Markovian Demand Inventory Models

259 Pages·2010·3.668 MB·English

by  Dirk Beyer, Feng Cheng, Suresh P. Sethi, Michael Taksar (auth.)

International Series in Operations
Research & Management Science
Volume 108
Forothertitlespublishedinthisseries,goto
http://www.springer.com/series/6161
· · ·
Dirk Beyer Feng Cheng Suresh P. Sethi
Michael Taksar
Markovian Demand
Inventory Models
123
DirkBeyer FengCheng
M-Factor FederalAviationAdministration
1400FashionIslandBlvd. OfficeofPerformanceAnalysisandStrategy
Suite602 800IndependenceAve.S.W.
SanMateoCA94404 Washington,D.C.20591
USA USA
[email protected] [email protected]
SureshP.Sethi MichaelTaksar
SchoolofManagement,M/SSM30 DepartmentofMathematics
TheUniversityofTexasatDallas UniversityofMissouri
800W.CampbellRoad,SM30 RollinsRoad
Richardson,TX75080-3021 ColumbiaMO65211
USA USA
[email protected] [email protected]
ISSN0884-8289
ISBN978-0-387-71603-9 e-ISBN978-0-387-71604-6
DOI10.1007/978-0-387-71604-6
SpringerNewYorkDordrechtHeidelbergLondon
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Table of Contents
List of Figures ix
List of Tables xi
Preface xiii
Notation xvii
Part I INTRODUCTION
1. INTRODUCTION 3
1.1 Characteristics of Inventory Systems 3
1.2 Brief Historical Overview of Inventory Theory 5
1.3 Examples of Markovian Demand Models 12
1.4 Contributions 15
1.5 Plan of the Book 16
Part II DISCOUNTED COST MODELS
2. DISCOUNTED COST MODELS WITH BACKORDERS 21
2.1 Introduction 21
2.2 Review of the Related Literature 22
2.3 Formulation of the Model 23
2.4 Dynamic Programming and Optimal Feedback Policy 26
2.5 Optimality of (s,S)-type Ordering Policies 31
2.6 Nonstationary Infinite Horizon Problem 33
2.7 Cyclic Demand Model 37
2.8 Constrained Models 37
2.9 Concluding Remarks and Notes 39
vi Table of Contents
3. DISCOUNT COST MODELS WITH POLYNOMIALLY
GROWING SURPLUS COST 41
3.1 Introduction 41
3.2 Formulation of the Model 42
3.3 Dynamic Programming and Optimal Feedback Policy 44
3.4 Nonstationary Discounted Infinite Horizon Problem 49
3.5 Optimality of (s,S)-type Ordering Policies 55
3.6 Stationary Infinite Horizon Problem 57
3.7 Concluding Remarks and Notes 57
4. DISCOUNTED COST MODELS WITH LOST SALES 59
4.1 Introduction 59
4.2 Formulation of the Model 60
4.3 Optimality of (s,S)-type Ordering Policies 63
4.4 Extensions 66
4.5 Numerical Results 69
4.6 Concluding Remarks and Notes 72
Part III AVERAGE COST MODELS
5. AVERAGE COST MODELS WITH BACKORDERS 83
5.1 Introduction 83
5.2 Formulation of the Model 86
5.3 Discounted Cost Model Results from Chapter 2 89
5.4 Limiting Behavior as the Discount Factor Approaches 1 90
5.5 Vanishing Discount Approach 98
5.6 Verification Theorem 102
5.7 Concluding Remarks and Notes 106
6. AVERAGE COST MODELS WITH POLYNOMIALLY
GROWING SURPLUS COST 107
6.1 Formulation of the Problem 107
6.2 Behavior of the Discounted Cost Model with Respect
to the Discount Factor 109
6.3 Vanishing Discount Approach 116
6.4 Verification Theorem 125
6.5 Concluding Remarks and Notes 130
MARKOVIAN DEMAND INVENTORY MODELS vii
7. AVERAGE COST MODELS WITH LOST SALES 133
7.1 Introduction 133
7.2 Formulation of the Model 133
7.3 Discounted Cost Model Results from Chapter 4 137
7.4 Limiting Behavior as the Discount Factor Approaches 1 138
7.5 Vanishing Discount Approach 142
7.6 Verification Theorem 145
7.7 Concluding Remarks and Notes 150
Part IV MISCELLANEOUS
8. MODELS WITH DEMAND INFLUENCED
BY PROMOTION 153
8.1 Introduction 153
8.2 Formulation of the Model 155
8.3 Assumptions and Preliminaries 162
8.4 Structural Results 164
8.5 Extensions 170
8.6 Numerical Results 173
8.7 Concluding Remarks and Notes 174
9. VANISHINGDISCOUNTAPPROACHVS.STATIONARY
DISTRIBUTION APPROACH 179
9.1 Introduction 179
9.2 Statement of the Problem 182
9.3 Review of Iglehart (1963b) 184
9.4 An Example 187
9.5 Asymptotic Bounds on the Optimal Cost Function 192
9.6 Review of the Veinott and Wagner Paper 195
9.7 Existence of Minimizing Values of s and S 197
9.8 Stationary Distribution Approach versus Dynamic
