OPERATIONS RESEARCH
informs
Vol. 56, No. 5, September–October 2008, pp. 1247–1255
issn 0030-364X eissn 1526-5463 08 5605 1247
®
doi 10.1287/opre.1080.0562
© 2008 INFORMS
Joint Inventory and Pricing Decisions for
an Assortment
Goker Aydin
Department of Industrial and Operations Engineering, The University of Michigan,
Ann Arbor, Michigan 48109, [email protected]
Evan L. Porteus
Graduate School of Business, Stanford University, Stanford, California 94305, [email protected]
We seek optimal inventory levels and prices of multiple products in a given assortment in a newsvendor model (single
period, stochastic demand) under price-based substitution, but not stockout-based substitution. We address a demand model
involving multiplicative uncertainty, motivated by market share models often used in marketing. The pricing problem that
arises is known not to be well behaved in the sense that, in its deterministic version, the objective function is not jointly
quasi-concave in prices. However, we find that the objective function is still reasonably well behaved in the sense that there
is a unique solution to the first-order conditions, and this solution is optimal for our problem.
Subject classifications: inventory/production: multi-item, pricing, stochastic; marketing: choice models, pricing, retailing,
wholesaling.
Area of review: Manufacturing, Service, and Supply Chain Operations.
History: Received October 2003; revisions received December 2005, December 2006; accepted August 2007.
1. Introduction
much attention is the effect of pricing on customer substitution within a set of similar products.” Indeed, even in
our simple single-period setting for an assortment, we do
not know the kinds of demand models that lead to tractable
problems. In this paper, we aim to fill some of this gap in
the literature.
The research on pricing decisions for an assortment has
remained primarily within the domain of marketing and
economics. The usual assumption made in this research is
that demand is deterministic or that production is to order,
and, therefore, there are no inventory decisions to make.
For example, Anderson and de Palma (1992), Besanko
et al. (1998), and Aydin and Ryan (2000) show that if the
demands for competing products are given by the multinomial logit (MNL) demand model, then the profit of the firm
is maximized when the firm uses the same profit margin for
all products. Starting with van Ryzin and Mahajan (1999),
who consider the assortment-planning problem in the presence of inventory considerations (for the case of fixed retail
prices), the MNL model has found frequent use within
the operations management literature, as a way of incorporating consumer choice into operational models. See, for
example, Cachon et al. (2005), Cachon and Kok (2007),
Aydin and Hausman (2008), and Hopp and Xu (2005). Our
demand model covers the MNL-type demand model as a
special case.
Under our demand model, the demand for any given product depends on the attractiveness of each and every product in the assortment, and each product’s attractiveness is
a function of its own price (in addition to what we loosely
More often than not, consumers make purchasing decisions
that require them to choose from an assortment of substitutable products, e.g., choosing a flavor and container size
when buying ice cream or choosing from different accessory packages when buying an automobile. Of course, the
prices of the products influence customer choice. Hence,
the prices of products in an assortment influence not only
the size of overall demand, but also how the demand will be
allocated among the products, thereby driving the inventory
decisions. Therefore, ideally, inventory and pricing decisions should be made jointly for the entire assortment. In
this paper, we consider such a pricing and inventory problem for an assortment of substitutable products. In particular, we consider the price-dependent newsvendor problem
for an assortment, where a single firm faces single-period,
stochastic demands for multiple products that compete on
price and must set the inventory levels and prices for those
products at the beginning of the period.
The pioneers of research on joint inventory and pricing decisions for a single product were Whitin (1955),
Mills (1959), and Karlin and Carr (1962). Petruzzi and
Dada (1999) review and extend the research stream on
the single-product price-dependent newsvendor problem,
and our paper builds on theirs by adding customer choice
among multiple substitutable products. In a recent review
of the literature on coordination of production and pricing decisions, Yano and Gilbert (2005, p. 96) state that
an “important practical consideration that has not received
1247
1248
call “the quality of the product”); hence, the demand for
any given product depends on the prices of all products in
the assortment. This demand model is motivated by market
share models, such as MNL and multiple competitive interactions (MCI) models (e.g., Urban 1969), that have been
tested empirically and used extensively in the marketing
literature.
We establish that the price-dependent newsvendor problem for an assortment is well behaved under our demand
model. Hanson and Martin (1996) consider the special case
of our problem where demand is deterministic, and they
construct an example in which the profit function is not
jointly quasi-concave in prices, which suggests that finding
the optimal prices may require sophisticated search techniques. In our setting, where demand is stochastic, the profit
function is not jointly quasi-concave in prices and inventory levels either. Nevertheless, we show that, assuming
that unmet demands become lost sales, our problem is well
behaved in the sense that there is a unique vector of prices
and inventory levels that satisfy the first-order conditions,
and this vector is optimal for our problem. Hence, simple
search techniques should be sufficient to find the optimal
solution.
