Decision Models for a Two-stage Supply Chain Planning under Uncertainty
with Time-Sensitive Shortages and Real Option Approach
by
Hwansik Lee
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
May 14, 2010
Keywords: shortages, time-dependent partial back-logging, optimizing lead-time, real
option, analytical inventory model, coordination
Copyright 2010 by Hwansik Lee
Approved by:
Chan S. Park, Co-Chair, Ginn Distinguished Professor of Industrial & Systems
Engineering
Emmett J. Lodree, Jr., Co-Chair, Assistant Professor of Industrial & Systems Engineering
Ming Liao, Professor of Mathematics & Statistics
Abstract
The primary objective of this research is to develop analytical models for typical supply
chain situations to help inventory decision-makers. We also derive closed form solutions
for each model and reveal several managerial insights from our models through numerical
examples. Additionally, this research gives decision-makers insights on how to implement
demand uncertainty and shortage into a mathematical model in a two-stage supply chain
and shows them what differences these proposed analytical models make as opposed to the
traditional models.
First, we model customer impatience in an inventory problem with stochastic demand
and time-sensitive shortages. This research explores various backorder rate functions in a
single period stochastic inventory problem in an effort to characterize a diversity of customer
responses to shortages. We use concepts from utility theory to formally classify customers
in terms of their willingness to wait for the supplier to replenish shortages. Additionally,
we introduce the notion of expected value of risk profile information (EVRPI), and then
conduct additional sensitivity analyses to determine the most and least opportune conditions
for distinguishing between customer risk-behaviors.
Second, we optimize backorder lead-time (response time) in a two-stage system with
time-dependent partial backlogging and stochastic demand. In this research, backorder cost
is characterized as a function of backorder response time. We also regard backorder rate as
a decreasing function of response time. We develop a representative expected cost function
and closed form optimal solutions for several demand distributions.
Third, we adopt an option approach to improve inventory decisions in a supply chain.
First of all, we apply a real option-pricing framework (e.g., straddle) for determining order
ii
quantity under partial backlogging and uncertain demand situation. We establish an op-
timal condition for the required order quantity when a firm has an desirable fill rate. We
develop a closed form solution for optimal order quantity to minimize the expected total
cost.
Finally, we implement an option contract to hedge the risk of the demand uncertainty.
We show that the option contract leads to an improvement in the overall supply chain prof-
its and product availability in the two-stage supply chain system. This research considers
a standard newsvendor problem with price dependent stochastic demand in a single man-
ufacturer and retailer channel. We derive closed form solutions for the appropriate option
prices set by the manufacturer as an incentive for the retailer to establish optimal pricing
and order quantity decisions for coordinating the channel.
iii
Acknowledgments
I could never have completed this work without supports and assistances from many
people. First and foremost, I would like to thank my wife (Yunhee) and children (James
& Jeana) for their unconditional love and support over these past several years, and for
putting up with me through all of it. Special thanks to Chan S. Park, my kind advisor
of my research and my life as well, who has given me constant advice, collaboration, and
most of all good relationship for many years now. Without his advice and support, I would
not be where I am today. Dr. Lodree has played a unique and important role, providing
countless opportunities for me to work on interesting and exciting research. His direction
has helped me to grow as a researcher, and for that I am very thankful. Dr. Liao has been
the best professor and advisor a student could hope for. He gave me a strong background
of mathematics including stochastic process. Without his helps, I could not complete my
research. I would also like to thank other our faculty members and staffs for their helps
and supports through the years. Their direction of my early efforts gave me the tools I
needed to complete this work. I appreciate INSY graduate students? supports in my Ph.D.
course works through all times. Also, special thanks to our AUKMAO members. Finally, I
appreciate their helps and supports to The Republic of Korea Army. The time in Auburn
was one of rememberable experiences in my whole life.
Great things are done by a series of small things brought together.
- Vincent Van Gogh
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Modeling Customer Impatience in an Inventory Problem with Stochastic De-
mand and Time-Sensitive Shortages . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 General Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 A Classification of Customer Impatience . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Risk-Neutral Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Risk-Seeking Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 Risk-Averse Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Sensitivity Analysis and Management Insights . . . . . . . . . . . . . . . . . 23
2.5 Value of Risk Profile Information . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Optimizing Backorder Lead-Time and Order Quantity in a Supply Chain with
Partial Backlogging and Stochastic Demand . . . . . . . . . . . . . . . . . . . . 36
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 The Newsvendor Problem with Time-dependent Partial Backlogging 38
v
3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 The Model with Backorder Setup Cost . . . . . . . . . . . . . . . . 46
3.3.2 Managerial Insights regarding the Response Time . . . . . . . . . . . 47
3.4 An Illusrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Insights from Table 3.4 and 3.5 . . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Sensitivity Analysis and Management Insights . . . . . . . . . . . . 49
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 The Effects of an Option Approach to Stochastic Inventory Decisions . . . . . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 The Traditional Newsvendor Approach with Partial Backlogging . . 57
4.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Model Formulation with an Option Pricing Framework . . . . . . . . 61
4.4 Inventory Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 Order Quantity with a Desirable Fill rate . . . . . . . . . . . . . . . 65
4.4.2 Optimal Fill rate and Order Quantity . . . . . . . . . . . . . . . . . 66
4.5 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Coordinating a Two-Stage Supply Chain Based On Option Contract . . . . . 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Channel Structure with an Option Contract . . . . . . . . . . . . . . . . . 78
5.4 Decisions for a Supply Chain Coordination . . . . . . . . . . . . . . . . . . 81
5.4.1 The Manufacturer?s Optimal Option Pricing Decisions . . . . . . . . 81
vi
5.4.2 The Retailer?s Optimal Order Quantity and Discount Retail price . 82
5.5 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A Modeling Customer Impatience in an Inventory Problem with Stochastic De-
mand and Time-Sensitive Shortages . . . . . . . . . . . . . . . . . . . . . . . . 100
A.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.2 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.3 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B Optimizing Backorder Lead-Time and Order Quantity in a Supply Chain with
Partial backlogging and Stochastic Demand . . . . . . . . . . . . . . . . . . . . 103
B.1 Derivations of Optimality for the Proposed Model . . . . . . . . . . . . . . 103
B.1.1 Optimal Order Quantity . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.1.2 Sufficient Condition for Convexity and Optimal Response Time . . . 103
B.1.3 Relationship between Optimal Backorder cost and Lost sales cost . . 104
B.2 Derivations of Optimality for the Proposed Model with Backorder Setup Cost105
B.2.1 Optimal Order Quantity . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.2.2 Sufficient Condition for Convexity and Optimal Response Time . . . 105
B.2.3 Relationship between Optimal Backorder cost and Lost sales cost . . 106
C The Effects of an Option Approach to Stochastic Inventory Decisions . . . . . . 107
C.1 Derivations of Optimality for the Partial Back logging Newsvendor Approach 107
C.2 Derivation of Total Cost Function with Real Option Framework . . . . . . . 107
vii
C.3 Derivation of Optimal Fill rate and Order Quantity . . . . . . . . . . . . . . 109
D Coordinating a Two-stage Supply Chain Based on Option Contract . . . . . . 111
D.1 Derivations of Profit Function for CS . . . . . . . . . . . . . . . . . . . . . . 111
D.2 Derivation of Profit Functions for DS . . . . . . . . . . . . . . . . . . . . . . 112
D.3 Derivation of Optimal Option Prices . . . . . . . . . . . . . . . . . . . . . . 112
D.4 Derivation of Optimal Order Quantity with Discounting . . . . . . . . . . . 113
D.4.1 Fixed Coefficient of Variance Case (FCVC) . . . . . . . . . . . . . . 113
D.4.2 Increasing Coefficient of Variance Case (ICVC) . . . . . . . . . . . . 115
viii
List of Figures
2.1 Customer impatience and lost sale thresholds. . . . . . . . . . . . . . . . . . 12
2.2 Customer impatience and backorder rates. . . . . . . . . . . . . . . . . . . . 13
2.3 The effect of cLS on the optimal order quantity. . . . . . . . . . . . . . . . . 24
2.4 The effect of cB on the optimal order quantity. . . . . . . . . . . . . . . . . 24
2.5 The effect of M on the optimal order quantity. . . . . . . . . . . . . . . . . 25
2.6 The effect of cH on the optimal order quantity. . . . . . . . . . . . . . . . . 25
2.7 The effect of cLS on EVRPI if customer is risk-averse. . . . . . . . . . . . . 29
2.8 The effect of cLS on EVRPI if customer is risk-neutral. . . . . . . . . . . . . 29
2.9 The effect of cLS on EVRPI if customer is risk-seeking. . . . . . . . . . . . . 30
2.10 The effect of cB on EVRPI if customer is risk-averse. . . . . . . . . . . . . . 30
2.11 The effect of cB on EVRPI if customer is risk-neutral. . . . . . . . . . . . . 31
2.12 The effect of cB on EVRPI if customer is risk-seeking. . . . . . . . . . . . . 31
2.13 The effect of M on EVRPI if customer is risk-averse. . . . . . . . . . . . . . 32
2.14 The effect of M on EVRPI if customer is risk-neutral. . . . . . . . . . . . . 32
2.15 The effect of M on EVRPI if customer is risk-seeking. . . . . . . . . . . . . 33
3.1 The effect of a on the optimal response time. . . . . . . . . . . . . . . . . . 51
3.2 The effect of b on the optimal response time. . . . . . . . . . . . . . . . . . 51
3.3 The effect of parameters on the optimal response time. . . . . . . . . . . . 52
3.4 The effect of parameters on the optimal order quantity. . . . . . . . . . . . 52
ix
3.5 The effect of t on the optimal order quantity. . . . . . . . . . . . . . . . . . 53
4.1 General problem description. . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Analogy between financial option and inventory decision. . . . . . . . . . . 60
4.3 A Straddle quoted from John C.Hull (2002). . . . . . . . . . . . . . . . . . . 62
4.4 Order quantity with traditional approach. . . . . . . . . . . . . . . . . . . . 69
4.5 Order quantity with option approach. . . . . . . . . . . . . . . . . . . . . . 69
5.1 Transactions between parties. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Profits in the arbitrary option prices. . . . . . . . . . . . . . . . . . . . . . . 86
5.3 ? in the arbitrary option prices. . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4 Profits in the optimal option price. . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Profits in the discount retail prices in FVC. . . . . . . . . . . . . . . . . . . 87
x
List of Tables
2.1 A summary of the related literature. . . . . . . . . . . . . . . . . . . . . . . 8
2.2 List of notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 A summary of the related literature. . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Differences between Lodree (2007) & Proposed Model. . . . . . . . . . . . . 41
3.3 List of notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Comparison of Optimal Order quantities. . . . . . . . . . . . . . . . . . . . 49
3.5 Comparison of Total costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 List of notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Comparison Optimal order quantities and Total costs. . . . . . . . . . . . . 70
5.1 A summary of the related literature. . . . . . . . . . . . . . . . . . . . . . . 75
5.2 List of notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xi
Chapter 1
Introduction
1.1 Problem Statement
Carrying inventories becomes inevitable in most businesses because the production and
consumption take place at different times in different locations and at different rates. In
fact, inventory is one of the single largest investments made by most businesses. Given
today?s global financial crisis, the supply chain inventory management could be more cru-
cial than before. The roughly 400 companies in the S&P Industrials have close to a36 500
billion invested in inventory. Inventory capital costs absorb a significant percentage of op-
erating profits for a company. For automotive and consumer products, these costs absorb
approximately 40% of operating profits. Therefore, inventory optimization is one of the
significant topics in supply chain circles today considering the dynamic state of markets
all over the world. As a growing number of organizations have proved, astute planning
and management can wring 20-30 % out of current inventories, saving ?several millions? in
direct costs and achieving huge gains in operational performance, all the while maintaining
or even improving product availability and customer service.
Indigent supply chain inventory management could spell disaster for any company. The
higher the inventory investment as a percentage of total assets of a company, the higher the
damage caused by poor inventory management. Consider the following situation recently
faced by the Japanese game company Nintendo, Inc. (Schlaffer, 2007):
Wii Shortages Could Cost Nintendo Billions In Sales:
The Nintendo Wii was one of the most popular consoles in 2006 and it is the most
1
popular in 2007. Only there is a problem, the company is still having problems meeting
demand. If you were hoping to provide your children, family or other loved one/friend
with a Wii this Christmas, I say that it is far too late to do so. No store will have
it in stock though you may be able to purchase one that has been jacked up in price
on eBay, chances of receiving it in time are slim. It seems a parts shortage has struck
Nintendo and it is going to hurt the company financially. Despite the shortage, the
company says it is doing everything it can to meet demand. But that?s not enough; it
may end up losing just slightly over a billion dollars if it cannot fix these problems.
To optimize the deployment of inventory, you need to manage the uncertainties, con-
straints, and complexities across a multi-stage supply chain on a continuous basis. There-
fore, many companies adopt inventory control systems, enabling them to handle many
variables and continuously update in order to optimize their multi-stage supply chain sys-
tems.
However, in many cases inventory decision-makers need analytical models to grasp
the big picture of supply chain inventory problems before making executive decisions or
implementing inventory control systems. In fact, a reliable analytical model is important
for practitioners (e.g., decision-maker) to make proper predictions of their field of interest.
Especially, this research develops appropriate models in order to abstract the features of a
supply chain system as a set of parameters or parameterized functions. These analytical
models are simple but provide effectively an overall view of the supply chain system.
In general, there are five basic blocks for inventory management activities: (1) De-
mand forecasting or demand management, (2) Sales and operations planning, (3) Produc-
tion planning, (4) Material requirements planning, (5) Inventory reduction and shortage
management.
2
Every activity is critical, but as shown in the Nintendo shortage case, demand fore-
casting and shortage management play a key role in supply chain inventory management.
Therefore, we focus our attention on the activities of demand forecasting and shortage man-
agement. In particular, this research focuses on how to implement the situation of demand
uncertainty and shortage into a mathematical model properly based on a two-stage supply
chain.
1.2 Research Objectives
The first objective of this study is to implement time-sensitive shortages into a con-
ventional analytical inventory model. We consider the fact that shortages are partially
backlogged; a fraction of shortages incur lost sales penalties while the remaining short-
ages are backlogged. Therefore, we examine the implications of incorporating the notion of
time-dependent partial backlogging in the single period stochastic inventory problem. More-
over, we explore linear and nonlinear decreasing backorder rate functions with respect to
lead-time in an inventory problem with time-dependent partial backlogging, demand uncer-
tainty, and emergency replenishment in an effort to characterize diverse customer responses
to shortages.
The second objective of this study is to consider demand uncertainty in an inventory
decision model. The conventional approach, such as the newsvendor problem, of inventory-
stocking decisions relies on a specific distribution of demand for the inventory item to
implement demand uncertainty. In dealing with market uncertainty, option-pricing models
have become a powerful tool in corporate finance. The literature that applies option-pricing
models to capital budgeting - often referred to as real options - is extensive. We explore an
option-pricing model to consider demand uncertainty in a supply chain.
3
The third objective of this research is to establish a coordination model of a two-stage
supply chain with a real option framework (e.g.,option contract). Actions taken by the two
parties in the supply chain often result in profits that are lower than what could be achieved
if the supply chain were to coordinate its actions with a common objective of maximizing
supply chain profits. This research develops an option contract in a newsvendor problem
with price dependent stochastic demand and shows that the option contract could lead to
coordination in the supply chain through improving the product availability and the overall
supply chain profits.
1.3 Research Plan
Chapter 2 develops a decision model considering customer impatience with stochastic
demand and time-sensitive shortages. We use concepts from utility theory to formally
classify customers in terms of their willingness to wait for the supplier to replenish shortages.
We conduct sensitivity analyses to determine the most and least opportune conditions for
distinguishing between customer risk-behaviors.
Chapter 3 establishes an additional model to optimize backorder lead-time (response
time) in a two-stage system with time-dependent partial backlogging and stochastic demand.
We consider backorder cost as a function of response time. A representative expected cost
function is derived and the closed form optimal solution is determined for a general demand
distribution.
Chapter 4 develops an inventory decision model with an option framework in a supply
chain. We apply a real option-pricing framework (e.g., straddle) for determining order quan-
tity under partial backlogging and demand uncertainty. We compare the results between
the traditional approach and the option approach with a numerical example.
4
Chapter 5 implements an option contract to improve an overall supply chain profits and
product availability in a two-stage supply chain system. We derive closed form solutions for
the appropriate option prices to coordinate a supply chain system. We illustrate our result
with numerical examples to help decision making in a supply chain coordination.
Chapter 6 presents a brief conclusion along with some suggestions for future research.
5
Chapter 2
Modeling Customer Impatience in an Inventory Problem with Stochastic Demand and
Time-Sensitive Shortages
Abstract
Customers across all stages of the supply chain often respond negatively to inventory short-
ages. One approach to modeling customer responses to shortages in the inventory control
literature is time-dependent partial backlogging. Partial backlogging refers to the case in
which a customer will backorder shortages with some probability, or will otherwise solicit
the supplier?s competitors to fulfill outstanding shortages. If the backorder rate (i.e., the
probability that a customer elects to backorder shortages) is assumed to be dependent on
the supplier?s backorder replenishment lead-time, then shortages are said to be represented
as time-dependent partial backlogging. This research explores various backorder rate func-
tions in a single period stochastic inventory problem in an effort to characterize a diversity
of customer responses to shortages. We use concepts from utility theory to formally classify
customers in terms of their willingness to wait for the supplier to replenish shortages. Under
assumptions, we verify the existence of a unique optimal solution that corresponds to each
customer type. Sensitivity analysis is conducted in order to compare the optimal actions
associated with each customer type under a variety of conditions. Additionally, we intro-
duce the notion of expected value of risk profile information (EVRPI), and then conduct
additional sensitivity analyses to determine the most and least opportune conditions for
distinguishing between customer risk-behaviors.
6
2.1 Introduction
Inventory shortages are often an indicator of suboptimal supply chain performance
caused by a mismatch between supply and demand. In general, shortages are classified as
either backorders or lost sales. Immediate consequences of backlogged shortages include
increased administrative costs, the cost of delayed revenue, emergency transportation costs,
and diminished customer perception (i.e., the loss of customer goodwill), while lost sales are
characterized by the opportunity cost of lost revenue and diminished customer perception.
In the long run, inventory shortages can compromise an organization?s market share and
negatively affect long term profitability.
Conventional stochastic inventory models such as the single period problem (or the
newsvendor problem) and its many variants (e.g., Khouja 1999) often assume that short-
ages are either completely backlogged or that all sales are lost. Although this assumption
is sometimes plausible in practice, there are situations in which an alternative approach to
modeling shortages is appropriate. This research explores the implications of incorporating
the notion of time-dependent partial backlogging in the single period stochastic inventory
problem. When shortages are partially backlogged, a fraction of shortages incur lost sales
penalties while the remaining shortages are backlogged. Therefore, time-dependent par-
tial backlogging implies that the backorder rate (i.e., the fraction of shortages backlogged)
depends on the time associated with replenishing the outstanding backorder. In many prac-
tical situations, customers are likely to fulfill shortages from a supplier?s competitor who has
inventory on hand if the backorder lead time is extensive. On the other hand, the supplier
is more likely to retain the customer?s business and possibly avoid the long-term conse-
quences of shortages if the backorder lead-time is reasonably short. From this perspective,
it is evident that the time-dependent partial backlogging approach to modeling shortages
is particularly useful to firms who compete in time-sensitive markets and embrace service
and responsiveness as a competitive strategy (e.g., Stalk and Hout 1990).
7
Table 2.1: A summary of the related literature.
Literature Demand Emergency Pricing Stock # backorder
uncertain replenish decision deteriorate rate functions
Zhou et al.(2004) x x x x 1
Skouri et al.(2003) x x x o 1
Dye et al.(2006) x x x o 1
Dye(2007) x x o o 1
Abad(1996) x x o o 1
Lodree(2007) o o x x 1
My Model o o x x 3
Several variations of inventory models with time-dependent partial backlogging have
been discussed in the research literature as shown in Table 2.1 including (i) models with
time-varying demand (Zou et al. 2004); (ii) models with time-varying demand and stock de-
terioration (e.g., Skouri and Papachristos 2003; Dye et al. 2006); (iii) models with stock de-
terioration, ordering decisions, and pricing decisions (Dye 2007); and (iv) models with time-
varying demand, ordering decisions, pricing decisions, and stock deterioration (Abad 1996).
In general, the backorder rate is assumed to be a piecewise linear function of the backo-
rder lead-time, except for Papachristos and Skouri (2000) and San Jos?e et al. (2006) who
consider exponential backorder rate functions. Additionally, the majority of the litera-
ture involves continuous review inventory policies in which shortages are replenished at the
time of the next scheduled delivery. However in practice, suppliers may attempt to re-
plenish backlogged shortages before the next scheduled delivery by engaging an emergency
replenishment process that involves emergency procurement of component parts, emergency
production runs, overtime labor, and expedited delivery. The time-dependent backlogging
literature also addresses various demand processes including time-varying, price-dependent,
and stock dependent; but the majority ignore demand uncertainty. The latter two issues
(demand uncertainty at the time of the inventory decision and emergency replenishment
after demand realization) are addressed in Lodree (2007), where the time-dependent partial
backlogging approach is used to model shortages in the newsvendor problem.
