assortment planning with inventory considerations in
TRANSCRIPT
Assortment Planning with Inventory
Considerations in Supply Chains
by
LEI XIE
Department of Business AdministrationDuke University
Date:
Approved:
Fernando Bernstein, Co-supervisor
Gurhan Kok, Co-supervisor
Jeannette Song
Serhan Ziya
Dissertation submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in the Department of Business Administration
in the Graduate School of Duke University2010
Abstract(Warlockery)
Assortment Planning with Inventory Considerations in
Supply Chains
by
LEI XIE
Department of Business AdministrationDuke University
Date:
Approved:
Fernando Bernstein, Co-supervisor
Gurhan Kok, Co-supervisor
Jeannette Song
Serhan Ziya
An abstract of a dissertation submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in the Department of Business Administration
in the Graduate School of Duke University2010
Copyright c© 2010 by LEI XIEAll rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial Licence
Abstract
The dissertation addresses optimization problems related to the management of prod-
uct variety with inventory considerations. The objective is to provide insights and
tools to help companies determine the optimal level of product variety they should
produce/stock and maximize revenues by strategically choosing assortments offered
to customers. The dissertation comprises of three chapters. The first chapter ex-
plores the role of component commonality in product assortment decisions. We study
how the use of component commonality in a production system affects the size of
the optimal assortment and the resulting optimal component stocking decisions. In
particular, we characterize the structure of the optimal assortment under various
configurations of the bill of materials and provide conditions that establish when
the optimal variety level increases or decreases relative to a system without common
components. We find that the effect of commonality on profit and on the level of
variety is stronger when demand exhibits negative correlation. The second chapter
explores dynamic assortment customization strategies with limited inventories, in
the contexts of a revenue management problem for a firm offering an assortment of
products. The goal of the chapter is to determine a dynamic assortment policy in a
setting with limited inventories and heterogeneous consumer preferences. The results
indicate that limiting the choice set to some customers, in consideration of the pre-
vailing inventory levels and the customers’ preferences, leads to increased revenues.
We find that, under the optimal dynamic assortment policy, each customer is al-
iv
ways offered its most preferred product variant, while it is also offered other product
variants provided that their inventory levels are large enough, i.e., above a certain
threshold level. These thresholds are increasing with inventory levels and with time.
The third chapter explores optimal assortment decisions with non-identical price and
cost parameters. We extend our results in the first chapter to a setting where prices
are weakly increasing in the net utility of the product. We also derive some novel
results related to the structure of the optimal assortment in the dedicated system
with general price and cost parameters. The goal of the project is to characterize the
optimal assortment and explore the role of component commonality with different
price and cost parameters.
v
Contents
Abstract iv
Acknowledgements viii
Introduction 1
1 The Role of Component Commonality in Product Assortment De-cisions 4
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Systems with Component Commonality . . . . . . . . . . . . . . . . . 13
1.4.1 Independent Population Model . . . . . . . . . . . . . . . . . 14
1.4.2 Trend-Following Demand Model . . . . . . . . . . . . . . . . . 16
1.4.3 Comparison of Independent Population and Trend-FollowingModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.4 Effect of Cost of Common Component . . . . . . . . . . . . . 18
1.5 General Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Dynamic Assortment Customization with Limited Inventories 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
vi
2.3.1 The Dynamic Assortment Optimization Problem . . . . . . . 34
2.4 The Optimal Assortment Policy . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Single Customer Segment and Multiple Product Variants . . . 37
2.4.2 Two Customer Segments with Two Product Variants . . . . . 38
2.5 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.1 Effect of Assortment Customization on Revenues . . . . . . . 41
2.5.2 Effect of Parameters on Threshold Levels . . . . . . . . . . . . 43
2.6 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6.1 Bounds on the Optimal Profit . . . . . . . . . . . . . . . . . . 45
2.6.2 Multiple Customer Segments and Multiple Product Variants . 47
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Assortment Decisions with Non-identical Price and Cost Parame-ters 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Extended Results of Chapter 1 with Non-Identical Price and CostParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Dedicated System with General Price and Cost Parameters . . . . . . 54
A Proofs of Main Results of Chapter 1 57
B Proofs of Supplementary Results of Chapter 1 69
C Proofs of Chapter 2 91
D Proofs of Chapter 3 107
Bibliography 113
Biography 118
vii
Acknowledgements
I owe my deepest gratitude to my supervisors, Professors Fernando Bernstein and
Gurhan Kok, whose encouragement, enthusiasm, strong support, and great patience
to explain things clearly and simply enabled me to complete this thesis.
I would also like to thank my committee members, Professors Jeannette Song
and Serhan Ziya. During my thesis-writing period, they have provided stimulating
discussions and valuable feedback.
This thesis would not have been possible without wonderful training and en-
couragement from Professors Paul Zipkin, Kevin Shang, Otis Jennings, Pranab Ma-
jumder, and Li Chen.
I offer my regards and blessings to many of my colleagues and friends to support
me during the completion of this thesis. Especially, I would like to show gratitude
to Zhengliang Xue, Fang Liu, Isilay Talay Degirmenci, and Suri Gurumurthi.
Finally, I wish to thank my family for the support they provided during my entire
life and in particular, I am forever grateful to my wife and children, Na Li, Boqian
Xie, and Lingxuan Xie, who offer a loving environment and unconditional support
all the time.
viii
Introduction
A wide assortment of variants allows customers to purchase products that more
closely match their preferences and helps retain customers for future purchases.
However, variety is costly from an operational perspective. Higher variety leads to
increased manufacturing complexity, e.g., in the form of more frequent switchovers in
production and to lower economies of scale. That is, adding more variants in the as-
sortment tends to reduce the demand for individual products and, at the same time,
increase their relative demand variability. Moreover, it requires increased ordering
and handling costs, i.e., more work is needed to display products. This reduces the
extent to which companies can take advantage of economies of scale. In addition,
larger demand variability due to a higher variety increases the likelihood of ending
the selling season with leftover inventory of several or all of the product variants.
The goal of this dissertation is to provide insights and tools to help companies deter-
mine the optimal level of product variety they should produce/stock and maximize
revenues by strategically choosing assortments offered to customers.
Using common components has been regarded as an effective operational strategy
to reduce manufacturing complexity and costs in academia and in industry. However,
little attention has been paid to the effect of commonality on product assortment
decisions. In Chapter 1, we analyze the role of component commonality in product
assortment decisions under various configurations of the bill of materials. We consider
a firm that produces multiple variants of a product. Products are assembled using
1
a combination of common and dedicated components. We characterize the optimal
assortment and derive the optimal inventory levels for the common and dedicated
components under various bill-of-material configurations. We investigate the effect
of commonality on product variety and compare its benefits under different demand
characteristics. While commonality always leads to increased profits, its effect on
the level of product variety depends on the type of commonality. If all common
components are used for the production of the entire set of products, then the optimal
variety level increases relative to the system with no commonality. However, if the
common components are used by a subset of the final products, then the optimal
variety level may decrease with commonality. We find that the effects of commonality
on profit and variety level are stronger under a demand model that exhibits negative
correlation relative to a model with independent demands.
Chapter 2 studies a revenue management problem in an online retail setting with
limited inventories and a set of substitutable products. The online retailers are able
to provide customized assortments to individual customers based on their personal
information and their browsing and purchasing history. In this chapter, we find that
an online retailer has the potential to increase revenues by strategically restricting
the set of product options it makes available to customers, even when all products
are in stock. We consider a retailer with limited inventories of substitutable products
in a retail category. Customers arrive sequentially and the firm decides which sub-
set of the products to offer to each arriving customer depending on the customer’s
preferences, the inventory levels and the time-to-go in the season. We show that the
optimal assortment policy is to offer all available products if customers are homoge-
nous. However, with multiple customer segments characterized by different product
preferences, it may be optimal to limit the choice set of some customers. That is, it
may be optimal to not offer products with low inventories to some customer segments
and reserve those units for future customers (who may have a stronger preference for
2
those products). The optimal assortment policy is a threshold policy under which a
product is offered to a customer segment if its inventory level is higher than a thresh-
old value. These thresholds are decreasing in time and increasing in the inventory
levels of other products. In a numerical study, we find that the revenue impact of
assortment customization can be significant, demonstrating its use as another lever
for revenue maximization in addition to pricing.
In Chapter 3, we extend the results of the first chapter to a system with non-
identical price and cost parameters. We assume price is linearly increasing in the
net utility of the product. We find that the optimal assortment in the dedicated
system under both demand models continues to be a popular set. The optimal
variety level increases as a common component is introduced to all products under
the trend-following demand model. The optimal assortment in the pooled system
is no longer to offer all products and turns to be a popular set which has a higher
variety level than that in the system with the common component. We also study
the structure of the optimal assortment in the dedicated system with general price
and cost parameters. For any product in the optimal assortment, all other products
with higher prices and utilities than this product are also included in the optimal
assortment.
3
1
The Role of Component Commonality in ProductAssortment Decisions
1.1 Introduction
Providing a wide assortment of variants in a product category is critical for most
manufacturers. A higher variety level tends to attract more customers, especially
in a competitive environment, as it allows them to find items that more closely
match their preferences. In turn, this helps retain customers for future purchases.
However, variety is costly from an operational perspective. Higher variety leads to
increased manufacturing complexity, e.g., in the form of more frequent switch-overs
in production, and to a more fragmented product line. That is, adding more variants
to the assortment tends to reduce the demand for individual products and, at the
same time, increase their relative demand variability. This reduces the extent to
which companies can take advantage of economies of scale. Thus, higher inventory
levels are required to maintain the same service level.
Companies employ different strategies to support a high level of variety, such as
using common components, investing in flexible capacity, and redesigning products
4
and processes to benefit from delayed differentiation – all of these are different forms
of operational flexibility. This generates a higher demand volume for the common
parts and therefore results in lower operational costs. While the benefits of imple-
menting some form of flexibility for a given line of products are well documented in
the literature, this paper explores the effect of commonality on assortment decisions.
There are three forces at play in our model: (1) Higher variety leads to increased total
demand; (2) This also leads to a higher relative demand variability for each product;
(3) At the same time, the demand aggregation effect for the common components
mitigates the increased costs associated with higher demand variability. In view of
these three effects, we investigate how the choice of an assembly configuration (or
bill of materials) affects a firm’s assortment and stocking decisions.
In particular, we consider an assemble-to-order system in which a manufacturer
produces a number of variants in one product category. Each variant is assembled
from several components. There are product-specific (i.e., dedicated) components
and components that are common to a subset of (or all) the product variants. We
consider two demand models, both introduced in van Ryzin and Mahajan (1999).
Under the independent population model, all consumers make purchasing decisions
independently from each other. Under the trend-following demand model, consumers
follow the choice made by the early buyers, which leads to a higher demand variance
for each product and to negative correlation between products, yielding the largest
risk pooling benefit from a common component. In these settings, we characterize
the optimal component inventory levels and the structure of the optimal assortment
for some special cases of the bill of materials. Based on these results, we compare a
system where all variants are produced using only dedicated components to systems
that incorporate flexibility in the form of component commonality.
We find that the effect of commonality on product assortment decisions depends
on how these components are integrated in the bill of materials. If each common
5
component used in manufacturing is shared by all product variants, then the optimal
level of variety increases with commonality. In this case, the demand pooling effect
results in high enough savings to allow the firm to introduce more variants. However,
this result does not necessarily hold for more general bills of materials. In fact, when
some of the common components are shared by a subset of the product variants, the
optimal assortment may decrease relative to that in a setting without commonality.
This reduction in the optimal assortment occurs when removing a variant that does
not use the common component leads to a sufficiently increased scale effect on the
variants that do share that component, making it more attractive to reduce the
assortment in a way that maximizes the demand pooling effect. In these cases, it
may be optimal to rationalize the product line in order to shift demand to products
that share common resources. In this respect, we explore conditions that lead to
increased or reduced assortment offerings in general systems. When commonality
does increase the depth of the optimal product assortment, we find that the extent
of this increase depends on the characteristics of market demand. Specifically, the
trend-following model, which exhibits higher demand variance and pair-wise negative
correlation between products, leads to higher increases in the set of products offered
relative to the independent population model.
Our results are relevant for product line managers (who decide which products to
offer), and supply chain managers and product designers (who make procurement or
design decisions that affect the bill-of-material configuration). If common compo-
nents are used to produce all (or most of) the products, then a wide range of variants
can be offered. If design limitations preclude the use of common components across
the entire product line, then it may be optimal to restrict the product offering to
benefit from an increased scale effect for the products that use the common compo-
nents, therefore leading to the exclusion of some products that do not share those
common components. Our results also suggest that there may be opportunities to
6
increase variety inexpensively by exploring new product designs that utilize as much
component commonality as feasible with the most popular products.
This paper builds on two streams of research: retail assortment planning and
component commonality/resource flexibility. The literature on retail assortment
planning generally focuses on the consumer choice aspect of assortment decisions
and works with the simplest production structure in which each variant is produced
using a dedicated component. van Ryzin and Mahajan (1999) consider such a
model and demonstrate that the optimal assortment consists of a certain number of
the most popular products. Other papers in this area include Smith and Agrawal
(2000), Cachon et al. (2005), Gaur and Honhon (2006), and Kok and Fisher (2007).
See Kok et al. (2008) for a detailed review of this stream of work. The literature
on component commonality (e.g., van Mieghem 1998, 2004, Bernstein et al. 2009)
and other forms of operational flexibility (e.g., Fine and Freund 1990, Tang and Lee
1997) studies inventory and common-component allocation decisions, and explores
the resulting value of flexibility for a given fixed assortment of products. Similarly,
the literature on assemble-to-order systems investigates optimal inventory policies
for a given assortment of products (Song and Zipkin 2003, Song and Zhao 2009). In
this paper, we consider a setting in which the set of products offered is an endogenous
decision based on the structure of the bill of materials that includes flexible resources
in the form of common components and on a consumer choice model like those
considered in the assortment planning literature.
Product variety has also been widely investigated in the marketing literature. For
example, Kahn (1998) provides a review of papers that explore how product variety
influences the revenue potential of a product line. The use of common components
in manufacturing may lead to lower costs, but also a to lower degree of product
differentiation from the consumers’ perspective. This tradeoff is investigated in the
product-line design literature (e.g., Desai et al. 2001 and Heese and Swaminathan
7
2006). A number of papers focus on how to create and implement product variety in
manufacturing settings (e.g., Ramdas et al. 2003). Hopp and Xu (2005) consider a
product selection problem and show that modularity always leads to higher variety.
The rest of the paper is organized as follows. Sections 1.2 describes the model set-
up and Section 1.3 present the results for the dedicated system. Section 1.4 presents
the analysis of systems with common and dedicated components, but in which the
common components are used in the production of all product variants. Section 1.5
contains an analysis of systems with more general bill of materials. Section 1.6
concludes the paper. All proofs and supplementary material are provided in an
Electronic Companion (Appendix A) and in Appendices B and C.
1.2 The Model
We consider a manufacturer that makes product line decisions regarding the set of
product variants to offer in the market. The manufacturer also decides the capacity
or inventory levels for the components used in production. Each product variant may
represent, for example, a color/size/design combination of a garment or a specific
configuration of a personal computer. The set of all variants is denoted by N =
{1, 2, · · · , N} and the firm offers a subset S ⊂ N . Each variant is produced according
to a bill of materials that dictates the components used in its fabrication. There are
product-specific (or dedicated) components and components that are common to a
subset of (or all) the variants. For example, Nike’s recently-designed “LunarLite”
foam is used in several of the company’s shoes, while a specific color may be common
to a subset of the sneakers, and a certain pattern may be specific to one product
variant. Similar examples apply to apparel and modular products, such as a personal
computer. The firm holds inventory of components and final production/assembly
time is negligible. This model structure applies to assemble-to-order systems, such
as Dell’s production of personal computers, where the final step of production takes
8
place after customers place their orders for specific product configurations. In the
case of Nike, the company has recently introduced an online system that links Nike
manufacturing partners to reduce lead times and reduce the percentage of shoes it
orders on speculation from 30% to 3%, essentially creating a make-to-order system
for the manufacturers (Holmes 2003). The model may also apply to settings in
which the manufacturer produces according to orders from retailers, provided that
final production or assembly takes place after the retailers’ orders are received.
We model demand using a consumer choice model similar to that in van Ryzin
and Mahajan (1999). The choice of a product variant within the offered assortment
is based on the Multinomial Logit (MNL) model. A consumer chooses a product
in the assortment or selects the no-purchase option (denoted by 0) to maximize her
utility. The expected utility derived from option i ∈ S ∪ {0} is given by ui. The
probability of a customer choosing option i is given by
qSi =θi∑
j∈S θj + θ0
, i ∈ S ∪ {0} ,
where θi = eui . (We refer to van Ryzin and Mahajan (1999) for details on the MNL
model.) For simplicity, we refer to θi as the utility for variant i and let Θ denote the
vector (θ1, θ2, · · · , θN). We assume that products are indexed in descending order
of their popularity (as measured by their utility parameters), i.e., θ1 ≥ θ2 · · · ≥ θN .
We assume that consumers choose a variant based on the offered assortment S
and if the selected product is not available, then the sale is backordered or lost.1 This
assumption implies a static or assortment-based substitution, therefore ignoring the
dynamics of product substitution that are based on the availability of products at
the time of the customer arrival. This model is particularly suitable for catalog
1 Because we consider a single-period setting, backorders and lost sales are equivalent from amodeling point of view and would lead to the same profit expression (except for a constant) if thebackorder penalty cost is set to equal the profit margin. It is possible to include a salvage valueand a shortage penalty cost by a simple redefinition of parameters.
9
retailers and for manufacturers selling customized products to retailers or directly
to consumers. In those settings, consumers choose from a menu of products in a
catalog or on a web site. Depending on the availability of components, the firm
either delivers that product immediately or the sale is lost. The same is the case
with apparel manufacturers that receive orders from retailers based on the offered
assortment.
Demand arrives over a single selling season. We consider two models of consumer
demand, both introduced in van Ryzin and Mahajan (1999). Under the indepen-
dent population model, denoted IP, the total number of customers interested in the
product category follows a Poisson distribution with rate λ and each customer se-
lects a variant independently of the choices made by all other customers. Then,
demand for variant i follows a Poisson process with rate λi = qSi λ. We approximate
this distribution with a Normal distribution with mean λi and standard deviation
σi = σλβi , with σ > 0 and 0 < β < 1.
We also consider the trend-following demand model, denoted TF. The trend-
following model is a stylized model construct that describes a rather extreme rep-
resentation of customer behavior for products where the trend is established by the
leader (early buyers, buyers with influence, reputable consumer reports, etc.), and
all other customers follow the trend. This model is appropriate for fashion goods,
where customers tend to follow trends set early in the season. Thus, for each vari-
ant i, demand Di is either zero (with probability 1 − qSi ) or equal to total market
demand (with probability qSi ). Total market demand is itself a Normal random vari-
able with mean λ and standard deviation σ. We assume that λ is sufficiently larger
than σ so that the probability of a negative demand realization is negligible. The
trend-following model exhibits pair-wise negative correlation between product de-
mands and a higher demand variance at the individual product level relative to the
10
IP model.2 In our setting, demand variance and correlation influence not only the
capacity decisions and the system’s profit, but also the assortment decision.
The bill of materials consists of a set of components M = {1, 2, · · · ,M} and a
production matrix matching components with end products. We assume, without
loss of generality, that a finished product requires one unit of each of its components.
We investigate special forms of the production matrix. In a dedicated system (de-
noted D), all components are product-specific. We then study a setting (denoted
C) in which each variant is produced using a dedicated component and a component
that is common to all products. (Although we assume that every product requires
two components – dedicated or common – any one of these components may repre-
sent a kit of parts or sub-assemblies.) We finally consider settings with more general
bill of materials.
As in van Ryzin and Mahajan (1999), we assume identical total production costs
and prices across all variants for tractability. The selling price of each variant is p.
In system D, the costs of the two dedicated components for each product are kd and
kc, respectively. In system C, the cost of each dedicated component is kd and the
cost of the common component is kc (this component replaces all of the dedicated
components with cost kc in system D). Under other bill of materials, the cost of
components is set so that the total unit production cost is kd + kc for all products.
The firm’s objective is to determine the assortment (which products to offer) and
the stocking levels of all components to maximize profit. We define a popular set
as the set of the n most popular products and denote it by An = {1, 2, .., n}. We
denote the optimal profit of the firm when it offers assortment S by ΠS, and define
Πndef= ΠAn . We denote the structure of the bill of materials (D or C) and the
2 Consider an assortment S. Under the TF model, Cov(Di, Dj) = −λ2qSi qSj < 0. Also, setting
σ = 1 and β = 1/2 in the IP model and σ = λ1/2 in the TF model (both corresponding to theNormal approximation to a Poisson random variable), var(Di) = λqSi under the IP model andvar(Di) = λqSi + λ2qSi (1− qSi ) under the TF model.
11
demand model (IP or TF) in the superscript of the relevant variables.
1.3 Preliminaries
We first explore the optimal assortment structure in the dedicated system. In this
setting, each product is manufactured using only dedicated components. Because a
product requires one unit of each component, the results below essentially assume a
single dedicated component per product, with an aggregate cost of k = kd + kc per
unit.
Given an assortment S, we let S(δ) = S ∪ {m} denote the set of variants in S
plus an additional variant m 6∈ S with utility δ. The resulting optimal profit is
ΠDS (δ)
def= ΠD
S(δ). van Ryzin and Mahajan (1999) show that ΠDS (δ) is a quasi-convex
function of δ (implying that, given an assortment S, it is optimal to either add the
next most popular product not in S or to not add any other variant). The paper also
shows that the optimal assortment is a popular set. (We actually prove the result
for a more general formulation of the TF model in which the total market demand
is random, whereas this is a constant in van Ryzin and Mahajan 1999. A detailed
analysis of this case and the proofs of these results are presented in Appendix A.)
This result provides a useful characterization of the optimal assortment as it reduces
the number of sets to be considered for optimization from 2N to N . Note that the
optimal assortment is not necessarily equal to the set of all possible products N .
While adding a variant increases total demand, the relative demand variability for
each variant in the assortment also increases, leading to higher inventory costs.
We next present and discuss two technical conditions that we impose throughout
the rest of the paper. Because ΠDS (δ) first decreases and then increases in δ, let δn
denote the value of δ > 0 such that ΠDn (0) = ΠD
n (δn). If the utility of the next most
popular product is greater than δn (i.e., θn+1 > δn), then it is profitable to include
12
variant n + 1 in the assortment. The quantity δn is a function of the utility vector
and of the cost and price parameters. If ΠDn (δ) is increasing, then we let δn = 0. If
ΠDn (δ) < ΠD
n (0) for all δ, then we let δn =∞. The conditions are as follows.
Condition 1. Given a set of utilities {θ1, · · · , θN}, the following holds for all n =
1, ..., N − 1:
If δn ≤ θn, then δn−1 − δn ≤ θn − θn+1.
Condition 2. δn is increasing in k as long as δn < θn.
Condition 1 implies that the profit function in the dedicated system is quasi-
concave in n (see Theorem A.1 in Appendix A). This result, together with the
optimality of a popular set, implies that the optimal assortment can be found it-
eratively by adding one product variant at a time (in a sequence given by their
popularity) starting with the empty set. Condition 2 implies that adding a product
to the assortment becomes less attractive as production costs increase. Both condi-
tions depend only on the parameters of the model. In our numerical experiments,
covering a wide range of parameter values, these conditions are always satisfied. The
conditions enable us to compare the size of the optimal assortment across various bill-
of-material configurations, but they are not required for the results on the structure
of the optimal assortment under any of the system configurations.
1.4 Systems with Component Commonality
In this section we explore settings in which the bill of materials includes dedicated
as well as common components and in which the common component is shared
by all offered product variants. This production structure features what we call
universal commonality. In its more general form, this system consists of a common
component, which can represent a kit of common components, with cost kc, and
a set of dedicated components, such that each product variant is produced using
13
the common component and a dedicated component (possibly representing a kit of
dedicated components), with cost kd. This setting is comparable to the dedicated
system, in the sense that the total cost associated with the production of a variant
is equal to k = kc + kd. Figure 1.5 (a) in Section 5 illustrates a system with this
bill-of-material configuration.
