Initial Shelf Space Considerations at New Grocery Stores: An

International Entrepreneurship and Management Journal 1, 183–202, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The United States.

Initial Shelf Space Considerations at New GroceryStores: An Allocation Problem with ProductSwitching and Substitution

PEDRO M. REYES pedro [email protected] University, Hankamer School of Business, P.O. Box 98006, Waco, Texas 76798, USA

GREGORY V. FRAZIER [email protected] Systems and Operations Management Department, College of Business Administration, The Uni-versity 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 andmanage shelf space is critical to grocery retail profitability. Moreover, this decision is escalated for initial shelfspace considerations at new grocery stores. Without loss to generality, this paper presents a new approachto the shelf space allocation problem that could be applied to new grocery stores for determining their initialshelf space consideration by incorporating consumer behavior actions based on the consumer’s decisionprocess.

Keywords: shelf space allocation, grocery retailing, product proliferation, modelling

With the growing self-service retailing industry and the proliferation of new prod-ucts, the management of scarce display areas continues to be an increasingly sensitivesubject. The decision of shelf space allocation and management is critical to effectivegrocery retail operations management and is escalated issue for those new (and oftenindependent) grocery stores. A well-managed shelf space not only improves customerservice by reducing out-of-stock occurrences; it can also improve the return on inven-tory investment by increasing sales and profit margins (Yang, 2001; Yang and Chen,1999). Ideally, the decision rules regarding shelf space allocation should consider theprofit contribution of each product in the category against the opportunity costs forcarrying the inventory (Cox, 1964, 1970). The theory within the context of self-servicegrocery retail stores is that the demand for a product is influenced by the quantity ofdisplay exposure, and it has been speculated that this structure of promotion is capableof shifting brand choices among consumers (Anderson, 1979; Urban, 2002).

In general, one of the primary concerns of grocery retail management involves de-termining 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 itemsfrom the entire category must be selected in order to maximize profits for the category,which is different from maximizing each product independently (Bawa, Landweher andKrishna, 1989; Judd and Vaught, 1988; McIntyre and Miller, 1999; Nielson, 1995).

Once the category assortment has been determined, the next step is the shelf spaceallocation. There are many approaches to the shelf space allocation problem. Amongthem are optimization models whose optimal decisions are obtained using some op-erational constraints with respect to the practical retail environment. The primaryargument in the literature posits that the allocation decision is critical to the opera-tional decision because it directly relates to profitability as it affects cost and revenues.The space allocation influences the buyers’ perceptibility and therefore demand, alongwith various costs that include ordering, holding, handling, and transportation.

Anderson and Amato (1974) addressed the fundamental short run resource allo-cation problem by proposing a mathematical model for simultaneously determiningthe most profitable short run brand mix concurrent with a determination of the op-timal shelf space allocation of a fixed display area among the available brand mix.The primary assumption was that the optimal displayed area allocation depends onthe composition of brand preference for potential demand—where the potential de-mand refers to the limit of the inventory of the product displayed that would be soldif all available product brands were displayed. The expected demand function consistsof three disjoint components: switching preference demand, non-switching preferencedemand, and random demand. The random demand represents the attractiveness ofthe displayed inventory to stimulate demand for those buyers that do not have a brandpreference. However, this model did not include the cost effects of inventory.

The mathematical programming model of Hansen and Heinsbroek (1979) proposeda model for maximizing profits by expressing the contributions to profits of all prod-ucts less the costs associated with replenishing the shelf stock. Incorporating the maindemand effect with the cost effect made the model more complete. One major assump-tion is that while the space elasticity for each unit sales is assumed to be constant, thesubstitution and complementary effects between products are not explicitly considered.

Corstjens and Doyle (1981) developed a more comprehensive model for optimizingretail space. Their general model proposed to maximize profits by incorporating the“real dimensions of the retailer’s optimization problem”—it is a model that incorpo-rates 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 andstore profitability. Whereas additional space given to a product increases its sales po-tential, it also affects the sales of other products in the assortment. Hence, ignoringthese cross-elasticities and considering only the main demand effects can lead to asub-optimization in the shelf space allocation procedure. The demand structure de-fined as a multiplicative power function of the displayed areas allocated to all of theproducts, which represents the direct elasticity with respect to a unit of shelf space, ascaling constant, the cross space elasticity between products, and the number of prod-ucts, which is necessary to include the individual space elasticities and the cross spaceelasticities 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 betweenthe 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 increasedshelf space. The cross-space elasticity is defined by the degree of substitution, wherethe more similar the products, the more likely the customer would consider them to besubstitutes.

