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PART Ⅰ．INTRODUCTION 1

1 Historical Background＆(Kenneth J．Arrow) 3

INTRODUCTION 3

THE TRANSACTION MOTIVE 4

THE PRECAUTIONARY MOTIVE 7

THE SPECULATIVE MOTIVE 8

RISKS IN MULTI-PERIOD ANALYSIS 13

CONCLUSION 14

2 The Nature and Structure of Inventory Problems＆(Kenneth J．Arrow,Samuel Karlin,and Herbert Scarf) 16

DECISION PROBLEMS 16

INVENTORY AND PRODUCTION PROBLEMS 18

COST AND REVENUE CONSIDERATIONS 19

DEMAND 23

DELIVERIES 24

THE GENERAL STRUCTURE OF INVENTORY MODELS 24

TYPES OF ANALYSIS 30

3 Summaries＆(Kenneth J．Arrow,Samuel Karlin,and Herbert Scarf) 37

CHAPTERS 4,5,AND 6: 37

CHAPTER 7: 41

CHAPTER 8: 42

CHAPTER 9: 44

CHAPTER 10: 45

CHAPTER 11: 46

CHAPTER 12: 48

CHAPTER 13: 48

CHAPTER 14: 49

CHAPTER 15: 52

CHAPTER 16: 55

CHAPTER 17: 56

PART Ⅱ．OPTIMAL POLICIES IN DETERMINISTIC INVENTORY PROCESSES 59

4 Production over Time with Increasing Marginal Costs＆(Kenneth J．Arrow and Samuel Karlin) 61

INTRODUCTION 61

A CONSISTENCY REQUIREMENT 62

BEGINNING A CONSTRUCTIVE SOLUTION 63

THE CASE OF t0 < T 64

THE CASE OF NO STORAGE COST 66

EXAMPLES 66

5 Smoothed Production Plans＆(Kenneth J．Arrow and Samuel Karlin) 70

INTRODUCTION 70

THE EVALUATION OF J(z) 71

CONVENIENT TRANSFORMATIONS 71

UPPER AND LOWER BOUNDS ON THE OPTIMAL POLICY 72

THE CASE IN WHICH s(t) IS CONVEX 74

THE CASE IN WHICH s(t) IS CONVEX-CONCAVE 78

THE CASE IN WHICH s(t) IS CONCAVE-CONVEX 82

THE CASE IN WHICH s(t) IS CONCAVE-CONVEX-CONCAVE 83

THE GENERAL CASE 83

6 Production Planning without Storage＆(Kenneth J．Arrow and Samuel Karlin) 86

7 The Optimal Expansion of the Capacity of a Firm＆(Kenneth J．Arrow,Martin J．Beckmann,and Samuel Karlin) 92

INTRODUCTION 92

MATHEMATICAL FORMULATION 94

DERIVATION OF ALGORITHM 96

THE SOLUTION WHEN INITIAL CAPACITY IS SMALLER THAN INITIAL DEMAND 100

DEMAND CONSTANT OVER INTERVALS 102

SUMMARY OF THE ALGORITHM 103

SOME ILLUSTRATIONS 104

PART Ⅲ．OPTIMAL POLICIES IN STOCHASTIC INVENTORY PROCESSES 107

8 One Stage Inventory Models with Uncertainty＆(Samuel Karlin) 109

MODEL I WITH LINEAR ORDERING COST FUNCTIONS 111

MODEL I WITH MORE GENERAL ORDERING COST FUNCTIONS 118

MODEL Ⅱ．FIXED ORDERING LEVELS 124

MODEL Ⅲ．RANDOM ORDERING LEVELS 126

EXAMPLES OF THE STATIC INVENTORY MODEL 131

9 Optimal Inventory Policy for the Arrow-Harris-Marschak Dynamic Model＆(Samuel Karlin) 135

THE DYNAMIC MODEL WITH A LINEAR ORDERING COST 137

THE DYNAMIC MODEL WITH A SET-UP COST 142

DYNAMIC MODELS WITH A CONVEX ORDERING COST FUNCTION 149

EXAMPLES 152

10 Inventory Models of the Arrow-Harris-Marschak Type with Time Lag＆(Samuel Karlin and Herbert Scarf) 155

