GIáo trình operations research an introduction 10th global edition by taha

www.ebookslides.com Operations Research An Introduction www.ebookslides.com This page intentionally left blank www.ebookslides.com Operations Research An Introduction Tenth Edition Global Edition Hamdy A Taha University of Arkansas, Fayetteville Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Dubai • Singapore • Hong Kong Tokyo • Seoul • Taipei • New Delhi • Cape Town • Sao Paulo • Mexico City • Madrid • Amsterdam • Munich • Paris • Milan www.ebookslides.com VP/Editorial Director, Engineering/ Computer Science: Marcia J Horton Editor in Chief: Julian Partridge Executive Editor: Holly Stark Editorial Assistant: Amanda Brands Assistant Acquisitions Editor, Global Edition: Aditee Agarwal Project Editor, Global Edition: Radhika Raheja Field Marketing Manager: Demetrius Hall Marketing Assistant: Jon Bryant Team Lead, Program Management: Scott Disanno Program Manager: Erin Ault Director of Operations: Nick Sklitsis Operations Specialist: Maura Zaldivar-Garcia Cover Designer: Lumina Datamatics Media Production Manager, Global Edition: Vikram Kumar Senior Manufacturing Controller, Global Edition: Angela Hawksbee Full-Service Project Management: Integra Software Services Pvt Ltd Cover Photo Credit: © Lightspring/Shutterstock Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2017 The rights of Hamdy A Taha to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, Operations Research An Introduction, 10th edition, ISBN 9780134444017, by Hamdy A Taha published by Pearson Education © 2017 All rights reserved 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 or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 ISBN 10: 1-292-16554-5 ISBN 13: 978-1-292-16554-7 Typeset in 10/12 Times Ten LT Std by Integra Software Services Private Ltd Printed and bound in Malaysia www.ebookslides.com To Karen Los ríos no llevan agua, el sol las fuentes secó ¡Yo sé donde hay una fuente que no de secar el sol! La fuente que no se agota es mi propio corazón —V Ruiz Aguilera (1862) www.ebookslides.com This page intentionally left blank www.ebookslides.com Contents What’s New in the Tenth Edition Acknowledgments About the Author Trademarks Chapter 27 29 31 31 1.1 Introduction 1.2 Operations Research Models 1.3 Solving the OR Model 1.4 Queuing and Simulation Models 1.5 Art of Modeling 1.6 More than Just Mathematics 1.7 Phases of an OR Study 1.8 About this Book Bibliography Chapter 25 What Is Operations Research? Problems 23 31 34 35 36 37 39 41 41 42 Modeling with Linear Programming 45 45 47 2.1 Two-Variable LP Model 2.2 Graphical LP Solution 2.2.1 Solution of a Maximization Model 48 2.2.2 Solution of a Minimization Model 50 2.3 Computer Solution with Solver and AMPL 2.3.1 LP Solution with Excel Solver 52 2.3.2 LP Solution with AMPL 56 2.4 Linear Programming Applications 59 2.4.1 Investment 60 2.4.2 Production Planning and Inventory Control 2.4.3 Workforce Planning 67 2.4.4 Urban Development Planning 70 2.4.5 Blending and Refining 73 2.4.6 Additional LP Applications 76 Bibliography Problems 52 62 76 76 www.ebookslides.com Contents Chapter The Simplex Method and Sensitivity Analysis 99 3.1 LP Model in Equation Form 3.2 Transition from Graphical to Algebraic Solution 3.3 The Simplex Method 103 3.3.1 Iterative Nature of the Simplex Method 103 3.3.2 Computational Details of the Simplex Algorithm 3.3.3 Summary of the Simplex Method 111 100 105 3.4 Artificial Starting Solution 112 3.4.1 M-Method 112 3.4.2 Two-Phase Method 115 3.5 Special Cases in the Simplex Method 3.5.1 Degeneracy 118 3.5.2 Alternative Optima 119 3.5.3 Unbounded Solution 121 3.5.4 