Pricing, Variety, and Inventory Decisions for Product Lines ... - CiteSeerX

Pricing, Variety, and Inventory Decisions for Product Lines of Substitutable Items Bacel Maddah1 , Ebru K. Bish2 , and Brenda Munroe3 1

Engineering Management Program, American University of Beirut, P.O. Box 11-0236, Riad El Solh, Beirut 1107 2020, Lebanon. Phone: +961 1 350 000 Ext.: 3551; fax: +961 1 744 462; email: [email protected]

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Grado Department of Industrial and Systems Engineering (0118), 250 Durham Hall, Virginia Tech, Blacksburg, VA 24061. Phone: (540) 231 7099; fax: (540) 231 3322; email:[email protected]

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Hannaford Bros., 145 Pleasant Hill Rd, Scarborough, ME 04074. Phone: (207) 885-2097; email: [email protected]

1 Introduction and Motivation Integrating operations and marketing decisions greatly benefits a firm. The interaction between operations management (OM) and marketing is clear. Marketing actions drive consumer demand, which significantly influences OM decisions in areas such as capacity planning and inventory control. On the other hand, the marketing department of a firm relies on OM cost estimates in making decisions concerning pricing, variety, promotions, etc. Therefore,

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Maddah, Bish, and Munroe

developing joint operations and marketing models is a research objective that arises naturally. In this chapter, we review recent works on pricing, assortment (or variety), and inventory decisions in retail operations management. Deciding on the prices and the breadth of items to be offered in a retail store is among the main functions of marketing. Moreover, inventory decisions that take into account demand uncertainty are the responsibility of OM. Therefore, the research reviewed here contributes to the growing literature on joint marketing/OM models (see, for example, Eliashberg and Steinberg [21], Griffin and Hauser [29], Karmarkar [38], and Porteus and Whang [60]). In this chapter, we focus on retailer settings because of the large number of recent works in retail operations management, and because of our own work in this area. In addition, the research in retail settings is connected to and related to manufacturing product design and production planning problems (see Section 3). Within the spirit of an integrated marketing/OM approach, one of the main contributions of the reviewed research is to study pricing, assortment, and inventory decisions jointly. Under this integrative framework, the retailer sets two or all of the above decisions simultaneously. This seems to be a successful business practice for several retailers. For example, JCPenney received the “Fusion Award” in supply chain management for “its innovation in integrating upstream to merchandising and allocation systems and then downstream to suppliers and sourcing.” A JCPenney vice president attributes this success to the fact that, at JCPenney, “assortments, allocations, markdown

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Pricing, Variety, and Inventory Decisions of Substitutable Items

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pricing are all linked and optimized together” (Frantz [23]). Canadian retailer Northern Group managed to get out of an unprofitable situation by implementing a merchandise optimization tool. Northern Group’s CFO credits this turnaround to “assortment planning” and the attempt to “sell out of every product in every quantity for full price” (Okun [57]). Moreover, our experience with Hannaford4 on various aspects of pricing, variety, and shelf inventory decisions attests to the strong inter-dependence among these decisions. More specifically, an important contribution of the research reviewed in this chapter is to include inventory costs within pricing and assortment optimization models. Most of the classical literature along this avenue assumes infinite inventory levels and, therefore, excludes inventory considerations (see, for example, Dobson and Kalish [18], Green and Krieger [26], Kaul and Rao [39], and the references therein). (We briefly review some of these works in Section 6 in order to better understand the effect of limited inventory on pricing and variety decisions.) We believe this is due, in part, to the complexities introduced by modeling inventory. For example, the review paper by Petruzzi and Dada [59] indicates a high level of difficulty associated with joint pricing and inventory optimization even for the single item case. These difficulties do not, however, justify ignoring inventory effects in modeling. For example, in 2003 the average End-of-Month capital invested in inventory of food retailers (grocery and liquor stores) in the U.S. was approximately 34.5 Billion dollars, with an Inventory/Sales ratio of approximately 82% (U.S. Census Bu4

One of the largest chains of grocery stores in New England.

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reau, [64]). On the other hand, the net 2003 profit margin in food retailing is estimated to be 0.95% (Food Marketing Institute, [24]). With an inventory cost of capital commonly estimated at 20% (annually) or higher, these numbers indicate that food retailers can significantly increase their profitability by reducing their inventory costs. Similar arguments apply to other retailing industries as well. In addition to the integrative approach explained above, the reviewed works adopt demand models from the marketing and economics literature that reflect the actual manner consumers make their buying decisions. These “consumer choice” models are based on the classical principle of utility maximization (see, for example, Anderson and de Palma [2], Ben-Akiva and Lerman [10], Manski and McFadden [49], and McFadden [50]). In this chapter, we focus on decisions involving a family of “substitutable” items, referred to as a “product line” or a “category.” More specifically, a retailer’s product line is a set of substitutable items that serve the same need for the consumer but that differ in some secondary aspects. Thus, a product line may consist of different brands as well as different variants of the same brand (such as different sizes, colors, or flavors). This definition of a product line applies to a wide selection of items, ranging from books and CDs to food items such as coffee, yogurt, ice-cream, cereals, soda, to other consumer products such as shampoo and toothpaste. When faced with a purchasing decision from a product line, a consumer selects her most preferred item, given the trade-off between price and quality. She may also choose not to buy any

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of the displayed items and postpone her purchase or seek a different retailer. Pricing has a major impact on the consumer’s choice among the available alternatives. However, other factors are also important. Such factors include the assortment or variety level, in terms of the composition of items offered in the product line, and the shelf inventory levels of these items. Given the complexity of the product line problem, most research focuses on two of the three essential decisions involved (pricing, variety, and inventory), with the exception of one recent work ([45]) that considers an integrated model involving all three decisions. Furthermore, to simplify the analysis, most works consider a single-period newsvendor-type inventory setting and “static choice” assumptions (i.e., consumers make their purchasing decisions independently of the inventory status at the moment of their arrival and leave the store emptyhanded if their preferred item is out of stock without considering stock-out based substitution). Not surprisingly, these assumptions simplify the analysis, making it possible to gain managerial insights into this complex problem. This chapter is structured along this line of research. In addition, we limit the scope to monopolistic settings involving a single retailer. The remainder of this chapter is organized as follows. In Section 2, we present a brief overview of the related literature. As stated above, all material is presented in the context of a retail setting. Therefore, in Section 3, we discuss how this research is related to the manufacturer’s product design, pricing, and production planning decisions. Then in Section 4, we present the key ideas of a set of consumer choice models that are commonly used in

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developing product line demand functions. After this background material, we present the core material. Specifically, in Section 5, we review recent works on product line inventory and variety decisions within a newsvendor setting under exogenous pricing. These works include Bish and Maddah [12], Cachon et al. [13], Gaur and Honhon [25], and van Ryzin and Mahajan [61]. In Section 6, we briefly review works on product line pricing and inventory decisions while assuming infinite inventory levels. These works include Aydin and Ryan [5] and Dobson and Kalish [18]. In Section 7, we review works on product line pricing and inventory decisions while assuming that the assortment of items in the product line is given. These works include Aydin and Porteus [6], Bish and Maddah [12], and Cattani et al. [14]. In Section 8, we review the work of Maddah and Bish [45] on product line joint pricing, inventory, and variety decisions. In Section 9, we summarize our observations on the current practice of retail pricing, inventory, and variety management. Finally, in Section 10 we conclude and provide suggestions for future research. We note that our chapter is not the first in its league. Mahajan and van Ryzin [46] wrote an excellent book chapter on a similar topic. However, we review works that mostly appeared after the publication of Mahajan and van Ryzin [46]. The background on the relevant economics, marketing, and operations management literature provided in [46] allows us to focus on recent works that address specific problems, with practical relevance and a wide potential for future research. We refer the interested readers to [46] for further background information.

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2 Related Literature The literature on this area is at the interface of economics, marketing, and OM disciplines. The economics literature approaches this topic from the point of view of product differentiation (see Lancaster [40] for a review). The focus of this literature is on developing consumer choice models that reflect the way consumers actually make their purchasing decision from a set of differentiated products (see, for example, Hotelling [36], Lancaster [41], and McFadden [50]). The Multinomial Logit Choice (MNL) model is among the most popular consumer choice models (see, for example, Anderson and de Palma [2] and Ben-Akiva and Lerman [10]). Interestingly, the MNL has its roots in Mathematical Psychology (see, for example, Luce [42] and Luce and Suppes [43]). It has also been widely used to model travel demand in transportation systems (see, for example, Domencich and McFadden [20]). The economics literature also utilizes the MNL and other consumer choice models in modeling variety within a market-equilibrium framework in a market with many firms selling differentiated products (see, for example, Anderson and de Palma [3] and [4]). The marketing literature emphasizes the process of collecting data and fitting appropriate choice models to it (see, for example, Besanko et al. [11], Guadagni and Little [30], and Jain et al. [31]). The data is typically compiled from scanner data (i.e., log of all sales transactions in a store) and panel data (obtained by tracking the buying habits of a selected group of customers), and represents the actual consumer behavior. A popular technique for de-

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termining consumer utilities from store data is “conjoint analysis” (see, for example, Green and Krieger [28]). Several works in the marketing literature address the problem of product line design (in terms of which items to offer to consumers, i.e., variety decisions) and pricing utilizing data obtained from conjoint analysis (see Green and Krieger [26] and Kaul and Rao [39] for reviews). A typical approach is to utilize deterministic estimates of utilities for each consumer segment and formulate the problem as a mixed integer mathematical programming problem, with the objective of maximizing the firm’s profit subject to consumer utility maximization constraints (see, for example, Dobson and Kalish [18] and [19], and Green and Krieger [27]). Other works on product line design and pricing include Moorthy [54], Mussa and Rosen [55], and Oren et al. [58]. In recent years, many works in the OM literature extend the marketing literature on product line design and pricing decisions by developing models that account for operational aspects (mostly inventory costs) as well as consumer choice. These works are reviewed in detail in the remainder of this chapter. As stated above, we limit our scope to monopolistic settings. We must note that there are a few recent works that study the product line pricing and/or inventory decisions under consumer choice processes while considering competition between retailers (see, for example, Anderson and de Palma [3], Besanko et al. [11], Hopp and Xu [35] and Mahajan and van Ryzin [48]). Finally, the works on single item inventory models with price dependent demand are also relevant to the research reviewed in this chapter. Examples of these works

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include Chen and Simchi-Levi [15], Federgruen and Heching [22], Karlin and Carr [37], Mills [53], Petruzzi and Dada [59], Whitin [65], and Young [67].

