operations research an introduction 8th edition

www.elsolucionario.net OPERATIONS RESEARCH \"< I 'TRODl!C'1l0'1~l)lTro" www.elsolucionario.net !:,ICIITll I lAM DY A TAl IA www.elsolucionario.net www.elsolucionario.net www.elsolucionario.net Operations Research: An Introduction Eighth Edition Hamdy A Taha University of Arkansas, Fayetteville Upper Saddle River, New Jersey 07458 www.elsolucionario.net www.elsolucionario.net www.elsolucionario.net Library of Congress Calaloging.in-Publicalion Data Taha, Hamdy A Operations research: an introduction I Hamdy A Taha.~8th ed p ern Includes bibliographical references and index_ ISBN 0-13-188923·0 Operations research Programming (Mathematics) Title T57.6.T3 199796-37160 003 -dc21 Vice President and Editorial Director, ECS: Marcia J Horton Senior Editor: Holly Stark Executive Managing Editor: Vince O'Brien Managing Editor: David A George Production Editor: Craig Little Director of Creative Services: Paul Belfanti Art Director: Jayne Conte Cover Designer: Bruce Kenselaar Art Editor: Greg Dulles Manufacturing Manager: Alexis HeydJ-Long Manufacturing Buyer: Lisa McDowell _ _ ã - â 2007 by Pearson Education, Inc Pearson Prentice Hall Pearson Education, Inc Upper Saddle River, NJ 07458 All rights reserved No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher Pearson Prentice Hall"" is a trademark of Pearson Education, Inc Preliminary edition, first, and second editions © 1968,1971 and 1976, respectively, by Hamdy A Taha Third, fourth, and fifth editions © 1982,1987, and 1992, respectively, by Macmillan Publishing Company Sixth and seventh editions © 1997 and 2003, respectively, by Pearson Education, Inc The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs Printed in the United States of America 10 ISBN 0-13-188923-0 Pearson Education Ltd., London Pearson Education Australia Pty Ltd., Sydney Pearson Education Singapore, Pte Ltd Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educaci6n de Mexico, S.A de C V Pearson Education-Japan, Tokyo Pearson Education Malaysia, Pte Ltd Pearson Education, Inc., Upper Saddle River, New Jersey www.elsolucionario.net 96-37160 www.elsolucionario.net Los rios no llevan agua, el sallas fuentes sec6 jYo se donde hay una fuente que no de secar el sol! La fuente que no se agota es mi propio coraz6n -li: RuizAguilera (1862) www.elsolucionario.net To Karen www.elsolucionario.net www.elsolucionario.net www.elsolucionario.net Contents xvii About the Author Trademarks Chapter xx What Is Operations Research? 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Chapter xix Operations Research Models Solving the OR Model Queuing and Simulation Mode!s Art of Modeling More Than Just Mathematics Phases of an OR Study About This Book 10 References 10 Modeling with Linear Programming www.elsolucionario.net Preface 11 2.1 Two-Variable LP Model 12 Graphical LP Solution 15 2.2.1 Solution of a Maximization Model 16 2.2.2 Solution of a Minimization Model 23 12.3 Selected LP Applications 27 2.3.1 Urban Planning 27 2.3.2 Currency Arbitrage 32 2.3.3 Investment 37 2.3.4 Production Planning and Inventory Control 42 2.3.5 Blending and Refining 51 2.3.6 Manpower Planning 57 2.3.7 Additional Applications 60 2.4 Computer Solution with Excel Solver and AMPL 68 2.4.1 lP Solution with Excel Solver 69 2.4.2 LP Solution with AMPl 73 References 80 2.2 Chapter The Simplex Method and Sensitivity Analysis 3.1 LP Model in Equation Form 82 Converting Inequalities into Equations with Nonnegative Right-Hand Side 82 3.1.2 Dealing with Unrestricted Variables '84 Transition from Graphical to Algebraic Solution 81 3.1.1 3.2 85 vii www.elsolucionario.net Contents 3.3 3.4 3.5 3.6 Chapter Duality and Post-Optimal Analysis 4.1 4.2 4.3 4.4 4.5 Chapter The Simplex Method 90 3.3.1 Iterative Nature of the Simplex Method 90 3.3.2 Computational Details of the Simplex