Description
Book SynopsisReflecting the current high level of interest in the application of the principle of mathematical induction to the solution of optimization problems, this book offers a comprehensive introduction to the field.
Trade Review"This textbook...draws on his many years of experience in teaching this topic as well as on his considerable professional expertise in the area. It is ideally suited to its stated purpose as a student text." (Short Book Reviews, Vol. 20. No. 3, December 2000)
"...I was impressed with this book..." (The Statistician, Vol.51, No.2 2002)
"...excellent for the audience to whom it is addressed, and it is to be hoped that the author will write a further textbook..." (Jnl of the Operational Research Society, Vol 54(10) 2003)
Table of ContentsPreface xi
1 Introduction 1
1.1 Mathematical Induction 1
1.2 Historical Background 2
1.3 Dynamic Programming 5
1.4 The Executioner’s Tale 8
1.5 Summary 8
Exercises 10
I Deterministic Models 11
2 Multi-Stage Decision Problems 13
2.1 Maximizing Utilities 13
2.2 A General Model 17
2.3 Applications 19
Exercises 25
3 Networks 27
3.1 Shortest Paths 27
3.2 Directed Networks 29
3.3 Critical Path Analysis 30
Exercises 37
4 Further Applications 39
4.1 Discrete Actions 39
4.2 The Knapsack Problem 39
4.3 A Simple Replacement Model 42
4.4 Scheduling Problems 44
4.5 Johnson’s Algorithm 45
Exercises 49
5 Convexity 51
5.1 Convex and Concave Functions 51
5.2 Allocation Problems 56
5.3 Concave Utility Functions 60
Exercises 64
II Stochastic Models 67
6 Markov Systems 69
6.1 Introduction 69
6.2 Stochastic Dynamic Programming 70
6.3 Applications 72
Exercises 78
7 Optimal Stopping 79
7.1 Introduction 79
7.2 Stopping Times and Stopping Sets 82
7.3 Applications 90
Exercises 94
8 Special Problems 97
8.1 Introduction 97
8.2 Selling an Asset 97
8.3 The Marriage Problem 104
8.4 Prophet Inequalities 109
Exercises 116
III Markov Decision Processes 119
9 General Theory 121
9.1 Introduction 121
9.2 Minimizing Discounted Expectations 122
9.3 Policy Improvements 130
9.4 A Machine Replacement Model 137
10 Minimizing Average Costs 145
10.1 Introduction 145
10.2 Long-Term Average Costs 148
10.3 Extension to Infinitely Many States 153
10.4 Optimal Inventory Policies 158
11 Statistical Decisions 165
11.1 Introduction 165
11.2 Testing Statistical Hypotheses 166
11.3 The Sequential Probability Ratio Test 170
Notes On the Exercises 177
Chapter 1 177
Chapter 2 177
Chapter 3 178
Chapter 4 179
Chapter 5 179
Chapter 6 180
Chapter 7 181
Chapter 8 183
References 185
Index 187
