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On Scheduling and Resource Allocation Problems with Uncertainty

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Title: On Scheduling and Resource Allocation Problems with Uncertainty
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Copyright Date: 2008

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Permanent Link: http://ufdc.ufl.edu/UFE0002303/00001

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Title: On Scheduling and Resource Allocation Problems with Uncertainty
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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ONSCHEDULINGANDRESOURCEALLOCATIONPROBLEMSWITHUNCERTAINTYByGANGCHENADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2003

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Copyright2003byGangChen

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Idedicatethisworktomyfamily:mywifeShu,mybrotherNing,andmyparents.

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ACKNOWLEDGMENTSFirstandforemost,IwanttothankmydissertationadvisorDr.Zuo-JunShenforhiscontinuousadvice,support,andencouragement.Hehasservedasoneofmyrolemodelsbecauseofhisextraordinarydiligenceanddevotiontohisresearch.IwishtothankDr.JosephP.Geunes,Dr.H.EdwinRomeijn,Dr.StanleySu,andDr.StanislavUryasevforservingonmycommittee;andfortheirhelpfulcommentsandsuggestionsleadingtoimprovementsinthisdissertation.IwishtothankmyfriendsattheISEdepartmentLihuiBai,JianLiu,WeiHuang,YangZhu,LianQi,andmanymorefortheirhelpandsupport;andforthegoodtimeswehavehadtogether.IespeciallywanttothankmywifeShu,forherlove,support,andsacrice.Withoutherbymyside,thisworkwouldnothavebeenpossible.Finally,IwishtothankmyparentsandmyelderbrotherNing,fortheirloveandsupport.Idedicatethisdissertationtothem. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... ix CHAPTER 1PROBABILISTICASYMPTOTICANALYSISONSTOCHASTICONLINESCHEDULINGPROBLEMS:BASICRESULTS ......... 1 1.1Introduction .............................. 1 1.2LiteratureReview ........................... 3 1.3SingleMachineProblem ....................... 6 1.4FlowShopProblem .......................... 8 1.5UniformParrallelMachineProblem ................. 13 1.6UniformParrallelMachineProblemwithBoundedProcessing Requirements ............................ 17 1.7ConcludingRemarksandDiscussions ................ 23 1.8Notes .................................. 23 2PROBABILISTICASYMPTOTICANALYSISONSTOCHASTICONLINESCHEDULINGPROBLEMS:EXTENDEDRESULTS ..... 25 2.1Introduction .............................. 25 2.2SingleMachineProblem ....................... 25 2.3UniformParallelMachineProblem ................. 34 2.4FlowShopProblem .......................... 38 2.5ConcludingRemarks ......................... 46 2.6Notes .................................. 47 3COMPREHENSIVESIMULATIONOFSTOCHASTICONLINESCHEDULINGPROBLEMS ............................. 48 3.1Introduction .............................. 48 3.2LiteratureReview ........................... 51 3.3TotalWeightedCompletionTimeMetric .............. 52 3.3.1LowerBounds ......................... 53 3.3.2ExperimentDesignandSimulationResults ......... 56 v

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3.4TotalWeightedFlowTimeandTotalWeightedStretchMetric .. 60 3.5ConcludingRemarks ......................... 66 4NEWMODELFORSTRATEGICSUPPLYCHAINLOCATION .... 68 4.1Introduction .............................. 68 4.2LiteratureReview ........................... 69 4.3Model ................................. 71 4.4ComputationalResults ........................ 77 4.5AMeanExcessBasedHeuristicforSolvingtheMinimaxModel .. 83 4.6DiscussionsandConclusion ...................... 86 REFERENCES ................................... 88 BIOGRAPHICALSKETCH ............................ 92 vi

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LISTOFTABLES Table page 3{1Simulationresultsfor1 j x j stoch;r j j P w j C j ............. 60 3{2Simulationresultsfor Qm j x j stoch;r j j P w j C j ............ 61 3{3Simulationresultsfor Qm j x j stoch;r j j P w j C j ............ 61 3{4Simulationresultsfor Fm j x ji stoch;r j j P w j C j ........... 62 3{5Simulationresultsfor Om j x ji stoch;r j j P w j C j ........... 62 3{6Simulationresultsfor Jm j x ji stoch;r j j P w j C j ........... 62 3{7Simulationresultsfor Qm j x j stoch;r j j P w j f j and Qm j x j stoch;r j j P w j R j .................... 65 3{8Simulationresultsfor Fm j x ji stoch;r j j P w j f j and Fm j x ji stoch;r j j P w j R j .................... 65 3{9Simulationresultsfor Jm j x ji stoch;r j j P w j f j and Jm j x ji stoch;r j j P w j R j .................... 66 4{1 SolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. .. 79 4{2 TotalSolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. 81 4{3 SolutionsofMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. ...... 82 vii

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LISTOFFIGURES Figure page 3{1Relativeperformancesof random FCFS ,and WSPRA to LP for Qm j x j stoch;r j j P w j C j ........... 63 3{2Relativeperformancesof random FCFS ,and WSPRA to LP for Fm j x ji stoch;r j j P w j C j ...................... 63 4{1SolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcess with =0.95. .............................. 80 4{2TotalSolutionTimesforMinimaxI,MinimaxII,MinimaxIIIand meanExcesswith =0.95. ....................... 81 viii

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyONSCHEDULINGANDRESOURCEALLOCATIONPROBLEMSWITHUNCERTAINTYByGangChenDecember2003Chair:Zuo-JunShenMajorDepartment:IndustrialandSystemsEngineeringWeanalyzeschedulingproblemsinthestochasticonlineschedulingenvironment.Inthisenvironment,informationonthefuturearrivalofajobisunknownuntilthejobarrivesatthesystem;andtheprocessingrequirementofajobremainsuncertainuntilthejobisnished.Ourgoalwastoidentifyonlinealgorithmswithattractiveasymptoticperformanceratios.Undersomemildprobabilisticassumptions,werstshowedthatanynondelayalgorithmisasymptoticallyoptimalforstochasticonlinesinglemachineproblem,uniformparallelmachineproblemandowshopproblemwiththeobjectiveofminimizingthetotalweightedcompletiontime.Wethenex-tendedtheseresultstoamoregeneralrealmandillustratedthesignicanceandpracticalusagebygivingexamples.Oursimulationstudiesontheseproblemsandthestochasticonlinejobshopandopenshopproblemsshowthattwogenericnonde-layalgorithmsconvergeveryfast.Thesimulationresultsalsosuggestthat,comparedwiththetotalweightedcompletiontimemetric,thetotalweightedowtimemetricandtotalweightedstretchmetricaremoresensitiveandmaybebetterperformancemeasuresintheonlineschedulingenvironment. ix

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Wealsopresentanewapproachtostrategicfacilitylocationplanning.Inthisapproach,decisionmakersidentifyanumberoffuturescenariosandestimatethelikelihoodofeachscenariooccurring.Wethenusedthemodeltondasolutionthatminimizestheexpectedregretwithrespecttoanendogenouslyselectedsubsetofworst-casescenarioswhosecollectiveprobabilityofoccurrenceisexactly1-.Ournewmodel,-reliablemeanAccessregret,hasanumberofadvantagesovertheexisting-reliablep-medianminimaxregretmodel.First,bydenition,-reliablemeanAccessregretisanupperboundofthecorresponding-reliablep-medianminimaxregret.Therefore,minimizing-reliablemeanAccessregretwillalsoleadtoalow-reliablep-medianminimaxregret.Second,minimizationof-reliablemeanAccessregretavoidsthedangerousincreaseintheworst-caseregret.Third,the-reliablemeanAccessmodeliscomputationallymucheasiertosolve.Alloftheseadvantageshavebeendemonstratedbyournumericalexperiments. x

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CHAPTER1PROBABILISTICASYMPTOTICANALYSISONSTOCHASTICONLINESCHEDULINGPROBLEMS:BASICRESULTS1.1IntroductionInthestochasticonlineschedulingenvironment,asetofjobsN=f1;2;ngarriveovertimeandmustbeprocessednonpreemptivelyononeormoreofmma-chines.Thereleasetimeandweightofeveryjobj2Nremainunknownuntiljobjarrives.Inaddition,theprocessingrequirementofeveryjobj2Nisarandomvari-ablewhoseactualvalueremainsunknownuntiljobjisnished.Theprocessingtimeofjobjisequaltoitsprocessingrequirementdividedbythespeedofthemachineonwhichitisprocessed.Inthischapterwestudythreedierentstochasticonlineschedulingproblems.Theyarethestochasticonlinesinglemachineproblem,uniformparallelmachineproblemandowshopproblem.Inthesinglemachineproblem,allthejobsmustbeprocessed,oneatatime,onasinglemachinewithaunitspeed.Intheuniformparallelmachineproblem,therearemmachineseachwithaconstantspeedsi>0i2f1;2;mgandeachjobjj2f1;2;nghastobeprocessedononeofthemmachines.Intheowshopproblem,eachofthemmachineshasaunitspeedandeachjobj2Nmustvisitthemachines1,2,;minthatsameorder.Withtheobjectiveofminimizingthetotalweightedcompletiontime,thestochas-ticonlinesinglemachineproblem,uniformparallelmachineproblemandowshopproblemcanbedenoted,instandardschedulingnotationsee,e.g.,Grahametal.,1979,by1jxjstoch;rjjPwjCj,Qmjxjstoch;rjjPwjCjandFmjxjistoch;rjjPwjCj,respectively,wherexjistheprocessingrequirementofjobj.Notethatintheowshopproblemxj=Pmi=1xji,wherexjiistheprocessingrequirementofjobjonmachinei.Thedeterministicvariantoftheseproblems,inwhichtheexact 1

