Assortment Planning for Vertically Differentiated Products
Xiajun Amy Pan
Department of Marketing and Supply Chain Management
Henry W. Bloch School of Management
University of Missouri at Kansas City
5110 Cherry St., Kansas City, MO 64110.
E-mail: [email protected]
Phone: 816-235-2892, Fax: 816-235-6506
Dorothée Honhon
Department of Information, Risk, and Operations Management
McCombs School of Business
University of Texas at Austin
CBA 3.440, 1 University Station, Austin, TX 78712.
E-mail: [email protected]
Phone: 512-471-4130, Fax: 816-235-6506
We consider the problem of a retailer managing a category of vertically differentiated products.
The retailer has to pay a fixed cost for each product included in the assortment and a variable cost
per product sold. Quality levels, fixed and variable costs are exogenously determined. Customers
differ in their valuation of quality and choose the product (if any) that maximizes their utility.
First, we consider a setting in which the selling prices are also fixed. We find that the optimal
set of products to offer depends on the distribution of customer valuations and might include
dominated products, i.e., products which are less attractive than at least one other product, on
every possible dimension. We develop an efficient algorithm to identify an optimal assortment.
Second, we consider a setting in which the retailer also determines the selling prices. We show that
in this case the optimal assortment does not include any dominated product and does not vary with
the distribution of customer valuations when there is no fixed cost. We develop several efficient
algorithms to identify an optimal assortment and optimally price the products. We also test the
applicability of our methods with realistic data for two product categories.
Keywords: Assortment planning, Vertical differentiation, Quality, Pricing.
History: Paper submitted in August 2010, first revision submitted in December 2010, second revision submitted in February 2011. Paper accepted in February 2011.
1 Introduction
When choosing what products to carry in a given product category, retailers typically have to choose
from hundreds, possibly thousands of variants, offered by dozens of different manufacturers. For instance, Newegg (www.newegg.com) carries more than 400 different USB flash drives and 1400 different
televisions but these only represent a fraction of the existing products in the market as the retailer
does not offer every brand and model. Selecting products to offer is a challenging problem for retailers, because customer choice depends on customer preferences as well as on the available assortment,
and profit depends on sales and the relative profitability of products. A naive way to obtain the
profit-maximizing assortment would be to enumerate all possible combinations of products and selling
prices and identify the most profitable one. However, this method is not practical for popular product
categories such as televisions and computers for which the number of possible products to choose from
is very large. Hence, developing efficient methods to obtain a profit-maximizing assortment is very
important and valuable for retailers.
This paper addresses the problem of a retailer managing a category of vertically differentiated
products. Each product is characterized by a quality level. All else equal, customers prefer a product
with a higher quality level to a product with a lower quality level, but they differ in how they value
quality. We assume that customers determine what product to buy by maximizing a linear utility
function which is increasing in quality and their valuation of quality and is decreasing in price. While
the retailer does not know how a specific customer values a unit of quality, she knows the distribution
of customer valuations. The quality levels of the products are exogenously determined by the one or
multiple manufacturers who supply the products to the retailer. The retailer pays the manufacturer(s)
a variable cost per product sold and incurs a fixed cost per product included in the assortment.
First, we examine the scenario where the selling prices are exogenous to the retailer and her only
decision is to choose the set of products to be included in the assortment. This setting is motivated by
product categories for which the manufacturers’ suggested retail prices (MSRP) are so prevalent that
retailers do not deviate from it. Carlton and Chevalier (2001) state that while “manufacturers may
not contractually bind the retailer to charge the MSRP [...] (they) may be willing to supply ‘exclusive’
products to retailers who adopt an across-the-board ‘no discounting’ policy”. The data that Carlton
and Chevalier (2001) collect about the fragrance market show that a number of stores (namely upscale
beauty and department stores) are charging the MSRP for the products they sell. We show that,
when prices are exogenous to the retailer, the optimal assortment is a function of the distribution of
1
customer valuations for quality and can be obtained by solving a shortest path problem.
Second, we consider the case where the retailer also decides the selling price of the products offered
in her assortment, i.e., selling prices are endogenous variables. We show that the optimal assortment
does not depend on the distribution of customer valuations for quality, as long as this distribution has
an increasing failure rate and the fixed cost is negligible. We propose a number of algorithms to obtain
the optimal assortment. The complexity of these algorithms depends on the value of the fixed cost
and on properties of the distribution of customer valuations for quality.
Third, we compare the optimal assortment in the two cases studied and obtain some interesting
insights. In particular, we show that the optimal assortment may contain dominated products when
prices are exogenous, but not when prices are endogenous. We say that a product is dominated if
there exists a product with a higher quality level, lower variable cost, and (in the exogenous case only)
higher selling price in the set of potential products to offer. We also demonstrate that the products
included in the optimal assortment are such that the selling prices, profit margins and price-to-quality
ratios are increasing in the quality level. Finally, we show that if the manufacturer(s) cannot supply
some of the offered products anymore, it may be optimal for the retailer to include products which
previously were not in the optimal assortment. However, it is never optimal to drop products which
were previously offered that the manufacturer(s) can still supply. We show that most of our results
are robust to the assumptions we made on the utility and profit functions.
Finally, we address two practical considerations. We show that in both the exogenous and endogenous prices case, there is a need for fast algorithms which provide the optimal solution because the
enumeration method quickly becomes unpractical as the number of products to choose from becomes
large (greater than 20) and simple heuristics can perform arbitrarily badly, i.e., lead to an optimality gap of 100%. In addition, we demonstrate the applicability of our model through two numerical
examples calibrated using realistic data.
In general our results apply when a retailer sells vertically differentiated products, with quality levels
exogenously determined by the manufacturer(s). The retailer has to determine the best assortment to
offer given the MRSP(s) provided by the manufacturer(s). If the retailer does not deviate from these
prices, then our results from Section 3 apply. If the retailer determines the selling prices as well as the
assortment, our results from Section 4 apply.
It is important to note that the exogenous price problem is not a special case of the endogenous
price problem and vice versa. First order conditions can be used to solve for prices in the endogenous
2
prices case, but the resulting prices may be a function of the chosen assortment so that it is generally
not possible to decouple the optimal prices and optimal assortment problems. As shown later, when
prices are decision variables, we are able to establish more properties about the optimal assortment
and develop more efficient algorithms to identify the optimal assortment.
The paper that is most closely related to ours is that of Barghava and Choudhary (2001) who also
consider the problem of selecting and pricing vertically differentiated products when quality levels are
exogenously determined. However, our work differs from theirs in numerous ways. First, we consider a
fixed cost and show that it is an important factor in determining the optimal assortment. Second, we
consider the case of exogenous selling prices. Third, we provide a full characterization of the optimal
assortment when prices are endogenous. In contrast, Barghava and Choudhary (2001) only provide
a partial characterization of the optimal assortment when there is no fixed cost. They focus only on
two special cases: when the optimal assortment contains only the highest quality product and when
it contains all the products. In other words, while their work addresses the question “when does the
optimal solution have a noteworthy structure?”, our work answers the more general question “how can
we obtain the optimal solution for any given scenario and what are its properties?”.
The rest of our paper is organized as follows. In Section 2 we review the related literature. In
Section 3 we present the exogenous prices model and develop an efficient algorithm to identify the
optimal assortment. Section 4 contains the endogenous prices model and the corresponding results.
In Section 5 we compare the two models, discuss interesting insights and the robustness of our model.
Section 6 shows the necessity of developing our solution methods and Section 7 demonstrates the
applicability of our model to a realistic setting. Section 8 concludes our work and provides directions
for future research. All proofs are presented in the Appendix.
2 Literature Review
Our work is at the intersection between two streams of research: the work on vertically differentiated
products and the work on assortment planning. We first review the work on vertically differentiated
products, and then discuss the relevant papers on assortment planning. To the best of our knowledge,
our work is the first paper that considers the problem of selecting products from a discrete set of
vertically differentiated options when prices are fixed and the problem of selecting and pricing vertically
differentiated products in the presence of a fixed cost when prices are decision variables.
3
Mussa and Rosen (1978) is, to our knowledge, the earliest work on vertical differentiated products.
In this paper, the authors capture the heterogeneity of customers using a continuous valuation parameter θ. The utility that a customer with valuation θ gets from a product of quality q is θq, i.e., the
utility function is linear. Assuming convex production cost, the authors show that price discrimination
by offering products of different quality levels is optimal. Moorthy (1984) generalizes the utility and
cost functions and uses discrete parameters to represent customer segments. He concludes that the
firm may reduce the number of product versions offered in order to mitigate the cannibalization effect.
Several other researchers, such as Green and Krieger (1985), and Dobson & Kalish (1988, 1993), use
a limited number of customer segments to represent the market. The information about the number
of customers and reservation price for products associated with each segment is assumed to be known.
They formulate the problem as a mathematical programming model and develop heuristics to solve it.
Assuming concave variable costs and considering a fixed setup cost for producing a batch of products,
Netessine and Taylor (2007) examine how production technology impacts the optimal product line
design. They demonstrate that cannibalization may lead to offering more products and higher quality
in the presence of production technology.
The papers mentioned above take the point of view of a manufacturer as it is assumed that the
decision maker can pick any quality level for their products and possibly provide a continuous menu
of quality-price combinations. In contrast, our work takes the point of view of a retailer and assumes
that there is only a discrete set of quality levels from which the retailer can choose. Like these papers,
we use a linear utility function but we do not make any assumption on the cost function and make
only limited assumptions on the distribution of customer valuations in the endogenous prices case.
Several other papers which study product differentiation problems incorporate issues we do not
explore here, such as the presence of outside opportunities (see Chen and Seshadri (2007)), network
effects on the versioning strategies (see, for example, Barghava and Choudhary (2004) and Jing (2007)),
the optimal time to introduce product versions, i.e., either simultaneously or sequentially (see, for
example, Moorthy and Png (1992) and Raghunathan (2000)) and the optimal pricing strategy in a
dynamic setting (see Akcay et al. (2010)). Finally, a number of papers consider the problem of offering
vertically differentiated products in a competitive setting (see, for example, Shaked and Sutton (1982),
Moorthy (1988), Shugan (1989), Rhee (1996), and Jing (2006)).
In assortment planning problems, a firm chooses what products to offer from a discrete set of
potential products and customers have heterogenous preferences for these products. The papers usually
differ in the consumer choice model that is used to represent customer preferences and the type of
4
substitution they assume. Most papers assume that products are horizontally rather than vertically
differentiated, which means that two customers can have a different favorite product even if all products
are offered at the same price. Since our work does not include the presence of inventory, we discuss the
most closely related work on assortment planning, namely work that assumes static, assortment-based
substitution. For a broader review of the topic (including papers that assume dynamic, stock-out based
substitution), see Kök et al. (2008).
Among the earliest papers on assortment planning, Pentico (1974) studies a one-dimensional assortment planning problem with downward substitution for stochastic demand and obtains the optimal
solution with an assumption regarding the sequence of customer arrivals and a ‘no crossover’ assumption, which preclude dynamic substitution. Van Ryzin and Mahajan (1999) use the multinomial logit
model to represent customer preferences for horizontally differentiated products and show that the
optimal assortment includes a subset of the most popular products. Gaur and Honhon (2006) consider
the same problem but use a locational choice model to represent customer preferences. They introduce
a unimodal distribution of customers on the attribute space, and show that the products in the optimal
assortment are equally spaced and need not include the most popular product. Smith and Agrawal
(2000) consider this problem under stock-out based substitution but provide a solution method which
assumes assortment-based substitution and use a choice model specified by first choice probabilities
and a substitution matrix. Cachon et al. (2005) generalize the consumer choice process to incorporate
search costs, and show that ignoring consumer search in demand estimation can result in an assortment
with less variety and lower expected profit than the optimal solution.
The papers mentioned so far all assume that prices are given. The following papers consider prices
as decision variables. Hopp and Xu (2005) use a Bayesian Logit model to study the impact of modular
design on the joint assortment planning and pricing problem under assortment-based substitution.
They show that the optimal assortment for a risk-averse retailer is composed of the variants with the
highest price markups. Maddah and Bish (2007) consider a similar setting and propose a dominance
relationship for the general case that simplifies the search for an optimal assortment. Aydin and Porteus
(2008) study the joint assortment and pricing problem under price-based substitution with a demand
model involving multiplicative uncertainty. Alptekinoglu and Corbett (2008) use the locational choice
model to study competitive product positioning and pricing.
5
3 The Exogenous Prices Case
As mentioned in the Introduction, this setting applies, for example, when the retailer is not able to or
not willing to deviate from the MSRP and hence, the selling prices are exogenous to the retailer. In
this section, we present the model, analyze the optimal assortment structure, and develop an efficient
algorithm to identify the optimal solution.
