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MATHEMATICS IN INDUSTRY REPORT SERIES Choice of a proper pricing policy in electric power supply is important for power companies in order to maximize their revenue and sales and to retain customers Recently many electric power companies are considering real time pricing policies although spot prices of wholesale power on market are very volatile In this paper we propose a stochastic optimization problem for determining a proper pricing policy assuming probabilistic behaviors of wholesale price and consumptions of customers By using sample average approximation method the optimization problem is numerically solved and results are compared with a constant pricing policy through a real data example of the Tokyo electric power company AN INVERSE PRICING PROBLEM OF REAL TIME PRICING IN ELECTRIC POWER MARKET AND A NUMERICAL SOLUTION TATSUO HATAKEYAMA MASAHIRO FUJISAWA HIROSHI KUBO TAKUYA UMEKI JUN KATO MASAHIRO TAKEI AND MASAHIDE YAMAZAKI Department of Information Engineering Faculty of Science and Technology Tokyo University of Science Ibaraki 2630 Toride Ibaraki 3028551 JAPAN tatsuohrskagutusacjp Department of Electrical and Electronic Engineering Tokyo Institute of Technology 2121 Ookayama Meguroku Tokyo 1528550 JAPAN Department of Electrical Engineering Tokyo University of Science Noda 2641 Noda Chiba 2788510 JAPAN jkatorskagutusacjp Mathematical Institute for Complex Systems Tokyo University of Science Noda 2641 Noda Chiba 2788510 JAPAN yamazakirskagutusacjp DOI 1012988ami20188128 Mathematics in Industry Vol 23 No 1 2018 7386 ISSN 22813318 printed ISSN 22813326 online httpswwwspringercom11129 Keywords and phrases optimal pricing problem real time pricing in electricity market sample average approximation stochastic optimization AMC Subject Classifications 62P20 90C15 90C59 90C90 1 INTRODUCTION Real time pricing RTP is an electricity pricing policy that reflects the hourly wholesale price in the power market into the price for consumers in principle while it is basically done on a dayahead basis In Japan a part of customers typically largescale industrial companies will be charged RTPbased high power price by the government after separating power distribution from power supply sectors In the United States RTP has already been put into practical use in several states for industrial and residential customers For the power companies price setting for their customers plays an important rule for expanding their sales and revenue and for customer retention Therefore a welldesigned price policy is quite important Model Building in Mathematical Programming H PAUL WILLIAMS Model Building in Mathematical Programming FIFTH EDITION WILEY Model Building in Mathematical Programming Fifth Edition H Paul Williams London School of Economics UK A John Wiley Sons Ltd Publication Copyright 1978 1985 1990 1993 1999 2013 by John Wiley Sons Ltd The Atrium Southern Gate Chichester West Sussex PO19 8SQ England Telephone 44 1243 779777 This edition first published 2013 Registered office John Wiley Sons Ltd The Atrium Southern Gate Chichester West Sussex PO19 8SQ United Kingdom For details of our global editorial offices for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at wwwwileycom The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise except as permitted by the UK Copyright Designs and Patents Act 1988 without the prior permission of the publisher Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names service marks trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required the services of a competent professional should be sought Library of Congress CataloginginPublication Data Williams H P Model building in mathematical programming H Paul Williams 5th ed p cm Includes bibliographical references and index ISBN 9781118443330 pbk 1 Programming Mathematics 2 Mathematical models I Title T577W55 2013 5197dc23 2012037292 A catalogue record for this book is available from the British Library ISBN 9781118443330 Set in 10pt12pt Times by Laserwords Private Limited Chennai India To Eileen Anna Alexander and Eleanor Contents Preface xvii Part I 1 1 Introduction 3 11 The concept of a model 3 12 Mathematical programming models 5 2 Solving mathematical programming models 11 21 Algorithms and packages 11 211 Reduction 12 212 Starting solutions 12 213 Simple bounding constraints 12 214 Ranged constraints 13 215 Generalized upper bounding constraints 13 216 Sensitivity analysis 13 22 Practical considerations 13 23 Decision support and expert systems 16 24 Constraint programming CP 17 3 Building linear programming models 21 31 The importance of linearity 21 32 Defining objectives 23 321 Single objectives 24 322 Multiple and conflicting objectives 26 323 Minimax objectives 27 324 Ratio objectives 28 325 Nonexistent and nonoptimizable objectives 29 33 Defining constraints 29 331 Productive capacity constraints 29 332 Raw material availabilities 30 333 Marketing demands and limitations 30 334 Material balance continuity constraints 30 viii CONTENTS 335 Quality stipulations 31 336 Hard and soft constraints 31 337 Chance constraints 32 338 Conflicting constraints 32 339 Redundant constraints 34 3310 Simple and generalized upper bounds 35 3311 Unusual constraints 35 34 How to build a good model 36 341 Ease of understanding the model 36 342 Ease of detecting errors in the model 37 343 Ease of computing the solution 37 344 Modal formulation 38 345 Units of measurement 40 35 The use of modelling languages 40 351 A more natural input format 41 352 Debugging is made easier 41 353 Modification is made easier 41 354 Repetition is automated 41 355 Special purpose generators using a high level language 41 356 Matrix block building systems 42 357 Data structuring systems 42 358 Mathematical languages 42 3581 SETs 43 3582 DATA 43 3583 VARIABLES 43 3584 OBJECTIVE 43 3585 CONSTRAINTS 43 4 Structured linear programming models 45 41 Multiple plant product and period models 45 42 Stochastic programmes 53 43 Decomposing a large model 55 431 The submodels 63 432 The restricted master model 64 5 Applications and special types of mathematical programming model 67 51 Typical applications 67 511 The petroleum industry 68 512 The chemical industry 68 513 Manufacturing industry 68 514 Transport and distribution 69 515 Finance 69 516 Agriculture 70 CONTENTS ix 517 Health 70 518 Mining 70 519 Manpower planning 71 5110 Food 71 5111 Energy 71 5112 Pulp and paper 72 5113 Advertising 72 5114 Defence 72 5115 The supply chain 72 5116 Other applications 73 52 Economic models 74 521 The static model 74 522 The dynamic model 80 523 Aggregation 81 53 Network models 81 531 The transportation problem 82 532 The assignment problem 87 533 The transhipment problem 88 534 The minimum cost flow problem 89 535 The shortest path problem 93 536 Maximum flow through a network 93 537 Critical path analysis 94 54 Converting linear programs to networks 98 6 Interpreting and using the solution of a linear programming model 103 61 Validating a model 103 611 Infeasible models 103 612 Unbounded models 104 613 Solvable models 105 62 Economic interpretations 107 621 The dual model 109 622 Shadow prices 112 623 Productive capacity constraints 114 624 Raw material availabilities 114 625 Marketing demands and limitations 114 626 Material balance continuity constraints 114 627 Quality stipulations 114 628 Reduced costs 116 63 Sensitivity analysis and the stability of a model 121 631 Righthand side ranges 121 632 Objective ranges 125 633 Ranges on interior coefficients 128 634 Marginal rates of substitution 131 635 Building stable models 132 x CONTENTS 64 Further investigations using a model 133 65 Presentation of the solutions 135 7 Nonlinear models 137 71 Typical applications 137 72 Local and global optima 140 73 Separable programming 147 74 Converting a problem to a separable model 153 8 Integer programming 155 81 Introduction 155 82 The applicability of integer programming 156 821 Problems with discrete inputs and outputs 156 822 Problems with logical conditions 158 823 Combinatorial problems 158 824 Nonlinear problems 160 825 Network problems 161 83 Solving integer programming models 162 831 Cutting planes methods 162 832 Enumerative methods 163 833 PseudoBoolean methods 163 834 Branch and bound methods 164 9 Building integer programming models I 165 91 The uses of discrete variables 165 911 Indivisible discrete quantities 165 912 Decision variables 165 913 Indicator variables 166 92 Logical conditions and 01 variables 172 93 Special ordered sets of variables 177 94 Extra conditions applied to linear programming models 182 941 Disjunctive constraints 183 942 Nonconvex regions 184 943 Limiting the number of variables in a solution 186 944 Sequentially dependent decisions 186 945 Economies of scale 187 946 Discrete capacity extensions 188 947 Maximax objectives 188 95 Special kinds of integer programming model 189 951 Set covering problems 189 952 Set packing problems 191 953 Set partitioning problems 193 954 The knapsack problem 195 955 The travelling salesman problem 195 956 The vehicle routing problem 198 CONTENTS xi 957 The quadratic assignment problem 199 96 Column generation 201 10 Building integer programming models II 207 101 Good and bad formulations 207 1011 The number of variables in an IP model 207 1012 The number of constraints in an IP model 211 102 Simplifying an integer programming model 218 1021 Tightening bounds 218 1022 Simplifying a single integer constraint to another single integer constraint 220 1023 Simplifying a single integer constraint to a collection of integer constraints 222 1024 Simplifying collections of constraints 226 1025 Discontinuous variables 228 1026 An alternative formulation for disjunctive constraints 229 1027 Symmetry 230 103 Economic information obtainable by integer programming 231 104 Sensitivity analysis and the stability of a model 238 1041 Sensitivity analysis and integer programming 238 1042 Building a stable model 239 105 When and how to use integer programming 240 11 The implementation of a mathematical programming system of planning 243 111 Acceptance and implementation 243 112 The unification of organizational functions 245 113 Centralization versus decentralization 247 114 The collection of data and the maintenance of a model 249 Part II 251 12 The problems 253 121 Food manufacture 1 253 122 Food manufacture 2 255 123 Factory planning 1 255 124 Factory planning 2 256 125 Manpower planning 256 1251 Recruitment 257 1252 Retraining 257 1253 Redundancy 258 1254 Overmanning 258 1255 Shorttime working 258 xii CONTENTS 126 Refinery optimisation 258 1261 Distillation 258 1262 Reforming 259 1263 Cracking 259 1264 Blending 260 127 Mining 261 128 Farm planning 262 129 Economic planning 263 1210 Decentralisation 265 1211 Curve fitting 266 1212 Logical design 266 1213 Market sharing 267 1214 Opencast mining 269 1215 Tariff rates power generation 270 1216 Hydro power 271 1217 Threedimensional noughts and crosses 272 1218 Optimising a constraint 273 1219 Distribution 1 273 1220 Depot location distribution 2 275 1221 Agricultural pricing 276 1222 Efficiency analysis 278 1223 Milk collection 278 1224 Yield management 282 1225 Car rental 1 284 1226 Car rental 2 287 1227 Lost baggage distribution 287 1228 Protein folding 289 1229 Protein comparison 290 Part III 293 13 Formulation and discussion of problems 295 131 Food manufacture 1 296 1311 The singleperiod problem 296 1312 The multiperiod problem 297 132 Food manufacture 2 299 133 Factory planning 1 300 1331 The singleperiod problem 300 1332 The multiperiod problem 301 134 Factory planning 2 302 1341 Extra variables 302 1342 Revised constraints 303 135 Manpower planning 303 1351 Variables 304 CONTENTS xiii 1352 Constraints 305 1353 Initial conditions 305 136 Refinery optimization 306 1361 Variables 307 1362 Constraints 308 1363 Objective 310 137 Mining 310 1371 Variables 310 1372 Constraints 311 1373 Objective 312 138 Farm planning 312 1381 Variables 312 1382 Constraints 313 1383 Objective function 315 139 Economic planning 316 1391 Variables 316 1392 Constraints 316 1393 Objective function 317 1310 Decentralization 317 13101 Variables 318 13102 Constraints 318 13103 Objective 319 1311 Curve fitting 319 1312 Logical design 320 1313 Market sharing 322 1314 Opencast mining 324 1315 Tariff rates power generation 325 13151 Variables 325 13152 Constraints 325 13153 Objective function to be minimized 326 1316 Hydro power 326 13161 Variables 326 13162 Constraints 326 13163 Objective function to be minimized 327 1317 Threedimensional noughts and crosses 327 13171 Variables 327 13172 Constraints 328 13173 Objective 328 1318 Optimizing a constraint 328 1319 Distribution 1 330 13191 Variables 331 13192 Constraints 331 13193 Objectives 332 1320 Depot location distribution 2 332 1321 Agricultural pricing 333 xiv CONTENTS 1322 Efficiency analysis 335 1323 Milk collection 336 13231 Variables 336 13232 Constraints 336 13233 Objective 337 1324 Yield management 337 13241 Variables 338 13242 Constraints 338 13243 Objective 340 1325 Car rental 1 340 13251 Indices 340 13252 Given data 340 13253 Variables 341 13254 Constraints 341 13255 Objective 342 1326 Car rental 2 342 1327 Lost baggage distribution 343 13271 Variables 343 13272 Objective 344 13273 Constraints 344 1328 Protein folding 344 1329 Protein comparison 345 Part IV 347 14 Solutions to problems 349 141 Food manufacture 1 349 142 Food manufacture 2 349 143 Factory planning 1 350 144 Factory planning 2 351 145 Manpower planning 354 146 Refinery optimization 356 147 Mining 357 148 Farm planning 358 149 Economic planning 359 1410 Decentralization 361 1411 Curve fitting 361 1412 Logical design 363 1413 Market sharing 363 1414 Opencast mining 364 1415 Tariff rates power generation 364 1416 Hydro power 366 1417 Threedimensional noughts and crosses 368 1418 Optimizing a constraint 369 CONTENTS xv 1419 Distribution 1 369 1420 Depot location distribution 2 371 1421 Agricultural pricing 371 1422 Efficiency analysis 372 1423 Milk collection 374 1424 Yield management 376 1425 Car rental 379 1426 Car rental 2 380 1427 Lost baggage distribution 380 1428 Protein folding 382 1429 Protein comparison 382 References 383 Author index 397 Subject index 401 Preface Mathematical programmes are among the most widely used models in operational research and management science In many cases their application has been so successful that their use has passed out of operational research departments to become an accepted routine planning tool It is therefore rather surprising that comparatively little attention has been paid in the literature to the problems of formulating and building mathematical programming models or even deciding when such a model is applicable Most published work has tended to be of two kinds Firstly case studies of particular applications have been described in the operational research journals and journals relating to specific industries Secondly research work on new algorithms for special classes of problems has provided much material for the more theoretical journals This book attempts to fill the gap by in Part I discussing the general principles of model building in mathematical programming In Part II 29 practical problems are presented to which mathematical programming can be applied By simplifying the problems much of the tedious institutional detail of case studies is avoided It is hoped however that the essence of the problems is preserved and easily understood Finally in Parts III and IV suggested formulations and solutions to the problems are given together with some computational experience Many books already exist on mathematical programming or in particular linear programming Most such books adopt the conventional approach of paying a great deal of attention to algorithms Since the algorithmic side has been so well and fully covered by other texts it is given much less attention in this book The concentration here is more on the building and interpreting of models rather than on the solution process Nevertheless it is hoped that this book may spur the reader to delve more deeply into the often challenging algorithmic side of the subject as well It is however the authors contention that the practical problems and model building aspect should come first This may then provide a motivation for finding out how to solve such models Although desirable knowledge of algorithms is no longer necessary if practical use is to be made of mathematical programming The solution of practical models is now largely automated by the use of commercial package programs that are discussed in Chapter 2 For the reader with some prior knowledge of mathematical programming parts of this book may seem trivial and can be skipped or read quickly Other parts are however rather more advanced and present fairly new material This is particularly true of the chapters on integer programming Indeed this book can xviii PREFACE be treated in a nonsequential manner There is much crossreferencing to enable the reader to pass from one relevant section to another This book is aimed at three types of readers 1 It is intended to provide students in universities and polytechnics with a solid foundation in the principles of model building as well as the more mathematical algorithmic side of the subject which is conventionally taught For students who finally go on to use mathematical programming to solve real problems the model building aspect is probably the more important The problems in Part II provide practical exercises in problem formulation By formulating models and solving them with the aid of a computer students learn the art of formulation in the most satisfying way possible They can compare their numerical solutions with those of other students obtained from differently build models In this way they learn how to validate a model It is also hoped that these problems will be of use to research students seeking new algorithms for solving mathematical programming problems Very often they have to rely on trivial or randomly generated models to test their computational procedures Such models are far from typical of those found in the real world Moreover they are one or more steps removed from practical situations They therefore obscure the need for efficient formulations as well as algorithms 2 This book is also intended to provide managers with a fairly nontechnical appreciation of the scope and limitations of mathematical programming In addition by looking at the practical problems described in Part II they may recognize a situation in their own organization to which they had not realized mathematical programming could be applied 3 Finally constructing a mathematical model of an organization provides one of the best methods of understanding that organization It is hoped that the general reader will be able to use the principles described in this book to build mathematical models and thereby learn about the functioning of systems which purely verbal descriptions fail to explain It has been the authors experience that the process of building a model of an organi zation can often be more beneficial even than the obtaining of a solution A greater understanding of the complex interconnections between different facets of an organization is forced upon anybody who realistically attempts to model that organization Part I of this book describes the principles of building mathematical program ming models and how they may arise in practice In particular linear program ming integer programming and separable programming models are described A discussion of the practical aspects of solving such models and a very full discussion of the interpretation of their solutions is included PREFACE xix Part II presents each of the 29 practical problems in sufficient detail to enable the reader to build a mathematical programming model using the numerical data In some cases the origin of the problem is mentioned Part III discusses each problem in detail and presents a possible formulation as a mathematical programming model Part IV gives the optimal solutions obtained from the formulations presented in Part III Some computational experience is also given in order to give the reader some feel of the computational difficulty of solving the particular type of model It is hoped that readers will attempt to formulate and possibly solve the problems for themselves before proceeding to Parts III and IV By presenting 29 problems from widely different contexts the power of the technique of mathematical programming in giving a method of tackling them all should be apparent Some problems are intentionally unusual in the hope that they may suggest the application of mathematical programming in rather novel areas Many references are given at the end of the book The list is not intended to provide a complete bibliography of the vast number of case studies published Many excellent case studies have been ignored The list should however provide a representative sample which can be used as a starting point for a deeper search into the literature Many people have both knowingly and unknowingly helped in the prepa ration of successive editions of this book with their suggestions and opinions In particular I would like to thank Gautam Appa Sheena and Robert Ashford Martin Beale Tony Brearley Ian Buchanan Colin Clayman Lewis Corner Martyn Jeffreys Bob Jeroslow Clifford Jones Bernard Kemp Ailsa Land Adolfo Fonseca Manjarres Kenneth McKinnon Gautam Mitra Heiner Muller Merbach Bjorn Nygreen Pat Rivett Richard Thomas Steven Vajda and Will Watkins I must express a great debt of gratitude to Robin Day of Edinburgh University whose deep computing knowledge and programming ability helped me immensely in building the models for early editions of this book and implementing our design of the modelling system MAGIC This has now been replaced by the system NEWMAGIC which has been written by George Skondras It is available with the optimizer EMSOL All the models in the book have been built and solved in this system The models formulated in Part III have been modelled in the NEWMAGIC language and are available on the website wwwwileycomgomodelbuildingmathematical programming Since the first edition was written computational power has increased immensely Solution times for the models are therefore in most cases of little relevance and are ignored A few of the models are still difficult to solve In these cases some computational experience is given The fourth edition has been enhanced by new sections or models on Constraint Logic Programming Data Envelopment Analysis Hydro Electricity Generation xx PREFACE Milk Distribution and Yield Management in an airline together with stateofthe art material and extra references on a number of topics and applications I am very grateful to Kenneth McKinnon for advice and help with these models It is gratifying to know how well this book has been received together with the demand for a fourth edition Preface to the Fifth Edition The fifth edition includes new sections on Stochastic Programming and Column Generation a subsection on the Vehicle Routing problem and an enhanced section on Constraint Logic Programming In addition the subsection on modelling non linear functions and constraints by integer variables has been improved with a more versatile formulation Many other small clarifications and improvements have also been included Five new problems have been added a Car Rental and Return problem and an extension an Airport Lost Baggage Distribution problem and two problems in Molecular Biology New references have been added although it is recognized that it is impossible to mention all the excellent papers that have been written since earlier editions of this book I apologize to the many authors of these papers who have not been cited A number of people have given me advice on improving and correcting the fourth edition to produce this fifth edition In particular I would like to mention Harvey Greenberg John Hooker Cormac Lucas Andrew McGee and Ken McKinnon They have all helped validate the new material Also a number of correspondents have pointed out small typos I am grateful to them all PAUL WILLIAMS Winchester England Part I SOLVING MATHEMATICAL PROGRAMMING MODELS 13 214 Ranged constraints It is sometimes necessary to place upper and lower bounds on the level of some activity represented by a linear expression This could be done by two constraints such as j a j x j b 1 and j a j x j b 2 A more compact and convenient way to do this is to specify only the first constraint above together with a range of b 1 b 2 on the constraint The effect of a range is to limit the slack variable which will be introduced into the constraint by the package to have an upper bound of b 1 b 2 so implying the second of the above constraints Most commercial packages have the facility for defining such ranges not to be confused with ranging in sensitivity analysis discussed below on constraints 215 Generalized upper bounding constraints Constraints representing a bound on a sum of variables such as x 1 x 2 x n M are very common in many LP models Such a constraint is sometimes referred to by saying that there is a generalized upper bound GUB of M on the set of variables x 1 x 2 x n If a considerable proportion of the constraints in a model are of this form and each such set of variables is exclusive of variables in any other set then it is efficient to use the socalled GUB extension of the revised simplex algorithm When this is used it is not necessary to specify these constraints as rows of the model but treat them in a slightly analogous way to simple bounds on single variables The use of this extension to the algorithm usually makes the solution of a model far quicker 216 Sensitivity analysis When the optimal solution of a model is obtained there is often interest in investigating the effects of changes in the objective and righthand side coefficients and sometimes other coefficients on this solution Ranging is the name of a method of finding limits ranges within which one of these coefficients can be changed to have a predicted effect on the solution Such information is very valuable in performing a sensitivity analysis on a model This topic is discussed at length in Section 63 for LP models Almost all commercial packages have a range facility 22 Practical considerations In order to demonstrate how a model is presented to a computer package and the form in which the solution is presented we will consider the second example 1 Introduction 11 The concept of a model Many applications of science make use of models The term model is usually used for a structure that has been built with the purpose of exhibiting features and characteristics of some other objects Generally only some of these features and characteristics will be retained in the model depending upon the use to which it is to be put Sometimes such models are concrete as is a model aircraft used for wind tunnel experiments More often in operational research we will be concerned with abstract models These models will usually be mathematical in that algebraic symbolism will be used to mirror the internal relationships in the object often an organization being modelled Our attention will mainly be confined to such mathematical models although the term model is sometimes used more widely to include purely descriptive models The essential feature of a mathematical model in operational research is that it involves a set of mathematical relationships such as equations inequalities and logical dependencies that correspond to some more downtoearth relation ships in the real world such as technological relationships physical laws and marketing constraints There are a number of motives for building such models 1 The actual exercise of building a model often reveals relationships that were not apparent to many people As a result a greater understanding is achieved of the object being modelled 2 Having built a model it is usually possible to analyse it mathematically to help suggest courses of action that might not otherwise be apparent 3 Experimentation is possible with a model whereas it is often not possible or desirable to experiment with the object being modelled It would Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 4 MODEL BUILDING IN MATHEMATICAL PROGRAMMING clearly be politically difficult as well as undesirable to experiment with unconventional economic measures in a country if there were a high probability of disastrous failure The pursuit of such courageous experiments would be more though not perhaps totally acceptable on a mathematical model It is important to realize that a model is really defined by the relationships that it incorporates These relationships are to a large extent independent of the data in the model A model may be used on many different occasions with differing data for example costs technological coefficients and resource availabilities We would usually still think of it as the same model even though some coefficients have been changed This distinction is not of course total Radical changes in the data would usually be thought of as a change in the relationships and therefore the model Many models used in operational research and other areas such as engineer ing and economics take standard forms The mathematical programming type of model that we consider in this book is probably the most commonly used standard type of model Other examples of some commonly used mathematical models are simulation models network planning models econometric models and time series models There are many other types of model all of which arise sufficiently often in practice to make them areas worthy of study in their own right It should be emphasized however that any such list of standard types of model is unlikely to be exhaustive or exclusive There are always practical situations that cannot be modelled in a standard way The building analysing and experimenting with such new types of model may still be a valuable activity Often practical problems can be modelled in more than one standard way as well as in nonstandard ways It has long been realized by operational research workers that the comparison and contrasting of results from different types of model can be extremely valuable Many misconceptions exist about the value of mathematical models partic ularly when used for planning purposes At one extreme there are people who deny that models have any value at all when put to such purposes Their criti cisms are often based on the impossibility of satisfactorily quantifying much of the required data for example attaching a cost or utility to a social value A less severe criticism surrounds the lack of precision of much of the data that may go into a mathematical model for example if there is doubt surrounding 100 000 of the coefficients in a model how can we have any confidence in an answer it produces The first of these criticisms is a difficult one to counter and has been tackled at much greater length by many defenders of costbenefit analysis It seems undeniable however that many decisions concerning unquan tifiable concepts however they are made involve an implicit quantification that cannot be avoided Making such a quantification explicit by incorporating it in a mathematical model seems more honest as well as scientific The second crit icism concerning accuracy of the data should be considered in relation to each specific model Although many coefficients in a model may be inaccurate it is INTRODUCTION 5 still possible that the structure of the model results in little inaccuracy in the solution This subject is mentioned in depth in Sections 42 and 63 At the opposite extreme to the people who utter the above criticisms are those who place an almost metaphysical faith in a mathematical model for decision making particularly if it involves using a computer The quality of the answers that a model produces obviously depends on the accuracy of the structure and data of the model For mathematical programming models the definition of the objec tive clearly affects the answer as well Uncritical faith in a model is obviously unwarranted and dangerous Such an attitude results from a total misconception of how a model should be used To accept the first answer produced by a math ematical model without further analysis and questioning should be very rare A model should be used as one of a number of tools for decisionmaking The answer that a model produces should be subjected to close scrutiny If it represents an unacceptable operating plan then the reasons for unacceptability should be spelled out and if possible incorporated in a modified model Should the answer be acceptable it might be wise only to regard it as an option The specification of another objective function in the case of a mathematical programming model might result in a different option By successive questioning of the answers and altering the model or its objective it should be possible to clarify the options available and obtain a greater understanding of what is possible 12 Mathematical programming models It should be pointed out immediately that mathematical programming is very different from computer programming Mathematical programming is program ming in the sense of planning As such it need have nothing to do with computers The confusion over the use of the word programming is widespread and unfortunate Inevitably mathematical programming becomes involved with computing as practical problems almost always involve large quantities of data and arithmetic that can only reasonably be tackled by the calculating power of a computer The correct relationship between computers and mathematical programming should however be understood The common feature that mathematical programming models have is that they all involve optimization We wish to maximize something or minimize something The quantity that we wish to maximize or minimize is known as an objective function Unfortunately the realization that mathematical programming is con cerned with optimizing an objective often leads people summarily to dismiss mathematical programming as being inapplicable in practical situations where there is no clear objective or there are a multiplicity of objectives Such an atti tude is often unwarranted because as we shall see in Chapter 3 there is often value in optimizing some aspect of a model when in real life there is no clearcut single objective In this book we confine our attention to some special types of mathematical programming model These can most easily be classified as linear programming 6 MODEL BUILDING IN MATHEMATICAL PROGRAMMING LP models nonlinear programming NLP models and integer programming IP models We begin by describing what an LP model is by means of two small examples Example 11 A Linear Programming LP Model Product Mix An engineering factory can produce five types of product PROD 1 PROD 2 PROD 5 by using two production processes grinding and drilling After deducting raw material costs each unit of each product yields the following contributions to profit PROD 1 PROD 2 PROD 3 PROD 4 PROD 5 550 600 350 400 200 Each unit requires a certain time on each process These are given below in hours A dash indicates when a process is not needed PROD 1 PROD 2 PROD 3 PROD 4 PROD 5 Grinding 12 20 25 15 Drilling 10 8 16 In addition the final assembly of each unit of each product uses 20 hours of an employees time The factory has three grinding machines and two drilling machines and works a sixday week with two shifts of 8 hours on each day Eight workers are employed in assembly each working one shift a day The problem is to find how much of each product is to be manufactured so as to maximize the total profit contribution This is a very simple example of the socalled product mix application of LP In order to create a mathematical model we introduce variables x1 x2 x5 representing the numbers of PROD 1 PROD 2 PROD 5 that should be produced in a week As each unit of PROD 1 yields 550 contribution to profit and each unit of PROD 2 yields 600 contribution to profit etc our total profit contribution will be represented by the expression 550x1 600x2 350x3 400x4 200x5 11 The objective of the factory is to choose x1 x2 x5 so as to make the value of this expression as high as possible that is Expression 11 is the objective function that we wish to maximize in this case INTRODUCTION 7 Clearly our processing and labour capacities to some extent limit the values that the xj can take Given that we have only three grinding machines working for a total of 96 hours a week each we have 288 hours of grinding capacity available Each unit of PROD 1 uses 12 hours grinding x1 units will therefore use 12x1 hours Similarly x2 units of PROD 2 will use 20x2 hours The total amount of grinding capacity that we use in a week is given by the expression on the lefthand side of Inequality 12 12x1 20x2 25x4 15x5 288 12 Inequality 12 is a mathematical way of saying that we cannot use up more than the 288 hours of grinding available per week Inequality 12 is known as a constraint It restricts or constrains the possible values that the variables xj can take The drilling capacity is 192 hours a week This gives rise to the following constraint 10x1 8x2 16x3 192 13 Finally the fact that we have only a total of eight assembly workers each working 48 hours a week gives us a labour capacity of 384 hours As each unit of each product uses 20 hours of this capacity we have the constraint 20x1 20x2 20x3 20x4 20x5 384 14 We have now expressed our original practical problem as a mathematical model The particular form that this model takes is that of an LP model This model is now a welldefined mathematical problem We wish to find values for the variables x1 x2 x5 that make expression 11 the objective function as large as possible but still satisfy constraints 1214 You should be aware of why the term linear is applied to this particular type of problem Expression 11 and the lefthand sides of constraints 1214 are all linear Nowhere do we get terms like x2 1 x1x2 or log x appearing There are a number of implicit assumptions in this model that we should be aware of Firstly we must obviously assume that the variables x1 x2 x5 are not allowed to be negative that is we do not make negative quantities of any product We might explicitly state these conditions by the extra constraints x1 x2 x5 0 15 In most LP models the nonnegativity constraints 15 are implicitly assumed to apply unless we state otherwise Secondly we have assumed that the variables x1 x2 x5 can take fractional values for example it is meaningful to make 236 units of PROD 1 This assumption may or may not be entirely warranted If for example PROD 1 represented gallons of beer fractional quantities would be acceptable On the other hand if it represented numbers of motor cars it would not be meaningful In practice the assumption that the variables can be 8 MODEL BUILDING IN MATHEMATICAL PROGRAMMING fractional is perfectly acceptable in this type of model if the errors involved in rounding to the nearest integer are not great If this is not the case we have to resort to IP The model above illustrates some of the essential features of an LP model 1 There is a single linear expression the objective function to be maximized or minimized 2 There is a series of constraints in the form of linear expressions which must not exceed some specified value LP constraints can also be of the form and indicating that the value of certain linear expressions must not fall below a specified value or must exactly equal a specified value 3 The set of coefficients 288 192 384 on the righthand sides of constraints 1214 is generally known as the righthand side column Practical models will of course be much bigger more variables and constraints and more complicated but they must always have the above three essential features The optimal solution to the above model is included in Section 62 In order to give a wider picture of how LP models can arise we give a second small example of a practical problem Example 12 A Linear Programming Model Blending A food is manufactured by refining raw oils and blending them together The raw oils come in two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 Vegetable oils and nonvegetable oils require different production lines for refin ing In any month it is not possible to refine more than 200 tons of vegetable oil and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored There is a technological restriction of hardness in the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly The costs per ton and hardness of the raw oils are VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 Cost 110 120 130 110 115 Hardness 88 61 20 42 50 The final product sells for 150 per ton INTRODUCTION 9 How should the food manufacturer make their product in order to maximize their net profit This is another very common type of application of LP although of course practical problems will be generally much bigger Variables are introduced to represent the unknown quantities x1 x2 x5 represent the quantities tons of VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 that should be bought refined and blended in a month y represents the quantity of the product that should be made Our objective is to maximize the net profit 110x1 120x2 130x3 110x4 115x5 150y 16 The refining capacities give the following two constraints x1 x2 200 17 x3 x4 x5 250 18 The hardness limitations on the final product are imposed by the following two constraints 88x1 61x2 2x3 42x4 5x5 6y 0 19 88x1 61x2 2x3 42x4 5x5 3y 0 110 Finally it is necessary to make sure that the weight of the final product is equal to the weight of the ingredients This is done by a continuity constraint x1 x2 x3 x4 x5 y 0 111 The objective function 16 to be maximized together with constraints 17111 make up our LP model The linearity assumption of LP is not always warranted in a practical problem although it makes any model computationally much easier to solve When we have to incorporate nonlinear terms in a model either in the objective function or the constraints we obtain an nonlinear programming NLP model In Chapter 7 we will see how such models may arise and a method of modelling a wide class of such problems using separable programming Nevertheless such models are usually far more difficult to solve Finally the assumption that variables can be allowed to take fractional values is not always warranted When we insist that some or all of the variables in an LP model must take integer whole number values we obtain an integer programming IP model Such models are again much more difficult to solve than conventional LP models We will see in Chapters 810 that IP opens up the possibility of modelling a surprisingly wide range of practical problems Another type of model that we discuss in Section 42 is known as a stochastic programming model This arises when some of the data are uncertain but can be specified by a probability distribution Although data in many LP models may be uncertain their representation by expected values alone may not be 10 MODEL BUILDING IN MATHEMATICAL PROGRAMMING sufficient Situations in which a more explicit recognition of the probabilistic nature of data may be made but the resultant model still converted to a linear program are described In Chapter 3 we mention chanceconstrained models and in Chapter 4 multistaged models with recourse both of which fall in the category of stochastic programming An example of the use of this latter type of model is given in Sections 1224 1324 and 1424 where it is applied to determining the price of airline tickets over successive periods in the face of uncertain demand A good reference to stochastic programming is Kall and Wallace 1994 2 Solving mathematical programming models 21 Algorithms and packages A set of mathematical rules for solving a particular class of problem or model is known as an algorithm We are interested in algorithms for solving linear programming LP separable programming and integer programming IP models An algorithm can be programmed into a set of computer routines for solving the corresponding type of model assuming the model is presented to the com puter in a specified format For algorithms that are used frequently it turns out to be worth writing very sophisticated and efficient computer programmes for use with many different models Such programmes usually consist of a num ber of algorithms collected together as a package of computer routines Many such package programmes are available commercially for solving mathematical programming models They usually contain algorithms for solving LP models separable programming models and IP models These packages are written by computer manufacturers consultancy firms and software houses They are fre quently very sophisticated and represent many personyears of programming effort When a mathematical programming model is built it is usually worth making use of an existing package to solve it rather than getting diverted onto the task of programming the computer to solve the model oneself The algorithms that are almost invariably used in commercially available packages are i the revised simplex algorithm for LP models ii the separable extension of the revised simplex algorithm for separable programming models and iii the branch and bound algorithm for IP models It is beyond the scope of this book to describe these algorithms in detail Algorithms i and ii are well described in Beale 1968 Algorithm iii is Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 12 MODEL BUILDING IN MATHEMATICAL PROGRAMMING outlined in Section 83 and well described in Nemhauser and Wolsey 1988 Although the above three algorithms are not the only methods of solving the corresponding models they have proved to be the most efficient general methods It should also be emphasized that the algorithms are not totally independent Hence the desirability of incorporating them in the same package Algorithm ii is simply a modification of i and would use the same computer programme that would make the necessary changes in execution on recognizing a separable model Algorithm iii uses i as its first phase and then performs a tree search procedure as described in Section 83 One of the advantages of package programmes is that they are generally very flexible to use They contain many procedures and options that may be used or ignored as the user thinks fit We outline some of the extra facilities which most packages offer besides the three basic algorithms mentioned above 211 Reduction Some packages have a procedure for detecting and removing redundancies in a model and so reducing its size and hence time to solve Such procedures usually go under the name REDUCE PRESOLVE or ANALYSE This topic is discussed further in Section 34 212 Starting solutions Most packages enable a user to specify a starting solution for a model if heshe wishes If this starting solution is reasonably close to the optimal solution the time to solve the model can be reduced considerably 213 Simple bounding constraints A particularly simple type of constraint that often occurs in a model is of the form x U where U is a constant For example if x represented a quantity of a product to be made U might represent a marketing limitation Instead of expressing such a constraint as a conventional constraint row in a model it is more efficient simply to regard the variable x as having an upper bound of U The revised simplex algorithm has been modified to cope with such a bound algorithmically the bounded variable version of the revised simplex Lower bound constraints such as x L need not be specified as conventional constraint rows either but may be dealt with analogously Most computer packages can deal with bounds on variables in this way 14 MODEL BUILDING IN MATHEMATICAL PROGRAMMING given in Section 12 This blending problem is obviously much smaller than most realistic models but serves to show the form in which a model might be presented This problem was converted into a model involving five constraints and six variables It is convenient to name the variables VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 and PROD The objective is conveniently named PROF profit and the constraints VVEG vegetable refining NVEG nonvegetable refining UHAR upper hardness LHAR lower hardness and CONT continuity The data are conveniently drawn up in the matrix presented in Table 21 It will be seen that the righthand side coefficients are regarded as a column and named CAP capacity Blank cells indicate a zero coefficient Table 21 VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 PROD CAP PROF 110 120 130 110 115 150 VVEG 1 1 200 NVEG 1 1 1 250 UHAR 88 61 20 42 50 60 LHAR 88 61 20 42 50 30 CONT 10 10 10 10 10 10 The information in Table 21 would generally be presented to the computer through a modelling language There is however a standard format for presenting such information to most computer packages and almost all modelling languages could convert a model to this format This is known as MPS mathematical programming system format Other format designs exist but MPS format is the most universal The presented data would be as in Table 21 These data are divided into three main sections the ROWS section the COLUMNS section and the RHS section After naming the problem BLEND the ROWS section consists of a listing of the rows in the model together with a designator N L G or E N stands for a nonconstraint row clearly the objective row must not be a constraint L stands for a lessthanorequal constraint G stands for a greaterthanorequal constraint E stands for an equality constraint The COLUMNS section contains the body of the matrix coefficients These are scanned column by column with up to two nonzero coefficients in a statement zero coefficients are ignored Each statement contains the column name row names and corresponding matrix coefficients Finally the RHS section is regarded as a column using the same format as the COLUMNS section The ENDATA entry indicates the end of the data Clearly it may sometimes be necessary to put in other data as well such as bounds The format for such data can always be found from the appropriate manual for a package With large models the solution procedures used will probably be more com plicated than the standard default methods of the package being used There are SOLVING MATHEMATICAL PROGRAMMING MODELS 15 many refinements to the basic algorithms which the user can exploit if he or she thinks it desirable It should be emphasized that there are few hardandfast rules concerning when these modifications should be used A mathematical program ming package should not be regarded as a black box to be used in the same way with every model on all computers Experience with solving the same model again and again on a particular computer with small modifications to the data should enable the user to understand what algorithmic refinements prove efficient or inefficient with the model and computer installation Experimentation with dif ferent strategies and even different packages is always desirable if a model is to be used frequently NAME BLEND ROWS N PROF L VVEG L NVEG L UHRD G LHRD E CONT COLUMNS VEG 01 PROF 110000000 VVEG 1000000 VEG 01 UHRD 8800000 LHRD 8800000 VEG 01 CONT 1000000 VEG 02 PROF 120000000 VVEG 1000000 VEG 02 UHRD 6100000 LHRD 6100000 VEG 02 CONT 1000000 OIL 01 PROF 130000000 NVEG 1000000 OIL 01 UHRD 2000000 LHRD 2000000 OIL 01 CONT 1000000 OIL 02 PROF 110000000 NVEG 1000000 OIL 02 UHRD 4200000 LHRD 4200000 OIL 02 CONT 1000000 OIL 03 PROF 115000000 NVEG 1000000 OIL 03 UHRD 5000000 LHRD 5000000 OIL 03 CONT 1000000 PROD PROF 150000000 UHRD 6000000 PROD LHRD 3000000 CONT 1000000 RHS RHS00001 VVEG 200000000 NVEG 250000000 ENDATA One computational consideration that is very important with large models is mentioned briefly This concerns starting the solution procedure at an intermediate stage There are two main reasons why one might wish to do this First one might 16 MODEL BUILDING IN MATHEMATICAL PROGRAMMING be resolving a model with slightly modified data Clearly it would be desirable to exploit ones knowledge of a previous optimal solution to save computation in obtaining the new optimal solution With linear and separable programming models it is usually fairly easy to do this with a package although it is much more difficult for IP models Most packages have the facility to SAVE a solution on a file Through the control programme it is usually possible to RESTORE or REINSTATE such a solution as the starting point for a new run A second reason for wishing to SAVE and RESTORE solutions is that one may wish to terminate a run prematurely Possibly the run may be taking a long time and a more urgent job has to go on the computer Alternatively the calculations may be running into numerical difficulty and have to be abandoned In order not to waste the sometimes considerable computer time already expended the intermediate nonoptimal solution obtained just before termination can be saved and used as a starting point for a subsequent run It is common to save intermediate solutions at regular intervals in the course of a run In this way the last such solution before termination is always available 23 Decision support and expert systems Some mathematical programming algorithms are incorporated in computer soft ware designed for specific applications Such systems are sometimes referred to as decision support systems Often they are incorporated into management information systems They usually perform a large number of other functions as well as possibly solving a model These functions probably include accessing databases and interacting with managers or decision makers in a user friendly manner If such systems do incorporate mathematical programming algorithms then many of the modelling and algorithmic aspects are removed from the user who can concentrate on their specific application In designing and writing such systems however it is obviously necessary to automate many of the model building and interpreting procedures discussed in this book As decision support systems become more sophisticated they are likely to present users with possible choices of decision besides the simpler tasks of stor ing structuring and presenting data Such choices may well necessitate the use of mathematical programming although this function will be hidden from the user Another related concept in computer applications software is that of expert systems These systems also pay great attention to the user interface They are for example sometimes designed to accept informal problem definitions and by interacting with users to help them build up a more precise definition of the problem possibly in the form of a model This information is combined with expert information built up in the past in order to help decision making The computational procedures used often involve mathematical programming concepts eg tree searches in IP as described in Section 83 While expert systems are beyond the scope of this book the design and writing of such systems should again depend on mathematical programming and modelling concepts SOLVING MATHEMATICAL PROGRAMMING MODELS 17 The use of mathematical programming in artificial intelligence and expert systems in particular is described by Jeroslow 1985 and Williams 1987 24 Constraint programming CP This is a different approach to solving many problems that can also be solved by IP It is also sometimes known as constraint satisfaction or finite domain programming While the methods used can be regarded as less sophisticated in a conventional mathematical sense they use more sophisticated computer science techniques They have representational advantages that can make problems easier to model and sometimes easier to solve in view of the more concise representation Therefore we discuss the subject here It seems likely that hybrid systems will eventually emerge in which the modelling capabilities of constraint programming CP are used in modelling languages for systems that use either CP or IP for solving the models The integration of the two approaches is discussed by Hooker 2011 who has helped to design such a system SIMPL Yunes et al 2010 In CP each variable has a finite domain of possible values Constraints connect or restrict the possible combinations of values that the variables can take These constraints are of a richer variety than the linear constraints of IP although these are also included and are usually expressed in the form of predicates which must be true or false They are also known as global or meta constraints since they are applied if used to the model as a whole and incorporated in the CP system used That is in contrast to local or procedural constraints which can be constructed by the user to aid the solution process Conventional IP models use only declarative constraints so making a clear distinction between the statement of the model and the solution procedure For illustration we give some of the more common global constraints that are used in CP systems often with different names alldifferent x1 x2 xn means that all the variables in the predicate must take distinct values While this condition can be modelled by a conventional IP formulation it is more cumbersome Once one of the variables has been set to one of the values in its domain either temporarily or permanently this predicate implies that the value must be taken out of the domain of the other variables a process known as constraint propagation In this way constraint propagation is used progressively to restrict the domain of the variables until a feasible set of values can be found for each variable or it can be shown that none exists This has similarities to REDUCE PRESOLVE and ANALYSE discussed in Section 34 when applied to bound reduction CP however allows domain reduction as well as bound reduction Other useful predicates for modelling are The predicate stipulated by a constraint of the form j aj xj b 18 MODEL BUILDING IN MATHEMATICAL PROGRAMMING The cardinality predicate written in a form such as cardm x1 x2 xnv meaning that exactly m of the variables x1 x2 xn must take the value v The cumulative t1 t2 tn D1 D2 Dn C 1 C 2 C n C predicate restricts n jobs to start at times t1 t2 tn when they have durations D1 D2 Dn consume a resource in amounts C 1 C 2 C n and cannot be using a total of more than C units of the resource at any time The circuit x1 x2 xn predicate stipulates that the numbers x1 x2 xn are a permutation of 1 2 n where xi is the number in the permutation occurring after i Whatsmore the permutation must be such that it represents a complete circuit that is there are no subcircuits see Chapter 9 in relation to the travelling salesman problem The element j x1 x2 xn z predicate sets z equal to the jth element in list of variables x1 x2 xn The lex greater predicate can be written in the form x1 x2 xn lex y1 y2 yn It is true if x1 y1 x2 y2 xr yr xr 1 yr 1 While all the above predicates can be modelled often in more than one way using conventional IP variables and constraints this may be cumbersome and difficult There are many more global constraintspredicates available with most com mercial CP systems described in the associated manual There are a number of situations where CP is more flexible than IP For example it is common when building an IP model Chapter 9 to wish to use assignment variables such as xij 1 if and only if i is assigned to j In CP this would be done by a function f i j Another common condition is to wish to specify the inverse function that is given j find i such that f i j This can be done by a version of the element predicate for example element j f i i which specifies whether the inverse of j is i The problems to which CP is applied often exhibit a large degree of sym metry This may be among the constraints predicates solutions or both For example i 1 i 2 i n may be identical entities that have to be assigned to entities j 1 j 2 j n Rather than enumerating all the equivalent solutions computation can be radically reduced by for example using lex greater to give a priority order to the solutions This problem also arises with conventional IP models and improved modelling can also be used there Chapter 10 CP is mainly useful where one is trying to find a solution from an often astro nomic number of possibilities This often happens with combinatorial problems Chapter 8 where one is trying to find a needle in a haystack that is a solu tion to a complicated set of conditions where few or none exist It is less useful where one is seeking an optimal solution to a problem with many feasible solu tions for example the travelling salesman problem Chapter 9 or if a problem SOLVING MATHEMATICAL PROGRAMMING MODELS 19 has no objective function or all that is sought is a feasible solution It usually lacks the ability to prove optimality apart from by imposing successively weaker constraints on the objective until feasibility is obtained Conventional IP uses the concept of a usually LP relaxation Section 83 to restrict the tree search for an optimal solution This is done since the relaxation gives a bound on the optimal objective value an upper bound for a maximization or a lower bound for a minimization This gives it great strength However CP uses more sophis ticated branching strategies than IP Such algorithmic considerations are beyond the scope of this book There is considerable interest in using predicates such as those used in CP for modelling in IP and then converting to a conventional IP formulation An early paper by McKinnon and Williams 1989 shows how all the constraints of any IP model can be formulated using the nested atleastm x1 x2 xnv meaning at least m of x1 x2 xn must take the value v predicate Comparisons and connections between IP and CP are discussed by Barth 1995 Bockmayr and Kasper 1998 Brailsford et al 1996 Proll and Smith 1998 DarbyDowman and Little 1998 Hooker 1998 and Wilson and Williams 1998 Different ways of formulating the alldifferent predicate using IP is discussed by Williams and Yan 2001 This page is blank 3 Building linear programming models 31 The importance of linearity It was pointed out in Section 12 that a linear programming model demands that the objective function and constraints involve linear expressions Nowhere can we have terms such as x3 1 ex1 or x1x2 appearing For many practical problems this is a considerable limitation and rules out the use of linear programming Non linear expressions can however sometimes be converted into a suitable linear form The reason why linear programming models are given so much attention in comparison with nonlinear programming models is that they are much easier to solve Care should also be taken however to make sure that a linear programming model is only fitted to situations where it represents a valid model or justified approximation It is easy to be influenced by the comparative ease with which linear programming models can be solved compared with nonlinear ones It is worth giving an indication of why linear programming models can be solved more easily than nonlinear ones In order to do this we use a twovariable model as it can be represented geometrically Maximize 3x1 2x2 subject to x1 x2 4 2x1 x2 5 x1 4x2 2 x1 x2 0 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 22 MODEL BUILDING IN MATHEMATICAL PROGRAMMING The values of the variables x1 and x2 can be regarded as the coordinates of the points in Figure 31 0 0 1 C A B 2 3 4 1 2 x1 x2 3 Figure 31 The optimal solution is represented by point A where 3x1 2x2 has a value of 9 Any point on the broken line in Figure 31 will give the objective 3x1 2x2 this value Other values of the objective correspond to a line parallel to this It should be obvious geometrically that in any twovariable example the optimal solution will always lie on the boundary of the feasible shaded region Usually it will occur at a vertex such as A It is possible however that the objective lines might be parallel to one of the sides of the feasible region For example if the objective function in the above example were 4x1 2x2 the objective lines would be parallel to AC in Figure 31 The point A would then still be an optimal solution but C would be also and any point between A and C We would have a case of BUILDING LINEAR PROGRAMMING MODELS 23 alternative solutions This topic is discussed at greater detail in Sections 62 and 63 Our focus here however is to show that the optimal solution if there is one always lies on the boundary of the feasible region In fact even if the case of alternative solutions does arise there will always be an optimal solution that lies at a vertex This last fact generalizes to problems with more variables which would need more dimensions to be represented geometrically It is this fact that makes linear programming models comparatively easy to solve The simplex algorithm works by only examining vertex solutions rather than the generally infinite set of feasible solutions It should be possible to appreciate that this simple property of linear program ming models may not apply well to nonlinear programming models For models with a nonlinear objective function the objective lines in two dimensions would no longer be straight lines If there were nonlinearities in the constraints the feasible region might not be bounded by straight lines either In these circum stances the optimal solution might well not lie at a vertex It might even lie in the interior of the feasible region Moreover having found a solution it may be rather difficult to be sure that it is optimal socalled local optima may exist All these considerations are described in Chapter 7 Our purpose here is simply to indicate the large extent to which linearity makes mathematical programming models easier to solve Finally it should not be suggested that a linear programming model always represents an approximation to a nonlinear situation There are many practical situations where a linear programming model is a totally respectable representation 32 Defining objectives With a given set of constraints different objectives will probably lead to different optimal solutions Nevertheless it should not automatically be assumed that this will always happen It is possible that two different objectives can lead to the same operating pattern As an extreme case of this it can happen that the objective is irrelevant The constraints of a problem may define a unique solution For example consider the following three constraints x1 x2 2 31 x1 1 32 x2 1 33 These force the solution x1 x2 1 no matter what the objective is Practical sit uations do arise where there is no freedom of action and only one feasible solution is possible If the model for such a situation is at all complicated this property may not be apparent Should different objective functions always yield the same optimal solution the property may be suspected and should be investigated as a 24 MODEL BUILDING IN MATHEMATICAL PROGRAMMING greater understanding of what is being modelled must surely result In fact such a discovery would result in there being no further need for linear programming We now assume however that we have a problem where the definition of a suitable objective is of genuine importance Possible objectives that might be suggested for optimization by an organization are as follows Maximize profit Minimize cost Maximize utility Maximize turnover Maximize return on investment Maximize net present value Maximize number of employees Minimize number of employees Minimize redundancy Maximize customer satisfaction Maximize probability of survival Maximize robustness of operating plan Many other objectives could be suggested It could well be that there is no desire to optimize anything at all Frequently a number of objectives will apply simultaneously Possibly some of these objectives will conflict It is how ever our contention that mathematical programming can be relevant in any of these situations that is in the case of optimizing single objectives multiple and conflicting objectives or problems where there is no optimization of the objective 321 Single objectives Most practical mathematical programming models used in operational research involve either maximizing profit or minimizing cost The profit that is max imized would usually be more accurately referred to as contribution to profit or the cost as variable cost In a cost minimization the cost incorporated in an objective function would normally only be a variable cost For example suppose each unit of a product produced cost C It would only be valid to assume that x units would cost Cx if C were a marginal cost that is the extra cost incurred for each extra unit produced It would normally be incorrect to add in fixed costs such as administration or capital costs of equipment An exception to this does arise with some integer programming models when we allow the model itself to decide whether or not a certain fixed cost should be incurred for example if we do not make anything we incur no fixed cost but if we make anything at all we do incur such a cost Normally however with standard linear programming models we are interested only in variable costs Indeed a common mistake in building a model is to use average costs rather than marginal costs Similarly when profit coefficients are calculated it is only normally correct to subtract BUILDING LINEAR PROGRAMMING MODELS 25 variable costs from incomes As a result of this the term profit contribution might be more suitable In the common situation where a linear programming model is being used to allocate productive capacity in some sort of optimal manner there is often a choice to be made over whether simply to minimize cost or maximize profit Normally a cost minimization involves allocating productive capacity to meet some specified known demand at minimum cost Such models will probably contain constraints such as j xij Di for all i 34 or something analogous where xij is the quantity of product i produced by process j and Di is the demand for product i Should such constraints be left out inadvertently as sometimes happens the minimum cost solution will often turn out to produce nothing On the other hand if a profit maximization model is built it allows one to be more ambitious Instead of specifying constant demands Di it is possible to allow the model to determine optimal production levels for each product The quantities Di would then become variables di representing these production levels Constraints 34 would become j xij di 0 for all i 35 In order that the model can determine optimal values for the variable di they must be given suitable unit profit contribution coefficients Pi in the objective function The model would then be able to weigh up the profits resulting from different production plans in comparison with the costs incurred and determine the optimal level of operations Clearly such a model is doing rather more than the simpler cost minimization model In practice it may be better to start with a cost minimization model and get it accepted as a planning tool before extending the model to be one of profit maximization In a profit maximization model the unit profit contribution figure Pi may itself depend on the value of the variable di The term Pi di in the objective function would no longer be linear If Pi could be expressed as a function of di a nonlinear model could be constructed An example of this idea is described by McDonald et al 1974 in a resource allocation model for the health service A complication may arise in defining a monetary objective when the model represents activities taking place over a period of time Some method has to be found for valuing profits or costs in the future in comparison with the present The most usual technique is to discount future money at some rate of interest Objective coefficients corresponding to the future will then be suitably reduced Models where this might be relevant are discussed in Section 41 and are known as multiperiod or dynamic models A number of problems presented in Part II give rise to such models A further complication is illustrated by the ECONOMIC PLANNING problem of Part II Here decisions have to be made regarding 26 MODEL BUILDING IN MATHEMATICAL PROGRAMMING whether or not to forego profit now in order to invest in new plant so as to make larger profits in the future The relative desirabilities of different growth patterns lead to alternative objective functions 322 Multiple and conflicting objectives A mathematical programming model involves a single objective function that is to be maximized or minimized This does not however imply that problems with multiple objectives cannot be tackled Various modelling techniques and solution strategies can be applied to such problems A first approach to a problem with multiple objectives is to solve the model a number of times with each objective in turn The comparison of the different results may suggest a satisfactory solution to the problem or indicate further investigations A fairly obvious example of two objectives each of which can be optimized in turn is given by the MANPOWER PLANNING problem of Part II Here either cost or redundancy can be minimized The computational task of using different objectives in turn is eased if each solution is used as a starting solution to a run with a new objective as mentioned in Section 23 Objectives and constraints can often be interchanged for example we may wish to pursue some desirable social objective so long as costs do not exceed a specified level or alternatively we may wish to minimize cost using the social consideration as a constraint on our operations This interplay between objectives and constraints is a feature of many mathematical programming models which is far too rarely recognized Once a model has been built it is extremely easy to convert an objective into a constraint or vice versa The proper use for such a model is to solve it a number of times making such changes An examination and discussion of the resultant solutions should lead to an understanding of the operating options available in the practical situation which has been modelled We therefore have one method of coping with multiple objectives through treat ing all but one objective as a constraint Experiments can then be carried out by varying the objective to be optimized and the righthand side values of the objectiveconstraints Another way of tackling multiple objectives is to take a suitable linear com bination of all the objective functions and optimize that It is clearly necessary to attach relative weightings or utilities to the different objectives What these weightings should be may often be a matter of personal judgment Again it will probably be necessary to experiment with different such composite objectives in order to map out a series of possible solutions Such solutions can then be presented to the decision makers as policy options Most commercial packages allow the user to define and vary the weightings of composite objective functions very easily It should not be thought that this approach to multiple objectives is completely distinct from the approach of treating all but one of the objectives as a constraint When a linear programming model is solved each constraint has associated with it a value known as a shadow price These values have considerable economic importance and are discussed at length in Section 62 If these shadow prices were to be used as the weightings to be given to the objectivesconstraints in the composite approach that we have described in this section then the same optimal solution could be obtained When objectives are in conflict as multiple objectives frequently will be to some extent any of the above approaches can be adopted Care must be taken however when objectives are replaced by constraints not to model conflicting constraints as such The resultant model would clearly be infeasible Conflicting constraints necessitate a relaxation in some or all of these constraints Constraints become goals which may or may not be achieved The degree of over or under achievement is represented in the objective function This way of allowing the model itself to determine how to allow a certain degree of relaxation is described in Section 33 and is sometimes known as goal programming There is no one obvious way of dealing with multiple objectives through mathematical programming Some or all of the above approaches should be used in particular circumstances Given a situation with multiple objectives in which there are no clearly defined weightings for the objectives no cutanddried approach can ever be possible Rather than being a cause for regret this is a healthy situation It would be desirable if alternative approaches were adopted more often in the case of single objective models rather than a once only solution being obtained and implemented 323 Minimax objectives The following type of objective arises in some situations Minimize Maximum 𝑖 𝑗𝑎𝑖𝑗𝑥𝑗 subject to conventional linear constraints This can be converted into a conventional linear programming form by introducing a variable z to represent the above objective In addition to the original constraints we express the transformed model as Minimize z subject to 𝑗𝑎𝑖𝑗𝑥𝑗 z 0 for all 𝑖 The new constraints guarantee that z will be greater than or equal to each of 𝑗𝑎𝑖𝑗𝑥𝑗 for all 𝑖 By minimizing z it will be driven down to the maximum of these expressions A special example of this type of formulation arises in goal programming and is discussed in Section 33 It also arises in the formulation of zerosum games as linear programmes Of course a maximin objective can easily be dealt with similarly It should however be pointed out that a maximax or minimin objective cannot be dealt with by linear programming and requires integer programming This is discussed in Section 94 324 Ratio objectives In some applications the following nonlinear objective arises Maximize or Minimize 𝑗𝑎𝑗𝑥𝑗𝑗𝑏𝑗𝑥𝑗 Rather surprisingly the resultant model can be converted into a linear programming form by the following transformations i Replace the expression 1𝑗𝑏𝑗𝑥𝑗 by a variable t ii Represent the products 𝑥𝑗𝑡 by variables 𝑤𝑗 The objective now becomes Maximize 𝑗𝑎𝑗𝑤𝑗 iii Introduce a constraint 𝑗𝑏𝑗𝑤𝑗1 in order to satisfy condition i Convert the original constraints of the form 𝑗𝑑𝑗𝑥𝑗 e to 𝑗𝑑𝑗𝑤𝑗 et 0 It must be pointed out that this transformation is only valid if the denominator 𝑗𝑏𝑗𝑥𝑗 is always of the same sign and nonzero If necessary and it is valid an extra constraint must be introduced to ensure this If 𝑗𝑏𝑗𝑥𝑗 always be negative the directions of the inequalities in the constraints above must of course be reversed Once the transformed model is solved the values of the 𝑥𝑗 variables can be found by dividing 𝑤𝑗 by t An interesting application involving this type of objective arises in devising performance measures for certain organizations where for example there is no BUILDING LINEAR PROGRAMMING MODELS 29 profit criterion that can be used This is described by Charnes et al 1978 The objective arises as a ratio of weighted inputs and outputs of the organization variables represent the weighting factors Each organization is allowed to choose the weighting factors so as to maximize its performance ratio subject to certain constraints This subject is now known as data envelopment analysis DEA An illustration and description of its use are given by the EFFICIENCY ANALYSIS problem in Part II 325 Nonexistent and nonoptimizable objectives The phrase nonoptimizable objectives might be regarded as selfcontradictory We will however take the word objective in this phrase in the nontechnical sense In many practical situations there is no wish to optimize anything Even if an organization has certain objectives such as survival there may be no question of optimizing them or possibly any meaning to be attached to the word optimization when so applied Mathematical programming is sometimes dis missed rather peremptorily as being concerned with optimization when many practical problems involve no optimization Such a dismissal of the technique is premature If the problem involves constraints finding a solution that satis fies the constraints may be by no means straightforward Solving a mathematical programming model of the situation with an arbitrary objective function will at least enable one to find a feasible solution if it exists that is a solution that satisfies the constraints The last remark is often very relevant to certain inte ger programming models where a very complex set of constraints may exist It is however sometimes relevant to the constraints of a conventional linear programming model The use of a contrived objective is also of value beyond simply enabling one to create a welldefined mathematical programming model By optimizing an objective or a series of objectives in turn extreme solu tions satisfying the constraints are obtained These extreme solutions can be of great value in indicating the accuracy or otherwise of the model Should any of these solutions be unacceptable from a practical point of view then the model is incorrect and should be modified if possible As stated before the validation of a model in this way is often as valuable if not more valuable an activity as that of obtaining solutions to be used in decision making 33 Defining constraints Some of the most common types of constraint that arise in linear programming models are classified below 331 Productive capacity constraints These are the sorts of constraints that arose in the product mix example of Section 12 If resources to be used in a productive operation are in limited supply then the relationship between this supply and the possible uses made of it by the different activities give rise to such a constraint The resources to be considered could be processing capacity or manpower 332 Raw material availabilities If certain activities such as producing products make use of raw materials that are in limited supply then this clearly should result in constraints 333 Marketing demands and limitations If there is a limitation on the amount of a product that can be sold which could well result in less of the product being manufactured than would be allowed by the other constraints this should be modelled Such constraints may be of the form 𝑥 M 36 where 𝑥 is the variable representing the quantity to be made and M is the market limitation Minimum marketing limitations might also be imposed if it were necessary to make at least a certain amount of a product to satisfy some demand Such constraints would be of the form 𝑥 L 37 Sometimes Inequalities 36 or 37 might be constraints instead if some demand had to be met exactly The constraints 36 and 37 or as equalities are especially simple When the simplex algorithm is used to solve a model such constraints are more efficiently dealt with as simple bounds on a variable This is discussed later in this chapter 334 Material balance continuity constraints It is often necessary to represent the fact that the sum total of the quantities going into some process equals the sum total coming out For example in the blending problem of Section 12 we had to ensure that the weight of the final product was equal to the total weight of the ingredients Such conditions are often easily overlooked Material balance constraints such as this will usually be of the form 𝑗𝑥𝑗 𝑘𝑦𝑘 0 38 showing that the total quantity weight or volume represented by the 𝑥𝑗 variables must be the same as the total quantity represented by the 𝑦𝑘 variables Sometimes some coefficients in such a constraint will not be unity but some value representing a loss or gain of weight or volume in a particular process 335 Quality stipulations These constraints usually arise in blending problems where certain ingredients have certain measurable