Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page i 1.9.2007 11:49am Compositor Name: BMani Optimization Modelling A Practical Approach Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page ii 1.9.2007 11:49am Compositor Name: BMani Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page iii 1.9.2007 11:49am Compositor Name: BMani Optimization Modelling A Practical Approach Ruhul A Sarker Charles S Newton Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page iv 1.9.2007 11:49am Compositor Name: BMani CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-13: 978-1-4200-4310-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Sarker, Ruhul A Optimization modelling : a practical introduction / Ruhul A Sarker and Charles S Newton p cm Includes bibliographical references and index ISBN 978-1-4200-4310-5 (alk paper) Mathematical models Mathematical optimization I Newton, Charles S (Charles Sinclair), 1942- II Title QA401.S266 2007 658.4’0352 dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2007013747 Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page v 1.9.2007 11:49am Compositor Name: BMani Table of Contents List of Figures xv List of Tables xxi List of Mathematical Notations xxiv Preface xxv Acknowledgments xxix Authors xxxi Section I Introduction to Optimization and Modelling Introduction 1.1 General Introduction .3 1.2 History of Optimization .4 1.3 Optimization Problems 1.4 Mathematical Model 1.4.1 Characteristics and Assumptions 1.5 Concept of Optimization 1.6 Classification of Optimization Problems 11 1.7 Organization of the Book 13 Exercises 14 References 15 The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Process of Optimization .17 Introduction 17 Decision Process 17 Problem Identification and Clarification 19 Problem Definition 20 Development of a Mathematical Model 21 2.5.1 Measure of Effectiveness 23 Deriving a Solution 25 Sensitivity Analysis .26 Testing the Solution 26 Implementation 27 v Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page vi 1.9.2007 11:49am Compositor Name: BMani vi 2.10 Summary .28 Exercises 29 Introduction to Modelling 31 3.1 Introduction 31 3.2 Components of a Mathematical Model 31 3.2.1 Decision Variables 32 3.2.2 Objective Function 32 3.2.3 Constraints 32 3.3 Simple Examples 32 3.4 Analyzing a Problem 34 3.4.1 A Nonmathematical Programming Problem 35 3.5 Modelling a Simple Problem .36 3.5.1 Defining the Variables 37 3.5.2 Objective Function 37 3.5.3 Constraints 37 3.6 Linear Programming Model 39 3.7 More Mathematical Models .39 3.8 Integer Programming 42 3.9 Multi-Objective Problem 45 3.9.1 Objective versus Goal 47 3.10 Goal Programming 47 3.11 Nonlinear Programming 49 3.12 Summary .52 Exercises 52 Section II Modelling Techniques Simple Modelling Techniques I 59 4.1 Introduction 59 4.2 Use of Subscripts in Variables 59 4.3 Simple Modelling Techniques 60 4.3.1 Additional Work Requirement in the Formulation 61 4.3.2 Variables as Fractions of Other Variables 64 4.3.3 Maintaining Certain Ratios among Different Variables 68 4.3.4 One Constraint Is a Fraction of Another Constraint 70 4.3.5 Maxi–Min or Mini–Max Objective Function 75 4.3.6 Multi-Period Modelling 77 4.3.7 Transforming Infeasible Solutions to Satisfactory Solutions 79 4.3.8 Single to Multiple Objectives 81 4.4 Special Types of Linear Programming 82 4.4.1 Transportation Problem 83 4.4.2 Assignment Problem 86 4.4.3 Transshipment Problem 88 4.4.4 Project Management Problem 91 Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page vii 1.9.2007 11:49am Compositor Name: BMani vii 4.5 Summary .98 Exercises 98 Bibliography 102 Simple Modelling Techniques II .103 5.1 Introduction 103 5.2 Precedence Constraints 103 5.3 Either–or Constraints 104 5.4 K out of N Constraints Must Hold 105 5.5 Yes-or-No Decisions 106 5.6 Functions with N Possible Values 108 5.7 Mutually Exclusive Alternatives and Contingent Decisions 109 5.8 Linking Constraints with the Objective Function 111 5.9 Piecewise Linear Functions 113 5.10 Nonlinear to Approximate Functions 116 5.11 Deterministic Models with Probability Terms 118 5.12 Alternate Objective Functions 121 