even if the divisibility property is violatedthat is for linear models with integerconstrained variablesbut Solver then embeds the simplex method in a more complex algorithm branch and bound in its solution procedure 355 Linear Models and Scaling12 In some cases you might be sure that a model is linear but when you check the Simplex option or the Assume Linear Model option and then solve you get a Solver message to the effect that the conditions for linearity are not satisfied This can indicate a logical error in your formulation so that the proportionality and additivity conditions are indeed not sat isfied However it can also indicate that Solver erroneously thinks the linearity conditions are not satisfied which is typically due to roundoff error in its calculationsnot any error on your part If the latter occurs and you are convinced that the model is correct you can try not using the simplex method to see whether that works If it does not you should con sult your instructor It is possible that the nonsimplex algorithm employed by Solver sim ply cannot find the solution to your problem In any case it always helps to have a wellscaled model In a wellscaled model all of the numbers are roughly the same magnitude If the model contains some very large numbers100000 or more sayand some very small numbers0001 or less say it is poorly scaled for the methods used by Solver and roundoff error is far more likely to be an issue not only in Solvers test for linearity conditions but in all of its algorithms 96 Chapter 3 Introduction to Optimization Modeling Reallife problems are almost never exactly linear However linear approximations often yield very useful results 12This section might seem overly technical However when you develop a model that you are sure is linear and Solver then tells you it doesnt satisfy the linear conditions you will appreciate this section Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it You can decrease the chance of getting an incorrect Conditions for Assume Linear Model are not satisfied message by changing Solvers Precision setting If you believe your model is poorly scaled there are three possible remedies The first is to check the Use Automatic Scaling option in Solver It is found by clicking on the Options button in the main Solver dialog box This might help and it might not we have had mixed success Frontline Systems the company that develops Solver has told us that the only drawback to checking this box is that the solution procedure can be slower The second option is to redefine the units in which the various quantities are defined Finally you can change the Precision setting in Solvers Options dialog box to a larger number such 000001 or 00001 The default has five zeros Excel Tip Rescaling a Model Suppose you have a whole range of input values expressed say in dollars and you would like to reexpress them in thousands of dollars that is you would like to divide each value by 1000 There is a simple copypaste way to do this Enter the value 1000 in some unused cell and copy it Then highlight the range you want to rescale and from the Paste dropdown menu select Paste Special and then the Divide option No formulas are required your original values are automatically rescaled and you can then delete the 1000 cell You can use this same method to add subtract or multiply by a constant 36 INFEASIBILITY AND UNBOUNDEDNESS In this section we discuss two of the things that can go wrong when you invoke Solver Both of these might indicate that there is a mistake in the model Therefore because mistakes are common in LP models you should be aware of the error messages you might encounter 361 Infeasibility The first problem is infeasibility Recall that a solution is feasible if it satisfies all of the constraints Among all of the feasible solutions you are looking for the one that optimizes the objective However it is possible that there are no feasible solutions to the model There are generally two reasons for this 1 there is a mistake in the model an input was entered incorrectly such as a symbol instead of a or 2 the problem has been so constrained that there are no solutions left In the former case a careful check of the model should find the error In the latter case you might need to change or even eliminate some of the constraints To show how an infeasible problem could occur suppose in PC Techs product mix problem you change the maximum sales constraints to minimum sales constraints and leave everything else unchanged That is you change these constraints from to If Solver is then used the message in Figure 317 appears indicating that Solver cannot find a feasible solution The reason is clear There is no way given the constraints on labor hours that the company can produce these minimum sales values The companys only choice is to set at least one of the minimum sales values lower In general there is no foolproof way to remedy the problem when a no feasible solution message appears Careful checking and rethinking are required 362 Unboundedness A second type of problem is unboundedness In this case the model has been formulated in such a way that the objective is unboundedthat is it can be made as large or as small for minimization problems as you like If this occurs you have probably entered a wrong input or forgotten some constraints To see how this could occur in the product mix problem Figure 317 No Feasible Solution Message Figure 318 Unbounded Solution Message suppose that you change all constraints to be instead of Now there is no upper bound on how much labor is available or how many PCs the company can sell If you make this change in the model and then use Solver the message in Figure 318 appears stating that the objective cell does not converge In other words the total net profit can grow without bound 363 Comparison of Infeasibility and Unboundedness Infeasibility and unboundedness are quite different in a practical sense It is quite possible for a reasonable model to have no feasible solutions For example the marketing department might impose several constraints the production department might add some more the engineering department might add even more and so on Together they might constrain the problem so much that there are no feasible solutions left The only way out is Except in very rare situations if Solver informs you that your model is unbounded you have made an error PROBLEMS Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 Other sensitivity analyses besides those discussed could be performed on the product mix model Use SolverTable to perform each of the following In each case keep track of the values in the changing cells and the objective cell and discuss your findings a Let the selling price for Basics vary from 220 to 350 in increments of 10 b Let the labor cost per hour for assembling vary from 5 to 20 in increments of 1 c Let the labor hours for testing a Basic vary from 05 to 30 in increments of 05 d Let the labor hours for assembling and testing an XP vary independently the first from 45 to 80 and the second from 15 to 30 both in increments of 05 2 In PC Techs product mix problem assume there is another PC model the VXP that the company can produce in addition to Basics and XPs Each VXP requires eight hours for assembling three hours for testing 275 for component parts and sells for 560 At most 50 VXPs can be sold a Modify the spreadsheet model to include this new product and use Solver to find the optimal product mix b You should find that the optimal solution is not integervalued If you round the values in the changing cells to the nearest integers is the resulting solution still feasible If not how might you obtain a feasible solution that is at least close to optimal 3 Continuing the previous problem perform a sensitivity analysis on the selling price of VXPs Let this price vary from 500 to 650 in increments of 10 and keep track of the values in the changing cells and the objective cell Discuss your findings 4 Again continuing Problem 2 suppose that you want to force the optimal solution to be integers Do this in Solver by adding a new constraint Select the changing cells for the left side of the constraint and in the middle dropdown list select the int option How does the optimal integer solution compare to the optimal noninteger solution in Problem 2 Are the changing cell values rounded versions of those in Problem 2 Is the objective value more or less than in Problem 2 5 If all of the inputs in PC Techs product mix problem are nonnegative as they should be for any realistic version of the problem are there any input values such that the resulting model has no feasible solutions Refer to the graphical solution 6 There are five corner points in the feasible region for the product mix problem We identified the coordinates of one of them 560 1200 Identify the coordinates of the others a Only one of these other corner points has positive values for both changing cells Discuss the changes in the selling prices of either or both models that would be necessary to make this corner point optimal b Two of the other corner points have one changing cell value positive and the other zero Discuss the changes in the selling prices of either or both models that would be necessary to make either of these corner points optimal SkillExtending Problems 7 Using the graphical solution of the product mix model as a guide suppose there are only 2800 testing hours available How do the answers to the previous problem change Is the previous solution still optimal Is it still feasible 8 Again continuing Problem 2 perform a sensitivity analysis where the selling prices of Basics and XPs simultaneously change by the same percentage but the selling price of VXPs remains at its original value Let the percentage change vary from 25 to 50 in increments of 5 and keep track of the values in the changing cells and the total profit Discuss your findings 9 Consider the graphical solution to the product mix problem Now imagine that another constraintany constraintis added Which of the following three things are possible 1 the feasible region shrinks 2 the feasible region stays the same 3 the feasible region expands Which of the following three things are possible 1 the optimal value in objective cell decreases 2 the optimal value in objective cell stays the same 3 the optimal value in objective cell increases Explain your answers Do they hold just for this particular model or do they hold in general 36 Infeasibility and Unboundedness 99 37 A LARGER PRODUCT MIX MODEL The problem we examine in this section is a direct extension of the product mix model in the previous section There are two modifications First the company makes eight com puter models not just two Second testing can be done on either of two lines and these two lines have different characteristics 100 Chapter 3 Introduction to Optimization Modeling E X A M P L E 32 PRODUCING COMPUTERS AT PC TECH A s in the previous example PC Tech must decide how many of each of its computer models to assemble and test but there are now eight available models not just two Each computer must be assembled and then tested but there are now two lines for testing The first line tends to test faster but its labor costs are slightly higher and each line has a certain number of hours available for testing Any computer can be tested on either line The inputs for the model are same as before 1 the hourly labor costs for assembling and testing 2 the required labor hours for assembling and testing any computer model 3 the cost of component parts for each model 4 the selling prices for each model 5 the maximum sales for each model and 6 labor availabilities These input values are listed in the file Product Mix 2xlsx As before the company wants to determine the prod uct mix that maximizes its total net profit Objective To use LP to find the mix of computer models that maximizes total net profit and stays within the labor hour availability and maximum sales constraints WHERE DO THE NUMBERS COME FROM The same comments as in Example 31 apply here Solution Table 32 lists the variables and constraints for this model You must choose the number of computers of each model to produce on each line the sum of which cannot be larger than the maximum that can be sold This choice determines the labor hours of each type used and all revenues and costs No more labor hours can be used than are available Table 32 Variables and Constraints for Larger Product Mix Model Input variables Hourly labor costs labor availabilities labor required for each computer costs of component parts unit selling prices and maximum sales Decision variables changing cells Numbers of computer of each model to test on each line Objective cell Total net profit Other calculated variables Number of each computer model produced hours of labor used for assembling and for each line of testing Constraints Computers produced Maximum sales Labor hours used Labor hours available It is probably not immediately obvious what the changing cells should be for this model at least not before you look at Table 32 You might think that the company simply needs to decide how many computers of each model to produce However because of the two Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it testing lines this is not enough information The company must also decide how many of each model to test on each line For example suppose they decide to test 100 model 4s on line 1 and 300 model 4s on line 2 This means they will need to assemble and ultimately sell 400 model 4s In other words given the detailed plan of how many to test on each line everything else is determined But without the detailed plan there is not enough information to complete the model This is the type of reasoning you must go through to determine the appropriate changing cells for any LP model An Algebraic Model We will not spell out the algebraic model for this expanded version of the product mix model because it is so similar to the twovariable product mix model However we will say that it is larger and hence probably more intimidating Now we need decision variables of the form xij the number of model j computers to test on line i and the total net profit and each labor availability constraint will include a long SUMPRODUCT formula involving these variables Instead of focusing on these algebraic expressions we turn directly to the spreadsheet model DEVELOPING THE SPREADSHEET MODEL The spreadsheet in Figure 319 illustrates the solution procedure for PC Techs product mix problem See the file Product Mix 2xlsx The first stage is to develop the spreadsheet model step by step ➊ Inputs Enter the various inputs in the blue ranges Again remember that our convention is to color all input cells blue Enter only numbers not formulas in input cells They should always be numbers directly from the problem statement In this case we supplied them in the spreadsheet template ➋ Range names Name the ranges indicated According to our convention there are enough named ranges so that the Solver dialog box contains only range names no cell addresses Of course you can name additional ranges if you like Note that you can again use the rangenaming shortcut explained in the Excel tip for the previous example That is you can take advantage of labels in adjacent cells except for the Profit cell ➌ Unit margins Note that two rows of these are required one for each testing line because the costs of testing on the two lines are not equal To calculate them enter the formula B13B3B9B4B10B12 in cell B14 and copy it to the range B14I15 ➍ Changing cells As discussed above the changing cells are the red cells in rows 19 and 20 You do not have to enter the values shown in Figure 319 You can use any trial values initially Solver will eventually find the optimal values Note that the four values shown in Figure 319 cannot be optimal because they do not satisfy all of the constraints Specifically this plan uses more labor hours for assembling than are available However you do not need to worry about satisfying constraints at this point Solver will take care of this later ➎ Labor use d Enter the formula SUMPRODUCTB9E9Totalcomputersproduced in cell B26 to calculate the number of assembling hours used Similarly enter the formulas SUMPRODUCTB10I10Numbertestedonline1 and SUMPRODUCTB11I11Numbertestedonline2 in cells B27 and B28 for the labor hours used on each testing line Excel Tip Copying formulas with range names When you enter a r ange name in an Excel formula and then copy the formula the r ange name reference acts like an absolute reference Therefore it wouldnt work to copy the formula in cell B27 to cell B28 Howe ver this would work if r ange names hadnt been used This is one potential disadvantage of range names that you should be aware of 102 Chapter 3 Introduction to Optimization Modeling Figure 319 Larger Product Mix Model with Infeasible Solution A B C D E F G H I J 1 Assembling and testing computers 2 3 Cost per labor hour assembling 11 4 Cost per labor hour testing line 1 19 5 Cost per labor hour testing line 2 17 6 7 Inputs for assembling and testing a computer 8 Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 9 Labor hours for assembly 4 5 5 5 55 55 55 6 10 Labor hours for testing line 1 15 2 2 2 25 25 25 3 11 Labor hours for testing line 2 2 25 25 25 3 3 35 35 12 Cost of component parts 150 225 225 225 250 250 250 300 13 Selling price 350 450 460 470 500 525 530 600 14 Unit margin tested on line 1 128 132 142 152 142 167 172 177 15 Unit margin tested on line 2 122 128 138 148 139 164 160 175 16 17 Assembling testing plan of computers 18 Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 19 Number tested on line 1 0 0 0 0 0 500 1000 800 20 Number tested on line 2 0 0 0 1250 0 0 0 0 21 Total computers produced 0 0 0 1250 0 500 1000 800 22 23 Maximum sales 1500 1250 1250 1250 1000 1000 1000 800 24 25 Constraints hours per month Hours used Hours available 26 Labor availability for assembling 19300 20000 27 Labor availability for testing line 1 6150 5000 28 Labor availability for testing line 2 3125 6000 29 30 Net profit per month Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Totals 31 Tested on line 1 0 0 0 0 0 83500 172000 141600 397100 32 Tested on line 2 0 0 0 184375 0 0 0 0 184375 33 581475 34 35 Range names used 36 Hoursavailable ModelD26D28 37 Hoursused ModelB26B28 38 Maximumsales ModelB23I23 39 Numbertestedonline1 ModelB19I19 40 Numbertestedonline2 ModelB20I20 41 Totalcomputersproduce d ModelB21I21 42 Totalprofit ModelJ33 and SUMPRODUCTB11I11Numbertestedonline2 in cells B27 and B28 for the labor hours used on each testing line Excel Tip Copying formulas with range names When you enter a r ange name in an Excel formula and then copy the formula the r ange name reference acts like an absolute reference Therefore it wouldnt work to copy the formula in cell B27 to cell B28 Howe ver this would work if r ange names hadnt been used This is one potential disadvantage of range names that you should be aware of 102 Chapter 3 Introduction to Optimization Modeling 6 Revenues costs and profits The area from row 30 down shows the summary of monetary values Actually only the total profit in cell J33 is needed but it is also useful to calculate the net profit from each computer model on each testing line To obtain these enter the formula B14B19 in cell B31 and copy it to the range B31I32 Then sum these to obtain the totals in column J The total in cell J33 is the objective to maximize Experimenting with Other Solutions Before going any further you might want to experiment with other values in the changing cells However it is a real challenge to guess the optimal solution It is tempting to fill up the changing cells corresponding to the largest unit margins However this totally ignores their use of the scarce labor hours If you can guess the optimal solution to this model you are better than we are USING SOLVER The Solver dialog box should be filled out as shown in Figure 320 Again note that there are enough named ranges so that only range names appear in this dialog box Except that this model has two rows of changing cells the Solver setup is identical to the one in Example 31 Figure 320 Solver Dialog Box You typically gain insights into a solution by checking which constraints are binding and which contain slack Discussion of the Solution When you click on Solve you obtain the optimal solution shown in Figure 321 The optimal plan is to produce computer models 1 4 6 and 7 only some on testing line 1 and others on testing line 2 This plan uses all of the available labor hours for assembling and testing on line 1 but about 1800 of the testing line 2 hours are not used Also maximum sales are achieved only for computer models 1 6 and 7 This is typical of an LP solution Some of the constraints are met exactlythey are bindingwhereas others contain a certain amount of slack The binding constraints prevent PC Tech from earning an even higher profit Figure 321 Optimal Solution to Larger Product Mix Model Excel Tip Roundoff Error Because of the way numbers are stored and calculated on a computer the optimal values in the changing cells and elsewhere can contain small roundoff errors For example the value that really appears in cell E20 on one of our Excel 2007 PCs is 475000002015897 not exactly 475 For all practical purposes this number can be treated as 475 and we have formatted it as such in the spreadsheet We have been told that roundoff in Solver results should be less of a problem in Excel 2010 Sensitivity Analysis If you want to experiment with different inputs to this problem you can simply change the inputs and then rerun Solver The second time you use Solver you do not have to specify the objective and changing cells or the constraints Excel remembers all of these settings and saves them when you save the file You can also use SolverTable to perform a more systematic sensitivity analysis on one or more input variables One possibility appears in Figure 322 where the number of available assembling labor hours is allowed to vary from 18000 to 25000 in increments of 1000 and the numbers of computers produced and profit are designated as outputs Figure 322 Sensitivity to Assembling Labor Hours There are several ways to interpret the output from this sensitivity analysis First you can look at columns B through I to see how the product mix changes as more assembling labor hours become available For assembling labor hours from 18000 to 23000 the only thing that changes is that more model 4s are produced Beyond 23000 however the company starts to produce model 3s and produces fewer model 7s Second you can see how extra labor hours add to the total profit Note exactly what this increased profit means For example when labor hours increase from 20000 to 21000 the model requires that the company must pay 11 apiece for these extra hours if it uses them But the net effect is that profit increases by 29500 or 2950 per extra hour In other words the labor cost increases by 11000 111000 but this is more than offset by the increase in revenue that comes from having the extra labor hours As column J illustrates it is worthwhile for the company to obtain extra assembling labor hours even though it has to to pay for them because its profit increases However the increase in profit per extra labor hourthe shadow price of assembling labor hours is not constant In the top part of the table it is 2950 per extra hour but it then decreases to 2044 and then 242 The accompanying SolverTable chart of column J illustrates this decreasing shadow price through its decreasing slope SolverTable Technical Tip Charts and Roundoff As SolverTable makes all of its Solver runs it reports and then charts the values found by Solver These can include small roundoff errors and slightly misleading charts For example the chart in Figure 323 shows one possibility where we varied the cost of testing on line 2 and charted the assembling hours used Throughout the range this output value was 20000 but because of slight roundoff 199999999999292 and 200000000003259 in two of the cells the chart doesnt appear to be flat If you see this behavior you can change it manually Figure 323 A Misleading SolverTable Chart Finally you can gain additional insight from Solvers sensitivity report shown in Figure 324 However you have to be very careful in interpreting this report Unlike Example 31 there are no upper bound maximum sales constraints on the changing cells The maximum sales constraints are on the total computers produced row 21 of the model not the changing cells Therefore the only nonzero reduced costs in the top part of the table are for changing cells currently at zero not those at their upper bounds as in the previous example Each nonzero reduced cost indicates how much the profit margin for this activity would have to change before this activity would be profitable Also there is a row in the bottom part of the table for each constraint including the maximum sales constraints The interesting values are again the shadow prices The first two indicate the amount the company would pay for an extra assembling or line 1 testing labor hour Does the 295 value look familiar Compare it to the SolverTable results above The shadow prices for all binding maximum sales constraints indicate how much more profit the company could make if it could increase its demand by one computer of that model Figure 324 Solvers Sensitivity Report The information in this sensitivity report is all relevant and definitely provides some insights if studied carefully However this really requires you to know the exact rules Solver uses to create this report That is it requires a fairly indepth knowledge of the theory behind LP sensitivity analysis more than we have provided here Fortunately we believe the same basic informationand morecan be obtained in a more intuitive way by creating several carefully chosen SolverTable reports SkillBuilding Problems Note All references to the product mix model in the following problems are to the larger product mix model in this section 10 Modify PC Techs product mix model so that there is no maximum sales constraint This is easy to do in the Solver dialog box Just highlight the constraint and click on the Delete button Does this make the problem unbounded Does it change the optimal solution at all Explain its effect 11 In the product mix model it makes sense to change the maximum sales constraint to a minimum sales constraint simply by changing the direction of the inequality Then the input values in row 23 can be considered customer demands that must be met Make this change and rerun Solver What do you find What do you find if you run Solver again this time making the values in row 23 onequarter of their current values 12 Use SolverTable to run a sensitivity analysis on the cost per assembling labor hour letting it vary from 5 to 20 in increments of 1 Keep track of the computers produced in row 21 the hours used in the range B26B28 and the total profit Discuss your findings Are they intuitively what you expected 13 Create a twoway SolverTable for the product mix model where total profit is the only output and the two inputs are the testing line 1 hours and testing line 2 hours available Let the former vary from 4000 to 6000 in increments of 500 and let the latter vary from 3000 to 5000 in increments of 500 Discuss the changes in profit you see as you look across the various rows of the table Discuss the changes in profit you see as you look down the various columns of the table 14 Model 8 has fairly high profit margins but it isnt included at all in the optimal mix Use SolverTable along with some experimentation on the correct range to find the approximate selling price required for model 8 before it enters the optimal product mix SkillExtending Problems 15 Suppose that you want to increase all three of the resource availabilities in the product mix model simultaneously by the same percentage You want this percentage to vary from 25 to 50 in increments of 5 Modify the spreadsheet model slightly so that this sensitivity analysis can be performed with a oneway SolverTable using the percentage change as the single input Keep track of the computers produced in row 21 the hours used in the range B26B28 and the total profit Discuss the results 16 Some analysts complain that spreadsheet models are difficult to resize You can be the judge of this Suppose the current product mix problem is changed so that there is an extra resource packaging labor hours and two additional PC models 9 and 10 What additional input data are required What modifications are necessary in the spreadsheet model including range name changes Make up values for any extra required input data and incorporate these into a modified spreadsheet model Then optimize with Solver Do you conclude that it is easy to resize a spreadsheet model By the way it turns out that algebraic models are typically much easier to resize 17 In Solvers sensitivity report for the product mix model the allowable decrease for available assembling hours is 2375 This means that something happens when assembling hours fall to 20000 2375 17625 See what this means by first running Solver with 17626 available hours and then again with 17624 available hours Explain how the two solutions compare to the original solution and to each other 38 A MULTIPERIOD PRODUCTION MODEL The product mix examples illustrate a very important type of LP model However LP models come in many forms For variety we now present a quite different type of model that can also be solved with LP In the next few chapters we provide other examples linear and otherwise The distinguishing feature of the following model is that it relates decisions made during several time periods This type of problem occurs when a company must make a decision now that will have ramifications in the future The company does not want to focus completely on the short run and forget about the long run 38 A Multiperiod Production Model 109 E X A M P L E 33 PRODUCING FOOTBALLS AT PIGSKIN T he Pigskin Company produces footballs Pigskin must decide how many footballs to pro duce each month The company has decided to use a sixmonth planning horizon The forecasted monthly demands for the next six months are 10000 15000 30000 35000 25000 and 10000 Pigskin wants to meet these demands on time knowing that it currently has 5000 footballs in inventory and that it can use a given months production to help meet the demand for that month For simplicity we assume that production occurs during the month and demand occurs at the end of the month During each month there is enough production capacity to produce up to 30000 footballs and there is enough storage capacity to store up to 10000 footballs at the end of the month after demand has occurred The forecasted produc tion costs per football for the next six months are 1250 1255 1270 1280 1285 and 1295 respectively The holding cost per football held in inventory at the end of any month is figured at 5 of the production cost for that month This cost includes the cost of storage and also the cost of money tied up in inventory The selling price for footballs is not considered relevant to the production decision because Pigskin will satisfy all customer demand exactly when it occursat whatever the selling price is Therefore Pigskin wants to determine the production schedule that minimizes the total production and holding costs Objective To use LP to find the production schedule that meets demand on time and minimizes total production and inventory holding costs WHERE DO THE NUMBERS COME FROM The input values for this problem are not all easy to find Here are some thoughts on where they might be obtained See Figure 325 The initial inventory in cell B4 should be available from the companys database sys tem or from a physical count The unit production costs in row 8 would probably be estimated in two steps First the company might ask its cost accountants to estimate the current unit production cost Then it could examine historical trends in costs to estimate inflation factors for future months The holding cost percentage in cell B5 is typically difficult to determine Depending on the type of inventory being held this cost can include storage and handling rent property taxes insurance spoilage and obsolescence It can also include capital coststhe cost of money that could be used for other investments The demands in row 18 are probably forecasts made by the marketing and sales department They might be seatofthepants forecasts or they might be the result of a formal quantitative forecasting procedure as discussed in Chapter 14 Of course if there are already some orders on the books for future months these are included in the demand figures The production and storage capacities in rows 14 and 22 are probably supplied by the production department They are based on the size of the workforce the available machinery availability of raw materials and physical space Solution The variables and constraints for this model are listed in Table 33 There are two keys to relating these variables First the months cannot be treated independently This is because Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it many months the ending inventory is at the upper limit On the other hand even when the holding cost percentage reaches 10 the company still continues to hold a maximum end ing inventory of 5000 A second possible sensitivity analysis is suggested by the way the optimal production schedule would probably be implemented The optimal solution to Pigskins model speci fies the production level for each of the next six months In reality however the company would probably implement the models recommendation only for the first month Then at the beginning of the second month it would gather new forecasts for the next six months months 2 through 7 solve a new sixmonth model and again implement the models rec ommendation for the first of these months month 2 If the company continues in this man ner we say that it is following a sixmonth rolling planning horizon The question then is whether the assumed demands really forecasts toward the end of the planning horizon have much effect on the optimal production quantity in month 1 You would hope not because these forecasts could be quite inaccurate The twoway Solver table in Figure 329 shows how the optimal month 1 production quantity varies with the forecasted demands in months 5 and 6 As you can see if the forecasted demands for months 5 and 6 remain fairly small the optimal month 1 production quantity remains at 5000 This is good news It means that the optimal production quantity in month 1 is fairly insensitive to the possibly inaccurate forecasts for months 5 and 6 116 Chapter 3 Introduction to Optimization Modeling 3 4 5 6 7 A B C D E F G H I J Month 5 demand cell F18 values along side Month 6 demand cell G18 values along top output cell in corner Unitsproduced1 10000 20000 30000 10000 5000 5000 5000 20000 5000 5000 5000 30000 5000 5000 5000 Figure 329 Sensitivity of Month 1 Production to Demand in Months 5 and 6 Solvers sensitivity report for this model appears in Figure 330 The bottom part of this report is fairly straightforward to interpret The first six rows are for sensitivity to changes in the storage capacity whereas the last six are for sensitivity to changes in the demand There are no rows for the production capacity constraints because these are sim ple upperbound constraints on the decision variables Recall that Solvers sensitivity report handles this type of constraint differently from normal constraints In contrast the top part of the report is virtually impossible to unravel This is because the objective coefficients of the decision variables are each based on multiple inputs Each is a combi nation of unit production costs and the holding cost percentage Therefore if you want to know how the solution will change if you change a single unit production cost or the hold ing cost percentage this report does not answer your question This is one case where a sensitivity analysis with SolverTable is much more straightforward and intuitive It allows you to change any of the models inputs and directly see the effects on the solution Modeling Issues We assume that Pigskin uses a sixmonth planning horizon Why six months In multi period models such as this the company has to make forecasts about the future such as the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 39 A COMPARISON OF ALGEBRAIC AND SPREADSHEET MODELS To this point you have seen three algebraic optimization models and three corresponding spreadsheet models How do they differ If you review the two product mix examples in this chapter we believe you will agree that 1 the algebraic models are quite straightforward and 2 the spreadsheet models are almost direct translations into Excel of the algebraic models In particular each algebraic model has a set of xs that corresponds to the changing cell range in the spreadsheet model In addition each objective and each left side of each constraint in the spreadsheet model corresponds to a linear expression involving xs in the algebraic model However the Pigskin production planning model is quite different The spreadsheet model includes one set of changing cells the production quantities and everything else is related to these through spreadsheet formulas In contrast the algebraic model has two sets of variables the Ps for the production quantities and the Is for the ending inventories and together these constitute the decision variables These two sets of variables must then be related algebraically and this is done through a series of balance equations This is a typical situation in algebraic models where one set of variables the produc tion quantities corresponds to the real decision variables and other sets of variables along with extra equations or inequalities are introduced to capture the logic We believeand this belief is reinforced by many years of teaching experiencethat this extra level of abstraction makes algebraic models much more difficult for typical users to develop and comprehend It is the primary reason we have decided to focus almost exclusively on spreadsheet models in this book 310 A DECISION SUPPORT SYSTEM If your job is to develop an LP spreadsheet model to solve a problem such as Pigskins pro duction problem then you will be considered the expert in LP Many people who need to use such models however are not experts They might understand the basic ideas behind LP and the types of problems it is intended to solve but they will not know the details In this case it is useful to provide these users with a decision support system DSS that can help them solve problems without having to worry about technical details 118 Chapter 3 Introduction to Optimization Modeling 21 In one modification of the Pigskin problem the maximum storage constraint and the holding cost are based on the average inventory not ending inventory for a given month where the average inventory is defined as the sum of beginning inventory and ending inventory divided by 2 and beginning inventory is before production or demand Modify the Pigskin model with this new assumption and use Solver to find the optimal solution How does this change the optimal production schedule How does it change the optimal total cost SkillExtending Problems 22 Modify the Pigskin spreadsheet model so that except for month 6 demand need not be met on time The only requirement is that all demand be met eventually by the end of month 6 How does this change the optimal production schedule How does it change the optimal total cost 23 Modify the Pigskin spreadsheet model so that demand in any of the first five months must be met no later than a month late whereas demand in month 6 must be met on time For example the demand in month 3 can be met partly in month 3 and partly in month 4 How does this change the optimal production schedule How does it change the optimal total cost 24 Modify the Pigskin spreadsheet model in the following way Assume that the timing of demand and production are such that only 70 of the production in a given month can be used to satisfy the demand in that month The other 30 occurs too late in that month and must be carried as inventory to help satisfy demand in later months How does this change the optimal production schedule How does it change the optimal total cost Then use SolverTable to see how the optimal production schedule and optimal cost vary as the percentage of production usable for this months demand now 70 is allowed to vary from 20 to 100 in increments of 10 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it We will not teach you in this book how to build a fullscale DSS but we will show you what a typical DSS looks like and what it can do14 We consider only DSSs built around spreadsheets There are many other platforms for developing DSSs that we will not con sider Basically a spreadsheetbased DSS contains a spreadsheet model of a problem such as the one in Figure 327 However as a user you will probably never even see this model Instead you will see a front end and a back end The front end allows you to select input values for your particular problem The user interface for this front end can include several features such as buttons dialog boxes toolbars and menusthe things you are used to seeing in Windows applications The back end will then produce a report that explains the solution in nontechnical terms We illustrate a DSS for a slight variation of the Pigskin problem in the file Decision Supportxlsm This file has three worksheets When you open the file you see the Explanation sheet shown in Figure 331 It contains two buttons one for setting up the prob lem getting the users inputs and one for solving the problem running Solver When you click on the Set Up Problem button you are asked for the inputs the initial inventory the forecasted demands for each month and others An example appears in Figure 332 These input boxes should be selfexplanatory so that all you need to do is enter the values you 310 A Decision Support System 119 Figure 331 Explanation Sheet for DSS Figure 332 Dialog Box for Obtaining User Inputs 14For readers interested in learning more about this DSS this textbooks essential resource Web site includes notes about its development in the file Developing the Decision Support A pplicationdocx under Chapter 3 Example Files If you are interested in learning more about spreadsheet DSSs in general Albright has written the book VBA for Modelers now in its third edition It contains a primer on the Visual Basic for Applications language and presents many applications and instructions for creating DSSs with VBA Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it want to try To speed up the process the inputs from the previous run are shown by default After you have entered all of these inputs you can take a look at the Model worksheet This sheet contains a spreadsheet model similar to the one in Figure 327 but with the inputs you just entered Now go back to the Explanation sheet and click on the Find Optimal Solution button This automatically sets up the Solver dialog box and runs Solver There are two possibili ties First it is possible that there is no feasible solution to the problem with the inputs you entered In this case you see a message to this effect as in Figure 333 In most cases how ever the problem has a feasible solution In this case you see the Report sheet which sum marizes the optimal solution in nontechnical terms Part of one sample output appears in Figure 334 120 Chapter 3 Introduction to Optimization Modeling Figure 333 Indication of No Feasible Solutions Figure 334 Optimal Solution Report After studying this report you can then click on the Solve Another Problem button which takes you back to the Explanation sheet so that you can solve a new problem All of this is done automatically with Excel macros These macros use Microsofts Visual Basic for Applications VBA programming language to automate various tasks In most Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it professional applications nontechnical people need only to enter inputs and look at reports Therefore the Model sheet and VBA code will most likely be hidden and pro tected from end users 311 CONCLUSION This chapter has provided a good start to LP modelingand to optimization modeling in general You have learned how to develop three basic LP spreadsheet models how to use Solver to find their optimal solutions and how to perform sensitivity analyses with Solvers sensitivity reports or with the SolverTable addin You have also learned how to recognize whether a mathematical programming model satisfies the linear assumptions In the next few chapters you will see a variety of other optimization models but the three basic steps of model development Solver optimization and sensitivity analysis remain the same 311 Conclusion 121 Summary of Key Terms Term Explanation Excel Page Linear programming An optimization model with a linear 68 model objective and linear constraints Objective The value such as profit to be optimized 69 in an optimization model Constraints Conditions that must be satisfied in 69 an optimization model Changing cells Cells that contain the values of the Specify in Solver 69 decision variables dialog box Objective cell Cell that contains the value Specify in 69 of the objective Solver dialog box Nonnegativity constraints Constraints that require the decision 69 variables to be nonnegative usually for physical reasons Feasible solution A solution that satisfies all of the constraints 70 Feasible region The set of all feasible solutions 70 Optimal solution The feasible solution that has 70 the best value of the objective Solver Addin that ships with Excel for Solver on 70 performing optimization Data ribbon Simplex method An efficient algorithm for finding the 70 optimal solution in a linear programming model Sensitivity analysis Seeing how the optimal solution changes 70 as various input values change Algebraic model A model that expresses the constraints 72 and the objective algebraically Graphical solution Shows the constraints and objective 72 graphically so that the optimal solution can be identified useful only when there are two decision variables Spreadsheet model A model that uses spreadsheet formulas 74 to express the logic of the model Binding constraint A constraint that holds as an equality 82 continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 122 Chapter 3 Introduction to Optimization Modeling Summary of Key Terms Continued Term Explanation Excel Page Nonbinding constraint A constraint where there is a difference the 82 slack slack between the two sides of the inequality Solvers sensitivity Report available from Solver that shows Available in Solver 83 report sensitivity to objective coefficients and dialog box right sides of constraints after Solver runs Reduced cost Amount the objective coefficient of a 85 variable currently equal to zero must change before it is optimal for that variable to be positive or the amount the objective of a variable currently at its upper bound must change before that variable decreases from its upper bound Shadow price The change in the objective for a change in 85 the right side of a constraint indicates amount a company would pay for more of a scarce resource SolverTable Addin that performs sensitivity analysis SolverTable ribbon 87 to any inputs and reports results in tabular and graphical form Selecting multiple ranges Useful when changing cells eg are in Pressing Ctrl key 89 noncontiguous ranges drag ranges one after the other Mathematical Any optimization model whether linear integer 94 programming model or nonlinear Proportionality Properties of optimization model that result 94 additivity divisibility in a linear programming model Infeasibility Condition where a model has no feasible solutions 97 Unboundedness Condition where there is no limit to the objective 97 almost always a sign of an error in the model Rolling planning horizon Multiperiod model where only the decision in the 116 first period is implemented and then a new multiperiod model is solved in succeeding periods Decision support system Userfriendly system where an end user can 118 enter inputs to a model and see outputs but need not be concerned with technical details P R O B L E M S SkillBuilding Problems 25 A chemical company manufactures three chemicals A B and C These chemicals are produced via two production processes 1 and 2 Running process 1 for an hour costs 400 and yields 300 units of A 100 units of B and 100 units of C Running process 2 for an hour costs 100 and yields 100 units of A and 100 units of B To meet customer demands at least 1000 units of A 500 units of B and 300 units of C must be produced daily a Use Solver to determine a daily production plan that minimizes the cost of meeting the companys daily demands b Confirm graphically that the daily production plan from part a minimizes the cost of meeting the companys daily demands c Use SolverTable to see what happens to the decision variables and the total cost when the hourly processing cost for process 2 increases in increments of 050 How large must this cost increase be before the decision variables change Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it possible to assemble both types in the assembly shop Each type 1 truck contributes 1000 to profit each type 2 truck contributes 1500 Use Solver to maximize the companys profit Hint One approach but not the only approach is to try a graphical procedure first and then deduce the constraints from the graph 36 A company manufactures mechanical heart valves from the heart valves of pigs Different heart operations require valves of different sizes The company purchases pig valves from three different suppliers The cost and size mix of the valves purchased from each supplier are given in the file P0336xlsx Each month the company places an order with each supplier At least 500 large 300 medium and 300 small valves must be purchased each month Because of the limited availability of pig valves at most 500 valves per month can be purchased from each supplier a Use Solver to determine how the company can minimize the cost of acquiring the needed valves b Use SolverTable to investigate the effect on total cost of increasing its minimal purchase requirements each month Specifically see how the total cost changes as the minimal purchase requirements of large medium and small valves all increase from their original values by the same percentage Revise your model so that SolverTable can be used to investigate these changes when the percentage increase varies from 2 to 20 in increments of 2 Explain intuitively what happens when this percentage is at least 16 37 A company that builds sailboats wants to determine how many sailboats to build during each of the next four quarters The demand during each of the next four quarters is as follows first quarter 160 sailboats second quarter 240 sailboats third quarter 300 sailboats fourth quarter 100 sailboats The company must meet demands on time At the beginning of the first quarter the company has an inventory of 40 sailboats At the beginning of each quarter the company must decide how many sailboats to build during that quarter For simplicity assume that sailboats built during a quarter can be used to meet demand for that quarter During each quarter the company can build up to 160 sailboats with regular time labor at a total cost of 1600 per sailboat By having employees work overtime during a quarter the company can build additional sailboats with overtime labor at a total cost of 1800 per sailboat At the end of each quarter after production has occurred and the current quarters demand has been satisfied a holding cost of 80 per sailboat is incurred a Determine a production schedule to minimize the sum of production and inventory holding costs during the next four quarters 311 Conclusion 125 b Use SolverTable to see whether any changes in the 80 holding cost per sailboat could induce the company to carry more or less inventory Revise your model so that SolverTable can be used to investigate the effects on ending inventory during the fourquarter period of systematic changes in the unit holding cost Assume that even though the unit holding cost changes it is still constant over the fourquarter period Are there any nonnegative unit holding costs that would induce the company to hold more inventory than it holds when the holding cost is 80 Are there any unit holding costs that would induce the company to hold less inventory than it holds when the holding cost is 80 38 During the next two months an automobile manufacturer must meet on time the following demands for trucks and cars month 1 400 trucks and 800 cars month 2 300 trucks and 300 cars During each month at most 1000 vehicles can be produced Each truck uses two tons of steel and each car uses one ton of steel During month 1 steel costs 700 per ton during month 2 steel is projected to cost 800 per ton At most 2500 tons of steel can be purchased each month Steel can be used only during the month in which it is purchased At the beginning of month 1 100 trucks and 200 cars are in the inventory At the end of each month a holding cost of 200 per vehicle is assessed Each car gets 20 miles per gallon mpg and each truck gets 10 mpg During each month the vehicles produced by the company must average at least 16 mpg a Determine how to meet the demand and mileage requirements at minimum total cost b Use SolverTable to see how sensitive the total cost is to the 16 mpg requirement Specifically let this requirement vary from 14 mpg to 18 mpg in increments of 025 mpg Explain intuitively what happens when the requirement is greater than 17 mpg 39 A textile company produces shirts and pants Each shirt requires two square yards of cloth and each pair of pants requires three square yards of cloth During the next two months the following demands for shirts and pants must be met on time month 1 1000 shirts and 1500 pairs of pants month 2 1200 shirts and 1400 pairs of pants During each month the following resources are available month 1 9000 square yards of cloth month 2 6000 square yards of cloth In addition cloth that is available during month 1 and is not used can be used during month 2 During each month it costs 8 to produce an article of clothing with regulartime labor and 16 with overtime labor During each month a total of at most 2500 articles of clothing can be produced with regulartime labor and an unlimited number of articles of clothing can be Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it this by changing both of these requirements from at least half to at least x percent where x can be any multiple of 5 from 0 to 50 Modeling Problems 44 Suppose you use Solver to find the optimal solution to a maximization model Then you remember that you omitted an important constraint After adding the constraint and running Solver again is the optimal value of the objective guaranteed to decrease Why or why not 45 Consider an optimization model with a number of resource constraints Each indicates that the amount of the resource used cannot exceed the amount available Why is the shadow price of such a resource constraint always zero when the amount used in the optimal solution is less than the amount available 46 If you add a constraint to an optimization model and the previously optimal solution satisfies the new constraint will this solution still be optimal with the new constraint added Why or why not 47 Why is it generally necessary to add nonnegativity constraints to an optimization model Wouldnt Solver automatically choose nonnegative values for the changing cells 48 Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit What does the additive assumption imply about the profit objective What does the proportionality assumption imply about the profit objective Be as specific as possible Can you think of any reasonable profit functions that would not be linear in the amounts of the products produced 49 In a typical product mix model where a company must decide how much of each product to produce to maximize profit discuss possible situations where 311 Conclusion 127 there might not be any feasible solutions Could these be realistic If you had such a situation in your company how might you proceed 50 In a typical product mix model where a company must decide how much of each product to produce to maximize profit there are sometimes customer demands for the products We used upperbound constraints for these Dont produce more than you can sell Would it be realistic to have lowerbound constraints instead Produce at least as much as is demanded Would it be realistic to have both where the upper bounds are greater than the lower bounds Would it be realistic to have equality constraints Produce exactly what is demanded 51 In a typical production scheduling model like Pigskins if there are no production capacity constraintsthe company can produce as much as it needs in any time periodbut there are storage capacity constraints and demand must be met on time is it possible that there will be no feasible solutions Why or why not 52 In a production scheduling problem like Pigskins suppose the company must produce several products to meet customer demands Would it suffice to solve a separate model for each product as we did for Pigskin or would one big model for all products be necessary If the latter discuss what this big model might look like 53 In any optimization model such as those in this chapter we say that the model is unbounded and Solver will indicate as such if there is no limit to the value of the objective For example if the objective is profit then for any dollar value no matter how large there is a feasible solution with profit at least this large In the real world why are there never any unbounded models If you run Solver on a model and get an unbounded message what should you do Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it APPENDIX INFORMATION ON SOLVERS Microsoft Office or Excel ships with a builtin version of Solver This version and all other versions of Solver have been developed by Frontline Systems not Microsoft When you install Office or Excel you have the option of installing or not installing Solver In most cases a typical install should install Solver To check whether Solver is installed on your system open Excel select the Office Button or the File tab in Excel 2010 select Excel Options select AddIns and click on Go If there is a Solver item in the list Solver has been installed To actually add it in make sure this item is checked Otherwise you need to run the Office Setup program with the AddRemove feature to install Solver Users of previous versions of Excel 2003 or earlier should note that the actual Solver addin file is a different one in Excel 2007 or Excel 2010 In previous versions it was Solverxla now it is Solverxlam However the basic functionality is the same If you have used versions of Solver in Excel 2007 or earlier you will see some changes in Solver for Excel 2010 First the user interface is slightly different as you have already seen in the screen shots of its main dialog box Second it now includes the Evolutionary algorithm which used to be available only in the Premium Solver product Because of this we no longer need to supply an educational version of Premium Solver with the book We will continue to use the Evolutionary algorithm extensively in Chapter 8 Third the Solver algorithms have been revised to work better Specifically we have very rarely seen the annoying message about a model not being linear when we know it is linear This mes sage can still occur in certain models but it is less likely to occur than in previous versions of Solver The builtin version of Solver is able to solve most problems you are likely to encounter However it has two important limitations you should be aware of First it allows only 200 changing cells This might sound like plenty but many realworld prob lems go well beyond 200 changing cells Second Solver for Excel 2010 allows only 100 constraints There was no such limit in previous versions For example if you specify a constraint such as B15B17D15D17 this counts as three constraints against the 100 constraint limit However simple upper or lower bound constraints such as B15100 or B1550 where B15 is a changing cell do not count against the 100constraint limit If you want to solve larger problems you will need to purchase one of Frontlines commer cial versions of Solver For more information check Frontline Systems Web site at wwwsolvercom 128 Chapter 3 Introduction to Optimization Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Case 32 Sonoma Valley Wines 131 C A S E A fter graduating from business school George Clark went to work for a Big Six accounting firm in San Francisco Because his hobby has always been wine making when he had the opportunity a few years later he purchased five acres plus an option to buy 35 additional acres of land in Sonoma Valley in Northern California He plans eventually to grow grapes on that land and make wine with them George knows that this is a big undertaking and that it will require more capital than he has at the present However he figures that if he persists he will be able to leave accounting and live full time from his winery earnings by the time he is 40 Because wine making is capitalintensive and because growing commercialquality grapes with a full yield of five tons per acre takes at least eight years George is planning to start smallThis is necessitated by both his lack of capital and his inexperience in wine making on a large scale although he has long made wine at home His plan is first to plant the grapes on his land to get the vines startedThen he needs to set up a small trailer where he can live on weekends while he installs the irrigation system and does the required work to the vines such as pruning and fertilizingTo help maintain a positive cash flow during the first few years he also plans to buy grapes from other nearby growers so he can make his own label wine He proposes to market it through a small tasting room that he will build on his land and keep open on weekends during the springsummer season To begin George is going to use 10000 in savings to finance the initial purchase of grapes from which he will make his first batch of wine He is also thinking about going to the Bank of Sonoma and asking for a loan He knows that if he goes to the bank the loan officer will ask for a business plan so he is trying to pull together some numbers for himself firstThis way he will have a rough notion of the profitability and cash flows associated with his ideas before he develops a formal plan with a pro forma income statement and balance sheet He has decided to make the preliminary planning horizon two years and would like to estimate the profit over that period His most immediate task is to decide how much of the 10000 should be allocated to purchasing grapes for the first year and how much to purchasing grapes for the second year In addition each year he must decide how much he should allocate to purchasing grapes to make his favorite Petite Sirah and how much to purchasing grapes to make the more popular Sauvignon Blanc that seems to have been capturing the attention of a wider market during the last few years in California In the first year each bottle of Petite Sirah requires 080 worth of grapes and each bottle of Sauvignon Blanc uses 070 worth of grapes For the second year the costs of the grapes per bottle are 075 and 085 respectively George anticipates that his Petite Sirah will sell for 800 a bottle in the first year and for 825 in the second year while his Sauvignon Blancs price remains the same in both years at 700 a bottle Besides the decisions about the amounts of grapes purchased in the two years George must make estimates of the sales levels for the two wines during the two yearsThe local winemaking association has told him that marketing is the key to success in any wine business generally demand is directly proportional to the amount of effort spent on marketingThus since George cannot afford to do any market research about sales levels due to his lack of capital he is pondering how much money he should spend to promote each wine each yearThe winemaking association has given him a rule of thumb that relates estimated demand to the amount of money spent on advertising For instance they estimate that for each dollar spent in the first year promoting the Petite Sirah a demand for five bottles will be created and for each dollar spent in the second year a demand for six bottles will result Similarly for each dollar spent on advertising for the Sauvignon Blanc in the first year up to eight bottles can be sold and for each dollar spent in the second year up to ten bottles can be sold 32 SONOMA VALLEY WINES15 15This case was written by William D Whisler California State University Hayward Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 132 Chapter 3 Introduction to Optimization Modeling The initial funds for the advertising will come from the 10000 savingsAssume that the cash earned from wine sales in the first year is available in the second year A personal concern George has is that he maintain a proper balance of wine products so that he will be well positioned to expand his marketing capabilities when he moves to the winery and makes it his fulltime jobThus in his mind it is important to ensure that the number of bottles of Petite Sirah sold each year falls in the range between 40 and 70 of the overall number of bottles sold Questions 1 George needs help to decide how many grapes to buy how much money to spend on advertising how many bottles of wine to sell and how much profit he can expect to earn over the twoyear period Develop a spreadsheet LP model to help him 2 Solve the linear programming model formulated in Question 1 The following questions should be attempted only after Questions 1 and 2 have been answered correctly 3 After showing the business plan to the Bank of Sonoma George learns that the loan officer is concerned about the market prices used in estimating the profitsrecently it has been forecasted that Chile and Australia will be flooding the market with highquality lowpriced white wines over the next couple of years In particular the loan officer estimates that the price used for the Sauvignon Blanc in the second year is highly speculative and realistically might be only half the price George calculatedThus the bank is nervous about lending the money because of the big effect such a decrease in price might have on estimated profitsWhat do you think 4 Another comment the loan officer of the Bank of Sonoma has after reviewing the business plan isI see that you do have an allowance in your calculations for the carryover of inventory of unsold wine from the first year to the second year but you do not have any cost associated with thisAll companies must charge something for holding inventory so you should redo your plans to allow for this If the holding charges are 010 per bottle per year how much if any does Georges plan change 5 The president of the local grape growers association mentions to George that there is likely to be a strike soon over the unionization of the grape workers Currently they are not represented by any union This means that the costs of the grapes might go up by anywhere from 50 to 100 How might this affect Georges plan 6 Before taking his business plan to the bank George had it reviewed by a colleague at the accounting firm where he worksAlthough his friend was excited about the plan and its prospects he was dismayed to learn that George had not used present value in determining his profitGeorge you are an accountant and must know that money has a time value and although you are only doing a twoyear planning problem it still is important to calculate the present value profit George repliesYes I know all about present value For big investments over long time periods it is important to consider But in this case for a small investment and only a twoyear time period it really doesnt matter Who is correct George or his colleague Why Use an 8 discount factor in answering this question Does the answer change if a 6 or 10 discount rate is used Use a spreadsheet to determine the coefficients of the objective function for the different discount rates 7 Suppose that the Bank of Sonoma is so excited about the prospects of Georges winegrowing business that they offer to lend him an extra 10000 at their best small business rate28 plus a 10 compensating balance16 Should he accept the banks offer Why or why not 8 Suppose that the rule of thumb George was given by the local winemaking association is incorrectAssume that the number of bottles of Petite Sirah sold in the first and second years is at most four for each dollar spent on advertisingAnd likewise for the Sauvignon Blanc assume that it can be at most only five in years 1 and 2 9 How much could profits be increased if Georges personal concerns that Petite Sirah sales should account for between 40 and 70 of overall sales are ignored 16The compensating balance requirement means that only 9000 of the 10000 loan is available to George the remaining 1000 remains with the bank Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C H A P T E R 4 Linear Programming Models PRODUCTIONINVENTORY AND DISTRIBUTION AT KELLOGG T he Kellogg Company is the largest cereal producer in the world and is a leading producer of convenience foods Its worldwide sales in 1999 were nearly 7 billion Kelloggs first product in 1906 was Corn Flakes and it developed a variety of readytoeat cereals over the years including Raisin Bran Rice Krispies Corn Pops and others Although the company continues to develop and market new cereals it has recently gone into convenience foods such as PopTarts and NutriGrain cereal bars and has also entered the healthfood market Kellogg produces hundreds of products and sells thousands of stockkeeping units SKUs Managing production inventory and distribution of thesethat is the daily operationsin a costeffective manner is a challenge By the late 1980s Kellogg realized that the increasing scale and complexity of its operations required optimization methods to coordinate its daily operations in a centralized manner As described in Brown et al 2001 a team of management scientists developed an optimization software system called KPS Kellogg Planning System This system was originally in tended for operational daily and weekly decisions but it expanded into a system for making tactical longerrange decisions about issues such as plant budgets capacity expansion and consolidation By the turn of the century KPS had been in use for about a decade Operational decisions made by ROB KIMLandov 133 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it KPS reduced production inventory and distribution costs by approximately 45 million per year Better yet the tactical side of KPS recently suggested a consolidation of pro duction capacity that saved the company approximately 35 million to 40 million annually Kellogg operates 5 plants in the United States and Canada has 7 distribution centers DCs in such areas as Los Angeles and Chicago and has about 15 copackers companies that contract to produce or pack some of Kelloggs products Customer demands are seen at the DCs and the plants In the cereal business alone Kellogg has to coordinate the packaging inventorying and distributing of 600 SKUs at about 27 locations with a total of about 90 productions lines and 180 packaging lines This requires a tremendous amount of daytoday coordination to meet customer demand at a low cost The KPS operational system that guides operational decisions is essentially a large linear program ming model that takes as its inputs the forecasted customer demands for the various products and specifies what should be produced held and shipped on a daily basis The resulting model is similar to the Pigskin model of football production discussed in the previous chapter except that it is much larger Specifically for each week of its 30week planning horizon the model specifies 1 how much of each product to make on each production line at each facility 2 how much of each SKU to pack on each packaging line at each facility 3 how much inven tory of each SKU to hold at each facility and 4 how much of each SKU to ship from each location to other locations In addition the model has to take constraints into ac count For example the production within a given plant in a week cannot exceed the processing line capacity in that plant Linear programming models such as Kelloggs tend to be very largethousands of decision variables and hundreds or thousands of con straintsbut the algorithms Kellogg uses are capable of optimizing such models very quickly Kellogg runs its KPS model each Sunday morning and uses its recommendations in the ensuing week Kelloggs KPS illustrates a common occurrence when companies turn to manage ment science for help As stated earlier the system was originally developed for making daily operational decisions Soon however the company developed a tactical version of KPS for longrange planning on the order of 12 to 24 months The tactical model is simi lar to the operational model except that time periods are now months not days or weeks and other considerations must be handled such as limited product shelf lives The point is however that when companies such as Kellogg become comfortable with man agement science methods in one part of their operations they often look for other areas to apply similar methods As with Kellogg such methods can save the company millions of dollars 134 Chapter 4 Linear Programming Models 41 INTRODUCTION In a recent survey of Fortune 500 firms 85 of those responding said that they used linear programming In this chapter we discuss some of the LP models that are most often ap plied to real applications In this chapters examples you will discover how to build optimization models to purchase television ads schedule postal workers create an aggregate labor and production plan at a shoe company create a blending plan to transform crude oils into end products Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it plan production of interdependent products at a drug company choose an investment strategy at a financial investment company manage a pension fund determine which of several hospitals use their inputs efficiently The two basic goals of this chapter are to illustrate the wide range of real applications that can take advantage of LP and to increase your facility in modeling LP problems in Excel We present a few principles that will help you model a wide variety of problems The best way to learn however is to see many examples and work through numerous prob lems In short mastering the art of LP spreadsheet modeling takes hard work and practice which you will find plenty of in this chapter Before continuing remember that all of the models in this chapter are linear models as described in the previous chapter This means that the target cell is ultimately possibly through a series of formulas in intervening cells a sum of products of constants and chang ing cells where a constant is defined by the fact that it does not depend on changing cells Similarly each side of each constraint is either a constant or a sum of products of constants and changing cells Also each changing cell except in a few cases where it is specified otherwise is allowed to contain a continuous range of values not just integer values These properties allow Solver to use its very efficient simplex method to find the optimal solution1 42 ADVERTISING MODELS Many companies spend enormous amounts of money to advertise their products They want to ensure that they are spending their money wisely Typically they want to reach large numbers of various groups of potential customers and keep their advertising costs as low as possible The following example illustrates a simple modeland a reasonable ex tension of this modelfor a company that purchases television ads 42 Advertising Models 135 E X A M P L E 41 PURCHASING TELEVISION ADS T he General Flakes Company sells a brand of lowfat breakfast cereal that appeals to people of all age groups and both genders The company advertises this cereal in a va riety of 30second television ads and these ads can be placed in a variety of television shows The ads in different shows vary by costsome 30second slots are much more ex pensive than othersand by the types of viewers they are likely to reach The company has segmented the potential viewers into six mutually exclusive categories males age 18 to 35 males age 36 to 55 males over 55 females age 18 to 35 females age 36 to 55 and females over 55 A rating service can supply data on the numbers of viewers in each of these cate gories who will watch a 30second ad on any particular television show Each such viewer is called an exposure The company has determined the required number of exposures it wants to obtain for each group It wants to know how many ads to place on each of several television shows to obtain these required exposures at minimum cost The data on costs per ad numbers of exposures per ad and minimal required exposures are listed in Table 41 where numbers of exposures are expressed in millions and costs are in thousands of dollars What should the company do 1In the special cases where integer constraints are imposed on some changing cells the Simplex LP option can still be chosen However Solver uses a somewhat different optimization algorithm when there are integerconstrained changing cells This is covered in more depth in Chapter 6 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 4 Total cost The quantities of ads purchased also determine the total cost of advertising Calculate this cost in cell B31 with the formula SUMPRODUCTB14I14Numberadspurchased USING SOLVER The main Solver dialog box appears in Figure 42 After filling it out as shown and check ing the NonNegative option and selecting the Simplex LP method click on the Solve button to obtain the solution shown in Figure 41 138 Chapter 4 Linear Programming Models Figure 42 Solver Dialog Box for the Advertising Model Discussion of the Solution The optimal solution is probably not the one you would have guessed With a set of ads that cost very different amounts and reach very different mixes of viewers it is difficult to guess the optimal strategy For comparison however we calculated the total number of viewers from each type of ad in row 12 and divided the costs in row 14 by the numbers of viewers in row 12 to obtain the cost per million viewers in row 15 You might expect the ads with low cost per million viewers to be chosen most frequently However this is not necessarily the case For example Monday Night Football MNF has the secondlowest cost per million viewers but the optimal solution doesnt include any ads for this show Sensitivity Anal ysis Solvers sensitivity report shown in Figure 43 is enlightening for this solution Here is a sample of the information it provides The company is not currently purchasing any ads on Desperate Housewives The reduced cost for this show implies that the cost per ad would have to decrease by at least 10 10000 before it would be optimal to purchase any ads on this show The company is currently purchasing about 20 ads on Sports Center The allowable increase and decrease for this show indicate how much the cost per ad would have to Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it such as this you typically use one of the objectives as the target cell and constrain the other Here the company is asking how many excess exposures it can get for a given budget There is no natural budget to use and it makes perfect sense to ask questions such as these How many exposures can the company get for 19 million How many for 20 million How many for 21 million Fortunately SolverTable is the perfect tool to answer all of these questions in one step You first develop the model as in Figure 44 using any budget such as 20 million in cell D32 and run Solver in the usual way Then you run a oneway SolverTable allowing the budget to vary over some desired range and keep track of selected output variables Typical results appear in Figure 46 which are based on the SolverTable settings in Figure 47 For low budget levels the problem is infeasiblethere is no way with this bud get to obtain the required exposures Above a certain budget level the problem becomes feasible and the optimal solutions are shown As the budget increases the company can clearly obtain larger numbers of excess exposures but the optimal advertising strategy in columns B through I changes in a somewhat unpredictable way The results of this sensitivity analysis can be shown graphically in a tradeoff curve as in Figure 48 To create this highlight the numbers in columns A and J of Figure 46 from row 43 down and insert a line chart This chart illustrates the rather obvious fact that when the company is allowed to spend more on advertising it can achieve more total excess exposures 42 Advertising Models 141 Figure 46 Sensitivity of Optimal Solution to the Advertising Budget For dualobjective models you optimize one objective and put a constraint on the otherThen you can use SolverTable to vary the righthand side of this constraint The result is a tradeoff curve 1 2 3 A B C D E F G H I J Oneway analysis for Solver model in Model worksheet Budget cell D32 values along side output cells along top hased1 hased2 hased3 hased4 hased5 hased6 hased7 hased8 osures 4 5 Numberadspurch Numberadspurch Numberadspurch Numberadspurch Numberadspurch Numberadspurch Numberadspurch Numberadspurch Totalexcessexpos 1800 Not feasible 5 6 7 8 9 10 11 12 1800 Not feasible 1850 Not feasible 1900 0000 0000 8208 0000 0000 1887 0000 8679 23717 1950 0000 0000 6934 0000 0000 8491 0000 9057 32726 2000 0000 0000 6030 0000 0000 12060 0000 9548 41688 2050 0000 0000 5653 0000 0000 11307 0000 10201 50583 2100 0000 0000 5276 0000 0000 10553 0000 10854 59477 2 150 0 000 0 000 4 899 0 000 0 000 9 799 0 000 11 508 68 372 12 13 14 15 16 17 18 2150 0000 0000 4899 0000 0000 9799 0000 11508 68372 2200 0000 0000 4523 0000 0000 9045 0000 12161 77266 2250 0000 0000 4146 0000 0000 8291 0000 12814 86161 2300 0000 0000 3769 0000 0000 7538 0000 13467 95055 2350 0000 0000 3392 0000 0000 6784 0000 14121 103950 2400 0000 0000 3015 0000 0000 6030 0000 14774 112844 2450 0000 0000 2638 0000 0000 5276 0000 15427 121739 19 2500 0000 0000 2261 0000 0000 4523 0000 16080 130633 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 43 Worker Scheduling Models 145 E X A M P L E 42 POSTAL EMPLOYEE SCHEDULING A post office requires different numbers of fulltime employees on different days of the week The number of fulltime employees required each day is given in Table 43 Union rules state that each fulltime employee must work five consecutive days and then receive two days off For example an employee who works Monday to Friday must be off on Saturday and Sunday The post office wants to meet its daily requirements using only fulltime employees Its objective is to minimize the number of fulltime employees on its payroll Table 43 Employee Requirements for Post Office Day of Week Minimum Number of Employees Required Monday 17 Tuesday 13 Wednesday 15 Thursday 19 Friday 14 Saturday 16 Sunday 11 In real employee scheduling problems much of the work involves forecasting and queueing analysis to obtain worker requirementsThis must be done before an optimal schedule can be found Objective To develop an LP spreadsheet model that relates fiveday shift schedules to daily numbers of employees available and to use Solver on this model to find a schedule that uses the fewest number of employees and meets all daily workforce requirements WHERE DO THE NUMBERS COME FROM The only inputs needed for this problem are the minimum employee requirements in Table 43 but these are not easy to obtain They would probably be obtained through a combination of two quantitative techniques forecasting Chapter 14 and queueing analysis Chapter 13 The post office would first use historical data to forecast customer and mail arrival patterns throughout a typical week It would then use queueing analysis to translate these arrival patterns into worker requirements on a daily basis Actually we have kept the problem relatively simple by considering only daily requirements In a realistic setting the organization might forecast worker requirements on an hourly or even a 15minute basis tradeoff curve from the results of the sensitivity analysis 6 Suppose there are three objectives not just two the total advertising cost the total number of excess exposures to men and the total number of excess exposures to women Continuing the approach sug gested in the previous problem how might you proceed Take it as far as you can including a sensitivity analysis and a tradeoff curve SkillExtending Problems 5 In the dualobjective advertising model we put a budget constraint on the total advertising cost and then maximized the total number of excess expo sures Do it the opposite way reversing the roles of the two objectives That is model it so that you put a lower limit on the total number of excess exposures and minimize the total advertising cost Then run a sensitivity analysis on this lower limit and create a 43 WORKER SCHEDULING MODELS Many organizations must determine how to schedule employees to provide adequate ser vice The following example illustrates how LP can be used to schedule employees Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 146 Chapter 4 Linear Programming Models FUNDAMENTAL INSIGHT Choosing the Changing Cells The changing cells which are really just the decision variablesshould always be chosen so that their values determine all required outputs in the model In other words their values should tell the compan y exactly how to run its business Sometimes the choice of changing cells is obvious but in many cases as in this worker scheduling model the pr oper choice of changing cells takes some deeper thinking about the problem An improper choice of changing cells typi cally leads to a dead end where their values do not supply enough information to calculate required out puts or implement certain constraints Note that this is a wraparound problem We assume that the daily requirements in Table 43 and the worker schedules continue week after week So for example if eight em ployees are assigned to the Thursday through Monday shift these employees always wrap around from one week to the next on their fiveday shift DEVELOPING THE SPREADSHEET MODEL The spreadsheet model for this problem is shown in Figure 411 See the file Worker Schedulingxlsx To form this spreadsheet proceed as follows 1 Inputs and range names Enter the number of employees needed on each day of the week from Table 43 in the blue cells and create the range names shown 2 Employees beginning each day Enter any trial values for the number of employees beginning work on each day of the week in the Employeesstarting range These beginning Table 44 Variables and Constraints for Postal Scheduling Problem Input variables Minimum required number of workers each day Decision variables changing cells Number of employees working each of the fiveday shifts defined by their first day of work Objective cell Total number of employees on the payroll Other calculated variables Number of employees working each day Constraints Employees working Employees required Ú The key to this model is choosing the correct changing cells Solution The variables and constraints for this problem appear in Table 44 The trickiest part is iden tifying the appropriate decision variables Many students believe the decision variables should be the numbers of employees working on the various days of the week Clearly these values must eventually be determined However it is not enough to specify say that 18 em ployees are working on Monday The problem is that this doesnt indicate when these 18 employees start their fiveday shifts Without this knowledge it is impossible to implement the fiveconsecutiveday twodayoff requirement If you dont believe this try developing your own model with the wrong decision variables You will eventually reach a dead end The trick is to define the decision variables as the numbers of employees working each of the seven possible fiveday shifts By knowing the values of these decision variables the other output variables can be calculated For example the number working on Thursday is the sum of those who begin their fiveday shifts on Sunday Monday Tuesday Wednesday and Thursday Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 148 Chapter 4 Linear Programming Models Figure 412 Solver Dialog Box for Worker Scheduling Model At this point you might want to experiment with the numbers in the changing cell range to see whether you can guess an optimal solution without looking at Figure 411 It is not that easy Each worker who starts on a given day works the next four days as well so when you find a solution that meets the minimal requirements for the various days you usually have a few more workers available on some days than are needed USING SOLVER Invoke Solver and fill out its main dialog box as shown in Figure 412 You dont need to include the integer constraints yet We will discuss them shortly Make sure you check the NonNegative option and use the Simplex LP method Discussion of the Solution The optimal solution shown in Figure 411 has one drawback It requires the number of em ployees starting work on some days to be a fraction Because parttime employees are not allowed this solution is unrealistic However it is simple to add an integer constraint on the changing cells This integer constraint appears in Figure 412 With this integer constraint the optimal solution appears in Figure 413 The changing cells in the optimal solution indicate the numbers of workers who start their fiveday shifts on the various days You can then look at the columns of the B14H20 range to see which employees are working on any given day This optimal solution is typi cal in scheduling problems Due to a labor constrainteach employee must work five consecutive days and then have two days offit is typically impossible to meet the mini mum employee requirements exactly To ensure that there are enough employees available on busy days it is necessary to have more than enough on hand on light days Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 43 Worker Scheduling Models 151 P R O B L E M S SkillBuilding Problems 7 Modify the post office model so that employees are paid 10 per hour on weekdays and 15 per hour on weekends Change the objective so that you now mini mize the weekly payroll You can assume that each employee works eight hours per day Is the previous optimal solution still optimal 8 How much influence can the worker requirements for one two or three days have on the weekly schedule in the post office example Explore this in the following questions a Let Mondays requirements change from 17 to 25 in increments of 1 Use SolverTable to see how the total number of employees changes b Suppose the Monday and Tuesday requirements can each independently of one another increase from 1 to 8 in increments of 1 Use a twoway SolverTable to see how the total number of employees changes 1 The postal employee scheduling example is called a static scheduling model because we assume that the post office faces the same situation each week In reality de mands change over time workers take vacations in the summer and so on so the post office does not face the same situation each week A dynamic scheduling model not covered here is necessary for such problems 2 In a weekly scheduling model for a supermarket or a fastfood restaurant the number of decision variables can grow quickly and optimization software such as Solver will have difficulty finding an optimal solution In such cases heuristic methods essentially clever trialanderror algorithms have been used to find good solutions to the problem Love and Hoey 1990 indicate how this was done for a particular staff scheduling problem 3 Our model can easily be expanded to handle parttime employees the use of overtime and alternative objectives such as maximizing the number of weekend days off received by employees You are asked to explore such extensions in the problems MODELING ISSUES Heuristic solutions are often close to optimal but they are never guaranteed to be optimal Scheduling Employees in Quebecs Liquor Stores The SAQ is a public corporation of the Province of Quebec that is responsible for distribu ting and selling alcoholbased products through a large network of more than 400 stores and warehouses Every week the SAQ has to schedule more than 3000 employees Until 2002 the scheduling of these employees was done manually incurring an annual expense of about Can 1300000 Gendron 2005 developed an integer programming model that is estimated to have saved the SAQ about Can 1000000 annually The model has to deal with complex union rules For example there is a rule that shifts of six hours or more can be split between two employees but it must be coupled with another rule that forces em ployees to take onehour unpaid lunch or dinner breaks ADDITIONAL APPLICATIONS Note that we did not use Solvers sensitivity report here for two reasons First Solver does not offer a sensitivity report for models with integer constraints Second even if the integer constraints are deleted Solvers sensitivity report does not answer questions about multiple input changes as we have asked here It can be used only for questions about one atatime changes to inputs such as a change to a specific days worker requirement In this sense SolverTable is a more flexible tool Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 160 Chapter 4 Linear Programming Models Linearizing the Backlo gging Model Although this nonlinear model with IF functions is natural the fact that it is not guaran teed to find the optimal solution is disturbing Fortunately it is possible to handle shortages with a linear model The method is illustrated in Figure 421 See the file Aggregate Plan ning 3xlsx To develop this modified spreadsheet model starting from the original model in the Aggregate Planning 1xlsx file follow these steps 1 Enter shortage cost Insert a new row below row 14 and enter the shortage cost per pair of shoes per month in cell B15 2 Rows for amounts held and short Insert five new rows which will now be rows 38 through 42 between the Demand and Ending inventory rows The range B39E40 will be changing cells The Leftover range in row 39 contains the amounts left in inventory if any whereas the Shortage range in row 40 contains the shortages if any Enter any values in these ranges FUNDAMENTAL INSIGHT Nonsmooth Functions and Solv er Excels Solv er as w ell as most other commer cial optimization softwar e packages has tr ouble with nonlinear functions that ar e not smooth These non smooth functions typically have sharp edges or discon tinuities that make them difficult to handle in optimiza tion models and in Excel the y ar e typicall y implemented with functions such as IF MAX MIN ABS and a f ew others There is nothing wrong with using such functions to implement complex logic in Excel optimization models The only problem is that Solver cannot handle models with these functions predictablyThis is not really the fault of Solver Such problems are inherently difficult indeed the optimal solution but we were lucky When certain functions including IF MIN MAX and ABS are used to relate the objective cell to the changing cells the resulting model becomes not only nonlinear but nonsmooth Essentially nonsmooth functions can have sharp edges or discontinuities Solvers GRG nonlinear algorithm can handle smooth nonlinearities as you will see in Chapter 7 but it has trouble with nonsmooth functions Sometimes it gets lucky as it did here and other times it finds a nonoptimal solution that is not even close to the optimal solution For example we changed the unit shortage cost from 20 to 40 and reran Solver Starting from a solu tion where all changing cells contain zero Solver stopped at a solution with total cost 726360 even though the optimal solution has total cost 692820 In other words we werent so lucky this time The moral is that you should avoid these nonsmooth functions in optimization models if at all possible If you do use them as we have done here you should run Solver several times starting from different initial solutions There is still no guarantee that you will get the optimal solution but you will see more evidence of how Solver is progressing Alternatively you can use Frontline Systemss Evolutionary Solver which became available in Excels Solver in Excel 2010 and is discussed in detail in Chapter 8 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 45 Blending Models 163 45 BLENDING MODELS In many situations various inputs must be blended together to produce desired outputs In many of these situations linear programming can find the optimal combination of outputs as well as the mix of inputs that are used to produce the desired outputs Some examples of blending problems are given in Table 46 to 10 per hour They will still work 20 days per month Will this change the optimal nobacklogging solution 15 The current solution to SureSteps nobacklogging aggregate planning model requires a lot of firing Run a oneway SolverTable with the firing cost as the input variable and the numbers fired as the outputs Let the firing cost increase from its current value to double that value in increments of 400 Do high fir ing costs eventually induce the company to fire fewer workers 16 Suppose SureStep could begin a machinery upgrade and training program to increase its worker productivity This program would result in the following values of labor hours per pair of shoes over the next four months 4 39 38 and 38 How much would this new program be worth to SureStep at least for this fourmonth planning horizon with no backlog ging How might you evaluate the programs worth beyond the next four months SkillExtending Problems 17 In the current nobacklogging problem SureStep doesnt hire any workers and uses almost no overtime This is evidently because of low demand Change the demands to 6000 8000 5000 and 3000 and reopti mize Is there now hiring and overtime With this new demand pattern explore the tradeoff between hiring and overtime by running a twoway SolverTable As inputs use the hiring cost per worker and the maximum overtime hours allowed per worker per month varied as you see fit As outputs use the total number of workers hired over the four months and the total number of overtime hours used over the four months Write up your results in a short memo to SureStep management 18 In the SureStep nobacklogging problem change the demands so that they become 6000 8000 5000 3000 Also change the problem slightly so that newly hired workers take six hours to produce a pair of shoes dur ing their first month of employment After that they take only four hours per pair of shoes Modify the model appropriately and use Solver to find the optimal solution 19 We saw that the natural way to model SureSteps backlogging model with IF functions leads to a nonsmooth model that Solver has difficulty handling Another version of the problem is also difficult for Solver Suppose SureStep wants to meet all demand on time no backlogging but it wants to keep its employment level as constant across time as possible To induce this it charges a cost of 1000 each month on the absolute difference between the beginning number of workers and the number after hiring and firingthat is the absolute difference between the values in rows 17 and 20 of the original spreadsheet model Implement this extra cost in the model in the natural way using the ABS function Using demands of 6000 8000 5000 and 3000 see how well Solver does in trying to solve this non smooth model Try several initial solutions and see whether Solver gets the same optimal solution from each of them Table 46 Examples of Blending Problems Inputs Outputs Meat filler water Different types of sausage Various types of oil Heating oil gasolines aviation fuels Carbon iron molybdenum Different types of steel Different types of pulp Different kinds of recycled paper The following example illustrates how to model a typical blending problem in Excel Although this example is small relative to blending problems in real applications it is still probably too complex for you to guess the optimal solution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 164 Chapter 4 Linear Programming Models 44 BLENDING AT CHANDLER OIL C handler Oil has 5000 barrels of crude oil 1 and 10000 barrels of crude oil 2 available Chandler sells gasoline and heating oil These products are produced by blending the two crude oils together Each barrel of crude oil 1 has a quality level of 10 and each barrel of crude oil 2 has a quality level of 56 Gasoline must have an average quality level of at least 8 whereas heating oil must have an average quality level of at least 6 Gasoline sells for 75 per barrel and heating oil sells for 60 per barrel We assume that demand for heat ing oil and gasoline is unlimited so that all of Chandlers production can be sold Chandler wants to maximize its revenue from selling gasoline and heating oil Objective To develop an LP spreadsheet model for finding the revenuemaximizing plan that meets quality constraints and stays within limits on crude oil availabilities WHERE DO THE NUMBERS COME FROM Most of the inputs for this problem should be easy to obtain The selling prices for outputs are dictated by market pressures The availabilities of inputs are based on crude supplies from the suppliers The quality levels of crude oils are known from chemical analysis whereas the required quality levels for outputs are specified by Chandler probably in response to competitive or regulatory pressures Solution The variables and constraints required for this blending model are listed in Table 47 The key is the selection of the appropriate decision variables Many students when asked what decision variables should be used specify the amounts of the two crude oils used and the amounts of the two products produced However this is not enough The problem is that this information doesnt tell Chandler how to make the outputs from the inputs The com pany instead requires a blending plan how much of each input to use in the production of a barrel of each output Once you understand that this blending plan is the basic decision all other output variables follow in a straightforward manner 6To avoid being overly technical we use the generic term quality level In real oil blending qualities of interest might be octane rating viscosity and others In typical blending problems the correct decision variables are the amounts of each input blended into each output Table 47 Variables and Constraints for Blending Model Input variables Unit selling prices availabilities of inputs quality levels of inputs required quality levels of outputs Decision variables changing cells Barrels of each input used to produce each output Objective cell Revenue from selling gasoline and heating oil Other calculated variables Barrels of inputs used barrels of outputs produced and sold quality obtained and quality required for outputs Constraints Barrels of inputs used Barrels available Quality of outputs obtained Ú Quality required A secondary but very important issue in typical blending models is how to implement the quality constraints The constraints here are in terms of quality In other blending E X A M P L E Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 45 Blending Models 167 Figure 423 Solver Dialog Box for Blending Model 1 2 3 A B C D E F G Oneway analysis for Solver model in Model worksheet Selling price gasoline cell B4 values along side output cells along top sold1 sold2 e e 4 5 6 7 8 Barrels Barrels Revenue Increase 50 0 15000 900000 55 0 15000 900000 0 60 5000 10000 900000 0 65 5000 10000 925000 25000 9 10 11 12 13 70 5000 10000 950000 25000 75 5000 10000 975000 25000 80 5000 10000 1000000 25000 85 5000 10000 1025000 25000 90 5000 10000 1050000 25000 Figure 424 Sensitivity to the Selling Price of Gasoline Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 168 Chapter 4 Linear Programming Models 1 2 3 A B C D E F G Oneway analysis for Solver model in Model worksheet Barrels available crude 1 cell F16 values along side output cells along top sold1 sold2 e e 4 5 6 7 8 Barrels Barrels Revenue Increase 9 10 11 12 13 14 15 12000 19000 3000 1 605 000 90 000 15 16 17 18 19 20 21 12000 19000 3000 1605000 90000 22 23 2000 0 10000 600000 3000 1000 12000 795000 195000 4000 3000 11000 885000 90000 5000 5000 10000 975000 90000 6000 7000 9000 1065000 90000 7000 9000 8000 1155000 90000 8000 11000 7000 1245000 90000 9000 13000 6000 1335000 90000 10000 15000 5000 1425000 90000 11000 17000 4000 1515000 90000 13000 21000 2000 1695000 90000 14000 23000 1000 1785000 90000 15000 25000 0 1875000 90000 16000 26000 0 1950000 75000 17000 27000 0 2025000 75000 18000 28000 0 2100000 75000 19000 29000 0 2175000 75000 20000 30000 0 2250000 75000 Figure 425 Sensitivity to the Availability of Crude 1 of interest First as the price of gasoline increases from 55 to 65 Chandler starts pro ducing gasoline and less heating oil exactly as you would expect Second the revenue can only increase or stay the same as the changes in column E calculated manually indicate In the second sensitivity analysis we vary the availability of crude 1 from 2000 barrels to 20000 barrels in increments of 1000 barrels The resulting SolverTable out put appears in Figure 425 These results make sense if you analyze them carefully First the revenue increases but at a decreasing rate as more crude 1 is available This is a common occurrence in LP models As more of a resource is made available rev enue can only increase or remain the same but each extra unit of the resource pro duces less or at least no more revenue than the previous unit Second the amount of gasoline produced increases whereas the amount of heating oil produced decreases Heres why Crude 1 has a higher quality than crude 2 and gasoline requires higher quality Gasoline also sells for a higher price Therefore as more crude 1 is available Chandler can produce more gasoline receive more revenue and still meet quality standards Could these sensitivity questions also be answered with Solvers sensitivity report shown in Figure 426 Consider the sensitivity to the change in the price of gasoline The first and third rows of the top table in this report are for sensitivity to the objective coeffi cients of decision variables involving gasoline The problem is that when the price of gaso line changes both of these coefficients change The reason is that the objective includes the sum of these two decision variables multiplied by the unit price of gasoline However Solvers sensitivity report is valid only for oneatatime coefficient changes Therefore it cannot answer our question Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 170 Chapter 4 Linear Programming Models FUNDAMENTAL INSIGHT Clearing Denominators Some constraints particularly those that arise in blending models are most naturall y expressed in terms of ratios For example the percentage of sulfur in a product is the ratio amount of sulfur in product total amount of product which could then be con strained to be less than or equal to 6 say This is a perfectly valid way to expr ess the constraint but it has the undesirable eff ect of making the model non linearThe fix is simpleTo make the model linear mul tiply through by the denominator of the ratioThis has the added benefit of ensuring that ther e division b y zero will not occur P R O B L E M S SkillBuilding Problems 20 Use SolverTable in Chandlers blending model to see whether by increasing the selling price of gasoline you can get an optimal solution that produces only gasoline no heating oil Then use SolverTable again to see whether by increasing the selling price of heating oil you can get an optimal solution that produces only heating oil no gasoline 21 Use SolverTable in Chandlers blending model to find the shadow price of crude oil 1that is the amount Chandler would be willing to spend to acquire more crude oil 1 Does this shadow price change as Chandler keeps getting more of crude oil 1 Answer the same questions for crude oil 2 22 How sensitive is the optimal solution barrels of each output sold and profit to the required quality points In reality a company using a blending model would run the model periodically each day say and set production on the basis of the current inventory of inputs and the current fore casts of demands and prices Then the forecasts and the input levels would be updated and the model would be run again to determine the next days production MODELING ISSUES Blending at Texaco Texaco in DeWitt et al 1989 uses a nonlinear programming model OMEGA to plan and schedule its blending applications Texacos model is nonlinear because blend volatil ities and octanes are nonlinear functions of the amount of each input used to produce a par ticular gasoline Blending in the Oil Industry Many oil companies use LP to optimize their refinery operations Magoulas and Marinos Kouris 1988 discuss one such blending model that has been used to maximize a refinerys profit ADDITIONAL APPLICATIONS Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 46 Production Process Models 171 Answer this by running a twoway SolverTable with these three outputs You can choose the values of the two inputs to vary 23 In Chandlers blending model suppose a chemical ingredient called CI is needed by both gasoline and heating oil At least 3 of every barrel of gasoline must be CI and at least 5 of every barrel of heating oil must be CI Suppose that 4 of all crude oil 1 is CI and 6 of all crude oil 2 is CI Modify the model to incorporate the constraints on CI and then optimize Dont forget to clear denominators 24 In the current blending model a barrel of any input re sults in a barrel of output However in a real blending problem there can be losses Suppose a barrel of input results in only a fraction of a barrel of output Specifi cally each barrel of either crude oil used for gasoline results in only 095 barrel of gasoline and each barrel of either crude used for heating oil results in only 097 barrel of heating oil Modify the model to incorporate these losses and reoptimize SkillExtending Problem 25 We warned you about clearing denominators in the quality constraints This problem illustrates what can happen if you dont do so a Implement the quality constraints as indicated in Inequality 43 of the text Then run Solver with the Simplex LP method What happens What if you use the GRG Nonlinear method instead b Repeat part a but increase the selling price of heating oil to 120 per barrel What happens now Does it matter whether you use the Simplex LP method as opposed to the GRG Nonlinear method Why 46 PRODUCTION PROCESS MODELS LP is often used to determine the optimal method of operating a production process In particular many oil refineries use LP to manage their production operations The models are often characterized by the fact that some of the products produced are inputs to the pro duction of other products The following example is typical E X A M P L E 45 DRUG PRODUCTION AT REPCO R epco produces three drugs A B and C and can sell these drugs in unlimited quanti ties at unit prices 8 70 and 100 respectively Producing a unit of drug A requires one hour of labor Producing a unit of drug B requires two hours of labor and two units of drug A Producing one unit of drug C requires three hours of labor and one unit of drug B Any drug A that is used to produce drug B cannot be sold separately and any drug B that is used to produce drug C cannot be sold separately A total of 4000 hours of labor are available Repco wants to use LP to maximize its sales revenue Objective To develop an LP spreadsheet model that relates production decisions to amounts required for production and amounts available for selling and to use Solver to maximize sales revenue subject to limited labor hours WHERE DO THE NUMBERSCOME FROM The inputs for this problem should be easy to obtain The company sets its selling prices which are probably dictated by the market The available labor hours are based on the size of the current workforce assigned to production of these drugs These might be flexible quantities depending on whether workers could be diverted from other duties to work on these drugs and whether new labor could be hired The labor and drug usage inputs for producing the various drugs are probably well known based on productivity levels and chemical requirements Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 174 Chapter 4 Linear Programming Models same method can be used to copy a column r ange to a row range However this method doesnt copy formulas correctly Excel Function TRANSPOSE and Other Array Functions The TRANSPOSE function is useful for linking a row to a column or vice versa It has the syntax TRANSPOSERange To implement it highlight the row or column range where the results will go type the formula and pr ess CtrlShiftEnter This function is one of several array functions in Excel which means that it fills an entire range not just a single cell all at once All array formulas require you to highlight the entire range where the re sults will go type the formula and then press CtrlShiftEnter After you do this you will notice curly brackets around the formula in the Formula Bar You should not actually type these curly brackets They simply indicate the presence of an array function 4 Units sold Referring to Equation 44 determine the units sold of each drug by sub traction Specifically enter the formula B16B18 in cell B19 and copy it to the range C19D19 5 Labor hours used Calculate the total number of labor hours used in cell B23 with the formula SUMPRODUCTB5D5Unitsproduced 6 Total revenue Calculate Repcos revenue from sales in cell B25 with the formula SUMPRODUCTB12D12Unitssold USING SOLVER To use Solver to maximize Repcos revenue fill in the Solver dialog box as shown in Figure 428 As usual check the NonNegative option and select the Simplex LP method before optimizing Note that the drugs produced are constrained to be greater than or equal to the drugs used in production of other drugs An equivalent alternative is to constrain the units sold to be nonnegative Figure 428 Solver Dialog Box for Repco Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 47 Financial Models 177 SkillExtending Problem 29 In a production process model such as Repcos certain inputs make no sense in the usage table the range B7D9 of the model For example suppose that in addition to current usages each unit of drug A requires one unit of drug C Why does this result in a nonsensical problem What happens if you run Solver on it anyway What happens if you run Solver on it after adding a constraint that the sum of the units pro duced over all three drugs must be at least 1 47 FINANCIAL MODELS The majority of optimization examples described in management science textbooks are in the area of operations scheduling blending logistics aggregate planning and others This is probably warranted because many of the most successful management science applications in the real world have been in these areas However optimization and other management science methods have also been applied successfully in a number of finan cial areas and they deserve recognition Several of these applications are discussed throughout this book In this section we begin the discussion with two typical applications of LP in finance The first involves investment strategy The second involves pension fund management E X A M P L E 46 FINDING AN OPTIMAL INVESTMENT STRATEGY AT BARNEYJONES A t the present time the beginning of year 1 the BarneyJones Investment Corporation has 100000 to invest for the next four years There are five possible investments la beled A through E The timing of cash outflows and cash inflows for these investments is somewhat irregular For example to take part in investment A cash must be invested at the beginning of year 1 and for every dollar invested there are returns of 050 and 100 at the beginnings of years 2 and 3 Information for the other investments follows where all re turns are per dollar invested Investment B Invest at the beginning of year 2 receive returns of 050 and 100 at the beginnings of years 3 and 4 Investment C Invest at the beginning of year 1 receive return of 120 at the beginning of year 2 Investment D Invest at the beginning of year 4 receive return of 190 at the beginning of year 5 Investment E Invest at the beginning of year 3 receive return of 150 at the beginning of year 4 We assume that any amounts can be invested in these strategies and that the returns are the same for each dollar invested However to create a diversified portfolio BarneyJones wants to limit the amount put into any investment to 75000 The company wants an in vestment strategy that maximizes the amount of cash on hand at the beginning of year 5 At the beginning of any year it can invest only cash on hand which includes returns from pre vious investments Any cash not invested in any year can be put in a shortterm money mar ket account that earns 3 annually Objective To develop an LP spreadsheet model that relates investment decisions to total ending cash and to use Solver to find the strategy that maximizes ending cash and invests no more than a given amount in any one investment Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 182 Chapter 4 Linear Programming Models The following example illustrates a common situation where fixed payments are due in the future and current funds must be allocated and invested so that their returns are sufficient to make the payments We place this in a pension fund context 3 4 5 6 7 8 9 10 11 12 13 A B C D E F G H I Interest on cash cell B6 values along side Max per investment cell B5 values along top output cell in corner Maximuminmoneymarket 75000 100000 125000 150000 175000 200000 225000 05 139420 126923 112500 87500 62500 37500 12500 10 139554 126923 112500 87500 62500 37500 12500 15 139688 126923 112500 87500 62500 37500 12500 20 139821 126923 112500 87500 62500 37500 12500 25 139955 126923 112500 87500 62500 37500 12500 30 140089 126923 112500 87500 62500 37500 12500 35 140223 126923 112500 87500 62500 37500 12500 40 140357 126923 112500 87500 62500 37500 12500 45 140491 126923 112500 87500 62500 37500 12500 Figure 434 Sensitivity of Maximum in Money Market to Two Inputs E X A M P L E 47 MANAGING A PENSION FUND AT ARMCO J ames Judson is the financial manager in charge of the company pension fund at Armco Incorporated James knows that the fund must be sufficient to make the payments listed in Table 410 Each payment must be made on the first day of each year James is going to finance these payments by purchasing bonds It is currently January 1 2010 and three bonds are available for immediate purchase The prices and coupons for the bonds are as follows All coupon payments are received on January 1 and arrive in time to meet cash demands for the date on which they arrive Bond 1 costs 980 and yields a 60 coupon in the years 2011 through 2014 and a 1060 payment on maturity in the year 2015 Bond 2 costs 970 and yields a 65 coupon in the years 2011 through 2020 and a 1065 payment on maturity in the year 2021 Bond 3 costs 1050 and yields a 75 coupon in the years 2011 through 2023 and a 1075 payment on maturity in the year 2024 James must decide how much cash to allocate from company coffers to meet the initial 11000 payment and buy enough bonds to make future payments He knows that any excess cash on hand can earn an annual rate of 4 in a fixedrate account How should he proceed Table 410 Payments for Pension Example Year Payment Year Payment Year Payment 2010 11000 2015 18000 2020 25000 2011 12000 2016 20000 2021 30000 2012 14000 2017 21000 2022 31000 2013 15000 2018 22000 2023 31000 2014 16000 2019 24000 2024 31000 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 47 Financial Models 185 USING SOLVER The main Solver dialog box should be filled out as shown in Figure 436 Once again notice that the Moneyallocated cell is both the objective cell and one of the changing cells Figure 436 Solver Dialog Box for Pension Fund Model Discussion of the Solution The optimal solution appears in Figure 435 You might argue that the numbers of bonds purchased should be constrained to integer values We tried this and the optimal solution changed very little The optimal numbers of bonds to purchase changed to 74 79 and 27 and the optimal money to allocate increased to 197887 With this integer solution shown in Figure 437 James sets aside 197887 initially Any less than this would not workhe couldnt make enough from bonds to meet future pension payments All but 20387 of this see cell B20 is spent on bonds and of the 20387 11000 is used to make the current pension payment After this the amounts in row 20 which are always sufficient to make the payments in row 22 are composed of returns from bonds and cash with interest from the previous year Even more so than in previous examples there is no way to guess this optimal solution The timing of bond returns and the irregular pension payments make a spreadsheet optimization model absolute necessary Sensitivity Analysis Because the bond information and pension payments are evidently fixed there is only one obvious direction for sensitivity analysis on the fixed interest rate in cell B9 We tried this Constraints always have the potential to penalize the objective to some extent SolverTable is a perfect tool for finding the magnitude of this penalty Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 47 Financial Models 187 P R O B L E M S SkillBuilding Problems 30 In the BarneyJones investment model increase the maximum amount allowed in any investment to 150000 Then run a oneway sensitivity analysis to the money market rate on cash Capture one output variable the maximum amount of cash ever put in the money market You can choose any reasonable range for varying the money market rate 31 Modify the BarneyJones investment model so that a minimum amount must be put into any investment although this minimum can vary by investment For example the minimum amount for investment A might be 0 whereas the minimum amount for invest ment D might be 50000 These minimum amounts should be inputs you can make up any values you like Run Solver on your modified model 32 We claimed that our model for BarneyJones is gener alizable Try generalizing it to the case where there are two more potential investments F and G Investment F requires a cash outlay in year 2 and returns 050 in each of the next four years for every dollar invested Investment G requires a cash outlay in year 3 and returns 075 in each of years 5 6 and 7 for every dol lar invested Modify the model as necessary making the objective the final cash after year 7 33 In the BarneyJones investment model we ran invest ments across columns and years down rows Many financial analysts seem to prefer the opposite Modify the spreadsheet model so that years go across columns and investments go down rows Run Solver to ensure that your modified model is correct There are two possible ways to do this and you can experiment to see which you prefer First you could basically start over on a blank worksheet Second you could use Excels TRANSPOSE function 34 In the pension fund model suppose there is an upper limit of 60 on the number of bonds of any particular type that can be purchased Modify the model to incorporate this extra constraint and then reoptimize How much more money does James need to allocate initially 35 In the pension fund model suppose there is a fourth bond bond 4 Its unit cost in 2010 is 1020 it returns coupons of 70 in years 2011 to 2014 and a payment of 1070 in 2015 Modify the model to incorporate this extra bond and reoptimize Does the solution changethat is should James purchase any of bond 4 36 In the pension fund model suppose James has been asked to see how the optimal solution will change if the required payments in years 2015 to 2024 all increase by the same percentage where this percent age could be anywhere from 5 to 25 Use an ap propriate oneway SolverTable to help him out and write a memo describing the results 37 The pension fund model is streamlined perhaps too much It does all of the calculations concerning cash flows in row 20 James decides he would like to break these out into several rows of calculations Beginning cash for 2010 this is the amount allocated for other years it is the unused cash plus interest from the pre vious year Amount spent on bonds positive in 2010 only Amount received from bonds positive for years 2011 to 2024 only Cash available for making pension fund payments and below the Amount required row Cash left over amount invested in the fixed interest rate Modify the model by inserting these rows enter the appropriate formulas and run Solver You should obtain the same result but get more detailed information SkillExtending Problems 38 Suppose the investments in the BarneyJones model sometimes require cash outlays in more than one year For example a 1 investment in invest ment B might require 025 to be spent in year 1 and 075 to be spent in year 2 Does the current model easily accommodate such investments Try it with some cash outlay data you make up run Solver and interpret the results 39 In the pension fund model if the amount of money initially is less than the amount found by Solver then James will not be able to meet all of the pension fund payments Use the current model to demonstrate that this is true To do so enter a value less than the opti mal value into cell B16 Then run Solver but remove the Moneyallocated cell as a changing cell and as the target cell If there is no target cell Solver simply tries to find a solution that satisfies all of the con straints What do you find 40 Continuing the previous problem in a slightly different direction continue to use the Moneyallocated cell as a changing cell and add a constraint that it must be less than or equal to any value such as 195000 that is less than its current optimal value With this constraint James will not be able to meet all of the pension fund payments Create a new target cell to minimize the total amount of payments not met The easiest way to do this is with IF functions Unfortu nately this makes the model nonsmooth and Solver might have trouble finding the optimal solution Try it and see Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 190 Chapter 4 Linear Programming Models DEVELOPINGTHE SPREADSHEETMODEL Figure 439 contains the DEA spreadsheet model used to determine the efficiency of hos pital 1 See the file Hospital DEAxlsx To develop this model proceed as follows 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 K J I H G F E D C B A DEA model for checking efficiency of a selected hospital Range names used Inputcosts ModelB14B16 Selected hospital 1 Outputvalues ModelD14D16 Selectedhospital ModelB3 Inputs used Input 1 Input 2 Outputs produced Output 1 Output 2 Output 3 Selectedhospitalinputcost ModelB19 Hospital la Hos tip 14 5 1 la outputv ula e Mod le B22 tc edhos tip S ele 16 4 9 1 Hospital la Hos tip 15 8 2 Mod le B1 0 C10 U tin co sts of ni pu st 10 7 5 2 Hospital la Hos tip 12 7 3 Mod le F1 0 H10 U tin p cir esofoutpu st 13 9 4 3 Unit costs of U tin 00 71 00 00 ni pu st prices of outputs 00000 00000 0063 Constraints that input costs must cover output values Hospital Input costs Output values 1 1000 1000 2 1071 0625 3 0857 0813 Constraint that selected hospitals input cost must equal a nominal value of 1 Selected hospital input cost 1000 1 Maximize selected hospitals output value to see if it is 1 hence efficient Selected hospital output value 1000 Figure 439 DEA Model for Hospital 1 1 Input given data and name ranges Enter the input and output information for each hospital in the ranges B6C8 and F6H8 and name the various ranges as indicated 2 Selected hospital Enter 1 2 or 3 in cell B3 depending on which hospital you want to analyze You will eventually analyze all three 3 Unit input costs and output prices Enter any trial values for the input costs and out put prices in the Unitcostsofinputs and Unitpricesofoutputs ranges 4 Total input costs and output values In the Inputcosts range calculate the cost of the inputs used by each hospital To do this enter the formula SUMPRODUCTUnitcostsofinputsB6C6 in cell B14 for hospital 1 and copy this to the rest of the Inputcosts range for the other hos pitals Similarly calculate the output values by entering the formula SUMPRODUCTUnitpricesofoutputsF6H6 in cell D14 and copying it to the rest of the Outputvalues range Note that even though the focus is currently on hospital 1s efficiency you still need to calculate input costs and output values for the other hospitals so that you have something to compare hospital 1 to 5 Total input cost and output value for the selected hospital In row 19 constrain the total input cost of the selected hospital to be 1 by entering the formula VLOOKUPSelectedhospitalA14B162 in cell B19 and enter a 1 in cell D19 Similarly enter the formula VLOOKUPSelectedhospitalA14D164 in cell B22 Make sure you understand how these VLOOKUP functions work Remember that because the selected hospitals input cost is constrained to be 1 its output value in cell B22 is automatically its efficiency USING SOLVERTO DETERMINEWHETHERHOSPITAL 1 IS EFFICIENT To determine whether hospital 1 is efficient use Solver as follows When you are finished the Solver dialog box should appear as shown in Figure 440 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 48 Data Envelopment Analysis DEA 191 1 Objective Select cell B22 as the target cell to maximize Because the cost of hospital 1 inputs is constrained to be 1 this causes Solver to maximize the efficiency of hospital 1 2 Changing cells Choose the Unitcostsofinputs and Unitpricesofoutputs ranges as the changing cells 3 Selected hospitals input cost constraint Add the constraint Selectedhospital inputcost1 This sets the total value of hospital 1s inputs equal to 1 4 Efficiency constraint Add the constraint InputcostsOutputvalues This ensures that no hospital is more than 100 efficient 5 Specify nonnegati vity and optimize Check the NonNegative option and the Simplex LP method and then solve to obtain the optimal solution shown in Figure 439 The 1 in cell B22 of this solution means that hospital 1 is efficient In words Solver has found a set of unit costs for the inputs and the unit prices for the outputs such that the total value of hospital 1s outputs equals the total cost of its inputs Figure 440 Solver Dialog Box for the DEA Model 1 2 K J I H G F E D C B A DEA model for checking efficiency of a selected hospital Range names used Inputcosts ModelB14B16 3 4 5 6 7 8 9 10 11 12 13 14 15 p Selected hospital 2 Outputvalues ModelD14D16 Selectedhospital ModelB3 Inputs used Input 1 Input 2 Outputs produced Output 1 Output 2 Output 3 Selectedhospitalinputcost ModelB19 Hospital la Hos tip 14 5 1 la outputv ula e Mod le B22 tc edhos tip S ele 16 4 9 1 Hospital la Hos tip 15 8 2 Mod le B1 0 C10 U tin co sts of ni pu st 10 7 5 2 Hospital la Hos tip 12 7 3 Mod le F1 0 H10 U tin p cir esofoutpu st 13 9 4 3 Unit costs of U tin 00 67 00 00 ni pu st prices of outputs 00800 00533 0000 Constraints that input costs must cover output values Hospital Input costs Output values 1 0933 0933 2 1000 0773 15 16 17 18 19 20 21 22 2 1000 0773 3 0800 0800 Constraint that selected hospitals input cost must equal a nominal value of 1 Selected hospital input cost 1000 1 Maximize selected hospitals output value to see if it is 1 hence efficient Selected hospital output value 0773 Figure 441 DEA Model for Hospital 2 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 192 Chapter 4 Linear Programming Models Determining Whether Hospitals 2 and 3 Are Efficient To determine whether hospital 2 is efficient simply replace the value in cell B3 by 2 and rerun Solver The Solver settings do not need to be modified The optimal solution appears in Figure 441 From the value of 0773 in cell B22 you can see that hospital 2 is not effi cient Similarly you can determine that hospital 3 is efficient by replacing the value in cell B3 by 3 and rerunning Solver see Figure 442 In summary the Solver results imply that hospitals 1 and 3 are efficient but hospital 2 is inefficient What Does It Mean to Be Efficient or Inefficient This idea of efficiency or inefficiency might still be a mystery so lets consider it further A hospital is efficient if the inputs and outputs can be priced in such a way that this hospital gets out all of the value that it puts in The pricing scheme depends on the hospital Each hospital tries to price inputs and outputs to put its operations in the best possible light In the example hospital 1 attaches 0 prices to input 1 hospital beds and output 3 patient days for patients over 65 and it attaches positive prices to the rest This makes hospital 1 look efficient Hospital 3 which is also efficient also attaches 0 prices to input 1 and output 3 but its prices for the others are somewhat different from hospital 1s prices If DEA finds that a hospital is inefficient there is no pricing scheme where that hospi tal can recover its entire input costs in output values Actually it can be shown that if a hos pital is inefficient then a combination of the efficient hospitals can be found that uses no more inputs than the inefficient hospital yet produces at least as much of each output as the inefficient hospital In this sense the hospital is inefficient To see how this combination can be found consider the spreadsheet model in Figure 443 Begin by entering any positive weights in the Weights range For any such weights they dont even need to sum to 1 consider the combination hospital as a fraction of hospital 1 and another fraction of hospital 3 For example with the weights shown the combination hospital uses about 26 of the inputs and produces about 26 of the outputs of hospital 1 and it uses about 66 of the inputs and produces about 66 of the outputs of hospital 3 When they are combined in row 6 with the SUMPRODUCT function for ex ample the formula in cell D6 is SUMPRODUCTWeightsD4D5 you can see the quantities of inputs this combination hospital uses and the quantities of outputs it produces 1 2 K J I H G F E D C B A DEA model for checking efficiency of a selected hospital Range names used Inputcosts ModelB14B16 3 4 5 6 7 8 9 10 11 12 13 14 15 p Selected hospital 3 Outputvalues ModelD14D16 Selectedhospital ModelB3 Inputs used Input 1 Input 2 Outputs produced Output 1 Output 2 Output 3 Selectedhospitalinputcost ModelB19 Hospital la Hos tip 14 5 1 la outputv ula e Mod le B22 tc edhos tip S ele 16 4 9 1 Hospital la Hos tip 15 8 2 Mod le B1 0 C10 U tin co sts of ni pu st 10 7 5 2 Hospital la Hos tip 12 7 3 Mod le F1 0 H10 U tin p cir esofoutpu st 13 9 4 3 Unit costs of U tin 00 83 00 00 ni pu st prices of outputs 01000 00667 0000 Constraints that input costs must cover output values Hospital Input costs Output values 1 1167 1167 2 1250 0967 15 16 17 18 19 20 21 22 2 1250 0967 3 1000 1000 Constraint that selected hospitals input cost must equal a nominal value of 1 Selected hospital input cost 1000 1 Maximize selected hospitals output value to see if it is 1 hence efficient Selected hospital output value 1000 Figure 442 DEA Model for Hospital 