Multi-objective Optimization Problems. Concepts and Self-adaptive Parameters with Mathematical and Engineering Applications

161 Pages·2017·6.07 MB·English

by  Fran Sérgio Lobato

Fran Sérgio Lobato • Valder Steffen Jr.
Multi-Objective
Optimization Problems
Concepts and Self-Adaptive Parameters with
Mathematical and Engineering Applications
123
FranSérgioLobato ValderSteffenJr.
SchoolofChemicalEngineering SchoolofMechanicalEngineering
FederalUniversityofUberlândia FederalUniversityofUberlândia
Uberlândia-Brazil Uberlândia-Brazil
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-319-58564-2 ISBN978-3-319-58565-9 (eBook)
DOI10.1007/978-3-319-58565-9
LibraryofCongressControlNumber:2017940795
©TheAuthor(s)2017
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Naturally, real-world problems involve the simultaneous optimization of two or
more (often-conflicting) objectives, called multi-objective optimization problems
(MOOP).Thesolutionofsuchproblemsisdifferentfromthatofasingle-objective
optimization one. The main difference is that the solution of multi-objective
optimization problems is represented by a curve (surface) that contains various
points, which have all the same importance from the mathematical point of view.
Traditionally, the treatment of such problems is done by transforming the original
MOOP into a single-objective problem. These methods follow a preference-based
approach, in which a relative preference vector is used to scalarize multiple
objectives.Sinceclassicalsearchingandoptimizationmethodsuseapoint-by-point
approachsothatthesolutionissuccessivelymodified,theoutcomeofthisclassical
optimizationmethodisasingleoptimizedsolution.Ontheotherhand,evolutionary
algorithms(EA)canfindmultipleoptimalsolutionsinonesinglesimulationrundue
to their population-based search approach. Thus, EA are ideally suited for multi-
objectiveoptimizationproblems.
AmongthevariousexistingEA,wecancitethedifferentialevolution(DE).The
crucial idea behind DE is the scheme used for generating trial parameter vectors.
DEaddstheweighteddifferencebetweentwopopulationvectorstoathirdvector.
The key control parameters for the DE are the following: the population size, the
crossoverprobability,theperturbationrate,andthestrategyconsideredtogenerate
potentialcandidates.
Inthecontextofmulti-objectiveoptimization,theDEparametersareconsidered
constant during the evolutionary process. This aspect simplifies the algorithm, but
representsaconstraintthatdoesnotfollowthebiologicalevolutionfoundinnature.
Thenaturalphenomenonincludescontinuousvariationinthenumberofindividuals,
which increases when there are highly fitted individuals and abundant resources,
and decreases otherwise. It may be beneficial to expand the population in early
generations when the phenotype diversity is high. However, the population can
be contracted when the unification of the individuals in terms of structure and
fitnessnolongerjustifiesthemaintenanceofalargepopulationthatleadstohigher
computationalcosts.Thisaspectofferstotheindividualstheopportunitytoexplore
the design space accordingly. On the other hand, from the optimization point of
view,intheendoftheevolutionaryprocess,thenaturaltendencyofthepopulation
isbecominghomogeneous,whichimpliesunnecessaryevaluationsoftheobjective
functionand,consequently,theincreaseofcomputationalcost,whenthepopulation
size is kept constant. In addition, the dynamic update of the required parameters
such as population size, crossover parameter, and perturbation rate can accelerate
theconvergencerateandavoidlocalminima.
In this contribution, the self-adaptive multi-objective optimization differential
evolution (SA-MODE) algorithm is proposed in order to reduce the number of
evaluations of the objective function and update dynamically the DE parameters
during the evolutionary process. In this strategy, the population size is updated
dynamicallybyusingtheconceptofconvergenceratetoevaluatethehomogeneity
ofthepopulation,andtheotherparameters(crossoverprobabilityandperturbation
rate) are dynamically updated by using the concept of population variance. The
proposedmethodologyisthenappliedbothtomathematicalfunctionswithdifferent
levelsofcomplexityandtoengineeringsystemdesign.Amongtheseapplications,
we can cite some test cases in the following areas: (1) cantilevered beam design;
(2)machinabilityofstainlesssteel;(3)optimizationofhydrocycloneperformance;
(4)alkylationprocessoptimization;(5)batchstirredtankreactor;(6)crystallization
process; (7) rotary dryers; and (8) rotor-dynamics design. The results obtained by
usingSA-MODEarecomparedwiththoseobtainedbyotherevolutionarystrategies.
Uberlândia,Brazil FranSérgioLobato
ValderSteffenJr.