qualities associated with them If it is necessary that the products of blending have qualities within certain limits then constraints will result The blending example of Section 12 gave an example of this Such constraints may involve for example quantities of nutrients in foods octane values for petrols and strengths of materials We now turn our attention to a number of more abstract considerations concerning constraints that we feel a model builder should be aware of 336 Hard and soft constraints A linear programming constraint such as sum over j of aj xj b 39 obviously rules out any solutions in which the sum over j exceeds the quantity b There are some situations where this is unrealistic For example if Inequality 39 represented a productive capacity limitation or a raw material availability there might be practical circumstances in which this limitation would be overruled It might sometimes be worthwhile or necessary to buy in extra capacity or raw materials at a high price In such circumstances 39 would be an unrealistic representation of the situation Other circumstances exist in which it might be impossible to violate 39 For example 39 might be a technological constraint imposed by the capacity of a pipe whose cross section could not be expanded Constraints such as 39 which cannot be violated are sometimes known as hard constraints It is often argued that what we need are soft constraints that can be violated at a certain cost If 39 were to be rewritten as sum over j of aj x u b 310 and u were given a suitable positive negative cost coefficient c for a minimization maximization problem we would have achieved our desired effect The variable b would represent a capacity or raw material availability that could be expanded to b u at a cost cu if the optimization of the model found this to be desirable Possibly the surplus variable u would be given a simple upper bound as well to prevent the increase exceeding a specified amount If Inequality 39 were a constraint an analogous effect could be achieved by a slack variable If Inequality 39 be an equality constraint it would be possible to allow the righthand side coefficient b to be overreached or underreached by modelling it as sum over j of aj xj u v b 311 and giving u and v appropriately weighted coefficients in the objective function It should be apparent that either u or v must be zero in the optimal solution Any solution in which u and v came out positive could be adjusted to produce a better solution by subtracting the smaller of u and v from both of them An alternative way of viewing such soft constraints is through fuzzy sets where a degree of membership corresponds to the amount by which the constraint is violated It has however been shown by Dyson 1980 that formulations in terms of fuzzy set theory can be reformulated as conventional LP models with minimax objectives 337 Chance constraints In some applications it is desired to specify that a certain constraint be satisfied with a given probability for example we may wish to be 95 confident that a constraint holds This situation is written as P sum over j of aj xj b β 312 where β is a probability In practice one might expect larger values of β to correspond to higher costs which should be reflected in the objective function This would create a more complicated model Should no relationship between the value of β and the cost be known as is often the case then a rather crude but sometimes satisfactory approach is to replace Inequality 312 by a deterministic equivalent We would then replace 312 by sum over j of aj xj b 313 where b is a number larger than b such that the satisfaction of 313 implies Inequality 312 This idea is due to Charnes and Cooper 1959 338 Conflicting constraints It sometimes happens that a problem involves a number of constraints not all of which can be satisfied simultaneously A conventional model would obviously be infeasible The objective in such a case is sometimes stipulated to be to satisfy all the constraints as nearly as possible We have the case of conflicting objectives referred to in Section 32 but postponed to this section The type of model that this situation gives rise to is sometimes known as a goal programming model This term was invented by Charnes and Cooper 1961b but the type of model that results is still a linear programming model Each constraint is regarded as a goal that must be satisfied as nearly as possible For example if we wished to impose the following constraints sum over j of aij xj bi for all i 314 but wanted to allow the possibility of them not all being satisfied exactly we would replace them by the soft constraints sum over j of aij xj ui vi bi 315 We are clearly using the same device described before Our objective would be to make sure that each such constraint 314 is as nearly satisfied as possible There are a number of alternative ways of making such an objective specific Two possibilities are as follows 1 Minimize the sum total of the deviations of the row activities sum over j of aij xj from the righthand side values bi 2 Minimize the maximum such deviation over the constraints For many practical problems which of the above objectives is used is not very important Objective 1 can be dealt with by defining an objective function consisting of the sum of the slack u and surplus v variables in the constraints such as Equation 315 Possibly these variables might be weighted in the objective with nonunit coefficients to reflect the relative importance of different constraints An example of the use of such an objective is given in the MARKET SHARING problem of Part II The fact that this is an integer programming model does not affect the argument as such models could equally well result from linear programming models A linear programming application arises in the CURVE FITTING problem Objective 2 is slightly more complicated to deal with but can surprisingly still be accomplished in a linear programming model An extra variable z is introduced This variable represents the maximum deviation We therefore have to impose extra constraints z ui 0 for all i 316 z vi 0 for all i 317 The objective function to be minimized is simply the variable z Clearly the optimal value of z will be no greater than the maximum of ui and vi by virtue of optimality Nor can it be smaller than any of ui or vi by virtue of the constraints 316 and 317 Therefore the optimal value of z will be both as small as possible and exactly equal to the maximum deviation of sum over j of aij xj from its corresponding righthand side bi This type of problem is sometimes known as a bottleneck problem Both the CURVE FITTING and MARKET SHARING problems illustrate this type of objective An interesting example of such a minimax objective is described by Redpath and Wright 1981 in an application of linear programming to deciding levels and directions for irradiating cancerous tumours As an alternative to minimizing the variance of radiation across a tumour they minimize the maximum difference of radiation levels This can clearly be done by computationally easier linear programming rather than quadratic programming Hooker and Williams 2012 show how equity in the form of a minimax objective can be combined with a utilitarian objective in an integer programming model 339 Redundant constraints Suppose we have a constraint such as j ajxj b 318 in a linear programming model If in the optimal solution the quantity j ajxj turns out to be less than b then the constraint 318 will be said to be nonbinding and have a zero shadow price Such a nonbinding constraint could just as well be left out of the model as this would not affect the optimal solution There may well however be good reasons for including such redundant constraints in a model First the redundancy may not be apparent until the model is solved The constraint must therefore be included in case it turns out to be binding Second if a model is to be used regularly with changes in the data then the constraint might become binding with some future data There would then be virtue in keeping the constraint to avoid a future remodelling Third ranging information which is discussed in Section 63 depends on constraints that may well be redundant in the sense of not affecting the optimal solution It should be noted that a constraint such as 318 can be nonbinding even if the quantity j ajxj is equal to b This happens if the removal of the constraint does not affect the optimal solution that is j ajxj must be equal to b for reasons other than the existence of constraint 318 Such nonbinding constraints are recognized by there being a zero shadow price on the constraint in the optimal solution Shadow prices are discussed in Section 62 For constraints analogous results hold If Inequality 318 were such a constraint it would be nonbinding if j ajxj exceeded b but possibly still nonbinding if j ajxj equalled b Finally it should be pointed out that with integer programming it would not be true to say that if j ajxj were less than b in Inequality 318 then Inequality 318 would be nonbinding Inequality 318 could well be binding and therefore not redundant This is discussed in Section 103 The advisability or otherwise of including redundant constraints in a model is discussed in Section 33 A quick method of detecting some such redundancies is also described 3310 Simple and generalized upper bounds It was pointed out that marketing constraints often take the particularly simple forms of Inequality 36 or 37 Simple bounds on a variable such as these are more efficiently dealt with through a modification of the simplex algorithm Most commercial package programmes use this modification Such bounds are not therefore specified as conventional constraints but specified as simple bounds for the appropriate variables A generalization of simple bounds known as generalized upper bounds GUBs also proves to be of great computational value in solving a model and therefore worth recognizing Such constraints are usually written as j xj b 319 The set of variables xj is said to have a GUB of b This means that the sum of these variables must be b If Equation 319 were a constraint instead of an equality the addition of a slack variable would obviously convert it to the form Equation 319 The fact that the coefficients of the variables in Equation 319 are unity is not very important as scaling can always convert any constraint with all nonnegative coefficients into this form What is however important is that when a number of constraints such as Equation 319 exist the variables in them form exclusive sets A set of variables is said to belong to the GUB set if it can belong to no others If variables are specified as belonging to GUB sets it is not necessary to specify the corresponding constraints such as Equation 319 A further modification of the simplex algorithm copes with the implied constraint in an analogous way to simple bounding constraints The diagrammatic representation of a model in Figure 32 shows three GUBtype constraints These constraints could be removed from the model and treated as GUB sets There is normally only virtue in using the GUB modification of the simplex algorithm if a large number of GUB sets can be isolated For example in the transportation problem which is discussed in Section 53 it can be seen that at least half of the constraints can be regarded as GUB sets The computational advantages of recognizing GUB constraints can be very great indeed with large models A way of detecting a large number of such sets if they exist in a model is described by Brearley et al 1975 3311 Unusual constraints In previous sections we have concentrated on constraints that can be modelled using linear programming It is important however not to dismiss unusual restrictions that may arise in a practical problem as not being able to be so 36 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Right hand side General constraints 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Objective function Figure 32 modelled By extending a model to be an integer programming model it is sometimes possible to model such restrictions For example a restriction such as We can only produce product 1 if product 2 is produced but neither of products 3 or 4 are produced could be modelled This topic is discussed much further in Chapter 9 34 How to build a good model Possible aims that a model builder has when constructing a model are the ease of understanding the model detecting errors in the model and computing the solution Ways of trying to achieve and resolve these aims are described below 341 Ease of understanding the model It is often possible to build a compact but realistic model when quantities appear implicitly rather than explicitly for example instead of a nonnegative quantity being represented by a variable y and being equated to an expression f x by the constraint f x y 0 320 the variable y does not appear but f x is substituted into all the expressions where it would appear Building such compact models often leads to great difficulty and extra calculation in interpreting the solution Even though a less compact model takes longer to solve it is often worth the extra time If however a report writer is used this may take care of interpretation difficulties and the use of compact models is desirable from the point of view of the third aim BUILDING LINEAR PROGRAMMING MODELS 37 It is also desirable to use mnemonic names for the variables and constraints in a problem to ease the interpretation of the solution The computer input to the small blending problem illustrated in Section 12 shows how such names can be constructed A very systematic approach to naming variables and constraints in a model is described by Beale et al 1974 342 Ease of detecting errors in the model This aim is clearly linked to the first Errors can be of two types i clerical errors such as bad typing and ii formulation errors To avoid the first type of error it is desirable to build any but very small models using a matrix generator MG or language There is also great value to be obtained from using a PRESOLVE or REDUCE procedure on a model for error detection Clerical or formulation errors often result in a model being unbounded or infeasible Such conditions can often be revealed easily using such a procedure A simple procedure of this kind which also simplifies models is outlined below Formulation can sometimes be done with error detection in mind This point is illustrated in Section 61 343 Ease of computing the solution LP models can use large amounts of computer time and it is desirable to build models that can be solved as quickly as possible This objective can conflict with the first In order to meet the first objective it is desirable to avoid compact models If a PRESOLVE or REDUCE procedure is applied after the model has been built but before solving it then dramatic reductions in size can sometimes be achieved An algorithm for doing this is described by Brearley et al 1975 Karwan et al 1983 provided a collection of methods for detecting redundancy The reduced problem can then be solved and this solution used to generate a solution to the original problem In order to illustrate the possibility of spotting redundancies in a linear pro gramming model we consider the following numerical example Maximize 2x1 3x2 x3 x4 321 subject to x1 x2 x3 2x4 4 322 x1 x2 x3 x4 1 323 x1 x4 3 324 x1 x2 x3 x4 0 325 As x3 has a negative objective coefficient and the problem is one of maximiza tion it is desirable to make x3 as small as possible x3 has positive coefficients 38 MODEL BUILDING IN MATHEMATICAL PROGRAMMING in constraints 322 and 323 As these constraints are both of the type there can be no restriction on making x3 as small as possible Therefore x3 can be reduced to its lower bound of zero and hence be regarded as a redundant variable Having removed the variable x3 from the model constraint 323 is worthy of examination All the coefficients in this constraint are now negative Therefore the value of the expression on the lefthand side of the inequality relation can never be positive This expression must therefore always be smaller than the righthand side value of 1 indicating that the constraint is redundant and may be removed from the problem The above model could therefore be reduced in size With large models such reductions could well lead to substantial reductions in the amount of computation needed to solve the model Infeasibilities and unboundedness can also sometimes be revealed by such a procedure A model is said to be infeasible if there is no solution that satisfies all the constraints including nonnegativity conditions on the variables If on the other hand there is no limit to the amount by which the objective function can be optimized the model is said to be unbounded Such conditions in a model usually suggest modelling errors and are discussed in detail in Section 61 Some package programmes have procedures for reducing models in this way Such procedures go under names such as REDUCE PRESOLVE and ANAL YSE Problem reduction can be taken considerably further Using simple bounds on variables and considering the dual model the dual of a linear programming model is described in Section 62 the above example can be reduced to noth ing completely solved A more complete treatment of reduction is beyond the scope of this book as such reduction procedures are usually programmed and carried out automatically It is not therefore always important that a model builder knows how to reduce his or her model although the fact that it can be simply reduced must have implications for the situation being modelled A much fuller treatment of the subject is given in Brearley et al 1975 and Karwan et al 1983 Substantial reduction in computing time can also often be achieved by exploit ing the special structure of a problem One such structure that has proved particularly valuable is the GUB as described in Section 33 It is obviously desir able that the model builder detects such structure if it be present in a problem although a few computer packages have facilities for doing this automatically through procedures such as those described by Brearley Mitra and Williams 344 Modal formulation In large LP problems reduction in the number of constraints can be achieved using modal formulations If a series of constraints involves only a few vari ables the feasible region for the space of these variables can be considered For example suppose that xA and xB are two of the nonnegative variables in an LP model and they occur in the following constraints BUILDING LINEAR PROGRAMMING MODELS 39 xA xB 7 326 3xA xB 15 327 xB 5 328 The situation can be modelled as demonstrated in Figure 33 by letting the activities for the extreme modes of operation be represented by variables instead of xA and xB If these variables are λ0 λ1 λ2 λ3 and λ4 we only have to specify the single constraint λ0 λ1 λ2 λ3 λ4 1 329 1 0 1 2 3 4 5 2 3 xA xB λ1 λ2 λ3 λ4 λ0 4 5 Figure 33 Whenever xA and xB occur in other parts of the model apart from con straints 326 327 and 328 that are now ignored we substitute the following expressions for xA 2λ2 4λ3 5λ4 330 for xB 5λ1 5λ2 3λ3 331 The coefficients for xA are the xA coordinates of λ1 λ2 in Figure 33 and the coefficients of xB are the xB coordinates In this case we have saved two constraints The ingenious use of this idea can result in substantial savings in the number of constraints in a model Experiments in the use of a special case of this technique are described by Knolmayer 1982 Smith 1973 discusses the use of this approach to modelling practical problems Modal formulations are quite common for particular processes in the petroleum industry The REFINERY OPTIMIZATION model demonstrates 40 MODEL BUILDING IN MATHEMATICAL PROGRAMMING a very simple case of this where a process has only one mode of operation with inputs and outputs in fixed proportions In this case it is better to model the level of the process as an activity to avoid having to represent the fixed relationships between the inputs and outputs by constraints A general methodology for modelling processes with different types of inputoutput relations is developed by MullerMerbach 1987 345 Units of measurement When a practical situation is modelled it is important to pay attention to the units in which quantities are measured Great disparity in the sizes of the coefficients in a linear programming model can make such a model difficult if not impossible to solve For example it would be stupid to measure profit in pounds if the unit profit coefficients in the objective function were of the order of millions of pounds Similarly if capacity constraints allowed quantities in thousands of tons it would be better to allow each variable to represent a quantity in thousands of tons rather than in tons Ideally units should be chosen so that each nonzero coefficient in a linear programming model is of a magnitude between 01 and 10 In practice this may not always be possible Most commercial package programmes have procedures for automatically scaling the coefficients of a model before it is solved The solution is then automatically unscaled before being printed out 35 The use of modelling languages A number of computer software aids exist for helping users to structure and input their problem into a package in the form of a model Programmes to do this are sometimes known as matrix generators MGs Some such systems are more usefully thought of as special purpose high level programming languages and are therefore known as modelling languages It is now recognized that the main hurdle to the successful application of a mathematical programming model is often no longer the problem of computing the solution Rather it lies with the interface between the user and the computer Some of the difficulties can be overcome by freeing the modeller from the specific needs of the package used to solve the model An ambitious attempt to overcome the user interface problems is described by Greenberg 1986 A survey of modelling systems is given by Greenberg and Murphy 1992 Other systems are described by Buchanan and McKinnon 1987 and Green berg et al 1987 These systems pay attention to the output interface as well as the input A number of quite distinct approaches to building MGslanguages are described below Before doing this however we specify the main advantages to be gained from using them BUILDING LINEAR PROGRAMMING MODELS 41 351 A more natural input format The input formats required by most packages are designed more with the package in mind than the user As described in Section 22 most commercial packages use the MPS format This orders the model by columns in a fixed format based on the internal ordering of the matrix in the computer This is often unnatural tedious and a source of clerical error In addition this format is far from concise requiring a separate data statement for every two coefficients in the model The use of a modelling language can overcome all these disadvantages 352 Debugging is made easier As with computer programming the debugging and verification of a model is an important and sometimes timeconsuming task The use of a format that the modeller finds natural makes this task much easier 353 Modification is made easier Models are often used on a regular basis with minor modifications or once built may be used experimentally many times with small changes in data The separation of the data which changes from the structure of the model in most modelling languages facilitates this task 354 Repetition is automated Large models usually arise from the combination or repetition of smaller models for example multiple periods plants or products as described in Section 41 In such circumstances there will be a lot of repetition of particular items of data and structure Formats such as MPS demand a repetition of this data which is both inefficient and a potential source of error Modelling languages generally take account of such repetition very easily through the indexing of time periods etc Some of the distinct approaches adopted by different modelling languages are outlined below Fourer 1983 also provides a very good and comprehensive discussion The methods can only be understood in detail through the manuals of the relevant systems 355 Special purpose generators using a high level language It is possible to write a different program in a general language for each model one wishes to build This approach ignores the similarities in structure that all mathematical programming models have however diverse their application area as well as the standard format required eg MPS which is tedious to remember and incorporate into a general program It seems more sensible to incorporate such similarities into a language 356 Matrix block building systems The often huge matrix of coefficients in a practical model always exhibits a high degree of structure The matrix tends to consist of small dense blocks of coefficients in an otherwise very sparse matrix These blocks are often repetition of each other and are linked together by simple matrices such as identity matrices The systematic combination and replication of submatrices into larger matrices in this way is the method used in certain systems Such systems have functions that for example concatenate and transpose matrices A defect of such systems is that they still require users to think in terms of matrices of coefficients even though they free them from some of the mechanics of assembling such matrices 357 Data structuring systems Some systems allow the user to structure a model in a diagrammatic form Flow diagrams are often particularly helpful For example a problem may involve flows of different raw materials into processes to give products that are then distributed via depots to customers The system will allow the user to translate this representation into the system directly avoiding algebraic notation Such systems can be useful for specific applications for example process industries They do have the defect that the user needs to learn quite a lot about the system before he or she can use it 358 Mathematical languages The above approaches all avoid the use of conventional notation that is notation and indexed sets Indeed an argument sometimes used in their favour is that mathematical notation is unnatural to certain users who prefer to think of their model in other ways This may well be the case with certain established applications eg the petroleum industry where mathematical programming is so established and important for a conceptual methodology of its own to have developed If more generality is required then conventional mathematical notation provides a language that is widely understood as well as providing a concise way of defining many models The fact that the notation is so widely understood by anyone with an elementary mathematical training makes learning to use such a system comparatively easy Most general modelling systems based on mathematical notation are very similar We describe their common features We then illustrate this by a formulation of one of the models in Part II using a typical language NEWMAGIC It is easy to translate such a model into any of the other similar modelling languages Such languages require the user to identify the following components of their model It is straightforward to see how these elements are used by means of the example below although different systems will have varied key words and syntax rules 3581 SETs These usually represent indices for certain classes of linear or integer programming variables For example PROD 1 PROD 2 PROD 57 might be the variables representing 57 different products that vary only in their profits and resource usages We could define a root name PROD followed by an index from a SET A 1 57 In most systems indices can be numbers or names eg towns or months of the year 3582 DATA These are usually coefficients for the model which may be defined in arrays or read from external files The items in the arrays or files are often indexed by the indices described above 3583 VARIABLES These are the linear or integer programming variables that appear in the model typically defined using a root name together with associated indices for example PRODA If the variable is of a special sort for example integer free binary etc this is usually specified with the variable 3584 OBJECTIVE This is the Objective Function of the model given with MAXIMIZE or MINIMIZE It is given a name and usually defined using notation over index sets defined above but by means of key words such as sum or sigma as most keyboards do not support the symbol 3585 CONSTRAINTS These are the conventional linear constraints They will probably also be indexed In addition such languages will have many other facilities and features that are specific to the language and are not discussed in this chapter but explained in the associated manuals We give the example in NEWMAGIC below As with other languages it is convenient to include comments that are done here by including them between the symbols or on a line preceded by MODEL FoodB DATA max t 6 max i 5 SET A 12 B 15 T 1max t I 1max i DATA costT I costdat 44 MODEL BUILDING IN MATHEMATICAL PROGRAMMING hardI 8861204250 VARIABLES bIT uIT sIT ProdT OBJECTIVE MAXIMIZE prof sumt in T 150 Prodt sumi in I costti bit 5 sit CONSTRAINTS loili in I 1 500 bi1 ui1 si1 0 loili in I t in T t 1 si t 1 bit uit sit 0 vvegt in T sumi in A uit 200 voilt in T sumi in B uit 250 lhrdt in T 3 Prodt sumi in I hardi uit uhrdt in T sumi in I hardi uit 6 Prodt contt in T sumi in I uit Prodt Specify bounds for the variables fori in I t in T t maxt sit 1000 fori in I si maxt 500 This is the end of the continuous part of the model The section below adds integer variables and extra constraints to model some logical conditions Separate VARIABLE and CONSTRAINT sections are not necessary they could be included in the VARIABLE and CONSTRAINT sections above The keyword BINARY means an integer variable that can only take the values 0 or 1 VARIABLES dIT BINARY CONSTRAINTS fort in T Formulate second condition fori in A UB on amount of veg oil refinedmonth 200 20 dit uit uit 200 dit fori in B UB on amount of nonveg oil refinedmonth 250 20 dit uit uit 250 dit Formulate first condition sumi in I dit 3 Formulate third condition d1t d5t d2t d5t END MODEL SOLVE FoodB PRINT SOLUTION FOR FoodB FoodBsol QUIT 4 Structured linear programming models 41 Multiple plant product and period models The purpose of this section is to show how large linear programming LP models can arise through the combining of smaller models Almost all very large models arise in this way Such models prove to be more powerful as decision making tools than the submodels from which they are constructed In order to illustrate how a multiplant model can arise in this way we take a very small illustrative example Example 41 A Multiplant Model A company operates in two factories A and B Each factory makes two products standard and deluxe A unit of standard gives a profit contribution of 10 while a unit of deluxe gives a profit contribution of 15 Each factory uses two processes grinding and polishing for producing its products Factory A has a grinding capacity of 80 hours per week and polish ing capacity of 60 hours per week For factory B these capacities are 60 and 75 hours per week respectively The grinding and polishing times in hours for a unit of each type of product in each factory are given in the table below Factory A Factory B Standard Deluxe Standard Deluxe Grinding 4 2 5 3 Polishing 2 5 5 6 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 46 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is possible for example that factory B has older machines than factory A resulting in higher unit processing times In addition each unit of each product uses 4 kg of a raw material which we refer to as raw The company has 120 kg of raw available per week To start with we will assume that factory A is allocated 75 kg of raw per week and factory B the remaining 45 kg per week Each factory can build a very simple LP model to maximize its profit contribu tion This is an obvious example of the product mix application of LP mentioned in Section 12 The following are the resultant models Factory As Model Maximize Profit A 10x1 15x2 subject to Raw A 4x1 4x2 75 Grinding A 4x1 2x2 80 Polishing A 2x1 5x2 60 x1 x2 0 where x1 is the quantity of standard to be produced in A and x2 is the quantity of deluxe to be produced in A Factory Bs Model Maximize Profit B 10x3 15x4 subject to Raw B 4x3 4x4 45 Grinding B 5x3 3x4 60 Polishing B 5x3 6x4 75 x3 x4 0 where x3 is the quantity of standard to be produced in B and x4 is the quantity of deluxe to be produced in B These two models can easily be solved graphically Our purpose is not how ever to concentrate on the mechanics of solving these individual models We do however give the optimal solutions below as these will be discussed later Optimal Solution to Factory As Model Profit is 225 obtained from making 1125 of standard and 75 of deluxe There is a surplus grinding capacity of 20 hours Optimal Solution to Factory Bs Model Profit is 16875 obtained from making 1125 deluxe There is a surplus grind ing capacity of 2625 hours and a surplus polishing capacity of 75 hours STRUCTURED LINEAR PROGRAMMING MODELS 47 Suppose now that a company model is built in order to maximize total profit We will assume that the factories remain distinct and geographically sep arated We will however no longer allocate 75 kg of raw to A and 45 kg to B Instead we will allow the model to decide this allocation There will now be a single raw material constraint limiting the company to 120 kg per week The resultant model is as follows Maximize Profit 10x1 15x2 10x3 15x4 subject to Raw 4x1 4x2 4x3 4x4 120 Grinding A 4x1 2x2 80 Polishing A 2x1 5x2 60 Grinding B 5x3 3x4 60 Polishing B 5x3 6x4 75 x1 x2 x3 x4 0 The Company Model The fact that the constraints raw A and raw B of the factory models have been combined into a single constraint for the company model is of crucial significance We are now asking the model to split the 120 kg of raw optimally between A and B rather than making an arbitrary allocation ourselves As a consequence we would expect a more efficient split resulting in a greater overall company profit This is borne out by the optimal solution Optimal Solution to the Company Model Total profit is 40415 obtained from making 917 of standard in A 833 of deluxe in A and 125 of deluxe in B There is a surplus grinding capacity in A of 2667 hours and a surplus grinding capacity in B of 225 hours A number of points are worth noting in comparing this solution with those for factories A and B individually 1 The total profit is 40414 which is greater than the combined profit 39375 from A and B acting independently 2 Factory A only contributes 21665 to the new total profit whereas it produced a profit of 225 before Factory B however now contributes 1875 to total profit whereas it only produced 16875 before 3 Factory A now uses 70 kg of raw and factory B uses 50 kg It is clear that the company model has biased production more toward factory B than before This has been done by allocating B 50 kg of raw instead of 45 kg 48 MODEL BUILDING IN MATHEMATICAL PROGRAMMING and so depriving A of 5 kg If it had been possible to decide this 7050 split before it would not have been necessary to build a company model This argument also applies to much larger more realistic multiplant models Normally however there will be a number of scarce resources that must be shared between plants rather than the single resource raw which we consider here An optimal split would have to be found for each of these resources Determining such splits would obviously be complex The needs of each plant have to be balanced against how efficiently they use the scarce resources In our example factory Bs older machinery results in it being allocated less of raw than A To start with however our 7545 split was over biased in As favour The above example is intended to show how a multiplant model can arise It is a method of using LP to cope with allocation problems between plants as well as help with decision making within plants The model that we built was a very simple example of a common sort of structure which arises in multiplant models This is known as a block angular structure If we detach the coefficients in the company model and present the problem in a diagrammatic form we obtain Figure 41 10 15 4 4 4 2 120 80 60 60 75 2 5 10 15 4 4 5 3 5 6 Figure 41 The first two rows are known as common rows Obviously one of the common rows will always be the objective row The two diagonally placed blocks of coefficients are known as submodels For a more general problem with a number of shared resources and n plants we would obtain the general block angular structure shown in Figure 42 STRUCTURED LINEAR PROGRAMMING MODELS 49 A0 b0 B1 A1 A2 An B2 b1 b2 bn Bn Figure 42 A0 A1 B1 B2 are blocks of coefficients b0 b1 are columns of coefficients forming the righthand side The block A0 may or may not exist but is sometimes conveniently represented A0 A1 represent the common rows Common constraints in multiplant models usually involve allocating scarce resources raw material processing capacity labour etc across plants They might sometimes represent transport relations between plants For example it might in certain circumstances be advantageous to transport the product of an intermediate process from one plant to another The model could conveniently allow for this if extra variables were introduced into the plants in question to represent amounts transported Suppose we simply wanted to allow for transport from plant 1 to plant 2 This would be accomplished by the constraint x1 x2 0 41 where x1 is the quantity transported from 1 into 2 and x2 is the quantity trans ported into 2 from 1 Apart from constraint 41 x1 would only be involved in constraints relating to the submodel for plant 1 Similarly x2 would only be involved in constraints relating to plant 2 x1 or x2 but not both would probably be given cost coeffi cients in the objective function representing unit transport costs Constraint 41 clearly gives another common row constraint Should a problem with a block angular structure have no common con straints it should be obvious that optimizing it simply amounts to optimizing each subproblem with its appropriate portion of the objective For our sim ple numerical example if there had been no raw material constraint we could have solved each factorys model separately and obtained an overall company optimum In fact as far as the company was concerned it would have been perfectly acceptable to treat each factory as autonomous Once we introduce common constraints however this will probably no longer be the case as the small example demonstrated The more common constraints there are the more interconnected the separate plants must be In Section 42 we discuss how a knowledge of the optimal solutions to the subproblems might be used to obtain an optimal solution to the total problem This can be quite important computationally as such structured problems can often be very large and take a long time to solve if treated as one large model Common sense would suggest that the number of common rows would be a good measure of how close the optimal solution to the total problem was to the sum total of the optimal solutions to the subproblems This is usually the case For problems with only a few common rows one might expect there to be virtue in taking account of the optimal solutions of the subproblems The block angular structure exhibited in Figure 42 does not arise only in multiplant models Multiproduct models arise quite frequently in blending problems Suppose for example that the blending problem that we presented in Section 12 represented only one of a number of products brands that a company manufactured If the different products used some of the same ingredients and processing capacity then it would be possible to take account of their limited supply through a structured model The individual blending models such as those presented in Section 12 would be unable to help make rational allocations of such scarce shared resources between products For example in Figure 42 B1 B2 would represent the individual blending constraints for each product These subproblems would contain variables such as xij representing the quantity of ingredient i in product j If ingredient i was in limited supply we would impose a common constraint j xij availability of ingredient i 42 If ingredient i used αij units of a particular processing capacity in blending product j we would have the common constraint j αij xij total processing capacity available for ingredient i 43 As with multiplant models multiproduct models almost always arise through combining existing submodels The submodels can be used to help make certain operational decisions By combining such submodels into multiple models further decisions can be brought into the realm of LP Another way in which the block angular structure of Figure 42 arises is in multiperiod models Suppose that in our blending problem of Section 12 we wanted to determine not just how to blend in a particular month but also how to purchase each month with the possibility of storing for later use It would then be necessary to distinguish between buying using and storing For each ingredient there would be three corresponding variables These would be linked STRUCTURED LINEAR PROGRAMMING MODELS 51 together by the relations Amount in store at end of period t 1 amount bought in period t amount used in period t amount in store at end of period t 44 These relations give equality constraints The block angular structure of Figure 42 arises through these equality constraints providing common rows linking consecutive time periods Each subproblem B1 consists of the original blending constraints involving only the using variables Such a multiperiod model arises from the FOOD MANUFACTURE problem of Part 2 This problem is the blending problem of Section 12 taken over six months with different raw oil prices for different months Full details of the formulation of this problem are given in Part 3 In a multiperiod model of the kind described above each period need not necessarily be of the same duration Some periods might be of a month while later months might be aggregated together with a corresponding increase in resources represented by righthand side coefficients It will however be very likely that B1 B2 in Figure 42 are the same submatrices representing the same blending constraints in each period The corresponding rows and columns of these matrices will of course be distinguished by different names The way in which multiperiod models should be used is important Such a model is usually run with the first period relating to the present times and subsequent periods relating to the future As a result only the operating decisions suggested by the model for the present month are acted on Operating decisions for future months will probably only be taken as provisional After a further month or the appropriate time period has elapsed the model will be rerun with updated data and the first period applying to the new present period In this way a multiperiod model is in constant use as both an operating tool for the present and a provisional planning tool for the future A further point of importance in multiperiod models concerns what happens at the end of the last time period in the model If the stocks at the end of the last period which occur in constraint 44 are included simply as variables the optimal solution will almost always decide that they should be zero From the point of view of the model this would be sensible as it would be the minimum cost or maximum profit solution In a practical situation however the model is unrealistic as operations will almost certainly continue beyond the end of the last period and stocks would probably not be allowed to run right down One possible way out is to set the final stocks to constant values representing sensible final levels It could be argued that the operating plans for the final period will be very provisional anyway and any inaccuracy that far ahead will not be serious This is the suggestion that is made for dealing with both the multiperiod FACTORY PLANNING and FOOD MANUFACTURE problems of Part 2 An alternative approach that is sometimes adopted is to value the final stocks in some way that is give the appropriate variables positive profits in a maximization model or negative costs in a minimization model In effect 52 MODEL BUILDING IN MATHEMATICAL PROGRAMMING such a valuation would cause the optimal solution to suggest producing final stocks to sell if it appeared profitable Although the organization might never consider the possibility of selling off final stocks the fact that they had been given realistic valuations would cause them to come out at sensible levels A description of a highly structured model that is potentially multiperiod multiproduct and multiplant is given by Williams and Redwood 1974 Another type of structure that often arises in multiperiod models is the stair case structure This is illustrated in Figure 43 A0 B1 A1 B2 b1 A2 B3 An1 Bn Objective function b2 b3 bn Figure 43 In fact a staircase structure such as this could be converted into a block angular structure If alternate steps such as A0 B1 A2 B3 were treated as subproblem constraints and the intermediate steps as common rows we would have a block angular structure The block angular multiperiod model which was described above could also easily be rearranged into a staircase structure Another structure that sometimes arises although less commonly than block angular and staircase structures is shown in Figure 44 Common columns Objective function B1 B2 b1 b2 bn Bn Figure 44 STRUCTURED LINEAR PROGRAMMING MODELS 53 In this type of model the subproblems are connected by common columns rather than rows Such a structure is the dual to the block angular structure the dual is defined in Section 62 The method of Benders 1962 is applicable to solving these types of model even if they are not integer programmes A way in which this type of structure arises is given in the next section 42 Stochastic programmes Stochastic programmes are a special type of LP model Although the title Stochastic Programming is sometimes used it can convey the misleading impression that this is another technique A stochastic programme is a type of LP which models uncertainty in a particular way Stochastic programming is a special case of robust optimization In many practical situations the data in a resultant LP model is not known with certainty This may be for a number of reasons One is an inability to obtain totally accurate data Another is that the model is time staged discussed in Section 41 where certain events which need to be modelled have not yet occurred Robust optimization encompasses both these situations even when we cannot quantify the uncertainty or associated risks It is concerned with obtaining solutions that remain reasonably stable in the face of the uncertainty whether it can be quantified or not The use of sensitivity analysis which is discussed in Section 63 gives information on how a solution changes with uncertainty in the data This is a rather limited analysis that can be misleading in some circumstances see Kall and Wallace 1994 A totally riskaverse approach might be to seek a maximin solution as discussed in Section 32 in order to make the worst possible result however unlikely as little bad as possible Often however this extreme approach is excessively conservative A very good discussion of robust optimization is given by Greenberg and Morrison 2008 In this section we confine ourselves to situations where the uncertainty can be quantified One way of modelling some such situations is by chance constraints as discussed in Section 33 We will confine ourselves here to modelling situa tions where some decisions stage 1 have to be made before other information becomes available The result of the stage 1 decisions and the subsequent avail ability of extra information allow stage 2 decisions to be made Unfortunately the relative merits of the stage 1 decisions depend on this uncertain information and the resultant stage 2 decisions For example it may be necessary to decide production stage 1 before the demand uncertain is known Then as a result of the production decisions and the subsequent actual demand stage 2 decisions must be made regarding selling any excess production at a lower price or extra production to make up a shortfall at a higher cost If it is realistic to consider the uncertain information as taking one of a small finite number of possible values with known probabilities a discrete probability distribution then a set of distinct stage 2 variables may be introduced to correspond to each uncertain value The objective becomes one of minimizing expected cost This model then becomes a linear program For example suppose that the production decisions are represented by stage 1 variables x1 x2 xn The excess production or shortfall is represented by stage 2 variables y1 y2 yn z1 z2 zn These stage 2 variables will be replicated m times according to each of the possible demand levels d1j d2j dmj This gives a model of the form Minimize j cj xj r pr j ej yrj j fj zrj subject to j aij xj bi for all production constraints i xj yrj zrj drj for all j and r xj yrj zrj 0 where pr are given probabilities and cj ej and fj are production excess production and shortfall costs respectively It is important to realize how such a model might be used Initial decisions would be made as a result of stage 1 variables Only the values of those stage 2 variables would ultimately be used which corresponded to the subsequent value of the uncertain information Also it is important to realize the extra information that is incorporated in such a model While a conventional model may contain future data in the form of a single point estimate a stochastic model represents such data by a series of estimates weighted by probabilities The result may be a solution that is not as good as the conventional model in terms of optimal objective but that is more likely to be realistic to be realizable and to reduce risk Another way of viewing such models is one of keeping ones options open that is not putting all ones eggs in one basket The coefficients of the xj variables above form the common columns of the LP model while the variables yrj and zrj each appear in only the rth block of the model The combination of this structure with the block angular structure sometimes occurs giving models of the form shown in Figure 45 An example of the use of a stochastic program is YIELD MANAGEMENT selling airline tickets which is given in Sections 1224 1324 and 1424 References to stochastic programming are Dempster 1980 Kall and Wallace 1994 and Greenberg and Morrison 2008 Stochastic programmes can become very large because of the multiplicity of scenarios resulting from successive periods that is from each scenario in the second stage there will be a number of scenarios into the third stage etc The resultant tree of scenarios into the STRUCTURED LINEAR PROGRAMMING MODELS 55 Common columns Common rows B1 B2 b1 b0 b2 bn Bn Figure 45 future rapidly becomes very large The largest LP ever built and solved to the authors belief is a stochastic program due to Gondzio and Grothey 2006 It has 353 million constraints and 1010 million variables 43 Decomposing a large model Common sense suggests that the optimal solution to a structured model should bear some relation to the optimal solutions to the submodels from which it is constructed This is usually the case and there may therefore be virtue in devising computational procedures to exploit this particularly in view of the large size of many structured models Ways of solving a structured model by way of solutions to the subproblems form the subject of decomposition It is sometimes mistakenly thought that decomposition is the actual act of splitting a model up into submodels Although computational procedures have been devised for doing this decomposition concerns the solution process on a structure that is usually already known to the modeler The importance of decomposition is not only computational but also economic A decomposition procedure applies to a mathematical model of a structured practical problem If the structured model arises by way of a structured organization such as a multiplant model the decomposition procedure mirrors a method of decentralized planning It is for this reason that decomposition is discussed in a book on model building The subject is best considered through this analogy with decentralized planning 56 MODEL BUILDING IN MATHEMATICAL PROGRAMMING We will again consider the small illustrative problem of Section 41 which gave rise to the following multiplant model Maximize Profit 10x1 15x2 10x3 15x4 subject to Raw 4x1 4x2 4x3 4x4 120 Grinding A 4x1 2x2 80 Polishing A 2x1 5x2 60 Grinding B 5x3 3x4 60 Polishing B 5x3 6x4 75 x1 x2 x3 x4 0 It was seen that splitting the 120 kg of raw between A and B in the ratio 75 45 led to a nonoptimal overall solution The optimal overall solution showed that this ratio should be 70 50 Unfortunately we had to solve the total model in order to find this optimal split If we had a method of predetermining this optimal split we would be able to solve the individual models for factories A and B and combine the solutions to give an optimal solution for the total model For a general block angular model as illustrated in Figure 42 we would need to find optimal splits in all the righthand side coefficients in b0 for the common rows Algorithms for the decomposition of a block angular structure based on this principle do exist Such methods are known as decomposition by allocation One such algorithm is the method of Rosen 1964 An alternative approach is decomposition by pricing In a block angular struc ture such as our small example above where common rows represent constraints on raw material availability we could try to seek valuations for the limited raw material These valuations could be used as internal prices to be charged to the submodels If accurate valuations could be obtained we might hope to get each submodel optimizing to the overall benefit of the total model One such approach to this is the DantzigWolfe decomposition algorithm A full description of the algorithm is given in Dantzig 1963 We provide a less rigorous description here paying attention to the economic analogy with decentralized planning To illustrate the method we again use the small twofactory example If it were not for the raw material being in limited supply we would have the following submodels for A and B Maximize Profit A 10x1 15x2 subject to Grinding A 4x1 2x2 80 Polishing A 2x1 5x2 60 x1 x2 0 STRUCTURED LINEAR PROGRAMMING MODELS 57 Maximize Profit B 10x3 15x4 subject to Grinding B 5x3 3x4 60 Polishing B 5x3 6x4 75 x3 x4 0 These submodels should not be confused with the submodels for the same problem in Section 41 The raw availability constraints were included in both submodels with a suitable allocation of raw material between them Here we are not including such constraints Instead an attempt is made to find a suitable internal price for raw and to incorporate this into the submodels Suppose that raw were to be internally priced at p per kilogram The objective functions for the above submodels would then become Profit A 104p x1 15 4p x2 45 and Profit B 104p x3 15 4p x4 46 In effect we have taken multiples of p times the raw material availability con straints and subtracted them from the objectives If p is set too low we might find that the combined solutions to the submodels use more raw than is available in which case p should be increased For example if p is taken as zero there is no internal charge for raw we obtain the following optimal solutions Factory As Optimal Solution with Raw Valued at 0 Profit is 250 obtained from making 175 standard and 5 of deluxe Factory Bs Optimal Solution with Raw Valued at 0 Profit is 1875 obtained from making 125 of deluxe These solutions are clearly unacceptable to the company as a whole as they demand 140 kg of raw which is more than the 120 kg available We therefore seek some way of estimating a more realistic value for the internal price p Whatever the value of p A and B will have optimal solutions that are vertex solutions of the submodels presented above As these models only involve two variables each they can be represented graphically This is shown in Figures 46 and 47 With the value of zero for p we can easily verify the optimal solutions above for A and B as being 175 5 in Figure 46 and 0 125 in Figure 47 Any feasible solution to the total problem must clearly be feasible with respect to both subproblems as well as additionally satisfying raw material availability limitation The values of x1 and x2 in any feasible solution to the total problem 58 MODEL BUILDING IN MATHEMATICAL PROGRAMMING must therefore be a convex linear combination of the vertices of the feasible region shown in Figure 46 that is x1x2 λ11 0 0 λ12 20 0 λ13 175 5 λ14 0 12 47 λ11 λ12 λ13 and λ14 are weights attached to the vertices They must be nonnegative and satisfy the convexity condition λ11 λ12 λ13 λ14 1 48 STRUCTURED LINEAR PROGRAMMING MODELS 59 The vector Equation 47 is a way of relating x1 and x2 to a new set of variables λij by the following two equations x1 0λ11 20λ12 175λ13 0λ14 49 x2 0λ11 0λ12 5λ13 12λ14 410 A similar argument can be applied to the second subproblem represented in Figure 47 This allows x3 and x4 to be related to yet more variables λ21 λ22 λ23 and λ24 by the following equations x3 0λ21 12λ22 9λ23 0λ24 411 x4 0λ21 0λ22 5λ23 125λ24 412 λ21 λ22 λ23 λ24 1 413 λ2j are weights for vertices in the second subproblem while λ2j are weights in the first subproblem It is worth pointing out that what we are really doing is giving modal formu lations as described in Section 34 for each subproblem A slight complication arises if the feasible regions of some of the submod els are open for example we have the situation shown in Figure 48 This complication is easily dealt with and fully explained in Dantzig 1963 We can use Equations 49412 to substitute for x1 x2 x3 and x4 in the objective and single common constraint raw of our total model The grinding and polishing constraints of the two subproblems will be satisfied so long as the λij are nonnegative and satisfy the convexity constraints Equations 48 x1 x2 Feasible region Figure 48 60 MODEL BUILDING IN MATHEMATICAL PROGRAMMING and 413 In this way our multiplant model can be reexpressed as Maximize Profit 200λ12 250λ13 180λ14 120λ22 165λ23 1875λ24 subject to Raw 80λ12 90λ13 48λ14 48λ22 56λ23 50λ24 120 conv 1 λ11 λ12 λ13 λ14 1 conv 2 λ21 λ22 λ23 λ24 1 The above model is known as the master model It can be interpreted as a model to find the optimum mix of vertex solutions of each of the submodels For any internal price p which is charged to A and B they will each produce vertex solutions Such vertex solutions give rise to what are known as proposals as they represent proposed solutions from the submodels given the provisional internal price p for raw The proposals are the columns of coefficients in the master model corresponding to a particular vertex of a submodel For example the proposal from the third vertex of the first subproblem is the column 250 90 1 0 This proposal is given a weight of λ13 in the master model The role of the master model is to choose the best combination of all the proposals that have been obtained The master model would generally have far fewer constraints than the original model There will be the same number of common rows as in the total model Each submodel will however have been condensed down into a single convexity constraint such as conv 1 in the example above Unfortunately the savingin constraints will generally be more than offset by a vast expansion in the number of variables We will have a λij variable for each vertex of each submodel This could be an astronomical number in a realistic problem In practice the great majority of proposals corresponding to these variables will be zero in an optimal solution For a master model with a comparatively small number of constraints but a very large number of variables the great majority of variables will never enter a solution We therefore resort to a practice that is used quite widely in mathematical programming Columns are generated in the course of optimization A column proposal is added to the master model only when it seems worthwhile This is an example of column generation which is discussed in Section 96 We therefore deal only with a subset of the possible proposals Such a truncated model is known as a restricted master model Proposals are STRUCTURED LINEAR PROGRAMMING MODELS 61 added to and sometimes deleted from the restricted master model in the course of optimization In general only a very small number of the potential proposals will ever be generated and added to the restricted master problem In order to describe how we proceed we will consider our small multiplant model Instead of using the master model which we were fortunate enough to be able to obtain from geometrical considerations in so small an example we will work with a restricted master problem To start with we will take only the proposals corresponding to λ11 and λ13 from submodel A and λ21 and λ24 from submodel B This choice is largely arbitrary How it is made is not important to our discussion In practice a number of good proposals from each submodel would be used to make up the first version of the restricted master model in order to have a reasonably realistic model with some substance Our first restricted master model is therefore Maximize Profit 250λ13 1875λ24 subject to Raw 90λ13 50λ24 120 conv 1 λ11 λ13 1 conv 2 λ21 λ24 1 When this model is optimized we can obtain a valuation for the raw mate rial This valuation is the marginal value of the raw constraint in the optimal solution Such marginal values for constraints are sometimes known as shadow prices They are discussed much more fully together with their economic inter pretation in Section 62 In fact the marginal value associated with a constraint such as raw is the rate at which the optimal profit would increase for small marginal increases in the righthand side coefficient It is sufficient for our purpose here simply to remark that such valuations for constraints are possible With any package programs these valuations shadow prices are presented with the optimal solution If our first restricted master model is optimized the shadow price on the raw constraint turns out to be 278 This can be taken as value p and used as an internal price which factories A and B are charged for each kilogram of raw that they wish to use When A is charged this internal price it will reform its objective function to take account of the new charge The new objective function comes from the expression 45 taking p as 278 This gives Profit A 112x1 318x2 414 If this objective is used with the constraints of the submodel for A we obtain the solution x1 0 x2 12 62 MODEL BUILDING IN MATHEMATICAL PROGRAMMING This clearly corresponds to the vertex 0 12 in Figure 46 The proposal corre sponding to this is the column for λ14 This is easily calculated to be 180 48 1 0 A new variable λ14 but with a different name is therefore added to the restricted master problem with this column of coefficients This new proposal represents factory As new provisional production plan given the new internal charge for raw Our attention is now turned to factory B When they are charged 278 per kilogram for raw the expression 46 gives their objective function as Profit B 112x3 318x4 415 When this objective is optimized with the constraints of the submodel for factory B we obtain the solution x3 0 x4 125 We have the vertex 0 125 in Figure 47 The proposal corresponding to this is the column for λ24 This proposal has already been included in our first restricted master model We therefore conclude that even if factory B is charged at the suggested rate of 278 per kilogram for raw they would not suggest a new proposal provisional production plan Having added only the proposal corresponding to λ14 to our restricted master model it now becomes Maximize Profit 250λ13 180λ14 1875λ24 subject to Raw 90λ13 48λ14 50λ24 120 conv 1 λ11 λ13 λ14 1 conv 2 λ21 λ24 1 Optimizing this model the shadow price on raw turns out to be 167 We see that our previous valuation of 278 appears to have been an over estimate The cycle is now repeated and each factory is internally charged 167 per kilogram for raw This gives the following new objectives for A and B Profit A 332x1 832x2 416 Profit B 332x3 832x4 417 When the objective 416 is used with the constraints of the submodel for factory A the optimal solution obtained is x1 175 x2 5 STRUCTURED LINEAR PROGRAMMING MODELS 63 This is the vertex 175 5 in Figure 46 and gives the proposal corresponding to λ13 As this proposal has already been incorporated in the restricted master problem factory A has no new proposal to offer as a result of the revised internal charge of 167 per kilogram for raw Factory B optimizes objective 417 subject to the constraints of its submodel This results in the solution x3 0 x4 125 This is the vertex 0 125 of Figure 47 which results in the proposal corresponding to λ24 As this proposal is already present in the restricted master model factory B also has no further useful proposal to add as a result of the revised charge for raw We therefore conclude that factories A and B have submitted all the useful proposals that they can The optimal solution to the latest version of the restricted master model gives the proportions in which these proposals should be used For our example the optimal solution to this restricted master model is λ13 052 λ14 048 λ24 1 This enables us to calculate the optimal values for x1 x2 x3 and x4 by considering the vertex solutions of the submodels corresponding to λ13 and λ14 and λ24 We obtain x1x2 052 175 5 048 0 12 x3x4 1 0 125 This gives us the optimal solution to the total model x1 917 x2 833 x4 125 giving an objective value of 40415 Notice however that we have obtained the optimal solution to the total model without solving it directly Instead we have dealt with what would generally be much smaller models The two types of model we have used are the submodels and the restricted master model We further discuss the significance of these models below 431 The submodels These contain the details relevant to the individual subproblems For a multiplant model such as that used in our example the coefficients in the constraints concern only the particular factory that is grinding and polishing times and capacities in each factory 64 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 432 The restricted master model This is an overall model for the whole organization but unlike the total model it contains none of the technological detail relating to the individual subproblems Such detail is left to the submodels Instead the constraints for each subproblem are accounted for by a simple convexity constraint In our example we had the constraints for factories A and B reducing to convexity constraints conv 1 and conv 2 respectively On the other hand the restricted master model does contain the common rows in full as its main purpose is to determine suitable valuations for the resources represented by these common rows By means of interactions between the submodels and the restricted master model it is possible eventually to obtain the optimal solution to the usually much larger total model without ever building and solving it directly The process that we describe here is diagrammatically represented in Figure 49 Restricted master model RMM Internal prices for common resources shadow prices on common rows of RMM Proposals new columns for RMM Sub model 1 Sub model 2 Sub model 3 Sub model 4 Figure 49 There is considerable attraction in such a scheme as it is never necessary to build solve and maintain the often huge total model which would result from a large structured organization In a multiplant organization the individual plants would probably be geographically separated This would make the avoidance of including all their technological details in one central model desirable Each plant might well build and maintain their own model inside their own plant and solve it on their own computer The head office could maintain a restricted master model on another computer which would be linked to the computers in the individual plants Each model could then be run independently but could supply the vital information of proposals and internal prices to the other models It would then be possible to use a system automatically to obtain an overall optimal solution for the organization Decomposition has interest for economists as it clearly represents a system of decentralized planning The existence of decomposition algorithms such as the STRUCTURED LINEAR PROGRAMMING MODELS 65 DantzigWolfe algorithm demonstrates that it is possible to devise a method of decentralized planning which achieves an optimal solution for an organization considered as a whole This is done by allowing the suborganizations to decide their own optimal policies given limited control from the centre In the case of the DantzigWolfe algorithm this control takes the form of internal prices For other methods it may take the form of allocations An informal version of such procedures takes place in many organizations A discussion of a large number of decomposition algorithms and their relation to decentralized planning in real life is given by Atkins 1974 Decomposition algorithms are also of considerable computational interest as they offer a possibility of avoiding the large amount of time needed to solve large models should those models be structured Unfortunately decomposition has only met with limited computational success The most widely used algorithm is the DantzigWolfe algorithm that applies to block angular structured models There are many circumstances however in which it is more efficient to solve the total model rather than to use decomposition The main advice to someone building a structured model should not be to attempt decomposition on other than an experimental basis As experience and knowledge grow decomposition may well become a more reliable tool but at present the computational experience with such methods has been disappointing If a model is to be used very frequently experimenting with decomposition might be considered worthwhile The larger the model and the smaller the proportion of common rows for block angular structures the more valuable this is likely to be Sometimes other aspects of the structure can be exploited to advantage An example of this is described by Williams and Redwood 1974 Computational experiences using DantzigWolfe decomposition are given by Beale et al 1965 and Ho and Loute 1981 A very full description of the computational side of decomposition is given by Lasdon 1970 Finally decomposition may commend itself for the purely organizational considerations resulting from the desirability of decentralized planning The use of communication networks for linking computers together makes it possible to implement a decomposition algorithm splitting the models between different computers eg the master model in the head office and the submodels at individual plants This idea has not to the authors knowledge yet been tried Visual content Two graphs Figure 46 shows a shaded polygon defined by vertices 00 012 1755 and 200 on a graph with x1 on the horizontal axis and x2 on the vertical Figure 47 shows a shaded polygon defined by vertices 00 0125 95 and 120 on a graph with x3 on the horizontal axis and x4 on the vertical 5 Applications and special types of mathematical programming model 51 Typical applications The purpose of this section is to create an awareness of the areas where linear pro gramming LP is applicable To categorize totally those industries and problems where LP can or cannot be used would be impossible Some problems clearly lend themselves to an LP model For other problems the use of an LP model may not provide a totally satisfactory solution but may be considered acceptable in the absence of other approaches The decision of when to use and when not to use LP is often a subjective one depending on an individuals experience This section can do no more than try to give a feel for those areas in which LP can be applied In order to do this a list of industries and areas in which the technique has been applied is given This list is by no means exhaustive but is intended to include most of the major users A short discussion is given of the types of LP models that are of use in each area References are given to some of the relevant published case studies In view of the very wide use that has been made of LP it would be almost impossible to seek out every reference to published case studies Nor would it be helpful to submerge the reader in a mass of often superfluous literature The intention is to give sufficient references and allowing the reader to follow up published case studies From the references given here it should be possible to find other references if necessary In many cases practical applications are illustrated by problems in Part II Although the intention is mainly to consider LP applications in this chapter the resultant models can very often naturally be extended by integer programming Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 68 MODEL BUILDING IN MATHEMATICAL PROGRAMMING or nonlinear programming models In this way more complicated or realistic situations can often be modelled These topics and further applications are con sidered more fully in Chapters 7 8 9 and 10 The subject of LP does not have clearly defined boundaries Other subjects impinge on and merge with LP Two types of model that have to some extent been studied independently of LP are considered further in this chapter Firstly economic models which are sometimes referred to as inputoutput or Leontief models are considered in Section 52 Such models can often be regarded as a special type of LP model Secondly network models which arise frequently in operational research are considered in Section 53 Such models are again often usefully considered as special types of LP model Another area that also has connections with LP is not considered in this book in view of its limited practical applicability to date This is the theory of games Game theory models can sometimes be converted into LP models and vice versa The following list of applications should indicate the surprisingly wide applicability of LP and in consequence its economic importance Many more references exist than are given The purpose of the references is to give a window into the literature 511 The petroleum industry This is by far the biggest user of LP Very large models involving thousands and occasionally tens of thousands of constraints have been built These models are used to help make a number of decisions starting with where and how to buy crude oil how to ship it and which products to produce out of it Such corporate models contain elements of distribution resource allocation blending and possibly marketing A typical example of the sort of model that arises in the industry is the REFINERY OPTIMIZATION problem of Part II Descriptions of the use of LP in the petroleum industry are given by Manne 1956 Catchpole 1962 and McColl 1969 512 The chemical industry The applications here are rather similar to those in the petroleum industry although the models are rarely as large Applications usually involve blending or resource allocation An application is described by Royce 1970 513 Manufacturing industry LP is frequently used here for resource allocation The product mix example described in Section 12 is an example of this type of application Resources to be allocated are usually processing capacity raw materials and manpower A multi period problem of this type considered in relation to the engineering industry is the FACTORY PLANNING problem of Part II Other common applications of LP APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 69 in manufacturing are blending and blast furnace burdening the steel industry Three references to the application of LP here are Lawrence and Flowerdew 1963 Fabian 1967 and Sutton and Coates 1981 514 Transport and distribution Problems of distribution can often be formulated as LP models Two classic examples are the transportation and transhipment problems which are consid ered in Section 53 as they involve networks A simple DISTRIBUTION problem of this type is presented in Part II An extension of this problem DISTRIBUTION 2 involves depot location as well and requires integer programming Scheduling problems eg lorries aircraft tankers trains and buses can often be tackled through integer programming One such problem is the MILK COLLECTION problem given in Part II Problems of assignment trains to mines and power stations arising in transport have also been tackled through mathematical pro gramming A comprehensive description of the use of mathematical programming in shipping is given in Christiansen et al 2007 Applications of mathematical programming in distribution in general are described by Eilon et al 1971 and Markland 1975 515 Finance A very early application of mathematical programming was in portfolio selection This was due to Markowitz 1959 Given a sum of money to invest the problem was how to spend it among a portfolio of shares and stocks The objective was to maintain a certain expected rate of return from the investment but to minimize the variance of that return The model that results is a quadratic programming model Agarwala and Goodson 1970 suggest how LP can be used by a government to design an optimum tax package to achieve some required aim in particular an improvement in the balance of payments LP is increasingly being used in accountancy The economic information that can be derived from the solution to an LP model can provide accountants with very useful costing information This sort of information is described in detail in Section 62 A description of how LP can be used in accountancy is given by Salkin and Kornbluth 1973 Spath et al 1975 describe how an LP model is applied by a mail order firm in order to minimize the total interest cost on all credits An interesting extension of a finance model to allow the user to set goals interactively rather than objectives and operate within the degrees of freedom permitted by the constraints is described by Jack 1985 A very important potential area of application is yield revenue management which is concerned with setting prices for goods at different times in order to maximize revenue It is particularly applicable to the hotel catering airline and train industries An example of its use is the YIELD MANAGEMENT problem in Part II 70 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 516 Agriculture LP has been used in agriculture for farm management Such models can be used to decide what to grow where how to rotate crops how to expand production and where to invest An example of such a problem is the FARM PLANNING problem of Part II Swart et al 1975 apply a multiperiod LP model to planning the expansion of a large dairy farm Other general references are Balm 1980 and Fokkens and Puylaert 1981 Blending models are often applicable to agriculture problems It is often desired to blend together livestock feeds or fertilizer at minimum cost Glen 1980 describes a model for beef cattle ration formulation Distribution problems often arise in this area The distribution of farm prod ucts in particular milk can be examined by the network type of LP model described in Section 53 Models for planning the growth and harvesting of a number of agricultural products are described by Glen 1980 1988 1995 1996 1997 Quadratic programming has been used for determining optimal prices for the sale of milk in the Netherlands This is described by Louwes et al 1963 The AGRICULTURAL PRICING problem of Part II is based on this study A mixed integer programming model for irrigation in a developing country is described by Rose 1973 517 Health The obvious application of mathematical programming in this area is in problems of resource allocation How are scarce resources for example doctors nurses hospitals etc to be used to best effect In such problems there will obviously be considerable doubt concerning the validity of the data for example how much of a nurses time does a particular type of treatment really need In spite of doubts concerning much of the data in such problems it has been possible to use math ematical programming models to suggest plausible policy options McDonald et al 1974 describe a nonlinear programming model for allocating resources in the UK health service Revelle et al 1969 describe how a nonlinear pro gramming model can be used for controlling tuberculosis in an underdeveloped country Warner and Prawda 1972 describe a mathematical programming model for scheduling nurses Redpath and Wright 1981 describe how LP is used to decide intensity and direction of beams for irradiating cancerous tumours 518 Mining A number of interesting applications of mathematical programming occur in mining The straightforward applications are simply ones of resource allocation that is how should manpower and machinery be deployed to best effect APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 71 Blending problems also occur when it is necessary to mix together ores to achieve some required quality Two examples of mining problems are given in Part II The MINING prob lem concerns what combination of mines a company should operate in successive years The OPENCAST MINING problem is to decide what the boundaries of an opencast mine should be References to the application of mathematical program ming in mining are Young et al 1963 Meyer 1969 and Boland et al 2009 519 Manpower planning The possible movement of people between different types of job and its control by recruitment promotion retraining etc can be examined by linear programming An example of such a problem MANPOWER PLANNING is given in Part II Applications of mathematical programming to manpower planning are described by Price and Piskor 1972 Davies 1973 Vajda 1975 Charnes et al 1975 and Lilien and Rao 1975 5110 Food The food industry makes considerable use of LP Blending sausages meat pies margarines ice cream etc is an obvious application often giving rise to very small and easily solved models Problems of distribution also arise in this industry giving rise to the network type models described in Section 53 As in other manufacturing industries problems of resource allocation arise which can be tackled through LP The FOOD MANUFACTURE problem of Part II is an example of a multi period blending problem in the food industry A more complicated version of this problem is described by Williams and Redwood 1974 Jones and Rope 1964 describe another LP model in the food industry 5111 Energy The electricity and gas supply industries both use mathematical programming to deal with problems of resource allocation The TARIFF RATES POWER GEN ERATION problem of Part II involves scheduling electric generators to meet varying loads at different times of day This problem is similar to that described by Garver 1963 This problem is extended to the HYDRO POWER model which illustrates another important application Archibald et al 1999 describe how a similar problem can be tackled by stochastic dynamic programming LP can also be applied to distribution problems involving