Resource-constrained project scheduling_ Notation, classific(6)

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8P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

applicationsschedulingareproblemshavebeen(e.g.achievedjobshopfordisjunctive celeratejpre currentlyjggoingtobescheduling)extendedandtomax.Theaimofsuche ortsisthroughheuristicsfeasibleanearlydetectionorbranch-and-boundofnonattractivemethodstoac-orin-constraintnodes.AcomprehensiveintroductionintoTsangsistency[195].propagationtestsareConstrainthasdescribedpropagation-basedbeenprovidedbyinAppendixA.con-e.g.3.1.Branch-and-boundmethods

varietyStartingdevelopedofbranch-and-boundwithanearlyworkofJohnson[101]apartialverticesschedulesfor jwhichpre jgalgorithmshavebeenaremax.associatedMostofthemwithuseprocessinanddi erentconsistsoftheenumerationtheways.oftree.ThebranchingDominanceextendingrules,thepartiallowerschedulebounds,numberimmediateschedule.ofalternativesselectionforallowtodecreasetheschemes®rst-searchandThemethodsuseextendingdi erentthebranchingpartialquirementsispruningusedinmethods.ordertoIngeneral,depth-proposedThePrecedencelow.

keepmemoryre-Tree:Pattersonetal.precedenceaningeachthedummytree.algorithmTheguidedbytheso-called[155]beginningprocedureactivitybeginsattimewithstart- tlevelgofthebranch-and-boundtree,the0.Atsetitgofthecurrentlyscheduledactivitiesandthesetthegdetermined.predecessorsoftheeligibleactivities,thatis,thoseactivitiesNowThenofanwhicheligibleareactivityalreadyscheduled,isstartassignedtimetheearliest precedenceandresourcejgisselected.feasiblejgthatisnotlessthanthecomputed.ondummyThenthepreviouswebranchleveltoofthethenextsearchstartlevel.treetimeIftheisscheduleterminationinguntestedtothehaspreviousbeenfound.activityiseligible,acompletelevelInoccurs.thiscase,Here,backtrack-activitieseligibleactivityischosen.Ifalltheeligiblenextback.precedenceEachhavebranchbeenfromtested,wetrackanotherstepthesetofactivitiestreecorrespondstherootwhichisprecedencetoatopermutationaleafofthefeasibleofintheathissmallersensethatsearchalgorithmindexeachpredecessorofanactivityjghashasinthebeensequenceenhancedthanwithjg.Recently,conceptDelaytreeAlternatives:reductiontechniquesThisalgorithmbySprecherpowerfulbaseson[185].etmeulemeesteral.[40]ofdelaywhichalternativeshasbeenusedbyChristo®desthethetheprecedenceandtreeHerroelenalgorithm,[48].enhancedIncontrastbyDe-to®xedbranch-and-boundhereeachlevelgoftivitiestimetreeisassociatedwithade®nitionmayinstantbestarted.tg(decisionConsequently,point)atawhichac-rithm:eligibleAcurrentlyofeligibleunscheduledactivitiesisactivityusedinthisdi erentalgo-scheduledattimetjiscalledgifallofitspredecessorsithermore,withacompletiontimegareiTtg.Fur-beTheinprocessanatactivitytimetjwithstarttime jissaidtogifwehave jTtg` branch-and-boundproceedingatthecurrentlevelgofj thepj.cisionpletionpointttreeisasfollows:Thenewde-gisdeterminedNotetimeoftheactivitiesasintheprocessearliestatcom-tgthescheduledrenewablethat,duetoÀ1.resources,theconstantavailabilitylevelsofstartingactivitiesunscheduledactivitiespoint,thatare®nishedones.needtoonlybe®nishconsideredtimesforofatUsingorbeforethesetptgoftheputed.thesetitoftheeligibleactivitiesthedecisiongiscom-addingHavingprocess,themtostartedthesetallts eligibleactivitiesbygofThus,computedthemaysetofhavecausedaresourcetheactivitiescon¯ict.indelayforalternativeaccordingtheminimaltothefollowingdelayalternativesde®nition:Ais

eachrenewableDAgisaresourcesubsetofkts PgRsuchitthatiscalledjPts gnDAgrjkdelayselectedalternative.minimalTif kno.AdelayalternativeDAgisAminimalpropersubsetdelayofalternativeDAgisisamovednofromandthethecurrentactivitiespartialtobedelayedarere-alternativeresourcetimesiscon¯icttheoccurs,theonlyschedule.minimalNote,delayifinformationofanactivityemptyjtobeset.delayedWestorethestarting.nextThenithasisbranchedtoberestoredduringbecausebacktrack-thiscompletedecisionpointiscomputed.tothenextIfthelevelscheduleandthenextminimalnow,delaybacktrackingalternativeisperformedistested.Clearly,

andtheis