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6/07 p:lC|RC/DateDue.indd—p.1

ESSAYS IN SUPPLY CHAIN CONTRACTS

By

Wenming Chung

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Marketing and Supply Chain Management

2007

ABSTRACT
ESSAYS IN SUPLPY CHAIN CONTRACTS
By

Wenming Chung

This dissertation investigates two supply chain contracts that are widely utilized
in various industries: the quantity flexibility contract and the price discount scheme.
Motivated by the need of a more through understanding of the implications of supply
chain contracts, we develop three supply chain contract models in three different
types of supply chain structures: (1) the quantity flexibility contract with incentives in
a one- supplier, one-buyer supply chain (2) the quantity flexibility contract and price
markdown scheme in a two-supplier, one-buyer supply chain (3) the price discount
scheme in a three-echelon supply chain consisting of a supplier, an original
equipment manufacturer (OEM), and a buyer. The first essay focuses on designing a
new quantity flexibility that seeks to balance the inventory risk between the
traditional QF contract and the price-only contract for both the buyer and the supplier.
The second essay analyzes the competition between the quantity flexibility contract
and the price discount scheme. The third essay investigates the impact of price
discount on retail price, capacity planning, and stocking policy decisions.

We found in essay 1 that the new quantity flexibility contract with incentives is
able to achieve supply chain coordination. It also allows firms to identify areas where
the new contract is able to Pareto—improve from the traditional quantity flexibility
contract under certain circumstances. In essay 2, we present the conditions under

which flexibility is more desirable and areas where the price discount has an

advantage. We found that in most cases the buyer will be better off simply using only
one supplier. In essay 3, we show that firms will need to increase the capacity and
stocking level to cope with the impact of price discount. We found that when price
difference is not significantly large and the component’s arrival time is uncertain,
firms will be better off keeping higher inventory beyond the effective time of the new
price. We address discussions, managerial implications, and provide directions for

future research opportunities in each of the three essays.

Copyright by
Wenming Chung
2007

ACKNOWLEDGMENTS

First and foremost, I would like to thank Professor Srinivas Talluri for his
guidance and support during the wonderful years I spent at Michigan State
University. He spent many hours meeting and discussing the details of this
dissertation with me. I am extremely grateful for the role he played throughout the
whole process.

I want to thank Professor Ram Narasirnhan for the care with which he reviewed
this dissertation. I would like to thank him for his assistance in enhancing the quality
of my research work and written presentations during my doctoral program. I am
also grateful to Professor Morgan Swink and Professor Anthony Ross for agreeing to
serve on my dissertation committee.

There have been many other faculty members, department staff, and Ph.D.
students who have helped me during my doctoral studies. I would like to express my
appreciation for their help.

I also have made several wonderful friends during my Ph.D. program. I want to
thank each of them for being part of my life for the past four years.

I want to especially thank my best friend, Dr. Douglas David, for his continuing
support. I would not be able to make it through these years without his
encouragement.

Finally, for the unconditional love they showed through this process, this
dissertation is dedicated to my mother, Alu Chiu and my late father, J ungchuan

Chung, who I know would have been so proud.

TABLE OF CONTENTS

LIST OF FIGURES viii

Chapter 1 Introduction ........................................................................ 1
Chapter 2 Quantity Flexibility Contract with Incentives ................................. 6
2.1 Introduction ........................................................................... 6
2.2 Literature Review .................................................................... 9

2.3 The Traditional QF Contract and The Performance Benchmark ............. 12
2.3.1 The centralized system: a benchmark ....................................... 13

2.4 The Quantity Flexibility Contract with incentives .............................. 15

2.5 The QF i Contract and Supply Chain Coordination ............................. 20
2.6 Comparative Analysis Between (a,a),w1,w2) and (Ct/,0), w) : the Supplier’s
Perspective .......................................................................... 24

2.6.1 The coordinated QFi/QF supply chains ................................... 24

2.6.2 The general QFi/QF supply chains ....................................... 28

2.7 Numerical Experiments ............................................................ 30

2.7.1 Comparing the QFi and the QF contracts: varying discount pricing. ...30
2.7.2 Comparing the QFi and the QF contracts: varying the

flexibility parameter .......................................................... 35

2.7.3 Comparing the QF i and the QF contracts when the supply chain is
coordinated ..................................................................... 39
2.8 Conclusions and Discussions ...................................................... 44
Appendix of Chapter 2 .................................................................. 47
Chapter 3 OF Contract in a Two-Supplier-One-Buyer Supply Chain... 53
3.1 Introduction and Literature Review .............................................. 53
3.2 Problem Setting ..................................................................... 57
3.3 The 2-1 QF Models ................................................................ 59
3.3.1 The system view: the benchmark .......................................... 63
3.4 Solutions and Optimality Conditions for Case I ............................... 64
3.5 Solutions and Optimality Conditions for Case II ................................ 70
3.6 Numerical Examples ................................................................. 73
3.6.1 Competitive pricing and its impact on the buyer’s decisions ............ 73
3.6.2 Flexibility and its impact on the buyer’s decisions ........................ 78
3.6.3 Discussion ....................................................................... 84
3.7 Conclusions and Future Research Directions. ................................. 85
Appendix of Chapter 3 .................................................................. 87
Chapter 4 Price markdown scheme in a three-echelon supply chain... 97
4.1 Introduction ........................................................................ 97
4.2 Literature Review ................................................................. 100

vi

4.3 Problem Setting ................................................................... 103
4.4 The Decentralized Supply Chain Model in The Presence of a Price

Discount ............................................................................ 104
4.4.1 Demand structure ............................................................. 106
4.4.2 The supplier’s model ........................................................ 107
4.4.3 The buyer’s model ........................................................... 109
4.4.4 Finding the optimal pricing and stocking policies ........................ 111

4.4.4.1 Optimal pricing and stocking policies for the buyer ............... 111
4.4.4.2 Optimal pricing and stocking policies for the supplier ............ 114

4.5 The OEM’S Model ................................................................ 118
4.5.1 Model I: deciding optimal stocking policy for the new period. . . . . . 119
4.5.2 Model 11: deciding optimal “ending” inventory policy .................. 120
4.5.2.1 ND>NW ................................................................. 123
4.5.2.2 NW>ND .................................................................. 124

4.6. Numerical Experiments .......................................................... 126
4.6.1 On the price sensitivity ...................................................... 126
4.6.2 On the characteristics of the random demand error ...................... 129
4.6.3 On the OEM’s ending inventory during price breaks ................... 132
4.6.4 Discussion ..................................................................... 135
4.7 Concluding Remarks and Discussion ........................................... 136
Appendix of Chapter 4 ................................................................ 138
Chapter 5 Concluding Remarks .................................................... 141
5.1 Summary of Contributions ....................................................... 141
5.1.1 Chapter 2 ....................................................................... 141
5.1.2 Chapter 3 ...................................................................... 142
5.1.3 Chapter 4 ...................................................................... 143
5.2 Future Research Directions ...................................................... 143
BIN LIOGRAPHY ........................................................................ 145

vii

Figure 2.1

Figure 2.2
Figure 2.3
Figure 2.4

Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8

Figure 2.9
Figure 2.10

Figure 2.11
Figure 2.12
Figure 3.1

Figure 3.2
Figure 3.3
Figure 3.4‘

Figure 3.5

Figure 3.6

Figure 3.7
Figure 3.8

Figure 3.9

Figure 4.1
Figure 4.2

LIST OF FIGURES

q, vs. q2 under QFi: W, =42 ................................................. 31
Supplier/buyer profits QFi vs. QF: W, =42 ............................... 32
Minimum flexibility for QF to remain executed ......................... 33
Supply chain performance: QFi vs. QF: W, =42 ......................... 34
q, vs. 612 under QFi: W, =42, W2 =40 ..................................... 37
Supplier/buyer profits QFi vs. QF: W, =42, W2 =40 ....................... 37
Supplier profit difference (QFi-QF): W, =42, W2 =40 .................... 38
Total profits QFi vs. QF: W, =42, W2 =40 .................................. 38
Supplier/buyer profits QFi vs. QF at coordination: WQF =38.85 ....... 40

q, vs. qz under QFi at coordination: W, = WQF .......................... 42
Supplier/buyer profits QFi vs. QF: W, =42 ................................ 43
Coordinating pricing in QFi and QF conuacts ........................... 43
Forecast/ordering quantities 2-1 QF vs. 1-1 QF: W, =42 ................ 75
Buyer/supplier profits 2-1 QF vs. 1-1 QF: W, =42 ....................... 75
Minimum flexibility for the QF mechanism: W, =42 .................... 77
Supply chain efficiency 2-1 QF vs. 1-1 QF: W, =42 ..................... 77
Forecast/ordering quantities 2-1 QF vs. 1-1 QF: W, =42, W2 =40,

varying a) ..................................................................... 80
Forecast/ordering quantities 2-1 QF vs. 1-1 QF: W, =42, W2 =40,

varying a ..................................................................... 80
Buyer profits 2-1 QF vs. 1-1 QF: W, =42, W2 =40 ....................... 82
Supplier profits 2-1 QF vs. 1-1 QF: W, =42, W2 =40 .................... 83
2-1 QF chain performance: W, =42, W2 =40 .............................. 83
Optimal pricing: varying price sensitivity ............................... 128

Optimal capacity and stocking policies: varying price sensitivity. . ..128

viii

Figure 4.2
Figure 4.3

Figure 4.4

Figure 4.5
Figure 4.6

Figure 4.7
Figure 4.8

Optimal capacity and stocking policies: varying price sensitivity. 128

Optimal Profits: varying price sensitivity ................................ 129

Optimal capacity and stocking policies: varying 0', at two levels

of ,us .......................................................................... 130
Optimal profits: varying 0’6. at two levels of ,U6 ...................... 131
Optimal pricing: varying 0",, ............................................. 131

Optimal stocking policies LTD vs. D: varying price difference ...... 133
OEM profits LTD vs. D: varying price difference ...................... 135

ix

CHAPTER 1 INTRODUCTION

This dissertation seeks to contribute to better understanding of how firms can improve
their profits in their best interests or in line with the system objective via proper design of
contractual agreements. It is well documented that when firms’ actions are aligned with the
global objective in the presence of a contract, a decentralized supply chain may possibly
achieve the same level of profits as that of a centralized supply chain. Thus in this case, the
supply chain is considered to be coordinated. When the agents in a supply chain act in
their best interests, 3 well-know phenomenon referred to as “double marginalization” occurs,
and supply chain efficiency decreases. Thus, supply chain coordination is not possible.
Nonetheless, research in both areas is equally important, because the former provides
guidance for firms to achieve higher supply chain performance and the latter describes
phenomena that are most commonly observed in real-world business practices.

The benefits of information sharing and the lengthened physical spans of supply chains
have allowed utilization of forecasting mechanisms between firms to mitigate the effect of
supply chain uncertainties. On the other hand, buffer stock remains an effective way for
firms to cope with unexpected demand. It is common that firms negotiate agreements on
both forecasting mechanisms and buffer stock level to facilitate their buy-and-sell activities.
This can be best described by the well—known “quantity flexibility” contract. Although it is
widely utilized by firms in various industries, the quantity flexibility contract has received
little attention in the supply chain contract literature.

On the other hand, rapid technology innovations have brought about benefits such as
frequent launch of new products and constant reduction of production costs to name a few.

Among the consequences is the frequent price discount scheme that is widely observed in the

high tech industry, e.g., the personal computer and cell phone industries. The margins in
the manufacturing sector have been known to be low. With short product lifecycles and
frequent price discounts, inventory management remains a major challenge for firms in
supply chains in today’s dynamic business environment. Although work in price discount
or quantity discount is rich, extant research does not thoroughly address the various issues
revealed by the presence of price discounts.

This dissertation intends to study the effects ofthe quantity flexibility contract and the
price discount scheme on the operational decisions in supply chains. The bulk of research
in supply chain contracts appeared in the past decade. However, the majority has focused
on a simplified supply chain that consists of one buyer and one supplier. More complex
systems that involve multiple suppliers or three echelons have received relatively little
attention. Specifically, supply chain contract research involved with competing suppliers is
rare.

This dissertation contains three separate, analytic essays on three different supply chain
structures. Each essay studies either the quantity flexibility contract or the price discount
scheme, or a mixture of both. Additionally, each essay takes a position from the buyer’s
view, the supplier’s perspective, or from both standpoints. The goal is to study supply chain
decisions in the presence of a contract from different perspectives under a specific supply
chain structure. It provides foundations to future research for expanding research problems
into more complicated supply chains. It adds a new design of the quantity flexibility
contract to the literature, in the presence of competition in the contractual setting, and in a
contract that involves multi—echelon supply chains.

Chapter 2, entitled “The Quantity Flexibility Contract with Incentives”, studies a quantity
flexibility problem in a one buyer, one supplier supply chain in which the supplier offers

discount price for any units sold above the contracted quantity. The purpose is 0 shift

portion of the inventory burden back to the buyer, thus creating higher profits than what the
original quantity flexibility contract would allow. Chapter 2 intends to identify areas
where it is most appropriate for the supplier to introduce the discount incentives. On the
other hand, the buyer will choose not to execute the discount scheme if not expecting higher
profits than the quantity flexibility contract would offer. As such, when the new contract is
executed, it needs to allow Pareto improvement from the traditional quantity flexibility
contract.

Another goal of this research is to examine whether the system can achieve supply chain
coordination under the new quantity flexibility contract. It is known that the quantity
flexibility contract allows a decentralized supply chain to reach the centralized efficiency.
This research intends to design a new quantity flexibility contract with incentixes that
performs equally well as the traditional version.

The analysis makes very general assumptions; it is applicable to any demand
distribution. The new contract offers “flexibility” in terms of the ability to identify the
coordinating conditions, yet this can be achieved with easy access to information needed.
Much emphasis is placed on comparing differences in inventory decisions and supply chain
performance between the original and the proposed new quantity flexibility contracts. It
aims to establish a framework that can assist managers to easily identify areas where either
one has an advantage over the other, thus guiding them to make the right decisions.

Chapter 3, entitled “ Quantity Flexibility Contract in a Two-Supplier-One-Buyer Supply
Chain”, considers supplier competition in a two-echelon supply chain. In this supply chain,
one supplier offers the quantity flexibility contract while the other offers better price for the
same, but not identical, component. This chapter analyzes the effects of the competition on
the buyer’s and supplier’s decisions. It intends to examine the tradeoffbetween the benefits

from the quantity flexibility and from cost saving. It takes the buyer’s standpoint and

investigates his decision making behavior in the presence of the two options. It also
provides insights to the suppliers as to whether the incentives that they offer are attractive
enough to the buyer.

Similarly, chapter 3 investigates the performance of the focused supply chain with
competing suppliers. One can expect that the buyer will take advantage of the competition
and make the most benefit out of it. However, from the system’s point of view, such a
multi-competing—supplier supply chain may likely result in supply chain deficiency. This
research intends to specify the optimal conditions and the corresponding profits for each
agents and the entire system, thus analyzing what contributes to the supply chain deficiency
should it occur. A further step is to propose managerial insights as to actions or strategies
that allow system performance to improve.

The analysis needs no specific demand distribution. However, certain types of
distributions may be favored due to constraints in computational power of computers.
Commonly, buyers have multiple sources of supply for the same components. Contract
research in this regard is rare primarily because of technical restrictions. AS such, this paper
not only adds a significant contribution to literature but also provides managerial insights to
practitioners. It enables managers to estimate how much flexibility is worth and the ideal
allocation between the two competing suppliers.

Chapter 4, entitled “Price Markdown Scheme in a Three-Echelon Supply Chain”,
considers a price discount scheme involved in a supply chain consisting of a supplier, an
original equipment manufacturer (OEM), and a buyer (retailer). The supplier is able to
offer price discount on a regular basis due to innovation in manufacturing technologies.
This research extends the discount literature by considering three agents in a supply chain as

well as the emerging issues of delivery and demand uncertainties that play a critical role in

agents’ decisions.

When a supplier takes the initiative and offers a new, lower, all-unit price markdown, it
resembles price-only contract rather than the quantity discount mechanism. Thus, such a
supply chain discount problem is known not being able to achieve supply chain coordination.
However, this does not degrade the importance of the problem, because the price markdown
scheme is executed frequently by many firms across various industries, especially in the high
tech industry. As such, this research is concerned with supply chain decisions at the
individual level rather than from the system’s view. It analyzes how the new pricing should
be specified and how the corresponding capacity planning and inventory stocking strategies
along the chain can be decided.

The development of this research is based on stochastic and price sensitive demand
functions. A variety of price-sensitive demand functions are available in the economic
literature, a common property of which is that demand decreases as price increases. The
only additional requirement for the model in chapter 4 is that the demand function needs to
be concave. Specifically, this research examines the effect of the price discount on the
inventory stocking level beyond the price break by explicitly considering the role of
lead-time uncertainty in such decisions. The analysis can facilitate manager’s decisions
when the material’s arrival time is not somewhat unpredictable.

The concluding remarks are provided in chapter 5, where we review the contributions of
this dissertation. The study of supply chain contracts enhances the understanding of the
impact of contractual agreements on firms’ financial performance, thus providing the
fiindamental know-how in designing efficiency-enhancing and Pareto-improving contracts.
This dissertation is most appropriate for a potential audience such as supply chain managers,
logistics managers, sales representatives, as well as academicians in the field of operations

management or management science.

CHAPTER 2 QUANTITY FLEXIBILITY CONTRACT WITH
INVENTICES

2.1 Introduction

Wistron Corporation, a Taiwan-based company that employs 20,000 workers worldwide,
is one of world’s largest PC original equipment manufacturer (OEM) based on revenue, with
a large customer base of global, branded IT companies. Wistron develops high-technology
products such as notebook and desktop computer systems, servers and storage systems, and
networking and communication systems. To better the relationship with its customers and
secure a high market share of the PC business, Wistron agrees to fully commit to the rolling
13-week forecast provided by its customers if material supply is not an issue. Furthermore,
Wistron is also willing to prepare additional buffer inventorybased on the forecast in case of
unforeseen demand. On the other hand, Wistron’s customers are not free of material
liability. If obsolete inventory occurs from the rolling forecast, Wistron’s customer is
obligated to absorb the material burden to a certain extent. Such acontractual mechanism is
called the Quantity Flexibility (QF) contract in the supply chain management literature.

The QF contract is widely observed in various industries. For example, Sun Microsystems,
IBM (High tech/PC), Toyota (automobile), Nippon Otis (Elevator) are all reported to employ
the QF contract with their suppliers or customers (Tsay, 1999).

In general terms, the QF contract can be described as following: the buyer first provides
forecast q to the supplier, the supplier then decides her production quantity Q. When the
demand is realized, the buyer will place a purchase order to the supplierat quantity r.

Based on the QF contract, the supplier will need to produce a certain percentage (1 of
buffer stock above the forecasted quantity, and the buyer needs to buy a certain percentage

(1 — (o) of the forecasted number, no matter what the demand is (Tsay, 1999).

It is clear that the QF contract works in favor of the buyer compared with the traditional
price-only business behavior. Under the price-only contract, the buyer places orders to the
supplier to fulfill demand observed from the market; the supplier produces exactly the
quantity that the buyer orders. As such, the buyer bears all the inventory riskwhile the
supplier’s profit is linear to the order size. With the QF contract in place, the buyer does not
need to assume 100% liability for the forecasted quantity, yet the supplier will have to ensure
that extra buffer stock is available for the buyer above the forecasted quantity. As a result,
the supplier shares certain level of risk under the QF contract. Whenthe level of
competition is high and the buyer has many sources of supply to choose from, the supplier is
likely to offer incentives such as the QF contract to secure business from the buyer.

In this research, we intend to study a revision of the QF contract that can potentially
partially reduce the burden of inventory risk for the supplier that she would have to bear
when offering the QF contract. We investigate a supply chain in which the buyer and the
supplier agree upon a QF contract for the buy-and-sell activities. To encourage the buyer to
purchase more, the supplier also offers a discount for any units sold above the QF-contracted
quantity. However, these units at the discount price are not protected by the QF contract;
they can not be returned even if they are unused eventually. As such, the supplier offers a
menu of contracts with different component (transaction) prices, and the buyer needs to
decide how many units to buy under the QF contract and how many to purchase at the
discount price. We name this contract as the quantity flexibility with incentives, hereafter,
the QF i contract. The lower price will encourage the buyer to consider to buy more than the
contracted quantity. However, having to bear all the risk of inventory of the discounted
units may prevent the buyer to purchase all that is needed at the discounted price.

Consequently, the tradeoff for the buyer is to value the cost saving for the additional, cheaper

units and the potential loss from carrying too much inventory should the demand be lower
than the estimated level.

The motivation of this study is to take the supplier’s perspective and identify the
conditions under which the supplier will be benefit more by offering the QFi contract
contract than by offering the traditional QF contract. When supplier offers theQF i contract,
She expects to stimulate the buyer to increase the order size, and thus raising her profit level.
However, as will be detailed in the later sections, the QFi contract can actually hurt the
supplier if not introduced at the right conditions. In such cases the supplier will be
better-off to stick to the traditional QF contract and not to offer any discount incentive. We
intend to identify the conditions under which the supplier should and should not offer the QFi
contract.

One common theme in the research stream of supply chain contracts is to seek the
conditions under which the individually motivated buyer and/or supplier can act in line with
the Optimal system-wide profit through proper designs of mutually-agreed contracts. It is
well-known that the whole chain’s profits level can reach the highest level when decisions
are made centrally in a supply chain. This condition is termed as supply chain coordination
in the supply chain contract literature (Pastemeck, 1985; Weng, 1995; Tsay, 1999; Taylor.
2002; Cachon and Lariviere, 2005; Berstein and Federgruen, 2005). The system-wide
optimal profit level normally is used as a benchmark to evaluate the performance of a supply
chain. Supply chain efficiency, defined as decentralized chain profit divided by the
centralized chain profit, is the most cited metric to estimate the performance of a supply
chain when there is a contract in place. It is easy to show that the total profit of a
decentralized supply chain is always smaller than or equal to the centralized profit. When a

supply chain efficiency reaches 100%, the supply chain is considered coordinated.

The QF contract is one of the several contracts in literature that have been proven to be
able to achieve supply chain coordination. Tsay (1999) showed that if the transaction price
between the buyer and the supplier is set under certain conditions, the supplier will be
induced to produce the quantity that will result in the highest expected profit level for the
entire chain. We will show that the revised QF i contract will be able to achieve supply
chain coordination as well. The remainder of the chapter is organized as follows. Section
2.2 discusses the related literature in the area of supply chain contracts. The problem
setting is laid out in section 2.3. The new QF i contract is forwarded in section 2.5. In
section 2.5, the topic of supply chain efficiency under the QF i contract is addressed. In
section 2.6, the QFi contract is compared with the traditional QF contract. In section 2.7,a
series of numerical examples are presented and managerial implications are discussed. The
chapter concludes in section 2.8 with the contributions of this research and suggests future

research opportunities.

2.2 Literature Review

It is well-documented that when supply chain’s decision making is aligned with the
global objectives of the entire chain, it is considered a coordinated supply chain (Pastemeck,
1985; Taylor. 2002; Sahin and Robinson, 2002; Cachon and Lariviere, 2005). This can be
achieved relatively easily when there is one single owner of the entire chain. However, it is
generally the case that a supply chain consists of independent agents who act only in their
own best interests, i.e., a decentralized supply chain. As such, their goal pursuing often
leads to sub-optimal results for the entire system, thus supply chain deficiency occurs. For
instance, a buyer and a supply in a supply chain may utilize inventory policies differently

from a central owner, resulting in the well-known phenomenon of “double marginalization”

(Spengler, 1950; Tirole, 1988).

A common cure for supply chain deficiency is to utilize supply chain contracts that could
possibly allow Pareto improvement in the supply chain or even achieve supply chain
coordination (Iyer and Bergen, 1997; Cachon, 2004). The general idea is to install rules that
guide business conduct between agents in a supply chain to be aligned with the best interest
of the system (Tsay, 1999). Supply chain contracts may consist of financial or
non-financial mechanisms. The former are the most studied. Common supply chain
contracts include mechanisms such as quantity discount (Chiang et al., 1994; Weng, 1995;
Cachon, 2004; Tomlin, 2003), price discount and rebate (Banstein and F edergruen, 2005),
sales rebate (Taylor, 2002; Krishnan et al., 2004), buybacks (Pastemact, 1985; Lariviere,
1999), revenue sharing (Cachon and Lariviere, 2005), quantity flexibility (Tsay, 1999),
forecast-commitment (Durango—Cohen and Yano, 2006), quick response (Iyer and Gergen,
1997), to name a few.

We focus on the QF contract in this research. As mentioned, the QF contract is widely
observed in various industries. However, it has received relatively little attention in
literature. The QF contract allows the buyer’s ultimate purchase quantity to deviate from
the original estimation (Lariviere, 1999). As such, the supplier agrees to bear the risk of
inventory to a certain extent that she would not be liable for if the QF contact is not
implemented. Such a mechanism functions similarly to the buyback contract in the sense
that in both cases, the supplier absorbs unsold units. But the extent to which the supplier
resumes inventory responsibility may differ between the QF contact and the buyback
contract, based on the contract agreements.

The seminal work in the QF contract is attributed to Tsay (1999) He considers a supply
chain consisting of one buyer and one supplier. The buyer provides aforecast to the
supplier and the supplier decides the production quantity based on the forecast information as

well as the contract restriction. Tsay (1999) formulates a general QF problem in which not

10

only the buyer is granted additional availability above the forecasted quantity, but alsohe is
liable for only portion of the estimated demand. He found that under the QF contract, the
supplier will produce exactly the contracted quantity. As a result, the forecast directly
affects the final availability of the component (product). Furthermore, Tsay showed that the
supply chain can be coordinated utilizing the QF mechanism. He identified the component
pricing scheme that will induce the buyer to respond in a manner aligned with the system’s
desirable outcome. However, identifying the component price that coordinaes the supply
chain via the QF contract requires the knowledge of the buyer’s demand distribution.

Sethi et a1. (2003) studied the QF contract in a setting with multi-periods and two sources
of component supplies. The buyer is allowed to purchase a two distinct time periods, with
the size of the second order being constrained by the QF contractual parameter. However,
these units for the second order are sold at a higher price than that in the first order. On the
other hand, the buyer will also have an option to buy from the spot market for any desired
quantity at the market price, which can be higher or lower than the supplier’s price. Under
these conditions, they identified the optimal order quantities from the supplier in both
periods, as well as the order size from the spot market in the second period While their
findings are not unexpected, they demonstrated how one can incorporate information updates
into the QF contract decisions. Wu (2005) conducted a similar research on the QF contract
with Bayesian updating. He followed Tsay’s (1999) framework and allowed multiple
updates of demand information before the ultimate purchase order. He concluded that QF
contract works in favor of the buyer. It allows both agents to share the benefits from
information updating.