3.1 Model
We consider a product category with n vertically differentiated products. Let qj denote the quality
of product j. The quality level can also be regarded as a combination of many of the product’s
characteristics if the product has many characteristics, which is a common assumption in the literature
on vertically differentiated products. Without loss of generality we assume that 0 ≤ q1 < q2 < ... < qn .1
Let rj ≥ 0 and cj ≥ 0 be the selling price and variable cost of product j respectively. Note that we
do not assume that cj ≥ cj−1 . Let K ≥ 0 be a fixed cost incurred for each product that is offered. In
practice K includes, for example, the cost of advertising the product. For notational convenience, let 0
be a fictitious product 0 with q0 = c0 = r0 = 0. Let ~r = (r1 , ..., rn ), ~c = (c1 , ..., cn ), and ~q = (q1 , ..., qn ).
We assume that customers are characterized by their willingness to pay for one unit of quality in
the product category, or valuation. A customer with valuation θ gets utility θqj − rj from buying one
unit of product j and zero for additional units. Without loss of generality, we assume that the utility
of buying nothing is equal to zero. A customer buys the product which gives him the highest utility as
long as it is positive. The retailer cannot identify the specific θ value for any customer, but knows the
distribution of θ. Let f (θ) and F (θ) with support [θ, θ̄] denote the probability density function and
cumulative density function of customer valuations respectively, where θ ≥ 0 and 0 < θ̄ ≤ ∞.
In the exogenous prices model, ~r, ~
q and ~c are given and fixed. It is assumed that cj < rj < θqj for
j = 1, ..., n; otherwise, it would be optimal not to stock product j since cj ≥ rj would imply a negative
profit margin and
rj
qj
≥ θ would imply that no customer gets a positive utility from product j. Note
that we do not exclude the possibility that two products i, j are such that ri < rj and qi > qj (better
quality for a lower price). The retailer’s decision is to determine which products to offer. Let S denote
the set of products that are offered, or assortment. We summarize our notation in Table 1.
1
All of our results would continue to hold if we had qj = qj+1 for some j but, for ease of exposition, we ignore this
case.
6
Symbol
n
j
qj
cj
rj
q~
~c
~
r
θ
f (θ)
Definition
Number of potential products in the category
Product index, j = 1, ..., n
Quality level of product j
Variable cost of product j
Selling price of product j
Quality vector
Variable cost vector
Selling price vector
Consumer valuation, ∈ [θ, θ̄]
Probability density fct of consumer valuation
Symbol
F (θ)
h(θ)
η(θ)
θj
Pj
S
S∗
K
EΠ
EΠ∗
Definition
Cumulative distribution function of consumer valuation
Failure rate of distribution F (θ)
Inverse failure rate
Valuation of consumer indifferent btw j − 1 and j
Purchase probability of product j
Assortment, i.e., set of products offered
Optimal assortment
Fixed cost
Retailer’s expected profit
Retailer’s optimal expected profit
Table 1: Notation
Let Pj (S) be the proportion of customers who purchase product j given assortment S, or purchase
R θ̄
probability. We have Pj (S) = θ I{qj θ − rj = maxi∈S (qi θ − ri ) and qj θ − rj > 0}dF (θ) if j ∈ S and 0
otherwise, where I{A} is the indicator function for event A. In theory, it is possible to have Pj (S) = 0
for j ∈ S, that is, product j is offered but no customer buys it. In this case, removing product j does
not affect the demand for other products and it decreases the total fixed cost; therefore, the optimal
assortment never includes a product with zero purchase probability. It follows that one can restrict
the search for the optimal assortment to sets S = {j1 , j2 , ..., jm } such that j1 < j2 < ... < jm and
θ≤
rj − rjm−1
rj 1
rj − rj1
< 2
< ... < m
< θ̄,
qj1
qj2 − qj1
qjm − qjm−1
(1)
which are necessary and sufficient conditions for Pji (S) > 0 for i = 1, ..., m since
rji+1 − rji
rji − rji−1
Pji (S) = F
−F
for i = 1, ..., m − 1,
qji+1 − qji
qj − qji−1
i
rjm − rjm−1
.
Pjm (S) = 1 − F
qjm − qjm−1
(2)
(3)
where j0 = 0. Figure 1 illustrates an example with three products. Consumers in Group A are such that
θ≤θ≤
r1
q1
and purchase nothing since they get a non-positive utility from every product. Consumers in
Group B are such that
r1
q1
<θ≤
r2 −r1
q2 −q1
and purchase product 1 because it gives them the highest utility.
Consumers in Group C are such that
r2 −r1
q2 −q1
<θ ≤
r3 −r2
q3 −q2
and buy product 2 although each product
−r2
gives them a positive utility. Finally, consumers in group D are such that qr33 −q
< θ ≤ θ and purchase
2 r1
r3 −r2
r2 −r1
1
−
F
,
F
−
F
product 3. Hence, the purchase probabilities are equal to F rq22 −r
−q1
q1
q3 −q2
q2 −q1
r3 −r2
and 1 − F q3 −q2 respectively for product 1, 2 and 3.
The expected profit is as follows:
EΠ(S) =
X
Pj (S)(rj − cj )µ − K|S|,
j∈S
7
(4)
Utility
Do not
purchase
Purchase
Product 1
Purchase
Product 2
Purchase
Product 3
A
B
C
D
θ
r3 − r2
q3 − q 2
r2 − r1
q2 − q1
r1
q1
θ
θ
Product 1
Product 2
Product 3
Figure 1: Purchase probabilities in an example with 3 products.
where µ is the expected market size and |S| denotes the size of set S. In what follows, we assume,
without loss of generality, that µ is equal to 1 and that the fixed cost K is scaled appropriately. The
firm’s objective is to find S ∗ such that
EΠ∗ = EΠ(S ∗ ) =
max
S⊆{1,...,n}
EΠ(S).
(5)
3.2 Results
To solve the retailer’s profit maximization problem (5), we can enumerate all possible assortments S,
compute their expected profit, and identify the optimal assortment with the highest expected profit.
However, the complexity of this naive enumeration method is O(2n ), which is not practical when n is
large as we show in Section 6. Consequently, we develop an efficient algorithm to obtain the optimal
assortment. This algorithm is based on some important properties of the optimal assortment which
we present first. Let S ∗ = {j1 , j2 , ..., jm } be the optimal set such that j1 < j2 < ... < jm .
Lemma 1. The products in an optimal assortment S ∗ satisfy the following three conditions:
rj1 < rj2 < ... < rjm ,
rj
rj
rj
θ ≤ 1 < 2 < ... < m < θ̄,
qj1
qj2
qjm
rj1 − cj1 ≤ rj2 − cj2 ≤ ... ≤ rjm − cjm .
8
(6)
(7)
(8)
Lemma 1 indicates that, in the optimal assortment, the product prices and price-quality ratios are
strictly increasing in the quality level and the profit margins are non-decreasing in the quality levels.
The first two conditions are necessary for all products in S ∗ to get a positive purchase probability. The
intuition behind the third property is that the retailer can get a higher expected profit by removing
product j if a product j has a lower profit margin than products of lower quality.
From (2), (3) and (4), notice that the contribution of product ji to the expected profit in assortment
S depends only on the adjacent products ji−1 and ji+1 , since products before ji−1 and after ji+1 have
no impact on the purchase probability of product ji . As a consequence, we are able to model this
problem as a shortest path problem and solve it in polynomial time. Moreover, we use the properties
of Lemma 1 to construct a parsimonious network.
We construct a graph G = (V, A), where V is the set of nodes and A is the arc set of G. Node
set V consists of pairs of products (i, j) such that product j has a higher quality and price-quality
ratio than product i and a profit margin which is no lower than that of product i. A node (i, j) ∈ V
indicates that products i and j could be offered together in the assortment. We also introduce two
fictitious nodes: a source node (0, 0) and a destination node (n + 1, n + 1). If nodes (i, j) and (j, k)
satisfy θ ≤
rj −ri
qj −qi
<
rk −rj
qk −qj
< θ̄, then the arc between these two nodes is a valid arc, which belongs to A.
A valid arc between (i, j) and (j, k) implies that product j could be offered along with products i and
k. We are able to compute the cost of the arc between (i, j) and (j, k) using the prices, variable costs,
and quality levels of these three products. We find the optimal assortment by solving the shortest path
problem from the source node to the destination node. We formalize the procedure as follows.
ALGORITHM: Shortest Path ‘Exo’
• Step 1. Construct the node set V , which consists of the following nodes:
V
rj
ri
< θ̄ ∪ {(j, n + 1) : 1 ≤ j ≤ n and
< θ̄} ∪
=
(0, i) : 1 ≤ i ≤ n and θ ≤
qi
qj
rj
ri
<
< θ̄ and ri − ci ≤ rj − cj
{(0, 0), (n + 1, n + 1)} ∪ (i, j) : 1 ≤ i < j ≤ n and
qi
qj
• Step 2. Construct the arc set by adding an arc from node (i, j) ∈ V to (l, k) ∈ V to set A if j = l = k or
j = l < k and θ ≤
rj −ri
qj −qi
r −r
rk −rl
qk −ql
< θ̄ if k < n + 1 and θ ≤ qjj −qii < θ̄ if k = n + 1.

h i
rk −rj
rj −ri

K
−
(r
−
c
)
F
−
F

j
j

qj −qi

i
hqk −qj r −r
• Step 3. Compute the arc costs: C(i,j),(j,k) =
K − (rj − cj ) 1 − F qjj −qii




0
<
• Step 4. Solve the shortest path problem from (0, 0) to (n + 1, n + 1).
9
if 0 < j < k < n + 1
if 0 < j < k = n + 1
otherwise
Theorem 1. The Shortest Path ‘Exo’ algorithm gives an optimal assortment.
Corollary 1. The complexity of the Shortest Path ‘Exo’ algorithm is O(n3 ).
Note that it is necessary to define the nodes as pairs of products as one needs to know which
product is to the left and which product is to the right of a given product in order to compute its
contribution to the expected profit. Also, note that our result does not make any assumption on the
distribution of customer valuation F . In the context of a firm determining the optimal tradeoff between
variety and leadtime for horizontally differentiated products, Alptekinoglu and Corbett (n.d.) also use
a shortest path formulation to solve for the optimal product line. The efficiency of the Shortest Path
‘Exo’ algorithm makes our method to identify the optimal assortment attractive. We use an example
in next section to illustrate how to use the algorithm.
3.3 Example and Properties of the Optimal Solution
The shortest path ‘Exo’ algorithm is most useful when the retailer faces a large number of candidate
products, but the small example below illustrates the underlying mechanism and provides valuable
insights.
Example 1. A retailer can choose from three vertically differentiated products with ~c = (5, 4.5, 50),
~q = (30, 36, 100) and ~r = (15, 15.5, 80). She knows that the distribution of customer valuations follows
distribution F (θ) = 1 − (1 − θ)b , where b > 0 with support [0, 1]. She needs to decide what products to
offer in order to maximize the profit.
The distribution of customer valuations F (θ) = 1 − (1 − θ)b is a common distribution to model
consumer preferences; see, for example, Debo et al. (2005) and Sundararajan (2004). It corresponds
to a beta distribution2 with parameter a = 1 and b > 0. Note that the uniform distribution on
[0, 1] is a special case of this distribution obtained by setting b = 1. If b > 1, then the distribution
function is concave, meaning that there are more customers with low valuations. If 0 < b < 1, then
the distribution function is convex, meaning that there are more customers with high valuations.
Using the enumeration method, one would need to consider seven assortments: {1}, {2}, {3}, {1, 2},
{1, 3}, {2, 3}, and {1, 2, 3} and compute the expected profit for each one. Figure 2 shows the graph.
(0,0) and (4,4) are the source and destination nodes respectively. Table 2 shows the correspondence
2
The probability density function is B(θ; a, b) =
θ a−1 (1−θ)b−1
,
β(a,b)
10
where β(a, b) =
R1
0
xa−1 (1 − x)b−1 dx.
between the valid path and assortment. The network contains four valid paths which correspond to
the following assortments: {1}, {2}, {3}, and {1, 3}. Therefore, compared to the enumeration method,
we are able to exclude three assortments. In Figure 2, the arc costs are computed using K = 0 and
b = 1. In this case, applying the Shortest Path ‘Exo’ algorithm gives the optimal path (0, 0) → (0, 1) →
(1, 3) → (3, 4) → (4, 4), which corresponds to the assortment {1, 3}, and gives an optimal expected
profit of $6.43.