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2 processingrequirementofeveryjobjisknownuponjobj'sarrivalattimerj,aredenotedrespectivelyby1jrjjPwjCj,QmjrjjPwjCjandFmj;rjjPwjCj.Inthispaperwesaythatajobiswaitingattimetifitisreleasedbutnotbeingprocessedattimet.Wesaythatajobisinthesystemattimetifithasbeenreleasedbutnotnishedbytimet.Wesaythatacertainamountofprocessingrequirementiswaitingattimetifithasbeenreleasedbutnotnishedbytimet.Furthermore,throughoutthispaperweassumethatalljobshaveboundedweights.Wealsoassumethatthemachinesineachproblemhashaveadequatecapacity.Thatis,inthelongrunthemeanrateatwhichjobsarriveisstrictlylessthanthemeanrateatwhichthemachinesisarecapableofprocessing.Wejustifythisassumptionbyobservingthat,withthisassumptionbeingunsatised,thenumberofjobsthatarewaitingforprocessingwillkeepincreasingandwilleventuallyapproachinnityinanyfeasibleschedule.Thismeansthat,afteracertainperiodoftime,therewillalwaysbeanextremelylargenumberofjobswaitingforprocessingandthevastmajorityofjobinformationisknownwheneveradecisionistobemade.Suchkindofaproblembearsmorecharacteristicsofanoineproblemthananonlineproblemandshouldberegardedmoreappropriatelyasanoineproblemandthuswillnotbeconsideredinthispaper.Forthedeterministiconlineschedulingproblems1jrjjPwjCjandQmjrjjPwjCjwithboundedprocessingrequirementsandboundedweights,Chouetal.2001showthattheWeightedShortestProcessingRequirementamongAvailablejobsWSPRAisasymptoticallyoptimal.ForthedeterministiconlineowshopproblemFmjrjjPwjCj,Liu001showthatthreeheuristicsextendedfromWSPRAareasymptoticallyoptimal,withtheassumptionsthatjobweightsareboundedandi.i.d.,processingrequirementsareboundedandi.i.d.acrossallthemachinesandjobs,andjobre-leasetimesincreaseintheorderofOn.Incontrast,inthispaperweshowthatanynondelayalgorithm,i.e.,algorithmthatkeepsthemachinesbusyaslongas

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3 thereisworkavailableforprocessingPinedo,1995,isasymptoticallyoptimalforthesinglemachineproblem1jxjstoch;rjjPwjCjandtheowshopproblemFmjxjstoch;rjjPwjCjaslongasthejobweightsarealwaysbounded,jobinter-arrivaltimesarei.i.d,processingrequirementsarei.i.d.acrossallthejobsandallthemachinesfortheowshopproblemonly,andmachinecapacityisadequate.FortheuniformparallelmachineproblemQmjxjstoch;rjjPwjCj,weshowthatanynondelayalgorithmisasymptoticallyoptimalwiththeadditionalassumptionthatalljobprocessingrequirementsarebounded.Therestofthischapterisorganizedasfollows.InSection1.2webrieyreviewrelatedresultsintheliterature.InSection1.3andSection1.4weprovetheasymptoticoptimalityresultsforthesinglemachineproblemandowshopproblemrespectively.InSection1.5weshowthattheFirst-Come-First-Serveruleisasymptoticallyoptimalfortheuniformparallelmachineproblem.InSection1.6weshowthatanynondelayalgorithmisasymptoticallyoptimalfortheuniformparallelmachineproblemwiththeadditionalassumptionthatalljobprocessingrequirementsarebounded.Finally,weconcludeourdiscussionandsuggestfutureresearchdirectionsinSection1.7.1.2LiteratureReviewInthissectionwebrieyreviewrelatedresultsintheliterature.Asymptoticper-formanceanalysisiswidelyusedtoevaluatetheperformanceofanalgorithmonlargesizeinstances.Gazmuri85studiesthesinglemachineproblem1jrjjPCjundertheassumptionsthatalljobprocessingrequirementsareboundedi.i.d.integers,allinterarrivaltimesarei.i.d.integers,andprocessingrequirementsandinterarrivaltimesareindependentofeachother.Heconsidertwodierentcaseswhereintherstcasetheexpectedjobprocessingrequirementisstrictlylessthantheexpectedinter-arrivaltimeandinthesecondcasetheexpectedjobprocessingrequirementisstrictlygreaterthantheexpectedinterarrivaltime.Ineachcase,adierentasymptoticallyoptimalalgorithmisgiven;intherstcase,anoinealgorithmwhileinthesecond

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4 case,anonlinealgorithm.Morerecently,KaminskyandSimchi-Levi01astudythesamesinglemachineproblem1jrjjPCjandshowthattheShortestProcessingTimeamongAvailablejobsSPTAruleisasymptoticallyoptimalforthisproblemaslongasalljobprocessingrequirementsarebounded.BuildingontheresultsofGoemans97andGoemansetal.999,Chouetal.001showthatagener-alizedversionofSPTArule,theWSPRA,isasymptoticallyoptimalfortheweightedversionofthesinglemachineproblem1jrjjPwjCjandtheuniformparallelmachineproblemQmjrjjPwjCjwithboundedweightsandprocessingrequirements.Inthisheuristic,wheneveramachineisavailable,thejobwiththelargestratiowj=xjamongallthewaitingjobsisselectedtobeprocessednext.Ifthereisnojobwaiting,thenthemachineremainsidleuntilthenextjobarrives.Chouetal.deriveanupperboundonthemaximumdelaythatanyamountofworkcanincurintheWSPRAschedule,relativetotheLPrelaxationpresentedbyGoemans1997.TheythenderivefromthisboundtheasymptoticoptimalityoftheWSPRAalgorithmforthesinglemachineanduniformparallelmachineproblems.Chou001alsoextendsthisresulttothestochasticversionofthesinglemachineproblem1jxjstoch;rjjE[PwjCj],wheretheobjectiveistominimizetheexpectedtotalweightedcompletiontimes,E[PwjCj].TheyprovethattheWeightedShortestExpectedProcessingTimeWSEPTruleisasymptoticallyoptimalfor1jxjstoch;rjjE[PwjCj]aslongasthejobweightsandprocessingrequirementsareboundedandtheprocessingrequirementsareindepen-dentlydistributedwithknownmeanvalues.Forshopproblems,KaminskyandSimchi-Levi01bstudytheowshopprob-lemFmjjPCjandshowthattheShortestProcessingTimeSPTruleisasymptot-icallyoptimalaslongasthejobprocessingrequirementsareindependentlyandiden-ticallydistributedi.i.d..KaminskyandSimchi-Levi99andXiaetal.00studythemoregeneralowshopproblemFmjjPwjCj.Theyuseprobabilisticanal-ysistoshowthattheWSPRAruleisasymptoticallyoptimalforFmjjPwjCjunder

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5 somemildprobabilisticassumptionsonthedistributionsofjobprocessingtimesandweights.BuildingontheresultsofChouetal.001,Liu01presenttwoonlineheuristicsandonesemi-onlineheuristicwhichareasymptoticallyoptimalfortheowshopproblemFmjrjjPwjCjwithboundedjobprocessingrequirementsandboundedweights.AlloftheseheuristicsarecloselyrelatedtotheWSPRAalgorithm.Itisalsoworthnotingthat,thereisadistinctionbetweenonlineschedulingproblemandonlineschedulingalgorithm.Inanoineschedulingproblem,allthejobinformation,especiallythejobreleasetimes,areknownapriori,sothataglobalmaybeoptimaldecisioncanbemadeattimezerobyconsideringallthejobs,in-cludingthosethathavenotbeenreleased.Inanonlineschedulingproblem,thejobinformationisusuallypresentedpiecebypieceanddecisionshavetobemadebasedonlyontheinformationthatisavailableatanygivenmoment.Typically,theexistenceofajobisnotknownuntilitisreleasedtothesystem.Anoineschedulingalgorithmconsidersallthejobs,includingthosethathavenotbeenreleased.Anonlinealgorithmconsidersonlyjobsthathavebeenreleased.Apparently,onlinealgorithmscanbeappliedtobothonlineandoineschedulingproblems,whileoinealgorithmscanonlybeappliedtooineschedulingproblems.Somepapers,includingKaminskyandSimchi-Levi001aandChou,Queyranne,andSimchi-Levi001,concernonlyonlinealgorithms.Theyarenotparticularaboutwhethertheproblemwasonlineoroine,sinceanonlinealgorithmcanbeappliedtobothonlineandoineschedulingproblems.Hence,theirresultscanbeunderstoodineitherofthefollowingtwomeanings:therstmeaningisthatthereisaclassofschedulingproblemswhoseparameters,suchasarrivaltimes,followagivenprobabilitydistribution,andanalgorithmgivesaperformanceguaranteeforallinstancesinthisproblemclass.Butforeachinstance,theschedulerknowstheparametersbeforemakingtheschedule.Theothermeaningisthattheparametersfollowadistributionandtheparametervaluesarenotknownaheadoftime.

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6 Inourpaper,werestrictourdiscussiontoonlinealgorithmsappliedtoonlineproblems.Hence,intheliteraturereview,wehavereviewedotherpapersbyputtingtheminthecontextofsolvingonlineschedulingproblems.1.3SingleMachineProblemInthestochasticonlinesinglemachineschedulingproblem1jxjstoch;rjjPwjCj,asetofjobsN=f1;2;;ngarriveovertimeandhavetobeprocessednonpreemp-tivelyonasinglemachinewithspeedofone.Jobprocessingrequirementsarerandomvariablesthatarerealizedonline.Thatis,thereleasetimerjandweightwjremainunknownuntiljobjisreleased.Theactualprocessingrequirement,xj,isarandomvariablewhosevalueremainsunknownuntiljobjisnished.Jobsareindexedintheorderoftheirarrivalsandthemachinecanprocessatmostonejobatatime.Theinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.301 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.TheowtimeofjobjisdenedtobethedierenceofitsreleasetimerjandcompletiontimeCj,i.e.,fj=Cj)]TJ/F22 11.955 Tf 11.821 0 Td[(rj.Thewaitingtimeofjobj,Wj,isdenedtobethedierenceofitsreleasetime,rj,andstarttime,Sj,i.e.,Wj=Sj)]TJ/F22 11.955 Tf 12.018 0 Td[(rjandfj=Wj+xj.Theobjectiveistominimizethetotalweightedcompletiontimeofallthejobs.Further,wehavethefollowingassumption: Assumption1 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1 xj,j=1;2;;n,arei.i.d.withmean0<<1andvariance0<2x<1 Ljandxj,j=1;2;;n,areindependentlydistributedwith=<1 thereexistconstantsww>0suchthatwwjwforalljobs.UnderAssumption 1 ,itcanbeshownthatthemeanwaitingtimeofalljobsobtainedbyapplyinganynondelayalgorithmisnite,asformalizedbythefollowinglemma: Lemma1 Withprobabilityonelimn!1Pnj=1fj n2L+2x 2)]TJ/F22 11.955 Tf 11.955 0 Td[(+; .1

