Porteus, Evan L.

Foundations of Stochastic Inventory Theory

Preface xv Conventions xviii Two Basic Models 1(26) The EOQ Model 1(6) The Newsvendor Model 7(20) Exercises 16(9) References 25(2) Recursion 27(14) Solving a Triangular System of Equations 27(1) Probabilistic Analysis of Models 28(1) Proof by Mathematical Induction 29(1) Shortest-Route Problems 29(3) Stochastic Shortest-Route Problems 32(2) Deterministic Production Planning 34(1) Knapsack Problems 35(6) Exercises 36(4) References 40(1) Finite-Horizon Markov Decision Processes 41(16) Example: e-Rite-Way 42(5) General Vocabulary and Basic Results 47(10) Exercises 54(2) References 56(1) Characterizing the Optimal Policy 57(20) Example: The Parking Problem 57(7) Dynamic Inventory Management 64(8) Preservation and Attainment 72(5) Exercises 73(3) References 76(1) Finite-Horizon Theory 77(14) Finite-State and -Action Theory 77(6) Proofs for the Finite-State and -Action Case 83(3) Generalizations 86(1) Optimality of Structured Policies 87(4) Exercises 88(2) References 90(1) Myopic Policies 91(12) General Approaches to Finding Solutions 92(1) Development 93(3) Application to Inventory Theory 96(1) Application to Reservoir Management 97(1) Extensions 98(5) Exercises 100(2) References 102(1) Dynamic Inventory Models 103(16) Optimality of (s, S) Inventory Policies 103(8) Linear-Quadratic Model 111(8) Exercises 115(3) References 118(1) Monotone Optimal Policies 119(14) Intuition 119(3) Lattices and Submodular Functions 122(4) A Dynamic Case 126(2) Capacitated Inventory Management 128(5) Exercises 131(1) References 132(1) Structured Probability Distributions 133(18) Some Interesting Distributions 133(4) Quasi-K-Convexity 137(2) A Variation of the (s, S) Inventory Model 139(4) Generalized (s, S) Policies 143(8) Exercises 148(2) References 150(1) Empirical Bayesian Inventory Models 151(16) Model Formulation 152(3) Conjugate Priors 155(4) Scalable Problems 159(2) Dimensionality Reduction 161(6) Exercises 163(3) References 166(1) Infinite-Horizon Theory 167(14) Problem Formulation 167(3) Mathematical Preparations 170(3) Finite State and Action Theory 173(5) Generalizations 178(3) Exercises 178(2) References 180(1) Bounds and Successive Approximations 181(12) Preliminary Results 182(2) Elimination of Nonoptimal Actions 184(4) Additional Topics 188(5) Exercises 190(2) References 192(1) Computational Markov Decision Processes 193(16) Policy Iteration 193(2) Use of Linear Programming 195(2) Preparations for Further Analysis 197(2) Convergence Rates for Value Iteration 199(2) Bounds on the Subradius 201(1) Transformations 202(7) Exercises 207(1) References 208(1) A Continuous Time Model 209(14) A Two-Product Production/Inventory Model 210(1) Formulation and Initial Analysis 211(5) Results 216(7) Exercises 221(1) References 221(2) Appendix A Convexity 223(18) A.1 Basic Definitions and Results 223(7) A.2 Role of the Hessian 230(4) A.3 Generalizations of Convexity 234(7) Exercises 235(4) References 239(2) Appendix B Duality 241(20) B.1 Basic Concepts 241(4) B.2 The Everett Result 245(5) b.3 Duality 250(11) Exercises 255(5) References 260(1) Appendix C Discounted Average Value 261(18) C.1 Net Present Value 262(2) C.2 Discounted Average Value 264(3) C.3 Alternatives with Different Time Horizons 267(1) C.4 Approximating the DAV 268(3) C.5 Application to the EOQ Model 271(2) C.6 Random Cycle Lengths 273(1) C.7 Random-Yield EOQ Problem 274(5) Exercises 275(3) References 278(1) Appendix D Preference Theory and Stochastic Dominance 279(14) D.1 Basic Concepts 280(2) D.2 Stochastic Dominance 282(4) D.3 How to Define Variability 286(1) D.4 Application to the Newsvendor Problem 287(6) Exercises 289(2) References 291(2) Index 293