pricing, variety, and inventory decisions for product ...bm05/research/retail.pdf · in section 7,...

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]

2 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]

3 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,

2 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, assort-

ment (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 mar-

keting/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 suc-

cessful business practice for several retailers. For example, JCPenney received

the “Fusion Award” in supply chain management for “its innovation in in-

tegrating upstream to merchandising and allocation systems and then down-

stream 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 3

pricing are all linked and optimized together” (Frantz [23]). Canadian retailer

Northern Group managed to get out of an unprofitable situation by imple-

menting 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 de-

cisions 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 op-

timization 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 pric-

ing 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 re-

tailers (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 Bu-

4 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 num-

bers 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 max-

imization (see, for example, Anderson and de Palma [2], Ben-Akiva and Ler-

man [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 empty-

handed 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 differenti-

ated products (see, for example, Hotelling [36], Lancaster [41], and McFadden

[50]). The Multinomial Logit Choice (MNL) model is among the most popu-

lar consumer choice models (see, for example, Anderson and de Palma [2] and

Ben-Akiva and Lerman [10]). Interestingly, the MNL has its roots in Mathe-

matical 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 re-

views). A typical approach is to utilize deterministic estimates of utilities for

each consumer segment and formulate the problem as a mixed integer math-

ematical 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 lit-

erature 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 inven-

tory 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 (as-

sortment), pricing, and production planning of their product lines (in terms

of which items to produce, how to price them, and in what quantities to pro-

duce), 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 prob-

lem can, with certain modifications, be applied to product design and pro-

duction planning problems in manufacturing settings. (In addition, some of

the reviewed works, e.g., Cattani et al. [14] and Hopp and Xu [34], are pre-

sented within a manufacturing framework; see also Alptekinoglu and Corbett

[1] and Hopp and Xu [35] for further work in manufacturing settings.) In par-

ticular, 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 manufactur-

ers’ 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 struc-

ture for a manufacturing firm may involve additional terms not considered

here such as product development costs (e.g., costs related to product de-

velopment, 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 inven-

tory 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 manufactur-

ers. 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 cus-

tomers 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 ac-

tivities 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 ca-

pacity 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 familiarwith 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 vari-

ants 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 cus-

tomer 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 dif-

ferent shopping occasions. Then, the probability that a consumer buys item

i ∈ S is qi(S) = PrUi = 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+εi, and the utility of the

no-purchase option is U0 = u0 + ε0, where ui is the expected utility for item i,

u0 is the expected utility for the no-purchase option, and εi, 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 inde-

pendent 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 param-

eters 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)qj(S)

=qi(T )qj(T )

=vi

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 chocolate-

based 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 pur-

chase 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 Ben-

Akiva [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.6 This sequence of choice applies in a monopolistic setting where one retailer man-

ages 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 + εik,

and the utility for the no-purchase option given that the customer chooses

nest k is U0k = u0k + ε0k, where εik, 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∪0Ujk

]= µ2ln

∑

j∈Nk∪0vjk

. (3)

The consumer’s utility for nest Nk ∈ N is Uk = Ak +εk, where εk, 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

qk(N ) =eAk/µ1∑nl=1 eAl/µ1

=

(∑j∈Nk∪0 vjk

)µ2/µ1

∑nl=1

(∑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 ) . (5)

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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] represent-

ing 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 con-

sumers 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 loca-

tion). 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 func-

tion.

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 purchas-

ing 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 re-

tailers that sell based on floor models. However, in many realistic situations

these static assumptions will not hold. In such cases, the static models pre-

sented 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 opti-

mal 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 sec-

ondary 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 proba-

bility 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 repre-

sents 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,

y∗i (S), and the expected profit from S at optimal inventory levels, Π(S), can

be written as

y∗i (S) = λqi(S) + Φ−1(1 − c/p)σ(λqi(S))β , i ∈ S , (6)

Π(S) =∑i∈S

Πi(S) =∑i∈S

[λqi(S)(p − c) − pθσ(λqi(S))β

], (7)

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∗) = maxS⊆Ω

Π(S) , (8)

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 as-

sortment 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 conse-

quently 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 popu-

lar 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 asymptotic7 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 addi-

tional 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) , (9)

Π(p, k) = k[λq(p, k)(p − c) − pθ(p)

√λq(p, k)

], (10)

where q(p, k) = e(α−p)/µ

v0+ke(α−p)/µ 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 approxi-

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

c

p

√λq(p, k)

]. (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,

∂Π(p,k)∂k

∣∣∣k=1

> 0, or equivalently, λ > a2c2

p2 (1 + v0e−(α−p)/µ)

(1 + e(α−p)/µ

2v0

)2

.

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 uni-

modal in k.