5.13 Constrained to Unconstrained Problem 122 5.14 Simplifying Cross Product of Binary Variables 124 5.15 Fractional Programming 126 5.16 Unrestricted Variables .128 5.17 Changing Constraint and Objective Type .129 5.17.1 From to ¼ Constraints 129 5.17.2 From ! to ¼ Constraints 130 5.17.3 From ! to Constraints 130 5.17.4 From to ! Constraints 130 5.17.5 From ¼ Constraint to ! and Constraints 130 5.17.6 Changing Objective Type 131 5.18 Conditional Constraints 132 5.19 Dual Formulation .133 5.20 Regression Model 136 5.21 Stochastic Programming 137 5.22 Constraint Programming 137 5.23 Summary .138 Exercises 138 Bibliography 142 References 143 Modelling Large-Scale and Well-Known Problems I 145 6.1 Introduction 145 6.2 Use of the Summation (S) Sign .145 6.3 Use of the Subset (2) Sign 147 6.4 Network Flow Problems 149 6.4.1 Shortest Path Problem 149 6.4.2 Maximum Flow Problem 150 6.4.3 Multi-Commodity Flow Problem 152 Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page viii 1.9.2007 11:49am Compositor Name: BMani viii 6.5 Knapsack Problem .154 6.5.1 Capital Budgeting Problem 154 6.5.2 Bin Packing Problem 155 6.5.3 Cutting Stock Problem 157 6.6 Facility Location and Layout .159 6.6.1 Facility Location Problem 159 6.6.2 Facility Layout Problem 161 6.7 Production Planning and Scheduling 164 6.7.1 Relevant Literature 165 6.8 Logistics and Transportation .167 6.8.1 Airlift Problem 167 6.8.2 Relevant Literature 168 6.9 Summary .170 Exercises 170 References 172 Modelling Well-Known Problems II .177 7.1 Introduction 177 7.2 Job and Machine Scheduling .177 7.2.1 Relevant Literature 179 7.3 Assignment and Routing 180 7.3.1 Generalized Assignment Problem 180 7.3.2 Traveling Salesperson Problem 181 7.3.3 Relevant Literature on Traveling Salesperson Problem 184 7.3.4 Vehicle Routing Problem 185 7.3.5 Relevant Literature on Vehicle Routing Problem 188 7.4 Staff Rostering and Scheduling 189 7.4.1 Staff Scheduling: A Weekly Problem 189 7.4.2 Daily Rostering Problem 191 7.4.3 Relevant Literature on General Staff Scheduling 192 7.4.4 Crew Planning=Scheduling Problem 193 7.5 Scheduling and Timetabling Problem 194 7.5.1 School Timetabling Problem 194 7.5.2 University Timetabling 196 7.5.3 Relevant Literature 197 7.6 Summary .199 Exercises 199 References 201 Alternative Modelling 205 8.1 Introduction 205 8.2 Modelling under Different Assumptions 205 8.2.1 A Coal Blending Problem 205 8.2.2 First Alternative Blending Model 207 8.2.3 Second Alternative Blending Model 209 Sarker/Optimization Modelling: A Practical Approach 43102_C000 Final Proof page ix 1.9.2007 11:49am Compositor Name: BMani ix 8.2.4 Comparing the Two Simple Alternative Models 210 8.2.5 A Crop Planning Problem 211 8.2.6 Crop Planning Model 212 8.2.7 Crop Planning Model 213 8.3 Hierarchical Modelling: An Introduction 214 8.3.1 Hierarchical Modelling in a Manufacturing Context 215 8.3.2 Aggregate Model 216 8.3.3 Family Scheduling Model 217 8.3.4 Individual Item Scheduling Model 218 8.4 Summary 219 References 220 Section III Model Solving Solution Approaches: An Overview 223 9.1 Introduction .223 9.2 Complexity and Complexity Classes 223 9.2.1 Complexity of Algorithms 223 9.2.2 Complexity Classes 224 9.3 Classical Optimization Techniques 225 9.3.1 Linear Programming 225 9.3.2 Integer Programming: The Curse of Dimensionality 227 9.3.3 Integer Linear Program: Solution Approaches 228 9.3.4 Special Linear Programming Models 230 9.3.5 Goal Programming 230 9.3.6 Nonlinear Programming 231 9.3.7 Multi-Objective Models 232 9.4 Heuristic Techniques 233 9.4.1 Hill Climbing 233 9.4.2 Simulated Annealing 233 9.4.3 Tabu Search 234 9.4.4 Genetic Algorithms 234 9.4.5 Ant Colony Optimization 235 9.4.6 Memetic Algorithms 236 9.4.7 Other Heuristics 236 9.5 Optimization Software 236 9.5.1 LINGO=LINDO 237 9.5.2 MPL with OptiMax 2000, CPLEX, and XPRESS 237 9.5.3 GAMS 237 9.5.4 Solver and Premium Solver 238 9.5.5 Win QSB 238 9.5.6 MINOS 238 9.6 Summary 239 Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof page 457 1.9.2007 11:48am Compositor Name: BMani Solving Practical Problems 457 LIMIT = 32.1 113.2 157 141.5 32.1 742.6 61.1 90.9 