Introduction to Dynamic Programming

Chapter 1. Introduction


1.1. Optimization


1.2. Separable Functions


1.3. Convex and Concave Functions


1.4. Optima of Convex and Concave Functions


1.5. Dynamic Programming


1.6. Dynamic Programming: Advantages and Limitations


1.7. The Development of Dynamic Programming

Exercises—Chapter 1

Chapter 2. Some Simple Examples


2.1. Introduction


2.2. The Wandering Applied Mathematician


2.3. The Wandering Applied Mathematician (Continued)


2.4. A Problem in "Division"


2.5. A Simple Equipment Replacement Problem


2.6. Summary

Exercises—Chapter 2

Chapter 3. Functional Equations: Basic Theory


3.1. Introduction


3.2. Sequential Decision Processes


3.3. Functional Equations and the Principle of Optimality


3.4. The Principle of Optimality—Necessary and Sufficient Conditions

Exercise—Chapter 3

Chapter 4. One-dimensional Dynamic Programming: Analytic Solutions


4.1. Introduction


4.2. A Prototype Problem


4.3. Some Variations of the Prototype Problem


4.4. Some Generalizations of the Prototype Problem


4.5. Some Generalizations


4.6. A Problem in Renewable Resources


4.7. Multiplicative Constraints and Functions


4.8. Some Variations on State Functions


4.9. A Minimax Objective Function

Exercises—Chapter 4

Chapter 5. One-dimensional Dynamic Programming: Computational Solutions


5.1. Introduction


5.2. A Prototype Problem


5.3. An Example of the Computational Process


5.4. The Computational Effectiveness of Dynamic Programming


5.5. An Integer Nonlinear Programming Problem


5.6. Computation with Continuous Variables


5.7. Convex and Concave Φj(xj)


5.8. Equipment Replacement Problems


5.9. Some Integer Constrained Problems


5.10. A Deterministic Inventory Problem—Forward and Backward Recursion

Exercises—Chapter 5

Chapter 6. Multidimensional Problems


6.1. Introduction


6.2. A Nonlinear Allocation Problem


6.3. A Nonlinear Allocation Problem with Several Decision Variables


6.4. An Equipment Replacement Problem


6.5. Some Investment Problems


6.6. A Stochastic Decision Problem


6.7. The Traveling Salesman Problem


6.8. A Multicomponent Reliability Problem


6.9. A Problem in Product Development and Planning


6.10. A Smoothing Problem


6.11. Operation of a Chemical Reactor

Exercises—Chapter 6

Chapter 7. Reduction of State Dimensionality and Approximations


7.1. Introduction


7.2. Lagrange Multipliers and Reduction of State Variables


7.3. Method of Successive Approximations


7.4. Approximation in Policy and Function Space


7.5. Polynomial Approximation in Dynamic Programming


7.6. Reduction of Dimensionality and Expanding Grids


7.7. A New Method for Reduction of Dimensionality

Exercises—Chapter 7

Chapter 8. Stochastic Processes and Dynamic Programming


8.1. Introduction


8.2. A Stochastic Allocation Problem—Discrete Case


8.3. A Stochastic Allocation Problem—Continuous Case


8.4. A General Stochastic Inventory Model


8.5. A Stochastic Production Scheduling and Inventory Control Problem


8.6. Markov Processes


8.7. Markovian Sequential Decision Processes


8.8. The Policy Iteration Method of Howard

Exercises—Chapter 8

Chapter 9. Dynamic Programming and the Calculus of Variations


9.1. Introduction


9.2. Necessary and Sufficient Conditions for Optimality


9.3. Boundary Conditions and Constraints


9.4. Practical Difficulties of the Calculus of Variations


9.5. Dynamic Programming in Variational Problems


9.6. Computational Solution of Variational Problems by Dynamic Programming


9.7. A Computational Example


9.8. Additional Variational Problems

Exercises—Chapter 9

Chapter 10. Applications of Dynamic Programming


10.1. Introduction


10.2. Municipal Bond Coupon Schedules


10.3. Expansion of Electric Power Systems


10.4. The Design of a Hospital Ward


10.5. Optimal Scheduling of Excess Cash Investment


10.6. Animal Feedlot Optimization


10.7. Optimal Investment in Human Capital


10.8. Optimal Crop Supply


10.9. A Style Goods Inventory Model

Appendix. Sets, Convexity, and ^-Dimensional Geometry

A.1. Sets and Set Notation

A.2. n-Dimensional Geometry and Sets

A.3. Convex Sets

References

Index