1.4.1 Independent Population Model
We begin with the independent population demand model. For a given assortment
S, let yi be the stocking level for the dedicated component corresponding to variant
i, i ∈ S, and yc the stocking level for the common component. The corresponding
profit is
ΠC,IPS ({yi}i∈S, yc) = pE[min{yc,
∑i∈S
min{yi, Di}}]−∑i∈S
kdyi − kcyc.
As demonstrated in Van Mieghem (1998), ΠC,IPS is jointly concave in ({yi}i∈S, yc) .
It is difficult to analytically compare the size of the optimal assortment between
systems D and C under the independent population demand model, primarily be-
cause of the lack of closed-form expressions for the optimal stocking levels in system
C. In the dedicated system, each component experiences the same demand as its
corresponding finished product. In contrast, in system C, the dedicated components
face the same demand stream as in system D, whereas the common component faces
a demand stream that is equal to the aggregate demand for all variants. As a result,
the profit function is not separable in the stocking quantities. By considering a sim-
plified version of the demand distribution under the IP model, the following result
shows that universal commonality increases the size of the optimal assortment.
Theorem 1.4.1. Consider a setting with N identical products (i.e., θi = θ for i =
1, ..., N and, for any assortment S with n = |S|, qSi = qS = θnθ+θ0
for any i ∈ S)
14
for sufficiently large N . Furthermore, consider the following two-point demand
distribution, defined for a given assortment S:
DSi =
{λqS + σ2, with probability λqS
λqS+σ2
0, otherwise.
(This distribution preserves the mean and standard deviation of the original Normal
distribution, with β = 1/2.) Then, ΠD,IPn+1 − ΠD,IP
n ≤ ΠC,IPn+1 − ΠC,IP
n , which implies
that the optimal assortment in the system with universal commonality contains at
least as many variants as the optimal assortment in the dedicated system.
Theorem 1.4.1 states that if the marginal gain from adding a new variant in the
dedicated system is positive, then that new variant should be included in the system
with commonality as well. This, together with the quasi-concavity of the profit
function in the dedicated system as a function of n (see Theorem A.1), allows us
to conclude that replacing a set of dedicated components with a component that
is common to all product variants results in an increase in the size of the optimal
assortment. Universal commonality leads to a broader product offering because
pooling resources for one component of each product reduces the realized overage
and underage costs for all products. This mitigates the cost of having a high level
of product variety.
For the original model with Normal demand distributions, an extensive numerical
study suggests that the optimal assortment in the system with commonality is a
popular set and that it is indeed no smaller than the optimal assortment in the
dedicated system. The study consists of 360 experiments, all with ten possible
variants and considering two possible demand rates, λ = 200 and λ = 400 (further
details are presented in Appendix A). Comparing the number of variants in the
optimal assortments in systems D and C, we find that component commonality leads
to an increase in the level of variety from an average of 6.67 variants to an average of
15
7.81 variants when λ = 200 and from an average of 7.65 variants to an average of 8.74
variants when λ = 400. The impact of commonality is larger at low demand levels
(λ = 200) because demand is more variable in those settings and inventory costs
represent a larger portion of total profit. We also find that the effect of commonality
on the level of product variety is larger in settings with a relatively high utility of the
no-purchase option and in settings in which product utilities have relatively similar
values.3 In all cases, universal component commonality effectively reduces inventory
costs, leading to higher variety levels. At the same time, in settings with larger θ0 or
with similar values of the products’ utilities, the gains in market share from adding
another product to an existing assortment are relatively larger. Hence, the firm has
a stronger incentive to increase variety in those settings, therefore amplifying the
effect of commonality on both variety level and profit. Finally, we find that when
the cost of the common component is higher relative to the corresponding cost in
the dedicated system, the impact of commonality in the increase of product variety
is more significant.
1.4.2 Trend-Following Demand Model
We now focus on the trend-following demand model. Under this model, all cus-
tomers purchase the product variant selected by the first customer. (Recall that,
at the time of making assortment and component capacity/stocking decisions, there
is uncertainty regarding which product variant will be the preferred choice in the
market.) The profit function for the firm is given by
ΠC,TFS ({yi}i∈S, yc) =
∑i∈S
qSi pEmin{yc,min{yi, D}} −∑i∈S
kdyi − kcyc.
3 Consider the following example with λ = 200, p = 10, kd = kc = 2, σ = 1, β = 0.5, θ0 = 10.For Θ = (20, 18, 16, 14, 12, 10, 8, 6, 4, 2), the optimal variety level increases from 6 in the dedicatedsystem to 7 in the system with commonality, accompanied by a 1.7% increase in optimal profit(from 953.9 to 970.1). On the other hand, for Θ = (20, 19, 18, 17, 16, 15, 14, 13, 12, 11), the optimalvariety level increases from 7 in the dedicated system to 9 in the system with commonality, leadingto a 2.1% increase in optimal profit (from 968.4 to 989.0).
16
Because it is optimal to stock less of the dedicated components than of the common
component, i.e., yi ≤ yc, we have that ΠC,TFS ({yi}i∈S, yc) =
∑i∈S(qSi pEmin{yi, D}−
kdyi) − kcyc. This function is jointly concave in ({yi}i∈S , yc). Proposition A.1
in Appendix A derives the optimal stocking quantities and the resulting optimal
profit in this setting. The result indicates that it is optimal to stock an equal
amount of the dedicated components corresponding to a subset of the most popular
variants and that that amount is itself equal to the stocking quantity of the common
component. The stocking quantities of the remaining variants decrease in order of
their popularity. Using the form of the optimal profit function and optimal stocking
levels, we characterize the effect of adding one more variant to an existing assortment,
which leads to the following result.
Theorem 1.4.2. (i) The function ΠC,TFS (δ) is quasi-convex in δ. Therefore, the
optimal assortment is a popular set. (ii) Moreover, ΠD,TFn+1 −ΠD,TF
n ≤ ΠC,TFn+1 −ΠC,TF
n ,
which implies that the optimal assortment in the system with universal commonality
is no smaller than that in the dedicated system.
This result shows that, under the trend-following model, the structure of the
optimal assortment is preserved with the introduction of universal commonality, but
the optimal level of product variety increases (weakly) relative to the dedicated
system.
1.4.3 Comparison of Independent Population and Trend-Following Models
We have shown that, under both demand models, replacing a set of dedicated com-
ponents by a common component shared by all product variants results in a larger
optimal assortment. At the same time, for a fixed set of products, the effect of
commonality on profit is more significant when the difference between the sum of
the individual products’ demand variabilities and the aggregate demand variability
17
is relatively larger, as is the case under the TF model compared to the IP model.
In our setting, however, changes in the demand model affect not only the benefits of
risk-pooling but also, possibly, the size of the optimal assortment. In this section,
we report the results of a numerical study that examines the effect of commonality
on the optimal variety level under the two demand models.
The study for the TF model is based on the same 360 experiments reported
in Section 1.4.1, with the exception that each of the nine possible cost vectors is
adjusted to ensure that the average number of variants (across all other 40 parameter
combinations) in the dedicated system is the same as that under the IP model. For
each instance, we compute the optimal assortment and profit in system D and in
system C. Table A.2 in Appendix A summarizes the results. The increase in the
optimal level of variety is generally higher in terms of frequency and average absolute
magnitude under the TF model than under the IP model. Under the IP model, the
level of variety increases by an average of 1.12 units. In contrast, under the TF
model, the level of variety increases by an average of 2.84 units. Similarly, the
percentage profit improvement due to commonality is higher under the TF model
(4.15% versus 3.55%). Based on these findings, we conclude that the effect of
commonality on the level of variety and on profit is larger under the TF model, even
when assortment decisions are endogenous.
1.4.4 Effect of Cost of Common Component
We have so far assumed that the per unit cost of a common component is the same as
its dedicated-component counterpart in the dedicated system (both equal to kc). In
many cases, designing or purchasing a common component may be more costly than
a dedicated component. Proposition B.2 in Appendix B shows that, in a system
with dedicated components, the level of variety decreases as unit costs increase.
Hence, the addition of commonality (weakly) increases the level of variety if the
18
common component is equal in cost or somewhat more expensive than the dedicated
component it replaces. However, the level of variety may decrease in the system
with commonality, relative to the dedicated system, if the common component is
significantly more expensive. Based on our numerical study, we find that the optimal
number of product variants is larger with commonality even for increases of up to
30% in the cost of the common components.
1.5 General Systems
In this section we explore systems with general bill of materials. In particular, we
consider systems in which common components may be used to produce some, but
not all, of the finished products. We consider a few representative structures of the
bill of materials, all of which are illustrated in Figure 1.5. The configuration in (a)
represents the system with universal commonality studied in Section 1.4. In (b)
and (c), there is only one common component shared by two products, while the
other products are built using dedicated components only. In (d), two products
use both dedicated and common components and the other two only use dedicated
components. In (e), all products share a common component and require an addi-
tional component, which is a dedicated component each for products 3 and 4 and
a common component for products 1 and 2. We refer to the systems in (b)-(e) as
systems with partial component commonality. The results obtained in this section
for the settings depicted in Figure 1.5 (b)-(e) allow us to derive insights regarding
the effect of commonality in product assortment decisions in systems with general
bill of materials.
We start by considering a simple setting with N product variants and M = N−1
components, in which two variants, say i and j, share a common component and all
other variants use dedicated components (as illustrated in Figure 1.5, (b) and (c)).
We compare the optimal assortment in this system with that in the system with
19
Figure 1.1: Sample structures of the bill of materials involving commonality
only dedicated components. To make the comparison meaningful, we maintain the
cost of producing each variant equal in both systems. That is, in the system with
dedicated components, the cost of each dedicated component is given by kd + kc,
while in the system with partial commonality the cost of the common component
is kd + kc and all other products continue to use the same dedicated components as
in the original system. (Recall that a popular set is defined according to the order
given by the utilities in the vector Θ.)
Proposition 1.5.1. Consider a system with partial component commonality in which
variants i and j are produced using a common component and all other variants use
dedicated components. For both demand models, the optimal assortment is either a
popular set or it consists of variants i and j plus a popular subset of the remaining
variants.
Proposition 1.5.1 implies that in a system with a general bill of materials, the
optimal assortment may not necessarily be a popular set. For example, in Figure
1.5(c), under the IP model with parameters p = 10, kd = kc = 3, λ = 150, σ = 1,
β = 0.5, θ0 = 5, and Θ = (16, 15, 8, 1), the optimal assortment is {1, 2, 4}. In
contrast, under the same parameter values, but in the setting in which all variants
use dedicated components, the optimal assortment is {1, 2, 3}. That is, pooling the
resources used by products 1 and 4 makes it more profitable to include the least
20
popular variant in the assortment, rather than keeping variant 3, to benefit from the
scale generated by aggregating the demands for these products. We next analyze the
size of the optimal assortment in settings with commonality, relative to the dedicated
system.
Proposition 1.5.2. Consider either the IP or the TF demand models. Let Ai+j be
the optimal assortment of a system in which variants i and j (i < j) are produced
using a common component and all other variants use dedicated components. Let An
be the optimal assortment in the corresponding dedicated system in which all variants
are produced using dedicated components.
1. If i, j ≤ n, then there exists a threshold t1 such that Ai+j is a strict subset of
An containing variants i and j if θn < t1, and Ai+j = An otherwise.
2. If i < n and j > n, then there exists a threshold t2 such that Ai+j is a strict
subset of An containing variants i and j if θn < t2, and Ai+j = An ∪ {j}
otherwise.
3. If i, j > n, then, there exist thresholds t3 > t4 such that
(a) Ai+j = An ∪ {i, j}, if t3 < θi + θj ≤ θn or t4 ≤ θn < θi + θj,
(b) Ai+j = An, if θi + θj < min (t3, θn) ,
(c) Ai+j = A′∪{i, j} where A′ is a strict subset of An, if θn < min(t4, θi +
θj).
This result provides a complete characterization of the optimal assortment in the
system with a common component for variants i and j. The optimal assortment
actually depends on the relative “popularity”of variants i and j (given that i < j).
In particular, if either variant i or both variants i and j are in the optimal assortment
of the original dedicated system, then both variants are in the optimal assortment
21
in the system with partial commonality as well (Proposition 1.5.2, parts 1 and 2).
However, in this setting, the optimal assortment may be smaller than that in the
dedicated system. If neither variant i nor variant j is in the optimal assortment of
the dedicated system, then it is optimal to offer these two variants in the system with
commonality only if the sum of their utilities is large enough (Proposition 1.5.2, part
3). At the same time, if that is the case, then one or more variants from the optimal
assortment of the dedicated system may be dropped in the optimal assortment of
the system with partial commonality (Proposition 1.5.2, part 3(c)).
Along the lines of the system studied in Proposition 1.5.2, if a product designer
had the liberty to choose the two products that should share a common component,
then the profit-maximizing solution would be achieved by pooling the components
of the two most popular products (see Proposition B.6 in Appendix B). At the
same time, Proposition 1.5.2 suggests that this may result in a smaller optimal
assortment. With partial commonality, the extent of product variety can only
increase if commonality is used with at least one product outside of the optimal
assortment in the dedicated system. In this way, new variants may be introduced
to the assortment at a relatively low cost, benefiting from the operational scale of
the products already in the assortment.
We conclude that, in sharp contrast to the results in Section 1.4, replacing a
strict subset of dedicated components by a common component may result in a
reduced number of variants in the optimal assortment and this set may no longer
be a popular set. To understand why partial commonality may reduce the optimal
assortment, let us consider the case with i < j < n. A setting with n = 3, i = 1,
and j = 2, is illustrated in Figure 1.5 below. The graphs in (a) show the profit
of each variant in the dedicated system, while the graphs in (b) show the combined
profit associated with variants i and j, and the profit for variant n, in the system with
partial commonality. Variant n is in the optimal assortment of the dedicated system
22
because the profit associated with this variant is higher than the aggregate increase
in profit that the other products in the assortment would experience in the absence
of variant n (recall that the choice probabilities – and therefore the expected demand
rates – of the other products in the assortment increase if variant n is removed from
the assortment, in this case from qi to qi and from qj to qj). That is, ∆n > ∆i + ∆j
in Figure 1.5 (a). On the other hand, in the system with partial commonality the
common component shared by variants i and j experiences a larger demand rate (and
therefore a more pronounced scale effect) than the dedicated component for variant i
does in the dedicated system. Note that the profit function of an individual product
variant is convex increasing in its demand rate (because of the economies of scale
associated with inventory costs). Hence, in the system with partial commonality,
the profit that variant n generates may no longer be larger than the profit impact of
the demand spill-over effect (in particular, onto the common component for variants
i and j) that occurs in the absence of variant n. In reference to Figure 1.5 (b),
we may have that ∆n < ∆ij. In that case, variant n would be excluded from the
assortment.
We next consider two generalizations of the system, represented in Figures 1.5
(d) and (e). In Figure 1.5 (d), variants i and j are assembled from one common
component and one dedicated component each (with costs kc and kd, respectively)
while, in the corresponding dedicated system, these variants are assembled using only
dedicated components (each with cost kd+kc). In Figure 1.5 (e), there is a component
common to all products and another component common to products i and j only.
All other variants require, in addition, a dedicated component. The results for
the system in Figure 1.5 (d) are similar to those in Proposition Proposition 1.5.2.
That is, the optimal assortment in this setting is either a popular set or a popular
set plus variants {i, j} . Also, the number of variants in the optimal assortment
associated with the system with partial commonality may decrease, remain the same,
23
Figure 1.2: Profit of variants in the dedicated and partial-commonality systems
or increase relative to the dedicated system. Regarding the system in Figure 1.5 (e),
we compare it to the system in Figure 1.5 (a) to examine the effect of introducing an
additional common component to a subset of the variants in a system with universal
commonality. The optimal assortment in the system shown in Figure 1.5 (a) is itself
no smaller than that in the related system with only dedicated components. In turn,
here too, the optimal assortment in the system with partial commonality shown in
Figure 1.5 (e) may be larger, equal or smaller than that in the system with universal
commonality. The existence of a common component shared by a strict subset
of the product variants again creates an imbalance in terms of the scale advantage
generated by demand aggregation – this benefit is higher for those products that use
a larger number of common components. As a result, it may be optimal to exclude
a product that makes less use of commonality to drive more demand to those that
use more common components and therefore increase the benefits associated with
demand pooling. (Propositions B.7 and B.8 in Appendix B contain formal statements
24
Figure 1.3: Role of component commonality in optimal assortment
and proofs of these results.)
To close, while introducing commonality to a manufacturing system always in-
creases total profit (as long as the costs of components do not change), the effect
on the optimal variety level is not necessarily monotone in the amount of com-
monality. Figure 1.5 shows an example based on the trend-following model with
p = 10, kd = kc = 0.3, λ = 150, θ0 = 5, and Θ = (12, 11, 5.8).
1.6 Conclusion
This chapter considers the role of component commonality in product assortment
and component inventory decisions in assemble-to-order systems. We characterize
and compare the structure of the optimal assortment and the optimal inventory levels
for various bill-of-material configurations. While component commonality increases
total system profit, we find that the effect of commonality on the optimal assortment
depends on the specific configuration of the bill of materials. With universal com-
monality (a common component is shared by all variants), the optimal assortment
is (weakly) larger than that in the system without commonality. Furthermore, the
assortment consists of a subset of the most popular variants (as in the case without
commonality). However, these results do not hold for more general bill-of-material
configurations. If the common component is shared by a strict subset of the vari-
25
ants, then the optimal assortment may no longer be a popular set and the optimal
level of variety may increase or decrease relative to the system without commonality.
In particular, the level of product variety (weakly) decreases when commonality is
introduced for products that are already in the optimal assortment corresponding to
the dedicated system (i.e., products with high demand volumes). This result may
hold even when commonality is introduced in a system that already involves other
common components shared by all product variants. These findings indicate that
product line and supply chain managers must be aware of the effect that introducing
commonality in manufacturing may have on optimal product line decisions, such as
the removal of products with relatively small demand. This is particularly important
if market share and market coverage (level of product variety) are critical to the com-
petitive position of a firm. Finally, we find that the effect of commonality on both
profit and variety level is more pronounced when demands are more variable and
exhibit pair-wise negative correlation. In an alternative interpretation of our model,
the results apply to the use of flexible resources and the related capacity decisions
(rather than common components and the corresponding inventory decisions).
26
2
Dynamic Assortment Customization with LimitedInventories
2.1 Introduction
Online retailing remains the retail industry’s growth engine even in the second biggest
recession of U.S. history. In the last decade, online retailers have heavily invested in
diverse technologies, such as sophisticated analytics and personalization tools. To-
gether with its low operating cost and the convenience of online shopping, e-commerce
has emerged as an effective way of improving customer service and retention. For
example, online retailers are able to send personalized promotions or display a cus-
tomized price for a specific product to individual customers based on their personal
information and browsing and purchasing history, which can effectively enhance the
customers’ shopping experience and influence their purchasing activities. In par-
ticular, as we argue in this paper, providing customized assortments to individual
customers has the potential to increase overall sales for online retailers. Compared
to traditional bricks-and-mortar stores, online retailers are able to easily collect per-
sonal information, record their customers’ purchase history, and control the selection
27
of products a customer is exposed to by strategically selecting the product variants
displayed on the web page. For example, Amazon.com’s web site, like many other
online retailers’ web sites, requires a customer to log in using her/his account be-
fore making a purchase. This allows the company to track the customer’s personal
information and purchase history. Based on this data, the company may be able
to estimate the customer’s preferences on styles and colors. An online retailer like
Amazon has several opportunities to use this information to restrict the set of prod-
ucts made available to customers. For example, suppose that a customer wants
to buy a pair of basketball shoes. He/she begins by searching for this item at the
company’s web page. The identity of the customer may be revealed through his/her
log in or through the browsing history recorded in the customer’s computer. The
company displays a selection of product options over multiple pages. The shoes on
the first page are likely to receive more attention. Therefore, the company may be
able to control the selection of shoes offered to a customer by restricting the set of
product variants displayed in the first few pages of the search results. In the next
step, the customer clicks on the product he/she prefers and the web site displays the
colors or sizes that are available. Here, the company has another opportunity to
customize the product offering.
In this paper, we find that a firm has the potential to increase revenues by strate-
gically restricting the set of product options it makes available to customers, even
when all products are in stock. In other words, the company may conceal a product
short on inventory or choose to show the product as not available, in anticipation of
future sales to other customers who may have a stronger preference for this product.
We formulate this as a dynamic assortment optimization problem where the assort-
ment decision depends on the inventory levels, the current customer’s preferences,
and the distribution of preferences of future customers.
We consider this problem in a setting with multiple products in a retail category.
28
There are limited inventories of the products to sell over a finite selling season. The
customer base is heterogeneous and characterized by multiple customer segments
with different product preferences. The retailer is able to limit the assortment
to each arriving customer without cost and every customer purchases at most one
product. In this setting, we first characterize the optimal dynamic assortment
policy for the case with a single customer segment and multiple product variants.
We next characterize the optimal policy in a setting with two customer segments
and two product variants. Next, we perform a numerical study to understand the
revenue impact of assortment customization and provide insights on how the policy
parameters depend on inventory levels and the heterogeneity within and between
customer segments. Finally, we propose upper and lower bounds for the retailer’s
profit function and develop heuristic solutions for the general case.
When there is a single customer segment, it is optimal to offer the entire assort-
ment to any arriving customer, because all future customers have the same prefer-
ences in expectation. In other words, we prove that the optimal assortment in each
period is to offer all products with positive inventories. In the setting with two
customer segments and two product variants, one customer segment is more likely to
choose one product rather than the other, and vice-versa. In this case, it may not
be optimal to offer all available products to all arriving customers. In particular, we
show that the optimal assortment policy is a threshold-type policy. Each customer
segment is always offered its most preferred product. The least preferred product
is offered only when its inventory level is above a threshold value. This threshold
is decreasing in time. In other words, as we approach the end of the season, it is
more likely that both products will be offered to both customer segments. In terms
of comparative statics, we find that as a product becomes relatively more popular
(i.e., with a larger utility) for any customer segment, the benefit of reserving this
product for future customers is higher – that is, it is less likely that this product will
29
be offered to a customer segment with a smaller utility for this product. We also
find that the revenue impact of assortment customization is significant in scenarios
where the inventory of one product is relatively large and the other one is relatively
small.
The rest of the chapter is organized as follows. Section 2.2 provides a review of
the relevant literature. Section 2.3 formulates the model. Section 2.4 characterizes
the optimal dynamic assortment policy for the case with a single customer segment
and multiple product variants. It also includes the analysis of a setting with two
customer segments and two product variants. Section 2.5 presents a numerical study
exploring the potential revenue benefits of assortment customization. In Section
2.6, we derive upper and lower bounds for the value function and develop a heuristic
solution for the general case with multiple segments and product variants. Section
2.7 concludes the chapter. All proofs are provided in Appendix D.
2.2 Literature Review
Our work is related to three streams of research. The first one is the literature
on retail assortment planning, with papers focusing on assortment decisions for a
single customer segment. Kok et al. (2008) provide a review of this literature.
van Ryzin and Mahajan (1999) derive the optimal assortment policy for a line of
products in a newsvendor setting. Mahajan and van Ryzin (2001) and Honhon
et al. (2008) consider models with dynamic customer substitution, while Cachon
et al. (2005) incorporate consumer search costs in a similar context. Smith and
Agrawal (2000) discuss an optimization approach for the assortment selection and
inventory management problems in a multi-item setting with demand substitution
and one replenishment cycle. Kok and Fisher (2007) focus on demand estimation
and assortment optimization in a retail chain. Caro and Gallien (2007) analyze a
30
model of dynamic assortment selection with demand learning during a single sell-
ing season. Chen and Bassok (2009) study the impact of choice uncertainty and
aggregate demand uncertainty on optimal assortment and inventory decisions.
Another related stream of research is that on choice-based network revenue man-
agement. Zhang and Cooper (2005) study a revenue management problem in a
setting with parallel flights and customer-choice behavior. The paper derives upper
and lower bounds for the value function. Talluri and van Ryzin (2004a) build up
a framework for choice-based network revenue management models with multiple
products and components. Each product consists of multiple components and cus-
tomers follow a choice process. The optimal assortment policy is characterized in
Talluri and van Ryzin (2004b) when all components share the same resource. For
general choice-based network revenue management models, Gallego et al. (2004) and
Liu and van Ryzin (2008) use a choice-based linear programming (CDLP) model to
approximate the dynamic control problem. In addition, Liu and van Ryzin (2008)
propose a dynamic programming decomposition heuristic and characterize the effi-
cient sets which are used in the optimal policy. Zhang and Adelman (2009) use an
affine function to approximate the value function and Chen and Homem de Mello
(2009) develop an approximation that consists of a sequence of two-stage stochastic
programs with simple recourse. Miranda Bront et al. (2009) show that the problem
is NP-hard and study a column generation algorithm on the CDLP of more gen-
eral models. van Ryzin and Vulcano (2008a, b) study virtual nesting policies in a
similar context where the demand process is a stochastic sequence of heterogeneous
customers.
