Inventory Theory

CONTENTS 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 7.2 Deterministic Demand Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

7.2.1 Continuous Review: Economic Order Quantity Model . . . . . . . . . . . . . . . . . . 353 7.2.2 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.2.2.1 Demand Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.2.2.2 Cost Parameters Vary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 7.2.2.3 Problem Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.2.2.4 Optimization Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.2.2.5 Perishing and Obsolescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 7.2.2.6 Delivery Times and Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.2.2.7 Nonlinear Holding Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.2.2.8 Allowed Shortages: Backordering and Lost Sales . . . . . . . . . . . . . . . 360 7.2.2.9 Waiting Time Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

7.2.3 Continuous Review: EPQ Model-Finite Replenishment Rates . . . . . . . . . . . . 363 7.2.3.1 Setup Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

7.2.4 Periodic Review: The Wagner-Whitin Model . . . . . . . . . . . . . . . . . . . . . . . . 364 7.2.4.1 Exact Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 7.2.4.2 Lot-Sizing Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 7.2.4.3 Capacitated Lot Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

7.3 Stochastic Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.3.1 Single Period: The Newsvendor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.3.2 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

7.3.2.1 Positive Initial Stock and Ordering Cost . . . . . . . . . . . . . . . . . . . . . . 369 7.3.2.2 Optimization Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7.3.2.3 Parameter Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

7.3.3 Periodic Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7.3.3.1 Positive-Order Lead Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 7.3.3.2 Positive-Setup Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 7.3.3.3 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 7.3.3.4 Perishable Inventories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

7.3.4 Continuous Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 7.3.4.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 7.3.4.2 Exact Analysis: A General Framework . . . . . . . . . . . . . . . . . . . . . . . 377 7.3.4.3 The (Q, r) Model with Poisson Demands . . . . . . . . . . . . . . . . . . . . . 380 7.3.4.4 The (s, S) Model with Compound Poisson Demands . . . . . . . . . . . . . 381

7.3.4.5 The (r , Nq) Model with Compound Poisson Demands . . . . . . . . . . . . 382 7.3.4.6 The (S − 1, S) Model with Poisson Demands . . . . . . . . . . . . . . . . . . 382 7.3.4.7 Approximate Analysis for the (Q, r) Model . . . . . . . . . . . . . . . . . . . 385 7.3.4.8 The (Q, r) Model with Fully or Partially Lost Sales . . . . . . . . . . . . . . 388 7.3.4.9 Perishable Inventories with Random Demands . . . . . . . . . . . . . . . . . 390 7.3.4.10 Multiple Modes of Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

7.4 Joint-Replenishment Models for Multiple Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 7.5 Multiechelon Inventory Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

7.5.1 Periodic-Review Echelon Stock Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 7.5.2 Periodic-Review Installation Stock Policies . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.5.3 Continuous-Review Installation Stock Policies: One-for-One

Replenishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.5.3.1 The Exact Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.5.3.2 The METRIC Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 7.5.3.3 The Two-Moment Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 398

7.5.4 Continuous-Review Installation Stock Policies: Batch-Ordering Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

7.5.5 Multiechelon Inventory-Practice Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

ABSTRACT Inventory theory is concerned with management of the quantity and timing of the replenishment of assets typically stored to satisfy future demands. In this chapter, we introduce the fundamentals of inventory theory and the basic models that constitute the technical core of supply chain management. The purpose of the chapter is to expose the reader to the basic concepts, models, and theoretical results that would serve as a foundation to build upon and to introduce the existing literature. The exposition follows the common classification of inventory models on the basis of echelon structures (single vs. multiple locations/levels), demand processes (deterministic vs. random demands), problem horizon lengths (finite vs. infinite), and perishability (nonperishables vs. perishables). Specifically, the chapter introduces the basic terminology and inventory-related costs and proceeds to construct the continuous and periodic review models with deterministic demands. The models considered herein establish the fundamental trade-offs between the cost components. In each category, the stylistic models are presented first to develop intuition, followed by relaxations of the basic assumptions that result in more realistic models and richer insights. Stochastic demands are treated extensively as they constitute not only the more general contexts but also a vaster portion of the existing literature. For the continuous-review, stochastic demand setting, three different modeling approaches are illustrated in detail to provide a foundation for different solution methodologies encountered in the literature. Special cases are also treated in detail to the same end. The multiple-item and multiple-echelon inventory systems are discussed concisely but all the basic models and results are presented. For such systems, extensive references are given. The modeling approaches assume a basic familiarity with probability and stochastic processes; in cases of specialized techniques, sufficient intuition is provided for the uninitiated. The emphasis is on the model development and codification of the existing knowledge. Where available, optimal control policies are established and presented. In the absence of such policies, commonly used approximations and/or heuristics are given. To illustrate some of the heuristic methods and models, simple numerical examples are also provided. The chapter concludes with some discussion of issues and practices encountered in the implementation of the discussed models.