Corstjens and Doyle (1983) later extended their static optimization model to a dy-namic model incorporating “essential determinants of retail performance under dy-namic trading conditions” focusing on strategic allocation of retail space by incor-porating 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 prod-uct in particular period and a retention rate that measuring how much of the “goodwilleffect” in one period is retained in the next. The shortcoming of this model is that theirconstraints were simplified and the cost functions from prior work were relaxed in thismodel formulation.

Zufryden (1986, 1987) proposed a dynamic programming model approach as a dy-namic extension of Corstjens and Doyle (1981). The primary focus was to choosea product selection from the entire set of products in the category assortment, andoptimally allocate them to scarce supermarket retail shelf space, while considering re-alistic constraints (i.e. supply availability and operational restrictions). The design ofthis approach considered a general maximization objective function based on revenues(demand) that accounted for space elasticity and potential demand related marketingvariables, and costs of sales. The demand function for each product is defined by theelasticities that are associated with each demand explanatory (marketing) variable. Theadvantages that are claimed by this approach over those previous proposed methodsare that non-space variables have not been included in previous model formulations.

The optimization model presented by Bultez and Naert (1988) introduced “demandinterdependencies prevailing across and within product groups” detailing a theoreticalshelf 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 affectwithin each homogeneous product category, which is modeled by an “attraction” of theitems 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 wasempirically tested on a sample data—concluding that variants of cannibalism produceasymmetric demand substitution patterns within the retail category assortment.

Borin, Farris and Freeland (1994) developed a category management model fordetermining 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 di-vision of the product’s market share into two components—uncompromised demandand compromised demand. The uncompromised demands are characterized by thecustomers’ preference for a particular product, the in-store merchandizing support,and the product availability. The compromised demand comes from customers that arewilling to compromise their choice when specific brands they wanted are not available.


Conceptually, their model formulation of the demand for a product within a store iscomprised of four factors: unmodified demand, modified demand, acquired demand,and stock out demand. The unmodified demand represents the customers’ intrinsicpreference effect for the product, which represents the brand’s market strength ex-clusive of the in-store merchandising support. The modified demand represents thein-store merchandising support effect. This factor is the interaction effect of the un-modified demand and the “in-store attractiveness” based on the in-store merchandisingsupport—where this demand could be either positive or negative. This is the same asCorstjens-Doyle’s cross-space elasticities. The acquired demand then captures the as-sortment effect, and is made up of two parts in the multiplicative relationship: therelative sales strength and a resistance to compromise (also known as brand loyaltyfactor). The effects of inventory levels can also positively (a stock out benefit) or neg-atively (a stock out loss) influence the stock out factor.

A generalized inventory-level-demand inventory model using the demand rate as afunction of displayed inventory is integrated into the product assortment and shelfspace allocation model by Urban (1998). The model presented is a deterministic andcontinuous-review inventory system, where the demand rate is constant as long as storeinventory levels (both back room and shelf space) exceed the shelf space allocation. Theassumption is that the shelves are kept fully stocked with continuous replenishmentserviced from the back room inventory stock. Once the back room inventory is usedup, 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 modelsis presented in the polynomial form. While this demand rate function considers thesubstitution effect from an un-stocked item (not carried in the assortment) to a stockeditem, this function does not consider ‘temporary’ out-of-stock items in the normallycarried assortment. Moreover, the grocery retail stores have little backroom space. Bymaintaining 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 allocationproblem specific to new grocery retail stores (and not necessarily affiliated with a grocerychain). Ideally the decision rules regarding the initial shelf space allocation shouldconsider the trade-off between profit contributions and operating costs of each productin the category. However (in this paper), a more complex allocation model wouldincorporate consumer behavior based on the consumer’s decision process. Assumingthat market research information about consumer behavior is available—a demandfunction that reflects consumer behavior actions can also developed and utilized in theinitial shelf space allocation model.