-2.14
(0,1)
0
(0,0)
0
(1,3)
-5
(2,3)
(0,2)
-6.26
(1,4)
0
-4.29
(2,4)
0
0
(4,4)
0
(0,3)
-6
(3,4)
Figure 2: Graph for Example 1 with K = 0 and b = 1.
Valid path
(0, 0) → (0, 1)
(0, 0) → (0, 1)
(0, 0) → (0, 2)
(0, 0) → (0, 3)
→ (1, 3) → (3, 4) → (4, 4)
→ (1, 4) → (4, 4)
→ (2, 4) → (4, 4)
→ (3, 4) → (4, 4)
Assortment
{1, 3}
{1}
{2}
{3}
Table 2: Mapping between path and assortment for Example 1.
We now compare the optimal assortments for different values of the fixed cost, i.e., K = 0, 0.1, or
0.5, and different distributions of customer valuations, i.e., b = 1/2, 1 or 2. Table 3 shows the optimal
assortment S ∗ for each possible combination of K and b. We see that the optimal assortment varies
with the customer valuation distribution F since it varies with b (we will show that this result is not
true when K = 0 in the endogenous prices case). In particular, consider the three cases when K = 0:
with b = 1/2, many customers have a high valuation of quality and it is optimal to offer only product
3, which is the most profitable product. When b = 1, the distribution of quality valuation is uniform
and it is optimal to add product 1 to the assortment in order to increase total demand, even though
it also leads to some customers switching from product 3 to the less profitable product 1. Finally
when b = 2 and most customers have low valuations of quality and it is most important to capture the
greatest possible market, so the optimal assortment is to offer only product 2, which has the lowest
price-to-quality ratio. Moreover, when b = 1, we see that, as the fixed cost changes from K = 0.1 to
K = 0.5, the optimal assortment changes from {1, 3} to {2}. Therefore, retailers need to be aware of
any change in the fixed cost as it can have a great impact on the optimal assortment. As expected, the
expected profit decreases with b since a decrease in b implies that more customers have high valuation
for quality.
11
K=0
K = 0.1
K = 0.5
b = 1/2
S ∗ = {3}
EΠ∗ = 13.41
S ∗ = {3}
EΠ∗ = 13.32
S ∗ = {3}
EΠ∗ = 12.91
b=1
S ∗ = {1, 3}
EΠ∗ = 6.43
S ∗ = {1, 3}
EΠ∗ = 6.23
S ∗ = {2}
EΠ∗ = 5.74
b=2
S ∗ = {2}
EΠ∗ = 3.57
S ∗ = {2}
EΠ∗ = 3.47
S ∗ = {2}
EΠ∗ = 3.07
Table 3: Optimal assortments and optimal expected profit values as a function of K
and b in Example 1.
Another interesting observation is with regard to the relative attractiveness of the products in the
optimal assortment. We say that product i dominates product j if ci < cj , qi > qj , ri > rj , and
ri
qi
<
rj
qj
and that a product is dominated if there exists at least one product in {1, ..., n} that dominates it.
Note that
ri
qi
<
rj
qj
implies that product i would claim a higher market share than dominated product
j if it was offered exclusively. In Example 1, product 1 is dominated by product 2 and yet product 1
is included in the optimal solution when b = 1 and K = 0 or 0.1. The intuition is as follows: product
1 is offered alongside product 3 which is the most lucrative product (i.e., the one with the highest
profit margin) because it brings in extra demand without cannibalizing the sales of product 3 too
much. While offering product 2 instead of product 1 would bring in more demand and a higher profit
margin on these customers, it would cannibalize the sales of product 3 too much and result in less
profit overall. Hence, a dominated product is never included in an assortment alongside the product
which dominates it, but it can be included instead of it, when doing so reduces the cannibalization of
a more profitable product. This example also suggests that one cannot eliminate dominated products
as they might be included in the optimal assortment (in the next section we show that this property
does not hold when prices are decision variables) and this is true whether K = 0 or K > 0.
4 The Endogenous Prices Case
This section presents the model when prices are endogenous. We discuss the optimal solution structure
and develop several efficient algorithms to identify the optimal assortment. The choice of which
algorithm to use depends on the value of the fixed cost and the nature of the distribution of customer
valuations.
12
4.1 Model
In the endogenous prices model, only ~
q and ~c are given and fixed. The retailer needs to determine
the assortment S and the selling price rj for product j ∈ S. Note that it is possible to set the selling
price of a product so high that no customer buys it, that is, such that the product has a zero purchase
probability. In this case, the product should not be included in the assortment so as to save on the fixed
cost. Therefore we regard ~r as the only decision variable in this problem and define the corresponding
assortment as the set of the products with positive purchasing probability given ~r. As in the exogenous
model, each customer observes ~r and ~
q then chooses the product that gives him the highest utility as
long as it is positive.
Let h(θ) =
f (θ)
1−F (θ)
be the failure (or hazard) rate of distribution F , and η(θ) =
1
h(θ)
be the inverse
failure (or hazard) rate. F is an increasing failure rate (IFR) distribution if h′ (θ) ≥ 0 or equivalently
η ′ (θ) ≤ 0 for all θ. In this section, we assume that F is an IFR distribution. This assumption is satisfied
by most common distributions, e.g., uniform, normal, logistic, chi-squared, exponential, Laplace, and
beta distributions with a = 1. Note that this assumption is not required in the exogenous prices case.
Let θj =
rj −rj−1
qj −qj−1
for j = 1, ..., n. A customer with valuation θj gets the same utility from products
j − 1 and j. Without loss of generality, we assume that the prices are set such that
θ ≤ θ1 ≤ ... ≤ θn ≤ θ̄.
(9)
because for any set of prices that does not satisfy this condition, there exists a set of prices that does,
with the same purchase probability for each product and the same total expected profit. Let Pj (~r)
be the purchase probability for product j. We have Pj (~r) = F (θj+1 ) − F (θj ) and Pn (~r) = 1 − F (θn ).
Let S(~r) denote the assortment, that is, the set of products with Pj (~r) > 0 given price vector ~r.
Given (θ1 , ..., θn ) satisfying (9), we have S(~r) = {j = 1, ..., n : θ ≤ θj < θj+1 < θ̄}. We write the
P
expected profit as EΠ(~r) = nj=1 Pj (~r)(rj − cj ) − K|S(~r)|. The firm’s objective is to find ~r∗ such that
EΠ∗ = EΠ(~r∗ ) = max~r EΠ(~r). We use S ∗ = S(~r∗ ) to denote the optimal assortment.
Note that there is a one-to-one correspondence between ~r and ~θ = (θ1 , ..., θn ). Let θ~ be the vector
corresponding to ~r, and S(~
θ) be the assortment corresponding to ~θ. Hence, we can rewrite the expected
P
profit function as a function of ~
θ only as EΠ(~θ) = nj=1 [1−F (θj )] [θj (qj − qj−1 ) − (cj − cj−1 )]−K|S(~
θ)|.
13
The retailer’s profit maximization problem is
maxθ~ EΠ(~θ)
(10)
s.t. θ ≤ θ1 ≤ θ2 ≤ ... ≤ θn ≤ θ̄
4.2 Results
In Section 3.2, we observe that, when prices are exogenous, the products in the optimal assortment
have some nice properties regarding prices and price-quality ratios. Those properties enable us to
develop an efficient algorithm to identify the optimal set of products to offer. Similarly, when prices
are decision variables, we obtain the properties of the optimal assortment then use them to develop
efficient solution methods. Let S ∗ = {j1 , ..., jm }, such that j1 < j2 < ... < jm be an optimal assortment
of products with positive purchase probability.
Lemma 2. The products in an optimal assortment S ∗ satisfy the following three conditions:
cj − cjm−1
cj1
cj − cj1
< 2
< ... < m
< θ̄,
qj1
qj2 − qj1
qjm − qjm−1
cj
cj
cj1
< 2 < ... < m < θ̄,
qj1
qj2
qjm
cj1 < cj2 < ... < cjm .
(11)
(12)
(13)
Lemma 2 shows that both the variable costs and cost-quality ratios of products in the optimal
assortment are strictly increasing in the quality levels. These conditions allow the retailer to price the
products so that she can extract maximum surplus from consumers. Lemma 3 provides a method to
compute the optimal prices r ∗ , once S ∗ is known.
Lemma 3. The values of θj∗i for i = 1, ..., m are obtained by solving:
θj∗i = η(θj∗i ) +
cji − cji−1
qji − qji−1
for i = 1, ..., m.
(14)
∗
if k < jm and θk∗ = θ̄ if k > jm . The optimal prices are
where j0 = 0. For k ∈
/ S ∗ , we have θk∗ = θk+1
obtained as follows:
rj∗i =
i
X
θj∗x (qjx − qjx−1 ), for i = 1, ..., m.
x=1
14
(15)
Note that products which are not in S ∗ are not offered and therefore Lemma 3 does not provide
a selling price for them. Using these optimal prices, we establish some properties of the optimal
assortment.
Corollary 2. The products in an optimal assortment S ∗ satisfy the following two conditions:
rj∗1 < rj∗2 < ... < rj∗m ,
(16)
rj∗1 − cj1 < rj∗2 − cj2 < ... < rj∗m − cjm .
(17)
Corollary 2 reveals that, in an optimal assortment, both product prices and profit margins are
strictly increasing in the quality levels. Intuitively, if the profit margin of a product is less than
another product with lower quality, the retailer can improve the profit by either changing the product
price or removing the product.
In the exogenous prices case, we notice that the optimal assortment may include dominated products
and therefore these products cannot be deleted before using the Shortest Path ‘Exo’ algorithm. We
now examine if the same result holds when prices are endogenous. Because prices are no longer fixed,
we have to modify the definition of dominated products slightly and say that product i dominates
product j if ci < cj , qi > qj , and a product is dominated if there exists at least one product that
dominates it. The following result shows that dominated products are never included in the optimal
assortment.
Lemma 4. An optimal assortment S ∗ does not contain any dominated product.
This result is particularly useful if the number of products n is very large, since one can eliminate
dominated products before using the algorithm below. Identifying dominated products can be done in
O(n2 ).
In a different context, Deneckere and McAfee (1996) show that, under some conditions, it is profitable to introduce a product with a lower quality and higher variable cost than an existing product.
However, they note in Lemma 3 on page 168, that this result does not hold when the ratio of how
a customer values the product with low quality to how he values the product with high quality is
nondecreasing in θ. In our model, that ratio is equal to
θql
θqh
=
ql
qh
where ql and qh are the value of
low quality and high quality respectively, which is nondecreasing in θ. Therefore, their result does not
apply to our setting. Using the properties from Lemma 2, we develop a method to find the optimal
solution in polynomial time by modeling the problem of finding S ∗ as a shortest path problem.
15
ALGORITHM: Shortest Path ‘Endo’
• Step 1. Construct node set V , which consists of the following nodes:
V
=
cj
ci
(0, i) : 1 ≤ i ≤ n and θ ≤
< θ̄ ∪ {(j, n + 1) : 1 ≤ j ≤ n and
< θ̄}
qi
qj
ci
cj
∪ {(0, 0), (n + 1, n + 1)} ∪ (i, j) : 1 ≤ i < j ≤ n and
<
< θ̄ ,
qi
qj
• Step 2. Construct arc set by adding an arc from node (i, j) ∈ V to (l, k) ∈ V to set A if j = l = k or
cj −ci
qj −qi
c −c
ck −cl
qk −ql
< θ̄ if k < n + 1 and qjj −qii < θ̄ if k = n + 1 and l < k.

 K − [θ (q − q ) − (c − c )] [1 − F (θ )]
k k
j
k
j
k
• Step 3. Compute the arc costs: C(i,j),(j,k) =

0
j = l < k and
<
where θk is the solution to θk = η(θk ) +
ck −cj
qk −qj
if k < n + 1
if k = n + 1
.
• Step 4. Solve the shortest path problem from (0, 0) to (n + 1, n + 1).
Theorem 2. Shortest Path ‘Endo’ Algorithm gives an optimal assortment.
Corollary 3. The complexity of the Shortest Path ‘Endo’ algorithm is O(n3 ).
Once S ∗ is obtained by solving Shortest Path ‘Endo’ Algorithm, we can use Lemma 3 to obtain
the optimal prices. Note that the Shortest Path ‘Endo’ Algorithm has the same complexity as the
Shortest Path ‘Exo’ Algorithm of Section 3. However, in practice, the number of nodes and arcs is
generally greater in the endogenous case since there are fewer conditions on the node set, and, as a
result, solving the Shortest Path ‘Endo’ Algorithm generally takes longer than solving the Shortest
Path ‘Exo’ Algorithm.
