Foundations of inventory management

1 General Introduction	p. 1
1.1 Inventories Are Important--So Are Foundations	p. 1
1.2 About the Book	p. 5
2 Systems and Models	p. 19
2.1 Introduction	p. 19
2.2 Contexts	p. 20
2.2.1 Computer Systems	p. 20
2.2.2 Monitoring	p. 21
2.2.3 Organizational Issues	p. 22
2.3 Models	p. 23
2.3.1 Overview	p. 23
2.3.2 Cost Estimation	p. 24
2.3.3 Demand Forecasting	p. 26
Notes	p. 27
3 One Item with a Constant Demand Rate	p. 29
3.1 Introduction	p. 29
3.2 The Economic-Order-Quantity Model	p. 30
3.2.1 Setting the Scene	p. 30
3.2.2 Performance Criteria	p. 33
3.2.3 The Optimal Policy	p. 36
3.2.4 Sensitivity Analysis	p. 37
3.3 Planned Backorders	p. 39
3.3.1 The Setting	p. 39
3.3.2 Reorder-Point/Order-Quantity Policies	p. 40
3.3.3 Performance Criteria	p. 41
3.3.4 The Optimal Policy and Sensitivity Analysis	p. 44
3.3.5 Constrained Stockouts	p. 46
3.3.6 Space and Time	p. 47
3.3.7 Costs for Backorder Occurrences	p. 48
3.3.8 Lost Sales	p. 49
3.4 A Finite Production Rate	p. 50
3.4.1 The Setting	p. 50
3.4.2 Performance Criteria	p. 50
3.4.3 The Optimal Policy and Sensitivity Analysis	p. 51
3.4.4 Setup Time	p. 52
3.4.5 Demand-Ready Output in Whole Batches	p. 54
3.5 Quantity Discounts	p. 55
3.5.1 Alternative Specifications of Purchase Costs	p. 55
3.5.2 Incremental Discounts	p. 55
3.5.3 All-Units Discounts	p. 57
3.5.4 Discussion	p. 57
3.6 Imperfect Quality	p. 58
3.6.1 Where Do Defects Occur and When Are They Found?	p. 58
3.6.2 The EOQ Setting	p. 58
3.6.3 Limited Capacity	p. 60
3.6.4 Perishable Products	p. 61
3.7 The Present-Value Criterion	p. 62
3.7.1 Objectives and Holding Costs	p. 62
3.7.2 Definition of Present Value	p. 62
3.7.3 Formulation and Analysis	p. 63
3.7.4 Comparison of Average-Cost and Present-Value Models	p. 64
Notes	p. 67
Problems	p. 67
4 Time-Varying Demands	p. 73
4.1 Introduction	p. 73
4.2 Extreme Cases	p. 74
4.2.1 Small Changes	p. 74
4.2.2 Fast Changes	p. 74
4.2.3 Slow Changes	p. 75
4.2.4 Discussion	p. 76
4.2.5 A Finite Production Rate	p. 77
4.3 The Dynamic Economic Lotsize Model	p. 78
4.3.1 Discrete-Time Formulation	p. 78
4.3.2 The Linear-Cost Case	p. 81
4.3.3 The Zero-Inventory Property	p. 82
4.3.4 Network Representation and Solution	p. 83
4.3.5 Examples--Opacity	p. 87
4.3.6 Heuristics	p. 90
4.3.7 Modeling and Implementation Issues	p. 91
4.4 Extensions	p. 93
4.4.1 Leadtimes	p. 94
4.4.2 Discounted Costs	p. 94
4.4.3 Continuously Accumulating Costs	p. 95
4.4.4 Backorders	p. 95
4.4.5 Limited Capacity	p. 97
4.4.6 Quantity Discounts	p. 98
4.5 Smoothing Models	p. 99
4.5.1 Formulation and Discussion	p. 99
4.5.2 Solution: The Linear Decision Rule	p. 100
4.5.3 Extensions	p. 101
Notes	p. 103
Problems	p. 104
5 Several Products and Locations	p. 107
5.1 Introduction	p. 107
5.1.1 Items: Product-Location Pairs	p. 107
5.1.2 Structural Complexity: Networks of Items	p. 108
5.1.3 Shared Supply Processes	p. 111
5.1.4 Time-Varying Demands	p. 111
5.2 Independent Items	p. 112
5.2.1 Aggregate Performance Measures	p. 112
5.2.2 The Inventory-Workload Tradeoff Curve	p. 114
5.2.3 Cost Estimation and Optimization	p. 116