3 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 49 Conclusion 197 a Determine how to minimize the bus companys salary hiring and firing costs over the next five years b Use SolverTable to determine how the total number hired total number fired and total cost change as the unit hiring and firing costs each increase by the same percentage 46 During each fourhour period the Smalltown police force requires the following number of onduty police officers eight from midnight to 4 AM seven from 4 AM to 8 AM six from 8 AM to noon six from noon to 4 PM five from 4 PM to 8 PM and four from 8 PM to midnight Each police officer works two consecutive fourhour shifts a Determine how to minimize the number of police offi cers needed to meet Smalltowns daily requirements b Use SolverTable to see how the number of police officers changes as the number of officers needed from midnight to 4 AM changes 47 Shoemakers of America forecasts the following demand for the next six months 5000 pairs in month 1 6000 pairs in month 2 7000 pairs in month 3 9000 pairs in month 4 6000 pairs in month 5 5000 pairs in month 6 It takes a shoemaker 20 minutes to produce a pair of shoes Each shoemaker works 150 hours per month plus up to 40 hours per month of overtime A shoemaker is paid a regular salary of 2000 per month plus 20 per hour for overtime At the beginning of each month Shoemakers can either hire or fire workers It costs the company 1000 to hire a worker and 1200 to fire a worker The monthly holding cost per pair of shoes is 5 of the cost of producing a pair of shoes with regulartime labor The raw materials in a pair of shoes cost 10 At the beginning of month 1 Shoemakers has 15 workers and 500 pairs of shoes in inventory Determine how to minimize the cost of meeting on time the demands of the next six months 48 NewAge Pharmaceuticals produces the drug NasaMist from four chemicals Today the company must pro duce 1000 pounds of the drug The three active ingre dients in NasaMist are A B and C By weight at least 8 of NasaMist must consist of A at least 4 of B and at least 2 of C The cost per pound of each chemical and the amount of each active ingredient in one pound of each chemical are given in the file P0448xlsx At least 100 pounds of chemical 2 must be used a Determine the cheapest way of producing todays batch of NasaMist b Use SolverTable to see how much the percentage of requirement of A is really costing NewAge Let the percentage required vary from 6 to 12 49 You have decided to enter the candy business You are considering producing two types of candies Slugger candy and Easy Out candy both of which consist solely of sugar nuts and chocolate At present you have in stock 10000 ounces of sugar 2000 ounces of nuts and 3000 ounces of chocolate The mixture used to make Easy Out candy must contain at least 20 nuts The mixture used to make Slugger candy must contain at least 10 nuts and 10 chocolate Each ounce of Easy Out candy can be sold for 120 and each ounce of Slugger candy for 140 a Determine how you can maximize your revenue from candy sales b Use SolverTable to determine how changes in the price of Easy Out change the optimal solution c Use SolverTable to determine how changes in the amount of available sugar change the optimal solution 50 Sunblessed Juice Company sells bags of oranges and cartons of orange juice Sunblessed grades oranges on a scale of 1 poor to 10 excellent At present Sunblessed has 100000 pounds of grade 9 oranges and 120000 pounds of grade 6 oranges on hand The average quality of oranges sold in bags must be at least 7 and the average quality of the oranges used to produce orange juice must be at least 8 Each pound of oranges that is used for juice yields a revenue of 150 and incurs a variable cost consisting of labor costs variable overhead costs inventory costs and so on of 105 Each pound of oranges sold in bags yields a revenue of 150 and incurs a variable cost of 070 a Determine how Sunblessed can maximize its profit b Use SolverTable to determine how a change in the cost per bag of oranges changes the optimal solution c Use SolverTable to determine how a change in the amount of grade 9 oranges available affects the optimal solution d Use SolverTable to determine how a change in the required average quality required for juice changes the optimal solution 51 A bank is attempting to determine where its assets should be invested during the current year At present 500000 is available for investment in bonds home loans auto loans and personal loans The annual rates of return on each type of investment are known to be the following bonds 10 home loans 16 auto loans 13 personal loans 20 To ensure that the banks portfolio is not too risky the banks investment manager has placed the following three restrictions on the banks portfolio The amount invested in personal loans cannot ex ceed the amount invested in bonds The amount invested in home loans cannot exceed the amount invested in auto loans No more than 25 of the total amount invested can be in personal loans Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 49 Conclusion 201 75 Based on Gaballa and Pearce 1979 Northwest Airlines has determined that it needs the number of ticket agents during each hour of the day listed in the file P0475xlsx Workers work ninehour shifts one hour of which is for lunch The lunch hour can be either the fourth or fifth hour of their shift What is the minimum number of workers needed by Northwest 76 A rock company uses five types of rocks to fill four orders The phosphate content availability of each type of rock and the production cost per pound for each rock are listed in the file P0476xlsx as well as the size of each order and the minimum and maximum phosphate percentage in each order What is the cheapest way to fill the orders 77 An automobile manufacturer needs to plan its produc tion for the next year Demands for the next 12 months are forecasted to be 940 790 360 720 270 130 160 300 990 290 280 and 790 Other relevant informa tion is as follows Workers are paid 5000 per month It costs 500 to hold a car in inventory for a month The holding cost is based on each months ending inventory It costs 4000 to hire a worker It costs 6000 to fire a worker Each worker can make up to eight cars a month Workers are hired and fired at the beginning of each month At the beginning of month 1 there are 500 cars in inventory and 60 workers How can the company minimize the cost of meeting demand for cars on time 78 An oil company produces gasoline from five inputs The cost density viscosity and sulfur content and the number of barrels available of each input are listed in the file P0478xlsx Gasoline sells for 72 per barrel Gasoline can have a density of at most 098 units per barrel a viscosity of at most 37 units per barrel and a sulfur content of at most 37 units per barrel a How can the company maximize its profit b Describe how the optimal solution to the problem changes as the price of gasoline ranges from 65 to 80 per barrel 79 The HiTech company produces BluRay disc players Estimated demands for the next four quarters are 5000 10000 8000 and 2000 At the beginning of quarter 1 HiTech has 60 workers It costs 2000 to hire a worker and 4000 to fire a worker Workers are paid 10000 per quarter plus 80 for each unit they make during overtime A new hire can make up to 60 units per quarter during regulartime whereas a previously hired worker can make up to 90 units per quarter Any worker can make up to 20 units per quarter during overtime Each disc player is sold for 160 It costs 20 to hold a disc player in inventory for a quarter Assume workers are hired and fired at the beginning of each quarter and that all of a quarters production is available to meet demand for that quarter Initial inven tory at the beginning of quarter 1 is 1000 disc players How can the company maximize its profit Assume that demand is lost if insufficient stock is available That is there is no backlogging of demand and there is no requirement that HiTech must satisfy all of its demand SkillExtending Problems 80 MusicTech manufactures and sells a portable music device called an mTune similar to an iPod At beginning of month 1 the company has 100000 and 15 employees Each machine the company owns has the capacity to make up to 900 mTunes per month and each worker can make up to 600 mTunes per month The company cannot use more labor or machine capacity than is available in any given month Also the company wants to have a nonnegative cash balance at all points in time The companys costs are the following Holding cost of 2 each month per mTune in ending inventory Cost in month 1 of buying machines 3000 per machine Raw material cost of 6 per mTune Monthly worker wage of 3500 Hiring cost of 4000 per worker Firing cost of 5000 per worker In the absence of advertising the monthly demands in months 1 through 6 are forecasted to be 5000 8000 7000 6000 5000 and 5000 However MusicTech can increase demand each month by advertising Every 10 up to a maximum of 50000 per month spent on advertising during a month increases demand for that month by one mTune The devices are sold for 75 each The sequence of events in any month is that the company buys machines month 1 only hires and fires workers makes the mTunes advertises pays all costs for the month and collects revenues for the month Develop a model to maximize profit total revenue minus total costs earned during the next six months 81 You want to take out a 300000 loan on a 20year mortgage with endofmonth payments The annual rate of interest is 6 Twenty years from now you will need to make a 40000 ending balloon payment Be cause you expect your income to increase you want to structure the loan so at the beginning of each year your monthly payments increase by 2 a Determine the amount of each years monthly payment You should use a lookup table to look up each years monthly payment and to look up the year based on the month eg month 13 is year 2 etc Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 204 Chapter 4 Linear Programming Models at 130 degrees Celsius ASTM is the American Society for Testing and Materials The attributes and daily availability in liters of each input are listed in the file P0491xlsx The requirements for each output are also listed in this file The daily demand in thou sands of liters for each product must be met but more can be produced if desired The RON and ASTM re quirements are minimums the RVP requirement is a maximum Regular gasoline sells for 0754 per liter premium gasoline for 0819 Before each product is ready for sale 015 gram per liter of lead must be removed The cost of removing 01 gram per liter is 0213 At most 38 of each type of gasoline can consist of FCG How can the company maximize its daily profit 92 Capsule Drugs manufactures two drugs The drugs are produced by blending two chemicals By weight drug 1 must contain at least 65 chemical 1 and drug 2 must contain at least 55 chemical 1 Drug 1 sells for 6 per ounce and drug 2 sells for 4 per ounce Chemicals 1 and 2 can be produced by one of two pro duction processes Running process 1 for an hour re quires 7 ounces of raw material and 2 hours skilled labor and it yields 3 ounces of each chemical Run ning process 2 for an hour requires 5 ounces of raw material and 3 hours of skilled labor and it yields 3 ounces of chemical 1 and 1 ounce of chemical 2 A total of 3000 hours of skilled labor and 5000 ounces of raw material are available Determine how to maximize Capsules sales revenues 93 Molecular Products produces three chemicals B C and D The company begins by purchasing chemical A for a cost of 650 per 100 liters For an additional cost of 320 and the use of three hours of skilled labor 100 liters of A can be transformed into 40 liters of C and 60 liters of B Chemical C can either be sold or processed further It costs 130 and one hour of skilled labor to process 100 liters of C into 60 liters of D and 40 liters of B For each chemical the selling price per 100 liters and the maximum amount in 100s of liters tha can be sold are listed in the file P0493xlsx A maximum of 200 labor hours is available Determine how Molecular can maximize its profit 94 Bexter Labs produces three products A B and C Bexter can sell up to 3000 units of product A up to 2000 units of product B and up to 2000 units of prod uct C Each unit of product C uses two units of A and three units of B and incurs 5 in processing costs Products A and B are produced from either raw mater ial 1 or raw material 2 It costs 6 to purchase and process one unit of raw material 1 Each processed unit of raw material 1 yields two units of A and three units of B It costs 3 to purchase and process a unit of raw material 2 Each processed unit of raw material 2 yields one unit of A and two units of B The unit prices for the products are A 5 B 4 C 25 The quality levels of each product are A 8 B 7 C 6 The average quality level of the units sold must be at least 7 Determine how to maximize Bexters profit 95 Mondo Motorcycles is determining its production schedule for the next four quarters Demands for motorcycles are forecasted to be 400 in quarter 1 700 in quarter 2 500 in quarter 3 200 in quarter 4 Mondo incurs four types of costs as described here It costs Mondo 800 to manufacture each motorcycle At the end of each quarter a holding cost of 100 per motorcycle left in inventory is incurred When production is increased from one quarter to the next a cost is incurred primarily for training employees If the increase in production is x motor cycles the cost is 700x When production is decreased from one quarter to the next a cost is incurred primarily for severance pay and decreased morale If the decrease in production is x motorcycles the cost is 600x All demands must be met on time and a quarters production can be used to meet demand for the current quarter as well as future quarters During the quarter immediately preceding quarter 1 500 Mondos were produced Assume that at the beginning of quarter 1 no Mondos are in inventory a Determine how to minimize Mondos total cost during the next four quarters b Use SolverTable to determine how Mondos optimal production schedule would be affected by a change in the cost of increasing production from one quarter to the next c Use SolverTable to determine how Mondos optimal production schedule would be affected by a change in the cost of decreasing production from one quarter to the next 96 An automobile manufacturing company has a 1500000 advertising budget To increase its automo bile sales the company is considering advertising in newspapers and on television The more the company uses a particular medium the less effective each additional ad is The file P0496xlsx lists the number of new customers reached by each ad Each newspaper ad costs 1000 and each television ad costs 10000 At most 30 newspaper ads and 15 television ads can be placed How can the company maximize the num ber of new customers created by advertising 97 Broker Sonya Wong is currently trying to maximize her profit in the bond market Four bonds are available for purchase and sale at the bid and ask prices shown in the file P0497xlsx Sonya can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price During each of the next three years the person who sells a bond will pay Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 206 Chapter 4 Linear Programming Models 100 to hold an air conditioner in inventory for a month At the beginning of month 1 the company has 200 air conditioners in stock Determine how the company can minimize the cost of meeting air condi tioner demands for the next three months 104 Gotham City National Bank is open Monday through Friday from 9 AM to 5 PM From past experience the bank knows that it needs the numbers of tellers listed in the file P04104xlsx Gotham City Bank hires two types of tellers Fulltime tellers work 9 AM to 5 PM five days a week with one hour off each day for lunch The bank determines when a full time employee takes his or her lunch hour but each teller must go between 12 PM and 1 PM or between 1 PM and 2 PM Fulltime employees are paid in cluding fringe benefits 15 per hour which includes payment for lunch hour The bank can also hire part time tellers Each parttime teller must work exactly four consecutive hours each day A parttime teller is paid 9 per hour and receives no fringe benefits To maintain adequate quality of service the bank has decided that at most five parttime tellers can be hired Determine how to meet the banks teller requirements at minimum cost 105 Based on Rothstein 1973 The Springfield City Police Department employs 30 police officers Each officer works five days per week The crime rate fluctuates with the day of the week so the number of police officers required each day depends on the day of the week as follows Saturday 28 Sunday 18 Monday 18 Tuesday 24 Wednesday 25 Thursday 16 Friday 21 The police department wants to schedule police officers to minimize the number whose days off are not consecutive Determine how to accomplish this goal 106 Based on Charnes and Cooper 1955 Alex Cornby makes his living buying and selling corn On January 1 he has 5000 bushels of corn and 10000 in cash On the first day of each month Alex can buy corn at the forecasted prices per bushel listed in the file P04106xlsx On the last day of each month Alex can sell corn at the forecasted prices listed in the same file Alex stores his corn in a warehouse that can hold 10000 bushels of corn He must be able to pay cash for all corn at the time of purchase Deter mine how Alex can maximize his cash on hand at the end of April 107 City 1 produces 500 tons of waste per day and city 2 produces 400 tons of waste per day Waste must be incinerated at incinerator 1 or 2 and each incinerator can process up to 500 tons of waste per day The cost to incinerate waste is 40 per ton at incinerator 1 and 30 per ton at incinerator 2 Incineration reduces each ton of waste to 02 ton of debris which must be dumped at one of two landfills Each landfill can receive at most 200 tons of debris per day It costs 3 per mile to transport a ton of material either de bris or waste Distances in miles between locations are listed in the file P04107xlsx Determine how to minimize the total cost of disposing of the waste from both cities 108 Based on Smith 1965 Silicon Valley Corporation Silvco manufactures transistors An important as pect of the manufacture of transistors is the melting of the element germanium a major component of a transistor in a furnace Unfortunately the melting process yields germanium of highly variable quality Two methods can be used to melt germanium Method 1 costs 50 per transistor and method 2 costs 70 per transistor The qualities of germanium obtained by methods 1 and 2 are listed in the file P04108xlsx Silvco can refire melted germanium in an attempt to improve its quality It costs 25 to refire the melted germanium for one transistor The results of the refiring process are also listed in the same file For example if grade 3 germanium is refired half of the resulting germanium will be grade 3 and the other half will be grade 4 Silvco has sufficient fur nace capacity to melt or refire germanium for at most 20000 transistors per month Silvcos monthly demands are for 1000 grade 4 transistors 2000 grade 3 transistors 3000 grade 2 transistors and 3000 grade 1 transistors Determine how to minimize the cost of producing the needed transistors 109 The Fresh Turkey Company produces two types of turkey cutlets for sale to fastfood restaurants Each type of cutlet consists of white meat and dark meat Cutlet 1 sells for 279 per pound and must consist of at least 70 white meat Cutlet 2 sells for 189 per pound and must consist of at least 60 white meat At most 10000 pounds of cutlet 1 and 4000 pounds of cutlet 2 can be sold The two types of turkey used to manufacture the cutlets are purchased from a turkey farm Each type 1 turkey costs 899 and yields six pounds of white meat and two pounds of dark meat Each type 2 turkey costs 599 and yields three pounds of white meat and three pounds of dark meat Determine how to maximize Fresh Turkeys profit 110 The production line employees at Grummins Engine work four days a week 10 hours a day Each day of the week the following minimum numbers of line employees are needed Monday through Friday 70 employees Saturday and Sunday 30 employees Grummins employs 110 line employees Determine how to maximize the number of consecutive days off received by these employees For example a worker who gets Sunday Monday and Wednesday off receives two consecutive days off 111 Based on Lanzenauer et al 1987 To process income tax forms the IRS first sends each form Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 49 Conclusion 207 through the data preparation DP department where information is coded for computer entry Then the form is sent to data entry DE where it is entered into the computer During the next 3 weeks the fol lowing quantities of forms will arrive week 1 40000 week 2 30000 week 3 60000 All employ ees work 40 hours per week and are paid 500 per week Data preparation of a form requires 15 min utes and data entry of a form requires 10 minutes Each week an employee is assigned to either data entry or data preparation The IRS must complete processing all forms by the end of week 5 and wants to minimize the cost of accomplishing this goal As sume that all workers are fulltime employees and that the IRS will have the same number of employees each week Assume that all employees are capable of performing data preparation and data entry Deter mine how many workers should be working and how the workers should allocate their hours during the next five weeks 112 Based on Robichek et al 1965 The Korvair Department Store has 100000 in available cash At the beginning of each of the next six months Korvair will receive revenues and pay bills as listed in the file P04112xlsx It is clear that Korvair will have a shortterm cash flow problem until the store receives revenues from the Christmas shopping season To solve this problem Korvair must borrow money At the beginning of July the company takes out a sixmonth loan Any money borrowed for a six month period must be paid back at the end of De cember along with 9 interest early payback does not reduce the total interest of the loan Korvair can also meet cash needs through monthtomonth bor rowing Any money borrowed for a onemonth pe riod incurs an interest cost of 4 per month Deter mine how Korvair can minimize the cost of paying its bills on time 113 Mackk Engine produces diesel trucks New govern ment emission standards have dictated that the aver age pollution emissions of all trucks produced in the next three years cannot exceed 10 grams per truck Mackk produces two types of trucks Each type 1 truck sells for 20000 costs 15000 to manufac ture and emits 15 grams of pollution Each type 2 truck sells for 17000 costs 14000 to manufac ture and emits 5 grams of pollution Production ca pacity limits total truck production during each year to at most 320 trucks The maximum numbers of each truck type that can be sold during each of the next three years are listed in the file P04113xlsx Demand can be met from previous production or the current years production It costs 2000 to hold one truck of any type in inventory for one year Deter mine how Mackk can maximize its profit during the next three years 114 Each hour from 10 AM to 7 PM Bank One receives checks and must process them Its goal is to process all checks the same day they are received The bank has 13 check processing machines each of which can process up to 500 checks per hour It takes one worker to operate each machine Bank One hires both fulltime and parttime workers Fulltime work ers work 10 AM to 6 PM 11 AM to 7 PM or 12 PM to 8 PM and are paid 160 per day Parttime workers work either 2 PM to 7 PM or 3 PM to 8 PM and are paid 75 per day The numbers of checks re ceived each hour are listed in the file P04114xlsx In the interest of maintaining continuity Bank One believes that it must have at least three fulltime workers under contract Develop a work schedule that processes all checks by 8 PM and minimizes daily labor costs 115 OwensWheat uses two production lines to produce three types of fiberglass mat The demand require ments in tons for each of the next four months are shown in the file P04115xlsx If it were dedicated entirely to the production of one product a line 1 machine could produce either 20 tons of type 1 mat or 30 tons of type 2 mat during a month Similarly a line 2 machine could produce either 25 tons of type 2 mat or 28 tons of type 3 mat It costs 5000 per month to operate a machine on line 1 and 5500 per month to operate a machine on line 2 A cost of 2000 is incurred each time a new machine is pur chased and a cost of 1000 is incurred if a machine is retired from service At the end of each month Owens would like to have at least 50 tons of each product in inventory At the beginning of month 1 Owens has five machines on line 1 and eight ma chines on line 2 Assume the perton cost of holding either product in inventory for one month is 5 a Determine a minimum cost production schedule for the next four months b There is an important aspect of this situation that cannot be modeled by linear programming What is it Hint If Owens makes product 1 and prod uct 2 on line 1 during a month is this as efficient as making just product 1 on line 1 116 Rylon Corporation manufactures Brute cologne and Chanelle perfume The raw material needed to manufacture each type of fragrance can be purchased for 60 per pound Processing 1 pound of raw mater ial requires 1 hour of laboratory time Each pound of processed raw material yields 3 ounces of Regular Brute cologne and 4 ounces of Regular Chanelle perfume Regular Brute can be sold for 140 per ounce and Regular Chanelle for 120 per ounce Rylon also has the option of further processing Regu lar Brute and Regular Chanelle to produce Luxury Brute sold at 360 per ounce and Luxury Chanelle sold at 280 per ounce Each ounce of Regular Brute Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 208 Chapter 4 Linear Programming Models processed further requires an additional 3 hours of laboratory time and a 40 processing cost and yields 1 ounce of Luxury Brute Each ounce of Regular Chanelle processed further requires an additional 2 hours of laboratory time and a 40 processing cost and yields 1 ounce of Luxury Chanelle Each year Rylon has 6000 hours of laboratory time available and can purchase up to 4000 pounds of raw material a Determine how Rylon can maximize its profit Assume that the cost of the laboratory hours is a fixed cost so that it can be ignored for this problem b Suppose that 1 pound of raw material can be used to produce either 3 ounces of Brute or 4 ounces of Chanelle How does your answer to part a change c Use SolverTable to determine how a change in the price of Luxury Chanelle changes the optimal profit d Use SolverTable to determine how simultaneous changes in lab time and raw material availability change the optimal profit e Use SolverTable to determine how a change in the extra lab time required to process Luxury Brute changes the optimal profit 117 Sunco Oil has three different processes that can be used to manufacture various types of gasoline Each process involves blending oils in the companys catalytic cracker Running process 1 for an hour costs 20 and requires two barrels of crude oil 1 and three barrels of crude oil 2 The output from running process 1 for an hour is two barrels of gas 1 and one barrel of gas 2 Running process 2 for an hour costs 30 and requires one barrel of crude 1 and three bar rels of crude 2 The output from running process 2 for an hour is three barrels of gas 2 Running process 3 for an hour costs 14 and requires two barrels of crude 2 and three barrels of gas 2 The output from running process 3 for an hour is two barrels of gas 3 Each month 4000 barrels of crude 1 at 45 per bar rel and 7000 barrels of crude 2 at 55 per barrel can be purchased All gas produced can be sold at the following perbarrel prices gas 1 85 gas 2 90 gas 3 95 Determine how to maximize Suncos profit revenues less costs Assume that only 2500 hours of time on the catalytic cracker are avail able each month 118 Flexco produces six products in the following man ner Each unit of raw material purchased yields 4 units of product 1 2 units of product 2 and 1 unit of product 3 Up to 1200 units of product 1 can be sold and up to 300 units of product 2 can be sold Demand for products 3 and 4 is unlimited Each unit of product 1 can be sold or processed further Each unit of product 1 that is processed further yields 1 unit of product 4 Each unit of product 2 can be sold or processed further Each unit of product 2 that is processed further yields 08 unit of product 5 and 03 unit of product 6 Up to 1000 units of product 5 can be sold and up to 800 units of product 6 can be sold Up to 3000 units of raw material can be purchased at 6 per unit Left over units of products 5 and 6 must be destroyed It costs 4 to destroy each leftover unit of product 5 and 3 to destroy each leftover unit of product 6 Ignoring raw material purchase costs the unit price and production cost for each product are listed in the file P04118xlsx Determine a profitmaximizing production schedule for Flexco 119 Each week Chemco can purchase unlimited quanti ties of raw material at 6 per pound Each pound of purchased raw material can be used to produce either input 1 or input 2 Each pound of raw material can yield 2 ounces of input 1 requiring 2 hours of processing time and incurring 2 in processing costs Each pound of raw material can yield 3 ounces of input 2 requiring 2 hours of processing time and incurring 4 in processing costs Two production processes are available It takes 2 hours to run process 1 requiring 2 ounces of input 1 and 1 ounce of input 2 It costs 1 to run process 1 Each time process 1 is run 1 ounce of product A and 1 ounce of liquid waste are produced Each time process 2 is run requires 3 hours of processing time 2 ounces of input 2 and 1 ounce of input 1 Each process 2 run yields 1 ounce of product B and 08 ounce of liquid waste Process 2 incurs 8 in costs Chemco can dispose of liquid waste in the Port Charles River or use the waste to produce product C or product D Government regula tions limit the amount of waste Chemco is allowed to dump into the river to 5000 ounces per week Each ounce of product C costs 4 to produce and sells for 18 Producing 1 ounce of product C re quires 1 hour of processing time 2 ounces of input 1 and 08 ounce of liquid waste Each ounce of product D costs 5 to produce and sells for 12 Producing 1 ounce of product D requires 1 hour of processing time 2 ounces of input 2 and 12 ounces of liquid waste At most 7000 ounces of product A and 5000 ounces of product B can be sold each week but weekly demand for products C and D is unlimited Product A sells for 22 per ounce and product B sells for 24 per ounce Each week 25000 hours of pro cessing time are available Determine how Chemco can maximize its weekly profit 120 Bexter Labs produces three products A B and C Bexter can sell up to 2000 units of product A up to 2500 units of product B and up to 800 units of prod uct C Each unit of product C uses two units of A and three units of B and incurs 5 in processing costs Products A and B are produced from either raw ma terial 1 or raw material 2 It costs 6 to purchase and Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 49 Conclusion 209 process one unit of raw material 1 Each processed unit of raw material 1 yields two units of A and three units of B It costs 3 to purchase and process a unit of raw material 2 Each processed unit of raw mater ial 2 yields one unit of A and two units of B The unit prices for the products are A 5 B 4 C 25 The quality levels of each product are A 8 B 7 C 6 The average quality level of the units sold must be at least 7 Determine how to maximize Bexters profit 121 Based on Franklin and Koenigsberg 1973 The city of Busville contains three school districts The numbers of minority and nonminority students in each district are given in the file P04121xlsx The local court has decided that each of the towns two high schools Cooley High and Walt Whitman High must have approximately the same percentage of minority students within 5 as the entire town The distances in miles between the school districts and the high schools are also given in the same file Each high school must have an enrollment of 300 to 500 students Determine an assignment of students to schools that minimizes the total distance students must travel to school 122 Based on Carino and Lenoir 1988 Brady Corporation produces cabinets Each week Brady requires 90000 cubic feet of processed lumber The company can obtain lumber in two ways First it can purchase lumber from an outside supplier and then dry it at the Brady kiln Second Brady can chop down trees on its land cut them into lumber at its sawmill and then dry the lumber at its kiln The company can purchase grade 1 or grade 2 lumber Grade 1 lumber costs 3 per cubic foot and when dried yields 07 cubic foot of useful lumber Grade 2 lumber costs 7 per cubic foot and when dried yields 09 cubic foot of useful lumber It costs the company 3 to chop down a tree After being cut and dried a log yields 08 cubic feet of lumber Brady incurs costs of 4 per cubic foot of lumber it dries It costs 250 per cubic foot of logs sent through the sawmill Each week the sawmill can process up to 35000 cubic feet of lumber Each week up to 40000 cubic feet of grade 1 lumber and up to 60000 cubic feet of grade 2 lumber can be purchased Each week 40 hours of time are available for drying lumber The time it takes to dry one cubic foot of lumber is as follows grade 1 2 seconds grade 2 08 second log 13 seconds Determine how Brady can mini mize the weekly cost of meeting its demand for processed lumber 123 Based on Dobson and Kalish 1988 Chandler Enterprises produces two competing products A and B The company wants to sell these products to two groups of customers The values each customer places on a unit of A and B are shown in the file P04123xlsx Each customer will buy either product A or product B but not both A customer is willing to buy product A if she believes that the premium of product A is greater than or equal to the premium of product B and premium of product A is greater than or equal to 0 Here the premium of a product is its value minus its price Similarly a customer is willing to buy B if she believes the premium of product B is greater than or equal to the premium of product A and the premium of product B is greater than or equal to 0 Group 1 has 1000 members and group 2 has 1500 members Chandler wants to set prices for each product to ensure that group 1 members purchase product A and group 2 members purchase product B Determine how Chandler can maximize its revenue 124 Based on Robichek et al 1965 At the beginning of month 1 Finco has 4500 in cash At the beginning of months 1 2 3 and 4 Finco receives certain revenues after which it pays bills See the file P04124xlsx Any money left over can be invested for one month at the interest rate of 025 per month for two months at 028 per month for three months at 033 per month or for four months at 037 per month Determine an investment strategy that maximizes cash on hand at the beginning of month 5 125 During each sixhour period of the day the Blooming ton Police Department needs at least the number of police officers shown in the file P04125xlsx Police officers can be hired to work either 12 consecutive hours or 18 consecutive hours Police officers are paid 15 per hour for each of the first 12 hours they work in a day and 23 per hour for each of the next six hours they work in a day Determine how to minimize the cost of meeting Bloomingtons daily police requirements 126 Based on Glassey and Gupta 1978 A paper recy cling plant processes box board tissue paper newsprint and book paper into pulp that can be used to produce three grades of recycled paper The prices per ton and the pulp contents of the four inputs are shown in the file P04126xlsx Two methods deinking and asphalt dispersion can be used to process the four inputs into pulp It costs 20 to deink a ton of any input The process of deinking removes 10 of the inputs pulp leaving 90 of the original pulp It costs 15 to apply as phalt dispersion to a ton of material The asphalt dispersion process removes 20 of the inputs pulp At most 3000 tons of input can be run through the asphalt dispersion process or the deinking process Grade 1 paper can be produced only with newsprint or book paper pulp grade 2 paper only with book paper tissue paper or box board pulp and grade 3 paper only with newsprint tissue paper or box board pulp To meet its current demands the company Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it needs 500 tons of pulp for grade 1 paper 500 tons of pulp for grade 2 paper and 600 tons of pulp for grade 3 paper Determine how to minimize the cost of meeting the demands for pulp 127 At the beginning of month 1 GE Capital has 50 mil lion accounts Of these 40 million are paid up 0 due 4 million are 1 month overdue 1due 4 mil lion are 2 months overdue 2due and 2 million are 3 months overdue 3due After an account is more than 3 months overdue it is written off as a bad debt For each overdue account GE Capital can either phone the cardholder send a letter or do nothing A letter requires an average of 005 hour of labor whereas a phone call requires an average of 010 hour of labor Each month 500000 hours of labor are available We assume that the average amount of a monthly payment is 30 Thus if a 2due account remains 2due it means that 1 months payment 30 has been received and if a 2due account becomes 0due it means that 3 months payments 90 have been received On the basis of thousands of accounts DMMs Delinquency Movement Matrices shown in the file P04127xlsx have been estimated For example the topleft 060 entry in the first table means that 60 of all 1due accounts that receive a letter become 0due by the next month The 010 and 030 values in this same row mean that 10 of all 1due accounts remain 1due after receiving a letter and 30 of all 1due accounts become 2due after receiving a letter Your goal is to determine how to allocate your workforce over the next four months to maximize the expected collection revenue received during that time Note 0due accounts are never contacted which accounts for the lack of 0due rows in the first two tables 128 Three bonds as listed in the file P04128xlsx are currently for sale Each bond has a face value of 100 Every six months starting six months from the current date and ending at the expiration date each bond pays 05coupon rateFace value At the expiration date the face value is paid For example the second bond pays 275 six months from now 10275 a year from now Given the current price structure the question is whether there is a way to make an infinite amount of money To answer this you need to look for an arbi trage An arbitrage exists if there is a combination of bond sales and purchases today that yields a positive cash flow today nonnegative cash flows at all future dates If such a strategy exists then it is possible to make an infinite amount of money For example if buying 10 units of bond 1 today and selling 5 units of bond 2 today yielded say 1 today and nothing at all future 210 Chapter 4 Linear Programming Models dates you could make k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today You could also cover all payments at future dates from money received on those dates a Show that an arbitrage opportunity exists for the bonds in the file P04128xlsx Hint Set up an LP that maximizes todays cash flow subject to constraints that cash flow at each future date is nonnegative You should get a no convergence message from Solver b Usually bonds are bought at an ask price and sold at a bid price Consider the same three bonds as before and suppose the ask and bid prices are as listed in the same file Show that these bond prices admit no arbitrage opportunities Modeling Problems 129 You have been assigned to develop a model that can be used to schedule employees at a local fastfood restaurant Assume that computer technology has advanced to the point where very large problems can be solved on a PC at the restaurant a What data would you collect as inputs to your model b Describe in words several appropriate objective functions for your model c Describe in words the constraints needed for your model 130 You have been assigned to develop a model that can be used to schedule the nurses working in a mater nity ward a What data would you collect as inputs to your model b Describe in words several appropriate objective functions for your model c Describe in words the constraints needed for your model 131 Keefer Paper produces recycled paper from paper purchased from local offices and universities The company sells three grades of paper highbrightness paper mediumbrightness paper and lowbrightness paper The highbrightness paper must have a bright ness level of at least 90 the mediumbrightness paper must have a brightness level of between 80 and 90 and the lowbrightness paper must have a brightness level no greater than 80 Discuss how Keefer might use a blending model to maximize its profit 132 In this chapter we give you the cost of producing a product and other inputs that are used in the analysis Do you think most companies find it easy to deter mine the cost of producing a product What difficul ties might arise 133 Discuss how the aggregate planning model could be extended to handle a company that produces Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it several products on several types of machines What information would you need to model this type of problem 134 A large CPA firm currently has 100 junior staff members and 20 partners In the long runsay 20 years from nowthe firm would like to consist of 130 junior staff members and 20 partners During a given year 10 of all partners and 30 of all ju nior staff members leave the firm The firm can control the number of hires each year and the fraction of junior employees who are promoted to partner each year Can you develop a personnel strategy that would meet the CPA firms goals 49 Conclusion 211 135 The worker scheduling model in this chapter was purposely made small only seven changing cells What would make a similar problem for a company like McDonalds much harder What types of con straints would be required How many changing cells approximately might there be 136 Explain why it is problematic to include a constraint such as the following in an LP model for a blending problem Total octane in gasoline 1 blend Barrels of gasoline 1 blended daily Ú 10 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E S audi Arabia is a kingdom in the Middle East with an area of 865000 square miles occupying about fourfifths of the Arabian Peninsula With a popula tion of about 10 million this Muslim and Arab state is generally recognized as being formed in 1927 when Ibn Saud united the country and was acknowledged as the sovereign independent ruler Summer heat is intense in the interior reaching 124F but it is dry and tolerable in contrast to coastal regions and some highlands which have high humidity during the summer Winters December through February are cool with the coldest weather occurring at high altitudes and in the far north A minimum tempera ture recorded at atTurayf in 1950 was 10F and it was accompanied by several inches of snow and an inch of ice on ponds Average winter temperatures are 74F at Jidda and 58F at Riyadh the capital city which has an annual precipitation of 25 to 3 inches After oil was discovered in Bahrain in 1932 many companies turned to Saudi Arabia and started explor ing Thus in 1937 the American Arabian Oil Com pany Inc AMARCO was formed as a joint venture between Standard Oil Company of California SOCAL and the Government of Saudi Arabia to ex plore produce and market any petroleum found in the country The year before a geologist from SOCAL had discovered a small quantity of oil in the Eastern Province at Dammam Dome on which the oil company town of Dhahran is now built It was just beginning to be developed when another discovery was madeof what was to prove to be the largest oil field in the world Called the Ghamar field it would start Saudi Arabia on the road to becoming a highly developed country in just a generation Located about 50 miles inland from the western shores of the Persian Gulf the Ghamar field is a structural accumu lation along 140 miles of a northsouth anticline The productive area covers approximately 900 square miles and the vertical oil column is about 1300 feet It is generally considered to have recoverable re serves of about 75 billion barrels of oil Total proven reserves in Saudi Arabia are estimated at more than 500 billion barrels enough for more than a hundred years of production 41 AMARCO INC9 Since 1950 Saudi Arabia has experienced greater and more rapid changes than it had in the several preceding centuries For example during this time as skilled nationals became available more and more of the exploration drilling refining and other produc tion activities came under the control of the country SOCAL was left primarily with the marketing and transportation functions outside the country During the 1960s AMARCO increased its profitability substantially by hiring Dr George Dantzig then of the University of California as a consultant He supervised the development and implementation of LP models to optimize the production of different types of crude oils their refining and the marketing of some of their principal products As a result of this effort an operations research OR department was started in the company with the responsibility of continuing to review the firms operations to find other areas where costs might be decreased or profits increased by applications of OR Now attention is being focused on another aspect of one of the companys small California refinery operations the production of three types of aviation gasoline from the Saudi Arabian crude oil available Recently the marketing of petroleum products to the airline industry has become a rather substantial portion of AMARCOs business As shown in Figure 445 the three aviation gasolines A B and C are made by blending four feedstocks Alkylate Catalytic Cracked Gasoline Straight Run Gasoline and Isopentane In Table 414TEL stands for tetraethyl lead which is measured in units of milliliters per gallon mlgal Thus a TEL of 05 means there is 05 milliliter of tetraethyl lead per gallon of feedstock Table 414 shows that TEL does influence the octane number but does not influence the Reid vapor pressure Each type of aviation gasoline has a maximum permissible Reid vapor pressure of 7 Aviation gasoline A has a TEL level of 05 mlgal and has a minimum octane number of 80 The TEL level of aviation gasolines B and C is 4 mlgal but the former has a minimum octane number of 91 whereas the latter has a minimum of 100 9 This case was written by William D Whisler California State Uni versity Hayward 212 Chapter 4 Linear Programming Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Assume that all feedstocks going into aviation gasoline A are leaded at a TEL level of 05 mlgal and that those going into aviation gasolines B and C are leaded at a TEL level of 4 mlgal Table 415 gives the Aviation Gas A Refinery Crude Oil Aviation Gas C Alkylate Catalytic Cracked Gasoline Straight Run Gasoline Isopentane Aviation Gas B Figure 445 The Production of Aviation Gasoline Table 414 Stock Availabilitiesa Feedstock Catalytic Straight Cracked Run Characteristic Alkylate Gasoline Gasoline Isopentane Reid Vapor Pressure 5 8 4 20 Octane Number If TEL is 05 94 83 74 95 If TEL is 40 1075 93 87 108 Available Bblday 14000 13000 14000 11000 Value Bbl 1700 1450 1350 1400 aSome of the data in this case have been adapted from Walter W Garvin Introduction to Linear Programming New York McGrawHill 1960 Chapter 5 Table 415 Aviation Gasoline Data Aviation Gasoline Characteristic A B C Minimum requirements Bblday 12000 13000 12000 Price Bbl 1500 1600 1650 aviation gasoline data A final condition is that marketing requires that the amount of aviation gas A produced be at least as great as the amount of aviation gas B Case 41 AMARCO Inc 213 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Questions 1 AMARCOs planners want to determine how the three grades of aviation gasoline should be blended from the available input streams so that the specifications are met and income is maximized Develop an LP spreadsheet model of the companys problem 2 Solve the linear programming model devel oped in Question 1 The following questions should be attempted only after Questions 1 and 2 have been answered correctly 3 Suppose that a potential supply shortage of Saudi Arabian petroleum products exists in the near future due to possible damage to AMARCOs oil production facilities from Iraqi attacks This could cause the prices of the three types of aviation gasolines to double while the values of the stocks remain the same because they are currently on hand How would this affect the refinerys opera tions If after current stocks are exhausted additional quantities must be obtained at values double those given in Table 414 how might AMARCOs plans be affected 4 Suppose that because of the new Iraqi crisis the supply of alkylate is decreased by 1800 bblday catalytic cracked gas is decreased by 2000 bblday and straight run gasoline is decreased by 5000 bblday How does this affect AMARCOs operations 5 AMARCO is considering trying to fill the avia tion gasoline shortage created by the new Iraqi crisis by increasing its own production If addi tional quantities of alkylate catalytic cracked gasoline straight run gasoline and isopentane are available should they be processed If so how much of them should be processed and how do their values affect the situation 6 Due to the uncertainty about both the US economy and the world economy resulting from the Iraqi crisis AMARCOs economists are considering doing a new market research study to reestimate the minimum requirement forecasts With the economy continually weakening it is felt that demand will decrease possibly drastically in the future However because such marketing research is expensive management is wondering whether it would be worthwhile That is do changes in the minimum requirements have a significant effect on AMARCOs operations What is the change in profit from an increase or a decrease in the minimum requirements Over what ranges of demand do these profit changes apply 7 Suppose that the Middle East crisis ends and a flood of oil fills the marketplace causing the prices of aviation gasoline to drop to 1000 1100 and 1150 respectively for A B and C How would this affect the companys plans 8 Suppose that the US government is considering mandating the elimination of lead from aviation gasoline to decrease air pollution This law would be based on new technology that allows jet engines to burn unleaded gasoline efficiently at any octane level Thus there would no longer be any need for constraints on octane level How would such a new law affect AMARCO 9 The Environmental Protection Agency is propos ing regulations to decrease air pollution It plans to improve the quality of aviation gasolines by decreasing the requirement on Reid vapor pressure from 7 to 6 Management is concerned about this regulation and wonders how it might affect AMARCOs profitability Analyze and make a recommendation 10 The Marketing Department indicates that AMARCO will be able to increase its share of the market substantially with a new contract being negotiated with a new customer The difficulty is that this contract will require that the amount of aviation gas A plus the amount of B must be at least as great as the amount of C produced Because aviation gasolines A and B are least profitable of the three this could cause a big decrease in profit for the company However marketing indicates that this is a shortrun view because the large increase in market share with the concomitant longrun profit increases will more than offset the temporary small decrease in profits because of the additional restrictionWhat do you recommend Why 214 Chapter 4 Linear Programming Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it A merican Office Systems Inc was established by the late R J Miller Sr in 1939 It started as an office supply store in MountainView California and expanded slowly over the years into the manufacture of small office equipment overhead projectors and bookkeeping machines In the 1950s computers started eroding its market for bookkeeping machines so the company diversified into the copy machine market However it never captured a large market share because bigger firms such as Xerox Canon Sharp and A B Dick were so firmly entrenched A few years ago American Office Systems engineering staff developed an adapter that links a standard copy machine to personal computers allowing a copy machine to be used as a laser printer scanner and fax The adapters show great promise for both home and office use However the company is not well known by either the financial community or the copy machine market principally due to its small size and rather lackluster record so it could secure only 15 million in initial financial backing for the adapters The 15 million was used to finance the construction of a small production facility and of administrative offices in 1994 and in 1995 produc tion and sales began Two versions of the adapter exist one for PCs and one for Apple computers The former sells for 175 and the latter for 200 At the beginning of December 1995 Dr R J Miller II the president convened a meeting about the coming years plans for the adapters Rob OlsenVice President of Production argued that production facilities should be expanded Until we have sufficient capacity to produce the adapters he saidthere is no use advertising SueWilliams Director of Marketing repliedOn the contrary without any demand for the adapters there is no reason to produce them We need to focus on advertising first JT Howell the Comptroller pointed out that Olsen and Williams were talking about the situation as if it only involved a decision between production and marketing Yes funds need to be allocated between production and advertising However more important than both is the cash flow difficulty that the company has been experiencing As you know it was only yesterday that C A S E 42 AMERICAN OFFICE SYSTEMS INC10 finally I was able to secure a 750000 line of credit for the coming year from Citibank I might add that it is at a very favorable interest rate of 16 This will partially solve our cash flow problems and it will have a big effect on both production and advertising deci sions In addition there are financial and accounting factors that must be allowed for in any decision about the adapters Olsen interjected Wow this is more complicated than I anticipated originally Before we make a decision I think we ought to use some modern management science techniques to be sure that all the relevant factors are considered Last week I hired Carlos Garcia from Stanford He has a Masters degree in Operations Research I think this would be a good project for him However Williams said that she thinks that an executive judgmental decision would be much betterLets not get carried away with any of the quantitative mumbojumbo that Rob is always suggesting Besides his studies always take too much time and are so technical that no one can understand them We need a decision by the end of next week After listening to the discussion Miller decided to appoint an executive action team to study the problem and make a recommendation at next weeks meetingRob and Sue I want both of you to document your arguments in more detail JT be more precise with your comments about the cash flow accounting and financial problems And by the way Rob have Carlos look into a model to see if it might produce some insights Most of the 15 million initial financing was used to build a fivestory building in Mountain View south of San Francisco Although currently only about 90 complete it is being used The first floor contains the production and shipping facilities plus a small storage area A larger warehouse already owned by the company is located across the street The other four floors of the building are for the engineering depart ment second floor a research lab third floor and administration top two floors The production facility operates two shifts per day and has a produc tion capacity of 30 PC adapters and 10 Apple adapters per hour Olsen uses 20 production days per month in his planning Usually there are a few more but these are reserved for maintenance and repairs The last stage of the initial construction will be finished by the beginning of the fourth quarter 10 This case was written by William D Whisler California State University Hayward Case 42 American Office Systems Inc 215 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 216 Chapter 4 Linear Programming Models making the building 100 finished This will increase the production capacity rates by 10 Howell normally does the companys financial planning monthly and he assumes that cash flows as sociated with all current operating expenses sales revenues taking collections into account advertising costs loans from the line of credit investments of excess cash in shortterm government securities and so forth occur at the end of the corresponding month Because he needs information for the meeting next week however he decides to do a rough plan on a quarterly basis This means that all the just men tioned cash flows and so on will be assumed to occur at the end of the quarter After the meeting when more time is available the plan will be expanded to a monthly basis To get started one of his senior financial analysts prepares the list of quarterly fixed operating expenses shown in Table 416 In addition the accounting department calculates that the variable costs of the adapters are 100 each for the PC version and 110 each for the Apple version Table 416 Quarterly Fixed Operating Expenses Expense Cost Administrative expense 1500000 Fixed manufacturing costs 750000 Sales agents salaries 750000 Depreciation 100000 At present American Office Systems is experiencing a cash flow squeeze due to the large cash requirements of the startup of the adapter production advertising and sales costs If excess cash is available in any quarter however Howell says that the company policy is to invest it in shortterm government securities such as treasury bills He estimates that during the coming year these investments will yield a return of 6 Olsen asks Garcia to look into the production and inventory aspects of the situation first because this area was his specialty at Stanford Then he says that he wants him to think about a programming model that might integrate all components of the problemproduction sales advertising inventory accounting and finance A mixedinteger programming model appears to be the most appropriatehowever he asks Garcia to use linear programming as an approximation due to the time limitations and Williamss concern about his ideas always being too technical There will be more time after next weeks meeting to refine the model he says After discussions with Olsen and Williams Garcia feels that something needs to be done to help the company handle the uncertainty surrounding future sales of the adapters He points out that it is impossible to guarantee that the company will never be out of stock However it is possible to decrease shortages so that any difficulties associated with them would be small and not cause major disrup tions or additional management problems such as excess time and cost spent expediting orders Thus Garcia formulates an inventory model To be able to solve the model he has to check the inventory levels of the adapters currently on hand in the warehouse From these quantities he calculates that there will be 10000 PC and 5000 Apple adapters on hand at the beginning of 1996 Based on the results of the model he recommends that a simple rule of thumb be used production plus the endofperiod inventory for the adapters should be at least 10 larger than the estimated sales for the next period This would be a safety cushion to help prevent shortages of the adapters In addition to provide a smooth transition to 1997 the inventory level plus production at the end of the fourth quarter of 1996 should be at least twice the maximum expected sales for that quarter Garcia says that using these rules of thumb will minimize annual inventory costs When explaining the inventory model to Olsen Garcia emphasizes the importance of including inventory carrying costs as part of any analysis even though such costs frequently are not outofpocket He says that his analysis of data provided by the accounting depart ment yielded a 1 per month inventory carry cost and this is what he used in his model Sales during the first year 1995 for the adapters are shown in Table 417 Next years sales are uncertain One reason for the uncertainty is that they depend on the advertising To begin the analysis Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Williams asks her marketing research analyst Debra Lu to estimate the maximum sales levels for the coming four quarters if no advertising is done Since last years sales of both models showed a steady increase throughout the year Lu projects a continuation of the trend She forecasts that the company will be able to sell any number of adapters up to the maximum expected sales amounts shown in Table 417 Table 417 1995 Adapter Sales and Maximum Expected 1996 Sales 1996 Maximum 1995 Sales Expected Sales PC Apple PC Apple Quarter Adapters Adapters Adapters Adapters 1 5000 1000 9000 1800 2 6000 1200 10000 2000 3 7000 1400 11000 2200 4 8000 1600 12000 2400 Miller suggests that advertising in magazines such as PC World and Home Office will increase consumer awareness of both the company and the adapters The next day Williams has a meeting with several staff members of a San Francisco advertising agency They show her recommendations for two types of ads one for the PC adapters and one for the Apple adapters give her cost information and the estimated effectiveness of an advertising cam paign Armed with this information and some data from Lu Williams prepares a brief report for Miller setting out her reasons for thinking that each 10 spent on advertising will sell an additional PC adapter the same relationship holds true for the Apple adapter Based on an analysis of 1995 sales and accounts receivable the accounting department determines that collection experience is as shown in Table 418 For example 75 of the PC adapters sold in a quarter are paid for during the quarter 20 are paid for during the following quarter and 3 are paid for during the third quarter The remaining 2 are written off and sold to a collection agency for 050 on the dollar Table 418 Collections Quarter PC Adapters Apple Adapters 1 075 080 2 020 011 3 003 005 Questions 1 Suppose that you are Garcia Develop an LP spreadsheet model of the situation to help the executive action team make a decision about how to allocate funds between production and advertising so that all the cash flow financial accounting marketing inventory and production considerations are taken into account and American Office Systems profits are maximized Use the data collected and the estimates made by the members of the executive action team 2 Solve the LP model formulated in Question 1 The executive action team has assembled to reconsider the plans for the adapters for the coming year Garcia who developed the LP model concludes his presentation by sayingAs everyone can see the model gives the optimal solution that maximizes profits Since I have incorporated the estimates and assumptions that all of you made clearly it is the best solution No other alternative can give a higher profit EvenWilliams who initially was skeptical of using quantitative models for making executivelevel decisions is impressed and indicates that she will go along with the results Miller saysGood work Carlos This is a complex problem but your presentation made it all seem so simple However remember that those figures you used were based on estimates made by all of us Some were little bet ter than guesses What happens if they are wrong In other words your presentation has helped me get a handle on the problem we are facing and I know that models are useful where hard accurate data exist However with all the uncertainty in our situation and the many rough esti mates made it seems to me that I will still have to make a judgment call when it comes down to making a final decision Also there has been a new development JT tells me that we might be able to get another 1 million Case 42 American Office Systems Inc 217 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 218 Chapter 4 Linear Programming Models line of credit from a Bahamian bank It will take a while to work out the details and maybe it will cost us a little I am wondering if it is worth it What would we do with the 1 million if we got it J T respondsWe really need the 1 million But it is a drop in the bucket My analysis shows that we really need another 8 million line of credit Analyze as Garcia is going to do the effect of uncer tainty and errors on the results of Questions 1 and 2 by answering the following questionsThey should be attempted only after Questions 1 and 2 have been answered correctly 3 One area where assumptions were made is adapter price a What happens if the prices for the adapters are a little weak and they decrease to 173 for the PC version and 198 for the Apple version Does this make any difference b What about decreases to 172 and 197 respectively for the PC and Apple versions Explain the answers in terms that Miller will understand c Suppose that American Office Systems can increase the price of the adapters to 180 and 205 How would this affect the original solution 4 Another potential variable is adapter production cost a Suppose that an error was made in determin ing the costs of the adapters and that they really should have been 102 for the PC version and 112 for the Apple versionWhat is the effect of this error b What about costs of 105 and 115 Explain the answers in terms that Miller will understand 5 Howell notes that one of the contributing factors to American Office Systems cash squeeze is the slow collection of accounts receivable He is con sidering adopting a new collection procedure recommended by a consulting company It will cost 100000 and will change the collection rates to those given in Table 419 a Analyze the effect of this new collection policy and make a recommendation to Howell about whether to implement the new proce dure As before any accounts receivable not collected by the end of the third quarter will be sold to a collection agency for 050 on the dollar b Howell wonders whether switching to selling adapters for all cash is worth the effort This would ameliorate the cash squeeze because it would eliminate not only the slow collections but also the use of the collection agency for accounts that remain unpaid after nine months It would cost about 90000 more than at present to implement the allcash policy because the accounting system would need to be modified and personnel would have to be retrained Analyze this possibility and make a recommendation to Howell Table 419 New Collections Quarter PC Adapters Apple Adapters 1 090 092 2 007 003 3 001 001 6 Yet another variable is advertising effectiveness a Suppose that Williams overestimated the effectiveness of advertising It now appears that 100 is needed to increase sales by one adapter How will this affect the original solution Explain the answer in terms that Miller will understand b What happens if the required advertising outlay is 1250 per additional adapter sold 7 Suppose that the line of credit from Citibank that Howell thought he had arranged did not work out because of the poor financial situation of the company The company can obtain one for the same amount from a small local bank however the interest rate is much higher 24 Analyze how this change affects American Office Systems 8 The safety cushion for inventory is subject to revision a Suppose that Garcia finds a bug in his original inventory model Correcting it results in a Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it safety cushion of 15 instead of the 10 he suggested previously Determine whether this is important b What if the error is 20 Explain the answers in terms that Miller will understand 9 Production capacity is scheduled to increase by 10 in the fourth quarter a Suppose that Miller is advised by the construction company that the work will not be finished until the following year How will this delay affect the companys plans b In addition to the delay in part a suppose that an accident in the production facility damages some of the equipment so that the capacity is decreased by 10 in the fourth quarter Analyze how this will affect the original solution 10 Williams is worried about the accuracy of Lus 1996 maximum expected sales forecasts If errors in these forecasts have a big effect on the company profits she is thinking about hiring a San Francisco marketing research firm to do a more detailed analysis They would charge 50000 for a study Help Williams by analyzing what would happen if Lus forecasts are in error by 1000 for PC adapters and 200 for Apple adapters each quarter Should she hire the mar keting research firm 11 a To determine whether the extra 1 million line of credit is needed analyze its effect on the original solution given in Question 2 b To fully understand the ramifications of the extra 1000000 line of credit redo 1 Question 3b 2 Question 4b 3 Question 6a and 4 Question 8b Summarize your results c What about Howells claim that an extra 8000000 line of credit is necessary Use that adjustment and redo Question 6a Case 42 American Office Systems Inc 219 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it L akefield Corporations oil trading desk buys and sells oil products crude oil and refined fuels options and futures in international markets The trading desk is responsible for buying raw material for Lakefields refining and blending operations and for selling final products In addition to trading for the companys operations the desk also takes speculative positions In speculative trades the desk attempts to profit from its knowledge and informa tion about conditions in the global oil markets One of the traders Lisa Davies is responsible for transactions in the cash market as opposed to the futures or options markets Lisa has been trading for several years and has seen the prices of oilrelated products fluctuate tremendously Figure 446 shows the prices of heating oil 2 and unleaded gasoline from January 1986 through July 1992 Although ex cessive volatility of oil prices is undesirable for most businesses Lakefields oil trading desk often makes substantial profits in periods of high volatility The prices of various oil products tend to move together over long periods of time Because finished oil products are refined from crude oil the prices of all finished products tend to rise if the price of crude increases Because finished oil products are not per fect substitutes the prices of individual products do not move in lockstep In fact over short time peri ods the price movements of two products can have a low correlation For example in late 1989 and early 1990 there was a severe cold wave in the north eastern United States The price of heating oil rose from 060 per gallon to over 1 per gallon In the same time period the price of gasoline rose just over 010 per gallon Davies believes that some mathematical analysis might be helpful to spot trading opportunities in the cash markets The next section provides background about a few important characteristics of fuel oils along with a discussion of the properties of blended fuels and some implications for pricing Characteristics of Hydr ocarbon Fuels The many varieties of hydrocarbon fuels include heating oil kerosene gasoline and diesel oil Each type of fuel has many characteristics for example heat content viscosity freezing point luminosity C A S E 43 LAKEFIELD CORPORATIONS OIL TRADING DESK 20 40 60 80 100 120 Jan86 Jan89 Jan92 Jan87 Jan88 Jan90 Jan91 Heating Oil 2 Unleaded Gasoline Date Price centsgallon Figure 446 Price of Heating Oil 2 and Unleaded Gasoline 220 Chapter 4 Linear Programming Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Questions 1 Suppose that 03 barrel of fuel 2 03 barrel of fuel 3 and 04 barrel of fuel 4 are blended togetherWhat is the cost of the blended fuel What are the linear properties of the blended fuel ie density linear viscosity sulfur content and linear flash point 2 Using the data from Table 421 check whether any of the fuels violate the noarbitrage pricing condition If no fuel violates the condition which fuels price comes the closest to the noarbitrage upper bound If there is a violation give the explicit recipe 3 What modifications would you make to the analysis to account for blending costs 4 What would be the important issues or steps involved in creating a real system for this problem 224 Chapter 4 Linear Programming Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 227 Network Models C H A P T E R RESTRUCTURING BASF NORTH AMERICAS DISTRIBUTION SYSTEM A quick look through Interfaces the journal that chronicles management science success stories from real applications indicates that many of these success stories involve network optimization the subject of this chapter A typical example appears in Sery et al 2001 The authors describe their efforts to restructure BASF North Americas distribution system The BASF Group with headquarters in Germany is one of the worlds leading chemical companies with annual sales over 30 billion and more than 100000 employees worldwide BASF offers a variety of chemical and chemicalbased products to customers in Europe the NAFTA region South America and Asia You probably know the company from its catchy sloganWe dont make a lot of the products you buy We make a lot of the products you buy better Its diverse product mix includes chemicals polymers automotive coatings colors dyes pharmaceuticals nylon fibers and agricultural products In the mid1990s BASF examined its distribution of packaged goods in the North America region and discovered that it shipped 16 billion pounds of finished goods annually to customers from a network of 135 locations at an annual cost including transportation and warehousing of nearly 100 million The majority 86 of the 135 locations were distribution 5 FRANK RUMPENHORSTDPALandov Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it centers DCs although almost a billion pounds were shipped directly from plants to customers Unfortunately there had never been any systematic attempt to optimize this network configuration it had just evolved over the years The authors of the study were asked to make recommendations that would 1 decrease logistics costs and 2 increase customer service defined as the percentage of shipments that reach the customer on the same day or the next day This percentage was about 77 before the study The authors developed a linear programming model that when implemented was able to 1 reduce the number of DCs from 86 to 12 2 reduce the annual trans port facility and inventory carrying costs by 6 3 achieve a onetime 9 improve ment in cash flows from a reduction in the working capital tied up in inventory and 4 increase the customer service measure to 90 The redesign worked so well that BASF later developed similar models for its European Scandinavian and Far East distribution systems The articles description of the study is a virtual textbook example of the modeling process described in Chapter 1 of this book The problem was first identified as fol lowsDefine the optimal number and location of warehouses and the corresponding material flows needed to meet anticipated customer demand and required delivery ser vice times at the lowest overall cost The project team next performed the arduous task of collecting the various demands and costs required for the optimization model Although we try to indicate Where Do the Numbers Come From in the examples in this book the authors of the study describe just how difficult data collection can be particularly when the data is stored in a variety of legacy systems that use a wide range of data definitions Next the authors developed a verbal statement of the model including all assumptions they made which was then translated in a straightforward manner into the network optimization model itself The next step was to build a deci sion support system to implement the model This userfriendly system allowed BASF management to become comfortable with the model and learn to trust it by running it repeatedly under different scenarios to answer all sorts of whatif questions Finally the models recommendations were used to redesign the distribution system in North America and an honest evaluation of its effectsreduced costs and increased customer servicewas made 228 Chapter 5 Network Models 51 INTRODUCTION Many important optimization models have a natural graphical network representation In this chapter we discuss some specific examples of network models There are several rea sons for distinguishing network models from other LP models The network structure of these models allows them to be represented graphically in a way that is intuitive to users This graphical representation can then be used as an aid in the spreadsheet model development In fact for a book at this level the best argu ment for singling out network problems for special consideration is the fact that they can be represented graphically Many companies have real problems often extremely large that can be represented as network models In fact many of the best management science success stories have involved large network models For example Delta Airlines developed a network model to schedule its entire fleet of passenger airplanes A few other real applications of networkbased models are listed throughout the chapter but the list is by no means exhaustive A quick scan of the articles in the Interfaces journal indicates that there are probably more networkbased applications reported than any other type Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Specialized solution techniques have been developed specifically for network models Although we do not discuss the details of these solution techniquesand they are not implemented in Excels Solverthey are important in realworld appli cations because they allow companies to solve huge problems that could not be solved by the usual LP algorithms 52 TRANSPORTATION MODELS In many situations a company produces products at locations called origins and ships these products to customer locations called destinations Typically each origin has a limited amount that it can ship and each customer destination must receive a required quantity of the product Spreadsheet optimization models can be used to determine the minimumcost shipping plan for satisfying customer demands For now we assume that the only possible shipments are those directly from an origin to a destination That is no shipments between origins or between destinations are possi ble This problemgenerally called the transportation pr oblemhas been studied extensively in management science In fact it was one of the first management science models developed more than a half century ago The following is a typical example of a small transportation problem 52 Transportation Models 229 E X A M P L E 51 SHIPPING CARS FROM PLANTS TO REGIONS OF THE COUNTRY T he Grand Prix Automobile Company manufactures automobiles in three plants and then ships them to four regions of the country The plants can supply the amounts listed in the right column of Table 51 The customer demands by region are listed in the bottom row of this table and the unit costs of shipping an automobile from each plant to each region are listed in the middle of the table Grand Prix wants to find the lowestcost shipping plan for meeting the demands of the four regions without exceeding the capacities of the plants Table 51 Input Data for Grand Prix Example Region 1 Region 2 Region 3 Region 4 Capacity Plant 1 131 218 266 120 450 Plant 2 250 116 263 278 600 Plant 3 178 132 122 180 500 Demand 450 200 300 300 Objective To develop a spreadsheet optimization model that finds the leastcost way of shipping the automobiles from plants to regions staying within plant capacities and meet ing regional demands WHERE DO THE NUMBERS COME FROM A typical transportation problem requires three sets of numbers capacities or supplies demands or requirements and unit shipping and possibly production costs We discuss each of these next The capacities indicate the most each plant can supply in a given amount of timea month sayunder current operating conditions In some cases it might be possible to Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it increase the base capacities by using overtime for example In such cases the model could be modified to determine the amounts of additional capacity to use and pay for The customer demands are typically estimated from some type of forecasting model as discussed in Chapter 14 The forecasts are often based on historical customer demand data The unit shipping costs come from a transportation cost analysiswhat does it really cost to send a single automobile from any plant to any region This is not an easy question to answer and it requires an analysis of the best mode of transportation such as railroad ship or truck However companies typically have the required data Actually the unit shipping cost can also include the unit production cost at each plant However if this cost is the same across all plants as we are tacitly assuming here it can be omitted from the model Solution The variables and constraints required for this model are listed in Table 52 The company must decide exactly the number of autos to send from each plant to each regiona ship ping plan Then it can calculate the total number of autos sent out of each plant and the total number received by each region 230 Chapter 5 Network Models Table 52 Variables and Constraints for Transportation Model Input variables Plant capacities regional demands unit shipping costs Decision variables changing cells Number of autos sent from each plant to each region Objective cell Total shipping cost Other calculated variables Number sent out of each plant number sent to each region Constraints Number sent out of each plant Plant capacity Number sent to each region Ú Region demand In a transportation problem all flows go from left to right from origins to destinationsYou will see more complex network structures in the next subsection Figure 51 Network Representation of Transportation Model Representing Transportation in a Network Model A network diagram of this model appears in Figure 51 This diagram is typical of network models It consists of nodes and arcs A node indicated by a circle generally represents a geographical location In this case the nodes on the left correspond to plants and the nodes on the right correspond to regions An arc indicated by an arrow generally represents a route for getting a product from one node to another Here the arcs all go from a plant node to a region nodefrom left to right Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This formula sums all products of unit shipping costs and amounts shipped You now see the benefit of placing unit shipping costs and amounts shipped in similarsize rectangular rangesyou can then use the SUMPRODUCT function USING SOLVER Invoke Solver with the settings shown in Figure 53 As usual check the NonNegative option and specify the Simplex LP method before optimizing 232 Chapter 5 Network Models It is typical in transportation models especially large models that only a relatively few of the possible routes are used Figure 53 Solver Dialog Box for Transportation Model Discussion of the Solution The Solver solution appears in Figure 52 and is illustrated graphically in Figure 54 The company incurs a total shipping cost of 176050 by using the shipments listed in Figure 54 Except for the six routes shown no other routes are used Most of the shipments occur on the lowcost routes but this is not always the case For example the route from plant 2 to region 1 is relatively expensive and it is used On the other hand the route from plant 3 to region 2 is relatively cheap but it is not used A good shipping plan tries to use cheap routes but it is constrained by capacities and demands Note that the available capacity is not all used The reason is that total capacity is 1550 whereas total demand is only 1250 Even though the demand constraints are of the type there is clearly no reason to send the regions more than they request because it only increases shipping costs Therefore the optimal plan sends them the minimal Ú Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it selling prices are large enough that every automobile sale adds to aftertax profit so the company sells as many as it can Of course this raises the question of how many automo biles each region can really sell It might be more realistic to keep the lower bounds on sales the current demand constraints but to impose upper limits on sales as well We ask you to explore this in one of the problems 240 Chapter 5 Network Models Figure 510 Solver Dialog Box for the Extended Grand Prix Model P R O B L E M S Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 In the original Grand Prix example the total capacity of the three plants is 1550 well above the total customer demand Would it help to have 100 more units of capacity at plant 1 What is the most Grand Prix would be willing to pay for this extra capacity Answer the same questions for plant 2 and for plant 3 Explain why extra capacity can be valuable even though the company already has more total capacity than it requires 2 The optimal solution to the original Grand Prix problem indicates that with a unit shipping cost of 132 the route from plant 3 to region 2 is evidently too expensiveno autos are shipped along this route Use SolverTable to see how much this unit shipping cost would have to be reduced before some autos would be shipped along this route 3 Suppose in the original Grand Prix example that the routes from plant 2 to region 1 and from plant 3 to region 3 are not allowed Perhaps there are no rail road lines for these routes How would you modify the original model Figure 52 to rule out these routes How would you modify the alternative model Figure 57 to do so Discuss the pros and cons of these two approaches 4 In the Grand Prix example with varying tax rates the optimal solution more than satisfies customer demands Modify the model so that regions have not only lower limits on the amounts they require but upper limits on the amounts they can sell Assume these upper limits are 50 autos above the required lower limits For example the lower and upper limits for region 1 are 450 and 500 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 53 Assignment Models 243 Figure 513 Solver Dialog Box for the Assignment Model E X A M P L E 53 ASSIGNING SCHOOL BUSES TO ROUTES AT SPRING VIEW T he city of Spring View is taking bids from six bus companies on the eight routes that must be driven in the surrounding school district Each company enters a bid of how much it will charge to drive selected routes although not all companies bid on all routes The data are listed in Table 55 If a company does not bid on a route the corresponding entry is blank The city must decide which companies to assign to which routes with the specifications that 1 if a company does not bid on a route it cannot be assigned to that route 2 exactly one company must be assigned to each route and 3 a company can be assigned to at most two routes The objective is to minimize the total cost of covering all routes Table 55 Bids on Bus Routes Company Route 1 Route 2 Route 3 Route 4 Route 5 Route 6 Route 7 Route 8 1 8200 7800 5400 3900 2 7800 8200 6300 3300 4900 3 4800 4400 5600 3600 4 8000 5000 6800 6700 4200 5 7200 6400 3900 6400 2800 3000 6 7000 5800 7500 4500 5600 6000 4200 Objective To use a network model to assign companies to bus routes so that each route is covered at minimum cost to the city and no company is assigned to more than two routes Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Discussion of the Solution The optimal solution in Figure 514 indicates that the city should make the following assignments company 1 covers bus route 3 company 2 covers bus routes 6 and 7 company 3 covers bus route 2 company 5 covers bus routes 4 and 8 and company 6 covers bus routes 1 and 5 The total cost to the city of this assignment is 40300 Note that company 4 is not assigned to any bus routes There is no constraint that every company must be assigned to at least one bus route and company 4 is evidently underbid by at least one company for all bus routes If the city wanted to require that all companies be assigned to at least one bus route you would simply add a lower bound of 1 on all of the outflows from the company nodes in rows 8 to 13 Of course this would probably increase the total cost to the city Sensitivity Analysis One interesting sensitivity analysis is to see what effect the upper bound constraint on the maximum routes has on the total cost Presumably if more bus routes per company are allowed assuming this is physically possible for the companies the companies who tend to bid lowest will be assigned to the bulk of the bus routes and the total cost will probably decrease Using SolverTable the analysis itself is straightforward with no modifications to the model necessary You should specify cell B4 as the single input cell allow it to vary say from 1 to 7 in increments of 1 and keep track of total cost The resulting output appears in Figure 516 If each company can be assigned to only one route there is no feasible solution But the reason for this is clear There are eight routes to cover and only six companies For larger values of the maximum routes allowed the total cost begins to decrease but only until this 246 Chapter 5 Network Models Figure 515 Solver Dialog Box for the Bus Route Assignment Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 54 Other Logistics Models 249 FUNDAMENTAL INSIGHT Flow Balance Constraints All network optimization models have some form of flow balance constraints at the various nodes of the network This flow balance r elates the amount that enters the node to the amount that leaves the node In man y netw ork models the simple structur e of these flo w balance constraints guarantees that the optimal solutions have integer values It also enables specialized network versions of the simplex method to solve the huge netw ork models typically encoun tered in real logistics applications E X A M P L E 54 PRODUCING AND SHIPPING TOMATO PRODUCTS AT REDBRAND T he RedBrand Company produces a tomato product at three plants This product can be shipped directly to the companys two customers or it can first be shipped to the companys two warehouses and then to the customers Figure 517 is a network representation of RedBrands problem Nodes 1 2 and 3 represent the plants these are the origins denoted by S for supplier nodes 4 and 5 represent the warehouses these are the transshipment points denoted by T and nodes 6 and 7 represent the customers these are the destinations denoted by D Note that some shipments are allowed among plants among warehouses and among customers Also some arcs have arrows on both ends This means that flow is allowed in either direction Figure 517 Graphical Representation of Logistics Model The cost of producing the product is the same at each plant so RedBrand is concerned with minimizing the total shipping cost incurred in meeting customer demands The pro duction capacity of each plant in tons per year and the demand of each customer are shown in Figure 517 For example plant 1 node 1 has a capacity of 200 and customer 1 node 6 has a demand of 400 In addition the cost in thousands of dollars of shipping a ton of the product between each pair of locations is listed in Table 57 where a blank indi cates that RedBrand cannot ship along that arc We also assume that at most 200 tons of the product can be shipped between any two nodes This is the common arc capacity RedBrand wants to determine a minimumcost shipping schedule Objective To find the minimumcost way to ship the tomato product from suppliers to customers possibly through warehouses so that customer demands are met and supplier capacities are not exceeded Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 252 Chapter 5 Network Models Figure 519 Solver Dialog Box for Logistics Model USING SOLVER The Solver dialog box should be set up as in Figure 519 The objective is to minimize total shipping costs subject to the three types of flow balance constraints and the arc capacity constraints Discussion of the Solution The optimal solution in Figure 518 indicates that RedBrands customer demand can be sat isfied with a shipping cost of 3260000 This solution appears graphically in Figure 520 Note in particular that plant 1 produces 180 tons under capacity and ships it all to plant 3 not directly to warehouses or customers Also note that all shipments from the warehouses go directly to customer 1 Then customer 1 ships 180 tons to customer 2 We purposely chose Figure 520 Optimal Flows for Logistics Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 54 Other Logistics Models 255 Figure 523 Solver Dialog Box for TwoProduct Logistics Model in cell I16 and copying to cell I17 This formula says that what goes out the first term is 90 of what goes in The other 10 perishes Of course shrinkage results in a larger total costabout 20 largerthan in the original RedBrand model Interestingly however some units are still sent to both warehouses and the entire capacity of all plants is now used Finally you can check that a feasible solution exists even for a shrinkage factor of 0 where everything sent to warehouses disappears As you might guess the solution then is to send everything directly from plants to customersat a steep cost 1 Excels Solver uses the simplex method to solve logistics models However the simplex method can be simplified dramatically for these types of models The simplified ver sion of the simplex method called the network simplex method is much more efficient than the ordinary simplex method Specialized computer codes have been written to implement the network simplex method and all large logistics problems are solved by using the network simplex method This is fortunate because real logistics models tend to be extremely large See Winston 2003 for a discussion of this method 2 If the given supplies and demands for the nodes are integers and all arc capacities are integers the logistics model always has an optimal solution with all integer flows Again this is very fortunate for large problemsyou get integer solutions for free without having to use an integer programming algorithm Note however that this integers for free benefit is guaranteed only for the basic logistics model as in MODELING ISSUES Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 55 Shortest Path Models 257 55 SHORTEST PATH MODELS In many applications the objective is to find the shortest path between two points in a net work Sometimes this problem occurs in a geographical context where for example the objective is to find the shortest path on interstate freeways from Seattle to Miami There are also problems that do not look like shortest path problems but can be modeled in the same way We look at one possibility where the objective is to find an optimal schedule for replacing equipment The typical shortest path problem is a special case of the network flow problem from the previous section To see why this is the case suppose that you want to find the shortest path between node 1 and node N in a network To find this shortest path you create a net work flow model where the supply for node 1 is 1 and the demand for node N is 1 All other nodes are transshipment nodes If an arc joins two nodes in the network the ship to each other Modify the model appropriately and reoptimize How much does the total cost increase because of these disallowed routes 21 In the original RedBrand problem the costs for ship ping from plants or warehouses to customer 2 were purposely made high so that it would be optimal to ship to customer 1 and then let customer 1 ship to customer 2 Use SolverTable appropriately to do the following Decrease the unit shipping costs from plants and warehouses to customer 1 all by the same amount until it is no longer optimal for customer 1 to ship to customer 2 Describe what happens to the optimal shipping plan at this point 22 In the original RedBrand problem we assume a con stant arc capacity the same for all allowable arcs Modify the model so that each arc has its own arc capacity You can make up the required arc capacities 23 Continuing the previous problem make the problem even more general by allowing upper bounds arc capacities and lower bounds for the flows on the allowable arcs Some of the upper bounds can be very large numbers effectively indicating that there is no arc capacity for these arcs and the lower bounds can be zero or positive If they are positive then they indi cate that some positive flow must occur on these arcs Modify the model appropriately to handle these upper and lower bounds You can make up the required bounds 24 Expand the RedBrand twoproduct spreadsheet model so that there are now three products competing for the arc capacity You can make up the required input data 25 In the RedBrand twoproduct problem we assumed that the unit shipping costs are the same for both products Modify the spreadsheet model so that each product has its own unit shipping costs You can assume that the original unit shipping costs apply to product 1 and you can make up new unit shipping costs for product 2 SkillExtending Problems 26 How difficult is it to expand the original RedBrand model Answer this by adding a new plant two new warehouses and three new customers and modify the spreadsheet model appropriately You can make up the required input data 27 In the RedBrand problem with shrinkage change the assumptions Now instead of assuming that there is some shrinkage at the warehouses assume that there is shrinkage in delivery along each route Specifically assume that a certain percentage of the units sent along each arc perish in transitfrom faulty refrigera tion sayand this percentage can differ from one arc to another Modify the model appropriately to take this type of behavior into account You can make up the shrinkage factors and you can assume that arc capacities apply to the amounts originally shipped not to the amounts after shrinkage Make sure your input data permit a feasible solution After all if there is too much shrinkage it will be impossi ble to meet demands with available plant capacity Increase the plant capacities if necessary 28 Consider a modification of the original RedBrand problem where there are N plants M warehouses and L customers Assume that the only allowable arcs are from plants to warehouses and from warehouses to customers If all such arcs are allowableall plants can ship to all warehouses and all warehouses can ship to all customershow many changing cells are in the spreadsheet model Keeping in mind that Excels Solver can handle at most 200 changing cells give some combinations of N M and L that will just barely stay within Solvers limit 29 Continuing the previous problem develop a sample model with your own choices of N M and L that barely stay within Solvers limit You can make up any input data The important point here is the layout and formulas of the spreadsheet model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it shipping cost is equal to the length of the arc The flow through each arc in the net work in the optimal solution is either 1 or 0 depending on whether the shortest path includes the arc No arc capacities are required in the model The value of the objective is then equal to the sum of the distances of the arcs involved in the path 258 Chapter 5 Network Models Geographical Shortest Path Models The following example illustrates the shortest path model in the context of a geographic network FUNDAMENTAL INSIGHT Shortest Path Problems as Network Flow Models Shortest route problems can be modeled as a special case of more general logistics modelsusing a supply of 1 at the origin node and a demand of 1 at the destination node Because specialized algorithms can solve these more general models very quickly short est r oute pr oblems inherit this attractiv e f eature This is a fa vorite trick of management scientists They always try to model a specific problem as a spe cial case of a mor e general pr oblem that has been well studied and can be solved relatively easily E X A M P L E 55 SHORTEST WALK ACROSS THE STATE M aude Jenkins a 90yearold woman is planning to walk across the state west to east to gain support for a political cause she favors6 She wants to travel the shortest dis tance to get from city 1 to city 10 using the arcs roads shown in Figure 525 The numbers on the arcs are miles Arcs with doubleheaded arrows indicate that travel is possible in both directions with the same distance in both directions What route should Maude take 6This is based on a real 90yearold woman who reportedly decided to walk across the country We assume she finished 1 79 18 54 69 70 63 56 19 29 25 50 73 67 72 17 31 72 87 97 69 15 51 52 69 61 67 45 85 2 4 3 6 9 5 7 8 10 Figure 525 Network for the Shortest Path Problem Objective To specialize the general network flow model so that a shortest path from node 1 to node 10 can be found WHERE DO THE NUMBERS COME FROM The distances on the arcs are presumably listed on state maps for the roads Maude is consid ering Note however that in shortest path problems such as this the objective is sometimes total cost not distance Although the cost of an arc might be proportional to its distance it might not be For example a steep uphill route might be more costly than a flat stretch of similar length In such cases the arc costs would be somewhat more difficult to obtain The distancesin shortest path models are sometimes costs Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 260 Chapter 5 Network Models Figure 527 Solver Dialog Box for the Shortest Path Model 1 Arc list There is an arc listed in columns A and B for each arc in the graphical net work If the arc goes in both directions it is listed twice 2 to 4 and 4 to 2 for example with the same distance in both directions 2 Net outflows All types of nodes are listed in a single block in the flow balance con straint section Node 1 is an origin with a supply of 1 and it has only outflows Similarly node 10 is a destination with demand 1 and it has only inflows The intermediate nodes are all transshipment nodes You can treat all of the nodes similarly by calculating the net outflow from each To do so enter the formula SUMIFOriginF5FlowSUMIFDestinationF5Flow in cell G5 and copy it down for the other nodes For node 1 this net outflow is really just the outflow so it should be 1 For node 10 this net outflow is really just the negative of the inflow so it should be 1 For all intermediate nodes the net outflow should be 0 This explains the values in column I 3 Total distance The objective to minimize is total distance calculated in cell B42 with the formula SUMPRODUCTDistanceFlow Discussion of the Solution After Solver finds the optimal flows which are 0s and 1s it is easy to identify the shortest pathjust follow the 1s According to Figure 526 Maude first goes from node 1 to node 4 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 55 Shortest Path Models 261 see row 7 then she goes from node 4 to node 6 see row 20 and finally she goes from node 6 to node 10 see row 31 Using this route from 1 to 10 Maude must walk 198 miles the sum of the distances on the three arcs she traverses Make sure you understand exactly how this model works There are really two parts the total distance and the balance of inflows and outflows For any solution of 0s and 1s in the Flow column the SUMPRODUCT for total distance simply sums the distances in col umn C corresponding to the arcs traversed This accurately reflects the total distance Maude walks For flow balance consider any intermediate node If Maudes route goes through it such as with node 6 the two SUMIF functions in column G for this node both evaluate to 1that is one of the arcs leading into node 6 has a flow of 1 and one of the arcs leading out of node 6 has a flow of 1 On the other hand if Maudes route doesnt go through the node such as with node 3 the two SUMIF functions for this node both evaluate to 0no flow in and no flow out Finally the flow balance constraints for nodes 1 and 10 ensure that exactly one arc leading out of node 1 has a flow of 1 and exactly one arc leading into node 10 has a flow of 1 Equipment Replacement Models Although shortest path problems often involve traveling through a network this is not always the case For example when should you trade your car in for a new car As a car gets older the maintenance cost per year increases and it might become worthwhile to buy a new car If your goal is to minimize the average annual cost of owning a car ignoring the time value of money then it is possible to set up a shortest path representation of this problem Actually the method we discuss can be used in any situation where equipment replacement is an issue Of course many people trade in a car because they like the feel of a new car This aspect is not modeled in the problem only the financial aspects are included The following is an example of how equipment replacement can be modeled as a shortest path problem All flows in a shortest path model are either 0 or 1 a route is either used or it isnt E X A M P L E 56 EQUIPMENT REPLACEMENT AT VANBUREN METALS V anBuren Metals is a manufacturing company that uses many large machines to work on metals These machines require frequent maintenance because of wear and tear and VanBuren finds that it is sometimes advantageous from a cost standpoint to replace machines rather than continue to maintain them For one particular class of machines the company has estimated the quarterly costs of maintenance the salvage value from reselling an old machine and the cost to purchase a new machine7 We assume that the maintenance cost and the salvage value depend on the age of the current machine at the beginning of the quarter However we assume that the maintenance costs the salvage val ues and the purchase cost do not depend on time In other words we assume no inflation Specifically we assume the following The purchase cost of a new machine is always 3530 The maintenance cost of a machine in its first quarter of use is 100 For each suc ceeding quarter the maintenance cost increases by 65 This reflects the fact that machines require more maintenance as they age The salvage value of a machine after one quarter of use is 1530 After each suc ceeding quarter of use the salvage value decreases by 110 7One issue in these types of models is the time period to use We assume that VanBuren uses quarters Therefore the only times it considers purchasing new machines are at beginnings of quarters Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 262 Chapter 5 Network Models VanBuren wants to devise a strategy for purchasing machines over the next five years As a matter of policy the company never sells a machine that is less than one year old and it never keeps a machine that is more than three years old Also the machine in use at the beginning of the current quarter is brand new Objective To find the optimal replacement strategy by modeling the problem as an equivalent shortest path problem WHERE DO THE NUMBERS COME FROM In general a company would gather historical data on maintenance costs and salvage val ues for similar machines and fit appropriate curves to the data probably using regression as discussed in Chapter 14 Solution The variables and constraints required for this machine replacement model appear in Table 510 We claimed that this problem can be modeled as a shortest path model which is probably far from obvious There are two keys to understanding why this is possible 1 the meaning of nodes and arcs and 2 the calculation of costs on arcs After you understand this the modeling details are exactly as in the previous example Table 510 Variables and Constraints for the Equipment Replacement Model Input variables Purchase cost maintenance costs as a function of age salvage values as a function of age Decision variables Flows on arcs 1 if arc is used 0 otherwise which determine the changing cells replacement schedule Objective target cell Total net cost Other output cells Flows into and out of arcs Constraints Flow balance at each node 1 5 6 7 13 17 21 Figure 528 Selected Nodes and Arcs for the Machine Replacement Network An arc from any node to a later node corre sponds to keeping a machine for a certain period of time and then trading it in for a new machine The network is constructed as follows There is a node for each quarter including the current quarter and the quarter exactly five years 20 quarters from now Remember that VanBuren uses a fiveyear planning horizon These nodes are labeled 1 through 21 where node 1 is the current quarter node 2 is the next quarter and so on There is an arc from each node to each later node that is at least 4 quarters ahead but no more than 12 quarters ahead This is because VanBuren never sells a machine less than one year old and never keeps a machine more than three years Several of these arcs are shown in Figure 528 Many nodes and arcs do not appear in this figure Consider the arc from node 9 to node 17 for example Using this arc on the shortest paththat is putting a flow of 1 on itcorresponds to starting with a new machine in quarter 9 keeping it for eight quarters selling it and purchasing another new machine at the beginning of quarter 17 An entire strategy for the fiveyear period is a string of such arcs For example with the path 191721 VanBuren keeps the first machine for eight quarters trades it in for a second machine at the beginning of quarter 9 keeps the second Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it node 21 includes the purchase cost in quarter 21 so this cost has no effect on which path is best The effect of including the salvage value in arcs into node 21 is to penal ize strategies that end with old machines after five years Regardless of how you model the problem you probably ought to penalize such strategies in some way In addition VanBuren will probably use a rolling planning horizonthat is it will implement only shortterm decisions from the model The way you model the end of the fiveyear horizon should have little effect on these early decisions 266 Chapter 5 Network Models Periodic Maintenance at Schindler Elevator Schindler the worlds largest escalator company and secondlargest elevator company maintains tens of thousands of elevators and escalators throughout North America Thousands of technicians work each day to maintain repair and help in emergencies Blakeley et al 2003 describe how they developed an automated routescheduling and planning system to optimize Schindlers preventive maintenance operations The system uses a series of algorithms to assign maintenance work to technicians and route them to where they are needed The estimated savings from the optimization system is more than 1 million annually ADDITIONAL APPLICATIONS P R O B L E M S SkillBuilding Problems 30 In Maudes shortest path problem suppose all arcs in the network are doublearrowed that is Maude can travel along each arc with the same distance in either direction Modify the spreadsheet model appropriately Is her optimal solution still the same 31 In Maudes shortest path problem suppose all arcs in the current network from highernumbered nodes to lowernumbered nodes such as from node 6 to node 5 are disallowed Modify the spreadsheet model and find the shortest path from node 1 to node 10 Is it the same as before Should you have known the answer to this question before making any changes to the original model Explain 32 Continuing the previous problem suppose again that all arcs go in both directions but suppose Maudes objective is to find the shortest path from node 1 to node 7 not node 10 Modify the spreadsheet model appropriately and solve 33 In the VanBuren machine replacement problem we assumed that the maintenance cost and salvage val ues are linear functions of age Suppose instead that the maintenance cost increases by 50 each quarter and that the salvage value decreases by 10 each quarter Rework the model with these assumptions What is the optimal replacement schedule 34 How difficult is it to add nodes and arcs to an existing shortest path model Answer by adding a new node node 11 to Maudes network Assume that node 11 is at the top of the network geographically with doublearrowed arcs joining it to nodes 2 5 and 7 with distances 45 22 and 10 Assume that Maudes objective is still to get from node 1 to node 10 Does the new optimal solution go through node 11 35 In the VanBuren machine replacement problem the companys current policy is to keep a machine at least 4 quarters but no more than 12 quarters Suppose this policy is instead to keep a machine at least 5 quarters but no more than 10 quarters Modify the spreadsheet model appropriately Is the new optimal solution the same as before 36 In the VanBuren machine replacement problem the companys current policy is to keep a machine at least four quarters but no more than 12 quarters Suppose instead that the company imposes no upper limit on how long it will keep a machine its only policy requirement is that a machine must be kept at least four quarters Modify the spreadsheet model appropriately Is the new optimal solution the same as before Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 56 NETWORK MODELS IN THE AIRLINE INDUSTRY We conclude this chapter with two network models that apply to the airline industry The airline industry is famous for using management science in a variety of ways to help man age operations and save on costs Neither of these problems looks like a network at first glance but some creative thinking reveals underlying network structures The first prob lem turns out to be an assignment model the second is similar to the RedBrand logistics model Note that these two examples are considerably more difficult than any covered so far in this chapter They indicate that it is not always straightforward to translate a real world problem into a spreadsheet model 56 Network Models in the Airline Industry 267 SkillExtending Problems 37 In the VanBuren machine replacement problem suppose the company starts with a machine that is eight quarters old at the beginning of the first quarter Modify the model appropriately keeping in mind that this initial machine must be sold no more than four quarters from now 38 We illustrated how a machine replacement problem can be modeled as a shortest path problem This is probably not the approach most people would think of when they first see a machine replacement problem In fact most people would probably never think in terms of a network How would you model the problem Does your approach result in an LP model E X A M P L E 57 CREW SCHEDULING AT BRANEAST AIRLINES B raneast Airlines must staff the daily flights between New York and Chicago shown in Table 5118 Braneasts crews live in either New York or Chicago Each day a crew must fly one New YorkChicago flight and one ChicagoNew York flight with at least one hour of downtime between flights For example a Chicagobased crew can fly the 911 ChicagoNew York flight and return on the 1214 New YorkChicago flight This incurs a downtime of one hour Braneast wants to schedule crews to cover all flights and minimize the total downtime 8All times in the spreadsheet model are represented as military time For example time 13 corresponds to 1 PM Also all times listed are Eastern Standard Time EST Table 511 Flight Data for Braneast Problem Flight Leave Chicago Arrive NY Flight Leave NY Arrive Chicago 1 6 AM 8 AM 1 7 AM 9 AM 2 9 AM 11 AM 2 8 AM 10 AM 3 Noon 2 PM 3 10 AM Noon 4 3 PM 5 PM 4 Noon 2 PM 5 5 PM 7 PM 5 2 PM 4 PM 6 7 PM 9 PM 6 4 PM 6 PM 7 8 PM 10 PM 7 7 PM 8 PM Objective To schedule crews without violating the onehour downtime restriction so that total downtime is minimized WHERE DO THE NUMBERS COME FROM The flight data are part of the airlines overall flight schedule The onehour downtime restriction is for safety reasons and is probably built into a union contract Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Solution The important insight is that this can be set up as an assignment model The variables and constraints required are listed in Table 512 The discussion following this table describes how the assignment model works 268 Chapter 5 Network Models Table 512 Variables and Constraints for the Crew Scheduling Model Input Variables Flight Schedule Data Decision variables Flows on arcs 01 variables indicating assignments of crews to pairs changing cells of flights Objective target cell Total downtime Other output cells Downtimes for crews assigned to flights flows in and out of nodes Constraints Flow balance The network is constructed as follows There are two sets of nodes one for flights departing from Chicago and one for flights departing from New York There is an arc from a Chicagobased node to a New Yorkbased node if the Chicago flight leaves early the New York flight leaves later and there is at least one hour of downtime if a crew is assigned to this pair of flights For example flight 1 from Chicago leaves at 6 AM and arrives at 8 AM in New York Therefore there is an arc from this flights node to the node of each New Yorkbased flight that leaves 9 AM or after This includes the last five flights leaving from New York see Figure 531 All such arcsthose that pair an early flight out of Chicago with a later flight out of New York that then flies back to Chicagomust be staffed by a Chicagobased crew A similar set of arcs go in the opposite direction from New York to Chicago and then back to New York which must be staffed by a New Yorkbased crew Chicagobased flights New Yorkbased flights Chicagobased flights New Yorkbased flights C1 C2 C3 C4 N5 N6 N7 N1 N2 N3 N4 N5 N6 N4 N3 C3 C4 C5 C6 C7 For example C3 indicates the third Chicagobased flight N2 indicates the second New Yorkbased flight Figure 531 Network for Airline Crew Scheduling Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 Enter inputs Enter the given flight information in the ranges B6C12 and F6G12 Because this information will be used with lookup functions later on the ranges A6C12 and E6G12 have been named CTable and NTable respectively The labels in columns A and E serve only to identify the various flights 2 Find feasible assignments To fill in the Chicagobased crews section find each early flight leaving from Chicago that can be paired with a later flight leaving from New York so that at least one hour of downtime occurs in between These correspond to the arcs in the top section of Figure 531 Then enter the flight codes of all such pairs of flights in columns A and B Do the same for the pairs that could be handled by New Yorkbased crews These correspond to the arcs in the bottom section of Figure 531 Note that all this information is entered manuallyno formulas are involved 3 Downtimes for feasible assignments Calculate the downtime for each feasible pair of flights by using lookup functions to extract the information from the flight schedules Specifically enter the formula VLOOKUPB17NTable2VLOOKUPA17CTable3 in cell C17 and copy it down for other flight pairs starting in Chicago This subtracts the beginning time of the second flight in the pair from the ending time of the first flight in the pair Do you see why military time is used Similarly enter the formula VLOOKUPB32CTable2VLOOKUPA32NTable3 in cell C32 and copy it down for other flight pairs starting in New York 4 Flows Enter any flows in the CFlow and NFlow ranges in column D Remember that these will eventually be 0s and 1s indicating that a crew is either assigned to a pair of flights or it isnt 5 Flow balance constraints There is a node in the network for each flight and a flow balance constraint for each nodehence 14 flow balance constraints However things get a bit tricky because each flight could be either the first or second flight in a given flight pair For example consider flight C3 From Figure 531 or Figure 532 flight C3 is the later flight for two flight pairs corresponding to rows 32 and 37 of the model and it is the ear lier flight for two flight pairs corresponding to rows 26 and 27 of the model Now comes the key observation for this particular model Flight C3 must be flown exactly once so exactly one of these arrows must have flow 1 and the others must have flow 0 Therefore you should add this nodes total inflow to its total outflow and constrain this sum to be 19 To implement this enter the formulas SUMIFCOriginF17CFlow and SUMIFCDestinationF17NFlow in cells G17 and H17 and copy them to the range G18H23 to take care of the flights leav ing from Chicago Then enter the formulas SUMIFNOriginF24NFlow and SUMIFNDestinationF24CFlow 270 Chapter 5 Network Models 9Admittedly this is not the usual flow balance constraint but it works here You might want to search for an alter native way of constructing the network Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 56 Network Models in the Airline Industry 273 E X A M P L E 58 SCHEDULING FLIGHTS AT TRICITIES AIRLINES T riCities Airlines flies several daily commuter flights to and from New York City Washington DC and Boston The company has been flying a fixed daily schedule of flights but it is now deciding whether to change this schedule Each potential flight has an estimated net revenue based on the typical number of passengers for the flight Look ahead to Figure 537 for a listing of all potential flights and their net revenues The com pany owns four airplanes and it does not anticipate buying any more There is a fixed cost of 1500 per plane per day that flies any flights However a plane that is not used does not incur this fixed cost We assume although this could be relaxed that there is no required delay time on the ground therefore if a flight arrives in Boston at time 10 it can leave on a new flight at time 10 Time is again measured in military time Also any plane that arrives in a city after its last flight of the day has two options It can sit overnight in that city or at a cost of 500 it can be flown empty to another city overnight The companys objective is to maximize its net profit per day which equals net revenues from flights flown minus fixed costs of flying planes minus overnight costs of flying empty Objective To develop a network model for scheduling the airlines flights given its available aircraft to maximize net profit from the flights WHERE DO THE NUMBERS COME FROM In a real setting the airline would first have to decide which flights including flight times to include in the potential list of flights This is presumably based on customer demands The financial inputs are obtained from accounting records For example the net revenue for a flight is based on the number of passengers who typically take the flight ticket prices personnel costs and fuel costs The fixed cost of operating a plane includes any costs that do not depend directly on the amount of time the plane spends in the air Solution We first discuss how this problem can be modeled as a network flow model which is cer tainly not obvious The trick is to have a node for each citytime combination Because flights are allowed on the halfhour this means having nodes of the form Boston8 Boston85 and so on up toWashDC20 assuming that the earliest flight leaves at time 8 and the latest flight arrives at time 20 There are three types of arcs The most obvious type is a flight arc For example if there is a flight from Boston at time 125 that arrives at Washington DC at time 14 then there is a flight arc from node Boston125 to node WashDC14 The flow on such an arc represents the number of planes that fly this flight Because each flight can be flown at most once a capacity of 1 is imposed on all such flight arcs The cost on a flight arc is the net revenue for flying the flight In this model it is more natural to use net revenues as the arc costs so that the objective will be to maximize net profit The other arcs are less obvious If a flight arrives in New York say at time 13 it might sit on the ground until time 145 at which time it leaves for another city This can be modeled with the ground arcs NY13NY135 NY135NY14 and NY14NY145 In general the flow on any ground arc represents the number of planes sitting on the ground in that city for that halfhour period These ground arcs have no capacities and no costs Finally it is important to relate one day to the next Suppose that one or more planes end up in New York at the end of the day at time 20 They can either sit overnight in Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Then calculate the various monetary values with the usual SUMPRODUCT functions For example the formula for total net revenue from flights is SUMPRODUCTFlightnetrevenueFlightflow Finally combine these into a profit objective in cell B156 with the formula B153B154B155 The Solver dialog box follows easilyand is remarkably compact for such a large and complex model Discussion of the Solution The optimal solution can be seen primarily from Figures 538 and 540 The former indi cates that TriCities should fly only 17 of the potential 23 flights The latter shows that no overnight flights should be flown It also shows that all four planes are used Two of these sit overnight in Boston and the other two sit overnight in Washington DC No overnight flights are flown evidently because the cost of doing so is too large The daily profit from this solution is 39600 Sensitivity Analysis You could run many interesting sensitivity analyses For example what if TriCities had more planes To answer this you can run SolverTable with cell B4 as the single input cell allowing it to vary from 4 to 8 in increments of 1 and keep track of the monetary values as well as the number of flights flown This latter output is calculated in cell B158 with the formula SUMFlightflow The results appear in Figure 544 As expected profit and the number of flights flown both increase when the company owns more planes but this analysis does not take the cost of purchasing more planes into account TriCities would need to trade off the cost of new planes with this increased profit 278 Chapter 5 Network Models 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A B C D E F G Oneway analysis for Solver model in Model worksheet Planes owned cell B4 values along side output cells along top B153 B154 B155 Netprofit B158 4 456 60 0 396 17 5 495 75 0 420 19 6 523 90 5 428 20 7 551 105 10 436 22 8 551 105 10 436 22 The first four outputs above are monetary values from the previous sheet The output in cell B158 is the number of flights flown Figure 544 Sensitivity to Planes Owned Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it From Figure 544 you can see that TriCities still does not fly all 23 potential flights even with eight planes Could it You can answer this question easily by changing the objective from maximizing profit to maximizing the number of flights flown in cell B158 and rerunning Solver If you do so you will find that the maximum is 23 Therefore TriCities could fly all 23 flights with eight planes but the cost structure makes it more profitable to fly only 22 The driving factor here is evidently the fixed cost per plane When TriCities owns eight planes the optimal profit solution uses only seven of these planes A final sensitivity analysis involves empty overnight flights When TriCities owns seven planes Figure 544 indicates see cell E165 that it flies two empty overnight flights These are both from Boston to Washington DC What happens to this solution if as a matter of company policy empty overnight flights are not allowed You can modify the model in three ways to answer this question First you can impose a huge cost on overnight flights effectively ruling them out Second you can impose capacities of zero on the overnight flight arcs in Figure 540 Third you can simply eliminate these arcs By using the first method you obtain the results shown in Figure 545 The solution changes fairly dramatically Now TriCities uses only five of its seven planes it flies only 19 instead of 22 flights and its profit decreases from 43600 to 42000 56 Network Models in the Airline Industry 279 148 149 150 151 152 153 154 155 156 157 158 D C B A Constraint on planes Number Number used owned 7 5 Monetary values Net 495 revenues Fixed 75 co sts Overnight 0 co sts Net 420 ro tif p Flights 19 wn olf Figure 545 Model with Overnight Flights Disallowed As stated previously airlines are heavy users of management science A quick look through recent issues of the Interfaces journal confirms this Here are some examples Virtually all of these examples describe optimization models that employ network and integer programming algorithms Improving Fractional Aircraft Ownership Operations at Flexjet Fractional aircraft ownership programs allow individuals to buy shares in a business jet at a fraction of the cost of full ownership The fractional aircraft market is the fastest growing segment of the business aircraft market Hicks et al 2005 describe how they used large scale mixedinteger nonlinear optimization models to maximize the use of aircraft crew and facilities for Flexjets fractional aircraft ownership operations Since inception the system has generated savings in excess of 54 million with projected additional savings of 27 million annually Optimizing Pilot Staffing and Training at Continental Airlines Yu et al 2004 describe how they developed the Crew ResourceSolver decisionsupport system for Continental Airlines This system employs advanced optimization modeling ADDITIONAL APPLICATIONS Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it and solution techniques to solve large complex pilot staffing and training problems The authors estimate that the system has saved Continental over 10 million annually UPS Optimizes Its Air Network Armacost et al 2004 describe how a team of operations research analysts at UPS and Massachusetts Institute of Technology created a system to optimize the design of service networks for delivering express packages The system determines aircraft routes fleet assignments and package routings to ensure overnight delivery at minimum cost UPS credits the system with savings in excess of 87 million between 2000 and 2002 and it anticipates future savings to be in the hundreds of millions of dollars Optimizing OnDemand Aircraft Schedules for Fractional Aircraft Operators Martin et al 2003 describe how Bitwise Solutions developed a flexible integrated decisionsupport system to help fractional management companies companies that man age fractional aircraft ownership programs optimize their fleet schedules The system handles all aspects of fractional fleet management reservations scheduling dispatch air craft maintenance and crew requirements In November 2000 Raytheon Travel Air began using the system and reported a 44 million savings in the first year of use Delta Optimizes ContinuingQualificationTraining Schedules for Pilots The downturn in airline business after the terrorist attacks of September 11 2001 forced air lines to modify their operations Sohoni et al 2003 describe modifications at Delta Airlines which had to reduce its workforce and modify its requirements for scheduling pilot training To minimize Deltas costs and automate the scheduling process under a rigid planning time line the authors developed an optimization system that builds and assigns training schedules based on individual pilots requirements Delta expects to save 75 million in annual operat ing costs by using the system to schedule continuing qualification training for its pilots Crew Recovery at Continental Airlines Due to unexpected events such as inclement weather airline crews may not be in position to service their remaining scheduled flights Airlines must reassign crews quickly to cover open flights and return them to their original schedules in a costeffective manner that honors vari ous regulations Yu et al 2003 describe how they developed a decisionsupport system for Continental Airlines to generate optimal or nearly optimal crewrecovery solutions Since its implementation the system has dealt successfully with several disruptive events including snowstorms a flood and the 911 terrorist attacks Continental estimates that the system was responsible for savings of approximately 40 million for major disruptions alone 280 Chapter 5 Network Models P R O B L E M S SkillBuilding Problems 39 In the crewscheduling problem suppose as in the sensitivity analysis we discussed that the first Chicago flight C1 is delayed by two hoursthat is its departure and arrival times move up to 8 AM and 10 AM respectively How does the model need to be modified What is the new optimal solution Is it the same as the solution indicated by SolverTable in Figure 535 If not why not 40 The required downtime in the crewscheduling problem is currently assumed to be one hour Suppose instead that it is required to be two hours How does the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 57 Conclusion 281 model need to be modified What is the new optimal solution 41 In the crewscheduling problem suppose that two extra flights are added to the current list The first leaves Chicago at 11 AM and arrives in New York at 1 PM The second leaves New York at 6 PM and arrives in Chicago at 8 PM Remember that all quoted times are EST Modify the model to incorporate these two new flights What is the new optimal solution 42 In the flightscheduling model use SolverTable to examine the effect of decreasing all net revenues by the same percentage assuming that the company owns six planes Let this percentage vary from 0 to 50 in increments of 10 Discuss the changes that occur in the optimal solution 43 In the flightscheduling model use SolverTable to examine the effect of increasing both the fixed cost per plane and the overnight cost by the same percentage assuming that the company owns eight planes Let this percentage vary from 0 to 50 in increments of 10 Discuss the changes that occur in the optimal solution SkillExtending Problems 44 One rather unrealistic assumption in the flight scheduling model is that a given plane can fly two consecutive flights with no downtime For example it could fly flight 5903 that gets into Washington DC at time 14 and then fly flight 7555 that leaves Washington DC at time 14 Modify the model so that there must be at least one hour of downtime between consecutive flights 45 In the crewscheduling model there are exactly as many flights departing from Chicago as departing from New York Suppose more flights are departing from one city than from the other How