Contents
1 Introduction .................................................................. 1
References..................................................................... 4
PartI BasicConcepts
2 Multi-objectiveOptimizationProblem.................................... 9
2.1 Introduction............................................................ 9
2.2 BasicConceptsandDefinitions....................................... 11
2.3 OptimalityConditions................................................. 17
2.4 MetricsforConvergenceandDiversity............................... 19
2.4.1 ErrorRate(ER)................................................ 19
2.4.2 ConvergenceMetric(‡)...................................... 19
2.4.3 GenerationalDistance(GD) .................................. 19
2.4.4 Spreading(Spc)................................................ 20
2.4.5 NumberofNiches(NC)....................................... 20
2.4.6 DiversityMetric((cid:2)) .......................................... 20
2.5 MethodologiestoSolvetheMOOP................................... 21
2.5.1 TypeofApproach ............................................. 21
2.5.2 ProblemFormulation.......................................... 22
2.6 Summary............................................................... 22
References..................................................................... 22
3 TreatmentofMulti-objectiveOptimizationProblem.................... 25
3.1 ClassicalAggregationMethods....................................... 26
3.1.1 WeightedSumMethod........................................ 27
3.1.2 "-ConstraintMethod .......................................... 29
3.1.3 GoalProgrammingMethod................................... 30
3.1.4 HierarchicalOptimizationMethod........................... 30
3.1.5 CompromiseOptimizationMethod .......................... 31
3.2 DeterministicandNon-DeterministicMethods...................... 31
3.2.1 DeterministicMethods........................................ 32
3.2.2 Non-DeterministicMethods .................................. 33
3.3 HandlingtheConstraints.............................................. 36
3.3.1 PenaltyFunctionsMethods................................... 36
3.3.2 InteriorPenaltyFunctionMethod ............................ 36
3.3.3 ExteriorPenaltyFunctionMethod............................ 37
3.3.4 AugmentedLagrangeMultiplierMethod.................... 37
3.3.5 DeathPenaltyMethod......................................... 38
3.3.6 MethodsBasedonthePreservationofViableSolutions .... 38
3.4 HeuristicMethodsAssociatedwithDominanceConcept ........... 38
3.4.1 Vector-EvaluatedGeneticAlgorithm......................... 39
3.4.2 Multi-objectiveGeneticAlgorithm........................... 39
3.4.3 Niched-ParetoGeneticAlgorithm............................ 40
3.4.4 Non-dominatedSortingGeneticAlgorithmIandII......... 40
3.4.5 StrengthParetoEvolutionaryAlgorithmIandII............ 40
3.4.6 Multi-objectiveOptimizationDifferentialEvolution........ 41
3.4.7 Multi-objectiveOptimizationBio-InspiredAlgorithm...... 41
3.5 Summary............................................................... 42
References..................................................................... 42
PartII Methodology
4 Self-adaptiveMulti-objectiveOptimizationDifferentialEvolution.... 47
4.1 DifferentialEvolution:ABriefReview .............................. 48
4.2 Multi-objectiveDifferentialEvolution:AReview................... 50
4.3 Self-adaptiveParameters:Motivation ................................ 53
4.4 Self-adaptiveMulti-objectiveOptimizationDifferential
Evolution............................................................... 59
4.4.1 UpdatingofthePopulationSize(IntegralOperator) ........ 61
4.4.2 FandCRUpdating............................................ 64
4.5 SA-MODE:ATestCase .............................................. 65
4.6 Summary............................................................... 70
References..................................................................... 70
PartIII Applications
5 Mathematical................................................................. 77
5.1 SCH2Function ........................................................ 77
5.2 FONFunction.......................................................... 81
5.3 KURFunction ......................................................... 82
5.4 GTPFunction.......................................................... 86
5.5 ZDTFunctions......................................................... 89
5.6 Min-ExFunction....................................................... 95
5.7 BNHFunction ......................................................... 99
5.8 SRNFunction.......................................................... 101
5.9 OSYFunction.......................................................... 103