8
This study explores linear and nonlinear backorder rate functions in an inventory prob-
lem with time-dependent partial backlogging, demand uncertainty, and emergency replen-
ishment in an effort to characterize a diversity of customer responses to shortages. Moreover,
we find it convenient to use concepts from utility theory to characterize customer responses
to shortage as either risk-averse, -neutral, or -seeking. We also compare the optimal inven-
tory levels associated with each customer type through sensitivity analysis and identify the
conditions in which there are minute or significant differences in the optimal levels. Finally,
we use this framework to determine the benefit of understanding the dominant market char-
acteristic in terms of being risk-averse, -neutral, or -seeking with respect to time sensitivity.
In particular, if a firm is uncertain about the characteristics of the market it serves, we
define the expected value of risk profile information (EVRPI) so that the firm can assess
the value of a study or survey whose results reveal the true dominant market characteristic.
2.2 General Mathematical Model
In this section, we present a mathematical model for the newsvendor problem with time-
dependent partial backlogging. To do so, let ? represent the backorder rate, which can be
interpreted as the fraction of shortages backlogged or the probability that a given customer
will choose to backlog shortages. Since ? is time-dependent, we have that ? : L mapsto? [0,1],
where L ? [0,?) is the backorder lead-time (refer to Table 2.2 for a list of notations used
repeatedly in this study). We assume that L is directly proportional to the magnitude of
an observed shortage. In other words, we assume that L is a linear function in x?Q for
x ? Q, where x is observed demand, Q is the inventory level before demand realization, and
max{x?Q,0} is the observed number of shortages. Without loss of generality, we assume
that L = x?Q so that the terms ?backorder lead-time? and ?magnitude of shortage? can
be used interchangeably for the purposes of this study. Therefore, the backorder rate is
expressed as the function ?(x?Q). Let M be the maximum allowable shortage in the sense
that the probability of backlogging is zero if x?Q ? M. Then assuming X is a continuous
9
Table 2.2: List of notations.
Q: Order quantity (the decision variable).
X: Supplier?s demand (the buyer?s order), a continuous random variable.
x: Actual value of demand.
f(x): Probability density function (pdf) of X.
F(x): Cumulative distribution function (cdf) of X.
cO: Unit ordering/production cost before demand realization.
cH: Unit cost of holding excess inventory at the of the season.
cB: Unit ordering/production cost after demand realization for backlogged shortages.
cLS: Unit cost of shortages that are lost sales.
?(x?Q): Fraction of shortages that are backlogged.
M: Lost sales threshold.
TC(Q): Total expected cost function.
Q?: Optimal quantity that minimizes TC(Q).
random variable that represents demand, the newsvendor problem with time-dependent
partial backlogging can be expressed as follows:
TC(Q) = Order Cost + Expected Holding Cost + Expected Backorder Cost
+ Expected Lost Sales Cost
= cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cB
integraldisplay Q+M
Q
(x?Q)?(x?Q)f(x)dx
+ cLS
integraldisplay ?
Q
(x?Q)[1??(x?Q)]f(x)dx (2.1)
The backorder rate function should satisfy the following properties.
Property 2.1 d?(x?Q)d(x?Q) < 0 ? x ? [Q,Q+M) and d?(x?Q)d(x?Q) = 0 ? x ? [Q+M,?).
Property 2.2 lim
(x?Q)?0+
?(x?Q) = 1.
Property 2.3 lim
(x?Q)??
?(x?Q) = 0.
10
Property 2.1 indicates that ?(x ? Q) is a continuous and decreasing function in the
number of observed shortages, which suggests that customers are more likely to wait for
backorder replenishment if the backorder lead-time is short, and are less likely to wait if the
lead-time is close to the lost sales threshold. The second part of Property 2.1 is a result of
the fact that the probability of backlogging remains zero if x?Q ? M. Properties 2.2 and
2.3 reiterate Property 2.1, but also ensure that ?(x?Q) ? [0,1].
We want to determine the inventory level Q? that minimizes TC(Q) given by Eq. (2.1),
provided an optimum exists. It turns out that the convexity of Eq. (2.1) cannot be guaran-
teed in general, but the following theorem identifies a sufficent (but not necessary) condition
for the existence of a unique minimizer. Before presenting the theorem, we introduce the
following assumption.
Assumption 2.1 cO < cB < cLS
The inequalities cO < cB and cO < cLS are assumed in order to avoid trivial cases and
are also reflective of practice. The inequality cB < cLS is also representative of practice,
although there are some situations where backorder costs exceed short term lost sale costs
such as lost revenue. Additionally, the latter inequality will enable us to prove the next
theorem as well as other results presented later.
Theorem 2.1 A sufficient condition for TC(Q) to be a convex function for all Q ? 0 is
(x?Q)??primeprime(x?Q) ? 2?prime(x?Q) (2.2)
where ?prime(x?Q) = d?(x?Q)dQ and ?primeprime(x?Q) = d?
prime(x?Q)
dQ
Proof: It is shown in Appendix A.1.
11
M
3
M
2
M
1
x?Q
?(x?Q)
1
Figure 2.1: Customer impatience and lost sale thresholds.
2.3 A Classification of Customer Impatience
An obvious candidate for measuring customer impatience is the lost sales threshold,
M. For example, if M1 < M2 (where M1 and M2 are lost sales thresholds for customer 1
and customer 2 respectively), then it seems reasonable to conclude that customer 1 is more
impatient than customer 2. However, Figure 2.1 tells a different story. In particular, one
can argue from Figure 2.1 that customer 3 is more impatient than customer 2 and that
customer 2 is more impatient than customer 1, even though M1 < M2 < M3. At the very
least, Figure 2.1 reveals that M alone does not distinctively measure customer impatience
and that the shape of ?(x?Q) should also be taken into account in distinguishing among
varying degrees of customer impatience.
Consider the case in which M1 = M2 = M3 = M as shown in Figure 2.2. According
to Figure 2.2, customer 3 is more impatient than customer 2, and customer 2 is more
impatient than customer 1. Figure 2.2 also suggests that customer 1 is patient at first, but
decreasingly patient because his rate of change in impatience is increasing. On the other
hand, customer 3 is impatient at first, but decreasingly impatient. The relative impatience
of these customers can be attributed to individual attitudes and personalities, but could
12
M
x?Q
?(x?Q)
1
Customer1
Customer2
Customer3
Figure 2.2: Customer impatience and backorder rates.
also be interpreted as one person?s attitude with respect to waiting for different classes of
products.
For instance, customer 3 could represent an individual?s willingness to wait for a prod-
uct, such as a dairy, party product or a computer mouse pad, in which a suitable substitute
can be easily identified. In this case, the customer will most likely purchase the suitable
substitute as opposed to waiting for backorder replenishment, even if the backorder lead-
time is relatively short. Customer 1 would represent this same customer?s willingness to
wait for a different product type, such as automobile parts, classic furniture, or specialized
computer software, in which a product of comparable utility cannot be as easily identified.
In this case, the customer is more likely to wait for backorder replenishment, especially if
the backorder lead-time is relatively short. These notions of impatience, increasing impa-
tience, and decreasing impatience for describing customer behavior with respect to waiting
for backorder replenishment can be defined more precisely using analogous concepts and
terminology from utility theory.
13
Let u(y) represent the utility of the value y. Then the following result is fundamental
to utility theory (for example, see Winkler 2003):
Theorem 2.2 Let u(y) be a continuous and twice differentiable function. Then a decision-
maker is
a136 Risk-averse if and only if uprimeprime(y) < 0.
a136 Risk-neutral if and only if uprimeprime(y) = 0.
a136 Risk-seeking if and only if uprimeprime(y) > 0.
We will classify customers as risk-averse, risk-neutral, or risk-seeking based on the
backorder rate function ?(x ? Q) as opposed to some utility function u(y). In order to
develop our classification scheme, we introduce the following definitions all of which are
special cases of classical utility theory.
Definition 2.1 A lottery with respect to waiting time, L = (t1,p1;...;tn,pn), consists
of a set of waiting times {t1,...,tn} and a set of probabilities {p1,...,pn} such that a
decision-maker waits for ti time units with probability pi, where i = 1,...,n.
Definition 2.2 The certainty equivalent with respect to waiting time of a lottery
L = (t1,p1;...;tn,pn), denoted CEW(L), is the waiting time such that the decision-maker
is indifferent between L and waiting for CEW(L) with certainty.
Definition 2.3 The risk premium with respect to waiting time of a lottery L =
(t1,p1;...;tn,pn), denoted RPW(L), is defined as
RPW(L) = EW[L]?CEW(L)
where EW[L] is the expected value of the lottery L.
14
Definition 2.4 If L = (t1,p1;...;tn,pn) is a lottery with respect to waiting time with
n > 1, then a decision-maker is
a136 Risk-averse with respect to waiting time if and only if RPW(L) < 0.
a136 Risk-neutral with respect to waiting time if and only if RPW(L) = 0.
a136 Risk-seeking with respect to waiting time if and only if RPW(L) > 0.
The difference in conventional utility theory and utility theory with respect to waiting
time as described by definitions 2.1 ? 2.4 is noticeably observable in Definition 2.4. In
particular, the definition of conventional risk aversion is RPW(L) > 0 (see Winkler 2003,
for example), but the definition of risk aversion with respect to waiting time is RPW(L) < 0.
Similarly, the inequalities are also reversed in the definitions of conventional risk-seeking and
risk-seeking with respect to waiting time. In order to illustrate Definition 2.4, consider two
decision-makers, DM1 and DM2, who are presented with a lottery with respect to waiting
time related to the time they wait to be seated in a restaurant. In particular, the lottery
is defined as L = (50-min.,0.25;10-min.,0.75). Suppose DM1 specifies CEW1(L) = 12
minutes and DM2 specifies CEW2(L) = 40 minutes. Since EW[L] = 20 minutes, we have
RPW1(L) = 20?12 = 8 minutes and RPW2(L) = 20?40 = ?20 minutes. Since CEW1(L)
is a shorter wait than EW[L], DM1 considers L to be RPW1(L) = 8 minutes better than
its expected value, which suggests that DM1 is influenced more by the possibility of the 10-
minute wait than the risk of a 50-minute wait. Therefore, DM1 is risk-seeking. On the other
hand, since CEW2(L) is a longer wait than EW[L], DM2 considers L to be RPW2(L) = ?20
minutes better (i.e., 20 minutes worse) than its expected value, which suggests that DM2 is
more influenced by the risk of a 50-minute wait than the chances of a 10-minute wait. Thus
DM2 prefers to avoid the lottery?s risk and is therefore risk-averse. This example gives some
insight into the logic behind Definition 2.4.
The following theorem relates the backorder rate function, ?(x ? Q), to risk-averse,
-neutral, and -seeking behavior with respect to waiting time.
15
Theorem 2.3 Let ? : y mapsto? [0,1], where y ? [0,?), be a continuous and twice differentiable
function. Then a decision-maker is
a136 Risk-averse with respect to waiting time if and only if ?primeprime(y) < 0.
a136 Risk-neutral with respect to waiting time if and only if ?primeprime(y) = 0.
a136 Risk-seeking with respect to waiting time if and only if ?primeprime(y) > 0.
Proof: Please refer to Appendix A.2.
Based on Theorem 2.3, customers 1, 2, and 3 in Figure 2.2 are risk-averse, -neutral,
and -seeking, respectively. Thus when customers all have the same lost sales threshold,
M, the risk-averse customer is more patient than the risk-neutral customer, and the risk
neutral customer is more patient than the risk-seeking customer since ?1 > ?2 > ?3 for all
x?Q ? [0,M], where ?i is the backorder rate for customer i.
2.3.1 Risk-Neutral Behavior
A backorder rate function ?(x?Q) that describes risk-neutral behavior with respect
to waiting time (see definitions 2.1 ? 2.4) should satisfy properties 2.1 ? 2.3 as well as the
second part of Theorem 2.3. The following proposition defines a representative backorder
rate function that satisfies these conditions.
Proposition 2.1 Suppose x?Q ? 0. Then
?(x?Q) =
??
?
??
1? x?QM , x ? [Q,Q+M)
0, x ? [Q+M,?)
(2.3)
satisfies properties 2.1 ? 2.3 and the second part of Theorem 2.3.
16
Proof: Since
d?(x?Q)
d(x?Q) =
??
?
??
? 1M < 0, x ? [Q,Q+M)
0, x ? [Q+M,?)
Property 2.1 holds. Also,
d2?(x?Q)
d(x?Q)2 = 0
shows that the second part of Theorem 2.3 holds. Now
lim
x?Q?0
max
braceleftbigg
1? x?QM ,0
bracerightbigg
= 1
shows that Property 2.2 holds, and
lim
x?Q??
max
braceleftbigg
1? x?QM ,0
bracerightbigg
= max{??,0} = 0
shows that Property 2.3 holds. Q.E.D.
The next result indicates the existence of Q? that minimizes TC(Q) given by Eq. (2.1)
when ?(x?Q) is defined as in Proposition 2.1.
Theorem 2.4 Suppose ?(x?Q) is defined as in Proposition 2.1. Then TC(Q) given by
Eq. (2.1) is a convex function.
Proof: If x ? [Q,Q + M), then ?prime(x?Q) = 1M and ?primeprime(x?Q) = 0. Thus the inequality
(x ? Q)?primeprime(x ? Q) ? 2?prime(x ? Q) given by Eq. (2.2) reduces to 2M ? 0. Since the last
inequality always holds, it follows that ?(x?Q) satisfies the condition in Theorem 2.1. If
x ? [Q + M,?), then ?(x ? Q) = 0, which also satisfies the condition in Theorem 2.1.
17
Therefore, Theorem 2.1 guarantees that TC(Q) is a convex function for all Q ? 0 whenever
?(x?Q) is defined as in Eq. (2.3).
This can be proven by showing the second derivation of TC(Q) is positive. When
?(x?Q) is defined as the linear equation, TC(Q) can be written as
TC(Q) = cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cLS
integraldisplay ?
Q
(x?Q)f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
(x?Q)
parenleftbigg
1? x?QM
parenrightbigg
f(x)dx (2.4)
Thus, the first order derivatives of TC(Q) are calculated as follows
dTC(Q)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
bracketleftbigg
?1+ 2(x?Q)M
bracketrightbigg
f(x)dx
(2.5)
and second derivatives of TC(Q) yields
d2TC(Q)
dQ2 = (cH +cLS)f(Q)+(cLS ?cB)f(Q)
+ 2(cLS ?cB)M
integraldisplay Q+M
Q
f(x)dx (2.6)
If Assumption 2.1 holds, it follows that cLS ?cB > 0 so that all terms of TC(Q) are defi-
nately non-negative and it means the convexity of TC(Q) can be ascertained for a general
demand distribution.Q.E.D.
18
2.3.2 Risk-Seeking Behavior
A backorder rate function ?(x?Q) that describes risk-seeking behavior with respect
to waiting time should satisfy properties 2.1 ? 2.3 as well as the third part of Theorem
2.3. The following proposition defines a representative backorder rate function that satisfies
these conditions.
Proposition 2.2 Suppose x?Q ? 0 and a > 0 is a constant. Then
?(x?Q) =
??
?
??
e?a(x?Q), x ? [Q,Q+M)
0, x ? [Q+M,?)
(2.7)
satisfies properties 2.1 ? 2.3 and the third part of Theorem 2.3.
Proof: The proof is similar to the proof of Proposition 2.1. Refer to Appendix A.3 for
details.
Theorem 2.5 Suppose ?(x?Q) is defined by Eq. (2.7) and a < 2M. Then TC(Q) given
by Eq. (2.1) is a convex function.
Proof: Let A = (0, 2a) and B = (M,?). It is straightforward to show that Eq. (2.7)
reduces to x ? Q ? A. Since ?(x ? Q) = 0 ? x ? Q ? B and Eq. (2.7) holds, it follows
from Theorem 2.1 that TC(Q) is convex ? x?Q ? A?B. In order to verify that TC(Q)
is convex ? Q ? 0, we need to show that convexity holds ? x?Q ? R+, which actually
reduces to showing that A?B = R+. Let C = (2a,M). Then A?B ?C = R+. However,
since the condition a ? 2M is given, we have that C = ? and A?B ?C = A?B = R+.
19
This can be proven by showing the second derivation of TC(Q) is positive. When
?(x?Q) is defined as a exponential decreasing function, TC(Q) can be written as
TC(Q) = cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cLS
integraldisplay ?
Q
(x?Q)f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
(x?Q)
parenleftBig
e?a(x?Q)
parenrightBig
f(x)dx (2.8)
Thus, the first order derivatives of TC(Q) are calculated as follows
dTC(Q)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
bracketleftBig
?e?a(x?Q) +(x?Q)ae?a(x?Q)
bracketrightBig
f(x)dx
(2.9)
and second derivatives of TC(Q) yields
d2TC(Q)
dQ2 = (cH +cLS)f(Q)+(cLS ?cB)f(Q)
+ (cLS ?cB)
integraldisplay Q+M
Q
bracketleftBig
2ae?a(x?Q) ?(x?Q)a2e?a(x?Q)
bracketrightBig
f(x)dx
(2.10)
If Assumption 2.1 holds, it follows that cLS ?cB > 0. We need one condition of
bracketleftbig2ae?a(x?Q) ?(x?Q)a2e?a(x?Q)bracketrightbig > 0. It can be reduced to a(x?Q) < 2 for
x ? [Q,Q+M). Therefore, all terms of TC(Q) under the condition of a < 2M are definately
non-negative. It means the convexity of TC(Q) can be ascertained for a general demand
distribution. Q.E.D.
20
Note that the parameter a controls the rate at which ?(x ? Q) given by Eq. (2.7)
decreases. In particular, a1 > a2 implies that customer 1 is more risk seeking (or more
impatient) than customer 2. The condition a < 2M suggests that Theorem 2.5 can only
guarantee convexity if ?(x?Q) does not decrease too quickly as x?Q approaches M.
2.3.3 Risk-Averse Behavior
A backorder rate function ?(x ? Q) that describes risk-averse behavior with respect
to waiting time should satisfy properties 2.1 ? 2.3 as well as the first part of Theorem
2.3. The following proposition defines a representative backorder rate function that satisfies
these conditions.
Proposition 2.3 Suppose x?Q ? 0. Then
?(x?Q) =
??
?
??
cos
bracketleftbigg(x?Q)pi
2M
bracketrightbigg
, x ? [Q,Q+M)
0, x ? [Q+M,?)
(2.11)
satisfies properties 2.1 ? 2.3 and the first part of Theorem 2.3.
Proof: The proof is similar to the proof of Proposition 2.1. Refer to Appendix A.4 for
details.
Theorem 2.6 Suppose ?(x?Q) is defined by Eq. (2.11). Then TC(Q) given by Eq. (2.1)
is a convex function.
Proof: If x ? [Q,Q+M), then
?prime(x?Q) = pi2M sin
parenleftbiggx?Q
2M pi
parenrightbigg
?primeprime(x?Q) = ? pi
2
4M2 cos
parenleftbiggx?Q
2M pi
parenrightbigg
21
Thus the inequality given by Eq. (2.2) reduces to
?(x?Q)pi4M cot
parenleftbiggx?Q
2M pi
parenrightbigg
? 1. (2.12)
If we let z = (pi/2M)(x?Q), then Eq. (2.12) becomes (?z/2)cot(z) ? 1, where z ? (0,pi/2).
Since cot(z) > 0 and ?z/2 < 0 for z ? (0,pi/2), we have (?z/2)cot(z) < 0 ? 1, which
implies that the inequality given by Eq. (2.12) holds. Thus it follows that ?(x?Q) satisfies
the condition in Theorem 2.1. If x ? [Q + M,?), then ?(x?Q) = 0, which also satisfies
the condition in Theorem 2.1. Therefore, Theorem 2.1 guarantees that TC(Q) is a convex
function for all Q ? 0 whenever ?(x?Q) is defined as in Eq. (2.11).
This can be proven by showing the second derivation of TC(Q) is positive. When
?(x?Q) is defined as a smoothly decreasing function, TC(Q) can be written as
TC(Q) = cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cLS
integraldisplay ?
Q
(x?Q)f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
(x?Q)cos(x?Q2M pi)f(x)dx (2.13)
Thus, the first order derivatives of TC(Q) are calculated as follows
dTC(Q)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx+(cB ?cLS)
?
integraldisplay Q+M
Q
bracketleftbigg
?cos(x?Q2M pi)+(x?Q) pi2M sin(x?Q2M pi)
bracketrightbigg
f(x)dx
(2.14)
and second derivatives of TC(Q) yields
d2TC(Q)
dQ2 = (cH +cLS)f(Q)+(cLS ?cB)f(Q)+(cLS ?cB)
?
integraldisplay Q+M
Q
bracketleftbigg pi
M sin(
x?Q
2M pi)+(x?Q)(
pi
2M)
2 cos(x?Q
2M pi)
bracketrightbigg
f(x)dx
22
(2.15)
If Assumption 2.1 holds, it follows that cLS?cB > 0. We need to verify piM sin(x?Q2M pi)+
(x ? Q)( pi2M)2 cos(x?Q2M pi) > 0. It can be reduced to ?(x?Q)pi4M cot
parenleftbiggx?Q
2M pi
parenrightbigg
< 1 for
x ? [Q,Q + M). Therefore, all terms of TC(Q) are definately non-negative and It means
the convexity of TC(Q) can be ascertained for a general demand distribution. Q.E.D.