The ability of a company to limit the assortment to its customers is a form
of inventory rationing. Therefore, our paper is also related to research on this
topic. Ha (1997a) considers a single-item, make-to-stock production system with
several demand classes and lost sales, and demonstrates that the optimal policy is
31
characterized by rationing levels for each demand class, Ha (1997b) studies a similar
system with two demand classes and backordering. de Vericourt et al. (2002) extend
this model to one with multiple demand classes characterized by different backorder
penalty costs.
2.3 Model Formulation
We consider a retailer that sells a set of product variants within a retail category
over a finite selling season. The retailer decides an assortment to offer to each ar-
riving customer. Each product variant represents a combination of features that
does not change the functionality of a product, such as a color/size/design combi-
nation as in the example of basketball shoes. The set of all products is denoted
as N = {1, · · · , N}. The selling season has T periods. There is no replenishment
during the season. The initial amount of stock for all products is denoted by an
N -dimensional vector y0 = (y01, · · · , y0N). This model is applicable, for example,
to end-of-season sales for any category of products and for short-life-cycle products
with long procurement lead times.
There are M customer segments characterized by different product preferences.
The choice process of each customer follows the Multinomial Logit (MNL) model.
Suppose that the offered assortment is given by a subset S ∈ N . The utility derived
from choosing product i (i ∈ S) for a customer in segment m is umi + ξmi, where
umi is the expected utility derived from product i and ξmi is a random variable
representing the heterogeneity of utilities across customers. In addition, customers
can always choose to not purchase any product, receiving a utility um0 + ξm0. Each
customer chooses the product that offers the maximum utility. We assume that
ξmi are i.i.d. random variables following a Gumbel distribution with mean zero and
variance π2/6. The probability of a customer choosing product i that arises from
32
this utility maximization problem is given by
qmi(S) =θmi∑
j∈S θmj + θm0
, i ∈ S ∪ {0}
where θmi = eumi . See Anderson et al. (1992) for more details on the MNL model
and Kok et al. (2008) for a comparison with other demand models. For simplicity,
we refer to θmi as the utility for product i of customer segment m and let Θm denote
the utility vector (θm1, θm2, · · · , θmN). Customers within a segment have identical
expected utilities for the same products. Therefore, there are in total M different
expected utility vectors, one for each of the M customer segments. The retailer
is able to estimate these utilities based on the customers’ purchase history. For
simplicity, we assume that the retailer knows the utility vector for each customer
segment. MNL models are commonly used by retailers and marketing firms to
identify multiple latent customer segments in the customer base and to estimate
purchasing behavior for each segment (see, e.g., Gupta and Chintagunta 1994 and
Wedel and Kamakura 1998).
We consider a Poisson arrival process and assume that at most one customer
arrives in each period. The sequence of events in each period is as follows: At
the beginning of the period, a consumer arrives with probability λ and the arriving
customer belongs to segment m with probability ρm. (Clearly,∑M
i=1 ρi = 1.) The
retailer has perfect information on the customer’s segment upon arrival. The retailer
offers an assortment (subset of the available products) to the customer. Next, the
customer makes a purchasing decision according to the choice process and the revenue
is received if a product is sold.
To isolate the effect of heterogeneous customers, we focus on a model with identi-
cal prices for all products, denoted by p. In a setting with non-identical prices, there
is a clear incentive for rationing (e.g., not offering a lower-priced product at certain
33
levels of inventory), even with a homogeneous customer base (as in Talluri and van
Ryzin 2004b). Finally, for expositional simplicity, we assume that the salvage value
for unsold units at the end of season is zero.
We use the following notation throughout the paper. We let ei denote the
ith unit vector. In addition, we let S(y) be the set of products with positive in-
ventory. Additionally, we denote the cardinality of set S as |S| and let y−i =
(y1, · · · , yi−1, yi+1, · · · , yN).
2.3.1 The Dynamic Assortment Optimization Problem
Define the value function in period t as Vt(y|m), given the vector of inventories y
and that the customer arriving in this period is in segment m. Taking expectation
across all customer segments, the value function at the beginning of period t is given
by
Vt(y) =∑m∈M
ρmVt(y|m)
Provided that the current arriving customer is in segment m, the retailer needs
to select an assortment to maximize revenues for the current period and for the rest
of the season. Therefore, the optimality equation is given by
Vt(y|m) = maxS⊂S(y)
{∑i∈S
λqmi(S)(p+ Vt+1(y − ei)) + λqm0(S)Vt+1(y)
}+(1−λ)Vt+1(y).
(2.3.1)
The term inside the curled brackets is the value function assuming that the ar-
riving customer is in segment m (an arrival occurs with probability λ). For a given
choice of assortment S, this term includes the probability of selling one unit of vari-
ant i in S, earning a revenue of p in this period plus the revenue-to-go function in
period t+ 1 evaluated at the current inventory level minus the unit sold in period t.
The term also accounts for the possibility that the customer purchases nothing, in
34
which case all inventories are left to the next period and the revenue is the profit-to-
go function in period t + 1 evaluated at the current inventory level. We maximize
this term over all possible subsets of variants with positive inventories. The last
term is the future value function if no customer arrives (with probability 1 − λ).
Therefore, in each period, the goal is to derive the optimal assortment of products
to offer to each customer segment so as to maximize current and future revenues.
The total optimal revenue over the selling season is given by V1(y0). In addition,
the terminal condition of this dynamic program is the value function in period T .
Because there are no more customers beyond the last period, the optimal policy is to
offer all products with positive inventory to any arriving customer regardless of the
segment the customer belongs to. Therefore, VT (y|m) =∑
i∈S(y)∪{0} λqmi(S(y))p.
The dimensionality of the action set in each period is 2|S(y)|M . Thus, the above
dynamic assortment optimization problem is intractable for large N and M .
Define ∆it+1(y) = Vt+1(y)− Vt+1(y− ei) as the marginal expected revenue of the
yi-th unit of inventory of product i in period t+1, where i ∈ S(y). Clearly, ∆it+1(y)
is nonnegative and cannot exceed the price p, i.e., p ≥ ∆it+1(y) ≥ 0. In addition,
∆it+1(y) = 0 when yi ≥ T − t+ 1, that is, when there is enough inventory to satisfy
demand throughout the remaining selling season. We can then rewrite (2.3.1), the
optimality equation, as
Vt(y|m) = maxS⊂S(y)
{∑i∈S
λqmi(S)(p−∆it+1(y))
}+ Vt+1(y) (2.3.2)
If product i is offered and sold in period t, the revenue is the price p minus the
lost opportunity revenue of this unit after period t, given by ∆it+1(y). Therefore, we
denote the effective marginal price in period t as pit(y) = p−∆it+1(y) if i ∈ S(y) and
pit(y) = 0 if i ∈ N \ S(y). Let us order these effective marginal prices in period t
given inventory levels y so that pi1t (y) ≥ · · · ≥ piNt (y). Note that the indeces (ij)Nj=1
35
depend on the inventory vector y and the period t. We denote a set that has the
products with the first k largest effective marginal prices as Ak(y) = {i1, · · · , ik}.
Then, if the value function Vt+1(·) is known for each inventory level, the problem in
(2.3.2) reduces to a one-period assortment optimization problem. This problem has
been solved by Talluri and van Ryzin (2004b) for a single customer segment. We
now show that their result extends to the setting with multiple customer segments
considered in this paper.
Proposition 2.3.1. Suppose that the inventory level in period t is given by y.
Then, the optimal assortment for each customer segment in period t is an element
of {A1(y), · · · , AN(y)}.
This result shows that the optimal assortment for each customer segment is re-
stricted to one of N possible sets Ai(y) if the effective marginal prices (pit(y))i∈N are
known (these can be characterized by value function Vt+1(·)). In particular, when
the inventory level is large enough, say yi ≥ T − t+ 1, it is always optimal to provide
product i regardless of the arriving customer’s segment because product i’s marginal
revenue is p in this case. In general, evaluating the value function in each period
is complicated and therefore so is the computation of the effective marginal prices.
Nevertheless, Proposition 2.3.1 provides a preliminary result about the structure of
the optimal assortment to offer to an arriving customer in period t. In particular,
this result implies that there is a product – product i1 – that will be offered to any
customer arriving in period t. In the following, we develop additional structural
properties of the optimal assortment policy.
2.4 The Optimal Assortment Policy
In this section, we turn our attention to the dynamic assortment policy. We first
analyze the optimal assortment policy in a setting with a single customer segment
36
and multiple product variants, and then move to a model with two customer segments
and two product variants.
2.4.1 Single Customer Segment and Multiple Product Variants
We drop the subscript denoting customer segment in this section and first present
results on the value function and marginal expected revenue.
Proposition 2.4.1. (i) ∆it(y) is decreasing in yj, j ∈ N .
(ii) If yi = yj ≥ 1 and θi ≥ θj, then ∆it(y) ≥ ∆j
t(y).
(iii) Both Vt(y) and ∆it(y) are decreasing in t.
The first property in Proposition 2.4.1 implies that the value function is concave
in yi and submodular in yi and y−i. In other words, the marginal expected profit of
a product decreases as the inventory level of any product increases, because a higher
inventory level lowers the probability that any existing unit will be sold over the
remaining selling season. The second property shows that a product with a higher
expected utility has a higher marginal profit than a product with a lower expected
utility, when both products have the same inventory level. The last part indicates
that both the value function and the marginal expected profit decrease as there is
less time remaining in the selling season, because the probability that a unit will be
sold decreases with time t.
Based on these results, we can characterize the optimal assortment in this setting.
Theorem 2.4.1. Consider a setting with a single customer segment and N products.
The optimal assortment in each period is to offer all products with positive inventory.
Theorem 2.4.1 shows that, in this setting, it is optimal to follow the myopic
policy, offering all products in each period thus maximizing the probability of selling a
37
product in that period. Because all product variants have identical prices, optimizing
the profit over the selling season is equivalent to maximizing total sales. In addition,
because customers are homogeneous, all future customers have the same preferences
as the current arriving customer in expectation, so it does not make sense to reserve
a product in anticipation of future sales.
2.4.2 Two Customer Segments with Two Product Variants
We now consider a setting with two customer segments and two product variants.
The utility vectors for customer segments 1 and 2 are denoted as (θ11, θ12, θ10) and
(θ21, θ22, θ20), respectively. We assume, without loss of generality, that both customer
segments have the same utility for the no-purchase option, i.e., θ0 = θ10 = θ20 > 0.
In addition, we restrict attention to the symmetric case where θ11 ≥ θ12, θ22 = θ11,
and θ21 = θ12. In other words, we consider a setting in which segment 1 customers
are more likely to purchase product 1 if both products are available, while segment
2 customers are more likely to purchase product 2 if both products are available. In
addition, when only product i is offered to both customer segments, a customer from
segment i purchases product i with a higher probability than a customer from the
other segment. Therefore, these two customer segments are distinctively different
in terms of their product preferences.
We begin by identifying some properties of the value function and marginal ex-
pected profits.
Proposition 2.4.2. (i) ∆it(y1, y2) is decreasing in yj, i, j ∈ {1, 2}.
(ii) Both Vt(y) and ∆it(y) decrease in t.
As shown for the case of a single customer segment, the value function is concave
in yi and submodular in y, and both the value function and the marginal expected
38
profit decrease as time approaches the end of the selling season. Subsequently, we
turn attention to the optimal assortment policy.
Theorem 2.4.2. Consider a setting with two customer segments and two product
variants. The following properties characterize the optimal assortment policy.
(i) Product i is always included in the optimal assortment of an arriving customer
of segment i.
(ii) In each period, given the inventory level yi for product i, there exists a thresh-
old level y∗j (yi) such that the optimal assortment for a segment i customer is
determined as follows:
If yj ≥ y∗j (yi), then the optimal assortment is {1, 2}; If yj < y∗j (yi), then the
optimal assortment is {i}.
The result shows that each customer segment is always offered its most preferred
product in the optimal dynamic assortment policy. Furthermore, a segment i cus-
tomer is also offered product j if the inventory level of that product is large enough.
Otherwise, the customer is only offered product i. This policy is then characterized
by a set of threshold levels y∗i and y∗j in each period t. Because the products prices
are equal, it is optimal to sell as many units of either product as possible throughout
the selling season. Then, for an arriving customer of segment i, if the inventory level
of product j is low, it may be optimal to reserve that inventory for future arriving
customers of segment j, since those customers are more likely to leave without pur-
chasing any product if product j is not available. On the other hand, if the inventory
level of product j is large enough, then it is optimal to offer both product variants
as future sales to segment j customers are not highly compromised. Furthermore,
it may be optimal to reduce the inventory level of product j, rather than increasing
the chances of consuming further units of product i.
39
As suggested by Theorem 2.4.2, in this setting with multiple customer segments,
it may not be optimal to offer all product variants to each arriving customer, even
when the selling prices are the same for all products. This indeed suggests a different
reason for rationing inventory that is not related to differences in prices. The
following result shows that in each period, at least one customer segment is offered
both products.
Proposition 2.4.3. In each period t, the optimal assortment for at least one cus-
tomer segment is {1, 2}.
To conclude the characterization of the optimal policy in the setting with two
customer segments and two products, we next discuss some monotonicity properties
of the threshold levels.
Proposition 2.4.4. There are two monotonicity properties of the threshold levels.
(i) For any given period, let y∗i (yj) and y∗j (yi) be the thresholds for inventory levels
yj and yi, respectively. If yi ≥ y∗i (yj) and yj ≤ y∗j (yi), then y∗i (yj) is decreasing
in yj.
(ii) For fixed inventory levels, the threshold levels decrease as time approaches the
end of the selling season.
The first part of Proposition 2.4.4 implies that, if in a given period and for given
inventory levels yi and yj, it is optimal to offer both products to customers from
segment j and to offer product i only to customers from segment i, then the same
policy continues to be optimal for lower inventory levels of product j. The second
part of Proposition 2.4.4 indicates that, as we get closer to the end of the selling
season, there is more incentive to offer both variants to any arriving customer. This
is because towards the end of the season, there is less concern about future sales
40
and therefore it is optimal to sell as many units as possible. Finally, we note that
because of the submodularity of the value function in y, the problem of choosing
the initial inventory levels y0 with consideration of the associated procurement costs
and optimal revenues can be easily solved to optimality.
2.5 Numerical Study
We conduct a numerical study to examine the effect of assortment customization on
revenues. We also study the impact of various parameters related to the customer
segmentation on the optimal threshold values. The numerical study considers gen-
eral parameter values for the two products and two customer segments, with θii ≥ θij
and θii ≥ θji for i, j = 1, 2, i 6= j. We find that the policy presented in Section 4 for
the symmetric case is also optimal in this setting.
The study consists of 735 experiments – all with two customer segments and two
product variants, p = λ = 1, and T = 50. The utility vector (θ11, θ12) for segment 1 is
equal to one of the following vectors: {(10, 2), (10, 4), (10, 6), (10, 8), (10, 10), (12, 6),
(15, 6)}. Similarly, the utility vector (θ21, θ22) for segment 2 is given by one of
the following vectors: {(2, 10), (4, 10), (6, 10), (8, 10), (10, 10), (6, 12), (6, 15)}. The
utility of the no-purchase option is given by one of the following five values: θ0 =
{5, 10, 15, 20, 25}. For each of these, the probability that an arriving customer
belongs to segment 1 takes one of the following three values: ρ1 = {0.2, 0.5, 0.8}.
2.5.1 Effect of Assortment Customization on Revenues
We first report the effect of assortment customization on revenues by calculating the
percentage revenue increase due to assortment customization relative to a policy that
always offers both products to any arriving customer.
41
Figure 2.1: Revenue impact of assortment customization
Figure 2.5.1 below shows the average percentage increase in revenue due to as-
sortment customization as a function of the inventory levels. When inventory levels
for both products are large, the retailer is less likely to run out of stock, so there is
no need for rationing inventory and therefore the revenue impact is small. When
inventory levels for both products are small relative to the time horizon, there is
enough time to sell all units, so both products are likely to be offered in the optimal
solution and again the impact is small. The revenue impact is significant in the
areas where the inventory level for one product is high while the inventory level for
the other product is low. These are the scenarios in which strategically selecting
assortments is most important.
We now consider the average revenue increase over the inventory levels under
which the percentage increase due to customization is greater than 1%. The revenue
impact is smaller when the customers of one segment represent a higher proportion of
the total customer base. This is because, in that case, most arriving customers be-
long to one of the segments and therefore the possibility of making use of customized
assortments is reduced. In the extreme case, when there is a single customer seg-
42
ment, it is always optimal to offer all available products as shown in Section 4.1,
thereby eliminating any customization. We also find that the revenue impact is
sensitive to significant changes in the utility of a customer segment’s least preferred
product. As the utility of a segment’s least preferred product increases, offering this
product will reduce the extent to which customers in that segment purchase their
most preferred product. In those cases, it may then be optimal to limit the assort-
ment offered to those customers, thus curtailing their access to the least preferred
product. Therefore, the revenue impact of assortment customization is larger. If
the utility of the least preferred product decreases, there is less incentive to offer
this product as it may be better to reserve the inventory of that product for the
future customers of the other segment. Again, the revenue impact of customization
increases. As the utility of the no-purchase option increases, i.e., customers are
more likely to leave without purchasing any product, total demand decreases and
the revenue impact of customization is lower.
2.5.2 Effect of Parameters on Threshold Levels
We next explore the impact of changes in the system parameters on the optimal
threshold levels. We focus on segment 1 customers. Similar observations can be
made for segment 2 customers.
Note that, for an arriving customer of segment 1, it is optimal to offer product
2 if the inventory level of that product is higher than the threshold level, i.e., if
y2 ≥ y∗2(y1). We first note that, for a fixed inventory level y1, this threshold is
increasing in θ22−θ21. For example, when θ22 increases and θ21 decreases, a segment
2 customer is even more likely to purchase product 2 than product 1. It thus makes
sense to reserve product 2 for future arriving customers of segment 2. Therefore,
the threshold level is increasing in this difference. In addition, as either θ12 or θ22
increases (one customer segment has higher relative preference for product 2), the
43
demand for that product becomes stochastically larger. Then, the advantage of
offering only product 1 to a customer in segment 1 becomes more significant, so that
more units of product 2 are reserved for customers in the other segment. That is,
the threshold level increases with θ12 and θ22. However, this threshold is decreasing
as either θ11 or θ21 increases (when customers in either segment are more likely to
purchase product 1 than product 2). In this case, making both products available
to segment 1 customers is less likely to have a severe impact on product 2’s future
inventory levels, and therefore on overall sales.
We also find that, as the proportion of customers from segment 1, i.e., ρ1, in-
creases, offering both products to segment 1 customers becomes more attractive as
this has the potential of increasing sales. Therefore, the threshold level y∗2(y1) de-
creases with ρ1. This observation is consistent with the effect of ρ1 on the revenue
impact of assortment customization discussed before. Similarly, the threshold level
decreases with the utility of the no-purchase option.
Finally, Figure 2.5.2 exhibits the monotonicity properties of the threshold levels
shown in Proposition 2.4.4. As we get closer to the end of the season (i.e., we
move from the top curve to the bottom curve in Figure 2.5.2), the threshold levels
decrease, that is, less rationing occurs as time approaches the end of the selling
season. Moreover, in any given time period, the threshold level y∗2 is increasing
in y1. That is, for a given amount of inventory of product 2, it is more likely to
offer that product to an arriving customer of segment 1 when the inventory level of
product 1 is low. This is because it may be optimal to re-direct some segment 1
customers to purchase product 2 so as to reduce the likelihood of running out of stock
of product 1 in the future. On the other hand, when the inventory level of product
1 is high, it may be optimal to limit the assortment to product 1 only, therefore
increasing demand for that product.
44
Figure 2.2: Monotonicity of threshold levels
2.6 General Case
In this section, we consider the general problem with N products and M customer
segments. We first develop upper and lower bounds on the optimal profit and then
propose a heuristic solution.
2.6.1 Bounds on the Optimal Profit
The dynamic assortment optimization problem with N products and M customer
segments can be described as a Markov decision process (MDP) problem. Assort-
ments over all periods are controlled through a control policy µ. The assortment
offered to a segment m customer in period t is Sµt (m|Ft) where Ft is the history up
to period t. We denote the profit under a control policy µ, given initial inventory
levels y0, as V µ1 (y0). The set of all feasible policies is denoted as A. Therefore,
V1(y0) = maxµ∈A
V µ1 (y0)
We first propose a lower bound to the optimal profit. Define a threshold policy
(µT ) as follows: let (bmt1, · · · , bmtN) a vector of threshold levels for customer segment
45
m in period t, depending on the prevailing inventory level y in period t, such that the
set of products to offer to an arriving customer of segment m is Smt = {i : yi ≥ bmti}.
The set of all feasible threshold policies is denoted as T . In particular, let µAll
represent the control policy under which any customer is offered all products with
positive inventory. Then, we have the following lower bounds of the optimal profit:
V µAll
1 (y0 ) ≤ maxµT∈T{V µT
1 (y0 )} ≤ V1(y0 )
Next, we propose an upper bound on the optimal profit by converting the stochas-
tic control problem into a choice-based deterministic linear programming problem
(CDLP). This approach has been used in Gallego et al. (2004) and Liu and van
Ryzin (2008) in the context of settings with a single customer segment. The cus-
tomer arrival and choice processes are assumed to be deterministic, the first with
rate λρm for customer segment m and the second with a proportion qmj(S) of cus-
tomers in segment m choosing product j ∈ S. We also assume that demand and
inventory are continuous variables. The total revenue for segment m under an as-
sortment S is then given by Rm(S) =∑
j∈S ρmpqmj(S) and the associated inventory
consumption rate is Qm(S) =∑
j∈S ρmqmj(S). Given a sequence of assortments
offered to segment m customers arriving over the selling season, any permutation of
this sequence leads to the same revenue and inventory consumption. Therefore, in
this approximation, we derive the length of time in which an assortment set is offered
to a given customer segment. Specifically, let tm(S) denote the interval of time over
46
which assortment S is offered to segment m customers. The CDLP is then given by
V CDLP (y0) = max∑m∈M
∑S⊂N
λRm(S)tm(S)
s.t.∑m∈M
∑S⊂N
λQm(S)tm(S) ≤ y0i, i ∈ N
∑S⊂N
tm(S) ≤ T, ∀m ∈M
tm(S) ≥ 0, ∀m ∈M and ∀S ⊂ N . (2.6.1)
The CDLP is not only simpler than the MDP problem, but also each feasible solution
to the MDP has a corresponding feasible solution in the CDLP.
Proposition 2.6.1. For each feasible policy in the MDP there is a corresponding
feasible solution in the CDLP with the same profit. Therefore,
V1(y0) = maxµ∈A
V µ1 (y0) ≤ V CDLP (y0).
This result implies that the optimal profit of the CDLP in (2.6.1) is an upper
bound to the MDP problem.
2.6.2 Multiple Customer Segments and Multiple Product Variants
In this section, we briefly introduce a heuristic for the general assortment customiza-
tion problem with M customer segments and N product variants. As shown in
Proposition 2.3.1, the optimal assortment for any customer segment in period t,
given an inventory vector y, is given by one of the sets Ak(y), defined based on
the effective marginal prices p1t (y), · · · , pNt (y). In other words, for known values of
Vt+1(y) and Vt+1(y − ei) for all i, the computation of the optimal assortment for a
given customer segment is reduced to a search over N possible product sets. Nev-
ertheless, the size of the state space still makes it intractable to derive the optimal
47
policy by solving the MDP. We therefore propose a heuristic based on the bounds
derived above.
Heuristic
Step 1: Consider the vector of inventory levels y in period t. Let S(y) be the set
of products with positive inventory level and denote s = |S(y)|.
Step 2: Calculate the bounds V boundt+1 (y) and V bound
t+1 (y − ej) for all j ∈ S(y).
