Effect of replenishment and backroom on retail shelf-space planning

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; Hübner 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 “space-elastic 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.

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; Hübner and Kuhn 2012; Kök et al. 2015). Existing models do not account for replenishment frequencies and costs, or options for leveraging backroom inventory (Hübner 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 in-store 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. 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 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. 6 has the conclusion and outlook.

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. Curşeu 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.

In this section, we develop the Capacitated Shelf-Space 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 \({\mathbb {N}}\) where \(N = |{\mathbb {N}}\)| and with the item index i, \(i \in {\mathbb {N}}\). For this set of items, the retailer simultaneously needs to decide how much shelf space to allocate to the items (i.e., the number of facings), whether to display them on the shelf lengthwise or crosswise (i.e., the display orientation) and how often to order them (i.e., 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-space and reorder planning and its consequences on direct and indirect replenishment, we follow the majority of contributions and consider a single showroom shelf (cf. e.g., Zufryden 1986; Corstjens and Doyle 1981; Irion et al. 2012), i.e., we do not account for different shelf levels and assume the shelf consists of one level with a one-dimensional 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 \(k_i\) 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 \(b_i\), defines whether an item i is displayed lengthwise or crosswise. This impacts whether the customer sees the item length (\(b_i=l_i\)) or width (\(b_i=w_i\)) when looking at the shelf from the front. Finally, the third type, order frequency \(f_i\), determines the integer number of orders per period, and consequently the number of direct store replenishments.

Three types of auxiliary variables, \(g_{ib_i}\) for the stock per facing, \(x_{ib_i}\) for the total shelf quantity and \(y_i\) for the backroom inventory, are used in addition: Behind each facing, a certain stock, \(g_{ib_i}\), 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 \(b_i\), a fixed number of units can be placed behind the facing. How many units fit behind the facing depends on the item dimensions (\(l_i\) or \(w_i\), determined by the display orientation chosen), and on the shelf depth. The second auxiliary variable, the total shelf quantity \(x_{ib_i}\), depends on the number of facings \(k_i\) and the stock per facing \(g_{ib_i}\), and is computed by \(x_{ib_i}=k_i \cdot g_{ib_i}\). To derive the third auxiliary variable, the backroom quantity \(y_i\), we consider the fact that the order frequency \(f_i\) divides the considered period into equal \(f_i\) subperiods. A subperiod demand needs to be covered during each of these subperiods. This subperiod demand corresponds to the total item demand \(D_i\) divided by the number of subperiods: \(D_i/f_i\). The subperiod demand is fulfilled by the shelf quantity \(x_{ib_i}\) and the backroom inventory \(y_i\), which we calculate as the part of the subperiod demand not covered by the shelf quantity: \(y_i = {\text {max}} [\lceil \frac{D_i}{f_i} - x_{ib_i} \rceil ; 0]\). This ensures that customer demand is always fulfilled. Total demand \(D_i\) for an item is assumed to be deterministic, which implies that out-of-stock 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 replenishment 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, i.e., if \(f=2\), a delivery occurs in a six day period, e.g., on days 1 and 4, but not on days 1 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 \(k_i\), visible width of a facing \(b_i\) and order frequencies \(f_i\) across all items, represented by the respective vectors \(\bar{k}, \bar{b}\) and \(\bar{f}\), with \(\bar{k} = \{k_1,k_2, \ldots , k_N\}\), \(\bar{b} = \{b_1,b_2, \ldots , b_N\}\) and \(\bar{f} = \{f_1,f_2, \ldots , f_N\}\) (cf. Eq. 1).To obtain the item profit \(p_i\), we deduct the total cost of direct replenishment \({\text {C}}^\mathrm{{DIR}}_{i}\) and the total cost of backroom replenishment \({\text {C}}^\mathrm{{BR}}_{i}\) (cf. Eq. 2) from the total gross margin of an item. The gross margin of an item is calculated as the product of its total demand \(D_i\) and its unit margin \(m_i\). The item unit margin \(m_i\) corresponds to the difference between its sales price \(r_i\) and its purchase cost \(c_i\).The total period demand \(D_i(k_i,b_i)\) of an item i is a composite function of the minimum demand \(d_i\) and the facing- and display orientation-dependent demand. The minimum demand rate \(d_i\) represents the retailer’s forecast for an item that is independent of the facing and visible facing width (cf. Hansen and Heinsbroek 1979; Hübner and Kuhn 2012; Bianchi-Aguiar 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 \(k_i\). 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. e.g., Hansen and Heinsbroek 1979; Irion et al. 2012), the facing- and display orientation-dependent demand rate is a polynomial function of the number of facings \(k_i\) allocated to an item, the visible facing width \(b_i\) and the space-elasticity \(\beta _i\) (with \(0 \le \beta _i \le 1\)). 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 \(b_i\). Eq. 3 summarizes the demand calculation applied.Total direct replenishment cost (\({\text {C}}^\mathrm{{DIR}}_i\)) comprises three parts, as shown in Eq. (4): fixed replenishment costs for each replenishment of an item (\(\mathrm{FC}_i^\mathrm{{DIR}}\)), variable replenishment costs for each unit replenished of an item (\(\mathrm{VC}_i^\mathrm{{DIR}}\)) and showroom inventory holding costs (\(h_i^\mathrm{{SR}}\)) per unit. Assuming continuous demand, the average showroom inventory used for calculating the related inventory costs is calculated as \(x_{i_i}/2\).The total backroom replenishment costs (\({\text {C}}^\mathrm{{BR}}_i\)) consist of the same three elements (cf. Eq. 5), where we also need to consider the number of shelf refills from the backroom (\(\lceil \frac{y_i}{x_{ib_i}} \rceil\)) during a subperiod for the fixed replenishment costs, the backroom inventory \(y_i\) instead of the showroom shelf quantity, and the backroom’s inventory holdings costs per unit (\(h_i^\mathrm{{BR}}\)). Solution approach Equations (3)–(5) contain several non-linear components relating to the decision variables (e.g., space-elastic demand with \((k_i \cdot b_i)^{\beta _i}\) or the division of two variables in the backroom replenishment costs \(\lceil \frac{y_i}{x_{ib_i}} \rceil\)). Therefore, it is a non-linear model. To handle the non-linearity, we precalculate the associated profit \(p_{i}(k_i,b_i,f_i)\) and space requirements in the show- and backroom (\(s^\mathrm{{SR}}_{i}(k_i,b_i,f_i)\), \(s^\mathrm{{BR}}_{i}(k_i,b_i,f_i)\)) 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 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 \(k_i \in {\mathbb {K}}_i\), 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., \(b_i = l_i\) for lengthwise and \(b_i = w_i\) for crosswise display orientation. In the MIP model, we decode the visible width of a facing of item i by o, where \(o=1\) if \(b_i = l_i\), and by \(o=2\) if \(b_i = w_i\). Third, the frequency of direct replenishments (i.e., orders) cannot exceed the maximum number of warehouse deliveries with \(f_i \in {\mathbb {F}}_i\), e.g., not more than six times per week. This allows us to precalculate the profit (denoted as \(\pi _\mathrm{{ikof}}\) in the MIP) for every item i and every possible combination of the three decision variables for the predefined ranges \(k_i \in {\mathbb {K}}_i\), \(o \in {\mathbb {O}}\) and \(f_i \in {\mathbb {F}}_i\), \(i \in {\mathbb {N}}\). This means that \(\pi _\mathrm{{ikof}}\) 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 \(s_\mathrm{{ikof}}^\mathrm{{SR}}\) for the showroom and \(s_\mathrm{{ikof}}^\mathrm{{BR}}\) for the backroom.