3 Connection to Manufacturer’s Product Design and Production Planning Problems While most of the research reviewed in this chapter is presented within the retailing context, this research is relevant to the manufacturer’s product design and production planning problems in two important aspects. First, manufacturers face similar product design and production planning problems in that they need to make decisions regarding the composition (assortment), pricing, and production planning of their product lines (in terms of which items to produce, how to price them, and in what quantities to produce), and these decisions revolve around similar trade-offs to those discussed in this chapter. As a result, manufacturers can benefit from the models and insights presented here in answering these questions. The research reviewed here that specifically focuses on the retailer’s problem can, with certain modifications, be applied to product design and production planning problems in manufacturing settings. (In addition, some of the reviewed works, e.g., Cattani et al. [14] and Hopp and Xu [34], are presented within a manufacturing framework; see also Alptekinoglu and Corbett [1] and Hopp and Xu [35] for further work in manufacturing settings.) In particular, the demand models that we review here, all of which are based on

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consumer choice theory, can be applied immediately to estimate manufacturers’ demand functions, as is done in several papers that specifically consider manufacturing settings (e.g., Alptekinoglu and Corbett [1], Cattani et al. [14], and Hopp and Xu [34, 35]). Indeed, such consumer choice models are gaining popularity among manufacturers in an attempt to more realistically model their demand. To give a few examples, we are aware of such efforts at several leading automotive manufacturers such as General Motors and Honda. On the cost side, implementing the models presented here to a manufacturer’s problem may necessitate certain adjustments. This is because the cost structure for a manufacturing firm may involve additional terms not considered here such as product development costs (e.g., costs related to product development, launch, and marketing) and fixed setup costs. Since these costs are generally not linear in production volume, their inclusion requires further analysis to understand how they impact the assortment, pricing, and inventory decisions that we consider here. In addition, a manufacturer will have capacity constraints for its production resources (e.g., plant, labor). Second, in supply chain settings, manufacturers’ and retailers’ product line design, pricing, and inventory decisions are intimately related in that they impact each other. These dependencies are also impacted by cooperation and contractual agreements on profit sharing between retailers and manufacturers. Although there are some works in this area, as far as we are aware, this dependence has not been fully explored in the OM literature and there are interesting potential avenues for research in this direction. For example, Ay-

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din and Porteus [7] investigate the dependence between the manufacturer’s rebates and the retail’s pricing and inventory decisions in a setting where the retailer faces a logit demand generated from an MNL choice model. In another paper, Aydin and Hausman [8] discuss supply chain coordination in assortment planning between a manufacturer and a retailer with end customers making their purchase decisions based on the MNL choice model. The channel selection problem widely studied in the marketing literature is also related, since it refers to the manufacturer’s problem of what type of retailers to select for her product line (i.e., “whether to vertically integrate retail activities or to use independent retailers and in the latter case, whether to use franchised dealers or to use common retail stores that sell competing brands”, Choi [16]). However, this research generally ignores the manufacturer’s capacity constraints when a product line (i.e., a set of differentiated items) is considered, see Yano [66] for a recent review of the literature in this area. The very recent paper by Yano [66] is an exception; it analyzes the role of the manufacturers’ capacity constraints in a setting where two manufacturers, each producing a differentiated but a competing product, sell their products through a common retailer. As can be seen, more research is needed that studies the complex problem of how the manufacturer’s and retailer’s product line design, pricing, and inventory decisions impact each other, and how these decisions should be made in a supply chain setting with multiple and competing manufacturers and

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retailers. We believe these decisions can benefit from more academic research in this area.

4 Overview of Consumer Choice Models In this section, we briefly review some of the discrete choice models that represent the consumer preference as a stochastic utility function. Our goal here is not to present an in-depth treatment of consumer choice theory, but rather to introduce the key ideas to readers not familiar with the choice theory. We will use these concepts subsequently in the chapter. We refer the readers interested in an in-depth treatment of the choice theory to Anderson et al. [2] and Ben-Akiva and Lerman [10] for more details on these and other choice models. We present choice models within the context of a retail setting since this is the focus of this chapter. Let Ω = {1, 2, . . . , n} be the set of possible variants from which the retailer can compose her product line. Let S ⊆ Ω denote the set of items stocked by the store. Demand for items in S is generated from a random number of customers arriving to the retailer’s store. A customer chooses to purchase at most one item from set S so as to maximize her utility. Thus, a consumer either purchases one item from S or chooses not to buy anything and leaves the store empty-handed. Consumers have a random utility, Ui , for each item i ∈ S, and a random utility, U0 , for the “no-purchase” option. The randomness in Ui , i ∈ S ∪ {0}, is due to differences

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in tastes among consumers and inconsistencies in consumer behavior on different shopping occasions. Then, the probability that a consumer buys item i ∈ S is qi (S) = P r{Ui = maxj∈S∪{0} Uj }. Several consumer choice models are derived based on the distribution of Ui , i ∈ S ∪ {0}. We discuss some of these models below.

4.1 The Multinomial Logit Model (MNL) Under the MNL, the utility for i ∈ S ⊆ Ω is Ui = ui +i , and the utility of the no-purchase option is U0 = u0 + 0 , where ui is the expected utility for item i, u0 is the expected utility for the no-purchase option, and i , i ∈ S ∪ {0}, are independent and identically distributed (i.i.d.) Gumbel (double exponential) random variables with mean 0 and shape factor µ. The cumulative distribution function for a Gumbel random variable is given by F (x) = e−e

−(x/µ+γ)

, where

γ is Euler’s constant (γ ≈ 0.5772). The Gumbel distribution is utilized mainly because it is closed under maximization (i.e., the maximum of several independent Gumbel random variables is also a Gumbel random variable). This property leads to closed form expressions for purchase probabilities, given by qi (S) =

vi j∈S∪{0} vj

, i ∈ S,

q0 (S) =

v0

j∈S∪{0} vj

,

(1)

where vj ≡ euj /µ , j ∈ S ∪ {0}, and q0 (S) is the probability that a customer buys nothing. We will refer to vj as the preference of item j (as in Mahajan and van Ryzin [46]), because it is increasing in the mean utility uj .

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These closed form expressions of the purchase probabilities lead to tractable analytical models. In addition, it is easy to statistically estimate the parameters of the MNL (and perform goodness of fit tests) using the actual store transaction data, especially with the wide use of information systems that track such transactions (see, for example, Guadagni and Little [30], Hauser [33], McFadden [50], and McFadden et al. [51]). These references indicate that the MNL predicts product line demand with high accuracy. As a result, it is not surprising that the MNL model is widely used both in academic research and in practice. One drawback of the MNL is that it suffers from the independence from irrelevant alternatives (IIA) property, which refers to the fact that the ratio of purchase probabilities of items i, j is the same regardless of the choice set containing i and j. Specifically, it follows from (1) that for any S ⊆ T ⊆ Ω, qi (S) qi (T ) vi . = = qj (S) qj (T ) vj To understand the limitations brought by the IIA property, suppose that an item l is removed from choice set T . Then the IIA property implies that the purchase probability of each item i ∈ T \ {l} will increase by the same per centage ( j∈T ∪{0} vj )/( j∈T \{l}∪{0} vj ) . Therefore, all items in T can be thought of as being “broadly similar.” This limits the applicability of the MNL model, since in reality a subset of the items in a product line will typically be closer substitutes relative to the other items. For example, the chocolatebased flavors are close substitutes in an ice-cream product line. Removing

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a chocolate-based variant from the product line will likely increase the purchase probabilities of other chocolate-flavored variants more than the purchase probabilities of vanilla-flavored variants.

4.2 The Nested Multinomial Logit Model (NMNL)

The NMNL has been proposed as a variation of the MNL to overcome its limitations brought by the IIA property. The NMNL is attributed to BenAkiva [9]. In the NMNL, items are ordered into groups or “nests.” Items within the same nest are close substitutes of one another, while those in different nests are substitutes to a lesser degree. Let N (Ω) be the set of nests from which the retailer can compose her product line5 , and N = {N1 , N2 , . . . , Nn˜ } ⊆ N (Ω) be the set of nests offered by the retailer, where n ˜ denotes the cardinality of N (i.e., the number of offered nests). The choice mechanism under the NMNL can be seen as a two-level decision process. A consumer first chooses a nest and then selects an item in that particular nest (or selects to buy nothing).6 5

N (Ω) forms a partition of Ω in the sense that ∪i:Ni∈N (Ω) Ni = Ω and Ni ∩ Nk = ∅, for Ni , Nk ∈ N (Ω), i = k.

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This sequence of choice applies in a monopolistic setting where one retailer manages all the nests. The consumer arrives to the store, chooses a nest, and then decides whether to buy or not based on the items available in the chosen nest. That is, a no-purchase option is associated with each nest. Another modeling alternative is to have one no-purchase option such that a customer may choose to buy nothing upon inspecting which nests are available, as in Hopp and Xu’s [35] competitive setting.

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The consumer’s utility for item i ∈ Nk , k = 1, . . . , n ˜ , is Uik = uik + ik , and the utility for the no-purchase option given that the customer chooses nest k is U0k = u0k + 0k , where ik , i ∈ Nk ∪ {0}, are i.i.d. Gumbel random variables with mean 0 and shape factor µ2 , and uik is the expected utility of i ∈ Nk ∪ {0}. Then, the conditional probability that a customer purchases item i in Nk or does not purchase anything, given that that the customer chooses Nk , are respectively given by qi|k (Nk ) =

vik j∈Nk ∪{0} vjk

, i ∈ Nk ,

q0|k (Nk ) =

v0k j∈Nk ∪{0} vjk

, (2)

where vjk ≡ eujk /µ2 , j ∈ Nk ∪ {0}, k = 1, . . . , n ˜. An arriving customer chooses among nests based on the “attractiveness” of each nest. The attractiveness, Ak , of nest Nk ∈ N is defined as the expectation of the maximum utility from Nk , Ak = E



max

j∈Nk ∪{0}

Ujk = µ2 ln 



vjk  .

