Effect of replenishment and backroom on retail shelf-space planning

Hübner, Alexander; Schaal, Kai Article Effect of replenishment and backroom on retail shelfspace planning Business Research Provided in Cooperation with: VHB - Verband der Hochschullehrer für Betriebswirtschaft, German Academic Association of Business Research Suggested Citation: Hübner, Alexander; Schaal, Kai (2017) : Effect of replenishment and backroom on retail shelf-space planning, Business Research, ISSN 2198-2627, Springer, Heidelberg, Vol. 10, Iss. 1, pp. 123-156, https://doi.org/10.1007/s40685-016-0043-6 This Version is available at: https://hdl.handle.net/10419/177268 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ ORIGINAL RESEARCH Effect of replenishment and backroom on retail shelf-space planning Alexander Hu ¨bner 1 •Kai Schaal 1 Received: 16 March 2016 / Accepted: 8 December 2016 / Published online: 9 January 2017 The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Shelf-space optimization models support retailers in making optimal shelf-space 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 in-store replenishment processes. However, the two areas of shelf-space planning and in-store 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. However, 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 shelf-space decisions and replenishment processes has not been sufficiently addressed in the existing literature. To quantify the cost associated with the relevant in-store 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 shelf-space decisions and replenishment processes. The model optimizes retail profits by determining the optimum number of facings, the optimum display orientation of items, and the optimum order frequencies, while accounting for space-elasticity effects as well as limited shelf and backroom space. Applying our model to the grocery retailer’s 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 large-scale problem instances. Finally, we use the model to show the impact of backroom space availability and &Alexander Hu ¨bner [email protected]e 1 Catholic University Eichstaett-Ingolstadt, Ingolstadt, Germany 123 Business Research (2017) 10:123–156 DOI 10.1007/s40685-016-0043-6 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 shelf-space decisions and in-store replenishment planning. Keywords Shelf space Backroom Space-elastic 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 (i.e., 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 (i.e., indirect replenishment from backroom). Both decisions, i.e., shelf space and reordering, are interrelated, because, e.g., 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 replenishments for this item, but also less space available for other items, which in turn must be reordered more frequently. Shelf-space 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;Hu ¨bner 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 retailer’s scarcest resource (cf. e.g., 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 ‘‘spaceelastic demand’’. Additionally, the costs associated with in-store replenishment are significant, because in-store 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 shelf-space planner should therefore also consider options for arranging items on the shelf, in-store replenishment frequencies and costs, and the availability of a backroom for replenishment. 124 Business Research (2017) 10:123–156 123 Current literature on shelf-space management mainly addresses the demand side by modeling the effect of space-elastic demand. In this case, a retailer’s profit is maximized under shelf-space constraints by defining the number of facings for each product (i.e., first visible unit of an item in the front row; Hu ¨bner and Kuhn 2012; Ko ¨k et al. 2015). Existing models do not account for replenishment frequencies and costs, or options for leveraging backroom inventory (Hu ¨bner and Kuhn 2012; Bianchi-Aguiar et al. 2016). To investigate the above-mentioned relationships, we conducted a time and motion study for a German grocery retailer and identified both the relevant in-store replenishment processes and the associated costs. Building on these insights, we then develop an optimization model that simultaneously optimizes shelf-space and in-store replenishment decisions while also accounting for space-elastic 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 shelf-space planning. We apply the model to show why an integrated perspective on shelf-space and instore replenishment optimization is worthwhile and demonstrate how retailers can apply the model to increase their profits. We address the trade-offs between shelf-space allocation and in-store replenishment (e.g., 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 shelf-space decisions and order frequencies. The remainder of this paper is organized as follows: Sect. 2provides the conceptual background of our paper and presents the related literature on shelfspace optimization. The time and motion study, and the description of identified replenishment processes, are presented in Sect. 3. Section 4explains the optimization 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. 