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