Infeasible Solution 122 3.6 Sensitivity Analysis 123 3.6.1 Graphical Sensitivity Analysis 124 3.6.2 Algebraic Sensitivity Analysis—Changes in the Right-Hand Side 128 3.6.3 Algebraic Sensitivity Analysis—Objective Function 3.6.4 Sensitivity Analysis with TORA, Solver, and AMPL 136 3.7 117 Computational Issues in Linear Programming Bibliography Problems 132 138 142 Case Study: Optimization of Heart Valves Production Chapter 99 142 145 Duality and Post-Optimal Analysis 169 169 4.1 Definition of the Dual Problem 4.2 Primal–Dual Relationships 172 4.2.1 Review of Simple Matrix Operations 172 4.2.2 Simplex Tableau Layout 173 4.2.3 Optimal Dual Solution 174 4.2.4 Simplex Tableau Computations 177 4.3 Economic Interpretation of Duality 178 4.3.1 Economic Interpretation of Dual Variables 179 4.3.2 Economic Interpretation of Dual Constraints 180 4.4 Additional Simplex Algorithms 182 4.4.1 Dual Simplex Algorithm 182 4.4.2 Generalized Simplex Algorithm 184 www.ebookslides.com Contents 4.5 Post-Optimal Analysis 185 4.5.1 Changes Affecting Feasibility 186 4.5.2 Changes Affecting Optimality 189 192 Problems 192 Bibliography Chapter Transportation Model and Its Variants 207 207 211 5.1 Definition of the Transportation Model 5.2 Nontraditional Transportation Models 5.3 The Transportation Algorithm 214 5.3.1 Determination of the Starting Solution 216 5.3.2 Iterative Computations of the Transportation Algorithm 220 5.3.3 Simplex Method Explanation of the Method of Multipliers 226 5.4 The Assignment Model 227 5.4.1 The Hungarian Method 227 5.4.2 Simplex Explanation of the Hungarian Method Bibliography 231 230 Case Study: Scheduling Appointments at Australian Tourist Commission Trade Events 232 Problems 236 Chapter Network Model 247 247 6.1 Scope and Definition of Network Models 6.2 Minimal Spanning Tree Algorithm 6.3 Shortest-Route Problem 251 6.3.1 Examples of the Shortest-Route Applications 6.3.2 Shortest-Route Algorithms 255 6.3.3 Linear Programming Formulation of the Shortest-Route Problem 261 250 6.4 Maximal Flow Model 265 6.4.1 Enumeration of Cuts 266 6.4.2 Maximal Flow Algorithm 267 6.4.3 Linear Programming Formulation of Maximal Flow Mode 272 6.5 CPM and PERT 273 6.5.1 Network Representation 274 6.5.2 Critical Path Method (CPM) Computations 6.5.3 Construction of the Time Schedule 279 252 276 www.ebookslides.com 834 Index B Babbage, Charles, 35 Backward pass in CPM, 276 Backward recursive equation in DP, 473 Balance equation in queues, 663 Balancing transportation model, 209–210 Balking in queues, 655 Barrier algorithm, 141 See also Interior point algorithm Basic solution, 101–102, 306–308 relationship to corner (extreme) point, 101, 306 Basic variable, 103, 307 Basis, 307 See also Inverse restricted, 772–774, 779 vector representation of, 307–308 Bayes’ probabilities, 562, 576–579 Bernoulli, Daniel, 579 Bernoulli, Nicolas, 579 Binomial distribution, 551 Poisson approximation of, 551–552 probability calculations with excelStatTables.xls, 551 Birthday problem, 543–544 Blending and refining model, 73–76 Bounded variables definition, 318 dual simplex algorithm for, 324 primal simplex algorithm for, 317–322 Box-Muller sampling method for normal distribution, 719–720 Branch-and-bound algorithm integer programming, 367–373 traveling salesperson (TSP), 441–444 Bridges of Königsberg, 249 Bus scheduling model, 68–70 C Capacitated network model, 22.1–11 conversion to uncapacitated, 22.33 LP equivalence, 22.2–4 simplex-based algorithm, 22.6–11 Capital budgeting, 360–361 Cargo-loading model See Knapsack model Case analysis AHP CIM facility layout, 26.45–54 assignment model scheduling trade events, 26.13–17 Bayes’ probabilities Casey’s medical test evaluation, 26.56–59 decision trees hotel booking limits, 26.53–55 dynamic programming Weyerhauser log cutting, 26.41–45 game theory Ryder Cup matches, 26.59–61 goal programming CIM facility layout, 26.45–54 Mount Sinai hospital, 26.29–33 heuristics bid lines generation at FedEx, 397 fuel tankering, 26.2–9 scheduling trade events, 26.13–17 integer programming Mount Sinai hospital, 26.29–33 PFG building glass, 26.33–40 Qantas telephone sales staffing, 26.74–80 ship routing, 26.21–28 inventory Dell’s supply chain, 26.65–69 Kroger pharmacy inventory management using spreadsheet simulation, linear programming, 501, 531–535, 26.61–65 fuel tankering, 26.2–9 heart valve production, 26.9–13 Markov chains Forest cover change prediction sub-Himalayan India, 629, 26.69–71 queuing internal transport system, 26.72–74 Qantas telephone sales staffing, 26.74–80 shortest route saving federal travel dollars, 26.17–21 transportation ship routing, 26.21–28 traveling salesperson high resolution imaging in Australia, 435 Case studies, E.1–34 decision theory, E.25–28 www.ebookslides.com Index dynamic programming, E.23, E.34 forecasting, E.34 goal programming, E.15–16 integer programming, E.23–25 inventory, E.23–25, E.28–30 linear programming, E.1–7, E.13–15 networks, E.11–13, E.33 queuing, E.30–33 transportation, E.7–11 CDF See Cumulative density function Central limit theorem, 554 Chance-constrained programming, 781–784 Chapman-Kolomogrov equations, 632 Chebyshev model for regression analysis, 356 Chi-square statistical table, 795 Chi-square test See Goodness-of-fit test Circling in LP See Cycling in LP Classical optimization constrained, 746–758 Jacobian method, 747–753 Karush-Khun-Tucker conditions, 754–758 Lagrangean method, 753–754 unconstrained, 741–746 Newton-Raphson method, 744–746 Column-dropping rule in goal programming, 345–350 Computational issues in LP, 138–142 Concave function, D.15 Conditional probability, 544–545 Connected network, 248 Constrained gradient, 749 Constraint programming, 423–425 constraint propagation, 424 Continuous probability distribution, 545 Continuous review in inventory, 505 Convex combination, 306 Convex function, D.15 Convex set, 305 Corner point in LP, 50 See also Extreme point in LP relationship to basic solution, 101 relationship to extreme point in LP, 305 835 Correlation coefficient, 23.6 Covariance, 549 CPM See Critical Path Method Critical activity in CPM: definition, 276 determination of, 277–278 Critical path method (CPM) calculations, 276–278 Cumulative distribution function (CDF), 545 Curse of dimensionality in DP, 489 Cuts in integer programming, 373–378 maximum flow network, 266–267 traveling salesperson problem, 444–445 Cutting plane algorithm ILP, 373–378 TSP, 444–445 Cycle See Loop Cycling in LP, 118–119, 141 D Dantzig, George B., 105, 250, 317, 378, 448, 590 Decision-making, types of, 567–609 certainty, 567–574 risk, 574–581 uncertainty, 581–584 Decision trees, 574–575 Decomposition algorithm, 22.13–21 Degeneracy, 118, 141 See also Cycling in LP Determinant of a square matrix, D.5–6 Deviational variables in goal programming, 342 Dichotomous search, 788 Die rolling experiment, 545, 548 Diet problem, 50–52 Difference Engine, Babbage’s, 35 Dijkstra’s algorithm, 255–258 See also Floyd’s algorithm Direct search method, 763–766 Discrete distribution, 547 Dual price algebraic determination of, 129, 323 graphical determination of, 125 relationship to dual variables, 180 www.ebookslides.com 836 Index Dual problem in LP definition of, 169–172, 322 economic interpretation dual constraint, 180–182 dual variable, 179–180 See also Dual