Post on 07-Feb-2022
The Effect of the Fast-Ship Option in Retail SupplyChains
A DISSERTATION
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Hao-Wei Chen
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Doctor of Philosophy
Diwakar Gupta
Janurary, 2011
c© Hao-Wei Chen 2011
ALL RIGHTS RESERVED
Acknowledgements
I would like to express my deepest gratitude to my advisor Professor Diwakar Gupta for
giving me the opportunity to join his research lab (SCORLAB). Without his mentoring
and support over the years, it would not have been possible for me to finish my doctoral
work.
I am very grateful to Professor Haresh Gurnani for giving me advice in research.
Without his constant encouragement and inspiration, it would not be possible for me to
develop and achieve some success, if any at all, in research. I would also like to thank
other committee members Professor William L. Cooper, Karen Donohue , and Arthur
Hill for their helpful suggestions and comments over these years.
Many thanks to my colleagues in SCORLAB, Kannapha Amarchkul Mai, Amy Pe-
terson, Dustin Kuchera, for their true friendship as well as their advice and assistance
over the years. I am also thankful to Chin-Yi Liu, Tsung-Yi Pan, Yi-Su Chen, Hung-
chung Su, David Zepeda, and all other friends in Minnesota for their cares for me and
the great times we had together.
Finally, I am especially grateful to my wife, Wen-Ya Wang, and my parents, Ching-
Fiui Chen and Chi-Herng Tai for their continuous caring for me. Their supports and
love give me the strength and power to face the challenges in my life.
i
Dedication
Dedicated to my loving wife, Wen-Ya, my supporting parents, Ching-Fiui and Chi-
Herng, my caring grandmother, Gui and in memory of my grandfather, Ching-Chiang.
ii
Abstract
To reduce loss of sales caused by demand uncertainty, retailers can offer a fast-ship
option to customers who experience stockout. The fast-ship option is a common practice
that serves as a secondary source of supply. When this option is offered, the supply chain
partners arrange to have out-of-stock items shipped directly from the supplier to willing
customers at no additional cost to the customers. The fast-ship option serves as a
mechanism by which inventory risk can be shared between the retailer and the supplier.
We investigate the retailers and the suppliers interactions when the fast-ship option
is offered under different scenarios. More specifically, we characterize the suppliers
and the retailers ordering policies and investigate how the supplier and the retailer
react to different levels of participation for fast-ship orders. We also study how the
supplier can manage its risk by using either price markup or supply commitment under
different supply contract structures. In addition, when the fast-ship option is offered,
we investigate how alliance or competition between retailers can affect the profitability
of the supplier and retailers.
iii
Contents
Acknowledgements i
Dedication ii
Abstract iii
List of Tables vii
List of Figures viii
1 Introduction 1
1.1 A Base-Case Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 A Multi-Period Model 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 The Retailer’s and the Supplier’s Decisions . . . . . . . . . . . . . . . . 22
2.3.1 Optimal Ordering Policies . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Problems with Stationary Demand . . . . . . . . . . . . . . . . . 25
2.3.3 Problems with Non-Stationary Demand . . . . . . . . . . . . . . 27
2.3.4 Optimal Policies for Two-Period Problems . . . . . . . . . . . . . 30
2.3.5 Choice of δ - Stationary Demand and Infinite Horizon . . . . . . 35
2.4 Effect of Customer Participation Rates . . . . . . . . . . . . . . . . . . . 38
iv
2.4.1 Effect of Customer Participation Rate α . . . . . . . . . . . . . . 39
2.4.2 Effect of Customer Participation Rate β . . . . . . . . . . . . . . 41
2.5 Insights & Model Extension . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Effect of Demand Variability . . . . . . . . . . . . . . . . . . . . 43
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Fast-Ship Commitment Contracts 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Parameter Optimization: Structures A and B . . . . . . . . . . . . . . . 53
3.4 Parameter Optimization: Structure C . . . . . . . . . . . . . . . . . . . 58
3.5 Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Supplier’s and Retailer’s Contract Structure Preferences . . . . . 60
3.5.2 Contract Structure Selection . . . . . . . . . . . . . . . . . . . . 64
3.5.3 The Effect of Customer Participation Rate . . . . . . . . . . . . 67
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Two-Retailer Structures 70
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Structure A – Two Independent Retailers . . . . . . . . . . . . . 78
4.2.2 Structure B – Two-Retailer Alliance . . . . . . . . . . . . . . . . 78
4.2.3 Structure C - Two Competing Retailers . . . . . . . . . . . . . . 81
4.3 Supplier’s and Retailers’ Operational Choices . . . . . . . . . . . . . . . 82
4.3.1 Retailers’ Ordering Decisions . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Supplier’s Ordering Decisions . . . . . . . . . . . . . . . . . . . . 85
4.4 Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1 The Effect of Customer Participation Rate . . . . . . . . . . . . 86
4.4.2 Performance Comparisons . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Conclusions 95
v
References 98
Appendix A. Proofs for Chapter 2 104
Appendix B. Proofs for Chapter 3 115
Appendix C. Proofs for Chapter 4 120
vi
List of Tables
2.1 The values of ς(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 The Retailer’s Profit under the Optimal Wholesale Price . . . . . . . . . 63
3.2 An Example of Conflict Resolution by Providing Modified Contracts . . 65
3.3 The Effect of Customer Participation Rate α . . . . . . . . . . . . . . . 67
4.1 The Effect of Customer Participation Rate α . . . . . . . . . . . . . . . 87
vii
List of Figures
2.1 The Effect of Participation Rate α. . . . . . . . . . . . . . . . . . . . . 39
2.2 The Effect of Participation Rate β. . . . . . . . . . . . . . . . . . . . . 42
2.3 The Effect of Demand Variability . . . . . . . . . . . . . . . . . . . . . . 44
4.1 The Three Sourcing Structures . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 The Effect of Customer Participation Rate . . . . . . . . . . . . . . . . . 89
4.3 The Retailer’s Profit Comparison . . . . . . . . . . . . . . . . . . . . . . 92
4.4 The Retailer’s Profit Comparison . . . . . . . . . . . . . . . . . . . . . . 93
viii
Chapter 1
Introduction
Stockouts occur when supply falls short of demand. Stockouts are not uncommon
in retail supply chains and often result in customer dissatisfaction (Grant and Fernie
2008). Gruen et al. (2002) reported a worldwide average out-of-stock rate of 8.3 percent.
In industries such as toys and apparel, matching supply with demand is especially
challenging due to changing customer preferences and market trends, which leads to
high inventory costs, markdowns, and lost sales (Johnson 2001). Because demand-
supply uncertainty is not entirely avoidable, the options available to customers when
they experience stockouts affect profits of supply chain partners.
When customers learn that the retail store they are in has stocked out of a desired
item, they may respond in a number of different ways. Some customers may switch
brands and buy a substitute product, others may buy from a different retail store, and
some others may postpone purchase decision or choose an entirely different product
(Emmelhainz et al. 1991). Note that although some customers purchase substitutes
when the item they desire is out of stock, stockout events can negatively affect the
overall sales of other products in the same category due to lack of selections available to
the customers (Kalyanam et al. 2007). That is, customers are increasingly intolerant to
stockouts, and the purchasing behavior triggered by stockouts may hurt retailers. Re-
tailers’ profit margins in many industry segments are low (for example, see Datamonitor
2008a and Datamonitor 2008b for electronics and food retail industry profiles and profit
margins of key players), which suggests that cost-effective means of capturing loss of
sales caused by stockouts would be of interest to retailers. Campo et al. (2003) state
1
2
that compared to the losses for the retailers, the losses for the suppliers could be even
greater. This is because in some cases customers may purchase a substitute product
from the same retailer, limiting the retailer’s losses. That is, suppliers also would be
interested in evaluating options to reduce stockouts.
About 50 percent of retailer-store stockouts are caused by either inaccurate forecasts
or bad ordering decisions by retailers and more than a half of the stockout situations last
more than a day (Gruen et al. 2002). Retailers’ responses to supply-demand mismatch
include two major themes — more accurate forecasts and speedier (more frequent)
replenishments. Management systems designed to achieve these goals are called Ef-
ficient Consumer Response (Kurt Salmon Associates 1993) and Quick Response (Iyer
and Bergen 1997), respectively. Because of the inherent uncertainty in demand, forecast
accuracy cannot be improved beyond a certain point. Therefore, many retailers use the
Quick Response strategy, either on its own, or in combination with frequent forecast
updates after portions of demand are observed. For instance, Italian designer Benet-
ton exemplifies faster replenishments through its use of superior logistics techniques to
replenish stock in its retail outlets as often as once a week (Meichtry 2007).
To reduce loss of sales during stockout periods, retailers may negotiate a supply
contract that allows them to either adjust the order size before the start of the selling
season or to place multiple orders during the selling season. The latter includes offering
a fast-ship option to customers, which is the focus of this study. A retailer that offers the
fast-ship option arranges to have out-of-stock items shipped directly from the supplier
to willing customers at no additional cost to the customers, thereby creating a hybrid
between traditional and drop-ship channels. Specifically, the fast-ship option allows the
channel to use the retailer-held inventory as the primary source of supply (traditional
approach) and supplier-held backup inventory as the secondary source of supply (drop-
ship approach). The latter is used only when the primary source is exhausted. This
contrasts with the two extremes of traditional and drop-ship channels in which all
inventory is kept either at the retailer location or at the supplier location (Wilson
2000). Drop-ship channels are commonly encountered in the context of Internet-based
retailers (e.g. Zappos, an Internet footwear store).
In the traditional approach, the retailer bears all of the inventory risk and its stocking
decision can affect channel performance. Use of drop-ship approach reduces retailer’s
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risk. However, because drop shipping is observed largely in the context of Internet sales,
in and of itself, this option does not meet the needs of those customers who prefer to
touch and feel the items before buying and those who do not want to wait. In fact,
depending on the product, between 47 to 92 percent of retail sales happen in “brick and
mortar” retail stores (Schonfeld 2010). The fast-ship option combines the advantages of
both traditional and drop-ship channels and provides a mechanism by which inventory
risk can be shared between the retailer and the supplier.
The fast-ship option is a common practice among retailers, especially for items that
are not substitutable. Many consumer electronic stores (e.g., Apple) help customers
place orders for out-of-stock items (such as laptop computers and cell phones) and have
them shipped directly from manufacturers to customers. This not only helps retain
demand but also increases customer satisfaction.
1.1. A Base-Case Model
In order to formalize how the fast-ship option affects the supply chain partners, and to
introduce common notation and assumptions used in the sequel, we present a base-case
model. In this model, the supply chain consists of a single supplier, denoted by S, and
a single retailer, denoted by R, and we focus on the problem of meeting demand for a
single product in a one-period setting.
R’s demand X ∈ R+ is continuous with probability density and distribution func-
tions f(·) and F (·), respectively. We also assume that f(·) > 0 over the support of X.
The shipping costs for regular order and fast-ship orders are τ1 and τ2, respectively. We
assume that τ1 ≤ τ2 because fast-ship orders utilize premium shipping with expedited
delivery whereas regular orders are sent to retailers utilizing an efficient transporta-
tion system. Both τ1 and τ2 are paid by the supplier to a third party logistics service
provider. Note that τ1 and τ2 are independent of origin and destination because all
orders are handled by a third-party logistics provider who charges a flat rate depend-
ing on the item’s size and/or weight. Such pricing schemes are common in the US;
see for example U.S. Postal Service’s (www.usps.com) shipping rate for standard sized
boxes of a certain maximum weights regardless of origin and destination. Furthermore,
because the expedited transportation cost is linear in the number of fast-ship items,
4
modeling several fast-ship orders as a single second replenishment does not affect the
profit functions of the two players.
The retailer first orders q items to be delivered before the start of the selling season.
Next, demand X is realized and in some instances customers may experience stock out.
The retailer offers to have the out-of-stock item shipped directly to the customer’s ad-
dress at no additional cost to the customer. A fraction α ∈ [0, 1] of customers decide to
utilize the fast-ship option. Others do not make a purchase at the retailer’s store. In
other words, the total fast-ship demand is α(x− q)+, where z+ = max(0, z). Hereafter,
α is also referred as the customer participation rate. Note that if the customer partici-
pation rate is a random variable A, so long as X and A are independent, our analysis
remains unchanged with α = E(A).
The supplier has two replenishment (or production) opportunities. The first occurs
after receiving the initial order from the retailer and the second occurs after receiving
fast-ship orders. The replenishment costs are c1 and c2, respectively. We assume c2 ≥ c1
because the second replenishment requires expedited procurement/production. Because
c2 ≥ c1, the supplier may produce extra y units during the first replenishment period
in anticipation of fast-ship orders.
The retailer sells items to customers at a unit retail price r regardless of whether
the item is sold from on-hand inventory or by using the fast-ship option. The supplier
sells items to the retailer at a unit wholesale price w for initial orders, and w2 for fast-
ship orders. The wholesale price for fast-ship items is obtained by adding a mark-up
to the base price w and markups of 10-20% are common (Scheel 1990). Let δ ≥ 0
denote the price markup. Then, we can also write w2 = w + δ. For the supplier,
the worst case additional cost of supplying fast-ship orders is τ2 − τ1 + c2 − c1. When
0 ≤ δ ≤ τ2 − τ1 + c2 − c1, the additional cost of fulfilling fast-ship orders is shared
between the supplier and the retailer. In contrast, when δ > τ2−τ1+c2−c1, the retailer
pays the additional production and transportation costs as well as a premium for the
fast-ship supply. The fast-ship option is guaranteed to be profitable for the supplier if
w + δ ≥ τ2 + c2. We do not model the specific arrangement between the supplier and
the retailer regarding who pays how much of the extra transportation (τ2 − τ1) and the
extra production (c2− c1) costs. Each markup includes many combinations of how such
costs are shared.
5
In all models, we assume that w and δ are exogenous parameters and their values
are chosen such that the fast-ship option is attractive to both the supplier and the
retailer. To ensure that both initial and fast-ship orders are profitable for the retailer,
we assume that w < r − δ. In absence of this condition, the retailer may choose not
to offer the fast-ship option to customers when a stockout occurs. Because w2 ≥ w
and the retail price does not change when items are supplied via the fast-ship option,
the retailer’s unit profit is greater if an item is supplied from stock. Similarly, when
w and δ are exogenous, we assume that w − τ1 − c1 ≥ α(w2 − τ2 − c1), which makes
fast-ship orders less profitable for the supplier as well. These choices of parameters are
reasonable because fast-ship option is not meant to be the primary means by which
customers make their purchases.
Subsequently, we allow either δ or w to be chosen by the supplier. For the base
case model, it can be argued that if the supplier were allowed to choose δ, it would set
δ = r−w and the retailer would make no additional profit from serving fast-ship orders
(see Gupta et al. 2010 for a formal proof). However, when δ is fixed and w is chosen by
the supplier, it may not choose the largest possible value of w (this would be the value
at which the retailer makes no more than its reservation profit, which is assumed to be
zero in this thesis). We do not consider cases in which both w and δ may be chosen by
the supplier because that may not be realistic. A supplier in such settings gets nearly
all of the supply chain profits, leaving little more than the reservation profit for the
supplier. The purpose of these variants is to study if the structural results obtained
when w and δ are assumed exogenous remain intact if each parameter were chosen by
the supplier.
With the fast-ship option, the retailer’s expected profit function can be written as
follows.
πR(q) = rE[q ∧X]− wq + (r −w2)E[α(X − q)+], (1.1)
where (a ∧ b) = min(a, b) and (a)+ = max(0, a). In (1.1), rE[q ∧X] − wq is the profit
from sales that use on-hand inventory, and (r − w2)αE[α(X − q)+] is the profit from
fast-ship orders. Similarly, the supplier’s expected profit with the fast-ship option is
πS(y | q) = (w−τ1)q−c1(q+y)+(w2−τ2)E[α(X−q)+]−c2E[(α(X−q)+−y)+], (1.2)
where (w−τ1)q−c1(q+y) is the profit from the first replenishment, (w2−τ2)E[α(X−q)+]
6
is the revenue from fast-ship demand, and c2E[(α(X − q)+ − y)+] is the additional
replenishment cost induced by the fast-ship orders.
For a given exogenous set of parameters (r, w, δ, τ1 , τ2, c1, c2) in a region in which both
the retailer and the supplier prefer to support fast-ship option, the retailer’s problem
is to find a q∗ = argmaxq
πR(q) and the supplier’s problem is to find a y∗ = argmaxy
=
πS(y | q). These problems are not hard in the base-case setting. However, the variants
of these problems studied in this thesis are significantly more challenging.
Three variants of the base-case model are studied in this thesis. Each scenario is
presented in a separate chapter. First, in Chapter 2, we focus on purely operational
issues — how much should the retailer order and how much should the supplier pro-
cure/produce — in a multi-period setting. It is assumed that a supply contract exists
between the two players and for a fixed set of parameters, both players agree to support
fast-ship option. In this problem, both players have two replenishment opportunities in
each period. Chapter 2 characterizes the retailer’s and the supplier’s optimal stocking
policies with both stationary and non-stationary demands.
In Chapter 3, we shift focus to study the relative performance of different contract
structures and the contract selection process. Because such models are more compli-
cated, we restrict attention to a single period setting in this case in order to keep the
models tractable. The motivation behind studying different contract structures is as
follows. When the fast-ship option is supported, it transfers some inventory risk from
the retailer to the supplier. Therefore, when wholesale price and markup are exoge-
nous, the supplier may want to consider other levers in a supply contract to reduce
uncertainty. One such lever is limited supply commitment for fast-ship orders. In a
supply commitment contract, the supplier may choose the fast-ship supply commitment
ζ(q), where ζ(q) can be either a function of q or independent of q. In this setting, only
(α(x− q)+ ∧ ζ(q)) customers who experience a stockout may be able to get the item.
We analyze three supply commitment contracts. In contract type A, the supplier
specifies a total supply commitment and allows the retailer to choose its split between
the initial order and the amount left to satisfy fast-ship orders. In contract types B
and C, the supplier agrees to fully supply the retailer’s initial order but restricts the
quantity available as fast-ship commitment. The difference between the second and the
third contracts is that in contract type B, the supplier moves first, whereas in contract
7
type C, the supplier determines its fast-ship commitment after observing the retailer’s
order.
In Chapter 4, we broaden the scope of study even more to consider supply chain
configurations involving two retailers. The retailers may act either as independent enti-
ties, or cooperate, or compete. When the retailers make their decisions independently,
the supplier is the sole supply source for initial orders and fast-ship orders. When the
retailers cooperate, one retailer can satisfy fast-ship demand from the other retailer’s
inventory, and vice versa. When the supply chain has two competing retailers, a fraction
of customers who experience stockout travel to the other retailer to buy the item. We
compare and contrast these structures from the supplier’s and the retailers’ viewpoints.
In summary, we study the supplier’s and the retailer’s operational decisions in Chap-
ter 2, the performance of different supply commitment contracts in Chapter 3, and the
effect of retailers’ alliance in Chapter 4. In each case, we also study how customer
participation rate affects supplier and retailer profits.
1.2. Related Literature
In this section, we briefly summarize the related literature. A more focused review
related to the problem variant introduced in each ensuing chapter is provided in that
chapter. In a broad sense, this research is related to the classical newsvendor model in
which the retailer has one opportunity to choose an order quantity and the replenishment
from the supplier arrives before the selling season begins. In supplier-retailer models
of this type, it is assumed that the supplier chooses a wholesale price. Wholesale price
contracts are studied because they are common in practice. For example, Lariviere and
Porteus (2001) investigate a price-only contract between a supplier and a newsvending
retailer. They show that the efficiency of the supply-chain increases as relative demand
variability decreases. However, lower relative variability may cause a higher wholesale
price in cases where the supplier has pricing power. In addition, the authors show
the supplier may choose a lower wholesale price than its individually optimal value
if some factors such as retailer’s market power or retailer’s effort in reducing demand
variability are considered. Anupindi and Bassok (1999) extend the model to a multi-
period environment in which leftover inventory can be carried to the next period. In
8
such scenarios, the retailer’s optimal ordering policy is an order-up-to-level policy.
These two models and our model share some common assumptions. First, all three
models assume that the retail price and demand are exogenous. Second, the supplier
is the first mover in the game and acts as a Stackelberg leader. However, the retailer
and the supplier in our model have a second chance to re-match supply and demand
while the models mentioned above do not. Our model is one among many that consider
multiple replenishments, which help reduce supply-demand mismatch costs. However,
the literature on multiple replenishments typically focuses on the buyer getting addi-
tional (but incomplete) information after the first replenishment. In contrast, in our
model, the fast-ship order is placed after the demand uncertainty is completely resolved.
In the ensuing discussion, we divide the literature on multiple replenishments into two
categories depending on the timing of the second replenishment relative to demand
realization and discuss each case separately.
Two replenishments (before demand realization)
Cachon (2004) studies a two-price model in which the retailer has a second replenish-
ment opportunity during the selling season with the same or a higher wholesale price.
The supplier has only one production opportunity, but it may produce more than the
retailer’s initial order. When the retailer bears all inventory risk, the resulting contract
is called a push contract, which is identical to the model studied by Lariviere and Por-
teus (2001). If the supplier carries all inventory risk, then it is called a pull contract.
When both firms share inventory risk, it is called an advance-purchase discount contract
because the advance purchase price is lower than the wholesale price for orders placed
later. There are similarities between our model and advance-purchase discount contract.
First, the retailer needs to pay a higher price for the second order (we call it a fast-ship
order and Cachon (2004) calls it an “at-once” order). Second, the supplier produces
more than the retailer’s first order. However, in Cachon’s model, the supplier produces
more because it has only one production opportunity. In contrast, the supplier in our
model produces/procures more because the second replenishment costs more.
Donohue (2000) analyzes a supply contract between a supplier and a distributor with
two-mode production where the first mode is cheaper but needs a longer lead time and
the second mode is more expensive but the production lead time is shorter. Gurnani and
9
Tang (1999) consider a model with two ordering opportunities where a retailer needs to
balance the tradeoff between a better demand forecast and an uncertain second order
cost to determine an optimal order quantity at the first ordering instant. Tagaras and
Vlachos (2001) consider an inventory system in which the retailer has an opportunity to
place an emergency order at a higher cost with a shorter lead time to avoid stockouts.
The authors develop a heuristic algorithm to calculate optimal order-up-to levels and
discover that the proposed system generates higher cost savings compared to a system
without emergency orders.
Each paper in this category has different focus. For example, the focus of Cachon
(2004) and Lariviere and Porteus (2001) is about the efficiency of the supply chain in
terms of total supply profit for different price mechanisms. Donohue (2000) focus on the
conditions in which efficiency can be achieved for different degree of demand forecast
improvement. Tagaras and Vlachos (2001) focus on developing algorithm for calculating
ordering policy for multi-period problems. However, our work is different because we
assume that only a fractions of customer can be satisfied by the second replenishment.
Also, we focus on the individual players’ performances instead of the efficiency of the
supply chain as a whole.
Two replenishments (after demand realization)
Huggins and Olsen (2003) investigate a two-stage centralized chain in which the up-
stream supplier always meets the requests from the downstream retailer. If the supplier
cannot meet the retailer’s demand using the on-hand inventory, then the supplier needs
to use costly overtime production to satisfy the retailer’s demand. The authors show
that the optimal inventory policy for the retailer depends on total inventory in the sys-
tem and the optimal inventory policy for the supplier is an echelon base-stock policy. In
their models, the supplier may produce more in each batch because there is a fixed pro-
duction cost. There is no fixed cost for production in our fast-ship model, and because
faster procurement is costlier, second procurement by the supplier is meant to satisfy
only fast-ship orders. Also, we allow the expedited delivery cost to be shared between
the supplier and the retailer.
Gupta et al. (2010) study the fast-ship option in a single-period setting where the
supplier decides wholesale price(s) and the retailer chooses the order quantity. The
10
events happen in the following sequence. First, the supplier announces wholesale
price(s). Knowing the price(s), the newsvending retailer places an initial order. If
demand exceeds the initial order quantity, a fraction of the excess demand is satisfied
via fast-ship orders. This model assumes that the supplier does not produce more in the
first batch. Therefore, the supplier does not bear any inventory risk. Two different price
mechanisms are used in this paper. In the single-price mechanism, the supplier sets a
single wholesale price for both initial and fast-ship orders; in the dual-price mechanism,
the supplier sets a wholesale price for the initial order and a different price for fast-ship
orders.
The findings of this article are as follows. In both single-price and dual-price mech-
anisms, most of the profits go to the supplier so long as the retailer accepts fast-ship
orders, especially when the customer participation rate is high. It is worth noting that
the supplier always chooses the highest possible fast-ship price. Therefore, if the re-
tailer can make the supplier choose a lower fast-ship order price, then both players may
benefit from fast-ship orders. Also, the authors show that the retailer’s profit increases
in demand variability up to a point because the supplier is forced to lower the price in
order to get a larger initial order. If the retailer can use alternate supply sources, such
as ordering through a distributor or providing a substitute product, to reduce the sup-
plier’s market power, then the retailer may influence the supplier to change its pricing
strategy and benefit more from fast-ship orders.
Our work is similar to Gupta et al. (2010). However, we analyze and compare several
possible contract structures in which the second replenishment can be utilized. We also
focus on supply chains with two retailers and study how different relationships between
retailers can affect the supply partners’ profits and decisions.
1.3. Key Findings
We briefly summarize the key findings of Chapters 2, 3 and 4 in this section. In Chapter
2, we analyze a multi-period model with a single supplier and a single retailer. We
identify certain demand structures for which problems with non-stationary demand are
solvable. Also, we show that there exists a critical markup price such that the retailer
supports the fast-ship option when δ is less than or equal to the critical value. In
11
addition, when the wholesale price for the fast-ship option is chosen by the supplier, the
supplier earns all additional profit from fast-ship option, which implies that the supplier
is the only party that benefits from a higher customer participation rate.
However, our analysis also shows that if the wholesale price for fast-ship orders is
exogenous such that the retailer can earn a strictly positive profit from fast-ship orders,
a greater customers participation rate may adversely affect the supplier’s profit. We
also show that the retailer’s profit is decreasing whereas the supplier’s profit may not
be monotone in demand variability.
In Chapter 3, we introduce three fast-ship supply commitment contracts in a single-
period setting with one supplier and one retailer. We solve the supplier’s optimal com-
mitment in each contract using Variation Diminishing Property if demand is a Polya
frequency function of order 2. We also show that the two players have different contract
preferences and no contract structure dominates others in terms of channel performance
with pre-negotiated wholesale prices. However, either party may propose a contingent
contract to eliminate the conflict and create a win-win resolution. Also, we show that
such conflict may not exist when w is chosen by the supplier within each structure. In
such cases, structure B can be the best contract for both parties.
In Chapter 4, we show that a unique pure strategy Nash equilibrium exists for
the retailers’ problem. When wholesale prices are chosen optimally, we show that the
supplier’s expected profits are increasing in customer participation rate for all three
sourcing modes. However, the retailer’s expected profit may not be monotone in cus-
tomer participation rate when two retailers compete or cooperate. Also, we find that the
supplier’s profit with two independent retailers is higher than the other two structures
whereas the retailers are better off in terms of profit when they cooperate or compete.
Moreover, when wholesale price is chosen by the supplier, competing retailers sometimes
earn a greater profit, which is somewhat counterintuitive. Overall conclusions and fu-
ture research directions are discussed in Chapter 5. The vast majority of the proofs are
presented in Appendices.
This dissertation studies mechanisms that can improve supplier/retailer efficiency
and promote activities that help more consumers to obtain products in a timely fash-
ion. The focus of this dissertation is on a mechanism called the fast-ship option. The
dissertation provides both analytical and numerical results under a variety of settings.
12
In summary, offering the fast-ship option is beneficial for the supply chain because it
reduces lost sales by keeping customers who experience stockouts. However, our results
suggest that finding the balance between the supplier and the retailer can be challenging.
If the supplier is given to much decision power, then it leaves no room for the
retailer to make the fast-ship option profitable. If the fast-ship option is profitable for
the retailer, then the supplier ends up facing too much uncertainty. We show that both
the supplier and the retailer can increase the profitability for the fast-ship option in some
ways. For instance, the supplier can commit to a limited number of supply to mitigate
the uncertainty whereas the retailer can reduce the dependency on the supplier by either
seeking other retail partners or attracting demand from other competitors. This provides
practitioners insights into how the fast-ship option changes the best production/ordering
decisions for the supplier and retailers, whether the supply chain benefits from this
option, and which types of contracts achieve a win-win outcome for all players.
