Optimization of Shelfspace and Replenishment in Retail

Business Research 2017 10123156 3 CrossMark DOI 101007s4068501600436 ORIGINALRESEARCH Effect of replenishment and backroom on retail shelfspace planning Alexander Hiibner Kai Schaal Received 16 March 2016 Accepted 8 December 2016 Published online 9 January 2017 The Authors 2017 This article is published with open access at Springerlinkcom Abstract Shelfspace optimization models support retailers in making optimal shelfspace decisions They determine the number of facings for each item included in an assortment One common characteristic of these models is that they do not account for instore replenishment processes However the two areas of shelfspace planning and instore replenishment are strongly interrelated Keeping more shelf stock of an item increases the demand for it due to higher visibility permits decreased replenishment frequencies and increases inventory holding costs How ever because space is limited it also requires the reduction of shelf space for other items which then deplete faster and must be reordered and replenished more often Furthermore the possibility of keeping stock of certain items in the backroom instead of the showroom allows for more showroom shelf space for other items but also generates additional replenishment costs for the items kept in the backroom The joint optimization of both shelfspace decisions and replenishment processes has not been sufficiently addressed in the existing literature To quantify the cost associated with the relevant instore replenishment processes we conducted a time and motion study for a German grocery retailer Based on these insights we propose an optimization model that addresses the mutual dependence of shelfspace deci sions and replenishment processes The model optimizes retail profits by deter mining the optimum number of facings the optimum display orientation of items and the optimum order frequencies while accounting for spaceelasticity effects as well as limited shelf and backroom space Applying our model to the grocery retailers canned foods category we found a profit potential of about 29 We further apply our model to randomly generated data and show that it can be solved to optimality within very short run times even for largescale problem instances Finally we use the model to show the impact of backroom space availability and Alexander Hiibner alexanderhuebner kude Catholic University EichstaettIngolstadt Ingolstadt Germany Q Springer 124 Business Research 2017 10123156 replenishment cost on retail profits and solution structures Based on the insights gained from the application of our model the grocery retailer has decided to change its current approach to shelfspace decisions and instore replenishment planning Keywords Shelf space Backroom Spaceelastic demand Optimization Replenishment 1 Introduction Retailers use shelves to offer their products to customers In doing so they must decide how much shelf space to allocate to which item Because shelf quantities assigned to retail shelves become depleted over time due to customer purchases retailers need to regularly refill shelves and reorder items Reordering directly impacts replenishment processes As soon as reorders arrive at the store the respective items are transported to the showroom where shelves are replenished ie direct replenishment As a result every order process triggers a direct replenishment process Items that do not fit onto the showroom shelf space are stored in the backroom from where shelves are later replenished ie indirect replenishment from backroom Both decisions ie shelf space and reordering are interrelated because eg to meet customer demand a retailer has the option of increasing the shelf quantity and decreasing the order frequency for a specific item or vice versa If space is limited a higher shelf quantity for one item implies less frequent reorders and replenish ments for this item but also less space available for other items which in turn must be reordered more frequently Shelfspace and reorder planning are of great importance to retailers for several reasons The increasing number of products is in conflict with limited shelf space Today up to 30 more products than ten years ago compete for scare space EHI Retail Institute 2014 Hiibner et al 2016 This puts retailers under pressure to manage profitability with narrow margins and to maintain space productivity Gutgeld et al 2009 In fact shelf space has been referred to as a retailers scarcest resource cf eg Lim et al 2004 Irion et al 2012 Geismar et al 2015 Bianchi Aguiar et al 2015a Above all changes in shelf space impact customer demand due to the higher visibility of items Eisend 2014 In other words the demand for an item grows the more shelf space is allocated to it This is referred to as space elastic demand Additionally the costs associated with instore replenishment are significant because instore logistics costs amount to up to 50 of total retail supply chain costs Kotzab and Teller 2005 Broekmeulen et al 2006 Reiner et al 2013 Kuhn and Sternbeck 2013 However the options for changing replenishment frequencies are subject to the availability of backroom inventory for intermediate storage Eroglu et al 2013 Pires et al 2016 and the degree of freedom to choose different delivery frequencies Sternbeck and Kuhn 2014 Besides product margins and demand effects the shelfspace planner should therefore also consider options for arranging items on the shelf instore replenishment frequencies and costs and the availability of a backroom for replenishment Q Springer Current literature on shelfspace management mainly addresses the demand side by modeling the effect of spaceelastic demand In this case a retailers profit is maximized under shelfspace constraints by defining the number of facings for each product ie first visible unit of an item in the front row Hubner and Kuhn 2012 Kok et al 2015 Existing models do not account for replenishment frequencies and costs or options for leveraging backroom inventory Hubner and Kuhn 2012 BianchiAguiar et al 2016 To investigate the abovementioned relationships we conducted a time and motion study for a German grocery retailer and identified both the relevant instore replenishment processes and the associated costs Building on these insights we then develop an optimization model that simultaneously optimizes shelfspace and instore replenishment decisions while also accounting for spaceelastic demand as well as limited showroom and backroom space The model accounts not only for product margins but also for the costs of direct shelf replenishment upon delivery of orders to the store and for replenishment from the backroom Furthermore we consider the cost of inventory kept in the showroom and the backroom This extended model addresses the research question of how different replenishment procedures and the opportunity to use backroom space impact shelfspace planning We apply the model to show why an integrated perspective on shelfspace and in store replenishment optimization is worthwhile and demonstrate how retailers can apply the model to increase their profits We address the tradeoffs between shelfspace allocation and instore replenish ment eg more space less frequent orders and replenishments Because retailers use backrooms as a planned buffer or for excess inventory after shelf replenishment we further investigate how the availability of a backroom impacts shelfspace decisions and order frequencies The remainder of this paper is organized as follows Sect 2 provides the conceptual background of our paper and presents the related literature on shelf space optimization The time and motion study and the description of identified replenishment processes are presented in Sect 3 Section 4 explains the optimiza tion model and presents a solution approach Numerical results for testing our model and the impact of backroom space and replenishment cost on objective values and solution structures are presented in Sect 5 Finally Sect 6 has the conclusion and outlook 2 Conceptual background and related literature 21 Conceptual background and decision problem In the following we analyze the basic decisions retailers need to make in shelf space and reorder planning namely 1 how much shelf space to allocate to items and 2 how often to reorder them 1 Shelfspace decision Shelfspace planning is a midterm task and typically executed every six months requiring a retailer to assign shelf space and shelf quantities to listed products under the constraints of limited shelf size Hubner and Business Research 2017 10123156 125 123 126 Business Research 2017 10123156 Kuhn 2012 Hiibner et al 2013 BianchiAguiar et al 2016 The results of these decisions are visualized in a planogram which displays the number of facings display orientation and position on the shelf for every item cf Fig 1 A facing is the first visible unit of an item in the front row of a shelf Behind each facing there is certain quantity of stock ie additional units of the respective item The number of facings and the stock per facing determine the total shelf quantity of an item Furthermore items can be displayed lengthwise or crosswise cf Dréze et al 1994 Particularly when single units of an item are stored in cartons the retailer must decide on the display orientation of the respective carton Figure 1 left shows the difference between facings and shelf quantities For example item a gets 2 facings with a stock of 4 units each resulting in a total quantity of 24 8 units Item b gets 1 facing with a stock of 6 units and a total shelf quantity of 6 The right of Fig explains the difference between length and crosswise display orientation Display orientation also impacts the stock per facing since more or fewer units of an item fit behind one facing depending on whether the item is positioned length or crosswise In Fig 1 fewer units would fit behind one facing lengthwise and more units behind a facing with crosswise display orientation The stock that can be placed behind one facing is determined by the depth of the shelf and the item dimensions Since the space behind one facing is always filled completely after a replenishment ie filled up until the shelf depth is fully occupied the stock per facing is not itself a decision of the retailer but is determined via shelf and item dimensions as well as the decision on the display orientation Finally the position of an item on the shelf is described by its vertical ie which shelf level and horizontal position ie which items are located next to each other We focus on the core demand effect of space elasticity and therefore do not account for these positioning effects in our model For models accounting for item positioning we refer the reader to eg Hwang et al 2009 Hansen et al 2010 Russell and Urban 