In related research, Mahajan and van Ryzin (2001) and
Netessine and Rudi (2003) address setting the inventory levels for a given assortment, allowing stockout-based substitution. We add selection of prices, but drop stockout-based
substitution. In addition, there has been some work on the
decentralized, competitive version of the problem we consider, where independent firms select the inventory and price
of one product each. Bernstein and Federgruen (2005) consider the decentralized problem under price-based, but not
stockout-based, substitution. Zhao and Atkins (2008) add
stockout-based substitution, which is independent of prices,
to the same problem, under additive uncertainty. Hopp and
Xu (2008) use an approximation to model both price- and
stockout-based substitution to analyze inventory, pricing,
and assortment decisions in centralized and decentralized
problems.
Zhu and Thonemann (2002) consider a multiperiod inventory and pricing problem for two products, whereas we consider a single-period problem for an arbitrary number of
products. They assume that the expected demand for each
product is linear in prices. In addition, they assume that all
unmet demand in a period is backlogged and the expected
holding and shortage cost does not depend on the retail
prices. In our single-period setting, where we assume all
unmet demands become lost sales, any induced holding and
shortage cost function would depend on the retail prices.
A closely related work is by Maddah and Bish (2004),
who also consider the price-dependent newsvendor problem for an assortment. In addition to inventory and pricing
decisions, they address the question of assortment selection. Their demand model is based on a normal approximation of Poisson arrivals of customers, each of whom
chooses one of the products in the assortment in accordance
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
with the MNL model. To alleviate some of the technical
difficulties associated with the inventory and pricing problem, they work with an approximation to the expected profit
of the firm, and obtain several insights into the assortmentplanning problem. We consider the assortment to be fixed,
and do not approximate the objective function.
In the next section, we describe the inventory and pricing problem for an assortment, and we discuss the demand
model in detail. Section 3 characterizes the optimal solution
and builds certain comparative statics. Finally, we conclude
in §4 with a discussion of future research directions.
2. Inventory and Pricing Problem
In our price-dependent newsvendor problem for an assortment, the firm orders and prices the products at the beginning of the period (selling season), and the stochastic
demand for a product is a function of the prices of all products. Unmet demands become lost sales, and leftover inventory has no value. Our model is more suitable for fashion
and seasonal items than it is for grocery staples where multiple replenishments and price adjustments take place over a
long period of time. We also assume no stockout-based substitution: once a product runs out of stock, customers who
prefer that product do not switch to another product, but
simply buy nothing. This is admittedly a restrictive assumption because we would expect to see some demand for the
out-of-stock product to spill over to the ones that are still in
stock. However, we believe that it is useful to characterize
the problem and its solution for the simpler case addressed
in this paper before introducing stockout-based substitution
as well as price-based substitution.
2.1. Problem Formulation
Let pi denote the unit selling price and ci the unit procurement cost of product i. Throughout the remainder of the
paper, let p = p1 pn denote the vector of prices and
y = y1 yn the vector of stock levels. We define Di p
to be the demand for product i given price vector p; i p x
and i p x are, respectively, the cumulative distribution
function (c.d.f.) and the probability density function (p.d.f.)
of Di p. We assume that i p x is strictly increasing
in x. Given a real-valued function f defined on Rn , we use
i f p to denote the partial derivative of f with respect to
the ith component of its argument evaluated at the point p.
Similarly, ij2 f p and ii2 f p denote the cross-partial and
second partial of f , respectively.
The firm wishes to maximize the following:
py =
n
Epi minDi pyi −ci yi i=1
=
n
i=1
pi
yi
0
xi pxdx +yi 1−i pyi −ci yi (1)
(2)
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
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Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
Because p y is separable and concave in the yi s, the
optimal stock level for product i, yi∗ p, is given for each
given price vector p as the usual critical fractile solution
pi − ci
pi
i p yi∗ p =
(3)
and the firm’s problem can be rewritten as maximizing
the following induced profit function (see, for example,
Porteus 2002):
p =
n
i=1
pi
0
yi∗ p
xi p x dx (4)
One question of interest is how the optimal stock levels
depend on the set of prices used. In the traditional newsvendor model (with a single product whose stochastic demand
does not depend on price), if the retail price increases, then
the stock level will increase as well because a price increase
causes the unit underage cost to increase. However, common
intuition suggests that the product’s demand will decrease
when its retail price increases, and it is not intuitively clear
which of the two opposite effects, the increase in underage
cost versus the decrease in demand, dominates. To address
this question, we work with a rather general model, summarized by Assumptions (A1)–(A3) below. (We shall specialize the model shortly to address additional questions.)
Assumption (A1). For each given p, Di p and Dj p are
(statistically) independent for each i = j.