6has the conclusion and outlook. 2 Conceptual background and related literature 2.1 Conceptual background and decision problem In the following, we analyze the basic decisions retailers need to make in shelfspace and reorder planning, namely (1) how much shelf space to allocate to items, and (2) how often to reorder them. (1) Shelf-space decision Shelf-space planning is a mid-term 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 (Hu ¨bner and Business Research (2017) 10:123–156 125 123 Kuhn 2012;Hu ¨bner et al. 2013; Bianchi-Aguiar 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, i.e., 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. Dre `ze 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 agets 2 facings with a stock of 4 units each, resulting in a total quantity of 2 4¼8 units. Item bgets 1 facing with a stock of 6 units and a total shelf quantity of 6. The right of Fig. 1explains 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 (i.e., 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 (i.e., which shelf level) and horizontal position (i.e., 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 e.g., Hwang et al. (2009), Hansen et al. (2010), Russell and Urban (2010) and Hu ¨bner and Schaal (2017). Shelf-space 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, i.e., the visible item width the customer sees. Empirical studies examine these so-called ‘‘spaceelasticity effects’’, (cf. e.g., Cox 1964; Frank and Massy 1970; Curhan 1972; Dre `ze et al. 1994; Desmet and Renaudin 1998). Chandon et al. (2009) show that the Fig. 1 Illustration of facing, shelf quantity and display orientation 126 Business Research (2017) 10:123–156 123 number of facings is the most important in-store factor affecting customer demand. Using a meta-analysis across empirical studies, Eisend (2014) quantified the average space-elasticity 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 ‘‘cross-space elasticity’’) is ambiguous in the pertinent literature. For example, Zufryden (1986) and Ko ¨k et al. (2015) state that there is no empirical evidence that product-level demand can be modeled with cross-space elasticity. In addition, the measured effects of cross-space elasticity appear to have only a limited influence on sales (Eisend 2014;Hu ¨bner and Schaal 2016). We therefore comply with these results in the literature and disregard cross-space elasticity effects in the remainder of our paper. Furthermore, shelf-space decisions also depend on assortment planning. However, in retail practice assortment and shelf-space decisions are typically two sequential planning steps of the category planning process (Hu ¨bner et al. 2013;Ko ¨k et al. 2015; Bianchi-Aguiar 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 e.g., 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 in-store logistics are not yet well aligned, and that this constitutes a new area of research. Reiner et al. (2013) identify opportunities for improving in-store logistics and show that it is important to not only consider customer needs and the demand side when designing store layout and taking shelf-space decisions, but also logistics requirements. Their process analysis reveals that the efficient design of in-store 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 in-store 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 in-store 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 in-store Business Research (2017) 10:123–156 127 123 logistics processes related to direct and backroom replenishment in more detail in Sect. 3. 2.2 Related literature on shelf-space optimization Following Seuring et al. (2005) and Kotzab et al. (2005), we first defined the scope of our contribution (as in Sect. 2.1), and then identified the related literature. The identification step included the material collection/selection 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 shelf-space modeling papers that contain considerations of replenishment, inventory holding and store operations. Further modeling papers dealing with shelfspace 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. Shelf-space models typically assume a given assortment with space-elastic demand, ignore substitution (because the assortment is predetermined), and account for limited shelf space. Our problem relates to the shelf-space literature, and we therefore focus on this area. For comprehensive overviews of shelf-space problems, we refer to Hu ¨bner and Kuhn (2012), Ko ¨k et al. (2015) and Bianchi-Aguiar 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 