International Entrepreneurship and Management Journal 1, 183 202, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The United States. Initial Shelf Space Considerations at New Grocery Stores: An Allocation Problem with Product Switching and Substitution PEDRO M. REYES pedro reyes@baylor.edu Baylor University, Hankamer School of Business, P.O. Box 98006, Waco, Texas 76798, USA GREGORY V. FRAZIER frazier@uta.edu Information Systems and Operations Management Department, College of Business Administration, The University of Texas at Arlington, Arlington, Texas 76019-0437, USA Abstract. Managing limited display areas is an increasingly challenging task in the grocery retail industry, especially given the current high levels of product proliferation. The decision of how to best allocate and manage shelf space is critical to grocery retail profitability. Moreover, this decision is escalated for initial shelf space considerations at new grocery stores. Without loss to generality, this paper presents a new approach to the shelf space allocation problem that could be applied to new grocery stores for determining their initial shelf space consideration by incorporating consumer behavior actions based on the consumer s decision process. Keywords: shelf space allocation, grocery retailing, product proliferation, modelling With the growing self-service retailing industry and the proliferation of new products, the management of scarce display areas continues to be an increasingly sensitive subject. The decision of shelf space allocation and management is critical to effective grocery retail operations management and is escalated issue for those new (and often independent) grocery stores. A well-managed shelf space not only improves customer service by reducing out-of-stock occurrences; it can also improve the return on inventory investment by increasing sales and profit margins (Yang, 2001; Yang and Chen, 1999). Ideally, the decision rules regarding shelf space allocation should consider the profit contribution of each product in the category against the opportunity costs for carrying the inventory (Cox, 1964, 1970). The theory within the context of self-service grocery retail stores is that the demand for a product is influenced by the quantity of display exposure, and it has been speculated that this structure of promotion is capable of shifting brand choices among consumers (Anderson, 1979; Urban, 2002). In general, one of the primary concerns of grocery retail management involves determining the variety of brands to be stocked, and the allocation of scare shelf space Corresponding author. Pedro M. Reyes, Baylor University, Hankamer School of Business, P.O. Box 98006, Waco, Texas 76798, USA, Tel.: (254) 710-7804.
184 REYES AND FRAZIER among these stocked brands so as to maximize the retail store s profits. A subset of items from the entire category must be selected in order to maximize profits for the category, which is different from maximizing each product independently (Bawa, Landweher and Krishna, 1989; Judd and Vaught, 1988; McIntyre and Miller, 1999; Nielson, 1995). Once the category assortment has been determined, the next step is the shelf space allocation. There are many approaches to the shelf space allocation problem. Among them are optimization models whose optimal decisions are obtained using some operational constraints with respect to the practical retail environment. The primary argument in the literature posits that the allocation decision is critical to the operational decision because it directly relates to profitability as it affects cost and revenues. The space allocation influences the buyers perceptibility and therefore demand, along with various costs that include ordering, holding, handling, and transportation. Anderson and Amato (1974) addressed the fundamental short run resource allocation problem by proposing a mathematical model for simultaneously determining the most profitable short run brand mix concurrent with a determination of the optimal shelf space allocation of a fixed display area among the available brand mix. The primary assumption was that the optimal displayed area allocation depends on the composition of brand preference for potential demand where the potential demand refers to the limit of the inventory of the product displayed that would be sold if all available product brands were displayed. The expected demand function consists of three disjoint components: switching preference demand, non-switching preference demand, and random demand. The random demand represents the attractiveness of the displayed inventory to stimulate demand for those buyers that do not have a brand preference. However, this model did not include the cost effects of inventory. The mathematical programming model of Hansen and Heinsbroek (1979) proposed a model for maximizing profits by expressing the contributions to profits of all products less the costs associated with replenishing the shelf stock. Incorporating the main demand effect with the