2010 and Hiibner and Schaal 2017 Shelfspace decisions impact customer demand Item demand depends on the visible quantity on the shelf and the display orientation The higher the visibility of an item the higher its demand The visibility of an item increases with the number of facings assigned to that item and its display orientation ie the visible item width the customer sees Empirical studies examine these socalled space elasticity effects cf eg Cox 1964 Frank and Massy 1970 Curhan 1972 Dréze et al 1994 Desmet and Renaudin 1998 Chandon et al 2009 show that the Facing and shelf quantity Display orientation Lengthwise Crosswise a ra oe fatal 4 Lower visibility Higher visibility Lower frontrow space Higher frontrow space Fig 1 Illustration of facing shelf quantity and display orientation Q Springer number of facings is the most important instore factor affecting customer demand Using a metaanalysis across empirical studies Eisend 2014 quantified the average spaceelasticity factor as 17 which implies a demand increase of 17 each time the number of facings is doubled The discussion of the demand effect on other items referred to as crossspace elasticity is ambiguous in the pertinent literature For example Zufryden 1986 and Kok et al 2015 state that there is no empirical evidence that productlevel demand can be modeled with crossspace elasticity In addition the measured effects of crossspace elasticity appear to have only a limited influence on sales Eisend 2014 Hubner and Schaal 2016 We therefore comply with these results in the literature and disregard crossspace elasticity effects in the remainder of our paper Furthermore shelfspace decisions also depend on assortment planning How ever in retail practice assortment and shelfspace decisions are typically two sequential planning steps of the category planning process Hubner et al 2013 Kok et al 2015 BianchiAguiar et al 2016 Assortment planning is usually executed in an overarching planning step by the marketing department whereas shelf planning is a subordinate planning problem and generally owned by the sales department However if the shelf space planner can also make assortment decisions and delist items due to shelf space constraints for instance or low profitability of items one needs to take into account potential substitutions due to demand switches from delisted to listed items 2 Ordering decision Since ordering decisions impact store operations thorough reorder planning is crucial for retailers see eg Fisher 2009 Zelst et al 2009 Donselaar et al 2010 DeHoratius and Zeynep 2015 Kuhn and Sternbeck 2013 use qualitative interviews to identify that space management and instore logistics are not yet well aligned and that this constitutes a new area of research Reiner et al 2013 identify opportunities for improving instore logistics and show that it is important to not only consider customer needs and the demand side when designing store layout and taking shelfspace decisions but also logistics requirements Their process analysis reveals that the efficient design of instore logistics processes leads to substantial service performance improvements Furthermore Kotzab et al 2005 and Kotzab et al 2007 use qualitative interviews with store managers to identify the relevant instore logistics and replenishment processes Their findings and process descriptions form the starting point of our research Because shelves are depleted over time due to customer purchases retailers need to reorder items and replenish shelves Consequently a retailer needs to decide how often to reorder an item This order frequency impacts instore logistic processes because each order triggers a delivery from the warehouse to the store which again results in a direct replenishment effort to transport the delivered items to the showroom shelves Beyond this the order frequency also impacts the number of replenishments from the backroom because the less frequently items are ordered from warehouses the higher the backroom quantity at the store that needs to be kept if shelf space is not sufficient to fulfill customer demand and the higher the number of replenishments of showroom shelves from the backroom We discuss the instore Business Research 2017 10123156 127 123 logistics processes related to direct and backroom replenishment in more detail in Sect 3 22 Related literature on shelfspace optimization Following Seuring et al 2005 and Kotzab et al 2005 we first defined the scope of our contribution as in Sect 21 and then identified the related literature The identification step included the material collectionselection and category selection Finally we completed a content analysis Because we focus on quantitative decision models we excluded literature that strictly covers general management marketing and service management issues and does not discuss modeling aspects and decision support systems at all Papers were assessed based on their decision modeling demand models and their relation to space and reorder planning In the following we introduce the fundamental modeling papers and analyze only the shelfspace modeling papers that contain considerations of replenishment inventory holding and store operations Further modeling papers dealing with shelf space problems that do not include any of these considerations or that are not related to our decision problem are not the focus and not further analyzed Shelfspace models typically assume a given assortment with spaceelastic demand ignore substitution because the assortment is predetermined and account for limited shelf space Our problem relates to the shelfspace literature and we therefore focus on this area For comprehensive overviews of shelfspace problems we refer to Hubner and Kuhn 2012 Kok et al 2015 and BianchiAguiar et al 2016 The models reviewed here all optimize the number of facings for a given set of items and limited shelf space The main demand effect considered is space elasticity To account for the nonlinearity arising from the polynomial demand function various solution approaches are applied and spaceelasticity effects either assumed to be linearly dependent on the number of facings approximated by piecewise linearization or nonlinear models applied and solved with heuristics Basic shelfspace management literature uses deterministic demand models to factor in spaceelasticity effects cf Kok et al 2015 One of the first models is proposed by Hansen and Heinsbroek 1979 who formulate a nonlinear model with various constraints such as minimum and integer shelf quantities and space elasticities of polynomial form To solve the problem they apply a Lagrangian relaxation Corstjens and Doyle 1981 propose a limited shelfspace model that considers space and crossspace elasticities of polynomial form Geometric programming is applied to solve the model for up to five product groups The model cannot be applied to largescale problems on an itemlevel and therefore works with product groups rather than SKUs Zufryden 1986 formulates a model with spaceelastic demand of polynomial form which is solved through dynamic programming for up to 40 products Borin et al 1994 propose a model that considers space and crossspace elasticities of polynomial form Substitution effects are integrated and the model is solved with a simulated annealing heuristic for six items Yang and Chen 1999 assumes a linear space elasticity function and solves the model through a multiknapsack heuristic Urban 1998 provides the first enhancement with available inventory and replenishment systems The polynomial 128 Business Research 2017 10123156 123 demand model takes into account restrictions in backroom capacity minimum order quantities and ensures that replenish quantities meet demand The decision variables for facings and order quantities are continuous values and thus violate integer requirements They are only rounded afterwards Furthermore the proposed solution heuristic is only applied to a small data set and no efficiency analysis is conducted in terms of solution quality Hwang et al 2005 develop a shelfspace optimization model with inventory control aspects Space elasticity is assumed to be polynomial and the model solved through a genetic algorithm and a gradient search heuristic Tests are limited to instances of up to four items on six shelves Hariga et al 2007 propose a model that simultaneously optimizes assortments shelf space store location and inventory replenishment frequencies The model accounts for space and crossspace elasticities of polynomial form but does not differentiate between direct and backroom replenishment costs The problem is tested with instances of four items and solved by a standard solver Ramaseshan et al 2008 2009 determine shelfspace allocation and inventory quantities Their decision model is implemented in Excel and generates an approximate solution for up to 14 items Murray et al 2010 present a model that considers pricing aspects and optimizes shelfspace allocations Best to our knowledge this is the only contribution so far additionally accounting for the display orientation of items The problem is solved through a nonlinear solver and tested on largescale instances with up to 100 items Hubner and Kuhn 2011 develop a MIP model to account for polynomial spaceelasticity and replenishment cost They show that demand effects have a significant impact on item profit The model balances the tradeoff between over and undersupply situations where either store staff need to refill shelves in between two regular shelf refills or where overstocks result in capital cost The order frequency decision is not explicitly taken Direct and indirect replenishment is not distinguished and backroom capacity is not accounted for Irion et al 2012 develop a nonlinear model for crossspace and space elasticities that is then solved by piecewise linear approximation which makes it possible to handle data sets of a size relevant in practice They include inventory holding costs in the model Hariga and AlAhmari 2013 develop an integrated space allocation and inventory model for a single item with stockdependent demand They analyze different setups regarding the supplierretailer relationship and optimize the order quantity reorder point and number of facings However this paper is restricted to one single product showroom and backroom replenishments are not distinguished and facing dependent spaceelasticity effects not considered BianchiAguiar et al 2015a use a MIP approach to develop a model that considers product grouping and display orientation constraints and therefore incorporates merchandising rules Tests for instances with up to 256 items are conducted 23 Summary and research contribution When retail shelf space is limited retailers need to thoroughly consider the tradeoff between shelfspace and ordering decisions The two