The following sections describe the development of the new demand function thatconsiders consumer behavior impacts on product demand and how it could be appliedfor initial shelf space allocation (and more specifically, at new grocery retail stores).A numerical example for this allocation model is provided, followed by a sensitivityanalysis on the flexibility parameter in the model. Finally, conclusions and suggestionsfor future research are provided.


Development of model

This section first discusses consumer behavior issues that affect the in-store demandfor a product. Next, a new demand function is developed that directly considers theseconsumer behavior impacts on product demand. Finally, an initial shelf space allocationmodel 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 retailstore is a function of two general factors: consumers’ preferences prior to entering thestore, and influences on these preferences due to in-store displays of products. Concep-tually, the resulting demand function consists of five consumer behavior componentsbased on four general elements: (1) natural demand for a product in the categoryassortment, (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 pref-erence. The conceptual framework used for the formulation of the resulting demandfunction is presented in Figure 1. Figure 2 that illustrates the consumer behavior anddecision process, and Figure 3 illustrates the sources of demand for product i . The de-mand 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 toout-of-stock conditions presented by Emmelhainz, Stock and Emmelhainz (1991) andCampo, Gijsbrechts and Nisol (2000), and is also a modification of the typical demandformulation used in space-allocation models as presented by Urban (1998, p. 25). Itis assumed that an analyst (or in the more general case: the entrepreneur) predefineswhich products are included in the category assortment (as is commonly done in theinitial stage) and that the consumer behavior probabilities are available from marketingresearch organizations. Terms that are used for the formulation of the resulting demandfunction are defined in Table 1.

Figure 1. Conceptual framework used for the resulting demand function.


Table 1. Definitions of variables for the resulting demand function.

Term Definition

N is the set of all products in the market for a categoryN+ is the set of products in a category that are included in the assortment selectionN− is the set of products in a category that are not included in the assortment selectionαi is the natural demand for product iλi is the probability of product loyalty for product i when i is in stockγi j is the probability of switching from product i to product j when i is in stockµi is the probability of product loyalty for product i when i is not carriedβi is the probability of substituting another product when i is not carrieddi is the expected resulting demand rate for product isN+ is the total shelf space allocated to the product categorysi 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 natu-ral demand rate for the product, the effects of product switching due to in-store productdisplays, substitution occurrences due to products not carried in the category assort-ment, and random selection when there is no preference. Each of these components isdiscussed next.

The natural demand rate αi is defined here as the demand rate for product i bycustomers when they arrive at the store, before they are influenced by what they see onthe shelves. This is an “out-of-store” decision prior to the shopping trip. However, this


Figure 3. Sources of demand for product i .

natural demand rate can be disrupted by the effects of the in-store environment, suchas the proliferation of the other category products in the assortment, their displayedinventory levels, and price. The consumer is then confronted with the “in-store” decisionto either purchase their initial preference or switch to another product in the categoryassortment that is carried by the grocery retail store. This buying behavior is contingentin part on the consumer’s degree of loyalty λi for product i , and therefore the respectiveprobability of switching from product i to product j (denoted as γi j ). The relationshipis mathematically expressed as

λi +∑

jj �=i

j∈N+

γi j = 1; i ∈ N+ (1)


where N is the set of all products in the market for a category, N+ is the set of productsin a category that are included in the store’s assortment selection, N− is the set ofproducts 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 theconsumer’s first preference is not normally stocked by the grocery retail store. In thiscase, the decision is to either remain loyal to their first preference and not buy, or sub-stitute to a second preference. Therefore, a second measure of the consumer’s degree ofloyalty (denoted as µi ) applies when product i is not in stock. Then the respective prob-ability of substituting another product for product i is denoted as βi . The relationshipfor these probabilities is mathematically expressed as

µi + βi = 1. (3)

There may also be consumers who do not have an a priori preference for a specificproduct in the category, but do have a demand for some product in the category. Thisbuying behavior is referred to here as randomness, and has been used in previousresearch (for example see Anderson and Amato (1974)).

By understanding these components of buying behavior, a resulting demand functionfor a specific product of the category can be formulated. It is reasonable to assume thatthe switching and substitution probabilities can be estimated by the use of marketingresearch to provide better insights into the demand components.