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7 wherefjistheowtimeofjobjobtainedbyapplyinganynondelayalgorithm.Proof.Thestochasticsinglemachineproblemwhichsatisestheaboveassump-tionhasbeenwellstudiedinthequeueingliteraturesee,e.g.,GrossandHarris1985astheSingle-SeverQueueingsystemwithGeneralInputandGeneralServicePatternsG/G/1.ItisaknownresultseeLindley1952that,withAssumption 1 satised,thejobwaitingtimeshavealimitingdistributionfunctionwhichisindepen-dentoftheinitialconditionofthesystem.Inaddition,asnapproachesinnity,themeanwaitingtimeofthejobs,denotedbyW,isboundedbythefollowinginequalitysee,e.g.,Marshall,1968andGrossandHarris1985W=limn!1Pnj=1Wj n2L+2x 2)]TJ/F22 11.955 Tf 11.955 0 Td[(; .2 regardlessofthequeueingdiscipline.Aslongasaconservationrule,i.e.,arulethatkeepstheserverbusyaslongasthereisatleastonejobwaiting,isadopted.Therefore,basedonAssumption 1 andtheLawofLargeNumberswehavelimn!1Pnj=1fj n=limn!1Pnj=1Wj+xj n=limn!1Pnj=1Wj n+limn!1Pnj=1xj n=W+Exj2L+2x 2)]TJ/F22 11.955 Tf 11.956 0 Td[(+ NowconsideranygivennondelayalgorithmAforthestochasticsinglemachineproblem1jxjstoch;rjjPwjCj.LetCjdenotethecompletiontimeofjobjob-tainedbyapplyingalgorithmAandletZIdenotetheoptimumobjectivevalue.Wehavethefollowingtheorem. Theorem1 IfAssumption 1 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:

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8 Proof.Werstclaimthatalmostsurelylimn!1nc Pnj=1rj=0;8c<1: .3 Notethat8j,rj=Pj)]TJ/F21 7.97 Tf 6.586 0 Td[(1=0r)]TJ/F22 11.955 Tf 13.2 0 Td[(r)]TJ/F21 7.97 Tf 6.586 0 Td[(1=Pj)]TJ/F21 7.97 Tf 6.587 0 Td[(1=0L,andLj's,j=1;2;;n,arei.i.d..ThereforebasedontheLawofLargeNumberswehavelimn!1nc Pnj=1rjlimn!1nc Pnj=dn 2erj=limn!1nc Pnj=dn 2ePj)]TJ/F21 7.97 Tf 6.587 0 Td[(1=0L=limn!1nc Pnj=dn 2ej=limn!1nc n+dn 2e 2n)-222(dn 2e+1limn!13nc 8n2=0; .4 Nowlet=2L+2x 2)]TJ/F22 11.955 Tf 11.955 0 Td[(+,wethenhavelimn!1Pnj=1wjCj ZIlimn!1Pnj=1wjCj Pnj=1wjrj=limn!1+Pnj=1wjfj Pnj=1wjrj1+limn!1Pnj=1wfj Pnj=1wrj=1+limn!1wPnj=1fj wPnj=1rj1+limn!1wn wPnj=1rj=1 1.4FlowShopProblemInthestochasticonlineowshopproblemFmjxjistoch;rjjPwjCj,asetofnjobshavetobeprocessednon-preemptivelyonmmachines.Eachmachineii21;2;mhasspeedsi=1andcanprocessatmostonejobatatime.Eachjobjj=1;2;;nmustvisitallthemmachinesinthesameorder:1;2;;m.Associatedwitheachjobjisareleasetimerj,aweightwjandaprocessingrequire-mentxjiofjobjonmachinei.Thereleasetimerjandweightwjremainunknownuntiljobjisreleased.Theprocessingrequirementofeachjobj,xji,j2f1;2;;ng,

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9 i2f1;2;;mgisarandomvariablewhoseactualvalueremainsunknownuntiljobjisnished.Thetotalprocessingrequirementsofjobjonallmachines,denotedbyxj,isequaltoPmi=1xji.Jobsareindexedintheorderoftheirarrivalsandtheinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.444 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.Theobjectiveistodetermineapermutationscheduleofthejobssothatthetotalweightedcompletiontimesofalljobsisminimum.Wehavethefollowingassumption: Assumption2 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1 xji,j=1;2;;n;i=1;2;;m,arei.i.d.withmean0<<1andvariance0<2x<1 Ljandxji,j=1;2;;n;i=1;2;;m,areindependentlydistributedwith=<1 thereexistconstantsww>0suchthatwwjwforalljobs.ConsideranygivennondelayalgorithmAandanyinstanceIofthestochasticowshopproblemFmjxjstoch;rjjPwjCj.LetCjidenotethecompletiontimeofjobjonmachineiobtainedbyapplyingalgorithmAtoI.Ifwere-indexthejobsinIintheincreasingorderoftheirstartingtimesontherstmachine,i.e.,S11
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10 NotethattheupperboundgivenEquation 2.14 hasthreecomponentsandtherstcomponentiscloselyrelatedtothestochasticonlinesinglemachineproblemwhichweanalyzedinSection 1.3 .Toseethis,weconsiderastochasticonlinesinglemachineproblem,referredtoas
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11 Proof.limn!1nPj=1wjCj ZI=limn!1nPj=1wjCjm ZIlimn!1nPj=1wjCjm nPj=1wjrjlimn!1nPj=1wjCjm wnPj=1rjlimn!1nPj=1wjmax1ljfrl+jPk=lxk1g wnPj=1rj+limn!1mPi=2max1abnbPk=axki)]TJ/F22 11.955 Tf 11.955 0 Td[(xk1nPj=1wj wnPj=1rj+limn!1mmaxj=1;2;;nxj1nPj=1wj wnPj=1rj; .7 wherethelastinequalityisbyLemma 2 .Notethattheright-hand-sideofEquation 2.15 hasthreecomponents.Fortherstcomponent,recallthatnXj=1wjmax1ljfrl+jXk=lxk1gcanbeinterpretedasthetotalweightedcompletiontimeobtainedbyapply-ingalgorithmAtothestochasticonlinesinglemachineproblemthatwasstudiedinSection 1.3 .Therefore,accordingtoTheorem 1 ,wehavewithprobabilityequaltoonelimn!1nPk=1wkmax1lkfrl+kPj=lxj1g wnPj=1rj=1:Forthesecondcomponent,Xiaetal00,alsoseeNote1provedthatunderAssumption 2 ,withprobabilityequaltoonelimn!11 nmax1abnbXk=axki)]TJ/F22 11.955 Tf 11.955 0 Td[(xk1=0;i=2;;m: .8 Thisimpliesthatwithprobabilityonelimn!1mPj=2max1abnbPj=axji)]TJ/F22 11.955 Tf 11.955 0 Td[(xj1nPk=1wk wnPj=1rjlimn!1nwmPj=2max1abnbPj=axji)]TJ/F22 11.955 Tf 11.955 0 Td[(xj1 wnPj=1rj=0:

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12 Finally,forthethirdcomponent,weusetheKolmogorov'sInequalitysee,e.g.,Williams1991andNote2toshowthatwithprobabilityonelimn!1mmaxj=1;2;;nxj1nPk=1wk wnPj=1rj=0:Todothis,werstdenerandomvariableszj=xj1)]TJ/F22 11.955 Tf 12.079 0 Td[(;8j2f1;2;;ng,sothatzjj2f1;2;;ng,arei.i.d.withmean0andvariance0<2z<1.Wethendenejsuchthatjzjj=max1jnjzjjandwehavezj=jXj=1zj)]TJ/F23 7.97 Tf 11.956 15.431 Td[(j)]TJ/F21 7.97 Tf 10.82 0 Td[(1Xj=1zjandjzjjjjXj=1zjj+jj)]TJ/F21 7.97 Tf 10.82 0 Td[(1Xj=1zjj2max1knjkXj=1zjj:Now,foranygivenwehavePrfmax1jnzj ng=Prfmax1jnzjngPrfmax1jnjzjjngPrf2max1knjkXj=1zjjngPrfmax1knjkXj=1zjjn 2g1 n 22nXj=1Varzj=1 n 22n2z=42z n2wherethelastinequalitywasbyKolmogorov'sInequality.Equation 2.18 impliesthatwithprobabilityonelimn!1max1jnxj1 n=limn!1max1jnzj n=0:Therefore,basedonEquation 1.4 ,withprobabilityonewehavelimn!1mmaxj=1;2;;nxj1nPk=1wk wnPj=1rjlimn!1mnwmaxj=1;2;;nxj1 wnPj=1rjlimn!13mnwmaxj=1;2;;nxj1 8wn2=0:

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13 1.5UniformParrallelMachineProblemInthestochasticonlineuniformparallelmachineschedulingproblemQmjxjstoch;rjjPwjCj,asetofjobsN=f1;2;;ngarrivingovertimemustbeprocessednonpreemptivelyononeofthemgivenmachines.Machineii2f1;;mghasconstantspeedsi>0andcanprocessatmostonejobatatime.Machinesareindexedinthenonincreasingorderoftheirspeeds,i.e.,s1s2sm)]TJ/F21 7.97 Tf 6.586 0 Td[(1sm>0.Foreachmachinei2f1;2;;mg,wedenesti=siifmachineiisbusyattimetandsti=0otherwise.Foreachjobj2f1;2;;ng,thereleasetimerjandweightwjremainunknownuntiljobjisreleased.Thejobprocessingrequirementxjisarandomvariablewhoseactualvalueremainsunknownuntiljobjisnished.Thetimeittakestoprocessjobjonmachineiisequaltoxj=si.Jobsareindexedintheorderoftheirarrivalsandtheinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.295 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.TheowtimeofjobjisdenedtobethedierenceofitsreleasetimerjandcompletiontimeCj,namely,fj=Cj)]TJ/F22 11.955 Tf 12.228 0 Td[(rj.Thewaitingtimeofjobj,Wj,isdenedtobethedierenceofitsreleasetime,rj,andstarttime,Sj,namely,Wj=Sj)]TJ/F22 11.955 Tf 12.887 0 Td[(rjandfj=Wj+xj=sj;i,wherej;iisthemachineonwhichjobjisprocessed.TheobjectiveistondafeasiblescheduleforallthejobsinNthatminimizesthetotalweightedcompletiontime.Wehavethefollowingassumption: Assumption3 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1 xj,j=1;2;;n,arei.i.d.withmean0<<1andvariance0<2x<1 Ljandxj,j=1;2;;n,areindependentlydistributedwith Pmi=1si<1 thereexistconstantsww>0suchthatwwjwforalljobsj.WerstdenethefollowingFirstComeFirstServeFCFSrule:wheneveranymachineisavailable,ndthemachineisuchthatsi)]TJ/F22 11.955 Tf 10.767 0 Td[(sti=maxfsi)]TJ/F22 11.955 Tf 10.767 0 Td[(stig,wheretisthecurrenttime,andstartprocessingonmachineithejobthathasbeenwaitingfor