would you model this Illustrate by assuming that there is an extra flight from Chicago that leaves at 11 AM and arrives at New York at 1 PM Remember that all quoted times are EST 57 CONCLUSION In this chapter you have seen a number of management science problems that can be for mulated as network models Often these problems are of a logistics natureshipping goods from one set of locations to another However you have also seen that problems that do not involve shipping or traveling along a physical network can sometimes be formu lated as network models Examples include the bus route assignment and machine replace ment problems Formulating a problem as a network model has at least two advantages First although Excels Solver doesnt employ them fast specialpurpose algorithms exist for various forms of network models These enable companies to solve extremely large problems that might not be solvable with ordinary LP algorithms Second the graphical representation of network models often makes them easier to visualize When a problem can be visualized graphically it is often simpler to model in a spreadsheet or otherwise and ultimately to optimize Summary of Key Management Science Terms Term Explanation Page Network models Class of optimization models that can be represented 228 graphically as a network typically but not always involves shipping goods from one set of locations to another at minimum cost Nodes Points in a network representation often correspond to locations 230 Arcs Arrows in a network representation often correspond to routes 230 connecting locations Flows Decision variables that represent the amounts sent along arcs 231 Arc capacities Upper bounds on flows on some or all arcs 231 continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Term Explanation Page Flow balance constraints Constraints that force the amount sent into a node to equal 235 the amount sent out except possibly for amounts that start out or end up at the node Assignment models Class of optimization models where members of one set like 241 workers must be assigned optimally to members of another set like jobs Shortest path models Network models where the goal is to get from an origin node 257 to a destination node at minimal distance or cost Summary of Key Excel Terms Term Explanation Excel Page SUMIF function Sums values in one range SUMIFcompareRange 236 corresponding to cells in a criterion sumRange related range that satisfy a criterion COUNTIF function Counts values in one range that COUNTIFrangecriterion 253 satisfy a criterion 282 Chapter 5 Network Models P R O B L E M S SkillBuilding Problems 46 The government is auctioning off oil leases at two sites At each site 100000 acres of land are to be auc tioned Cliff Ewing Blake Barnes and Alexis Pickens are bidding for the oil Government rules state that no bidder can receive more than 40 of the land being auctioned Cliff has bid 1000 per acre for site 1 land and 2000 per acre for site 2 land Blake has bid 900 per acre for site 1 land and 2200 per acre for site 2 land Alexis has bid 1100 per acre for site 1 land and 1900 per acre for site 2 land a Determine how to maximize the governments revenue with a transportation model b Use SolverTable to see how changes in the governments rule on 40 of all land being auctioned affect the optimal revenue Why can the optimal revenue not decrease if this percentage required increases Why can the optimal revenue not increase if this percentage required decreases 47 The 7th National Bank has two checkprocessing sites Site 1 can process 10000 checks per day and site 2 can process 6000 checks per day The bank processes three types of checks vendor salary and personal The processing cost per check depends on the site as listed in the file P0547xlsx Each day 5000 checks of each type must be processed Develop a network model to determine how to minimize the daily cost of processing checks 48 The Amorco Oil Company controls two oil fields Field 1 can produce up to 20 million barrels of oil per day and field 2 can produce up to 15 million barrels of oil per day At field 1 it costs 3750 to extract and refine a barrel of oil at field 2 the cost is 4120 Amorco sells oil to two countries United Kingdom and Japan The shipping costs per barrel are shown in the file P0548xlsx Each day the United Kingdom is willing to buy up to 10 million barrels at 6580 per barrel and Japan is willing to buy up to 25 million barrels at 6840 per barrel Determine how to maxi mize Amorcos profit 49 Touche Young has eight auditors Each can work up to 160 hours during the next month during which time six projects must be completed The hours required for each project and the amounts each auditor can be billed for each project are given in the file P0549xlsx Note that more than one auditor can work on a given project in which case their hours add to the total for the project Determine how to maximize total billings during the next month 50 Five employees are available to perform four jobs The time it takes each person to perform each job is given in the file P0550xlsx Determine the assignment of employees to jobs that minimizes the total time required to perform the four jobs A blank indicates that a person cannot do that particular job Also assume that no person can do more than one job 51 Based on Machol 1970 A swimming coach is putting together a relay team for the 400meter relay Each swimmer must swim 100 meters of breaststroke Summary of Key Management Science Terms Continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it backstroke butterfly or freestyle and each swimmer can swim only one race The coach believes that each swimmer can attain the times given in the file P0551xlsx To minimize the teams total time for the race which swimmers should swim which strokes 52 A company is taking bids on four construction jobs Three contractors have placed bids on the jobs Their bids in thousands of dollars are given in the file P0552xlsx A blank indicates that the contractor did not bid on the given job Contractor 1 can do only one job but contractors 2 and 3 can each do up to two jobs Determine the minimum cost assignment of contractors to jobs 53 A company manufactures widgets at two factories one in Memphis and one in Denver The Memphis factory can produce up to 150 widgets per day and the Denver factory can produce up to 200 widgets per day The company are shipped by air to customers in Los Angeles and Boston The customers in each city require 130 widgets per day Because of the deregula tion of airfares the company believes that it might be cheaper to first fly some widgets to New York or Chicago and then fly them to their final destinations The costs of flying a widget are shown in the file P0553xlsx a Determine how to minimize the total cost of ship ping the required widgets to the customers b Suppose the capacities of both factories are reduced in increments of 10 widgets per day Use SolverTable to see how much the common reduc tion can be before the total cost increases and how much it must be before there is no feasible solution 54 General Ford produces cars in Los Angeles and Detroit and has a warehouse in Atlanta The company supplies cars to customers in Houston and Tampa The costs of shipping a car between various points are listed in the file P0554xlsx where a blank means that a shipment is not allowed Los Angeles can produce up to 1100 cars and Detroit can produce up to 2900 cars Houston must receive 2400 cars and Tampa must receive 1500 cars a Determine how to minimize the cost of meeting demands in Houston and Tampa b Modify the answer to part a if shipments between Los Angeles and Detroit are not allowed c Modify the answer to part a if shipments between Houston and Tampa are allowed at a cost of 75 per car 55 Sunco Oil produces oil at two wells Well 1 can pro duce up to 150000 barrels per day and well 2 can produce up to 200000 barrels per day It is possible to ship oil directly from the wells to Suncos customers in Los Angeles and New York Alternatively Sunco could transport oil to the ports of Mobile and Galveston and then ship it by tanker to New York or 57 Conclusion 283 Los Angeles Los Angeles requires 160000 barrels per day and New York requires 140000 barrels per day The costs of shipping 1000 barrels between various locations are shown in the file P0555xlsx where a blank indicates shipments that are not allowed Determine how to minimize the transport costs in meet ing the oil demands of Los Angeles and New York 56 Nash Auto has two plants two warehouses and three customers The plants are in Detroit and Atlanta the warehouses are in Denver and New York and the cus tomers are in Los Angeles Chicago and Philadelphia Cars are produced at plants then shipped to ware houses and finally shipped to customers Detroit can produce 200 cars per week and Atlanta can produce 160 cars per week Los Angeles requires 80 cars per week Chicago requires 70 and Philadelphia requires 60 It costs 8000 to produce a car at each plant The costs of shipping a car between various cities are listed in the file P0556xlsx Assume that during a week at most 75 cars can be shipped from a warehouse to any particular city Determine how to meet Nashs weekly demands at minimum cost 57 Edsel Motors produces cars in Detroit and Dallas The Detroit plant can produce up to 8500 cars and the Dallas plant can produce up to 4000 cars Producing a car costs 2000 in Detroit and 1800 in Dallas Cars must be shipped to 12 cities The costs of shipping a car from each plant to each city and the city require ments are given in the file P0557xlsx At most 1000 cars can be sent from a given plant to a given city Determine how to minimize the cost of meeting all demands 58 Each year Data Corporal produces up to 5000 comput ers in Boston and up to 3500 computers in Charlotte There are customers in Los Angeles New York and Seattle who must receive 2300 3700 and 1300 computers respectively Producing a computer costs 250 in Boston and 275 in Charlotte Computers are transported by plane and can be sent through Chicago The costs of sending a computer between pairs of cities are shown in the file P0558xlsx a Determine how to minimize the total production plus shipping cost of meeting Data Corporals annual demand Why doesnt it make sense to ship any computers through Chicago b Modify the model so that no more than 1250 com puters can be shipped between any two cities and find the optimal solution to this modified model Why are computers now shipped through Chicago 59 It costs 300 to buy a lawn mower from a lawn supply store Assume that you can keep a lawn mower for at most five years and that the estimated maintenance cost each year of operation is as follows year 1 90 year 2 135 year 3 175 year 4 200 year 5 250 You have just purchased a new lawn mower Assuming Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it that a lawn mower has no salvage value determine the strategy that minimizes the total cost of purchasing and operating a lawn mower for the next 10 years 60 Suppose it costs 20000 to purchase a new car The annual operating cost and resale value of a used car are shown in the file P0560xlsx Assume that you presently have a new car Determine a replacement policy that minimizes your net costs of owning and operating a car for the next six years 61 At the beginning of year 1 a new machine must be purchased The cost of maintaining a machine depending on its age is given in the file P0561xlsx The cost of purchasing a machine at the beginning of each year is given in this same file There is no trade in value when a machine is replaced The goal is to minimize the total purchase plus maintenance cost of having a machine for five years Determine the years in which a new machine should be purchased 62 Delko is considering hiring people for four types of jobs The company would like to hire the number of people listed in the file P0562xlsx for each type of job Delko can hire four types of people Each type is qualified to perform two types of jobs as shown in this same file A total of 20 type 1 30 type 2 40 type 3 and 20 type 4 people have applied for jobs Determine how Delko can maximize the number of employees assigned to suitable jobs assuming that each person can be assigned to at most one job Hint Set this up as a transportation model where the supplies are the applicants 63 The town of Busville has three school districts The numbers of black students and white students in each district are shown in the file P0563xlsx The Supreme Court requires the schools in Busville to be racially balanced Thus each school must have exactly 300 students and each school must have the same number of black students The distances between dis tricts are shown in the same file Determine how to minimize the total distance that students must be bussed while still satisfying the Supreme Courts requirements Assume that a student who remains in his or her own district does not need to be bussed 284 Chapter 5 Network Models 64 A truck must travel from New York to Los Angeles As shown in Figure 546 several routes are available The number associated with each arc is the number of gallons of fuel required by the truck to traverse the arc Determine the route from New York to Los Angeles that uses the minimum amount of gas 65 You are trying to help the MCSCC Monroe County School Corporation determine the appropriate high school district for each housing development in Bloomington For each development you are given the number of students the mean family income the per centage of minorities and the distance to each high school South and North These data are listed in the file P0565xlsx In assigning the students MCSCC wants to minimize total distance traveled subject to the following constraints Each school must have at least 1500 students The mean family income must be at least 85000 for students of each school Each school must have at least 10 minorities Determine an optimal assignment of students to schools Then provide a oneparagraph summary of how the optimal solution changes as the required minority percentage varies from 5 to 11 66 A school system has 16 bus drivers that must cover 12 bus routes Each driver can cover at most one route The drivers bids for the various routes are listed in the file P0566xlsx Each bid indicates the amount the driver will charge the school system to drive that route How should the drivers be assigned to the routes to minimize the school systems cost After you find the optimal assignments use conditional formatting so that the cost the school system pays for each route is highlighted in red and whenever the cheapest bid is not used for a route that bid is highlighted in green SkillExtending Problems 67 Allied Freight supplies goods to three customers who each require 30 units The company has two ware houses In warehouse 1 40 units are available and in warehouse 2 30 units are available The costs of 1800 400 1300 900 600 950 800 600 1200 1000 600 1100 400 900 Cleveland Phoenix Los Angeles Salt Lake City Nashville St Louis New York Dallas Figure 546 Network for Truck Problem Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 286 Chapter 5 Network Models sent through Indianapolis or Cleveland then through Dallas or Denver and finally to LA The number of calls that can be routed between any pair of cities is shown in the file P0575xlsx The phone company wants to know how many of the 70000 calls originat ing in New York and Philadelphia can be routed to LA Set this up as a network flow modelthat is specify the nodes arcs shipping costs and arc capaci ties Then solve it 76 Eight students need to be assigned to four dorm rooms at Faber College Based on incompatibility measure ments the cost incurred for any pair of students room ing together is shown in the file P0576xlsx How should the students be assigned to the four rooms to minimize the total amount of incompatibility 77 Based on Ravindran 1971 A library must build shelving to shelve 200 4inchhigh books 600 8inch high books and 500 12inchhigh books Each book is 05 inch thick The library has several ways to store the books For example an 8inchhigh shelf can be built to store all books of height less than or equal to 8 inches and a 12inchhigh shelf can be built for the 12inch books Alternatively a 12inchhigh shelf can be built to store all books The library believes it costs 2300 to build a shelf and that a cost of 5 per square inch is incurred for book storage Assume that the area required to store a book is given by the height of the storage area multiplied by the books thickness Determine how to shelve the books at minimum cost Hint We agree that this is not a very realistic problem in terms of how a library operates but it is a good modeling challenge Create nodes 0 4 8 and 12 and make the cost associated with the arc joining nodes i and j equal to the total cost of shelving all books of height greater than i and less than or equal to j on a single shelf 78 In the original RedBrand problem Example 54 suppose that the company could add up to 100 tons of capacity in increments of 10 tons to any single plant Use SolverTable to determine the yearly savings in cost from having extra capacity at the various plants Assume that the capacity will cost 28000 per ton right now Also assume that the annual cost savings from having the extra capacity will extend over 10 years and that the total 10year savings will be discounted at an annual 10 interest rate How much extra capacity should the company purchase and which plant should be expanded Hint Use the PV function to find the present value of the total cost sav ing over the 10year period You can assume that the costs occur at the ends of the respective years 79 Based on Jacobs 1954 The Carter Caterer Company must have the following number of clean napkins available at the beginning of each of the next four days day 1 1500 day 2 1200 day 3 1800 day 4 600 After being used a napkin can be cleaned by one of two methods fast service or slow service Fast service costs 50 cents per napkin and a napkin cleaned via fast service is available for use the day after it is last used Slow service costs 30 cents per napkin and these napkins can be reused two days after they are last used New napkins can be purchased for a cost of 95 cents per napkin Determine how to minimize the cost of meeting the demand for napkins during the next four days Note There are at least two possible modeling approaches one network and one nonnet work See if you can model it each way 80 Kellwood a company that produces a single product has three plants and four customers The three plants will produce 3000 5000 and 5000 units respectively during the next time period Kellwood has made a commitment to sell 4000 units to customer 1 3000 units to customer 2 and at least 3000 units to customer 3 Both customers 3 and 4 also want to buy as many of the remaining units as possible The profit associated with shipping a unit from each plant to each customer is given in the file P0580xlsx Determine how to maximize Kellwoods profit 81 You have put four valuable paintings up for sale Four customers are bidding for the paintings Customer 1 is willing to buy two paintings but each other cus tomer is willing to purchase at most one painting The prices that each customer is willing to pay are given in the file P0581xlsx Determine how to maximize the total revenue you receive from the sale of the paintings 82 Powerhouse produces capacitors at three locations Los Angeles Chicago and New York Capacitors are shipped from these locations to public utilities in five regions of the country northeast NE northwest NW midwest MW southeast SE and southwest SW The cost of producing and shipping a capacitor from each plant to each region of the country is given in the file P0582xlsx Each plant has an annual pro duction capacity of 100000 capacitors Each year each region of the country must receive the following number of capacitors NE 55000 NW 50000 MW 60000 SE 60000 SW 45000 Powerhouse believes that shipping costs are too high and it is therefore considering building one or two more production plants Possible sites are Atlanta and Houston The costs of producing a capacitor and shipping it to each region of the country are given in the same file It costs 3 million in current dollars to build a new plant and operating each plant incurs a fixed cost in addition to variable shipping and production costs of 50000 per year A plant at Atlanta or Houston will have the capacity to produce 100000 capacitors per year Assume that future demand patterns and produc tion costs will remain unchanged If costs are dis counted at a rate of 12 per year how can Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 288 Chapter 5 Network Models I nternational Textile Company Ltd is a Hong Kongbased firm that distributes textiles world wide The company is owned by the Lao family Should the Peoples Republic of China continue its economic renaissance the company hopes to use its current base to expand operations to the mainland International Textile has mills in the Bahamas Hong Kong Korea Nigeria and Venezuela each weaving fabrics out of two or more raw fibers cotton poly ester andor silkThe mills service eight company dis tribution centers located near the customers geographical centers of activity Because transportation costs historically have been less than 10 of total expenses management has paid little attention to extracting savings through judicious routing of shipments Ching Lao is returning from the United States where he has just completed his bachelors degree in marketing He believes that each year he can save International Textile hundreds C A S E 51 INTERNATIONAL TEXTILE COMPANY LTD10 of thousands of dollarsperhaps millionsjust by better routing of fabrics from mills to distribution centers One glaring example of poor routing is the current assignment of fabric output to the Mexico City distribution center from Nigeria instead of from Venezuela less than a third the distance Similarly the Manila center now gets most of its textiles from Nigeria and Venezuela although the mills in Hong Kong itself are much closer Of course the cost of shipping a bolt of cloth does not depend on distance aloneTable 514 provides the actual costs supplied to Lao from com pany headquarters Distribution center demands are seasonal so a new shipment plan must be made each month Table 515 provides the fabric requirements for the month of March International Textiles mills have varying capacities for producing the various types of clothTable 516 provides the quantities that apply during March 10This case was written by Lawrence L Lapin San Jose State University Table 514 Shipping Cost Data Dollars Per Bolt Distribution Center Mill Los Angeles Chicago London Mexico City Manila Rome Tokyo New York Bahamas 2 2 3 3 7 4 7 1 Hong Kong 6 7 8 10 2 9 4 8 Korea 5 6 8 11 4 9 1 7 Nigeria 14 12 6 9 11 7 5 10 Venezuela 4 3 5 1 9 6 11 4 Table 515 Fabric Demands for March Bolts Distribution Center Fabric Los Angeles Chicago London Mexico City Manila Rome Tokyo New York Cotton 500 800 900 900 800 100 200 700 Polyester 1000 2000 3000 1500 400 700 900 2500 Silk 100 100 200 50 400 200 700 200 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Case 51 International Textile Company Ltd 289 Table 516 March Production Capacities Bolts Production Capacity Mill Cotton Polyester Silk Bahamas 1000 3000 0 Hong Kong 2000 2500 1000 Korea 1000 3500 500 Nigeria 2000 0 0 Venezuela 1000 2000 0 Lao wants to schedule production and shipments in such a way that the most costly customers are shorted when there is insufficient capacity and the leastefficient plants operate at less than full capacity when demand falls below maximum production capacity You have been retained by International to assist Lao Questions 1 Find the optimal March shipment schedule and its total transportation cost for each of the following a cotton b polyester c silk 2 The company will be opening a silkmaking department in the Nigeria mill Although it will not be completed for several months a current capacity of 1000 bolts for that fabric might be used during March for an added one time cost of 2000 Find the new optimal shipment schedule and the total cost for that fabric Should the Nigeria mill process silk in March 3 Lao learns that changes might have to be made to the March plans If a new customer is obtained the cotton demand in Manila and in Mexico City will increase by 10 at each loca tion Meanwhile a big New York customer might cut back which would reduce polyester demand by 10 in both New York and Chicago Find the contingent optimal schedules and total costs a for cotton and b for polyester 4 InternationalTextile loses a profit of 10 for each bolt of cotton it falls short of meeting the distribu tion centers demand For polyesterthe loss is 20 per bolt for silkit is a whopping 50 per bolt By running the mills on overtimethe com pany can produce additional bolts at the addi tional costs shown inTable 517 Using only the original data from Tables 514 through 516 and the information in Table 517determine new pro duction schedules to maximize overall profit for successively a cottonb polyester and c silk Which fabrics and locations involve overtime productionand what are the overtime quantities 5 Without making any calculations offer Lao other suggestions for reducing costs of transportation Table 517 Overtime Production Costs Cost per Bolt Mill Cotton Polyester Silk Bahamas 10 10 NA Hong Kong 15 12 25 Korea 5 8 22 Nigeria 6 NA NA Venezuela 7 6 NA Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 290 Chapter 5 Network Models A typical paper mill might produce 1200 tons of paper per day to fill orders from 250 customers Sending 100 truckload shipments per day would not be unusual for a mill served by 20 motor carriers The carriers will generally accept shipments to any destination that they serve subject to daily volume commitments and equipment availability Each carrier has a different and somewhat complex rate struc ture Given a pool of orders that must be shipped on a given day the mills problem is to assign truckloads to carriers to minimize its total shipping cost Westvaco Company Overview Each year Westvaco sells more than 2 billion worth of manufactured paper paperboard and specialty chemicals Production occurs at five domestic paper mills and four chemical plants In addition Westvaco has many converting locations which manufacture liquid packaging envelopes folding cartons and corrugated boxes Some of Westvacos products include the following Fine papers often used in printing applications magazines and annual reports Bleached paperboard used in packaging milk and juice cartons freezer to oven entrees and so forth Kraft paper used for corrugated boxes and deco rative laminates such as Formica Chemicals including activated carbon printing ink resins Transportation Function The corporate transportation function has a dual role at Westvaco It supports the operating locations by negotiating freight rates and service commitments with rail truck and ocean carriers In addition it serves as an internal consulting group for reviewing operations in the field and making recommendations on streamlining tasks making organizational changes to support changing customer requirements and supporting the implementation of new technology C A S E 52 OPTIMIZED MOTOR CARRIER SELECTION AT WESTVACO Local traffic departments are responsible for daytoday operations of mills and plants including carrier assignments dispatching and switching lists for the railroads Production Overview The production cycle is summarized in Figure 547 Customer Service Orders received Scheduling Orders scheduled to meet delivery date Manufacturing Orders produced on papermaking machines Delivery Order delivered to customer Distribution Loads assigned to truck carriers rail and ocean vessels Load Planning Lessthantruckload quantities consolidated Figure 547 Production Cycle Overview Orders The majority of paper orders are for rolls where customers request a specific grade and size of paper diameter and width amount pounds or linear or square feet and delivery date The orders typically range in width from 8 to 70 inches With greater emphasis on justintime production by Westvacos customers delivery dates are sometimes specified in halfhour time windows Orders that arrive before or after the time window are not accepted Scheduling After orders are received they are scheduled on paper machines up to 200 inches wide The paper business is heavily capital intensive new machines can cost more than 400 million each Machines usually run 24 hours a day and scheduling is done to minimize waste while meeting shipping date requirements After production of a parent roll the orders are cut on a rewinder into the exact order size Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 292 Chapter 5 Network Models Table 518 Current Distribution Data for Westvaco Case Study Carrier Destination State Trips Stops Miles ABCT IRST LAST MRST NEST PSST Atlanta GA 4 0 612 088 115 087 095 105 Everett MA 1 3 612 118 127 139 135 128 Ephrata PA 3 0 190 342 173 171 182 200 Riverview MI 5 0 383 079 101 125 096 095 111 Carson CA 1 2 3063 080 087 100 Chamblee GA 1 0 429 123 161 122 133 147 Roseville MN 1 3 600 124 113 189 132 141 141 Hanover PA 1 0 136 478 223 239 226 257 Sparks NV 2 0 2439 145 120 Parsippany NJ 1 1 355 162 136 139 103 176 Effingham IL 5 0 570 087 087 125 087 090 131 Kearny NJ 7 0 324 201 154 153 128 195 Minimum charge per truckload 350 400 350 300 350 300 Stopoff charge 50 75 50 35 50 50 Available pulls 4 8 7 7 3 4 Commitment 1 7 6 0 0 4 Note Asterisks indicate carrier does not travel to the destination rates in dollarsmile on weekends where revolving coverage added significant variability to the carrier selection process The technique adds accountability to the transporta tion planners position and tied to a reason code for changing the carrier offers a clear answer to manage ment questions regarding carrier selection Finally the time savings have also been significant The carrier assignment portion of the transportation planners job can be done much faster than before11 11This case was coauthored with Dave Rimple who identified and implemented this project at Westvaco Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 293 Optimization Models with Integer Variables C H A P T E R USAIR FORCE SP ACE COMMANDS LONGTERM INVESTMENT IN S P A CE SYSTEMS T he US Air Force created Space Command in 1982 to enhance defense in the United States through space superiority and to protect the coun try from weapons of mass destruction Space Command spends billions of dollars each year procuring and deploying launch vehicles and space systems required for mission area tasks Space Command included a space and mis sile optimization analysis SAMOA group to determine the best use of funds to satisfy requirements over a 24year time horizon Brown et al 2003 describe their role within SAMOA to develop a strategic plan that was pre sented to Congress in 1999 as part of the militarys overall strategic plan The authors of the plan developed an integer programming model similar to the capital budgeting model in this chapter but much larger in scale to determine the best set of space projects to undertake over the planning horizon This plan tries to achieve the various missions of Space Command as fully as possible while staying within budget Like everything in the military the model has an acronym SCOUT space command optimizer of utility toolkit The overall planning process within SAMOA is extremely complex The process consists of five steps 1 mission area assessment 2 mission needs analysis 3 mission solution analysis 4 portfolio selection and 5 refined portfolio selection The first three steps are essentially steps 1 and 2 of the sevenstep modeling process described in Chapter 1 They define the tasks that Space Command needs to accomplish to achieve its missions the 6 JOSHUA GATES WEISBERGEPALandov Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it current and future needsover and above what already existsto accomplish these tasks and the required data on candidate systems being considered This data includes 1 scores for how each system or combination of systems accomplishes various tasks 2 possible starting and ending times for the system where the possible starting times can be several years in the future due to the time required for RD 3 expected sys tem costs including development and operating costs over a multiyear period 4 vari ous precedence relations and side constraints for example system B cant be selected unless project A is selected 5 launch requirements and perlaunch costs and 6 bud getary restrictions The last two steps build the integer programming model and then refine it based on nonquantitative considerations such as political pressures The model itself has a large number of integer decision variables There is a binary variable for each combination of system and starting and ending years For example if a given system can be started any year from 2005 until 2010 and then end 12 years later there will be six binary variables one for each potential starting year There are also integer variables for the number of launches by each selected system each year The constraints are mostly of the logical type For example they enforce all precedence relations and side constraints and they allow a given system to be selected for only one startend time combination The authors use a penalty type of objectiveThat is the objective is total discounted penalty dollars with penalties for not completely achieving task performance and for violating budget constraints This allows solutions to violate constraints slightly they can be slightly over budget say which provides more flexibility The discounting is done in the usual financial sense to make violations in the distant future less important The strategic master plan the result of the SCOUT model and its refinements was submitted to Congress in 1999 The plan included planned investments totaling about 310 billion As the authors state This planning effort is the beststaffed and most scrupulously managed example of optimizationbased capital planning that we have ever seen Since 1999 Space Command and several other military units have used SAMOA to help create their strategic master plans We recommend both this article and a some what more general article about military capital planning by Brown et al 2004 They are both excellent examples of how integer programming can be used to make impor tant and costly capital budgeting decisions They also indicate the differences between capital budgeting in the military versus capital budgeting in civilian organizations 294 Chapter 6 Optimization Models with Integer Variables 61 INTRODUCTION In this chapter we show how many complex problems can be modeled using 01 variables and other variables that are constrained to have integer values A 01 variable is a decision variable that must equal 0 or 1 Usually a 01 variable corresponds to an activity that either is or is not undertaken If the 01 variable corresponding to the activity equals 1 the activity is undertaken if it equals 0 the activity is not undertaken A 01 variable is also called a binary variable Optimization models in which some or all of the variables must be integers are known as integer programming IP models1 In this chapter we illustrate many of the modeling techniques that are needed to formulate IP models of complex situations You should be aware that any optimization software including Excels Solver typically has a much 1Many problems in the literature are described as mixed integer linear programming MILP models which indi cates that some of the changing cells are constrained to be integers and others are not Although we do use this acronym some of our models are of this type Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it harder time solving an IP problem than an LP problem In fact optimization software is sometimes unable to solve an IP problem even if the IP problem has an optimal solution The reason is that these problems are inherently difficult to solve no matter what software package is used However as you will see in this chapter your ability to model complex problems increases tremendously when you use binary variables IP models come in many forms You saw examples in Chapter 4 where the decision variables are naturally integervalued For example when scheduling postal workers Example 42 it is natural to require the numbers of workers to be integers In examples like this where you do not want certain decision variables to have fractional values the problems are basically LP models with integer constraints added at the last minute In many such examples if you ignore the integer constraints optimize with Solver and then round to the nearest integers the resulting integer solution will probably be close to optimalalthough admittedly the rounded solution is sometimes not optimal The integer models in Chapter 4 are not the types of IP models discussed in this chapter If it were simply a matter of adding integer constraints to decision variables such as the numbers of workers this chapter wouldnt be necessary However many inherently nonlinear problems can be transformed into linear models with the use of binary variables These are the types of models discussed here The clever use of binary variables allows you to solve many interesting and difficult problems that LP algorithms are incapable of solving All the models we analyze in this chapter are aside from binary or integer changing cells linear models As in previous chapters this means that the target cell is ultimately a sum of products of constants and changing cells The same goes for both sides of all con straints In other words the models in this chapter look much like the models in the previ ous three chapters The only difference is that some of the changing cells are now constrained to be binary or integer Although the basic algorithm that Solver uses for such models is fundamentally differentbecause of the binary or integer variablesit still helps that the models are linear They would present even more of a challenge to Solver if they were nonlinear 62 OVERVIEW OF OPTIMIZATION WITH INTEGER VARIABLES When Excels Solver solves a linear model without integer constraints it uses a very effi cient algorithm the simplex method to perform the optimization As discussed in Chapter 3 this method examines the corner points of the feasible region and returns the best corner point as the optimal solution The simplex method is efficient because it typi cally examines only a very small fraction of the hundreds thousands or even millions of possible corner points before determining the best corner point The main difference between LP and IP models is that LP models allow fractional val ues such as 0137 and 53246 in the changing cells whereas IP models allow only integer values in integerconstrained changing cells In fact if changing cells are constrained to be binary the only allowable values are 0 and 1 This suggests that IP models should be easier to solve After all there are many fewer integer values in a given region than there are con tinuous values so searching through the integers should be quickerespecially if their only possible values are 0 and 1 However IP models are actually much more difficult to solve than LP models Although several solution methods have been suggested by researchersand new methods for specialized problems are still being developedthe solution procedure used by Solver is called branch and bound Although we do not go into the details of the algorithms we discuss briefly what Solver is doing This way you can appreciate some of the difficulties with IP models and you might also understand some of the messages you see in the status bar as Solver performs its optimization 62 Overview of Optimization with Integer Variables 295 Except for binary or integer constraints on some changing cells all models in this chapter are linear The branch and bound algorithm is a general approach to searching through all of the pos sibly millions of solu tions in an efficient manner Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it time The lower the upper bounds you can produce the quicker you can prune branches and the faster the algorithm will be The procedures used to find good upper bounds for branches are beyond the level of this book Fortunately Solver takes care of the details However you should now under stand some of the messages you will see in the status bar when you run Solver on IP mod els For example try running Solver on the cutting stock model in Example 67 with a tolerance of 0 see below You will see plenty of these messages where the incumbent objective value and the current subproblem or branch quickly flash by For this particular cutting stock model Solver quickly finds an incumbent solution that is optimal but it must examine literally thousands of branches before it can guarantee that the incumbent is opti mal After a minute or two of computing we had seen results for 10000 branches and there was no end in sight The Solver Tolerance Setting The Solver Options dialog box contains a Tolerance setting which is relevant for integer constrained models Excels default tolerance is 5 In Excel 2010 this setting listed as Integer Optimality is found under Solver Options in the dialog box shown in Figure 61 In earlier versions it was also under Solver Options but in a slightly differ ent dialog box To explain the Tolerance option we must first define the LP relaxation of an IP model This is the same model as the IP model except that all integer constraints are omitted In particular cells that are originally constrained to be binary are allowed under the LP relaxation to have any fractional values between 0 and 1 including 0 and 1 The LP relaxation is typically easy to solve using the simplex method and it provides a bound for the IP model For example consider a maximization problem where the optimal solution to the LP relaxation has an optimal objective value of 48214 Then the optimal objective for the original integerconstrained problem can be no larger than 48214 so this value represents an upper bound for the original problem 62 Overview of Optimization with Integer Variables 297 Figure 61 Solver Tolerance Setting Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it A tolerance setting of 5 means that Solver stops as soon as it finds a feasible inte ger solution to the IP model that is within 5 of the current upper bound Initially the optimal objective value of the LP relaxation serves as the upper bound As Solver proceeds to find solutions that satisfy the integer constraints it keeps updating the upper bound The exact details need not concern you The important point is that when Solver stops it guar antees an integer solution that is within at least 5 of the optimal integer solution The implication is that if you set the tolerance to 0 Solver will in theory run until it finds the optimal integer solution So why isnt a tolerance setting of 0 always used The reason is that for many IP models especially large models it can take Solver a long time to find the optimal solution or guarantee that the best solution found so far is opti mal On the other hand a solution that is close to optimalwithin 5 saycan often be found quickly This explains why Frontline Systems the developer of Solver chose the default tolerance setting of 5 We use a tolerance of 0 for all the models in this chapter simply to guarantee an optimal solution Therefore if you use the default tolerance of 5 you might get a solu tion that is slightly worse than ours 298 Chapter 6 Optimization Models with Integer Variables To guarantee an opti mal integer solution change the Solver tol erance setting to 0 The disadvantage of this approach is that Solver can run consid erably longer on large models FUNDAMENTAL INSIGHT Recognizing the Optimal Integer Solution IP algorithms such as brand and bound often find a very good integer solution v ery quickly So why do they sometimes run so longThis is due to the implicit enumeration aspect of the algorithms They have diffi culty ruling out large n umbers of potential solutions until they have searched all r egions of the solution space In other words they have difficulty recognizing that the y might ha ve f ound the optimal solution because ther e ar e man y potential solutions the y havent yet exploredWhen you run Solver on a rea sonably large IP model watch the status bar Often a very g ood incumbent solution the best solution found so far is found within seconds but then Solver spins its wheels f or minutes or even hours trying to verify that this solution is optimal Solver Messages Until now the only Solver message you have probably seen is the final one that says an optimal solution has been found When you run Solver on some of the difficult problems in this chapter however you might see a few other messages such as those in Figures 62 and 63 These are due to Solver running a long time and bumping into the limits in the Options dialog box in Figure 61 If you see one of these types of messages you have two options First you can change the options in Figure 61 You would have to make this change before the Solver run For example you could increase the Max Subproblems setting to a number greater than 5000 Second you can simply click on Continue to let Solver run Figure 62 Max Subproblems Warning Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it longer We recommend the second option Actually if you are tired of waiting and believe the incumbent solution is good enough you can also click on Stop in which case Solver gives you the option of saving the best solution so far 63 CAPITAL BUDGETING MODELS Perhaps the simplest binary IP model is the following capital budgeting example which illustrates the gonogo nature of many IP models 63 Capital Budgeting Models 299 Figure 63 Time Limit Warning E X A M P L E 61 SELECTING INVESTMENTS AT TATHAM T he Tatham Company is considering seven investments The cash required for each investment and the net present value NPV each investment adds to the firm are listed in Table 61 The cash available for investment is 15000 Tatham wants to find the invest ment policy that maximizes its NPV The crucial assumption here is that if Tatham wants to take part in any of these investments it must go all the way It cannot for example go halfway in investment 1 by investing 2500 and realizing an NPV of 8000 In fact if par tial investments were allowed you wouldnt need IP you could use LP Table 61 Data for the Capital Budgeting Example Investment Cash Required NPV 1 5000 16000 2 2500 8000 3 3500 10000 4 6000 19500 5 7000 22000 6 4500 12000 7 3000 7500 Objective To use binary IP to find the set of investments that stays within budget and maximizes total NPV WHERE DO THE NUMBERS COME FROM The initial required cash and the available budget are easy to obtain Obtaining the NPV for each investment is undoubtedly harder A time sequence of anticipated cash inflows from the investments and a discount factor are required Simulation might even be used to estimate these NPVs In any case financial analysts must provide the estimations of the required NPVs Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it USING SOLVER The Solver dialog box appears in Figure 65 The objective is to maximize the total NPV sub ject to staying within the budget However the changing cells must be constrained to be 01 Fortunately Solver makes this simple as shown in Figure 66 You add a constraint with Investmentslevels in the left box and choose the bin option in the middle box The word binary in the right box is then added automatically Note that if all changing cells are binary you do not need to check the NonNegative option because 0 and 1 are certainly nonnega tive but you should still choose the Simplex LP method if the model is linear as it is here2 63 Capital Budgeting Models 301 Solver makes it easy to specify binary con straints by clicking on the bin option Figure 65 Solver Dialog Box for the Capital Budgeting Model Figure 66 Specifying a Binary Constraint 2 All the models in this chapter satisfy two of the three properties of linear models in Chapter 3 proportionality and additivity Even though they clearly violate the third assumption divisibility which precludes integer constraints they are still considered linear by Solver Therefore you should still choose the Simplex LP method Discussion of the Solution The optimal solution in Figure 64 indicates that Tatham can obtain a maximum NPV of 46000 by selecting investments 1 2 and 5 These three investments consume only 14500 of the available budget with 500 left over However this 500 is not enoughbecause of the investing all the way requirementto invest in any of the remaining investments If Tathams investments are ranked on the basis of NPV per dollar invested see row 7 of Figure 64 the ranking from best to worst is 4 1 2 5 3 6 7 Using your economic Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it total NPV for the model Then manually choose the one that stays within the budget and has the largest NPV 10 Make up an example as described in Problem 8 with 20 possible investments However do it so the ratios of NPV to cash requirement are in a very tight range from 30 to 32 Then use Solver to find the optimal 306 Chapter 6 Optimization Models with Integer Variables solution when the Solver tolerance is set to its default value of 5 and record the solution Next solve again with the tolerance set to 0 Do you get the same solution Try this on a few more instances of the model where you keep changing the inputs The ques tion is whether the tolerance setting matters in these types of close call problems 64 FIXEDCOST MODELS In many situations a cost is incurred if an activity is undertaken at any positive level This cost is independent of the level of the activity and is known as a fixed cost or fixed charge Here are three examples of fixed costs The construction of a warehouse incurs a fixed cost that is the same whether the warehouse is built with a low or a highcapacity level A cash withdrawal from a bank incurs a fixed cost independent of the size of the withdrawal A machine that is used to produce several products must be set up for the production of each product Regardless of the batch size produced the same fixed cost lost pro duction due to the setup time is incurred In these examples a fixed cost is incurred if an activity is undertaken at any positive level whereas no fixed cost is incurred if the activity is not undertaken at all Although it might not be obvious this feature makes the problem inherently nonlinear which means that a straightforward application of LP is not possible However a clever use of 01 vari ables can result in a model with linear constraints and a linear objective It is important to realize that the type of model discussed here and throughout the rest of the chapter except for Example 67 is fundamentally different from the previous capital bud geting model and the integerconstrained models in Chapter 4YoudonotsimplycreateanLP model and then add integer constraints Instead you use 01 variables to model the logic The logic in this section is that if a certain activity is done at any positive level a fixed cost is incurred However no fixed cost is incurred if the activity is not done at all Your first instinct might be to handle such logic with IF functions However Solver cannot handle IF functions predictably This is not really a weakness of Solver These types of problems are inherently dif ficult Fortunately Solver is able to handle linear models with binary variables so this is the approach you should take whenever possible The appropriate use of 01 variables allows you to solve a whole new class of difficult problems The following example is typical FUNDAMENTAL INSIGHT Binary Variables for Modeling Binary variables are often used to transform a nonlin ear model into a linear integer model For example a fixed cost is not a linear function of the le vel of some activity it is either incurred or it isnt incurred This type of onoff beha vior is difficult f or nonlinear solvers to handle However this behavior can often be handled easil y when binar y variables ar e used to make the model linear Still large models with man y binary variables can be difficult to solveOne approach is to solve the model without integer constraints and then round fractional values to the near est integer 0 or 1 Unfortunately this approach is typically not very g ood because the r ounded solution is often infeasible and even if it is f easible its objective value can be considerably worse than the optimal objective value Unless you use binary variables to handle the logic fixedcost models are nonlinear and diffi cult to solve Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Although Solver finds the optimal solution automatically you should understand the effect of the logical upper bound constraint on production It rules out a solution such as the one shown in Figure 612 This solution calls for a positive production level of pants but does not incur the fixed cost of the pants equipment The logical upper bound con straint rules this out because it prevents a positive value in row 16 if the corresponding binary value in row 14 is 0 In other words if the company wants to produce some pants the constraint in Inequality 61 forces the associated binary variable to be 1 thus incur ring the fixed cost for pants Note that Inequality 61 does not rule out the situation you see for skirts in Figure 612 where the binary value is 1 and the production level is 0 However Solver will never choose this type of solution as optimal Solver recognizes that the binary value in this case can be changed to 0 so that no skirt equipment is rented and its fixed cost is not incurred Discussion of the Solution The optimal solution in Figure 610 indicates that Great Threads should produce about 966 shorts and 379 jackets but no shirts pants or skirts The total profit is 54614 Note that the 01 variables for shirts pants and skirts are all 0 which forces production of these products to be 0 However the 01 variables for shorts and jackets the products that are produced are 1 This ensures that the fixed cost of producing shorts and jackets is included in the total cost It might be helpful to think of this solution as occurring in two stages In the first stage Solver determines which products to producein this case shorts and jackets only Then in the second stage Solver determines how many shorts and jackets to produce If you know that the company plans to produce shorts and jackets only you could then ignore the fixed costs and determine the best production quantities with the same types of product mix models discussed in Chapter 3 Of course these two stagesdeciding which products 310 Chapter 6 Optimization Models with Integer Variables Figure 611 Solver Dialog Box for the FixedCost Model There is no point to set ting a binary variable equal to 1and Solver will never do itunless there is positive produc tion of that product Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it of the file The resulting model looks the same as in Figure 611 but it incorporates the fol lowing changes The binary range is no longer part of the changing cells range Instead cell B14 con tains the formula IFB16010 which is copied across to cell F14 Logically this probably appears more natural If a production quantity is positive a 1 is entered in row 14 which means that the fixed cost is incurred The effective capacities in row 18 are modeled with IF functions Specifically cell B18 contains the formula IFB160MIND22B5D23B60 which is copied across to cell F18 Actually this constraint isnt even necessary now Why The Solver dialog box is now set up as shown in Figure 615 The Rentequipment range is not part of the changing cells range and there is no binary constraint The GRG Nonlinear method is selected because the IF functions make the model nonlinear When we ran Solver on this modified model we found inconsistent results depending on the initial production quantities entered in row 16 For example when we entered initial values all equal to 0 the Solver solution was exactly thatall 0s Of course this solution is terrible because it leads to a profit of 0 However when we entered initial production quantities all equal to 100 Solver found the correct optimal solution the same as in Figure 610 Was this just lucky To check we tried another initial solution where the pro duction quantities for shorts and jackets were 0 and the production quantities for shirts pants and skirts were all 500 In this case Solver found a solution where only skirts are produced Of course we know this is not optimal The moral is that the IFfunction approach is not the way to go Its success depends strongly on the initial values entered in the changing cells and this requires you to make very good guesses In contrast the binary approach ensures that you get the correct solu tion regardless of the initial values in the changing cells 64 FixedCost Models 313 Figure 615 Solver Dialog Box When IF Functions Are Used You can try modeling the logic with IF func tions but depending on the initial values in the changing cells Solver is likely to get the wrong solution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The following example is similar to the Great Threads example in that there is a fixed cost for any positive level of production of a given product However an additional requirement states that if the company produces any of a given product then possibly because of economies of scale it must produce at least some minimal level such as 1000 This is a typical example of a problem with eitheror constraints The companys level of production must either be 0 or at least 1000 In the next example we show how the use of binary variables allows you to model the eitheror constraints in a linear manner 314 Chapter 6 Optimization Models with Integer Variables E X A M P L E 63 MANUFACTURING AT DORIAN AUTO D orian Auto is considering manufacturing three types of cars compact midsize and large and two types of minivans midsize and large The resources required and the profit contributions yielded by each type of vehicle are shown in Table 65 At present 6500 tons of steel and 65000 hours of labor are available If any vehicles of a given type are produced production of that type of vehicle is economically feasible only if at least a minimal number of that type are produced These minimal numbers are also listed in Table 65 Dorian wants to find a production schedule that maximizes its profit Table 65 Data for the Dorian Car Example Vehicle Compact Midsize Large Midsize Large Type Car Car Car Minivan Minivan Steel tonsunit 15 3 5 6 8 Labor hoursunit 30 25 40 45 55 Minimum production if any 1000 1000 1000 200 200 Profit contributionunit 2000 2500 3000 5500 7000 Objective To use a binary model to determine which types of vehicles to produce above their minimal requirements and in what quantities to maximize profit WHERE DO THE NUMBERS COME FROM This is basically a product mix problem similar to those in Chapter 3 Therefore the same comments about inputs discussed there apply here as well The only new inputs in this problem are the minimal production quantities These might be policy decisions determined by Dorianmanagement sees no reason to produce midsize minivans unless it can produce at least 200 of them saybut these policy decisions are undoubt edly based on costs Presumably the fixed costs of product design manufacturing and marketing are prohibitive unless a minimal number of any vehicle type is produced Solution The variables and constraints for the Dorian model are listed in Table 66 Dorian must decide not only how many of each type of vehicle to produce but also which types to pro duce Of course after it decides to produce midsize minivans say then it must produce at least 200 of them The constraints include the usual resource availability constraints In addi tion there are lower and upper limits on the production quantities of any vehicle type The lower limit is zero or the minimal production quantity depending on whether that vehicle type is produced The upper limit is similar to the upper limit in the Great Threads fixedcost Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it minimum number 200 of each type of minivan More specifically the company should produce just enough compact cars and midsize minivans to meet the minimal production quantities These vehicle types are relatively profitable given the resources they use However they are evidently not as profitable as large minivans The company should make as many of these as it can after producing the compact cars and midsize minivans until it runs out of labor hours This solution is certainly not intuitive For example if large minivans are so prof itable why doesnt the company produce all large minivans and nothing else Do you see why Also this solution appears to be very sensitive to the inputs Although we do not present any formal sensitivity analysis with SolverTable we urge you to try different val ues for the minimal production quantities the unit profit contributions andor the resource availabilities We found that even small changes in these can yield a very different optimal production policy For example you can check that if the availability of steel decreases to 6000 tons only compact cars and midsize minivans are produced both above their mini mal levels and no large minivans are produced 64 FixedCost Models 317 Figure 617 Solver Dialog Box for the Dorian Production Model Locating Distribution Centers When Dow Consumer Products a manufacturer of foodcare products acquired the Texize homecare product lines of Morton Thiokol in 1985 to form DowBrands Inc the distribution channels of the two organizations remained for the most part separate Each had its own district and regional distribution centers for storing and then shipping products to the customer regions This led to possible inefficiencies in a business where keeping logistics costs low is the key to survival Robinson et al 1993 acting as consultants for ADDITIONAL APPLICATIONS Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it DowBrands modeled the problem as a fixedcost network problemwhich distribution centers to keep open and which routes to use to satisfy which customers with which prod ucts The study was highly successful and convinced DowBrands to close a significant number of distribution centers to reduce costs 318 Chapter 6 Optimization Models with Integer Variables P R O B L E M S SkillBuilding Problems 11 How difficult is it to expand the Great Threads model to accommodate another type of clothing Answer by assuming that the company can also produce sweat shirts The rental cost for sweatshirt equipment is 1100 the variable cost per unit and the selling price are 15 and 45 respectively and each sweatshirt requires one labor hour and 35 square yards of cloth 12 Referring to the previous problem if it is optimal for the company to produce sweatshirts use SolverTable to see how much larger the fixed cost of sweatshirt machinery would have to be before the company would not produce any sweatshirts However if the solution to the previous problem calls for no sweat shirts to be produced use SolverTable to see how much lower the fixed cost of sweatshirt machinery would have to be before the company would start producing sweatshirts 13 In the Great Threads model we didnt constrain the production quantities in row 16 to be integers arguing that any fractional values could be safely rounded to integers See whether this is true Constrain these quantities to be integers and then run Solver Are the optimal integer values the same as the rounded frac tional values in Figure 610 14 In the optimal solution to the Great Threads model the labor hour and cloth constraints are both bindingthe company is using all it has a Use SolverTable to see what happens to the opti mal solution when the amount of available cloth increases from its current value You can choose the range of input values to use Capture all of the changing cells the labor hours and cloth used and the profit as outputs in the table The real issue here is whether the company can profitably use more cloth when it is already constrained by labor hours b Repeat part a but reverse the roles of labor hours and cloth That is use the available labor hours as the input for SolverTable 15 In the optimal solution to the Great Threads model no pants are produced Suppose Great Threads has an order for 300 pairs of pants that must be produced Modify the model appropriately and use Solver to find the new optimal solution Is it enough to put a lower bound of 300 on the production quantity in cell D16 Will this automatically force the binary value in cell D14 to be 1 Explain How much profit does the company lose because of having to produce pants 16 In the Dorian production model the optimal solution calls for the minimum number of compact cars and midsize minivans to be produced but for more than the minimum number of large minivans to be pro duced If the large minivans are evidently that prof itable why doesnt Dorian discontinue making compact cars and midsize minivans and instead pro duce even more large minivans 17 As the Dorian production model is currently stated each vehicle type has a minimum production level if this type is produced at all its production quantity must be at least this minimum Suppose that for large minivans there is also a maximum production level of 400 If large minivans are produced the production level must be from 200 to 400 Modify the model as necessary and use Solver to find the new optimal solu tion How do you know that the current optimal solu tion is not optimal for the modified model 18 The optimal solution to the Dorian production model appears to be sensitive to the model inputs For each of the following inputs create a oneway Solver Table that captures all changing cells and the target cell as outputs You can choose the ranges of these inputs to make the results interesting Comment on your results a The steel available b The labor hours available c The unit profit contribution of large minivans d The minimum production level currently 200 of large minivans e The minimum production level currently 1000 of compact cars 19 If Solver could handle IF functions correctly how would you use them in the Dorian production example to create an arguably more natural modelwithout binary variables Run Solver on your modified model Do you get the correct solution Note You will have to use the GRG Nonlinear method Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SkillExtending Problems 20 In the Great Threads model you found an upper bound on production of any clothing type by calculat ing the amount that could be produced if all of the resources were devoted to this clothing type a What if you instead used a very large value such as 1000000 for this upper bound Try it and see whether you get the same optimal solution b Explain why any such upper bound is required Exactly what role does it play in the model devel oped in this section 21 In the last sheet of the file Fixed Cost Manufacturingxlsx we illustrated one way to model the Great Threads problem with IF functions that didnt work Try a slightly different approach here Eliminate the binary variables in row 14 altogether and eliminate the upper bounds in row 18 and the corresponding upper bound constraints in the Solver dialog box The only constraints will now be the resource availability con straints However use IF functions to calculate the total fixed cost of renting equipment so that if the amount of any clothing type is positive its fixed cost 65 SetCovering and LocationAssignment Models 319 is added to the total fixed cost Is Solver able to handle this model Does it depend on the initial values in the changing cells Dont forget to use the GRG Nonlinear method 22 In the Dorian production model suppose that the pro duction quantity of compact cars must either be less than or equal to 100 a small batch or greater than or equal to 1000 a large batch The same statements hold for the other vehicle types as well except that the small and large batch limits for both sizes of minivans are 50 and 200 Modify the model appropriately and use Solver to find the optimal solution 23 Suppose in the Dorian production model that no minimum production limits are placed on the individ ual vehicle types However minimum production limits are placed on all cars and on all minivans Specifically if Dorian produces any cars regardless of size it must produce at least 1500 cars total Similarly if the company produces any minivans it must produce at least 1000 minivans total Modify the model appropriately and use Solver to find the optimal solution 65 SETCOVERING AND LOCATIONASSIGNMENT MODELS Many companies have geographically dispersed customers that they must service in some way To do this they create service center facilities at selected locations and then assign each customer to one of the service centers Various costs are incurred including 1 fixed costs of locating service centers in particular locations 2 operating costs depending on the service centers locations and 3 transportation costs depending on the distances between customers and their assigned service centers In this section we illustrate several examples of this basic problem We first examine a particular type of location model called a setcovering model In a setcovering model each member of a given set set 1 must be covered by an acceptable member of another set set 2 The usual objective in a setcovering problem is to minimize the number of members in set 2 that are needed to cover all the members in set 1 For example set 1 might consist of all cities in a county and set 2 might consist of the cities where a fire station is located A fire station covers a city if the fire station is located say within 10 minutes of the city The goal is to minimize the number of fire stations needed to cover all cities Setcovering models have been applied to areas as diverse as airline crew scheduling truck dispatching political redistricting and capital investment The following example presents a typical setcovering model E X A M P L E 64 HUB LOCATION AT WESTERN AIRLINES W estern Airlines wants to design a hub system in the United States Each hub is used for connecting flights to and from cities within 1000 miles of the hub Western runs flights among the following cities Atlanta Boston Chicago Denver Houston Los Angeles New Orleans New York Pittsburgh Salt Lake City San Francisco and Seattle The company wants to determine the smallest number of hubs it needs to cover all these Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it USING SOLVER The Solver dialog box is shown in Figure 619 The objective is to minimize the total num ber of hubs subject to covering each city by at least one hub and ensuring that the chang ing cells are binary As usual you should select the Simplex LP method 322 Chapter 6 Optimization Models with Integer Variables Figure 619 Solver Dialog Box for the SetCovering Model Discussion of the Solution Figure 620 is a graphical representation of the optimal solution where the double ovals indicate hub locations and the large circles indicate ranges covered by the hubs These large circles are not drawn to scale In reality they should be circles of radius 1000 miles centered at the hubs Three hubsin Houston New York and Salt Lake Cityare needed6 Would you have guessed this The Houston hub covers Houston Atlanta and New Orleans The New York hub covers Atlanta Pittsburgh Boston New York and Chicago The Salt Lake City hub covers Denver Los Angeles Salt Lake City San Francisco and Seattle Note that Atlanta is the only city covered by two hubs it can be serviced by New York or Houston Sensitivity Analysis An interesting sensitivity analysis for Westerns problem is to see how the solution is affected by the mile limit Currently a hub can service all cities within 1000 miles What if the limit were 800 or 1200 miles say To answer this question data on actual distances among all the cities must be collected After you have a matrix of these distances you can build the 01 matrix corresponding to the range B6M17 in Figure 618 with IF functions 6 Multiple optimal solutions exist for this model all requiring three hubs so you might obtain a different solution from ours Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The modified model appears in Figure 621 See the file Locating Hubs 2xlsx The typi cal formula in B24 is IFB8B410 which is then copied to the rest of the B24M35 range7 You can then run SolverTable selecting cell B4 as the single input cell letting it vary from 800 to 1200 in increments of 100 and keeping track of where the hubs are located and the number of hubs The SolverTable results at the bottom show the effect of the mile limit When this limit is lowered to 800 miles four hubs are required but when it is increased to 1100 or 1200 only two hubs are required By the way the solution shown for the 1000mile limit is different from the previous solution in Figure 618 because of multiple optimal solutions but it still requires three hubs 65 SetCovering and LocationAssignment Models 323 Bos Chi Atl NO Hou Sea LA SL Den SF Pit NY Figure 620 Graphical Solution to the SetCovering Model 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N Oneway analysis for Solver model in Model worksheet Mile limit cell B4 values along side output cells along top AT BO CH DE HO LA NO NY PI SL SF SE Totalhubs 800 1 1 0 0 0 0 0 0 0 1 0 1 4 900 1 1 0 0 0 0 0 0 0 1 0 0 3 1000 1 1 0 0 0 0 0 0 0 1 0 0 3 1100 0 0 1 0 0 0 0 0 0 1 0 0 2 1200 0 0 1 0 0 1 0 0 0 0 0 0 2 Figure 621 Sensitivity to Mile Limit 7We have warned you about using IF functions in Solver models However the current use affects only the inputs to the problem not quantities that depend on the changing cells Therefore it causes no problems Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 324 Chapter 6 Optimization Models with Integer Variables Locating Florida Disaster Recovery Centers In 2001 the Federal Emergency Management Agency FEMA required every Florida county to identify potential locations for disaster recovery centers DRCs Dekle et al 2005 describe a study sponsored by Alachua County in northcentral Florida to identify potential DRC sites The authors developed a version of the setcovering model with a twostage approach The first stage required each resident to be within 20 miles of the clos est DRC It identified a solution with three DRC locations The second stage then refined this solution to relax the 20mile requirement and include evaluation criteria not included in stage 1 The final results provided significant improvements over the original FEMA location criteria and it maintained acceptable travel distances to the nearest DRC Selecting Receiver Locations for Automated Meter Reading Gavirneni et al 2004 developed and solved a setcovering model for Schlumberger a utility company The company needed to deploy its receivers on utility poles so that all wireless meters in the region can transmit their readings to at least one receiver The authors solved a largescale model with 116600 meters and 20636 utility poles The following example is similar to a setcovering model but it also has an assign ment component ADDITIONAL APPLICATIONS E X A M P L E 65 LOCATING AND ASSIGNING SERVICE CENTERS AT UNITED COPIERS U nited Copiers sells and services copy machines to customers in 11 cities throughout the country The company wants to set up service centers in three of these cities After United Copiers chooses the location of the service centers it must assign customers in each city to one of the service centers For example if it decides to locate a service cen ter in New York and then assigns its Boston customers to the New York service center a service representative from New York will travel from Boston when services are required there The distances in miles between the cities are listed in Table 69 The estimated annual numbers of trips to the various customers are listed in Table 610 What Table 69 Distances for the Service Center Example Los New San Boston Chicago Dallas Denver Angeles Miami York Phoenix Pittsburgh Francisco Seattle Boston 0 983 1815 1991 3036 1539 213 2664 792 2385 2612 Chicago 983 0 1205 1050 2112 1390 840 1729 457 2212 2052 Dallas 1815 1205 0 801 1425 1332 1604 1027 1237 1765 2404 Denver 1991 1050 801 0 1174 2041 1780 836 1411 1765 1373 Los Angeles 3036 2112 1425 1174 0 2757 2825 398 2456 403 1909 Miami 1539 1390 1332 2041 2757 0 1258 2359 1250 3097 3389 New York 213 840 1604 1780 2825 1258 0 2442 386 3036 2900 Phoenix 2664 1729 1027 836 398 2359 2442 0 2073 800 1482 Pittsburgh 792 457 1237 1411 2456 1250 386 2073 0 2653 2517 San Francisco 2385 2212 1765 1765 403 3097 3036 800 2653 0 817 Seattle 2612 2052 2404 1373 1909 3389 2900 1482 2517 817 0 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 65 SetCovering and LocationAssignment Models 327 4 Number of ser vice centers Calculate the number of service centers with the formula SUMIncludeservicecenter in cell M19 This just sums the 01 range so it equals the number of 1s 5 Number of service centers assigned to each city Calculate the number of service centers assigned to each city with row sums in the Totalassignments range in column M That is enter the formula SUMB23L23 in cell M23 and copy it down to cell M33 These row sums will eventually be constrained to equal 1 to ensure that exactly one service center is assigned to each city 6 Total annual distances Calculate the total annual distance traveled in 1000s of miles to each city by entering the formula B40SUMPRODUCTB5L5B23L231000 in cell C40 for Boston and copying it down to cell C50 for the other cities Note that this SUMPRODUCT includes a row of distances from Boston and a row of assignments to cus tomers in Boston This row of assignments will eventually include only a single 1only a single service center will be assigned to customers in Boston Therefore this SUMPROD UCT will be the distance between Boston and the service center assigned to Boston It is multiplied by the annual trips to Boston cell B40 to obtain the total annual distance trav eled to Boston and it is divided by 1000 to convert to thousands of miles 7 Logical capacities You need to ensure that only existing service locations can be assigned to customers One way to ensure this is to calculate column sums of the binary variables in row 34 For example the 2 in cell D34 indicates that two cities are serviced by Dallas Dallas and Denver Then create the logical capacities in row 36 by entering the formula 11B19 in cell B36 and copying it across row 36 The effect is that if a binary value in row 19 is 0 then no cities can be serviced by the corresponding city For example this is the case for Boston However if the binary value in row 19 is 1 then the logical capacity is 11 the number of cities and this capacity constraint is essentially irrelevant 8 Total annual distance tra veled Calculate the total distance traveled annually in 1000s of miles in cell B53 with the formula SUMC40C50 USING SOLVER The completed Solver dialog box is shown in Figure 623 You should also set the Solver tolerance to 0 There is no need to check the NonNegative option because all changing cells are binary and hence nonnegative Discussion of the Solution The optimal solution in Figure 622 indicates that United Copiers should locate service centers in Dallas New York and San Francisco Of course each of these centers services the customers in its own city In addition the Dallas center services customers in Denver the New York center services customers in Boston Chicago Miami and Pittsburgh Always be careful to convert to appropriate units of measurement if necessaryA factor such as 100 or 1000 in a formula is often evidence of a measure ment conversion Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The final example in this section is structurally similar to the service center location model but it arises in a slightly different business context9 65 SetCovering and LocationAssignment Models 329 E X A M P L E 66 MANUFACTURING AND DISTRIBUTING FERTILIZER AT GREEN GRASS Like the previous example this example is basically a fixedcost locationassignment model However one difference here is that not all customers need to be assigned T he Green Grass Company manufactures and distributes a fertilizer product The company sells its product to highvolume customers in various US cities where it has manufacturing plants but it can decide to operate only some of these plants in any given month The fixed monthly cost for operating any plant is 60000 the plant capacity for any operating plant is 2500 pounds per month and the production cost at any operating plant is 1025 per pound After the product is manufactured it is shipped to customers at a rate of 002 per pound per mile The cities and the dis tances between them are listed in Table 612 The customers submit order sizes and price bids to Green Grass as listed in Table 613 For example the customer in Boston requires an order of 1430 pounds this month and is willing to pay 75740 for it Green Grass can decide to fill this order or not If not you can assume that the cus tomer takes its business to another company For the current month Green Grass must decide which plants to operate and which customers to service from which operating plants to maximize its monthly profit 9This example is based on a real problem Winston was asked to solve during a consulting experience with a major US manufacturing company Table 612 Distances Between Cities for the Green Grass Example Boston Chicago Dallas Denver Los Angeles Miami New York Phoenix Boston 0 983 1815 1991 3036 1539 213 2664 Chicago 983 0 1205 1050 2112 1390 840 1729 Dallas 1815 1205 0 801 1425 1332 1604 1027 Denver 1991 1050 801 0 1174 2065 1780 836 Los Angeles 3036 2112 1425 1174 0 2757 2825 398 Miami 1539 1390 1332 2065 2757 0 1258 2359 New York 213 840 1604 1780 2825 1258 0 2442 Phoenix 2664 1729 1027 836 398 2359 2442 0 Table 613 Orders and Price Bids for the Green Grass Example Quantity Price Boston 1430 75740 Chicago 870 44370 Dallas 770 46320 Denver 1140 87780 Los Angeles 700 43850 Miami 830 21000 New York 1230 74850 Phoenix 1070 83980 Objective To develop a binary model to help Green Grass decide which manufacturing plants to operate and which customer orders to fill from which operating plants WHERE DO THE NUMBERS COME FROM The distances in Table 612 are well known and the customers can supply the data in Table 613 Cost accountants can supply the fixed cost of operating a plant the variable production cost per pound and the unit shipping cost per mile Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 8 Fixed costs Each 1 in the Openplant range adds a fixed cost To calculate the total fixed cost enter the formula B6SUMOpenplant in cell B50 This is the number of open plants multiplied by the fixed cost per plant 9 Monthly profit Calculate the monthly profit in cell B51 with the formula SUMB41I48B50 USING SOLVER The Solver dialog box is shown in Figure 626 As usual you should select the Simplex LP method but you do not need to check the NonNegative option because all changing cells are constrained to be binary hence nonnegative The last constraint ensures that each plant produces nothing if it isnt open and no more than its capacity if it is open The second constraint ensures that each customers demand is satisfied by at most one plant This allows the possibility that a customers demand is not satisfied by Green Grass at all 332 Chapter 6 Optimization Models with Integer Variables Figure 626 Solver Dialog Box for the Green Grass Model Discussion of the Solution The optimal solution in Figure 625 indicates that the company should open four plants Boston to supply the Boston customer Denver to supply the Denver and Dallas cus tomers New York to supply the New York and Chicago customers and Phoenix to sup ply the Phoenix and Los Angeles customers In addition the model indicates that Green Grass should not supply the Miami customer at all You can see the main reason for this if you calculate the ratio of order size to price bid for each customer Miamis ratio is well below the others Therefore it is evidently not profitable to supply the Miami customer Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Sensitivity Analysis One possible sensitivity analysis is to see how much larger Miamis price bid needs to be before Green Grass supplies it We tried this varying Miamis price bid and keeping track of the row sum in cell J31 that indicates whether Miami is supplied The results after some trial and error to find an interesting price bid range appear in Figure 627 When the Miami price bid increases to some value between 31000 and 32000 it becomes prof itable to supply Miami You can check by rerunning Solver that Miami is then supplied by New York 65 SetCovering and LocationAssignment Models 333 1 2 3 4 5 6 7 8 9 10 11 12 A B C D E F G Oneway analysis for Solver model in Model worksheet Miami bid price cell M16 values along side output cells along top NumberservicedbyMiami 28000 0 29000 0 30000 0 31000 0 32000 1 33000 1 34000 1 35000 1 Figure 627 Sensitivity to Miamis Price Bid Another possible sensitivity analysis is on the common plant capacity currently 2500 pounds The optimal solution in Figure 625 indicates that capacity is not currently a constraining factor Four of the plants are open and all are operating well under capacity Therefore an increase in the common capacity has absolutely no effect and a slight decrease down to 2100 the highest plant production also has no effect However any decrease below 2100 should have an effect This is explored in Figure 628 where the common plant capacity is varied and the optimal total fixed cost and profit are outputs As you can see if the capacity is below 2100 the total profit decreases However the total fixed cost remains constant at least for this range of capacities This implies that all of 1 2 3 4 5 6 7 8 9 A B C D E F Oneway analysis for Solver model in Model worksheet Plant capacity cell B7 values along side output cells along top Monthly fixed cost Totalmonthlyprofit 1500 240000 32433 1750 240000 32433 2000 240000 89628 2250 240000 110464 2500 240000 110464 Figure 628 Sensitivity to Common Plant Capacity Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it these solutions keep four plants open How does the optimal solution change Although the results in Figure 628 do not provide the answer you can rerun Solver with any of these capacities to find out It turns out that the same four plants stay open but supply fewer cus tomers For example when the common capacity is 1500 or 1750 the four plants supply only the customers in their respective cities If you run these sensitivity analyses with SolverTable you will immediately notice the longer computing times These are difficult problems even for Solver and you wont get the immediate solutions you are accustomed to Each problem has 272 possible binary solutions because there are 72 binary changing cells which is an enormous number of potential solutions for Solver to sort through with its branch and bound algorithm Although a binary model of this type and size is still well within Solvers capabilities this example should con vince you that not all management science optimization models are easy to solve 334 Chapter 6 Optimization Models with Integer Variables 1 We have assumed that all possible plant locations are in the same cities as the customers This is not necessary There could be any number of customers at one set of locations and any other number of plant locations at another set of locations As long as the dis tances from each plant to each customer are known the model changes hardly at all 2 We have assumed that the inputs in the range B4B7 see Figure 625 are constant the same for each plant or plantcustomer pair This is also not necessary If these inputs differ across plants or plantcustomer pairs more input values must be esti mated by the cost accountants but modifications to the model itself are minimal 3 We currently assume that the plants in the various locations are already built and it is just a matter of which to open each month Suppose instead that the company is expand ing and must decide where or whether to build new plants Then there is a onetime fixed cost of building each new plant in addition to the fixed cost of opening an existing plant in the example Unfortunately combining these costs is not a trivial matter The fixed cost of building must be amortized over some period of time so that it can be com bined correctly with the monthly revenues and costs in the current model MODELING ISSUES P R O B L E M S SkillBuilding Problems 24 In the original Western setcovering model in Figure 618 we used the number of hubs as the objec tive to minimize Suppose instead that there is a fixed cost of locating a hub in any city where these fixed costs can possibly vary across cities Make up some reasonable fixed costs modify the model appropri ately and use Solver to find the solution that mini mizes the sum of fixed costs 25 In the original Western setcovering model in Figure 618 we assumed that each city must be cov ered by at least one hub Suppose that for added flexi bility in flight routing Western requires that each city must be covered by at least two hubs How do the model and optimal solution change 26 Setcovering models such as the original Western model in Figure 618 often have multiple optimal solutions See how many alternative optimal solutions you can find Of course each must use three hubs because this is optimal Hint Use various initial values in the changing cells and then run Solver repeatedly10 27 How hard is it to expand a setcovering model to ac commodate new cities Answer this by modifying the model in Figure 621 See the file Locating Hubs 2xlsx Add several cities that must be served Memphis 10One of our colleagues at Indiana University Vic Cabot now deceased worked for years trying to develop a general algorithm other than trial and error for finding all alternative optimal solutions to optimization models It turns out that this is a very difficult problemand one that Vic never totally solved Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Dallas Tucson Philadelphia Cleveland and Buffalo You can look up the distances from these cities to each other and to the other cities in a reference book or on the Web or you can make up approximate distances a Modify the model appropriately assuming that these new cities must be covered and are candi dates for hub locations b Modify the model appropriately assuming that these new cities must be covered but are not candidates for hub locations 28 In the United Copiers service center model we assumed that the potential locations of service centers are the same as existing customer locations Change the model so that the customer locations are the ones given but the only potential service center locations are in Memphis Houston Cleveland Buffalo Minneapolis St Louis and Kansas City You can look up the distances from these cities to the customer cities in a reference book or on the Web or you can make up approximate distances Use Solver to find the optimal solution 29 In the United Copiers service center model we used total distance traveled as the objective to minimize Suppose in addition that there is an annual fixed cost of locating a service center in any city where this fixed cost can vary across cities There is also a cost per mile of traveling Modify the current model to make total annual cost the objective to minimize You can make up reasonable fixed costs and unit traveling costs 30 In the Green Grass shipping model we assumed that certain inputs see the range B4B7 in Figure 625 are the same for all plants or plantcustomer combina tions Change this so that the unit production cost the monthly fixed cost and the monthly capacity can vary by plant and the unit shipping cost can vary by plantcustomer combination You can make up data that vary around the values in the B4B7 range Use Solver to find the new optimal solution 31 In the optimal solution to the Green Grass shipping model the Miami customers order is not satisfied Suppose that Green Grass decides as a matter of pol icy to satisfy each customers order at the customers bid price How much profit will the company lose from this policy decision 32 In the Green Grass shipping model use SolverTable to perform a sensitivity analysis on the fixed cost of opening a plant letting it vary over some reasonable 66 Cutting Stock Models 335 range that extends below and above the current value of 60000 Keep track of enough outputs so that you can see the effect on the plants that are opened and the customers whose orders are satisfied as well as on the total profit Summarize your findings in words SkillExtending Problems 33 In the United Copiers service center model we assumed that a customer is serviced totally by a single service center Suppose a customer can be serviced partly by multiple service centers For example the customer in Denver could get half of its service from Dallas and the other half from San Francisco In this case you can assume that half of Denvers annual trips would be made from Dallas reps and half by San Francisco reps Modify the model appropriately and then solve it with Solver How do you interpret the optimal solution Hint Allow the changing cells in the Assignments range to be fractional values between 0 and 1 34 In the Green Grass shipping model we assumed that the plants are already built so that in each month the only decision is whether to open particular plants at a monthly fixed cost Consider instead a general locationshipping model of this type where the plants are not yet built The company must first decide where to build plants then how much to produce at the plants and finally which customers to service from them The problem is that the building costs are onetime costs whereas other costs are monthly How can you recon cile these two types of costs What should you use as an objective to minimize Illustrate your procedure on the Green Grass example where the plant opening fixed costs are ignoredwe assume that all plants that are built will remain openbut building costs which you can make up are given 35 In the Green Grass shipping model we currently assume that if a customers order is satisfied it must be satisfied from a single plant Suppose instead that it can be satisfied from more than one plant For exam ple if the company decides to satisfy Dallass order it could ship part of this order from Denver and part from Phoenix or some other combination of open plants Continue to assume however that the com pany must satisfy either all or none of each customers order Modify the model appropriately and use Solver to solve it Does the solution change 66 CUTTING STOCK MODELS The final model we discuss in this chapter has found many realworld applications espe cially in manufacturing The model is relevant in situations where a product is produced in a standard size which must then be cut into one of several patterns to satisfy customer orders In contrast to the other models in this chapter this cutting stock model does not Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it available width If the waste were 12 or greater it would be possible to get another usable cut from the pattern For this particular roll width and this particular set of available widths there are 26 feasible patterns You have to be careful when listing them It is easy to miss some 3 Decision variables Enter any values into the Rollscut range These are the decision variables in this model They indicate how many rolls to cut into the various patterns 4 Widths obtained Calculate the number of each width obtained by entering the formula SUMPRODUCTRollscutB10B35 in cell B40 and copying it to the rest of the Obtained range For example the value in cell B40 is the number of rolls of width 12 inches obtained from all possible patterns 5 Rolls cut Calculate the number of rolls cut in cell B45 with the formula SUMRollscut USING SOLVER Fill out the Solver dialog box as indicated in Figure 630 The objective is to minimize the number of rolls produced subject to meeting customer orders Also the number cut according to each pattern must be an integer but not binary As usual you should check the NonNegative option and choose the Simplex LP method Discussion of the Solution The solution indicates that Rheem can meet its customer orders this week with 47 rolls cut as specified in rows 10 through 35 For example 12 of the 47 rolls should be cut according to pattern 4 each with three 12inch rolls and one 24inch roll There is at least one other optimal solution with objective value 47 that you might find Note that there are two 338 Chapter 6 Optimization Models with Integer Variables Figure 630 Solver Dialog Box for the Cutting Stock Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it sources of waste in this solution First there is the unusable waste from all leftover rolls with width less than 12 inches For example there are two 1inch rolls left over from the two rolls cut into pattern 20 Second there is some waste from the usable rolls that are not needed in this weeks orders Fortunately it is minimalonly one 15inch roll is left over Actually if Rheem solves this model on a weekly basis the model could easily incorporate the inventory of usable leftover rolls from previous weeks Solver Tolerance Setting Until now we have suggested setting the Solver tolerance to 0 This guarantees the opti mal solution However this example illustrates why the default tolerance setting is 5 or at least not 0 If you set the tolerance to 0 and click on Solve you will see that Solver quickly gets to a solution that requires 47 rolls but then it runs and runs and runs We got tired of waiting so we pressed the CtrlBreak key combination to stop it prematurely After some experimenting we found that with the tolerance set at 2 or above the solu tion is obtained almost instantaneously but with the tolerance set at 1 or 0 it runs seemingly forever This behavior is not at all uncommon in IP models Solver often finds a very good or even optimal solution very quickly but then it takes a long time to verify that it is optimal or to find something slightly better The moral is clear If you set the toler ance to a low value and find that the Solver is taking forever without getting anywhere press CtrlBreak to get out By that time you probably already have a very good or even optimal solution 66 Cutting Stock Models 339 We did not perform any sensitivity analysis on this model because there is no obvious sen sitivity analysis to perform The only inputs are the roll width the set of available widths and the order amounts Although it would make sense to perform sensitivity analysis on the order amounts it would make more sense in a realistic setting to wait for next weeks orders and simply solve the problem again Note that the model is not set up to perform sensitivity analysis with SolverTable on the roll width or the set of available widths If these change the entire table of patterns must be recreated manually For example if the roll width changes to 64 inches patterns 2 9 11 14 16 and 23 are no longer in the list why not and several new patterns enter the list what are they MODELING ISSUES P R O B L E M S SkillBuilding Problems 36 In the cutting stock example we minimized the total number of rolls cut Do you get the same solution if you minimize the total inches of waste For example given the solution in Figure 629 this waste includes 2 inches from pattern 6 12 inches from the extra 12inch roll produced in cell B40 and a couple of others Solve the problem with this objective 37 Woodco sells 3foot 5foot and 9foot pieces of lum ber Woodcos customers demand 25 3foot boards 20 5foot boards and 15 9foot boards Woodco meets its demands by cutting up 17foot boards How can it satisfy its customers demands with the least amount of waste Assume that all boards are the same width and thickness SkillExtending Problem 38 The Mayfree Appliance Company requires sheet metal for its appliances The company can purchase long coils of sheet metal in two different widths 65 inches and 40 inches The company must purchase the coils by linear foot of length 120 per foot for a 64inch coil and 100 per foot for a 40inch coil This implies that a square foot say of the wider coil is less Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it expensive Up to 4000 feet of the 65inch coil is available and up to 6000 feet of the 40inch coil is available There are manufacturing requirements for six different widths 50 45 40 35 20 and 10 inches Mayfrees requirements are expressed as lengths of the various widths The company requires 1000 feet of 50inch width 2500 feet of 45inch width 3000 feet of 40inch width 2300 feet of 35inch width 1300 feet of 20inch width and 2000 feet of 10inch width Determine how much of each width coil Mayfree should purchase and how it should cut the coils into various widths to meet its requirements at minimal cost Hint First list all patterns that can be cut from a 65inch coil and do the same for a 40inch coil Then have a changing cell for each pattern that designates the number of linear feet to be cut in this pattern 340 Chapter 6 Optimization Models with Integer Variables 67 CONCLUSION Three important points emerge from this chapter A wide variety of important problems can be modeled as IP problems with binary variables These can generally be identified as problems where at least some of the activities such as making a particular investment opening a particular plant or supplying a customer from a particular plant must be done or not done there is no inbetween Regular LP models cannot handle these problems IP models with binary variables often can Some IP models are simply LP models with integer constraints on the variables For example you might constrain the number of refrigerators produced to be an integer These problems can often be solved by solving the associated LP model and then rounding the solution to integer values Although there is no guarantee that the rounded solution is optimal it is often close enough In contrast most of the problems discussed in this chapter introduce binary decision variables that specify whether an activity is done or not If you ignore the binary constraints and only constrain these variables to be between 0 and 1 it is generally impossible to find the optimal solution by rounding The solution approach required for IP problems especially those with 01 variables is inherently more difficult than the simplex method for LP problems The relatively small examples in this chapter might give the impression that a spreadsheet Solver can handle IP models just as easily as it handles LP models but this is definitely not the case In fact even with the most sophisticated IP computer codes on the most power ful computers there are IP problemsfrom real applicationsthat defy solution Analysts typically employ heuristic methods on these really difficult problems Summary of Key Management Science Terms Term Explanation Page Binary variables Variables constrained to have values 1 or 0 usually used to indicate 294 whether an activity is undertaken or not Also called 01 variables IP models Optimization models where some or all of the decision variables 294 are constrained to have integer values Branch and bound algorithm A general algorithm for searching through all integer solutions in an 295 IP model Complete enumeration An exhaustive method of checking the objective value of every 296 feasible integer solution Implicit enumeration A clever way of checking that no feasible integer solution can possibly 296 be better than the optimal solution without explicitly looking at each feasible integer solution Incumbent solution The best feasible solution found so far 296 continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Term Explanation Page LP relaxation The same linear model but without the integer constraints 297 Fixedcost models Difficulttosolve models where certain costs are fixed at some positive 306 level if an activity is undertaken at any level and are 0 otherwise Eitheror constraints Constraints where one of two mutually exclusive conditions must be satisfied 314 Setcovering models Models where members of one set such as ambulances must be located 319 so that they cover members of another set such as city districts Location models Models where items such as branch offices must be located to provide 319 required services at minimal cost Summary of Key Excel Terms Term Explanation Excel Page Solver Tolerance setting Setting that specifies whether Specify under Solver Options 297 Solver will stop at a nearoptimal default 5 doesnt guarantee integer solution or will continue optimality 0 does to optimality 67 Conclusion 341 P R O B L E M S SkillBuilding Problems 39 Four projects are available for investment The pro jects require the cash flows and yield the net present values in millions shown in the file P0639xlsx If 6 million is available now for investment find the investment plan that maximizes NPV 40 You are given a group of possible investment projects for your companys capital For each project you are given the NPV the project would add to the firm as well as the cash outflow required by each project dur ing each year Given the information in the file P0640xlsx determine the investments that maximize the firms NPV The firm has 30 million available during each of the next five years All numbers are in millions of dollars 41 You are moving from New Jersey to Indiana and have rented a truck that can haul up to 1100 cubic feet of furniture The volume and value of each item you are considering moving on the truck are given in the file P0641xlsx Which items should you bring to Indiana 42 NASA must determine how many of three types of objects to bring on board the space shuttle The weight and benefit of each of the items are given in the file P0642xlsx If the space shuttle can carry up to 2600 pounds of items 1 through 3 how many of each item should be taken on the space shuttle assuming that at least one of each is necessary 43 Coach Night is trying to choose the starting lineup for the basketball team The team consists of seven play ers who have been rated on a scale of 1 poor to 3 excellent according to their ball handling shooting rebounding and defensive abilities The positions that each player is allowed to play and the players abilities are listed in the file P0643xlsx The fiveplayer start ing lineup must satisfy the following restrictions At least four members must be able to play guard G at least two members must be able to play for ward F and at least one member must be able to play center C The average ballhandling shooting and rebound ing level of the starting lineup must each be at least 18 Either player 2 or player 3 or both must start Given these constraints Coach Night wants to maximize the total defensive ability of the starting team Use Solver to determine his starting team 44 To graduate from Southeastern University with a major in operations research OR a student must complete at least two math courses at least two OR courses and at least two computer courses Some courses can be used to fulfill more than one require ment Calculus can fulfill the math requirement Operations Research can fulfill the math and OR requirements Data Structures can fulfill the computer and math requirements Business Statistics can fulfill the math and OR requirements Computer Simulation can fulfill the OR and computer requirements Introduction to Computer Programming can fulfill the computer requirement and Forecasting can fulfill the OR and math requirements Some courses are prereq uisites for others Calculus is a prerequisite for Business Statistics Introduction to Computer Programming is a prerequisite for Computer Simulation and for Data Structures and Business Statistics is a prerequisite for Forecasting Determine how to minimize the number of courses needed to Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it satisfy the major requirements Hint Because Calculus is a prerequisite for Business Statistics for example you will need a constraint that ensures that the changing cell for Calculus is greater than or equal to the changing cell for Business Statistics 45 Based on Bean et al 1987 Boris Milkems firm owns six assets The expected selling price in millions of dollars for each asset is given in the file P0645xlsx For example if asset 1 is sold in year 2 the firm receives 20 million To maintain a regular cash flow Milkem must sell at least 20 million of assets during year 1 at least 30 million worth during year 2 and at least 35 million worth during year 3 Determine how Milkem can maximize his total rev enue from assets sold during the next three years 46 The Cubs are trying to determine which of the follow ing freeagent pitchers should be signed Rick Sutcliffe RS Bruce Sutter BS Dennis Eckersley DE Steve Trout ST or Tim Stoddard TS Feel free to substitute your own set of players for these old guys The cost of signing each pitcher and the predicted number of victories each pitcher will add to the Cubs are listed in the file P0646xlsx The Cubs want to sign the pitchers who will add the most victo ries to the team Determine who the Cubs should sign based on the following restrictions At most 25 million can be spent At most two righthanded pitchers can be signed The Cubs cannot sign both BS and RS 47 Based on Sonderman and Abrahamson 1985 In treating a brain tumor with radiation physicians want the maximum amount of radiation possible to bom bard the tissue containing the tumors The constraint is however that there is a maximum amount of radia tion that normal tissue can handle without suffering tissue damage Physicians must therefore decide how to aim the radiation to maximize the radiation that hits the tumor tissue subject to the constraint of not damaging the normal tissue As a simple example of this situation suppose six types of radiation beams beams differ in where they are aimed and their inten sity can be aimed at a tumor The region containing the tumor has been divided into six regions three regions contain tumors and three contain normal tis sue The amount of radiation delivered to each region by each type of beam is shown in the file P0647xlsx If each region of normal tissue can handle at most 60 units of radiation which beams should be used to maximize the total amount of radiation received by the tumors 48 Because of excessive pollution on the Momiss River the state of Momiss is going to build some pollution control stations Three sites are under consideration Momiss is interested in controlling the pollution levels of two pollutants The state legislature requires that at least 80000 tons of pollutant 1 and at least 60000 tons of pollutant 2 be removed from the river The relevant data for this problem are shown in the file P0648xlsx The last two rows indicate the number of tons of pollutants removed per ton treated a Determine how to minimize the cost of meeting the state legislatures goals b Use SolverTable to analyze how a change in the requirement for pollutant 1 changes the optimal solution Do the same for pollutant 2 49 A manufacturer can sell product 1 at a profit of 20 per unit and product 2 at a profit of 40 per unit Three units of raw material are needed to manufacture one unit of product 1 and six units of raw material are needed to manufacture one unit of product 2 A total of 15000 units of raw material are available If any product 1 is produced a setup cost of 20000 is incurred if any product 2 is produced a setup cost of 35000 is incurred a Determine how to maximize the manufacturers profit b If either of the products is not produced in the opti mal solution use SolverTable to see how much this products unit profit must be before it will be pro duced and then use SolverTable again to see how much this products fixed cost must be decreased before it will be produced 50 A company is considering opening warehouses in four cities New York Los Angeles Chicago and Atlanta Each warehouse can ship 10000 units per week The weekly fixed cost of keeping each warehouse open is 40000 for New York 50000 for Los Angeles 30000 for Chicago and 25000 for Atlanta Region 1 of the country requires 8000 units per week region 2 requires 7000 units per week and region 3 requires 4000 units per week The costs including production and shipping costs of sending one unit from a ware house to a region are shown in the file P0650xlsx The company wants to meet weekly demands at mini mum cost subject to the preceding information and the following restrictions If the New York warehouse is opened then the Los Angeles warehouse must be opened At most two warehouses can be opened Either the Atlanta or the Los Angeles warehouse must be opened 51 Glueco produces three types of glue on two different production lines Each line can be used by up to 20 workers at a time Workers are paid 500 per week on production line 1 and 900 per week on production line 2 For a week of production it costs 5000 to set up production line 1 and 4000 to set up production line 2 During a week on a production line each worker produces the number of units of glue shown in the file P0651xlsx Each week at least 800 units of 342 Chapter 6 Optimization Models with Integer Variables Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 61 Heinsco produces tomato sauce at five different plants The tomato sauce is then shipped to one of three ware houses where it is stored until it is shipped to one of the companys four customers The following inputs for the problem are given in the file P0661xlsx The plant capacities in tons The cost per ton of producing tomato sauce at each plant and shipping it to each warehouse The cost of shipping a ton of sauce from each warehouse to each customer The customer requirements in tons of sauce The fixed annual cost of operating each plant and warehouse Heinsco must decide which plants and warehouses to open and which routes from plants to warehouses and from warehouses to customers to use All customer demand must be met A given customers demand can be met from more than one warehouse and a given plant can ship to more than one warehouse a Determine the minimumcost method for meeting customer demands b Use SolverTable to see how a change in the capac ity of plant 1 affects the total cost c Use SolverTable to see how a change in the cus tomer 2 demand affects the total cost d Suppose that each customers demand must be met from a single warehouse Solve the problem with this restriction 62 Eight jobs need to be completed Each job can be com pleted on any of six machines and each machine can complete any number of jobs If a machine is assigned to at least one job the setup time listed in the file P0662xlsx is required All times are in hours The time required for each machine to complete each job excluding the setup time is also listed in the same file How should the jobs be assigned to machines to mini mize the sum of setup times and job completion times 63 Based on Walker 1974 The Smalltown Fire Department currently has seven conventional ladder companies and seven alarm boxes The two closest ladder companies to each alarm box are listed in the file P0663xlsx The town council wants to maximize the number of conventional ladder companies that can be replaced with tower ladder companies Unfortunately political considerations dictate that a conventional company can be replaced only if after replacement at least one of the two closest companies to each alarm box is still a conventional company Determine how to maximize the number of conven tional companies that can be replaced by tower companies 64 State University must purchase 1100 computers from three vendors Vendor 1 charges 500 per computer plus a total delivery charge of 5000 Vendor 2 charges 350 per computer plus a total delivery charge of 4000 Vendor 3 charges 250 per computer plus a total delivery charge of 6000 Vendor 1 will sell the university at most 500 computers vendor 2 at most 900 and vendor 3 at most 400 The minimum order from any vendor is 200 computers Determine how to minimize the cost of purchasing the needed computers 65 At Blair General Hospital six types of surgical opera tions are performed The types of operations each sur geon is qualified to perform indicated by an X are listed in the file P0665xlsx Suppose that surgeons 1 and 2 dislike each other and cannot be on duty at the same time Determine the minimum number of sur geons required so that the hospital can perform all types of surgery 66 Eastinghouse ships 12000 capacitors per month to its customers The capacitors can be produced at three different plants The production capacity fixed monthly cost of operation and variable cost of produc ing a capacitor at each plant are given in the file P0666xlsx The fixed cost for a plant is incurred only if the plant is used to make any capacitors If a plant is used at all at least 3000 capacitors per month must be produced at the plant Determine how to minimize the companys monthly costs of meeting its customers demands 67 Based on Liggett 1973 A court decision has stated that the enrollment of each high school in Metropolis must be at least 20 black The numbers of black students and white students in each of the citys five school districts are listed in the file P0667xlsx The distance in miles that a student in each district must travel to each high school is shown in the same file School board policy requires that all students in a given district must attend the same school Assuming that each school must have an enrollment of at least 150 students determine how to minimize the total distance that Metropolis students must travel to high school 68 Based on Westerberg Bjorklund and Hultman 1977 Newcors steel mill has received an order for 150 tons of steel The steel must be 5 carbon and 5 molyb denum by weight The steel is manufactured by com bining three types of metal steel ingots scrap steel and alloys Four individual steel ingots are available At most one of each can be purchased The weight in tons cost per ton and the carbon and molybdenum content of each ingot are given in the file P0668xlsx Three types of alloys can be purchased The cost per ton and chemical makeup of each alloy are given in the same file Steel scrap can be purchased at a cost of 100 per ton Steel scrap contains 3 carbon and 9 molybdenum Determine how Newcor can mini mize the cost of filling its order 69 Based on Boykin 1985 Chemco annually produces 359 million pounds of the chemical maleic anhydride 344 Chapter 6 Optimization Models with Integer Variables Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it A total of four reactors are available to produce maleic anhydride Each reactor can be run on one of three settings The cost in thousands of dollars and pounds produced in millions annually for each reactor and each setting are given in the file P0669xlsx A reactor can be run on only one set ting for the entire year Determine how Chemco can minimize the cost of meeting its annual demand for maleic anhydride 70 Based on Zangwill 1992 Hallco runs a day shift and a night shift Regardless of the number of units pro duced the only production cost during a shift is a setup cost It costs 8000 to run the day shift and 4500 to run the night shift Demand for the next two days is as follows day 1 2000 night 1 3000 day 2 2000 night 2 3000 It costs 1 per unit to hold a unit in inventory for a shift a Determine a production schedule that minimizes the sum of setup and inventory costs All demand must be met on time Note Not all shifts have to be run b After listening to a seminar on the virtues of the Japanese theory of production Hallco has cut the setup cost of its day shift to 1000 per shift and the setup cost of its night shift to 3500 per shift Now determine a production schedule that minimizes the sum of setup and inventory costs All demand must be met on time Show that the decrease in setup costs has actually raised the average inventory level Is this reasonable 71 Based on Fitzsimmons and Allen 1983 The State of Texas frequently audits companies doing business in Texas Because these companies often have head quarters located outside the state auditors must be sent to outofstate locations Each year auditors must make 500 trips to cities in the Northeast 400 trips to cities in the Midwest 300 trips to cities in the West and 400 trips to cities in the South Texas is considering basing auditors in Chicago New York Atlanta and Los Angeles The annual cost of basing auditors in any city is 100000 The cost of sending an auditor from any of these cities to a given region of the country is given in the file P0671xlsx Determine how to minimize the annual cost of conducting outofstate audits SkillExtending Problems 72 Suppose you own 11 bronze coins worth a total of 150 11 silver coins worth a total of 160 and 11 gold coins worth a total of 170 Develop a linear integer model to find a combination of coins worth exactly 110 73 Cousin Bruzie of radio station WABC schedules radio commercials in 60second blocks This hour the sta tion has sold time for commercials of 15 16 20 25 30 35 40 and 50 seconds Determine the minimum number of 60second blocks of commercials that must be scheduled to fit in all the current hours commercials 74 Based on Bean et al 1988 Simons Mall has 10000 square feet of space to rent and wants to deter mine the types of stores that should occupy the mall The minimum number and maximum number of each type of store along with the square footage of each type are given in the file P0674xlsx The annual profit made by each type of store depends on how many stores of that type are in the mall This depen dence is given in the same file For example if two department stores are in the mall each department store will earn 210000 profit per year Each store pays 5 of its annual profit as rent to Simons Determine how Simon can maximize its rental income from the mall 75 Indiana Universitys Business School has two rooms that seat 50 students one room that seats 100 students and one room that seats 150 students Classes are held five hours a day At present the four types of requests for rooms are listed in the file P0675xlsx The busi ness school must decide how many requests of each type to assign to each type of room Suppose that classes that cannot be assigned to a business school room are assigned to another campus building Determine how to assign classes to minimize the num ber of hours students spend each week outside the business building 76 Based on Efroymson and Ray 1966 Stonecutters is a new bakery chain that sells bread to customers throughout the state of Indiana Stonecutters is consid ering building bakeries in three locations Evansville Indianapolis and South Bend Each bakery can bake up to 900000 loaves of bread each year The cost of building a bakery at each site is 5 million in Evansville 4 million in Indianapolis and 45 mil lion in South Bend To simplify the problem we assume that Stonecutters has only three customers Their demands each year are 700000 loaves cus tomer 1 400000 loaves customer 2 and 300000 loaves customer 3 The total cost of baking and ship ping a loaf of bread to a customer is given in the file P0676xlsx Assume that future shipping and produc tion costs are discounted at a rate of 12 per year Assume that once built a bakery lasts forever How would you minimize the companys total cost of meeting demand present and future 77 On Monday morning you have 3000 in cash on hand For the next seven days the following cash requirements must be met Monday 5000 Tuesday 6000 Wednesday 9000 Thursday 2000 Friday 7000 Saturday 2000 Sunday 3000 At the begin ning of each day you must decide how much money if any to withdraw from the bank It costs 10 to make a withdrawal of any size You believe that the 67 Conclusion 345 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it from any of these cities to any region of the country You also know the hourly wage that you must pay workers in each city This information is listed in the file P0685xlsx Assume that an average call requires four minutes of labor You make calls 250 days per year and the average number of calls made per day to each region of the country is listed in the same file The cost of building a calling center in each possible location is also listed in this file Each calling center can make up to 5000 calls per day Given this informa tion how can you minimize the discounted cost at 10 per year of running the telemarketing opera tion for 10 years Assume all wage and calling costs are paid at the ends of the respective years 86 State University is scheduling 24 sections of a large computer skills course in the Fall semester There are eight time slots for these sections four on Monday Wednesday MW and four on TuesdayThursday TR In each time slot three sections are scheduled These are shown in the file P0686xlsx The sec tions will be taught by six instructors Instructors 1 to 3 must teach at least three sections and no more than four sections each Instructors 4 to 6 must teach at least four sections and no more than five sections each The instructors have submitted their top four prefer ences for time slots as shown in the file Four points are awarded for satisfying an instructors first prefer ence three for second preference two for third preference and one for fourth preference These points appear in the file For example instructor 1s preferences are in decreasing order MW 910 MW 11noon MW 12 and TR 11noon Find an assignment of instructors to sections that maximizes the points from satisfying preferences Of course no instructor can teach more than one section in the same time slot 87 Hoosier Power needs to determine a capacity expan sion plan to meet Bloomingtons power needs for the next 20 years The current capacity is 5000 kwh The demand for the current year is 4000 kwh and demand is expected to increase by 1000 kwh in each succeed ing year At the beginning of each year Hoosier Power must determine the amount of capacity to add given the following inputs Any year in which capacity is added a fixed cost of 120000 is incurred plus a cost of 120 per kwh of capacity At most 10000 kwh of capacity can be added in a single year It costs 25 per year to maintain a unit of capacity It costs 12 per year to produce a kwh If production does not meet demand a shortage cost of 80 per kwh short is incurred Develop a linear integer model to help Hoosier Power minimize its costs for the next 20 years 88 Based on Angel et al 2003 A fertilizer company is trying to determine the cheapest fertilizer mix that provides desired amounts of nutrients The mix is made by combining the following fertilizers SSA SPO GUR TSP KCI FERT and SPF The mix can not contain both GUR and TSP The percentage of potassium K sulfur S calcium Ca sodium Na and phosphorus P in each fertilizer is listed in the file P0688xlsx For example a pound of SSA is 16 K and 26 Na The mix must contain at least 600 pounds of K 550 pounds of S 750 pounds of Ca 900 pounds of Na and 750 pounds of P The mix cannot contain both GUR and TSP because if both are present in the mix the affect of other fertilizers is nullified The cost per pound in cents of each fertilizer is listed in the same file Develop a linear integer model to find the minimumcost fertilizer mixture that meets the stated chemical requirements 89 Sam is in his final year of college and is trying to schedule his courses for the year He has narrowed his search to 16 courses each of which is offered in at least one time slot out of a possible five time slots in each semester The file P0689xlsx lists the courses and when they are offered For example course C1 is offered in time slots T4 and T5 during semester S1 and in time slot T3 in semester S2 The course also lists the values Sam attaches to the various coursetime slotsemester combinations on a 1 to 10 scale Assuming that Sam must take exactly five courses each semester find the combination that maximizes the total value of the courses he takes Of course he cant take the same course more than once and he cant take more than one course at the same time 90 A medical supply company has customers in eight cities It is trying to decide how many salespeople it needs to service these customers Each salesperson needs to be located in one of the eight cities and needs to be assigned to a subset of the customers For example the company might base a salesperson in New York and have this person service customers in New York Boston and Philadelphia Each salesper son receives an annual salary of 50000 and can work as many as 230 days per year This includes days working with customers and days traveling to and from customers The file P0690xlsx contains data on the annual travel costs for example 15900 for a salesperson based in New York traveling for customers in Orlando the annual number of days of work required for the customers and the annual num ber of days traveling to and from customers Find an assignment that minimizes the total cost of salaries and traveling The solution should indicate the num ber of salespeople to employ where they should be based and which cities they should serve Assume that customers in a given city must be serviced by a single salesperson 67 Conclusion 347 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 67 Conclusion 349 is a binary for customer segment c buying product j and M is a large number M equal to the largest prod uct utility will work This constraint ensures that the ycj binary can equal 1 only if the binary xj equals 1 that is customer segment c can buy product j only if it is included in the product line Note that if ycj is 0 then this inequality is automatically satisfied Modeling Problems 97 Suppose that you want to divide a state containing 12 cities into five congressional districts How might you use IP to assign cities to districts 98 An insurance company has hired you to determine the number of sales divisions into which the country should be divided Each division will need a presi dent a vice president and a divisional staff The time needed to call on a client will depend on the distance of the salesperson from the client Discuss how you would determine the optimal number of sales divisions and the allocation of the companys sales force to the various divisions 99 Ten different types of brownies are sold You are think ing of developing a new brownie for sale Brownies are rated on the basis of five qualities price chocolate fla vor chewiness sweetness and ease of preparation You want to group the 10 brownies on the market into three clusters Each cluster should contain brownies that are relatively similar a Why would this be useful to you b How would you do it 100 Telco a national telemarketing firm usually picks a number of sites around the country from which it makes its calls As a service ADDs telecommunica tion marketing department wants to help Telco choose the number and location of its sites How can IP be used to approach this problem Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 350 Chapter 6 Optimization Models with Integer Variables T his case deals with strategic planning issues for a large company The main issue is planning the companys production capacity for the coming year At issue is the overall level of capacity and the type of capacityfor example the degree of flexibility in the manufacturing system The main tool used to aid the companys planning process is a mixed integer linear programming MILP model A mixed integer program has both integer and continuous variables Problem Statement The Giant Motor Company GMC produces three lines of cars for the domestic US market Lyras Libras and Hydras The Lyra is a relatively inexpen sive subcompact car that appeals mainly to firsttime car owners and to households using it as a second car for commuting The Libra is a sporty compact car that is sleeker faster and roomier than the LyraWithout any options the Libra costs slightly more than the Lyra additional options increase the price The Hydra is the luxury car of the GMC line It is significantly more expensive than the Lyra and Libra and it has the highest profit margin of the three cars Retooling Options f or Capacity Expansion Currently GMC has three manufacturing plants in the United States Each plant is dedicated to producing a single line of cars In its planning for the coming year GMC is considering the retooling of its Lyra andor Libra plants Retooling either plant would C A S E 61 GIANT MOTOR COMPANY represent a major expense for the company The retooled plants would have significantly increased production capacities Although having greater fixed costs the retooled plants would be more efficient and have lower marginal production coststhat is higher marginal profit contributions In addition the retooled plants would be flexiblethey would have the capability of producing more than one line of cars The characteristics of the current plants and the retooled plants are given in Table 616 The retooled Lyra and Libra plants are prefaced by the word new The fixed costs and capacities in Table 616 are given on an annual basis A dash in the profit margin sec tion indicates that the plant cannot manufacture that line of car For example the new Lyra plant would be capable of producing both Lyras and Libras but not Hydras The new Libra plant would be capable of producing any of the three lines of cars Note how ever that the new Libra plant has a slightly lower profit margin for producing Hydras than the Hydra plant The flexible new Libra plant is capable of pro ducing the luxury Hydra model but is not as efficient as the current Hydra plant that is dedicated to Hydra production The fixed costs are annual costs incurred by GMC independent of the number of cars produced by the plant For the current plant configurations the fixed costs include property taxes insurance pay ments on the loan that was taken out to construct the plant and so on If a plant is retooled the fixed costs will include the previous fixed costs plus the additional cost of the renovation The additional Table 616 Plant Characteristics Lyra Libra Hydra New Lyra New Libra Capacity in 1000s 1000 800 900 1600 1800 Fixed cost in millions 2000 2000 2600 3400 3700 Profit Margin by Car Line in 1000s Lyra 2 25 23 Libra 3 30 35 Hydra 5 48 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 353 Nonlinear Optimization Models C H A P T E R POR TFOLIO OPTIMIZA TION AT GE P ortfolio optimization one of the models discussed in this chapter is big business This is illustrated in the article by Chalermkraivuth et al 2005 They describe how GE Asset Management Incorporated GEAM a wholly owned subsidiary of General Electric Company GE manages investment portfolios on behalf of various GE units and more than 200 unaffiliated clients worldwide worth billions of dollars GEAM manages portfolios of assets pro duced by various insurance businesses and its investments are primarily in corporate and government bonds The authors developed a specialpurpose algorithm for finding optimal portfolios Since 2003 their algorithm has been used to optimize more than 30 portfolios valued at over 30 billion They esti mate thatbased on 100 billion of assetsthe present value of potential bene fits from their approach is approximately 75 million over a fiveyear period As in most