4.3 Example and Properties of the Optimal Solution
To illustrate how the Shortest Path ‘Endo’ Algorithm works, we show how to obtain the optimal
assortment using the same data as in Example 1.
Example 2. (Cont’d from Example 1) Let ~c, ~q and the customer valuation distribution F be as in
Example 1. The retailer needs to decide what products to offer and how to price the offered products.
In this case, the network, shown in Figure 3, contains ten nodes and five valid paths. Table 4 maps
the valid paths to assortments {1}, {2}, {3}, {1, 3},and {2, 3}, so that there is one more valid path than
in the exogenous case. Compared to the enumeration method, we are able to exclude two assortments.
16
(1,3)
-2.32
0
(0,1)
0
-5.21
-1.34
(0,0)
-6.89
(1,4)
0
(2,3)
(0,2)
0
(0,3)
0
0
(2,4)
0
(4,4)
-6.25
0
(3,4)
Figure 3: Graph for Example 2 with K = 0 and b = 1.
Valid path
(0, 0) → (0, 1)
(0, 0) → (0, 2)
(0, 0) → (0, 1)
(0, 0) → (0, 2)
(0, 0) → (0, 3)
→ (1, 3) → (3, 4) → (4, 4)
→ (2, 3) → (3, 4) → (4, 4)
→ (1, 4) → (4, 4)
→ (2, 4) → (4, 4)
→ (3, 4) → (4, 4)
Assortment
{1, 3}
{2, 3}
{1}
{2}
{3}
Table 4: Mapping between paths and assortments in Example 2.
In Figure 3, the arc costs are computed using K = 0 and b = 1. In this case, applying the
Shortest Path ‘Endo’ algorithm gives the optimal path (0, 0) → (0, 2) → (2, 3) → (3, 4) → (4, 4) which
corresponds to the assortment {2, 3} and gives an optimal expected profit of $8.23.
As in exogenous prices case, we now compare the optimal assortments for different values of the
fixed cost, i.e., K = 0 or 0.5 and different distributions of customer valuations, i.e., b = 1/2, 1 or 2.
Table 5 shows the optimal assortment S ∗ for each possible combination of K and b.
K=0
K = 0.5
b = 1/2
S ∗ = {2, 3}
r2∗ = 20.25, r3∗ = 75
EΠ∗ = 13.93
S ∗ = {2, 3}
r2∗ = 20.25, r3∗ = 75
EΠ∗ = 12.93
b=1
S ∗ = {2, 3}
r2∗ = 20.25, r3∗ = 75
EΠ∗ = 8.23
S ∗ = {2, 3}
r2∗ = 20.25, r3∗ = 75
EΠ∗ = 7.23
b=2
S ∗ = {2, 3}
r2∗ = 20.25, r3∗ = 75
EΠ∗ = 3.21
S ∗ = {2}
r2∗ = 20.25
EΠ∗ = 2.51
Table 5: Optimal assortments, price vectors, and expected profit values as a function of
K and b in Example 2.
As expected, we see that expected profit is decreasing in K and b. Notice that the optimal assortment varies with b when K > 0, but not when K = 0. Lemma 5 in Section 4.4 shows that this result
for K = 0 is true in general.
4.4 Special Cases
Advances in information technology has made it possible for some online retailers to sell the their
products with a negligible fixed cost. Moreover, the distribution of customer valuations F (θ) = 1 −
(1 − θ)b , which is a common distribution to model consumer preferences, has a linear inverse hazard
17
rate function. In this section, we explore more results for the following special cases of zero fixed cost
and/or linear inverse hazard rate function.
4.4.1 Special case 1: Zero Fixed Cost
When K = 0 we are able to obtain stronger conditions on the cost and quality levels of the products
included in the optimal assortment.
Lemma 5. If K = 0, S ∗ is optimal if and only if the conditions from Lemma 2 are satisfied along with,
cj − ck
ck − cji
,
≥ i+1
qk − qji
qji+1 − qk
ck − cjm
≥ θ̄,
qk − qjm
for i = 1, ..., m − 1 and k = ji + 1, ..., ji+1 − 1,
(18)
for k = jm + 1, ..., n.
(19)
Further, the optimal assortment S ∗ is unique.
Lemma 5 implies that the products which are not included in S ∗ have a higher cost-quality ratio
than products in S ∗ with a lower quality level.
While more than one assortment may satisfy the conditions of Lemma 2 (in particular, if an
assortment does, so do all of its subsets), only one assortment satisfies these conditions along with (18)
and (19) (remember that S ∗ is defined as the optimal set of products with strictly positive purchase
probability). Also, the optimal assortment remains the same whatever the distribution of customers’
valuation is (as long as it is IFR) as stated in Corollary 4.
Corollary 4. If K = 0, S ∗ does not depend on the distribution of customer valuation F .
This is a surprising result since the non-zero arc costs depend on F in the Shortest path ‘Endo’
algorithm, Therefore, one would expect the shortest path and the corresponding assortment to vary
with F as in the exogenous prices case. We provide an intuition for this result in next section.
For the case when K = 0, we use the properties in Lemma 5 to develop a more efficient algorithm.
For each product i, starting with a fictitious product 0, we identify the next product to be included in
the optimal assortment by looking for a product which satisfies conditions (11) and (18). We examine
candidate products one by one, starting from the highest quality product (i.e., product n) down to
product i + 1. After identifying the optimal set S ∗ , we compute the optimal prices using Lemma 3.
18
Note that the algorithm assumes that dominated products have been deleted from the list of possible
products to offer. The algorithm is formally stated as follows.
ALGORITHM: Zero Fixed Cost Algorithm
• Step 0. S ∗ = ∅, i = 0.
• Step 1. If i < n, {
For j := n down to i + 1, {
If
cj −ci
qj −qi
< θ̄ AND
ck −ci
qk −qi
>
cj −ck
qj −qk
for k = i + 1, ..., j − 1 {
S ∗ := S ∗ ∪ {j}, i := j and back to step 1. } } }
• Step 2: Use Lemma 3 to obtain ~r∗ .
Proposition 1. The set S ∗ obtained from Zero Fixed Cost Algorithm is optimal when K = 0.
Corollary 5. The complexity of the Zero Fixed Cost Algorithm is O(n2 ).
In practice the Zero Fixed Cost algorithm can be used whenever the advertising costs and other
fixed cost are negligible, which is more likely to be true for online retailers. Barghava and Choudhary
(2001) previously studied this setting. They provide conditions under which the optimal assortment
contains all n products, i.e., S ∗ = {1, ..., n} and conditions under which it contains only the product
with the highest quality, i.e., S ∗ = {n}. In contrast, our work provides an efficient algorithm to identify
the optimal solution for any (~c, ~
q ). Further, we show in Section 6 that our Zero Fixed Cost Algorithm
is very fast.
There is a nice graphical interpretation for the optimal solution when K = 0: on a two dimensional
graph the functions θqj − cj for j = 1, ..., n are drawn as a function of θ, the optimal assortment
corresponds to the set of products that belong to the upper envelope of the lines in the positive
quadrant for θ ∈ [θ, θ]. Figure 4 shows the corresponding graph for Example 2 with K = 0 and b = 1.
From the graph, we can see the optimal assortment is {2, 3} since these two products belong to the
upper envelope. We formalize this observation in the following Corollary.
Corollary 6. When K = 0, j ∈ S ∗ if and only if there exist at least two values of θ ∈ [θ, θ] such that
θqj − cj = maxi=1,...,n(θqi − ci ).
19
θq −c
Product 1
Product 2
Product 3
0
1 θ
Figure 4: Graphical interpretation for S ∗ in Example 2 with K = 0 and b = 1.
4.4.2 Special case 2: Linear Inverse Rate Function
From Example 2 above, we observe that the prices of products 2 and 3 are the same whether K = 0 or
K = 0.5, provided they are being offered in the optimal assortment. Next we formally prove that this
property holds whenever the inverse hazard rate function η(θ) is linear in θ. Examples of distributions
having linear inverse hazard rate include exponential distributions and F = 1 − (1 − θ)b with b > 0.
As mentioned earlier, the distribution of F = 1 − (1 − θ)b is a common distribution to model consumer
preference and the uniform distribution is a special case of this distribution.
Lemma 6. Suppose η(θ) = αθ + β, where α ≤ 0 and β are constants. The optimal price of any product
in S ∗ = {j1 , ..., jm } with j1 < ... < jm depends only on its own cost and quality, i.e.,
rj∗i =
cji + βqji
1−α
for i = 1, ..., m.
(20)
We use this result to develop a faster algorithm for the case of linear η(θ).
Proposition 2. Suppose η(θ) = αθ + β, where α ≤ 0 and β are constants. Solving (10) is equivalent to
solving (5) with selling prices rj =
cj +βqj
1−α
for j = 1, ..., n.
By Proposition 2, the problem with endogenous prices can be solved using the algorithm for the
exogenous prices case when the inverse hazard rate function is linear. This result is useful because our
solution method for the exogenous case is generally faster than that for the endogenous case (despite
the fact that they have the same theoretical complexity). This is because the set of nodes in Shortest
Path ‘Exo’ is generally smaller than the set of nodes in Shortest Path ‘Endo’.
20
4.4.3 Special case 3: Zero Fixed Cost and Linear Inverse Rate Function
In the special case where K = 0 and η(θ) is linear, the following proposition provides an another
method to obtain the optimal solution.
Proposition 3. When K = 0 and η(θ) = αθ + β, where α ≤ 0 and β are constants, j ∈ S ∗ if and only
if there exist at least two values of θ ∈ [θ, θ] such that θqj − rj = maxi=1,...,n(θqi − ri ) where ri =
ci +βqi
1−α
for i = 1, ..., n.
Proposition 3 shows when K = 0 and η(θ) is linear, the optimal assortment can also be obtained
graphically by looking for the upper envelope of the utility curves drawn in the positive quadrant for
θ ∈ [θ, θ] for each product using prices rj =
cj +βqj
1−α
for j = 1, ..., n as shown in the following example.
Note that this method has the same complexity as the Zero Fixed Cost Algorithm. The following
example illustrates how it works.
Example 3. (Cont’d from Examples 1 and 2) Let ~c, ~q and the customer valuation distribution F (θ) be
as in Examples 1 and 2. We have η(θ) =
1−θ
b
so the inverse hazard rate is linear in θ (with α = − 1b
and β = 1b ). When b = 1, we obtain $17.5, $20.25 and $75 respectively for the prices of products 1,
2 and 3. In Figure 5, each line corresponds to the utility that customers get from each product when
they are offered at these prices. The upper envelope of the three lines in the positive quadrant only
intersects the lines from products 2 and 3, therefore {2, 3} is the optimal assortment.
Utility
Product 1
Product 2
Product 3
0
1 θ
Figure 5: Upper envelope graph for Example 3.
In summary, we have developed a number of efficient algorithms to identify the optimal assortment
when prices are endogenous. Table 6 provides a summary of our solution methods as a function of K and
η(θ). Note that all the solution methods can be prefaced by the elimination of dominated products
from the set of products to consider since an optimal assortment does not contain any dominated
product.
21
K>0
K=0
Non-linear inverse rate function η(θ)
Shortest Path ‘Endo’
O(n3 )
Zero Fixed Cost Algorithm
O(n2 )
Linear inverse rate function η(θ)
use Shortest Path ‘Exo’ with (20)
O(n3 )
Upper envelope with (20)
O(n2 )
Table 6: Solution methods to identify the optimal assortment.
5 Discussion
5.1 Insights
In Sections 3 and 4 we show how to obtain the optimal solution efficiently for the exogenous and
endogenous prices settings. In this section, we discuss some interesting properties of these solutions
and compare these two cases.
We have shown that the optimal assortment in the exogenous prices case may contain dominated
products while the optimal assortment in the endogenous prices case does not. This is because the
retailer who sets prices is able to increase the expected profit by pricing dominated products high
enough so that no customer buys them. Including a product with zero purchase probability in the
assortment does not increase the profit, therefore, the optimal assortment does not contain dominated
products. In contrast, the retailer who does not set prices may want to include dominated products
because the prices she is working with are generally sub-optimal (in the sense that they are different
from the prices that she would choose if she could) and it may be wise to offer a dominated product
instead of the product that dominates it when the dominating product would cannibalize the sales of a
more profitable product too much. Note that such sub-optimal prices may not arise if the manufacturers
are strategic in setting their MRSP and sell their products only to that one retailer.
Another interesting observation is about how the distribution of customer valuations affects the
optimal assortment. When the fixed cost is negligible (i.e., K = 0) and prices are endogenous, the
optimal assortment does not vary with the distribution of customer valuations (as long as it is IFR).
