Resource-constrained project scheduling_ Notation, classific(18)

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20P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±41

constraintsoptimal scheduleforthe itfollowingholdsthatreasons: Foreach0 0.Moreover,any0 0forallSinceheuristicfeasiblethereallparametersmethodschedulesddiscussed constructedinSectionusing6.3.ij iYj PiprovidedalwaysSections6.2thatexistsanintegeroptimal areintegers,scheduleandS Y.AllmethodsdiscussedindecompositionForapproximately6.3constructsolvinginteger jtempschedules.jgexpedient.isAcycleapproachoftenturnsoutmaxto,beanodes.astrongcomponentstructurewhichofcontainstheprojectatnetworkrateForeachcyclestructuretreatedasleastasepa-twoandsubprojectcorrespondingstartedattimewithzero,originalaschedulingresourcecapacitieswhosesubschedules(feasible)toproblemssolutions(4)±(6)arecanbeproblemstatedproventhefollowing.Neumanntheorem.

andZhancalled[140](feasible)workThereisafeasiblesched-foreachifcycleandonlystructure.ifthereisaly)Severalstrictsolvingheuristicprojectschedulingproceduresproblemsfor(approximate-requirelengthordertheofa0longestinnodepathsetfrom (cf.node[70]).iLettodaijbethepathprojectifrominetwork,toj.ForwhereiYdtherenodeisjnoinij ÀIifand0jdifandonlyifeitherjP(a) Ydi j,0weorthen(b)dde®neijbij 0ji`0.

6.2.Branch-and-boundmethods

forThemalsolvingbasic ideajofbranch-and-boundalgorithms example,jtempsolutiontempjgtothejgmaxresourceisasfollows.relaxationAnopti-ofmax(i.e.problemsi theearliestschedulei (4) and i (5)),forj jP withStartingj d0j,atwithcanschedulebefound i inpolynomialtime.

pointsintimet,thatis,,resourcecon¯ictsrjkb kforsomekPRandp A Yt

jPp

7

cantionalbeseveraltemporalresolvedsuccessivelyconstraintsbywhichintroducingdelayoneaddi-biddenclusion,setactivities.SetpinEq.(7)iscalledafor-orReyckBartuschit.isIftermedpisminimalwithrespecttosetin-etal.[12],aminimalDeReyckforbidden[51],setand.

precedenceandHerroelenwhichpcorrespondconstraints[54]toaddingofhaveDethearcstypeused ``ordinary''jP i pi,disjunctivetothenetwork.precedenceSchwindt iYj withweighticonstraints[174]hasintroduced jPiPminpnfjg

i pi Y

8

wheredelayingpbeonlyisaoneminimalactivityforbiddenjset.Insteadofminimaldelayedatthesametimewhich,severalformactivitiesacan(Eq.(8))isdelayingreplacedalternativeby(cf.[51]).so-calledThenminjPw2

jPminiPw1

i pi

withwminimaldelayingalternativew2andcontaining1X A Yt nw2.w2forbiddensetatleastoneiselementaninclusion-minimalofeachminimalsetconsideredTosolveproblemtwo p jA Yt .

partialtempjgproblems.max,SchwindtThesequencing[174]haswhichsuchschedulingthatsatisfyconsistsY disjunctiveof®ndingSprecedenceaset ofconstraintsschedules S.Thesubjectordinarytogionprecedence problemPSconsistsofminimizingcorresponding n 1 .IncontrasttothecaseofconvexSconstraints,thefeasiblere- ofthelatterproblemused,dra.butifrepresentsdisjunctivetheprecedenceisnolongerunionconstraintsareforvisedsolvingApseudopolynomialofconvexpolyhe-bySchwindtthescheduling[174].

problem®xed-pointhasalgorithmbeende-of appropriatelyThebranch-and-boundalgorithmthenconsistsYFFFY rwith enumeratingrsequencingsuchsolutions1m 1 S m Sthatmin PS

n 1 m min

1YFFFYr PminS m

n 1X

Preprocessingboundsand-boundandproceduresaswellasgoodlowermethodfathoming(see[174]).rulesspeedAnoverviewuptheofbranch-recent