(3)

j∈Nk ∪{0}

˜, The consumer’s utility for nest Nk ∈ N is Uk = Ak +k , where k , k = 1, . . . n are i.i.d. Gumbel random variables with mean 0 and shape factor µ1 . Then, the probability that a customer chooses Nk ∈ N is

eAk /µ1

qk (N ) = n˜

l=1

eAl /µ1

=

j∈Nk ∪{0} vjk

n˜ l=1

µ2 /µ1

j∈Nl ∪{0} vjl

µ2 /µ1 .

(4)

Finally, the probability that a customer purchases item i ∈ Nk ∈ N is qik (N ) = qi|k (Nk )qk (N ) .

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4.3 Locational Choice Model

This model is attributed to Lancaster (see, for instance, Lancaster [40]). Items in Ω = {1, 2, . . ., n} are assumed to be located on the interval [0, 1] representing the attribute space. The location of product i is bi . Consumers have an ideal product in mind with location X. (This location may vary among consumers and is therefore considered a random variable.) Then, the consumer’s utility for item i is Ui = U − g(|X − bi |), where U represents the utility of a product at the ideal location and g(.) is a strictly increasing function representing the disutility associated with deviation from the ideal location (|X − bi | is the distance between the location of item i and the ideal location). The purchase probabilities of items in Ω are then derived based on the probability distribution of X. In the remainder of this chapter we present several product line models that utilize the above consumer choice models to generate the demand function.

5 Product Line Variety and Inventory Decisions under Exogenous Pricing In this section, we review models that assume that prices of items in the choice set Ω = {1, 2, . . ., n} are exogenously set. The retailer’s problem is to decide on the subset of items, S ⊆ Ω, to offer in her product line together with

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the inventory levels for items in S. The works we review make the following modeling assumptions. Demand is generated from customers arriving to the retailer’s store in a single selling period. The demand function is developed based on one of the consumer choice models discussed in Section 4. On the supply side, inventory costs are derived based on the classical newsvendor model. In addition, the reviewed models derive demand and cost functions under the following “static choice” assumptions: (i) Consumers make their purchasing decisions independently of the inventory status at the moment of their arrival, and (ii) they will leave the store empty-handed if their preferred item (in S) is out of stock (i.e., there is no stock-out based substitution). These assumptions simplify the analysis. Without these assumptions, the models are not analytically tractable. Mahajan and van Ryzin [46] argue that static choice assumptions hold in certain situations such as catalog retailers and retailers that sell based on floor models. However, in many realistic situations these static assumptions will not hold. In such cases, the static models presented below may be seen as an approximation of reality. In fact, Gaur and Honhon [25] argue that static models lead to a lower bound on the expected profit under dynamic substitution. In their numerical results, the static model leads to an expected profit within 2% of the optimal profit (accounting for dynamic substitutions), which suggests that the static model is a reasonable approximation. However, their numerical results are limited to the locational choice model that they utilize.

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Mahajan and van Ryzin [47] study the problem of determining the optimal inventory levels of an assortment under a general consumer choice model within a newsvendor setting under dynamic choice. They propose a sample path gradient algorithm for this problem, and develop numerical examples with both MNL and locational choice models. They observe through these examples that dynamic substitution generally leads to lower inventory (in terms of the total assortment inventory) than what static models suggest, with higher inventories for popular items (as these items are subject to secondary stock-out based demand). They also observe that, under MNL choice, static assumptions lead to an expected profit which is close to the optimal expected profit under dynamic substitution when the profit margins of items in the assortment are equal. When profit margins are not equal, one example in [47] indicates a somewhat significant loss of profitability of around 20% due to static assumptions; see also Agrawal and Smith [63] and Netessine and Rudi [56] for models under dynamic choice assumptions. In the following we present detailed reviews of recent works on joint variety and inventory decisions under the above assumptions. The first works in this line of research are those of van Ryzin and Mahajan [61] and Smith and Agrawal [63]. Many recent papers build on the work of van Ryzin and Mahajan [61].

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5.1 The Van Ryzin and Mahajan Model

Van Ryzin and Mahajan (VRM) [61] consider a product line under the MNL consumer choice model within a newsvendor inventory setting. The probability that a customer buys item i ∈ S ⊆ Ω is given by qi (S) in (1). The mean number of customers visiting the store in the selling period is λ. Then, the demand for item i ∈ S is assumed to be a Normal random variable, Xi , with mean λqi (S) and standard deviation σ(λqi (S, p))β , where σ > 0 and 0 ≤ β < 1. The logic behind this choice of parameters is to have a coefficient of variation of Xi that is decreasing in the mean store volume λ (this seems to be the case in practice). The special case with σ = 1 and β = 1/2 represents a Normal approximation to demand generated from customers arriving according to a Poisson process with rate λ. VRM refer to this model as the “independent population model,” since it assumes that customers make their choice independently of each other. They also suggest a more simplified “trend following model” where all consumers choose the same item of the product line. Here, we restrict our attention to the independent population model. VRM assume that all items in Ω have the same unit cost, c, and are sold at the same price, p (or have the same c/p ratio). They argue that this assumption may hold in certain situations (such as the case of a product line having different flavors or colors of the same variant). On the cost side, items of the product line do not have a salvage value and no additional holding or shortage costs apply. This cost structure captures the essence of inventory

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costs in terms of overage and underage costs. By utilizing the well-known results for the newsvendor model under Normal demand (see, for example, Silver et al. [62], p. 404-408), the optimal inventory level for item i ∈ S, yi∗ (S), and the expected profit from S at optimal inventory levels, Π(S), can be written as yi∗ (S) = λqi (S) + Φ−1 (1 − c/p)σ(λqi (S))β , i ∈ S , Π(S) =

i∈S

Πi (S) =

(6)

λqi (S)(p − c) − pθσ(λqi (S))β ,

(7)

i∈S

where θ ≡ φ(Φ−1 (1 − c/p)), φ(·) and Φ(·) are the probability density function and the cumulative distribution function of the standard Normal distribution, respectively, and Πi (S) = λqi (S)(p − c) − pθσ(λqi (S))β is the expected profit from item i ∈ S. Observe that in (7), the first term is the “riskless” expected profit (assuming an infinite supply of items), while the second term involving the demand standard deviation represents the inventory cost. The retailer’s objective is to find the assortment yielding the maximum profit, Π ∗ : Π ∗ = Π(S ∗ ) = max{Π(S)} ,

(8)

S⊆Ω

where S ∗ is an optimal assortment. The main factor involved in determining the optimal assortment is the trade-off between the sales revenue and the inventory cost. It can be easily seen from (1) the expected total demand for an assortment S,

j∈S

λqj (S),

increases if an item is added to S. That is, higher variety increases the assortment demand, and consequently leads to a higher sales revenue from the

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assortment. In addition, it can be seen that qi (S) decreases if a new item is added to S. That is, higher variety leads to the “thinning” of each individual item’s demand, which results in a higher demand variability7, and consequently to a higher inventory cost. In brief, high variety leads to a high sales revenue as well as a high inventory cost. As a result, the optimal assortment should not be too small (to generate enough sales revenue) nor too large (to avoid the excessive inventory cost). The following result from [61] is a consequence of the trade-off between the sales revenue and the inventory cost. Lemma 1. Consider an assortment S ⊆ Ω. Then, the expected profit from S, Π(S, vi ), is quasiconvex in vi , the preference of item i ∈ S. Lemma 1 allows deriving the structure of the optimal assortment, the main result in [61]. Theorem 1. Assume that the items in Ω are ordered such that v1 ≥ v2 ≥ . . . vn . Then, an optimal assortment is S ∗ = {1, 2, . . ., k}, for some k ≤ n. Theorem 1 states that an optimal assortment contains the k most popular items for some k ≤ n. Thus, the structure of the optimal assortment is quite simple. VRM then study the factors that affect the variety level. Assuming v1 ≥ v2 ≥ . . . vn , they consider assortment of the optimal form Sk = {1, 2, . . ., k}, and use k as a measure of variety. They derive asymptotic 7

The coefficient of variation of an item demand is σ(λqi (S))β−1 , which is decreasing in qi (S).

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results stating that (i) Π(Sk+1 ) > Π(Sk ) for sufficiently high selling price, p; (ii) Π(Sk+1 ) < Π(Sk ) for sufficiently low no-purchase preference, v0 ; and (iii) Π(Sk+1 ) > Π(Sk ) for sufficiently high store volume, λ. For example, (iii) implies that stores with high volume such as “Super Stores” should offer a high variety. Finally, VRM utilize the majorization ordering theory and derive a measure of “fashion” that allows comparing the profitability of two or more product lines.

5.2 The Bish and Maddah Variety Model Bish and Maddah (BM) [12] consider the model in VRM [61] under the additional assumption that v1 = v2 = . . . = vn = v. This is a stylized model with “similar” items. It may apply in cases such as a product line with different colors or flavors of the same variant, where consumer preferences for the items in the product line are quite similar. The main research question here is to characterize the optimal assortment size (i.e. the number of similar items to carry in the store). In addition, this simple setting allows a comprehensive study of the factors that affect the variety level through a comparative statics analysis. To understand the effect of pricing, the expected utility is modeled as v = α − p, where α may be seen as the mean reservation price or the quality index of an item. Such a structure is common in the literature (e.g., Guadagni and Little [30]). In addition, to simplify the exposition, BM consider a demand function that is a special case of that in VRM [61] where the parameters

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characterizing the demand standard deviation are σ = 1 and β = 1/2 (which, as aforementioned, represents a Normal approximation to demand generated from a Poisson process). However, all the results in [12] hold under the more general demand model with σ > 0 and 0 ≤ β ≤ 1. Then, for an assortment of k items, the optimal inventory level of each item and the expected profit at optimal inventory level in (6) and (7) reduce to

y∗ (p, k) = λq(p, k) + Φ−1 (1 − c/p) λq(p, k) ,

Π(p, k) = k λq(p, k)(p − c) − pθ(p) λq(p, k) , where q(p, k) =

e(α−p)/µ v0 +ke(α−p)/µ

(9) (10)

and θ(p) ≡ φ(Φ−1 (1 − c/p)).8

Despite the simplified form of the expected profit in (10), it is still difficult to analyze it because of the complicating term θ(p).9 BM develop the following approximation to simplify the analysis: θ(x) ≈ ax(1 − x) ,

(11)

where a > 0. With a = 1.66, the approximation is reasonably accurate with an average error of 8.6%. (See Maddah [44] for more details on this approximation.) With this approximation, Π(p, k) in (10) simplifies to the following: 8

We write y∗ (p, k) and Π(p, k) as functions of both p and k because we will refer to this model later, in Section 7.2, to present the pricing analysis.