Programming and Vanishing Discount Approach 203
9.9 Concluding Remarks and Notes 206
viii Table of Contents
Part V CONCLUSIONS AND OPEN RESEARCH
PROBLEMS
10. CONCLUSIONS AND OPEN RESEARCH PROBLEMS 211
Part VI APPENDICES
A. ANALYSIS 217
A.1 Continuous Functions on Metric Spaces 217
A.2 Convergence of a Sequence of Functions 219
A.3 The Arzel`a-Ascoli Theorems 220
A.4 Linear Operators 222
A.5 Miscellany 223
B. PROBABILITY 225
B.1 Integrability 225
B.2 Conditional Expectation 226
B.3 Renewal Theorem 227
B.4 Renewal Reward Processes 227
B.5 Stochastic Dominance 228
B.6 Markov Chains 229
C. CONVEX, QUASI-CONVEX AND K-CONVEX
FUNCTIONS 233
C.1 PF Density and Quasi-convex Functions 233
2
C.2 Convex and K-convex Functions 234
References 241
Copyright Permissions 247
Author Index 249
Subject Index 251
List of Figures
1.1 Relationships between different chapters. 18
2.1 Temporal conventions used for the discrete-time
inventory problem. 25
4.1 Results for Case 1.1. 73
4.2 Results for Case 1.2. 74
4.3 Results for Case 1.3. 75
4.4 Results for Case 1.4. 76
4.5 Results for Case 2.1. 77
4.6 Results for Case 2.2. 78
4.7 Results for Case 2.3. 79
4.8 Results for Case 2.4. 80
8.1 Numerical results for Case 3.1. 175
8.2 Numerical results for Case 3.2. 176
8.3 Numerical results for Case 3.3. 177
8.4 Numerical results for Case 3.4. 178
9.1 Definitions of Sl and Su. 197
List of Tables
4.1 Numericalresultsforthelostsalesmodelwithuni-
form demand distribution. 71
4.2 Numericalresultsforthelostsalesmodelwithtrun-
cated normal demand distribution. 71
8.1 NumericalresultsfortheMDPmodelwithuniform
demand distribution. 172
8.2 Numerical results for the MDP model with trun-
cated normal demand distribution. 173
Preface
Inventory management is among the most important topics in op-
erations management and operations research. It is perhaps also the
earliest, as evidenced by the economic order quantity (EOQ) formula,
which dates back to Ford W. Harris (1913). Inventory management is
concerned with matching supply with demand. The problem is to find
the amount to be produced or purchased in order to maximize the total
expected profit or minimize the total expected cost.
In the two decades following Harris, several variations of the formula
appeared, mostly in trade journals written by and for inventory man-
agers. These were first collected in a full-length book on inventory man-
agement by Raymond (1931). Erlenkotter (1989,1990) has provided a
fascinating account of this early history of inventory management.
Another classic formula in the inventory literature is the newsvendor
formula. It can be attributed to Edgeworth (1888), although it was not
until the work of Arrow, Harris and Marschak (1951) and Dvoretzky,
Kiefer and Wolfowitz (1953) that a systematic study of inventory mod-
els, incorporating uncertainty as well as dynamics, began. These early
papers were followed by the well-known treatise by Arrow, Karlin and
Scarf (1958) as well as the seminal work of Scarf (1960). These works
established the optimality of base-stock and (s,S) policies, the most
well-known policies in dynamic stochastic inventory modeling.
An important assumption in this vast literature has been that the de-
mands in different periods are independent and identically distributed
(i.i.d.). In real life, demands may depend on environmental considera-
tions or the states of the world, such as the weather, the state of the
economy, etc. Moreover, these states of the world are represented by
stochastic processes – exogenous or controlled.
In this book, we are concerned with inventory models where these
world states are modeled by Markov processes. We then show that we
obtain the optimality of (s,S)-type policies, or base-stock policies (i.e.,
s = S)whentherearenofixedorderingcostswiththeprovisionthatthe
policy parameters s and S depend on the current state of the Markov
processrepresentingtheenvironment. Modelsallowingbackorderswhen
the entire demand cannot be filled from the available inventory, as well