2.4 Sensitivity Analysis and Management Insights
This section investigates the effects that various problem parameters have on optimal
ordering decisions. The following example data was used for the analysis:
X ? N(500,500),cO = 50,cH = 20,cB = 75,cLS = 100,M = 1000 (2.16)
Based on the results shown in Figure 2.3 through 2.6, we observe the following:
Observation 2.1 Let QA,QN, and QS be the optimal order quantity associated with the
risk-averse, -neutral, and -seeking cases, respectively. Then QS ? QN ? QA.
This is intuitive since according to Figure 2.2, the risk-averse customer is more patient
than the risk-neutral customer, and the risk-neutral customer is more patient than the risk-
seeking customer. Therefore, it is reasonable to expect the optimal order quantity to be an
increasing function of customer impatience.
Observation 2.2 Optimal order quantities are always non-decreasing in cLS and cB, and
always non-increasing in cH and M.
These results are intuitive and consistent with the results reported in Lodree (2007).
23
0
200
400
600
800
1000
1200
100 200 300 400 500
Risk neutral
Risk averse
Risk seeking
Q
?
c
LS
Figure 2.3: The effect of cLS on the optimal order quantity.
200
250
300
350
400
450
55 65 75 85 95
Risk neutral
Risk averse
Risk seeking
Q
?
c
B
Figure 2.4: The effect of cB on the optimal order quantity.
24
330
340
350
360
370
380
390
400
410
420
430
800 900 1000 1100 1200
Risk nuetral
Risk averse
Risk seeking
Q
?
M
Figure 2.5: The effect of M on the optimal order quantity.
0
50
100
150
200
250
300
350
400
450
20 120 220 320 420
Risk neutral
Risk averse
Risk seeking
Q
?
c
H
Figure 2.6: The effect of cH on the optimal order quantity.
25
Observation 2.3 According to Figure 2.3 and Figure 2.5, differences in optimal order
quantities among the three customer types are increasing functions of cLS and M.
From a managerial perspective, this means that it is in the decision-maker?s best interest
to be astute with respect to customer risk profiles when (i) the lost sales cost is expensive
(i.e., the product is expensive) and (ii) when the backorder / lost sales threshold is large.
These effects can be explained mathematically by observing Eq. (A.1) from Appendix A.
In particular, the effect of the backorder rate function ?(x?Q) on the total expected cost
function TC(Q) is magnified when either cLS or M is increased. Thus it is reasonable
to expect increasing differences in Q? among the three cases as ?(x?Q) assumes a more
dominant role in TC(Q). Finally, note that of these two parameters, it can be observed
from Figure 2.3 and 2.5 that the optimal decision is more sensitive to cLS.
Observation 2.4 According to Figure 2.4 and Figure 2.6, differences in optimal order
quantities among the three customer types are decreasing functions of cB and cH.
From a managerial perspective, this means that the decision-maker should keenly observe
customer risk profiles when (i) the backorder cost is small relative to the lost sales cost
and (ii) the holding cost is small relative to the ordering cost, backorder cost, and lost
sales cost. These effects can also be explained mathematically by observing Eq. (A.1) from
Appendix A. Since cB ? cLS based on Assumption 2.1, our analysis involves increasing
cB only up until it reaches cLS. Thus as cB approaches cLS, the term involving ?(x?Q)
in Eq. (A.1) approaches zero, and the effects of customer risk profiles become increasingly
negligible (in fact, TC(Q) becomes the newsvendor problem). As for cH, it is increased
beyond cO, cB, and cLS for the purpose of our analysis, although this is not likely to
occur in practice. However, the results suggest that differences in optimal order quantities
among the three customer types become increasingly insignificant as holding costs become
increasingly dominant in TC(Q) (or equivalently, as ?(x?Q) becomes less dominant in the
expected cost function).
26
Observation 2.5 Differences in optimal order quantities between the risk-seeking and risk-
neutral cases are always greater than the differences in optimal order quantities between the
risk-averse and risk-neutral cases.
This is necessarily a consequence of the specific backorder rate functions studied in this study
for the risk-averse and risk-seeking cases. More specifically, the risk-seeking backorder rate
function given by Eq. (2.7) is more different than the risk neutral case when compared to the
risk-averse backorder rate function given by Eq.(2.11). To illustrate this point, consider the
data in Eq. (2.16) and suppose x?Q = 300. Then using equations (2.3), (2.7), and (2.11),
?A(300)??N(300) = 0.191 and ?N(300)??S(300) = 0.698, where ?N(x?Q), ?S(x?Q),
and ?A(x?Q) are equations (2.3), (2.7), and (2.11), respectively. Since ?N(300)??S(300) >
?A(300)??N(300), it is reasonable to expected that QN ?QS > QA ?QN.
2.5 Value of Risk Profile Information
Suppose a decision-maker can conduct a study to obtain more information about the
risk profile of a customer or market segment. This section explores the expected value of
conducting such a study. To carry out the analysis, let us first assume that the decision-
maker currently orders QN, which is the optimal order quantity associated with the risk-
neutral case. If the customer is actually risk-averse, then the expected value of a market
study is
V = TCA(QN)?TCA(QA),
where TCA(?) and QA are the expected cost function and optimal order quantity, respec-
tively, for the risk-averse case. In general, let Vij equal the value of the market study if the
decision-maker orders Qi and the customer risk profile is actually j, where i,j ?{A,N,S}
(Averse, Neutral, Seeking). Also, let TCi(?) and Qi represent the expected total cost func-
tion and optimal order quantity, respectively, for case i ? {A,N,S}. If i = j, then clearly
27
Vij = 0 for all i ?{A,N,S}. Otherwise if i negationslash= j, then
VNA = TCA(QN)?TCA(QA)
VNS = TCS(QN)?TCS(QS)
VAN = TCN(QA)?TCN(QN)
VAS = TCS(QA)?TCS(QS)
VSA = TCA(QS)?TCA(QA)
VSN = TCN(QS)?TCN(QN).
The example data from Eq. (2.16) was used to construct the graphs shown in Figure
2.7 through 2.15. The following insights can be interpreted from these figures:
Observation 2.6 The expected value of risk profile information (EVRPI) is an increasing
function of cLS and M, and a decreasing function of cB and cH.
This observation is intuitive since differences in optimal order quantities among the three
customer types are increasing in cLS and M, and decreasing in cB and cH (see Section 3.4.2).
Observation 2.7 EVRPI is most valuable when cLS is large and when the decision-maker
assumes a risk-seeking profile, but the customer is actually risk-averse (see Figure 2.7:
EVRPI is nearly 4,000). On the other hand, the risk profile information is least valuable
whenever the value of cB is very close to the value of cLS (see Figure 2.7 through 2.12).
More generally, the larger values of EVRPI occur when either the actual risk profile is
risk-seeking and the decision-maker assumes risk-averse, or vice-versa.
Observation 2.8 Based on the Observation 2.7, the decision that minimizes the worst
possible outcome (a min-max criterion) assumes that the risk profile is risk-neutral (see
Figure 2.7, 2.9, 2.10, 2.12, 2.13, and 2.15).
28
0
500
1000
1500
2000
2500
3000
3500
4000
100 200 300 400 500
Assume Risk Neutral
Assume Risk Seeking
EVRPI
c
LS
Figure 2.7: The effect of cLS on EVRPI if customer is risk-averse.
0
200
400
600
800
1000
1200
1400
1600
100 200 300 400 500
Assume Risk Averse
Assume Risk Seeking
c
LS
EVRPI
Figure 2.8: The effect of cLS on EVRPI if customer is risk-neutral.
29
0
500
1000
1500
2000
2500
3000
3500
100 200 300 400 500
Assume Risk Neutral
Assume Risk Averse
EVRPI
c
LS
Figure 2.9: The effect of cLS on EVRPI if customer is risk-seeking.
0
50
100
150
200
250
55 65 75 85 95
Assume Risk Neutral
Assume Risk Seeking
c
B
EVRPI
Figure 2.10: The effect of cB on EVRPI if customer is risk-averse.
30
0
20
40
60
80
100
120
55 65 75 85 95
Assume Risk Averse
Assume Risk Seeking
EVRPI
c
B
Figure 2.11: The effect of cB on EVRPI if customer is risk-neutral.
0
50
100
150
200
250
300
55 65 75 85 95
Assume Risk Neutral
Assume Risk Averse
c
B
EVRPI
Figure 2.12: The effect of cB on EVRPI if customer is risk-seeking.
31
0
20
40
60
80
100
120
140
160
800 900 1000 1100 1200
Assume Risk Neutral
Assume Risk Seeking
EVRPI
M
Figure 2.13: The effect of M on EVRPI if customer is risk-averse.
0
10
20
30
40
50
60
70
800 900 1000 1100 1200
Assume Risk Averse
Assume Risk seeking
M
EVRPI
Figure 2.14: The effect of M on EVRPI if customer is risk-neutral.
32
0
20
40
60
80
100
120
140
160
800 900 1000 1100 1200
Assume Risk Neutral
Assume Risk Averse
EVRPI
M
Figure 2.15: The effect of M on EVRPI if customer is risk-seeking.
On the other hand, if the customer?s risk profile is not likely to be risk-seeking, then it is
better to assume that the customer is risk-averse. To see this, consider that if we ignore the
risk seeking case (Figure 2.9), then the penalty for assuming risk-neutral if the customer
is actually risk-averse is slightly greater than the penalty for assuming risk-averse and the
customer is actually risk-neutral (see Figure 2.7 and 2.8). Similar arguments can be used
to determine optimal decisions if the customer is not likely to be risk-averse or -neutral.
Observation 2.9 As previously mentioned, the largest values of EVRPI occur when cLS is
large. However, the rate of change in EVRPI is more sensitive to cB (compare Figure 2.7
through 2.9 to Figure 2.10 through 2.12).
2.6 Conclusion
Shortages are often represented as either backorders or lost sales in conventional inven-
tory models and in practice. The time-dependent backlogging approach to characterizing
33
inventory shortages acknowledges that there is some probability associated with whether
or not a customer will backorder a shortage, and that this probability (i.e., the backorder
rate) is related to the lead-time associated with replenishing the outstanding backorder.
While the majority of the research literature considers time-dependent backlogging within
the context of continuous review models with deterministic demand, this research studies
time-dependent partial backlogging in the single period inventory problem with stochastic
demand.
The backorder rate function is used to classify customers as either risk-averse, -neutral,
or -seeking with respect to their willingness to wait for the supplier to replenish shortages.
Representative backorder rate functions are presented, and the existence of a unique order
quantity that minimizes total expected costs is shown for each case. Sensitivity analysis
experiments are conducted to examine the similarities and differences in optimal order
quantities among the three customer types as a function of various problem parameters.
Additionally, the expected value of risk profile information (EVRPI) is defined in order
to assess the expected benefit of knowing the risk behavior of a customer or market segment.
Our results suggest that EVRPI is most significant when the difference between the unit
lost sales cost cLS and the unit backorder cost cB is large. Our results also indicate that
EVRPI increases as the difference cLS ? cB increases and as the lost sales threshold M
increases.
This study can be extended in several ways. A useful extension would be to relax
Assumption 1 such that cB > cLS is a possibility. This cost structure is known to happen in
practice based on one of the author?s interactions with the production manager of a major
manufacturing firm. In particular, this production manager?s primary objective in the event
of a shortage is to maintain a positive rapport with the client (especially a major client).
He is much less concerned with the short term cost inefficiencies associated with emergency
procurement, production, and delivery in light of the long term implications related to
compromising a business relationshp. Therefore, this manager would replenish backorders at
34
unit cost cB, even in the event that cB far exceeds cLS. This manufacturing firm?s strategic
approach to managing shortages suggests another promising extension of this study, namely
an extension that entails both short term and long term management of inventory shortages
in a multi-period setting. Finally, consider that we have verified the existence of a unique
optimal order quantity for each of a specific set of backorder rate functions, and also that
some of the results presented in sections 2.4 and 2.5 are specific to these functions and their
interrelationships. Thus the development of other backorder rate functions, analysis of their
mathematical properties, and the validity of our management insight results based on these
functions and their interrelationships also warrants exploration.
35
Chapter 3
Optimizing Backorder Lead-Time and Order Quantity in a Supply Chain
with Partial Backlogging and Stochastic Demand
Abstract
We examine a backorder lead-time (i.e., response time) optimization in a two-stage system
with time-dependent partial backlogging and stochastic demand. From the partial backlog-
ging point of view, the objective of representative mathematical models is to minimize the
expected total cost related to ordering, inventory holding, and shortages. Backordering cost
is often regarded as a constant in literature. However, the backorder cost is characterized
as a function of response time in a real situation. Moreover, the backorder rate (i.e., the
probability that a customer elects to backorder shortages) is a decreasing function of sup-
plier response time. We investigate a supply chain system with a time-dependent backorder
cost and rate in an uncertain demand setting. A representative expected cost function and
closed form solution are derived for specific demand distributions. We also illustrate our
results (e.g., cost savings) with an example.
3.1 Introduction
A firm achieving strategic fit shows the right balance between responsiveness and ef-
ficiency. Therefore, their supply chain should be structured to provide responsiveness to
customers while improving the overall efficiency (Chopra et al. 2007). Optimizing the re-
sponse time is one of the critical factors when companies are competing with each other
under a highly uncertain demand situation. The company works to optimize the response
time not only to reduce the expected cost as a faster response requires a higher cost, but
also to increase responsiveness to customers. Many companies often get opportunities to
36
improve customer satisfaction and reduce the expected total cost by optimizing the response
time, which is evident in changes such as lean manufacturing as a philosophy of production
that emphasizes minimizing the amount of all resources (including time) used in various
enterprise activities.
Toaddressthissituation, wedevelopedandsolvednewsvendormodelsinvolvingstochas-
tic demand, time-dependent backorder rate and cost. A tactical planning approach is taken
by viewing a two-stage supply chain system consisting of one supplier and one retailer with
time-dependent partial backlogging and backorder cost. The objective of the supplier is to
optimize order quantity and response time to replenish backorders in order to minimize its
expected total cost. The decision variables considered are the order quantity Q and the
response time (backorder lead-time) t. It turns out that determining the response time is
equivalent to deciding how much financial capital should be invested in the future backo-
rders. Simply stated, our problem is to decide how many products should be ordered before
the selling season and how much financial capital should be invested into backorders during
mid-season so that both backorder cost and lost sales cost are minimized.
More specifically, we consider a single period model consisting of various backorder
processing time modes. A supplier determines the number of items Q to order such that
Q items are available at the beginning of the season. For the time being, assume that the
optimal value Q? is determined based on the information about the demand distribution X,
where X is a random variable with the probability density function f(x) and the cumulative
distribution function F(x). If Q is larger than the realized demand, then the supplier incurs
a unit inventory holding cost of cH for the season. If Q is less than the realized demand,
both backorder cost cB(t) and/or lost sales cost cLS are incurred. Backorder cost cB(t) is
one of the convex and decreasing functions for t > 0. The cost incurred during the backorder
process is based on the supplier?s selected backorder lead-time t. This assumption regarding
the form of cB(t) implies that the response time (i.e., backorder lead-time) t can be reduced
by increasing the backorder processing cost.
37
Applications of this problem can be found in the auto industry, seasonal fashion prod-
ucts, and school supplies. In each of these examples, a number of products are available at
the beginning of the season where the inventory and other costs are incurred for ordering
too much, and backorders are used to satisfy the unfulfilled demand, and some amount of
unfulfilled demand becomes lost sales. Our objective as a supplier is to minimize the sum
of inventory holding costs, ordering costs, backorder, and lost sales costs.
3.1.1 The Newsvendor Problem with Time-dependent Partial Backlogging
Recall the newsvendor problem with time-dependent partial backlogging of Lodree
(2007). In this newsvendor problem, ? represents the backorder rate, which can be inter-
preted as the fraction of shortages backlogged or the probability that a given customer will
choose to backlog shortages. Since ? is time-dependent, ? : L mapsto? [0,1], where L ? [0,?)
is the backorder lead-time. L is directly proportional to the magnitude of an observed
shortage. In other words, L is a linear function in x ? Q for x ? Q, where x is the ob-
served demand, Q is the inventory level before demand realization, and max{x?Q,0} is
the observed number of shortages. Without loss of generality, L = x?Q so that the terms
backorder lead-time and ?magnitude of shortage? can be used interchangeably. Therefore,
the backorder rate is expressed as a function ?(x ? Q). Then X is a continuous random
variable that represents demand. The newsvendor problem with a time-dependent partial
backlogging can be expressed as follows:
TC(Q) = Order Cost + Expected Holding Cost + Expected Backorder Cost
+ Expected Lost Sales Cost
= cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cB
integraldisplay ?
Q
(x?Q)?(x?Q)f(x)dx
+ cLS
integraldisplay ?
Q
(x?Q)[1??(x?Q)]f(x)dx (3.1)
38
?(x ? Q) ? [0,1] is a continuous and decreasing function in the number of observed
shortages, which suggests that the customers are more likely to wait for backorder replen-
ishment if the backorder lead-time is short, and are less likely to wait if the lead-time is too
long. It determines the inventory level Q? that minimizes TC(Q), provided an optimum
exists. It turns out that the convexity is guaranteed under a sufficient (but not necessary)
condition for the existence of a unique minimizer.
3.2 Literature Review
Our problem is a choice of approaches to integrate two variations of stochastic inven-
tory models that have been studied in literature. These variations can be classified as (1)
stochastic inventory models with variable lead-time, and (2) stochastic inventory models
with time-dependent partial backlogging. We now briefly review research in each of the
two aforementioned types of the stochastic inventory models. Table 3.1 summarizes some
literature related to stochastic inventory models.
First, Liao and Shyu (1991) present a probabilistic model in which lead-time is a
unique decision variable and the order quantity is predetermined to minimize the sum of
the expected holding cost and the additional cost. Moreover, in their model, the demand
follows a normal distribution and the lead-time consists of n components each having a
different cost for reduced lead-time. In many practical situations, lead-time is controllable;
that is, the lead-time can be shortened, at the expense of extra costs, so as to improve
customer service, reduce inventory investment in the safety stocks, and improve system
responsiveness. Ben-Daya and Raouf (1994) extend the Liao and Shyu (1991) model by
allowing both the lead-time and the order quantity as decision variables where the shortages
are ignored. Ouyang et al. (1996) assume that the shortages and stockout cost are allowed.
In addition, a mixture of backorders and lost sales is considered to generalize Ben-Daya and
Raouf?s (1994) model, where the backorder rate is fixed.
39
Table 3.1: A summary of the related literature.
Stochastic Inventory Model Literature
Liao and Shyu (1991), Ben-Daya and Raouf (1994),
Variable Lead-time Ouyang et al. (1996), Wu (2001),
Lodree and Jang (2004)
Abad (1996), Chang and Dye (1999),
Time-dependent Papachristos and Skouri (2000),
Partial Backlogging Abad (2001), Wang (2002),
Papachristos and Skouri (2003), Lodree (2007)
Wu (2001) extends the model of Ouyang et al. (1996) by considering the mixtures
of normal distribution and allowing shortages. Moreover, the total amount of stock-out
is considered as a mixture of backorders and lost sales during the stock-out period. In a
practical situation, because the demands of the different customers do not have identical
lead-times, then they use the mixtures of normal distribution to describe the lead-time.
Lodree and Jang (2004) consider the customer response time minimization in a two-
stage system facing stochastic demand. They investigate a supply chain system in an
uncertain demand setting that encompasses the customer waiting costs as well as the tra-
ditional plant costs (i.e., production and inventory costs). In this study, they assume all
shortages are backlogged and replenished within a single period.
Second, some literature tackles the partial backlogging situation in an inventory deci-
sion problem. In many real-life situations, customers are likely to fulfill the shortages from a
supply competitor who has inventory on hand if the backorder lead-time is extensive. On the
other hand, the supplier is more likely to retain the customer?s business and possibly avoid
the long-term consequences of the shortages if the backorder lead-time is reasonably short.
Especially for fashion commodities and seasonal products, the willingness of a customer to
wait for backlogging during a shortage period declines with the length of the waiting time.
When shortages are partially backlogged, a fraction of shortages incurs lost sales penalties
40
Table 3.2: Differences between Lodree (2007) & Proposed Model.
Lodree (2007) Proposed Model
Backorder cB : cB(t):
cost Constant Fuction of t
Backorder ?(x?Q) = 1? x?QM ?(t) = e?at
rate Linear fuction of x?Q Exponential function of t
Decision variable Q Q,t
Total cost fuction TC[Q] TC[Q,t]
while the remaining shortages are backlogged. Therefore, time-dependent partial backlog-
ging implies that the backorder rate (i.e., the fraction of shortages backlogged) depends on
the time associated with replenishing the outstanding backorder.