price optimal solution, 174–177, 320–324 use in transportation algorithm, 226–227 weak duality theory, 322 Dual simplex method, 140, 182–184, 324 See also Generalized simplex algorithm artificial constraints in, 184, 185 bounded variables, 337 motivation for, 324 revised matrix form, 317–322 Dual variable optimal value of, 174–177 relationship to dual price, 179 Dynamic programming, 469–500 applications equipment replacement, 482–485 inventory deterministic, 521–527 probabilistic, 587–589 investment, 485–488 knapsack problem, 475–480 mill operation, 26.71–75 shortest route model, 469–473 workforce size, 480–482 backward recursion, 473 deterministic models, 469–500 dimensionality problem, 488 forward recursion, 473 Markovian decision process, 25.2–5 optimality principle, 473 probabilistic models, 24.1–11 recursive equation, 472 stage in DP, 470, 475 state in DP, 472, 475 E Economic order quantity See EOQ Edge in LP solution space, 104 Either-or constraint, 364–366 Elevator problem, 39 Empirical distribution, 555–560 Employment scheduling model, 22.4–6 EOQ constrained, 515 dynamic no setup model, 518–520 setup model, 521–530 probabilistic, 611–617 static classical, 507–511 price-breaks, 511–514 storage limitation, 514–518 Equation form of LP, 99–100 Equipment replacement model, 252–253, 482–485 Ergodic Markov chain, 634 See also Markov Chains Euler, Leonard, 249, 250 Event in probability, 543 simulation, 715 Excel Solver See Solver (Excel-based) Expected value, definition of, 547 joint random variables, 548–550 Experiment, statistical, 543 Exponential (negative) distribution, 552–553, 656–657 forgetfulness property, 656 probability calculations with excelStatTables.xls, 553 Exponential smoothing, 23.3–4 Extreme point in LP definition of, 305 relationship to basic solution, 306–308 viewed graphically as corner point, 50 F Fathoming solutions in B&B algorithm, 369, 373 Feasible solution, 34 FIFO See Queue discipline First passage time See Markov chains Fixed-charge problem, 362–364 Floats in CPM, 280 Floyd’s algorithm, 258–261 See also Dijkstra’s algorithm Fly-away kit model See Knapsack model Forecasting models, 23.1–10 www.ebookslides.com Index Forgetfulness of the exponential, 656 Forward pass in CPM, 276–277 Forward recursion in DP, 473 Fractional cut, 375 Franklin, Benjamin, 398 Franklin rule, 398 Full-rank matrix See Nonsingular matrix G Game theory, zero-sum, 571–577 non-cooperative, (aha!), 589 optimal solution graphical, 587–590 linear programming, 590–592 saddle point, 586 value, 586 Gauss-Jordan method, 108, 111, D.8 Generalized simplex algorithm, 184–185 Genetic algorithm, 411–415 crossover, 411, 412 gene coding, 411 ILP application, 420–423 mutation, 411 TSP application, 454–457 Goal programming, 341–350 column-dropping rule, 345–346, 347–350 deviatinal variables, 342 preemptive method, 343, 345–350 weights method, 343–345 Golden-section search method, 763 Goodness-of-fit test, 557–560 Gradient method, 733–736 Graphical solution games, 587–590 LP maximization, 47 LP minimization, 50 Greedy search heuristic, 398–403 H Hamming, Richard, 437 Hamming distance, 437 Harris EOQ formula See also Inventory models Harris, Ford, 511 837 Heuristic definition, 34 Silver-Meal, 527–530 TSP, 445–448 types of greedy, 398–403 meta, 404–415 Histograms, 556 Hitchcock, Frank, 211 Hungarian method See Assignment model Hurwicz criterion, 583, 584 I If-then constraint, 364 Imputed cost, 180 See also Dual price Index of optimism, 583 Inequalities, conversion to equations, 99 Infeasible solution in LP, 122 Insufficient reason, principle of, 582 Integer programming algorithms branch-and-bound, 367–373 bounding, 