Step 3: Set pjt(y) = p − V boundt+1 (y) + V bound
t+1 (y − ej) for each j ∈ S(y). Rearrange
(pjt(y))j∈N in decreasing order so that pi1t (y) ≥ · · · ≥ piNt (y) and let Aj =
{i1, · · · , ij} for all j ≤∣∣S(y)
∣∣.Step 4: Determine the optimal assortment for each customer segment m by compar-
ing the one-period profit values∑
i∈Aj λqmi(Aj)pit for each j ≤
∣∣S(y)∣∣. Offer
the assortment with the highest profit to an arriving customer of segment m.
The performance of the heuristic depends on the gap between the solution given
by the bound and the optimal solution to the MDP. We consider two versions of
the heuristic, heuristic one using the upper bound CDLP and heuristic two using
the lower bound that involves offering all products with positive inventory to each
arriving customer. In this section, we conduct a numerical study to examine the per-
formance of the bounds and heuristics. The study consists of 48 cases – all with four
customer segments, four product variants, p = 10, λ = 1, and θ0 = 1. The probabil-
ity vectors ρ = (ρ1, ρ2, ρ3, ρ4) are equal to one of the six vectors: ρ = (ρ1, ρ2, ρ3, ρ4) ∈
{(0.1, 0.1, 0.4, 0.4), (0.2, 0.2, 0.3, 0.3), (0.1, 0.2, 0.3, 0.4), (0.1, 0.1, 0.1, 0.7),
(0.2, 0.2, 0.2, 0.4), (0.3, 0.3, 0.3, 0.1)}. The preference vector for segment i(∈ {1, 2, 3, 4})
customers takes one of the following four different forms : Θi = {(θii = 10, θij = 1),
48
(θii = 10, θi,i+1 = 9, θij = 1), (θii = 10, θi,i+2 = 9, θij = 1), (θii = 10, θi,i+3 = 9, θij =
1)} where j ∈ {1, 2, 3, 4}\{i} and θi,i+1(θi,i+2 or θi,i+3) is the preference of segment i
customers for product i− 3(i− 2 or i− 1) if i > 3(i > 2 or i > 1). In another group
of experiments, the preference vectors of segments 1, 2, and 3 customers take one of
these four forms, however, segment 4 customers are almost indifferent among the four
products, i.e., Θ4 = (9, 9, 9, 10). For each case, we compute the optimal profit under
the optimal dynamic assortment policy, the profits derived from upper and lower
bounds, and the profits based on both heuristics. We report the average maximum
gap and the median across all cases. The average percentage maximum gap and the
median between the upper bound CDLP and the optimal profit are 7.3% and 1.5%,
respectively, while these numbers are −4.1% and −1.2% between the lower bound
and the optimal profit, respectively. The lower bound is closer to the optimal profit
than the upper bound. Regarding the heuristics, heuristic two, based on the lower
bound, performs much better than heuristic one based on the CDLP. In particular,
the average percentage maximum gap and the median between the lower bound and
the optimal profit are 0.3% and 0.0%, respectively.
2.7 Conclusion
We consider a revenue management problem for a firm that offers multiple product
variants in a retail category. Customers arrive according to a Poisson process and
they are segmented according to their expected preferences for the product variants.
The firm has information on which segment each arriving customer belongs to and
makes customized assortment decisions so as to maximize revenues over the selling
horizon. We prove that it is optimal to offer all product variants in the case with a
single customer segment. However, when there are multiple customer segments with
distinctly different preferences for the products, it may not be optimal to offer all
49
the available products to each customer. In particular, in the case of two customer
segments and two products, the optimal assortment policy is a threshold-type policy.
Each customer is always offered its most preferred product, while the other product
is offered only if that product’s inventory level is larger than a threshold value.
We assess the impact of offering customized assortments in a numerical study and
characterize the settings in which assortment customization is most valuable. The
revenue impact is significant when the inventory level of one product is relatively high
and that of the other product is relatively low. We also find that the revenue impact
is highest when there is a relatively equal split of customers in both segments. We
also study the effect of changes in the system parameters on the optimal threshold
values.
We derive upper and lower bounds on the value function and use these to pro-
pose two heuristics for the general case of multiple products and multiple customer
segments.
50
3
Assortment Decisions with Non-identical Price andCost Parameters
3.1 Introduction
In Chapter 1, we assume that the price and cost parameters of the product variants
are identical. While this may be applicable to certain product categories (such as
similar style shoes or apparel), in many cases products may have different selling
prices and costs of components. In this section, we discuss the generalization of our
results to the case of non-identical costs and prices.
Let us start with the system with dedicated components. van Ryzin and Mahajan
(1999) claim that they “can allow different prices and costs, provided the ratio p/c
is the same for all variants.” This claim is not correct. In the absence of inventory
costs (therefore no economies of scale), the optimal assortment in a setting with non-
identical margin consists of the k highest-margin items (Liu and van Ryzin 2008).
With inventory costs and identical margins, the optimal assortment consists of a
subset of the most popular products (van Ryzin and Mahajan 1999). Hence, with
both explicit inventory costs and non-identical margins, there are two forces at play
51
and the optimal assortment may not necessarily have any particular structure. That
is, it is possible that the optimal assortment contains a product that is neither the
highest-margin product nor the one with the highest level of popularity, even if the
ratios p/c are the same for all variants.
The following example illustrates the difficulty of obtaining structural results
for products in a setting in which popularity is decreasing with prices: N = 3,
λ = 100, β = 1/2, σ = 3, cj/pj = r = 0.7, uj = aj − pj, (aj)j=0,1,2,3 = (10, 13, 14, 15) ,
(pj)j=0,1,2,3 = (0, 6, 8, 11) , (uj)j=0,1,2,3 = (10, 7, 6, 4) , where uj is the net utility, aj is
the inherent utility, and pj is the selling price. Note that the net utility is decreasing
in j and the price is increasing in j for j ≥ 1. The optimal assortment is S∗ = {2} ,
which consists of a product that is neither the most popular, nor the highest margin
product.
Hence, with non-identical prices, it is generally not possible to derive structural
results regarding the optimal assortment, even in the system with dedicated compo-
nents. In this chapter, our first goal is to demonstrate the impact of commonality
on variety, and we are able to show that our results continue to hold for some set-
tings with non-identical prices, namely, for settings in which prices (and margins)
are weakly increasing in the net utility of the product. This is the only range of
non-identical cost and price parameters in which there exists a clear measure of level
of variety and, therefore, meaningful comparisons across systems are possible. We
then derive some results related to the structure of the optimal assortment with
general price and cost parameters in the system with dedicated components.
3.2 Extended Results of Chapter 1 with Non-Identical Price and CostParameters
It is reasonable to assume that a product with a higher inherent utility has a higher
production cost and selling price. However, the choice probabilities (and popularity)
52
are determined by the average net utility, which equals inherent utility minus the
selling price. The relation between net utilities and prices depends on the setting.
Casual observation suggests that there is a non-monotone relationship between prices
and popularity: the most popular products are usually those with medium ranged
prices. For example, a quick online search shows that Toyota Camry and Honda
Accord are the top selling cars in 2009 in the U.S. Note that a product that has a
higher price and a lower utility than another product may still attract some customers
due to the heterogeneity of customer preferences.
Below we state our results for the independent population model and the trend-
following model. For expositional simplicity, we assume linear functions, pj = p0 +
γuj, where p0, γ ≥ 0. The total cost of components for variant i is ki = kdi + kci .
We assume that the cost/price ratios are identical across all products, i.e., ki/pi = r
for r ∈ (0, 1) and any i ∈ N .
Theorem 3.2.1. Under the independent and trend-following population demand
models, the optimal assortment in the dedicated model is a popular set.
Because unit costs are different across products, we need to replace Condition 2
in the paper with the following analogous condition, which implies that the optimal
assortment becomes larger as r decreases or, equivalently, as the unit costs of all
products decrease.
Condition 3. δn is increasing in r as long as δn < θn.
We now describe the cost structure in the system with commonality. For the
cost of the dedicated components, assume that kdi/pi = τr, where τ, r ∈ (0, 1). The
cost of the common component is given by kc and suppose that kc ≤ kc1 . Note
that kc1 is the cost of the most expensive version of that component. Similar to
the case with identical prices, we can solve the profit maximization problem for the
trend-following demand model by using the KKT conditions.
53
Theorem 3.2.2. In the system with commonality under the trend-following model,
the optimal assortment is a popular set. For any popular set An, we have that if
ΠD,TFAn+1
≥ ΠD,TFAn
then ΠC,TFAn+1
> ΠC,TFAn
. Thus, the optimal assortment in the system
with commonality is no smaller than that in the dedicated system.
Finally, we consider the pooled system, in which there are no dedicated compo-
nents and all products use the same common component. Note that, in the case of
non-identical prices, it is no longer optimal to offer all products in the pooled system.
Assume that the total cost of common components is k = kd + kc, with kd ≤ kd1 .
Theorem 3.2.3. In the pooled system under the trend-following demand model, the
optimal assortment is a popular set. Furthermore, the optimal assortment in pooled
system is no smaller than that in the dedicated system ifkqNN∑i∈N piqi
≤ r. The optimal
assortment in the pooled system is no smaller than that in the system with universal
commonality if k ≤ kd1 + kc.
This results show that in the pooled system it is optimal to offer more prod-
ucts than in the system with commonality as long as the cost of the new common
component in the pooled system (the one that is not common in the commonality
system) is not higher than the most expensive dedicated component the system with
commonality .
3.3 Dedicated System with General Price and Cost Parameters
The retailing price of product i is pi and its utility is ui. Since all prices are given,
we denote the utility of product i as θi = eui−pi . The production cost of product i is
ki = αpi where 1 > α > 0. The total demand is a Poisson process with rate λ. For
54
any assortment S, the profit function is
ΠS = (1− α)λ∑i∈S
piqSi − φ(Φ−1(1− α))λβ
∑i∈S
pi(qSi )β
= pi
[(1− α)λ
∑i∈S
qSi − φ(Φ−1(1− α))λβ∑i∈S
(qSi )β
]
From Lemma B.1, we have that ΠS is increasing and convex in qSi and is linearly
increasing in pi. Suppose the new variant is denoted as product m with utility δ and
price pm. Define δm the value of δ > 0 such that ΠS+{δm},pm = ΠS. Then it is easy
to get the following proposition.
Proposition 3.3.1. If i, j ∈ S and pi ≥ pj, then δi ≤ δj.
This proposition implies that it is better to include a product with a higher price
if they have the same utility. However, if prices and utilities are ordered differently,
it is unclear which one is better to be included. We then define assortment S as an
efficient set, if S satisfies the following three conditions.
1. ΠS ≥ ΠS+{i} for any i ∈ S.
2. ΠS ≥ ΠS−{j} for any j ∈ S.
3. ΠS ≥ ΠS−{j}+{i} for any j ∈ S and i ∈ S.
According to the definition, it is not optimal to add, remove or replace any
product from an efficient set. Obviously, the optimal assortment will be an efficient
set. Moreover, we have the following property for these efficient sets.
Proposition 3.3.2. In any efficient set S, for any i ∈ S, if pj > pi and θj ≥ θi or
pj ≥ pi and θj > θi, then j ∈ S.
55
Figure 3.1: Efficient set
This proposition says that, for any product in an efficient set, all other products
who both have a larger price and utility than this product are contained in this set as
well. As show in Figure 3.3, if product A is included in an efficient set, then product
B and C are also in this efficient set.
We then can get the results below for two special cases easily by using Proposition
3.3.2.
Proposition 3.3.3. 1. If all products have the same utility, the optimal assort-
ment is a set which includes the most profitable products.
2. (van Ryzin and Mahajan (1999)) If all products have the same price, the opti-
mal assortment is a set which includes the most popular products.
56
Appendix A
Proofs of Main Results of Chapter 1
Analysis of the Dedicated System
In a system with dedicated components, the profit function is separable. For
a given assortment S, the profit function under the independent population model
is ΠD,IPS =
∑i∈S π
D,IPi (qSi )
def=∑
i∈S maxyi{E[pmin{yi, Di}] − kyi}, where πi is the
profit function and yi is the stocking quantity for variant i. The optimal stocking
quantities are given by y∗i = λi+zIPσi, where zIP = Φ−1(
1− kp
). The resulting op-
timal profit is ΠD,IPS =
∑i∈S(p−k)λi−pφ(zIP )σi. Under the trend-following model,
the optimal expected profit, given an assortment S, is ΠD,TFS =
∑i∈S π
D,TFi (qSi )
def=∑
i∈S maxyi{E[pqSi min{yi, D}]−kyi}. The optimal stocking quantity for variant i is
given by y∗i = λ+ zTF (qSi )σ, where the value of zTF (qSi ) depends on the profitability
of variant i as follows:
zTF (qSi ) =
{Φ−1
(1− k
qSi p
), if qSi p ≥ k
−µσ, otherwise.
If qSi p < k, then the optimal stocking level for variant i is zero. As a result, the
57
profit for variant i is (qSi p− k)λ− qSi pσφ(zTF (qSi )) if qSi p ≥ k, and zero otherwise.
Given an assortment S, we let S(δ) = S ∪ {m} denote the set of variants in
S plus an additional variant m 6∈ S with utility δ. The choice probability for
i ∈ S is qS(δ)i = θi∑
j∈S θj+δ+θ0and q
S(δ)m = δ∑
j∈S θj+δ+θ0. The resulting optimal profit is
ΠDS (δ) =
∑i∈S(δ) π
Di (q
S(δ)i ) under either demand model. Proposition B.1 in Appendix
B shows that ΠDS (δ) is a quasi-convex and decreasing-increasing function and that
the optimal assortment is a popular set. The proof is similar to that in van Ryzin
and Mahajan (1999).
Theorem A.1. ΠDn is quasi-concave in n.
Proof of Theorem A.1: Quasi-concavity of ΠDn in n is established if the following
are satisfied: 1) If ΠDn+1 ≥ ΠD
n , then ΠDn ≥ ΠD
n−1; and 2) If ΠDn+1 ≤ ΠD
n , then ΠDn+2 ≤
ΠDn+1. For (1), ΠD
n+1 ≥ ΠDn implies that δn ≤ θn+1 ≤ θn. Under Condition 1 we have
δn−1− δn ≤ θn− θn+1. Combining the two, we have that δn−1 ≤ δn + θn− θn+1 ≤ θn,
which implies that ΠDn ≥ ΠD
n−1. For (2), ΠDn+1 ≤ ΠD
n implies that δn ≥ θn+1. There
are two possibilities. If δn+1 > θn+1, then ΠDn+2 ≤ ΠD
n+1 because θn+1 ≥ θn+2. If
δn+1 ≤ θn+1, then Condition 1 implies that δn+1 ≥ δn − θn+1 + θn+2 ≥ θn+2, which
implies that ΠDn+2 ≤ ΠD
n+1. �
Proof of Theorem 1.4.1: Because all variants are symmetric with identical util-
ities, the optimal order quantities are the same for all variants under model D. In
model D, ΠD,IPn = maxy{pnEmin{y,Di} − n(kc + kd)y} where the optimal order
quantity y∗ satisfies Pr(Di ≤ y∗) ≥ 1 − kc+kdp
. We consider settings with y∗ > 0,
which can be guaranteed by assuming that λ >> σ. In turn, this implies that
Pr{DSi = 0} = σ2
λqS+σ2 < 1 − kc+kdp≤ Pr{DS
i = λqS + σ2} = λqS
λqS+σ2 and therefore
y∗ = λqS + σ2. Hence, ΠD,IPn = n(pλqS − (kc + kd)(λq
S + σ2)). Now consider
model C. The optimal order quantities for the dedicated components are equal to
y∗ in model C as well, because unit cost is smaller than that in model D due to
58
risk pooling effect of the common component and the dedicated components face
the same two-point demand distribution as in model D. The common component,
however, faces the total demand. Moreover, y∗c = m∗y∗ where m∗ is an integer
between 1 and n, since the demand for each variant is either zero or y∗. Thus,
ΠC,IPn = (λqS + σ2) maxm{pmin{m,
∑ni=1 min{1, DS
i /(λqS + σ2)}} −mkc − nkd}.
Define Nn =∑n
i=1 min{1, DSi /(λq
S + σ2)} which subjects to a binomial distribu-
tion with parameters n and λqS
λqS+σ2 . Then the optimal m∗ satisfies Pr(Nn ≤ m∗) ≥
1− kcp
. We now use a normal distribution with mean E[Nn] = nλqS
λqS+σ2 and standard
deviation σ(Nn) =√n λqSσ2
(λqS+σ2)2to approximate the binomial distribution. The
optimal profit function in model C then can be expressed as
ΠC,IPn = (λqS + σ2)[(p− kc)E[Nn]− pφ(Φ−1(1− kc/p))σ(Nn)− nkd]
To ensure that the normal approximation of the binomial distribution is accurate,
we focus on cases that satisfy ΠC,IPn ≥ ΠD,IP
n . This inequality states that, for a
given assortment, the profit in the system with commonality is no smaller than that
in the system with dedicated components. This inequality is equivalent to
kcσ√λθpφ(Φ−1(1− kc/p))
≥
√1
n(nθ + θ0). (A.1)
This condition is satisfied for sufficiently large n. We now proceed to prove that
ΠC,IPn+1 − ΠC,IP
n > ΠD,IPn+1 − ΠD,IP
n .
ΠC,IPn+1 − ΠC,IP
n − ΠD,IPn+1 + ΠD,IP
n
= σpφ(Φ−1(1−kc/p))√λθ
[√n
nθ + θ0
−
√n+ 1
(n+ 1)θ + θ0
+kcσ√
λθpφ(Φ−1(1− kc/p))
](A.2)
59
Using the inequality (A.1) in expression (A.2), we have that
(A.2) ≥ σpφ(Φ−1(1− kc/p))√λθ
[√n
nθ + θ0
−
√n+ 1
(n+ 1)θ + θ0
+
√1
n(nθ + θ0)
]
= σpφ(Φ−1(1− kc/p))√λθn(nθ + θ0)
[n+ 1−
√(n+ 1)n(nθ + θ0)
(n+ 1)θ + θ0
]> 0.
This completes the proof. �
Proposition A.1. For a given assortment S, there exists a popular subset Se ⊂ S
such that y∗i = y∗c for all i ∈ Se and the optimal stocking quantities of the remaining
variants decrease in order of their popularity (i.e., y∗ir ≤ y∗is < y∗c for ir, is ∈ Sldef=
S \ Se with θir ≤ θis). Hence, the optimal profit is given by
ΠC,TFS =
∑i∈Sl
[(qSi p− kd)µ− qSi pσφ(zSi )
]
+
[(∑i∈Se
(qSi p− kd)− kc)µ−∑i∈Se
qSi pσφ(Φ−1(t∗c(S)))
]
with zSi = 1− kdpqSi
and t∗c(S) = 1− kc+|Se|kdp∑i∈Se q
Si
.
Proof of Proposition A.1: The following are the KKT optimality conditions for
the profit maximization problem subject to the constraints 0 ≤ yi ≤ yc for i ∈ S
(with dual variables αi):
αi ≥ 0, αi(y∗c − y∗i ) = 0,
∑j∈S
αj = kc
Φ
(y∗i − λσ
)= 1− kd + αi
pqSi. (A.3)
Based on the KKT conditions, we partition the variants in S into two sets, Sl and Se.
The set Sl corresponds to the product variants i for which αi = 0. A variant i ∈ Se
60
has αi > 0, implying that y∗c = y∗i . Because∑
j∈S αj = kc, we have that Se 6= ∅.
This and (A.3) imply that there exists a fractile t∗c(S) such that y∗c = λ+Φ−1(t∗c(S))σ.
In other words, 0 < αi = pqSi (1− t∗c(S))− kd ≤ kc for all i ∈ Se. Using the fact that∑j∈Se αj = kc, we have that t∗c(S) = 1− kc+|Se|kd
p∑i∈Se q
Si
.
Suppose that S = {i1, · · · , ir} with i1 < · · · < ir, so that θi1 > · · · > θir and
therefore
1− kdpqSi1
> · · · > 1− kdpqSir
> 0.
Then, there exists a variant ia such that
1− kdpqSi1
> · · · > 1− kdpqSia
> t∗c(S) ≥ 1− kdpqSia+1
> · · · > 1− kdpqSir
and the assortment is partitioned as Se = {i1, · · · , ia} and Sl = {ia+1, · · · , ir}. (That
is, the set Se is a popular subset of S.) To see this, note that for i ∈ Sl, αi = 0 and
y∗i = λ+ Φ−1(
1− kd+αipqSi
)σ = λ+ Φ−1
(1− kd
pqSi
)σ ≤ λ+ Φ−1(t∗c(S))σ = y∗c (because
y∗i ≤ y∗c ). This implies that t∗c(S) ≥ 1 − kdpqSi
for all i ∈ Sl. Finally, note that for
i ∈ Se, the values of αi are such that
1− kd + αi1pqSi1
= · · · = 1− kd + αiapqSia
= t∗c(S). �
Proof of Theorem 1.4.2: Quasi-convexity of ΠC,TFS (δ) in δ can be demonstrated
by showing that the first-order derivative of ΠC,TFS (δ) is continuous and crosses zero
(from negative to positive) only once. (The function ΠC,TFS (δ) is evaluated at the
optimal stocking quantities for all variants in S.) As δ increases, we have from
Lemma B.4 that Se(δ) and Sl(δ) change as well, leading to different expressions of
the first-order derivative of ΠC,TFS (δ) with respect to δ. Consider the threshold values
δi, i = 1, 2, 3, 4, defined in Lemma B.4.
61
If 0 ≤ δ < δ1, then it is not profitable to include the new variant in the assortment.
Therefore,
∂ΠC,TFS (δ)
∂δ=∑i∈S
∂ΠC,TFS (δ)
∂qS(δ)i
∂qS(δ)i
∂δ=
1
(∑
i∈S θi + δ + θ0)2
[−∑i∈S
θi∂ΠC,TF
S (δ)
∂qS(δ)i
](A.4)
If δ1 ≤ δ < δ2, then the new variant enters the set Sl, and
∂ΠC,TFS (δ)
∂δ=∑i∈S
∂ΠC,TFS (δ)
∂qS(δ)i
∂qS(δ)i
∂δ+∂ΠC,TF
S (δ)
∂qS(δ)m
∂qS(δ)m
∂δ
=1
(∑
i∈S θi + δ + θ0)2
[−∑i∈S
θi∂ΠC,TF
S (δ)
∂qS(δ)i
+∂ΠC,TF
S (δ)
∂qS(δ)m
(∑i∈S
θi + θ0
)](A.5)
If δ2 ≤ δ < δ3, then the new variant enters the set Se, and
∂ΠC,TFS (δ)
∂δ=∑i∈S
∂ΠC,TFS (δ)
∂qS(δ)i
∂qS(δ)i
∂δ+∂ΠC,TF
S (δ)
∂qS(δ)m
∂qS(δ)m
∂δ
=1
(∑
i∈S θi + δ + θ0)2
[−∑i∈S
θi∂ΠC,TF
S (δ)
∂qS(δ)i
+∂ΠC,TF
S (δ)
∂qS(δ)m
(∑i∈S
θi + θ0
)](A.6)
Note that, because m ∈ Se for δ2 ≤ δ < δ3, the expression for∂ΠC,TFS (δ)
∂qS(δ)i
in (A.6) is
different from that in (A.5). For δ ≥ δ3, the general form of the derivative is as
in (A.6), again with different expressions for∂ΠC,TFS (δ)
∂qS(δ)i
. From Lemma B.5, ΠC,TFS is
increasing and convex in qSi . In addition, for i ∈ S, qSi (δ) is decreasing in δ and qSm(δ)
is increasing in δ. Therefore, the terms within square brackets in equations (A.4),
(A.5) and (A.6) are continuous and increasing in δ. We then have that ΠC,TFS (δ)
is quasi-convex in δ for each of the δ-intervals. Also, note that the expression in
(A.4) is negative. In addition, for large δ, we have that Se = {m} and Sl = ∅, so
that∂ΠC,TFS (δ)
∂δis positive. It only remains to prove that the first-order derivative is
62
continuous at each threshold δi. That will establish the result that the derivative
crosses zero from negative to positive only once. In what follows, we prove that
the derivative is continuous at δ1 and δ2, and similar results can be obtained for
δ ≥ δ3. For δ = δ1, note that y∗m(δ1) = 0, which implies from (B.2) of Lemma B.1
that∂ΠC,TFS (qSm(δ))
∂qSm(δ)|δ=δ1 = 0. Thus, the first-order derivative is continuous at δ1. For
δ = δ2, we have that y∗m(δ2) = y∗ia(δ2) = y∗c , which shows that (A.5) and (A.6) are
equal at δ2.
