Resource-constrained project scheduling_ Notation, classific(16)

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18P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

ful®llsofthatthechangedtheprecedenceactivitiesconstraints.keepsitsNotethateachassignment.is,thismodi®cationassignedmode,positionThendoesnotchangethemodetermineofmqanewmodefortheactivityon2s jissrandomlychosen,thatis,werede-q2 activityf1YFFFYMbydrawingarandomintegeroutjsqg.Whilethe®rststepmaycreate

ducedmaycurrentintroducebysequences2

thecrossoverthatcouldnothavebeenpro-amodethatoperator,hasnottheoccurredsecondinstepformingpopulation.necessarilyamutationItonshouldbenotedthatper-thetothechangetherelatedanindividualschedule.ThisdoesisduenotbeenTworedundancyvariantsofinthethegeneticselectionrepresentation.operatorhavevival-of-the-®ttestconsidered.Thetionmethod:®rstvariantTheoriginalisasimplesur-individualssizeistheandrestoredremovingbykeepingthe y popula-bestsecondpopulationtheremainingonesfromsurvival-of-the-®ttestvariant(tiesisaarerandomizedbrokenarbitrarily).versionofThethewhichAnumberofs vtechnique.

islandstakeswithplace.thearti®cialareconsideredonOneachevolutionisland,theasdescribedevolutionabovestartstion.begenerationdenotedLetanindependentlythegeneratedinitialpopula-asislandiwithcurrently1Tiunderconsiderationprespeci®edbeandmigrationdenotedprobabilityas`gs vwith,and1letthecurrentwTgTqix.AmigrationistrolarandomnumberqP 0Y1 isusedqtionTwthemigrationbetweenthedrawnislands:tocon-Ifwheregmigrationg.

itleaves,thenisaddedislandthetoi®ttestindividualofgenera-theandpopulationmigratestoofislandgenerationi 1speci®edThestoppingcriterioniseithertoreachaalternatively,numbertimetomeetofislandsaasdescribedabovepre-or,thewithoutboundingthegivennumberlimitofonislands.theCPUIncompletedlatterweandcase,theiftimeqixlimitgenerationshavebeention.skips vClearly,totheifnextthenumberislandandhasofstartnotyetbeenmet,islandsanewisgivenevolu-byare,calculated.

atmosts vÁ y Áqixdi erentindividualsproblemThegeneticspeci®calgorithmlocalsearchismethodaugmentedtoimprove

byatheproachscheduleleftal.shiftiswhichbasedrelatedtoanindividual.Theap-hasonbeenthede®nitionofamulti-modebound[187]inordertoaccelerateintroducedtheirbySprecheretleftscheduleshiftalgorithmofanoutlinedabove.Abranch-and-multi-modewithoutotherchangingwhichactivityreducesjisanoperationonagiventhemodesthe®nishortimeofactivityjstraints.activitiesandwithoutviolating®nishtimestheofcon-thechanged.

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jectAsetoftestproblemsconstructedbythepro-byavailableKolischgeneratoretProGenal.[116]whichhasbeendevelopedPSPLIB.intheprojectschedulinghasbeenproblemused.Theyarereferredalso).toForKolischdetailedandinformationSprecher[115]thereaderlibrary(cf.[114]isinstancesThemulti-modeproblemsetscontainingtivitiesmayhavewithbeen10,12,14,and16nondummyac-durationbeperformedused.ods.ofamodeinvariesoneEachoutoftherealactivitiesbetweenofthree1modes.TheableTherearetworenewableandtwoandnonrenew-10peri-instancesresources.fourwasForeachproblemsize,asetoftheparameters,generatedthatis,bythesystematicallyresourcefactorvaryingcomparisonInresourceHartmannstrengthandofeachresourcecategory.andcombinationofthethreeDrexlbranching[86]acomputationalschemesinTheisprecedencewithtreeboundingalgorithmrulescanbefound.timesthedelayfasterfastestthanprocedurethealgorithmonthewithbasedaverage.thecutsetruleonmodeItistwoandandties,sevenalternativesincreasingthatis,timesthecomparisonfasterwhenfor10activitiesprojectsareconsideredfactorwith16activi-basedtimesextensionfasteronmodenumberofactivities.increasesThealgorithmwithanthanandthedelayalternativesisatmost1.4performedalternatives,algorithmhence,thebasedlatterononemodeandtobytheothertwoalgorithmswithrespectisout-toaverage``maximal''thefactcomputationthattimes.Thisseemstobedueextensionbranchingalternatives.maynotTheberestrictedprecedence

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