cost effect made the model more complete. One major assumption is that while the space elasticity for each unit sales is assumed to be constant, the substitution and complementary effects between products are not explicitly considered. Corstjens and Doyle (1981) developed a more comprehensive model for optimizing retail space. Their general model proposed to maximize profits by incorporating the real dimensions of the retailer s optimization problem it is a model that incorporates both the main demand effects with the cost effects of alternative space allocations. The significance of this model is the true nature of the relationship between space and store profitability. Whereas additional space given to a product increases its sales potential, it also affects the sales of other products in the assortment. Hence, ignoring these cross-elasticities and considering only the main demand effects can lead to a sub-optimization in the shelf space allocation procedure. The demand structure defined as a multiplicative power function of the displayed areas allocated to all of the products, which represents the direct elasticity with respect to a unit of shelf space, a scaling constant, the cross space elasticity between products, and the number of products, which is necessary to include the individual space elasticities and the cross space elasticities that exist between products within the store. This provided an advantage
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 185 over previous models because the explicit consideration of cross elasticities between the products accounts for substitutability and complementarity of product offerings. The individual space elasticity is defined as the main effects (demand) of the product, where the potential for positive elasticity of unit sales exists with respect to increased shelf space. The cross-space elasticity is defined by the degree of substitution, where the more similar the products, the more likely the customer would consider them to be substitutes. Corstjens and Doyle (1983) later extended their static optimization model to a dynamic model incorporating essential determinants of retail performance under dynamic trading conditions focusing on strategic allocation of retail space by incorporating the supply side inflexibilities and the demand side goodwill effects. The goodwill effects is formulated into the demand structure by using the sales of a product in particular period and a retention rate that measuring how much of the goodwill effect in one period is retained in the next. The shortcoming of this model is that their constraints were simplified and the cost functions from prior work were relaxed in this model formulation. Zufryden (1986, 1987) proposed a dynamic programming model approach as a dynamic extension of Corstjens and Doyle (1981). The primary focus was to choose a product selection from the entire set of products in the category assortment, and optimally allocate them to scarce supermarket retail shelf space, while considering realistic constraints (i.e. supply availability and operational restrictions). The design of this approach considered a general maximization objective function based on revenues (demand) that accounted for space elasticity and potential demand related marketing variables, and costs of sales. The demand function for each product is defined by the elasticities that are associated with each demand explanatory (marketing) variable. The advantages that are claimed by this approach over those previous proposed methods are that non-space variables have not been included in previous model formulations. The optimization model presented by Bultez and Naert (1988) introduced demand interdependencies prevailing across and within product groups detailing a theoretical shelf space allocation model called Shelf Allocation for Retailers Profits (SH.A.R.P.) as an extension to Corstjens and Doyle (1981). Essentially, this is a cannibalism affect within each homogeneous product category, which is modeled by an attraction of the items shared in the assortment of products. This model was then extended by Bultez, Naert, Gijsbrechts and vanden Abeele (1989), now referred as SH.A.R.P. II, and was empirically tested on a sample data concluding that variants of cannibalism produce asymmetric demand substitution patterns within the retail category assortment. Borin, Farris and Freeland (1994) developed a category management model for determining retail category product assortment and shelf space allocation based on in-store support as a function of space. A key characteristic of this model is the division of the product s market share into two components uncompromised demand and compromised demand. The uncompromised demands are characterized by the customers preference for a particular product, the in-store merchandizing support, and the product availability. The compromised demand comes from customers that are willing to compromise their choice when specific brands they wanted are not available.