In formulating demand functions, economists in past research have employed variousmeans for estimating demand for differentiated products. However, as pointed outby Stavins (1997); recent studies by Trajtenberg (1990), Berry (1994), Feenstra andLevinsohn (1995), and Berry, Levinsohn and Pakes (1995) offer no agreement on thebest method to find estimated demand elasticities. In fact, the proliferation of productsin a category has resulted in analysts placing strong restrictions on demand in order toavoid thousands of elasticities having to be estimated. In other models, it is assumed thatproducts 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 func-tion to represent two types of consumers’ behavior for switching away from their firstpreference when it is in stock. The first type, cross elasticity, is the likelihood of switch-ing between two products due to differences in packaging, price, features, or othercharacteristics inherent in the products. The more similar two products are, the greateris 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 thatcross elasticities can be obtained through marketing research and would be availablefrom marketing research firms.

The second type, cross-space elasticity, is the likelihood of switching between twoproducts 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


by larger displays of products (i.e., more shelf space) (Anderson, 1979; Bultez andNaert, 1988; Corstjens and Doyle, 1981, 1983; Urban, 1998, 2002). In the currentstudy, cross-space elasticity is replaced by a ratio of the relative difference in shelf spaceallocated to two products. For example:

1 + si − sk

sN+= sN+ + si − sk

sN+(4)

represents the relative difference of shelf space allocated to product i with respect toshelf space allocated to product k, expressed as a percentage of total category shelfspace. This yields a multiplicative factor centered around 1.0, which is used to adjustthe cross elasticity to account for differences in shelf space allocations. The benefit ofthis 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 functionis explained next. The first component is the natural demand, or the consumer’s firstpreference prior to the shopping trip, and is represented by

αi . (5)

The next two components represent switching to or from another product when bothproducts are in stock. Switching may occur as a result of in-store product support. Thein-store product support is the support given to the shelf space, such as the displayedinventory level and/or the frequency of stock replenishment. The second componentrepresents switching from i to l . This is represented by multiplying γi l , which representsthe probability of switching from product i to product l, by the shelf space allocationratio, and then multiplying by the natural demand for the first preference

∑

ll∈N+

γi lsN+ + sl − si

sN+αi . (6)

Similarly, the third component represents switching from product l (the first preference)to product i.

∑

ll∈N+

γlisN+ + si − sl

sN+αl . (7)

The significance of these two components is that the switching effects either reduce oradd to the resulting demand for a product.

The fourth component addresses the situation where a consumer may make a pre-buying decision to purchase a product from the category, but does not have any prefer-ence for a particular brand. This no preference effect (randomness) assumes that each


product carried in the grocery retail store category assortment has a probability ofbeing selected in proportion to its relative amount of shelf space, as represented by

si

sN+αR, (8)

where αR 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 bestocked. So for those products that are not carried in the assortment, there will alsobe a portion of their respective natural demands that is substituted or transferred tothose products that are carried in the category assortment by the grocery retail store.This fifth component is represented by

∑

jj∈N−

β jsi

sN+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 com-

ponents:

1. The natural demand from the consumer’s first preference2. The potential switching to another product that is carried in the category assortment3. The potential switching from another product that is carried in the category assort-

ment4. The randomness effect due to no initial first preference5. 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 asingle product within a category is mathematically expressed by the additive function:Component:

(1) (2) (3)

di = αi −∑

ll∈N+i �=l

γi lsN+ + sl − si

sN+αi +

∑

ll∈N+l �=i

γlisN+ + si − sl

sN+αl

(4) (5)

+ si

sN+αR +

∑

jj∈N−

β jsi

sN+α j . (10)


Table 2. Definitions of variables for the shelf space allocation model.

Term Definition

ci Purchase cost for product iCi Inventory operating cost for product iwi Gross margin contribution for product iPi Selling price to the consumer for product iO Ordering cost rate expressed as $/orderH Holding cost rate expressed as daily % of unit cost�N+ Category profits for the products included in the assortment selectionx 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. Termsused in this model are defined in Table 2.

