Table Of ContentSarker/OptimizationModelling:APracticalApproach 43102_C000 FinalProof page i 1.9.2007 11:49am CompositorName:BMani
Optimization
Modelling
A Practical Approach
Sarker/OptimizationModelling:APracticalApproach 43102_C000 FinalProof page ii 1.9.2007 11:49am CompositorName:BMani
Sarker/OptimizationModelling:APracticalApproach 43102_C000 FinalProof page iii 1.9.2007 11:49am CompositorName: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
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CRC Press
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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)
1. Mathematical models. 2. Mathematical optimization. I. Newton, Charles S.
(Charles Sinclair), 1942- II. Title.
QA401.S266 2007
658.4’0352--dc22 2007013747
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
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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
1 Introduction......................................................................................................3
1.1 GeneralIntroduction.............................................................................3
1.2 History of Optimization.......................................................................4
1.3 Optimization Problems.........................................................................5
1.4 Mathematical Model..............................................................................6
1.4.1 Characteristics and Assumptions...........................................6
1.5 Concept ofOptimization......................................................................8
1.6 Classification ofOptimization Problems.........................................11
1.7 Organization of the Book...................................................................13
Exercises...........................................................................................................14
References........................................................................................................15
2 The Process of Optimization.......................................................................17
2.1 Introduction..........................................................................................17
2.2 DecisionProcess...................................................................................17
2.3 Problem Identification and Clarification.........................................19
2.4 Problem Definition..............................................................................20
2.5 Development of aMathematical Model..........................................21
2.5.1 Measure ofEffectiveness........................................................23
2.6 Deriving a Solution..............................................................................25
2.7 SensitivityAnalysis.............................................................................26
2.8 Testing the Solution.............................................................................26
2.9 Implementation....................................................................................27
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2.10 Summary...............................................................................................28
Exercises...........................................................................................................29
3 Introduction to Modelling...........................................................................31
3.1 Introduction..........................................................................................31
3.2 Components of a Mathematical Model............................................31
3.2.1 DecisionVariables...................................................................32
3.2.2 Objective Function...................................................................32
3.2.3 Constraints................................................................................32
3.3 SimpleExamples..................................................................................32
3.4 Analyzing a Problem...........................................................................34
3.4.1 ANonmathematical Programming Problem......................35
3.5 Modelling aSimpleProblem.............................................................36
3.5.1 Defining theVariables............................................................37
3.5.2 Objective Function...................................................................37
3.5.3 Constraints................................................................................37
3.6 LinearProgrammingModel..............................................................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
4 Simple ModellingTechniques I................................................................59
4.1 Introduction..........................................................................................59
4.2 Use of Subscripts in Variables...........................................................59
4.3 SimpleModelling Techniques...........................................................60
4.3.1 Additional Work Requirement inthe Formulation...........61
4.3.2 Variablesas Fractions of OtherVariables...........................64
4.3.3 MaintainingCertainRatiosamongDifferentVariables....68
4.3.4 One Constraint Is a Fraction ofAnother Constraint.........70
4.3.5 Maxi–Min orMini–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 TransportationProblem.........................................................83
4.4.2 Assignment Problem..............................................................86
4.4.3 TransshipmentProblem.........................................................88
4.4.4 Project ManagementProblem...............................................91
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4.5 Summary...............................................................................................98
Exercises...........................................................................................................98
Bibliography..................................................................................................102
5 SimpleModelling Techniques II.............................................................103
5.1 Introduction........................................................................................103
5.2 PrecedenceConstraints.....................................................................103
5.3 Either–or Constraints........................................................................104
5.4 K out of NConstraintsMustHold.................................................105
5.5 Yes-or-NoDecisions..........................................................................106
5.6 Functions with N PossibleValues...................................................108
5.7 MutuallyExclusiveAlternatives and ContingentDecisions......109
5.8 Linking Constraintswith the Objective Function........................111
5.9 PiecewiseLinearFunctions..............................................................113
5.10 Nonlinear to Approximate Functions............................................116
5.11 Deterministic Modelswith 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 (cid:1) to¼Constraints.....................................................129