186 REYES AND FRAZIER Conceptually, their model formulation of the demand for a product within a store is comprised of four factors: unmodified demand, modified demand, acquired demand, and stock out demand. The unmodified demand represents the customers intrinsic preference effect for the product, which represents the brand s market strength exclusive of the in-store merchandising support. The modified demand represents the in-store merchandising support effect. This factor is the interaction effect of the unmodified demand and the in-store attractiveness based on the in-store merchandising support where this demand could be either positive or negative. This is the same as Corstjens-Doyle s cross-space elasticities. The acquired demand then captures the assortment effect, and is made up of two parts in the multiplicative relationship: the relative sales strength and a resistance to compromise (also known as brand loyalty factor). The effects of inventory levels can also positively (a stock out benefit) or negatively (a stock out loss) influence the stock out factor. A generalized inventory-level-demand inventory model using the demand rate as a function of displayed inventory is integrated into the product assortment and shelf space allocation model by Urban (1998). The model presented is a deterministic and continuous-review inventory system, where the demand rate is constant as long as store inventory levels (both back room and shelf space) exceed the shelf space allocation. The assumption is that the shelves are kept fully stocked with continuous replenishment serviced from the back room inventory stock. Once the back room inventory is used up, and as the inventory on the shelf decreases, then the demand rate will decrease for which the typical demand function that is used in the space allocation models is presented in the polynomial form. While this demand rate function considers the substitution effect from an un-stocked item (not carried in the assortment) to a stocked item, this function does not consider temporary out-of-stock items in the normally carried assortment. Moreover, the grocery retail stores have little backroom space. By maintaining a backroom stock increases the holding cost (e.g. storing and handling), and therefore the grocery retailers would prefer zero backroom stock. In this paper a new approach is developed to address the shelf space allocation problem specific to new grocery retail stores (and not necessarily affiliated with a grocery chain). Ideally the decision rules regarding the initial shelf space allocation should consider the trade-off between profit contributions and operating costs of each product in the category. However (in this paper), a more complex allocation model would incorporate consumer behavior based on the consumer s decision process. Assuming that market research information about consumer behavior is available a demand function that reflects consumer behavior actions can also developed and utilized in the initial shelf space allocation model. The following sections describe the development of the new demand function that considers consumer behavior impacts on product demand and how it could be applied for initial shelf space allocation (and more specifically, at new grocery retail stores). A numerical example for this allocation model is provided, followed by a sensitivity analysis on the flexibility parameter in the model. Finally, conclusions and suggestions for future research are provided.
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 187 Development of model This section first discusses consumer behavior issues that affect the in-store demand for aproduct. Next, a new demand function is developed that directly considers these consumer behavior impacts on product demand. Finally, an initial shelf space allocation model is developed that utilizes the new demand function. Consumer behavior issues The actual or final demand (referred here as resulting demand) for a product at retail store is a function of two general factors: consumers preferences prior to entering the store, and influences on these preferences due to in-store displays of products. Conceptually, the resulting demand function consists of five consumer behavior components based on four general elements: (1) natural demand for a product in the category assortment, (2) switching effects due to display effects and product proliferation, (3) substitution effects due to products not carried, and (4) randomness due to lack of preference. The conceptual framework used for the formulation of the resulting demand function is presented in Figure 1. Figure 2 that illustrates the consumer behavior and decision process, and Figure 3 illustrates the sources of demand for product i. The demand function in this study extends the demand model presented by Borin et al. (1994) by including the effects of product variety and identified likely consumer responses to out-of-stock conditions presented by Emmelhainz, Stock and Emmelhainz (1991) and Campo, Gijsbrechts and Nisol (2000), and is also a modification of the typical demand formulation used in space-allocation models as presented by Urban (1998, p. 25). It is assumed that an analyst (or in the more general case: the entrepreneur) predefines which products are included in the category assortment (as is commonly done in the initial stage) and that the consumer behavior probabilities are available from marketing research organizations. Terms that are used for the formulation of the resulting demand function are defined in Table 1. Figure 1. Conceptual framework used for the resulting demand function.