Each component of the objective function is first discussed before the function issummarized. The first component represents total gross profit per day for the entirecategory of n products, and is given as

n∑

i=1

wi di , (11)

where wi is the gross margin contribution for product i. The marginal rate per unit isdefined by

wi = pi − ci , (12)

and the assumption is that the selling price to the consumer (pi ) and the productpurchase cost (ci ) are fixed.

The cost side of the objective function considers the fixed cost and the variable costsassociated with product i. There are four primary factors that contribute to operatingexpense 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 inthe gross margin contribution. Since the objective function is to allocate shelf spaceconsidering the potential substitution and switching effects, we assume that the stockout 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 thecategory and is given in dollars per order. Multiplying the ordering cost rate by theexpected number of orders per day di/si (assuming no safety stock is used) gives

Odi

si, (13)

resulting in an estimated ordering cost stated in dollars per day.


The daily holding cost rate (H) for the category is given as a percentage of the purchasecost of an item. Multiplying the holding cost rate by the shelf space allocation ratiogives us

Hci si , (14)

which results in an estimated holding cost that is higher for products that are morecostly and have a greater shelf space allocation. The total operating costs per day (Ci )for product i is then expressed as

Ci = Odi

si+ Hci si . (15)

Combining (11) and (15) results in the objective function summarized as

�N+ =n∑

i=1

wi di −n∑

i=1

Ci (16)

or

�N+ =n∑

i=1

(pi − ci ) di −n∑

i=1

(O

di

si+ Hci si

). (17)

Formulating the constraints

Three types of constraints are included in the problem formulation. The first constraintaddresses the shelf space allocated to the entire category. This space constraint requiresthat the space allocated to all products must equal the total available shelf space forthe category, sN+ .

n∑

i=1

si = sN+ (18)

The second type of constraint addresses lower and upper limits, which is commonin retailing. The reason for this is to maintain the store’s image by stocking at leastreasonable minimum quantities of each product, and not have only the most profitableitem stocked. Since allocating shelf space strictly in proportion to demand rates maynot be optimal due to varying profit margins and operating costs, a level of flexibilityis included for the retailer. Hence, base-level shelf space allocations in proportion tonatural demand are considered and then the retailer is provided some flexibility inthe shelf space allocation for maximizing category profits. The base-level shelf space


allocation is expressed mathematically as

s0i = αi∑n

i=1 αi∗ sN+ . (19)

Note that if a constant demand rate (di ) is used instead of the resulting demand functionpresented in the previous section, then the constant rate di would replace αi in equation(19) to obtain s0

i . From this base-level allocation, we can then set the lower and upperconstraints on the shelf space allocation at some desired range (i.e., x = 20%) from thebase level. These are represented as

si ≥ (1 − x) s0i (20)

and

si ≤ (1 + x) s0i (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 allocatedfor the category and allowing some degree of flexibility to the grocery manager in theallocation assignment.

In summary, the shelf space allocation problem is mathematically expressed as

Max �N+ =n∑

i=1

wi di −n∑

i=1

Ci

Subject to:

n∑

i=1

si = sN+

si ≥ (1 − x) s0i

si ≤ (1 + x) s0i

si > 0, Integer.

This is a non-linear integer programming optimization problem that can be solved us-ing widely available numerical search approaches. It is important to note that althoughthe resulting demand function from the previous section was used in the objectivefunction above, any demand function could be used instead.


Table 3. Input variables and parameters for products in the category.

Product Product Product Natural demand Loyaltyselling price purchase cost rate probability

1 p1 = 2.99 c1 = 0.59 α1 = 5 λ1 = 0.902 p2 = 2.49 c2 = 0.49 α2 = 4 λ2 = 0.853 p3 = 2.79 c3 = 0.56 α3 = 3 λ3 = 0.804 p4 = 2.29 c4 = 0.46 α4 = 2 λ4 = 0.755 — — α5 = 3 µ5 = 0.606 — — α6 = 2 µ6 = 0.80Random — — αR = 2 -

Numerical example

Consider an example where the category contains six products, for which only fourare 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., naturaldemand rates and loyalty probabilities) are found in Table 3. Note that the loyaltyprobabilities are λi for those products that are carried in the category assortment andµ j for those products that are not carried in the category assortment at the grocery retailstore. The probability of substitution (β j ) when a product is not carried in the categoryassortment can then be determined (e.g.; β5 = 1 −µ5 = 0.40 and β6 = 1 −µ6 = 0.20).