131.5 106.4 84 400.6 93.5 121.1 183.4 146 127.3 524.1 129.6 154 236.1 196 203.4 697.5 173.9 191.8 295.6 242.6 295.8 888.8 SLOPE = 6.735 3.778 3.749 2.500 5.0 0.0 0.936 1.303 1.395 0.874 2.233 0.243 0.457 0.760 0.841 0.521 1.004 0.136 0.229 0.124 0.4214 0.223 0.476 0.259 0.3193 0.159 0.594 0.339 0.064 0.035 230.9 292.3 358.4 231.9 280.2 347.6 346.4 439.5 530.1 306.8 388.3 455.4 400.6 524.1 697.5 1144.3 1446.5 0; 0.071 0.028 −0.0003 0.094 0.026 0.005 0.093 0.055 0.009 0.086 0.045 0.002 0.178 0.085 0.045 0.049 −0.004 −0.001; ENDDATA END FIGURE 15.8 (continued) TABLE 15.16 Summary of Solutions Well Gas Used Oil Produced Total ¼ 1,013.400 1,183.100 2,255.800 1,527.600 2,364.800 3,655.300 12,000.00 391.9329 774.8991 1,175.803 667.8241 844.2198 300.4421 4,155.121 For a limit of 12,000 units of gas, the solution of the model suggests 4155.121 units of oil production The well-wise gas consumption and oil production can be summarized as shown in Table 15.16 15.7 Summary In this chapter, we solved a number of representative practical problems and discussed their solutions The problems considered were a sanitary-wares product-mix problem, a crop planning problem, a two-stage transportation problem, a power generation and planning problem, and a gas-lift optimization problem The data preparation, LINGO codes, model solving difficulties, model validation, and practical model solving issues were also discussed Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof 458 page 458 1.9.2007 11:48am Compositor Name: BMani Optimization Modelling: A Practical Approach References Begum, S., Determination of optimum generation mix for the power sector of Bangladesh, Unpublished M.Sc Thesis, Industrial and Production Engineering, Bangladesh University of Engineering and technology, Dhaka, 2004 Buitrago, S., Rodriguez, E., and Espin, D., Global optimization techniques in gas allocation for continuous flow gas lift systems, SPE Gas Technology Conference, Calgary, Canada, SPE35616, 375, 1996 Sarker, R and Haque, A., Industrial engineering (IE) techniques in production decision, Journal of the Institution of Engineers, Bangladesh, 13, 67, 1985 Sarker, R., Talukder, S., and Haque, A., Determination of optimum crop-mix for crop cultivation in Bangladesh, Applied Mathematical Modelling, 21, 621, 1997 Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof page 459 1.9.2007 11:48am Compositor Name: BMani Appendix-15A Crop Planning Linear Programming Model Objective function: Maximize Z ẳ 1134 X_1_1_1 ỵ 116 X_2_2_1 ỵ 2773 X_3_3_1 ỵ 2195 X_4_4_1 ỵ 291 X_5_5_1 ] 702 X_6_6_1 ỵ 8125 X_7_7_1 ỵ 7090 X_8_8_1 ỵ 47722 X_9_9_1 ỵ 16761 X_10_10_1 þ 1134 X_1_1_2 þ 2773 X_3_1_2 þ 1134 X_1_2_2 þ 2195 X_4_2_2 ỵ 1134 X_1_3_2 ỵ 291 X_5_3_2 ỵ 1134 X_1_4_2 ỵ 8225 X_7_4_2 ỵ 116 X_2_5_2 ỵ 2195 X_4_5_2 þ 116 X_2_6_2 þ 291 X_5_6_2 þ 116 X_2_7_2 þ 8125 X_7_7_2 ỵ 2773 X_3_8_2 ỵ 2195 X_4_8_2 ỵ 2773 X_3_9_2 ỵ 291 X_5_9_2 ỵ 2773 X_3_10_2 ] 702 X_6_10_2 þ 2773 X_3_11_2 þ 8125 X_7_11_2 þ 2195 X_4_12_2 ] 702 X_6_12_2 ỵ 2195 X_4_13_2 ỵ 7090 X_8_13_2 ỵ 291 X_5_14_2 ] 702 X_6_14_2 ỵ 291 X_5_15_2 ỵ 7090 X_8_15_2 ] 702 X_6_16_2 ỵ 8125 X_7_16_2 ỵ 8125 X_7_17_2 ỵ 7090 X_8_17_2 ỵ 1134 X_1_1_3 ỵ 2773 X_3_1_3 ỵ 2195 X_4_1_3 ỵ 1134 X_1_2_3 ỵ 2773 X_3_2_3 ỵ 291 X_5_2_3 þ 1134 X_1_3_3 þ 2773 X_3_3_3 þ 8125 X_7_3_3 þ 2773 X_3_4_3 ỵ 2195 X_4_4_3 ] 702 X_6_4_3 ỵ 2773 X_3_5_3 ỵ 291 X_5_5_3 ] 702 X_6_5_3 ] 702 X_6_6_3 þ 2773 X_3_6_3 þ 8125 X_7_6_3 ] 5940 I_1 ] 5940 I_2 ] 5940 I_3 ] 5940 I_4 ] 4609 I_5 ] 12341 I_7 Constraints: (1) ỵ 0.46 X_1_1_1 ỵ 0.46 X_1_1_2 ỵ 0.46 X_1_2_2 ỵ 0.46 X_1_3_2 ỵ 0.46 X_1_4_2 ỵ 0.46 X_1_1_3 ỵ 0.46 X_1_2_3 ỵ 0.46 X_1_3_3 þ I_1 ! 2274566 (2) þ 0.41 X_2_2_1 þ 0.41 X_2_5_2 ỵ 0.41 X_2_6_2 ỵ 0.41 X_2_7_2 ỵ I_2 ! 892498 ỵ 0.7 X_3_3_1 ỵ 0.7 X_3_1_2 ỵ 0.7 X_3_8_2 þ 0.7 X_3_9_2 þ 0.7 X_3_10_2 þ 0.7 X_3_11_2 þ 0.7 X_3_1_3 ỵ 0.7 X_3_2_3 ỵ 0.7 X_3_3_3 ỵ 0.7 X_3_4_3 ỵ 0.7 X_3_5_3 ỵ 0.7 X_3_6_3 ỵ I_3 ! 8785105 (3) (4) ỵ 1.056 X_4_4_1 ỵ 1.056 X_4_2_2 ỵ 1.056 X_4_5_2 ỵ 1.056 X_4_8_2 ỵ 1.056 X_4_12_2 ỵ 1.056 X_4_13_2 ỵ 1.056 X_4_1_3 ỵ 1.056 X_4_4_3 ỵ I_4 ! 