Chapter 2
A Multi-Period Model
2.1. Introduction
In this chapter, we consider a retailer that receives periodic replenishments from a
supplier for an item. In each period, only few customers would wait until the next
regular replenishment arrives if the retailer stocks out. What should the retailer do if
it runs out of stock during a selling period? The retailer is sure to lose virtually all
remaining demand in that period if it does nothing. However, if the retailer can obtain
fast-ship orders from its supplier during the current period, then it may be able to satisfy
demand from customers who agree to wait the short amount of time it takes to receive
the fast-ship orders. For example, heavier and frequent snow fall in late November and
early December can lead to greater demand for snow tires earlier in the winter season
and cause a stock out situation at a retailer for popular brand/size of snow tires. If the
retailer is not scheduled to receive its next regular shipment for several weeks, it may
find that few customers would be willing to wait that long for snow tires. However,
the retailer may be able to convince its loyal customers to wait for a few days. Before
implementing such a strategy, the retailer would need to have an agreement with its
supplier that makes it possible to obtain replenishments fast. It needs to know whether
the supplier will support its efforts to reduce lost sales and and how would the additional
cost of filling fast-ship orders be shared. Finally, both players need to know the ranges
of parameter for which the fast-ship option would increase their profits. We address all
these issues in this chapter.
13
14
Specifically, we imbed the supplier-retailer interaction in a multi-period procure-
ment model where the supplier and the retailer would support the fast-ship option with
pre-determined prices. Notwithstanding retailer efforts, only a fraction of customers
(referred to as the fast-ship participation rate) who experience a stockout prefer to wait
for the product. Others choose not to wait and forgo making the purchase. Similarly,
if this option is not offered, a fraction of customers (referred to as the backorder par-
ticipation rate) facing out-of-stock situation would reserve a product to be purchased
in the next period (i.e. backorder) and the rest do not make a purchase. For both the
supplier and the retailer, the decision to serve customers who agree to wait is assumed
irreversible. That is, once a decision is made in favor of the fast-ship option, contract
terms require that both players must continue to honor their commitment in future
periods to avoid loss of goodwill from changing an established business practice.
Related Literature
This chapter is related to previous works involving more than one replenishment op-
portunity; see, for example, Eppen and Iyer (1997a,b), Gurnani and Tang (1999), and
Donohue (2000). These authors have studied the use of two ordering opportunities for
fashion products when both opportunities arise prior to the start of the selling season.
The retailer, after placing an initial order, observes a signal that is correlated with the
demand during the selling period. With this new information, the demand forecast is
updated and the second replenishment is used to lower supply-demand mismatch costs.
The focus of the papers cited above is to model the effect of the retailer getting addi-
tional (but incomplete) demand information after placing its first order. In contrast,
in our setting, no early demand signal is observed by the retailer. The purpose of the
second replenishment (which takes place after demand realization in each selling period)
is to serve customers that agree to wait for out-of-stock items.
Another difference in the modeling approaches is that all previous papers except
Donohue (2000) consider a centralized setting, whereas we model a decentralized supply
chain. Donohue (2000)’s focus is on designing coordinating contracts, whereas we are
interested in understanding the role of consumers’ willingness-to-wait on supplier’s and
retailer’s profits. Our findings also diverge from the earlier work. Whereas in the
latter, the second order is unequivocally beneficial to the buyer, the provision of a
15
second replenishment may lead to lower expected profits for the retailer/supplier in our
setting.
Cachon (2004) and Dong and Zhu (2007) also consider up to two opportunities for
the retailer to procure products from a supplier. In these papers, the first ordering
opportunity occurs prior to the selling season and the second occurs during the selling
season. In anticipation of a possible second order, the supplier stocks (produces) more
than the retailer’s initial order. The authors study the use of different wholesale price
contracts to improve supply chain efficiency. Although at first glance, our model appears
closely related to those of Cachon (2004) and Dong and Zhu (2007), closer inspection
reveals significant differences. These differences are highlighted next. Cachon (2004)
and Dong and Zhu (2007) do not model consumer response. If the retailer pre books
supply q before the selling season begins and the realized demand is X, then the entire
excess demand (X− q)+ constitutes the size of the second order. There is no additional
cost of supplying the second order – neither to the supplier nor to the retailer. The
supplier is not obligated to serve the excess demand and incurs no additional penalty
if it does not have enough stock to meet the demand. The supplier has no option to
replenish its stock during a selling season.
In contrast, in our setting, the retailer and the supplier agree on a set of wholesale
prices that make the fast-ship option profitable for both players. Note, fast-ship orders
entail an additional expedited-delivery charge (Hausman 2005). If both players agree to
support the fast-ship option, then they are obligated to satisfy demand from customers
who are willing to wait. The supplier can use a second replenishment during the selling
season at a higher cost.
The focus of Cachon (2004) and Dong and Zhu (2007) is on studying supply chain
efficiency under different wholesale price mechanisms, whereas our focus is on under-
standing the impact on supplier’s and retailer’s profits of the consumers willingness to
wait. Finally, we model a multi-period replenishment problem whereas Cachon (2004)
and Dong and Zhu (2007) and all other works cited above consider newsvendor envi-
ronments.
This chapter is also related to Bhargava et al. (2006) who study a retailer’s prob-
lem of simultaneously determining optimal order quantity, selling price, and stockout
compensation (retail-price discount) offered to the customers. Bhargava et al. (2006)
16
assume a deterministic price-dependent demand and a single decision maker who deter-
mines the optimal ordering and pricing policy. In contrast, we consider random demand,
exogenous retail price, and extra cost to retailer (rather than a retail-price discount)
of second orders. Moreover, our goal is to shine light on the supplier-retailer interac-
tions in a two-player (decentralized) supply chain. In summary, we focus on a setting
that has not been studied before and that has the potential to improve supply chain
performance.
Many papers focus on identifying an optimal ordering policy in a multi-period set-
ting. For example, Lovejoy (1990) studies conditions under which a myopic solution is
near-optimal. Levi et al. (2007b) and Levi et al. (2007a) develop algorithms for solving
multi-period newsvendor problems with non-stationary demand. We also summarize
conditions such that optimal ordering policies can be obtained for the supplier and the
retailer. However, we focus on how supply chain parameters affect the supplier and the
retailer’s profit when there are two replenishment opportunities in each selling period.
Developing heuristic for solving problems of this kind is not our primary goal.
The importance of managing stockouts has motivated a stream of empirical studies
that analyze its impact on consumer behavior; see Anupindi et al. (1998), Campo et al.
(2000), and Campo et al. (2003). However, there is no parallel literature dealing with
the supplier-retailer interactions with multiple replenishments. We address this gap by
investigating the following issues in the context of a decentralized supply chain.
• What are the optimal ordering policies for the supplier and the retailer when the
fast-ship option is supported?
• How does the supplier’s markup decision affect the retailer’s decision to offer
customers the option to wait?
• How do different values of customer participation rates affect profits of the supplier
and the retailer?
• How does increasing demand variability affect profits of the supplier and the re-
tailer?
An important finding of this chapter is that the retailer’s optimal ordering policy has
a myopic order-up-to level structure, whereas the supplier follows an echelon base-stock
17
policy. A similar result has been shown in the literature for problems with no second
replenishment (Zipkin 2000); our model is however different as both the retailer and
the supplier can replenish stock twice in each period. Retailer profits are shown to be
decreasing in the level of markup and there exists a critical markup price that makes
the retailer indifferent between offering the fast-ship option and the backorder option.
A higher fast-ship participation rate, that is, a greater fraction of customers willing
to wait for the out-of-stock product, is beneficial for the supplier when it can choose
markup price to increase profitability for fast-ship orders. However, when it is not able
to do so, a higher fast-ship participation rate is not always beneficial for the supplier.
This is because a greater participation rate not only increases sales but also uncertainty
faced by the supplier, producing mixed overall results. In contrast, the retailer earns a
higher profit under a higher customer participation rate only when wholesale prices are
exogenous. This is because when either wholesale price or markup price is set by the
supplier, the supplier gets most or all profit from fast-ship orders.
Similarly, the effect of backorder participation rate on expected profits is also not
monotone and both the supplier and the retailer may not benefit from higher values
of backorder participation rate depending on how prices are set. Also, we observe that
when wholesale price is set by the supplier, a higher backorder participation rate may
reduce the retailer’s profit with the fast-ship option. This is because the retailer’s profit
with the backorder option may be lower under a higher backorder participation rate.
Therefore, the supplier can charge a higher wholesale price with the fast-ship option
and still make the fast-ship option attractive to the retailer.
The critical markup price (which makes the retailer indifferent between the fast-ship
option and backorders) is shown to be unaffected by demand variability. The supplier
can earn either a higher or lower profit whereas the retailer earns lower profit when
demand variability increases. In addition, the fast-ship option helps manage risk from
demand variability better for the supplier and the retailer because the profit difference
between the fast-ship option and the backorder option increases in demand variability.
The remainder of this chapter is organized as follows. In the next section, we present
the model formulation. The optimal operational decisions for the supplier and the
retailer are derived in section 2.3. In section 2.4, we study the effect of customer
participation rates on the expected profits for both players. Then, in section 2.5, we
18
consider the effect of demand variability on profits. Conclusions can be found in section
2.6 and the proofs are presented in Appendix B.
2.2. Model Formulation
We use alphabets S and R to denote the supplier and the retailer, respectively, and index
t to denote time. In each period, S sells products to R at a pre-negotiated wholesale
price w for regular orders. If the fast-ship option is supported, each fast-ship order is
sold to R at wholesale price w2 = w + δ, where δ ≥ 0 is the negotiated markup price.
The model described in this chapter concerns decisions by S and R about feasible value
of w and δ that make the fast-ship option profitable. Note that the price markup offsets
the additional transportation and delivery costs associated with expedited orders, either
in part or in its entirety. The sequence of events is as follows. S first chooses to support
the fast-ship option at a pre-negotiated w and δ. Then, R decides the regular order
quantity qtR in each period. Finally, the supplier decides how many additional units yt
to produce.
R’s demand Xt ∈ ℜ+ is continuous with probability density and distribution func-
tions ft(·) and Ft(·), respectively. We assume that ft(·) > 0 over the support of Xt to
make the exposition clearer. R is a price taker and the retail price is r. R observes
its beginning-of-period inventory and places its regular order before observing demand.
Thereafter, replenishments arrive, supply is matched with demand, and financial trans-
actions occur. When customers are given the fast-ship option, a fraction α of those who
find the product out of stock would decide to place an order. The rest do not make a
purchase. Similarly, if there is no fast-ship option, a fraction β would reserve a product
to be purchased in the next period (i.e. backorder) and the rest do not make a purchase.
In this chapter, we refer to α and β as customer participation rates. The difference
between the fast-ship option and backordering is that items that are ordered through
the fast-ship option in period t arrive in the same period, whereas items that are back
ordered in period t arrive along with the regular replenishment in period (t+1). Clearly,
for the vast majority of products, the relationship between α and β is characterized by
0 ≤ β ≤ α ≤ 1. We assume this relationship throughout the chapter.
The reason for having two separate models (the fast-ship option and the backorder
19
option) is that the supplier in our model must satisfy all fast-ship demand. If the
supplier and the retailer negotiate a maximum fast-ship commitment ζ, then the two
options may be offered at the same time — a fraction of customers who cannot obtain
the item through the fast-ship option would backorder. In such cases, the number of
fast-ship orders is α(Xt − a)+ ∧ ζ and the size of backordering is β(α(Xt − a)+ − ζ)+,
where a is the retailer’s on-hand inventory. We do not consider such scenarios in this
chapter. However, such scenarios in a single period setting are the focus of Chapter 3.
We use it and ut to denote S’s and R’s beginning-of-period inventory in period t.
If S and R have leftover stock in either the backorder or the fast-ship scenario, i.e. if
it, ut > 0, then they carry inventory and incur carrying charges. Carrying costs for S
and R are hS and hR per unit per unit time, and their discount rates are denoted by
λS and λR, respectively.
Each regular order (resp. fast-ship order) is shipped to the retailer with a unit
shipping cost τ1 (resp. τ2), which is paid by the supplier to a third party logistics
provider. Without the fast-ship option, S either produces or orders from its supplier
once in each period whose size qtS is the minimum necessary to satisfy qtR plus any
backordered demand from the previous period. It carries inventory only until its initial
stock runs out. In contrast, when the fast-ship option is offered, S has two opportunities
to order from its supplier. The first replenishment quantity qtS can be used to satisfy
both the regular order and possibly a portion of the fast-ship order. S would need a
second replenishment only if the fast-ship order exceeds the amount leftover with S
after it supplies R’s regular order. The latter equals qtS + it− qtR. The unit procurement
costs for S are c1 and c2, respectively, with c1 ≤ c2. This makes sense because S
has relatively shorter time available within which to procure items ordered through
the fast-ship option. We use the bar notation to denote opposites of fractions; thus,
e.g. α = 1 − α, β = 1 − β, and so on. All problem parameters are assumed known to
both players.
The Retailer’s Profit Functions
Consider first the case when R decides not to offer the fast-ship option and let ρtR,B(u, q)
denote R’s period-t reward function when its starting inventory in period t is u and it
chooses to order q. We use alphabets B (for backorders) and Q (for fast-ship orders)
20
throughout this chapter to identify backorder and fast-ship scenarios. Then,
ρtR,B(u, q) = −hRu− wq + rE[min(u+ q,Xt)] + βλR(r − w)E[(Xt − u− q)+]. (2.1)
The first term on the right-hand side of (2.1) is the cost of carrying u units of inventory,
wq is the order cost, rE[min(u+q,Xt)] is the expected revenue from the initial sales and
βλR(r−w)E[(Xt−u−q)+] is the present value of the expected profit from backordered
demand. Because backordered items are delivered in the next period, this revenue is
discounted by λR.
Similarly, let ρtR,Q(u, q | δ) denote R’s period-t reward function when the fast-ship
option is offered, markup price is δ, starting inventory in period t is u, and order quantity
is q. Then,
ρtR,Q(u, q | δ) = −hRu− wq + rE[min(u+ q,Xt)] + α(r −w2)E[(Xt − u− q)+]. (2.2)
The difference between (2.1) and (2.2) is that in (2.2), the term α(r−w2)E[(Xt−u−q)+]
accounts for the expected profit from the fast-ship order, replacing βλR(r−w)E[(Xt −
u− q)+] in (2.1). Because the fast-ship order is received in the same period, its revenue
is not discounted.
Suppose the retailer chooses a sequence of order quantities qR = (q(1)R , q
(2)R , · · · , q
(N)R ).
Define R’s expected total discounted profits as πBR (qR) and πS
R(qR | δ) for option B and
Q, respectively. Then,
πBR(qR) =
N∑
t=1
(λR)t−1E(ρtR,B(u
t, qtR)), (2.3)
and
πQR(qR | δ) =
N∑
t=1
(λR)t−1E(ρtR,Q(u
t, qtR | δ)). (2.4)
Note that u(1), the starting inventory in period 1, is a known constant. The expec-
tation in the total profit expressions is over realized values of ut, ∀ t > 1, which can be
determined upon knowing the order quantity and the realized demand. The retailer’s
problem is to find an ordering policy that determines the profit maximizing sequence of
order quantities under options B and Q for each possible value of δ.
21
The Supplier’s Profit Functions
For S, we start by writing its single-period expected profit under scenario B. In this
case, S’s sales equal the sum of R’s regular order and the realized backordered demand
from the previous period. Moreover, because S has no incentive to order more than the
minimum necessary to meet retailer’s order in each period, it orders an amount equal
to the positive part of R’s regular order size plus backorders minus its inventory level in
period t. For t ≥ 2, this leads to the following expression for S’s period-t profit function.
ρtS,B = −htSit + (w − τ1)[q
tR + βt−1{(xt−1 − (ut−1 + qt−1
R ))+}]
−c1[qtR + βt−1{(xt−1 − (ut−1 + qt−1
R ))+} − it]+, (2.5)
where xt−1 is the realized demand in period t− 1. Similarly, when t = 1,
ρ(1)S,B = −htSi
(1) + (w − τ1)q(1)R − c1(q
(1)R − i(1))+. (2.6)
We turn next to the case in which the fast-ship option is offered. In this case S
chooses a sequence of order quantities (q(1)S , q
(2)S , · · · , q
(N)S ). In period t, S’s expected
profit upon receiving a regular order for qtR items from R, observing inventory levels it
and ut, and choosing to order qtS ≥ (qtR − it)+, can be written as follows.
ρtS,Q(it, ut, qtS | δ, qtR) = −ithS − qtSc1 + (w − τ1)q
tR
+(w + δ − τ2)αE[(Xt − (ut + qtR))+]
−c2E[{α(Xt − (ut + qtR))+ − (it + qtS − qtR)}
+]. (2.7)
The first three terms in (2.7) represent, respectively, the inventory carrying charges,
the cost of first replenishment, and the revenue from regular sales to the retailer. The
fourth term captures the expected sales revenue from the fast-ship order and the last
term contains the expected second replenishment cost.
In summary, if the fast-ship option is not offered, the supplier’s profit is determined
entirely by the retailer’s decisions qR = (q(1)R , q
(2)R , · · · , q
(N)R ). S does not make any
operational choices and its expected total profit is given by
πBS =
N∑
t=1
(λS)t−1E(ρtS,B). (2.8)
22
If the fast-ship is offered and S chooses qS = (q(1)S , q
(2)S , · · · , q
(N)S ), then its expected
total profit is given by
πQS (qS | δ,qR) =
N∑
t=1
(λS)t−1E(ρtS,Q(i
t, ut, qtS | δ, qtR)). (2.9)
The supplier’s problem is to find qS that maximize πQS (qS | δ,qR) for each possible
value of δ.
2.3. The Retailer’s and the Supplier’s Decisions
We first present the retailer and the supplier optimal ordering policies for scenarios B
and Q (Section 2.3.1). In addition in Section 2.3.2 and 2.3.3, we identify conditions
under which the parameter of the optimal policies can be obtained. Finally, we show
how the retailer and the supplier may choose w and δ such that the fast-ship option is
supported for problem instances in which demand is stationary in Section 2.3.5.
2.3.1 Optimal Ordering Policies
The retailer’s problem is to find a policy that would form the basis for choosing qtR upon
knowing δ, to maximize either πBR (qR) or πF
R(qR | δ) depending on whether fast-ship
option is supported. Let a = q + u denote R’s on-hand inventory after ordering q.
Equations (2.1) and (2.2) can be rewritten as
ρtR,B(u, a) = −hRu− w(a− u) + rE[min(a,Xt)] + βλR(r − w)E[(Xt − a)+], (2.10)
and
ρtR,Q(u, a | δ) = −hRu−w(a− u)+ rE[min(a,Xt)] +α(r−w− δ)E[(Xt − a)+]. (2.11)
Let vtR,B(u, a) and vtR,Q(u, a | δ) denote the value functions when a set of optimal actions
is implemented from period t onwards under options B and Q, respectively. Then, R’s
period-t problems under options B and S are
atB = argmaxa≥u
vtR,B(u, a) =[
ρtR,B(u, a) + λRE[ maxat+1≥ut+1
vt+1R,B(u
t+1, at+1)]]
, (2.12)
23
and
atQ = argmaxa≥u
vtR,Q(u, a | δ) =[
ρtR,Q(u, a | δ) + λRE[ maxat+1≥ut+1
vt+1R,Q(u
t+1, at+1 | δ)]]
.
(2.13)
Proposition 2.1. Both vtR,B(u, a) and vtR,Q(u, a | δ) are concave in a for every t.
Proposition 2.1 is proved by induction. Let atB and atQ be unconstrained maximizers
for vtR,B(u, a) and vtR,Q(u, a | δ), respectively. Proposition 2.1 then implies that R’s opti-
mal policy is to order up to base-stock levels atB = max(ut, atB) and atQ = max(ut, atQ) for
scenario B and Q respectively for period t. From (2.12), we observe that vtR,B(u, a) is the
sum of separable functions of u and a because vt+1R,B(u
t+1, at+1) = vt+1R,B((X
t − a)+, at+1)
is independent of u and ρtR,B(u, a) is an additive function of u and a. Similarly,
vtR,Q(u, a | δ) is also the sum of separable function of u and a. Therefore, when we
take the derivative of vtR,B(u, a) or vtR,Q(u, a | δ) with respect to a, functions of u are
eliminated and both atB and atQ do not depend on u.
Similarly, the supplier’s problem is to find a policy for choosing qtS , upon observing
qtR, it and ut, that maximizes πB
S (qS | qR) and πQS (qS | δ,qR) for each δ. Subsequently,
we also identify the range of values of delta within which the fast-ship option would be
attractive to the retailer.
In scenario B, when a stockout occurs in period t, the supplier delivers the period-t
backorder items in period t+1. Because the unit cost for procuring period-t backorder
demand in period t+ 1 is less that that in period t (e.g., hS + c1 < λSc1), the supplier
does not have incentive to prepare items in advance. That is, S does not carry inven-
tory after its initial inventory i(1) runs out and that qtS = [(atB − ut)+ + βt−1(xt−1 −
max(ut−1, atB))+− it]+, which is R’s regular order size plus backorders minus its inven-
tory level in period t.
In scenario Q, S’s optimal first replenishment quantity decision has two components.
The first component, which we call the non-discretionary replenishment amount, equals
the amount that S must order to cover R’s order. The non-discretionary replenishment
amount may be zero in some periods, but S does not choose this component of its order
quantity. Because S knows that R orders up to a base-stock level atS in each period, qtR =
(atQ−ut)+ and the non-discretionary replenishment amount equals max{0, (atQ−ut)+−
it}. The second component, which we call discretionary replenishment quantity, allows
24
S to build up inventory in anticipation of the fast-ship order during each selling period.
S’s choice of the discretionary replenishment quantity is tantamount to choosing its on-
hand inventory level gt ≥ 0 after supplying R’s order, where gt = it − (atQ − ut)+ + qtS .
That is, S’s first-batch replenishment is specified by g = (g(1), g(2), · · · , g(N)).
Recall that atQ = max(atQ, ut). Let zt = it+ut, gt = gt+αatQ, and ςt = (zt−atQ)
++
αatQ. Because there is a one-to-one correspondence between gt and gt, we hereafter
use gt to denote S’s discretionary replenishment decision. This transformation makes
it possible to prove the main result of this section shown in Proposition 2.2 below. We
also define
φt(atQ) =αhSλS(atQ + E[(Xt − atQ)
+]) + λSc1(−E[Xt] + αE[(Xt − atQ)+])
− (1− λS)c1αatQ + λS(w − τ1)E[(at+1
Q − (atQ −Xt)+)+]
+ α(w + δ − τ2)E[(Xt − atQ)+], (2.14)
and
ρtS,Q(ςt, gt | δ) = (λSc1 − htSλS)[g
t +E[(αXt − gt)+]]− ct1gt − ct2E[(αXt − gt)+]. (2.15)
With these notation in hand, we can obtain a convenient decomposition of S’s ex-
pected profit function as shown in Lemma 2.1 below.
Lemma 2.1. Let g = (g(1), g(2), · · · , g(N)) denote transformed discretionary replenish-
ment quantities. Then, S’s expected discounted profit with the fast-ship option as a
function of δ and g equals
πQS (δ, g) = −i(1)hS + c1(u
(1) + i(1)) + (w − τ1)(a(1)Q − u(1))+ +
N∑
t=1
(λS)t−1φt(atQ)
+
N∑
t=1
(λS)t−1E
(
ρtS,Q(ςt, gt | δ)
)
. (2.16)
Lemma 2.1 is obtained after several simplifying steps. It shows that (2.9) can be
decomposed into separable functions of state variables ςt and atS , and that functions
involving the action variable gt are independent of atS . Because ρtS,Q(ςt, gt | δ) is the
only term in (2.16) that depends on the sequence g, we can ignore other terms in (2.16)
25
and focus on∑∞
t=1(λS)t−1E
(
ρtS,Q(ςt, gt | δ)
)
when maximizing πQS (δ, g). That is, S’s
period-t problem under scenario Q can be simplified as follows.
gt = argmaxg≥ς
vtS,Q(ς, g) = ρtS,Q(ς, g | δ) + λSE[ maxgt+1≥ςt+1
vtS,Q(ςt+1, gt+1)] (2.17)
Proposition 2.2. The function vtS,Q(ς, g) is concave in g for every t.
Similar to Proposition 2.1, Proposition 2.2 can also be proved by induction. In the
operations management literature, the sum of supply chain inventories at a particular
stocking point and all stocking points that are downstream from it (i.e. closer to the
customer) is referred to as the echelon stock at that point. Let ptS be the echelon stock
at the start of period-t and αg be the unconstrained maximizer for vtS,Q(ς, g). We obtain
gt = α(g − atQ)+ and ptQ = max(α(g − atS)
+ + atS , zt). Note that similar to retailer’s
problem, g does not depend on ς. Proposition 2.2 implies that an echelon base-stock
policy is optimal for S. A similar result has been obtained in the literature for the model
with no fast shipping; see, for example, Zipkin (2000), p. 302-308. Our model is different
because both the retailer and the supplier can replenish their stock twice in each period.
However, due to curse of dimensionality, calculating the parameters of the optimal
ordering policy for the supplier and the retailer can be difficult. In the next two sec-
tions, we identify conditions under which optimal policy parameters can be calculated
efficiently.
2.3.2 Problems with Stationary Demand
In this section, we show that it is optimal for R to use a myopic order-up-to policy under
both scenarios when demand is stationary. Similarly, under scenario B, S procures
an amount that is precisely equal to the positive part of R’s regular order size plus
backorders minus S’s inventory level in each period, whereas for scenario Q, S’s optimal
procurement policy is a myopic order-up-to policy.
The Retailer’s Policy
We define quantities aB and aQ(δ) below that we show in Proposition 2.3 to be the
order-up-to levels without and with the fast-ship option, respectively.
aB = F−1
(
w + βλR(r − w)− r
βλR(r − w)− r + λR(w − hR)
)
. (2.18)
26
aQ(δ) = F−1
(
w + α(r − w − δ)− r
α(r − w − δ)− r + λR(w − hR)
)
. (2.19)
Proposition 2.3. When demand is stationary, R’s optimal policy is a myopic order-
up-to level policy. Furthermore, if ut = u is the inventory level at the start of period
t, then the optimal order-up-to-level atB = max(aB , u) without the fast-ship option, and
atQ = max(aQ(δ), u) with the fast-ship option.
Proposition 2.3 can be proved by induction. The intuition behind the result in Propo-
sition 2.3 is that because demand is stationary, R faces the same holding and shortage
costs in each period, which in turn depend only on the total stock level (order-up-to
level) and not on the starting inventory. Therefore, R’s ordering decision does not
depend on future periods and a myopic policy is optimal.
An immediate consequence of Proposition 2.3 is that if u(1), the inventory level at
the start of period 1, is less than a, the optimal order-up-to level, then qtR = (a− ut)+,
where a is aB under scenario B and aQ(δ) under scenario Q. In contrast, when u(1) > a,
no orders are placed until a period in which the starting inventory drops below a for
the first time. That is, if u(1) > a, we would need to keep track of the index of the first
period in which the inventory level drops below the order-up-to level. This introduces
additional notation when writing the retailer’s profit function, but does not affect the
ensuing analysis presented in this section. Therefore, in order to keep the exposition
simple, we assume in the remainder of this section that u(1) ≤ aB in scenario B, and
u(1) ≤ aQ(δ) in scenario S.
The Supplier’s Policy
The optimal order-up-to level for S under scenario Q is obtained in Proposition 2.4.
Before presenting the result, we define
η = F−1
(
c2 − c1c2 − λSc1 + λShS
)
. (2.20)
Proposition 2.4. When demand is stationary, S’s optimal first-batch order quantity
with the fast-ship option is determined by a myopic order-up-to level. Furthermore,
if atQ = max{aQ(δ), ut} is R’s target inventory level at the start of period t, then the
optimal order-up-to level pt(δ) = max(α(η − atQ)+ + atQ, z
t).