Assumption (A2). There exists a strictly increasing c.d.f.,
, and for each i, a function ti p x such that i p x =
ti p x i = 1 n. Furthermore, we assume that
ti p x is strictly increasing in x.
Assumptions (A1) and (A2) will hold, for example, when
Di p = i pi + i p and the i s are independent and
identically distributed (i.i.d.) with c.d.f. . In such a case,
ti p x = x − i p/i p. In this instance, i p and
i p can be interpreted as scaling the market size for product i given the vector of selling prices, p, and i is a random perturbation of that potential market. Note that the
additive and multiplicative demand models follow as special cases. The following assumption imposes a structure on
how demand distributions change as prices change.
Assumption (A3). For each i, ti p x > 0, i ti p x > 0,
and j ti p x < 0 for j = i. Furthermore, ti p x is supermodular in each pi x; i.e., 2 ti p x/xpi 0.
The first part of (A3) is equivalent to assuming that Di
is stochastically decreasing in pi and stochastically increasing in pj . The second part says that ti p x becomes more
sensitive to changes in price at higher demand levels. When
Di = fi p + i , the second part of (A3) is readily satisfied. When Di = fi pi , it is satisfied if and only if fi p
is decreasing in pi , which is a very reasonable assumption.
When pi increases, Di stochastically decreases, which
tends to decrease yi∗ p and the critical fractile of product i increases, which tends to increase yi∗ p. In general,
we do not know which of these effects dominates. However, the following proposition states that if is IFR, then
yi∗ p is decreasing in pi at price points that are candidates for an optimal solution. (Recall that a c.d.f. (of
a nonnegative random variable) is IFR if its failure rate
hx = x/1 − x is increasing in x. This property
is satisfied by many distributions including the normal distribution and the Weibull distributions with shape parameter
greater than one. For more, see, for example, Barlow and
Proschan 1965.)
Proposition 1. Suppose that p satisfies i p = 0. If Assumptions (A1)–(A3) hold and is IFR, then i yi∗ p < 0.
This result is useful when dealing with joint inventory and
pricing problems. For example, in the context of a singleproduct inventory and pricing problem, Aydin and Porteus
(2008) use this result to analyze the effect of rebates on the
supply chain.
Unfortunately, as we briefly discuss in our conclusion section, if the general model satisfies only Assumptions (A1)–(A3), then the inventory and pricing problem
has some behavioral abnormalities. To further analyze the
inventory and pricing problem for an assortment, we will
work with a more specific demand model, described in the
next section.
2.2. The Demand Model
This demand model, defined by Assumptions (B1)–(B5),
is motivated by demand functions commonly used in marketing, as we will illustrate in §2.3. It satisfies Assumptions (A1)–(A3) as well. See Appendix B in the online supplement for a proof. An electronic companion to this paper
is available as part of the online version that can be found
at http://or.journal.informs.org/.
Assumption (B1). Di p = !qi pi for i = 1 n,
where i s are i.i.d. and nonnegative with c.d.f. that is IFR.
Assumption (B1) says that the realized demand for a
product is a multiplicative random perturbation of the
expected demand given by !qi p. The use of multiplicative randomness implies that the coefficient of variation of
the demand for each product is constant. If the coefficient
of variation of demand changes with price, then multiplicative randomness may not be appropriate. Throughout the
remainder of the paper, we assume without loss of generality (using normalization) that ! = 1 and E = 1, so that
qi p can be interpreted as the expected demand for product i at price vector p.
n
Assumption (B2). qi p = vi pi /v0 +
j=1 vj pj vi · 0 for i = 1 n.
1250
Given Assumption (B2), vi pi can be interpreted as a
positive measure of the attractiveness of product i, and the
expected demand for each product is proportional to the
product’s attractiveness. Furthermore, v0 can be interpreted
as the attractiveness of the no-purchase option. This model
is in line with the Luce choice model (Luce 1959). Throughout the remainder of the paper, we normalize the value of
v0 and let v0 = 1.
Assumpton (B3). vi pi is strictly decreasing in pi .
In light of the interpretation of vi pi offered above,
Assumption (B3) says that the attractiveness of a product is
decreasing in the product’s price. This assumption implies
that qi p is decreasing in pi and increasing in pj for j = i,
which can be interpreted as saying that the expected demand
of product i is decreasing in its own price and increasing in
the prices of other products.
Assumption (B4). pi i pi /i pi −1, where i pi =
vi pi /vi pi .
This assumption is rather technical and not very restrictive. For example, the assumption is satisfied whenever
vi pi is log-concave in pi , which allows a large set of
choices for vi functions. As we illustrate in §2.3, this
assumption is satisfied for at least two different demand
functions commonly used in the marketing literature, without imposing any constraints on the parameter values of
those functions, thereby allowing a large set of choices for
own-price and cross-price elasticities of demand.