188 REYES AND FRAZIER Table 1. Term Definitions of variables for the resulting demand function. Definition N is the set of all products in the market for a category N + is the set of products in a category that are included in the assortment selection N is the set of products in a category that are not included in the assortment selection α i is the natural demand for product i λ i is the probability of product loyalty for product i when i is in stock γ ij is the probability of switching from product i to product j when i is in stock µ i is the probability of product loyalty for product i when i is not carried β i is the probability of substituting another product when i is not carried d i is the expected resulting demand rate for product i s N + is the total shelf space allocated to the product category s i is the shelf space allocated to a product i, i N + Figure 2. Consumer behavior and decision process. As mentioned above, a product s sales within the store are a combination of the natural demand rate for the product, the effects of product switching due to in-store product displays, substitution occurrences due to products not carried in the category assortment, and random selection when there is no preference. Each of these components is discussed next. The natural demand rate α i is defined here as the demand rate for product i by customers when they arrive at the store, before they are influenced by what they see on the shelves. This is an out-of-store decision prior to the shopping trip. However, this
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 189 Figure 3. Sources of demand for product i. natural demand rate can be disrupted by the effects of the in-store environment, such as the proliferation of the other category products in the assortment, their displayed inventory levels, and price. The consumer is then confronted with the in-store decision to either purchase their initial preference or switch to another product in the category assortment that is carried by the grocery retail store. This buying behavior is contingent in part on the consumer s degree of loyalty λ i forproduct i, and therefore the respective probability of switching from product i to product j (denoted as γ ij ). The relationship is mathematically expressed as λ i + j j i j N + γ ij = 1; i N + (1)
190 REYES AND FRAZIER where N is the set of all products in the market for a category, N + is the set of products in a category that are included in the store s assortment selection, N is the set of products in a category that are not included in the store s assortment selection, and N = N + + N. (2) Another in-store decision due to the proliferation of product effects is when the consumer s first preference is not normally stocked by the grocery retail store. In this case, the decision is to either remain loyal to their first preference and not buy, or substitute to a second preference. Therefore, a second measure of the consumer s degree of loyalty (denoted as µ i )applies when product i is not in stock. Then the respective probability of substituting another product for product i is denoted as β i. The relationship for these probabilities is mathematically expressed as µ i + β i = 1. (3) There may also be consumers who do not have an a priori preference for a specific product in the category, but do have a demand for some product in the category. This buying behavior is referred to here as randomness, and has been used in previous research (for example see Anderson and Amato (1974)). By understanding these components of buying behavior, a resulting demand function for a specific product of the category can be formulated. It is reasonable to assume that the switching and substitution probabilities can be estimated by the use of marketing research to provide better insights into the demand components. In formulating demand functions, economists in past research have employed various means for estimating demand for differentiated products. However, as pointed out by Stavins (1997); recent studies by Trajtenberg (1990), Berry (1994), Feenstra and Levinsohn (1995), and Berry, Levinsohn and Pakes (1995) offer no agreement on the best method to find estimated demand elasticities. In fact, the proliferation of products in a category has resulted in analysts placing strong restrictions on demand in order to avoid thousands of elasticities having to be estimated. In other models, it is assumed that products are only competing with one or two of their nearest neighboring competitors (Feenstra and Levinsohn, 1995). Two types of product elasticities are incorporated into the proposed demand function to represent two types of consumers behavior for switching away from their first preference when it is in stock. The first type, cross elasticity, is the likelihood of switching between two products due to differences in packaging, price, features, or other characteristics inherent in the products. The more similar two products are, the greater is their cross elasticity. Cross-elasticity has been used in demand models by Curhan (1973), Hansen and Heinsbrock (1979), and Borin et al. (1994). It is assumed here that cross elasticities can be obtained through marketing research and would be available from marketing research firms. The second type, cross-space elasticity, is the likelihood of switching between two products due to differences in the amount of shelf space allocated to each product. Marketing research studies in the literature suggests that consumers are more attracted