9

Although θ(p) does not depend on k, one has to differentiate θ(p) when studying how the optimal assortment size varies in terms of p and c. This differentiation can be cumbersome.

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c Π(p, k) = k(p − c) λq(p, k) − a λq(p, k) . p

25

(12)

The following assumption is used by BM in order to eliminate some trivial cases where demand is too low and the retailer is better off selling nothing. (A1): The expected profit, Π(p, k), is increasing in k at k = 1, that is, 2 ∂Π(p,k) a2 c2 e(α−p)/µ −(α−p)/µ > 0, or equivalently, λ > (1 + v e ) 1 + . 2 0 ∂k p 2v0 k=1

Under (A1), the expected profit in (12) is well-behaved in the assortment size k, as the following theorem indicates. Theorem 2. The expected profit Π(p, k) is strictly pseudoconcave and unimodal in k. Let kp∗ ≡ arg maxk Π(p, k). Theorem 2 states that the expected profit increases with variety (k) up to k = kp∗ . For k > kp∗ , adding more items to the product line will only diminish the expected profit. Thus, Theorem 2 implies that kp∗ < ∞ (i.e., there exists an upper limit on the variety level). On the other hand, in the riskless case (which assumes infinite inventory levels), the expected profit, k(p − c)λq(p, k), is increasing in k, and there is no upper bound on variety in the product line. That is, inventory cost limits the variety level of the product line. Theorem 2 formally proves this last statement. The intuition behind this result is linked to the trade-off between the sales revenue and the inventory cost and their implications on variety level, as discussed in Section 5.1. Another consequence of Theorem 2 is that one can perform a comparative static analysis on the optimal assortment size, kp∗ , as done in the following theorem.

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Theorem 3. The optimal assortment size kp∗ is: (i) Decreasing in the unit cost per item, c; (ii) Increasing in the expected store volume (arrival rate), λ. Theorem 3 states that the higher the unit cost per item, the lower the optimal variety level. That is, retailers selling expensive items should not offer a wide variety. On the other hand, retailers with low cost items should diversify their assortments. This kind of practice is adopted by many retailers. Theorem 3 also indicates that a higher store volume allows the retailer to offer a wide variety. This extends the asymptotic result in VRM [61] discussed in Section 5.1. In general, monotonicity properties similar to Theorem 3 do not seem to hold for other model parameters. However, the following theorem establishes monotonicity results under a fairly mild condition. Theorem 4. If q(p, 1) >

1 2

(equivalently, u0 < α − p), then the optimal as-

sortment size kp∗ is: (i) Increasing in the price, p (in the range where p < α − u0 ); (ii) Decreasing in the mean reservation price, α (in the range where α > u0 + p); (iii) Increasing in the utility of the no-purchase option, u0 (in the range where u0 < α − p). The condition in Theorem 4 simply states that, on average, the no-purchase option is less appealing than buying from the retailer’s product line, even when

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it consists of a single item (observe that kq(p, k) ≥ q(p, 1), for k ≥ 1). Theorem 4 (i) indicates that a higher price increases the cost of underage, so variety is increased to reduce the risk of losing a customer. This holds if the price is not too high. Interestingly, at high prices the asymptotic result in VRM [61] reveals similar insights. Theorem 4 (ii) indicates that as the quality of the items increases, there will be less need for variety. Finally, Theorem 4 (iii) states that a high no-purchase utility (possibly indicating a fierce competitive environment) forces the retailer to increase the breadth of her product line in order to reduce the number of unsatisfied customers. This is also in line with the VRM [61] asymptotic results.

5.3 The Cachon et al. Model

Cachon et al. (CTX) [13] extend the VRM [61] model by accounting for “consumer search” and considering a slightly more general inventory cost function. Consumer search refers to the phenomenon that consumers may not purchase their most preferred item in the retailer’s product line if it is possible for them to search other retailers for, perhaps, “better” items. The expected profit function considered in [13] is a slight modification of that in (7), and is given by Π(S) =

i∈S

Πi (S) =

λqi (S)(p − c) − h(qi (S)) , i∈S

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where the inventory cost function h(.) is concave and increasing. As CTX argue, this cost structure covers inventory costs for common situations such as the classical EOQ and newsvendor settings. CTX first analyze the “no-search” model with the expected profit given by (13) and the purchase probabilities, qi (S), i ∈ S, given by (1). They show that the main result from VRM [61] in Theorem 1 continues to hold under the more general inventory cost function in (13). That is, an optimal assortment under (13) consists of the k most popular variants for some integer k. CTX then present two consumer search models that differ from the “nosearch” model in their formulation of the purchase probabilities. In the “independent assortment” search model, it is assumed that no other retailer in the market carries any of the items offered in the product line of the retailer under consideration. It is argued that this applies, for example, to product lines of jewelry or antiques. In this case, the consumer’s expected value from search is independent of the retailer’s assortment. In the “overlapping assortment” model, a limited number of variants are available in the market, and the same variant can be offered by many retailers (this applies, for example, to product lines of digital cameras). In this case, the consumer’s expected value from search decreases with the assortment size. Offering more items in an assortment reduces the search value for the consumer.10

10

We are using the term “assortment size” loosely here to refer to variety level in terms of number of items in an assortment. CTX use a more precise measure.

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The purchase probabilities in the independent assortment model are derived as follows. Similar to VRM [61], the utility for items in S and the no-purchase option are given by Ui = ui + i , i ∈ S ∪ {0}, where i are i.i.d. Gumbel random variables with mean zero and scale factor µ. The consumer’s expected search utility is Ur = ur + r , where ur is the mean utility of the search option and r is a Gumbel random variable with mean zero and scale factor µ. In addition, the search cost is b. It can be shown that a consumer purchases item i ∈ S only if Ui = maxj∈S∪{0} Uj and Ui > U , where U ≡ ur − b is the “search threshold.” Then, the purchase probabilities under the independent assortment search model, qisi (S), are derived as a function of the no-search purchase probabilities, qi (S) in (1), as qisi (S) = qi (S)(1 − H(U , S)) , i ∈ S,

q0si (S) = 1 −

qisi (S) , (14)

i∈S

−(U /µ+γ )

− v0 +

where H(U , S) = e

j∈S

vj e

. (The parameter γ is defined in

Section 4.1.) That is, the purchase probability of i ∈ S under the independent assortment search model is a fraction (1 −H(U, S)) of its purchase probability under the no-search model. In the overlapping assortment model, it can be shown that item i ∈ S is purchased only if Ui = maxj∈S∪{0} Uj and Ui > U (S), where the search threshold U (S) is the unique solution to

∞ U (S)

(x − U (S))w(x, S)dx = b, and

w(x, S) is the density function of the maximum utility from S = Ω \ S (i.e., w(x, S) is the density function of U max = maxj ∈S / Uj ). That is, item i is purchased only if it generates the highest utility for the consumer in S ∪ {0}

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and if its utility is not lower than the maximum utility from S minus the search cost b. Then, the purchase probabilities under the overlapping assortment search model, qiso (S), are derived as a function of the no-search purchase probabilities, qi (S), as qiso (S) = qi (S)(1 − H(U (S), S)) , i ∈ S,

q0so (S) = 1 −

qiso (S) , (15)

i∈S

where H(.) is as defined in (14). CTX show that the independent consumer search model does not change the structure of the optimal assortment in the no-search model in that it consists of the k most popular items, for some value of k, as given by Theorem 1. However, this result does not necessarily hold under the overlapping assortment search model, where CTX report finding optimal assortments not having the structure in Theorem 1. (They point out, however, that restricting the search to “popular assortments” having structure given by Theorem 1 provides reasonable results in most of the cases they have tested.) CTX also report finding optimal assortments having items with negative expected profit under the overlapping assortment search, where the unprofitable items are offered in order to decrease consumer search. They also show that if the search cost, b, is small enough, then it is optimal to offer the entire choice set Ω under the overlapping assortment search model. Finally, extensive numerical results are presented in [13] on comparing the no-search model with the two search models. These results indicate that ignoring consumer search generally leads to less variety (i.e., assortment with

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fewer variants) and loss of profitability. The main insight here is that large assortments prevent consumer search.

5.4 The Gaur and Honhon Model Gaur and Honhon (GH) [25] consider a model similar to that of VRM [61], but they utilize a locational choice model (see Section 4.3) instead of the MNL. Rather than choosing from a finite set of variants Ω in composing the product line, GH determine the locations and the number of items to be offered in the [0, 1] interval. Specifically, the problem in [25] is to determine the number, n, and the locations of items in an assortment given by b = (b1 , . . . , bn ) where bj ∈ [0, 1], bj < bj+1 , is the location of item j. Items are considered “horizontally differentiated, i.e., they differ by characteristics that affect quality or price.” All items have the same unit cost, c, and are sold at the same price, p (as in [61]). The demand function under the locational choice model is derived as follows. The consumer utility for item j is given by Uj = Z − p − g(|X − bj |), where Z is a constant, g(.) is an increasing real-valued function, and X is the location of the ideal item preferred by the consumer. A consumer purchases the item that maximizes her utility. The utility of the no-purchase option is assumed to be zero. The coverage distance of item j is defined as L = maxx {|x−bj | : Z −p−g(|x−bj |) > 0}. The first choice interval containing the ideal item locations for costumers who purchase item j (i.e., customers + who obtain maximum positive utility from j) is then given by [b− j , bj ], where

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+ b− j = max{bj − L, (bj + bj−1 )/2} and bj = min{bj + L, (bj + bj+1 )/2}. Finally,

the purchase probability of item j is given by + + − qj (b) = P r{b− j ≤ X ≤ bj } = FX (bj ) − FX (bj ) ,

where FX (.) is the cumulative distribution function of X. GH consider that the distribution of X is such that its density function is either unimodal or uniform. Then, similar to BM [12], demand for item j is considered to be a Normal random variable with mean λqj (b) and standard deviation

λqj (b).