369, 373 branching, 369, 373 fathoming, 369, 373 cutting plane, 373–378 implicit enumeration See Additive algorithm traveling salesperson branch and bound, 441–443 cutting plane, 444–445 Intensification and diversification in tabu search, 407 Interval programming, E.13–14 Inventory, case study deterministic models EOQ, 507–516 constrained, 515–516 price breaks, 511–514 spreadsheet solution of, 514 dynamic, spreadsheet solution of, 523–524 heuristic (Silver-Meal), spreadsheet solution of, 529–530 static, 507–508 probabilistic models EOQ, 611–617 multiple-period, 623–624 www.ebookslides.com 838 Index Inventory, case study (Continued) newsvendor problem, 618–620 s-S policy, 620–623 spreadsheet simulation of, 617, 620 Inventory policy, 504 Inventory ratio, 502–503 Interval of uncertainty, 763 Inverse of a matrix, D.7 computing methods adjoint, D.8 partitioned matrix, D.11–D12 product form, D.9 row (Gauss-Jordan) operations, D.8 determinant of, D.5 location in the simplex tableau, 177 Investment model, 60–62, 485–488 Iteration, definition of, 34 J Jacobian method, 747–753 relationship to Lagrangean method, 753 Job sequencing model, 364–366 Jockeying, 655 Joint probability distribution, 548–551 K Kamarkar algorithm See Interior point algorithm Kantorovich, Leonid, 105, 211 Karush-Khun-Tucker (KKT) conditions, 754–758 Kendall notation, 666 Kepler, Johannes, 472–473 Knapsack problem, 292, 475–480 Kolmogrov-Smirnov test, 558 Koopmans, Tjalling, 211 L Lack of memory property See Forgetfulness property Lagrangean method, 753 Lagrangean multipliers, 753 Laplace criterion, 582 Lead time in inventory models, 508 Least-cost transportation starting solution, 216–217 Leonardo da Vinci, 448 Leontief, Wassily, 105 LIFO See Queue discipline Linear combinations method, 770 Linear independence of vectors, 307 Linear programming applications, 67–76 See also Case analysis corner-point solution, 141 See also Extremepoint solution feasible solution, 47 graphical solution of a two-variable model maximization, 48 minimization, 50 infeasible solution, 49 optimum feasible solution, 184 post-optimal analysis, 169–192 See also Linear programming; sensitivity analysis additional constraint, 188–189 additional variable, 182 feasibility (right-hand side) changes, 186–187 optimality (objective function) changes, 182–184 sensitivity analysis See also Post-optimal analysis algebraic, 132–136 using AMPL, 143–144 dual price, 132, 136, 200, 377 graphical, 124–132 reduced cost, 177, 180–184, 189, 311 using Solver, 141–142 using TORA, 137–138 Little’s queuing formula, 667 Loop in a network, 248 Lottery in a utility function, 579 Lovelace, Ada, 35 M M-method, 112–115 See also Two-phase method M/D/1 queue See Pollaczek-Khintchine formula M/M/1 queue, 670–672 M/M/c queue, 675–680 M/M/R queue, 680–681 www.ebookslides.com Index Machine repair queuing model, 680–681 Manpower planning model, 68–70 See also Workforce size Marginal probability distribution, 549 Mark Twain, 559 Markov chain, 629–642 absolute probabilities, 632–633 absorption, probability of, 640 closed set, 633 cost-based decision model, 636 first passage time, 636–638 initial probabilities, 632 mean return time, 634–636 n-step transition matrix, 632–633 Spam filter, use in,steady state probabilities, 631 state classification in Markov chains, 633–634 Markov process, definition of, 630 Markovian decision process, 25.1–25.16 Exhaustive enumeration solution, 25.8, 25.11 linear programming solution, 25.13–25.15 policy iteration method, 25.11 Marriage problem, 472–473 Materials requirement planning, 517–518 Mathematical model, definition of, 34, 40–41 Matrices, D.1–D.5 addition