portfolio optimization problemsGEAM wants to find portfolios that provide appropriate riskreturn tradeoffs preferably higher expected returns and lower risk Howeverthe insurance industry is more complex than this portfolio managers must choose the assets within a port folio so that their characteristics match those of the firms liabilities They try to do this in such a way that the bonds and other financial instruments in the portfolio areimmunized against changes in the interest ratesone main source of risk in bond portfolios This can be done through a welldeveloped financial theory of matching theduration andconvexity of the assets and wavebreakmedia ltd2010Used under license from Shutterstockcom 7 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it liabilities within an acceptable toleranceSee Luenberger 1997for examplefor a dis cussion of the financial theory Using this theorythe authors formulated a portfolio opti mization model using the variance of economic surplus as a measure of riskwhere economic surplus is the difference between the market value of assets and liabilities Unfortunately most GEAM portfolios consist of a large number of securities and the risk measure is inherently nonlinear This combinationa large model with inherent nonlinearityis extremely difficult for even the best commercial optimization software Therefore the authors developed their own algorithm to locate the efficient frontier that is the set of portfolios that provide the maximum expected return for a given level of risk This approach is typical in the management science field If analysts encounter a problem that is either too large or too difficult to solve with existing algorithms they try to develop a new algorithm usually specific to the problem which can do the job In the authors algorithm they first find the point on the efficient frontier that maximizes the expected return without any regard for risk The result is typically a very risky portfolio Then in general given a set of portfolios on the efficient frontier they find a nearby port folio with slightly less risk and slightly less expected return than the previous one To do this they approximate the nonlinear portfolio variance by a linear function This approxi mation has two properties that recommend it 1 it is a very good approximation in the area of the previous optimal portfolio and 2 it yields a linear programming model that can be solved fairly quickly In the modern spirit of management science the authors went one step further They not only developed an algorithm that could be used to solve GEAMs large prob lems but they also developed a Webbased implementation that is easy for their clients to use With this system which has been in place for several years users do not need to install software on their desktops They can instead interact via the Web which provides the user interface The Web application processes user inputs and requests and displays results An optimization engine called MATLAB handles all of the heavy number crunch ing on a centralized application server and the required data is stored and accessed from an Oracle database Obviously this is a complex setup and months went into its devel opment But this is a small price to pay for the benefits the portfolio optimization model provides to GE and its customers 71 INTRODUCTION In many complex optimization problems the objective andor the constraints are nonlinear functions of the decision variables Such optimization problems are called nonlinear programming NLP problems In this chapter we discuss a variety of interesting problems with inherent nonlinearities from product pricing to portfolio optimization to rating sports teams A model can become nonlinear for several reasons including the following There are nonconstant returns to scale which means that the effect of some input on some output is nonlinear For example consider the effect of advertising on sales Advertising typically creates a saturation effect so that beyond some level extra advertising dollars have very little effect on salesmuch less than the effect of initial advertising dollars This violates the proportionality assumption of linear models discussed in Chapter 3 In pricing models where the goal is to maximize revenue or profit revenue is price multiplied by quantity sold and price is typically the decision variable Because 354 Chapter 7 Nonlinear Optimization Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it quantity sold is related to price through a demand function revenue is really price multiplied by a function of price and this product is a nonlinear function of price For example even if the demand function is linear in price the product of price and demand is quadratic in price because it includes a squared price term Analysts often try to find the model that best fits observed data To measure the goodness of the fit they typically sum the squared differences between the observed values and the models predicted values Then they attempt to minimize this sum of squared differences The squaring introduces nonlinearity In one of the most used financial models the portfolio optimization model financial analysts try to invest in various securities to achieve high return and low risk The risk is typically measured as the variance or standard deviation of the portfolio and it is inherently a nonlinear function of the decision variables the investment amounts As these examples illustrate nonlinear models are common in the real world In fact it is probably more accurate to state that truly linear models are hard to find The real world often behaves in a nonlinear manner so when you model a problem with LP you are typically approximating reality By allowing nonlinearities in your models you can often create more realistic models Unfortunately this comes at a pricenonlinear optimization models are more difficult to solve 72 BASIC IDEAS OF NONLINEAR OPTIMIZATION When you solve an LP problem with Solver you are guaranteed that the Solver solution is optimal When you solve an NLP problem however Solver sometimes obtains a subopti mal solution For example if you use Solver to maximize the function in Figure 71 it might have difficulty For the function graphed in this figure points A and C are called local maxima because the function is larger at A and C than at nearby points However only point A actually maximizes the function it is called the global maximum The prob lem is that Solver can get stuck near point C concluding that C maximizes the function and not find point A Similarly points B and D are local minima because the function has a lower value at B and D than at nearby points However only point D is a global minimum If you ask Solver to minimize this function it might concludeincorrectlythat point B is optimal 72 Basic Ideas of Nonlinear Optimization 355 Nonlinear models are often more realistic than linear models but they are also more difficult to solve A local optimum is better than all nearby points A global optimum is the best point in the entire feasible region For some NLP problems Solver can get stuck at a local optimum and never find the global optimum A C B D Figure 71 Function with Local Maxima and Minima Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 360 Chapter 7 Nonlinear Optimization Models Figure 77 Multistart Option in Excel 2010 only downside to this option is that it takes longer so if you know that no local optima are not globally optimal there is no need to use this option To use the Multistart option select the GRG Nonlinear method in the Solver dialog box click on Options and then on the GRG Nonlinear tab You can then check the Use Multistart box as shown in Figure 77 As an example we tried Multistart on the model in Figure 76 Regardless of the starting value in cell E5 Solver always found the globally optimal solution 1355567 The three options within the Multistart box can be useful The Population Size is the number of starting solutions chosen It must be at least 10 and 100 is suggested The Random Seed determines whether the starting solutions are the same from one run to the next If it is 0 the starting solutions are selected randomly but if it is positive they are always the same This might be useful when several users are testing the same model Finally if the Require Bounds on Variables box is checked you must include explicit lower and upper bound constraints on all changing cells Although this can be a nuisance Solvers online help indicates that the Multistart option works much better if such bounds exist In fact the tighter the bounds are the less searching Solver needs to perform Note that if this box is checked and you do not have explicit lower and upper bounds you will get the error message in Figure 78 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 73 PRICING MODELS Setting prices on products and services is becoming a critical decision for many compa nies A good example is pricing hotel rooms and airline tickets To many airline customers ticket pricing appears to be madness on the part of the airlines how can it cost less to fly thousands of miles to London than to fly a couple of hundred miles within the United States but there is a method to the madness In this section we examine several pricing problems that can be modeled as NLPs 73 Pricing Models 361 Figure 78 Error Message about Lack of Bounds on Variables T he Madison Company manufactures and retails a certain product The company wants to determine the price that maximizes its profit from this product The unit cost of pro ducing and marketing the product is 50 Madison will certainly charge at least 50 for the product to ensure that it makes some profit However there is a very competitive market for this product so that Madisons demand falls sharply when it increases its price How should the company proceed4 Objective To use a demand function in a nonlinear model to find the price that maximizes the companys profit WHERE DO THE NUMBERS COME FROM Cost accountants should be able to supply the unit cost Historical data on demands and prices of the product are needed to estimate the demand function as discussed next Solution The variables and constraints for this model are listed in Table 71 The unit price drives everything Through a demand function price determines demand and these combine to E X A M P L E 71 PRICING DECISIONS AT MADISON 4This example and the next two are based on Dolan and Simon 1996 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 5 Monetary v alues Calculate the daily revenue cost of capacity and profit in the corresponding cells with the formulas SUMPRODUCTDemandsPrices CapacityB9 and B24B25 USING SOLVER The Solver dialog box should be filled in as shown in Figure 720 The goal is to maximize profit by setting appropriate prices and capacity and ensuring that demand never exceeds capacity You should also check the NonNegative option prices and capacity cannot be negative and you should select the GRG Nonlinear method Again this is because prices are multiplied by demands which are functions of prices so that profit is a nonlinear function of the prices Discussion of the Solution The Solver solution in Figure 719 indicates that FPL should charge 7031 per kwh during the peakload period and 2653 during the offpeakload period These prices generate demands of 275 peak load and 205 off peak so that a capacity of 275 kwh is required The cost of this capacity is 275 When this is subtracted from the revenue of 247730 the daily profit becomes 220230 73 Pricing Models 375 Figure 720 Solver Dialog Box for the PeakLoad Pricing Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it To gain some insight into this solution consider what happens if FPL changes the peak load price slightly from its optimal value of 7031 If FPL decreases the price to 70 say you can check that the peakload demand increases to 2765 and the offpeak demand decreases to 2047 The net effect is that revenue increases slightly to 247878 However the peakload demand is now greater than capacity so FPL must increase its capacity from 2750 to 2765 This costs an extra 150 which more than offsets the increase in revenue A similar chain of effects occurs if FPL increases the peak price to 71 In this case peak load demand decreases offpeak demand increases and total revenue decreases Although FPL can get by with lower capacity the net effect is slightly less profit Fortunately Solver evaluates all of these tradeoffs for you when it finds the optimal solution Is the Solver Solution Optimal All of the constraints in this example are linear so they certainly meet the assumptions for a maximization problem Also it can be shown that the objective daily profit is a concave function of peakload price offpeak price and capacity levelalthough this is far from obvious It requires calculus to verify Algebraically this objective function is called quadratic meaning that it is a sum of linear terms such as Pp squared terms such as Pp 2 and crossproduct terms such as PpPo Not all quadratic functions are concave but there is a test to check whether a given quadratic function is concave Although the details of this test are not presented here we assure you that the quadratic function for this example passes the test Therefore the assumptions for a maximization problem are satisfied and the Solver solution without the Multistart option is guaran teed to be optimal Sensitivity Analysis To gain even more insight you can use SolverTable to see the effects of changing the unit cost of capacity allowing it to vary from 5 to 15 in increments of 1 The results appear in Figure 721 They indicate that as the cost of capacity increases the peakload price increases the offpeak price stays constant the amount of capacity decreases and profit decreases The latter two effects are probably intuitive but we challenge you to explain the effects on price In particular why does the peakload price increase and why doesnt the offpeak price increase as well 376 Chapter 7 Nonlinear Optimization Models Varying the changing cells slightly from their optimal values sometimes provides insight into the optimal solution Figure 721 Sensitivity to Cost of Capacity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A B C D E F G Oneway analysis for Solver model in Model worksheet Cost of capacity cell B9 values along side output cells along top Prices1 Prices2 Capacity Profit 5 6781 2653 2875 234292 6 6831 2653 2850 231430 7 6881 2653 2825 228592 8 6931 2653 2800 225780 9 6981 2653 2775 222992 10 7031 2653 2750 220230 11 7081 2653 2725 217492 12 7131 2653 2700 214780 13 7181 2653 2675 212092 14 7231 2653 2650 209430 15 7281 2653 2625 206792 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 73 Pricing Models 377 Pricing Analysis at Merrill Lynch In the late 1990s Merrill Lynch and other fullservice financial service firms were losing business due to electronic trading and the commoditization of trading Management decided to offer investors more choices for doing business with Merrill Lynch A cross functional team evaluated various alternatives including pricing strategies and constructed models to assess individual clients choice behavior The results enabled Merrill Lynch to change the financial services landscape and mitigate its revenue risk By the end of the year 2000 net new assets to the firm totaled 22 billion and incremental revenue had grown to 80 million ADDITIONAL APPLICATIONS P R O B L E M S Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 In Example 71 two points on the demand curve were given see Figure 710 a Suppose three additional points are estimated by Madison 1 demand of 460 when price is 65 2 demand of 355 when price is 75 and 3 demand of 275 when price is 85 With these new points and the original two points estimate and interpret the bestfitting linear demand curve do the same for the bestfitting constant elasticity demand curve b Calculate the mean absolute percentage error MAPE for each of the two fits linear and constant elasticity where each MAPE is the average of the absolute percentage errors for the five points On the basis of MAPE which curve provides the better fit 2 In Example 71 one demand function is linear and the other is called a constant elasticity demand function Using data tables show that the price elasticity in the linear demand function is not constant in price and show that the price elasticity is constant in the constant elasticity demand function 3 In the pricing model in Example 71 with the constant elasticity demand function the assumption is that all units demanded are sold Suppose the company has the capacity to produce only 200 units If demand is less than capacity all of demand is sold If demand is greater than or equal to capacity only 200 units are sold Use Solver to find the optimal price and the corresponding profit Then use SolverTable to see how sensitive these answers are to the production capacity letting it vary from 170 to 230 in increments of 10 Discuss your findings relative to the original solution in Example 71 In other words what is the effect of capacity on the optimal price and profit 4 Continuing the previous problem create a twoway data table similar to the oneway data table in Figure 711 This time however allow price to vary down a column and allow the capacity to vary across a row Each cell of the data table should capture the corresponding profit Explain how the values in the data table confirm the findings from SolverTable in the previous problem 5 Continuing Problem 3 in a slightly different direction create a twoway SolverTable where the inputs are the elasticity and the production capacity and the outputs are the optimal price and the optimal profit This actually creates two tables one for each output Discuss your findings 6 In the exchange rate model in Example 72 suppose the company continues to manufacture its product in the United States but now it sells its product in the United States the United Kingdom and possibly other countries The company can independently set its price in each country where it sells For example the price could be 150 in the United States and 110 in the United Kingdom You can assume that the demand function in each country is of the constant elasticity form each with its own parameters The question is whether the company can use Solver independently in each country to find the optimal price in this country You should be able to answer this question without actually running any Solver models but you might want to experiment just to verify your reasoning 7 Change the exchange rate model in Example 72 slightly so that the company is now a UK manufactur ing company producing for a US market Assume that the unit cost is now 75 the demand function has the same parameters as before although the price Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it b The assumption that customers will always buy on average the same number of shirts and ties per suit purchase regardless of the prices of shirts and ties is not very realistic How might you change this assumption and change your model from part a accordingly to make it more realistic 15 Continuing the previous problem the model in part a one step further assume that shirts and ties are also 74 Advertising Response and Selection Models 379 complementary Specifically assume that each time a shirt is purchased and is not accompanied by a suit purchase 13 ties on average and regardless of the price of ties are also purchased Modify the model from part a of the previous problem to find the prices of suits shirts and ties to maximize overall profit 74 ADVERTISING RESPONSE AND SELECTION MODELS In Chapter 4 we discussed an advertising allocation model Example 41 where the problem was basically to decide how many ads to place on various television shows to reach the required number of viewers One assumption of that model was that the adver tising responsethat is the number of exposuresis linear in the number of ads This means that if one ad gains say one million exposures then 10 ads will gain 10 million exposures This is a questionable assumption at best More likely there is a decreasing marginal effect at work where each extra ad gains fewer exposures than the previous ad In fact there might even be a saturation effect where there is an upper limit on the number of exposures possible and after sufficiently many ads this saturation level is reached In this section we look at two related examples In the first example a company uses historical data to estimate its advertising response functionthe number of exposures it gains from a given number of ads This is a nonlinear optimization model This type of advertising response function is used in the second example to solve a nonlinear version of the advertising selection problem from Chapter 4 Because the advertising response functions are nonlinear the advertising selection problem is also nonlinear E X A M P L E 75 ESTIMATING AN ADVERTISING RESPONSE FUNCTION R ecall that the General Flakes Company from Example 41 of Chapter 4 sells a brand of lowfat breakfast cereal that appeals to people of all age groups and both genders The company has advertised this product in various media for a number of years and has accumulated data on its advertising effectiveness For example the company has tracked the number of exposures to young men from ads placed on a particular television show for five different time periods In each of these time periods a different number of ads was used Specifically the numbers of ads were 1 8 20 50 and 100 The corresponding numbers of exposures in millions were 47 221 487 903 and 1305 What type of nonlinear response function might fit these data well Objective To use nonlinear optimization to find the response function from a given class of functions that best fits the historical data WHERE DO THE NUMBERS COME FROM The question here is how the company measures the number of exposures a given num ber of ads has achieved In particular what does the company mean by exposures If one person sees the same ad 10 times does this mean 10 exposures Is it the same thing as 10 people seeing the same ad once each Although we defer to the marketing experts here we suggest that one person seeing the same ad 10 times results in fewer Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Recall that this works best if lower and upper bounds are imposed on the changing cells Although there are no obvious bounds for a and b you can try 0001 and 1 for a and 50 and 250 for b This leads to the solution in Figure 723 Alternatively instead of using Multistart you could run Solver repeatedly from different starting solutions You should see that Solver finds the solution in Figure 723 for some starting solutions but not for really bad ones This is typical of many nonlinear optimization models Unless the starting solution is rea sonably close to the optimal solution Solver can go to a completely wrong solution This is the reason why the Multistart option is such a welcome addition to Solver 74 Advertising Response and Selection Models 383 0000 20000 40000 60000 80000 100000 120000 140000 160000 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Ads Predicted exposures Figure 725 Estimated Response Function We used the popular sumofsquarederrors measure or its RMSE equivalent to find the bestfitting response function Another possibility is to use the sum or average of the absolute errors Still another possibility is to use the maximum of the absolute errors All of these have been used in estimation problems and all lead to nonlinear optimization models They typically lead to similar but not necessarily identical solu tions We used the sumofsquarederrors measure because it has historically been the most frequently used measure and leads to a smooth nonlinear modelthe kind that Solver handles best Now that you know how a company can estimate the advertising response function for any type of ad to any group of customers you can use this type of response function in an advertising selection model MODELING ISSUES E X A M P L E 76 ADVERTISING SELECTION WITH NONLINEAR RESPONSE FUNCTIONS I n this example we revisit the problem faced by the General Flakes Company in Example 41 of Chapter 4 The company must decide how many ads to place on each of several television shows to meet exposure constraints for each of six groups of customers Refer to Figure 726 and the file Advertising Selectionxlsx for the specific inputs The differ ence now is that each combination of television show and customer group has its own In some nonlinear models such as this one Solver finds the optimal solution only if the starting solution is reasonably close to the optimal solution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it maximum limits When one of these coefficients increases fewer ads are needed to approach the saturation level Together these two sets of constants indicate which types of ads are most effective to the various customer groups Solver uses this information in its intricate algorithm to decide how many ads to place on each show Perhaps surprisingly no ads are placed on Monday Night Football although many exposures to men under 55 would be achieved from these ads Evidently these ads are too expensive and exposures to men in these groups can be achieved with cheaper ads on other shows Note also that the women in the 36 to 55 group are evidently the bottleneck group Check the differences between the two sides of the exposure constraints To achieve the required exposures for this group many more ads are required than are needed to achieve the required exposures to the other groups Is the Solver Solution Optimal It can be shown with calculus that this model satisfies the conditions necessary to ensure a single local minimum Therefore the Solver solution is optimal If you didnt know this however you could use the Multistart option Sensitivity Analysis An interesting sensitivity analysis for this nonlinear model is to see how the optimal cost varies if all of the required exposures change by the same percentage If you did this in a linear model and there were no other constraints to worry about the optimal cost would change by the same percentage due to the proportionality property of linear models For example if you increased the righthand sides of all constraints by 10 you would expect the optimal cost to increase by 10 in a linear model However this is not true in a nonlinear model as Figure 728 indicates Here you should change the model slightly so that you can vary a single percentage See the Sensitivity worksheet in the finished ver sion of the file for details The values in column C indicate the percentage increase in total 386 Chapter 7 Nonlinear Optimization Models Figure 727 Solver Dialog Box for the Advertising Selection Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 76 MODELS FOR RATING SPORTS TEAMS Sports fans always wonder which team is best in a given sport Was USC LSU or Oklahoma number one during the 2003 NCAA football season You might be surprised to learn that Solver can be used to rate sports teams We illustrate one method for doing this in the following example 76 Models for Rating Sports Teams 393 from 1 to 10 in increments of 1 Keep track of the changing cells and the target cell SkillExtending Problem 27 Modify the warehouse location model as suggested in Modeling Issue 2 Specifically assume that the same four customers have the same annual shipments but now there are only two possible warehouse locations each with distances to the various customers These distances along with other inputs are in the file P0727xlsx The company can build either or both of these warehouses The cost to build a warehouse is 50000 You can assume that this cost has been annualized That is the company incurs a building cost that is equivalent to 50000 per year If only one warehouse is built it will ship to all customers However if both warehouses are built then the com pany must decide which warehouse will ship to each customer There is a traveling cost of 1 per mile a Develop an appropriate model to minimize total annual cost and then use Solver to optimize it Is this model an NLP or an IP model or both b Use SolverTable with a single input the traveling cost per mile to see how large this cost must be before the company builds both warehouses rather than just one E X A M P L E 78 RATING NFL TEAMS9 We obtained the results of the 256 regularseason NFL games from the 2009 season and entered the data into a spreadsheet shown at the bottom of Figure 733 see the file NFL Ratingsxlsx Some of these results are hidden in Figure 733 to conserve space The teams are indexed 1 to 32 as shown at the top of the sheet For example team 1 is Arizon a team 2 is Atlanta and so on The first game entered row 6 is team 25 Pittsburgh versus team 31 Tennessee played at Pittsburgh Pittsburgh won the game by a score of 13 to 10 and the point spread home team score minus visitor team score is calculated in column J A positive point spread in column J means that the home team won a negative point spread indicates that the visiting team won The goal is to determine a set of ratings for the 32 NFL teams that most accurately predicts the actual outcomes of the games played Objective To use NLP to find the ratings that best predict the actual point spreads observed WHERE DO THE NUMBERS COME FROM Sports fans thank heaven for the Web The results of NFL games as well as NBA MLB and other sporting games can be found on a number of Web sites We got this data from httpwwwprofootballreferencecomyears2009gameshtm To see much more about sports ratings go to Jeff Sagarins page at httpwwwusatodaycomsportssagarinhtm Of course if you are an avid sports fan you probably already know the good Web sites 9The procedure used in this example is practically identical to the procedure used by the nationally syndicated Jeff Sagarin to rate various sports teams You can see his ratings at httpwwwusatodaycomsportssagarinhtm Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it DEVELOPING THE SPREADSHEET MODEL To produce the model in Figure 733 proceed as follows 1 Input game data If you want to determine the ratings for another NFL or NBA or MLB season you have to get the data from the Web We are fortunate to have an inside contactWinstons best friend is Jeff Sagarin 2 Changing cells Enter any value for the home field advantage and the 32 team ratings in the Hometeamadvantage and Rating ranges These are the changing cells Note that it would be possible to use a given value for the home team advantage such as 3 but the model will let Solver choose the home team advantage that best fits the data 3 Average rating Enter the nominal average rating in cell B43 and average the ratings in cell B41 with the formula AVERAGERating 4 Actual point spr eads Enter the actual point spreads in column J as differences between columns H and I 5 Predictions The data on games played refer to the team index numbers This allows you to use lookup functions to predict the point spreads To do this enter the formula HometeamadvantageVLOOKUPF6RatingTable3 VLOOKUPG6RatingTable3 in cell K6 for the first game and copy it down column K for the rest of the games The VLOOKUP functions simply look up the ratings of the home and visiting teams The range name RatingTable refers to the range A5C36 6 Prediction errors The objective is to minimize the sum of squared prediction errors Therefore enter the formula J6K62 in cell L6 and copy it down Then sum the squared errors in cell F2 USING SOLVER The completed Solver dialog box is shown in Figure 734 The objective is to find the rat ings and home field advantage that minimize the sum of squared prediction errors The only constraint is to make the ratings average to the nominal rating Because of the squared errors this is a nonlinear model so the GRG Nonlinear method should be used Also there is no need to check the NonNegative option Discussion of the Solution The solution in Figure 733 shows that a home team advantage of 217 provides the best fit at least for the 2009 season To provide a better picture of the ratings the teams are sorted from best to worst in Figure 735 You might recall that New Orleans won the Super Bowl beating Indianapolis The ratings ranked New Orleans number 2 almost 5point favorites over Indianapolis based on regularseason games only The ratings support the playoff picture fairly well The 12 playoff teams are shown with color shading Most of the highly rated teams made the playoffs Arizona being the lowest ranked team to make it in Of course the NFL has its own way of deciding which teams make the playoffs It doesnt just go according to the Sagarin ratings Remember that the actual values of the ratings are not as important as the differences between teams ratings For example if Green Bay plays Dallas at Green Bay Green Bay 76 Models for Rating Sports Teams 395 The VLOOKUP functions let you find the ratings to use for the predicted point spread Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Excel Function MMULT The MMULT and TRANSPOSE functions are useful for matrix operations They are called array functions because the y return results to an entir e range not just a single cell The MMULT function multiplies two matrices and has the syntax MMUL Trange1range2 where r ange1 must have as many columns as r ange2 has r ows To use this function highlight a range that has as many r ows as range1 and as many columns as r ange2 type the formula and pr ess CtrlShiftEnter The r esulting formula will have curly br ackets around it in the Excel Formula Bar You should not type these curly brackets Excel enters them automatically to remind you that this is an array formula The Portfolio Selection Model Most investors have two objectives in forming portfolios to obtain a large expected return and to obtain a small variance to minimize risk The problem is inherently nonlinear because variance is a nonlinear function of the investment amounts The most common way of handling this twoobjective problem is to require a minimal expected return and then minimize the variance subject to the constraint on the expected return The following example illustrates how to accomplish this in Excel 77 Portfolio Optimization Models 401 E X A M P L E 79 PORTFOLIO SELECTION AT PERLMAN BROTHERS P erlman Brothers an investment company intends to invest a given amount of money in three stocks From past data the means and standard deviations of annual returns have been estimated as shown in Table 77 The correlations among the annual returns on the stocks are listed in Table 78 The company wants to find a minimumvariance portfo lio that yields an expected annual return of at least 012 Table 78 Estimated Correlations Among Stock Returns Combination Correlation Stocks 1 and 2 06 Stocks 1 and 3 04 Stocks 2 and 3 07 Table 77 Estimated Means and Standard Deviations of Stock Returns Stock Mean Standard Deviation 1 014 020 2 011 015 3 010 008 Objective To use NLP to find the portfolio of the three stocks that minimizes the risk measured by portfolio variance subject to achieving an expected return of at least 012 WHERE DO THE NUMBERS COME FROM Financial analysts typically estimate the required means standard deviations and correla tions for stock returns from historical data as discussed at the beginning of this section However you should be aware that there is no guarantee that these estimates based on his torical return data are relevant for future returns If analysts have new information about the stocks they should incorporate this new information into their estimates Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it most risky but also having the highest expected return However the correlations play an important role in portfolio selection so it is not usually easy to guess the optimal portfolio on the basis of the means and standard deviations of stock returns alone The portfolio standard deviation of 01217 can be interpreted in a probabilistic sense Specifically if stock returns are approximately normally distributed then the probability is about 068 that the actual portfolio return will be within one standard deviation of the expected return and the probability is about 095 that the actual portfolio return will be within two standard deviations of the expected return Given that the expected return is 012 this implies a lot of risktwo standard deviations below this mean is a negative return or loss of slightly more than 12 Is the Solver Solution Optimal The constraints for this model are linear and it can be shown that the portfolio variance is a convex function of the investment fractions Therefore the Solver solution is guaranteed to be optimal Sensitivity Analysis This model begs for a sensitivity analysis on the minimum required expected return When the company requires a larger expected return it must incur a larger risk as shown in Figure 739 You can use SolverTable with cell D23 as the single input cell allowing it to vary from 010 to 014 in increments of 0005 Note that values outside this range are of no interest Stock 3 has the lowest expected return 010 and stock 1 has the highest expected return 014 so no portfolio can have an expected return outside of this range 404 Chapter 7 Nonlinear Optimization Models Figure 738 Solver Dialog Box for the Basic Portfolio Model Guessing the best allocation in portfolio optimization models is difficult because it depends not only on expected returns and standard deviations of returns but also on correlations between returns Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 5 Errors squared errors and absolute errors The error in any row is the actual stock return minus the predicted stock return Therefore enter the formulas C10D10 E102 ABSE10 in cells E10 F10 and G10 respectively and copy these down 6 Weights This is for the weighted sum of squares criterion only Enter a desired weighting constant in cell B7 Then enter 1 in cell H10 enter the formula B7H10 in cell H11 and copy this formula down column H This makes each weight a constant fraction of the previous weight so that more recent data are weighted more heavily 7 Objectives To set up eight possible objectives in the range B117C120 enter the formulas SUMF10F45 SUMPRODUCTF10F45H10H45 SUMG10G45 MAXG10G45 in cells E4 through E7 and enter similar formulas using all of the data in columns F to H in cells F4 through F7 USING SOLVER The completed Solver dialog box should look similar to Figure 742 except that any of the eight possible objective cells can be used as the target cell There are no constraints not even nonnegativity constraints and the GRG Nonlinear method should be chosen Discussion of the Solution The solution in Figure 741 indicates that McDonalds is not very sensitive to the market having a beta less than 1 for the sum of squared errors criterion when the most recent three years of data are used The solution shown in the alpha and beta cells is for mini mizing the sum of squared errors for the previous three years If you change the objective the beta for McDonalds ranges from about 036 to 053 across the four criteria using the weight 0995 for weighted sum of squares when the most recent three years of data are used and it ranges from about 074 to 083 when all of the data are used These results are shown in the top right of Figure 741 where each is the optimal beta for a different Solver run each using a different objective Clearly a stocks beta can depend not only on which optimality criterion is used but also on the time period selected To run this analysis for any other stock copy its returns to column C of the Model sheet and rerun Solver with one of the possible objectives You will find that the betas for differ ent companies can vary widely Alternative Modeling Approaches You might have noticed that we ignored one of our own warnings in this example Specifically the SAE and minimax objectives depend on the ABS and MAX functions Does Solver provide the correct solution for these two criteria The answer is not a definitive 78 Estimating the Beta of a Stock 411 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it yes but it appears that the solutions are correct for the problems we solved Basically Solver has difficulty with ABS and MAX functions when the objective or constraints are not suffi ciently smooth but it appears that the objectives used here pass the smoothness test However it is possible to develop alternative models for these two objectives that are linear The advantage of course is that the Simplex LP method can then be used which means that it is guaranteed to find the optimal solution In the interest of space a full discussion of these alternative models is not presented here but you can see them in the files Stock Beta 3 Alternative Finishedxlsx and Stock Beta 4 Alter native Finishedxlsx The only draw back to these models is that they rely on modeling tricks that are not obvious 412 Chapter 7 Nonlinear Optimization Models Figure 742 Solver Dialog Box for the Beta Estimation Model P R O B L E M S SkillBuilding Problems 43 Given the data in the file Stock Betaxlsx estimate the beta and alpha for Microsoft MSFT Do this for each criterion and each period of time to obtain a table analogous to that in the top right of Figure 741 What do you conclude about Microsoft 44 Repeat the previous problem but analyze GE instead of Microsoft 79 CONCLUSION A large number of realworld problems can be approximated well by linear models However many problems are also inherently nonlinear We have illustrated several such problems in this chapter including the important class of portfolio selection problems where the risk usually measured by portfolio variance is a nonlinear function of the decision variables We have purposely neglected much of the mathematics behind Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it nonlinear optimization because of its technical difficulty However it is important for you to realize that nonlinear models present many more hazards for spreadsheet Solvers or any other software than linear models Unless you can verify that the assumptions for a minimization or maximization problem are satisfiedand this can be difficult to do there is no guarantee that Solver will converge to the optimal solution or even converge at all The examples in this chapter were purposely kept small and relatively simple so that Solver could handle them and produce optimal solutions Larger and more complex non linear models are not always so accommodating and frequently require solution methods well beyond the level of this book 79 Conclusion 413 Summary of Key Management Science Terms Term Explanation Page Nonlinear Models with nonlinearities in the objective andor the 354 programming NLP constraints models Global optimum Solution that is guaranteed to be the optimal solution 355 Local optimum Solution that is better than all nearby solutions but might not 355 be the best overall Convex function Function with a nondecreasing slope 356 Concave function Function with a nonincreasing slope 356 Optimality guarantee No package including Solver can guarantee that the solution 358 for NLP models it stops at will be the global optimum unless certain convexityconcavity conditions are satisfied Multistart option A new option in Solver for Excel 2010 that automatically optimizes 359 from a number of starting points and returns the best solution found Demand function A function that relates demand for a product to its price 361 Constant elasticity A demand function where elasticity percent change in demand for 362 demand function a 1 change in price is constant for any price Minimizing sum of A popular method of fitting a curve of some form to a set of 380 squared errors points the errors are the differences between observed and predicted values Unconstrained models An optimization model with no constraints 382 Weighted sum of An important quantity in financial portfolio analysis random 398 random variables variables are returns from investments weights are fractions put in investments Return risk measures of Portfolio models try to maximize expected return and minimize 398 portfolio models variance of return risk formulas for these involve correlations or covariances among investment returns Matrix A rectangular array of numbers often useful for simplifying 399 complex summation formulas Efficient frontier Curve that shows the largest expected portfolio return possible 405 for a given level of risk Beta of a stock A value that indicates the responsiveness of a stocks return to 407 changes in the return of the market Sum of absolute errors An alternative criterion to sum of squared errors for making 408 SAE errors small Minimax An alternative criterion for making errors small minimizes the 409 largest error Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 418 Chapter 7 Nonlinear Optimization Models Modeling Problems 80 For the product mix examples Examples 31 and 32 in Chapter 3 discuss where you think the assump tions of a linear model are most likely to break down How might an NLP model look in this situation 81 For the oil blending example Example 44 in Chapter 4 discuss where you think the assumptions of a linear model are most likely to break down How might an NLP model look in this situation 82 For the aggregate planning example Example 43 in Chapter 4 is it likely that the cost per worker of changing the size of the workforce during a month would be constant as we assumed How could an NLP model account for a situation in which the cost per worker of changing the size of the workforce is not constant 83 Consider the sports ratings model in section 76 If you were going to give more recent games more weight how might you determine whether the weight given to a game from k weeks ago should be say 095k or 09k 84 Consider the sports ratings model in section 76 If you were going to use the approach used there to forecast future sports contests what problems might you encounter early in the season How might you resolve these problems 85 UE is going to invest 400 million to acquire compa nies in the auto andor electronics industry How would you apply portfolio optimization to determine which companies should be purchased 86 Your family owns a large farm that can grow wheat corn cotton alfalfa barley pears and apples Each product requires a certain amount of labor each month and a certain number of hours of machine time You have just studied portfolio optimization and want to help your family run its farm What would you do 87 Your company is about to market a new golf club You have convened a focus group of 100 golfers and asked them to compare your club to the clubs produced by your competitors You have found for example that 30 customers in the focus group would purchase your club if you charged 120 28 customers would pur chase your club if you charged 130 and so on How could you use this information to determine the price at which your club should be sold Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 421 Evolutionary Solver An Alternative Optimization Procedure C H A P T E R DEVELOPINGA N OPERA TINGPLAN MODEL AT SANTA FE RAILWAY L ike many other companies Santa Fe Railway faces increasing demands for customer service cost pressures and changing market conditions This is particularly true in its intermodal business area in which traffic moves on some combination of ship or truck and train The company averaged almost 8 growth per year in intermodal traffic handled during the period from 1989 to 1996 This increased growth and changing patterns of customer traffic created difficult problems for Santa Fe as described in Gorman 1998 The company needed to use its trains and rail lines efficiently from a cost standpoint but it also had to provide customers with highquality service In addition the company had to be flexible to change its operating plan quickly in response to changing customer traffic patterns Historically Santa Fes service design was rather myopic The service designers tried their best to make incremental refinements to current operations but their thinking was based too much on historical procedures and could not adapt sufficiently to changing customer needs They eventually decided to create an operatingplan model capable of building an operating plan for the intermodal business unit from scratch one that could best adapt to the current and expected traffic patterns and would not be constrained by traditional patterns or historical schedules As inputs this model required Soleg1974 Dreamstimecom 8 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it customer service requirements engineering capabilities and physical plant constraints The outputs included a weekly train timetable traffictotrain assignments yard and railway line schedules and equipment and locomotive flowsThe objective was to simultaneously allocate physical rail network resources to trains and allocate scarce train space to traffic flows to minimize operating costs while meeting customer requirements The operatingplan problem was decomposed into two problems the train timetable problem and the traffic assignment problem The former prescribes which trains will travel on which lines at which times Given this information the latter problem prescribes which customer loads are assigned to which trains Each problem is huge and much ingenuity was required to model and solve these problems For the timetable problem the original model represented each hour of the week for every possible train as a binary decision variable where 1 indicates a train and 0 indicates no trainThis model was impossibly large so the service design team reduced its size by specifying a menu of allowable train routes about 200 from which the model could choose Even this reduced problem was much too large for traditional integer programming algo rithms to solve so the analysts did what is becoming more common in large optimiza tion models they turned to newer emerging types of algorithms In particular they tried the genetic survival of the fittest algorithms discussed in this chapter where they mixed schedules from a given population of schedules to carry over the best characteristics of these schedules to the next generation of schedules Unfortunately genetic algorithms alone were painfully slow at producing useful populations of train schedules for this large problem Therefore the authors combined genetic algorithms with another type of algorithm called tabu search to speed up the process Tabu search uses information from previous iterations to search in a promising direction However a tabu list prohibits the algorithm from undoing recent changes to the schedule or revisit ing recent solutions This method of combining algorithms worked and enabled Santa Fe to solve the timetable problem reasonably quickly The company was then able to solve the traffic assignment problem by a clever prioritybased shortestpath heuristic Santa Fe Intermodal used its operatingplan model to study many major changes in rail operations to predict train volumes based on longterm forecasts to quantify the impact of containerization of intermodal business on train operations and to develop a cost basis in contract negotiations for large amounts of incremental business The model has shown the potential to improve global service by 4 while reducing costs by 6 over the previous operating plan As R Mark Schmidt an analyst at Santa Fe stated Obviously as with any major deviation from traditional processes the acceptance of the operatingplan model has been a gradual one Recent successes of the model are building confidences and as a result the model is being interwoven into the intermodal service design process at Santa Fe 422 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure 81 INTRODUCTION In Chapters 3 through 7 we used Excels Solver to solve many interesting and important problems Unfortunately there are many optimization problems where Solvers Simplex LP and GRG Nonlinear algorithms are unable to find optimal solutions However genetic algorithms often perform well on optimization problems where Solvers other algorithms perform poorly The purpose of this chapter is to illustrate some interesting models that cannot be solved by the Solver algorithms discussed in previous chapters at least not easily or without tricks but can be solved with genetic algorithms in a reasonably straightforward Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it manner In short the methods in this chapter enable you to solve a much wider range of optimization models Fortunately Solver for Excel 2010 includes the Evolutionary algorithm which was previously available only in Premium Solver included with previous versions of the book Therefore Premium Solver is no longer necessary In fact we were told by Frontline Systems the developer of Solver that Solver for Excel 2010 is essentially the old Premium Solver The following summarizes the three algorithms included with Solver for Excel 2010 To avoid confusion from here on we will refer to the three Solver algorithms avail able with Excel 2010 as Simplex LP Solver GRG Nonlinear Solver and Evolutionary Solver Simplex LP Solver is used to solve linear models including models where some or all of the changing cells are restricted to be binary andor integer GRG Nonlinear Solver is used to solve nonlinear models when the objective cell and constraints are smooth functions of the changing cells Evolutionary Solver uses genetic algorithms to find good close to optimal solutions to more difficult problems including those where the objective cell andor constraints are nonsmooth functions of the changing cells Several times in previous chapters we stated that the first two Solvers cannot handle models with IF MAX MIN and several other Excel functions The problem is that such models often contain nonsmooth functions in the objective cell andor the constraint cells Technically a nonsmooth function has discontinuities or points where its derivatives do not exist It is sometimes possible to make these models linear so that the Simplex LP Solver can be used but nonobvious tricks are usually necessary to do so Fortunately this is not necessary with Evolutionary Solver as illustrated in this chapter Evolutionary Solver uses a type of algorithm called a genetic algorithm which is much more flexible Before discussing genetic algorithms and Evolutionary Solver we review the strengths and weaknesses of the Solvers used in previous chapters Recall that an optimization model is linear if the objective cell is a linear function of the changing cells the left and right sides of all constraints are linear functions of the changing cells and all changing cells are allowed to contain fractional valuesthat is there are no integer constraints For such models Simplex LP Solver is guaranteed to find an optimal solution if an optimal solution exists We have discussed many linear models in Chapters 3 through 5 Simplex LP Solver is an excellent method to use for any opti mization problem that can be set up as a linear model provided that the model does not exceed Solvers size restrictionsup to 200 changing cells and 100 constraints not count ing simple upper or lower bounds on changing cells Most larger linear models are diffi cult to handle in a spreadsheet format These larger models are often solved using a modeling language such as LINGO GAMS or AMPL With a modeling language a user can generate say 10000 supply constraints for a transportation model with one line of computer code This makes it easy to compactly represent and solve large models We should also mention that Frontline Systems has developed commercial largescale Solvers that are capable of solving very large spreadsheet models In Chapter 6 we considered linear models where some or all of the changing cells are constrained to be integers In theory Simplex LP Solver should be able to find optimal solutions to these problems but in practice it can take hours days or even weeks to find opti mal solutions to difficult integerconstrained models This is not necessarily a weakness of Solverintegerconstrained models are inherently difficult for any optimization software packagebut there are algorithms other than the algorithm used by Solver that work bet ter for some integer models 81 Introduction 423 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it In the previous chapter we discussed nonlinear models and saw that GRG Nonlinear Solver is capable of solving many of these However nonlinear models present two prob lems First as discussed in section 72 of Chapter 7 GRG Nonlinear Solver can get stuck at a local maximum or a local minimum and never find the global maximum or minimum The function shown in Figure 71 illustrates this situation In this example GRG Nonlinear Solver fails to find the global optimal solution for certain starting solutions Fortunately as discussed in Chapter 7 GRG Nonlinear Solver for Excel 2010 has a Multistart option that increases the chances of finding the global optimal solution in problems like this one Second if a spreadsheet model uses IF ABS MAX or MIN functions that depend on any of the models changing cells the model is typically nonsmooth and GRG Nonlinear Solver can have difficulty finding an optimal solution One possibility that could be caused by an IF function is illustrated in Figure 81 The context here is ordering a product with a quantity discount so that the order quantity is on the horizontal axis and the total cost ordering cost plus inventory holding cost is on the vertical axis The IF function specifies that if the order quantity is less than A one function specifies the total cost If the order quantity is between A and B another function specifies the total cost Finally if the order quantity is greater than B a third function specifies the total cost The resulting graph is not only nonlinear but it has discontinuities at A and B where the total cost jumps from one value to another The overall costminimizing order quantity is to the right of B If you select an initial solution to the right of B GRG Nonlinear Solver will locate the correct optimal solution However if you start at a point to the left of B GRG Nonlinear Solver will almost certainly not find the optimal solution 424 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Figure 81 A Cost Function with Discontinuities The point of this discussion is that although Simplex LP Solver and GRG Nonlinear Solver can handle many models with no difficulty they are not well suited to finding opti mal solutions for certain types of models We now discuss a completely different solution method that is sometimes more successful at solving these difficult problems The standard Solver cannot handle functions with discontinuities reliably Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 3 Select Evolutionary Solver Click on the dropdown list of available algorithms to select Evolutionary Solver see Figure 84 This is the option used throughout this chapter but you can also experiment with GRG Nonlinear Solver especially after Evolutionary Solver finds a good solution 83 Introduction to Evolutionary Solver 429 Figure 84 Selecting Evolutionary Solver Figure 85 Solvers All Methods Options 4 Solver Options Click on the Options button and then the All Methods tab to see the dialog box in Figure 85 The bottom section of this dialog box relevant for all Solver algo rithms allows you to change some limits to higher values The main reason for doing so is to keep Evolutionary Solver from repeatedly beeping at you as it reaches these limits Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Next click on the Evolutionary tab to see the dialog box in Figure 86 These are the settings that control Evolutionary Solver The following information about them is available in online help Convergence measures the rate of change of the objective You can leave this at its default value Mutation rate governs the frequency at which mutations are introduced into the pop ulation of solutions Mutations shouldnt be introduced too often but by introducing them every now and then the GA gets a chance to explore a completely different area of the feasible region You can leave this setting at its default value 0075 but we have sometimes had success by increasing it to 025 Population size is the number of candidate solutions chromosomes at any point in time and the default value of 100 should work well although we sometimes increase it to 150 Note that the initial population is chosen randomly but it includes at least one instance of the starting solution you specify in the changing cells Evolutionary Solver uses a random mechanism to perform its search but you can make it go through exactly the same calculations on two separate runs if you use the same random seed any integer on each run You can leave this box blank in which case Evolutionary Solver bases the seed on the system clock You should check the Require Bounds on Variables option This forces you to enter explicit upper and lower bounds on all changing cells which aids Evolutionary Solver in its search process Maximum Time without Improvement measured in seconds indicates the stopping rule for the algorithm If it doesnt find a meaningful improvement in this amount of time it will stop and report the best solution so far 430 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Figure 86 Solvers Evolutionary Options Some experimentation with Evolutionary Solvers settings may be necessary No single group of settings works best on every problem Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 Inputs Enter the inputs in the blue ranges Note that the large blue range is the price sensitivity table from Figure 87 2 Price The only decision variable in this model is the single price charged for every pack of Menthos sold Enter any value in this Unitprice cell 3 Total value table The values in the shaded price sensitivity range are marginal val ues the most each customer would pay for the next pack of Menthos In the range H6K15 calculate the total value of n packs for each customer for n from 1 to 10 First enter the formula B6 in cell H6 and copy it across row 6 Then enter the formula H6B7 in cell H7 and copy it to the range H7K15 4 Total cost column Using the singleprice scheme each customer must pay np for n packs if the price is p Calculate these amounts in the range E19E28 by entering the formula UnitpriceD19 in cell E19 and copying down 5 Surplus table This is the key to the model You need to calculate the surplus for any customer from buying n packs as the total value of n packs minus the total cost of n packs and you assume that the customer buys the number of packs with the largest surplus This makes sense economically If a customer places more value on n packs than it costs to buy n packs then presumably the customer will consider purchasing n packs But a customer will not purchase n packs if they cost more than she values them To calculate these surplus values enter the formula H6E19 in cell H19 and copy it to the range H19K28 6 Maximum surplus Calculate the maximum surplus for each customer by entering the formula MAXH19H28 in cell B32 and copying it across row 32 7 Packs purchased For each customer you need to find the number of packs that cor responds to the maximum surplus This can be done best with Excels MATCH function Specifically enter the formula IFB3200MATCHB32H19H280 in cell B33 and copy it across row 33 This formula says that if the maximum surplus is nega tive the customer will not purchase any packs at all Otherwise it matches the maximum surplus to the entries in the range H19H28 and returns the index of the cell where the match occurs In this example the match for customer 1 occurs in the fourth cell of the range H19H28 so the MATCH function returns 4 Note that the last argument of the MATCH function is 0 if you want an exact match as you do here Then calculate the total number of packs purchased by all customers with the formula SUMPRODUCTB34E34B33E33 in cell B36 434 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Excel Function MATCH The MATCH function with the syntax MATCHValueRangeType returns the position as an integer of the first match to Value in the given Range For example if Value is 6 and the values in the given Range are 8 7 6 5 6 5 8 the MA TCH function returns 3 The Type argument is usually set to 0 whic h returns an e xact match Other options for the Type parameter can be found in Excels online help 8 Profit Calculate the profit in cell B37 with the formula UnitpriceUnitcostB36 USING EVOLUTIONARY SOLVER First note that GRG Nonlinear Solver has trouble with this model because of the MAX IF and MATCH functions However these functions present no difficulties to Evolutionary Solver It should be set up as shown in Figure 89 using the same values for the Evolutionary options as in the previous example Note that an upper limit of 150 has been used for the unit price This suffices because the most any customer will pay for any pack of Menthos is 149 84 Nonlinear Pricing Models 435 Figure 89 Solver Dialog Box for the SinglePrice Model Discussion of the Solution Again Evolutionary Solver converges to the solution in Figure 88 quickly and then tries for a long timeunsuccessfullyto find a better solution You can be fairly certain that this solution is optimal but this is not guaranteed The single price of 080 produces a profit of 62000 It strikes the best balance for these four market segments A lower price would needlessly sacrifice revenue whereas a higher price would cause at least one market segment to buy fewer packs Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 85 COMBINATORIAL MODELS Consider the following situations Xerox must determine where to place maintenance facilities The more maintenance facilities selected the more copiers the company will sell due to better availability of maintenance How can the company locate maintenance facilities to maximize total profit A gasoline company is loading three different products on a tanker truck with five compartments Each compartment can handle at most one product How should the company load the truck to come as close as possible to meeting its delivery requirements Fox has 30 different ads of different lengths that must be assigned to 10 different twominute commercial breaks How should the company assign ads to maximize its total ad revenue John Deere must schedule its production of lawn mowers over the next four weeks The company wants to meet its forecasted demands keep production hours fairly constant from week to week and avoid model changeovers as much as possible How should the company schedule its production Each of these problems is a combinatorial optimization problem that requires a com pany to choose the best of many different combinations available Although combinatorial optimization problems can often be handled as Solver models with 01 changing cells it is often difficult to develop the constraints in a way that keeps the model linear You saw examples of the tricks required in Chapter 6 With Evolutionary Solver however it doesnt matter whether the constraints or the objective are linear The SUMIF and COUNTIF functions are often useful in such problems The two examples in this section illustrate typical combinatorial optimization problems Loading Products on a Truck The following example might appear simple when you first read it but it is not The num ber of possible solutions is enormous and it can take a Solver even Evolutionary Solver a long time to find an optimal or nearly optimal solution 438 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Combinatorial problems have only a finite num ber of feasible solutions However they can still be very difficult because this finite number is often enormous P R O B L E M S SkillBuilding Problems 3 In Example 82 determine the best pricing policy if quantity discounts with a singleprice breakpoint are used 4 In Example 82 determine the optimal pricing policy if Menthos are sold in only a onepack or a sixpack 5 Based on Schrage 1997 The file P0805xlsx lists the size of the four main markets for Excel Word and the bundle of Excel and Word We assume that Microsoft is willing to sell Excel or Word separately and it is willing to sell a package with Excel and Word only It also shows how much members of each group are willing to pay for each product combination How can Microsoft maximize the revenue earned from these products You should consider the following options No bundling where Word and Excel are sold separately Pure bundling where purchasers can buy only Word and Excel together Mixed bundling where purchasers can buy Word or Excel separately or they can buy them as a bundle Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 85 Combinatorial Models 439 E X A M P L E 83 LOADING A GAS STORAGE TRUCK A gas truck contains five compartments with the capacities listed in Table 81 Three products must be shipped on the truck and there can be only one product per com partment The demand for each product the shortage cost per gallon and the maximum allowable shortage for each product are listed in Table 82 How should the truck be loaded to minimize the shortage costs Table 81 Truck Capacities Compartment Capacity Gallons 1 2700 2 2800 3 1100 4 1800 5 3400 Table 82 Demand and Shortage Data Product Demand Max Shortage Allowed Cost per Gallon Short 1 2900 900 10 2 4000 900 8 3 4900 900 6 Objective To use Evolutionary Solver to find the combination of products to load in compartments that minimizes the total shortage cost WHERE DO THE NUMBERS COME FROM The data would be based on the truck dimensions and presumably on contracts the com pany has with its customers Solution The objective in this problem is to minimize the total shortage cost The decision variables indicate the type of product stored in each compartment and the amount of that product to load in the compartment The constraints must ensure that no compartment is overfilled and that the maximum allowable shortage is not exceeded DEVELOPING THE SPREADSHEET MODEL The completed model appears in Figure 812 See the file Loading Truckxlsx It can be developed as follows 1 Inputs Enter the inputs from Tables 81 and 82 into the shaded ranges 2 Decision variables Enter any integer values from 1 to 3 in the Product range and any values integer or noninteger in the Amount range These two ranges represent the changing cells Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in cell F21 and copying it down The solution shown in Figure 812 does not have any vio lations but the values in column F would be positive if any shortages in column D were greater than 900 6 Costs Calculate the total shortage cost in cell B26 with the formula SUMPRODUCTB5B7D21D23 Then calculate the penalty cost from maximum shortage violations in B27 with the formula B9SUMF21F23 Note that a penalty of 100 per unit shortage above the maximum allowed was chosen Any large dollar value would suffice here Finally calculate the total cost in cell B28 by sum ming the values in cells B26 and B27 USING EVOLUTIONARY SOLVER The Solver setup for this model is straightforward as shown in Figure 813 Unlike some previous models there are now natural lower limits and upper limits for the changing cells The Product range must be between 1 and 3 and they must be integers because there are only three products The Amount range must be between 0 and the given capacities of the compartments Discussion of the Solution The solution in Figure 812 shows that product 1 should be stored in compartment 2 prod uct 2 should be stored in compartments 1 and 3 and product 3 should be stored in com partments 4 and 5 the only compartments that end up with excess capacity The demands for products 1 and 2 are not quite met and the total shortage cost is 2600 but the short ages are well below the maximum shortages allowed Therefore there is no penalty cost for violating the maximum shortage constraints 85 Combinatorial Models 441 This example illus trates how violations of constraints can be incorporated into the objective in the form of penalties Figure 813 Solver Dialog Box for the Truck Loading Model This model is not easy for Evolutionary Solver in spite of its rather small size and its success depends a lot on the starting solution For example we tried one solution with all 3s in the Product range and all 1000s in the Amount range It got to a solution with objec tive value 3200 fairly quickly but then it spun its wheels for a long time and never Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 2 Models produced To calculate the number of different models produced each week which are needed for the model changeover objective enter the formula COUNTIFB22H220 in cell I22 and copy it down 3 Deviations from forecasts To calculate the total monthy production levels for each model and see how much they deviate from the monthly forecasts enter the formulas SUMB22B25 and ABSB6B26 in cells B26 and B27 for model 1 and copy these across for the other models Recall that ABS is Excels absolute value function 4 Pickup shortages To see how much week 1 production of each model is short if any of the pickup demand enter the formula IFB22B5B5B220 in cell B28 and copy it across 5 Hourly smoothing This is the trickiest objective The production hours at each machine center should remain as constant as possible across weeks Although there are undoubtedly other ways to implement this we suggest the following approach First calculate the weekly average hours required at each machine center if the company pro duces exactly enough in the month to meet monthly forecasts To do this enter the formula SUMPRODUCTB6H6B10H104 in cell B31 for center 1 and copy it down for the other two centers Note that division by 4 is used to obtain a weekly average These weekly averages become the targets Next cal culate the actual hours used at each center each week in the range B37E39 Unfortunately there is no way to enter a single formula and then copy it to the rest of the range However you can try the following Enter the formula SUMPRODUCTB22H22B10H10 in cell B37 and copy it down to cell B39 Then copy the range B37B39 to the range C37E39 The resulting formulas for weeks 2 to 4 in columns C to E will not be quite correct but you can modify them easily Specifically change each 22 in the column C formulas to 23 to 24 in column D and to 25 in column E The point is that when copying is not possible sometimes copying a formula and then modifying it is easier than entering new formulas from scratch Finally calculate the deviations from targets in the range H37K39 by entering the formula ABSB37B31 in cell H37 and copying it to the rest of the range Here copying is possible 6 Penalties Calculate the various penalties in the range B42B45 with the formulas B15SUM122125 B16SUMB28H28 B17SUMH37K39 85 Combinatorial Models 445 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it and B18SUMB27H27 Then calculate the total penalty as their sum in cell B46 USING EVOLUTIONARY SOLVER The Solver setup for this model appears in Figure 817 The objective is to minimize the total of penalties the changing cells are the production levels and there are no constraints other than lower and upper bounds and integer constraints on the production levels As for the upper bounds 150 is fairly arbitrary The largest monthly forecast for any model is 115 but the company might want production to exceed this forecast Therefore you can build in some padding with the upper limit of 150 446 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Figure 817 Solver Setup for the Lawn Mower Production Model After some experimenting you will see that this is a difficult problem even for Evolutionary Solver Depending on the starting solution it can take some time to find as good a solution as the one in Figure 816 Therefore it helps to enter large values in the Solver Options dialog boxes for Max Time Max Subproblems Max Feasible Solutions and Maximum Time without Improvement this latter setting under the Evolutionary tab Otherwise Evolutionary Solver might quit prematurely at a solution far from optimal Another possible strategy is to drop the integer constraint by checking the box in Figure 818 This will find a good noninteger solution relatively quickly Then you can run the Solver again starting from this noninteger solution with the box unchecked to find a good integer solution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Discussion of the Solution The solution in Figure 816 represents the best compromise we could find It produces all seven models during week 1 to keep the pickup shortages low In fact it has no pickup shortages After that it produces only three separate models each week to keep the changeover penalties low This solution produces exactly to the monthly forecasts Finally all of this is done in a way to keep the production hours as constant as possible across weeks Even so the chart in Figure 816 based on the data in the range B37E39 shows that the production hours still vary to some extent across weeks at each machine center Of course if you change the unit penalties to reflect different priorities on the objectives and then rerun Evolutionary Solver you could get a much different solution For example if EasyRide decides that pickup shortages are not such an important concern it could reduce the unit shortage penalty from 50 to say 25 or even 5 Then the production schedule might change so that all seven models are not produced in week 1 85 Combinatorial Models 447 Figure 818 Option to Ignore Integer Constraints P R O B L E M S SkillBuilding Problems 6 In the truckloading problem in Example 83 we assumed that any product could be loaded into any compartment Suppose the following are not allowed product 1 in compartment 2 product 2 in compartment 1 and product 3 in compartment 4 Modify the model appropriately and then use Evolutionary Solver to find the new optimal solution Hint Add a penalty to the objective for violating these new constraints 7 In the lawn mower production problem in Example 84 the model changeover cost dominates in the optimal objective value Is this because we assumed such a large unit penalty cost 200 for each model changeover Explore this question by changing this unit penalty cost to lower values such as 100 and 50 or even smaller What happens to the optimal solution 8 In the lawn mower production problem in Example 84 experiment with the penalty cost for unsatisfied pick ups in week 1 If this cost is sufficiently small does the company ever produce fewer than seven models in week 1 and allow some week 1 pickups to be unsatisfied Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 4 Objective Calculate the fraction of months during the earliest four years where the portfolio beats the SP 500 Do this in cell G8 with the formula COUNTIFAG13AG60Yes48 This is the objective to maximize Note that it contains the COUNTIF function This is the feature that necessitates Evolutionary Solver For comparison calculate the similar frac tion for the most recent fourplus years in cell G9 with the formula COUNTIFAG61AG116Yes56 USING EVOLUTIONARY SOLVER The Solver setup appears in Figure 824 You should constrain the sum of the weights to be 1 so that all of the money is invested and you should constrain the weights to be between 0 and 1 so that the investment in each stock is a positive fraction of the total investment You can allow negative weights if you want to permit short selling 454 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Figure 824 Solver Dialog Box for the Portfolio Optimization Model Discussion of the Solution There are several things to note about the optimal solution found in Figure 823 First this portfolio puts most of the weight on four companies 3M 227 Alcoa 247 American Express 220 and Procter Gamble 213 The rest of the weight is divided among four other companies and the rest of the companies are not in the portfolio at all Second this solution represents the portfolio that beats the SP 500 most frequently in the optimization periodthat is the earliest four years Whenever an optimization is based on a historical period there is no guarantee that this solution will work as well in a later time period The calculation in cell G9 shows how well the portfolio does in the most recent fourplus years of the data set Clearly it does not do as well The portfolio beats the SP 500 about 71 of the time during the earliest four years but only about 48 of the time during the most recent fourplus years Any time historical data is used to forecast what might happen in the future the implicit assumption is that historical patterns will repeat themselves As many forecasters have discovered to their dismay this assumption is not always correct Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Finally this is the best solution we found after experimenting with several random number seeds and several starting solutions for the weights Some of these converged to a solution with an objective less than 75 which is clearly suboptimal This is due to the randomness component built into GAs Different runs can have varying levels of success depending on the luck of the draw Is this method of portfolio optimization any better or worse than the variance minimizing method discussed in the previous chapter The answer probably depends on the investors attitude toward risk There is no guarantee that the probabilitymaximizing model in this chapter will achieve any particular expected return although if it beats the market index consistently it seems that it should provide a decent return Also there is no guarantee that this portfolio will provide an acceptable riskmeasured by a small variance Nevertheless this model might have an intuitive appeal to many investors If you can beat the SP 500 consistently you must be doing a good job 88 Cluster Analysis 455 P R O B L E M S SkillBuilding Problems 13 Visit httpbizyahoocomr Under Research Tools click on Historical Quotes and then download the monthly returns on at least four stocks for the preced ing 60 months Use this data to determine the portfolio that maximizes the chance of beating the SP 500 for these years Note that the ticker symbol for the SP 500 is GSPC Also this Web site gives closing prices which you will need to convert to returns 14 Continuing the previous problem determine the portfolio that minimizes the chance that you will lose money during any month subject to a lower bound constraint on your expected monthly return The lower bound will depend on your data It must not be above the largest average return of your stocks For example if you require the mean portfolio return to be greater than 1 and all stocks average less than 1 the constraint cant possibly be satisfied 88 CLUSTER ANALYSIS Marketers often want to group objects into clusters of similar objects For example identi fying similar customers can help a company identify market segments Identifying a cluster of similar products can help a company identify its main competitors Here are two actual examples of how the United States is divided into clusters5 Claritas divides each block of the United States into one of 62 clusters These include Blue Blood Estates New Homesteaders Middle America Gods Country and so on For example Blue Blood Estates consists primarily of Americas richest suburbs Over 1 in 10 residents of Blue Blood Estates is a millionaire This is valuable infor mation for marketers For example Blue Blood Estates residents consume imported beer at a rate nearly three times the national average SRI clusters families based on their financial status and demographics For example the cluster Bank Traditionalists consists of uppermiddleclass families of larger than average size with schoolage children This cluster is a natural prospecting ground for life insurance salespeople The following example illustrates how Evolutionary Solver can be used to cluster cities The same method could be use to cluster people products or other entities6 5The book by Johnson and Wichern 2002 has an excellent although somewhat mathematically advanced dis cussion of cluster analysis and the topic of the next section discriminant analysis 6This example is for illustration only There are many software packages other than Excel that are much more powerful for data mining tasks such as cluster analysis or discriminant analysis the subject of the next example Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it INDEXClustercenterU151 in cell V15 and copy it down This formula returns the name in the second row and first only column of the Clustercenter range in Figure 827 Excel Function INDEX The function INDEX using the syntax INDEXRangeRowColumn is usually used to return the value in a given row and column of a specified range For example INDEXA5C103 2 returns the value in the third row and second column of the range A5C10 that is the value in cell B7 If the given range is a single row the row argument can be omitted If the given range is a single column the column argument can be omitted 6 Sum of distances The objective is to minimize the sum of distances from all cities to the cluster centers to which they are assigned Calculate this objective in cell B11 in Figure 827 with the formula SUMT15T63 USING EVOLUTIONARY SOLVER The Solver dialog box should be set up as shown in Figure 829 Because the changing cells represent indexes of cluster centers they must be integerconstrained and suitable lower and upper limits are 1 and 49 the number of cities This problem is considerably more difficult to solve so you should allow Evolutionary Solver plenty of time to search through a lot of potential solutions 88 Cluster Analysis 459 Figure 829 Solver Dialog Box for Cluster Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 89 DISCRIMINANT ANALYSIS Discriminant analysis is a statistical tool used by analysts in marketing and other fields of business Although somewhat similar to cluster analysis it is also quite different In clus ter analysis there are no predefined clusters You look at the information on the different members of the population cities products or whatever to see which members should be clustered together because of similar characteristics You do not even know the number of clusters to use In discriminant analysis however the clusters usually called groups are predefined For example there might be two groups users of a particular product and nonusers You collect data on a sample often called a training sample of users and nonuserstheir income their ages and other possibly relevant dataand use this data to classify the customers as users or nonusers The analysis is successful if a large percentage of the customers in the training sample are classified correctly Of course the group mem bership of each customer in the training sample is already known Therefore the real pur pose is to see whether a large percentage of customers outside of the training sample can be classified correctly on the basis of their income age and other relevant variables Discriminant analysis has been used in many situations including the following Based on gender age income and residential location classify a consumer as a user or nonuser of a new breakfast cereal Based on income type of residence credit card debt and other information classify a consumer as a good or bad credit risk Based on financial ratios classify a company as a likely or unlikely candidate for bankruptcy In general discriminant analysis can be used to classify members of two or more groups We focus only on twogroup discriminant analysis In this case the approach is to find a weighted combination of the data for each member called a discriminant score and then to classify the member into group 1 or group 2 depending on which side of a cutoff score the members discriminant score falls The problem is to find the appropriate weights for the dis criminant scores and the appropriate cutoff score that maximize the percentage of correct classifications in the training sample The following example illustrates the procedure 89 Discriminant Analysis 461 In classification exam ples such as these you typically create an optimization model on a training data set and then apply it to a new data set to predict group membership E X A M P L E 88 CLASSIFYING SUBSCRIBERS AND NONSUBSCRIBERS TO THE WALL STREET JOURNAL T he file WSJ Subscribersxlsx contains the annual income and size of investment port folio both in thousands of dollars for 84 people It also indicates whether or not each of these people subscribes to the Wall Street Journal Using income and size of investment portfolio determine a classification rule that maximizes the number of people correctly classified as subscribers or nonsubscribers Objective To use Evolutionary Solver to find a function of income and investment that does the best job of classifying subscribers and nonsubscribers WHERE DO THE NUMBERS COME FROM In a general discriminant analysis you collect as much relevant financial and demographic data as possible about the people or companies to be classified Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 5 Tallies It is customary to tally the classifications in a classification matrix as shown in the range H12J14 The easiest way to find these tallies is to use the COUNTIFS func tion new in Excel 2007 Specifically enter the formula COUNTIFSD12D95H13F12F95I12 in cell I13 and copy it to the range I13J14 Then calculate the percent correctly classified in cell I17 with the formula I13J14SUMI13J14 This is the objective to minimize Excel Function COUNTIFS The function COUNTIFS new to Excel 2007 enables you to count the number of values that satisfy multiple criteria The arguments come in pairs The first member of each pair is a range and the second is a criterion In the e xample above there are two pairs The first r equires a matc h between the values in column D and one of the values in the H13H14 range The second requires a match between column F and one of the values in the I12J12 range For example the value in cell I13 means that 23 of the data r ows have Yes in column D and in column F USING EVOLUTIONARY SOLVER First note that Evolutionary Solver is required because of the IF and COUNTIFS functions used to make and tally the classifications The completed Solver dialog box appears in Figure 832 and is straightforward except for the lower and upper limits on the changing cells There are no natural weights or cutoff values to use However the weights can always be constrained to be between 1 and 1 The reasoning is that if you solve the prob lem with weights equal to say 15 and 15 you can divide them and the resulting cutoff score by 15 and obtain exactly the same classifications To obtain lower and upper limits on the cutoff value we first calculated the maximum sum of income and investment amount for any customer which is slightly less than 160 This means that the largest dis criminant score using weights of 1 is no larger than 160 and the smallest discriminant score using weights of 1 is no less than 160 Therefore there is no need to consider cut off values below 160 or above 160 89 Discriminant Analysis 463 Figure 832 Solver Dialog Box for Discriminant Analysis Model Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it USING EVOLUTIONARY SOLVER The Solver dialog box should be set up as shown in Figure 835 The objective is to mini mize the total distance traveled subject to the constraints that all indexes on the route other than Bostons are between 1 and 10 and they must be all different Specifying this AllDifferent constraint is similar to specifying an integer or binary constraint When you choose Evolutionary Solver a dif option is available when you add a constraint See Figure 836 It is useful in exactly this type of model where the numbers in a permutation must all be different 810 The Traveling Salesperson Problem 467 Figure 835 Solver Dialog Box with the AllDifferent Constraint Figure 836 Specifying an AllDifferent Constraint Discussion of the Solution The optimal solution appears in Figure 834 Willie should go from Boston to New York to Pittsburgh to Chicago to Denver to Seattle to San Francisco to Los Angeles to Phoenix to Dallas to Miami and finally to Boston Essentially Willie should travel around the country in a counter clockwise manner The distance of this route is 8995 miles Is this solution unique It definitely is not Willie could travel in a clockwise direction instead Boston to Miami to Dallas and so on Because the distance matrix is symmetric this clockwise route is bound to have the same total distance as the counterclockwise route Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 24 The 30 teams in the NBA are each assigned to one of six divisions where each division has five teams Suppose the goal is to assign the teams to divisions so that the average distance among teams in the divisions is minimized In other words the goal is to make the assignments so that teams within a division are close to one another The file P0824xlsx contains dis tances between all NBA cities Actually this was before the Seattle SuperSonics switched to Oklahoma City Use Evolutionary Solver to find an optimal assignment of teams to divisions Does it turn out that your assignments to divisions are the same as the NBAs Hint Arrange the 30 teams into six contigu ous blocks of five teams each Each block will have five team indexes With an AllDifferent constraint you can ensure that the 30 team indexes are all different For each block use lookups to find all distances between pairs of teams in that block and average these Then average these averages over the six divi sions to obtain the objective value 811 Conclusion 469 811 CONCLUSION This chapter contains cuttingedge material The Simplex LP and GRG Nonlinear Solvers have been available for several years to solve many linear integer and nonlinear problems However they have not been able to solve the types of problems discussed in this chapter except possibly by employing tricks or by using a lucky initial solution With Evolutionary Solver now available to a large audience a much wider variety of problems can be solved and the spreadsheet models are usually straightforwardthey do not require tricks Evolutionary Solver is typically much slower than other Solver algorithms especially for linear models with many constraints because it uses a totally different search procedure Because of this we do not recommend that you try Evolutionary Solver unless your model contains functions such as IF COUNT COUNTIF SUMIF MIN MAX and ABS that the other Solvers cannot handle reliably But if your model is formulated more naturally by using such functions or if you can think of no other way of formulating it then Evolutionary Solver can be very useful Summary of Key Management Science Terms Term Explanation Page Genetic algorithm GA Optimization search procedure that mimics the theory of 425 evolution using crossovers mutations and the survival of the fittest Penalties Often used in Evolutionary Solver models to handle 428 constraints penalties are included in objective for violating constraints Twopart tariff One of several pricing schemes that can be used to 432 increase revenue includes a fixed price and a variable price Surplus value Value to customer of purchasing product minus purchase cost 433 to customer assumption is that customer purchases the amount that maximizes surplus value Combinatorial problems Optimization problems where there are a finite number of 438 feasible solutions combinations often difficult because this finite number is huge Cluster analysis General method of grouping people products cities and so 455 on so that members within a cluster are similar and members in different clusters are dissimilar continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Summary of Key Management Science Terms Continued Term Explanation Page Discriminant analysis One of several methods used to classify people products cities 461 and so on into welldefined groups based on data about the members Traveling salesperson Famously difficult management science problem tries to find 464 problem optimal route for a salesperson who starts and ends in a given city and visits all other cities exactly once Summary of Key Excel Terms Term Explanation Excel Page Evolutionary Solver Solvers implementation Start up Solver choose 426 of GA in Excel 2010 only Evolutionary item Evolutionary Solver Various settings that control the Choose Solver Options 430 settings way the GA works see text for details then Evolutionary tab AllDifferent constraint Type of constraint available in One of several options for 465 Evolutionary Solver useful for constraint type in Add models where potential solutions are Constraint dialog box permutations of integers 1 through n 470 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure P R O B L E M S SkillBuilding Problems 25 Fourteen jobs must be assigned to one of three identical machines The goal is to minimize the total time needed to complete all 14 jobs The machine capacities and times needed for the jobs are given in file P0825xlsx For example job 8 requires three units of capacity on a machine for two hours At any given time a machine has five units of capacity How should the jobs be assigned to machines to achieve the earliest possible completion of all jobs 26 Nine jobs need to be completed within eight weeks The number of weeks required to complete each job is given in the file P0826xlsx For example job 2 requires three weeks Each job requires 40 hours of labor per week Each week 160 hours of regular time labor are available Up to 40 hours of overtime labor can be purchased each week at a cost of 10 per hour Additional overtime hours cost 20 per hour a Determine how to minimize the overtime cost incurred in completing the jobs within eight weeks b The same file also lists the due date for each job For example job 2 should be completed within six weeks A penalty of 500 is incurred for each day a job is late Determine how to minimize the sum of overtime and due date penalties 27 Eight students need to be assigned to four dorm rooms two students to a room at State University Based on incompatibility measures the cost incurred if two students room together is shown in the file P0827xlsx How should these students be assigned to rooms to minimize the total incompatibility 28 The costs of producing product A product B or prod ucts A and B bundled together are 50 90 and 140 respectively The file P0828xlsx lists the sizes of the three market segments for these products and how much each of the segments is willing to pay for A alone B alone or the bundle Under the assump tions that a market segment will buy the product combination that yields the maximum nonnegative surplus value minus cost and a segment will buy no product if no product has a nonnegative surplus determine an optimal set of product prices Should the company offer all products for sale 29 Cook County needs to build two hospitals There are nine cities where the hospitals can be built The num ber of hospital visits made annually by the inhabitants of each city and the x and y coordinates of each city are listed in the file P0829xlsx To minimize the total distance that patients must travel to hospitals where should the hospitals be located Solve the prob lem when people can travel in straight lines as the crow flies between cities Then solve it when people must travel along a horizontalvertical grid of roads Hint Use lookup functions to generate the distances between each pair of cities 30 The file P0830xlsx contains quarterly revenue for Nike for the years 1991 to 1998 It also contains Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it handle at most 1600 packages Where should the hubs be located they must be located in three of the 29 cities and which cities should be assigned to which hubs to minimize the total distance the shipments travel Modeling Problems 46 The discussion at the beginning of section 88 men tions Claritas If you were in the directmail business how would you use the information sold by Claritas to improve your profitability 47 How would you use cluster analysis to help test mar ket a consumer goods product 48 Your company sells credit card services and you are concerned with churn Churn occurs when your cus tomers go to a different company Describe how you could use discriminant analysis to learn what distin guishes the customers who switch to another company from those who stay loyal to your company How might you use such a model 49 Your company provides credit to customers Some of these customers default on their loans with very nega tive implications for you Describe how you could use discriminant analysis to learn what distinguishes the customers who default on their loans from those who pay back their loans How might you use such a model 811 Conclusion 473 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E T he MBA program at State University has approxi mately 260 incoming students each fall semester These students are divided into cohorts of approxi mately 65 students each and the students in each cohort sit through exactly the same set of fall courses together Much of the work in these courses is done in teams To ensure that the teams are com parable the MBA Office tries to divide the students in each cohort into 14 teams so that each team has the following qualities Four or five members At least one member with a CPA At least one member with quantitative expertise At least one female At least one minority student At least one international student The file MBA Teamsxlsxindicates the charac teristics of the students in a particular cohort of this years incoming class Your job is to use the Evolutionary Solver to see if you can create teams that have all of the desired properties It is not clear whether this will be possiblefor example there might not be enough minority students for each teamso you should create penalties for failing to meet the various goals where the penalties can be different for different goals 81 ASSIGNING MBA STUDENTS TO TEAMS 474 Chapter 8 Evolutionary Solver An Alternative Optimization Procedure Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 475 Decision Making under Uncertainty C H A P T E R DECIDING W AT BAYER HETHER T O DEVELOP NEW DR UGS T he formal decisionmaking process discussed in this chapter is often used to make difficult decisions in the face of much uncertaintylarge monetary valuesand longterm consequencesStonebraker 2002 chronicles one such decisionmaking process he performed for Bayer Pharmaceuticals in 1999 The development of a new drug is a timeconsuming and expensive process that is filled with risks along the way A pharmaceutical company must first get the proposed drug through preclinical trialswhere the drug is tested on animals Assuming this stage is successful and only about half arethe company can then file an application with the Food and Drug Administration FDA to conduct clinical trials on humansThese clinical trials have three phasesPhase 1 is designed to test the safety of the drug on a small sample of healthy patientsPhase 2 is designed to identify the optimal dose of the new drug on patients with the diseasePhase 3 is a statistically designed study to prove the efficacy and safety of the new drug on a larger sample of patients with the diseaseFailure at any one of these phases means that further testing stops and the drug is never brought to Reicadenwwwshutterstockcom 9 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it marketOf coursethis means that all costs up to the failure point are lostIf the drug makes it through the clinical tests and only about 25 of all drugs do sothe company can then apply to the FDA for permission to manufacture and market its drug in the United States Assuming that FDA approvesthe company is then free to launch the drug in the marketplace The study involved the evaluation of a new drug for busting blood clots called BAY 579602and it commenced at a time just prior to the first decision pointwhether to conduct preclinical testsThis was the companys first formal use of decision making for evaluating a new drugso to convince the company of the worth of such a studyStonebraker did exactly what a successful management scientist should doHe formulated the problem and its objectiveshe identified riskscostsand benefitshe involved key people in the organization to help provide the data needed for the decision analysisandbecause much of the resulting data consisted of educated guesses at besthe performed a thorough sensitivity analysis on the inputs Although we are not told in the article how everything turned outthe analysis did persuade Bayer management to proceed in January 2000 with preclinical testing of the drug The article provides a fascinating look at how such a study should proceed Because there is so much uncertainty the key is determining probabilities and probability distribu tions for the various inputs First there are uncertainties in the various phases of testing Each of these can be modeled with a probability of success For example the chance of making it through preclinical testing was assessed to be about 65 for BAY 579602 although management preferred to use the more conservative benchmark of 50 based on historical data on other drugs for the decision analysis Many of the other uncertain quantities such as the eventual market share are continuous random variables Because the decision tree approach discussed in this chapter requires discrete random variables usually with only a few possible values Stonebraker used a popular threepoint approximation for all continuous quantities He asked experts to assess the 10th percentile the 50th per centile and the 90th percentile and he assigned probabilities 03 04 and 03 to these three values The validity of such an approximation is discussed in Keefer and Bodily 1983 After getting all such estimates of uncertain quantities from the company experts the author examined the expected net present value NPV of all costs and benefits from developing the new drugTo see which of the various uncertain quantities affected the expected NPV most he varied each such quantity one at a time from its 10th percentile to its 90th percentile leaving the other inputs at their base 50th percentile valuesThis identified several quantities that the expected NPV was most sensitive to including the peak product share the price per treatment in the United States and the annual growth rateThe expected NPV was not nearly as sensitive to other uncertain inputs including the product launch date and the production process yieldTherefore in the final decision analysis Stonebraker treated the sensitive inputs as uncertain and the less sensitive inputs as certain at their base values He also calculated the risk profile from developing the drug This indicates the probability distribution of NPV taking all sources of uncertainty into account Although this risk profile was not exactly optimistic 90 chance of losing money using the conservative probabilities of success 67 chance of losing money with the more optimistic productspecific probabilities of success this risk profile compared favorably with Bayers other potential projectsThis evaluation plus the rigor and defensibility of the study led Bayer management to give the goahead on preclinical testing 476 Chapter 9 Decision Making under Uncertainty 91 INTRODUCTION This chapter provides a formal framework for analyzing decision problems that involve uncertainty Our discussion includes the following criteria for choosing among alternative decisions how probabilities are used in the decisionmaking process Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it how early decisions affect decisions made at a later stage how a decision maker can quantify the value of information how attitudes toward risk can affect the analysis Throughout we employ a powerful graphical toola decision treeto guide the analysis A decision tree enables a decision maker to view all important aspects of the problem at once the decision alternatives the uncertain outcomes and their probabilities the economic consequences and the chronological order of events We show how to implement decision trees in Excel by taking advantage of a very powerful and flexible addin from Palisade called PrecisionTree Many examples of decision making under uncertainty exist in the business world including the following Companies routinely place bids for contracts to complete a certain project within a fixed time frame Often these are sealed bids where each company presents a bid for complet ing the project in a sealed envelope Then the envelopes are opened and the low bidder is awarded the bid amount to complete the project Any particular company in the bid ding competition must deal with the uncertainty of the other companies bids as well as possible uncertainty regarding their cost to complete the project if they win the bid The tradeoff is between bidding low to win the bid and bidding high to make a larger profit Whenever a company contemplates introducing a new product into the market there are a number of uncertainties that affect the decision probably the most important being the customers reaction to this product If the product generates high customer demand the company will make a large profit But if demand is lowand after all the vast majority of new products do poorlythe company could fail to recoup its development costs Because the level of customer demand is critical the company might try to gauge this level by test marketing the product in one region of the country If this test market is a success the company can then be more optimistic that a fullscale national marketing of the product will also be successful But if the test market is a failure the company can cut its losses by abandoning the product Whenever manufacturing companies make capacity expansion decisions they face uncertain consequences First they must decide whether to build new plants If they dont expand and demand for their products is higher than expected they will lose revenue because of insufficient capacity If they do expand and demand for their products is lower than expected they will be stuck with expensive underutilized capacity Of course in todays global economy companies also need to decide where to build new plants This decision involves a whole new set of uncertainties including exchange rates labor availability social stability competition from local businesses and others Banks must continually make decisions on whether to grant loans to businesses or individuals As we all know many banks made many very poor decisions especially on mortgage loans during the years leading up to the financial crisis in 2008 They fooled themselves into thinking that housing prices would only increase never decrease When the bottom fell out of the housing market banks were stuck with loans that could never be repaid Utility companies must make many decisions that have significant environmental and economic consequences For these companies it is not necessarily enough to conform to federal or state environmental regulations Recent court decisions have found companies liablefor huge settlementswhen accidents occurred even though the companies followed all existing regulations Therefore when utility companies decide say whether to replace equipment or mitigate the effects of environmental pollution they must take into account the possible environmental consequences such as injuries to people as 91 Introduction 477 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it At the other extreme the decision maker might choose the decision that maximizes the best payoff This criterion called the maximax criterion is appropriate for a risk taker or optimist The best payoffs for the three decisions are the maximums in the three rows 10 30 and 80 The maximax decision maker chooses the decision corresponding to the best of these decision D3 with payoff 80 This criterion looks tempting because it focuses on large gains but its very serious downside is that it ignores possible losses Because this type of decision making could eventually bankrupt a company the maximax criterion is also seldom used 480 Chapter 9 Decision Making under Uncertainty The maximin and maximax criteria make sense in some situa tions but they are generally not used in real decisionmaking problems The maximax criterion finds the best payoff in each row of the payoff table and chooses the decision corresponding to the best of these The expected monetary value or EMV for any decision is a weighted average of the possible payoffs for this decision weighted by the probabilities of the outcomes Using the EMV criterion you choose the decision with the largest EMV This is sometimes called playing the averages 923 Expected Monetary Value EMV We have introduced the maximin and maximax criteria because 1 they are occasionally used to make decisions and 2 they illustrate that there are several reasonable criteria for making decisions In fact there are other possible criteria that we will not discuss although a couple are explored in the problems Instead we now focus on a criterion that is generally regarded as the preferred criterion in most decision problems It is called the expected monetary v alue or EMV criterion To motivate the EMV criterion we first note that the maximin and maximax criteria make no reference to how likely the various outcomes are However decision makers typically have at least some idea of these likeli hoods and they ought to use this information in the decisionmaking process After all if outcome O1 in our problem is extremely unlikely then the pessimist who uses maximin is being overly conservative Similarly if outcome O3 is quite unlikely then the optimist who uses maximax is taking an unnecessary risk The EMV approach assesses probabilities for each outcome of each decision and then calculates the expected payoff from each decision based on these probabilities This expected payoff or EMV is a weighted average of the payoffs in any given row of the payoff table weighted by the probabilities of the outcomes You calculate the EMV for each decision then choose the decision with the largest EMV Note that the terms expected payoff and mean payoff are equivalent We will use them interchangeably Where do the probabilities come from This is a difficult question to answer in general because it depends on each specific situation In some cases the current decision problem is similar to those a decision maker has faced many times in the past Then the probabili ties can be estimated from the knowledge of previous outcomes If a certain type of outcome occurred say in about 30 of previous situations an estimate of its current prob ability might be 030 However there are many decision problems that have no parallels in the past In such cases a decision maker must use whatever information is available plus some intuition to assess the probabilities For example if the problem involves a new product decision and one possible outcome is that a competitor will introduce a similar product in the coming year the decision maker will have to rely on any knowledge of the market and the competitors situation to assess the probability of this outcome It is important to note that Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it To explain this decision tree we introduce a number of decision tree conventions that have become standard 92 Elements of Decision Analysis 483 10 13 15 10 20 30 3 5 2 D1 D2 D3 O1 O2 O3 15 30 30 80 5 2 3 O1 O2 O3 Figure 92 Decision Tree for Simple Decision Problem Decision Tree Conventions 1 Decision trees are composed of nodes circles squares and triangles and branches lines 2 The nodes represent points in time A decision node a square represents a time when the decision maker makes a decision A probability node a circle represents a time when the result of an uncertain outcome becomes known An end node a triangle indicates that the problem is completed all decisions have been made all uncertainty has been resolved and all payoffs and costs have been incurred When people draw decision trees by hand they often omit the actual triangles as we have done in Figure 92 However we still refer to the righthand tips of the branches as the end nodes 3 Time proceeds from left to right This means that any branches leading into a node from the left have already occurred Any branches leading out of a node to the right have not yet occurred 4 Branches leading out of a decision node represent the possible decisions the decision maker can choose the preferred branch Branches leading out of proba bility nodes represent the possible outcomes of uncertain events the decision maker has no control over which of these will occur 5 Probabilities are listed on probability branches These probabilities are conditional on the events that have already been observed those to the left Also the probabilities on branches leading out of any probability node must sum to 1 6 Monetary values are shown to the right of the end nodes As we discuss shortly some monetary values are also placed under the branches where they occur in time 7 EMVs are calculated through a foldingback process discussed next They are shown above the various nodes It is then customary to mark the optimal decision branches in some way We have marked ours with a small notch Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The decision tree in Figure 92 follows these conventions The decision node comes first to the left because the decision maker must make a decision before observing the uncertain outcome The probability nodes then follow the decision branches and the probabilities appear above their branches Actually there is no need for a probability node after the D1 branch because its monetary value is a sure 10 The ultimate payoffs appear next to the end nodes to the right of the probability branches The EMVs above the probability nodes are for the various decisions For example the EMV for the D2 branch is 13 The maximum of the EMVs is for the D2 branch written above the decision node Because it corresponds to D3 we put a notch on the D3 branch to indicate that this decision is optimal This decision tree is almost a direct translation of the spreadsheet model in Figure 91 Indeed the decision tree is overkill for such a simple problem the spreadsheet model provides all of the required information However decision trees are very useful in business problems First they provide a graphical view of the whole problem This can be useful in its own right for the insights it provides especially in more complex problems Second the decision tree provides a framework for doing all of the EMV calculations Specifically it allows you to use the following foldingback procedure to find the EMVs and the optimal decision 484 Chapter 9 Decision Making under Uncertainty FoldingBack Procedure Starting from the right of the decision tree and working back to the left 1 At each probability node calculate an EMVa sum of products of monetary values and probabilities 2 At each decision node take a maximum of EMVs to identify the optimal decision This is exactly what we did in Figure 92 At each probability node we calculated EMVs in the usual way sums of products and wrote them above the nodes Then at the decision node we took the maximum of the three EMVs and wrote it above this node Although this procedure entails more work for more complex decision trees the same two stepstaking EMVs at probability nodes and taking maximums at decision nodesare the only arithmetic operations required In addition the PrecisionTree addin in the next section does the foldingback calculations for you 926 Risk Profiles In our small example each decision leads to three possible monetary payoffs with various probabilities In more complex problems the number of outcomes could be larger maybe considerably larger It is then useful to represent the probability distribution of the monetary values for any decision graphically Specifically we show a spike chart where the spikes are located at the possible monetary values and the heights of the spikes correspond to the probabilities In decisionmaking contexts this type of chart is called a risk profile By looking at the risk profile for a particular decision you can see the risks and rewards involved By comparing risk profiles for different decisions you can gain more insight into their relative strengths and weaknesses The foldingback process is a systematic way of calculating EMVs in a decision tree and thereby identifying the optimal decision strategy The risk profile for a decision is a spike chart that represents the probability distribution of monetary outcomes for this decision Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 486 Chapter 9 Decision Making under Uncertainty Table 92 Data for Bidding Example Low Bid Probability Less than 115000 02 Between 115000 and 120000 04 Between 120000 and 125000 03 Greater than 125000 01 WHERE DO THE NUMBERS COME FROM The company has probably done a thorough cost analysis to estimate its cost to prepare a bid and its cost to manufacture the instruments if it wins the contract Actually even if there is uncertainty in the manufacturing cost the only value required for the decision problem is the mean manufacturing cost The companys estimates of whether or how the competition will bid are probably based on previous bidding experience and some subjec tivity This is discussed in more detail next Solution Lets examine the three elements of SciToolss problem First SciTools has two basic strategies submit a bid or do not submit a bid If SciTools submits a bid then it must decide how much to bid Based on the cost to SciTools to prepare the bid and supply the instruments there is clearly no point in bidding less than 100000SciTools wouldnt make a profit even if it won the bid Actually this isnt totally true Looking ahead to future contracts SciTools might make a low bid just to get in the game and gain experi ence However we wont consider such a possibility here Although any bid amount over 100000 might be considered the data in Table 92 suggest that SciTools might limit its choices to 115000 120000 and 1250003 The next element of the problem involves the uncertain outcomes and their probabili ties We have assumed that SciTools knows exactly how much it will cost to prepare a bid and how much it will cost to supply the instruments if it wins the bid In reality these are probably only estimates of the actual costs and a followup study could perform a sensitivity analysis on these quantities Therefore the only source of uncertainty is the behavior of the competitorswill they bid and if so how much From SciToolss stand point this is difficult information to obtain The behavior of the competitors depends on 1 how many competitors are likely to bid and 2 how the competitors assess their costs of supplying the instruments Nevertheless we assume that SciTools has been involved in similar bidding contests in the past and can reasonably predict competitor behavior from past competitor behavior The result of such prediction is the assessed probability distribu tion in Table 92 and the 30 estimate of the probability of no competing bids The last element of the problem is the value model that transforms decisions and outcomes into monetary values for SciTools The value model is straightforward in this example If SciTools decides not to bid its monetary value is 0no gain no loss If it makes a bid and is underbid by a competitor it loses 5000 the cost of preparing the bid If it bids B dollars and wins the contract it makes a profit of B minus 100000 that is B dollars for winning the bid minus 5000 for preparing the bid and 95000 for supplying the instruments For example if it bids 115000 and the lowest competing bid if any is greater than 115000 then SciTools wins the bid and makes a profit of 15000 3The problem with a bid such as 117000 is that the data in Table 92 make it impossible to calculate the proba bility of SciTools winning the contract if it bids this amount Other than this however there is nothing that rules out such inbetween bids Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 492 Chapter 9 Decision Making under Uncertainty 93 THE PRECISIONTREE ADDIN Decision trees present a challenge for Excel We must somehow take advantage of Excels calculating capabilities to calculate EMVs for example and its graphical capabilities to depict the decision tree Fortunately there is a powerful addin PrecisionTree developed by Palisade Corporation that makes the process relatively straightforward This addin not only enables you to draw and label a decision tree but it performs the foldingback procedure automatically and then allows you to perform sensitivity analysis on key input parameters The first thing you must do to use PrecisionTree is to add it in We assume you have already installed the Palisade DecisionTools suite Then to run PrecisionTree you have two options If Excel is not currently running you can launch Excel and PrecisionTree by clicking on the Windows Start button and selecting the PrecisionTree item from the Palisade Decision Tools group in the list of Programs If Excel is currently running the first procedure will launch PrecisionTree on top of Excel You will know that PrecisionTree is ready for use when you see its tab and the associ ated ribbon shown in Figure 98 If you want to unload PrecisionTree without closing Excel you can do so from its Utilities dropdown list in the Tools group Figure 98 PrecisionTree Ribbon The Decision Tree Model PrecisionTree is quite easy to useat least its most basic items are We will lead you through the steps for the SciTools example Figure 99 shows the results of this procedure just so that you can see what you are working toward See the file SciTools Bidding Decision 2xlsx a Apply this criterion to the example in Simple Decision Problemxlsx Which decision do you choose b Repeat part a for the SciTools example c In general discuss potential strengths and weaknesses of this decision criterion 10 Referring to the previous problem another possible criterion is called expected regret Here you calculate the regret for each cell take a weighted average of these regrets in each row weighted by the probabilities of the outcomes and choose the decision with the smallest expected regret a Apply this criterion to the SciTools example Which decision do you choose b The expected regret criterion is actually equivalent to the EMV criterion in that they always lead to the same decisions Argue why this is true 11 In the SciTools example you might argue that there is a continuum of possible low competitor bids given that there is at least one competing bid not just four possibilities In fact assume the low competitor bid in this case is normally distributed with mean 118000 and standard deviation 4500 Also assume that SciTools will still either not bid or bid 115000 120000 or 125000 Use Excels NORMDIST function to find the EMV for each alternative Which is the best decision now Why cant this be represented in a decision tree Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 93 The PrecisionTree AddIn 495 Figure 913 Dialog Box for Adding or Labeling Branches Figure 915 Tree with All Decision Nodes and Branches Figure 914 Decision Tree with Decision Branches Labeled 14 15 16 17 18 19 C B A TRUE 1000 0 0 Bid 0 FALSE 00 0 0 SciTools Bidding Branch 1 Branch 2 14 15 C B A TRUE 1000 0 0 No 15 16 17 18 19 0 0 Bid 0 FALSE 00 5000 5000 SciTools Bidding Yes 14 15 D C B A TRUE 1000 0 0 No 16 17 18 19 20 21 22 23 Bid 0 TRUE 00 0 5000 FALSE How much to bid 5000 5000 FALSE 00 0 5000 SciTools Bidding 115K 120K Yes 23 24 25 0 5000 FALSE 00 0 5000 125K PrecisionTree Tip Allowable Entries On your computer screen you will note the colorcoding PrecisionTree uses If you inves tigate any colored nonblack cells you will see str ange formulas that PrecisionTree uses for its own purposes Y ou should not modify these formulas Y ou should enter your own probabilities and monetary values only in the cells with black font 4 More decision branches The top branch is completed if SciTools does not bid there is nothing left to do So click on the bottom end node the triangle following SciToolss deci sion to bid and proceed as in the previous step to add and label the decision node and three decision branches for the amount to bid Again refer to Figure 99 The tree to this point should appear as in Figure 915 Note that there are no monetary values below these decision branches because no immediate payoffs or costs are associated with the bid amount decision Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Interpreting the Decision Tree You are finished The completed tree in Figure 99 shows the best strategy and its associated EMV as we discussed previously In fact a comparison of the decision tree in Figure 96 that was created manually and the tree from PrecisionTree in Figure 99 indi cates virtually identical results The best decision strategy is now indicated by the TRUE and FALSE labels above the decision branches rather than the notches we entered by hand Each TRUE corresponds to the optimal decision out of a decision node whereas each FALSE corresponds to a suboptimal decision Therefore you simply follow the TRUE labels In this case the company should bid and its bid amount should be 115000 Note that you do not have to perform the foldingback procedure manually PrecisionTree does this for you Essentially the tree is completed as soon as you finish entering the relevant inputs In addition if you change any of the inputs the tree reacts automatically For example try changing the bid cost in cell B4 from 5000 to some large value such as 20000 You will see that the tree calculations update automatically and the best decision is then not to bid with an associated EMV of 0 PrecisionTree Tip Values at End Nodes You will notice that there are two values following each triangle end node The bottom value is the sum of all monetary values on branches leading to this end node The top value is the prob ability of getting to this end node when the optimal strategy is used This explains why many of these probabilities are 0 the optimal strategy will never lead to these end nodes Policy Suggestion and Risk Profile for Optimal Strategy Once the decision tree is completed PrecisionTree has several tools you can use to gain more information about the decision analysis First you can see a subtree called a Policy Suggestion for the optimal decision To do so choose Policy Suggestion from the Decision Analysis dropdown list and fill in the resulting dialog box as shown in Figure 918 You can experiment with other options The Policy Suggestion option shows only the part of the tree that corresponds to the best decision as shown in Figure 919 498 Chapter 9 Decision Making under Uncertainty To find the optimal decision strategy in any PrecisionTree tree follow the TRUE labels The Policy Suggestion shows only the subtree corresponding to the optimal decision strategy Figure 918 Dialog Box for Information about Optimal Decision You can also obtain a graphical risk profile of the optimal decision by selecting Risk Profile from the Decision Analysis dropdown list and filling in the resulting dialog box as shown in Figure 920 Again you can experiment with the other options As the risk pro file in Figure 921 indicates there are only two possible monetary outcomes if SciTools bids 115000 It either wins 15000 or loses 5000 and the former is much more likely Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The associated probabilities are 086 and 014 respectively This graphical information is even more useful when there are a larger number of possible monetary outcomes You can see what they are and how likely they are Sensitivity Analysis We have already stressed the importance of a followup sensitivity analysis to any decision problem and PrecisionTree makes this relatively easy to perform Of course you can enter any values in the input cells and watch how the tree changes but you can obtain more sys tematic information by clicking on PrecisionTrees Sensitivity Analysis button This brings up the dialog box in Figure 922 Although it has a lot of options it is easy to use once you understand the ideas behind it Here are the main options and how to use them 500 Chapter 9 Decision Making under Uncertainty It takes some practice and experimenting to get used to PrecisionTrees sensitivity analysis tools However they are powerful and worth learning Figure 922 Sensitivity Analysis Dialog Box The Analysis Type dropdown list allows you to vary one input OneWay Sensitivity or two inputs TwoWay Sensitivity simultaneously The Starting Node dropdown list lets you choose any node in the tree and the sensitivity analysis is then performed for the EMV from that node to the right In other words it assumes you have gotten to that node and are now interested in what will happen from then on The node selected in the figure C29 is the leftmost node so by selecting it the sensitivity analysis is on the EMV of the entire tree This is the most common setting You add inputs to vary in the Inputs section You can add as many as you like and all of the checked inputs are included in any particular sensitivity analysis When you add an input to this section you can specify the range over which you want it to vary For example you can vary it by plus or minus 10 in 10 steps from a selected base Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 94 BAYES RULE The examples to this point have required a single decision We now examine multistage problems where a decision maker must make at least two decisions that are separated in time such as when a company must first decide whether to buy information that will help it make a second decision In multistage decision problems there are typically alternating sets of decision nodes and probability nodes The decision maker makes a decision some uncertainty is resolved the decision maker makes another decision more uncertainty is resolved and so on Before analyzing such problems we must discuss one important prob ability issue In a multistage decision tree all probability branches at the right of the tree are con ditional on outcomes that have occurred earlier to their left Therefore the probabilities on these branches are of the form PAB read A given B where A is an event corre sponding to a current probability branch and B is an event that occurs before event A in time However when gathering data for the problem it is sometimes more natural to assess conditional probabilities in the opposite order that is PBA Whenever this is the case Bayes rule must be used to obtain the probabilities needed on the tree Essentially Bayes rule is a mechanism for revising probabilities as new information becomes available To develop Bayes rule let A1 through An be any outcomes Without any further infor mation we believe the probabilities of the As are PA1 through PAn These are called prior probabilities We then have the possibility of gaining some information There are several information outcomes we might observe a typical one of which is labeled B We assume the probabilities of B given that any of the As will occur are known These probabilities labeled PBA1 through PBAn are often called likelihoods Because an information outcome might influence our thinking about the probabilities of the As we need to find the conditional probability PAiB for each outcome Ai This is called the posterior probability of Ai This is where Bayes rule enters the picture It states that we can calculate posterior probabilities from the following formula 94 Bayes Rule 