Tsay and Lovejoy (1999) studied the QF contract under rolling-horizon planning. In
this research, they focus on understanding how the greatest benefits can be achieved under

the QF contract along the planning horizon and how much to pay for it. They explored

11

issues such as the impact of the system flexibility on inventory characteristics and the
patterns by which forecast and order variability propagate along the supply chain. They
found that, all else equal, increasing flexibility will reduce the buyer’s cost but at the expense
of the supplier’s cost. They also found that the QF contract may dampen the transmission
of order variability throughout the chain, thus potentially retarding the well-known “bullwhip
effect”.

Motivated by these findings, we study a variation of the QF contract by adding the
discount scheme on top of the QF mechanism from the supplier’s perspective, in the hope of
reducing the increased inventory risk that the supplier needs to absorb inthe QF contract
agreements. The purpose of this research is to design a new QFi contract that can
potentially reduce the inventory risk level and inorease profits for the supplier compared with
that from a traditional QF contract setting. With this new QFi contract, the supplier will
have an option as to whether to offer the new QFi contract or to stick to the original QF
agreements. We will show in our analyses that this new QFi contract is able to achieve the
supply chain coordination, thus performing equally well when compared to the traditional QF
contract. But the two QF contracts achieve 100% supply chain efficiency in different
conditions. The ultimate goal is to identify the conditions under which the supplier should

propose to offer the new contract and the conditions under which the supplier should not.

2.3 The Traditional QF Contract and The Performance Benchmark

We consider a supply chain that consists of one supplier and one buyer. The supplier
produces and sells components to the buyer, and the buyer makes the final product and sells
it to the end customers. The QF contract, (a, w) ,is in place in this supply chain. As such,
the buyer observes the market demand D, which is assumed stochastic, and releases forecast

q to the supplier; the supplier will then decide the production quantity Q. Based on the QF

12

contract, the supplier is obligated to produce at least (1+ a)q units and the buyer is liable
for buying at least (1— w)q units. To simplify the analysis, we consider only one
component and one product in single period.

Tsay (1999) showed that under the QF contract, the supplier’s optimal production
quantity is (1+ a)q . In other words, the supplier will produce only the minimum
contracted quantity and no more than that. As such, the buyer is able to decide the total
availability of the component via the forecasted quantity. So under the traditional QF
contract, the task for the buyer is to decide the optimal forecast q so as to lead the supplier
to produce (1+ a)q so that the buyer’s expected profit can be maximized. Consequently,

the buyer is clearly the leader of this Stackelberg game.

2.3.1 The centralized system: a benchmark

The centralized system is normally used as a benchmark to evaluate the efficiency of a
decentralized supply chain. When both the supplier and the buyer are owned bya single
firm, the entire supply chain has a single decision-maker. The total expected profitof the

centralized chain can be expressed as following:

H. =(ID-C)Q-(ID-V)15‘3(Q-D)+
where Q is the quantity produced, p is the retail price, c is the production cost, v is

the salvage value, D is the random demand, and F is the cumulative density firnction

(c.d.f.) of D . It is well documented that when the firm produces

Q: = F -1 (p — C / p - V), the centralized supply chain achieves the highest expected profit:

Qc
He : (P - V) deF (x) (Taylor, 2002). Any production/ordering quantity other than
0

Q: will disadvantage either buyer or supplier, or both, thus decreasing the entirechain’s

13

profit. Notice that the transaction cost W plays no role when there is a single owner for
the supply chain, because a markup of the component price between the two parties no longer
exists. Therefore, unlike the decentralized supply chain, the centralized chain is free of the
issue of “double marginalization” (Spengler, 1950).

As mentioned earlier, the most cited metric for evaluating a supply chain’s performance

is the supply chain efficiency, defined as: (II b + H5 VII: . When the supply chain

efficiency is equal to 1, the decentralized chain achieves supply chain coordination
(Pastemeck, 1985; Weng, 1995; Tsay, 1999; Donohue, 2000;Taylor, 2002; Cachon and
Lariviere, 2005; Krishanan et al., 2004; Berstein and F edergruen, 2005). In the next
section, we will describe and develop the new QFi contract model Throughout chapter 2,

we will utilize the following notation:

Notation

IIC : Total expected profit of a centralized supply chain
H, : Supplier’s total expected profit in a decentralized supply chain

II, : Buyer’s total expected profit in a decentralized supply chain

: Demand, a random variable

D
F : Cumulative density function (c.d.f.) of the random demand
f : Probability density function (p.d.f.) of the random demand

q : Forecast released by the buyer to the supplier under the traditional QF contract

q, : Forecast released by the buyer to the supplier under the QF contract with

incentives

l4

q 2 : Order size at the discount price under the QFi contract

p : Retail price of the final product

W, : Original component price under the new QF contract, W, = W.

w2 : Discounted component price

v : Salvage revenue of the component per unit for both the buyer and the supplier

c : Supplier’s production cost

Note: p>W,>w2>c>v>0

2.4 The Quantity Flexibility Contract with Incentives

Consider the QF contract with incentives. In order to earn more than she could under

the traditional QF contract, the supplier proposes that any unit sold above the contracted

quantity will be sold at a discounted price W2. However, these additional units can not be

returned no matter what the final demand. In other words, the supplier provides a menu of
contracts that combines the traditional QF contract and the discount scheme, presented as
(a, a), w, , w2) . The motivation is to encourage the buyer to buy more so that the supplier

can earn higher profits. Under this QFi contract, the buyer gets to decide the forecast

quantity q, under the QF contract and the additional firm order qz that he wants to

purchase at the discount price. Based on the contract, the supplier will produce

(1+ a)q, + q2 and the buyer is obligated to buy (1- 0))q, + qz. Any unit within the QF
contracted quantity (1+ a)q, is sold at W, and any unit of q2 is sold at W2; w, > W2 ,

a > 0 ,1 2 a) > 0. Notice that the buyer can choose to buy all that is needed via qz as

well. However, the full responsibility of inventory risk may prevent the buyer from doing

15

so. If the buyer places only a firm order q2 and chooses not to execute the QF contract,

i.e., q, = 0 , the problem becomes a price-only contract and it is well-known that
underproduction will occur. Besides, the entire chain will suffer from supply chain
deficiency and supply chain coordination can not be achieved. It is also straightforward that
when q2 =0, the problem becomes exactly the traditional QF contract problem. As a result,

the QFi contract seeks a balance of inventory risk somewhere in the middle between the

traditional QF contract and the price-only contract for both the buyer and the supplier.
Notice that the buyer remains the leader of the Stackelberg game under the new QFi

contract and has the control over the forecasting and purchasing decision making. As such,

we formulate the problem by maximizing the buyer’s total expected profit as follows:
H. = (p - Wi)(1+ 6061. + (p - W2)q2
- (p - wi)[E((1+ a)qi + qz - D)+ - E((1- w)q1+ ‘12 - D)+]
-(p-V)E((1-w)qi +612 -D)+ (2.1)
The first two terms describe the buyer’s profits if everything is sold. Under the QF contract,

the buyer can return as many as (a + COM, units. The second term describes the potential

unsold units falling within this limit that is eligible forretum and full refund from the
supplier. For any unsold units that exceed the returnable allowance, the buyer will have to
absorb the loss himself. These items are to be salvaged and the loss is described by the

third term. All unsold item’s revenue are deducted as indicated in last two terms. The

decision variables are q, and q, . Notice that q, represents the forecast under the QF

contract agreement while q, refers to a firm order at the discount price. Next, we explore

the property of this unconstrained nonlinear objective function.

16

Theorem 2.1

(1) The buyer’s expected profit function is jointly concave in q, and ‘12 .

1+a (Wl-VXP—Wz) .
(2) Let Z = T——' If 2’ > Z = , buyer wrll release a non-zero forecast
— (t)

_ (W2 T VXP - W1)
q, to the supplier.

(3)F 11 ' (2) 'r > h ' lf ’ F_1(M)_F—I(N) h
o owrn ,1 Z ,‘t’ ,t eo trma orecast q = ,w ere
g — p ' (z-l)(1-w)

 

M=(P—wr)2"'(P-W2) and N: 1(W1-W2) ,
(Z—1XP—Wi) (Z ‘1)(W1-V)

 

(i) if q: 2 qQF , then q: = qQF , the optimal discount order q; = O

. zF“‘(N)—F"(M)

(ii) otherwise, the optimal discount order q 2 = Z _ 1

 

a t _ — W
(4) Following (2)&(3), if I S 1 , then q, = 0 and q; = F l(LP:‘VL).

Proof. See Appendix of chapter 2.

Theorem 2.1(1) guarantees that if optimal solution q: and q; exist, they are unique to
maximize the buyer’s profit. However, technically, there is no guarantee that both q: and

q; will be positive at the optimum. Theorem 2.1(2) employs the measure of flexibility

introduced by Tsay (1999). Both a and a) are contractual parameters and are exogenous

to the model. It is straightforward to see that I must not be smaller than 1. Theorem

2.1(2) suggests that the flexibility needs to be large enough to attract the buyer to remain his

17

interest in using the QF contract, i.e., q : >0. Otherwise, the buyer will choose to purchase

all that is needed at the discount price. In addition, there are two extreme sets of
parameters. First, when (a, (o) = O , the QF contract does not exist and the problem

becomes a price-only contract problem. It is easy to Show that under the price-only

contract, the order quantity is always less than Q: , thus underproduction will occur and it

will not achieve the maximum profit for the entire chain (Tsay, 1999). In this case the
buyer assumes the full risk of inventory. Second, when (0 =1 , Z —> 00 , over-forecasting
will occur, thus leading to overproduction. In this case, the buyer does not have any
liability for the forecast quantity so he will be tempted to exaggerate the forecast that will
force the supplier to produce more than what is possibly needed. In either case supply chain

deficiency occurs. Detail discussion can also be found in Tsay (1999).

Finally, Theorems 2.1(3) and (4) present the solution forms of q: and q; . As
discussed, q ,* is the optimal forecast under the QF contract and q; is the optimal firm
order at the discount price. Note that q," and q S are both non-negative. So should

negative values occur for either q: and q; , they should be set to zero. When q; = 0 ,

. . , t _ _1 p — W2
the problem rs equivalent to the price only contract, thus q 2 - F (——). When
p — v

q ; = 0 , the problem degenerates into the traditional QF contract, thus q: will be the same

as the optimal qQF identified in Tsay (1999). Notice that the total availability is

(1+ a)q, + q; and the final purchase quantity can be expressed as

r. = D _L [(1 - (1))q,t + q; , (1 + a)q: + q;] . The expression denotes the point in the

18

interval that is closest toD. So when D < (1 —a))q; , r. = (1 -a))q,. +q;;
whenD > (1+ a)q; + q; , r‘ 2 (1+ a)q,. + q;; otherwise, r. = D.

The result of Theorem 2.1(2) provides a lower bound of the discount price W2 as shown

in the following Corollary.

"W V+ W —v
Corollary2.l If W2>y_v2:Z(P 1) P( 1 )
Z(p—W|)+(W,—V)

 

, the buyer will remain interested

in the executing the QF contract.

When the discounted price is lower than the lower bound W2 , the price discount becomes so

attractive to the buyer so that he will choose to execute his orders all at the discount price.

Additionally, W2 must also be greater than the production cost C. Otherwise, the

discounted units will cause a loss per unit sold and the purpose of making higher profits via

the combination of the QF contract and discount pricing will not be met. Note that W,

serves as the upper bound for W2 .

Corollary 2.2

(1)M is increasing in I

(2) N is decreasing in Z .

l9

w, —w2
(2.” IMP" W1)

 

Proof. M can be rewritten as 1- so it is clearly decreasing as

 

1 increases. On the other hand, it is straightforward that is decreasing in Z .

Z—

Thus, N is decreasing in Z. 0

In theorem 2.1 we solve for the equilibrium solution for the optimal forecast and optimal
discount order for the buyer and identify conditions under which optimal forecast is feasible
and non-zero. Since the supplier is the follower, she simply produces the quantity that
fulfills the contracted quantity and the discount order. In the next section we discuss the

coordination condition under the QFi contract.

2.5 The QFi Contract and Supply Chain Coordination

As indicated in section 2.3, Q: indicates the quantity that will lead to the highest

possible profit level for the entire supply chain. In this section, we investigate a pricing
scheme set by the supplier that incentivizes the buyer to choose a combination of forecast

and firm order quantities in line with the system-wide optimal solution.

Theorem 2.2 Under the QFi contract (a, to, WI , wz) , when the discount price w2 is set by

the following equation, the buyer will be induced to release (q ,c , q§) so that the total

availability Qép, will be equal to Q: , thus the individually motivated supply chain will be

coordinated:

20

_(z-1)(p-wi)(c-V)
p-V

 

W2 2W1

Proof. The total availability given (qf , q§) is Q5“- : (1+ 0!)q,c + q§ = Q: at supply

chain coordination. Thus, from (A23) in the Appendix we have

. (p—w,)Z—(p—w2) p—c .

 

 

 

 

:1_ w,—w2 : _c—v
(P-Wi)(,t’-1) P-v
z) w, _w2 : (p—w,)(z—1)(c-v)
p—v
3 w,=w,_(P‘W1)(Z‘1XC‘V) D (,2)

p-V

Condition (2.2) defines the discount pricing scheme for the QFi contract. Given the
flexibility negotiated as well as the retail price, salvage value and the supplier’s production
cost, w2 in (2.2) will be able to induce the buyer to respond with optimal inventory policies
that are in line with the centralized decision. As such, the decentralized supply chain will
be able to achieve the same profit level as the centralized chair does, thus achieving the state
of supply chain coordination. Consequently, the supply chain efficiency is at 100% under
the QFi contract at the pricing scheme in condition (2.2). In addition, it can be observed
that w2 is decreasing in flexibility measure I . When I is large, the buyer is in a more
powerful position. So the supplier is willing to sacrifice by assuming higher inventory risk

in order to gain orders from the buyer. As such, the supplier will be inclined to offer a

21

larger discount (lower w,) to encourage the buyer to enlarge the firm order size q 2 . By
doing so the inventory burden shifis partially back to the buyer’s side.

We can also observe that there is one-to—one mapping between the discount price w2
and flexibility 1. As such, when 2’ is agreed upon by both parties, there will be a unique
w2 that will achieve the supply chain coordination. Conversely, when w2 is given, there

will be a unique 2’ that achieves supply chain coordination.

Proposition 2.1

W —W —v ,
+( 1 2)(p ) such that QQF, =QC

(6 - V)(p - W1)

 

(1) Given w2 , there exists a unique ZC =

(2) When the supply chain is coordinated under the QFi contract, the QF mechanism must be
active between the buyer and supplier.
Proof. Prop. 2.1(1) is straightforward. Rearranging the terms in (2.2) we have

(W1 TW2XPTV)
(CTVXPT W1) '

 

1c = 1+ For Prop.2.1(2) it suffices to Show )(c 2 3:: .

Rearrange R.H.S:

(w, -V)(p-W2) = (W2 -V)(p-Wi)+(wi -W2)(p-V) 2,, (w, -w2)(p-V)
(W2 T VXP T W1) (W2 TVXP T W1) (W2 TVXPT W1)

 

 

(WI—W2)_(W1TW2)

w2>c :> (c—v) (wz—v) >0
(W, T VXP T W2)
(W2 T VXP T W1)

 

22> =1.

Thus by Theorem 2.1, q,c must be non-zero. D

22

Proposition 2.1 indicates that the coordination status fulfills the conditions that is necessary
for the unique optimal solutions described in theorem 2.1 and corollary 2.1.Although there is
a unique 2', when achieving the optimal system profits, the associated (01,60) do not need
to be unique. There will be a menu of (a,a)) that can result in the same 2’.-
Consequently, the pricing decision and the flexibility negotiation can work two ways:
flexibility parameters can be decided first, followed by the pricing scheme, or price can be
decided first, followed by the decisions of the flexibility parameters. Either way here is a
chance to achieve 100% supply chain efficiency.

Finally, both the traditional QF contract ((2,0), w) and the revised QFi contract
(a, a), w, , wz) can achieve supply chain coordination via the pricing schemes identified in
Tsay (1999) and in (2.2), respectively. Notice that w in Tsay’s price scheme equals w,
in our model. A major difference between these two contracts is that we have two
transactional prices (w, , wz) while there is only one under the traditional QF contract (Tsay,
1999). More importantly, our pricing scheme does NOT require knowledge of the demand
distribution for achieving the coordination status, while such knowledge is a must under the
traditional QF contract. This implies that the QFi contract’s coordinating price scheme will
achieve system-wide profit no matter what the demand distribution is, while the QF contract
needs to decide the demand distribution before one can identify the coordinating pricing.
Furthermore, the QF i contract can identify the coordinating discount price given any QF
contract price. But the QF contract may result in a coordinating price that deviates radically
from original QF component price. As such, our QFi contract has the advantage of
requiring relatively less information than the traditional QF contract, and a wide range of
coordinating pricing schemes available which are not possible under the traditional QF
contract. These are critical factors for successful implementation of any supply chain

contract.

23

2.6 Comparative Analysis Between (a,w,w,,w,) and (a,a), w): The

Supplier’s Perspective

We have thus far identified the optimal conditions that allow the decentralized supply
chain to achieve the highest profit level that a centralized chain can achieve. However, the
QFi contract (agar, w, ,w,) does not necessarily always benefit the supplier, even though
the intention of the supplier’s offering the QFi contract is to increase her profit level. In this
section we analyze the conditions under which the supplier will benefit from offering the QFi
contract and the conditions under which the supplier should stick to the traditional QF

contract ((1,0), w). We analyze these conditions both in the coordinated QFi/QF supply

chains and the general non-coordinated QFi/QF supply chains.

2.6.1 The coordinated QFi/QF supply chains

Define HSQF and HSQFi the supplier’s total expected profit when coordinating under
the QF contract and under the QFi contract, respectively. Let (wc,Qc, qc) be the
component price, the production quantity, and the forecast quantity under the traditional QF
contract that coordinates the supply chain. Readers can refer to Tsay (1999) and (Lariviere,

1999) for the details of the coordination pricing scheme and the optimal quantities. Notice

that Q = (l + a)qc, Q: = (1+ a)q,c + qzc . To facilitate the comparison, we let w, = we.
w, can be determined by the pricing scheme in Theorem 2.2. By definition, IT?” and

.8

113”” can be calculated as following:

Hf” = (WI — C)Q.- - (W1 - V)lE(Q. - D)+ - E((1 - a061. - 13)]

Q;-
= WIQ: — cQ: — (W, -— v) IF(x)dx (2.3)

l 140M?

24

H?” = (Wi -C)(1+a)qf +(W2 -C)q§

-(Wi - v)[E((1+ a)qf + 615 - DY - E((1- w)qf + (15 - D)l
Q2
=(w. (1+a1qr+w.q§)—cQ: —(w. -v> lF<x>dx (2.4)

(l-w)qic+q§~

The following properties in Corollary 2.3 are useful to facilitate the comparisons.

Corollary 2.3

(1) w.Q.‘ 2 w.(1+a)qf + mi

9.: Q;
(2, jF(x)dx2 [Fmdx

(1-w)qc (I-w)qf+q§

PrOOf- (1) WIQ: T W,(l+a)q,c +W1q; 2 W1(1+a)qrc +W2q;

O C C C 1 O
(2) Q. =(1+a)qi +612 =(1+a)q..so qi +—qz =61.
1+a

c 1—0) c
:>(1-w)q, +—-q2 =(1—w)q.
1+a

. . 1 — a) . .
From proposrtron 2.1, we know -1—— = 1, at coordination
+ a

C 1—0) C C 1 C c C
:> (l—wMi' +1—+Taqu =(1Tw)q1+T—q2 S(1T(‘))q1+q2 D

C

Corollary 2.3 indicates that there is a mixed relationship between TISQF and H39”; II?”

can be larger than, equal to, or smaller than IT?” . This suggests that providing the QFi

contract (a, a), w, , w,) to the buyer does not necessarily always benefit the supplier. In

some circumstances, the supplier will be better off to simply stay withthe traditional QF

contract (61,0), w). To better understand this phenomenon, define AH, = II?” — [1sz

25

Q5 (25
AH. -- (w. (1+a1qr + wzq§)-wiQZ —<w. —v>[ lF<x>dx— lF(x)dx1

(1-w)qf+qf 0-ch

95 Q2
—<w, — Wm; + (w, — v11 lF<x>dx— lF<x>dx1

] _ C C
(]_m)qlc+__qzc (1 (0)91 +02

ZC
(I-wiqfwf
=—(w. —w.)q§ +<w. —v) IF<x>dx (2,,
(I—w)qF+—'—q§
ZC
Recall that under the QF contract, the supplier needs to bearthe risk of unsold units as high

as (a + aa)qC . With the new QFi contract, the highest possible unsold inventory that the
supplier will claim for a loss is (a + co)q,‘. Because qc 2 q,‘ , (a + a2)qc 2 (a + w)q,‘ .

Therefore, the loss from expected unsold units that belongs to supplier’s responsibility is less
in the QFi contract than in the traditional QF contract. From (2.5), we can clearly observe
the tradeoff between the loss of profits from discounted units and the benefit from lesser
expected unsold units.

The first term refers to the total loss of profit (revenue) for units sold at the discount price.
The second term indicates the savings from lesser responsibility if the volume of units
produced is greater than the final demand. As such, when the expected saving from lesser
responsibility on the unsold units outweighs the loss caused by the discounted units,

i.e., AII, > 0 , the supplier will favor the new QFi contract. In contrast, when the expected
saving from lesser responsibility on the unsold items is not significant enough b cover the
loss of profits by offering discount, , i.e.,AI'I, < O , the supplier should stick to the traditional
QF contract and offer no discount options to the buyer. When AH, = 0 , the supplier is

indifferent between the two QF contracts.

26

Proposition 2.2 As the flexibility increases under the traditional QF contract, the supplier

will be inclined toward offering discount units above the contracted quantity.

dAII .
Proof. It suffices to show that ’ > 0. Altematrvely, (1 —a))q,‘ 4.ng decreases

 

dz. 1.
(l—w)qc+q§
in Zc- Thus, JLFOde increases in 2’. , so is AH, . D
(MIMIC +iqzc
ZC
(1—w1qf+q§
C
Theorem 2.3. If (W1 '- v) jF(x)dx > (W _ W2 )qz , there exists a unique
0—me

minimum flexibility 1m,“ , such that when I. > 1min , the supplier should offer the QFi

contract (a, a), w, , wz). Otherwise, the supplier Should remain offering the traditional QF

contract (a, a), w) .

 

 

 

dAH - c 1 c -1 c
Proof. S =(W1 Tv)[TF((lTW)q1 +—q2 )(__2_)q2]> 0
dz. 1.. z.
d’AI‘I - 2 C c 1 c c
2 s = (w. — v)( 32 )F((1- w1q. + —q. > - (w. — v)(g’7)’f(x) < 0
dz. 1. Z.- I.

So AI'IS rs nondecreasrng 1n 2’.

When zc=1, An, = —(W, — W2 )q§ < O. In this case qf will be 0.

(I—w)qf+q§
When/(C —-) 1=1, AH: : "(W1 T W2 )ch + (WI T V) IF(x)dx >0 under the
(I—w)qf

assumption. As a result, ATI, will intersect with AH, =0 only once.

Let 1",," be the value of I at the intersection. Then AH, 20, if

I. 2 1mm; AU, < 0 , otherwise. [3

27

Once the buyer and supplier agree upon the contract parameters(a, (u) , thus the flexibility

I under the QF contract, the supplier can calculate the discount pricing w2 via (2.2),
followed by finding 1, and 1",,“ indicated in Theorem 2.3. The final decision as to
whether to offer the discount prices above the contracted quantity or not can then be decided
through the rule stated in Theorem 2.3. This determines when and under what conditions
the supplier should propose discounted units on the top of the QF contract. When

(I-wyjfwf

F(x)dx > (W1 T W2 )q 2 , 1",," can be solved by setting AH, in (2.5)
(1—w)qf

(W1 T v)

(l-wichf
C
jF(-x)dx -<- (W1 T W2 )qz , it is the buyer
(1—w1qf

equal to zero. However, when (W1 T V)

who is going to benefit more from the QFi contract than from the traditional QF contract.
In this case, there is no reason for the supplier to offer thediscount incentives and she should

stick to offering only the traditional QF contract.

2.6.2 The general QFi/QF supply chains
A more general decision rule can be developed in a similar manner for the QFi/QF
supply chain when the supply chain efficiency is not specified at 100%. Let

QFi

S

Ans = 7:59“ — flQF , where 7! and ”50F are the supplier’s expected profit under the QFi

5

contract and the traditional QF contract, respectively. So, 7!?” and IZ'SQF can be expressed

as following:

(1+a)q’+q£
2r?“=(w.(1+a>q:+w2q;)-c(<1+a)q{+q;>—(w.—v) lF<x>dx (2.6)
(1—w1q.‘+q§

(I+a)q’
7r.” = (w. —c)(1+a)q‘ -(w. —v) lF<x>dx (2.7)

(I—w1q‘

28

_ QFi QF
Thus, Afls—fl-S _fl-s

= (Wi(1+a)qi +W2q;)-C((1+a)qi +q;)—(w, —C)(1+a)q‘

(1+a)q' (1+a)qi+q§
+(w. -v)r jF<x>dx— [Freda (2.8)
(l-w)q' (l-wMiWi

So, when A715 > 0 , supplier will benefit from providing the new QF i contract; when

Airs S 0 , the supplier should stick to the traditional QF contract. Notice that q. is the

optimal forecast quantity under the traditional QF contract; refer to Tsay (1999) for the

solution method. When the supply chain is not coordinated, (1+ a)q; + q;

and (1+ a)q. do not necessarily have a relationship as they do in the coordination
condition. As such, (2.8) can not be rearranged into afonnat similar to (2.5) in which the
difference of profits can be presented as a function of the flexibility measure 1 in stead of
(a,a)) . Consequently, there is no 2’ analogous to 1min in (2.5) that can be identified in
the general case.

Note q ,. and q; must be non-negative. Since the buyer decides the optimal

forecasting and purchasing quantity under the QFi contract, it is easy to see that the solutions
presented in Theorems 2.1(3) and (4) will bring the buyer at least the profit level that he can

earn under the QF contract. On the other hand, Ans must be positive to provide the

supplier incentive to propose the QFi contract. As such, when the optimal solution q: and

q2 are both positive and will result in positive Ans, the QFi contract achieves Pareto

improvement from the QF contract. In this cam, the QF i contract creates a win-win

situation for both the buyer and the supplier. We formally state this in the next proposition.

29

Proposition 2.3 When q: > O , q; > 0 , and A715 > 0 are true, the QFi contract

achieves Pareto improvement from the QF contract.

In the next section, we introduce numerical examples to add to the development of the

theoretical grounding of the new QFi contract and discuss managerial implications.

