
Foundations of Stochastic Inventory Theory
Stanford University Press, 2002 Cloth: 9780804743990 Library of Congress Classification TS160.P67 2002 Dewey Decimal Classification 658.7/87
ABOUT THIS BOOK  AUTHOR BIOGRAPHY  REVIEWS  TOC  REQUEST ACCESSIBLE FILE
ABOUT THIS BOOK
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 supplychain 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 aftersales 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 infinitehorizon 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. AUTHOR BIOGRAPHY
Evan L. Porteus is the Sanwa Bank Professor of Management Science at the Stanford Graduate School of Business. REVIEWS
“This book provides a comprehensive foundation for inventory theory. Amazingly, Evan Porteus is able to introduce and explain very complex concepts in simple and interesting ways. In providing the intuitions behind many of the theories and results, he brings difficult concepts to life. This is a mustread for anyone who wants to learn, apply, or conduct research in stochastic inventory theories.”—Hau Lee, Stanford University “Every organization is striving to match supply with demand. Foundations of Stochastic Inventory Theory introduces the fundamental theories for tackling this challenging management task. Emphasizing simple, intuitive, and practical inventory policies rather than complex theories for general settings, Evan Porteus has written both a great textbook for graduate students in management, as well as a great reference book for anyone interested in inventory theory. Simply put, this is one of the best books on inventory theory available.”—Lode Li, Yale School of Management “Evan Porteus has organized the important foundations that are essential not only to appreciate this area and apply known results effectively, but that are crucial to fuel further progress that inevitably will be required to meet new practical challenges.”—Sridhar Tayur, Graduate School of Industrial Administration, Carnegie Mellon University TABLE OF CONTENTS
1 Two Basic Models 1 1.1 The EOQ Model ......................... 1 1.2 The Newsvendor Model ..................... 7 Exercises ................... ......... 16 References ................... ......... 25 2 Recursion 27 2.1 Solving a Triangular System of Equations ........... 27 2.2 Probabilistic Analysis of Models ................ .. 28 2.3 Proof by Mathematical Induction . . . . . . . . . . . . . . . . 29 2.4 ShortestRoute Problems ...... . . . . . . . . . . . . . . 29 2.5 Stochastic ShortestRoute Problems . . . . . . . . . ..... 32 2.6 Deterministic Production Planning . . . . . . . . . . ..... 34 2.7 Knapsack Problems ..... . ... .. . .. . . . .. 35 Exercises ............... . ........... 36 References ..... .. ....... . . . . .. . . .. . . . . . 40 3 FiniteHorizon Markov Decision Processes 41 3.1 Example: eRiteWay ...................... ..42 3.2 General Vocabulary and Basic Results . . . . . . . . . . . . . 47 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References ......... ............. ... . .. 56 4 Characterizing the Optimal Policy 57 4.1 Example: The Parking Problem . . . . . . . . . . . . . . ... 57 4.2 Dynamic Inventory Management . . . . . . . . . . . . . ... 64 4.3 Preservation and Attainment . . . . . . . . . . . . . . . ... 72 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 76 5 FiniteHorizon Theory 77 5.1 FiniteState and Action Theory ...... . . . . . . . . . . 77 5.2 Proofs for the FiniteState and Action Case . . . . . . . . . . 83 5.3 Generalizations ......................... .86 5.4 Optimality of Structured Policies . . . . . . . . . . . .... 87 Exercises ............................ . . 88 References ......... ................. . 90 6 Myopic Policies 91 6.1 General Approaches to Finding Solutions . . . . . . . . . . . 92 6.2 Development ....................... . . 93 6.3 Application to Inventory Theory . . . . . . . . . . . . . . . . 96 6.4 Application to Reservoir Management . . . . . . . . ..... 97 6.5 Extensions .................. ........ . 98 Exercises ................. ........... . 