the design and use of supply networks Applications of mathematical programming in these areas are also described by Babayer 1975 Fanshel and Lynes 1964 Muckstadt and Koenig 1977 and Khodaverdian et al 1986 72 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 5112 Pulp and paper Problems of resource allocation in the manufacture of paper give rise to LP models Such models frequently involve an element of blending In addition the possibility of recycling waste paper has also been examined by LP as described by Glassey and Gupta 1974 A totally different type of problem arising in the paper industry and the glass industry is the trimloss problem This is the problem of arranging orders for rolls of paper of different widths so as to minimize the wastage This problem can be tackled by linear or sometimes integer programming This problem is described by Eisemann 1957 It has also been considered by Gilmore and Gomory 1961 1963 It is used in Section 96 to demonstrate column generation 5113 Advertising The problem of spreading ones advertising budget over possible advertising outlets television commercials newspaper advertisements etc has been approached through mathematical programming These problems are known as media scheduling problems Authors differ over the usefulness of mathematical programming in tackling this type of problem Selected references are Engel and Warshaw 1964 Bass and Lonsdale 1966 and Charnes et al 1968 The lastmentioned reference gives a very full list of references itself 5114 Defence Problems of resource allocation give rise to military applications of LP Such an application is described by Beard and McIndoe 1970 The siting of missile sites is described by Miercort and Soland 1971 5115 The supply chain It is apparent from the discussion of applications above that many aspects of an organization are amenable to being modelled using mathematical programming This is particularly true of the manufacturing industry but is also true of other industries such as finance and telecommunications When optimizing the oper ations of only one aspect or division of an organization one is in danger of losing sight of the bigger picture Furthermore there can sometimes be a danger of damaging part of an organizations operations by optimizing the operations of another part For example concentrating entirely on maximizing production could result in policies that are infeasible from the point of view of distribu tion or marketing What are needed are integrated models that link together such different aspects There is of course a danger that such models could become unwieldy and unusable if approached in the wrong way the aborting of APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 73 the projected huge IT system for the British National Health Service is a stark reminder Such major considerations form the subject of information systems and are beyond the scope of this book It is usually the case that such integrated models evolve from existing models of smaller specialist functions Some of the discussion in Chapter 4 is relevant to this Advances in information technology for example communications software database technology and userfriendly interfaces have made it easier to create and use such models Although applicable in other areas we illustrate our discussion of supply chain models by using examples from the manufacturing sector The following functions are amenable to optimization Purchasing for example deciding the best suppliers and quantities of raw material to buy at the right time Inward Distribution for example deciding what vehicles to hire of what capacities and how to route them Inventory Planning for example deciding how much to store of both inward materials and intermediate or final products Financial Planning for example deciding on whether to finance some activ ities by loans and what debt to incur Production for example how much to make of each product using which productionmanpower capacities and whether to take on extra capacity Marketing for example what marketing outlets to use and where to deploy marketing effort In addition other nonmathematical programming models may be relevant for example forecasting and accounting models A supply chain model integrates models for all or some of these functions taking outputs from some as inputs to others and performs a global optimization for example to maximize total profit 5116 Other applications There are numerous other applications of mathematical programming A few of the less usual ones are given here as they might otherwise go unnoticed Heroux and Wallace 1973 describe a multiperiod LP model for land devel opment Souder 1973 discusses the effectiveness of a number of mathematical pro gramming models for research and development Feuerman and Weiss 1973 show how a knapsack integer programming model can be used to help design multiple choicetype examinations Kalvaitis and Posgay 1974 apply integer programming to the problem of selecting the most promising kind of mailing list Wardle 1965 applies LP to forestry management 74 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Problems of pollution control have been tackled through mathematical pro gramming Applications are described by Loucks et al 1968 Kraft and Hill 1973 describe a 01 integer programming model for selecting journals for a university library Junger et al 1989 and Ferreira et al 1993 describe integer programming models arising in computer design Bollapragada et al 2001 apply integer programming to the civil engineering problem of truss design that is constructing an optimal structure of trusses to support a given weight modelling the logical dependencies of trusses Trick has applied integer programming extensively to sports scheduling see eg Nemhauser and Trick 1998 Chang and Sahinidis 2011 among others have applied integer programming to DNA sequencing This has similarities to the easier archaeological seriation problem which has been considered by Laporte 1976 and Wilkinson 1971 who formulate this problem as a version of the Travelling Salesman Problem see Chapter 9 The use of LP for performance evaluation in a number of organizations par ticularly in the public sector through the use of a type of model known as Data Envelopment Analysis has become important This is described through the EFFI CIENCY ANALYSIS problem in Part II References are Charnes et al 1978 Farrell 1957 Land 1991 and Thanassoulis et al 1987 A much fuller list of papers on mathematical programming applications has been compiled by Riley and Gass 1958 The special editions of the journal Mathematical Programming Studies Nos 9 1975 and 20 1982 are devoted to applications 52 Economic models A widely used type of national economic model is the inputoutput model representing the interrelationships between the different sectors of a countrys economy Such models are often referred to as Leontief models after their orig inator who built such a model of the American economy This is described by Leontief 1951 Inputoutput models are often regarded usefully as a special type of LP model 521 The static model The output from a particular industry or a sector of an economy is often used for two purposes i for immediate consumption for example coal to be sold to domestic consumers ii as an input to other industries or sectors of the econ omy for example coal to provide power for the steel industry As the outputs from many industries can be split in this way between exogenous consump tion and as endogenous inputs into these same industries a complex set of interrelationships will exist The inputoutput type of model provides about the simplest way of representing these relationships A number of strong and usually APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 75 oversimplified assumptions are made regarding the interindustry relationships The two major assumptions are as follows 1 The output from each industry is directly proportional to its inputs for example doubling all the inputs to an industry will double its outputs 2 The inputs to a particular industry are all in fixed proportions for example it is not possible to decrease one input and compensate for this by increasing another input These fixed proportions are determined by the technology of the production process In other words there is nonsubstitutability of inputs In order to demonstrate such a model we will consider a very simple example Example 51 A Threeindustry Economy We suppose that we have an economy made up of only three types of industry coal steel and transport Part of the outputs from these industries are needed as inputs to others for example coal is needed to fire the blast furnaces that produce steel steel is needed in the machinery for extracting coal etc The necessary inputs to produce 1 unit of output for each industry are given in the inputoutput matrix in Table 51 Table 51 An inputoutput matrix Inputs Outputs Coal Steel Transport Coal 01 05 04 Steel 01 01 02 Transport 02 01 02 Labour 06 03 02 It is usual in such tables to measure all units of production in monetary terms We then see that for example to produce 1 in worth of coal requires 01 of coal to provide the necessary power 01 of steel the steel used up in the wear and tear on the machinery and 02 of transport for moving the coal from the mine In addition 06 of labour is required Similarly the other columns of Table 51 give the inputs required s for each pound of steel and each pound of transport lorries cars trains etc Notice that the value of each unit of output is exactly matched by the sum of the values of its inputs This economy is assumed to be open in the sense that some of the output from the above three industries is used for exogenous consumption We will assume that these external requirements are in millions 76 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Coal 20 Steel 5 Transport 25 Such a set of exogenous demands is known as a bill of goods A number of questions naturally arise concerning our economy which a mathematical model might be used to answer 1 How much should each industry produce in total in order to satisfy a given bill of goods 2 How much labour would this require 3 What should the price of each product be If variables xc xs and xt are used to represent the total quantities of coal steel and transport produced in a year we get the following relationships xc 20 01xc 05xs 04xt 51 xs 5 01xc 01xs 02xt 52 xt 25 02xc 01xs 02xt 53 For example Equation 51 tells us that we must produce enough coal to satisfy external demand 20m input to the coal industry 01xc input to the steel industry 05xs and input to the transport industry 04xt Equations 51 52 and 53 can conveniently be rewritten as 09xc 05xs 04xt 20 54 01xc 09xs 02xt 5 55 02xc 01xs 08xt 25 56 Such a set of equations in the same number of unknowns can generally be solved uniquely In this case we would obtain the solution xc 561 xs 224 xt 481 The total labour requirement can then easily be obtained as 06 561 03 224 02 481 50 Clearly Equations 5456 could be regarded as the constraints of an LP model An objective function could be constructed and we could maximize it or minimize it subject to the constraints As the model stands however there would be little point in doing this as there is generally only one feasible solution The objective function would therefore have no influence on the solution APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 77 Once however we extend this very simple type of inputoutput model we frequently obtain a genuine LP model The model described above is unrealistic in a number of respects Equations 5456 give no real limitation to the productive capacity of the economy It can fairly easily be shown that so long as a particular positive bill of goods can be produced the economy is a productive one then these relationships guarantee that any bill of goods however large can be produced This is clearly unrealistic Firstly we would expect some limitation on productive capacity pre venting more than a certain amount of output from each industry in a given period of time Secondly we would expect the output from an industry only to be effective as the input to another industry after a certain time has elapsed This second consideration leads to dynamic inputoutput models which we consider below Before doing this however we will consider the problem of modelling limited productive capacity in the case of a static model In our small example we assumed that once we had decided how much each industry should produce in order to meet a specified bill of goods we could provide the labour required If labour were in short supply it might limit our productive capacity There would then be interest in seeing what bills of goods are or are not producible in a particular period of time Returning to our example if we were to limit labour to 40 m per year we could not produce our previous bill of goods But what bill of goods could we produce Answers to this question can be explored through LP Variables will now represent our bill of goods Coal yc Steel ys Transport yt Equations 5456 will give the constraints 09xc 05xs 04xt yc 0 57 01xc 09xs 02xt ys 0 58 02xc 01xs 08xt yt 0 59 The labour limitation gives the constraint 06xc 03xs 02xt 40 510 Achievable bills of goods will be represented by the values of yc yr and yt in feasible solutions to Equations 57510 Specific solutions can be found by introducing an objective function For example we might wish to maximize the total output xc xs xt 511 78 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Alternatively we might weight some outputs more heavily than others by giving xc xs and xt different objective coefficients We might wish simply to maximize production in one particular sector of the economy such as steel and simply maximize xs This is clearly a situation of the type referred to in Section 32 in which it is of interest to experiment with a number of different objectives rather than simply concentrate on one We have considered only labour as a limiting factor in productive capacity In practice there could well be other resource limitations such as processing capac ity raw material etc Such limitations could of course easily be incorporated in a model by extra constraints Limited resources of this sort are sometimes known as primary goods Primary goods only provide inputs to the economy They are not produced as outputs as well A major advantage of treating such models as LP models is that a lot of subsidiary economic information is also obtained from solving such a model Such information is described very fully in Section 62 In particular valuations are obtained for the constraints of a model These valuations are known as shadow prices For the type of model considered here we would obtain meaningful valuations for the primary goods In this way a pricing system could be introduced into our model This would give suitable prices for the outputs from all the industries Although any number of primary goods can be considered in an LP formu lation of an inputoutput model it is quite common only to consider labour In practice particularly in the relatively simple economies of developing countries it may well not be unreasonable to consider labour as the overall limitation If this can be done there is another less obvious advantage to be gained in the appli cability of such a model It has already been pointed out that an inputoutput model assumes nonsubstitutability of the inputs that is it is not possible to vary the relative proportions in which all the inputs are used to produce the output of a particular industry In practice this might well be a far from realistic assump tion For example we might well be able to produce each unit of coal by using more power more coal and less machinery less steel To model this possibility would require a variation in the coefficients of the inputoutput matrix It has been shown that if there is only one primary good usually labour then it will only be worthwhile to concentrate on one production process for each indus try This is the result of the Samuelson substitution theorem which we will not prove Such theoretical results and a fuller description of inputoutput models are given in Dorfman et al 1958 Another good reference is Shapiro 1979 The importance of this result is that we need not worry about the apparent non substitutability limitation so long as we only have one primary good There will be one and only one best set of inputs production processes for each industry This best production process will remain the best no matter what bill of goods we are producing There is of course the problem of finding for each industry the production process which should be used Once however this has been done no matter what the bill of goods we need only incorporate this one production process column of the inputoutput matrix into all future models In fact the finding of the best production process for each industry can be done by LP APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 79 Table 52 An inputoutput matrix with alternative production processes Inputs Outputs Coal Steel Transport Coal 01 02 05 06 04 06 Steel 01 01 01 02 02 Transport 02 01 01 02 005 Labour 06 07 03 03 02 015 To illustrate how this may be done as well as illuminating the import of the Samuelson substitution theorem we will extend our small example Table 52 gives two possible sets of inputs production processes to produce 1 unit of the three industries In practice there might be many more possibly an infinite number than two processes for each industry On the face of it we might think it advantageous to use some combination of the two processes for producing coal The first process is more economical on coal but uses some steel as well which the second process does not Similarly some mixture of the two processes for producing steel and the two processes for producing transport might seem appropriate Moreover which processes are used might seem likely to depend upon the particular bill of goods We repeat however that our intuitive idea would be false There will be exactly one best process for each industry and this will be used whatever bill of goods we have Instead of the variables xc xs and xt in our original model we can introduce variables xc1 xc2 xs1 xs2 xt1 and xt2 to represent the total quantities of coal steel and transport produced by each process Using the same bill of goods as before instead of constraints 5153 we obtain xc1 xc2 20 01xc1 02xc2 05xs1 06xs2 04xt1 06xt2 512 xs1 xs2 5 01xc1 00xc2 01xs1 01xs2 02xt1 02xt2 513 xt1 xt2 25 02xc1 01xc2 01xs1 00xs2 02xt1 005xt2 514 These equations can be written as 09xc1 08xc2 05xs1 06xs2 04xt1 06xt2 20 515 01xc1 00xc2 09xs1 09xs2 02xt1 02xt2 5 516 02xc1 01xc2 01xs1 00xs2 08xt1 095xt2 25 517 We are considering labour as the only primary good and will limit ourselves to 60 m This gives the constraint 06xc1 07xc2 03xs1 03xs2 02xt1 015xt2 60 518 80 MODEL BUILDING IN MATHEMATICAL PROGRAMMING There will generally be more than one solution to a system such as this In order to find the best solution we will define an objective function One possible objective function would of course be to ignore constraint 518 and minimize the expression on the lefthand side representing labour usage Alternatively we might specify another objective function For this example we will do this and simply maximize total output xc1 xc2 xs1 xs2 xt1 xt2 519 Our resultant optimal solution gives xc1 646 xc2 0 xs1 226 xs2 0 xt1 0 xt2 446 Notice that the Samuelson substitution theorem has worked in this case One process in each industry is the best to the total exclusion of all the others More over it could be shown that these processes will be the best no matter what the bill of goods is The optimal solution will be made up of only the variables xc1 xs1 and xt2 no matter what the righthand side coefficients in constraints 515518 are We could therefore confine all our attention to these processes and ignore the others It should be noted however that these best processes are only the best because of the objective function 519 that we have chosen If instead of maximizing output we were to choose another objective it might be preferable to switch to another process in some cases It will never however be worth mixing processes Once we consider more than a single primary good such mixing may well however become desirable 522 The dynamic model We have pointed out that our static model assumed that we could ignore time lags between an output being produced and used as an input to another or the same industry This unrealistic assumption can be avoided by introducing a dynamic model It has already been shown in Section 41 that LP models can often be extended to multiperiod models We can do much the same thing with the static type of inputoutput model In practice some of the output from an economy will immediately be consumed eg cars for private motoring while some will go to increase productive capacity eg factory machinery Such alternative uses for the output will result in different possible growth patterns for the economy that is we can live well now but neglect to invest for the future or we can sacrifice APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 81 present day consumption in the interests of future wealthproducing capacity A simple example of such a problem ECONOMIC PLANNING leading to a dynamic inputoutput model is given in Part II Rather than discuss Dynamic inputoutput models the discussion is taken up with the formulation of this problem in Part III A description of dynamic inputoutput models of this type is given by Wagner 1957 523 Aggregation To sum up the characteristics of a whole industry or sector of an economy in one column of an inputoutput matrix obviously requires a large amount of sim plification of the real situation It is necessary to group together many different industries into one This aggregation is necessary in order to obtain a reasonable size of problem Most inputoutput models are aggregated into less than 1000 industries The problem of aggregation is obviously of paramount concern to the model builder Unfortunately very little theoretical work has been done to indicate mathematically when aggregation is and is not justified Three criteria that common sense would suggest to be good grounds for aggregating particu lar industries are i substitutability ii complementarity and iii similarity of production processes Problems of this sort are discussed more fully by Stone 1960 In view of their sophistication and efficiency commercial mathematical pro gramming packages provide a useful way of solving inputoutput models Even if the model is of the simplest kind described above and only requires the solu tion of a set of simultaneous equations such packages are of use Almost all packages contain an inversion routine which is very useful for inverting large matrices sets of simultaneous equations It should however be pointed out that inputoutput models are often quite dense In this respect they are untypical of general LP models As already mentioned in Section 21 in a model with 1000 constraints one would expect only about 1 of the coefficients to be nonzero For an inputoutput model this figure could well be as high as 50 As a result inputoutput models can take a long time to solve on a computer and run into numerical difficulties It is sometimes worth exploiting the special structure of an inputoutput LP model and using a special purpose algorithm Dantzig 1955 describes how the simplex algorithm can be adapted to this purpose 53 Network models The use of models involving networks is very widespread in operational research Problems involving distribution assignment and planning critical path analysis and PERT frequently give rise to the analysis of networks Many of the resultant problems can be regarded as special types of LP problem It is often more efficient to use special purpose algorithms rather than the revised simplex algorithm Nevertheless it is important for the model builder to be aware when heshe is 82 MODEL BUILDING IN MATHEMATICAL PROGRAMMING dealing with a special kind of LP model In order to solve the model it may be useful for the builder to adapt the simplex algorithm to suit the special structure It may even sometimes be worth ignoring the special structure and using a general purpose package program As such programs are often highly efficient and well designed their speeds outweigh the algorithmic efficiency of less welldesigned but more specialized programs It is not intended that the coverage of this topic be comprehensive The main aim is simply to show the connection between network models and LP models References are given to much fuller treatments of the subject 531 The transportation problem This famous type of problem first described by Hitchcock 1941 is usefully regarded as one of obtaining the minimum cost flow through a special type of network Suppose that a number of suppliers S1 S2 Sm are to provide a number of customers T1 T2 Tn with a commodity The transportation problem is how to meet each customers requirement while not exceeding the capacity of any supplier at minimum cost Costs are known for supplying 1 unit of the commodity from each Si to each Tj In some cases it may not be possible to supply a particular customer Tj from a particular supplier Si It is sometimes useful to regard these costs as infinite in such cases In distribution problems these costs will often be related to the distances between Si and Tj It is assumed that the capacity of each supplier over some period such as a year is known and the requirement of each customer Tj is also known In order to describe the problem further we will consider a small numerical example Example 52 A Transportation Problem Three suppliers S1 S2 S3 are used to provide four customers T1 T2 T3 T4 with their requirements for a particular commodity over a year The yearly capac ities of the suppliers and requirements of the customers are given below in suitable units Suppliers S1 S2 S3 Capacities per year 135 56 93 Customers T1 T2 T3 T4 Requirements per year 62 83 39 91 The unit costs for supplying each customer from each supplier are given in Table 53 in pounds per unit We can easily formulate this problem as a conventional LP model by intro ducing variables xij to represent the quantity of the commodity sent from Si to APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 83 Table 53 Supplier Customer T1 T2 T3 T4 S1 132 a 97 103 S2 85 91 S3 106 89 100 98 aA dash indicates the impossibility of certain suppliers for cer tain depots or customers Tj in a year The resultant model is Minimize 132x11 Mx12 97x13 103x14 85x21 91x22 Mx23 Mx24 106x31 89x32 100x33 98x34 520 subject to x11 x12x13 x14 135 521 x21 x22x23 x24 56 522 x31 x32x33 x34 93 523 x11 x21 x31 62 524 x12 x22 x32 83 525 x13 x23 x33 39 526 x14 x24 x34 91 527 xij 0 all i j This model obviously has a very special structure to which we will refer later Notice that we have included variables for nonallowed routes in the model with objective coefficients M some very large number This has been done simply to preserve the pattern of the model In practice if we were to solve the model as an LP problem of this form we would simply leave these variables out Constraints 521523 are known as availability constraints There is one such constraint for each of the three suppliers These constraints ensure that the total quantity out of a supplier in a year does not exceed his capacity Constraints 524527 are known as requirement constraints These constraints ensure that each customer obtains his requirement In some formulations of the transportation problem constraints 521523 are treated as instead of If the sum total of the availabilities exactly matches the sum total of the requirements then this is acceptable as all capacities must obviously be completely exhausted In a case such as our numerical example however this is not so Total capacity 284 exceeds total demand 275 This can be coped with by introducing a dummy 84 MODEL BUILDING IN MATHEMATICAL PROGRAMMING customer T5 with a requirement for the excess of 9 If the cost of meeting this requirement of T5 from each supplier Si is made zero we have equated total capacity to total demand with no inaccuracy in our modified model The three constraints 521523 could then be made When special algorithms are used to solve the transportation problem the employment of devices such as this is sometimes necessary For a conventional LP formulation of the problem this is not necessary For a general transportation problem with m suppliers S1 S2 Sm and n customers T1 T2 Tn there will be m availability constraints and n requirement constraints giving a total of m n constraints If each supplier can be potentially used for each customer there will be mn variables in the LP model Clearly for practical problems involving large numbers of suppliers and customers the LP model could be very large This is one motive for using special algorithms The above problem can be looked at graphically as illustrated in Figure 51 In the network of Figure 51 we have the suppliers S1 S2 and S3 and the five customers T1 T2 T3 T4 and T5 including the dummy customer Si and Tj provide the nodes of the network to which we have attached the positive capac ities or negative requirements The possible supply patterns Si to Tj provide the arcs of the network to which we have attached the unit supply costs Our problem can now be regarded more abstractly as one where we wish to obtain the T1 62 135 56 93 83 39 91 9 T2 T3 T4 S1 S2 S3 T5 Figure 51 APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 85 minimum cost flow through the network The Si nodes are sources for the flow entering the system and the Tj nodes are sinks where flow leaves the system We must ensure that there is continuity of flow at each node total flow in equals total flow out These conditions give rise to material balance constraints of the type discussed in Section 33 If xij represents the quantity of flow in the arc i to j we obtain the following constraints x11 x13 x14 x15 135 x21 x22 x25 56 x31 x32 x33 x34 x35 93 x11 x21 x31 62 x22 x32 83 x13 x33 39 x14 x34 91 x15 x25 x35 9 528 529 530 531 532 533 534 535 These constraints are clearly equivalent to the constraints 521 to 527 We have however added the dummy customer T5 This has resulted in the additional variables x15 x25 and x35 and an added constraint 535 but allowed us to deal entirely with constraints We have also revised the signs on both sides of the availability constraints For more general minimum cost flow problems which we consider later it is convenient to give negative coefficients to flows out of a node and positive coefficients to flows in Therefore this convention has been applied here The transportation problem also arises in less obvious contexts than distribu tion We give a numerical example below of a production planning problem Example 53 Production Planning A company produces a commodity in two shifts regular working and overtime to meet known demands for the present and future Over the next four months the production capacities and demands in thousands of units producible are January February March April Regular working 100 150 140 160 Overtime 50 75 70 80 Demand 80 200 300 200 The cost of production of each unit is 1 if done in regular working or 150 if done in overtime Units produced can be stored before delivery at a cost of 030 per month per unit The problem is how much to produce each month to satisfy present and future demand 86 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is convenient to summarize the costs in Table 54 in Table 54 Production Demand January February March April January Regular 1 13 16 19 Overtime 15 18 21 24 February Regular 1 13 16 Overtime 15 18 21 March Regular 1 13 Overtime 15 18 April Regular 1 Overtime 15 Clearly it is impossible to produce for demand of an earlier month This is represented by a dash in the positions indicated an infinite unit cost The other unit costs arise from a combination of production and storage costs for example production in January by overtime working for delivery in March gives a unit cost of 150 production 060 storage 210 This cost matrix is of the same form as that given for the transportation problem in Table 53 Although the problem here is not one of distribution it can still therefore be regarded as a transportation problem In this case there are eight sources and five sinks including the surplus demand of 45 units Transportation problems are obviously expressed much more compactly in a square array such as Tables 53 and 54 rather than as an LP matrix This is one virtue of using a special purpose algorithm Dantzig 1951 uses the simplex algorithm but works within this compact format The special structure results in the algorithm taking a particularly simple form An alternative algorithm for the transportation problem is due to Ford and Fulkerson 1956 This algorithm is usefully thought of as a special case of a general algorithm for finding the minimum cost flow through a network Such problems are considered below and described in Ford and Fulkerson 1962 As a result of their special structure transportation problems are particularly easy to solve in comparison with other LP problems of comparable size They also have together with some other network flow problems the very important property that so long as the availabilities and requirements at the sources and sinks are integral the values of the variables in the optimal solution will also be so For example so long as the righthand side coefficients in the constraints 521527 of the LP problem of Example 52 are integers the variable values in the optimal solution will be as well This rather surprising property of the trans portation problem is computationally very important in many circumstances as it avoids the necessity of using integer programming to ensure that variables take integer values As will be discussed in Chapters 8 9 and 10 integer programming models are generally much more difficult to solve than LP models Sufficient conditions for a model to be expressible as a network flow problem are discussed in Section 101 The recognition of such conditions is important as it allows the use of specialized efficient algorithms and avoids the use of computationally expensive integer programming A further constraint that sometimes applies to transportation problems is that there are limits to the possible flow from a source to a sink This gives rise to the capacitated transportation problem There may be both lower and upper limits for the flow in each arc For the LP formulation of the transportation problem such as exemplified in Example 52 above such limits can be accommodated by simple bounds on the variables 0 lij xij uij Frequently lij will be 0 Capacitated transportation problems can like the ordinary transportation problem be solved by straightforward extensions to the special purpose algorithms mentioned above Another nondistribution example of the transportation problem is described by Stanley et al 1954 who describe how the problem arises in deciding how a government should award contracts 532 The assignment problem This is the problem of assigning n people to n jobs so as to maximize some overall level of competence For example person i might take an average time tij to do job j In order to assign each person to a job and to fill each job so as to minimize total time for all tasks our problem would be Minimize tij xij subject to xij 1 for all j 536 xij 1 for all i 537 where xij 1 if person i is assigned to job j 0 otherwise This can obviously be regarded as a special case of the transportation problem We can regard it as a problem with n sources and n sinks Each source has an availability of 1 unit and each sink has a demand of 1 unit Constraints 88 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 536 impose the condition that each job be filled Constraints 537 impose the condition that every person be assigned a job It might appear that this problem demands integer programming in order to ensure that xij can only take the values 0 or 1 Fortunately however because this problem is a special case of the transportation problem the integrality property mentioned above holds If we solve an assignment problem as a conventional LP model we can be certain that the optimal solution will give integer values to the xij 0 or 1 If marriage is regarded as an assignment problem of this kind Dantzig has suggested that the integrality property shows that monogamy leads to greatest overall happiness Obviously assignment problems could be solved as LP models although the resultant models could be very large For example the assigning of 100 people to 100 jobs would lead to a model with 10 000 variables It is much more efficient to use a specialized algorithm One of the specialized algorithms for the transportation problem could obviously be applied The most efficient method known is one allied to the Ford and Fulkerson algorithm but more specialized This is known as the Hungarian method and is described by Kuhn 1955 533 The transhipment problem This is an extension of the transportation problem due to Orden 1956 In this problem it is possible to distribute the commodity through intermediate sources and through intermediate sinks as well as from sources to sinks In Example 52 we could allow flow at a certain cost between suppliers S1 S2 and S3 as well as between customers T1 T2 T3 and T4 It might be advantageous sometimes to send a commodity from one supplier to another before dispatching it to the customer Similarly it might be advantageous to send a commodity to a customer via another customer first The transhipment problem allows for these possibilities If we extend Example 52 to allow the use of certain intermediate sources and sinks our graphical representation would be of the form of Figure 52 Costs have now been attached to the arcs between sources and the arcs between sinks Notice that it is sometimes possible to go either way between sources or sinks at not necessarily the same cost It is possible to convert a transhipment problem into a transportation problem To do this the sources and sinks are considered firstly as being all sources and then as all sinks When considered as sinks sources have no availabilities and when considered as sources sinks have no requirements Flow from a sink to a source is not allowed For the transhipment extension of Example 52 illustrated in Figure 52 we can draw up the unit cost array of Table 55 Sources T1 T2 T3 and T5will have zero availabilities and sinks S1 S2 and S3 zero requirements Transhipment problems can obviously be formulated as LP models just as the transportation problem can Again it is often desirable to use a specialized algorithm such as that described by Dantzig 1951 or by Ford and Fulkerson 1962 APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 89 T1 S1 S2 55 25 35 12 12 10 25 20 15 20 S3 T2 T3 T4 T5 Figure 52 Table 55 Sources Sinks S1 S2 S3 T1 T2 T3 T4 T5 S1 20 55 132 97 103 0 S2 35 85 91 0 S3 25 106 89 100 98 0 T1 20 10 T2 15 12 T3 25 12 T4 T5 As with transportation problems transhipment problems can be extended to capacitated transhipment problems where the arcs have upper and lower capacity limitations These can also be solved by specialized algorithms An application of the transhipment problem outside the field of distribution is described by Srinivason 1974 534 The minimum cost flow problem The transportation transhipment and assignment problems are all special cases of the general problem of finding a minimum cost flow through a network 90 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Such problems may have upper and lower capacities attached to the arcs in the capacitated case The uncapacitated case will be considered here Example 54 Minimum Cost Flow The network in Figure 53 has two sources 0 and 1 with availabilities of 10 and 15 There are three sinks 5 6 and 7 with requirements 9 10 and 6 respectively Each arc has a unit cost of flow associated with it 0 5 4 2 1 2 4 3 5 4 6 9 10 10 15 6 6 1 2 5 6 7 4 3 Figure 53 The arcs are directed in the sense that only flow in the direction marked by the arrow is allowed If flow is allowable in the opposite direction as well this is indicated by another arc in the reverse direction This happens in the case of the two arcs between node 2 and node 4 The problem is simply to satisfy the requirements at the sinks by flow through the network from the sources at total minimum cost In this case the total avail ability exactly equals the total requirement This can always be made possible by the use of a dummy sink if necessary as described for the transportation problem in Example 52 The LP formulation of Example 54 is Minimize 5x02 4x13 2x23 6x24 5x25 x34 2x37 4x42 6x45 3x46 4x76 subject to x02 10 x13 15 x02 x23 x24 x25 x42 0 x13 x23 x34 x37 0 x24 x34 x42 x45 x46 0 x25 x45 9 x46 x76 10 x37 x76 6 538 539 540 541 542 543 544 545 APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 91 In order to be systematic about this formulation it is convenient to regard each constraint as arising from the material balance requirement at each node For example at node 2 it is necessary to ensure that the total flow in x02 x42 is the same as the total flow out x23 x24 x25 This is achieved by constraint 540 At node 7 which is a sink the total flow in x37 must again be the same as the total flow out x76 6 This gives constraint 545 The matrix of coefficients in constraints 538545 of the model above is known as the incidence matrix of the network in Figure 53 It clearly has a very special structure This structure is further discussed in Section 101 as like the transportation problem the minimum cost flow problem whether capacitated or not can be guaranteed to yield an optimal integer solution so long as the availabilities requirements and arc capacities are integer values As with the other types of model so far discussed in this section it is generally more efficient to use specialized algorithms Those due to Dantzig 1951 and Ford and Fulkerson 1962 are also applicable here A comprehensive survey of applications of the minimum cost network flow problem is given by Bradley 1975 Other useful references are Glover and Klingman 1977 and Jensen and Barnes 1980 It is sometimes possible to convert an LP model into a form that is imme diately convertible into a network flow model A procedure for doing this or showing such a conversion to be impossible is given in Section 54 If arcs have lower or upper bounds or both on their capacities then as with the special case of the transportation problem it is possible to adapt the special algorithms to cope with this It is however worth pointing out that such models can be converted to the uncapacitated case This might be necessary if a program was being used which could not deal with such bounds Suppose the flow from node i to node j had a lower bound of l and cost cij An extra node i can be added with a new arc from i to i If there is an external flow l out of i and into i as shown in Figure 54 this provides the necessary restriction Similarly Figure 55 demonstrates how an upper bound of u can be imposed on the flow in arc i j i cij l l i j Figure 54 i cij u u j j Figure 55 92 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is important to ensure that a minimum cost network flow problem is well defined For example the unit flow costs are generally nonnegative If nega tive costs are allowed it is important to ensure that cost cannot be minimized indefinitely giving an unbounded problem This could happen for example if arc 24 were given in a unit cost of 6 instead of 6 Going round the loop indefinitely would continuously reduce the cost A minimum cost network flow problem DISTRIBUTION is given in Part II An extension of the problem of finding the minimum cost flow of a single commodity through a network is the problem of minimizing the cost of the flows of several commodities through a network This is the minimum cost multi commodity network flow problem There will be capacity limitations on the flows of individual commodities through certain arcs as well as capacity limitations on the total flow of all commodities through individual arcs For example in the network of Figure 53 one commodity might flow between source 0 and sink 5 and a second commodity flow between source 1 and sinks 6 and 7 This type of problem can again be formulated as an LP model The resultant model has a block angular structure of the type discussed in Section 41 The block angular structure makes the decomposition procedure of Dantzig and Wolfe which is discussed in Section 42 applicable In fact this leads to another LP formulation of the problem This aspect of the minimum cost multicommodity network flow problem is discussed by Tomlin 1966 Apart from decomposition there are no special algorithms applicable to the general minimum cost multicommodity network flow problem The often large LP model resulting from such a problem is best solved by the standard revised simplex algorithm using a package programme Charnes and Cooper 1961a formulate a traffic flow problem as a minimum cost multicommodity network flow model This extension of the singlecommodity network flow LP model to more than one commodity destroys the property that guarantees an integral optimal solution Fractional values for the flows may result from the optimum solution to the LP model even if all capacities availabilities and requirements are integral If the nature of the problem requires an optimal integer solution it is necessary to resort to integer programming Another important extension of the minimum cost network flow model is the generalized network flow model This is sometimes known as the network flow with gains model In this extension the flow in an arc may alter between the two nodes A multiplier is then associated with each arc which gives the factor by which flow is altered Situations that require this modification result from for example evaporation wastage or application of interest rates Glover and Klingman 1977 give applications If it is necessary that the flows be integer values then this can no longer be guaranteed from an LP solution It is necessary to use integer programming methods Nevertheless it is possible to exploit this simple structure to good effect in the algorithms used Glover et al 1978 describe such a method In fact any 01 integer programming problem can be APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 93 converted into such a generalized network model where flows must be integer values This is shown by Glover and Mulvey 1980 535 The shortest path problem This is the problem of finding a shortest path between two nodes through a network Rather surprisingly this problem can be regarded as a special case of the minimum cost flow problem Example 55 Finding the Shortest Path Through a Network In the network in Figure 56 we wish to find the shortest path between node 0 and node 8 The lengths of each arc are marked 0 1 1 1 1 1 4 2 3 3 4 1 1 1 2 2 2 1 2 5 7 8 6 4 3 Figure 56 We can reduce this problem to one of finding a minimum cost flow through the network by giving node 0 an availability of 1 unit a source and giving node 8 a requirement of 1 unit a sink Because of the property that minimum cost flow as with transportation transhipment and assignment problems have of guaranteeing integral optimal flows when solved as LP models we can be sure that this minimal cost flow through each arc in Figure 56 will be 0 or 1 Exactly one of the arcs out of node 0 will therefore have a flow of 1 and exactly one of the flows into node 8 will have a flow of 1 Similarly intermediate nodes on the flow path will have exactly one arc with flow in and one with flow out The cost of the optimal flow path will give the shortest route between 0 and 8 Although it is possible to use conventional LP to solve shortest path problems it would be more efficient to use a specialized algorithm One of the most efficient such algorithms is due to Dijkstra 1959 536 Maximum flow through a network When a network has capacity limitations on the flow through arcs there is often interest in finding the maximum flow of some commodity between sources and sinks We will again consider the network of Example 54 but our objective will 94 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 0 12 20 Sinks Sources 6 7 9 4 8 6 2 3 5 1 2 5 6 7 4 3 Figure 57 now be to maximize the flows into the sources and out of the sinks rather than these quantities being given Each arc has now been given an upper capacity which is the figure attached to it in Figure 57 Example 56 Maximizing the Flow Through a Network This problem can again be formulated as an LP model The variables and con straints will be the same as those in Example 54 apart from the introduction of five new variables xS0 xS1 x5T x6T and x7T representing the flows into sources 0 and 1 and out of sinks 5 6 and 7 The resultant model is Maximize xS0 xS1 subject to xS0 x02 0 xS1 x13 0 x02 x23 x24 x25 x42 0 x13 x23 x34 x37 0 x24 x34 x42 x45 x46 0 x25 x45 x5T 0 x46 x76 x6T 0 x37 x76 x7T 0 x02 12 x13 20 x23 6 x24 3 x25 6 x34 7 x37 9 x42 2 x45 5 x46 8 x76 4 546 547 548 549 550 551 552 553 Again this type of model has the property that the optimal solution will give integer flows so long as the capacities are integer It is again more efficient to use a specialized algorithm for this type of problem Such an algorithm is described by Ford and Fulkerson 1962 537 Critical path analysis This is a method of planning projects often in the construction industry that can be represented by a network The arcs of the network represent activities occupying a duration of time for example building the walls of a house and the nodes are used to indicate the termination and beginning of activities Once APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 95 a project has been represented by such a network model the network can be analysed to answer a number of questions such as 1 How long will it take to complete the project 2 Which activities can be delayed if necessary and by how long without delaying the overall project Such a mathematical analysis of this kind of network is known as critical path analysis The arcs for those activities in the network that cannot be delayed without affecting the overall completion time of the project can be shown to lie on a path This critical path is in fact the longest path through the network The problem of finding the critical path is a special kind of LP problem although the special structure of the problem makes a specialized algorithm appropriate Example 57 Finding the Critical Path in a Network The network in Figure 58 represents a project of building a house Each arc represents some activity forming part of the project The durations days of the activities are attached to the corresponding arcs The arc 42 marked with a broken line is a dummy activity having no duration Its only purpose is to prevent activity 25 starting before activity 34 has finished 1 0 2 3 4 5 6 Obtain bricks 10 2 7 4 12 5 3 4 Obtain tiles Lay foundations Dig foundations Walls Wiring Painting Roofing Figure 58 In order to formulate this problem as an LP model we can introduce the following variables t0 start time for activities 01 03 and 02 t1 start time for activity 13 t2 start time for activity 25 t3 start time for activity 34 t4 start time for activities 42 and 45 t5 start time for activity 56 z finish time for the project 96 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Our model is then Minimize z subject to t0 t1 4 t0 t2 12 t0 t3 7 t1 t3 2 t3 t4 10 t2 t4 0 t2 t5 5 t4 t5 3 t5 z 4 554 555 556 557 558 559 560 561 562 Each constraint represents a sequencing relation between certain activities For example activity 34 cannot start before activity 13 has finished This gives t3 t1 2 which leads to constraint 557 Finally as the project cannot be completed before activity 56 is finished we get constraint 562 On solving this model we obtain the following results Project completion time z 26 days t0 0 t1 4 t2 17 t3 7 t4 17 t5 22 The critical path is clearly 034256 Building and conventionally solving the above LP model would be an inef ficient method of finding the critical path Special algorithms exist and there are widely used package programs for doing critical path analysis Many extensions of the problem of scheduling a project in this way can be considered but are beyond the scope of this book A full discussion of this subject is contained in Lockyer 1967 One very practical extension of the problem is that of allocating resources to the activities in a project network For example in the network of Figure 58 both activity 45 wiring and activity 25 roofing may require people although in this contrived example they would probably have different skills If activity 45 requires three people and activity 25 requires six and there are only eight available the optimal schedule given above is unattainable and one of those activities will have to be delayed or extended The problem is then how to reschedule to achieve some objective For example the objective might well be APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 97 to delay the overall completion time as little as possible Alternatively there might be a desire to smooth the usage of this and other resources over time This extension to the problem is mentioned again in Section 95 as it gives rise to an integer programming extension to the LP problem of the type above Nevertheless integer programming would be generally far too costly in computer time to justify solving this type of problem in this way A problem that gives rise to a very simple network is jobshop scheduling This problem of scheduling jobs on machines can be regarded as a problem of allocating resources to activities in a network The operations in the jobshop eg machining give the activities Sequencing relations between those operations give a simple network structure The resources to be allocated are the limited machines All the problems described in this section apart from the minimum cost multicommodity network flow problem are best tackled through specialized algorithms rather than the revised simplex algorithm available on commercial package programs The reason for describing these problems and showing how they can if necessary be modelled as linear programs is that many practical prob lems are made up in part of network problems Such problems often however contain additional complications that make it impossible to use a pure network model This is where the conventional LP formulation becomes important Many practical LP models have a very large network component Such a feature usually makes very large models easy to solve using package programs Some package programs have special features to take advantage of some of the network struc ture within a model An example of this is the generalized upper bound GUB type of constraint which was mentioned in Section 33 In the LP formulation of the transportation problem in Example 52 constraints 521523 could be regarded as GUB constraints and not explicitly represented as constraints if a package with this facility was used Alternatively and preferably because there are more of them constraints 524527 could be represented as GUB con straints The use of the GUB facility makes the solution of many network flow problems or problems with a network flow component particularly easy by conventional LP Another virtue in recognizing a network flow component in an LP model is that many variables are likely to come out at integer values in the optimal solution It has been pointed out that this happens for all the variables in most of the network flow problems described in this section so long as the righthand side coefficients are integers If a model is not quite of a network flow kind it is probable that the great majority of variables will still take integer values in the optimal LP solution The computational difficulties of forcing all these variables to be integer values by integer programming will be much reduced This topic is discussed much more fully in Chapter 10 There is also great virtue to be gained from remodelling a problem in order to get it into the form of a network flow model This then opens up the possibility of using a special purpose algorithm An example of how this can sometimes be done for a practical problem is given by Veinott and Wagner 1962 Dantzig 1969 shows how a hospital admissions scheduling program can be remodelled 98 MODEL BUILDING IN MATHEMATICAL PROGRAMMING to give an LP model with a large network flow component He then exploits this structure by use of the GUB facility Other examples of reformulations into network flow models are Daniel 1973 Wilson and Willis 1983 and Cheshire et al 1984 An automatic way of either converting an LP model to a network flow model or showing such a conversion to be impossible is given in Section 54 The recognition or conversion of a model as a network structure relieves the need to use the computationally much more costly methods of integer programming In Section 62 the concept of the dual of an LP model is described Every LP model has a corresponding model known as the dual model The optimal solution to the dual model is very closely related to the optimal solution of the original model In fact the optimal solution to either one can be derived very easily from the optimal solution to the other It turns out that many practical problems give rise to an LP model that is the dual of a network flow model In such circumstances it could well be worth using a specialized algorithm on the corresponding network flow model Moreover the dual of any of the types of network flow mode mentioned here apart from the minimum cost multi commodity network flow model also has the property of guaranteeing optimal integer solutions so long as the objective coefficients of the original model are integers The recognition of this type of model can again be of great practi cal importance for this reason This topic is further discussed in Sections 101 and 102 In Part II the OPENCAST MINING problem can be formulated as the dual of a network flow problem The formulation is discussed in Part III The MINING problem of Part II can be formulated as an integer programming model a large proportion of which is the dual of a network flow model Some examples of such models are given in Williams 1982 One famous network problem that has not been discussed in this section is the travelling salesman problem This is the problem of finding a minimum distance cost route round a given set of cities This problem cannot generally be solved by an LP model in spite of its apparent similarity to the assignment problem It can however be modelled as an integer programming extension to the assignment problem and is fully discussed in Section 95 54 Converting linear programs to networks The advantages of converting linear programs to minimum cost network flow models if possible have already been discussed in Section 53 They are further discussed in Section 101 as minimum cost network flow models have inte ger optimal solutions so long as external flows in and out are integer values This relieves the need to use the much more expensive procedures of integer programming We outline a method described by Baston et al 1991 for converting linear programs to network flow models Another procedure but expressed APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 99 in the more abstract language of matroid theory is given by Bixby and Cunningham 1980 In order to illustrate the method we will take a numerical example Minimize c1x1 c2x2 c3x3 c4x4 subject to 2x1 x5 b1 6x1 9x3 x6 b2 8x1 4x2 8x4 x7 b3 x2 3x2 2x4 x8 b4 x2 3x3 x9 b5 x1 x2 x3 x4 x5 x6 x7 x8 x9 0 As the conversion does not depend on the objective or righthand side coef ficients we give these in a general form In this example we assume all the original constraints were of the form and that slack variables have been added to make them equations For constraints surplus variables would be subtracted If any of the original constraints were equations then we would add artificial variables variables constrained to take the value zero These logical slack surplus or artificial variables will represent arcs in the network created For the case of artificial variables these arcs will finally be deleted We carry out the following transformations 1 Scale the rows and columns in order to make the constraint coefficients 0 or 1 if possible This may not be possible in which case the conversion to a network is not possible In most practical problems such as those referenced in Section 53 for which it is worth attempting a conversion these coefficients will already be 0 or 1 In this example the resultant scaled coefficients are given below 6 7 8 9 1 2 3 4 5 1 2c1 c2 1 3c3 1 2c4 1 1 b1 1 1 1 1 3b2 1 1 1 1 1 4b3 1 1 1 1 b4 1 1 1 b5 It is also convenient to number the variables The logical variables are numbered 1 to 5 and the original variables 6 to 9 2 In this step the signs of the nonzero coefficients 1 are ignored The arcs corresponding to the logical variables are arranged in the form of a 100 MODEL BUILDING IN MATHEMATICAL PROGRAMMING spanning tree of the network This spanning tree must be compatible with the original variables of the model in the following sense the original variables each form a polygon with some of the arcs of the spanning tree Figure 59 illustrates how this is possible with the example The arc corresponding to variable 6 forms a polygon with the arcs of the tree corresponding to variables 1 2 and 3 as variable 6 has nonzero entries in rows 1 2 and 3 Similarly as variable 7 has nonzero entries in rows 3 4 and 5 arc 7 forms a polygon with arcs 3 4 and 5 Arcs 8 and 9 are similarly compatible with the tree It will not always be possible to find an arrangement of the logical arcs in the form of a spanning tree which is compatible with the other arcs in the manner demonstrated above In such a case the network conversion is not possible A systematic way of investigating whether it is possible to construct such a spanning tree or showing it to be impossible is described by Baston et al 1991 Having created such an undirected network the arcs are orientated in the following manner 3 For each polygon the arcs of the tree in the polygon are given a direction opposite to that of the nontree arc when going round the polygon if the entry in the column corresponding to this arc is 1 If the entry is 1 the arcs are given the same direction For this example the resulting orientations are shown in Figure 510 Arc 6 for example has an opposite orientation to arcs 1 and 2 variable 6 has 1 coefficients in rows 1 and 2 and the same orientation to arc 3 variable 6 has a 1 coefficient in rows 3 when going round the polygon formed by arcs 1 2 3 and 6 Other arcs are orientated similarly according to this rule It may not be possible to orientate the arcs in any manner compatible with the signs of the coefficients In such a case a directed network is not constructible 4 The nontree arcs in the network are given unit costs equal to the scaled objective coefficients of corresponding variables 5 Each node is given an external flow in equal to the sum of the scaled righthand side coefficients of the rows corresponding to those treearcs A 6 3 1 2 4 8 5 9 7 B C D E F Figure 59 APPLICATIONS AND TYPES OF MATHEMATICAL PROGRAMMING 101 A 6 3 1 2 4 8 5 9 7 B C D E F 1 c1 2 c2 1 c4 2 1 b3 4 b1 1 b2 3 1 b3 4 b4 1 b2 3 1 c3 3 b4 b5 b1 b5 Figure 510 leaving the node less the sum of the scaled righthand side coefficients of the rows corresponding to those treearcs entering the node In the example in Figure 510 node C has treearc 4 leaving it row 4 has scaled righthand side b4 and treearcs 2 and 3 entering it rows 2 and 3 have scaled righthand sides of 13b2 and 14b3 respectively Hence the external flow into node C is b4 13b2 14b3 A negative external flow would of course be regarded as a positive flow out The other nodes have external flows calculated in a similar manner The resulting directed network with its external flows gives the required minimum cost network flow model equivalent to the original linear program When solved the values of the flows in the arcs may have to be unscaled according to the scaling factors applied no text in the image 6 Interpreting and using the solution of a linear programming model 61 Validating a model Having built a linear programming model we should be very careful before we rely too heavily on the answers it produces Once a model has been built and converted into the format necessary for the computer program we will wish to attempt to solve it Assuming that there are no obvious clerical or keying errors which are usually detected by package programs there are three possible outcomes i the model is infeasible ii the model is unbounded iii the model is solvable 611 Infeasible models A linear programming model is infeasible if the constraints are selfcontradictory For example a model which contained the following two constraints would be infeasible x1 x2 1 61 x1 x2 2 62 In practice the infeasibility would probably be more disguised unless it arose through a simple keying error The program will probably go some way towards trying to solve the model until it detects that it is infeasible Most package programs will print out the infeasible solution obtained at the point when the program gives up Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 104 MODEL BUILDING IN MATHEMATICAL PROGRAMMING In most situations an infeasible model indicates an error in the mathematical formulation of the problem It is of course possible that we are trying to devise some plan that is technologically impossible but more usually we are modelling a situation where we know there should be a feasible solution The detection of why a model is infeasible can be very difficult If we have got the infeasible solution where the program gave up we can see which constraints are unsatisfied or which variables are negative in this solution This may enable us to find the cause of infeasibility fairly easily It is quite possible however that it will not help The cause of infeasibility may be fairly subtle For example it is impossible to satisfy each of the following constraints in the presence of the other two x1 x2 1 63 x2 x3 1 64 x1 x3 1 65 It might however be wrong to single out any one of Inequalities 6365 as being an infeasible constraint What is wrong is the mutual incompatibility of all three constraints Assuming that we are modelling a situation which we know to be realizable it should be possible to construct a feasible but probably nonoptimal solution to the situation For example in a product mix model such as that presented in Section 12 we could take a previous weeks mix of products assuming no capacity has since been reduced If our model were a correct representation of the situation it should be possible to substitute this constructed or known solution into all the constraints without breaking them As our model admits no feasible solution this substitution must violate some of the constraints in the model Any constraints violated in this way must have been modelled wrongly They are too restrictive and should be reconsidered 612 Unbounded models A linear programming model is said to be unbounded if its objective function can be optimized without limit that is for a maximization problem the objective function can be made as large as one wishes or for a minimization problem as small as one wishes For example the following trivial linear programming problem is unbounded as x1 x2 can be made as large as we like without violating the constraint Maximize x1 x2 66 subject to 2x1 x2 1 67 While a correctly formulated infeasible model is just possible as we might be trying to attain the unattainable a correctly formulated unbounded model is very unlikely INTERPRETING LINEAR PROGRAMMING MODELS 105 As with infeasible models most package programs print a solution to an unbounded model before the optimization is terminated on it being seen that the model is unbounded This solution can be of help in detecting what is wrong with the model On the other hand detection of unboundedness may well be as difficult as the detection of infeasibility An infeasible model has constraints which are over restrictive whereas an unbounded model either has vital constraints unrep resented or insufficiently restrictive Usually certain physical constraints have not been modelled These constraints may well be so obvious as to be forgotten A very common type of constraint to omit is a material balance constraint such as that described in Section 33 This sort of constraint is easily overlooked If such a constraint has been inadvertently overlooked the solution which the package program prints before giving up can be of use in detecting it A com mon sense examination of this solution may reveal that things do not add up for example we may find we are producing something without using any raw material In Section 62 we define an associated linear programming model for every model known as the dual model This model can have important economic inter pretations If a model is unbounded the corresponding dual model is infeasible It is therefore conceivable that a practical unbounded model might arise as the dual to an infeasible model where we were testing whether a certain course of action was or was not possible Should a model be infeasible we should not assume that its dual will necessarily be unbounded It is possible that the dual will also be infeasible 613 Solvable models Should a linear programming model be neither infeasible nor unbounded we will refer to it as solvable When we obtain the optimal solution to such a model we clearly want to know if the answer is sensible If it is not sensible then there must be something wrong with our model The first approach should be to examine the optimal solution critically simply using common sense This may well reveal an obvious nonsense which should enable us to detect and correct a modelling error Should the solution still appear sensible we could compare the optimal objec tive value with what we might expect in practice If in a maximization problem this value is lower than we expect we might suspect that our model is over restrictive that is some constraints are too severe On the other hand if the optimal objective is higher than we expect we might suspect that our model is insufficiently restrictive that is that some constraints are too weak or have been left out For minimization problems these conclusions would obviously be reversed To give an example suppose we were considering a product mix application such as that considered in Section 12 On solving the model we would obtain a maximum profit contribution Suppose this profit contribution was lower than what we already knew we could attain for example suppose it was lower than what we obtained in a previous week assuming there had been no subsequent cutback in productive capacity In this circumstance we would obviously suspect our model to be over restrictive To find where the over severe constraints lay we could substitute our known better solution into the constraints of our model Some of these constraints will obviously be violated and must therefore have been modelled incorrectly Possibly we may find that there is some productive capacity which we were unaware of but is being used by the workforce Our model even though not yet correct will already be proving valuable in helping to discover things we were unaware of Thinking again about our product mix model suppose we find the reverse situation that the optimal objective value was greater than anything we knew we could possibly achieve In this situation our model would obviously be underrestrictive The approach here would be to subject the optimal solution proposed by the model to a very critical examination This solution could be presented in an entirely nontechnical manner as a proposed operating policy and the appropriate management asked to spell out why it was impossible It should then be possible to use this information to modify constraints or add new constraints For situations such as the above where we are modelling an existing situation we obviously have the great advantage of the feasible solutions suggested by the existing mode of operation These solutions can be used to test and modify our model For situations where we are using linear programming or any other sort of model to design a new situation for example decide where to set up new plant there may be no obvious feasible solutions to use Testing the model may therefore be more difficult It is still a good approach to try to obtain good common sense solutions by rule of thumb methods These solutions may well reveal errors in our model and show how they may be corrected The value of optimizing an objective should be apparent in this discussion on the validation of linear programming models By optimizing some quantity we would expect to be using certain resources processing capacity raw materials manpower etc to their limit The resultant optimal solution suggested by the model would then be fairly likely to violate certain physical restrictions which had been ignored or modelled incorrectly and thereby highlight them It is often desirable to solve the model a number of times with different possibly contrived objectives in order to test out as many constraints as possible The value of optimizing an objective and so using mathematical programming where there is no real life objective should be apparent in validating and modifying a model It should be obvious from the foregoing discussion that the building and validation of a model should be a twoway process gradually converging on a more and more accurate representation of the situation being modelled Unfortunately this process is often ignored or neglected It can be an extremely valuable activity leading to a much clearer understanding of what is being modelled In many situations this greater understanding may be more valuable than even the optimal solution to the validated model It is often possible to build a model with error detection in mind For example suppose we wished to define the following constraint a j x j b 68 INTERPRETING LINEAR PROGRAMMING MODELS If there were any danger of this constraint being too severe and making the model infeasible we could allow it to be violated at a certain high cost by rewriting it as j aj xj u b 69 and giving u a negative cost coefficient in the objective function assuming the problem to be a maximization For a minimization problem u would be given a positive profit coefficient In this way the constraint could no longer cause the model to be infeasible but the violation of the original constraint 68 would be indicated by u appearing in the solution For certain applications it might be argued that 69 was a more correct formulation of the constraint anyway by for example allowing us to buy in more resources if really needed at a certain cost rather than restricting us absolutely This topic was discussed in more detail in Section 33 Another device can be used to avoid unboundedness occurring in a model Each variable can be given a possibly very large finite simple upper bound in the formulation No variable could then exceed its upper bound but any variable which rose to its bound would be open to suspicion This might then lead one more quickly to the cause of unboundedness in the model Finally the desirability of using matrix generators should be reemphasized here By automatically generating a model the possibility of error is greatly reduced The validation process itself is greatly simplified as the input to the matrix generator is usually presented in a much more physically meaningful form than for a linear programming package 62 Economic interpretations It will be useful to relate the discussion in this section to a specific type of model rather than present the material more abstractly We will use the small product mix example of Section 12 for this purpose The model was Maximize 550x1 600x2 350x3 400x4 200x5 subject to Grinding 12x1 20x2 25x4 15x5 288 Drilling 10x1 8x2 16x3 192 Manpower 20x1 20x2 20x3 20x4 20x5 384 x1 x2 x3 x4 x5 0 This problem was that of finding how much to make of five products PROD 1 PROD 2 PROD 5 subject to two processing capacity limitations grinding and drilling and a manpower limitation Practical problems will of course usually be much bigger and more complex The interpretation of the solution and the derivation of extra economic information is however well illustrated by 108 MODEL BUILDING IN MATHEMATICAL PROGRAMMING this example Other types of application eg blending will of course result in different interpretations being placed on this information It should however be possible to relate this information to any real life application after following the discussion applied to the small example above The practical problems presented in Part II provide an excellent means of relating such information to real life situations If the model above is solved by for example the simplex algorithm the optimal solution turns out to be x1 12 x2 72 x3 x4 x5 0 giving an objective value of 10920 that is we should make 12 of PROD 1 72 of PROD 2 and none of the other products This results in a total profit contribution over a week of 10920 It can easily be verified that the grinding and manpower capacities are fully exhausted but that there is slack drilling capacity This information is usually given along with the rest of the solution when a package program is used It can fairly easily be shown although it is beyond the scope of this book that in a model with m linear constraints including possibly simple upper bounds and a linear objective one would never expect more than m of the variables to be nonzero in the optimal solution In the example above one could therefore be sure that one need never produce more than three products In a problem of the above kind there is a considerable amount of extra economic information which might be of interest For example we can obtain answers to the following questions 1 Presumably products 35 are underpriced in comparison with products 1 and 2 How much more expensive should we make them in order for it to be worth manufacturing them 2 What is the value of an extra hour of grinding drilling or manpower capacity Strictly speaking we are interested in the marginal values of each of these capacities that is the effect of very small increases or decreases in capacity This extra information is usually presented with the ordinary solution when a package program is used The variables have associated with them quantities known as reduced costs which can be interpreted in this application as the necessary price increases Each constraint has associated with it a quantity known as the shadow price which can be interpreted as the marginal effect of increases or decreases in the capacities It should be emphasized that when the simplex algorithm or one of its vari ants is used to solve a model reduced costs and shadow prices arise naturally out of the optimal solution and most package programs print this information There is however an alternative and very illustrative way of deriving this economic information which helps to clarify its true meaning This is through another INTERPRETING LINEAR PROGRAMMING MODELS 109 associated linear programming model known as the dual model which will now be described 621 The dual model We will again use the product mix problem above to illustrate this discussion Suppose that an accountant were trying to value each of the resources of this problem grinding drilling and manpower capacities in some way so as to give a minimal overall valuation to the factory compatible with the optimal production plan Let us suppose that the valuations for each hour of each of the capacities were y1 y2 and y3 measured in The objective would be Minimize 288y1 192y2 384y3 610 The accountant wishes to obtain values for y1 y2 and y3 which totally explain the optimal production pattern It should be possible to impute the profit contribution obtained from each product produced to its usage of the three resources We must ensure that the profit contribution for each unit of each product is totally covered by its imputed value For example each unit of PROD 1 has a profit contribution of 550 This must be totally accounted for by the value of the 12 hours grinding capacity 10 hours drilling capacity and 20 hours manpower capacity used in making this unit As the values of an hour of each of these capacities are y1 y2 and y3 in we have 12y1 10y2 20y3 550 611 The reason why is used rather than in the above constraint will not be totally obvious to start with It should however become apparent later Similar arguments will relate the hourly values y1 y2 and y3 to the unit profit contributions of each of the other products This gives constraints 412415 20y1 8y2 20y3 600 612 16y2 20y3 350 613 25y1 20y3 400 614 15y1 y3 200 615 The objective 610 together with the constraints 611615 give us another linear programming model As with most linear programming models we will implicitly assume that the variables y1 y2 and y3 can only be nonnegative This new linear programming model is referred to as the dual of the original product mix model In contrast the original model is usually referred to as the primal model The derivation of this model may appear somewhat contrived at first but should become more plausible once we examine its solution 110 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Solving this dual model by a suitable algorithm we obtain the optimal solution y1 625 y2 0 y3 2375 giving an objective value of 10 920 that is we should value each hour of grinding capacity at 625 each hour of drilling capacity at nothing and each hour of manpower capacity at 2375 The total valuation of the factory is then 10 920 We can immediately see some connections between this result and the optimal production plan for the original product mix model 1 The total valuation of the factory over a week is the same as the optimal objective value of the original model This seems plausible The total value of a factory is equal to the value of its optimal productive output It follows from the duality theorem of linear programming that this result will always be true 2 Drilling capacity was not totally utilized in the optimal solution to the original primal model We see that it has been given a zero valuation This again seems plausible As we do not use all the capacity we have we are not likely to place much value on it The result here is another consequence of the duality theorem of linear programming If a constraint is not binding in the optimal primal solution the corresponding dual variable is zero in the optimal solution to the dual model Economists would refer to the drilling capacity as a free good that is in one sense it is not worth anything Let us examine the optimal solution to the dual problem further and see what it might suggest to an accountant about the production policy which the factory should pursue Each unit of PROD 1 contributes 550 to profit It uses up however 12 hours of grinding capacity valued at 625 per hour 10 hours of drilling capacity valued at nothing and 20 hours of manpower capacity valued at 2375 per hour The total value imputed to each unit of PROD 1 is therefore 12 625 10 0 20 2375 550 that is the profit of 550 which each unit of PROD 1 contributes is exactly explained by the value imputed to it by virtue of its usage of resources If we regarded the dual variables y1 y2 and y3 as costs that is we charged PROD 1 for its usage of scarce resources then we would come to the conclusion that PROD 