decisions are interdependent and impact instore logistics processes for shelf replenishment since every order triggers direct replenishment of shelves and since items that do not fit onto the Business Research 2017 10123156 129 123 showroom shelf must be indirectly replenished from the backroom Despite these interdependencies in instore logistics and space assignment an integrated optimization model is lacking in the existing literature The contributions on shelfspace planning mentioned all focus on optimizing the number of facings Demand is assumed to be facingdependent ie spaceelastic Nonlinearities arising from this are dealt with either via linear approximations or solution heuristics that are limited in their capability to solve instances of practicerelevant size To close this research gap our contribution is threefold 1 Based on a detailed time and motion study for a retailer we first quantify costs associated with direct and backroom replenishment processes 2 Using the insights from the time and motion study we propose an integrated model to optimize planograms ie facings and display orientations and order frequencies Our model accounts for space elastic demand assumes limited showroom and backroom space and differentiates between direct and indirect replenishment cost as well as showroom and backroom inventory holding cost By means of our modeling approach we obtain optimal solutions within very short runtimes even for largescale instances 3 By applying our model to a real data set we show how the retailer can increase profits with optimized shelfspace decisions and order frequencies Furthermore we use our model to show how backroom availability impacts profits and solution structures and prove the advantage of our model over other approaches that do not account for the relevant costs 3 Time and motion study for instore replenishment processes To accurately quantify the costs associated with the different replenishment processes we conducted a time and motion study Current literature serves as a starting point for defining the process steps for instore logistics eg Kotzab and Teller 2005 Zelst et al 2009 Reiner et al 2013 but does not sufficiently detail the costs affected by shelfspace planning and reordering Curseu et al 2009 conduct a similar time and motion study on parts of the direct replenishment process They measure process times for shelf refilling and waste disposal but ignore transport to the shelves from the receiving area Moreover backroom replenishment is not part of the investigation Hence our investigations serve to further specify replenishment processes and the interdependencies with shelfspace decisions 31 Replenishment processes observed during time and motion study Figure 2 provides an overview of the instore logistics processes and the associated subprocesses for replenishment It visualizes all shelf replenishment processes from the unloading of a store delivery from the truck to shelf restocking and waste disposal The processes can be distinguished by the respective store locations where they are executed namely the 1 receiving area the 2 showroom and the 3 backroom The relevant processes are described below 130 Business Research 2017 10123156 123 1 Preparation processes in the receiving area Are the starting point of related instore logistics When store deliveries arrive at the receiving area they are unloaded and brought to a presorting area where they are sorted by category prior to actual shelf replenishment and finally brought to the showroom to a central category location cf Kotzab and Teller 2005 From there the actual shelf replenishment starts This implies that a pallet roll cage or other means of transport is placed at this central category location from which stock clerks take individual items and transport them to the shelves Based on the process flow in Fig 2 a distinction can basically be made between two types of replenishment direct and backroom ie indirect replenishment 2 Direct replenishment Occurs for every order delivered from the warehouse and describes the processes after the items arrive in the showroom Upon arrival at the central category location items are individually picked transported to the specific shelf location and then refilled Finally waste from the packaging of refilled items is disposed of The fact that every order and store delivery induces a direct replenishment process implies that the number of orders equals the number of direct replenishments for an item 3 Backroom ie indirect replenishment Refers to the processes that occur when items that did not fit on the shelf are returned to the backroom where they are stored for later replenishment As soon as items are depleted on showroom shelves employees restock them from the backroom To do so a stock clerk searches for the relevant items in the backroom transports them through the store to the shelves and refills them Because the backroom replenishment process includes both the returning of excess units from the showroom after a direct replenishment as well as the refilling of shelves from the backroom indirect replenishment typically is at least twice as expensive as direct replenishment Retailers do not have a dedicated ordering process for backroom inventory Only surplus items from the showroom are temporarily stored in the backroom Fig 2 Overview of related instore logistics processes Business Research 2017 10123156 131 123 32 Identification of decisionrelevant replenishment costs Data collection through time and motion study To assign the cost to the instore replenishment processes described in the previous section we conducted a time and motion study for a German grocery retailer Stock clerks were accompanied to identify the different steps involved in the instore replenishment process To do so we made use of the methodstime measurement concept Maynard et al 1948 Following Barnes 1949 and Niebel 1988 we analyzed the replenishment process systematically by first identifying subprocesses and the most efficient way of executing them then categorizing and standardizing the subprocesses identified and ultimately determining the standard time required by a qualified stock clerk to execute each subprocess To level the effects of potential outliers in observed process times due to factors such as time staff capability or store location we measured the process times across two stores on all weekdays and for various employees Product groups were from the ambient assortment including both fast and slowmovers The division into subprocesses was as granular as possible to accurately differentiate between constant and variable elements Barnes 1949 Process mapping was used to detect potential process improvements but also to calculate the standard time required for a task During the data collection period the stores were asked to have their most qualified and properly trained personnel perform replenishment Moreover the observation days were selected such that they did not include any periods of major demand changes eg holidays Moreover we observed replenishments of single units as well as whole packages cartons with different case pack sizes cf Zelst et al 2009 Decisionrelevant replenishment cost Retailers make use of delivery patterns which define specific days for store deliveries from the warehouse for each store and each product group Hubner et al 2013 These delivery patterns mainly depend on given network structures and product groups eg fresh products are delivered more often than dry foods These fixed midterm delivery patterns allow for higher stability in warehousing transportation and in stores especially in terms of capacity management and workforce planning such that external stock clerks for instance can be scheduled for the day a specific category is delivered Holzapfel et al 2016 The costs associated with each delivery ie cost of transportation to the store unloading at the store transport of a pallet to the receiving area can only be avoided if no single product among all the categories is ordered for a certain delivery day The same holds true for the costs associated with each replenishment of an entire category ie presorting by category in the receiving area transporting categoryspecific pallets to the category location in the showroom Without compromising on the general applicability of our approach we assume that these fixed delivery costs and fixed replenishment costs per category are not decision relevant in shelfspace planning In contrast costs arising from the replenishment of single items are decision relevant for our problem context Direct and backroom replenishment processes generate decisionrelevant costs of this kind due to the subprocesses described above eg from the transportation of items to shelves to replenishment from the 132 Business Research 2017 10123156 123 backroom on depletion The decisionrelevant costs can be divided into variable costs incurred for every unit refilled and fixed costs incurred for every replenishment procedure Fixed costs for each direct replenishment are incurred for every replenishment process of an item This includes further sorting in the showroom eg readjusting the placement of pallets for better accessibility positioning roll cages or shopping carts instore transportation from the central category location to the shelf the search time to find the shelf location of the items rearranging existing stock and finally waste disposal Variable costs for direct replenishment are incurred for every delivered unit of an item These are quantitydependent costs for actual shelfstacking and shelf refilling activities per unit eg positioning the delivered units unpacking Fixed costs for backroom replenishment are incurred for every replenishment of shelves with items from the backroom These costs include the return transport of excess items from direct replenishment These units did not fit on the shelf and are stored in the backroom Furthermore these costs include searching for the items in the backroom instore transport to showroom shelves searching for the shelf locations preparing the shelves for refilling and disposing of waste Variable costs for backroom replenishment are incurred per unit refilled from the backroom to the showroom These are quantitydependent costs for actual shelfstacking and shelfrefilling activities per unit eg positioning the delivered units unpacking during the replenishment process Finally depending on the quantity kept on shelves and in the backroom inventory holding costs are incurred Typically holding inventory in the showroom is slightly more costly because shelves cannot be used as efficiently as a storage location like the backroom One reason is that showroom shelves have to look appealing which is not necessarily a requirement for backrooms In summary the relevant costs for