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14 thelongest.Ifthereisnojobwaiting,thenthemachineremainsidleuntilthenextjobisavailable.WedenotetheuniformparallelmachineproblemwiththeFCFSrulebyQmjxjstoch;rj;FCFSjPwjCj.NowwederiveanupperboundonthemeanowtimeofjobsinQmjxjstoch;rj;FCFSjPwjCjbydecomposingthisuniformparallelmachineproblemintomsinglemachineproblems.Supposetheschedulingenvironmentischangedandinsteadofhavingasinglequeueofjobsforallthemmachines,wehavemqueues,oneforeachofthemmachines.Furthermore,eacharrivingjobwillberoutedintothequeueofmachineiwithaprobabilityPri=si=Pmk=1skandeachmachinei2f1;2;;mgwillservethejobsthatareassignedtoitaccordingtotheFCFSrulewithnojockeyingallowedbetweenanytwoqueues.Wedenotethisnewproblemby
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15 a.alltheLij'sj=1;2;;niarei.i.d.;b.allthexijsj=1;2;;niarei.i.d;c.allthepairsofLijandxijj2f1;2;;nigaremutuallyindependent.Theneverymachinei2f1;2;;mgtogetherwiththesetofjobsthatareas-signedtoitconstituteasub-problemof
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16 SubstitutingE[Lij];Var[Lij];E[xij]andVar[xij]inintermsof;2L;and2x,wehaveE[WQi]Pris2i2L+)]TJ/F22 11.955 Tf 11.956 0 Td[(Pri2s2i+Pr2i2x 2Prisisi)]TJ/F22 11.955 Tf 11.955 0 Td[(Pri .13 andE[WQ]=E[mXi=1PriWQi]=mXi=1E[PriWQi]mXi=1PriPris2i2L+)]TJ/F22 11.955 Tf 11.955 0 Td[(Pri2s2i+Pr2i2x 2Prisisi)]TJ/F22 11.955 Tf 11.955 0 Td[(Pri=mXi=1i:Therefore,basedoninequality 1.2 andtheLawofLargeNumberslimn!1Pnj=1fj n=limn!1Pnj=1Wj+xj=sj;i n=limn!1Pnj=1Wj n+limn!1Pnj=1xj=sj;i nE[WQ]+limn!1Pnj=1xj=sm n=E[WQ]+1 smExjmXi=1i+ sm Theorem3 IfAssumption 3 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:Proof.Let=mXi=1i+ sm;wethenhavelimn!1Pnj=1wjCj ZIlimn!1Pnj=1wjCj Pnj=1wjrj=limn!1+Pnj=1wjfj Pnj=1wjrj

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17 1+limn!1Pnj=1wfj Pnj=1wrj=1+limn!1wPnj=1fj wPnj=1rj1+limn!1wn wPnj=1rj=1wherethelastequalityisbasedoninequality 1.3 1.6UniformParrallelMachineProblemwithBoundedProcessingRequirementsIntheprevioussectionwehaveshownthattheFCFSruleisasymptoticallyoptimalforthestochasticonlineuniformparallelmachineproblem.Inthissectionweshowthat,withtheadditionalassumptionthatalljobshaveboundedprocessingrequirements,anynondelayalgorithmisasymptoticallyoptimalforthestochasticonlineuniformparallelmachineproblem.Specically,weassumethatthereexistconstantspp>0suchthatpxjpforalljobsj2f1;2;;ngineveryinstance.ConsideranygivennondelayalgorithmAforthestochasticuniformparallelmachineproblemQmjxjstoch;rjjPwjCj.LetAdenotethescheduleobtainedbyapplyingalgorithmAtoQmjxjstoch;rjjPwjCj.LetPAtdenotethetotalwaitingprocessingrequirementattimetinA.LetYAtdenotethetotalnumberofwaitingjobsattimetinAandletYAt=maxfYAt:t0g.Foreachmachinei2f1;2;;mgwedeneitsactivespeedsAit=siifmachineiisbusyattimetinAandsAit=0otherwise.WecharacterizetherelationshipbetweenanytwonondelayalgorithmswiththefollowingLemma. Lemma4 Consideranytwonondelayalgorithms,A1andA2,forthestochasticuniformparallelmachineproblemQmjxjstoch;rjjPwjCj.WehavejPA2t)]TJ/F22 11.955 Tf 11.955 0 Td[(PA1tjm)]TJ/F15 11.955 Tf 11.956 0 Td[(1p;8t0

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18 Proof.Assume,withoutlossofgenerality,thatattimet1,PA2t1>PA1t1.Thentheremustbeatleastonetimepoint,t,beforet1suchthatPmi=1sA2itPmi=1sA1it.Considerthelatestofsuchtimepoints,t0,i.e.,t0=maxft:Pmi=1sA2itPmi=1sA1itandtt1g.Thenattimet0thereexitsatleastonemachinei0whichisbusyinA1butidleinA2.Notethatmachinei0canbeidleattimet0onlyifallthejobsinthesystematt0arebeingprocessedononeormoreoftheotherm)]TJ/F15 11.955 Tf 12.073 0 Td[(1machinesattimet0.Thisimpliesthatthereareatmostm)]TJ/F15 11.955 Tf 10.605 0 Td[(1jobsinthesystemattimet0inA2andthereforePA2t0m)]TJ/F15 11.955 Tf 11.955 0 Td[(1p.WethenhavePA2t1)]TJ/F22 11.955 Tf 11.955 0 Td[(PA1t1PA2t0)]TJ/F22 11.955 Tf 11.955 0 Td[(PA1t0PA2t0m)]TJ/F15 11.955 Tf 11.956 0 Td[(1p: Lemma 4 statesthatthedierenceinthetotalamountofwaitingprocessingre-quirementsbetweenanytwonondelayschedulesforanygiveninstanceoftheuniformparallelmachineproblemisalwaysboundedfromabovebyaconstant.Thisimpliesthat,inthelongrun,anynondelayalgorithmforQmjxjstoch;rjjPwjCjisasfastasanyothernondelayalgorithm.Basedonthisobservation,wecanshowthefollowing: Lemma5 ConsideranynondelayalgorithmAforthestochasticuniformparallelmachineproblemQmjxjstoch;rjjPwjCjwithboundedjobprocessingrequire-ments.IfAssumption 3 issatised,thenwithprobabilityoneYAt<1;8t>0:Proof.TheproofisbasedonLemma 4 ,i.e.,thedierenceinthetotalamountofwaitingprocessingrequirementsbetweenanytwonondelayschedulesforanygiveninstanceofQmjxjstoch;rjjPwjCjisalwaysboundedbyaconstant.WerstshowthatwithprobabilityonethetotalamountofwaitingprocessingrequirementobtainedbyapplyingtheFCFSrule,whichisaspecialnondelayalgorithm,toQmjxjstoch;rjjPwjCjisniteatanytimet>0.Then,basedonLemma 4 ,theresultfollowseasily.

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19 LetPFCFStdenotethetotalamountofwaitingprocessingrequirementsattimetobtainedbyapplyingtheFCFSruletoQmjxjstoch;rjjPwjCj.RecallfromSec-tion 1.5 thatthisproblemisdenotedbyQmjxjstoch;rj;FCFSjPwjCj.Again,weconsiderthemodiedversionofQmjxjstoch;rj;FCFSjPwjCj,0,sincetheconditionwhichweaddedinordertotransformQmjxjstoch;rj;FCFSjPwjCjinto0;whereiandiare,resp.,themeanvaluesoftheinterarrivaltimesandprocessingrequirementsofthejobsassignedtomachinei.Accordingtoinequality 1.11 ,i i<18i2f1;2;;mgandwehavePrfWQi=1g=0.ThuswithprobabilityequaltooneWQi<18i2f1;2;;mg.Moreover,observethatineachsub-problem
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20 PFCFStQi<18i2f1;2;;mg;8t=r1;r2;;rn.Alsoobservethatthetotalamountofwaitingprocessingrequirementneverincreasesduringthetimeintervalstj;tj+1j=1;2;;n,whichimpliesPFCFStQi<18i2f1;2;;mg;8t>0andPFCFStmXi=1PFCFStQi<1;8t>0 .14 Nowwehaveshownthatwithprobabilityonethetotalamountofwaitingpro-cessingrequirementatanytimeobtainedbyapplyingtheFCFSruletoQmjxjstoch;rjjPwjCjisnite.AccordingtoLemma 4 ,thedierenceinthetotalamountofwaitingprocessingrequirementsobtainedbyapplyingFCFSandanyothernonde-layalgorithmAisboundedbytheconstanttermm)]TJ/F15 11.955 Tf 11.7 0 Td[(1p.Thereforewehave,withprobabilityequaltoone,thatYAtPAt pPFCFSt+m)]TJ/F15 11.955 Tf 11.955 0 Td[(1p p<1;8t>0;wherePAtisthetotalamountofwaitingprocessingrequirementsattimetobtainedbyapplyingalgorithmA. Lemma6 ConsideranynondelayscheduleoftheuniformparallelmachineproblemQmjxjstoch;rjjPwjCj,wehavenXj=1WjYAt)]TJ/F15 11.955 Tf 11.955 0 Td[(1nXj=1xj Pmi=1siandnXj=1fjYAtnXj=1xj Pmi=1siProof.Letusrstconsiderageneralschedulingenvironment.Ajobjcanincurawaitingtimeduringanytimeinterval[t;t+t]ifandonlyifbothofthefollowingtwoconditionshold:a.jobjisinthesystemduring[t;t+t];

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21 b.jobjisnotbeingprocessedduring[t;t+t];Thisimpliesthattheamountofwaitingtimeincurredbyallthejobsinanytimeinterval[t;t+t]isequaltoYAtt.HencethetotalamountofwaitingtimeincurredbyallthejobsinanydelayornondelayscheduleofanyschedulingproblemisequaltoZ10YAtdt.Now,consideranynondelayschedule.Ajobcanbewaitingattimetonlyifatleastoneoftheotherjobsisbeingprocessedattimet.Therefore,denet=8><>:1:atleastonejobisbeingprocessedattimet;0:otherwise:wethenhavenXj=1Wj=Z10YAtdt=Z10YAttdtYAtnXj=1xj sj;i .15 NotethatthelowerboundgiveninEquation 1.15 isveryloosesinceweareimplicitlyassumingthatajobj2f1;2;;ngcanwaitforitself.HenceatighterboundcanbeobtainedbysubtractingthetotalprocessingtimesofallthejobsfromthelowerboundgiveninEquation 1.15 .NamelynXj=1WjYAtnXj=1xj sj;i)]TJ/F23 7.97 Tf 18.021 14.944 Td[(nXj=1xj sj;i=YAt)]TJ/F15 11.955 Tf 11.955 0 Td[(1nXj=1xj sj;i;andnXj=1fj=nXj=1Wj+nXj=1xj sj;iYAtnXj=1xj sj;i;Inthespecialcaseofthenondelayuniformparallelmachineproblemoranyothernondelayschedulingproblemwithnoprecedenceconstraintsajobcanbewaitingattimetonlyifallthemachinesarebusyattimet.Hence,dene!t=8><>:1:allthemachinesarebusyattimet;0:otherwise:

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22 wethenhavenXj=1Wj=Z10YAtdt=Z10YAt!tdtYAtnXj=1xj Pmi=1si)]TJ/F23 7.97 Tf 18.02 14.944 Td[(nXj=1xj sj;iYAt)]TJ/F15 11.955 Tf 11.956 0 Td[(1nXj=1xj Pmi=1si;andnXj=1fj=nXj=1Wj+nXj=1xj sj;iYAtnXj=1xj Pmi=1si: Theorem4 ConsideranynondelayalgorithmAfortheuniformparallelmachineproblemQmjxjstoch;rjjPwjCjwithboundedjobprocessingrequirements.IfAssumption 3 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:Proof.Werstshowthatwithprobabilityonelimn!1Pnj=1fj n<1,wherefjistheowtimeofjobj2f1;2;;ng.AccordingtoLemma 6 limn!1Pnj=1fj nlimn!1YAtPnj=1xj=Pmi=1si nlimn!1YAtPnj=1p=Pmi=1si n=YAtp Pmi=1si:Sincep Pmi=1siisaconstantand,accordingtoLemma 5 ,almostsurelyYAt<1,thereforewithprobabilityonelimn!1Pnj=1fj n<1.Nowwehavelimn!1Pnj=1wjCj ZIlimn!1Pnj=1wjCj Pnj=1wjrj=limn!1+Pnj=1wjfj Pnj=1wjrj1+limn!1Pnj=1wfj Pnj=1wrj=1+limn!1wPnj=1fj wPnj=1rj

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23 Sincewithprobabilityonelimn!1Pnj=1fj n<1,basedonequalityEquation 1.3 wehave,withprobabilityonethat,limn!1Pnj=1wjCj ZI1+limn!1wPnj=1fj wPnj=1rj=1+limn!1wnPnj=1fj n wPnj=1rj=1;wherethelastequalityisbasedonEquation 1.3 1.7ConcludingRemarksandDiscussionsInthischapterwehaveshownthatanynondelayalgorithmisasymptoticallyoptimalfortheonlinestochasticsinglemachineproblem,uniformparallelmachineproblemandowshopproblemundersomemildassumptions.Someargumentsinthischaptermightbeusedtoprovesimilarasymptoticoptimalityresultsforotherstochasticonlineschedulingproblem,e.g.,thestochasticonlineopenshopproblem.OurnumericalstudiesseeChapter3,alsoseeChenandShen2003aandChenandShen2003balsoshowthattheseasymptoticoptimalityresultsmaybeextendabletothestochasticonlineopenshopandjobshopproblems.1.8Notes 1. Xiaetal00showthatequalityEquation 2.17 holdsaslongasthenjobsarescheduledbasedonlyontheinformationassociatedwitheachjobj'sj=1;2;;ntotalprocessingrequirementsonthemmachinesandnotoneachjobj'sj=1;2;;nprocessingrequirementoneachofthemmachines.Sinceinthestochasticonlineschedulingenvironmenttheprocessingrequire-mentsofeachjobremainunknownuntilthejobisnished,Xiaelal'sassump-tionisautomaticallysatisedbyanynondelayscheduleforanystochasticonlineschedulingproblem.Therefore,equality 2.17 holdsforanynondelayscheduleforthestochasticonlineowshopproblemFmjxjistoch;rjjPwjCj. 2. Kolmogorov'sInequality:LetX1;X2;;XnbeindependentrandomvariablessuchthatE[Xk]=0andVar[Xk]<1fork=1;2;;n.Thenforeach>0:Pmax1knjSkj1 2nXk=1Var[Xk]

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24 whereSk=X1+X2++Xk.

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CHAPTER2PROBABILISTICASYMPTOTICANALYSISONSTOCHASTICONLINESCHEDULINGPROBLEMS:EXTENDEDRESULTS2.1IntroductionInthischapterweextendtheresultsinchapter1toamuchmoregeneralrealmforeachofthethreestochasticonlineschedulingproblems.Inparticular,wewillseehowthei.i.d.assumptionontheprocessingrequirementsacrossthejobsareappli-cableinamoregeneralcontext,suchasmulti-classjobarrivalsandbatcharrivals.Inaddition,Themutualindependencerequirementbetweentheinterarrivaltimesandprocessingrequirementisreplacedwithamuchweakerassumption.Further,forthestochasticonlineowshopproblem,jobsneednottohavei.i.d.processingrequirementsacrossthemachines.Instead,theyareassumedtobenon-decreasinglytransferabllyexchangeable,amuchweakerassumptioncomparedwiththei.i.d.as-sumption.Wealsoprovideexamplestoillustratethesignicanceandpracticalusageoftheseextensions.Therestofthischapterisorganizedasfollows.InSection2.2,2.3and2.4wepresenttheasymptoticoptimalityresultsforthestochasticonlinesinglemachineproblem,uniformparallelmachineproblemandowshopproblem,respectively.WeconcludeourdiscussionandsuggestfutureresearchdirectionsinSection2.5.2.2SingleMachineProblemInthestochasticonlinesinglemachineschedulingproblem1jxjstoch;rjjPwjCj,asetofjobsN=f1;2;;ngarriveovertimeandhavetobeprocessednonpreemp-tivelyonasinglemachinewithspeedofone.Jobprocessingrequirementsarerandomvariablesthatarerealizedonline.Thatis,thereleasetimerjandweightwjremain 25

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26 unknownuntiljobjisreleased.Theactualprocessingrequirement,xj,isarandomvariablewhosevalueremainsunknownuntiljobjisnished.Jobsareindexedintheorderoftheirarrivals.Theinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.795 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.TheowtimeofjobjisdenedtobethedierenceofitsreleasetimerjandcompletiontimeCj,i.e.,fj=Cj)]TJ/F22 11.955 Tf 12.922 0 Td[(rj.Thewaitingtimeofjobj,Wj,isdenedtobethedierenceofitsreleasetime,rj,andstarttime,Sj,i.e.,Wj=Sj)]TJ/F22 11.955 Tf 12.435 0 Td[(rjandfj=Wj+xj.Theobjectiveistominimizethetotalweightedcompletiontimeofallthejobs.Tofacilitateourdiscussionsinthissectionandthefollowingsection,wedeneLtobeagenericrandomvariablerepresentinganarbitraryinterarrivaltimeandxtobeagenericrandomvariablerepresentinganarbitraryjobprocessingrequirement.Wealsodenerandomvari-ableuj=xj)]TJ/F22 11.955 Tf 12.935 0 Td[(Lj;j21;2;;n.WesaythatjobarrivalsareindependentifLj'sj21;2;;nareindependent,xj'sj21;2;;nareindependentand,8j16=j2,Lj1andxj2areindependent.Further,wehavethefollowingassumptionforthestochasticonlinesinglemachineproblem: Assumption4 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1, xj,j=1;2;;n,arei.i.d.withmean0<<1andvariance0<2x<1and=<1, uj,j=0;1;2;;n,areindependent, thereexistconstantsww>0suchthatwwjwforalljobs.Notethat,inAssumption 4 ,jobprocessingrequirementsandinterarrivaltimesneednottobemutuallyindependent.ThisissignicantsinceinreallifesituationstheprocessingrequirementandinterarrivaltimeofajobareoftencorrelatedasillustratedinExample 1 below.Further,theindependenceofuj'sisguaranteedifjobarrivalsareindependent.UnderAssumption 4 ,wecanshowthatalargeclass

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27 ofnondelayalgorithms,whichwedenetobeProcessingRequirementIndependentPRInondelayalgorithmsinthefollowingdenition,areasymptoticallyoptimalforthestochasticonlinesinglemachineproblem. Denition1 AnondelayalgorithmisProcessingRequirementIndependentPRIifitdoesnotuseanyinformationrelatedtotheprocessingrequirementsofthejobs.Itshouldbenotedthat,requiringanalgorithmtobePRIisequivalenttosayingthat,fromthepointofviewofjobprocessingrequirements,thejobsareindistin-guishableseeKingman1962a.Inparticular,ifthejobprocessingrequirementsandinterarrivaltimesaremutuallyindependent,thenanynondelayalgorithmisPRI.Ifthejobprocessingrequirementsandinterarrivaltimesarecorrelated,thenanynon-delayalgorithmwhichdoesnotusetheinformationrelatedtotheinterarrivaltimesisPRI.Therefore,simplealgorithmssuchasFirstComeFirstServe,LastComeFirstServeandRandomServiceareincluded.NowconsideranygivennondelayalgorithmAforthestochasticonlinesinglemachineproblem1jxjstoch;rjjPwjCj.LetCjdenotethecompletiontimeofjobjobtainedbyapplyingalgorithmAandletZIdenotetheoptimumobjectivevalue.UnderAssumption 4 wecanprovethefollowingtheorem. Theorem5 ConsideranyPRInondelayalgorithmforthestochasticonlinesinglemachineproblem,ifAssumption 4 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:Proof.WewillrstshowthatthemeanwaitingtimeofallthejobsobtainedbyapplyinganyPRInondelayalgorithmisnite,i.e.,limn!1Pnj=1fj n<1:ItisknownthatseeLoynes1962andNote 2 ,underAssumption 4 ,thejobwaitingtimesobtainedbyapplyinganyPRInondelayalgorithmhasalimitingdistributionfunction.Namely,letWdenotethethemeanwaitingtimeofthejobsobtainedby

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28 applyinganyPRInondelayalgorithm,thenW=limn!1Pnj=1Wj n: .1 Further,Kingman962a,alsoseeKingman1962bandNote 1 showsthatWVar[L]+Var[x])]TJ/F22 11.955 Tf 11.955 0 Td[(Cov[L;x] 2E[L])]TJ/F22 11.955 Tf 11.955 0 Td[(E[x]=2L+2x)]TJ/F22 11.955 Tf 11.955 0 Td[(Cov[L;x] 2)]TJ/F22 11.955 Tf 11.955 0 Td[(<1; .2 whereCov[L;x]isthecovarianceofLandx.Now,basedonAssumption 4 andtheLawofLargeNumberswehavelimn!1Pnj=1fj n=limn!1Pnj=1Wj+xj n=limn!1Pnj=1Wj n+limn!1Pnj=1xj n=W+Exj2L+2x)]TJ/F22 11.955 Tf 11.955 0 Td[(Cov[L;x] 2)]TJ/F22 11.955 Tf 11.956 0 Td[(+<1: .3 Wenowuseinequality 2.3 toproveTheorem 5 .InChapter1weshowedthatlimn!1nc Pnj=1rjlimn!13nc 8n2=0;8c<1: .4 Therefore,let=2L+2x)]TJ/F22 11.955 Tf 11.956 0 Td[(Cov[L;x] 2)]TJ/F22 11.955 Tf 11.955 0 Td[(+,wethenhavelimn!1Pnj=1wjCj ZIlimn!1Pnj=1wjCj Pnj=1wjrj=limn!1+Pnj=1wjfj Pnj=1wjrj1+limn!1Pnj=1wfj Pnj=1wrj=1+limn!1wPnj=1fj wPnj=1rj1+limn!1wn wPnj=1rj=1: .5 Now,weextendsimilarasymptoticoptimalityresultstomoregeneralandpracticalcasesuchasmulti-classjobsandbatcharrivals,asillustratedbythefollowingexamples.