1 produced zero profit In accounting terms this does not matter as these costs are purely internal accounting devices Similarly we find that each unit of PROD 2 has an imputed value or extra cost of 20 65 8 0 20 2375 600 showing that the 600 contribution to profit is exactly accounted for INTERPRETING LINEAR PROGRAMMING MODELS 111 For each unit of PROD 3 we get an imputed value or extra cost of 0 625 16 0 20 2375 475 This cost exceeds its profit contribution by 125 An accountant would conclude that PROD 3 would cost more in terms of usage of scarce resources than it would contribute to profit He or she would therefore suggest that PROD 3 not be manufactured We came to the same conclusion using our original primal model It can easily be verified that PROD 4 and PROD 5 have costs which exceed their unit profit contributions by 23125 and 36875 respectively and should therefore not be produced We are now in a position to see why the rather than constraints in the dual model are acceptable If the total activity in the lefthand side the cost of a constraint in the dual model strictly exceeds the righthand side coefficient the profit contribution we do not incur this excess cost by simply not manufacturing the corresponding product This gives us a third connection between the optimal solutions to the dual and primal models 3 If a product has a negative resultant profit after subtracting its imputed costs we do not make it This is again a consequence of the duality theorem in linear programming If a constraint in the dual model is not binding in the optimal solution to the dual model then the corresponding variable is zero in the optimal solution to the primal model The symmetry between this property and property 2 should be obvious This result is sometimes referred to as the equilibrium theorem Economically it simply means that nonbinding constraints have zero valuation Returning to our accountants analysis of the problem using the valuations derived from the dual model we conclude that a At most PROD 1 and PROD 2 should be manufactured at zero internal profit b Drilling capacity is not a binding constraint having a zero valuation Strictly speaking the deduction of b is not quite valid It is certainly true that a nonbinding constraint implies a zero valuation The converse cannot immediately be concluded although it presents no difficulty here This complication is referred to later The accountant could therefore eliminate x3 x4 and x5 from the primal prob lem ignore the second drilling constraint and treat the remaining two constraints as equations This gives 12x1 20x2 288 616 20x1 20x2 384 617 Solving this pair of simultaneous equations gives us x1 12 and x2 72 that is the accountant deduces the same production plan using his or her valuations for the three resources as we do from our ordinary primal linear programming model In practice the derivation of these valuations values of the dual variables by the method we have suggested building and solving the dual model might be just as difficult as or more difficult than building and solving the original primal model Our purpose is however to explain a useful concept In practice one would not normally build or solve the dual model although in some circumstances this model is easier to solve computationally and might be used for this reason The product mix model for which we constructed a dual model was a maximization with all constraints For completeness we should define the dual corresponding to a more general model It is convenient to regard all problems as maximizations In order to cope with a minimization we can negate the objective function and maximize it It is also possible to convert all the constraints to We do not choose to do this as it is helpful to keep the original model close to its original form The dual variable corresponding to a constraint was a conventional nonnegative linear programming variable A constraint can be dealt with by only allowing the dual variable to be nonpositive For an constraint we will allow the dual variable to be unrestricted in sign Such a variable is sometimes known as a free variable There is a symmetry concerning duality which should be mentioned although its main interest is purely mathematical The dual of a dual model gives the original model 622 Shadow prices The valuations values of the dual variables which we have obtained by this roundabout means are in fact the shadow prices which we referred to in the earlier part of this section They arise naturally as subsidiary information out of the optimal solution to the primal model if the simplex algorithm is used It would often in fact be possible to deduce the shadow prices from simply the optimal solution values of the variables by using an argument similar to the accountancy argument which we used above We will not however discuss the derivation of the shadow prices any longer as most package programs present these values in the output of the optimal solution It can fairly easily be seen although we do not do so in this book that the values of the dual variables the shadow prices represent the effects of small changes on the righthand side coefficients that is they are marginal valuations For example if we were to increase grinding capacity by a small amount Δ then the resultant increase in total profit after rearranging our optimal production plan would be 625 Δ Similarly the decrease in total profit by reducing grinding capacity would be 625 Δ There will usually be limits within which Δ can lie These limits ranges are discussed in the next section Strictly speaking it is INTERPRETING LINEAR PROGRAMMING MODELS 113 possible that one or both of these limits be zero This complication is discussed later It will also only be valid to interpret the shadow prices as referring to the effect of small changes on one of the righthand side coefficients at a time that is we could not make small changes in two righthand side coefficients simultaneously and conclude that the effect on total profit will be the sum of the shadow prices Shadow prices can be of considerable value in making investment decisions For example each extra hour of grinding capacity is worth 625 per week to the factory This interpretation remains valid as long as we are permitted to increase our grinding capacity by a sufficient amount In the next section we will see that this capacity can be increased up to 384 hours per week its upper range with each extra unit resulting in an extra 625 per week The effect of increasing capacity beyond this upper range will result in a smaller though not immediately predictable extra profit per unit of increase As we can increase grinding capacity up to 384 hours per week enabling us to make an extra 600 of profit we could decide whether it is worth investing in or hiring more grinding machines We might compare this with the 2375 which would result from each extra hour of manpower capacity within the permitted range and decide where limited funds might be invested to best effect The shadow price on a nonbinding constraint such as that representing the drilling capacity is zero As we might expect there is no value in increasing this capacity as we do not use all we have already Shadow prices are an example of opportunity costs This is a concept which accountants are increasingly although still too rarely using For example an increase in grinding capacity results in an increased opportunity to make more profit Similarly a decrease in grinding capacity loses us some opportunity to make a profit The shadow price represents the cost of the lost opportunity Unlike some of the other costs which accountants use such as average costs opportunity costs are quite a sophisticated concept They result from a careful weighing up of the demands which each product makes on the scarce resources and contributes to profit in return As a consequence they take into account the alternative uses to which the resources may be put and the comparative val ues of these alternative uses One would obviously expect such costs to be of more value than less sophisticated costs Accountants are becoming increasingly interested in linear programming as a result A good description of the appli cation of linear programming to accountancy is given by Salkin and Kornbluth 1973 Our discussion of the interpretation of shadow prices has been confined to our product mix application For other applications it should be possible to deduce the correct interpretation by relating small changes in the righthand side coeffi cients to the physical situation represented by the model To help with any such interpretation we suggest meanings which might be attached to the different types of constraint described in Section 33 114 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 623 Productive capacity constraints These are the sort of constraints we have discussed above where the capacity may represent limited processing or manpower The interpretation of the shadow prices on such constraints has been fully dealt with above 624 Raw material availabilities Suppose we have modelled limited raw material availabilities by constraints Those raw materials which the model suggests be used to their limit will be represented by constraints generally but not always having a nonzero shadow price This shadow price will indicate the value of acquiring more of the raw material within the permitted ranges Similarly it will represent the cost of cutting back on raw material This shadow price may be very useful in helping to decide whether to purchase more of the raw material or not at a certain cost 625 Marketing demands and limitations The interpretation of the shadow prices here will be the effect on the value of the objective of altering demands or limitations of the market for example forcing the factory to produce more or less or increasing or decreasing the maximum market size Such a figure can often usefully be compared with the cost of extra sales effort Frequently such constraints will take the form of simple upper bounds as discussed in Section 33 If such simple upper bounds are treated as such in the model they will not appear as constraints and therefore have no shadow price The desired interpretation can however be obtained from the reduced cost of the bounded variable as described below 626 Material balance continuity constraints The shadow price on such constraints may well have no useful interpretation For example the small blending problem Example 12 of Section 12 has a material balance constraint to make sure that the weight of the final product equals the total weight of the ingredients The righthand side value is zero The shadow price predicts the effect of altering this zero value It is hard to see a useful interpretation for this In some circumstances the shadow price on a material balance constraint may be of interest For example we may be equating our initial or final stocks to some expression involving the variables of the model The shadow price will then indicate the effect on the optimal objective value of changing these stocks A good example of this is the FOOD MANUFACTURE model of Part II 627 Quality stipulations Any model which involves blending as part of the total model usually involves quality stipulations for example the proportion of vitamins must not fall below INTERPRETING LINEAR PROGRAMMING MODELS a certain value or the octane rating of the petrol must not fall below some value As an example the blending problem in Section 12 has two quality constraints indicating a limitation on hardness The shadow prices of these constraints can be used to predict the effect on total revenue of relaxing or tightening up on these hardness stipulations In some models where the righthand side coefficient is itself a quality parameter the interpretation is straightforward For our small example we had a righthand side value of 0 The upper hardness constraint is 88x1 61x2 2x3 42x4 5x6 6y 0 618 Suppose this righthand side coefficient were not 0 but were Δ We could rewrite the constraint as 88x1 61x2 2x3 42x4 5x5 6 Δy y 0 619 In order to interpret the effect of a relaxing or tightening of the upper hardness limitation of 6 we would have to take into account the value of y in the optimal solution A unit increase or decrease in the hardness 6 would result in an increase or decrease of y in the righthand side and the interpretation of the shadow price would have to be adjusted accordingly The derivation of ranges within which the hardness parameter could validly be altered with this interpretation would be difficult as the value of y would probably alter as well To decide whether a small change in a righthand side coefficient results in an increase or decrease in the optimal objective value we should decide whether the change results in a relaxation or a tightening of the problem If we relax a problem slightly for example increase the righthand side coefficient on a constraint or decrease the righthand side coefficient on a constraint we would expect to at worse have no effect on the optimal objective value and perhaps improve it Therefore the optimal objective value in a maximization problem would possibly get larger and in a minimization problem possibly get smaller that is we would expect to do no worse possibly to do better in view of the lessening of the restrictions For a tightening of the constraints for example reducing the righthand side coefficients in a constraint and increasing them in a constraint we would expect to do worse if anything that is possibly degrade the optimal objective value In the case of constraints the improving or degrading effect of small changes in the righthand side coefficient will probably be apparent from the meaning of the model Rather than formulate mathematical rules for interpreting whether the shadow price should be added or subtracted from the objective for each unit change in the corresponding righthand coefficient it is probably better to follow the above scheme as different package programs vary in their sign conventions regarding shadow prices The suggested formulation of the TARIFF RATES POWER GENERATION problem in Part III causes the required rates for the sale of electricity to arise as shadow prices on demand constraints 116 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 628 Reduced costs In our small product mix example we saw that the unit profit contributions of PROD 1 and PROD 2 are exactly accounted for by the imputed costs derived from the shadow prices on each constraint For PROD 3 PROD 4 and PROD 5 however the imputed costs exceed the unit profit contribution The amount by which these unit profit contributions are exceeded in each case is of interest These quantities are the reduced costs of these appropriate variables Note that any variable which comes out at a nonzero level in the optimal solution has a zero reduced cost this result is modified if the variable has a simple upper bound the modification is dealt with later For our example the reduced costs which we derived earlier for PROD 3 and could also derive for PROD 4 and PROD 5 are PROD 3 125 PROD 4 23125 PROD 5 36875 If we wanted to make PROD 3 we would have to increase its unit price by 125 before it was just worth making This price increase would then allow the profit contribution of PROD 3 to balance the costs imputed to it by way of its usage of scarce resources Similarly the reduced costs of PROD 4 and PROD 5 indicate price increases necessary for those products The reduced costs are usually printed out with the solution values of the variables when a pack age program is used and there is no necessity to calculate them by way of the shadow prices as we have done Our purpose in using this derivation has been to indicate the correct interpretation which should be placed on reduced costs For other applications the interpretation of reduced costs will be different In a blending problem for example such as that in Section 12 the reduced costs might represent price decreases necessary before it became worth buying and incorporating an ingredient in a blend The above interpretation of reduced costs can be viewed the other way round Suppose we insisted on making a small amount of PROD 3 As this will deny processing and manpower capacity to some of the other products who could use it to better effect we would expect to degrade our total profit The amount by which this profit will be degraded for each unit of PROD 3 we make will be given by the reduced cost 125 of PROD 3 In fact there will be a limit to the level at which PROD 3 can be made with this interpretation holding This limit is another type of range which will be discussed in the next section It is therefore possible to cost a nonoptimal decision such as making PROD 3 This cost results from the lost opportunity to make other more profitable products It is again an opportunity cost We should mention the slight variation in the interpretation of the reduced costs on variables with a finite simple upper bound as discussed in Section 33 INTERPRETING LINEAR PROGRAMMING MODELS 117 If in the optimal solution to a model such a variable comes out at a value below its simple upper bound there is no difficulty The interpretation of the reduced cost will be exactly the same as that above Suppose however that the variable were to come out at a value equal to its simple upper bound This simple upper bound could well in this case be acting as a binding constraint If the simple upper bound had been modelled as a conventional constraint it would then have a nonzero shadow price which would be interpreted in the usual way The variable would being at a nonzero level have a zero reduced cost By modelling this constraint as a simple upper bound we will have lost the shadow price but it will appear as the reduced cost of the variable The correct interpretation of this reduced cost will be the effect on the objective of forcing the variable down below its upper bound degrading the objective or increasing the upper bound improving the objective We showed above how the shadow prices values of the dual variables could be used as accounting costs in order to derive the optimal production plan in the product mix model It is important to point out that such a procedure will not always work This discussion also leads us on to an important feature of some linear programming models alternative optimal solutions Let us consider the following small linear programming model Maximize 3x1 15x2 620 subject to x1 x2 4 621 2x1 x2 5 622 x1 4x2 2 623 x1 x2 0 If the dual model is formulated with dual variables y1 y2 and y3 corresponding to the three constraints and then solved we obtain the following result y1 0 y2 15 y3 0 An accountant could then use the values of y1 y2 and y3 as valuations for the constraints This would lead the accountant to the following conclusions 1 The accounting cost attributed to each unit of x1 is 3 exactly equalling its objective coefficient Similarly the accounting cost attributed to x2 is 15 exactly equalling its objective coefficient Therefore x1 and x2 should both be allowed to be in the solution of the original primal model 2 The second constraint 622 has a nonzero valuation and must therefore be binding ie can be treated as an equation It would be wrong however as was mentioned in applying the same procedure to the product mix example to immediately conclude that the constraints 621 and 623 are nonbinding as they have zero dual variables shadow prices 118 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Suppose for the moment we did ignore constraints 621 and 623 We deduce that x1 and x2 must satisfy the equation 2x1 x2 5 624 This equation obviously does not determine x1 and x2 uniquely It is by no means obvious how the accountant should proceed from here In fact it is impossible to deduce a unique solution to the above model from the solution to the dual model as the original primal model does not possess a unique solution This is easily seen geometrically in Figure 61 Different values of the objective function 620 correspond to different lines parallel to PQ It so happens that PQ is parallel to the edge of AC of the feasible region created by constraint 622 We therefore see that any point on the line 0 0 1 C Q P A 2 3 4 1 2 x1 x2 3 Figure 61 INTERPRETING LINEAR PROGRAMMING MODELS 119 AC between and including A and C gives an optimal solution to our model objective 75 All that our accounting information has told us if we ignore constraints 621 and 623 is that 2x1 x2 5 that is that we must choose points lying on or beyond the line AC We still have to pay some attention to the constraints 621 and 623 If we were to treat constraint 621 as binding we would arrive at the point A Treating 623 as binding would give us point C As mentioned in Section 22 the simplex algorithm only examines vertex solutions and would therefore choose either the solution at A x1 1 x2 3 or the solution at C x1 2 x2 1 Most package programs indicate when there are alternative solu tions They recognize this result by noticing one of the following properties of the optimal solutions 1 A binding constraint with a zero shadow price 2 A variable at zero level with a zero reduced cost In our example if the optimal solution at A had been selected we would note that constraint 621 was binding but had a zero shadow price At C it would be constraint 623 that was binding with a zero shadow price It might seem rather unlikely that alternative solutions would arise in a prac tical linear programming model In fact this is quite a common occurrence For problems with more than two variables the situation will obviously be more complex The characterization of all the optimal solutions all vertex optimal solutions and various combinations of them is often very difficult It is how ever important to be able to recognize the phenomenon as if one optimal solution is unacceptable for some reason we have a certain flexibility to look for another solution without degrading our objective One of the purposes of this discussion has been to indicate that applying valuations shadow prices to the constraints of a linear programming model sometimes does not lead us to a unique solution operating plan The converse situation can also happen We can have a unique solution operating plan deter mined by more than one set of valuations For example consider the following small problem Maximize 3x1 2x2 625 subject to x1 x2 3 626 2x1 x2 4 627 4x1 3x2 10 628 x1 x2 0 Let us attach valuations y1 y2 and y3 to each of the constraints 626628 There are a number of valuations which will lead us to the optimal solution For example 120 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1 y1 1 y2 1 y3 0 2 y1 0 y2 1 2 y3 1 2 Both these sets of valuations will lead us to the unique optimal solution to this problem x1 1 x2 2 giving an objective value of 7 In a practical problem the question that would naturally arise is how should we derive a shadow price for our scarce resources given this ambiguity for example what would the value be of increasing the righthand side value of 3 to 4 Would it be 1 or 0 This problem is again best illustrated by a diagram Figure 62 P Q C E B D A 3 2 1 0 1 2 3 X2 X1 Figure 62 The constraints 626628 give rise to the lines AB BC and DBE respec tively Different objective values give rise to lines parallel to PQ showing that B represents the optimal solution The reason for the ambiguity in shadow price is that we have the accidental result of three constraints going through the same point Normally in two dimensions we would only expect two constraints to go through B We can therefore regard either one of the constraints 626 or 628 as nonbinding having a zero valuation in the presence of the other constraint This leads to the ambiguity in our shadow prices INTERPRETING LINEAR PROGRAMMING MODELS 121 In a situation such as this it would not be correct to increase the righthand side value of 3 in constraint 626 at all using the shadow price of 1 The upper range of this righthand side value with this shadow price will be 3 The coefficient is already at its upper range and any further increase will not alter the objective at the rate suggested by the valuation In these situations the shadow price is usually defined as being the rate of change of value of the objective function with respect to the righthand side value Then it can be shown that the upper shadow price for increases in right hand side value is the minimum of all the possible valuations and the lower shadow price for decreases in righthand side value is the maximum of all the possible valuations Defining y1 y2 and y3 as shadow prices we have upper shadow prices y1 0 y2 1 2 y3 0 lower shadow prices y1 1 y2 1 y3 1 2 When a computer package is used the shadow prices printed out are usually only the values of the dual variables corresponding to one dual solution They are therefore misleading if interpreted as shadow prices according to the above definition In fact to evaluate all the alternative dual solutions and therefore obtain the true shadow price would generally be computationally prohibitive It is however possible to evaluate ranges within which the valuations associated with any particular dual solution chosen are valid For example the upper shadow price on constraint 626 is not 1 If we take the dual solution i above then the associated upper range on the righthand side coefficient of 3 is also 3 showing there is no range within which this coefficient can be increased to reflect a rate ofchange increase of 1 in the objective This example is considered again in Section 63 when ranges are discussed A good discussion of the problem is given by Aucamp and Steinberg 1982 It should be pointed out that the two complications we have just discussed are dual situations On the one hand we had a unique dual solution leading to more than one primal solution The first phenomenon is referred to as alternative solutions in the primal model while the second phenomenon is referred to as degeneracy in the primal model 63 Sensitivity analysis and the stability of a model 631 Righthand side ranges In the last section we described how shadow prices can be useful in predicting the effect of small changes in the righthand side coefficients on the optimal value of the objective function It was pointed out however that this interpretation is only valid in a certain range In this case the range is known as a righthand side range We will again use the product mix problem of Section 12 to illustrate our discussion This gave the model Maximize 550x1 600x2 350x3 400x4 200x5 629 subject to 12x1 20x2 25x4 15x5 288 630 10x1 8x2 16x3 192 631 20x1 20x2 20x3 20x4 20x5 384 632 We saw that the optimal solution told us to make 12 of PROD 1 and 72 of PROD 2 and nothing of PROD 3 PROD 4 and PROD 5 This resulted in a profit of 10920 and exhausted the grinding capacity constraint 630 and manpower capacity constraint 632 The shadow prices on constraints 630632 turned out to be 625 0 and 2375 Increasing the grinding capacity by Δ hours per week would therefore result in a weekly increase in a total profit of 625 Δ Similarly cutting back on grinding capacity by Δ hours per week would result in a weekly decrease in the total profit of 625 Δ We are interested in the limits within which the change Δ can take place for this interpretation to apply For this example these ranges are as follows lower range 2304 upper range 384 that is we can increase grinding capacity by up to 96 hours a week or decrease it by up to 576 hours per week with the predicted effect on profit Changes outside these ranges are unpredictable in their effect and require further analysis The information that we have got here is however of some if limited usefulness It tells us for instance that adding one grinding machine would improve profit by 96 625 600 per week We could not however predict the effect on total profit of taking away a grinding machine as this would take us below the lower range of the grinding capacity It would nevertheless be correct to say that 600 is if anything an underestimate for the resultant decrease in profit It is important to restate that this interpretation of the effect on the objective of decreasing or increasing a righthand side coefficient is only valid if one coefficient is changed at a time within the permitted ranges The effect of changing more than one coefficient at a time as well as changes outside the permitted ranges are efficiently examined by parametric programming which is discussed in the next section It is beyond the scope of this book to describe how the righthand side ranges are calculated Most computer packages provide these ranges with their solution output For completeness we will give the upper and lower ranges on the other two capacities drilling and manpower The ranges on the drilling capacity of 192 hours per week are fairly trivial to obtain Remember that we are not using all our drilling capacity In fact we INTERPRETING LINEAR PROGRAMMING MODELS 123 have a slack capacity of 144 hours This immediately enables us to calculate the lower range by subtracting this figure from the existing capacity As we are not using all our drilling capacity increasing it without limit can have no effect on the solution The ranges are therefore lower range 1776 upper range Within these ranges there will be no change at all in the objective or optimal solution as the shadow price on the nonbinding drilling constraint is zero The ranges on the manpower capacity of 384 hours per week are again non trivial to calculate They turn out to be as follows lower range 288 upper range 4061 For changes within these ranges the objective changes by 2375 for each unit of change For example an extra worker 48 hours per week improves profit by 2375 48 1140 per week This might prove quite valuable information when compared with the fact that an extra grinding machine is worth only 600 per week The meaning of righthand side ranges is well illustrated geometrically using a twovariable example Maximize 3x1 2x2 633 subject to x1 x2 4 634 2x1 x2 5 635 x1 4x2 2 636 x1 x2 0 Geometrically we have the situation in Figure 63 The optimal solution is rep resented by the point A giving an objective value of 9 The shadow price on constraint 634 turns out to be 1 If constraint 634 is relaxed by increasing the righthand side coefficient from 4 to 45 we move the boundary AB of the feasible region to AB The resultant increase in the objective is 1 05 05 We can continue to increase the righthand side coefficient to 5 when this boundary of the feasible region shrinks to the point D There is clearly no virtue in further increasing the right hand side coefficient as the point D will continue to represent the optimal solution The upper range on the righthand side coefficient in constraint 634 therefore is obviously 5 If constraint 634 is tightened by decreasing the righthand side coefficient the boundary AB of the feasible region progressively moves down until 124 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 5 4 3 x2 x1 2 1 0 1 2 3 D B B E F A A C Figure 63 it becomes CE This happens when the righthand side coefficient is 3 Any decreases in this coefficient between 4 and 3 result in a degradation of the objective by 1 the shadow price for each unit of decrease The value 3 is the lower range For decreases in coefficient values below 3 the optimal objective value will decrease at a greater rate This rate will not immediately be predictable from our knowledge of the solution at A For decreases down to 3 the optimal objective value is represented by points on the line AC Decreases below 3 give points on the line CF So far we have only suggested using righthand side ranges together with shadow prices to investigate the effect which changes on the righthand side coefficients have on the optimal objective value Such investigations sometimes give rise to the expression postoptimal analysis Another purpose to which such information can be put is in investigating the sensitivity of the model to the righthand side data This forms part of a subject known as sensitivity analysis We will consider this new use to which righthand side ranging information can be put in relation to the product mix model Suppose we were rather doubtful about the accuracy of the grinding capacity of 288 hours per week constraint 630 In practice it is quite likely that such a figure would be open to doubt INTERPRETING LINEAR PROGRAMMING MODELS 125 as machines could well be down for maintenance or repair for durations which were not wholly predictable How confident should we be in suggesting that the factory dispenses with making PROD 3 PROD 4 and PROD 5 and only concentrates on PROD 1 and PROD 2 From our range information we can immediately say that this should remain the optimal policy as long as the true capacity figure does not lie outside the limits 2304384 Although the policy in terms of what is and is not made does not change within these limits the levels at which PROD 1 and PROD 2 are made and the resultant profit obviously will change This information is therefore of limited but nevertheless sometimes significant usefulness Obviously if there is doubt about one capacity figure there will probably also be doubt about others and ranging information can only strictly be applied when one change only is made Nevertheless such information does give some indication of the sensitivity of the solution to accuracy in the righthand side data If for example the ranges on the righthand side coefficient in constraint 630 had been very close say 2872885 we would have to be very careful in applying our suggested solution as it clearly would depend very critically on the accuracy of the grinding capacity figure It should be pointed out that the sensitivity analysis interpretation of the easily obtained ranges of 1776 and on the nonbinding drilling constraint is rather stronger As long as this capacity lies within these figures not only will our optimal solution be no different but the levels of the variables in that optimal solution will be unchanged as well Changing the righthand side coefficient on a binding constraint within the permitted ranges does of course alter the values of the variables in the optimal solution as well as the objective value The rates at which the values of the variables change are quite easily obtained using most package programs Such values are known as marginal rates of substitution and discussed below Before doing this however we will consider other types of ranging information to be obtained from a linear programming model 632 Objective ranges It is often useful to know the effects of changes in the objective coefficients on the optimal solution Suppose we change an objective coefficient eg a unit profit contribution or cost How will this affect the value of the objective function In an analogous manner to righthand side ranges we can define objective ranges If a single objective coefficient is changed within these ranges then the optimal solution values of the variables will not change although the optimal value of the objective may change This is a rather stronger result than in the righthand side ranging case where the solution values could change The result is not altogether intuitive If a particular item became more profitable or costly we might expect to bias our solution towards or away from that item In fact our solution will not change at all as long as we remain within the permitted ranges The situation is 126 MODEL BUILDING IN MATHEMATICAL PROGRAMMING well described geometrically by means of the twovariable model we used before Maximize 3x1 2x2 637 subject to x1 x2 4 638 2x1 x2 5 639 x1 4x2 2 640 x1 x2 0 This gives rise to Figure 64 The optimal solution x1 1 x2 3 objective 9 is represented by the point A The line PQ represents the points giving an objec tive value of 9 0 1 C Q P B A 2 3 4 1 2 x1 x2 3 Figure 64 Suppose that the objective coefficient 3 of variable x1 were to increase The steepness of the line PQ would increase in a negative sense but A would still represent the optimal solution as long as PQ did not rotate around A in a clockwise direction beyond the direction AC When the objective coefficient of x1 became 4 the objective line would coincide with the line AC This provides us with an upper range of 4 for this coefficient Similarly a lower range is provided INTERPRETING LINEAR PROGRAMMING MODELS 127 by decreasing this coefficient until the objective line coincides with AB This gives a lower range of 2 Note that as long as the objective coefficient of x1 remains within the range 24 the optimal solution values of the variables will be unchanged and represented by point A This is the slightly unintuitive result mentioned before The optimal objective value does of course change if we alter this coefficient within its ranges As x1 has a value of 1 in the optimal solution each unit of increase or decrease in its objective coefficient will obviously provide an increase or decrease of 1 in the objective value Ranges on the objective coefficient of 2 for x2 can be discovered in a similar manner We will now return to our product mix example and present the objective ranges for that problem To describe how these ranges are calculated in a problem which cannot be represented geometrically is beyond the scope of this book Most package programs do however provide this information Our main concern is with the correct interpretation rather than the derivation For PROD 1 the ranges on the objective coefficient are lower range 500 upper range 600 This means that if we vary the objective coefficient of PROD 1 within these limits the values of the variables in our optimal solution will not change We should still continue to produce 12 of PROD 1 72 of PROD 2 and nothing else As remarked before such a result is often difficult for people with little understanding to accept We might expect there to be a gradual and continuous shift towards making more of PROD 1 the more expensive we make it In fact the production plan should not alter at all until PROD 1 produces a unit profit contribution of more than 600 When this happens there will be a new optimal production plan almost certainly involving more of PROD 1 the investigation of which would require further analysis The same restrictions apply to the interpretation of objective ranges as applied in the case of righthand side ranges Our interpretation is only valid within the permitted ranges and if only one change is made in an objective coefficient This clearly limits the value of this information somewhat Parametric programming provides an efficient means of investigating the effects of more than one change possibly outside the ranges Although the optimal solution values of the variable do not change within the ranges the objective value obviously will For each 1 that we add to the price of PROD 1 we obviously improve the objective by 12 as the optimal solution involves producing 12 of PROD 1 Similarly objective ranges can be obtained for PROD 2 These turn out to be lower range 550 upper range 6833 128 MODEL BUILDING IN MATHEMATICAL PROGRAMMING For PROD 3 PROD 4 and PROD 5 which should not be made in the optimal production plan the derivation of objective ranges is simple We have already seen that PROD 3 must be increased in price by 125 its reduced cost before it is worth making This gives us its upper range As it is already too cheap to be worth making a reduction in price cannot affect the solution The objective ranges for PROD 3 are therefore lower range upper range 475 For PROD 4 we have lower range upper range 63125 For PROD 5 we have lower range upper range 56875 Objective ranges can be put to a similar use to that which righthand side ranges are put in sensitivity analysis If there were doubt surrounding the true profit contribution of 550 from PROD 1 we could be sure of our production plan as long as we knew that this figure was between 500 and 600 The width or narrowness of the objective ranges enables one to obtain a good feel for the sensitivity of the solution to changes or errors in the objective data In fact this interpretation is rather stronger than is the case with righthand side ranges Not only will the optimal production policy not change if a coefficient is altered within its permitted ranges the values of the variable quantities in that plan will not alter either 633 Ranges on interior coefficients It is also possible to obtain ranges for other coefficients in a linear program ming model Sometimes these may provide valuable information in much the same way that righthand side and objective ranges do Generally however such information is far less useful When a model is built often the only data which are changed are the objective and righthand side coefficients taken together these are sometimes referred to as the rim of the model In consequence many package programs do not provide the facility for obtaining ranges on coefficients other than in the rim When such information is obtainable its interpretation and use is very similar to that in the case of righthand side and objective ranges We must now consider the complication which we described in Section 62 when there was ambiguity concerning the valuations shadow prices of the con straints This ambiguity also extends to the ranging information Alternative sets of valuations corresponding to a unique optimal solution lead to alternative sets of INTERPRETING LINEAR PROGRAMMING MODELS 129 ranges We will repeat the small example we used before Maximize 3x1 2x2 641 subject to x1 x2 3 642 2x1 x2 4 643 4x1 3x2 10 644 x1 x2 0 There are a number of possible valuations for the constraints which give rise to the optimal solution We presented the following two where y1 y2 and y3 are the dual values on constraints 642644 respectively i y1 1 y2 1 y3 0 ii y1 0 y2 1 2 y3 1 2 i arises when we regard constraints 642 and 643 only as binding ii arises when we regard constraints 643 and 644 only as binding Table 61 gives the righthand side ranges within which the set i can be used as shadow prices Table 61 Constraint Lower Range Upper Range 642 2 3 643 3 4 644 10 Each range extends on one side of the righthand side coefficient only For example constraint 642 has an upper range of 3 indicating that it would be incorrect to use a shadow price of 1 on this constraint to predict the effect of increasing the righthand side coefficient It would however be correct to use this as a shadow price to predict the effect of decreasing this coefficient Similar arguments apply to the other two constraints For constraint 644 we can increase the righthand side coefficient of 10 indefinitely without affecting the optimal solution or objective value shadow price of 0 We cannot however decrease this coefficient at all without altering the objective This last result is obvious from studying Figure 62 The clue to changes in the righthand side coefficients in the other directions is given by the set of valuations ii together with their corresponding ranges These ranges are given in Table 62 For example increasing the righthand side coefficient of 3 has no effect shadow price of 0 on the objective whereas 130 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 62 Constraint Lower Range Upper Range 642 3 643 4 5 644 8 10 decreasing this coefficient does have an effect shadow price of 1 Similarly the second set of ranges on the other constraints are in opposite directions indicating the ranges of interpretation of the second set as shadow prices We have demonstrated how twovalued shadow prices can exist for constraints with different marginal values for decreasing or increasing a righthand side coefficient In solving a practical model using a package program the user will only be presented with one set of shadow prices corresponding to his or her optimal solution Which set of dual values this is will be very arbitrary and depend upon the manner in which the model was solved Solving the same model with exactly the same data could well result in a different set of dual values The clue to the limited onesided interpretation which can be placed on these shadow prices lies in the righthand side ranges If some of these ranges lie on one side of their corresponding righthand side coefficient only then this limits the shadow price to being interpreted for changes in one direction only To investigate the effects of changes in the opposite direction alternate sets of dual values and their corresponding ranges should be sought Alternatively parametric programming could be used as discussed in the next section It may well seem to the reader that a disproportionate amount of atten tion has been paid in both this and the last section to the apparently rather trivial complication concerning alternative valuations shadow prices for the constraints of a linear programming model As has already been pointed out however this complication is a very common occurrence in practical linear programming models It is well known in the computational side of linear pro gramming as the phenomenon of degeneracy We have here been concerned with the economic interpretations concerning degenerate solutions In view of the difficulty which many people have in understanding this we have given it considerable attention A further rather different final point should be made concerning the unique ness of the ranging information from a model The inclusion or exclusion of redundant constraints from a model will obviously not affect the optimal solu tion This could well however affect the values of the ranges Figure 63 and its associated model illustrate this point Constraint 636 gives rise to the edge CF of the feasible region The ignoring of constraint 636 would obviously not affect the optimal solution represented by the point A In this sense constraint 636 is redundant and might never have been modelled in the first place in a practical situation if its redundancy had been obvious The presence or absence of constraint 636 will however obviously affect the value of the lower righthand INTERPRETING LINEAR PROGRAMMING MODELS 131 side range on constraint 634 With constraint 636 present this lower range is 3 If constraint 636 were removed it would be 25 634 Marginal rates of substitution We have seen that there is a certain flexibility in altering a righthand side coef ficient of a model As long as we remain with the permitted range the effect on the objective value is governed for each unit of change by the shadow price of the constraint This change in the objective value occurs because the values of the variables in the optimal solution change The relative rates at which they change are given by the marginal rates of substitution It is beyond the scope of this book to describe how these figures can be calculated but they are obtainable from the solution output of most package programs A small geometric example will illustrate the meaning which should be attached to these figures Our example is the same as that used before Maximize 3x1 2x2 subject to x1 x2 4 645 2x1 x2 5 646 x1 4x2 2 647 x1 x2 0 The feasible region is represented by Figure 65 The optimal solution is repre sented by point A x1 1 x2 3 objective 9 We have already seen that the righthand side coefficient of 4 on constraint 645 can be changed within the range 35 causing the objective to change by 1 the shadow price on constraint 645 for each unit of change The values of the variables x1 and x2 will also obviously change Their rates of change per unit increase in this righthand side coefficient are x1 1 x2 2 that is x1 will get smaller by one and x2 will get larger by two for each unit of increase Decreasing the righthand side coefficient will obviously produce the opposite effect Such figures can be very valuable in many applications For example they might indicate how a production plan should alter as a certain resource becomes scarcer or more plentiful In a blending application they could indicate the rate at which one ingredient should be substituted for another as it became more plentiful 132 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 0 1 A 2 3 4 1 2 x1 x2 3 Figure 65 The same limitations apply to the interpretation of marginal rates of substitu tion as apply to shadow prices We can only consider one change at a time within the permitted ranges If a range is onesided we can only apply the marginal rates of substitution for changes in one direction A different set of marginal rates of substitution will exist although it will require more computation to derive for changes in the opposite direction The subjects of sensitivity and postoptimal analysis are fully covered in the papers of Greenberg 1993a 1993b 1993c 1994 635 Building stable models We have shown that quite a lot of useful information can be obtained to suggest how sensitive the optimal solution to a linear programming model is to changes or inaccuracies in the data In many practical situations stable solutions are more valuable than optimal solutions Having obtained an optimal solution to a model we might be very reluctant to change our operating plan if only small changes occurred in certain parameters for example capacities or costs To some extent we can incorporate this reluctance in our model Suppose we were to run out of a particular resource eg processing capacity raw material and manpower It might well be that in practice we would buy in more in this resource at a cost until this increased cost became prohibitive As was pointed out in Section 61 this situation can easily be modelled For example suppose the following constraint represented a resource limitation j aj xj b 648 The alternative representation j aj xj u b 649 would allow the resources availability to be expanded at a cost per unit of c if u was given this objective cost profit of c in the objective function Sometimes 649 is referred to as a soft constraint in contrast to the hard constraint 648 as 649 does not provide the total barrier that 648 does 64 Further investigations using a model The previous two sections described how a lot of useful information could be derived concerning the effects on the optimal solution of changes in the coefficients of a linear programming model Such information was however limited in value by the fact that i only one coefficient could be changed at a time and ii that coefficient could only be changed within certain ranges Parametric programming is a convenient method of investigating the effects of more radical changes It is beyond the scope of this book to describe how the method works There is virtue however in describing how such investigations can be organized and presented to a computer package Parametric programming is computationally quick and therefore provides a cheap means of determining how the optimal solution changes with alterations in objective and righthand side coefficients It is useful to illustrate the description again by means of a numerical example We will again use the product mix model introduced in Section 12 In the last section we saw how the righthand side coefficients could be varied individually within their permitted ranges Suppose that instead we wished to investigate the possibility of increasing both grinding capacity and manpower simultaneously Parametric programming on the righthand side allows us to carry out such an investigation as long as the coefficients increase or decrease by constant relative amounts For example we could investigate the effect of employing an extra worker on assembly for each new grinding machine bought Each increase of 96 hours per week in grinding capacity will therefore be accompanied by an increase of 48 hours per week in manpower capacity If θ new grinding machines are bought the righthand side vector of coefficients would become 288 192 384 θ 96 0 48 134 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Parametric programming allows one to specify a change column such as 96 0 48 and limits within which θ can vary The procedure will then allow θ to vary between these limits and calculate the value of the objective function at various intervals For example the parameter θ was allowed to vary between 0 the original capacities and 10 when the new capacities become 1248 192 and 864 respectively Such information can be very valuable in examining the effect of expan sions or contractions in the availability of different resources The result is well expressed graphically as is done for the example we have just discussed in Figure 66 25000 20000 15000 10000 5000 2 4 6 q 8 10 0 Total profit Figure 66 Analogous to parametric programming on the righthand side is paramet ric programming on the objective function For example in the product mix model if we wished to increase the prices of all products by the same amount simultaneously we would specify a change row 1 1 1 1 1 A parameter θ would be allowed to vary between specified limits adding θ times this row to the row of objective coefficients The effect on the optimal objective view would again be mapped out INTERPRETING LINEAR PROGRAMMING MODELS 135 65 Presentation of the solutions This chapter has described a lot of useful information which can be derived from linear programming models It is often easy to lose sight of the difficulty which nonspecialists may have in relating such information to the real life situation which has been modelled Obscure presentation of information derived from a linear programming model can lead to complete rejection of the technique Com puter output is often obscure as a package program is designed to solve any model no matter what the application is The output must obviously therefore relate to the mathematical aspects of the model One way of overcoming the dif ficulty is for an intermediate person possibly the model builder to convert the computer output into a written report which can easily be understood For mod els which are run frequently this can be a cumbersome and slow process Many organizations have computer programs which read the solution output from the linear programming package and convert it into a form which relates to the appli cation Computer programs which do this are known as report writers Clearly a report writer program must be related to the application As a consequence such programs are usually purpose written for the type of model solved Frequently they are written by the organization using the model rather than by the producer of the package A high level language is usually used It is necessary for the program to read the solution output from the package With most packages it is quite easy to do this by getting the package to write its output on to a file Many practitioners would argue in favour of such purposebuilt report writers There do however exist general purpose report writers which sometimes accompany particular packages Sometimes these report writers can be used satisfactorily for the application required Report writers are often considered in conjunction with matrix generators which were discussed in Section 35 A matrix generator relates the input for a linear programming model to the practical problem while a report writer so relates the output In order to show the advantage of a report writer the information is presented through a purposebuilt report writer for the application The output should be selfexplanatory 136 MODEL BUILDING IN MATHEMATICAL PROGRAMMING DATE 1 JANUARY 1999 OPTIMAL SOLUTION TO BLEND PROBLEM TOTAL REVENUE 17 593 MANUFACTURE 450 TONS OF PROD USE 159 TONS OF VEG1 USE 41 TONS OF VEG2 USE 250 TONS OF OIL2 HARDNESS OF PROD 60 UNITS PRICE REDUCTION NECESSARY BEFORE PURCHASE OIL1 12TON OIL3 8TON MARGINAL VALUE OF EXTRA REFINING CAPACITY VEGETABLE OILS 30TON NONVEGETABLE OILS 47TON 7 Nonlinear models 71 Typical applications It has already been pointed out that many mathematical programming models contain variables representing activities that compete for the use of limited resources For a linear programming model to be applicable the following must apply 1 there must be constant returns to scale 2 use of a resource by an activity is proportional to the level of the activity 3 the total use of a resource by a number of activities is the sum of the uses by the individual activities These conditions are clearly applied in the product mix example of Section 12 and the result was the linear programming model given there All the expressions in that model are linear Nowhere do we get expressions such as x2 1 x1x2 and log x1 Suppose however that the first of the above conditions did not apply Instead of each unit of PROD 1 produced contributing 550 to profit we suppose that the unit profit contribution depends on the quantity of PROD 1 produced If this unit profit contribution increases with the quantity produced we are said to have increasing returns to scale For decreases in the unit profit contribution there is said to be decreasing returns to scale These two situations together with the case of a constant return to scale are illustrated in Figures 7173 respectively In our product mix model each unit of PROD 1 produced contributes 550 to profit This gives rise to a term 550x1 in the objective function The whole objective function is made up of the sum of similar terms and is said to be linear As an example of increasing returns to scale we could suppose that the unit profit contribution depended on x1 and was 550 2x1 This would give Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 138 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Total profit contribution Quantity Figure 71 Total profit contribution Quantity Figure 72 rise to the nonlinear term 2x2 1 in the objective function We would now have a nonlinear model although the constraints would still be linear expressions In practice the nonlinear term might not be known explicitly and might simply be represented graphically as shown in Figure 71 or 72 As an example of decreasing returns to scale we could suppose that the unit profit contribution was 5501 x1 This would give rise to the nonlinear term 550x11 x1 in the objective function The above two cases of nonlinearities in the objective function arose through profit margins being affected by increasing or decreasing unit costs arising from increased production In a similar way unit profit margins will be altered if unit selling price is affected by the volume of production It frequently happens in practice that the unit price which a product can command increases with the NONLINEAR MODELS 139 Total profit contribution Quantity Figure 73 demand for it In this case it may be convenient to treat unit selling price as a variable to be determined by the model For example if the quantity to be produced is x the unit cost is c and the unit selling price is px which depends on x we have profit contribution p x c x p x x cx The term pxx introduces a nonlinearity into the objective function An obvi ous first approximation is to take px as a linear function of x If this is done quadratic terms are introduced into the objective function resulting in a quadratic programming model An example of such a model results from the AGRICULTURAL PRICING problem of Part II A sometimes rather more accurate way to reflect the relationship between px and x is through a price elasticity We have the following definition of the price elasticity of demand for a good x Ex percentage change in quantity of x demanded percentage change in p x It is possible to regard E x as constant for the range of values of x and px considered The above definition then leads to p x k x1Ex where k is the price above which the demand is reduced to unity For a particular volume of production x px gives the unit price at which this level will exactly balance the demand The resultant nonlinear term in the objective function will be p x x kx11Ex 140 MODEL BUILDING IN MATHEMATICAL PROGRAMMING A way in which price elasticities are incorporated in a nonlinear programming model of the British National Health Service is described by McDonald et al 1974 So far we have described how mathematical programming models with non linear objective functions can arise Nonlinear terms also sometimes arise in the constraints of a mathematical programming model For example in a blending problem such as that described in Example 12 a quality such as hardness might not depend linearly on the ingredient propor tions Restrictions on such qualities would then give rise to nonlinear constraints One type of model that has received a certain amount of attention where nonlinearities occur in the constraints is the geometric programming type of model Here the nonlinear expressions are all polynomials For example we might have the following constraint x1x2x3 2x2 2x3 32 Such models do arise often in engineering problems They are however fairly rare in managerial applications A full treatment of the subject is given by Duffin et al 1968 Nonlinear programming models are usually far more difficult to solve than correspondingly sized linear models It is however often possible to solve an approximation to the model through either linear programming or an extension of linear programming known as separable programming Which case is applicable depends upon an important classification of nonlinear models into convex and nonconvex problems This distinction is explained in the next section 72 Local and global optima Nonlinear programming can be divided usefully into convex programming and nonconvex programming Definition A region of space is said to be convex if the portion of the straight line between any two points in the region also lies in the region For example the area shaded in Figure 74 is a convex region in two dimensional space On the other hand the area shaded in Figure 75 is a non convex region since A and B are both points within the region yet the line AB between them does not lie entirely in the region Definition A function f x is said to be convex if the set of points x y where y f x forms a convex region NONLINEAR MODELS 141 Figure 74 A B Figure 75 3 2 1 1 2 3 0 x y 9 8 6 4 3 1 5 2 7 Figure 76 For example the function x2 is convex as is demonstrated in Figure 76 because the shaded area is a convex region On the other hand the function 2x2 is nonconvex as is demonstrated in Figure 77 It should be intuitive that the concepts of convex and nonconvex regions and functions apply in as many dimensions as required Definition A mathematical programming model is said to be convex if it involves the minimization of a convex function over a convex feasible region 142 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1 1 2 3 4 5 6 7 1 0 y x 1 3 2 2 3 3 2 Figure 77 Clearly minimizing a convex function is equivalent to maximizing the nega tion of a convex function Such a maximization problem will also therefore be convex Example 71 A Convex Programming Model Minimize x2 1 4x1 2x2 subject to x1 x2 4 2x1 x2 5 x1 4x2 2 x1 x2 0 The function x2 1 4x1 2x2 is represented in Figure 78 and easily seen to be convex This model is represented geometrically in Figure 79 with different objective values represented by the curved lines which arise from contours of the surface in Figure 78 Clearly the optimal solution is represented by point A It should be obvious that linear programming LP is a special case of convex programming All LP models can be expressed as minimizations if necessary of linear functions A linear function obviously satisfies the definition of a convex function Moreover the feasible region defined by a set of linear constraints can easily be shown to be convex NONLINEAR MODELS 143 Function X2 X1 Figure 78 5 4 3 2 1 0 1 2 x1 3 4 x2 A Objective 11 10 9 Figure 79 144 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Example 72 A Nonconvex Programming Model Minimize 4x3 1 3x1 6x2 subject to x1 x2 4 2x1 x2 5 x1 4x2 2 x1 x2 0 The function 4x3 1 3x1 6x2 is represented in Figure 710 and seen to be nonconvex although of course the feasible region of the model in Figure 79 is still convex Function x2 x1 Figure 710 The model is represented geometrically in Figure 711 Again different objective values give rise to the curved lines which are con tours of the surface in Figure 710 Clearly the optimal solution is at C For a larger problem however we would not have the geometrical intuition which we have here As far as many algorithms including the separable program ming algorithm are concerned the point A would also appear as an optimal solution At A the curved line representing this objective value of 19 devi ates away from the feasible region in both directions This would be taken as evidence in a convex including linear programming model that A were the optimal solution Figure 79 demonstrates how the objective contours for a con vex problem curve away from the feasible region For our nonconvex example however Figure 711 demonstrates how the objective contours curve in towards NONLINEAR MODELS 145 Objective 32 Objective 24 Objective 19 6 5 4 3 2 1 0 1 x1 2 B A C x2 Figure 711 the feasible region and may reenter it This in fact happens in the example here The fact that the objective contour passing through A deviates away from the feasible region at A in both directions cannot therefore be taken as evidence that it will not reenter it In fact as we can see it is possible to move out to a better smaller objective contour passing through B Even then we can move out to a still better objective contour passing through C Points A and B repre sent what are known as local optima Many computational procedures such as separable programming can guarantee no more than local optima Clearly what is really required is a true global optimum such as that represented by point C in Figure 711 A fairly graphic way of describing the situation is to consider the problem of a team of mountaineers on a range of mountains in a thick fog It is easy for them to determine when they are at a local optimum that is a moun tain peak The ground will begin to drop in height whichever way they walk This does not guarantee however that there is not another higher mountain peak hidden in the fog The situation of the mountaineers is similar to that of many nonlinear programming algorithms The possibility of local optima arising from nonconvex programming models when certain algorithms are used is what makes such models much 146 MODEL BUILDING IN MATHEMATICAL PROGRAMMING more difficult to solve than convex programming models With a convex programming model any optimum found must be a global optimum To find a guaranteed global optimum to a nonconvex model requires more sophisticated algorithms than the separable programming algorithm described in the next section A satisfactory though often computationally expensive way of tackling such problems is through integer programming as described in Section 93 A good illustration of the distinction between convex and nonconvex pro gramming models is given by the nonlinear programming models mentioned in Section 71 which arise when there are diseconomies of scale and economies of scale The former type of model is convex and the latter nonconvex To see why this is so we first consider the type of cost function that arises when we have diseconomies of scale This is illustrated in Figure 712 If x1 and x2 represent quantities of two products to be made diseconomies of scale will result in the total cost rising faster and faster with increasing values of x1 and x2 The result is clearly a convex cost function to be minimized Total cost X2 X1 Figure 712 For the case of economies of scale we have the situation represented in Figure 713 x1 and x2 again represent quantities of two products to be made As x1 and x2 increase decreasing unit costs result in the total cost rising less and less quickly The result is clearly a nonconvex cost function to be minimized NONLINEAR MODELS 147 Total cost X2 X1 Figure 713 73 Separable programming Definition A separable function is a function that can be expressed as the sum of functions of a single variable For example the function x2 1 2x2 ex3 is separable because each of terms x2 1 2x2 and ex3 is a function of a single variable On the other hand the function x1x2 x2 1 x1 x3 is not separable because the terms x1x2 and x21 x1 are each functions of more than one variable The importance of separable functions in a mathematical programming model lies in the fact that they can be approximated to by piecewise linear functions It is then possible to use separable programming to obtain a global optimum to a convex problem or possibly only a local optimum for a nonconvex problem Although the class of separable functions might seem to be a rather restrictive one it is often possible to convert mathematical programming models with nonseparable functions into ones with only separable functions Ways in 148 MODEL BUILDING IN MATHEMATICAL PROGRAMMING which this may be done are discussed in Section 74 In this way a surprisingly wide class of nonlinear programming problems can be converted into separable programming models In order to convert a nonlinear programming model into a suitable form for separable programming it is necessary to make piecewise linear approximations to each of the nonlinear functions of a single variable As will become apparent it does not matter whether the nonlinearities are in the objective function or the constraints or both To illustrate the procedure we will consider the nonlinear model given in Example 71 The only nonlinear term occurring in the model is x2 1 A piecewise linear approximation to this function is illustrated in Figure 714 It is easy to see that x1 can never exceed 25 from the second constraint of the problem The piecewise linear approximation to x2 1 need therefore only be considered for values of x1 between 0 and 25 The curve between 0 and C has been divided into three straight line portions This inevitably introduces some inaccuracy into the problem For example when x1 is 15 the transformed model will regard x2 1 as 25 instead of 225 Such inaccuracy can obviously be reduced by a more refined grid involving more straight line portions For our purpose however we will content ourselves with the grid indicated in Figure 714 If such an inaccuracy is considered serious one approach would be to take the value of x obtained from the optimal solution refine the grid in the neighbourhood of this value and reoptimize Some package programs allow one to do this automatically a number of times all within one computer run 7 6 5 4 3 2 1 0 y x1 2 05 1 15 x1 2 25 A B C Figure 714 Our aim is to eliminate the nonlinear term x12 from our model We can do this by replacing it by the single linear term y It is now possible to relate y to x1 by the following relationships x1 0λ1 1λ2 2λ3 25λ4 71 y 0λ1 1λ2 4λ3 625λ4 72 λ1 λ2 λ3 λ4 1 73 The λi are new variables which we introduce into the model They can be interpreted as weights to be attached to the vertices 0 A B and C It is however necessary to add another stipulation regarding the λi at most two adjacent λi can be nonzero 74 The stipulation 74 guarantees that corresponding values of x1 and y lie on one of the straight line segments 0A AB or BC For example if λ2 05 and λ3 05 other λi being zero we could get x1 15 and y 25 Clearly ignoring stipulation 74 would incorrectly allow the possibility of values x1 and y off the piecewise straight line 0ABC Equations 7173 give rise to constraints that can be added to our original model Example 71 The term x12 is replaced by y This results in the following model Minimize y 4x1 2x2 subject to x1 x2 4 2x1 x2 5 x1 4x2 2 x1 λ2 2λ3 25λ4 0 y λ2 4λ3 625λ4 0 λ1 λ2 λ3 λ4 1 y x1 x2 λ1 λ2 λ3 λ4 0 It is important to remember that stipulation 74 must apply to the set of variables λi A solution in which for example we had λ1 13 and λ3 23 would not be acceptable because this results in the wrong relationship between x1 and y x12 In general stipulation 74 cannot be modelled using linear programming constraints It can however be regarded as a logical condition on the variables λi and be modelled using integer programming This is described in Section 93 Fortunately in our example here no difficulty arises over stipulation 74 This is because the original model was convex Suppose for example we were to take as the set of values λ1 05 λ2 025 λ3 025 and λ4 0 This clearly breaks stipulation 74 From the relations 71 and 72 it clearly leads to the point x1 075 and y 125 in Figure 714 This is above the piecewise straight line As our objective involves minimizing y x12 we would expect to get a better solution by taking x1 075 and y 075 when we drop on to the piecewise straight line In view of the convex shape of the graph we cannot obtain values 150 MODEL BUILDING IN MATHEMATICAL PROGRAMMING for the λi which give us points below the piecewise straight line Therefore we must always obtain corresponding values of x1 and y which lie on one of the line segments by virtue of optimality Stipulation 74 is therefore guaranteed in this case We can therefore solve our transformed model by linear programming and obtain a satisfactory optimal solution It is not even necessary to resort to the separable extension of the simplex algorithm that is discussed below This happens however only because our problem is convex For a nonconvex problem stipulation 74 would generally not be satisfied automatically In order to guarantee that it be satisfied we could resort to the sepa rable programming modification of the simplex algorithm In order to demonstrate the difficulty that a nonconvex problem presents we will make a piecewise lin ear approximation to the nonlinear term x3 1 in the nonconvex model of Example 72 This is demonstrated in Figure 715 16 12 8 4 0 05 1 15 2 25 y x1 3 x1 A B C D Figure 715 This gives us the relationships x1 0λ1 05λ2 1λ3 15λ4 25λ5 y 0λ1 0125λ2 1λ3 3375λ4 15625λ5 As before the λi variables can be interpreted as weights attached to the vertices in Figure 715 The λi variables must again satisfy the stipulation that at most two adja cent λi are nonzero This time this stipulation is not automatically guaranteed by optimality Suppose for example that we were to obtain the set of values λ2 04 λ3 05 and λ4 01 This would give x1 085 and y 13375 The point with these coordinates lies above the piecewise line in Figure 715 As our objective function to be minimized is dominated by the term 4x3 1 our NONLINEAR MODELS 151 optimization will tend to maximize y This will tend to take us away from the piecewise line rather than down on to it In this nonconvex case it is therefore necessary to use an algorithm which does not allow more than two adjacent λi to be nonzero The separable programming extension of the simplex algorithm due to Miller 1963 never allows more than two adjacent λi into the solution as a result it restricts corresponding values of xi and y to the coordinates of points lying on the desired piecewise straight line Unfortunately with nonconvex problems such as Example 72 which was modelled above restricting the values of λi to at most two adjacent ones being nonzero does not guarantee any more than a local optimum We could easily end up at point A or B in Figure 711 rather than C In both our examples the nonlinear functions of a single variable appeared in the objective function Should such nonlinear functions also appear in the constraints the analysis is just the same They are replaced by linear terms and a piecewise linear approximation made to the nonlinear function Variables λi are then introduced in order to relate the new term to the old It may not always be easy to decide whether a problem is convex or not For known convex problems such as problems that are linear apart from dis economies of scale consisting only of separable functions piecewise linear approximations are all that are needed It is not necessary here to use the separa ble programming algorithm For problems that are nonconvex such as problems with economies of scale or where it is not known whether or not they are con vex linear programming is not sufficient Separable programming can be used but no more than a local optimum can be guaranteed It is often possible to solve such a model a number of times using different strategies to obtain different local optima The best of these may then have some chance of being a global optimum Such computational considerations are however beyond the scope of this book but are sometimes described in manuals associated with particular packages The only really satisfactory way of being sure of avoiding local optima when a prob lem is not known to be convex is to resort to integer programming which is generally much more costly in computer time This is discussed in Sections 92 and 93 Before the end of this section an alternative way of modelling a piecewise linear approximation to a separable function will be described The formulation method just described is usually known as the λform for separable programming where variables λi are interpreted as weights attached to the vertices in the piece wise straight line There is an alternative formulation known as the δform In order to demonstrate the δform we reconsider the piecewise linear approxima tion to the function y x2 1 shown in Figure 74 This approximation is redrawn in Figure 716 Variables δ1 δ2 and δ3 are introduced to represent proportions of the intervals 0P PQ and QR which are used to make up the value of x1 We then get x1 δ1 δ2 05δ3 75 152 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 7 6 5 4 3 2 1 0 05 1 15 2 25 U T S A B C P Q R y x1 2 x1 Figure 716 where 0 δ1 δ2 δ3 1 As 0P and PQ are each of length 1 the coefficients of δ1 and δ2 in Equation 75 are 1 The coefficient of δ3 is 05 reflecting the length of the interval QR Similarly y δ1 3δ2 225δ3 76 where the coefficients of δ1 δ2 and δ3 are now the lengths of the intervals 0S ST and TU In order to ensure that x1 and y are the coordinates of points on the piecewise lines 0C we must make the following stipulation if any δi is nonzero all the preceding δi must take the value 1 and all the succeeding δi must take the value 0 77 Stipulation 77 clearly ensures that x1 and y truly represent distances along their respective axes It can be shown that the relaxation associated with the λformulation is more constrained than that associated with the δformulation This is the relaxation obtained by dropping the logical conditions 74 and 77 on the λ and δ variables as opposed to the LP relaxation obtained by relaxing the IP formula tions see Section 83 making it generally easier to solve This was pointed out early on by Williams 1985 When the extra logical conditions are imposed by NONLINEAR MODELS 153 integer variables as described in Section 93 the λformulation can be improved in a way described there 74 Converting a problem to a separable model The restriction of only allowing nonlinear functions to be separable might seem to impose a severe limitation on the class of problems that can be tackled by separable programming Rather surprisingly it is possible to convert a very large class of nonlinear programming problems into separable models When nonseparable functions occur in a model it is often possible to transform the model into one with only separable functions A very common nonseparable function that occurs in practice is the product of two or more variables For example if a term such as x 1 x 2 occurs the model is not immediately a separable one because this is a nonlinear function of more than one variable The model is easily converted into a separable form however by the following transformation 1 introduce two new variables u 1 and u 2 into the model 2 relate u 1 and u 2 to x 1 and x 2 by the relations u112x1x278 u212x1x279 3 replace the term x 1 x 2 in the model by u21u22 It is easy to see by elementary algebra that u21u22 is the same as the product x 1 x 2 as long as 78 and 79 are added to the model in the form of equality constraints The model now contains nonlinear functions u21 and u22 of single variables and is therefore separable These nonlinear terms can be dealt with by piecewise linear approximations It is important to remember that u 2 might need to take negative values When the possible ranges of values for u 1 and u 2 are considered it may be necessary to either translate u 2 by an appropriate amount or treat it as a free variable A free variable in linear programming is one which is not restricted to nonnegative values Should the product of more than two variables occur in a model as might happen for example in a geometric programming model then the above procedure could be repeated successively to reduce the model to a separable form An alternative way of dealing with product terms in a nonlinear model is to use logarithms We will again consider the example where a product x 1 x 2 of two variables occurs in a model although the method clearly generalizes to larger products The following transformation can be carried out 154 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1 replace x1x2 by a new variable y 2 relate y to x1 and x2 by the equation log y log x1 log x2 710 Equation 710 gives a nonlinear equality constraint to be added to the model The expression in this constraint is however separable as we only have non linear functions log y log x1 and log x2 of single variables Care must be taken however when this transformation is made to ensure that neither x1 nor x2 and consequently y ever take the value 0 If this were to happen their logarithms would become It may be necessary to translate x1 and x2 by certain amounts to avoid this ever happening The use of logarithms to convert a nonlinear model to a suitable form does however sometimes lead to computational problems Beale 1975 suggests that this can happen if both a variable and its logarithm occur together in the same model If these are likely to be of widely different orders of magnitude numerical inaccuracy may lead to computational difficulties Nonlinear functions of more than one variable such as product terms can be dealt with in yet another way by generalizing a piecewise linear approximation to more dimensions A more complex relationship then has to be specified between the λi As with nonconvex separable models this is only really satisfactorily dealt with through integer programming and is described in Section 93 Many other nonlinear functions of more than one variable can be reduced to nonlinear functions of a single variable by the addition of extra variables and constraints Ingenuity is often required but the range of nonlinear programming problems that can be made separable in this way is vast Such transformations do however often greatly increase the size of a model and hence the time to solve it 8 Integer programming 81 Introduction A surprisingly wide class of practical problems can be modelled using integer variables and linear constraints Sometimes such a model consists solely of integer variables That is a pure integer programming PIP model More commonly there are both conventional continuous variables together with integer variables present Such a model is said to be a mixed integer programming MIP model The wide applicability of integer programming IP sometimes known as discrete programming as a method of modelling is not obvious Clearly we can think of situations where it is only meaningful to make integral quantities of certain goods for example cars aeroplanes or houses or use integral quantities of some resource for example employees In these cases we might use an IP model instead of an LP model Although such obvious applications of IP do occur they are not common In fact in such situations it is often more desirable to use conventional LP and round off the optimal solution values to the nearest integers The obvious type of application described above obscures the real power of IP as a method of modelling Most practical IP models restrict the integer variables to two values 0 or 1 Such 01 variables are used