27
Similar to R’s problems, Proposition 2.4 is also proved by induction. When demand
is stationary, S’s optimal decisions for future periods do not depend on current decision
if that decision is chosen optimally. Therefore, the supplier’s optimal period-t decision
may be chosen without considering future periods.
2.3.3 Problems with Non-Stationary Demand
It can be shown that under certain conditions, R’s and S’s optimal policies are myopic
policy even when demand is not stationary, although the order-up-to level for each
period may not remain fixed. We identify such scenarios in this section. The proof for
each example follows similar arguments in Proposition 2.3 and 2.4. Therefore, instead
of presenting the proof in detail, we only point out the key step in the proof for each
case by showing that the optimal decision in period-(t+1) is always feasible and is not
affected by the optimal decision in period-t.
Note that we only deal with the case in which demand is non-stationary. Before
presenting results, we first define the following equalities, which will be used repeatedly
in this section.
atB = F−1t
(
w + βλR(r − w)− r
βλR(r − w)− r + λR(w − hR)
)
. (2.21)
atQ(δ) = F−1t
(
w + α(r − w − δ)− r
α(r − w − δ)− r + λR(w − hR)
)
. (2.22)
ηt = F−1t
(
c2 − c1
c2 − λSct+11 + λSh
t+1S
)
. (2.23)
Stochastically Increasing Demand
In the first scenario, we assume that demand process {Xt} is stochastically increasing
in t such that Ft(x) ≥ Ft+1(x) (denoted as Xt ≤st Xt+1). In such cases, we observe
that the optimal atB , atQ, and αηt are exactly as defined in equation (2.21), (2.22),
(2.23). Note that atQ is a function of δ. However, we omit argument δ to simplify
notation. To prove that atB and atQ are indeed optimal decisions, we only need to show
that at+1B , at+1
Q ≥ ut+1 when atB and atQ are chosen in period-t and follow all other steps
shown in the proofs of Proposition 2.3 and 2.4. Because Xt ≤st Xt+1 and the definition
28
of atB and atQ in (2.18) and (2.19), we observe that atB ≤ at+1B and atQ ≤ at+1
Q . Therefore,
we obtain ut+1 = (atB − xt)+ ≤ at+1B for scenario B and ut+1 = (atQ − xt)+ ≤ at+1
Q for
scenario Q.
Similarly, for the supplier, we need to show that αηt+1 ≥ ςt+1 when αηt is chosen
in period-t. Because
ςt+1 = (zt+1 − at+1Q )+ + αat+1
Q
= {(gt − α(atQ + (Xt − atQ)+))+ + (atQ −Xt)+ − (at+1(δ) ∨ (atQ −Xt)+)}+
+(αat+1 ∨ α(atQ −Xt)+)
≤ {(gt − αatQ)+ + (atQ −Xt)+ − (at+1 ∨ (atQ −Xt)+)}+
+(αat+1 ∨ α(atQ −Xt)+,
we observe that
ςt+1 ≤
{
(gt − αatQ)+ + α(atQ −Xt) if atQ −Xt ≥ at+1
Q ,
((gt − αatQ)+ + (atQ −Xt)+ − at+1
Q )+ + αat+1Q otherwise.
(2.24)
Suppose that the optimal gt = αηt, it implies that αηt ≥ αatQ and αηt+1 ≥ αat+1Q .
Hence, we observe that ςt+1 defined in (2.24) is always less than or equal to αηt+1.
Non-Stationary Demand (Xt = mt + Y t)
Suppose that demand is defined as Xt = mt + Y t. The myopic policy holds if {Yt} is
stationary, mt are constants that change over time and mt+Y t has a lower bound. For
example, if mt and Y t ≥ 0, then the optimal (unconstrained) order-up-to levels are
atB = atB +mt, (2.25)
atQ = atQ +mt, and (2.26)
gt = α(ηt +mt). (2.27)
Again, we only need to show that ut+1 ≤ at+1B , atQ when at = at+1
B , atQ, respectively,
and ςt+1 ≤ gt+1 when gt = gt. First consider retailer’s problems under scenario B.
Because ut+1 = (atB −Xt)+ = (atB −mt − Y t)+, we observe that
ut+1 =[
atQ − Y t]+
≤ at+1Q +mt+1 = at+1
B . (2.28)
29
By applying similar arguments, we obtain ut+1 ≤ atQ for scenario Q as well. For the
supplier, we obtain ςt+1 ≤ gt+1 by following the same arguments provided after equation
(2.24) because inequality (2.24) holds in this example after mt and mt+1 are canceled
out. Hence, details are omitted.
Mean Preserving Demand (Xt = ιtE[Y t] + (1− ιt)Y t)
Suppose that demand Xt = ιtE[Y t] + (1 − ιt)Y t where Y t are i. i. d. We observe that
the retailer has a myopic order-up-to-level policy when either (1) ιt is decreasing in t
(that is, V ar(Xt) is increasing in t) and atB (or atQ) is greater than or equal to E[Y t],
(2) or ιt is increasing in t (V ar(Xt) is decreasing in t) and atB (or atQ) is less than E[Y t].
In addition, the unconstrained optimal decisions are defined as follows.
atB = (1− ιt)atB + ιtE[Y t], and (2.29)
atQ = (1− ιt)atQ + ιtE[Y t]. (2.30)
Similarly, the feasibility of at+1B and at+1
Q can be checked by the following inequalities
ut+1 = (atB − xt)+ ≤ atB ≤ at+1B for B and ut+1 = (atQ − xt)+ ≤ atQ ≤ at+1
Q and Q.
However, the supplier has a myopic policy only when ιt is decreasing in t and ηt ≥
E[Y t]. In addition, unconstrained optimal gt for period-t is
gt = α((1 − ιt)ηt + ιtE[Y t]). (2.31)
Because
ςt+1 ≤
{
(gt − αatQ)+ + α(atQ −Xt) if atQ −Xt ≥ at+1
Q ,
((gt − αatQ)+ + (atQ −Xt)+ − at+1
Q )+ + αat+1Q otherwise,
(2.32)
we observe that ςt+1 ≤ gt+1 when gt = gt from the fact that gt ≤ gt+1 and αat+1Q ≤ gt+1.
This example shows that the condition under which a myopic policy is optimal for the
supplier may be more restrictive than those for the retailer.
When demand is non-stationary and does not belong to any of the categories above,
we are able to obtain the optimal policy parameters for the supplier and the retailer in
a two-period problem. Details can be found in the next section.
30
2.3.4 Optimal Policies for Two-Period Problems
We first consider two-period scenarios in which demand is non-stationary and conditions
for myopic solutions do not hold. Suppose that after the end of period 2, the backordered
items are sold to R at a unit price w and unsold items are salvaged at a unit price w−hR.
We obtain the optimal policies for both the supplier and the retailer.
The Retailer’s Policy
Based on (2.1) and (2.2), we obtain the optimal order-up-to levels a(2)B = max(a
(2)B , u(2))
without the fast-ship option and a(2)Q = max(a
(2)Q (γ), u(2)) with the fast-ship option,
where u(2) is the start-of-period-2 inventory. Note that a(2)B , u(2)) and a
(2)Q are defined
as in (2.18) and (2.19).
Moreover, base on (2.12) and (2.13), we can write R’s period-1 problem as
a(1)B =argmax
a≥u
{
ρ(1)R,B(u
(1), a) + λR
[
− h(2)R E(a−X(1))+ − wE(a
(2)B − (a−X(1))+)+)
+ rE(X(2))− (r − βλR(r − w))E(X(2) − (a(2)B ∨ (a−X(1))+))+
]}
, (2.33)
and
a(1)S (γ) = argmax
a≥u
{
ρ(1)R,S(u
(1), a | γ)
+ λR
[
− hRE(a−X(1))+ − wE(a(2)S − (a−X(1))+)+)
+ rE(X(2))− (r − α(r − w2))E(X(2) − (a(2)S ∨ (a−X(1))+))+
]}
. (2.34)
Because R’s expected profits for option S and B from period 1 onward are concave in
a (Proposition 2.1), we obtain a(1)B = max(a
(1)B , u(1)) and a
(1)S (γ) = max(a
(1)S (γ), u(1)),
where a(1)B and a
(1)S (γ) are solutions to the following equations,
w = [r − λRβ(r − w)]F1(a)− λR[hRF1(a)− w(F1(a)− F1(a− a(2)B ))]
−λR(r − βλR(r − w))[−
∫ a−a(2)B
0F2(a− x)f1(x)dx],
and
w = [r − α(r − w(1)2 )]F1(a)− λR[hRF1(a)− w(F1(a)− F1(a− a
(2)S ))]
−λR(r − α(r − w(2)2 ))[−
∫ a−a(2)S
0F2(a− x)f1(x)dx],
respectively.
31
The Supplier’s Policy
Similarly, suppose that the backordered items at the end of period-2 are replenished at
a unit price c1 and S’s leftover inventory is salvaged at a unit price c1 − hS . Based on
(2.16), solving maxg
πQS (γ, g) is equivalent to solving max
g
∑nt=1(λS)
t−1E(
ρtS,Q(ςt, gt | γ)
)
,
which are the only terms that depends on S’s decisions gt. Because ρtS,Q(ςt, gt | γ) is
concave in gt and the fact that gt ≥ ςt , we obtain the second-period (the last-period)
optimal g(2) = max(αη(2), ς(2)), where η(2) is defined in (2.20). Next, in order to solve
the period-1 problem, we rewrite
maxg
2∑
t=1
(λS)t−1E
(
ρtS,Q(ςt, gt | γ)
)
= maxg
ρ(1)S,Q(ς
(1), g | γ) + λSE(
ρ(2)S,Q(ς
(2), g(2) | γ))
(2.35)
Suppose that g and a(1)Q are S and R’s optimal decisions in period-1, respectively. Then
ς(2) = (z(2) − a(2)Q )+ + αa
(2)Q can be written as
ς(2) = {(g − α(a(1)Q + (X(1) − a
(1)Q )+))+ + (a
(1)Q −X(1))− (a(2)(γ) ∨ (a
(1)Q −X(1))+)}+
+(αa(2)(γ) ∨ α(a(1)Q −X(1))+), (2.36)
because u(2) = (a(1)S −X(1)), i(2) = (g−αα(a
(1)S +(X(1)−a
(1)S )+))+, and a
(2)S = (a(2)(γ)∨
(a(1)S −X(1))+). From equation (2.36), we summarize the value of ς(2) for different ranges
of g and X(1) in Table 2.1
Range of X(1) Value of ς(2)
X(1) ≤ a(1)Q − a
(2)Q ς(2) =
{
α(a(1)Q −X(1)) if g < αa
(1)Q ,
g − αX(1) otherwise
a(1)Q − a
(2)Q < X(1) ≤ a
(1)Q ς(2) =
{
αa(2)Q if g < αa
(1)Q
(g + (1− α)a(1)Q −X(1) − a
(2)Q )+ + αa
(2)Q otherwise
a(1)Q < X(1) ς(2) = ((g − αX(1))+ − a
(2)Q )+ + αa
(2)Q
Table 2.1: The values of ς(2)
Let I(·) denote the indicator function. Based on the fact that g(2) = max(αη(2), ς(2))
and the results shown in Table 2.1, we obtain E(
ρ(2)S,Q(ς
(2), g(2) | γ))
as follows.
32
Case 1: when g < αa(1)Q ,
E(
ρ(2)S,Q(ς
(2), g(2) | γ))
= I(a(2)Q < a
(1)Q )
∫ a(1)Q −a
(2)Q
0ρ(2)S,Q(α(a
(1)Q − x), (α(a
(1)Q − x) ∨ αη(2)) | γ)dF1(x)
+
∫ a(1)Q
(a(1)Q
−a(2)Q
)+ρ(2)S,Q(αa
(2)Q , (αa
(2)Q ∨ αη(2)) | γ)dF1(x)
+
∫ ∞
a(1)Q
ρ(2)S,Q(((g − αX(1))+ − a
(2)Q )+ + αa
(2)Q , (((g − αX(1))+ − a
(2)Q )+
+ αa(2)Q ∨ αη(2)) | γ)dF1(x) (2.37)
Case 2: when g ≥ αa(1)Q ,
E(
ρ(2)S,Q(ς
(2), g(2) | γ))
= +I(a(2)Q < a
(1)Q )
∫ a(1)Q −a
(2)Q
0ρ(2)S,Q(g − αx, (g − αx) ∨ αη(2)) | γ)dF1(x)
+
∫ a(1)Q
(a(1)Q −a
(2)Q )+
ρ(2)S,Q((g + (1− α)a
(1)Q − x− a
(2)Q )+ + αa
(2)Q , ((g + (1− α)a
(1)Q
− x− a(2)Q )+ + a
(2)Q ∨ αη(2)) | γ)dF1(x)
+
∫ ∞
a(1)Q
ρ(2)S,Q(((g − αX(1))+ − a
(2)Q )+ + αa
(2)Q , (((g − αX(1))+ − a
(2)Q )+
+ αa(2)Q ∨ αη(2)) | γ)dF1(x) (2.38)
Define µ(1)(ς(1), g | γ).= ρ
(1)S,Q(ς
(1), g | γ) + λSE(
ρ(2)S,Q(ς
(2), g(2) | γ). For each term in
(2.37) and ((2.38)), we identify the range of g in which g(2)S depends on g and obtain
∂µ(1)(ς(1), g | γ)/∂g in the following equation.
∂µ(1)(ς(1), g | γ)
∂g= (λSc
t1 − htSλS)F
(
g
α
)
− c1 + c2F
(
g
α
)
+ λS
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g,
(2.39)
Let ωt(g).= ρ
(2)S,Q(ς, g | γ) and ∇a
(1)Q = a
(1)Q − q
(2)Q . ∂E
(
ρ(2)S,Q(ς
(2), g(2) | γ))
/∂g in (2.39)
can be obtained from follows.
33
Case 1: when g < αa(1)Q ,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g= 0.
Case 2: when αa(1)Q ≤ g < αa
(1)Q + a
(2)Q ,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))
∂
∂g
∫ ∇a(1)Q
0ω(2)(g − αx)dF1(x)
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
∂
∂g
{
∫g−αη(2)
α
0ω(2)(g − αx)dF1(x)
+
∫ ∇a(1)Q
g−αη(2)
α
ω(2)(αη(2))dF1(x)}
+∂
∂g
{
∫ g+(1−α)a(1)Q −a
(2)Q −α(η(2)−a
(2)Q )+
(∇a(1)Q )+
ω(2)(g + α(∇a(1)Q )− x)dF1(x)
+
∫ a(1)Q
g+(1−α)a(1)Q −a
(2)Q −α(η(2)−a
(2)Q )+
ω(2)(αa(2)Q ∨ αη(2))dF1(x)
}
.
Case 3: when αa(1)Q + a
(2)Q ≤ g < αa
(1)Q + a
(2)Q + α(η(2) − a
(2)Q )+,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))
∂
∂g
∫ ∇a(1)Q
0ω(2)(g − αx)dF1(x)
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
∂
∂g
{
∫g−αη(2)
α
0ω(2)(g − αx)dF1(x)
+
∫ ∇a(1)Q
g−αη(2)
α
ω(2)(αη(2))dF1(x)}
+∂
∂g
{
∫ g+α∇a(1)Q −αη(2)
(∇a(1)Q )+
ω(2)(g + (1− α)a(1)Q − x)dF1(x)
+
∫ a(1)Q
g+α∇a(1)Q −αη(2)
ω(2)(αη(2))dF1(x)}
34
Case 4: when g ≥ αa(1)Q + a
(2)Q + α(η(2) − a
(2)Q )+,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))
∂
∂g
∫ ∇a(1)Q
0ω(2)(g − αx)dF1(x)
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
∂
∂g
{
∫g−αη(2)
α
0ω(2)(g − αx)dF1(x)
+
∫ ∇a(1)Q
g−αη(2)
α
ω(2)(αη(2))dF1(x)}
+∂
∂g
∫ a(1)Q
(∇a(1)Q )+
ω(2)(g + α∇a(1)Q − x)dF1(x)
+∂
∂g
{
∫
g−a(2)Q
−α(η(2)−a(2)Q
)+
α
a(1)Q
ω(2)(g − αx− (1− α)a(2)Q )dF1(x)
+
∫ ∞
g−a(2)Q
−α(η(2)−a(2)Q
)+
α
ω(2)(a(2)Q ∨ αη(2))dF1(x)
}
.
After applying Leibniz integral rule to each term above, we obtain
Case 1: when g < αa(1)Q ,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g= 0.
Case 2: when αa(1)Q ≤ g < αa
(1)Q + a
(2)Q ,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))[−c1F1(∇a
(1)Q ) + c2
∫ ∇a(1)Q
0F2(
g − αx
α)dF1(x)]
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
{
− c1F1(g − αη(2))
+ c2
∫g−αη(2)
α
0F2(
g − αx
α)dF1(x)
}
+{
− c1(F1(g + (1− α)a(1)Q − a
(2)Q − α(η(2) − a
(2)Q )+)− F1((∇a
(1)Q )+))
+ c2
∫ g+(1−α)a(1)Q −a
(2)Q −α(η(2)−a
(2)Q )+
(∇a(1)Q )+)
F2(g + α(∇a
(1)Q )− x
α)dF1(x))
}
35
Case 3: when αa(1)Q + a
(2)Q ≤ g < αa
(1)Q + a
(2)Q + α(η(2) − a
(2)Q )+,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))[−c1F1(∇a
(1)Q ) + c2
∫ ∇a(1)Q
0F2(
g − αx
α)dF1(x)]
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
{
− c1F1(g − αη(2))
+ c2
∫g−αη(2)
α
0F2(
g − αx
α)dF1(x)
}
+{
− c1(F1(g + α∇a(1)Q − αη(2))− F1((∇a
(1)Q )+)))
+ c2
∫ g+α∇a(1)Q −αη(2)
(∇a(1)Q )+
F2(g − a
(2)Q − α(η(2) − a
(2)Q )+
α)dF1(x)
Case 4: when g ≥ αa(1)Q + a
(2)Q + α(η(2) − a
(2)Q )+,
∂E(
ρ(2)S,Q(ς
(2), g(2) | γ))
∂g
= I(a(2)Q < a
(1)Q )I(g < αη(2) + α(∇a
(1)Q ))[−c1F1(∇a
(1)Q ) + c2
∫ ∇a(1)Q
0F2(
g − αx
α)dF1(x)]
+ I(a(2)Q < a
(1)Q )I(g ≥ αη(2) + α(∇a
(1)Q ))
{
− c1F1(g − αη(2))
+ c2
∫g−αη(2)
α
0F2(
g − αx
α)dF1(x)
}
+{
− c1(F1(a(1)Q )− F1((∇a
(1)Q )+))) + c2
∫ a(1)Q
(∇a(1)Q )+
F2(g + α(∇a
(1)Q )− x
α)dF1(x)
+{
− c1(F1(g − a
(2)Q
α)− F1(a
(1)Q )) + c2
∫
g−a(2)Q
−α(η(2)−a(2)Q
)+
α
a(1)Q
F2(g − αx− αa
(2)Q
α)dF1(x)
}
Let αη(1) be the optimal g such that ∂µ(1)(ς(1),αη(1)|γ)∂g
= 0. We obtain the optimal
g(1) = α(η(1) − a(1)Q )+.
2.3.5 Choice of δ - Stationary Demand and Infinite Horizon
So far in this section, we have dealt with how S chooses its ordering policy for a given
δ. Now we consider S’s problem of choosing δ. For this purpose, we shall further
36
assume that demand is stationary, u(1) ≤ max{aQ(δ), aB} and z(1) ≤ α(η − a(1)Q )+ =
α(η − aQ(δ))+. That is, atQ = aQ(δ), a
tB = aB and pt(δ) = α(η − aQ(δ))
+ for all t.
These assumptions result in simpler notation, but they are not needed for the ensuing
analysis to hold.
For each fixed u(1) ≤ a, we define πBR(a) =
∞∑
t=1(λR)
t−1E(ρtR,B(ut, a)) to be the total
expected profit for R without the fast-ship option, given that it orders up to a in each
period, and obtain
πBR (a) = (w − hR)u
(1) +1
1− λR
(
− wa+ rE[X] + (βλR(r − w)− r)E[(X − a)+])
+λR
1− λR(w − hR)E[(a −X)+]. (2.40)
The order-up-to level aB in equation (2.18) can be obtained alternatively from (2.40) and
the first-order optimality equation. Similarly, using πQR(a | δ) =
∞∑
t=1(λR)
t−1E(ρtR,S(ut, a |
δ)) to denote R’s total expected profit with the fast-ship option, and performing similar
operations that led to (2.40), we get
πQR(a | δ) = (w − hR)u
(1) +1
1− λR
(
rE[X] + (α(r − w − δ)− r)E[(X − a)+]
−wa)
+λR
1− λR(w − hR)E[(a−X)+]. (2.41)
Note that u(1) in Equations (2.40) and (2.41) is a known parameter.
Next, we characterize R’s response to different markup price in the situation where
S supports fast shipping. To simplify notation, we hereafter use πQR(aQ(δ)) instead
of πQR(aQ(δ) | δ) to denote the optimal retailer profit when price markup is δ. Upon
comparing Equations (2.40) and (2.41), we find that πQR(a | 0) ≥ πB
R (a) for each fixed a
because α ≥ β. Therefore, this inequality must also hold when R picks its optimal order-
up-to level under each scenario. This means that S can incentivize R to support the fast-
ship option by setting the value of markup equal to zero. The inequality πQR(aQ(0)) ≥
πBR (aB) makes sense on an intuitive level because the fast-ship option reduces the risk
of shortage by allowing the retailer a second chance to procure supply.
We also see from equation (2.41) that for each fixed a, R’s profit is decreasing in δ
when R offers the fast-ship option. Therefore, given 0 ≤ δ1 < δ2 < r−w, πQR(aQ(δ2)) ≤
πQR(aQ(δ2) | δ1) < πQ
R(aQ(δ1)). That is, R’s profit is also decreasing in δ when R picks
37
the optimal order-up-to level corresponding to each δ. Define δc =[
(α−βλR)(r−w)/α]
.
This observation helps obtain the following proposition.
Proposition 2.5. R strictly prefers the fast-ship option only if δ < δc. It does not offer
the fast-ship option when δ > δc, and is indifferent between these two options when
δ = δc.
Proposition 2.5 can be proved as follows. Upon comparing (2.40) and (2.41), we
observe that aQ(δc) = aB when δ = δc. Note that δc ≥ 0 (because α > β). To remove
any ambiguity, we hereafter assume that R always offers the fast-ship option when it is
indifferent between the two options. Therefore, if δ∗ is Supplier-chosen markup, then
δ∗ must be less than or equal to δc.
Because atQ = aQ(δ) for all t, we obtain gt = gt + αatQ = pt(δ) − atQ + αatQ =
α(η − atQ)+ + αatQ = α(η − aQ(δ))
+ + αaQ(δ). Let πQS (δ) denote S’s total expected
profit resulting from an optimal choice of replenishment quantities when R chooses to
offer the fast-ship option. Let πQS (δ) =
∞∑
t=1(λS)
t−1E(ρtS,Q(it, gt | δ)) and the fact that
gt = α(η − aQ(δ))+ + αaQ(δ), we obtain
πQS (δ) =
[
λS(w − τ1 − c1)E(X) + (1− λS)(−u(1)(w − τ1)− i(1)hS + (u(1) + i(1))c1)
+(w − τ1 − c1)(1 − λS)aQ(δ)− α(η − aQ(δ))+[c1(1− λS) + λShS ]
+E[(X − aQ(δ))+]{α(w + δ − τ2) + αλS(hS − c1)− λS(w − τ1 − c1)}
−E[(X − aQ(δ)− (η − aQ(δ))+)+]{c2 + λS(hS − c1)}
]
(1− λS)−1. (2.42)
Proposition 2.6. πQS (δ) is non-decreasing in δ.
Proposition 2.6 is consistent with intuition – so long as S supports the fast-ship option,
its profit is greater when it offers a higher markup price, i.e. higher δ. Since πQS (δ) is
non-decreasing in δ, there exists a δl such that πQS (δ) ≥ πB
S for all δ ≥ δl.
Next, let δ∗ be S’s optimal action under the fast-ship option. S would support the
fast-ship option only if πQS (δ
∗) ≥ πBS . Because S does not know if it prefers the fast-ship
option without calculating δ∗ first, we obtain δ∗ by solving the following optimization
38
problem
maxδ
πQS (δ) (2.43)
subject to πQR(aQ(δ) | δ) ≥ πB
R (aB) (2.44)
πQR(aQ(δ) | δ) ≥ 0 (2.45)
0 ≤ δ ≤ r − w. (2.46)
Note that when δ ≤ δc, πQR(aQ(δ)) ≥ πB
R (aB) by the definition of δc. Moreover, because
πBR (aB) ≥ 0 by assumption, it follows that constraints (2.44) and (2.45) are satisfied
when δ ≤ δc. This means that all three constraints can be simplified and reduced to
0 ≤ δ ≤ δc. Because of Proposition 2.6, a solution to (2.43) – (2.46) can be obtained in a
straightforward manner. Knowing that R will offer fast shipping so long as δ ≤ δ∗.= δc
and that its own profit is non-decreasing in δ, if πQS (δ
∗) ≥ πBS , then S chooses δ∗ and R
offers the fast-ship option. However, it may be possible that δc < δl (i.e.,πQS (δ
∗) < πBS ).
In that case, there is no feasible δ in which the fast-ship option can be supported by
the supplier and the retailer at the same time. Such scenarios are excluded from our
analysis.
2.4. Effect of Customer Participation Rates
Customer participation rates may be influenced by brand loyalty, retailer’s sales effort,
and advertising (marketing) strategies used by both players (Mishra and Raghunathan
2004). For instance, supplier’s product branding effort can affect customer loyalty and
increase backorder participation rate, Similarly, the retailer’s sales effort can induce
more customers to use the fast-ship option (a higher α). Therefore, understanding the
effect of participation rates α and β provides insights into how S and R could make the
fast-ship option more profitable. It is of particular interest to find out if either player
could be worse off on account of increased customer participation.
Three scenarios are considered in this section — (1) scenarios with exogenous w
and δ, (2) scenarios with exogenous w and supplier-selected δ, and (3) scenarios with
supplier-selected w and exogenous δ
39
2.4.1 Effect of Customer Participation Rate α
Scenarios with Exogenous w and δ
When w and δ are exogenous, the effect of customer participation rate for the retailer
is presented in the following proposition.
Proposition 2.7. For fixed w and δ, πQR(δ) is strictly increasing in α.
Proposition 2.7 can be proved as follows. For a fixed order-up-to level a, the retailer’s
expected profit πQR(δ) shown in (2.41) is higher under a higher α. Hence, the same result
holds when a is chosen optimally according to the value of α. Note that when r−w2 = 0,
πQR(δ) does not change in α. In other words, if the fast-ship orders is strictly profitable
for the retailer at pre-negotiated prices, the retailer earns more under a higher customer
participation rate.
However, the supplier’s profit can be either higher or lower with the fast-ship option
when α increases. To illustrate the effect of changing α, we assume that demand X
follows a gamma distribution with E[X] = 2, 500 and Var(X) = 125, 000. Moreover,
we set r = 300, w = 100, δ = 50, hR = 30, hS = 5, τ1 = 0.1, τ2 = 50, β = 0.2,
λR = λS = 0.9, c1 = 10, and c2 = 20. Then we change α in steps of size 0.05 from 0.3
to 1 and plot the supplier’s and the retailer’s profits are in Figure 2.1. The solid lines in
Figure 2.1(a) show S’s optimal profit when it chooses the best strategy (Q or B). The
corresponding profit for R is shown in Figure 2.1(b).
0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.25
2.251
2.252
2.253
2.254x 10
6
Customer Participation Rate α
The
Sup
plie
r’s E
xpec
ted
Pro
fit
πBS
πQS
(δ)
(a) Supplier’s Profit
0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.4
4.6
4.8
x 106
Customer Participation Rate α
The
Ret
aile
r’s E
xpec
ted
Pro
fit
πQR
(aQ(δ))
πBR (aB)
(b) Retailer’s Profit.
Figure 2.1: The Effect of Participation Rate α.
In Figure 2.1(a), we observe that S’s profit function with the fast-ship option is not
40
monotone. It first decreases then increases in α. This is because the retailer chooses a
smaller order-up-to level when α is higher. This not only increases the inventory risk
for the supplier but also reduces the profit from the initial order. Hence, the supplier’s
profit can decrease. However, when α becomes even higher, the supplier’s profit can
increase in customer participation rate because it receives sufficient fast-ship demand.