7102647 459 Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof page 460 1.9.2007 11:48am Compositor Name: BMani Optimization Modelling: A Practical Approach 460 (5) ỵ 0.75 X_5_5_1 ỵ 0.75 X_5_3_2 þ 0.75 X_5_6_2 þ 0.75 X_5_9_2 þ 0.75 X_5_14_2 þ 0.75 X_5_15_2 ỵ 0.75 X_5_2_3 ỵ 0.75 X_5_5_3 ỵ I_5 ! 1180728 (6) ỵ 0.658 X_6_6_1 ỵ 0.658 X_6_10_2 ỵ 0.658 X_6_12_2 X_6_14_2 ỵ 0.658 X_6_16_2 ỵ 0.658 X_6_4_3 ỵ 0.658 ỵ 0.658 X_6_6_3 ! 287016 ỵ 0.921 X_7_7_1 ỵ 0.921 X_7_4_2 ỵ 0.921 X_7_7_2 X_7_11_2 ỵ 0.921 X_7_16_2 ỵ 0.921 X_7_17_2 ỵ 0.921 ỵ 0.921 X_7_6_3 ỵ I_7 ! 4074360 (7) (8) (9) ỵ 0.658 X_6_5_3 ỵ 0.921 X_7_3_3 þ 0.935 X_8_8_1 þ 0.935 X_8_13_2 þ 0.935 X_8_15_2 þ 0.935 X_8_17_2 ! 417513 ỵ 9.453 X_9_9_1 ! 5942200 (10) (11) ỵ 0.506 X_10_10_1 ! 138600 ỵ X_1_1_1 ỵ X_2_2_1 þ X_3_3_1 þ X_4_4_1 þ X_5_5_1 þ X_6_6_1 þ X_7_7_1 þ X_8_8_1 þ X_9_9_1 þ X_10_10_1 7727000 (12) 0.5 X_1_1_2 þ 0.5 X_3_1_2 þ 0.5 X_1_2_2 þ 0.5 X_4_2_2 þ 0.5 X_1_3_2 ỵ 0.5 X_5_3_2 ỵ 0.5 X_1_4_2 ỵ 0.5 X_7_4_2 ỵ 0.5 X_2_5_2 ỵ 0.5 X_4_5_2 ỵ 0.5 X_2_6_2 þ 0.5 X_5_6_2 þ 0.5 X_2_7_2 þ 0.5 X_7_7_2 þ 0.5 X_3_8_2 ỵ 0.5 X_4_8_2 ỵ 0.5 X_3_9_2 ỵ 0.5 X_5_9_2 ỵ 0.5 X_3_10_2 ỵ 0.5 X_6_10_2 ỵ 0.5 X_3_11_2 þ 0.5 X_7_11_2 þ 0.5 X_4_12_2 þ 0.5 X_6_12_2 þ 0.5 X_4_13_2 ỵ 0.5 X_8_13_2 ỵ 0.5 X_5_14_2 ỵ 0.5 X_6_14_2 ỵ 0.5 X_5_15_2 ỵ 0.5 X_8_15_2 ỵ 0.5 X_6_16_2 þ 0.5 X_7_16_2 þ 0.5 X_7_17_2 9615000 (13) þ þ þ þ þ þ (14) 0.333333 X_1_1_3 þ 0.333333 X_3_1_3 þ 0.333333 X_4_1_3 0.333333 X_1_2_3 þ 0.333333 X_3_2_3 þ 0.333333 X_5_2_3 0.333333 X_1_3_3 ỵ 0.333333 X_3_3_3 ỵ 0.333333 X_7_3_3 0.333333 X_3_4_3 ỵ 0.333333 X_4_4_3 ỵ 0.333333 X_6_4_3 0.333333 X_3_5_3 ỵ 0.333333 X_5_5_3 ỵ 0.333333 X_6_5_3 0.333333 X_6_6_3 ỵ 0.333333 X_3_6_3 þ 0.333333 X_7_6_3 2387000 þ 4313 X_1_1_1 þ 3450 X_2_2_1 þ 4702 X_3_3_1 þ 7557 X_4_4_1 þ 4518 X_5_5_1 þ 5233 X_6_6_1 ỵ 2090 X_7_7_1 ỵ 1355 X_8_8_1 ỵ 1680 X_9_9_1 ỵ 12486 X_10_10_1 ỵ 4313 X_1_1_2 ỵ 4702 X_3_1_2 þ 4313 X_1_2_2 þ 7557 X_4_2_2 þ 4313 X_1_3_2 þ 4518 X_5_3_2 ỵ 4313 X_1_4_2 ỵ 2090 X_7_4_2 ỵ 3450 X_2_5_2 ỵ 7557 X_4_5_2 ỵ 3450 X_2_6_2 ỵ 4518 X_5_6_2 þ 3450 X_2_7_2 þ 2090 X_7_7_2 þ 4702 X_3_8_2 þ 7557 X_4_8_2 ỵ 4702 X_3_9_2 ỵ 4518 X_5_9_2 ỵ 4702 X_3_10_2 ỵ 5233 X_6_10_2 ỵ 4702 X_3_11_2 ỵ 2090 X_7_11_2 þ 7557 X_4_12_2 þ 5233 X_6_12_2 þ 7557 X_4_13_2 þ 1355 X_8_13_2 ỵ 4518 X_5_14_2 ỵ 5233 X_6_14_2 ỵ 4518 X_5_15_2 ỵ 1355 X_8_15_2 ỵ 5233 X_6_16_2 ỵ 2090 X_7_16_2 þ 2090 X_7_17_2 þ 1355 X_8_17_2 þ 4313 X_1_1_3 þ 4702 X_3_1_3 ỵ 7557 X_4_1_3 ỵ 4313 X_1_2_3 ỵ 4702 X_3_2_3 Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof Solving Practical Problems ỵ ỵ ỵ ỵ X_1_1_2 ] X_3_1_2 ẳ X_1_2_2 ] X_4_2_2 ¼ X_1_3_2 ] X_5_3_2 ¼ (18) (19) X_1_4_2 ] X_7_4_2 ¼ X_2_5_2 ] X_4_5_2 ¼ (20) (21) X_2_6_2 ] X_5_6_2 ¼ X_2_7_2 ] X_7_7_2 ¼ (22) X_3_8_2 ] X_4_8_2 ¼ X_3_9_2 ] X_5_9_2 ¼ X_3_10_2 ] X_6_10_2 ¼ (23) (24) (25) (26) (27) 461 4518 X_5_2_3 ỵ 4313 X_1_3_3 ỵ 4702 X_3_3_3 ỵ 2090 X_7_3_3 4702 X_3_4_3 þ 7557 X_4_4_3 5233 X_6_4_3 þ 4702 X_3_5_3 4518 X_5_5_3 þ 5233 X_6_5_3 þ 5233 X_6_6_3 þ 4702 X_3_6_3 2090 X_7_6_3 1.7E ỵ 11 (16) (17) (15) X_3_11_2 ] X_7_11_2 ¼ X_4_12_2 ] X_6_12_2 ¼ X_4_13_2 ] X_8_13_2 ¼ (30) (31) X_5_14_2 ] X_6_14_2 ¼ X_5_15_2 ] X_8_15_2 ¼ X_6_16_2 ] X_7_16_2 ¼ X_7_17_2 ] X_8_17_2 ¼ (32) X_1_1_3 ] X_3_1_3 ¼ (33) (34) X_3_1_3 ] X_4_1_3 ¼ X_1_2_3 ] X_3_2_3 ¼ (35) (36) X_3_2_3 ] X_5_2_3 ¼ X_1_3_3 ] X_3_3_3 ¼ (37) (38) (39) X_3_3_3 ] X_7_3_3 ¼ X_3_4_3 ] X_4_4_3 ¼ X_4_4_3 ] X_6_4_3 ¼ (40) (41) X_3_5_3 ] X_5_5_3 ¼ X_5_5_3 ] X_6_5_3 ¼ (42) X_6_6_3 ] X_3_6_3 ¼ (43) (44) X_3_6_3 ] X_7_6_3 ẳ I_1 ỵ I_2 þ I_3 þ I_4 þ I_5 (45) X_9_9_1 þ X_10_10_1 (28) (29) page 461 1.9.2007 11:48am Compositor Name: BMani 1969350 Sarker/Optimization Modelling: A Practical Approach 43102_C015 Final Proof page 462 1.9.2007 11:48am Compositor Name: BMani Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof page 463 1.9.2007 11:50am Compositor Name: BMani Index A Activity on arc network, 92 Activity on node network, 92 Adding constraints in solver, 267 Addition of new variable, 331 Additional problem solving, 309–324 Aggregate model, 216 Airlift problem, 167 Alternate objective function, 121 Alternative modelling, 205 Analog model, 22 Ant colony optimization, 235 AOA network, 92 AON network, 92 Approximating nonlinear function, 