100 References . . . . .. . .. . .. . . . . .. . . . . . . . .. . .102 7 Dynamic Inventory Models 103 7.1 Optimality of (s, S) Inventory Policies . . . . . . . . . .... 103 7.2 LinearQuadratic Model ..................... 111 Exercises .............................. 115 References ..... ......... ............. . . . . 118 8 Monotone Optimal Policies 119 8.1 Intuition ................ ................. 119 8.2 Lattices and Submodular Functions . . . . . . . . . . . . ... 122 8.3 A Dynamic Case ......................... 126 8.4 Capacitated Inventory Management . . . . . . . . . . . . . 128 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .. . 131 References . . . . . . .. ...... .... ... .. ...132 9 Structured Probability Distributions 133 9.1 Some Interesting Distributions . . . . . . . . . . . . .... 133 9.2 QuasiKConvexity ......... ...............137 9.3 A Variation of the (s, S) Inventory Model . . . . . . . . . . . 139 9.4 Generalized (s, S) Policies ....................143 Exercises ..... ........................148 References . .. .. ... .. .... ... ............150 10 Empirical Bayesian Inventory Models 151 10.1 Model Formulation ........................152 10.2 Conjugate Priors .........................155 10.3 Scalable Problems ........................159 10.4 Dimensionality Reduction . . . . . . . . . . . . . . ..... . 161 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163 References . . . ..... ........... ...... ....166 11 InfiniteHorizon Theory 167 11.1 Problem Formulation ..... ..................167 11.2 Mathematical Preparations . . . . . . . . . . . . . . . . . . . 170 11.3 Finite State and Action Theory . . . . . . . . . . . . .... 173 11.4 Generalizations ......... .................178 Exercises .......... ...................178 References . .. .. ............. ...........180 12 Bounds and Successive Approximations 181 12.1 Preliminary Results ....................... 182 12.2 Elimination of Nonoptimal Actions . . . . . . . . . . . . ... 184 12.3 Additional Topics .........................188 Exercises ............................. 190 References . ... ....... ... .. ... ..... .....192 13 Computational Markov Decision Processes 193 13.1 Policy Iteration ..........................193 13.2 Use of Linear Programming . . . . . . . . . . . . . . . . ... 195 13.3 Preparations for Further Analysis . . . . . . . . . . . . . ... 197 13.4 Convergence Rates for Value Iteration . . . . . . . . ..... 199 13.5 Bounds on the Subradius ................... . 201 13.6 Transformations .......................... 202 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 References .. ............. ..............208 14 A Continuous Time Model 209 14.1 A TwoProduct Production/Inventory Model . . . . . . ... 210 14.2 Formulation and Initial Analysis . . . . . . . . . . . ..... 211 14.3 Results . ...... ... ................. .... 216 Exercises .......... .................... 221 References.......................... . ..... 221 Appendix A Convexity 223 A.1 Basic Definitions and Results . . . . . . . . . . ...... .. . 223 A.2 Role of the Hessian ............... . .. . ... 230 A.3 Generalizations of Convexity . . . . ..... . . . . . . ... . . 234 Exercises . .... .. .. .... .. . ............. 235 References .... ........... .............. 239 Appendix B Duality 241 B.1 Basic Concepts ...... .......... . . . .. . .....241 B.2 The Everett Result ................... . . . .245 B.3 Duality . ................. . ............. 250 Exercises ................... . . . ... . ......255 References.......................... . ..... 260 Appendix C Discounted Average Value 261 C.1 Net Present Value ....................... . 262 C.2 Discounted Average Value .......... ......... 264 C.3 Alternatives with Different Time Horizons . . . . . . . . . . . 267 C.4 Approximating the DAV ........ . . . . . ..268 C.5 Application to the EOQ Model . . . . . . . . . . . . ..... 271 C.6 Random Cycle Lengths ................... . .273 C.7 RandomYield EOQ Problem . . . . . . . . . . . . . .... 274 Exercises .............. ...... . . . .... 275 References . . . . .........................278
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