Algorithm 93 3.3.3 Summary of the Simplex Method 99 Artificial Starting Solution 103 3.4.1 M-Method 104 3.4.2 Two-Phase Method 108 Special Cases in the Simplex Method 113 3.5.1 Degeneracy 113 3.5.2 Alternative Optima 116 3.5.3 Unbounded Solution 119 3.5.4 Infeasible Solution 121 Sensitivity Analysis 123 3.6.1 Graphical Sensitivity Analysis 123 3.6.2 Algebraic Sensitivity Analysis-Changes in the RightHand Side 129 3.6.3 Algebraic Sensitivity Analysis-Objective Function 139 3.6.4 Sensitivity Analysis with TORA, Solver, and AMPL 146 References 150 151 Definition of the Dual Problem 151 Primal-Dual Relationships 156 4.2.1 Review of Simple Matrix Operations 156 4.2.2 Simplex Tableau Layout 158 4.2.3 Optimal Dual Solution 159 4.2.4 Simplex Tableau Computations 165 Economic Interpretation of Duality 169 4.3.1 Economic Interpretation of Dual Variables 170 4.3.2 Economic Interpretation of Dual Constraints 172 Additional Simplex Algorithms 174 4.4.1 Dual Simplex Method 174 4.4.2 Generalized Simplex Algorithm 180 Post-Optimal Analysis 181 4.5.1 Changes Affecting Feasibility 182 4.5.2 Changes Affecting Optimality 187 References 191 Transportation Model and Its Variants 5.1 5.2 5.3 193 Definition of the Transportation Model 194 Nontraditional Transportation Models 201 The Transportation Algorithm 206 5.3.1 Determination of the Starting Solution 207 5.3.2 Iterative Computations of the Transportation Algorithm 211 www.elsolucionario.net viii www.elsolucionario.net Contents 5.5 Chapter 5.4.2 Simplex Explanation of the Hungarian Method The Transshipment Model 229 References 234 Network Models 6.1 6.2 6.3 6.4 6.5 Chapter 5.3.3 Simplex Method Explanation of the Method of Multipliers 220 The Assignment Model 221 5.4.1 The Hungarian Method 222 235 Scope and Definition of Network Models 236 Minimal Spanning Tree Algorithm 239 Shortest-Route Problem 243 6.3.1 Examples of the Shortest-Route Applications 243 6.3.2 Shortest-Route Algorithms 248 6.3.3 Linear Programming Formulation of the Shortest-Route Problem 257 Maximal flow model 263 6.4.1 Enumeration of Cuts 263 6.4.2 Maximal-Flow Algorithm 264 6.4.3 Linear Programming Formulation of Maximal Flow Mode 273 (PM and PERT 275 6.5.1 Network Representation 277 6.5.2 Critical Path (CPM) Computations 282 6.5.3 Construction of the Time Schedule 285 6.5.4 Linear Programming Formulation of CPM 292 6.5.5 PERT Calculations 293 References 296 Advanced Linear Programming 7.1 7.3 7.4 7.5 297 Simplex Method Fundamentals 298 From Extreme Points to Basic Solutions 300 7.1.2 Generalized Simplex Tableau in Matrix Form 303 Revised Simplex Method 306 7.2.1 Development of the Optimality and Feasibility Conditions 307 7.2.2 Revised Simplex Algorithm 309 Bounded-Variables Algorithm 315 Duality 321 7.4.1 Matrix Definition of the Dual Problem 322 1.4.2 Optimal Dual Solution 322 Parametric Linear Programming 326 7.5.1 Parametric Changes in ( 327 7.5.2 Parametric Changes in b 330 References 332 7.1.1 7.2 228 www.elsolucionario.net 5.4 ix www.elsolucionario.net Index AMPL (Continued) subsets using conditions, 735-736 union, diff, and inter, 735 table statement, 740 trunc function, 77 unfix command, 394, 745 union set operator, 735 var, 75, 723, 725 variable defined, 725 bounds on, 727 initial value using :=, 706 See also let command write output data to spreadsheet, 744 table, 742 wri te table statement, 742, 744 Analytic Hierarchy Process (AHP),490-497 comparison matrix, 492 consistency, 494-495 Excel-based calculations of, 496-497 normalizing a comparison matrix, 493 Applications of OR, selected See Case analyses Art of modeling,S Artificial constraints in dual simplex method, 178 Artificial variable in simplex method, 104 See also M-method Aspiration level criterion in queues, 602 Assignment model, 221-229 relationship to simplex method, 228 traveling salesperson problem, use in, 385-386,392 