Assumption (B5). limpi →
vi pi = 0, limpi →
pi vi pi = 0, limpi →
i pi > −
, and Ei < for i = 1 n.
This technical assumption guarantees that the expected
demand is finite and a product’s contribution becomes zero
as its price becomes arbitrarily large.
Next, we discuss a number of attractive examples of the
demand model specified by Assumptions (B1)–(B5).
2.3. Special Cases of the Demand Model
Assumptions (B3)–(B5) govern the relationship between the
price and the attractiveness of a product. The attractiveness functions of all products then combine, as specified
by (B2), to determine qi p, the expected demand of product i. Although Assumptions (B2)–(B5) may seem restrictive, they are satisfied by two forms of demand functions
used commonly in the marketing and economics literature,
the logit demand function (based on the multinomial logit
(MNL) consumer choice model) and the multiplicative competitive interactions (MCI) demand function. Below we state
the attractiveness functions, i.e., vi pi s, that correspond to
the logit and MCI demand functions. In addition, we offer
a third form for vi pi that satisfies our assumptions, which
is a plausible way of modeling a consumer population with
multiple customer segments. See Appendix B in the online
supplement for a proof that the following models satisfy
Assumptions (B3)–(B5).
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
(D1) Logit Demand
vi pi = exp$i − pi $i > 0 i = 1 n
This demand function is based on the MNL consumer
choice model used extensively in marketing. (See, for example, Guadagni and Little 1983.) Under the MNL model,
the stochastic surplus of a customer for product i is given
by $i − pi + %i , where $i can be interpreted as the customer’s expected utility for product i and %i is a Gumbel
error term with shape parameter one. Then, with vi pi as
defined by (D1), qi p gives the probability that a consumersurplus-maximizing individual will choose product i (out of
all n products), which we treat as the expected demand for
product i. For a detailed discussion of the use of consumer
choice models in inventory management and their limitations, see van Ryzin and Mahajan (1998).
(D2) MCI Demand
−&i
vi pi = $i pi
$i > 0 &i > 1 i = 1 n
Here, $i can be loosely interpreted as the quality of product i and &i as a measure of the consumer population’s price
sensitivity for product i. The MCI demand function has
been used extensively in the marketing literature to model
market shares. See, for example, Urban (1969), who offers
MCI demand as a model of market shares in dealing with
product-line decisions. Nakanishi and Cooper (1974) discuss parameter estimation for MCI demand and provide
empirical support.
(D3) A Demand Model with Multiple Customer
Segments
vi pi =
m
k=1
m
k=1
&k = 1
&k exp$ik − pi $ik > 0 &k > 0
i = 1 n
The form of vi pi given by (D3) is a plausible way of
incorporating customer segments into the demand model.
Suppose that there are m customer segments, and a customer
belongs to segment k with probability &k . Furthermore,
assume that the attractiveness of product i for a customer in
segment k is given by exp$ik − pi . Then, vi pi as defined
by (D3) yields the expected attractiveness of product i.
In the next section, we provide structural results on the
joint inventory and pricing problem, assuming that (B1)–
(B5) hold.
3. Characterizing the Optimal Solution
In general, the difficulty in dealing with inventory and pricing problems for multiple products is that neither the profit
function, p y in (2), nor the induced profit function,
p in (4), is separately concave in the pi s, let alone jointly
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
1251
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
450
400
350
300
250
200
150
100
i p
where
8.7
10.0
22.0
11.3
12.6
13.9
15.2
41.8
33.0
6.1
1
500
450
(7)
In the rest of this section, we use the representation
above to show a number of structural results on p under
Assumptions (B1)–(B5).
Proposition 2. Suppose that Assumptions B1–B5 hold.
(a) If p satisfies i p = 0 for i = 1 n, then
ii2 p < 0.
(b) If p satisfies i p = 0 for i = 1 n, then
ij2 p = 0 for j = i.
(c) There exists a unique p that satisfies i p = 0 for
i = 1 n. Furthermore, this p maximizes ·.
Proposition 2 generalizes Petruzzi and Dada’s (1999)
single-product version of this result, using a similar
approach. Part (a) implies that p is strictly quasiconcave in each price. However, the joint optimization over
the prices of multiple products is further complicated by the
existence of cross-price effects. In fact, in the deterministic
logit demand case (i.e., the case where vi pi is given by
(D1) and i = 1 for i = 1 n), Hanson and Martin (1996)
construct an example in which the profit function is not
jointly quasi-concave in prices. Indeed, consider the twoproduct inventory and pricing problem with stochastic logit
demand in which $1 = 10, $2 = 25, c1 = 6, c2 = 20, and
the random error terms, 1 and 2 , are uniformly distributed
between 100 and 500. Figure 1 depicts (from two different angles) the expected profit function for this two-product
400
350
300
250
200
150
100
50
0
13.
