# Mathematical Techniques of Operational Research

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Mathematical Techniques of Operational Research is a seven-chapter text that covers the principles and applications of various mathematical tools and models to for operational research. Chapter I provides the basic mathematical ideas used in later chapters. Chapters II and III deal with linear programming, including the special cases of transportation and assignment, as well as their applications such as the Trim Problem. Chapters IV and V discuss the theory of queues and describe the general stationary properties of the single-channel queue, and of simple queues in series and in parallel. These chapters also examine some transient properties of queues. Chapter VI focuses on machine interference, which is an aspect of queueing theory, while Chapter VII deals with the important and mathematically subject of Stock Control or Inventory Theory. This book is intended primarily to graduate mathematicians, business manages, and industrial leaders.

## Table of Contents

Preface

I. Mathematical Introduction

1. Algebra. Matrices and Vectors

2. Systems of Linear Equations

3. Analysis. Introduction

4. The Stieltjes' Integral

5. The Dirac delta Function

6. Bessel Functions

7. The Incomplete Gamma Function

8. Integral Equations

9. The Laplace Transform

10. Probability. Introduction

11. Conditional Probability

12. Random Variables and Probability Distributions

13. Probability Generating Functions

14. The Addition of Random Variables: Convolutions

15. The Laplace Transform of a Probability Distribution

16. The Poisson Process

17. Some Problems of Waiting Time

18. The Solution of a Type of Partial Differential Equation

II. Linear Programming

1. Introduction

2. The Problem of Linear Programming

3. The Simplex Method

4. Remarks on the Simplex Method

5. Example of the use of the Simplex Method

6. The Caterer Problem

7. The Trim Problem

III. Transportation and Assignment

1. Introduction

2. The Problem of Transportation. The Initial Solution

3. Testing a Solution

4. Improvement of a Solution

5. Degenerate Solutions

6. Alternative Optimal Solutions

7. Basic and Derived Solutions

8. The Problem of Assignment. The theorem of König

9. Solutions to the Problem of Assignment

10. The Algorithm of Munkres

11. The Complete Solution in a Particular Case

12. General Remarks

IV. Queueing Theory: The Single Channel Queue

1. Introduction

2. General Concepts and Definitions

3. Types of Distributions and a Notation for Queues

4. The Problems of Queueing Theory

5. The Queue M/G/1: Formulae for E(n) and E(w)

6. The Queue M/M/1: Differential-Difference Equations for the Queue Length

7. Use of the Laplace Transform and Probability Generating Function

8. Use of Integral Equations

9. Analysis of Transient Behaviour

10. Queue Disciplines: Random Selection, Bulk Service and Priority

V. Queueing Theory: Channels in Series or Parallel

1. Introduction

2. Channels in Parallel with Random Input

3. Parallel Channels with General Input: The Queue G/M/c

4. Channels in Series

5. Channels in Series: Various Restricted Cases

VI. Machine Interference

1. Introduction

2. The Case of One Operator (r = 1)

3. Determination of the Average Length (xm) of a Repair Period

4. System Characteristics

5. The Case of r Operators (r > 1)

6. System Characteristics in the Case of Several Operators

7. The Case of an Arbitrary Distribution of Repair Periods

8. General Remarks

VII. Problems of Stock Control

1. Introduction

2. Some Elementary Problems of Optimization

3. Operating Characteristics of a Simple Stockpile

4. The Problem of Hammersley

5. The Problem of Finch

6. Types of Replenishment Policies

7. A Variant of the Problem of Hammersley

8. Problems Involving a Lead Time

9. Problem 1

10. Problem 2

Index

## Product details

- No. of pages: 240
- Language: English
- Copyright: © Pergamon 1963
- Published: January 1, 1963
- Imprint: Pergamon
- eBook ISBN: 9781483180601

## About the Editors

### I. N. Sneddon

### S. Ulam

### M. Stark

## About the Author

### L. S. Goddard

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