non-linearity arising from the polynomial demand function, various solution approaches are applied and space-elasticity effects either assumed to be linearly dependent on the number of facings, approximated by piecewise linearization, or non-linear models applied and solved with heuristics. Basic shelf-space management literature uses deterministic demand models to factor in space-elasticity effects (cf. Ko ¨k et al. 2015). One of the first models is proposed by Hansen and Heinsbroek (1979), who formulate a non-linear 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 shelf-space model that considers space and cross-space elasticities of polynomial form. Geometric programming is applied to solve the model for up to five product groups. The model cannot be applied to large-scale problems on an item-level, and therefore, works with product groups rather than SKUs. Zufryden (1986) formulates a model with space-elastic 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 cross-space 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 multi-knapsack heuristic. Urban (1998) provides the first enhancement with available inventory and replenishment systems. The polynomial 128 Business Research (2017) 10:123–156 123 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 shelf-space 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, shelfspace, store location and inventory replenishment frequencies. The model accounts for space- and cross-space 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 shelf-space 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 shelf-space 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 non-linear solver and tested on large-scale instances with up to 100 items. Hu ¨bner and Kuhn (2011) develop a MIP model to account for polynomial space-elasticity and replenishment cost. They show that demand effects have a significant impact on item profit. The model balances the trade-off 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 non-linear model for cross-space 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 Al-Ahmari (2013) develop an integrated space allocation and inventory model for a single item with stock-dependent demand. They analyze different setups regarding the supplier-retailer 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 facingdependent space-elasticity effects not considered. Bianchi-Aguiar 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. 2.3 Summary and research contribution When retail shelf space is limited, retailers need to thoroughly consider the trade-off between shelf-space and ordering decisions. The two decisions are interdependent and impact in-store 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) 10:123–156 129 123 showroom shelf must be indirectly replenished from the backroom. Despite these interdependencies in in-store logistics and space assignment, an integrated optimization model is lacking in the existing literature. The contributions on shelf-space planning mentioned all focus on optimizing the number of facings. Demand is assumed to be facing-dependent (i.e., space-elastic). Non-linearities arising from this are dealt with either via linear approximations or solution heuristics that are limited in their capability to solve instances of practice-relevant 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 (i.e., facings and display orientations) and order frequencies. Our model accounts for spaceelastic 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 large-scale instances. (3) By applying our model to a real data set, we show how the retailer can increase profits with optimized shelf-space 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 in-store 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 in-store logistics (e.g., Kotzab and Teller 2005; Zelst et al. 2009; Reiner et al. 2013), but does not sufficiently detail the costs affected by shelf-space planning and reordering. Curs¸eu 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 shelf-space decisions. 3.1 Replenishment processes observed during time and motion study Figure 2provides an overview of the in-store 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) 10:123–156 123 piðki;bi;fiÞand space requirements in the show- and backroom (sSR iðki;bi;fiÞ, sBR iðki;bi;fiÞ) for each item iand 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 non-linear model terms outside the optimization model using precalculation. Suboptimal heuristics are therefore not required. The MIP can be solved optimally in a time-efficient manner (see runtime tests in Sect. 5.2.1). 