OPERATIONAL RESEARCH BY EXAMPLE By the same authors COLIN F. PALMER Total Quality Control (Edgard Blucher Ltd, Sao Paulo, Brazil) Ergonomia (Funda~ao Getulio Vargas, Rio de Janeiro, Brazil, 1976) ALEXANDER E. INNES Business Statistics by Example (Macmillan, revised edition 1979) Business Mathematics by Example (Macmillan, 1977) OPERATIONAL RESEARCH BY EXAMPLE Colin F. Palmer M.Sc., C.Eng., M.B.I.M., M.I.Prod.E. Lecturer in Quantitative Methods University of Birmingham and Alexander E. Innes B.Sc. (Econ.) (Hons), D.P.A. (Lond.), Cert. Ed. (Leeds) Senior Lecturer in Business Statistics Liverpool Polytechnic M © Colin F. Palmer and Alexander E. Innes 1980 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1980 by THE MACMILLAN PRESS LTD London and Baslngstoke Associated companies in Delhi Dublin Hong Kong Johannesburg Lagos Melbourne New York Singapore and Tokyo Typeset by Preface Ltd, Salisbury, Wilts British Library Cataloguing in Publication Data Palmer, Colin F Operational research by example. 1. Operations research I. Title II. Innes, Alexander Edward 658.4'034 T57.6 ISBN 978-0-333-22690-2 ISBN 978-1-349-16254-3 (eBook) DOI 10.1007/978-1-349-16254-3 This book is sold subject to the standard conditions of the Net Book Agreement The paperback edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent pur chaser Contents Preface vii Acknowledgements Vlll 1 Introducing operational research 1 2 Queueing and waiting-time problems 5 3 Simulation 26 4 Forecasting 50 5 Stock and inventory control 85 6 Linear programming 98 7 The transportation method 121 8 Assignment 142 9 Replacement 150 10 Network control 162 Answers, hints and solutions to exercises 178 Tables 1 Logarithms 202 2 Antilogarithms 204 3 Squares 206 4 Square roots 208 5 Random numbers 210 6 Random standardised normal deviates 212 7 Present values 214 8 Areas in tail of normal distribution 220 9 Exponential functions 222 Index 223 Preface This book introduces the nature of operational research and some basic techniques to students in higher and further education and to people work ing in business and public administration. Explanation is mainly through carefully chosen examples and exercises, occasionally simplified, but otherwise intended to be realistic in context. Most readers are likely to have the basic knowledge needed to follow the calculations given in the text and to attempt the exercises; those with no calculus or statistics should be able to master most of the material; and those without may find the two books by A. E. Innes mentioned on page ii helpful. Examples and exercises have been kept fairly simple. Application of the principles they use to business and to administration requires more rigor ous and formalised methods, because the problems are more complex. For a complete use of operational research in a large organisation a master model will be constructed, co-ordinating the work of separate models, in each one of the specialised techniques described in this book. Each chapter lays a simple foundation for a wide and important topic, and each bibliography contains key titles for further study. Readers will find that deeper penetration into specialised branches often depends upon more complex mathematical methods. The authors are glad to acknowledge the great help received in planning and writing the book. Mr Shaie Selzer, one of the publisher's editors, was concerned with the birth of the book, and its early upbringing, and his successor, Mr Nicholas Brealey, saw it into publication. Mrs Sonia Yuan, B.A., M.Sc., Senior Lecturer in Statistics at Oxford Polytechnic, discussed with the authors the general proposals and saw some of the manuscript, and her advice has proved most valuable. Our three typists served in this taxing field most competently; they were Mrs Joan Jones and Miss Anne Westover, both of the Institute of Local Government Studies, Birmingham University, and Mrs Beryl Perry. The authors have worked closely together to make the book useful, accurate and up to date, ideals in a developing quantitative field easier to set than to achieve; for any shortcomings they take full responsibility. c. F. PALMER A. E. INNES Acknowledgements The authors and publisher wish to thank the following, who have kindly given permission for the use of copyright material: The Bank of England for an extract from Bank of England Statistical Abstract, no. 1, 1970. The Controller of Her Majesty's Stationery Office for tables from Monthly Digest of Statistics, no. 375 (1975); United Kingdom Energy Statistics (1973); Annual Abstract of Statistics (1975), and Housing and Construc tion Statistics, no. 3. The London and Cambridge Economic Service, University of Cambridge, for tables from The British Economy Key Statistics 1900-1970. McGraw-Hill Book Company (U.K.) Ltd for a table from Discounted Cash Flow, 2nd ed. by M. G. Wright, ©1973. Every effort has been made to trace all the copyright-holders, but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangement at the first opportunity. 