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29 Example1 ConsiderastochasticonlinesinglemachineproblemwithKdierentclassesofjobs.WithprobabilityPrk>0,anewarrivalisofthekthclass,k21;2;;K.Jobsofthekthclasshaveprocessingrequirementsthatarei.i.d.withmeank<1andvariance2Bk<1andinterarrivaltimesthatarei.i.d.withmeank<1andvariance2Ak<1.Allthejobshaveboundedweights.Inaddition,PKk=1Prkk
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30 andPKk=1E[yk]=1.WethenhaveVar[X]=Var[KXk=1ykzk]=KXk=1Var[ykzk]+2Xk1
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31 boundedweights.Inaddition,E[]
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32 arandomselectedbatchwethenhaveE[pb]=E[E[pbj]]=E[]=E[]<<1Var[pb]=E[Var[pbj]]+Var[E[pbj]]=E[]2x+2Var[]<1: .8 Then,accordingtoinequality 2.2 ,wehaveWa<1.Now,letsconsiderWb.NotethatWb=E[qPl=1xfjb0;lg],wheretherandomintegervariableq+1isthepositioninlineoftherandomlyselectedjobj0withinitsbatchjb0andfjb0;lgistheindexofthejobwhichisprocessedlthwithinthebatchjb0.ToanalyzeWb,wehavetoanalyzethedistributionofqrst.ObservethattheprobabilityPrq=l0isequaltotheproportionofthejobswhichareprocessedq+1stwithintheirownbatchesinthelongrun.Thisproportionisequaltothetotalnumberofjobswhichareprocessedq+1stwithintheirownbatchesdividedbythetotalnumberofjobsinallthebatches.Further,thetotalnumberofjobswhichareprocessedq+1stwithintheirownbatchesisequaltothenumberofbatcheseachofwhichhasatleastq+1jobsinit.NamelywehavePrq=l0=limnb!1nbPr>l0 nbP=1=limnb!1nbPr>l0 nbE[]=Pr>l0 E[] .9 andE[q]=1Xl=0lPrq=l=1 E[]1Xl=0lPr>l=1 E[]1Xl=0fl1Xk>lPr=kg=1Xk=1fPr=kk)]TJ/F21 7.97 Tf 6.587 0 Td[(1Xl=0lg=1Xk=1fPr=kk)]TJ/F15 11.955 Tf 11.956 0 Td[(1+1 2k)]TJ/F15 11.955 Tf 11.955 0 Td[(1g=E[)]TJ/F15 11.955 Tf 11.956 0 Td[(1] 2E[]<1: .10

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33 ThereforeWb=E[qXl=1xfjb0;lg]=E[E[qXl=1xfjb0;lgjq]]=E[q]=E[q]E[]=E[)]TJ/F15 11.955 Tf 11.955 0 Td[(1] 2<1 .11 NowwehaveshownthatW=Wa+Wb<1.Sincetheinter-batchdelayhasalimitingdistributionfunctionseeLoynes1962andtheNote 2 ,itiseasytoseethatlimn!1Pnj=1Wj n=limn!1Pnj=1Wja n+limn!1Pnj=1Wjb n=Wa+Wb=W;whereWja,WjbandWjdenote,resp.,theinter-batch,withinbatchandtotalwaitingtimesofjobj.Then,BasedonAssumption 4 andtheLawofLargeNumberswehavelimn!1Pnj=1fj n=limn!1Pnj=1Wj+xj n=limn!1Pnj=1Wj n+limn!1Pnj=1xj n=W+<1: GivenLemma 8 ,wecanusethesameargumentsasthoseusedintheproofofTheorem 5 toshowthatanyalgorithmwhichschedulesthebatchesinanondelaymannerisasymptoticallyoptimalfortheproblemdescribedinExample 2 .WealsowanttopointoutthatexamplesmoregeneralthanExamples 1 and 2 existforwhichmostnondelayalgorithmsareasymptoticallyoptimal.Forexample,considerastochasticonlinesinglemachineproblemwithKdierentclassesofjobsthatarriveandareprocessedinbatches.WithprobabilityPrk>0,anarrivingbatchisfromthekthclass,k21;2;;K.Batchesfromthekthclasshavei.i.d.interarrivaltimeswithk<1andvariance2Ak<1andi.i.d.batchsizeswithmeanE[k]<1and2k<1.Jobswithinthebatchesfromthekthclasshavei.i.d.processingrequirementswithmeank<1andvariance2Bk<1.Inaddition,PKk=1PrkE[k]k
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34 argumentssimilartothosewehavepresentedinthetheproofsofLemma 7 andLemma 8 ,itcanbeeasilyshownthatanyalgorithmwhichschedulethebatchesinanondelaymanneranddoesnotusetheinformationrelatedtotheclassicationsandinterarrivaltimesisasymptoticallyoptimalforthisexample.2.3UniformParallelMachineProblemInthestochasticonlineuniformparallelmachineschedulingproblem,asetofjobsarrivingovertimemustbeprocessednonpreemptivelyononeofthemgivenmachines.Machineii2f1;;mghasconstantspeedsi>0andcanprocessatmostonejobatatime.Machinesareindexedinthenonincreasingorderoftheirspeeds,i.e.,s1s2sm)]TJ/F21 7.97 Tf 6.587 0 Td[(1sm>0.Foreachmachinei2f1;2;;mg,wedenesti=siifmachineiisbusyattimetandsti=0otherwise.Foreachjobj2f1;2;;ng,thereleasetimerjandweightwjremainunknownuntiljobjisreleased.Thejobprocessingrequirementxjisarandomvariablewhoseactualvalueremainsunknownuntiljobjisnished.Thetimeittakestoprocessjobjonmachineiisequaltoxj=si.Jobsareindexedintheorderoftheirarrivalsandtheinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.011 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.TheowtimeofjobjisdenedtobethedierenceofitsreleasetimerjandcompletiontimeCj,namely,fj=Cj)]TJ/F22 11.955 Tf 10.307 0 Td[(rj.Thewaitingtimeofjobj,Wj,isdenedtobethedierenceofitsreleasetime,rj,andstarttime,Sj,namely,Wj=Sj)]TJ/F22 11.955 Tf 10.253 0 Td[(rjandfj=Wj+xj=sj;i,wherej;iisthemachineonwhichjobjisprocessed.TheobjectiveistondafeasiblescheduleforallthejobsinNthatminimizesthetotalweightedcompletiontime.Ourassumptionsforthestochasticonlineuniformparallelmachineproblemcanbesummarizedasthefollowing: Assumption5 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1, xj,j=1;2;;n,arei.i.d.withmean0<<1andvariance0<2x<1,

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35 uj,j=0;1;2;;n,areindependentand Pmi=1si<1, thereexistconstantsww>0suchthatwwjwforalljobsj.Notethat,inAssumption 5 ,jobprocessingrequirementsandinterarrivaltimesneednottobemutuallyindependent.Themuchweakerassumptionthatuj'sareindependentissatisedaslongasthejobarrivalsareindependent.Underassumption 5 wecanshowthatasimpleFirstComeFirstServeFCFSruleisasymptoticallyoptimalforthestochasticonlineuniformparallelmachineproblem.TheFCFSrulecanbedescribedasfollows:wheneveranymachineisavailableattimet,ndthemachineisuchthatsi)]TJ/F22 11.955 Tf 12.487 0 Td[(sti=maxfsi)]TJ/F22 11.955 Tf 12.487 0 Td[(stig,andstartprocessingonmachineithejobthathasbeenwaitingforthelongest.Ifthereisnojobwaiting,thenthemachineremainsidleuntilthenextjobisavailable.LetCjdenotethecompletiontimeofjobjobtainedbyapplyingtheFCFSruletothestochasticonlineuniformparallelmachineproblemandletZIdenotetheoptimumobjectivevalue,wethenhavethefollowingtheorem. Theorem6 ConsidertheFCFSrule,ifAssumption 5 issatised,thenwithproba-bilityonelimn!1Pnj=1wjCj ZI=1:Proof.DenotetheuniformparallelmachineproblemwiththeFCFSrulebyQmjxjstoch;rj;FCFSjPwjCj.WerstderiveanupperboundonthemeanowtimeofjobsinQmjxjstoch;rj;FCFSjPwjCjbydecomposingthisuniformparallelmachineproblemintomsinglemachineproblemsasfollows.Supposetheschedulingenvironmentischangedandinsteadofhavingasinglequeueofjobsforallthemmachines,wehavemqueues,oneforeachofthemmachines.Furthermore,eacharrivingjobwillberoutedintothequeueofmachineiwithaprobabilityPri=si=Pmk=1skandeachmachinei2f1;2;;mgwillservethejobsthatareassignedtoitaccordingtotheFCFSrulewithnojockeyingallowedbetweenanytwoqueues.Wedenotethisnewproblemby
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36 Qmjxjstoch;rj;FCFSjPwjCj,sincetheconditionwhichweaddedinordertotransformQmjxjstoch;rj;FCFSjPwjCjinto
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37 q Var[Lij]Var[xij]<1andlimn!1Pnj=1Wj n0suchthatppjpforalljobsj2f1;2;;ng,wecanshowthatanyPRInondelayalgorithmisasymptoticallyoptimalforthestochasticonlineuniformparallelmachineproblemunderassumption 5 .ConsideranynondelayalgorithmAfor1jxjstoch;rjjPwjCj.LetCjdenotethecompletiontimeofjobjobtainedbyapplyingalgorithmAandletZIdenotetheoptimumobjectivevalue.Wethenhavethefollowingtheorem