to represent yes or no decisions Logical connections between such decisions can often be imposed using linear constraints Such methods of modelling are described in the next chapters Before discussing the building of IP models something must be said about the way in which they are solved Mathematically IP models involve many times as much calculation in solution as similarsized LP models The difficulty of integer problems compared with continuous problems involving real or rational numbers is well known in other branches of mathematics While an LP model involving thousands of constraints and variables can almost certainly be solved in a reasonable amount of time using a modern computer and package program Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 156 MODEL BUILDING IN MATHEMATICAL PROGRAMMING a similar situation does not hold for IP models There is a considerable danger in building an IP model only to find no way of solving it in a reasonable time Ways of avoiding this unfortunate and embarrassing experience are described in the next two chapters In view of this danger as well as the computational difficulty of IP Section 83 is devoted to methods of solving IP models This section purposely outlines in any detail only the most successful method of solving IP models the branch and bound method It is felt necessary that a model builder should have at least this minimum level of understanding as he or she can often influence the exact way in which the calculation proceeds to great advantage Moreover this most successful method to date of solving IP models receives surprisingly little attention in theoretical mathematical programming books possibly because of its lack of mathematical sophistication 82 The applicability of integer programming The purpose of this section is to classify loosely the different types of problem for which IP models may be built Inevitably there is a certain amount of overlap in this classification where particular applications do not fit neatly into any category Many practical problems will combine aspects from a number of categories By this loose classification it is hoped to convey a feeling for the type of problem to which IP is applicable In some ways the name discrete programming conveys what this sort of problem is rather better One of the reasons why IP has not been applied anywhere near as widely as it might to practical situations is the failure to recognize when a problem can be cast in this mould This section is purposely rather superficial Little indication is given of the way in which particular problems may be formulated as IP models This is left to the next chapter or in particular cases to the detailed formulations of Part III of this book A precise definition of those situations which can be formulated by IP models is given by Meyer 1975 and Jeroslow 1987 821 Problems with discrete inputs and outputs This class of problems includes the most obvious IP applications mentioned earlier where it is only possible to make whole numbers of products or use integral units of a resource Economists sometimes refer to such problems as having lumpy inputs or outputs To see why IP is sometimes necessary here consider the following very small and contrived model Maximize x1 x2 subject to 2x1 2x2 1 8x1 10x2 13 x1 x2 0 INTEGER PROGRAMMING 157 If the variables represent quantities of two different goods to be made it is important to be clear if these outputs should be integral for example represent indivisible goods such as aeroplanes Should this not be the case and they repre sent divisible goods such as gallons of beer we would be content with treating the problem as an LP model and taking the continuous optimum which is x1 4 x2 41 2 81 On the other hand restricting the variables to be integer forces us to accept the integer optimum which is x1 1 x2 2 82 It is very difficult to see how one could arrive at the solution 82 from the continuous optimum 81 Rounding the values in 81 to the nearest integers gives an infeasible solution In some circumstances such as this it is therefore necessary to solve such a problem as an IP model This is obviously likely to happen when the values of the variables will be small say less than 5 For most problems of this type however the values of the variables are likely to be much larger than this and the errors involved in rounding the LP fractional solution will not be serious Indeed solving such a problem as an IP model could well take a great deal of time in view of the many combinations of integer solutions that could be considered An extreme case of this was once seen where a national agricultural IP model was built On examination this model was found to be an IP model only because there were an integral number of cows hens pigs etc in the country considered Similar considerations to those above apply to problems where the inputs rather than or as well as the outputs are discrete Frequently such an input will be manpower capacity which if sufficiently large may be treated as continuous infinitely divisible An application that is quite common occurs when an input usually a pro cessing capacity or a resource only occurs at certain discrete values Apart from this the model may be a conventional LP model For example processing capac ity might be measured in machine hours per week By buying extra machines it might be possible to expand processing capacity It will however only be correct to allow this capacity to expand in steps equal to the machine hours per week resulting from extra whole machines IP can be used to model this type of situation A related situation to this is presented in the FACTORY PLANNING 2 problem in Part II Another particular type of problem in this category where IP must be used is the knapsack problem This is an IP problem with a single constraint A particular instance where it might arise is in stocking a warehouse The problem is as follows given a limited warehouse capacity stock the warehouse with goods of different volumes to maximize some objective such as the total value of goods in the warehouse It will generally not be possible to use the LP solution to this 158 MODEL BUILDING IN MATHEMATICAL PROGRAMMING problem which is trivial and simply involves stacking the warehouse to capacity with the most valuable good per unit volume thereby ignoring the discrete nature of the goods Extensions of the knapsack problem arise where extra simple upper bounding constraints also apply to the variables of the problem It is again usually necessary to use IP rather than LP to obtain a meaningful solution Knapsack problems do arise in practice but they occur most commonly as subproblems that have to be solved as part of a much larger LP or IP problem An example of this arises in the cutting stock problem discussed in Section 96 822 Problems with logical conditions It frequently happens that it is desired to impose extra conditions on an LP model These conditions are sometimes of a logical nature that cannot be modelled by conventional LP For example an LP model might be used to decide how much to make of each of the possible products in a factory subject to capacity limitations the product mix application It might be desirable to add an extra condition such as If product A is made then product B or C must be made as well By introducing some extra integer variables into the model together with extra constraints conditions such as this easily can be imposed The resultant model is a mixed integer problem Any such logical condition as that above can be imposed on an LP model using IP This is illustrated in the FOOD MANUFACTURE 2 problem In addition that problem also illustrates another common use of IP to extend an LP model limiting the number of ingredients in a blend The correct formulation of logical conditions sometimes involves considerable ingenuity and can be done in a number of different ways Methods of approaching such a formulation systematically are described in Chapter 9 Williams 2009 and Hooker 2000 cover the relationship between logic and IP at length 823 Combinatorial problems Many operational research problems have the characteristic of a very large num ber of feasible solutions often an astronomical number arising from different orders of carrying out operations or of allocating items or people to different positions Such problems are loosely referred to as combinatorial It is use ful to further subdivide this category into sequencing problems and allocation problems A particularly difficult type of sequencing problem arises in jobshop scheduling where an optimal ordering of operations on different machines in a jobshop is desired IP gives a method of modelling this type of situation There are a number of possible formulations Unfortunately IP has not proved a very successful way of tackling this problem to date Another very wellknown sequencing problem is the travelling salesman problem This is the problem of finding the optimal order in which to visit a set of cities and return home covering a minimum distance There are other problems which take the same form For example the problem of sequencing operations on a machine so as to minimize total setup cost takes this form INTEGER PROGRAMMING 159 Obviously the setup cost for an operation will depend on the preceding oper ation and can be regarded as the distance between operations The travelling salesman problem is again a very difficult type of problem for which different IP formulations have been attempted A very straightforward allocation problem is given in Part II This is the MARKET SHARE problem The problem involves allocating customers to divi sions in a company for their services Although the formulation is comparatively straightforward problems of this type are not always easy to solve Problems of a very similar form arise in project selection and capital budget allocation The class of allocation problems includes two problems already mentioned for which IP is not needed It has been pointed out in Section 53 that in the transportation problem it is not necessary to impose an integrality requirement The LP optimal solution will automatically be integer because of the structure of the problem As the assignment problem can be regarded as a special case of the transportation problem this property holds for it also Fortunately these two problems although apparently IP problems can therefore be treated as LP problems and solved fairly easily Other apparently IP problems also have this property or can be formulated to have this property with great computational advantage This topic is treated further in Sections 101 and 102 A complicated extension of the assignment problem is the quadratic assignment problem This problem occurs where the cost of an assignment is not independent of other assignments The resulting problem can be regarded as an assignment problem with a quadratic objective function The quadratic terms can be converted into lin ear expressions reducing the problem to an IP problem The quadratic assignment problem is one of the most difficult combinatorial problems known in mathemat ical programming Fairly small problems can be tackled by reducing the problem to a linear IP model The DECENTRALIZATION example of Part II is a spe cial sort of quadratic assignment problem The quadratic assignment problem is discussed further in Section 95 A practical problem that arises and can be regarded as falling into the allo cation category is the assembly line balancing problem This is the problem of assigning workers to tasks on a production line to achieve a given production rate It is possible to formulate this problem as an IP problem and solve it fairly easily The usual formulation results in a special sort of IP model known as a set partitioning problem This type of problem is discussed further in Section 95 Another set partitioning problem is the aircrew scheduling problem This is the problem of assigning aircrews to sets of flights rotations or rosters In prac tice this type of problem frequently involves an enormous number of potential rosters and is difficult to solve as an IP problem in consequence This is discussed further in Section 95 Problems of logical design involving switching circuits or logical gates can be tackled through IP Unfortunately such problems often turn out to be very large when formulated in this way and so difficult to solve A small problem of this kind LOGICAL DESIGN is given in Part II 160 MODEL BUILDING IN MATHEMATICAL PROGRAMMING The political districting problem is also a set partitioning problem when regarded as an IP problem This is the problem of designing constituencies or electoral districts in order to equalize as near as possible political representation IP has been used in the United States to solve practical problems of this nature A fairly common application of IP is to the depot location problem This is the problem of deciding where to locate depots or warehouses or even factories in order to supply customers Two types of costs may enter the problem the capital costs of building the depots and the distribution costs resulting from particular sites of the depots This type of problem can be modelled using IP Frequently the resultant model is a mixed integer problem The DEPOT LOCATION problem of Part II is an example 824 Nonlinear problems As was mentioned in Chapter 7 nonlinear problems can sometimes be treated as IP problems with advantage If the problem can be expressed in a separable pro gramming form it can be solved using either separable programming as described in Section 73 or by IP Should the problem be convex this term is explained in Section 72 it may be treated by LP and no difficulty arises On the other hand special methods have to be used for nonconvex problems where separable programming has the disadvantage of producing possibly local optima this is again explained in Section 72 IP overcomes this difficulty and produces a true global optimal solution although possibly with considerably more expenditure in computer time Problems to which this method is relevant have already been mentioned in Chapter 7 They include problems involving economies of scale quadratic programming problems and geometric programming problems as well as much more general nonlinear programming problems The way in which such problems can be converted into a separable form is described in Chapter 7 How to convert the resultant separable problems into an IP form is described in Chapter 9 There is however an alternative way of approaching such problems by IP This is through the use of special ordered sets of variables A number of package programs have facilities for dealing with IP problems in this way to considerable computational advantage The concept of special ordered sets and how they may be applied to nonlinear problems as well as other types of problem is described by Beale and Tomlin 1969 A very common application of IP is the fixed charge problem This occurs when the cost of an activity involves a threshold setup cost as well as the usual costs that rise in proportion to the level of the activity In this way the problem can be regarded as nonlinear For example if it is decided to produce any amount of a product at all it may be necessary to set up a piece of machinery This setup cost is independent of the quantity produced It is not possible to model this situation using conventional LP but it can be modelled very easily using IP This is described in Example 94 INTEGER PROGRAMMING 161 825 Network problems Many problems in operational research involve networks A lot of these problems can be modelled using LP or IP Those problems which give rise to LP models have already been considered in Section 53 It has already been pointed out in Section 53 that the problem of finding the critical path in a PERT network can be viewed as an LP problem A secondary problem often arises in practice This is the problem of resource allocation on a PERT network It may be necessary to alter the order in which certain activities arcs are carried out in view of the limited resources available for example we cannot simultaneously build the walls and lay the floors in a house if there are not enough workers available The problem of optimally allocating these limited resources to the arcs of the PERT network so as to for example minimize the total completion time for a project can be formulated as an IP problem Although an IP model is a method of tackling this problem it is not to be recommended except in very simple cases The computational difficulties of solving a complex problem of this kind by IP can be very great In practice nonrigorous heuristic means are used to obtain useful but nonoptimal solutions Many IP problems arise in graph theory A wellknown problem to which IP is relevant is the fourcolour problem It has recently been proved that at most four colours are needed to colour every country on a map differently from its neighbouring countries For any map IP could be used to find the mini mum number of colours necessary The above problem can be represented as the problem of colouring the vertices of a graph so that vertices jointed by an edge are coloured differently Other colouring problems arise in graph theory For example it is possible to devise problems involving colouring the edges of a graph Although many such problems exist and can be solved using IP such considerations are largely beyond the scope of this book The concern here is mainly with practical problems Graph theory and IP has been extensively treated elsewhere for example by Christofides 1975 The above five categories encompass most of the different types of problem which arise to which IP is applicable In practice most problems that one meets fall into the second category They are LP problems on which it is desired to impose extra conditions These extra conditions are frequently of a logical type One therefore extends the LP model by adding integer variables and extra constraints The extra constraints applied to the integer variables sometimes have a combinatorial flavour Sometimes these extra constraints serve the purpose of modelling nonlinearities in an otherwise LP model PIP models arise less frequently in practice They are usually combinato rial problems The comparatively large number of combinatorial problems listed above should not disguise the fact that the majority of practical IP models are in the second category and may arise as extensions to almost any application of LP Combinatorial problems do arise in practice however and are sometimes 162 MODEL BUILDING IN MATHEMATICAL PROGRAMMING satisfactorily solved through IP Great care must be exercised however when applying IP to such problems While IP offers an apparently attractive way of modelling a combinatorial problem experience has shown that such models can be very difficult to solve It is often desirable to experiment with smallscale versions of the problem before embarking on a large model There are also good and bad ways of formulating such problems from the point of view of ease of solution this is discussed in Section 101 Also it is hoped that the comparative solution times given with the solutions to the models in Part IV will be indicators of the difficulty of certain types of model 83 Solving integer programming models This section is in no way intended to be a full description of IP algorithms A much fuller description is given in Williams 1993 Instead it is an attempt to indicate different ways in which IP models may be solved and to suggest how a model builder may use existing packages in an efficient manner Some references to fuller expositions of the algorithmic side of IP are given The main approaches to solving IP problems are categorized below Unlike LP with the simplex algorithm no one good IP algorithm has emerged Different algorithms prove better with different types of problem often by exploiting the structure of special classes of problem It seems unlikely that a universal IP algo rithm will ever emerge If it did it would open up the possibility of solving a very wide class of problems Some of these problems such as the travelling salesman problem and the quadratic assignment problem have defied many attempts to find powerful algorithms for their solution There is now even some theoretical evidence resulting from the theory of computational complexity to suggest that a universal IP algorithm is unlikely The most successful algorithm so far found to solve practical general IP problems is the branch and bound method described below Considering its apparent crudeness the success of this method is surpris ing Almost all commercial packages offering an MIP facility use the algorithm In fact the algorithm is little more than an approach to solving IP problems There is great flexibility in the way it can be used This is one of the reasons why it is briefly described in a book on model building Using the branch and bound method in a way suited to the problem can show dramatic improvements over less intelligent strategies Most methods of solving IP problems fall into one of four broad categories There is some overlap between the categories and some particularly successful approaches to large problems have exploited features of a number of methods 831 Cutting planes methods These methods can be applied to general MIP problems They usually start by solving an IP problem as if it were an LP problem by dropping the inte grality requirements known as the LP relaxation If the resultant LP solution INTEGER PROGRAMMING 163 the continuous optimum is integer this solution will also be an integer optimum otherwise extra constraints cutting planes are systematically added to the prob lem further constraining it The new solution to the further constrained problem may or may not be an integer By continuing the process until an integer solution is found or the problem is shown to be infeasible the IP problem can be solved Although cutting plane methods may appear mathematically fairly elegant they have not proved very successful on large problems although combined with Branch and Bound methods they can prove very powerful The original method of this type is described by Gomory 1958 Further references are given with the exposition in Chapter 5 of Garfinkel and Nemhauser 1972 832 Enumerative methods These are generally applied to the special class of 01 PIP problems In theory there are only a finite though extremely large number of possible solutions to a problem in this class Although it would be prohibitive to examine all these possibilities by use of a tree search it is possible to examine only some solutions and systematically rule out many others as being infeasible or nonoptimal Such methods together with their variants and extensions have proved very successful with certain types of problem and not very successful on others Commercial package programs do exist for such methods but are not widely used The best known of these methods is Balass additive algorithm described by Balas 1965 Other methods are given by Geoffrion 1969 A good overall exposition is given in Chapter 4 of Garfinkel and Nemhauser 1972 833 PseudoBoolean methods Attempts have been made to exploit the obvious analogy between Boolean alge bra and 01 PIP problems A number of algorithms have been developed As with other algorithms they work well on some types of problem but less well on others This approach is entirely unlike any other for solving IP problems The constraints of a problem are expressed not as equations or inequalities but through Boolean algebra In some cases this can give a very concise statement of the constraints but in others it is large and unwieldy As far as the author is aware no commercial packages capable of accepting practical problems use any of these methods These approaches have been developed by Hammer and are described in Hammer and Rudeanu 1968 Granot and Hammer 1972 and Hammer and Peled 1972 It should not be thought that the specialized 01 PIP algorithms are only of relevance to 01 PIP problems Methods have been developed for partitioning a MIP problem with 01 variables into its continuous and integer portions It then becomes necessary at stages in the optimization to solve a 164 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 01 PIP problem as a subproblem Clearly these specialized algorithms might be applicable Any discussion of this topic is beyond the scope of this book The main reference to this partitioning method is Benders 1962 834 Branch and bound methods It is these methods that have proved most successful in general on practical MIP problems Such methods are also sometimes classed as enumerative but we choose to distinguish them from the enumerative methods described earlier As with cutting plane methods the IP problem is first solved as an LP problem by relaxing the integrality conditions If the resultant solution the continuous optimum is an integer the problem is solved otherwise we perform a tree search Full details are given in Williams 1993 A very good discussion of efficient solution strategies to use with the branch and bound method on practical models is given by Forrest et al 1974 Geoffrion and Marsten 1972 put the branch and bound method together with enumeration methods which also use a tree search into a general framework that makes the basic principles easy to understand References to the various forms of the branch and bound method are given in that paper More recent incorporation of Cutting Plane methods into Branch and Bound Branch and Cut has proved very powerful A comprehensive survey of IP packages is given by Land and Powell 1979 9 Building integer programming models I 91 The uses of discrete variables When integer variables are used in a mathematical programming model they may serve a number of purposes These are described below 911 Indivisible discrete quantities This is the obvious use mentioned at the beginning of Chapter 8 where we wish to use a variable to represent a quantity that can only come in whole numbers such as aeroplanes cars houses or people 912 Decision variables Variables are frequently used in integer programming IP to indicate which of a number of possible decisions should be made Usually these variables can only take the two values zero or one Such variables are known as zeroone 01 or binary variables For example δ 1 indicates that a depot should be built and δ 0 indicates that a depot should not be built We usually adopt the convention of using the Greek letter δ for 0 1 variables and reserve Latin letters for continuous real or rational variables It is easy to ensure that a variable which is also specified to be integer can only take the two values 0 or 1 by giving the variable a simple upper bound SUB of 1 All variables are assumed to have a simple lower bound of 0 unless it is stated to the contrary Although decision variables are usually 0 1 they need not always be For example we might have γ 0 indicates that no depot should be built γ 1 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 166 MODEL BUILDING IN MATHEMATICAL PROGRAMMING indicates that a depot of type A should be built γ 2 indicates that a depot of type B should be built 913 Indicator variables When extra conditions are imposed on a linear programming LP model 0 1 variables are usually introduced and linked to some of the continuous variables in the problem to indicate certain states For example suppose that x represents the quantity of an ingredient to be included in a blend We may well wish to use an indicator variable δ to distinguish between the state where x 0 and the state where x 0 By introducing the following constraint we can force δ to take the value 1 when x 0 x Mδ 0 91 where M is a constant coefficient representing a known upper bound for x Logically we have achieved the condition x 0 δ 1 92 where stands for implies In many applications Equation 92 provides a sufficient link between x and δ eg Example 91 below There are applications eg Example 92 below however where we also wish to impose the condition x 0 δ 0 93 Equation 93 is another way of saying δ 1 x 0 94 Together Equations 92 and 93 or Equation 94 impose the condition δ 1 x 0 95 where stands for if and only if It is not possible totally to represent Equation 93 or Equation 94 by a constraint On reflection this is not surprising Equation 94 gives the condition if δ 1 the ingredient represented by x must appear in the blend Would we really want to distinguish in practice between no usage of the ingredient and say one molecule of the ingredient It would be much more realistic to define some threshold level m below which we regard the ingredient as unused Equation 94 can now be rewritten as δ 1 x m 96 This condition can be imposed by the constraint x mδ 0 97 BUILDING INTEGER PROGRAMMING MODELS I 167 Example 91 The Fixed Charge Problem x represents the quantity of a product to be manufactured at a marginal cost of C 1 per unit In addition if the product is manufactured at all there is a setup cost of C 2 The position is summarized as follows x 0 total cost 0 x 0 total cost C1x C2 The situation can be represented graphically as in Figure 91 C2 C1 Total cost 0 1 2 Quantity manufactured x 3 4 Figure 91 Clearly the total cost is not a linear function of x It is not even a continuous function as there is a discontinuity at the origin Conventional LP is not capable of handling this situation In order to use IP we introduce an indicator variable δ so that if any of the product is manufactured δ 1 This can be achieved by constraint 91 above The variable δ is given a cost of C 2 in the objective function giving the following expression for the total cost Total cost C1x C2δ By the introduction of 01 variables such as δ and extra constraints such as 91 to link these variables to the continuous variables such as x fixed charges can be introduced into a model if the δ variables are given objective coefficients equal to the fixed charges 168 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is worth pointing out that this is a situation where it is not generally necessary to model the condition 93 This condition will automatically be satisfied at optimality if the objective of the model has the effect of minimizing cost Although a solution x 0 δ 1 does not violate the constraints it is clearly nonoptimal as x 0 δ 0 will not violate the constraints either but will result in a smaller total cost It would certainly not be invalid to impose condition 93 explicitly by a constraint such as in Equation 96 as long as m was sufficiently small In certain circumstances it might even be computationally desirable Example 92 Blending This example is relevant to the FOOD MANUFACTURE 2 problem in Part II xA represents the proportion of ingredient A to be included in a blend xB represents the proportion of ingredient B to be included in a blend In addition to the conventional quality constraints for which LP can be used connecting these and other variables in the model it is wished to impose the following extra condition if A is included in the blend B must be included also IP must be used to model this extra condition A 01 indicator variable δ is introduced which will take the value 1 if xA 0 This is linked to variable xA by the following constraint of type 14 xA δ 0 98 Here the coefficient M of constraint 91 can conveniently be taken as 1 as we are dealing with proportions We are now in a position to use the new 01 variable δ to impose the condition δ 1 xB 0 99 In order to impose this condition of Equation 94 we must choose some proportionate level m say 1100 below which we regard B as out of the blend This gives us the following constraint xB 001δ 0 910 We have now imposed the extra condition on the LP model by introducing a 01 variable δ with two extra constraints 98 and 910 Notice that here unlike Example 91 it was necessary to introduce a con straint to represent a condition of type 94 The satisfaction of such a condition could not be guaranteed by optimality An extension of the extra condition that we have imposed might be the following if A is included in the blend B must be included also and vice versa This requires two extra constraints that the reader might like to formulate BUILDING INTEGER PROGRAMMING MODELS I 169 It should be pointed out that any constant coefficient M can be chosen in constraints of type in Equation 91 so long as M is sufficiently big not to restrict the value of x to an extent not desired in the problem being modelled In practical situations it is usually possible to specify such a value for M Although theoretically any sufficiently large value of M will suffice there is computational advantage in making M as realistic as possible This point is explained further in Section 101 of Chapter 10 Similar considerations apply to the coefficient m in constraints of type in Inequality 97 It is possible to use indicator variables in a similar way to show whether an inequality holds or does not hold First suppose that we wish to indicate whether the following inequality holds by means of an indicator variable δ j a j x j b The following condition is fairly straightforward to formulate We therefore model it first δ 1 j a j x j b911 Equation 911 can be represented by the constraint j a j x j M δ M b912 where M is an upper bound for the expression j a j x j b It is easy to verify that Inequality 912 has the desired effect that is when δ 1 the original constraint is forced to hold and when δ 0 no constraint is implied A convenient way of constructing Inequality 912 from condition 911 is to pursue the following train of reasoning If δ 1 we wish to have j a j x j b 0 that is if 1δ 0 we wish to have j a j x j b 0 This condition is imposed if j a j x j b M 1 δ where M is a sufficiently large number In order to find how large M must be we consider the case δ 0 giving j a j x j b M This shows that we must choose M sufficiently large that this does not give an undesired constraint Clearly M must be chosen to be an upper bound for the expression j a j x j b Rearranging the constraint we have obtained with the variables on the left we obtain Inequality 912 We now consider how to model the reverse of constraint 911 that is j a j x j b δ 1913 170 MODEL BUILDING IN MATHEMATICAL PROGRAMMING This is conveniently expressed as δ 0 j a j x j b914 that is δ 0 j a j x j b915 In dealing with the expression j a j x j b we run into the same difficulties that we met with the expression x 0 We must rewrite j a j x j b as j a j x j b ε where ε is some small tolerance value beyond which we will regard the constraint as having been broken Should the coefficients a j be integers as well as the variables x j as often happens in this type of situation there is no difficulty as ε can be taken as 1 Equation 915 may now be rewritten as δ 0 j a j x j b ε 0916 Using an argument similar to that above we can represent this condition by the constraint j a j x j m ε δ b ε917 where m is a lower bound for the expression j a j x j b Should we wish to indicate whether a inequality such as j a j x j b holds or not by means of an indicator variable δ the required constraints can easily be obtained by transforming the above constraint into a form The corresponding constraints to Inequalities 912 and 917 above are j a j x j mδ m b918 j a j x j M ε δ b ε919 BUILDING INTEGER PROGRAMMING MODELS I 171 where m and M are again lower and upper bounds respectively on the expression j aj xj bi Finally to use an indicator variable δ for an constraint such as j aj xj b is slightly more complicated We can use δ 1 to indicate that the and cases hold simultaneously This is done by stating both the constraints 912 and 918 together If δ 0 we want to force either the or the constraint to be broken This may be done by expressing Inequalities 917919 with two variables δ and δ giving j aj xj m ε δ b ε 920 j aj xj M ε δ b ε 921 The indicator variable δ forces the required condition by the extra constraint δ δ δ 1 922 In some circumstances we wish to impose a condition of the type 911 Alternatively we may wish to impose a condition of the type 913 or impose both conditions together These conditions can be dealt with by the linear constraints 912 or 917 taken individually or together Example 93 Use a 01 variable δ to indicate whether or not the following constraint is satisfied 2x1 3x2 1 x1 and x2 are nonnegative continuous variables that cannot exceed 1 We wish to impose the following conditions δ 1 2x1 3x2 1 923 2x1 3x2 1 δ 1 924 Using Inequality 912 M may be taken as 4 2 3 1 This gives the following constraint representation of condition 923 2x1 3x2 4δ 5 925 172 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Using Inequality 917 m may be taken as 1 0 0 1 We take ε as 01 This gives the following constraint representation of Equation 924 2x1 3x2 11δ 11 926 The reader should verify that Inequalities 925 and 926 have the desired effect by substituting 0 and 1 for δ In all the constraints derived in this section it is computationally desirable to make m and M as realistic as possible 92 Logical conditions and 01 variables In Section 91 it was pointed out that 01 variables are often introduced into an LP or sometimes an IP model as decision variables or indicator variables Having introduced such variables it is then possible to represent logical connec tions between different decisions or states by linear constraints involving these 01 variables It is at first sight rather surprising that so many different types of logical condition can be imposed in this way Some typical examples of logical conditions that can be so modelled are given below Further examples are given by Williams 1977 1 If no depot is sited here then it will not be possible to supply any of the customers from the depot 2 If the librarys subscription to this journal is cancelled then we must retain at least one subscription to another journal in this class 3 If we manufacture product A we must also manufacture product B or at least one of products C and D 4 If this station is closed then both branch lines terminating at the station must also be closed 5 No more than five of the ingredients in this class may be included in the blend at any one time 6 If we do not place an electronic module in this position then no wires can connect into this position 7 Either operation A must be finished before operation B starts or vice versa It will be convenient to use some notation from Boolean algebra in this section This is the socalled set of connectives given below means or this is inclusive ie A or B or both means and BUILDING INTEGER PROGRAMMING MODELS I 173 means not means implies or if then means if and only if These connectives are used to connect propositions denoted by P Q R etc x 0 x 0 δ 1 etc For example if P stands for the proposition I will miss the bus and Q stands for the proposition I will be late then P Q stands for the proposition If I miss the bus then I will be late P stands for the proposition I will not miss the bus As another example suppose that X i stands for the proposition Ingredient i is in the blend i ranges over the ingredients A B and C Then X A X B X C stands for the proposition If ingredient A is in the blend then ingredient B or C or both must also be in the blend This expression could also be written as X A X B X A X C It is possible to define all these connectives in terms of a subset of them For example they can all be defined in terms of the set Such a subset is known as a complete set of connectives We do not choose to do this and will retain the flexibility of using all the connectives listed above It is important how ever to realize that certain expressions are equivalent to expressions involving other connectives We give all the equivalences below that are sufficient for our purpose To avoid unnecessary brackets we consider the symbols and as each being more binding than their successor when written in this order For example P Q R can be written as P Q R P Q R can be written as P Q R P is the same as P 927 P Q is the same as P Q 928 P Q R is the same as P Q P R 929 P Q R is the same as P Q P R 930 P Q R is the same as P R Q R 931 P Q R is the same as P R Q R 932 P Q is the same as P Q 933 P Q is the same as P Q 934 Expressions 933 and 934 are sometimes known as De Morgans laws Although Boolean algebra provides a convenient means of expressing and manipulating logical relationships our purpose here is to express these rela tionships in terms of the familiar equations and inequalities of mathematical 174 MODEL BUILDING IN MATHEMATICAL PROGRAMMING programming In one sense we are performing the opposite process to that used in the pseudoBoolean approach to 01 programming mentioned in Section 83 We will suppose that indicator variables have already been introduced in the manner described in Section 91 to represent the decisions or states that we want logically to relate It is important to distinguish propositions and variables at this stage We use Xi to stand for the proposition δi 1 where δi is a 01 indicator variable The following propositions and constraints can easily be seen to be equivalent X1 X2 is equivalent to δ1 δ2 1 935 X1 X2 is equivalent to δ1 1 δ2 1 936 X1 is equivalent to δ1 0 or 1 δ1 1 937 X1 X2 is equivalent to δ1 δ2 0 938 X1 X2 is equivalent to δ1 δ2 0 939 To illustrate the conversion of a logical condition into a constraint we consider an example Example 94 Manufacturing If either of products A or B or both are manufactured then at least one of products C D or E must also be manufactured Let Xi stand for the proposition Product i is manufactured i is A B C D or E We wish to impose the logical condition XA XB XC XD XE 940 Indicator variables are introduced to perform the following functions δi 1 if and only if product i is manufactured δ 1 if the proposition XA XB holds The proposition XA XB can be represented by the inequality δA δB 1 941 The proposition XC XD XE can be represented by the inequality δC δD δE 1 942 Firstly we wish to impose the following condition δA δB 1 δ 1 943 Using Inequality 919 of Section 91 we impose this condition by the constraint δA δB 2δ 0 944 BUILDING INTEGER PROGRAMMING MODELS I 175 Secondly we wish to impose the condition δ 1 δC δD δE 1 945 Using Inequality 918 of Section 91 this is achieved by the constraint δC δD δE δ 0 946 Hence the required extra condition can be imposed on the original model LP or IP by the following 1 Introduce 01 variables δA δB δC δD and δE and link them to the original probably continuous variables by constraints of type 91 and 97 of Section 91 It is not strictly necessary to include constraints of type 97 for the variables δA and δB as it is not necessary to have the conditions 94 of Section 91 in these cases 2 Add the additional constraints 944 and 946 above This is not the only way to model this logical condition Using the Boolean identity 932 above it is possible to show that condition 940 can be reexpressed as XA XC XD XE XB XC XD XE 947 The reader should verify that an analysis similar to that above results in the constraint 946 together with the following two constraints in place of Inequality 944 δA δ 0 948 δB δ 0 949 Both ways of modelling the condition are correct There are computational advantages in Inequalities 948 and 949 over 944 This is discussed further in Section 101 of Chapter 10 It is sometimes suggested that polynomial expressions in 01 variables are useful for expressing logical conditions Such polynomial expressions can always be replaced by linear expressions with linear constraints possibly with a considerable increase in the number of 01 variables For example the constraint δ1 δ2 0 950 represents the condition δ1 0 δ2 0 951 More generally if a product term such as δ1 δ2 were to appear anywhere in a model the model could be made linear by the following steps 176 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1 Replace δ1δ2 by a 01 variable δ3 2 Impose the logical condition δ3 1 δ1 1 δ2 1 952 by means of the extra constraints δ1 δ3 0 δ2 δ3 0 953 δ1 δ2 δ3 1 An example of the need to linearize products of 01 variables in this way arises in the DECENTRALIZATION problem in Part III Products involving more than two variables can be progressively reduced to single variables in a similar manner A major defect of the above formulation is that if there are n original 01 variables there can be up to nn 1 new 01 variables and 3nn 1 extra constraints A much more compact formulation is due to Glover 1975 Sup pose we consider all the terms involving δ1 in an expression and group them together for example δ12δ2 3δ3 δ4 We represent this expression by a new continuous variable w Then the relations between w and the δi are as follows δ1 1 w 2δ2 3δ3 δ4 δ1 0 w 0 We model this by w 5δ1 2δ2 3δ3 δ4 5 w 2δ2 3δ3 δ4 w 5δ1 It should be noted however that if there relatively few connections between the variables that is each variable forms few products with more than one other variable there may be little saving in the size of model It is even possible to linearize terms involving a product of a 01 variable with a continuous variable For example the term xδ where x is continuous and δ is 01 can be treated in the following way 1 Replace xδ by a continuous variable y 2 Impose the logical conditions δ 0 y 0 δ 1 y x 954 BUILDING INTEGER PROGRAMMING MODELS I 177 by the extra constraints y Mδ 0 x y 0 x y Mδ M 955 where M is an upper bound for x and hence also y Other nonlinear expressions such as ratios of polynomials involving 01 variables can also be made linear in similar ways Such expressions tend to occur fairly rarely and are not therefore considered further They do however provide interesting problems of logical formulation using the principles described in this section and can provide useful exercises for the reader The purpose of this section together with Example 94 has been to demonstrate a method of imposing logical conditions on a model This is by no means the only way of approaching this kind of modelling Different ruleofthumb methods exist for imposing the desired conditions Experienced modellers may feel that they could derive the constraints described here by easier methods It has been the authors experience however that the following are true 1 Many people are unaware of the possibility of modelling logical conditions with 01 variables 2 Among those people who realize that this is possible many find themselves unable to capture the required restrictions by 01 variables with logical constraints 3 It is very easy to model a restriction incorrectly By using concepts from Boolean algebra and approaching the modelling in the above manner it should be possible satisfactorily to impose the desired logical conditions A system for automating the formulation of logical conditions within standard predicates and implementing them within the language PROLOG is described by McKinnon and Williams 1989 Logical conditions are sometimes expressed within a constraint logic pro gramming language as discussed in Section 24 The tightest way this concept is explained in Section 83 of expressing certain examples using linear program ming constraints is described by Hooker and Yan 1999 and Williams and Yan 2001 There is a close relationship between logic and 01 integer programming which is explained in Williams 1995 2009 Williams and Brailsford 1999 and Chandru and Hooker 1999 93 Special ordered sets of variables Two very common types of restriction arise in mathematical programming prob lems for which the concept special ordered set of type 1 SOS1 and special 178 MODEL BUILDING IN MATHEMATICAL PROGRAMMING ordered set of type 2 SOS2 have been developed This concept is due to Beale and Tomlin 1969 An SOS1 is a set of variables continuous or integer within which exactly one variable must be nonzero An SOS2 is a set of variables within which at most two can be nonzero The two variables must be adjacent in the ordering given to the set It is perfectly possible to model the restrictions that a set of variables belongs to an SOS1 set or an SOS2 set using integer variables and constraints The way in which this can be done is described below There is great computational advantage to be gained however from treating these restrictions algorithmically The way in which the branch and bound algorithm can be modified to deal with SOS1 and SOS2 sets is beyond the scope of this book It is described in Beale and Tomlin Some examples are given on how SOS1 sets and SOS2 sets can arise Example 95 Depot Siting A depot can be sited at any one of the positions A B C D or E Only one depot can be built If 01 indicator variables δi are used to perform the following purpose δi 1 if and only if the depot is sited at i i is A B C D or E then the set of variables δ1 δ2 δ3 δ4 δ5 can be regarded as an SOS1 set The SOS1 condition together with the constraint δ1 δ2 δ3 δ4 δ5 1 956 guarantees integrality and it is not necessary to stipulate that the δi be integral Only if the sites have a natural ordering is there great advantage to be gained in the SOS formulation Example 96 Capacity Extension The capacity C of a plant can be extended in discrete amounts by increasing levels of investment I If the set of variables δ0 δ1 δ2 δ3 δ4 δ5 is regarded as an SOS1 set then we can model C C1δ1 C2δ2 C3δ3 C4δ4 C5δ5 957 I I1δ1 I2δ2 I3δ3 I4δ4 I5δ5 958 δ0 δ1 δ2 δ3 δ4 δ5 1 959 It is not necessary to treat the δi as integer variables as the SOS1 condition together with Equation 959 forces integrality Conceptually it is important to regard an SOS set as an entity We can then regard C as a quantity that is a discrete function of I This can be regarded as a generalization of a 01 variable to more than two discrete values Such a generalization is often more useful than the conventional general integer variable Although the most common application of SOS1 sets is to modelling what would otherwise be 01 integer variables with a constraint such as in Equation 959 there are other applications The most common application of SOS2 sets is to modelling nonlinear functions as described by the following example Example 97 Nonlinear Functions In Section 73 the concept of a separable set was introduced in order to make a piecewise linear approximation to a nonlinear function of a single variable Using the λconvention for such a separable formulation we obtained the following convexity constraint λ1 λ2 λn 1 960 In addition in order that the coordinates of x and y should lie on the piecewise linear curve in Figure 715 it was necessary to impose the following extra restriction At most two adjacent λs can be nonzero 961 Instead of approaching this restriction through separable programming with the danger of local rather than global optima as described in Section 72 we can use an SOS2 The restriction in 961 need not be modelled explicitly Instead we can say that the set of variables λ1 λ2 λn is an SOS2 The formulation of a nonlinear function in Example 97 demands that the nonlinear function be separable that is the sum of nonlinear functions of a single variable It was demonstrated in Section 74 how models with nonseparable functions may sometimes be converted into models where the nonlinearities are all functions of a single variable While this is often possible it can be cumbersome increasing the size of the model considerably as well as the computational difficulty An alternative is to extend the concept of an SOS set to that of a chain of linked SOS sets This has been done by Beale 1980 The idea is best illustrated by a further example Example 98 Nonlinear Functions of Two or More Variables Suppose z gx y is a nonlinear function of x and y We define a grid of values of x y not necessarily equidistant and associate nonnegative weightings λij with each point in the grid as shown in Figure 92 If the values of x y at the grid points are denoted by Xs Yk we can approximate the function z gx y by means of the following relations x s k Xsλsk 962 y s k Ykλsk 963 z s k g Xs Yk λsk 964 s k λsk 1 965 In addition it is necessary to impose the following restriction on the λ variables At most four neighbouring λs can be non zero 966 This last condition is clearly a generalization of an SOS2 set We can impose condition 966 in the following way Let ξs k λsk ηk s λsk for all s k ξ1 ξ2 ξ3 and η1 η2 η3 with each taken as SOS2 sets The SOS2 condition for the first set allows λs to be nonzero in at most two neighbouring rows in Figure 92 For the second set the SOS2 condition allows λs to be nonzero in at most two neighbouring columns For example we might have ξ213 ξ323 η514 η634 y x λ14 λ24 λ34 λ44 λ54 λ64 λ74 λ13 λ23 λ33 λ43 λ53 λ63 λ73 λ12 λ22 λ32 λ42 λ52 λ62 λ72 λ11 λ21 λ31 λ41 λ51 λ61 λ71 Figure 92 The values of ξ and η above could arise from λ2516 λ2616 λ35112 λ36712 all other λs being zero They could however also arise from other values of the λs for example λ2514 λ26112 λ3623 with all other λs being zero In order to get round this nonuniqueness we can restrict the nonzero λs to vertices of a triangle such as in the second instance above A lengthy way of doing this is to impose the extra constraints ζt s λsts 967 and treat the ζt as a further SOS2 set BUILDING INTEGER PROGRAMMING MODELS I 181 If however we are content to restrict the x or y to grid values ie not interpolate in that direction then the problem does not arise Indeed we can also avoid introducing the sets ξ s so long as within each set λsk with the same s the member of which is nonzero has the same index k The sets λsk are then known as a chain of linked SOS sets as described by Beale 1980 and the restriction can be dealt with algorithmically Some of the problems presented in Part II can be formulated to take advan tage of special ordered sets In particular DECENTRALIZATION and LOGIC DESIGN can exploit SOS1 While it is desirable to treat SOS sets algorithmically if this facility exists in the package being used the restrictions that they imply can be imposed using 01 variables and linear constraints This is now demonstrated Suppose x1 x2 xn is an SOS1 set If the variables are not 01 we introduce 01 indicator variables δ1 δ2 δn and link them to the xi variables in the conventional way by constraints xi Miδi 0 i 1 2 n 968 xi miδi 0 i 1 2 n 969 where M i and mi are constant coefficients being upper and lower bounds respec tively for xi The following constraint is then imposed on the δi variables δ1 δ2 δn 1 970 If the xi variables are 01 we can immediately regard them as the δi variables above and only need impose the constraint 970 To model an SOS2 set using 01 variables is more complicated Suppose λ1 λ2 λn is an SOS2 set We introduce 01 variables δ1 δ2 δn1 together with the following constraints λ1 δ1 0 λ2 δ1 δ2 0 λ3 δ2 δ3 0 λn1 δn2 δn1 0 λn δn1 0 971 and δ1 δ2 δn1 1 972 This formulation suggests the relationship between SOS1 and SOS2 sets as Equation 972 could be dispensed with by regarding the δi as belonging to an SOS1 set as long as the δi each have an upper bound of 1 An improvement to the IP modelling of the λformulation has been proposed by Sherali 2001 The formulation gives a tighter LP relaxation and also allows the modelling of discontinuous piecewise linear expressions such as that illustrated in Figure 93 In order to define the function uniquely at the point of discontinuity we must make it upper or lower semicontinuous at the point of discontinuity that is define the left or righthand interval to be open at this point y b1 b4 b2 b3 b0 λ2 λ3 λ5 λ4 λ1 a0 a1 a2 a3 x Figure 93 In order to model a piecewise linear function we associate two weighting variables λL λR which must sum to 1 with each end of each section Then we stipulate that exactly one pair of weights is positive confining the function to exactly one of the sections The IP formulation of the function illustrated in Figure 93 is δ1 δ2 δ3 1 δi 01 973 λ1 λ2 δ1 λ3 λ4 δ2 λ4 λ5 δ3 974 x a0λ1 a1 λ2 λ3 a2λ4 a3λ5 975 y b0λ1 b1λ2 b2λ3 b3λ4 b4λ5 976 94 Extra conditions applied to linear programming models As the majority of practical applications of IP give rise to mixed integer programming models where extra conditions have been applied to an otherwise LP model this subject is considered further in this section A number of the commonest applications are briefly outlined BUILDING INTEGER PROGRAMMING MODELS I 183 941 Disjunctive constraints Suppose that for an LP problem we do not require all the constraints to hold simultaneously We do however require at least one subset of constraints to hold This could be stated as R₁ R₂ Rₙ 977 where Rᵢ is the proposition The constraints in subset i are satisfied and constraints 1 2 N form the subset in question Expression 977 is known as a disjunction of constraints Following the principles of Section 91 we introduce N indicator variables δᵢ to indicate whether the Rᵢ are satisfied In this case it is only necessary to impose the conditions δᵢ 1 Rᵢ 978 This may be done by constraints of type 912 or 918 Section 91 taken singly or together according to whether Rᵢ are or constraints We can then impose condition 977 by the constraint δ₁ δ₂ δₙ 1 979 An alternative formulation of Expression 977 is possible This is discussed in Section 102 and is due to Jeroslow and Lowe 1984 who report promising computational results in Jeroslow and Lowe 1985 A generalization of Expression 977 that can arise is the condition At least k of R₁ R₂ Rₙ must be satisfied 980 This is modelled in a similar way but using the constraint below in place of Inequality 979 δ₁ δ₂ δₙ k 981 A variation of expression 980 is the condition At most k of R₁ R₂ Rₙ must be satisfied 982 To model Expression 982 using indicator variables δᵢ it is only necessary to impose the conditions Rᵢ δᵢ 1 983 This may be done by constraints of type 917 or 919 Section 91 taken together or singly according to whether Rᵢ is a or constraint Condition 982 can then be imposed by the constraint δ₁ δ₂ δₙ k 984 Disjunctions of constraints involve the logical connective or and necessitate IP models 184 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is worth pointing out that the connective and can obviously be coped with through conventional LP as a conjunction of constraints simply involves a series of constraints holding simultaneously In this sense one can regard and as corresponding to LP and or as corresponding to IP 942 Nonconvex regions As an application of disjunctive constraints we show how restrictions correspond ing to a nonconvex region may be imposed using IP It is well known that the feasible region of an LP model is convex Convexity is defined in Section 72 There are circumstances however in nonlinear programming problems where we wish to have a nonconvex feasible region For example we consider the feasible region ABCDEFGO of Figure 94 This is a nonconvex region bounded by a series of straight lines Such a region may have arisen through the problem considered or represent a piecewise linear approximation to a nonconvex region bounded by curves 3 A B C D E G J F 4 2 1 0 1 2 3 4 5 H K x2 x1 Figure 94 We may conveniently think of the region ABCDEFGO as made up of the union of the three convex regions ABJO ODH and KFGO as shown in Figure 94 The fact these regions overlap does not matter Region ABJO is defined by the constraints x2 3 x1 x2 4 985 Region ODH is defined by the constraints x1 x2 0 3x1 x2 8 986 BUILDING INTEGER PROGRAMMING MODELS I 185 Region KFGO is defined by the constraints x₂ 1 x₁ 5 987 We introduce indicator variables δ₁ δ₂ and δ₃ to use in the following conditions δ₁ 1 x₂ 3 x₁ x₂ 4 988 δ₂ 1 x₁ x₂ 0 3x₁ x₂ 8 989 δ₃ 1 x₂ 1 x₁ 5 990 Equations 988990 are respectively imposed by the following constraints x₂ δ₁ 4 x₁ x₂ 5δ₁ 9 991 x₁ x₂ 4δ₂ 4 3x₁ x₂ 7δ₂ 15 992 x₂ 3δ₃ 4 x₁ 5 993 It is now only necessary to impose the condition that at least one of the set of Inequalities 985 986 or 987 must hold This is done by the constraint δ₁ δ₂ δ₃ 1 994 It would also be possible to cope with a situation in which the feasible region was disconnected in this way There is an alternative formulation for a connected nonconvex region such as that above so long as the line joining the origin to any feasible point lies entirely within the feasible region Seven weighting variables λA λB λG are associated with the vertices A B G and incorporated in the following constraints λB 2λC 4λD 3λE 5λF 5λG x₁ 0 995 3λA 3λB 2λC 4λD λE λF x₂ 0 996 λA λB λC λD λE λE λG 1 997 The λ variables are then restricted to form an SOS2 Notice that this is a generalization of the use of an SOS2 set to model a piecewise linear function such as that represented by the line ABCDEFG In that case constraint 997 would become an equation that is the λs would sum to 1 For the example here we simply relax this restriction to give a constraint 186 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 943 Limiting the number of variables in a solution This is another application of disjunctive constraints It is well known that in LP the optimal solution need never have more variables at a nonzero value than there are constraints in the problem Sometimes it is required however to restrict this number still further to k To do this requires IP Indicator variables δᵢ are introduced to link with each of the n continuous variables xᵢ in the LP problem by the condition xᵢ 0 δ₁ 1 998 As before this condition is imposed by the constraint xᵢ Mᵢδᵢ 0 999 where Mᵢ is an upper bound on xᵢ We then impose the condition that at most k of the variables xᵢ can be nonzero by the constraint δ₁ δ₂ δₙ k A very common application of this type of condition is in limiting the number of ingredients in a blend The FOOD MANUFACTURE 2 example of Part II is an example of this Another situation in which the condition might arise is where it is desired to limit the range of products produced in a product mix type LP model 944 Sequentially dependent decisions It sometimes happens that we wish to model a situation in which decisions made at a particular time will affect decisions made later Suppose for example that in a multiperiod LP model n periods we have introduced a decision variable γₜ into each period to show how a decision should be made in each period We let γₜ represent the following decisions γₜ 0 means the depot should be permanently closed down γₜ 1 means the depot should be temporarily closed this period only γₜ 2 means the depot should be used in this period Clearly we would wish to impose among others the conditions γₜ 0 γₜ₁ 0 γₜ₂ 0 γₙ 0 9100 This may be done by the following constraints 2γ₁ γ₂ 0 2γ₂ γ₃ 0 2γₙ₁ γₙ 0 9101 In this case the decision variable γₜ can take three values More usually it will be a 01 variable BUILDING INTEGER PROGRAMMING MODELS I 187 A case of sequentially dependent decisions arises in the MINING problem of Part II 945 Economies of scale It was pointed out in Chapter 7 that economies of scale lead to a nonlinear programming problem where the objective is equivalent to minimizing a non convex function In this situation it is not possible to reduce the problem to an LP problem by piecewise linear approximations alone Nor is it possible to rely on separable programming as local optima might result Suppose for example that we have a model where the objective is to mini mize cost The amount to be manufactured of a particular product is represented by a variable x For increasing x the unit marginal costs decrease Diagrammatically we have the situation shown in Figure 95 This may be the true cost curve or be a piecewise linear approximation c3 c2 c1 b1 0 Cost b2 b3 Quantity produced x A B C D Figure 95 The unit marginal costs are successively c1 b1 c2 c1 b2 b1 c3 c2 b3 b2 Using the λformulation of separable programming as described in Chapter 7 we introduce n 1 variables λi i 0 1 2 n that may be interpreted as weights attached to the vertices A B C D etc We then have x b1λ1 b2λ2 bnλn 9102 cost c1λ1 c2λ2 cnλn 9103 The set of variables λ0 λ1 λn is now regarded as a special ordered set of type 2 If IP is used a global optimal solution can be obtained It is also of course possible to model this situation using the δformulation of separable programming 946 Discrete capacity extensions It is sometimes unrealistic to regard an LP constraint usually a capacity constraint as applying in all circumstances In real life it is often possible to violate the constraint at a certain cost This topic has already been mentioned in Section 33 There however we allowed this constraint to be relaxed continuously Often this is not possible Should the constraint be successively relaxed it may be possible that it can only be done in finite jumps for example we buy in whole new machines or whole new storage tanks Suppose the initial righthand side RHS value is b0 and that this may be successively increased to b1 b2 bn We have j aj xj bi 9104 also cost 0 if i0 ci otherwise where 0 c1 c2 cn This situation may be modelled by introducing 01 variables δ0 δ1 δ2 etc to represent the successive possible RHS values applying We then have j aj xj b0 δ0 b1 δ1 bn δn 0 9105 The following expression is added to the objective function c1 δ1 c2 δ2 cn δn 9106 The set of variables δ0 δ1 δn may be treated as an SOS1 set If this is done then the δi can be regarded as continuous variables having a generalized upper bound of 1 that is the integrality requirement may be ignored 947 Maximax objectives Suppose we had the following situation Maximize Maximumi j aij xj subject to conventional linear constraints This is analogous to the minimax objective discussed in Section 32 but unlike that case it cannot be modelled by linear programming We can however treat it as a case of a disjunctive constraint and use integer programming The model can be expressed as Maximize z subject to j a1j xj z 0 or j a2j xj z 0 or 95 Special kinds of integer programming model The purpose of this section is to describe some wellknown special types of IP model that have received considerable theoretical attention By describing the structure of these types of model it is hoped that the model builder may be able to recognize when he or she has such a model Reference to the extensive literature and computational experience on such models may then be of value It should however be emphasized that most practical IP models do not fall into any of these categories but arise as MIP models often extending an existing LP model The considerable attention that has been paid to the IP problems described below should not disguise their limited but nevertheless sometimes significant practical importance 951 Set covering problems These problems derive their name from the following type of abstract problem We are given a set of objects that we number as the set S 1 2 3 m We are also given a class ℑ of subsets of S Each of these subsets has a cost associated with it The problem is to cover all members of S at minimum cost using members of ℑ For example suppose S1 2 3 4 5 and ℑ 1 2 1 3 5 2 4 5 3 1 4 5 and all members of ℑ have cost 1 A cover for S would be 1 2 1 3 5 and 2 4 5 In order to obtain the minimum cost cover of S in this example we could build a 01 PIP model The variables δi have the following interpretation δi1 if the ith member of ℑ is in the cover 0 otherwise Constraints are introduced to ensure that each member of S is covered For example in order to cover the member 1 of S we must have at least one of the members 1 2 1 3 5 or 1 of ℑ in the cover This condition is imposed by constraint 9107 below The other constraints ensure a similar condition for the other members of S If the objective is simply to minimize the number of members of S used in the cover the resultant model is as follows Minimize δ1 δ2 δ3 δ4 δ5 δ6 subject to δ1 δ2 δ5 1 9107 δ1 δ3 1 9108 δ2 δ4 1 9109 δ3 δ6 1 9110 δ2 δ3 δ6 1 9111 Alternatively the variables could be given nonunit cost coefficients in the objective This model has a number of important properties Property 91 The problem is a minimization and all constraints are Property 92 All RHS coefficients are 1 Property 93 All other matrix coefficients are 0 or 1 As a result of the abstract set covering application described above all 01 PIP models with the above three properties are known as set covering problems Generalizations of the problem exist If property 92 is relaxed and we allow larger positive integers than 1 for some RHS coefficients we obtain the weighted set covering problem An interpretation of this is that some of the members of S are to be given greater weight in the covering than others that is they must be covered a certain number of times Another generalization that sometimes occurs is when we relax property 92 above but also relax property 93 to allow matrix coefficients 0 or 1 This gives rise to the generalized set covering problem The best known application of set covering problems is to aircrew scheduling Here the members of S can be regarded as legs that an airline has to cover and the members of ℑ are possible rosters or rotations involving particular combinations of flights A common requirement is to cover all legs over some period of time using the minimum number of crews Each crew is assigned to a roster A comprehensive list of applications of set covering problems is given by Balas and Padberg 1975 Special algorithms do exist for set covering problems Two such algorithms are described in Chapter 4 of Garfinkel and Nemhauser 1972 Set covering problems have an important property that often makes them comparatively easy to solve by the branch and bound method It can be shown that the optimal solution to a set covering problem must be a vertex solution in the same sense as for LP problems Unfortunately this vertex solution will not generally be but sometimes is the optimal vertex solution to the corresponding LP model It is however often possible to move from this continuous optimum to the integer optimum in comparatively few steps The difficulty of solving set covering problems usually arises not from their structure but from their size In practice models frequently have comparatively few constraints but an enormous number of variables There is often considerable virtue in generating columns for the model in the course of optimization Column generation forms the subject of Section 96 952 Set packing problems These problems are closely related to set covering problems The name again derives from an abstract problem that we first describe We are given a set of objects which we number as the set S 1 2 3 m We are also given a class S of subsets of S each of which has a certain value associated with it The problem is to pack as many of the members of S into S as possible to maximize total value but without any overlap For example suppose S 1 2 3 4 5 6 and S 1 2 5 1 3 2 4 3 6 2 3 6 A pack for S would be 1 2 5 and 3 6 Again we could build a 0 1 PIP model to help solve the problem The variables δi have the following interpretation δi 1 if the ith member of S is in the pack 0 otherwise Constraints are introduced to ensure that no member of S is included in more than one member of S in the pack that is there shall be no overlap For example in order that the member 2 shall not be included more than once we cannot have more than one of 1 2 5 2 4 and 2 3 6 in the pack This condition gives rise to the constraint 9113 below The other constraints ensure similar conditions for the other members of S The objective here is to maximize the number of members of S we can pack in For this example the model is Maximize δ1 δ2 δ3 δ4 δ5 subject to δ1 δ2 1 9112 δ1 δ3 δ5 1 9113 δ2 δ4 δ5 1 9114 δ3 1 9115 δ1 1 9116 δ4 δ5 1 9117 Constraints 9115 and 9116 are obviously redundant Like the set covering problem this model has a number of important properties Property 94 The problem is a maximization and all constraints are Property 95 All RHS coefficients are 1 Property 96 All other matrix coefficients are 0 or 1 An interesting observation is that the LP problem associated with a set packing problem with objective coefficients of 1 is the dual of the LP problem associated with a set covering problem with objective coefficients of 1 This result is of little obvious significance as far as the optimal solutions to the IP problems are concerned as there may be a duality gap between these solutions This term is explained in Section 103 Although a set packing problem with objective coefficients of 1 is the dual of a set covering problem with objective coefficients of 1 in this sense it should be realized that the set S that we wish to cover with members of S is not the same as the set S that we wish to pack with members of another class of subsets S The two examples used above to illustrate set covering and set packing problems are so chosen as to be dual problems in this sense but the set S and class of subsets S are different As with the set covering problem there are generalizations of the set packing problem obtained by relaxing some of the properties 9496 above If property 95 is relaxed and we allow positive integral RHS coefficients greater than 1 we obtain the weighted set packing problem If we relax property 95 as above together with property 96 to allow matrix coefficients of 0 or 1 we obtain the generalized set packing problem A special sort of packing problem is known as the matching problem This problem can be represented by a graph in which the objects S to be packed are nodes Each subset from S consists of two members of S and is represented by an arc of the graph whose end points represent the two members of S The problem then becomes one of matching or pairing as many vertices as possible together Extensive work on this problem has been done by Edmonds 1965 Set packing problems are equivalent to the class of set partitioning problems which we consider next Applications and references are described there 953 Set partitioning problems In this case we are as before given a set of objects that we number as the set S 1 2 3 m and a class S of subsets of S The problem this time is both to cover all the members of S using members of S and also to have no overlap We in this sense have a combined covering and packing problem There is no useful purpose in distinguishing between maximization and minimization problems here Both may arise As an example we consider the same S and S used in the example to describe the set covering problem The difference is that we must now impose stricter conditions to ensure that each member of S is in exactly one member of the partition or cover and pack combined This is done by making the constraints 91079111 instead of giving δ1 δ2 δ5 1 9118 δ1 δ3 1 9119 δ2 δ4 1 9120 δ3 δ6 1 9121 δ2 δ3 δ6 1 9122 For example a feasible partition of S consists of 1 2 3 and 4 5 An easily understood way in which the set partitioning problem can arise is again in aircrew scheduling Suppose that we did not allow aircrews to travel as passengers on other flights It would then be necessary that each member of S a leg be covered by exactly one member of S a roster We would then have a set partitioning problem in place of the set covering problem Another application of the set partitioning problem is to political districting In this problem there is usually an extra constraint fixing the total number of political districts or constituencies The objective is usually to minimize the maximum deviation of the electorate in a district from the average electorate in a district This minimax objective can be dealt with in the way described in Section 33 A reference to tackling the problem by IP is Garfinkel and Nemhauser 1970 We now show the essential equivalence between set partitioning and set packing problems If we introduce slack variables into each of the constraints of a set packing problem we obtain constraints These slack variables can only take the values 0 or 1 and could therefore be regarded with all the other variables in the problems as 0 1 variables The result would clearly be a set partitioning problem To show the converse is slightly more complicated Suppose we have a set partitioning problem Each constraint will be of the form a1 δ1 a2 δ2 an δn 1 9123 We introduce an extra 0 1 variable δ into constraint 9123 giving a1 δ1 a2 δ2 an δn δ 1 9124 194 MODEL BUILDING IN MATHEMATICAL PROGRAMMING If the problem is one of minimization δ can be given a sufficiently high positive cost M in the objective to force δ to be zero in the optimal solution For a maximization problem making M a negative number with a sufficiently high absolute value has the same effect Instead of introducing S explicitly into the objective function we can substitute the expression below derived from Equation 9124 1 a1δ1 a2δ2 anδn 9125 There is no loss of generality in replacing Equation 9124 by the constraint a1δ1 a2δ2 anδn 1 9126 If similar transformations are performed for all the other constraints the set partitioning problem can be transformed into a set packing problem Hence we obtain the result that set packing problems and set partitioning problems can easily be transformed from one to the other Both types of problem possess the property mentioned in connection with the set covering problems that is the optimal solution is always a vertex solution to the corresponding LP problem The set partitioning or packing problem is generally even easier to solve than the set covering problem It is easy to see from the small numerical example that a set partitioning problem by using constraints is more constrained than the corresponding set covering problem Likewise the corresponding LP problem is more constrained It will be seen in Section 101 that constraining the corresponding LP problem to an IP problem as much as possible is computationally very desirable A comprehensive list of applications and references to the set packing and par titioning problems is given by Balas and Padberg 1975 Chapter 4 of Garfinkel and Nemhauser 1972 treats these problems very fully and gives a special pur pose algorithm Ryan 1992 describes how the set partitioning problem can be applied to aircrew scheduling In view of its close similarity to set partitioning and packing problems it is rather surprising that the set covering problem has some important differences Although the set covering problem is an essentially easy type of problem to solve it is distinctly more difficult than the former problems It is possible to transform a set partitioning or packing problem into a set covering problem by introducing an extra 0 1 variable with a negative coefficient in Equation 9123 and performing analogous substitutions to those described above The reverse transformation of a set covering problem to a set partitioning or pack ing problem is not however generally possible revealing the distinctness of the problems BUILDING INTEGER PROGRAMMING MODELS I 195 954 The knapsack problem A PIP model with a single constraint is known as a knapsack problem Such a problem can take the form Maximize p1γ1 p2γ2 pnγn subject to a1γ1 a2γ2 anγn b 9127 where γ 1 γ 2 γ n 0 and take integer values The name knapsack arises from the rather contrived application of a hiker trying to fill her knapsack to maximum total value Each item she considers taking with her has a certain value and a certain weight An overall weight limitation gives the single constraint An obvious extension of the problem is where there are additional upper bounding constraints on the variables Most commonly these upper bounds will all be 1 giving a 0 1 knapsack problem Practical applications occasionally arise in project selection and capital bud geting allocation problems if there is only one constraint The problem of stocking a warehouse to maximum value given that the goods stored come in indivisible units also gives rise to an obvious knapsack problem The occurrence of such immediately obvious applications is however fairly rare Much more commonly the knapsack problem arises in LP and IP problems where one generates the columns of the model in the course of optimization As such techniques are intimately linked with the algorithmic side of mathematical programming they are beyond the scope of this book The original application of the knapsack problem in this way was to the cutting stock problem This is described by Gilmore and Gomory 1963 1965 and in Section 96 In practice knapsack problems are comparatively easy to solve The branch and bound method is not however an efficient method to use If it is desired to solve a large number of knapsack problems eg when generating columns for an LP or IP model it is better to use a more efficient method Dynamic programming proves an efficient method of solving knapsack problems This method and further references are given in Chapter 6 of Garfinkel and Nemhauser 1972 955 The travelling salesman problem This problem has received a lot of attention largely because of its conceptual simplicity The simplicity with which the problem can be stated is in marked contrast to the difficulty of solving such problems in practice The name travelling salesman arises from the following application A salesman has to set out from home to visit a number of customers before finally returning home The problem is to find the order in which he or she should visit all the customers in order to minimize the total distance covered Generalizations and special cases of the problem have been considered For example sometimes it is not necessary that the salesman return home Often the problem itself has special properties for example the distance from A to B is the same as the distance from B to A The vehicle routing problem can be reduced to a travelling salesman problem This is the problem of organizing deliveries to customers using a number of vehicles of different sizes The reverse problem of picking up deliveries eg letters from post office boxes given a number of vehicles or postal workers can also be treated in this way This problem is considered below Job sequencing problems in order to minimize setup costs can be treated as travelling salesman problems where the distance between cities represents the setup cost between operations A not very obvious nor probably practical application of this sort is to sequencing the colours that are used for a single brush in a series of painting operations Obviously the transition between certain colours will require a more thorough cleaning of the brush than a transition between other colours The problem of sequencing the colours to minimize the total cleaning time gives rise to a travelling salesman problem The travelling salesman problem can be formulated as an IP model in a number of ways We present one such formulation here Any solution not necessarily optimal to the problem is referred to as a tour Suppose the cities to be visited are numbered 0 1 2 n 01 integer variables are introduced with the following interpretation if the tour goes from i to j direct otherwise The objective is simply to minimize ij cij δij where cij is the distance or cost between i and j There are two obvious conditions that must be fulfilled Exactly one city must be visited immediately after city i 9128 Exactly one city must be visited immediately before city j 9129 Condition 9128 is achieved by the constraints ij δij 1 i 0 1 n 9130 Condition 9129 is achieved by the constraints ij δij 1 j 0 1 n 9131 BUILDING INTEGER PROGRAMMING MODELS I 197 As the model has so far been presented we have an assignment problem as described in Section 53 Unfortunately the constraints 9130 and 9131 are not sufficient Suppose for example we were considering a problem with eight cities n 7 The solution drawn in Figure 96 would satisfy all the conditions 9130 and 9131 Clearly we cannot allow subtours such as those shown here The problem of adding extra constraints so as to avoid subtours proves quite difficult In practice it is often desirable to add these constraints in the course of optimization as subtours arise For example as a result of the solution to the relaxed problem exhibited by Figure 96 we would add the extra constraint x46 x64 x47 x74 x67 x76 2 9132 1 5 6 4 7 2 3 0 Figure 96 This would