Let k∗p ≡ arg maxk Π(p, k). Theorem 2 states that the expected profit increases

with variety (k) up to k = k∗p. For k > k∗

p, adding more items to the product

line will only diminish the expected profit. Thus, Theorem 2 implies that k∗p <

∞ (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, k∗p, as done in the following

theorem.

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Theorem 3. The optimal assortment size k∗p 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) > 12 (equivalently, u0 < α − p), then the optimal as-

sortment size k∗p 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 “con-

sumer 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) =∑i∈S

[λqi(S)(p − c) − h(qi(S))

], (13)

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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 “no-

search” model in their formulation of the purchase probabilities. In the “in-

dependent 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 assort-

ment” 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 de-

rived as follows. Similar to VRM [61], the utility for items in S and the

no-purchase option are given by Ui = ui + εi, i ∈ S ∪ 0, where εi are

i.i.d. Gumbel random variables with mean zero and scale factor µ. The con-

sumer’s expected search utility is Ur = ur + εr, where ur is the mean utility

of the search option and εr 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 con-

sumer 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, qsii (S), are derived as a function of

the no-search purchase probabilities, qi(S) in (1), as

qsii (S) = qi(S)(1 − H(U, S)) , i ∈ S, qsi

0 (S) = 1 −∑i∈S

qsii (S) , (14)

where H(U, S) = e−(v0+

∑j∈S

vj

)e−(U/µ+γ)

. (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 Umax = 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, qsoi (S), are derived as a function of the no-search purchase

probabilities, qi(S), as

qsoi (S) = qi(S)(1 − H(U(S), S)) , i ∈ S, qso

0 (S) = 1 −∑i∈S

qsoi (S) , (15)

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 fol-

lows. 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 pur-

chases 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 , b+j ], where

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b−j = maxbj −L, (bj + bj−1)/2 and b+j = minbj +L, (bj + bj+1)/2. Finally,

the purchase probability of item j is given by

qj(b) = Prb−j ≤ X ≤ b+j = FX(b+

j ) − FX(b−j ) ,

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∑

i=1

[λqi(b)(p − c) − pθ

√λqi(b)

]−nf . (16)

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.

maximizenc∑i=1

n∑j=1

(pj − cj)xij −n∑

j=1

fjzj

subject ton∑

k=1

(αik − pk)xik ≥ (αij − pj)zj , i = 1, . . . nc, j = 1, . . . n (17)

n∑k=1

(αik − pk)xik ≥ 0 , i = 1, . . .nc (18)

n∑j=0

xij = 1 , i = 1, . . . nc (19)

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

qi(S, p) =e(αi−pi)/µ

v0 +∑

j∈S e(αj−pj)/µ, i ∈ S, (20)

q0(S, p) =v0

v0 +∑

j∈S e(αj−pj)/µ,

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

Πi(S, p) =∑i∈S

λqi(S, p)(pi − ci) . (21)

12 Referring back to the MNL formulation in Section 4.1, AR derive the MNL pur-

chase probabilities by assuming that Uj = αj − pj + εj , j ∈ S, where εj are

i.i.d. Gumbel random variables, as we do here, but they assume that U0 = 0

rather than U0 = u0 + ε0, where ε0 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∗) = maxS⊆Ω

maxp∈ΓS

Π(S, p) , (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 con-

tinue 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 price-

setting 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 a13 In a “multiplicative” demand model, demand is of the form D(p) = f(p)ε, where

ε 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) = Mqi(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|∑i=1

Πi(S, p) =|S|∑i=1

pi

∫ y∗i (S,p)

0

xfDi (S, p, x)dx , (23)

where Πi(S, p) =∫ y∗

i (S,p)

0xfDi (S, p, x)dx is the expected profit from item

i ∈ S, y∗i (S, p) is the optimal inventory level for item i ∈ S, i.e., y∗i (S, p) =

F−1Di

(S, p, 1 − ci/pi), with F−1Di

(S, p, ·) and fDi (S, p, ·) respectively denoting

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∗) = maxp∈ΓS

Π(S, p) , (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)+ε. In a “mixed multiplicative-

additive” demand model, D(p) = g(p) + f(p)ε, 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 prod-

uct 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 c

p

√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)∂p

∣∣∣p=c

> 0, or equivalently, λ > a2(k + v0e

−(α−c)/µ).

(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 ≥ 32 c for some k ∈ Z+, then p∗k is increasing in k for all

k ≥ k. In particular, if p∗1 ≥ 32c, 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 in-

ventory 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 re-

lationship 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 ≤ p0

k. Otherwise, p∗k ≥ p0k.