of, D.1 product of, D.3–D.4 simple arithmetic operations, review of, 172–173 Maximal flow model, 265–273 algorithm, 267–272 AMPL solution of, 273 cuts in, 267 LP formulation, 272–273 Solver solution of, 272–273 Maximin criterion, 582 Maximization, conversion to minimization, 111 Mean return time See Markov chains Mean value, 547 See also Expected value, definition of Metaheuristics, 404–415 algorithms genetic, 411–415 simulated annealing, 408–410 tabu, 404–408 839 applications cartographic label placement, 428–429 job sequencing, 405–407, 409–410 map coloring, 430–432 minimal spanning tree, constrained, 428 timetable scheduling, 430 warehouse allocation, 427 Military planning, 96 Minimal spanning tree algorithm, 250–251 constrained, 428 Mixed cut, 377 Mixed integer problem, 360 Model, elements of an OR, 34 abstraction levels, 36 Modeling art of, 36–37 levels of abstraction in, 36 Monge, Gaspard, 211 Monte Carlo simulation, 711–732 MRP, See Material requirement planning Multiplicative congruential method for random numbers, 720 Multipliers, method of, 220 See also Transportation algorithm N N queens problem as ILP, 390 Nash Equilibrium, See Game theory, non-cooperative Nash, John, 589 Needle spinning experiment, 548 Neighborhood in heuristics, definition of, 416 Network definitions, 247–250 Networks LP representation capacitated network, 266 critical path method, 273–274 maximum flow, 272 shortest route, 282–283 News vendor problem, 618–620 Newton-Raphson method, 744–746 Non-Poisson queues, 682 Nonbasic variable, 103, 309 Nonlinear programming algorithms, 763–788 Nonnegativity restriction, 54 Nonsingular matrix, 307, D.6 www.ebookslides.com 840 Index Normal distribution, 553–555 calculations with excelStatTables.xls, 554–555 statistical tables, 781–782 Northwest-corner starting solution, 216–217, 219 O Observation-based variable in simulation, 726 Optimal solution, 59, 181 OR study, phases of, 37, 39, 40 OR techniques, 34 P Parametric programming, 325–329 See also Linear programming; sensitivity analysis Partitioned matrices inverse, 173, D.11–D.12 product of, 176, D.9–D11 Path in networks, 248 pdf See Probability density function Penalty method in LP See M-method Periodic review in inventory, 505 PERT See Program evaluation and review technique Petersburg paradox, 579 Petrie, Flinders, 436–437 Poisson distribution, 551–552, 658–659 approximation of binomial, 551 calculations with excelStatTables.xls, 548 truncated, 661 Poisson queuing model, generalized, 665 Policy iteration, 55.41 Pollaczek-Khintchine formula, 682 Post-optimal analysis, 185–192 See also Parametric programming Posterior probabilities See Bayes’ probabilities Pre-solver, 142 Preemptive method in goal programming, 345–347 Price breaks in inventory, 510–514 Pricing in LP, hybrid, 139 Primal simplex algorithm See Simplex algorithm Primal-dual relationships in LP, 172–178, 309 Principle of optimality in DP, 472 Prior probabilities, 576 See also Bayes’ probabilities Prisoner’s Dilemma, 589–590 Pro-con list, See Franklin rule Probability density function definition of, 545 joint, 548–549 marginal, 548–550 Probability laws addition, 544 conditional, 544–545 Probability theory, review of, 473 Product form of inverse, 317, D.9–D.11 in the revised simplex method, 316 Production-inventory control multiple period, 64 with production smoothing, 65 shortest route model, viewed as a single period, 62–63 Program evaluation and review technique (PERT), 273–285 Pseudorandom numbers, 720 Pure birth model, 657–660 Pure death model, 661–662 Pure integer problem, 360 Q Quadratic forms, 778, D.14–15 Quadratic programming, 777–778, 781, 786, 790 