117 Assignment and routing, 180 Assignment problem, 86 B Basic optimization techniques, 347 Basis, 357 Batch sizing problem, 122 Bin packing problem, 155 Binary variables, 107, 124 Binding constraint, 328 Blending model, 207 Branch-and-bound method, 228, 365–374 Branch-and-cut, 228 C Capacity planning problem, 107 Capital budgeting problem, 43, 154 Changes in constraint coefficient, 332 Changes in objective coefficients, 331 Changes in RHS values, 332 Changing cell, 268 Changing constraint type, 129–131 Changing objective type, 131 Classical optimization techniques, 225 Coal bank scheduling problem, 423 dynamic model, 426 static model, 424 Coal blending problem, 205, 207–210 Coal production and marketing problem, 398 multi-objective model, 403 multi-period model, 403 single-period model, 399 Combat logistics problem, 411 Company registers, 278 Complexity and complexity classes, 223 Complexity classes, 225 Complexity of algorithms, 223 Conditional constraint, 132 Conference organizer problem, 34, 63–64 Constrained problem, 122, 232 Constrained to unconstrained, 122 Constrained nonlinear model, 123 Constraint and variable reduction, 448 Constraint programming, 137 Constraints, 31, 32 Contingent decision, 109 Convexity, 12 Cooperative search, 236 CPLEX, 237 Crew planning, 193 Crew scheduling, 194 Crop mix problem, 71 Crop planning model, 212–214 Crop planning problem, 211, 284, 381, 445–452 constraint and variable reduction, 448 GP model, 385 LP model, 382 multi-objective model, 451 scaling the model, 450 solution, 445–452 working with solutions, 450 Cross product of variables, 124 463 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof 464 page 464 1.9.2007 11:50am Compositor Name: BMani Optimization Modelling: A Practical Approach Cultural algorithms, 236 Curtain material trim loss problem, 33, 61–63 Cutting plane, 228 Cutting stock problem, 157 D Daily rostering problem, 191 Data, 277 Data aggregation, 283 crop planning, 284 hierarchical production planning, 287 power generation planning, 286 production planning, 285 Data collection, 277 Data collection methods, 278 Data from interviews, 280 Data preparation, 280 data requirements, 282 logistics problem, 281 Data preprocessing, 287 Data quality, 278 Data type, 279 data from interviews, 280 forecasted data, 279 historical data, 280 stationary data, 279 stochastic data, 280 survey data, 280 text data, 280 time varying data, 279 Data-driven model, 292 Decision process, 17, 19 Decision variables, 31, 32 Defence application, 119, 121, 167, 411 airlift problem, 167 combat logistics problem, 411 nonlinear war planning problem, 121 war planning problem, 119 Descriptive model, 22 Deterministic model, 22, 118 with probability terms, 118 Deviational variables, 47 Diet problem, 33, 40–41 Differentiability, 13 Dual formulation, 133 Dummy activity, 93 Dynamic model, 426 E Economic order quantity, Either–or constraint, 104 Entering variables, 361 Example problem approximating nonlinear function, 117 assignment problem, 86 batch sizing problem, 122 capacity planning problem, 107 capital budgeting problem, 43 conference organizer problem, 34 constrained nonlinear model, 123 crop mix problem, 71 curtain material trim loss problem, 33 diet problem, 33 dual formulation, 133, 134 facility layout problem, 159, 161 financial management problem, 65 fixed charge problem, 111 goal programming problem, 47 job sequencing problem, 104 knapsack problem, 44 location problem, 50, 109 manufacturing planning problem, 73, 75 multi-objective product mix problem, 82 multi-period production planning problem, 77 mutually exclusive constraint, 105 nonlinear war planning problem, 121 network flow problem, 89 oil blending problem, 66 piecewise linear function, 113 product mix problem, 32, 69, 79 production planning problem, 45, 75, 108 productivity maximization, 127 project management problem, 95 sales prediction, 136 sequencing problem, 104 transportation problem, 83 vehicle mix problem, 33 war planning problem, 119 Excel solver, 264 adding constraints, 267 changing cell, 266 important options, 269 solution approaches, 270 