Abad (1996) investigates optimal ordering and pricing policies for a perishable product
with a deterministic and time-varying demand. The model is examined through an example
with constant demand rate, exponential backorder rate, and exponential stock decay. Chang
and Dye (1999) develop an inventory model in which the proportion of customers willing to
accept backlogging is the reciprocal of a linear function of the waiting time. They examine
the effect of the backlogging rate on the economic order quantity decision. Papachristos and
Skouri (2000) establish a continuous review inventory model over a finite-planning horizon
with deterministic varying demand and constant deterioration rate. The model allows for
shortages, which are partially backlogged at a rate which varies exponentially with time.
Later, several related articles are presented, dealing with such inventory problems. For
example, Abad (2001), Papachristos and Skouri (2003), and Wang (2002). Recently Lodree
(2007) investigates a supply chain system in which a supplier prepares for the selling season
by building stock levels prior to the beginning of the season and the shortages realized at
the beginning of the season are represented as mixtures of the backorders and the lost sales.
In this research, we consider the concept of variable lead-time and time-dependent
partial backlogging simultaneously in a stochastic inventory model unlike literature. The
backorder cost is regarded as a constant in Lodree (2007) recently. However, the backorder
41
cost is characterized as a function of backorder lead-time in many real situations. Table
3.2 summarizes differences between Lodree (2007) and our proposed model. Our proposed
model extends the basic newsvendor problem to consider the feature of time-dependent
partial backlogging when a backorder occurs. The supplier fulfils the remaining customer
demand at an optimal backorder cost and backorder rate, even though some amount of
remaining customer demand becomes lost sales as the response time gets delayed further.
3.3 Model Formulation
We consider backorder lead-time as a decision variable, and we incorporate the possibil-
ity of an emergency replenishment during the selling season in the name of backorder after
demand is realized. We apply a time-dependent partial backlogging approach to optimize
the backorder lead-time and the order quantity to minimize the expected total cost.
Assumption 3.1 All items during the expedited back ordering process are shipped together
in one shipment.
Assumption 3.2 All variables, parameters, and costs (except for Q ? 0 and f(x) ? 0) are
strictly greater than zero.
Assumption 3.3 cO < cB(t?) < cLS
Assumption 3.4 The plant produces a single product type.
The goal of our model is to determine the number of items Q to order during the
off-season and the backorder lead-time t to be used to replenish the backorders during the
season to minimize the expected total costs. The notation of Table 3.3 will be used through-
out this research. We assume backorder cost is the unit backorder processing cost function
42
Table 3.3: List of notations.
cO: Unit ordering cost
cH: Unit cost of the holding excess inventory
cB(t): Unit back order cost
where cB(t) is strictly convex and decreasing for t > 0
cLS: Unit cost of shortages that are lost sales
?(t): Fraction of shortages that are backlogged
where ?(t) is strictly convex and decreasing for t > 0
K(y): Setup cost after demand realization
where K(y) is increasing for y > 0
t: the unit back order processing time (the decision variable)
Q: Order quantity (the decision variable)
Ne?bt: Special case of cB(t) for b > 0
where N is a maximum value of the back order processing cost
e?at: Special case of ?(t) for a > 0
?y: Special case of K(y)
X: A continuous random variable of demand
f(x): Probability density function (pdf) of X
F(x): Cumulative distribution function (cdf) of X
TC(?): Total expected cost function
of the backorder lead-time t. Therefore, the backorder cost can be represented as a function
of cB(t). Thus the expected cost function is given by
TC(Q,t) = Order(production) Cost + Expected Holding Cost
+ Expected Back order Cost + Expected Lost Sales Cost
= cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cB(t)
integraldisplay ?
Q
(x?Q)?(t)f(x)dx
+ cLS
integraldisplay ?
Q
(x?Q)[1??(t)]f(x)dx
= cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cLS
integraldisplay ?
Q
(x?Q)f(x)dx
+ (cB(t)?cLS)
integraldisplay ?
Q
(x?Q)?(t)f(x)dx. (3.2)
43
Our goal is to derive the optimal solution (Q?,t?) that minimizes Eq.(3.2). Although
the expected cost function Eq.(3.2) is not convex for all feasible values of Q and t, its
marginal properties allow us to find the globally optimal solution. To derive the solution,
consider the following derivatives of TC(Q,t). If the total cost function is a convex function,
then the unique minimizer Q? can be obtained by equating the first derivative to zero. First
of all, we verify whether the total cost function is a convex function or not. The convexity
of the total cost function is ensured since the second derivation is always positive as follows:
dTC(Q,t)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx+(cLS ?cB(t))
integraldisplay ?
Q
?(t)f(x)dx
d2TC(Q,t)
dQ2 = {cH +?(t)cB(t)+(1??(t))cLS}f(Q) (3.3)
We have a convex objective function and we can get a feasible point, a unique minimizer
of order quantity, at which the first-order conditions hold that first derivative equal to zero.
Q? = F?1
bracketleftbigg?(t)c
B(t)+(1??(t))cLS ?cO
?(t)cB(t)+(1??(t))cLS +cH
bracketrightbigg
(3.4)
The first and second order derivatives of TC(Q,t) are then
dTC(Q,t)
dt = {c
prime
B(t)?(t)+cB(t)?
prime(t)?cLS?prime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
d2TC(Q,t)
dt2 = {c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)+cB(t)?primeprime(t)?cLS?primeprime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
= {cprimeprimeB(t)?(t)+2cprimeB(t)?prime(t)+(cB(t)?cLS)?primeprime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
(3.5)
44
Consider the second part of Eq.(3.5). Clearly the second part is non-negative if the
condition of Eq.(3.6) is satisfied. The detailed derivation is in Eq.(B.3). The sufficient
condition is simplified as follows:
cprimeprimeB(t)?(t)+2cprimeB(t)?prime(t) + (cB(t)?cLS)?primeprime(t) ? 0
cLS ?cB(t) ? c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)
?primeprime(t)
cLS ?cB(t) ? 2ab+b
2
a2 Ne
?bt
cLS ?
parenleftbigga+b
a
parenrightbigg2
cB(t)
(3.6)
Thus, TC(Q,t) is convex in t for any fixed Q where the condition of Eq.(3.6) is satisfied,
and the optimal value t? corresponding to a fixed value Q is obtained by setting the first
part of Eq.(3.5) equal to zero and solving for t, which yields Eq.(3.7) The value t? is unique
because Eq.(3.5) shows TC(Q,t) is strictly convex in t for any fixed Q.
cprimeB(t)?(t) = ?cB(t)?prime(t)+cLS?prime(t)
?(t) = {cLS ?cB(t)}?
prime(t)
cprimeB(t)
t? = ln
parenleftbigg(a+b)N
a?cLS
parenrightbigg
b?1 (3.7)
We need to determine the optimal regular season production quantity Q?, which can be
accomplished by examining Eq.(3.2) as a function of Q only. Because the second part
of Eq.(3.3) is non-negative, TC(Q,t) is convex in Q so that the optimal value Q? (for a
fixed value t) satisfies the resulting equation when Eq.(3.3) is set to zero. Since t? given
by Eq.(3.7) is independent of Q, Q? is also optimal for t = t?. Therefore it follows that
45
the global optimal solution (Q?,t?) satisfies the first order conditions. Now equating the
first part of Eq.(3.3) to zero and solving, we attain the optimal value Q? as a closed form.
By observing Eq.(3.7), we see that the optimal backorder processing policy is such that
optimal backorder lead-time increases as N, a maximum value of back order processing cost,
increases and decreases as cLS increases. According to Eq.(3.7), we define the relationship
between cB(t?) and cLS. The details are in Eq.(B.5)
cB(t?) = Ne?b?b?1ln
?(a+b)N
a?cLS
?
= Neln
? a?c
LS
(a+b)N
?
= N a?cLS(a+b)N
= a?cLS(a+b)
(3.8)
3.3.1 The Model with Backorder Setup Cost
The backorder setup cost is incurred in setting up a work center, assembly line, or ship-
ment to switch from a regular order process to a backorder process. We add the backorder
setup cost to the proposed expected total cost function. Thus the expected cost function is
given as follows:
TC(Q,t) = Order(production) Cost + Expected Back order Cost
+ Expected Lost Sales + Expected Holding Cost
= cOQ+
integraldisplay ?
Q
[K((x?Q)?(t))+cB(t)(x?Q)?(t)]f(x)dx
+ cLS
integraldisplay ?
Q
(x?Q)[1??(t)]f(x)dx+cH
integraldisplay Q
0
(Q?x)f(x)dx (3.9)
46
We use a similar process to attain optimal decisions for this model as in the former
model considering no backorder setup cost. Finally, we attain the optimal value t? and Q?
as a closed form and define the relationship between cB(t?) and cLS in Appendix B.2.
3.3.2 Managerial Insights regarding the Response Time
By observing Eq.(3.7), we see that the optimal response policy is such that the response
time decreases as the customer impatience factor(a), the backorder cost parameter(b), and
the lost sales cost(cLS) increase and response time increases as the maximum value of the
backorder cost(N) increases. Eq.(3.8) implies that the optimal backorder cost is related to
a proportion a(a+b) of the lost sales cost. It is the same as our intuition that the suppliers
want to reduce the amount of lost sales in case of increasing the customer impatience
factor(a), increasing the backorder cost parameter(b), and/or increasing the lost sales cost.
Additionally, by observing Eq.(B.9), we see that the response time increases as the backorder
setup cost parameter(?) increases. It is also reasonable that increasing the backorder setup
cost parameter(?) causes a decrease in the backorder rate and it causes the optimal response
time to increase consequently.
In subsequent sections, we determine the optimal order quantity for various demand
distributions with numerical examples and illustrate how these solutions can be interpreted.
3.4 An Illusrative Example
An example problem will be constructed in this section using the set of data. The
example will be given for each of the following demand distributions: exponential and nor-
mal. First, the optimal backorder lead-time will be determined using Eq.(3.7)and Eq.(B.9).
Recall that the optimal backorder lead-time (t?) is independent of the demand distribu-
tion and the order quantity Q. Thus for the example problems, the optimal lead-time (t?)
is computed once while the optimal order quantity(Q?) is determined for each different
demand distribution. The following data will be used for the example problems.
47
E(X) = 2000,?X = 2000,cO = $10,cH = $75,cLS = $200,a = 0.005,b = 0.02,N = $1000
In this section, the results of the numerical examples are studied in further detail so as
to validate the mathematical model. We examine our results and then compare this solution
to the result of the prior newsvendor model. We now illustrate that the prior newsvendor
models solution produces higher expected cost than our solutions when backorder lead-
time is optimized. The illustration includes the cases of the normal and the exponential
distribution of demand.
Based on the results shown in Table 3.4 and 3.5, we infer the following insights. With
regular ordering cost to the backorder processing, the prior newsvendor model?s optimal
order quantities associated with normal distribution and exponential distribution case are
2,530 for normal and 1,853 for the exponential case. Detailed comparisons of the prior
newsvendor and the proposed solution are shown in Table 3.4 and 3.5. Tables show the
differences of the optimal order quantities and the total costs between both models. Thus
a negative value indicates a reduction of optimal order quantity or total cost. A study of
Table 3.4 and 3.5 reveals the following insights:
3.4.1 Insights from Table 3.4 and 3.5
The data used have satisfied the sufficient condition of Eq.(3.6) and the optimal back-
order lead-time is determined using Eq.(3.7). The optimal backorder lead-time is 160.94
hours (6.7days). The backorder lead-time determines a backorder rate of 0.447 and a back-
order cost of a3640. When we assume the demand is normally distributed, the optimal fill
rate is 58.2% and the optimal order quantity is 2,415 and the total cost is a36137,634. The
difference between the prior and the proposed model shows 115 items for the order quantity
and a365,324 for the total cost. In the exponentially distributed demand case, the optimal fill
rate is the same and the optimal order quantity is 1,743 and the total cost is a36168,366. The
48
difference between the prior and the proposed model shows 110 items for order quantity
and a369,325 for total cost.
Table 3.4: Comparison of Optimal Order quantities.
Dist. Model Optimal Q* Difference
Norm. Prior 2,530 -115
Proposed 2,415
Expo. Prior 1,853 -110
Proposed 1,743
Table 3.5: Comparison of Total costs.
Dist. Model Cost Total cost Difference
Order Holding Backorder Lostsales
Norm. Prior a3625,300 a3639,203 a361,773 a3676,682 a36142,958 -a365,324
Proposed a3624,150 a3635,455 a3610,867 a3667,162 a36137,634
Expo. Prior a3618,525 a3648,344 a362,504 a36108,318 a36177,691 -a369,325
Proposed a3617,432 a3643,485 a3614,964 a3692,485 a36168,366
When we do not optimize the backorder lead-time, the prior newsvendor solution results
in significant expected loss (over a365,300 per season for normal demand and over a369,300 per
season for exponential demand).
3.4.2 Sensitivity Analysis and Management Insights
This section investigates the effects that various problem parameters have on the op-
timal order quantity or the backorder lead-time decisions. Based on the results shown in
Figure 3.1 through Figure 3.5, we observe the following:
Observation 3.1 According to Figure 3.1, optimal response times are non-increasing in
the customer?s impatience factor(a). As the customer?s impatience factor(a) increases to
infinity, optimal response time converges at a certain point.
Increasing the customer?s impatience factor(a) means that more lost sales occur in
shortages. The suppliers can prevent the amount of lost sales by decreasing the response
49
time to backorder. Assuming the extreme case, when the customer?s impatience factor(a)
is relatively large, every shortage becomes lost sales. The case should be the same as an
entire lost sales case. Otherwise, when the customer?s impatience factor(a) is very small,
every shortage becomes a backorder. This case is regarded as an entire backorder case.
Observation 3.2 According to Figure 3.2, optimal response times are non-increasing in
the backorder cost parameter(b). As the backorder cost parameter(b) increases, optimal
response time converges at a certain point.
Increasing the backorder cost parameter(b) means that the backorder cost is decreased
more sharply as the response time increases. Assuming the extreme case, when the back-
order cost parameter(b) is relatively large, the backorder cost is very sensitive (a large
difference in costs either way) to the response time and to the supplier?s efforts to decrease
the response time as much as possible. Otherwise, when the backorder cost parameter(b)
becomes very small (but not zero), the backorder cost is not sensitive (not much difference
in the costs either way) to the response time and to the supplier?s efforts to increase the
response time as much as possible.
Observation 3.3 According to Figure 3.3, optimal response times are non-decreasing in
N,? and non increasing in cLS.
When the maximum value of backorder cost(N) increases, the supplier will increase the
optimal response time to reduce the backorder cost. Similarly, as the backorder setup cost
parameter(?) increases, the supplier will increase the optimal response time to reduce the
backorder cost. Otherwise, when the lost sales cost(cLS ) increases, the supplier decreases
the optimal response time to reduce the amount of shortages.
Observation 3.4 According to Figure 3.4, optimal order quantities are non-decreasing in
cLS and non-increasing in cO,cH.
50
0
20
40
60
80
100
120
140
160
180
0.0050.0100.0150.0200.0250.0300.0350.0400.0450.0500.0550.0600.0650.070
a
t*
b=0.02
b=0.025
b=0.03
b=0.035
b=0.04
Figure 3.1: The effect of a on the optimal response time.
0
20
40
60
80
100
120
140
160
180
0
.
0
2
0
.
0
2
5
0
.
0
3
0
.
0
3
5
0
.
0
4
0
.
0
4
5
0
.
0
5
0
.
0
5
5
0
.
0
6
0
.
0
6
5
0
.
0
7
0
.
0
7
5
0
.
0
8
0
.
0
8
5
b
t*
a=0.005
a=0.01
a=0.015
a=0.02
a=0.025
Figure 3.2: The effect of b on the optimal response time.
51
100
120
140
160
180
200
220
240
260
280
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Category
t*
N
c_LS
alpha
Figure 3.3: The effect of parameters on the optimal response time.
500
1000
1500
2000
2500
3000
3500
4000
4500
1 2 3 4 5 6 7
Category
Q*
cO
cH
cLS
Figure 3.4: The effect of parameters on the optimal order quantity.
52
2000
2400
2800
3200
3600
4000
40 80 120 160 200 240 280
t
Q*
Figure 3.5: The effect of t on the optimal order quantity.
Optimal order quantity increases with the lost sales cost as the company is not willing to
have lost sales when the lost sales cost increases. Otherwise, the ordering and the holding
cost show an opposite phenomenon. These results are intuitive and consistent with the
results reported in Lodree (2007).
Observation 3.5 According to Figure 3.5, the optimal order quantity shows the minimum
at the point of the optimal backorder lead-time.
A fast backorder response costs so much that the company tries to reduce the shortages
and increase the optimal order quantity. On the other hand, a delayed backorder response
causes more lost sales and the company works to reduce the shortages by increasing the
order quantity. At the optimal backorder lead-time, all trade offs between the backorders
and the lost sales costs were mitigated and the optimal order quantity shows the minimum
among other optimal order quantities.
53
3.5 Conclusion
The time-dependent backlogging approach is prevalent in characterizing inventory
shortages in an inventory decision model. This approach acknowledges that there is some
probability associated with whether or not a customer will backorder a shortage, and that
this probability (i.e., the backorder rate) is related to the backorder lead-time associated
with replenishing the outstanding backorder.
In this study, we consider the fact that the backorder cost is also related to the back-
order lead-time on a stochastic inventory model for a two-stage supply chain system. We
simultaneously consider the order quantity and the backorder lead-time as decision vari-
ables. Closed form optimal solutions are derived for the specific demand distribution, and
intuitive insights were acknowledged by examining the solution for the normally distributed
demand. Our example problem suggests that significant cost savings can be achieved by
implementing the derived optimal solution as opposed to the prior model solution.
A limitation of the model arises from the selection of specific functions for (e.g., back-
order cost function) which contain many cost behaviors as the parameters change, but not
enough for the general case. For instance, the function cB(t) as Ne?btwas selected because
it logically represents the behavior of the time-cost trade-offs (i.e., the cost increases as
the processing time decreases) and it exhibits analytic properties such that the analysis is
tractable. However, a variety of the backorder cost functions should be explored. Practi-
tioners should investigate these functions to implement this model into their specific supply
chain situation. Even though the setup cost shows piece-wise increasing behaviors in prac-
tice, we approximate the backorder setup cost function as the linearly increasing continuous
function in our model. Further research is needed to see how the results could change with
a better approximation for the behavior of the setup cost.
54
Chapter 4
The Effects of an Option Approach to Stochastic Inventory Decisions
Abstract
This study presents an option approach (e.g., straddle) for determining the order quantity
under partial backlogging and demand uncertainty. First, we establish an optimal condition
for the required order quantity when a firm has a desirable fill rate. Second, we develop
a closed form solution for optimal order quantity to minimize the total cost. Finally, we
compare the results between the traditional and the option approaches with a numerical
example. The comparison could lead to several inventory decision implications and shows
the advantages of an inventory decision with an option framework over the traditional
inventory decision under high uncertainty.
4.1 Introduction
Almost every decision-maker needs to take a forecast. If he has an idea of what will
happen in the future, he can make appropriate management decisions. He also needs to
assess the effect of his present decisions on the future so that the right decisions are made
today to create a desired condition in the future.
Inventory is capital intensive and, therefore, cost sensitive. The size of the investment
in inventory is typically between 10% and 40% of the total assets of most large corporations
(Stowe 1997). If we know how many items are likely to be demanded, we can improve
the quality of decisions concerning production, procurement, placement and promotion.
Consequently, we can minimize the expected total cost tied up in inventories, avoid running
out of stock and, generally, increase the sales and improve the profits.
55
D0
Lead time(T)
Decision point
(Q)
DT
The beginning of
a season
Figure 4.1: General problem description.
For decades, most studies on traditional inventory management under demand uncer-
tainty analyze the detrimental impacts of demand distributions (i.e., normal distribution).
Companies have typically estimated the demand using the normal distribution with its clas-
sic bell-shaped curve.
It is not appropriate to forecast demand as a normal distribution where negative demand
is possible to generate in a highly uncertain situation. In addition, a traditional approach
with demand distribution does not consider the information about a current demand that
significantly affects the future demand.
Many companies receive orders continuously from its customers. At a decision point,
A company could attain current demand information from their reservation systems (i.e.,
airlines industry). There will be additional customer orders for units or the cancellation
of orders during the planning horizon after the decision point as shown in Figure 4.1. In
a highly uncertain situation, the decision-makers are not sure whether the current demand
56
will go up or down. This situation is closely related to the financial option-pricing frame-
work. We adopt the framework of option pricing to implement a demand process in an
uncertain situation.
This study develops a variant of the option pricing theory as applied to the inventory
planning and reevaluates the basic newsvendor problem with partial backlogging to show
that the tool of financial options can be used to make inventory decisions in supply chain
management.
4.1.1 The Traditional Newsvendor Approach with Partial Backlogging
The newsvendor problem is not complex, but it can give sufficient information that
enables us to compare between the traditional and the option approaches in a two-stage
supply chain. First of all, we figure out how to decide the optimal order quantity in a
traditional newsvendor approach. In this newsvendor problem, a unit inventory holding
cost of cH is incurred if Q is larger than the demand(x), and the unit back order cost
??cB or/and unit lost sales cost (1??)cLS is incurred if Q is smaller than the demand(x).
