Resource-constrained project scheduling_ Notation, classific(11)

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P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±4113

opt X minfgmax p jpjPMjY p T ggivingbudgetthe ,andminimumthefunction

makespanasafunctionofthefopt X minf p jpjPMjYgmax p TgYgivingdeadlinethe(theminimumcostasafunctionofthew Thebudgetprojectproblemcostiscurvea).

specialcaseofWhile1jpre jgmaxwhichiscoveredinSection5.sourcesinsource,andthisonlysectionwehavenorenewablere-version.Section5considersonesinglethegeneralnonrenewablemulti-modere-characteristicsSimilarly,andofthethemulti-modedeadlineproblemcase(Sectioncombines5)Inofw ourthe1jpre notation,resourcelevellingj

ritcouldproblembedenoted(Section7).bykk Yt .

4.2.Exactalgorithms

getKelleycontextproblemandandWalkerthedeadline[104]discussproblemboththebud-In[15]),thisthethesettingofthelineartime±costtradeo withinproblem.theproject(andcostalsoinamoregeneralone,seesiderinverseonlyffunctionofcurve foptcanbeobtainedasopt,soitsu cestocon-opt.

pressedForevery®xeddeadlinefopt canresemblesasUsingaaspecialmin-costlinear¯owprogramproblemwhosebe(see[74]).dualex-optimizationstandardresultsfromparametriclinearpiecewiserameterlinearithenceandconvexfollowsfunctionthatfoptof the ispa-adevelopedFulkerson[74]andprojectactivity-on-arccostthecurvesamealgorithmKelley[102]toindependentlycomputetheiterativelyrepresentationfopt.Thisofalgorithmtheusesan``cheap''activitiescutscalculatesinasequenceofnetworklessandandlessbreakpointbywhichthethecurrentmakespannetworkisreduced.ofcriticaltoone.

achangeofofthethecurrentprojectcutcosttocurveamorecorrespondsEveryexpensive¯owEverycomputationsuchcutincanwhichbedeterminedthecapacitiesbyaaremax-de-

rivedoffromtheslopesofthelinearcostfunctions jquentlythecritical[159,160].

beenimprovedactivities.byThesePhillipsideasandhaveDessoukysubse- [79]),O jThis j2logalgorithmj j byispolynomialpercutbepleslarge.butSkutellathenumberstandard[178]ofprovidescutsto¯owbemethods(cf.acomputedclassofmaynentiallyforwhichtheprojectcostcurvehasexam-expo-exponentialmanybreakpoints,thusrequiringanareThenumberofcutcalculations.

cently,discretecase Deetiswhereal.quitethe[46]commonpossibleprocessingtimesshowedthat,inpractice.Onlyre-there,itisgisstronglyarealizationNP-completepsuchtogiventhatdecidethebudget p Twhether andatmaxjMmost p Ttwo2.Thisprocessingholdsalreadytimeforalternatives,activitieswithi.e.,manyDuejjT2.

been(exponentialtothepracticaltime)importanceexactalgorithmsoftheproblem,havegrammingproposed.[96]approachesEarlyexamplesbyHindelangaredynamicpro-gorithmandRobinson[164],andanenumerationandMuthal-onThecurrentlybyHarveybestandknownPattersonalgorithms[87].

stilladditiondynamicunderlyingthedecompositionprogramming,putationThedecompositionisknownasmodularthatofthefa-hascombinatorialmanyapplicationsorsubstitutiondecompositionandcomprehensiveoptimizationinnetworkandothermacherarticlebyM oproblems,hringandseeRader-the

lemIts[129].

Rothfarbwasusefulness®rstparalleletal.observedforthe[165]forbytime±costtradeo prob-theFrankspecialetcaseal.of[71]series±andtiondue[130]).

totheoremdecompositions.Thegeneraldecomposi-BillsteinthatandinvolvesRadermacherarbitrary[15]modules(seealsoisprojectBecausecertaincostofthemodulardecomposition,theinindecomposablecurveneedssubnetworksonlytobeevaluated(theforThisacompositioncomposableisdoneseries)oftheoriginalnetwork.factorsnetworkby``transforming''toaseries±parallelsuchannetwork

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