This page intentionally left blank AN INTRODUCTION TO LINEAR PROGRAMMING AND GAME THEORY This page intentionally left blank AN INTRODUCTION TO LINEAR PROGRAMMING AND GAME THEORY Third Edition Paul R Thie G E Keough WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2008 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic format For information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Thie, Paul R., 1936An introduction to linear programming and game theory / Paul R Thie, G E Keough — 3rd ed p cm Includes bibliographical references and index ISBN 978-0-470-23286-6 (cloth) Linear programming Game theory I Keough, G E II Title T57.74.T44 2008 519.7'2—dc22 2008004933 Printed in the United States of America 10 To OUR W I V E S , MARY LOU AND DIANNE and IN MEMORY OF A GENTLE IRISHMAN OF GIFTED W I T AND CHARM This page intentionally left blank Contents Preface xi Mathematical Models 1.1 Applying Mathematics 1.2 The Diet Problem 1.3 The Prisoner's Dilemma 1.4 The Roles of Linear Programming and Game Theory 1 The Linear Programming Model 2.1 History 2.2 The Blending Model 2.3 The Production Model 2.4 The Transportation Model 2.5 The Dynamic Planning Model 2.6 Summary The Simplex Method 3.1 The General Problem 3.2 Linear Equations and Basic Feasible Solutions 3.3 Introduction to the Simplex Method 3.4 Theory of the Simplex Method 3.5 The Simplex Tableau and Examples 3.6 Artificial Variables 3.7 Redundant Systems 3.8 A Convergence Proof 3.9 Linear Programming and Convexity 3.10 Spreadsheet Solution of a Linear Programming Problem 57 57 63 72 77 85 93 101 106 110 115 Duality 4.1 Introduction to Duality 4.2 Definition of the Dual Problem 4.3 Examples and Interpretations 4.4 The Duality Theorem 4.5 The Complementary Slackness Theorem 121 121 123 132 138 154 Sensitivity Analysis 5.1 Examples in Sensitivity Analysis 5.2 Matrix Representation of the Simplex Algorithm 161 161 175 An Introduction to Linear Programming and Game Theory, Third Edition By P R Thie and G E Keough Copyright © 2008 John Wiley & Sons, inc 9 10 21 34 38 47 CONTENTS Vlll 5.3 5.4 5.5 5.6 5.7 Changes in the Objective Function Addition of a New Variable Changes in the Constant-Term Column Vector The Dual Simplex Algorithm Addition of a Constraint 183 189 192 196 204 Integer Programming 6.1 Introduction to Integer Programming 6.2 Models with Integer Programming Formulations 6.3 Gomory's Cutting Plane Algorithm 6.4 A Branch and Bound Algorithm 6.5 Spreadsheet Solution of an Integer Programming Problem 211 211 214 228 237 244 The Transportation Problem 7.1 A Distribution Problem 7.2 The Transportation Problem 7.3 Applications 251 251 264 282 Other Topics in Linear Programming 8.1 An Example Involving Uncertainty 8.2 An Example with Multiple Goals 8.3 An Example Using Decomposition 8.4 An Example in Data Envelopment Analysis 299 299 306 314 325 Two-Person, Zero-Sum Games 9.1 Introduction to Game Theory 9.2 Some Principles of Decision Making in Game Theory 9.3 Saddle Points 9.4 Mixed Strategies 9.5 The Fundamental Theorem 9.6 Computational Techniques 9.7 Games People Play 337 337 345 350 353 360 370 382 10 Other Topics in Game Theory 10.1 Utility Theory 10.2 Two-Person, Non-Zero-Sum Games 10.3 Noncooperative Two-Person Games 10.4 Cooperative Two-Person Games 10.5 The Axioms of Nash 10.6 An Example 391 391 393 397 404 408 414 A Vectors and Matrices 417 B An Example of Cycling 421 C Efficiency of the Simplex Method 423 SOLUTIONS TO SELECTED PROBLEMS 446 (d) Minimize -6xj + 2x2 - 2x% - 9x3 - 300 subject to 2xi — Xi + = 100 = 200 = 50 = -60 6x + 6x — X3 + X4 X2 — X2 + 9X3 + *5 X\ + X6 x — x2 - x-i - X8 = X3 Xi,X ,X / ,X3,X4,X5,X6,X7,X > (a) {(0,0,A,0):A> 11} (b) {(5,0,6,0)} (c) Problem Set 3.2 (a) (1,2,-3) (b) Arbitrarily selecting x\ and X2 to use as basic variables, two pivot steps yield the following equivalent system: X2 + YjXj, = Xl 17 X3 13 17 17 Thus the solution set is {( - T7 + Ỵ7^'Ï7 - Ï ^ > ^ ) : ^ e R l The system is equivalent to various systems of equations in canonical form For example, an equivalent system with basic variables x\ and X3 is the system xi - 8x2 = -41 —3x2 +X3 = —16 (a) x2 = Xi - X = (b) No (c) b = (17,4)' can be expressed as a linear combination of A^ = (2, 1)' and A^ = (1,0)', but not as a linear combination of AO andA^3' = ( - , - ) ' (b) (0,6,2,0) and (0,0,2,2) (d) The minimum value of the objective function is 8, attained at (0,0,2,2) M i n / = i5 attained a t ( f ,0,0, §) SOLUTIONS TO SELECTED PROBLEMS 447 Problem Set 3.3 (a) (b) (c) (d) X] = — 2x4,X2 — — 3x4,xj — 18 — 6x4 < x4 < x2 We should extract X2 from the basis; therefore, pivot at the 3x4 term of the second equation Pivoting here yields: (e) x\ — 1*2 = 5X2 + X4 = — 2x2 + X3 =6 The associated basic solution, (4,0,6,2), is feasible (f) The minimum of | , f, and ^ is f, attained with the data from the second equation Pivoting at the 2x4 term of the first constraint yields the equivalent problem of minimizing z with 5X2 — 3x3 + X4 = x\ + 5X2 — X3 = 3x2 - 14x3 Xi,X2,X3,X4 > = 18 + z The expression for z suggests putting X3 into the basis, but there is no positive X3 coefficient in the constraints In fact, from this representation of the constraints, we see that the set of feasible solutions contains the set {(8+x ,0,x ,3 + x ) : * > } What happens to z on this set? Problem Set 3.4 Minz = - attained at (0, f ,0, ^ , f ) (a) (b) (c) (d) (e) (f) (g) Minz = attained at (5,10,0,0) No pivots necessary Minz = attained at (5,10,0,0) No pivots necessary Unbounded objective function Unbounded objective function Minz = - attained at (5,0,5,0) Minz = attained at (0,10,0,0) One pivot necessary Unbounded objective function No pivots necessary When the Min{è,-/a,-,s : als > 0} is attained in more than one row Problem Set 3.5 (a) Minz = - 0 attained at (0,0,50,0) SOLUTIONS TO SELECTED PROBLEMS 448 (c) Unbounded objective function (d) Maxz = 90 attained at (250,10,0,40,0,0) See Example D.l on page 427 (a) In the final tableau, c\ = and at least one a*2 > Thus X2 can be inserted into the basis Similarly forx7 (b) (0,0,0,25,0,15,15) (c) (10,30,0,20,0,0,0) Maximum income is $7020, attained by producing 240 radios, 85 televisions, and stereos Problem Set 3.6 (a) Applying the simplex algorithm to the problem of Minimizing w — XA, + X5 subject to X\ - X2 + X4 2x\ + X2 — X3 = +^ = Xi,X2,X3,X4,X5 > generates the solution point ( | , | , ) to the original system (a) Minz = \ attained at (0, ^ , ^ ) (b) Minz = —^ attained at (0, ^ , ^ ) (Only one artificial variable required.) (c) No feasible solutions The row corresponds to the expression for the function w = x^+x^ in terms of the nonbasic variables for that tableau, namely, X2, X4, x$, and X& Follows from the definition of w and from Problem of Section 3.4 on page 84 Minimal cost is $1950 attained by using Process for | hr and Process for | hr Problem Set 3.7 (a) Minz = 50 attained at (50,0,0,0) No redundant equations (c) Minz = — I attained at (0,0, | , | ) One redundant equation (d) Maxz = —6 attained at (0,1,2,0) No redundant equations True If any artificial variables remained in the basis, they would be at zero level The elimination of these variables from the basis would lead to a degenerate solution to the original system Problem Set 3.8 (a) Changing the constant-term column entries to in the tableaux of Table 3.4, we have Maxz = attained at (0,0,0) SOLUTIONS TO SELECTED PROBLEMS 449 (b) From the modified tableaux of Table 3.5, the objective function is unbounded Problem Set 4.1 Maximum gain is $475, attained at (25,100) (a) Minimum cost is $475, attained at (0, |,-j^) Problem Set 4.2 (a) Minimize 100y, + 90y2 + 500j subject to 5yi n > 20 -4yi + \2y2 + y3 > 30 yi,y2,y3 > o (b) Maximize -30yi - 50y2 - 80y3 subject to 6ji - 2y2 < lyi + lyi - yi < - y\,y2,y3 > o (c) Minimize 60yi — 10j2 + 20^3 subject to 5y\ - 3^2 + w > - l y\ + 8^2 + 7y3 > yi,3'2 > 0, )>3 unrestricted (f) Maximize 50xi — 70x2 — 15*3 subject to 4xi < 2x2 > —X\ — X2 + X3 > x\ unrestricted, X2,x3 > 0, (b) Mine-F is 411, attained at ( | , | ) (c) Maxc-X is l | , attained at ( ^ , , ^ ) Problem Set 4.5 (a) (1,1,0,0) optimal; complementary slackness generates (2,2,0), a feasible solution to the dual (b) (0,4,0,2) optimal; complementary slackness generates (3,2,0), a feasible solution to the dual 450 SOLUTIONS TO SELECTED PROBLEMS (c) (3,0,1,0,5) not optimal; complementary slackness generates (0,1,3), but this point is not a feasible solution to the dual Problem Set 5.1 (b) Upper limit equals 14.25 No Since | < ||g < i°-, the daily minimum cost of an adequate diet would be 124 + 2(120), that is, $3.64 This is an increase of $1.20 over the original daily minimum cost of $2.44 and so, over weeks, would cost $16.80 10 An increase of % in the bluegrass requirement should increase the cost of producing 100 lb of the composition by $0.50; and an increase of 1% in the fescue requirement should have no effect on costs Problem Set 5.2 -2 R-l , cB -6 > B - 24 (b) b*=B-lb=[5,8]' Thereforeb = 5A^ + 8A 0