505 The whole purpose of Bayes rule is to revise probabilities as new information becomes available Bayes Rule 91 PAt B PB AtPAt PB A1PA1 Á PB AnPAn Denominator of Bayes Rule Law of Total Probability 92 PB PB A1PA1 Á PB A1PAn In words Bayes rule says that the posterior is the likelihood times the prior divided by a sum of likelihoods times priors As a side benefit the denominator in Bayes rule is also useful in multistage decision trees It is the probability PB of the information out come This formula is important in its own right For B to occur it must occur along with one of the As Formula 92 simply decomposes the probability of B into all of these possibili ties It is sometimes called the law of total probability Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Of the products that eventually did fair in the national market 18 did great in the test market 57 did fair in the test market and 25 did awful in the test market Of the products that eventually did awful in the national market 9 did great in the test market 48 did fair in the test market and 43 did awful in the test market8 The company wants to use a decision tree approach to find the best strategy It also wants to find the expected value of the information provided by the test market Objective To develop a decision tree to find the best strategy for Acme to perform a sen sitivity analysis on the results and to find EVSI and EVPI WHERE DO THE NUMBERS COME FROM The fixed costs of the test market and the national market are probably accurate estimates based on planned advertising and overhead expenses The unit margin is just the difference between the anticipated selling price and the known unit cost of the product The sales volume estimates are clearly approximations to reality because the sales from any new product would form a continuum of possible values Here the company has discretized the problem into three possible outcomes for the national market and it has estimated the sales for each of these discrete outcomes The conditional probabilities of nationalmarket results given testmarket results are probably based on results from previous products that went through test markets and then national markets Solution We begin by discussing the three basic elements of this decision problem the possible strategies the possible outcomes and their probabilities and the value model The possible strategies are clear Acme must first decide whether to run a test market Then it must decide whether to introduce the product nationally However it is important to realize that if Acme decides to run a test market it can base the national market decision on the results of the test market In this case its final strategy will be a contingency plan where it con ducts the test market then introduces the product nationally if it receives sufficiently posi tive testmarket results but abandons the product if it receives sufficiently negative testmarket results The optimal strategies from many multistage decision problems involve similar contingency plans 95 Multistage Decision Problems 515 8You can question why the company ever marketed products nationally after awful testmarket results but we will assume that for whatever reason the company made a few such decisionsand that a few even turned out to be winners In a contingency plan later decisions can depend on earlier decisions and information received FUNDAMENTAL INSIGHT Making Sequential Decisions Whenever you have a chance to make several sequential decisions and you will learn useful information between decision points the decision you make initially depends on the decisions you plan to make in the future and these depend on the information you will learn in the meantime In other words when you decide what to do initially you should look ahead to see what your future options will be and what your decision will be under each option Such a contingency plan is typically superior to a myopic shortsighted plan that doesnt take into account future options in the initial decision making Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 524 Chapter 9 Decision Making under Uncertainty P R O B L E M S SkillBuilding Problems 24 In deciding whether to perform mandatory drug testing we claimed that it is difficult to justify such testing under reasonable conditions Check this yourself in the following questions a Drug testing ought to be more attractive if the test is more reliable Keeping the costs the same as in the example use PrecisionTrees twoway sensitivity tool to see whether the optimal decision test or not test changes as the probability of a false positive and the probability of a false negative both change You can let them vary through some reasonable ranges Explain the results b Repeat part a but first double the two monetary values that make the test more attractive the benefit of identifying a user and the cost of not identifying a user How do your results differ from those in part a c In this part keep the probabilities of false positives and false negatives the same but let the benefits and costs vary Specifically let the benefit of identifying a user and the cost of not identifying a user be of the form 25a and 20a where a is some factor that you can vary Similarly let the cost of barring a nonuser and the cost of violating privacy be of the form 50b and 2b The cost of the test is still 1 The idea is that large values of a andor small values of b will make the testing more attractive Use PrecisionTrees twoway sensitivity tool to see whether the optimal decision test or not test changes for a reasonable range of values of a and b Discuss your results 25 In the drug testing decision find and interpret EVSI and EVPI Here sample information refers to the information from the imperfect drug test whereas perfect information refers to completely reliable information on whether the athlete uses drugs 26 Explain in general why EVSI is the same regardless of the actual cost of the information For example in the Acme problem EVSI is the same regardless of whether the actual cost of the test market is 100000 200000 or any other value Then explain how EVSI together with the actual cost of the information leads to the decision about whether to purchase the information 27 Following up on the previous problem the expected net gain from information is defined as the expected amount gained by having access to the information at its given cost as opposed to not having access to the information Explain how you would calculate this in general What is its value for the Acme problem 28 Prior probabilities are often educated guesses at best so it is worth performing a sensitivity analysis on their values However you must make sure that they are varied so that all probabilities are nonnegative and sum to 1 For the Acme problem perform the following sensitivity analyses on the three prior probabilities and comment on the results a Vary the probability of a great national market in a oneway sensitivity analysis from 0 to 06 in increments of 01 Do this in such a way that the probabilities of the two other outcomes fair and awful stay in the same ratio as they are currently 7 to 4 b Vary the probabilities of a great and a fair national market independently in a twoway sensitivity analysis You can choose the ranges over which these vary but you must ensure that the three prior probabilities continue to be nonnegative and sum to 1 For example you couldnt choose ranges where the probabilities of great and fair are 06 and 05 29 In the Acme problem perform a sensitivity analysis on the quantity sold from a great national market the value in cell B11 Let this value vary over a range of values greater than the current value of 600 so that a great national market is even more attractive than before Does this ever change the optimal strategy If so in what way 30 Using trial and error on the prior probabilities in the Acme problem find values of them that make EVSI equal to 0 These are values where Acme will make the same decision regardless of the testmarket results it observes Comment on why the test market is worthless for your particular prior probabilities SkillExtending Problems 31 We related EVPI to the value of an envelope that contains the true ultimate outcome This concept can be extended to less than perfect information For example in the Acme problem suppose that the company could purchase information that would indicate with certainty that one of the following two outcomes will occur 1 the national market will be great or 2 the national market will not be great Note that outcome 2 doesnt say whether the national market will be fair or awful it just says that it wont be great How much should Acme be willing to pay for such information 32 The concept behind EVPI is that you purchase perfect information the envelope then open the envelope to see which outcome occurs and then make an easy decision You do not however get to choose what Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 961 Utility Functions We begin by discussing an individuals utility function This is a mathematical function that transforms monetary valuespayoffs and costsinto utility values Essentially an individuals utility function specifies the individuals preferences for various monetary payoffs and costs and in doing so it automatically encodes the individuals attitudes toward risk Most individuals are risk averse which means intuitively that they are willing to sacrifice some EMV to avoid risky gambles In terms of the utility function this means that every extra dollar of payoff is worth slightly less to the individual than the previous dollar and every extra dollar of cost is considered slightly more costly in terms of utility than the previous dollar The resulting utility functions are shaped as shown in Figure 941 Mathematically these functions are said to be increasing and concave The increasing part means that they go uphilleveryone prefers more money to less money The concave part means that they increase at a decreasing rate This is the riskaverse behavior 526 Chapter 9 Decision Making under Uncertainty 8 6 4 2 0 2 5 0 5 Monetary Value millions Utility Figure 941 RiskAverse Utility Function There are two aspects of implementing expected utility maximization in a real deci sion analysis First an individuals or companys utility function must be assessed This is a timeconsuming task that typically involves many tradeoffs It is usually carried out by experts in the field and we do not discuss the details of the process here Second the resulting utility function is used to find the best decision This second step is relatively straightforward You substitute utility values for monetary values in the decision tree and then fold back as usual That is you calculate expected utilities at probability branches and take maximums of expected utilities at decision branches We will look at a numerical example later in this section 962 Exponential Utility As we have indicated utility assessment is tedious Even in the best of circumstances when a trained consultant attempts to assess the utility function of a single person the process requires the person to make a series of choices between hypothetical alternatives involving uncertain outcomes Unless the person has some training in probability these choices will probably be difficult to understand let alone make and it is unlikely that the person will answer consistently as the questioning proceeds The process is even more difficult when a companys utility function is being assessed Because different company executives typically have different attitudes toward risk it can be difficult for these people to reach a consensus on a common utility function For these reasons classes of readymade utility functions have been developed One important class is called exponential utility and has been used in many financial investment decisions An exponential utility function has only one adjustable numerical parameter called the risk tolerance and there are straightforward ways to discover an appropriate value of this parameter for a particular individual or company So the advantage of using an exponential utility function is that it is relatively easy to assess The drawback is that Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it a 05 million loss a 01 million gain and a 1 million gain The probabilities of these outcomes are 025 050 and 025 respectively The possible outcomes of the more risky venture are a 1 million loss a 1 million gain and a 3 million gain The probabilities of these outcomes are 035 060 and 005 respectively If Venture Limited must decide on exactly one of these alternatives what should it do Objective To see how the companys risk averseness determined by its risk tolerance in an exponential utility function affects its decision WHERE DO THE NUMBERS COME FROM The outcomes for each of the risky alternatives probably form a continuum of possible values However as in Example 94 the company has classified these into a few possibili ties and made intelligent estimates of the monetary consequences and probabilities of these discrete possibilities Solution We assume that Venture Limited has an exponential utility function Also based on Howards guidelines we assume that the companys risk tolerance is 64 of its net sales or 192 million A sensitivity analysis on this parameter will be performed later on You can substitute into Equation 96 to find the utility of any monetary outcome For example the gain from the riskless alternative in thousands of dollars is 125 and its utility is As another example the utility of a 1 million loss is These are the values we use instead of monetary values in the decision tree DEVELOPING THE DECISION TREE MODEL Fortunately PrecisionTree takes care of the details After building a decision tree and labeling it with monetary values in the usual way click on the name of the tree the box on the far left of the tree to open the dialog box shown in Figure 942 Then fill in the information under the Utility Function tab as shown in the figure This says to use an exponential utility function with risk tolerance 1920 the value in cell B5 As indicated in the spreadsheet all monetary values are measured in 1000s It also indicates that expected utilities as opposed to EMVs should appear in the decision tree The completed tree for this example is shown in Figure 943 See the file Using Exponential Utilityxlsx You build it in exactly the same way as usual and link probabilities and monetary values to its branches in the usual way For example there is a link in cell C22 to the monetary value in cell B12 However the expected values shown in the tree those shown in color on a computer screen are expected utilities and the optimal decision is the one with the largest expected utility In this case the expected utilities for the riskless option investing in the less risky venture and investing in the more risky venture U1000 1 e10001920 1 16834 06834 U125 1 e1251920 1 09370 00630 528 Chapter 9 Decision Making under Uncertainty Dont worry about the actual utility values for example whether they are positive or negative Only the relative magnitudes matter in terms of decision making The tree is built and labeled with monetary values exactly as before PrecisionTree then takes care of calculating the expected utilities Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it decisions are 125000 175000 and 400000 The latter two of these are calculated in row 15 as the usual SUMPRODUCT of monetary values and probabilities So from an EMV point of view the more risky venture is definitely best In fact the ordering of the three alternatives using the EMV criterion is exactly the opposite of the ordering using expected utility But because Venture Limited is sufficiently risk averse and the monetary values are sufficiently large the company is willing to sacrifice 275000 of EMV to avoid risk Sensitivity Analysis How sensitive is the optimal decision to the key parameter the risk tolerance You can answer this by changing the risk tolerance and watching how the decision tree changes You can check that when the company becomes more risk tolerant the more risky venture eventually becomes optimal In fact this occurs when the risk tolerance increases to approximately 2210 million In the other direction of course when the company becomes less risk tolerant the riskless decision continues to be optimal The middle decision the less risky alternative is evidently not optimal for any value of the risk toler ance The bottom line is that the decision considered optimal depends entirely on the attitudes toward risk of Venture Limiteds top management 530 Chapter 9 Decision Making under Uncertainty 963 Certainty Equivalents Now lets change the problem slightly so that Venture Limited has only two options It can either enter the less risky venture or receive a certain dollar amount x and avoid the gamble altogether We want to find the dollar amount x so that the company is indifferent between these two options If it enters the risky venture its expected utility is 00525 calculated ear lier If it receives x dollars for certain its utility is To find the value x where the company is indifferent between the two options set equal to 00525 or and solve for x Taking natural loga rithms of both sides and multiplying by 1920 the result is Because of the units of measure this is really 104000 This value is called the certainty equivalent of the risky venture The company is indifferent between entering the less risky venture and receiving 104000 to avoid it Although the EMV of the less risky venture is 175000 the company acts as if it is equivalent to a sure 104000 In this sense the company is willing to give up the difference in EMV 71000 to avoid a gamble By a similar calculation the certainty equivalent of the more risky venture is approxi mately 86000 That is the company acts as if this more risky venture is equivalent to a sure 86000 when in fact its EMV is a hefty 400000 In this case the company is willing to give up the difference in EMV 314000 to avoid this particular gamble Again the reason is that the company wants to avoid risk You can see these certainty equivalents in PrecisionTree by changing the Display box in Figure 942 to show Certainty Equivalent The resulting tree is shown in Figure 944 The certainty equivalents we just discussed appear in cells C24 and C32 Note that we rounded the values in the text to the nearest 1000 The values in the figure are more exact x 1920 ln09475 104 ex1920 09475 1 ex1920 Ux 1 ex1920 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 96 Incorporating Attitudes Toward Risk 531 17 18 19 20 D C B A TRUE 1000 125 1250000 Which venture 1250000 Ventures Riskless Bad 21 22 23 24 25 250 00 500 5000000 FALSE Outcome 0 1035447 500 00 Ventures Riskless Less risky Bad Fair 26 27 28 29 30 100 1000000 250 00 1000 10000000 350 00 1000 10000000 Ventures Riskless Less risky Bad Fair Good More risky Bad 31 32 33 34 35 FALSE Outcome 0 862017 600 00 1000 10000000 50 00 Ventures Riskless Less risky Bad Fair Good More risky Bad Fair 36 3000 30000000 Good Figure 944 Certainty Equivalents in Tree E X A M P L E 94 MARKETING A NEW PRODUCT AT ACME CONTINUED B efore concluding this section we take a last look at the Acme marketing decision from the previous section Suppose Acme decides to use expected utility as its criterion with an exponential utility function Is the EMVmaximizing decision still optimal Remember that this strategy first performed the test market and then marketed nationally only if the testmarket results were great Objective To see how risk aversion affects Acmes strategy Solution There is very little work to do You first enter a risk tolerance value in a blank cell Then starting with the tree from Figure 932 fill out the dialog box in Figure 942 with a link to the risk tolerance cell See the finished version of the file Acme Marketing Decisions 2xlsxfor the details It is then interesting to perform a sensitivity analysis on the risk tolerance We tried this letting the risk tolerance vary from 1000 to 10000 remember that these are in thousands of dollars and seeing whether the decision to run a test market changes The results appear in Figure 945 Do you understand why it is better to run the test market only if the risk tolerance is sufficiently large It is not really because of the cost of the test market When the risk tol erance is small the company is so risk averse that it never markets nationallyon any of the National market decision nodes So information from the test market is worthless However as R increases the company becomes less risk averse and in some scenarios its Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 534 Chapter 9 Decision Making under Uncertainty Summary of Key Terms Continued Term Explanation Excel Page Equation Maximax criterion The optimists criterion find the best 480 possible payoff for each decision and choose the decision with the best of these Expected monetary The weighted average of the possible 480 value EMV payoffs from a decision weighted by their probabilities EMV criterion Choose the decision with the 480 maximum EMV Decision tree A graphical device for illustrating all 482 of the aspects of the decision problem and for finding the optimal decision or decision strategy Foldingback Calculation method for decision tree 484 procedure starting at the right take EMVs at probability nodes maximums of EMVs at decision nodes Risk profile Chart that represents the probability 484 distribution of monetary outcomes for any decision PrecisionTree Useful Excel addin developed Has its 492 by Palisade for building and own ribbon analyzing decision trees in Excel PrecisionTree Useful for seeing how the optimal Use PrecisionTree 501 strategy region chart decision changes as selected Sensitivity inputs vary Analysis button PrecisionTree Useful for seeing which inputs Use PrecisionTree 501 tornado and affect a selected EMV Sensitivity Analysis spider charts the most button Bayes rule Formula for updating probabilities 505 91 as new information becomes available prior probabilities are transformed into posterior probabilities Law of total The denominator in Bayes rule 505 92 probability for calculating the unconditional probability of an information outcome Expected value of The most the imperfect sample information 513 94 sample information such as the results of a test market would EVSI be worth Expected value of The most perfect information on some 513 95 perfect information uncertain outcome would be worth EVPI represents an upper bound on any EVSI Contingency plan A decision strategy where later decisions 515 depend on earlier decisions and outcomes observed in the meantime Expected utility Choosing the decision that maximizes the 526 maximization expected utility typically sacrifices EMV to avoid risk when large monetary amounts are at stake Utility function A mathematical function that encodes an 526 individuals or companys attitudes toward risk continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 97 Conclusion 535 Term Explanation Excel Page Equation Exponential utility A popular class of utility functions 527 96 function risk where only a single parameter the risk tolerance tolerance has to be specified Certainty equivalent The sure dollar value equivalent to the 530 expected utility of a gamble P R O B L E M S SkillBuilding Problems 37 The SweetTooth Candy Company knows it will need 10 tons of sugar six months from now to implement its production plans Jean Dobson SweetTooths purchasing manager has essentially two options for acquiring the needed sugar She can either buy the sugar at the going market price when she needs it six months from now or she can buy a futures contract now The contract guarantees delivery of the sugar in six months but the cost of purchasing it will be based on todays market price Assume that possible sugar futures contracts available for purchase are for five tons or ten tons only No futures contracts can be purchased or sold in the intervening months Thus SweetTooths possible decisions are to 1 purchase a futures contract for ten tons of sugar now 2 purchase a futures contract for five tons of sugar now and purchase five tons of sugar in six months or 3 purchase all ten tons of needed sugar in six months The price of sugar bought now for delivery in six months is 00851 per pound The transaction costs for fiveton and tenton futures contracts are 65 and 110 respectively Finally Ms Dobson has assessed the probability distribution for the possible prices of sugar six months from now in dollars per pound The file P0937xlsx contains these possible prices and their corresponding probabilities a Given that SweetTooth wants to acquire the needed sugar in the least costly way create a cost table that specifies the cost in dollars associated with each possible decision and possible sugar price in the future b Use PrecisionTree to identify the decision that minimizes SweetTooths expected cost of meeting its sugar demand c Perform a sensitivity analysis on the optimal decision letting each of the three currency inputs vary one at a time plus or minus 25 from its base value and summarize your findings In response to which of these inputs is the expected cost value most sensitive 38 Carlisle Tire and Rubber Inc is considering expanding production to meet potential increases in the demand for one of its tire products Carlisles alternatives are to construct a new plant expand the existing plant or do nothing in the short run The market for this particular tire product may expand remain stable or contract Carlisles marketing department estimates the probabilities of these market outcomes as 025 035 and 040 respectively The file P0938xlsx contains Carlisles estimated payoff in dollars table a Use PrecisionTree to identify the strategy that maximizes this tire manufacturers expected profit b Perform a sensitivity analysis on the optimal decision letting each of the monetary inputs vary one at a time plus or minus 10 from its base value and summarize your findings In response to which monetary inputs is the expected profit value most sensitive 39 A local energy provider offers a landowner 180000 for the exploration rights to natural gas on a certain site and the option for future development This option if exercised is worth an additional 1800000 to the landowner but this will occur only if natural gas is discovered during the exploration phase The landowner believing that the energy companys interest in the site is a good indication that gas is present is tempted to develop the field herself To do so she must contract with local experts in natural gas exploration and development The initial cost for such a contract is 300000 which is lost forever if no gas is found on the site If gas is discovered however the landowner expects to earn a net profit of 6000000 The landowner estimates the probability of finding gas on this site to be 60 a Create a payoff table that specifies the landowners payoff in dollars associated with each possible decision and each outcome with respect to finding natural gas on the site b Use PrecisionTree to identify the strategy that maximizes the landowners expected net earnings from this opportunity c Perform a sensitivity analysis on the optimal decision letting each of the inputs vary one at a time plus or minus 25 from its base value and summarize your findings In response to which model inputs is the expected profit value most sensitive Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 40 Techware Incorporated is considering the introduction of two new software products to the market In particular the company has four options regarding these two proposed products introduce neither product introduce product 1 only introduce product 2 only or introduce both products Research and development costs for products 1 and 2 are 180000 and 150000 respectively Note that the first option entails no costs because research and development efforts have not yet begun The success of these software products depends on the trend of the national economy in the coming year and on the consumers reaction to these products The companys revenues earned by introducing product 1 only product 2 only or both products in various states of the national economy are given in the file P0940xlsx The probabilities of observing a strong fair or weak trend in the national economy in the coming year are assessed to be 030 050 and 020 respectively a Create a payoff table that specifies Techwares net revenue in dollars for each possible decision and each outcome with respect to the trend in the national economy b Use PrecisionTree to identify the strategy that maximizes Techwares expected net revenue from the given marketing opportunities c Perform a sensitivity analysis on the optimal decision letting each of the inputs vary one at a time plus or minus 25 from its base value and summarize your findings In response to which model inputs is the expected net revenue value most sensitive 41 Consider an investor with 10000 available to invest He has the following options regarding the allocation of his available funds 1 he can invest in a riskfree savings account with a guaranteed 3 annual rate of return 2 he can invest in a fairly safe stock where the possible annual rates of return are 6 8 or 10 or 3 he can invest in a more risky stock where the possible annual rates of return are 1 9 or 17 Note that the investor can place all of his available funds in any one of these options or he can split his 10000 into two 5000 investments in any two of these options The joint probability distribution of the possible return rates for the two stocks is given in the file P0941xlsx a Create a payoff table that specifies this investors return in dollars in one year for each possible decision and each outcome with respect to the two stock returns b Use PrecisionTree to identify the strategy that maximizes the investors expected earnings in one year from the given investment opportunities c Perform a sensitivity analysis on the optimal decision letting the amount available to invest and the riskfree return both vary one at a time plus or minus 100 from their base values and summarize your findings 42 A buyer for a large department store chain must place orders with an athletic shoe manufacturer six months prior to the time the shoes will be sold in the department stores In particular the buyer must decide on November 1 how many pairs of the manufacturers newest model of tennis shoes to order for sale during the coming summer season Assume that each pair of this new brand of tennis shoes costs the department store chain 45 per pair Furthermore assume that each pair of these shoes can then be sold to the chains customers for 70 per pair Any pairs of these shoes remaining unsold at the end of the summer season will be sold in a closeout sale next fall for 35 each The probability distribution of consumer demand for these tennis shoes during the coming summer season has been assessed by market research specialists and is provided in the file P0942xlsx Finally assume that the department store chain must purchase these tennis shoes from the manufacturer in lots of 100 pairs a Create a payoff table that specifies the contribution to profit in dollars from the sale of the tennis shoes by this department store chain for each possible purchase decision and each outcome with respect to consumer demand b Use PrecisionTree to identify the strategy that maximizes the department store chains expected profit earned by purchasing and subsequently selling pairs of the new tennis shoes c Perform a sensitivity analysis on the optimal decision letting the three monetary inputs vary one at a time over reasonable ranges and summarize your findings In response to which model inputs is the expected earnings value most sensitive 43 Each day the manager of a local bookstore must decide how many copies of the community newspaper to order for sale in her shop She must pay the newspapers publisher 040 for each copy and she sells the news papers to local residents for 075 each Newspapers that are unsold at the end of day are considered worth less The probability distribution of the number of copies of the newspaper purchased daily at her shop is provided in the file P0943xlsx Create a payoff table that lists the profit from each order quantity multiples of 1000 only and each demand and use it to find the order quantity that maximizes expected profit Why is this an easier approach than a decision tree for this particular problem 44 Two construction companies are bidding against one another for the right to construct a new community center building in Bloomington Indiana The first construction company Fine Line Homes believes that its competitor Buffalo Valley Construction will place 536 Chapter 9 Decision Making under Uncertainty Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it a bid for this project according to the distribution shown in the file P0944xlsx Furthermore Fine Line Homes estimates that it will cost 160000 for its own company to construct this building Given its fine reputation and longstanding service within the local community Fine Line Homes believes that it will likely be awarded the project in the event that it and Buffalo Valley Construction submit exactly the same bids Create a payoff table that lists the profit from each Fine Line bid and each competing bid and use it to find the bid that maximizes Fine Lines expected profit Why is this an easier approach than a decision tree for this particular problem 45 Suppose that you have sued your employer for damages suffered when you recently slipped and fell on an icy surface that should have been treated by your companys physical plant department Specifically your injury resulting from this accident was sufficiently serious that you in consultation with your attorney decided to sue your company for 500000 Your companys insurance provider has offered to settle this suit with you out of court If you decide to reject the settlement and go to court your attorney is confident that you will win the case but is uncertain about the amount the court will award you in damages He has provided his assessment of the probability distribution of the courts award to you in the file P0945xlsx In addition there are extra legal fees of 10000 you will have to pay if you go to court Let S be the insurance providers proposed outofcourt settlement in dollars For which values of S will you decide to accept the settlement For which values of S will you choose to take your chances in court Assume that you are seeking to maximize the expected net payoff from this litigation 46 One of your colleagues has 2000 available to invest Assume that all of this money must be placed in one of three investments a particular money market fund a stock or gold Each dollar your colleague invests in the money market fund earns a virtually guaranteed 3 annual return Each dollar he invests in the stock earns an annual return characterized by the probability distribution provided in the file P0946xlsx Finally each dollar he invests in gold earns an annual return characterized by the probability distribution given in the same file a If your colleague must place all of his available funds in a single investment which investment should he choose to maximize his expected earnings over the next year b Suppose now that your colleague can place all of his available funds in one of these three investments as before or he can invest 1000 in one alternative and 1000 in another Assuming that he seeks to maximize his expected total earnings in one year how should he allocate his 2000 47 Consider a population of 2000 individuals 800 of whom are women Assume that 300 of the women in this population earn at least 60000 per year and 200 of the men earn at least 60000 per year a What is the probability that a randomly selected individual from this population earns less than 60000 per year b If a randomly selected individual is observed to earn less than 60000 per year what is the probability that this person is a man c If a randomly selected individual is observed to earn at least 60000 per year what is the probability that this person is a woman 48 Yearly automobile inspections are required for residents of the state of Pennsylvania Suppose that 18 of all inspected cars in Pennsylvania have problems that need to be corrected Unfortunately Pennsylvania state inspections fail to detect these problems 12 of the time On the other hand assume that an inspection never detects a problem when there is no problem Consider a car that is inspected and is found to be free of problems What is the probability that there is indeed something wrong that the inspection has failed to uncover 49 Consider again the landowners decision problem described in Problem 39 Suppose now that at a cost of 90000 the landowner can request that a soundings test be performed on the site where natural gas is believed to be present The company that conducts the soundings concedes that 30 of the time the test will indicate that no gas is present when it actually is When natural gas is not present in a particular site the soundings test is accurate 90 of the time a Given that the landowner pays for the soundings test and the test indicates that gas is present what is the landowners revised estimate of the probability of finding gas on this site b Given that the landowner pays for the soundings test and the test indicates that gas is not present what is the landowners revised estimate of the probability of not finding gas on this site c Should the landowner request the given soundings test at a cost of 90000 Explain why or why not If not at what price if any would the landowner choose to obtain the soundings test 50 The chief executive officer of a firm in a highly competitive industry believes that one of her key employees is providing confidential information to the competition She is 90 certain that this informer is the vice president of finance whose contacts have been extremely valuable in obtaining financing for the company If she decides to fire this vice president and he is the informer she estimates that the company will gain 500000 If she decides to fire this vice president but he is not the informer the company will lose his 97 Conclusion 537 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 85 If your company makes a particular decision in the face of uncertainty you estimate that it will either gain 10000 gain 1000 or lose 5000 with probabilities 040 030 and 030 respectively You correctly calculate the EMV as 2800 However you distrust the use of this EMV for decision making purposes After all you reason that you will never receive 2800 you will receive 10000 1000 or lose 5000 Discuss this reasoning 86 In the previous question suppose you have the option of receiving a check for 2700 instead of making the risky decision described Would you make the risky decision where you could lose 5000 or would you take the sure 2700 What would influence your decision 87 In a classic oildrilling example you are trying to decide whether to drill for oil on a field that might or might not contain any oil Before making this decision you have the option of hiring a geologist to perform some seismic tests and then predict whether there is any oil or not You assess that if there is actually oil the geologist will predict there is oil with probability 085 You also assess that if there is no oil the geologist will predict there is no oil with probability 090 Why will these two probabilities not appear on the decision tree Which probabilities will be on the decision tree 88 Your company has signed a contract with a good customer to ship the customer an order no later than 20 days from now The contract indicates that the customer will accept the order even if it is late but instead of paying the full price of 10000 it will be allowed to pay 10 less 9000 due to lateness You estimate that it will take anywhere from 17 to 22 days to ship the order and each of these is equally likely You believe you are in good shape reasoning that the expected days to ship is the average of 17 through 22 or 195 days Because this is less than 20 you will get your full 10000 What is wrong with your reasoning 89 You must make one of two decisions each with possible gains and possible losses One of these decisions is much riskier than the other having much larger possible gains but also much larger possible losses and it has a larger EMV than the safer decision Because you are risk averse and the monetary values are large relative to your wealth you base your decision on expected utility and it indicates that you should make the safer decision It also indicates that the certainty equivalent for the risky decision is 210000 whereas its EMV is 540000 What do these two numbers mean What do you know about the certainty equivalent of the safer decision 90 A potentially huge hurricane is forming in the Caribbean and there is some chance that it might make a direct hit on Hilton Head Island South Carolina where you are in charge of emergency preparedness You have made plans for evacuating everyone from the island but such an evacuation is obviously costly and upsetting for all involved so the decision to evacuate shouldnt be made lightly Discuss how you would make such a decision Is EMV a relevant concept in this situation How would you evaluate the consequences of uncertain outcomes 91 It seems obvious that if you can purchase information before making an ultimate decision this information should generally be worth something but explain exactly why and when it is sometimes worth nothing 92 Insurance companies wouldnt exist unless customers were willing to pay the price of the insurance and the insurance companies were making a profit So explain how insurance is a winwin proposition for customers and the company 93 You often hear about the tradeoff between risk and reward Is this tradeoff part of decision making under uncertainty when the decision maker uses the EMV criterion For example how does this work in investment decisions 94 Can you ever use the material in this chapter to help you make your own reallife decisions Consider the following You are about to take an important and difficult exam in one of your MBA courses and you see an opportunity to cheat Obviously from an ethical point of view you shouldnt cheat but from a purely monetary point of view could it also be the wrong decision To model this consider the longterm monetary consequences of all possible outcomes 546 Chapter 9 Decision Making under Uncertainty Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it T he Jogger Shoe Company is trying to decide whether to make a change in its most popular brand of running shoes The new style would cost the same to produce and be priced the same but it would incorporate a new kind of lacing system that according to its marketing research people would make it more popular There is a fixed cost of 300000 for changing over to the new style The unit contribution to beforetax profit for either style is 8 The tax rate is 35 Also because the fixed cost can be depreciated and will therefore affect the aftertax cash flow a depreciation method is needed You can assume it is straightline depreciation The current demand for these shoes is 190000 pairs annually The company assumes this demand will continue for the next three years if the current style is retained However there is uncertainty about demand for the new style if it is introduced The company models this uncertainty by assuming a normal distribution in year 1 with mean 220000 and standard deviation 20000 The company also assumes that this demand whatever it is will remain constant for the next three years However if demand in year 1 for the new style is sufficiently low the company can always switch back to the current style and realize an annual demand of 190000 The company wants a strategy that will maximize the expected net present value NPV of total cash flow for the next three years where a 15 interest rate is used for the purpose of calculating NPV 91 JOGGER SHOE COMPANY Case 91 Jogger Shoe Company 547 C A S E Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E T he Westhouser Paper Company in the state of Washington currently has an option to purchase a piece of land with good timber forest on it It is now May 1 and the current price of the land is 22 million Westhouser does not actually need the timber from this land until the beginning of July but its top executives fear that another company might buy the land between now and the beginning of July They assess that there is a 5 chance that a competitor will buy the land during May If this does not occur they assess that there is a 10 chance that the competitor will buy the land during June If Westhouser does not take advantage of its current option it can attempt to buy the land at the beginning of June or the beginning of July provided that it is still available Westhousers incentive for delaying the purchase is that its financial experts believe there is a good chance that the price of the land will fall significantly in one or both of the next two months They assess the possible price decreases and their probabilities in Table 97 and Table 98 Table 97 shows the probabilities of the possible price decreases during May Table 98 lists the conditional probabilities of the possible price decreases in June given the price decrease in May For example it indicates that if the price decrease in May is 60000 then the possible price decreases in June are 0 30000 and 60000 with respective probabilities 06 02 and 02 If Westhouser purchases the land it believes that it can gross 3 million This does not count the cost of purchasing the land But if it does not purchase the landWesthouser believes that it can make 650000 from alternative investments What should the company do Table 97 Distribution of Price Decrease in May Price Decrease Probability 0 05 60000 03 120000 02 92 WESTHOUSER PAPER COMPANY 548 Chapter 9 Decision Making under Uncertainty Table 98 Distribution of Price Decrease in June Price Decrease in May 0 60000 120000 June Decrease Probability June Decrease Probability June Decrease Probability 0 03 0 06 0 07 60000 06 30000 02 20000 02 120000 01 60000 02 40000 01 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E B iotechnical Engineering specializes in developing new chemicals for agricultural applications The company is a pioneer in using the sterilemale pro cedure to control insect infestations It operates several laboratories around the world that raise insects and expose them to extralarge doses of radiation making them sterile As an alternative to chlorinated hydrocarbon pesticides such as DDT the sterilemale procedure has been used frequently with a good track record of success most notably with the Mediterranean fruit fly or Medfly That pest was controlled in California through the release of treated flies on the premise that the sterile male flies would compete with fertile wild males for mating opportunities Any female that has mated with a sterile fly will lay eggs that do not hatch The California Medfly campaigns required about five successive releases of sterile malesat intervals timed to coincide with the time for newly hatched flies to reach adulthoodbefore the Medfly was virtually eliminated Only sterile flies were subsequently caught in survey traps The effectiveness of the sterilemale procedure was enhanced by the release of malathion poisonous bait just a few days before each release cutting down on the number of viable wild adults More recently Biotechnical Engineering has had particular success in using genetic engineering to duplicate various insect hormones and pheromones scent attractants Of particular interest is the application of such methods against the Gypsy Moth a notorious pest that attacks trees The company has developed synthetic versions of both hormones and pheromones for that moth It has a synthetic sexual attractant that male moths can detect at great distances Most promising is the synthetic juvenile hormone The juvenile hormone controls moth meta morphosis determining the timing for the trans formation of a caterpillar into a chrysalis and then into an adult Too much juvenile hormone wreaks havoc with this process causing caterpillars to turn into freak adults that cannot reproduce Biotechnical Engineering has received a government contract to test its new technology in an actual eradication campaign The company will participate in a smallscale campaign against the Gypsy Moth in the state of Oregon Because the pest is so damaging Dr June Scribner the administrator in charge is considering using DDT as an alternative procedure Of course that banned substance is only available for government emergency use because of the environmental damage it may cause In addition to spraying with DDT two other procedures may be employed 1 using Biotechnicals scent lure followed by the release of sterile males and 2 spraying with the companys juvenile hormone to prevent larvae from developing into adults Dr Scribner wants to select the method that yields the best expected payoff described below Although both of the newer procedures are known to work under laboratory conditions there is some uncertainty about successful propagation of the chemicals in the wild and about the efficacy of the sterilemale procedure with moths If the scentlure program is launched at a cost of 5 million Biotechnical claims that it will have a fifty fifty chance of leaving a low number of native males versus a high number Once the results of that phase are known a later choice must be made to spray with DDT or to release sterile malesthe cost of the sterilization and delivery of the insects to the countrsi de is an additional 5 million But if this twophase program is successful the net present value of the worth of trees saved is 30 million including the benefit of avoiding all other forms of environmental damage The indigenous moth population would be destroyed and a new infestation could occur only from migrants Biotechnicals experience with other eradication programs indicates that if the scent lure leaves a small native male population there is a 90 chance for a successful eradication by using sterile malesotherwise there is only a 10 chance for success by using sterile males A failure results in no savings 9This case was written by Lawrence L Lapin San Jose State University 93 BIOTECHNICAL ENGINEERING9 Case 93 Biotechnical Engineering 549 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 551 DEVELOPING BOARDING STRATEGIES AT AMERICA W EST M anagement science often attempts to solve problems that we all experience One such problem is the boarding process for airline flights As customers we all hate to wait while travelers boarding ahead of us store their luggage and block the aisles But this is also a big problem for the airlines Airlines lose money when their airplanes are on the ground so they have a real incentive to reduce the turnaround time from when a plane lands until it departs on its next flight Of course the turnaround time is influenced by several factors including passenger deplaning baggage unloading fueling cargo unloading airplane maintenance cargo loading baggage loading and passenger boarding Airlines try to perform all of these tasks as efficiently as possible but passenger boarding is particularly difficult to shorten Although the airlines want passengers to board as quickly as possible they dont want to use measures that might antagonize their passengers One study by van den Briel et al 2005 indicates how a combination of management science methods including simulation was used to make passenger boarding more efficient at America West Airlines America West which merged with US Airways in 2006 was a major US carrier based in Phoenix Arizona It served more destinations nonstop than any other airline Image SourceJupiter Images Introduction to Simulation Modeling C H A P T E R 10 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The airlines fleet consisted of Airbus A320s Airbus A319s Boeing 757s Boeing 737s and Airbus A318s At the time of the study airlines used a variety of boarding strategies but the predominant strategy was the backtofront BF strategy where after boarding first class passengers and passengers with special needs the rest of the passengers are boarded in groups starting with rows in the back of the plane As the authors suspected and most of us have experienced this strategy still results in significant congestion Within a given section of the plane the back say passengers storing luggage in over head compartments can block an aisle Also people in the aisle or middle seat often need to get back into the aisle to let windowseat passengers be seatedThe authors developed an integer programming IP model to minimize the number of such aisle blockagesThe decision variables determined which groups of seats should be boarded in which order Of course the BF strategy was one possible feasible solution but it turned out to be a suboptimal solutionThe IP model suggested that the best solution was an outsidein OI strategy where groups of passengers in window seats board first then groups in the middle seats and finally groups in aisle seats with all of these groups going essentially in a backtofront order The authors recognized that their IP model was at best an idealized model of how passengers actually behave Its biggest drawback is that it ignores the inherent random ness in passenger behaviorTherefore they followed up their optimization model with a simulation model As they state We used simulation to validate the analytical model and to obtain a finer level of detail This validation of an approximate or idealized analytical model is a common use for simulationTo make the simulation as realistic as possible they used two cameras one inside the plane and one inside the bridge leading to the plane to tape customer behavior By analyzing the tapes they were able to estimate the required inputs to their simulation model such as the time between passengers walking speed blocking time and time to store luggage in overhead com partments After the basic simulation model was developed it was used as a tool to evaluate various boarding strategies suggested by the IP model It also allowed the authors to experiment with changes to the overall boarding process that might be beneficial For example reducing congestion inside the airplane is not very helpful if the gate agent at the entrance to the bridge processes passengers too slowlyTheir final recommendation based on a series of simulation experiments was to add a second gate agent there had been only one before and to board passengers in six groups using an OI strategyThe simulation model suggested that this could reduce the board ing time by about 37 The authors recommendations were implemented first as a pilot project and then systemwideThe pilot results were impressive with a 39 reduction in boarding times By September 2003 the new boarding strategies had been implemented in 80 of America Wests airports with a decrease in departure delays as much as 601 Besides this obvious benefit to the airline customers also appear to be happier Now they can easily understand when to queue up for boarding and they experience less blocking after they get inside the plane 552 Chapter 10 Introduction to Simulation Modeling 101 INTRODUCTION A simulation model is a computer model that imitates a reallife situation It is like other mathematical models but it explicitly incorporates uncertainty in one or more input vari ables When you run a simulation you allow these random input variables to take on Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it various values and you keep track of any resulting output variables of interest In this way you are able to see how the outputs vary as a function of the varying inputs The fundamental advantage of a simulation model is that it provides an entire distrib ution of results not simply a single bottomline result As an example suppose an automobile manufacturer is planning to develop and market a new model car The com pany is ultimately interested in the net present value NPV of the profits from this car over the next 10 years However there are many uncertainties surrounding this car including the yearly customer demands for it the cost of developing it and others The company could develop a spreadsheet model for the 10year NPV using its best guesses for these uncertain quantities It could then report the NPV based on these best guesses However this analysis would be incomplete and probably misleadingthere is no guarantee that the NPV based on bestguess inputs is representative of the NPV that will actually occur It is much better to treat the uncertainty explicitly with a simulation model This involves enter ing probability distributions for the uncertain quantities and seeing how the NPV varies as the uncertain quantities vary Each different set of values for the uncertain quantities can be considered a scenario Simulation allows the company to generate many scenarios each leading to a particular NPV In the end it sees a whole distribution of NPVs not a single best guess The company can see what the NPV will be on average and it can also see worstcase and bestcase results These approaches are summarized in Figures 101 and 102 Figure 101 indicates that the deterministic nonsimulation approach using best guesses for the uncertain inputs is generally not the appropriate method It leads to the flaw of averages as we will discuss later in the chapter The problem is that the outputs from the deterministic model are often not representative of the true outputs The appropriate method is shown in Figure 102 Here the uncertainty is modeled explicitly with random inputs and the end result is a prob ability distribution for each of the important outputs 101 Introduction 553 Figure 101 Inappropriate Deterministic Model Figure 102 Appropriate Simulation Model Simulation models are also useful for determining how sensitive a system is to changes in operating conditions For example the operations of a supermarket could be simulated Once the simulation model has been developed it could then be run with suit able modifications to ask a number of whatif questions For example if the supermarket experiences a 20 increase in business what will happen to the average time customers must wait for service A huge benefit of computer simulation is that it enables managers to answer these types of whatif questions without actually changing or building a physical system For example the supermarket might want to experiment with the number of open registers to see the effect on customer waiting times The only way it can physically experiment with Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it more registers than it currently owns is to purchase more equipment Then if it determines that this equipment is not a good investmentcustomer waiting times do not decrease appreciablythe company is stuck with expensive equipment it doesnt need Computer simulation is a much less expensive alternative It provides the company with an electronic replica of what would happen if the new equipment were purchased Then if the simula tion indicates that the new equipment is worth the cost the company can be confident that purchasing it is the right decision Otherwise it can abandon the idea of the new equipment before the equipment has been purchased Spreadsheet simulation modeling is quite similar to the other modeling applications in this book You begin with input variables and then relate these with appropriate Excel formulas to produce output variables of interest The main difference is that simulation uses random numbers to drive the whole process These random numbers are generated with special functions that we will discuss in detail Each time the spreadsheet recalcu lates all of the random numbers change This provides the ability to model the logical process once and then use Excels recalculation ability to generate many different scenar ios By collecting the data from these scenarios you can see the most likely values of the outputs and the bestcase and worstcase values of the outputs In this chapter we begin by illustrating spreadsheet models that can be developed with builtin Excel functionality However because simulation is becoming such an important tool for analyzing real problems addins to Excel have been developed to streamline the process of developing and analyzing simulation models Therefore we then introduce RISK one of the most popular simulation addins This addin not only augments the simulation capabili ties of Excel but it also enables you to analyze models much more quickly and easily The purpose of this chapter is to introduce basic simulation concepts show how sim ulation models can be developed in Excel and demonstrate the capabilities of the RISK addin Then in the next chapter armed with the necessary simulation tools we will explore a number of interesting and useful simulation models Before proceeding you might ask whether simulation is really used in the business world The answer is a resounding yes The chapter opener described an airline example and many other examples can be found online For example if you visit wwwpalisadecom you will see descriptions of interesting RISK applications from companies that regu larly use this addin Simulation has always been a powerful tool but it had limited use for several reasons It typically required specialized software that was either expensive or difficult to learn or it required a lot of tedious computer programming Fortunately in the past two decades spreadsheet simulation together with Excel addins such as RISK has put this powerful methodology in the hands of the massespeople like you and the companies you are likely to work for Many businesses now understand that there is no longer any reason to ignore uncertainty they can model it directly with spreadsheet simulation 102 PROBABILITY DISTRIBUTIONS FOR INPUT VARIABLES In this section we discuss the building blocks of spreadsheet simulation models All spreadsheet simulation models are similar to the spreadsheet models from previous chapters They have a number of cells that contain values of input variables The other cells then contain formulas that embed the logic of the model and eventually lead to the output variables of interest The primary difference between the spreadsheet models you have developed so far and simulation models is that at least one of the input variable cells in a simulation model contains random numbers Each time the spreadsheet recalculates the random numbers change and the new random values of the inputs produce new values of 554 Chapter 10 Introduction to Simulation Modeling In spreadsheet simulation models input cells can contain random numbersAny output cells then vary as these random inputs change Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the outputs This is the essence of simulationit enables you to see how outputs vary as random inputs change Excel Tip Recalculation Key The easiest way to mak e a spr eadsheet recalculate is to pr ess the F9 key This is often called the recalc key Technically speaking input cells do not contain random numbers they contain probability distributions In general a probability distribution indicates the possible values of a variable and the probabilities of these values As a very simple example you might indicate by an appropriate formula to be described later that you want a probability dis tribution with possible values 50 and 100 and corresponding probabilities 07 and 03 If you force the sheet to recalculate repeatedly and watch this input cell you will see the value 50 about 70 of the time and the value 100 about 30 of the time No other values besides 50 and 100 will appear When you enter a given probability distribution in a random input cell you are describ ing the possible values and the probabilities of these values that you believe mirror reality There are many probability distributions to choose from and you should always attempt to choose an appropriate distribution for each specific problem This is not necessarily an easy task Therefore we address it in this section by answering several key questions What types of probability distributions are available and why do you choose one probability distribution rather than another in an actual simulation model Which probability distributions can you use in simulation models and how do you invoke them with Excel formulas In later sections we address one additional question Does the choice of input probability distribution really matterthat is are the outputs from the simulation sensitive to this choice 102 Probability Distributions for Input Variables 555 FUNDAMENTAL INSIGHT Basic Elements of Spr eadsheet Simulation A spr eadsheet sim ulation model r equires thr ee elements 1 a method for entering random quantities from specified pr obability distributions in input cells 2 the usual types of Excel f ormulas f or r elating outputs to inputs and 3 the ability to mak e the spreadsheet r ecalculate man y times and ca pture the resulting outputs f or statistical anal ysis Excel has some capabilities for performing these steps but Excel addins such as RISK provide much better tools for automating the process 1021 Types of Probability Distributions Imagine a toolbox that contains the probability distributions you know and understand As you obtain more experience in simulation modeling you will naturally add probability distri butions to your toolbox that you can then use in future simulation models We begin by adding a few useful probability distributions to this toolbox However before adding any spe cific distributions it is useful to provide a brief review of some important general character istics of probability distributions These include the following distinctions Discrete versus continuous Symmetric versus skewed Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Bounded versus unbounded Nonnegative versus unrestricted Discrete Versus Continuous A probability distribution is discrete if it has a finite number of possible values1 For example if you throw two dice and look at the sum of the faces showing there are only 11 discrete possibilities the integers 2 through 12 In contrast a probability distribution is continuous if its possible values are essentially some continuum An example is the amount of rain that falls during a month in Indiana It could be any decimal value from 0 to say 15 inches The graph of a discrete distribution is a series of spikes as shown in Figure 1032 The height of each spike is the probability of the corresponding value 556 Chapter 10 Introduction to Simulation Modeling FUNDAMENTAL INSIGHT Choosing Probability Distributions for Uncertain Inputs In sim ulation models it is impor tant to choose appropriate probability distributions for all uncertain inputsThese choices can str ongly affect the results Unfortunately there are no right answers You need to choose the pr obability distributions that best encode your uncertainty and this is not necessaril y easy However the properties discussed in this sec tion pr ovide y ou with useful guidelines f or making reasonable choices 1Actually it is possible for a discrete variable to have a countably infinite number of possible values such as all the nonnegative integers 0 1 2 and so on However this is not an important distinction for practical applications 2This figure and several later figures have been captured from Palisades RISK addin Figure 103 A Typical Discrete Probability Distribution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it In contrast a continuous distribution is characterized by a density function a smooth curve as shown in Figure 104 There are two important properties of density functions First the height of the density function above any value indicates the relative likelihood of that value and second probabilities can be calculated as areas under the curve 102 Probability Distributions for Input Variables 557 Figure 104 A Typical Continuous Probability Distribution The heights above a density function are not probabilities but they still indicate relative likelihoods of the possible values Sometimes it is convenient to treat a discrete probability distribution as continuous and vice versa For example consider a students random score on an exam that has 1000 possible points If the grader scores each exam to the nearest integer then even though the score is really discrete with many possible integer values it is probably more convenient to model its distribution as a continuum Continuous probability distributions are typically more intuitive and easier to work with than discrete distributions in cases such as this where there are many possible values In contrast continuous distributions are sometimes discretized for simplicity Symmetric Versus Skewed A probability distribution can either be symmetric or skewed to the left or right Figures 104 105 106 provide examples of each of these You typically choose between a symmetric and skewed distribution on the basis of realism For example if you want to model a students score on a 100point exam you will probably choose a leftskewed distribution This is because a few poorly prepared students typically pull down the curve On the other hand if you want to model the time it takes to serve a customer at a bank you will probably choose a rightskewed distribution This is because most customers take only a minute or two but a few customers take a long time Finally if you want to model the monthly return on a stock you might choose a distribution symmetric around zero reasoning that the stock return is just as likely to be positive as negative and there is no obvious reason for skewness in either direction Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Bounded Versus Unbounded A probability distribution is bounded if there are values A and B such that no possible value can be less than A or greater than B The value A is then the minimum possible value and the value B is the maximum possible value The distribution is unbounded if there are no such bounds Actually it is possible for a distribution to be bounded in one direction but not the other As an example the distribution of scores on a 100point exam is bounded between 0 and 100 In contrast the distribution of the amount of damages Mr Jones 558 Chapter 10 Introduction to Simulation Modeling Figure 105 A Positively Skewed Probability Distribution Figure 106 A Negatively Skewed Probability Distribution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it submits to his insurance company in a year is bounded on the left by 0 but there is no natural upper bound Therefore you might model this amount with a distribution that is bounded by 0 on the left but is unbounded on the right Alternatively if you believe that no damage amount larger than 20000 can occur you could model this amount with a distribution that is bounded in both directions Nonnegative Versus Unrestricted One important special case of bounded distributions is when the only possible values are nonnegative For example if you want to model the random cost of manufacturing a new product you know for sure that this cost must be nonnegative There are many other such examples In such cases you should model the randomness with a probability distribution that is bounded below by 0 This rules out negative values that make no practical sense 1022 Common Probability Distributions Now that you know the types of probability distributions available you can add some common probability distributions to your toolbox The file Probability Distributionsxlsx was developed to help you learn and explore these Each sheet in this file illustrates a particular probability distribution It describes the general characteristics of the distribu tion indicates how you can generate random numbers from the distribution either with Excels builtin functions or with RISK functions and it includes histograms of these distributions from simulated data to illustrate their shapes3 It is important to realize that each of the following distributions is really a family of distributions Each member of the family is specified by one or more parameters For example there is not a single normal distribution there is a normal distribution for each possible mean and standard deviation you specify Therefore when you try to find an appropriate input probability distribution in a simulation model you first have to choose an appropriate family and then you have to select the appropriate parameters for that family Uniform Distribution The uniform distribution is the flat distribution illustrated in Figure 107 It is bounded by a minimum and a maximum and all values between these two extremes are equally likely You can think of this as the I have no idea distribution For example a manager might realize that a building cost is uncertain If she can state only that I know the cost will be between 20000 and 30000 but other than this I have no idea what the cost will be then a uniform distribution from 20000 to 30000 is a natural choice However even though some people do use the uniform distribution in such cases these situations are arguably not very common or realistic If the manager really thinks about it she can prob ably provide more information about the uncertain cost such as The cost is more likely to be close to 25000 than to either of the extremes Then some distribution other than the uniform is more appropriate Regardless of whether the uniform distribution is an appropriate candidate as an input distribution it is important for another reason All simulation software packages including Excel are capable of generating random numbers uniformly distributed between 0 and 1 These are the building blocks of most simulated random numbers in that random numbers from other probability distributions are generated from them 102 Probability Distributions for Input Variables 559 Think of the Probability Distributionsxlsx file as a dictionary of the most commonly used distributions Keep it handy for reference 3In later sections of this chapter and all through the next chapter we discuss much of RISKs functionality For this section the only functionality we use is RISKs collection of functions such as RISKNORMAL and RISKTRIANG for generating random numbers from various probability distributions You can skim the details of these functions for now and refer back to them as necessary in later sections A family of distribu tions has a common name such as nor mal Each member of the family is specified by one or more numer ical parameters Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it FREEZING RANDOM NUMBERS The automatic recalculation of random numbers can be useful sometimes and annoying at other times There are situations when you want the random numbers to stay fixedthat is you want to freeze them at their current values The following threestep method does this 1 Select the range that you want to freeze such as A4A503 in Figure 108 2 Press Ctrlc to copy this range 3 With the same range still selected select the Paste Values option from the Paste dropdown menu on the Home ribbon This procedure pastes a copy of the range onto itself except that the entries are now numbers not formulas Therefore when ever the spreadsheet recalculates these numbers do not change Each sheet in the Probability Distributionsxlsx file has a list of 500 random numbers that have been frozen The histograms in the sheets are based on the frozen random numbers However we encourage you to enter live random numbers in column B over the frozen ones and see how the histogram changes when you press F9 1023 Using RISK to Explore Probability Distributions5 The Probability Distributionsxlsx file illustrates a few frequently used probability distribu tions and it shows the formulas required to generate random numbers from these distribu tions Another option is to use Palisades RISK addin which allows you to experiment with probability distributions Essentially it allows you to see the shapes of various distributions and to calculate probabilities for them all in a userfriendly graphical interface To run RISK click on the Windows Start button go to the Programs tab locate the Palisades DecisionTools suite and select RISK After a few seconds you will see the welcome screen which you can close At this point you should have an RISK tab and corresponding ribbon Select a blank cell in your worksheet and then click on Define Distributions on left of the RISK ribbon see Figure 1011 You will see one of several galleries of distributions depending on the tab you select For example Figure 1012 102 Probability Distributions for Input Variables 563 Figure 1011 RISK Ribbon RISK Function RISKUNIFORM To g enerate a r andom number fr om any uniform distrib ution enter the formula RISKUNIFORMMinValMaxVal in any cell Her e MinVal and MaxVal are the minimum and maximum possible values Note that if MinVal is 0 and MaxVal is 1 this function is equivalent to Excels RAND function Random numbers that have been frozen do not change when you press the F9 key 5Palisade previously offered a standalone program called RISKview for exploring probability distributions and we discussed it in the previous edition However Palisade discontinued RISKview and instead incorporates its functionality in RISK Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it shows the gallery of continuous distributions Highlight one of the distributions and click on Select Distribution For example choose the uniform distribution with minimum 75 and maximum 150 You will see the shape of the distribution and a few summary measures to the right as shown in Figure 1013 For example it indicates that the mean and standard deviation of this uniform distribution are 1125 and 2165 Everything in this window is interactive Suppose you want to find the probability that a value from this distribution is less than 95 You can drag the lefthand slider in the diagram the vertical line with the triangle at the top to the position 95 as shown in Figure 1013 564 Chapter 10 Introduction to Simulation Modeling Figure 1012 Gallery of Continuous Distributions Figure 1013 RISK Illustration of Uniform Distribution Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it You see immediately that the lefthand probability is 0267 Similarly if you want the proba bility that a value from this distribution is greater than 125 you can drag the righthand slider to the position 125 to see that the required probability is 03333 Rather than sliding you can enter the numbers such as 95 and 125 directly into the areas above the sliders You can also enter probabilities instead of values For example if you want the value such that there is probability 010 to the left of itthe 10th percentileenter 10 in the left space above the chart You will see that the corresponding value is 825 Similarly if you want the value such that there is probability 010 to the right of it enter 10 in the right space above the chart and you will see that the corresponding value is 1425 The Define Distributions window in RISK is quick and easy We urge you to use it and experiment with some of its options By the way you can click on the third button from the left at the bottom of the window to copy the chart into an Excel worksheet However you then lose the interactive capabilities such as moving the sliders Discrete Distribution A discrete distribution is useful for many situations either when the uncertain quantity is not really continuous the number of televisions demanded for example or when you want a discrete approximation to a continuous variable All you need to specify are the possible values and their probabilities making sure that the probabilities sum to 1 Because of this flexibility in specifying values and probabilities discrete distributions can have practically any shape As an example suppose a manager estimates that the demand for a particular brand of television during the coming month will be 10 15 20 or 25 with respective probabilities 01 03 04 and 02 This typical discrete distribution is illustrated in Figure 1014 102 Probability Distributions for Input Variables 565 Figure 1014 Discrete Distribution from RISK The interactive capabilities of RISKs Define Distributions window with its sliders make it perfect for finding probabilities or percentiles for any given distribution The Discrete sheet of the Probability Distributionsxlsx file indicates how to work with a discrete distribution See Figure 1015 As you can see there are two quite differ ent ways to generate a random number from this distribution We discuss the Excel way in detail in section 104 For now we simply mention that this is one case of many where it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it parameters justify your choice in words and use RISK to draw your chosen distribution a Company experts have no idea what the distribution of the development cost is All they can state is we are 95 sure it will be at least 450000 and we are 95 sure it will be no more than 650000 b Company experts can still make the same statement as in part a but now they can also state We believe the distribution is symmetric reasonably bellshaped and its most likely value is about 550000 c Company experts can still make the same statement as in part a but now they can also state We believe the distribution is skewed to the right and its most likely value is about 500000 10 Continuing the preceding problem suppose that another key uncertain input is the development time which is measured in an integer number of months For each of the following scenarios choose an appropriate distribution together with its parameters justify your choice in words and use RISK to draw your chosen distribution a Company experts believe the development time will be from 6 to 10 months but they have absolutely no idea which of these will result b Company experts believe the development time will be from 6 to 10 months They believe the probabilities of these five possible values will increase linearly to a most likely value at 8 months and will then decrease linearly c Company experts believe the development time will be from 6 to 10 months They believe that 8 months is twice as likely as either 7 months or 9 months and that either of these latter possibilities is three times as likely as either 6 months or 10 months 103 Simulation and the Flaw of Averages 573 103 SIMULATION AND THE FLAW OF AVERAGES To help motivate simulation modeling in general we present a simple example in this sec tion It will clearly show the distinction between Figure 101 a deterministic model with bestguess inputs and Figure 102 an appropriate simulation model In doing so it will illustrate a pitfall called the flaw of averages that you should always try to avoid6 6As far as we know the term flaw of averages was coined by Sam Savage the same Stanford professor quoted earlier E X A M P L E 101 ORDERING CALENDARS AT WALTON BOOKSTORE I n August Walton Bookstore must decide how many of next years nature calendars to order Each calendar costs the bookstore 750 and sells for 10 After January 1 all unsold calendars will be returned to the publisher for a refund of 250 per calendar Walton believes that the number of calendars it can sell by January 1 follows some proba bility distribution with mean 200 Walton believes that ordering to the average demand that is ordering 200 calendars is a good decision Is it Objective To illustrate the difference between a deterministic model with a best guess for uncertain inputs and a simulation model that incorporates uncertainty explicitly WHERE DO THE NUMBERS COME FROM The monetary values are straightforward The mean demand is probably an estimate based on historical demands for similar calendars Solution A deterministic model appears in Figure 1022 See the file Walton Bookstore 1xlsx Assuming the best guess for demand Walton orders to this average value and it appears Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 500 in profit on average Absolutely not The situation isnt symmetric The largest profit you can get is 500 which occurs about half the time whenever demand is greater than 200 A typical such situation appears in the figure where the excess demand of 63 is simply lost However when demand is less than 200 the profit is less than 500 and it keeps decreasing as demand decreases We ran RISK with 1000 iterations which will be explained in detail in section 105 and found the resulting histogram of 1000 simulated profits shown in Figure 1024 The large spike on the right is due to the cases where demand is 200 or more and profit is 500 All the little spikes to the left are where demand is less than 200 and profit is less than 500 sometimes considerably less You can see on the right that the mean profit the average of the 1000 simulated profits is only about 380 well less than the 500 suggested by the deterministic model 103 Simulation and the Flaw of Averages 575 Figure 1024 Histogram of Simulated Profits The point of this simple example is that a deterministic model can be very misleading In particular the output from a deterministic model that uses best guesses for uncertain inputs is not necessarily equal to or even close to the average of the output from a simula tion This is exactly what the flaw of averages means FUNDAMENTAL INSIGHT The Flaw of Averages If a model contains uncertain inputsit can be very mis leading to build a deterministic model b y using the means of the inputs to predict an output The resulting output value can be considerabl y differentlower or higherthan the mean of the output values obtained from running a sim ulation with uncer tainty incorpo rated explicitly Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 104 SIMULATION WITH BUILTIN EXCEL TOOLS In this section we show how spreadsheet simulation models can be developed and ana lyzed with Excels builtin tools without using addins As you will see this is certainly possible but it presents two problems First the RISK functions illustrated in the Probability Distributionsxlsx file are not available You are able to use only Excels RAND function and transformations of it to generate random numbers from various prob ability distributions Second there is a bookkeeping problem Once you build an Excel model with output cells linked to appropriate random input cells you can press the F9 key as often as you like to see how the outputs vary However there is no quick way to keep track of these output values and summarize them This bookkeeping feature is the real strength of a simulation addin such as RISK It can be done with Excel usually with data tables but the summarization of the resulting data is completely up to the useryou Therefore we strongly recommend that you use the Excelonly method described in this section only if you dont have an addin such as RISK To illustrate the Excelonly procedure we continue analyzing the calendar problem from Example 101 This general problem occurs when a company such as a news vendor must make a onetime purchase of a product such as a newspaper to meet customer demands for a certain period of time If the company orders too few newspapers it will lose potential profit by not having enough on hand to satisfy its customers If it orders too many it will have news papers left over at the end of the day that at best can be sold at a loss More generally the problem is to match supply to an uncertain demand a very common problem in business In much of the rest of this chapter we will discuss variations of this problem 576 Chapter 10 Introduction to Simulation Modeling E X A M P L E 102 SIMULATING WITH EXCEL ONLY AT WALTON BOOKSTORE R ecall that Walton Bookstore must decide how many of next years nature calendars to order Each calendar costs the bookstore 750 and sells for 10 After January 1 all unsold calendars will be returned to the publisher for a refund of 250 per calendar In this version we assume that demand for calendars at the full price is given by the probability distribution shown in Table 101 Walton wants to develop a simulation model to help it decide how many calendars to order Table 101 Probability Distribution of Demand for Walton Example Demand Probability 100 030 150 020 200 030 250 015 300 005 Objective To use builtin Excel toolsincluding the RAND function and data tables but no addinsto simulate profit for several order quantities and ultimately choose the best order quantity WHERE DO THE NUMBERS COME FROM The numbers in Table 101 are the key to the simulation model They are discussed in more detail next Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Notes about Confidence Intervals It is common in computer simulations to estimate the mean of some distribution by the average of the simulated observations The usual practice is then to accompany this esti mate with a confidence interval which indicates the accuracy of the estimate You might recall from statistics that to obtain a confidence interval for the mean you start with the estimated mean and then add and subtract a multiple of the standard error of the estimated mean If the estimated mean that is the average is the confidence interval is given in the following formula X 104 Simulation with BuiltIn Excel Tools 581 The confidence interval provides a measure of accuracy of the mean profit as estimated from the simulation Confidence Interval for the Mean X Multiple Standard Error of X Approximate 95 Confidence Interval for the Mean X 2s1n Sample Size Determination n 4 Estimated standard deviation2 B2 Standard Error of s1n X The standard error of is the standard deviation of the observations divided by the square root of n the number of observations X Here s is the symbol for the standard deviation of the observations You can obtain it with the STDEV function in Excel The multiple in the confidence interval formula depends on the confidence level and the number of observations If the confidence level is 95 for example then the multiple is usually very close to 2 so a good guideline is to go out two standard errors on either side of the average to obtain an approximate 95 confidence interval for the mean To be more precise if n is reasonably large which is almost always the case in simula tions the central limit theorem implies that the correct multiple is the number from the stan dard normal distribution that cuts off probability 0025 in each tail This is a famous number in statistics 196 Because 196 is very close to 2 it is acceptable for all practical purposes to use 2 instead of 196 in the confidence interval formula Note that you should use a different mul tiple if you want a 90 or a 99 confidence level rather than a 95 level Analysts often plan a simulation so that the confidence interval for the mean of some important output will be sufficiently narrow The reasoning is that narrow confidence intervals imply more precision about the estimated mean of the output variable If the con fidence level is fixed at some value such as 95 the only way to narrow the confidence interval is to simulate more replications Assuming that the confidence level is 95 the following value of n is required to ensure that the resulting confidence interval will have a halflength approximately equal to some specified value B The idea is to choose the number of itera tions large enough so that the resulting confidence interval will be sufficiently narrow Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it By now you should appreciate the usefulness of data tables in spreadsheet simulations They allow you to take a prototype simulation and replicate its key results as often as you like This method makes summary statistics over the entire group of replications and corresponding charts fairly easy to obtain Nevertheless it takes some work to create the data tables and charts In the next section you will see how the RISK addin does a lot of this work for you 586 Chapter 10 Introduction to Simulation Modeling P R O B L E M S SkillBuilding Problems 11 Suppose you own an expensive car and purchase auto insurance This insurance has a 1000 deductible so that if you have an accident and the damage is less than 1000 you pay for it out of your pocket However if the damage is greater than 1000 you pay the first 1000 and the insurance pays the rest In the current year there is probability 0025 that you will have an accident If you have an accident the damage amount is normally distributed with mean 3000 and standard deviation 750 a Use Excel to simulate the amount you have to pay for damages to your car This should be a oneline simulation so run 5000 iterations by copying it down Then find the average amount you pay the standard deviation of the amounts you pay and a 95 confidence interval for the average amount you pay Note that many of the amounts you pay will be 0 because you have no accidents b Continue the simulation in part a by creating a twoway data table where the row input is the deductible amount varied from 500 to 2000 in multiples of 500 Now find the average amount you pay the standard deviation of the amounts you pay and a 95 confidence interval for the average amount you pay for each deductible amount c Do you think it is reasonable to assume that damage amounts are normally distributed What would you criticize about this assumption What might you suggest instead 12 In August of the current year a car dealer is trying to determine how many cars of the next model year to order Each car ordered in August costs 20000 The demand for the dealers next year models has the probability distribution shown in the file P1012xlsx Each car sells for 25000 If demand for next years cars exceeds the number of cars ordered in August the dealer must reorder at a cost of 22000 per car Excess cars can be disposed of at 17000 per car Use simulation to determine how many cars to order in August For your optimal order quantity find a 95 confidence interval for the expected profit 13 In the Walton Bookstore example suppose that Walton receives no money for the first 50 excess calendars returned but receives 250 for every calendar after the first 50 returned Does this change the optimal order quantity 14 A sweatshirt supplier is trying to decide how many sweatshirts to print for the upcoming NCAA basketball championships The final four teams have emerged from the quarterfinal round and there is now a week left until the semifinals which are then followed in a couple of days by the finals Each sweatshirt costs 10 to produce and sells for 25 However in three weeks any leftover sweatshirts will be put on sale for half price 1250 The supplier assumes that the demand for his sweatshirts during the next three weeks when interest in the tournament is at its highest has the distribution shown in the file P1014xlsx The residual demand after the sweat shirts have been put on sale has the distribution also shown in this file The supplier being a profit maxi mizer realizes that every sweatshirt sold even at the sale price yields a profit However he also realizes that any sweatshirts produced but not sold even at the sale price must be thrown away resulting in a 10 loss per sweatshirt Analyze the suppliers problem with a simulation model SkillExtending Problems 15 In the Walton Bookstore example with a discrete demand distribution explain why an order quantity other than one of the possible demands cannot maxi mize the expected profit Hint Consider an order of 190 calendars for example If this maximizes expected profit then it must yield a higher expected profit than an order of 150 or 100 But then an order of 200 calendars must also yield a larger expected profit than 190 calendars Why Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 105 INTRODUCTION TO RISK Spreadsheet simulation modeling has become extremely popular in recent years both in the academic and corporate communities Much of the reason for this popularity is due to simulation addins such as RISK There are two primary advantages to using such an addin First an addin gives you easy access to many probability distributions you might want to use in your simulation models You already saw in section 102 how the RISKDIS CRETE RISKNORMAL and RISKTRIANG functions among others are easy to use and remember Second an addin allows you to perform simulations much more easily than is possible with Excel alone To replicate a simulation in Excel you typically need to build a data table Then you have to calculate summary statistics such as averages stan dard deviations and percentiles with builtin Excel functions If you want graphs to enhance the analysis you have to create them In short you have to perform a number of timeconsuming steps for each simulation Simulation addins such as RISK perform much of this work automatically Although we will focus only on RISK in this book it is not the only simulation add in available for Excel Two worthy competitors are Crystal Ball developed by Decisioneering wwwdecisioneeringcom and Risk Solver Platform developed by Frontline Systems the developer of Solver wwwfrontsyscom Both Crystal Ball and Risk Solver Platform have much of the same functionality as RISK However the authors have a natural bias for RISKwe have been permitted by its developer Palisade Corporation wwwpalisadecom to provide the academic version free with this book If it were not included you would have to purchase it from Palisade at a fairly steep price Indeed Microsoft Office does not include RISK Crystal Ball Risk Solver Platform or any other simulation addinyou must purchase them separately 1051 RISK Features Here is an overview of some of RISKs features We will discuss all of these in more detail in this section 1 RISK contains a number of functions such as RISKNORMAL and RISKDIS CRETE that make it easy to generate observations from a wide variety of probability distributions You saw some of these in section 102 2 You can designate any cell or range of cells in your simulation model as output cells When you run the simulation RISK automatically keeps summary measures averages standard deviations percentiles and others from the values generated in these output cells across the replications It also creates graphs such as histograms based on these values In other words RISK takes care of tedious bookkeeping operations for you 3 RISK has a special function RISKSIMTABLE that allows you to run the same simulation several times using a different value of some key input variable each time This input variable is often a decision variable For example suppose that you would like to simulate an inventory ordering policy as in the Walton Bookstore example Your ultimate purpose is to compare simulation outputs across a number of possible order quantities such as 100 150 200 250 and 300 If you use an appropriate formula involving the RISKSIMTABLE function the entire simulation is performed for each of these order quantities separatelywith one click of a button You can then compare the outputs to choose the best order quantity 105 Introduction to RISK 587 RISK provides a number of functions for simulating from various distributions and it takes care of all the bookkeeping in spreadsheet simula tions Excel simulations without RISK require much more work for the user Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is the same Walton Bookstore model as before except that a triangular distribution for demand is used 1052 Loading RISK To build simulation models with RISK you need to have Excel open with RISK added in The first step if you have not already done so is to install the Palisade DecisionTools suite with the Setup program Then you can load RISK by clicking on the Windows Start button selecting the Programs group selecting the Palisade DecisionTools group and finally selecting the RISK item If Excel is already open this loads RISK inside Excel If Excel is not yet open this launches Excel and RISK simultaneously8 After RISK is loaded you see an RISK tab and the corresponding RISK ribbon in Figure 10309 588 Chapter 10 Introduction to Simulation Modeling 8We have had the best luck when we 1 close other applications we are not currently using and 2 launch Excel and RISK together by starting RISK However it is also possible to start RISK after Excel is already running 9If you have been using version 50 of RISK you will see only minor changes in the newer versions 551 or 57 now available However if you have been using version 45 you will see major changes in the user interface Figure 1030 RISK Ribbon 1053 RISK Models with a Single Random Input Variable In the remainder of this section we illustrate some of RISKs functionality by revisiting the Walton Bookstore example The next chapter demonstrates the use of RISK in a number of interesting simulation models Throughout our discussion you should keep one very important idea in mind The development of a simulation model is basically a two step procedure The first step is to build the model itself This step requires you to enter all of the logic that transforms inputs including RISK functions such as RISKDISCRETE into outputs such as profit This is where most of the work and thinking go exactly as in models from previous chapters and RISK cannot do this for you It is your job to enter the formulas that link inputs to outputs appropriately However once this logic has been incorporated RISK takes over in the second step It automatically replicates your model with different random numbers on each replication and it reports any summary measures that you request in tabular or graphical form Therefore RISK greatly decreases the amount of busy work you need to do but it is not a magic bullet We begin by analyzing an example with a single random input variable The majority of the work and thinking goes into developing the model Setting up RISK and then running it are relatively easy E X A M P L E 103 USING RISK AT WALTON BOOKSTORE R ecall that Walton Bookstore buys calendars for 750 sells them at the regular price of 10 and gets a refund of 250 for all calendars that cannot be sold In contrast to Example 102 assume now that Walton estimates a triangular probability distribution for demand where the minimum most likely and maximum values of demand are 100 175 and 300 respectively The company wants to use this probability distribution together with RISK to simulate the profit for any particular order quantity with the ultimate goal of finding the best order quantity Objective To learn about RISKs basic functionality by revisiting the Walton Bookstore problem Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 Simulation settings You must first choose some simulation settings To do so the buttons on the left in the Simulation group see Figure 1032 are useful We typically do the following Set Iterations to a number such as 1000 RISK calls replications iterations Any number can be used but because the academic version of RISK allows only 1000 uninterrupted iterations we typically choose 1000 Set Simulations to 1 In a later section we will explain why you might want to request multiple simulations Click on the dice button so that it becomes orange This button is actually a toggle for what appears in your worksheet If it is orange the setting is called Monte Carlo and all random cells appear random they change when you press the F9 key If it is blue only the means appear in random input cells and the F9 key has no effect We prefer the Monte Carlo setting but both settings have exactly the same effect when you run the simulation Many more settings are available by clicking on the button to the left of the dice button but the ones we mentioned should suffice In addition more permanent set tings can be chosen from Application Settings under Utilities on the RISK ribbon You can experiment with these but the only one we like to change is the Place Reports In setting The default is to place reports in a new workbook If you like the reports to be in the same workbook as your model you can change this setting to Active Workbook 105 Introduction to RISK 591 Figure 1032 Simulation Group on RISK Ribbon RISK TECHNICAL ISSUES Latin Hypercube Sampling and Mersenne Twister Generator Two settings you shouldn t change are the Sampling T ype and Gener ator settings avail able from the b utton to the left of the dice b utton and then the Sampling tab The y should r emain at the default Latin Hyper cube and Mer senne T wister settings The Mersenne Twister is one algorithm of many for g enerating random numbers and it has been shown to have very good statistical pr operties Not all r andom number g enerators do Latin Hypercube sampling is a more efficient way of sampling than the other option Monte Carlo because it produces a more accurate estimate of the output distribution In fact we wer e surprised how accur ate it is In r epeated runs of this model always using different random numbers we virtually always got a mean pr ofit within a fe w pennies of 33750 It turns out that this is the true mean profit for this input distribution of demand Amazingly simulation estimates it corr ectlyalmost e xactlyon virtually e very run Unfortunately this means that a conf idence interval for the mean based on RISK s outputs and the usual confidence interval formula which assumes Monte Carlo sampling is much wider mor e pessimistic than it should be Therefore we do not e ven calculate such confidence intervals from here on 2 Run the simulation To run the simulation simply click on the Start Simulation on the RISK ribbon When you do so RISK repeatedly generates a random number for Leave Latin Hyper cube sampling on It produces more accurate results Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it each random input cell recalculates the worksheet and keeps track of all output cell values You can watch the progress at the bottom left of the screen 3 Examine the results The big questions are 1 which results you want and 2 where you want them RISK provides a lot of possibilities and we mention only our favorites You can ask for summary measures in your model worksheet by using the RISK statistical functions such as RISKMEAN discussed earlier The quickest way to get results is to select an input or output cell we chose the profit cell F13 and then click on the Browse Results button on the RISK ribbon See Figure 1033 This provides an interactive histogram of the selected value as shown in Figure 1034 You can move the sliders on this histogram to see probabili ties of various outcomes Note that the window you see from Browse Results is temporaryit goes away when you click on Close You can make a permanent copy of the chart by clicking on the third button from the left see the bottom of Figure 1034 and choosing one of the copy options 592 Chapter 10 Introduction to Simulation Modeling For a quick histogram of an output or input select the output or input cell and click on RISKs Browse Results button Figure 1033 Results and Tools Groups on RISK Ribbon Figure 1034 Interactive Histogram of Profit Output RISK Tip Percentiles Displayed on Charts When we displayed the c hart in F igure 1034 the f irst time it had the right slider on 500 but showed 5 to the right of it By default RISK puts the sliders at the 5th and 95th percentiles so that 5 is on either side of them F or this e xample 500 is indeed the 95th percentile why but the picture is a bit misleading because there is no chance of a profit greater than 500 When we manually moved the right slider away from 500 and back again it displayed as in F igure 1034 correctly indicating that ther e is no pr obability to the right of 500 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Remember that the results in these cells are meaningless or show up as errors until you run the simulation You can select the profit cell and click on Browse Results to see a histogram of profits as shown in Figure 1038 By default the histogram shown is for the first simulation where the order quantity is 150 However if you click on the red his togram button with the pound sign you can select any of the simulations As an example Figure 1039 shows the histogram of profits for the fifth simulation where the order quantity is 250 Do you see why these two histograms are so different When the order quantity is 150 there is a high probability of selling out hence the spike on the right is large But the probability of selling out with an order quantity of 250 is much lower hence its spike on the right is much less dominant 596 Chapter 10 Introduction to Simulation Modeling Figure 1038 Histogram of Profit with Order Quantity 150 Figure 1039 Histogram of Profit with Order Quantity 250 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it You can click on the Summary button to get the results from all simulations shown in Figure 1040 These results match those in Figure 1037 You can click on Excel Reports to get any of a number of reports on permanent worksheets Specifically Quick Reports is a good choice This produces several graphs and summary measures for each simulation each on a different worksheet This provides a lot of information with almost no work For this particular example the results in Figures 10371040 are illuminating You can see that an order quantity of 175 provides the largest mean profit However is this neces sarily the optimal order quantity This depends on the companys attitude toward risk Certainly larger order quantities incur more risk their histograms are more spread out their 5th and 95th percentiles are more extreme but they also have more upside potential On the other hand a smaller order quantity while having a somewhat smaller mean might be preferable because of less variability It is not an easy choice but at least the simulation results provide plenty of information for making the decision 105 Introduction to RISK 597 Figure 1040 Summary Report for All Five Simulations 1054 Some Limitations of RISK The academic version of RISK has some limitations you should be aware of The com mercial version of RISK doesnt have these limitations Also the exact limitations could change as newer academic versions become available The simulation model must be contained in a single workbook with at most four worksheets and each worksheet is limited to 300 rows and 100 columns The number of RISK input probability distribution functions such as RISKNORMAL is limited to 100 The number of unattended iterations is limited to 1000 You can request more than 1000 but you have to click a button after each 1000 iterations All RISK graphs contain a watermark The Distribution Fitting tool can handle only 150 observations The first limitation shouldnt cause problems at least not for the fairly small models discussed in this book However we strongly urge you to close all other workbooks when Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the left and choose Correlation Coefficients This produces the chart in Figure 1043 The Regression option produces similar results but we believe the Correlation option is easier to understand This figure shows graphically and numerically how each of the random inputs correlates with profit the higher the magnitude of the correlation the stronger the rela tionship between that input and profit In this sense you can see that the regularprice demand has by far the largest effect on profit The other two inputs maximum supply and saleprice demand are nearly uncorrelated with profit so they are much less impor tant Identifying important input variables is important for real applications If a random input is highly correlated with an important output then it is probably worth the time and money to learn more about this input and possibly reduce the amount of uncertainty involving it 105 Introduction to RISK 601 Figure 1042 Histogram of Simulated Profits for Order Quantity 200 Figure 1043 Tornado Graph for Sensitivity Analysis A tornado chart indicates which of the random inputs have large effects on an output Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 106 THE EFFECTS OF INPUT DISTRIBUTIONS ON RESULTS In section 102 we discussed input distributions The randomness in input variables causes the variability in the output variables We now briefly explore whether the choice of input distributions makes much difference in the distribution of an output variable such as profit This is an important question If the choice of input distributions doesnt matter much then you do not need to agonize over this choice However if it does make a difference then you have to be more careful about choosing the most appropriate input distribution for any particular situation Unfortunately it is impossible to answer the ques tion definitively The best we can say in general is It depends Some models are more sensitive to changes in the shape or parameters of input distributions than others Still the issue is worth exploring We discuss two types of sensitivity analysis in this section First we check whether the shape of the input distribution matters In the Walton Bookstore example we assumed a triangularly distributed demand with some skewness Are the results basically the same if a symmetric distribution such as the normal distribution is used instead Second we check whether the independence of input variables that have been assumed implicitly to this point is crucial to the output results Many random quantities in real situations are not inde pendent they are positively or negatively correlated Fortunately RISK enables you to build correlation into a model We analyze the effect of this correlation 1061 Effect of the Shape of the Input Distributions We first explore the effect of the shape of the input distributions As the following exam ple indicates if parameters that allow for a fair comparison are used the shape can have a relatively minor effect 106 The Effects of Input Distributions on Results 603 E X A M P L E 105 EFFECT OF DEMAND DISTRIBUTION AT WALTONS W e continue to explore the demand for calendars at Walton Bookstore We keep the same unit cost unit price and unit refund for leftovers as in Example 103 However in that example we assumed a triangular distribution for demand with parame ters 100 175 and 300 Assuming that Walton orders 200 calendars is the distribution of profit affected if a normal distribution of demand is used instead Objective To see whether a triangular distribution with some skewness gives the same profit distribution as a normal distribution for demand WHERE DO THE NUMBERS COME FROM The numbers here are the same as in Example 103 However as discussed next the parameters of the normal distribution are chosen to provide a fair comparison with the triangular distribution used earlier Solution It is important in this type of analysis to make a fair comparison When you select a normal distribution for demand you must choose a mean and standard deviation for this distribu tion Which values should you choose It seems only fair to choose the same mean and For a fair comparison of alternative input distributions the distributions should have at least approximately equal means and standard deviations Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it It is probably safe to conclude that the profit distribution in this model is not greatly affected by the choice of demand distribution at least not when 1 the candidate input distributions have the same mean and standard deviation and 2 their shapes are not too dissimilar We would venture to guess that this general conclusion about insensitivity of output distributions to shapes of input distributions can be made in many simulation models However it is always worth checking as we have done here especially when there is a lot of money at stake 606 Chapter 10 Introduction to Simulation Modeling Figure 1047 Graphical Results for Comparison Model FUNDAMENTAL INSIGHT Shape of the Output Distribution Predicting the sha pe of the output distribution fr om the shapes of the input distributions is difficult For example normally distributed inputs don t necessarily produce normally distributed outputsIt is also difficult to predict how sensitive the shape of the output distri bution is to the sha pes of the input distributions For example normally and triangularl y distributed inputs with the same means and standar d deviations are likely to lead to similar output distributions but there could be differencessayin the tails of the output distributionsIn any caseyou should examine the entire output distribution carefully not just a few of its sum mary measures 1062 Effect of Correlated Input Variables Until now all of the random numbers generated with RISK functions have been proba bilistically independent This means for example that if a random value in one cell is much larger than its mean the random values in other cells are completely unaffected They are no more likely to be abnormally large or small than if the first value had been average or below average Sometimes however independence is unrealistic In such cases the random numbers should be correlated in some way If they are positively correlated then large numbers will tend to go with large numbers and small with small If they are negatively correlated then large will tend to go with small and small with large As an example you might expect daily stock price changes for two companies in the same industry to be positively correlated If the price of one oil company increases you might expect the price of another oil company to increase as well RISK enables you to build in this correlated behavior with the RISKCORRMAT function as we illustrate in the follow ing continuation of the Walton example Input variables in real world problems are often correlated which makes the material in this section particularly important Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 5 Other formulas The other formulas in rows 14 and 15 are identical to ones devel oped in previous examples so they arent presented again here The quantities in row 16 are simply sums of rows 14 and 15 Also the only RISK output we designated is the total profit in cell F16 but you can designate others as output cells if you like Running the Simulation You should set up and run RISK exactly as before For this example set the number of iterations to 1000 and the number of simulations to 3 because three different correlations are being tested Discussion of the Simulation Results Selected numerical and graphical results are shown in Figures 1049 and 1050 You will probably be surprised to see that the mean total profit is the same regardless of the corre lation This is no coincidence In each of the three simulations RISK uses the same ran dom numbers but shuffles them in different orders to get the correct correlations This means that averages are unaffected The idea is that the average of the numbers 30 26 and 48 is the same as the average of the numbers 48 30 and 26 106 The Effects of Input Distributions on Results 609 Figure 1049 Summary Results for Correlated Model Figure 1050 Graphical Results for Correlated Model However the correlation has a definite effect on the distribution of total profit You can see this in Figure 1049 for example where the standard deviation of total profit increases as the correlation goes from negative to zero to positive This same increase in variability is apparent in the histograms in Figure 1050 Do you see intuitively why this increase in variability occurs It is basically the Dont put all of your eggs in one basket effect When the correlation is negative high demands for one product tend to cancel low Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it to see what this means The file P1044xlsx gets you started There is a single output cell B5 You can enter any random value in this cell such as RISKNORMAL500100 There are already RISK statistical formulas in rows 912 to calculate summary measures of the output for each of 10 simulations On the RISK ribbon click on the button to the left of the dice button to bring up the Simulation Settings dialog box click on the Sampling tab and make sure the Sampling Type is Latin Hypercube Run 10 simulations with at least 1000 iterations each and then paste the results in rows 912 as values in rows 1720 Next get back in Simulations Settings and change the Sampling Type to Monte Carlo run the 10 simulations again and paste the results in rows 912 as values into rows 2326 For each row 1720 and 2326 summa rize the 10 numbers in that row with AVERAGE and STDEV What do you find Why do we say that Latin Hypercube sampling is more efficient Thanks to Harvey Wagner at University of North Carolina for suggesting this problem 45 We are continually hearing reports on the nightly news about natural disastersdroughts in Texas hurricanes in Florida floods in California and so on We often hear that one of these was the worst in over 30 years or some such statement Are natural disasters getting worse these days or does it just appear so How might you use simulation to answer this question Here is one possible approach Imagine that there are N areas of the country or the world that tend to have to some extent various types of weather phenomena each year For example hurricanes are always a potential problem for Florida and fires are always a potential problem in southern California You might model the severity of the problem for any area in any year by a normally distributed random number with mean 0 and standard deviation 1 where negative values are interpreted as good years and positive values are interpreted as bad years We suggest the normal distribution but there is no reason other distributions couldnt be used instead Then you could simulate such values for all areas over a period of several years and keep track say of whether any of the areas have worse conditions in the current year than they have had in the past several years where several could be 10 20 30 or any other number of years you want to test What might you keep track of How might you interpret your results Modeling Problems 46 You are making several runs of a simulation model each with a different value of some decision variable such as the order quantity in the Walton calendar model to see which decision value achieves the largest mean profit Is it possible that one value beats another simply by random luck What can you do to minimize the chance of a better value losing out to a poorer value 47 If you want to replicate the results of a simulation model with Excel functions only not RISK you can build a data table and let the column input cell be any blank cell Explain why this works 48 Suppose you simulate a gambling situation where you place many bets On each bet the distribution of your net winnings loss if negative is highly skewed to the left because there are some possibilities of really large losses but not much upside potential Your only simu lation output is the average of the results of all the bets If you run RISK with many iterations and look at the resulting histogram of this output what will it look like Why 49 You plan to simulate a portfolio of investments over a multiyear period so for each investment which could be a particular stock or bond for example you need to simulate the change in its value for each of the years How would you simulate these changes in a realistic way Would you base it on historical data What about correlations Do you think the changes for different investments in a particular year would be correlated Do you think changes for a particular investment in different years would be correlated Do you think correlations would play a significant role in your simulation in terms of realism 50 Big Hit Video must determine how many copies of a new video to purchase Assume that the companys goal is to purchase a number of copies that maximizes its expected profit from the video during the next year Describe how you would use simulation to shed light on this problem Assume that each time a video is rented it is rented for one day 51 Many people who are involved in a small auto acci dent do not file a claim because they are afraid their insurance premiums will be raised Suppose that City Farm Insurance has three rates If you file a claim you are moved to the next higher rate How might you use simulation to determine whether a particular claim should be filed 52 A building contains 1000 lightbulbs Each bulb lasts at most five months The company maintaining the building is trying to decide whether it is worthwhile to practice a group replacement policy Under a group replacement policy all bulbs are replaced every T months where T is to be determined Also bulbs are replaced when they burn out Assume that it costs 005 to replace each bulb during a group replacement and 020 to replace each burnedout bulb if it is replaced individually How would you use simulation to determine whether a group replacement policy is worthwhile 107 Conclusion 617 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 53 Why is the RISKCORRMAT function necessary How does RISK generate random inputs by default that is when RISKCORRMAT is not used 54 Consider the claim that normally distributed inputs in a simulation model are bound to lead to normally dis tributed outputs Do you agree or disagree with this claim Defend your answer 55 It is very possible that when you use a correlation matrix as input to the RISKCORRMAT function in an RISK model the program will inform you that this is an invalid correlation matrix Provide an example of an obviously invalid correlation matrix involving at least three variables and explain why it is invalid 56 When you use a RISKSIMTABLE function for a decision variable such as the order quantity in the Walton model explain how this provides a fair comparison across the different values tested 57 Consider a situation where there is a cost that is either incurred or not It is incurred only if the value of some random input is less than a specified cutoff value Why might a simulation of this situation give a very different average value of the cost incurred than a deterministic model that treats the random input as fixed at its mean What does this have to do with the flaw of averages 618 Chapter 10 Introduction to Simulation Modeling Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 621 Simulation Models C H A P T E R MERRILL L YNCH IMPR OVES LIQUIDITY RISK MANA GEMENT FOR REV OLVING CREDIT LINES T he Merrill Lynch banking group comprises several Merrill Lynch affiliates including Merrill Lynch Bank USA ML Bank USA Its parent company is Bank of America ML Bank USA has assets of more than 60 billion as of June 30 2005 when the following article was written closer to 70 billion by 2010 The bank acts as an intermediary accepting deposits from Merrill Lynch retail customers and using the deposits to fund loans and make investments One way ML Bank USA uses these assets is to provide revolving credit lines to institutional and large corporate borrowers Currently it has a portfolio of about 13 billion in creditline commitments with more than 100 companiesWhen it makes these commitments it must be aware of the liquidity risk defined as the ability to meet all cash obligations when due In other words if a borrower asks for funds as part of its revolving creditline agreement the bank must have the funds available to honor the request typically on the same day the request is made This liquidity requirement AP PhotoMary Altaffer 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it poses a huge risk to the bank The bank must keep enough cash or liquid investments ie investments that can be converted to cash quickly in reserve to honor its customers requests whenever they occur If the bank knew when and in what quantities these requests would occur it could manage its cash reserves more prudently essentially holding a smaller amount in liquid investments for credit requests and investing the rest in other more illiquid and profitable investments Duffy et al 2005 discuss their role as members of Merrill Lynchs Banking Group and Management Science Group in developing a model to manage the liquidity risk for ML Bank USAs revolving credit lines The revolving credit lines give borrowers access to a specified amount of cash on demand for shortterm funding needs in return for a fee paid to the bankThe bank also earns an interest rate on advances that compensates it for the liquidity and other risks it takes These credit lines are therefore profitable for the bank but they are not the borrowers primary sources of funding Customers typically use these credit lines to retire maturing commercial paper available at cheaper interest rates during the process of rolling it over ie attempting to reissue new commercial paper notes andor when their credit rating falls The essence of the problem is that when a customers credit ratings measured by the Moody rating scale for example fall the customers are less likely to obtain funds from cheaper sources such as commercial paper so they then tend to rely on their credit lines from ML Bank USA and other banks This poses problems for ML Bank USA It must honor its commitments to the borrowers as spelled out in the creditline agreements but customers with low credit ratings are the ones most likely to default on their loans Two other aspects of the problem are important First the creditline agreements often have a termout option which allows the borrower to use funds for an additional period after expiration typically for one year A customer that is experiencing financial difficulties and has seen its credit rating fall is the type most likely to use its termout option Second movements in credit ratings for customers in the same industry or even in different industries tend to be positively correlated because they can all be affected by movements in their industry or the overall economy This increases the liquidity risk for ML Bank USA because it increases the chance that poor economic conditions will lead many customers to request additional credit The authors built a rather complex simulation model to track the demand for usage of these credit facilities The model simulates monthly creditline usage for each customer over a fiveyear period During this period some credit lines are renewed some expire and are not renewed and some customers exercise their termout options The model has several significant features 1 It models the probabilistic changes in credit ratings for its customers where a customers credit rating can move from one level to another level in a given month with specified probabilities 2 these probabilities are chosen in such a way that movements in credit ratings are positively correlated across customers and 3 expertsystem business rules are used to determine whether the company will renew or terminate expiring lines of credit and whether customers will exercise their termout options For example a typical rule is that the bank does not renew a credit line if the borrowers credit rating is below a certain threshold The authors developed a userfriendly Excelbased system to run their model It actually invokes and executes the simulation behind the scenes in a simulation package called Arena Users of the system can change many of the parameters of the model such as the businessrule cutoffs to customize the simulation The model has helped ML Bank USA manage its revolving credit linesThe output of the model provides a scientific and robust measure of liquidity risk that the bank has confidence inand therefore usesThe model has led to two tangible financial benefits First the model reduced the banks liquidity requirement from 50 to 20 of outstanding commitments thus freeing up about 4 billion of liquidity for other 622 Chapter 11 Simulation Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 111 INTRODUCTION In the previous chapter we introduced most of the important concepts for developing and analyzing spreadsheet simulation models We also discussed many of the features available in the powerful simulation addin RISK that you receive with this book Now we apply the tools to a wide variety of problems that can be analyzed with simulation For conve nience we group the applications into four general areas 1 operations models 2 finan cial models 3 marketing models and 4 games of chance The only overriding theme in this chapter is that simulation models can yield important insights in all of these areas You do not need to cover all of the models in this chapter or cover them in any particular order You can cover the ones of most interest to you in practically any order 112 OPERATIONS MODELS Whether we are discussing the operations of a manufacturing or a service company there is likely to be uncertainty that can be modeled with simulation In this section we look at examples of bidding for a government contract uncertainty in the bids by competitors warranty costs uncertainty in the time until failure of an appliance and drug production uncertainty in the yield and timing 1121 Bidding for Contracts In situations where a company must bid against competitors simulation can often be used to determine the companys optimal bid Usually the company does not know what its competitors will bid but it might have an idea about the range of the bids its competitors will choose In this section we show how to use simulation to determine a bid that maxi mizes the companys expected profit profitable illiquid investments Second during the first 21 months after the system was implemented the banks portfolio expanded from 8 billion in commitments and 80 customers to 13 billion and more than 100 customersThe bank continues to use the model for its longrange planning E X A M P L E 111 BIDDING FOR A GOVERNMENT CONTRACT T he Miller Construction Company must decide whether to make a bid on a construction project Miller believes it will cost the company 10000 to complete the project if it wins the contract and it will cost 350 to prepare a bid However there is uncertainty about each of these Upon further reflection Miller assesses that the cost to complete the project has a triangular distribution with minimum most likely and maximum values 9000 10000 and 15000 Similarly Miller assesses that the cost to prepare a bid has a triangular distribution with parameters 300 350 and 500 Note the skewness in these distributions Miller recognizes that cost overruns are much more likely than cost under runs Four potential competitors are going to bid against Miller The lowest bid wins the contract and the winner is then given the winning bid amount to complete the project Based on past history Miller believes that each potential competitor will bid indepen dently of the others with probability 05 Miller also believes that each competitors bid 112 Operations Models 623 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Running the Simulation Set the number of iterations to 1000 and set the number of simulations to 10 because there are 10 bid amounts Miller wants to test Discussion of the Simulation Results The summary results appear in Figure 112 For each simulationthat is each bid amountthere are two outputs 1 or 0 to indicate whether Miller wins the contract and Millers profit The only interesting results for the 01 output are in the Mean column which shows the fraction of iterations that resulted in 1s So you can see for example that if Miller bids 12000 simulation 4 the probability of winning the bid is estimated to be 0581 This probability clearly decreases as Millers bid increases 626 Chapter 11 Simulation Models Figure 112 Summary Results for Bidding Simulation In terms of net profit if you concentrate only on the Mean column a bid amount of 13000 