Genetic algorithms using multi-objectives(7)

We are interested in a job-shop scheduling problem corresponding to an industrial problem. Gantt diagram’s optimization can be considered as an NP-difficult problem. Determining an optimal solution is almost impossible, but trying to improve the current s

A.Cardonetal./RoboticsandAutonomousSystems33(2000)179–190183

isavectorofnrealvaluescomingfromobjectivefunctions,xisavectorofnvariablesofdecisionandX={x|x∈Rm,gk(x) 0,

k=1,...,m,andx∈S}

(3)

isasetofpossiblesolutions.gkisarealfunctionvaluerepresentingmthekthconstraintandSisasubsetofRrepresentingalltheotherformsofconstraints.Theidealsolutiontosuchaproblemisapointwhereeachobjectivefunctioncorrespondstothebest(minimum)possiblevalue.Theidealsolutioninmostcases,doesnotexistinviewofcontradictoryobjectivefunctionsandhencecompromiseshavetobemade.Adiffer-entconceptofoptimalityhastobeintroduced.Solv-inganMOPgenerallyrequirestheidenti cationofParetooptimalsolutions[33],aconceptintroducedbyV.Pareto,aprominentItalianeconomistattheendofthelastcentury.AsolutionissaidtobeParetoopti-mal,ornondominated,ifstartingfromthatpointinthedesignspace,thevalueofanyoftheobjectivefunctionscannotbeimprovedwithoutdeterioratingatleastoneoftheothers.AllpotentialsolutionstotheMOPcanthusbeclassi edintodominatedandnon-dominated(Paretooptimal)solutions,andthesetofnondominatedsolutionstoanMOPiscalledParetofront.The rstandmostimportantstepinsolvinganMOPisto ndthissetorarepresentativesubset.Af-terwardsthedecisionmaker’spreferencemaybeap-pliedtochoosethebestcompromisesolutionfromthegeneratedset.Thenaturalorderingofvectorval-uedquantitiesisbasicforParetooptimality.Tode nethenotionofdomination,letf=(fg=(g)betworeal-valuedvectors1,...,offnn)and1,...,gmele-ments:fispartiallysmallerthangif: i∈1,...,nand k∈1,...,m,fg,wei≤saygkand i|fthatfdominatesi≤gk,wenotef<pg.Iff<pg.Con-sequently,afeasiblesolutionx issaidtobeaParetooptimaloftheproblemifandonlyifanotherx∈Xdoesnotexistsuchthatf(x)<pf(x ).6.DevelopmentofParetooptimalsolutionsTwodifferentstrategiesareeffectiveingeneratingParetooptimalsolutions[12,16].Inthe rststrategy,anappropriatescalaroptimizationproblem(SOP)[42]isset-upinparametricform,sothatthesolutiontothe

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