PROOF OF THEOREMS 1 AND 2 159

CHARACTERIZATIONS OF THE OPTIMAL POLICY FOR MODEL II 162

SOLUTION OF ONE STAGE LAG,MODEL I 169

THE ANALYSIS OF ＂SIMPLE＂ POLICIES IN AN INVENTORY MODEL WITH TIME LAG 172

11 Optimal Policy for Hydroelectric Operations＆(John Gessford and Samuel Karlin) 179

A HYDROELECTRIC MODEL WITH LINEAR COST FUNCTIONS 180

A HYDROELECTRIC MODEL WITH A CONVEX COST FUNCTION 194

12 A Min-Max Solution of an Inventory Problem＆(Herbert Scarf) 201

INTRODUCTION 201

THE MATHEMATICAL SOLUTION OF THE PROBLEM 203

SOME OBSERVATIONS ON THE SOLUTION 207

THE RESULT WITH SALVAGE VALUE 209

13 On the Two-Bin Inventory Policy:An Application of the Arrow-Harris-Marschak Model＆(Martin J．Beckmann and Richard F．Muth) 210

INTRODUCTION 210

DISTRIBUTION OF DEMAND FOR SOME MACHINE REPAIR PARTS 211

SOLUTION OF THE INVENTORY EQUATION FOR EXPONENTIALLY DISTRIBUTED DEMAND 215

PART Ⅳ．OPERATING CHARACTERISTICS OF INVENTORY POLICIES 221

14 Steady State Solutions＆(Samuel Karlin) 223

INVENTORY MODEL WITH A LAG 227

STATIONARY INVENTORY MODEL WITH (s,S) POLICIES 229

THE EXISTENCE OF THE LIMITING DISTRIBUTION FOR (s,S) POLICIES 234

INVENTORY MODEL WITH RANDOM SUPPLY 237

INVENTORY MODEL WITH GENERAL RANDOM SUPPLY AND EXPONENTIAL DEMAND 239

INVENTORY MODEL WITH RANDOM SUPPLY AND DEMAND DISTRIBUTION A MEMBER OF THE GAMMA FAMILY 243

UNIQUENESS AND AVERAGE CONVERGENCE 260

EXTENSIONS 267

15 The Application of Renewal Theory to the Study of Inventory Policies＆(Samuel Karlin) 270

SOME RESULTS CONCERNING RENEWAL PROCESSES 272

A SIMPLE MODEL INVOLVING COMPARISON OF TWO PROCEDURES 277

PROBABILITY EVALUATION OF LOSSES FOR ORDERING RULES WHICH ARE OF (s,S) TYPE 280

AN INVENTORY MODEL WITH POISSON INPUT 285

DISTRIBUTIONS RELATED TO RENEWAL PROCESSES 288

16 Stationary Operating Characteristics of an Inventory Model with Time Lag＆(Herbert Scarf) 298

THE CASE D = 1 (POISSON DEMAND) 304

THE INFINITE MODEL WITH AN ARBITRARY INTERARRIVAL DISTRIBUTION AND A NEGATIVE EXPONENTIAL DISTRIBUTION FOR THE TIME LAG 307

THE FINITE MODEL WITH AN ARBITRARY INTERARRIVAL DISTRIBUTION AND A NEGATIVE EXPONENTIAL DISTRIBUTION FOR THE TIME LAG 311

A DETERMINATION OF CERTAIN AVERAGES 316

17 Inventory Models and Related Stochastic Processes＆(Samuel Karlin and Herbert Scarf) 319

INTRODUCTION 319

THE DISTRIBUTION OF THE NUMBER OF OUTSTANDING ORDERS FOR GENERAL INPUT AND SERVICE DISTRIBUTIONS 320

THE MAXIMUM TIME FOR SERVICE TO BE COMPLETED 325

EXTENSIONS 328

TRUNCATED MODEL WITH POISSON INPUT 330

POISSON INPUT VARYING WITH TIME 334

Bibliography of Inventory Theory 337