2.7 Numerical Experiments

We consider a base dataset for the numerical experiments as follows: p = 50 , c = 30 ,

v = 20, w, = 42 , w2 = 40 . For convenience of the analysis and to enable closed forms

for decisions variables in the QFi contract, we consider the market demand to follow a
uniform distribution over the interval [400, 800]. Our analysis mainly focuses on
comparison between the the QFi and the traditional QF contract. We examine the inventory
decisions and the profit levels for both the buyer and the supplier across a range of discount
pricing and flexibility agreements. We identify the conditions under which the supplier will
benefit from offering additional discount units and conditions under which the supplier will
be better off sticking to the QF contract. Then we analyze these areas of interest at the

coordination condition.

2.7 .1 Comparing the QFi and the QF contracts: varying discount pricing

We set the flexibility parameters at a =O.2 , 0) =0.25 in this experiment. The
component price is $42 under the QF agreement. Intended to earn more, the supplier

proposes that the buyer can purchase any additional units above the contracted quantity

Q, = (1+ a)q, at a cheaper price W2. Figure 2.1 presents the buyer’s decisions when the

30

supplier decides to offer discount incentives in addition to the QF contract. As one can
observe, it will not take much discount for the buyer to decide to simply go for the discount
scheme for all that is needed. As the discount price W2 decreases from 842, it takes about

$1 of discount for the buyer to be interested in taking the offer. Between $39 and $41, the

order size of the discount order q2 increases and the forecast quantity decreases

tremendously, and the buyer loses interest in the flexibility very quickly. When he
discount price is below $39, the buyer will discard the flexibility and only purchase discount

units.

 

 

 

 

l
{-
’ l
1 °. I l
a, 1 ’3 1' l
E 400,1. '\ 1' i
3 l, ‘. I’ l
300. ‘11" l
r )1 i
l ‘
f ,— —_i we #- A. 1' 1 l
..... q I '\_
200 , 1 ; I” ‘. ,,
\
3 - ...... qZ , I, '\.
100}, , —QFi total availability, I, ‘-, l
‘ ' ---------- QF total availability | ' '\, l
l L _ __1 ’ '
__ __ .4. I ‘ l
0' _l_ .1 .L _L I 1 I, _W 1_ _ _. ‘1...._..i
32 33 34 35 36 37 38 39 40 41 42

Figure 2.1 q, v.s.q2 under QFi: W, =42

From Figure 2.2, we can clearly see that the supplier can actually benefit from the QFi

contract when she offers the discount price in the range between $40.3 and $41. In this

31

range, the supplier is expected to earn more than what she could. from the QF contract. On

the other hand, we found that when 40.3S W2 S41, the buyer can increase his profit by
ordering nonzero q, and q2 at the same time. In other words, with W, =$41 and

$40.3 S W2 S$4l, the QFi contract is able to create the Pareto improvement from the QF

contract. The supplier’s profit decreases as the discount price decreases. If she offers the
discount price below $39, the buyer will only be interested in the discount scheme and the
supplier will be worse off compared with her expected profit when there is no discount, i.e.,
the QF contract. Note that between $41 and $42, the buyer will only be interested in
executing the QF mechanism. So the result suggests that the supplier can propose the QFi
contract, but the discount price should be somewhat close to the QF price. Otherwise, the

QFi could only benefit the buyer more at the expense of the supplier.

10000 ._...-... A .2 A, L_,_.f,. _,_,,_

 

 

 

‘ , -——QFi supplier—d
9000 - W...‘ , - ...... QFi buyer ,1
\‘x ------ QF supplier l

8000- '~-., , .......... or: buyer ll

 

\.‘. - ___,___,_____ '4,

Profits

 

 

32 -. 33 34 7 35 T3367 377—38L— 39 ’40 4‘177—"(4‘2

Figure 2.2 Supplier/buyer profits QFi v.s. QF: W, =42

32

Figure 2.3 presents the minimum level of flexibility against the discount pricing so that the
buyer remains interested in executing the QF mechanism. As one can expect, the higher the
discount, the higher the flexibility needed. Note that at each discount price, the supplier
will be able to estimate the expected profit if all units are sold underthe discount price. As
such, the supplier will be able to understand the “value” of the flexibility and how much it
will cost her by agreeing upon a set of flexibility parameter values (a , a) ). This helps the
supplier to estimate the magnitude of the flexibility offered and thus, leads to decisions as to

whether to offer the QFi contract or not.

Flexibility

 

32’" I 33 W I 34” 35 35— 374—38 ‘39 70 41 W 42

Figure 2.3 Minimum flexibility for QF to remain executed

Figure 2.4 presents the supply chain efficiency considering the buyer and the supplier’s

profits combined. The result indicates that there is a mixed performance compared w'th the

33

QF contract for the entire chain, as indicated in section 2.6. When the discount price is
between $39.8 and $41, the QFi contract outperforms the QF contract. However, as the
price continues to go down to $38.9, the entire chain’s profit starts declining down to an
efficiency of 95.05% before it starts to climb up again. Combining this with the result in
Figure 2.2, we can see that although the supply chain gets higher efficiency when the
discount price continues to decrease from $39.8, it is the buyer who gains all the benefits
from the lower price. As the discount price decreases towards the component cost of $30,
the supply chain efficiency moves towards 100%, which is literately equivalent to a
centralized chain in which the supplier makes no markup of the component cost when selling
it to the buyer. The readers can also observe that the QB contract achieves supply chain
coordination when the discount price is at $40.4, which is within the interval of the Pareto

improvement. It is in this range that the supplier should target at setting the discount price.

 

 

 

 

 

 

Supply chain efficiency

 

 

l
ELM—s; __.L_.___._.__ 1 1- 1
N _ A v

'33 " 34‘ 35 "36 3'7 38 39" 4‘0"“41
w2

Figure 2.4 Supply chain performance: QFi vs. QF: W, =42

34

2.7.2 Comparing the QFi and the QF contracts: varying the flexibility parameter

The magnitude of flexibility in the QF contract is another key factor that affects the
supplier’s decisions in offering the discount on additional units. Flexibility defined in the
QF contract contains two parameters a and a). We analyzed both parameters’ impact
on the QFi contract decisions. We found that increasing a and increasing a) , both
enlarging flexibility, imposes similar impact on the inventory decisions in this experiment.

Thus, we present only results from varying the a) value in this section. In this
experiment, we set W, = 42 , W2 = 40 , and a = 0.1 .
Figure 2.5 presents the QFi decisions varied with the magnitude of QF flexibility

indicated by a) . When (0 increases, flexibility increases as well. As one can observe,

when (0 is small, the $2 discount is more attractive. So the buyer will only purchase firm

order q, at the discount price W2 = $40. Recall that these units are non-returnable. As

(0 increases, i.e., flexibility increases, the buyer gains interest in executing the QF contract
as the protection of inventory risk provided will outweigh the cost savings from the discount
scheme. However, this result simply presents the buyer’s decisions across the various
levels of flexibility. Results from the previous section clearly indicate that when only the
discount scheme is executed, the supplier is actually worse-off. Our next experiment
presents the ideal range of flexibility to offer the discount.

From Figure 2.6, we can observe that the supplier can benefit from offering discount at

W2 =$40 when (020.33. As we know, when flexibility is high, the supplier’s

inventory responsibility is high under the traditional QF contract. So, with the units sold at
the discount price, the inventory risk for the supplier is mitigated and its expected financial
impact (gain) is greater than the sales loss of $2 per unit. Therefore, the supplier will

benefit from offering the QFi contract in the high flexibility range as indicated in the figure.

35

On the other hand, one can also observe that the buyer can always benefit if the discount
scheme is also offered. As such, when a) 2 0.33 , both supplier and. buyer can benefit
from the QFi contract, yielding the Pareto improvement from the QF contract. Although the
numerical results may vary with different parameter values, this observation supports that in
some circumstances, the QFi contract can mostly outperform the traditional QF contract
when the flexibility is high. Figure 2.7 shows explicitly the profit difference between the
two contracts for the supplier.

Finally, Figure 2.8, provides a clear view to support the benefits of the QFi contract.
Both the QFi contract and the QF contract achieve 100% efficiency at a point within the
presented range of a) . From the system profit point of view, the QFi contract outperforms
the QF contract when a) 2 0.3 in this specific experiment. Note that from Figure 2.6,
when (I) Z 0.33 , the QFi contract achieves Pareto improvement fiom the QF contract. On

the other hand, when a) < 0.3, the QF contract achieves higher system profit for the entire

chain. Specifically, when a) S 0.2 , under the QFi contract, the buyer will buy only qz

and is not interested in executing the QF mechanism. In this range, the buyer gains profit,
but the supplier will suffer from financial loss, the magnitude of which is so large that it
causes the entire chain’s efficiency to decline. In this case, the system will favor the QF

contract over the QFi contract.

36

Units

profit

 

l ,—‘ T 2.1,

, --- ‘ .......... q1 ,
500. \\ , ..... q2

l ‘, T T

I \

. \

\

400 .- \

| \

' \

    

 

 

 

 

    

 

 

3001-
, ‘~~-____--_--v’—T'
200 . ,
l l
I
100.
1 l
= 1
0|._ -' I i ._ 1___.__1___ _._.. ; _,_ 1, #l
0 0.05 0.1 0.15 0.2 0.25 0. 0.35 0.4 0.45 0.5
Flexibiltiy(Omega)
Figure 2.5 q, VS. ‘12 under QFi: W, =42, W2 =40
6500 TT ‘ 'T "T TT" ’ 'T ‘TTTT‘ T T T T ‘ TTT_T Tira: T :T, ,
, , QFi supplier, ,
l """"""""""""""""""""""""""""" , ---- QFi buyer l
............. ' QF supplier ,
6000 1. .......... , _ ______ QF buyer J ,
. ...... ,
5500' i
‘ l
l “a’T‘i
5000 , ,--—v"' .,,
, “",‘,—"‘ ..................... J
l TTTT ‘:°’.‘.' ''''''' ,
4500. I, - —" 4
l .x'“ l
.r'TIT.’.’ i
4000 ‘.. , . ’" l
l " ,
l l
3500 L- l

o 0.05T 0.11 T of15T 0,2"T'0f25 T 0.3T 0.35T‘0i4 T045 T 0.5

Flexibiltiy(Omega)

Figure 2.6 Supplier/buyer profits QFi v.s. QF: W, =42, W2 =40

37

     

 

profit

Supply chain efficiency

 

0.05 T 0.1 ' 0.?5’02T71.§5TT 03 "0.55 " 04‘ ”0.45 " 0.5
Flexibiltiy(Omega)

Figure 2.7 Supplier profit difference (QFi-QF): W, =42, W2 =40

 

 

0.921. f-._ . , .3
. . QFi ;
0.91 l. ; ..... QF 1

 

 

'K,___ _-_L__ l . L; .1 __ I

0. -,. ,r I L _ 1. a
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Flexibiltiy(Omega)

 

Figure 2.8 Total profits QFiv.s. QF: W, =42, w2 =40

38

2.7.3 Comparing the QFi and the QF contracts when the supply chain is coordinated

Recall that the QFi can achieve supply chain coordination by identifying w2 via (2.2),
given w, . So, we vary W, in this experiment and calculate the corresponding
coordinating W2 via (2.2). The benefit of the QFi contract is that it allows coordination

literately for any given W, , without loss of generality, although the coordination might come

at the discount price that is undesirable for the supplier. In these experiments, we set
a=0.1 and (0:01.

Figure 2.9 presents the buyer and the supplier’s profits under both the QFi and QF
contracts when both coordinate the supply chain. With the given set of parameter values,
the coordinating price scheme under the QF contract is $35.85. The coordinated QF

contract brings the buyer a profit close to $8000 and the supplier around $3000. As

indicated in the figure, as W, increases, supplier’s profit can also increase. Intuitively, the

supplier’s profit will increase when both W, and W2 increase. However, there is an

important managerial implication here as discussed below.

In our initial problem setting, the component price W, is set at $42. To achieve the

supply chain efficiency under the QF contract, supplier has to agree to sellthem at $35.85 so
that the entire supply chain’s expected profit can reach its maximum. However, the
improved supply chain performance comes at the expense of the supplier, because she has to
provide the same level of flexibility, yet lowering the price to induce the buyer to act in line
with the global objective. As such, the supply chain coordination becomes an ideal

condition and is unlikely to happen when the buyer is taking all the benefits. The QFi

contract provides a solution for this. The supplier can offer any price W, in between the

39

original price $42 and the QF coordinating price $35.85, and offer a discount price W2 via

(2.2) to increase her profit, yet without causing a deficiency for the whole chain. The profit

allocation between the buyer and the supplier is clearly presented in Figure 2.9. Notice that

each W, is associated with a coordinating discount scheme W2; these discount prices W2

are not presented in Figure 2.9.

 

 

12000,.e—m — .~-=--——--.—— —-—-— —.- = —* -—-1—-—— - =1 —‘ = _T‘I
l l QF i supplierji ,

, s‘ . ,I

\\ ----- QFl buyer 3

10000, N.“ .......... QF supplier ii,
, \\ , - ...... QF buyer ,

__J

 

 

 

 

 

 

E
Q l
4,
l
l
i
l
1
t. _. ' 1 - ___—i— ___—__l______ - ______. _J
30 32 34 36 38 40 42

w1

Figure 2.9 Supplier/buyer profits QFi v.s. QF at coordinationzng =38.85

Finally, we examine the performance of the QFi contract at coordination when the

magnitude of flexibility ((0) changes. Both the coordinating component price WQF under

the QF contract and the discount price W2 under the QFi contract vary with flexibility. As

such, WQF and W2 are not constant in this experiment. But W, is fixed at $42. As

40

shown in Figure 2.10, when (0 S022, the QFi contract will not result in non-zero q; at

coordination. As such, the buyer and the suppliers’ profit curves are overlap in Figure 2.11.
However, once (0 exceeds 0.22, the buyer will execute both the QF and price discount
scheme to achieve the 100% efficiency. However, Figure 2.11 exhibits that the supplier is
actually worse-off from the QF contract when coordinating in the high a) region. This is
opposite to the results from previous experiments in Figure 2.6 when supply chain is not
coordinated. So, does the QFi contract result in undesirable discount scheme for achieving
system-wide profit for the supplier when flexibility is high? Not really. A deeper

investigation on the coordinating pricings (Figure 2.12) reveals that under the current
experiment setting, the coordinating WQF under the QF contract increases toward the retail

price ($50) as (0 increases. In other words, when the supplier offers large flexibility
under the QF contract, the coordinating component price needs to be very close to the retail

price, and the supplier expects the majority of profit as indicated in Figure 2.11. However,

due to its capability of coordinating the supply chain at any (reasonable) W, , the QFi
contract can identify a discount scheme W2 so that, together with W, , it can reach the 100%

supply chain efficiency. Although W2 varies with a) , it is relatively stable across the

various level of flexibility, as indicated in Figure 2.12. As a result, the QFi contract does
not result in radical change in pricing scheme as the QF contract does, when the systemwide
profit is the goal.

Furthermore, we can observe in Figure 2.10 that as (0 exceeds 0.22, the decisions
under the QFi contract start deviating from those under the QF contract. However, the
resulting buyer and suppliers’ profits in Figure 2.11 do not vary much. In contrast, under

the QF contract, it is obvious that one party (in this case, the supplier) tends to get all the

41

benefit to allow an efficient supply chain to exist. In other words, the profit allocation
between the two parties changes dramatically as flexibility grows. Consequently, when
flexibility is relatively high, we found that the QF i contract is a more desirable contract from
the system’s perspective. It can easily achieve global optimal profit at the original price
combined with a relative stable and reasonable discount scheme. It also avoids dramatic

changes in profit split between the two agents in the supply chain.

 

units

 

0.05 01 0315 04.2
Flexibiltiy(Omega)

Figure 2.10 q, v.s.q2 under QFi at coordination: W, = WQF

42

profit

12000_ __ _ _L _ _ _,_ L_.._.._-_-_,_L ___ LL LLNI,

,._

, , ....:;:'dl="ialaslia,

, , - ------ QFi buyer
10000, , —— OF supplier

i ~\ l ----- QF buyer l

\ L .__ M.___._.__..._..._ ._4

 

l

l

i m__ .4 ,__ L_,.A_ L_ ._1,______Jfl_h_____L______E_Ln____JL I
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

 

 

 

 

 

O _ .
Flexibiltiy(Omega)
Figure 2.11 Supplier/buyer profits QFi v.s. QF: W, =42
50 “ # Ti 7 i I A ‘ m r m r— —-1 ‘num — w —— 7
I x I
l , '351 j 1’ i
48’ I l I’ i
, nnnnnnnnnnn W2 , ”’ ,
461.. , ------ wQF l ,” J
‘ _ __._ __ I
44, ”’I 1
I ,/ I
g, 42, ................................... x‘ I
_ ..................... I
,2 , "It" .................................. '
o- 40 , ”’ ..... ‘,
l ’I’ i
38, / ,
I ’,” ,
36 i ’x’
l ”” ,
34i I”” ,
I I”’ ,
32lu__, A“ 1 ALADl ____ 1 a 1 1"___, 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Flexibiltiy(Omega)

Figure 2.12 Coordinating pricing in QFi and QF contracts

43

2.8 Conclusions and Discussions

We considered a decentralized supply chain in which the traditional QF contract is in
employed to govern the buy and sell activities between the buyer and the supplier in this
supply chain. Under the QF contract, the supplier agrees to share the burden of inventory
risk with the buyer to a certain extent. To mitigate the impact of the returnable stock on her
total expected profit, the supplier proposes to offer a discount price for any unit sold above
the QF-contracted quantity. The purpose is to reduce the supplier’s inventory burden (loss)
imposed by the QF contract so that her total expected profit may increase from the QF
contract settings. We termed this contract as the QF contract with incentives, the QFi
contract.

We analyzed how the buyer reacts to the QFi contract so that the conditions under which
the QF i contract can work in favor of the supplier are identified. We have shown that when
the QF i contract is executed, it can actually benefit both the supplier and the buyer at the
same time. Specifically, we found that the QFi contract works the best when the supplier
offers a slightly lower price than the QF contract price for the same component. In this
case, the QFi contract can create Pareto improvement compared with the QF contract, thus
resulting in a win-win situation for both the buyer and the supplier.

We have shown that the QFi contract is able to achieve supply chain coordination.

More importantly, it can achieve the system-wide profit literally at any component price
agreed upon by both parties under the QF agreements. In contract to the QF contract, the
QFi contract does not require the knowledge of demand distribution for obtaining the
coordinating price scheme (Tsay, 1999; Cachon and Lariviere, 2005).

We conducted a series of numerical experiments to demonstrate various strategies that
can be utilized to improve the supplier’s profit via the QFi contract. We identified the areas

where the QFi contract achieves Pareto improvement compared with the QF contract. We

44

showed how the supplier can estimate the “worth” of flexibility that she offers to the buyer.
More importantly, we showed that the QFi achieves supply chain coordination in a much
more “flexible” manner than the QF contract.

In summary, the main contributions of this chapter are threefold. First, we designed a
new QFi contract that can work as well as the QF contract, yet it provides an option for the
buyer and supplier to compromise the inventory risk level between the QF contract and the
price-only contract. We developed the theoretical foundation of the QFi contract, and
demonstrated a series of numerical experiments to demonstrate how the QFi contract can be
utilized to improve the supplier’s profit from the QF contract. Although we developed this
contract from the supplier’s perspective, we have shown that at the correctly chosen discount
pricing, both parties can benefit when QFi is executed. The supplier can stick to the QF
contract, or introduce the QFi contract when appropriate.

Second, we are able to identify areas where it is more beneficial to implement the QFi
contract than the QF contract from the supplier’s point of view. We conducted thorough
comparisons and analyses between the QFi contract and the QF contract. We presented
conditions where the QFi contract outperforms the traditional QF contract, at both the
individual and the system level. We provided a framework for the supplier to evaluate the
value of flexibility. More importantly, we showed that the QFi contract can possibly result
in Pareto improvement.

Third, we have shown that QFi contract has the “flexibility” to achieve supply chain
coordination. As mentioned earlier, the QFi contract has the advantage of achieving
system-wide profit at multiple pricing schemes, yet it is not restricted by the need for the
knowledge of demand distribution. We consider this a major improvement from the QF
contract. Specifically, under the QF contract the coordination might heavily favor one

specific agent in the supply chain. Thus, the disadvantaged party will be reluctant to act in

45

line with the system’s goal under the QF contract. With the QF i contract, such a problem
will easily be fixed. As shown in our experiments, implementation of the QF i contract
towards the 100% supply chain efficiency does not result in radical change of system profit
allocation between the buyer and the supplier. The coordination can be achieved at the very
component price that is agreed upon by the two parties under the QF contract. As such, the
coordinating profits for both parties will not deviate Significantly from where they stand
when there is no discount incentive. As such, the buyer and supplier should not be against
strongly aligning their actions with the system’s goal. Consequently, from the system’s
perspective, we believe that the QFi contract is more desirable than the QF contract due to
these advantages.

From the numerical experiments, we found that the supplier should offer small
incentives off the QF contract price rather than offering a large discount, or buyer will be
tempted to the cost saving and lose interest in the benefit of flexibility very quickly. As the
discount enlarges, the buyer will not hesitate to simply buy everything at the discount price,
if no other restriction apply. In this case, the QFi contract becomes a price-only contract
and the discount price will likely hurt the supplier more than expected. As such, the
supplier should not “overdo” the discount; a more conservative discount scheme is shown
most favorable for the supplier to achieve the desirable results.

The QFi contract is designed to allow the supplier to pose no restriction to the buyer in
terms of forecasting under the QF agreement and the firm order under the price discount
scheme. As mentioned, it might not take much discount before the buyer becomes
completely uninterested in. executing the QF mechanisms. Posing a minimum required
forecast under the QF contract may dampen the effects that the discount pricing may have on
the buyer’s loss of interest in the QF contract and on the‘sudden decrease of supply chain

efficiency , as observed in our results. This provides an opportunity for future research to

46

further investigate the performance of the QFi contract. Additionally, future research can
apply the QFi contract on a multiple period problem setting or a more complicated supply
chain structure. When multi- period scenario is considered, the need for offering discount
might not be at the same level as it is in the single period. Furthermore, when there are
multiple buyers in the supply chain, the supplier’s behavior is likely to change as well.
Finally, the inventory decisions and the coordinating discount scheme in the QFi contract
will change if the demand is price-sensitive. More research on the areas of the QFi/QF

contracts is warranted.

Appendix of Chapter 2

Proof of Theorem 2.1 (1)

 

:1ng = (p-w,)(1+a)-(p—w.)lF((1+0061. +q.>0 +a>—F((1 —w>q. +q2)(1—w)l
-(p-v)F((1- 61% +q2)(1—w)
= (p - w.)(1+ a)F((1+a)qi)+(p- w.)(1- w)F((1— com, + q.)
- ([9 -V)(1- w)F ((1 - (0)61. + q.)
= (p-w.)(1+a)’F‘(0 +a1q, +q.)-(w. -v)(1 —w)F((1—w)q. +q.)
6:? = —(p—wl )(1+a)2f((1+a)ql +q.)

-(Wl -V)(1-60)2f((1-€0)ql +q2)<0

C3111" =(p-w2)-(p-wl)[F((1+05)611+‘12)4’“((1T“’)‘11 +92”

 

T(p—V)F((1T(0)CI1 +612)

47

= (w, — w.)+(p— w. )F((1+a)q. +q.)—(w. — v)F((1—w>ql +q.»

 

ddgb =—(p_W1)f((1+a)q1+q2)-(W,-v)f((1—a))q, +q2)<0
2
d211,, _ d211,

 

dqldqz " dqqu, :T(PTWl)(1+a)f((1+a)ql +q2)

T(Wl TV)(1T(0)f((1T(0)q1 +612) < 0

So by examining the Hessian matrix IHI and its leading principle minors IH,| and iHZI

2

 

 

 

wehave: IH,,= dqzb <0 and
l
d211,, dZHb
dqf dqidqz
lel=lHl= d211,, d211,,
dqqul dqzz

 

 

=(p-wi)2(1+a)2fz((1+a)q.+q2)+(p-w1)(w,—v)(1-w)2f((1-w)qi+q2)
+ (p - wi)(1+ a)2(wl - v)f ((1 +a)ql +q2)f ((1 - (0).], + qz)
+(1-w)’(w, -v)2f2((1-w)q1 +612) -(p-w,)2(1+a)2f’((1+a)ql +612)
- 2(1) — wl)(1+ a)(wl - v)(1- w)f ((1 + a)q. + q; )f ((1 — w)q. + q.)
-(w, —v)2(1-w)2f2((1-w)ql HI.)
= (p - w. )(wl - v)f ((1 + 6061. + q2)f ((1 — (0)61, + qz)l(1+ 0:)2
+(l—a))2 —2(1+a)(1—a))]

= (p - Wl)(Wl - V)f((1+a)ql +qz)f((1- (0)611 + quaZ + 602 + 2W)

48

=(p—w,)(w, —v)f((1+a)q, +q2)f((1_w)ql +q2)(a+w)2 > 0

So ‘Hl is negative-definite, thus II, is concave in q, and q2 0

Proof of Theorem 2.1 (2)

Here we consider the non-extreme cases only, i.e. ((1,0)) ¢ 0 and a) rt 1 . At optimum, we

 

 

have the first order conditions: b = O and b = 0 . Thus,

dql dqz
(P - Wl)(1+ a)F((1+Ot)ql)--(Wl - V)(1- a))F((1- wlql +612) = 0 (AZ-1)
(WI T W2) + (P T W1)F((1+ a)q1+ 92) T (“’1 T V)F((1T 0))‘11 + 92)) = O (AZ-2)

Operate (A2.2)* (1 - (0) - (A2.l), then divided by (l — (0) we get
(w. — w.)+ (p — w. W1 + 6% + chi—(p — 110mm — wlq. + ‘12)): o

3 (W1 TWz) = (lTl)(PTWl)F((1+a)ql +92)

 

:5 F((1+az)ql +q,) = (W‘ —W2) =1—F((1+a)ql +q2)
(ITIXPTWI)

(W1 Twz) ___ (PTWI)ZT(PTW2)

F 1 1 2 :1—
=> ((+a)q +61) (z-IXP‘WI) (Z—IXp-wl)

: (PTWI)ZT(pTW2)
(.0 - wl )1 - (p - W1) (A23)

Operate (A2.2)* (1+ 01) — (A2.1), then divided by (1 — (0) we get

Z(W1 TW2)TZ(W1 TV)F((1Tw)ql +q2)+(Wl TV)F((1Tw)ql +q2» =0

3 (1T1)(W1TV)F((1T(0)‘I1 +612): Z(W1TW2)

49

_ _ 1(Wl T W2)
:5 F((1 w)ql +q.) — 05—1101». _v) 012.4)

 

Since F is the p.d.f. of the random variable D , the following must be true:
05 F((1-w)ql +42) -<- F((1+a)ql +612) 51
From (A23) and (A24), it is easy to verify that F((1+ a)q, + qz) _<_1 and

F((l — co)q, + q,) 2 0. From (A23) and (A24) we also have

1(W1TW2) S (PTW1)ZT(PTW2) = (PTW1)ZT(PTW2)
(Z-IXWl—V) (P—Wl)Z—(P‘Wl) (Z-IXP—Wl)

 

 

 

Z(WI_W2): _Z(W2TV)<(PTW1)ZT(PTW2): _(PTwz)
(WlTV) (WlTV) — (PTWl) (PTwl)
(PT W2 )(WI TV)
2 =
T” 1 (p-w.)(w.—v> (A25)

Proof of Theorem 2.1 (3) & (4)

 

 

(PTW1)ZT(PTW2) 1(W1TW2)
M = N =
Let (Z ‘1)(P _ W1) and (Z _ 1)(W1 _ v) . From (A23) and (A24) we
have
(1+a)q, +q2 = F"(M) (A26)
(1 —w)ql + q. = F" (N) (A27)

(l-w)(F"(M)-F“(N))
(z-l)

 

Thus, q.‘ = —1—(F“<M)—F"(N» =
(1+0)

50

q; =—1——((1+a)F—I(N)_(1_w)F-1(M»= ZF- (N)TF_ (M)
a+co ZT’I

 

To Show that q, is constrained by qQF , let q; > 0 optimal firm order under the QFi

contract. Thus, we can express the buyer’s expected profit as
Ham. .q;) = (p — w.>(1+ alq. + (p — Wm;
-(p- W.)[E((1+a)qi +61; -D)+ - E((1- (0)91 +61; — DY]
— (p — v)E((1- w)q, + q; - DY (A28)
Let D. = D - q; , (A28) can then be rewritten as
r1..(q,.q;) = (p — w,>(1+ a)q. + (p — wzlq;
- (p - w, )[E((1 + 00% - D')+ - E((1- w)qi - D'F]
— (p — v)E((1— (0)11, — D)“ (A29)
It is clear that the q, of (A29) follows the QF contract solution, only the demand function

is D' rather than D. Since D. <D, q: <qQF must betrue when q; >0.