the problem considered here must include all instore replenishment processes after an item has reached the showroom The decisionrelevant costs can be separated into fixed and variable replenishment costs for direct and backroom replenishment as well as inventory holding costs for items in the showroom and backroom 4 Model development In this section we develop the Capacitated ShelfSpace and Reorder Problem with Backroom Space which addresses the decision problem described above It is abbreviated by CSRPBS below A retailer considers a category with a given set of items N where N ¼ jN and with the item index i i 2 N For this set of items the retailer simultaneously needs to decide how much shelf space to allocate to the items ie the number of facings whether to display them on the shelf lengthwise or crosswise ie the display orientation and how often to order them ie reorder frequency We assume a limited showroom shelf space of S and a limited backroom space of B Because we aim to investigate the interdependencies between shelf Business Research 2017 10123156 133 123 134 Business Research 2017 10123156 space and reorder planning and its consequences on direct and indirect replenish ment we follow the majority of contributions and consider a single showroom shelf cf eg Zufryden 1986 Corstjens and Doyle 1981 Irion et al 2012 ie we do not account for different shelf levels and assume the shelf consists of one level with a onedimensional space S Accordingly we focus on space elasticity as the dominant demand effect cf Chandon et al 2009 Table 1 provides an overview of the notation used The model optimizes three types of decisions variables The first type k determines the integer number of facings on the showroom shelf for each item i The second type the visible width of a facing of an item b defines whether an item i is displayed lengthwise or crosswise This impacts whether the customer sees the item length b or width b w when looking at the shelf from the front Finally the third type order frequency f determines the integer number of orders per period and consequently the number of direct store replenishments Three types of auxiliary variables g for the stock per facing xj for the total shelf quantity and y for the backroom inventory are used in addition Behind each facing a certain stock gjp can be put onto the shelf This stock depends on the visible width of a facing determined through the display orientation For each of the two possible widths b a fixed number of units can be placed behind the facing How many units fit behind the facing depends on the item dimensions J or wi determined by the display orientation chosen and on the shelf depth The second auxiliary variable the total shelf quantity x depends on the number of facings k and the stock per facing gj and is computed by xj k gip To derive the third auxiliary variable the backroom quantity y we consider the fact that the order frequency f divides the considered period into equal f subperiods A subperiod demand needs to be covered during each of these subperiods This subperiod demand corresponds to the total item demand D divided by the number of subperiods Df The subperiod demand is fulfilled by the shelf quantity x and the backroom inventory y which we calculate as the part of the subperiod demand not covered by the shelf quantity y max7 Xip0 This ensures that customer demand is always fulfilled Total demand D for an item is assumed to be deterministic which implies that outofstock situations cannot arise The demand is only dependent on the initial facing assignment and does not change between replenishments when one of several facings are empty We do not consider joint replenishment effects which would allow the fixed direct or backroom replenish ment cost to be spread across several items Furthermore we do not account for delivery patters with unequal intervals and assume that the time in between two replenishments is always the same ie if f 2 a delivery occurs in a six day period eg on days and 4 but not on days and 5 We also do not model further cost savings via a joint vehicle routing and delivery frequency selection across stores The retailer pursues the objective of maximizing the total profit through selecting the optimal number of facings k visible width of a facing b and order frequencies f across all items represented by the respective vectors kb and f with k kikokw b b1 bobw and f fiftfw cf Eq 1 Q Springer Business Research 2017 10123156 135 Table 1 Notation Sets and indices i Item index N Set of items a retailer must assign to the shelf N 12iN K Set of facings a retailer can select for item i IK k k EF Set of order frequencies a retailer can select for item i F ff O Set of display orientations a retailer can select from O 12 Parameters B Available backroom space measured in number of space units eg m Cj Unit purchasing costs of item i CPR CBR Total direct DIR and backroom BR replenishment cost of item i d Minimum demand rate of item i if it is assigned one facing only and has visible width of 1 D Total demand rate of item i including spaceelasticity effects frit fe Lower and upper bound on the order frequency of item i FCP FCPRR Fixed costs per replenishment of item i for direct replenishment DIR and replenishment from backroom BR HSK ABR Inventory holding costs per unit of item 7 in the showroom SR and in the backroom BR kemin pmax Lower and upper bound on the number of facings of item i I Length of item i relevant if item i is displayed lengthwise m Gross margin per unit of item i rj Sales price per unit of item i S Available showroom shelf space onedimensional front row vePR vcpR Variable costs per replenishment of one unit of item i for direct replenishment DIR and replenishment from backroom BR Wi Width of item i relevant if item 7 is displayed crosswise B Spaceelasticity factor of item i Decision variables kj Integer variable number of facings assigned to item i on the showroom shelf i N Dj Visible width of a facing of item i depending on its display orienation ie lengthwise b1 or crosswise bw i N Si Integer variable order frequency for item i ie the number of times per period an item is ordered and directly replenished i N Auxiliary variables Sib Number of units of item i per facing ie stock per facing of item i depending on b Xib Integer variable total shelf quantity for each item i on the showroom shelf with Xip Ki Bi Yi Integer variable backroom inventory for each item i with y max 7 Xin 0 maxII1k bf S pilki bisfi 1 icN To obtain the item profit p we deduct the total cost of direct replenishment cP and the total cost of backroom replenishment CPR cf Eq 2 from the total gross margin of an item The gross margin of an item is calculated as the product of its Q Springer 136 Business Research 2017 10123156 total demand D and its unit margin m The item unit margin m corresponds to the difference between its sales price r and its purchase cost c Pilki bifi Diki bi mi CP ki binfi CPS ki bin fi 2 The total period demand Dkb of an item i is a composite function of the minimum demand d and the facing and display orientationdependent demand The minimum demand rate d represents the retailers forecast for an item that is independent of the facing and visible facing width cf Hansen and Heinsbroek 1979 Hiibner and Kuhn 2012 BianchiAguiar et al 2015a The forecast may be based on historical sales but may also incorporate further demand effects such as shelf location in the store or other marketing effects The higher the visibility of an item the higher is its demand The visibility increases with the number of facings k Furthermore it increases when the item dimension visible to the customer increases which is either the item width or the item length In accordance with prior research cf eg Hansen and Heinsbroek 1979 Irion et al 2012 the facing and display orientationdependent demand rate is a polynomial function of the number of fac ings k allocated to an item the visible facing width b and the spaceelasticity f with 0 f 1 Most existing models assume that items have a fixed item width as they can be displayed in one display orientation only We use the demand model assumed by Irion et al 2012 to factor in the visible facing width b Eq 3 sum marizes the demand calculation applied Dik bi dj ki bi 3 Total direct replenishment cost CP comprises three parts as shown in Eq 4 fixed replenishment costs for each replenishment of an item FCP variable replenishment costs for each unit replenished of an item cvcP and showroom inventory holding costs 8 per unit Assuming continuous demand the average showroom inventory used for calculating the related inventory costs is calculated as Xi 2 Xib cP ki fi Piles k n FCPIR Si vepR Xib fi nse o 4 The total backroom replenishment costs CP consist of the same three elements cf Eq 5 where we also need to consider the number of shelf refills from the backroom during a subperiod for the fixed replenishment costs the backroom inventory y instead of the showroom shelf quantity and the backrooms inventory holdings costs per unit h ce 4 fi bi kig FCPR7 4 VOPR y ff mPR 21 5 1 Xib Ki Sib 1 Xib l l 2 Solution approach Equations 35 contain several nonlinear components relating to the decision variables eg spaceelastic demand with k b4 or the division of two variables in the backroom replenishment costs Therefore it is a non linear model To handle the nonlinearity we precalculate the associated profit Q Springer Business Research 2017 10123156 137 pikibif and space requirements in the show and backroom sR ki bi fi sPkbf for each item i and each possible combination of facings visible facing width and order frequencies The precalculated data is then used in a MIP to ultimately choose the optimal combination and thus globally optimize profits The usage of a MIP comes with various computational conveniences We handle the nonlinear model terms outside the optimization model using precalculation Suboptimal heuristics are therefore not required The MIP can be solved optimally in a timeefficient manner see runtime tests in Sect 521 Finally the MIP offers the possibility of adding further model constraints in case these are required from a practical perspective An example of this is the restriction that a certain item must be positioned with a predefined display orientation This precalculation approach can be applied because in practice all decision variables have upper limits First a retailer will only assign a certain number of facings to an item with k KK typi cally not more than 2025 facings Second only two values are possible for the visible width of a facing of item i ie b 1 for lengthwise and b w for crosswise display orientation In the MIP model we decode the visible width of a facing of item i by 0 where o 1 if bj 1 and by o 2 if b w Third the frequency of direct replenishments