GH derive a newsvendor type expected profit function similar to VRM [61] but they include a fixed cost, f, for adding an item to an assortment. The expected profit at optimal inventory level is then given by Π(b) =

n i=1

Πi (b) =

n

λqi (b)(p − c) − pθ

λqi (b) −nf .

(16)

i=1

GH [25] determine the product locations b = (b1 , . . . , bn ), bj ∈ [0, 1] together with the number of items to offer, n, that maximize the expected profit in (16). Their main result is that items should be equally spaced on a subinterval of [0, 1], with the distance between any two adjacent items equal to 2L. That is, b∗j = b∗1 + 2L(j − 1), j = 2, . . . , n∗ . They also derive the optimal number of items to offer, n∗ , as a function of b∗1 and propose a line search method to find b∗1 . Gaur and Honhon [25] also consider dynamic stock-out based substitution.

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6 Product Line Variety and Pricing Decisions under Ample Inventory In this section, we review models that assume that there is ample supply of all items in the product line (i.e., inventory levels of all items are infinite), and consider the retailer’s problem of deciding on which subset of items to offer in her product line (S ⊆ Ω) and at what prices to sell these products (pj , j ∈ S). This is a problem which has been extensively studied in the Marketing and Economics literature, especially in competitive settings. The assumption of ample supply holds in certain settings such as retailers that have a policy of maintaining a full shelf. As before, we restrict our attention to monopolistic settings. The classical approach in this setting is to estimate consumer utilities (which are assumed to be deterministic) and formulate the problem as a mathematical programming problem that we briefly present below. (See, for example, Dobson and Kalish [18] and [19] and Green and Krieger [27].) Recent works in this area consider stochastic utilities based on the MNL choice model (e.g., Aydin and Ryan [5] and Hopp and Xu [34]).

6.1 The Mathematical Programming Approach (Dobson and Kalish [18])

Consider a market with nc consumers. Each customer purchases at most one item from Ω in a way as to maximize her surplus (utility). The reservation

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price of consumer i for product j is αij . The utility of the no-purchase option, which is denoted by j = 0, is zero. Adding item i to the product line incurs a fixed cost fi . Let xij = 1 if consumer i purchases item j, and xij = 0 otherwise, and zj = 1 if item j is offered in the product line, and zj = 0 otherwise. The problem is then to determine pj and zj , j ∈ Ω, which are the solution to the following mixed integer programming problem ([18])11 . maximize

nc n

(pj − cj )xij −

i=1 j=1

n

fj zj

j=1

subject to n

(αik − pk )xik ≥ (αij − pj )zj , i = 1, . . . nc, j = 1, . . . n

k=1 n

(αik − pk )xik ≥ 0 , i = 1, . . . nc

k=1 n

xij = 1 , i = 1, . . . nc

(17) (18) (19)

j=0

xij = 0, 1, i = 1, . . . nc , j = 0, . . . n zj = 0, 1, j = 1, . . . n . Constraints (17)-(19) ensure that a customer purchases the item that gives her maximum positive utility or, otherwise, purchases nothing. Dobson and Kalish [18] show that this problem is NP-complete and propose a heuristic solution method.

11

The formulation we present here is a slight modification of the formulation in Dobson and Kalish [18], with one less decision variable, z0 .

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6.2 Variety and Pricing Decisions under MNL choice (The Aydin and Ryan Model) This problem is studied in Aydin and Ryan (AR) [5]. However, AR utilize a variation of the MNL model which leads to a more complex form of the purchase probabilities.12 In the following, we present similar results to AR [5], but under a more common form of the purchase probabilities, which is a natural extension of (1). (We have verified that the AR results continue to hold under the form of the purchase probabilities that we use here.) Assuming that uj = αj − pj , where αj may be seen as the quality index or the mean reservation price of item j, the purchase probabilities in (1) are now given by e(αi −pi )/µ , i ∈ S, v0 + j∈S e(αj −pj )/µ v 0 , q0 (S, p) = v0 + j∈S e(αj −pj )/µ qi (S, p) =

(20)

where p = (p1 , . . . , p|S|) is the price vector corresponding to items in S, with |S| denoting the cardinality of set S. Assuming infinite inventory levels and no operational costs, the expected profit in (7) reduces to Π(S, p) =

i∈S

12

Πi (S, p) =

λqi (S, p)(pi − ci ) .

(21)

i∈S

Referring back to the MNL formulation in Section 4.1, AR derive the MNL purchase probabilities by assuming that Uj = αj − pj + j , j ∈ S, where j are i.i.d. Gumbel random variables, as we do here, but they assume that U0 = 0 rather than U0 = u0 + 0, where 0 is also a Gumbel random variable as it is commonly done.

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The retailer’s objective of maximizing the expected profit is now given by Π ∗ = Π(S ∗ , p∗) = max max {Π(S, p)} , S⊆Ω p∈ΓS

(22)

where S ∗ is an optimal assortment, p∗ is the corresponding optimal price vector, and ΓS = {(p1 , . . . , p|S|) | p1 > c1 , . . . , p|S| > c|S| }. In this case, the optimal prices (that maximize the expected profit from a given assortment) have a special structure characterized by equal profit margins, as indicated in the following theorem. Theorem 5. Consider an assortment S ⊆ Ω. Then, the optimal prices of any two items i, j ∈ S are characterized by p∗i − ci = p∗j − cj = m∗ . Furthermore, the expected profit from S, Π(S, m), is unimodal in m. In Theorem 5, the expected profit, Π(S, m), is obtained from (21) by setting pi = ci + m, i ∈ S. The intuition behind this theorem is mainly related to the special structure of the MNL model and to the IIA property implying that items are broadly similar. The following lemma shows how the optimal expected profit from a given assortment varies as a function of an item’s parameter. Lemma 2. Consider an assortment S ⊆ Ω. Then, the optimal profit from S is increasing in αi − ci , the “average margin” of item i ∈ S. Lemma 2 is intuitive but it has important implications on the structure of the optimal assortment defined in (22). Specifically, it can be shown that

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the optimal assortment of size k has the k items with the largest values of the average margin, αi − ci . In addition, Lemma 2 also implies that the optimal assortment with no size restriction is Ω. That is, in the absence of operational costs (e.g. inventory), the retailer should offer as much variety as possible. Finally, we note that Hopp and Xu [34] show that the above results continue to hold when the demand follows a logit choice model with random brand effects (a generalization of the MNL model we consider here) and for a risk averse retailer having an exponential utility function.

7 Product Line Inventory and Pricing Decisions for a Given Assortment In this section, we review models that assume that the retailer’s assortment is exogenously determined, and consider the retailer’s problem of determining the prices and inventory levels for the items in the assortment. All these models consider a single-period setting and static choice assumptions, as discussed in Section 5. The works reviewed here can be seen as extensions to the pricesetting newsvendor model (see, for example, Petruzzi and Dada [59]).

7.1 The Aydin and Porteus Model Aydin and Porteus AP [6] consider the inventory and pricing decisions for a given assortment under a multiplicative13 logit demand model within a 13

In a “multiplicative” demand model, demand is of the form D(p) = f (p), where is a random variable and f (p) is a function of the price, p. In an “additive”

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newsvendor setting. In particular, the demand for item i ∈ S is given by Di (S, p) = M qi (S, p)ξi , where M is a positive constant, qi (S, p) is the MNL purchase probability given in (20), and ξi , i ∈ S, are i.i.d. random variables with positive support and with cumulative distribution function Fξi (.), which is IFR14. That is, AP consider the demand Di (S, p) as a random perturbation of the expected demand given by the product of the expected market size, M , and the purchase probability, qi (S, p). AP write the expected profit at the optimal inventory levels as Π(S, p) =

|S|

Πi (S, p) =

i=1

where Πi (S, p) =

yi∗ (S,p) 0

|S|

pi

i=1

yi∗ (S,p)

0

xfDi (S, p, x)dx ,

(23)

xfDi (S, p, x)dx is the expected profit from item

i ∈ S, yi∗ (S, p) is the optimal inventory level for item i ∈ S, i.e., yi∗ (S, p) = −1 −1 FD (S, p, 1 − ci /pi ), with FD (S, p, ·) and fDi (S, p, ·) respectively denoting i i

the cumulative distribution function and the density function of Di (S, p). The objective is then to find the optimal prices, Π ∗ = Π(S, p∗ ) = max Π(S, p) , p∈ΓS

(24)

where ΓS is as defined in Section 6.2. AP make the following assumption, which guarantees that the optimal price vector is an internal point solution. demand model, demand is of the form D(p) = f (p)+. In a “mixed multiplicativeadditive” demand model, D(p) = g(p) + f (p), where g(p) is also a function of the price. 14

F (.) is IFR if its failure rate f (x)/(1 − F (x)) is increasing in x, where f (x) is the corresponding density function.

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(A2): The expected profit, Π(S, p), is increasing in pi , i ∈ S, at pi = ci , and is decreasing in pi as pi → ∞. Under (A2), AP [6] derive the following result on the structure of Π(S, p) in (23). Theorem 6. There exists a unique price vector p∗ that satisfies

∂Π(S,p) ∂pi

=

0, i ∈ S. Furthermore, p∗ maximizes Π(S, p). Theorem 6 states that the expected profit is well-behaved in the sense that the optimal prices are the unique solution to the first-order optimality conditions. Note, however, that Theorem 6 does not imply that the expected profit is jointly quasiconcave in the prices. In fact, AP present numerical examples indicating that this is not necessarily the case. AP also develop the following comparative statics results on the behavior of the optimal prices as a function of the unit costs. Lemma 3. The optimal price of item i ∈ S, p∗i , is: (i) Increasing in item i’s own unit cost, ci ; (ii) Decreasing in the unit cost of item j, cj , j = i, j ∈ S. Lemma 3 is intuitive. Increasing the unit cost of an item increases its own price and decreases the prices of other items in the assortment.

