Resource-constrained project scheduling_ Notation, classific(8)

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10P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

schemeonejschedule3ofiPtheifforanytwodi erentactivitiesiYjexactlyCfollowingrelationsholds:i3jPCorfeasibleschemeoriÀjP CDYDorYNikjPNori$jPU.Aallschedules(whichmayYU bede®nesempty),anamelysetofinCfeasibleYDYNscheduleswhichsatisfyalltherelationsDIfC.

0isthesetofallgivenprecedencerelations,be0straints,processedisthesetofinallparallelpairsofactivitieswhichcannotthenschedules. CandUduetotheresourcepairsicon-0thesetofallremaining$j,0YD0YYofthe CYYUD0 representsthesetofallfeasible00YYYU0 correspondstotherootitForenumerationascheduleschemetree.

oftheform CYDsatisfyingcanbeshownthateithernofeasiblescheduleYNYY feasiblecanscheduleallthecanrelationsbecalculatedexistsor(cf.a[117]).dominatingofbedoneinO n3 time.Thus,scheduleschemesBoththetheform CYDYNYY canbetreatedasplacingenumerationa¯exibilitytreeandonecanbranchleavesbyre-ofschemesTheinclusionrelationi$jbyiÀjorikj.nodeallowstoformofconjunctionsatemporalanalysisinscheduleineachconjunctions,oftheenumerationaredisjunctionstree.orByparallelitythisanalysisrelationsnewwindowsdeduced.boundcanFurthermore,becalculatedforwhichtheimproveactivitieslowertimemacherMinimalcalculations.

scheme[99,100]forbiddenhavesets:introducedIgelmundaandbranchingRader-detailedbasedinSectionsdiscussiononminimaloftheseforbiddenconceptscansets.beAmoreSectionBesidelower6.2andbounds8.2offoundwhichthissurvey.

arewithinDemeulemeestere.g.3.2,thedominancepartialenumerationrulesaresuccessfullydescribedusedinSprechersearch[185]inorderandtoHerroelenalgorithmsprunelarge[48,49]paresDuringaboutthealreadyevaluatedpartialstoredobtainabledata.theIfcurrentpartialsearchscheduleprocesswiththerulethenotfromitthecancurrentbeprovenpartialthatscheduleanysolutioncan-previouslybebetterevaluatedthanasolutionpartialscheduleobtainablethefrominfor-

amationtrackingofrulestheisskippedmaywhichbehasbeenstored,thenback-hereperformed.fortheAsakedescriptionofsuchrulesreaderareoutlined.

isreferredtoSectionof5.2shortnesswheresomeand3.2.Lowerbounds

valueUsuallylowerboundsfortheoptimallaxingof jpre jgsolutionmaxcanbecalculatedlaxedsomeoftheconstraintsandsolvingthebyre-re-resourceproblemlengthconstraintstooptimality.leadstoRelaxationthecriticaloftheStinsonwhichanetal.[188]providesasimplelowerbound.pathpathactivitytimeg .Theyiwhichimprovecalculatedoesanotthismaximalbelongboundbyaddingnumbertoacriticalewithpathg units.Byactivityaddingimaxcanfbe0YpprocessedinparalleliofiÀesteadlengthanewlowerboundiisgtothecriticalandpathHerroelenofaddingaugmentasingleactivity,Demeulemeesterprovided.In-bound node-disjointprocedure.forg usingwithaag criticaldynamicandcalculatepathg byaprogrammingalowercoincideg ation (cf.withIngeneral[172]).theHowever,optimalthislowersolutionbounddoesnotthetwo-pathvaluerelax-formethodcantwooptimaljobs.developedbesolvedThisapproachfortotheoptimalityjobbyagraphicalalsoshopallowsproblemtowithwindowssolutionforg whichrespects®ndtimeanfollowingAboundforoftheactivitiesing (cf.[28]).thelinearMingozziprogrametwhichal.[126]isbasedontheTheyprecedencebeconsidermaximalconstraintssetsandallowsrelaxespreemption.partiallycharacteristicprocessedinparallel.Letof activities whichcan1Y2YFFFY qbethelinearprogrammingvectorsrelaxationofallthesehassets.theformThenthemin qxj

1 j 1 qsXtX

ijxjPpiYi 1YFFFYnY

2 j 1

xjP0Yj 1YFFFYqY

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