5.10 Summary............................................................... 106
References..................................................................... 107
6 Engineering................................................................... 109
6.1 BeamwithSectionI................................................... 109
6.2 WeldedBeam .......................................................... 113
6.3 MachinabilityofStainlessSteel ...................................... 117
6.4 OptimizationofHydrocyclonePerformance......................... 120
6.5 AlkylationProcessOptimization ..................................... 125
6.6 BatchStirredTankReactor(Biochemical)........................... 128
6.7 CatalystMixing........................................................ 131
6.8 CrystallizationProcess ................................................ 134
6.9 RotaryDryer........................................................... 140
6.10 Rotor-DynamicsDesign............................................... 145
6.11 Summary............................................................... 150
References..................................................................... 151
PartIV FinalConsiderations
7 Conclusions................................................................... 155
Reference...................................................................... 157
Index............................................................................... 159
Figures
Fig.2.1 PossiblePareto’sCurvesforabi-objectiveoptimization........... 10
Fig.2.2 ConvergenceanddiversityinthePareto’sCurve.................... 11
Fig.2.3 ObjectivefunctionversusindependentvariablesforEq.(2.1)...... 12
Fig.2.4 ObjectivefunctionversusindependentvariablesforEq.(2.4)...... 13
Fig.2.5 Representationofdesignspaceandobjectivespacefora
bi-objectiveoptimizationproblem................................... 15
Fig.2.6 Non-dominatedsolutionsfortheexampleproposed................ 16
Fig.3.1 ResolutionofMOOP’susingclassicalaggregationmethods....... 26
Fig.3.2 Geometrical interpretation of the WSM for the convex
Pareto’sCurvecase................................................... 28
Fig.3.3 GeometricalinterpretationoftheWSMforthenon-convex
Pareto’sCurvecase................................................... 29
Fig.4.1 Genericmulti-objectivealgorithmflowchart........................ 50
Fig.4.2 MODEalgorithmflowchart.......................................... 54
Fig.4.3 Evolutionofthepopulationduringtheevolutionaryprocess
forthemathematicalfunctionproposedbyHauptandHaupt...... 55
Fig.4.4 Influence of DE parameters on the number of required
generations for the mathematical function proposed by
HauptandHaupt...................................................... 56
Fig.4.5 Influenceoftheperturbationrateontheconvergencemetric
fortheZDT1function................................................ 57
Fig.4.6 Influenceoftheperturbationrateonthediversitymetricfor
theZDT1function.................................................... 57
Fig.4.7 Influenceofthecrossoverprobabilityontheconvergence
metricfortheZDT1function ........................................ 57
Fig.4.8 Influenceofthecrossoverprobabilityonthediversitymetric
fortheZDT1function................................................ 58
Fig.4.9 SA-MODEalgorithmflowchart...................................... 60
Fig.4.10 InitialpopulationfortheZDT1function ............................ 62
Fig.4.11 Pareto’sCurvefortheZDT1functionusingMODE................ 62
Fig.4.12 AreaasafunctionofthegenerationnumberfortheZDT1
function................................................................ 63
Fig.4.13 Convergence rate ((cid:3)) as a function of the generation
i
numberfortheZDT1function....................................... 64
Fig.4.14 UpdateDEparametersflowchart..................................... 66
Fig.4.15 Pareto’sCurvefortheZDT1functionbyusingtheSA-MODE.... 67
Fig.4.16 CumulativenumberofobjectiveevaluationsfortheZDT1
functionusingNSGAII,MODE,andSA-MODE.................. 68
Fig.4.17 ConvergencerateandpopulationsizefortheZDT1function
byusingtheSA-MODE.............................................. 68
Fig.4.18 EvolutionoftheparametersFandCRfortheZDT1function
byusingtheSA-MODE.............................................. 69
Fig.5.1 Pareto’sCurvesfortheSCH2function.............................. 79
Fig.5.2 ConvergencerateandpopulationsizefortheSCH2function...... 80
Fig.5.3 EvolutionoftheDEparametersfortheSCH2function ............ 80
Fig.5.4 Pareto’sCurvesfortheFONfunction ............................... 82
Fig.5.5 ConvergencerateandpopulationsizefortheFONfunction....... 83
Fig.5.6 EvolutionoftheDEparametersfortheFONfunction ............. 83
Fig.5.7 Pareto’sCurvesfortheKURfunction............................... 84