Next, we prove part (ii) . Given a popular set An, the first-order profit difference
in system D is
ΠD,TFAn+1
(k)− ΠD,TFAn
(k) =∑i∈An
[πD,TF (pqAn+1
i , k)− πD,TF (pqAni , k)] + πD,TF (pqAn+1
n+1 , k)
For system C, Lemma B.6 suggests that we need to consider the following three
cases: (1) If An ⊂ Ne, then (An)e = An and (An+1)e = An+1; (2) If An = Ne, then
(An)e = An and (An+1)e = An; (3) If An ⊃ Ne, then (An)e = Ne and (An+1)e = Ne.
For simplicity, we only analyze the first case here (the other two cases can be analyzed
similarly to obtain the same results). The first-order profit difference in model C is
ΠC,TFAn+1
− ΠC,TFAn
=∑i∈An
[πD,TF (pqAn+1
i , kd + αAn+1
i )− πD,TF (pqAni , kd + αAni )]
+ πD,TF (pqAn+1
n+1 , kd + αAn+1
n+1 ) (A.7)
where αAni and αAn+1
i are the KKT multipliers from the optimization in (A.3). (Re-
call also that∑
i∈An αAni =
∑i∈An+1
αAn+1
i = kc.) Note that kc > αAni > αAn+1
i > 0
for i ∈ An. To see this, suppose that αAni ≤ αAn+1
i for some i. Because the stock-
ing quantities for all products in (An)e = An are equal, we have thatkd+αAni
θi=
kd+αAnjθj
for all i, j ∈ An. Similarly,kd+α
An+1i
θi=
kd+αAn+1j
θjfor all i, j ∈ An+1.
63
Then, αAni ≤ αAn+1
i implies that αAnj ≤ αAn+1
j for any other j ∈ An. Thus,
kc =∑
i∈An αAni ≤
∑i∈An α
An+1
i <∑
i∈An+1αAn+1
i = kc, a contradiction. It also
follows that αAn+1
j > αAn+1
i if θi ≤ θj.
From Lemma B.1, because πD,TF (x, k) is decreasing in k, we have that (A.7) > (A.8),
where
∑i∈An
[πD,TF (pqAn+1
i , kd+αAn+1
i )−πD,TF (pqAni , kd+αAn+1
i )]+πD,TF (pqAn+1
n+1 , kd+αAn+1
n+1 ).
(A.8)
Because πD,TF (x, k) is submodular in (x, k), we also have that (A.8) > (A.9), where
∑i∈An
[πD,TF (pqAn+1
i , kd+αAn+1
n+1 )−πD,TF (pqAni , kd+αAn+1
n+1 )]+πD,TF (pqAn+1
n+1 , kd+αAn+1
n+1 ).
(A.9)
Note that the expression in (A.9) is equal to ΠD,TFAn+1
(kd +αAn+1
n+1 )−ΠD,TFAn
(kd +αAn+1
n+1 )
and that kd + αAn+1
n+1 < k. From Proposition B.2, the optimal assortment in system
D with unit cost k1 is smaller than that with unit cost k2. In other words, if
ΠD,TFAn+1
(k1)−ΠD,TFAn
(k1) ≥ 0, then ΠD,TFAn+1
(k2)−ΠD,TFAn
(k2) ≥ 0. Therefore, if ΠD,TFAn+1
(k)−
ΠD,TFAn
(k) ≥ 0, then ΠC,TFAn+1
− ΠC,TFAn
> ΠD,TFAn+1
(kd + αAn+1
n+1 ) − ΠD,TFAn
(kd + αAn+1
n+1 ) > 0,
which implies that the optimal assortment in system C is no smaller than that in
system D. �
Proof of Proposition 1.5.1: Because ΠDS (δ) is quasi-convex and θi + θj >
max {θi, θj}, if it is optimal to include variant i or j, then it is optimal to include both
of them in the optimal assortment. Thus, the optimal assortment includes either all
the variants that share a common component or none of them. If these variants are
not in the optimal assortment, then the results for the dedicated system show that
the optimal assortment is a popular set. Otherwise, note that the profit function
for any assortment S not containing variants i and j, but containing a new variant
64
m with utility δ, is quasi-convex in δ. We then have that the optimal assortment,
excluding variants i and j, is a popular subset of the remaining variants. �
Proof of Proposition 1.5.2: We prove the result for the IP model. A similar
argument applies to the TF model.
1. If i, j ≤ n, then i, j ∈ Ai+j. To see this, note that from part 1 we have that
either both variants or neither of them are in Ai+j. Suppose that i, j 6∈ Ai+j. Then,
ΠDAn
< ΠPCAn
< ΠPCAi+j = ΠD
Ai+j , where the superscript ‘PC’ denote the profit in the
system with partial commonality. The first inequality follows because commonality
increases the profit for a given assortment, the second inequality follows because i
and j are not in the optimal assortment in the system with partial commonality,
and the equality follows because, in that case, Ai+j only contains products that
use dedicated components. Because An is the optimal assortment in the dedicated
system, we reach a contradiction. Therefore, i, j ∈ Ai+j. Also, the profit function
of any assortment that excludes variants i and j, but contains a new variant m with
utility δ, is quasi-convex in δ. Therefore, Ai+j \ {i, j} is a popular subset of the
remaining variants. Note that1
∂ΠD,IPAn−1
(δ)
∂δ=
1∑l∈An−1
θl + δ + θ0
− ∑l∈An−1
qAn−1
l
∂πD,IP (λqAn−1
l , k)
∂qAn−1
l
+
∑l∈An−1
qAn−1
l
∂πD,IP (λqAn−1m , k)
∂qAn−1m
(A.10)
A similar derivative is obtained when the vector of utilities is given by Θi+j:
∂ΠD,IPAn−1
(δ/Θi+j)
∂δ=
1∑l∈An−1
θl + δ + θ0
− ∑l∈An−1\{i,j}
qAn−1
l
∂πD,IP (λqAn−1
l , k)
∂qAn−1
l
1 Recall that An−1(δ) = An−1 ∪ {m}. We omit the dependence of An−1 on δ to simplify thenotation in the expressions below.
65
−(qAn−1
i + qAn−1
j )∂πD,IP (λq
An−1
i + λqAn−1
j , k)
∂(qAn−1
i + qAn−1
j )+
∑l∈An−1
qAn−1
l
∂πD,IP (λqAn−1m , k)
∂qAn−1m
.(A.11)
From equation (B.1) of Lemma B.1, we have that
x∂πD,IP (x, k)
∂x= (p− k)x− pφ(z(p, k))σβxβ.
Moreover, Lemma B.2 implies that (qAn−1
i +qAn−1
j )β < (qAn−1
i )β+(qAn−1
j )β. Therefore,
(qAn−1
i + qAn−1
j )∂πD,IP (λq
An−1
i + λqAn−1
j , k)
∂(qAn−1
i + qAn−1
j )>
qAn−1
i
∂πD,IP (λqAn−1
i , k)
∂qAn−1
i
+ qAn−1
j
∂πD,IP (λqAn−1
j , k)
∂qAn−1
j
,
which implies that (A.10) > (A.11). Define t1(θ1, · · · , θi, · · · , θj, · · · , θn−1)
= δn−1(Θi+j). (Recall that, under the vector of utilities Θi+j, this value of δ is
defined so that ΠDn−1(0) = ΠD
n−1(δn−1(Θi+j)).) Because (A.10) > (A.11), we have
that δn−1(Θi+j) > δn−1(Θ) (≤ θn). Therefore, if θn ≥ δn−1(Θi+j) > δn−1(Θ), we
have that Ai+j = An. Otherwise, Ai+j is a strict subset of An. Similar expressions
as in (A.10) and (A.11) as well as a similar argument apply to the set An, which
proves the result for the case j = n.
2. If i < n and j > n, then, with a similar argument as in case 1, we have that
i, j ∈ Ai+j. Define t2(θ1, · · · , θi, · · · , θn−1, θj) = δn−1(Θi+j). From Lemma B.3, we
have that t2(θ1, · · · , θi, · · · , θn−1, θj) ≥ δn−1(Θ). Thus, from part (1) of Proposition
B.5, we conclude that if θn < t2(θ1, · · · , θi, · · · , θn−1, θj), then Ai+j \ {j} is a strict
subset of An, and otherwise, Ai+j = An ∪{j}. A similar argument proves part 3. �
66
Numerical Study
The parameters of the numerical study are as follows. N = 10, p = 10, and
β = 0.5. We consider two demand rates, λ = 200 and λ = 400. The costs of dedi-
cated and common components are given by one of the following nine combinations:
(kd, kc) = {(1, 7) , (4, 4) , (7, 1) , (1, 5) , (3, 3) , (5, 1) , (1, 3) , (2, 2) , (3, 1)}. The utilities
are given by one of the following five vectors:
Θ1 = (20, 18, 16, 14, 12, 10, 8, 6, 4, 2), Θ2 = (20, 18, 16, 14, 12, 5, 4, 3, 2, 1),
Θ3 = (20, 19, 12, 11, 10, 9, 8, 3, 2, 1), Θ4 = (20, 19, 18, 10, 9, 8, 4, 3, 2, 1),
Θ5 = (20, 19, 18, 17, 16, 15, 14, 13, 12, 11). For each of these, the no-purchase utility
takes one of the following four values: θ0 = {10, 20, 40, 60} . Tables A.1 and A.2
summarize the results2.
Table A.1: Optimal profit and variety levels in systems with commonality relativeto the corresponding systems with dedicated components under the IP model.
Average change in Average change in Cases with a strictoptimal profit optimal variety level increase in variety
Θ λ = 200 λ = 400 λ = 200 λ = 400 λ = 200 λ = 400Θ1 4.03% 3.32% 1.31 0.89 69.44% 55.56%Θ2 3.41% 2.91% 1.19 1.22 61.11% 61.11%Θ3 3.91% 2.94% 0.94 1.36 47.22% 69.44%Θ4 3.49% 2.90% 1.25 1.33 69.44% 69.44%Θ5 5.19% 3.42% 1.00 0.67 47.22% 33.33%
Average 4.01% 3.10% 1.14 1.09 58.89% 57.78%
2 The “Average change in optimal variety level” denotes the average increase in the optimalnumber of variants offered in the system with commonality relative to the system with dedicatedcomponents.
67
Table A.2: The effect of component commonality on optimal profit and variety levelsunder IP and TF demand models.
Independent Population Model Trend-Following ModelAve. change in Ave. change in Ave. change in Ave. change in
Θ opti. profit opti. variety level opti. profit opti. variety levelΘ1 3.68% 1.10 4.51% 2.67Θ2 3.16% 1.21 3.42% 3.79Θ3 3.42% 1.15 3.85% 3.08Θ4 3.20% 1.29 3.59% 3.5Θ5 4.31% 0.83 5.38% 1.17
Ave. 3.55% 1.12 4.15% 2.84
68
Appendix B
Proofs of Supplementary Results of Chapter 1
This appendix contains 1) definition and properties of some functions used through-
out the proofs, 2) intermediate results that are used in the proofs in Appendix A,
and 3) a few supplementary results that were not included in the paper.
Let Φ(·) and φ(·) denote the cumulative and probability density functions of the
standard normal distribution. Define z(x, k) = Φ−1(1 − kx). Differentiating with
respect to x and k, we get the following.
φ(z(x, k))∂z(x, k)
∂x=
k
x2,
φ(z(x, k))∂z(x, k)
∂k= −1
x.
Using the property φ′ (z) = −φ(z)z, we get
∂φ(z(x, k))
∂x= −φ(z(x, k))z(x, k)
∂z(x, k)
∂x= −z(x, k)
k
x2,
∂φ(z(x, k))
∂k= −φ(z(x, k))z(x, k)
∂z(x, k)
∂k= z(x, k)
1
x.
Following the profit functions for the two demand models in Section 1.3, let k = kd+kc
69
and redefine
πD,IP (x, k) = (p− k)x− pφ(z(p, k))σxβ, where x = λqi and
πD,TF (x, k) = (x− k)λ− xφ(z(x, k))σ, where x = pqi.
The profit expressions are different because the choice model determines the propor-
tion of demand allocated to variants under the IP demand model and the probability
of having the full demand and revenue for a particular variant under the TF demand
model. Given assortment S, the optimal profit functions for the IP and TF models
in system D are given by
ΠD,IPS =
∑i∈S
πD,IP (λqSi , kd + kc)and ΠD,TFS =
∑i∈S
πD,TF (pqSi , kd + kc).
As mentioned, we use the notation ΠD,IPS (δ) = ΠD,IP
S∪{m} and ΠD,TFS (δ) = ΠD,TF
S∪{m},
where θm = δ.
Lemma B.1. Both ΠD,IP (x, k) and ΠD,TF (x, k) are increasing and convex in x,
decreasing and convex in k, and submodular in (x, k).
Proof of Lemma B.1: The first-order derivatives of ΠD,IP (x, k) and ΠD,TF (x, k)
are
∂ΠD,IP (x, k)
∂x= (p− k)− pφ(z(p, k))σβxβ−1 > 0, (B.1)
∂ΠD,TF (x, k)
∂x= λ− σφ(z(x, k)) + σz(x, k)
k
x
=1
x(πD,TF (x, k) + k(λ+ σz(x, k)))
= Emin {y∗, D} ≥ 0, (B.2)
∂ΠD,IP (x, k)
∂k= −x− z(p, k)σxβ < 0,
∂ΠD,TF (x, k)
∂k= −y∗ = −λ− σz(x, k) < 0. (B.3)
70
Thus, both ΠD,IP (x, k) and ΠD,TF (x, k) are increasing in x and decreasing in k. The
second-order derivatives are given by:
∂2ΠD,IP (x, k)
∂x2= (1− β)pφ(z(p, k))σβxβ−2 > 0,
∂2ΠD,TF (x, k)
∂x2= σ
1
φ(z(x, k))
k2
x3> 0.
∂2ΠD,IP (x, k)
∂k2= −∂z(p, k)
∂kσxβ > 0,
∂2ΠD,TF (x, k)
∂k2= −∂z(x, k)
∂kσ > 0.
∂2ΠD,IP (x, k)
∂x∂k= −1− βz(p, k)σxβ−1 < 0,
∂2ΠD,TF (x, k)
∂x∂k= −σ∂z(x, k)
∂x< 0.
Thus, ΠD,IP (x, k) and ΠD,TF (x, k) are convex in x and k and submodular in (x, k).�
Proposition B.1. ΠDS (δ) is quasi-convex in δ. As a result, the optimal assortment
set under either demand model in the system with dedicated components is a popular
set.
Proof of Proposition B.1: See van Ryzin and Mahajan (1999) for the proof of
the IP demand model. For the TF demand model,
ΠD,TFS (δ) =
∑i∈S
πD,TF (pqS(δ)i , k) + πD,TF (pqS(δ)
m , k).
Differentiating with respect to δ, we obtain
∂ΠD,TFS (δ)
∂δ=
∑i∈S
∂πD,TF (pqS(δ)i , k)
∂qS(δ)i
∂qS(δ)i
∂δ+∂πD,TF (pq
S(δ)m , k)
∂qS(δ)m
∂qS(δ)m
∂δ
=
[−∑i∈S
θi∂πD,TF (pq
S(δ)i , k)
∂qS(δ)i
+ (∑i∈S
θi + θ0)∂πD,TF (pq
S(δ)m , k)
∂qS(δ)m
]/(∑
)2.
71
where∑
=∑
i∈S θi + δ + θ0. Define LD,TFS (δ) as the expression inside the square
brackets. From Lemma B.1, the first term of LD,TFS (δ) is itself increasing in qS(δ)i
and qS(δ)i is itself decreasing in δ for i ∈ S. The second term is increasing in q
S(δ)m
and qS(δ)m is increasing in δ. Thus, LD,TFS (δ) is increasing in δ, which implies that
ΠD,TFS (δ) is quasi-convex with respect to δ. Hence, for any starting assortment that
is not a popular set, it is possible to improve the profit by dropping the least popular
item in the assortment and possibly including the most popular product outside the
assortment. With an argument similar to the proof of the IP case in van Ryzin and
Mahajan(1999), this suggests that the optimal assortment is a popular set. �
Proposition B.2. The optimal assortment set becomes smaller and the optimal
profit decreases as k increases. Furthermore, ΠS(k) is convex in k.
Proof of Proposition B.2: Given an assortment S, ΠD,IPS =
∑i∈S π
D,IP (λqSi , k)
and ΠD,TFS =
∑i∈S π
D,TF (pqSi , k) are decreasing and convex in k by Lemma 1. Next,
we prove that the optimal assortment set becomes smaller as k increases for both
demand models. Assume that k1 > k2 and that the optimal assortments are An1 and
An2 for the settings with k1 and k2, respectively. By definition of n1, δn1(k1) > θn1+1
and θn1 ≥ δn1−1(k1). Furthermore, from Condition 2, θn1 ≥ δn1−1(k1) ≥ δn1−1(k2),
which implies that An1 ⊂ An2 . If δn1(k2) > θn1+1, then n1 = n2. Otherwise, n2 > n1.
Therefore the optimal assortment becomes smaller as k increases. �
The next result derives upper and lower bounds for the optimal inventory levels
in system C under IP demand model. These bounds are useful for the optimiza-
tion of the profit function under the independent population model in the numerical
experiments. In particular, using the bounds on the inventory values, we use an iter-
ative optimization algorithm (optimizing along each variable, one at a time) which is
72
guaranteed to converge to the optimal inventory levels because of the joint concavity
of the profit function. ΠP,IPS denotes the total profit in the pooled system.
Proposition B.3. Consider system C under IP demand model. The following
comparisons hold for any given assortment S.
(i) y∗c ≥ y∗i > λqSi + zIPσ(λqSi )β, which is the stocking level for components
associated with product variant i in the dedicated system.
(ii) y∗c <∑
i∈S y∗i .
(iii) y∗c ≤ λ∑
i∈S qSi + Φ−1(1 − kc
p)σλβ(
∑i∈S q
Si )β, which is the stocking level of
the common component in the pooled system with unit cost kc.
(iv) ΠD,IPS ≤ ΠC,IP
S ≤ ΠP,IPS .
Proof of Proposition B.3: The optimal stocking levels satisfy the following first-
order conditions:
Pr{y∗i ≤ Di, y∗c >
∑i∈S
min{y∗i , Di}} = kd/p (B.4)
Pr{∑i∈S
min{y∗i , Di} > y∗c} = kc/p (B.5)
For part (i), because Pr{y∗i ≤ Di or y∗c >∑
i∈S min{y∗i , Di}} ≤ 1, then
Pr{y∗i ≤ Di}+ Pr{y∗c >∑i∈S
min{y∗i , Di}} − Pr{y∗i ≤ Di, y∗c >
∑i∈S
min{y∗i , Di}} ≤ 1.
With some algebra, we get
Pr{y∗i ≤ Di} ≤ Pr{y∗i ≤ Di, y∗c >
∑i∈S
min{y∗i , Di}}+ Pr{y∗c <∑i∈S
min{y∗i , Di}}.
Using (B.4) and (B.5), we then have that
Pr{y∗i ≤ Di} ≤k
p,
73
which implies that y∗i > λqSi +zIPσ(λqSi )β. The proof of part (ii) is by contradiction.
Suppose that y∗c ≥∑
i∈S y∗i . Then, y∗c ≥
∑i∈S y
∗i ≥
∑i∈S min{y∗i , Di}, which implies
that Pr{∑
i∈S min{y∗i , Di} > y∗c} = 0. This contradicts equation (B.5). For part
(iii), we have that
Pr{y∗c >∑i∈S
Di} ≤ Pr{y∗c >∑i∈S
min{y∗i , Di}} = 1− kc/p,
where the equality follows by (B.5). Hence, y∗c should be larger than the stocking
quantity for the sum of the demand for all the components, which is the stocking
level in the pooled system. That is, y∗c < λ∑
i∈S qSi + Φ−1(1 − kc
p)σλβ(
∑i∈S q
Si )β.
Finally, for part (iv), let ΠCS denote the profit in system C with the optimal stocking
quantities of system D (i.e., y∗i = λqSi + zIPσ(λqSi )β, y∗c =∑
i∈S λqSi + zIPσ(λqSi )β)
and ΠCS denote the optimal profit in system C. Clearly, ΠD,IP
S ≤ ΠC,IPS ≤ ΠC,IP
S .
With a similar argument, we obtain ΠC,IPS ≤ ΠP,IP
S . �
Lemma B.2.
(∑i∈S
θi)β <
∑i∈S
θβi , for any S and β ∈ [0, 1] .
Proof of Lemma B.2: We show the result for the case with |S| = 2. Define
f(x) = aβ + xβ − (a + x)β where a, x ≥ 0. Since ∂f(x)∂x
= β[ 1x1−β− 1
(a+x)1−β] > 0, we
have that f(x) > f(0) = 0 for any x > 0. Therefore aβ + xβ > (a + x)β. This can
be extended to the cases with |S| > 2. �
Lemma B.3. For any n and i ≤ n, if δn−1 ≤ θn, then δn is increasing in θi.
Proof of Lemma B.3: By the definition of δn, ΠDn (δn) = ΠD
n (0). Note that δn is
a function of (θ1, · · · , θn) and independent of (θn+1, · · · , θN). Also, both ΠDn (0) and
74
ΠDn (δn) are functions of (θ1, · · · , θn) through the choice probabilities (qAni )i∈An and
(qAn(δn)i )i∈An(δn), respectively. Differentiate both sides of the equality with respect
to θi, for i ∈ An, to obtain
∂ΠDn (δn)
∂δ
∂δn∂θi
+∑
j∈An(δn)
∂ΠDn (δn)
∂qAn(δn)j
∂qAn(δn)j
∂θi=∑j∈An
∂ΠDn
∂qAnj
∂qAnj∂θi
(B.6)
(Recall that An(δn) denotes the set An plus a variant m with utility δn.) Because
ΠDn (δ) is quasi-convex, its first-order derivative at δn is positive from the definition
of δn, i.e., ∂ΠDn (δ)/∂δ
∣∣δ=δn
> 0. Hence, ∂δn/∂θi > 0 if we show that (B.7) > (B.8)
below:
∑j∈An
∂ΠDn
∂qAnj
∂qAnj∂θi
=1∑
j∈An θj + θ0
[∂πD(aqAni , k)
∂qAni−∑j∈An
qAnj∂πD(aqAnj , k)
∂qAnj
](B.7)
∑j∈An(δn)
∂ΠDn (δn)
∂qAn(δn)j
∂qAn(δn)j
∂θi=
1∑j∈An(δn) θj + θ0
[∂πD(aq
An(δn)i , k)
∂(qAn(δn)i
)
−∑
j∈An(δn)
qAnj (δn)∂πD(aq
An(δn)j , k)
∂(qAn(δn)j
) ], (B.8)
where a = λ for the IP model and a = p for the TF model. We prove that the
expression in (B.7) is positive by contradiction. If it was negative, then that would
imply that ΠDAn\{i} > ΠD
An. However, Condition 1 and δn−1 < θn (assumed in the
statement of the Lemma) imply that An is the optimal assortment when the set
of possible products is restricted to the n most popular variants. This leads to a
contradiction. In addition, we have that∑
j∈An(δn) θj + θ0 >∑
j∈An θj + θ0. Hence,
we can obtain the desired result by comparing the terms in the brackets in equations
(B.7) and (B.8). Convexity of πD(x, k) in x together with ∂qAni (δ) /∂δ < 0 imply
that the first term inside the bracket in (B.7) is larger than the first term inside the
75
bracket in (B.8) . Next, we compare the second terms in each equation for each
demand model separately.
For the IP model, from (B.1), we have
∑j∈An
qAnjπD,IP (λqAnj , k)
∂qAnj=
∑j∈An
qAnj λ((p− k)− pφ(z(p, k))σβ(λqAnj )β−1
)= (p− k)λ
∑j∈An
qAnj − βpσφ(z)∑j∈An
(λqAnj )β, (B.9)
∑j∈An(δn)
qAn(δn)j
πD,IP (λqAn(δn)j , k)
∂(qAn(δn)j
) = (p−k)λ∑
j∈An(δn)
qAn(δn)j −βpσφ(z)
∑j∈An(δn)
(λqAn(δn)j )β.