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof page 465 1.9.2007 11:50am Compositor Name: BMani Index solving LP, 264 target cell, 266 F Facility layout problem, 159, 161 Facility location, 159 Family scheduling model, 217 Fixed charge problem, 111 Force effectiveness, 24 Forecasted data, 279 Fractional programming, 126 Function approximation, 116 Function with N possible values, 108 G GAMS, 237, 260 Gantt charts, Gap between solution and outcomes, 342 GAP, 180 Gas-lift optimization problem, 429, 455–457 General blending problem, 404 GP model, 408 LP model, 407 General staff scheduling, 192 Generalized assignment problem, 180 Genetic algorithms, 234 Goal constraint, 47, 80 Goal programming, 47, 80, 81, 336 deviational variables, 47 goal constraint, 47 hierarchy of priority, 231 nonpreemptive, 230 overachievement deviation, 47 preemptive, 230 soft constraint, 47 solution approaches, 230 underachievement deviation, 48 undesirable deviation, 47 GP model, 385, 394, 408 Graphical method of LP, 347 example, 348–355 graphing objective function, 352–353 graphing the feasible region, 348 optimal solution, 354 Graphical method, 225 Graphing objective function, 352–353 Graphing the feasible region, 348 465 H Hard constraint, 13 Heuristic model, 22 Heuristic techniques, 233 Hierarchical modelling, 214 aggregate model, 216 family scheduling model, 217 item scheduling model, 218 level of planning, 215–216 product structure, 215–216 sub-problems, 215–216 Hierarchical production planning, 215, 287 Hierarchy of priority, 231 Hill climbing, 233 Historical data, 280 History of optimization, HPP, 215 Hungarian method, 230 I Iconic model, 22 Immune system, 236 Implementing the solution, 27–28 Important options in solver, 269 Incidence matrix, 183 Infeasible solution, 79 Initial solution, 340, 356 Input preparation, 277 Integer linear program branch-and-bound, 228 branch-and-cut, 228 cutting plane, 228 Integer programming, 42; see also IP branch-and-bound method, 365–374 curse of dimensionality, 227 Interior point method, 226 IP, 228; see also Integer programming Item scheduling model, 218 J Job and machine scheduling, 177 Job sequencing problem, 104 Joint lot sizing and transportation problem, 420 K Knapsack problem, 44, 154 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof 466 page 466 1.9.2007 11:50am Compositor Name: BMani Optimization Modelling: A Practical Approach L Leaving variable, 361 Lexicographic simplex method, 226 Linear programming, 4, 12; see also LP model basic assumptions, 39 graphical method, 225, 347–355 interior point method, 226 simplex method, 226, 355–365 solution approaches, 225 special types, 82 Linear vs integer model, 340 Linear vs nonlinear relationships, 339, 340 LINGO=LINDO, 237, 241 additional problem solving, 311–326 example, 246–251 inputting model, 241 solver status window, 243 solving the model, 243 special features, 244 syntax, 252 LINGO example, 302–304, 310, 313 modified transportation model, 313 transportation model, 310 two-stage transportation problem, 302–304 Linking constraint, 111 Location problem, 50, 109 Logistics and transportation, 167 Logistics problem, 281 Lot sizing problem, 415 finished product inventory, 417 raw material inventory, 418 total cost function, 419 LP model, 382, 387, 393, 407 M Management issues in implementation, 309 Manufacturing planning problem, 73, 75 Mathematical model, 6, 18, 21, 22, 31 components, 31 constraints, 31 decision variables, 31 objective function, 31 subscripts in variables, 59 use of subset sign, 147 use of summation sign, 145 Mathematical programming model, Mathematical properties, 12 convexity, 12 differentiability, 13 multimodal, 13 Maximal flow problem, 150 Maxi-min objective, 75 Measure of effectiveness, 23 Memetic algorithms, 236 Military environment, 26 Mini-max objective, 75 MINOS, 238 Model, 22 Model development, 19, 21 Model solving example Excel solver, 264 LINGO and MPL, 295 nonlinear model, 298 product mix problem, 292 two-stage transportation problem, 300 Model solving, 277, 292 Model validation, 452 Model-driven data, 292 Modelling simple problem conference organizer problem, 63–64 curtain