Pr
ice
2
9.1
20.6
zi p % d%
6.1
7.6
15.
1
6
12.
1
10.
1
6
ce
i
Pr
22.0
0
pi −ci /pi
25.8
36.2
i p = pi
ice
Pr
(6)
31.0
i=1
41.4
n
e2
Profit
p =
Pric
37.4
0
With this definition, zi p % is the demand that corresponds
to the % fractile of i , given the vector p of selling prices,
and, thus, yi∗ p = zi p pi − ci /pi Using this representation, the objective function in (4) can be written as
7.4
50
24.2
(5)
500
28.6
i p zi p % = %
The expected profit from a two-product
assortment, as a function of the prices, where
the stock levels are chosen optimally at each
price pair.
19.8
and substitute this in (4). This approach results in an induced
profit function that is inconvenient to work with. In this
paper, we find that the induced profit function takes on a
more convenient form when we use the inverse demand
functions. The inverse demand function zi p % is defined
for i = 1 n (implicitly) by
Figure 1.
Profit
concave. To obtain structural results on the maximization
problem in (4), the standard approach would be to write
yi∗ p in terms of the inverse of and substitute the resulting expression in p in (4). For example, when Assumption (B1) holds, using (3), we would write
∗
−1 pi − ci
yi p = qi p
pi
inventory and pricing problem. (Recall that inventory levels at any combination of prices are determined uniquely
by the critical fractiles.) In this example, which is typical
of what we see with different parameter sets, there are two
nearly perpendicular ridges in the profit function. If we pick
a price point out on each ridge, then along the line that
connects these two price points, the profit function would
first decrease and then increase. That is, as Hanson and
Martin (1996) observed in their example with deterministic
demand, the expected profit function is not jointly quasiconcave in this stochastic demand example either. Nevertheless, Proposition 2(c) states that there is a unique price
1252
vector that satisfies the first-order conditions (FOCs), and
that the price vector yields the optimal solution.
The significance of Proposition 2 is that any hill-climbing
algorithm will find the unique optimal price vector. (Such
an algorithm stops only when the FOCs are satisfied, and
Proposition 2(c) states that only the optimal point satisfies them.) In short, our problem is well behaved in the
sense that the FOCs are necessary and sufficient for optimality even though the expected profit function is not jointly
quasi-concave in prices. Given the recent popularity of price
optimization tools (see, for example, Johnson et al. 2001 and
Gustke 2002), this result is rather encouraging; it implies
that logit and MCI demand functions, which provide attractive explanations for consumer behavior, result in reasonably well-behaved profit functions that can be optimized
without too much difficulty.
Another point of interest is how the optimal prices
respond to changes in unit procurement costs and product
quality. Here, we loosely define quality as a product attribute
of which customers prefer more. Note that in the logit and
MCI demand models given by (D1) and (D2), the parameter
$i can be interpreted as the quality of product i. It is intuitively clear that when either cj or $j increases, the price of
product j, pj , will increase as well, in one case as a reaction
to the cost increase, and, in the other case, to take advantage
of the increased quality of the product. How other products’ prices will respond to these changes is not so obvious. One can show that, under Assumptions (B1)–(B5), if
the unit cost of one product increases, the optimal prices
of all other products will decrease. Furthermore, under an
additional nonrestrictive, technical assumption on how the
attractiveness function vi pi depends on quality $i , one can
show that if the quality of one product increases, then the
optimal prices of all products will increase.1
4. Conclusion
In this paper, we investigated a price-dependent newsvendor
problem for an assortment. We worked with profit expressions written in terms of the inverse demand functions.
This approach alleviates some of the technical challenges
posed by price-dependent demand distributions. Under a
demand model that follows from an attractive explanation
of consumer choice behavior, we showed that the inventory
and pricing problem is well behaved (i.e., optimized at the
unique vector of prices and inventory levels that correspond
to the stationary point).
One line of analysis for future research is to extend the
results to other demand models. For example, one might
want to check if Proposition 2 holds under different forms
of qi p, the expected demand for product i as a function of price vector p. We should note here that Proposition 2 does not hold under some commonly used alternative
forms for qi p. Consider, for example, the Cobb-Douglas
−& demand (i.e., qi p = $i pi i j=i pj ij &i ij > 0 i j =
1 n). Under Cobb-Douglas demand, one can construct
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
numerical examples that show that the profit function is
not even separately quasi-concave in prices; i.e., Proposition 2(a) does not hold. Alternatively, one could check
whether Proposition 2 continues to hold under additive randomness. If one assumes that the expected demand is given
by the logit demand function and the additive random shock
has zero mean, then one can show that Proposition 2 continues to hold. However, this introduces a new problem to the
model: when the price of a product is set high, its expected
demand will be low, resulting in a nonnegligible probability of negative demand due to the zero-mean additive
shock.