Attribute in simulation, 612 "" B Backward pass in CPM; 283 Balking in queues, 552 Balance equation in queues, 564 Balancing transportation model, 196-197 Basic solution, 86, 88, 300 relationship to corner (extreme) point, 86, 300 Basic variable, 88, 300 Basis, 300 See also Inverse vector representation of, 301-302 restricted, 701, 710 Bayes' probabilities, 467, 506-510 Excel-based calculations of,510 Binomial distribution, 475 Poisson approximation of, 477 probability calculations with excelStatTables.xls,477 Blending and refining model, 51-54 Box-Muller sampling method for normal distribution, 619 Bounded variables definition, 315 dual simplex algorithm for, 321 primal simplex algorithm for, 315-319 Branch-and-bound algorithm, integer programming, 370-375 traveling salesperson (TSP),392-394 Bridges of Konigsberg, 237 Bus scheduling model, 58 c Capacitated network model, CDI-14 of Ch 20 AMPL solution, CD14 of Ch 20 conversion to uncapacitated, CD9 of Ch 20 Solver solution, CD14 of Ch 20 LP equivalence, CD4-6 of Ch 20 simplex-based algorithm, CD9-14 of Ch 20 Capital budgeting, 350 Cargo-loading model See Knapsack model Case analysis: AHP CIM facility layout, CD118-125 of Ch 24 assignment model scheduling trade events, CD89-93 of Ch 24 Bayes' probabilities 9lsey's medical test evaluation,CD128-131 ofCh.24 decision trees hotel booking limits, CD125-128 of Ch 24 dynamic programming, Weyerhauser log cutting, CD113-118 of Ch 24 www.elsolucionario.net 804 www.elsolucionario.net ) ) 24 l31 game theory Ryder Cup matches, CD131-133 of Ch 24 goal programming CIM facility layout, CD123-126 of Ch 24 Mount Sinai hospital, CDl04-108 of Ch 24 heuristics fuel tankering, 12, CD79-85 of Ch 24 scheduling trade events, CD89-93 of Ch 24 integer programming Mount Sinai hospital, CD103-l07 of Ch 24 PFG building glass, CD107-113 of Ch 24 Qantas telephone sales staffing, 57, CD139-l44 of Ch 24 ship routing,CD97-103 of Ch 24 inventory Dell's supply chain, CD133-136 of Ch 24 linear programming fuel tankering, 12, CD79-85 of Ch 24 heart valve production, 82, CD86-89 ofCh.24 queumg internal transport system, CD 136-138 of Ch 24 Qantas telephone sales staffing, 57, CD139-144 of Ch 24 shortest route saving federal travel dollars, CD93-97 ofCh.24 transportation ship routing, CD97-103 of Ch 24 Case studies, CD161-197 of App E decision theory, CD188-190 of App E dynamic programming, CDl85, CD197 ofApp.E goal programming, CDI77-178 of App E integer programming, CD178-185 of App E inventory, CD 186-187 of App E linear programming, CD161-I66, CD173-176 of App E networks, CD 171-I?3, CD195 of App E transportation, CD167-I71 of App E queuing, CD192-195 of App E CDF See Cumulative density function Central limit theorem, 479 Chance-constrained programming, 713 Chapman-Kolomogrov equations, 644 805 Chebyshev model for regression analysis, 338 Chi-square statistical table, 751 Chi-square test See Goodness-of-fit test Circling in LP See Cycling in LP Classical optimization: constrained, 665-672 Newton-Raphson method, 670 unconstrained,672-689 Jacobian method, 673 Karush-Khun-Tucker conditions, 685 Lagrangean method, 683 Column-dropping rule in goal programming, 343-345 Column-generation algorithm, CD174 of App E CPM See Critical Path Method Concave function, CD161 of App D Conditional probability, 465 Connected network, 237 Constrained gradient, 675 Continuous probability distribution, 467 Continuous review in inventory, 428 Convex combination, 298 Convex function, CD161 of App D Convex set, 298 Corner point in LP, 18 See also Extreme point in LP relationship to basic solution, 86 relationship to extreme point in LP, 298 Correlation coefficient, CD44 of Ch 21 Covariance, 472 Critical activity in CPM: definition, 282 determination of, 283 Critical path method (CPM) calculations, 282-284 AMPL-based,289-291 Cumulative density function (CDF),467 Currency arbitrage model, 32-36 Curse of dimensionality in Dp, 424 Cuts in: integer programming, 379-383 maximum flow network, 264 traveling salesperson problem, 395-396 Cutting plane algorithm, ILP, 379-383 TSp, 395-396 www.elsolucionario.net Index www.elsolucionario.net Index Cycle See loop Cycling in LP, 114,116 o Decision-making, types of: certainty, 490-497 risk, 500-514 uncertainty, 515-519 Decision trees, 501,507 Decomposition algorithm, CDl6-25 of Ch 20 Degeneracy, 113 See also Cycling in LP Determinant of a square matrix, CD150 ofApp.D Deviational variables in goal programming, 335 Dichotomous search, 691 Die rolling experiment, 468, 471 Diet problem, 23 Dijkstra's algorithm, 248-250 See also Floyd's algorithm Direct search method, 691-694 Discrete distribution, 467 Discrete-event simulation: languages, 638 mechanics of, 624-629 sampling, 613 621 steady state, 633 statistical observations, gathering of 633-638 regenerative method, 636 replication method, 635 subinterval method, 634 transient state, 633 Dual price, algebraic determination o~ 130,159,323 graphical determination of, 124 relationship to dual variables, 170 Dual problem in LP: economic interpretation: dual constraint, 172 dual variable, 170 See also Dual price definition of, 151-155, 322 optimal solution, 159, 161,323 use in transportation algorithm, 220 weak duality theory, 322 Dual simplex method, 174-177 See also Generalized simplex algorithm artificial constraints in, 178 bounded variables, 321 feasibility condition, 174 optimality condition, 174 revised matrix form, 314 Dual variable, optimal value o~ 159 relationship to dual price, 170,324 Dynamic programming, 399-426, applications: equipment replacement, 416-419 inventory deterministic, 450-457 probabilistic, 545-547 investment, 420-423 knapsack problem, 405-408 mill operation, CD1l4 of eh 24 shortest route model, 400 work-force size, 413-415 backward recursion, 403 deterministic models, 399-426 dimensionality problem, 424 forward recursion, 403 Markovian decision process, CD59-78 of Ch 23 optimality principle, 403 probabilistic models, CD47-57 of Ch 22 recursive equation, 402 E Economic order quantity See EOQ Edge in LP solution space, 91 Efficient solution in goal programming 340 Either-or constraint, 364 Elevator problem, Empirical distribution, 481 EGQ: constrained,440 dynamic, no setup model,445 setup model, 449-453 static, classic, 430-434 www.elsolucionario.net 806 www.elsolucionario.net www.elsolucionario.net www.elsolucionario.net Index Integer programming algorithms {Continued) implicit enumeration See Additive algorithm traveling salesperson, 385-396 branch and bound, 392':·-394 cutting plane, 394-396 heuristic, 389-391 Interior point algorithm, CD27-35 of Ch 20 Interval of uncertainty, 691 Interval programming, CD175 of App E Inventory models: application, CD135-138 of Ch 24 deterministic, EOQ,430-434 constrained EOQ, 440 static, 430-440 dynamic, 445-453 heuristic (Silver-Meal), 457-460 probabilistic,528-548 EOQ,532-537 newsvendor problem, 539-542 s-S policy, 543-545 multiple-period, 545-547 Inventory policy, 427 Inverse of a matrix, CD150-151 of App D computing methods, adjoint, CD151 of App D Excel-based, CD156 of App D partitioned matrix, CD 155-156 ofApp.D product form, CD153-154 of App D row (Gauss-Jordan) operations, CD152 ofApp.D determinant of, CD148 of App D location in the simplex tableau, 158 Investment model, 37-39, 420-423 Iteration, definition of, J Jacobian method, 673-680 relationship to Lp, 680 relationship to Lagrangean method, 683 Job sequencing model, 364, 367 Jockeying, 552 Joint probability distribution, 472 K Kamarkar algorithm See Interior point algorithm Kendall notation, 569 Knapsack problem, 247,405-408 Karush-Khun-Tucker (KKT) conditions, 685 Kolmogrov-Smirnov test, 485 L L L L Lack of memory property See Forgetfulness