# Introduction to Dynamic Programming

### International Series in Modern Applied Mathematics and Computer Science, Volume 1

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Introduction to Dynamic Programming introduces the reader to dynamic programming and presents the underlying mathematical ideas and results, as well as the application of these ideas to various problem areas. A large number of solved practical problems and computational examples are included to clarify the way dynamic programming is used to solve problems. A consistent notation is applied throughout the text for the expression of quantities such as state variables and decision variables. This monograph consists of 10 chapters and opens with an overview of dynamic programming as a particular approach to optimization, along with the basic components of any mathematical optimization model. The following chapters discuss the application of dynamic programming to variational problems; functional equations and the principle of optimality; reduction of state dimensionality and approximations; and stochastic processes and the calculus of variations. The final chapter looks at several actual applications of dynamic programming to practical problems, such as animal feedlot optimization and optimal scheduling of excess cash investment. This book should be suitable for self-study or for use as a text in a one-semester course on dynamic programming at the senior or first-year, graduate level for students of mathematics, statistics, operations research, economics, business, industrial engineering, or other engineering fields.

## Table of Contents

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

## Product details

- No. of pages: 300
- Language: English
- Copyright: © Pergamon 1981
- Published: January 1, 1981
- Imprint: Pergamon
- eBook ISBN: 9781483136622

## About the Authors

### Leon Cooper

### Mary W. Cooper

## About the Editor

### E. Y. Rodin

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