0 _ — W
Next, we show that ‘12 is constrained by qwz : F l(“p—’-""2').
— v

Let q, > 0 the optimal

forecast under the QFi contract. Then we can express the buyer’s expected profit as

HQF,(qj,q2)= (p—w,)(1+a)q,’ +(p—W2)q2
‘(P—Wl)[E((1+a)4i +42 —D)+ —E((1—w)qj +€12 —D)+l

—(p—v)E((1—w)q.‘ +q. —D)*

51

=(p- w2)q2 - ([7 -V)E(q2 -Dz)+
-(p-wl)[E(q2 -Di)+ -E(q2 TD2)+I
+(P-Wl)(1+a)qi (A2.10)
where D, = D—(1+a)q,' and D, = D—(l —w)q,‘ . Also, the last term is a constant.
Define H‘Qqu) = (p- wz)q2 -(p-V)E(q2 TD2)+
~(p-w,)[E(q2 -Dl)" -E(q2 TD2)+]

”En-(612% (p - W2)q2 -(p-V)E(q2 -Dz)+

 

It is clear that the optimal solution for H0Fi(q2) is F5“

- W
pp v2 ) , where F02 is the

 

 

p-W2)<F-l(p—w2

c.d.f. of D2. Thus, F5“ ) must be true because D2 <D.

HIQF,-(q2) Shares the same optimal solution with HQF, (q: ,qz) . Note that
[E(q2 — D, )+ — E(q2 — D2 )+] in Ehrlich) is always greater than or equal to 0. It is
also increasing in qz . So the optimal solution of HDFIMZ) is always less than or equal

_ —w
to FDi(p 2

 

) , because a larger q2 will result in reduced value of both

(p—w2)q2 T(PTV)E(q2TD2)+ and —(p—w,)[E(q2—D,)+—E(q2—D2)+],

Thus, we conclude that the optimal q; in Ugh-(m. ,qz) of (A2.10) is always less than

.. ._w ‘1
qw2=F «pp-1f) when q, >0. 0

 

52

CHAPTER 3 QUANTITY F LXIBILITY CONTRACT IN A
TWO-SUPPLIER-ONE-BUYER SUPPLY CHAIN

3.1 Introduction and Literature Review

Recent trends in business partnerships and reduction in the size of supply base in various
industries has increased the need for developing solutions and directions for improving a
supply chain’s performance from a system’s view. Among them is the stream of supply
chain contract research. Utilizing the newsvendor framework as backbone, many have
reported that through proper design of supply chain contract(s), the decentralized decision
making can be aligned with the global goal of the entire supply chain. Through
mutually-agreed contractual mechanisms, buyers and suppliers can possibly coordinate their
business and create a win-win situation compared with the transactional—based relationship.

Although numerous studies on supply chain contracts can be found in literature, the
majority have a focus on either one-buyer-one-supplier (1-1) supply chains or one
supplier-many-buyer(retailer) (l-N) supply chains. However, a firm can very likely have
more than one supplier for the same set of components. F orexample, a PC manufacturer
can normally buy Hard Disk Drive (HDD) from more than two HDD suppliers; an
automobile manufacturer can have multiple sources for car lamps; and finally, a grocery store
has different brands of cereal on Shelves. These suppliers are competing suppliers and their
components are “substitutable”. Additionally, the buyer may have different types of
relationships with these suppliers that produce the same (substitutable) components. For
example, Acer, a PC maker from Taiwan, announced its partnership withan original
equipment manufacturer, Wistron Corporation, yet Acer employees two other OEMs as

alternative sources at the same time. The supply chain contract literature has given relatively

53

little attention to the larger segments of manufacturing and most consumer goods in which
multiple competing suppliers are available to the buying firm (Choi, 1999).

Tomlin (2003) investigated the price-only contracts considering capacity investments
between a component supplier and a manufacturer. He identified a quantity-premium
price-only contract that coordinates one-supplier-one-manufacturer supply chains, and
extended the results to an N-supplier assembly system. Each of the N suppliers produces a
different component and the manufacturer assembles the N components into the end product.
In other words, they are complimentary suppliers, rather than competing suppliers. Gerchak
and Wang (2004) studied the revenue sharing contract with incentive to suppliers and the
wholesale price contracts with buyback, each under a supply chain consisting of multiple
complementary suppliers and one assembler. They identified conditions under which the
channel is coordinated for each contract. They found that the channel performance of the
latter can be degraded with the number of suppliers, while it is not the case for the former
contract. Zou et al. (2004) argued that, due to variations of order processing time and cost
structures, supply chains consisting of multiple suppliers and one assembler need to be
synchronized to improve system efficiency. The N suppliers in their research are also
complementary suppliers. They found that risk sharing and proper safety stock placement
lead to better system coordination and improve system performance.

Several researchers dealt with competing suppliers in difference problem settings.’ Choi
(1991) analyzed three channel competition problems in a supply chain that consist of two
competing suppliers and single powerful buyer. Two of the three problems resemble the
Stackelberg game and one resembles the Nash game, resulting from difference in the power
structure of the buyer and the two suppliers. He investigated the effect of cost differences
on equilibrium prices and profits. He found that the forms of the demand function play an

important role in such decisions: the buyer has an incentive to use multiple suppliers when

54

demand is linear while it is more desirable for the buyer to use only one supplier if the
demand function is non-linear.

A more “contract-oriented” mutli-supplier and single buyer problem can be seen in Serel
et a1. (2003) and Sethi et al (2003). Serel et al. (2003) studied sourcing decisions of a buyer
that employs the “capacity reservation contract” with a long-term supplier and an option to
buy the same component from the spot market. Under the contract, the buyer essentially
buys the rights to order up to a certain number of units from the supplier in each period.
They concluded that inclusion of the spot market alternative significantly reduces the
capacity commitments from the contracted supplier. Sethi et al. (2003) studied the QF
contract in a multiperiod, two sources of supply setting. The buyer is allowed to purchase at
two distinct time periods, with the size of the second order being constrained by the QF
contractual parameter. However, the units for the second order are sold at a higher price
than that in the first order. On the other hand, the buyer will also have an option to buy
from the spot market in the next period for any desired quantity at the market price, which
can be higher or lower than the supplier’s price. In other words, the spot market is not
considered a source of supply in the first period. They solved for the optimal orders from
the supplier for both periods, as well as the order from the spot market. They also examined
the impact of information quality and the flexibility on optimal decisions.

The problem of interest in this research has a similar structure to Sethi et al’s (2003)
work. The key difference is that in our model, we intend to examine the buyer’s inventory
policies in a supply chain consisting of two competing suppliers that can supply the
components in the same time period. Specifically, one supplier offers the QF contract as an
incentive to share the inventory risk while the other supplier offers financial incentives with a
cheaper component price. The components from both suppliers have the same functionality

and features; they are not identical but are substitutable. As a result, the buyer can use

55

either one to assemble the final product, which will be sold at the same retail price to the end
customers no matter what the source of the componentis. The research question then is,
what is the buyer’s optimal allocation of forecasting and firm orders for the two suppliers?
Will using the alternative supplier always benefit the buyer? How much flexibility should
the supplier offer, facing the price competition? Will such a supply chain achieve the optimal
system-wide profit?

The main contributions of this research are threefold. First, our study examines the
tradeoff between the level of flexibility and the cost saving benefits for the buyer, which
sheds some light on the value of flexibility. As will be indicated in later sections, when
flexibility is high, the buyer will be inclined towards increasing the forecast to the QF
supplier and decreasing the size of firm order to the price-only supplier. Our models can
assist buyers to make such decisions when having multiple sources. Second, our analysis
indicates that from the system’s standpoint, the 1-1 QF supply chain is favored over the 2-1
QF supply chain. Our results indicate that a 2-1 QF supply chain will fail to achieve the
global profit level. We found that in general, the system profit in a 2-1 QF supply chain is
always lower than or equal to that in a 1-1 QF supply chain. This finding provides support
to the recent trend of firms’ shrinking their supply base and maintaining closer relationship
with selected suppliers. Third, our results also provide insights to the suppliers: when
facing competition, how should they adapt their pricing policies or flexibility allowance so
that they can win more business from the buyer over their competitor. It is important for the
suppliers to understand how they could better initiate business strategies and production
plans to cope with competition.

The remainder of the paper is organized as follows. We first discuss the problem in
section 3.2. Then we formulate the 2-1 QF contract problem with two criteria in section

3.3. We then develop the solution and identify the optimal conditions for the two cases in

56

sections 3.4 and 3.5, respectively. We provide numerical examples and discuss managerial
implications in section 3.6. Finally, in section 3.7,we conclude this research and suggest

future research opportunities.

3.2 Problem Setting

We consider a supply chain that consists of two suppliers and one buyer. The two
suppliers produce substitutable components that provide same functionality and same
features. As such, there is no difference in the final product using either one. The
components are substitutable, however, they are not identical. They may differ in price,
production cost, salvage value, quality performance, etc. To simplify the analysis, we
consider only the difference in financial terms and assume everything else as equal.

The buyer has a stronger relationship with one of the supplier, say, supplier 1, and she
has agreed upon a QF contract to govern joint business activities. As a result, the buyer
provides forecast to supplier l and the supplierl makes the component available complying
with the QF contract agreement. On the other hand, the buyer has atransaction-based
relationship with supplier 2, and uses it as an alternative source. Owing to the production
lead-time, the buyer needs to provide forecast to supplier 1 or places a purchase order to
supplier 2, or both, in advance. Both suppliers will not be able to fulfill sudden orders
should final demand exceed the previous estimation.

The component prices between these two suppliers are different When supplier 2’s
price is higher than or equal to supplier 1’s, it is not difficult to see that the buyer will
disregard supplier 2 and release his order (forecast) to supplier 1 only, as supplier 1 provides
flexibility and shares portion of the inventory risk, yet her component price is cheaper. In
this case, the problem degenerates into the traditional 1-1 QF supply chain problem.

However, when supplier 2 can offer a cheaper price to the buyer, the problem becomes more

57

interesting because there is clearly a tradeoff between the cost saving and risk sharing for the
buyer. As such, the buyer needs to decide the forecast quantity to supplier 1 as well as the
ordering quantity to supplier 2 so that the total availability of the components will best
benefit the buyer. Consequently, we will only analyze the 2-1 QF supply chain problem
when supplier 2’s price is lower than supplier 1’s price. This is most reasonable because
the flexibility may not come free of charge to the buyer. Throughout this chapter we use the

following notation:
H b : Buyer’s total expected profit in a decentralized supply chain
D : Demand, a random variable

q, : Buyer’s forecast to supplier 1 (the QF supplier)

q 2 : Buyer’s order to supplier 2

p : Retail price of the final product

W, : Component price from supplier 1.

W2 : Component price from supplier 2.

v, : Salvage revenue of the component from supplier 1
v2 : Salvage revenue of the component from supplier 2
c, : Supplier 1’s production cost

c2 : Supplier 2’s production cost
F : c.d.f. of the random demand

f : p.d.f. of the random demand

Note: p>w,>W2>c,>v,>0, p>w,>w2>c2>v2>0

58

3.3 The 2-1 QF Models

Under this QF contract, the buyer releases forecast q, to supplier 1. Supplier 1 will
then produce (1+ a)q, and the buyer needs to purchase at least (1 — (0)q,;
a 2 0 ,1 2 a) > 0. Supplier 1 sells the component to the buyer at W, per unit. On the
other hand, the buyer may place a firm order q2 to supplier 2 at W2. This order is not
returnable, so the buyer bears all the inventory risk for any unit in the order 92 . As a
result, the total availability of the component is (1+ a)q, + q2 and the supplier is obligated

to own (1— COM, + qz. Notice that W, > W2 , but the component price difference does not

affect the final product’s retail price p . On the other hand, should overstocking occur, the

buyer can return at most (a + COM, to supplier 1 and salvage the rest of excessive

inventory if there is any. Consequently, the buyer’s priority is to use up either

(1— COM, units from supplier 1 or qz units from supplier 2 before further consuming the
remaining units of (a + 0))q, from supplier 1. We assume that the cost of such a
prioritizing control is marginal and thus can be ignored. As for whether to feed (1 — 0))q,

or q, first to the production depends on the salvage values of the two components.

Although the two components are substitutable, they are not identical. As such, they

may have difference salvage values V, and v2 , respectively. For example, a PC make

can use either Seagate’s or Maxtor’s 6OG hard drive disk (HDD) in a specific model. A PC
equipped with either one will sell at the same retail price, but the two HDDs may differ in the

component price as well as in the resale value if obsolete inventory should occur.

59

Furthermore, there is no guarantee that the resale value of the more expensive component is
absolutely going to be higher than the cheaper one. Asa result, V, can be greater than,
equal to, or smaller than V2 in our analysis. When realized demand is lower than expected
and there are more unsold units than the returnable allowance (a + (1))q, , the higher the
salvage value is, the higher the priority is for salvage, thus the lower the priority for usage.
So when v, > v2 , qz units should be used for production; when V, S v2 , (1 — (0)q, units
from supplier 1 should be consumed first, followed by q, . In both cases, the buyer should
save (a + (0)q, units from supplier 1 before using any 92 from supplier 2.

Consequently, we formulate the 2-1 QF problems for the two cases:

Case I: v, > v2
H. = (p - W1 )(1 + 60611 + (p - wz)q2

- (p - W1 )[E((1 + 6061i + 612 - DY - E((1-60M + qz - DY]

T(P TV1)[E((1 T (0)q1 +612 T Dr T E(qz TD)+]

—(p—v2)E(q.-D)* (3.1)
The first term in (3.1) describes the total profit if inventory is sold out. The second term
deducts the profits from the potentially unsold units that are returnable under the QF contract.
The third term subtracts revenue of the potentially unsold, non-retumable units from supplier
l and adds the salvage revenue of those units. Finally, the last term frrrther deductsthe

revenue of potential unsold, non-retumable units purchased from supplier 2 and adds the

salvage revenue. As such, the priority for component return and salvage follows the order

of the terms subtracted in (3.1), because of the fact that W, > v, > v2. Conversely, the

60

priority of component usage follows the reverse order; they all create the same revenue p

per unit, if consumed, but the one that creates the least salvage revenue should be used first.
So in this case, the buyer should use up all q, before using any (1+ a)q,:
Case II: V, S v2
F1. = (p - Wi)(1+ 60% + (p - W2)q2

- (p - W1 )[E((1 + 00611 + qz — DY - E((1 - ((0% + (I: - DY]

- (.0 — V2)lE((1- 00% + qz - DY - E((1- (0)611 -D)+]

-(p-vl)E((1-w)ql -D)" (3.2)
The difference between (3.2) and (3.1) is that any units in ‘12 have higher priority for
salvage than any non-retumable (1 - (0)q, , because of the higher salvage value. As such,
the component usage priority follows (1 — w)q, —) q2 —) (a + (0)q, . In other words,
after consuming (1— 0))q, , the buyer should start using qz before he sends any of

remaining (a + w)q, units into production.

Our models are formulated with the newsvendor problem (NVP) framework. The
salvage values are attributed to part of the overstocking cost. To simplify the analysis, we

omit inclusion of the understocking cost in our models. As one can observe, there are two

decision variables (q, , qz) in the models. Our first step is to analyze the property of the

objective functions for profit maximization.

61

Theorem 3.1 The buyer’s total expected profit function in either case is jointly concave in

q, and q2.

Proof. See Appendix of chapter 3

Theorem 3.1 assures that the optimal solution q: and q; are unique if they do exist.

The readers may observe the similarity between the models in (3.1), (3.2) and (2.1)
However they differ in at least four different aspects. First, in this research, the components
are from two different suppliers and they are substitutable but not identical while in the QFi
model they are identical and from the same supplier. Second, the suppliers have control
over their own price only whereas in the QFi contract the supplier decides both prices. The
suppliers do not know each other’s price, either. Third, the components from the two
suppliers have different salvage revenue. Fourth, this research is analyzed from the buyer’s
perspective while the previous chapter is taking the supplier’s position. Nonetheless, the
buyer maintains leader’s position in this problem as well.

On the other hand, the traditional QF contract and the price-only contract in a 1-1 supply

chain provide boundaries for the optimal solution of the 2-1 QF contract problem. When

the buyer decides not to purchase any units from supplier 2, i.e., q; = 0 , then the optimal
forecast quantity q: should follow the optimal forecast qQF identified in Tsay (1999).
On the other hand, if the buyer decides to not buy from the QF supplier, i.e., q: = 0 , then

PTcz

 

following the classic newsvendor problem solution, q S = F _1( ) = qw2 . Thus,

2

q]. + q; IS bounded by qQF + qu '

62

3.3.1. The system view: the benchmark

In a 1-1 supply chain, the centralized expected profit can be expressed as
H. = (p -c)Q-(p -V)(Q-D)+
It follows the same notation as that in section 2.3.1. The optimal system profit H: occurs

when QC= F _l (p - C/ p - V) . In a 2-1 supply chain, the system-wide expected profit can

be calculated in a similar manner. The key difference is that when there are two suppliers,

we have two pairs of (c, ,v,) and (62 ,V2) . It is easy to see that HC increases in V,
and decreases in C, . So ideally, in a 2-1 supply chain, He will reach optimal with
min(c, ,c2) and max(v, , v2). However, it is more reasonable to use the pair of (c, v)
from the same supplier. AS such, the system-wide profit will be either TIC (c, , V,) or

TIC (c2 , v2 ). To simplify the analysis, we assume both suppliers have the same production
cost, i.e., c, = c2 = c. As such, when V, > v2 , H, = Hc(v,) and QC =

F-'(p—c/p—v,). When v, S v2, ["16 = Hc(v2) and QC = F-l(p—c/p—v2),
Finally, our research has a focus on the QF contract and intends to analyze inventory
policies and system performance with the existence of the QF contract in a 2-1 supply chain.

We are specifically interested in comparing the supply chain decisions and performance
between the 1-1 QF and 2-1 QF supply chains. Consequently, we will discuss the topic of

supply chain performance only for case I, because in this case the benchmark profit
TIC 2 H6 (V,) shares the same form as in the 1-1 QF chain. We develop the models of the

two cases separately.

63

3.4 Solutions and Optimality Conditions for Case I
As mentioned earlier, buyer decides the forecasting and ordering quantities. Buyer’s

total expected was expressed in (3.1). We also have shown that the objective function is
jointly concave in (q, ,qz) . Next, we discuss the optimality conditions.

1+a
Lemma3.1 Let Z=T—.

(1) If optimal solution exists, (W, - W2 ) - (V, — v2 )F(q;) Z 0 must be true.

1 _ e l _ e
(2) Atoptimum, —(F l(I‘D-CIDT—‘(F 1(B)-q2) mustbe true, where
l+a l—a)

T(Wl TW2)+(ZT1)(PTW1)+(VI -v2)F(q;)
(ITIXPTW1)

A:

B _ Z[(W1TW2)T(V1TV2)F(q;)I

 

 

(Z T 1)(W1 _VI)
1 w — v - w .
(3) If 2’ > _2: = ( I IXP 2) , buyer will release a non-zero forecast q, to the
(W2 T VIXP T W1)
supplier.

Proof. See Appendix of chapter 3.

Lemma 3.1 presents the necessary conditions for optimality if it exists. Both q: and q;

must be greater than or equal to zero. If negative optimal solution occurs, the optimal

solution should be either (qQF ,0) if q; S 0 or (0, q W2 ) if q, S 0 , as discussed previously.
Note that both A and B are a function of 92 . Rearranging the terms in the equality in

Lemma 3.1 (2) we obtain the following equality: (Z —1)q2 = If“! (B) — F—1 (A) .

64

Theorem 3.2 Let m(q,) = 1F"'(B)—F"'(A) . Given ,7;
(1) If m(0) > 0 , there exists optimal positiveq; that solves

(I —1)q2 = [FT] (B) — F-1 (A). But q; must fulfill the boundary constraints:

(i)ifq.o $613. 61: =0, q§=qwz
1

(ii) if 0 < q; s q... q? = ——(F“ (MD) — qS)
1+ (1
1 __ O C
= ———(F ‘(B(q2)-q2)
1— a)
(2) If m(0) S 0 , the optimal solution is (qQF , 0).
Proof. m(q2) is obvious continuous in q2 . Given 1 , m(q2) is monotonically
decreasing in qz , because B decreases in qz and A increases in qz . Thus, as long as
m(0) > 0 , the two functions m(q2) = [FT] (B) — F—1 (A) and m(q2) = (Z —1)q2 must

intersect at a point where qz is positive, since m(q2) = (Z —1)q2 is linearly increasing in

q, . The fact that q; is bounded by qwz leads to (i) and (ii). Theorem 3.1 assures

uniqueness of the optimal solutions. If m(0) S 0 , the two functions intersect at negative
q2 , thus q; = 0 , the problem degenerates into the 1-1 QF problem. Thus, q: = qQF

must be true. 0

Theorem 3.2 presents the unique optimal solution for the model 1. Because of the

inverse function of the demand distribution, close form solution q; is not possible.

Nonetheless, when optimality conditions in Theorem 3.2 are met, one can solve for q; and

65

q; accordingly. As a result, the model is able to assist the buyer to identity an optimal

solution as to how much forecast to release to supplier l and how many units to buy from
supplier 2 at the cheaper price. In addition, Lemma 3.1(3) serves as a useful tool for the
suppliers to analyze the relationship between the flexibility offered and the discount price.

It provides the minimum level of flexibility that supplier 1 should offer to be able to compete
against the price discount offered by supplier 2. It allows the supplier l to examine whether

the QF contract is attractive enough, and supplier 2 to understand if the price difference is
good enough to the buyer. The total availability is (1 + a)q, + q; and the final purchase

quantity can be expressed as r3 = D _L [(1— w)q: + q;,(1+ Ct')q,t + qg]. The

expression on the high hand side denotes the point in the interval that is closest toD .

Next, we discuss the 2-1QF supply chain from the system’s perspective. We intend to

investigate the supply chain efficiency that a 2-1 QF supply chain can achieve.

Theorem 3.3 In a 2-1 QF supply chain, if the buyer purchases components from both
suppliers, this supply chain will not be able to achieve the system-wide profit.

Proof.

HS, =(w, —c)(l+a)q,
T(Wl TVl)[E((1+a)ql +q2 TD)+ TE((1Tw)q1+q2 TD)+] (3-3)

“51 :(W2 -c)q2 (3-4)

Add (3.1),(3.3), (3.4) we get the total profit of the 2-1 QF supply chain as following:

HZ—IQF : (P —c)((1+a)q, +42)

-(P—Vl)[E((1+a)ql +q2 —D)+ -E(q2 “DH

66

T(PTV2)E(qz TD)+ (3-5)
It can be Shown that HHQF is jointly concave in (q, ,qz). We omit the derivation of this

property. Then exploring the First Order Conditions (FOCS), we get

 

M=<rwl>0+a>
91
+(p-V.)[F((1+a)ql+q2)(1+a)-F((1-w)ql+q2)(1-w)=0
:> F((1+a)q. +q2) = pf (3.6)
alea F
—q—Q—=(p-c)-(p-v1)[F((1+a)qi+q2)-F(q2)]-(p-v2)qu)=0
:(p—c)—(p—v,)F((1+a/)q,+q2)—(v,-v2)F(q2)=0 (3-7)

Substituting (3.6) in (3.7) results in F(q2) = 0 , so q; = 0 . Thus,

TC 1.1 :1
F((1+a)ql)= p =F(Q.),whiohleads ql =qQF. Soif q2>0,the2-1QF

l

 

chain fails to achieve the system-wide profit 0

From Theorem 3.3, we see that only if the buyer does not buy from supplier 2 it will
allow the system to reach the centralized profit level. In this case, the 2-1 QF supply chain
becomes the traditional 1-1 QF supply chain. As a result, from the system’s perspective, the
2-1QF supply chain is less efficient than a 1-1 QF supply chain. We will analyze this
phenomenon in greater detail via numerical examples in section 3.7. In brief, we found that

the competition between the two suppliers can only benefit the buyer. The system’s

67

efficiency maybe at a high level, however, majority of the system’s profit goes to the buyer.
The suppliers suffer from competition badly.

In the proof of Theorem 3.3, we show that when at optimum, the following is true:

— W a all
F((1+ a)q,) = T‘g—VL = F(Qc) , i.e., (q, ,q2) = (qQF,O). However, we found that
T 1

(qQF ,0) is not the only optimal solution for the 2-1 QF chain that results in the total

availability at QC .