property Lagrangean method, 683 Lagrangean multipliers, 684 Laplace criterion, 515 Lead time in inventory models, 432 Least-cost transportation method, 208 Linear combinations method, 718 Linear independence of vectors, 300 Linear programming: additivity property, 15 applications, 27-68 See also Case analysis corner-point solution, 18 See also Extremepoint solution feasible solution, 14 graphical solution of a two-variable model maximization, 16 minimization, 23 infeasible solution, 14 linearity properties, 14-15 optimum feasible solution, 14 sensitivity analysis See also Post-optimal analysis graphical, objective function, 126-128 right-hand side, 123-126 algebraic, objective function, 139-144 right-hand side, 129-134 dual price, 124, 130, 170,324 reduced cost, 140, 172, 307 using AMPL See AMPL using Solver See Solver (Excel-based) using TORA, 146 www.elsolucionario.net 808 Iv Iv Iv Iv Iv Iv N tv N Iv :tv Iv www.elsolucionario.net post-optimal analysis See also Linear programming sensitivity analysis additional constraint, 185-186 additional variable, 189-190 feasibility (right-hand side) changes, 182-184 optimality (objective function) changes, 187-189 Little's queuing formula, 570 Loop in a network, 237 Lottery in a utility function, 512-514 M MIDII queue See Pollaczek-Khintchine formula MIMI1 queue, 573-582 MIMIc queue, 582-588 MIMIR queue, 592 M-method, 104 See also Two-phase method Machine repair queuing model, 592 Majorizing function, 620 Manpower planning model, 57-59 See also Workforce size Marginal probability distribution, 472 Markov process, definition of, 642 Markov chains, 641 664 absolute probabilities, 644 absorption, probability of, 659 closed set, 647 cost-based decision model, 651 Excel-based calculations, 650 first passage -time, 654 initial probabilities, 644 mean return time, 649 n-step transition matrix, 644 steady state probabilities, 646 state classification in Markov chains: absorbing, 646 transient, 646 recurrent, 646 ergodic, 647 periodic, 646 Markovian decision process, CD58-77 of Ch 23 Exhaustive enumeration solution, CD64 of Ch 23 809 linear programming solution, CD73-77 ofCh.23 policy iteration method, CD67-73, of Ch 23 Materials requirement planning See MRP Mathematical model, definition of, 3, 12 Matrices, CD145-157 of App D addition of, CD147 of App D product of, CD147-148 of App D simple arithmetic operations, review of, 156-157 Excel-based manipulations, CD156 of App D Maximal flow model, 264-269 algorithm, 264 AMPL solution of, 273 cuts in, 264 Solver solution of, 273 LP formulation, 273 Maximization, conversion to minimization, 100 Maximin criterion, 516 Mean return time See Markov chains Mean value, 471 See also Expected value, definition of Military planning, 66 Minimal spanning tree algorithm, 239 Mixed cut, 383 Mixed integer problem, 350 Modeling art of, levels of abstraction in, Monte Carlo simulation, 60S 609 Moving average technique, CD37-39 of Ch 21 MRp,444 Multipliers, method of, 212 See also Transportation algorithm Multiplicative congruential method for random numbers, 622 N Needle spinning experiment, 468, 471 Network definitions, 236 Networks LP representation, capacitated network, CD1-3 of Ch 20 critical path method, 292 www.elsolucionario.net Index www.elsolucionario.net Index Networks LP representation (Continued) maximum flow, 273 shortest route, 257 Newsvendor problem, 539-542 Newton-Raphson method, 670 Nonbasic variable, 88 Nonlinear programming algorithms, 691-722 Nonnegativity restriction, 14 Non-Poisson queues, 595, 597 Nonsingular matrix, 300, CD152 of App D Normal distribution, 478 calculations with exceIStatTables.xls, 480 statistical tables, 785 Northwest-corner method, 208 o Observation-based variable in simulation, 628 Optimal solution, 3,14 OR study, phases of, OR techniques, 4-5 p Parametric programming, 326-332 See also