Resource-constrained project scheduling_ Notation, classific(19)

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P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±4121

preprocessingdestructivefoundinHeilmannlowertechniquesboundsasandSchwindtforwell jastempconstructiveand[89].

jgmaxcanbestraintsTheationmarkedlyconceptofreducesdisjunctivethenumberprecedenceofenumer-con-comparisonnodesofbywiththethesearchtreetobeinvestigatedinanalysisDeReyckproblembySchwindt[51].Anbranch-and-boundexperimentalmethod[174]performancesourcesProGen/maxeachinstances(generatedwith100bytheactivitiesbasedproblemandupongenerator®ve1080re-Schwindt'sbySchwindt[174])hasshownthatmalitys(usingwithinmethod10sthansolvesDemoreReyck'sinstancestoopti- AnotheranIBM-compatiblemethodin100etjtempjgbranch-and-boundPCPentiumprocedure200).maxhasbeeninvestigatedby hringfor

resolvingal.[135].Themaindi erenceliesintheMwayo

ofcedures[51],proposedresourcebycon¯icts.BartuschContraryetal.tothepro-[174],DeconstraintswhereReyckandHerroelen[54][12],andDeSchwindtReyckideaThatisareadditionalintroduced(disjunctive)toresolveacon¯ict,precedencetheresolvedis,toaintroducetivitiesbyresourceordinaryincreasingcon¯ictthereleaseatreleaseacertaindatesinstead.datesdtimetis0j0andjjPPwwofac-2(i.e.thetimelagsbetweenactivities2)accordingto

d0newjX miniPw1

i pi

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whereww2isatreeX datesisAthen Yt nminimalwdelayingalternativeand12.Everynodeoftheenumerationtheunchangedvalues(orstartrepresenteddtimes,respectively),onlybyavectorand,exceptofreleasefor0inj,thethecoursepathlengthsofdij iYjP remaintheOnobservedenumerationtheonehand,thismaytheinalgorithm.

principleenlargerelationbySchwindttreeconsiderably[174].Sinceasnohasprecedencealsobeendisjunctive,iscon¯ictitintroduced,mayhappenneitherthatthe``ordinary''samenorexistencehashand,markablethisoftoberesolvedseveraltimes,dueresourcetothewaymaximalofbranchingtimelags.givesButontheotherfeasibleformed.schedulesspeedupinthecomputationriseoftoatime-re-Moreprecisely,onceathebranchingcomputationhasbeenofopti-

per-mallowertime-feasiblelinearboundsdateinthenumberforschedules,newlyandthecorresponding

ofgeneratednodesisthenvantagethatcedenceoverhasbeentheproceduresincreased.activitiesThisforeveryreleasethatisintroduceamajorad-computationconstraints,wherethecomplexityforpre-thedraticprecedenceinthenumberoftime-feasibleofschedulesisqua-andconstraint(seeactivitiese.g.Bartuschforeveryetaddedprecedencepseudopolynomialinthecaseofdisjunctiveal.[12]),schedulingconstraints(seetheabove-mentionedsameTheproblemand[174]).

growthresourcedisadvantagecon¯icts,ofmultipleandoccurrenceofthesmalloftheenumerationtreetheistriedcorrespondingtowellbyperformingimmediateselectionrulesbekeptasComputationalasa(surprisinglyimplementationresultsindicatesimple)that,dominancerule.bounds,proposedtheprocedureofmoresophisticatedevenwithoutlowerHerroelenbySchwindtis[174]competitiveandwiththeonespropagationRecently,[54].

DeReyckandDorndorfetal.[57]showedonthattechniquesanintegrationfor ofjfurthertempusedjgconstraintmax.Theywithinthestarttimesintheaforementionedconstraintssenseprovidesaverynewpromisingtime-orientedresults.branchingscheme6.3.Heuristicprocedures

proximately,Tosolvelargeinstancesof jtempjgmaxap-niquespriority-rulebasedtruncatedbranch-and-boundtech-truncatedmethodsuponhaveSchwindt'sbeendeveloped.algorithmAsandbeamrithm,searchbranch-and-boundtechnique,procedures,a®lteredtoproposedandadecompositionane-approximatemethodalgo-methodSectionexploitsbySchwindt[174].Thedecompositionhavebeennetwork,6.1.theanFirst,theoptimalforDecompositionscheduleeachcycleTheoremfrom CstructureCofthecyclebranch-and-boundiscomputedbyofusinglengthstructurealgorithm.Second,eachzeroCisreplacedbyanequivalentcycleapplied Cto.Third,whosearcweightsaredeterminedtheresultingthee-approximatenetwork.

algorithmis