Then, X is a continuous random variable that represents the demand and ? represents the
backorder rate. The objective is to determine the quantity (Q) that minimizes the following
expected cost function.
E[TC(Q)] = Order Cost + Expected Holding Cost + Expected Back order Cost
+ Expected Lost Sales Cost
= cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cB
integraldisplay ?
Q
(x?Q)?f(x)dx
+cLS
integraldisplay ?
Q
(x?Q)(1??)f(x)dx (4.1)
If the total cost function is a convex function, then a unique minimizer (Q?) can be
obtained. And then, we determine whether the total cost function is a convex function.
57
The convexity of the total cost function is guaranteed since the second derivation is always
positive. Refer to Appendix C.1 for details. Because we have a convex objective function,
we can get an optimal order quantity at which the first-order condition is equal to zero.
Q? = F?1
bracketleftbigg?c
B +(1??)cLS ?cO
?cB +(1??)cLS +cH
bracketrightbigg
(4.2)
Therefore, the optimal demand fill rate is represented by the following:
?? = ?cB +(1??)cLS ?cO?c
B +(1??)cLS +cH
(4.3)
The traditional approach, such as the basic newsvendor problem, assumes demand
as a specific distribution (e.g., normal distribution) to implement demand uncertainty as
shown in this model. The optimal order quantity totally depends on the assumed specific
distribution in the traditional approach.
4.2 Literature Review
There has been limited literature which applies financial theory to inventory decisions.
Kim and Chung (1989) and Morris and Chang (1991) use the capital-asset pricing model
(CAPM) as an alternative to the conventional profit-maximization approach. A key insight
of this approach is that a demand beta (sensitivity of demand to the market index) is
inversely related to the stocking level.
Several studies have applied the option-pricing approach to the inventory problem since
1990. Stowe and Gehr (1991) provide two financial approaches to handling the inventory
problem. The first is a replicating portfolio approach comparing the value of a portfolio
of securities to the inventory investment. The second approach applies an option-pricing
58
solution to the discrete sales problem. Chung (1990) also shows another option-pricing
solution for inventory payoffs that is a linear function of the demand when the demand and
the investors? discount rate are jointly log-normally distributed. Becker (1994) solves for
the inventory reorder point using a binomial option-pricing approach.
Agrawal et al. (2000) show a single-period inventory model in which a risk-averse
retailer faces uncertain customer demand and makes a purchasing-order quantity and a
selling-price decision with the objective of maximizing the expected utility. Birge and Zhang
(1999) derive an optimal policy with an option valuation model. Berling and Rosling (2005)
apply a real options framework considering the stochastic demand and the purchase costs.
They present the two inventory models, a single-period model of the newsboy type and an
infinite-horizon model with a fixed set-up cost.
Our approach is an extension of the earlier studies. Our intuition is that if any inventory
payoff function can be mapped onto an underlying state variable, the payoff can be replicated
by a portfolio of options (e.g., straddle strategy). Straddle is one of the most popular options
trading strategies in buying both a call and a put option at the same strike price and the
same maturity date in the case of a volatile market. We also refer to the discrete time
option pricing model of Cox et al. (1979) for the evaluation of the firm?s inventory decision.
The option approach presented here associates the demand for an inventory item with
an underlying state variable. This approach is well chosen for the analysis of the firm?s
inventory decisions under uncertainty because the payoff from the firm?s inventory decision
is an option with its value depending on the uncertain future demands.
This study is organized as follows. The next section introduces the model and presents
a total cost valuation using the option-pricing model. The following section then presents
the optimality for the required order quantity when a firm has a given fill rate and yields
a closed form solution for the optimal order quantity to minimize the total cost. The final
section is a summary and has concluding remarks.
59
1
Q
T
?
D
0Current stock price
Exercise price
Time to maturity
Stock volatility
Current demand
Order quantity
Time to the end
of season
Demand volatility
K
T
?
S
0
Figure 4.2: Analogy between financial option and inventory decision.
4.3 Model formulation
The underlying asset price can fluctuate continuously over time. A commonly used
process to describe the evolution of the demand (asset price) is the lognormal diffusion
process. A key feature of this process is that the demand (asset price) never drops below
zero. Another feature of the lognormal diffusion process that is similar to the binomial is
that the variance of the demand (asset price) grows with time. In fact, the variance grows
proportionally to the length of the time horizon (Chopra 2007). We substitute the demand
for the asset price in the option valuation because we assume that the characteristics of
demand are the same as the asset price of the option in this study. The decision problem
which the firm is facing at the decision point is to determine and to notify its supplier of
the order quantity(Q), in order to meet the unknown accumulated orders (i.e., demand) of
units from its customers at the end of the period. We consider the change of demand during
a considerable lead-time to determine the order quantity at the decision point. Here we
discuss a single period inventory problem. This approach is closely related to the principles
of an asset pricing model, such as the Black-Scholes option-pricing model. Then we set up
60
Table 4.1: List of notations.
cO : Unit ordering cost
cH : Unit cost of the holding excess inventory
cB: Unit cost of the backlogged shortages
cLS: Unit cost of the shortages that are lost sales
? : Fraction of shortages that are backlogged
rf: Risk free interest rate
T: Lead-time
?: Demand fill rate
Q : Order quantity
Q? : Order quantity with a given demand fill rate
Q?: Optimal order quantity that minimizes the total cost
D0: Initial demand(i.e., number of reservations by the decision point)
DT : Demand (the customers order) at time T
X: Random variable of demand
f(x): p.d.f. of the demand distribution
F(x): c.d.f. of the demand distribution
TC(?): Total expected cost function
an analogy between the financial option and the inventory decision as shown in Figure 4.2.
The notation of Table 4.1 will be used throughout this study.
4.3.1 Model Formulation with an Option Pricing Framework
The total cost may be represented as the order cost, holding cost, backorder cost,
and lost sales cost. If the demand (D) is greater than the quantity ordered (Q), the cost
is assumed to be the profit lost on unsatisfied demand. If the demand is less than the
quantity ordered, unsold units incur a holding cost for storage at a warehouse. The total
cost function can be represented as follows:
TC(Q) =
??
?
??
Order cost+Backorder cost+Lost sales cost if D ? Q
Order cost+Holding cost if D < Q
(4.4)
61
K
ST
Profit
K-STK-ST0ST< K
Total PayoffPay off from putPayoff form call
Range of
Stock price
ST-K
0ST-KST> K
Figure 4.3: A Straddle quoted from John C.Hull (2002).
With the end-of-period convention, the cash flows at the end of the period will be
cOQ+cB?(D?Q)+cLS(1??)(D?Q) if the demand is greater than the quantity ordered,
or cOQ+cH(Q?D) if the demand is less than the quantity ordered. To consider a general
situation, we assume cB < cLS, although there are some situations where backorder costs
exceed the short term lost sale costs.
TC(Q) =
??
?
??
cOQ+cB?(D?Q)+cLS(1??)(D?Q) if D ? Q
cOQ+cH(Q?D) if D < Q
(4.5)
This total cost combination closely imitates a straddle strategy in financial options
as shown in Figure 4.3. Straddle is one of the most popular options trading strategies in
buying both a call and a put option at the same strike price and on the same maturity
date in the case of a volatile market. If you buy a straddle, you expect the price of the
62
underlying asset to move significantly, but you?re not sure whether it will go up or down.
Your risk in buying a straddle is limited to the premium you pay. The first part of Eq.
(4.5) represents a call option situation and the second part of Eq. (4.5) represents a put
option situation. Then the present total cost of cash flows can be expressed as the following
conditional expectation. Refer to Appendix C.4 for details.
E[TC(Q)] = e?rfT ?E[{cB? +(1??)cLS}D
?{cB? +(1??)cLS ?cO}Q|D ? Q]?P(D ? Q)
+e?rfT ?E[(cO +cH)Q?cHD|D < Q]?P(D < Q)
(4.6)
We convert the end of the period?s conditional cash flow into a certain present value
with a continuous discount factor. We can divide the former equation into two components.
E[TC(Q)] = E[TCa(Q)]+E[TCb(Q)]
E[TCa(Q)] = e?rfT ?E[{cB? +(1??)cLS}D
?{cB? +(1??)cLS ?cO}Q|D ? Q]?P(D ? Q)
E[TCb(Q)] = e?rfT ?E[(cO +cH)Q?cHD|D < Q]?P(D < Q) (4.7)
We denote the current demand by D0 and the demand at the end of the lead-time by
DT. The ratio DTD0 is a random variable with a lognormal distribution where ln( DiDi?1) = ri
is the rate of demand change in the ith period and is a random variable with normal
distribution under our assumptions. Let ri have the expected value ?r and variance ?2r
for each i . Then r1 + r2 + r3 + ??? + rT is a normal random variable with the expected
63
value and variance . Thus, we can define the expected value of DTD0 = er1+r2+r3+???+rT as
E[DTD0 ] = eT?r+T?
2r
2 .
The risk neutral approach to valuation, introduced by Cox et al. (1979), is based on
the same arguments that underlie the option valuation. Under this assumption that the
expected demand change for one period (e.g.,D1D0 = erf ) is regarded as the exponential
value of the risk free interest rate, the expected demand change rate E[DTD0 ] is assumed to
be erf?T where rf is the risk free interest rate. In other words, since we assume that DTD0 is
log-normally distributed, we can define that the rf as ?r+?2r2 . Therefore, ln(DTD0 ) is normally
distributed using the relationship between the lognormal and the normal distribution. We
can rewrite the first part of Eq. (4.7) as follows. Refer to Appendix C.6 for details.
E[TCa(Q)] = e?rfT ?D0 ?{cB? +(1??)cLS}
integraldisplay ?
ln( QD0 )
rf(r)dr
?e?rfT ?Q{cB? +(1??)cLS ?cO}
integraldisplay ?
ln( QD0 )
f(r)dr
= D0{cB? +(1??)cLS}N(d1)?e?rfT ?Q{cB? +(1??)cLS ?cO}N(d2)
where d1 = ln(
D0
Q )+(rf +
?2r
2 )T
?r?T , d2 =
ln(D0Q )+(rf ? ?2r2 )T
?r?T (4.8)
We also rewrite the second part of Eq. (4.7) using similar procedures as follows. Refer
to Appendix C.7 for details.
E[TCb(Q)] = e?rfT ?Q(cO +cH)
integraldisplay ln( Q
D0 )
0
f(r)dr?e?rfT ?D0cH
integraldisplay ln( Q
D0 )
0
rf(r)dr
= e?rfT ?Q(cO +cH)N(?d2)?D0 ?cHN(?d1)
where d1 = ln(
D0
Q )+(rf +
?2r
2 )T
?r?T , d2 =
ln(D0Q )+(rf ? ?2r2 )T
?r?T (4.9)
64
Then we can rewrite the Eq. (4.7) simply as follows where N(?) denotes the cumulative
distribution function of the standard normal distribution.
E[TC(Q)] = E[TCa(Q)]+E[TCb(Q)])
= D0{cB? +(1??)cLS}N(d1)?e?rfT ?Q{cB? +(1??)cLS ?cO}N(d2)
+e?rfT ?Q(cO +cH)N(?d2)?D0 ?cHN(?d1)
where d1 = ln(
D0
Q )+(rf +
?2r
2 )T
?r?T , d2 =
ln(D0Q )+(rf ? ?2r2 )T
?r?T (4.10)
4.4 Inventory Decisions
We develop two closed form solutions for the inventory problem with this model. First,
we consider a firm that has an desirable fill rate to meet a certain level of customer satisfac-
tion. Next, we consider a firm wants to minimize the expected total costs with an optimal
fill rate.
4.4.1 Order Quantity with a Desirable Fill rate
Suppose that the decision-maker of the firm has a desirable fill rate, ?. The firm desires
to satisfy this level of fill rate. N(?d2) is interpreted as the fill rate, demand being less
than the quantity ordered, as shown in the following equation.
P(D < Q) = P( DD
0
< QD
0
)
= P(ln( DD
0
) < ln( QD
0
))
= 1?P(r ? ln( QD
0
))
= 1?
integraldisplay ?
ln( QD0 )
f(r)dr = 1?N(d2)
= N(?d2)
65
where ln(Dt+1D
t
) = r ? Normal(?r,?r) (4.11)
Then the minimum required order quantity can be obtained by letting N(?d2) equal
to ?. Finally, we develop the closed-form solution for the minimum required order quantity.
Refer to Appendix C.10 for details.
? = N(?d2)
Q? =
?
? D0e(rf?
?2r
2 )T
eN?1(1??)?r
?T
?
? (4.12)
N?1(?) denotes the inverse cumulative distribution function of the standard normal distri-
bution.
4.4.2 Optimal Fill rate and Order Quantity
Now suppose that the objective is to minimize the present total cost cash flows. Then,
we could attain optimal stock out probability and order quantity fromd[E[TC(Q)]]dQ = 0.
d[E[TC(Q)]]
dQ = e
?rfT(cO +cH)N(?d2)?e?rfT{cB? +(1??)cLS ?cO}N(d2) = 0
e?rfT(cO +cH)N(?d2) = e?rfT{cB? +(1??)cLS ?cO}N(d2)
(4.13)
Note that cB?+(1??)cLS ?cO represents the penalty cost of the shortage (i.e., lost profit
per unit), and N(d2) represents the probability of stock-out. Thus the right-hand side of
Eq. (4.13) measures the expected cost of shortages. Similarly, note that cO + cH is the
implicit cost of each unsold unit, and N(?d2) represents the fill rate. Thus the left-hand
66
side of Eq. (4.13) measures the expected cost of an overstock. The Eq. (4.13) can be
simplified as follows:
?? = N(?d2) = ?cB +(1??)cLS ?cO?c
B +(1??)cLS +cH
(4.14)
If we have a convex objective function and a feasible point at which the first-order
conditions hold, then we know that we have found a unique minimizer. We need to verify
the convexity of Eq. (4.10) as the convexity of the function guarantees the expected total
cost function has a unique minimizer. The convexity of Eq. (4.10) is guaranteed since the
second derivation is always positive as follows. Refer to Appendix C.9 for details.
d[E[TC(Q)]]2
d2Q = e
?rfT{cH +cB? +(1??)cLS}(Q?r?T)?1f(d2) (4.15)
The optimal order quantity occurs when the expected shortage penalty cost equals the
expected cost of the overstock. We derive the optimal order quantity (Q?) as follows. Refer
to Appendix C.10 for details.
N(?d2) = ??
Q? = D0e
(rf??2r2 )T
e?r
?TN?1(1???) (4.16)
The value of ?? can be interpreted as the optimal fill rate to minimize the expected
total cost. Then, we also obtain the closed-form solution for the optimal order quantity.
The optimal fill rate is the same as the partial backlogging newsvendor problem. It is
67
reasonable to make the option framework approach applicable as the equation of optimal
fill rate is composed with determined parameters.
4.5 An Illustrative Example
Comparisons between the traditional and option approaches will be given in this section
using the same set of data. The following example data was used for the analysis:
cO = 10,cH = 5,cB = 20,cLS = 40,D0 = 500,rf = 0.05,? = 0.3,T = 1 (4.17)
We adopt the relationship proposed by Dixit et al. (1994) between the volatility of
demand (?r) and the volatility of future demand at the end of the study period (?T) to
compare both the approaches.
?r =
radicaltpradicalvertex
radicalvertexradicalvertex
radicalbtln
parenleftbigg
?2T
D20erfT +1
parenrightbigg
T (4.18)
Order Quantity with an Desirable Fill rate
The required order quantities of both the approaches with an desirable fill rate are at-
tained according to the change of the volatility of demand (?r). We illustrate the differences
between both the approaches with graphs to find some implications.
As shown in Figure 4.4, optimal order quantities are exponentially increased as the
volatility of demand (?r) is increased with the fill rate greater than 50%. Otherwise, optimal
order quantities are exponentially decreased as the volatility of demand (?r) is increased
with the fill rate less than 50%. Optimal order quantities are the same regardless of the
change of the volatility of demand (?r) with the fill rate 50%. However, the optimal order
68
-6000
-4000
-2000
0
2000
4000
6000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Qs
p=0.9
p=0.7
p=0.5
p=0.3
p=0.1
?
r
Figure 4.4: Order quantity with traditional approach.
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Qs
p=0.9
p=0.7
p=0.5
p=0.3
p=0.1
?r
Figure 4.5: Order quantity with option approach.
69
Table 4.2: Comparison Optimal order quantities and Total costs.
Q* TC(Q*)
?r Traditional Option app. Difference Traditional Option app. Difference
0 526 526 - a36 5,000 a36 5,000 -
0.2 557 546 (10) a36 6,843 a36 6,525 (a36 318)
0.4 590 546 (44) a36 8,496 a36 8,078 (a36 418)
0.6 627 524 (104) a36 10,167 a36 9,598 (a36 569)
0.8 672 483 (189) a36 12,003 a36 11,029 (a36 974)
1 728 428 (300) a36 14,277 a36 12,322 (a36 1,955)
1.2 802 364 (439) a36 17,310 a36 13,445 (a36 3,865)
1.4 906 297 (609) a36 21,569 a36 14,383 (a36 7,186)
1.6 1058 234 (825) a36 27,824 a36 15,134 (a36 12,689)
1.8 1289 176 (1,113) a36 37,369 a36 15,714 (a36 21,655)
2 1655 128 (1,527) a36 52,491 a36 16,143 (a36 36,348)
quantities show different shapes when using the option approach as shown in Figure 4.5.
It shows that optimal order quantities are increasing in a nonlinear fashion to some point
and then decreasing as the volatility of demand (?r) is increased with the fill rate greater
than 50%. Otherwise, optimal order quantities are decreasing in a nonlinear fashion as the
volatility of demand (?r) is increased with the fill rate less than or equal to 50%.
The traditional approach gives negative order quantities, which are not applicable in
practice as shown in Figure 4.4. The optimal order quantities show a different pattern when
we apply the option approach as the volatility of demand (?r) increase as shown in Figure
4.5. In the traditional approach with high volatility of demand (?r), the decision-maker can
increase the order quantity beyond an acceptable amount. Consequently, the traditional
approach does not address risk in an economically meaningful way under high uncertainty
because it provides information of optimal order quantity as negative or beyond acceptable
amount. The option approach shows that optimal order quantities are positive and have
upper limits even in a highly uncertain situation.
70
Optimal Fill rate and Order Quantity
When we attain the optimal fill rate with data for an illustrative example, the optimal
fill rate is determined by constant parameters. The optimal fill rate is the same, 61.54%, in
this numerical example for both the approaches using Eq. (4.3) and Eq. (4.14). We check
the optimal order quantities for both approaches as the volatility of demand (?r) increases.
We change the volatility of demand (?r) within some intervals for both the approaches under
the optimal fill rate (61.54%). Table 4.2 shows the comparison of the order quantities and
total costs between the two approaches. As shown in Table 4.2, the optimal order quantities
of the traditional approach are increased as the volatility of demand (?r) increases with the
optimal fill rate. However, the optimal order quantities represent a different pattern when
using the option approach. It shows that the pattern of optimal order quantities is definitely
different from the traditional approach when the volatility of demand (?r) increases with the
same optimal fill rate. In the option approach, the probability density function of demand
under high volatility shows a more skewed shape towards the side of small demands. The
option approach suggests us to reduce order quantities in a highly volatile demand situation
unlike the traditional approach. The difference in optimal order quantities of both the
approaches is not significant at a low volatility of demand (?r) but the difference can be
significant at a high volatility of demand (?r). Table 4.2 shows us the difference of total
costs is a3636,348 when the volatility of demand (?r) is equal to 2.
4.6 Conclusion
Our research presents an option approach to make a decision about the order quantity.
This approach incorporates the economic principles of the asset pricing models, such as the
Black-Scholes option-pricing model, to replace the expected total cost minimization sense
of the traditional approach. Asset-pricing models have been incorporated into many of the
other investing and financing pricing decisions of the firms, and we have shown that an
option-pricing approach looks promising for determining the inventory decisions as well.
71
This study presents an option-pricing framework to decide the order quantity under
demand uncertainty during non-negligible lead-time. First, we establish an optimal condi-
tion for required order quantity when a firm has an desirable fill rate. Second, we develop
a closed-form solution for optimal order quantity to minimize the total cost. Moreover, we
compare the results between the traditional and option approaches with different values
of volatility of the demand (?r). The comparison could lead to several inventory decision
implications and shows the advantages (i.e., total cost reduction) of an inventory decision
with a real option framework. The option approach addresses risk in an economically mean-
ingful way. Specifically, it does not make any negative order quantity when the fill rate is
less than 50% with a high volatility of demand (?r) but the traditional approach gives us a
negative order quantity.
The usefulness of the option approach relies on the identification of financially traded
assets that can be used to forecast the customer demand, allowing for the specification of
payoff functions for the inventory problem. Another requirement for the application of the
option approach is the willingness of decision-makers to adapt option-pricing models to spe-
cific inventory situations. If these requirements are met, there should be more opportunities
for successfully applying the option approach to inventory decisions.