5.2.4 Aggregate Sensitivity Analysis	p. 117
5.2.5 Aggregate Sensitivity Analysis in Action	p. 118
5.2.6 ABC Analysis	p. 119
5.3 Series Systems	p. 120
5.3.1 The Setting	p. 120
5.3.2 Echelons and Echelon Inventories	p. 121
5.3.3 Policy Characteristics	p. 124
5.3.4 The Relaxed Problem	p. 125
5.3.5 Constructing a Feasible Policy Using a Base Period	p. 130
5.3.6 Constructing a Feasible Policy with No Base Period	p. 131
5.3.7 Discussion: Coordination and Sensitivity	p. 132
5.3.8 Proofs	p. 135
5.4 Tree Systems	p. 139
5.4.1 Network Structure	p. 139
5.4.2 Leadtimes and Quantity Units	p. 140
5.4.3 Echelons and Echelon Inventories	p. 143
5.4.4 Policy Characteristics	p. 144
5.4.5 The Relaxed Problem	p. 145
5.4.6 Constructing a Feasible Policy	p. 147
5.4.7 Non-Nested Systems	p. 147
5.4.8 Coordination and Sensitivity Revisited	p. 149
5.5 Coordinated Supply: Economies of Scope	p. 149
5.5.1 The Joint-Replenishment Problem	p. 149
5.5.2 An Equivalent Distribution System	p. 151
5.5.3 Complex Economies of Scope	p. 152
5.5.4 The Inventory-Routing Problem	p. 153
5.6 Shared Production Capacity: Economic Lot Scheduling	p. 154
5.6.1 Average Resource Usage	p. 154
5.6.2 The Economic Lot Scheduling Problem (ELSP)	p. 154
5.6.3 Extensions	p. 157
5.7 Time-Varying Demand	p. 158
5.7.1 Supply-Demand Networks: Extreme Cases	p. 158
5.7.2 Supply-Demand Networks: Discrete-Time Formulation	p. 158
5.7.3 Limited Capacity	p. 162
5.8 Material Requirements Planning (MRP)	p. 163
5.8.1 Overview	p. 163
5.8.2 The Core Model and the Heuristic	p. 164
5.8.3 The Larger Context	p. 165
5.8.4 Evaluation: MRP versus OPT and JIT	p. 166
Notes	p. 168
Problems	p. 170
6 Stochastic Demand: One Item with Constant Leadtimes	p. 175
6.1 Introduction	p. 175
6.1.1 Major Themes	p. 175
6.1.2 Summary	p. 176
6.1.3 Demand Models	p. 177
6.1.4 Policies	p. 177
6.2 Policy Evaluation: Poisson Demand	p. 178
6.2.1 Preliminaries	p. 178
6.2.2 Base-Stock Policies--A Unit Batch Size	p. 181
6.2.3 General Batch Sizes	p. 186
6.2.4 Backorders and Waiting Time	p. 191
6.2.5 Proofs	p. 193
6.3 World-Driven Demand	p. 196
6.3.1 Discussion	p. 196
6.3.2 The MCDC Process	p. 198
6.3.3 Analysis	p. 201
6.3.4 Behavior of Performance Measures	p. 202
6.3.5 Technical Issues and Proofs	p. 203
6.4 Approximations	p. 205
6.4.1 Base-Stock Policies--Normal Approximation	p. 205
6.4.2 Base-Stock Policies--Other Continuous Approximations	p. 209
6.4.3 General (r,q) Policies	p. 210
6.4.4 Approximate Performance Measures	p. 211
6.5 Optimization	p. 213
6.5.1 Formulation	p. 213
6.5.2 The Base-Stock Model	p. 214
6.5.3 The General (r,q) Model: Continuous Approximation	p. 217
6.5.4 The General (r,q) Model: Discrete Formulation	p. 223
6.5.5 A Service-Level Constraint	p. 226
6.6 Lumpy Demand	p. 227
6.6.1 Discussion	p. 227
6.6.2 Policy Evaluation	p. 228
6.6.3 Optimization	p. 231
6.7 Other Extensions	p. 231
6.7.1 The Expected-Present-Value Criterion	p. 231
6.7.2 Using Current Information about Demand	p. 233
6.7.3 Periodic Review	p. 235
Notes	p. 237
Problems	p. 238
7 Stochastic Leadtimes: The Structure of the Supply System	p. 243
