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

3

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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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.