Another line of future research would be to compare the
optimal prices under deterministic and stochastic demand
scenarios (with identical expected demand functions). Previous research addressed this question in detail for the singleperiod inventory and pricing problem for a single product.
See Petruzzi and Dada (1999) for a review. For example,
Karlin and Carr (1962) showed that in the single-product
case with multiplicative uncertainty, the firm’s optimal price
under stochastic demand exceeds that for deterministic
demand. In our model with multiple products, one can construct numerical examples to show that no such ordering
exists. Nevertheless, it would be interesting to further analyze the effect of demand variability on prices.
Although we model price-based substitution, we do not
allow stockout-based substitution. Under stockout-based
substitution, even when prices are fixed, the optimal stock
levels are no longer independent from one another. (See, for
example, Netessine and Rudi 2003.) Therefore, even when
the prices of the products are fixed, the optimal stock levels
are hard to characterize, which makes the joint inventory
and pricing problem all the more difficult. It would be interesting to determine whether there are reasonable conditions
under which the result of Proposition 2 continues to hold
even when stockout-based substitution is allowed.
Another simplifying assumption we made was to ignore
external competition faced by a firm selling an assortment.
In a more general setting, there would be multiple retailers,
each of whom sells an assortment. Such a model could be
used to address not only price competition, but also competition on the depth of assortment.
Throughout the paper, we assumed that the retail prices
were uniform for the entire duration of the selling season.
Of course, in most retail settings, the retail price will change
over time. Also, for many retail situations, a model where
the retailer can place multiple orders over time would be
more realistic. It would therefore be interesting to consider
the inventory and pricing problem that arises with dynamic
pricing and multiple ordering opportunities.
5. Electronic Companion
An electronic companion to this paper is available as
part of the online version that can be found at http://or.
journal.informs.org/.
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
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Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
Appendix
−
We first state Lemmas 1–4 that are used for the proofs of
the propositions. We then prove Propositions 1 and 2. See
Appendix A in the online supplement for the proofs of the
lemmas.
Lemma 1. Suppose that (A2) holds. For yi∗ p implicitly
defined by (3), we have
1
a i yi∗ p =
pi n+1 ti p yi∗ phti p yi∗ p
· 1 − pi i ti p yi∗ phti p yi∗ p b
j yi∗ p = −
j ti p yi∗ p
>0
n+1 ti p yi∗ p
for j = i
a i qi p = i pi qi p1 − qi p < 0
b i qk p = −i pi qi pqk p > 0
d
for k = i
ii2 qk p i qi pi qk p
−
qk p
qi pqk p
i qk p i pi − i pi 2 qi p
=
qk p
i pi ij2 qk p
qk p
i qj pj qk p j qi pi qk p
−
= 0
qj pqk p
qi pqk p
−
a i p =
pi −ci /pi
0
zi p % d% +
ci ∗
y p
pi i
q p pk −ck /pk
+ pk i k
zk p % d%
qk p 0
k=1
n
b
ii2 p =
i qi p
q p
i p + i i
qi p
qi p
pi −ci /pi
c
·
zi p % d% + i i yi∗ p
pi
0
2
n
ii qk p i qi pi qk p
+ pk
−
qk p
qi pqk p
k=1
·
c ij2 p =
0
pk −ck /pk
zk p % d%
j qi p
i qj p
i p +
p
qi p
qj p j
2
n
ij qk p i qj pj qk p
+ pk
−
qj pqk p
qk p
k=1
pk −ck /pk
0
zk p % d%
Proof of Proposition 1. First, we use integration by parts
and Assumption (A2) to rewrite p in (4) as follows:
p =
n
pj − cj yj∗ p − pj
0
yj∗ p
tj p x dx Now, taking the partial derivative of p with respect to pi
and using (3) to eliminate the terms involving i yj∗ p yields
yi∗ p
ti p x
i p = yi∗ p −
0
yj∗ p
n −
pj
i tj p xtj p x dx j=1
0
Using the definition of hx = x/1 − x, we can
rearrange i p to yield
i p =
yi∗ p
0
1 − ti p x
· 1 − pi i ti p xhti p x dx
y∗ p
j
−
pj
i tj p xtj p x dx Lemma 3. Suppose that Assumptions B1–B4 hold.
Then, for j = i
Lemma 4. Suppose that Assumptions B1–B5 hold.