(B.10)
On the other hand, ΠDAn
(δn) = ΠDAn
implies that
(p− k)λ∑j∈An
qAnj − pσφ(z)∑j∈An
(λqAnj )β = (p− k)λ∑
j∈An(δn)
qAnj (δn)
−pσφ(z)∑
j∈An(δn)
(λqAnj (δn))β
Hence,
(B.9)− (B.10) = (1− β) (p− k)λ[∑j∈An
qAnj −∑
j∈An(δn)
qAn(δn)j ] < 0.
Thus, (B.7) > (B.8) for the IP model.
For the TF model, recall that ΠD,TFn =
∑j∈An q
Anj E[min{yAnj , D}] − kyAnj , where
yAnj = y∗j given qAnj . Note that δn and ΠDn are functions of k. By definition of δn,
we have ΠD,TFn (δn) = ΠD,TF
n . Differentiating both sides of the equation with respect
to k, we obtain
∂ΠD,TFAn
(δ)
∂δ
∣∣∣∣∣x=δn
∂δn∂k
+∑
j∈An(δn)
∂πD,TF (pqAn(δn)j , k)
∂k=∑j∈An
∂πD,TF (pqAn(δn)j , k)
∂k.
76
The first term on the left is positive because of the definition of δn and Condition 2.
Thus, we have
∑j∈An
∂πD,TF (pqAn(δn)j , k)
∂k>
∑j∈An(δn)
∂πD,TF (pqAn(δn)j , k)
∂k.
By (B.3) , this is equivalent to ∑j∈An(δn)
yAn(δn)j >
∑j∈An
yAnj
At the same time, ΠD,TFn (δn) = ΠD,TF
n is equivalent to∑j∈An(δn)
pqAn(δn)j E[min{yAn(δn)
j , D}]−∑
j∈An(δn)
yAn(δn)j =
∑j∈An
pqAnj E[min{yAni , D}]−∑j∈An
yAnj .
As a result, we have∑j∈An(δn)
qAn(δn)j E[min{yAn(δn)
j , D}] >∑j∈An
qAnj E[min{yAnj , D}].
This implies the desired inequality:
∑j∈An(δn)
qAn(δn)j
∂πD,TF (pqAn(δn)j , k)
∂qAn(δn)j
>∑j∈An
qAnj∂πD,TF (pqAnj , k)
∂qAnj
because, by (B.2), E[min{yAn , D}] = ∂πD,TF (x,k)∂x
|x=pqAnj. This completes the proof.
�
Lemma B.4. Se(δ) is a popular subset of S that decreases (in terms of the “ ⊂ ”
order) to the set {m} as δ increases. At the same time, Sl(δ) = S ∪ {m} − Se(δ)
increases to the set S as δ increases. The common-component fractile t∗c(S ∪ {m})
is continuous and first decreasing and then increasing in δ.
77
Proof of Lemma B.2: This lemma explores the structure of the sets Se and Sl
when a new variant m, with utility δ, is added to the assortment S. We let Se(δ)
and Sl(δ) denote the sets (S ∪ {m})e and (S ∪ {m})l, respectively. We study how
Sl(δ), Se(δ), and t∗c(S ∪ {m}) change as δ increases. Let a be the cardinality of Se.
We define the following threshold values of δ:
δ1def=
kd(∑
i∈S θi + θ0)
p− kd, δ2
def=
kd∑
i∈Se θi
(kc + akd), δ3
def=
kc + (a+ 1)kdkd
θia −∑i∈Se
θi,
and δ4def=
kc + akdkd
θia−1 −∑i∈Se
θi − θia .
Case 1: The fractile 1− kd
pqS(δ)m
< 0, or equivalently δ < δ1. In this case, the new variant
m is in the assortment S, but with a stocking quantity equal to zero. Therefore
Sl(δ) = Sl ∪ {m} and Se(δ) = Se. Also, t∗c(S ∪ {m}) = 1 − (kc+akd)(∑i∈S θi+δ+θ0)
p∑al=1 θil
is
continuous and decreasing in δ.
Case 2: When 0 ≤ 1 − kd
pqS(δ)m
< t∗c(S ∪ {m}), or equivalently δ1 ≤ δ < δ2, the
new variant m has a positive optimal stocking level which is smaller than that of
the common component. Therefore, Sl(δ) = Sl ∪ {m} and Se(δ) = Se. Again,
t∗c(S ∪ {m}) = 1− (kc+akd)(∑i∈S θi+δ+θ0)
p∑al=1 θil
, which is continuous and decreasing in δ.
Case 3: When 1− kd
pqS(δ)m
> t∗c(S ∪ {m}) and 1− kd
pqS(δ)ia
≥ t∗c(S ∪ {m}), or equivalently
δ2 ≤ δ < δ3, the new variant m has the same optimal stocking level as that of the
common component. Therefore, Se(δ) = Se ∪{m} (which continues to be a popular
subset of S ∪ {m}) and Sl(δ) = Sl. Also, t∗c(S ∪ {m}) = 1− (kc+(a+1)kd)(∑i∈S θi+δ+θ0)
p∑al=1 θil+pδ
is continuous and increasing in δ.
Case 4: When 1 − kd
pqS(δ)m
> t∗c(S ∪ {m}) > 1 − kd
pqS(δ)ia
and 1 − kd
pqS(δ)ia−1
≥ t∗c(S ∪ {m}),
or equivalently δ3 ≤ δ < δ4, the new variant m is still in Se(δ) and variant ia,
78
which has the lowest utility in Se, moves to Sl(δ). (Note that δ ≥ δ3 implies that
δ ≥ θia .) Therefore, Se(δ) = Se∪{m}\{ia}, which continues to be a popular subset
of S∪{m} and Sl(δ) = Sl∪{ia}. Also, t∗c(S∪{m}) = 1− (kc+akd)(∑i∈S θi+δ+θ0)
p∑al=1 θil+pδ−pθia
, which
is continuous and increasing in δ.
Following a similar reasoning, as δ increases beyond δ4, the set Se(δ) (which includes
m) continues to be a popular subset of S ∪ {m}. For large enough δ, the set Se(δ)
reduces to {m}. On the other hand, Sl(δ) = S∪{m}\Se(δ) increases in δ, eventually
becoming equal to S. The threshold t∗c(S ∪ {m}) is continuous and increasing in δ.
�
Lemma B.5. For any i ∈ S, ΠC,TFS is increasing and convex in qSi .
Proof of Lemma B.5: Based on equation (B.2) of Lemma B.1, the result is
established if we prove that y∗i is continuous and increasing in qSi regardless of how
the sets Se and Sl change with qSi . (Note that, based on the expression of ΠC,TFS ,
the form of the derivative in (B.2) also applies in system C.) Assume that i ∈ Sl.
Then, y∗i is continuous and increasing qSi beyond the point at which variant i enters
the set Se because the common-component fractile is continuous and increasing in
qSi as implied by the proof of Lemma B.4. Before that point, the stocking level is
y∗i = λ+σz(pqSi , kd). This is continuous and increasing in qSi . Moreover, it is easy to
verify that y∗i is continuous at the point in which variant i enters the set Se. Thus,
y∗i is increasing and convex in qSi . For i ∈ Se, the result can be proved similarly. �
The following lemma presents a technical property regarding the set of variants
Se that results from the stocking optimization problem when the assortment is given
by set S. The result is used to prove Theorem 1.4.2. For a popular set An, (An)e
is itself a popular set (following Proposition B.4). Note also that, when S = N , the
set of all products, we denote Se as Ne. Finally, let |S| denote the cardinality of set
79
S.
Lemma B.6. Consider a popular set An. If n ≤ |Ne|, then (An)e = An. Otherwise,
(An)e = Ne.
Proof of Lemma B.6: Let r = |Ne| so that Ne = Ar. By definition of Ne, r
satisfies the following:
1− kd∑N
i=0 θipθj
|j∈Ar ≥ t∗c(N ) = 1− (kc + rkd)∑N
i=0 θip∑r
i=1 θi> 1− kd
∑Ni=0 θi
pθr+1
. (B.11)
We complete the proof by contradiction. If n < r, then (An)e ⊂ An, but suppose
that (An)e 6= An. Let z = |(An)e| < n < r, which satisfies the following:
1− kd∑n
i=0 θipθj
|j∈Az ≥ t∗c(An) = 1− (kc + zkd)∑n
i=0 θip∑z
i=1 θi> 1− kd
∑ni=0 θi
pθz+1
. (B.12)
By combining (B.11) and (B.12), we have
kc + rkd∑ri=1 θi
>kdθr> · · · > kd
θz+1
>kc + zkd∑z
i=1 θi.
Inverting all the terms, reversing the order of the inequalities, and taking the average
of all the terms with θr to θz + 1, we have that
kc + zkd∑zi=1 θi
<(r − z)kd∑r
i=z+1 θi<kc + rkd∑r
i=1 θi(B.13)
Let a = kc + zkd, b =∑z
i=1 θi, c = (r − z)kd and d =∑r
i=z+1 θi. The inequality
(B.13) can then be expressed as
a
b<c
d<a+ c
b+ d
80
The first inequality shows that ad < bc. However, the second inequality implies that
bc < ad, a contradiction. Therefore (An)e = An when n < r. Similarly, we can
prove that (An)e = Ne when n ≥ r. �
Proposition B.4. The optimal profit is decreasing in the unit costs kc and kd under
both demand models. Moreover, the optimal profit is more sensitive to changes in kd
than in kc.
Proof of Proposition B.4: Under the IP model, the optimal stocking levels y∗i and
y∗c , with i ∈ S, are derived from equations (B.4) and (B.5) of Proposition B.3. The
optimal profit function is given by
ΠC,IPS ({y∗i }i∈S, y∗c , kc, kd) = pE[min{y∗c ,
∑i∈S
min{y∗i , Di}}]−∑i∈S
kdy∗i − kcy∗c .
Differentiating with respect to kc and kd, respectively, we obtain
∂ΠC,IPS ({y∗i }i∈S, y∗c , kc, kd)
∂kc= −y∗c ≤ 0, (B.14)
∂ΠC,IPS ({y∗i }i∈S, y∗c , kc, kd)
∂kd= −
∑i∈S
y∗i ≤ 0. (B.15)
Hence, the optimal profit function is decreasing in kc and kd. Furthermore, from
Proposition B.3, (B.14) > (B.15). Therefore, the rate of change in the optimal profit
with respect to kc is less significant than with respect to kd. Under the TF model,
the optimal profit function is given by
ΠC,TFS ({y∗i }i∈S, y∗c , kc, kd) =
∑i∈Se
[pqiEmin{y∗c , D}−kdy∗c ]+∑i∈Sl
[pqiEmin{y∗i , D}−kdy∗i ]
−kcy∗c .
81
Then, we have
∂ΠC,TFS ({y∗i }i∈S, y∗c , kc, kd)
∂kc= −y∗c , (B.16)
∂ΠC,TFS ({y∗i }i∈S, y∗c , kc, kd)
∂kd= −ay∗c −
∑i∈Sl
y∗i (B.17)
where a = |Se|. Therefore the optimal profit function is decreasing in kc and kd.
Because (B.16) > (B.17), we again have that the rate of change in the optimal profit
with respect to kc is less significant than with respect to kd. �
We now proceed with comparative statics results regarding the structure of the
optimal assortment in the dedicated system. These results will be instrumental
in characterizing the optimal assortment structure under general bill-of-materials
configurations. Suppose that the utility for variant j increases from θj to θ′j, with
θ′j > θj. Let Θ = (θ1, · · · , θN) be the original vector of utilities and Θ′ be the
vector Θ with θj replaced by θ′j and reorganized in decreasing order. Suppose that
An is the optimal popular set under the original utilities and let A′ be the optimal
popular set under the utilities in Θ′. The following result characterizes the set A′
both for the independent population and trend-following population models. The
result states that if the utility of a variant in the optimal assortment increases, then
the resulting optimal assortment may contain fewer variants than the original. On
the other hand, if the utility of a variant outside the optimal assortment increases,
then it will be included in the optimal assortment as long as the increase is large
enough.
Proposition B.5. Consider the dedicated system.
(i) If j < n, then j ∈ A′. In addition, if δ′n−1 > θn then A′ is a strict subset of
An. Otherwise, A′ = An. (The quantity δ′n−1 corresponds to the vector of
utilities Θ′.)
82
(ii) If j = n, then A′ = An when θ′n ≤ θn−1 or when θ′n > θn−1 and δ′n−1 ≤ θn−1.
When θ′n > θn−1 and δ′n−1 > θn−1, A′ is a strict subset of An.
(iii) If j ≥ n + 1, then An ∪ {j} ⊂ A′ when θn > θ′j ≥ δn > θn+1, and j ∈ A′ and
A′ \ {j} ⊂ An when θ′j ≥ θn.
(iv) If j ≥ n+ 1 and δn > θ′j, then A′ = An.
Proof of Proposition B.5: For (i), because j < n and θ′j > θj, variant j is still in
the optimal assortment, i.e., j ∈ A′. The n− 1 most popular variants are the same
as An−1 which includes variant j. From Lemma B.3, δ′n−1 > δn−1. In addition,
θn ≥ δn−1 because An is the optimal assortment under the original utilities. Thus,
if δ′n−1 which is a function of (θ1, · · · , θ′j, · · · , θn−1) is larger than θn, then A′ is a
strict subset of An. Otherwise, A′ = An. For (ii), if j = n and θ′n ≤ θn−1,
then θ′n > θn ≥ δn−1, Moreover, Lemma B.3 says that δ′n > δn > θn+1. Thus,
A′ = An. If θ′n > θn−1, then the n − 1 most popular variants are (1, · · · , n − 2, n).
From Lemma B.3, δ′n−1 which is a function of (θ1, · · · , θn−2, θ′n) is larger than δn−1.
Then A′ is a strict subset of An if δ′n−1 > θn−1. For (iii), because j ≥ n + 1 and
θn > θ′j ≥ δn > θn+1, variant j as well as An are included in the optimal assortment
A′. When θ′j ≥ θn, similarly as case (i) and (ii), we have j ∈ A′ and A′ \ {j} ⊂ An.
For (iv), because j ≥ n+ 1 and δn > θ′j, it is apparent that A′ = An. �
Proposition B.6. Let Πi+j denote the optimal profit of a system in which variants i
and j (i < j) are produced using a set of common components and all other variants
use dedicated components. Under both the IP and the TF demand models, Π1+2 ≥
Πi+j for any i, j.
Proof of Proposition B.6: Let Θi+j denote the utility vector of a system in which
variants i and j (i < j) are produced using a set of common components and all
83
other variants use dedicated components. For variants i and j, Θ1+2 � Θi+j in a
majorization order. Van Ryzin and Mahajan (1999) shows that Π1+2 ≥ Πi+j since
Θ1+2 � Θi+j for the IP demand model. With a similar argument, the result also
holds for the TF demand model. �
Proposition B.7. Consider the trend-following demand model. Let Ai+j be the op-
timal assortment of a system with partial component commonality in which variants
i and j use a common component and a dedicated component each, and all other
variants use only dedicated components, as depicted in Figure 1(d). Let An be the
optimal (popular) assortment in the corresponding dedicated system. The following
results hold.
1. If i, j ≤ n, then Ai+j = A′ ∪ {i, j} where A′ is a popular subset of An\ {i, j}.
There exists a threshold value t7 such that A′ is a strict subset of An\ {i, j} if
and only if θn < t7,.
2. If i < n and j > n, then Ai+j = A′∪{i, j} where A′ is a popular subset of An.
3. If i > n and j > n, and assuming that θj >kd
kd+kcθi, then there exists a threshold
value t8 such that Ai+j = A′ ∪ {i, j} if θi + θj ≥ t8, and Ai+j = An otherwise
(A′ is a popular subset of An).
Proof of Proposition B.7: We will denote by ‘PC’ the system described in the
statement of the proposition.
1. If i, j < n, then i, j ∈ Ai+j (following a similar argument to that in Proposition
1.5.2). Consider any assortment S that contains variants i and j, and let m be a
variant not in S with utility δ. Because πD,TF (x, k) is submodular in (x, k) from
Lemma B.1, we have
84
∂πD,TF (pqS(δ)i , k)
∂qS(δ)i
<∂πD,TF (pq
S(δ)i , kd + αi(θi, θj))
∂qS(δ)i
∂πD,TF (pqS(δ)j , k)
∂qS(δ)j
<∂πD,TF (pq
S(δ)j , kd + αj(θi, θj))
∂qS(δ)j
Therefore,
∂ΠD,TFS (δ)
∂δ
=p
(∑
l∈S θl + δ + θ0)2
[−∑l∈S
θl∂πD,TF (pq
S(δ)l , k)
∂qS(δ)l
+
(∑l∈S
θl + θ0
)∂πD,TF (pq
S(δ)m , k)
∂qS(δ)m
]
>−p
(∑
l∈S θl + δ + θ0)2
∑l∈S\{i,j}
θl∂πD,TF (pq
S(δ)l , k)
∂qS(δ)l
+θi∂π
D,TF (pqS(δ)i , kd + αi(θi, θj))
∂qS(δ)i
+θj∂πD,TF (pq
S(δ)j , kd + αj(θi, θj))
∂qS(δ)j
−
(∑l∈S
θl + θ0
)∂πD,TF (pq
S(δ)m , k)
∂qS(δ)m
]=∂ΠPC,TF
S (δ)
∂δ.
(Note that ΠPC,TFS (δ) is quasi-convex in δ which implies that Ai+j \{i, j} is a popular
subset of the remaining variants.) The inequality above implies that δl(Θ|PC) >
δl(Θ) for any l ≥ j. We then define t7(θ1, · · · , θi, · · · , θj, · · · , θn−1) = δn−1(Θ|PC).
If θn < t7(θ1, · · · , θi, · · · , θj, · · · , θn−1), then Ai+j is a strict subset of An. Otherwise,
Ai+j = An. The case with j = n follows similarly.
2. If i < n, then, following a similar argument as before, we can show that i ∈ Ai+j.
If j > n is not in Ai+j, we then have that Ai+j = An. Suppose now that j ∈ Ai+j.
For a given assortment S and a variant m 6∈ S, we have proved that ΠPC,TFS (δ) is
quasi-convex in δ. Therefore, Ai+j = A′ ∪ {i, j} where A′ is a popular subset of An.
3. Consider now the case with i > n and j > n (and θj >kd
kd+kcθi). We first note
that δn(Θ) > θn+1 ≥ θi, θj. The threshold value δn(Θ) arises in the setting in which
all variants have unit production costs equal to k. Because θj >kd
kd+kcθi, we have
85
from Lemma B.7 that αi, αj > 0. Then, we can write the combined profit function
for variants i and j as πTFi,j , being a function of p(qi + qj) and 2kd + kc (essentially,
the two variants jointly play the role of a single variant with utility θi + θj and total
cost 2kd + kc). Let S be any assortment not containing variants i and j, and let
ΠPC,TFS∪{i,j} =
(∑l∈S π
D,TFl
)+ πTFi,j . If m 6∈ S with utility δ, using similar arguments
as before, we have that ΠPC,TFS (δ) is quasi-convex in δ. We then define δ(Θ|PC)
such that ΠPC,TFS (0) = ΠPC,TF
S (δ(Θ|PC)). (Note that now δ(Θ|PC) may depend
on 2kd + kc.) Let m denote a new variant with utility δ and An(δ) = An ∪ {m}.
In system D, this new variant has a cost of k = kd + kc and utility δ, while in the
system with partial commonality, this new variant represents the combined variants
i and j with cost 2kd + kc and utility δ. Because πD,TF (x, k) is increasing in x and
submodular in (x, k) (from Lemma B.1), we have
∂ΠD,TFAn
(δ)
∂δ=
p
(∑
l∈An θl + δ + θ0)2
[−∑l∈An
θl∂πD,TF (pq
An(δ)l , k)
∂qAn(δ)l
+
(∑l∈An
θl + θ0
)∂πD,TF (pq
An(δ)m , k)
∂qAn(δ)m
]
≥∂ΠPC,TF
An(δ)
∂δ=
p
(∑
l∈An θl + δ + θ0)2
[−∑l∈An
θl∂πD,TF (pq
An(δ)l , k)
∂qAn(δ)l
+
(∑l∈An
θl + θ0
)∂πD,TF (pq
An(δ)m , 2kd + kc)
∂qAn(δ)m
].
This implies that δn(Θ|PC) > δn(Θ). Therefore, we define t8(θ1, · · · , θn, αi) =
δn(Θ|PC). If t8(θ1, · · · , θn, αi) ≤ θi + θj, then i, j ∈ Ai+j and Ai+j = A′ ∪ {i, j}
where A′ is a popular subset of An. Otherwise, Ai+j = An. �
86
Lemma B.7. Consider the system described in Proposition B.7. Let S be a subset
of N , and assume that variants i > j are both included in S. There exist positive
values αi(θi, θj) and αj(θi, θj) such that αi(θi, θj) + αj(θi, θj) = kc. If θj ≤ kdkd+kc
θi,
then αi(θi, θj) = kc and αj(θi, θj) = 0. Otherwise,kd+αi(θi,θj)
θi=
kd+αj(θi,θj)
θj.
Proof of Lemma B.7: Suppose that θj ≤ kdkd+kc
θi, which implies that 1− kd+kcθi≥
1 − kdθj
. From the results in system C, we have that if the KKT multipliers αi > 0
and αj > 0, then αi + αj = kc and kd+αiθi
=kd+αjθj
. This equality, together with
θj ≤ kdkd+kc
θi can only occur if αi = kc and αj = 0, a contradiction. We then have
that αi = kc and αj = 0 (and both values are independent of all other parameters,
except for θi and θj). If θj >kd
kd+kcθi, then 1 − kd+kc
θi< 1 − kd
θj. In this case, there
exist positive values αi(θi, θj) and αj(θi, θj) from the solution of the KKT conditions,
such that αi(θi, θj) +αj(θi, θj) = kc andkd+αi(θi,θj)
θi=
kd+αj(θi,θj)
θj. Here again, αi and
αj only depend on kc, kd, θi, and θj. �
We also study a system in which there is a component common to all products and
another component common to only products i and j. (All other variants require,
in addition, a dedicated component.) Figure 1.5(e) illustrates an example of such
system. In the next proposition, we compare this system to the related system C
(illustrated in Figure 1.5(a)).
Proposition B.8. Consider a system with partial component commonality in which
there is a component common to all products and another one common to only prod-
ucts i and j, while all other variants also require a dedicated component, as depicted
in Figure 1(e). Consider the trend-following model. Let Ai+j be the optimal assort-
ment in that system. Let An be the optimal assortment in the corresponding system
C in which all variants are produced using a dedicated component and a common
87
component. Assume that Ne ⊂ An.1
1. If i, j ≤ n, then Ai+j = A′ ∪ {i, j} where A′ is a popular subset of An\ {i, j}.
There exists a threshold value t5 such that A′ is a strict subset of An\ {i, j} if
θn < t5.
2. If i < n and j > n, then Ai+j = A′∪{i, j} where A′ is a popular subset of An.
3. If i > n and j > n, then there exists a threshold value t6 such that Ai+j =
A′ ∪ {i, j} if θi ≥ t6, and Ai+j = An otherwise. A′ is a popular subset of An.
Proof of Proposition B.8: Based on the assumption that Ne ⊂ An, we consider
the case in which i ∈ Ne and j ∈ N \ Ne (applicable to parts 1 and 2). The
other cases can be proved similarly. From the results of the pooled system, it is
optimal to include both variants i and j in the optimal assortment. In this setting
with partial commonality, we have that the (unordered) utility vector is Θi+j =
(θ1, · · · , θi + θj, · · · , 0, · · · , θn). For an assortment S, we denote the corresponding
set of equal stocking quantities in this setting as SPCe . From Lemma B.4, we have
that N PCe ⊆ Ne.
1. If i, j < n, then i, j ∈ Ai+j. Given any assortment S including variants i and j,
ΠPC,TFS (δ) is quasi-convex in δ by Lemma B.5. This implies that Ai+j \ {i, j} is a
popular subset of the remaining variants. Consider An−1 = {1, 2, · · · , n − 1}, with
i, j ∈ An−1. Then, we have
∂ΠPC,TFS (δ)
∂δ
1 Recall thatNe is the popular subset of variants with equal stocking levels as the stocking quantityof the common component that results from the profit maximization problem with respect to thestocking quantities assuming an assortment that contains all possible variants in N .