material trim loss problem, 61–63 diet problem, 40–41 product mix problem, 36–39 vehicle mix problem, 41–42 Models for practical problem, 381, 411 Modified transportation model, 313 Monolithic model, 215 MPL, 237, 253 product-mix model, 256 use of MPL, 253 using vectors and indexes, 255 Multi-commodity flow problem, 152 Multi-dimensional search, 231 Multimodal, 13 Multi-objective model, 391, 403, 451 Pareto frontier, 233 Pareto-optimal, 232 simultaneous optimization, 233 solution approaches, 232 trade-off surface, 232 weighting method, 232 Multi-objective optimization, 336 Multi-objective problem, 45 Multi-objective product mix problem, 82 Multi-period model, 401 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof page 467 1.9.2007 11:50am Compositor Name: BMani Index Multi-period modelling, 77 Multi-period production planning problem, 77 Multiple objective, 81 Multiple shift planning problem, 432 Mutually exclusive alternative, 109 Mutually exclusive constraint, 105 N Negative RHS, 336 Network flow problem, 149 maximal flow problem, 150 multi-commodity flow problem, 152 shortest path problem, 149 Node-arc incidence matrix, 183 Nonbinding constraint, 328, 329 Nonlinear model, 298 Nonlinear programming, 49 constrained, 232 multi-dimensional search, 231 one-dimensional search, 231 penalty function, 232 solution approaches, 231 unconstrained, 231 Nonlinear war planning problem, 121 Nonpreemptive GP, 230 Non-smooth relationships, 339 Nontechnical report, 341 Normative model, 22 Network flow problem, 89 Number of basic variables, 337 O Objective function, 31, 32 Objective versus goal, 47 Observations, 278 Oil blending problem, 66 One-dimensional search, 231 Operational research, Optimal solution, 353 OptiMax, 237 Optimization model, 22 Optimization problems, Optimization process, 17 Optimization software, 236 CPLEX, 237 GAMS, 237 LINGO=LINDO, 237 467 MINOS, 238 MPL, 237 Premium solver, 238 Solver, 238 WinQSB, 238 XPRESS, 237 Optimization technique, 4, 10, 17 Optimization, 4, Output analysis, 325 Overachievement deviation, 47 P Pareto frontier, 233 Pareto optimal, 232 Penalty function, 232 Piecewise linear function, 113 Pivot element, 362 Power generation planning, 286, 387, 452–455 LP model, 387 multi-objective model, 391 model validation, 452 solution, 452-455 Practical issues and tips, 336 gap between solution and outcomes, 342 goal programming, 336 initial solution, 340 linear vs integer model, 340 linear vs nonlinear relationships, 339, 340 management issues in implementation, 309 multi-objective optimization, 336 negative RHS, 338 non-smooth relationships, 339 nontechnical report, 342 number of basic variables, 337 reduction of constraints, 336 reduction of variables, 336 scaling factor in modelling, 339 special cases in LP models, 342 unrestricted variables, 338 variable bounds, 340 Practical issues, 327 Practical problem coal bank scheduling, 423 coal production and marketing, 398 combat logistics, 411 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof 468 Optimization Modelling: A Practical Approach crop planning, 381 general blending, 404 joint lot sizing and transportation, 420 lot sizing, 415 multiple shift planning, 432 power generation, 387 scaffolding system, 427 supply chain, 395 water supply, 392 Practical problem solving, 437 crop planning, 445–452 gas-lift optimization, 455–457 power generation planning, 452–455 product mix, 437–443 two-stage transportation, 443–445 Precedence constraint, 92–95, 103 Precedence relationship, 92–95 Predictive model, 22 Preemptive GP, 230 Premium solver, 238 Prescriptive model, 22 Problem classification, 11–12 Problem definition, 19, 20 Problem identification, 19 Product mix problem, 32, 36–39, 69, 79, 292, 437–443 number of products required, 440, 442–443 scaling problem, 439–441 solution, 437–443 Product structure, 215–216 Production planning, 45, 75, 285 Production planning and scheduling, 164 Production planning problem, 108 Productivity maximization problem, 127 Project, 92 Project management problem, 91, 95 Q Qualitative approach, 17 Quantitative approach, 17 Queuing, R Reduced cost, 328 Reduction of