Resource-constrained project scheduling_ Notation, classific(10)

项目进度管理

12P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

activitiesMbytewithinaveragetimeof12.33susing24theusingsamememory,problemsthewithalgorithmaverageoftimeSprecherof12.85solvessal.The400but[32]branch-and-boundKbytememory.

solveslessproblemsalgorithmthanotherofalgorithms,Bruckeretquirement.itsmainusesproblemsatmostForadvantageissmallermemoryre-10problemswithupto90activitiesit®rst20/801timewithinwith60Mbyte.a1activities326ofhtimelimitwerethe480veri®edbenchmarkfortheconcludedComparingworkstation.

onaSUN/SparctheheuristicmethodsitKolischbasedandthattheadaptivesearchalgorithmcanbeofgeneticonlatedalgorithmscheduleDrexlofschemes[110],theHartmannoftabuBaarsearchmethod[85]andetal.[5],thecocqannealingalgorithmofBouleimentheandsimu-Le-Kolisch[24]parisonandprovideHartmannthe[112]bestperformedresults.Recently,rithmsofmostoftheavailableheuristicacom-algo-results,thetheforgeneticlargeralgorithminstances.ofAccordingtotheseandsimulatedLecocq[24]annealingarethemostalgorithmHartmannpromisingofcandidates.Bouleimen[85]and4.Time±costtradeo problems

worksSofar,generalizationwithwe®xedhaveonlyconsideredprojectnet-ministicof(andthisknown)settingprocessingthatistimes.Aobtained(andaccordingbyassumescompleteinformation)stilldeter-ispaytopermittinghowmuchprocessingtheplannertimestovaryonlocationtheforprocessingit.Inviewoftimesthecannextiswillingtobesection,interpretedthiscontrolasties,ofanonrenewableresourcetotheactivi-al-higherwherecostalargerallocationtoanactivity(i.e.,aprojectTheplannerinput)thenreducesaimsatitseitherprocessingtime.onmakespansubjecttoa®xedminimizingupperboundthelemtoproblema),thegivenornonrenewableatminimizingresource(thebudgetprob-boundonthethemakespantotalallocationsubjectmoney,).Astheallocationisusually(themeasureddeadlineinastime±costthesetradeo problemsproblems.

arecommonlyreferredto4.1.Model

lemsAactivities,consistsformalmodelofasetfor time±cost f0Y1YFFtradeo FYnYn prob-denceadirectedgraphq Yi of1gofeachtimesactivityconstraintsj,aamongsetMtheactivities,and,prece-forfunctionpPMjofpossibleprocessingjj,togetherwithanonincreasingtradeo andbetween jXMj3R thatmodelstheindividualcatedthecost(oramountprocessingoftimeresource)pjof activityjj pj allo-w toit.Section5coversthemodeldealtjasimilarwithpre wayinjgthismax,ageneralizationofthemodeltosection.denotetheThere,setMjwillbeusedincostDi erentcostfunctionsassumptions leadtodi erentontheofsubcasessetsmodes.

Mjofandtime±thejMFortradeo instance,problems.

ifeveryMjonproblemM isaclosedintervalj jY j andjisa nelinearanddecreasingj,wehavethelinear(i.e.,If,the®nite)onintroducedthesetotherbyand hand,Kelleyeveryandtime±costMWalkertradeo [104].jisadiscretejisdecreasingonMbyMuthHarveydiscreteandtime±costj,wehavePattersontradeo problemintroducedsignmentArealization[96].

[87]andHindelangand 2

oftheprojectisanas-jgivenP .TheoftotalprocessingpPRn

costtimespjPMjtoactivities p ofgearliestp byofthe p therealizationpisjP j pj .Themakespanmax hasstartschedulerealizationofthepisprojectthemakespanwhenactivityofthepathprocessingjPinq timeYi pwithi.e.,pthelengthofalongestjj,jas``length''ofvertexoptimizationFixing .

eithermizeproblemscostortime,withweobtaintworelatedgetBudgettheothertheobjectivetomini-minimizes P0,problem:parameter.

®ndarealizationForagivenpwithnonnegative p T bud-makespan,Deadlinethethat®ndproblem:makespanarealizationForgthatamaxgiven p .

pdeadlinewithgonthemax p allUsually,minimizesonethewantstotalcost p .

Tfunction

possiblebudgetsortodeadlines.solvetheseThisproblemsleadstoforthe