The four products that are being carried in the category assortment are to be allocatedto a finite space. In this example, the entire shelf space for the category will contain90-units (i.e., SN+ = 90). The lower and upper shelf space constraints are ±20% of thebase-level allocation, and are si ≥ 0.8s0

i and si ≤ 1.2s0i , respectively.

Additional input parameters are the potential switching probabilities of an “in-stock” product. They are found in Table 4. Notice that the loyalty probabilities are inthe main diagonal.

It is also assumed that ordering cost (O) and holding cost (H) are known. Thesecosts were estimated based on first-hand interviews with the dry grocery manager at alocal grocery store (Houser, 2002; Salmon, 2002). Hence in this example, the categoryordering 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

FromProduct 1 0.90 0.05 0.03 0.02Product 2 0.08 0.85 0.05 0.02Product 3 0.10 0.08 0.80 0.02Product 4 0.12 0.08 0.05 0.75


transportation costs, and the handling costs of re-stocking the shelves. The categoryordering costs is estimated at $3 and the category holding costs is estimated as 25% ofthe purchase unit cost.

Then by expanding the summation, substituting for the input variables, and the inputparameters, the shelf-space allocation model can be rewritten as:

Max �N+

=(

2.99 − 0.59 − 3s1

)

∗[

5 − 590

[90(1 − 0.90) + 0.05s2 + 0.03s3 + 0.02s4 − s1(1 − 0.90)]

+ 190

[(90 + s1)(0.08 ∗ 4 + 0.10 ∗ 3 + 0.12 ∗ 2) − (0.08 ∗ 4s2 + 0.10

∗ 3s3 + 0.12 ∗ 2s4)] + s1

90(2) + s1

90(0.40 ∗ 3 + 0.20 ∗ 2)

]− (0.25 ∗ 0.59s1)

+(

2.49 − 0.49 − 3s2

)∗

[4 − 4

90[90(1 − 0.85) + 0.08s1 + 0.05s3

∗ + 0.02s4 − s2(1 − 0.85)] + 190

[(90 + s2)(0.05 ∗ 5 + 0.08 ∗ 3 + 0.08 ∗ 2)

− (0.05 ∗ 5s1 + 0.08 ∗ 3s3 + 0.08 ∗ 2s4)]

+ s2

90(2) + s2

90(0.40 ∗ 3 + 0.20 ∗ 2)

]− (0.25 ∗ 0.49s2)

+(

2.79 − 0.56 − 3s3

)∗

[3 − 3

90[90(1 − 0.80) + 0.10s1 + 0.08s2

+ 0.02s4 − s3(1 − 0.80)] + 190

[(90 + s3)(0.03 ∗ 5 + 0.05 ∗ 4 + 0.05 ∗ 2)

− (0.03 ∗ 5s1 + 0.05 ∗ 4s2 + 0.05 ∗ 2s4)] − (0.25 ∗ 0.49s2)

+ s3

90(2) + s3

90(0.40 ∗ 3 + 0.20 ∗ 2)

]− (0.25 ∗ 0.56s3)

+(

2.29 − 0.46 − 3s4

)∗

[2 − 2

90[90(1 − 0.75) + 0.12s1 + 0.08s2

+ 0.05s3 − s4(1 − 0.75)] + 190

[(90 + s4)(0.02 ∗ 5 + 0.02 ∗ 4 + 0.02 ∗ 3)

− (0.02 ∗ 5s1 + 0.02 ∗ 4s2 + 0.02 ∗ 3s3)] + s4

90(2) + s4

90(0.40 ∗ 3 + 0.20 ∗ 2)

]

− (0.25 ∗ 0.56s4)

Subject to:

s1 + s2 + s3 + s4 = 90

s1 ≥ 0.8 ∗ 32.14


s2 ≥ 0.8 ∗ 25.71

s3 ≥ 0.8 ∗ 19.29

s4 ≥ 0.8 ∗ 12.86

s1 ≤ 1.2 ∗ 32.14

s2 ≤ 1.2 ∗ 25.71

s3 ≤ 1.2 ∗ 19.29

s4 ≤ 1.2 ∗ 12.86

s1 ≤ 0, s2 > 0, s3 > 0, s4 > 0

s1, s2, s3, s4, Integers.