Also, we notice that pre-negotiated prices may not guarantee participation in the
fast-ship option for all values of α. For example, when α is between 0.4 and 0.96, the
supplier earns more profit by not supporting the fast-ship option. In other words, prices
need to be re-negotiated in such regions so that both parties can benefits from the
fast-ship option.
Scenarios with Exogenous w and Supplier-Selected δ
When δ can be chosen by the supplier, the effects of α on the supplier’s and the retailer’s
profits are presented in Proposition 2.8 and 2.9.
Proposition 2.8. If δ is chosen by the supplier, the retailer’s expected profit with the
fast-ship option remains invariant regardless of the value of α.
The intuition behind Proposition 2.8 is as follows. When the supplier supports the
fast-ship option, it always sets δ∗ = δc such that πQR(qQ(δ
∗)) = πBR (qB). Since πB
R (qB)
is independent of α, whether or not the fast-ship option is supported by the supplier,
the retailer’s profit remains the same regardless of the value of α.
Next, we show that the supplier’s profit becomes higher or remains the same if the
value of α is higher.
Proposition 2.9. The supplier’s profit with the fast-ship option is increasing in α when
δ is chosen by the supplier.
Proposition 2.9 can be explained from the fact that δ∗ = δc = (α−βλR)(r−w)/α. Note
that δc is increasing in α. It means that the supplier charges a higher wholesale price
for fast-ship orders when α is higher. Recall that qB = qQ(δc), the fast-ship demand is
higher when α is higher. Thus, the supplier earns a higher profit under a higher α when
fast-ship is supported by both parties.
41
Scenarios with Supplier-Selected w and Exogenous δ
Suppose that wholesale price w is chosen by the supplier. The results are similar to
that with supplier-chosen δ. We illustrate the results through a numerical example.
We assume that demand X follows a gamma distribution with E[X] = 2, 500 and
Var(X) = 125, 000. Moreover, we set hR = 30, hS = 5, τ1 = 0.1, τ2 = 50, β = 0.2,
λR = λS = 0.9, c1 = 10, and c2 = 20.
When we set α = [0.05, 0.95] and , δ = [0, 50], we observe that the supplier’s profit
is increasing in α, whereas the retailer’s profit is either decreasing or invariant in α.
The intuition behind this result is follows. The supplier profit increases because the
supplier can charge a higher w under a higher α. For the same reason, the retailer’s
profit decreases because ordering costs are higher with a higher α. In addition, its profit
remains invariant when w is chosen such that the retailer is indifferent between offering
or not offering the fast-ship option. Note that our numerical results also hold with other
distributions such as log-normal and uniform.
2.4.2 Effect of Customer Participation Rate β
Scenarios with Exogenous w and δ
Next, when w and δ are exogenous, it is clear that the suppliers’ and the retailers’
profits with fast-ship option are not affected by β. However, different value of β may
make pre-negotiated prices become invalid. This is demonstrated with the help of
numerical example below. We assume that demand X follows a gamma distribution
with E[X] = 625 and Var(X) = 15, 625 (CoV = 1/5). Moreover, we set r = 300,
w = 150, δ = 10, hR = 30, hS = 5, τ1 = 0.1, τ2 = 10, α = 0.9, λR = λS = 0.9, c1 = 10,
and c2 = 30. In this example, both the supplier’s and the retailer’s profits with the
fast-ship option are invariant in β because δ and α are fixed.
We can see from Figure 2.2 (a) that S’s profit function without the fast-ship option
first decreases then increases as β increases. Therefore, the pre-determined prices work
only when β is between 0.7 and 0.85. The supplier does not support the fast-ship option
in other regions unless prices are renegotiated.
42
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.98.6
8.61
8.62
8.63
8.64
8.65
8.66x 10
5
The
Sup
plie
r’s E
xpec
ted
Pro
fit
πBS
πQS
(δ)
(a) Supplier’s Profit
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.96.5
7.5
8.5
9.5
10.5x 10
5
Cusomer Participation Rate β
The
Ret
aile
r’s E
xpec
ted
Pro
fit
πBR (aB)
πQR
(aQ(δ))
(b) Retailer’s Profit.
Figure 2.2: The Effect of Participation Rate β.
Scenarios with Supplier-Selected δ
We next study the effect of β when δ is chosen by the supplier. In such cases, we observe
that the retailer’s profit can be higher with βH as compared to that with βL. This result
is shown in Proposition 2.10.
Proposition 2.10. The retailer’s profit with the fast-ship option is increasing in β
when δ is chosen by the supplier.
Proposition 2.10 can be explained on an intuitive level with the following argument.
Because R’s profit without the fast-ship option is increasing in β, S is forced to offer a
lower δ. Consequently, R’s profit becomes higher with a higher β.
However, the supplier’s profit with fast-ship option is lower under a higher β. The
results are presented in Proposition 2.11.
Proposition 2.11. When δ is chosen by the supplier, the supplier’s profit is lower
under a higher β.
Proposition 2.11 can be explained as follows. If the supplier supports the fast-ship
option for both βL and βH , where βH > βL then the supplier must choose a smaller δ
under βH . This makes the fast-ship option less profitable for the supplier.
Scenarios with Supplier-Selected w and Exogenous δ
When w is chosen by the supplier within each option, the results are different from those
with supplier-selected δ. We assume that demand X follows a gamma distribution with
43
E[X] = 625 and Var(X) = 15, 625 (CoV = 1/5). Moreover, we set r = 300, δ = 10,
hR = 30, hS = 5, τ1 = 0.1, τ2 = 10, α = 0.9, λR = λS = 0.9, c1 = 10, and c2 = 30.
When we set β = [0.05, 0.9] and δ = [0, 30], we observe that the supplier profit with
fast-ship option is increasing in β, whereas the retailer’s profit with fast-ship option is
decreasing in β. We next explain why the results are different from those with supplier-
chosen δ. First, when β increase, the supplier can charge a higher wholesale price for
backorder option. As a result, the retailer’s profit without the fast-ship option decreases.
In order to let the retailer support the fast-ship option, the supplier needs to make sure
that the retailer earns a higher profit with the fast-ship option than without the fast-
ship option. Since the retailer’s profit with backorder option is decreasing in β, the
supplier can charge a higher wholesale price with the fast-ship option and still make
the fast-ship option attractive to the retailer. In other words, when w is chosen by the
supplier, a higher β can effectively lower the retailer’s reservation profit. Consequently,
the supplier earns a higher profit and the retailer earns a lower profit under a higher β.
2.5. Insights & Model Extension
We study how demand variability affects each player’s profit in this section.
2.5.1 Effect of Demand Variability
Some products have highly variable demand, e.g. fashion goods (Eppen and Iyer 1997a),
whereas others such as consumer staples have steady demand patterns over time. On an
intuitive level, higher variability would be undesirable because it increases the expected
cost of supply and demand mismatch. It turns out that in the presence of fast shipping,
it is possible for either S or R to benefit from higher demand variability.
Scenarios with Exogenous w and δ
When w and δ are exogenous, An illustrative example is provided next to underscore
this point. Suppose r = 300, w = 100, δ = 50, hR = 30, hS = 5, τ1 = 0.1, τ2 = 50,
α = 0.8, β = 0.2, λR = λS = 0.9, c1 = 10, c2 = 19, and demand X follows a gamma
distribution with E[X] = 90. We vary Var(X) from 180 to 810 and show its impact on
R’s and S’s profits in Figure 2.3.
44
We observe that both the supplier’s and the retailer’s profit are decreasing in demand
variability in this example. However, if we set s = 100 instead of s = 50 in the example
above, S’s profit can increase. Also, we observe that the supplier supports the fast-
ship option only when demand variability is high in this example. Similarly, when
fast-ship is supported, the profit difference between supporting the fast-ship option and
not supporting the fast-ship option becomes greater in demand variability for both the
retailer and the supplier.
In other words, both the supplier and the retailer are more likely to mitigate the
risk caused by demand variability through offering the fast-ship option.
200 300 400 500 600 700 8008.08
8.085
8.09
8.095
8.1
8.105
8.11x 10
4
The Effect of Demand Variability
M’s
Exp
ecte
d P
rofit
πBS
πQS
(δ∗)
(a) Supplier’s Profit
200 300 400 500 600 700 8001.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7x 10
5
The Effect of Demand Variability
R’s
Exp
ecte
d P
rofit
πQR
(aQ(δ∗))
πBR (aB)
(b) Retailer’s Profit.
Figure 2.3: The Effect of Demand Variability
Scenarios with Optimal δ
We consider demands that are related according to the convex order, denoted as “≤cx”.
For random variables V and Y , we say that V ≤cx Y if E[ω(V )] ≤ E[ω(Y )] for all convex
functions ω(·) for which the expectations exist. It is straightforward to confirm that
X ≤cx X implies that E[X] = E[X ], Var(X) ≤ Var(X), E[(X − a)+] ≤ E[(X − a)+],
and E[(a−X)+] ≤ E[(a−X)+]; see details in Shaked and Shanthikumar (1994), pp. 56-
57. We consider only those changes in demand variability that do not cause S’s and
R’s profits to drop below their reservation levels under the default backorder option.
This is because S and R may negotiate a different set of basic parameters, such as the
45
wholesale price, if demand variability is too large.
Before introducing our main result, it is worthwhile to note that under the as-
sumption about the reservation profit levels stated above, the markup price δ∗ of-
fered by S does not change in demand variability. This is because the critical point
δc =[
(α − βλR)(r − w)]
/α does not depend on X. In what follows, we affix a tilde
to the order-up-to levels and to the expected profit functions to delineate that they
correspond to demand X.
Proposition 2.12. The retailer’s profit with either the backorder option or the fast-ship
option is non-increasing in demand variability.
To prove Proposition 2.12, we first show that if demand X ≤cx X, then (1) πQR(a(δ) |
δ) ≤ πQR(a(δ) | δ) for any δ ≤ δc, and (2) πB
R (aB) ≤ πBR(aB). That is, R’s profit with the
fast-ship option for a fixed markup price, and R’s profit without the fast-ship option, are
non-increasing in demand variability. To see that statement (1) is true, let πQR(a, x | δ)
be R’s profit with fast shipping, where x is realized demand. From (2.41), we get
πQR(a, x | δ) = (w − hR)u
(1) +λR
1− λR(w − hR)(a− x) +
1
1− λR(−wa+ rx)
+1
1− λR(λR(w − hR) + α(r − w − δ) − r) (x− a)+. (2.47)
Because λR(w − hR) + α(r − w − δ) − r ≤ −(1 − α)(r − w) − λRhR − αδ ≤ 0 and
(x − a)+ is convex in x, it follows that πQR(a, x | δ) is concave in x. Therefore, πQ
R(a, |
δ) = E[πQR (a,X | δ)] ≥ E[πQ
R (a, X | δ)] = πQR(a | δ) where the inequality comes from the
fact that X ≤cx X. Furthermore, πQR(aQ(δ) | δ) ≤ πQ
R(aQ(δ) | δ) ≤ πQR(aQ(δ) | δ) where
the first inequality comes from the previous argument and the second inequality comes
from the fact that πQR(a(δ) | δ) is R’s optimal profit when demand is X and δ ≤ δc.
The arguments needed to show that πBR(aB) ≤ πB
R (aB) above are similar. Moreover,
because R’s profit with the fast-ship option equals its profit without the fast-ship option
when δ is chosen by the supplier, it follows that R’s profit is non-increasing in demand
variability. In light of these arguments, a formal proof of Proposition 2.12 is omitted.
For the supplier, the results with supplier-selected δ is a special case for that with
exogenous δ because the critical markup price does not change in demand variability.
Therefore, similar to example shown in the previous section, the supplier can either earn
a higher or lower profit under a higher demand variability.
46
When w is chosen by the supplier, the results are similar to above two scenarios,
Hence, we omit the details because it does not provide additional insights.
2.6. Conclusions
The globalization of supply chains and increasing competition for customers has in-
creased the importance of the procurement function. Many variants of standard pro-
curement contracts are being investigated. We added to this literature by considering
a multi-period inventory model of interactions between a supplier and a retailer, when
both players have a second opportunity to replenish stock.
Retailers’ efforts to have the right products on shelves to meet uncertain demand
include a plethora of approaches including sophisticated demand forecasting methods
and faster replenishments. In this chapter, we studied one such strategy that is com-
monly adopted by some retailers where they offer customers to order items that are out
of stock through the fast-ship option. The retailers incur an additional cost to have the
items shipped on an expedited basis, but reduce the amount of lost sales. A premise
that is supported by intuitive reasoning is that the provision of the fast-ship option will
lead to higher profits for the supplier and the retailer when the additional costs of fast
shipping are not high.
To study when this premise holds, we considered formal models of supplier–retailer
interactions permitting each party to decide whether or not to support the fast-ship
option in a multi-period setting. We identified structures underlying optimal opera-
tional decisions of the two players and characterized a feasible markup price to make
the fast-ship option profitable for both the supplier and the retailer. We also studied
the effect of changes in certain key parameters – e.g. customer participation rates and
demand variability, on the two players’ profits and their willingness to support the fast-
ship option. In some cases, parametric comparisons yielded counter-intuitive results —
e.g. the two players could be worse off with higher customer participation rates. Our
models provide guidance to suppliers and retailers on how to make ordering decisions
and to evaluate their options with respect to supporting the fast-ship option.
Chapter 3
Fast-Ship Commitment Contracts
3.1. Introduction
In Chapter 2, we prove that when the supplier can choose the markup price optimally,
it gets all addition profit from offering the fast-ship option to customers. In practice,
the negotiated markup price may be lower than the supplier’s optimal in order to make
the fast-ship option more attractive to the retailer. However, in such cases, the fast-ship
option may be less profitable for the supplier because the supplier not only receives a
smaller initial order size but also faces a greater chance of procuring the fast-ship orders
at a higher cost. In such scenarios, does the supplier have alternative levers to encourage
the retailer to order more up front?
One alternative available to supplier is to limit the fast-ship commitment via the
terms of a supply contract. In this chapter, we compare three possible supply commit-
ment contract structures between a single supplier (S) and a single retailer (R) that
support the fast-ship option for a product with a short selling season. Both the retailer
and the supplier have two available replenishment options. Facing a random demand
X, the retailer orders q before the start of the selling season and several fast-ship orders
that occur later in the selling season. The fast-ship orders are placed, as needed, if in-
ventory at the retail store runs out. Similarly, the supplier procures a certain quantity
of items before the selling season, which equals at least q, and may procure additional
items during the selling season as needed. The retailer purchases items from the supplier
at pre-negotiated unit wholesale prices w and w2 for the initial order and the fast-ship
47
48
orders, respectively, where w2 = w + δ and δ ≥ 0 is the markup. The retailer sells
products to customers at a unit retail price r.
Other parameters are identical to those defined in the base model in Chapter 1. A
fraction α ∈ [0, 1] of customers who find products out of stock respond to the availability
of the fast-ship option by placing orders and the rest do not make a purchase. The
shipping costs for regular order and fast-ship orders are τ1 and τ2, respectively, which
are paid by the supplier to a third-party logistics provider for expedited delivery. In
addition, we assume that r ≥ w2 so that parameter values belong to a region in which
the fast-ship option is attractive to the retailer.
The supplier faces replenishment costs c1 and c2 for the two replenishment options:
c1 per unit for items procured before the start of the selling season, and c2 for items
procured during the selling season, where c1 ≤ c2. Consequently, the supplier may
procure q + y, where y ≥ 0, in response to retailer’s firm order q and replenish during
the selling season only when y cannot satisfy all promised fast-ship orders.
Within each structure, a particular set of values of the retailer’s and the supplier’s
parameters is referred to as a contract. The first structure leads to a flexible total
commitment contract, referred to as Type-A contract. In this contract, the supplier
commits to a maximum total quantity p ≥ 0. The retailer can then choose any initial
order quantity and place any number of fast-ship requests so long as the total amount
ordered does not exceed p. The second structure, referred to as Type B, limits only the
supplier’s fast-ship commitment. That is, the supplier commits to supply no more than
z ≥ 0 via the fast-ship option. It also supplies any amount q ordered by the retailer
before the start of the selling season.
Both A and B are supplier-driven structures because the supplier makes its choice
first and the retailer orders q after learning p or z. The third structure, in contrast, is a
retailer-driven structure. It is referred to as Type C and it may be viewed as a retailer-
led analog of Type B structure because the supplier chooses its fast-ship commitment
γ after receiving R’s initial order q. The actual number of items that are fast shipped
depends on the parameter values chosen by the two players in each supply structure.
From a retailer’s perspective, the fast-ship option may be particularly attractive for
high-value items for which the obsolescence cost is high and the additional cost of direct
shipping to customers is relatively small. This is because the fast-ship option can reduce
49
not only the magnitude of lost sales but also the retailer’s inventory risk. A supplier
who cooperates with the retailer to support the fast-ship option may also benefit from
this practice because the total sales may be higher. However, because the fast-ship
option transfers some inventory from the retailer to the supplier, careful analysis and
selection of a contracting mechanism is necessary before it could be implemented.
We develop mathematical models that help explain how the supplier and the retailer
would choose values of their parameters within each contract structure when they max-
imize their individual profits. We establish structural properties of the retailer’s and
the supplier’s parameter optimization problems, which allow us to solve these problems
using nonlinear optimization techniques. We show that from the supplier’s viewpoint,
B is the most preferred structure and A is the least preferred when players make in-
dividually optimal decisions. This is because the retailer orders less up front under
contract Type-A and shifts more inventory/procurement responsibility to the supplier.
As a result, among the two supplier-led structures, the supplier will not offer Structure
A, even though it may provide greater flexibility to the retailer.
From the retailer’s perspective, structure A is usually preferred, except in cases
where the total promised supply (p) is smaller than the promised supply under other
contract structures (i.e. p is less than either q+z or q+γ). However, since structure A will
not be chosen voluntarily by the supplier, it is appropriate to compare only structures
B and C from the retailer’s perspective. We show that when the retailer faces a choice
between structures B and C, it prefers structure C with a fixed pre-negotiated price.
We also show that it is possible to resolve the conflict by proposing a counter offer or
by establishing a new pre-negotiated profit allocation contract.
We also study the effect of customer participation rates. Consistent with intuition,
examples reveal that both the retailer and the supplier can realize higher profits as
a result of higher customer participation rate if c1 is much smaller than c2, and that
both players’ profits can be lower when this is not true. Overall, the main contribution
of this chapter lies in presenting mathematically rigorous approaches for computing
contract parameters for each mechanism and for comparing the three mechanisms from
individual players’ and channel perspectives.
50
Related Literature
Flexibility is a common theme in supply contracts literature (e.g. White et al. 2005) and
several articles study the interactions between a supplier and a retailer under flexible
contracts; see, for example, Van Mieghem (2003), Wu et al. (2005), and Stevenson and
Spring (2007). In particular, quantity flexibility (QF) contracts are closely related to our
work. Quantity flexibility allows the buyer to adjust the purchase quantity in a certain
range without penalty, improving risk sharing and supply chain’s ability to respond to
uncertainty (see, e.g., Wu 2005). Similarly, fast-ship commitment contracts provide the
retailer the second replenish opportunity to response to stockout events.
In a QF contract, the buyer announces an early tentative order qT before the pro-
duction period begins. Knowing qT , the supplier commits to supply qS. After receiving
a more accurate demand forecast, which occurs before the selling season starts, the
buyer then adjusts its order size and comes up with a final (firm) order qF . The buyer
(resp. supplier) is not penalized if qF ≥ qT − a (resp. qS ≥ min{qF , (qT + b)}), where a
and b are called flexibility parameters (see, e.g., Tsay 1999). In summary, the buyer in
a QF contract commits to purchasing no less than a certain amount/percent below the
forecast. In return, the seller commits to supply up to a certain amount/percent above
the forecast.
The contracts we study are different from QF contracts. Supply flexibility parame-
ters are often exogenously determined in QF contracts; for example, a ≥ 0 and b ≥ −a
are exogenous in Tsay (1999). In our setting, within each contract structure, supply
commitment is determined by parameters p, z, or γ, which are chosen by the supplier,
and both players pick individually optimal parameters. Our setting, particularly Type-
A contract, is also related to Eppen and Iyer’s (1997a) two-period stochastic dynamic
programming model of a backup agreement contract. In the first period, the buyer
commits to buy up to some amount qT for the selling season and claims immediate
ownership of (1− σ)qT units where σ is exogenous. After period-1 demand is realized,
the buyer can adjust its inventory by placing a second order of up to σqT units at the
original price in period 2. In each period, a small portion of sales is returned and a
constant fraction of returned units can be reused to satisfy demand. In addition, the
buyer pays a penalty ℓ for any reserved units that are not purchased.
Our approach is similar because we also allow the retailer to place a second order
51
up to some pre-determined total quantity commitment by the supplier. However, our
problem is different because (1) the total supply commitment is a decision made by
the supplier in our models and consequently the buyer does not pay a penalty for not
purchasing all of the promised supply, (2) we model both the supplier’s and the retailer’s
problems and obtain their optimal parameters, whereas Eppen and Iyer do not address
the supplier’s problem, and (3) Eppen and Iyer focus on the the impact of backup
fraction σ and penalty ℓ on the buyer’s expected profit and commitment qT , whereas we
study the interactions between the supplier and the retailer for three different contract
structures when both players make individually-optimal decisions within each structure.
Netessine and Rudi (2006) paper also models multiple replenishments — each retailer
uses its stockpile as the primary source of items needed to satisfy demand and drop
shipping as a backup source when its stock runs out. Although the dual strategy in
Netessine and Rudi (2006) is similar to the fast-ship option considered in this chapter,
there are important differences between the two approaches. First, all customer demand
is satisfied in the dual-strategy model, which corresponds to setting α = 1 in our
model. Second, the supplier in Netessine and Rudi (2006) has a single replenishment
opportunity and its inventory decision is chosen simultaneously with the retailer’s order
quantity. Third, Netessine and Rudi (2006) compares the dual strategy with both pure
traditional (i.e. where z or γ = 0) and pure drop-ship (i.e. where q = 0) environments. In
contrast, we analyze different contract structures when fast-ship option is offered. That
is, Netessine and Rudi (2006) paper identifies the best channel strategy for different
supply chain characteristics whereas we provide insights on how contract structure and
parameter selection affects the performance of supply chain partners when fast-ship
option is offered to customers.
The rest of this chapter is organized as follows. Notation and model formulations for
the three contract structures are introduced in Section 3.2. We analyze the two player’s
optimal decisions for structures A and B in Section 3.3 and structure C in Section 3.4.
In Section 3.5, we contrast the three structures from the retailer’s, the supplier’s and
the supply chain’s perspectives. Section 3.6 summarizes this chapter. Some proofs are
presented in Appendix C.
52
3.2. Model Formulation
Index i ∈ {A,B,C} denotes contract structure. In each of the three structures, the
retailer selects the size of its initial order q and the supplier selects the value of extra
procurement quantity y ≥ 0. In addition, the supplier also selects a supply commitment,
which is denoted by p, z, or γ, depending on the structure. In expressions that apply
to all contract structures, we use parameter j ∈ {p, z, γ} to denote supply commitment.
The three contract structures belong to a family of affine supply commitment contracts
in which the supplier’s total commitment is an affine function of the form aq+ b, and a
and b are contract parameters. Different values of a and b give rise to different relation-
ships between the initial order size and the fast-ship supply commitment. Specifically,
Type-A structure arises when a = 1 and b = p− q, whereas Types B and C arise when
a = 1 and b is either z or γ.
R’s demand X ∈ R+ is continuous with probability density and distribution func-
tions f(·) and F (·), respectively. We assume a continuous demand distribution and
f(·) > 0 over the support of X. In addition, we assume that w−τ1−c1 ≥ α(w2−τ2−c1)
so that the expected marginal profit from the fast-ship order is smaller than that from
regular orders for the supplier. Similar assumption applies to the retailer. That is,
r − w ≥ α(r − w2).
The retailer’s expected profit if contract structure i is used, supplier commits j, and
retailer orders q, can be written as follows.
πiR(q, j) = rE[X ∧ q]− wq + (r − w2)E[α(X − q)+ ∧ ζ ij(q)], (3.1)
where (X ∧ q) denotes min(X, q), and ζ ij(q) is the maximum fast-ship supply committed
by S. That is, ζ ij(q) = p−q, or z, or γ when (i, j) = (A, p), (B, z), and (C, γ), respectively.
Moreover, rE[X∧q]−wq is the expected profit from the initial order, (α(X−q)+∧ζ ij(q))
is the magnitude of fast-ship demand and (r−w2)E[α(X − q)+ ∧ ζ ij(q)] is the expected
profit from the fast-ship orders.
Similarly, when contract structure i is used, the retailer orders q, and the supplier
chooses y and j, the supplier’s expected profit is given by
πiS(y, j, q) = (w − τ1 − c1)q − c1y + (w2 − τ2)E[α(X − q)+ ∧ ζ ij(q)]
−c2E[(α(X − q)+ − y)+ ∧ (ζ ij(q)− y)+]. (3.2)
53
In (3.2), (w−τ1−c1)q−c1y is the profit from R’s initial order, (w2−τ2)E[α(X−q)+∧ζ ij(q)]
is the revenue from fast-ship demand. The last term comes from the fact that S has an
uncovered commitment of (ζ ij(q)− y)+ and the leftover fast-ship demand after stockpile
y is exhausted equals (α(X−q)+−y)+. Therefore, c2E[(α(X−q)+−y)+∧(ζ ij(q)−y)+] is
the extra procurement cost for the fast-ship orders that are not served from the amount
stocked by the supplier in response to the retailer’s initial order.
With expressions (3.1) and (3.2) in hand, we are ready to find optimal parameter
values for each player under each contract structure. In the ensuing analysis, we use
eij(q).= q + ζ ij(q)/α for notational convenience and assume, without loss of general-
ity, that both the retailer and the supplier pick the smallest among possible optimal
parameter values when multiple such values exist. Because structures A and B are sup-
plier driven and structure C is retailer driven, we combine the analysis of the first two
structures in the same section.
Note that the supplier may not have the second replenishment opportunity for some
product categories and must prepare all items during the first replenishment period.
Such scenario is just a special case for our models and can be covered by setting c2 = ∞.
3.3. Parameter Optimization: Structures A and B
Let qi(j) = argmaxπiR(q, j), (i, j) ∈ {(A, p), (B, z)}, denote an optimal order quantity
for the retailer. Because πiR(q, j) is concave in q when r ≥ w2, R’s optimal order
quantities under contract structures A and B can be obtained from the first-order-
optimality equations. The results are shown below.
Structure A
qA(p) =
p if p < F−1(
ww2
)
, and
F−1(
w+(1−α)(r−w2)F (eAp (qA(p)))
r−α(r−w2)
)
otherwise.(3.3)
Structure B
qB(z) = F−1
(
w − α(r − w2)F (eBz (qB(z)))
r − α(r − w2)
)
. (3.4)
Expression (3.4) is a straightforward analog of the expression one would obtain in a
54
newsvendor model and requires no further explanation. However, (3.3) can be explained
further. When p is small, the retailer would prefer to have all item sold from the initial
stockpile This is because the marginal benefit of satisfying a demand from the initial
stockpile is higher than or equal to that of satisfying demand by taking advantage of
the fast-ship option (because w ≤ w2).
Let πiR(j) = max
qπiR(q, j) denote R’s optimal expected profit as a function of j when
i ∈ {A,B}. We can show that πiR(j) is increasing in j. This makes sense on an intuitive
level. A higher value of j implies greater supply flexibility for the retailer. As a result, it
incurs a smaller risk from demand uncertainty because it is able to satisfy more demand
after inventory runs out. Others have observed a similar result (e.g. Tsay 1999 and Wu
2005).
The supplier’s expected profit πiS(y, j, q
i(j)) shown in (3.2) is concave in y (details
are not provided). Therefore, we obtain an optimal yi(j) = argmaxπiS(y, j, q
i(j)) as
follows.
yi(j) = [α(ηS − qi(j))+ ∧ ζ ij(qi(j))], (3.5)
where ηS = F−1(c1/c2). The quantity ηS has a straightforward intuitive explanation.
