Resource-constrained project scheduling_ Notation, classific(9)

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P.Bruckeretal./EuropeanJournalofOperationalResearch112(1999)3±4111

wheretivitiesxintegerrepresentedjdenotesthebynumber oftimeunitsallac-jareprocessedjointly.packingversionofthedualofEqs.(1)±(3)isThelowerproblem.Mingozzietal.[126]provideasetschedulingboundsproblemproblemforthebyresource-constrainedsolvingthesetpackingprojectdirectlyBaaretheuristically.

al.[5]solvetheniques.byapplyingcolumnlineargenerationprogram(1)±(3)tech-growingAlthoughtiesexponentiallythewithnumbertheofcolumnsis[31]thetimeenhancedmethodtheisapproachquitefast.byBruckernumberofactivi-takingintoandaccountKnustdowswindowsusingarefortheactivities.Thesetimewin-Nowa®ctitiousderivedupperfromboundtheprecedenceconstraintswhichthewindow.cancolumnsbeprocessedcorrespond jointlytoforinsetsthemakespan.aofgivenactivitiesscheduleTheobjectiveisto®ndapreemptivetimeschedulesearchdoesrespectingnotalltimeswindows.Ifsucha``destructiveprovidesexist, isalowerbound.Binarybeenimprovement''largest withtechniquethisproperty.hasThesevarioususedbyKleinandScholl[105]whotestedalsoupperboundmethods forprovinginfeasibilityofanbeenOtherand-boundintroducedlinear.

programmingalgorithminconnectionbasedboundshaveofChristo®deswithettheal.branch-[40].

3.3.Heuristicmethods

basedThe®rstheuristicmethodswerepriority-rulemultitudeschedulingmethods(cf.[103]).Uptonow,atestedority-basedexperimentallyofpriority(cf.rules[4,22,42,153,194]).wereproposedandintuitive,computationaleasyheuristicstohavetheadvantageofbeingPri-withoptimalrespecttoe ort.implement,theaverageHowever,andfastintermsofdeviationtheydonotexcelresearchobjectiveristicsintegerlikeinterestsfunctionvalue.Hence,fromrecentthetruncatedshiftedbranch-and-boundtomoreelaborate(cf.heu-disjunctiveprogrammingstraint-basedarcanalysisconceptsbasedheuristics(cf.[148]),[4]),(cf.(cf.[196]),[4,14]),samplinglocaltech-

con-niques[5,24,85,108,119,169]).(cf.[110]),andlocalsearchtechniques(putationalresults

branch-and-boundDemeulemeester(cf.algorithmandHerroelenon[48]testedtheirup[154])whichconsistsof110testtheproblemsPatterson-setanto51activities.TheysolvedtheseproblemswithwithModelaveragerithmof70computationtimeof0.21s(IBMPS/2StinsonA21,25etal.MHz)outperformingthealgo-drivenKolischetal.[116][188]developedbyafactortheofparameter-nearly12.hasprojectgeneratorProGenwhichthereafteralgorithmsbeenwidelyusedasatoolfortheevaluationofproject90,scheduling.proposedMeanwhileforresource-constrainedtestsetswith30,consistingand120activitieshavebeengeneratedeach60,activitybytestofset480instancesofvarioustypes.The30sonalDemeulemeesterwasalso15computer(IBMandusedtotesttheapproachPS/2HerroelenModel[48]55SX,onaper-solvedMHz).480timeproblemswithinWhereaslimitof1000instances1.06sonthePatterson-sethas386SX,beensperhaveaverage,beenonlysolved415withinoftheaprogrammingMingozzietbetterformulational.[126]problem.

reportthattheirlinearbythanthecriticalsequencebasedboundboundsintroducedperformcompetitiveStinsonetal.[188],andthattheiralgorithmismeulemeestertoknownandtheHerroelenprocedure[48],presentedbyDe-[49]uptothen.DemeulemeesterandtheHerroelenbestoneboundenhancedfourunsignedresourcesofMingozzitheirof8etapproachbyadaptingalowerbital.[126]sizeandbyrepresentingtheyinteger.Allowingasthroughmuchasone32bitsetcouldsolvetheentire30activitybenchmark-24Mbyte34forSprechersonthea®rsttime.TheCPU-timeaveragesaboutalgorithm[185]personalcomparedcomputerhis(80486,25MHz).rithmwiththeenhancedversionbranch-and-boundofthealgo-outmemorytheoftradeo DemeulemeesterbetweenandcomputationHerroelentimespointingrithmsolves479ofrequirements.ofDemeulemeesterWhilethe480benchmarkandtheenhancedalgo-andproblemsHerroelenwith[49]30