Markovian Demand Inventory Models (eBook)

Table of Contents 5
List of Figures 9
List of Tables 10
Preface 11
Notation 14
I Introduction 18
1. Introduction 19
1.1 Characteristics of Inventory Systems 19
1.2 Brief Historical Overview of Inventory Theory 21
1.3 Examples of Markovian Demand Models 28
1.4 Contributions 31
1.5 Plan of the Book 32
II Discounted Cost Models 35
2. Discounted Cost Models with Backorders 36
2.1 Introduction 36
2.2 Review of the Related Literature 37
2.3 Formulation of the Model 38
2.4 Dynamic Programming and Optimal Feedback Policy 41
2.5 Optimality of (s,S)-type Ordering Policies 46
2.6 Nonstationary Infinite Horizon Problem 48
2.7 Cyclic Demand Model 52
2.8 Constrained Models 52
2.9 Concluding Remarks and Notes 54
3. Discount Cost Models with Polynomially Growing Surplus Cost 56
3.1 Introduction 56
3.2 Formulation of the Model 57
3.3 Dynamic Programming and Optimal Feedback Policy 59
3.4 Nonstationary Discounted Infinite Horizon Problem 64
3.5 Optimality of (s,S)-type Ordering Policies 70
3.6 Stationary Infinite Horizon Problem 72
3.7 Concluding Remarks and Notes 72
4. Discounted Cost Models with Lost Sales 74
4.1 Introduction 74
4.2 Formulation of the Model 75
4.3 Optimality of (s,S)-type Ordering Policies 78
4.4 Extensions 81
4.5 Numerical Results 84
4.6 Concluding Remarks and Notes 87
III Average Cost Models 96
5. Average Cost Models with Backorders 97
5.1 Introduction 97
5.2 Formulation of the Model 100
5.3 Discounted Cost Model Results from Chapter 2 103
5.4 Limiting Behavior as the Discount Factor Approaches 1 104
5.5 Vanishing Discount Approach 112
5.6 Verification Theorem 116
5.7 Concluding Remarks and Notes 120
6. Average Cost Models with Polynomially Growing Surplus Cost 121
6.1 Formulation of the Problem 121
6.2 Behavior of the Discounted Cost Model with Respectto the Discount Factor 123
6.3 Vanishing Discount Approach 130
6.4 Verification Theorem 139
6.5 Concluding Remarks and Notes 144
7. Average Cost Models with Lost Sales 146
7.1 Introduction 146
7.2 Formulation of the Model 146
7.3 Discounted Cost Model Results from Chapter 4 150
7.4 Limiting Behavior as the Discount Factor Approaches 1 151
7.5 Vanishing Discount Approach 155
7.6 Verification Theorem 158
7.7 Concluding Remarks and Notes 163
IV Miscellaneous 164
8. Models with Demand Influenced by Promotion 165
8.1 Introduction 165
8.2 Formulation of the Model 167
8.3 Assumptions and Preliminaries 174
8.4 Structural Results 176
8.5 Extensions 182
8.6 Numerical Results 185
8.7 Concluding Remarks and Notes 186
9. Vanishing Discount Approach vs. Stationary Distribution Approach 191
9.1 Introduction 191
9.2 Statement of the Problem 194
9.3 Review of Iglehart (1963b) 196
9.4 An Example 199
9.5 Asymptotic Bounds on the Optimal Cost Function 204
9.6 Review of the Veinott and Wagner Paper 207
9.7 Existence of Minimizing Values of s and S 209
9.8 Stationary Distribution Approach versus Dynamic Programming and Vanishing Discount Approach 215
9.9 Concluding Remarks and Notes 218
V Conclusions and Open Research Problems 220
10. Conclusions and Open Research Problems 221
VI Appendices 224
A. ANALYSIS 225
A.1 Continuous Functions on Metric Spaces 225
A.2 Convergence of a Sequence of Functions 227
A.3 The Arzelà-Ascoli Theorems 228
A.4 Linear Operators 230
A.5 Miscellany 231
B. PROBABILITY 233
B.1 Integrability 233
B.2 Conditional Expectation 234
B.3 Renewal Theorem 235
B.4 Renewal Reward Processes 235
B.5 Stochastic Dominance 236
B.6 Markov Chains 237
C. CONVEX, QUASI-CONVEX AND K-CONVEX FUNCTIONS 240
C.1 PF2 Density and Quasi-convex Functions 240
C.2 Convex and K-convex Functions 241
References 248
Copyright Permissions 254
Author Index 256
Subject Index 258