Theorem 3.4

When the following conditions holds, the total availability in the 2-1 QF supply chain

reaches Q :

C

(11 zF“(K>-F"(—§:—:‘l=(z—1)F"(Hl,whe.e

l

K: 1(PTWlXCl TV1) H— WI Twz _(Z-IXP-WlXCl TVl)

 

 

3

(p—v,)(w, Tvl) V, Tvz (V1 TVzXPTvl)

(2) The flexibility I must be in the interval [1min , Ema, ], where

 

 

 

I =(W1Tvl)[(pTWl)(ClTv1)+(PTVl)(WlTW2)I T :(PTC)(W1TV1)
_min (,0 T "’1 )(Cl T V1 )(WI T V2) , max (P T Wl )(C T V1)
(3, w, S w,_(z-1)(p-Wl)(cl —v.)

PTVl

Proof. See Appendix of chapter 3.

68

Corollary 3.1

t 1 —l p-‘Cl _] . _1
=—F —— —F H 2F H.
ql l+a[ (p—vl) ( )] and q2 ( )

Proof. This is a direct result from Theorem 3.4(1).

Theorem 3.4 presents the conditions that need to hold when the total availability

(1+ a)q1‘ + q; equals QC in the 2-1 QF supply chain. Note that W2 needs to be

decided by the equality shown in Theorem 3.4 (1). This pricing scheme also needs to fulfill

the constraint presented in Theorem 3.4 (3). w2 that does not fulfill this constraint will
lead the buyer to decide q; = qQF and q; = 0.
Two key areas differentiate the meaning that QC carries between the 2-1 QF chain and

the 1-1 QF i contract models. First, QC allows a 1-1 supply chain to achieve supply chain

coordination. However, it will not achieve the same profit level in a 2-lQF supply chain.

It can be easily shown that when the total availability (1+ a)q: + q ; equals QC , the 2-1

QF system’s profit is (V! off from the centralized profit. As one can

— v2 )qu(x)dx

observe, the higher q; is, the greater the 2-1 system profit will deviate from the centralized

profit. Second, in the 1-1 QFi model, the supplier has total control over both WI and w2 .

However, in the 2-1 QF problem, suppliers can decide only their own component price. As
such, they both will determine the pricing policy in their own interest, the result of which will

not necessarily fulfill the conditions in Theorem 3.4.

69

3.5 Solutions and Optimality Conditions for Case II

In this case supplier 2’s component is cheaper, yet it has a higher salvage revenue than
the components from supplier 1. Therefore, supplier 2’s component is more attractive to the
buyer in this case than in the previous case. As a result, supplier 1’s flexibility needs to be
raised to an even higher level than that in case 2 so that the buyer will be willing to continue
to purchase from supplier 1.

The buyer’s total expected profit function was presented in (3.2). We also have shown

that it is jointly concave in (q1 ,q2) . We follow similar procedures for case 1 to explore

the solution for case 11.

Lemma 3.2

(1) If optimal solution exists, the following must be true:

F((1—w)q‘)s 2“P‘WIX‘Wz”en-(1v-szm-vz)
' (p—vzxvz -v.>

(2) At optimum, (a + (1))q: = F“](A') — F"](B') , where

 

A. :1_ (W1 TW2)+(V2 —v1)F((l-(o)q:)
(Z-IXP—Wl)

9

 

B. = 1(wl -w2)+(V2 —V.)F((1“0)qi)
(z-1XW1—V2)

Proof. See Appendix of chapter 3.

Lemma 3.2 presents the necessary conditions for optimality if it exists. Notice that both

A. and B. are a function of q]. Similar to model I, let m(ql) = F'1(A') — F—l (B) ,

we have the following properties:

70

Corollary 3.2
(1) A. is decreasing in ql;B' is increasing in ql
(2) A. is increasing in X;B' is decreasing in I

(3) m(q,) is increasing in Z and decreasing in ql

We can observe that (a + (1))ql is a linearly increasing function of ql while
F ‘1 (A) — F _l (B) is non-linear and decreasing function of q]. Similarly, if the two
functions intersect in the area where ql is positive, non-zero Optimal q; exists. Otherwise

the buyer will not provide any forecast q1 to the supplier and will buy all that is needed

from supplier 2. We discuss the optimality conditions and solution in Theorem 3.5.

Theorem 3.5

' _(PTW2)(W1TV2) , , t
(1) When I > 1min - , there ex13ts optimal ql > 0 that solves
(P T W1)(W2 T V2)

 

(a + a))ql = F_1(A') - F-1(B'). q: must fulfill the boundary constraints:
(0 if 0 s q: < qQF . q; = F"(A')—(1+a>q,‘ = F“(B')—(1—w>qf
(ii) if qi Z qQF, 61? = qQF, q; = 0

(2) When 131...... qi =0 and q; =q...2-

71

Proof. With properties in Corollary 3.1, when m(0) > 0 , the two functions

m(ql) = F'1(A') — F‘1 (B) and m(q,) = (a + 0))ql will intersect and intersect only
once. When m(0) > O, A. > B. must be true ,so

(WI-W2) >B.(0)T Z(W1TW2)

A'(O)=1— _
(z-l)(p- w.) (z-l)(w1-v2)

 

 

' - w w — v
Rearrange terms will lead to Z > Z min = (p 2)( l 2)
(p T W1)(W2 T V2)

 

If )5 S 1;,“ , the two

function will not intersect at a positive ql . Thus q; = 0 , q; = qu . Other conditions

are direct result of the boundaries for q: and q; 0

Theorem 3.5 suggests that when there is a competitor who can offer a cheaper price for the

component, and the component has a better resale value, supplier 1 has to offer flexibility at

least at ,finin for the buyer to be interested in placing an forecast (order) to her.

Otherwise, supplier 2’s cheaper price and the higher resale value will lead the buyer to take

the full inventory risk and buy everything from supplier 2. As the flexibility level increases,

m(q1) = F Tl (A) — F _l (B) increases. This implies that the intersection of

m(q,) = F-1 (A) — F_1 (B) and m(q,) = (a + (1))ql will move up towards the right,
thus resulting in a larger q: . Note that q: is bounded by the qQF , given QF
parameters (a, 0)). Again, the total availability is (1+ a)q1. + q; and the final purchase

quantity is r‘ = D i[(1— mg," +q;,(1+a)q,‘ + q;].

72

In summary, supplier 1 can affect the buyer’s decision by changing the flexibility level

under the QF contract and supplier 2 can affect the buyer’s decision by reducing its price.

Nonetheless, the buyer can utilize our models to decide the best combination of q: and q;

to make the most profit when there are multiple sources of component supply. In the next

section, we conduct a series of numerical experiments to supplement our theoretical models.

3.6 Numerical Examples

We consider a base dataset for the numerical experiments as follows: p = 50 ,
01:62 =30, a =0.l,a) =0.1,wl =42,w2 =40 ,vl =20 ,v2 =18. For convenience

of analysis and to enable closed forms for decisions variables, we consider the market
demand to follow a uniform distribution over the interval [400, 800]. We also present only
the results of numerical experiments for the first model in which supplier 1’s component is
more expensive but has a higher salvage value. Our analysis mainly focuses on how the
price difference between the two suppliers affects the buyer’s decision and how the
magnitude of the flexibility offered by supplier 1 can affect such decisions. The key areas
we will examine include the profit for each party, the optimal ordering and forecasting
policies for the buyer, and most importantly, the advantages and disadvantages of having an

alternative supplier from the buyer’s perspective and from the system’s perspective.

3.6.1 Competitive pricing and its impact on the buyer’s decisions

In this experiment, we examine the situation when the component price from supplier 1 is

set at Wl =$42 in both the 1-1 QF and the 2-1 QF supply chains. Supplier 2’s price w2 is

lower than W1. We vary w2 and investigate how the price difference would affect the

73

buyer’s decision. From Figure 3.1 we observe that as supplier 2’s price changes to below

$41.2, the cost saving of 80 cents per unit will start to outweigh the benefit of the flexibility

provided by supplier 1. As w2 continues to decrease, supplier 2 starts to gain orders
rapidly. Once w2 reaches $40.6, the buyer will favor buying only from supplier 2 even if

the buyer has to take inventory burden. As w2 continues to decrease, the buyer will order

more units from supplier 2. As one can observe, it takesa small amount of price difference
for the buyer to be interested only in buying from supplier 2. Notice that the buyer’s

decision also depends on the salvage value of the component from supplier 2. The

intersection of ql and q2 curves in Figure 3.1 will move towards the left if v2 is higher

than the current value of $18. Comparing ql with qQF, we can see that supplier 1’s

business fades away rapidly when she faces a competitor that takes the discount strategy
against her flexibility strategy.

Figure 3.2 presents the profits of each party in this supply chain. As we can see, the
buyer will truly enjoy the competition between the two suppliers. If the price discount is
not large enough, he stays with supplier 1’s QF contract. But when the price difference is
high, buyer can only benefit from buying more or buying solely from supplier 2. For
supplier 2, our result suggests that its best discount price is at around $40.6, which will
secure all the buyer’s business. As one can see, any further discount below $40.6 will only
worsen supplier 2’s profit; it will only benefit the buyer even if supplier 2 gets the whole pie.
On the other hand, supplier 1 will need to offer a larger flexibility to induce the buyer to
remain interested in the flexibility that she offers. Notice that the magnitude of flexibility in

this experiment is set at 1.222.

74

 

 

 

 

 

 

 

 

 
   
   
 

 

 

 

 

 

 

 

 

400, 1
9 l
3 1 1
300‘. 1
1 1 _____ 2-1:q1 ; j i
200 . j ..q-.- 2'1-q2 l “ I _.
1 1 2-1: Total availability . i, ,' 1
! 1 1-1- 1 g 1
100 1 .......... -qQ: 1 'I a 1'
. 1 —— 1-1: Total availabilitjd ,' a .
1 P I Q 1
01 'r . 1
32 33 34 33 36 37 38 39 40 T 41 42
w2
Figure 3.1 Forecast/ordering quantities 2-1 QF v.s. l-l QF: w1=42
10000 * "— _7—‘ _‘r— 'r i '1 ‘1—— —T 7*. 1’ r _;f _._ ‘1
1—2-1zbuyerprofit ;1
9000' 1 ..... 1-1: buyer profit 1'1
1 - ...... 2-1: supplier1 profit ‘1
80001 i .......... 2-1: supplier 2 profit .11
I - ____ _ _- 1
7000 L 1
|
scoot i 1
.. 1 .'
C: = .
3’ 5°°°1 1
4000 1 ........ q 1
, .......... 1
30001 ........... 1
......... 1
20001 ........ E 1
1 ....... i I
10001" g 1
1 i
1 u I 1
01 J. _. _ L = 3 1 I 1 I ,'
32 33 34 35 36 37 38 39 4O 41 42
w2
Figure 3.2 Buyer/supplier profits 2-1 QF v.s. 1-1 QF: wl =42

75

Figure 3.3 presents the minimum required flexibility for supplier 1 to at least gain some
business from the buyer against the various levels of the discount price offered by supplier 2.
When W2 is fairly close to WI, the minimum-required flexibility is relatively low. As

the price difference increases, the required flexibility goes up. Our numerical results

indicates that when w2 is at $39.4, i.e., $2.6 or 6.2% cheaper than w], the minimum
required flexibility for supplier l is 1.5 to stay in business with the buyer. When w2 is at

$37.4, i.e., $4.6 or 11% cheaper than W1: the minimum required flexibility for supplier 1 is

2. These results assist supplier 1 to understand better the value of flexibility she offers.
From the system’s standpoint, such a competition between the two suppliers actually
leads to supply chain deficiency, indicated in Figure 3.4. If using only one supplier under
the QF contract, the entire chain’s profit can reach $10,492 at the current setting, which
achieves 98.36% supply chain efficiency. In this case the buyer’s estimated profit is
$41728 and the supplier’s expected profit is $6,319.1. As one can see from previous
results, when supplier 2 is joining the game and offers a lower price, the buyer’s profit starts
roaring as supplier 2’s price decreases. As indicated in Figure 3.2, supplier 2’s highest
profit is lower than what supplier 1 can earn under the 1-1 QF contract without competition.
Meanwhile, the buyer’s profit actually increases slower than the loss of suppliers’ total

profits at that point. This contributes to the sharp decrease in efficiency that we observe in

Figure 3.4. As W2 continues to decrease, the entire chain’s efficiency starts to increase.

But as mentioned, in the low discount price range, it is the buyer who gains all the benefits.

If W2 drops down to $30, which is equivalent to the production cost C2 , it reaches the

condition that is equivalent to the centralized chain, thus will achieve centralized profit.

76

 

3:936:

Figure 3.3 Minimum flexibility for the QF mechanism: wl =42

 

 

 

 

_111111141114111111311311 4
.F F U
11
.1. ..r 4
2 4|
‘ . .
_ _
._ -.0
a -— _4
.u
_11_
9
3
18
3
17
3
-6
3
35
,3
.4
x3
_
_
_
..3
3
1 11 1 11 1 11 113111150“
5 8 5 7 5 6
8 0. 7 0 6 0.
0 0 O

 

35656 520 >395

=42

w2

Figure 3.4 Supply chain efficiency 2-1 QF v.s. 1-1 QF: W,

77

These results clearly describe the buyer’s behavior when he can choose from two
suppliers for the substitutable components and the benefit of having a second source of
material supply. In contrast to the previous model in chapter 2, where the single supplier
has full control of the two pricing schemes, the two suppliers in this chapter can only control
their own prices. However, the buyer will take the advantage of getting both price
quotations from the two suppliers. As mentioned, under the 1-1 QF contract, the buyer’s
profit may not be at his desired level. Unless supplier 2’s price is actually higher than
supplier 1’s, having an alternative choice will always benefit the buyer. In the next section

we investigate the magnitude of flexibility and its impact on the buyer’s decisions.

3.6.2 Flexibility and its impact on the buyer’s decisions

In this experiment, we intend to investigate the impact of flexibility on the 2-1 QF supply
chain decisions. We examine the situation where the component price from supplier 1 is set
wl =$42 in both 2-1 QF and 1-1 QF supply chain and supplier 2’s price is W2: $40. We
vary the magnitude of flexibility by fixing (1 =0.1 and changing the a) value. From
Figure 3.5, we observe that as flexibility increases, the buyer will start to appreciate the
flexibility and is willingness to share the inventory burden offered by supplier 1. With the
$2 difference in the component cost, if supplier 1 agrees that she prepares 10% of buffer
stock above forecast and the buyer is only liable for around 79% of his forecasted, i.e., a
flexibility of 1.39, supplier 1 will be able to acquire the buyer’s business for at least 79% of
the forecasted quantity. On the other hand, if the buyer needs to be responsible for 80% of
the forecast, the buyer will not be interested in what the QF contract can offer and will switch

all orders to supplier 2. Notice that in the previous experiment we found that when

W2 5 $40.6, the buyer will not place any forecast to supplier 1. In this experiment, we

78

demonstrate that supplier 1 can cope with the supplier 2’s low pricing strategy (w2 = $40) by

increasing the level of flexibility.

The result in Figure 3.2 indicates that supplier 1 can either increase the buffer stock a
or lower the buyer’s liability (0 to be able to compete against supplier 2, if keeping the
same price. However, as a) continues to increase beyond 0.48, supplier 1 actually will
suffer from offering too much flexibility and the buyer will begin placing some orders to

supplier 2. This is attributed to the structure of the buyer’s expected profit function.

Technically, when a) is large, a combination of large (0 and positive q, and q2 will

achieve higher expected profit than a combination of large a) , positive ql and zero 42 .

This is somewhat counterintuitive because one would think that as flexibility increases, the
buyer will definitely favor the flexibility over the price discount. A further investigation
reveals that when the a) value is high, the buyer’s responsibility from his forecasted
quantity is so low that it is safe for him to buy some non-returnable units from the supplier 2
to a certain extent. The final demand is likely to be close to the mean of previous
estimation, so these non-returnable units will likely be consumed. This explains why the
supplier 1 will, surprisingly, suffer from offering too much flexibility, as indicated in Figure
3.5.

Additionally, Figure 3.5 indicates that when compared with the 1-1 QF supply chain,
having two suppliers with differenct prices is likely to result in less total available units to
fulfill the end-customer demand. This might cause problems to the buyer if the stockout
cost is considered by the buyer for decision making. A high stockout cost will lower the
total expected profit. One can firrther investigate the impact of the stockout cost on the 2-1

QF problem by incorporating stockouts into our framework.

79

Units

Figure 3.5

Units

Figure 3.6

0 ,2 _ s _ _ __ J.
o“””°8.?”””0.§" ' ' ' "01’3' ' ' ' ° ' '01' ' 'T" 613 0.6

 

 

 

30..........

3‘ [.QZéIméfilarqf1 1
| ..... p ..... 2'11Cl1 1 J

‘ ..... ' ..... 2-1:qZ '1 '..'+

1 ...... 1-1: Total availability 1 c..." 1

1. 2-1: Total availability 1 _-' 1
T" T" T‘TTTT'T’ 1
1

 

 

Flexibility/(Omega)

Forecast/ordering quantities 2—1 QF v.s. 1-1 QF: wl =42, w2 =40, varying a)

 

 

.::::===9filflm-.w==;:.u

 
  

 

 

 

 

M.QO.“ ' 11
'r
1 _._°_,_ 2-1:(1+a|p’trla)"q1 1 l. j
l ‘ : I '
1 __‘-- 2-1'Q1 1 gflja‘j
- ' 1
1 "nu..." 2 1'Q2 1 ..‘Or
1 ..... 1-1:Tota| availability 1 -' 1
' 2-1:Total availability 1 1

 

.Wmir—fi

 

.4 . .6
Flexibility(Alpha)

Forecast/ordering quantities 2-1 QF v.s. 1-1 QF: W,=42, w2 =40, varying a

80

Figure 3.6 shows results from the same experiment except here, we fixa) value at 0.1

and vary the a value. It reveals similar patterns to those in Figure 3.5. However, while

(l+a)q, reveals the same patterns in both figures, ql does not. In Figure 3.5, ql is

moving upwards in the middle segment but it is moving downwards in Figure 3.6. This is

due to the fact that in both experiments we have the same frontier (1+ a)q, , but enlarging

a allows smaller ql to reach the same (1+ Ct!)ql value.

Figure 3.7 presents the financial consequences to the buyer due to the magnitude of the
flexibility, with everything else being equal. As we can see, with the optional QF supplier 1
and price-competitive supplier 2, the buyer will always enjoy higher profit than he could in
the 1-1 QF supply chain. Interestingly, under the 1-1 QF contract, the buyer’s total
expected profit goes down as to increases to 1. This is due to the characteristics of the
buyer’s expected function under the QF contract. When a) increases, supplier’s (supplier
1’s) inventory responsibility increases, which is being deducted from both the buyer’s and
supplier 1’s total expected profits. As a result, the buyer’s expected profits decreases as (0
increases under the QF contract, so does the QF supplier’s expected profit.

On the other hand, the suppliers’ profits are presented in Figure 3.8. As one can
observe, when a) is below 0.2, supplier 2’s lower pricing captures the business. When
a) is greater than 0.2, supplier 1’s flexibility has an advantage. As a) exceeds 0.48,
combing with Figure 3.8, we observe that both the buyer’s and suppliers’ total profits in the
2-1 QF chain increase. The higher total suppliers’ profit comes from supplier 2’s increasing
profit.

Finally, Figure 3.9 summarizes the experiment results observed in this section. When
flexibility is low ( (0 <0.2), the buyer obtains his components only from supplier 2, due to the

cheaper price. The sum of the buyer and supplier 2’s profit will be lower than the total

81

profits in the 1-1 QF chain. Thus, in this area, the 1-1 QF supply chain outperforms the 2-1
QF supply chain, from the system’s perspective. When a) value falls in between 0.2 and
0.48, the two supply chains perform equally well. However, when a) exceeds 0.48, the

2-1QF supply chain starts outperforming the 1-1 QF supply chain.

smog—a a a a _ .— .0 .__- ,_~_ aka—.__.-.

 

 

1 1_— 2- 1 buyer profit1
1 ..... 1 -1 buyer profit 1

     

 

 

 

proflt
A
O)
O
E

O

O

1.

\

\

a’
1....1_.__ 4

 

_1—_ . ' .L_ .
0.2 0.3 0.4 0.5 0.6 0.7
Flexibility(0mega)

Figure 3.7 Buyer profits 2-1 QF v.s. 1-1 QF: w1=42, w2 =40

82

profit

Supply chain performance

 

 

 

 

 

 

1

_ _-_, _ .___,fl., 1

1 —— 2-1: suppller total profit at“ |

3000' 1 -—¢—- 2-1: supplier 1 profit 1 a '
1 ---o--- 2-1: supplier 2 profit 1 \1

. 1 ‘0

2000 1 1 1 .......... 1-1: spppller 1profit 1 "(cl 1
1. ___.-_._ .__fl— __ , (Cl _

1 1
1000. 1

1 1
i .' 1
oo””"‘67‘“”’6?§"""‘6f3’“””0‘1"""of5“ ~ 06

Flexibility(0mega)

 

Figure 3.8 Supplier profits 2-1 QF v.s. l-l QF: w1=42, w2 =40

 

 

0.851 \1
. 1
08- 1
1 1
0.75‘r 1
1 1
0.7. a— _1
1 ,‘ _ 2-1 QF' 1
1 1 _ 1 1
0‘65’ 1 "“11 1 QFr 1
1 .
0.6- , _ t __s ._____ 1 _ . xx.

0 0.1 0.2 0.3 0.4 0.5 0.6

F lexibility(0mega)

Figure 3.9 2-1 QF chain performance:W1=42, W2 =40

83

profit

Supply chain performance

 

 

 

 

1 : i1: supplier total profit. \u\ 1
3000. ".__ 2-1: supplier1 profit 1. It _1
1 ---o--- 2-1: supplier 2 profit 1 \i
20001, 91" sppp"°'_‘£’:"f‘L __1 xv”?
1 r l

l 1 1 1
10001 :1 5 ,.~
. {g ! 1

1 n l

1 i l 1

01, , ‘ ' _¢L___W __..
o"’"”6'.1"""'6. """‘o’.’:¥""". "0.5 0.6

Flexibility(0mega)

Figure 3.8 Supplier profits 2-1 QF v.s. 1-1 QF: w1=42, w2 =40

 

 

Figure 3.9

 

 

 

\\““ ‘1

ss“‘1

1

1

“1:71 01:7 1

1 ..... 1-1 QF1 1

' 057” 0:4 0.5 ' "'0.
Flexibility(0mega)

2-1 QF chain performance: w1 =42, w2 =40

83

3.6.3 Discussion

One important observation from these two sets of experiments is that, surprisingly, the
results seem to suggest that the buyer will most likely be better off simply keeping one
supplier. From Figure 3.1 and 3.5, we can see that in most cases our model suggess that
the buyer purchases from either supplier 1 or supplier 2. The area that suggests buying
from both suppliers simultaneously is relatively small. In the past decade, there is a trend
observed in business practices that firms are downsizing the supply base and maintaining a
closer relationship with selected suppliers. Our study provides theoretical explanation of
this phenomenon and offers support to this strategy. Notice that in our model, we examined
tradeoffs between two forces: price and (volume) flexibility. Other key factors to the
supply chain sourcing decisions such as quality and delivery capabilities will play a role if
incorporated into the model. This provides an avenue for fixture research on the topic of
supply chain contracts.

Additionally, we found that from the systems’ view, a multi-supplier system is likely to
be outperformed by the single-supplier supply chain, unless the flexibility is extremely high.
In a reasonable range of price difference and small magnitude of flexibility, existence of
multiple competing suppliers is not favored by the system, as shown in Figures3.4 and 3.9.
This finding suggests that a more complex supply chain structure may not perform as well as
a simple, streamlined supply chain. This, again, supports the benefit of a small supply base
and close partnerships from the system’s perspective. However, as mentioned previously,
the buyer is likely to enjoy the benefits brought about from the competition between
suppliers. As such, the supplier’s task is to convince the buyer tobe willing to sacrifice a
little to allow the supplier to earn a reasonable profit, thus enhancing the system’s

performance and creating a win-win situation.

84

When flexibility increases to a certain level, a) > 0.48 in Figure 3.8, the 2-1 QF supply
chain efficiency is observed to increase. However, supplier 1 does not benefit from offering
higher flexibility that improves the system’s performance. As a matter of fact, Figure 3.8
shows that supplier 1’s expected profit is decreasing; it is supplier 2 who gets the benefit
from receiving orders from the buyer in the range of higher flexibility. Although we
assumed that the suppliers only know their own prices in our models, commonly in
real-world setting, firms may actually gather information of their competitor’s prices in the
same industry. For example, AMD has information about Intel’s CPU prices, and Micron is
aware that its memory module is a little more expensive than Hynix’s. So if supplier I
somehow is able to find out (estimate) that supplier 2’s price is at $40, there is no incentive
for supplier 1 to further increase flexibility ((0) above 0.21, where she starts to win business
over supplier 2. Therefore, if the QF supplier knows her competitor’s price, the better
system performance to the right in Figure 3.9 is unlikely to occur. Thus, the system
performance of the 2-1 QF chain will likely be no better than that of the 1-1 QF chain, if not
worse.

Finally, our numerical experiments were conducted based on a uniformly distributed
demand function. A different distribution may result in different decision patterns and
characteristics for the buyer and suppliers. A random demand variable with a complex
probability density function may cause computational issues such as failing to identify close
form solutions. Nonetheless, our models are developed on a strong theoretical ground and

conceptually, should work for all distributions.
3.7 Conclusions and Future Research Directions

We study a decentralized supply chain in which there are two competing suppliers and

one single buyer. One supplier offers the QF contract while the other provides a lower

85

price. We study this problem from the buyer’s perspective and solve for the buyer’s
optimal buying and forecasting decisions between the two suppliers. We investigate how
well the QF contract can compete against the price discount scheme, everything else being
equal. We identify areas where flexibility will be favored and areas where the discount
price will receive more attention by the buyer. We found that the buyer will be better-off
simply using either supplier in most cases. Compared with the situation where the buyer
has the QF supplier as the single source, the buyer can only improve his profit having an
alternative supplier who can offer a cheaper price.

However, from the system point of view, we found that the 2-1 QF supply chain is
always outperformed by the 1-1 QF supply chain. The competition for the buyer’s business
between the two suppliers will result in a lower total profit that suppliers can earn than that in
the 1-1 QF supply chain. Furthermore, the loss of the supplier’s total profit is larger than
the buyer’s gain in the 2-1 QF supply chain. As a result, the system’s profit declines and
the supply chain performance worsens from the 1-1 QF supply chain. Although from the
buyer’s perspective, having an alternative source of supply is always beneficial. Oar
finding suggests that supply chain deficiency occurs with the existence of multiple suppliers.

We successfully developed models that are able to examine a two-supplier-one
-buyer problem in a supply chain contract setting. Past research in the supply chain contract
literature primarily focused on single-buyer-single-supplier, single-supplier
-mu1tiple-buyer, or multi-(complementary)supplier-single-buyer supply chain problems.

