Library of Mathematics

LIBRARY OF MATHEMATICS EDITED BY WALTER LEDERMANN The aim of this series is to provide short introductory text-books for the topics which are normally covered in the first two years of mathematics courses at Universities, Polytechnics, Colleges of Education and Colleges of Technology. Each volume is made as nearly self-contained as possible, with exercises and answers, and contains an amount of material that can be covered in about twenty lectures. Thus each student will be able to build up a collection of text-books which is adapted to the syllabus he has to follow. The exposition is kept at an elementary level with due regard to modern standards of rigour. When it is not feasible to give a complete treatment, because this would go beyond the scope of the book, the assumptions are fully explained and the reader is referred to appropriate works in the literature. 'The authors obviously understand the difficulties of undergraduates. Their treatment is more rigorous than what students will have been used to at school, and yet it is remarkably clear. All the books contain worked examples in the text and exercises at the ends of the chapters. They will be invaluable to undergraduates. Pupils in their last year at school, too, will find them useful and stimulating. They will learn the university approach to work they have already done, and will gain a foretaste of what awaits them in the future.' - The Times Educational Supplement 'It will prove a valuable corpus. A great improvement on many works published in the past with a similar objective.' - The Times Literary Supplement 'These are all useful little books, and topics suitable for similar treatment are doubtless under consideration by the editor of the series.' - T. A. A. Broad bent, Nature A complete list of books in the series appears on the inside back cover. £1'75 net LINEAR PROGRAMMING LIBRARY OF MATHEMATICS edited by WALTER LEDERMANN D.Sc., Ph.D., F.R.S.Ed., Professor of Mathematics, University of Sussex Linear Equations P. M. Cohn Sequences and Series J. A. Green Differential Calculus P. J. Hilton Elementary Differential Equations and Operators G. E. H. Reuter Partial Derivatives P. J. Hilton Complex Numbers W. Ledermann Principles of· Dynamics M. B. Glauert E1ectrical and Mechanical Oscillations D. S. Jones Vibrating Systems R. F. Chisnell Vibrating Strings D. R. Bland Fourier Series I. N. Sneddon Solutions of Laplace's Equation D. R. Bland Solid Geometry P. M. Cohn Numerical Approximation B. R. Morton Integral Calculus W. Ledermann Sets and Groups J. A. Green Differential Geometry K. L. Wardle Probability Theory A. M. Arthurs Multiple Integrals W. Ledermann Fourier and Laplace Transforms P. D. Robinson Introduction to Abstract Algebra C. R. J. Clapham Functions of a Complex Variable, 2 vols D. O. Tall LINEAR PROGRAMMING BY KATHLEEN TRUSTRUM ROUTIEDGE & KEGAN PAUL LONDON, HENLEY AND BOSTON First published 1971 in Great Britain by Routledge & Kegan Paul Ltd 39 Store Street London WC1E 7DD, Broadway House, Newtown Road Henley-on-Thames Oxon RG9 lEN and 9 Park Street Boston, Mass. 02108, USA Whitstable Litho Ltd, Whitstable, Kent © Kathleen Trustrum 1971 No part of this book may be reproduced in any form without permission from the publisher, except for the quotation of brief passages in criticism ISBN-13: 978-0-7100-6779-1 e-ISBN-13: 978-94-010-9462-7 001: 10.1007/978-94-010-9462-7 Contents page Preface vii Chapter One: Convex Sets 1. Convex hulls, polytopes and vertices I 2. Basic solutions of equations 4 3. Theorem of the separating hyperplane 8 4. Alternative solutions of linear inequalities 10 Exercises 12 Chapter Two: The Theory of Linear Programming I. Examples and classes of linear programmes 14 2. Fundamental duality theorem 18 3. Equilibrium theorems 22 4. Basic optimal vectors 25 5. Graphical method of solution 27 Exercises 28 Chapter Three: The Transportation Problem 1. Formulation of problem and dual 31 2. Theorems concerning optimal solutions 35 3. Method of solution with modifications for degeneracy 36 4. Other problems of transportation type 41 Exercises 44 Chapter Four: The Simplex Method 1. Preliminary discussion and rules 46 2. Theory of the simplex method 53 v 3. Further techniques and extensions 58 Exercises 66 Chapter Five: Game Theory 1. Two-person zero-sum games 68 2. Solution of games: saddle points 70 3. Solution of games: mixed strategies 72 4. Dominated and essential strategies 74 5. Minimax theorem 77 6. Solution of matrix games by simplex method 79 Exercises 81 Suggestions for Further Reading 83 Solutions to Exercises 84 Index 87 vi Preface Linear programming is a relatively modern branch of Mathe matics, which is a result of the more scientific approach to management and planning of the post-war era. The purpose of this book is to present a mathematical theory of the subject, whilst emphasising the applications and the techniques of solution. An introduction to the theory of games is given in chapter five and the relationship between matrix games and linear programmes is established. The book assumes that the reader is familiar with matrix algebra and the background knowledge required is covered in the book, Linear Equations by P.M. Cohn, of this series. In fact the notation used in this text conforms with that intro duced by Cohn. The book is based on a course of about 18 lectures given to Mathematics and Physics undergraduates. Several examples are worked out in the text and each chapter is followed by a set of examples. I am grateful to my husband for many valuable suggestions and advice, and also to Professor W. Ledermann, for encourag ing me to write this book. University of Sussex Kathleen Trustrum vii .

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