7.3 The Cattani et al. Model

Cattani et al. (CDS) [14] consider the pricing and inventory (capacity) deci-

sions 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 sim-

ilar 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−ck

i )qi(S, p).17

Set qki (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) =∑

i∈S λqki (S)(pk

i − ci)−

17 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(pk

i )√

λqki (S), where θi(pi) ≡ φ(Φ−1(1 − ci/pi)). Set ck+1

i = cki +

pki θi(p

ki )√

λqki(S)

.18

Step 3: If |ck+1i − ck

i | < δ ∀ i ∈ S, then stop, set the prices equal to pCT =

pk, and find the corresponding optimal inventory levels yCTi = λqk

i (S) +

Φ−1(1− cki /pCT

i )√

λqki (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 yCTi , 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+1i is the equivalent unit cost of a marketing model that yields the same

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 inven-

tory 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 im-

portant 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, y∗i (S, p), and

the expected profit from S at optimal inventory levels, Π(S, p), in (6) and (7)

reduce to

y∗i (S, p) = λqi(S, p) + Φ−1(1 − ci/pi)√

λqi(S, p), i ∈ S , (25)

Π(S, p) =∑i∈S

Πi(S, p) (26)

=∑i∈S

[λqi(S, p)(pi − ci) − piθi(pi)

√λqi(S, p)

],

where qi(S, p) is given by (20).The retailer’s objective of maximizing the ex-

pected profit is then given by

Π∗ = Π(S∗, p∗) = maxS⊆Ω

maxp∈ΓS

Π(S, p) . (27)

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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∈Ωαj − cj. There exists p0i

> ci such that

Π(i, p0i) > 0.

Under (A4), MB develop the following result on the behavior of the ex-

pected 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:

1µ (p∗i − ci)

(1 − θi(p

∗i )

2√

λqi(S∗,p∗)

)+ θi(p

∗i )[1−(ci/2µ)]−(ci/p∗

i )Φ−1(1−ci/p∗i )√

λqi(S∗,p∗)

= 1µ(p∗j − cj)

(1 − θj(p

∗j )

2√

λqj(S∗,p∗)

)+ θj(p∗

j )[1−(cj/2µ)]−(cj/p∗j )Φ−1(1−cj/p∗

j )√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 opti-

mal 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,

y∗i , i ∈ Ω, and the no-purchase probability, q∗0 = q∗0(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 y∗

1 m∗2 y∗

2 m∗3 y∗

3 q∗0 S∗ Π∗

1 None 2.572 17.405 2.567 17.769 2.563 18.249 0.370 1, 2, 3 116.313

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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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 y∗

1 m∗2 y∗

2 m∗3 y∗

3 m∗4 y∗

4 q∗0 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

α1 = 26.02

9 α4 = 21.4, ∞ 0 3.168 10.166 3.159 10.467 2.940 64.764 0.312 2, 3, 4 230.665

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 prof-

itable 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,

eliminate any item such that m√

λ e(αi−ci−m)/µ

v0+e(αi−ci−m)/µ −(ci+m)θi(ci +m) ≤ 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

ΠHk (Sk, mk) = max

m>0

∑i∈Sk

[mλqi(Sk, m) − (ci + m)θi(ci + m)

√λqi(Sk, m)

],

where qi(Sk, m) = e(αi−ci−m)/µ

v0+∑

j∈Sk

e(αj−cj −m)/µ .

Step 3: Find kH such that kH = arg maxk=1,...,n

ΠHk (Sk, mk). Set SH = SkH ,

mH = mkH , and ΠH = ΠHkH(SkH , mkH ). Set prices and inventory levels

of items in SH to

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pHi = ci + mH ,

yHi = λqi(SH , mH) + Φ−1(1 − ci/(ci + mH))

√λqi(SH , 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 rel-

ative to the effort required to find the optimal solution: The EMH generates

at most n assortments, each requiring a single variable search (over the com-

mon 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 gen-

erates solutions that are very close to the optimal solution, with the ratio of

heuristic expected profit to the optimal expected profit, ΠH/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 yH1 yH

2 yH3 qH

0 SH ΠH ΠH/Π∗

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 0.318 1 181.072 99.72%

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 inter-

est 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 perfor-

mance (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 yH1 yH

2 yH3 yH

4 qH0 SH ΠH ΠH/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%

α1 = 26.02

9 α4 = 21.4, 3.004 12.004 0 12.400 60.225 0.310 1, 3, 4 230.018 99.72%

c1 = 23.98

approaches rather than optimization-based methodology to make these deci-

sions. 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 com-

plexity 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 super-

market, 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 complex-

ities 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 Oper-

ations 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 ap-

ply 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 ex-

cept 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 ef-

fect 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 assump-

tion. 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, nu-

merically 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 opti-

mal assortment and prices for the joint problem studied by Maddah and Bish

[45] deserve more study.

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