Linear programming: sensitivity analysis Partitioned matrices, inverse, CD155 of App D product of, CD148 of App D Path in networks, 237 pdf See Probability density function Penalty method in LP See M-method Periodic review in inventory, 428 PERT See Program evaluation and review technique Poisson distribution, 476, 556-558 approximation of binomial, 476 calculations with exceIStatTables.xls, 477 truncated, 561 Poisson queuing model, generalized, 563 Policy iteration, CD65, 68,71 of Ch 23 Pollaczek-Khintchine formula, 595 Posterior probabilities See Bayes' probabilities Post-optimal analysis, 181-190 See also Parametric programming Preemptive method in goal programming, 341 Price breaks in inventory, 436 Primal-dual relationships in LP, 156-161, 303-305 Primal simplex algorithm See Simplex algorithm Principle of optimality, 403 Prior probabilities, 506 See also Bayes' probabilities Probability density function: definition of,467 joint, 472 marginal,472 Product form of inverse, CD155 of App D in the revised simplex method, 310 Production-inventory control multiple period, 44, 201 with production smoothing, 46 shortest route model, viewed as a, 247 single period, 42 Probability laws addition, 464 conditional, 465 Probability theory, review of, 463-488 Program evaluation and review technique (PERT),293-296 Pseudo-random numbers, 623 Pure birth model, 556-559 Pure death model, 560 562 Pure integer problem, 350 Q Quadratic forms, CD157-158 of App D Quadratic programming, 708-713 Queue discipline, 552 Queuing models., 549-604 decision models, 597-604 aspiration level, 602 cost, 598 generalized model, 563-566 machine service model, 592 www.elsolucionario.net 810 www.elsolucionario.net multiple-server models, 582-592 single-server models, 573-582,595 non-Poisson models, 595,597 R Random variables: definition of, 467, expected value, 469, 472 standard deviation, 470 variance, 470 Random number generator, 622 Reddy Mikks model, 12 Reduced cost, 140, 172, 307 Regression analysis, CD42-44 of Ch 21 using mathematical programming, 65,338 Regret (Savage) criterion, 516 Reneging in queues, 552 Reorder point in inventory, 432 Residuals in network, 264 Resource, types of: scarce, 98 abundant, 98 Restricted basis, 701, 710 Revised simplex method, dual, 314, 321 primal, 309-313 Risk, types of, averse, 512 neutral, 512 seeker, 512 Roundoff error in simplex method, 105, 109 s s-S policy, 543-545 Saddle point, 522 Sample space in probability, 463 Sampling in simulation, methods of: acceptance-rejection, 620 convolution, 616 inverse, 613 ;! L normal distribution transformation, Box~Muller, 619 Sampling from distributions: beta, 621 discrete, 614 Erlang (gamma), 617 exponential,614 geometric, 616 normal,618 Poisson, 617 triangular, 616 uniform, 615 Weibull,616 Savage criterion See Regret criterion Secondary constraints, 186 Seed of a random number generator, 623 Self-service queuing model, 590 Sensitivity analysis in: dynamic programming, 408 Jacobian method, 679 linear programming See Linear programming Separable programming, 699-707 convex, 704 Set covering problem, 354 Shadow price See Dual price Shortest-route problem algorithms Dijkstra's,248 DP,400 Floyds's, 251 LP,257 transshipment, 231 applications, 243-246 computer solution using AMPL,261 Soiver,258 TORA, 250,255 Silver-Meal heuristic, 457 Simplex algorithm See also Generalized simplex algorithm entering variable, 92,94, 307 feasibility condition, 95,99,307 Gauss-Jordan row operations, 95 leaving variable, 92, 95, 307 811 www.elsolucionario.net Index www.elsolucionario.net 'Index Simplex algorithm (Continued) ratios, 94 optimality condition, 94, 99, 307 steps of, 100, 309 Simplex method, types of, dual, 174,314,321 generalized, 180 primal,93-100 revised, 309 Simplex multiplier, 212 See also Dual price Simplex tableau, 93 layout of, 158 matrix computation