The results were found using Lingo non-linear optimization software, which uses abranch-and-bound algorithm. The shelf space assignment solution for maximizing netcategory profits per day is , with the shelf space allocated as follows:

s1 = 38

s2 = 25

s3 = 16

s4 = 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 thecategory. A sensitivity analysis was conducted to determine distribution of shelf spaceand profits due to the lower and upper bound constraints. The optimization model wasprocessed with varying lower and upper bounds depending on the percentage valueused for x in equation (20) and (21). The six different model runs were in increments of10 percentage points, starting with the base of ±10% and going up to ±50%. Figure 4presents 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 thedaily category profits. The shelf space allocation changes with each ±10% -incrementby 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 desireto have a “reasonable” mix of products on the shelves. The retailer must decide howmuch flexibility should be “reasonable” in this case. A sensitivity analysis, such aspresented here, can assist the retailer in making the best decision.

This 4-product example required 4 decision variables with 9 constraints. The shelf-space allocation model could be easily applied to a much larger product category. For


Table 5. Shelf space allocation and daily category net profits for each ±10%.

±0% ±10% ±20% ±30% ±40% ±50%

s1 32 35 38 41 45 48s2 26 25 25 24 22 21s3 19 18 16 16 15 14s4 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%.


example, a category with 40 products carried in the category assortment would require40 decision variables and 81 constraints.

Conclusion and future research

In this paper the formulation of a theoretical resulting demand function began byidentifying five consumer behavior components. Conceptually, these components aremade up of four general elements: (1) the natural demand rate for a product in thecategory assortment, (2) the switching effect due to product proliferation, (3) thesubstitution effect as a result of items that are not carried by the store, and (4) ran-domness. The natural demand rate was defined as an “out-of-store” decision priorto the shopping trip, while the other elements are “in-store” decisions that may dis-rupt the natural demand rate, and they were defined as effects of the environment.The environmental effects were described as the switching and substitution effectsand some degree of randomness. The advantage of this approach is simple. In to-day’s information age environment, the category manager (or the grocery retail en-trepreneur) increasingly has access to consumer response information (i.e. marketingresearch).

The environmental effects are influenced by the products that are carried in thecategory assortment, the shelf space allocation to those products, and the degree ofconsumer loyalty for a particular product. The decision-maker generally makes thefirst two decisions once or twice a year, whereas the degree of consumer loyalty mayvary 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 naturaldemand rates, (2) shelf space allocation, (3) the probability of product loyalty when itis 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 intothe resulting demand.

An initial shelf space allocation model that incorporates the resulting demand func-tion was formulated. This non-linear integer optimization model seeks to assign shelfspace to each product in order to maximize the daily category profits. The objectivefunction included the gross marginal contribution, the resulting demand function pro-posed, and the operating costs for all products within a category.

Three sets of constraints were used in formulating this model. The first was to ensurethat all of the available shelf space is. The second and third sets of constraints wereused to allow the retailer some flexibility in lower and upper bounds on the shelf spaceassignments.

It should be pointed out that the usefulness of the shelf space allocation modelpresented here does not depend on the form of the demand function—that is that anyform 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 substitutionbuying behavior components in the demand function, which provides better insightsinto the true sources of demand for a product.


One limitation of this study is that only a single product category was considered.One possible extension for future research would be to develop demand functions forseveral related product categories, which might include complementary products (suchas razors and shaving cream). Another extension would be to consider potential stock-outs and time-dependent variables in managing the shelf space. Dynamic order pointscould be developed that consider product substitution and current inventory levels ofother products.

To summarize, the contributions of this study are both theoretical and practical. Abetter understanding of the consumer buying behavior and decision making processand then incorporating these sources of demand into an allocation problem shouldlead to more robust decision-making tools. This research provides a tool for groceryentrepreneurs to make good shelf space allocation decisions and yet have some degreeof flexibility to both maximize category profits and maintain a reasonable product mix.

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