88
=p
(∑
l∈S θl + δ + θ0)
[−∑l∈S
qS(δ)l
∂ΠPC,TF (δ)
∂qS(δ)l
+ (∑l∈S
qS(δ)l + q
S(δ)0 )
∂ΠPC,TF (δ)
∂qS(δ)m
](B.18)
∂ΠC,TFS (δ)
∂δ=
p
(∑
l∈S θl + δ + θ0)
[−∑l∈S
qS(δ)l
∂ΠC,TF (δ)
∂qS(δ)l
+ (∑l∈S
qS(δ)l + q
S(δ)0 )
∂ΠC,TF (δ)
∂qS(δ)m
]. (B.19)
We want to prove that (B.19) > (B.18) when δ < δn−1(Θ) (< θn), so that
δn−1(Θi+j) > δn−1(Θ). For δ < δn−1(Θ) (< θn), the new variant m is not in Se(δ),
and therefore not in SPCe (δ) either as SPCe (δ) ⊂ Se(δ). Therefore, the second terms
in the square brackets in (B.19) and (B.18) are equal. We then compare the first
terms in the square brackets. We denote the optimal stocking levels in the system
with partial commonality as yPC∗l . The proof of Lemma B.4 implies that yPC∗l > y∗l
for any l ∈ Se ∪ {j} and yPC∗l = y∗l for any l ∈ S \ (Se ∪ {j}). Moreover, from (B.2)
of Lemma B.1, we have that
−∑l∈S
qS(δ)l
∂ΠPC,TF (δ)
∂qS(δ)l
= −∑l∈S
qS(δ)l Emin{yPC∗l , D}
and
−∑l∈S
qS(δ)l
∂ΠC,TF (δ)
∂qS(δ)l
= −∑l∈S
qS(δ)l Emin{y∗l , D}.
Therefore, (B.19)> (B.18), which implies that δn−1(Θi+j) > δn−1(Θ). Then, defining
t5(θ1, · · · , θi, · · · , θj, · · · , θn−1) = δn−1(Θi+j),
we have that if t5(θ1, · · · , θi, · · · , θj, · · · , θn−1) > θn, then A′ is a strict subset of An.
Otherwise, A′ = An.
2. If i < n, then i ∈ Ai+j. Because it is optimal to include both variants i and j
89
together, we also have that j ∈ Ai+j. Given assortment S including both i and j,
we have proved that ΠPC,TFS (δ) is quasi-convex in δ. Therefore, Ai+j = A′ ∪ {i, j},
where A′ is a popular subset of An.
3. Because i > n and j > n, we have that δn(Θ) > θn+1 ≥ θi, θj. Let t6(θ1, · · · , θn) =
δn(Θ). If δn(Θ) > θi + θj, then Ai+j = An. Otherwise, Ai+j = A′ ∪ {i, j} where A′
is a popular subset of An. �
90
Appendix C
Proofs of Chapter 2
Proposition 2.3.1
Proof: The optimality equation (2.3.2) can be rewritten as
Vt(y|m) = maxS⊂S(y)
{∑i∈S
λqmi(S)pit(y)
}+ Vt+1(y) (B.1)
If Vt+1(·), the value function in period t + 1, is known, then (pit(y))i∈N can be
derived. Without loss of generality, assume pi1t ≥ · · · ≥ piNt and S(y) = N . Note
that these marginal revenues are the same for all customer segments. Suppose the
arriving customer belongs to segment m. We prove the result by contradiction.
Suppose the optimal assortment is S∗ such that there exists k and l where ik ∈ S∗,
il ∈ N \S∗, and pilt (y) > pikt (y). Then pilt (y) > pikt (y) ≥∑
j∈S∗ qmj(S)pijt (y) because
ik ∈ S∗. For simplicity, let a =∑
j∈S∗ θmjpijt (y) and b =
∑j∈S∗ θmj + θm0. Because
(a+ θx)/(b+ θ) is larger than a/b for any positive θ if and only if x ≥ a/b, then we
have ∑j∈S∗\{ik}
qmj(S∗ ∪ {il})p
ijt (y) + qmil(S
∗ ∪ {il})pilt (y) >∑j∈S∗
qmj(S)pijt (y).
91
Hence, assortment S∗ ∪ {il} is strictly better than S∗. This is a contradiction. �
Parts (i) and (ii) of Proposition 2.4.1 and Theorem 2.4.1 follow from these tech-
nical properties:
Property 1 Vt(y − ej)− Vt(y − ei − ej)− Vt(y) + Vt(y − ei) ≥ 0, i, j ∈ N .
Property 2 Vt(y − ei) ≤ Vt(y − ej) where yi = yj ≥ 1 and θi ≥ θj.
Property 3 For any S ⊂ N \ {i}, p−∆it(y) ≥
∑j∈S αj(p−∆j
t (y))∑j∈S αj+θ0
. In other words,
θ0p+ (∑j∈S
αj + θ0)Vt(y − ei)−∑j∈S
αjVt(y − ej)− θ0Vt(y) ≥ 0
Properties 1 and 2 are (i) and (ii) of Proposition 2.4.1, respectively. From the
proof of Proposition 2.3.1, Property 3 claims that the optimal assortment in period
t−1 is to offer all available products, which is Theorem 2.4.1. We prove these results
together by induction. First, because the optimal assortment in period T is to offer
all available products, it is easy to verify that the results hold for period T . Second,
we assume these three results hold in period t + 1. Note that Property 3 in period
t+1 implies that the optimal assortment in period t is to offer all available products.
Finally, we prove Properties 1 and 2 in period t in the proof of Proposition 2.4.1
and Property 3 in period t in the proof of Theorem 2.4.1. Then the entire proof is
completed by induction. Without loss of generality, assume S(y) = N .
Proposition 2.4.1
Proof:
(i) and (ii): We begin with the proof of Property 1. There are in total six cases.
When i = j, there are two cases: (1) yi = 2; (2) yi > 2. When i 6= j, there are
four cases: (3) yi ≥ 2 and yj ≥ 2; (4) yi = 1 and yj ≥ 2; (5) yi ≥ 2 and yj = 1; (6)
92
yi = 1 and yj = 1. Here, we prove case 6 only. The other cases are either easier or
similar. Define∑N = (
∑k∈N θk + θ0),
∑N\{i} = (
∑k∈N\{i} θk + θ0), ei,j = ei + ej,
and ei,j,k = ei + ej + ek.
Vt(y − ej)− Vt(y − ei,j)− Vt(y) + Vt(y − ei) =
θ0θiθj(∑N\{i}+
∑N\{j})∑
N∑N\{i}
∑N\{j}
∑N\{i,j}
p
+
∑k∈N\{j} θkVt+1(y − ej,k)∑
N\{j}−∑
k∈N\{i,j} θkVt+1(y − ei,j,k)∑N\{i,j}
−∑
k∈N θkVt+1(y − ek)∑N
+
∑k∈N\{i} θkVt+1(y − ei,k)∑
N\{i}+θ0Vt+1(y − ej)∑
N\{j}
−θ0Vt+1(y − ei,j)∑N\{i,j}
− θ0Vt+1(y)∑N
+θ0Vt+1(y − ei)∑
N\{i}
=
∑k∈N\{i,j} θk (Vt+1(y − ej,k)− Vt+1(y − ei,j,k)− Vt+1(y − ek) + Vt+1(y − ei,k))∑
N
+θ0 (Vt+1(y − ej)− Vt+1(y − ei,j)− Vt+1(y) + Vt+1(y − ei))∑
N
+θj∑
k∈N\{i,j} θkVt+1(y − ej,k)∑N\{j}
∑N
−(θi + θj)
∑k∈N\{i,j} θkVt+1(y − ei,j,k)∑N\{i,j}
∑N
+θi∑
k∈N\{i,j} θkVt+1(y − ei,k)∑N\{i}
∑N
+θjθ0Vt+1(y − ej)∑
N\{j}∑N
−(θi + θj)θ0Vt+1(y − ei,j)∑N\{i,j}
∑N
+θiθ0Vt+1(y − ei)∑
N\{i}∑N
+θiVt+1(y − ei,j)∑
N\{j}+θjVt+1(y − ei,j)∑
N\{i}− θiVt+1(y − ei) + θjVt+1(y − ej)∑
N
+θiθjθ0p∑
N∑N\{j}
∑N\{i,j}
+θiθjθ0p∑
N∑N\{i}
∑N\{i,j}
(B.2)
93
The first two terms are positive due to Property 1 for period t+ 1, then
(B.2) ≥
θj∑
k∈N\{i,j} θk(Vt+1(y − ei,j)− Vt+1(y − ei,j,k)− Vt+1(y − ej) + Vt+1(y − ej,k)∑N\{j}
∑N
+θi∑
k∈N\{i,j} θk(Vt+1(y − ei,j)− Vt+1(y − ei,j,k)− Vt+1(y − ei) + Vt+1(y − ei,k))∑N\{i}
∑N
− θiθjθ0Vt+1(y − ei)∑N\{i,j}
∑N\{i}
∑N− θiθjθ0Vt+1(y − ej)∑
N\{i,j}∑N\{j}
∑N
+θiθj
∑k∈N\{i,j} θk(Vt+1(y − ei,j)− Vt+1(y − ei,j,k)− Vt+1(y − ei,j,k))∑
N\{i,j}∑N\{i}
∑N
+θjθi
∑k∈N\{i,j} θk(Vt+1(y − ei,j)− Vt+1(y − ei,j,k)− Vt+1(y − ei,j,k))∑
N\{i,j}∑N\{j}
∑N
+θiθj∑
N∑N\{j}
∑N\{i,j}
[θ0p+ (∑
k∈N\{i,j}
θk + θ0)Vt+1(y − ei,j)]
+θiθj∑
N∑N\{i}
∑N\{i,j}
[θ0p+ (∑
k∈N\{i,j}
θk + θ0)Vt+1(y − ei,j)] (B.3)
The first two terms are positive due to Property 1 for period t + 1. For the
last two terms we use the following two inequalities from Property 3 for period
t + 1 : θ0p + (∑N\{i,j})Vt(y − ei,j) ≥
∑k∈N\{i,j} θkVt(y − ej,k) + θ0Vt(y − ej) and
θ0p+ (∑N\{i,j})Vt(y − ei,j) ≥
∑k∈N\{i,j} θkVt(y − ei,k) + θ0Vt(y − ei). Then,
(B.3) ≥
θiθj∑
k∈N\{i,j} θk(Vt+1(y − ei,j)− 2Vt+1(y − ei,j,k) + Vt+1(y − ei,k))∑N\{i,j}
∑N\{i}
∑N
+θjθi
∑k∈N\{i,j} θk(Vt+1(y − ei,j)− 2Vt+1(y − ei,j,k) + Vt+1(y − ej,k))∑
N\{i,j}∑N\{j}
∑N
94
This is positive.
We next prove Property 2. We focus on the case where yi = yj = 1. The case
where yi = yj > 1 is easier to prove.
Vt(y − ej)− Vt(y − ei) =(θi − θj)θ0p∑N\{j}
∑N\{i}
+
∑k∈N∪{0}\{j} θkVt+1(y − ej − ek)∑
N\{j}−∑
k∈N∪{0}\{i} θkVt+1(y − ei − ek)∑N\{i}
=(θi − θj)[θ0p+ (
∑k∈N\{i,j} θk + θ0)Vt+1(y − ei,j)−
∑k∈N∪{0}\{i,j} θkVt+1(y − ei,k)]∑
N\{j}∑N\{i}
+
∑k∈N∪{0}\{i,j} θk[Vt+1(y − ej,k)− Vt+1(y − ei,k)]∑
N\{j}
This is positive, because θ0p+(∑N\{i,j})Vt+1(y−ei,j)−
∑k∈N∪{0}\{i,j} θkVt+1(y−
ei,k) ≥ 0 from Property 3 for period t+ 1 and Vt+1(y− ej,k)−Vt+1(y− ei,k) ≥ 0 from
Property 2 for period t+ 1.
(iii): Because we have one more period to sell products in period t compared to
period t + 1, we can easily prove that Vt(y) and ∆it(y) are decreasing in time by
sample path analysis. �
Theorem 2.4.1
Proof: We now focus on Property 3. For simplicity, define S ′ as follows: j ∈ S ′
if and only if yj = 1 and j ∈ S. There are two cases: (1) yi > 1; (2) yi = 1. We
only show the proof of case 2 here and case 1 can be proved similarly. Moreover, we
assume S ′ 6= ∅ and θi > θj for any j ∈ S ′.
θ0p+ (∑S
)Vt(y − ei)−∑j∈S
θjVt(y − ej)− θ0Vt(y)
95
=
∑k∈N\S\{i} θk[θ0p+ (
∑S)Vt+1(y − ei,k)−
∑j∈S θjVt+1(y − ej,k) + θ0Vt+1(y − ek)]∑
N
+
∑k∈N\S\{i} θkθi(
∑S)Vt+1(y − ei,k)∑
N∑N\{i}
−∑
k∈N\S\{i} θk∑N
[∑j∈S′
θ2j
Vt+1(y − ej,k)∑N\{j}
]
+
∑k∈S\S′ θk[θ0p+ (
∑S)Vt+1(y − ei,k)−
∑j∈S θjVt+1(y − ej,k) + θ0Vt+1(y − ek)]∑
N
+
∑k∈S\S′ θkθi(
∑S)Vt+1(y − ei,k)∑
N∑N\{i}
−∑
k∈S\S′ θk∑N
[∑j∈S′
θ2j
Vt+1(y − ej − ek)∑N\{j}
]
+
∑k∈S′ θk[θ0p+ (
∑S\{k})Vt+1(y − ei,k)−
∑j∈S\{k} θjVt+1(y − ej,k) + θ0Vt+1(y − ek)]∑N
+
∑k∈S′ θkθi(
∑S\{k})Vt+1(y − ei,k)∑N∑N\{i}
−∑
k∈S′ θk∑N
[∑
j∈S′\{k}
θ2j
Vt+1(y − ej,k)∑N\{j}
]
+
∑j∈S′ θ
2j (Vt+1(y − ei,j)∑N\{i}
+θ0[θ0p+ (
∑S)Vt+1(y − ei)−
∑j∈S θjVt+1(y − ej) + θ0Vt+1(y)]∑N
+θ0θi(
∑S)Vt+1(y − ei)∑N∑N\{i}
− θ0∑k∈N θk + θ0
[∑j∈S′
θ2j
Vt+1(y − ej)∑N\{j}
]
+θiθ0
∑k∈N\S\{i} θkp∑N∑N\{i}
+θ0p∑
k∈N θk + θ0
[∑j∈S′
θ2j∑N\{j}
]
−θi∑
j∈S\S′ θjVt+1(y − ei,j)∑N
−∑j∈S′
θiθjVt+1(y − ei,j)∑N\{j}
− θiθ0Vt+1(y − ei)∑N
(B.4)
The first, fourth, seventh, and eleventh terms are positive due to Property 3 for
period t+ 1, then
96
(B.4) ≥θiθ0
∑k∈N\S\{i} θkp∑N∑N\{i}
+
∑k∈N\S\{i} θkθi(
∑S)Vt+1(y − ei,k)∑
N∑N\{i}
−∑
k∈N\S\{i} θk∑N
[∑j∈S′
θ2j
Vt+1(y − ej,k)∑N\{j}
]
+
∑k∈S\S′ θkθi(
∑S)Vt+1(y − ei,k)∑
N∑N\{i}
−∑
k∈S\S′ θk∑N
[∑j∈S′
θ2j
Vt+1(y − ej,k)∑N\{j}
]
+
∑k∈S′ θkθi(
∑S\{k})Vt+1(y − ei,k)
(∑N∑N\{i}
−∑
k∈S′ θk∑N
[∑
j∈S′\{k}
θ2j
Vt+1(y − ej,k)∑N\{j}
]
+
∑j∈S′ θ
2j (Vt+1(y − ei,j)∑N\{i}
+θ0θi(
∑S)Vt+1(y − ei)∑N∑N\{i}
− θ0∑N
[∑j∈S′
θ2j
Vt+1(y − ej)∑N\{j}
]
+θ0p∑
k∈N θk + θ0
[∑j∈S′
θ2j∑N\{j}
]
−θi∑
j∈S\S′ θjVt+1(y − ei,j)∑N
−∑j∈S′
θiθjVt+1(y − ei,j)∑N\{j}
− θiθ0Vt+1(y − ei)∑N
(B.5)
Using θ0p + (∑S)Vt+1(y − ei,k) ≥
∑j∈S θjVt+1(y − ei,j) + θ0Vt+1(y − ei) from
Property 3 in period t+ 1 in the first two terms, we get
(B.5) ≥
∑j∈S′
θ2j (θ0p+ (
∑N\{j})Vt+1(y − ei,j)−
∑k∈N\{j} θkVt+1(y − ej,k) + θ0Vt+1(y − ej))∑N\{j}
∑N
This is positive from Property 3 in period t + 1. This completes the proof by
induction. �
97
Parts (i) and (ii) of Proposition 2.4.2, Theorem 2.4.2, and part (i) of Proposition
2.4.4 follow from the properties described below. Note that these results are pre-
sented for segment 2 customers only – we can similarly prove the results for segment
1.
Property 4 ∆2t (y1, y2) is decreasing in yi, i ∈ {1, 2}.
Property 5 ∆2t (y1, y2) is decreasing in t.
Property 6 p − ∆2t (y1, y2) ≥ θ21
θ21+θ0(p − ∆1
t (y1, y2)). In other words, θ0p + (θ0 +
θ21)Vt(y1, y2 − 1) ≥ θ0Vt(y1, y2) + θ21Vt(y1 − 1, y2).
Property 7
∆1t (y1, y2)−∆1
t (y1 + 1, y2) ≥ θ22
θ22 + θ0
(∆2t (y1, y2)−∆2
t (y1 + 1, y2)).
In other words,
(θ22 + 2θ0)Vt(y1, y2) + θ22Vt(y1, y2 − 1) ≥
(θ22 + θ0)Vt(y1 − 1, y2) + θ22Vt(y1 + 1, y2 − 1) + θ0Vt(y1 + 1, y2)
Property 8 If p−∆1t (y1, y2) ≥ p−∆2
t (y1, y2), then p−∆1t (y1, y2−1) ≥ p−∆2
t (y1, y2−
1). In other words, If Vt(y1 − 1, y2) ≥ Vt(y1, y2 − 1), then Vt(y1 − 1, y2 − 1) ≥
Vt(y1, y2 − 2).
Property 9 Vt(y1, y2 + 1) > Vt(y1 + 1, y2) if and only if y2 < y1. In addition, if
y2 = y1, Vt(y1, y2 + 1) = Vt(y1 + 1, y2).
Properties 4 and 5 are parts (i) and (ii) of Proposition 2.4.2, respectively. Prop-
erty 6 implies that product 2 is always in the optimal assortment of segment 2 and
Property 7 implies the threshold policy, which are the results in Theorem 2.4.2.
98
Property 8 indicates that, if it is optimal to offer {1, 2} for customer segment 2 and
{1} for customer segment 1 under inventory (y1, y2), the optimal assortment for cus-
tomer segment 2 remains to be {1, 2} under inventory (y1, y2−1), which implies that,
if y1 ≥ y∗1(y2) and y2 ≤ y∗2(y1), then y1 ≥ y∗1(y2 − 1). This is part (i) of Proposition
2.4.4. These results are proved together by induction. First, because the optimal
assortment in period T is to offer all available products, it is easy to verify that the
results hold for period T . Second, we assume these four results hold in period t+ 1.
Note that Properties 6 and 7 in period t+ 1 implies that the optimal assortment in
period t is of threshold-type. Finally, we prove Properties 4 and 5 in period t in the
proof of Proposition 2.4.2, Properties 6 and 7 in period t in the proof of Theorem
2.4.2, and Properties 8 and 9 in period t in the proof of Proposition 2.4.4. Then, the
entire proof is completed by induction. Since Vt(y1, y2) is a linear combination of
Vt(y1, y2|m)(m = 1, 2), if we can show that the results hold for Vt(y1, y2|m)(m = 1, 2),
then they are also true for Vt(y1, y2). Without loss of generality, assume S(y) = N
and p−∆1t+1(y1, y2) ≥ p−∆2
t+1(y1, y2).
Proposition 2.4.2
Proof: We start with Property 4 and focus on proving the case where Vt(y1, y2) −
Vt(y1 − 1, y2) − Vt(y1, y2 + 1) + Vt(y1 − 1, y2 + 1) ≥ 0. The other cases are similar.
There are in total twenty four cases. The optimal assortments for both customer
segments can be summarized as follows.
(y1, y2) (y1 − 1, y2) (y1, y2 + 1) (y1 − 1, y2 + 1)S∗t (1) {1, 2} or {1} {1, 2} or {1} {1, 2} or {1} {1, 2} or {1}S∗t (2) {1, 2} {1, 2} or {2} {1, 2} or {2} {1, 2} or {2}
Here, we just show the proof of the case for segment 1 customers with the optimal
assortments below and the other cases are either easier or similar.
(y1, y2) (y1 − 1, y2) (y1, y2 + 1) (y1 − 1, y2 + 1)S∗t (1) {1} {1, 2} {1, 2} {1, 2}
99
According to these optimal assortments, we have p − ∆2t+1(y1, y2) < θ11
θ11+θ0[p −
∆1t+1(y1, y2)], p−∆2
t+1(y1−1, y2) ≥ θ11θ11+θ0
[p−∆1t+1(y1−1, y2)], p−∆2
t+1(y1, y2 +1) ≥
θ11θ11+θ0
[p−∆1t+1(y1, y2+1)], and p−∆2
t+1(y1−1, y2+1) ≥ θ11θ11+θ0
[p−∆1t+1(y1−1, y2+1)].
Then
Vt(y1, y2|1)− Vt(y1 − 1, y2|1)− Vt(y1, y2 + 1|1) + Vt(y1 − 1, y2 + 1|1)
=θ12[θ11Vt+1(y1 − 1, y2) + θ0Vt+1(y1, y2)− θ0p− (θ11 + θ0)Vt+1(y1, y2 − 1)]
(θ11 + θ0)(θ11 + θ12 + θ0)
+θ11[Vt+1(y1 − 1, y2)− Vt+1(y1 − 2, y2)− Vt+1(y1 − 1, y2 + 1) + Vt+1(y1 − 2, y2 + 1))]
θ11 + θ12 + θ0
+θ12[Vt+1(y1, y2 − 1)− Vt+1(y1 − 1, y2 − 1)− Vt+1(y1, y2) + Vt+1(y1 − 1, y2)]
θ11 + θ12 + θ0
+θ0[Vt+1(y1, y2)− Vt+1(y1 − 1, y2)− Vt+1(y1, y2 + 1) + Vt+1(y1 − 1, y2 + 1)]
θ11 + θ12 + θ0
(B.6)
The first term is positive because p−∆2t+1(y1, y2) < θ11
θ11+θ0[p−∆1
t+1(y1, y2)] which is
equivalent to θ0p+(θ11 +θ0)Vt+1(y1, y2−1) < θ11Vt+1(y1−1, y2)+θ0Vt+1(y1, y2). The
last three terms are positive from Property 4 in period t+ 1. Therefore, (B.6) ≥ 0.
We then turn our attention to Property 5. There are in total four cases and the
optimal assortments for both customer segments can be summarized as follows.
(y1, y2) (y1, y2 − 1)S∗t (1) {1, 2} or {1} {1, 2} or {1}S∗t (2) {1, 2} {1, 2}
We focus on the case below, the others are similar.