simulation 6 is the best But as the other numbers in this figure indicate the mean doesnt tell the whole story For example if Miller bids 13000 it could win the bid but still lose a considerable amount of money because of cost overruns The histogram of profit in Figure 113 indicates this more clearly It shows that in spite of the positive mean most outcomes are negative So what should Miller do If it doesnt bid at all its profit is a certain 0 If Miller is an expected profit maximizer then the fact that several of the means in Figure 112 are pos itive indicates that bidding is better than not bidding with a bid of 13000 being the best bid However potential cost overruns and the corresponding losses are certainly a concern Depending on Millers degree of risk aversion the company might decide to 1 not bid at all or 2 bid higher than 13000 to minimize its worse loss Still we would caution Miller not to be too conservative Rather than focusing on the Min worst case column in Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 112 we would suggest focusing on the 5 column This shows nearly how bad things could get 5 of the time it would be worse than this and this 5th percentile remains fairly constant for higher bids Figure 113 Histogram of Profit with 13000 Bid 1122 Warranty Costs When you buy a new product it usually carries a warranty A typical warranty might state that if the product fails within a certain period such as one year you will receive a new product at no cost and it will carry the same warranty However if the product fails after the warranty period you have to bear the cost of replacing the product Due to random lifetimes of products we need a way to estimate the warranty costs to the manufacturer of a product The next example illustrates how this can be accomplished with simulation E X A M P L E 112 WARRANTY COSTS FOR A CAMERA T he Yakkon Company sells a popular camera for 400 This camera carries a warranty such that if the camera fails within 15 years the company gives the customer a new camera for free If the camera fails after 15 years the warranty is no longer in effect Every replacement camera carries exactly the same warranty as the original camera and the cost to the company of supplying a new camera is always 225 Use simulation to esti mate for a given sale the number of replacements under warranty and the NPV of profit from the sale using a discount rate of 8 Objective To use simulation to estimate the number of replacements under warranty and the total NPV of profit from a given sale WHERE DO THE NUMBERS COME FROM The warranty information is a policy decision made by the company The hardest input to estimate is the probability distribution of the lifetime of the product We discuss this next 112 Operations Models 627 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 118 Histogram of NPV of Profit Figure 117 Histogram of Number of Failures Figure 116 Summary Statistics for Warranty Model 112 Operations Models 631 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it that if the camera fails within a year the customer gets a new camera for free whereas if the time to failure is between 1 and 15 years the customer pays some pro rata share of the replacement cost Finally it could try to sell the customer an extended warrantyat a hefty price We ask you to explore these possibilities in the problems 1123 Drug Production with Uncertain Yield In many manufacturing settings products are produced in batches and the usable yields from these batches are uncertain This is particularly true in the drug industry The follow ing example illustrates how a drug manufacturer can take this uncertainty into account when planning production 632 Chapter 11 Simulation Models E X A M P L E 113 TRYING TO MEET AN ORDER DUE DATE AT WOZAC T he Wozac Company is a drug manufacturer Wozac has recently accepted an order from its best customer for 8000 ounces of a new miracle drug and Wozac wants to plan its production schedule to meet the customers promised delivery date of December 1 2010 There are three sources of uncertainty that make planning difficult First the drug must be produced in batches and there is uncertainty in the time required to produce a batch which could be anywhere from 5 to 11 days This uncertainty is described by the discrete distribu tion in Table 111 Second the yield usable quantity from any batch is uncertain Based on historical data Wozac believes the yield can be modeled by a triangular distribution with minimum most likely and maximum values equal to 600 1000 and 1100 ounces respec tively Third all batches must go through a rigorous inspection once they are completed The probability that a typical batch passes inspection is only 08 With probability 02 the batch fails inspection and none of it can be used to help fill the order Wozac wants to use simulation to help decide how many days prior to the due date it should begin production Table 111 Distribution of Days to Complete a Batch Days Probability 5 005 6 010 7 020 8 030 9 020 10 010 11 005 Objective To use simulation to determine when Wozac should begin production for this order so that there is a high probability of completing it by the due date WHERE DO THE NUMBERS COME FROM The important inputs here are the probability distributions of the time to produce a batch the yield from a batch and the inspection result The probabilities we have assumed would undoubtedly be based on previous production data For example the company might have observed that about 80 of all batches in the past passed inspection Of course a discrete distribution is natural for the number of days to produce a batch and a continuous distrib ution is appropriate for the yield from a batch Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 640 Chapter 11 Simulation Models in cell B8 and copying it to the range B8C51 Note how this formula references the loca tion of the previous drop The IF function captures the logic for all four rules 3 Simulate drops Simulate the positions of the drops by entering the formula RISKNORMALB71 in cell D7 and copying it to the range D7E51 This says that the balls drop position is nor mally distributed with mean equal to the funnels position and standard deviation 1 4 Distance Calculate the final distance from the target in cell K14 with the formula SQRTSUMSQD51E51 Here we have used the SUMSQ function to get the sum of squares for the distance formula Then designate this cell as an RISK output cell Running the Simulation We set the number of iterations to 1000 and the number of simulations to 4 because of simulating the four rules simultaneously Selected summary measures for the final dis tance from the target for all four rules are shown in Figure 1113 We also show histograms of this distance for rules 1 2 and 3 in Figures 1114 1115 and 1116 The histogram for rule 4 isnt shown because it is practically identical to the one for rule 3 Figure 1114 Histogram of Distance from Target for Rule 1 Figure 1113 Summary Results for All Rules Discussion of the Sim ulation Results These results prove Demings point about tampering Rule 2 might not appear to be much worse than rule 1 but its mean distance and standard deviation of distances are both about 45 higher than for rule 1 Rules 3 and 4 are disastrous Their mean distances are more than six times larger than for rule 1 and their standard deviations are also much larger The reason is that the drops for rule 3 tend to swing back and forthfirst to the left then to the right Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it then to the left and so onand the swings tend to increase through time In contrast the drops for rule 4 tend to drift away from the target over time The moral of the story as Deming preached is that you should not tamper with a stable process If the process is not behaving as desired then fundamental changes to the process are required not a lot of tinkering 112 Operations Models 641 P R O B L E M S Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 If the number of competitors in Example 111 doubles how does the optimal bid change 2 In Example 111 the possible profits vary from negative to positive for each of the 10 possible bids examined a For each of these use RISKs RISKTARGET function to find the probability that Millers profit is positive Do you believe these results should have any bearing on Millers choice of bid b Use RISKs RISKPERCENTILE function to find the 10th percentile for each of these bids Can you explain why the percentiles have the values you obtain 3 Referring to Example 111 if the average bid for each competitor stays the same but their bids exhibit less variability does Millers optimal bid increase or decrease To study this question assume that each competitors bid expressed as a multiple of Millers cost to complete the project follows each of the following distributions a Triangular with parameters 10 13 and 24 b Triangular with parameters 12 13 and 22 c Use RISKs Define Distributions window to check that the distributions in parts a and b have the same mean as the original triangular distribution in the example but smaller standard deviations What is the common mean Why is it not the same as the most likely value 13 Figure 1115 Histogram of Distance from Target for Rule 2 Figure 1116 Histogram of Distance from Target for Rule 3 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 113 FINANCIAL MODELS There are many financial applications where simulation can be applied Future cash flows future stock prices and future interest rates are some of the many uncertain variables finan cial analysts must deal with In every direction they turn they see uncertainty In this section we analyze a few typical financial applications that can benefit from simulation modeling 1131 Financial Planning Models Many companies such as GM Eli Lilly Procter Gamble and Pfizer use simulation in their capital budgeting and financial planning processes Simulation can be used to model the uncertainty associated with future cash flows In particular simulation can be used to answer questions such as the following What are the mean and variance of a projects net present value NPV What is the probability that a project will have a negative NPV 642 Chapter 11 Simulation Models 4 See how sensitive the results in Example 112 are to the following changes For each part make the change indicated run the simulation and comment on any differences between your outputs and the outputs in the example a The cost of a new camera is increased to 300 b The warranty period is decreased to one year c The terms of the warranty are changed If the camera fails within one year the customer gets a new camera for free However if the camera fails between 1 year and 15 years the customer pays a pro rata share of the new camera increasing linearly from 0 to full price For example if it fails at 12 years which is 40 of the way from 1 to 15 the customer pays 40 of the full price d The customer pays 50 up front for an extended warranty This extends the warranty to three years This extended warranty is just like the original so that if the camera fails within three years the customer gets a new camera for free 5 In Example 112 the gamma distribution was used to model the skewness to the right of the lifetime distribution Experiment to see whether the triangular distribution could have been used instead Let its minimum value be 0 and choose its most likely and maximum values so that this triangular distribution has approximately the same mean and standard deviation as the gamma distribution in the example Use RISKs Define Distributions window and trial and error to do this Then run the simulation and comment on similarities or differences between your outputs and the outputs in the example 6 In Example 113 we commented on the 95th percentile on days required in cell I35 and the corresponding date in cell J35 If the company begins production on this date then it is 95 sure to complete the order by the due date We found this date to be August 2 Do you always get this answer Find out by 1 running the simulation 10 more times each with 1000 iterations and finding the 95th percentile and corresponding date in each and 2 running the simulation once more but with 10000 iterations Comment on the difference between simulations 1 and 2 in terms of accuracy Given these results when would you recommend that production should begin 7 In Example 113 suppose you want to run five simulations where the probability of passing inspection is varied from 06 to 10 in increments of 01 Use the RISKSIMTABLE function appropriately to do this Comment on the effect of this parameter on the key outputs In particular does the probability of passing inspection have a large effect on when production should start Note When this probability is low it might be necessary to produce more than 25 batches the maximum built into the model Check whether this maximum should be increased 8 In the simulation of Demings funnel experiment the RISK outputs show how tampering leads to poor results at least in terms of the mean and standard deviation of the distance of the final drop from the target However the results we presented dont show how the tampering rules particularly rules 3 and 4 go wrong To get a better idea of this create two scatter charts one of the xcoordinate in column D versus the drop number in column A and one of the y coordinate in column E versus the xcoordinate in column D You could also create a third scatter chart of the ycoordinate versus the drop number but it would be about the same as the first Use the chart subtype that connects the dots for each scatter chart To go from one rule to another enter a number from 1 to 4 in cell B3 not a formula Then press the F9 key several times to see how the scatter charts change Describe how the drops seem to evolve over time according to the various rules Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it What are the mean and variance of a companys profit during the next fiscal year What is the probability that a company will have to borrow more than 2 million during the next year The following example illustrates how simulation can be used to evaluate an investment opportunity E X A M P L E 115 DEVELOPING A NEW CAR AT GF AUTO G eneral Ford GF Auto Corporation is developing a new model of compact car This car is assumed to generate sales for the next five years GF has gathered information about the following quantities through focus groups with the marketing and engineering departments Fixed cost of developing car This cost is assumed to 700 million The fixed cost is incurred at the beginning of year 1 before any sales are recorded Margin per car This is the unit selling price minus the variable cost of producing a car GF assumes that in year 1 the margin will be 4000 Every other year GF assumes the margin will decrease by 41 Sales The demand for the car is the uncertain quantity In its first year GF assumes salesnumber of cars soldwill be triangularly distributed with parameters 50000 75000 and 85000 Every year after that the company assumes that sales will decrease by some percentage where this percentage is triangularly distributed with parameters 5 8 and 10 GF also assumes that the percentage decreases in successive years are independent of one another Depreciation and taxes The company will depreciate its development cost on a straightline basis over the lifetime of the car The corporate tax rate is 40 Discount rate GF figures its cost of capital at 10 Given these assumptions GF wants to develop a simulation model that will evaluate its NPV of aftertax cash flows for this new car over the fiveyear time horizon Objective To simulate the cash flows from the new car model from the development time to the end of its life cycle so that GF can estimate the NPV of aftertax cash flows from this car WHERE DO THE NUMBERS COME FROM There are many inputs to this problem As we indicated they are probably obtained from experts within the company and from focus groups of potential customers Solution This model is like most financial multiyear spreadsheet models The completed model extends several years to the right but most of the work is for the first year or two From that point you can copy to the other years to complete the model 1The margin decreases because the company assumes variable costs tend to increase through time whereas sell ing prices tend to remain fairly constant through time 113 Financial Models 643 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it What is most responsible for this huge variability in NPV the variability in firstyear sales or the variability in annual sales decreases This can be answered with RISKs tornado chart See Figure 1119 To get this chart click on the tornado button below the histogram shown in Figure 1118 and select the Correlation option This chart answers the question emphatically Variability in firstyear sales is by far the largest influence on NPV It correlates almost perfectly with NPV The annual decreases in sales are not unimportant but they have much less effect on NPV If GF wants to get a more favorable NPV distribution it should do all it can to boost firstyear salesand make the firstyear sales distribution less variable 646 Chapter 11 Simulation Models Financial analysts typically look at VAR 5 to see how bad or more precisely almost how bad things could get Figure 1119 Tornado Chart for NPV Before finishing this example we revisit the flaw of averages What if GF used a deterministic model to estimate NPV Would the results match those from the simulation We tried this two ways once by entering the most likely values of the inputs instead of the random numbers and once by entering the means instead of the random numbers The results appear in Figure 1120 The mean of a triangular distribution is the average of its three parameters These means appear in cells H5 and H6 Now there are no random num bers in rows 12 and 24 only most likely values or means The difference between the two NPVs is huge In this case the NPV by using means is very close to the mean NPV from the simulation about 31 million But if the company used most likely values for the inputs in its deterministic model which certainly seems sensible the NPV would be off by a factor of more than two another variation of the flaw of averages Besides this problem neither deterministic model provides even a hint that the company has about a 29 chance of a negative NPV2 A tornado chart lets you see which random inputs have the most effect on a specified output If you create a deterministic model using the most likely values of the uncer tain inputs you can possibly get an output value that is nowhere near the mean of that output 2It turns out that the NPV in this model is linear in the two random inputs When an output is linear in the inputs the deterministic model using means of inputs always gives the correct mean output so that the flaw of averages in the form from the previous chapter does not occur Even so a deterministic model still provides no indication of how bad or how good things could get Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1133 Investment Models Individual investors typically want to choose investment strategies that meet some pre specified goal The following example is typical Here a person wants to meet a retirement goal starting at an early age 652 Chapter 11 Simulation Models 04 06 08 1 12 14 5 95 02 0 02 04 06 08 1 12 14 Jan Feb Mar Apr May Jun Values in Thousands Loan amount from Jan to Jun 1 StdDev Mean Figure 1123 Summary Chart of Loans over Time E X A M P L E 117 INVESTING FOR RETIREMENT A ttorney Sally Evans has just begun her career At age 25 she has 40 years until retirement but she realizes that now is the time to start investing She plans to invest 1000 at the beginning of each of the next 40 years Each year she plans to put fixed percentagesthe same each yearof this 1000 into stocks Treasury bonds Tbonds and Treasury bills Tbills However she is not sure which percentages to use We call these percentages investment weights She does have historical annual returns from stocks Tbonds and Tbills from 1946 to 2007 These are listed in the file Retirement Planningxlsx This file also includes inflation rates for these years For example for 1993 the annual returns for stocks Tbonds and Tbills were 999 1824 and 290 respectively and the inflation rate was 275 Sally would like to use simulation to help decide what investment weights to use with the objective of achieving a large investment value in todays dollars at the end of 40 years Objective To use simulation to estimate the value of Sallys future investments in todays dollars from several investment strategies in Tbills Tbonds and stocks WHERE DO THE NUMBERS COME FROM Historical returns and inflation rates such as those quoted here are widely available on the Web Solution The most difficult modeling aspect is settling on a way to use historical returns and infla tion factors to generate future values of these quantities We suggest using a scenario approach You can think of each historical year as a possible scenario where the scenario You can simulate future scenarios by randomly choosing past scenarios giving higher probabilities to more recent scenarios Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Running the Simulation Set the number of iterations to 1000 and the number of simulations to 3 one for each set of investment weights to be tested Then run the simulation as usual Discussion of the Simulation Results Summary results appear in Figure 1126 The first simulation which invests the most heav ily in stocks is easily the winner Its mean final cash slightly more than 153000 in todays dollars is much greater than the means for the other two sets of weights The first simulation also has a much larger upside potential its 95th percentile is close to 360000 and even its downside is slightly better than the others Its 5th percentile is the best and its minimum is only slightly worse than the minimum for the other sets of weights 656 Chapter 11 Simulation Models Figure 1126 Summary Results for Retirement Simulation Nevertheless the histogram for simulation 1 put 80 in stocks shown in Figure 1127 indicates a lot of variabilityand skewnessin the distribution of final cash As in Example 115 the concept of value at risk VAR is useful Recall that VAR 5 is defined as the 5th percentile of a distribution and is often the value investors worry about Perhaps Sally should rerun the simulation with different investment weights with an eye on the weights that increase her VAR 5 Right now it is slightly more than 40000not too good considering that she invests 40000 total She might not like the prospect of a 5 chance of ending up with no more than this We also encourage you to try running this simulation with other investment weights both for the 40year horizon and after modifying the spreadsheet model Figure 1127 Histogram of Final Cash with 80 in Stocks Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the standard deviation of stock returns as the volatility and the time until the exercise date as the duration 2 Simulated stock price at exer cise date Using Equation 112 with µ replaced by the riskfree rate simulate the stock price in six months by entering the formula CurrentpriceEXPRiskfreerate05Volatility2Duration VolatilityRiskNormal01SQRTDuration in cell B12 3 Cash flow from option Calculate the cash flow from the option by entering the formula 100MAXB12Exerciseprice0 in cell B13 This says that if the future price in cell B12 is greater than the exercise price in cell B5 you make the difference otherwise you never exercise the option and make nothing Note that you multiply by 100 because the option is for 100 shares of the stock 4 Discount the cash flow Discount the cash flow in cell B14 with the formula EXPDurationRiskfreerateB13 This represents the NPV of the cash flow if any realized at the exercise date Because the price of the option is the average of this discounted value you should designate it as an RISK output cell 5 Average of output cell You can take advantage of RISKs RISKMEAN function to obtain the eventual price of the option on the spreadsheet itself To do this enter the formula RISKMEANB14 in cell B16 Running the Simulation Because this is a small simulation model and you want an accurate average in cell B16 you can afford to run a lot of iterations Therefore set the number of iterations to 10000 and the number of simulations to 1 After running RISK the value 47594 appears in cell B16 According to the result of Cox et al this average is an estimate of the fair price for the option It turns out from the BlackScholes formula that 47594 is very close to the correct price for this option In other words the simulation got it almost exactly right This surprised us initially After all from basic statistical inference it is difficult to estimate a mean exactly The estimated mean is usually surrounded by 95 confidence limits to indicate the level of accuracy However the effect of using Latin Hypercube sam pling is that means can be estimated much more accurately With 10000 iterations the cor rect answer is evidently obtained to within a few pennies We now extend the previous example by simulating a portfolio that includes a com panys stock and a call option on that stock 660 Chapter 11 Simulation Models E X A M P L E 119 RETURN ON A PORTFOLIO WITH A STOCK AND AN OPTION ON THE STOCK S uppose the investor buys 100 shares of AnTech stock at the current price and one call option on this stock for 47594 the fair price found in the previous example Use sim ulation to find the return on the investors portfolio as of the exercise date Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 3 Option cash flo w Calculate the cash flow from the option exactly as before by entering the formula 100MAXB15Exerciseprice0 in cell B16 4 Portfolio value In six months the portfolio will be worth the value of the stock plus the cash flow from the option Calculate this in cell B18 with the formula SUMPRODUCTB10B11B15B16 Then in cells B19 and B20 calculate the amount paid for the portfolio and its return the percentage change with the formulas SharespurchasedCurrentpriceOptionspurchasedOptionprice and B18B19B19 Then designate cell B20 as an RISK output cell 5 RISK summary statistics You can again show the basic summary results from RISKonthespreadsheetbyusingitsRISKMEANRISKSTDDEVRISKMINRISKMAX RISKPERCENTILE and RISKTARGET functions For example the formulas in cells B27 and B29 are RISKPERCENTILEB20005 and 1RISKTARGETB200 USING RISK After running RISK for 10000 iterations we obtain the values in the range B23B29 of Figure 1130 The mean return from this portfolio is about 94 but there is considerable variability There is a 5 chance that it will lose at least 24 and there is a 5 chance that it will gain at least 564 The probability that it will provide a positive return is about 059 If you have any intuition for financial portfolios you have probably noticed that this investor is putting all her eggs in one basket If the stock price increases she gains by owning the shares of stock and she also gains from holding the options because she is more likely to be in the money However if the price of the stock decreases she loses money on her shares of stock and her options are worthless A safer strategy is to hedge her bets She can purchase 100 shares of the stock and purchase one put option on the stock A put option allows her to sell shares of stock for the exercise price at the exercise date With a put option the investor hopes the stock price will decrease because she can then sell her shares at the exercise price and immediately buy them back at the decreased stock price thus earning a profit Therefore a portfolio consisting of shares of stock and put options on the stock covers the investor in both directions It has less upside potential but it decreases the downside risk Valuing a More Exotic Call Option The European call option is fairly simple A variety of other derivative securities are cur rently available In fact their variety and complexity are what make them attractiveand dangerous for the unsuspecting investor We illustrate one variation of the basic call 662 Chapter 11 Simulation Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 22 In the financial world there are many types of complex instruments called derivatives that derive their value from the value of an underlying asset Consider the following simple derivative A stocks current price is 80 per share You purchase a derivative whose value to you becomes known a month from now Specifically let P be the price of the stock in a month If P is between 75 and 85 the derivative is worth nothing to you If P is less than 75 the derivative results in a loss of 10075P dollars to you The factor of 100 is because many derivatives involve 100 shares If P is greater than 85 the derivative results in a gain of 100P85 dollars to you Assume that the distribution of the change in the stock price from now to a month from now is normally distributed with mean 1 and standard deviation 8 Let EMV be the expected gainloss from this derivative It is a weighted average of all the possible losses and gains weighted by their likelihoods Of course any loss should be expressed as a negative number For example a loss of 1500 should be expressed as 1500 Unfortunately this is a difficult probability calculation but EMV can be estimated by an RISK simulation Perform this simulation with at least 1000 iterations What is your best estimate of EMV 23 Suppose you currently have a portfolio of three stocks A B and C You own 500 shares of A 300 of B and 1000 of C The current share prices are 4276 8133 and 5822 respectively You plan to hold this portfolio for at least a year During the coming year economists have predicted that the national economy will be awful stable or great with probabilities 02 05 and 03 Given the state of the economy the returns oneyear percentage changes of the three stocks are independent and normally distributed However the means and standard deviations of these returns depend on the state of the economy as indicated in the file P1123xlsx a Use RISK to simulate the value of the portfolio and the portfolio return in the next year How likely is it that you will have a negative return How likely is it that you will have a return of at least 25 b Suppose you had a crystal ball where you could predict the state of the economy with certainty The stock returns would still be uncertain but you would know whether your means and standard deviations come from row 6 7 or 8 of the P1123xlsx file If you learn with certainty that the economy is going to be great in the next year run the appropriate simulation to answer the same questions as in part a Repeat this if you learn that the economy is going to be awful How do these results compare with those in part a 24 If you own a stock buying a put option on the stock will greatly reduce your risk This is the idea behind portfolio insurance To illustrate consider a stock that currently sells for 56 and has an annual volatility of 30 Assume the riskfree rate is 8 and you esti mate that the stocks annual growth rate is 12 a Suppose you own 100 shares of this stock Use simulation to estimate the probability distribution of the percentage return earned on this stock during a oneyear period b Now suppose you also buy a put option for 238 on the stock The option has an exercise price of 50 and an exercise date one year from now Use simulation to estimate the probability distribution of the percentage return on your portfolio over a oneyear period Can you see why this strategy is called a portfolio insurance strategy c Use simulation to show that the put option should indeed sell for about 238 25 For the data in the previous problem the following is an example of a butterfly spread sell two calls with an exercise price of 50 buy one call with an exercise price of 40 and buy one call with an exercise price of 60 Simulate the cash flows from this portfolio 26 A stock currently sells for 69 The annual growth rate of the stock is 15 and the stocks annual volatility is 35 The riskfree rate is currently 5 You have bought a sixmonth European put option on this stock with an exercise price of 70 a Use RISK to value this option b Use RISK to analyze the distribution of percentage returns for a sixmonth horizon for the following portfolios Portfolio 1 Own 100 shares of the stock Portfolio 2 Own 100 shares of the stock and buy the put described in part a Which portfolio has the larger expected return Explain why portfolio 2 is known as portfolio insurance 27 A knockout call option loses all value at the instant the price of the stock drops below a given knockout level Determine a fair price for a knockout call option when the current stock price is 20 the exercise price is 21 the knockout price is 1950 the mean annual growth rate of the stock is 12 the annual volatility is 40 the riskfree rate is 10 and the exercise date is one month from now where you can assume there are 21 trading days in the month and 250 in a year 28 Suppose an investor has the opportunity to buy the fol lowing contract a stock call option on March 1 The contract allows him to buy 100 shares of ABC stock at the end of March April or May at a guaranteed price of 50 per share He can exercise this option at most once For example if he purchases the stock at the end of March he cannot purchase more in April or May at the guaranteed price If the investor buys the contract he is hoping that the stock price will go up 666 Chapter 11 Simulation Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 114 Marketing Models 667 The reasoning is that if he buys the contract the price goes up to 51 and he buys the stock that is he exer cises his option for 50 he can then sell the stock for 51 and make a profit of 1 per share Of course if the stock price goes down he doesnt have to exercise his option he can just throw the contract away Assume that the stock price change each month is normally distributed with mean 0 and standard deviation 2 The investor uses the following strategy At the end of March he exercises the option only if the stock price is above 5150 At the end of April he exercises the option assuming he hasnt exercised it yet only if the price is above 5075 At the end of May he exercises the option assuming he hasnt exercised it yet only if the price is above 5000 This isnt necessarily his best strategy but it is a rea sonable one Simulate 250 replications of this strategy and answer the following a Estimate the probability that he will exercise his option b Estimate his net profit with this strategy This doesnt include the price of the contract c Estimate the probability that he will net over 300 d Estimate the worth of this contract to him 114 MARKETING MODELS There are plenty of opportunities for marketing departments to use simulation They face uncertainty in the brandswitching behavior of customers the entry of new brands into the mar ket customer preferences for different attributes of products the effects of advertising on sales and so on We examine some interesting marketing applications of simulation in this section 1141 Models of Customer Loyalty What is a loyal customer worth to a company This is an extremely important question for companies It is an important part of customer relationship management or CRM cur rently one of the hottest topics in marketing Companies know that if customers become dissatisfied with the companys product they are likely to switch and never return Marketers refer to this customer loss as churn The loss in profit from churn can be large particularly because longstanding customers tend to be more profitable in any given year than new customers The following example uses a reasonable model of customer loyalty and simulation to estimate the worth of a customer to a company It is based on the excel lent discussion of customer loyalty in Reichheld 1996 E X A M P L E 1111 THE LONGTERM VALUE OF A CUSTOMER AT CCAMERICA C CAmerica is a credit card company that does its best to gain customers and keep their business in a highly competitive industry The first year a customer signs up for service typically results in a loss to the company because of various administrative expenses However after the first year the profit from a customer is typically positive and this profit tends to increase through the years The company has estimated the mean profit from a typical customer to be as shown in column B of Figure 1132 For example the company expects to lose 40 in the customers first year but to gain 87 in the fifth yearprovided that the cus tomer stays loyal that long For modeling purposes we assume that the actual profit from a customer in the customers nth year of service is normally distributed with mean shown in Figure 1132 and standard deviation equal to 10 of the mean At the end of each year the customer leaves the company never to return with probability 015 the churn r ate Alternatively the customer stays with probability 085 the retention rate The company wants to estimate the NPV of the net profit from any such customer who has just signed up for ser vice at the beginning of year 1 at a discount rate of 15 assuming that the cash flow occurs in the middle of the year5 It also wants to see how sensitive this NPV is to the retention rate 5This assumption makes the NPV calculation slightly more complex but it is probably more realistic than the usual assumption that cash flows occur at the ends of the years Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Regardless of which version is more realistic and an argument can be made for either an advantage of the model with only a few random numbers is that you can use RISKs tornado chart to see which source of randomness is most highly correlated with NPV This tornado chart appears in Figure 1142 It is for simulation 2 with free maintenance agree ment but the chart for simulation 1 is virtually the same Perhaps surprisingly it is not the switching behavior that drives NPV it is driven more by the percentage of customers who purchase As this example illustrates it is sometimes an advantage to keep the models simple Key insights are then more apparent than when there is more complexity 1142 Marketing and Sales Models We conclude this marketing section with a model of marketing and selling condos The main issue is the timing of sales and we demonstrate how a deterministic model of this timing can provide very misleading results 676 Chapter 11 Simulation Models Figure 1141 Summary Results for Modified Model Figure 1142 Tornado Chart for NPV E X A M P L E 1113 MARKETING AND SELLING CONDOS T he Blackstone Development Company has just finished building 120 highend condos each priced at 300000 Blackstone has hired another company Pletcher Marketing to market and sell these condos Pletcher will incur all of the marketing and maintenance costs assumed to be 800 per unsold condo per month and it will receive a 10 commission Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 114 Marketing Models 677 30000 from Blackstone at the time of each condo sale Because Blackstone wants these condos to be sold in a timely manner it has offered Pletcher a 200000 bonus at the end of the first year if at least half of the condos have been sold and an extra 500000 bonus at the end of the second year if all of the condos have been sold Pletcher estimates that it can sell five condos per month on average so that it should be able to collect the bonuses However Pletcher also realizes that there is some uncertainty about the number of sales per month How should this uncertainty be modeled and will the resulting simulation model give different qualitative results than a deterministic model where exactly five con dos are sold per month Objective To develop a simulation model that allows us to see how the uncertain timing affects the monetary outcomes for Pletcher and to compare this simulation model to a deterministic model with no uncertainty about the timing of sales WHERE DO THE NUMBERS COME FROM The inputs are straightforward from Blackstones agreement with Pletcher The only diffi culty is determining an appropriate probability model for the timing of sales which we dis cuss next Solution To make a fair comparison between a deterministic model with five sales per month and a simulation model with uncertainty in the timing of sales we need a discrete distribution for monthly sales that has mean 5 One attractive possibility is to use the Poisson distribution It is a commonly used discrete distribution with only one parameter the mean The Poisson distribution has one theoretical drawback in that it allows all nonnegative integers to occur but this has no practical effect As shown in Figure 1143 the Poisson distribution with mean 5 has virtually no probability of values larger than say 15 Figure 1143 Poisson Distribution with Mean 5 DEVELOPING THE SIMULATION MODEL The deterministic model is very straightforward and is not shown here By selling a sure five condos per month Pletcher sells all condos by the end of year 2 receives both Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 680 Chapter 11 Simulation Models Figure 1146 Histogram of Total Bonus Received Figure 1147 Histogram of NPV peaks is influenced heavily by the bonuses or lack of them On average the NPV is only about 239 million much less than estimated by the deterministic model This is still one more examplea dramatic oneof the flaw of averages P R O B L E M S SkillBuilding Problems 29 Suppose that Coke and Pepsi are fighting for the cola market Each week each person in the market buys one case of Coke or Pepsi If the persons last purchase was Coke there is a 090 probability that this persons next purchase will be Coke otherwise it will be Pepsi You can assume that there are only two brands in the market Similarly if the persons last purchase was Pepsi there is a 080 probability that this persons next purchase will be Pepsi otherwise it will be Coke Currently half of all Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 115 SIMULATING GAMES OF CHANCE We realize that this is a book about business applications However it is instructive and fun to see how simulation can be used to analyze games of chance including sports contests Indeed many analysts refer to Monte Carlo simulation and you can guess where that name comes fromthe gambling casinos of Monte Carlo 1151 Simulating the Game of Craps Most games of chance are great candidates for simulation because they are by their very nature driven by randomness In this section we examine one such game that is extremely popular in the gambling casinos the game of craps In its most basic form the game of 682 Chapter 11 Simulation Models This example is based on one such offer We assume that a mobile provider named Syncit is willing to give a customer a free laptop computer at a cost of 300 to Syncit if the customer signs up for a guaranteed two years of service During that time the cost of service to the customer is a constant 60 per month or 720 annually After two years we assume the cost of service increases by 2 annually We assume that in any year after the guaranteed two years the probability is 07 that the customer will stay with Syncit This probability is the retention rate We also assume that if a customer has switched to another mobile service there is always a probability of 01 that the customer will without any free laptop offer willingly rejoin Syncit The company wants to see whether this offer makes financial sense in terms of NPV using a 10 discount rate It also wants to see how the NPV varies with the retention rate Simulate a 15year time horizon both with and without the free offer to estimate the difference For the situation without the free offer assume the customer has probability 05 of signing up with Syncit during year 1 34 Suppose that GLC earns a 2000 profit each time a person buys a car We want to determine how the expected profit earned from a customer depends on the quality of GLCs cars We assume a typical customer will purchase 10 cars during her lifetime She will purchase a car now year 1 and then purchase a car every five yearsduring year 6 year 11 and so on For simplicity we assume that Hundo is GLCs only competitor We also assume that if the consumer is satisfied with the car she purchases she will buy her next car from the same company but if she is not satisfied she will buy her next car from the other company Hundo produces cars that satisfy 80 of its customers Currently GLC produces cars that also satisfy 80 of its customers Consider a customer whose first car is a GLC car If profits are discounted at 10 annually use simulation to estimate the value of this customer to GLC Also estimate the value of a customer to GLC if it can raise its customer satisfaction rating to 85 to 90 or to 95 You can interpret the satisfaction value as the probability that a customer will not switch companies 35 The Mutron Company is thinking of marketing a new drug used to make pigs healthier At the beginning of the current year there are 1000000 pigs that could use the product Each pig will use Mutrons drug or a competitors drug once a year The number of pigs is forecast to grow by an average of 5 per year However this growth rate is not a sure thing Mutron assumes that each years growth rate is an independent draw from a normal distribution with probability 095 that the growth rate will be between 3 and 7 Assuming it enters the market Mutron is not sure what its share of the market will be during year 1 so it models this with a triangular distribution Its worst case share is 20 its most likely share is 40 and its bestcase share is 70 In the absence of any new competitors entering this market in addition to itself Mutron believes its market share will remain the same in succeeding years However there are three potential entrants in addition to Mutron At the beginning of each year each entrant that has not already entered the market has a 40 chance of entering the market The year after a competitor enters Mutrons market share will drop by 20 for each new competitor who entered For example if two competitors enter the market in year 1 Mutrons market share in year 2 will be reduced by 40 from what it would have been with no entrants Note that if all three entrants have entered there will be no more entrants Each unit of the drug sells for 220 and incurs a variable cost of 040 Profits are discounted by 10 annually a Assuming that Mutron enters the market use simulation to find its NPV for the next 10 years from the drug b Again assuming that Mutron enters the market it can be 95 certain that its actual NPV from the drug is between what two values Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it announced we did not know which team would win6 All we knew were the pairings which teams would play which other teams and the team ratings based on Jeff Sagarins nationally syndicated rating system We show how to simulate the tournament and keep a tally of the winners Objective To simulate the 64team NCAA basketball tournament and keep a tally on the number of times each team wins the tournament WHERE DO THE NUMBERS COME FROM As soon as you learn the pairings for the next NCAA tournament you can visit Sagarins site at wwwusatodaycomsportssagarinhtmhoop for the latest ratings Solution We need to make one probabilistic assumption From that point it is a matter of playing out the games and doing the required bookkeeping To understand this probabilistic assumption suppose team A plays team B and Sagarins ratings for these teams are say 85 and 78 Then Sagarin predicts that the actual point differential in the game team As score minus team Bs score will be the difference between the ratings or 77 We take this one step further We assume that the actual point differential is normally distributed with mean equal to Sagarins prediction 7 and standard deviation 10 Why 10 This is an esti mate based on an extensive analysis of historical data Then if the actual point differential is positive team A wins If it is negative team B wins DEVELOPING THE SIMULATION MODEL We provide only an outline of the simulation model You can see the full details in the file March Madness Men 2010xlsm Remember that an xlsm file contains macros When you open it you need to enable the macros This file includes the data for the 2010 tournament but you can easily modify it for future tournaments by following the direc tions on the sheet We have also included the March Madness Women 2010xlsm file The womens tournament was won by the University of Connecticut The entire simula tion is on a single Model sheet Columns A through C list team indexes team names and Sagarin ratings If two teams are paired in the first round they are placed next to one another in the list Also all teams in a given region are listed together The regions are colorcoded Columns K through Q contain the simulation The firstround results are at the top the secondround results are below these and so on Winners from one round are automatically carried over to the next round with appropriate formulas Selected portions of the Model sheet appear in Figures 1149 and 1150 We now describe the essential features of the model 1 Teams and ratings We first enter the teams and their ratings as shown in Figure 1149 Most of the teams shown here were in the East region in the 2010 tournament Kentucky played East Tennessee State in the first round Texas played Wake Forest and so on 686 Chapter 11 Simulation Models We model the point spread as normally distributed with mean equal to the difference between the Sagarin ratings and standard deviation 10 6Actually 65 teams are announced and an early playoff game occurs to see which of two lowly rated teams gets to play a 1 seed This has no effect on the simulation because neither lowly ranked team has much chance of winning against the 1 seed 7In general there is also a homecourt advantage but we assume all games in the tournament are on neutral courts so that there is no advantage to either team Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it corresponding variation of any selected output You can then see usually through one of several charts which inputs are most critical At this point you can either conclude the analysis or switch to RISK and model the key inputs with appropriate probability dis tributions The following example which illustrates how TopRank and RISK can work in tan dem is an extremely important one Simulation in the business world is often used to ana lyze potential products The profitability of a new product is highly uncertain because it depends on many uncertain quantities Many companies we have worked with including General Motors and Eli Lilly begin the analysis of every new product by determining the uncertain quantities that can affect the profitability of the product This analysis is often the deciding factor in whether the product is developed and marketed 692 Chapter 11 Simulation Models E X A M P L E 1116 NEW PRODUCT DEVELOPMENT AT SIMTEX S imTex a pharmaceutical company is in the early stages of developing a new drug called Biathnon As with most new drugs the future of Biathnon is highly uncertain For example its introduction into the market could be delayed pending tests by the Food and Drug Administration FDA Also its market could be diminished by a potential rival product from SimTexs competition SimTex has identified the following key inputs that will affect Biathnons future profitability Number of years after product is developed until it is produced due to potential FDA delays Number of years the product sells Initial cost incurred in developing the product Salvage value obtained from equipment after production of the product has been discontinued Fixed production cost incurred during years in which the product is manufactured Unit cost of producing the product Unit price of the product Initial demand for the product during the first year it is sold Annual percentage growth in demand for the product Percentage of demand for the product that is lost to the competition Discount rate used to discount cash flows from the product These are the inputs to a profitability model for Biathnon A natural question is how changes in the inputs affect the key outputthe NPV of Biathnon over its lifetime How can SimTex use TopRank and RISK to analyze this NPV Objective To use TopRank to identify the inputs that affect NPV most and then to use RISK to model these inputs with probability distributions WHERE DO THE NUMBERS COME FROM Most of the inputs in the preceding list are difficult to estimate However this is exactly why TopRank is being used to see how sensitive NPV is to the various input values Then the company can spend more energy trying to estimate the inputs that really matter Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 2 Financials and other formulas The formulas in the other cells are then straightfor ward For year 1 column C the formulas in rows 21 to 27 are IFC20YesAnnualfixedcost0 IFANDB20NoC20YesInitialdemandIFC20YesB221Annual demandgrowth0 IFC2200C221Lostsales IFC2300C23Unitcost IFC2300C23UnitPrice IFANDC20YesD20NoSalvagevalue0 and C19C21C24C25C26 The second of these formulas in cell C22 might require some explanation The first IF checks whether production occurs this year but not the previous year If so this must be the first year of production so that the demand is the initial demand Otherwise the second IF checks whether production is still occurring If so then demand is the previous years demand plus the growth percentage Similarly the formula for salvage value in cell C26 checks whether production occurs this year but not next year If so then this must be the year when the salvage value is obtained 3 NPV Calculate the NPV discounted to the beginning of year 0 in cell B29 with the formula NPVDiscountrateC27AF27B27 Note that the fixed cost in cell B27 is not discounted Now that the model has been developed you could use trial and error or data tables to see how the NPV reacts to changes in the inputs However TopRank does this for you Actually it can be used in a number of ways We discuss only one of them although it appears to us to be the most useful USING TOPRANK To use TopRank all you need to modify is the input section10 Instead of entering constants in the input cells you should enter TopRanks RISKVARY function This function has the syntax RISKVARYbaseminimummaximumrangetypestepsdistribution where base is the base case best guess for the input minimum is the smallest possible value for the input maximum is the largest possible value for the input rangetype is 0 1 or 2 and determines the way minimum and maximum should be entered even though 0 is the default value we use rangetype 2see the TopRank manual for more details steps is the number of values from minimum to maximum to use for this input distribution is an optional argument that we omit 694 Chapter 11 Simulation Models 10 This discussion assumes that TopRank is open within Excel It can be opened exactly like RISK from the Start button of Windows Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 706 Chapter 11 Simulation Models increase in spending by credit card users presumably on liquor but maybe also on more expensive food The restaurant wants to simulate a fiveyear horizon Its base case is not to accept credit cards at all in which case it expects to earn 234 million in revenue each year It wants to use simulation to explore other options where it will accept credit cards in year 1 and then discontinue them in years 25 if the bank fee is less than or equal to some cutoff value For example one possibility is to accept credit cards in year 1 and then discontinue them only if the bank fee is less than or equal to 3 You should explore the cutoffs 2 to 4 in increments of 05 Which policy provides with the largest mean increase in revenue over the fiveyear horizon relative to never using credit cards 72 The Ryder Cup is a threeday golf tournament played every other year with 12 of the best US golfers against 12 of the best European golfers They play 16 team matches each match has two US golfers against two European golfers on Friday and Saturday and they play 12 singles matches each match has a single US golfer against a European golfer on Sunday Each match is either won or tied A win yields 1 point for the winning team and 0 points for the losing team A tie yields 05 point for each team A team needs 145 points to win the Cup If each team gets 14 points the tournament is a tie but the preceding winner gets to keep the Cup In 1999 the US was behind 10 points to 6 after the team matches To win the Cup the US needed at least 85 points on Sunday a very unlikely outcome but they pulled off the miracle and won Use simulation to estimate the probability of the US scoring at least 85 points in the 12 singles matches assuming all golfers in the tournament are essentially equal Proceed as follows a Use simulation to estimate the probability call it h for half that a given match ends in a tie To do this you can assume that any of the 18 holes is tied with probability 0475 and won with probability 0525 These are the historical fractions of holes that have been tied and won in singles matches in the past few Ryder Cups Note that each match is match play so the only thing that counts on each hole is whether a golfer has fewer strokes than the other golferwinning a hole by one stroke is equivalent to winning the hole by two or more strokes in match play The player winning the most holes wins the match unless they tie b Run another simulation using the estimated probability h as an input to estimate the probability that the US will score at least 85 points in the 12 singles matches 73 Based on Bukiet et al 1997 Many Major League teams including Oakland Boston LA Dodgers and Toronto use mathematical models to evaluate baseball players A common measure of a players offensive effectiveness is the number of runs generated per inning RPI if a team were made up of nine identical copies of this player For example which team would score more runs a team with nine copies of Ichiro Suzuki or a team with nine copies of Manny Ramirez We can use simulation to answer this question Lets consider a simplified baseball game in which each plate appearance results in one of six outcomes Out Runners do not advance Walk Runners advance if forced Single Runner on first moves to second All other runners score Double Runner on first moves to third All other runners score Triple All runners on base score Home Run All runners and batter score A team gets three outs per inning You are given the data in the file P1173xlsx on Ichiro Suzuki and Manny Ramirez from the 2004 season Use simulation to determine which hitter is more valuable according to the RPI criterion 74 In this version of dice blackjack you toss a single die repeatedly and add up the sum of your dice tosses Your goal is to come as close as possible to a total of 7 without going over You may stop at any time If your total is 8 or more you lose If your total is 7 or less the house then tosses the die repeatedly The house stops as soon as its total is 4 or more If the house totals 8 or more you win Otherwise the higher total wins If there is a tie the house wins Consider the fol lowing strategies Keep tossing until your total is 3 or more Keep tossing until your total is 4 or more Keep tossing until your total is 5 or more Keep tossing until your total is 6 or more Keep tossing until your total is 7 or more For example suppose you keep tossing until your total is 4 or more Here are some examples of how the game might go You toss a 2 and then a 3 and stop for total of 5 The house tosses a 3 and then a 2 You lose because a tie goes to the house You toss a 3 and then a 6 You lose You toss a 6 and stop The house tosses a 3 and then a 2 You win You toss a 3 and then a 4 for total of 7 The house tosses a 3 and then a 5 You win Note that only 4 tosses need to be generated for the house but more tosses might need to be generated for you depending on your strategy Develop a simulation and run it for at least 1000 iterations for each of the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it strategies listed previously For each strategy what are the two values so that you are 95 sure that your probability of winning is between these two values Which of the five strategies appears to be best 75 It is now May 1 of year 0 and GM is deciding whether to produce a new car The following information is relevant The fixed cost of developing the car is incurred on January 1 of year 1 and is assumed to follow a tri angular distribution with smallest possible cost 300 million most likely cost 400 million and largest possible cost 700 million The fixed cost is depreciated on a straightline base during years 2 to 5 The tax rate is 40 The car will first come to market during year 2 and is equally likely to sell for 6 7 or 8 years The market size during year 2 will be between 20000 and 90000 cars There is a 25 chance that the market size will be less than or equal to 50000 cars a 50 chance that it will be less than or equal to 70000 cars and a 75 chance that it will be less than or equal to 80000 cars After year 2 the market size is assumed to grow by 5 per year The market share during year 2 is assumed to fol low a triangular distribution with most likely value 40 There is a 5 chance that market share will be 20 or less and a 5 chance that it will be 50 or more The market share during later years will remain unchanged unless RD makes a design improvement There is a 50 chance that RD will make a design improvement during year 3 a 20 chance that it will make a design improvement during year 4 and a 30 chance that no design improvement will occur There will be at most one design improve ment During the year if any in which a design improvement occurs GMs market share will increase to 50 above its current value For exam ple suppose GMs market share at the beginning of year 3 is 30 If a design improvement occurs dur ing year 3 its market share during year 3 and all later years will be 45 The car sells for 15000 each year The cost of producing the first x cars is 10000x09 dollars This builds a learning curve into the cost structure During year 2 and later years cash flows are assumed to occur midyear GM discounts its cash flows at 15 per year Use simulation to model GMs situation Based on the simulation output GM can be 95 sure that the NPV generated by the car is between what two values Should GM produce this car Explain why or why not What are the two key drivers of the cars NPV Hint The way the uncertainty about the market size in year 2 is stated suggests using the Cumul distribution implemented with the RISKCUMUL function Look it up in RISKs online help 76 It is January 1 of year 0 and Lilly is considering developing a new drug called Dialis We are given the following information On March 15 of year 0 Lilly incurs a fixed cost that is assumed to follow a triangular distribution with best case 10 million most likely case 35 million and worst case 50 million This cost will be depreciated on a straightline basis during years 1 to 6 The product will be sold during years 1 to 6 In years 1 and 2 the product will be sold only in the United States but starting in year 3 Lilly might sell the product overseas The year 1 market size in the United States is assumed to be between 500000 and 3000000 units A market size of 1000000 units is assumed to be twice as likely as a market size of 700000 and a market size of 2000000 units is assumed to be three times as likely as a market size of 700000 Lillys year 1 market share is assumed to follow a triangular distribution with worst case 10 most likely case 20 and best case 30 Lilly assumes that its market share will remain the same unless a competitor enters the market The growth rate in market size in later years is assumed to be the same each year In year 1 it is assumed to follow a triangular distribution with worst case 5 annual growth most likely case 12 annual growth and best case 14 annual growth A single competitor might enter the market Each year the competitor has a 30 chance of entering the market assuming it has not already entered The year after entering the market a competitor causes a permanent loss of 40 of Lillys market share For example suppose the competitor enters in year 2 and Lillys share was 20 Then in the years 3 to 6 its market share will be 12 At the beginning of year 3 Lilly will decide whether to sell Dialis overseas If no competitor has entered the market by the end of year 2 there is a 70 chance that Lilly will sell the product over seas If a competitor has entered the market by the end of year 2 there is only a 30 chance that Lilly will sell the product overseas Lillys market share overseas will equal its market share in the United States It estimates that the overseas market is 25 of world sales for drugs of this type The other 75 is US sales Each year the product sells for 120 and incurs a unit cost of 80 Cash flows are discounted at 15 annually and profits are taxed at 40 Cash flows for years 1 to 6 take place midyear 118 Conclusion 707 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Use simulation to model Lillys situation Based on the simulation output Lilly can be 95 sure the NPV for this project is between what two numbers Would you go ahead with this project Explain why or why not Hint The way the uncertainty about the market size in year 1 is stated suggests using the General distribution implemented with the RISKGENERAL function Look it up in RISKs online help 77 It is January 1 of year 0 and Merck is trying to deter mine whether to continue development of a new drug The following information is relevant You can assume that all cash flows occur at the ends of the respective years Clinical trials the trials where the drug is tested on humans are equally likely to be completed in year 1 or 2 There is an 80 chance that clinical trials will succeed If these trials fail the FDA will not allow the drug to be marketed The cost of clinical trials is assumed to follow a triangular distribution with best case 100 million most likely case 150 million and worst case 250 million Clinical trial costs are incurred at the end of the year clinical trials are completed If clinical trials succeed the drug will be sold for five years earning a profit of 6 per unit sold If clinical trials succeed a plant will be built during the same year trials are completed The cost of the plant is assumed to follow a triangular distri bution with best case 1 billion most likely case 15 billion and worst case 25 billion The plant cost will be depreciated on a straightline basis during the five years of sales Sales begin the year after successful clinical trials Of course if the clinical trials fail there are no sales During the first year of sales Merck believe sales will be between 100 million and 200 million units Sales of 140 million units are assumed to be three times as likely as sales of 120 million units and sales of 160 million units are assumed to be twice as likely as sales of 120 million units Merck assumes that for years 2 to 5 that the drug is on the market the growth rate will be the same each year The annual growth in sales will be between 5 and 15 There is a 25 chance that the annual growth will be 7 or less a 50 chance that it will be 9 or less and a 75 chance that it will be 12 or less Cash flows are discounted 15 per year and the tax rate is 40 Use simulation to model Mercks situation Based on the simulation output would you recommend that Merck continue developing Explain your reasoning What are the three key drivers of the projects NPV Hint The way the uncertainty about the first year sales is stated suggests using the General distribution implemented with the RISKGENERAL function Similarly the way the uncertainty about the annual growth rate is stated suggests using the Cumul distribu tion implemented with the RISKCUMUL function Look these functions up in RISKs online help 78 Nucleon is trying to determine whether to produce a new drug that makes pigs healthier The product will be sold in years 1 to 5 The following information is relevant A fixed cost is incurred on January 1 of year 0 and will be between 1 billion and 5 billion There is a 20 chance the fixed cost will be less than or equal to 2 billion a 60 chance that it will be less than or equal to 3 billion and a 90 chance that it will be less than or equal to 4 billion The fixed cost is depreciated on a straightline basis during years 1 to 5 The weighted average cost of capital is 15 This is the rate Nucleon uses for discounting cash flows The market size in year 1 is 10 million pigs During each of years 2 to 5 the market size will grow at the same rate This growth rate is assumed to follow a triangular distribution with best case 15 most likely case 6 and worst case 1 The selling price is always 100 per unit and the unit cost of production is always 16 per unit In year 1 the average number of units of the drug sold for each pig will be between 1 and 2 with 13 and 17 being equally likely and 15 being twice as likely as 13 There are three potential competitors During each of years 1 to 5 a competitor who has not entered the market has a 60 chance of entering the market The year after a competitor enters the market the average units sold per pig of the Nucleon drug drops by 20 for each competitor entering For example suppose that sales per pig are 15 units in year 1 If two competitors enter the market in year 1 Nucleon sales per pig drop to 09 in year 2 All cash flows other than the fixed cost on January 1 of year 0 are incurred midyear Use simulation to model Nucleons situation Based on the simulation output would you go ahead with this project Explain why or why not What are the three key drivers of the projects NPV Hint The way the uncertainty about the fixed cost is stated suggests using the Cumul distribution implemented with the RISKCUMUL function Similarly the way the uncer tainty about the units sold per pig in year 1 is stated suggests using the General distribution implemented with the RISKGENERAL function Look these func tions up in RISKs online help 708 Chapter 11 Simulation Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E Y our nextdoor neighbor Scott Jansen has a 12 yearold daughter and he intends to pay the tuition for her first year of college six years from nowThe tuition for the first year will be 17500 Scott has gone through his budget and finds that he can invest 200 per month for the next six years Scott has opened accounts at two mutual fundsThe first fund follows an investment strategy designed to match the return of the SP 500The second fund invests in shortterm Treasury bills Both funds have very low fees Scott has decided to follow a strategy in which he contributes a fixed fraction of the 200 to each fund An adviser from the first fund suggested that in each month he should invest 80 of the 200 in the SP 500 fund and the other 20 in the Tbill fund The adviser explained that the SP 500 has averaged much larger returns than the Tbill fund Even though stock returns are risky investments in the short run the risk should be fairly minimal over the longer sixyear period An adviser from the second fund recommended just the opposite invest 20 in the SP 500 fund and 80 in Tbills because treasury bills are backed by the United States government If you follow this allocation he said your average return will be lower but at least you will have enough to reach your 17500 target in six years Not knowing which adviser to believe Scott has come to you for help Questions 1 The file Investing for Collegexlsxcontains 261 monthly returns of the SP 500 and Treasury bills from January 1970 through September 1991 If you can find more recent data on the Web feel free to use it Suppose that in each of the next 72 months six years it is equally likely that any of the historical returns will occur Develop a spreadsheet model to simulate the two suggested investment strategies over the sixyear period Plot the value of each strategy over time for a single iteration of the simulationWhat is the total value of each strategy after six years Do either of the strategies reach the target 2 Simulate 1000 iterations of the two strategies over the sixyear period Create a histogram of the final fund values Based on your simulation results which of the two strategies would you recommend Why 3 Suppose that Scott needs to have 19500 to pay for the first years tuition Based on the same simulation results which of the two strategies would you recommend now Why 4 What other realworld factors might be important to consider in designing the simulation and making a recommendation 111 COLLEGE FUND INVESTMENT 710 Chapter 11 Simulation Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 713 Inventory Models C H A P T E R INVENT OR Y DECISIONS IN DELL S SUPPL Y CHAIN D ell is the largest computersystems company based on estimates of global market share and it is also the fastest growing of the major computersystems companies competing in the business education govern ment and consumer markets Dells key to success is its strategy of bypassing retailers and selling its products directly to customers Inventory manage ment is extremely important to a company such as Dell It not only incurs the usual costs for holding inventoryloss of interest from capital tied up in inventory and storage costsbut it also incurs huge costs from obsoles cence Because of the rapid changes in technology many computer compo nents lose from 05 to 20 of their value per week so that a supply chain filled with yesterdays technology is practically worthless Although Dell was aware of the costs of holding too much inventory it didnt employ the types of mathematical models discussed in this chapter for managing its inventory until 1999 when it hired a group from the University of Michigan to study the problem The results of this study appear in Kapuscinski et al 2004 Due to direct sales Dell actually carries very little inventory It assembles computer systems at its manufacturing plants in AustinTexas and ships them to customers in just a few days Therefore the plants carry virtually no inven tory of finished goods The inventory of computer components held at Dells suppliers is a different story Many of its suppliers are located in Southeast Asia Because transportation of components from Asia to Texas can take any where from a week to a month Dell requires its suppliers to keep inventory Gerry BroomeAP Photo 12 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it on hand in revolvers small warehouses located within a few miles of Dells assembly plants in Austin Each revolver is shared by several suppliers who pay rents for using them The key problem is to reduce inventory at the revolvers while maintaining an adequate ser vice level Dells service level is about 988 meaning that the components it needs are available about 988 of the time Dell shares its data on demand forecasts and actual demands with its suppliers and provides guidelines on how to manage their inventory lev els at the revolvers The authors recommended using an R Q ordering policy at the revolvers for one particular important component called XDX in the paper to conceal its identity This means that when inventory of XDX reaches the reorder point R the supplier orders an amount Q When this type of ordering policy is discussed later in this chapter you will see that the difficult part is finding the appropriate reorder point R During the time it takes an order to arrive at the revolver called the lead time Dell experiences demand for the com ponent To guard against stockouts in case this lead time demand is larger than expected R contains some safety stock The amount of safety stock to hold depends on several fac tors 1 the variance of demand during lead time 2 the variance of the length of the lead time and 3 the desired service levels The first two of these are caused by uncer tainty whereas the third is based on costs The authors performed a careful study of the causes of uncertainty They broke demand into two parts the aggregate demand for com puter systems and the percentage of this aggregate demand for particular components such as XDX Another source of uncertainty at least to the suppliers is the pull vari ance This occurs when multiple suppliers supply the same component in their revolvers Dell doesnt pull from these suppliers at a uniform rate It might use supplier As compo nents for a few days and then use supplier Bs for a few days The authors examined how each of these sources of uncertainty affects the amount of safety stock and hence excess inventory prescribed by the model and suggested how better forecasting methods and information sharing can lead to improved results In terms of service level the authors used a critical fractile analysis to determine an optimal service level This critical fractile also discussed later in this chapter is a ratio of the cost of having too little inventory for example lost profit from a canceled order and increased shipping cost for not having a component when needed to the cost of having too much inventory for example cost of capital tied up in excess inventory and price erosion from having obsolescent components The authors recommendations went into effect in 1999 and to our knowledge are still being implemented They estimated that Dell could reduce the current inventory from 105 days by about 38 Dell thinks of inventory in terms of days of supply rather than units on hand By removing approximately four days of safetystock inventory they estimate that the NPV of savings in XDX passing through the revolvers is about 43 million Of course as the authors system is used for other important components the savings will only increase 714 Chapter 12 Inventory Models 121 INTRODUCTION Inventory management is one of the most important decisions faced by many companies These companies include not only retailers that stock products for sale to customers like you but also companies that supply other companies They all face two competing pressures The first is the pressure to have enough inventory on hand The most obvious reason for this is that they do not want to run out of products that customers demand Another promi nent reason however is the fixed cost of ordering or producing as discussed throughout this chapter If a fixed cost is incurred each time the company orders from its supplier or a Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it fixed cost is incurred each time a manufacturer produces a batch where this cost does not depend on the order or batch size the company has an incentive to place large orders or produce large batches to minimize its annual fixed costs1 The second pressure related to inventory management is the pressure to carry as little inventory as possible The most obvious reasons for this are the cost of storing items and the interest costs involved in tying up money in inventory If the company has to pay cash for items that end up sitting on the shelf for long periods of time it loses potential interest on this money that could be invested elsewhere Storage space is sometimes an issue as well Some companies simply do not have the space to store as much inventory as they might like For example there is fierce competition for shelf space in supermarkets These two competing pressures are at the heart of most inventory models Companies want to order enough but they do not want to order too much The balance is typically not easy to find so they need models to determine the best ordering or production policy An inventory problem can usually be broken up into two parts 1 how much to order on each ordering opportunity and 2 when to order When customer demand is assumed to be known the resulting models are called deterministic models If customer demand is known and the order quantity has been determined then specifying when the orders should be placed is relatively easy A more realistic situation occurs when customer demand is uncer tain In this case the decision on when to place orders becomes more difficult Orders should be placed early enough so that the chance of running out before they arrive is fairly small These more difficult problems require probabilistic inventory models Inventory management as an academic subject falls somewhere between management science and operations management We have been told that many instructors who use this book for a management science class do not cover this chapter because it is covered in the operations management course However inventory management has long held an impor tant place in management science both in theory and in practice There is plenty of evidence to support this claim For example a quick scan of Interfaces articles indicates there are many real applications of inventory management and supply chain management To name a few three articles by Billington et al 2004 Guide et al 2005 and Laval et al 2005 describe supply chain management at HewlettPackard de Kok et al 2005 describe how Philips Electronics synchronizes its supply chain to minimize the socalled bullwhip effect Troyer et al 2005 discuss inventory management and order fulfillment at Deeres Commercial and Consumer Equipment Division and Bangash et al 2004 discuss inventory requirements planning at Lucent Technologies Four of these articles appeared in the prize winning issues of Interfaces So regardless of whether inventory management is discussed in a management science course or an operations management course this topic is extremely important for todays global organizations Inventory management also uses a variety of management science tools many of which are described in this chapter 122 CATEGORIES OF INVENTORY MODELS Researchers have analyzed many inventory models both deterministic and probabilistic We discuss only the most basic of these models which have been used extensively in real applications We begin by discussing several important issues and introducing some termi nology2 Keep in mind however that the possible number of realworld situations that 122 Categories of Inventory Models 715 1Some companies order products from vendors whereas other companies produce the products they need They both face similar inventory decisions Throughout most of the chapter we focus on companies that order from vendors and we talk about order quantities but similar models apply to companies that produce They must decide on production quantities often called batch sizes 2Entire books such as Cachon and Terwiesch 2009 discuss the general topic of matching supply with demand in much more depth than we provide here Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it require inventory management is virtually unlimited We list only some of the factors that are common to these situations Deterministic versus Probabilistic Models We have already mentioned the distinction between deterministic and probabilistic inven tory models In deterministic models all inputs to the problem particularly customer demand are assumed to be known when the decisions are made In reality a company must always forecast future demands with some type of forecasting model The outputs of this forecasting model might include a mean demand and a standard deviation of demand In deterministic models however only the mean is used and any information about the uncertainty such as the standard deviation is ignored This makes the resulting models simpler but usually less realistic Probabilistic models use this information about uncer tainty explicitly They are typically more difficult to analyze but they tend to produce bet ter decisions especially when the level of uncertainty is high External versus Internal Demand A second factor in inventory modeling is whether demand for the product is generated externally or internally External demand or independent demand occurs when the com pany that sells the product cannot directly control the extent or the timing of customer demand For example a retailer who orders products from a supplier and then waits to see how many customers request these products faces external demand In these situations we usually assume that ordering decisions are influenced by but do not affect customer demand In contrast internal demand or dependent demand occurs in most assembly and manufacturing processes Consider for example a company that manufactures laptop computers The external demand is for the finished product but the internal demand is for the components that go into the finished product After the company forecasts the number of laptops its customers will demand say in the next month it must then determine an appropriate production schedule for producing them This production schedule will neces sitate having inventories of the laptops component parts and subassemblies on hand at the right time In short the production schedule determines in large part the inventory required for all of the individual parts and subassemblies The coordination of all of theseensuring that everything is on hand when it is neededis a complex problem that we do not discuss in this book However it is a big part of supply chain management a topic that is receiving more attention than ever from both academics and practitioners The supply chain needs to ensure that the parts and subassemblies are available at the right time and the right place and at the cheapest cost for manufacturers to compete in todays busi ness environment Ordering versus Production A third factor in inventory modeling is whether the company orders the products from a supplier or produces them internally If the products are ordered then there is typically an order lead time the time elapsed from when the order is placed until it arrives In ordering models there is also usually a fixed cost also called a setup or ordering cost each time an order is placed where this cost is independent of the order quantity In contrast if prod ucts are produced internally there is also a lead time the time it takes to produce a batch of items This time is determined by a production rate such as 10 units per hour and possibly by a setup time the fixed time necessary to set up any machinery to produce a specific type 716 Chapter 12 Inventory Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it of product As in ordering models there can also be a setup cost each time a batch is pro duced where this cost is independent of the batch size Continuous versus Periodic Review A fourth factor in inventory modeling is whether inventory is reviewed continuously or periodically In continuous review models the inventory is monitored continually and orders can be placed at any time Typically there is a reorder pointa specific inventory levelso that when the inventory on hand reaches this reorder point an order is placed immediately This could happen Wednesday afternoon Friday morning or any other time In contrast in periodic review models there is some standard time such as every Monday morning when the inventory is reviewed and ordering decisions are made Except possibly for emergency orders these are the only times when orders are placed Continuous review models can certainly be implemented given todays computerized access to inventory lev els in real time and these models can result in lower annual costs than periodic review models However when a company stocks many products hundreds or even thousands it is often more convenient to order these say only on Monday mornings SingleProduct versus MultipleProduct Models A final factor in inventory modeling concerns the number of products involved Models that consider only a single product are conceptually and mathematically simpler so we ini tially analyze singleproduct models However most companies have many different prod ucts that must be considered simultaneously If the company orders these items from a supplier it may be wise to synchronize the orders in some way to minimize ordering costs We look at one such synchronization model in section 124 123 TYPES OF COSTS IN INVENTORY MODELS Companies face a number of costs when they manage inventories Although the types of costs vary depending on the company and the situation the following costs are typical Ordering or Setup Cost We have already mentioned the ordering or setup cost This is the fixed cost incurred every time an order is placed or a batch is produced independent of the amount ordered or produced This ordering cost includes the cost of paperwork and billing each time an order is placed and could include other costs as well such as paying a truck driver to deliver the order to the companys warehouse If the product is produced rather than ordered this cost can include the cost to set up equipment Unit Purchasing or Production Cost The unit purchasing or production cost is the cost for each additional unit purchased or produced often referred to as the variable cost For example to order 100 units the com pany might have to pay a setup cost of 500 plus 3 per unit for a total of 800 Here 3 is the unit purchasing cost If the company produces the product the unit production cost includes the cost of raw materials and the labor cost for each unit produced Sometimes the unit purchasing cost is not constant but changes according to a quantity discount schedule We consider a quantity discount model in section 124 123 Types of Costs in Inventory Models 717 The setup cost is inde pendent of the order or production batch size Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Holding or Carrying Cost The holding or carrying cost is the cost that motivates the company to keep less inven tory on hand This cost generally has two components the financial holding cost and the nonfinancial holding cost The nonfinancial holding cost is usually the cost of storing the product For example this might be the cost of renting warehouse space The financial holding cost is the opportunity cost of having money tied up in inventory when that money could instead be earning interest in other investments There can be other holding costs such as spoilage insurance and overhead which vary according to the amount and type of inventory on hand Shortage or Penalty Cost It is often important to measure the cost of running out of inventory This shortage or penalty cost is a difficult cost to measure For one thing it depends on how the company handles shortages At one extreme there are lost sales models where any demands that occur when inventory is zero are lost these customers take their business elsewhere At the other extreme there are complete backlogging models where demands that occur when inventory is zero are satisfied as soon as a new order arrives3 Both of these modelsor any in between called partial backlogging modelshave negative effects for the com pany There is lost revenue loss of goodwill and possibly expedited shipments with higher costs Unfortunately it can be difficult to put a dollar value on the cost of running out of inventory An alternative is to specify a service level such as meeting at least 95 of the demand on time Revenue Finally there is the selling price of the product and the resulting revenue to the company In many situations the revenue is a fixed amount that is not affected by any ordering deci sions This occurs when the selling price remains constant and the company intends to sat isfy all demand eventually In such cases the total revenue can be added to the relevant costs but it does not affect any ordering or production decisions On the other hand there are times such as in lost sales models when the selling price affects ordering decisions Here the shortage cost depends on the amount of revenue lost by not having enough inven tory on hand and this clearly depends on the selling price 124 ECONOMIC ORDER QUANTITY EOQ MODELS We first examine a class of models called economic order quantity EOQ models These are the most basic of all the inventory planning models Developed originally in 1915 by F W Harris of Westinghouse Corporation they are also among the earliest management science models Despite their simplicity numerous companies have applied these models and they continue to play a prominent role in inventory management We begin by studying the most basic EOQ model Then we examine several interest ing variations of this basic model All of these models make the following assumptions A company orders a single product from a supplier and sells this product to its customers 718 Chapter 12 Inventory Models A large part of the holding cost is the cost of capital tied up in inventory 3We also say the excess demand is backordered Both terms backlog and backorder mean that these orders are kept on the books and are satisfied when additional shipments arrive Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Orders can be placed at any time continuous review There is a constant known demand rate for the product usually expressed in units per year annual demand There is a constant known lead time for delivery of the product from the supplier There is a fixed ordering cost each time the product is ordered independent of the size of the order The price the company charges for the product is fixed The annual holding cost is proportional to the average amount of inventory on hand The constant demand rate means for example that if the yearly demand is 52000 units then each weeks demand is approximately 1000 unitsthere are no peaks or valleys during the year The known lead time means that if the company places an order on Monday and the lead time is three days then the order arrives with certainty on Thursday We discuss the holding cost in more detail shortly The Basic EOQ Model The most basic EOQ model adds the following two assumptions No stockouts are allowed that is the company never allows itself to run out of inventory The unit cost of purchasing the product from the supplier is constant In particular no quantity discounts are available These assumptions have important implications Because the demand rate and lead time are assumed to be known the company can ensure that it always has enough on hand to meet demand on time The main decision is whether to order small amounts frequently or to order large amounts infrequently The former results in large fixed costs and small holding costs less inventory on hand whereas the latter results in the opposite The EOQ analysis bal ances these two competing forces We now analyze this basic EOQ model in the following example 124 Economic Order Quantity EOQ Models 719 A crucial assumption of the basic EOQ model is that demand occurs at a constant known rate through time FUNDAMENTAL INSIGHT Importance of EOQ The basic EOQ model and its variations are among the simplest models discussed in this book and they have been known for close to a century However they cap ture the essence of man y companies pr oblems and they are still in wide use today As with most models for managing inventory they balance the costs of or dering too frequently and not ordering frequently enough E X A M P L E 121 ORDERING CAMERAS AT MACHEYS M acheys Department Store sells 1200 cameras per year and the demand pattern throughout the year is very steady The store orders its cameras from a regional warehouse and it usually takes one week for the cameras to arrive after an order has been placed Each time an order is placed an ordering cost of 125 is incurred The store pays 100 for each camera and sells them for 130 apiece There is no physical storage cost but the stores annual cost of capital is estimated at 8 per yearthat is it can earn 8 on any excess cash it invests The store wants to determine how often it should order cameras when it should place orders and how many cameras it should order in each order Objective To determine when to order and how much to order so that the store never runs out of cameras and profit is maximized Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 2 Unit purchase cost For any order quantity in cell B15 find the relevant unit pur chase cost by entering the formula VLOOKUPOrderquantityLookupTable2 in cell B20 Note that this returns the correct cost even at the breakpoints For example if the order quantity is 800 as in the figure the unit purchase cost is 26 as it should be 3 Basic EOQ Given the unit purchase cost in cell B20 develop the rest of the EOQ model exactly as in the previous example This time however note that there is no reve nue Everything is in terms of costs so that the objective is to minimize USING SOLVER Solver should be set up as shown in Figure 124 Note that an upper bound of 2000 has been placed on the order quantity although any large value could be used Also because the quantity discounts lead to a nonsmooth objective it is a good idea to use the Multistart option as discussed in Chapter 7 Alternatively Evolutionary Solver could be used but it doesnt appear to be necessary GRG Nonlinear Solver with the Multistart option finds the optimal solution quickly 724 Chapter 12 Inventory Models Figure 124 Solver Dialog Box for the Quantity Discount Model Discussion of the Solution The Solver solution indicates that the company should order just enough units 800 to achieve the lowest unit purchase cost You can check that if the order quantity is only 799 the total annual cost increases by about 10000 due mostly to the much larger annual purchasing cost In the other direction if the order quantity increases to 801 the annual purchasing cost doesnt change at all why but the net effect of a slightly smaller annual fixed ordering cost and a slightly larger annual holding cost is a slightly larger total annual cost Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The critical fractile analysis discussed here is in terms of the newsvendor model where a company orders exactly once As discussed in the chapter opener about Dells sup ply chain this same critical fractile analysis can be used to determine an optimal service level for a company As you will see service levels play an important role in the RQ ordering policies discussed next The RQ Ordering Policy The previous subsection analyzed a onetime ordering decision which is relevant for a product such as a newspaper or a fashion item that quickly goes out of style We now exa mine an ordering decision for a product with sales that continue into the indefinite future As with the EOQ model we assume that demand is more or less constant through timeno upward or downward trends and no seasonalitybut that it is random That is the proba bility distribution of demand in any month say is always the same but the actual demands in different months can be different because of randomness As with the deterministic EOQ model the company must make two decisions when to order and how much to order We assume that it uses a popular type of policy called an RQ policy where R is the reorder point and Q is the order quantity Under this policy the company continually monitors its inventory When inventory drops to R or below the company places an order for Q units When a company chooses the reorder point R it must take into account the effects of run ning out of inventory If the company believes shortages are very expensive or undesirable it should choose a relatively large value of R This leads to a relatively large level of safety stock the expected amount of inventory left overthe cushionby the time the next order arrives On the other hand if shortages are not con sidered too expensive or undesirable the company can afford to use a lower value of R with a smaller resulting level of safety stock As in the newsvendor model we show how to determine an appropriate tradeoff between leftovers and shortages To specify an RQ policy we must also deter mine the appropriate order quantity Q It turns out that the choices of R and Q can be made almost independently The choice of R depends largely on how shortage costs or customer service are mea sured whereas the choice of Q depends mostly on the same cost factors considered in the determinis tic EOQ models Specifically the company wants to order enough to avoid frequent fixed ordering costs but as little as possible to avoid excessive holding costs Fortunately it is possible to develop a Solver model that determines Q and R simultane ously as illustrated in the following example 740 Chapter 12 Inventory Models FUNDAMENTAL INSIGHT Ordering with Uncertain Demand When future demand is uncer tain and can be f ore casted only approximately a company has to deter mine the tradeoff betw een ordering too much and having excess inventory costs and ordering too little and having shortage costs and a lo w service level This often results in safety stock the extra inventory that is used as a cushion in case demand during lead time is gr eater than expected In todays computer ized worldwhere companies share more information about inventories and demands with their suppliers better f orecasting and cooperation betw een the members of the supply chain can often reduce safety stock and the resulting cost E X A M P L E 127 ORDERING CAMERAS WITH UNCERTAIN DEMAND AT MACHEYS I n Example 121 we considered Macheys department store which sells on average 1200 cameras per year The store pays a setup cost of 125 per order and the holding cost is 8 per camera per year It takes one week for an order to arrive after it is placed In that example the optimal order quantity Q was found to be 194 cameras Now we assume that the annual demand is normally distributed with mean 1200 and standard deviation 70 Macheys wants to know when to order and how many cameras to order at each ordering opportunity Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 127 Supply Chain Models 755 E X A M P L E 129 PLANNING PRODUCTION OF BLOUSES AT SHIRTTAILS S hirtTails is a clothing manufacturer that operates its own chain of discount retail stores At the beginning of November 2011 ShirtTails is trying to plan its production of a new blouse that is worn primarily in the warmer months Based on production constraints from other products the company knows it has two opportunities to produce this blousein November 2011 and later in April 2012 The production capacity for this blouse is 1200 in November In April the capacity will increase to 2500 By April demand for the blouses produced in November will be known Using this information ShirtTails will then be able to plan its production in April The unit cost of producing a blouse is 12 and the selling price will be 14 These remain constant There is a 1 holding cost per blouse still in inventory after the preApril demand By November 2012 any remaining blouses in inventory will be sold at a mark down price of 4 This is because ShirtTails plans to introduce a new blouse the next year Demand for the blouses before April is not known with any certainty but ShirtTails believes it should be somewhere between 100 and 1000 After April the demand for blouses is expected to be anywhere from 3 to 75 times as large as the demand before April What production plan should the company use to maximize the expected profit from these blouses Objective To develop an optimization model that specifies production quantities of blouses in two time periods where the second production quantity can be based on demand information from the first period Solution You first need to recognize that a production plan is really a contingency plan This means that the company will determine a production quantity in November but it will not commit to a production quantity in April until after it observes the preApril demand In other words the contingency plan will specify a single production quantity in November and a production quantity in April for each preApril demand that might be observed Before solving anything numerically specific probability distributions of demand are required We will eventually try several but we initially assume unimodal symmetric discrete distributionsessentially the discrete analog of a normal distribution where the probabilities increase and then decrease We spell out the details shortly Finally we point out explicitly that this is not a simulation model despite the uncer tainty The plan is to calculate an expected profit for any given production plan and then use Evolutionary Solver as in Chapter 8 to maximize this expected profit DEVELOPING THE SPREADSHEET MODEL The completed model appears in Figures 1222 and 1223 See the file Fashion Productionxlsx It can be developed with the following steps 1 Inputs Enter the inputs in the blue ranges in Figure 1222 These include the given costs the capacities and the probability distributions we are initially assuming Regarding these distributions rows 13 and 14 indicate the distribution of preApril demand which can be any value from 100 to 1000 in increments of 100 Note that the probabilities increase gradually and then decreasethe unimodal property The table in rows 18 to 27 then specifies the distribution of postApril demand given the preApril demand For example if preApril demand is 400 in column E then postApril demand will be one of the values in the range E18E27 with the corresponding probabilities in column L which Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in cell B51 and copy it to the range B51K60 These cells show the markdown revenue for each demand combination Then calculate the expected markdown revenues given pre April demand by entering the formula SUMPRODUCTB51B60L18L27 in cell K61 and copying it across row 61 8 Expected revenues costs and profits At this point rows 35 37 49 and 61 contain revenues and costs for each possible value of preApril demand To get overall expected values you must SUMPRODUCT these with the row of preApril demand probabilities For example calculate the overall expected sales revenue in cell B66 with the formula SUMPRODUCTB49K49B14K14 The others are calculated similarly and the expected profit is the sum of expected revenues minus the sum of expected costs These are the values ShirtTails can expect as it looks ahead from November 2011that is before any demands have been observed USING SOLVER Solver should be set up as shown in Figure 1224 The objective cell is the expected profit the changing cells are the production quantities and they must be constrained to be within capacity Of course the production quantities must also be nonnegative Note that Evolutionary Solver is used because of the various MAX and MIN functions in the cell formulas Recall that the other Solvers have problems with such functions whereas Evolutionary Solver handles them nicely 758 Chapter 12 Inventory Models Figure 1224 Solver Dialog Box for the Fashion Model Discussion of the Solution The solution in Figure 1223 is fairly intuitive ShirtTails could produce up to 1200 units in November but it holds production to 600 because it is not sure whether these blouses will be popular After observing the preApril demand the company then produces more or less depending on the success of the blouses to that point If preApril demand is its mini mum value 100 then there are already 500 of these dogs left in inventory and the company does not produce any more But if preApril demand is sufficiently large the company recognizes that it has a hot item and produces to capacity in April Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it first of these Ushaped because the probabilities are large on either end but decrease in the middle This is reasonable if ShirtTails believes the blouse will be either very popular or very unpopular The second distribution in Figure 1226 has equal probabilities for all demand values This equally likely case is reasonable if ShirtTails has no idea how popular the blouses will be In comparison with the unimodal scenario there are some clear differ ences between the optimal solutions The equally likely scenario calls for less production in November generally less production in April and a somewhat lower expected profit This pattern is even more evident with the Ushaped scenario which has the lowest pro duction levels and the lowest expected profit These differences make intuitive sense With a unimodal distribution the company has the most assurance of what demand is likely to be and it can plan accordingly Planning is more difficult with the equally likely no idea distribution and it is even more difficult with the Ushaped distribution With this latter distribution the company isnt sure whether to produce a lot in case demand is strong or to produce very little in case demand is weak It stands to lose no matter what it does Of course the company cannot simply choose one dis tribution over another because one produces a larger expected profit It should choose the distribution most in line with its realistic assessment of future demand Excel Tip Scenario Manager As the text box in Figure 1222 indicates we used Excels Scenario feature to save each of the three scenarios under the names Unimodal Ushaped and Equally Likely This feature is useful if you want to store several named scenarios in a single workbook To do so enter key input values in your spreadsheet that constitute a scenario including the probabilities and the values in the red cells after running Solver Then use the Scenario Manager under WhatIf Analysis on the Data ribbon This gives you a chance to name a scenario and des ignate the cells unfortunately called Changing Cells b ut not at all the same concept as Solvers Changing Cells that include the key inputs If you ever want to view this scenario later on just use the Scenario Mana ger select the scenario you want fr om the list of sce narios and click on View The following example illustrates inventory management in a multiechelon setting that is in a setting where inventory is held at multiple locations Although many versions of this general problem exist in both academic articles and in real companies we illustrate the sit uation where a central warehouse holds and distributes inventory to several retailers each of which has uncertain demand The problem is complicated as it usually is in real situations by ordering lead times and the way inventory should be managed is far from obvious 760 Chapter 12 Inventory Models E X A M P L E 1210 MANAGING INVENTORY AT LEE SUPPLY Lee Supply has three retail stores that are supplied by a central warehouse For this exam ple the focus is on a single product sold at the stores At the beginning of each week each store requests a quantity of this product from the warehouse and such shipments arrive at the beginning of the following week oneweek lead time Similarly at the beginning of each week the warehouse orders a quantity of this product from an overseas manufac turer and such shipments arrive in three weeks threeweek lead time Weekly demands at each retailer are independent normally distributed random variables and any demands that cannot be met from onhand inventory are backordered and satisfied as soon as pos sible The means and standard deviations of demand can vary across retailers but they are constant through time All ordering policies are characterized by an orderupto quantity Q where each retailer and the warehouse can have a different Q For a retailer this works as follows At the beginning of a week the retailer checks the beginning inventory after Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in cell H34 copy it to cells I34 and J34 and copy these down The reason for the MAX is that if the retailers inventory position after subtracting the expected demand is already at or above the orderupto quantity no order is placed 5 Amounts sent to retailers The total requested by the retailers is compared to the on hand inventory at the warehouse If there is enough the requests are satisfied If not they are satisfied proportionally To implement this logic enter the formula IFSUMH34J34E34H34ROUNDH34SUMH34J34E340 in cell K34 copy it to cells L34 and M34 and copy these down The ROUND function ensures that integer amounts are sent 6 Warehouse order quantity Given the orderupto policy by the warehouse enter the formula MAXE22SUME34G34SUMK34M340 in cell N34 and copy it down Again the MAX is for the case where the warehouse net inventory including pipeline inventory but subtracting shipments to retailers is already at or above the orderupto quantity Then no order is placed 7 Retailer demand To generate the normally distributed demands enter the formula ROUNDNORMINVRAND B17B180 in cell O34 copy it to cells P34 and Q34 and copy these down Again the ROUND func tion ensures integer demands Also the Excel way of generating normally distributed random values rather than RISKNORMAL has been used Otherwise the 52week simu lation would use more RISK functions than are allowed in the academic version 8 Ending r etailer in ventories To calculate the ending retailer inventories before warehouse requests at the beginning of this week arrive enter the formula B34O34 in cell R34 copy it to S34 and T34 and copy these down Note that a negative value in any of these cells indicates the amount backordered 9 Backorders Finding the backorders for a retailer in a given week is tricky If the retailers beginning inventory is nonnegative and demand is greater than this then the excess demand is backordered However if the retailers beginning inventory is negative indicating that it cannot completely satisfy the previous weeks backorders then all demand this week will be backordered To implement this logic enter the formula IFB340O34IFO34B34O34B340 in cell U34 copy it to V34 and W34 and copy these down Then find the fill rate over all 52 weeks with the formula 1SUMU34W85SUMO34Q85 in cell B25 The ratio of sums is total backorders divided by total demands so one minus this is the fraction of demand satisfied on time 10 System inventory System inventory in any week is defined as the sum of beginning inventories at the retailers plus warehouse inventory on hand or in the pipeline so enter the formula SUMB34G34 in cell X34 and copy it down Then average these values in cell B26 127 Supply Chain Models 763 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 11 RISK summary measures Designate cells B25 and B26 as RISK output cells Then summarize then in cells B29 and B30 with the formulas RISKTARGETB25095 and RISKMEANB26 The first of these finds the probability that the overall fill rate is less than or equal to a tar get value 95 The orderupto quantities should be chosen to make this rather small we chose 02 while making the mean in cell B30 as small as possible USING RISK AND RISK OPTIMIZER You now have two options First you can run RISK for 1000 iterations say with any chosen values for the orderupto quantities You will then see the key outputs in rows 25 26 29 and 30 However it is fairly difficult to guess orderupto quantities that achieve a given probability such as 02 in cell B29 and make the mean average system inventory in cell B30 small Therefore your second option is to use RISKOptimizer a companion Palisade addin to RISK We tried this setting up RISKOptimizer to minimize the mean in cell B30 while constraining the probability in cell B29 to be less than or equal to 02 As with Solver RISKOptimizer provides a dialog box for setting up the optimization model see Figure 1229 but it has more options because of the simulation context We will not pursue the details here except to say that RISKOptimizer is a very powerful tool in con junction with RISK simulations and that it leads to the orderupto quantities in Figure 1228 Note that total weekly expected demand at the retailers is 450 but to ensure that 95 of the demand is met on time with fairly high probability 08 the total system inventory has to average around 595 You can probably guess the reasons uncertainty in demand complicated by lead times in deliveries The result is the large level of safety stock held in this modeland by most companies 764 Chapter 12 Inventory Models Figure 1229 RISKOptimizer Model Definition Dialog Box Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 128 CONCLUSION We have examined a variety of inventoryordering models in this chapter The general theme is the balance companies try to find between competing costs If they order frequent small quantities they keep inventory low but they incur large fixed ordering costs In con trast if they order infrequent large quantities they minimize ordering costs but they incur large holding costs The basic EOQ model and its many variations are able to achieve the right balance between these costs These EOQ models are relatively straightforward and find many uses in todays business world However as we introduce complications that real companies face such as multiple products uncertain demand uncertain delivery lead times and complex supply chain considerations the models can become extremely difficult In this case simulation is often the best alternative sometimes it is the only alternative Summary of Key Management Science Terms Term Explanation Page Deterministic inventory Model where all inputs including demands and lead times are 715 model assumed to be known Probabilistic inventory Model where demands and possibly other inputs are uncertain 715 model and must be estimated with probability distributions Lead time The time between placement of an order and receiving it 716 Setup cost or ordering cost Fixed cost of placing an order independent of the size of the order 716 Continuous review model Model where order can be placed at any point in time 717 Reorder point Inventory level that triggers an order to replenish stock 717 Periodic review model Model where order is placed only at discrete points in time 717 such as the beginning of a week Holding or carrying cost Cost of holding inventory could be cost of physical storage 718 or cost of money tied up in inventory Shortage or penalty cost Cost of not having enough on hand to meet customer demand 718 could be a dollar cost or a loss of goodwill Economic order quantity Commonly used models that find the order quantity that trades 718 EOQ models off setup cost versus holding cost plus possibly other costs typified by the famous square root formula continued 128 Conclusion 765 P R O B L E M SkillBuilding Problem 25 The problem in Example 129 assumes that the heaviest demand occurs in the second postApril phase of sell ing It also assumes that capacity is higher in the second production opportunity than in the first Suppose the sit uation is reversed so that the higher capacity and most of the demand occur in the first phase Make some rea sonable assumptions for the resulting input parameters and then solve for the optimal production plan Do you get qualitatively different results Which situation would you rather face if you were ShirtTails 26 The multiechelon inventory model in Example 1210 requires about 595 items of onhand or pipeline inven tory on average to satisfy the fill rate constraint even though the mean total demand per week is only 450 See how this changes as the amount of uncertainty decreases Specifically make the standard deviations of demand smaller and then run RISKOptimizer with exactly the same settings You can make the standard deviations as small as you like Does the mean total system inventory get closer to 450 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C H A P T E R 13 Queueing Models REDUCING WORKINPROGRES S LEVEL S AT WOODWARD AIRCRAFT ENGINE SY STEMS T he previous chapter was all about inventory management where companies try to achieve the correct balance between holding too much inventory and not holding enough inventory to meet demands A type of inventory that is particularly important in the manufacturing industry is called workinprocess WIP inventory As its name implies this is inventory that is partway through the manufacturing process and is not yet a finished good Manufacturing companies try to keep WIP low for reasons of space and financial concerns but they need a certain amount of WIP to keep their processes running smoothly Srinivasan et al 2003 discuss a study they per formed at Woodward Aircraft Engine Systems to achieve appropriate levels of WIP Woodward is a leading producer of fuelcontrol systems and com ponents for aircraft and industrial engines and turbines With headquarters in Fort Collins Colorado Woodward serves a global market and has about 5500 employees Their Rockford Illinois plant manufactures a large variety of products at low volumes some as low as 100 per year As these products are manufactured they flow through cells groups of machines that perform similar operations and the various products require different routings through these cells depending on their specifications The company knows or forecasts its demands for the various products so it knows how many of each product it needs to manufacture per time period the throughput to meet demands The problem is to determine the amount of WIP required to achieve the desired throughputs Tonis ValingUsed under license from Shutterstockcom 773 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 774 Chapter 13 Queueing Models The authors model the manufacturing system as a closed queueing network CQN A queueing network is a sequence of cells or machines that partially completed products must pass through as they are being manufactured into finished products Products typically form a queue in front of the machines on their routings and congestion is very possible especially when certain machines are bottlenecks A closed queueing network means that there are a constant number of partially completed products of a given type in the network at all timesThis type of model is often used when a new product of a given type is introduced into the network as soon as a part of that type finishes and leaves the network Researchers have done much analytical work in the area of queueing networks and various approximations exist for calculating performance measures of CQNs At Woodward there are essentially two decision variables for any given product type The first is the batch size the number of parts on a pallet A given batch goes through the manufacturing process that is through its routing of cells and machines as a unit At any machine along the route there can be a setup time and a processing time per unit Therefore larger batch sizes are sometimes beneficial for reducing setups The second decision variable is the number of batches in the system at any point in time Because the overall system is modeled as a CQN this number of batches for any given product type is constant Together these two decision variables determine the amount of WIP in the system at all times The problem is to adjust these two decision variables for all product types so that the throughputs of all products match the demands for them as closely as possible The authors developed an approximate algorithm using results from the vast queueing literature to do this Then they implemented this algo rithm in Excel with a userfriendly interface so that Woodward employees could use it easily to answer various whatif questions Although the details of the algorithm are quite complex they rely on a very basic formula called Littles formula which is discussed in this chapter Littles formula states that the expected number of parts in a system is equal to the arrival rate of parts to the system multiplied by the average time a part spends in the system Littles formula can be applied in an amazing variety of situations the only trick is to see how it applies In Woodwards situation the number of parts in the system is fixed because of the CQN assumption it is the number of pallets of a given product type in the system at all times The arrival rate of parts to the system is the throughput of a given product type The reasoning is that the rate at which products leave the system the throughput rate must equal the rate at which new products of this type enter the system Finally the average time a part spends in the system is known in manufacturing as the cycle time the time it takes to manufacture a typical product So Littles law relates cycle time to throughput and the number of pallets to use The authors algorithm and spreadsheet implementation have helped Woodward immensely by enabling the company to reduce its WIP inventory from about three weeks of inventory to less than one week of inventory As Director of Manufacturing Steven J Ebbing states The spreadsheet software tool presented in this paper has enabled a smooth flow of products through the various operations in the cells at Woodward with significant reduction in WIP levels The whatif analysis is invaluable for setting WIP levels for different products as well as for individual machines 131 INTRODUCTION A basic fact of life is that we all spend a great deal of time waiting in lines queues We wait in line at a bank at a supermarket at a fastfood restaurant at a stoplight and so on Actually people are not the only entities that wait in queues Televisions at a television repair shop other than the ones being repaired are essentially waiting in line to be Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it repaired Also when messages are sent through a computer network they often must wait in a queue before being processed Mathematically it does not really matter whether the entities waiting are people or televisions or computer messages The same type of analysis applies to all of these The purpose of such an analysis is generally twofold The first objective is to examine an exist ing system to quantify its operating characteristics For example if a fastfood restaurant currently employs 12 people in various jobs the manager might be interested in determin ing the amount of time a typical customer must wait in line or how many customers are typically waiting in line The second objective is to learn how to make a system better The manager might find for example that the fastfood restaurant would do better from an economic standpoint by employing only 10 workers and deploying them in a different manner The first objective analyzing the characteristics of a given system is difficult from a mathematical point of view The two basic modeling approaches are analytical and simulation The analytical approach searches for mathematical formulas that describe the operating characteristics of the system usually in steady state The mathematical mod els are typically too complex to solve unless simplifying and sometimes unrealistic assumptions are made For example at a supermarket customers typically join one of several lines probably the shortest possibly switch lines if they see that another line is moving faster and eventually get served by one of the checkout people Although this behavior is commonand is simple to describe in wordsit is very difficult to analyze analytically With the second approach simulation much more complex systems can be analyzed without making many simplifying assumptions However the drawback to queueing simu lation is that it usually requires specialized software packages or trained computer pro grammers to implement In this chapter we employ both the analytical approach and simulation For the for mer we discuss several wellknown queueing models that describe somebut certainly not allqueueing situations in the real world These models illustrate how to calculate such operating characteristics as the average waiting time per customer the average num ber of customers in line and the fraction of time servers are busy These analytical models generally require simplifying assumptions and even then they can be difficult to under stand Therefore we also discuss queueing simulations Unfortunately queueing simula tions are not nearly as straightforward as the simulations discussed in previous chapters It is necessary to generate random times between customer arrivals and random service times and then play out the events This playing out of events is far from easy in a spreadsheet We provide only a taste of what can be doneand show why commercial software pack ages are usually used instead of spreadsheets The second objective in many queueing studies is optimization where the goal is to find the best system Of course to find the best system each of several competing sys tems must be analyzed either analytically or by simulation But beyond this difficult choices must be made For example if the fastfood restaurant wants to decide how many employees to hire for various times of day it must analyze the tradeoff between more employees better service higher wages and fewer employees worse service lower wages The cost of extra employees is fairly easy to quantifythe marginal cost of one extra employee is the wage rate However estimating the cost of making a customer wait an extra two minutes in line for instance is difficult In terms of immediate outofpocket costs it costs the restaurant nothing However it can have longrange implications fewer customers will bring their business to this restaurant To find the optimal number of employees the restaurant must estimate the dollar cost of having customers wait in line Only by estimating this cost can it make an economic choice between the cost of waiting and the cost of more efficient service 131 Introduction 775 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The examples in this chapter highlight these two objectives We show how to find important characteristics such as expected waiting times of specific systems and to a lesser extent we illustrate how to search for economically optimal systems This chapter is very different from earlier chapters because of the nature of queueing systems The models in previous chapters could almost always be developed from first principles By using relatively simple formulas involving functions such as SUM SUMPRODUCT IF and so on it was fairly straightforward to convert inputs into outputs This is no longer possible with queueing models The inputs are typically mean customer arrival rates and mean service times The required outputs are typically mean waiting times in queues mean queue lengths the fraction of time servers are busy and possibly others Deriving the formulas that relate the inputs to the outputs is mathematically very difficult well beyond the level of this book Therefore many times in this chapter you will have to take our word for it Nevertheless the models we illustrate are very valuable for the impor tant insights they provide 776 Chapter 13 Queueing Models 132 ELEMENTS OF QUEUEING MODELS We begin by listing some of the features of queueing systems that distinguish one system from another Almost all queueing systems are alike in that customers enter a system pos sibly wait in one or more queues get served and then depart1 This general description of a queueing systemcustomers entering waiting in line and being servedhardly sug gests the variety of queueing systems that exist We now discuss some of the key features and their variations Characteristics of Arrivals First the customer arrival process must be specified This includes the timing of arrivals as well as the types of arrivals Regarding timing specifying the probability distribution of interarrival times the times between successive customer arrivals is most common These interarrival times might be knownthat is nonrandom For example the arrivals at some doctors offices are scheduled fairly precisely Much more commonly however interarrival times are random with a probability distribution In real applications this prob ability distribution must be estimated from observed customer arrival times Also this dis tribution can vary through time For example the rate of arrivals to McDonalds is certainly higher around noon than in the middle of the afternoon Regarding the types of arrivals there are at least two issues First customers can arrive one at a time or in batchescarloads for example The simplest system is when customers arrive one at a time an assumption made in all of the models in this chapter Second customers can all be essentially alike or they can be separated into priority classes At a computer center for example certain jobs might receive higher priority and run first whereas the lowerpriority jobs might be sent to the back of the line and run only after midnight Throughout this chapter all customers are assumed to have the same priority Another issue is whether or how long customers will wait in line A customer might arrive to the system see that too many customers are waiting in line and decide not to The formulas that relate queueing inputs to queueing outputs are difficult to derive mathematicallyA few of these formulas are presented but they are not derived Interarrival times are the times between successive customer arrivals We assume customers arrive one at a time and all have the same priority 1From here on we refer to the entities requesting service as customers regardless of whether they are actually people Also we refer to servers performing service on these customers regardless of the type of work being per formed and whether the servers are people machines or other types of technology Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it enter the system at all This is called balking A variation of balking occurs when the choice is made by the system not the customer In this case we assume there is a waiting room size so that if the number of customers in the system equals the waiting room size newly arriving customers are not allowed to enter the system We call this a limited waiting room system Another type of behavior called reneging occurs when a customer already in line becomes impatient and leaves the system before starting service Systems with balking and reneging are difficult to analyze so no such systems are considered in this chapter However we do discuss limited waiting room systems Service Discipline When customers enter the system they might have to wait in line until a server becomes available In this case the service discipline must be specified The service discipline is the rule that states which customer from all who are waiting goes into service next The most common service discipline is firstcomefirstserved FCFS where customers are served in the order of their arrival All of the models in this chapter use the FCFS discipline However other service disciplines are possible including serviceinrandomorder SRO lastcomefirstserved LCFS and various priority disciplines if there are customer classes with different priorities For example a type of priority discipline used in some manufacturing plants is called the shortestprocessingtime SPT discipline In this case the jobs that are waiting to be processed are ranked according to their eventual processing service times which are assumed to be known Then the job with the shortest processing time is processed next One other aspect of the waiting process is whether there is a single line or multiple lines For example most banks now have a single line An arriving customer joins the end of the line When any teller finishes service the customer at the head of the line goes to that teller In contrast most supermarkets have multiple lines When a customer goes to a checkout counter she must choose which of several lines to enter Presumably she will choose the shortest line but she might use other criteria in her decision After she joins a line she might decide to move to another line that seems to be moving faster Service Characteristics In the simplest systems each customer is served by exactly one server even when the system contains multiple servers For example when you enter a bank you are eventually served by a single teller even though several tellers are working The service times typically vary in some random manner although constant