Then, for p defined by (6) and (7):
(a) i p > 0 when pi = ci ,
(b) i p → 0− as pi → .
j=1
Lemma 2. Suppose that B2 and B3 hold, and let
i pi = vi pi /vi pi . Then
c
·
j qi pi qk p
qi pqk p
j=i
0
(8)
By (A3), i tj p x < 0 for j = i. Suppose now that p satisfies i p = 0. Then, it follows from (8) that we must
have
0
yi∗ p
1 − ti p x
· 1 − pi i ti p xhti p x dx < 0
(9)
For convenience, fix i and p, and define G temporarily by
Gx = 1 − pi i ti p xhti p x. By (9), we must have
Gx < 0 for some x between 0 and yi∗ p. (Otherwise, the
integrand in (9) would be positive for all x between 0 and
yi∗ p, leading to a contradiction.) Now, note that, by (A2),
ti p x is increasing in x. Also, by (A3), i ti p x > 0
and 2 ti p x/pi x > 0. Furthermore, h· is increasing
in its argument because we assume that is IFR. It follows that Gx is decreasing in x. Finally, using the facts
that Gx < 0 for some x between 0 and yi∗ p, and Gx is
decreasing in x, we conclude that Gyi∗ p < 0. Now, the
result follows from part (a) of Lemma 1.
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
1254
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
Proof of Proposition 2. (a) and (b). Substituting from
Lemma 2(c) and (d) in Lemma 3(a) and (b), we obtain
ii2 p =
ij2 p =
i qi p
q p pi −ci /pi
i p + i i
zi p % d%
qi p
qi p 0
i pi − i pi 2 qi p
ci
∗
+ i yi p +
pi
i pi (10)
n
p
−c
/p
k
k
k
q p
· pk i k
zk p % d%
qk p 0
k=1
j qi p
i qj p
p +
p
qi p i
qj p j
j = i
Part (b) of the proposition follows directly from the above
expression for ij2 p. To prove part (a), consider the following two cases:
Case (i). i pi − i pi 2 qi p/i pi 0. We make
the following observations:
Observation 1. i qi p < 0 by Lemma 2(a).
Observation 2. By Proposition 1, i yi∗ p < 0 when
i p = 0.
Observation 3. When i p = 0, from Lemma 3(a),
it follows that
n
q p pk −ck /pk
pk i k
zk p % d%
qk p 0
k=1
pi −ci /pi
c
zi p % d% − i yi∗ p < 0
(11)
=−
p
0
i
Applying Observations 1–3 to (10) yields part (a) for
Case (i).
Case (ii). i pi − i pi 2 qi p/i pi < 0. We make
the following observations:
Observation 1. For k = i, we have i qk p > 0 by
Lemma 2(b).
Observation 2. By Proposition 1, i yi∗ p < 0 when
i p = 0.
Observation 3. We note that
i qi p pi −ci /pi
zi p % d%
qi p 0
p − i pi 2 qi p i qi p pi −ci /pi
+ i i
pi
zi p % d%
i pi qi p 0
i qi p
i pi − i pi 2 qi p
=
1+
pi
qi p
i pi pi −ci /pi
·
zi p % d%
0
q p
p = i i
1 + i i pi − i pi qi ppi
qi p
i pi pi −ci /pi
·
zi p % d% < 0
0
where the inequality follows from 1 + i pi /i pi pi 0
and i < 0 (see Assumptions (B3) and (B4)) and
i qi p < 0 (by Lemma 2(a)). Now, applying Observations 1–3 to (10) yields part (a) for Case (ii).
(c). The idea underlying the proof of part (c) was used
previously by Petruzzi and Dada (1999) in their analysis
of the single-product price-dependent newsvendor problem.
We formalize that argument and extend it to our n-product
problem. Here, we provide only a sketch of the proof. See
Appendix A in the online supplement for a formal proof.
First, note that if p is a maximizer of p, then i p = 0
for i = 1 n due to Lemma 4. To conclude the proof,
we will need to show that there exists a unique p such
that i p = 0 for i = 1 n, and p maximizes p.
Parts (a) and (b) together imply that the Hessian of p
is negative definite in the neighborhood of any p such that
i p = 0 for i = 1 n. Hence, any stationary point is
a strict local maximum. Suppose now that there exist more
than one, say two, stationary points of the function p.
Because both those points need to be local maxima, the
function should also have a local minimum somewhere in
between, which is a contradiction to the result that all stationary points are local maxima.
Endnote
1. The proofs are available from the authors upon request.
Acknowledgments
The first author’s research was supported in part by Ericsson
AB under the Ericsson Supply Chain Research Grant to
the Department of Management Science and Engineering
at Stanford University. The authors thank the associate editor and two anonymous referees for their comments, which
helped improve the paper.
References
Anderson, S. P., A. de Palma. 1992. Multi-product firms: A nested logit
approach. J. Indust. Econom. 40 261–276.
Aydin, G., W. H. Hausman. 2008. The role of slotting fees in the coordination of assortment decisions. Working paper, Department of Industrial
and Operations Engineering, University of Michigan, Ann Arbor.