constraints, 336 Reduction of variables, 336 Regression model, 136 page 468 1.9.2007 11:50am Compositor Name: BMani Risk analysis in modelling, 344 Rostering problem, 287 S Sales prediction model, 136 Scaffolding system, 427 Scaling factor in modelling, 339 Scaling problem, 439–441, 450 Scanning, 278 Scheduling and timetabling, 194 School timetabling problem, 194 Sensitivity analysis, 19, 26 addition of new variable, 331 changes in constraint coefficient, 332 changes in objective coefficients, 331 changes in RHS values, 332 integer and nonlinear models, 333 linear programming, 331 Sensors, 278 Sequencing problem, 104 Shadow price, 327 Shortest path problem, 149 Simple modelling techniques, 60 additional work required, 61 alternate objective function, 121 changing constraint type, 129–131 changing objective type, 131 conditional constraint, 132 constrained to unconstrained, 122 contingent decision, 109 cross product of variables, 124 dual formulation, 133 either–or constraint, 104 fractional programming, 126 function approximation, 116 function with N possible values, 108 linking constraint, 111 maintaining certain ratios among different variables, 68 mutually exclusive alternative, 109 mutually exclusive constraint, 105 one constraint is a fraction of another constraint, 70 peicewise linear function, 113 precedence constraint, 103 regression model, 136 unrestriced variables, 128 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof page 469 1.9.2007 11:50am Compositor Name: BMani Index variables as fraction of other variables, 64 yes-or-no decision, 106 Simplex algorithm, Simplex method for LP, 4, 226, 355–365 basis, 357 entering variables, 361 example, 346–365 initial solution, 356 leaving variable, 361 pivot element, 362 Simulated annealing, 234 Simultaneous optimization, 233 Single-period model, 397 Slack, 327 Soft constraint, 13, 47 Soliciting, 278 Solution and reports, 325 Solution approaches, 223 Solution, 437–443 Solver, 238, 264 Special cases in LP models, 342 Special types LP, 230 assignment problem, 86 network flow problem, 89 project management problem, 91 transportation problem, 83 transshipment problem, 88 Staff rostering and scheduling, 189 Staff scheduling problem, 189 Static model, 424 Stationary data, 279 Stochastic data, 280 Stochastic model, 22 Stochastic programming, 137 Subscripts in variables, 59 Subset sign, 147 Subtour, 183 Summation sign, 145 Supply chain problem, 395 Surplus, 329 Survey data, 280 Surveys, 278 Swarm optimization, 236 Symbolic model, 22 T Tabu search, 234 Target cell in solver, 266 469 Technological constraint, 80 Testing the solution, 26 Text data, 280 Time series study, 278 Time varying data, 279 Trade-off surface, 232 Transportation model, 83, 310 Transportation simplex, 230 Transshipment problem, 88 Travelling salesperson problem, 181 TSP, 181 Subtour in TSP, 183 Two-stage transportation problem, 300, 302–304, 443–445 U Unconstrained problem, 122, 231 Underachievement deviation, 48 Undesirable deviation, 47 Unimodal, 13 University timetabling, 196 Unrestricted variables, 128, 338 V Validation, 19 Variable bounds, 340 Variables, 42 Vehicle mix problem, 33, 41–42 Vehicle routing problem, 185 VRP, 185 W Waiting line, War planning problem, 119 Water supply problem, 392 GP model, 394 LP model, 393 Weighting method, 232 WinQSB, 238, 273 Working with solutions, 450 X XPRESS, 237 Y Yes-or-no decision, 106 Sarker/Optimization Modelling: A Practical Approach 43102_C016 Final Proof page 470 1.9.2007 11:50am Compositor Name: BMani ... Zohrul Kabir, IIU, Dhaka, Bangladesh Bob McKay, Seoul National University, Korea Abu Mamun, Qantas Airlines, Australia M Quaddus, CUT, Australia Tapabrata Ray, UNSW, at ADFA, Australia K.C Tan, National...Sarker /Optimization Modelling: A Practical Approach 43102_C000 Final Proof page i 1.9.2007 11:49am Compositor Name: BMani Optimization Modelling A Practical Approach Sarker /Optimization Modelling: ... Kingdom Graham Freeman, UNSW, at ADFA, Australia Eldon Gunn, Dalhousie University, Canada Aman Haque, Pennsylvania State University, United States of America Anwarul Haque, VC, RUET, Bangladesh Zohrul