Queue discipline, 655 FIFO, 655, 666 GD, 666 LIFO, 655, 666 SIRO, 655, 666 Queuing models, 655–684 decision models, 684–686 aspiration level, 686–688 cost, 654, 684–686 generalized model, 662–665 machine service model, 680–681 multiple-server models, 674–676 non-Poisson models, 682–683 single-server models, 670–674 simulation using spreadsheet, 726–728 www.ebookslides.com Index R Random number generator, 713, 722 Random variables definition of, 546 expected value, 547 standard deviation, 547–548 variance, 547–548 Reddy Mikks model, 45–53, 55–58 Reduced cost, 132–140, 144–145, 190–191, 311 Regression analysis, 23.4, 23.8 using mathematical programming, 94–95, 356 Regret (Savage) criterion, 582 Reneging in queues, 655 Reorder point in inventory, 505, 508–509 Residuals in network, 267 Resource, types of, 110 Restricted basis, 772–774, 779 Revised simplex method dual, 322–325 primal, 322–325 Risk, types of, 574 Roundoff error in simplex method, 112, 113, 115 S s-S policy, 620–623 Saddle point, 586 Sample space in probability, 543–544 Sampling from distributions discrete, 722–728 Erlang (gamma), 718 exponential, 717–718 normal, 719 Poisson, 718–719 triangular, 727–728 uniform, 727 Sampling in simulation, methods of acceptance-rejection, 716–717 convolution, 716, 718–719 inverse, 717–718 normal distribution transformation, Box-Muller, 719–720 Savage criterion See Regret criterion Secondary constraints, 205 Secretary problem, See Marriage problem Seed of a random number generator, 713 Self-service queuing model, 679 Sensitivity analysis in dynamic programming, 477 Jacobian method, 752–753 linear programming See Linear programming Separable programming, 770–777 convex, 774–777 Set covering problem, 362 Shadow price See Dual price Shortest-route problem algorithms Dijkstra’s, 255–256 DP, 478 Floyds’s, 258 LP, 261–263 transshipment, 224 applications, 252–255 computer solution using AMPL, 265 Solver, 263–265 TORA, 258, 261 Silver-Meal heuristic, 527–529 Simon, Herbert, 344 Simplex algorithm See also Generalized simplex algorithm entering variable, 107–109, 111, 312 feasibility condition, 111, 114, 312 Gauss-Jordan row operations, 108–111 leaving variable, 110, 319 optimality condition, 111, 119, 181, 312 ratios, 107 steps of, 117, 312–313 Simplex method algorithms dual, 182–184, 309 generalized, 182 primal, 140, 149, 178, 312–313 Simplex multiplier, 220 See also Dual price Simplex tableau, 106, 172 layout of, 173–174 matrix computation of, 175–176 matrix form of, 172, 309–311 841 www.ebookslides.com 842 Index Simulated annealing algorithm, 408–410 acceptance condition, 408 ILP application, 359–366 temperature schedule, 408 TSP application, 435–436 Simulation discrete-event animation, 732 languages, 731 mechanics of, 722–728 sampling, 716–720 spreadsheet, 726–728 inventory, 727, 738 queues, 684, 715–716, 727 steady state, 728 statistical observations, gathering of, 728–731 regenerative method, 729 replication method, 730 subinterval method, 729 transient state, 728–730 Simultaneous linear equations, types of solutions, 306–307 Slack variable, 99 Solver, commercial, 141–142 Solver (Excel-based), 52–56 Spanning tree, definition of, 248 basic solution in capacitated network, 52.37 Stage in DP, definition of, 470 State in DP, definition of, 472 State classification See Markov chains Statistical tables, 793–795 chi-square, 795 Excel-based (16 pdfs), 548, 551, 552, 553, 555 normal, 793–794 student t, 794–795 Steady-state in Markov chains See Markov Chains queuing See Queuing models simulation See Discrete event simulation Steepest ascent method See Gradient method Strategies in games, mixed and pure, 586–587 