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38 Theorem7 ConsideranyPRInondelayalgorithmforthestochasticuniformparallelmachineproblemwithboundedprocessingrequirements,ifAssumption 5 issatised,thenwithprobabilityonelimn!1Pnj=1wjCj ZI=1:Proof.BasedonLemma1.6.1,Lemma1.6.2andLemma1.6.3inChapter1andTheorem 6 ,Theresultimmediatelyfollows. WealsowanttomentionthatexamplessimilartoExample 1 andExample 2 canbeconstructedtoshowhowsimilarasymptoticoptimalityresultscanbeextendedtomoregeneralsituations,e.g.,problemswithmulti-classjobsandbatcharrivals.2.4FlowShopProblemInthestochasticonlineowshopproblemFmjxjistoch;rjjPwjCj,asetofnjobshavetobeprocessednon-preemptivelyonmmachines.Eachmachineii21;2;mhasspeedsi=1andcanprocessatmostonejobatatime.Eachjobjj=1;2;;nmustvisitallthemmachinesinthesameorder:1;2;;m.Associatedwitheachjobjisareleasetimerj,aweightwjandaprocessingrequire-mentxjiofjobjonmachinei.Thereleasetimerjandweightwjremainunknownuntiljobjisreleased.Theprocessingrequirementofeachjobj,xji,j2f1;2;;ng,i2f1;2;;mgisarandomvariablewhoseactualvalueremainsunknownuntiljobjisnished.Thetotalprocessingrequirementsofjobjonallmachines,denotedbyxj,isequaltoPmi=1xji.Jobsareindexedintheorderoftheirarrivalsandtheinter-arrivaltimeisdenedtobeLj=rj+1)]TJ/F22 11.955 Tf 12.444 0 Td[(rj8j2f0;1;2;;ng,whereL0=r1.Theobjectiveistodetermineapermutationscheduleofthejobssothatthetotalweightedcompletiontimesofalljobsisminimum.Tofacilitateourdiscussions,wedeneLtobeagenericrandomvariablerepre-sentinganarbitraryinterarrivaltimeandxtobeagenericrandomvariablerepresent-inganarbitraryjobprocessingrequirementontherstmachine.WealsodeneXjtobetherandomvectorfxj1;xj2;;xjnganddenerandomvariableuj=xj1)]TJ/F22 11.955 Tf 10.657 0 Td[(Lj,

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39 j21;2;;n.WesaythatjobarrivalsareindependentifLj'sj21;2;;nareindependent,Xj'sj21;2;;nareindependentand,8j16=j2,Lj1andXj2areindependent.Ourassumptionsforthestochasticonlineowshopproblemcanbesummarizedasfollowing: Assumption6 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1 xj1,j=1;2;;narei.i.d.withmean0<<1andvariance0<2x<1 xj1;xj2;;xjm,j2f1;2;;ng,areexchangeableseetheNote 3 uj,j=1;2;;n,areindependentlywith=<1 thereexistconstantsww>0suchthatwwjwforalljobs.NotethatAssumption 6 representstwosignicantextensionscomparedwithourresultsinChapter1,whichassumethattheinterarrivaltimesandprocessingrequirementsaremutuallyindependentandtheprocessingrequirementsarei.i.d.acrossallthejobsandmachines.InAssumption 6 ,jobprocessingrequirementsnolongerneedtobeindependentofinterarrivaltimes.Themuchweakerassumptionthatuj'sareindependentissatisedaslongasthejobarrivalsareindependent.Further,theprocessingrequirementsofeachjobjonthemmachines,xj1;xj2;;xjm,nolongerneedtobei.i.dbutonlyneedtobeexchangeable.Thesignicanceoftheseextensionsisthree-fold.First,inreallifesituationstheprocessingrequirementandinterarrivaltimeofajobareoftencorrelatedasillustratedinExample 1 .Second,i.i.d.,randomvariablesareonlyaspecialclassofexchangeablevariable.Finally,inpractice,processingtimesofthesamejobondierentmachinesareoftencorrelated.UnderAssumption 6 wecanshowanyPRInondelayalgorithmsisasymptoticallyoptimalforthestochasticonlineowshopproblem.ConsideranynondelayalgorithmAforthestochasticonlineowshopproblemFmjxjstoch;rjjPwjCj.LetCjdenotethecompletiontimeofjobjobtainedbyapplyingalgorithmAandletZIdenotetheoptimumobjectivevalue.Wethenhavethefollowingtheorem.

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40 Theorem8 ConsideranyPRInondelayalgorithmforthestochasticonlineowshopproblem,ifAssumption 6 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:Proof.Tofacilitatetheproof,were-indexthejobsintheincreasingorderoftheirstartingtimesontherstmachine,i.e.,S11
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41 thecompletiontimeofjobjinIFthenitiseasytoverifythatCj1=max1ljfrl+jXk=lxk1g .16 Therefore,thenumeratoroftherstcomponentintheupperboundgiveninEquation 2.14 canbeinterpretedasthetotalweightedcompletiontimeobtainedbyapplyingalgorithmAtothestochasticsinglemachineproblem
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42 Todothis,werstdenerandomvariableszj=xj1)]TJ/F22 11.955 Tf 12.161 0 Td[(;8j2f1;2;;ng,sothatzjj2f1;2;;ng,arei.i.d.withmean0andvariance0<2z<1.Wethendenejsuchthatjzjj=max1jnjzjjandwehavezj=jXj=1zj)]TJ/F23 7.97 Tf 11.955 15.431 Td[(j)]TJ/F21 7.97 Tf 10.821 0 Td[(1Xj=1zjandjzjjjjXj=1zjj+jj)]TJ/F21 7.97 Tf 10.821 0 Td[(1Xj=1zjj2max1knjkXj=1zjj:Now,foranygivenwehavePrfmax1jnzj ng=Prfmax1jnzjngPrfmax1jnjzjjngPrf2max1knjkXj=1zjjngPrfmax1knjkXj=1zjjn 2g1 n 22nXj=1Varzj=1 n 22n2z=42z n2wherethelastinequalitywasbyKolmogorov'sInequality.Equation 2.18 impliesthatwithprobabilityonelimn!1max1jnxj1 n=limn!1max1jnzj n=0: .18 Thereforewithprobabilityonewehavelimn!1mmaxj=1;2;;nxj1nPk=1wk ZIlimn!1mnwmaxj=1;2;;nxj1 wnPj=1rjlimn!13mnwmaxj=1;2;;nxj1 8wn2=0:wherethelastinequalityisduetoinequalityEquation 2.4 Theorem 8 immediatelyleadstomoregeneralextensions.Toseethis,letusrstlookatthefollowingtwoexamples. Example3 Considerastochasticonlinetwo-machineowshopproblemwherejobsarrivewithconstantinterarrivaltimeL=6.Eachjobj,j2f1;2;;ng,has

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43 processingrequirementsonthetwomachineswhichsatisfy:xj1=xj2andxj1Uniform[4;6]. Example4 Considerastochasticonlinetwo-machineowshopproblemwherejobsarrivewithconstantinterarrivaltimeL=6.Eachjobj,j2f1;2;;ng,hasprocessingrequirementsonthetwomachineswhichsatisfy:xj2=xj1=0:75andxj1Uniform[4;6].CompareExample 3 andExample 4 andweobservethatthesetwoexamplesarealmostidenticalexceptthatinExample 4 theworkloadonthesecondmachineislighterthantherstmachine.Inaddition,inbothexamples,themachineshaveenoughcapacity.Observethat,inExample 3 ,eachjobhasexchangeablebutnoti.i.d.processingrequirementsonthetwomachines.Basedonourpreviousdiscus-sionsitcanbeconcludedthatanynondelayalgorithmwhichgeneratesapermutationscheduleisasymptoticallyoptimalforExample 3 .WhenwecompareExample 3 andExample 4 ,itseemsintuitivethatanynondelayalgorithmisalsoasymptoticallyop-timalforExample 4 .Inthefollowing,wewillprovethatthisintuitionisindeedtrue.Notethat,foreachjobjinExample 4 ,xj1andxj2arenotexchangeable.How-ever,ifwedenerandomvariablevj2=4 3xj2,thenxj1andvj2areexchangeable.Wecallthetransformationfromxj2tovj2anon-decreasinglytransformation.Con-sequently,wesaythatxj1andxj2arenon-decreasinglytransformabllyexchange-able,sincethetransformationisnon-decreasingfornon-negativevariablesandthetransformedvariablesareexchangeable.WealsoobservethatthistransformationismachinecapacitypreservinginthesensethatthetransformedproblembecomesidenticaltoExample 3 andthemachinecapacityremainsenoughinthetransformedproblem.RecallthatanynondelayalgorithmisasymptoticallyoptimalforExample

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44 3 and,therefore,soasthetransformedproblemofExample 4 .WewillthenusethetransformedproblemofExample 4 andargumentssimilartothoseusedpreviouslytoprovethatanynondelayalgorithmisasymptoticallyoptimalforExample 4 andtheclassofproblemsrepresentedbyit.Therelaxedassumptionforthestochasticonlineowshopproblemcanbesummarizedasthefollowing: Assumption7 Lj,j=0;1;2;;n,arei.i.d.withmean0<<1andvariance0<2L<1, foreachjobj,j=1;2;;n,thereexistsatransformationTj=fTj1;Tj2;;Tjm;g,suchthat: { Tjiisnondecreasing,i.e.,Tji;8iand0; { thetransformedrandomvariables,Tj1xj1;Tj2xj2;;Tjmxjm,areex-changeable; { thetransformedprocessingrequirementsontherstmachineofalljobs,T11x11;T21x21;;Tn1xn1,arei.i.d.withmean0<<1andvariance0<2x<1, uj,j=0;1;2;;n,areindependentand<, thereexistconstantsww>0suchthatwwjwforalljobs.UnderAssumption 7 wecanshowanyPRInondelayalgorithmsisasymptoticallyoptimalforthestochasticonlineowshopproblem.ConsideranynondelayalgorithmAforthestochasticonlineowshopproblemFmjxjstoch;rjjPwjCj.LetCjdenotethecompletiontimeofjobjobtainedbyapplyingalgorithmAandletZIdenotetheoptimumobjectivevalue.Wethenhavethefollowingtheorem. Theorem9 ConsideranyPRInondelayalgorithmforthestochasticonlineowshopproblem,ifAssumption 7 issatised,thenwithprobabilityonewehavelimn!1Pnj=1wjCj ZI=1:

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45 Proof.ConsideraowshopproblemwhichsatisesAssumption 7 ,weconstructitstransformedproblemusinganon-decreasinglyandmachinecapacitypreservingtransformationasdescribedbefore.Specically,toeachnondelayscheduleoftheoriginalproblemFmjxjstoch;rjjPwjCj,weassociateanondelayscheduleTofthetransformedproblem,denotedbyPTF,suchthatinbothandT,jobsareprocessedinthesameorder.Sincethetransformationisnondecreasing,itiseasytoverifythattherealwaysexitssuchatransformednondelayscheduleTforeachnondelayscheduleoftheoriginalproblem.Further,thetotalweightedcompletiontimeof,Pnj=1wjCj,isnomorethanthetotalweightedcompletiontimeofT,whichisdenotedbyZT.Namely,wehavePnj=1wjCjZT.Now,letxTji=Tjixji8j;idenotethetransformedjobprocessingrequirements,thenforeachjobj2f1;2;;ng,xTj1;xTj2;;xTjmareexchangeablerandomvariables.Inaddition,thetransformedprocessingrequirementsontherstmachineofalljobs,xT11;xT21;;xTn1,arei.i.d.withnitevarianceandnitemean<.Therefore,thetransformedproblemsatisesAssumption 5 ,and,accordingtoTheorem 8 ,wehavelimn!1ZT nPj=1wjrj=0;a:s::Hencelimn!1Pnj=1wjCj ZIlimn!1ZT ZIlimn!1ZT nPj=1wjrj=0;a:s:: Theseextensionsmakeourasymptoticaloptimalityresultsapplicabletoamuchwiderrangeofproblems.Further,wegivethefollowingtwoexamples,forwhichitcanbeeasilyshownthatanynondelayalgorithmwhichdoesnotusetheinformationrelatedtotheclassicationsofthejobsortheinterarrivaltimesisasymptoticallyoptimal.