7.1 Introduction	p. 243
7.1.1 Discussion	p. 243
7.1.2 Taxonomy	p. 244
7.1.3 Preview of the Analysis	p. 245
7.2 Independent, Stochastic Leadtimes	p. 246
7.2.1 Poisson Demand, Base-Stock Policy	p. 247
7.2.2 World-Driven Demand	p. 248
7.2.3 Lost Sales	p. 249
7.2.4 Larger Batches	p. 251
7.2.5 Lumpy Demand	p. 253
7.2.6 Imperfect Quality	p. 253
7.2.7 Processing Networks	p. 254
7.3 Limited-Capacity Supply Systems	p. 256
7.3.1 Discussion	p. 256
7.3.2 One Processor, Poisson Demand, Base-Stock Policy	p. 256
7.3.3 Flexible Capacity	p. 261
7.3.4 Lost Sales	p. 263
7.3.5 Imperfect Quality	p. 263
7.3.6 Processing Networks	p. 265
7.3.7 Multiple Input Sources	p. 266
7.3.8 Customer Waiting Time	p. 267
7.3.9 World-Driven Demand	p. 268
7.3.10 Larger Batches	p. 269
7.3.11 The Exponential Approximation	p. 271
7.4 Exogenous, Sequential Supply Systems	p. 273
7.4.1 Description	p. 273
7.4.2 Analysis	p. 276
7.4.3 (Approximate) Extensions	p. 278
7.5 Leadtime-Demand Distributions	p. 279
7.5.1 Poisson Demand	p. 279
7.5.2 World-Driven Demand	p. 282
7.5.3 Approximations	p. 284
7.5.4 Lumpy Demand	p. 285
7.5.5 Proofs	p. 286
Notes	p. 289
Problems	p. 290
8 Several Items with Stochastic Demands	p. 295
8.1 Introduction	p. 295
8.2 Independent Items	p. 298
8.2.1 Base-Stock Policies	p. 298
8.2.2 General (r,q) Policies	p. 301
8.3 Series Systems	p. 302
8.3.1 Base-Stock Policy Evaluation: Poisson Demand, Constant Leadtimes	p. 302
8.3.2 Echelon-Based Calculations	p. 306
8.3.3 Base-Stock Policy Optimization	p. 308
8.3.4 Other Demand and Supply Processes	p. 310
8.3.5 Illustrations	p. 312
8.3.6 The Two-Stage System	p. 316
8.3.7 Economies of Scale	p. 319
8.3.8 A Service-Level Constraint	p. 322
8.4 Assembly Systems	p. 323
8.4.1 The Model	p. 323
8.4.2 Echelon Analysis and Inventory Balance	p. 324
8.4.3 The Balanced Echelon Base-Stock Policy	p. 327
8.4.4 Policy Evaluation and Optimization	p. 328
8.4.5 Local Base-Stock Policy Evaluation	p. 329
8.4.6 Proof of Theorem 8.4.1	p. 330
8.5 Distribution Systems: Local Control	p. 331
8.5.1 The Basic Model	p. 331
8.5.2 Policy Evaluation	p. 332
8.5.3 Optimization	p. 333
8.5.4 Extensions	p. 336
8.5.5 Economies of Scale	p. 338
8.6 Distribution Systems: Central Control	p. 339
8.6.1 Echelon Cost Accounting	p. 339
8.6.2 Myopic Allocation and the Balanced Approximation	p. 340
8.6.3 Discussion	p. 344
8.6.4 Coupled Systems	p. 345
8.6.5 Extensions	p. 347
8.7 Shared Supply Processes	p. 348
8.7.1 The Joint-Replenishment Problem	p. 348
8.7.2 Shared Production Capacity	p. 349
8.8 Kanban Systems	p. 352
8.8.1 Definition	p. 352
8.8.2 The Generalized Kanban Policy	p. 354
8.8.3 Analysis	p. 358
8.8.4 The Larger Context: The Just-In-Time Approach	p. 358
Notes	p. 359
Problems	p. 361
9 Time-Varying, Stochastic Demand: Policy Optimization	p. 367
9.1 Introduction	p. 367
9.2 Extreme Cases	p. 368
9.2.1 Small or Fast Changes	p. 368
9.2.2 Slow Changes	p. 369
9.2.3 Other Supply Processes	p. 370
9.3 Discrete-Time Formulation	p. 371
9.3.1 Dynamics	p. 371
9.3.2 Costs	p. 372
9.3.3 Salvage Value	p. 373
9.4 Linear Order Costs	p. 374
9.4.1 The Single-Period Problem	p. 374