5.17.2 From (cid:2) to¼Constraints.....................................................130
5.17.3 From (cid:2) to (cid:1)Constraints....................................................130
5.17.4 From (cid:1) to (cid:2)Constraints....................................................130
5.17.5 From¼Constraint to (cid:2) and(cid:1) Constraints....................130
5.17.6 Changing ObjectiveType...................................................131
5.18 ConditionalConstraints....................................................................132
5.19 Dual Formulation...............................................................................133
5.20 RegressionModel..............................................................................136
5.21 StochasticProgramming...................................................................137
5.22 Constraint Programming..................................................................137
5.23 Summary.............................................................................................138
Exercises.........................................................................................................138
Bibliography..................................................................................................142
References......................................................................................................143
6 Modelling Large-Scaleand Well-KnownProblems I..........................145
6.1 Introduction........................................................................................145
6.2 Use ofthe Summation (S)Sign.......................................................145
6.3 Use ofthe Subset(2)Sign................................................................147
6.4 Network Flow Problems...................................................................149
6.4.1 Shortest Path Problem........................................................149
6.4.2 MaximumFlow Problem...................................................150
6.4.3 Multi-Commodity Flow Problem.....................................152
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6.5 KnapsackProblem...............................................................................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 Locationand 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 Logisticsand Transportation.............................................................167
6.8.1 AirliftProblem........................................................................167
6.8.2 Relevant Literature.................................................................168
6.9 Summary...............................................................................................170
Exercises.........................................................................................................170
References......................................................................................................172
7 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 andRouting....................................................................180
7.3.1 Generalized Assignment Problem.......................................180
7.3.2 Traveling Salesperson Problem............................................181
7.3.3 Relevant Literature onTraveling Salesperson
Problem.....................................................................................184
7.3.4 VehicleRoutingProblem.......................................................185
7.3.5 Relevant Literature onVehicle Routing Problem.............188
7.4 Staff Rostering andScheduling.........................................................189
7.4.1 Staff Scheduling: A Weekly Problem..................................189
7.4.2 Daily RosteringProblem.......................................................191
7.4.3 Relevant Literature onGeneralStaffScheduling..............192
7.4.4 Crew Planning=Scheduling Problem...................................193
7.5 Scheduling andTimetablingProblem..............................................194
7.5.1 SchoolTimetabling Problem.................................................194
7.5.2 University Timetabling..........................................................196
7.5.3 Relevant Literature.................................................................197
7.6 Summary...............................................................................................199
Exercises.........................................................................................................199
References......................................................................................................201
8 Alternative Modelling................................................................................205
8.1 Introduction..........................................................................................205
8.2 Modelling under Different Assumptions........................................205
8.2.1 ACoal Blending Problem......................................................205
8.2.2 FirstAlternative Blending Model........................................207
8.2.3 SecondAlternative Blending Model....................................209
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8.2.4 Comparing the Two SimpleAlternative Models...........210
8.2.5 A CropPlanning Problem..................................................211
8.2.6 Crop PlanningModel1......................................................212
8.2.7 Crop PlanningModel2......................................................213
8.3 Hierarchical Modelling: An Introduction.....................................214
8.3.1 Hierarchical Modelling ina Manufacturing Context....215
8.3.2 Aggregate Model.................................................................216
8.3.3 FamilyScheduling Model..................................................217
8.3.4 IndividualItem Scheduling Model...................................218
8.4 Summary............................................................................................219
References......................................................................................................220
Section III Model Solving
9 Solution Approaches: AnOverview........................................................223
9.1 Introduction.......................................................................................223
9.2 Complexity and ComplexityClasses............................................223
9.2.1 Complexity ofAlgorithms..................................................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 LinearProgram: 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,
andXPRESS...........................................................................237
9.5.3 GAMS.....................................................................................237
9.5.4 Solver and PremiumSolver................................................238
9.5.5 Win QSB.................................................................................238
9.5.6 MINOS...................................................................................238
9.6 Summary............................................................................................239