Few looked at the multi-competing-supplier scenario that is common in a variety of business.
Our analysis yields insights not only in multiple sourcing decisions in the presence of the QF
contract, but also has implications in the technique to modeling supply chain contract

problems involving competing suppliers.

86

In this research, we focused on supply chain sourcing decisions regarding the tradeoff
between discount pricing and the quantity flexibility from two competing suppliers.
However, such decisions may involve other important factors such as quality performance,
logistics capability, cost sharing, promotional efforts, or service agreements to name a few.
Our study indicates that a simple 1-1 supply chain is more desirable from the systems’ point
of View. The extra benefits that a buyer expects to receive from having competing suppliers
may lead the supply chain to become less efficient. As such, a fruitful avenue for future
research is to put more efforts to studying the supply chain contract problems that involve
multiple competing suppliers. In addition, in our research, we assumed that the component
supply is not an issue. However, this may not be true in a real-world setting. Two key
factors that will affect the component’s availability is lead-time and capacity. When a
buyer has multiple customers, the component availability issue becomes even more critical.
Furthermore, we constrained our focus to a single period problem. When buyers and
suppliers continue their business relationship on the same product over alonger period, the
models need to be modified and the problems need to be revisited. Future research efforts

can target these areas.

Appendix of Chapter 3
Proof of Theorem 3.1

It suffices to show that the Hessian matrix is negative definite for both cases.
(1) Case I: V1> v2

:11" =(p—w1><1+a>
Qt

 

- (p- W1)[F ((1 +0061. +612 )(1 +a)-F ((1 -w)q1 +q2)(1—w)]

87

—(p-v1)F((1-w)q1+qz)(1-w)
= (p - w1)(1+ a)F((1+a)q1+q2)-(W1 -v1)(1-w)F((1-w)q1 + qz) (A3.1)

erI,

dqz =—(p—w1)(l+a)2f((1+a)q1 +q2)—(Wl _V1)(1—w)2f((1"w)qi +q2)<0

 

 

:21], :(P-Wz)‘(P—Wi)[F((1+a)qi+42)—F((1—w)q1+q2)]

_(P"vl)[F((1_w)q1+qz)_F(qz)]_(p”V2)F(q2)

=(w1—w.>+<p—w.>F((1+a>q.+q2)

 

—(Wl—V)F((1—60)ql+q2))—(V1_V2)F(q2) (A32)
dzflb
dqzz = —(p—wl)f((1+a)ql +q2)—(Wl —V1)f((1—w)qi +q2)
‘(V1_V2)f(q2)<0
din, _ d211,

 

dqldqz " dqqu1 =‘(P”W1)(1+a)f((1+a)q1+q2)

-(W1— V1)(l- w)f((1- 00611 + 612)
So by examining the Hessian matrix 1H1 and its leading principle minors 1H ,1 and 1H 2I

dZII

 

 

 

we have: 1Hll=jdq—2b-<O and
l
d211,, din,
dq? dqidqz
iH2i=iHi= dznb d’H,
dqqu1 a’qz2

 

 

88

=(P—W1)2(1+a)2f2((1+a)q1+612)
+(P‘ W1)(W1-V1)(1-w)2f((1+a)q1+qz)f((1—(0)q1+42)
+(P— W1)(1+a)2(W1 -V1)f((1+a)qi +qz)f((1-w)q1 +612)
+(1-(0)2(W1—V1)2f2((1-€0)q1+qz)
+(p- wi)(v1-v2)(1+a)2f((1+a)q.+q2)f(qz)
+(W1-V1)(V1“V2)(1-w)2f((1-w)q1+qz)f(qz)
-(p-w1)2(1+a)2f2((1+a)q1+612)
—2(P— W1)(1+a)(W1-V1)(1-(0)f((1+a)q1+qz)f((1—€0)q1+q2)
-(w1-v1)2(1-w)2f2((1-w)q1+612)

= (p- w1)(w1 -v1)(a +w)2f((1+a)q. +q2)f((1-w)r11 +q2)
+(p-w1)(v1-v2)(1+a)2f((1+a)q1+q2)f(qz)
+(w1-v1)(v1-v2)(1-w)2f((1-w)q1+q2)f(q2)>0

So 1H 1 is negative-definite.

(2) Case 11: vlsv,

6117:,
dql

 

=(P—W1)(1+a)

”(I)" W1)[F((1+a)q1+q2)(1+(1)—F((1—w)qi “hm—(0)]
—(P- V2)[F((1—60)qi +Q2)(1—€0)— F((1_w)ql)(1_w)]

- (P - V; )F ((1 - w)qi)(1- (a)

89

= (p - w1)(1+ a)F((1+ a)q1 +q2)-(w1 - v2)(1- w)F ((1 - (0)611 + qz)
‘ (V2 _ V1)(1- (0)F((1— (0)611) (A33)

d211,
dqi

 

= —(p— w1)(1+a)’f((1+a)q1+q2)-(wl-v2)(1-w)2f((1—w)qu+q2)

"(vz _V1)(1—w)2f((1-(0)q1) <0

d111,

dq =(p-w2)—(p-w1)[F((l+a)ql+q2)—F((1"w)q1+q2)]

 

‘(P—V2)F((1_w)qi+q2)
=(Wi'W2)+(P—W1)F((1+a)q1+92)‘(Wi'V2)F((1"w)q1+q2» (A34)

d211,,

dqz = _(p_ W1)f((1+a)q1+q2)—(W1—V2)f((1‘w)q1+q2)<0

 

dZIIb _ d211,,

dqldqz — dqqul =‘(P—W1)(1+05)f((1+a)qi+42)

 

“(W1 ”V2)(1‘w)f((1-w)qi +612)

 

 

 

d211,,
So we have: 1H11=—2- < 0 and
dql
d211,, d211,,
dqf 61(11qu
iH2i=iHl = d211,, d211,,
dqzdcn dqzz

 

 

=(p-W1)2(1+a)2f2((1+a)q1+42)

+(p—W1XW1 _v2)(1+a)2f((1+a)ql +q2)f((1—w)q1 +612)

90

+(p-w1)(w1-v2)(1—w)2f((l+a)q1+q2)f((1-w)q1+q2)
+(w1-v2)2(1-w)2f2((1-w)q1+612)

+(p - W1)(v2 -v1 )(1 -w)2f ((1 +6061] +q2)f ((1 -w)q1)

+(Wi —V2)(V2 —V1)(1-w)2f((1—w)qi +q2)f((1—a))ql)
-(p-w1)2(1+a)2f2((1+a)q1+612)

-2(P- W1)(W1-V2)(1+0!)(1-60)f((1+a)q1+qz)f((1-€0)qi+q2)
-(w1-v2)2(1-w)2f2((1-w)q.+612)

= (p- w1 )(w1 —v2 )(a+w)2f ((1 +a)q1 +q2)f((1-w)q. +612)

+(P - W1)(V2 -V1)(1-€0)2f((1+a)qi +q2)f((1_w)ql)
+(W1-V2)(V2 —V1)(1-€0)2f((1-60)qi +qz)f((1—w)q1) > 0

So 1H1 is negative-definite, thus [1,, is concave in ql and q2 D

Proof of Lemma 3.1

Exploring the FOCs, we have:
(p — w,)(1+ affm + an. + q2)-(w1 — v. )(1 — w)F<(1— mm. + q.) = 0 (A35)
(w1— w.)+ (p — w.)F((1+ a)q. + q.)— (w. — v.)F((1— anq, + q.»

— (vl — v2)F(q2) = 0 (A3.6)

Operate (A3.6)* (1 - (0) - (A35), then divided by (1 — (0) we get

91

(W1_W2)‘(Z-1)(P_Wi)+(Z-1)(P"W1)F((1+a)q1+q2)—(Vi_v2)F(q2)=O

(W1_ W2)+(l_1)(p-Wl)+(vl —V2)F(q2)
(Z‘IXP-Wi)

=A

 

3 F((1+a)q1+q2): —

=> (1+a)ql +612 =F—‘(A)

1 _
:> q1=—(F ‘(A)—q2) (A37)
1+a

Operate (A3.6)* (l + (1)—(A3.5), then divided by (l - (0) we get

10"] —W2)—(Z’1)(Wi’V1)F((1_(0)qi+qz)‘Z(V1—V2)F(qz)=0

 

D F((1—(0)q1 +q2): [(Wl-W2)—Z(V1_V2)F(q2):B

(Z — 1)(W1 _ vl)
3(1-w)qi+qz = F409)

1 _
:> q1= —(F ‘(B)—q2) (A33)
l—a)
Notice that OSBSASI,so we obtain
(a) OSB

Z(W1 —w2)—Z(V1—V2)F(q2)
(Z—I)(W1“V1)

20

 

Z) Z(W1—W2)—Z(V1_v2)F(q2)ZO
3 W1 ‘W2 2(V1"V2)F(qz)

 

=> F(q2)-<-

must be true at optimum

v1 V2

92

(b) A31

3 _(W1 _w2)+(I—1)(P-W1)+(V1"V2)F(Q2)
(Z-l)(P—W1)

Sl

 

3 —(W1 _W2)+(Z_1)(P—Wi)+(vi _V2)F(q2)S(Z-1)(P-Wl)
3 W1 —W2 2("'1“’2)F(92)
(c) 133A

Z(W1‘W2)_Z(V1_V2)F(42)
(Z—1XW1 - V1)

 

S _(Wl —W2)+(Z—1)(P_Wi)+(vi —V2)F(qz)
(Z-IXP-Wi)

 

2(W1_V1)[(P—W2)"'(V1‘V2)F(q2)]
(P—W1)[(W2 _V1)+(V1-V2)F(q2)]

 

DZ

(detailed derivation skipped)

F(q2)_<__’1’l___w_2

1V2

(W1 ‘V1)[(P"W2)-(V1 ‘V2)F(92)] >
(P-W1)[(W2 'V1)+(V1"V2)F(q2)]—

 

 

(W1 _V1)[(P' W2)_(V1_V2)Ml
(v1 _v2) :1
<p—w.)1(w2—v.)+(v. wawi

(V1 _ V2)

wl—w

 

So we conclude that at the optimal condition, F (qz) S v
— v
1 2

The second part of the Lemma is a direct result (2) is a by letting (A3.7)=(A3.8).

93

(A39)

2 must be true for all ,1/ .

When

(P — W2)(W1 ‘- V1)

q; = 0, from (A39) we get 1 2 . Thus if

 

(P ‘ W1)(W2 — v1)
1 — w w —v ,
Z>£=(p 2X 1 1), B<A,thus ql >0 mustbettrue. D
(p - W1)(W2 -v1)
Proof of Theorem 3.4

-C
When(1+ a)ql +q2 = Q,, F ((1 + a)q1 +q2) = F (Q) = 15—7. Thus from (A3.5)
"' 1

and (A3.6) we have

 

_ _Z(p—Wl)(C-vl)_
F((1 w)ql+q2)- (P‘v1)(wi_v1) —K

F(q ): W1 "W2 _ (l-l)(p-W1)(C-v1)
2 V1 'V2 (V1 ’V2)(P_Vi)

S.) q. = F"(H), q, = fiiF“ (Io—F“ (11)] = 73—01-117“ (iii—F“ (H)1

l

_ _ -c _
Rearrange the terms we get IF 1(10‘ F l(%) = (Z - 1)F 1(H).

l

p—c
P—VI.

 

Notice that 0 S H S K S

(a) OSH

W1 —W2 _(Z_1)(p_wl)(C—V|) >0
___) _
v1 —v2 (V1 "Vz)(p_vi)

(z-l)(p-W1)(c-v1)
P—vl

 

(b) HSK

94

: W1_W2 _(Z-IXP-Wixc—Vi) < 1(p‘W1XC-V1)
V1_V2 (vl—v2)(p—v1) — (p-viXW1—V1)

 

 

 

 

 

 

Wi—Wz (P—W1)(C—V1) < 1(P—W1XC—V1) Z(P—Wi)(C-V1)
:> + _ +
V1_V2 (V1 —v2)(p—v1) (p_vi)(W1-v1) (v1 ‘V2)(P—Vi)

_ (W1"V1)[(P‘Wi)(C—V1)+(P‘Vixwi_W2)]
:> z—

 

=1

 

 

 

(P _ W1)(C - V1)(Wl — V2) —min
(c) Ks 17—6
p_v1
Z(p—w,)(c—v1)< p—c 2) 5(P’C)(W1—V1):E D
(iv-V,)(w,—v1)"'p—pl (p-W,)(c—v,) max

Proof of Lemma 3.2

Exploring the FOCs, we have:
(p — w,)(1+ a)F((1+ a)q. + q.) - (w. - v.)(1— w)F((1— w)q, + .12)

—(v. —v.)(1-w)F((1—w)q1)=o (A310)
(w1- w2)+(p — w1)F((1+a)q1+q.)—(w.— v2)F((1—w)qi +q.)) = 0 (A3.) 1)
Operate (3.8)* (1 — (a) - (3.7), then divided by (1 — m) we get

(w1— w.)— (2: -1)(p — w.)F((1 + a)q, + q.) +(v. — v.)F((1— w)q.) = 0

(w! _W2)+(V2 ‘V1)F((1—0))qz)= A.
(l-IXP-Wi)

 

:> F((1+a)qi +q2)=1- (A3.12)

=> (1+a)q1+q2 =F"(A')

3 q2 = F-1(AI)_(1+a)q1

Now operate (3.8)*(1+ (1)—(3.7), then divided by (1 - 0)) we get

95

Z(WlTW2)—(ZT1)(W1TV2)F((1"w)ql+qz)+(V2 —v1)F((1—a))q1)=0

Z(W1TW2)+(V2 —v1)F((1—a))q1)

=B' A3.13
(I -1)(Wi T V2) ( )

 

3 F((1Ta’)ql +612):

=> (1-w)q1+q2 =F“‘(B')

=> qz =F"(B')-(1-w)q1
Thus, F"(A')—(1+ot)q,=F"(B')—(1-a))ql leads to
(a+w)q{ = F“(A')—F"(B')
Also, OsB'sA'slz

(a) Its straightforward that 0 S B. and A' 5.1 are always true..

(b) B' s A'

 

 

1(WlTW2)+(V2TVI)F((1Tm)ql) <1_(WlTW2)+(V2TVI)F((1Tw)q2)
(l-l)(Wi Tvz) — (ITIXPT W1)

F((1-w)q') s ICU—WIXWZ TV2)—(P-W2)(w1—v2)
l (P"\('2)(v2 -v1)

 

I)

96

CHAPTER 4 PRICE MARKDOWN SCHEME IN A
THREE-ECHELON SUPPLY CHAIN

4.1 Introduction

Quantity discount pricing, or price-break schemes are fundamental strategies common to
the high-technology sector (PC assembly, semiconductor manufacturing), the service sector
(cell phone services, transportation services), and the consumer packaged goods sector
(breakfast cereals, dairy products) among others. The topic of quantity discount hasbeen
the subject of both managerial debate and academic research. These include: 1) suppliers’
motivation to offer price discount incentives in the hope of stimulating sales volume, 2)
excess inventories that lead to high carrying costs and obsolescence, 3) buying organizations’
imperative to reduce procurement spending while contributing to corporate profitability, and
4) buying organizations’ tendency to place large-sized special orders when offered
cost-saving price-breaks by suppliers (Ramasesh and Rachamadugu, 2001). In general, the
main task has been to find the total cost minimizing order quantity when the seller offers a
price discount schedule.

The three most common price-break structures in the literature are the all-units
price-break, the incremental quantity price-break, and price markdown (Hu and Munson,
2002; Weng, 1995; Gupta, 1988; Madan et al., 1993; Christoph and LaForge, 1989; Diaby
and Martel, 1993; Arcelus and Srinivasan, 1995; Wee and Yu, 1997; Fazel et al., 1998;
Abad, 2003; Lin & Kroll, 1997; Khouja, 1995). In an all-units scheme, the buyer pays the
same unit price for every unit ordered with this unit price being determined by the quantity
range into which the order fits (Madan et al., 1993). In the incremental discounting scheme,
discounts are offered only on the additional units ordered beyond a specified quantity (Gupta,

1988). In the price markdown scheme, the buyer pays the same unit prices regardless of the

97

ordering quantity. All three discount schemes are in nature a quantity discount scheme such
that the greater the purchasing volume, the higher the saving.

In this chapter we study a price markdown contract in a supply chain that consists of a
supplier, an original equipment manufacturer (OEM), and a buyer (retailer). The supplier
produces the component that is needed for the product that the buyer sells to end-customers
and the OEM is hired by the buyer to produce or assemble the product. A supply chain
characterized with an OEM can be found in various industries such as the PC, cell phone,
high-tech, and grocery/retailing. The component price is negotiated between the supplier
and buyer. The supplier sells the component directly to the OEM and the buyer pays tothe
OEM the component costs and a fixed rate per unitof final product made. In this supply
chain, the supplier is able to develop new technologies so that it can cut down the production
cost. Demand is price sensitive; it increases as price decreases. Thus, the supplier takes
the initiative to offer a price discount and the buyer will adjust the retail price downwards
accordingly. Both parties expect to benefit from higher demand induced by a lower price.
Analogous to the Stackelberg game, the supplier is a leader of this game and the buyer plays
the role of a follower.

Price markdown problems in a three-echelon supply chain can be found in many
industries. For example, in the personal computer assembly sector of the computer
industry, PC makers routinely negotiate price reductions with their suppliers on a quarterly or
even monthly basis, because of the relatively short life-cycles of the components. The price
reduction is made possible because component suppliers are able to develop new
technologies so that they can produce the same components at a lower cost. An excellent
example of the price markdown scheme that we consider is Intel’s CPU/Chipset price-breaks.
Intel offers price markdown roughly once every two months. The price difference between

the current price and the new price can approach 35% of the original price of the same CPUs.

98

Intel’s price breaks are typically announced well in advance and the new prices become
effective on a pre-determined date. As a result, the PC makers also reduce the prices of PCs
to reflect the CPU price breaks. Many other electronic component manufacturers, such as
Seagate, Western Digital (Hard drive disks), Hitachi, Sony (optical drives), Micron, Infenion,
Hynix, Samsung (Semiconductor memory chips/ modules), and LG/Philips (LCD) also offer
the same type of price markdown schemes to the PC makers.

When a supplier decides to offer a price discount, her task is to specify the new pricing so
that she can benefit the most from the increased demand. In addition, the supplier needs to
reserve enough capacity to cope with the higher demand. On the other hand, the buyerwill
adjust the retail price to reflect the lowered component cost in order to induce higher
demand, hoping to create higher profits. The increasing demand will also induce the need
to adjust the stocking level of the final product. These decisions are effective in the next
period with the new, lower pricing for both the supplier and buyer.

In contrast, the OEM’s problem deals with when the new price becomes effective. The
OEM’s profits per unit of product made is not affected by the price changes, however, the
price difference will affect its holding cost and stockout cost for any units of components
carried beyond the price break point. In addition, new components’ arrival time is uncertain
due to weather condition, airport congestion, or custom inspection just to name a few.
Moreover, the delivery uncertainty is filrther complicated by a mismatch between the buyer’s
operating schedules and the carrier’s delivery schedules. In this context, the OEM is mainly
concerned with the safety inventory level to be carried beyond the price break because the
price difference may cause substantial finance loss if a wrong stocking decision is made.

In this chapter, we develop a price-break model that involves a complete supply chain.
This decision problem has not been considered thoroughly in previous discount contract

research. The model discussed in the paper extends the price-break literature. The results

99

offer significant managerial insights to practitioners in industries where the
“cost-price-performance squeeze” creates slim profit margins and in lean procurement
environments.

The remainder of the paper is organized as follows. We begin with reviewing the
related literature. We then describe the research problem of interest. Next, we formulate
the supplier’s and buyer’s problems, followed by the OEM’s problem. Weprovide a
numerical example to demonstrate the usefulness of the model. The data utilized in the
numerical illustration were obtained from a major OEM firm in the PC industry. We also
present a series of sensitivity analyses, which yield several interesting managerial insights
The final section summarizes our results and presents several extensions to consider in future

research.

4.2 Literature Review

Research on the price-break problem is rich and has been addressed in such diverse
fields as economics, marketing, and procurement. We focus primarily on those studies in
the inventory and procurement literature. In this research stream, several researchers have
studied the problem from the buyer’s perspective which seeks to minimize the total cost by
determining the optimal ordering quantity (Gupta, 1988; Ardalan, 1988; Christoph and
LaForge, 1989; Aull-Hyde, 1992; Madan et al., 1993; Diaby and Martel, 1993; Arcelus and
Srinivasan, 1995; Wee and Yu, 1997; F azel et el., 2003; Rubin and Benton, 2003). In
contrast, several researchers have examined the problem of determining an Optimal schedule
for price discounts from the supplier perspective. This perspective seeks to maximize the
profit for the supplier (Kim and Hwang, 1988; Wang and Wu, 2000; Klastorin et al., 2002;
Rubin and Benton, 2003; Bumetas et al., 2007). Yet another stream of research in quantity

discount has paid much attention to how buyer and supplier can jointly determine the optimal

100

discount and ordering policies that achieves the system profit; see Weng (1995), Corbett and
Groote (2000), Wang (2005).

While various research aim at designing a quantity discount mechanism that can achieve
the system-wide profit for the individually motivated buyer and supplier, such a goal may not
be viable in a decentralized, wholesale-price-based supply chain, which is prevalent in
numerous industries (Wang, 2005). A price—markdown scheme is virtually a price-only
contract. A more thorough understanding of the decision making processes in the presence
of the price markdown scheme in a decentralized supply chain is warranted. The majority
of work in the discount literature considered a two-echelon supply chain that consists of a
supplier and a buyer (retailer). However, it is common that a supply chain contains more
than two agents. In this case, the price discount’s impact is beyond the two echelons. In
this chapter, we study a price markdown problem on a three-echelon supply chain in which
an OEM provides production service to the buyer. Such a price markdown scheme is
induced by manufacturing technology innovation and is widely observed in the hi-tech
industry (Lee et al., 2000)

Various forms of the demand function have been utilized in the quantity discount research
stream. In those focusing on ordering cost and lot-sizing in the presence of quantity discount,
market demand was assumed to be a constant that is independent of the price discount (Chiang,
et al., 1994). On the other hand, those studying how the joint profits can be affected by
quantity discounts have considered demand as a deterministic decreasing function of price
(Weng, 1995; Viswanathan and Wang, 2003; Wang, 2005; Yue et al., 2006). In contrast,
others have studied quantity discount problems by considering stochastic demand; for a
detailed discussion of stochastic demand functions and their assumptions,refer to Petruzzi and
Dada (1999), Ray et al. (2006) and Zhou (2007), Bumetas et al (2007), and Lau et al. ( 2007).

Finally, stochastic demand that incorporate delivery uncertainty has been well documented,

101

i.e., the lead-time demand (Zipkin, 2000). However, work in the discount literature that
considers both demand and lead-time uncertainties is rare.

Similar to most work in the supply chain contract and mechanism literature, he price
markdown scheme that we consider in this paper has a model structure equivalent to the
newsvendor problem. Interest in the newsvendor problem continues unabated, as many
extensions to it have been proposed (Lau and Lau, 198 8a,b; Nahmias and Smith, 1994).
These extensions have considered alternative pricing schedules and lot-sizing rules, multiple
locations, and progressive multiple discounts. These characterizations of the problem have
been observed not only in the consumer packaged goods and airline industries, but also in retail
apparel and automotive industries (Khouja, 1995; Cherikh, 2000). A comprehensive
taxonomy of the newsvendor problem is available in Khouja (1999).

In this chapter, we study a price markdown problem with price-sensitive, stochastic
demand from a supplier, the buyer, and the OEM perspectives. We formulate our models
following the Stackelberg game framework, with the supplier being the leader of the game.
Our model has a similar structure to Ray et al’s (2006) research. Ray et al. (2005) considered
a pricing and stocking problem in a supply chain that consists of a manufacture, a distributor,
and a retailer. But they focused on the logistics side of this supply chain; they modeled and
solved for the optimal pricing and stocking policies for the distributor and the retailer, omitting
the manufacturer’s problem. Our research, on the other hand, considers the entire supply
chain in which we study each party’s problem in this chain. We study the supplier’s optimal
component pricing and capacity planning decisions, the buyer’s optimal retail pricing and
stocking policies, and the OEM’s optimal stocking policy in the presence of the price
markdown scheme. Additionally, we incorporate the interplay of the OEM’s operational
hours and local carrier’s delivery schedules in our models and examine its impact on the

OEM’s optimal ending inventory decisions when offered a price markdown. The problem we

102

study is common in the hi—tech industry and an emerging practice in other industries, however,
little attention has been paid to this discount problem in the literature. Extant literature does
not consider the emerging issue of delivery uncertainty in the optimum ordering policy models
with price-breaks. We seek to fill this gap. In the next section, we describe the general

problem setting and include our assumptions specific to the price—break models.

4.3 Problem Setting

We consider a supply chain that consists of a supplier, a retailer (buyer), and an OEM for
the buyer. The OEM buys component(s) from the supplier and manufactures the product
for the buyer. The buyer pays the OEM a fixed rate per unit made as well as the cost of the
components. In this supply chain, the supplier decides the component price and the buyer
decides the retail price, given the component price The OEM charges the buyer at a fixed
rate per unit built; this service rate is not affected by the prices of the component and the
product. The supplier ships material(s) to the OEM. The OEM then produces and ships
the product(s) to the buyer. To simplify our analysis, we assume that there is one
component and one product in this supply chain.

Demand is assumed price-sensitive in this supply chain. Specifically, demand increases
as price decreases. As discussed previously, the supplier is able cutdown her production
cost by innovation in manufacturing technologies (Lee et al., 2000) Therefore, the supplier
offers a price markdown scheme to the buyer on a regular basis. As the component price
decreases, the buyer will adjust the retail price accordingly. The goal of cutting down the
retail price is to induce a higher level of demand in the hope of achieving a higher level of
profits. On the other hand, the OEM’s service charge is assumed not affected by the price

changes of the component and the product; OEM earns the same fixed rate per unit built for

the buyer.

103

We assume that the buyer shares full information of market demand observation with the
supplier and the OEM, i.e., they are aware of the demand patterns and distribution. So the
supplier’s challenge is to decide the optimal component price to maximize her profit level
and the planned capacity to cope with increased demand. On the other hand, the buyer’s
challenge is to decide the optimal retail pricing policy and the Optimal stocking policies to
maximize his own profit. Notice that both supplier’s and buyer’s problems are associated
with the whole next period with the new pricing. Although not explicitly specified in our
model, the length of the next period with the new pricing can be two to four months which is,
for example, the case in the PC industry.