If the supplier stocks out (relative to its commitment), then it incurs a unit shortage
cost of (c2 − c1). If, in contrast, it stocks too much, then its overage cost is c1. Thus,
F (ηS) = (c1/(c1 + c2 − c1)) represents the fractile of demand that the supplier should
stock in absence of constraints. However, its commitment is limited to ζ ij(qi(j)), only α
fraction of customers use fast-ship option, and y is needed only after the initial stockpile
qi(j) runs out. This explains expression (3.5).
Let πiS(j)
.= πi
S(yi(j), j, qi(j)) and j∗ = argmaxj π
iS(j) for each (i, j) ∈ {(A, p), (B, z)}.
We are now ready to solve for j∗. We first point out a special case in Proposition 3.1 in
which the supplier does not restrict its total commitment under contract structure A.
This happens when w2 ≥ c2 + τ2.
Proposition 3.1. If w2 − τ2 ≥ c2, then p∗ is unbounded.
When w2−τ2 ≥ c2, the supplier can earn a positive profit from fast-ship orders even
when it does not produce any extra quantity up front (i.e. y = 0). As a result, there is
no economic reason for the supplier to limit the size of its commitment. One may be
55
tempted to extend this intuition to contract structure B. That is, to expect that when
w2 − τ2 ≥ c2, the supplier would choose z∗ = ∞. As we show below, the above result
does not hold for structure B.
In the sequel, we show that the supplier’s profit under structuresA andB is unimodal
in p and z for many distribution families. A profit maximizing value of p can be
unbounded as seen in Proposition 3.1, but optimal values of z are always finite. Before
presenting this results, we first introduce the Variation Diminishing Property (VDP) of
PF2, where PF2 stands for Polya frequency function of order 2. Details of the VDP can
be found in Karlin (1968) and Li et al. (2009). Let M(u) be a function on (−∞,∞), and
f be PF2 on [0,∞) and zero on (−∞, 0]. According to VDP of PF2, ifM(u) changes sign
at most once on (−∞,∞), then the transformation g(v) =∫∞−∞M(u)f(v − u)du also
changes sign at most once. In addition, their sign changes occur in the same order. A
density function is PF2 if and only if it is log-concave. Some well-known distributions in
PF2 class include Gaussian, Double exponential, Gamma (with shape parameter ≥ 1),
Beta, Weibull (with shape parameter ≥ 1), and Normal (see details in Pal et al. 2007).
Proposition 3.2. If the demand distribution is PF2. then
1. the supplier’s profit under a Type-A contract is either increasing in p or has at
most one local maximum.
2. the supplier’s profit under a Type-B contract is either decreasing in z or has at
most one local maximum.
Proof. We provide proof for structure A only. Similar arguments can be applied to
contract structure B. Based on Proposition 1, the optimal p∗ = ∞ when w2 − τ2 ≥ c2.
Hence, in what follows we focus only on cases when w2 − τ2 < c2.
To establish the Proposition statement, we have to show that ∂πAS (p)/∂p changes sign
at most twice. Moreover, the first sign change must be from positive to negative and
the second from negative to 0. Let
pL = F−1
(
w
w2
)
(3.6)
We first discuss case when 0 ≤ p < pL. It is easy to check that πAS (p)
′ = (w − τ1 −
c1) > 0 from the fact that qA(p) = p for 0 ≤ p < pL
56
We next discuss three cases for the situation in which p ≥ pL. Upon replacing
F(
eAp (qA(p))
)
by[
(r − α(r − w2))F(
qA(p))
− w]
/(1 − α)(r −w2), and defining k.=
qA(p), h(k).= qA(p)′,
σ(1)(k).= (w − τ1 − c1)h(k) −
(w2 − τ2 − c2)w(1 − (1− α)h(k))
(1− α)(r − w2)
−(w2 − τ2 − c2)F (k)
(
αh(k) −(r − α(r − w2))(1 − (1− α)h(k))
(1− α)(r − w2)
)
,
(3.7)
σ(2)(k).= (w − τ1 − (1− α)c1)h(k) −
(w2 − τ2 − c2)w(1 − (1− α)h(k))
(1− α)(r − w2)
− F (k)
(
α(w2 − τ2)h(k) −(w2 − τ2 − c2)(r − α(r − w2))(1 − (1− α)h(k))
(1− α)(r − w2)
)
,
(3.8)
and
σ(3)(k).= (w − τ1)h(k)− c1 −
w(w2 − τ2)(1− (1− α)h(k))
(1− α)(r − w2)
−(w2 − τ2)F (k)
(
αh(k) −(r − α(r − w2))(1− (1− α)h(k))
(1− α)(r −w2)
)
. (3.9)
We obtain
πAS (p)
′ =
σ(1)(k) if p is in a region such that yA(p) = 0,
σ(2)(k) if p is in a region such that yA(p) = α(ηS − qA(p)), and
σ(3)(k) if p is in a region such that yA(p) = p− qA(p).
(3.10)
Showing the sign changes in p for (3.10) is equivalent to showing the sign changes in
k for the corresponding σ(i)(k). Note that F (k) =∫∞k
f(x)dx. Upon using x = k − u,
we get F (k) =∫ 0−∞ f(k − u)du. Therefore, we can rewrite σ(ℓ)(k) where ℓ ∈ {1, 2, 3} in
(3.7) – (3.9) as σ(ℓ)(k) =∫∞−∞ υ(ℓ)(u)f(k − u)du, where
υ(1)(u) =
(w − τ1 − c1)h(k) −(w2−τ2−c2)w(1−(1−α)h(k))
(1−α)(r−w2)
−(w2 − τ2 − c2)(
αh(k) − (r−α(r−w2))(1−(1−α)h(k))(1−α)(r−w2)
)
if u < 0, and
(w − τ1 − c1)h(k) −(w2−τ2−c2)w(1−(1−α)h(k))
(1−α)(r−w2)otherwise,
(3.11)
57
υ(2)(u) =
(w − τ1 − (1− α)c1)h(k) −(w2−τ2−c2)w(1−(1−α)h(k))
(1−α)(r−w2)
−α(w2 − τ2)h(k) +(w2−τ2−c2)(r−α(r−w2))(1−(1−α)h(k))
(1−α)(r−w2)if u < 0, and
(w − τ1 − (1− α)c1)h(k) −(w2−τ2−c2)w(1−(1−α)h(k))
(1−α)(r−w2)otherwise,
(3.12)
and
υ(3)(u) =
(w − τ1)h(k) − c1 −w(w2−τ2)(1−(1−α)h(k))
(1−α)(r−w2)
−(w2 − τ2)(
αh(k) − (r−α(r−w2))(1−(1−α)h(k))(1−α)(r−w2)
)
if u < 0, and
(w − τ1)h(k) − c1 −w(w2−τ2)(1−(1−α)h(k))
(1−α)(r−w2)otherwise.
(3.13)
Because 0 ≤ h(k) ≤ (1 − α)−1, both υ(1)(u) and υ(2)(u) could possibly be less than
0 when u < 0. In addition, υ(1)(u), υ(2)(u) ≥ 0 when u ≥ 0. Therefore, both υ(1)(u) and
υ(2)(u) may change sign in u ∈ (−∞,∞) at most once from negative to positive. Based
on variation diminishing property of PF2, this implies that both σ1(k) and σ2(k) may
change sign in k ∈ (−∞,∞) at most once and that this sign change is from negative
to positive. That is, when p is in a region such that yA(p) = α(ηS − qA(p))+, πAS (p)
′
changes sign at most once from negative to positive. Similarly, we observe that the
υ(3)(u) does not depend on u. That is, υ(3)(u) is a constant when u < 0 and a different
constant when u ≥ 0. Therefore, υ(3)(u) can change sign at most once and it can be
either from negative to positive or from positive to negative depending on parameters.
Based on VDP, πAS (p)
′ also changes sign at most once when p is in the range such that
yA(p) = p− qA(p).
Note that yA(p) = [α(ηS − qA(p))+ ∧ p − qA(p)], which can be obtained from (3.5)
and the fact that ζji (qA(0)) = p−qA(p). In addition, because 0 ≤ qA(p)′ ≤ (1−α)−1, we
observe that p− qA(p)−α(ηS − qA(p)) is increasing in p and α(ηS − qA(p)) is decreasing
in p. Therefore, the value of yA(p) can only change from p − qA(p) to α(ηS − qA(p))+
when p increases, but not in the other direction. In addition, because πAS (p)
′ |p=0> 0,
the first sign change from positive to negative (if any) must occur when p ≥ pL and
yA(p) = p− qA(p). If the second sign change exists, it must be from negative to positive
when α(ηS − qA(p))+. However, since limp→∞
πAS (p)
′ = 0, the second sign change must
58
be from negative to 0. This is because if the second sign change were from negative to
positive, there must exist the third sign change from positive to 0, which was shown not
to occur in this problem.
In summary, πAS (p)
′ has at most two sign changes: the first from positive to negative,
and the second from negative to 0. If no sign change occurs, πAS (p) is increasing in p.
Otherwise, there is only one local maximum, which is also the global optimal. Hence,
proved.
Proposition 3.2 implies that the optimal p and z can be obtained efficiently through
simple line searches if demand distribution is PF2.
Before closing this section, we present a comparison of structures A and B in terms of
their impact on retailer’s stocking decision. For this purpose, let qi(j)′ denote the rate of
change in q as a function of j. From (3.3)-(3.4), we observe that 0 ≤ qA(p)′ ≤ (1−α)−1
and −α−1 ≤ qB(z)′ ≤ 0, which shows that R responds differently within the two
structures if the supplier were to increase available supply — q is non-decreasing in p
and non-increasing in z. The different responses come from different ways in which the
retailer can react to changes in supply commitments under structures A and B. These
observations also provide greater insight into the relative size of initial orders proved in
Proposition 3.3 below.
Proposition 3.3. For any p and z, qA(p) ≤ qB(z).
Proposition 3.3 can be proved by observing that
limp→∞
qA(p) = limz→∞
qB(z) = F−1 (w/(r − α(r − w2)))
That combined with the fact that qA(p)′ ≥ 0 and qB(z)′ ≤ 0 implies that qA(p) ≤ qB(z)
for any p and z. This means that the supplier receives a larger initial order under
structure B as compared to A. This result is utilized in Proposition 6 (Section 3.5) in
which we show that the supplier prefers structure B to A.
3.4. Parameter Optimization: Structure C
In structure C, the supplier chooses γ after knowing q. Recall that the extra supply y
is chosen by the supplier after observing q in all three structures. It can be shown that
59
the supplier chooses yC according to
yC(q) = [α(ηS − q)+ ∧ γ], (3.14)
and the optimal γ(q) = argmaxγ πCS (y
C(q), γ, q) is obtained from
γ(q) =
{
∞ if w2 − τ2 ≥ c2, and
α(ηR − q)+ otherwise,(3.15)
where ηR = F−1(c1/w2 − τ2). The quantity ηR can be explained in a manner similar to
ηS . The reason why c2 does not appear in the expression for ηR is that when w2 < c2,
the supplier always selects y = γ and there is no need to obtain more items at unit
cost c2 (details are omitted). Equations (3.14) and (3.15) follow from the facts that
πCS (y, γ, q) is concave in y and πC
S (yC(q), γ, q) is concave in γ. Detailed arguments
showing concavity are omitted in the interest of brevity.
Next, we obtain an optimal order quantity, qC = argmaxq πCR(q, γ(q)), as shown in
Proposition 3.4 below. Hereafter, we use πCR(q) = πC
R(q, γ(q)) and πCS (q) = πC
S (yC(q), γ(q), q)
for convenience.
Proposition 3.4. The retailer’s profit is bimodal in q and there exists a c1 ∈ [w(w2 −
τ2)/r,w(w2 − τ2)/(r − α(r − w2))] such that the optimal qC can be obtained as follows.
qC =
F−1(
wr−α(r−w2)
)
if either w2 − τ2 ≥ c2, or w2 − τ2 < c2 and c1 ≤ c1, and
F−1(
wr
)
if w2 − τ2 < c2 and c1 > c1.
(3.16)
The intuition behind Proposition 3.4 is as follows. When either the wholesale price
is sufficiently large (w2 − τ2 ≥ c2), or the unit cost of supplier’s initial purchase is
sufficiently small (w2 − τ2 < c2 and c1 ≤ c1), the supplier makes an ample fast-ship
commitment to the retailer. In such cases, the retailer’s decision is based upon an
assumption of ample availability of fast-ship supply. That is, in this case, all customers
who exercise the fast-ship option are served. However, when w2 − τ2 < c2 and c1 > c1,
the supplier chooses a conservative value of γ(q) because its second replenishment cost
is higher. Anticipating this response, the retailer orders more up front.
60
Note that when qC = F−1 (w/(r)) and γ(qC) = 0, a Type-C contract is identical to
a Type-B contract with z∗ = 0. Similarly, when w ≤ c2, contract Type-A and C are
identical because qA = qC and p − qA = γ(qC). These results show that in some cases,
the ability to be the first to choose contract parameters (also called market leadership)
does not affect either party’s expected profit. Equation (3.15) and Proposition 3.4 also
help obtain the following inequalities.
Proposition 3.5. For a fixed pair of (w, δ) values, the following inequalities hold: (1)
qA(p) ≤ qC for any p, (2) γ(q) ≥ z∗ when q = qB(z∗), and (3) qB(z) ≥ qC when
z = γ(qC).
Proofs of Proposition 3.5 and all subsequent propositions can be found in the Ap-
pendix. The arguments that lead to Part 1 of Proposition 3.5 are similar to those pre-
sented immediately after Proposition 3.3. Because the retailer enjoys greater freedom
to adjust the supply between initial order and fast-ship orders under contract structure
A, it is not required to commit to an order quantity as large as in contract structure C.
The intuition behind Part 2 of Proposition 3.5 is that because the supplier chooses γ
after knowing qC , it can commit to a higher supply than that under structure B without
worrying about the possibility that a higher supply commitment may induce the retailer
to order less up front. For similar reasons, the retailer chooses a smaller order quantity
under structure C when the supply commitment under structure C is the same as that
under structure B (Part 3 of Proposition 3.5). Proposition 3.5 is important because it
leads to key results related to contract preference (Proposition 3.6) and the possibility
of resolving conflict (Proposition 3.9) in Section 3.5.
3.5. Insights
Next, we use formal arguments to analyze the three contract structures from the re-
tailer’s, the supplier’s, and the supply chain’s perspectives.
3.5.1 Supplier’s and Retailer’s Contract Structure Preferences
We first investigate which contract structures are preferred by each player. Three sce-
narios are included in this section — (1) exogenous w and δ, (2) exogenous w and
61
supplier-selected δ, and (3) supplier-selectedw and exogenous δ.
Scenarios with exogenous w and δ
In Proposition 3.6, we show that the supplier weakly prefers Type-B contracts and
the retailer weakly prefers Type-A contracts unless the total promised supply under
Type-A is lower than that under the other two contract types. In Proposition 3.6, the
relationship “<” denotes a weak preference.
Proposition 3.6. For a fixed pair of (w, δ) values, the following statements are true.
1. The supplier’s preference ordering of contract structures is B < C < A.
2. If the total promised supply under structure A is at least as much as that under
contract structures B and C, then the retailer prefers A.
3. When structure A is unavailable, the retailer prefers C < B.
Proposition 3.6 can be explained by first observing that for the same total supply
commitment, the supplier’s profit is higher within a contract structure that induces
the retailer to order more up front. This is because higher initial purchase quantity
simultaneously increases initial sales revenue and reduces the need for fast-ship supply,
which can be costly to the supplier. Conversely, the retailer’s profit is higher when
a contract structure allows it to order slightly less up front without sacrificing supply
commitment, or else when a structure allows it to obtain a greater fast-ship supply
commitment for the same initial purchase quantity. From Proposition 3 and Part 1 of
Proposition 3.5, we observe that the retailer orders less when structure A is utilized,
regardless of supplier’s total commitment. Therefore, it is clear that structure A is the
least preferred structure for the supplier. Moreover, from Part 3 of Proposition 3.5, we
observe that when supply commitment is held the same, the retailer orders more under
structure B than structure C. This explains the preference ordering from the supplier’s
viewpoint.
62
We consider the retailer’s viewpoint next. If the total supply under structure A is no
less than the other two structures, it is clear that this would be the preferred structure
for the retailer because it can choose to order less up front. We also observe that the
retailer prefers structure C over B because it can secure greater supply commitment for
the same initial purchase quantity. Observe that when the choice is between structures
A and C, leadership in choosing contract parameters does not render an advantage to
the leader, as one might expect in two player interactions of this type. In fact, both the
supplier and the retailer prefer a contract in which they do not move first in such cases.
Note that when the value of p under structure A leads to a smaller total supply than
under structures B and C, the retailer may prefer contract structures B and C over A.
In other words, by choosing a small p, the supplier can make structure A unattractive
to the retailer.
Scenarios with Supplier-Selected δ and Exogenous w
When δ is chosen optimally, we first show that the supplier always choose δ = r −w in
all three structures.
Proposition 3.7. The supplier’s profit is increasing in δ in all three structures.
The reason behind Proposition 3.7 is as follows. The supplier earns a higher margin
from the initial order than that from fast-ship orders. Increasing δ leads to a higher
initial order quantity, which causes the supplier’s profit to increase. This result helps
obtain the supplier’s and the retailer’s contract preferences in the following proposition.
Proposition 3.8. When δ is chosen optimally by the supplier, the supplier and the
retailer are indifferent among the three contract structures.
Proposition 3.8 is given without a formal proof. Because the supplier always chooses
δ = r − w, the additional profit from fast-ship order is zero for the retailer. Therefore,
the retailer’s order decision is independent of α and ζji . As a result, the retailer’s
decisions, profit functions and corresponding profits for the three structures are the
63
same. Similar reasoning applies to the supplier as well. The supplier’s profits for the
three structures are also identical because the commitment does not affect the retailer’s
ordering decision.
Scenarios with Supplier-Selected w and Exogenous δ
Suppose that the supplier can optimally choose w within each contract structure. In
Proposition 3.6, we show that the supplier prefers structure B over C and C over A
for a fixed w. It is straightforward to prove that the same ordering will hold when
w can be chosen optimally by the supplier. However, the preference for the retailer
can change when w is optimally chosen in each structure. We observe two possibilities
upon performing many numerical experiments. The two possible orders are presented
via a numerical example reported in Table 3.1. Suppose that X is Gamma distributed
with E[X] = 225 and V ar(X) = 3375. Other problem parameters are r = 12, δ = 1,
τ1 = 0.1, τ2 = 2, c1 = 1, c2 = 10, and α ∈ {0.1, 0.5}.
α wA wB wC πAR πB
R πCR Preference
0.1 10.664 10.623 10.661 185.86 191.89 188.52 B ≻ C ≻ A0.5 11.000 11.000 11.000 131.37 131.37 131.37 A = B = C
Table 3.1: The Retailer’s Profit under the Optimal Wholesale Price
When α = 0.1, the retailer’s preference is identical to the supplier’s. This is because
the wholesale price for structure B is the lowest and the wholesale price for structure
A is the highest. When α = 0.5, the retailer’s profits under the three structures are
identical. This is because the supplier chooses a wholesale price such that r = w2 in
all three structures. Because the retailer earns zero profit from fast-ship orders when
r = w2, there is no difference among the three structures. In summary, when w is chosen
optimally, the retailer may either prefer structure B or be indifferent among the three
structures.
64
3.5.2 Contract Structure Selection
We show in Section 3.5.1 that among supplier-driven structures, the supplier prefers
B over A regardless whether w is exogenous or chosen by the supplier. Therefore, a
supplier will not select A so long as the option to select B is available. We also observed
that among structures B and C, the supplier prefers B, whereas the retailer prefers C
for a fixed w. This creates a potential conflict when the supplier lacks pricing power. In
this section, we discuss how such conflict may be resolved. Note that this conflict may
not exist when w is chosen by the supplier because both the supplier and the retailer
may prefer structure B in that case.
We discuss conflict resolution from two approaches. First, we show when there is a
dominant player, the other player can coffer a modified contract to increase its profit
without hurting the dominant player’s profit. Second, we show that when there is no
dominant player, the two parties can use a bargaining framework to decide the split
of profit between the two players such that both players’ profits are higher than their
disagreement profits.
Scenarios with a Dominant Player
If the channel has a dominant player, then a structure that maximizes its individual
profit is likely to be the only one selected. However, this may lead to a lower profit from
the other players. We argue next that this conflict can be resolved if either the supplier
or the retailer is willing to offer a modified contract and create a win-win situation. We
show below that such win-win resolution is always feasible.
Proposition 3.9.
1. There exists a z ≥ z∗ such that πBR (z) ≥ πC
R(qC) and πB
S (z) ≥ πCS (q
C).
2. There exists a q ≥ qC such that πBR(z
∗) ≤ πCR(q) and πB
S (z∗) ≤ πC
S (q).
The results in Proposition 3.9 can be explained as follows. If the retailer is the
dominant player and it strictly prefers Type-C contract, the supplier may offer a higher
65
supply commitment only if the retailer agrees to the choice of structure B and ensure
that the retailer earns a slightly higher profit under B than that under C. In addition,
the order quantity under the modified Type-B contract (accepted by the retailer) can
be proven to be higher than that under Type-C contract (arguments are similar to those
underlying Part 3 of Proposition 3.5). Therefore, the modified Type-B contract is still
a better choice for the supplier. Similarly, if the supplier is the dominant player, the
retailer can always find a q ≥ qC such that the modified Type-C contract is preferred
by both the supplier and the retailer. This happens because the supply commitment
under modified Type-C contract remains higher than z∗ (arguments are similar to Part
2 of Proposition 3.5).
In the following, we use an example to illustrate the results shown in Proposition 3.9.
Consider a case in whichX is Gamma distributed with E[X] = 400 and V ar(X) = 8000.
Other problem parameters are r = 12, w = 8, w2 = 10.5, τ1 = 0.1, τ2 = 2, c1 = 1,
c2 = 9, and α = 0.5. The results are shown in Table 3.2.
z qB(z) γ(qC) qC πBS (z) πB
R (z) πCS (q
C) πCR(q
C)
Original 70.5 348.5 81.0 346.0 2610.7 1265.8 2601.0 1267.8
R’s Offer – – 79.52 349.0 – – 2614.4 1267.6
S’s Offer 85 347.6 – – 2608.0 1268.6 – –
Table 3.2: An Example of Conflict Resolution by Providing Modified Contracts
The first row of Table 3.2 shows both parties’ profits and preferences when each
makes individually optimal decision. As shown in Proposition 3.6, the supplier prefers
B and the retailer prefers C. Next, in the second row, we compare the original Type-B
contract and the modified Type-C contract when the retailer commits to a higher-than-
optimal order quantity. We see that when the retailer sets a greater qC , both parties
prefer the modified C contract over B. Similarly, if the supplier offers a modified B
contract by choosing a higher z ≥ z∗, shown in row 3 of Table 3.2, then both prefer
modified B.
66
Scenarios with no Dominant Player
Suppose that there is no dominant player. Because the supplier’s profit is greater under
contract structure B and the retailer’s profit is greater under structure C, the disagree-
ment profits (minimum profit each party can earn) are πBR(z
∗) and πCS (q
C) for the
retailer and the supplier, respectively. We call these disagreement profits because the
retailer (resp. the supplier) can earn a minimum of πBR (z
∗) (resp. πCS (q
c)) by agreeing to
the selection of contract structure B (resp. C). Let πGT (q, y, j) denote the total supply
chain profit in a negotiated settlement for structure where i ∈ {B,C} and 0 < σS < 1
(resp. σR = 1 − σS) denote the supplier’s (resp. retailer’s) relative bargaining power.
The supplier’s and the retailer’s profits in a negotiated settlement for structure B can be
written as πGS (q, y, j) = σSπ
GT (q, y, j) and πG
R(q, y, j) = σRπGT (q, y, j), respectively. Be-
cause σS and σR do not depend on (q, y, j), maximizing individual profit in a negotiated
settlement contract is equivalent to maximizing πGT (q, y, j). That is, let
q = argmaxq
πGR(q, y, j), (3.17)
and
(y, j) = argmaxy,j
πGS (q, y, j). (3.18)
we obtain πGT (q, y, j) ≥ πG
T (q, y, j). Note that the values of (q, y, j) do not change in
σR. Also, because the only difference between structure B and C is the sequence of
decisions, it is easy to check that πGT (q, y, z) = πG
T (q, y, γ) and z = γ. A key result is
presented in the following proposition.
Proposition 3.10. There exists some (σR, σS) such that πGS (q, y, j) ≥ πC
S (qC) and
πGR(q, y, j) ≥ πB
R (z∗).
We show that both players can always find a set of (σR, σS) such that a negotiated
settlement contract generates a higher individual profit than each player’s disagreement
profit. Therefore, when there is no dominant player, a settlement contract can help
resolve the conflict. We demonstrate how such settlement works in a example. Using
67
parameters in the example provided in section with dominant player, we observe that
πGS (q, y, j) = 7218.7. Hence, any negotiated settlement contract with σS = [0.61, 0.7]
should be acceptable to the two players when there is no dominant player.
3.5.3 The Effect of Customer Participation Rate
In this section, we investigate the effect of customer participation rate through a series
of numerical examples.
Scenarios with Exogenous w and δ
We focus only on structures B and C because we argued in Section 3.5.2 that a supplier
would not select structure A. The parameters of the problem analyzed in the example
are as follows: X is Gamma distributed with E[X] = 400 and V ar(X) = 8000, r = 12,
w = 8, w2 = 11.5, τ1 = 0.1, τ2 = 3, and c2 = 9.
Structure B Structure C
c1 πBS (z
∗) πBR (z
∗) πCS (q
C) πCR(q
C)
1 ↑ ↑ ↑ ↑
3 ↑ ↓ ↑ ↓
5 ↑ ↓ ↓ ↓
Table 3.3: The Effect of Customer Participation Rate α
Table 3.3 reveals that, depending on the value of c1, the supplier’s and the retailer’s
expected profits may be either higher or lower as a function of the proportion of cus-
tomers willing to exercise the fast-ship option. In general, the supplier faces a greater
uncertainty when α is high. When c1 is small, the supplier is more willing to commit to
a greater amount of supply because the fast-ship option can be more profitable by utiliz-
ing advance procurement (y). Hence, both parties may benefit from having a higher α.
In contrast, when c1 is relatively large, the supplier makes less fast-ship supply available
due to the fact that advance procurement is less effective. In such cases, the retailer’s
68
profits can decline in α due to reduced supply availability. The supplier’s profit may
also decline because of a higher uncertainty.
Scenario with Supplier-Selected w
However, if w is chosen optimally within each structure, we observe that the supplier
always earns higher profit under a higher α whereas the retailer earns a lower profit
under a higher α regardless of the structure. This is because the supplier can charge
a higher price to increase the profitability for the fast-ship option when the customer
participation rate is higher.
3.6. Conclusions
In decentralized supply chains, retailers make stocking decisions to meet uncertain de-
mand. However, retailers often experience stockouts leading to costly opportunity loss
as well as future goodwill loss. To address this issue, some retailers may offer a fast-ship
option (directly ship out-of-stock items) to customers. Offering such an option can be
beneficial to both the supplier and the retailer because the total sales can increase. How-
ever, the incentives for the two players may be different since the provision of fast ship
reduces (resp. increases) inventory responsibility for the retailer (resp. the supplier).
In this chapter, we studied three different contract structures to provide insights
into the effect of different contract types and parameters values on each player’s per-
formance. We proved that structure A is dominated by B both from the supplier’s
perspectives. Therefore, we argued that a supplier will not offer Type-A contracts even
though they are preferred by the retailer. Among the remaining two structures, we
showed that the supplier prefers structure B whereas the retailer prefers structure C
with exogenous wholesale price and markup. Finally, our results show that there exist
cost parameters for which a higher customer participation rate can benefit both the sup-
plier and the retailer, even though that shifts more of the inventory from the retailer to
the supplier. The main contribution of the chapter lies in presenting a mathematically
69
rigorous framework for comparing different contract structures.
Chapter 4
Two-Retailer Structures
4.1. Introduction
In this chapter, we focus on the fast-ship option in a single-period setting with a single
supplier and two retailers. In a two-retailer supply chain, both the retailers and cus-
tomers have more options when experiencing a stockout. For example, one retailer may
use its excess inventory to support the other retailer’s fast-ship demand. Alternatively,
customers who experience stockout may decide to buy the item from the other retail
store. In other words, the supplier is no longer the only source of supply for fast-ship
orders.