7.2 The Bish and Maddah Pricing Model Bish and Maddah (BM) [12] study the retailer’s pricing decision of a product line within the setting of their similar items model presented in Section

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5.2. Recall from (10) that the expected profit at optimal inventory levels in

[12] levels is approximated by Π(p, k) = k(p − c) λq(p, k) − a pc λq(p, k) . In this section, we discuss the analytical properties of the optimal price, p∗k = arg maxp>c Π(p, k), assuming that the assortment size, k, is fixed. The objective is to understand the structure of the optimal pricing decision and its interaction with inventory and variety decisions within a simple framework. BM make the following assumption, which ensures that the retailer will not be better off by not selling anything. (A3): The expected profit, Π(p, k), is increasing in p at p = c, that is, ∂Π(p,k) > 0, or equivalently, λ > a2 k + v0 e−(α−c)/µ . ∂p p=c

(Note that Assumption (A3) is similar to assumption (A2) of Aydin and Porteus [6].) BM observe, numerically, that, under (A3), the expected profit, Π(p, k), is well behaved (pseudoconcave) in the price, p, for reasonably low prices where Π(p, k) > 0.15 However, this result did not lend itself to analytical proof.16 The following lemma is the main result in [12] on the structure of the expected profit as a function of p.

15

BM also prove that under (A3) there exists pk < ∞ such that Π(p, k) > 0 for p ∈ (c, pk ), and Π(p, k) < 0 for p > pk .

16

This is an indication of the complexity of the joint pricing and inventory problem, even within a simple setting as in [12]. One main reason behind this complexity is the demand model adopted in [12], which may be seen as mixed-multiplicative additive, see footnote 9. This is further discussed at the end of this section.

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Lemma 4. The expected profit Π(p, k) is pseudoconcave in p ≥ c in the region where Π(p, k) > 0 and q(p, k) ≥ 1/3. Lemma 4 shows that pseudoconcavity holds under a somewhat weak condition (q(p, k) ≥ 1/3), which is likely to be satisfied if the assortment size k is small. Bish and Maddah [12] develop the following comparative statics results on the behavior of the optimal price function of the assortment size and the store volume. Lemma 5. If p∗k ≥

3 2c

for some k ∈ Z+ , then p∗k is increasing in k for all

k ≥ k. In particular, if p∗1 ≥ 32 c, then p∗k is increasing in k for all k ∈ Z+ . Lemma 5 states that if the optimal price is relatively high (p∗k ≥ 3c/2) at a given variety level, then increasing variety will also increase the optimal price. The condition, p∗k ≥ 3c/2, can be seen as an indicator that consumers tolerate high prices so that the retailer is induced to increase the price if the breadth of the assortment is enlarged. This could be the case of a store located in an upscale neighborhood. Numerical results in [12] suggest that in environments where consumers do not tolerate high prices (with p∗k < 3c/2), the retailer may expand the breadth of the assortment, while decreasing the price.

Lemma 6. If p∗k > 2c (p∗k < 2c) at some λ = λ0 , then p∗k is decreasing (increasing) in λ for all λ with limλ→∞ p∗k (λ) ≥ 2c (limλ→∞ p∗k (λ) ≤ 2c). Lemma 6 asserts that the optimal price as a function of the expected store volume moves in one direction only, all else held constant. This might be the

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case of an expensive store with a low volume where the price decreases as a result of an increase in volume, or the case of a low-price high-volume store where the price increases with volume. The condition, p∗k > 2c, may also be seen as an indicator of the nature of the marketplace and the store. BM also compare the optimal “riskless” price when assuming infinite inventory levels, p0k , to the optimal “risky” price, p∗k . This is an important question in the literature on single item joint pricing and inventory problem. For an additive demand function, Mills [53] finds that p∗ ≤ p0 , where p∗ and p0 respectively denote the risky and riskless price. On the other hand, for the multiplicative demand case, Karlin and Carr [37] prove that p∗ ≥ p0 . In the case of BM model, the demand is mixed multiplicative-additive, with variance

(λq(p, k)) decreasing in p, and demand coefficient of variation (1/ λq(p, k)) increasing in p. For such cases, Petruzzi and Dada [59] conjecture, on the relationship between p∗ and p0 , that “either the price dependency of demand variance or of demand coefficient of variation will take precedence, thereby ensuring a determinable direction for the relationship.” The following result confirms Petruzzi and Dada’s conjecture and suggests the criterion, p0k < 2c, with which to determine the direction of the relationship. Lemma 7. If p0k < 2c, then p∗k ≤ p0k . Otherwise, p∗k ≥ p0k .

7.3 The Cattani et al. Model Cattani et al. (CDS) [14] consider the pricing and inventory (capacity) decisions for two substitutable products, which can be produced either by a single

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flexible resource or by two dedicated resources. Similar to [12], they assume that demand is generated from customers arriving according to a Poisson process and making purchase decisions according to an MNL choice model, within a newsvendor inventory setting. Their expected profit function is similar to that of VRM [61] in (7), but the purchase probabilities are replaced by those in (20). The main finding of CDS relevant to our discussion here is a is a heuristic for setting the prices and inventory levels of a given assortment, referred to as “cooperative tattonement” (CT). The idea of the CT heuristic is to develop a near-optimal solution by iterating between a marketing model, which sets prices, and a production model, which determines the inventory cost. In the following, we illustrate the application of the CT method for an assortment S. CT Heuristic Step 0: Set k = 0 and c0i = ci , i ∈ S. Step 1: Starting with the AR marketing model in Section 6.2, obtain initial estimates for the optimal prices, pk = arg maxp

i∈S

λ(pi − cki )qi (S, p).17

Set qik (S) = qi (S, pk ). Step 2: Find the expected profit at the optimal inventory levels using the VRM operations model in Section 5.1 as Π(S) = 17

i∈S

λqik (S)(pki − ci ) −

This problem can be solved by a single variable search since optimal prices have equal profit margins as shown in Section 6.2.

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pki θi (pki ) λqik (S), where θi (pi ) ≡ φ(Φ−1 (1 − ci /pi )). Set ck+1 = cki + i k

k

pi θi (pi ) 18 √ . k λqi (S)

Step 3: If |ck+1 − cki | < δ ∀ i ∈ S, then stop, set the prices equal to pCT = i pk , and find the corresponding optimal inventory levels yiCT = λqik (S) +

k Φ−1 (1 − cki /pCT i ) λqi (S), i ∈ S (as in (6)). Otherwise, set k = k + 1 and go to Step 1. In Step 3 of CT, δ is a small number determining a stopping rule for the heuristic, and pCT and yiCT , i ∈ S, are the near-optimal prices and inventory levels developed by CT. CDS report a good performance of the CT procedure in obtaining near-optimal solutions. Based on five iterations of the CT, the expected profit from CT is found to be within 0.1% of the optimal expected profit in many cases, all involving two-item assortments. In Section 8, we discuss another heuristic, the Equal Margins Heuristic, that finds a near-optimal assortment in addition to prices and inventories, and that is also found to perform quite well.

8 Joint Variety, Pricing, and Inventory Decisions Finally, we consider a retailer that jointly sets the three key decisions for her product line: variety, pricing, and inventory. This is a realistic integrative setting, which is sought to enhance retailers’ profitability. To the best of our 18

Here ck+1 is the equivalent unit cost of a marketing model that yields the same i expected profit as the operations model.

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knowledge, the only work that addresses this joint setting is the recent paper by Maddah and Bish (MB) [45] which consider this joint decision making under an MNL choice model within a newsvendor setting. MB also adopt the static choice assumptions discussed in Section 5. We devote the remainder of this section to the MB model [45]. The MB model can be seen as an extension to the VRM [61] model by endogenizing the prices and to the AR [5] model by considering finite inventory levels. For ease of exposition, MB develop their results for a special case of the demand model utilized in VRM [61] by considering a demand standard deviation equal to σ(λqi (S, p))β , with σ = 1 and β = 1/2. This covers the important case of demand generated from Poisson arrivals. However, MB results hold for any value of σ > 0 and 0 ≤ β ≤ 1. With this demand model, given an assortment S ⊆ Ω, the optimal inventory level for item i ∈ S, yi∗ (S, p), and the expected profit from S at optimal inventory levels, Π(S, p), in (6) and (7) reduce to

yi∗ (S, p) = λqi (S, p) + Φ−1 (1 − ci /pi ) λqi (S, p), i ∈ S , Π(S, p) =

i∈S

=

Πi (S, p)

(25) (26)

λqi (S, p)(pi − ci ) − pi θi (pi ) λqi (S, p) ,

i∈S

where qi (S, p) is given by (20).The retailer’s objective of maximizing the expected profit is then given by Π ∗ = Π(S ∗ , p∗) = max max {Π(S, p)} . S⊆Ω p∈ΓS

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(27)

46


MB make the following assumption, which guarantees that the retailer will not be better off not selling anything (hence, the optimal expected profit is positive). (A4): Let i ≡ arg maxj∈Ω {αj − cj }. There exists p0i > ci such that Π({i}, p0i ) > 0. Under (A4), MB develop the following result on the behavior of the expected profit as a function of the mean reservation price of an item. Lemma 8. Consider an assortment S ⊆ Ω. Assume that prices of items in S are fixed at some price vector p. Then, the expected profit from S, Π(S, p, αi), is strictly pseudoconvex in αi , the mean reservation price of item i. Lemma 8 extends the result of VRM in Lemma 1. The intuition behind this lemma is related to the trade-off between the sales revenue and the inventory cost discussed in Section 5.1. MB also investigate the behavior of the expected profit as a function of the unit cost of an item, ci , i ∈ S. They find that decreasing the unit cost of an item in an optimal assortment increases the expected profit (an intuitive result). This together with Lemma 8 leads to MB’s main “dominance result” presented below. Lemma 9. Consider two items i, k ∈ Ω such that αi ≤ αk and ci ≥ ck , with at least one of the two inequalities being strict, that is, item k “dominates” item i. Then, an optimal assortment cannot contain item i and not contain item k.