However, the OEM’s challenge is not only to identify the optimal stocking level for the
next period, but also the optimal “ending” inventory level right before the price break. The
price difference of the components will tremendously increase the carrying cost and decrease
the stockout cost for the OEM, which will lead to a different stocking level from that in the
regular time. After the new (lower) price becomes effective, the OEM’s carrying cost will
be back to the original level. Therefore, the OEM’s decision making focuses on the very
short period of time when the price markdown becomes effective. OEM is known to have a
low margin. Proper ending inventory control can possibly create substantial savings for the
OEM. Both the supplier and the buyer can decide the timing of price discount. The
supplier can announce a price discount after she produces the component at the lower cost;
the buyer can announce a price markdown after he uses up the existing inventory. So the

ending inventory control is not as critical to the supplier and buyer as it is to the OEM.

4.4 The Decentralized Supply Chain Model in The Presence of a Price

Discount

104

We model our problem via a wholesale-price-only discount contract, following the
Stackelberg framework (Ray et al., 2005). We first focus on the supplier’s and the buyer’s
problem and solve for the Optimal (reduced) pricing for both parties as well as the optimal
capacity investment for the supplier and Optimal stocking policy for the buyer. We then
turn our attention to the OEM’s problem, given the new (lower) component and retail prices.

The following notation will be utilized throughout this chapter.

Notation:

H s : Supplier’s total expected profit

H 3 : Buyer’s total expected profit

D : Demand

p,- : retail price in period 1', i=0,l. 0: current period; 1: next period
w. : Component price in period 1'

w : OEM’s service charge per unit of product made

0. : Supplier’s production cost in period 1'

ck : cost per unit of planned capacity

c : OEM’s production cost per unit

: Planned capacity in period 2'

K
¢ : p.d.f. of standard normal distribution
(I)

: c.d.f. of standard normal distribution

(25,) : p.d.f. of random variableD

105

(D D : c.d.f. of random variable D
h : buyer’s holding cost per unit of product
It : OEM’s holding cost per unit of component

S : supplier’s understocking cost per unit

3 : OEM’s understocking cost per unit

S a : Buyer’s understocking cost per unit
I. : Buyer’s stocking level per unit time in period 1'

x. : OEM’s stocking level per unit time in period 1'

4.4.1 Demand structure

We consider a price-sensitive, end-customer demand D( p) arriving at the buyer per
unit time, where p is the retail price of the product. D( p) consists of a deterministic term
and a stochastic error term (Zipkin, 2000; Ray et al., 2005) Two types ofprice-sensitive
demand functions have been widely utilized in the economic and management science
literature: the additive and the multiplicative demand functions (Petruzzi and Dada, 1999).
To simplify the development of the models, we consider only the addictive demand function;
the Optimal policies in our models can easily be revised to adapt the multiplicative demand

functions.
The typical additive demand function has the following format: D( p) = y( p) + 8 ,
where y(p) is deterministic, and decreasing and concave in p (Ray et al., 2005), and 8

is a continuous random variable that follows a normal distribution with mean u.g and

variance 0',2 , i.e., N01,, 0': ) . As a result, D( p) also follows the normal distribution

106

with mean 1(1)) = y(p) + u, and variance 0': , i.e., N(/i(p),0':) . Examples ofthe
forms of y(p)include(1) (A—Bp")’,A > 0,3 > 0,k 2,1,7 31 (2)

A—Bpk,A>0,B>1 (3) ln[(A-Bp)”],720,B>O (Rayetal.,2005). Detailed

discussions of additive and multiplicative D( p) can be found in Mills (1959), Karlin and
Carr (1962), Petruzzi and Dada (1999), Ray et al. (2005), Arcelus et al. (2005), and Bernstein

and Federgruen (2005).

4.4.2 The supplier’s model

Let ( w0 , CO ) be the supplier’s component price and production cost in the current period,
say, period 0; p0 is the retail price set by the buyer, given W0 . The supplier is able to
produce the same component at a lower cost 01 and is planning to reduce the price to WI in
the next period, say, period 1. As a result, the buyer will reduce the retail price from p0
to pl to induce the demand to increase from D(po) to D( p1) . To cope with the
increased demand, the supplier will need to decide a capacity level K], with current
capacity being K 0 . The supplier’s task is to decide the Optimal pricing w: and optimal
capacity K 1‘ so that the expected profit at the new price will increase from the expected

profit when the component price is kept at WO . In other words, the supplier’s Objective is

to maximize the following:

MaxAl'IS = l—IS(w1,cl,K1)-l—IS(w0,cl,K0), where,wO > W], KO <Kl,&

107

H5(w1.,c1,K1.)=(w1-cl)E(D(p1))—ckK1.—sE(D(p1)—K1)+

Lemma 4.1 Maximizing A115 is equivalent to maximizing AU 5 (W1 ,cl) .
Proof. The newer, lower-cost manufacturing technology has been developed, therefore, Cl
is treated as a parameter in the profit function. One can clearly see that H 3 (W0 ,61 ,K 0) is

independent of the decision variables(w1,K1) . As such,

AU 5 = l‘IS(w1,cl , K1) - constant. This completes the proof 0

As a result of lemma 4.1, the all-unit price discount model is virtually a wholesale-price-only
model, because in our model we have only one reduced component price, regardless of order
quantity. Market demand at the buyer’s location will be passed onto the OEM and the
supplier under the assumption of full information sharing. Therefore, the supplier’s model

can then be expressed as following:

Maxns(wlac]aK1):(wl -Cl)2’(pl)—ckKl —SE(D(pl)—Kl)+ (41)

The first terms indicates the expected profit per unit time wih the new component price and

the new retail price. The second term is the cost of capacity per unit time planned for the

next period (period 1). Notice that there is difference between C] and ck . The former

refers to the cost associated with production activities; cl will not occur if there is no

production, e.g., material cost, machine time, etc. On the other hand, ck is the cost

108

associated with reserving the capacity; the planned capacity will cost the supplier even if
there is no production for the component. Examples of C]: can be labor hired, machine

purchased, and facility depreciation. Finally, the model is structured in a manner that
supplier will fulfill all demand. When the demand rate exceeds the planned capacity rate, it
costs the supplier extra cost S per unit to satisfy buyer’s orders. Examples of S include

working overtime, rescheduling production, and expediting a shipment.

Corollary 4.1 HS(WI,C,,K,) is jointly concave in WI and K].

 

 

 

 

 

 

dUS dzl'IS dp
. S = :1. -—1—<O
Proof It can be easrly shown that dwl 2(1),), dwlz (P1) dWl ,

m, <1> (K) ‘1an ¢(K) 0 SSHS
= —c — —=— < --—=

dK, S k S D " dKf S D ‘ ’dwldK,
derS o

d2H _ _ alw2
Therefore, lHlizHll: dwlzs <0and IHZI‘IHI- l dZHS >0.

(IKE

 

 

So the Hessian matrix H is negative definite. This completes the proof. 0

4.4.3 The buyer’s model

The buyer (retailer) directly faces the end-customer and market demand. He sells the
product that is made by the OEM he hires. He bears the material costs and pays the OEM at
a fixed rate per unit of product made. The buyer decides the retail price that maximizes his
expected profit, given the material cost and the OEM service charges. The buyer also needs

to decide a stocking policy so that he will be expecting the highest benefit considering the

109

understocking and overstocking costs. As such, the buyer’s problem can be modeled as

following:

MaxUB(p.,Wp11)=(p1- W1 - Wm)/1(p1)-[hE((11- D(P1))+ + SBE(D(pl) - 10+]
(4.2)

The first term refers to the expected profits from selling the product to endcustomers. The

second terms includes the carrying cost and undertocking penalty from the discrepancy

between the safety stocking level and demand rate per unit time. Notice that the decision
variables in the buyer’s model are(pl , I l) . W1 is exogenous and is decided by the

supplier. Since the buyer is the follower of the Stackelberg game, his goal is to maximize

the expect profit in the next period by finding the optimal (p1, I l) , given W1. Therefore,

the buyer’s model also has the same structure as a wholesaleprice-only contract model.

Parameter values (p0 , wO , 10) in current period serve as a boundary of the optimal solution

and will not affect the decision of (171,11) , unless (191,11) are biding to the boundary

value.

Corollary 4.2

(1)H3(pl,wl,I,) isjointlyconcave in p, and II

(2) Given component price wl , there is a unique retail price pl (W1) and II that will

maximize the buyer’s expected profit.
Proof. (1) The proof is skipped as it is similar to the proof of corollary 4.1.

(2) This is the direct result of corollary 42(1).

110

4.4.4. Finding the optimal pricing and stocking policies
The solution method follows the backward induction that is well documented in the
contract literature (Tsay, 1999; Ray et al., 2005). We will solve the buyer’s model,

followed by solving the supplier’s model.

4.4.4.1 Optimal pricing and stocking policies for the buyer

Lemma4.2 I: =y(P1)+#g+Z:0},Whefe 2: =¢-l(——SB )
h+sB
Proof.

To solve I I. , we first rearrange the terms in the bracket in (42) as follows:
hEUl - D(PO)+ + SBE(D(.01) - 11 )+
= hE(11—)’(P1)—3)+ +SB[13(11”yum-8)+ +fl. -(11-y(p1))]
= (h + SB)E(11- y(p1) - 8T + Sig/1. - 33(11- y(p1))

Applying the standard normalization procedure, let I I — y(p1) = #5 + 2105

53—11

£)+

 

2 EU. -y<p.>—e>* = E(zla. —(a—u.»: = o.E<z. —

6

a—uc

 

Notice that ~ N(0,1). So the objective fimction can be expressed as follows:

8

H3

—#5

E

 

8 +
=(p1—W1 _Wm)’1(pl)—(h+SB)O-€E(Zl _ ) ‘53.“; +5301; +2105)

111

due
.__) dz,

 

= —(h+SB)O'£CD(Zl)+SBO'£ = O

2.2: =<I>"<—SB—)
h+sB

=> If = mm + y. + 2&7. a (4.3)

The advantage of utilizing the standard normal 21 is that when 8 is not normal, the
solution process can be used to approximate the optimal stocking policy, as long as the mean
and variance of a are known. In addition, one can observe from (4.3) that buyer’s optimal

stocking policy per unit time is affected by the retail price as well as bythe distribution

parameters of the error term in the demand function. Finally, once I 1‘ is identified, the

buyer will know how much change in the stocking level the new, lower price has caused.

Corollary 4.3 A1 = I: — It; > 0.
Proof. It’s straightforward to show that I g = y(p0) + #6 + 2305 , where

SB

)=Z;.Therefore, AI'=IlS-Ig=y(pl)—y(p0)>0 [:1
h+sB

 

zO =CD'1(

Next we solve for the optimal pricing policy.

Theorem 4.1 Given component price w,, the buyer’s optimal retail price is the unique

 

solution to the following: ,01 +

112

Proof. First we rearrange the terms in the bracket in (4.2) as following:
hE(11_D(p1))+ +SaE(D(rvr)-Ir)+
= hE[(€ — (Ir -y(Pr)))+ +(11- y(Pr)) -#.]+SBE(8 - (Ir - y(Pr)))+
= (h +Sa)E(8 ~(11- y(Pr)))+ + 11(11 — y(Pr))- hfla
Substituting in I 1. = y( P1)+ lug + 2:08 we get

hE(I, _D(pr))+ +SBE(D(p1)_11)+

 

8— .+
= (h+s.>o-.E< a”: —z.) +h<I. —y(p.))—Iw.

6

=(h+SB)0-51N(Z:)+h(lus +Zf0-e)_h#53

where I~(z:) = M) — z;'(1— wt» and z.‘ = ch“ (1L)
h + SB

(Porteus, 2002)

So we can rewrite the buyer’s expected profit function as following:
H3 2 (pl _ W1_ Wnr)/S'(pl) _ (h + SB)0£IN (2i) — h(#a + 2:06)+ hr“;
= (p. — w. — w.,)/1(p.)— (h + s. )a.t¢(zf ) — z:<1— <1><z:»1— hzfa.

:(pl - WI _ Wm)/l~(P1)-(h +53 )0'£¢(ZIS)

 

dn
FOC: de = (p. - W1_ Wm )l'(pr)+/1(pr) = 0
l

:>The optimal retail price p; solves P1 +— D (4.4)

113

Notice that p: 2 W1 + W... due to the fact that A'(p,° ) < 0. It is also clear that the retail

price pl is increasing in the component price w,. Once the supplier decides the
component price wI , the buyer will be able to obtain the optimal retail price via (4.4).
Interestingly, during the standardization procedure of the normal random variable, the terms
that are dependent on price pl the terms of overstocking and understanding costs of (4.2)

get eliminated. As such, the optimal stocking policy I I is affected by the retail price (4.3),

but the expected total overstocking and understocking costs are not.

4.4.4.2 Optimal pricing and stocking policies for the supplier
Now we turn our attention to (4.1). Similarly to the solution procedure of the buyer’s
model, we first solve for the optimal capacity level for the supplier, then we analyze the

optimal pricing policy and the optimal conditions.

t t o _ S—C
Lemma14.3 If s>ck, Kl =y(p1)+,u£+220'£,where 22 =(D l(——k).
3

Otherwise, K: = y(p1)+,u£ .
Proof. Supplier’s expected profit function can be rewritten as
Hs(w|9C13Kl) : (W1 —Cl)’i(p1)+(s —Ck)K1-SE(K1—y(p1)_£)+ _Slug

Let K1 -y(pl)=,u£ +220£,wehave

—#£

8

 

r1. = (w. —c.)2<p.)+(s -c.)(y(pr)+#. + z.a.)—so-.E(zz - S ): -sa.

dH
FOC; S =(s-ck)—s(l>(zz)=0
dz2

 

114

e _ S—C t
:> If S-Ck >0, z2 =(D l(———"—)=0. Otherwise, 22 =0
5

=> K: = y(pl)+,u£ +z;0'£ a (4.5)

As one can see, the optimal capacity is also dependent on the retail price, as the retail price
affects the demand level, which in turn affects the supplier’s planned capacity. Additionally,

it is clear that the optimal capacity is increasing in the retail price. Furthermore, the

capacity needs to be greater than or equal to zero. When S < Ck , the 2; will approach
negative infinity, thus resulting in K 1‘ to be negative infinite. In this case,we simply reset

the K 1. to be zero. Finally, we expect that K It to increase from K ; because when a

price discount is offered, the demand is expected to increase, thus resulting in a larger

capacity reservation to accommodate the potentially larger demand.

Corollary 4.4 AK =KlS—Kg >0.

The next step is to solve the optimal component pricing for the next period. Solving

w: directly from the first order condition of H S can be problematic because we will not
be able to convert pl to WI in a clean format without knowing the exact form of 1(1),).
So instead of solving for optimal w: , we convert wI to pl via (4.4) and rewrite the

supplier’s expected profit as a function of pl . By doing so, we can solve for the optimal

p: that benefits the supplier the most. Through (4.4), we can then identify the optimal

115

 

 

w; that will induce the buyer to decide the retail price as p: . This technique has been

also employed by Ray et al. (2005).

2(1’1)
1071)

 

Lemma 4.4 MaxHS is equivalent to Alpaflpl + — Wm ‘Cl ‘01. )1(P1)
P1 1

Proof. Utilize (4.5), (4.6) and rewrite (4.1) as

Mpr)

I_IS : (pl +—'—_—-Wm _Cl)/S'(pl)_ck(y(pl)+fls +Z;O-£)—S0-£[N(Z;)
3(171)
xi . .
=(P1+1%%—Wm 'Cl _Ck )A(pl)_ckZZO-e _S051N(ZZ) (4-6)
1

t e t a o _ S_C
where [N (22) =¢(zz)—z2 (1—<D(zz)) and 22 = (D l(-—S-L).It is clear that the

last two terms are independent of p1 , thus the proof is completed. 0

 

 

Theorem 4.2
.3 ..2 . ..
2." 4,1 32. 5M . , _
(1)If > ’12 + 1' — A ,thenthere ex18tsaun1que pl such that H5(pl,zz) lS
minimized

(2) Buyer’s desired p: solves the following:

_ 22(1). mm + 34001)
13(1)!) 1(101)

 

l =Cl+Ck+Wm

Proof. To prove the uniqueness of optimal pl , it suffices to show that H s is unimodal,

 

116

 

 

 

 

 

 

 

 

 

aTIS =“d—’[(P1+it—wm "CI ‘Ck)’l~]
dp, dpl ’1
A . A .
=(pl SPY—Wm —C1—Ck) 1+(pl+7_wm —cl —clt)’1
=(2—%M+(pr +i—wm-cr—c.)2
,1 2.
2.2)." . .
=3l—7+(p1 —wm —c] —ck)/l =0
31 2.2/l"
(P1 _Wm _Cl —Ck):—_/1_'+ Z3
2 -2 u 2 I m— 2 n
d st :4,._2M 2. +22.» 21/1 +(pr-wm—cr-CM"
dp, 3.
l2 n 2 I "I— 2 n2 2 n
241-le l +2.2;Z 22/1 +(_fi+2/31)l.
,t' w A '
. 22/1” 311" 22121" +2311” 4,122.”
=41 + '3 — , — '3
A. ,1 A
2
Since it. and 21" are both negative, it is straightforward that ZS < 0 requires
P1
42” 31” 5M"
2. > ’12 + 1— A tobetrue. 0

When 2m satisfies the inequality, there is an unique p: such that Us is maximized.

This optimal p: is desired by the supplier, and anticipated by (4.4). As such, by

specifying the desirable p: , the supplier will set the optimal component price w: through

(4.4), which will consequently induce the buyer to set the retail price at p; . Notice that

117

each term on the right hand side of the inequality above is negative, under the assumption
that xi. is decreasing and concave. So as long as it" 2 0, H s is guaranteed unimodal,

thus having a unique optimal solution.

We have thus far solved for supplier’s optimal pricing and capacity planning policies, as
well as the buyer’s optimal retail pricing and stocking policies. Both buyer and supplier’s
decisions are effective for the next whole period. In contrast, the OEM’s problem has a
slightly different focus. As mentioned previously, the OEM’s problem falls on deciding the
optimal stocking level to be carried beyond the price break. The price difference does
change tremendously both the holding and stockout costs specifically to the inventory being
carried beyond the price break point, which is termed “ending inventory” hereafter. Once
the new (lower) component price becomes effective, both carrying cost and stockout cost wil
be back to the normal level. Furthermore, observations from the PC industry suggest that
lead-time variation needs to be incorporated into the ending inventory decision, because the
supplier tends not to make component available before the price break. We present the

OEM’s model in the next section.

4.5 The OEM’s Model

The OEM’s problem is twofold. We first present the model that aims to solve the
optimal “regular” stocking policy of the component for the OEM for the next period with the
new pricing. We then present the second model which facilitates the OEM’s decision in
deciding the “ending” inventory level to cope with the impact of price difference and the
delivery uncertainty. Furthermore, we analyze how the match and mismatch of the carrier’s
schedule and the OEM’s operation schedule, which further complicates the delivery

uncertainty, will affect such ending inventory decisions.

118

4.5.1. Model I: deciding optimal stocking policy for the new period
The OEM’s expected profit function has the same structure as the buyer’s expected profit

function:

Um (W... .6... ,x1) = (W... - Cm)/1(pr ) - [hmE((xr - 130% ))+ + S..E(D(Pr ) - x1)+]

As mentioned, the OEM is not involved in deciding pI , thus 1(pl) is exogenous to the

OEM’s model. Additionally, W comes from negotiation between the OEM and the buyer,

m

and the production cost Cm is assumed not changing with the component price. As a result,

the first term in the OEM’s expected profit function is considered exogenous.
Consequently, OEM’s goal to decide the optimal stocking policy can be done by minimizing

the holding and stockout costs. Therefore, the objective fimction is as follows:

Min Cm = hmE[(xl —D(p,)]+ +smE[D(p,)-xl]+ (4.7)

The expected total cost function follows the typical newsvendor problem framework, thus it
is convex and has a unique optimal solution that minimizes the objective function.

Following the same solution approach presented in section 4.4.4, it can be easily shown that

Sm

h+s

m m

 

x: = y(pl)+,u£ + 23.06 , where Z; = (D—l( ) . One can observe that x: is also

dependent on p1 , which is similar to buyer’s I 1. . When there is a price discount on the

component price, retail price is expected to be lowered. Thus, the OEM’s stocking policy

119

should be adjusted accordingly in order to accommodate the increased demand. Notice that

x]. is the optimal stocking policy at regular time after the price break. Next, we turn our

attention to the optimal “ending inventory” to be carried beyond the price break. The
ending inventory is purchased in the current period at the current component price in order to

fulfill any demand incurred before the new component’s arrival after the price break.

4.5.2. Model 11: deciding optimal “ending” inventory policy

Inventory to be carried beyond the price break are purchased at the old price wo. As
mentioned the new component price will be w1 . So each unit of component carried beyond
the price break will result in a loss of A = w0 — wl . As a result, the carrying cost for the
ending inventory is increased from hm to hm + A. On the other hand, the stockout cost is

actually decreased from Sm to Sm — A , because each unit of product not fulfilled avoids a

loss of price difference A. As seen in the previous section, both carrying cost and stockout
cost are key elements to the optimal stocking policy, given the retail price. When price
difference A is large, its impact on the stocking policy can be substantial.

Additionally, it is common in the PC industry that the component suppliers will not make
the new components at the new (lower) price available before the price break point. They
tend to make aggressive shipments of the new component to target at arriving on the price
break effective day. Component suppliers such as Intel, AMD, and Seagate all have
practiced this way for years. However, shipments of new components normally come from
oversea countries and the arrival time can be easily delayed due to weather, airport
congestion, or custom inspection. As a result, the arrival time is uncertain and this

uncertainty is critical to the safety stock decision right before the price break. Therefore, in

120

this model we consider both demand and delivery uncertainties. As such, the objective

function can be expressed as follows:

M" C... = (h... + A)E[(x- D(PpLHS +(S... * A)E[D(P1,L)-x]+ (4.8)

Stochastic demand that incorporates lead-time uncertainty is well documented in literature
(Zipkin, 2000; Ray et al., 2005); such demand has been termed “lead-time demand” (LTD).
We employ the treatment for the lead-time demand, however, ourL is slightly different from
that in literature. While L refers to a random variable “lead-time” in literature, it is referred
as the random variable “arrival time” (after price break point) in our model. In other words,
what we consider in this model is the randomness of the component’s arrival time after price
break, rather than the variation of the length of time in transit. Nonetheless, thisdoes not
cause any change of the treatment for the lead-time demand technically.

Assume that L has mean E (L) and variance Var(L) , but its distribution is unknown.

Hence the lead-time demand D( p, ,L) can be expressed as y, = M p, )E (L) and
of = ,t( p, )2 Var(L) + a§E(L) (Ross, 1989; Zipkin, 2000; Ray eta1., 2005).

sm—A

Corollary 4.5 x. = y(p,)+,uL +Z;0'L, where Z; = (1)—Kn)

Here we follow the same procedure as that presented in section 4.4.4 and 4.5.1 to obtain the

optimal ending inventory x'. However, in section 4.4.1 and 4.5.1 we dealt with normal

random variable 8. Here we deal with a random variable with an unknown distribution.
As a result, the standardization process presents an “approximated” solution to the true

optimal stocking policy.

121

It is clear that x. is decreasing in A; the price break induces the OEM to lower the

inventory level due to the lower stockout cost. Interestingly, the increased carrying cost

does not have any effect in the optimal ending inventory; it gets balanced out by the effect of

the decreased stockout cost, thus resulting in the same denominator in 24 as that in 23 .
Although the price difference A ensures 23 > 24 , x. can be greater than, equal to, or

smaller then x, , depending on the value of ,Lt, and a, . As the mean and variance
increase, xS will increase to cope with the delivery uncertainty. Finally, itis easy to see

that if one considers the lead-time uncertainty for the entire next period, i.e., model I, x,

will be expressed as y(p,) + ,uL + 2:0", . In this case x. < x, will always be true.

So far we have developed the OEM’s price break model by assuming that the shipment of
the new component at the new (lower) price can arrive any time after the price break.
However, in realty this tends not to be the case. It is common in practice for the freight
companies to co-locate with the OEM (for example, in an industrial park). The components
shipped from the supplier first arrive at the freight company’s local depotbefore they are
delivered to the OEM’s facility. The carrier and the OEM may have different operating
hours on each working day. Deliveries are not possible during the carrier’s non-delivery
(ND) hours, nor are they possible during the OEM’s non-working (NW) hours, assuming
receiving is not possible during the NW hours. As a result, the mismatch of the OEM’s and
carrier’s schedules is worth studying as it affects the size of total demand over time, thus
affecting the optimal ending inventory level.

Without loss of generality, and to simplify the presentation, we assume that the OEM is
near the freight company’s Loci] depot such that the transportation lead-time between these

two facilities is short enough to be ignored. Under this assumption, the component’s

122

arrival time at the OEM is the same as the freight company’s time of delivery from its local
depot. This assumption is not restrictive and can be easily relaxed by adding a constant
local transportation lead-time into the model.

In the next sections we analyze two mismatch cases: NW>ND and ND>NW. The

model we have developed in section 4.5.2 is considered a “full-time” model in which both

carrier and OEM operate 24 hours a day.

4.5.2.1 ND>NW
In the non-24-hour cases, the random variableL is defined only on the carrier’s operating
hours; no delivery is available during the ND hours. On the other hand, demand D(p,)

exists only during the working hours. In other words, we assume that there will be no
demand when the manufacturer is not operating. In addition, we assume that ND and NW
are fixed on each working day and that both the carrier and OEM have the same starting time

on each working day.

Let L0=ND-NW. Demand associated withLand L0 are labeled D, and 0,0, and

. 2 . .
each has mean and variance of (,ll, , 0' L) and (11,0 ,0'20) , respectively. Notice that D,

and D ,0 are no longer unit-time demand; they are total demand for L and L0. Additionally,

D, and D ,0 are clearly independent.

Corollary 4.6 ,u, = Mp, )E(L) ,0: = Mp, )2 Var(L) + 0:E(L) , 21,0 = Mp, )LO,

and 0'20 = L00",2 . (Proof skipped)

123

The total demand, D , prior to the arrival of the new shipment of new components will be
the sum of D, and 0,0. Letting the time of anival fall on the i-th day after the

price-break’s effective date, the demand by the arrival time can be expressed as
D = D, + (i -—1)D,0 . As such, the mean and variance ofD, (,uD , 0,2,) can be derived as

follows:
IUD : #1. +(i—1)1u1.0 : 1(P1)[E(L) + (i — D140]
of, = of + (i — 1)2 of, = 1(1), )2 Var(L) + ojE(L) + (i — 1)2 Loo;

= ,r( p, )2 Var(L) + 03ml.) + (i—1)2L0] (4.9)

Now we have the mean and variance of the demand calculated, we can then approximate the
solution by using the normalization technique. Similar to solutions in corollary 4.5, we

have the following:

‘ t t —l Sm _A
Corollary 4.7 x = y(p,)+,uD +2400, where 24 = (D (27:)

To facilitate the task of solving the optimal ending inventory when ND>NW, we develop ar

heuristic algorithm to find the optimal solution x'. See the Appendix of chapter 4 for the

details.