Based on the distance between retailers and their relationship, we consider three
sourcing structures in this chapter: the independent mode (structure A), the alliance
mode (structure B), and the competing mode (structure C). Illustrations of the three
structures are shown in Figure 4.1.
In (A), the two retailers operate independently and obtain all fast-ship items from
the supplier. That is, each retailer first places an initial order before demand realization
and subsequent fast-ship orders from the same supplier if its inventory runs out. Un-
der structure A, each retailer’s ordering decision is independent of the other retailer’s
decision. This structure occurs when the two retailers are geographically far apart and
neither views the other as either a competitor or a partner in meeting market demand.
70
71
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183"
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143"
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4?0+59)/0"@=5":5"+5"+=)"5+=)/"/)+,-.)/"
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,'("*,/6B-*,+)"-'";<!"5*65'"
#% #%# #%##$#$ #$
Figure 4.1: The Three Sourcing Structures
In (B), the two retailers are aware of each other and choose to form an alliance that
allows one retailer to fast ship its leftover inventory to fulfill the other retailer’s fast-ship
demand and vice versa. Any unmet fast-ship demand that cannot be fulfilled by the
retailers can be satisfied by the supplier. Under this structure, the extra profits arising
from the alliance are allocated between the two retailers. This structure is usually seen
when the two retailers are not too close to each other and it is inconvenient for customers
to travel from one retailer to another. Under structure B, the two retailers’ decisions
are interdependent.
In structure C, the two retailers are located geographically close to each other and
customers can travel easily between the two retailers. That is, when customers expe-
rience stockout in one store, a fraction of customers who want the item immediately
choose to go to the other store and the rest do not make a purchase. If customers who
travel to the other store once again experience a stockout, then they place a fast-ship
order at the second store (i.e. the last store visited).
The main difference between structures B and C comes from the manner in which
profit from fast-ship orders is allocated between the retailers. When the two retailers
72
form an alliance (structure B), the extra profit from fast-ship orders is divided between
the retailers according to a predetermined allocation rule, regardless of which retailer
completes the sale. To ensure that the alliance in structure B is stable and efficient,
we assume that this allocation rule is based on Shapley value (Shapley 1953). Shapley
value has been used in many previous papers on supply chain management (Sosic 2006).
We choose Shapley value not only because of ease of calculation, but also because its
monotonicity properties ensure that a player who contributes more to profit receives a
greater portion of allocation.
In this chapter, we develop mathematical models to address the following questions:
• In a two-retailer chain, do retailers earn higher profits if they form an alliance?
• Which structure is preferred by the supplier, and by the retailers?
• How does customer participation rate affect the supplier’ and the retailers’ profits?
We characterize the supplier’s and the retailers’ problems and show that a unique pure
strategy Nash equilibrium exists for the retailers’ ordering decisions. We show that the
supplier’s profit can be higher under a higher customer participation rate if prices are
exogenous and the fast-ship markup is high. Otherwise, the supplier’s profit can be
lower under a higher customer participation rate. Also, when prices are exogenous, the
retailers earn higher profits under a higher participation rate.
However, when the wholesale price is chosen by the supplier, the supplier always
earns a higher profit under a higher customer participation rate. The results for the
retailers are mixed. In general, in this case, the retailers’ profits are decreasing in
customer participation rate. When the supplier’s wholesale price reaches its upper
bound (i.e. a wholesale price that makes the retailers’ marginal benefits from supplier-
supported fast-ship orders zero), the retailers’ profit can be increased by competing or
cooperating with each other. This usually happens when customer participation rate is
high.
73
Also, we observe that the supplier prefers structure A over C and C over B when
customer participation rates are the same under the three structures. The former is
caused by the fact that the supplier receives a greater portion of fast-ship demand in A
as compared to C whereas the latter comes from the fact that the retailers tend to order
more when they compete than when they form an alliance. For the retailers, we observe
that both structures B and C can generate greater profits than A with exogenous prices.
Whether B or C is better depends on parameters, and structure C is usually better than
B when markup is high. When wholesale price is chosen by the supplier, we observe
that structure C is more profitable than either B or A if the shipping cost for fast-ship
orders is high. This is because the supplier tends to charge a higher wholesale price in
structure B than in C in such cases.
Related Literature
As mentioned in previous chapters, our work is related to papers that involve more than
one replenishment opportunity; some of this literature focuses on updating the demand
distribution or improving cost estimates upon getting additional (but incomplete) infor-
mation after the first replenishment. For example, Eppen and Iyer (1997a,b), Gurnani
and Tang (1999), and Donohue (2000) belong to this stream. After the new information
becomes available, the second order is used to reduce the supply-demand mismatch. In
contrast, in our model, fast-ship orders are designed to serve customers who are will-
ing to wait for out-of-stock items. Therefore, the fast-ship orders are placed after the
demand uncertainty is completely resolved.
Papers such as Cachon (2004) and Dong and Zhu (2007) are related to our work
because the second order is place at the end of the selling season. However, these
papers do not model consumer responses and assume that the entire excess demand
(up to available supply) is the size of the second order. In our setting, only a fraction
of customers would participate in the fast-ship option. In addition, the supplier in our
setting has a second opportunity to procure additional items to satisfy unmet fast-ship
74
demand at a higher cost. This option is not provided in Cachon (2004) and Dong and
Zhu (2007).
Our structure B is also related to works on transshipment between retailers. For
example, Tagaras (1989), Herer and Rashit (1999) and Dong and Rudi (2004) study
transshipment under a centralized control. Anupindi et al. (2001), Rudi et al. (2001),
Granot and Sosic (2003), and Sosic (2006) study transshipment when each retailer makes
individually optimal ordering decisions. The two-retailer alliance model in this chapter
belongs to the latter category.
Some papers assume that the transshipped item is sold by the sending retailer to
the receiving retailer at a fixed transshipment price and the others assume that the
additional profit is divided based on some allocation rules. For example, Rudi et al.
(2001) belongs to the former group and the profit allocation between two retailers is
based on a transshipment price chosen either by the retailer with surplus, or the retailer
with shortage, or negotiated between the two players. It shows that a transshipment
price can be chosen to achieve joint-profit maximum.
Many papers, including structure B in our paper, assume that profit is distributed
according to an allocation rule instead of setting a transshipment price. Anupindi et al.
(2001) study a two-stage distribution problem with n retailers. In the first noncoop-
erative stage, each retailer places an order to satisfy its own demand. In the second
cooperative stage, retailers transship items to satisfy unmet demand and allocate ad-
ditional profit. Anupindi et al. (2001) shows that an allocation rule based on a dual
solution can induce retailers’ decisions to coincide with those in a centralized system.
Granot and Sosic (2003) introduces a three-stage system (in the third stage, each
retailer decides how much to share) and shows that an allocation rule based on a dual
solution may not provide sufficient incentive to retailers to share all excess inventory.
In contrast, a monotone allocation (e.g., Shapley value or fractional allocation) does
provide such an incentive and maximizes additional profit from transshipment. Sosic
75
(2006) further shows that such monotone allocation rules are stable for farsighted re-
tailers even though they do not belong to the core (the set of feasible allocations that
cannot be improved upon by a coalition formed by a subset of the players.). A farsighted
retailer considers entire sequence of the chain reactions by other players when it makes
its decisions. Therefore, although some alternatives’ immediate payoff may be higher, a
farsighted retailer does not change its decision because the long-run consequences may
make it worse off.
Our setting with a retailer alliance is similar to Anupindi et al.’s (2001) two-stage
distribution problem. That is, we assume that retailers agree to share all excess in-
ventory. Consistent with this assumption, we use Shapley value as the allocation rule
because it provides incentive to maximizes profit from fast-ship orders as explained in
the previous paragraph. In addition, the supplier’s decisions are also considered in our
models. Instead of identifying allocation rules that achieve total profit equal to that
in a centralized chain, we focus on how alliance affects different players’ profits under
different sourcing modes.
In summary, our alliance model (structure B) and previous works on transshipment
are different in the following ways. First, because only a fraction of customers par-
ticipate in the fast-ship option in our models, the retailers’ problems are technically
more challenge to solve in our setting. Second, all unmet demand after transshipment
is satisfied by the supplier in our models. As a result, the supplier still bears some
inventory/procurement risk when retailers form a coalition. Finally, instead of studying
benefits of transshipment from each player’s perspective, we investigate how different
sourcing modes affect the retailers’ and the supplier’s profits.
Structure C formulation is related to papers that model competing newsvendors.
In these models, a fraction of customers who experience stockout try to get the item
from a different store. Depending on the demand model, these paper can be divided
into two categories. In the first category, demands for two retailers are allocated from
a random market demand. Paper of this kind includes Lippman and McCardle (1997),
76
Nagarajan and Rajagopalan (2009), and Caro and Martinez-de Albeniz (2010). In Na-
garajan and Rajagopalan (2009), the demand for each retailer is allocated randomly
from a deterministic demand d. The paper shows that with reasonable cost parameters,
the equilibrium inventory level can be sufficiently high and competition may be ignored.
Lippman and McCardle (1997) and Caro and Martinez-de Albeniz (2010) use a differ-
ent approach. They assume that the market demand D is random but the allocation
between the two retailers is deterministic. In those paper, the existence of a unique pure
strategy Nash equilibrium can be shown under some conditions. For example, Lippman
and McCardle (1997) requires symmetric retailers.
In the second category, the two retailers face independent demands (see, for example,
Parlar 1988 and Avsar and Baykal-Gursoy 2002). In Parlar (1988), the author studies
a single period problem with two independent retailers. Our model also belongs to
this category. That is, retailer-1 faces random demand D1 and retailer-2 faces random
demand D2. Avsar and Baykal-Gursoy (2002) extends the model to an infinite horizon
problem. In these papers, a unique pure strategy Nash equilibrium is shown to exist
when commutative demand is strictly increasing.
Two model features make our work substantially different from the above-mentioned
papers. First, we include the supplier in our analysis and compare different sourcing
structures for the retailers and suppliers. Second, the retailers in our model can obtain
additional replenishments from the supplier at a possibly higher price if their inventory
is not enough to support excess demand.
The rest of this chapter is organized as follows: We provide model formulation in
Section 4.2. In section 4.3, we present the supplier’s and the retailers’ optimal decisions
for all three structures. In Section 4.4, we compare different sourcing modes and analyze
the effect of participation rate. Section 4.5 concludes this chapter.
77
4.2. Model Formulation
In this section, we model a supply chain with two retailers (denoted by Ri, i ∈ {1, 2})
and a single supplier (denoted by S). Both Ri and S are risk-neutral decision makers. Ri
faces independently random demand Xi ∈ R+ with density and distribution functions
fi(·) and Fi(·), respectively. We assume fi(·) > 0 over the support of Xi to avoid cases
in which each retailer has multiple optima for its decision problem.
Each retailer sells products at a unit retail price r. The shipping costs for regular
orders and fast-ship orders are τ1 and τ2, respectively, which are paid by the source of
the supply to a third party logistics provider. The supplier also has two production
opportunities and it’s production costs are c1 for the initial order and c2 for fast-ship
orders that cannot be fulfilled from the first production lot, where c2 ≥ c1. The wholesale
prices are w and w2 = w + δ for initial orders and fast-ship orders, respectively, where
w and δ are exogenous. We also study how our results might change if either δ or w
were a decision variable for the supplier. Due to the complexity of the problem, we are
unable to derive results concerning the effect of the choice of w and δ on the supplier’s
and the retailers’ profits. Therefore, such comparisons are mostly carried out with the
help of numerical examples.
The sequence of decision is as follows. Upon knowing w and w2, Ri places an initial
order of size qi and S produces an amount (y+q1+q2) in response to the retailers’ order
decisions. Finally, the demand is realized and the total fast-ship demand generated
from Ri is αi(Xi − qi)+, where αi is the fraction of customers who wish to obtain the
item immediately. Parameter αi is also referred to as retailer-i customer participation
rate. All fast-ship demand in structure A is supported by S. However, in structures B
and C, S only fulfills partial fast-ship demand because a portion of fast-ship demand
can be satisfied by the other retailer. Note that S must purchase/produce items at a
higher cost c2 if there are any orders that cannot be satisfied by y. Hereafter, we use
πkRi
and πkS to denote the retailers’ and the supplier’s expected profits under structure
k ∈ {A,B,C}, respectively.
78
4.2.1 Structure A – Two Independent Retailers
We first present the model for structure A in which the two retailers in the supply chain
are unaware of each other and act independently. In structure A, all fast-ship orders are
supplied by S with price w2. Therefore, Ri’s expected profit can be written as follows
for any given q.
πARi(q) = rE[min(Xi, q)]− wq + αi(r − w2)E(Xi − q)+. (4.1)
Here, each retailer’s problem is to choose a qAi such that
qAi = argmaxq
πARi(q).
When the two retailers order q1 and q2, respectively, the supplier’s expected profit
is
πAS (y, q1, q2) = −c2E[α1(X1 − q1)
+ + α2(X2 − q2)+ − y]+ + (w − τ1 − c1)(q1 + q2)
−yc1 + (w2 − τ2)[α1E(X1 − q1)+ + α2E(X2 − q2)
+], (4.2)
where (w− τ1− c1)(q1+ q2) is the profit from regular order, yc1 is the cost of producing
extra items in advance, (w2 − τ2)[α1E(X1 − q1)+ + α2E(X2 − q2)
+] is the revenue
from fast-ship orders, and c2E[α1(X1 − q1)+ + α2(X2 − q2)
+ − y]+ is the additional
procurement cost of fulfilling fast-ship demand. Knowing qA1 and qA2 , the supplier’s
problem is to choose a yA such that
yA = argmaxy
πAS (y, q
A1 , q
A2 ).
4.2.2 Structure B – Two-Retailer Alliance
When the two retailers form an alliance, each retailer’s fast-ship demand is first fulfilled
by the other’s excess inventory. Any remaining unmet fast-ship demand is then satisfied
by the supplier. As a result, when the two retailers order qi and qj, where j = {1, 2}\i,
respectively, the expected profit for Ri is
πBRi(qi, qj) = rE[min(qi,Xi)]− wqi + E[φP
i (qi, qj)] + (r − w2)E[αi(Xi − qi)+]. (4.3)
79
Note that the retailer earns (r −w2) for each fast-ship order supported by the supplier
and earns (r−w2) as well for each fast-ship order supported by the other retailer before
receiving allocated profit from the alliance. Hence, (r−w2)E[αi(Xi−qi)+] in (4.3) is the
minimum profit from fast-ship demand. E[φPi (·)] denotes the expected profit allocation
that player-i receives when a set of players P are in an alliance. The allocation rule
for calculating φPi (·) is based on Shapley value (Shapley 1953), which is defined as the
average marginal contribution for all possible orderings (each ordering is a particular
sequence in which the retailers join the alliance). Because there are two retailers in
our problems, only two orderings are possible — either R1 joins after R2 or vice versa.
Therefore, the allocation for Ri can be calculated from follows.
φ(1,2)i (qi, qj) =
(v{i}(qi, qj)− v{∅}(qi, qj)) + (v{i,j}(qi, qj)− v{j}(qi, qj))
2. (4.4)
In (4.4), v{i,j}(qi, qj) denotes the profit for fast-ship sales when both Ri and Rj are
in the coalition and v{i}(qi, qj) denotes the profit when Ri is the only player in the
alliance. In other words, (v{i}(qi, qj) − v{∅}(qi, qj)) is the marginal contribution to the
supply chain when Ri joins the coalition first and (v{i,j}(qi, qj) − v{j}(qi, qj)) is the
marginal contribution to the supply chain when Ri joins the coalition later. Based on
the definition of v{i,j}(qi, qj), we obtain
v{i,j}(qi, qj) = (w2 − τ2){
[αi(xi− qi)+ ∧ (qj −xj)
+] + [αj(xj − qj)+ ∧ (qi−xi)
+]}
. (4.5)
In (4.5), v{1,2}(q1, q2) > 0 only when one retailer has excess inventory to fully or partially
meet the other retailer’s fast-ship demand. In contrast, v{1,2}(q1, q2) = 0 when the two
retailers both have either shortage or overage at the same time. In addition, because the
extra benefit for fast-ship orders without an alliance is 0 regardless of order quantities,
v{i}(qi, qj)− v{∅}(qi, qj) = 0 and therefore, φ(1,2)i (qi, qj) = v{i,j}(qi, qj)/2.
80
With the allocation rule shown in (4.4) and (4.5) on hand, we obtain
E[φi(qi, qj)] =w2 − τ2
2
{
E[αi(Xi − qi)+ ∧ (qj −Xj)
+] + E[αj(Xj − qj)+ ∧ (qi −Xi)
+]}
=w2 − τ2
2
{∫ ∞
0P (αi(Xi − qi) ≥ z)P (qj −Xj ≥ z)dz
+
∫ ∞
0P (αj(Xj − qj) ≥ z)P (qi −Xi ≥ z)dz
}
=w2 − τ2
2
{∫ qj
0Fi(qi +
z
αi)Fj(qj − z)dz +
∫ qi
0Fj(qj +
z
αj)Fi(qi − z)dz
}
.
(4.6)
Note that other allocation rules can also be applied to our problems. However, sev-
eral properties make Shapley value more desirable for our settings. First, this allocation
rule is efficient because∑
i∈PφPi (·) = vP(·), where P ⊆ A and A is the set of all players.
Second, if any two players i, j /∈ P are equivalent such that vP⋃{i}(·) = vP
⋃{j}(·), then
the allocation for these two player are equal (i.e., φP
⋃{i}
i (·) = φP
⋃{j}
j (·)). Finally, in
the coalition, each retailer earns at least as much as that when it is not in the coalition
(i.e., φPi (·) ≥ v{i}(·)).
For structure B, Ri anticipates Rj ’s optimal order quantity qBj and select a qBi such
that
qBi = argmaxq
πBRi(qi, q
Bj ).
Similarly, when the two retailers order q1 and q2, respectively, the supplier’s expect
profit function is
πBS (y, q1, q2) = (w−τ1−c1)(q1+q2)−c1y+(w2−τ2) (E[ς1] + E[ς2])−c2E (ς1 + ς2 − y)+ ,
(4.7)
where (w2−τ2) (E[ς1] + E[ς2]) is the revenue from fast-ship demand and c2E (ς1 + ς2 − y)+
is the additional cost of procuring fast-ship demand. S’s problem is to anticipate (qB1 , qB2 )
and choose a yB such that
yB = argmaxy
πBS (y, q
B1 , q
B2 ).
81
4.2.3 Structure C - Two Competing Retailers
The model formulation for a supply chain with two competing retailers is as follows. A
customer who experiences a stockout in one store will travel to get the item from the
other retailer. If the item is not available in the other store, then the customer will
place a fast-ship order from the second store. Note that the main difference between
structure B and structure C is that the profit goes to the retailer who completes the
sale in structure C whereas the profit is allocated based on Shapley value in structure
B. When the two retailers order qi and qj, where j = {1, 2}\i, respectively, the expected
profit for Ri is
πCRi(qi, qj) = rE[min(qi,Xi)]−wqi+rE[αj(Xj−qj)
+∧(qi−Xi)+]+(r−w2)E[ςj ]. (4.8)
In (4.8), rE[αj(Xj − qj)+ ∧ (qi −Xi)
+] is the sales revenue from customers who experi-
enced a stockout upon visiting store j first. Ri anticipates Rj ’s optimal order quantity
qCj and then choose a qCi such that
qCi = argmaxq
πCRi(qi, q
Cj ).
The supplier does not see the difference between B and C in terms of profit structure.
Therefore, when the two retailers order q1 and q2, the supplier’s expect profit function
is
πCS (y, q1, q2) = πB
S (y, q1, q2). (4.9)
S’s problem is to anticipate (qC1 , qC2 ) and choose a yC such that
yC = argmaxy
πCS (y, q
C1 , q
C2 ).
With the retailers’ and the supplier’s expected profit functions on hand, we obtain
their optimal decisions in the next section.
82
4.3. Supplier’s and Retailers’ Operational Choices
4.3.1 Retailers’ Ordering Decisions
Structure A
From (4.1), we observe that πARi(q) is concave in q. As a result, the optimal order
quantity qBi can be obtained by setting ∂πARi(q)/∂q to 0. That is,
qAi = F−1i
(
w
r − αi(r − w2)
)
. (4.10)
Based on qAi shown in (4.10), we can further show that qAi is decreasing in both w and
αi. This is reasonable because in either case, retailers may depend more on fast-ship
orders to meet their customers’ demands.
Structure B
We are ready to solve the retailers’ problems when the two retailers form an alliance.
It is easy to show that πBRi(qi, qj , w) is concave in qi if allocated profits E[φ
(1,2)i (qi, qj)]
were concave in qi. However, the functions E[φ(1,2)i (qi, qj)] in (4.6) are neither concave or
convex in qi. As a result, to find the optimal qi, we first need to prove that πBRi(qi, qj , w)
are well-behaved in qi.
By substituting (4.6) in (4.3) and taking derivatives of πBRi(qi, qj, w) with respect to
qi, we obtain
∂πBRi(qi, qj, w)
∂qi=w2 − τ2
2
[
−
∫ qj
0fi(qi +
z
αi)Fj(qj − z)dz +
∫ qi
0Fj(qj +
z
αj)fi(qi − z)dz
]
− w + (r − αi(r − w2))Fi(qi), (4.11)
and
∂2πBRi(qi, qj , w)
∂q2i=− (r − αi(r −w2))fi(qi) +
(w2 − τ2)
2
[
−
∫ qj
0f ′i(qi +
z
αi)Fj(qj − z)dz
+ Fj(qj +qiαj
)fi(0) +
∫ qi
0Fj(qj +
z
αj)f ′
i(qi − z)dz]
. (4.12)
83
Next,
∫ qj
0f ′i(qi +
z
αi)Fj(qj − z)dz = −αifi(qi)Fj(qj) + αi
∫ qj
0fi(qi +
z
αi)fj(qj − z)dz
= −αifi(qi)Fj(qj) + αi
∫ qj
0fi
(
qi +qj − xj
αi
)
fj(xj)dxj ,
where the first equality comes from integration by parts and the second comes from
change of variables xj = qj − z. Similarly,
∫ qi
0Fj(qj +
z
αj)f ′
i(qi − z)dz = Fj(qj)fi(qi)−1
αj
∫ qi
0fi(qi − z)fj(qj +
z
αj)dz
−Fj(qj +qiαj
)fi(0)
= Fj(qj)fi(qi)−
∫ ∞
qj
fi(qi − αj(xj − qj))fj(xj)dxj ,
−Fj(qj +qiαj
)fi(0)
where the first equality comes from integration by parts and the second comes from
change of variables xj = qj + y/αj and the fact that f(x)=0 for x < 0. Using the
equalities above, we can rewrite (4.12) as
∂2πBRi(qi, qj)
∂q2i= −(r − αi(r − w2))fi(qi) +
w2 − τ22
[
(1− (1− αi)Fj(qj))fi(qi)
− αi
∫ qj
0fi
(
qi +qj − xj
αi
)
dFj(xj)−
∫ ∞
qj
fi (qi − αj(xj − qj)) dFj(xj)]
< −(r −w2 − τ2
2− αi(r − w2))fi(qi) +
w2 − τ22
[
− αi
∫ qj
0fi
(
qi +qj − xj
αi
)
dFj(xj)−
∫ ∞
qj
fi (qi − αj(xj − qj)) dFj(xj)]
,
(4.13)
where the inequality comes from that (1 − (1 − αi)Fj(qj))fi(qi) ≤ fi(qi). Because
f(·) > 0, it follows that equation (4.13) that ∂2πBRi(qi, qj)/∂q
2i < 0. Therefore, πB
Riis
strictly concave in qi and a unique optimal qi exists for any given qj. Let (qB1 , q
B2 ) be a
pure strategy Nash equilibrium for retailers. qBi can be obtained by setting (4.11) to 0.
84
That is, (qB1 , qB2 ) must satisfy the following equalities simultanuously.
w = (r − α(r − w2))Fi(qBi ) +
w2 − τ22
[
−
∫ qBj
0fi(q
Bi +
z
αi)Fj(q
Bj − z)dz
+
∫ qBi
0Fj(q
Bj +
z
αj)fi(q
Bi − z)dz
]
, (4.14)
where (i, j) = {(1, 2), (2, 1)}.
Proposition 4.1. A unique pure strategy Nash equilibrium (qB1 , qB2 ) exists for structure
B.
Proposition 4.1 can be proved by showing that the best response correspondences
for the two retailers have exactly one intersection. The uniqueness of Nash equilibrium
is useful when analyzing differences in profits across supply modes.
Structure C
By following similar steps that lead to equation (4.6), we observe that E[αj(Xj − qj)+∧
(qi −Xi)+] =
∫ qi0 Fj(qj +
zαj)Fi(qi − z)dz. Therefore, from (4.8), we obtain
∂πCRi(qi, qj)
∂qi= −w + rFi(qi) + (r − αj(r − w2))
∫ qi
0Fj(qj +
z
αj)fi(qi − z)dz
(4.15)
and
∂2πCRi(qi, qj)
∂q2i= −rfi(qi) + (r − αj(r − w2))[Fj(qj +
qiαj
)fi(0)
+
∫ qi
0Fj(qj +
z
αj)f ′
i(qi − z)dz]
= −rfi(qi) + (r − αj(r − w2))[Fj(qj)fi(qi)
−
∫ ∞
qj
fi(qi − αj(xj − qj))dFj(xj)],
(4.16)
where the second equality comes from integration by parts and change of variable xj =
qj − z. From (4.16), we observe that ∂2πCRi(qi, qj)/∂q
2i < 0 and therefore, πC
Riis strictly
85
concave in qi. Let (qC1 , q
C2 ) be the Nash equilibrium order quantities under structure C.
We obtain (qC1 , qC2 ) from the following equalities.
w = rFi(qCi ) + (r − α(r − w2))
∫ qCi
0Fj(q
Cj +
z
αj)fi(q
Ci − z)dz, (4.17)
where (i, j) = {(1, 2), (2, 1)}.
Proposition 4.2. A unique pure strategy Nash equilibrium (qC1 , qC2 ) exists for structure
C.
Next, suppose that the two retailers are identical. In Proposition 4.3, we show that
the initial order quantity in structure C is the highest among the three structures.
Proposition 4.3. When the two retailers are identical, qCi > qAi and qCi > qBi for each
fixed w and δ.
This result is reasonable because when customers experience a stockout in one re-
tailer’s store under structure C, they either go to the other retailer’s or forego making a
purchase. As a result, each retailer orders more up front because it cannot turn excess
demand into sales in structure C.
4.3.2 Supplier’s Ordering Decisions
Structure A
From (4.2), we observe that πAS (w, y, q1, q2) is concave in y for any given (w, q1, q2).
However, the optimal y can be obtained by setting ∂πAS (w, y, q1, q2)/∂y to 0. That is,
yA = max(0, ηA), where ηA must satisfy the following equality.
c1c2
= F2(q2)F1
(
q1 +ηA
α1
)
+F1(q1)F2
(
q2 +ηA
α2
)
+
∫ ∞
q2
F1
(
q1 +ηA − α2(x− q2)
α1
)
dF2(x).
(4.18)
Structure B
From (4.7), we also observe that πBS (y, q1, q2) is concave in y for each fixed (q1, q2).
Similar to structure A, the optimal y can be obtained by setting ∂πBS (w, y, q1, q2)/∂y to
86
0. That is, yB = max(0, ηB), where ηB must satisfy the following equality.
c1c2
=
∫ q2
0F1(q1 +
ηB + q2 − x
α1)f2(x)dx+
∫ q1
0F2(q2 +
ηB + q1 − x
α2)f1(x)dx
+
∫ ∞
q2
F1
(
q1 +ηB − α2(x− q2)
α1
)
dF2(x). (4.19)
Structure C
For the supplier, there is no difference between structures B and C except that qBi and
qCi may be different for each fixed wholesale price w and δ. As a result, yC = max(0, ηC),
where ηC must satisfy equality
c1c2
=
∫ q2
0F1(q1 +
ηC + q2 − x
α1)f2(x)dx+
∫ q1
0F2(q2 +
ηC + q1 − x
α2)f1(x)dx
+
∫ ∞
q2
F1
(
q1 +ηC − α2(x− q2)
α1
)
dF2(x). (4.20)
4.4. Insights
In this section, we compare the supplier and the retailers’ performances as well as the
effect of customer participation rate under different sourcing modes. This helps identify
which sourcing mode is more profitable for each player. It also helps identify under what
conditions, each player earns a higher profit under a higher customers participation level.