1 Introducing Operational Research Operational research, or operations research (to use the American term), describes the application of specialised quantitative techniques to solving problems met in industry, in commerce and in administration. For many years separate statistical and mathematical methods had been used to guide decision-making in these fields, but it was the impetus of the Second World War in Great Britain and America that began to bring together teams of mathematicians, statisticians, psychologists, physicists and other scientists to tackle problems demanding over-all strategies of enquiry and appli cation. Calculations of manpower and material needed to land and maintain combatant forces on hostile territory and of civilian food requirements in a siege economy were two examples of the fields in which operational research developed. Peaceful applications have been made in most coun tries of the world on lines described in the remaining chapters of this book. Operational research studies systems, a term readily recognised but not easy to define. A fleet of lorries regularly delivering goods from a ware house to a firm's customers constitutes a system. The term could properly be applied to surgeons working together in a hospital, with the anaesthe tists, sisters and nurses working with them, and the equipment they use. In. business, study of the behaviour of a system usually involves study of related sub-systems: the production-line of a factory is an identifiable sys tem; but a change in its working would involve changes in the supplies of raw materials, in the employment of workers and perhaps in the storage of finished products. Essentially, therefore, a system is a group of people engaged in joint, purposeful activity, together with material means used to achieving it, within the general context of industry, commerce or adminis tration; and such a system is likely to be supported by, or associated with, other systems called sub-systems. Groups and sub-groups thus described rarely lend themselves to direct experiment of the kind carried out in laboratories in the natural sciences. A Birmingham firm, for example, with a new product to export may need to choose between Liverpool, Bristol and London as its outlet. It cannot set up three separate dock offices and in the light of experience choose the most suitable. A government may need to decide whether £100 million is better spent on electrifying the railway between two major cities or upon improving the road system between them. Not only are the issues at stake 2 Operational Research by Example very complicated, but for the government there is little scope for crucial experiment, as the scientist would understand it. Both kind of problem are suitable for operational research, and insight would be obtained by setting up models. Iconic models, i.e. those which give a physical resemblance to the ori ginal, are often used in engineering, architecture and other branches of study. The nautical properties of a proposed oil-tanker, for example, can be studied by constructing a small-scale model and simulating stormy weather in a laboratory. Operational research uses a variety of quantitative models and formulae which are mathematical or statistical in origin. A firm deciding upon its stock-ordering policy could use the simple model derived on page 90: je~;) Q = where Q is the most economical size of batch in which to order stock, D is the total annual demand, P is the cost of placing one order, S is a measure of stock-holding costs and C is the unit cost of the items. This is an a priori model, a general description constructed from first principles. The expression 80 P = Q + 65 where Q is the number of units of a commodity demanded, and P is the unit price, is a simple model showing how prices change as demand changes, and which is likely to have been obtained by observing a number of pairs of values of P and Q for this commodity and fitting a relationship which best suits them. Such a model is empirical or a posteriori, though the general shape of the model, and in particular the positions of P and Q, reflect simple economic theory. The letters in both models indicate vari ables. Those which are determined by factors outside the defined system are called exogenous; for example, in the stock model, D, total demand, and the other factors on the right-hand side are determined by the size of the firm's market, salaries of order clerks, etc. Q, in this context, is an endogenous variable, because it influences the system from within. The batch size, for example, will affect the firm's storage and transport policies. In the second model Q will be exogenous and P endogenous. The reason for constructing an operational research model is optimisa tion, i.e. the calculation of the best value for a particular set of conditions. In the stock model Q gives the size of batch which will give the lowest average figure of unit cost when the combined effect of purchase, ordering and holding are taken into account. An optimum may be a maximum value: we may, for example, in planning vehicle routes between a series of