4.5.2.2 NW>ND

Let Lo=NW-ND. During Lo , delivery of materials is possible but manufacturer has

no demand. However, random variable L needs to be considered in the non-ND hours, the
NW does not prevent the material’s arrival at the carrier’s local depot. To avoid changing

the definition of the random variable L, we assume that there is there is pseudo demand

124

D(p,) per unit time during Lo. Total pseudo demand during Lo is labeled as DZ.‘

Assuming the component’s arrival time falls on the i-th day after the price-break’s effective

date, the total demand prior to the arrival of the new shipment of materials with lower price

r—l
can be expressed as D = D, — ZDZO, . Note that if the arrival time falls on the i-th Lo ,
o

it will automatically be considered as arriving on the (1' +1 )—th day, which is the earliest

possible day for the OEM to receive the materials . D, has ,u, = M pl )E (L) and
0,2 = Mp,)2Var(L)+0'£SE(L). For each D201” 'uZOi = #20 = l(p,)Z0 and

2

ULOi

2 _ 2
= 0Z0 = L00: . So based on the newsvendor problem framework, we have

i—l t S -A
PD — D-Sx =$—
( L g [.01 ) h +S (4.10)

m m

We express (4.10) differently from all previous models in this section due to the fact that

D20 is part of D, and they should NOT be independent of each other. The expressions

of ,uD and 0,2) can be obtained, but they are somewhat complex and the derivation is
tedious. As a result, we omit the expression of #0 and a; and present only the heuristic
algorithm for solving x. in (4.10), the details of which can be found in the appendix of this
chapter.

In summary, we have developed the OEM’s model in response to the price break,
considering the lead-time demand as well as the mismatch of schedules. The delivery
uncertainty in our OEM’s model has two facets the variation of arrival time and the

match/mismatch of the operating schedules of the carrier and the OEM. Price difference

125

 

would lead the OEM to lower the ending inventory, however, the delivery uncertainty would
bring the inventory up. Our model can assist the OEM to understand the impact of these
two forces on the optimal ending inventory decision more thoroughly and thus identify the
optimal ending policy when there is a price break announced. In the next section we
present a numerical example to demonstrate the usefulness of our models and discuss the

managerial implications.

4.6. Numerical Experiments
We present the numerical experiments and results of for the buyer, supplier and the OEM

in the supplier chain. For the convenience of analysis, we consider a base dataset for the

numerical experiments as follows: a=2000, b=2, Wm =30, 6,, =5 , Cm =15, 11:10, hm =4,

s=15, Sm =5, SB =5, #8 =20, 0'E =10, unless otherwise specified.

Our analysis mainly focuses on the impact of price sensitivity, supplier’s production cost,
characteristics of the random demand error on the optimal retail and component prices, and
capacity planning and stocking policies. Additionally, we investigate how the magnitude of
price difference, combined with the consideration of the lead-time uncertainty, affects the

OEM’s optimal ending inventory decisions.

4.6.1 On the price sensitivity

Demand is a key factor that affects inventorylevels and capacity planning. Our
research considers price-sensitivity, stochastic demand. As such, in the first experiment, we
vary the magnitude of price sensitivity in the demand function to see how it affects the
pricing and inventory/capacity decisions. Figure 4.1 presents the resulton the retail and

component pricing decisions. We can observe that both the optimal retail and transaction

126

prices decrease more rapidly when price sensitivity is low. Nonetheless, both prices
continue to decrease as price sensitivity increases. When price sensitivity is high, the price
needs to be lowered, as price affects the demand to a greater extent than it does when price
sensitivity is low. Additionally, we incorporate two separate experiments in which
supplier’s production cost is set at difference values. The results confirm that both retail
and component prices need to be higher when production cost is higher.

Figure 4.2 shows how the price sensitivity and supplier’s production affect the optimal
capacity planning and stocking policies for the supplier, the buyer, and the OEM. All of the
three policies have a downward slope with respect to the price sensitivity, with each varying
in the same direction as the supplier’s production cost. Notice that in each level of the
production cost, the supplier has the highest capacity, followed by the OEM and the buyer in
this specific experimental setting. If we decrease the supplier’s stockoutcost, the optimal

capacity will decrease accordingly. On the other hand, one can observe that the set of lines

at high production cost (6,, =100) has a steeper slope. This suggests that higher production

cost accelerates the effects of higher price sensitivity on lowering the optimal capacity and
stocking policies in the supply chain. Notice that higher production cost results in higher
retail price and component price as shown in Figure 4.1.

The corresponding profits for each party in this supply chain are presented in Figure 4.3.
In our experiment, the supplier earns the highest profit, followed by the buyer. The OEM
only earns the production service fee, thus has more stable profits across the selected range of
price sensitivity. As the price sensitivity increases, the supplier’s and buyer’s profits go

down, and the difference of all the profits decreases as the same time.

127

14oole-; e if a .22..--“ 7-- e- w e. _ e. 7r-
1

——~— ——— l

 

 

 

 

 

'T fiétaifegso l
l l _____ Retail: c1=100 El
1200 -
i I ,,,,,,,,,, Component: c1=50 ~ 1
1 l _ ,,,,,, Component: c1=100 ,
1000?. '1 ————- —= ~— —— J
G)
.2
ct
.L._ _L _,,. ' JA__- 1.
1.5 2 2.5 3 3.5 4 4.5 5
Price sensitivity
Figure 4.1 Optimal pricing: varying price sensitivity
:2
'E
D

 

   

Supplier: «$1150 S1 ‘ .~‘-.

1 "°

,"1 _____ Supplien c1=100 '~-..

 

550

ll .......... Buyer. c1=50 l x,“
5001" ' """ Buyer, c1=100 i "' -._.. ..1
I; aunt‘s... OEM: 01:50 i 1]
,1 _.'_ OEM. C1=1OO !

4501:— " , “___ 'T_“_'_.“_-_—' ‘ __ , _ z 1 7-,

 

 

 

1.5 2 2.5 3 3.5 4 4.5 5
Price sensitivity

Figure 4.2 Optimal capacity and stocking policies: varying price sensitivity

128

 

 

3.5L \\ 1 _____ Supplier: c1=100 .,
\ .

l \\ i ,,,,,,,,,, Buyer: 01:50 i ,1

3i \ i _ ...... Buyer: c1=100 1,

l ‘\ . — l

2.5 \ l ......... OEM. c1-50 ,,

I;

l , 1 _,,_OEM: c1=100
, ,
l

Profit

 

 

 

 

Price sensitivity
Figure 4.3 Optimal Profits: varying price sensitivity
4.6.2 On the characteristics of the random demand error
In this experiment, we examine the effects of the stOchastic element of the demand
function on the optimal pricing and stocking/capacity policies. We fix the price sensitivity
b at 2 , and the supplier’s production cost C, at 100. The solution format of the optimal
capacity and stocking policies lead us to expect a positive linear relationship. Interestingly,

we found the buyer’s stocking decision does not follow that pattern. Afurther investigation

reveals the root cause is when h=10 and S 3 =5, the 2,. value in I ,S = y( p, ) + ,u,_ + 2:0",E

is actually negative. So when y( p, ) + ,u£ is not large, 1,. may decrease as 0',

increases, which was somewhat unexpected before conducting this experiment. Overall, the
results resemble that in Figure 4.2 in that the supplier’s capacity is largest when the buyer’s

stocking level is the lowest.

129

Figure 4.5 indicates that both the retail and component pricing decisions are not affected
by the size of the standard deviation of the demand error, so is not the profit level
Although the random demand error is included in all the profit functions, it gets eliminated
during the normalization process for the buyer and the OEM, thus disappearing from the final
expressions of estimated profits. However, it is not the case for the supplier, due to the lack

of the expected stockout cost in the supplier’s profit function. So one can observe that in

Figure 4.5, the buyer’s and the OEM’s profits do not vary with 0', while the supplier’s

profit functions decrease as 0', increases.

 

 

 

 

 

 

 

 

 

 

1800. ., , , , , , . f 1 fl
1 S" Supplier: 11%—CT ”xi,
1600,: ----- Supplier. u=30 ‘ /,.v ,
“ I .......... Buyer: u=10 i ”’I’ .
1400 , i - ------ Buyer: u=30 l x’x’ .+-‘ .++
' ......... OEM; u=10 ’I” “FIJI.- l
| .....+.. OEM: “=30 : I,“ ,p-‘d'...
1200 i' i‘" ———-—-——~—-— ”” *...+p" 2
’a” ”JP". ‘
£9 , _,.+
E 1000.
3 |
800,
600 +—
400, ........................................................ l
200 .. 1 .L- .l. ___- _1 1 r

 

 

 

 

1o 12 14 1'6 18 20 ‘22 24 26 23 30
Std. of random error

Figure 4.4 Optimal capacity/stocking policies: varying 0', at two levels of ,us

130

Profit

Pnce

1.41
1.2

1..

0.8,

0.6.
i
0.4,
0.2;
i

10

 

 

 

 

 

 

i

l .......... Buyer: u=10
", - ...... Buyer: u=30

sir—polic-IrTJfio—l
------ Supplier: u=30 ‘ ,
I
1
i

 

 

Std. of random error

.'.:32::.'.:::=:::323............................fi...............................§‘.2fi:3:.‘2:.‘::.‘ll

i

i

i
0...?".+..+...§...+m.....L........§...+...Ql....+...f.......+l...§...+.......+
—'"‘12”“’ 14" "'1‘6 18 20 22 24““26 “"728“,” 30

Figure 4.5 Optimal profits: varying 0', at two levels of #5

500
10

 

. ----.. Component I

12 14 16 1‘8 20 22 2'4 26
Std. of random error

Figure 4.6 Optimal pricing: varying 0",.

131

fieiail '

28

‘7
I
i
l

30

On the other hand, [1, does affect both pricing decisions as it is included in fi(p,)

that appears in both optimal pricing schemes. Figure 4.6 presents the relationship between

the optimal pricing decisions and the mean of the random demand error (#5 ). One can

observe that the effect of ,u, on the optimal prices is relatively minor compared with that

from price sensitivity, as indicated in Figure 4.1.

4.6.3 On the OEM’s ending inventory during price breaks

In this experiment, we examine the effects of the price discount on the ending inventory
decisions prior to a price break. We introduce the lead-time uncertainty into the experiment
and set the mean of the arrival time distribution E (L) =0.5 and the variance V ar(L) =O.5.
If the “unit time” demand function represents daily demand function, arrival time of the
component from the supplier at the OEM site is in average 0.5 days after the price break,
with a standard deviation of 0.25. All the otherparameters follow the same value utilized in
the previous experiments.

Figure 4.7 shows the effect of the price difference on the ending inventory decisions.
As one can see, the ending inventory decreases as theprice difference increases. This is
because the understocking cost decreases and the holding cost increases tremendously when
price difference is considered as a portion of “benefit” for not holding inventory. From the

solution of the optimal ending inventory shown in corollary 4.5, we can see clearly that when

the price difference approaches towards the original understocking cost (Sm =15), the 2;
value in corollary 4.5 will be approaching negative infinity so that the optimal ending

a . O O I ‘ O I
inventory x becomes negative, in whlch case we set x =0. So in our experiments, one

can observe that as the price difference nears 15, the ending inventory solution moves

132

towards zero. In this case, the price difference is so significant that it balancesoff the
impact of the original understocking cost. Thus,the OEM should not keep any inventory
right before the price break.

In Figure 4.7 we also compare the ending inventory decisions between the two cases
when we consider stochastic lead-time demand (LTD) and when we consider only stochastic
demand. We can observe that when lead-time uncertainty is also considered, the inventory

level will be much higher than in the regular demand cases. This is due to the fact that

incorporating the lead-time uncertainty results in a very large LTD variance 0,2 , which
directly leads to a high optimal ending inventory value x. . However, when we only
consider stochastic demand, we can clearly see that price markdown resuls in a
lower-than-regular inventory level. As the price difference reaches 15, the OEM should

decide not to keep any inventory.

 

 

 

i— 1_—— Enotng‘ihventory (LTD) 1|
..... Ending inventory (D) H
, .......... Regular inventory , ,'

 
 
  
   

 

Units

 

 

 

0 g___.__¥" .r_,__ __ 1 l . _,_ i
12 12.5 13 13.5 14 14.5 15
Price difference

Figure 4.7 Optimal stocking policies LT D vs. D: varying price difference

133

Finally, Figure 4.8 presents the expected profits at price break. Recall that the ending
inventory decision is to cope with the price difference as well as the arrival uncertainly of the
very first shipment of new components after price break becomes effective. Our results
suggest that, if without considering the lead time uncertainty, lowering inventory as indicated
in Figure 4.7 will be able to achieve higher profit for the OEM. The difference of the
expected profits increases as the price discount increases. However, the potential profit is
relatively insignificant compared with the expected profit for OEM if lead-time uncertainty is
known to exist. When the price difference is small, the saving per unit is not worth therisk
and cost of running out of inventory due to the variation of new material’s arrival time. As
such, experiment results convey one signal to the OEM: if delivery uncertainty is a concern
and the price break does not result in a significant amount of price difference, the OEM will
be better-off simply having higher ending inventory right before the price break. But if the
material’s arrival time is very stable and predictable, the OEM may want to consider to

taking the lower inventory approach.

134

 

 

Prof It
or
CD
0‘)

*0 ._...

   
 

 

 

F— Ending inventory (LTD) 4
‘ ..... Ending inventory (D) i
.......... Regular inventory J .

 

 

6780i -_ _, l.,—__ -1 --L__l____ l .1 -_
12 12.5 13 13.5 14 14.5 15

Price difference

 

Figure 4.8 OEM profits LTD v.s. D: varying price difference

4.6.4 Discussion

The treatment of incorporating both lead-time and demand uncertainties seems to create a
very large gap up from not considering the lead-time uncertainty. Combing both
uncertainties make the inventory decision very sensitive to the price difference. As one can
observe from Figure 4.7, the curve of the LTD ending inventory has a much steeper slope
than the other ending inventory curve that considers only demand uncertainty. Intuitively,
one would try to reduce the inventory level when there is a price break to come. We expect
that lead time variation will recommend the OEM to increase stocking level. However, the
experiment result indicates that the resulting optimal solutions could be actually several times
higher if the price difference is small. Of course, factors such as the choice of demand and

lead-time distributions matter. However, our results cast doubts onwhether the treatment of

135

lead-time demand is truly capable of describing the combined effect of both uncertainties.
As such, we recommend that firms should use the lead-time demand to make inventory
decisions with discretion. A comparison between using lead-time demand and using only
the stochastic demand is a must.

Note that the computational experiments are based on the selected demand functions as
well as on random error distribution. Despite the advantage of easy implementation and
analyses, the linear demand function may not be ideal for capturing the true demand pattern.
An example is that the optimal retail pricing and the component pricing which seem to be set
much higher than the corresponding production cost and OEM’s service charge in our
experiments. Additionally, the selection of the parameters of lead-time uncertainty also
results in higher-than-expected ending inventory decisions as mentioned. The demand
firnction and the lead time distribution are key to acquiring these optimal decisions. The
purpose of these experiments is to demonstrate the effectiveness of the model’s capabilities
in facilitating managers’ decisions. We nonetheless note the importance of being able to
identify the demand and lead-time random patterns so that one can take full advantage of the

development of this research.

4.7 Concluding Remarks and Discussion

We investigate a problem involving price discount and corresponding inventory
decisions in a decentralized supply chain that consists of a supplier, an OEM and a buyer
(retailer). The supplier offers a new, lower component price to the buyer, due to innovation
of manufacturing technologies that allow lower production cost. Such a price markdown
will lead buyer to reduce his retail price of the final product. On the other hand, demand is
price-sensitive; the reduced price will induce higher demand, thus affecting all agents’

capacity planning and stocking policies in this supply chain. We develop a supply chain

136

model that can identify the optimal pricing scheme for both the component and the retail
prices, as well as the optimal capacity reservation and stocking level for the supplier, buyer
and the OEM.

Our models allow the entire supply chain to make decisions on capacity planning,
component pricing, stocking policies, and the retail pricing. We show how these decisions
are affected by the demand patterns such as price sensitivity andparameters of random
demand error. We also found that increasing the standard deviation of the random demand
error does not necessarily lead to increase of stocking level as one would predict.
Furthermore, we develop models and solution algorithms specifically to facility theOEM’s
ending inventory decision making right before the price break. We found that when the
new price is not much lower than the old price and if the material’s arrival time is known to
be varying, the OEM should keep higher inventory than usual to cope with the lead-time
uncertainty. But when the price difference is large and more importantly, the material’s
arrival is highly predictable, the OEM will benefit from lowering the inventory, thus enjoy
the benefit of profit gain by cost saving.

The main contribution is that we develop a full supply chain model in the presence of
price discount. Past supply chain contract research primarily focused on a two-echelon
supply chain. Our research incorporates the OEM’s problem and develops a three-echelon
supply chain model. Furthermore, we specifically develop models to describe the effects of
the price discount on the inventory decisions beyond the price break point for the OEM.

We incorporate the demand uncertainty and consider the mismatch of schedules for the OEM
to more accurately predict such impact. We demonstrate that lead-time uncertainty can
substantially affect the ending inventory decisions via numerical examples.

An immediate extension is to incorporate quantity-sensitive discount scheme into this

model. In this study, we only consider the price markdown scheme and the price-sensitivity

137

demand. Inclusion of quantity-dependent discount contracts can possibly bring more
fruitful insights to supply chain managers and facilitate their decision making. Furthermore,
there is need to consider also the “time” factor as part of the demand pattern. Demand will
be simulated by lower price. However, after a period of time, demand may decrease even
when price does not change. A good example is from the PC market or cell phone industry.
When a product is launched at a lower price, the demand is normally strong. Once
end-users expect newer, better product to be available in the future, demand forthe same
product starts declining. Thus, a demand fimction that can capture both the priceand time
elements will be most suitable for the problems studied in our models Furthermore, the
decisions in capacity planning and inventory stocking policies, though all linked to the same
demand function, are decided individually. When gaps in these decisions exist, it’s easy to
see that the agent that has the lowest capacity or inventory level will become a bottleneck for
other agents should demand be higher than its capacity or stocking level. As such, an
“integrated” decision making in capacity planning and stocking policies should be desirable.
Finally, we omit to consider the fact that the discount timing may not be aligned across the
entire supply chain. For example, the supplier offers a price markdown, however, the buyer
does not take any action until two weeks later. The model will be more complete if the
synchronization of the discount schedules at each echelon of the supply chain is considered.

Future research needs to investigate these issues.

Appendix of Chapter 4
Algorithms
( 1) ND>NW

Step 1. Let i=1, &

138

sm—A
h +3

"I m

 

x‘ = ,u. + 2;ch = up. )E(L) + (bk Mp. )2 Var(L) + 0.3m».

Calculate the mean and variance of lead-time demand whenL=24-ND:

#01 = 1(Pl)E(L) 2 1(Pl)(24 — ND)

0,2,, = of = 2( p, )2 Var(L) + o§E(L) = o§(24— ND)

ChCCk ifx S #0, + 24001?

(a) If true, x. is the optimal solution. Go to Step 3.
(b) Otherwise, proceed to Step 2, because it contradicts the
assumption that shipment falls in day 1.

Step 2. Let i=i+l , then

x‘ = #0 + 2.10-D = 1(pr)[E(L)+(i-1)Lol

 

+<I>" (lim—If-h/MPJZVCIKL)+03[E(L)+(i-1)2Lol

m M

Calculate the mean and variance of lead-time demand whenL = i(24-ND):

#0.- = #r +(i-1),u1,o = l(p.)[E (L) + (i - 1)L0]
= 2( p, )[i(24 — ND) + (i —1)L,]
of, = of + (i— 1)2 of, = 2( p, )2 Var(L) + 0312(2) + (i — 1)2L,oj
= o§[i(24 — ND) + (i — 1)2 L0]

Check if x S #0,- +2400, ?

(a) If true, x. is the optimal solution. Go to Step 3.
(b) Otherwise, goto Step 2.

Step 3. Stop.

139

(2) ND<NW

Step 1. Let i=1 &

1: m ' A meal Var(L) + 0315(1)]
+ S

m m

XL = #D + 2:00 = 1(P1)E(L) +(D—l(

 

Calculate the mean and variance of lead-time demand whenL' =24-NW:

#01 = 1(pr)E(L‘) = 1(1).)(24 - NW)
0,2,, = 2( p, )2 Var(L') + (film) = 03(24 — NW)

Check ifx, 3 pm + 2,100, ?
(a) If true, x. = x, is the optimal solution. Go to Step 3.
(b) Otherwise, proceed to Step 2, because it contradicts the
assumption that shipment falls in day 1.
Step 2. Let i=i+1, 0

Calculate the mean and variance of lead-time demand when L' = 241' — NW

#01 : 1(pl)E(L') : 4(1),)(241' " NW)
012).- = 1(1).)2 Var(L') + a§E(L') = of (241' - NW)
Check if x, 3 #0,. +40”?

(a) If true, x. = x, -(i-1),uZO is the optimal solution. Go to Step 3.

(b) (lherwise, go to Step 2.

Step 3. Stop.

140

CHAPTER 5 CONCLUDING REMARKS

This dissertation studied the quantity flexibility contract and price markdown scheme in
three types of supply chains. Chapter 2 designed a new quantity flexibility contract with
price discount incentives, from the supplier’s perspective, in a one-buyer—one-supplier supply
chain. Chapter 3 investigated the competition between the quantity flexibility contract and
the price discount scheme from the buyer’s perspective in a two-compding-supplier,
one-buyer supply chain. Chapter 4 extended the price markdown scheme to a threeechelon
supply chain and analyzed pricing decisions and inventory policies for each agent in the
supply chain. We summarize the contributions of each of thesechapters and revisit

directions for future research.

5.1 Summary of Contributions
5.1.1 Chapter 2
In this research we found that QF i contract combined with the QF contract can create the
most benefit for this 1-1 decentralized supply chain. We showed that QFi creates Pareto
improvement from the OP contract under certain conditions. When executed, the QR
contract enables both agents to enhance profits from where they are at under the OP contract.
Our results indicated that QFi is most appropriate when the discount is only slightly off
from the OP contractual price. As such, we suggest that the supplier should not “overdo”
the discount. The QF i contract enables agents in a supply chain to estimate the “worth” of
flexibility offered. We presented conditions where the QFicontract outperforms the

traditional QF contract and where the QF contract has an advantage.

141

We showed that the QFi contract is able to achieve supply chain coordination. The
coordination pricing schemes do not require the knowledge of demand distribution as the QF
contract does. The coordination can be achieved, without loss of generality, at the very
component price that is agreed upon by the two parties under the OP contract. As such,
achieving 100% supply chain efficiency needs not to radically change the QF contract price.
The QF i contract can work equally well as the QF contract from the system point of view.

It provides an option for the supply chain to redistribution the inventory risk burden.

5.1.2 Chapter 3

This research considered competing suppliers in the presence of the quantity flexibility
contract and the price discount scheme. We specified the optimal decisions for the buyer in
allocating the forecast and purchase orders to the two suppliers. The development of this
research contributed to the literature by adding knowledge to supply chain contract problems
involved with two competing sources of supply. I

Our analysis suggested that the 1-1 QF supply chain is favored over the 2-1 QF supply
chain from the system’s perspective. We found that in general, the system profit in 1-1 QF
supply chain always outperforms a 2-1 QF supply chain. Although having an alternative
source of supply is always beneficial to the buyer, our finding suggested that supply chain
deficiency occurs in the presence of multiple suppliers

The analysis identified areas where flexibility will be favored and areas where the
discount price is winning the competition. Our results provided insights to suppliers in that it
enabled them to better initiate business strategies and production plans to cope with
competition. Interestingly, we found that the buyer should simply use either supplier in
most cases. Areas where the buyer utilizing both suppliers simultaneously creating higher

profits are somewhat limited when compared to simply using a single supplier.

142

5.1.3 Chapter 4

We developed a complete supply chain model in the presence of price discount in this
research. We developed an approach to specify the optimal pricing decisions, capability
planning and inventory stocking policies affected by the new, lower pricing. Specifically,
we analyzed the effects of the price discount and the delivery uncertainties on the inventory
decision beyond the price break point for the OEM. We found that lead-time uncertainty
can change the inventory decisions tremendously if it is considered.

We found that the effects of higher price sensitivity on lowering the optimal capacity and
stocking policies will be fueled by higher production cost. Our results also suggest that the
impact of random error on inventory decisions is not as significant as one would expect.
Numerical experiments lead us to conclude that unless the price difference is large, the
benefit of lowering ending inventory to cope with price discount effects is piecemeal.

The main contribution is that we develop a full supply chain model in the presence of
price discount. Past supply chain contract research primarily focuses a two-echelon supply

chain

5.2 Future Research Directions

The three essays in this dissertation all raise interesting questions that provides future
research directions. First, in essay 1, the QR contract allows the buyer to choose freely the
desirable combination of forecast under the OP mechanism and firm order at the discount
price. However, if the price difference is large enough, the buyer may only execute the
discount scheme of the QFi contract. An extension that is worth pursing may involve
imposing a minimum required quantity on the OP mechanism before the buyer can consider

firm order at the discount price. Another potential topic is to incorporatea price-sensitive

143

demand function and examine the QFi contract performance when demand varies with price.
Finally, it will be interesting to examine how well the QFi contract performs in more
complex supply chains or in a multi-period setting.

Second, essay 2 looks at the competition between flexibility and price. Other factors,
such as quality performance, logistics capability, cost sharing, promotional efforts, or service
agreements to name a few, can also be incorporated into a contractual setting. Future
research can borrow our developments in essay 2 and study the tradeoffs between these
competing forces. We considered two competing suppliers in essay 2, one can expand the
problem to consider more buyers and/or more competing suppliers, thus allowing the
analysis and results to more closely resemble the real-world setting and provide fruitful
managerial insights. Moreover, further investigation of the competition between flexibility
and price discount in the long run is also warranted.

Finally, in essay 3, we considered a price markdown scheme. An immediate extension
is to incorporate a more complex quantity discount schane into the model and examine how
a different price discount scheme can affect the supply chain decisions. On the other hand,
in our model the supplier takes the leading role. It is possible that the buyer is the one who
retains all the power. Future research can look at this problem and analyze how firms will
behave differently when the buyer takes charge of the pricing negotiation. Furthermore, an
integrated capacity planning and stocking policy is more desirable from the system point of
view. This can provide a benchmark for evaluating the performance of the decentralized
supply chain. Finally, when the discount timing of the supplier and buyer is not
synchronized, price discount may affect inventory stocking policies in a different manner.

Further investigation that considers the desynchronizing discount schedules along a supply

chain is worth pursuing.

144

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