Pricing, Variety, and Inventory Decisions for Product Lines of Substitutable Items Bacel Maddah 1 , Ebru K. Bish 2 , and Brenda Munroe 3 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,
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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
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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4 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 5
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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6 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 7
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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8 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 9
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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10 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 11
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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12 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 13
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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14 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 15
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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16 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 17
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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18 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 19
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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20 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 21
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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22 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 23
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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24 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 25
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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26 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 27
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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28 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 29
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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30 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 31
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
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 37
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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38 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 39
(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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40 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 41
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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42 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 43
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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44 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 45
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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46 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 47
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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48 Maddah, Bish, and Munroe
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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Pricing, Variety, and Inventory Decisions of Substitutable Items 49
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