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 191 by larger displays of products (i.e., more shelf space) (Anderson, 1979; Bultez and Naert, 1988; Corstjens and Doyle, 1981, 1983; Urban, 1998, 2002). In the current study, cross-space elasticity is replaced by a ratio of the relative difference in shelf space allocated to two products. For example: 1 + s i s k s N + = s N + + s i s k s N + (4) represents the relative difference of shelf space allocated to product i with respect to shelf space allocated to product k, expressed as a percentage of total category shelf space. This yields a multiplicative factor centered around 1.0, which is used to adjust the cross elasticity to account for differences in shelf space allocations. The benefit of this ratio is the ease of estimating cross-space elasticities. Development of the resulting demand function Before summarizing the resulting demand function, each component of the function is explained next. The first component is the natural demand, or the consumer s first preference prior to the shopping trip, and is represented by α i. (5) The next two components represent switching to or from another product when both products are in stock. Switching may occur as a result of in-store product support. The in-store product support is the support given to the shelf space, such as the displayed inventory level and/or the frequency of stock replenishment. The second component represents switching from i to l. This is represented by multiplying γ il,which represents the probability of switching from product i to product l, by the shelf space allocation ratio, and then multiplying by the natural demand for the first preference γ il l l N + s N + + s l s i α i. (6) s N + Similarly, the third component represents switching from product l (the first preference) to product i. γ li l l N + s N + + s i s l α l. (7) s N + The significance of these two components is that the switching effects either reduce or add to the resulting demand for a product. The fourth component addresses the situation where a consumer may make a prebuying decision to purchase a product from the category, but does not have any preference for a particular brand. This no preference effect (randomness) assumes that each
192 REYES AND FRAZIER product carried in the grocery retail store category assortment has a probability of being selected in proportion to its relative amount of shelf space, as represented by s i s N + α R, (8) where α R is the demand rate from this group of customers. Because of limited shelf space, a grocery retail store cannot possibly carry the entire set of possible products in a category. Therefore only a limited assortment will be stocked. So for those products that are not carried in the assortment, there will also be a portion of their respective natural demands that is substituted or transferred to those products that are carried in the category assortment by the grocery retail store. This fifth component is represented by β j j j N s i s N + α j, (9) where product j is not carried in the category assortment. Hence the resulting demand for a product consists of five consumer behavior components: 1. The natural demand from the consumer s first preference 2. The potential switching to another product that is carried in the category assortment 3. The potential switching from another product that is carried in the category assortment 4. The randomness effect due to no initial first preference 5. The substitution effect due to demand for a product that is not carried in the category assortment These five components are also shown in the formula component column in Figure 3. From these five components (5) (9) for consumer behavior, the resulting demand for a single product within a category is mathematically expressed by the additive function: Component: (1) (2) (3) d i = α i l l N + i l s N + + s l s i γ il α i + s N + l l N + l i s N + + s i s l γ li α l s N + (4) (5) + s i α R + s i β j α j. (10) s N + s j N + j N
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 193 Table 2. Term c i C i w i P i O H N + x Definitions of variables for the shelf space allocation model. Definition Purchase cost for product i Inventory operating cost for product i Gross margin contribution for product i Selling price to the consumer for product i Ordering cost rate expressed as $/order Holding cost rate expressed as daily % of unit cost Category profits for the products included in the assortment selection Range of flexibility for shelf space allocation given to grocery manager Formulating the objective function This section develops the objective function of the shelf space allocation model. Terms used in this model are defined in Table 2. Each component of the objective function is first discussed before the function is summarized. The first component represents total gross profit per day for the entire category of n products, and is given as n w i d i, (11) i=1 where w i is the gross margin contribution for product i. The marginal rate per unit is defined by w i = p i c i, (12) and the assumption is that the selling price to the consumer (p i ) and the product purchase cost (c i )arefixed. The cost side of the objective function considers the fixed cost and the variable costs associated with product i. There are four primary factors that contribute to operating expense associated with the inventory costs for the category. They are purchase cost, ordering cost, holding cost, and stock out cost. The purchase cost is accounted for in the gross margin contribution. Since the objective function is to allocate shelf space considering the potential substitution and switching effects, we assume that the stock out costs are near zero and therefore not included in the inventory cost function. The ordering cost rate (O) is assumed to be a constant rate for all products in the category and is given in dollars per order. Multiplying the ordering cost rate by the expected number of orders per day d i /s i (assuming no safety stock is used) gives O d i s i, (13) resulting in an estimated ordering cost stated in dollars per day.