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 ki2Ki, typically not more than 20–25 facings. Second, only two values are possible for the visible width of a facing of item i, i.e., bi¼lifor lengthwise and bi¼wifor crosswise display orientation. In the MIP model, we decode the visible width of a facing of item iby o, where o¼1ifbi¼li, and by o¼2ifbi¼wi. Third, the frequency of direct replenishments (i.e., orders) cannot exceed the maximum number of warehouse deliveries with fi2Fi, e.g., not more than six times per week. This allows us to precalculate the profit (denoted as pikof in the MIP) for every item iand every possible combination of the three decision variables for the predefined ranges ki2Ki,o2Oand fi2Fi,i2N. This means that pikof is the profit for item i if it gets kfacings, is given the display orientation oand is ordered ftimes a period. Similarly, we precalculate how much showroom (backroom) space item iconsumes for every combination of k,oand f. The respective space consumption is denoted as sSR ikof for the showroom and sBR ikof for the backroom. Using the precalculated profits as data input, the MIP model then selects the binary variable cikof to indicate how many facings item ishould 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: Max!Pð cÞ¼X i2NX k2KiX o2OX f2Fi pikof cikof ð6Þ Subject to: X i2NX k2KiX o2OX f2Fi sSR ikof cikof Sð7Þ X i2NX k2KiX o2OX f2Fi sBR ikof cikof Bð8Þ X k2KiX o2OX f2Fi cikof ¼18i2Nð9Þ cikof 2½0;18i2N;k2Ki;o2O;f2Fið10Þ Equation (6) is the objective function and is the summation of all item-specific profits. Equations (7) and (8) ensure that showroom shelf Sand backroom space Business Research (2017) 10:123–156 137 123 Brestrictions are met. Finally, Eq. (9) ensures that each item igets exactly one combination of facings, item display orientations, and order frequencies. Equation (10) declares cikof as a binary variable. Note that the available showroom space Sis the one-dimensional shelf length (front row) available for the placement of facings, e.g., measured in centimeters or meters. In contrast, the size of the backroom Bis measured in space units, e.g., in m 2 , 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, sSR ikof is a one-dimensional length and sBR ikof is a two-dimensional area. Model complexity The MIP model developed belongs to the class of knapsack problems, which are known to be NP-hard (cf. Kellerer et al. 2004; Pisinger 2005). In our case, the model complexity is driven not only by the number of combinations for allocating Nitems to a shelf of size S, but also by the fact that each item can be ordered up to Ftimes and get one of two different display orientations. The resulting model complexity can therefore be calculated by Eq. (11): YðN;S;FÞ¼ S1 N1  FN2Nð11Þ The binomial coefficient calculates the number of possible combinations for allocating Nitems to a showroom shelf of size S. The second term accounts for the fact that each item can be ordered up to Ftimes, and finally, the third term accounts for the display orientation. For example, for N¼50, a showroom shelf of S¼100, up to daily deliveries (F¼6) and one of two possible display orientations, the number of possible configurations is 4:59 1082. Our modeling approach based on precalculated profits helps to significantly reduce this complexity, because instead of the Ycombinations, we only need to precalculate NKF2 profits (assuming Kis the number of elements in Ki, with ki¼1;...K; similarly for F). These are then provided as input into the MIP choosing the optimal combination. For the example above, the number of required precalculations corresponds to NKF2¼50 25 62¼15;000, if we assume an upper limit for the number of facings per item of K¼25. 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. 5.1. To generalize the results, Sect. 5.2 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 5.3 summarizes the findings from the numerical results. All numerical tests were conducted on a Windows 7 32-bit Intel Core i5-2520 with 2.5 GHz and 4 GB memory. The tests were implemented in VB.net (Visual Studio 2013) and GAMS 24.1 to use the CPLEX solver for the MIP. 138 Business Research (2017) 10:123–156 123 5.1 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 (i.e., cartons) and not in single customer units (i.e., cans). We treat one carton as one facing and each carton contains a quantity of six or twelve units. The display orientation-dependent stock per facing gibiis 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 space-elasticity effect is reflected in Eq. (3) by the b-value for the crosswise/lengthwise display orientation of the facing, i.e., demand increases when the larger of the two carton dimensions is displayed. We consider a sales period of one week. The average minimum demand diof the items i2Nis between 1 and 110 units per week. For determining di, we first measured the average total demand Diacross 10 months with the current number of facings ki, the current visible width of a facing bi, an assumed space elasticity of bi and then recalculated the average minimum demand diusing 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 0.50 €ri2.49 €. For reasons of confidentiality, the corresponding values for unit, replenishment and inventory cost cannot be provided. Showroom shelf space is limited to S¼5920 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 1 and 21 facings to the items, which all have a lengthwise display orientation. The number of facings was determined according to a sales-proportional allocation (SPA) rule, which assigns shelf space to items based on their share of category sales, but ignores replenishment cost (cf. Hu ¨bner and Kuhn 2012). The item length liis 14.8 cm li40.0 cm and item width wiis 22.4 cm wi41.7 cm. All items are currently ordered twice a week (fi¼2). Approaches analyzed To show the extent to which the retailer benefits from the model, we investigate the following modeling approaches: [1] ‘‘Status quo’’—which represents the number of facings and display orientation as observed in the current shelf-space assignment and a given order frequency of f¼2 for all items. To ensure comparability, we evaluate the observed values of all decision variables using the objective function (cf. Eq. 6) of the CSRPBS-model. To compute margins and replenishment costs ‘‘a posteriori’’, we apply the respective ‘‘a posteriori’’ model, denoted as CSRP*. Furthermore, we ensure that all constraints (7–10) 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 CSRPBS*ð fÞ)or facings and display orientations (approach [3], model CSRPBS*ð k; bÞ) are optimized. In [2] we keep kand bas per current and in [3] we do so for f. All Business Research (2017) 10:123–156 139 123 profit components are evaluated by applying the respective ‘‘a posteriori’’ calculation for the non-optimized variables. Finally, in approach [4] ‘‘Integrated optimization’’, the CSRPBSð k; b; fÞmodel optimizes k,band fsimultaneously, and therefore, shows the full potential. This model corresponds to the CSRPBS introduced in Sect. 4. To highlight the differences to approaches [1]–[3], we add here the superscript for the decision variables. Table 2summarizes the respective assumptions for each modeling approach: Results Figure 3illustrates the advantage of the fully integrated model (CSRPBSð k; b; fÞ) 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, i.e., 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 (CSRPBSð k; b; fÞ) over pure shelf-space optimization (CSRPBS*ð k; bÞ) diminishes with increasing space elasticity. This is due to the fact that the importance of assigning the right amount of space to high-margin 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 shelf-space optimization model. To better understand the latter, Fig. 4shows the absolute values for total profits (Pi2Npi), total gross margins (Pi2NDimi) and total replenishment cost (Pi2NCDIR iand Pi2NCBR i) 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 CSRPBS*ð fÞ,kiand biare determined regardless of the space elasticity, and are therefore, identical across all b-values. 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 frequency [3] Facings/Displ. orient. [4] Integrated Model applied CSRP* CSRPBS*ð fÞCSRPBS*ð k; bÞCSRPBSð k; b; fÞ Optimization None Order frequency fFacings kFacings k Variables (all as per current) Visible facing width bVisible facing width b Order frequency f 140 Business Research (2017) 10:123–156 123 demand realized increases with an increase in the assumed space elasticity (see Eq. 2). Table 3shows the changes in solution structure, i.e., 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 CSRPfBS* advantage over CSRP* CSRPk,bBS* advantage over CSRP* CSRPk,b,fBS advantage over CSRP* Fig. 3 Relative profit advantage of integrated and partial optimization models over status quo 20% 2,250 1,750 1,250 1,000 500 250 30% 1,500 2,000 750 25%10%5% 15%0% Total gross margin EUR Space elasticity Percent 2,000 1,250 1,000 20%5%0% 0 30%15% 25%10% 1,750 1,500 750 500 250 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 CSRPk,bBS* CSRPfBS* CSRPk,b,fBS CSRP* Fig. 4 Total profits, total gross margins and total replenishment cost for each modeling approach Business Research (2017) 10:123–156 141 123 with the status quo shows significant differences in all three optimization variables, e.g., 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 (21.4% with different order frequencies, [2] vs. [4]) or facings and display orientations (15.7% with different facings, 2.9% with different display orientation, [3] vs. [4]). 5.2 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 5.2.1 describes the data used and the test setting. Section 5.2.2 then provides runtime tests. Section 5.2.3 investigates the impact of backroom availability on profits and solution structures and Sect. 5.2.4 analyzes the profit advantages retailers can achieve when thoroughly accounting for replenishment processes. We compare our model to a sales-proportional allocation rule in Sect. 5.2.5 and develop an extension to account for assortment decisions in Sect. 5.2.6. 