Mathematical Techniques of Operational Research

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