194 REYES AND FRAZIER The daily holding cost rate (H) for the category is given as a percentage of the purchase cost of an item. Multiplying the holding cost rate by the shelf space allocation ratio gives us Hc i s i, (14) which results in an estimated holding cost that is higher for products that are more costly and have a greater shelf space allocation. The total operating costs per day (C i ) forproduct i is then expressed as C i = O d i s i + Hc i s i. (15) Combining (11) and (15) results in the objective function summarized as or N + = N + = n w i d i i=1 n C i (16) i=1 n (p i c i ) d i i=1 n i=1 ( O d ) i + Hc i s i. (17) s i Formulating the constraints Three types of constraints are included in the problem formulation. The first constraint addresses the shelf space allocated to the entire category. This space constraint requires that the space allocated to all products must equal the total available shelf space for the category, s N +. n s i = s N + (18) i=1 The second type of constraint addresses lower and upper limits, which is common in retailing. The reason for this is to maintain the store s image by stocking at least reasonable minimum quantities of each product, and not have only the most profitable item stocked. Since allocating shelf space strictly in proportion to demand rates may not be optimal due to varying profit margins and operating costs, a level of flexibility is included for the retailer. Hence, base-level shelf space allocations in proportion to natural demand are considered and then the retailer is provided some flexibility in the shelf space allocation for maximizing category profits. The base-level shelf space
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 195 allocation is expressed mathematically as s 0 i = α i n i=1 α i s N +. (19) Note that if a constant demand rate (d i )isused instead of the resulting demand function presented in the previous section, then the constant rate d i would replace α i in equation (19) to obtain s 0 i.from this base-level allocation, we can then set the lower and upper constraints on the shelf space allocation at some desired range (i.e., x = 20%) from the base level. These are represented as and s i (1 x) s 0 i (20) s i (1 + x) s 0 i (21) respectively. Finally, the last type of constraints requires that the decision variables be nonnegative integers. The model is then formulated by combining formulas (16), (18), (20), and (21). The objective of the shelf space allocation problem is to maximize the net category profits ( N +), providing that the total shelf space assigned equals the shelf space allocated for the category and allowing some degree of flexibility to the grocery manager in the allocation assignment. In summary, the shelf space allocation problem is mathematically expressed as Max N + = n w i d i n C i i=1 i=1 Subject to: n s i = s N + i=1 s i (1 x) s 0 i s i (1 + x) s 0 i s i > 0, Integer. This is a non-linear integer programming optimization problem that can be solved using widely available numerical search approaches. It is important to note that although the resulting demand function from the previous section was used in the objective function above, any demand function could be used instead.
196 REYES AND FRAZIER Table 3. Input variables and parameters for products in the category. Product Product Product Natural demand Loyalty selling price purchase cost rate probability 1 p 1 = 2.99 c 1 = 0.59 α 1 = 5 λ 1 = 0.90 2 p 2 = 2.49 c 2 = 0.49 α 2 = 4 λ 2 = 0.85 3 p 3 = 2.79 c 3 = 0.56 α 3 = 3 λ 3 = 0.80 4 p 4 = 2.29 c 4 = 0.46 α 4 = 2 λ 4 = 0.75 5 α 5 = 3 µ 5 = 0.60 6 α 6 = 2 µ 6 = 0.80 Random α R = 2 - Numerical example Consider an example where the category contains six products, for which only four are carried in the product assortment by the grocery retailer. The input variables (e.g., product selling price and product purchase cost) and the input parameters (e.g., natural demand rates and loyalty probabilities) are found in Table 3. Note that the loyalty probabilities are λ i for those products that are carried in the category assortment and µ j for those products that are not carried in the category assortment at the grocery retail store. The