On the contrary, the optimal assortment may vary with the distribution of customer valuations in
the exogenous prices case. The underlying reason is as follows. When prices are decision variables,
Lemma 5 gives a unique optimal assortment. The optimal selling prices of these products vary with
the distribution of customer valuations and as a result, so do their purchasing probabilities and profit
margins. This is done in such a way that maximizes the total expected profit. In contrast, in the
exogenous case, only the purchasing probabilities change when the distribution of customer valuations
changes. Since the prices are often suboptimal, the retailer needs to modify the assortment in order
22
to maximize the total expected profit. Note that, in the endogenous prices case, the retailer who only
has incomplete information about the distribution of customer valuations can still identify the optimal
assortment as long as the distribution is known to be an IFR distribution. However, setting optimal
prices requires knowledge of the distribution.
We use Example 4 below to further analyze how the optimal assortments in the two cases differ.
Example 4. A retailer can choose from two vertically differentiated products with ~c = (1, 0.5) and
~q = (20, 40). The distribution is F (θ) = 1 − (1 − θ)b with support [0, 1], b = 6 and K = 0.
When the prices are fixed with ~r = (2, 19), it is optimal to offer both products. However when the
retailer is free to set prices, it is optimal to offer only product 2 at a price of $6.56.
This example shows that the optimal assortment in the exogenous prices case can contain two
products i and j such that ci > cj and qi < qj , that is, one product has a higher quality and a lower
price than another product. By Lemma 2, this is not true in the endogenous price case.
From the comparison of the optimal assortments in Examples 1 and 2, along with Example 4,
we find that the optimal assortment in the exogenous prices case can be larger or smaller than the
optimal assortment in the endogenous prices case and that one is not necessarily a subset of the other.
In particular, a product which is included in the optimal assortment when prices are fixed may be
dropped when the retailer is able to set her own prices. It is therefore necessary for a retailer who
acquires the freedom of setting selling prices to re-evaluate her whole assortment and re-optimize using
the appropriate algorithm.
Example 5 illustrates some interesting properties of the optimal assortment as the set of candidate
products to choose from shrinks.
Example 5. A retailer can choose from three vertically differentiated products with ~c = (2, 6, 10) and
~q = (10, 14, 20). The distribution of customer valuations is uniform over [0, 1] and K = 0. The optimal
assortment when prices are endogenous is S ∗ = {1, 3} and optimal prices are r1∗ = 6 and r3∗ = 15. Now
suppose that product 1 is no longer available, that is, the retailer can only offer a subset of {2, 3}. In
that case, the optimal assortment is {2, 3} and the optimal prices are r2∗ = 10 and r3∗ = 15.
From this example, we find that, if a product from the optimal assortment becomes unavailable, for
example, because the manufacturer discontinues its production, then it may be optimal for the retailer
to include new products in the assortment. This result provides a possible explanation for why some
23
manufacturers would continue to offer products which are not included in the optimal assortment when
the retailers can choose from the full list of potential products (another possible explanation is that
the manufacturer also sells his products to other retailers operating in different markets and for whom
the optimal assortment is different because their cost structure is different). Note that this property is
also true in the exogenous prices case. To see this, consider the counterpart of example 5 where prices
are exogenous and the price vector ~r = (6, 10, 15) and notice that the optimal assortments are the
same. Further, Proposition 4 shows that, when prices are endogenous and K = 0, it is never optimal
to drop a product from the optimal assortment which has not been discontinued.
∗ be the optimal assortment when the set of potential products to
Proposition 4. Suppose K = 0. Let SN
∗ denote the optimal assortment when the set of potential products to choose
choose from is N . Let SN
′
∗ ∩ N ′) ⊆ S ∗ .
from is N ′ ⊆ N . We have (SN
N′
5.2 Robustness of the results
In this section, we discuss the robustness of our results to some of our modeling assumptions, namely,
our choice of utility and fixed cost functions.
First we examine our assumption regarding the utility function. We assume that a customer with
valuation θ gets a linear utility θqj −rj from buying one unit of product j. All of our results continue to
hold if the utility function has the following generalized form: θφ(qj ) − rj , where φ(·) is increasing and
concave, as suggested by Mussa and Rosen (1978). This is because this simply involves a redefinition
of the units in which ‘quality’ is measured as noted by them. If the utility function is generalized to
the form ξ(θ)qj − rj , where ξ(·) is increasing and concave, our results apply as well. This is because
the problem can be solved using a linear utility function θ ′ qj − rj , where θ ′ has distribution F ′ defined
as follows: F ′ (θ ′ ) = F (ξ −1 (θ ′ )), for θ ′ ∈ [ξ(θ), ξ(θ)]. For our results from the endogenous prices case
to apply, we need to further show that F ′ is IFR when F is IFR. The hazard rate h′ of F ′ is given
−1
′
)
by h′ (θ ′ ) = h(ξ −1 (θ ′ )) dξ dθ(θ
where h(ξ −1 (θ ′ )) is non-decreasing in θ ′ since F is IFR and
′
dξ −1 (θ ′ )
dθ ′
is
increasing in θ ′ since ξ(·) is increasing and concave. Hence h′ (θ ′ ) is non-decreasing and F ′ is also IFR.
Second, we examine our assumption regarding the fixed cost. We assume that the fixed cost is
linear, i.e., of the form K|S|. In practice, it is possible that the fixed cost is concave in the number of
products in the assortment, i.e., of the form c(|S|) where c′ ≥ 0 and c′′ ≤ 0. In that case, all our results
continue to hold except Theorems 1 and 2 as it is no longer possible to solve the problem as a shortest
path problem. This is because the contribution of a product to the expected profit now depends on
24
the total number of products in the assortment.
Our results also partially extend if we consider the following expected profit function as in Cachon
et al. (2005):
EΠ(S) =
X
(rj − cj )Pj (S) − c(Pj (S)),
(21)
j∈S
where c(·) is such that c′ ≥ 0 and c′′ ≤ 0 and can be interpreted as the cost of stocking a product. All
of our results in the exogenous prices case continue to hold (see Lemma 7 in the Appendix for a proof)
with the following straightforward modification to the arc costs in the shortest path algorithm:
C(i,j),(j,k)
 h i
rk −rj
rj −ri
rk −rj
rj −ri

c
F
−
F
−
(r
−
c
)
F
−
F

j
j
qk −qj
qk −qj


qj −qi h
i qj −qi
rj −ri
rj −ri
=
c 1 − F qj −qi
− (rj − cj ) 1 − F qj −qi




0
if 0 < j < k < n + 1
if 0 < j < k = n + 1
otherwise
However our results in the endogenous prices case do not hold anymore. The solution from the
FOCs do not necessarily constitute a local maximum and Lemma 2 can be violated. Moreover, the
shortest path algorithm does not apply. Even if the FOCs give the optimal solution, the optimal prices
of the products in the assortment depend on the parameters of all the products which are included,
not only the adjacent ones. The intuition is that under (21), the retailer has an incentive to price the
products to distribute the purchasing probabilities unevenly so as to save on the stocking costs. For
example, it is more profitable to offer two products with purchasing probabilities 0.4 and 0.1 rather
than two products with purchasing probabilities 0.25 each. The study of the optimal assortment with
profit function (21) requires a different methodology and will be the subject of our future research.
6 Performance analysis
The purpose of this section is to demonstrate the necessity of developing our solution methods both in
terms of speed and performance. First we conduct a numerical study to compute the computational
time of the solution methods we have proposed for the exogenous and endogenous prices case and
compare them with that of the enumeration method as described below. The results show that the
enumeration method becomes quickly impractical as n gets large while our algorithms always give the
optimal solutions within a short time. Second, we show that simple heuristics can perform arbitrarily
25
badly, i.e., they can lead to an optimality gap of 100%.
In the exogenous prices case, the only decision variable is the set of products to offer. The enumeration method is to list the 2n possible assortments to find the optimal one. To speed up the search
we check for each possible assortment if the conditions of Lemma 1 are satisfied and if so, we compute
expected profit for that assortment. The optimal assortment is the one that yields the highest expected
profit of all the ones which satisfy the conditions of Lemma 1. In the endogenous prices case, we need
to determine the assortment and the selling prices. The enumeration method consists in listing the 2n
possible assortments and checking if the conditions of Lemma 2 are satisfied. If so, we use Lemma 3 to
set the prices and compute expected profit. The optimal assortment is the one that yields the highest
expected profit of all the ones which satisfy the conditions of Lemma 2.
We conduct a numerical study in order to compare the computational time of the enumeration
method with that of the solution methods we have proposed. We implement the enumeration method
and our algorithms in Matlab 7.6 on a Dell Precision T5500 Workstation with a 64bit Quad Core Intel
Xeon Processor E5530 with 2.4GHz and 6 Gb of RAM.
In all problem instances we assume that the distribution of customer valuations has a uniform
distribution on [0, 1]. We first vary the number of products n between 4 and 10 and vary the quality,
cost and price vectors are as follows. We set q = (20, 20 + β, 20 + 2β) where β ∈ {1, 2, ..., 10}, ci = αqi2
n
o
0.002
0.01
for i = 1, ..., n where α ∈ 0.001
, ri = γci for i = 1, ...n where γ ∈ {1.1, ..., 2} and
,
,
...,
qn
qn
qn
K = kc1 where k ∈ {0, 0.1, ..., 0.9}. These values are chosen to guarantee the following conditions are
satisfied: (i)
ci
qi
≤ 1 for i = 1, ..., n in the endogenous prices case, (ii)
ri
qi
≤ 1 for i = 1, ..., n and ri > ci
for i = 1, ..., n in the exogenous prices case. These are necessary (but not sufficient) conditions for the
products to be included in the optimal assortment.
The results show that the coefficient of variation of computational time is small (< 4%), meaning
that the computational time does not vary much from one problem instance to the other. Therefore
we only run one problem instance for large values of n. We take β = 1, α =
0.001
qn ,
K = 0, and γ = 2
for n = 11 to 22.
Table 7 reports the computational time (in seconds) of the different methods for this one problem
instance. We see that the computational time of the enumeration is totally impractical for n larger
than 20, e.g., over 16 days in the exogenous prices case when n = 22, while our optimal algorithms
give the optimal solutions in less than one second.
26
n
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Exogenous case
Enumeration
Shortest path Alg
0.007
0.003
0.017
0.006
0.039
0.009
0.089
0.013
0.205
0.018
0.447
0.027
0.989
0.035
2.204
0.045
4.953
0.056
11.733
0.071
31.102
0.088
99.469
0.108
304.229
0.128
1,093.058
0.153
4,214.674
0.181
16,947.186
0.214
69,902.111
0.246
292,917.660
0.282
1,421,247.700
0.378
Enumeration
0.020
0.051
0.119
0.283
0.655
1.474
3.275
7.216
15.802
34.215
73.679
158.058
336.449
715.170
1,517.688
3,199.214
6,756.454
14,180.477
31,438.924
Endogenous case
Shortest path Alg
0.002
0.003
0.005
0.008
0.011
0.015
0.020
0.026
0.033
0.041
0.052
0.069
0.077
0.092
0.108
0.127
0.149
0.173
0.297
Zero FC Alg
0.002
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.011
0.012
0.014
0.015
0.017
0.019
0.022
0.023
0.025
0.244
Table 7: Computational time (in seconds) of enumeration method, shortest path
method and zero fixed cost algorithm in exogenous and endogenous prices case.
Barghava and Choudhary (2001) studied the conditions under which it is optimal to stock only the
product with the highest quality level and the conditions under which it is optimal to offer all n products
in the endogenous prices case. These two solutions can be used as heuristics when the optimality
conditions described by Barghava and Choudhary (2001) are not satisfied. In the endogenous prices
case, we set the prices according to Lemma 3. Note that the heuristic which offers all n products may
give a solution in which some of the products get a zero purchase probability, so that these products are
effectively not included in the assortment. In the event that the solution given by a heuristic generates
negative expected profit, we set the profit equal to zero. We measure the performance of a heuristic
using the optimality gap, computed as OGH =
EΠ∗ −EΠH
,
EΠ∗
where ΠH is the profit obtained with the
heuristic. The following example shows that the optimality gap for the two heuristics can be as high
as 100%.
Example 6. Let n = 2, q = (2.10, 5.54), c = (6.15, 9.68) and K = 0.5. It is optimal to offer only product
1 and price it at 4.12 in order to get an expected profit of 0.17. The two heuristics return an expected
profit of zero because offering only product 2 or offering both products 1 and 2 yields a negative value
of expected profit.
In most practical settings, the number of potential products to consider is substantially greater
than 20 and simple heuristics may lead to a very large optimality gap; therefore there is a need for
fast, optimal solution methods like ours.
