Foundations of Stochastic Inventory Theory

Foundations of Stochastic Inventory Theory

by Evan Porteus
     
 

ISBN-10: 0804743991

ISBN-13: 9780804743990

Pub. Date: 08/28/2002

Publisher: Stanford University Press

In 1958, Stanford University Press published Studies in the Mathematical Theory of Inventory and Production (edited by Kenneth J. Arrow, Samuel Karlin, and Herbert Scarf), which became the pioneering road map for the next forty years of research in this area. One of the outgrowths of this research was development of the field of supply-chain management,

Overview

In 1958, Stanford University Press published Studies in the Mathematical Theory of Inventory and Production (edited by Kenneth J. Arrow, Samuel Karlin, and Herbert Scarf), which became the pioneering road map for the next forty years of research in this area. One of the outgrowths of this research was development of the field of supply-chain management, which deals with the ways organizations can achieve competitive advantage by coordinating the activities involved in creating products—including designing, procuring, transforming, moving, storing, selling, providing after-sales service, and recycling. Following in this tradition, Foundations of Stochastic Inventory Theory has a dual purpose, serving as an advanced textbook designed to prepare doctoral students to do research on the mathematical foundations of inventory theory and as a reference work for those already engaged in such research.

The author begins by presenting two basic inventory models: the economic order quantity model, which deals with "cycle stocks," and the newsvendor model, which deals with "safety stocks." He then describes foundational concepts, methods, and tools that prepare the reader to analyze inventory problems in which uncertainty plays a key role. Dynamic optimization is an important part of this preparation, which emphasizes insights gained from studying the role of uncertainty, rather than focusing on the derivation of numerical solutions and algorithms (with the exception of two chapters on computational issues in infinite-horizon models).

All fourteen chapters in the book, and four of the five appendixes, conclude with exercises that either solidify or extend the concepts introduced. Some of these exercises have served as Ph.D. qualifying examination questions in the Operations, Information, and Technology area of the Stanford Graduate School of Business.

Product Details

ISBN-13:
9780804743990
Publisher:
Stanford University Press
Publication date:
08/28/2002
Edition description:
1
Pages:
320
Product dimensions:
6.38(w) x 9.25(h) x 0.90(d)

Table of Contents

Prefacexv
Conventionsxviii
1Two Basic Models1
1.1The EOQ Model1
1.2The Newsvendor Model7
Exercises16
References25
2Recursion27
2.1Solving a Triangular System of Equations27
2.2Probabilistic Analysis of Models28
2.3Proof by Mathematical Induction29
2.4Shortest-Route Problems29
2.5Stochastic Shortest-Route Problems32
2.6Deterministic Production Planning34
2.7Knapsack Problems35
Exercises36
References40
3Finite-Horizon Markov Decision Processes41
3.1Example: e-Rite-Way42
3.2General Vocabulary and Basic Results47
Exercises54
References56
4Characterizing the Optimal Policy57
4.1Example: The Parking Problem57
4.2Dynamic Inventory Management64
4.3Preservation and Attainment72
Exercises73
References76
5Finite-Horizon Theory77
5.1Finite-State and -Action Theory77
5.2Proofs for the Finite-State and -Action Case83
5.3Generalizations86
5.4Optimality of Structured Policies87
Exercises88
References90
6Myopic Policies91
6.1General Approaches to Finding Solutions92
6.2Development93
6.3Application to Inventory Theory96
6.4Application to Reservoir Management97
6.5Extensions98
Exercises100
References102
7Dynamic Inventory Models103
7.1Optimality of (s, S) Inventory Policies103
7.2Linear-Quadratic Model111
Exercises115
References118
8Monotone Optimal Policies119
8.1Intuition119
8.2Lattices and Submodular Functions122
8.3A Dynamic Case126
8.4Capacitated Inventory Management128
Exercises131
References132
9Structured Probability Distributions133
9.1Some Interesting Distributions133
9.2Quasi-K-Convexity137
9.3A Variation of the (s, S) Inventory Model139
9.4Generalized (s, S) Policies143
Exercises148
References150
10Empirical Bayesian Inventory Models151
10.1Model Formulation152
10.2Conjugate Priors155
10.3Scalable Problems159
10.4Dimensionality Reduction161
Exercises163
References166
11Infinite-Horizon Theory167
11.1Problem Formulation167
11.2Mathematical Preparations170
11.3Finite State and Action Theory173
11.4Generalizations178
Exercises178
References180
12Bounds and Successive Approximations181
12.1Preliminary Results182
12.2Elimination of Nonoptimal Actions184
12.3Additional Topics188
Exercises190
References192
13Computational Markov Decision Processes193
13.1Policy Iteration193
13.2Use of Linear Programming195
13.3Preparations for Further Analysis197
13.4Convergence Rates for Value Iteration199
13.5Bounds on the Subradius201
13.6Transformations202
Exercises207
References208
14A Continuous Time Model209
14.1A Two-Product Production/Inventory Model210
14.2Formulation and Initial Analysis211
14.3Results216
Exercises221
References221
Appendix AConvexity223
A.1Basic Definitions and Results223
A.2Role of the Hessian230
A.3Generalizations of Convexity234
Exercises235
References239
Appendix BDuality241
B.1Basic Concepts241
B.2The Everett Result245
B.3Duality250
Exercises255
References260
Appendix CDiscounted Average Value261
C.1Net Present Value262
C.2Discounted Average Value264
C.3Alternatives with Different Time Horizons267
C.4Approximating the DAV268
C.5Application to the EOQ Model271
C.6Random Cycle Lengths273
C.7Random-Yield EOQ Problem274
Exercises275
References278
Appendix DPreference Theory and Stochastic Dominance279
D.1Basic Concepts280
D.2Stochastic Dominance282
D.3How to Define Variability286
D.4Application to the Newsvendor Problem287
Exercises289
References291
Index293

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