probability of substitution (β j )when a product is not carried in the category assortment can then be determined (e.g.; β 5 = 1 µ 5 = 0.40 and β 6 = 1 µ 6 = 0.20). The four products that are being carried in the category assortment are to be allocated to a finite space. In this example, the entire shelf space for the category will contain 90-units (i.e., S N + = 90). The lower and upper shelf space constraints are ±20% of the base-level allocation, and are s i 0.8si 0 and s i 1.2si 0,respectively. Additional input parameters are the potential switching probabilities of an instock product. They are found in Table 4. Notice that the loyalty probabilities are in the main diagonal. It is also assumed that ordering cost (O) and holding cost (H) are known. These costs were estimated based on first-hand interviews with the dry grocery manager at a local grocery store (Houser, 2002; Salmon, 2002). Hence in this example, the category ordering cost is estimated from the time to review inventories and place orders, the Table 4. Potential switching of an In-Stock product. To Product 1 Product 2 Product 3 Product 4 From Product 1 0.90 0.05 0.03 0.02 Product 2 0.08 0.85 0.05 0.02 Product 3 0.10 0.08 0.80 0.02 Product 4 0.12 0.08 0.05 0.75
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 197 transportation costs, and the handling costs of re-stocking the shelves. The category ordering costs is estimated at $3 and the category holding costs is estimated as 25% of the purchase unit cost. Then by expanding the summation, substituting for the input variables, and the input parameters, the shelf-space allocation model can be rewritten as: Subject to: Max N + ) = (2.99 0.59 3s1 [ 5 5 90 [90(1 0.90) + 0.05s 2 + 0.03s 3 + 0.02s 4 s 1 (1 0.90)] + 1 90 [(90 + s 1)(0.08 4 + 0.10 3 + 0.12 2) (0.08 4s 2 + 0.10 3s 3 + 0.12 2s 4 )] + s 1 90 (2) + s ] 1 (0.40 3 + 0.20 2) (0.25 0.59s 1 ) 90 ) [ + (2.49 0.49 3s2 4 4 90 [90(1 0.85) + 0.08s 1 + 0.05s 3 + 0.02s 4 s 2 (1 0.85)] + 1 90 [(90 + s 2)(0.05 5 + 0.08 3 + 0.08 2) (0.05 5s 1 + 0.08 3s 3 + 0.08 2s 4 )] + s 2 90 (2) + s ] 2 (0.40 3 + 0.20 2) (0.25 0.49s 2 ) 90 ) [ + (2.79 0.56 3s3 3 3 90 [90(1 0.80) + 0.10s 1 + 0.08s 2 + 0.02s 4 s 3 (1 0.80)] + 1 90 [(90 + s 3)(0.03 5 + 0.05 4 + 0.05 2) (0.03 5s 1 + 0.05 4s 2 + 0.05 2s 4 )] (0.25 0.49s 2 ) + s 3 90 (2) + s ] 3 (0.40 3 + 0.20 2) (0.25 0.56s 3 ) 90 ) [ + (2.29 0.46 3s4 2 2 90 [90(1 0.75) + 0.12s 1 + 0.08s 2 + 0.05s 3 s 4 (1 0.75)] + 1 90 [(90 + s 4)(0.02 5 + 0.02 4 + 0.02 3) (0.02 5s 1 + 0.02 4s 2 + 0.02 3s 3 )] + s 4 90 (2) + s ] 4 (0.40 3 + 0.20 2) 90 (0.25 0.56s 4 ) s 1 + s 2 + s 3 + s 4 = 90 s 1 0.8 32.14
198 REYES AND FRAZIER s 2 0.8 25.71 s 3 0.8 19.29 s 4 0.8 12.86 s 1 1.2 32.14 s 2 1.2 25.71 s 3 1.2 19.29 s 4 1.2 12.86 s 1 0, s 2 > 0, s 3 > 0, s 4 > 0 s 1, s 2, s 3, s 4, Integers. The results were found using Lingo non-linear optimization software, which uses a branch-and-bound algorithm. The shelf space assignment solution for maximizing net category profits per day is, with the shelf space allocated as follows: s 1 = 38 s 2 = 25 s 3 = 16 s 4 = 11. Sensitivity analysis The lower and upper bound constraints in equations (20) and (21) allow the retailer (or the entrepreneur) some flexibility in allocating space to the various products in the category. A sensitivity analysis was conducted to determine distribution of shelf space and profits due to the lower and upper bound constraints. The optimization model was processed with varying lower and upper bounds depending on the percentage value used for x in equation (20) and (21). The six different model runs were in increments of 10 percentage points, starting with the base of ±10% and going up to ±50%. Figure 4 presents how the shelf space allocation changes with each set of lower and upper bounds. The shelf space allocation numerical results are presented in Table 5, along with the daily category profits. The shelf space allocation changes with each ±10% -increment by allocating more space to the product(s) with a higher gross margin contribution (w).the net changes for each ±10%-increment is shown in Table 5, and in Figure 5. Although increasing the amount of flexibility (x)results in increasing category profits, the associated imbalance in shelf space allocations must be weighted against the desire to have a reasonable mix of products on the shelves. The retailer must decide how much flexibility should be reasonable in this case. A sensitivity analysis, such as presented here, can assist the retailer in making the best decision. This 4-product example required 4 decision variables with 9 constraints. The shelfspace allocation model could be easily applied to a much larger product category. For