ie orders cannot exceed the maximum number of warehouse deliveries with f F eg not more than six times per week This allows us to precalculate the profit denoted as zjxo in the MIP for every item i and every possible combination of the three decision variables for the predefined ranges k K 0 O and fj F i N This means that zjxo is the profit for item i if it gets k facings is given the display orientation o and is ordered f times a period Similarly we precalculate how much showroom backroom space item i consumes for every combination of k o and f The respective space consumption is denoted as ss for the showroom and s8 for the backroom Using the precalculated profits as data input the MIP model then selects the binary variable to indicate how many facings item i should be given how it is displayed and how often it should be ordered The objective function and the constraints for the resulting model CSRPBS can be formulated as follows MaxII7 S S S S Tikof Vikof 6 iN kek oO fEF Subject to Siust Vikor S 7 iN kek 0oO fEF Sikor Vikot B 8 iN kek oO fEF So SES e vivor 1 wie N 9 kek 0O fCF Vikof 0 1 VIEN KE Ko O fe 10 Equation 6 is the objective function and is the summation of all itemspecific profits Equations 7 and 8 ensure that showroom shelf S and backroom space Q Springer 138 Business Research 2017 10123156 B restrictions are met Finally Eq 9 ensures that each item i gets exactly one combination of facings item display orientations and order frequencies Equa tion 10 declares as a binary variable Note that the available showroom space S is the onedimensional shelf length front row available for the placement of facings eg measured in centimeters or meters In contrast the size of the back room B is measured in space units eg in m because in the backroom items can theoretically be stored behind each other whereas items need to be placed next to each other on showroom shelves for reasons of visibility Analoguously to that sok is a onedimensional length and sB8 is a twodimensional area Model complexity The MIP model developed belongs to the class of knapsack problems which are known to be NPhard cf Kellerer et al 2004 Pisinger 2005 In our case the model complexity is driven not only by the number of combinations for allocating N items to a shelf of size S but also by the fact that each item can be ordered up to F times and get one of two different display orientations The resulting model complexity can therefore be calculated by Eq 11 vinsF 4 PY 28 11 N1 The binomial coefficient calculates the number of possible combinations for allo cating N items to a showroom shelf of size S The second term accounts for the fact that each item can be ordered up to F times and finally the third term accounts for the display orientation For example for N 50 a showroom shelf of S 100 up to daily deliveries F 6 and one of two possible display orientations the number of possible configurations is 459 108 Our modeling approach based on precal culated profits helps to significantly reduce this complexity because instead of the Y combinations we only need to precalculate N K F 2 profits assuming K is the number of elements in K with k 1K similarly for F These are then pro vided as input into the MIP choosing the optimal combination For the example above the number of required precalculations corresponds to NKF2502562 15000 if we assume an upper limit for the number of facings per item of K 25 Note that our MIP is always solved to optimality within these assumed limits 5 Numerical results The numerical results are presented in this section Our model is first applied to a case study in Sect 51 To generalize the results Sect 52 uses randomly generated data to test the runtime performance and investigate the impact of backroom space and replenishment processes on objective values and solutions Section 53 summarizes the findings from the numerical results All numerical tests were conducted on a Windows 7 32bit Intel Core i52520 with 25 GHz and 4 GB memory The tests were implemented in VBnet Visual Studio 2013 and GAMS 241 to use the CPLEX solver for the MIP Q Springer Business Research 2017 10123156 139 51 Application to real data case study This section applies our model to the canned foods assortment of a German grocery retailer Data applied We consider the product category for which we also obtained the process descriptions and cost structures from our time and motion study This category encompasses 70 different items In this category the retailer only puts the items onto the shelf in case packs ie cartons and not in single customer units ie cans We treat one carton as one facing and each carton contains a quantity of six or twelve units The display orientationdependent stock per facing gip is derived here as follows As the shelf depth does not allow case packs to be put behind each other regardless of a crosswise or lengthwise positioning the stock per facing is given by the quantity per carton Please note that here the spaceelasticity effect is reflected in Eq 3 by the bvalue for the crosswiselengthwise display orientation of the facing ie demand increases when the larger of the two carton dimensions is displayed We consider a sales period of one week The average minimum demand d of the items i N is between 1 and 110 units per week For determining dj we first measured the average total demand D across 10 months with the current number of facings k the current visible width of a facing b an assumed space elasticity of f and then recalculated the average minimum demand d using Eq 3 We assume that space elasticity is equal for all items within the canned foods category The items are sold for a price of 050 r 249 For reasons of confidentiality the corresponding values for unit replenishment and inventory cost cannot be provided Showroom shelf space is limited to S 5920 cm and backroom utilization is currently very low which is why we can consider backroom space capacity to be unlimited At the time of data collection the retailer assigned between and 21 facings to the items which all have a lengthwise display orientation The number of facings was determined according to a salesproportional allocation SPA rule which assigns shelf space to items based on their share of category sales but ignores replenishment cost cf Hiibner and Kuhn 2012 The item length J is 148 cm J1 400 cm and item width w is 224cm w 417 cm All items are currently ordered twice a week f 2 Approaches analyzed To show the extent to which the retailer benefits from the model we investigate the following modeling approaches 1 Status quowhich represents the number of facings and display orientation as observed in the current shelfspace assignment and a given order frequency of f 2 for all items To ensure comparability we evaluate the observed values of all decision variables using the objective function cf Eq 6 of the CSRPBSmodel To compute margins and replenishment costs a posteriori we apply the respective a posteriori model denoted as CSRP Furthermore we ensure that all constraints 710 are fulfilled From here we derive the profit potential in steps Approaches 2 and 3 are partial optimizations where either order frequencies approach 2 model CSRPBSf or facings and display orientations approach 3 model CSRPBSkb are optimized In 2 we keep k and b as per current and in 3 we do so for f All Q Springer 140 Business Research 2017 10123156 profit components are evaluated by applying the respective a posteriori calculation for the nonoptimized variables Finally in approach 4 Integrated optimization the CSRPBSk bf model optimizes k b and f simultaneously and therefore shows the full potential This model corresponds to the CSRPBS introduced in Sect 4 To highlight the differences to approaches 13 we add here the superscript for the decision variables Table 2 summarizes the respective assumptions for each modeling approach Results Figure 3 illustrates the advantage of the fully integrated model CSRPBSkbf over the partial optimization models and the status quo for different values of the space elasticity The analysis reveals several insights First independent of space elasticity a significant opportunity exists for the retailer to improve the status quo If we assume a space elasticity of 15 cf Eisend 2014 who identified an average of 17 the full potential of integrated optimization compared to the status quo approach 4 vs 1 amounts to approximately 29 Second the retailer also profits from partial optimization ie only optimizing order frequencies 2 vs 1 or facings and display orientations 3 vs 1 The higher potential clearly lies in the optimization of facings and display orientations However the results from 4 show that an integrated perspective is still better than partial optimization Finally we see that the advantage of the integrated model CSRPBSk bf over pure shelfspace optimization CSRPBSk b diminishes with increasing space elasticity This is due to the fact that the importance of assigning the right amount of space to highmargin items increases as space elasticity and the connected demand increase which is achieved by both models Simultaneously the magnitude of replenishment cost decreases and so does the advantage of the fully integrated over the shelfspace optimization model To better understand the latter Fig 4 shows the absolute values for total profits QCicn Pi total gross margins 0Dm and total replenishment cost Sic CP and 57 C for the four approaches The profit increase is mainly driven by the increase in gross margins while replenishment costs are much lower in magnitude With increasing space elasticity the impact of the gross margin effect rises even higher Note that in CSRP and CSRPBSf k and b are determined regardless of the space elasticity and are therefore identical across all fvalues However the gross margin increases also for these two approaches because the Table 2 Different approaches for case study Approach 1 Status quo Partial optimization Full optimization 2 Order 3 FacingsDispl 4 Integrated frequency orient Model CSRP CSRPBSf CSRPBSk b CSRPBSk bf applied Optimization None Order frequency f Facings k Facings k Variables all as per Visible facing width b Visible facing width current b Order frequency f Q Springer demand realized increases with an increase in the assumed space elasticity see Eq 2 Table 3 shows the changes in solution structure ie facings visible facing width and order frequencies The comparison of the partial and full optimization models 25 20 35 45 10 0 30 5 5 50 30 20 40 15 25 10 0 15 Space elasticity Percent Profit advantage Percent CSRPf BS advantage over CSRP CSRPkb BS advantage over CSRP CSRPkbf BS advantage over CSRP Fig 3 Relative profit advantage of integrated and partial optimization models over status quo 20 2250 1750 1250 1000 500 250 30 1500 2000 750 25 10 5 15 0 Total gross margin EUR