27
7 Case Study
The goal of this section is to demonstrate the applicability of our model in practice. We present two
numerical examples which are calibrated using realistic data. Because of the restrictive assumptions
of our model (i.e., one quality dimension, no competition) and the lack of data on all the relevant
parameters (in particular the variable and fixed costs), we recognize the limitations of this case study.
Nevertheless we believe that this section provides a roadmap on how one might conduct an in-depth
study of the assortment decisions with vertically differentiated products. In the two examples we
present, we focus on one product line from a single manufacturer, such that the products only differ
in one or two vertically differentiated attributes.
Exogenous prices example: Apple’s iPad sold at Target or Best Buy
We consider the problem of a retailer (such as Target or Best Buy) who carries Apple’s iPad product
line (first generation). On September 24, 2010 it was announced that Target would start selling Apple’s
iPad at “the same price that Apple sells the device for” except for “special sales promotion” (e.g.,
purchases with a Target credit card).3 Hence, this is an example of an exogenous prices case. As of
November 2010, the iPad comes in six different models and Target and Best Buy are selling all six
models. Table 8 shows their respective selling prices:
Index
1
2
3
4
5
6
Model
wifi 16GB
wifi 32GB
wifi 64GB
wifi+3G 16GB
wifi+3G 32GB
wifi+3G 64GB
Selling price
$499
$599
$699
$629
$729
$829
Variable cost
$449
$539
$629
$566
$656
$746
Percentage of sales
22.85%
18.17%
12.98%
9.36%
14.16%
22.47%
Estimated quality level
74.16
88.44
102.02
92.58
105.94
118.42
Table 8: Selling prices, variable costs and percentage of sales and estimated
quality levels for Apple’s iPad product line given α = 10% and b = 3.
In order to apply our model to this product category, we need to estimate the following parameters:
rj , cj , qj for j = 1, ..., 6, µ, F and K. For the selling prices rj , we take the values given in column 3
(Selling price) of Table 8. In order to calculate the variable cost cj we used the fact that Apple’s gross
profit margin on the iPad varies between 42.9% and 55.1%, which gives us an upper bound of 25% on
the profit margin of the retailer given that Apple is unlikely to give away more than half of its profit
margin.4 Let α denote the retailer’s profit margin (as a percentage of the selling price). We consider α
3
4
“Target to begin selling iPad in October” http://news.cnet.com/8301-13506 3-20017559-17.html?tag=mncol;4n
Source: Broadpoint AmTech estimates.
28
in {5%, 10%, 15%, 20%, 25%}. In column 4 (Variable cost) of Table 8, we present the values of variable
cost that result from assuming α = 10%. We use µ = 200 million units for the market size, where
the term ‘market’ refers to the number of people (in the United States) who would gladly take an
iPad if given to them for free.5 This number is an estimate of the American adult population size and
was obtained using the total US population size in 2010 (310 million) and the proportion of people
between the age of 15 and 65 (67%).6 We use the following distribution of customers valuation for
quality F (θ) = 1 −
1
b
b (θ − θ)
θ
on support [0, θ] because of its flexibility, i.e., by varying b, we can easily
vary the relative proportion of customers with a low and high valuation for quality. We set θ = 10 and
considered values for b ∈ {1, ..., 6} so that the dollar value of one unit of quality is between 0 and 10,
and is skewed to the left, meaning that more customers have a lower valuation of quality. We believe
that this is reasonable: relatively few customers - the ones who are devoted fans of Apple products have a valuation of quality close to 10, most consumers have a valuation closer to zero. In what follows
we present our results for the case of b = 3.
Estimating the quality levels qj is a tricky task since the quality of the iPad is not solely a function
of the memory size and whether or not it has 3G capabilities. We also need to evaluate the intrinsic
quality of the product, which is based on its design, functionalities, etc. Because our model allows for
only one dimension for quality, we combine all of these features into a single one measure. Our approach
is to look for values of qj , j = 1, ..., 6 such that the purchasing probabilities given by our model match
the actual percentages of the population buying each iPad model. Table 8 shows the actual percentage
of sales from each model.7 As of November 2010, Apple has sold over 7 million iPads in the United
States, this corresponds to about 3.5% of the total American market given our estimate of µ, hence
we multiply the percentages from Table 8 by 3.5%. We obtain the values in column 6 (Estimated
quality level) of Table 8. First we check that the quality levels make sense: the values we obtained
are increasing in the memory size for each model and the 3G version always has a quality level higher
than the wifi only version of the same size. Also we believe these quality levels are consistent with the
value of the iPad coming mostly from the design, convenience of use and functionality, irrespective of
the memory size.
Then we use our results from Section 3 to obtain the the optimal assortment given all the parameters
mentioned above. When K = 0 we find that the optimal assortment is the one that contains all six
5
This is different from the traditional use of the term ‘market’ as the number of potential buyers given the current
price.
6
Source: http://www.census.gov/main/www/popclock.html and www.google.com/publicdata. Retrieved November
17, 2010
7
Source: http://gizmodo.com/5530111/which-ipad-did-you-buy. Retrieved November 17, 2010.
29
models of iPads, hence justifying the assortment chosen by Target and Best Buy. This assortment
is optimal for 0 ≤ K ≤ 50, 652, which means that unless the fixed cost of adding a product in the
assortment is greater than $50,652, it is optimal to offer all six versions of the iPad. Note that this
value was obtained assuming a market size µ of 200 million units, so, more generally the threshold
value of K above which it is optimal to stop offering product 4 is given by 0.00025326µ. We believe
that in the case of retailers such as Target or Best Buy, it is likely that their fixed cost is minimal,
given that most of the advertising is conducted by Apple themselves. The following table shows the
optimal assortment as a function of the fixed cost K.
K
[0; 50, 652]
[50, 652; 199, 306]
[199, 307; 613, 023]
[613, 024; 1, 830, 195]
[1, 830, 196; 13, 233, 455]
[13, 233, 456; 434, 251, 809]
Optimal assortment
{1, 2, 3, 4, 5, 6}
{1, 2, 3, 5, 6}
{1, 2, 5, 6}
{2, 5, 6}
{2, 6}
{6}
Table 9: Optimal assortment as a function of K with α = 10% and b = 3.
We find that our results are robust to our choice of parameters. In all cases, the values of qj we
obtain make sense as described above and the optimal assortment with K = 0 was to offer all 6 versions
of the iPad.
Endogenous prices example: Lexar Flash Drives sold at three online retailers
Next we consider “TwistTurn USB 2.0 JumpDrive” product category by the manufacturer Lexar.
Table 10 shows the selling prices from the manufacturer as well as three online retailers who carry the
product line.8
Index
1
2
3
4
5
6
Model number
LJDTT2GBASBNA
LJDTT4GBASBNA
LJDTT8GBASBNA
LJDTT16GASBNA
LJDTT32GASBNA
LJDTT64GASBNA
Memory size
2GB
4GB
8GB
16GB
32GB
64GB
Lexar
13.99
19.99
29.99
49.99
99.99
199.99
Selling
NewEgg
12.79
10.99
23.79
24.13
83.99
128.79
prices
J&R
9.99
10.99
19.99
29.99
59.99
129.99
Variable cost
Amazon
11.10
11.58
19.93
26.83
60.23
118.29
6.30
9.00
13.50
22.50
45.00
90.00
Table 10: Selling prices and variable costs given α = 45%.
We study the problem of the three retailers determining their assortment and prices for the product
category. Since the retailers have the freedom to charge a different price than the one suggested by
8
The data was collected directly from the firms’ websites (www.lexar.com, www.newegg.com, www.jr.com and
www.amazon.com) Retrieved November 17, 2010
30
the manufacturer, this is an example of an endogenous prices case.
For the variable costs cj , we assume that the manufacturer charges a percentage α of his own
selling price to each retailer. We use α ∈ {40%, 45%, 50%}. With these values, the average gross profit
margin of Amazon across model number, calculated as (selling price - variable cost)/variable cost, is
equal to 35.30%, 27.20% and 19.12% respectively for α = 40%, 45% and 50%. These numbers are
in the same order of magnitude as the company’s gross profit margin, as calculated from their 2010
income statement (24.00%). In the last column (Variable cost) of Table 10, we show the values for
cj obtained when assuming α = 45%. As in the iPad example, we use the following distribution of
customers valuation for quality F (θ) = 1 −
1
b (θ
θ
− θ)b on support [0, θ] and considered b ∈ {5, 7, 9, 11}
and θ in {6, 8, 10, 12}. We set the quality levels using qj = q0 + memory size of product j in gigabytes,
with q0 ∈ {0, 2, 4, 6}. So for example, q0 = 0 implies that q1 = 2, q2 = 4, q3 = 8, q4 = 16, q5 = 32
and q6 = 64. Finally we assume K = 0, which is reasonable given that we are talking about online
retailers.9 In total we considered 192 parameters sets.
In what follows, we present our results given α = 45%, θ = 6, b = 9 and q0 = 0. We find that the
optimal assortment is to offer only the 16GB and 64GB versions, i.e., S ∗ = {4, 6}. With a market size
µ normalized to 1, the optimal expected profit is equal to 1.03. Even though each online retailer offers
all six versions of the product on their website, their selling prices may be such that some models are
not purchased by any customer (this would not hurt their profit since K = 0). We refer to the set
of products that get a positive purchasing probability given our model as the ‘effective assortment’
of a retailer (see Table 11, row 2). The selling prices of the products in the effective assortments
are the only ones that matter when computing expected profit (see Table 11, row 3). We compute
expected profit generated by these effective assortments and selling prices assuming a market size µ
equal to 1 (see Table 11, row 4). Finally, for each retailer, we calculate the percentage of the optimal
expected profit they achieved (see Table 11, row 5). Interestingly we find that, NewEgg and Amazon
effectively offer the optimal assortment, but because Amazon’s pricing is closer to the optimal values,
they achieve the highest expected profit. Despite not offering the optimal assortment, J&R achieves a
high expected profit value because of their carefully chosen prices. All three retailers are within 73%
of the optimal expected profit as calculated by our model. Again, our results are robust to our choice
of parameters. In all 192 cases, we found it was optimal to offer products 4 and 6 in the assortment
and it was never optimal to offer products 2 and 3. The average percentage of the optimal profit across
9
Estimating the value of K for a brick-and-mortar retailer is tricky as it includes advertising and fixed shelf-space
related costs. Moreover the value of K must be evaluated relative to the potential market size, which is one more
parameter to estimate. For these reasons we choose to study only the endogenous prices case with K = 0 and consider
only the assortment of online retailers for which we believe it is a reasonable assumption.
31
all 192 parameter sets was 76.25%, 86.70% and 82.50% respectively for NewEgg, J&R and Amazon,
which seems to indicate that J&R’s pricing is the closest to optimal.
Effective assortment
Corresponding prices
Expected profit
% of the optimal
Optimal
{4, 6}
(25.60, 110.39)
1.03
100%
NewEgg
{4, 6}
(24.13, 128.79)
0.76
73.69%
J&R
{4, 5, 6}
(29.99, 59.99, 129.99)
0.94
90.99%
Amazon
{4, 6}
(26.83, 118.29)
1.00
96.72%
Table 11: Profit comparison given α = 45%, θ = 6, b = 9 and q0 = 0.
Note that our analysis does not take competition into account. In essence, we assume that each
retailer faces a population of loyal customers who only consider that retailer’s assortment. Also we
ignore the fact that the retailers sell other brands of USB flash drives and even other models from the
Lexar brand. Finally we assume that all three retailers face the same population of customers and pay
the same variable cost to the manufacturer for the products they purchase. All these assumptions may
not be valid in practice, which could explain the difference in assortments and prices chosen by the
retailers. We plan to explore how relaxing these assumptions impacts assortment choice in our future
research.
8 Conclusion
We study the problem of a retailer offering an assortment of vertically differentiated products to
customers who differ in their valuation of quality. We first consider the scenario where prices are
exogenously determined and the retailer’s only decision is to decide the set of products to offer. We
show that the problem of finding the optimal assortment can be modeled as a shortest path problem,
which has complexity O(n3 ). Interestingly we show that the optimal assortment may contain dominated
products, i.e., products which have a lower quality, lower selling price, and higher cost than one other
product. Second, we examine the setting where the retailer can also determine the selling prices of
products which are included in the assortment. We show that this problem can also be modeled as a
shortest path with complexity O(n3 ). However, in practice, this problem usually takes longer to solve
because the network generally contains more nodes and arcs. When the fixed cost associated with
each offered product is negligible (i.e., K = 0), we develop a more efficient algorithm, with complexity
O(n2 ). If the distribution of customer valuations has a linear inverse hazard rate we show that the
optimal prices can be obtained independently of the optimal assortment so that the optimal assortment
can be found by using the solution method developed for the exogenous prices case. We show that
32
most of our results are robust to the assumptions we made on the utility and profit functions. Finally,
our numerical studies demonstrate the need for our efficient optimal algorithms and the applicability of
our model in practice. In summary, we provide efficient methods to obtain the optimal assortment for
any quality levels and variable cost, in the presence of fixed cost and also consider the exogenous prices
case. Further, we provide a number of interesting insights and guidelines to practitioners regarding
their product mix strategy.