5.2.1 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 itemspecific parameters which are within the following intervals: di2½50;70; ri2½10;20;ci2½75%ri;80%ri;gibi2½3;5;VC DIR i¼½0:02;0:06; VCBR i¼½0:06;0:10;FC DIR i¼½0:08;0:12;FC BR i¼½0:16;0:24;hSR i2½2:5% ri;3:5%riand hBR i2½1:5%ci;2:0%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 biis assumed to vary between 0 and 35%. To focus on the core effects, we assume wi¼li¼1, if not stated otherwise. Furthermore, we set Table 3 Changes in solution structure Models Approaches Change of kbf CSRP* versus CSRPBS*ð fÞ[1] versus [2] – – 95.7% CSRP* versus CSRPBS*ð k; bÞ[1] versus [3] 88.6% 61.4% – CSRP* versus CSRPBSð k; b; fÞ[1] versus [4] 90.0% 58.6% 88.6% CSRPBS*ð fÞversus CSRPBS*ð k; bÞ[2] versus [3] 88.6% 61.4% 95.7% CSRPBS*ð fÞversus CSRPBSð k; b; fÞ[2] versus [4] 90.0% 57.1% 21.4% CSRPBS*ð k; bÞversus CSRPBSð k; b; fÞ[3] versus [4] 15.7% 2.9% 88.6% Scenario with bi¼15% 142 Business Research (2017) 10:123–156 123 K¼15 and F¼6 for the precalculations, were K(F) corresponds to the number of elements in Kiwith ki¼1;...;K(Fiwith fi¼1;...;F). 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. 6–10). The CSRP* model is used to evaluate the impact of ignoring specific effects (e.g., replenishment cost) ‘‘a posteriori’’. The measured effects are averages across 100 randomly generated instances of Nitems. 5.2.2 Runtime test Table 4shows the average runtime for different problem instances and shows that our model can efficiently generate optimal results even for large-scale problem instances. While we assumed wi¼li¼1 for all instances up to N¼400 items, for the instance with N¼2000, S¼60;000 and B¼30;000 we randomly generated item dimensions with wi2½2;10and li2½5;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 N¼2000, S¼60;000 and B¼30;000, where unit margins and space occupied correlate with R2¼0:9. The average runtime for these 100 instances is 78.06 s, with a minimum of 62.34 s and a maximum of 93.47 s, which shows that our approach can also handle hard knapsack problems efficiently. 5.2.3 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 2-item example below and then extend the analysis to a more comprehensive set of randomly generated data. 2-item example We consider two items (1 and 2) with identical demand and cost parameters. Both items have a length of li¼1 and a width of wi¼2. The stock per facing gibidepends 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 (b1¼30%;b2¼0%). Below, we analyze how facings, display orientations, order Table 4 Runtime tests for different problem sizes, in seconds Number of items N5 50 100 200 400 2000 Showroom space S20 200 400 800 1000 60,000 Backroom space B5 100 200 400 500 30,000 Ø Runtime 0.86 1.48 1.91 3.21 6.62 45.93 Average of 100 examples Business Research (2017) 10:123–156 143 123 frequencies and backroom quantities change if showroom and backroom space increase. Figure 5shows that with a showroom space of S¼6 and no backroom space (B¼0), the highly space-elastic item 1 receives k1¼5 facings and is replenished f1¼2 times a week. Item 2 receives only the minimum of k2¼1 facing because its space elasticity is zero. Item 2 needs to be ordered more often (at f2¼6) 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 f2¼6(B1) to f2¼2(B4) and instead replenish it indirectly from the backroom where inventory holding cost is lower (y2¼2). If showroom space Sis doubled from six to twelve (and no backroom exists), item 1 now receives only k1¼3 facings, which are displayed crosswise. Item 2 is also positioned crosswise and receives k2¼1 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 space-elastic item 1, but since this would induce additional space-elastic 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 S¼12 and B¼4, where y1¼2). Since no backroom exists, item 1 remains at three facings. Second, the crosswise S B Optimal shelf configuration (showroom shelf) f* y* Item 1 Space 12 High space elasticity item No space elasticity item 60/1 1 1 1 1 1 2 1 1 1 1 1 2 2/3 1 1 1 1 1 2 4-6 a 1a 1a 1a 2 12 0/1 a 1 a 1 a 12 2/3 a 1 a 1 a 1 4/5 a 1 a 1a 2 a 1a 1a 1 6/7 a 1a 12 a 1 a 1 a 1 8-11 a 1 a 12 a 1a 1a 1 12 a 1a 1a 2 f* y* Item 2 Showroom usage Backroom usage 6 0 20 60 622031 642022 8 0 20 30 722031 12 4 12 30 11 6 1 2 3 1 11 8 1 2 22 12 12 1 2 14 Profit 100.0% 105.2% 106.7% 102.2% 105.6% 116.6% 120.0% 121.5% 122.0% Fig. 5 Analysis of availability of showroom and backroom space on profit and solution structure, 2-item example (changes of decision variables between subsequent scenarios are in bold) 144 Business Research (2017) 10:123–156 123 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 f2¼6 (at S¼6;B¼0) to f2¼3 (at S¼12;B¼0). Because b2¼0%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 B¼2 (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 (y2¼1). If backroom space increases to B¼4 (or 5), priority is immediately given to item 1, since now the additional space-elastic demand caused by an increased number of facings (k1¼5) can be served from the backroom, which is completely occupied with item 1 (y1¼2). 