This work can be extended to various studies for inventory decisions in several ways. A
useful extension would be to apply this approach to a multi-period problem. A multi-period
problem is more likely to be applicable to practitioners. Another extension we suggest is
to use a time-dependent backorder rate function and not a deterministic backorder rate (?)
as used in this study. For example, the backorder rate is expressed as the function of the
magnitude of the shortages (Lodree 2007).
72
Chapter 5
Coordinating a Two-Stage Supply Chain Based On Option Contract
Abstract
Manufacturers and retailers are implementing option contracts to improve overall supply
chain profits and product availability. This research considers a standard newsvendor prob-
lem with price dependent stochastic demand in a single manufacturer-retailer channel and
shows the expected profits that each party receives under an option contract designed to
eliminate the double marginalization problem. We derive closed form solutions for the ap-
propriate option prices by the manufacturer as an incentive for the retailer to make optimal
pricing and order quantity decisions for coordinating the channel. The option contract not
only coordinates the supply chain; it also can divide the profit between the manufacturer
and the retailer. The findings of this research are illustrated in a numerical example for a
normal demand distribution.
5.1 Introduction
Recent years have seen growing interest in the area of supply chain coordination. One of
the features of supply chain is the existence of the multiple decision-makers. Optimal supply
chain performance requires the execution of a precise set of actions. Unfortunately, those
actions are not always in the best interest of all the members of the supply chain. Actions
taken by the two parties in the supply chain often result in profits that are lower than what
could be achieved if the supply chain were to coordinate its actions with a common objective
of maximizing supply chain profits. We consider a product whose demand is significantly
affected by the retail price. And the retailer decides its price based on its margin. The
retailer?s margin is only a fraction of the supply chain margin, which could lead to a retail
73
price that is higher than the optimal retail price and an actual order quantity that is lower
than the optimal amount for the supply chain. To improve overall profits, the manufacturer
designs a contract that encourages the retailer to purchase more and increase the level of
product availability. This requires the manufacturer to share in some of the retailer?s risk
of the demand uncertainty.
We suggest an option contract to hedge the risk of the demand uncertainty. This can
improve the product availability before the season and maximize the overall supply chain
profits at the end of the season through coordination of the option prices and the order
quantity. One of the most important benefits gained from trading in the option contract is
that the retailer can order a wide range of items with a relatively small initial investment
(e.g. option premium). When a retailer assumes a demand position in the future, he or she
locks in a price for the commodity from the manufacturer. This fixed price is the option
strike price at which the contract is bought or sold. Subsequently, as the demand for the
commodity rises or falls, the option strike price follows suit, making or losing money. The
option contract provides protection for each party (e.g.,manufacturer and retailer) against
dangerous demand swings.
We use a newsvendor model to illustrate the benefits of an option contract in the
supply chain. The newsvendor model is not complex, but it is sufficiently rich to offer
many implications for the manufacturer and the retailer in supply chain coordination. In
the newsvendor model the retailer orders a single product from the manufacturer well in
advance of the selling season with a stochastic demand. The manufacturer produces the
item after receiving the retailer?s order and delivers their production to the retailer at the
start of the selling season. The retailer has no additional replenishment opportunity. How
much the retailer chooses to order depends on the terms of the trade, i.e., the contract,
between the retailer and the manufacturer. Our model allows the retailer to choose his
discounted retail price to increase the product demand. Coordination is more complex in
74
Table 5.1: A summary of the related literature.
Contract Literature
Buyback Padmanabhan and Png (1997), Emmons and Gilbert (1998),
Donohue (2000), Taylor (2001),
Lee, Padmanabhan, Taylor and Wang (2000)
Revenue Mortimer (2000), Dana and Spier (2001),
sharing Gerchak, Cho and Ray (2001)
Quantity Pasternack (1985), Eppen and Iyer (1997), Tsay (1999),
flexibility Cachon and Lariviere (2001), Lariviere (2002),
Plambeck and Taylor (2002)
Rebates Krishan, Kapuscinski and Butz (2001), Taylor (2002)
Quantity Jucker and Rosenblatt (1985), Pantumsinchai and Knowles (1991),
discounts Khouja (1996), Lin and Kroll (1997),
Chen, Federgruen, and Zheng (2001), Weng (1995)
Option Erkoc and Wu (2002), Barnes-Schuster, Bassok and Anupindi (2002),
Cheng, Ettl, Lin, Schwarz and Yao (2003), Kleindorfer and Wu (2003),
Kamrad and Siddique (2004),Martnez-de-Albenz and Simchi-Levi (2003),
Ozer and Wei (2004), Burnetas and Ritchken (2005)
this setting because the incentives provided to align one action (e.g., order quantity) may
cause distortions with the other action (e.g., option price).
5.2 Literature Review
Several different types of contract are shown to coordinate the supply chain and divide
its profit; buy back, revenue sharing, quantity flexibility, sales rebate, and quantity discount
contracts. In recent years, researchers have examined option contracts to help coordinate a
supply chain. Table 5.1 summarizes literature about the contract policies in supply chain
coordination.
Taylor (2000) incorporates a buy back contract with a sales rebate contract to coordi-
nate the newsvendor with an effort dependent demand. Donohue (2000) studies buy back
contracts in a model with multiple production opportunities and improving the demand
forecasts. For the literature related to revenue sharing contracts, Mortimer (2000) provides
75
a detailed econometric study of the impact of revenue sharing contracts in the video rental
industry. Dana and Spier (2001) study these contracts in the context of a perfectly com-
petitive retail market. Gerchak, Cho and Ray (2001) consider a video retailer that decides
how many tapes to purchase and how long to keep them.
In addition, there are a number of papers about quantity- flexibility contracts. Cachon and
Lariviere (2001) and Lariviere (2002) study the interaction between quantity flexibility con-
tracts and forecast sharing. Plambeck and Taylor (2002) study quantity flexibility contracts
with more than one downstream firm and ex-post renegotiation. The sales rebate contract
is studied in Taylor (2000) and Krishan, Kapuscinski and Butz (2001). In Taylor (2000)
effort is chosen simultaneously with the order quantity, whereas Krishan, Kapuscinski and
Butz (2001) focus on the case in which the retailer chooses an order quantity, a signal of
demand is observed and then effort is exerted.
The literature about the option contracts is studied by Erkoc and Wu (2002). They
propose two channel coordination contracts, and discuss how such contracts can be tai-
lored for situations where the supplier has the option of not complying with the contract,
and when the buyer?s demand information is only partially updated during the supplier?s
capacity lead-time. Cheng, Ettl, Lin, Schwarz and Yao (2003) develop an option model
to quantify and price a flexible supply contract, by which the buyer (a manufacturer), in
addition to a committed order quantity, can purchase option contracts and decide whether
or not to exercise them after demand is realized. However, unlike our study, they assume
the retail price is constant. Martnez-de-Albenz and Simchi-Levi (2003) consider the impact
of a supply option contract on the newsvendor and derive an optimal replenishment policy
for a portfolio consisting of long-term and option contracts. Burnetas and Ritchken (2005)
investigate the role of option contracts in a supply chain when the demand curve is sloping
downward. They show that the introduction of the option contracts causes the wholesale
price to increase and the volatility of the retail price to decrease.
76
Before
theseason
After
theseason
v
e
v
0
p
Q
Q
v
0
Q
[Q?X]
+
v
e
X
Manufacturer Retailer
Figure 5.1: Transactions between parties.
Our study is to consider contracting arrangements in a supply chain for a newsvendor
environment. Our approach, however, differs in that we do not assume the retail price
and the demand distribution as given; rather, we assume demand distribution is influenced
by the discount of the retail price, because, in practice, the retailer usually has a right to
discount the retail price to some extent and the demand is relative to the retail price.
The rest of this study is organized as follows. In section 5.3, we describe the channel
structure including the meanings of notations and the model assumptions for this study.
Section 5.4 describes the types of supply chain decisions as closed form solutions (i.e.,
optimal option purchase and strike prices, optimal order quantity) for the manufacturer
and the retailer respectively. In section 5.5, we demonstrate the results with numerical
examples. Finally, we conclude in Section 5.6 with possible extensions.
77
Table 5.2: List of notations.
po : Original selling price.
p : Discounted selling price.
h: Holding cost.
c: Production cost.
X : r.v. of Demand.
Q : Order quantity.
x : Value of demand.
f(x): pdf of X with discounted retail price
f0(x): pdf of X with initial retail price
F(x): cdf of X with discounted retail price
F0(x): cdf of X with initial retail price
v0: Option Purchase price (e.g.,premium).
ve: Option Strike price.
?: Proportion of the channel profit got by retailer
Vc: Profit in the centralized system.
Vr: Profit of retailer in the decentralized system.
Vm: Profit of manufacturer in the decentralized system.
5.3 Channel Structure with an Option Contract
We consider a single period two-stage supply chain option contract between a manu-
facturer and a retailer who sells a short life-cycle product to the customers. The retailer
can discount a retail price in an appropriate range. The interaction takes place between
the retailer and the manufacturer in a principal agent setting where the manufacturer is
the leader. Before the season the manufacturer provides the option contract consisting of
the option purchase and the exercise price, then the retailer determines the retail price and
the order quantity. Under the forced compliance environment, the manufacturer must build
enough quantity to satisfy the final order up to the number of the order quantity.
After the demand is realized the retailer exercises the option accordingly. So in this
setting there is no remainder, for the retailer has a right to return leftovers by return policy
as shown in Figure 5.1. In this study it is also assumed that the unsatisfied demand becomes
lost sales and there is no penalty cost for lost sales. Throughout the study, the parameters
and the decision variables of the model are given in Table 5.2.
78
A discounted retail price could affect the expected demand and the variance of the de-
mand. Generally, with discounting of the retail price, the expected demand and the variance
will increase. We make the following assumptions concerning the model parameters:
c < v0 +ve
c < p < p0 (5.1)
These conditions rule out the cases where the contract cannot establish between the two
parties. The first condition ensures that the manufacturers marginal profit is nonnegative.
The second condition regulates the limits of the discounted retail price.
We compare two channel structures. In the Centralized System (CS), we assume that
there is one decision-maker who controls the channel and hence makes decisions to optimize
the total system profits. Observe that under this scenario, the option and the exercise prices
play no role in determining the system profits. The only appropriate decision variables are
the order quantity and the retail price decisions for the single period at the beginning of
the season. The optimal solution to the CS is called as the first-best solution.
In the Decentralized System (DS), the retailer and the manufacturer play in a principal
agent setting where the manufacturer is the leader and the retailer, the follower. This is
reasonable whenever the manufacturer has more influence in the channel than the retailer.
In this situation, for a given set of option prices announced by the manufacturer, the retailer
places orders and determines the appropriate discounted retail price in the single period,
which maximizes their expected profits. These orders act as implicit demand functions for
the manufacturer. The CS is used as a benchmark case to investigate whether there exists
an appropriate decentralization mechanism (through prices and/or quantities) such that
the DS will achieve the first-best solution.
79
To establish a benchmark for the performance comparison, we analyze the centralized
system and the details are in Appendix D.1. In CS, the expected profit function of the
system and optimal order quantity are expressed as follows:
VC(Q) = p?E[[X ?Q]]?h?E[[Q?X]+]?cQ
= (p?c)Q?(p+h)?
integraldisplay Q
0
F0(x)dx
where [x]+ = max{x,0},[x?y] = min{x,y}
Q? = F?10
parenleftbiggp?c
p+h
parenrightbigg
(5.2)
This reduces the problem to an optimization problem over the single variable p . Using
the single variable optimal theory, we can find the optimal retail price p? or demonstrate
that VC(Q?,p?) is uni-modal or concave. The value for p? can be found by using a one-
dimensional search algorithm within a closed interval [c,p0] in this research.
To analyze the DS, first we need to develop the expressions for the expected profits of
each system. In our setting, the retailer has a right to return the leftovers to the manufac-
turer without penalty. For clarity of exposition, we derive the individual profit function of
the manufacturer and the retailer respectively in the DS as follows:
Vm(v0,ve) = (v0 ?c)Q+ve ?E[X ?Q]]?h?E[[Q?X]+]
= (v0 +ve ?c)Q?(ve +h)?
integraldisplay Q
0
F(x)dx
Vr(Q,p) = (p?ve)?E[[X ?Q]]?v0Q
= (p?ve ?v0)Q?(p?ve)?
integraldisplay Q
0
F(x)dx
(5.3)
80
5.4 Decisions for a Supply Chain Coordination
We want to provide closed form solutions for the option prices, the order quantity,
and the discounted retail price. We consider that demand distribution is influenced by the
retail price. It demonstrates that the options redistribute the risk between two parties in
shifting part of the retailer?s risk due to demand uncertainty to the manufacturer. The
manufacturer is compensated by the additional revenue obtained from the option contract.
Any share of this profit improvement can be represented by an option contract through
a suitable choice of the contract parameters. Under certain conditions, this profit sharing
mechanism with an option contract will achieve the channel coordination by eliminating
the double marginalization problem in the supply chain.
5.4.1 The Manufacturer?s Optimal Option Pricing Decisions
In this study we provide an option contract to coordinate the decentralized supply chain
with stochastic demand. By setting proper contract parameters, channel coordination is
achieved. Moreover the manufacturer who shares the risk of demand uncertainty can be
compensated by establishing favorable contract parameters in order to take a larger portion
of the channel profit. The contractual arrangements we designed between the manufacturer
and the retailer allow the decentralized supply chain to perform as well as a centralized
one. In order to eliminate the double marginalization problem and to induce the perfect
coordination, the relationship between the profit functions is established as follows:
Vr = ?VC
Vm = (1??)VC
(5.4)
81
where ? is a retailer?s portion of the channel profit. Through derivations of Eq.(5.4) in
Appendix D.3, the manufacturer should provide the explicit contract parameters as follows:
v0 = ?(h+c)
ve = (1??)p? ??h
(5.5)
The Eq.(5.5) represents the optimal option purchase and the strike price for the man-
ufacturer. The result shows that the manufacturer?s optimal option purchase price is not
dependent on retail price while the optimal strike price is dictated by the optimal discounted
retail price.
Furthermore, the result of Eq.(5.5) can support a division of the channel profit between
the manufacturer and the retailer by varying the contract parameters (i.e., the option
purchase or the strike price).
? = p
? ?ve
p? +h
? = v0h+c
(5.6)
5.4.2 The Retailer?s Optimal Order Quantity and Discount Retail price
We attain the retailer?s optimal order quantity and the discount retail price by solving
the system optimal problem. Sometimes the retailer insists on no discounting in his retail
price so as to keep the initial retail price p0 to get maximum margin from each sale. With a
contractual arrangement for coordination, the retailer could agree to a discount retail price
to increase expected demand for the product to improve the profitability of the overall
82
supply chain. In that case, we assume that the demand (X) is a random variable, which
depends on the discount retail price p. The expected value of X is a concave decreasing
function in p where c < p < p0.
We can express the expected demand (?) with discount as ?0 +?0?(p0 ?p)? where ?
and ? are empirically determined constants which indicate the effectiveness of the discount.
A condition of ? = 0 indicates that the expected demand is independent of the retail price.
For ? > 0, the larger the value of ?, the more effective is the discounting. Three cases can
be identified for the variance of X:
1. A retail price discount increases the mean demand but does not change the demand
variance. Thus ?2 = ?20 : This case will be referred to as the fixed variance case (FVC).
2. A retail price discount increases both the mean and the variance of demand in a
proportional fashion. Thus, the coefficient of variation is a constant given by CV = CV0 :
This case will be referred to as the fixed coefficient of variation case (FCVC). The derivation
is in Appendix D.4.1.
3. A retail price discount increases the variance of demand at a faster rate than it
increases the mean demand. Thus, CV is an increasing function of the (p0 ?p) : This case
will be referred to as the increasing coefficient of variation case (ICVC). The ICVC reflects
the stronger effects of other factors on the demand at a big discount. For example, the
effects of unanticipated economic conditions or competitor?s actions may lead to very large
deviations from the expected demand at a big discount. The derivation is in Appendix D.4.2.
83
Fixed Variance Case (FVC)
We can express the expected demand and variance with discounted retail price as
follows:
? = ?0 +?0?(p0 ?p)?
?2 = ?20
where ?,? > 0 (5.7)
The profit function and the optimal order quantity of a centralized system can be
expressed as follows:
VC(Q) = (p?c)Q?(p+h)?
integraldisplay Q
0
F(x)dx
= (p?c)Q?(p+h)?
integraldisplay Q
0
F0(x??0?(p0 ?p)?)dx
Q? = F?10 (p?cp+h)+?0?(p0 ?p)?
(5.8)
For example, we apply a uniform distribution of demand on a centralized system with
FVC and get an optimal order quantity solution as a consistent form with Eq.(5.8) above.
VC(Q) = (p?c)Q?(p+h)?
integraldisplay Q
0
x??0?(p0 ?p)? ?a
b?a dx
?VC(Q)
?Q = (p?c)?
p+h
b?a(Q??0?(p0 ?p)
? ?a) = 0
Q? = p?cp+h(b?a)+a+?0?(p0 ?p)?
84
= F?10 (p?cp+h)+?0?(p0 ?p)?
= Q?0 +?0?(p0 ?p)?
(5.9)
5.5 An Illustrative Example
In this section, the numerical example shows a mechanism of supply chain coordination
with an option contract. The following example data is used for the analysis.
c = 55,h = 25,p0 = 95,? = 0.125,?0 = 5,000,?0 = 1,000,? = 1.5,? = 0 (5.10)
The illustration will consider a normal distribution of the demand. Based on the results
shown in the following figures, we observe the following insights.
First, we consider the case that each party agrees to a total option price (i.e., v0+ve =
90) and the retailer could decide his order quantity without discounting his retail price.
Observation 5.1 According to Figure 5.2, optimal order quantity to maximize the supply
chain profit is independent of the option prices between parties.
Figure 5.2 shows us a consistency between Eq.(5.8), of the optimal order quantity and
the result of this numerical analysis. To maximize the overall supply chain profit, the
retailer should decide his order quantity based on Eq.(5.8). In these arbitrary combinations
of option purchase price and strike price, the retailer?s portion of the channel profit (?)
fluctuates between 1.2% to 13.9% even if the retailer does order the optimal order quantity
for coordination. Therefore, the next question is what combination of option purchase price
and strike price can lead to guarantee the negotiated retailer?s portion of the channel profit
(?).
85
$(120,000)
$(70,000)
$(20,000)
$30,000
$80,000
$130,000
$180,000
2570 3570 4570 5570 6570
Profit
Vm(90,0)
Vm(0,90)
Vm(45,45)
Vc
Vr(0,90)
Vr(45,45)
Vr(90,0)
Figure 5.2: Profits in the arbitrary option prices.
0.012
0.055
0.097
0.125
0.139
0.000
0.100
0.200
1 2 3 4 5
Case
(vo,ve)
Gamma
(90,0) (60,30) (30,60) (10,80) (0,90)
Figure 5.3: ? in the arbitrary option prices.
86
$-
$20,000
$40,000
$60,000
$80,000
$100,000
$120,000
$140,000
$160,000
$180,000
2570 3570 4570 5570 6570
Profit
Vc
)Vm(10,80
)Vr(10,80
Figure 5.4: Profits in the optimal option price.
Profit
$(50,000)
$50,000
$150,000
$250,000
$350,000
$450,000
$550,000
1 2 3 4 5
Case
(vo*,ve*)
(p,Q*)
Vc
Vm
Vr
(10,80)
(95,4569)
(10,76)
(90,14836)
(10,71)
(85,16282)
(10,63)
(75,17813)
(10,67)
(80,17178)
Figure 5.5: Profits in the discount retail prices in FVC.
87
Observation 5.2 According to Figure 5.3 , optimal combination of the option purchase
price and the strike price (i.e., v0 = 10,ve = 80) from Eq.(5.5) guarantees the negotiated
retailer?s portion of the channel profit (0.125) when the retailer does order the optimal order
quantity for coordination.
The total option price (i.e., v0 +ve = 90) regulates a division of the channel profit between
the manufacturer and the retailer. In this case, we get 12.5% of the retailer?s portion of the
channel profit (?) based on the Eq.(5.6). On the other hand, when we know the retailer?s
portion of the channel profit (?), we can get an optimal combination of the option purchase
price and the strike price (i.e., v0 = 10,ve = 80) with Eq.(5.5) to coordinate the supply
chain.
Observation 5.3 According to Figure 5.4, with the optimal combination of the option pur-
chase price and the strike price (i.e., v0 = 10,ve = 80), the negotiated retailer?s portion of
the channel profit (?) is protected even if the retailer does not order the optimal order
quantity for coordination.
Second, we consider the discount retail price with FVC (Fixed Variance Case). The
retailer decreases a retail price from the initial retail price p0 to a discount price p . A
retail price discount increases the mean demand but does not change the demand variance
in FVC. A discount retail price affects the optimal order quantity and the profits of each
party.
Observation 5.4 According to Figure 5.5, we can get an optimal state of decision variables
to maximize the profit of the centralized supply chain. In this example the optimal price is
decided at the point of a365 discount from the initial retail price of a3695 with optimal order
quantity (14,836) and v0 = 10,ve = 76.