rule out any subtour around the cities 4 6 and 7 and implicitly around 0 1 2 3 and 5 if these were the only other cities Potentially there are an exponential number of such subtour elimination constraints and it would be impractical to state them all explicitly Therefore they are added on an as needed basis as above In practice one rarely needs to add more than a very small fraction of this exponential number before a solution without subtours is obtained When this happens we clearly have the optimal solution There are ingenious ways of producing nonexponential sized formulations by the introduction of new variables or replacing the variables by new ones with a different interpretation However all except one of these formulations have weaker LP relaxations than the conventional formulation which is due to Dantzig et al 1954 Orman and Williams 2006 give eight different formulations and show surprisingly that one can rank the strengths of their LP relaxations independently of prob lem instance They classify the nonexponential formulations into three distinct forms A sequential formulation is due to Miller et al 1960 who introduce extra variables representing the sequence numbers in which cities are visited and use these variables in constraints that prevent subtours Gavish and Graves 1978 introduce flow variables which are only allowed in arcs that are used in the tour These together with the material balance constraints Section 53 of the network flow problem prevent subtours There are a number of variants of this formu lation for example Finke et al 1983 In particular Wong 1980 and Claus 1984 give a multicommodity network flow formulation which remarkably 198 MODEL BUILDING IN MATHEMATICAL PROGRAMMING has the same strength LP relaxation as the conventional formulation However it has of the order of n3 variables and constraints which still makes it impractical to solve the full model in one optimization Finally timestaged formulations are given by Vajda 1961 and Fox et al 1980 who use a third index t so as to let the new variables δijt represent whether the salesman goes from city i to city j at stage time t These variables can then be incorporated in constraints to rule out subtours A comprehensive survey of the travelling salesman problem is Lawler et al 1995 There have been dramatic advances in solving the problem fairly recently Applegate et al 2006 discuss the computational side Special algorithms have been developed One of the most successful is that of Held and Karp 1971 There are also variants and extensions of the travelling salesman problem One such is the MILK COLLECTION problem described in Part II This is based on a paper by Butler et al 1997 Another is the LOST BAGGAGE DISTRIBUTION problem also described in Part II 956 The vehicle routing problem This is practically the most important extension of the travelling salesman prob lem where one has to schedule a number of vehicles of limited capacity around a number of customers Hence in addition to the sequencing of the customers to be visited the travelling salesman problem one has to decide which vehicles visit which customers Each customer has a known demand assumed to be of one commodity although it is possible easily to extend to more than one com modity Hence the limited vehicle capacities have to be taken into account In addition there are often time window conditions requiring that certain customers can only receive deliveries between certain times We give a formulation of this problem as an extension of that of the travelling salesman problem mentioned above As with that formulation it is generally necessary to add extra constraints to break subtours in the course of optimization Let δijk 1 iff vehicle k goes directly from customer i to customer j γ ik 1 iff vehicle k visits customer i apart from the depot τ i time at which customer i is visited We assume that each customer apart from the depot regarded as customer 0 is visited by exactly one vehicle that is there are no split deliveries to a customer For data we assume there are m vehicles and that vehicle k has capacity Qk and customer i has requirement qi Also we assume that the time taken between customers i and j is dij and that customer i must take delivery between times ai and bi There are a number of possible objectives for example minimize the number of vehicles needed minimize total cost or minimize maximum time taken by a vehicle For illustration we consider the second objective and assume that the cost not necessarily proportional to the time of going from i to j directly is cij The model is Minimize ijk cijk δijk subject to j δijk j δjik γik if a vehicle k visits customer i it is entered once and left once by k k γik 1 if customer i not 0 is visited by exactly one vehicle i γik γ1k if vehicle k is used then it must visit town 1 i qi γik Qk the combined requirements of customers visited by vehicle k must lie within the capacity of k τi τj M 1 δijk dij if customer j is visited immediately after customer i then this must be at a time at least dij greater than the time at which customer i is visited ai τi b each customer must be visited within its time window Such a formulation could prove very difficult to solve for reasonablesized problem instances Another approach is to use column generation which is discussed in Section 96 see also Desrosiers et al 1984 In this approach feasible routes for a vehicle that is obeying the capacity and time window constraints are generated separately from the main model which then selects and combines them in a feasible and optimal manner Variants of this problem are the MILK COLLECTION and LOST BAGGAGE DISTRIBUTION problems given in Part II 957 The quadratic assignment problem This is a generalization of the assignment problem described in Section 53 Whereas that problem could be treated as an LP problem and was therefore comparatively easy to solve the quadratic assignment problem is a genuine IP problem and often very difficult to solve As with the assignment problem we consider the problem as related to two sets of objects S and T S and T both have the same number of members which are be indexed 1 to n The problem is to assign each member of S to exactly one member of T in order to achieve some objective There are two sorts of conditions we must fulfil Each member of S must be assigned to exactly one member of T 9133 Each member of T must have exactly one member of S assigned to it 9134 0 1 variables δij can be introduced with the following interpretations δij 1 if i a member of S is assigned to j a member of T 0 otherwise Conditions 9133 and 9134 are imposed by the following two types of constraint j1 δij 1 i 1 2 n 9135 i1 δij 1 j 1 2 n 9136 The objective is more complex than with the assignment problem We have cost coefficients cijkl which have the following interpretations cijkl is the cost incurred by assigning i a member of S to j a member of T at the same time as assigning k a member of S to l a member of T This cost will clearly be incurred only if δij 1 and δkl 1 that is if the product δij δkl 1 The objective then becomes a quadratic expression in 0 1 variables Minimize ijkl1 ki to n cijkl δij δkl 9137 The condition k i on the indices prevents the cost of each pair of assignments being counted twice It is very common for the coefficients cijkl to be derived from the product of other coefficients tik and djl so that cijkl tik djl 9138 In order to understand this rather complicated model it is worth considering two applications Firstly we consider S to be a set of n factories and T to be a set of n cities The problem is to locate one factory in each city and to minimize total communication costs between factories The communication costs depend on i the frequency of communication between each pair of factories and ii the distances between the two cities where each pair of factories is located Clearly some factories will have little to do with each other and can be located far apart at little cost On the other hand some factories may need to communicate a lot The cost of communication will depend on the distance BUILDING INTEGER PROGRAMMING MODELS I 201 apart In this application we can interpret the coefficients tik and djl in Equation 9138 as follows tik is the frequency of communication between factories i and k measured in appropriate units djl is the cost per unit of communication between cities j and l clearly this will be related to the distance between j and l Obviously the cost of communication between the factories i and k located in cities j and l will be given by Equation 9138 The total cost is therefore represented by the objective function 9137 Another application concerns the placement of n electronic modules in n predetermined positions on a backplate S represents the set of modules and T represents the set of positions on the backplate The modules have to be connected to each other by a series of wires If the objective is to minimize the total length of wire used we have a quadratic assignment problem similar to the above model The coefficients tik and djl have the following interpretations tik is the number of wires that must connect module i to the module k and djl is the distance between position j and position l on the backplate Many modifications of the quadratic assignment problem described above exist Frequently conditions 9133 and 9134 are relaxed to allow more than one member of S or possibly none to be assigned to a member of T where S and T may not have the same number of members A modified problem of this sort is the DECENTRALIZATION example in Part II One way of tackling the quadratic assignment problem is to reduce the prob lem to a linear 0 1 PIP problem It is necessary to remove the quadratic terms in the objective function Two ways of transforming products of 0 1 variables into linear expressions in an IP model were described in Section 92 For fairly small models the first transformation is possible but for larger models the result can be an enormous expansion in the number of variables and constraints in the model The solution of a practical problem of this sort by using a similar type of transformation is described by Beale and Tomlin 1972 A survey of practical applications as well as other methods of tackling the problem is given by Lawler 1974 He also shows how a different formula tion of the travelling salesman problem to that given here leads to a quadratic assignment problem It should be pointed out however that although a travelling salesman problem is often difficult to solve a quadratic assignment problem of comparable dimensions is usually even harder 96 Column generation The generation of columns in the course of optimization has already been referred to in the context of decomposition in Section 43 and the set covering and vehicle routing problems in Section 95 All the relevant models there are characterized by a very large often astronomic number of variables only a small proportion of which will feature in the optimal solution Therefore it seems more efficient only to append them to the model in the course of optimization if it seems likely they will be in the optimal solution Although the method of generation 202 MODEL BUILDING IN MATHEMATICAL PROGRAMMING will depend on the nature of the model and the optimization method used it is still appropriate to consider this general approach in a book on model building In principle but not in practice all the possible variables could be created at the beginning Column generation is frequently applied to special types of integer programming models Hence its inclusion in this chapter Possible solutions or components of solutions are generated separately from the main master model sometimes by optimizations and sometimes by heuris tics and appended to it in the course of optimization In the case of decompo sition these components are the solutions of subproblems In the case of the set covering problem applied to aircrew scheduling they might be possible rosters generated as paths through a spacetime diagram of an airline schedule For the vehicle routing problem they would be routes covered by individual vehicles respecting capacity constaints and time windows We illustrate column generation by means of the cutting stock problem which was one of the original applications decribed by Gilmore and Gomory 1961 and 1963 This can be applied to cutting ordered widths of wallpaper from standard widths to minimize waste or ordered lengths of pipe from standard lengths in order to minimize the number of standard lengths that need to be cut The cutting stock problem can also be regarded as a special case of the bin packing problem where it is desired to pack a number of items of different sizes into standard size bins so as to minimize the number of bins used Suppose a plumber stocks standard lengths of pipe all of length 19 m He has orders for 12 lengths of 4 m 15 lengths of 5 m 22 lengths of 6 m How should he cut these lengths out of his standard lengths so as to minimize the number of standard lengths used We firstly give a bad formulation of this problem as an integer programme in order to illustrate the advantage of one based on column generation Let xij number of pipes of length i 4 5 or 6 m cut from standard pipe j numbering the standard stocked pipes 1 2 3 etc δj 1 if and only if standard pipe j used The constraints are x1j 4δj x2j 3δj x3j 3δj standard pipe j only used if it appears in an order 4x1j 5x2j 6x3j 19 for j 1 2 3 orders must be within length of each standard pipe used j x1j 12 j x2j 15 j x3j 22 orders must be met The objective is Minimize j δj A major defect of this model is that there will be many symmetrically equivalent solutions as the standard pipes are indistinguishable Even if symmetry breaking constraints Section 102 are appended this model would take a very long time to solve the reader might like to try solving this tiny example using a computer package We therefore present a much better model based on column generation The main constraint above is the second knapsack one The solutions to this constraint are the possible patterns that can be made up out of the standard lengths We give some of the possible patterns below including the waste W There are obviously more patterns and in a larger example there will be an astronomic number most of which will not be used Pattern 1 4455W Pattern 2 4456 Pattern 3 4555 A better formulation using column generation which finds the minimum number of such patterns needed is γj number of times pattern j is used Minimize γ1 γ2 γ3 subject to 2γ1 2γ2 γ3 12 2γ1 γ2 3γ3 15 γ2 22 Note that each variable has a column of constraint coefficients corresponding to the number of times each of the order lengths occurs in the pattern The RHS coefficients are the demands for each length Solving this model as an IP yields γ1 0 γ2 22 γ3 0 requiring 22 standard pipes Clearly there are 204 MODEL BUILDING IN MATHEMATICAL PROGRAMMING better solutions if other patterns are used as well It is possible to devise other useful patterns using the shadow prices from the optimal LP solution to the current model To start with these shadow prices turn out to be 0 0 1 If we were to have a pattern with a lengths of 4 m b lengths of 5 m and c lengths of 6 m its reduced cost value in reducing the objective would be c 1 We therefore seek a pattern that maximizes this quantity subject to fitting the pattern into the standard length that is we wish to Maximize c 1 subject to 4a 5b 6c 19 Clearly a b c are integer variables giving a knapsack problem This instance has the optimal solution a b 0 c 3 with objective 2 indicating that this pattern will reduce the number of pipes used in the LP relaxation giving 6 Pattern 4 6 6 W Hence we have generated a new column that can be appended to the model to give Minimize γ1 γ2 γ3 γ4 subject to 2γ1 2γ2 γ3 12 2γ1 γ2 3γ3 15 γ2 3γ4 22 This model has optimal IP solution γ 1 0 γ 2 4 γ 3 4 γ 4 6 requiring 14 standard pipes The shadow prices are 13 0 13 which in order to find another pattern with maximum reduced cost leads to Maximize 13 a 13 c 1 subject to 4a 5b 6c 19 This has the optimal solution a 4 b 0 c 0 with objective value 13 giving 4 Pattern 5 4 4 4 W BUILDING INTEGER PROGRAMMING MODELS I 205 Appending the column corresponding to this pattern gives Minimize γ1 γ2 γ3 γ4 γ5 subject to 2γ1 2γ2 γ3 4γ5 12 2γ1 γ2 3γ3 15 γ2 3γ4 22 This model gives the same solution as before In fact this is the overall optimal solution which has therefore been obtained without considering many of the potential patterns blank page 10 Building integer programming models II 101 Good and bad formulations Most of the considerations of Section 34 concerning linear programming LP models will also apply to integer programming IP models and will not be reconsidered here There are however some important additional considerations that must be taken into account when building IP models The primary additional consideration is the much greater computational difficulty of solving IP models over LP models It is fairly common to build an IP model only to find the cost of solving it prohibitive Frequently it is possible to reformulate the problem giving another model that is easier to solve Such reformulations must often be considered in conjunction with the solution strategy to be employed It will be assumed throughout that the branch and bound method described in Section 83 is to be used In some respects much greater flexibility is possible in building IP models than in building LP models The flexibility results in a greater divergence between good and bad models of a practical problem The purpose of this chapter is to suggest ways in which good models may be constructed It is convenient to consider variables and constraints separately There is often the possibility of using many or few variables and many or few constraints in a model The considerations governing this will be considered 1011 The number of variables in an IP model We confine our attention here to the number of integer variables in an IP model as this is often regarded as a good indicator of the computational difficulty Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 208 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Suppose we had a 01 IP model either mixed integer or pure integer If the model had n 01 variables this would indicate 2n possible settings for the variables and hence 2n potential nodes hanging at the bottom of the solution tree In total there would be 2n 1 1 nodes in such a tree One might therefore expect the solution time to go up exponentially with the number of 01 variables For quite modest values of n 2n is very large for example 2100 is greater than one million raised to the power 5 The situation is not of course anywhere near as bad as this as many of the 2n potential nodes will never be examined The branch and bound method rules out large sections of the potential tree from examination as being infeasible or worse than solutions already known It is however worth pausing to consider the fact that one may sometimes solve 100 01 variable IP problems in a few hundred nodes This represents only about 000 001 per cent of the potential total where there are 28 zeros after the decimal point In view of this very surprising efficiency that the branch and bound method exhibits over the potential amount of computation the number of 01 variables is often a very poor indicator of the difficulty of an IP model We however suggest one circumstance in which the number of such variables might be usefully reduced later in this section Before doing that we indicate ways in which the number of integer variables in a model might be increased to good effect It is convenient here to describe a wellknown device for expanding any general integer variable in a model to a number of 01 variables Suppose γ is a general nonnegative integer variable with a known upper bound of u an upper bound is required for all integer variables in an IP model if the branch and bound method is to be used that is 0 γ u γ may be replaced in this model by the expression δ0 2δ1 4δ2 8δ3 2rδr 101 where the δi are 01 variables and 2r is the smallest power of 2 greater than or equal to u It is easy to see that the expression 101 can take any possible integral value between 0 and u by different combinations of values for the δi variables Clearly the number of 01 variables required in an expansion like this is roughly log2u In practice u will probably be fairly small and the number of 01 variables produced not too large If however this device were employed on a large number of the variables in the model the result might be a great expansion in model size Generally there will be little virtue in an expansion of this sort except to facilitate the use of some specialized algorithm applying only to 01 problems This is beyond the scope of this book Although there is some virtue in keeping an LP model compact any such advantages that this may imply for the corresponding IP model are usually drowned by other much more important considerations Using the branch and bound method there is sometimes virtue in introducing extra 01 variables as useful variables in the branching process Such 01 variables represent dichotomies in the system being modelled To make such dichotomies explicit can be valuable as is demonstrated in the following example due to Jeffreys 1974 Example 101 One new factory is to be built The possible decisions are represented by 01 variables δnb δnc δsb and δsc δnb 1 if the factory is in the north and uses a batch process 0 otherwise δnc 1 if the factory is in the north and uses a continuous process 0 otherwise δsb 1 if the factory is in the south and uses a batch process 0 otherwise δsc 1 if the factory is in the south and uses a continuous process 0 otherwise The condition that only one factory be built can be represented by the constraint δnb δnc δsb δsc 1 102 It is not possible using constraint 102 to express the dichotomy either we site the factory in the north or we site it in the south by a single 01 variable As this is clearly an important decision it would be advantageous to have a 01 variable indicating the decision By adding an extra 01 variable δ to represent this decision together with the extra constraints δnb δnc δ 0 103 δsb δsc δ 1 104 this is possible δ is a valuable variable to have at our disposal as use can be made of it as a variable to branch on in the tree search The dichotomy either we use a batch process or we use a continuous process could also be represented by another 01 decision variable in a similar way Another use of extra integer variables in a model is to specify the slack variable in a constraint made up of only integer variables as itself being integer For example if all the variables coefficients and righthand side are integers in the constraint j aj xj b 105 BUILDING INTEGER PROGRAMMING MODELS II 211 1012 The number of constraints in an IP model It was pointed out in Chapter 3 that the difficulty of an LP model is very dependent on the number of constraints Here we show that in an IP model this effect is often completely drowned by other considerations In fact an IP model is often made easier to solve by expanding the number of constraints In an LP model we search for vertex solutions on the boundary of the feasible region For the corresponding IP model we are interested in integer points that may well lie in the interior of the feasible region This is illustrated in Figure 101 ABCD is the feasible region of the LP problem but we must confine our attention to the lattice of integer points For an IP model the corresponding LP model is known as the LP relaxation 5 4 3 2 1 0 1 2 3 4 5 D V U C A T S R B Q P x2 x1 Figure 101 In this diagram we assume both x1 and x2 to be integer variables For mixed integer problems we are interested in points in an diagram analogous to Figure 101 some of whose coordinates are integers but where other coordinates are allowed to be continuous Clearly this situation is difficult to picture in a few dimensions Suppose however that there were a further continuous variable x3 to be considered in Figure 101 This would give rise to a coordinate coming out at right angles to the page Feasible solutions to the mixed integer problem would consist of lines parallel to the x3axis coming out of the page from the integer points in Figure 101 These lines would have of course to lie inside the threedimensional feasible region of the corresponding LP problem 212 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Ideally we would like to reformulate the IP model so that the feasible region of the corresponding LP model becomes PQRSTUV This is known as the convex hull of feasible integer points in ABCD It is the smallest convex set containing all the feasible integer points If it were possible to reformulate the IP problem in this way we could solve the problem as an LP problem as the integrality requirement would automatically be satisfied Such a formulation where the LP relaxation gives the convex hull of IP solutions is known as sharp Each vertex and hence optimal solution of the new feasible region PQRSTUV is an integer point Unfortunately in many practical problems the effort required to obtain the convex hull of integer points would be enormous and far outweigh the computation needed to solve the original formulation of the IP problem There are however important classes of problems where one of the following can happen i The straightforward formulation results in an IP model where the feasible region is already the convex hull of integer points ii The problem can fairly easily be reformulated to give a feasible region corresponding to the convex hull of integer points iii By reformulating it is possible to reduce the feasible region of the LP problem to nearer that of the convex hull of integer points We consider each of these classes of problems in turn Case i concerns some problems that have already been considered in Section 53 Although superficially these problems might appear to give rise to PIP pure integer programming models the optimal solution of the corresponding LP problem always results in integer values for the integer variables The model may therefore be treated as an LP problem Problems falling into this category include the transportation problem the minimum cost network flow problem and the assignment problem It is sometimes possible to recognize when an IP model has a structure that guarantees that the corresponding LP model will have an integer optimal solution Clearly it is very useful to be able to recognize this property as the high computational cost of IP need not be incurred Consider the following LP model Maximize c x subject to Ax b and x 0 Here we assume that slack variables have been added to the constraints if necessary to make them all equalities For the above model to yield an optimal solution with all variables integer for every objective coefficient vector c and integer righthand side b the matrix A must have a property known as total unimodularity Definition A matrix A is totally unimodular if every square submatrix of A has its deter minant equal to 0 or 1 BUILDING INTEGER PROGRAMMING MODELS II 213 The fact that this property guarantees that there will be an integer optimal solution to the LP model for any c and integer b is proved in Garfinkel and Nemhauser 1972 p 67 Unfortunately the above definition of total unimod ularity is of little help in detecting the property To evaluate the determinant of every square submatrix would be prohibitive There is however a property which we call P that is easier to detect which guarantees total unimodularity It is however only a sufficient condition not a necessary condition Matrices without property P may still be totally unimodular Property P 1 Each element of A is 0 1 or 1 2 No more than two nonzero elements appear in each column 3 The rows can be partitioned into two subsets P1 and P2 such that a if a column contains two nonzero elements of the same sign one element is in each of the subsets b if a column contains two nonzero elements of opposite sign both elements are in the same subset A particular case of the property P is when subset P1 is empty and P2 consists of all the rows of A Then for the property to hold we must have all columns consisting of either one nonzero element 1 or two nonzero elements 1 and 1 As an example of this property holding we can consider the small transporta tion problem of Section 53 x11 x13 x14 x15 135 x21 x22 x25 56 x31 x32 x33 x34 x35 93 x11 x21 x31 62 x22 x32 84 x13 x33 39 x14 x34 91 x15 x25 x35 9 1010 Each column clearly contains a 1 and a 1 showing the property to hold and hence guaranteeing total unimodularity There is often virtue in trying to reformulate a model in order to try to capture this easily detected property The automation of such reformulation where possible was described in Section 54 In case ii below the dual situation is illustrated Although the property above refers to a partitioning of the rows of a matrix A into two subsets the partitioning could equally well apply to the columns If A is totally unimodular its transpose A must also be totally unimodular Again a particular instance of this property guaranteeing total unimodularity is when each row of the matrix A of an LP model contains a 1 and a 1 214 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Finally it should be pointed out that total unimodularity is a strong property that guarantees integer optimal solutions to an LP problem for all c and integer b Many IP models for which the matrix A is not totally unimodular frequently although not always produce integer solutions to the optimal solution of the corresponding LP problem In particular this often happens with the set pack ing partitioning and covering problems discussed in Section 95 There are good reasons why this is likely to happen for these types of problems Such consid erations are however fairly technical and beyond the scope of this book They are discussed in Chapter 8 of Garfinkel and Nemhauser 1972 Properties of A that guarantee integer LP solutions for a specific righthand side b are discussed by Padberg 1974 for the case of the set partitioning problem The discussion of total unimodularity above applies only to PIP models Clearly there are corresponding considerations for MIP mixed integer program ming models where integer values for the integer variables in the optimal LP solution are guaranteed Such considerations are however very difficult and there is little theoretical work as yet that is of value to practical model builders Case ii above concerns problems where with a little thought a reformu lation can result in a model with the total unimodularity property Consider a generalization of constraint 944 δ1 δ2 δn nδ 0 1011 where δi and δ are 01 variables This kind of constraint arises fairly frequently in IP models and represents the logical condition δ1 1 δ2 1 δn 1 δ 1 1012 Sometimes this condition is more easily thought of as the logical equivalent condition δ 0 δ1 0 δ2 0 δn 0 1013 It was shown in Section 92 that by a different argument constraint 944 of that section can be reformulated using two constraints A similar reformulation can be applied here giving the n constraints δ1 δ 0 δ2 δ 0 δn δ 0 1014 Should all the constraints in the model be similar to the constraints of Inequality 1014 then the dual problem has the property P described above BUILDING INTEGER PROGRAMMING MODELS II 215 which guarantees total unimodularity There is therefore great virtue in such a reformulation as the high computational costs associated with an IP problem over an LP problem is avoided An example of a reformulation of a problem in this way is described by Rhys 1970 He also demonstrates another advantage of the reformulation as yielding a more meaningful economic interpretation of the shadow prices This topic is considered in Section 103 A practical example of a formulation such as that described above is given in Part III where the formulation of the OPENCAST MINING problem is discussed Constraint 1011 above shows the possibility of sometimes reformulating a PIP problem that is not totally unimodular in order to make it totally unimodular There is also virtue in reformulating an already totally unimodular problem which we do not know to be totally unimodular if by so doing we convert it into a form where property P applies Examples of this are given by Veinott and Wagner 1962 and Daniel 1973 As with case i the above discussion of case ii only applies to PIP problems Again it must be possible although difficult to generalize these ideas to MIP problems Case iii concerns problems where there is either no obvious totally uni modular reformulation or where the problem gives an MIP model In cases i and ii we reduced the feasible region to the convex hull of feasible integer points even though this was not obvious from the algebraic treatment given It is sometimes possible to go part way towards this aim Suppose for example as might frequently happen in a MIP model that only some of the constraints were of the form 1011 By expanding these constraints into the series of constraints 1014 we would reduce the size of the feasible region of the LP Even though the existence of other constraints in the problem might result in some integer variables taking fractional values in the LP optimal solution this solution should be nearer the integer optimal solution than would be the case with the original model The term nearer is purposely vague A reformulation such as this might result in the objective value at node 1 of the solution tree eg Figure 81 being closer to the objective value of the optimal integer solution when found On the other hand it might result in there being less fractional solution values in the LP optimum Whatever the result of the reformulation one would normally expect the solution time for the reformulated model to be less than for the orig inal model Constraints involving just two coefficients 1 and 1 also arise in models involving sequentially dependent decisions as described in Section 94 Such constraints are always to be desired even if their derivation results in an expansion of the constraints of a model An example of this is the suggested formulation in Part III for the MINING problem It is worth indicating in another way why a series of constraints such as in Inequalities 1014 is preferable to the single constraint 1011 Although constraints 1011 and 1014 are exactly equivalent in an IP sense they are certainly not equivalent in an LP sense In fact Inequality 1011 is the sum of all the constraints in Inequalities 1014 By adding together constraints in 216 MODEL BUILDING IN MATHEMATICAL PROGRAMMING an LP problem one generally weakens their effect This is what happens here Inequality 1011 admits fractional solutions which Inequality 1014 would not admit For example the solution δ1 1 2 δ2 δ3 1 4 δ 1 n all other δi 0 1015 satisfies Inequality 1011 but breaks Inequalities 1014 for n 3 Hence Inequalities 1014 are more effective at ruling out unwanted frac tional solutions The ideas discussed above are relevant to the FOOD MANUFACTURE 2 problem which gives rise to an MIP model and to the DECENTRALIZATION and LOGICAL DESIGN problems which give rise to PIP models Some of the material so far presented in this section was first published by Williams 1974 A discussion of some very similar ideas applied to a more complicated version of the DECENTRALIZATION problem is given in Beale and Tomlin 1972 It is also relevant here to discuss the value of the coefficient M when linking indicator variables to continuous variables by constraints such as in Equations 91 912 919 and 921 These types of constraints usually but not always arise in MIP models We will consider the simplest way in which such a constraint arises when we are using a 01 variable δ to indicate the condition below on the continuous variable x x 0 δ 1 1016 This condition is represented by the constraint x Mδ 0 1017 So long as M is a true upper bound for x condition 1016 is imposed however large we make M There is virtue however in making M as small as possible without imposing a spurious restriction on x This is because by making M smaller we reduce the size of the feasible region of the LP problem corresponding to the MIP problem Suppose for example we took M as 1000 when it was known that x would never exceed 100 The following fractional solution would satisfy Inequality 1017 x 70 δ 1 2 1018 but would violate it if M were taken as 100 There are other good reasons for making M as realistic as possible For example if M were again taken as 1000 the following fractional solution would satisfy Inequality 1017 x 5 δ 0005 1019 218 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 2 To make the LP problem corresponding to the IP problem as constrained as possible 3 To use special ordered sets as described in Section 93 if it is possible and the computer package used is capable of dealing with them This is a final objective not yet mentioned in this section Further ways of reformulating IP models in order to ease their solution are described in the next section Before a large IP model is built it is often a very good idea to build a small version of the model first Experimentation with different solution strategies possibly with reformulation can give one valuable experience before one embarks on the much larger model Sometimes by examining the structure of a model it is possible to make observations that lead one to a tightening of the constraints A dramatic example of this even when the application was unknown is described by Daniel 1978 resulting in the solution of a reformulated model in 171 nodes where previously the tree search had been abandoned after 4757 nodes The automatic reformulation of IP models in order to tighten the LP relaxation is described by Crowder et al 1983 and Van Roy and Wolsey 1984 The derivation of facetdefining constraints is discussed by Wolsey 1976 1989 102 Simplifying an integer programming model In Section 101 it was shown that it is often possible to reformulate an IP model in order to create another model that is easier to solve This is sometimes made possible by considering the practical situation being modelled In this section we are concerned with rather less obvious transformations of an IP model Again the aim is to make the model easier to solve 1021 Tightening bounds In Section 34 part of the procedure of Brearley et al 1975 for simplifying LP models was outlined The full application of that procedure involves removing redundant simple bounds in an LP model It is not however generally worth while removing redundant bounds on an integer variable Instead it is better to tighten the bounds if possible The argument for doing this is similar to some of the reformulation arguments used in Section 101 By tightening bounds the corresponding LP problem may be made more constrained resulting in the opti mal solution to the LP problem being closer to the optimal IP solution In order to illustrate the procedure a small example from Balas 1965 is used This example was also used in the description of the procedure given by Brearley Mitra and Williams BUILDING INTEGER PROGRAMMING MODELS II 219 Example 104 Minimize 5δ1 7δ2 10δ3 3δ4 δ5 subject to δ1 3δ2 5δ3 δ4 δ5 2 R1 2δ1 6δ2 3δ3 2δ4 2δ5 0 R2 δ2 2δ3 2δ4 δ5 1 R3 The δi are all 01 variables 1 By constraint R3 2δ3 1 δ2 δ4 δ5 1 Hence δ3 1 2 As δ3 is an integer variable this implied lower bound may be tightened In this case as δ3 is 01 δ3 may be set to 1 and removed from the problem 2 By constraint R2 6δ2 3 2δ1 2δ4 2δ5 1 Hence δ2 1 6 Similarly the lower bound of δ2 may be tightened to 1 so fixing δ2 at 1 3 By constraint R3 δ4 δ5 0 Hence δ4 0 Therefore δ4 can be fixed at 0 4 By constraint R3 δ5 0 Therefore δ5 can be fixed at 0 5 All the constraints now turn out to be redundant and may be removed The only remaining variable is δ1 which must obviously be set to 0 This example is obviously an extreme case of the effect of tightening bounds in an IP model as this procedure alone completely solves the problem 220 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1022 Simplifying a single integer constraint to another single integer constraint Consider the integer constraint 4γ1 6γ2 9 1023 where γ 1 and γ 2 are general integer variables By looking at this constraint geometrically in Figure 102 it is easy to see that it may also be written as γ1 2γ2 2 1024 In Figure 102 the original constraint 1023 indicates the feasible points must lie to the left of AB By shifting the line AB to CD no integer points are excluded from and no new integer points are included in the feasible region CD gives rise to the new constraint 1024 Clearly there are advantages in using constraint 1024 rather than 1023 as the feasible region of the corresponding LP problem has been reduced While constraints such as 1023 involving general integer variables do not arise very frequently such constraints can occur involving only 01 variables We therefore confine our attention to the 01 case Should more general integer variables be involved it is of course always possible to expand them into 01 variables as described in Section 101 Our problem is now given a constraint such as a1δ1 a2δ2 anδn a0 1025 where the δi are 01 variables to reexpress it as an equivalent constraint b1δ1 b2δ2 bnδn b0 1026 2 1 0 1 2 y2 y1 A B C D Figure 102 BUILDING INTEGER PROGRAMMING MODELS II 221 where Inequality 1026 is more constrained than Inequality 1025 in the corresponding LP problem There is no loss of generality in assuming all the coefficients of Inequalities 1025 and 1026 to be nonnegative as should a negative coefficient ai occur in Inequality 1025 the corresponding variable δi may be complemented by the substitution δi 1 δ i making the new coefficient of δ i positive Clearly constraints can be converted to Equality constraints are most conveniently dealt with by converting them into a together with a constraint The result will be two simplified constraints in place of the original single constraint The simplification of a pure 01 constraint such as in Inequality 1025 in order to produce the one in Inequality 1026 can itself be formulated and solved as an LP problem Rather than present the technical details of the procedure here a specific problem OPTIMIZING A CONSTRAINT is given in Part II The formulation and discussion in Part III should make the general procedure clear If the original constraint to be simplified involved general integer variables rather than 01 integer variables and it is desired to replace it by a constraint in the same variables then it will be necessary to restrict the new coefficients of the 01 form For example suppose we had a general integer variable γ 7 If its original coefficient were a in the 01 form of the constraint we would have the term aγ1 2aγ2 4aγ3 1027 with γ 1 2γ 2 4γ 3 representing variable γ where γ 1 γ 2 and γ 3 are 01 variables The resultant simplification would produce an equivalent term b1γ1 b2γ2 b3γ3 1028 For this term to be replaceable by a term bγ we would have to ensure that b3 2b2 b2 2b1 1029 giving b b1 Using the LP formulation described for the 01 example in Part II it would be necessary to impose conditions such as 1029 as extra LP constraints for the general integer case The effort of trying to simplify single integer constraints in this way is often not worthwhile In many models these constraints represent logical conditions and are already in their simplest form as single constraints Applications where such simplification could possibly prove worthwhile are project selection and cap ital budgeting problems It also proves worthwhile to simplify single constraints in this way in the MARKET SHARING problem presented in Part II The simplification of a single 01 constraint into another single 01 con straint considered here has been described by Bradley et al 1974 222 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1023 Simplifying a single integer constraint to a collection of integer constraints We again confine our attention to pure 01 constraints given that general integer variables can be expanded if necessary into 01 variables It may often be advantageous to express a single 01 constraint as a collection of 01 constraints We have already seen this in Section 101 where constraint 1011 was reexpressed as the constraints 1014 Here we present a general procedure for expanding any pure 01 constraint into a collection of constraints Ideally we would like to be able to reexpress the constraint by a collection of constraints defining the convex hull of feasible 01 solutions In order to make this clear we present an example Example 105 3δ1 3δ2 2δ3 2δ4 2δ5 4 1030 δi are 01 variables The convex hull of 01 solutions that are feasible according to Expression 1027 is given by the constraints δ1 δ2 δ3 δ4 1 1031 δ1 δ2 δ3 δ5 1 1032 δ1 δ2 δ4 δ5 2 1033 2δ1 δ2 δ3 δ4 δ5 2 1034 δ1 2δ2 δ3 δ4 δ5 2 1035 together with the trivial constraints δi 0 and δi 1 Unfortunately no practical procedure has yet been devised for obtaining con straints defining the convex hull of integer solutions corresponding to a single 01 constraint The faces that form the boundary of the feasible region defined by an LP problem are known as facets For twovariable problems these facets are onedimensional lines and can easily be visualized in a diagram such as Figure 102 With threevariable problems the facets are twodimensional planes For nvariable problems these facets are n 1dimensional hyper planes Hammer et al 1975 give a table of the facets of the convex hulls for all 01 constraints involving up to five variables In order to use this table it is first necessary to simplify the constraint to another single constraint using the procedure mentioned above It is also possible to obtain the convex hull for par ticular types of constraint involving more than five variables The expansion of constraint 1011 into constraints 1014 in Section 101 is obviously an instance of this Although a general procedure for producing the convex hull constraints for a single 01 constraint does not yet exist Balas 1975 gives a procedure for producing some of the facets of the convex hull Hammer et al 1975 and BUILDING INTEGER PROGRAMMING MODELS II 223 Wolsey 1975 also give similar procedures They are able to obtain those facets represented by constraints containing only coefficients 0 and 1 For example the procedure would obtain the constraints 10311033 of Example 105 We now describe this procedure of Balas Consider the following pure 01 constraint a1δ1 a2δ2 a3δ3 anδn a0 1036 As before there is no loss of generality in considering a constraint with all coefficients nonnegative Definition A subset i 1 i 2 i r of the indices 1 2 n of the coefficients in Inequality 1033 will be called a cover if ai1 ai2 air a0 Clearly it is not possible for all the δi corresponding to indices i within a cover to be 1 simultaneously This condition may be expressed by the constraint δi1 δi2 δir r 1 1037 Definition A cover i 1 i 2 i r is said to be a minimal cover if no proper subset of i 1 i 2 i r is also a cover Definition A minimal cover i 1 i 2 i r can be extended in the following way 1 Choose the largest coefficient aij where i j is a member of the minimal cover 2 Take the set of indices i r 1 i r 2 i r s not in the minimal cover corresponding to coefficients airk such that airk aij 3 Add these set of indices to the minimal cover giving i 1 i 2 i r s This new cover is known as an extended cover If i 1 i 2 i r is a minimal cover it can be augmented to give an extended cover i 1 i 2 i r s The constraint 1037 corresponding to the minimal cover can correspondingly be extended to δi1 δi2 δirs r 1 1038 Definition If a minimal cover gives rise to an extended cover that is not a proper subset of any other extended cover arising from a minimal cover with the same number of indices the original minimal cover is known as a strong cover 224 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Balas shows that if an extended cover arises from a strong cover then the constraints such as in Inequality 1038 encompass all the facets of the convex hull of feasible 01 points of Inequality 1033 that have coefficients 0 or 1 It is obviously fairly easy to devise a systematic and computationally quick method of generating all strong covers corresponding to a 01 constraint and therefore obtaining facet constraints such as Inequality 1038 In this way any facet of the convex hull of feasible 01 points of a constraint can be defined when the facet is represented by a constraint involving only coefficients 0 and 1 Two examples are presented in order to make the process clear Example 106 This is the same as Example 105 only we here show how the constraints 10311033 involving only coefficients 0 and 1 are obtained It is more convenient to write constraint 1030 with positive coefficients as 3δ1 3δ2 2δ 3 2δ4 2δ5 6 1039 where δ 3 1 δ3 The minimal covers of constraint 1039 are 1 2 3 1040 1 2 4 1041 1 2 5 1042 1 3 4 1043 1 3 5 1044 1 4 5 1045 2 3 4 1046 2 3 5 1047 2 4 5 1048 These minimal covers can be augmented to produce the extended covers 1 2 3 from the cover 1040 1049 1 2 4 from the cover 1041 1050 1 2 5 from the cover 1042 1051 1 2 3 4 from the covers 1043 and 1046 1052 1 2 3 5 from the covers 1044 and 1047 1053 1 2 4 5 from the covers 1046 and 1048 1054 226 MODEL BUILDING IN MATHEMATICAL PROGRAMMING All the covers 1062 do extend to the same extended cover 1 2 n n 1 1063 The minimal covers 1061 and 1062 in general are of a different size if r 1 Therefore covers 1061 and 1063 are all extensions of strong covers and give rise after substituting 1 δn 1 for δ n1 to the constraints δi δn1 0 i 1 2 n 1064 and δ1 δ2 δn δn1 r 1 1065 These constraints do not necessarily all represent facets but they are particularly restrictive constraints on the corresponding LP problem and include all the facets with coefficients 0 or 1 It would therefore be advantageous to append constraints 1064 and 1065 to the original constraint 1058 in a model This way of obtaining particularly strong extra constraints to add to an IP model could prove of value in the MARKET SHARING problem of Part II 1024 Simplifying collections of constraints So far we have described ways of simplifying individual constraints involving 01 variables What we would ideally like to do would be to simplify the collec tion of all the constraints into a collection of constraints defining the convex hull of feasible integer points It should be pointed out that simplifying constraints individually will not generally suffice although it may be computationally help ful towards the ultimate aim of solving the IP problem more easily This is demonstrated by an example Example 108 Maximize δ1 2δ2 δ3 subject to 2δ1 3δ2 2δ3 3 1066 δ1 δ2 2δ3 0 1067 where δ1 δ2 δ3 are 01 variables The optimal solution to the associated LP problem is δ1 0 δ2 3 4 δ3 3 8 giving an objective of 15 8 BUILDING INTEGER PROGRAMMING MODELS II 227 If constraint 1066 is replaced by the constraints representing its facets we obtain constraint 1068 in the model below Constraints 1069 and 1070 come from the two facets of constraint 1067 The simplified model is then Maximize δ1 2δ2 δ3 subject to δ1 δ2 δ3 1 1068 δ1 δ3 0 1069 δ2 δ3 0 1070 The optimal solution to the associated LP problem for this reformulated model is δ1 0 δ2 1 2 δ3 1 2 giving an objective of 3 2 Clearly simplifying constraints individually has not constrained the whole model sufficiently to guarantee an integer solution to the LP model although the objec tive value is closer to the integer optimum δ1 0 δ2 0 δ3 1 giving an objective of 1 Unfortunately no practical procedure is known for producing the convex hull of the feasible 01 points corresponding to a general collection of pure 01 constraints In fact if such a computationally efficient procedure were known we could reduce all PIP problems to LP problems and dispense with PIP Procedures of this sort do exist for special restricted classes of PIP problems The best known is for the matching problem that was mentioned in Section 95 The main reference to the problem is found in Edmonds 1965 Partial results exist enabling one to obtain some of the facets of the convex hull for some types of PIP problems Hammer et al 1975 have a procedure for generating the facet constraints involving only coefficients 0 and 1 for a class of PIP problems they call regular Within this class of problems are the set covering problem and the knapsack problem Clearly by being able to cope with the knapsack problem they can also obtain the facets obtained by Balas for a single constraint Apart from what has already been described none of these partial results seems as yet to give a valuable formulation tool for practical problems and they will not therefore be described further It should also be pointed out that procedures have been devised for doing the reverse of what we have described here A collection of pure integer equality constraints can be progressively combined with one another to obtain a single equality constraint giving a knapsack problem Bradley 1971 describes such a 228 MODEL BUILDING IN MATHEMATICAL PROGRAMMING procedure Unfortunately the resulting coefficients are often enormous There is little interest in such a procedure if the corresponding LP problem is to be used as a starting point for solving the IP problem The general effect of aggregating constraints in this way will be to weaken rather than restrict the corresponding LP problem Chvatal and Hammer 1975 also describe a procedure for combin ing pure 01 inequality constraints A more recent paper is by Bienstock and McClosky 2012 Most of the discussion in this section has concerned adding further restrictions to an IP model in order to restrict the corresponding LP model Such extra restrictions can be viewed in the context of cutting planes algorithms for IP We are adding cuts to a model in order to eliminate some possible fractional solutions Most algorithms which make use of cutting planes generate the extra constraints in the course of optimization Our interest here in a book on model building is only in adding cuts to the initial model 1025 Discontinuous variables It is sometimes necessary to restrict a continuous variable to segments of con tinuous values for example x 0 or a x b or x c 1071 where 0 a b c A straightforward approach to follow is that described for disjunctive con straints in Section 94 where 01 variables are used to indicate each of the three or more possibilities A constraint of type 979 forces x to satisfy the condition There is an alternative formulation that has been suggested by Brearley 1975 following a more conventional formulation of a blending problem with logical restrictions by Thomas et al 1978 This is x ay1 by2 cδ2 1072 δ1 y1 y2 δ2 1 1073 where δ1 and δ2 are 01 integer variables and y1 and y2 are nonnegative continuous variables Condition 1071 can clearly be generalized or specialized A common spe cial case is that of a semicontinuous variable that is x 0 or x a a 0 1074 In order to model this an upper bound M must be specified for x giving the formulation x ay1 My2 1075 δ y1 y2 1 1076 where δ is a 01 variable and y1 and y2 are nonnegative continuous variables BUILDING INTEGER PROGRAMMING MODELS II 231 An alternative would be to collect these variables together as nj representing the number of vehicles that visit customer j However in some circumstances such an aggregation of variables can weaken the LP relaxation Such methods of reducing the generation of symmetrically equivalent solutions will generally be dependent on the particular model but it is important to recognize the com putational disadvantages of symmetry in IP unlike many other branches of mathematics 103 Economic information obtainable by integer programming We saw in Section 62 that in addition to the optimal solution values of an LP problem important additional economic information can also be obtained from such quantities as the shadow prices and reduced costs The dual LP model was also shown to have an important economic interpretation in many situations In addition a close relationship between the solution to the original model and its dual exist It is worth just briefly pointing out how the duality relationship fails in IP Suppose we have an IP maximization problem P Corresponding to P we have the LP problem P As long as P is feasible and not unbounded we have a solvable dual problem Q From duality in LP we know that maximum objective of P minimum objective of Q By imposing extra integrality requirements on P we obtain the IP problem P Clearly as P is more constrained than P we have maximum objective of P P minimum objective of Q The minimum objective of Q is the smallest upper bound we can obtain for the objective of P by a set of valuations on the constraints of P This contrasts with the LP case where the dual values provide a strict upper bound ie a bound that is obtained by the optimum for the objective In consequence the difference between the maximum objective value of P and the maximum objective of P or minimum objective of Q is sometimes known as a duality gap It can be regarded rather loosely as a measure of how inadequate any dual values will be when used as shadow prices In this section we attempt to obtain corresponding economic information from an IP model to that obtainable from an LP model It will be seen that this information is much more difficult to come by in the case of IP and is in some cases rather ambiguous In order to demonstrate the difficulties we will consider a product mix problem where the variables in the model represent quantities of different products to be made and the constraints represent limitations on productive capacity It is only meaningful to make integral numbers of each product 232 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Example 109 Maximize 12γ1 5γ2 15γ3 10γ4 subject to 5γ1 γ2 9γ3 12γ4 15 1085 2γ1 3γ2 4γ3 γ4 10 1086 3γ1 2γ2 4γ3 10γ4 8 1087 γ1 γ2 γ3 γ4 0 The γ 1 are general integer variables The optimal integer solution is γ 1 2 γ 2 1 γ 3 0 and γ 4 0 giving an objective value of 29 For comparison the optimal solution to the corresponding LP problem is γ 2 2 3 γ 2 0 γ 3 0 and γ 4 0 giving an objective value of 32 In addition to the fractional solution of the LP problem we would be able to obtain answers to such questions as the following Q1 What is the marginal value of increasing fully utilized capacities Q2 How much should a nonmanufactured product be increased in price to make it worth manufacturing We saw in Section 62 that the answers to Q1 came from the shadow prices on the corresponding constraints These shadow prices represented valuations that could be placed on the capacities Once these optimal valuations have been obtained the optimal manufacturing policy can be deduced by simple accounting Moreover the total valuation for all the capacities implied by these shadow prices is the same as the profit obtainable by the optimal manufacturing policy Unfortunately there are no such neat values that may be placed on capacities in the case of an IP model For example the capacity represented by constraint 1085 is not fully used up in the IP optimal solution In an LP problem if a constraint has slack capacity as in this case it represents a free good as described in Section 62 and has a zero shadow price Such a constraint could be omitted from the LP model and the optimal solution would be unchanged In this case constraint 1085 cannot be omitted without changing the optimal solution We would therefore like to give the constraint some economic valuation This gives the first important difference that must exist between any valuations of constraints in IP and shadow prices in LP A If a constraint has positive slack it does not necessarily represent a free good and may therefore have a positive economic value Why this should be so is fairly easy to see as although there is no virtue in slightly increasing the righthand side value of 15 in constraint 1085 there clearly is virtue in increasing it by at least 3 as we could then bring two of γ 3 into the solution in place of two of γ 1 and one of γ 2 BUILDING INTEGER PROGRAMMING MODELS II 233 Even if we admit positive valuations on unsatisfied as well as satisfied capacities it may still be impossible to arrive at a method of decisionmaking through pricing in a similar way to that described in Section 62 for LP This is demonstrated by the following example Example 1010 Maximize 4γ1 3γ2 γ3 subject to 2γ1 2γ2 γ3 7 1088 where γ 1 γ 2 γ 3 0 and take integer values The optimal solution is γ 1 3 γ 2 0 γ 3 1 We will attempt to find a valuation for the constraint that will produce this answer Suppose we give the constraint a shadow price or accounting value of π then to make γ 3 profitable we must have π 1 This however implies 2π 3 making γ 3 also profitable We know from the optimal solution that γ 3 is worth making but that γ 2 is not This example demonstrates a second difference between any economic valu ation of constraints in IP in comparison with shadow prices in LP B For general IP problems no valuations will necessarily exist for the constraints that allow the optimal solution to be obtained in a similar manner to the LP case By constraints in B we must as usual exclude the feasibility constraints γ 1 0 One way out of the dilemma posed by the IP model above is to obtain constraints representing the convex hull of feasible integer solutions For MIP models we would of course consider the convex hull of points with integer coordinates in the dimensions representing integer variables as mentioned in Section 101 The model can then be treated as an LP problem and shadow prices obtained with desirable properties Although a procedure for obtaining the convex hull of integer solutions is computationally often impractical as discussed in the last section it can be applied to certain models making useful economic information possible This is demonstrated on a particular type of problem the shared fixed cost problem in Example 1012 below In spite of the general computational difficulties it is still worth considering this as a theoretical solution to our dilemma For the purposes of explanation we have reformulated the IP model in Example 109 using constraints for the convex hull of feasible integer points This gives the model below Example 1011 Maximize 12γ1 5γ2 15γ3 subject to γ1 γ3 2 1089 γ1 γ2 γ3 3 1090 2γ1 γ2 3γ3 5 1091 γ3 1 1092 234 MODEL BUILDING IN MATHEMATICAL PROGRAMMING where γ 1 γ 2 γ 3 0 and take integer values When it takes an integer value γ 4 is clearly forced to be zero by constraint 1087 The shadow prices on the new constraints are constraint 1089 shadow price 2 constraint 1090 shadow price 0 constraint 1091 shadow price 5 constraint 1092 shadow price 0 Note that as Example 1011 is degenerate there are alternative shadow prices as explained in Section 62 Unfortunately it is not clear how the new constraints 10891092 defining the convex hull of Example 109 can be related back to the original constraints 10851087 It is therefore difficult to apply the shadow prices above to give meaningful valuations for the physical constraints of the original model An attempt has been made to do this by Gomory and Baumol 1960 They apply a cutting planes algorithm to the IP model successively adding constraints until an integer solution is obtained Then shadow prices are obtained for the original constraints and for the added constraints The shadow prices for the added constraints are imputed back to the original constraints from which they were derived Unfortunately the valuations they obtain for the original constraints are not unique and depend on the way in which the cutting planes algorithm is applied In addition they have to include the feasibility conditions γ i 0 among their original constraints As a result these constraints may end up being given nonzero economic valuations In LP such valuations could be regarded as the reduced costs on the variables γ i and no difficulty would arise Variables with positive reduced costs would be out of the optimal solution With IP using this procedure it would be perfectly possible for the feasibility condition γ i 0 to have nonzero economic values suggesting that in some sense γ i 0 was a binding constraint and for γ i to be in the optimal solution at a positive level One way of justifying this would be to regard the economic valuation given to the feasibility constraint as a cost associated with the indivisibility of γ i Not only should γ i be charged according to the use it makes of scarce capacities it should also be charged an extra amount in view of the fact it can only come in integral quantities The fact that a constraint such as γ i 0 may have a nonzero valuation in the GomoryBaumol system yet γ i may not be at zero level is a special case of a more general difference between the GomoryBaumol prices in IP and the shadow prices in LP C A free good as represented by a constraint in IP does not necessarily have a zero GomoryBaumol price attached to it This difference is not as serious as it might first seem as a similar situation can happen in LP We saw in Section 62 that with degeneracy there are alternate BUILDING INTEGER PROGRAMMING MODELS II 235 dual solutions some of which may give nonzero shadow prices to alternatively redundant constraints This also indicates that the problem of nonuniqueness in the GomoryBaumol prices is not confined to IP It clearly also happens although to a much less serious extent with degenerate problems in LP The exact way of obtaining limited economic information from an MIP model that is used quite widely in practice should be mentioned This is simply to take this information from the LP subproblem at the node in the solution tree that gave the optimal integer solution Such information may well be unreliable as the integer variables should only change by discrete amounts while the economic information results from the effect of marginal changes Other integer variables particularly 01 variables will have become fixed by the bounds imposed in the course of evaluating the solution tree and it will not be possible to evaluate the effect of any changes on these variables A variation of this way of obtaining economic information from an MIP model is to fix all integer variables at their optimal values and only consider the effect of marginal changes on the continuous variables This procedure has something to recommend it as the integer variables usually represent major operating decisions Given that these decisions have been accepted the economic effects of marginal changes within the basic operating pattern may be of interest An example of the need to value the constraints of an MIP model is provided by the TARIFF RATES POWER GENERATION problem in Part II The rates at which electricity is sold on different tariffs implicitly value the constraints Different ways of doing this are discussed with the solution of the model in Part IV In spite of the difficulties in getting meaningful subsidiary economic infor mation out of an IP model there are circumstances where useful information can be obtained by reformulation of the model This is discussed by Williams 1981 We illustrate this in the example below by reformulating a model using the ideas contained in Section 102 The problem considered involves shared fixed costs and is described by Rhys 1970 to whom these ideas are due Example 1012 In the network of Figure 103 the nodes represent capital investments with the costs associated with them The arcs represent moneymaking activities with the estimated revenues associated with them To carry out any activity arc it is necessary to use both the resources represented by the nodes at either end of the arc The problem is to share the capital investment fixed cost associ ated with a node in some optimal way among the activities associated with the arcs joining the node for example how should the capital cost of node D be shared among the arcs AD BD CD and DE An illustrative way of viewing this problem is to think of the nodes as stations with the arcs as railways between those stations In order to show how the required economic information can arise through reformulating an IP model we first consider a rather different problem BUILDING INTEGER PROGRAMMING MODELS II 237 Geometrically we have specified the convex hull of feasible integer points by the constraints 1096 As we now have an LP problem we obtain welldefined shadow prices on the constraints In this example the shadow prices have the following interpretation The shadow price on δi δij 0 is the amount of the capital cost of node i that should be met by revenue from arc ij Similarly the shadow price on δi δij 0 is the amount of the capital cost of node j that should be met by revenue from arc ij We have clearly found a way of sharing the capital costs of the nodes among the arcs Should any activity not be able to meet the capital cost demanded of it it should be cut out This allocation of capital costs will be such as to lead to the most profitable network Using the numbers given on the network in Figure 103 the following shadow prices result as shown in Table 101 Table 101 Constraint Shadow Price δA δAB 0 5 δB δAB 0 2 δA δAC 0 0 δC δAC 0 3 δA δAD 0 0 δD δAD 0 3 δB δBD 0 2 δD δBD 0 6 δC δCD 0 4 δD δCD 0 0 δC δCE 0 0 δE δCE 0 2 δD δDE 0 0 δE δDE 0 1 It will be seen that for example node C will receive 3 from AC 4 from CD and 0 from CE Clearly node C is no longer viable and must be cut Applying similar arguments to all the other nodes we are left with the optimal network shown in Figure 104 238 MODEL BUILDING IN MATHEMATICAL PROGRAMMING A D B 5 3 8 4 8 7 Figure 104 An interesting observation following from duality in LP is that dividing up the capital costs of the nodes in other ways among the arcs could not lead to a more profitable network and could well lead to a less profitable one It should have become apparent from all the discussion in this section that there is no generally satisfactory way of getting the subsidiary economic infor mation from an IP model that often proves so valuable in the case of LP This topic represents a considerable gap in mathematical programming theory The subject is more fully discussed in Williams 1979 1997 Appa 1997 shows that after fixing the values of integer variables in a mixed integer programming model the structural duality of the resulting LP model can sometimes be used Greenberg 1998 gives a bibliography 104 Sensitivity analysis and the stability of a model We saw in Section 63 that having built and solved an LP model it was very important to see how sensitive the answer was to changes in the data from which the model was constructed Using ranging procedures on the objective and right hand side coefficients some insight into this could be obtained In addition it was shown how a model could to some extent be built in order to behave in a stable fashion Our considerations here are exactly the same as those in Section 63 only they concern IP models As in the Section 103 we will see that IP models present considerable extra difficulties This section falls into two parts Firstly we consider ways of testing the sensitivity of the solution of an IP model Secondly we consider how models may be built in order that they may exhibit stability 1041 Sensitivity analysis and integer programming A theoretical way of doing sensitivity analysis on the objective coefficients of a model would be to replace the constraints by constraints representing the convex hull of feasible integer points The model could then be treated as an LP model and objective ranging performed as described in Section 63 For those PIP models where a reformulation easily yields the constraints for the convex hull this is fairly straightforward Otherwise it is not a practical way BUILDING INTEGER PROGRAMMING MODELS II 239 of approaching the problem Nor does it give a way of performing righthand side ranging For MIP models solved by the branch and bound method a sensitivity anal ysis can be performed on the LP subproblem at the node giving the optimal integer solution Alternatively the integer variables can be fixed at their optimal values and a sensitivity analysis performed on the continuous part of the prob lem These approaches clearly have the same drawbacks as those apparent when using similar approaches to derive economic information from an IP model as described in Section 103 The only really satisfactory method of sensitivity analysis in IP involves solv ing the model again with changed coefficients and comparing optimal solutions Obviously the subsequent time to solve the model should be able to be reduced by exploiting the knowledge of the previous solution 1042 Building a stable model In an LP model the optimal value of the objective function varies continuously as the righthand side and objective coefficients are changed In an IP model this may not happen We consider the following very simple example Example 1013 Maximize 40δ1 35δ2 15δ3 8δ4 9δ5 subject to 8δ1 8δ2 5δ3 4δ4 3δ5 16 1097 where δ1 δ2 δ3 δ4 δ5 are 01 variables The optimal solution to this model is δ1 1 and δ2 1 giving an objective value of 75 If however the righthand side value of 16 is reduced by a small amount the optimal solution changes to δ1 1 δ4 1 and δ5 1 giving an objective value of 57 The optimal value of the objective function is obviously not a continuous function of the righthand side coefficient In many practical situations this is unrealistic Suppose constraint 1097 represented a budgetary limitation and the 01 variables referred to capital investments It is very unlikely that a small decrease in the budget would cause us radically to alter our plans A more likely occurrence is that the budget would be stretched slightly at some increased cost or the cost of one of the capital investments would be trimmed slightly It is important that if this is the case this should be represented in the model As it stands the above example represents a poor model of the situation One way to remodel constraint 1097 is to use the device described in Section 33 in order to allow constraints to be violated at a certain cost A surplus 240 MODEL BUILDING IN MATHEMATICAL PROGRAMMING continuous variable u is added into constraint 1097 and given a cost say 20 in the objective This results in the following model Maximize 40δ1 35δ2 15δ3 8δ4 9δ5 20u subject to 8δ1 8δ2 5δ3 4δ4 3δ5 u 16 1098 For a righthand side of 16 the optimal solution is still δ1 δ2 1 giving an objective value of 75 If the righthand side value of 16 is reduced slightly the effect of u will be to top the budget up to 16 at a unit cost of 20 For example if the righthand side falls to 15 1 2 u will become 1 2 We will retain the same optimal solution but the objective will fall by 10 to 65 As the righthand side is further reduced the optimal value of the objective will continue gradually to fall until we reach a righthand side value of 15 1 10 We will then have the solution δ1 1 δ4 1 and δ5 1 as an alternative optimum If the righthand side is further reduced we will transfer to this alternative optimum It can be seen that this device of adding a surplus variable to the problem with a certain cost has two desirable effects on the model 1 The optimal objective value becomes a continuous function of the right hand side coefficient 2 The optimal solution values do not change suddenly as the righthand side coefficient changes They are said to be semicontinuous functions of the righthand side In some applications we might wish to add a slack continuous variable v as well giving v a cost in the objective of say 8 This would result in the following model Maximize 40δ1 35δ2 15δ3 8δ4 9δ5 20u 8v subject to 8δ1 8δ2 5δ3 4δ4 3δ5 u v 16 1099 This topic is not pursued further here as it has been fully covered for LP problems in Section 63 It is important to notice however the desirability of forcing the optimal solution of an IP model to vary continuously with the data coefficients Clearly for many logical type constraints which appear in MIP models it is meaningless to use a device such as that above There are some classes of MIP models where the continuity property can be shown to hold without further reformulation This subject is discussed in greater depth by A C Williams 1989 105 When and how to use integer programming In this section we try very briefly to summarize some of the points made in the preceding three chapters as a quick guide to using IP BUILDING INTEGER PROGRAMMING MODELS II 241 1 If a practical problem has any of the characteristics described in Section 82 it is worth considering the use of an IP model 2 Before a decision is made to build an IP model an estimate should be made of the potential size If the number of integer variables is more than a few hundred then unless the problem has a special structure it is probable that IP will be computationally too costly 3 A close examination of the structure of the IP model that would result from the problem is always worthwhile Should the model be a PIP model and have a totally unimodular structure then LP can be used and models involving thousands of constraints and variables solved in a reasonable period of time If the structure is a PIP model but not totally unimodular it is worth seeing if it can easily be transformed into a known totally uni modular structure as described in Section 102 For MIP models or models where there is no apparent way of producing a unimodular structure it is often possible to constrain the corresponding LP problem more tightly If it is apparent that the IP model will have one of the other special struc tures mentioned in Section 95 it is worth examining the literature and computational experience with the class of problem to get an impression of the difficulty 4 Before embarking on the fullscale model it is worth building a model for a much smaller version of the problem Experiments should be performed on this model to find out how easy it is to solve If necessary reformulations such as those described in Section 102 should be tried Different solution strategies as mentioned in Section 83 should also be experimented with 5 If the problem appears too difficult to solve as an IP model after carry ing out the above investigations some heuristic method will have to be used Much literature exists on different heuristic algorithms for opera tional research problems but this topic is beyond the scope of this book For an apparently difficult problem where an IP model still seems worth while it may also be worth some time being spent on a heuristic approach to get a fairly good though probably not optimal solution This good solution can then be exploited in the tree search as a cutoff value for the objective function as described in Section 83 6 Having built an IP model it is very important to use an intelligent solution strategy using if possible ones knowledge of the practical problem This has been mentioned briefly in Section 83 but is in the main beyond the scope of a book on model building An extremely good description of this subject is given by Forrest et al 1974 Finally it should be pointed out that theoretical and computational progress in IP is being made all the time making it possible to solve larger and more complex models 11 The implementation of a mathematical programming system of planning 111 Acceptance and implementation Most of this book has been concerned with the problems of formulating and interpreting the solution of mathematical programming models There is how ever generally another phase to be gone through before the solution of a model influences the making of real decisions This final phase is that of gaining accep tance for and implementing the solution Many people who have been involved with all the phases of formulating a model solving it interpreting the solution gaining acceptance for the solution and then implementing it have found the last two phases to be the most difficult In some cases they may have stumbled fatally at this point There are a number of lessons to be learnt from such expe riences that are considered in this chapter Obviously the problem of acceptance and implementation will depend on the type of organization involved as well as the type of application It is useful here to classify mathematical programming models for planning into short medium and longterm planning models Shortterm planning models may be simply one off models used to decide the answer to a specific question for example do we build a new factory or not what is the optimum design for this communications network If these questions are unlikely to recur then there is generally little to be said about the acceptance and implementation The solution produced by the model is either used or not used For other shortterm planning questions that do arise regularly at daily or weekly intervals there may be an implementation problem Firstly it should be pointed out that in some cases a mathematical programming model may be Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 244 MODEL BUILDING IN MATHEMATICAL PROGRAMMING applicable but unworkable For example a complicated distribution or jobshop scheduling problem may be essentially dynamic Changes may be taking place all the time new orders may be coming in and machines may be breaking down It might be impossible to define a once and for all schedule To adapt such a schedule to each change might involve a rerun of the model Unless the changes were infrequent and the model comparatively quick to solve such an approach would be impossible In such cases special purpose adaptive and probably non optimal quick decision rules would probably be used Some shortterm decision problems that occur regularly can often be tackled through a mathematical pro gramming model Day to day blending problems for example may be of this nature Sometimes fertilizer suppliers or food manufacturers run small linear pro gramming models for individual orders or blends In such cases acceptance of the method must have already been achieved An organizational problem will also have had to be solved probably by automating the conversion of the data into a standard type of model for quick solution by a computer probably through a terminal The use of such shortterm operational models creates few other imple mentation problems because once accepted their use is fairly straightforward Mediumterm planning is usually considered to involve periods of a month up to one or two years It is for these problems that mathematical programming has been most widely used to date Once a mediumterm planning model has been implemented and is being used regularly it may be incorporated into or form a starting point for a longer term planning model typically looking up to about six years ahead The problems of getting acceptance of such models are similar but usually more acute in the case of longer term planning Both are considered together Much has been written on the more general problem of how to gain acceptance for and ensure the implementation of the results of any operational research study A clear and useful account of the considerations that should be made is given by Rivett 1968 The main lesson to be learnt is to involve the potential decision makers in a model building project at an early stage If they have lived with the problems of defining and building a model they will be much more likely to accept and understand its final use than if it is sprung on them at a late stage Early involvement by top management can have its dangers It is possible that the development of a model may get bogged down in detailed technical arguments between managers who would normally never be concerned with such technicalities More seriously a model building project might be rejected at an early stage