There are a number of interesting extensions to our work. First, we assume that the products
differ with respect to only one attribute which can be regarded a combination of a product’s many
characteristics. We are interested in explicitly considering multiple attributes to capture the complexity
of consumer choice. Second, we do not consider the case in which the retailer may have constraints when
determining the selling prices of products. For example, the retailer does not want to set a price that
is higher than a competitor’s selling price. Third, our model assumes that the retailer is a monopolist
and that the product quality levels and variable costs are fixed and determined by the manufacturer(s).
An interesting extension would be to consider a setting in which the manufacturer chooses the product
quality levels and transfer costs in anticipation of the retailer’s assortment choice or a setting in which
two or more retailers compete in prices when selling the same manufacturer’s products. Finally, our
model does not include the presence of inventory and therefore does not incorporate the impact of
stock-outs and the resulting substitution behavior of customers. We plan to explore these extensions
in our future research.
Appendix A: Proofs
Proof of Lemma 1. Notice that (6) and (7) follow directly from (1). We prove (8) by contradiction.
Let k be the smallest integer such that rjk − cjk > rjk+1 − cjk+1 . There are two cases: (1) k = m − 1
or rjk − cjk > rji − cji for i = k + 1, ..., m, or (2) otherwise. In case (1), let S = {j1 , ..., jk }. We have:
≥
>
EΠ(S) − EΠ(S ∗ )
rjk − rjk−1
(rjk − cjk )
1−F
qjk − qjk−1
"m−1 X
rji+1 − rji
rji
−
F
−F
qji+1 − qji
qji
i=k
rjk − rjk−1
(rjk − cjk )
1−F
qjk − qjk−1
"m−1 X
rji
rji+1 − rji
−F
−
F
qji+1 − qji
qji
i=k
− rji−1
− qji−1
− rji−1
− qji−1
#
rjm − rjm−1
(rji − cji ) + 1 − F
(rjm − cjm ) ,
qjm − qjm−1
(rjk
33
#
rjm − rjm−1
− cjk ) + 1 − F
(rjk − cjk ) = 0.
qjm − qjm−1
Hence, S ∗ cannot be optimal. In case (2), let l ∈ {k+2, ..., m} be the smallest integer rjk −cjk ≤ rjl −cjl
and let S = {j1 , ..., jk , jl , ..., jm }.
≥
>
EΠ(S) − EΠ(S ∗ )
rjk − rjk−1
rjk+1 − rjk
rjk − rjk−1
rjl − rjk
F
−F
(rjk − cjk ) − F
−F
(rjk − cjk )
qjl − qjk
qjk − qjk−1
qjk+1 − qjk
qjk − qjk−1
l−1 X
rji − rji−1
rji+1 − rji
−F
(rji − cji ),
−
F
qji+1 − qji
qji − qji−1
i=k+1
rjk − rjk−1
rjk+1 − rjk
rjk − rjk−1
rjl − rjk
F
−F
(rjk − cjk ) − F
−F
(rjk − cjk )
qjl − qjk
qjk − qjk−1
qjk+1 − qjk
qjk − qjk−1
rjk+1 − rjk
rjl − rjl−1
rjl − rjk
rjl − rjk
−F
(rjk − cjk ) − F
−F
(rjl − cjl ) = 0
− F
qjl − qjk
qjk+1 − qjk
qjl − qjl−1
qjl − qjk
Where the inequality comes from
rjk −rjk−1
qjk −qjk−1
<
rjl −rjk
qjl −qjk
<
rjl+1 −rjl
qjl+1 −qjl
and the fact that (rji − cji ) <
rjk − cjk < rjl − cjl for i = k + 1, ..., l − 1. Hence S ∗ cannot be optimal and we have a contradiction.
Proof of Theorem 1. Let S = {j1 , . . . , jm } with j1 < ... < jm and p(S) be the path that
corresponds to S, where p(S) = (0, 0) → (0, j1 ) → (j1 , j2 ) → ... → (jm−1 , jm ) → (jm , n + 1) →
(n + 1, n + 1). Let P be the set of paths. Every set S that satisfies the condition of Lemma 1 corresponds to a path in P and vice versa. The cost of path p(S), C(p(S)) is equal to C(0,0),(0,j1 ) +
P
C(0,j1 ),(j1 ,j2 ) + . . . + C(jm−1 ,jm ),(jm ,n+1) + C(jm ,n+1),(n+1,n+1) , which is equal to mK − m−1
i=1 (rji −
r −r
i
h
r −r
i
h r −r j
j
j
j
j
j
− (rjm − cjm ) 1 − F qjm −qjm−1 , which is equal to −EΠ(S).
cji ) F qji+1 −qji − F qji −qji−1
i+1
i
i
m
i−1
m−1
Hence, min C(p(S)) = min [−EΠ(S)] = max EΠ(S).
p∈P
S
S
Proof of Corollary 1. The complexity of a shortest path problem in an acyclic network is
bounded by the number of arcs (see Ahuja et al. (1993) page 107). The graph has a special structure,
because there is possibly an arc between two nodes (i, j) to (l, k) only if j = l. There are at most j
nodes that end with product j ∈ {1, ...n} and these are connected to at most n + 1 − j nodes that
P
start with product j. Therefore the maximum number of arcs is equal to 2n + nj=1 (n + 1 − j)j, where
2n is the maximum number of nodes leaving the source or ending in the destination node. Hence, the
maximum number of arcs is O(n3 ).
P
Proof of Lemma 2. We can write the expected profit function as EΠ(~θ) = m
i=1 [1−F (θji )][θji (qji −
qji−1 )−(cji −cji−1 )]−mK. Taking the derivative of the expected profit with respect to rji for i = 1, ..., m,
i
h
cji −cji−1
we get ∂EΠ
. At the first order conditions (FOC), we have
=
f
(θ
)
η(θ
)
−
θ
+
ji
ji
ji
∂rj
qj −qj
i
i
i−1
θji = η(θji ) +
cji − cji−1
qji − qji−1
34
for i = 1, ..., m.
(22)
Also,
∂EΠ2
∂rji ∂rjk F OC
=


0







for k ∈
/ {i − 1, i}
−f (θji )
′
qji −qji−1 [η (θji )
f (θji )
′
qji −qji−1 [η (θji )
− 1] > 0
for k = i − 1
− 1] < 0
for k = i
. Therefore, the Hessian matrix
is negative definite and the solutions to (22) determine the maximum. Since F is an IFR distribution,
η(θ) is a decreasing function and therefore, each equation in (22) gives a unique solution. Let θj∗i , for
i = 1, ..., m, denote the solutions to (22). By (9) and the definition of S ∗ , the solution must satisfy
θj∗1 < θj∗2 < ... < θj∗m < θ̄, therefore we need
cj1
qj1
<
cj2 −cj1
qj2 −qj1
< ... <
cjm −cjm−1
qjm −qjm−1
< θ̄, which proves that
(11) is a necessary condition. Finally, (12) and (13) follow directly from (11).
Proof of Lemma 3. The proof of Lemma 2 shows that the first order conditions determine the
optimal solutions; therefore, the values of θj∗i for i = 1, ..., m can be obtained by solving (14). Moreover,
we can get the prices ~r∗ by using θj∗ =
∗
rj∗ −rj−1
qj −qj−1
for j = 1, ..., n.
Proof of Corollary 2. Notice that (16) follows directly from (15). To prove (17), notice that
P
rj∗i − cji = ik=1 (qjk − qjk−1 )η(θj∗k ) for i = 1, ..., m which is strictly increasing in i.
Proof of Lemma 4. First we show that S ∗ = {j1 , ..., jm } with j1 < ... < jm cannot contain
a dominated product along with a product that dominates it. Suppose not (contradiction), then
there must exist ji and ji+1 such that ji+1 dominates ji . In this case we would have
cji+1 −cji
qji+1 −qji
<
0 which contradicts (11) from Lemma 2. Now, suppose S ∗ contains a dominated product but the
product(s) that dominate(s) it are not in S ∗ . In this case, there must exists ji for some i = 1, ..., m
which is dominated by k where ji < k < ji+1 . Given that ck < cji and qk > qji ,
ck −cji
qk −qji
< 0.
Let S = {j1 , ..., ji−1 , k, ji+1 , ..., jm }. We know from Lemma 3 that, in S ∗ , θj∗i are obtained using
(14). In S, let θjx = θj∗x for x = 1, ..., i − 1, i + 1, ...., m and θk = θj∗i for i = 1, ..., m. We have
EΠ(S)− EΠ(S ∗ ) = [1− F (θj∗i )][θj∗i (qk − qji )− (ck − cji )]− [1− F (θj∗i+1 )][θj∗i+1 (qk − qji )− (ck − cji )] = (qk −
i
h
i
h
c −c
c −c
c −c
qji ) [1 − F (θj∗i )] θj∗i − qkk −qjji − [1 − F (θj∗i+1 )] θj∗i+1 − qkk −qjji . Let ψ(θ) = [1 − F (θ)] θ − qkk −qjji .
i
i
i
i
h
c −c
c −c
We have ψ ′ (θ) = f (θ) η(θ) + qkk −qjji − θ < 0 given that qkk −qjji < 0 and the FOC (22). Therefore,
i
i
EΠ(S) − EΠ(S ∗ ) > 0 due to the fact that θj∗i+1 > θj∗i and qk > qji . Hence, S ∗ is not optimal, which is
a contradiction.
Proof of Theorem 2. The proof is similar to that of Theorem 1 and is therefore omitted.
Proof of Corollary 3. The proof is similar to that of Corollary 1 and is therefore omitted.
Proof of Lemma 5. Let j0 = 0. We first show that (18)-(19) are necessary conditions. We prove
(18) by contradiction, that is, suppose that S ∗ is optimal but there exists k ∈ {ji + 1, ..., ji+1 − 1}
35
cji+1 −ck
ck −cji
qk −qji < qji+1 −qk . Using the
cj
−cj
c −c
that qkk −qjji < qji+1 −qji . Let S
i
i+1
i
a
b
c
d
a
b
a+c
b+d
such that
fact that
implies
= {j1 , ..., ji , k, ji+1 , ..., jm }. We know from Lemma 3 that, in
<
implies that
<
ck −cji
qk −qji
S ∗ , θj∗i are obtained using (14). In S, let θji = θj∗i for i = 1, ..., m. If
ck −cji
qk −qji ;
the solution to θk = η(θk ) +
when a, b, c, d > 0, this
>
cji −cji−1
qji −qji−1 ,
let θk be
otherwise, let θk be a value such that θj∗i < θk < θj∗i+1 . We have
EΠ(S) − EΠ(S ∗ ) = [1 − F (θk )][θk (qk − qji ) − (ck − cji )] − [1 − F (θj∗i+1 )][θj∗i+1 (qk − qji ) − (ck − cji )] = (qk −
i
h
i
h
c −c
c −c
c −c
qji ) [1 − F (θk )] θk − qkk −qjji − [1 − F (θj∗i+1 )] θj∗i+1 − qkk −qjji . Let ψ(θ) = [1 − F (θ)] θ − qkk −qjji .
i
i
i
h i
cj −cj
c −c
c −c
We have ψ ′ (θ) = f (θ) η(θ) + qkk −qjji − θ . If qkk −qjji > qji −qji−1 , then ψ ′ (θk ) = 0 and ψ ′ (θ) < 0
i
i
i
i−1
for θ > θk ; otherwise, ψ ′ (θ) ≤ 0 for θ ≥ θk , so ψ(θ) is decreasing in θ for θ ≥ θk . Therefore
EΠ(S) − EΠ(S ∗ ) > 0 due to the fact that θj∗i+1 > θk and qk > qji . Hence, S ∗ is not optimal, which is a
contradiction. Finally we prove that (19) is a necessary condition. Suppose (contradiction) that S ∗ is
ck −cjm
qk −qjm
optimal with jm < n and there exists k ∈ {jm+1 , ..., n} such that
< θ̄. Let S = {j1 , ..., jm , k}.
