Resource-constrained project scheduling_ Notation, classific(7)

项目进度管理

P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±419

thisalgorithmproceduregle)branch-and-boundactivitiesinthatisdi erentaresetsfromtheprecedencetreestartedofactivitiesateachinsteadlevelofof(sin-instantterminedatwhichactivitiestree.Moreover,mayherethetimethelected.algorithm,Finally,beforetheactivitiesthemselvesbestartedareisde-se-schedulingthisinapproachcontrasttoallowstheprecedencetowithdrawtreebeenmadedecisionsataloweratthecurrentlevelthathaveproposedExtensionpartialtouseAlternatives:level.

extensionalternativesStinsonettoal.[188]eachsociatedlevelschedules.goftheAsinthepreviousalgorithm,constructactivitieswithtivities,inprocess,adecisionbranch-and-boundasetpointpttasetts treeisas-g,gofthegcurrentandasetitofthe®nishedac-gofeligibleactivities.subsetwithoutofpartialthescheduleisextendedbystartingThentheaprecisely,violatingeligibletheactivitiesresourceatconstraints.thedecisionMorepointofholdstheeligibleanextensionsetforalternativewhich EAgisasubsetjPts g EAgrjkTEAforeachresourcekPRand,moreover, ktheg Yifts g Y.Note,inorderemptyalgorithmprocess.extensionterminates,wemayonlytosecurehavenon-thatinHowever,alternativesifthereareifnocurrentlyactivitiesareinalternativeprocess,theguaranteewhichemptymustsetisalwaysanextensionactivitiesbranch-and-boundoptimality.AtthebecurrenttestedlevelinordergtoDeterminesetthenewtreedecisiontheprocedureisasfollows:ofthealternatives.oftheeligibleactivitiesandpointtheandsetcomputeoftheEAbranchingandstartFinally,thecorrespondingselectanextensionalternativeextensiongmechanismtorithm.theNoteequalsthenextactivitiesbeforethatthisthelevel.Thebacktrackingprocedureoneoftheispreviousdi erentalgo-fromcludespreviousbeenthepossibilityalgorithm:todelayWhereasactivitiestheformerin-latterstartedonalowerthanthecurrentthatlevel,havethedecisiondoesmayofnotalowerallowlevel.towithdrawaschedulingalternativesnotrestrictthesearchtoAs``maximal''aconsequence,extensionweconsideringStinsononlywhileminimalwedonotdelaylosealternatives.optimalityNote,whenbymeansetofal.an[188]example.

introducedtheproceduresolelyaingslightlyBlockExtensions:di erentapproachMingozzibasedetal.on[126]theconsidertimes

ideas.Thereexistsanoptimalschedulede®ningfollow-t0 0`t1`t2`ÁÁÁ`tl

andthat

correspondingsetsofactivitiese1YFFFYelsuch(i)tivity,

eachti ib0 isthe®nishingtimeofsomeac-(ii)duringallactivities(iii)itifan tineicanbeprocessedjointlyiÀactivity1Yti i jP1YeFFFYl ,

iisnot(iv)willalsobeprocessedin t®nishedin tiÀ1Yti iYti 1 ,and

ataAtimeallpredecessorsblocktofanyactivitywhichstarticonsistsarescheduledofsuchbeforeanintervaltime tti.

iÀFurthermoresete1Yti withiofactivitieswhichcanquenceThenofblocksapartialschedulebeisprocessedde®nedbyjointly.ase-vidingitisbranchedsatisfyingbyaddingconditionsnew(iii)blocksandpro-(iv).gorithmScheduleagainizesdevelopedschemes:partialschedules.

byTheBruckerbranch-and-boundetal.[32]al-schedulingbranch-and-boundschedulingproblemandmethodsthemultiprocessorforthejobgeneral-shoptaskconcepts[12].whichproblemcanbe(cf.found[30,118]).inBartuschItalsoetusesfeasibleInsteadscheduleschedulesofusingarerepresentedpartialschedules,al.bysetsofvatedschemes.Scheduleschemescantheso-calledeitherForasbemoti-twofollows.

arbitraryconjunctionsaparallelityifi3jorrelationactivitiesj3i.iik3jascheduleorjoneof theinducestwoitimeandi®nishesjarebeforeprocessedthestartinparalleltimeofholdsjfor.ikjifandonlyatmeansleastthatonetheseunit.disjunctionrelations.WegetiiÀji.3setsiÀjorofj3schedulesiarerelaxedbyrelaxingbytheparallelity3jorj3relationsrelationsi.FurthermorejmeansthatwehaveeitherikjcanbedisjunctionsrelaxedtoiÀjandwhichi$j.i$jmeansthatitisundecided¯exibilityCofthetworelationsiÀjorikjholds.yetdisjunctions,YDYNandrelations,respectively.parallelityUdenotethe Crelations,setsofYDYNYU andconjunctions,isa¯exibilityschedule