Hereafter, we assume that the two retailers are identical and omit the indices i and j
when the meaning is clear from the context.
4.4.1 The Effect of Customer Participation Rate
The effect of customer participation depends on the values of w and δ. We present results
for three scenarios — (1) exogenous w and δ, (2) supplier-selected δ and exogenous w,
and (3) supplier-selected w and exogenous δ.
87
Scenarios with Exogenous w and δ
When w and δ are exogenous, we observe that the retailers earn a higher profit under
a higher α. However, the supplier’s profit may not be monotone in α. We illustrate
these through an example. We set Xi ∼ U(0, 150). Other parameters are r = 15, c1 =
0.9, c2 = 9, δ = 1, τ1 = 0.1, τ2 = 2, and α = [0.05, 0.95]. The results are shown in Table
4.1, where we use arrows to denote increasing/decreasing trend (e.g., ↑/ ↓ implies that
the profit first increases then decreases).
Structure A Structure B Structure C
(w, δ) Supplier Retailer Supplier Retailer Supplier Retailer
(5, 5) ↓/↑ ↑ ↓ ↑ ↑/↓ ↑
(5, 8) ↑ ↑ ↓ ↑ ↑ ↑
(12, 14) ↑ ↑ ↑ ↑ ↑ ↑
Table 4.1: The Effect of Customer Participation Rate α
The retailers’ profits are increasing α. This is because the retailers can receive a
greater fast-ship demand or sell excess inventory to customers from the other retailer
more easily when α is high. Since the fast-ship order is profitable for the retailers,
they earns a higher profit under a higher α. Note that the retailers may order less up
front with a higher α. This hurts the supplier, especially in structure B or C, in which
the supplier only receives a small portion of total fast-ship demand. For structure A,
although the supplier gets all fast-ship demand, it may be worse off under a higher α
when δ is low. This is because the revenue from fast-ship orders is smaller when δ is
small.
Scenarios with Exogenous w and Supplier-Selected δ
When δ is chosen by the supplier, we observe that the supplier sets δ = r − w. This is
not provable in this chapter because we do not have closed form solutions for optimal
y and qi in all three structures. However, the reason behind this result is similar to
88
what we observed in Chapter 2 and 3. Because the marginal profit from the initial
order is higher than that from fast-ship orders and qi is increasing in δ, the supplier
earns a higher profit for a greater δ. As a result, the effect of customer participation
rate with optimal δ is similar to that with exogenous w and δ. However, there are
two minor differences. First, the retailers’ profits under structure A is independent of
α because profit from fast-ship orders is zero for the retailer. Second, the supplier’s
profit is increasing in α. We are not able to find a case such that the supplier’s profit
is decreasing in α because when δ is chosen optimally, supporting fast-ship orders is
profitable for the supplier.
Scenarios with Supplier-Selected w and Exogenous δ
We next show that the supplier earns a higher profit under a higher α when w is
chosen by the supplier. We observe that each retailer’s profit is decreasing in α under
structure A. However, under structure B or C, each retailer’s profit first increases then
decreases as α becomes higher. We demonstrate the results through an example. We
set Xi ∼ U(0, 150), r = 10, c1 = 3, c2 = 6, δ = 1, τ1 = 0, and τ2 = 1. The results are
shown in Figure 4.2.
In Figure 4.2(a), we observe that the supplier earns a higher profit from a higher
α for structures A, B, and C. When α is higher, it means that the supplier is more
likely to receive a higher fast-ship demand. As a result, the supplier chooses a higher
wholesale price and because a higher w also implies a higher w2, this move makes the
supplier earn a even higher profit from fast-ship orders when α is high.
In Figure 4.2(b), we observe that each retailer’s profit is decreasing in α under
structure A because the supplier charges a higher w when α is higher. However, when
α becomes even higher, R’s profit stops decreasing. This is because in these range of α,
w is set such that w = r− δ, which makes R’s profit independent of α. In structures B
and C, R’s profit first increases then decreases as α becomes higher. These results can
be explained as follows. When α is in a lower range, R’s profit decreases in α because
89
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
400
600
800
1000
1200
1400
Customer Participation Rate
The
Sup
plie
r’s E
xpec
ted
Pro
fit
πAS
πBS
πCS
(a) The Supplier’s Expected Profits
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
40
50
60
70
80
90
Customer Participation Rate
The
Ret
aile
r’s E
xpec
ted
Pro
fit
πARi
πBRi
πCRi
(b) The Retailer’s Expected Profits
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9400
600
800
1000
1200
1400
1600
Customer Participation Rate
The
Sup
ply
Cha
in E
xpec
ted
Pro
fit
A
B
C
(c) The Supply Chain’s Expected Profits
Figure 4.2: The Effect of Customer Participation Rate
w is increasing in α. When α is in a higher range, in order to keep w ≤ r− δ, S cannot
increase w any further. As a result, R also benefits from a higher fast-ship demand.
For similar reasons, the supply chain’s expected profit may not be monotone in
customer participation rate in structures B and C (see Figure 4.2(c)).
4.4.2 Performance Comparisons
The Supplier’s Profit
In this section, we compare the performance of the three structures in terms of expected
profit from the supplier’s and the retailers’ perspectives. Note that to emphasize how
the supplier’s and the retailer’s performances are affected by the shipping cost and
90
markup price, we assume that customer participation rates for the three structures are
identical, though it might be different in practice. In Proposition 4.4, we show that
the supplier’s profit is higher when facing two competing retailers as compared to two
cooperative retailers. This result holds either when w and δ are exogenous or when one
of these parameters is chosen by the supplier.
Proposition 4.4. The supplier’s expected profit for structure C is higher than that for
structure B.
The results shown in Proposition 4.4 is intuitive and can be explained as follows.
First consider cases in which w and δ are exogenous. Because the retailers orders more
in structure C as compared to structure B for the same wholesale price (see Proposition
4.3) and the order quantity in structure C is decreasing in w, we know that the supplier
can charge a higher price under structure C and still obtain the same amount of initial
order size and fast-ship demand. Hence, the supplier’s profit is higher for structure C
as compared to B when wholesale prices are chosen by the supplier. Because structure
C generates a higher profit than B for a fixed w and δ, the same result must holds when
either w or δ is chosen by the supplier.
The comparison between A and B or A and C is done numerically. Intuitively, we
think that the supplier may sometimes earn a higher profit under structures B and C
as compared to A because the initial order size can be greater under structures B and
C than that in A. In addition, although the supplier receives all fast-ship demand from
the retailers under structure A, it may make the supplier worse off when w2 − τ2 < c2.
However, numerical examples show that this intuition may not be true. When we set
Xi ∼ U(0, 150), r = 15, c1 = [1, 3], τ1 = 0.1, and vary c2 ∈ [c1, 10], α ∈ [0.05, 0.95],
δ ∈ [0, 7], and τ2 ∈ [0.5, 7], we observe that the supplier’s profit for structure A is always
the highest among the three structures because it has the largest fast-ship demand
regardless of whether w and δ are exogenous or chosen by the supplier.
91
The Retailers’ Profits
The comparison of the retailer’s profit among the three structures depends on parame-
ters. We provide some insights from each scenario.
Scenarios with Exogenous w and δ
When w and δ are exogenous, we show in the following proposition that structure A
generates the lowest profit among the three contracts for retailers.
Proposition 4.5. For a fixed w, each retailer’s expected profit in structure A is the
lowest among the three structures.
The results show in Proposition 4.5 can be explained as follows. Because the ad-
ditional profit from alliance is allocated using Shapley value, the retailer earns r −
(w2 + τ2)/2 from serving fast-ship orders from the other retailer, which is higher than
r − w2, the revenue from fast-ship orders from the supplier. In addition, the retailer
earns (w2 − τ2)/2 for each unit of excess inventory fast shipped to the other retailer in
B instead of salvaging at 0 in A. These two reasons make structure B more profitable
than A for a fixed w. For the second part of the proposition, the retailers can earn a
higher profit under structure C than that in A because the retailer is able to get extra
demand (from the other retailer) regardless of its inventory level.
The ordering between structure B and C depends on parameters. We discuss it
through numerical examples. Suppose that Xi ∼ U(0, 150), r = 15, w = 12, c1 =
1, c2 = 9, α = 0.35, τ1 = 0.1 and τ2 = 2. When we vary α ∈ [0.45, 0.95] and δ ∈ [0, 3],
the results are shown in Figure 4.3.
In Figure 4.3, we observe that the retailers prefer structure C to B when δ is high.
This is because when δ is high, the retailers’ marginal benefit for fast-ship order is lower.
Hence, the retailers find it more attractive to compete because selling items to customers
from the other retailer in structure C gives rise to a greater margin. However, when δ
is low, the retailers’ marginal benefit for fast-ship order is higher. Therefore, structure
92
0.5 0.6 0.7 0.8 0.90
0.5
1
1.5
2
2.5
3
Customer Participation Rate
Who
lesa
le P
rice
Mar
kup
πBR < πC
R
πCR < πB
R
Figure 4.3: The Retailer’s Profit Comparison
B can be more attractive to the retailers because they can earn additional profits when
either shortage or overage occurs.
Scenarios with Exogenous w and Supplier-Selected δ
When δ is chosen by the supplier, we argued earlier that the supplier would set δ = r−w.
Therefore, this case is a special case for scenarios with exogenous w and δ with δ = r−w.
Hence, all results in the previous section apply to this section as well. However, since
δ is always high when it is chosen by the supplier, we observe that retailers may prefer
structure C to B. This can be observed in Figure 4.3 as well (when δ = 3).
Scenarios with Supplier-Selected w and Exogenous δ
Next, we use numerical examples to discuss scenarios with supplier-selected w. Suppose
that Xi ∼ U(0, 150), r = 15, c1 = 1, c2 = 8, α = 0.35, δ ∈ [0, 5] τ1 = 0.1 and τ2 ∈ [0.5, 5].
The retailer’s profit comparison is shown in Figure 4.4.
Some interesting results can be observed in Figure 4.4. First, we notice that the
retailers can generate a higher profit under C as compared to B when shipping cost τ2
is high. This makes sense because a greater τ2 lowers the marginal benefit from fast-
ship orders served by transshipping between retailers. Therefore, the retailers prefer to
compete with each other instead of forming an alliance. Second, structure A is more
93
0 1 2 3 4 50
1
2
3
4
5
Shipping Cost of Fast−Ship Orders
Who
lesa
le P
rice
Mar
kup
πAR < πC
R < πBR
πAR < πB
R < πCR
πBR < πA
R < πCR
πBR < πC
R < πAR
(a) c1 = 1, c2 = 8, α = 0.35
Figure 4.4: The Retailer’s Profit Comparison
profitable than B and/or C for a lower wholesale price markup (δ). This can be explained
by using two sets of arguments. When δ is high, the supplier tends to charge a higher
price under structures B and C because competing retailers or cooperative retailers
tend to order more than that under structure A. This makes structures B and C less
profitable than A for retailers. However, when δ is high, it is likely that wA = wB = wC
because the condition of w2 ≤ r needs to be satisfied. When this happens, structure A
is the least profitable structure among the three structures as shown in Proposition 4.5.
4.5. Conclusions
In this chapter, we investigated a two-retailer supply chain under three different sourcing
structures. Each structure represents a unique relationship between the two retailers. In
structure A, the two retailers are independent and rely on the suppler’s supports for both
regular orders and fast-ship orders. In structure B, the two retailers form an alliance
and agree to support fast-ship orders for each other using their leftover inventory. In
structure C, there are two competing retailers that also sell items to customers who
travel from the other store.
We showed that regardless of the structure, retailers’ expected profit can be higher
under a higher customer participation rate whereas the supplier may not earn a higher
94
profit under a higher customer participation rate when the wholesale price is not chosen
by the supplier. When wholesale price is set by the supplier, the retailers’ profits in
structure A is always decreasing in customer participation rate. However, their profits
may either decrease or increase in customer participation rate when they form an alliance
or compete with each other. This happens because w is bounded by r−δ, which ensures
the retailers’ profitability from each fast-ship order.
We also observed that structure A is more profitable than the other two structures
for the supplier. This is especially true when customer participation rate is high (e.g.,
a greater fast-ship demand). This result does not change when the supplier’s second
replenishment cost is high because the supplier can procure items in advance to mitigate
the chance of selling at a high replenishment cost.
For the retailers, we showed that structure C is more profitable than B when shipping
cost is high because the marginal benefit from selling items to customers who experience
a stockout is higher in C. We also discovered that operating independently (structure
A) may be more profitable for the retailers when the wholesale price markup is low
because the supplier can charge a higher wholesale price for structures B and C in such
scenarios.
Chapter 5
Conclusions
Many retailers carry substantial in-store inventory even when the holding cost of the
product is high (due to high obsolescence). This is because when a customer finds that
an item is out of stock, (s)he is likely to walk away and buy either from a different store
or from a competitor. Due to demand uncertainty and short product cycles, retailers
can end up carrying too much inventory, leading to mark downs and losses. Finding
a way to effectively reduce inventory costs without affecting sales is a high priority for
such retailers.
Many supply chains may use multiple replenishments to reduce inventory level and
the impact of stockout incidences. Offering the fast-ship option is one of many practices
of this kind. The fast-ship option helps reduce lost sales by sourcing out-of-stock items
through a backup channel without additional cost to customer. With the fast-ship
option, the retailer can ensure that the product will be shipped to the customer in
a short period of time which potentially reduces the number of customers who leave
without purchasing.
We have anecdotal evidence that the fast-ship ordering is used by many retailers in
practice. However, academic research on this topic is lacking, leaving room for future
work along these lines. This dissertation presents an initial attempt to address gaps in
the literature about models dealing with the fast-ship option. We investigate the role
95
96
of the fast-ship option under a variety of supplier chain environments. Particularly, we
identified ordering policies for the supplier and the retailer. Also, we studied how the
supplier and the retailer can benefit from the fast-ship option under different contract
structures, price decisions, retailer-relationships and customer participation parameters.
One common result is observed in all three chapters. That is, when the fast-ship
option is strictly profitable for the retailer at pre-determined prices, retailers may benefit
from a higher customer participate rate, but not the supplier. In contrast, when either
wholesale price or markup value is chosen by the supplier, then the supplier earns a larger
portion of profit from fast-ship orders, which usually makes the retailer worse off under a
higher customer participation rate. In practice, this may not be realistic because prices
should be determined such that both the supplier and the retailers could benefit from
the fast-ship option. We also showed that in a multiple-retailer environment, retailers
may increase the profitability for the fast-ship option and weaken the supplier’s power
by either forming an alliance or competing with each other. However, is there other
alternatives to balance the supply chain power? In other words, it would be interesting
to learn how the profit can be more reasonably distributed among supply chain partners.
Several research directions can be pursued in the future to strengthen this line of
research:
1. When solving the supplier’s operational decision, it is assumed that the supplier
knows all parameters associated with the retailer, including demand forecasts and
retailer’s inventory level. Does the retailer benefit from not reporting its demand
forecasts and inventory level truthfully? How does the supplier’s performance
change in such cases? Also, what would happen if the supplier and the retailer
have different demand forecasts?
2. How do the supplier and the retailer benefit from the fast-ship option if additional
benefits and costs of the fast-ship orders are shared between the two parties based
on some allocation rules? Can a contract of this type help both parties earn higher
profits from a higher customer participation rate?
97
3. The fast-ship option offers a superior service to customers. However, there might
be some front end costs of setting up the fast-ship option, such as training sales-
persons and monitoring supply’s reliability. Suppose that the fast-ship option
may lead to increased customer loyalty and greater sales over time, should the
supply chain offer the fast-ship option to customers if it incurs a front end cost of
providing such service?
4. In all our models, we assume that the customer participation rate is independent of
supply operations. What would happen if the customer participation rate depends
on one or more decisions chosen by the supplier or the retailer? For example, the
customer participation rate can be sensitive to the lead time. Fewer customers
may be willing to participate in the fast-ship option if they expect to wait for a
longer time. If the retailer/supplier can change from different shipping services
that offer different lead times, how do our results change?
5. In practice, a retailer may carry multiple substitute products in store. In such en-
vironments, a customer who experiences stockout has three options — (1) utilize
the fast-ship option for the item s/he originally intended to buy, (2) purchase the
less preferred substitute item, and (3) leave the store without making a purchase.
Therefore, it is interesting to know how the retailer benefit from the fast-ship op-
tion in a substitutable-product environment and how does that affect the retailer’s
assortment strategy is affected.
6. When wholesales prices are set by the supplier, the retailer may adjust retail price
to maximize its profit. Facing a downward sloping demand, does empowering the
retailer to choose retail price help balance the profit allocation between the two
parties? We believe that additional insights may be formed if above directions are
pursued.
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Appendix A
Proofs for Chapter 2
Proof of Proposition 2.1. We show that vtR,Q(u, a | δ) is concave in a by induction.
Similar steps can also be applied to vtR,B(u, a). Hence, we omit the details for the latter
case.
Let µtR,Q(u) = max
a≥uvtR,Q(u, a | δ) and µN+1
R,Q (u) = 0. Based on equation (2.11) and
(2.13), we observe that vNR,S(u, a | δ) is concave in a and µNR,Q(u) is concave in u. Now,
suppose that vt+1R,S(u, a | δ) is concave in a and µt+1
R,Q(u) is concave in u. We show that
they also hold in period-t < N . Because ut+1 = (at −Xt)+, we obtain
∂vtR,Q(u, a | δ)
∂a= −w + (r − α(r − w2))Ft(a) + λR
∫ ∞
a
µ′t+1R,Q(0)dFt(x)
+λR
∫ a
0µ′t+1
R,Q(a− x)dFt(x),
and
∂2vtR,Q(u, a | δ)
∂a2= −(r − α(r − w2))f
t(a) + λR
∫ ∞
a
µ′′t+1R,Q(0 | δ)dFt(x)
+λR
∫ a
0µ′′t+1
R,Q(a− x | δ)dFt(x) ≤ 0,
where the inequality hold because v′′t+1R,Q(u) ≤ 0. Therefore, vtR,Q(u, a | δ) is concave in
a.
We next show that µ′′tR,Q(u) ≤ 0 so the induction arguments work for period-1
to (t − 1). Let a = argmaxa≥0
vtR,Q(u, a | δ) and a = argmaxa≥u
vtR,Q(u, a | δ). Because
104
105
vtR,Q(u, a | δ) is concave in a, a = max(a, u). Therefore, we obtain
µtR,Q(u) = ρtR,Q(u, a | δ) + λRE[µt+1
R,Q((a−Xt)+)].
In addition, we obtain ∂2µtR,Q(u)/∂u
2 as follows.
• If u < a,
∂2µtR,Q(u)
∂u2= 0, and (A.1)
• if u ≥ a,
∂2µtR,Q(u)
∂u2= −(r − α(r − w2))ft(u) +
∫ ∞
u
µ′′t+1R,Q(0 | δ)dFt(x)
+λR
∫ u
0µ′′t+1
R,Q(u− x, u− x | δ)dFt(x) ≤ 0,
where the inequality comes from v′′t+1R,Q(u, a | δ) ≤ 0. By induction, we conclude that
vjR,S(u, a | δ) is concave in a for all j < t. Hence proved.
Proof of Lemma 2.1. We first simplify the single-period profit function in (2.7) via a
series of steps. For t ≥ 2, the terms in the right hand side of (2.7) can be written in terms
of it, ut, and gt as follows — it = [gt−1 − α(Xt−1 − at−1Q )+]+, ut = (at−1
Q −Xt−1)+,
qtS = gt− it+(aQ(δ)−ut)+ = gt+atQ−zt, qtR = (aS(δ)− (at−1Q −Xt−1)+)+, and α(Xt−
(ut + qtR))+ − (it + qtS − qtR) = α(Xt − atQ)
+ − gt. Upon applying these transformations
to (2.7), we notice that certain terms depend on period (t − 1)’s decisions and others
depend on period t’s decisions. Similarly, there are terms involving Xt−1, which is
known at the time of choosing period-t order quantity, and Xt, which is unknown. In
order to isolate terms involving period t’s state and decision variables, we move terms
with indices (t− 1) into the expression for ρt−1S,Q and bring terms involving index t from
period (t + 1) into the expression for ρtS,Q. This is simply an accounting change that
leads to more elegant presentation. It does not change S’s expected total profit for
each given sequence of replenishment quantities. With this accounting method, and
106
in a slight abuse of notation, continuing to use ρtS,Q to denote S’s profit function with
transformed variables in any period t ≥ 1 , we obtain from (2.7)
ρtS,Q(atQ, z
t, gt | δ) = −hSλSE[(gt − α(Xt − atQ)+)+]− c1[g
t + atQ − zt]
+λS(w − τ1)E[(aQ(δ) − (atQ −Xt)+)+]
+α(w2 − τ2)E[(Xt − atQ)+]− c2E[(α(Xt − atQ)
+ − gt)+]
Because (gt − α(Xt − atQ)+)+ = (gt − α(Xt − atQ)
+) + (α(Xt − atQ)+ − gt)+, and
(α(Xt − atQ)+ − gt)+ = (αXt − (αatQ + gt))+, the terms involving gt in the above
expression can be rewritten in terms of gt = gt + αatQ. Furthermore, because for t ≥ 2,
zt = (gt−1 − α(Xt−1 − at−1Q )+)+ + (at−1
Q − Xt−1)+ depends on terms with time index
(t − 1), we move these terms to the one-period reward function for period (t − 1) and
move the term λSc1zt+1 into the expression for period-t reward function. The one-
period reward function depends on zt only through the fact that gt ≥ zt − atQ, and we
obtain the following.
ρtS,Q(atQ, z
t, gt | δ) = φt(atQ) + ρtS,Q(ςt, gt | δ), (A.2)
where gt ≥ ςt = (zt − atQ)+ + αatQ, and φt(atQ) and ρtS,Q(ς
t, gt | δ) are as defined in
(2.14) and (2.15).
The decision concerning the magnitude of gt is taken after observing atQ. Therefore,
designating gt as S’s action (discretionary replenishment quantity) leads to a completely
equivalent description of S’s per-period reward function. We obtain S’s total profit
function by adding financial transactions that occur only in period 1, but that do not
affect its choice of gts. Put differently, after replacing ρ1S,Q(u1, i1, q1S | δ) in (2.9) by
−i1hS + c1(u1 + i1) + w(aS(δ) − u1)+ρ1S,Q(a
1Q, z
1, g1 | δ) and ρtS,Q(ut, it, qtS | δ) in (2.9)
by (A.2) for all t ≥ 2, we obtain (2.16). Hence proved.
Proof of Proposition 2.2. Similar to Proposition 2.1, we prove this proposition by
induction. Let µtS,Q(ς) = max
g≥ςvtS,Q(ς, g) and µN+1
S,Q (ς) = 0. Based on equation (2.15)
and (2.17), we observe that vNS,Q(ς, g) is concave in g and µNS,Q(ς) is concave in ς. Now,
107
suppose that vt+1R,Q(u, a | δ) is concave in a and µt+1
R,Q(u) is concave in u. We show that
they also hold in period-t where t < N . Then, we obtain
∂vtS,Q(ς, g)
∂g= λS(c1 − hS)Ft
(
g
α
)
− c1 + c2Ft
(
g
α
)
+∂
∂g
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x),
and
∂2vtS,Q(ς, g)
∂g2=
λS(c1 − hS)
αft
(
g
α
)
−c2αft
(
g
α
)
+∂2
∂g2
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x). (A.3)
Let atQ be the constrained optimal order-up-to level and atQ be the unconstrained optimal
order-up-to level for R in period-t. That is atQ = max(ut, atQ). In addition, let I denote
a indicator function. Because
ςt+1 = (zt+1 − at+1Q )+ + αat+1
Q
= {(g − α(atQ + (Xt − atQ)+))+ + (atQ −Xt)− (at+1(δ) ∨ (at+1
Q −Xt)+)}+
+(αat+1(δ) ∨t (atQ −Xt)+), (A.4)
from the fact that ut+1 = (atQ − Xt), it+1 = (g − α(atQ + (Xt − atQ)+))+, and at+1
Q =
(at+1(δ) ∨ (atQ −Xt)+), we observe
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x)
= I(at+1Q < atQ)I(g < αatQ)
∫ atQ−at+1Q
0µt+1S,Q(α(a
tQ − x)) | δ)dFt(x)
+ I(at+1Q < atQ)I(g ≥ αatQ)
∫ atQ−at+1Q
0µt+1S,Q(g − αx)dFt(x)
+ I(g < αatQ)
∫ atQ
(atQ−at+1Q )+
µt+1S,Q(a
t+1Q )dFt(x)
+ I(g ≥ αatQ)
∫ atQ
(atQ−at+1Q )+
µt+1S,Q((g + (1− α)atQ − x− at+1
Q )+ − αat+1Q )dFt(x)
+
∫ ∞
atQ
µt+1S,Q(((g − αXt)+ − at+1
Q )+ + αat+1Q )dFt(x), (A.5)
108
∂
∂g
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x)
= I(at+1Q < atQ)I(g ≥ αatQ)
∫ atQ−at+1Q
0µ′t+1
S,Q(g − αx)dFt(x)
+ I(g ≥ αatQ)[
∫ g+αatQ−at+1Q
(atQ−at+1
Q)+
µ′t+1S,Q(g + (1− α)atQ − x)dFt(x)
+
∫ atQ
g+αatQ−at+1
Q
µ′t+1S,Q(αa
t+1Q )dFt(x)
]
+ I(g ≥ at+1Q + αatQ)
[
∫ ∞
g−at+1Q
α
µ′t+1S,Q(αa
t+1Q )dFt(x) +
∫
g−at+1Q
α
atQ
µ′t+1S,Q((g − αXt)
− αat+1Q )dFt(x)
]
, (A.6)
and
∂2
∂g2
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x)
= I(at+1Q < atQ)I(g ≥ αatQ)
∫ atQ−at+1Q
0µ′′t+1
S,Q(g − αx)dF1(x)
+ I(g ≥ αatQ)[
∫ g+αatQ−at+1Q
(atQ−at+1Q )+
µ′′t+1S,Q(g + (1− α)atQ − x)dFt(x)
+
∫ atQ
g+αatQ−at+1Q
µ′′t+1S,Q(αa
t+1Q )dFt(x)
]
+ I(g ≥ at+1Q + αatQ)
[
∫ ∞
g−at+1Q
α
µ′′t+1S,Q(αa
t+1Q )dFt(x) +
∫
g−at+1Q
α
atQ
µ′′t+1S,Q((g − αXt)
− αat+1Q )dFt(x)
]
. (A.7)
Because µ′′t+1S,Q(·) ≤ 0, we observe ∂
∫∞0 µt+1
S,Q(ςt+1)dFt(x)∂g ≤ 0. Therefore, we observe
from (A.3) that ∂2vtS,Q(ς, g)/∂g2 ≤ 0
We next show that µ′′tS,Q(ς) ≤ 0 so the induction arguments work for period-1
to (t − 1). Let g = argmaxg≥0
vtS,Q(u, a | δ) and g = argmaxg≥ς
vtR,Q(ς, g | δ). Because
vtS,Q(ς, g | δ) is concave in g, g = max(g, ς). Therefore, we obtain
µtS,Q(ς) = ρtS,Q(ς, g | δ) + λRE[µt+1
S,Q(ςt+1)].
109
In addition, ∂µtS,Q(ς)/∂ς and ∂2µt
S,Q(ς)/∂ς2 can be written as follows.