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Observe that the number of assortments to be considered in the search for an optimal assortment can be significantly reduced if there are a few dominance relations like the one described in Lemma 9. For example, with exactly one pair of items satisfying the dominance relationship given in the lemma, the number of assortments to be considered is reduced by more than 25%. In addition, Lemma 9 allows the development of the structure of an optimal assortment in a special case, as stated in the following theorem. Theorem 7. Assume that the items in Ω are such that α1 ≥ α2 ≥ . . . αn , and c1 ≤ c2 ≤ . . . cn . Then, an optimal assortment is S ∗ = {1, 2, . . ., k}, for some k ≤ n. The most important situation where Theorem 7 applies is the case in which all items in Ω have the same unit cost. In this case, the items may be seen as horizontally differentiated in the sense of broadly having equivalent qualities (this holds, for example, for a product line composed of different colors or flavors of the same variant). In this case, Theorem 7 implies that an optimal assortment has the k, k ≤ n, items with the largest values of αi . Theorem 7 extends the result of van Ryzin and Mahajan in Theorem 1 to a product line with items having distinct endogenous prices. In addition to its application to the important case of horizontally differentiated items, Theorem 7 provides motivation for an efficient heuristic discussed below. Theorem 7 greatly simplifies the search for an optimal assortment in the special case where it applies, as it suffices to consider only n assortments

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out of (2n − 1) possible assortments. For cases where Theorem 7 does not apply, one may expect an optimal assortment to have the k, k ≤ n, items with the largest average margins αi − ci , similar to the structure of the optimal assortment in Theorem 7 and to the result discussed in Section 6.2 under the assumption of infinite inventory levels. However, MB report finding several counter-examples of optimal assortments not having items with the largest average margins, indicating that a result similar to Theorem 7 does not hold in general. Nevertheless, MB [45] also report that assortments consisting of items with the largest average margins return expected profits that are very close to the optimal expected profit. This last observation is one of the main motivations for the heuristic procedure discussed below, which can be utilized in cases where Theorem 7 does not hold. MB also analyze the structure of optimal prices by exploiting the first order optimality conditions. Lemma 10. Consider an optimal assortment S ∗ ⊆ Ω. Then, the optimal prices of any two items i, j ∈ S ∗ satisfy the following equation: −1 θi (p∗ θ (p∗ )[1−(ci /2µ)]−(ci /p∗ (1−ci /p∗ i) i )Φ i) √ √ + i i − ci ) 1 − 2 λqi (S ∗ ,p∗ ) λqi (S ∗ ,p∗ ) ∗ −1 θj (p∗ θj (p∗ (1−cj /p∗ j) j )[1−(cj /2µ)]−(cj /pj )Φ j) √ + = µ1 (p∗j − cj ) 1 − √ . ∗ ∗ ∗ ∗

1 ∗ µ (pi

2

λqj (S ,p )

λqj (S ,p )

The main insight from Lemma 10 is that in an optimal assortment where the mean item demands, λqi (S ∗ , p∗), are reasonably large, the optimal profit margins are approximately equal. That is, (p∗i − ci ) ≈ (p∗j − cj ), i, j ∈ S ∗ . Thus, the equal margins result in the riskless case (which assumes infinite

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inventory levels) in Theorem 5 continues to hold approximately under finite inventories. This observation holds in the numerical results presented in [45]. A sample of these results is presented below.

8.1 Numerical Results

MB develop numerical results for three-item and four-item assortments. A sample of these results is shown in Tables 1 and 2. In addition to the optimal assortment, S ∗ , and its expected profit, Π ∗ , Tables 1 and 2 report the optimal profit margins, m∗i ≡ p∗i − ci , i ∈ Ω, the optimal inventory levels, yi∗ , i ∈ Ω, and the no-purchase probability, q0∗ = q0∗ (S ∗ , p∗) (i.e., the fraction of customers who leave the store empty-handed). Note that an infinite profit margin indicates that the item is not included in S ∗ . The second column of Tables 1 and 2 shows the modification from the “base case,” described in the table heading. Each modification involves changing the parameters given in the second column of the table only, while keeping other parameters at their base values.

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Table 1. Optimal Solution (n = 3). Base case: λ = 100, α1 = 11, α2 = 10, α3 = 9, c1 = 9, c2 = 8, c3 = 7, v0 = 1, µ = 1 Case Modification

m∗1

y1∗

m∗2

y2∗

m∗3

y3∗

q0∗

S∗

1

None

2.572 17.405 2.567 17.769 2.563 18.249 0.370

{1, 2, 3}

2

α1 = 12

2.815 35.796 2.946 10.794 2.943 11.215 0.335

{1, 2, 3} 141.744

3

c1 = 8

2.816 36.271 2.952 10.750 2.940 11.215 0.336

{1, 2, 3} 142.686

4

α1 = 12.8

3.114 56.378 ∞

0

3.411 6.324

0.310

{1, 3}

176.263

5

c1 = 7.2

3.118 57.401 ∞

0

3.429 6.239

0.311

{1, 3}

178.694

6

α1 = 15

4.682 75.280 ∞

0

∞

0

0.211

{1}

324.703

7

c1 = 5

4.693 78.356 ∞

0

∞

0

0.213

{1}

335.063

8

α1 = 12.85, c2 = 7.98

3.143 57.100 ∞

0

3.461 5.950

0.307

{1, 3}

179.039

9

c1 = 7.15, c2 = 7.98

3.148 58.161 ∞

0

3.480 5.863

0.308

{1, 3}

181.583

Tables 1 and 2 reveal three important insights. 1. Items in S ∗ have approximately equal profit margins, m∗i , i ∈ S ∗ . This finding is not surprising given Lemma 10. 2. An optimal assortment need not have the items with the largest values of αi − ci (which shows that a result similar to Theorem 7 does not hold in general). For example, in Case 8 of Table 1 the two items with the largest values of αi − ci are items 1 and 2, while the optimal assortment contains items 1 and 3 only. A similar observation holds in Case 9. MB [45] also observe, however, that assortments containing items with the largest

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51

Table 2. Optimal Solution (n = 4). Base case: λ = 150, α1 = 26, α2 = 24, α3 = 22, α4 = 20 c1 = 24, c2 = 22, c3 = 20, c4 = 18,v0 = 1, µ = 1 Case Modification m∗1

y1∗

m∗2

y2∗

m∗3

y3∗

m∗4

y4∗

q0∗

S∗

Π∗

1

None

2.700 18.508 2.698 18.790 2.695 19.103 2.691 19.454 0.334 {1, 2, 3, 4} 178.240

2

α4 = 21.4

∞

0

3.168 10.166 3.159 10.467 2.940 64.764 0.312 {2, 3, 4}

230.665

3

c4 = 16.6

∞

0

3.173 10.133 3.164 10.434 2.942 65.155 0.312 {2, 3, 4}

231.566

4

α4 = 21.6

∞

0

∞

0

3.224 8.456

2.974 78.558 0.316 {3, 4}

245.074

5

c4 = 16.4

∞

0

∞

0

3.230 9.842

2.976 79.011 0.317 {3, 4}

246.221

6

α4 = 21.8

∞

0

∞

0

∞

0

3.034 91.661 0.317 {4}

262.334

7

c1 = 16.2

∞

0

∞

0

∞

0

3.037 92.160 0.318 {4}

263.759

8

α4 = 21.4,

∞

0

3.168 10.166 3.159 10.467 2.940 64.764 0.312 {2, 3, 4}

230.665

∞

0

3.168 10.166 3.159 10.467 2.940 64.764 0.312 {2, 3, 4}

230.665

α1 = 26.02 9

α4 = 21.4, c1 = 23.98

values of αi − ci yield expected profits that are very close to the optimal expected profits. 3. Reducing an item’s unit cost by a certain amount is slightly more profitable than increasing its mean reservation price by the same amount. For example, while item 1 has the same value of α1 − c1 = 3 in Cases 2 and 3 of Table 1 (with items 2 and 3 having the same parameters in both cases), Case 3 (where c1 is smaller) yields a slightly higher expected profit than Case 2. The same observation is valid when comparing Cases 4 and 5, or Cases 6 and 7, or Cases 8 and 9.

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8.2 EMH Heuristic

Based on the properties of an optimal assortment discussed above, Maddah and Bish [45] develop the “Equal Margins Heuristic (EMH),” presented below. Equal Margins Heuristic (EMH) Step 0: Eliminate any item i ∈ Ω that cannot return a positive expected profit (assuming equal profit margins, with a common margin m). That is, (αi −ci −m)/µ eliminate any item such that m λ v e+e(α −(ci +m)θi (ci +m) ≤ 0 i −ci −m)/µ 0

for all m ∈ (0, ∞). Let Ω ⊆ Ω, with cardinality n ≤ n, be the set of items which are not eliminated in this step. Step 1: Sort items in Ω in nonincreasing order of αi −ci . Break ties according to the smaller value of ci . Number items in Ω such that item 1 is the item with the largest αi − ci , item 2 is the item with the second largest αi − ci , and so on. Step 2: Assuming equal profit margins, find the common margin, mk , that would yield the highest profit from Sk = {1, 2, . . ., k}, k = 1, 2, . . . n . That is, find ΠkH (Sk , mk ) = max m>0

where qi (Sk , m) =

mλqi (Sk , m) − (ci + m)θi (ci + m) λqi (Sk , m) ,

i∈Sk

v0 +

(αi −ci −m)/µ e . e(αj −cj −m)/µ

j∈Sk

Step 3: Find k H such that k H = arg max ΠkH (Sk , mk ). Set S H = Sk H , k=1,...,˜ n

mH = mk H , and Π H = ΠkHH (Sk H , mk H ). Set prices and inventory levels of items in S H to

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H pH i = ci + m ,

yiH = λqi (S H , mH ) + Φ−1 (1 − ci /(ci + mH )) λqi (S H , mH ) . In the EMH Algorithm, Step 0 is a consequence of a result in [45] aimed at eliminating items with very low demand from consideration. The tie breaking rule in Step 1 is motivated by the observation that decreasing ci is more profitable than increasing αi , as discussed above. The remaining steps then find the most profitable assortment consisting of the k ≤ n items with the largest values of αi − ci , and under the restriction of equal profit margins. The advantage of EMH is that it requires little computational effort relative to the effort required to find the optimal solution: The EMH generates at most n assortments, each requiring a single variable search (over the common margin), while determining the optimal solution requires generating up to 2n − 1 assortments (when Theorem 7 does not hold), with a multi-variable search (over the price vector) for each assortment. Moreover, the EMH generates solutions that are very close to the optimal solution, with the ratio of heuristic expected profit to the optimal expected profit, Π H /Π ∗ , being larger than 99.5% in all tested cases; see the last column in Tables 3 and 4 which report this ratio for the examples in Tables 1 and 2 respectively. These results indicate an excellent performance for the EMH.