nonrandom service times are sometimes possible When service times are random the probability distribution of a typi cal service time must be specified This probability distribution can be the same for all cus tomers and servers or it can depend on the server andor the customer As with interarrival times service time distributions must typically be estimated from service time data in real applications In a situation like the typical bank where customers join a single line and are then served by the first available teller the servers tellers are said to be in parallel see Figure 131 A different type of service process is found in many manufacturing settings For example various types of parts the customers enter a system with several types of machines the servers Each part type then follows a certain machine routing such as machine 1 then machine 4 and then machine 2 Each machine has its own service time distribution and a typical part might have to wait in line behind any or all of the machines on its routing This type of system is called a queueing network The simplest type of queueing network is a series system where all parts go through the machines in numerical 132 Elements of Queueing Models 777 We always assume a FCFS discipline Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it order first machine 1 then machine 2 then machine 3 and so on see Figure 132 We examine mostly parallel systems in this chapter However we discuss the simulation of a series system toward the end of the chapter 778 Chapter 13 Queueing Models Customers in line Servers Figure 131 Queueing System with Servers in Parallel Customers waiting in line Servers Figure 132 Queueing System with Servers in Series ShortRun versus SteadyState Behavior If you run a fastfood restaurant you are particularly interested in the queueing behavior during your peak lunchtime period The customer arrival rate during this period increases sharply and you probably employ more workers to meet the increased customer load In this case your primary interest is in the shortrun behavior of the systemthe next hour or two Unfortunately shortrun behavior is the most difficult to analyze at least with analyti cal models Although we show in section 136 that shortrun behavior can be approxi mated analytically analysts usually resort to simulation to understand what happens in the short run But where is the line drawn between the short run and the long run The answer depends on how long the effects of initial conditions persist In the restaurant example the initial conditions are determined by the number of customers already in line at the begin ning of the lunch periodsay at 1130 Suppose the restaurant manager is interested in the average number of customers waiting in line over a twohour peak period The question then is how much this average is affected by the number of customers in line at 1130 Specifically do the effects of the initial conditions get washed out in a period as long as two hours Ultimately the only way to answer this question is with empirical evidence A lunch period starting with no people in line at 1130 might be compared to one where 10 people are already in line at 1130 If the average levels of congestion over the entire twohour lunch period are approximately the same in each case then the initial conditions at 1130 evidently make little difference and a longrun analysis is permitted However if the lunch period that starts with many people in line is never able to overcome this initial loadthat is it tends to stay crowdedthen the initial conditions are important and a shortrun analysis is required Analytical models are best suited for studying longrun behavior This type of analysis is called steadystate analysis and is the focus of much of the chapter One requirement for steadystate analysis is that the parameters of the system remain constant for the entire time period In particular the arrival rate must remain constant In the restaurant example Steadystate analysis is relevant for the long run but the long run can sometimes be as short as an hour or two Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the steadystate results for the period from 100 to 200 indicate very short queue lengths whereas the transient results indicate much larger queue lengths The reason is very sim ple The steadystate results fail to account for the customers who are still in line at 100 These customers who are left over from the rush the hour before are the reason the system doesnt converge to steady state during the next hour Therefore the steadystate results give the manager an overly optimistic picture of the hour from 100 to 200 In contrast the transient results take this leftover congestion into account so they give the manager a much more realistic view of this last hour Keep in mind that this approach is not simulation No random numbers are involved and nothing will change if you press the F9 key However because we calculate the proba bility distributions only on every fivesecond interval the results are only approximately correct We could make them more exact by using a onesecond interval say but this would require five times as many calculations and rows of output Because not too much can happen in a fivesecond interval this time interval should yield sufficiently accurate results in most situations 814 Chapter 13 Queueing Models 10 000 12000 14000 16000 18000 20000 Expected queue length 0000 2000 4000 6000 8000 10000 110000 AM 110835 AM 111710 AM 112545 AM 113420 AM 114255 AM 115130 AM 120005 PM 120840 PM 121715 PM 122550 PM 123425 PM 124300 PM 125135 PM 10010 PM 10845 PM 11720 PM 12555 PM 13430 PM 14305 PM 15140 PM ExpQ LowerQ UpperQ SSExpQ Figure 1321 Chart of Expected Line Length P R O B L E M S 38 In the lunchtime rush example we assumed that the system starts empty and idle at 11 AM Assume now that the restaurant opens earlier than 11 AM but we are still interested only in the period from 11 AM to 2 PM How does the initial number of customers present at 11 AM affect the results Run the model six times varying the initial number of customers from 0 to 10 in increments of 2 You will need to run the macro for each of these Write a short report on your findings 39 In the lunchtime rush example the arrival rate changed fairly gradually throughout the period of interest Assume now that the arrival rate first increases and then decreases in a more abrupt manner Specifically replace the arrival rates in the example by the following 15 20 70 85 30 and 20 Note that the sum of these rates is the same as the sum of the rates in the example so that we expect the same total num ber of arrivals but now they are more concentrated in Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 137 QUEUEING SIMULATION MODELS A popular alternative to using the analytical models from the previous two sections is to develop queueing simulations There are several advantages to using simulation Probably the most important advantage is that you are not restricted to the assumptions required by the standard analytical queueing models These models typically require that interarrival times and service times are exponentially distributed customers wait in a single queue and are served in FCFS fashion all servers are identical in terms of their service time dis tributions there are no customer types with higher priority than others and so on8 When you use simulation anything goes If you want nonexponential service times they are easy to build in If you want customers to wait in several lines one behind each server and even allow them to switch queues as they might in a supermarket simulation can handle it If you want higherpriority customers to be able to bump lowerpriority cus tomers out of service this is no problem with simulation Just about any queueing situa tion can be simulated A second advantage of queueing simulation is that you get to see the action through time Simulation outputs typically include not only summary measures such as the average queue length for some period of time but they can also include time series graphs of important quantities such as the number of servers busy or the number of customers wait ing in line In this way you can see how queues build from time to time In addition you can run a simulation many times each time using different random numbers to see how one day might differ from another The downside of queueing simulation is that it has traditionally required a clever computer programmer a specialized software package or both Generating all of the ran dom quantities interarrival times and service times say required by a simulation is easy The difficult part is essentially a bookkeeping problem Imagine that you are given a list of customer arrival times and their corresponding service times and you must then play out the events as they would then occur through time Say customer 17 arrives at 947 sees that four customers are ahead of her in line and all three of the servers in the system are currently busy with customers How do you know when customer 17 will enter ser vice and with which server This is the biggest challenge in a queueing simulation keeping track of the state of the system as events occur through time Special queueing software packages are available to do all of the bookkeeping for you but this software is often expensive and far from trivial to master Therefore some people write their own pro grams in C Visual Basic or some other language to keep track of the events Unfortunately even good programmers sometimes struggle when writing queueing simulations There are 137 Queueing Simulation Models 815 the noon to 1 PM hour Compare the results with these arrival rates to the results in the example Write a short report on your findings SkillExtending Problem 40 Using the arrival rates from the lunchtime rush exam ple it seems sensible to vary the number of servers so that more servers work during the busy hours In par ticular suppose management wants to have an average of three servers working in parallel in any halfhour period but the number working can vary across peri ods Also assume that each server has a service rate of 16 customers per hour Experiment with ways to deploy the servers assuming that at least one server must be working each halfhour period For example at one extreme you could have three servers working each halfhour period At the other extreme you could have a single server working all but one of the half hour periods and 13 servers working during the other halfhour period Defend the deployment you think works best in a brief report 8There are analytical models for many nonstandard queueing systems but they are mathematically too complex for most users to understand Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 822 Chapter 13 Queueing Models E X A M P L E 1310 QUEUEING FOR HELP AT HYTEX H yTex is a software company that offers technical support for its customers over the phone The demand for help is fairly constant throughout the day with calls arriving at a rate of approximately 10 per minute HyTex keeps 35 technical support lines open at all times and it takes 35 minutes on average to answer a customers question Customers who call when all technical support people are busy face two possible situations If there are fewer than 20 customers already on hold the phone version of waiting in line then a new caller is also put on hold But if 20 customers are already on hold a new caller gets a busy signal and must hang up The service timesthe times to answer customers questionsare highly variable HyTex wants to know how much it is suffering because of this variability Objective To use simulation to analyze the affect of the shape of the service time distri bution on customer waiting times WHERE DO THE NUMBERS COME FROM These inputs are estimated from the extensive call data available However a subtle issue concerns the arrival rate of 10 per minute Estimating the arrival rate of all calls is not easy because of the difficulty associated with tracking calls that receive a busy signal and are therefore lost Solution This example is important because it illustrates how we can use a simulation model as a tool to study system behavior with various input parameters Selection of Inputs If the service times are highly variable a histogram of them might resemble an exponential distributionthat is a lot of short calls but a few really long ones Therefore we first sim ulate the system with exponential service times The arrival rate is 10 the mean service time is 35 the number of servers is 35 and the maximum allowable queue size is 20 With these parameters we used a warmup period of 1000 minutes and a runtime period of 2000 minutes for each simulation you can think of this as several days strung together and we made five runs with different random number seeds We then changed the service time distribution to a gamma distribution with mean 35 and standard deviation 28 This distribution has a squared coefficient of variation 064 so it is not as variable as the expo nential distribution which has squared coefficient of variation 1 Finally we changed the service time distribution to be constant with value 35 For both the gamma and constant distributions we made five runs using the same seeds as in the exponential runs If you want to mimic our results you should use the seeds 111 222 333 444 and 555 Discussion of the Results Selected results appear in Table 134 For each simulation run two quantities are listed the average time in queue for the customers who did not receive busy signals and the fraction of callers who received busy signals and were therefore lost If you look only at the aver age times in queue the results sometimes go in the opposite direction from what was pre dicted The most variable distribution the exponential sometimes has the smallest times whereas the least variable distribution the constant always has the largest times However Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it between arrivals are exponentially distributed Similarly the processing service times for the different stations can differ and each can have either a constant nonrandom distribu tion or an exponential distribution The simulation starts in the empty and idle state there can be a warmup period where no statistics are collected and then the simulation runs for a prescribed number of minutes Guessing how this type of system might behave is very difficult In fact this is the whole purpose of the simulation It allows a manufacturer to analyze many whatif scenar ios without actually making changes to the physical system We illustrate how this might work in the following example 824 Chapter 13 Queueing Models E X A M P L E 1311 PROCESSING PARTS AT STREAMLINING T he Streamlining Company manufactures various types of automobile parts Its factory has several production lines all versions of the series system in Figure 1330 with varying numbers of stations and machines In an effort to improve operations the company wants to gain some insights into how average throughput times and other output measures are affected by various inputs The throughput time is the elapsed time from when a part enters the system until it finishes processing at all stations Specific questions of interest include the following Is it better to have a single fast machine at each station or multiple slower machines How much does the variability of the arrival process to station 1 affect outputs What about the variability of processing times at machines The company has experimented with 0 buffers and has found that the resulting block ing can be disastrous It now wants to create some buffers which entails a significant cost Where should it place the buffers Objective To use simulation to learn how the inputs to the system including the config uration of buffers affect such output measures as throughput times WHERE DO THE NUMBERS COME FROM The company should use reasonable inputs for the simulation based on historical observa tions However the whole point of the simulation is to use it as a tool to learn how outputs are affected by varying inputs Solution The simulation model in the file Series Simulationxlsm allows you to experiment as much as you like by changing inputs running the simulation and examining the outputs Buffers Buffers Arrivals Finished M1 M3 M21 M22 Station 1 Station 2 3station system with multiple machines at station 2 Station 3 Figure 1330 A Series System with Possible Blocking Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it down the line and the other should be placed about twothirds of the way down breaking the line into three approximately equal sections Similarly when there are three buffers they should be placed to break the line into four approximately equal sections The bottom section of Figure 1336 indicates the saturation effect of adding more buffers The company gets a lot from its money from the first few buffers but after the first few blocking becomes a minor problem and more buffers fail to make much of an improvement If buffers entail significant costs Streamlining must trade off these costs against lower average throughput times and possibly other considerations 830 Chapter 13 Queueing Models Improving Car Body Production at PSA Peugeot Citroen In 1998 the new CEO of PSA Peugeot Citroen the French carmaker decided to set ambi tious targets for growth innovation and profitability To meet these targets PSA decided to focus on the carbody shops the bottlenecks at its plants An RD team conducted a man agement science study of carbody production using a number of analytic tools including a simulation model of seriesparallel systems They used this simulation to analyze a number of different configurations of manufacturing stations and buffers in the manufacturing line and they were able to persuade PSA to implement the best of these configurations They estimate that their study contributed 130 million to the bottom line in 2001 alone with minimal capital investment and no compromise in quality P ADDITIONAL APPLICATIONS P R O B L E M S SkillBuilding Problems 41 The Smalltown Credit Union experiences its greatest congestion on paydays from 1130 AM until 100 PM During these rush periods customers arrive according to a Poisson process at rate 21 per minute The credit union employs 10 tellers for these rush periods and each takes 47 minutes to service a customer Customers who arrive to the credit union wait in a single queue if necessary unless 15 customers are already in the queue In this latter case arriving customers are too impatient to wait and they leave the system Simulate this system to find the average wait in queue for the customers who enter the average number in queue the percentage of time a typical teller is busy and the percentage of arrivals who do not enter the system Try this simulation under the following conditions and comment on your results For each condition make three separate runs using a different random number seed on each run a Try a warmup time of two hours Then try no warmup time Use exponentially distributed service times for each b Try exponentially distributed service times Then try gammadistributed service times where the standard deviation of a service time is 24 minutes Use a warmup period of one hour for each c Try 10 tellers as in the statement of the problem Then try 11 then 12 Use exponentially distributed service times and a warmup period of one hour for each d Why might the use of a long warmup time bias the results toward worse system behavior than would actually be experienced If you could ask the pro grammer of the simulation to provide another option concerning the warmup period what would it be Hint The real rush doesnt begin until 1130 42 How long does it take to reach steady state Use simu lation with the Multiserver Simulationxlsm file to experiment with the effect of warmup time and run time on the key outputs For each of the following assume a fiveserver system with a Poisson arrival rate of one per minute and gammadistributed service times with mean 40 minutes and standard deviation 31 minutes For each part make three separate runs using a different random number seed on each run a Use a warmup time of 0 and a run time of 30 minutes b Use a warmup time of 0 and a run time of 180 minutes c Use a warmup time of 120 minutes and a run time of 30 minutes d Use a warmup time of 120 minutes and a run time of 180 minutes Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 53 On average 50 customers arrive per hour at a small post office Interarrival times are exponentially distrib uted Each window can serve an average of 25 cus tomers per hour Service times are exponentially distributed It costs 25 per hour to open a window and the post office values the time a customer spends waiting in line at 15 per customer hour To minimize expected hourly costs how many postal windows should be opened 54 On average 300 customers arrive per hour at a huge branch of Bank 2 It takes an average of two minutes to serve each customer It costs 10 per hour to keep a teller window open and the bank estimates that it will lose 50 in future profits for each hour that a customer waits in line How many teller windows should Bank 2 open 55 Ships arrive at a port facility at an average rate of two ships every three days On average it takes a single crew one day to unload a ship Assume that interarrival and service times are exponential The shipping company owns the port facility as well as the ships using that facility The company estimates that it costs 1000 per day for each day that a ship spends in port The crew ser vicing the ships consists of 100 workers each of whom is paid an average of 30 per day A consultant has rec ommended that the shipping company hire an additional 40 workers and split the employees into two equalsize crews of 70 each This would give each crew an average unloading or loading time of 15 days Which crew arrangement would you recommend to the company 56 A printing shop receives an average of one order per day The average length of time required to complete an order is half a day At any given time the print shop can work on at most one job Interarrival times and service times are exponentially distributed a On average how many jobs are present in the print shop b On average how long will a person who places an order have to wait until it is finished c What is the probability that an order will begin work within two days of its arrival 57 On average 40 jobs arrive per day at a factory The time between arrivals of jobs is exponentially distrib uted The factory can process an average of 42 jobs per day and the time to process a job is exponentially distributed a On average how long does it take before a job is completed measured from the time the job arrives at the factory b What fraction of the time is the factory idle c What is the probability that work on a job will begin within two days of its arrival at the factory 58 At the Franklin Post Office patrons wait in a single line for the first open window On average 100 patrons enter the post office per hour and each 834 Chapter 13 Queueing Models window can serve an average of 45 patrons per hour The post office estimates a cost of 010 for each minute a patron waits in line and believes that it costs 20 per hour to keep a window open Interarrival times and service times are exponential a To minimize the total expected hourly cost how many windows should be open b If the post offices goal is to ensure that at most 5 of all patrons will spend more than five minutes in line how many windows should be open 59 The manager of a large group of employees must decide whether she needs another photocopying machine The cost of a machine is 40 per eighthour day regardless of whether the machine is in use On average four peo ple need to use the copying machine per hour Each per son uses the copier for an average of 10 minutes Interarrival times and copying times are exponentially distributed Employees are paid 8 per hour and we assume that a waiting cost is incurred when a worker is waiting in line or is using the copying machine How many copying machines should be rented 60 The Newcoat Painting Company has for some time been experiencing high demand for its automobile repainting service Because it has had to turn away business management is concerned that the limited space available to store cars awaiting painting has cost them in lost revenue A small vacant lot next to the painting facility has recently been made available for rental on a longterm basis at a cost of 10 per day Management believes that each lost customer costs 20 in profit Current demand is estimated to be 21 cars per day with exponential interarrival times including those turned away and the facility can service at an expo nential rate of 24 cars per day Cars are processed on a FCFS basis Waiting space is now limited to 9 cars but can be increased to 20 cars with the lease of the vacant lot Newcoat wants to determine whether the vacant lot should be leased Management also wants to know the expected daily lost profit due to turning away cus tomers if the lot is leased Only one car can be painted at a time Try using the Limited Queue Templatexlsm file for an analytical solution and the Multiserver Simulationxlsm file for a simulation solution 61 On average 90 patrons arrive per hour at a hotel lobby interarrival times are exponential waiting to check in At present there are five clerks and patrons wait in a single line for the first available clerk The average time for a clerk to service a patron is three minutes exponentially distributed Clerks earn 10 per hour and the hotel assesses a waiting time cost of 20 for each hour that a patron waits in line a Compute the expected cost per hour of the current system b The hotel is considering replacing one clerk with an Automatic Clerk Machine ACM Management estimates that 20 of all patrons will use an ACM Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 836 Chapter 13 Queueing Models C A S E T he Catalog Company is a mail and phoneorder company that sells generic brands of houseware items and clothing Approximately 95 of customer orders are received by phone the remaining 5 are received in the mail Phone orders are accepted at Catalog Companys tollfree 800 number 800SAVE NOW The number is available nine hours per day 8 AM to 5 PM five days a week Sarah Walters a recent graduate of Columbia Business School has just been hired by Catalog to improve its operations Sarah would like to impress her boss Ben Gleason the president of Catalog Company with some ideas that would quickly improve the companys bottom line After spending a week learning about Catalogs operations Sarah feels that a substantial impact can be made by a closer evaluation of the phone order system Currently Catalog employs a single fulltime operator to take orders over the phone Sarah won ders whether additional operators should be hired to take phone orders Ben feels that Sarahs time might be better spent studying the catalog mailing lists Ben reasons that the mailing lists are where customers are generated and improving the list will bring in more revenue And besides Ben says Catalogs phone operator Betty Wrangle seems to be doing nothing more than half of the time that I walk by Hiring more operators to do nothing will just waste more money Although Sarah knows the mailing lists are important she thinks that a study of the mailing lists will take far more time than a quick evaluation of the phone order system Forging ahead Sarah discovers the following information about the phone order system The phone operator Betty Wrangle is paid 9 per hour in wages and benefits The average cost to Catalog for a completed 800 number call is 150 With only one phone line any incoming calls that arrive when Betty is on the phone to another customer get a busy signal The cost of the phone line is 40 per monthThe phone company can immediately add up to four additional phone lines using the same 800 number each at a cost of 40 per month per line Catalogs phone system is such that it cannot be upgraded in the near future to allow incoming calls to be placed on hold The average profit on an order 131 CATALOG COMPANY PHONE ORDERS not including the cost of the operator or phone call is 40 of the amount of the order For example an order of 100 brings a profit of 40 to Catalog Sarah decided that additional information needed to be collected about the frequency of incoming calls the length of the calls and so on After talking to the phone company Sarah learned that she could borrow equipment for one day that could detect when a call was coming in even when Betty was on the phone The caller would still get a busy signal and be lost but Sarah would know that a call had been attempted Sarah collected almost nine hours of data the next day these data are presented in the file Catalog Ordersxlsx Sarah believes that most of the callers who receive a busy signal take their business elsewhere and are totally lost to Catalog Sarah does not feel that extending the hours of operation of the 800 number would be beneficial because the hours of operation are printed prominently in all of the catalogs The first call arrives 0036 hour into the day It takes Betty 0054 hour to process the call and record the order for 6521 worth of merchandise Callers 5 and 6 get busy signals when they call because Betty was still processing caller 4 Because calls 5 and 6 were lost no call length information was available and no orders were placed Data collection was stopped at call number 80 Questions Use the complete information in the file Catalog Ordersxlsxto answer the following questions 1 Approximately what fraction of the time is Betty idle Is Bens estimate correct 2 Approximately how many calls are lost in an average hour due to a busy signal 3 Use the data to estimate the average arrival rate of all attempted calls to Catalog Give an approxi mate 95 confidence interval for the estimate Plot a frequency histogram of interarrival times Does the distribution of interarrival times appear to be exponential Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Case 131 Catalog Company Phone Orders 837 4 Use the data to estimate the average service rate of all completed calls Give an approximate 95 confidence interval for the estimate Plot a frequency histogram of service times Does the service time distribution appear to be exponen tial Give an approximate 95 confidence inter val for the average revenue per call 5 Would you recommend that Catalog acquire additional phone lines and operators If so how many If not why not Justify your answer in enough detail so that Ben Gleason would be convinced of your recommendation Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 838 Chapter 13 Queueing Models C A S E P acific National Bank is a mediumsize bank with 21 branches in the San Francisco Bay Area Until very recently Pacific did not operate its own ATMs instead it relied on an outside vendor to operate them Ninety percent of the ATM customers obtained cash advances with nonPacific credit cards so the ATMs did little to directly improve Pacifics own banking business Operations Vice President Nancy Meisterhaus wants to change that by having Pacific offer a broader mix of banking services with its own machines tied into its own dataprocessing network The industry consensus is that the ATM appeals to customers in much the same way as the super market express line It minimizes the amount of wait ing But for Pacific the 24hour ATM would also have the broader appeal of providing essential banking services at all hours reaching a segment of the mar ket not currently served Historically customers who find standard banking hours inconvenient have been lost to Pacific so the ATM will increase the banks market share Besides attracting more customers and servicing existing customers better the ATM operation should offer substantial cost advantages Fewer human tellers would be required for the same volume of transactions as before The per transaction cost of the machine which does need some human attention for restocking and maintenance should be substan tially less But even if that were not so its 24hour readiness would be extremely expensive to duplicate with human tellers who would have to be given extra protection for dangerous latenight work MsMeisterhaus selected theWalnut Creek office as the test branch for a captive ATMCustomers from that branch were recruited to sign up for a Pacific ATM card All residents within the neighboring ZIP codes were offered an incentive to open free checking accounts at Pacific when they also signed up for the card After a critical mass ofATM card holders was establishedbut before the banking ATM was installedstatistics were kept The arrival times in Table 135 were determined for various times of the week 132 PACIFIC NATIONAL BANK10 The bank opens at 10 AM and closes at 3 PM except on Friday when it closes at 6 PM Past study shows that over each period customers arrive ran domly at a stable mean rate so the assumption of a Poisson process is valid The mean time required to complete customer transactions is two minutes and the individual service times have a frequency distribu tion with a pronounced positive skew so an exponen tial distribution is a reasonable approximation to reality Tellers all work parttime and cost 10 per bank hour Pacifics experience has established that there will be a significant dropoff in clientele soon after a bout when customers suffer lengthy delays in getting teller access The supplier of the ATM equipment claims that other banks of comparable size have experienced a 30 diversion of regular business away from human tellers to the ATM which pro duced a further 20 expansion beyond the previous level of overall client transactionsall absorbed by the ATM half of it outside regular banking hours The supplier also maintains that ATM traffic is fairly uniform except between 11 PM and 6 AM when it is negligible Ms Meisterhaus believes that the ATM busyperiod arrivals will constitute a single Poisson process Industry experience is that the mean service time at an ATM is onehalf minute with an exponential distribution serving as an adequate approximation to the unknown positively skewed unimodal distribution that actually applies Ms Meisterhaus believes that once the ATM is installed the Walnut Creek human tellers will be left with a greater proportion of the more involved and Table 135 Customer Arrivals at the Walnut Creek OfficeBefore ATM Installation Daily Average Number Period of Arrivals 1 MondayFriday 10 AM12 PM 155 2 MondayFriday 121 PM 242 3 MondayFriday 13 PM 290 4 Friday 36 PM 554 10This case was written by Lawrence L Lapin San Jose State University Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C H A P T E R 14 Regression and Forecasting Models REVENUE MANAGEMENT AT HARRAHS CHEROKEE CASINO HOTEL R eal applications of forecasting are almost never done in isolationThey are typically one parta crucial partof an overall quantitative solution to a business problemThis is certainly the case at Harrahs Cherokee Casino Hotel in North Carolina as explained in an article by Metters et al 2008 This particular casino uses revenue management RM on a daily basis to increase its revenue from its gambling customers As customers call to request reservations at the casinos hotel the essential problem is to de cide which reservations to accept and which to deny The idea is that there is an opportunity cost from accepting early requests from lowervalued customers because highervalued customers might request the same rooms later on MonkeybusinessimagesDreamstimecom 841 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it As the article explains there are several unique features about casinos and this casino in particular that make a quantitative approach to RM successful First the detailed behaviors of customers can be tracked via electronic cards they use while placing bets in the electronic gambling machines so that the casino can create a large database of individ ual customers gambling patterns This allows the casino to segment the customers into different groups based on how much they typically bet in a given night For example one segment might contain all customers who bet between 500 and 600 per night When a customer calls for a room reservation and provides his card number the casino can im mediately look up his information in the database and see which segment he is in A second reason for the successful use of RM is that customers differ substantially in the price they are willing to pay for the same commodity a stay at the casinos hotel Actually many dont pay anything for the room or the foodthese are frequently com plimentary from the casinobut they pay by losing money at gambling Some customers typically gamble thousands of dollars per night while others gamble much less This is quite different from the disparities in other hotels or in air travel where a business trav eler might pay twice as much as a vacationer but not much more Because some cus tomers are much more valuable than others there are real opportunity costs from treating all customers alike A third reason for the success of RM at this casino is that the casino can afford to hold out for the bestpaying customers until the last minuteThe reason is that a significant percentage of the customers from all segments wait until the last minute to make their reservations In fact they often make them while driving say from Atlanta to the casino Therefore the casino can afford to deny requests for reservations to lowervalued cus tomers made a day or two in advance knowing that lastminute reservations very possibly from highervalued customers will fill up the casinos rooms Indeed the occupancy rate is virtually always 98 or above The overall RM solution includes 1 data collection and customer segmentation as explained above 2 forecasting demand for reservations from each customer segment 3 a linear programming LP optimization model that is run frequently to decide which reservations to accept and 4 a customer relationship management model to entice loyal customers to book rooms on nights with lower demandThe forecasting model is very similar to the Winters exponential smoothing model dis cussed later in this chapter Specifically the model uses the large volume of historical data to forecast customer demand by each customer segment for any particular night in the futureThese forecasts include information about timerelated or seasonal pat terns weekends are busier for example and any special events that are scheduled Also the forecasts are updated daily as the night in question approachesThese fore casts are then used in an LP optimization model to determine which requests to approve For example the LP model might indicate that given the current status of bookings and three nights to go requests for rooms on the specified night should be accepted only for the four most valuable customer segmentsAs the given night ap proaches and the number of booked rooms changes the LP model is rerun many times and provides staff with the necessary information for realtime decisions By the way a customer who is refused a room at the casino is often given a free room at another nearby hotelAfter all this customer can still be valuable enough to offset the price of the room at the other hotel It is difficult to measure the effect of this entire RM system because it has always been in place since the casino opened But there is no doubt that it is effective Despite the fact that it serves no alcohol and has only electronic games not the traditional gaming tables the casino has nearly full occupancy and returns a 60 profit margin on gross revenuedouble the industry norm 842 Chapter 14 Regression and Forecasting Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 141 INTRODUCTION Many decisionmaking applications depend on a forecast of some quantity Here are several examples When a service organization such as a fastfood restaurant plans its staffing over some time period it must forecast the customer demand as a function of time This might be done at a very detailed level such as the demand in succes sive quarterhour periods or at a more aggregate level such as the demand in successive weeks When a company plans its ordering or production schedule for a product it must forecast the customer demand for this product so that it can stock appropriate quantitiesneither too much nor too little When an organization plans to invest in stocks bonds or other financial instruments it typically attempts to forecast movements in stock prices and interest rates When government representatives plan policy they attempt to forecast movements in macroeconomic variables such as inflation interest rates and unemployment Many forecasting methods are available and all practitioners have their favorites To say the least there is little agreement among practitioners or theoreticians as to the best forecasting method The methods can generally be divided into three groups 1 judg mental methods 2 regression methods and 3 extrapolation methods The first of these is basically nonquantitative and is not discussed here Regression models also called causal models forecast a variable by estimating its re lationship with other variables For example a company might use a regression model to estimate the relationship between its sales and its advertising level the population income level the interest rate and possibly others The technique of regression is extremely popu lar due to its flexibility and power Regression can estimate relationships between time se ries variables or crosssectional variables those that are observed at a single point in time and it can estimate linear or nonlinear relationships Extrapolation methods also called time series methods use past data of a time series variableand nothing elseto forecast future values of the variable Many extrapo lation methods are available including the two we discuss here moving averages and exponential smoothing All extrapolation methods search for patterns in the historical se ries and then attempt to extrapolate these patterns into the future Some try to track long term upward or downward trends and then project these Some try to track the seasonal patterns sales up in November and December down in other months for example and then project these Much academic research has been devoted to forecasting methods in the past few decades and with the advances in computing power many of the methods described in the academic literature have been implemented in software packages Interestingly however there is not complete agreement even among academics that we can obtain better forecasts today than we could say in 1970 An article by Franses 2004 describes a survey of 76 members of the editorial boards of academic journals associated with forecasting The survey asked several questions about the status of forecasting methods today versus a few decades ago Most of the respondents believe that the advances in theory and software have resulted in better forecasts but they are not unanimous in this opinion They appear to rec ognize that quantitative forecasting methods can go only so far Many of the respondents believe that the opinions of experts in the subject area should be used to complement the forecasts from software packages In other words they dont believe that human judgment should be omitted from the forecasting process 141 Introduction 843 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Regression analysis and time series analysis are both very broad topics with many en tire books and thousands of research articles devoted to them We can only scratch the sur face of these topics in a single chapter However a little can go a long way By the time you have read this chapter you will be able to apply some very powerful techniques 142 OVERVIEW OF REGRESSION MODELS Regression analysis is the study of relationships between variables It is one of the most useful tools for a business analyst because it applies to so many situations Some potential uses of regression analysis in business address the following questions How do wages of employees depend on years of experience years of education and gender How does the current price of a stock depend on its own past values as well as the current and past values of a market index How does a companys current sales level depend on its current and past advertising levels the advertising levels of its competitors the companys own past sales levels and the general level of the market How does the unit cost of producing an item depend on the total quantity of items that have been produced How does the selling price of a house depend on such factors as the square footage of the house the number of bedrooms in the house and perhaps others Each of these questions asks how a single variable such as selling price or employee wages depends on other relevant variables If you can estimate this relationship you can better understand how the world operates and also do a better job of predicting the variable in question For example you can understand how a companys sales are affected by its advertising and also use the companys records of current and past advertising levels to pre dict future sales Regression analysis can be categorized in several ways One categorization is based on the type of data being analyzed There are two basic types crosssectional data and time series data Crosssectional data are usually data gathered from approximately the same period of time from a cross section of a population The housing and wage examples mentioned previously are typical crosssectional studies The first concerns a sample of houses presumably sold during a short period of time such as houses sold in Blooming ton Indiana during the first quarter of 2011 The second concerns a sample of employees observed at a particular point in time such as a sample of automobile workers observed at the beginning of 2010 In contrast time series studies involve one or more variables that are observed at several usually equally spaced points in time The stock price example men tioned previously fits this description The price of a particular stock and possibly the price of a market index are observed at the beginning of every week say and regression can then be used to explain the movement of the stocks price through time A second categorization of regression analysis involves the number of explanatory variables in the analysis First we must introduce some terms In every regression study the goal is to explain or predict a particular variable This is called the dependent variable or the response variable and is often denoted generically as Y To help explain or predict the dependent variable one or more explanatory variables are used These variables are also called independent variables or predictor variables and they are often denoted generically as Xs If there is a single explanatory variable the analysis is called simple regression If there are several explanatory variables it is called multiple regression 844 Chapter 14 Regression and Forecasting Models Regression is capable of dealing with cross sectional data and time series data Regression uses one or more explanatory variables to explain a single dependent variable Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it There are important differences between simple and multiple regression The primary difference as the name implies is that simple regression is simpler The calculations are simpler the interpretation of output is somewhat simpler and fewer complications can occur We will begin with a simple regression example to introduce the ideas of regression Then we will move on to the more general topic of multiple regression of which simple regression is a special case You will learn how to estimate regression equations that describe relationships be tween variables You will also learn how to interpret these equations explain numerical measures that indicate the goodnessoffit of the estimated equations and describe how to use the regression equations for prediction1 The LeastSquares Line The basis for regression is a fairly simple idea If you create a scatterplot of one variable Y versus another variable X you obtain a swarm of points that indicates any possible relationship between these two variables The terms scatterplot scatter chart and XY chart are all used to describe the same thing We use scatterplot in this chapter To quan tify this relationship you try to find the bestfitting line or curve through the points in the graph But what does bestfitting really mean Consider the scatterplot in Figure 141 The line shown is one possible fit It appears to be a reasonably good fit but a numerical measure of goodnessoffit is needed so that this fit can be compared with the fits of other possible lines 142 Overview of Regression Models 845 1The terms prediction and forecasting are practically synonyms Some analysts reserve the term forecasting for future values of a time series variable and use the term prediction for any type of variable time series or other wise However we do not make this distinction 10 20 A B X0 30 40 50 Positive residual Negative residual Height of line above X0 is predicted fitted value for X0 60 70 0 20 40 60 80 100 120 140 0 Scatterplot of Y vs X Figure 141 Scatterplot with Proposed Regression Line A residual is a prediction error It is the difference between an observed Y and the predicted Y from the regression line The measure commonly used is the sum of squared residuals Here a residual is de fined as the vertical distance from a point to the line as illustrated for points A and B If the point is above the line point A the residual is positive if the point is below the line point B the residual is negative The most commonly used measure of goodnessoffit is the sum of squared residuals Intuitively a good fit should have a small sum of squared residuals In fact the goal in regression is to find the line with the minimum sum of squared residuals where the minimum is over all possible lines This is called the leastsquares line and is the line found by regression Why are the residuals squared One reason is to make them all positive Another is to severely penalize large residuals The most compelling reason how ever is that this is the way it has been done by statisticians for many years Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Excel Tool Creating a Scatterplot with Excel To create a scatterplot in Excel select the two series of data and then select a Scatter c hart of some type from the Insert ribbon By default the range on the left will be on the horizontal axis and the range on the right will be on the vertical axis If this isnt what you want select the chart and use the Select Data Source option on the Chart Tools Design ribbon to switch the roles of the two series This is the key step You can experiment with other options but they are mainly for formatting the c hart If you want to use the StatT ools addin which will be used in other examples shortly it is even easier to create one or more scatterplots Fitting a Linear Trend Line To superimpose a linear trend line on any scatterplot rightclick on any point on the chart and then select the Add Trendline menu item This brings up the dialog box in 143 Simple Regression Models 849 1 2 3 4 5 6 7 8 9 10 11 12 A B Historical data Year Sales 1 1345000 2 1352000 3 1463000 4 1511000 5 1610000 6 1649000 7 1713000 8 1850000 9 2051000 10 2203000 Figure 142 Historical Sales at Best Chips 1200000 1400000 1600000 1800000 2000000 2200000 2400000 1 2 3 4 5 6 7 8 9 10 Year 1 2000 Sales versus Year Figure 143 Time Series Plot of Sales Solution A good place to start any regression analysis is with a scatterplot of Y versus X where X is time in this example See Figure 143 Sales are clearly increasing over time but it is not absolutely clear whether they are increasing at a constant rate which would favor a linear trend line or at an increasing rate which would favor an exponential trend line Therefore you can try fitting both of these Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Figure 144 You can select any of six types of trend lines For now select the default Linear option Also check the Display Equation on Chart option You can also elect to display the Rsquare value if you like The result appears in Figure 145 850 Chapter 14 Regression and Forecasting Models Figure 144 Dialog Box for Adding a Trendline y 92091x 1168200 1200000 1400000 1600000 1800000 2000000 2200000 2400000 1 2 3 4 5 6 7 8 9 10 Year 1 2000 Sales versus Year Figure 145 Plot with Superim posed Linear Trend Line Excel Tool Add Trendline It is easy to f it any of several types of trend lines to a scatterplot of some variable ver sus time To do so rightclic k on any point on the c hart and select Add T rendline from the menu This brings up a dialog box where you can select one of several types of trend lines In addition you can elect to display an equation of the trend line andor the Rsquare value on the chart This equation andor the Rsquare value appear in a text box You can select this text box and move it change its font size or change its number format as you like Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 854 Chapter 14 Regression and Forecasting Models Objective To use simple regression to estimate the relationship between Units Produced and Total Cost and to use this relationship to predict future total costs Solution When you try to relate two variables with regression it is always a good idea to create a scatterplot of the two variables first just to see whether there is any relationship worth pur suing This can be done with Excels chart tools in the usual way or it can be done easily with Palisades StatTools addin We will rely on StatTools for the rest of the statistical analysis in this chapter so this is a good place to start Excel AddIn StatTools from Palisade The StatTools addin implements many statistical procedures including regression analysis and forecasting It is part of the P alisade DecisionTools suite you pr obably already in stalled for the use of RISK andor PrecisionTools in previous chapters As with the other addins in the suite you can load StatT ools from the Windows Start button selecting All Programs and then StatT ools from the Palisade DecisionTools group If Excel is not already running this will launch Excel StatTools is very easy to use There is one basic thing you need to know To get started with any statistical analysis on any Excel data set you must first use Data Set Manager from the StatTools ribbon see Figure 149 to designate a StatTools data set The idea is that StatTools can analyze data only after it has been designated as a StatTools data set You need to do this only once per data set although a given Excel file can have multiple StatTools data sets To do so for this example put the cursor on any cell in the data set se lect Data Set Manager click on Yes that you want a new StatTools data set and fill out the resulting dialog box as in Figure 1410 Usually you can accept the defaults in this di alog box and click directly on OK However you can change the name of the data set to something more meaningful than the default Data Set 1 and you can override the data set range We did the latter so that only data through year 16 row 17 is part of the data set The future years with blank data for Total Cost shouldnt be part of the data set used for regression 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A B C D E F G Month Units Produced Total Cost Month Units Produced Total Cost 1 500 131000 17 400 2 600 135000 18 800 3 400 104000 4 300 76000 5 800 186000 6 900 190100 7 600 150000 8 400 98000 9 300 78000 10 200 60000 11 400 108000 12 600 152000 13 700 158000 14 500 134380 15 300 86000 16 200 60000 Figure 148 Cost and Production Data for a Single Product A scatterplot of Y ver sus X is always a good place to start in any regression analysis Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 143 Simple Regression Models 855 Once you have designated a StatTools data set again the first step in any analysis you can then create a scatterplot from the Summary Graphs dropdown list This leads to the dialog box in Figure 1411 where you can select the Y and X variables Actually you can select multiple Ys and Xs You will then get a scatterplot of each YX pair You can also control where the results go for the scatterplot procedure or any of the other StatTools pro cedures by clicking on the doublecheck button at the bottom This leads to the dialog box in Figure 1412 where you can select from the four Placement options shown We tend to favor either the Active Workbook option which places the results on a new worksheet or the Query for Starting Cell option where you can designate where you want the results to start You can experiment with these options The resulting scatterplot for this example appears in Figure 1413 This plot indicates a clear linear relationship where Total Cost increases as Units Produced increases Although this chart was created with StatTools it like other StatTools charts is a regular Excel chart so you can modify it just as you can modify any other Excel chart In particular you can superimpose a trend line along with the equation of the line and the Rsquare value as shown in the figure Figure 149 StatTools Ribbon Figure 1410 StatTools Data Set Manager Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Discussion of the Results The equation of the straight line has a slope 19847 and an intercept 23643 For this example both numbers have a natural interpretation The slope corresponds to the unit variable cost of production Each extra unit produced contributes an estimated 19847 to total cost The intercept corresponds to the fixed cost of production The estimate of the fixed cost is 23643 regardless of the production level 856 Chapter 14 Regression and Forecasting Models Figure 1411 Scatterplot Dialog Box Figure 1412 Results Placement Options in StatTools Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The regression results appear in Figures 1415 and 1416 and the future predictions appear in Figure 1417 There is a lot of information here but the good news is that the regression output from StatTools includes the same items mostly even in the same format as the regression output from any other statistical package The most important aspects of the output are the following The estimated regression line is specified by the values in the Coefficients column of Figure 1415 In particular the value in cell B19 implies that each additional unit produced adds about 198 to total cost The large Rsquare and multiple R values at the top of Figure 1415 confirm exactly what the scatterplot indicatesthat a very strong linear relationship exists between Total Cost and Units Produced The standard error of estimate at the top of Figure 1415 indicates that the prediction errors based on this regression equation will be in the neighborhood of 7000many prediction errors will be less than this value and a few will be more This large an error might sound like a lot but it is not all that large compared to the magnitudes of total costs which are often well over 100000 858 Chapter 14 Regression and Forecasting Models Figure 1414 Regression Dialog Box Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 860 Chapter 14 Regression and Forecasting Models The scatterplot of residuals versus fitted values in Figure 1416 is a diagnostic tool used to see if there are peculiar points or patterns The shapeless swarm seen here is an indication that no regression assumptions are violated This plot is based on the data below it You can check that each fitted value can be found by plugging its X into the regression equation and each residual is the difference between the actual Total Cost value and the predicted fitted value The predictions in Figure 1417 are also found by plugging the known X values into the regression equation but no residuals are possible because the actual Total Cost values are not yet known for these months Instead StatTools provides the limits for a 95 prediction interval around each prediction Essentially if you make a lot of predictions based on a regression equation about 95 of the actual Y values will be inside their respective 95 prediction intervals 1 2 3 E F G H I Month Units Produced Total Cost LowerLimit95 UpperLimit95 17 400 1030296 868983 1191610 18 800 1824160 1652118 1996202 Figure 1417 Prediction of Future Values P R O B L E M S Solutions for problems whose numbers appear within a colored box can be found in the Student Solutions Files Refer to this books preface for purchase information SkillBuilding Problems 1 The file P1401xlsx contains the monthly number of airline tickets sold by a travel agency a Does a linear trend appear to fit these data well If so estimate and interpret the linear trend model for this time series Also interpret the R2 and se values b Provide an indication of the typical forecast error generated by the estimated model in part a c Is there evidence of some seasonal pattern in these sales data If so characterize the seasonal pattern 2 The file P1402xlsx contains the daily closing prices of Walmart stock for a oneyear period Does a linear or exponential trend fit these data well If so estimate and interpret the best trend model for this time series Also interpret the R2 and se values 3 The file P1403xlsx contains monthly data on produc tion levels and production costs during a fouryear pe riod for a company that produces a single product Use simple regression on all of the data to see how Total Cost is related to Units Produced Use the resulting equation to predict total cost in month 49 given that the proposed production level for that month is 450 units Do you see anything wrong with the analysis How should you modify your analysis if your main task is to find an equation useful for predicting future costs and you know that the company installed new machinery at the end of month 18 Write a concise memo to management that describes your findings 4 The file P1404xlsx lists the monthly sales for a com pany in millions of dollars for a 10year period a Fit an exponential trend line to these data b By what percentage do you estimate that the com pany will grow each month c Why cant a high rate of exponential growth con tinue for a long time d Rather than an exponential curve what type of curve might better represent the growth of a new technology 5 Management of a home appliance store wants to understand the growth pattern of the monthly sales of a new technology device over the past two years The managers have recorded the relevant data in the file P1405xlsx Have the sales of this device been growing linearly over the past 24 months By examining the results of a linear trend line explain why or why not 6 Do the sales prices of houses in a given community vary systematically with their sizes as measured in square feet Answer this question by estimating a simple regression equation where the sales price of the house is the dependent variable and the size of the house is the explanatory variable Use the sample data Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it In Equation 147 a is again the Yintercept and b1 through bk are the slopes Collec tively a and the bs are called the regression coefficients Each slope coefficient is the ex pected change in Y when that particular X increases by one unit and the other Xs in the equation remain constant For example b1 is the expected change in Y when X1 increases by one unit and the other Xs in the equation X2 through Xk remain constant The intercept a is typically less important Literally it is the expected value of Y when all of the Xs equal 0 However this makes sense only if it is practical for all of the Xs to equal 0 which is rarely the case 862 Chapter 14 Regression and Forecasting Models The regression coefficients are the intercept and slopes of the regression equation We illustrate these ideas in the following extension of Example 142 E X A M P L E 143 ESTIMATING TOTAL COST FOR SEVERAL PRODUCTS S uppose the company in Example 142 now produces three different products A B and C The company has kept track of the number of units produced of each product and the total production cost for the past 15 months These data appear in Figure 1418 and in the file Cost Regression 2xlsx What does multiple regression say about the relationship be tween these variables How can multiple regression be used to predict future production costs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B C D E Month Units A Units B Units C Total Cost 1 696 819 895 58789 2 627 512 925 50276 3 122 323 814 43703 4 313 981 670 50857 5 340 884 356 46397 6 462 599 673 46731 7 269 302 737 40328 8 343 495 878 42368 9 986 191 592 44617 10 555 314 467 40515 11 908 593 749 55546 12 595 115 458 36856 13 557 369 160 35697 14 271 550 457 40130 15 878 750 983 59929 Figure 1418 Cost and Production Data for Multiple Products Objective To use multiple regression to estimate the relationship between units produced of three products and the total production cost and to use this relationship to predict future total costs Solution The dependent variable Y is again Total Cost but there are now three potential Xs Units A Units B and Units C It is not necessary to use all three of these but we do so here In fact it is again a good idea to begin with scatterplots of Y versus each X to see which Xs are indeed related to Y You can do this in one step with StatTools selecting Total Cost as the Y variable and Units A B and C as the X variables A typical scatterplot appears in Figure 1419 This scatterplotand the ones for products A and C are similarindicates a fairly strong linear relationship between Total Cost and Units B A useful first step in multiple regression is to create a scatterplot of Y versus each of the Xs Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 866 Chapter 14 Regression and Forecasting Models The standard error of estimate has exactly the same interpretation as before It is a ballpark estimate of the magnitude of the prediction errors you are likely to make based on the regression equation Here this value is about 1981not too bad considering that the total costs vary around 50000 As before the fitted values in Figure 1422 are found by substituting each set of Xs into the regression equation and the residuals are the differences between actual total costs and fitted values As indicated by the standard error of estimate most of the residuals are no more than about 2000 in magnitude and quite a few are considerably less than this Also the scatterplot of residuals versus fitted values in Figure 1422 is a shapeless swarm a promising indication that no regression assump tions have been violated The predictions of future values in Figure 1423 are found by plugging the known X values into the regression equation As before StatTools provides a 95 prediction interval for each of these predictions StatTools provides outputs with more decimal places than shown in the figures We believe it is a good idea to round these Dont be fooled into thinking that regression can be accurate to 10 decimal places or however many just because the software shows this many decimal places It is not that exact a science especially not with data from the business world A Note about Adjusted Rsquare You are probably wondering what the adjusted Rsquare value means in the multiple regression output Although it has no simple interpretation like Rsquare percentage of variation explained it is useful for comparing regression equations The problem with Rsquare is that it can never decrease when extra explanatory variables are added to a regres sion equation However there ought to be some penalty for adding variables that dont re ally belong This is the purpose of adjusted Rsquare which acts as a monitor If you add one or more extra explanatory variables to an already existing equation adjusted Rsquare can decrease If this occurs it is evidence that the extra variables dont really belong in the equation and should probably be deleted Incorporating Categorical Variables The goal of regression analysis is to find good explanatory variables that explain some dependent variable Y Often these explanatory variables are quantitative such as the Units Produced variables in the two previous examples However there are often useful qualita tive categorical variables that help explain Y such as gender male or female region of country east south west or north quarter of year Q1 Q2 Q3 or Q4 and so on Because regression works entirely with numbers categorical variables must typically be trans formed into numeric variables that can be used in a regression equation This is usually done by creating dummy variables also called 01 variables or indicator variables For any categorical variable you create a dummy variable for each possible category Its value is 1 for each observation in that category and it is 0 otherwise The interpretation of regression output for multiple regression is similar to that for sim ple regression In par ticular Rsquare multi ple R the standard error of estimate the fitted values and the residuals mean exactly the same thing in both cases If adjusted Rsquare decreases when extra explanatory variables are added to a regres sion equation these variables are not useful and should probably be deleted A dummy variable for any category equals 1 for all observations in that category and 0 for all observations not in that category For example the variable Gender has two possible values Male and Female so you can create two dummy variables Male and Female Male equals 1 for all males and 0 for Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it not room here for a complete discussion of these assumptions and their ramifications we briefly state a few cautions you should be aware of Multicollinearity In the best of worlds the explanatory variables the Xs should provide nonoverlapping in formation about the dependent variable Y They should not provide redundant information However sometimes redundancy is difficult to avoid For example in trying to explain em ployee salaries three potential explanatory variables are age years of seniority with this company and years of experience with this type of job These three variables are likely to be highly correlated with one another as well as with salary and it is not clear whether all three should be included in a regression equation for salary When you do include Xs that are highly correlated with one another you introduce a problem called multicollinearity The problem is that when Xs are highly correlated with one another it is virtually impossible to sort out their separate influences on Y This inabil ity to sort out separate effects can even lead to wrong signs on the regression coefficients For example if age years of seniority and years of experience are all entered in an equa tion for salary it is possible that one of the three regression coefficients will be negative even though all three variables are positively correlated to salary Therefore the presence of multicollinearity makes regression equations difficult to interpret Fortunately however multicollinearity is not a problem if you are concerned only with prediction of new Ys Nonlinear Relationships If scatterplots of Y versus the various Xs indicate any nonlinear relationships a linear rela tionship will almost certainly lead to a poor fit and poor predictions Fortunately as with the exponential trend line there are often nonlinear transformations of Y andor the Xs that straighten out the scatterplots and allow you to use linear regression We will not discuss such transformations here We simply warn you that if the scatterplots of the original vari ables do not appear to be linear you should not blindly proceed to estimate a linear relationship Nonconstant Error Variance One assumption of regression is that the variation of the Y values above any values of the Xs is the same regardless of the particular values of the Xs chosen Sometimes this as sumption is clearly violated For example if Y is a households annual amount spent on vacations and X is the households annual income it is very possible that the variation of Y values for lowincome households is considerably less than that for highincome households The lowincome households dont have much to spend on vacations so their vacation spending is likely to be tightly bunched at low values In contrast the highincome households have a lot to spend but they might or might not elect to spend it on vacations Typically nonconstant error variance appears in a scatterplot as a fanshaped swarm of points We simply alert you to this possibility and suggest that you obtain expert help if you spot an obvious fan shape Autocorrelation of Residuals Autocorrelation means that a variables values are correlated with its own previous values This typically occurs in time series variables For example regression might be used to forecast monthly sales If the residuals are autocorrelated then an overprediction in January is likely to be followed by an overprediction in February and an underprediction in June is likely to be followed by an underprediction in July It is not difficult to detect 144 Multiple Regression Models 871 Multicollinearity makes it difficult to interpret individual regression coefficients but it does not have a negative effect on predictions Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it autocorrelation of residuals although we will not discuss the measures for doing so but it is much more difficult to deal with autocorrelation appropriately Again you should con sult an expert if you believe your time series analysis is subject to autocorrelation 872 Chapter 14 Regression and Forecasting Models FUNDAMENTAL INSIGHT Cautions about Regr ession Regression is a very powerful method for discovering relationships between variables and with the soft ware available in todays world it is very easy to use Unfortunately it is also v ery easy to use incor rectly Many people are not aware of the assumptions behind the regression model how to check whether these assumptions holdor how to modify the analysis if the assumptions do not hold This has led to many incor rect interpretations of r egression output Like most powerful tools regression is easy to misuse if y ou dont understand some of the theor y behind it Be cause this theory is fairly complex dont be afraid to ask a statistical expert for help if you are conducting an important regression analysis P R O B L E M S SkillBuilding Problems 12 Suppose you are an analyst for a company that pro duces four products and you are trying to decide how much of each product to produce next month To model this decision problem you need the unit vari able production cost for each product After some dig ging you find the historical data on production levels and costs in the file P1412xlsx Use these data to find estimates of the unit costs you need You should also find an estimate of the fixed cost of production Will this be of any use to you in deciding how much of each product to produce Why or why not 13 A trucking company wants to predict the yearly main tenance expense Y for a truck using the number of miles driven during the year X1 and the age of the truck X2 in years at the beginning of the year The company has gathered the data given in the file P1413xlsx Note that each observation corresponds to a particular truck Estimate a multiple regression equation using the given data Interpret each of the estimated regression coefficients Also interpret the standard error of estimate and the Rsquare value for these data 14 An antique collector believes that the price received for a particular item increases with its age and with the number of bidders The file P1414xlsx contains data on these three variables for 32 recently auctioned com parable items Estimate a multiple regression equation using the given data Interpret each of the estimated regression coefficients Is the antique collector correct in believing that the price received for the item in creases with its age and with the number of bidders Interpret the standard error of estimate and the Rsquare value for these data 15 Stock market analysts are continually looking for re liable predictors of stock prices Consider the prob lem of modeling the price per share of electric utility stocks Y Two variables thought to influence this stock price are return on average equity X1 and an nual dividend rate X2 The stock price returns on equity and dividend rates on a randomly selected day for 16 electric utility stocks are provided in the file P1415xlsx Estimate a multiple regression equation using the given data Interpret each of the estimated regression coefficients Also interpret the standard error of estimate and the Rsquare value for these data 16 The manager of a commuter rail transportation system was recently asked by her governing board to deter mine which factors have a significant impact on the demand for rides in the large city served by the trans portation network The system manager collected data on variables thought to be possibly related to the num ber of weekly riders on the citys rail system The file P1416xlsx contain these data a What do you expect the signs of the coefficients of the explanatory variables in this multiple regres sion equation to be Why Answer this before running the regression Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it b Estimate a multiple regression equation using the given data Interpret each of the estimated regres sion coefficients Are the signs of the estimated coefficients consistent with your expectations in part a c What proportion of the total variation in the num ber of weekly riders is not explained by this esti mated multiple regression equation 17 Consider the enrollment data for Business Weeks top US graduate business programs in the file P1417xlsx Use the data in the MBA Data sheet to estimate a multiple regression equation to assess whether there is a relationship between the total num ber of fulltime students Enrollment and the follow ing explanatory variables a the proportion of female students b the proportion of minority students and c the proportion of international students enrolled at these business schools a Interpret the coefficients of the estimated regres sion equation Do any of these results surprise you Explain b How well does the estimated regression equation fit the given data 18 Suppose that a regional express delivery service com pany wants to estimate the cost of shipping a package Y as a function of cargo type where cargo type in cludes the following possibilities fragile semifragile and durable Costs for 15 randomly chosen packages of approximately the same weight and same distance shipped but of different cargo types are provided in the file P1418xlsx a Estimate a regression equation using the given sample data and interpret the estimated regression coefficients b According to the estimated regression equation which cargo type is the most costly to ship Which cargo type is the least costly to ship c How well does the estimated equation fit the given sample data How might the fit be improved d Given the estimated regression equation predict the cost of shipping a package with semifragile cargo SkillExtending Problems 19 The owner of a restaurant in Bloomington Indiana has recorded sales data for the past 19 years He has also recorded data on potentially relevant variables The data are listed in the file P1419xlsx a Estimate a simple regression equation involving annual sales the dependent variable and the size of the population residing within 10 miles of the restaurant the explanatory variable Interpret Rsquare for this regression 144 Multiple Regression Models 873 b Add another explanatory variableannual adver tising expendituresto the regression equation in part a Estimate and interpret this expanded equa tion How does the Rsquare value for this multiple regression equation compare to that of the simple regression equation estimated in part a Explain any difference between the two Rsquare values How can you use the adjusted Rsquares for a com parison of the two equations c Add one more explanatory variable to the multiple regression equation estimated in part b In particu lar estimate and interpret the coefficients of a mul tiple regression equation that includes the previous years advertising expenditure How does the in clusion of this third explanatory variable affect the Rsquare compared to the corresponding values for the equation of part b Explain any changes in this value What does the adjusted Rsquare for the new equation tell you 20 Does the rate of violent crime acts vary across differ ent regions of the United States Answer this with the somewhat old 1999 data in the file P1420xlsx as requested below a Estimate an appropriate regression model to ex plain the variation in violent crime rate across the four given regions of the United States Interpret the estimated equation Rank the four regions from highest to lowest according to their mean violent crime rate Could you have done this without re gression Explain b How would you modify the regression model in part a to account for possible differences in the vi olent crime rate across the various subdivisions of the given regions Estimate your revised regres sion equation and interpret your findings Rank the nine subdivisions from highest to lowest according to their mean violent crime rate 21 The file P1421xlsx contains data on over 200 movies that came out in 2006 and 2007 Create a new variable Total Revenue that is the sum of Total US Gross Inter national Gross and US DVD Sales How well can this new variable be predicted from the data in columns CF For Distributor relabel the categories so that there are only two Large Distributor and Small Dis tributor The former is any distributor that had at least 12 movies in this period and the latter is all the rest For Genre relabel the categories to be Comedy Drama Adventure Action ThrillerSuspense and Other Other includes Black Comedy Documentary Horror Musical and Romantic Comedy Interpret the coefficients of the estimated regression equation How would you explain the results to someone in the movie business Do you think that predictions of total revenue from this regression equation will be very accurate Why Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 145 OVERVIEW OF TIME SERIES MODELS To this point we have discussed regression as a method of forecasting Because of its flexi bility regression can be used equally well for time series variables and for crosssectional variables From here on however we focus exclusively on time series variables and we discuss nonregression approaches to forecasting All of these approaches fall under the general umbrella of extrapolation methods With an extrapolation method you form a time series plot of the variable Y that you want to forecast analyze any patterns inherent in this time series plot and extrapolate these patterns into the future You do not use any other variablesthe Xs from the previous sectionto forecast Y you use only past values of Y to forecast future values of Y The idea is that history tends to repeat itself Therefore if you can discover the patterns in the his torical data you ought to obtain reasonably good forecasts by projecting these historical patterns into the future Before examining specific extrapolation techniques we discuss the types of patterns that are common in time series data We also briefly discuss the measures that are typically used to judge how well forecasting methods track the historical data Components of Time Series A time series variable Y typically contains one or more components These components are called the trend component the seasonal component the cyclic component and the ran dom or noise component We provide a brief discussion of these components here We start with a very simple time series in which every observation is the same as shown in Figure 1429 The graph in this figure shows time t on the horizontal axis and the observation value Y on the vertical axis We assume that Y is measured at regularly spaced intervals usually days weeks months quarters or years The value of Y in period t is de noted as Yt As indicated in the figure the individual points are usually joined by straight lines to make any patterns in the time series more apparent Because all observations in this series are equal the resulting plot is a horizontal line We refer to this series as the base se ries Then we build more interesting times series from this base series Trend Component If the observations increase or decrease regularly over time we say that the time series has a trend The graphs in Figure 1430 illustrate several possible trends We already discussed the linear trend in Figure 1430a and the exponential trend in Figure 1430b in section 143 The curve in Figure 1430c is an Sshaped trend As an example this type of trend curve is 874 Chapter 14 Regression and Forecasting Models 1 Yt t 2 3 4 5 6 7 Figure 1429 The Base Series A trend implies a consistent upward or downward movement of the series over time Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it appropriate for a new product that takes a while to catch on then exhibits a rapid increase in sales as the public becomes aware of it and finally tapers off to a fairly constant level The curves in Figure 1430 all represent upward trends Of course there are downward trends of the same types Seasonal Component Many time series have a seasonal component For example a companys sales of swim ming pool equipment increase every spring then stay relatively high during the summer and then drop off until next spring at which time the yearly pattern repeats itself An im portant aspect of the seasonal component is that it tends to be predictable from one year to the next That is the same seasonal pattern tends to repeat itself every year In Figure 1431 we show two possible seasonal patterns Figure 1431a shows nothing but the seasonal component That is if there were no seasonal variation this would be the base series from Figure 1429 In Figure 1431b we show a seasonal pattern superimposed on an upwardsloping trend line Cyclic Component The third component of a time series is the cyclic component By studying past movements of many business and economic variables it becomes apparent that business cycles affect many variables in similar ways For example during a recession housing starts generally go down unemployment goes up stock prices go down and so on But when the recession is over all of these variables tend to move in the opposite direction 145 Overview of Time Series Models 875 Yt t Yt t Yt t a Linear trend b Exponential trend c Sshaped trend Figure 1430 Series with Trends Yt t Yt t a Seasonal component only b Seasonal component with trend Figure 1431 Series with Seasonality In a seasonal pattern some seasons are regularly higher than others each year Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it We know that the cyclic component exists for many time series because it is visible as the periodic swings in the levels of the time series graphs However the cyclic component is harder to predict than the seasonal component The reason is that seasonal variation is much more regular For example swimming pool supplies sales always start to increase during the spring Cyclic variation on the other hand is more irregular because the busi ness cycle does not always have the same length A further distinction is that the length of a seasonal cycle is generally one year whereas the length of a business cycle is generally much longer than one year The graphs in Figure 1432 illustrate the cyclic component of a time series In Fig ure 1432a cyclic variation is superimposed on the base series from Figure 1429 In Figure 1432b this same cyclic variation is superimposed on the series from Figure 1431b The resulting graph has trend seasonal variation and cyclic variation Random Noise Component The final component in a time series is called the random component or simply noise This unpredictable component gives most time series graphs their irregular zigzag appear ance Usually a time series can be determined only to a certain extent by its trend seasonal and cyclic components Then other factors determine the rest These other factors might be inherent randomness unpredictable shocks to the system the unpredictable behavior of human beings who interact with the system and others Figures 1433 and 1434 show the affect that noise can have on a time series graph The graph on the left of each figure shows the random component only superimposed on the base series Then on the right of each figure the random component is superimposed on the graph of trend with seasonal component from Figure 1431b The difference between Figure 1433 876 Chapter 14 Regression and Forecasting Models Yt t Yt t a Cyclic component only b Cyclic component with seasonality and trend Figure 1432 Series with Cyclic Component Yt t Yt t a Noise only b Noise superimposed on trend and seasonal components Figure 1433 Series with Noise By definition noise is unpredictable It often makes trends and sea sonal patterns more difficult to recognize Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it RMSE is similar to a standard deviation in that the errors are squared because of the square root its units are the same as those of the original variable MAE is similar to RMSE except that absolute values of errors are used instead of squared errors MAPE the same measure we introduced in section 143 is probably the easiest measure to understand be cause it does not depend on the units of the original variable it is always stated as a per centage For example the statement that the forecasts are off on average by 2 has a clear meaning even if you do not know the units of the variable being forecasted Depending on the forecasting software used one or more of these measures will typically be reported Fortunately models that make any one of these measures small tend to make the others small as well so that you can choose whichever measure you want to focus on One caution is in order however The measures MAE RMSE or MAPE are used to see how well the forecasting model tracks historical data But even if these measures are small there is no guarantee that future forecasts will be accurate As stated previously extrapolation methods all make the implicit assumption that history will repeat itself How ever history does not always repeat itself When this is the case a model that closely tracks historical data can yield poor forecasts of the future In addition there is a danger of track ing a historical series too closely Tracking every little up and down is pointless if these movements represent random noise that will not repeat in the future 878 Chapter 14 Regression and Forecasting Models FUNDAMENTAL INSIGHT Limitations of Extra polation Methods All extrapolation forecasting methods such as the moving averages and exponential smoothing methods discussed next make the crucial assumption that his torical patterns are likely to repeat themselves If an unexpected shock occurs such as a disruption in oil supplies from the Mid East or a gr oundbreaking dis covery in biotechnolog y extrapolation methods can fail miserably in the period after the shock In addi tion extrapolation methods can be too finely tuned If they are optimized to follow all of the ups and downs of a time seriesthey might just be learning patterns of noise patterns that are unlikely to continue in the fu tureThis is why smoothed forecasts that follow the basic underlying patterns are usually preferred 146 MOVING AVERAGES MODELS Perhaps the simplest and one of the most frequently used extrapolation methods is the method of moving averages Very simply the forecast for any period with this method is the average of the observations from the past few periods To implement the moving aver ages method you must first choose a span the number of terms in each moving average Lets say that the data are monthly and a span of six months is used Then the forecast of next months value is the average of the previous six months values For example you average the January to June values to forecast July you average the February to July values to forecast August and so on This is the reason for the term moving averages A good forecasting model typically makes all three measures of forecast errors small The larger the span the smoother the fore cast series will be The span in the moving averages method is the number of observations in each average The role of the span is important If the span is largesay 12 monthsthen many ob servations go into each average and extreme values have relatively little effect on the aver ages The resulting series of forecasts will be much smoother than the original series For this reason the moving average method is called a smoothing method In contrast if the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it span is smallsay three monthsthen extreme observations have a larger effect on the averages and the forecast series will be much less smooth In the extreme if the span is one month there is no smoothing effect at all The method simply forecasts next months value to be the same as this months value What span should you use This requires some judgment If you believe the ups and downs in the series are random noise then you do not want future forecasts to react too quickly to these ups and downs and you should use a relatively large span But if you want to track most of the ups and downsunder the belief that these ups and downs are pre dictablethen you should use a smaller span You should not be fooled however by a graph of the forecast seriesthat is a graph of the averagessuperimposed on the origi nal series This graph will almost always look better when a small span is used because the forecast series will appear to track the original series better But this does not mean it will provide better future forecasts Again tracking random ups and downs closely is pointless if the ups and downs represent unpredictable noise The following example illustrates the use of moving averages on a series of weekly sales We continue to take advantage of the StatTools addin which includes procedures for creating time series graphs and implementing moving averages and exponential smoothing methods 146 Moving Averages Models 879 E X A M P L E 145 FORECASTING WEEKLY SALES OF HARDWARE AT LEES L ees is a local discount store that sells a variety of merchandise much like Kmart Walmart and Target In particular Lees sells a full line of hardware The company has kept track of weekly total dollar sales of hardware items for the past 104 weeks These data ap pear in the file Hardware Salesxlsx Lees is planning to use moving averages with an ap propriate span to forecast future weekly hardware sales Does this appear to be a good idea Objective To judge the effectiveness of the moving averages method with different spans to forecast weekly hardware sales at Lees Solution A time series graph of weekly sales appears in Figure 1435 You can create this easily from Excels builtin charting tools as a line chart or you can use the StatTools time series graph procedure available under the Time Series and Forecasting dropdown list We did 0 500 1000 1500 2000 2500 3000 3500 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 103 Time Series of Sales Weekly Data Figure 1435 Time Series Plot of Hardware Sales Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the latter after remembering that the first step is always to designate a StatTools data set This series appears to meander with no obvious trend or seasonality Evidently sales of hardware at Lees are relatively constant throughout each year This type of series is a good candidate for moving averages However it is not clear which span to use We tried spans of 3 6 and 12 weeks Spans of 3 and 6 give similar results whereas a span of 12 gives less good results We illustrate the calculations for a span of 3 you can check the calculations for the other spans in the finished version of Hardware Salesxlsx DEVELOPING THE SPREADSHEET MODEL Using a span of 3 the forecast for week 4 is the average of the observed sales in weeks 1 to 3 the forecast for week 5 is the average of the observed sales in weeks 2 to 4 and so on The calculations are straightforward in Excel However they can be performed much more quickly by using the forecasting procedure in StatTools To do so select Forecast from the Time Series and Forecasting dropdown list on the StatTools ribbon This leads to a dialog box with three tabs in the lower section The Time Scale tab shown in Figure 1436 is used to identify the type of data annual monthly and so on and the starting date or index The Graphs to Display tab shown in Figure 1437 allows you to check which graphs you want in the output We typically choose the first and third Finally the important Forecast 880 Chapter 14 Regression and Forecasting Models Figure 1436 Time Scale Tab in Forecasting Dialog Box Figure 1437 Graphs to Display Tab in Forecasting Dialog Box A series that mean ders with no obvious trend or seasonality is a good candidate for moving averages Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 886 Chapter 14 Regression and Forecasting Models E X A M P L E 146 FORECASTING HARDWARE SALES AT LEES I n the previous example you saw that the moving averages method was able to provide only fair forecasts of weekly hardware sales at Lees Using the best of three potential spans its forecasts were still off by about 139 on average The company would now like to try sim ple exponential smoothing to see whether this method with an appropriate smoothing con stant can outperform the moving averages method How should the company proceed Objective To see whether simple exponential smoothing with an appropriate smoothing constant can provide more accurate forecasts of weekly hardware sales than the moving averages forecasts Solution You already saw in Example 145 that the hardware sales series meanders through time with no apparent trends or seasonality Therefore this series is a good candidate for simple exponential smoothing This is no guarantee that the method will provide accurate fore casts but at least it cannot be ruled out as a promising forecasting method DEVELOPING THE SPREADSHEET MODEL Using Equation 1411 it is fairly easy to implement simple exponential smoothing with copyable Excel formulas but as with moving averages it is much easier to use StatTools In fact you can use the same settings in the forecasting dialog box as with moving aver ages The only exception is in the Forecast Settings section As shown in Figure 1443 you should check the Exponential Smoothing Simple option and enter a value of alpha on the right We chose 01 Alternatively you can check the Optimize Parameters option in which case StatTools finds the value of alpha that minimizes RMSE Discussion of the Results The simple exponential smoothing calculations are shown in Figure 1444 You can check that Equation 1411 is implemented in the Level column and that each forecast is the previous level It is common to use the first observation as the first level Note that the last level is Figure 1443 Forecast Settings for Simple Exponential Smoothing Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it about any business analyst because virtually all business analysts need to relate variables discover trends and seasonal patterns and make forecasts Fortunately the basic tools we have presented are reasonably easy to understand and use especially given the builtin ca pabilities of Excel and the available statistical addins for Excel These tools are extremely widespread flexible and powerful We suspect that most of you will use them at some point in your careers Summary of Key Management Science Terms Term Explanation Page Regression models Statistical models that estimate an equation to relate one 843 variable to one or more explanatory variables Extrapolation Statistical models that relate a time series variable to previous 843 time series models values of that same variable Dependent variable The variable being explained in a regression model 844 typically denoted by Y Explanatory variables The variables used to explain the dependent variable in a 844 regression model typically denoted by Xs also called independent or predictor variables Simple regression A regression model with a single explanatory variable 844 Multiple regression A regression model with multiple explanatory variables 844 Leastsquares line The regression line that minimizes the sum of squared 845 residuals the resulting line from a typical regression analysis Residual The difference between an actual Y value and the value 845 predicted by the regression equation Fitted value A predicted value of Y as predicted by the regression equation 846 Standard error of Essentially the standard deviation of the residuals an estimate 846 estimate of the magnitude of prediction errors made from the regression equation Multiple R The correlation between the actual Ys and the fitted Ys 847 Rsquare The percentage of variation of the Ys explained by the regression 847 Linear trend A trend usually through time where a variable changes by 848 a constant amount each time period Exponential trend A trend usually through time where a variable changes 848 by a constant percentage each time period Dummy variables 01 variables that are used in regression equations to encode 866 a categorical variable such as Gender or Quarter Regression coefficients The estimated intercept and slope terms in a regression output that 862 define the regression equation Multicollinearity Occurs when Xs are highly correlated with one another makes 871 interpretation of the regression coefficients difficult Autocorrelation of residuals Occurs when nearby residuals are correlated with one another 871 usually with time series data Extrapolation methods Forecasting methods where past patterns of a time series variable 874 are discovered and extrapolated into the future Time series components The items including trend seasonality cyclic behavior and 874 noise that produce the patterns observed in most time series variables MAE RMSE MAPE Three popular measures of forecast errors in time series analysis 878 898 Chapter 14 Regression and Forecasting Models continued Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Term Explanation Page Moving averages method A forecasting method where the forecast for any period is the 878 average of the several most recent periods Span The number of terms in each average in moving averages 878 larger spans produce a smoother forecast series Exponential smoothing A forecasting method where the forecast for any period is a 884 method weighted average of previous periods with more recent periods getting more weight Smoothing constants One or more constants all between 0 and 1 that drive the 884 exponential smoothing equations lower values produce a smoother forecast series Simple exponential Version of exponential smoothing appropriate when there is no 884 smoothing obvious trend or seasonality Holts method Version of exponential smoothing appropriate when there is a trend 884 but no obvious seasonality Winters method Version of exponential smoothing appropriate when there is 884 seasonality and possibly a trend Summary of Key Excel Terms Term Explanation Excel Page Creating a scatterplot Useful for identifying a relationship Create a scatter chart from 849 between two variables Insert ribbon can also use StatTools addin Superimposing Useful for identifying a linear or Create a scatterplot then use the 850 a trend line exponential trend through a scatterplot Trendline tool EXP function Used to raise the special number e to a EXPvalue 852 power also called the antilog function StatTools addin A powerful and easytouse statistical Has its own ribbon 854 addin developed by Palisade Analysis ToolPak A statistical addin that comes with Use Data Analysis from Data ribbon 857 Excel useful for regression and several other statistical procedures Creating a time Useful for seeing how a time series Create a line chart from 876 series graph variable behaves through time Insert ribbon can also use StatTools addin 148 Conclusion 899 P R O B L E M S SkillBuilding Problems 41 Many companies manufacture products that are at least partially produced using chemicals eg paint gasoline and steel In many cases the quality of the finished product is a function of the temperature and pressure at which the chemical reactions take place Suppose that a particular manufacturer wants to model the quality Y of a product as a function of the temperature X1 and the pressure X2 at which it is produced The file P1441xlsx contains data obtained from a carefully designed experiment involving these variables Note that the assigned quality score can range from a minimum of 0 to a maximum of 100 for each manufactured product a Estimate a multiple regression equation that includes the two given explanatory variables Does the estimated equation fit the data well b Add an interaction term between temperature and pressure the product of these two variables and run the regression again Does the inclusion of the interaction term improve the models goodness of fit c Interpret each of the estimated coefficients in the two equations How are they different How do you interpret the coefficient for the interaction term in the second equation Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 902 Chapter 14 Regression and Forecasting Models and that months sales The file P1455xlsx contains data on these two accounts for the last 36 months It also shows the sales levels two months prior to month 1 a Is there any statistical evidence to suggest a rela tionship between the monthly sales level and accounts receivable b Referring to part a would the relationship be de scribed any better by including this months sales and the previous months sales called lagged sales in the equation for accounts receivable What about adding the sales from more than a month ago to the equation For this problem why might it make accounting sense to include lagged sales vari ables in the equation How do you interpret their coefficients c During month 37 which is a fiscal yearend month sales were 1800000 The reported accounts receivable balance was 3000000 Does this reported amount seem consistent with past experience Explain 56 Based on an actual court case in Philadelphia In the 1994 congressional election the Republican candidate outpolled the Democratic candidate by 400 votes excluding absentee ballots The Democratic candi date outpolled the Republican candidate by 500 absen tee votes The Republican candidate sued and won claiming that vote fraud must have played a role in the absentee ballot count The Republicans lawyer ran a regression to predict based on past elections how the absentee ballot margin could be predicted from the votes tabulated on voting machines Selected results are given in the file P1456xlsx Show how this re gression could be used by the Republican to support his claim of vote fraud Hint Does the 1994 observa tion fall outside the general pattern That is in statisti cal terms is it an outlier 57 Confederate Express is attempting to determine how its monthly shipping costs depend on the number of units shipped during a month The file P1457xlsx contains the number of units shipped and total shipping costs for the past 15 months a Use regression to determine a relationship between units shipped and monthly shipping costs b Plot the errors for the predictions in order of time sequence Is there any unusual pattern c Suppose there was a trucking strike during months 11 to 15 and we believe that this might have influ enced shipping costs How could the answer to part a be modified to account for the effects of the strike After accounting for the effects of the strike does the unusual pattern in part b disappear Hint Use a dummy variable 58 The file P1458xlsx contains monthly cost accounting data on overhead costs machine hours and direct material costs This problem will help you explore the meaning of R2 and the relationship between R2 and correlations a Create a table of correlations between the individual variables b If you ignore the two explanatory variables Machine Hours and Direct Material Cost and predict each Overhead Cost as the mean of Overhead Cost then a typical error is Overhead Cost minus the mean of Overhead Cost Find the sum of squared errors using this form of prediction where the sum is over all observations c Now run three regressions 1 Overhead Cost OHCost versus Machine Hours 2 OHCost versus Direct Material Cost and 3 OHCost versus both Machine Hours and Direct Material Cost The first two are simple regressions the third is a multiple regression For each find the sum of squared residuals and divide this by the sum of squared errors from part b What is the relationship between this ratio and the associated R2 for that equation Now do you see why R2 is referred to as the percentage of variation explained d For the first two regressions in part c what is the relationship between R2 and the corresponding correlation between the dependent and explanatory variable For the third regression it turns out that the R2 can be expressed as a complicated function of all three correlations in part a That is the function involves not just the correlations between the dependent variable and each explanatory variable but also the correlation between the explanatory variables Note that this R2 is not just the sum of the R2 values from the first two regressions in part c Why do you think this is true intuitively However R2 for the multiple regression is still the square of a correlationnamely the correlation between the observed and predicted values of OHCost Verify that this is the case for these data 59 The Wilhoit Company has observed that there is a linear relationship between indirect labor expense and direct labor hours Data for direct labor hours and indirect labor expense for 18 months are given in the file P1459xlsx At the start of month 7 all cost categories in the Wilhoit Company increased by 10 and they stayed at this level for months 7 through 12 Then at the start of month 13 another 10 acrosstheboard increase in all costs occurred and the company operated at this price level for months 13 through 18 a Plot the data Verify that the relationship between indirect labor expense and direct labor hours is approximately linear within each sixmonth period Use regression three times to estimate the slope Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 148 Conclusion 903 and intercept during months 1 through 6 during months 7 through 12 and during months 13 through 18 b Use regression to fit a straight line to all 18 data points simultaneously What values of the slope and intercept do you obtain c Perform a price level adjustment to the data and reestimate the slope and intercept using all 18 data points Assuming no cost increases for month 19 what is your prediction for indirect labor expense if there are 35000 direct labor hours in month 19 d Interpret your results What causes the difference in the linear relationship estimated in parts b and c 60 Pernavik Dairy produces and sells a wide range of dairy products Because a government regulatory board sets most of the dairys costs and prices most of the competition between the dairy and its competitors takes place through advertising The controller of Per navik has developed the sales and advertising levels for the past 52 weeks These appear in the file P1460xlsx Note that the advertising levels for the three weeks prior to week 1 are also listed The con troller wonders whether Pernavik is spending too much money on advertising He argues that the com panys contributionmargin ratio is about 10 That is 10 of each sales dollar goes toward covering fixed costs This means that each advertising dollar has to generate at least 10 of sales or the advertising is not costeffective Use regression to determine whether advertising dollars are generating this type of sales re sponse Hint The sales value in any week might be affected not only by advertising this week but also by advertising levels in the past one two or three weeks These are called lagged values of advertising Try re gression models with lagged values of advertising in cluded and see whether you get better results 61 The file P1461xlsx contains five years of monthly data for a company The first variable is Time 160 The second variable Sales1 has data on sales of a product Note that Sales1 increases linearly throughout the period with only a minor amount of noise The third variable Sales2 will be used in the next prob lem For this problem use the Sales1 variable to see how the following forecasting methods are able to track a linear trend a Forecast this series with the moving averages method with various spans such as 3 6 and 12 What can you conclude b Forecast this series with simple exponential smoothing with various smoothing constants such as 01 03 05 and 07 What can you conclude c Repeat part b with Holts method again for various smoothing constants Can you do much better than in parts a and b 62 The Sales2 variable in the file from the previous prob lem was created from the Sales1 variable by multiply ing by monthly seasonal factors Basically the sum mer months are high and the winter months are low This might represent the sales of a product that has a linear trend and seasonality a Repeat parts a to c from the previous problem to see how well these forecasting methods can deal with trend and seasonality b Use Winters method with various values of the three smoothing constants to forecast the series Can you do much better Which smoothing con stants work well c What can you conclude from your findings in parts a and b about forecasting this type of series 63 The file P1463xlsx contains data on a motel chains revenue and advertising a Use these data and multiple regression to make pre dictions of the motel chains revenues during the next four quarters Assume that advertising during each of the next four quarters is 50000 Hint Try using advertising lagged by one period as an explanatory variable See the Problem 60 for an explanation of a lagged variable Also use dummy variables for the quarters to account for possible seasonality b Use simple exponential smoothing to make predic tions for the motel chains revenues during the next four quarters Experiment with the smoothing constant c Use Holts method to make forecasts for the motel chains revenues during the next four quarters Experiment with the smoothing constants d Use Winters method to determine predictions for the motel chains revenues during the next four quarters Experiment with the smoothing constants e Which forecasts from parts a to d would you ex pect to be the most reliable Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E H owies Bakery is one of the most popular bakeries in townand the favorite at Howies is French breadEach day of the weekHowies bakes a number of loaves of French breadmore or less according to a daily schedule To maintain its fine reputation Howies gives to charity any loaves not sold on the day they are baked Although this occurs frequentlyit is also com mon for Howies to run out of French bread on any given daymore demand than supplyIn this caseno extra loaves are baked that daythe customers have to go elsewhere or come back to Howies the next day for their French bread Although French bread at Howies is always popularHowies stimulates demand by running occasional 10 off sales Howies has collected data for 20 consecutive weeks 140 days in all These data are listed in the file Howies Bakeryxlsx The variables are Day MondaySunday Supply number of loaves baked that day OnSale whether French bread is on sale that day and Demand loaves actually sold that day Howies wants to see whether regression can be used successfully to estimate Demand from the other data in the file Howie reasons that if these other variables can be used to predict Demand then he might be able to determine his daily supply num ber of loaves to bake in a more costeffective way How successful is regression with these data Is Howie correct that regression can help him deter mine his daily supply Is any information missing that would be useful How would you obtain it How would you use it Is this extra information really necessary 141 DEMAND FOR FRENCH BREAD AT HOWIES BAKERY 904 Chapter 14 Regression and Forecasting Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E T he Indiana University Credit Union Eastland Plaza branch was having trouble getting the correct staffing levels to match customer arrival patterns On some days the number of tellers was too high relative to the customer traffic so that tellers were often idle On other days the opposite occurred long customer waiting lines formed be cause the relatively few tellers could not keep up with the number of customers The credit union manager James Chilton knew that there was a prob lem but he had little of the quantitative training he believed would be necessary to find a better staffing solution James figured that the problem could be broken down into three parts First he needed a reliable forecast of each days number of customer arrivals Second he needed to translate these fore casts into staffing levels that would make an adequate tradeoff between teller idleness and customer waiting Third he needed to translate these staffing levels into individual teller work assignmentswho should come to work when The last two parts of the problem require analy sis tools queueing and scheduling that we will not pursue here However you can help James with the first partforecasting The file Credit Union Arrivalsxlsxlists the number of customers enter ing this credit union branch each day of the past year It also lists other information the day of the week whether the day was a staff or faculty payday and whether the day was the day before or after a holi day Use this data set to develop one or more fore casting models that James could use to help solve his problem Based on your models make any recom mendations about staffing that appear reasonable 143 ARRIVALS AT THE CREDIT UNION 906 Chapter 14 Regression and Forecasting Models Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 151 Project Management C H A P T E R SCHEDULING THE N EWPR ODUCT DEVELOPMENT PR OCESS AT DOW AGROSCIENCES D ow AgroSciences a wholly owned subsidiary of The Dow Chemical Company is in the business of developing new agricultural products It subjects product candidates to tests covering safety efficacy and environmental impact as well as other tests to validate the biology and confirm that the prod ucts will do well in the business market To beat the competition to market the company is under pressure to do its testing and use its resources as effi ciently as possible The development schedule is the key At any time around 30 products can be going through testing each of which consists of tens to hundreds of tasks that must be performed The scheduling of these tasks must take the following data into account 1 the net present value NPV of the cash flows each candidate is expected to generate depending on its launch date 2 the costs of tasks in the development process 3 the technical prece dence relationships for tasks 4 the durations of the tasks 5 the probability that the candidate will fail a task resulting in the cancellation of the develop ment process for that candidate 6 resource requirements and capacities and others Many of the required inputs are uncertain so that probability distribu tions are needed to model them correctly Bassett et al 2004 describe a simulationbased optimization model they developed to help generate good schedules in this complex environment CORBIS 15 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Unlike the rather simple project scheduling examples described in this chapter the situa tion at Dow AgroSciences is much larger and considerably more complex First there are several projects in process at a given time not a single project and they are all com peting for scarce resources such as lineitem budgets Second some tasks can fail for some projects in which case these projects do not continue in the development process and therefore free up resources for other projects Third precedence relationships are only partly fixed There are often fixed precedence relationships of the type discussed in this chapter where for example task C cannot start until tasks A and B are finished However other precedence relationships can be introduced for strategic reasons For example suppose tasks E and F can begin at the same time but there is a probability that task E will fail Then it might be better to allow task F to start only when task E is suc cessfully completedThe reason is that if task E fails the cost of performing task F will be savedAlso the schedule can impose precedence relationships across projects to reduce the simultaneous use of scarce resources Finally due to the seasonal nature of agricul tural products a delay of one month that causes a product to miss the growing season might be just as costly as a delay of 10 months The authors first tried to formulate their problem as an integer programming IP model as has often been done in the project scheduling literature However they found that the size and complexity of the problem made the resulting IP model too difficult to solve in a reasonable amount of timeTherefore they turned to simulation and heuristic methods for optimizing using precedence relationships as decision variables For any proposed solution that is any set of precedence relationships within and across projects they simulate the development of these projects over a multiyear period The simulation output contains the value of the objective they want to maximize expected NPV They then experiment with several heuristic methods including the genetic algorithms dis cussed in Chapter 8 to find solutions with larger values of the objective Of course each new solution must be simulated to find its value of the objective There is no guarantee that this methodology will find an optimal solution but it appears to produce very good solutions in an acceptable amount of computing time The authors implemented their solution method in a system with an Excelbased userfriendly front end In the background the system uses a simulation package AweSim plus the authors own C computer code to implement the simulation and heuristic algorithms Dow AgroSciences put this system into practice via their Six Sigma project in Research and Development From 1998 to 2004 the company verified savings of several million dollars based on the schedules determined by the system As Beth Swisher Manager of RD Effectiveness at Dow AgroSciences statesI feel comfortable stating that more than one million dollars have been saved due to our possession of the technology In addition to these hard savings the improved understanding of the overall newproduct development process across all the functions in the company has been invaluable 152 Chapter 15 Project Management 151 INTRODUCTION All organizations have ongoing activities and they have projects The distinction is that a project has a beginning an end and one or more welldefined goals The project could be the development of a software program the building of a house or an office building the development of a new drug a marketing campaign for a new product and many others Typically a team of employees is assigned to a project and one member of the team is des ignated as the project manager The team is assigned to complete the project within a cer tain time within a certain budget and within certain specifications At some point in the Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it future the team will complete the project or deem it a failure and the projects life cycle will be finished The purpose of this chapter is to discuss ways to manage projects suc cessfully This is an extremely important topic for real organizations There can be serious consequences when a project is not finished on time runs over budget or fails to meet specifications As an academic discipline project management is discussed in management opera tions management and management science The discussion in management tends to focus on the soft skills necessary to manage projects successfully The project manager must be an effective leader and team members must communicate successfully agree on goals cooperate report progress clearly and so on Although the importance of these peo ple skills is clear it is not the focus of this chapter Management science and operations management tends to focus on the quantitative tools that have been developed to manage projects These go under the twin acronyms of PERT Program Evaluation and Review Technique and CPM Critical Path Method These methods were developed independently about a halfcentury ago PERT was devel oped jointly by the US Navy Lockheed and the consulting firm of Booz Allen and Hamilton in their work on the Polaris nuclear missile CPM was developed at DuPont and RemingtonRand to improve the construction of new production facilities and the shut down of existing facilities The main difference between PERT and CPM is that CPM was developed for projects with a set of commonly performed tasks where the task times are fairly well known In contrast PERT was developed for projects with tasks where scien tists had little experience and could not estimate their times with much certainty In short the CPM model did not include uncertainty in task times but the PERT model did Over the years the two methods have tended to merge so that people now often speak of PERTCPM models In either case the emphasis is on a project that starts at some point and ends some time later The project consists of a number of tasks that must be completed for the project to be completed These tasks have durations the time it takes to complete them assumed known for CPM random for PERT they typically cost money and they often require nonfinancial resources such as people and facilities They also have prece dence relationships For example task G might not be able to start until tasks B D and F are finished These precedence relationships put constraints on what can be done when In addition limited resources can place constraints on the tasks that can be done simultane ously A wellestablished methodology has been developed to analyze such projects It involves various charts and some reasonably simple calculations We explain how it works in this chapter As you will see most of it can be accomplished in Excel However you should be aware that there is another package in the Microsoft Office family called Microsoft Project1 This powerful package is devoted exclusively to managing projects Of course power usually implies complexity and Project is very complex We discuss it briefly at the end of the chapter However a thorough discussion of the Project software is well beyond the scope of this book The calculations discussed in this chapter are performed in Excel Projects have three dimensions time resources and scope2 The usual discussion of PERTCPM focuses primarily on the time dimension How long will the project take to complete if everything goes according to schedule which tasks form bottlenecks that pre vent the project from being completed earlier and which tasks have some slack in the sense that they can be delayed to some extent without delaying the project These ques tions are the usual focus of PERTCPM models and we too focus primarily on the time 151 Introduction 153 CPM usually implies known activity times and PERT usually implies uncertain activity times 1We tend to think of Microsoft Office as including Excel Word Access PowerPoint Outlook and a few others which make up the package you get when you purchase Office However Microsoft includes other packages such as Project when it discusses its Office family Unfortunately you have to purchase these other packages separately 2Some people add a fourth dimension quality However quality can be encompassed within scope The focus of most PERTCPM discussions is time but resource usage money people facilities and so on is also very important Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it dimension However we also discuss the resource dimension The tasks in a project almost always compete for resources whether dollars or nonfinancial resources and no real project management application can afford to ignore these resources For example one version of the problem we analyze is the crashing problem In this problem you decide how to spend money optimally to speed up crash the completion of the project For example if you find that the project will not be completed until 16 weeks from now but you have a deadline of 14 weeks from now you need to find a way to crash critical tasks to save a couple of weeks The third dimension scope is the most difficult to model quantitatively Scope involves the deliverable itselfwhat it is intended to do and what features it should include For example if the purpose of the project is to deliver a new version of Excel the software developers at Microsoft have to control scope It is all too easy to keep adding features refining existing features and generally adding to the scope This is undoubtedly why Microsofts software products often come out later than originally advertised And Microsoft is certainly not alone If the project manager doesnt keep a constant eye on scope cr eep the project can easily run over budget andor fail to meet its deadline Unfortunately scope is not easy to model so we do not discuss it any further here This chapter provides an introduction to project management In particular it dis cusses the basic deterministic CPM model where task times are assumed to be known and it uses simulation to analyze a version of the PERT model where task times are assumed to be random However the opener to this chapter indicates how complex project manage ment can be in the real world A company such as Dow AgroSciences often needs to jug gle many projects simultaneously the timing of eventual revenues needs to be considered possible failures in testing at some stage along the way can terminate projects and result in lost costs extra precedence relationships can be introduced to manage costs and other resources and so on The problems can quickly become complex which is all the more reason to employ management science techniques to solve them as companies such as Dow AgroSciences have learned to their benefit Before continuing we note that many entire books are devoted to project manage ment and the material we include here is typically found in two or three chapters of such books This material is certainly an important aspect of project management but it is not the only aspect Other aspects include selecting the project in the first place setting goals and specifications for the project properly managing people involved in the pro ject including adequate communication monitoring the progress of the project and making changes to the original plan when necessary knowing when to pull the plug on a project that is not making adequate progress and others All of these aspects are important for determining whether a realworld project is successful or not and the fail ure to manage them properly is the reason why so many projects have been unsuccessful One notable failure occurred in the 1990s when Health Care Financing Administration the agency that administers Medicare spent at least 50 million developing a Medicare Transactions system that never became a reality This failure of this project is described in Friel 2000 If you are interested in learning more about project management we recommend the following books Klastorin 2004 Marchewka 2006 and Gido Clements 2006 152 THE BASIC CPM MODEL In this section we describe the basic CPM procedure for finding the length of time required to complete a project This approach assumes that we know 1 the activities that comprise the project 2 the precedence relationships among activities and 3 the time 154 Chapter 15 Project Management Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it required to complete each activity3 This time called the activity duration is assumed to be known with certainty However even when we relax this assumption in a later section and assume there is a probability distribution for each activity duration the same basic pro cedure explained here can still be used as part of a simulation model To proceed we need a list of the activities that make up the project The project is complete when all of the activities have been completed Each activity has a set of activi ties called its immediate predecessors that must be completed before the activity begins It also has a set of activities called its immediate successors that cannot start until it has finished The word immediate is sometimes omitted A project network diagram is usu ally used to represent the precedence relationships among activities Two types of dia grams do this activityonnode AON networks and activityonarc AOA networks and proponents of each type have rather strong feelings We favor AON networks because we believe they are more intuitive so we do not discuss AOA networks in this book In the AON representation of a project there is a node for each activity Then there is an arc from node i to node j if node i is an immediate predecessor of node j To illustrate this consider a project that consists of five activities labeled A B C D and E Activities A and B can start immediately Activity C cannot start until activity B is finished activity D cannot start until activity A is finished and activity E cannot start until activities A and C are both finished The project is finished when all activities are finished The precedence relationships are listed in Table 151 and the AON network appears in Figure 151 Table 151 also includes the duration for each activity In an AON network these durations are placed next to the nodes In addition there is typically a Start node and a Finish node in the diagram These indicate the start and the finish of the project Note that activity E illustrates the meaning of the term immediate predecessor Clearly activity B is also a predecessor of activity Eit must be finished before activity E can startbut it is not an immediate predecessor because it will be finished before another predecessor of activity E activity C can even begin 152 The Basic CPM Model 155 3Activities are also called tasks in the projectmanagement literature The two terms activities and tasks are synonymous AON networks use nodes for activities and arcs to indicate prece dence relationships 6 12 3 8 10 Start A D C B E Finish Figure 151 AON Network for a FiveActivity Project Table 151 Data for a FiveActivity Project Immediate Immediate Node Predecessors Successors Duration A None D E 8 B None C 10 C B E 3 D A None 12 E A C None 6 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The following example illustrates how to implement this method 158 Chapter 15 Project Management The earliest start time and earliest finish time for any activity are the earliest the activity can start or finish given precedence relationships and durations The latest start time and latest finish time for any activity are the latest the activity can start or finish without delaying the project as a whole The slack of any activity is the amount of time