It is rather difficult to get a closed form solution for the optimal discount retail price
because it is complicated to derive analytically. With simulations and other tools, we can
88
attain the solutions. We show one solution with a numerical example. The result could be
varied with changes of parameters (i.e., ?,?). We can verify our derivations of Appendix
D.4 for the cases of FCVC and ICVC.
5.6 Conclusion
We develop an option contract to hedge the risk of the demand uncertainty. This
option contract leads to an improved product availability by hedging the risk in the supply
chain and to maximize the overall supply chain profits at the end of the season through
coordination of the option prices and the order quantity. We consider the discount of retail
price. In addition, we examine that the discounted retail price affects the expected product
demand and the variance in several cases (i.e., FVC, FCVC, ICVC).
In this study we investigate the role of the option contract in a two-stage supply chain
system. We use a single period model with and without discounting the retail price and
the stochastic demand. We derive a closed form solution for the option purchase and strike
price for the manufacturer and the optimal order quantity for the retailer. Furthermore,
the analysis shows that the manufacturer?s optimal option purchase price is independent of
the retail price while the optimal strike price is dictated by the optimal discounted retail
price.
Several extensions of our research could be investigated. Current models are too de-
pendent on single shot contracting. Most supply chain interactions occur over long periods
of time with many opportunities to renegotiate or to interact with the spot markets. More
research is needed on how multiple suppliers compete for the affection of the multiple re-
tailers, i.e., additional emphasis is needed on many-to-one or many-to-many supply chain
structures.
89
Chapter 6
Conclusion
This research developed several analytical models for typical supply chain situations
to help inventory decision-makers who need mathematical models to grasp the big picture
of supply chain inventory problems before making executive decisions. Additionally, we
derived closed form solutions for each model and found several managerial insights from
our models through sensitivity analysis of numerical examples.
First, we developed a decision model considering customer impatience with stochastic
demand and time-sensitive shortages. While the majority of the research literature con-
sidered time-dependent backlogging within the context of continuous review models with
deterministic demand, this research studied time-dependent partial backlogging in the single
period inventory problem with stochastic demand. We used concepts from utility theory to
formally classify customers in terms of their willingness to wait for the supplier to replenish
shortages. We conducted sensitivity analyses to determine the most and least opportune
conditions for distinguishing between customer risk-behaviors. Our results suggested that
the expected value of risk profile information (EVRPI) is most significant when the dif-
ference between the unit lost sales cost cLS and the unit backorder cost cB is large. Our
results also indicated that EVRPI increases as the lost sales threshold M increases.
Second, we established an additional model to optimize backorder lead-time (response
time) in a two-stage system with time-dependent partial backlogging and stochastic demand.
We considered backorder cost as a function of response time while other literature regarded
the backorder cost as a constant. A representative expected total cost function was derived
and the closed form optimal solution was determined for a demand distribution. Our result
showed that significant cost savings can be achieved by implementing the derived optimal
solution as opposed to the prior model solution.
90
Third, we developed an inventory decision model applying an option-pricing framework
(e.g., straddle) for determining order quantity under the situation of partial backlogging and
uncertain demand. We compared the results between the traditional and option approaches
with a numerical example. The result showed the advantages (i.e., total cost reduction) of
an inventory decision with the real option framework under a highly uncertain demand
situation. Specifically, the option approach provided decision-makers with positive optimal
order quantity decisions even under the situation of high volatility of demand (?r) and low
fill rate while the traditional approach gave them a negative optimal order quantity decision.
We have shown that an option- pricing approach looks promising for determining inventory
decisions under a highly uncertain demand situation as well.
Finally, we implemented an option contract to hedge the risk of the demand uncer-
tainty. Our result showed that the option contract could lead to improvement of product
availability by hedging the risk in the supply chain and maximizing overall supply chain
profits at the end of the season through coordination of option prices and order quantity.
We considered the discount of retail price. In addition, we examined the case where the
discounted retail price affects the expected product demand and the variance in several
cases (i.e., FVC, FCVC, ICVC). We derived closed form solutions of the appropriate option
prices for the manufacturer as an incentive for the retailer to establish optimal pricing and
order quantity decisions for coordinating the channel. Furthermore, the result showed that
the manufacturer?s optimal option purchase price is independent of retail price while the
optimal strike price is dictated by the optimal discounted retail price.
This research gave decision-makers insights into how to implement the situation of
demand uncertainty and shortage into a mathematical model in a two-stage supply chain
and showed them what differences these proposed analytical models make as opposed to
the traditional models. Even though each analytical model is simple but each provided an
effective overall view of the supply chain system by abstracting the features of a supply
chain system as a set of parameterized functions. We implemented time-sensitive shortages
91
into an inventory model under emergency replenishment. In an effort to characterize di-
verse customer responses to shortages, we explored several types of decreasing backorder
rate functions. In dealing with demand uncertainty, we adopted an option-pricing model.
This showed that the tool of financial economics can be used to help make inventory stock-
ing decisions. This research also developed an option contract in a newsvendor problem
with price dependent stochastic demand and showed that the option contract could help
coordination of a supply chain through improving the product availability and the overall
supply chain profits.
This study can be extended in several ways. A useful extension would be to apply
this approach to a multi-period problem. A multi-period problem is more likely to be
applicable to real problems. Current models are too dependent on single shot contracting.
Most supply chain interactions occur over long periods of time with many opportunities to
renegotiate or to interact with the spot markets. More research is needed on how multiple
suppliers compete for the affection of the multiple retailers (e.g., many-to-one or many-to-
many structure).
92
Bibliography
[1] Abad, P. 1996. Optimal pricing and lot-sizing under conditions of perishability and
partial backordering. Management Science 42 (8), 1093?1104.
[2] Abad, P. 2001. Optimal price and order size for a reseller under partial backlogging.
Computers and Operations Research28 (1), 53?65.
[3] Agrawal, V., Seshadri, S. 2000. Impact of Uncertainty and Risk Aversion on Price and
Order Quantity in the News vendor Problem. Manufacturing and Service Operations
Management 4 (October), 410?423.
[4] Barnes-Schuster, D., Bassok Y., Anupindi R. 2002. Coordination and flexibility in
supply contracts with options. Manufacturing and Service Operations Management 4,
171?207.
[5] Becker, M. 1994. The Reorder Point as an Option on Future Sales. Paper Presented
at the International Symposium on Cash, Treasury, and Working Capital Management
(October). St. Louis, MO.
[6] Burnetas, A., Ritchken 2005. Option Pricing with Downward Sloping Demand Curves:
The Case of Supply Chain Options. Management Science 51, 566?580.
[7] Ben-Daya,M., Raouf, A. 1994. Inventory models involving lead time as decision vari-
able. Journal of the Operational Research Society 45, 579?582.
[8] Berling, P., Rosling, K. 2005. The Effect of Financial Risks on Inventory Policy. Man-
agement Science 51, 1804?1815.
[9] Birge, J.R., Zhang, R.Q. 1999. Risk-nerutral Option Pricing Methods for Adjusting
Constrained Cash Flows. The Engineering Economist 44, 36?49.
[10] Brennan, M. J., Eduardo, S. S. 1983. Optimal Financial Policy and Firm Valuation.
Unpublished Manuscript, University of British Columbia.
[11] Cachon, G. P., Lariviere, M. A. 2001. Turning the supply chain into a revenue chain.
Harvard Business Review 79(3), 20?21.
[12] Chang, H.J., Dye, C.Y. 1999. An EOQ model for deteriorating items with time varying
demand and partial backlogging. Journal of the Operational Research Society 50 (11),
1176?1182.
93
[13] Chen, F., Federgruen, A., Zheng, Y. S. 2001. Coordination mechanisms for a distri-
bution system with one supplier and multiple retailers. Management Science 47(5),
693?708.
[14] Chen, F. 2003. Information sharing and supply chain coordination. Supply chain man-
agement: Design, coordination and operation. Amsterdam: North Holland, 341?421.
[15] Chopra, S., Meindl, P. 2007. Supply Chain Management: Strategy, Planning, and Op-
eration (3rd ed.),Upper Saddle River, New Jersey: Pearson Education, Inc.
[16] Chung, K.H. 1990. Inventory decision under demand uncertainty : a contingent claims
approach. Financial Review, 623?40.
[17] Cox, J.C., Ross, S.A., Rubinstein, M. 1979. Option pricing : A simplified approach.
Journal of Financial Economics 7, 229?263.
[18] Dana, J. D., Spier, K. E. 2001. Revenue sharing and vertical control in the video rental
industry. The Journal of Industrial Economics XLIX(3), 223?245.
[19] Dye, C.-Y. 2007. Joint pricing and ordering policy for a deteriorating inventory with
partial backlogging. Omega: International Journal of Management Science 35 (2),
184?189.
[20] Dye, C.Y. Chang, H.J., Teng, J.T. 2006. A deteriorating inventory model with time-
varying demand and shortage-dependent partial backlogging. European Journal of Op-
erational Research 172 (2), 417?429.
[21] Dixit, A.K., Pindyck, R.S. 1994. Investment under Uncertainty Princeton University
Press, Princeton, NJ.
[22] Donohue, K. L. 2000. Efficient supply contracts for fashion goods with forecast updating
and two production models. Management Science 46(11), 1397?1411.
[23] Emmons, H., Gilbert, S. M. 1998. Note: The role of returns policies in pricing and
inventory decisions for catalogue goods. Management Science 44(2), 276?283.
[24] Eppen, G., Iyer, A. 1997. Backup agreements in fashion buying - the value of upstream
flexibility. Management Science 43(11), 1469?84.
[25] Erkoc, M., Wu, S. D. 2002. Managing high-tech capacity expansion via reservation
contracts. Lehigh University working paper.
[26] Gallego, G., Moon, I. 1993. The distribution free newsboy problem: review and exten-
sions. Journal of the Operational Research Society 44, 825?834.
94
[27] Gerchak, Y., Cho, R., Ray, S. 2001. Coordination and dynamic shelf-space management
of video movie rentals. University of Waterloo working paper.
[28] Hull, J. C. 2002. Fundamentals of Futures and Options Markets, Prentice-Hall, Upper
Saddle River, NewJersey.
[29] Jeter, M.W. 1986. Mathematical Programming: An Introduction to Optimization, Mar-
cel Dekker, New York, NY.
[30] Jucker, J. V., Rosenblatt, M. J. 1985. Single-period inventory models with demand
uncertainty and quantity discounts: Behavioral implications and a new solution pro-
cedure. Naval Research Logistics Quarterly 32(4), 537?550.
[31] Kamrad, B., Siddique, A. 2004. Supply contracts, profit sharing, switching, and reac-
tion options. Management Science 50, 64?82.
[32] Khouja, M. 1996. The Newsboy Problem with multiple discount offered by suppliers
and retailers. Decision Sciences 27(3), 589?599.
[33] Khouja, M. 1996. A note on the newsboy problem with an emergency supply option.
Journal of the Operational Research Society 47, 1530?1534.
[34] Khouja, M. 1999. The single-period (news-vendor) problem: Literature review and
suggestions for future research. Omega: International Journal of Management Science
27, 537?553.
[35] Kim, Y.H., Chung, K.H. 1989. Inventory Management Under Uncertainty: A Financial
Theory for the Transactions Motive. Managerial and Decision Economics (December),
291?298.
[36] Kleindorfer, P. R., Wu, D. J. 2003. Integrating long-and short-term contracting via
business-to-business exchanges for capital intensive industries. Management Science49,
1597?1615.
[37] Krishnan, H., Kapuscinski, R., Butz, D. 2001. Coordinating contracts for decentralized
supply chains with retailer promotional effort. University of Michigan working paper.
[38] Lariviere, M. 2002. Inducing forecast revelation through restricted returns. Northwest-
ern University working paper.
[39] Lee, H., Padmanabhan, V., Taylor, T., Whang, S. 2000. Price protection in the personal
computer industry. Management Science 46(4), 467?82.
[40] Liao,C.J., Shyu,C.H. 1991. An analytical determination of lead time with normal de-
mand. International Journal of Operations Production Management 11, 72?78.
95
[41] Lin, C., Kroll, D. E. 1997. The single-item newsboy problem with dual performance
measures and quantity discounts. European Journal of Operational Research 100(3),
562?565.
[42] Lodree, E., Klein, C., Jang, W. 2004. Minimizing customer response time in a two-
stage supply chain system with variable lead time and stochastic demand. International
Journal of Production Research 42(11), 2263?2278.
[43] Lodree, E.J. 2007. Advanced supply chain planning with mixtures of backorders, lost
sales, and lost contract. European Journal of Operational Research 181(1), 168?183.
[44] Lodree, E., Kim, Y., Jang, W. 2008. Time and quantity dependent waiting costs in a
newsvendor problem with backlogged shortages. Mathematical and Computer Modeling
47, 60?71.
[45] Martinez-de-Albeniz, V., Simchi-Levi, D. 2003. Competition in the Supply Option
Market. MIT working paper.
[46] Morris, J.R., Chang, J.S.K. 1991. Inventory Policies in the CAPM Valuation Frame-
work. Advances in Working Capital Management 227?239.
[47] Mortimer, J. H. 2000. The effects of revenue-sharing contracts on welfare in vertically
separated markets: evidence from the video rental industry. University of California at
Los Angeles working paper.
[48] Ouyang,L.Y., Yeh,N.C., Wu,K.S. 1996. Mixture inventory model with backorders and
lost sales for variable lead time. Journal of the Operational Research Society 47, 829?
832.
[49] Ozer, O., Wei, W. 2004. Inventory Control with Limited Capacity and Advance De-
mand Information. Operations Research 52(6), 988 ?1000.
[50] mPadmanabhan, V., Png, I. P. L. 1997. Manufacturers returns policy and retail com-
petition. Marketing Science 16(1), 81?94.
[51] Pantumsinchai, P., Knowles, T.W. 1991. Standard container size discounts and the
single-period inventory problem. Decision Sciences 22(3), 612?619.
[52] Papachristos, S., Skouri, K. 2000. Optimal replenishment policy for deteriorating items
with time-varying demand and partial-exponential type - backlogging. Operations Re-
search Letters 27(4), 175?184.
[53] Papachristos, S., Skouri, K. 2003. An inventory model with deteriorating items, quan-
tity discount, pricing, and time-dependent partial backlogging. International Journal
of Production Economics 83(3), 247?256.
96
[54] Pasternack, B. A. 1985. Optimal pricing and return policies for perishable commodities.
Marketing Science 4(2), 166?176.
[55] Plambeck, E., Taylor, T. 2002. Implications of renegotiation for optimal contract flex-
ibility and investment. Stanford University working paper.
[56] San Jos?e, L.A., Sicilia, J., Garc??a-Lagyna, J. 2006. Analysis of an inventory system
with exponential partial backordering. International Journal of Production Economics
100, 76?86.
[57] Schlaffer, J. 2007. Wii shortage could cost Nintendo billions Retrieved December 19,
2008, from Blorge Web site: http://gamer.blorge.com
[58] Skouri, K., Papachristos, S. 2003. Four inventory models for deteriorating items with
time varying demand and partial backlogging: A cost comparison. Optimal Control
Applications and Methods 24, 6, 315?330.
[59] Stalk, G., Hout, T. 1990. Competing Against Time: How Time-based Competition Is
Reshaping Global Markets, Macmillan, New York, NY.
[60] Stowe, J.D., Gehr, A.K. Jr. 1991. A Finance-Theoretic Approach to the Basic Inventory
Decision. Advances in Working Capital Management, 213?225.
[61] Stowe, J.D., Tie, S. 1997. A Contingent-Claims Approach to the Inventory-Stocking
Decision. Financial Management 26(4), Winter.
[62] Taylor, T. A. 2001. Channel coordination under price protection, mid-life returns, and
end-of-life returns in dynamic markets. Management Science 47(9), 1220?1234.
[63] Taylor, T. A. 2002. Supply chain coordination under channel rebates with sales effort
effects. Management Science 48(8), 992?1007.
[64] Tsay, A. A., Nahmias, S., Agrawal, N. 1999. Modeling supply chain contracts: Quanti-
tative models of supply chain management, Boston: Kluwer Academic Publishers.
[65] Tsay, A. A. 1999. The quantity flexibility contract and supplier-customer incentives.
Management Science 45(10), 1339?1358.
[66] Wang, S.P. 2002. An inventory replenishment policy for deteriorating items with short-
ages and partial backlogging. Computers and Operations Research 29 (14), 2043?2051.
[67] Weng,Z.K. 1995. Channel coordination and quantity discounts. Management Science
41(9), 1509?1522.
[68] Winkler, R.L. 2003. An Introduction to Bayesian Inference and Decision (Second Edi-
tion), Probabilistic Publishing, Gainesville, FL.
97
[69] Wu, J.W., Tsai,H.Y. 2001. Mixture inventory model with back orders and lost sales
for variable lead time demand with the mixtures of normal distribution. International
Journal of Systems Science 32, 259?268.
[70] Zou ,Y-W., Lau, H-.S., Yang, S.-L. 2004. A finite horizon lot-sizing problem with
time-varying deterministic demand and waiting-time-dependent partial backlogging.
International Journal of Production Economics 91(2), 109?119.
98
Appendices
99
Appendix A
Modeling Customer Impatience in an Inventory Problem with Stochastic Demand and
Time-Sensitive Shortages
A.1 Proof of Theorem 2.1
TC(Q) given by Eq. (2.1) can be written as
TC(Q) = cOQ+cH
integraldisplay Q
0
(Q?x)f(x)dx+cLS
integraldisplay ?
Q
(x?Q)f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
(x?Q)?(x?Q)f(x)dx. (A.1)
The first and second order derivatives of TC(Q) are then
dTC(Q)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx
+ (cB ?cLS)
integraldisplay Q+M
Q
bracketleftbig??(x?Q)+(x?Q)?prime(x?Q)bracketrightbigf(x)dx (A.2)
d2TC(Q)
dQ2 = (cH +cLS)f(Q)+(cLS ?cB)f(Q)
+ (cLS ?cB)
integraldisplay Q+M
Q
bracketleftbig2?prime(x?Q)?(x?Q)?primeprime(x?Q)bracketrightbigf(x)dx. (A.3)
The first term in Eq.(A.3) is obviously nonnegative. Since cLS ? cB > 0 based on
Assumption 2.1, it follows that the second term is also nonnegative. Thus the third term is
nonnegative if and only if the integrand is nonnegative. That is, the optimality condition
holds if and only if
2?prime(x?Q)?(x?Q)?primeprime(x?Q) ? 0 for all x ? [Q,Q+M). (A.4)
A.2 Proof of Theorem 2.3
For illustration, we will show that a decision-maker is risk-seeking if and only if his back
order rate function is convex. The proofs of parts 1 and 2 of Theorem 2.3 are similar.
(?) By definitions 2.3 and 2.4, we have RPW(L) > 0 and EW[L]?CEW(L) > 0. Now
?(CEW(L)) is the convex hull of {t1,...,tn}, which can be expressed as ?(CEW(L)) =
100
p1?(t1) + ... + pn?(tn), where p1 + ??? + pn = 1. Since ?(y) is a decreasing function by
Property 2.1 and EW[L] ?CEW(L) > 0, it follows that ?(EW[L]) ??(CEW(L)) < 0, or
equivalently,
?(p1t1 +???+pntn) < p1?(t1)+???+pn?(tn). (A.5)
Since Eq. (A.5) holds, it follows that ?(y) is a convex function and ?primeprime(y) > 0 (see, for
example, Jeter 1994 page 217).
(?)
Since ?(y) is a convex function, it follows that Eq. (A.5) holds by Proposition 2.1
shown in Jeter 1994 (page 217). Since CEW(L) is the convex hull of {t1,...,tn} and
p1t1 + ??? + pntn = EW[L], Eq. (A.5) becomes ?(EW[L]) ? ?(CEW(L)) < 0. Based on
the last inequality, we know that EW[L] ? CEW(L) > 0 by Property 2.1. Therefore by
definitions 2.3 and 2.4, the latter inequality implies risk seeking.
A.3 Proof of Proposition 2.2
Since
d?(x?Q)
d(x?Q) =
braceleftbigg ?ae?a(x?Q) < 0, x ? [Q,Q+M)
0, x ? [Q+M,?)
Property 2.1 holds. Also,
d2?(x?Q)
d(x?Q)2 = a
2e?a(x?Q) > 0
shows that the third part of Theorem 2.3 holds. Now
lim
x?Q?0+
max
braceleftBig
e?a(x?Q),0
bracerightBig
= 1
shows that Property 2.2 holds, and
lim
x?Q??
max
braceleftBig
e?a(x?Q),0
bracerightBig
= 0
shows that Property 2.3 holds.
101
A.4 Proof of Proposition 2.3
The derivative is computed as
d?(x?Q)
d(x?Q) =
??
?
? pi2M sin
parenleftbiggx?Q
2M pi
parenrightbigg
, x ? [Q,Q+M)
0, x ? [Q+M,?)
If we let z = (pi/2M)(x?Q), then d?(x?Q)/d(x?Q) becomes ?z(x?Q)sinz, where z ?
