Markovian Demand Inventory Models

by Beyer, Dirk; Cheng, Feng; Sethi, Suresh P.; Taksar, Michael

• ISBN13: 9780387716039
• ISBN10: 0387716033
• Format: Hardcover
• Copyright: 2009-09-30
• Publisher: Springer Verlag

Summary

Inventory management is concerned with matching supply with demand and a central problem in Operations Management. 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. Over the past two decades, several variations of the formula appeared, mostly in trade journals written by and for inventory managers. A critical assumption in the inventory literature is that the demands in different periods are independent and identically distributed. However, in real life, demands may depend on environmental considerations or the events in the world such as the weather, the state of economy, etc. Moreover, these events are represented by stochastic processes - exogenous or controlled. In Markovian Demand Inventory Models, the authors are concerned with inventory models where these world events are modeled by Markov processes. Their research on Markovian demand inventory models was carried out over a period of ten years beginning in the early nineties. They demonstrate that the optimality of (s, S)-type policies, or base-stock policies (i.e., s = S) when there are no fixed ordering costs with the provision that the policy parameters s and S depend on the current state of the Markov process representing the environment. Models allowing backorders when the entire demand cannot be filled from the available inventory as well as those when the current demand is lost are considered. As for cost criteria, we treat both the minimization of the expected total discounted cost and the long-run average cost. The average-cost criterion is mathematically more difficult than the discounted cost criteria. Finally, we generalize the usual assumptions on holding and shortage costs and on demands that are made in the literature.

Table of Contents

List of Figures	p. ix
List of Tables	p. xi
Preface	p. xiii
Notation	p. xvii
Introduction
Introduction	p. 3
Characteristics of Inventory Systems	p. 3
Brief Historical Overview of Inventory Theory	p. 5
Examples of Markovian Demand Models	p. 12
Contributions	p. 15
Plan of the Book	p. 16
Discounted Cost Models
Discounted Cost Models with Backorders	p. 21
Introduction	p. 21
Review of the Related Literature	p. 22
Formulation of the Model	p. 23
Dynamic Programming and Optimal Feedback Policy	p. 26
Optimality of (s, S)-type Ordering Policies	p. 31
Nonstationary Infinite Horizon Problem	p. 33
Cyclic Demand Model	p. 37
Constrained Models	p. 37
Concluding Remarks and Notes	p. 39
Discounted Cost Models with Polynomially Growing Surplus Cost	p. 41
Introduction	p. 41
Formulation of the Model	p. 42
Dynamic Programming and Optimal Feedback Policy	p. 44
Nonstationary Discounted Infinite Horizon Problem	p. 49
Optimality of (s, S)-type Ordering Policies	p. 55
Stationary Infinite Horizon Problem	p. 57
Concluding Remarks and Notes	p. 57
Discounted Cost Models with Lost Sales	p. 59
Introduction	p. 59
Formulation of the Model	p. 60
Optimality of (s,S)-type Ordering Policies	p. 63
Extensions	p. 66
Numerical Results	p. 69
Concluding Remarks and Notes	p. 72
Average Cost Models
Average Cost Models with Backorders	p. 83
Introduction	p. 83
Formulation of the Model	p. 86
Discounted Cost Model Results from Chapter 2	p. 89
Limiting Behavior as the Discount Factor Approaches 1	p. 90
Vanishing Discount Approach	p. 98
Verification Theorem	p. 102
Concluding Remarks and Notes	p. 106
Average Cost Models with Polynomially Growing Surplus Cost	p. 107
Formulation of the Problem	p. 107
Behavior of the Discounted Cost Model with Respect to the Discount Factor	p. 109
Vanishing Discount Approach	p. 116
Verification Theorem	p. 125
Concluding Remarks and Notes	p. 130
Average Cost Models with Lost Sales	p. 133
Introduction	p. 133
Formulation of the Model	p. 133
Discounted Cost Model Results from Chapter 4	p. 137
Limiting Behavior as the Discount Factor Approaches 1	p. 138
Vanishing Discount Approach	p. 142
Verification Theorem	p. 145
Concluding Remarks and Notes	p. 150
Miscellaneous
Models With Demand Influenced By Promotion	p. 153
Introduction	p. 153
Formulation of the Model	p. 155
Assumptions and Preliminaries	p. 162
Structural Results	p. 164
Extensions	p. 170
Numerical Results	p. 173
Concluding Remarks and Notes	p. 174
Vanishing Discount Approach Vs. Stationary Distribution Approach	p. 179
Introduction	p. 179
Statement of the Problem	p. 182
Review of Iglehart (1963b)	p. 184
An Example	p. 187
Asymptotic Bounds on the Optimal Cost Function	p. 192
Review of the Veinott and Wagner Paper	p. 195
Existence of Minimizing Values of s and S	p. 197
Stationary Distribution Approach versus Dynamic Programming and Vanishing Discount Approach	p. 203
Concluding Remarks and Notes 206 1Part V	p. 206
Conclusions and Open Research Problems	p. 211
Appendices
Analysis	p. 217
Continuous Functions on Metric Spaces	p. 217
Convergence of a Sequence of Functions	p. 219
The Arzela-Ascoli Theorems	p. 220
Linear Operators	p. 222
Miscellany	p. 223
Probability	p. 225
Integrability	p. 225
Conditional Expectation	p. 226
Renewal Theorem	p. 227
Renewal Reward Processes	p. 227
Stochastic Dominance	p. 228
Markov Chains	p. 229
Convex, Quasi-Convex and K-Convex Functions	p. 233
PF2 Density and Quasi-convex Functions	p. 233
Convex and K-convex Functions	p. 234
References	p. 241
Copyright Permissions	p. 247
Author Index	p. 249
Subject Index	p. 251
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