the activity can be delayed beyond its earliest start time without delaying the project as a whole An activity is critical only if its slack is 0 E X A M P L E 151 CREATING AN OFFICE LAN A n insurance company has decided to construct a local area network LAN in one of its large offices so that its employees can share printers files and other conveniences The project consists of 15 activities labeled A through O as listed in Table 152 This table indicates the immediate predecessors and immediate successors of each activity along with each activitys expected duration At this point these durations are assumed known Note that activity A is the only activity that can start right away and activity O is the last activity to be completed This table implies the AON network in Figure 152 The company wants to know how long the project will take to complete and it also wants to know which activities are on the critical path Table 152 Data on LAN Activities Immediate Immediate Description Activity Predecessors Successors Duration days Perform needs analysis A None B 10 Develop specifications B A C D 6 Select server C B E G 6 Select software D B F G 12 Select cables E C F 4 Purchase equipment F D E H I 3 Develop user manuals G C D J 6 Wire offices H F L 12 Set up server I F K 3 Develop training program J G M 14 Install software K I L 4 Connect network L H K M N 3 Train users M J L O 8 Test and debug system N L O 12 Get management acceptance O M N None 4 Objective To develop a spreadsheet model of the LAN project so that we can calculate the time required to complete the project and identify the critical activities WHERE DO THE NUMBERS COME FROM The computer systems people should be able to obtain the data in the first four columns of Table 152 They would know what needs to be done and in which order However the data in the last column the durations are probably guesses at best There is usually uncertainty regarding activity times due to workers not showing up unavailable components software The lists of activities and their immediate predecessors in such a table are enough to determine the list of immediate successors Try listing the succes sors on your own Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it bugs and so on We ignore this uncertainty here but we will deal with it explicitly in section 154 when we discuss simulation of a project Solution To implement the method you use Equations 151 and 152 to find the earliest start and finish times of all activities Equation 153 to find the project completion time Equations 154 and 155 to find the latest start and finish times and finally Equation 156 to find the slacks and hence the critical activities DEVELOPING THE SPREADSHEET MODEL The completed spreadsheet model is shown in Figure 153 see the file Project Scheduling xlsx and can be developed with the following steps 1 Input data Enter the predecessors successors and durations in the shaded range Note how we have entered data for the Start and Finish nodes in rows 5 and 21 2 Earliest start and f inish times Here you implement the forward pass of the algo rithm with Equations 151 and 152 To implement Equation 151 enter the formula B25E5 in cell C25 and copy it down to cell C41 To implement Equation 152 begin by entering 0 in cell B25 This is because the Start node can begin immediately Then every other ear liest start time is the maximum of the earliest finish times of its predecessors Unfortunately there is no way to enter a single formula and copy it down You need to specialize each for mula to each activitys particular predecessors For example the formulas for activities D and G in cells D29 and D32 are C27 and MAXC28C29 This is because activity D has a single predecessor whereas activity G has two predeces sors The other formulas in column B are similar 3 Project completion time The project completion time is given in Equation 153 as the earliest start time of the Finish node Record it in cell B43 with the formula B41 152 The Basic CPM Model 159 A Start Finish D F 10 12 6 12 6 6 14 4 4 4 8 3 12 3 3 B C J M E K H I G L N O Figure 152 AON Diagram for LAN Project Each earliest start time is the maximum of the earliest finish times of its predecessors Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 153 MODELING ALLOCATION OF RESOURCES The basic CPM model presented in the previous section is concerned solely with timing Given the known durations the activities are scheduled so that the project is completed as soon as possible In this section we discuss another aspect of project scheduling the allo cation of resources The activities in a project always consume resources including money people and possibly others When we say that an activity has a duration of 10 days we are implicitly assuming that certain resources have been allocated to this activity For example it might be that five engineers working at 300 per day per engineer can complete the activity in 10 days It is possible however that if more or fewer than five engineers were assigned to the activity or maybe they were paid more or less than 300 per day the activity would be finished sooner or later than 10 days These are tradeoffs that must typically be made when scheduling a project If you suspect that this is a multiobjective optimization problem discussed in Chapter 16 you are absolutely correct There are typically three primary objectives 1 to finish the project quickly 2 to consume as few resources as possible especially to mini mize costs and 3 to produce a highquality project Because of these three objectives there are many potential optimization models for project scheduling and the academic research in this area has explored many of them including some that are quite complex We set our sights considerably lower here We first indicate how a project manager can at least monitor resource usage This is not actually optimization but optimization models could be built upon it We then discuss one of the most popular optimization models for project scheduling called crashing In the crashing model it is possible to shorten the activity durations by spending extra money on themthat is it is possible to crash the activities The problem is to spend as little extra money as possible to complete the project within a given deadline We say extra because money is presumably already being spent to achieve the given activity durations Now we want to spend extra money to speed them up Monitoring the Use of Resources Almost all projects require money and people Therefore we focus on these two resources here Of course other resources such as facilities or equipment could also be monitored The following extension of the LAN project example from the previous section illustrates how the money and people devoted to the project can be monitored over time in Excel Admittedly this is somewhat tedious A software package that is devoted to project schedul ing such as Microsoft Project has much better tools for monitoring resource usage 1514 Chapter 15 Project Management E X A M P L E 152 MONITORING RESOURCES FOR THE LAN PROJECT R ecall from Example 151 that an insurance company is creating a LAN for one of its large offices In that example we provided activity durations for the 15 activities in the project and we showed that with these durations the project can be completed in 62 days We now make some assumptions about the money and people resources that are implicit in these activity durations First we assume that the various activities require dif ferent technical expertise which comes from five groups of people engineering systems purchasing installers and training To achieve the durations used in Example 151 we assume the numbers of people required per day for the various activities are those shown in Table 153 For example to perform the needs analysis in 10 days six engineers are required per day Note that connecting the network is the only activity that requires two Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it different types of people three systems people and five installers for each of the three days this activity takes to complete Also note that the last activity getting management accep tance doesnt show any people requirements In reality this activity is probably the responsibility of the project manager who is busy throughout the entire project Almost all projects have a project manager In addition to these people the various activities require money It certainly costs money to pay the people and there are probably other costs as well We assume the costs per day for the various activities are those shown in Table 154 The company wants to see how its people and money are used over time Also because some of the activities have some slack the company wants to see how the resource usages are affected by adjusting the starting times of the noncritical activities Objective To create time series charts of the money and people usages and to see how these are affected by the starting times of the noncritical activities 153 Modeling Allocation of Resources 1515 Table 153 People Required per Day for Various Activities Activity Duration Engineering Systems Purchasing Installers Training Perform needs analysis 10 6 Develop specifications 6 8 Select server 6 5 Select software 12 7 Select cables 4 3 Purchase equipment 3 4 Develop user manuals 6 5 Wire offices 12 8 Set up server 3 4 Develop training 14 9 program Install software 4 6 Connect network 3 3 5 Train users 8 8 Test and debug system 12 5 Get management acceptance 4 Table 154 Costs per Day for the Various Activities Activity Cost per Day Perform needs analysis 500 Develop specifications 500 Select server 400 Select software 400 Select cables 400 Purchase equipment 300 Develop user manuals 300 Wire offices 450 Set up server 400 Develop training program 300 Install software 400 Connect network 450 Train users 300 Test and debug system 400 Get management acceptance 250 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it systems people could be hired or borrowed from another project In any case the value of such a chart is that it shows potential infeasibilities in the current schedule so that alterna tive schedules can be pursued In a practical sense charts such as these are monitored throughout the lifetime of the project As we all know from experience cost estimates often change they usually increase as the project unfolds and estimates of other resource requirements can change as well Therefore the project manager needs to monitor requirements continually to ensure that they stay within allowable limits Crashing the Activities The objective in many projectscheduling analyses is to find a minimumcost method of reducing activity times to meet a deadline The term crashing the activities is often used to mean reducing the activity times Of course it typically costs money to crash activities hiring extra workers using extra equipment using overtime and so onso the problem becomes one of crashing just the right activities in just the right amounts to meet a deadline at minimum cost We now illustrate how Solver can be used to solve this problem 1520 Chapter 15 Project Management E X A M P L E 153 MEETING A DEADLINE FOR THE LAN PROJECT F rom the CPM calculations in Example 151 the insurance company knows that if the LAN activities continue to take as long as listed in Table 152 the entire project will take 62 working days to complete However the project manager is under pressure to fin ish the job in 56 working days He estimates that each activity could be crashed by a cer tain amount at a certain cost Specifically he estimates the cost per day of activity time reduction and the maximum possible days of reduction for each activity as shown in Table 155 For example activity As duration could be reduced from 10 days to 9 days at cost 600 or it could be reduced from 10 days to 8 days at cost 1200 It is even possible to have a fractional reduction such as from 10 days to 85 days at cost 900 On the other hand note that three of the activities cannot be crashed at all probably due to technical considerations How can the deadline be met at minimum cost Table 155 Crashing Inputs Maximum Description Activity Cost per Day Reduction Perform needs analysis A 600 2 Develop specifications B 600 1 Select server C 480 1 Select software D 480 3 Select cables E 480 1 Purchase equipment F 0 Develop user manuals G 360 1 Wire offices H 540 4 Set up server I 0 Develop training program J 360 4 Install software K 480 1 Connect network L 0 Train users M 360 2 Test and debug system N 480 3 Get management acceptance O 300 1 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the linear model in Project Crashing Linearxlsx could be modified to have a non linear objective Then GRG Nonlinear Solver could be used 2 There might only be discrete crashing opportunities available For example there might be two types of equipment that can be purchased to reduce some activitys duration each involving a certain cost and leading to a certain reduction This kind of discrete choice can be handled with binary 01 variables as in Chapter 6 Scheduling Multiple Projects Many organizations have limited labor resources and multiple projects that can or must be completed Selecting the projects to undertake is a very important problem for any com pany The company must select a portfolio of projects that is consistent with its overall goals and strategy provides desired diversification maintains adequate cash flows does not exceed resource availabilities and does not exceed a reasonable level of risk In this section we illustrate one possible model for project portfolio selection In this model we assume that each potential project has a worker requirement over some duration and a deadline If the project is completed by the deadline the company receives a reward otherwise it receives no reward We use Evolutionary Solver to determine the projects to undertake and the optimal start time for each project undertaken To simplify the example we consider each project as a single activity rather than as a series of activities as in other sections of this chapter 153 Modeling Allocation of Resources 1525 E X A M P L E 154 SCHEDULING PROJECTS AT TIMBURTON T imburton Construction has 10 projects that it can if desired complete within the next 10 months Each project earns a certain revenue when it is completed but only if it is completed within the next 10 months Otherwise the project earns no revenue The num ber of workers needed each month the number of months needed to complete each project and the revenue earned from each completed project are listed in Table 157 We assume that after the company begins working on a project it must work on the project during con secutive months until the project is completed Timburton has 220 workers available each month How can it maximize the revenue earned during the next 10 months Objective To find starting times for the projects so that total revenue is maximized and worker utilization each month is no greater than worker availability Table 157 Worker Requirements and Revenues Workers per Project Month Months Revenue 1 74 5 4800 2 98 2 3330 3 91 3 4100 4 95 4 6840 5 59 2 1650 6 81 3 3880 7 84 4 6380 8 78 3 4200 9 95 3 4860 10 58 5 5220 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in cell G7 and copying it down Note the effect of subtracting 1 The projects finish at the ends of the months in column G For example using the values in Figure 1516 project 4 starts in month 4 and finishes at the end of month 7 for a duration of four months 3 Revenues The revenue for a project is obtained only if the project is finished by the deadline so enter the formula IFG7DeadlineD70 in cell H7 and copy it down This is one of several places where IF functions are required This explains why Evolutionary Solver is required 4 Worker utilization The table in the middle of the model uses 01 values to indicate which months workers are used or not used by the various projects To fill it in enter the formula IFANDF7B19B19G710 in cell B20 and copy it to the range B20K29 Then to find the number of workers used each month enter the formula SUMPRODUCTB7B16B20B29 in cell B30 and copy it across row 30 This formula is based on the assumption that each project uses the same number of workers for its entire duration It wouldnt be difficult to change this assumption so that worker utilization could change during the projects duration 5 Penalties As discussed in Chapter 8 Evolutionary Solver does better with penalties for violating constraints than with explicit constraints Therefore check in row 31 whether each months worker availability is violated with the formula IFB30B410 in cell B31 copied across row 31 Then calculate a total penalty for worker constraint vio lations in cell B34 with the formula 100000SUMB31K31 Any suitably large constant could be used here It should be large relative to the magni tudes of the revenues 6 Objective Sum the revenues in column H to obtain the total revenue earned in cell B33 and calculate the objective to maximize in cell B35 with the formula B33B34 The penalty for violating constraints is subtracted from the real objective USING EVOLUTIONARY SOLVER The setup for Evolutionary Solver is shown in Figure 1517 Note that there are no explicit constraints on worker availabilities because these have been incorporated as penalties in the objective The only explicit constraints are that the start times must be integers between 1 and the deadline plus 1 Again the interpretation of a start time equal to 11 is that this project isnt undertaken at all Discussion of the Solution It took us a number of tries using various starting solutions in the changing cells and vari ous Evolutionary Solver settings to obtain the solution shown in Figure 1516 This is evi dently a difficult combinatorial problem even though there are only 10 changing cells 153 Modeling Allocation of Resources 1527 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it each with only 11 possible values Keep in mind that this implies 1011 possible solutions a very large number Dont be surprised if you obtain a solution with a slightly smaller objective than we obtained In fact there may even be a slightly better solution than ours In any case our solution indicates that the company can complete all but two of the proj ects within the deadline without violating worker availability in any month To achieve this it has to stagger the starting times of the projects so that they dont overlap too much You can see that the maximum number of projects ever in process at any time is three If you compare the worker requirements in the input section to the number of workers avail able each month 220 the solution makes sensefour projects never fit in a single month but some combinations of three projects do fit 1528 Chapter 15 Project Management Figure 1517 Evolutionary Solver Dialog Box P R O B L E M S SkillBuilding Problems 10 Suppose after doing the analysis in Example 152 the project manager sees a problem with the current setup Activity C selecting the server requires five systems peo ple and activity D selecting the software requires seven systems people The problem is that these two activities are scheduled concurrently even though it turns out that four of the five systems people for activity C and four of the seven systems people for activity D are the same peo ple Assuming that a given person can work on only one activity at a time some changes need to be made a One possible change is to assign two of the four people in common to activity C and the other two to activity D Now three people will be assigned to activity C and five people will be assigned to activ ity D Unfortunately with fewer people assigned the durations of these activities will increase from 6 days to 9 days for activity C and from 12 days to 14 days for activity D How much will these changes delay the project b Another possible change is to make activity D a suc cessor to activity C so that the four common people can continue to be assigned to both activities How should the AON diagram for the project be redrawn How much will this change delay the project c What other changes might you suggest 11 In the Monitoring Costs sheet of the Project Monitoringxlsx file we created two tables of daily Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 154 MODELS WITH UNCERTAIN ACTIVITY TIMES In section 152 we discussed how to calculate the required time to complete a project that consists of several activities We also saw that the critical path consists of the bottleneck activities those activities that cannot be delayed without delaying the project as a whole In that section we assumed that the individual activity times are known with certainty We now make the more realistic assumption that the activity times are random with given probability distributions and we find the distribution of the time needed to complete the project Because of randomness we can no longer identify the critical path We can only determine the probability that any activity is critical To illustrate this latter statement suppose that activities A and B can begin immedi ately Activity C can then begin as soon as activities A and B are both completed and the project is completed as soon as activity C is completed see Figure 1518 Activity C is clearly on the critical path but what about A and B Suppose that the expected activity times of A and B are 10 and 12 respectively If you use these expected times and ignore any uncertainty about the actual timesthat is if you proceed as in section 152then activity B is definitely a critical activity because its duration is definitely longer than activ ity As duration However suppose there is some positive probability that A can have dura tion 12 and B can have duration 11 Under this scenario A is a critical activity Therefore you cannot say in advance which of the activities A or B will be critical However you can use simulation to see how likely it is that each of these activities is critical You can also see how long the entire project is likely to take 1530 Chapter 15 Project Management When activity times are random you typi cally cannot say for certain whether a given activity will be on the critical path E X A M P L E 155 LAN PROJECT WITH UNCERTAIN ACTIVITY TIMES W e again analyze the LAN project from Example 151 but we now assume that the activity durations are uncertain with given probability distributions The company realizes that the actual activity times can vary due to unexpected delays worker illnesses and so on Assuming that the company has a deadline of 60 days it wants to use simulation to see 1 how long the project is likely to take 2 how likely it is that the project will be completed by the deadline and 3 which activities are likely to be critical Objective To simulate the time to complete the LAN project and to estimate the proba bility that any given activity will be part of the critical path WHERE DO THE NUMBERS COME FROM All of the data are the same as in Example 151 except for the probability distributions for activity times We discuss these in some detail here Start Finish A C B Figure 1518 A Simple Project Network We illustrate the procedure in the following example which is the same example that we have been discussing without crashing We repeat the story here for your convenience Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it in cell G26 and copy it down This records a 1 for any activity with 0 slackthat is for any critical activity However if you press the F9 key to generate new random durations you will see that the critical activities can change from one iteration to another It is convenient to calculate averages of these 01 values in column H To do so enter the formula RISKMEANG26 in cell H26 and copy it down Initially the values in this column are meaningless However after running the simulation they indicate the fraction of iterations that result in 1 This fraction is an estimate of the probability that the activity is critical 4 Summary measures Enter RISK statistical functions in column K for the project completion time For example enter RISKMEANB43 in cell K27 and RISKPERCENTILEB43J30 in cell K30 Running the Simulation You should set the number of iterations to 1000 and the number of simulations to 1 and then run the simulation in the usual way Discussion of the Simulation Results After running the simulation you can request the histogram of project times shown in Figure 1521 Recall from Example 151 that when the activity times are not random the project time is 62 days Now it varies from a low of 5445 days to a high of 7567 days with an average of 6283 days5 Because the company is interested in the probability of fin ishing the project within 60 days we moved the left slider in the graph to 60 This indicates that there is only about a 237 chance of achieving the deadline In the other direction you can see that there is about a 5 chance that the project will take longer than 6927 days This is certainly not good news for the company and it might have to resort to the crashing discussed in the previous section 154 Models with Uncertain Activity Times 1533 Figure 1521 Histogram of Project Completion Time 5It can be shown mathematically that the expected project time is always greater than when the expected activity times are used to calculate the project time as in Example 151 In other words an assumption of certainty always leads to an overly optimistic underestimation of the true expected project time Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it The RISK averages of 01 values in the range H26H40 of Figure 1520 indicate the fraction of iterations where each activity was critical Several of these fractions notably for activities A B and O are very close or equal to 1 This means that these activities are almost always or always critical Evidently only very unusual values for the random durations can make these activities noncritical Similarly activities I and K are never criti cal their fractions are equal to 0 The fractions for the other activities are less extreme Any one of them could easily be on the critical path Therefore there is no single critical path It depends completely on the random durations that happen to be observed One last observation is that the 01 values in column G are all or nothing That is if the slack changes from 0 to a very small positive number such as 000023 then the 01 variable in column G changes from 0 to 1 and indicates that the corresponding activity is noncritical This evidently happened in the few iterations where activities A and B were not critical They were still very close to being critical 1534 Chapter 15 Project Management MODELING ISSUES The traditional PERT approach to project scheduling with uncertain activity times does not involve simulation Instead it starts with a minimum most likely and maximum estimate of each activitys time just as we did with the PERT distribution Then it uses formulas to find the mean and standard deviation of each activity time distribution and finally it uses an approximate method to calculate the mean and standard deviation of the time to com plete the project This method has been in use for many years and it is found in many text books on project management However we favor the simulation approach used here because it has the following benefits over the traditional approach 1 it is more straight forward and easier to understand 2 it permits any distributions for the activity times not just the PERT distributions we used 3 it provides estimates of the probabilities that the various activities are critical and 4 it even allows you to build correlation with the RISKCORRMAT function into the activity times In short the simulation approach is more flexible and it can be implemented easily with Excel and RISK P R O B L E M S SkillBuilding Problems 20 In the model in Example 155 suppose bonuses and penalties are incurred for earliness or lateness Specifically suppose a bonus of 2000 is received if the project is completed within 60 days an extra bonus of 1000 is received if the project is completed within 58 days and a penalty of 1000 is incurred for every full day past a project completion of 64 days For example if the project is completed in 667 days the penalty is 2000two full days late Modify the model appropri ately and then run the simulation to find the distribution of the net monetary outcome negative if a penalty posi tive if a bonus What is the expected value of this net amount What is the probability of a 3000 total bonus What is the probability of a penalty of at least 4000 21 We indicated in Example 155 that the mean project length from the simulation is greater than the project length of 62 days from substituting the mean activity durations the ones used in earlier sections Note that the PERT distributions we used in the example with the exception of activity D are either symmetric around the most likely value or skewed to the right Could this skewness to the right lead to the rather large mean pro ject length from the simulation Experiment with the parameters of the PERT distributions in the example always keeping the same mean durations For example you could change the parameters of activity A from 8 9 16 to 7 10 13 to make it symmetric or to 4 11 12 to make it skewed to the left Each of these has the same mean 10 and there are many other combinations that have mean 10 that you could try Run the simulation Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it discussing MS Project allows you to save your work in an mpp file The file for this project is LAN Projectmpp Note that the start time for each project is 1302006 the day this section was originally written and the finish times are automatically entered as the start times plus the durations7 Note also that MS Project does not require Start and Finish nodes although you can add these with 0 durations if you like The next step is to enter the immediate predecessors of the tasks in the Predecessors column These appear in the nexttolast column of Figure 1523 For example the prede cessors of activity 6 purchase equipment are activities 3 and 4 and they are entered as 34 in the Predecessors column Note that we have now specified that the project can start on 912006 and working is permitted on weekends As soon as this information on durations and immediate predecessors is entered MS Project does the required CPM cal culations behind the scenes By rightclicking in the gray row at the top of the window you can ask for various columns of information to be inserted As Figure 1523 indicates we asked for the early start and finish times the late start and finish times the free and total slacks and the immediate successors You do not need to do anything to create these columns all you need to do is ask for them Note that MS Project shows each Start time 1536 Chapter 15 Project Management 7By default Project skips over the weekends For example note that the first activity with duration 10 days goes from Monday through Friday and then the next Monday through the next Friday However it is possible to change a setting so that work is performed over weekends as we do in later figures Figure 1522 Tasks for the LAN Project Figure 1523 Tasks and Precedence Relations for LAN Project Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it third column as the Early Start time by default However we know that tasks with slack can start anywhere between their early and late start times Technical Note Free and Total Slacks The quantity we have called slac k is often called total slac k A task s total slack is the amount of time the task can be delayed before the project finish date is delayed For exam ple task 3 selecting the server has a total slack of 2 days It can start as early as Sunday 9172006 but if it is delayed by 2 days and doesn t start until the following Tuesday the project as a whole will not be delayed There is also another slack called the free slack A tasks free slack is defined as the amount of time a task can be delayed without delaying its successor tasks For example note that task 3s successors tasks 5 and 7 have early start times Satur day 9232006 and F riday 9292006 T ask 3 s early f inish time is F riday 9222006 so if task 3 is delayed at all the early start time of one of its successors task 5 will be delayed This explains the free slack of 0 for task 3 However this free slack for task 3 is probably less relevant than its total slack because task 5 itself has slack and can there fore be delayed without delaying the project MS Project automatically creates a fairly large number of charts that you can view We show two of them in Figures 1524 and 1525 The Gantt chart in Figure 1524 is essentially the same as the one we constructed in Excel except that the order of tasks from top to bottom is reversed You can hover the cursor over any of these bars to see more information about the associated tasks The AON project diagram part of which appears in Figure 1525 shows the precedence relationships as well as the start and finish times the durations and information about resources used which we havent specified for this 155 A Brief Look at Microsoft Project 1537 Figure 1524 Gantt Chart for the LAN Project Figure 1525 Network Diagram for the LAN Project Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it project Although it is not visible in this black and white drawing the critical activities and noncritical activities appear in different colors on a computer screen In addition you can request many reports One possibility not shown here lists information about the activities on the critical path Of course MS Project figures out which activities are on the critical path All you need to do is ask for the report MS Project is a very powerful and featurerich software package however it does have some drawbacks First it has no builtin optimizer such as Solver to perform any optimiza tion such as crashing to meet a deadline Second it assumes a deterministic world where the durations of the activities are known with certainty Of course you can change any durations manually to see how the project as a whole is affected but you cannot run a simulation with random durations as we did with RISK In spite of these drawbacks MS Project and other project management software packages play a prominent role at many organizations and we wouldnt be surprised if some of you end up using one of these packages in your jobs 156 CONCLUSION As we have indicated in this chapter project management is an area all in itself This is due to the importance of managing large and costly projects in most organizations Many entire books have been written about the various aspects of project management and the topics we have covered here form only a relatively small percentage of the material in these books Nevertheless you have seen that management science offers a number of tools that are useful in scheduling and allocating resources to projects Among others these tools include 1 the CPM calculations used to determine the length of a project and its critical path 2 optimization models for crashing activities to meet a deadline at minimum cost and 3 simulation models for determining how the length of a project is affected by uncertain task times Finally you have seen that a number of software packages such as MS Project are devoted entirely to project management Although these packages lack some of the features available with Excel notably optimization and simulation they can be very effective for managing the timing and required resources of realworld projects 1538 Chapter 15 Project Management Summary of Key Management Science Terms Term Explanation Page CPM Critical Path Method used to analyze projects with known activity times 153 PERT Program and Evaluation Review Technique used to analyze projects with 153 random activity times Duration Time to complete an activity in a project 155 Immediate predecessor Activity that must be completed before a given activity can begin 155 Immediate successor Activity that cant start until a given activity is completed 155 Critical activity Activity whose delay will necessarily delay the completion of the project 156 Critical path Set of all critical activities also called the bottleneck path 156 Slack Amount a noncritical activity can be delayed without delaying the project 157 Earliest and latest Earliest and latest times an activity can start and finish given the 158 starting times precedence relationships in the project Earliest and latest Earliest and latest times an activity can start and finish without delaying 158 finish times the project Gantt chart Chart that shows the schedule of activities 1511 Crashing Reducing activity times at a cost to meet a deadline 1514 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 156 Conclusion 1539 P R O B L E M S SkillBuilding Problems 24 A company has a project that consists of 11 activities described in the file P1524xlsx Draw an AON pro ject network and then find the critical path and the minimum number of days required to complete this project Also create the associated Gantt chart 25 Before a new product can be introduced at Kehls the activities shown in the file P1525xlsx must be com pleted where all times are in weeks a Draw the AON project network and determine a critical path and the minimum number of weeks required before the new product can be introduced b The duration of each activity can be reduced by up to two weeks at the following cost per week A 80 B 60 C 30 D 60 E 40 F 30 G 20 Assume that activity H cannot be crashed Determine how to minimize the cost of getting the product into the stores for the peak Christmas sales period assuming that it is now 12 weeks before this period begins 26 The promoters of a rock concert in Indianapolis must perform the tasks shown in the file P1526xlsx before the concert can be held All durations are in days Draw the AON project network Then find the critical path and the minimum number of days needed to prepare for the concert and create the associated Gantt chart 27 Consider the simplified list of activities and prede cessors that are involved in building a house as shown in the file P1527xlsx a Draw an AON project network and find the critical path and the minimum number of days needed to build the house Also create the associated Gantt chart b Suppose that by hiring additional workers the duration of each activity can be reduced The costs per day of reducing the duration of the activities are also given in the file P1527xlsx Find the strategy that minimizes the cost of completing the project within 20 days 28 A company is planning to manufacture a product that consists of three parts labeled A B and C The com pany anticipates that it will take five weeks to design the three parts and determine the way in which these parts must be assembled to make the final product Then the company estimates that it will take four weeks to make part A five weeks to make part B and three weeks to make part C The company must test part A after it is completed and the testing takes two weeks The assembly line process will then proceed as follows assemble parts A and B two weeks and then attach part C one week Then the final product must undergo one week of testing Draw the AON project network Then find the critical path and the minimum amount of time needed to complete the project and create the associated Gantt chart 29 Horizon Cable is about to expand its cable TV offer ings in Smalltown by adding MTV and other stations The activities listed in the file P1529xlsx must be completed before the service expansion can be com pleted Draw the AON project network and find the critical path and the minimum number of weeks needed to complete the project Also create the associated Gantt chart 30 When an accounting firm audits a corporation the first phase of the audit involves obtaining knowledge of the business This phase of the audit requires the activities listed in the file P1530xlsx a Draw the AON project network and determine the critical path and the minimum number of days needed to complete the first phase of the audit Also create the associated Gantt chart Summary of Key Excel Terms Term Explanation Excel Page Gantt chart Way to show activity durations through See Excel Tip 1512 time in a meaningful way PERT distribution Useful for simulating activity times Use RISKPERT function in RISK 1532 Microsoft Project Separate from Excel but a useful package 1535 for analyzing multiactivity projects Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it b Assume that the first phase must be completed within 30 days The duration of each activity can be reduced by incurring the costs listed in the same file Find the strategy that minimizes the cost of meeting this deadline 31 The city of Bloomington is about to build a new water treatment plant After the plant is designed D we can select the site S the building contractor C and the operating personnel P After the site is selected we can erect the building B We can order the water treatment machine W and prepare the operations manual M only after the contractor is selected We can begin train ing T the operators when both the operations manual and operating personnel selection are completed When the treatment plant and the building are finished we can install the treatment machine I After the treatment machine is installed and operators are trained we can obtain an operating license L Assume that the time in months needed to complete each activity is normally distributed with the means and standard deviations given in the file P1531xlsx Use simulation to estimate the probability that the project will be completed in a under 50 days and b more than 55 days Also estimate the probabilities that B I and T are critical activities 32 To build Indiana Universitys new law building the activities in the file P1532xlsx must be completed all times are in months Assume that all activity times are normally distributed with the means and standard deviations given in the file a Estimate the probability that the project will take less than 30 months to complete b Estimate the probability that the project will take more than three years to complete c For each of the activities A B C and G estimate the probability that it is a critical activity 33 To complete an addition to the Business Building the activities in the file P1533xlsx must be completed all times are in months Assume that all activity times are normally distributed with the means and standard deviations given in the file The project is completed after Room 111 has been destroyed and the main structure has been built a Estimate the probability that it will take at least three years to complete the addition b For each activity estimate the probability that it will be a critical activity 34 Tom Jacobs an independent contractor has agreed to build a new room on an existing house He plans to begin work on Monday morning June 1 The main concern is when he will complete the project given that he works only on weekdays The work proceeds in stages labeled A through J as summarized in the table in the file P1534xlsx Three of these activities wiring plumbing and duct work will be done by sep arate independent subcontractors 1540 Chapter 15 Project Management a How long will the project take to complete given the activity times durations in the table Which are the critical activities b Use a oneway data table to see how sensitive the project completion time is to the duration of activ ity H hanging dry wall Let the duration vary from 2 to 8 days in increments of 05 day c Use a twoway data table to see how sensitive the project completion time is to the duration of activi ties E and F electrical wiring and plumbing Let the durations of each of these activities vary from 2 to 6 days in increments of 05 day d Tom is currently subcontracting the electrical wiring plumbing and duct work This explains why these three activities can be performed simultaneously Suppose instead that Tom plans to do the first two of these by himself and he can work on only one activity at a timeelectrical wiring and then plumbing Modify the critical path model appropriately How much does the project comple tion time increase What is the new critical path e Continuing part d where electrical wiring must be done before plumbing suppose Tom must com plete the project within a deadline of 17 days You are given the crashing data in the file P1534xlsx What should he do f How difficult is it to add new activities to an exist ing project scheduling model Answer this ques tion by assuming that Tom must also install bookshelves in the room and these can be installed only after the drywall has been hung It typically takes 25 days to install the bookshelves However he has been instructed to make these bookshelves from a special type of wood which must be custom ordered He can place the order right away and it is likely to take 10 working days to arrive In addi tion he has been instructed to install a wet bar in the room This cannot be started until the plumbing and electrical wiring are finished and this wet bar takes an estimated 35 days to finish Find the new project completion time Does the critical path change because of the new activities 35 In the previous problem all of Toms activities have fixed durations Now assume they have PERT distribu tions with the parameters listed in the file P1535xlsx a Use RISK to simulate this project What is the mean length of time required to complete the pro ject What is the probability that it will be com pleted within 20 days What is the probability that it will require more than 23 days to complete b Are there activities that are always or almost always critical Are there activities that are never or almost never critical For each other activity what is the probability that it is critical c For any activities that are never or almost never critical you might expect that the durations of Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it these activities are not highly correlated with the total project time Use RISKs sensitivity analy sis with the correlation option to see whether this is the case What correlations between the inputs and the output do you find Can you explain why they turn out as they do SkillExtending Problems 36 Realworld projects often have milestones where costs are incurred or payments are received Usually the costs are incurred relatively early and the payments are received relatively late Because of the time value of money it is advantageous to incur the costs as late as possible and receive the payments as early as possi ble Consider the AON diagram in Figure 1526 As before the circles denote activities the arrows denote precedence relationships and the numbers next to the circles are durations in months The diamonds denote milestones and the number next to each milestone denotes the cost incurred if negative or the payment received if positive when that milestone is reached The problem is to maximize the NPV of all cash flows payments minus costs by choosing the starting times of the activities appropriately Develop a Solver model to do so using an annual discount rate of 10 For dis counting purposes you can assume that if a milestone is reached after say 10 months of work then the cost or payment is incurred at the end of month 10 37 Based on LeBlanc et al 2000 A construction com pany has eight project managers and has 14 projects scheduled for the next 12 weeks Each project must be assigned a project manager The start and finish week for each project as well as the hours per week each project manager would need to spend on a project are given in file P1537xlsx For example project 1 starts at the beginning of week 4 and finishes at the end of 156 Conclusion 1541 week 10 for a duration of seven weeks Also note that if manager 2 is assigned to project 1 he will work 50 hours per week on the project In assigning man agers the company has a policy of not allowing a manager to work more than 70 hours a week Given this constraint and the fact that all projects must be done the company wants to minimize the total number of weeks during which managers work more than 50 or less than 30 hours Note that given the data for the problem working fewer than 30 hours in a week means not working that week at all How would you assign managers to projects Hint This problem is conceptually fairly simple but the bookkeeping is difficult Here is one possibility Let the changing cells be a column of indexes of the managers assigned to the various projects For example the changing cell for project 1 is 4 if we assign manager 4 to project 1 Based on the values in these changing cells use a lookup function to find the number of hours used by each project For example again assuming manager 4 is assigned to project 1 this lookup should return 38 for project 1 Now create a table with weeks along the top and projects along the side Each entry in the table should indicate how many hours are spent on each project each week IF functions work here Finally create one more table with weeks along the top and managers along the side and use SUMIF functions based on the data in the previous table to calculate the number of hours each manager is working each week As you can probably guess you will need to use Evolutionary Solver if you set it up this way Also you might have to let Evolutionary Solver run for a long time This is not an easy problem 38 Consider a project with six activities The CPM method has already been implemented with the results shown in the file P1538xlsx All times are in months This file also shows the number of workers Milestone 1 Start A B C D E Milestone 2 Milestone 3 Milestone 4 Finish 2 months 8 months 4 months 8 months 3 months 4000 9000 3000 5000 Figure 1526 AON Diagram for a Project with Milestones Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it of type A the number of workers of type B and the material costs per month for each of the activities Workers of type A receive 1600 per month and workers of type B receive 2400 per month a Create a table and then an associated line chart that shows the monthly cash flows through the end of the project when each activity is started at its earliest start time and when it is started at its latest start time That is two series should be plotted on the chart b Suppose the company in charge of this project wants to find the start times for the activities so that the NPV of the cash flows is minimized using an annual discount rate of 10 Create a Solver model to do this The only constraints are that the start times must be within their earliest and latest start time ranges 39 One problem with our Excel implementation of the CPM method is that the maximum and minimum for mulas for the earliest start time and the latest finish times have to be tailored to the specific AON network 1542 Chapter 15 Project Management That is you cant enter formulas for a typical activity and then copy them down for the other activities However there is a clever way of doing this if you are willing to use some advanced Excel functions8 This method is illustrated in the file P1539xlsx for the LAN project from Example 151 The text box in this file explains a few things about the new formulas including the fact that they deliberately create circular references a Use online help to learn exactly what the formulas for the earliest start times and latest finish times are doing and why one formula fits all for each Then explain in words how they work b Implement this method for the project in Figure 1527 You can make up any durations for the activities 8We thank Cliff Ragsdale a fellow textbook author for discovering this method A D F B H I E C G Start Finish J K M O T U V L N Q S P R Figure 1527 The AON Project Network Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 161 Multiobjective Decision Making C H A P T E R EVALU ATING AND P RIORITIZING PR OJECTS A T NASA M ore public pressure than ever before is on NASA to justify its choice of projects to undertake There is demand for accountability pressure to cut costs and an increasing number of potential projects to choose from In the past a committee of 15 members from NASA met once a year to review the 30 to 50 proposals submitted by contractors and divisions with the Kennedy Space Center The five voting members the decision makers or DMs gave each proposal a score from 1 to 10 the scores were averaged over the five DMs and the top scoring proposals were selected until the budget was exceeded Because the process was viewed as intuitive manage ment expressed concern about its subjectivity and consistency It wanted to replace this process with a more comprehensive and structured process Tavana 2003 describes the system he developed to meet these needs He calls it consensusranking organizationalsupport system CROSS The selection of projects at NASA is clearly a multiobjective decision making problem As Tavana describes there are a number of stakeholders for each project Essentially they are the different departments within NASAincluding Safety Systems Engineering Reliability and othersand each has its own criteria for a successful project For example Safety might be concerned about eliminating the possibility of death or serious injury Systems Engineering might be concerned about eliminating reliance on iden tified obsolete technology and Reliability might be concerned about increas ing the mean time between failures CROSS uses AHP Analytic Hierarchy Process discussed later in this chapter to obtain the information each DM Nosepress Dreamstimecom 16 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it needs to obtain a score for each project It then combines the DMs scores to get an overall consensus ranking of projects Finally it uses this consensus ranking along with project costs and the overall budget to select the projects to be funded More specifically the system first asks each DM to use AHP to evaluate the impor tance of the various stakeholders For example one DM might give Safety an importance weight of 05 whereas another might give Safety a weight of 04 In the next step each stakeholder is asked to use AHP to evaluate the importance of its various criteria This leads to a set of weights for each stakeholdercriterion combination The stakeholders are also asked to estimate the probability that each potential project will be successful in satisfying each criterion The system uses these probabilities to adjust the previous weights Next all of the weights from AHP are used to calculate a projectsuccess factor for each project as assessed by each DM and these factors are used to obtain each DMs rankings of the projects Finally the system attempts to reach consensus in the rankings using another nonAHP methodology The system is now being used successfully to select NASA projects As a measure of its perceived quality71 projects were submitted during the first two years of implementa tion of CROSSUsing this systemthe DMs chose 21 projects of the 71and management subsequently approved all 21 choices 162 Chapter 16 Multiobjective Decision Making 161 INTRODUCTION In many of your classes you have probably discussed how to make good decisions Usually you assume that the correct decision optimizes a single objective such as profit maximization or cost minimization In most situations you encounter in business and life however more than one relevant objective exists For example when you graduate many of you will receive several job offers Which should you accept Before deciding which job offer to accept you might consider how each job scores on several objectives such as salary interest in work quality of life in the city you will live in and nearness to family In this situation combining your multiple objectives into a single objective is difficult Similarly in determining an optimal investment portfolio you want to maximize expected return but you also want to minimize risk How do you reconcile these conflicting objec tives In this chapter we discuss three tools goal programming tradeoff curves and the Analytic Hierarchy Process that decision makers can use to solve multiobjective prob lems We show how to implement all three of these tools in a spreadsheet FUNDAMENTAL INSIGHT Optimizing with Multiple Objectiv es When there are multiple objectives you can proceed in several fundamental ways First you can prioritize your objectiv es This is done in g oal pr ogramming where the highest priority objective is optimized first then the second and so on Second you can optimize one objective while constraining the others to be no worse than specified values This approach is used to find tradeoff cur ves between the objectiv es Finally you can attempt to weight the objectives to measure their importance relative to one anotherThis is the approach taken b y the Analytic Hierarchy Process All of these a pproaches have their critics but they can all be used to mak e difficult decision pr oblems manageable Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 162 GOAL PROGRAMMING In many situations a company wants to achieve several objectives Given limited resources it may prove impossible to meet all objectives simultaneously If the company can prioritize its objectives then goal programming can be used to make good decisions The following media selection problem is typical of the situations in which goal program ming is useful This example presents a variation of the advertising model discussed in Chapters 4 and 7 162 Goal Programming 163 E X A M P L E 161 DETERMINING AN ADVERTISING SCHEDULE AT LEON BURNIT T he Leon Burnit Ad Agency is trying to determine a TV advertising schedule for a client The client has three goals listed here in descending order of importance con cerning whom it wants its ads to be seen by Goal 1 at least 65 million highincome men HIM Goal 2 at least 72 million highincome women HIW Goal 3 at least 70 million lowincome people LIP Burnit can purchase several types of TV ads ads shown on live sports shows on game shows on news shows on sitcoms on dramas and on soap operas At most 700000 total can be spent on ads The advertising costs and potential audiences in millions of viewers of a oneminute ad of each type are shown in Table 161 As a matter of policy the client requires that at least two ads each be placed on sports shows news shows and dramas Also it requires that no more than 10 ads be placed on any single type of show Burnit wants to find the advertising plan that best meets its clients goals Table 161 Data for the Advertising Example Ad Type HIM HIW LIP Cost Sports show 7 4 8 120000 Game show 3 5 6 40000 News 6 5 3 50000 Sitcom 4 5 7 40000 Drama 6 8 6 60000 Soap opera 3 4 5 40000 Objective To use goal programming to meet the companys goals of reaching various target audiences as much as possible while staying within an advertising budget WHERE DO THE NUMBERS COME FROM As in previous advertising models the company needs to estimate the number of viewers reached by each type of ad and it needs to know the cost of each ad Beyond this however management determines the goals They can set whatever goals they believe are in the companys best interests and they can prioritize these goals Solution The variables and constraints for this advertising model are shown in Table 162 Most of this is the same as in optimization models in previous chapters However the objective is not obvious and the table includes deviations from goals and balances for goals You Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 4 Exposures obtained Calculate the number of people in millions in each group that the ads reach in the Exposures range Specifically enter the formula SUMPRODUCTB7G7Numberpurchased in cell B26 for the HIM group and copy this to the rest of the Exposures range for the other two groups USING SOLVER The completed Solver dialog box is shown in Figure 162 At this point there is no objec tive to maximize or minimize The goal at this point is to find any solution that meets all of the constraints When you click on Solve you get the message that there is no feasible solution because it is impossible to meet all of the clients goals and stay within the budget To see how large the budget must be to meet all goals you can run SolverTable with the Budget cell as the single input cell varied from 700 to 850 and any cells as the output cells We chose the numbers of exposures cells as output cells The results appear in Figure 163 They show that unless the budget is greater than 750000 it is impossible to meet all of the clients goals 162 Goal Programming 165 Figure 162 Solver Dialog Box for Finding a Feasible Solution 1 2 3 4 5 6 7 8 9 10 11 A B C D E F Oneway analysis for Solver model in LP Model worksheet Budget cell D22 values along side output cells along top Exposures1 Exposures2 Exposures3 700 Not feasible 725 Not feasible 750 Not feasible 775 65000 72000 70000 800 65000 72000 70000 825 65000 72000 70000 850 65000 72000 70000 Figure 163 Checking How Large the Budget Must Be Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Use goal programming to determine an optimal pro duction schedule 6 Based on Steuer 1984 Deancorp produces sausage by blending beef head pork chuck mutton and water The cost per pound fat per pound and protein per pound for these ingredients are listed in the file P1606xlsx Deancorp needs to produce 1000 pounds of sausage and has set the following goals listed in order of priority Goal 1 Sausage should consist of at least 15 protein Goal 2 Sausage should consist of at most 8 fat Goal 3 Cost per pound of sausage should not exceed 006 Use a goal programming model to determine the com position of sausage 7 Based on Welling 1977 The Touche Young account ing firm must complete three jobs during the next month Job 1 will require 500 hours of work job 2 will require 300 hours and job 3 will require 100 hours At present the firm consists of five partners five senior employees and five junior employees each of whom can work up to 40 hours per week The dol lar amount per hour that the company can bill depends on the type of accountant assigned to each job as shown in the file P1607xlsx The X indi cates that a junior employee does not have enough experience to work on job 1 All jobs must be com pleted Touche Young has also set the following goals listed in order of priority Goal 1 Monthly billings should exceed 74000 Goal 2 At most one partner should be hired Goal 3 At most three senior employees should be hired Goal 4 At most one junior employee should be hired Use goal programming to help Touche solve its problem 8 There are four teachers in the Faber College Business School Each semester 200 students take each of the following courses Marketing Finance Production and Statistics The effectiveness of each teacher in teaching each course is given in the file P1608xlsx Each teacher can teach a total of 200 students during the semester The dean has set a goal of obtaining an average teaching effectiveness level of at least 6 in each course Deviations from this goal in any course are considered equally important Determine the semesters teaching assignments 9 The city of Bloomington has 17 neighborhoods The number of high school students in each neighborhood and the time required to drive from each neighborhood to each of the citys two high schools North and South are listed in the file P1609xlsx The Bloomington Board of Education needs to determine 162 Goal Programming 1611 how to assign students to high schools All students in a given neighborhood must be assigned to the same high school The Board has set in order of priority from highest to lowest the following goals Goal 1 Ensure that the difference in enrollment at the two high schools differs by at most 50 Goal 2 Ensure that average student travel time is at most 13 minutes Goal 3 Ensure that at most 4 of the students must travel at least 25 minutes to school a Determine an optimal assignment of students to high schools b If the enrollment at the two high schools can differ by at most 100 a change in goal 1 how does your answer change SkillExtending Problems 10 Based on Lee and Moore 1974 Faber College is admitting students for the class of 2007 Data on its applicants are shown in the file P1610xlsx Each row indicates the number of instate or outofstate appli cants with a given SAT score who plan to be business or nonbusiness majors For example 1900 of its in state applicants have a 700 SAT score and 1500 of these applicants plan to major in business Faber has set four goals for this class listed in order of priority Goal 1 The entering class should include at least 5000 students Goal 2 The entering class should have an average SAT score of at least 640 Goal 3 The entering class should consist of at least 25 outofstate students Goal 4 At least 2000 members of the entering class should not be business majors Use goal programming to determine how many appli cants of each type to admit Assume that all applicants who are admitted will decide to attend Faber 11 During the next four quarters Wivco faces the follow ing demands for globots quarter 1 13 quarter 2 14 quarter 3 12 quarter 4 15 Globots can be produced by regulartime labor or by overtime labor Production capacity number of globots and production costs during the next four quarters are shown in the file P1611xlsx Wivco has set the following goals in order of importance Goal 1 Each quarters demand should be met on time Goal 2 Inventory at the end of each quarter should not exceed three units Goal 3 Total production cost should be no greater than 250 Use a goal programming model to determine Wivcos production schedule for the next four quarters Assume that at the beginning of the first quarter one globot is in inventory Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 12 Lucys Music Store at present employs five fulltime employees and three parttime employees The normal workload is 40 hours per week for fulltime employees and 20 hours per week for parttime employees Each fulltime employee is paid 6 per hour for work up to 40 hours per week and can sell five recordings per hour A fulltime employee who works overtime is paid 10 per hour Each parttime employee is paid 3 per hour and can sell three recordings per hour It costs Lucy 6 to buy a recording and each recording sells for 9 Lucy has weekly fixed expenses of 500 She has established the following weekly goals in order of priority Goal 1 Sell at least 1600 recordings per week Goal 2 Earn a profit of at least 2200 per week Goal 3 Fulltime employees should work at most 100 hours of overtime Goal 4 To promote a sense of job security the number of hours by which each fulltime employee fails to work 40 hours should be minimized Use a goal programming model to determine how many hours per week each employee should work 13 Based on Taylor and Keown 1984 Gotham City is trying to determine the type and location of recre ational facilities to build during the next decade Four types of facilities are under consideration golf courses swimming pools gymnasiums and tennis courts Six sites are under consideration If a golf course is built it must be built at either site 1 or site 6 Other facilities can be built at sites 2 through 5 The amounts of available land in thousands of square feet at sites 2 through 5 are given in the file P1613xlsx The cost of building each facility in thousands of dollars the annual maintenance cost in thousands of dollars for each facility and the land in thousands of square feet required for each facility are also given in the same file The number of userdays in thousands for each type of facility also shown in this file depends on where it is built a Consider the following set of priorities Priority 1 The amount of land used at each site should be no greater than the amount of land available Priority 2 Construction costs should not exceed 12 million Priority 3 Userdays should exceed 200000 Priority 4 Annual maintenance costs should not exceed 200000 For this set of priorities use goal programming to determine the type and location of recreation facili ties in Gotham City b Consider the following set of priorities Priority 1 The amount of land used at each site should be no greater than the amount of land available 1612 Chapter 16 Multiobjective Decision Making Priority 2 Userdays should exceed 200000 Priority 3 Construction costs should not exceed 12 million Priority 4 Annual maintenance costs should not exceed 200000 For this set of priorities use goal programming to determine the type and location of recreation facili ties in Gotham City 14 A small aerospace company is considering eight projects Project 1 Develop an automated test facility Project 2 Bar code all inventory and machinery Project 3 Introduce a CADCAM system Project 4 Buy a new lathe and deburring system Project 5 Institute an FMS Flexible Manufacturing System Project 6 Install a LAN Local Area Network Project 7 Develop an AIS Artificial Intelligence Simulation Project 8 Set up a TQM Total Quality Management program Each project has been rated on five attributes return on investment ROI cost productivity improvement workforce requirements and degree of technological risk These ratings are given in the file P1614xlsx The company has set the following five goals listed in order of priority Goal 1 Achieve an ROI of at least 3250 Goal 2 Limit cost to 1300 Goal 3 Achieve a productivity improvement of at least 6 Goal 4 Limit workforce use to 108 Goal 5 Limit technological risk to a total of 4 Use goal programming to determine which projects should be undertaken 15 A new president has just been elected and has set the following economic goals listed from highest to lowest priority Goal 1 Balance the budget this means revenues are at least as large as costs Goal 2 Cut spending by at most 150 billion Goal 3 Raise at most 550 billion in taxes from the upper class Goal 4 Raise at most 350 billion in taxes from the lower class Currently the government spends 1 trillion per year Revenue can be raised in two ways through a gas tax and through an income tax You must determine G the pergallon tax rate in cents T1 the tax rate charged on the first 30000 of income T2 the tax rate charged on any income earned over 30000 and C the cut in spending in billions If the government chooses G T1 and T2 then we assume that the revenue given in the file P1615xlsx in billions of dollars is raised Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Of course the tax rate on income over 30000 must be at least as large as the tax rate on the first 30000 of income Use goal programming to help the president meet his goals 16 The HAL computer must determine which of eight research and development RD projects to undertake For each project four quantities are of interest 1 the net present value NPV in millions of dollars of the project 2 the annual growth rate in sales generated by the project 3 the probability that the project will succeed and 4 the cost in millions of dollars of the project The relevant information is given in the file P1616xlsx HAL has set the following four goals Goal 1 The total NPV of all chosen projects should be at least 200 million Goal 2 The average probability of success for all projects chosen should be at least 075 Goal 3 The average growth rate of all projects chosen should be at least 15 Goal 4 The total cost of all chosen projects should be at most 1 billion For the following sets of priorities use integer goal programming to determine the projects that should be selected a Goal 2 Goal 4 Goal 1 Goal 3 b Goal 1 Goal 3 Goal 4 Goal 2 17 Based on Klingman and Phillips 1984 The Marines need to fill three types of jobs in two cities Los Angeles and Chicago The numbers of jobs of each type that must be filled in each city are shown in the file P1617xlsx The Marines available to fill these jobs have been classified into six groups according to the types of jobs each person is capable of doing the 163 Pareto Optimality and Tradeoff Curves 1613 type of job each person prefers and the city in which each person prefers to live The data for each of these six groups are also listed in this file The Marines have the following three goals listed from highest priority to lowest priority Goal 1 Ensure that all jobs are filled by qualified workers Goal 2 Ensure that at least 8000 employees are assigned to the jobs they prefer Goal 3 Ensure that at least 8000 employees are assigned to their preferred cities Determine how the Marines should assign their work ers Note You may allow fractional assignments of workers 18 Based on Vasko et al 1987 Bethlehem Steel can fill orders using five different types of steel molds Up to three different molds of each type can be purchased Each individual mold can be used to fill up to 100 orders per year Six different types of orders must be filled during the coming year The waste in tons incurred if a type of mold is used to fill an order is shown in the file P1618xlsx where an x indicates that a type of mold cannot be used to fill an order The number of each order type that must be filled during the coming year is also shown in this file Bethlehem Steel has the following two goals listed in order of priority Goal 1 Because molds are very expensive Bethlehem wants to use at most five molds Goal 2 Bethlehem wants to have at most 600 tons of total waste Use goal programming to determine how Bethlehem should fill the coming years orders 163 PARETO OPTIMALITY AND TRADEOFF CURVES In a multiobjective problem with no uncertainty it is common to search for Pareto optimal solutions We assume that the decision maker has exactly two objectives and that the set of feasible points under consideration must satisfy a prescribed set of constraints First we need to define some terms A solution call it A to a multiobjective problem is called Pareto optimal if no other feasible solution is at least as good as A with respect to every objective and strictly better than A with respect to at least one objective A related concept is domination A feasible solution B dominates a feasible solution A to a multi objective problem if B is at least as good as A on every objective and is strictly better than A on at least one objective From this definition it follows that Pareto optimal solutions are feasible solutions that are not dominated If the score of all Pareto optimal solutions is graphed in the xy plane with the xaxis score being the score on objective 1 and the yaxis score being the score on objective 2 the graph is called a tradeoff curve It is also called the efficient frontier To illustrate sup pose that the set of feasible solutions for a multiobjective problem is the shaded region bounded by the curve AB and the axes in Figure 168 If the goal is to maximize both Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it objectives 1 and 2 then the curve AB is the set of Pareto optimal points All points below the AB curve are dominated by points on the curve As another illustration suppose the set of feasible solutions for a multipleobjective problem is all shaded points in the first quadrant bounded from below by the curve AB in Figure 169 If the goal is to maximize objective 1 and minimize objective 2 then the curve AB is the set of Pareto optimal points In this case all points to the left of the curve are dominated by points on the curve Finding a Tradeoff Curve To find a tradeoff curve you can proceed according to the following steps 1 Choose an objective say objective 1 and determine its best attainable value V1 For the solution attaining V1 find the value of objective 2 and label it V2 Then V1 V2 is a point on the tradeoff curve 2 For values V of objective 2 that are better than V2 solve the optimization problem in step 1 with the additional constraint that the value of objective 2 is at least as good as V Varying V over values of V preferred to V2 yields other points on the tradeoff curve 3 Step 1 located one endpoint of the tradeoff curve Now determine the best value of objective 2 that can be attained to obtain the other endpoint of the tradeoff curve We illustrate the concept of Pareto optimality and how to determine Pareto optimal solutions with the following example 1614 Chapter 16 Multiobjective Decision Making Dominated solutions Objective 1 Objective 2 B A Figure 168 Tradeoff Curve for Maximizing Two Objectives Dominated solutions Objective 1 Objective 2 B A Figure 169 Tradeoff Curve for Maximizing Objective 1 and Minimizing Objective 2 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Objectives To find the tradeoff curve between pollution and profit by solving a number of LP problems WHERE DO THE NUMBERS COME FROM The required data here is basically the same as in the product mix problems from Chapter 3 Of course the company also needs to find how much pollution each product is responsible for which requires some scientific investigation Solution The model itself is a straightforward version of the product mix models from Chapter 3 The objective is to find the product mix that stays within the lower and upper production limits uses no more labor or raw material than are available keeps pollution low and keeps profit high None of the formulas in the spreadsheet model see Figure 1610 and the file Pollution Tradeoffxlsx presents anything new so we focus instead on the solution procedure Referring to the general threestep procedure for finding the tradeoff curve let profit be objective 1 and pollution be objective 2 To obtain one endpoint of the curve step 1 you maximize profit and ignore pollution That is you maximize the Profit cell and delete the constraint indicated in row 26 from the Solver dialog box You can check that the solu tion has profit 20089 and pollution level 90051 This is not the solution shown in the fig ure At the other end of the spectrum step 3 you minimize the pollution in cell B26 and ignore any constraint on profit You can check that this solution has pollution level 3560 and profit 8360 In other words profit can get as high as 20089 by ignoring pollution or as low as 8360 and pollution can get as low as 3560 or as high as 9005 These establish the extremes Now you can search for points in between step 2 163 Pareto Optimality and Tradeoff Curves 1615 E X A M P L E 162 MAXIMIZING PROFIT AND MINIMIZING POLLUTION AT CHEMCON C hemcon plans to produce eight products The profit per unit the labor and raw mater ial used per unit produced and the pollution emitted per unit produced are given in Table 163 This table also includes lower and upper limits on production that Chemcon has imposed Currently 1300 labor hours and 1000 units of raw material are available Chemcons two objectives are to maximize profit and minimize pollution produced Chemcon wants to graph the tradeoff curve for this problem Table 163 Data for the Chemcon Example Product 1 2 3 4 5 6 7 8 Labor hrsunit 5 5 1 4 35 4 2 35 Raw materialunit 3 45 5 5 45 2 35 3 Pollutionunit 25 29 35 26 17 25 28 6 Profitunit 53 69 73 69 51 49 71 40 Min production 0 30 0 10 20 50 30 0 Max production 190 110 140 140 190 190 110 150 Get the two extreme points on the trade off curve by maximiz ing profit ignoring pollution and then minimizing pollution ignoring profit 1Actually this is not quite true as one user pointed out If you maximize profit and ignore pollution the result ing pollution level is 8980 To find the maximum possible pollution level you need to maximize pollution The resulting pollution level is 9005 Surprisingly the profit from this solution is less than the maximum profit 20089 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it an upper limit constraint in the previous example There the objective was to make pollu tion low Here the objective is to make exposures to women high The lower limit cell D30 becomes the single input cell for SolverTable which can vary from slightly greater than 79392 to slightly less than 89220 with suitable values in between The results appear in table form in Figure 1615 and in graphical form in Figure 1616 1620 Chapter 16 Multiobjective Decision Making 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A B C D E F G H I Oneway analysis for Solver model in Model worksheet Women lower bound cell D30 values along side output cells along top Numberpurchased1 Numberpurchased2 Numberpurchased3 Numberpurchased4 Numberpurchased5 Numberpurchased6 Exposurestowomen Exposurestomen 79393 4839 1744 6072 5000 5508 0776 79393 89515 80 4715 1835 6100 5000 5620 0928 80000 89506 81 4503 1994 6143 5000 5807 1215 81000 89449 82 4280 2163 6178 5000 5997 1555 82000 89336 83 4048 2347 6204 5000 6186 1954 83000 89156 84 3801 2538 6220 5000 6383 2421 84000 88900 85 3540 2745 6228 5000 6578 2969 85000 88554 86 3262 2976 6217 5000 6777 3604 86000 88096 87 2964 3225 6189 5000 6979 4357 87000 87500 88 2600 3580 6173 5000 7269 5000 88000 86713 89 2057 4276 6207 5000 7863 5000 89000 85478 89219 2000 5000 5387 5000 8177 5000 89219 84934 Figure 1615 SolverTable Results for the Advertising Tradeoff Model 88 89 90 o men Tradeoff of Men versus Women 84 85 86 87 78 80 82 84 86 88 90 Exposures t Exposures to women Figure 1616 Tradeoff Curve for the Advertising Example Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it is 6 Construct a tradeoff curve between the objec tives of maximizing profit and maximizing the number of units sold 22 GMCO produces three types of cars compact medium and large The variable cost per car and pro duction capacity per year for each type of car are given in the file P1622xlsx The annual demand for each type of car depends on the prices of the three types of cars also given in this file In this latter table PC is the price charged for a compact car in thousands of dollars The variables PM and PL are defined simi larly for medium and large cars Suppose that each compact car gets 30 mpg each medium car gets 25 mpg and each large car gets 18 mpg GMCO wants to keep the planet pollution free so in addition to maxi mizing profit it wants to maximize the average miles 1622 Chapter 16 Multiobjective Decision Making per gallon attained by the cars it sells Construct a tradeoff curve between these two objectives 23 In the capital budgeting example from Chapter 6 see Example 61 we maximized NPV for a given budget Now find a tradeoff curve for NPV versus budget Specifically minimize the amount invested with a lower bound constraint on the NPV obtained What lower bounds should you use Do you get the same tradeoff curve as in Figure 64 24 The portfolio optimization example from Chapter 7 see Example 79 found the efficient frontier by minimizing portfolio variance with a lower bound constraint on the expected return Do it the opposite way That is calculate the efficient frontier by maximizing the expected return with an upper bound on the portfolio standard deviation Do you get the same results as in Example 79 164 THE ANALYTIC HIERARCHY PROCESS AHP When multiple objectives are important to a decision maker choosing between alternatives can be difficult For example if you are choosing a job one job might offer the highest starting salary but rate poorly on other objectives such as quality of life in the city where the job is located and the nearness of the job to your family Another job offer might rate highly on these latter objectives but have a relatively low starting salary In this case it can be difficult for you to choose between job offers The Analytic Hierar chy Pr ocess AHP developed originally by Thomas Saaty is a powerful tool that can be used to make decisions in situations where multiple objectives are present We present an example to illustrate such a case3 Note Matrix notation and matrix multiplication are used in this section You may need to review the discussion of matrices in section 77 E X A M P L E 164 USING AHP TO SELECT A JOB J ane is about to graduate from college and is trying to determine which job to accept She plans to choose among the offers by determining how well each job offer meets the fol lowing four objectives Objective 1 High starting salary Objective 2 Quality of life in city where job is located Objective 3 Interest of work Objective 4 Nearness of job to family Objective To use the AHP method to help Jane select a job that is best in terms of the various job criteria WHERE DO THE NUMBERS COME FROM As discussed shortly Jane must make a number of tradeoffs during the implementation of AHP In this case the decision maker supplies the data 3The leading software package for implementing AHP is Expert Choice developed by Expert Choice Inc Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 Inputs Enter the pairwise comparison matrices in the shaded ranges Note that you can enter fractions such as 17 in cell C24 and have them appear as fractions by formatting the cells with the Fraction option 2 Normalized matrix Calculate the normalized matrix for the first pairwise compari son matrix in the range G5J8 This can be done quickly as follows Starting with the cursor in cell G5 highlight the range G5J8 Then type the formula B5SUMB5B8 and press ControlEnter both keys at once We introduced this really useful shortcut in an earlier chapter as a quick way to enter the same formula in an entire range 3 Weights of objectives In the range L5L8 calculate the weights for each objective Again do this the quick way Starting with the cursor in cell L5 highlight the range L5L8 Then type the formula AVERAGEG5J5 and press ControlEnter 4 Scores for jobs on objecti ves Repeat the same calculations in steps 2 and 3 for the other pairwise comparison matrices to obtain the normalized matrices in columns G through I and scores vectors in column L 5 Overall job scores In the range B37E39 form a matrix of job scores on the various objectives To get the score vector in the range L12L14 into the range B37B39 for exam ple highlight this latter range type the formula L12 and press ControlEnter Do likewise for the other three scores vectors in column L Then to obtain the overall job scores from the matrix product Sw highlight the range G37G39 type the formula MMULTB37E39L5L8 and press ControlShiftEnter Remember that ControlShiftEnter is used to enter a matrix function In contrast ControlEnter is equivalent to copying a formula to a high lighted range Again you can see that job 2 is the most preferred of the three jobs Calculating the Consistency Index We now show how to compute the consistency index CI for each of the pairwise compari son matrices See Figure 1618 which is also part of the file AHPJobsxlsx Note that columns G through K have been hidden to save space These contain the normalized matri ces from step 2 in the previous section The following steps are relevant for the first pair wise comparison matrix The others are done in analogous fashion 1 Product of comparison matrix and v ector of weights or scor es Calculate the product of the first pairwise comparison matrix and the weights vector in the range N5N8 by highlighting this range typing MMULTB5E8L5L8 and pressing CtrlShiftEnter 2 Ratios In cell O5 calculate the ratio of the two cells to its left with the formula N5L5 and copy this to the range O6O8 164 The Analytic Hierarchy Process AHP 1629 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it handle a hierarchy of objectives and subobjectiveshence the term hierarchy in the name of the procedure 2 Although the finished version of the Choosing Jobsxlsx file can be used as a tem plate for other AHP problems it is clear by now that typical users would not want to go to all this trouble to create a spreadsheet model certainly not from scratch If you intend to make any real decisions with AHP you will want to acquire specialpurpose software such as Expert Choice Alternatively you can use the file Choosing Jobs with VBAxlsm mentioned earlier 164 The Analytic Hierarchy Process AHP 1631 Automated Manufacturing Decisions Using AHP Weber 1993 reports the successful use of AHP in deciding which of several technologies to purchase for automated manufacturing As he discusses these decisions can have sev eral types of impacts quantitative financial such as purchase cost quantitative nonfinan cial such as throughput cycle time and scrap which are difficult to translate directly into dollars and qualitative such as product quality and manufacturing flexibility which are also difficult to translate into dollars When the decision maker is trying to rate the differ ent technologies along nonmonetary criteria then he or she should use the method dis cussed in this section For example how much more do you prefer technology 1 to technology 2 in the area of product quality However he advises that when quantitative financial data are available for example technology 1 costs twice as much as technology 2 then this objective information should be used in the AHP preference matrices Weber developed a software package called AutoMan to implement the AHP method This soft ware has been purchased by more than 800 customers since its first release in 1989 AHP in Saudi Arabia Bahurmoz 2003 designed and implemented a system based on AHP to select the best candidates to send overseas to do graduate studies and eventually become teachers at the Dar AlHekma womens college in Saudi Arabia Other Applications of AHP AHP has been used by companies in many areas including accounting finance market ing energy resource planning microcomputer selection sociology architecture and polit ical science See Zahedi 1986 Golden et al 1989 and Saaty 1988 for a discussion of applications of AHP ADDITIONAL APPLICATIONS P R O B L E M S SkillBuilding Problems 25 Each professors annual salary increase is determined by his or her performance in three areas teaching research and service to the university The administration has assessed the pairwise comparison matrix for these objec tives as shown in the file P1625xlsx The administra tion has compared two professors with regard to their teaching research and service over the past year The pairwise comparison matrices are also shown in this file a Which professor should receive a bigger raise b Does AHP indicate how large a raise each professor should be given c Check the pairwise comparison matrix for consistency 26 Your company is about to purchase a new PC Three objectives are important in determining which com puter you should purchase cost user friendliness and software availability The pairwise comparison matrix Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it for these objectives is shown in the file P1626xlsx Three computers are being considered for purchase The performance of each computer with regard to each objective is indicated by the pairwise comparison matrices also shown in this file a Which computer should you purchase b Check the pairwise comparison matrices for consistency 27 You are ready to select your mate for life and have determined that physical attractiveness intelligence and personality are key factors in selecting a satisfac tory mate Your pairwise comparison matrix for these objectives is shown in the file P1627xlsx Three people Chris Jamie and Pat are begging to be your mate This problem attempts to be genderneutral Your view of these peoples attractiveness intelli gence and personality is given in the pairwise com parison matrices also shown in this file a Who should you choose as your lifetime mate b Evaluate all pairwise comparison matrices for consistency 28 In determining where to invest your money two objectives expected rate of return and degree of risk are considered to be equally important Two invest ments 1 and 2 have the pairwise comparison matri ces shown in the file P1628xlsx a How would you rank these investments b Now suppose another investment investment 3 is available The pairwise comparison matrices for these investments are also shown in this file Observe that the entries in the comparison matri ces for investments 1 and 2 have not changed How would you now rank the investments Contrast your ranking of investments 1 and 2 with your answer from part a 29 You are trying to determine which MBA program to attend You have been accepted at two schools Indiana and Northwestern You have chosen three attributes to use in helping you make your decision cost starting salary for graduates and ambience of school can we party there Your pairwise compari son matrix for these attributes is shown in the file P1629xlsx For each attribute the pairwise compari son matrix for Indiana and Northwestern is also shown in this file Which MBA program should you attend 1632 Chapter 16 Multiobjective Decision Making 30 You are trying to determine which of two secretarial candidates John or Sharon to hire The three objec tives that are important to your decision are personal ity typing ability and intelligence You have assessed the pairwise comparison matrix for the three objec tives in the file P1630xlsx The score of each employee on each objective is also shown in this file If you follow the AHP method which employee should you hire SkillExtending Problems 31 A consumer is trying to determine which type of frozen dinner to eat She considers three attributes to be important taste nutritional value and price Nutritional value is considered to be determined by cholesterol and sodium level Three types of dinners are under consideration The pairwise comparison matrix for the three attributes is shown in the file P1631xlsx Among the three frozen dinners the pairwise comparison matrix for each attribute is also shown in this file To determine how each dinner rates on nutrition you will need the pairwise com parison matrix for cholesterol and sodium also shown in this file Which frozen dinner would the consumer prefer Hint The nutrition score for a dinner equals the score of the dinner on sodium multiplied by the weight for sodium plus the score for the dinner on cholesterol multiplied by the weight for cholesterol 32 Based on Lin et al 1984 You have been hired by Arthur Ross to determine which of the following accounts receivable methods should be used in an audit of the Keating Five and Dime Store analytic review method 1 confirmations method 2 or test of subsequent collections method 3 The three crite ria used to distinguish among the methods are reliabil ity cost and validity The pairwise comparison matrix for the three criteria is shown in the file P1632xlsx The pairwise comparison matrices of the three accounting methods for the three criteria are shown in this file Use AHP to determine which auditing proce dure should be used Also check the first pairwise comparison matrix for consistency 165 CONCLUSION Whenever you face a problem with multiple competing objectives as is the case in many realworld problems you are forced to make tradeoffs among these objectives This is usually a very difficult task and not all management scientists agree on the best way to proceed When the objectives are very different in nature no method can disguise the inherent complexity of comparing apples to oranges Although one method finding Pareto optimal solutions and drawing the resulting tradeoff curve locates solutions that Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it are not dominated by any others you still face the problem of choosing one of the many Pareto optimal solutions to implement The other two methods discussed in this chapter goal programming and AHP make tradeoffs and ultimately locate an optimal solution These methods have their critics but when they are used carefully they have the potential to help solve some difficult and important realworld problems Summary of Key Management Science Terms Term Explanation Page Goal programming Optimization method that prioritizes multiple objectives 163 goals tries to achieve higher priority goals before considering lower priority goals Hard constraint A constraint that must be satisfied 166 Soft constraint A constraint you would like to satisfy but dont absolutely have 166 to satisfy Pareto optimal solution Solution that is not dominated that is no other solution 1613 is at least as good on all objectives and better on at least one objective Tradeoff curve Efficient Curve showing Pareto optimal solutions used primarily 1613 frontier to show the tradeoffs between two competing objectives Analytical Hierarchy Method used to find best decision when a decision maker faces 1622 Process AHP multiple criteria requires a series of pairwise comparisons between criteria and between alternative decisions for each criterion 165 Conclusion 1633 P R O B L E M S SkillBuilding Problems 33 The Pine Valley Board of Education must hire teachers for the coming school year The types of teachers and the salaries that must be paid are given in the file P1633xlsx For example 20 teachers who are quali fied to teach history and science have applied for jobs and each of these teachers must be paid an annual salary of 21000 Each teacher who is hired teaches the two subjects he or she is qualified to teach Pine Valley needs to hire 35 teachers qualified to teach history 30 teachers qualified to teach science 40 teachers qualified to teach math and 32 teachers qualified to teach English The board has 14 million to spend on teach ers salaries A penalty cost of 1 is incurred for each dollar the board goes over budget For each teacher by which Pine Valleys goals are unmet the following costs are incurred because of the lower quality of edu cation science 30000 math 28000 history 26000 and English 24000 Determine how the board can minimize its total cost due to unmet goals 34 Stockco fills orders for three products for a local ware house Stockco must determine how many of each product should be ordered at the beginning of the current month This month 400 units of product 1 500 units of product 2 and 300 units of product 3 will be demanded The cost and space taken up by one unit of each product are shown in the file P1634xlsx If Stockco runs out of stock before the end of the month the stockout costs also shown in this file are incurred Stockco has 17000 to spend on ordering products and has 3700 square feet of warehouse space A 1 penalty is assessed for each dollar spent over the bud get limit and a 10 cost is assessed for every square foot of warehouse space needed a Determine Stockcos optimal ordering policy b Suppose that Stockco has set the following goals listed in order of priority Goal 1 Spend at most 17000 Goal 2 Use at most 3700 square feet of ware house space Goal 3 Meet demand for product 1 Goal 4 Meet demand for product 2 Goal 5 Meet demand for product 3 Develop a goal programming model for Stockco 35 BeatTrop Foods is trying to choose one of three com panies to merge with Seven factors are important in this decision Factor 1 Contribution to profitability Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it the week at least the following number of employees must be working Monday through Friday seven employ ees Saturday and Sunday three employees Lummins has set the following goals listed in order of priority Goal 1 Meet employee requirements with 11 workers Goal 2 The average number of weekend days off per employee should be at least 15 days Goal 3 The average number of consecutive days off an employee gets during the week should not exceed 28 days Use goal programming to determine how to schedule Lummins employees 41 You are the mayor of Gotham City and you must determine a tax policy for the city Five types of taxes are used to raise money Property taxes Let p be the property tax rate A sales tax on all items except food drugs and durable goods Let s be the sales tax rate A sales tax on durable goods Let d be the durable goods sales tax rate A gasoline sales tax Let g be the gasoline sales tax rate A sales tax on food and drugs Let f be the sales tax on food and drugs The city consists of three groups of people low income LI middle income MI and high income HI The amount of revenue in millions of dollars raised from each group by setting a particular tax at a 1 level is given in the file P1641xlsx For example a 3 tax on durable good sales will raise 360 million dollars from lowincome people Your tax policy must satisfy the following restrictions Restriction 1 The tax burden on MI people cannot exceed 28 billion Restriction 2 The tax burden on HI people cannot exceed 24 billion Restriction 3 The total revenue raised must exceed the current level of 65 billion Restriction 4 s must be between 1 and 3 Given these restrictions the city council has set the following three goals listed in order of priority Goal 1 Limit the tax burden on LI people to 2 billion Goal 2 Keep the property tax rate under 3 Goal 3 If their tax burden becomes too high 20 of the LI people 20 of the MI people and 40 of the HI people may consider moving to the sub urbs Suppose that this will happen if their total tax burden exceeds 15 billion To discourage this exodus goal 3 is to keep the total tax burden on these people below 15 billion Use goal programming to determine an optimal tax policy 165 Conclusion 1635 42 Based on Sartoris and Spruill 1974 Wivco produces two products which it sells for both cash and credit Revenues from credit sales will not have been received but are included in determining profit earned during the current sixmonth period Sales during the next six months can be made either from units produced during the next six months or from beginning inventory Relevant information about products 1 and 2 is as follows During the next six months at most 150 units of product 1 can be sold on a cash basis and at most 100 units of product 1 can be sold on a credit basis It costs 35 to produce each unit of product 1 and each sells for 40 A credit sale of a unit of product 1 yields 050 less profit than a cash sale because of delays in receiving payment Two hours of production time are needed to produce each unit of product 1 At the beginning of the sixmonth period 60 units of product 1 are in inventory During the next six months at most 175 units of product 2 can be sold on a cash basis and at most 250 units of product 2 can be sold on a credit basis It costs 45 to produce each unit of product 2 and each sells for 5250 A credit sale of a unit of product 2 yields 100 less profit than a cash sale Four hours of production time are needed to pro duce each unit of product 2 At the beginning of the sixmonth period 30 units of product 2 are in inventory During the next six months Wivco has 1000 hours for production available At the end of the next six months Wivco incurs a 10 holding cost on the value of ending inventory measured relative to production cost An opportunity cost of 5 is also assessed against any cash on hand at the end of the sixmonth period a Develop and solve an LP model that yields Wivcos maximum profit during the next six months What is Wivcos ending inventory position Assuming an initial cash balance of 0 what is Wivcos ending cash balance b Because an ending inventory and cash position of 0 is undesirable for ongoing operations Wivco is considering other options At the beginning of the sixmonth period Wivco can obtain a loan secured by ending inventory that incurs an interest cost equal to 5 of the value of the loan The maxi mum value of the loan is 75 of the value of the ending inventory The loan will be repaid one year from now Wivco has the following goals listed in order of priority Goal 1 Make the ending cash balance of Wivco come as close as possible to 75 Goal 2 Make profit come as close as possible to the profit level obtained in part a Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it C A S E P lay Time Toy faces a highly seasonal pattern of sales In the past Play Time has used a seasonal production schedule where the amount produced each month matches the sales for that month Under this production plan inventory is maintained at a constant level The production manager Thomas Lindop is proposing a switch to a level or constant production schedule This schedule would result in significant savings in production costs but would have higher storage and handling costs fluctuating levels of inventories and implications for financing Jonathan King president of Play Time Toy has been reviewing pro forma income statements cash bud gets and balance sheets for the coming year under the two production scenarios Table 169 shows the pro forma analysis under seasonal production and Table 1610 shows the pro forma analysis under level production 161 PLAY TIME TOY COMPANY Case 161 Play Time Toy Company 1637 Table 169 Seasonal Production Annual net profit 237 Play Time Toy Company Projected for 2011 Actual Dec 2010 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Total Production 850 108 126 145 125 125 125 145 1458 1655 1925 2057 1006 9000 sales value Inventory 813 813 813 813 813 813 813 813 813 813 813 813 813 sales value INCOME STATEMENT Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Total Net sales 108 126 145 125 125 125 145 1458 1655 1925 2057 1006 9000 Cost of goods sold Materials and 70 82 94 81 81 81 94 950 1079 1254 1340 656 5865 regular wages Overtime wages 0 0 0 0 0 0 0 61 91 131 151 0 435 Gross profit 38 44 51 44 44 44 51 447 486 539 565 350 2700 Operating expenses 188 188 188 188 188 188 188 188 188 188 188 188 2256 Inventory cost 0 0 0 0 0 0 0 0 0 0 0 0 0 Profit before 150 144 137 144 144 144 137 259 298 351 377 162 444 interest and taxes Net interest 10 2 1 1 2 2 2 3 7 18 19 19 86 payments Profit before taxes 160 146 138 146 146 147 140 256 290 333 359 144 358 Taxes 55 50 47 50 50 50 48 87 99 113 122 49 122 Net profit 106 97 91 96 97 97 92 169 192 220 237 95 237 Projected for 2011 BALANCE Actual SHEET Dec 2010 Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Cash 175 782 1365 1116 934 808 604 450 175 175 175 175 175 Accts receivable 2628 958 234 271 270 250 250 270 1603 3113 3580 3982 3063 Inventory 530 530 530 530 530 530 530 530 530 530 530 530 530 Net PE 1070 1070 1070 1070 1070 1070 1070 1070 1070 1070 1070 1070 1070 Total Assets 4403 3340 3199 2987 2804 2658 2454 2320 3378 4888 5355 5757 4838 Accts payable 255 32 38 44 38 38 38 44 437 497 578 617 302 Notes payable 680 0 0 0 0 0 0 0 408 1600 1653 1656 966 Accrued taxes 80 25 24 151 232 282 363 411 324 256 143 21 4 Long term debt 450 450 450 450 450 450 425 425 425 425 425 425 400 Equity 2938 2832 2736 2644 2548 2452 2355 2263 2431 2623 2843 3080 3175 Total liability 4403 3340 3199 2987 2804 2658 2454 2320 3378 4888 5355 5757 4838 and equity Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Altschuler S D Batavia J Bennett R Labe B Liao R Nigam and J Oh Pricing Analysis for Merrill Lynch Integrated Choice Interfaces 32 no 1 2002 519 Angel A L Taladriz and R Weber Soquimich Uses a System Based on MixedInteger Linear Programming and Expert Systems to Improve Customer Service Interfaces 33 no 4 2003 4152 Apte A U Apte R Beatty I Sarkar and J Semple The Impact of Check 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Supply Chain to End the Bullwhip Effect Interfaces 35 no 1 2005 3748 Dekle J M Lavieri E Martin H EmirFarinas and R Francis A Florida County Locates Disaster Recovery Centers Interfaces 35 no 2 2005 113139 Deming E Out of the Crisis Cambridge MA MIT Center for Advanced Engineering Study 1986 Denardo E U Rothblum and A Swersey Transportation Problem in Which Costs Depend on Order of Arrival Management Science 34 1988 774784 DeWitt C L Lasdon A Waren D Brenner and S Melhem OMEGA An Improved Gasoline Blending System for Texaco Interfaces 19 no 1 1989 85101 Dobson G and S Kalish Positioning and Pricing a Product Line Marketing Science 7 1988 107126 Dolan R and H Simon Power Pricing New York The Free Press 1996 Duffy T M Hatzakis W Hsu R Labe B Liao X Luo J Oh A Setya and L Yang Merrill Lynch Improves Liquidity Risk Management for Revolving Credit Lines Interfaces 35 no 5 2005 353369 Dunning D S Lockfort Q Ross P Beccue and J Stonebraker New York Power Authority Uses Decision Analysis to Schedule Refueling of Its Indian Point 3 Nuclear Power Plant Interfaces 31 no 5 2001 121135 Eaton D M Daskin D Simmons B Bulloch and G Jasma Determining Emergency Medical Service Vehicle Deployment in Austin Texas Interfaces 15 no 1 1985 96108 Efroymson M and T Ray A Branch and Bound Algorithm for Factory Location Operations Research 14 1966 361368 Evans J The Factored Transportation Problem Management Science 30 1984 10211024 Feinstein C Deciding Whether to Test Student Athletes for Drug Use Interfaces 20 no 3 1990 8087 Fitzsimmons J and L Allen A Warehouse Location Model Helps Texas Comptroller Select OutofState Audit Offices Interfaces 13 no 5 1983 4046 Fleischmann M J van Nunen and B Grave Integrating ClosedLoop Supply Chains and SpareParts Management at IBM Interfaces 33 no 6 2003 4456 Franklin A and E Koenigsberg Computer School Assignments in a Large District Operations Research 21 1973 413426 Franses P Do We Think We Make Better Forecasts Than in the Past A Survey of Academics Interfaces 34 no 6 2004 466468 Friel B Medicare Transactions A 50 Million Lesson in Project Management Government Executive April 2000 Gaballa A and W Pearce Telephone Sales Manpower Planning at Qantas Interfaces 9 no 3 1979 19 Garvin W W Introduction to Linear Programming New York McGrawHill 1960 Gavirneni S D Morrice and P Mullarkey Simulation Helps Maxager Shorten Its Sales Cycle Interfaces 34 no 1 2004 8796 L Clark and G Pataki Schlumberger Optimizes Receiver Location for Automated Meter Reading Interfaces 34 no 3 2004 208214 Gendron B Scheduling Employees in Quebecs Liquor Stores with Integer Programming Interfaces 35 no 5 2005 402410 Gido J and G Clements Successful Project Management 3rd edition Mason OH Thomson SouthWestern 2006 Glassey R and V Gupta A Linear Programming Analysis of Paper Recycling Studies in Mathematical Programming Ed H Salkin and J Saha New York NorthHolland 1978 Glover F and D Klingman Network Applications in Industry and Government AIIE Transactions 9 1977 363376 Goldberg D Genetic Algorithms in Search Optimization and Machine Learning Boston AddisonWesley 1989 Golden B E Wasil and P Harker The Analytic Hierarchy Process New York SpringerVerlag 1989 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it References 909 Gorman M Santa Fe Railway Uses an OperatingPlan Model to Improve Its Service Design Interfaces 28 no 4 1998 88103 Green P A Krieger and Y Wind Thirty Years of Conjoint Analysis Reflections and Prospects Interfaces 31 no 3 2 2001 S56S73 Grossman S and O Hart An Analysis of the Principal Agent Problem Econometrica 51 1983 745 Guide V D L Muyldermans and L van Wassenhove HewlettPackard Company Unlocks the Value Potential from TimeSensitive Returns Interfaces 35 no 4 2005 281293 Hansen P and R Wendell A Note on Airline Commuting Interfaces 11 no 12 1982 8587 Heady E and A Egbert Regional Planning of Efficient Agricultural Patterns Econometrica 32 1964 374386 Heyman D and S Stidham The Relation between Customer and Time Averages in Queues Operations Research 28 1980 983984 Hicks R R Madrid C Milligan R Pruneau M Kanaley Y Dumas B Lacroix J Desrosiers and F Soumis Bombardier Flexjet Significantly Improves Its Fractional Aircraft Ownership Operations Interfaces 35 no 1 2005 4960 Holland J Adaptation in Natural and Artificial Systems Ann Arbor MI University of Michigan Press 1975 Holland J Adaptation in Natural and Artificial Systems Cambridge MA MIT Press 1992 Hoppensteadt F and C Peskin Mathematics in Medicine and the Life Sciences New York SpringerVerlag 1992 Howard R Decision Analysis Practice and Promise Management Science 34 no 6 1988 679695 Huerter J and W Swart An Integrated LaborManagement System for Taco Bell Interfaces 28 no 1 1998 7591 Ignall E and P Kolesar Operating Characteristics of a Simple Shuttle under Local Dispatching Rules Operations Research 20 1972 10771088 Jacobs W The Caterer Problem Naval Logistics Research Quarterly 1 1954 154165 Johnson R and D Wichern Applied Multivariate Statistical Analysis 5th ed Upper Saddle River NJ Prentice Hall 2002 Kahn J M Brandeau and J DunnMortimer OR Modeling and AIDS Policy From Theory to Practice Interfaces 28 No 3 1998 322 Kalvaitishi R and A Posgay An Application of Mixed Integer Programming in the Direct Mail Industry Management Science 20 no 5 1974 788792 Kapuscinski R R Zhang P Carbonneau R Moore and B Reeves Inventory Decisions in Dells Supply Chain Interfaces 34 no 3 2004 191205 Keefer D and S Bodily ThreePoint Approximations for Continuous Random Variables Management Science 29 no 5 1983 595609 Keisler J W Buehring P McLaughlin M Robershotte and R Whitfield Allocating Vendor Risks in the Hanford Waste Cleanup Interfaces 34 no 3 2004 180190 Kekre S U Rao J Swaminathan and J Zhang Reconfiguring a Remanufacturing Line at Visteon Mexico Interfaces 33 no 6 2003 3043 Kelly J A New Interpretation of Information Rate Bell System Technical Journal 35 1956 917926 Kimbrough S and F Murphy A Study of the Philadelphia Knowledge Economy Interfaces 35 no 3 2005 248259 Kirkwood C An Overview of Methods for Applied Decision Analysis Interfaces 22 no 6 1992 2839 Klastorin T Project Management Tools and TradeOffs New York Wiley 2004 Klingman D and N Phillips Topological and Computations Aspects of Preemptive Multicriteria Military Personnel Assignment Problems Management Science 30 no 11 1984 13621375 Kolesar P and E Blum Square Root Laws for Fire Engine Response Distances Management Science 19 1973 13681378 T Crabill K Rider and W Walker A Queueing Linear Programming Approach to Scheduling Police Patrol Cars Operations Research 23 1974 10451062 Koschat M G Berk J Blatt N Kunz M LePore and S Blyakher Newsvendors Tackle the Newsvendor Problem Interfaces 33 no 3 2003 7284 Lancaster L The Evolution of the Diet Model in Managing Food Systems Interfaces 22 no 5 1992 5968 Lanzenauer C E Harbauer B Johnston and D Shuttleworth RRSP Flood LP to the Rescue Interfaces 17 no 4 1987 2740 Laval C M Feyhl and S Kakouros HewlettPackard Combined OR and Expert Knowledge to Design Its Supply Chains Interfaces 35 no 3 2005 238247 LeBlanc L D Randels Jr and K Swann Heery Internationals Spreadsheet Optimization Model for Assigning Managers to Construction Projects Interfaces 30 No 6 2000 95106 J Hill G Greenwell and A Czesnat Nukotes Spreadsheet Linear Programming Models for Optimizing Transportation Interfaces 34 No 2 2004 139146 and M Galbreth Designing LargeScale Supply Chain Linear Programs in Spreadsheets Communications of the ACM 50 no 8 2007a 5964 and M Galbreth Implementing LargeScale Optimization Models in Excel Using VBA Interfaces 37 no 4 2007b 370382 Lee S and L Moore Optimizing University Admissions Planning Decision Sciences 5 1974 405414 Liggett R The Application of an Implicit Enumeration Algorithm to the School Desegregation Problem Management Science 20 1973 159168 Lin W T Mock and A Wright The Use of AHP as an Aid in Planning the Nature and Extent of Audit Procedures Auditing A Journal of Practice and Theory 4 no 1 1984 8999 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 913 01 variables 294 866 A Absolute addresses 29 Absolute percentage error APE 851 Activityonarc AOA 155 Activityonnode AON 155 Additivity 95 Adjusted Rsquare 866 Advertising models 135145 advertising response function 379383 with nonlinear response functions 383387 Aggregate planning models 152162 Aircraft schedules 280 Airline crew recovery 280 Airline models crew scheduling 267272 flight scheduling 273280 Algebraic models 72 vs spreadsheet models 118121 AllDifferent constraints 465 AllenCunneen approximation 802806 Allocation of resources modeling crashing the activities 15201525 monitoring use of resources 15141520 scheduling multiple projects 15251528 AMARCO Inc 212214 American Office Systems Inc 215219 America West Airlines 551552 Analysis ToolPak addin 857 Arc capacities 231 Arcs 230 Array functions 174 Asset allocation 398 Assignment models 241247 Autocorrelation of residuals 871872 B Backlogging 158 160162 718 Backordering 158 Balking 777 BASF North America 227228 Bayer Pharmaceuticals 475476 Bayes Rule 505508 Benefitcost tables 510511 Binary variables 294 306 Binding constraints 82 83 Binomial distributions 569571 Biotechnical Engineering 549550 Blending constraints 169 Blending models 163170 Bond investment strategy 711 Bond portfolio optimization 186 Bounded probability distributions 558559 Branch and bound algorithms 144 295297 Breakeven analysis 3139 Butterfly spread 666 C Calculation settings with data tables 583 Capital asset pricing model CAPM 55 408 Carrying costs 718 Cash balance models 647652 Categorical variables 866 Causal forecasting models 843 Cell comments 34 66 Certainty equivalents 530 Changing cells 36 69 146 183 Chromosomes 425 Churn 464 667 Clearing denominators 170 Closed queueing networks CQN 774 Cluster analysis 455460 College fund investment 710 Collusive duopoly models 415 Column vector 399 Combinatorial models 438447 Complete enumeration 296 Concave functions 356 Conditional Formatting tool 51 Confidence interval for the mean 581 Conjoint analysis 432 Constant elasticity demand function 362 Constraints 69 AllDifferent 465 binding 82 83 blending 169 effect on objective 86 eitheror 314 flow balance 235 249 in groups 80 integer 143144 157 nonbinding 82 83 nonnegativity 69 Continental Airlines 279280 Contingency plans 515 Continuous probability distributions 556 Continuous review model 717 Contract bidding 623627 Control charts 638 Convergence 430 Convex functions 356 Correlated inputs 610 Cost projections 2631 Cost table 478 COUNTIF function 253 COUNTIFS function 463 Covariance matrix 403 Covariances 399 CPM See Critical path model CPM model Craps game 682685 Crashing activities 154 1514 CRITBINOM 681 Critical activities 156 Critical fractile 737 Critical path model CPM model 1541512 Critical paths 156 Curve fitting 45 Customer averages 784 Customer loyalty models 667676 Cutting stock models 335339 Cyclic component 874 875876 D Damping factors 653 Data envelopment analysis DEA 188194 banking industry 194 hospital industry 188194 school bus transportation 194 Data tables calculation settings with 583 oneway 3536 repeating simulations 583 twoway 4244 584586 DEA See Data envelopment analysis DEA Decision making under uncertainty elements of 478491 introduction to 476478 Decision support systems DSS 118121 Decision trees 482484 Decision variables 22 69 Delta Airlines 280 Demand during lead time 741 Demand forecasting 904 Demand function estimating 362 Deming Edwards 637 Density function 557 Dependent demand 716 Dependent variable 844 Descriptive models 46 Deterministic checks 579 Deterministic inventory models 715 716 Discontinuities 424 Discount factor 55 Discrete distributions 565566 Discriminant analysis 461464 INDEX Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 914 Index Divisibility 95 Dow AgroSciences 151 Dow Consumer Products 317318 Drug production with uncertain yield 632637 Dualobjective optimization models 139141 Dummy variables 866 Duration 155 Dynamic scheduling models 151 E Earliest finish time 158 Earliest start time 158 Ebony Bath Soap 620 Economic order quantity EOQ models 718734 basic model 719 more than two products 734 order synchronization 731734 with quantity discounts 722725 reducing setup cost 728729 728731 with shortages allowed 725728 Efficiency with DEA 192194 Efficient frontier 405 Eitheror constraints 314 Elasticity of demand 362 EMV See Expected monetary value EMV EMV criterion 480 EMV maximizers 481 Enumeration 296 EOQ models See Economic order quantity EOQ models Equipment replacement models 261266 Erlang loss models 799 Evolutionary Solver 423 introduction to 426431 portfolio optimization and 452455 settings for 430 with SolverTable 442 EVPI See Expected value of perfect information EVPI EVSI See Expected value of sample information EVSI Excel calculation settings with data tables 583 charts in 2931 creating time series graphs 876 CTRLEnter shortcut 147 F9 recalc key 555 Formula Auditing tool 3739 Goal Seek tool 3637 Paste Special Transpose 173 recalculation F9 key 555 roundoff errors 104 row and column sums shortcut 156 scatterplots and 849 Scenario Manager 760 Excel addins Analysis ToolPak 18 857 Evolver 18 NeuralTools 18 PrecisionTree 18 492504 RISK 17 RISKOptimizer 18 Solver 17 70 128 Evolutionary algorithm 423 nonsmooth functions and 160 tolerance settings in 149 SolverTable 17 87 StatTools 1718 854 TopRank 18 691699 Excel functions array 174 COUNTIF 253 COUNTIFS 463 CRITBINOM 681 EXP 852 IF 24 INDEX 459 MATCH 435 matrix 399 MMULT 400401 NPV 58 RAND 560 RANDBETWEEN 560 SUMIF 235237 SUMPRODUCT 44 7778 SUMXMY2 390 TRANSPOSE 173 174 400 VLOOKUP 42 Excel tools Add Trendline 850 Conditional Formatting 51 Trendline 47 362 Exchange rate considerations 366377 Exercise date 658 Exercise price 658 Expected monetary value EMV 480 646 Expected payoff 480 Expected utility 525 Expected utility maximization 526 Expected value of perfect information EVPI 513 523 Expected value of sample information EVSI 513 522 EXP function 852 Explanatory variables 844 other than time 853860 Exponential curves 47 Exponential distribution memoryless property 779 Exponential smoothing methods 884895 Exponential trend 848 Exponential trend lines 853 Exponential utility 526527 Exponential utility functions 527 External demand 716 Extrapolation models See also Time series models 843 874 limitations of 878 moving averages models 878883 F F9 key 555 Facility location models 388393 Feasible regions 70 Feasible solutions 70 Financial holding costs 718 Financial models 177186 cash balance models 647652 financial planning 642647 investment models 652657 stock prices and options simulation 657664 Finishtofinish relationships 1512 Finishtostart relationships 1512 Firstcomefirstserved FCFS 777 Fitness functions 425 Fitted values 846 Fixed cost models 306318 Flaw of averages 573575 Flow balance constraints 235 249 Flows 231 Foldingback procedures 484 Forecast error measures of 877878 Forecasting models 843 Foreign currency trading 225 Formula Auditing tool 3739 Free slack 1537 Freezing random numbers 563 fx button 42 G Games of chance simulations game of craps 682685 NCAA basketball tournaments 685689 Gamma distribution 628 Gantt chart 1511 General Electric Company GE 67 353354 Genetic algorithms GA 425 penalties 428 strengths and weaknesses of 426 GE Plastics GEP 67 GGs models 802 Giant Motor Company GMC 350351 Global maximum 355 Global minimum 355 Global optimum 355 359 Gold mining stock GMS hedging 419420 Goodnessoffit measures of 380 846 Graphical solutions 7274 H Heuristic 151 Holding costs 718 Holland John 425 Holts exponential smoothing method for trend 884 888892 Hospital efficiency 188194 I IF functions 312314 Immediate predecessors 155 Immediate successors 155 Implicit enumeration 296 Incumbent solutions 296 Independent demand 716 Independent variables 844 INDEX function 459 Indicator variables 866 Inefficiency in DEA 192194 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Index 915 Infeasibility 9799 Infeasible solutions 70 Inflows 234 Information value of 513 514 Initial conditions 778 Input distributions 603610 Inputs 4 22 Input variables probability distributions for 554571 Integer constraints 143144 157 Integer programming IP models 294 capital budgeting 299305 cutting stock models 335339 difficulty of 296 fixedcost 306318 locationassignment models 324334 LP relaxation of 297 setcovering models 319324 Interarrival times 776 Interfaces journal 14 Internal demand 716 International Textile Company Ltd 288289 Inventory models categories of 715717 cost types in 717718 economic order quantity EOQ models 718734 introduction to 714715 ordering simulation models 749754 probabilistic models newsvendor model 737740 RQ ordering policy 740747 supply chain models 754764 Inventory position 749 Investment models 652657 J Jogger Shoe Company 547 Judgmental forecasting models 843 K Kellogg Company 133134 Kendalls notation 787 Knockout call options 666 L Lakefield Corporation 220224 Lastcomefirstserved LCFS 777 Latest finish time 158 Latest start time 158 Latin hypercube sampling 591 Law of total probability 505 Lead time 716 747 Leastsquares estimation 381 408 Leastsquares regression line 846 Level of series 884 Likelihoods 505 Limited source models 799 Limited waiting room models 777 Linear programming models 68 advertising models 135145 aggregate planning 152162 blending 163170 data envelopment analysis DEA 188194 financial 177186 pension funds 182186 production processess 171176 properties of 9497 scaling and 9697 worker scheduling 145151 Linear trend 848 Littles formula 785786 Local maxima 355 Local minimum 355 Local optimum 355 359 Locationassignment models 324 Logistics models other 248256 LP relaxation 297 M MAE mean absolute error 878 MAPE See Mean absolute percentage error MAPE Marketing models customer loyalty 667676 marketing and sales models 676680 Markowitz Harry 398 MATCH function 435 Mathematical models 3 Mathematical programming models 94 Matrixmatrices 399 Matrix product 399 Maximax criterion 480 Maximin criterion 479 Maximization problems conditions for 358 Maximum time without improvement 430 Mean absolute percentage error MAPE 49 852 878 Mean payoff 480 Measures of forecast error 877878 Memoryless property 780 Merrill Lynch 377 621623 Mersenne twister 591 Microsoft Project 15351538 Minimax criterion 409 Minimization problems conditions for 358 Minimizing sum of squared errors 380 Mixed integer linear programming MILP model 350351 MM1 model 787791 MMs model 791796 MMULT functions 400401 Morton Thiokol 317318 Motor carrier selection 290292 Moving averages method 878883 Multicollinearity 871 Multiperiod production models 108117 Multiple optimal solutions 149 Multipleproduct models 717 Multiple R 847 Multiple ranges selecting 89 Multiple regression 844 Multiple regression models 861872 Multistage decision problems 509524 Multistart option 359361 Mutation rate 430 Mutations 426 N NCAA basketball tournament simulation 685689 Net present value NPV 55 Network models airline industry and 267280 assignment models 241247 introduction 228229 other logistic models 248256 shortest path models 257266 transportation models 229240 Network simplex method 255 Newsvendor models 737740 Noarbitrage pricing principle 223 Nodes 230 Noise in forecasting 876877 Nonbinding constraints 82 83 slack 82 Nonconstant error variance 871 Nonfinancial holding costs 718 Nonlinear pricing models 431438 Nonlinear programming NLP models basic ideas of 355361 facility location models 388393 introduction to 354355 optimality guarantee for 358 portfolio optimization models 398406 pricing models 361377 sports teams rating models 393397 stock beta estimating 407412 Nonlinear relationships 871 Nonnegativity constraints 69 Nonsmooth functions 160 Normal distributions 566568 Normal loss function 743 NPV function 58 O Objective cells 69 183 Objective functions 69 Oneway data tables 36 Operations simulation models bidding for contracts 623627 drug production with uncertain yield 632637 warranty costs 627632 Operations research OR 2 Optimality guarantee 358 Optimal solutions 70 149 Optimization models 4 68 54 dualobjective 139141 with integer variables 294299 introduction to 67121 nonlinear programming models NLP 354355 Option pricing result 658659 OR See Operations research OR Ordering 716 Ordering costs 716 717718 Outflows 234 Overhead forecasting 905 P Pacific National Bank 838839 Palisade Decision Tools Suite 17 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 916 Index Parallel queueing systems 777 Partial backlogging 718 Payoff tables 478479 Penalties 428 Penalty costs 718 Pension fund models 182186 Perfect information 513 Periodic reviews 717 PERT 1532 PERT distribution 15311534 Piecewise linear 432 Poisson process model 783 Population size 430 Porteus Evan 728 Portfolio optimization models 398406 Evolutionary Solver and 452455 return measures 398 risk measures of 398 Portfolio standard deviation 403 Portfolio variance 403 Posterior probabilities 505 Power curves 47 PrecisionTree addin 492504 allowable entries 495 copying subtrees 497 entering monetary values probabilities 496 spider charts 501 strategy region chart 501 tornado charts 501 values at end nodes 498 Prediction and fitted values 846 Prediction errors 845 Predictor variables 844 Present value 55 Price and demand estimating relationship between 4555 Pricing analysis 377 Pricing models 361377 exchange rate considerations 366377 Prior probabilities 505 Probabilistic inventory models 715 716 Probability distributions common types 559562 types of 555556 for uncertain inputs 556 Production costs 717 Production process modeling 171176 Product mix models advertising models and 136137 twovariable 7083 Project management allocation of resources 15141528 CPM model 1541512 crashing activities 154 1514 introduction to 152154 PERT distribution 15311534 uncertain activity times models with 15301534 Proportionality 95 Pseudorandom numbers 562 Q Quantity discounts and demand uncertainty 4045 Queueing models analytical steadystate 787806 characteristics of arrivals 776777 exponential distribution 779784 important relationships 783786 introduction to 48 774776 service characteristics 777778 service discipline 777 simulation models 815830 series systems with blocking 823830 Queueing networks 777 R RANDBETWEEN function 560 RAND function 560 Random noise component 874 876877 Random numbers freezing 563 Random seeds 430 Random variables weighted sums of 398399 Range names 33 66 pasting 33 shortcuts 137 Ranges selecting multiples 87 Reduced costs 85 Reference base category 867 Regression assumptions 870871 Regressionbased trend models 848853 Regression coefficients 862 Regression models 843 introduction to 843844 leastsquares line 845846 overview of 844848 prediction and fitted values 846 simple 848860 trend models 848 Relative addresses 29 Reneging 777 Reorder point 717 Replicating with Excel only 583 Residuals 845 846 Response variables 844 Retention rate 667 Risk 525532 RISK addin 17 automated template for 690691 features of 587 introduction to 587601 Latin hypercube and Mersenne twister settings 591 limitations of 597598 loading 588 models with several random input variables 598601 models with single random input variables 588597 probability distributions and 563 RISKBINOMIAL 570 RISKCORRMAT 610 RISKDISCRETE 566 RISKGAMMA 630 RISKNORMAL 567 617 RISKOUTPUT 590 RISKPERT 15311532 RISKSIMTABLE 587 594597 RISKTARGET 637 RISKTRIANG 569 RISKUNIFORM 563 saving graphs and tables 593 TopRank addin and 691699 Risk aversion 525 526 RISKCORRMAT 610 RISKGAMMA 630 Risk index 203 Risk measures 398 RISKNORMAL 567 RISKPERT 15311532 Risk profiles 484485 RISKSIMTABLE 594597 RISKTARGET 637 Risk tolerance 526527 RISKTRIANG 569 RISKUNIFORM 563 RISKVARY 694 RMSE See Root mean square error RMSE Rolling planning horizons approach 116 aggregate planning model and 158162 Root mean square error RMSE 381 878 Roundoff errors 104 Row vector 399 RQ ordering policy 740747 Rsquare 847848 Rsquare adjusted 866 S SAE See Sum of absolute errors SAE Safety stock 740 741 Sales models 676680 Sample information 513 Sample size determination 581 Santa Fe Railway 421422 Saturation effect 830 Scaling in optimization models 9697 Scatterplots 845 with Excel 849 Scenario approach 406 Scope creep in projects 154 Screen splitting 57 Seasonal component 874 875 Sensitivity analysis 70 8394 482 Sequential decisions 515 Series systems 777 with blocking 823830 Server utilization 786 Service discipline 777 Serviceinrandomorder SRO 777 Service level constraints 745 Setcovering models 319324 Setup costs 716 717 Sevenstep modeling process 814 Shadow prices 85 Shortage costs 718 742 Shortest path models 257266 Shortestprocessingtime SPT 777 Shortrun behavior 778 analytical approximation of 809814 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Index 917 Simple exponential smoothing 884 equivalent formulas for 885 Simple regression 844 Simplex method 70 geometry of 74 Simulation models 9 775 financial models cash balance models 647652 financial planning 642647 investment models 652657 stock prices and options simulation 657664 flaw of averages 573575 games of chance simulations game of craps 682685 NCAA basketball tournaments 685689 introduction to 552554 marketing models customer loyalty 667676 marketing and sales 676680 operations models bidding for contracts 623627 Demings funnel experiment 637641 drug production with uncertain yield 632637 warranty costs 627632 probabilty distributions for input variables 554571 using builtin Excel tools 576586 Skewed probability distributions 557558 Slack in constraint 82 Slack in CPM models 158 Smoothing constants 884 Smoothing methods 878 Solver addin 70 7983 Evolutionary 423 GRG Nonlinear method 364 integer constraints 143 messages from 81 Multistart option 359361 nonsmooth functions and 160 tolerance settings in 149 SolverTable addin roundoff with 106 with Evolutionary Solver 442 sensitivity reports and 87 Span in moving averages 878 Spider charts 502 Splitting screen 57 Sports teams rating models 393397 Spreadsheet modeling 7478 breakeven analysis 3139 cost projections 2631 inequality and equality labels in 80 introduction to 2227 layout and documentation 26 vs algebraic models 118121 Squared coefficient of variation 803 Sshaped curves fitting 448452 Sshaped trend 874 Stable queueing systems 779 Standard error of estimate 846847 Standard errors of X 581 Starttofinish relationships 1512 Starttostart relationships 1512 StatTools 1718 854 Steadystate analysis 778 Stock beta estimation 407412 Stock prices and options simulation Asian options 662664 European call options 657660 portfolio returns with stocks and options 660662 stock prices 657658 Strategy region charts 501 Strike price 658 Subway token hoarding 772 SUMIF function 235237 Sum of absolute errors SAE 408 Sum of squared prediction errors 383 408 minimization of 380 weighted 408 Sum of squared residuals 845 SUMPRODUCT 44 SUMXMY2 function 390 Supply chain models 754764 Surplus values 433 Symmetric probability distributions 557558 T Tabu search 422 Tampering 638 Telecommunication discounts 352 Texaco OMEGA linear programming model 170 Text boxes 66 Time averages 784 Time series graphs 876 Time series models 843 components of 874 cyclic components of 875876 overview 874883 random noise components 876877 seasonal components of 875 trend component 874875 Time value of money 5560 TopRank addin key inputs 699 RISK and 691699 RISKVARY 694 Tornado charts 501 Total slack 1537 Tradeoff curves 141 Traffic intensity 787 797799 Training samples 461 Transient probability distributions 810 Transportation models 229240 234237 Transportation simplex method 237 TRANSPOSE function 173 400 Transshipment points 248 Traveling salesperson models 464468 Trend component 874 Trend line superimposing 850 Trendline tool 47 Trend models regressionbased 848853 Triangular distribution 568569 Twopart tariffs 432 Twovariable product mix model 7083 algebraic model for 72 graphical solutions for 7274 spreadsheet models for 7478 Twoway data tables 4244 584586 Twoway sensitivity charts 503504 U Unbounded solutions 9799 Unbounded probability distributions 558559 Uncertain demand ordering with 740 Uncertain timing dealing with 635 Unconstrained models 382 Uniform distribution 559 Unit purchasing cost 717 Unit shipping costs 230 Unrestricted probability distributions 559 US Air Force Space Command 293294 Utility functions 526 V Value at risk at the 5 level VAR 5 645 Variable costs 717 VLOOKUP 42 W Waiting line models See Queueing models Warranty costs 627632 Weighted sum of random variables 398399 Weighted sums of squared errors 408 Westhouser Paper Company 548 Westvaco 290292 Winters exponential smoothing method for seasonality 892896 Worker scheduling models 145151 Workinprocess WIP inventory 773774 X X predictor variable 844 Y Y response variable 844 Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2011 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook andor eChapters Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it