(y1, y2) (y1, y2 − 1)S∗t (1) {1, 2} {1}
100
According to the optimal assortments, we have p−∆2t+1(y1, y2 − 1) < θ11
θ11+θ0[p−
∆1t+1(y1, y2 − 1)]. Then
Vt(y1, y2|1)− Vt(y1, y2 − 1|1)− Vt+1(y1, y2) + Vt+1(y1, y2 − 1)
=θ12
(θ11 + θ12 + θ0)(θ11 + θ0)[θ0p+ (θ11 + θ0)Vt+1(y1, y2 − 1)] +
θ11Vt+1(y1 − 1, y2)
θ11 + θ12 + θ0
− θ11
θ11 + θ0
Vt+1(y1 − 1, y2 − 1)− θ11 + θ12
θ11 + θ12 + θ0
Vt+1(y1, y2) +θ11Vt+1(y1, y2 − 1)
θ11 + θ0
≥ θ11[Vt+1(y1 − 1, y2) + Vt+1(y1, y2 − 1)− Vt+1(y1 − 1, y2 − 1)− Vt+1(y1, y2)]
θ11 + θ0
≥ 0
The first inequality follows from using θ0p+(θ11+θ0)Vt+1(y1, y2−1) ≥ θ11Vt+1(y1−
1, y2) + θ0Vt+1(y1, y2) in the first term, which in turn follows from p−∆2t+1(y1, y2) ≥
θ11θ11+θ0
[p−∆1t+1(y1, y2)]. The last inequality is due to Property 4 for period t+ 1. �
Theorem 2.4.2
Proof: We now prove Property 6. There are in total twelve cases and the opti-
mal assortments for both customer segments under different inventory levels can be
summarized as follows.
(y1, y2) (y1 − 1, y2) (y1, y2 − 1)S∗t (1) {1, 2} or {1} {1, 2} or {1} {1, 2} or {1}S∗t (2) {1, 2} {1, 2} or {2} {1, 2} or {2}
Here, we just show the proof for the case below. The other cases are similar.
(y1, y2) (y1 − 1, y2) (y1, y2 − 1)S∗t (1) {1} {1, 2} {1}
According to the optimal assortments, we have p − ∆2t+1(y1, y2) < θ11
θ11+θ0[p −
∆1t+1(y1, y2)], p−∆2
t+1(y1− 1, y2) ≥ θ11θ11+θ0
[p−∆1t+1(y1− 1, y2)], and p−∆2
t+1(y1, y2−
101
1) < θ11θ11+θ0
[p−∆1t+1(y1, y2 − 1)]. Then
θ0p+ (θ0 + θ21)Vt(y1, y2 − 1|1)− θ0Vt(y1, y2|1)− θ21Vt(y1 − 1, y2|1)
= θ0p+θ11θ21p
θ11 + θ0
− θ21(θ11 + θ12)p
θ11 + θ12 + θ0
− θ0[θ11Vt+1(y1 − 1, y2) + θ0Vt+1(y1, y2)]
θ0 + θ11
+(θ0 + θ21)[θ11Vt+1(y1 − 1, y2 − 1) + θ0Vt+1(y1, y2 − 1)]
θ0 + θ11
− θ21[θ11Vt+1(y1 − 2, y2) + θ12Vt+1(y1 − 1, y2 − 1) + θ0Vt+1(y1 − 1, y2)]
θ11 + θ12 + θ0
(B.7)
Because (θ0 +θ21)Vt+1(y1, y2−1) ≥ θ0Vt+1(y1, y2)+θ21Vt+1(y1−1, y2)−θ0p, which
follows from p−∆2t+1(y1, y2) < θ11
θ11+θ0[p−∆1
t+1(y1, y2)], then
(B.7) ≥
θ11[θ0p+ (θ0 + θ21)Vt+1(y1 − 1, y2 − 1)− θ0Vt+1(y1 − 1, y2)− θ21Vt+1(y1 − 2, y2)]
θ11 + θ12 + θ0
+θ0θ12(θ11 − θ21)[p− Vt+1(y1 − 1, y2) + Vt+1(y1 − 1, y2 − 1)]
(θ0 + θ11)(θ11 + θ12 + θ0)
The first term is positive because p−∆2t+1(y1−1, y2) ≥ θ11
θ11+θ0[p−∆1
t+1(y1−1, y2)]
and the second term is also positive. Therefore, the proof is finished.
Finally, we move to Property 7. There are in total twelve cases and the opti-
mal assortments for both customer segments under different inventory levels can be
summarized as follows.
(y1, y2) (y1, y2 − 1) (y1 − 1, y2) (y1 + 1, y2 − 1) (y1 + 1, y2)S∗t (1) {1, 2}/{1} {1, 2} / {1} {1, 2} / {1} {1, 2} /{1} {1, 2} /{1}S∗t (2) {1, 2} {1, 2} / {2} {1, 2} /{2} {1, 2} /{2} {1, 2} / {2}
Here, we just show the proof for the case below. The other cases are similar.
(y1, y2) (y1, y2 − 1) (y1 − 1, y2) (y1 + 1, y2 − 1) (y1 + 1, y2)S∗t (1) {1, 2} {1} {1, 2} {1} {1}
102
According to the optimal assortments, we have p−∆2t+1(y1, y2 − 1) < θ11
θ11+θ0[p−
∆1t+1(y1, y2 − 1)], p − ∆2
t+1(y1 + 1, y2 − 1) < θ11θ11+θ0
[p − ∆1t+1(y1 + 1, y2 − 1)], and
p−∆2t+1(y1 + 1, y2) < θ11
θ11+θ0[p−∆1
t+1(y1 + 1, y2)]. Then,
(θ22 + 2θ0)Vt(y1, y2|1) + θ22Vt(y1, y2 − 1|1)− (θ22 + θ0)Vt(y1 − 1, y2|1)
−θ22Vt(y1 + 1, y2 − 1|1)− θ0Vt(y1 + 1, y2|1)
=θ12θ
20p
(θ11 + θ12 + θ0)(θ11 + θ0)+
θ11
θ11 + θ12 + θ0
[(θ22 + 2θ0)Vt+1(y1 − 1, y2)
+θ22Vt+1(y1−1, y2−1)−(θ22 +θ0)Vt+1(y1−2, y2)−θ22Vt+1(y1, y2−1)−θ0Vt+1(y1, y2)]
+θ12
θ11 + θ12 + θ0
[(θ22 + 2θ0)Vt+1(y1, y2 − 1)− (θ22 + θ0)Vt+1(y1 − 1, y2 − 1)]
+θ0
θ11 + θ12 + θ0
[(θ22 + 2θ0)Vt+1(y1, y2) + θ22Vt+1(y1, y2 − 1)
−(θ22 + θ0)Vt+1(y1 − 1, y2)− θ22Vt+1(y1 + 1, y2 − 1)− θ0Vt+1(y1 + 1, y2)]
+θ11θ12
(θ11 + θ0)(θ11 + θ12 + θ0)[θ22Vt+1(y1−1, y2−1)−θ22Vt+1(y1, y2−1)−θ0Vt+1(y1, y2)]
+θ0θ12
(θ11 + θ0)(θ11 + θ12 + θ0)[θ22Vt+1(y1, y2−1)−θ22Vt+1(y1+1, y2−1)−θ0Vt+1(y1+1, y2)]
(B.8)
In the second and fourth term, by using Property 6 in period t+ 1, we get
(B.8) ≥ θ12θ20p
(θ11 + θ12 + θ0)(θ11 + θ0)+
θ12
θ11 + θ12 + θ0
(θ22 + θ0)Vt+1(y1, y2 − 1)
+θ12
θ11 + θ12 + θ0
[θ0Vt+1(y1, y2 − 1)− (θ22 + θ0)Vt+1(y1 − 1, y2 − 1)]
+θ11θ12
(θ11 + θ0)(θ11 + θ12 + θ0)[θ22Vt+1(y1−1, y2−1)−θ22Vt+1(y1, y2−1)−θ0Vt+1(y1, y2)]
103
+θ0θ12
(θ11 + θ0)(θ11 + θ12 + θ0)[θ22Vt+1(y1, y2−1)−θ22Vt+1(y1+1, y2−1)−θ0Vt+1(y1+1, y2)]
(B.9)
Using p − ∆2t+1(y1, y2) ≥ θ11
θ11+θ0[p − ∆1
t+1(y1, y2)], which is equivalent to (θ11 +
θ0)Vt+1(y1, y2 − 1) ≥ θ0Vt+1(y1, y2) + θ11Vt+1(y1 − 1, y2)− θ0p in the second term, we
get
(B.9) ≥
θ12θ0θ22[2Vt+1(y1, y2 − 1)− Vt+1(y1 − 1, y2 − 1)− Vt+1(y1 + 1, y2 − 1)]
(θ11 + θ12 + θ0)(θ11 + θ0)
+θ12θ0θ11[Vt+1(y1, y2 − 1)− Vt+1(y1 − 1, y2 − 1)− Vt+1(y1, y2) + Vt+1(y1 − 1, y2)]
(θ11 + θ12 + θ0)(θ11 + θ0)
+θ12θ
20[Vt+1(y1, y2 − 1)− Vt+1(y1 − 1, y2 − 1) + Vt+1(y1, y2)− Vt+1(y1 + 1, y2)]
(θ11 + θ12 + θ0)(θ11 + θ0)
All three terms are positive from Property 4 in period t+ 1. �
Proposition 2.4.3
Proof: For any inventory y, if p1t (y) ≥ p2
t (y) , from Proposition 2.3.1, the optimal
assortments for both customer segments then are either {1} or {1, 2}. However,
product 2 is always offered to customer segment 2. Therefore, the optimal as-
sortment for customer segment 2 is {1, 2}. With a similar argument, the optimal
assortment for customer segment 1 is {1, 2} if p1t (y) < p2
t (y). In conclusion, the
optimal assortment for at least one customer segment is {1, 2}. �.
Proposition 2.4.4
Proof: (i) We now prove Properties 8 and 9. We first show the proof of Property
9 and then go to Property 8. For Property 9, if y2 = y1, because of the symmetry of
both customer segments, adding one more unit to any product leads to the same profit
104
functions, that is , Vt(y1, y2+1) = Vt(y1+1, y2). If y2 < y1, consider a new assortment
policy for inventory (y1, y2 + 1) as follows. In period t, the assortment policy follows
the optimal assortment policy of period t under inventory (y1 + 1, y2). From period
t+1 to the end of season, we use the optimal assortment policy. We denote the profit
function under this new assortment policy as V t(y1, y2+1). Therefore, Vt(y1, y2+1) ≥
V t(y1, y2 + 1). In the following, we prove V t(y1, y2 + 1) > Vt(y1 + 1, y2) by induction
and then we have Vt(y1, y2 + 1) > Vt(y1 + 1, y2). When comparing V t(y1, y2 + 1)
and Vt(y1 + 1, y2), if the customer arriving in period t chooses not to purchase, then
V t(y1, y2 + 1) = V t+1(y1, y2 + 1) > Vt+1(y1 + 1, y2) = Vt(y1 + 1, y2) by induction; if
a purchase is made, then V t(y1, y2 + 1) ≥ Vt(y1 + 1, y2) regardless of which product
is sold or which assortment is offered. Therefore, V t(y1, y2 + 1) > Vt(y1 + 1, y2) in
expectation. On the other hand, we prove that Vt(y1, y2 + 1) > Vt(y1 + 1, y2), only
if y2 < y1. Because, y2 ≥ y1 implies that Vt(y1 + 1, y2) ≥ Vt(y1, y2 + 1) based on our
previous argument.
Based on Property 9 in period t, p − ∆1t (y1, y2) ≥ p − ∆2
t (y1, y2) if and only if
y2 ≤ y1 and p−∆1t (y1, y2−1) ≥ p−∆2
t (y1, y2−1) if and only if y2−1 ≤ y1. Property
8 then can be restated as if y2 ≤ y1, then y2 − 1 ≤ y1. This is obviously correct.
(ii) We only consider segment 2 customers without loss of generality. Assume
the optimal assortment is {1, 2} in period t with the inventory level (y1, y2), i.e.,
θ0(p−∆1t+1(y1, y2)) + θ22(∆2
t+1(y1, y2)−∆1t+1(y1, y2)) ≥ 0. If we can prove that the
optimal assortment still is {1, 2} in period t + 1 with the same inventory level, i.e.,
θ0(p − ∆1t+2(y1, y2)) + θ22(∆2
t+2(y1, y2) − ∆1t+2(y1, y2)) ≥ 0., then the proof is com-
pleted. If y1 ≥ y2, then ∆2t+2(y1, y2)−∆1
t+2(y1, y2) = Vt+2(y1 − 1, y2)− Vt+2(y1, y2 −
1) ≥ 0 in this symmetric case. If y1 < y2, Vt+1(y1, y2 − 1) − Vt+1(y1 − 1, y2) ≥
Vt+2(y1, y2 − 1) − Vt+2(y1 − 1, y2), which implies that ∆2t+2(y1, y2) − ∆1
t+2(y1, y2) ≥
∆2t+1(y1, y2)−∆1
t+1(y1, y2). Moreover, from part 2 of Proposition 2.4.2, ∆it+1(y1, y2) >
∆it+2(y1, y2). Therefore, θ0(p − ∆1
t+2(y1, y2)) + θ22(∆2t+2(y1, y2) − ∆1
t+2(y1, y2)) ≥
105
θ0(p−∆1t+1(y1, y2)) + θ22(∆2
t+1(y1, y2)−∆1t+1(y1, y2)) ≥ 0. �
Proposition 2.6.1
Proof: Assume µ is a feasible policy in the MDP problem. Let Sµmt(Ft) denote
the assortment controlled by policy µ in period t for segment m customer and an N-
dimensional vector D(Sµmt(Ft)) denote the random demand in period t given segment
m consumer and assortment Sµmt(Ft). To be more specific, Dj(Sµmt(Ft)) = 1 indicates
that one unit of product j is sold and Dj(Sµmt(Ft)) = 0 indicates no sale for product
j. In addition, if j ∈ N \ Sµmt(Ft), Dj(Sµmt(Ft)) = 0. Because of the feasibility of
policy µ, we have
T∑t=1
∑m∈M
D(Sµmt(Ft)) ≤ y0, a.s.
We take the expectation and have∑T
t=1
∑m∈MED(Sµmt(Ft)) ≤ y0. Further-
more,
T∑t=1
∑m∈M
ED(Sµmt(Ft)) =∑m∈M
∑S⊂N
λQm(S)E[tµm(S)]
where tµm(S) indicates the random amount of time for which assortment S is
offered to segment m consumers under policy µ. The objective function then is
V µ1 (y0) =
∑m∈M
∑S⊂N
λRm(S)E[tµm(S)]
Then (E[tµm(S)])m∈M are feasible solution in CDLP and the profits are the same.
Based on this result, we then have V1(y0) = maxµ∈A Vµ
1 (y0) ≤ V CDLP (y0). �
106
Appendix D
Proofs of Chapter 3
Proof of Theorem 3.2.1: We start by proving the result for the IP model. Because
cost/price ratios are identical across all products, their percentiles denoted by zIP =
Φ−1(1−r) = Φ−1(1−ki/pi) are identical as well. Define f1(q) = (1−r)q−φ(zIP )σqβ.
The optimal profit for product i then is πD,IP (qi) = pif1(qi). From Lemma B.1,
f1(qi) is increasing and convex in qi. In addition, qi = ui∑j∈S uj+u0
implies that pi =
p0 + γqi(
∑i∈S\{i} uj)
1−qi which is increasing and convex in qi. Therefore,
∂πD,IP (qi)
∂qi= p′if1(qi) + pif
′1(qi) > 0,
∂2πD,IP (qi)
∂2qi= p′′i f1(qi) + 2p′if
′1(qi) + pif
′′1 (qi) > 0.
In other words, πD,IP (qi) is increasing and convex in qi. Hence, as in the proof of
Proposition B.1, the optimal assortment is a popular set for the IP model.
For the TF model, all products have identical percentiles given by zTF = Φ−1(1 −rq). Define f2(q) = (q − r) − qφ(zTF )σ. The optimal profit for product i then is
πD,TF (qi) = pif2(qi). From Lemma B.1, f1(qi) is increasing and convex in qi. Similar
107
as the previous proof for the IP model, the optimal assortment is a popular set for
the TF model. �
Proof of Theorem 3.2.2: We first prove that ΠC,TFS is increasing and convex in qi,
which implies that the optimal assortment is a popular set. With a similar argument
to that in the identical price case, the variants in Se have the same inventory levels
as the the common component, whose critical fractile is t∗c(S) = 1 − kc+∑i∈Se kdi∑
i∈Se piqSi
=
1− kc+τr∑i∈Se pi∑
i∈Se piqSi
. The optimal profit of set Se is
ΠC,TFSe
= πD,TF (∑i∈Se
piqSi , kc +
∑i∈Se
kdi)
=∑i∈Se
piqi[t∗c(S)− σφ(Φ−1(t∗c(S)))]
If we can prove that ΠC,TFSe
is increasing and convex in qi where i ∈ Se, then the
optimal assortment is a popular set. This is because ΠC,TFSl
=∑
i∈Sl πD,TF (piq
Si , kdi)
is increasing and convex in qi where i ∈ Sl, the proof of which is similar to the proof
of Theorem 3.2.1. In other words, we need to prove that f(qi) =∑
j∈Se pjqj[t∗c(S)−
σφ(Φ−1(t∗c(S)))] is increasing and convex in qi. First, we show that the common
fractile t∗c(S) = 1 − kc+τr∑j∈Se pj∑
j∈Se pjqjsatisfies two properties below which leads to the
result that f(qi) is increasing and convex in qi. Based on KKT conditions, we have
a KKT multiplier αSi for product i which is in Se. Moreover,
1− t∗c(S) =kdj + αSjpjqj
|j∈Se =kdi + αSipiqi
>kdipiqi
Therefore,
kc =∑j∈Se
αSj ≥ τr
∑j∈Se pjqj
qi− τr
∑j∈Se
pj
108
In other words,
qi(kc + τr∑j∈Se
pj) ≥ τr(∑j∈Se
pjqj)
The first order-derivative of t∗c(S) is given by
∂t∗c(S)
∂qi=
(kc + τr∑
j∈Se pj)(pi + p′iqi)− τrp′i(∑
j∈Se pjqj)
(∑
j∈Se pjqj)2
> 0
Moreover,
2(pi + qip′i)∂t∗c(S)
∂qi+ [∑j∈Se
pjqj]∂2t∗c(S)
∂2qi
=(kc + τr
∑j∈Se pj)(2p
′i + p′′i qi)− τrp′′i (
∑j∈Se pjqj)∑
j∈Se pjqj> 0 (B.1)
The first and second order derivatives of f(qi) are as follows.
∂f(qi)
∂qi= (pi+qip
′i)(t∗c(S)µ−σφ(Φ−1(t∗c(S))))+
∑j∈Se
pjqj∂t∗c(S)
∂qi(µ+σΦ−1(t∗c(S))) > 0
∂2f(qi)
∂2qi=
(2p′i + p′′i qi)(t∗c(S)µ− σφ(Φ−1(t∗c(S)))) + 2(pi + qip
′i)∂t∗c(S)
∂qi(µ+ σΦ−1(t∗c(S)))
+[∑j∈Se
pjqj]∂2t∗c(S)
∂2qi(µ+ σΦ−1(t∗c(S))) +
∑j∈Se
pjqj
(∂t∗c(S)
∂qi
)2
σ1
φ(Φ−1(t∗c(S)))> 0
The second inequality is due to (B.1). Thus, f(qi) is increasing and convex in qi.
Similarly, results analogous to Theorem 1.4.2 hold in the case of increasing prices.
Therefore, the optimal assortment is a popular set and the optimal assortment in
the system with commonality is no smaller than that in the dedicated system. �
109
Proof of Theorem 3.2.3: The profit function with order quantity Q in model P is
ΠP (S,Q) =∑i∈S
piqiEmin{D,Q} − kQ.
After optimizing the order quantity, the optimal profit function is
ΠP (S) = πD,TF (∑
i∈S piqi, k). From Lemma B.1, this optimal profit function is
increasing and convex in qi. Thus, the optimal assortment is a popular set. The
proof of the result that the optimal assortment in model P is no smaller than that in
model D is similar to and easier than the proof of the comparison between models P
and C. Next, we prove that the optimal assortment in model P is no smaller than
that in model C. In other words, we need prove that ΠP,TFAn+1−ΠP,TF
An≥ ΠC,TF
An+1−ΠC,TF
An
for any popular set An.
When An+1 ⊂ Ne,
ΠP,TFAn+1− ΠP,TF
An= πD,TF (
∑i∈An+1
piqi, k)− πD,TF (∑i∈An
piqi, k)
≥ πD,TF (∑
i∈An+1
piqi, kc +∑
i∈An+1
kdi)− πD,TF (∑i∈An
piqi, kc +∑i∈An
kdi) = ΠC,TFAn+1
− ΠC,TFAn
This is because πD,TF (x, k) is decreasing in k and is submodular in (x, k). When
Ne ⊂ An,
ΠP,TFAn+1− ΠP,TF
An=
πD,TF (∑i∈Ne
piqAn+1
i , k
∑i∈Ne piq
An+1
i∑i∈An+1
piqAn+1
i
)− πD,TF (∑i∈Ne
piqAni , k
∑i∈Ne piq
Ani∑
i∈An piqAni
)
+∑
i∈An\Ne
[πD,TF (piqAn+1
i , kpiq
An+1
i∑i∈An+1
piqAn+1
i
)− πD,TF (piqAni , k
piqAni∑
i∈An piqAni
)]
+ πD,TF (pn+1qAn+1
n+1 , kpn+1q
An+1
n+1∑i∈An+1
piqAn+1
i
) (B.2)
110
Because k ≤ kc+kd1 , k∑i∈Ne piq
An+1i∑
i∈An+1piq
An+1i
≤ k∑i∈Ne piq
Ani∑
i∈An piqAni
, kpiq
An+1i∑
i∈An+1piq
An+1i
≤ kpiq
Ani∑
i∈An piqAni
when i ∈ An \ Ne, and kpn+1q
An+1n+1∑
i∈An+1piq
An+1i
< kdn+1 . Moreover, since πD,TF (x, k) is
decreasing in k, then
(B.2) > πD,TF (∑i∈Ne
piqAn+1
i , k
∑i∈Ne piq
Ani∑
i∈An piqAni
)− πD,TF (∑i∈Ne
piqAni , k
∑i∈Ne piq
Ani∑
i∈An piqAni
)
+∑
i∈An\Ne
[πD,TF (piqAn+1
i , kpiq
Ani∑
i∈An piqAni
)− πD,TF (piqAni , k
piqAni∑
i∈An piqAni
)]
+πD,TF (pn+1qAn+1
n+1 , kdn+1) (B.3)
Because πD,TF (x, k) is submodular in (x, k), k∑i∈Ne piq
Ani∑
i∈An piqAni
≤ kc +∑
i∈Ne kdi , and
kpiq
Ani∑
i∈An piqAni
≤ kdi when i ∈ An \ Ne, we have
(B.3) ≥ πD,TF (∑i∈Ne
piqAn+1
i , kc +∑i∈Ne
kdi)− πD,TF (∑i∈Ne
piqAni , kc +
∑i∈Ne
kdi)
+∑
i∈An\Ne
[πD,TF (piqAn+1
i , kdi)− πD,TF (piqAni , kdi)] + πD,TF (pn+1q
An+1
n+1 , kdn+1) (B.4)
Note that the expression in (B.4) is equal to ΠC,TFAn+1−ΠC,TF
An. Thus, we have ΠP,TF
An+1−
ΠP,TFAn
≥ ΠC,TFAn+1−ΠC,TF
An, which implies that the optimal assortment in model P is no
smaller than that in model C. �
Proof of Proposition 3.3.1: According to the definition, ΠS = ΠS+{δj},pj =
ΠS+{δi},pi . Since the profit function is linear and increasing in price, ΠS+{δj},pi ≥
ΠS+{δj},pj = ΠS+{δi},pi . From Lemma B.1, then δj ≥ δi.�
Proof of Proposition 3.3.2: We prove the result by contradiction. Suppose
product j is not in assortment S. We include product j instead of product i. Then
ΠS−{i}+{j} > ΠS. This is a contradiction.�
111
Proof of Proposition 3.3.3: The result follows from Proposition 3.3.2.�
112
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Biography
Lei Xie was born at Hunan province of P.R. China on 06 July, 1981. He entered
University of Science and Technology of China in 1999. After four-year study, he
gained his bachelor degree of science in Mathematics and Applied Mathematics. In
2003, he was admitted to the department of System Engineering and Engineering
Management in Chinese University of Hong Kong with postgraduate scholarship. It
took his two years to get a master degree of philosophy in Operations Management.
In 2005, Lei entered Duke University as a doctoral student and majored in Operations
Management in the department of Business Administration. His research interests
include product variety, revenue management, and the interface between supply chain
and finance. After he graduates in September 2010, Lei plans to move to Montreal,
Canada, where he will continue his research as a post-doc research fellow in McGill
University.
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