Aydin, G., E. L. Porteus. 2008. Manufacturer-to-retailer versus
manufacturer-to-consumer rebates in a supply chain. N. Agrawal,
S. Smith, eds. Retail Supply Chain Management. Springer, Boston.
Forthcoming.
Aydin, G., J. K. Ryan. 2000. Product line selection and pricing under the
multinomial logit choice model. Working paper, School of Industrial
Engineering, Purdue University, West Lafayette, IN.
Barlow, R. E., F. Proschan. 1965. Mathematical Theory of Reliability.
Wiley, New York.
Bernstein, F., A. Federgruen. 2005. Decentralized supply chains with
competing retailers under demand uncertainty. Management Sci. 51
18–29.
Besanko, D., S. Gupta, D. C. Jain. 1998. Logit demand estimation under
competitive pricing behavior: An equilibrium framework. Management Sci. 44 1533–1547.
Cachon, G., A. G. Kok. 2007. Category management and coordination
in retail assortment planning in the presence of basket shopping
consumers. Management Sci. 53 934–951.
Cachon, G., C. Terwiesch, Y. Xu. 2005. Retail assortment planning in the
presence of consumer search. Manufacturing Service Oper. Management 7 330–346.
Aydin and Porteus: Joint Inventory and Pricing Decisions for an Assortment
Operations Research 56(5), pp. 1247–1255, © 2008 INFORMS
Guadagni, P. M., J. D. C. Little. 1983. A logit model of brand choice
calibrated on scanner data. Marketing Sci. 2 203–238.
Gustke, C. 2002. Prices and possibilities. Progressive Grocer. 18(1) 66–68.
Hanson, W., K. Martin. 1996. Optimizing multinomial logit profit functions. Management Sci. 42 992–1003.
Hopp, W. J., X. Xu. 2005. Product line selection and pricing with modularity in design. Manufacturing Service Oper. Management 7 172–187.
Hopp, W. J., X. Xu. 2008. A static approximation for dynamic demand
substitution with applications in a competitive market. Oper. Res.
56(3) 630–645.
Johnson, C. A., L. Allen, A. Dash. 2001. Retail revenue management.
The Forrester Report.
Karlin, S., C. R. Carr. 1962. Prices and optimal inventory policies. K.
J. Arrow, S. Karlin, H. Scarf, eds. Studies in Applied Probability
and Management Science. Stanford University Press, Stanford, CA,
59–172.
Luce, R. D. 1959. Individual Choice Behavior: A Theoretical Analysis.
Wiley, New York.
Maddah, B., E. K. Bish. 2007. Joint pricing, assortment, and inventory
decisions for a retailer’s product line. Naval Res. Logist. 54 315–330.
Mahajan, S., G. van Ryzin. 1998. Retail inventories and consumer choice.
S. Tayur, R. Ganeshan, M. J. Magazine, eds. Quantitative Models for
Supply Chain Management. Kluwer Academic Publishers, Boston,
491–552.
Mahajan, S., G. van Ryzin. 2001. Stocking retail assortments under
dynamic consumer substitution. Oper. Res. 49 334–351.
Mills, E. S. 1959. Uncertainty and price theory. Quart. J. Econom. 73
116–130.
1255
Nakanishi, M., W. G. Cooper. 1974. Parameter estimation for a multiplicative competitive interaction model—Least squares approach.
J. Marketing Res. 11 303–311.
Netessine, S., N. Rudi. 2003. Centralized and competitive inventory models with demand substitution. Oper. Res. 51 329–335.
Petruzzi, N. C., M. Dada. 1999. Pricing and the newsvendor problem:
A review with extensions. Oper. Res. 47 183–194.
Porteus, E. L. 2002. Foundations of Stochastic Inventory Theory. Stanford
University Press, Stanford, CA.
Urban, G. L. 1969. A mathematical modeling approach to product line
decisions. J. Marketing Res. 6 40–47.
van Ryzin, G., S. Mahajan. 1998. On the relationship between inventory
costs and variety benefits in retail assortments. Management Sci. 45
1496–1515.
Whitin, T. M. 1955. Inventory control and price theory. Management Sci.
2 61–68.
Yano, C., S. M. Gilbert. 2005. Coordinated pricing and production/procurement decisions: A review. A. Chakravarty, J. Eliashberg,
eds. Managing Business Interfaces: Marketing, and Engineering
Issues in the Supply Chain and Internet Domains. Springer, Boston,
65–103.
Zhao, X., D. Atkins. 2008. Newsvendors under price and inventory competition. Manufacturing Service Oper. Management. 10(3) 539–546.
Zhu, K., U. Thonemann. 2002. Coordinating pricing and inventory control across products. Working paper, Department of Industrial and
Engineering Management, Hong Kong University of Science and
Technology, Kowloon, Hong Kong.