INITIAL SHELF SPACE CONSIDERATIONS AT NEW GROCERY STORES 199 Table 5. Shelf space allocation and daily category net profits for each ±10%. ±0% ±10% ±20% ±30% ±40% ±50% s 1 32 35 38 41 45 48 s 2 26 25 25 24 22 21 s 3 19 18 16 16 15 14 s 4 13 12 11 9 8 7 N + 27.00829 27.40992 27.79900 28.21210 28.65491 28.96826 ( N + ) 0.40163 0.38908 0.41310 0.44281 0.31338 ( N + )% 1.49 1.42 1.49 1.57 1.09 Figure 4. Shelf space allocation and daily category net profits for each ±10%. Figure 5. Daily category net profits for each ±10%.
200 REYES AND FRAZIER example, a category with 40 products carried in the category assortment would require 40 decision variables and 81 constraints. Conclusion and future research In this paper the formulation of a theoretical resulting demand function began by identifying five consumer behavior components. Conceptually, these components are made up of four general elements: (1) the natural demand rate for a product in the category assortment, (2) the switching effect due to product proliferation, (3) the substitution effect as a result of items that are not carried by the store, and (4) randomness. The natural demand rate was defined as an out-of-store decision prior to the shopping trip, while the other elements are in-store decisions that may disrupt the natural demand rate, and they were defined as effects of the environment. The environmental effects were described as the switching and substitution effects and some degree of randomness. The advantage of this approach is simple. In today s information age environment, the category manager (or the grocery retail entrepreneur) increasingly has access to consumer response information (i.e. marketing research). The environmental effects are influenced by the products that are carried in the category assortment, the shelf space allocation to those products, and the degree of consumer loyalty for a particular product. The decision-maker generally makes the first two decisions once or twice a year, whereas the degree of consumer loyalty may vary throughout the year. These environmental effects were kept constant in this study. Hence, the inputs used to determine the resulting demand function were (1) the natural demand rates, (2) shelf space allocation, (3) the probability of product loyalty when it is in stock, and (4) the probability of product loyalty when the product is not in stock. Therefore, understanding these input components should provide better insights into the resulting demand. An initial shelf space allocation model that incorporates the resulting demand function was formulated. This non-linear integer optimization model seeks to assign shelf space to each product in order to maximize the daily category profits. The objective function included the gross marginal contribution, the resulting demand function proposed, and the operating costs for all products within a category. Three sets of constraints were used in formulating this model. The first was to ensure that all of the available shelf space is. The second and third sets of constraints were used to allow the retailer some flexibility in lower and upper bounds on the shelf space assignments. It should be pointed out that the usefulness of the shelf space allocation model presented here does not depend on the form of the demand function that is that any form can be used for the demand function, even a constant demand rate. However, one contribution of this study is the separate treatment of switching and substitution buying behavior components in the demand function, which provides better insights into the true sources of demand for a product.
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