Space elasticity Percent 2000 1250 1000 20 5 0 0 30 15 25 10 1750 1500 750 500 250 Total profit EUR Space elasticity Percent 30 15 20 25 5 10 0 5 10 0 30 25 20 15 Space elasticity Percent Total backroom replenishment cost EUR 40 120 140 100 80 60 5 20 30 15 25 20 10 0 0 Total direct replenishment cost EUR Space elasticity Percent CSRPkb BS CSRPf BS CSRPkbf BS CSRP Fig 4 Total profits total gross margins and total replenishment cost for each modeling approach Business Research 2017 10123156 141 123 142 Business Research 2017 10123156 Table 3 Changes in solution structure Models Approaches Change of k b f CSRP versus CSRPBSf 1 versus 2 957 CSRP versus CSRPBSk 5 1 versus 3 886 614 CSRP versus CSRPBSk bf 1 versus 4 900 586 886 CSRPBSf versus CSRPBSk b 2 versus 3 886 614 957 CSRPBSf versus CSRPBSk bf 2 versus 4 900 571 214 CSRPBSk b versus CSRPBSk bf 3 versus 4 157 29 886 Scenario with B 15 with the status quo shows significant differences in all three optimization variables eg 90 of all items get a different number of facings in the full optimization compared to the status quo Furthermore the comparison of the full optimization to the partial optimizations shows that there is still a significant share of items with differences in either order frequencies 214 with different order frequencies 2 vs 4 or facings and display orientations 157 with different facings 29 with different display orientation 3 vs 4 52 Generalization using randomly generated data After having shown how the retailer can use our model to increase profits we can now generalize these insights by conducting more extensive analyses with randomly generated data Section 521 describes the data used and the test setting Section 522 then provides runtime tests Section 523 investigates the impact of backroom availability on profits and solution structures and Sect 524 analyzes the profit advantages retailers can achieve when thoroughly accounting for replenish ment processes We compare our model to a salesproportional allocation rule in Sect 525 and develop an extension to account for assortment decisions in Sect 526 521 Data applied models and test bed The data generation process is based on the data obtained from the case study If not stated otherwise we use a uniform distribution to randomly generate the item specific parameters which are within the following intervals d 5070 r 10 20 ci 75 r 80 rij Sin 35 VCPR 002 006 VCBR 006 010 FCP 008012 FCP 016 024 ASR 25 735 rj and APR 15 c20 ci In other words replenishment from the backroom is twice as expensive as direct replenishment on average and inventory holding costs are lower in the showroom than in the backroom Space elasticity of the single items f is assumed to vary between 0 and 35 To focus on the core effects we assume w 1 if not stated otherwise Furthermore we set Q Springer Business Research 2017 10123156 143 Table 4 Runtime tests for different problem sizes in seconds Number of items NV 5 50 100 200 400 2000 Showroom space S 20 200 400 800 1000 60000 Backroom space B 5 100 200 400 500 30000 Runtime 086 148 191 321 662 4593 Average of 100 examples K 15 and F 6 for the precalculations were Kf corresponds to the number of elements in K with kj 1K F with fj 1F We use the model CSRPBS to optimize facings visible facing width and order frequencies and solve it using the MIP introduced in Sect 4 cf Eqs 610 The CSRP model is used to evaluate the impact of ignoring specific effects eg replenishment cost a posteriori The measured effects are averages across 100 randomly generated instances of N items 522 Runtime test Table 4 shows the average runtime for different problem instances and shows that our model can efficiently generate optimal results even for largescale problem instances While we assumed w 1 for all instances up to N 400 items for the instance with N 2000 S 60000 and B 30000 we randomly generated item dimensions with w 2 10 and 5 15 to test runtime performance under more complex assumptions The model still generates optimal results in a minimum amount of time in less than a minute runtime on average The CSRPBS is a knapsack problem It becomes a hard knapsack problem when item weight in our case the shelf space occupied by the item and the item contribution in our case the unit margin are strongly correlated cf Pisinger 2005 To test the performance of our approach on hard knapsack problems we run a further test on instances with N 2000 S 60000 and B 30000 where unit margins and space occupied correlate with R 09 The average runtime for these 100 instances is 7806 s with a minimum of 6234 s and a maximum of 9347 s which shows that our approach can also handle hard knapsack problems efficiently 523 Impact of backroom space on profits and solution structures To investigate the impact of backroom availability on profits and solutions structures we first present a 2item example below and then extend the analysis to a more comprehensive set of randomly generated data 2item example We consider two items 1 and 2 with identical demand and cost parameters Both items have a length of J and a width of w 2 The stock per facing gi depends on the visible facing width and is 1 in case of a lengthwise orientation and 2 in case of a crosswise orientation The only difference between the two items is that item 1 has a high space elasticity and item 2 has none B 30 B 0 Below we analyze how facings display orientations order Q Springer 144 Business Research 2017 10123156 frequencies and backroom quantities change if showroom and backroom space increase Figure 5 shows that with a showroom space of S 6 and no backroom space B 0 the highly spaceelastic item receives kj 5 facings and is replenished Ji 2 times a week Item 2 receives only the minimum of k facing because its space elasticity is zero Item 2 needs to be ordered more often at 6 to still satisfy the demand for it Both items are displayed lengthwise and therefore have a stock per facing of only one unit If backroom space now becomes available while showroom space remains the same it is beneficial to decrease order frequencies for item 2 from f 6 B 1 to fp 2 B4 and instead replenish it indirectly from the backroom where inventory holding cost is lower yz 2 If showroom space Sis doubled from six to twelve and no backroom exists item 1 now receives only k 3 facings which are displayed crosswise Item 2 is also positioned crosswise and receives k 1 facings Both items receive a stock of two units per facing due to the crosswise orientations Two interesting observations are to be made First a showroom shelf space of only eight out of a total of twelve is occupied The reason why the shelf space is not fully occupied is the following Further space could theoretically be assigned to the highly spaceelastic item 1 but since this would induce additional spaceelastic demand that cannot fully be supplied by the showroom inventory backroom quantities would need to be kept for item 1 to satisfy the demand for it compare scenario with S 12 and B 4 where y 2 Since no backroom exists item remains at three facings Second the crosswise 1 High space elasticity item No space elasticity item Space Showroom Backroom Item 1 Item 2 SS B Optimal shelf configuration showroom shelf Profit usage usage f y f y 6 0 G tthe 1000 6 0 2 0 6 oO 23 afi tk Kt ir 1052 6 2 2 0 3 41 48 Ghia 1067 6 4 2 0 2 2 2 of 1022 8 0 2 0 3 0 273 1056 7 2 2 0 3 1 4 45 A i A A i SS 1166 12 4 1 2 3 0 er F F 1200 1 6 1 2 3 4 B11 1215 1 8 1 2 2 2 A 12 1 1 1 1 1 SS 1220 12 12 1 2 1 4 Fig 5 Analysis of availability of showroom and backroom space on profit and solution structure 2item example changes of decision variables between subsequent scenarios are in bold Q Springer display orientation of item 2 allows for a stock of 2 units to be placed on the shelf This allows for a decrease in the order frequency from f2 ¼ 6 at S ¼ 6 B ¼ 0 to f2 ¼ 3 at S ¼ 12 B ¼ 0 Because b2 ¼ 0 and due to showroom inventory holding cost it is also not beneficial to use the remaining shelf space for further units of item 2 Obviously a backroom space of B ¼ 2 or 3 is not yet sufficient to further decrease f2 but it can be used to reduce the inventory held in the showroom The display orientation of item 2 therefore changes back to lengthwise which results in a stock of only one Inventory moves to the backroom where it is cheaper to keep stock y2 ¼ 1 If backroom space increases to B ¼ 4 or 5 priority is immediately given to item 1 since now the additional spaceelastic demand caused by an increased number of facings k1 ¼ 5 can be served from the backroom which is completely occupied with item 1 y1 ¼ 2 This requires putting item 2 back in a crosswise orientation since this allows a shelf quantity of two units that could not be kept in the backroom fully occupied by item 1 y2 ¼ 0 At B ¼ 6 or 7 item 2s display orientation can again be changed to lengthwise which shifts inventory holding cost from the expensive showroom to the less expensive backroom y2 ¼ 1 At B ¼ 8 or 91011 the additional backroom space is used to lower f2 to f2 ¼ 2 again similarly to the three scenarios with S ¼ 6 Finally at B ¼ 12 f2 can even decrease to f2 ¼ 1 Obviously backroom space is not enough to keep the lengthwise orientation Item 2 is placed crosswise and a stock of g2 ¼ 2 units must be kept on the showroom shelf The example shows that items with a high space elasticity should clearly be given priority in shelfspace assignment Furthermore the tradeoffs between availability of showroom and backroom space facings order frequencies and display orientations are illustrated Even with this stylized example it generally becomes evident that the availability of backroom space impacts optimal facings display orientations and order frequencies We can conclude if retailers have the opportunity to use backrooms for intermediate storage they should leverage them because backroom space allows for more flexibility in planning showroom shelf space and instore replenishment processes Extended analysis To underline the impact of backroom space B on profit and solution structures and to generalize the findings above we analyzed additional randomly generated data sets Each set contains N ¼ 50 items To focus on the main effects we ignore the display orientation and set li ¼ wi ¼ 1 We set F ¼ 6 and K ¼ 15 and assume a showroom shelf space of S ¼ 200 and in the basic scenario a backroom space of B ¼ 100 which is varied below For each analysis we report the average of 100 randomly generated data sets Figure 6 shows the impact of changing backroom size on financial performance ie total profits total gross margins total direct and backroom replenishment cost As seen above in the 2item example an increase in backroom space results in increased total profits This is due to an increase in demand as well as lower total direct replenishment costs ie