Using the precalculated profits as data input, the MIP model then selects the binary variable \(\gamma _\mathrm{{ikof}}\) 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:Subject to: Equation (6) is the objective function and is the summation of all item-specific profits. Equations (7) and (8) ensure that showroom shelf S and backroom space B restrictions are met. Finally, Eq. (9) ensures that each item i gets exactly one combination of facings, item display orientations, and order frequencies. Equation (10) declares \(\gamma _\mathrm{{ikof}}\) as a binary variable. Note that the available showroom space S is 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 B is measured in space units, e.g., in m², 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, \(s_\mathrm{{ikof}}^\mathrm{{SR}}\) is a one-dimensional length and \(s_\mathrm{{ikof}}^\mathrm{{BR}}\) 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 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):The binomial coefficient calculates the number of possible combinations for allocating 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 \(4.59\cdot 10^{82}\). Our modeling approach based on precalculated profits helps to significantly reduce this complexity, because instead of the Y combinations, we only need to precalculate \(N \cdot K \cdot F \cdot 2\) profits (assuming K is the number of elements in \({\mathbb {K}}_i\), with \(k_i=1, \ldots K\); 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 \(N \cdot K \cdot F \cdot 2 = 50 \cdot 25 \cdot 6 \cdot 2 = 15,000\), 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.

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.

In this paper, we presented a capacitated shelf-space optimization model that contributes to the existing literature by accounting for in-store 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 space-elastic demand. Retailers are provided with additional flexibility from the optimized display orientations of items. We have quantified the relevant in-store 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 in-store processes and cost types identified in the existing literature. To solve the resulting non-linear problem, we developed a mixed-integer model. Even for large-scale problems, our model yields optimal results efficiently within a feasible amount of time. We applied our model to the retailer’s 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 shelf-space and in-store replenishment planning by applying our model. Furthermore, we have shown that an integrated perspective on shelf-space and replenishment optimization is crucial for retailers, because backroom space and replenishment cost have a significant impact on retail profits and shelf-space planning. An integrated perspective for shelf planning is specifically important, since in practice shelf-space decisions are made by a central sales planning unit which oftentimes ignores the consequences of shelf planning on in-store 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 shelf-space 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 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., Hübner 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., Kök and Fisher 2007; Hübner 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.

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; Martínez-de Albéniz 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.

The model and solution approach proposed within this paper will be a good starting point to address the open areas of research mentioned above.