(0,pi/2). Since ?z(x?Q) < 0 and sinz > 0 in this interval, we have d?(x?Q)/d(x?Q) < 0
which shows that Property 2.1 holds.Also,
d2?(x?Q)
d(x?Q)2 = ?
pi2
4M2cos(
x?Q
2M pi) < 0
shows that the first part of Theorem 2.3 holds. Now
lim
x?Q?0+
max
braceleftbigg
cos
parenleftbiggx?Q
2M pi
parenrightbigg
,0
bracerightbigg
= 1
shows that Property 2.2 holds, and
lim
x?Q??
max
braceleftbigg
cos
parenleftbiggx?Q
2M pi
parenrightbigg
,0
bracerightbigg
= lim
x?Q?M
cos
parenleftbiggx?Q
2M pi
parenrightbigg
= 0
shows that Property 2.3 holds.
102
Appendix B
Optimizing Backorder Lead-Time and Order Quantity in a Supply Chain
with Partial backlogging and Stochastic Demand
B.1 Derivations of Optimality for the Proposed Model
B.1.1 Optimal Order Quantity
We can get a feasible point, a unique minimizer, at which the first-order conditions
hold dTC(Q,t)dQ = 0.
dTC(Q)
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx+(cLS ?cB(t))
integraldisplay ?
Q
?(t)f(x)dx = 0
= cO +cHF(Q)?cLS(1?F(Q))+(cLS ?cB(t))?(t)(1?F(Q)) = 0
F(Q?) = ?(t)cB(t)+(1??(t))cLS ?cO?(t)c
B(t)+(1??(t))cLS +cH
Q? = F?1
bracketleftbigg?(t)c
B(t)+(1??(t))cLS ?cO
?(t)cB(t)+(1??(t))cLS +cH
bracketrightbigg
(B.1)
B.1.2 Sufficient Condition for Convexity and Optimal Response Time
The first and second order derivatives of TC(Q,t) are then
dTC(Q,t)
dt = {c
prime
B(t)?(t)+cB(t)?
prime(t)?cLS?prime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
d2TC(Q,t)
dt2 = {c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)+cB(t)?primeprime(t)?cLS?primeprime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
= {cprimeprimeB(t)?(t)+2cprimeB(t)?prime(t)+(cB(t)?cLS)?primeprime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
(B.2)
Clearly the second part of Eq.(B.2) is non-negative if the condition of Eq.(B.3) is
satisfied.
cprimeprimeB(t)?(t)+2cprimeB(t)?prime(t) + (cB(t)?cLS)?primeprime(t) ? 0
103
cLS ?cB(t) ? c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)
?primeprime(t)
cLS ?cB(t) ? b
2Ne?bte?at +2bNe?btae?at
a2e?at
where cB(t) = Ne?bt,?(t) = e?at
cLS ?cB(t) ? 2ab+b
2
a2 Ne
?bt
cLS ?cB(t) ? 2ab+b
2
a2 cB(t)
cLS ?
parenleftbigga+b
a
parenrightbigg2
cB(t)
(B.3)
Thus, TC(Q,t) is convex in t for any fixed Q where the condition of Eq.(B.3) is satisfied.
The optimal value t? corresponding to a fixed value Q is obtained by setting the first
derivative part of Eq.(B.2) equal to zero and solving for t, which yields Eq.(B.4) The value
t? is unique because Eq.(B.2) shows TC(Q, t) is strictly convex in t for any fixed Q.
cprimeB(t)?(t) = ?cB(t)?prime(t)+cLS?prime(t)
?(t) = {cLS ?cB(t)}?
prime(t)
cprimeB(t)
e?at = ?{cLS ?Ne
?bt}ae?at
?bNe?bt
bNe?bt = a{cLS ?Ne?bt}
(a+b)Ne?bt = acLS
e?bt = acLS(a+b)N
?bt = ln acLS(a+b)N
t? = ln
parenleftbigg(a+b)N
a?cLS
parenrightbigg
b?1 (B.4)
B.1.3 Relationship between Optimal Backorder cost and Lost sales cost
cB(t?) = Ne?bt?
= Ne?b?b?1ln
?(a+b)N
a?cLS
?
104
= Neln
? a?c
LS
(a+b)N
?
= N a?cLS(a+b)N
= a?cLS(a+b)
(B.5)
B.2 Derivations of Optimality for the Proposed Model with Backorder Setup
Cost
B.2.1 Optimal Order Quantity
We can get a feasible point, a unique minimizer, at which the first-order conditions
hold dTC(Q,t)dQ = 0.
dTC(Q)
dQ = cO +cHF(Q)?
bracketleftbigc
B(t)?(t)+cLS(1??(t))???2(t)
bracketrightbig(1?F(Q)) = 0
F(Q?) = ?(t)cB(t)+(1??(t))cLS ???
2(t)?cO
?(t)cB(t)+(1??(t))cLS ???2(t)+cH
Q? = F?1
bracketleftbigg?(t)c
B(t)+(1??(t))cLS ???2(t)?cO
?(t)cB(t)+(1??(t))cLS ???2(t)+cH
bracketrightbigg
(B.6)
B.2.2 Sufficient Condition for Convexity and Optimal Response Time
The first and second order derivatives of TC(Q,t) are then
dTC(Q,t)
dt = {c
prime
B(t)?(t)+cB(t)?
prime(t)?cLS?prime(t)+??prime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
d2TC(Q,t)
dt2 = {c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)+(cB(t)?cLS +?)?primeprime(t)}
integraldisplay ?
Q
(x?Q)f(x)dx
(B.7)
Clearly the second part of Eq.(B.7) is non-negative if the condition of Eq.(B.8) is
satisfied.
cprimeprimeB(t)?(t)+2cprimeB(t)?prime(t) + (cB(t)?cLS +?)?primeprime(t) ? 0
cLS ?cB(t) ? c
primeprime
B(t)?(t)+2c
prime
B(t)?
prime(t)
?primeprime(t) +?
105
cLS ?cB(t) ? 2ab+b
2
a2 Ne
?bt +?
cLS ?
parenleftbigga+b
a
parenrightbigg2
cB(t)+?
(B.8)
Thus, TC(Q,t) is convex in t for any fixed Q where the condition of Eq.(B.8) is satisfied,
and the optimal value t? corresponding to a fixed value Q is obtained by setting the first
part of Eq.(B.7) equal to zero and solving for t, which yields Eq.(B.9) The value t? is unique
because Eq.(B.7) shows TC(Q,t) is strictly convex in t for any fixed Q.
cprimeB(t)?(t) = ?cB(t)?prime(t)+cLS?prime(t)???prime(t)
?(t) = {cLS ?cB(t)??}?
prime(t)
cprimeB(t)
t? = ln
parenleftbigg (a+b)N
a?(cLS ??)
parenrightbigg
b?1 (B.9)
B.2.3 Relationship between Optimal Backorder cost and Lost sales cost
cB(t?) = Ne?b?b?1ln
? (a+b)N
a?(cLS??)
?
= Neln
?a?(c
LS??)
(a+b)N
?
= Na?(cLS ??)(a+b)N
= a?(cLS ??)(a+b)
(B.10)
106
Appendix C
The Effects of an Option Approach to Stochastic Inventory Decisions
C.1 Derivations of Optimality for the Partial Back logging Newsvendor Ap-
proach
We need to check the total cost function is convex function or not. The convexity of the
total cost function is insured since second derivation is always positive as all terms are non
negatives.
dE[TC(Q)]
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx+(cLS ?cB)
integraldisplay ?
Q
?f(x)dx
d2E[TC(Q)]
dQ2 = {cH +?cB +(1??)cLS}f(Q) (C.1)
We have a convex objective function and we can get a feasible point, unique minimizer,
at which the first-order conditions hold dE[TC(Q)]dQ = 0.
dE[TC(Q)]
dQ = cO +cH
integraldisplay Q
0
f(x)dx?cLS
integraldisplay ?
Q
f(x)dx+(cLS ?cB)
integraldisplay ?
Q
?f(x)dx = 0
= cO +cHF(Q)?cLS(1?F(Q))+(cLS ?cB)?(1?F(Q)) = 0
F(Q) = ?cB +(1??)cLS ?cO?c
B +(1??)cLS +cH
Q? = F?1[?cB +(1??)cLS ?cO?c
B +(1??)cLS +cH
] (C.2)
?cB+(1??)cLS?cO
?cB+(1??)cLS+cH is an optimal over stocking probability and then optimal stock out
probability is equal to{1??cB+(1??)cLS?cO?cB+(1??)cLS+cH}and that we can represent stock out probability
is following.
P?s = cO +cH?c
B +(1??)cLS +cH
(C.3)
C.2 Derivation of Total Cost Function with Real Option Framework
Eq.(4.5) can be rearranged as follows:
TC(Q) =
braceleftbigg {c
B? +(1??)cLS}D?{cB? +(1??)cLS ?cO}Q if D ? Q
(cO +cH)Q?cHD if D < Q (C.4)
107
Then, Based on Eq.(C.4), the present total cost of cash flows can be expressed as Eq.(4.6).
E[TC(Q)] = e?rfT ?E[{cB? +(1??)cLS}D
?{cB? +(1??)cLS ?cO}Q|D ? Q]?P(D ? Q)
+e?rfT ?E[(cO +cH)Q?cHD|D < Q]?P(D < Q)
(C.5)
Eq.(4.8) can be derived as following
E[TCa(Q)] = e?rfT ?E[{cB? +(1??)CLS}D|D ? Q]?P(D ? Q)]
?e?rfT ?E[{cB? +(1??)CLS ?cO}Q|D ? Q]?P(D ? Q)
= e?rfT ?D0 ?{cB? +(1??)cLS}
integraldisplay ?
ln( QD0 )
rf(r)dr
?e?rfT ?D0 ?{cB? +(1??)cLS ?cO} QD
0
integraldisplay ?
ln( QD0 )
f(r)dr
= e?rfT ?D0 ?{cB? +(1??)cLS}erfTN(d1)
?e?rfT ?Q{cB? +(1??)cLS ?cO}N(d2)
= D0{cB? +(1??)cLS}N(d1)?e?rfT ?Q{cB? +(1??)cLS ?cO}N(d2)
where d1 = ln(
D0
Q )+(rf +
?2r
2 )T
?r?T , d2 =
ln(D0Q )+(rf ? ?2r2 )T
?r?T (C.6)
Eq.(4.9) can be derived as following
E[TCb(Q)] = e?rfT ?[E[(cO +cH)Q|D < Q]P(D < Q)]?E[{cHD|D < Q]?P(D < Q)]
= e?rfT ?D0(cO +cH)E[ QD
0
|ln( DD
0
) < ln( QD
0
)]P(ln( DD
0
) < ln( QD
0
))
?e?rfT ?D0cHE[ln( DD
0
)|ln( DD
0
) < ln( QD
0
)]P(ln( DD
0
) < ln( QD
0
))
= e?rfT ?Q(cO +cH)
integraldisplay ln( Q
D0 )
0
f(r)dr?e?rfT ?D0cH
integraldisplay ln( Q
D0 )
0
rf(r)dr
= e?rfT ?Q(cO +cH)N(?d2)?e?rfT ?D0 ?cHerfTN(?d1)
= e?rfT ?Q(cO +cH)N(?d2)?D0 ?cHN(?d1)
where d1 = ln(
D0
Q )+(rf +
?2r
2 )T
?r?T , d2 =
ln(D0Q )+(rf ? ?2r2 )T
?r?T (C.7)
108
C.3 Derivation of Optimal Fill rate and Order Quantity
Eq.(4.14) can be derived as following.
d[E[TC(Q)]]
dQ = e
?rfT(cO +cH)N(?d2)?e?rfT{cB? +(1??)cLS ?cO}N(d2) = 0
e?rfT[(cO +cH)?{cH +cB? +(1??)cLS}N(d2)] = 0
{cH +cB? +(1??)cLS}N(d2) = cO +cH
N(d2) = cO +cH{c
B? +(1??)cLS +cH}
N(?d2) = ?cB +(1??)cLS ?cO?c
B +(1??)cLS +cH
(C.8)
Eq.(4.15) is derived as following.
d[E[TC(Q)]]2
d2Q =
d
dQ[?e
?rfT[(cO +cH)?{cH +cB? +(1??)cLS}N(d2)]]
= [?e?rfT{cH +cB? +(1??)cLS}d[N(d2)]dQ
= [?e?rfT{cH +cB? +(1??)cLS} ddQ[N{ln(
D0
Q )+(rf ?
?2r
2 )T
?r?T }
= [?e?rfT{cH +cB? +(1??)cLS}f(d2) ddQ[ln(
D0
Q )
?r?T ]
= [?e?rfT{cH +cB? +(1??)cLS}f(d2) ddQ[?lnQ?
r
?T ]
= [?e?rfT{cH +cB? +(1??)cLS}f(d2){?(Q?r?T)?1}
= [e?rfT{cH +cB? +(1??)cLS}(Q?r?T)?1f(d2)
(C.9)
Eq.(4.16) is derived as following.
d2 = N?1[ cO +cH{c
B? +(1??)cLS +cH}
]
ln(D0Q )+(rf ? ?2r2 )T
?r?T = N
?1[ cO +cH
cB? +(1??)cLS +cH ]
ln(D0Q ) = ?(rf ? ?
2r
2 )T +?r
?TN?1[ cO +cH
cB? +(1??)cLS +cH
109
D0
Q = Exp[?(rf ?
?2r
2 )T +?r
?TN?1[ cO +cH
cB? +(1??)cLS +cH )]]
Q? = D0e
(rf??2r2 )T
e?r
?TN?1[ cO+cH
cB?+(1??)cLS+cH ]
(C.10)
110
Appendix D
Coordinating a Two-stage Supply Chain Based on Option Contract
D.1 Derivations of Profit Function for CS
VC(Q) = p?E[[X ?Q]]?h?E[[Q?X]+]?cQ
= p?E[Q?[Q?X]+]?h?E[[Q?X]+]?cQ
= pQ?(p+h)?E[[Q?X]+]?cQ
= (p?c)Q?(p+h)?E[[Q?X]+]
= (p?c)Q?(p+h)?
integraldisplay Q
0
F0(x)dx
where E[[Q?X]+] = E[max[Q?X,0]]
= E[?[X ?Q?0]]
= E[?[X ?Q]+Q]
= E[?[X ?Q]]+Q
= ?E[X ?Q]+Q
= ?E[Q?[Q?X]+]+Q
= ?Q+E[[Q?X]+]+Q
= ?Q+
integraldisplay Q
0
(Q?x)f0(x)dx+Q
= ?
bracketleftbigg
Q?
integraldisplay Q
0
F0(x)dx
bracketrightbigg
+Q
= ?
bracketleftbiggintegraldisplay Q
0
1dx?
integraldisplay Q
0
F0(x)dx
bracketrightbigg
+Q
= ?
bracketleftbiggintegraldisplay Q
0
1?F0(x)dx
bracketrightbigg
+Q
=
integraldisplay Q
0
F0(x)dx
?VC(Q)
?Q = (p?c)?(p+h)F0(Q) = 0
Q?0 = F?10
parenleftbiggp?c
p+h
parenrightbigg
(D.1)
111
D.2 Derivation of Profit Functions for DS
Vr(Q,p) = (p?ve)?E[[X ?Q]]?v0Q
= (p?ve)?E[Q?[Q?X]+]?v0Q
= (p?ve ?v0)Q?(p?ve)?E[[Q?X]+]
= (p?ve ?v0)Q?(p?ve)?
integraldisplay Q
0
(Q?x)f(x)dx
= (p?ve ?v0)Q?(p?ve)?
integraldisplay Q
0
F(x)dx
(D.2)
Vm(v0,ve) = (v0 ?c)Q+ve ?E[X ?Q]]?h?E[[Q?X]+]
= (v0 ?c)Q+ve ?E[Q?[Q?X]+]?h?E[[Q?X]+]
= (v0 ?c)Q+veQ?ve ?E[[Q?X]+]?h?E[[Q?X]+]
= (v0 +ve ?c)Q?(ve +h)?E[[Q?X]+]
= (v0 +ve ?c)Q?(ve +h)?
integraldisplay Q
0
(Q?x)f(x)dx
= (v0 +ve ?c)Q?(ve +h)?
integraldisplay Q
0
F(x)dx
(D.3)
D.3 Derivation of Optimal Option Prices
Vr = ?VC
(p?ve ?v0)Q?(p?ve)?
integraldisplay Q
0
F(x)dx = ?
bracketleftbigg
(p?c)Q?(p+h)?
integraldisplay Q
0
F0(x)dx
bracketrightbigg
{p?ve ?v0 ??(p?c)}Q+{?(p?ve)+?(p+h)}?
integraldisplay Q
0
F0(x)dx = 0
p?ve ?v0 ??(p?c) = 0
(p?ve)??(p+h) = 0
112
ve = p??(p+h)
v0 = (p?ve)??(p?c)
= p?{p??(p+h)}??(p?c)
= ?(h+c)
(D.4)
Vm = (1??)VC
(v0 +ve ?c)Q?(ve +h)?
integraldisplay Q
0
F(x)dx = (1??)
bracketleftbigg
(p?c)Q?(p+h)?
integraldisplay Q
0
F0(x)dx
bracketrightbigg
(v0 +ve ?p+?p??c)Q?(ve ?p+?p+?h)?
integraldisplay Q
0
F(x)dx = 0
v0 +ve ?p+?p??c = 0
ve ?p+?p+?h = 0
ve = p??(p+h)
v0 = (p?ve)??(p?c)
= p?{p??(p+h)}??(p?c)
= ?(h+c)
(D.5)
D.4 Derivation of Optimal Order Quantity with Discounting
D.4.1 Fixed Coefficient of Variance Case (FCVC)
CV(X) = CV(X0) = ?0?0 = ??
? = ?0 +?0?(p0 ?p)?
? = ?0 +?0?(p0 ?p)?
Uniform Distribution of Demand Case:
VC(Q) = (p?c)Q?(p+h)?
integraldisplay Q
0
x?a
b?adx
113
= (p?c)Q? (p+h)b?a
integraldisplay Q
0
(x?a)dx
?VC(Q)
?Q = (p?c)?
p+h
b?a(Q?a) = 0
Q? = p?cp+h(b?a)+a
= p?cp+h(b0 ?a0)(1+?(p0 ?p)?)+a0(1+?(p0 ?p)?)
=
parenleftbiggp?c
p+h(b0 ?a0)+a0
parenrightbigg
(1+?(p0 ?p)?)
= F?10 (p?cp+h)(1+?(p0 ?p)?)
= Q?0(1+?(p0 ?p)?)
(D.6)
Exponential Distribution of Demand Case:
VC(Q) = (p?c)Q?(p+h)?
integraldisplay Q
0
1?e?1?xdx
?VC(Q)
?Q = (p?c)?(p+h)(1?e
?1?Q) = 0
e?1?Q = 1? p?cp+h
?1?Q = ln
parenleftbigg
1? p?cp+h
parenrightbigg
Q? = ??ln
parenleftbiggh+c
p+h
parenrightbigg
= ??0(1+?(p0 ?p)?)ln
parenleftbiggh+c
p+h
parenrightbigg
= ??0 ln
parenleftbiggh+c
p+h
parenrightbigg
(1+?(p0 ?p)?)
= F?10
parenleftbiggp?c
p+h
parenrightbigg
(1+?(p0 ?p)?)
= Q?0 ?(1+?(p0 ?p)?)
(D.7)
114
D.4.2 Increasing Coefficient of Variance Case (ICVC)
CV(X) = (1+?p?)CV(X0)
? = ?0(1+?(p0 ?p)?)
?
? = (1+?(p0 ?p)
?)?0
?0
Uniform Distribution of Demand Case:
b+a
2 =
b0 +a0
2 (1+?(p0 ?p)
?)
b?a
b+a = (1+?(p0 ?p)
?)b0 ?a0
b0 +a0
a = (1+?(p0 ?p)?)
braceleftbigg
a0 ? ?(p0 ?p)
?
2 (b0 ?a0)
bracerightbigg
b = (1+?(p0 ?p)?)
braceleftbigg
b0 + ?(p0 ?p)
?
2 (b0 ?a0)
bracerightbigg
VC(Q) = (p?c)Q?(p+h)?
integraldisplay Q
0
x?a
b?adx
= (p?c)Q? (p+h)b?a
integraldisplay Q
0
(x?a)dx
?VC(Q)
?Q = (p?c)?
p+h
b?a(Q?a) = 0
Q? = p?cp+h(b?a)+a
= p?cp+h(1+?(p0 ?p)?)
braceleftBig
(b0 ?a0)+?(p0 ?p)?(b0 ?a0)
bracerightBig
+ (1+?(p0 ?p)?)
braceleftbigg
a0 ? ?(p0 ?p)
?
2 (b0 ?a0)
bracerightbigg
= (1+?(p0 ?p)?)
bracketleftbigg
Q?0 +?(p0 ?p)?(b0 ?a0)
braceleftbiggp?c
p+h ?
1
2
bracerightbiggbracketrightbigg
(D.8)
115
Exponential Distribution of Demand Case:
Exponential distribution is not applicable as its CV is always equal to one, which is
valid only for the FCVC.
116