direct replenishment and showroom inventory holding cost By providing more flexibility in the form of additional backroom space B more space for beneficial items can be reserved on the showroom shelf space to generate more sales resulting in higher total gross margins and the Business Research 2017 10123156 145 123 backroom can be more extensively used to refill showroom shelves if this is cost beneficial On the other hand this induces an increase in total backroom replenishment costs ie backroom replenishment and backroom inventory holding cost which is shown in the righthand graph Figure 7 provides the changes in the solution structure Up to 80 of the items are given a different number of facings k if backroom space B decreases An increase in backroom space B analogously induces changes in shelfspace assignment but the upper limit of K ¼ 15 limits the magnitude of this effect Up to 20 of the items are given a different number of facings k if backroom space B is doubled In terms of order frequencies f similar observations apply Up to 54 of the items have a change in order frequencies f if backroom space B changes The right hand graph shows that the average number of orders per week decreases as more backroom space B becomes available A larger backroom space B allows for decreased order frequencies f because backroom space can be used to store items and then replenish shelves from there Note that as soon as backroom space B becomes available order frequencies f increase slightly at first before gradually decreasing This is because less profitable and less spaceelastic items can be moved 3 2 1 0 1 100 50 0 50 100 Change in total profit Percent Change in backroom space Percent 4 3 2 1 0 1 100 50 0 50 100 Change in backroom space Percent Change in total gross margin Percent 15 10 05 0 05 10 15 20 25 30 100 50 0 50 100 Change in total direct replenishment cost Percent Change in backroom space Percent 100 80 60 40 20 0 20 100 50 0 50 100 Change in total backroom replenishment cost Percent Change in backroom space Percent Fig 6 Impact of backroom availability on financial performance 146 Business Research 2017 10123156 123 Business Research 2017 10123156 147 Share of items with changes in Share of items with changes in showroom facings order frequencies number of ordersweek 80 60 57 707 50 56 b 60 4 55 50 54 Sq 40 sof 53 30 20 x 20 t7r 7 XN awn 52 ee 10 vO 0 54 Nv 0 0 50 100 50 0 50 100 100 50 0 50 100 100 50 0 50 100 Change in backroom space Change in backroom space Change in backroom space Note Figures show average of 100 randomly generated data sets Fig 7 Impact of backroom availability on solution structure Table 5 Impact of neglecting replenishment and inventory costs on profits and solution structure for varying backroom sizes Total profit increase 138215 Share of items with changes in facings k 681691 Share of items with changes in visible facing width b 429452 Share of items with changes in order frequency f 773871 Backroom usage A posteriori CSRP 798853 Optimal CSRPBS 450633 S 800 B 0 100 100 randomly generated data sets evaluated for each scenario to the backroom to free up additional space in the showroom This space is used to allocate more facings to highprofit and high spaceelasticity items which generates additional demand Because showroom shelf S is scarce this demand increase enforces an increase in order frequencies These effects decrease in magnitude the more backroom space B becomes available because it can be used to fulfill the additional demand without further increases in order frequencies 524 Impact of replenishment and inventory costs on profit and solution structure To analyze the impact of replenishment and inventory costs we conducted an a posteriori analysis We considered a retailer who neglects the relevant cost elements and just accounts for item demand and margins ie VCPIR VCPR FCPIR FCPR h3 BR 0 Based on these assumptions we ran the CSRPmodel and a posteriori evaluated the solution structure k b and f assuming the actual costs We set S 800 and vary the available backroom space from B 0 100 Item dimensions vary as follows w 210 and 1 515 For these different backroom sizes Table 5 presents the advantage a retailer has when correctly accounting for relevant cost elements and considering facings display orientation and order frequency optimization from an integrated Q Springer 148 Business Research 2017 10123156 Table 6 Profit advantage of CSRPBS over CSRP and changes in solution structure Order frequency times per week 1 2 3 4 5 6 Profit advantage of CSRPBS over CSRP 533 533 691 858 1030 1207 Share of items wfacing changes 0 9800 Share of items worder frequency changes 5800 6200 9200 9400 9800 9800 S 1000 B 100 100 randomly generated data sets evaluated for each scenario perspective Depending on the size of the available backroom the profit is up to around 22 higher and solution structures change significantly 70 of the items are given a different number of facings k 7787 a different order frequency f and 4345 a different visible facing width b Moreover we see that backroom utilization is much higher if not all relevant costs are properly accounted for Backroom utilization is 8085 versus 4563 if replenishment and inventory holding costs are properly incorporated 525 Comparison to salesproportional allocation rule This section compares the results of the CSRPBS to those generated with a sales proportional allocation rule SPA For the SPA we again use an a posteriori evaluation This benchmark is denoted as CSRP like in Sect 51 The SPA rule does not define the display orientation To allow a good comparison we assume that 1 w and no display orientation is required We use the same N 50 item data sets as before In the SPA the overall showroom space S is allocated to the items in proportion to their sales share and order frequencies are fixed for all items We analyze six scenarios with f 16 Table 6 shows the profit advantage of the optimization model that optimizes for f and k the magnitude increases as order frequencies rise Furthermore we see the significant differences in solution structures Note that the profit potential here is lower than in the case study Sect 51 because the unit margins we assumed here are lower than the unit margins in the case 526 Impact of assortment decisions In this section we show how assortment decisions can be incorporated into the CSRPBS The resulting model is denoted as CASRPBS whereas the additional A represents the assortment decision So far we have assumed that the assortment is determined in a previous planning step and that the retailer must assign all items of set N to the shelf ie we did not allow zero facings k 0 Including the assortment decision allows more flexibility for two reasons 1 Solutions for situations with SWN can now be generated and items delisted 2 Even if S JN it might be beneficial to delist specific items and use the shelf space for more beneficial items eg items with a higher margin andor space elasticity The inclusion of assortment decisions requires an adaptation of the demand function cf Eq 3 to account for additional demand arousing from outofassortment OOA Q Springer Business Research 2017 10123156 149 situations Smith and Agrawal 2000 K6k and Fisher 2007 OOA substitution expresses the customers willingness to buy an alternative item if the preferred item is not listed By taking substitution into account demand and the profit for an item i also depend on the availability of all other items jj 4 i This extension increases the combinatorial complexity of our model since the crossproduct interdependen cies result in a quadratic problem and the isolated precalculation of the itemspecific profits as input to the MIP see Sect 4 does not capture these product interlinks anymore The model complexity Y of the CSRPBS with YNS F y 1 FN 2N NS1 N 4N increases to YNSF s F 2 for the CASRPBS when includ ing the assortment decision Due to the increased complexity and product interlinks we need a heuristic to iteratively determine the total demand D for an item considering the availability of other items We therefore first develop the extended solution approach for the CASRPBS and then complete numerical examples to show its efficiency and the impact of assortment decisions Extension of model and solution approach To incorporate assortment decisions we differentiate the total set of items N into the set of listed items N and the set of delisted items N with NtN7 CN NUN7 N and N QN7 Equa tion 3 is extended in the following way to account for OOA substitution Djk bi dj ki bi d k 12 Total demand D for item i now not only consists of its own spaceelastic demand dj ki biP but also of the OOA demand caused by the delisting of other items d k We calculate this new demand component as follows dP k Sodje yy Vie N 13 jen Equation 13 shows that if an item j is delisted j N and therefore kj 0 a certain share OF of its minimum demand dj is substituted by item i Vii is the substitution rate between items j and i We allow one round of substitution as in Smith and Agrawal 2000 K6k and Fisher 2007 or Hiibner et al 2016 ie sales are lost if the substitute is not available either In the following we use the aggregated substitution rate 6 for an item which corresponds to the likelihood that a delisted item gets substituted at all and then assume that this aggregated rate is split equally among the remaining items Le yj Vi Zi cf Kok et al 2015 To account for assortment decisions and to thoroughly compute the asso ciated demand function we apply the following heuristic e Step 1 Ignore OOA substitution by setting d4k 0 in Eq 12 and solve the MIP as introduced in Eqs 6 to 10 including k 0 in K Vie N e Step 2 Update the demand for all items using Eqs 12 and 13 with the values for k and b obtained in Step 1 Q Springer 150 Business Research 2017 10123156 Solution quality Percent heuristic profitFE profit e N58S3 4 N78S5 N10S7 1000 joonmegegs reeerengyrereererrere oy go5 44 4 o aa arn ome ae 990 4 au o e 4 95a o o o o 980 975 s s 970 a 965 4 960 e 955 e 950 945 e 940 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Data set Fig 8 Solution quality of heuristic in comparison to FE Table 7 Runtime performance of heuristic Number of items NV 5 7 10 50 100 500 2000 Showroom space S 3 5 7 45 80 400 1500 Average runtime heuristic s 466 541 597 754 945 4319 12566 Average runtime FE s 001 014 1001 14400 B 100 6 03 100 data sets evaluated for each scenario e Step 3 Solve the MIP with the updated demand from Step 2 again including k 0 afterwards update demand using Eqs 12 and 13 e Step 4 Repeat Step 3 until there is no more change in solutions Heuristic performance We investigate the performance of the heuristic in terms of solution quality and runtime To investigate the solution quality we compare the profits generated by our