through disagreement over detail These risks are however normally worth taking in comparison with the risk of rejection at the final hurdle Involving top management will often be by no means easy They will need to understand the potentialities of a mathematical programming model but will probably lack both detailed technical knowledge as well as detailed departmental knowledge of some aspects of their company Some of this detail may well need to be incorporated into the model The solution is probably to involve one or two such managers in the model building project as well as give regular presentations to the others There is then of course another danger to be avoided It is important not MATHEMATICAL PROGRAMMING SYSTEM OF PLANNING 245 to oversell the model If it is thought that the model will answer every question then later disillusion may be in store Many people have found that a new mathematical programming model has gained acceptance when the answers which it produces confirm a decision that has already been made For example an important investment decision may have been made in some other way If the answer to the model confirms this the ultimate regular use of mathematical programming may be assured It has already been suggested that sometimes the very exercise of building a model leads to the explicit recognition of relationships that were not realized before In such circumstances the modelling exercise may be as valuable as the answers produced When this happens the effect on management may well be to make the use of mathematical programming more acceptable The ultimate acceptance of a regular mathematical programming system of planning usually requires organizational changes that are considered in the next section A very full and useful description of the experiences in developing and gain ing acceptance for a large longterm planning mathematical programming model in British Petroleum is described by Stewart 1971 An analysis of the factors that make for success or failure in implementing the results of corporate models is given by Harvey 1970 On the result of a survey he isolates the characteristics of both management and decision problems that lead to success or failure in the implementation of a model Problems of implementation are also discussed very fully in the book by Miller and Starr 1960 A perceptive analysis of three industrial problems where linear programming has yielded disappointing results is given by Corner 1979 He draws conclusions and presents advice on the basis of these experiences Finally as was mentioned in Section 23 it seems quite likely that math ematical programming methods will be incorporated in much larger pieces of software known as decision support systems The resultant systems will usually be tailormade to specific applications In this way much of the effort of model building will be transferred to the computer system It will then be necessary to include model building expertise in the designing and writing of such systems 112 The unification of organizational functions One off the virtues of a corporate planning model is that many interconnections between different departments and functions in an organization have to be rep resented explicitly The obvious example is production and marketing In many manufacturing industries these two functions are kept very separate The result is sometimes a divergence of objectives Production may be trying to satisfy orders with a reasonable level of productivity Marketing may be trying to maximize the total volume of sales rather than concentrating on those that make the great est contribution to profit One of the greatest virtues of a product mix type of model is that it forces marketing to take account of production costs The result 246 MODEL BUILDING IN MATHEMATICAL PROGRAMMING is almost always a reduction in the range of products produced to those that can be produced most efficiently In practice with this type of model the incorpo ration of the marketing function can present difficulties As was suggested in Section 32 it may sometimes be politically more acceptable to first of all leave out the marketing aspect A model is built simply to meet all market estimates at minimum production cost When this type of model has come into regular use it can fairly easily be extended also to decide quantities to be marketed taking into account their profit contributions The marketing division of a company is often less able to quantify its data and more reluctant to accept the answers produced by a model Indeed their individual objective may be at variance with that of the company Stewart 1971 gives a case history of a linear programming model built for the Gas Council which was considered to be of little use for marketing In order to give an idea of the different company functions that can be incor porated in a mathematical programming model Figure 111 demonstrates how purchasing operating and planning functions can become involved in a model Storage availability and prices Production capacities and restrictions Production planner Production capacities and restrictions Monthly production plan Raw availabilities and costs Raw oil purchases Monthly production plan Production planner The buyer Operations planning manager Market demands all brands Figure 111 The model here is described by Williams and Redwood 1974 and takes in all the functions in the box enclosed by a dotted line It is a multiperiod multibrand model for aiding decisions of the purchase of edible oils on the commodity market and their blending together into foods The involvement of different functions and departments in the building and use of a model such as MATHEMATICAL PROGRAMMING SYSTEM OF PLANNING 247 this can force some degree of centralization on an organization This aspect is considered in Section 113 While the explicit recognition of interrelationships in an organization through a model is desirable it usually makes extra coordination necessary for the supply of data to and the use of a model Often this requires the creation of a special job or even a small department Such a coordinating role is often effectively done through the management accounting department They are one of the few departments with an overview of the whole organization One of the possibly contentious results of decisions which may be reached through a corporate model should be recognized Because such a model will have a corporate objective this may conflict within the objectives of individual departments It has already been pointed out that a marketing division may force them to reduce their product range and therefore their total volume in the interests of greater company profit The disputes that may result in consequence are added reason for obtaining top management backing for this method of planning In Section 41 it was demonstrated through a very simple example how a corporate objective could result in an individual factory having its individual profit reduced in the interests of overall company profit Again the possible political implications are obvious Although there may be obvious difficulties in getting different departments to work more closely together there can be advantages resulting from a planning model The combination of a large amount of data and information in one com puter model can lead to greater convenience Each department can be given the relevant portions of a computer run Problems of communication and incompat ibility of information will be reduced in consequence One of the byproducts of a corporate model is usually greater contact between departments Each depart ment will want to know what information other departments have fed into the model because this may affect the suggested plans for their own department The correct use of a multiperiod model for planning has already been men tioned in Section 41 Such a model may well provide operating information for the current time period as well as provisional information for future periods Such a model would generally be rerun at least once in each period giving the same combination of operating information and planning information based on the best available future estimates that have been fed into the model 113 Centralization versus decentralization A corporate model obviously brings into its orbit many aspects of an organi zation which might normally be separated into different divisions Sometimes these divisions may be separated geographically In some respects the resulting greater centralization of planning may seem desirable The small example of Section 41 demonstrated how individual factories in a company might produce suboptimal plans when working independently In the last section it was pointed out how the recognition of the interdependence of different departments in an 248 MODEL BUILDING IN MATHEMATICAL PROGRAMMING organization often results from a model Although many of these features are to some degree desirable when carried to excess they may be far from desirable In fact the increasing disadvantages of greater and greater centralization have been a major factor in discouraging the continual growth in the size of mathematical programming models In theory decomposition which was discussed in Section 42 offers a way out by avoiding the need to include all the details of an organization in one total model Unfortunately the computational problems of using a decomposi tion method in general have not yet been fully resolved At present it cannot be considered to be a sufficiently developed technique to be always usable in practice Moreover mathematical methods of decomposition do not always cor respond to acceptable decentralized planning methods This aspect of the subject has been discussed by Atkins 1974 Stewart 1971 rather surprisingly suggests that the use of a longterm planning model in British Petroleum led to some degree of decentralization In particular it was then possible for a division to obtain complete information before making a plan themselves rather than the information being restricted to head office This enabled the division to take on more responsibility On the other hand the division now had to work within some centrally defined objective rather than choose its own objective People now however felt that they were working to an independent system in the form of a model rather than being simply dictated to by head office This to some extent made centralization more acceptable As was pointed out in Chapter 6 a wealth of information can be obtained from the solution to a linear programming model For large corporate models this excess of information can create difficulties It is very important to ensure that departments only get relevant information otherwise they can be submerged The use of an automatic report writer as discussed in Section 65 is obviously desirable Such a report writer can be designed to produce the information in different sections for the use of different departments It is necessary to strike a correct balance between the model builders the contributors of data and the managers who use the results There is a great danger in a large organization of them losing touch with one another Relevant data may not be used at the right time and inaccurate results may be obtained Ideally management will become accustomed to using a model on a what if basis They will ask questions which the model builder will be able to answer quickly by new runs or postoptimal analysis The degree of centralization or decentralization which the use of a model should be allowed to impose on an organization must obviously depend very much on the organization and its ultimate objectives In consequence few general guidelines can be given The recognition of the problem of centralization is however important It is interesting that mathematical programming was used quite widely in the former communist countries for national economic planning In the West this is far less common Models are usually limited to the level of individual companies MATHEMATICAL PROGRAMMING SYSTEM OF PLANNING 249 114 The collection of data and the maintenance of a model The collection of data is a major task in building a large mathematical program ming model for the first time Once this has been done the organization should be geared to providing and updating the data regularly Inevitably much work will be involved in collecting suitable data Coming from different departments it will often need to be standardized Again such standardization and coordination will ultimately need to be the responsibility of a special person or department if the model is to be used regularly As has already been pointed out a man agement accounting department is often a suitable place to which to entrust this responsibility The person entrusted with the regular collection and standardization of the data must be in a position where heshe is kept up to date with all changes These changes will come from different departments There may be marketing estimates changes in productive capacity technological changes in production processes and changes in raw materials costs Ensuring that all this information gets incorporated in the model and it remains an uptodate representation is no mean task It may sometimes be necessary to do preliminary processing of the data For example sales estimates may be the result of a regular forecast ing exercise This may require preliminary computation It may be necessary to convert costing data to a suitable form for example to ensure that only variable costs are used With some packages it is now possible to prepare manipulate and present the data though spreadsheets The solutions are sometimes conveniently presented in this form also The use and regular updating of databases to include all relevant statistical information is of value when used in conjunction with a mathematical program ming model Such databases may also be used for other purposes as well within an organization It is important however to ensure that their organization and design are compatible with the mathematical programming model used Most commercial package programs for mathematical programming can be incorpo rated as part of a much larger computer software system eg a decision support system using a database As with so many computer applications their use does not reduce the need for employees instead it changes the type of employee required and sometimes increases the total number of people It should be obvious that the problems of collecting data and maintaining a model deserve considerable attention and effort if its use is to be effective The gain should come from the wider range of policy options that can be considered as well as their greater reliability 12 The problems There is no significance in the order in which the following 29 problems are presented Some are easy to formulate and present no computational difficulties in solution while others are more difficult in either or both of these respects Some can be solved with linear programming others require integer programming or separable programming It is suggested that readers attempt to formulate those problems that interest them before consulting the formulations and solutions proposed in Parts III and IV A computer package may be used on the model or an intuitive heuristic approach to some of the problems may be attempted using original methods of the readers own choosing answers can then be compared with the optimal solutions given in Part IV Readers wishing to follow the recommended course of solving the models using a computer package are strongly advised to use a matrix generatorlanguage as discussed in Sections 35 and 43 This will enable them to concentrate on the structure of the model as well as facilitating error detection and greatly reducing data preparation 121 Food manufacture 1 A food is manufactured by refining raw oils and blending them together The raw oils are of two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 254 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Each oil may be purchased for immediate delivery January or bought on the futures market for delivery in a subsequent month Prices at present and in the futures market are given below in ton VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 The final product sells at 150 per ton Vegetable oils and nonvegetable oils require different production lines for refining In any month it is not possible to refine more than 200 tons of vegetable oils and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored It is possible to store up to 1000 tons of each raw oil for use later The cost of storage for vegetable and nonvegetable oil is 5 per ton per month The final product cannot be stored nor can refined oils be stored There is a technological restriction of hardness on the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly and that the hardnesses of the raw oils are VEG 1 88 VEG 2 61 OIL 1 20 OIL 2 42 OIL 3 50 What buying and manufacturing policy should the company pursue in order to maximise profit At present there are 500 tons of each type of raw oil in storage It is required that these stocks will also exist at the end of June This problem and the subsequent problem is based on a larger model built for the margarine producer Van den Bergs and Jurgens and discussed in Williams and Redwood 1974 THE PROBLEMS 255 122 Food manufacture 2 It is wished to impose the following extra conditions on the food manufacture problem 1 The food may never be made up of more than three oils in any month 2 If an oil is used in a month at least 20 tons must be used 3 If either of VEG 1 or VEG 2 are used in a month then OIL 3 must also be used Extend the food manufacture model to encompass these restrictions and find the new optimal solution 123 Factory planning 1 An engineering factory makes seven products PROD 1 to PROD 7 on the following machines four grinders two vertical drills three horizontal drills one borer and one planer Each product yields a certain contribution to profit defined as unit selling price minus cost of raw materials These quantities in unit together with the unit production times hours required on each process are given below A dash indicates that a product does not require a process PROD 1 PROD 2 PROD 3 PROD 4 PROD 5 PROD 6 PROD 7 Contribution to profit 10 6 8 4 11 9 3 Grinding 05 07 03 02 05 Vertical drilling 01 02 03 06 Horizontal drilling 02 08 06 Boring 005 003 007 01 008 Planing 001 005 005 In the present month January and the five subsequent months certain machines will be down for maintenance These machines will be as follows January 1 Grinder February 2 Horizontal drills March 1 Borer April 1 Vertical drill May 1 Grinder and 1 Vertical drill June 1 Planer and 1 Horizontal drill 256 MODEL BUILDING IN MATHEMATICAL PROGRAMMING There are marketing limitations on each product in each month These are given in the following table 1 2 3 4 5 6 7 January 500 1000 300 300 800 200 100 February 600 500 200 0 400 300 150 March 300 600 0 0 500 400 100 April 200 300 400 500 200 0 100 May 0 100 500 100 1000 300 0 June 500 500 100 300 1100 500 60 It is possible to store up to 100 of each product at a time at a cost of 05 per unit per month There are no stocks at present but it is desired to have a stock of 50 of each type of product at the end of June The factory works a six days a week with two shifts of 8 h each day No sequencing problems need to be considered When and what should the factory make in order to maximise the total profit Recommend any price increases and the value of acquiring any new machines NB It may be assumed that each month consists of only 24 working days This problem and the subsequent problem is based on a larger model built for the Cornish engineering company of Holman Brothers which no longer exists 124 Factory planning 2 Instead of stipulating when each machine is down for maintenance in the factory planning problem it is desired to find the best month for each machine to be down Each machine must be down for maintenance in one month of the six apart from the grinding machines only two of which need be down in any six months Extend the model to allow it to make these extra decisions How much is the extra flexibility of allowing down times to be chosen worth 125 Manpower planning A company is undergoing a number of changes that will affect its manpower requirements in future years Owing to the installation of new machinery fewer unskilled but more skilled and semiskilled workers will be required In addition to this a downturn in trade is expected in the next year which will reduce the need for workers in all categories The estimated manpower requirements for the next three years are as follows THE PROBLEMS 257 Unskilled Semiskilled Skilled Current strength 2000 1500 1000 Year 1 1000 1400 1000 Year 2 500 2000 1500 Year 3 0 2500 2000 The company wishes to decide its policy with regard to the following over the next three years 1 Recruitment 2 Retraining 3 Redundancy 4 Shorttime working There is a natural wastage of labour A fairly large number of workers leave during their first year After this the rate is much smaller Taking this into account the wastage rates can be taken as follows Unskilled Semiskilled Skilled Less than one years service 25 20 10 More than one years service 10 5 5 There has been no recent recruitment and all workers in the current labour force have been employed for more than one year 1251 Recruitment It is possible to recruit a limited number of workers from outside In any one year the numbers that can be recruited in each category are as follows Unskilled Semiskilled Skilled 500 800 500 1252 Retraining It is possible to retrain up to 200 unskilled workers per year to make them semi skilled This costs 400 per worker The retraining of semiskilled workers to make them skilled is limited to no more than one quarter of the skilled labour force at the time as some training is done on the job Retraining a semiskilled worker in this way costs 500 258 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Downgrading of workers to a lower skill is possible but 50 of such workers leave although it costs the company nothing This wastage is additional to the natural wastage described above 1253 Redundancy The redundancy payment to an unskilled worker is 200 and to a semiskilled or skilled worker is 500 1254 Overmanning It is possible to employ up to 150 more workers over the whole company than are needed but the extra costs per employee per year are as follows Unskilled Semiskilled Skilled 1500 2000 3000 1255 Shorttime working Up to 50 workers in each category of skill can be put on shorttime working The cost of this per employee per year is as follows Unskilled Semiskilled Skilled 500 400 400 An employee on shorttime working meets the production requirements of half a fulltime employee The companys declared objective is to minimise redundancy How should they operate in order to do this If their policy were to minimise costs how much extra would this save Deduce the cost of saving each type of job each year 126 Refinery optimisation An oil refinery purchases two crude oils crude 1 and crude 2 These crude oils are put through four processes distillation reforming cracking and blending to produce petrols and fuels that are sold 1261 Distillation Distillation separates each crude oil into fractions known as light naphtha medium naphtha heavy naphtha light oil heavy oil and residuum according to THE PROBLEMS 259 their boiling points Light medium and heavy naphthas have octane numbers of 90 80 and 70 respectively The fractions into which one barrel of each type of crude splits are given in the following table Light naphtha Medium naphtha Heavy naphtha Light oil Heavy oil Residuum Crude 1 01 02 02 012 02 013 Crude 2 015 025 018 008 019 012 NB There is a small amount of wastage in distillation 1262 Reforming The naphthas can be used immediately for blending into different grades of petrol or can go through a process known as reforming Reforming produces a product known as reformed gasoline with an octane number of 115 The yields of reformed gasoline from each barrel of the different naphthas are given as follows 1 barrel of light naphtha yields 06 barrels of reformed gasoline 1 barrel of medium naphtha yields 052 barrels of reformed gasoline 1 barrel of heavy naphtha yields 045 barrels of reformed gasoline 1263 Cracking The oils light and heavy can either be used directly for blending into jet fuel or fuel oil or be put through a process known as catalytic cracking The catalytic cracker produces cracked oil and cracked gasoline Cracked gasoline has an octane number of 105 1 barrel of light oil yields 068 barrels of cracked oil and 028 barrels of cracked gasoline 1 barrel of heavy oil yields 075 barrels of cracked oil and 02 barrels of cracked gasoline Cracked oil is used for blending fuel oil and jet fuel cracked gasoline is used for blending petrol Residuum can be used for either producing lubeoil or blending into jet fuel and fuel oil 1 barrel of residuum yields 05 barrels of lubeoil 260 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1264 Blending 12641 Petrols motor fuel There are two sorts of petrol regular and premium obtained by blending the naphtha reformed gasoline and cracked gasoline The only stipulations concern ing them are that regular must have an octane number of at least 84 and that premium must have an octane number of at least 94 It is assumed that octane numbers blend linearly by volume 12642 Jet fuel The stipulation concerning jet fuel is that its vapour pressure must not exceed 1 kg cm2 The vapour pressures for light heavy cracked oils and residuum are 10 06 15 and 005 kg cm2 respectively It may again be assumed that vapour pressures blend linearly by volume 12643 Fuel oil To produce fuel oil light oil cracked oil heavy oil and residuum must be blended in the ratio 10431 There are availability and capacity limitations on the quantities and processes used as follows 1 The daily availability of crude 1 is 20 000 barrels 2 The daily availability of crude 2 is 30 000 barrels 3 At most 45 000 barrels of crude can be distilled per day 4 At most 10 000 barrels of naphtha can be reformed per day 5 At most 8000 barrels of oil can be cracked per day 6 The daily production of lube oil must be between 500 and 1000 barrels 7 Premium motor fuel production must be at least 40 of regular motor fuel production The profit contributions from the sale of the final products are in pence per barrel as follows Premium petrol 700 Regular petrol 600 Jet fuel 400 Fuel oil 350 Lubeoil 150 How should the operations of the refinery be planned in order to maximise total profit THE PROBLEMS 261 127 Mining A mining company is going to continue operating in a certain area for the next five years There are four mines in this area but it can operate at most three in any one year Although a mine may not operate in a certain year it is still necessary to keep it open in the sense that royalties are payable if it be operated in a future year Clearly if a mine is not going to be worked again it can be permanently closed down and no more royalties need be paid The yearly royalties payable on each mine kept open are as follows Mine 1 5 million Mine 2 4 million Mine 3 4 million Mine 4 5 million There is an upper limit to the amount of ore which can be extracted from each mine in a year These upper limits are as follows Mine 1 2 106 tons Mine 2 25 106 tons Mine 3 13 106 tons Mine 4 3 106 tons The ore from the different mines is of varying quality This quality is mea sured on a scale so that blending ores together results in a linear combination of the quality measurements for example if equal quantities of two ores were combined the resultant ore would have a quality measurement half way between that of the ingredient ores Measured in these units the qualities of the ores from the mines are given as follows Mine 1 10 Mine 2 07 Mine 3 15 Mine 4 05 In each year it is necessary to combine the total outputs from each mine to produce a blended ore of exactly some stipulated quality For each year these qualities are as follows 262 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Year 1 09 Year 2 08 Year 3 12 Year 4 06 Year 5 10 The final blended ore sells for 10 ton each year Revenue and expenditure for future years must be discounted at a rate of 10 per annum Which mines should be operated each year and how much should they produce This problem is based on a larger one arising in deciding which pits to work in the firm of English China Clays In that problem in the 1970s it was wished to work up to 4 mines out of 20 in each year The model proved very difficult to solve 128 Farm planning A farmer wishes to plan production on his 200 acre farm over the next five years At present he has a herd of 120 cows This is made up of 20 heifers and 100 milkproducing cows Each heifer needs 23 acre to support it and each dairy cow 1 acre A dairy cow produces an average of 11 calves per year Half of these calves will be bullocks which are sold almost immediately for an average of 30 each The remaining heifers can be either sold almost immediately for 40 or reared to become milkproducing cows at two years old It is intended that all dairy cows be sold at 12 years old for an average of 120 each although there will probably be an annual loss of 5 per year among heifers and 2 among dairy cows At present there are 10 cows each aged from newborn to 11 years old The decision of how many heifers to sell in the current year has already been taken and implemented The milk from a cow yields an annual revenue of 370 A maximum of 130 cows can be housed at the present time To provide accommodation for each cow beyond this number will entail a capital outlay of 200 per cow Each milk producing cow requires 06 tons of grain and 07 tons of sugar beet per year Grain and sugar beet can both be grown on the farm Each acre yields 15 tons of sugar beet Only 80 acres are suitable for growing grain They can be divided into four groups whose yields are as follows Group 1 20 acres 11 tons per acre Group 2 30 acres 09 tons per acre Group 3 20 acres 08 tons per acre Group 4 10 acres 065 tons per acre THE PROBLEMS 263 Grain can be bought for 90 per ton and sold for 75 per ton Sugar beet can be bought for 70 per ton and sold for 58 per ton The labour requirements are as follows Each heifer 10 h per year Each milkproducing cow 42 h per year Each acre put to grain 4 h per year Each acre put to sugar beet 14 h per year Other costs are as follows Each heifer 50 per year Each milkproducing cow 100 per year Each acre put to grain 15 per year Each acre put to sugar beet 10 per year Labour costs for the farm are at present 4000 per year and provide 5500 h labour Any labour needed above this will cost 120 per hour How should the farmer operate over the next five years to maximise profit Any capital expenditure would be financed by a 10year loan at 15 annual interest The interest and capital repayment would be paid in 10 equally sized yearly instalments In no year can the cash flow be negative Finally the farmer would neither wish to reduce the total number of dairy cows at the end of the fiveyear period by more than 50 nor wish to increase the number by more than 75 129 Economic planning An economy consists of three industries coal steel and transport Each unit pro duced by one of the industries a unit will be taken as 1s worth of value of production requires inputs from possibly its own industry as well as other indus tries The required inputs as well as the manpower requirements also measured in are given in Table 121 There is a time lag in the economy so that the output in year t 1 requires an input in year t Output from an industry may also be used to build productive capacity for itself or other industries in future years The inputs required to give unit increases capacity for 1s worth of extra production in productive capacity are given in Table 122 Input from an industry in year t results in a permanent increase in productive capacity in year t 2 Stocks of goods may be held from year to year At present year 0 the stocks and productive capacities per year are given in Table 123 in m There is a limited yearly manpower capacity of 470 m 264 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 121 Inputs year t Outputs year t 1 production Coal Steel Transport Coal 01 05 04 Steel 01 01 02 Transport 02 01 02 Manpower 06 03 02 Table 122 Inputs year t Outputs year t 2 productive capacity Coal Steel Transport Coal 00 07 09 Steel 01 01 02 Transport 02 01 02 Manpower 04 02 01 Table 123 Year 0 Stocks Productive capacity Coal 150 300 Steel 80 350 Transport 100 280 It is wished to investigate different possible growth patterns for the economy over the next five years In particular it is desirable to know the growth patterns that would result from pursuing the following objectives 1 Maximising total productive capacity at the end of the five years while meeting an exogenous consumption requirement of 60 m of coal 60 m of steel and 30 m of transport in every year apart from year 0 2 Maximising total production rather than productive capacity in the fourth and fifth years but ignoring exogenous demand in each year 3 Maximising the total manpower requirement ignoring the manpower capacity limitation over the period while meeting the yearly exogenous demands of 1 THE PROBLEMS 265 1210 Decentralisation A large company wishes to move some of its departments out of London There are benefits to be derived from doing this cheaper housing government incen tives easier recruitment etc which have been costed Also however there will be greater costs of communication between departments These have also been costed for all possible locations of each department Where should each department be located so as to minimise overall yearly cost The company comprises five departments A B C D and E The possible cities for relocation are Bristol and Brighton or a department may be kept in London None of these cities including London may be the location for more than three of the departments Benefits to be derived from each relocation are given in thousands of pounds per year as follows A B C D E Bristol 10 15 10 20 5 Brighton 10 20 15 15 15 Communication costs are of the form C ikDjl where C ik is the quantity of communication between departments i and k per year and Djl is the cost per unit of communication between cities j and l C ik and Djl are given by the following tables Quantities of communication C ik in thousands of units A B C D E A 00 10 15 00 B 14 12 00 C 00 20 D 07 Costs per unit of communication Djl in Bristol Brighton London Bristol 5 14 13 Brighton 5 9 London 10 266 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1211 Curve fitting A quantity y is known to depend on another quantity x A set of corresponding values has been collected for x and y and is presented in Table 124 1 Fit the best straight line y bx a to this set of data points The objec tive is to minimise the sum of absolute deviations of each observed value of y from the value predicted by the linear relationship 2 Fit the best straight line where the objective is to minimise the maximum deviation of all the observed values of y from the value predicted by the linear relationship 3 Fit the best quadratic curve y cx2 bx a to this set of data points using the same objectives as in 1 and 2 Table 124 x 00 05 10 15 19 25 30 35 40 45 y 10 09 07 15 20 24 32 20 27 35 x 50 55 60 66 70 76 85 90 100 y 10 40 36 27 57 46 60 68 73 1212 Logical design Logical circuits have a given number of inputs and one output Impulses may be applied to the inputs of a given logical circuit and it will respond by giving either an output signal 1 or no output signal 0 The input impulses are of the same kind as the outputs that is 1 positive input or 0 no input In this example a logical circuit is to be built up of NOR gates A NOR gate is a device with two inputs and one output It has the property that there is positive output signal 1 if and only if neither input is positive that is both inputs have the value 0 By connecting such gates together with outputs from one gate possibly being inputs into another gate it is possible to construct a circuit to perform any desired logical function For example the circuit illustrated in Figure 121 will respond to the inputs A and B in the way indicated by the truth table The problem here is to construct a circuit using the minimum number of NOR gates that will perform the logical function specified by the truth table in Figure 122 This problem together with further references to it is discussed in Williams 1974 Fanin and fanout are not permitted That is more than one output from a NOR gate cannot lead into one input nor can one output lead into more than one input It may be assumed throughout that the optimal design is a subnet of the maximal net shown in Figure 123 THE PROBLEMS 267 NOR NOR Output A NOR B A 0 0 1 1 B 0 1 0 1 Output 0 0 0 1 Figure 121 A 0 Inputs Output 0 1 1 B 0 1 0 1 0 1 1 0 Figure 122 NOR 4 NOR 2 NOR 5 NOR 6 NOR 3 NOR 1 NOR 7 Figure 123 1213 Market sharing A large company has two divisions D1 and D2 The company supplies retail ers with oil and spirit This is a much smaller version of the problem British Petroleum and Shell faced when they were forced to demerge one of the largest demergers in history The original model proved impossible to solve in 1972 It is desired to allocate each retailer to either division D1 or division D2 This division will be the retailers supplier As far as possible this division must be made so that D1 controls 40 of the market and D2 the remaining 60 The retailers are listed below as M1 to M23 Each retailer has an estimated market for oil and spirit Retailers M1 to M8 are in region 1 retailers M9 to M18 are in region 2 and retailers M19 to M23 are in region 3 Certain retailers are considered to have good growth prospects and categorised as group A and the others are in group B Each retailer has a certain number of delivery points as given below It is desired to make the 4060 split between D1 and D2 in each of the following respects 268 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1 Total number of delivery points 2 Control of spirit market 3 Control of oil market in region 1 4 Control of oil market in region 2 5 Control of oil market in region 3 6 Number of retailers in group A 7 Number of retailers in group B There is a certain flexibility in that any share may vary by 5 That is the share can vary between the limits 3565 and 4555 The primary aim is to find a feasible solution If however there is some choice then possible objectives are i to minimise the sum of the percentage deviations from the 4060 split and ii to minimise the maximum such deviation Build a model to see if the problem has a feasible solution and if so find the optimal solutions The numerical data are given in Table 125 Table 125 Retailer Oil market 106 gallons Delivery points Spirit market 106 gallons Growth category Region 1 M1 9 11 34 A M2 13 47 411 A M3 14 44 82 A M4 17 25 157 B M5 18 10 5 A M6 19 26 183 A M7 23 26 14 B M8 21 54 215 B Region 2 M9 9 18 102 B M10 11 51 21 A M11 17 20 54 B M12 18 105 0 B M13 18 7 6 B M14 17 16 96 B M15 22 34 118 A M16 24 100 112 B M17 36 50 535 B M18 43 21 8 B Region 3 M19 6 11 53 B M20 15 19 28 A M21 15 14 69 B M22 25 10 65 B M23 39 11 27 B THE PROBLEMS 269 1214 Opencast mining A company has obtained permission to opencast mine within a square plot 200 ft 200 ft The angle of slip of the soil is such that it is not possible for the sides of the excavation to be steeper than 45 The company has obtained estimates for the value of the ore in various places at various depths Bearing in mind the restrictions imposed by the angle of slip the company decides to consider the problem as one of the extracting of rectangular blocks Each block has horizontal dimensions 50 ft 50 ft and a vertical dimension of 25 ft If the blocks are chosen to lie above one another as illustrated in vertical section in Figure 124 then it is only possible to excavate blocks forming an upturned pyramid In a three dimensional representation Figure 124 would show four blocks lying above each lower block Figure 124 If the estimates of ore value are applied to give values in percentage of pure metal for each block in the maximum pyramid which can be extracted then the following values are obtained Level 1 surface 15 15 15 075 Level 2 25 ft depth 40 40 20 15 20 15 075 30 30 10 10 10 075 05 20 20 05 075 075 05 025 Level 3 50 ft depth 120 60 Level 4 75 ft depth 60 50 40 270 MODEL BUILDING IN MATHEMATICAL PROGRAMMING The cost of extraction increases with depth At successive levels the cost of extracting a block is as follows Level 1 3000 Level 2 6000 Level 3 8000 Level 4 10 000 The revenue obtained from a 100 value block would be 200 000 For each block here the revenue is proportional to ore value Build a model to help decide the best blocks to extract The objective is to maximise revenuecost The larger version of this problem arose with opencast iron mining in South Africa 1215 Tariff rates power generation A number of power stations are committed to meeting the following electricity load demands over a day 12 pm to 6 am 15 000 MW 6 am to 9 am 30 000 MW 9 am to 3 pm 25 000 MW 3 pm to 6 pm 40 000 MW 6 pm to 12 pm 27 000 MW There are three types of generating unit available 12 of type 1 10 of type 2 and five of type 3 Each generator has to work between a minimum and a maximum level There is an hourly cost of running each generator at minimum level In addition there is an extra hourly cost for each megawatt at which a unit is operated above the minimum level Starting up a generator also involves a cost All this information is given in Table 126 with costs in In addition to meeting the estimated load demands there must be sufficient generators working at any time to make it possible to meet an increase in load of up to 15 This increase would have to be accomplished by adjusting the output of generators already operating within their permitted limits THE PROBLEMS 271 Table 126 Minimum level Maximum level Cost per hour at minimum Cost per hour per megawatt above minimum Cost Type 1 850 MW 2000 MW 1000 2 2000 Type 2 1250 MW 1750 MW 2600 130 1000 Type 3 1500 MW 4000 MW 3000 3 500 Which generators should be working in which periods of the day to minimise total cost What is the marginal cost of production of electricity in each period of the day that is what tariffs should be charged What would be the saving of lowering the 15 reserve output guarantee that is what does this security of supply guarantee cost 1216 Hydro power This is an extension of the Tariff Rates Power Generation problem of Section 1215 In addition to the thermal generators a reservoir powers two hydro generators one of type A and one of type B When a hydro generator is run ning it operates at a fixed level and the depth of the reservoir decreases The costs associated with each hydro generator are a fixed startup cost and a run ning cost per hour The characteristics of each type of generator are shown in Table 127 For environmental reasons the reservoir must be maintained at a depth of between 15 and 20 m Also at midnight each night the reservoir must be 16 m deep Thermal generators can be used to pump water into the reservoir To increase the level of the reservoir by 1 m it requires 3000 MWh of electricity You may assume that rainfall does not affect the reservoir level At any time it must be possible to meet an increase in demand for electricity of up to 15 This can be achieved by any combination of the following switch ing on a hydro generator even if this would cause the reservoir depth to fall below 15 m using the output of a thermal generator which is used for pumping water into the reservoir and increasing the operating level of a thermal generator to its maximum Thermal generators cannot be switched on instantaneously to meet increased demand although hydro generators can be 272 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 127 Operating level Cost per hour Reservoir depth reduction per hour m Startup cost Hydro A 900 MW 90 031 1500 Hydro B 1400 MW 150 047 1200 Which generators should be working in which periods of the day and how should the reservoir be maintained to minimise the total cost 1217 Threedimensional noughts and crosses Twentyseven cells are arranged 3 3 3 in a threedimensional array as shown in Figure 125 Figure 125 Three cells are regarded as lying in the same line if they are on the same horizontal or vertical line or the same diagonal Diagonals exist on each horizontal and vertical section and connecting opposite vertices of the cube There are 49 lines altogether Given 13 white balls noughts and 14 black balls crosses arrange them one to a cell so as to minimise the number of lines with balls all of one colour THE PROBLEMS 273 1218 Optimising a constraint In an integer programming problem the following constraint occurs 9x1 13x2 14x3 17x4 13x5 19x6 23x7 21x8 37 All the variables occurring in this constraint are 01 variables that is they can only take the value of 0 or 1 Find the simplest version of this constraint The objective is to find another constraint involving these variables that is logically equivalent to the original constraint but that has the smallest possible absolute value of the righthand side with all coefficients of similar signs to the original coefficients If the objective were to find an equivalent constraint where the sum of the absolute values of the coefficients apart from the righthand side coefficient were a minimum what would be the result 1219 Distribution 1 A company has two factories one at Liverpool and one at Brighton In addition it has four depots with storage facilities at Newcastle Birmingham London and Exeter The company sells its product to six customers C1 C2 C6 Cus tomers can be supplied from either a depot or the factory directly see Figure 126 Table 128 Liverpool Newcastle London Exeter Birmingham Brighton Factories Depots C1 C2 C3 C4 C5 C6 Customers Figure 126 274 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 128 Supplied to Supplier Liverpool factory Brighton factory Newcastle depot Birmingham depot London depot Exeter depot Depots Newcastle 05 Birmingham 05 03 London 10 05 Exeter 02 02 Customers C1 10 20 10 C2 15 05 15 C3 15 05 05 20 02 C4 20 15 10 15 C5 05 05 05 C6 10 10 15 15 aA dash indicates the impossibility of certain suppliers for certain depots or customers The distribution costs which are borne by the company are known they are given in Table 128 in per ton delivered Certain customers have expressed preferences for being supplied from fac tories or depots which they are used to The preferred suppliers are as follows C1 Liverpool factory C2 Newcastle depot C3 No preferences C4 No preferences C5 Birmingham depot C6 Exeter or London depots Each factory has a monthly capacity given as follows which cannot be exceeded Liverpool 150 000 tons Brighton 200 000 tons Each depot has a maximum monthly throughput given as follows which cannot be exceeded Newcastle 70 000 tons Birmingham 50 000 tons London 100 000 tons Exeter 40 000 tons THE PROBLEMS 275 Each customer has a monthly requirement given as follows which must be met C1 50 000 tons C2 10 000 tons C3 40 000 tons C4 35 000 tons CS 60 000 tons C6 20 000 tons The company would like to determine the following 1 What distribution pattern would minimise overall cost 2 What the effect of increasing factory and depot capacities would be on distribution costs 3 What the effects of small changes in costs capacities and requirements would be on the distribution pattern 4 Would it be possible to meet all customer preferences regarding suppliers and if so what would the extra cost of doing this be 1220 Depot location distribution 2 In the distribution problem there is a possibility of opening new depots at Bristol and Northampton as well as of enlarging the Birmingham depot It is not considered desirable to have more than four depots and if necessary Newcastle or Exeter or both can be closed down The monthly costs in interest charges of the possible new depots and expan sion at Birmingham are given in Table 129 together with the potential monthly throughputs Table 129 Cost 1000 Throughput 1000 tons Bristol 12 30 Northampton 4 25 Birmingham expansion 3 20 The monthly savings of closing down the Newcastle and Exeter depots are given in Table 1210 The distribution costs involving the new depots are given in Table 1211 in per ton delivered 276 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 1210 Saving 1000 Newcastle 10 Exeter 5 Table 1211 Supplied to Supplier Liverpool factory Brighton factory Bristol depot Northampton depot New depots Bristol 06 04 Northampton 04 03 Customers C1 12 C2 06 04 C3 As given for distribution problem 05 C4 05 C5 03 06 C6 08 09 Which new depots should be built Should Birmingham be expanded Should Exeter or Newcastle be closed down What would be the best resultant distribu tion pattern to minimise overall costs 1221 Agricultural pricing The government of a country wants to decide what prices should be charged for its dairy products milk butter and cheese All these products arise directly or indirectly from the countrys raw milk production This raw milk is usefully divided into the two components as fat and dry matter After subtracting the quantities of fat and dry matter which are used for making products for export or consumption on the farms there is a total yearly availability of 600 000 tons of fat and 750 000 tons of dry matter This is all available for producing milk butter and two kinds of cheese for domestic consumption The percentage compositions of the products are given in Table 1212 For the previous year the domestic consumption and prices for the products are given in Table 1213 THE PROBLEMS 277 Table 1212 Fat Dry matter Water Milk 4 9 87 Butter 80 2 18 Cheese 1 35 30 35 Cheese 2 25 40 35 Table 1213 Milk Butter Cheese 1 Cheese 2 Domestic consumption 1000 tons 4820 320 210 70 Price ton 297 720 1050 815 Price elasticities of demand relating consumer demand to the prices of each product have been calculated on the basis of past statistics The price elasticity E of a product is defined by E Percentage decrease in demand Percentage increase in price For the two makes of cheese there will be some degree of substitution in con sumer demand depending on relative prices This is measured by crosselasticity of demand with respect to price The crosselasticity E AB from a product A to a product B is defined by EAB Percentage increase in demand for A Percentage increase in price of B The elasticities and crosselasticities are given in Table 1214 Table 1214 Milk Butter Cheese 1 Cheese 2 Cheese 1 to Cheese 2 Cheese 2 to Cheese 1 04 27 11 04 01 04 The objective is to determine what prices and resultant demand will maximise the total revenue 278 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It is however politically unacceptable to allow a certain price index to rise As a result of the way this index is calculated this limitation simply demands that the new prices must be such that the total cost of last years consumption would not be increased A particularly important additional requirement is to quantify the economic cost of this political limitation 1222 Efficiency analysis A car manufacturer wants to evaluate the efficiencies of different garages who have received a franchise to sell its cars The method to be used is data envel opment analysis DEA References to this technique are given in Section 32 Each garage has a certain number of measurable inputs These are taken to be Staff Showroom Space Catchment Population in different economic categories and annual Enquiries for different brands of car Each garage also has a certain number of measurable outputs These are taken to be Number Sold of different brands of car and annual Profit Table 1215 gives the inputs and outputs for each of the 28 franchised garages A central assumption of DEA although modified models can be built to alter this assumption is that constant returns to scale are possible that is doubling a garages inputs should lead to a doubling of all its outputs A garage is deemed to be efficient if it is not possible to find a mixture of proportions of other garages whose combined inputs do not exceed those of the garage being considered but whose outputs are equal to or exceed those of the garage Should this not be possible then the garage is deemed to be inefficient and the comparator garages can be identified A linear programming model can be built to identify efficient and inefficient garages and their comparators 1223 Milk collection A small milk processing company is committed to collecting milk from 20 farms and taking it back to the depot for processing The company has one tanker lorry with a capacity for carrying 80 000 litres of milk Eleven of the farms are small and need a collection only every other day The other nine farms need a collection every day The positions of the farms in relation to the depot numbered 1 are given in Table 1216 together with their collection requirements Find the optimal route for the tanker lorry on each day bearing in mind that it has to i visit all the every day farms ii visit some of the every other day farms and iii work within its capacity On alternate days it must again visit the every day farms and also visit the every other day farms not visited on the previous day For convenience a map of the area considered is given in Figure 127 THE PROBLEMS 279 Table 1215 Garage Inputs Outputs Staff Show room space 100 m2 Population in category 1 1000 s Population in category 2 1000 s Enquiries Alpha model 100 s Enquiries Beta model 100 s Alpha sales 1000 s Beta sales 1000 s Profit millions 1 Winchester 7 8 10 12 85 4 2 06 15 2 Andover 6 6 20 30 9 45 23 07 16 3 Basingstoke 2 3 40 40 2 15 08 025 05 4 Poole 14 9 20 25 10 6 26 086 19 5 Woking 10 9 10 10 11 5 24 1 2 6 Newbury 24 15 15 13 25 19 8 26 45 7 Portsmouth 6 7 50 40 85 3 25 09 16 8 Alresford 8 75 5 8 9 4 21 085 2 9 Salisbury 5 5 10 10 5 25 2 065 09 10 Guildford 8 10 30 35 95 45 205 075 17 11 Alton 7 8 7 8 3 2 19 070 05 12 Weybridge 5 65 9 12 8 45 18 063 14 13 Dorchester 6 75 10 10 75 4 15 045 145 14 Bridport 11 8 8 10 10 6 22 065 22 15 Weymouth 4 5 10 10 75 35 18 062 16 16 Portland 3 35 3 20 2 15 09 035 05 17 Chichester 5 55 8 10 7 35 12 045 13 18 Petersfield 21 12 6 6 15 8 6 025 29 19 Petworth 6 55 2 2 8 5 15 055 155 continued overleaf 280 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 1215 continued Garage Inputs Outputs Staff Show room space 100 m2 Population in category 1 1000 s Population in category 2 1000 s Enquiries Alpha model 100 s Enquiries Beta model 100 s Alpha sales 1000 s Beta sales 1000 s Profit millions 20 Midhurst 3 36 3 3 25 15 08 020 045 21 Reading 30 29 120 80 35 20 7 25 8 22 Southampton 25 16 110 80 27 12 65 35 54 23 Bournemouth 19 10 90 22 25 13 55 31 45 24 Henley 7 6 5 7 85 45 12 048 2 25 Maidenhead 12 8 7 10 12 7 45 2 23 26 Fareham 4 6 1 1 75 35 11 048 17 27 Romsey 2 25 1 1 25 1 04 01 055 28 Ringwood 2 35 2 2 19 12 03 009 04 THE PROBLEMS 281 Table 1216 Farm Position 10 miles Collection Collection frequency requirement East North 1000 l 1 Depot 0 0 2 3 3 Every day 5 3 1 11 Every day 4 4 4 7 Every day 3 5 5 9 Every day 6 6 5 2 Every day 7 7 4 7 Every day 3 8 6 0 Every day 4 9 3 6 Every day 6 10 1 3 Every day 5 11 0 6 Every other day 4 12 6 4 Every other day 7 13 2 5 Every other day 3 14 2 8 Every other day 4 15 6 10 Every other day 5 16 1 8 Every other day 6 17 3 1 Every other day 8 18 6 5 Every other day 5 19 2 9 Every other day 7 20 6 5 Every other day 6 21 5 4 Every other day 6 5 2 17 14 13 18 16 19 15 12 21 20 8 10 11 1 Depot 9 4 3 6 7 Every day farms Every other day farms Figure 127 282 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1224 Yield management An airline is selling tickets for flights to a particular destination The flight will depart in three weeks time It can use up to six planes each costing 50 000 to hire Each plane has the following 37 First Class seats 38 Business Class seats 47 Economy Class seats Up to 10 of seats in any one category can be transferred to an adjacent category It wishes to decide a price for each of these seats There will be further opportunities to update these prices after one week and two weeks Once a customer has purchased a ticket there is no cancellation option For administrative simplicity three price level options are possible in each class one of which must be chosen The same option need not be chosen for each class These are given in Table 1217 for the current period period 1 and two future periods Table 1217 Option 1 Option 2 Option 3 First 1200 1000 950 Period 1 Business 900 800 600 Economy 500 300 200 First 1400 1300 1150 Period 2 Business 1100 900 750 Economy 700 400 350 First 1500 900 850 Period 3 Business 820 800 500 Economy 480 470 450 Demand is uncertain but will be affected by price Forecasts have been made of these demands according to a probability distribution that divides the demand levels into three scenarios for each period The probabilities of the three scenarios in each period are as follows Scenario 1 01 Scenario 2 07 Scenario 3 02 The forecast demands are shown in Table 1218 THE PROBLEMS 283 Table 1218 Price option 1 Price option 2 Price option 3 First 10 15 20 Period 1 Business 20 25 35 Scenario 1 Economy 45 55 60 First 20 25 35 Period 1 Business 40 42 45 Scenario 2 Economy 50 52 63 First 45 50 60 Period 1 Business 45 46 47 Scenario 3 Economy 55 56 64 First 20 25 35 Period 2 Business 42 45 46 Scenario 1 Economy 50 52 60 First 10 40 50 Period 2 Business 50 60 80 Scenario 2 Economy 60 65 90 First 50 55 80 Period 2 Business 20 30 50 Scenario 3 Economy 10 40 60 First 30 35 40 Period 3 Business 40 50 55 Scenario 1 Economy 50 60 80 First 30 40 60 Period 3 Business 10 40 45 Scenario 2 Economy 50 60 70 First 50 70 80 Period 3 Business 40 45 60 Scenario 3 Economy 60 65 70 Decide price levels for the current period how many seats to sell in each class depending on demand the provisional number of planes to book and provisional price levels and seats to sell in future periods in order to maximise expected yield You should schedule to be able to meet commitments under all possible combinations of scenarios With hindsight ie not known until the beginning of the next period it turned out that demand in each period depending on the price level you chose was as shown in Table 1219 284 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 1219 Price option 1 Price option 2 Price option 3 First 25 30 40 Period 1 Business 50 40 45 Economy 50 53 65 First 22 45 50 Period 2 Business 45 55 75 Economy 50 60 80 First 45 60 75 Period 3 Business 20 40 50 Economy 55 60 75 Use the actual demands that resulted from the prices you set in period 1 to rerun the model at the beginning of period 2 to set price levels for period 2 and provisional price levels for period 3 Repeat this procedure with a rerun at the beginning of period 3 Give the final operational solution Contrast this solution to one obtained at the beginning of period 1 by pricing to maximise yield based on expected demands 1225 Car rental 1 A small cut price car rental company renting one type of car has depots in Glasgow Manchester Birmingham and Plymouth There is an estimated demand for each day of the week except Sunday when the company is closed These estimates are given in Table 1220 It is not necessary to meet all demand Table 1220 Glasgow Manchester Birmingham Plymouth Monday 100 250 95 160 Tuesday 150 143 195 99 Wednesday 135 80 242 55 Thursday 83 225 111 96 Friday 120 210 70 115 Saturday 230 98 124 80 Cars can be rented for one two or three days and returned to either the depot from which rented or another depot at the start of the next morning For example a 2day rental on Thursday means that the car has to be returned on Saturday morning a 3day rental on Friday means that the car has to be returned on Tuesday morning A 1day rental on Saturday means that the car has to be returned on Monday morning and a 2day rental on Tuesday morning THE PROBLEMS 285 Table 1221 From To Glasgow Manchester Birmingham Plymouth Glasgow 60 20 10 10 Manchester 15 55 25 5 Birmingham 15 20 54 11 Plymouth 8 12 27 53 Table 1222 From To Glasgow Manchester Birmingham Plymouth Glasgow 20 30 50 Manchester 20 15 35 Birmingham 30 15 25 Plymouth 50 35 25 The rental period is independent of the origin and destination From past data the company knows the distribution of rental periods 55 of cars are hired for one day 20 for two days and 25 for three days The current estimates of percentages of cars hired from each depot and returned to a given depot independent of day are given in Table 1221 The marginal cost to the company of renting out a car wear and tear administration etc is estimated as follows 1Day hire 20 2Day hire 25 3Day hire 30 The opportunity cost interest on capital storage servicing etc of owning a car is 15 per week It is possible to transfer undamaged cars from one depot to another depot irrespective of distance Cars cannot be rented out during the day in which they are transferred The costs per car of transfer are given in Table 1222 Ten percent of cars returned by customers are damaged When this happens the customer is charged an excess of 100 irrespective of the amount of damage that the company completely covers by its insurance In addition the car has to be transferred to a repair depot where it will be repaired the following day The cost of transferring a damaged car is the same as transferring an undamaged one except when the repair depot is the current depot when it is zero Again 286 MODEL BUILDING IN MATHEMATICAL PROGRAMMING the transfer of a damaged car takes a day unless it is already at a repair depot Having arrived at a repair depot all types of repair or replacement take a day Only two of the depots have repair capacity These are carsday as follows Manchester 12 Birmingham 20 Having been repaired the car is available for rental at the depot the next day or may be transferred to another depot taking a day Thus a car that is returned damaged on a Wednesday morning is transferred to a repair depot if not the current depot during Wednesday repaired on Thursday and is available for hire at the repair depot on Friday morning The rental price depends on the number of days for which the car is hired and whether it is returned to the same depot or not The prices are given in Table 1223 in Table 1223 Return to Same Depot Return to Another Depot 1Day hire 50 70 2Day hire 70 100 3Day hire 120 150 There is a discount of 20 for hiring on a Saturday so long as the car is returned on Monday morning This is regarded as a 1day hire For simplicity we assume the following at the beginning of each day 1 Customers return cars that are due that day 2 Damaged cars are sent to the repair depot 3 Cars that were transferred from other depots arrive 4 Transfers are sent out 5 Cars are rented out 6 If it is a repair depot then the repaired cars are available for rental In order to maximise weekly profit the company wants a steady state solu tion in which the same expected number will be located at the same depot on the same day of subsequent weeks How many cars should the company own and where should they be located at the start of each day This is a case where the integrality of the cars is not worth modelling Rounded fractional solutions are acceptable THE PROBLEMS 287 1226 Car rental 2 In the light of the solution to the problem stated in Section 1225 the company wants to consider where it might be most worthwhile to expand repair capacity The weekly fixed costs given below include interest payments on the necessary loans for expansion The options are as follows 1 Expand repair capacity at Birmingham by 5 cars per day at a fixed cost per week of 18 000 2 Further expand repair capacity at Birmingham by 5 cars per day at a fixed cost per week of 8000 3 Expand repair capacity at Manchester by 5 cars per day at a fixed cost per week of 20 000 4 Further expand repair capacity at Manchester by 5 cars per day at a fixed cost per week of 5000 5 Create repair capacity at Plymouth of 5 cars per day at a fixed cost per week of 19 000 If any of these options is chosen it must be carried out in its entirety that is there can be no partial expansion Also a further expansion at a depot can be carried out only if the first expansion is also carried out so for example option 2 at Birmingham cannot be chosen unless option 1 is also chosen If option 2 is chosen thereby also choosing option 1 these count as two options Similar stipulations apply regarding the expansions at Manchester At most three of the options can be carried out 1227 Lost baggage distribution A small company with six vans has a contract with a number of airlines to pick up lost or delayed baggage belonging to customers in the London area from Heathrow airport at 6 pm each evening The contract stipulates that each customer must have their baggage delivered by 8 pm The company requires a model which they can solve quickly each evening to advise them what is the minimum number of vans they need to use and to which customers each van should deliver and in what order There is no practical capacity limitation on each van All baggage that needs to be delivered in a twohour period can be accommodated in a van Having ascertained the minimum number of vans needed a solution is then sought which minimises the maximum time taken by any van On a particular evening the places where deliveries need to be made and the times to travel between them in minutes are given in Table 1224 No allowance 288 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 1224 Heathrow 20 25 35 65 90 85 80 86 25 35 20 44 35 82 Harrow 15 35 60 55 57 85 90 25 35 30 37 20 40 Ealing 30 50 70 55 50 65 10 25 15 24 20 90 Holborn 45 60 53 55 47 12 22 20 12 10 21 Sutton 46 15 45 75 25 11 19 15 25 25 Dartford 15 15 25 45 65 53 43 63 70 Bromley 17 25 41 25 33 27 45 30 Greenwich 25 40 34 32 20 30 10 Barking 65 70 72 61 45 13 Hammersmith 20 8 7 15 25 Kingston 5 12 45 65 Richmond 14 34 56 Battersea 30 40 Islington 27 Woolwich THE PROBLEMS 289 is made for drop off times For convenience Heathrow will be regarded as the first location Formulate optimisation models that will minimise the number of vans that need to be used and within this minimum minimise the time taken for the longest time delivery 1228 Protein folding This problem is based on one in the paper by Forrester and Greenberg 2008 It is a simplification of a problem in molecular biology We take a protein as consisting of a chain of amino acids For the purpose of this problem the amino acids come in two forms hydrophilic waterloving and hydrophobic water hating An example of such a chain is given in Figure 128 with the hydrophobic acids marked in bold Figure 128 Such a chain naturally folds so as to bring as many hydrophobic acids as possible close together An optimum folding for the chain in two dimensions is given in Figure 129 with the new matches marked by dashed lines The problem is to predict the optimum folding Forrester and Greenberg also impose a condition that the resultant protein be confined to a given lattice of points We do not impose that condition here This problem can be modelled by a number of integer programming formulations Some of these are discussed in the above reference Another formulation is suggested in section 1328 The problem posed here is to find the optimum folding for a chain of 50 amino acids with hydrophobic acids at positions 2 4 5 6 11 12 17 20 21 25 27 28 30 31 33 37 44 and 46 as shown in Figure 1210 Figure 129 290 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Figure 1210 1229 Protein comparison This problem is also based on one in the paper by Forrester and Greenberg 2008 It is concerned with measuring the similarities of two proteins A protein can be represented by an undirected graph with the acids represented by the nodes and the edges being present when two acids are within a threshold distance of each other This graphical representation is known as the contact map of the protein Given two contact maps representing proteins we would like to find the largest measured by number of corresponding edges isomorphic subgraphs in each graph The acids in each of the proteins are ordered We need to preserve this ordering in each of the subgraphs which implies that there can be no crossovers in the comparison This is illustrated in Figure 1211 If i k in the contact map i j k l Figure 1211 THE PROBLEMS 291 for the first protein then we cannot have l j in the second protein if i is to be associated with j and k with l in the comparison This problem is well known for being very difficult to solve for even modestly sized proteins In Figure 1212 we give an optimal comparison between two small contact maps leading to 5 corresponding edges Map 1 Map 2 Figure 1212 The problem we present here is to compare the contact maps given in Figures 1213 and 1214 Figure 1213 Figure 1214 13 Formulation and discussion of problems A suggested way of formulating each of the problems of Part II as a mathematical programming model is described Each of these formulations is good in the sense that it proved possible to solve the resultant model in a reasonable time on the computer system used With some of the problems other formulations were tried only to show that computation times were prohibitive It must be emphasized that although the formulations presented here are the best known to the author there are probably better formulations possible for some of the problems Indeed one of the purposes of the latter part of this book is to present concrete problems that may help advance the art of formulation All the models described here assume that a computer program is to be employed using one of the following algorithms 1 The revised simplex algorithm for linear programming problems 2 The branch and bound algorithm for integer programming problems 3 The separable extension to the revised simplex algorithm for separable programming problems These are the three algorithms that are most widely incorporated into commercial package programs For some of the problems more specialized algorithms might be more efficient In practice the employment of such algorithms is often made difficult by the absence of efficient computer packages for handling large models Where there is however obvious advantage to be gained from a specialized algorithm this is pointed out in the discussion associated with the formulation description Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 296 MODEL BUILDING IN MATHEMATICAL PROGRAMMING This is particularly true where the model or its dual admits a network formulation In such cases it should still however be possible to solve the model reasonably efficiently using one of the above three methods Sometimes it is desirable to build a model in a particular way to suit a particular algorithm particularly for integer programming problems In the formulations here the models are all built on the assumption that one of the above three algorithms will be used Again especially with integer programming models it is often desirable to build a model with a particular branch and bound solution strategy in mind When this is done the suggested strategy is explained Most of the models are very easy to solve For the more difficult ones there is some computational discussion 131 Food manufacture 1 Blending problems are frequently solved using linear programming Linear pro gramming has been used to find minimum cost blends of fertilizer metal alloys clays and many food products to name only a few Applications are described in Fisher and Schruben 1953 and Williams and Redwood 1974 for example The problem presented here has two aspects Firstly it is a series of simple blending problems Secondly there is a purchasing and storing problem To understand how this problem may be formulated it is convenient to consider first the blending problem for only one month This is the singleperiod problem that has already been presented as the second example in Section 12 1311 The singleperiod problem If no storage of raw oils were allowed the problem of what to buy and how to blend in January could be formulated as follows Maximize PROFIT 110x1 120x2 130x3 110x4 115x5 150y subject to VVEG x1 x2 200 NVEG x3 x4 x5 250 UHRD 88x1 61x2 2x3 42x4 5x5 6y 0 LHRD 88x1 61x2 2x3 42x4 5x5 3y 0 CONT x1 x2 x3 x4 x5 y 0 The variables x1 x2 x3 x4 x5 represent the quantities of the raw oils that should be bought respectively that is VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 y represents the quantity of PROD that should be made FORMULATION AND DISCUSSION OF PROBLEMS 297 The objective is to maximize profit which represents the income derived from selling PROD minus the cost of the raw oils The first two constraints represent the limited production capacities for refin ing vegetable and nonvegetable oils The next two constraints force the hardness of PROD to lie between its upper limit of 6 and its lower limit of 3 It is important to model these restrictions correctly A frequent mistake is to model them as 88x1 61x2 2x3 42x4 5x5 6 and 88x1 61x1 2x3 42x4 5x5 3 Such constraints are clearly dimensionally wrong The expressions on the left have the dimension of hardness multiplied by quantity whereas the figures on the right have the dimensions of hardness Instead of the variables xi in the above two inequalities expressions xiy are needed to represent proportions of the ingredients rather than the absolute quantities xi When such replacements are made the resultant inequalities can easily be reexpressed in a linear form as the constraints UHRD and LHRD Finally it is necessary to make sure that the weight of the final product PROD is equal to the weight of the ingredients This is done by the last constraint CONT which imposes this continuity of weight The singleperiod problems for the other months would be similar to that for January apart from the objective coefficients representing the raw oil costs 1312 The multiperiod problem The decisions of how much to buy each month with a view to storing for use later can be incorporated into a linear programming model To do this a multi period model is built It is necessary each month to distinguish the quantities of each raw oil bought used and stored These quantities must be represented by different variables We suppose the quantities of VEG 1 bought used and stored in each successive month are represented by variables with the following names BVEG 11 BVEG 12 and so on UVEG 11 UVEG 12 and so on SVEG 11 SVEG 12 and so on It is necessary to link these variables together by the relation quantity stored in month t 1 quantity bought in montht quantity used in montht quantity stored in montht 298 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Initially month 0 and finally month 6 the quantities in store are constants 500 The relation above involving VEG 1 gives rise to the following constraints BVEG 11 UVEG 11 SVEG 11 500 SVEG 11 BVEG 12 UVEG 12 SVEG 12 0 SVEG 12 BVEG 13 UVEG 13 SVEG 13 0 SVEG 13 BVEG 14 UVEG 14 SVEG 14 0 SVEG 14 BVEG 15 UVEG 15 SVEG 15 0 SVEG 15 BVEG 16 UVEG 16 500 Similar constraints must be specified for the other four raw oils It may be more convenient to introduce variables SVEG 10 and so on and SVEG 16 and so on into the model and fix them at the value 500 In the objective function the buying variables will be given the appropriate raw oil costs in each month The storage variables will be given the cost of 5 or profit of 5 Separate variables PROD 1 PROD 2 and so on must be defined to represent the quantity of PROD to be made in each month These variables will each have a profit of 150 The resulting model will have the following dimensions as well as the single objective function 6 5 30 buying variables 6 5 30 using variables 5 5 25 storing variables 6 product variables Total 91 variables 6 5 30 blending constraints as in the singleperiod model 6 5 30 storage linking constraints Total 60 constraints It is also important to realize the use to which a model such as this might be put for mediumterm planning By solving the model in January buying and blending plans could be determined for January together with provisional plans for the succeeding months In February the model would probably be resolved with revised figures to give firm plans for February together with provisional plans for succeeding months up to and including July By this means the best use is made of the information for succeeding months to derive an operating policy for the current month FORMULATION AND DISCUSSION OF PROBLEMS 299 132 Food manufacture 2 The extra restrictions stipulated are quite common in blending problems It is often desired to i limit the number of ingredients in a blend ii rule out small quantities of one ingredient and iii impose logical conditions on the combinations of ingredients These restrictions cannot be modelled by conventional linear programming Integer programming is the obvious way of imposing the extra restrictions In order to do this it is necessary to introduce 01 integer variables into the problem as described in Section 92 For each using variable in the problem a corre sponding 01 variable is also introduced This variable is used as an indicator of whether the corresponding ingredient appears in the blend or not For example corresponding to variable UVEG 11 a 01 variable DVEG 11 is introduced These variables are linked together by two constraints Supposing x1 represents UVEG 11 and δ1 represents DVEG 11 the following extra constraints are added to the model x1 200δ1 0 x1 20δ1 0 As δ1 is only allowed to take the integer values 0 and 1 x1 can only be nonzero ie VEG 1 in the blend in month 1 if δ1 1 and then it must be at a level of at least 20 tons The constant 200 in the first of the above inequalities is a known upper limit to the level of UVEG 11 the combined quantities of vegetable oils used in a month cannot exceed 200 Similar 01 variables and corresponding linkage constraints are introduced for the other ingredients Suppose x2 x3 x4 and x5 represent UVEG 21 UOIL 11 UOIL 21 and UOIL 31 then the following constraints and 01 variables are also introduced x2 200δ2 0 x2 20δ2 0 x3 250δ3 0 x3 20δ3 0 x4 250δ4 0 x4 20δ4 0 x5 250δ5 0 x5 20δ5 0 All these variables and constraints are repeated for all six months In this way condition 2 in the statement of the problem is automatically imposed Condition 1 can be imposed by the constraint δ1 δ2 δ3 δ4 δ5 3 and the corresponding constraints for the other five months Condition 3 can be imposed in two possible ways by the single constraints δ1 δ2 2δ5 0 300 MODEL BUILDING IN MATHEMATICAL PROGRAMMING or by the pairs of constraints δ1 δ5 0 δ2 δ5 0 There is computational advantage to be gained by using the second pair of con straints as they are tighter in the continuous problem see Section 101 Similar constraints are of course imposed for the other five months The model has now been augmented in the following way 6 5 30 01 variables Total 30 extra variables all integer 2 6 5 60 linking constraints 6 constraints for condition 1 2 6 12 constraints for condition 3 Total 78 extra constraints It is also necessary to impose upper bounds of 1 on all the 30 integer variables There is probably advantage to be gained from the user specifying a priority order for the variables in order to control the tree search in the branch and bound algorithm The six δ5 variables one for each month were given priority in choosing branching variables 133 Factory planning 1 This example is typical of some very common applications of linear program ming The objective is to find the optimum product mix subject to the production capacity and the marketing limitations As with the food manufacture problem there are two aspects Firstly there is the singleperiod problem then the exten sion to six months 1331 The singleperiod problem If storage of finished products is not allowed the model for January can be formulated as follows Maximize PROFIT 10x1 6x2 8x3 4x4 11x5 9x6 3x7 subject to GR 05x1 07x2 03x5 02x6 05x7 1152 VD 01x1 02x2 03x4 06x6 768 HD 02x1 08x3 06x7 1152 FORMULATION AND DISCUSSION OF PROBLEMS 301 BR 005x1 003x2 007x4 01x5 008x7 384 PL 001x3 005x5 005x7 384 Market 500 1000 300 300 800 200 100 upper bounds where GR VD HD BR and PL stand for grinding vertical drilling horizontal drilling boring and planing respectively The variables xi represent the quantities of PROD i to be made The singleperiod problems for the other months would be similar apart from different market bounds and different capacity figures for the different types of machine 1332 The multiperiod problem It is necessary each month to distinguish the quantities of each product manu factured from the quantities sold and held over in storage These quantities must be represented by different variables Suppose the quantities of PROD 1 manu factured sold and held over in successive months are represented by variables with the following names MPROD 11 MPROD 12 and so on SPROD 11 SPROD 12 and so on HPROD 11 HPROD 12 and so on It is necessary to link these variables together by the relation quantity held in month t 1 quantity manufactured in montht quantity sold in montht quantity held in montht Initially month 0 there is nothing held but finally month 6 there are 50 of each product held This relation involving PROD 1 gives rise to the following constraints MPROD 11 SPROD 11 HPROD 11 0 HPROD 11 MPROD 12 SPROD 12 HPROD 12 0 HPROD 12 MPROD 13 SPROD 13 HPROD 13 0 HPROD 13 MPROD 14 SPROD 14 HPROD 14 0 HPROD 14 MPROD 15 SPROD 15 HPROD 15 0 HPROD 15 MPROD 16 SPROD 16 50 Similar constraints must be specified for the other six products 302 MODEL BUILDING IN MATHEMATICAL PROGRAMMING It may be more convenient to define also variables HPROD 16 HPROD 26 and so on and fix them at the value 50 In the objective function the selling variables are given the appropriate unit profit figures and the holding variables coefficients of 05 The resulting model has the following dimensions 6 7 42 manufacturing variables 6 7 42 selling variables 6 7 42 holding variables Total 126 variables 6 5 30 capacity constraints 6 7 42 monthly linking constraints Total 72 constraints In addition in each month there are market bounds on the seven products and bounds on the holding quantities This gives a total of 84 upper bounds 134 Factory planning 2 The extra decisions that this problem requires over the factory planning problem requires the use of integer programming This is a clear case of extending a linear programming model by adding integer variables with extra constraints It is convenient to number the different types of machine as below Type 1 Grinders Type 2 Vertical drills Type 3 Horizontal drills Type 4 Borer Type 5 Planer 1341 Extra variables Integer variables γ it are introduced with the following interpretations γit number of machines of type i down from maintenance in montht for i 4 5 γit will have upper bounds of 1 1 2 γit will have upper bounds of 2 3 γit will have upper bounds of 3 There are 30 such integer variables 304 MODEL BUILDING IN MATHEMATICAL PROGRAMMING In order to formulate the problem presented here it will be assumed that everything happens on the first day of each year Clearly this assumption is far from the truth It is necessary to make some such assumption as it is only possible to represent quantities at discrete points of time if linear programming is to be applied On the first day of each year the following changes will take place simulta neously 1 Workers will be recruited into all categories 2 A certain proportion of these will leave immediately less than one years service 3 A certain proportion of last years labour force will leave more than one years service 4 A certain number of workers will be simultaneously