Student t statistical tables, 794–795 Suboptimal solution, 34 Sudoku puzzle as ILP, 382 SUMT algorithm, 787–788 Supply chains, 501 Surplus variable, 100 T Tabu search algorithm, 404–408 aspiration criterion, 407 ILP application, 415–418 intensification and diversification, 407 tabu list, 404 tabu tenure period, 404 TSP application, 449–451 Tankering (fuel), 45, 26.2–9 Time-based variable in simulation, 725 Tool sharpening model, 211–214 TOYCO model, 128 Traffic light control, 95 Transient period in simulation, 729 Transition probability See Markov chains Transition-rate diagram in queues 662 Transportation model algorithm, 214–227 applications, 207, 211–214 balancing of, 209–210 definition, 207 LP equivalence, 208 solution using, AMPL, 226 Solver, 225 Starting solution, 214–219 tableau, 209 Transpose of a matrix, D.3 Transshipment model, 22.12–13 Traveling salesperson problem, 435–468 algorithms, exact branch and bound, 441–444 cutting-plane, 444–445 algorithms, heuristics nearest neighbor, 445–446 reversal, 446–448 algorithms, metaheuristics genetic, 454–457 simulated annealing, 452–454 tabu, 449–451 applications automatic guided vehicles, 459, 463 celestial objects imaging, 459, 463 www.ebookslides.com Index DNA sequencing, 459, 462–463 high resolution imaging, 435 integrated circuit board, 459, 462 Mona Lisa art, 459 paint product sequencing, 438 protein clustering, 459, 460–461 wallpaper cutting, 463–464 warehouse order picking, 464–465 assignment model, relationship to, 438, 440 asymmetric distance matrix, 437 lower bound, 440–441 open tour solution, 440 solution of, 439 subtours, 437 symmetric distance matrix, 437 Tree, definition of, 248 Triple operation (Floyd’s algorithm), 258 TSP See Traveling salesperson problem Two-person zero-sum game, 585 Two-phase method, 115–117 See also M-method U Unbounded solution in LP, 121–122, 313, 323 Uniform distribution, 546, 711, 735 Unit worth of a resource See Dual price Unrestricted variable, 57, 66 in goal programming See Deviational variables Upper-bounded variables, 318 Urban renewal model, 70–72 Utility functions, 580–581 V Value of a game, 586 VAM See Vogel approximation method Variables, types of artificial, 112 basic, 103 binary, 359, 360 bounded, 317–319 deviational, 342 integer, 359 nonbasic, 103 slack, 99–100 surplus, 100 unrestricted, 57, 66 Variance of a random variable, 547–548 Vectors, D.1–2 linear independence, 307, D.2 Vogel approximation method (VAM), 218 W Waiting line models See Queuing models Waiting time distribution, first-come first-serve, 655 Warm-up period, See Transient period Water quality management, 96 Weak duality theory, 322 Weights method in goal programming, 343–345 Wilson EOQ formula See Harris’s EOQ formula Workforce size model using DP, 480–482 Z Zero-one integer problem, conversion to, 360 Zero-sum game, 585 843 www.ebookslides.com This page intentionally left blank www.ebookslides.com This page intentionally left blank www.ebookslides.com This page intentionally left blank www.ebookslides.com This page intentionally left blank www.ebookslides.com This page intentionally left blank ...www.ebookslides.com Operations Research An Introduction www.ebookslides.com This page intentionally left blank www.ebookslides.com Operations Research An Introduction Tenth Edition Global Edition Hamdy A Taha. .. been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, Operations Research An Introduction, 10th edition, ISBN... 544 14.2 Random Variables and Probability Distributions 545 14.3 Expectation of a Random Variable 547 14.3.1 Mean and Variance (Standard Deviation) of a Random Variable 547 14.3.2 Joint Random Variables