of, 165-166 matrix form of, 303 Simultaneous linear equations, types of solutions, 300-302 Simulation, See also Discrete-event simulation Slack variable, 82 Solver (Excel-based), 69-73 application models, IBC Spanning tree, definition of,237 basic solution in capacitated network, CD10 of Ch 20 State classification See Markov chains Statistical tables, 749-751 chi-square, 751 Excel-based (16 pdfs), 471, 475, 477, 478,480 normal, 749 student t, 750 Steepest ascent method See Gradient method Stage in DP, definition of, 400 State in DP, definition of, 402 Steady-state in Markov chains See Markov Chains queuing See Queuing models simulation See Discrete event simulation Stock-slitting problem See Trim-loss problem Strategies in games, mixed and pure, 522 Student t statistical tables, 750 Suboptimal solution, SUMT algorithm, 721 Surplus variable, 83 T Tankering (fuel), 12, CD83 of Ch 24 Time-based variable in simulation, 628 Tool sharpening model, 202-204 TORA models, IBC TOYCO model, 129 Traffic light control, 65 Transient period in simulation, 633 Transition probability See Markov chains Transition-rate diagram in queues, 564 Transportation model: algorithm, 206-215 applications, 194,201-204 balancing of, 196 definition, 194 LP equivalence, 195 solution using, AMPL,216 Solver, 216 tableau, 195 Transpose of a matrix, CD147 of App D Excel-based calculations, CD156 of App D Transshipment model, 229-230 Traveling salesperson problem, 385-397 algorithm, B&B,392-394 cutting plane, 395-396 heuristics, 389-390 subtour, 386 tour, 386 Tree, definition of, 237 Trim-loss problem, 60 See also Column generation model Triple operation (Floyd's algorithm), 252 TSP See Traveling salesperson problem Two-person zero-sum game, 521 Two-phase method, 108 See also M-method www.elsolucionario.net 812 u Unbounded solution in LP, 119,323 Unit worth of a resource See Dual price ,i L www.elsolucionario.net Index v VAM See Vogel approximation method Value of a game, 522 Variables, types of: artificial, 104 basic,88 binary, 370 deviational, 335 integer, 350 nonbasic, 88 slack,82 surplus, 83 bounded, 315 unrestricted, 84 J.D od ,i L Variance of a random variable, 470-471 Vectors, CD145 of App D linear independence, 300, CD146 of App D Vogel approximation method (VAM), 209 w Waiting line models See Queuing models Waiting time distribution, first-come first-serve, 577 Warm-up period, see Transient period Water quality management, 66 Weak duality theory, 322 Weights method in goal programming, 338 Wilson's economic lot size See EOQ Workforce size model using DP, 413-415 z Zero-one integer problem, conversion to, 370 Zero-sum game, 520 www.elsolucionario.net Uniform distribution, 468 Unrestricted variable, 84 in goal programming See Deviational variables Upper-bounded variables, 315 Urban renewal model, 27-30 Utility functions, 512-514 813 www.elsolucionario.net READ THIS LICENSE CAREFULLY BEFORE OPENING THIS PACKAGE BY OPENING THIS PACKAGE, YOU ARE AGREEING TO THE TERMS AND CONDITIONS OF THIS LICENSE IF YOU DO NOT AGREE, DO NOT OPEN THE PACKAGE PROMPTLY RETURN THE UNOPENED PACKAGE AND ALL ACCOMPANYING ITEMS TO THE PLACE YOU OBTAINED THEM [FOR A FULL REFUND OF ANY SUMS YOU HAVE PAID FOR THE SOFTWARE] THESE TERMS APPLY TO ALL LICENSED SOFTWARE ON THE DISC EXCEPT THAT THE TERMS FOR USE OF ANYSHAREWARE OR FREEWARE ON THE DISCS AREAS SET FORTH IN THE ELECTRONIC LICENSE LOCATED ON THE DISC: GRANT OF LICENSE and OWNERSHIP: The enclosed computer programs and data ("Software") are licensed, not sold, to you by Pearson Education, Inc ("We" or the "Company") and in consideration [of your payment of the license fee, which is part of the price you paid] [of 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REVERSE SIDE FOR INSTRUCTIONS TO OPEN JPERATICJ1~ RESEARCH l'-\N [NT'J.{C)l)l.fCTJC}N EIGHTH EDITION HAMDY A TAHA ISBN 0-13-173885-2 "