heuristic to the profits generated by an optimal full enumeration FE The optimal results via FE can only be obtained for small problem instances due to the combinatorial complexity of the problem Figure 8 shows that with an average solution quality of 995 the heuristic yields near optimal profits for three different problem instances with N S and data generated as described in Sect 521 Regarding runtime Table 7 shows that the heuristic is efficient While the FE is faster for small problem instances it is not capable of generating results for instances with N 50 within four hours while the heuristic still solves instances with N 2000 and 1500 in about two minutes Note that in all numerical examples the number of delisted items always corresponds to the difference between N and S which shows that it is always more profitable to list an item than delisting it and using the respective space for additional OOA demand of substitutes 2item example To show under which circumstances the delisting of items is beneficial and to investigate how the possibility of delisting items impacts facing Q Springer Business Research 2017 10123156 151 Table 8 Assortment changes depending on margin space elasticity and cost Scenario Description Facings Order frequency ky ky fi f 1 Both items equal 1 1 2 2 2 B 35 By 0 1 1 2 2 3 Margin 50 Margin 5 2 0 2 0 4a Cost 2 Cost 1 1 2 2 4b Cost 5 Cost 1 1 2 2 4c Cost 10 Cost 1 1 2 2 4d Cost 100 Cost 2 0 2 0 S2B10 and order decisions we again use the 2item example Items 1 and 2 compete for a showroom space of S 2 ie the retailer can split this space or allocate it exclusively to either one of the items In the base scenario 1 both items are completely identical and we choose item parameters as in the 2item example in Sect 523 To focus on the core effects we again assume J w 1 i 12 From here we investigate the extended problem if item 1 is highly spaceelastic and item 2 is not Scenario 2 if item 1 has a high and item 2 a low margin 7 cci Scenario 3 and if replenishment cost factors Cost ie VCP VCBR FCP and FCB of item 2 are 2100 times as high as for item 1 Scenarios 4a4d Table 8 shows that the showroom space is allocated equally k kz 1 if both items are identical Sc 1 This only changes across the seven scenarios if either margins are extremely different 3 where then only the highmargin item is listed k 2 and k 0 or if replenishment costs are very unrealistically different Sc 4d where item 2 with the 100 times higher replenishment costs is again delisted Significant differences in space elasticities do not result in the delisting of items Sc 2 In summary if space is sufficient to potentially list all items assortment decisions are only likely to be impacted by the joint shelfspace and reorder planning if items significantly differ in their margin The results from this 2item example are intuitive and the findings do apply to larger instances in the same way 53 Summary of numerical results The application of our model to real and randomly generated data sets reveal several important insights First we show in a case study that a retailer can increase profits by around 29 compared to the status quo if our model is applied in full Simultaneously optimizing facings display orientations and order frequencies is superior to the strictly shelfspacebased optimization of facings and display orientation without optimizing order frequencies This underlines the importance of an integrated perspective on shelfspace optimization and instore replenishment Q Springer processes Second our model can be efficiently solved by a MIP model The results consistently are optimal and even generated within a minimum amount of time for largescale instances Third if backroom space is available retailers can make use of it by assigning more showroom space to highly profitable items and moving less profitable items to the backroom from where they are replenished more frequently In other words the available backroom space gives more degrees of freedom to the retailer in terms of shelfspace optimization Fourth our model can be extended to account for assortment decisions which is especially relevant for situations where the available shelf space is not sufficient to list all possible items of a category Finally the profit advantage a retailer has when taking an integrated approach to facing display orientation and order frequency optimization can amount up to around 22 compared to a case where facings and display orientation are optimized without considering the related costs for instore replenishment This is because significantly different solution structures result in terms of facings display orientation and order frequencies if replenishment processes are neglected 6 Conclusion and outlook In this paper we presented a capacitated shelfspace optimization model that contributes to the existing literature by accounting for instore replenishment and the availability of backroom space The model maximizes retail profits while considering costs for direct and backroom replenishment cost for inventory limited showroom and backroom space as well as spaceelastic demand Retailers are provided with additional flexibility from the optimized display orientations of items We have quantified the relevant instore processes cost by means of a time and motion study for a German retailer Our process descriptions serve to further define in greater detail the instore processes and cost types identified in the existing literature To solve the resulting nonlinear problem we developed a mixedinteger model Even for largescale problems our model yields optimal results efficiently within a feasible amount of time We applied our model to the retailers canned foods category and showed how profits can be increased significantly by applying our model After the results were presented to the retailer he decided to change his current approach to shelfspace and instore replenishment planning by applying our model Furthermore we have shown that an integrated perspective on shelfspace and replenishment optimization is crucial for retailers because backroom space and replenishment cost have a significant impact on retail profits and shelfspace planning An integrated perspective for shelf planning is specifically important since in practice shelfspace decisions are made by a central sales planning unit which oftentimes ignores the consequences of shelf planning on instore operations Our model will help retailers to develop this integrated perspective Limitations and future areas of research The limitations of our model point to a variety of future areas of research We follow the general literature on shelfspace management and assume a deterministic and stationary demand for the tactical problem Because of this demand is always satisfied Hence one area is to further generalize the demand modeling Some authors argue that demand volatility can be 152 Business Research 2017 10123156 123 handled with exogenously determined safety stocks The resulting shelf space for the safety stock needs to be deducted from the total shelf space and only the remaining space can be distributed However our modeling approach has the advantage of being flexible enough to determine safety stocks endogenously As safety stocks protect against uncertainty in demand demand volatility and supply lead time volatility the impact of both decision variables ie the impact of the number of facings on the demand and the impact of the order frequency on supply need to be taken into account Hence for all precalculated combinations of the decisions variables one can calculate the safety stocks accordingly within the model Furthermore our model and solution approach is a good starting point to account for further demand effects Focusing on demand volatility would imply the development of a stochastic model for our decision problem with replenishment costs to account for demand variations cf eg Hubner and Schaal 2016a In such cases outofstock substitutions resulting from potentially insufficient shelf and backroom quantities for specific items would need to be taken into consideration as well cf eg Kok and Fisher 2007 Hubner et al 2016 A stochastic model would need to balance the tradeoffs between understock and overstock situations which is specifically relevant in the case of perishable items These additional costs can be included in the precalculations Apart from stochastic demand further demand effects such as item positioning cf eg Lim et al 2004 BianchiAguiar et al 2015b or crossspace elasticities cf eg Corstjens and Doyle 1981 would be worth considering when the model is applied to certain categories with these demand effects Our model concentrates on the cost associated with direct and indirect replenishment of shelves Future models could incorporate further decisions and associated cost such as upstream supply chain decisions and the cost of deliveries from warehouses to stores cf Sternbeck and Kuhn 2014 Holzapfel et al 2016 Moreover retail managers typically try to keep shelves as filled as possible since empty space is generally believed to have a negative impact on sales cf Baron et al 2011 This may result in differentiated refill costs We have shown that our solution approach is capable of solving a problem with up to 2000 items within less than a minute Although shelfspace and reordering decisions are typically made for each category separately our model could be extended for storewide shelfspace optimization across all categories where common order patterns for different categories would also be considered Finally the investigation of multistore environments can be considered A corresponding model would support retailers in deciding whether planograms should be more standardized or adjusted to storespecific needs Such a model would need to balance the tradeoff between storespecific demand fulfillment and the efficiency of upstream logistics processes Our optimization model takes the perspective of a retailer who wants to optimize category profit In contrast a manufacturer follows the objective of brand profit optimization which raises the topic of category captainship cf eg Kurtulus and Toktay 2011 Martınezde Albeniz and Roels 2011 A comprehensive study will need to address all the relevant subjects of negotiation between manufacturers and retailers such as assortment prices and shelf space Business Research 2017 10123156 153 123 The model and solution approach proposed within this paper will be a good starting point to address the open areas of research mentioned above Open Access This article is distributed under the terms of the Creative Commons Attribution 40 International License httpcreativecommonsorglicensesby40 which permits unrestricted use dis tribution and reproduction in any medium 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