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 (y2¼0). At B¼6 (or 7), item 2’s display orientation can again be changed to lengthwise, which shifts inventory holding cost from the expensive showroom to the less expensive backroom (y2¼1). At B¼8 (or 9,10,11), the additional backroom space is used to lower f2to f2¼2 again, similarly to the three scenarios with S¼6. Finally, at B¼12, f2can even decrease to f2¼1. Obviously backroom space is not enough to keep the lengthwise orientation. Item 2 is placed crosswise, and a stock of g2¼2 units must be kept on the showroom shelf. The example shows that items with a high space elasticity should clearly be given priority in shelf-space assignment. Furthermore, the trade-offs 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 shelfspace and in-store replenishment processes. Extended analysis To underline the impact of backroom space Bon profit and solution structures, and to generalize the findings above, we analyzed additional randomly generated data sets. Each set contains N¼50 items. To focus on the main effects, we ignore the display orientation and set li¼wi¼1. We set F¼6 and K¼15 and assume a showroom shelf space of S¼200 and in the basic scenario a backroom space of B¼100, which is varied below. For each analysis, we report the average of 100 randomly generated data sets. Figure 6shows the impact of changing backroom size on financial performance (i.e., total profits, total gross margins, total direct and backroom replenishment cost). As seen above in the 2-item 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, i.e., 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) 10:123–156 145 123 backroom can be more extensively used to refill showroom shelves if this is costbeneficial. On the other hand, this induces an increase in total backroom replenishment costs, i.e., backroom replenishment and backroom inventory holding cost, which is shown in the right-hand graph. Figure 7provides the changes in the solution structure: Up to 80% of the items are given a different number of facings kif backroom space Bdecreases. An increase in backroom space Banalogously induces changes in shelf-space assignment, but the upper limit of K¼15 limits the magnitude of this effect: Up to 20% of the items are given a different number of facings kif backroom space Bis doubled. In terms of order frequencies f, similar observations apply. Up to 54% of the items have a change in order frequencies fif backroom space Bchanges. The righthand graph shows that the average number of orders per week decreases, as more backroom space Bbecomes available. A larger backroom space Ballows 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 Bbecomes available, order frequencies fincrease slightly at first before gradually decreasing. This is because less profitable and less space-elastic 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 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 -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) 10:123–156 123 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 (i.e., 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. e.g., Hu ¨bner and Schaal 2016a). In such cases, out-of-stock substitutions resulting from potentially insufficient shelf and backroom quantities for specific items would need to be taken into consideration as well (cf. e.g., Ko ¨k and Fisher 2007;Hu ¨bner et al. 2016). A stochastic model would need to balance the trade-offs 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. e.g., Lim et al. 2004; Bianchi-Aguiar et al. 2015b) or cross-space elasticities (cf. e.g., 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 shelf-space and reordering decisions are typically made for each category separately, our model could be extended for store-wide shelf-space optimization across all categories, where common order patterns for different categories would also be considered. Finally, the investigation of multi-store environments can be considered. A corresponding model would support retailers in deciding whether planograms should be more standardized or adjusted to store-specific needs. Such a model would need to balance the trade-off between store-specific 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 e.g., Kurtulus and Toktay 2011; Martı ´nez-de Albe ´niz 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) 10:123–156 153 123 The model and solution approach proposed within this paper will be a good starting point to address the open areas of research mentioned above. 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