• If ς < g,∂µt
S,Q(ς)
∂ς= 0, and (A.8)
∂2µtS,Q(ς)
∂ς2= 0. (A.9)
• if ς ≥ g,
∂µtS,Q(ς)
∂ς= ρtS,Q(ς, ς | δ) + λRE[µt+1
S,Q(ςt+1)]
= I(at+1Q < atQ)I(ς ≥ αatQ)
∫ atQ−at+1Q
0µ′t+1
S,Q(ς − αx)dF1(x)
+ I(ς ≥ αatQ)[
∫ ς+αatQ−at+1Q
(atQ−at+1Q )+
µ′t+1S,Q(ς + (1− α)atQ − x)dFt(x)
+
∫ atQ
ς+αatQ−at+1Q
µ′t+1S,Q(αa
t+1Q )dFt(x)
]
+ I(ς ≥ at+1Q + αatQ)
[
∫ ∞
ς−at+1Qα
µ′t+1S,Q(αa
t+1Q )dFt(x) +
∫
ς−at+1Qα
atQ
µ′t+1S,Q((ς − αXt)
− αat+1Q )dFt(x)
]
, and (A.10)
∂2µtS,Q(ς)
∂ς2
= I(at+1Q < atQ)I(ς ≥ αatQ)
∫ atQ−at+1Q
0µ′′t+1
S,Q(ς − αx)dFt(x)
+ I(ς ≥ αatQ)[
∫ ς+αatQ−at+1Q
(atQ−at+1Q )+
µ′′t+1S,Q(ς + (1− α)atQ − x)dFt(x)
+
∫ atQ
ς+αatQ−at+1Q
µ′′t+1S,Q(αa
t+1Q )dFt(x)
]
+ I(ς ≥ at+1Q + αatQ)
[
∫ ∞
ς−at+1Qα
µ′′t+1S,Q(αa
t+1Q )dFt(x) +
∫
ς−at+1Qα
atQ
µ′′t+1S,Q((ς − αXt)
− αat+1Q )dFt(x)
]
. (A.11)
110
Because µ′′t+1S,Q((ς − αXt), ∂2µt
S,Q(ς)/∂ς2 ≤ 0. By induction, vjS,Q(ς, g | δ) is concave in
g for all j < t. Hence proved.
Proof of Proposition 2.3. We prove this by induction arguments. More specifically,
suppose that at+1Q (δ) = max(at+1
Q , ut+1) is the optimal order-up-to level for period (t+
1) where at+1Q is the unconstrained optimal order-up-to level as defined in equation
(2.19), then atQ(δ) = max(atQ, ut) is the optimal order-up-to level for period t. Similar
arguments can also be applied to scenario B. Hence, we omit the details for the latter
case.
Let µtR,Q(u) = max
a≥uvtR,Q(u, a | δ). Because ut+1 = (at −Xt)+, we obtain
∂vtR,Q(u, atQ | δ)
∂a= −w + (r − α(r − w2))Ft(a
tQ) + λR
∫ ∞
atQ
µ′tR,Q(0)dFt(x)
+λR
∫ atQ
0µ′t+1
R,Q(atQ − x)dFt(x). (A.12)
Recall that µt+1R,Q(u
t+1) = maxa≥u
vt+1R,Q(u
t+1, a | δ) = vt+1R,Q(u
t+1,max(at+1Q , ut+1) | δ). When
R orders up to atQ, we observe that at+1Q ≥ ut+1 = (at+1
Q −Xt)+. Therefore, µt+1R,Q(u
t+1) =
vt+1R,Q(u
t+1, at+1Q | δ) and
µ′t+1R,Q(u
t+1) =
{
w − hR if ut+1 > 0,
0 otherwise.(A.13)
By replacing (A.13) in (A.12), we obtain
∂vtR,Q(u, atQ | δ)
∂a= −w + (r − α(r − w2))Ft(a
tQ) + λR(w − hR)Ft(a
tQ).
By the definition of atQ in (2.19), we get ∂vtR,Q(u, atQ | δ)/∂a = 0, which implies that
atQ is the optimal because vtR,Q(u, a | δ) is concave in a (Proposition 2.1). In addition,
because at(δ)Q has to be greater than ut, we have atQ(δ) = max(atQ, ut). Similar steps
can also be applied to period 1 to (t− 1). Hence proved.
Proof of Proposition 2.4. We prove this by induction arguments. More specifically,
suppose that αηt+1 is the optimal g for period (t+1) where ηt+1 is defined in equation
111
(2.20), then αηt is the optimal g for period t. Let µtS,Q(ς) = max
g≥ςvtS,Q(ς, g). We obtain
∂vtS,Q(ς, g)
∂g= λS(c1 − hS)Ft
(
g
α
)
− c1 + c2Ft
(
g
α
)
+∂
∂g
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x).
Let atQ be the constrained optimal order-up-to level and atQ be the unconstrained
optimal order-up-to level for R in period-t. Note that
ςt+1 = (zt+1 − at+1Q )+ + αat+1
Q
= {(g − α(atQ + (Xt − atQ)+))+ + (atQ −Xt)+ − (at+1(δ) ∨ (atQ −Xt)+)}+
+α(at+1(δ) ∨ (atQ −Xt)+), (A.14)
from the fact that ut+1 = (atQ −Xt)+, it+1 = (g − α(atQ + (Xt − atQ)+))+, and at+1
Q =
(at+1(δ) ∨ (atQ − Xt)+). When g = αηt, we observe that ςt+1 ≤ (αηt − αatQ)+ +
αat+1S ≤ αηt+1 where the second inequality comes from the fact that αηt+1 ≥ αatS ≥
αat+1Q and αηt+1 = αηt. This implies that when g = αηt, µt+1
S,Q(ς) = maxg≥ς
vt+1S,Q(ς, g) =
vt+1S,Q(ς, αη
t+1), which is independent of either ς or gt. As a results, we obtain
∂vtS,Q(ς, αηt)
∂g= λS(c1 − hS)Ft
(
ηt)
− c1 + c2Ft
(
ηt)
+∂
∂g
∫ ∞
0µt+1S,Q(ς
t+1)dFt(x)
= λS(c1 − hS)Ft
(
ηt)
− c1 + c2Ft
(
ηt)
According to the definition of ηt in (2.20), we observe that ∂vtS,Q(ς, αηt)/∂g = 0. That
is, αηt is the optimal gt for period t because vtS,Q(ς, g) is concave in g(Proposition 2.2).
In addition, because gt = gt + αatQ where gt ≥ 0, we get the optimal gt = α(ηt − atQ)+
and echelon stock at the start of period-t ptQ = max(gt+atS , zt). Similar arguments can
also be applied to period-1 to (t− 1). Hence proved.
Proof of Proposition 2.6. Observe from (2.19) that a′S(δ) = (∂aQ(δ)/∂δ) ≥ 0; that
is, aQ(δ) is non-decreasing in δ. This makes sense on an intuitive level because when
fast shipping costs more, R would be inclined to stock to a higher level in order to avoid
procuring fast shipping. Next, we re-arrange terms in (2.42), separate terms that are
not a function of δ from those that are, and obtain πQS (δ) as follows.
πQS (δ) = (1− λS)[π
QS (δ) − κ]
112
where κ = −u1w − i1hS + (u1 + i1)c1 + (λS/(1 − λS))(w − τ1 − c1)E(X) is a constant.
It is easy to see that∂π
QS (δ)∂δ
= (1 − λS)∂π
QS (δ)∂δ
and because 0 ≤ λS ≤ 1, the two partial
derivatives have the same sign. Two cases now arise depending on whether η ≤ aQ(δ)
or η > aQ(δ). We deal with each case separately in the ensuing analysis.
CASE I: η ≤ aQ(δ)
∂πQS (δ)
∂δ= (w − τ1 − c1)(1− λS)a
′S(δ)− a′S(δ)F (aQ(δ))[α(w2 − τ2)− αc2
−λS(w − τ1 − c1)] + αsE[(X − aQ(δ))+]
≥ a′S(δ){(w − τ1 − c1)(1 − λSF (aQ(δ))) − α(w2 − τ2 − c2)F (aQ(δ))}
≥ a′S(δ)(w2 − τ2 − c2){1− λSF (aQ(δ)) − αF (aQ(δ))}
≥ 0. (A.15)
The first inequality above follows from the fact that αsE[(X − aQ(δ))+] ≥ 0. The sec-
ond inequality follows from the observation that (w − τ1 − c1) ≥ αF (·)(w2 − τ2 − c2).
Finally, the last inequality comes from the fact that 1 − λSF (aQ(δ)) − αF (aQ(δ)) =
1− λS − (α− λS)F (aQ(δ)) ≥ 0 because α ≤ 1 and F (·) ≤ 1. Therefore,∂π
QS (δ)∂δ
≥ 0 and
this completes the proof of Case I.
CASE II: η > aQ(δ).
From a series of arguments similar to Case I, we obtain
∂πQS (δ)
∂δ= (w − τ1 − c1)(1− λS)a
′S(δ) + a′S(δ)[c1(1− λS) + hSλS ]
−a′S(δ)F (aQ(δ))[α(w2 − τ2) + αλS(hS − c1)− λS(w − τ1 − c1)]
= a′S(δ){(w − c1)[1 − λSF (aQ(δ))] + α[c1(1− λS) + λShS ]F (aQ(δ))
−α(w2 − τ2 − c1)F (aQ(δ))
≥ a′S(δ){(w2 − τ2 − c2)[1− λSF (aQ(δ)) − αF (aQ(δ))]
≥ 0. (A.16)
The first inequality above comes from the fact that (w− τ1 − c1) ≥ αF (·)(w2 − τ2 − c1)
113
and that α[c1(1 − λS) + λShS ]F (aQ(δ)) ≥ 0. The last inequality holds for the same
reason that the last inequality in (A.15) holds. Hence proved.
Proof of Proposition 2.8. S can induce R to offer fast shipping by letting δ∗j = δcj
where j ∈ {αL, αH} and αL < αH . If S selects δ∗αL> δcαL
(resp. δ∗αH> δcαH
), then R will
choose not to offer fast shipping. Note that R’s profit with fast shipping equals πBR (aB)
if δ∗αL= δcαL
or δ∗αH= δcαH
. Since πBR (a(B) does not depend on α (see Equation 2.40),
R’s profit is invariant in α regardless of whether S decides to support fast shipping or
not. Hence proved.
Proof of Proposition 2.9. Let αL < αH . We first claim that if δ∗αLis a feasible
markup for S, then δ∗αHis also feasible. That is, πQ
S,αL(δ∗αL
) ≤ πQS,αH
(δ∗αH). To prove
this statement, recall that S’s profit with fast shipping is πQS (δ
∗) and R’s order-up-to
level aQ(δ∗) = aB where δ∗ = δc. Subtracting πBS from πQ
S (δ∗), we get
πQS (δ
∗)− πBS =
α
1− λSσ(aB)−
λSβ
1− λSk(aB), (A.17)
where
σ(aB) = −(c1 + λS(hS − c1)(z − aB)+)− (c2 + λS(hS − c1))E[(X −max(z, aB))+]
+(w2 − τ2 + λS(hS − c1))E[(X − aB)+], (A.18)
and
k(aB) = (w − τ1 − c1)E[(X − aB)+]. (A.19)
Note that neither σ(aB) nor k(aB) depends on α. If δ∗αLis feasible when α = αL,
then πQS,αL
(δ∗αL) − πB
S ≥ 0, and this furthermore implies that σ(aB) ≥ 0 . The latter
comes from the fact that αL
1−λSσ(aB) − λSβ
1−λSk(aB) ≥ 0 and k(aB) ≥ 0. Then from
the right hand side of (A.17), we conclude that πQS (δ
∗) − πBS is increasing in α, which
immediately implies that πQS,αH
(δ∗αH) ≥ πQ
S,αL(δ∗αL
). Hence proved.
114
Proof of Proposition 2.10. Regardless of R’s decision concerning fast shipping, its
profit is πQR(aQ(δ
∗βL
)) = πBR,βL
(aBβL) when β = βL, and πQ
R(aQ(δ∗βH
)) = πBR,βH
(aBβH)
when β = βH . That is, for a fixed β, R’s profit is not a function of whether the supply
chain supports fast shipping or not. However, because δ∗βH< δ∗βL
, πQR(aQ(δ
∗βL
)) <
πQR(aQ(δ
∗βH
)). Therefore, R’s profit is strictly increasing in β. Hence proved.
Appendix B
Proofs for Chapter 3
Proof of Proposition 3.1. Recall that yA(p) = [α(ηS − qA(p))+ ∧ (p − qA(p))] de-
pending on value of p. Because w2− τ2 ≥ c2, πA(p)′
.=
∂πAS (p)∂p
corresponding to different
ranges of p are
• When p is in a region such that yA(p) = 0,
πA(p)′ = qA(p)′((w − τ1 − c1)− αF (qA(p))(w2 − τ2 − c2))
+ (w2 − τ2 − c2)F (eAp (qA(p)))(1 − (1− α)qA(p)′)
≥ qA(p)′((w − τ1 − c1)− αF (qA(p))(w2 − τ2 − c2))
≥ qA(p)′((w − τ1 − c1)− αF (qA(p))(w2 − τ2 − c1)) ≥ 0 (B.1)
• When p is in a region such that yA(p) = α(ηS − qA(p)),
πA(p)′ = qA(p)′(w − τ1 − c1(1− α)− α(w2 − τ2)F (qA(p)))
+ (w2 − τ2 − c2)F (eAp (qA(p)))(1 − (1− α)qA(p)′)
≥ qA(p)′(w − τ1 − α(w2 − τ2)− c1(1− α)) ≥ 0. (B.2)
115
116
• When p is in a region such that yA(p) = p− qA(p),
πA(p)′ = qA(p)′(w − τ1 − c1 − α(w2 − τ2)F (qA(p))) − c1(1− qA(p)′)
+ (w2 − τ2)F (eAp (qA(p)))(1 − (1− α)qA(p)′)
≥ qA(p)′(w − τ1 − c1 − α(w2 − τ2)F (qA(p))) − c1(1− qA(p)′)
+ (w2 − τ2)c1c2(1− (1− α)qA(p)′)
≥ qA(p)′(w − τ1 − c1 − α(w2 − τ2)F (qA(p))) − c1(1− qA(p)′)
+ c1(1− (1− α)qA(p)′)
= qA(p)′(w − τ1 − (1− α)c1 − α(w − τ2)F (qA(p)))
≥ qA(p)′(w − τ1 − (1− α)c1 − α(w2 − τ2)) ≥ 0 (B.3)
Because w2−τ2 ≥ c2 and 0 ≤ q′A(p) ≤ (1−α)−1, it follows that (w2−τ2−c2)F (eAp (qA(p)))
(1 − (1 − α)q′A(p)) ≥ 0. Therefore, πA(p)′ in (B.1) and (B.2) are non-negative. When
yA(p) = p − qA(p), it implies that p − qA(p) ≤ α(ηS − qA(p)). Consequently, the first
inequality in (B.3) holds because eAp (qA(p)) = p−(1−α)qA(p)
α≥ c1
c2. The second inequality
in (B.3) holds because w2 − τ2 ≥ c2. Hence, proved.
Proof of Proposition 3.4. As shown in (3.15), γ(q) = ∞ if w2 − τ2 ≥ c2. In this
section, we also make use of the fact that πCR(q) is concave in q (proof is not presented
in the interest of brevity).
When w2 − τ2 ≥ c2, we observe from (3.1) that
∂πCR(q)
∂q= −w + (r − α(r −w2))F (q). (B.4)
Hence, the optimal qC = F−1(w/(r − α(r − w))). Similarly, if w2 − τ2 < c2, then
γ(q) = α(ηR − q)+, and we obtain
∂πCR(q)
∂q=
{
−w + (r − α(r − w2))F (q) when q ≤ ηR
−w + rF (q) when q > ηR. (B.5)
Let q1 and q2 be solutions to the equation∂πC
R(q)∂q
= 0 in the regions q < ηR and q ≥ ηR,
respectively. Note that this implies that γ(q1) = α(ηR − q1) and γ(q2) = 0.
117
First consider cases where wr−α(r−w) <
c1w2−τ2
. Because ∂πCR(q)/∂q ≥ 0 when q ≤ qR,
this implies that q1 = ηR, and γ(q1) = 0. Based on that we observe from (3.1) that
πCR(q1, γ(q1)) = πC
R(q1, γ(q2)) ≤ πCR(q2, γ(q2)) because γ(q1) = γ(q2) = 0 and q2 is the
optimal q when q ≥ qR. Therefore, the optimal qC = F−1(
wr
)
when wr−α(r−w) <
c1w2−τ2
.
Next consider cases when wr≥ c1
w2−τ2. It is easy to check that q1 = F−1
(
wr−α(r−w2)
)
and q2 = ηR. Therefore, we observe from (3.1) that πCR(q1, γ(q1) ≥ πC
R(q2, γ(q2)). Hence,
the optimal qC = F−1(
wr−α(r−w2)
)
.
Proof of Proposition 3.5.
Part 1: From (3.3) and Proposition 3.4, it is easy to check that
qA(p) ≤ F−1
(
w
r − α(r −w2)
)
≤ qC . (B.6)
Part 2: If c2 < w2 − τ2, then γ(q) = ∞ for any q. Therefore, γ(qB(z)) ≥ z for any z.
If c2 ≥ w2 − τ2, it can be shown that z∗ must satisfy the following equality.
qB(z∗)′(w−τ1−c1−α(w2−τ2)F (qB(z∗)))−c1+(w2−τ2)F (qB(z∗)+z∗/α)(1+αqB(z∗)′) = 0
(B.7)
Define a z such that z = α(ηR − qB(z)). We observe that
qB(z)′(w − τ1 − c1 − α(w2 − τ2)F (qB(z)))− c1
+(w2 − τ2)F (qB(z) + z/α)(1 + αqB(z)′)
= qB(z)′(w − τ1 − c1 − α(w2 − τ2)F (qB(z))) + αqB(z)′c1 ≤ 0, (B.8)
because ηR = qB(z) + z/α = F−1(c1/w2 − τ2). Therefore, z∗ must be less than z from
the fact that πBM is decreasing in z ≥ z∗. In addition, qB(z)′ > α−1, γ(qB(z∗)) ≥ z∗.
Hence proved.
Part 3: Based on Proposition 3.4, we know that qC = F−1(
wr−α(r−w2)
)
or F−1(
wr
)
.
When qC = F−1(
wr−α(r−w2)
)
, the corresponding γ(qC) ≤ ∞. Since qB(z)′ ≤ 0 and
limz→∞ qB(z) = F−1(
wr
)
, it follows that qB(z) ≥ qC . When qC = F−1(
wr
)
, the
corresponding γ(qC) = 0. Based on expression in (3.4), we observe that qB(z) =
qB(0) = F−1(
wr
)
= qC . Hence proved.
118
Proof of Proposition 3.6.
The supplier’s preference: Define πiS(i, j) = maxy π
iS(y, j, q). We first show that C <
A. Let γ = p∗−qA(p∗). It is easy to check that πAS (p
∗) = πCS (q
A(p∗), γ) ≤ πCS (q
C , γ) be-
cause qC ≥ qA(p∗) (Part 1 of Proposition 3.5) and ∂πBS (z, q)/∂q ≥ 0. From the fact that
γ(qC) is the optimum, it follows that πAS (p
∗) ≤ πCS (q
C , γ) ≤ πBS (q
C , γ(qC)). Next, we
show B < C by letting z = γ(qC). It is easy to check that πBS (q
B(z), z) ≥ πCS (q
C , γ(qC))
because qB(z) ≥ qC (see Proposition 3.5, Part 3). From the fact that z∗ is the opti-
mal z and the previous argument, it follows that πBS (q
B(z∗), z∗) ≥ πBS (q
B(z), z) ≥
πCS (q
C , γ(qC)). Hence proved.
The retailer’s preference: Let p = qB(z∗)+z∗. We get πAR(q
A(p), p) ≥ πAR(q
B(z∗), p) =
πBR (q
B(z∗), z∗) because qA(p) is the optimum when p = p. In addition, if p∗ ≥ p (based
on proposition statement), we get πAR(q
A(p∗), p∗) ≥ πAR(q
A(p), p) ≥ πBR (q
B(z∗), z∗) be-
cause ∂πAR(p)/∂ ≥ 0. Hence A < B holds. We use similar argument to show that A < C.
Let p = qC + γ(qC). We observe that πAR(q
A(p∗), p∗) ≥ πAR(q
A(p), p) ≥ πBR (q
C , γ(qC)).
Finally, we show that C < B by letting q = qB(z∗) first. Based on Part 2 of Proposition
3.5, we know that γ(q) ≥ z∗. As a result, we get πBR(z
∗) ≤ πCR(q) ≤ πC
R(qC) where
the first inequality comes from γ(q) ≥ z∗ and the second from the fact that qC is the
retailer’s optimal decision. Hence proved.
Proof of Proposition 3.7. We prove this for structure B. Similar arguments can be
applied to A and C. Recall that yB(z) = [α(ηS − qB(z))+ ∧ z] depending on value of p
and qB(z). Because q′.= ∂qB(z)
∂δπB(z)′
.=
∂πBS (z)∂δ
corresponding to different ranges of p
are
• When p is in a region such that yB(z) = 0,
πB(z)′ = q′((w − τ1 − c1)− α(F (qB(z)) − F (eBz (qB(z))))(w2 − τ2 − c2))
+ E[α(X − qB(z))+ ∧ ζ ij(qB(z))]
≥ q′((w − τ1 − c1)− α(F (qB(z)) − F (eBz (qB(z))))(w2 − τ2 − c2)) ≥ 0
(B.9)
119
• When p is in a region such that yB(z) = α(ηS − qB(z)),
πB(z)′ = q′(w − τ1 − c1(1− α)− α(w2 − τ2)(F (qB(z))− F (eBz (qB(z)))))
+ E[α(X − qB(z))+ ∧ ζ ij(qB(z))]
≥ q′(w − τ1 − c1 − α(w2 − τ2 − c1)(F (qB(z)) − F (eBz (qB(z))))) ≥ 0.
(B.10)
• When p is in a region such that yB(z) = z,
πB(z)′ = q′(w − τ1 − α(w2 − τ2)(F (qB(z)) − F (eBz (qB(z)))))
+ E[α(X − qB(z))+ ∧ ζ ij(qB(z))]
≥ q′(w − τ1 − α(w2 − τ2)(F (qB(z))) − F (eBz (qB(z)))) ≥ 0. (B.11)
In (B.9) to (B.11), the last inequality comes from our assumption w− τ1 − c1 ≥ α(w2 −
τ2 − c1). Hence, S’s profit is increasing in δ.
Proof of Proposition 3.9.
Part 1: Let z = γ(qC), we observe from Part 3 of Proposition 3.5 that qB(z) ≥ qC . As
a result, πBS (z) ≥ πC
S (γ(qC)). In addition, we observe that πB
R (qB(z), z) ≥ πB
R(qC , z) =
πCR(q
C) because qB(z) is the retailer’s optimal decision.
Part 2: Define πiS(i, j) = maxy π
iS(y, j, q). Based on Part 2 of Proposition 3.5, we
know that γ(q) ≥ z∗ if we let q = qB(z∗). As a result, we observe that πBR(z
∗) ≤
πCR(q). Similarly, because γ(q) is the supplier’s optimal decision, πB
S (z∗) = πC
S (q, z∗) ≤
πCS (q, γ(q)). Hence proved.
Proof of Proposition 3.10. Proposition 3.10 can be proved by utilizing results shown
in Proposition 3.9. According to Proposition 3.9, there exists a z ≥ z∗ such that
πBR (z) ≥ πC
R(qC) and πB
S (z) ≥ πCS (q
C). Therefore, there exists a z∗ ≤ z ≤ z such
that πBR (z) ≥ πB
R (z∗) and πB
S (z)πBS (z) ≥ πC
S (qC). Let σS = πB
S (z)/[πBR (z) + πB
S (z)].
Because πGT (q, y, z) = πB
R (z) + πBS (z), it is clear that π
GS (q, y, z) = πB
S (z) ≥ πCS (q
C) and
πGR(q, y, z) = πB
R (z) ≥ πBR(z
∗). Based on the definitions of πGS and πG
R , πGS (q, y, z) ≥
πCS (q
C) and πGR(q, y, z) ≥ πB
R (z∗) also hold because πG
S (q, y, z) ≥ πGS (q, y, z).
Appendix C
Proofs for Chapter 4
Proof of Proposition 4.1. Note that πBRi(qi, qj) is continuous in (qi, qj). In addition,
πBRi(qi, qj) is strictly concave in qi because ∂2πB
Ri(qi, qj)/∂q
2i < 0. These two conditions
along with the fact that qi belongs to a compact subset of Euclidean space, we know
there exists a pure strategy Nash Equilibrium for retailers’ problems based on a theorem
attributed to Debreu (1952), Glicksberg (1952), and Fan (1952) (see Fudenberg and
Tirole 1991 for details).
Note that this alone does not guarantee the uniqueness of the Nash Equilibrium. To
prove the uniqueness, we first define I1(q1, q2) = ∂πBR1
(q1, q2)/∂q1 = 0 and I2(q1,q2) =
∂πBR2
(q2, q1)/∂q2 be the reaction curves of retailer 1 and 2, respectively. The uniqueness
can then be proved by showing that reaction curve I1(q1, q2) = 0 and I2(q1, q2) = 0 have
exactly one intersection.
Let (qL1 , qH1 ) and (qL2 , q
H2 ) be the lower/higher bound for q1 and q2. To ensure that
I1(q1, q2) = 0, it is easy to check that q1 = qL1 when q2 → ∞ and q1 = qH1 when
q2 → 0. Similarly, to ensure that I2(q1, q2) = 0, q2 = qL2 can be obtained when q1 → ∞
and q2) = 0, q2 = qH2 can be obtained when q1 → 0. Define q(1)2 and q
(2)2 such that
I1(qL1 , q
(1)2 ) = 0 and I2(q
L1 , q
(2)2 ) = 0. It is easy to check that q
(1)2 = ∞ > qH2 > q
(2)2 .
Similarly, define q(1)2 and q
(2)2 such that I1(q
L1 , q
(1)2 ) = 0 and I2(q
L1 , q
(2)2 ) = 0. One can
show that q(2)2 > q
(1)2 .
120
121
Next, define ∂q(1)2 /∂q1 (resp. ∂q
(2)2 /∂q1) be the derivative of I1(q1, q2) = 0 (resp. I1(q1,
q2) = 0) at (q1, q2). Based on (4.14), we obtain ∂q(1)2 /∂q1 < −1 and ∂q
(2)2 /∂q1 > −1.
Because ∂q(1)2 /∂q1 < ∂q
(2)2 /∂q1 and the fact that q
(2)2 > q
(1)2 and q
(1)2 > q
(2)2 from the
previous arguments, we concluded that the two reaction curves have exactly one inter-
section. Hence proved.
Proof of Proposition 4.2. Similar to Proposition 4.1, We first argue that a pure
strategy Nash Equilibrium exists based on a theorem attributed to Debreu (1952),
Glicksberg (1952), and Fan (1952) (see Fudenberg and Tirole 1991 for details) This is
based on the facts that πCRi(qi, qj) is continuous in (qi, qj) and is strictly concave in qi ,
where qi belongs to a compact subset of Euclidean space. Then we show that the two
reaction curves have exactly one intersection. Hence proved.
Proof of Proposition 4.3. By the definition of qAi shown in equation (4.10) and plug
it into (4.15), we observe that ∂πCRi(qAi , q
Cj )/∂qi > 0. Because πC
Riis concave in qi, it
follows that qCi > qAi . Similar arguments can be applied to show qCi > qBi . Hence
proved.
Proof of Proposition 4.4. Recall that in Proposition 4.3, we show that for any given
w, qCi (w) ≥ qBi (w). Because qCi is decreasing in w, we can find a w > wB such that
qBi (wB) = qCi (w). Because πB
S (wB , yB, qB1 (w
B), qB2 (wB)) ≤ πC
S (w, yC , qC1 (w), q
C2 (w))
from the fact that w > wB , we obtain πBS (w
B , yB , qB1 (wB), qB2 (w
B)) ≤ πCS (w, y
C , qC1 (w),
qC2 (w)) ≤ πCS (w
C , yC , qC1 (wC), qC2 (w
C))
Proof of Proposition 4.5. From (4.1) and (4.3), we observe that πARi(qA, w) ≤ πB
Ri(qAi ,
qBj ) because E[φi(i, j)] ≥ 0. In addition, we have πBRi(qAi , q
Bj ) ≤ πB
Ri(qBi , q
Bj ) from the
fact that qBi is retailer-i’s optimal order quantity when retailer-j orders qBj in B. Similar
arguments can be applied to show that πARi(qA, w) ≤ πC
Ri(qCi , q
Cj ) when the condition in
the proposition description hold. Hence proved.