Resource-constrained project scheduling_ Notation, classification, models, and methods(14)

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16P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

branch-and-boundModeanddelayalternatives:WesummarizethehermodeetapproachproposedbySprec-atactivitiestimemal.[187].Aneligibleactivityjscheduledinjtwithstarthavetime jissaidtobeinprocessgifwejTtg` j pjmj.Eligiblepointpreviousthatarehave(temporarily)alreadybeenstartedassignedattheadecisiongiblelevelofthesearchtree.Iftheremodeareateli-amode,activitiessetternativeofmodethatis,alternativesifthatithavenitnotyetbeenassignedaggisÀ1isnotempty,thenthejalternative,Pitisamappingwhichcomputed:assignseachAmodeactivityal-gnitgÀ1amodemjPMj(temporarily)theHavingstartedremaining.Selectingamodeateligibleactivitiescanbethemhavetothestartedalleligiblethedecisionactivitiespointbyasaddingwell.minimalcausedsetaresourcets gofthecon¯ict.activitiesinprocess,maytoDAthefollowingdelayalternativesThus,thesetofthede®nition:iscomputedAaccordingnewablegisa qresourcesubsetofkPts Rq

gitsuchis

thatdelayforalternative

eachre-jPts qgnDAgrjkmjTternativek.Observethateachcombinationofamodeal-correspondsandarelatedminimaldelayalternativethetoadescendantofthecurrentnodetheModebranch-and-boundinandextensiontree.

ternativesconcept[86]areofintroducedmodealternativesalternatives:UsingagainbyHartmannextensional-cisely,totheholdseligibleanconstructextensionpartialsetforalternativeschedules.MoreandDrexlpre-which EAgisasubsetofjPqq

moreover,foreachts g EAgrjkmjT kbranch-and-boundEArenewableresourcekPRqand,g Yifts decisiontreeweg determineY.Atlevelthegoftheties.tivesThenpointwecomputeandthesetoftheeligibleactivi-newthatfor®xingthemodestheofsettheofeligiblemodeactivitiesalterna-activitieshave®xed.thenotmodesbeeneligibleofwhichbefore,havethatis,thosetheextensionsetAfterofextensionselectingamodealternative,notyetcomputebeenspondinglevel.activitiesalternativealternatives.beforeEAFinally,selectangandstartthecorre-andaaEachrelatedcombinationofbranchingamodetoalternativethenextand-bounddescendantextensiontree.

ofthecurrentalternativenodecorrespondsinthebranch-tow Recently,straintsjtempjgthemoregeneralproblemmaxstartgivenbyminimumwithgeneralandmaximumtemporalstart±con-subjecttimeexactofresearchlagsbetweeninactivitieshasbeentheo Acombinationbranch-and-boundHeilmann[88],whereanofprocedureispresented.modeproblemstudiedcasecoveredbyAhndealtandwithintheSectiondiscreteErengin uthis4c[1].sectionandtime±costofthetrade-hasmulti-been

5.2.Dominancerules

severalInHartmannthemwillboundingandberevisitedrulesDrexl[86]adescriptionof

incanwhatbefound.SomeofcannotNon-delayabilitycurrentbefeasiblyscheduledrule:Ifanfollows.

ineligibleactivityestpartialschedulewithoutanyexceedingmodeitsinlat-thetobe®nishexaminedtime,thennoothereligibleactivityneedsstartedLocalleftshiftonrule:thisIflevel.

anactivitythatboundchangingtreeatthecurrentlevelofthebranch-and-hasbeenscheduleitscanmode,belocallythentheleftshiftedcurrentwithoutunscheduledMulti-modeneedsnotrule:beAssumecompleted.

partialthatnocurrently®nishrentleftpartialtimeofactivityscheduleascheduledwillbestartedbeforetheiscompleted.activityjIfwhenthecur-resultingshiftonmodeoramodemH,1TreductionmHofactivityamulti-modejwithjjrmthecurrentpartialscheduleTjwjj,and,canbemoreover,performedifkjkmHmjcompleted.

,thenTrjkmthejholdscurrentforpartialeachnonrenewablescheduleneedresourcenotbetheOrderswaprule:start®nishthetimetimethatofmaywhichConsiderbeisassignedlessthanascheduledactivitywhenorequaltoanythiscurrentpartialschedule.Ifanordercompletingswaponthatactivitythecurrent®nishattogetherpartialitsstartwithanyofthoseactivitiesscheduletimecanneedbenotperformed,thenscheduleCutsetinPSrule:De®ningacutsetbeofcompleted.apartialfollowingPS,Sprecherastherule.LetandsetoftheactivitiesscheduledDrexldenote[186]apreviouslyproposedeval-

the

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