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Table 3. EMH Solution (n = 3). Base case: λ = 100, α1 = 11, α2 = 10, α3 = 9, c1 = 9, c2 = 8, c3 = 7, v0 = 1, µ = 1 Case Modification

mH

y1H

y2H

y3H

1

None

2.567 17.486 17.801 18.163 0.370 {1, 2, 3} 116.313 100.00%

2

α1 = 12

2.864 33.990 11.728 12.028 0.335 {1, 2, 3} 141.647 99.93%

3

c1 = 8

2.867 34.388 11.718 12.018 0.335 {1, 2, 3} 142.582 99.93 %

4

α1 = 12.80

3.147 54.340 0

8.254 0.309 {1, 3}

176.035 99.87%

5

c1 = 7.20

3.153 55.272 0

8.241 0.310 {1, 3}

178.447 99.86%

6

α1 = 15

4.682 75.282 0

0

0.211 {1}

324.703 100.00%

7

c1 = 5

4.693 78.356 0

0

0.213 {1}

335.066 100.00%

8

α1 = 12.85, c2 = 7.98

3.177 54.845 7.838 0

0.306 {1, 2}

178.332 99.61%

9

c1 = 7.15, c2 = 7.98

3.088 63.888 0

0.318 {1}

181.072 99.72%

0

q0H

SH

ΠH

Π H /Π ∗

9 Current Practice While a few retailers have started integrating their assortment, pricing, and inventory decisions for their product lines (see Section 1), most retailers still make these decisions separately. However, there has been an increase in interest in this type of integrated decision-making both in the academic community and the retailing industry. This interest is due to (i) advances in information technology, which makes it possible to collect detailed data on store performance (e.g., sales, inventory levels) and (ii) progress in academic research at the interface of marketing, economics, and OM. In addition to a sequential approach to these decisions, most retailers still use heuristic, rule-of-thumb

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Table 4. EMH Solution (n = 4). Base case: λ = 150, α1 = 26, α2 = 24, α3 = 22, α4 = 20, c1 = 24, c2 = 22, c3 = 20, c4 = 18,v0 = 1, µ = 1 Case Modification mH

y1H

y2H

y3H

y4H

q0H

SH

ΠH

Π H /Π ∗

1

None

2.964 18.600 18.825 19.074 19.354 0.334 {1, 2, 3, 4} 178.240 100.00%

2

α4 = 21.4

3.003 0

10.571 10.768 65.544 0.310 {2, 3, 4}

230.278 99.83%

3

c4 = 16.6

3.005 0

12.221 12.430 60.724 0.311 {2, 3, 4}

231.168 99.83%

4

α4 = 21.6

3.008 0

0

17.257 75.616 0.315 {3, 4}

244.792 99.88%

5

c4 = 16.4

3.010 0

0

12.582 76.027 0.316 {3, 4}

245.931 99.88%

6

α4 = 21.8

3.034 0

0

0

91.661 0.285 {4}

262.334 100.00%

7

c4 = 16.2

3.037 0

0

0

92.160 0.321 {4}

263.759 100.00%

8

α4 = 21.4,

3.004 12.002 0

12.400 60.226 0.310 {1, 3, 4}

239.013 99.72%

3.004 12.004 0

12.400 60.225 0.310 {1, 3, 4}

230.018 99.72%

α1 = 26.02 9

α4 = 21.4, c1 = 23.98

approaches rather than optimization-based methodology to make these decisions. This is because of the complexities involved (e.g., assessing competition effects, developing demand and cost functions, managing a large number of items, difficulties in obtaining reliable data, etc.). In this section, we briefly summarize our observations on pricing, inventory, and variety management based on our experience with Hannaford, a large chain of grocery stores in the North East. In Hannaford, currently variety, pricing, and inventory management are separated in most stores. However, some stores, with sophisticated inventory

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systems, are in the process of incorporating their inventory data into the pricing decision. We first discuss the pricing practice. Pricing strategy in a supermarket can be very challenging. The main complexity is due to the size of the problem: There are thousands of items to price and the number of new items continues to increase. Product lines also have a large number of items, which may be available in different forms in the supermarket, such as frozen biscuits, refrigerated biscuits, biscuit mix, and bakery fresh biscuits. The large size of the problem makes the decision set confusing to both consumers and retailers. On one hand, the pricing strategy needs to be tailored to each individual product. On the other hand, it should be scalable such that it is a manageable task. Furthermore, there are additional complexities not considered in any of the stylized models discussed above. One such consideration is that private-branded products (i.e., the store brand) need to represent a better value than the national brand, but not so discounted that quality is questionable. Another major consideration is competition coming from the multiple channels which carry similar products, i.e., super-centers, dollar stores, other supermarkets, and C-stores. As a result, pricing strategy in most retailers is done in a rule-of-thumb manner and different approaches are used to reduce the size of the problem. For example, it is typical for retailers to utilize more sophisticated pricing strategies only for a small portion (100-300) of their items; these include the best selling and price sensitive items. These items are divided into groups (e.g., meat, bread, milk), and competitive checks information (i.e., the prices

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of these items at other stores considered to be in direct competition with the retailer) as well as other available data (e.g., opening of a new competitor store, weather condition, supply interruptions) are compiled to determine the prices of items in each group. Items which are not considered best sellers are typically priced according to a simple rule (e.g., by applying a constant percentage mark-up or by using a simple constant price elasticity model). Other pricing practices are as follows. Line pricing, which refers to the process of grouping hundreds of SKU’s into one master item and charging a common price for the group, is another means of managing like items. For example, all Pepsi 2-Liter products may be priced the same regardless of their individual costs. Private label retails can be managed simply by using the national brand as a benchmark. For example, a retailer may decide to price its coffee at 10% below the national brand. Fresh foods (e.g., produce, deli, and bakery) have the additional consideration of “shrink” (i.e., loss of inventory mainly due to deterioration and spoilage). Some supermarkets use shrink estimates in pricing. However, it has been our experience that it is quite difficult to predict shrink values especially at the individual-item level. As mentioned above, pricing and inventory decisions are typically managed separately. However, some stores with sophisticated inventory systems that track inventory levels continuously and present useful statistics are attempting to leverage inventory data in pricing. In addition, these inventory management systems can help the retailer reduce supply chain costs, which can then be reflected in the price. In each store, there are vendor-managed items, such as

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beer and pharmacy products, for which the vendor is responsible for inventory management. Inventory decisions for other items are managed in a heuristic manner, depending on whether they are fast or slow movers and whether they have short or long lead-times. Regarding variety management, this decision is separated from pricing and inventory management. Store assortments are typically determined based on the store size and layout (there are few formats that are adopted by stores in the chain). The store format in turn is determined based on the demographical composition of the area where the store is located as well as the type of competitor stores available.

10 Conclusions In this chapter, we review recent works on pricing, variety and inventory decisions for a retailer’s product line composed of substitutable items. This stream of literature contributes to both the theory and the practice of Operations Management and Marketing in two aspects. First, demand models are developed in a natural (and realistic) way from consumer choice models, which are based on the classical principal of utility maximization (a widely accepted principal of human behavior). Second, decisions pertaining to traditionally separated departments in a firm (e.g., inventory and variety, inventory and pricing) are integrated and optimized jointly, an approach that can eliminate inefficiencies due to lack of coordination between marketing and operations.

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We must note, however, that the current retail practice, for the most part, has not yet caught up with this integrative approach that utilizes sophisticated demand models. Nevertheless, the wide spread of information systems and the recent progress made in the academic literature are fueling an interest in this approach. The area of research considered in this chapter is relatively new and there remain many questions to be exploited along the lines discussed here. One area for further research is to use refined (more applicable) consumer choice models. Most of the reviewed works utilize the popular MNL logit model. However, due to the IIA property (see Section 4), the logit model implicitly assumes that items in a product line are broadly similar. This may not apply in many practical situations. We believe that the remedy is to adopt a nested logit choice (NMNL) model, which, as discussed in Section 4, overcomes the shortcoming of the IIA while maintaining a reasonable level of analytical tractability. We have purposely presented the NMNL in detail in Section 4, although none of the relevant recent works (to our knowledge) utilize it except for the paper by Hopp and Xu [35]. However, Hopp and Xu [35] utilize NMNL in a competitive setting with the objective of understanding the effect of modular design on variety and pricing competition within a stylized duopoly environment. They do not address the issue of utilizing the NMNL as a refined choice model of a single retailer’s product line, which we believe is an important direction for future research.

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Another important area for further research (that would make this line of research more applicable) is developing effective decision tools under fairly general assumptions. For example, most of the reviewed works assume a “static choice” setting where consumers do not substitute another item for their preferred item in the event of a stock out. This is a restrictive assumption. While the limited results discussed in Section 5 indicate that solutions based on static choice assumptions do not perform too badly in handling the dynamic situation, more research is needed in this direction. Specifically, numerically efficient heuristics that can generate good solutions for the dynamic situation and that can be used in practice are needed. A promising area for future research involves studying the coordination of retailers’ and manufacturers’ pricing, variety, and inventory decisions in supply chains. The problem of coordinating order quantities has been widely studied. It would be of interest to investigate how order quantities can be coordinated jointly with pricing and variety. As discussed in Section 3, some research has already started in this direction, but more research is needed. Finally, further investigation of the analytical properties of the existing models is also worthwhile. For example, the structural properties of the optimal assortment and prices for the joint problem studied by Maddah and Bish [45] deserve more study.

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