retrained 5 A certain number of workers will be declared redundant 6 A certain number of workers will be put on short time 1351 Variables Strength of Labour Force tSKi number of skilled workers employed in year i tSSi number of semiskilled workers employed in year i tUSi number of unskilled workers employed in year i Recruitment uSKi number of skilled workers recruited in year i uSSi number of semiskilled workers recruited in year i uUSi number of unskilled workers recruited in year i Retraining vUSSSi number of unskilled workers retrained to semiskilled in year i vSSSKi number of semiskilled workers retrained to skilled in year i Downgrading vSKSSi number of skilled workers downgraded to semiskilled in year i uSKUSi number of skilled workers downgraded to unskilled in year i vSSUSi number of semiskilled workers downgraded to unskilled in year i Redundancy wSKi number of skilled workers made redundant in year i FORMULATION AND DISCUSSION OF PROBLEMS 305 wSSi number of semiskilled workers made redundant in year i wUSi number of unskilled workers made redundant in year i Shorttime Working xSKi number of skilled workers on shorttime working in year i xSSi number of semiskilled workers on shorttime working in year i xUSi number of unskilled workers on shorttime working in year i Overmanning ySKi number of superfluous skilled workers employed in year i ySSi number of superfluous semiskilled workers employed in year i yUSi number of superfluous unskilled workers employed in year i 1352 Constraints Continuity tSKi 095tSKi1t 09uSKi 095vSSSKi vSKSSi vSKUSi wSKi tSSi 095tSSi1 08uSSi 095vUSSSi vSSSKi 05vSKSSi vSSUSi wSSi tUSi 09tUSi1 075uUSi vUSSSi 05vSKUSi 05vSSUSi wUSi Retraining Semiskilled Workers vSSSKi 025tSKi 0 Overmanning ySKi ySSi yUSi 150 Requirements tSKi ySKi 05xSKi 1000 1500 2000 i 1 2 3 tSSi ySSi 05xSSi 1400 2000 2500 i 1 2 3 tUSi yUSi 05xUSi 1000 500 0 i 1 2 3 1353 Initial conditions The initial conditions are tSK0 1000 tSS0 1500 tUS0 2000 FORMULATION AND DISCUSSION OF PROBLEMS 307 1361 Variables In view of the many different sorts of variables in a model of this sort it is convenient to use mnemonic names in this description of the formulation The following variables are used to represent quantities of the materials measured in barrels CRA crude 1 CRB crude 2 LN light naphtha MN medium naphtha HN heavy naphtha LO light oil HO heavy oil R residuum LNRG light naphtha used to produce reformed gasoline MNRG medium naphtha used to produce reformed gasoline HNRG heavy naphtha used to produce reformed gasoline RG reformed gasoline LOCGO light oil used to produce cracked oil and cracked gasoline HOCGO heavy oil used to produce cracked oil and cracked gasoline CG cracked gasoline CO cracked oil LNPMF light naphtha used to produce premium motor fuel LNRMF light naphtha used to produce regular motor fuel MNPMF medium naphtha used to produce premium motor fuel MNRMF medium naphtha used to produce regular motor fuel HNPMF heavy naphtha used to produce premium motor fuel HNRMF heavy naphtha used to produce regular motor fuel RGPMF reformed gasoline used to produce premium motor fuel RGRMF reformed gasoline used to produce regular motor fuel CGPMF cracked gasoline used to produce premium motor fuel CGRMF cracked gasoline used to produce regular motor fuel LOJF light oil used to produce jet fuel HOJF heavy oil used to produce jet fuel RJF residuum used to produce jet fuel COJF cracked oil used to produce jet fuel RLBO residuum used to produce lubeoil PMF premium motor fuel RMF regular motor fuel JF jet fuel 308 MODEL BUILDING IN MATHEMATICAL PROGRAMMING FO fuel oil LBO lubeoil There are 36 such variables 1362 Constraints Availabilities The limited availability of the crude oils gives simple upper bounding constraints CRA 20 000 CRB 30 000 Capacities The distillation capacity constraint is CRA CRB 45 000 The reforming capacity constraint is LNRG MNRG HNRG 10 000 The cracking capacity constraint is LOCGO HOCGO 8000 The stipulation concerning production of lubeoil gives the following lower and upper bounding constraints LBO 500 LBO 1000 Continuities The quantity of light naphtha produced depends on the quantities of the crude oil used taking into account the way in which each crude splits under distillation This gives 01CRA 015CRB LN 0 Similar constraints exist for MN HN LO HO and R The quantity of reformed gasoline produced depends on the quantities of the naphthas used in the reforming process This gives the constraint 06 LNRG 052 MNRG 045 HNRG RG 0 FORMULATION AND DISCUSSION OF PROBLEMS 309 The quantities of cracked oil and cracked gasoline produced depend on the quantities of light and heavy oil used This gives the constraints 068 LOCGO 075 HOCGO CO 0 028 LOCGO 02 HOCGO CG 0 The quantity of lubeoil produced and sold is 05 times the quantity of residuum used This gives 05RLBO LBO 0 The quantities of light naphtha used for reforming and blending are equal to the quantities available This gives LN LNRG LNPMF LNRMF 0 Similar constraints exist for MN and HN The quantities of light oil used for cracking and blending are equal to the quantities available For the blending of fuel oil the proportion of light oil is fixed at 1018 Therefore separate variables have not been introduced for this proportion as it is determined by the variable LO This gives LO LOCGO LOJF 055 FO 0 Similar constraints exist for HO CO and R also involving fixed proportions of FO and for CG and RG The quantity of premium motor fuel produced is equal to the total quantity of its ingredients This gives LNPMF MNPMF HNPMF RGPMF CGPMF PMF 0 Similar constraints exist for RMF and JF Premium motor fuel production must be at least 40 of regular motor fuel production giving PMF 04 RMF 0 Qualities It is necessary to stipulate that the octane number of premium motor fuel does not drop below 94 This is done by the constraint 90 LNPMF 80 MNPMF 70 HNPMF 115 RGPMF 105 CGPMF 94 PMF 0 There is a similar constraint for RMF 322 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1313 Market sharing This problem can be formulated as an integer programming model where each of the 23 retailers is represented by a 01 variable δi If δi 1 retailer i is assigned to division D1 Otherwise the retailer is assigned to division D2 Slack and surplus variables can be introduced into each constraint to provide the required objective which is to minimize the sum of the proportionate deviations from the desired goals of the 4060 splits For example in the oil market in region 3 we have a total market of 100 106 gallons In order to split this 4060 we have the constraint 6δ19 15δ20 15δ21 25δ22 39δ23 y1 y2 40 y1 and y2 are slack and surplus variables which are given a cost of 001 in the first objective function y1 and y2 are also given simple upper bounds of 5 The other goal constraints can be treated in a similar fashion In order to define the second minimax objective we need to introduce a variable z together with the constraints z yi wi 0 for all slack and surplus variables yi where wi is the total quantity for the 40 goal constraint with which yi is associated The second objective is then simply to minimize z The resultant model has 18 constraints and 36 variables of which 23 are 01 As it stands this model is a correct representation of the problem but could prove difficult to solve as one feasible integer solution need bear no resemblance to another one In consequence the tree search for the optimal integer solution could be fairly random As the results in Section 1413 suggest this model did prove difficult to solve optimally One approach to reformulating the problem is to replace some of the constraints by facet constraints in the manner described in Section 102 For example the goal constraint above is equivalent to 6δ19 15δ20 15δ21 25δ22 39δ23 35 6δ19 15δ20 15δ21 25δ22 39δ23 45 The first of these constraints can be augmented by the following constraints obtained from strong covers δ20 δ22 δ23 1 δ21 δ22 δ23 1 δ19 δ22 δ23 1 δ20 δ21 δ23 1 FORMULATION AND DISCUSSION OF PROBLEMS 323 The second of the constraints can be augmented by the constraints δ20 δ23 1 δ21 δ23 1 δ22 δ23 1 δ19 δ20 δ22 δ23 2 δ19 δ21 δ22 δ23 2 δ20 δ21 δ22 δ23 2 All the other constraints could be treated in a similar fashion Unfortunately the number of such facet constraints derivable from each of the goal constraints for 1 and 2 in the statement of the problem in Section 1213 proved prohibitively large There are however a large but manageable number which can be derived from 3 and 4 together with those above for 5 In total the number of such constraints turned out to be 228 They were enumerated in a trivial amount of time using the method described in Section 102 It is still necessary to retain the original constraints with their slack and surplus variables providing an objective function as we are only adding in some of the facet constraints for each goal constraint Unfortunately this augmented formulation proved difficult to solve largely as a result of its increased size A third possibility is to replace some of the goal constraints by single tighter constraints How this may be done for a general 01 constraint is described by Bradley et al 1974 An example of such a single constraint simplification is provided by the problem OPTIMIZING A CONSTRAINT in Section 1318 The three goal constraints for 35 in Section 1213 can be written as three pairs of constraints 9δ1 13δ2 14δ3 17δ4 18δ5 19δ6 23δ7 21δ8 60 9δ1 13δ2 14δ3 17δ4 18δ5 19δ6 23δ7 21δ8 46 9δ9 11δ10 17δ11 18δ12 18δ13 17δ14 22δ15 24δ16 36δ17 43δ18 96 9δ9 11δ10 17δ11 18δ12 18δ13 17δ14 22δ15 24δ16 36δ17 43δ18 75 6δ19 15δ20 15δ21 25δ22 39δ23 45 6δ19 15δ20 15δ21 25δ22 39δ23 35 Using the method referenced above it is possible to give six constraints that have equivalent sets of 01 solutions but are tighter in a linear programming sense These are 5δ1 7δ2 8δ3 9δ4 10δ5 10δ6 13δ7 12δ8 33 2δ1 3δ2 3δ3 4δ4 4δ5 4δ6 5δ7 5δ8 11 8δ9 8δ10 13δ11 14δ12 14δ13 13δ14 17δ15 19δ16 28δ17 34δ18 75 FORMULATION AND DISCUSSION OF PROBLEMS 329 model It is convenient to consider the constraint in a standard form with posi tive coefficients in descending order of magnitude This can be achieved by the transformation y1 x7 y2 x8 y3 1 x6 y4 x4 y5 1 x3 y6 x5 y7 x2 y8 x1 giving 23y1 21y2 19y3 17y4 14y5 13y6 13y7 9y8 70 We wish to find another equivalent constraint of the form a1y1 a2y2 a3y3 a4y4 a5y5 a6y6 a7y7 a8y8 a0 The ai coefficients become variables in the linear programming model In order to capture the total logical import of the original constraint we search for subsets of the indices known as roofs and ceilings Ceilings are maximal subsets of the indices of the variables for which the sum of the corresponding coefficients does not exceed the righthand side coefficient Such a subset is maximal in the sense that no subset properly containing it or to the left in the implied lexicographical ordering can also be a ceiling For example the subset 1 2 4 8 is a ceiling 23 21 17 9 70 but any subset properly containing it eg 1 2 4 7 8 or to the left of it eg 1 2 4 7 is not a ceiling Roofs are minimal subsets of the indices for which the sum of the corresponding coefficients exceeds the righthand side coefficient Such a subset is minimal in the same sense as a subset is maximal For example 2 3 4 5 is a roof 21 19 17 14 70 but any subset properly contained in it eg 3 4 5 or to the right of it eg 2 3 4 6 is not a roof If i 1 i 2 i r is a ceiling the following condition among the new coeffi cients ai is implied ai1 ai2 air a0 If i 1 i 2 i r is a roof the following condition among the new coefficients ai is implied ai1 ai2 air a0 1 It is also necessary to guarantee the ordering of the coefficients This can be done by the series of constraints a1 a2 a3 a8 If these constraints are given together with each constraint corresponding to a roof or ceiling then this is a sufficient set of conditions to guarantee that the new 01 constraint has exactly the same set of feasible 01 solutions as the original 01 constraint 334 MODEL BUILDING IN MATHEMATICAL PROGRAMMING In addition the x variables are related to the p variables through the price elasticity relationships dxM xM EM dpM pM dxB xB EB dpB pB dxC1 xC1 EC1 dpC1 pC1 EC1C2 dpC2 pC2 dxC2 xC2 EC2 dpC2 pC2 EC2C1 dpC1 pC1 These differential equations can easily be integrated to give the x variables as expressions involving the p variables If these expressions are substituted in the above constraints and the objective function nonlinearities are introduced into the first two constraints as well as the objective function The nonlinearities could be separated and approximated to by piecewise linear functions as described in Section 74 In order to reduce the number of nonlinearities in the model the relationships implied by the differential equations above can be approximated to by the linear relationships xM xM xM EM pM pM pM xB xB xB EB pB pB pB xC1 xC1 xC1 EC1 pC1 pC1 pC1 EC1C2 pC2 pC2 pC2 xC2 xC2 xC2 EC2 pC2 pC2 pC2 EC2C1 pC1 pC1 pC1 x and p are the known quantities consumed with their prices for the previous year The approximation can be regarded as warranted if the resultant values of x and p do not differ significantly from x and p Using the above relationships to substitute for the x variables in the first two constraints and the objective function gives the model Maximize 6492p2 M 1200p2 B 220p2 C1 34p2 C2 53pC1pC2 6748pM 1184pB 420pC1 70pC2 subject to 260pM 960pB 7025pC1 06pC2 782 584pM 24pB 552pC1 58pC2 35 482pM 032pB 021pC1 007pC2 1939 In addition it is necessary to represent explicitly the nonnegativity conditions on the x variables These give pM 1039 pB 0987 220pC1 26pC2 420 27pC1 34pC2 70 FORMULATION AND DISCUSSION OF PROBLEMS 335 This is a quadratic programming model as there are quadratic terms in the objec tive function Special algorithms exist for obtaining a possibly local optimum such as that of Beale 1959 In order to solve the model with a standard pack age if this does not have a quadratic programming facility we can convert it into a separable form and approximate the nonlinear terms by piecewise linear expressions In order to put this model into a separable form it is necessary to remove the term pC1pC2 This may be done by introducing a new variable q together with the constraint pC1 pC2 0194q 0 It is important to allow q to be negative if necessary by incorporating it in the model as a free variable The objective function can then be written in the separable form a sum of nonlinear functions of single variables 6492p2 M 1200p2 B 194p2 C1 8p2 C2 q2 6748pM 1184pB 420pC1 70pC2 This transformation also demonstrates that the model is convex as described in Section 72 Using a piecewise linear approximation to the nonlinearities there is therefore no danger of obtaining a local optimum with separable programming In fact it is sufficient to use the conventional simplex algorithm This will enable one to obtain a true global optimum Nevertheless there is no guarantee that models of this sort if the price elastic ities had been different will always be convex Then it would be necessary to use integer programming to guarantee a global optimum or separable programming to produce a local optimum The solution given in Part IV is based on grids where pi q and p2 i and q2 are defined in intervals of width 005 It is only necessary to define the grids within the possible ranges of values which pi and q can take These ranges can be found by examining the constraints of the model or by using linear programming to successively maximize and minimize the individual variables pi and q subject to the constraints 1322 Efficiency analysis There are a number of linear programming formulations of the DEA problem Fuller coverage of the subject can be found in Farrell 1957 Charnes et al 1978 and Thanassoulis et al 1987 The formulation we give here is well described in Land 1991 It should be pointed out that there is also a dual for mulation that is commonly used and relies on finding weighted ratios of outputs to inputs using the technique mentioned in Section 32 This formulation can be found in the above references 338 MODEL BUILDING IN MATHEMATICAL PROGRAMMING departure and recommended price levels and sales for subsequent weeks under all possible scenarios Account will be taken of the probabilities of these scenarios in order to maximize expected yield A week later the model will be rerun taking into account the committed sales and revenue in the first week to redetermine recommended prices and sales for the second week ie with recourse and the third week under all possible scenarios The procedure will be repeated again a week later 13241 Variables p1ch 1 if price option h chosen for class c in week 1 0 otherwise c 1 2 3 h 1 2 3 p2ich 1 if price option h chosen for class c in week 2 as a result of scenario i in week 1 0 otherwise c 1 2 3 h 1 2 3 i 1 2 3 p3ijch 1 if price option h chosen for class c in week 3 as a result of scenario i in week 1 and scenario j in week 2 0 otherwise s1ich number of tickets sold in week 1 for class c under price option h and scenario i s2ijch number of tickets sold in week 2 for class c under price option h if scenario i holds in week 1 and scenario j in week 2 s3ijkch number of tickets sold in week 3 for class c under price option if scenario i holds in week 1 scenario j in week 2 and scenario k in week 3 r1ich revenue in week 1 from class c under price option h and scenario i r2ijch revenue in week 2 from class c under price option h if scenario i holds in week 1 and scenario j in week 2 r3ijkch revenue in week 3 from class c under price option h if scenario i holds in week 1 scenario j in week 2 and scenario k in week 3 uijkc slack capacity in class c under scenarios i j k in successive weeks vijkc excess capacity in class c under scenarios i j k in successive weeks n number of planes to fly While it is necessary to make the p variables 01 integer and n integer the s variables can be treated as continuous and their values rounded 13242 Constraints If a particular price option is chosen under certain scenarios then the sales cannot exceed the estimated demand and the revenue must be the product of the price and sales These conditions can be imposed using the device described in FORMULATION AND DISCUSSION OF PROBLEMS 343 δB1 1 if the Birmingham repair capacity is expanded by 5 cars per day δB2 1 if the Birmingham repair capacity is expanded by a further 5 cars per day δM 1 1 if the Manchester repair capacity is expanded by 5 cars per day δM 2 1 if the Manchester repair capacity is expanded by a further 5 cars per day δP 1 if the Plymouth repair capacity is expanded by 5 cars per day The following expression is added to the objective function 18 000δB1 8 000δB2 20 000δM1 5 000δM2 19 000δP together with the following extra constraints δB1 δB2 δM1 δM2 δB1 δB2 δM1 δM2 δP 3 and the repair capacity constraints for Birmingham Manchester and Plymouth have 5δB1 5δB2 5δM1 5δM2 and 5δP respectively added 1327 Lost baggage distribution We formulate this problem as an integer programming model It is an extension of the travelling salesman problem discussed in section 95 It can be regarded as a simple case of the vehicle routing problem In the problem here there are no vehicle capacites and no time windows for the delivery to each location apart from the overall 2 hours guarantee Nevertheless it is potentially a very difficult problem to solve for more than a modest number of locations Also it can give rise to a very large model There are a number of different ways of formulating a model which vary greatly in their size and difficuly of solution We suggest a model that is practicable in both size and computational tractibility Although the problem is symmetric in the sense that the distance from X to Y is the same as that from Y to X we extend the asymmetric formulation of the travelling salesman problem in order to give the time to return to Heathrow after a van has completed all its deliveries as 0 In this way only the total time taken to reach the last delivery is counted within the 2 hours time limit We are therefore seeking a Hamiltoniam path as opposed to a circuit for each van starting from Heathrow 13271 Variables All variables are 01 and integer xijk 1 iff van k visits and goes directly from location i to j for all i j and k yik 1 iff location i is visited by van k for all i k δk 1 iff van k is used 14 Solutions to problems Optimal solutions to the problems presented in Part II are given here The main purpose in giving the solutions is to allow the reader to check solutions heshe may have obtained to the same problems All the solutions given here result from the formulations presented in Part III Sometimes alternative optimal solu tions are possible as described in Section 62 although the optimal value of the objective function will be unique If the reader obtains a different solution to a problem from that given here heshe should attempt to validate hisher solution using the principles described in Section 61 In many cases it is realistic to take the solution given here to represent the current operating pattern in the situation being modelled The different solution obtained can then be validated relative to this solution 141 Food manufacture 1 The optimal policy is given in Table 141 The profit income from sales cost of raw oils derived from this policy is 107 843 This figure includes storage costs for the last month There are alternative optimal solutions 142 Food manufacture 2 This model proved comparatively hard to solve The optimal solution found is given in Table 142 The profit income from sales cost of raw oils derived from this pol icy is 100 279 There are alternative equally good solutions This solution was obtained after 645 nodes A total of 2007 nodes were examined to prove optimality Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd 350 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 141 Buy Use Store January Nothing 222 tons VEG 1 4778 tons VEG 1 1778 tons VEG 2 3222 tons VEG 2 250 tons OIL 3 500 tons OIL 1 500 tons OIL 2 250 tons OIL 3 February 250 tons OIL 2 200 tons VEG 2 4778 tons VEG 1 250 tons OIL 3 1222 tons VEG 2 500 tons OIL 1 750 tons OIL 2 March Nothing 1593 tons VEG 1 3185 tons VEG 1 407 tons VEG 2 815 tons VEG 2 250 tons OIL 2 500 tons OIL 1 500 tons OIL 2 April Nothing 1593 tons VEG 1 1593 tons VEG 1 407 tons VEG 2 407 tons VEG 2 250 tons OIL 2 500 tons OIL 1 250 tons OIL 2 May 500 tons OIL 3 1593 tons VEG 1 500 tons OIL 1 407 tons VEG 2 500 tons OIL 3 250 tons OIL 2 June 6593 tons VEG 1 1593 tons VEG 1 500 tons 5407 tons VEG 2 407 tons VEG 2 each oil 750 tons OIL 2 250 tons OIL 2 stipulated 143 Factory planning 1 The optimal policy is given in Table 143 This policy yields a total profit of 93 715 Information on the value of acquiring new machines can be obtained from the shadow prices on the appropriate constraints The value of an extra hour in the particular month when a particular type of machine is used to capacity is given below Machines used to capacity Value January Grinders 857 February Horizontal drills 0625 March Borer 200a aThe borer is not in use in March This figure indicates the high value of putting it into use As explained in Section 62 these figures only give the marginal value of increases in capacity and can only therefore be used as indicators SOLUTIONS TO PROBLEMS 351 Table 142 Buy Use Store January Nothing 852 tons VEG 1 4148 tons VEG 1 1148 tons VEG 2 3852 tons VEG 2 250 tons OIL 3 500 tons OIL 1 500 tons OIL 2 250 tons OIL 3 February 190 tons OIL 2 200 tons VEG 2 4148 tons VEG 1 230 tons OIL 2 1852 tons VEG 2 20 tons OIL 3 500 tons OIL 1 460 tons OIL 2 230 tons OIL 3 March 580 tons OIL 3 852 tons VEG 1 3296 tons VEG 1 1148 tons VEG 2 704 tons VEG 2 250 tons OIL 3 500 tons OIL 1 460 tons OIL 2 560 tons OIL 3 April Nothing 155 tons VEG 1 1746 tons VEG 1 230 tons OIL 2 704 tons VEG 2 20 tons OIL 3 500 tons OIL 1 230 tons OIL 2 540 tons OIL 3 May Nothing 250 tons VEG 1 196 tons VEG 1 Nothing 230 tons OIL 2 704 tons VEG 2 20 tons OIL 3 500 tons OIL 1 520 tons OIL 3 June 4804 tons VEG 1 200 tons VEG 2 500 tons 6296 tons VEG 2 230 tons OIL 2 each oil 730 tons OIL 2 20 tons OIL 3 stipulated 144 Factory planning 2 There are a number of alternative optimal maintenance schedules One such is to put the following machines down for maintenance in the following months January February One vertical drill one horizontal drill March One grinder two horizontal drills April One grinder one borer one planer May One vertical drill June 352 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Table 143 Manufacture Sell Hold January 500 PROD 1 500 PROD 1 8886 PROD 2 8886 PROD 2 3825 PROD 3 300 PROD 3 825 PROD 3 300 PROD 4 300 PROD 4 800 PROD 5 800 PROD 5 200 PROD 6 200 PROD 6 February 700 PROD 1 600 PROD 1 100 PROD 1 600 PROD 2 500 PROD 2 100 PROD 2 1175 PROD 3 200 PROD 3 100 PROD 5 500 PROD 5 400 PROD 5 100 PROD 7 300 PROD 6 300 PROD 6 250 PROD 7 150 PROD 7 March 400 PROD 6 100 PROD 1 Nothing 100 PROD 2 100 PROD 5 400 PROD 6 100 PROD 7 April 200 PROD 1 200 PROD 1 Nothing 300 PROD 2 300 PROD 2 400 PROD 3 400 PROD 3 500 PROD 4 500 PROD 4 200 PROD 5 200 PROD 5 100 PROD 7 100 PROD 7 May 100 PROD 2 100 PROD 2 100 PROD 3 600 PROD 3 500 PROD 3 100 PROD 5 100 PROD 4 100 PROD 4 100 PROD 7 1100 PROD 5 1000 PROD 5 300 PROD 6 300 PROD 6 100 PROD 7 June 550 PROD 1 500 PROD 1 50 of every 550 PROD 2 500 PROD 2 product 350 PROD 4 50 PROD 3 stipulated 550 PROD 6 300 PROD 4 50 PROD 5 500 PROD 6 50 PROD 7 This results in the production and marketing plans are shown in Table 144 These plans yield a total profit of 108 855 There is therefore a gain of 15 140 over the six months by seeking an optimal maintenance schedule instead of the one imposed in the FACTORY PLANNING solution SOLUTIONS TO PROBLEMS 353 Table 144 Manufacture Sell Hold January 500 PROD 1 500 PROD 1 Nothing 1000 PROD 2 1000 PROD 2 300 PROD 3 300 PROD 3 300 PROD 4 300 PROD 4 800 PROD 5 800 PROD 5 200 PROD 6 200 PROD 6 100 PROD 7 100 PROD 7 February 600 PROD 1 600 PROD 1 Nothing 500 PROD 2 500 PROD 2 200 PROD 3 200 PROD 3 400 PROD 5 400 PROD 5 300 PROD 6 300 PROD 6 150 PROD 7 150 PROD 7 March 400 PROD 1 300 PROD 1 100 PROD 1 700 PROD 2 600 PROD 2 100 PROD 2 100 PROD 3 100 PROD 3 100 PROD 4 100 PROD 4 600 PROD 5 500 PROD 5 100 PROD 5 400 PROD 6 400 PROD 6 200 PROD 7 100 PROD 7 100 PROD 7 April Nothing 100 PROD 1 Nothing 100 PROD 2 100 PROD 2 100 PROD 3 100 PROD 3 100 PROD 4 100 PROD 4 100 PROD 5 100 PROD 5 100 PROD 7 100 PROD 7 May 100 PROD 2 100 PROD 2 Nothing 500 PROD 3 500 PROD 3 100 PROD 4 100 PROD 4 1000 PROD 5 1000 PROD 5 300 PROD 6 300 PROD 6 June 550 PROD 1 500 PROD 1 50 PROD 1 550 PROD 2 500 PROD 2 50 PROD 2 150 PROD 3 100 PROD 3 50 PROD 3 350 PROD 4 300 PROD 4 50 PROD 4 1150 PROD 5 1100 PROD 5 50 PROD 5 550 PROD 6 500 PROD 6 50 PROD 6 110 PROD 7 60 PROD 7 50 PROD 7 354 MODEL BUILDING IN MATHEMATICAL PROGRAMMING This solution was obtained in 191 nodes The optimal objective value of 93 715 for the factory planning solution can be used as an objective cutoff because this clearly corresponds to a known integer solution A total of 3320 nodes were needed to prove optimality 145 Manpower planning With the objective of minimizing redundancy the optimal policies to pursue are given below Recruitment Unskilled Semiskilled Skilled Year 1 0 0 0 Year 2 0 649 500 Year 3 0 677 500 Retraining and downgrading Unskilled to semiskilled Semiskilled to skilled Semiskilled to unskilled Skilled to unskilled Skilled to semiskilled Year 1 200 256 0 0 168 Year 2 200 80 0 0 0 Year 3 200 132 0 0 0 Redundancy Unskilled Semiskilled Skilled Year 1 445 0 0 Year 2 164 0 0 Year 3 232 0 0 Shorttime working Unskilled Semiskilled Skilled Year 1 50 50 50 Year 2 50 0 0 Year 3 50 0 0 SOLUTIONS TO PROBLEMS 355 Overmanning Unskilled Semiskilled Skilled Year 1 130 18 0 Year 2 150 0 0 Year 3 150 0 0 These policies result in a total redundancy of 842 workers over the three years The total cost of pursuing these policies is 1 438 383 In order to obtain this total cost it is important to recognize that there are alternative solutions and we should choose that with minimum cost If the objective is to minimize the cost the optimal policies are those given below Recruitment Unskilled Semiskilled Skilled Year 1 0 0 55 Year 2 0 800 500 Year 3 0 800 500 Retraining and downgrading Unskilled to semiskilled Semiskilled to skilled Semiskilled to unskilled Skilled to unskilled Skilled to semiskilled Year 1 0 0 25 0 0 Year 2 142 105 0 0 0 Year 3 96 132 0 0 0 Redundancy Unskilled Semiskilled Skilled Year 1 812 0 0 Year 2 258 0 0 Year 3 354 0 0 356 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Shorttime working Unskilled Semiskilled Skilled Year 1 0 0 0 Year 2 0 0 0 Year 3 0 0 0 Overmanning Unskilled Semiskilled Skilled Year 1 0 0 0 Year 2 0 0 0 Year 3 0 0 0 These policies cost 498 677 over the three years and result in a total redun dancy of 1424 workers Again alternative solutions should be considered if necessary to ensure that this redundancy is the minimum possible for this min imum level of cost Clearly minimizing costs instead of redundancy saves 939 706 but results in 582 extra redundancies The cost of saving each job when minimizing redundancy could therefore be regarded as 1615 146 Refinery optimization The optimal solution results in a profit of 211 365 The optimal values of the variables defined in Part III are given below CRA 15 000 MNPMF 3537 CRB 30 000 MNRMF 6962 LN 6000 HNPMF 0 MN 10 500 HNRMF 2993 HN 8400 RGPMF 1344 LO 4200 RGRMF 1089 HO 8700 CGPMF 1936 R 5550 CGRMF 0 LNRG 0 LOJF 0 MNRG 0 HOJF 4900 HNRG 5407 RJF 4550 SOLUTIONS TO PROBLEMS 357 RG 2433 COJF 5706 LOCGO 4200 RLBO 1000 HOCGO 3800 PMF 6818 CG 1936 RMF 17 044 CO 5706 JF 15 156 LNPMF 0 FO 0 LNRMF 6000 LBO 500 147 Mining The optimal solution is as follows work the following mines in each year Year 1 Mines 1 3 4 Year 2 Mines 2 3 4 Year 3 Mines 1 3 Year 4 Mines 1 2 4 Year 5 Mines 1 2 3 Keep every mine open each year apart from mine 4 in year 5 Produce the following quantities of ore in millions of tons from each mine every year Mine 1 Mine 2 Mine 3 Mine 4 Year 1 20 13 245 Year 2 25 13 22 Year 3 195 13 Year 4 012 25 30 Year 5 20 217 13 Produce the following quantities of blended ore in millions of tons each year Year 1 575 Year 2 600 Year 3 325 Year 4 562 Year 5 547 This solution results in a total profit of 14684 millions 358 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 148 Farm planning The optimal plan results in a total profit of 121 719 over the five years Each years detailed plan should be as follows Year 1 Grow 22 tons of grain on group 1 land Grow 91 tons of sugar beet Buy 37 tons of grain Sell 23 tons of sugar beet Sell 31 heifers leaving 23 The profit on the year will be 21 906 Year 2 Grow 22 tons of grain on group 1 land Grow 94 tons of sugar beet Buy 35 tons of grain Sell 27 tons of sugar beet Sell 41 heifers leaving 12 The profit on the year will be 21 888 Year 3 Grow 22 tons of grain on group 1 land Grow 3 tons of grain on group 2 land Grow 98 tons of sugar beet Buy 38 tons of grain Sell 25 tons of sugar beet Sell 57 heifers leaving none The profit on the year will be 25 816 Year 4 Grow 22 tons of grain on group 1 land Grow 115 tons of sugar beet Buy 40 tons of grain Sell 42 tons of sugar beet Sell 57 heifers leaving none SOLUTIONS TO PROBLEMS 359 The profit on the year will be 26 826 Year 5 Grow 22 tons of grain on group 1 land Grow 131 tons of sugar beet Buy 33 tons of grain Sell 67 tons of sugar beet Sell 51 heifers leaving none The profit on the year will be 25 283 Only in year 1 the accommodation limits the total number of cows No investment should be made in further accommodation At the end of the fiveyear period there will be 92 dairy cows As pointed out in Part III the numbers of cows of successively increasing ages in successive years are effectively fixed by the continuity constraints This allows the model to be reduced using the procedures mentioned in Section 34 When this is done a further 18 variables can be fixed besides the 20 in the original model and 38 constraints can be removed 149 Economic planning With the first objective maximizing total productive capacity in year 5 the growth pattern shown in Table 145 results in a total productive capacity of 2142 million in year 5 Quantities are in million pounds Table 145 Year 1 Year 2 Year 3 Year 4 Year 5 Coal capacity 300 300 300 489 1512 Steel capacity 350 350 350 350 350 Transport capacity 280 280 280 280 280 Coal output 2604 2934 300 179 1664 Steel output 1353 1817 1931 1057 1057 Transport output 1407 2006 2672 923 923 Manpower requirement 2706 367 470 150 150 End of the year Coal stock 0 0 0 1484 0 Steel stock 0 0 0 0 0 Transport stock 0 0 0 0 0 360 MODEL BUILDING IN MATHEMATICAL PROGRAMMING With the second objective maximizing total production in the fourth and fifth years the growth pattern shown in Table 146 results in a total output of 2619 millions in those years Table 146 Year 1 Year 2 Year 3 Year 4 Year 5 Coal capacity 300 4305 4305 4305 4305 Steel capacity 350 350 350 3594 3594 Transport capacity 280 280 280 5194 5194 Coal output 1848 4305 4305 4305 4305 Steel output 867 1533 1829 3594 3594 Transport output 1413 1984 2259 5194 5194 Manpower requirement 3446 3842 470 470 150 End of the year Coal stock 316 164 0 0 0 Steel stock 115 0 0 0 1765 Transport stock 0 0 0 0 0 With the third objective maximizing manpower requirements the growth pattern shown in Table 147 results in a total manpower usage of 2450 millions Table 147 Year 1 Year 2 Year 3 Year 4 Year 5 Coal capacity 300 316 320 366 8594 Steel capacity 350 350 350 350 350 Transport capacity 280 280 280 280 280 Coal output 2518 316 3198 3663 8594 Steel output 1348 179 2241 2231 220 Transport output 1436 1817 280 2791 276 Manpower requirement 2811 3332 5397 6368 6597 End of the year Coal stock 0 0 0 0 0 Steel stock 11 0 0 0 0 Transport stock 42 0 0 0 0 The first objective obviously results in effort being concentrated on building up capacity in the coal industry This happens because the production of extra coal capacity uses comparatively little output from the other industries With the second objective more effort is put onto the transport industry This results in part from the fact that transport uses less manpower than other industries With the third objective the coal industry is again boosted in view of its heavy manpower requirement SOLUTIONS TO PROBLEMS 361 1410 Decentralization The optimal solution is locate departments A and D in Bristol locate departments B C and E in Brighton This results in a yearly benefit of 80 000 but communications costs of 65 100 It is interesting to note that communication costs are also reduced by moving out of London in this problem because they would have been 78 000 if each department had remained in London The net yearly benefit benefits less communication costs is therefore 14 900 1411 Curve fitting 1 The best straight line that minimizes the sum of absolute deviations is y 06375x 05812 This is line 1 shown in Figure 141 The sum of absolute deviations result ing from this line is 1146 1 0 1 2 3 4 5 6 7 2 3 4 5 x y 6 Line 1 Line 2 7 8 9 10 Figure 141 362 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 2 The best straight line that minimizes the maximum absolute deviation is y 0625x 04 This is line 2 shown in Figure 141 The maximum absolute deviation resulting from this line is 1725 Points 30 32 50 10 and 70 57 all have this absolute deviation from the line In contrast line 1 allows point 50 10 to have an absolute deviation of 277 On the other hand although line 2 allows no point to have an absolute deviation of more than 1725 the sum of absolute deviations is 1995 compared with the 1147 resulting from line 1 3 The best quadratic curve that minimizes the sum of absolute deviations is y 00337x2 02945x 09823 This is curve 1 shown in Figure 142 The sum of absolute deviations resulting from this curve is 1045 4 The best quadratic curve that minimizes the maximum absolute devia tion is y 0125x2 0625x 2475 1 0 1 2 3 4 5 6 7 2 3 4 5 x y 6 Curve 1 Curve 2 7 8 9 10 Figure 142 SOLUTIONS TO PROBLEMS 363 This is curve 2 shown in Figure 142 The maximum absolute deviation resulting from this curve is 1475 Points 00 11 30 32 50 10 and 70 57 all have this absolute deviation from the curve A way of obtaining analytical solutions to these sorts of problems is described by Williams and Munford 1999 1412 Logical design The optimal solution is shown in Figure 143 There are of course symmetric alternatives NOR 6 A A B B NOR 3 NOR 1 NOR 2 NOR 7 Figure 143 1413 Market sharing This model proves difficult to solve optimally although a feasible solution is comparatively easy to obtain Using the original formulation with no extra con straints a feasible solution was found after 21 nodes by minimizing the sum of percentage deviations This solution gave a sum of percentage deviations of 849 with the maximum such deviation being 373 the Region 3 OIL goal The best solution obtained in this run was after 360 nodes with a sum of percentage deviations of 453 the maximum such deviation being 25 the DELIVERY POINTS goal After a total of 1995 nodes the search was completed Minimizing the maximum percentage deviation produced a feasible solution after 37 nodes This solution gave a maximum deviation of 25 the OIL in region 3 goal The sum of percentage deviations was 882 This was proved to be the optimal solution after 3158 nodes The second formulation was obtained by adding 228 facets for the OIL goals in the three regions The increased size of this model drastically reduced the number of nodes that could be explored in a given time After 10 s with the objective of minimizing the sum of percentage deviations 75 nodes had been 364 MODEL BUILDING IN MATHEMATICAL PROGRAMMING explored with a minimum sum of percentage deviations of 1128 obtained at node 61 The maximum deviation associated with this solution was 277 the SPIRITS goal When the objective was to minimize the maximum deviation 36 nodes were explored with no feasible solution being found The most successful formulation was the third one where six single tighter constraints logically equivalent to the six constraints implied by the OIL goals in the three regions were added With the objective of minimizing the sum of percentage deviations a feasible solution was obtained after 35 nodes The sum of percentage deviations was 883 and the maximum deviation was 25 in the GROWTH prospects goals This run was terminated after 394 nodes with no other feasible solutions being found With the objective of minimizing the maximum deviation a feasible solution was found after 44 nodes with a maximum deviation of 315 in the SPIRITS goal The associated sum of percentage deviations was 1214 A better solution was found at node 200 with a maximum deviation of 25 in the GROWTH prospects goals The associated sum of percentage deviations was 97 No better solutions were found after 303 nodes With hindsight it is possible to observe that the minimax objective cannot be reduced below 25 because the slack variable in the GROWTH goal cannot be less than 02 given a righthand side of 32 Therefore the slack in this constraint was fixed at 02 and the surplus at 0 The third formulation of the problem was then rerun in order to minimize the sum of percentage deviations At node 681 the optimal integer solution was found where the sum of percentage deviations was 7806 The solution was proved optimal after 889 nodes This optimal solution allocates the following retailers to D1 M1 M2 M3 M4 M11 M16 M18 M21 M22 All other retailers are to be assigned to division D2 This problem has proved of considerable interest in view of its computational difficulty A remodelling approach using basis reduction is described by Aardal et al 1999 1414 Opencast mining The optimal solution is to extract the shaded blocks shown in Figure 144 This results in a profit of 17 500 This solution was obtained in 28 iterations 1415 Tariff rates power generation The following generators should be working in each period giving the following outputs SOLUTIONS TO PROBLEMS 365 Period 1 12 of type 1 output 10 200 MW 3 of type 2 output 4 800 MW Period 2 12 of type 1 output 16 000 MW 8 of type 2 output 14 000 MW Period 3 12 of type 1 output 11 000 MW 8 of type 2 output 14 000 MW Period 4 12 of type 1 output 21 250 MW 9 of type 2 output 15 750 MW 2 of type 3 output 3 000 MW Period 5 12 of type 1 output 11 250 MW 9 of type 2 output 15 750 MW Level 4 75 ft depth Level 3 50 ft depth Level 1 Surface 1 3 2 5 4 8 Level 2 25 ft depth 7 6 11 10 9 14 13 12 17 18 16 15 22 20 19 25 26 24 23 29 30 28 27 21 Figure 144 The total daily cost of this operating pattern is 988 540 Deriving the marginal cost of production from a mixed integer programming model such as this encounters the difficulties discussed in Section 103 We could adopt the approach of fixing the integer variables at the optimal integer values and obtaining this economic information from the resulting linear programming model The marginal costs then result from any changes within the optimal oper ating pattern that is without altering the numbers of different types of generator working in each period although their levels of operation could vary The marginal costs of production per hour are then obtained from the shadow prices on the demand constraints divided by the number of hours in the period These give Period 1 13 per megawatt hour Period 2 2 per megawatt hour Period 3 2 per megawatt hour Period 4 2 per megawatt hour Period 5 2 per megawatt hour 366 MODEL BUILDING IN MATHEMATICAL PROGRAMMING With this method of obtaining marginal valuations there will clearly be no values associated with the reserve output guarantee constraints because all the variables have been fixed in these constraints As an alternative to using the above method of obtaining marginal valuations on the constraints we could take the shadow prices corresponding to the contin uous optimal solution In this model this solution the continuous optimum does not differ radically from the integer optimum It gives the following operating pattern that is clearly unacceptable in practice because of the fractional number of generators working Period 1 12 of type 1 output 10 200 MW 275 of type 2 output 4 800 MW Period 2 12 of type 1 output 15 200 MW 846 of type 2 output 14 800 MW Period 3 12 of type 1 output 10 200 MW 846 of type 2 output 14 800 MW Period 4 12 of type 1 output 21 250 MW 96 of type 2 output 16 800 MW 13 of type 3 output 1 950 MW Period 5 12 of type 1 output 10 200 MW 96 of type 2 output 16 800 MW The resulting objective value cost is 985 164 The shadow prices on the demand constraints imply the following marginal costs of production Period 1 176 per megawatt hour Period 2 2 per megawatt hour Period 3 179 per megawatt hour Period 4 2 per megawatt hour Period 5 186 per megawatt hour In this case we can obtain a meaningful valuation for the 15 reserve output guarantee from the shadow prices on the appropriate constraints The only non zero valuation is on the constraint for period 4 This indicates that the cost of each guaranteed hour is 0042 The range on the righthand side coefficient of this constraint indicates that the 15 output guarantee can change between 2 and 52 with the marginal cost of each extra hour being 0042 1416 Hydro power The following thermal generators should be working in each period giving the following outputs SOLUTIONS TO PROBLEMS 367 Period 1 12 of type 1 output 10 565 MW 3 of type 2 output 5 250 MW Period 2 12 of type 1 output 14 250 MW 9 of type 2 output 15 750 MW Period 3 12 of type 1 output 10 200 MW 9 of type 2 output 15 750 MW Period 4 12 of type 1 output 21 350 MW 9 of type 2 output 15 750 MW 1 of type 3 output 1 500 MW Period 5 12 of type 1 output 10 200 MW 9 of type 2 output 15 750 MW The only hydro generator to work is B in periods 4 and 5 producing an output of 1400 MW Pumping should take place in the following periods at the given levels Period 1 815 MW Period 3 950 MW Period 5 350 MW Although it may seem paradoxical to both pump and run Hydro B in period 5 this is necessary to meet the requirement of the reservoir being at 16 m at the beginning of period 1 given that the Hydro can work only at a fixed level It would be possible to use the model to cost this environmental requirement The height of the reservoir at the beginning of each period should be Period 1 12 m Period 2 1763 m Period 3 1763 m Period 4 1953 m Period 5 1812 m The cost of these operations is 986 630 In solving this model it is valuable to exploit the fact that the optimal objective value must be less than or equal to that reported in Section 1415 This is for two reasons First the optimal solution in Section 1415 using only thermal generators is a feasible solution to this model Second the 15 extra output guarantee can be met at no startup cost using the hydro generators When solving this model by the branchandbound method the optimal objective value in Section 1415 could be used as an objective cutoff to prune the tree search 368 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Planning the use of hydro power by means of Stochastic Programming in order to model uncertainty is described by Archibald et al 1999 1417 Threedimensional noughts and crosses The minimum number of lines of the same colour is four There are many alter native solutions one of which is given in Figure 145 where the top middle and bottom sections of the cube are given Cells with black balls are shaded This solution was obtained in 15 nodes A total of 1367 nodes were needed to prove optimality Figure 145 SOLUTIONS TO PROBLEMS 369 1418 Optimizing a constraint The simplest version of this constraint with minimum righthand side coeffi cient is 6x1 9x2 10x3 12x4 9x5 13x6 16x7 14x8 25 This is also the equivalent constraint with the minimum sum of absolute values of the coefficients 1419 Distribution 1 The minimum cost distribution pattern is shown in Figure 146 with quantities in thousands of tons Newcastle Birmingham London 5 10 50 50 35 40 55 20 40 55 Brighton Liverpool Exeter C1 C2 C3 C4 C5 C6 Figure 146 There is an alternative optimal solution in which the 40 000 ton from Brighton to Exeter comes from Liverpool instead This distribution pattern costs 198 500 per month Depot capacity is exhausted at Birmingham and Exeter The value in reduc ing distribution costs of an extra ton per month capacity in these depots is 020 and 030 respectively 370 MODEL BUILDING IN MATHEMATICAL PROGRAMMING This distribution pattern will remain the same as long as the unit distribution costs remain within certain ranges These are given below for routes which are to be used Route Cost range Liverpool to C1 to 15 Liverpool to C6 to 12 Brighton to Birmingham to 05 Brighton to London 03 to 08 Brighton to Exeter to 02 Birmingham to C2 to 12 Birmingham to C4 to 12 Birmingham to C5 03 to 07 London to C5 03 to 08 Exeter to C3 0 to 05 Depot capacities can be altered within certain limits For the not fully utilized depots of Newcastle and London changing capacity within these limits has no effect on the optimal distribution pattern For Birmingham and Exeter the effect on total cost will be 02 and 03 per ton per month within the limits Outside certain limits the prediction of the effect requires resolving the problem The limits are Depot Capacity range Birmingham 45 000105 000 tons Exeter 40 00095 000 tons NB All the above effects of changes are only valid if one thing is changed at a time within the permitted ranges Clearly the above solution does not satisfy the customer preferences for suppliers By minimizing the second objective it is possible to reduce the number of goods sent by nonpreferred suppliers to a customer to a minimum This was done and revealed that it is impossible to satisfy all preferences The best that could be done resulted in the distribution pattern shown in Figure 147 where customer C5 receives 10 000 tons from his nonpreferred depot of London This is the minimum cost such distribution pattern There are alternative patterns which also minimize the number of nonpreferences but which cost more The minimum cost here is 246 000 showing that the extra cost of satisfying more customers preferences is 47 500 SOLUTIONS TO PROBLEMS 371 Newcastle London C1 10 10 10 50 35 50 30 40 40 20 50 C2 C3 C4 C5 C6 Brighton Liverpool Exeter Birmingham Figure 147 1420 Depot location distribution 2 The minimum cost solution is to close down the Newcastle depot and open a depot in Northampton The Birmingham depot should be expanded The total monthly cost taking account of the saving from closing down Newcastle result ing from these changes and the new distribution pattern is 174 000 Figure 148 shows the new distribution pattern with quantities in thousands of tons This solution was obtained in 40 iterations The continuous optimal solution was integer Therefore no tree search was necessary 1421 Agricultural pricing The optimal prices are Milk 303 per ton Butter 667 per ton Cheese 1 900 per ton Cheese 2 1085 per ton The resultant yearly revenue will be 1992 million It is straightforward to calculate the yearly demands that will result from these prices They are 372 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Milk 4 781 000 tons Butter 384 000 tons Cheese 1 250 000 tons Cheese 2 57 000 tons Liverpool Brighton Exeter London Northampton Birmingham C1 50 50 70 40 40 25 25 20 10 10 10 10 C2 C3 C4 C5 C6 Figure 148 The economic cost of imposing a constraint on the price index can be obtained from the shadow price on the constraint For this example this shadow price in the optimal solution indicates that each 1 by which the new prices are allowed to increase the cost of past years consumption would result in an increased revenue of 061 1422 Efficiency analysis The efficient garages turn out to be 3 Basingstoke 6 Newbury 7 Portsmouth 8 Alresford 9 Salisbury 11 Alton 15 Weymouth 16 SOLUTIONS TO PROBLEMS 373 Portland 18 Petersfield 22 Southampton 23 Bournemouth 24 Henley 25 Maidenhead 26 Fareham and 27 Romsey It should be observed that these garages may be efficient for different reasons For example Newbury has 12 times the staff of Basingstoke but only five times as much showroom space It sells 10 times as many Alphas and 104 times as many Betas and makes nine times as much profit This suggests that it makes more efficient use of showroom space but less of staff The other garages are deemed inefficient They are listed in Table 148 in decreasing order of efficiency together with the multiples of the efficient garages that demonstrate them to be inefficient Table 148 Garage Efficiency number Multiples of efficient garages 19 Petworth 0988 00666 001518 003425 067526 21 Reading 0982 12693 054415 119916 28624 13725 14 Bridport 0971 00333 047016 0 78324 019525 2 Andover 0917 085715 021525 28 Ringwood 0876 00083 032016 014624 5 Woking 0867 09528 002111 000922 014825 4 Poole 0862 03293 075716 043424 034525 12 Weybridge 0854 079715 014525 001826 1 Winchester 0840 00057 04168 04039 033315 009616 13 Dorchester 0839 01343 01048 011915 075216 003524 047926 20 Midhurst 0829 00599 006615 047216 004318 000925 17 Chichester 0824 00583 00978 033515 016616 023624 015426 10 Guildford 0814 04253 01507 06238 019215 016816 For example the comparators to Petworth taken in the multiples given below use inputs of Staff 502 Showroom space 550 m2 Category 1 population 2 1000s Category 2 population 2 1000s Alpha enquiries 735 100s Beta enquiries 398 100s 374 MODEL BUILDING IN MATHEMATICAL PROGRAMMING to produce outputs of 1518 1000s Alpha sales 0568 1000s Beta sales 1568 million pounds Profit Clearly this uses no more than the inputs used by Petworth but produces outputs at least 10119 times as great There is also a useful interpretation of the dual values on the input and output constraints These can be regarded as weightings that the particular garage would like to place on its inputs and outputs so as to maximize this weighted ratio of outputs to inputs but keep the corresponding ratios for the other garages above 1 ie not allow them to appear inefficient In the case of Petworth the dual values from solving the model turn out to be 0 01557 00618 00158 0 and 0 for the inputs and 01551 0 and 0495 for the outputs This gives a weighted ratio of 01551 15 0495 155 01557 55 00618 2 00158 2 0988 This is clearly the efficiency number Petworth weighs most heavily high outputs that bring it out in the best light and least heavily high inputs The dual formulation referred to in Sections 32 and 1322 chooses the weights so as to maximize this ratio while not allowing the ratios for other garages with these weights to fall below 1 1423 Milk collection The optimal solution is given in Figure 149 with dashes representing the first days routes and dotted lines those of the second day The total distance covered is 1229 miles In the formulation given in Section 1323 this first solution produced subtours on day 1 around farms 2 5 and 18 around 3 16 13 4 and 19 and around 1 8 21 9 11 7 6 and 10 and on day 2 around farms 6 7 and 20 and around all the other farms This infeasible solution covers 1214 miles there are alternative equally good solutions Subtour elimination constraints were then introduced to prevent the subtours on both days to prevent them rearising on the other day resulting in another solution with no subtours on day 1 but subtours on day 2 around farms 1 2 17 6 7 and 10 around 4 15 3 5 14 16 and 13 and around 8 9 and 21 Subtour elimination constraints were introduced to prevent these subtours on both days This resulted in the optimal solution no subtours given in Figure 149 SOLUTIONS TO PROBLEMS 375 Every day farms 3 15 19 4 13 16 14 18 2 17 12 8 10 11 20 21 9 7 6 1 Depot 5 Every other day farms Figure 149 These stages required 34 13 and 272 nodes respectively Out of interest not introducing the disaggregated constraints described in Section 1323 resulted in the first stage taking 1707 nodes This problem is based on a larger one described by Butler Williams and Yarrow 1997 That larger problem required a more sophisticated solution approach using generalizations of known results concerning the structure of the travelling salesman polytope 376 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1424 Yield management Numbers of seats have been rounded to nearest integers where necessary Period 1 Sell tickets at the following prices up to what is available First 1200 Business 900 Economy 500 Set provisional prices for period 2 First 1150 if scenario 1 in period 1 1150 if scenario 2 in period 1 1300 if scenario 3 in period 1 Business 1100 for all scenarios in period 1 Economy 700 for all scenarios in period 1 Set provisional prices for period 3 First 1500 for all scenarios in periods 1 and 2 Business 800 for all scenarios in periods 1 and 2 Economy 480 for all scenarios in period 1 and scenarios 1 and 2 in period 2 450 for all scenarios in period 1 and scenario 3 in period 2 Provisionally book three planes Expected revenue is 169 544 Period 2 Rerunning the model with the demand given with hindsight for the price levels decided for period 1 results in the following recommended decisions for period 2 Sell tickets at the following prices up to what is available First 1150 Business 1100 Economy 700 SOLUTIONS TO PROBLEMS 377 Set provisional prices for period 3 First 1500 for all scenarios in period 2 Business 800 for all scenarios in period 2 Economy 480 for scenarios 1 and 2 in period 2 450 for scenario 3 in period 2 Provisionally still book three planes Expected total revenue is now 172 969 Period 3 Rerunning the model with the known demands and price levels for periods 1 and 2 results in the following recommended decisions for period 3 Sell tickets at the following prices up to what is available First 1500 Business 800 Economy 480 Provisionally still book three planes Expected total revenue is now 176 392 Solution The resultant solution at takeoff using demands during the final week will therefore be Period 1 First Class 25 seats sold at 1200 Yield 30 000 Business Class 45 seats sold at 900 Yield 40 500 Economy Class 50 seats sold at 500 Yield 25 000 Period 2 First Class 50 seats sold at 1150 Yield 57 500 Business Class 45 seats sold at 1100 Yield 49 500 Economy Class 50 seats sold at 700 Yield 35 000 378 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Period 3 First Class 40 seats sold at 1500 Yield 60 000 Business Class 25 seats sold at 800 Yield 20 000 Economy Class 36 seats sold at 480 Yield 17 280 Three planes will be needed Total yield subtracting the costs of the planes will therefore be 184 780 It will be necessary to reallocate four seats from Business to First Class and five seats from Economy to Business Class If the model is altered to maximize yield subject to expected demand without recourse the resultant solution allowing for the given demands falling below those expected is Period 1 First Class 24 seats sold at 1200 Yield 28 800 Business Class 39 seats sold at 900 Yield 35 100 Economy Class 50 seats sold at 500 Yield 25 000 Period 2 First Class 55 seats sold at 1150 Yield 63 250 Business Class 43 seats sold at 1100 Yield 47 300 Economy Class 49 seats sold at 700 Yield 34 300 Period 3 First Class 34 seats sold at 01500 Yield 51 000 Business Class 35 seats sold at 800 Yield 28 000 Economy Class 37 seats sold at 480 Yield 17 760 Three planes are needed Total yield is 180 210 This can clearly be improved by in the last period toppingup to fill vacant capacity in the most beneficial way that is selling 39 Economy Class instead of 37 so filling the three planes This increases yield to 181 170 which is still considerably short of the yield produced by running the Stochastic Program with recourse SOLUTIONS TO PROBLEMS 379 1425 Car rental The numbers of cars in all aspects of the following solution have been rounded from the fractional answers that result from the linear programming model The company should own 624 cars and pursue the following policies that will result in a weekly profit of 122 398 The estimated number of undamaged cars in each depot at the beginning of each day will be as follows there will also be hired out cars not in any depot Glasgow Manchester Birmingham Plymouth Monday 68 99 145 46 Tuesday 66 94 154 39 Wednesday 70 100 125 47 Thursday 69 115 117 44 Friday 71 102 126 44 Saturday 65 96 154 73 The estimated number of damaged cars in each depot at the beginning of each day will be as follows Glasgow Manchester Birmingham Plymouth Monday 11 12 20 6 Tuesday 7 12 20 4 Wednesday 8 12 20 6 Thursday 9 12 20 5 Friday 11 12 20 5 Saturday 7 12 22 3 Of the undamaged cars the following should be rented out each day for the periods and destinations in the proportions given in the statement of the problem Glasgow Manchester Birmingham Plymouth Monday 68 99 95 46 Tuesday 66 94 154 39 Wednesday 70 80 125 47 Thursday 69 115 111 44 Friday 71 102 70 0 Saturday 65 93 124 73 380 MODEL BUILDING IN MATHEMATICAL PROGRAMMING No transfers of undamaged cars should be made but the following transfers of damaged cars should be made arriving the following day Glasgow to Manchester Monday Tuesday Wednesday Thursday Friday Saturday 3 2 3 2 3 2 Glasgow to Birmingham Monday Tuesday Wednesday Thursday Friday Saturday 5 5 3 4 9 5 Plymouth to Birmingham Monday Tuesday Wednesday Thursday Friday Saturday 5 3 6 5 5 3 The two repair depots of Manchester and Birmingham are fully occupied on all six days repairing 12 and 20 cars respectively each day Repair capacity is clearly a limiting factor on the operation of the company This is reflected by the highshadow prices on the repair capacity constraints which vary between 617 and 646 per car per day in both repairing depots A plan to increase repair capacity forms the subject of problem 1227 The solution above was obtained in 153 iterations 1426 Car rental 2 Only the Birmingham repair capacity should be increased using both expansion options to expand to 22 cars per day Repair capacities in all depots are again fully used This allows the company to expand its fleet to 895 cars resulting in a new weekly profit of 135 511 All demands still cannot be fully met This model solved in 5 nodes 1427 Lost baggage distribution Two vans are needed Solving the model defined in Section 1327 a relaxation of the final model leads to the solution given in Figure 1410 This required 2263 nodes to solve SOLUTIONS TO PROBLEMS 381 1 2 3 12 11 5 10 4 13 8 15 9 7 6 14 Heathrow Harrow Ealing Richmond Kingston Sutton Hammersmith Battersea Holborn Islington Greenwich Bromley Barking Woolwich Dartford Figure 1410 One van uses the bold lines and the other the dashed lines Both take 120 min Clearly there are unacceptable subtours A total of 31 subtour elimination constraints needed to be appended in 14 stages in order to obtain a true solution with no subtours there are a number of such solutions This solution required 1296 nodes a few seconds on a notepad computer Fixing the number of vans needed at two and minimizing the maxi mum time needed by the vans took a further 561 nodes to solve no additional subtour elimination constraints were needed It produced the two routes illus trated in Figure 1411 a bold line and a dashed line The times taken by the vans to reach their furthest locations are 99 and 100 min respectively 1 2 3 12 11 5 10 4 13 8 15 9 7 6 14 Heathrow Harrow Ealing Richmond Kingston Sutton Hammersmith Battersea Holborn Islington Greenwich Bromley Barking Woolwich Dartford 11 Figure 1411 382 MODEL BUILDING IN MATHEMATICAL PROGRAMMING 1428 Protein folding 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Beale E M L 11 37 65 154 160 1789 181 201 216 317 319 335 Beard C N 72 Beare G C 32 337 Beasley J 345 Benders J F 53 164 Bienstock D 228 Bixby R E 99 Bockmayr A 19 Boland N 71 Bollapragada S 74 Boot J L G 62 300 343 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd Bradley G H 91 221 227 323 328 330 Brailsford S C 19 177 230 Brameller A 64 342 Brearley A L 35 378 218 228 Buchanan C S 383 Buchanan J T 40 Butler M 198 375 Catchpole A R 68 Chang Y 74 Chandru V 177 Charnes A 29 32 712 74 92 335 Cheshire M 98 Christiansen M 69 Christofides N 161 Chvatal V 228 Claus A 197 Coates P A 69 Cook W J 383 Cooper W W 32 92 Corbishley B 63 345 Corner L J 245 Crowder H 218 Cuddeford G C 21 62 128 343 Cunningham W H 99 Daniel R C 98 215 Dantzig G B 56 59 65 81 86 88 912 97 197 DarbyDowman K 19 Davies G S 71 303 398 AUTHOR INDEX Dempster M 54 Desrosiers J 386 Desrosiers M 199 Devon J K 64 338 Dijkstra E W 93 Dorfman R P A 78 Duffin R J 140 Dunnett R M 64 342 Dyson R G 32 Dumitrescu I 384 Edmonds J 192 227 Eilon S 69 Eisemann K 72 Engel J F 72 Fabian T 69 Fagerholt K 385 Fanshel S 71 Farrell M J 74 335 Feldmann C F 62 344 Fergusson J G 63 345 Feuerman M 73 Ferreira C E 74 Finke G 197 Fisher W D 296 Flowerdew A D J 69 Fokkens B 70 Ford L R 86 88 91 94 330 Forrest J J H 164 241 Forrester R J 28990 345 Foster M J 65 302 344 Fourer R 41 Fox K R 198 Froyland G 384 Fulkerson D R 86 88 91 94 330 Gainen L 78 344 Garfinkel R S 163 190 19395 21314 Garver L L 71 325 Gass S I 74 Gavish B 197 Geoffrion A M 1634 Gilmore P C 72 195 202 Glassey R 72 Glen J J 70 Glover F 913 176 330 Gomory R E 72 163 195 202 2345 Goodson G C 69 Granot F 163 Graves S C 197 Greenberg H 40 534 132 238 28990 345 Ghattas O 384 Gondzio J 55 Greixer G 384 Grothey J 55 Grotschel M 387 Grun G 61 344 Gunn E 184 340 Gupta V 72 Gutgesell W 61 344 Hammer P L 163 222 2278 Harvey A 245 Held M 198 Heroux R L 73 Hill T W 74 Hirst J P H 153 220 340 Hitchcock F L 82 Ho J K 65 Hodson F R 394 Holderby T 62 277 344 Honig D 78 344 Hooker J N 17 19 34 158 177 Hubbard P M 16 338 Hughes P A B 38 338 Hultz J 84 341 Hurkens C A J 323 337 Jack W 69 Jeffreys M 209 Jennings J C 207 344 Jensen P A 91 330 Jeroslow R G 17 156 183 22930 Johnson E L 197 202 206 339 341 Johnson S M 184 339 Jones W G 71 Junger M 74 390 AUTHOR INDEX 399 Kall P 10 534 Kalvaitis R 73 Karp R M 198 Karwan M H 378 Kasper T 19 Kendal D G 394 Khodaverdian E 71 Kiefl S 387 Klingman D 912 330 Knolmayer G 39 Koenig S 71 Kornbluth J 69 113 Kraft D H 74 Kuhn H W 88 Land A 74 164 335 Laporte G 74 Lasdon L S 65 Lawler E 198 201 317 Lawrence J R 69 Learner D B 64 338 Lemarechal C 383 Lenstra J K 184 323 337 342 Leontief W 74 Lilien G L 71 Little J 19 Lockyer K G 96 Lonsdale R T 72 Lotfi V 334 342 Loucks O P 74 Loute E 65 Louwes S L 70 333 Lowe J 183 229 Lucas C 36 341 343 Lynes E S 71 Lynn W R 62 65 3434 McClosky B 228 McColl W H S 68 McDonald A G 25 70 140 McIndoe C T 72 McKinnon K I M 19 40 177 Manne A 68 306 Markland R E 69 Markowitz H 69 Marsten R E 164 Martin A 390 Martin R K 230 Meyer M 71 Meyer R R 156 Miercort F A 72 Milano M 389 Miller C E 150 197 Miller D W 245 Mitra G 38 210 218 Morrison T 534 Muckstadt J A 71 MullerMerbach H 35 40 343 Mulvey J 93 Murphy F 40 Munford A G 363 Nemhauser G L 12 74 163 190 1935 2134 Niehaus R J 63 338 Nygreen B 385 Orden A 88 Orman A J 197 Padberg M W 190 194 214 Peled U N 163 Peterson E L 129 339 Piskor W G 71 303 Posgay A G 73 Powell S 164 Prawda J 70 Price W L 71 303 Proll L 19 Puylaert M 70 Rahmouni M K 9091 337 Rao A G 71 Redpath A T 34 70 Redwood A C 52 65 71 246 254 296 Reinecke W 64 338 Revelle C S 70 Rhodes E 24 65 302 338 Rhys J M W 215 235 Riley V 74 Rinnooy Kan A H G 184 342 Rivett B H P 244 Ronen D 385 Rope C M 71 400 AUTHOR INDEX Rose C J 70 Rosen J B 56 Royce N J 68 Rudeanu S 163 Ryan D 194 Sahinidis N V 74 Salkin G 69 113 Samuelson P A 7880 Sholtz D 63 338 Schruben L W 296 Shapiro J F 78 Schittkowski K 394 Sherali H D 182 Shmoys D B 184 342 Small R E 58 338 Smeltink J 323 337 Smith B 19 Smith C 393 Smith D 39 Soland R M 72 Solow R M 69 339 Souder W E 73 Soumis F 386 Spath H 69 Srinivason V 89 Stanley E D 87 Starr M K 245 Steinberg D I 121 Stewart R 2456 248 Stone R 81 Stutz J 84 341 Sutton D W 69 Swart W 70 312 Tatham P B 32 337 Taut P 394 Telgen J 334 342 Thanassoulis E 74 335 Thomas G S 228 Thomas L C 63 326 337 Tomlin J A 92 160 178 201 216 317 319 Trick M A 74 Tucker A W 184 343 Vajda S 71 198 303 Van Hentenryck P 389 Van Roy T J 218 Veinott A F 97 215 Wage S 62 300 343 Wagner H M 81 97 215 316 Wallace W A 10 534 73 Wallace S W 10 342 Wardle P A 73 Warner D M 70 Warshaw M R 72 WatsonGandy C D T 61 184 339 Weismantel R 390 Weiss H 73 Williams A C 19 177 240 Williams H P 34 52 65 71 172 197 218 238 246 254 294 327 363 375 Wilkinson E M 74 394 Willis R J 98 Wilson E G J 98 Wilson J M 19 Wolfe P 92 Wolsey L A 12 218 223 Wong R T 197 Woodruff D L 340 Wright D H 34 70 Yan H 19 177 Yarrow LA 375 Young W 71 Yunes T 17 Zemlin R A 184 343 Zener C 129 339 Zionts S 334 342 Subject index acceptance problem of 2445 accountancy 69 11213 advertising industry 72 aggregation problem of 81 231 AGRICULTURAL PRICING problem 70 139 2767 3335 3712 agriculture 70 aircrew scheduling 159 190 1934 202 airline ticket sales see yield management algorithms 11 branch and bound 11 156 162 164 178 191 195 2078 210 217 230 239 2956 300 310 327 367 382 cutting planes 1624 210 234 decomposition 556 645 92 2012 248 enumeration 164 pseudoBoolean 1634 174 revised simplex 1113 81 92 97 295 330 separable extension of revised simplex 11 149 295 alldifferent predicate 17 19 allocation problems 48 1589 195 see also resource allocation Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd alternative optima 117 349 351 369 applications of mathematical programming 6971 73 approximations piecewise linear 1478 150151 1534 179 184 187 335 artificial intelligence 17 archaeological seriation 74 assembly line balancing problem 159 assignment problem 878 93 98 159 162 197 199201 212 317 see also quadratic assignment problem assignment variable 18 atleast predicate 19 basis reduction 364 blast furnace burdening 68 blending problems 14 3031 37 5051 71 11416 140 228 244 296 299 block angular structure 4850 524 56 65 92 Boolean algebra 163 172 177 230 bottleneck problems 34 bound reduction 17 bounded variable version of revised simplex algorithm 12 402 SUBJECT INDEX bounds simple see generalized upper bounds simple bounds generalized upper branch and bound algorithm 11 178 230 295 300 British Petroleum 245 248 267 cardinality predicate 18 cancer irradiation of 34 70 capital budgeting 195 221 CAR RENTAL PROBLEM 2847 340343 37980 centralized planning resulting from use of mathematical programming 2478 chains of linked special ordered sets 17782 218 318 chance constrained models 10 chemical industry 68 circuit predicate 18 coefficients changes in see ranging parametric programming objective 13 78 83 98 100 117 1257 134 167 192 212 2389 297 righthand side 1314 31 51 56 61 80 86 97 99101 11113 115 1215 12831 133 23940 273 324 32930 366 369 technological 4 combinatorial problems 18 15861 computer communication networks 58 computer package programs 10 144 153 computer programming 5 41 computers use of 124 227 231 computing the solution ease of 378 constraint logic programming 177 constraint satisfaction see constraint logic programming constraints 222 availability 57 835 binding 111 119 125 234 chance 32 53 common in a structured model 4950 59 conflicting 27 324 continuity see material balance convexity 5960 64 179 declarative 18 definition of 25 disjunctive 1834 186 189 22830 facet see facets formulation of 67 generalized upper bounding see generalized upper bounds global 18 hard and soft 312 133 labour see manpower local 17 manpower 49 779 316 marketing 3 30 35 114 material balance 30 85 105 114 197 meta 17 procedural 17 productive capacity 2930 77 114 quality 31 11415 raw material availability 301 47 49 114 redundant 345 130 235 3212 relaxation of 105 191 requirement 834 significance of number in an IP model 190 SUBJECT INDEX 403 simple bounding see simple bounds simplification of in an IP model 199 201 205 265 287 292 325 tightening of 115 218 unusual 356 valuation of see shadow prices construction industry 94 convex functions 1412 187 convex hull 212 215 222 224 2267 22930 2334 2378 convex models 146 150 convex programming 140 142 1446 convex regions 140141 1845 corporate models 68 245 248 costbenefit analysis 4 costs fixed 24 233 235 287 see also fixed charge problem variable shared fixed cost problem covering problem see set covering problem critical path analysis 81 948 crop rotation 70 cumulative predicate 18 CURVE FITTING problem 33 cutting stock problem 158 195 203 DantzigWolfe decomposition algorithm 56 65 data 3 collection of 249 format for presentation to package 14 41 databases 16 249 data envelopment analysis 29 74 278 DEA see data envelopment analysis DECENTRALIZATION problem 176 216 decentralized planning 556 645 248 see also decomposition decision support systems 16 245 decomposition DantzigWolfe 56 65 benders 53 of models 54 parallel with decentralized planning 645 Rosens algorithm 56 decreasing returns to scale see diseconomies of scale defence 72 degeneracy in a model 121 130 234 depot location problem 69 160 dichotomies 209 217 see also zeroone variables discontinuous variables 228 discrete programming see integer programming diseconomies of scale 146 151 disjunction of constraints see constraints disjunctive disjunctive normal form 230 distribution 6971 73 82 86 89 92 DISTRIBUTION 1 problem 230 2735 330331 36971 DISTRIBUTION 2 problem see DEPOT LOCATION problem DNA sequencing 74 domain reduction 17 dual model 38 46 56 98 105 10912 11718 dual variables see dual model duality theorem 110111 dynamic models 25 dynamic programming 71 195 economic interpretations of a model 105 107 130 economic models 68 74 404 SUBJECT INDEX ECONOMIC PLANNING problem 25 economics 4 economies of scale 146 151 160 1878 efficiency analysis see data envelopment analysis EFFICIENCY ANALYSIS problem 74 elasticity of demand 139 277 electricity industry 71 115 235 270271 element predicate 18 equilibrium theorem 112 errors in a model ease of detection 367 exclusive sets of variables 13 35 expert systems 1617 facets 218 2227 3223 363 FACTORY PLANNING problem 255 300302 350351 FACTORY PLANNING 2 problem 68 157 256 3023 3514 farm management 70 2623 3589 FARM PLANNING problem 31215 finance 69 finite domain programming see constraint logic generation fixed charge problem 160 1678 fixed costs 24 233 235 287 see also shared fixed cost problem fixed charge problem flows in networks 934 98 101 food industry 71 FOOD MANUFACTURE problem 51 71 2534 FOOD MANUFACTURE 2 problem 255 forestry 73 format for presentation of model to computer package 14 41 fourcolour problem 161 free goods 110 232 234 functions convex 1412 linear 139 142 167 182 185 334 nonconvex 133 136 172 nonlinear 147 151 1534 1792 335 objective 59 18 229 323 38 40 49 57 61 767 80 104 107 112 118 121 125 1334 13740 147 150151 159 167 188 194 201 239 241 298 302 306 315 317 319 3234 3267 3335 343 separable 1467 151 153 fuzzy sets 32 games theory of 68 garages efficiency of 278 336 3724 generalized network 923 generalized upper bounds GUBs 13 35 97 188 geometric programming 140 153 160 glass industry 72 global constraintspredicates 1719 goal programming 27 32 303 319 graph theory 161 Hamiltonian path 343 health 25 70 heuristic methods 241 Hungarian method 88 hydro electric generation 71 2712 3267 3668 SUBJECT INDEX 405 hydro power see hydro electric generation hydro power problem 3267 3668 implementation problem of 2439 incidence matrix 91 increasing returns to scale see economies of scale infeasibility detection of 1035 inputoutput models see Leontief models integer programming 6 11 24 289 334 36 43 67 6970 724 878 92 978 146 149 151 154205 20741 253 273 289 2956 299 302 310 317 322 325 327 335 343 345 365 integrality property 88 159 162 164 178 188 212 231 286 interpretation of solution 107 accuracy of 111 see also ranging importance of making easy 37 investment problems 178 235 239 245 359 irrigation 70 job sequencing problems 196 job shop scheduling 97 158 244 knapsack problem 1578 195 204 227 Leontief models 68 74 lexgreater predicate 18 libraries journal selection in 74 linear expressions 8 linear programming LP 5 7 131 169 linear programming relaxation 337 linearity importance of 213 linearity piecewise 147 logical conditions see relationships logical logical design 159 2667 320321 363 LOGICAL DESIGN problem 216 LOST BAGGAGE DISTRIBUTION problem 1989 2878 3434 380381 mailing lists construction of 73 maintenance of a model 249 management information systems 16 manpower planning 26 71 2567 3036 3546 MANPOWER PLANNING problem 26 manufacturing industry 689 72 marginal rates of substitution 125 1312 marginal values see shadow prices maritime transportation see shipping MARKET SHARING problem 334 221 226 marketing 12 30 68 723 114 2457 256 300 352 marketing constraints 3 35 master model in decomposition 6064 master problem see master model matching problem 192 227 mathematical model 37 55 76 mathematical programming 5 mathematical relationships 3 matrix totally unimodular 325 incidence 91 inputoutput see Leontief models matrix generators 40 107 135 matroids 99 maximax objectives 1889 406 SUBJECT INDEX maximum flow problem 934 media scheduling 72 milk distribution and collection of pricing of 278 MILK COLLECTION problem 69 1989 278 3367 3745 military applications 72 minimax objective 278 189 193 322 364 minimum cost flow problem 85 89 91 93 mining industry 7071 MINING problem 71 98 187 215 missile sites siting of 72 mixed integer programming MIP see programming integer mnemonic names usefulness of 37 307 modal formulations 389 59 310 model 3 abstract 3 compact 367 312 concrete 3 convex 1426 corporate 68 245 2478 dual 38 98 105 10912 11718 231 dynamic 25 8081 316 ease of understanding 367 econometric 4 infeasible 32 378 72 1034 inputoutput see Leontief model integer programming 6 11 24 289 334 36 43 67 6970 724 878 92 978 146 149 151 154205 20741 253 273 289 2956 299 302 310 317 322 325 327 335 343 345 365 interpreting solution of 10335 Leontief 68 74 31617 linear programming 610 2144 166 1829 maintenance of 249 master in decomposition 645 mathematical 3 6 mathematical programming 56 metapredicates 17 multiperiod 5052 80 247 303 multiplant 39 multiproduct 50 52 network 4 68 8198 network planning 4 68 8198 nonconvex 146 150 nonlinear programming 13754 nonstandard 4 simulation 4 solvable 1057 stability of 1323 23940 standard 4 static 745 77 80 structured 4565 time series 4 unbounded 1045 zeroone programming 144 152 154 160 190 modelling language 14 17 4042 molecular biology 289 MPS format 14 41 multicommodity network flow problem 92 97 197 multiperiod models 5052 80 247 303 use of for decision making 5052 247 multiplant models 39 multiproduct models 50 52 multiple choice examinations construction of 73 network models 68 8198 use of specialized algorithms for 261 297 network planning see network models SUBJECT INDEX 407 networks see network models NEWMAGIC modelling language 423 nonconvex functions nonconvex models 146 150 nonconvex programming 140 1446 nonconvex regions 141 1845 nonlinear programming 6 21 23 68 70 140 1457 153 160 184 187 objectives see objective functions objective functions 59 18 229 323 38 40 49 57 61 767 80 104 107 112 118 121 125 1334 13740 147 150151 159 167 188 194 201 239 241 298 302 306 315 317 319 3234 3267 3335 343 choice of 19 coefficients of 25 37 78 83 98 100 117 1257 134 167 192 212 2389 297 conflicting 267 formulation of 6 minimax 278 189 193 322 364 multiple 267 nonexistent 29 nonoptimizable 29 ratio 289 single 246 oil industry see petroleum industry oil refining see petroleum industry OPENCAST MINING problem 71 98 215 26870 3245 364 operational research 34 24 68 81 158 161 241 244 opportunity costs see shadow prices reduced costs optima alternative 117 349 351 369 continuous 366 371 global 140 179 188 integer 98 212 21415 236 local 23 145 151 160 187 optimal solutions see optima optimization 5 24 29 39 53 6061 68 73 105 150 163 191 195 1978 2012 30610 3367 344 3567 OPTIMIZING A CONSTRAINT problem 221 3234 32830 369 package programs 61 967 103 105 108 112 115 119 128 131 packing problem see set packing problem paper industry see pulp and paper industry parametric programming 122 127 130 1334 306 partitioning an IP model 163 partitioning problem see set partitioning problem performance measures 28 PERT 81 161 petroleum industry 39 42 68 306 political districting problem 160 pollution 74 portfolio selection 69 postoptimal analysis 124 132 248 see also parametric programming ranging sensitivity analysis power generation see electricity industry POWER GENERATION problem see TARIFF RATES problem predicate calculus 165 408 SUBJECT INDEX predicates in constraint logic programming in modelling 179 177 presentation of solutions 135 price elasticity see elasticity of demand prices shadow 267 34 612 64 78 108 11214 12832 204 215 2315 237 350 3656 372 380 pricing of products see reduced costs of resources shadow prices primal model 10912 1178 121 primary goods 78 products of variables 176 201 product mix problem 109 121 231 production planning 857 see also product mix problem production processes representation of 6 789 249 productive capacity constraints 6 789 249 programming computer 5 convex 140 142 145 dynamic 71 195 geometric 140 453 160 goal 27 32 303 319 integer 6 11 24 289 334 36 43 67 6970 724 878 92 978 146 149 151 154205 20741 253 273 288 2956 299 302 310 317 322 325 327 335 343 345 365 mathematical 419 234 267 29 41 60 nonconvex 140 1446 nonlinear 6 21 23 68 70 140 1457 1534 160 184 187 parametric 122 127 130 1334 306 quadratic 34 6970 139 160 335 separable 9 11 16 140 1447 150151 153 160 179 1878 253 295 335 stochastic 910 535 368 zeroone 144 152 154 160 165 190 project planning 948 project selection problem 159 195 221 PROLOG 177 proposals in decomposition 60 propositional calculus 152 161 163 165 PROTEIN COMPARISON problem 28990 345 382 PROTEIN FOLDING problem 290291 3445 382 pulp and paper 72 recycling of 72 pure integer programming PIP see integer programming quadratic assignment problem 15962 199201 317 quadratic programming 34 6970 139 160 335 ranges on constraints 13 130131 ranging of coefficients 12831 ratio objectives 289 raw materials availability of 30 114 reduced costs 108 11621 231 234 reduction of size of models 11 32 278 318 SUBJECT INDEX 409 redundancy in models 26 367 130 258 354 among constraints 29 120 detection of 37 removal of 11 REFINERY OPTIMIZATION problem 39 68 30610 3567 refining oil see petroleum industry relationships logical 155 173 mathematical 3 sequencing 967 technological 3 relaxation see linear programming relaxation report writers 135 research and development planning of 73 reservoir management see hydro electric generation resource allocation 25 68 7072 161 on a planning network 89 150 resource constraints 3 42 resources indivisible 146 154 174 labour 49 779 316 productive capacity 25 2930 77 114 231 raw material 77 114 316 scarce 49 114 valuation of 74 78 111 117 119 revenue management see yield management righthand side coefficients 1314 51 56 80 86 97 99101 11213 115 121 124 1289 133 330 robust optimisation 53 Samuelson substitution theorem 7880 scaling the coefficients of a model 40 scarce resources 489 70 110111 113 116 120 scenarios in stochastic programming 304 scheduling aircrew 159 190 1934 202 jobshop 97 158 244 media 72 semicontinuous variables 207 sensitivity analysis 13 53 12132 2389 separable functions 1467 151 153 separable programming 9 11 16 140 14452 160 179 1878 253 295 335 sequencing problems 158 196 256 relations 967 set covering problem 18991 194 202 227 set packing problem 1914 set partitioning problem 15960 1924 214 setup costs 15960 167 196 shadow prices 267 34 612 64 78 108 11224 12832 204 215 2315 237 350 3656 372 380 two valued 130 shared fixed cost problem 233 sharpness of integer programs 22930 shipping 69 shortest path problem 93 SIMPL program 17 simple bounding constraints see simple bounds simple bounds 1213 30 35 38 87 218 310 tightening of 2189 simplification of integer constraints 220226 410 SUBJECT INDEX slack variables 13 31 33 35 99 193 20910 212 240 3224 344 364 solutions alternative see optima alternative feasible 18 23 29 57 767 104 106 158 211 268 3634 367 optimal 8 1213 1819 223 27 32 34 467 502 557 6065 80 86 88 92 94 978 105106 108 110112 11517 11920 1223 12533 144 1489 155 15960 186 188 19192 194 197 201 204 21215 218 2278 2314 236 23941 253 255 268 325 330 349 3567 361 3634 367 369 372 374 stable see models stable starting 12 26 spanning trees 100 special ordered sets of variables 17782 218 318 spreadsheets 249 stability of a model 132133 23940 staircase structure of a model 52 starting solutions 12 26 static models 745 77 80 steel industry 68 74 76 stochastic programming models 910 535 368 submodels of a structured model 48 50 5557 59 634 230 substitution in Leontief models 69 72 surplus variables 31 33 99 240 3224 switching circuits 159 symmetry breaking of 18 203 210 TARIFF RATES problem 63 105 214 247 270271 324 taxation 69 theory of games 68 THREEDIMENSIONAL NOUGHTS AND CROSSES problem 272 3278 368 totally unimodular matrix 325 transhipment problem 69 889 93 transport industry 76 360 transportation problem 35 829 91 97 159 2123 travelling salesman problem 12 16 19 1634 20910 218 241 300 303 322 327 367 371 tree searches 12 16 19 1634 20910 218 241 300 303 322 327 367 371 trees spanning 100 trimloss problem 72 Tuberculosis control of 70 unboundedness detection of 38 105 107 validation of a model 29 106 variables buying 43 binary 44 continuous 155 1667 171 1756 186 188 210211 216 229 235 240 311 325 333 decision 1656 209 discrete see integer variables discontinuous 228 SUBJECT INDEX 411 dual see dual model exclusive sets of 35 free 112 153 319 335 342 345 indicator 16672 integer 43 53 165 products of 176 201 significance of number in an IP model 187 219 slack 13 31 33 35 99 193 20910 212 240 3224 344 364 storing 50 298 surplus 31 33 99 240 3224 zeroone 165 1727 vehicle routing problem 196 1989 202 343 warehouse location see depot location problem yield management 54 27884 33740 3768 YIELD MANAGEMENT problem 69 zeroone programming 144 152 154 160 190 zeroone variables 165 1727