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Foundations of Stochastic Inventory Theory
by Evan Porteus
Stanford University Press, 2002
Cloth: 978-0-8047-4399-0

ABOUT THIS BOOK | AUTHOR BIOGRAPHY | REVIEWS | TOC | REQUEST ACCESSIBLE FILE
ABOUT THIS BOOK
This book 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. All chapters conclude with exercises that either soidify or extend the concepts introduced.





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Evan L. Porteus is the Sanwa Bank Professor of Management Science at the Stanford Graduate School of Business.


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Foundations of Stochastic Inventory Theory





Evan L. Porteus





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.








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"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 must-read 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





"This book provides a powerful and insightful approach to the analysis and control of stochastic dynamic systems. The introduction to dynamic optimization is focused and efficient with emphasis on how the theory can be applied to operational control settings such as inventory management and many others. At last, with this book every student and researcher interested in optimal operational control can benefit!"--Jan A. Van Mieghem, Kellogg School of Management, Northwestern University
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 must-read 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  Shortest-Route Problems  ...... . . . . . . . . . . .  . . .  29
    2.5 Stochastic Shortest-Route Problems . . . . . . . . . .....  32
    2.6  Deterministic Production Planning . . . . . . . . . . .....  34
    2.7 Knapsack Problems ..... . ... .. . .. . . . ..         35
    Exercises  ...............       .  ...........        36
    References ..... ..     ....... . . .  .  .. .  .  ..  .  .   . . .  40
    3 Finite-Horizon Markov Decision Processes                    41
    3.1 Example: e-Rite-Way  ...................... ..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 Finite-Horizon Theory                                        77
    5.1 Finite-State and -Action Theory  ...... . . . . . . . . . .  77
    5.2 Proofs for the Finite-State 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  Linear-Quadratic 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 Quasi-K-Convexity .........    ...............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 Infinite-Horizon 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 Two-Product 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 Random-Yield EOQ Problem . . . . . . . . . . . . . ....  274
    Exercises  ..............      ...... . . . ....  275
    References  .  .  .  . .........................278
    
    
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