Fig.5.8 ConvergencerateandpopulationsizefortheKURfunction....... 85
Fig.5.9 EvolutionoftheDEparametersfortheKURfunction ............. 86
Fig.5.10 Pareto’sCurvesfortheGTPfunction................................ 87
Fig.5.11 ConvergencerateandpopulationsizefortheGTPfunction ....... 88
Fig.5.12 EvolutionoftheDEparametersfortheGTPfunction.............. 88
Fig.5.13 Pareto’sCurvesfortheZDT2function.............................. 90
Fig.5.14 Pareto’sCurvesfortheZDT3function.............................. 91
Fig.5.15 Pareto’sCurvesfortheZDT4function.............................. 91
Fig.5.16 Pareto’sCurvesfortheZDT6function.............................. 92
Fig.5.17 ConvergencerateandpopulationsizefortheZDT2function...... 93
Fig.5.18 ConvergencerateandpopulationsizefortheZDT3function...... 93
Fig.5.19 ConvergencerateandpopulationsizefortheZDT4function...... 94
Fig.5.20 ConvergencerateandpopulationsizefortheZDT6function...... 94
Fig.5.21 EvolutionoftheDEparametersfortheZDT2function ............ 95
Fig.5.22 EvolutionoftheDEparametersfortheZDT3function ............ 95
Fig.5.23 EvolutionoftheDEparametersfortheZDT4function ............ 96
Fig.5.24 EvolutionoftheDEparametersfortheZDT6function ............ 96
Fig.5.25 Pareto’sCurvesfortheMin-Exfunction ............................ 97
Fig.5.26 ConvergencerateandpopulationsizefortheMin-Exfunction.... 98
Fig.5.27 EvolutionoftheDEparametersfortheMin-Exfunction .......... 98
Fig.5.28 Pareto’sCurvesfortheBNHfunction............................... 100
Fig.5.29 ConvergencerateandpopulationsizefortheBNHfunction....... 100
Fig.5.30 EvolutionoftheDEparametersfortheBNHfunction ............. 101
Fig.5.31 Pareto’sCurvesfortheSRNfunction ............................... 102
Fig.5.32 ConvergencerateandpopulationsizefortheSRNfunction ....... 103
Fig.5.33 EvolutionoftheDEparametersfortheSRNfunction.............. 103
Fig.5.34 Pareto’sCurvesfortheOSYfunction ............................... 105
Fig.5.35 ConvergencerateandpopulationsizefortheOSYfunction....... 106
Fig.5.36 EvolutionoftheDEparametersfortheOSYfunction ............. 106
Fig.6.1 BeamwithsectionI................................................... 110
Fig.6.2 Pareto’sCurvesfortheI-beamproblem............................. 111
Fig.6.3 ConvergencerateandpopulationsizefortheI-beamproblem..... 112
Fig.6.4 EvolutionoftheDEparametersfortheI-beamproblem ........... 113
Fig.6.5 Weldedbeam ......................................................... 113
Fig.6.6 Pareto’sCurvesobtainedfortheproblemoftheweldedbeam..... 115
Fig.6.7 Convergencerateandpopulationsizefortheweldedbeam........ 116
Fig.6.8 EvolutionoftheDEparametersfortheweldedbeam .............. 116
Fig.6.9 Relationbetweentheinputfactorsandtheobservedoutput
response............................................................... 117
Fig.6.10 Pareto’s Curves obtained for the machinability of the
stainlesssteelproblem................................................ 119
Fig.6.11 Convergencerateandpopulationsizeforthemachinability
ofthestainlesssteelproblem......................................... 120
Fig.6.12 EvolutionoftheDEparametersforthemachinabilityofthe
stainlesssteelproblem................................................ 120
Fig.6.13 Schematicrepresentationofahydrocyclone ........................ 121
Fig.6.14 Pareto’s Curves obtained for the optimization of the
hydrocycloneperformanceproblem................................. 123
Fig.6.15 Convergencerateandpopulationsizefortheoptimization
ofthehydrocycloneperformanceproblem.......................... 124
Fig.6.16 EvolutionoftheDEparametersfortheoptimizationof
hydrocycloneperformanceproblem................................. 124
Fig.6.17 Simplifiedschematicofthealkylationprocess...................... 125
Fig.6.18 Pareto’sCurvesforthealkylationprocessproblem................. 127
Fig.6.19 Convergencerateandpopulationsizeforthealkylation
processproblem....................................................... 127
Fig.6.20 EvolutionoftheDEparametersforthealkylationprocess
problem................................................................ 128
Fig.6.21 Pareto’sCurvesforthebatchstirredtankreactorproblem ......... 130
Fig.6.22 Convergencerateandpopulationsizeforthebatchstirred
tankreactorproblem.................................................. 131
Fig.6.23 EvolutionoftheDEparametersforthebatchstirredtank
reactorproblem ....................................................... 131
Fig.6.24 Pareto’sCurvesforthecatalystmixingproblem.................... 133
Fig.6.25 ControlprofilesconsideringthepointsAandBforthe
catalystmixingproblem ............................................. 134
Fig.6.26 Convergencerateandpopulationsizeforthecatalystmixing
problem................................................................ 134
Fig.6.27 EvolutionoftheDEparametersforthecatalystmixingproblem.. 135