9.4.2 The Finite-Horizon Problem: Dynamic Programming	p. 375
9.4.3 The Myopic Policy	p. 378
9.4.4 The Infinite-Horizon Problem: Stationary Data	p. 382
9.4.5 The Infinite-Horizon Problem: Nonstationary Data	p. 385
9.4.6 Lost Sales	p. 386
9.4.7 Profit Maximization	p. 387
9.4.8 Imperfect Quality	p. 390
9.5 Fixed-Plus-Linear Order Costs	p. 395
9.5.1 The Single-Period Problem	p. 396
9.5.2 The Finite-Horizon Problem	p. 400
9.5.3 The Infinite-Horizon Problem	p. 402
9.5.4 Extensions	p. 404
9.6 Leadtimes	p. 404
9.6.1 Constant Leadtime	p. 404
9.6.2 Constant Leadtime: Rigorous Argument	p. 407
9.6.3 Other Supply Systems	p. 408
9.6.4 Relation to Continuous-Time Models	p. 410
9.6.5 Lost Sales and the Curse of Dimensionality	p. 411
9.6.6 Profit Maximization	p. 414
9.6.7 Imperfect Quality	p. 414
9.7 World-Driven Demand	p. 415
9.7.1 The Model	p. 415
9.7.2 Formulation and Analysis	p. 416
9.7.3 The Myopic Policy and Cost Approximation	p. 418
9.8 Series Systems	p. 420
9.8.1 Formulation	p. 420
9.8.2 The Form and Calculation of the Optimal Policy	p. 422
9.8.3 Proof	p. 424
9.8.4 Extensions	p. 426
Notes	p. 427
Problems	p. 428
Appendix A Optimization and Convexity	p. 433
A.1 Introduction	p. 433
A.2 Optimization	p. 433
A.2.1 Definitions	p. 433
A.2.2 Optimality Conditions	p. 434
A.3 Convexity	p. 434
A.3.1 Definitions and Properties	p. 434
A.3.2 Implications	p. 436
Notes	p. 437
Appendix B Dynamical Systems	p. 439
B.1 Definitions and Examples	p. 439
B.1.1 Vocabulary	p. 439
B.1.2 Block Diagrams	p. 441
B.2 Discrete-Time Systems	p. 442
B.2.1 Formulation	p. 442
B.2.2 The Optimality Principle of Dynamic Programming	p. 442
B.2.3 Linear-Quadratic Systems	p. 443
B.3 Continuous-Time Systems	p. 444
B.3.1 Formulation	p. 444
Notes	p. 444
Appendix C Probability and Stochastic Processes	p. 445
C.1 Introduction	p. 445
C.2 Probability and Random Variables	p. 445
C.2.1 Discussion	p. 445
C.2.2 Discrete Random Variables: General Properties	p. 446
C.2.3 Discrete Random Variables: Examples	p. 448
C.2.4 Continuous Random Variables: General Properties	p. 454
C.2.5 Continuous Random Variables: Examples	p. 456
C.3 Stochastic Processes	p. 460
C.3.1 Definition	p. 460
C.3.2 Classification	p. 460
C.4 Markov Chains: Discrete Time	p. 463
C.4.1 Definition	p. 463
C.4.2 The Dynamics of State Probabilities	p. 464
C.4.3 Classification and Analysis	p. 466
C.4.4 Discrete Phase-Type Distribution	p. 469
C.5 Markov Chains: Continuous Time	p. 470
C.5.1 Definition and Dynamics	p. 470
C.5.2 Analysis	p. 471
C.5.3 Joint Independent Processes	p. 473
C.5.4 Birth-Death Processes	p. 473
C.5.5 Continuous Phase-Type Distribution	p. 474
C.5.6 Poisson Processes and Renewal Processes	p. 475
C.6 Stochastic Linear Systems in Discrete Time	p. 476
C.6.1 Formulation	p. 476
C.6.2 Analysis	p. 477
C.6.3 Stochastic Coefficients	p. 478
C.6.4 Applications to Demand Forecasting	p. 478
Notes	p. 480
Appendix D Notational Conventions	p. 481
D.1 Objectives and Conventions	p. 481
D.2 Symbols Used	p. 482
Bibliography	p. 487
Index	p. 503

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