In S, let θji = θj∗i for i = 1, ..., m and let θk be any value such that θk <
ck −cjm
qk −qjm
< θ. We have
c −c
EΠ(S) − EΠ(S ∗ ) = [1 − F (θk )] [θk (qk − qjm ) − (ck − cjm )] = [1 − F (θk )](qk − qjm ) θk − qkk −qjjm > 0.
Therefore
S∗
cannot be optimal and we have a contradiction.
m
Now we prove that (11), (18) and (19) are sufficient conditions by showing only one set S ∗ satisfies
∗
1 } such that j 1 < ... < j 1
these conditions. Suppose we have two sets S1∗ = {j11 , ..., jm
m1 and S2 =
1
1
2 } such that j 2 < ... < j 2 satisfying (11), (18) and (19). If we have j 1 = j 2 for k = 1, ..., m
{j12 , ..., jm
1
m2
1
k
k
2
and m2 > m1 , then (19) applied to S1∗ implies that
−cj 2
m1
m1 +1
qj 2
−qj 2
m1
m1 +1
cj 2
≥ θ̄. However this contradicts (11)
for S2∗ . Therefore, we exclude this case. Without loss of generality, let i be the smallest integer such
1 ≥ j 2 , in this case let l be the
that jk1 = jk2 for k = 1, ..., i − 1 and ji1 < ji2 . We have two cases: (1) jm
i
1
1 < j2.
smallest integer such that jl1 ≥ ji2 ; (2) jm
i
1
Let us consider Case (1) first. Since S1∗ satisfies (11), we have
cj 1
−cj 1
qj 1
−qj 1
l−1
l−1
set
i−1
<
i−1
S2∗
cj 1 −cj 1
l
l−1
qj 1 −qj 1
l
l−1
ck −cj 2
i−1
qk −qj 2
<
i−1
that
1
jl−1
satisfies (18) and
and (ii) together imply
to
cj 2 −ck
i
qj 2 −qk
i
>
ck −cj 2
i−1
qk −qj 2
1
qj 1
1
< ... <
cj 1 −cj 1
l
l−1
qj 1 −qj 1
l
, which implies,(i)
l−1
1
2 , if j 1 = j 2 , this contradicts with (18) for
for i = 1, ..., l − 2. Due to ji−1
= ji−1
i
l
as it is equivalent to
S1∗
cj 1
<
l−1
qj 2 −qj 1
i
i
1 . So let us assume that j 1 > j 2 . The fact
with k = jl−1
i
l
i
cj 2 −cj 1
cj 1 −cj 1
cj 2 −cj 1
ji2 , q i2 −q l−1
j
j1
>
>
, which contradicts (18) for set S2∗ because it is equivalent
i
cj 2 −cj 1
i
cj 2 −ck
qj 2 −qk ,
l−1
cj 1 −cj 1
i−1
l−1
qj 1
l−1
l−1
−qj 1
cj 1 −cj 2
i
l
qj 1 −qj 2
i
l
, implies (ii)
i
l−1
qj 2 −qj 1
i
>
l−1
l
l−1
qj 1 −qj 1
l
. Equations (i)
l−1
i−1
1 .
with k = jl−1
i−1
1 < j 2 . From (19) of set S ∗ , we get
Let us now consider Case (2), i.e., jm
1
i
1
for set S2∗ we have
cj 1 −cj 2
m1
i−1
m1
i−1
qj 1 −qj 2
≥
cj 2 −cj 1
i
m1
qj 2 −qj 1
i
.. This implies that
m1
36
cj 1 −cj 1
m1
i−1
m1
i−1
qj 1 −qj 1
cj 2 −cj 1
i
m1
qj 2 −qj 1
i
> θ̄. From (18)
m1
> θ̄ However, by (11) for S1∗ ,
we must have
cj 1 −cj 1
m1
i−1
qj 1 −qj 1
m1
i−1
<
cj 1 −cj 1
m1
m1 −1
qj 1 −qj 1
m1
< θ̄, which is a contradiction. Therefore, there exists only one
m1 −1
set satisfying (11), (18) and (19), and thus, if a set S ∗ satisfies these conditions, then this set must be
the only optimal set.
Proof of corollary 4. Follows directly from Lemma 5.
Proof of Proposition 1. Let S ∗ = {j1 , ..., jm } with j1 < ... < jm . To prove the optimality of S ∗
we show that the conditions of Lemma 5 are satisfied. Directly from the second inequality in Step 1
in the algorithm, we obtain that (18) of Lemma 5 is satisfied.
We now prove by induction that (19) of Lemma 5 also holds. When the algorithm stops, i = jm .
cjm +1 −cjm
qjm +1 −qjm
c −c
that qyy −qjjm
m
≥ θ̄, since otherwise the
If jm = n, then (19) is trivially true. If jm < n, it must be that
algorithm would have added jm + 1 to set S ∗ . Now let us assume
and x ∈ {jm + 1, ..., n − 1} and prove that
cx+1 −cjm
qx+1 −qjm
≥ θ̄ for y ∈ {jm + 1, ..., x}
≥ θ̄. Because x + 1 was not added to set S ∗ when
i = jm , it must be that at least one of the following condition is true:
cx+1 −cjm
qx+1 −qjm
≥ θ̄ or
ck −cjm
qk −qjm
≤
cx+1 −ck
qx+1 −qk
for some k ∈ {jm + 1, ..., x}. If the first condition is true, then we are done. If the second condition
ck −cjm
qk −qjm
is true, then
cx+1 −cjm
qx+1 −qjm
≤
cx+1 −cjm
qx+1 −qjm .
By the induction hypothesis, we know that
ck −cjm
qk −qjm
≥ θ̄, therefore
≥ θ̄. Hence, (19) of Lemma 5 also holds.
We now prove that (11) of Lemma 2 holds. Since, the algorithm adds jm , it must be that
cjm −cjm−1
qjm −qjm−1
< θ̄. Now assume (contradiction) that (i)
cjx −cjx−1
qjx −qjx−1
cjx −cjx−1
qjx −qjx−1
≥
cjx+1 −cjx
qjx+1 −qjx
for some x ∈ {1, ..., m − 1}.
cj
−cj
This implies that
≥ qjx+1 −qjx−1 . Since jx was added by the algorithm, it must be that
x+1
x−1
cjx −cjx−1
cjx+1 −cjx−1
∗
<
θ̄.
Therefore,
(ii)
qjx −qjx−1
qjx+1 −qjx−1 < θ̄. The fact that the algorithm added jx to set S also
ck −cj
cj −cj
ck −cj
c −c
implies that qk −qjx−1 > qjjx −qkk for k = jx−1 + 1, ..., jx − 1, which implies qk −qjx−1 > qjx −qjx−1 for k =
x
x
x−1
x−1
x−1
cj
−cj
ck −cj
jx−1 + 1, ..., jx − 1 and therefore using (i) we have qk −qjx−1 > qjx+1 −qjx for k = jx−1 + 1, ..., jx − 1.
x
x−1
x+1
Combining the first and third sets of equations and using the fact that
that
a
b
>
c+e
d+f
when a, b, c, d, e, f > 0, we obtain (iii)
ck −cjx−1
qk −qjx−1
>
cjx+1 −ck
qjx+1 −qk
Similarly, the fact that the algorithm added jx+1 to set S ∗ implies that
jx + 1, ..., jx+1 − 1, which implies
(i) we have
cjx −cjx−1
qjx −qjx−1
>
cjx+1 −ck
qjx+1 −qk
cjx+1 −ck
qjx+1 −qk for k = jx +
cj
−ck
ck −cj
(iv) qk −qjx−1 > qjx+1 −qk
x−1
x+1
>
inequalities, we obtain
cjx+1 −cjx
qjx+1 −qjx
a
b
>
c
d
and
a
b
>
e
f
implies
for k = jx−1 + 1, ..., jx − 1.
ck −cjx
qk −qjx
>
cjx+1 −ck
qjx+1 −qk
for k =
for k = jx + 1, ..., jx+1 − 1 and therefore using
1, ..., jx+1 − 1 Combining the first and third sets of
for k = jx + 1, ..., jx+1 − 1. However, if (i), (ii), (iii)
and (iv) were true, then the algorithm would have added jx+1 instead of jx when i = jx−1 . Therefore
we have a contradiction. Since (12) and (13) of Lemma 2 follow from (11), we have proven that all five
conditions of Lemma 5 are satisfied and therefore S ∗ is optimal.
37
Proof of Corollary 5. In the worst-case scenario, Step 1 has to be repeated for i going from 0
to n. In each iteration we may have to consider up to n − i values of j.
Proof of Corollary 6. First we show that for every k which satisfies (18), there cannot be at least
two θ values such that θqk − ck = maxi=1,...,n(θqi − ci ). Since qk > qji , we have θqk − ck < θqji − cji for
ck −cji
qk −qji . Since qk < qji+1 , we have θqk − ck
cj
−ck
c −c
(18), qkk −qjji ≥ qji+1 −qk . Next we show that
i
i+1
cji+1 −ck
qji+1 −qk .
θ<
< θqji+1 − cji+1 for θ >
by
for every k that satisfies (19), there cannot be at least two
We get the result since,
values of θ < θ such that θqk −ck = maxi=1,...,n (θqi −ci ). Since qk > qjm , we have θqk −ck < θqjm −cjm for
θ<
ck −cjm
qk −qjm .
We get the result since by (19),
ck −cjm
qk −qjm
≥ θ. Finally we show that for every ji , i = 1, ..., m,
there exists at least two values of θ ∈ [θ, θ] such that θqji − cji = maxk=1,...,n(θqk − ck ). From (11),
c −c
cji+1 −cji
cji −cji−1
ji
ji−1 cji+1 −cji
<
and
therefore,
θq
−
c
=
max
(θq
−
c
)
for
θ
∈
,
j
j
k=1,...,n
k
k
i
i
qj −qj
qj
−qj
qj −qj
qj
−qj .
i
i−1
i+1
i
i
Proof of Lemma 6. From (14), we get θj∗i =
cj −cj
i
i−1
ji −qji−1
β+ q
1−α
i−1
i+1
i
. Substituting this expression into (15)
gives (20), which implies that rj∗i only depends on cji and qji .
Proof of proposition 2. Lemma 6 indicates that the prices of products in the optimal assortment are determined by (20), i.e., such that each price is a function of its own cost and quality level
only. Therefore, we can solve the assortment planning problem with exogenous prices defined by
(20).
Proof of Proposition 3.
From Proposition 2, we know that the products that are offered
are priced using (20). Let S = {j1 , ..., jm } with j1 < ... < jm be the set of products such that
Pji ({1, ..., n}) > 0 when rji =
cji +bqji
1−a .
In other words, we have Pk ({1, ..., n}) = 0 for k ∈
/ S. We show
that S satisfies the conditions of Lemmas 2 and 5. First, from the definition of S it must be that
rj1
qj1
<
rj2 −rj1
qj2 −qj1
< ... <
rjm −rjm−1
qjm −qjm−1
< θ̄. Using condition (20), we get
cj1
qj1
<
cj2 −cj1
qj2 −qj1
< ... <
cjm −cjm−1
qjm −qjm−1
< θ̄,
which is (11) from Lemma 2. The other conditions from Lemma 2 follow from this one. For k such
that ji < k < ji+1 , By definition of S, it must be that
ck −cji
qk −qj
>
cji+1 −ck
qji+1 −qk ,
S, it must be that
rk −rji
qk −qji
>
rji+1 −rk
qji+1 −qk
Using condition (20), we get
which is (18) from Lemma 5. Now consider k such that k > jm . By definition of
rk −rjm
qk −qjm
≥ θ̄. Using condition (20), we get
ck −cjm
qk −qjm
≥ θ̄, which is condition (19) from
Lemma 5. It follows that S satisfies all the conditions of Lemma 5 and therefore it is optimal.
∗ is such that for j ∈ S ∗ , there exists at least two
Proof of Proposition 4. By Corollary 6, SN
N
∗ ∩ N ′ , it must be true
values of θ ∈ [θ, θ] such that θqj − cj = maxi∈N (θqi − ci ). Consider j ∈ SN
that there exists at least two values of θ ∈ [θ, θ] such that θqj − cj = maxi∈N ′ (θqi − ci ). Therefore,
∗ .
j ∈ SN
′
38
Lemma 7. Lemma 1 holds when EΠ is given by (21).
Proof. The proof follows the same steps as that of Lemma 1 but also makes use of the following results
P
for c′ ≥ 0, c′′ ≤ 0: c(x)+c(y) ≥ c(x+y) and ni=1 c(xi ) ≥ c(x1 +...+xk α)+c(xk (1−α)+xk+1 +...+xn )
when 0 ≤ α ≤ 1, and 1 < k < n.
Acknowledgment
The authors are grateful for the feedback received in the informal seminar series of the University of
Texas at Austin.
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