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46 Example5 ConsiderastochasticonlineowshopproblemwithKdierentclassesofjobs.WithprobabilityPrk,anewarrivalisfromthekthclassk21;2;;K.Jobsofthekthclasshavei.i.d.interarrivaltimeswithk<1andvariance2Ak<1andi.i.d.processingrequirementsontherstmachinewithmeank1<1andvariance2Bk1<1.Alljobshaveboundedweights.Inaddition,PKk=1Prkk1
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47 2.6Notes 1. Kingman962aprovesinequalityEquation 2.2 underasubsetofAssump-tion 4 fortheFCFSrule.Kingman962bthenshowsthatthemeanwaitingtimeofjobsisindependentoftheschedulingalgorithmused,aslongastheschedulingalgorithmisnondelayanddoesnotuseanyinformationrelatedtotheprocessingrequirements. 2. Loynes62showsthat,undersomeassumptionsweakerthanAssumption 4 ,thejobwaitingtimeshavealimitingdistributionfunctionundertheFCFSrule,whichimpliesthatinequality 2.1 holdsundertheFCFSrule.Kingman962bthenshowsthatthemeanwaitingtimeofjobsisindependentoftheschedulingalgorithmused,aslongastheschedulingalgorithmisnondelayanddoesnotuseanyinformationrelatedtotheprocessingrequirements. 3. ExchangeableVariables:Aniteordenumerablyinnitesequenceofrandomvariablesx1;x2;;xkaresaidtobeexchangeableif,foranyk>1andforallrelevantindices1i10:Pmax1knjSkj1 2nXk=1Var[Xk]whereSk=X1+X2++Xk.

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CHAPTER3COMPREHENSIVESIMULATIONOFSTOCHASTICONLINESCHEDULINGPROBLEMS3.1IntroductionInthestochasticonlineschedulingenvironmentasetofjobsN=f1;2;ngarriveovertimeandmustbeprocessednonpreemptivelyononeormoreofmma-chines.Thereleasetimeandweightofeveryjobj2Nremainunknownuntiljobjarrives.Inaddition,theprocessingrequirementofeveryjobj2Nisarandomvariablewhoseactualvalueremainsunknownuntiljobjisnished.Theprocessingtimeofjobjisequaltoitsprocessingrequirementdividedbythespeedofthema-chineonwhichitisprocessed.Inthischapterwewillstudyvedierentstochasticonlineschedulingproblemsusingthreedierentperformancemeasures.Specically,westudythesinglemachineproblem,theuniformparallelmachineproblem,theowshopproblem,theopenshopproblemandthejobshopprobleminthestochasticonlineenvironment.Theperformancemetricsthatweusearethetotalweightedcompletiontime,totalweightedowtimeandtotalweightedstretch.Theowtimeofajobisdenedtobethedierenceofitsreleasetimeanditscompletiontime.Thestretchofajobistheratioofitsowtimetoitstotalprocessingrequirement.Ourworkismotivatedbythefactthat,althoughvariousasymptoticallyoptimalalgorithmshavebeenproposedfortheonlinesinglemachineproblem,uniformparallelmachineproblemandowshopproblemwiththeobjectiveofminimizingthetotalweightedcompletiontime,littlehasbeendonetoevaluatethespeedofconvergence.Inaddition,uptothisdate,noasymptoticperformanceresultorevaluationisknownforthetotalweightedowtimemetricorthetotalweightstretchmetric,whicharetwoofthemostnatural 48

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49 metricsforonlineschedulingproblems.Wealsocomparethesensitivityofthethreeperformancemetricsbyevaluatingtheperformancesofseveralalgorithmsunderthesethreemetrics.Firstwegiveabriefdescriptionoftheveproblemswestudy.Inthesin-glemachineproblem,allthejobsmustbeprocessed,oneatatime,onasinglemachinewithaunitspeed.Intheuniformparallelmachineproblem,therearemmachineseachwithaconstantspeedsi>0i2f1;2;mgandeachjobjj2f1;2;nghastobeprocessedononeofthemmachines.Theshopprob-lems,namelytheowshop,openshopandjobshop,aresimilarinthesensethateachofthemmachinesi2f1;2;;mghasaunitspeedandeachjobj2Nmustvisiteachofthemmachines.Thedierencesarethatintheowshopeveryjobmustvisitthemachines1,2,;minthatsameorder,whileintheopenshopthesequenceinwhicheachjobvisitsthemmachinesisarbitraryandinthejobshop,eachjobhastovisitthemachinesinaprespeciedorderwhichcanbedif-ferentfromjobtojob.Withtheobjectiveofminimizingthetotalweightedcom-pletiontime,thestochasticonlinesinglemachineproblem,uniformparallelma-chineproblem,owshopproblem,openshopproblemandjobshopproblemcanbedenoted,instandardschedulingnotationsee,e.g.,Grahametal.,1979,by1jxjstoch;rjjPwjCj,Qmjxjstoch;rjjPwjCj,Fmjxjistoch;rjjPwjCj,Omjxjistoch;rjjPwjCj,andJmjxjistoch;rjjPwjCj,respectively,wherexjistheprocessingrequirementofjobj.Notethatintheshopproblemsxj=Pmi=1xji,wherexjiistheprocessingrequirementofjobjonmachinei.Thedeterministicvariantoftheseproblems,inwhichtheexactprocessingrequirementofeveryjobjisknownuponjobj'sarrivalattimerj,aredenotedrespectivelyby1jrjjPwjCj,QmjrjjPwjCj,Fmj;rjjPwjCj,OmjrjjPwjCj,andJmjrjjPwjCj.LetfjandRjdenote,resp.,theowtimeandstretchofjobj2N,thenthetotalweightedow

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50 timemetricandtotalweightedstretchmetriccanbedenoted,resp.,byPwjfjandPwjRj.Inthischapterwesaythatajobiswaitingattimetifitisreleasedbutnotbeingprocessedattimet.Wesaythatajobisinthesystemattimetifithasbeenreleasedbutnotnishedbytimet.Wesaythatacertainamountofprocessingrequirementiswaitingattimetifithasbeenreleasedbutnotnishedbytimet.Furthermore,throughoutthispaperweassumethatalljobshaveboundedweights.Wealsoassumethatthemachinesineachproblemhashaveadequatecapacity.Thatis,inthelongrunthemeanrateatwhichjobsarriveisstrictlylessthanthemeanrateatwhichthemachinesisarecapableofprocessing.Wejustifythisassumptionbyobservingthat,withthisassumptionbeingunsatised,thenumberofjobsthatarewaitingforprocessingwillkeepincreasingandwilleventuallyapproachinnityinanyfeasibleschedule.Thismeansthat,afteracertainperiodoftime,therewillalwaysbeanextremelylargenumberofjobswaitingforprocessingandthevastmajorityofjobinformationisknownwheneveradecisionistobemade.Suchkindofaproblembearsmorecharacteristicsofanoineproblemthananonlineproblemandshouldberegardedmoreappropriatelyasanoineproblemandthuswillnotbeconsideredinthispaper.Therestofthischapterisorganizedasfollows.InSection3.2webrieyreviewrelatedresultsintheliterature.InSection3.3wepresentoursimulationstudiesusingArena,throughwhichwedemonstratethattwogenericnondelayalgorithmsconvergeveryfasttotheoptimalsolutionsunderthetotalweightedcompletiontimemetric.OursimulationresultsalsosuggestthatthesameasymptoticoptimalityresultsprovedinChapter1forthestochasticonlinesinglemachineproblem,uni-formparallelmachineproblemandowshopproblemarepossiblyextendabletothestochasticonlineopenshopproblemandjobshopproblem.Thesimulationresults

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51 alsosuggestthat,comparedwiththetotalweightedcompletiontimemetric,theto-talweightedowtimemetricandtotalweightedstretchmetricaremoresensitivetochangesinthesystemparametersanddierentalgorithmsandmaybebetterper-formancemeasurestobeusedintheonlineschedulingenvironment.WenallydrawourconclusionsandsuggestfutureresearchdirectionsinSection3.4.3.2LiteratureReviewInthissectionwebrieyreviewsomerelatedresultsintheliterature.Asymp-toticperformanceanalysisevaluatestheperformanceofanalgorithmonlargesizeinstances.Uptothisdatemostknownasymptoticresultsforonlineschedulingprob-lemsarewiththetotalweightedcompletiontimemetric.KaminskyandSimchi-Levi001astudythesinglemachineproblem1jrjjPCjandshowthattheShortestPro-cessingTimeamongAvailablejobsSPTAruleisasymptoticallyoptimalforthisproblem.BuildingontheresultsofGoemans1997andGoemansetal.999,Chouetal.01showthatageneralizedversionofSPTArule,theWSPRA,isasymp-toticallyoptimalfortheweightedversionofthesinglemachineproblem1jrjjPwjCjandtheuniformparallelmachineproblemQmjrjjPwjCjwithboundedweightsandprocessingrequirements.Inthisheuristic,wheneveramachineisavailable,thejobwiththelargestratiowj=xjamongallthewaitingjobsisselectedtobepro-cessednext.Ifthereisnojobwaiting,thenthemachineremainsidleuntilthenextjobarrives.Chouetal.deriveanupperboundonthemaximumdelaythatanyamountofworkcanincurintheWSPRAschedule,relativetotheLPrelaxationpresentedbyGoemans97.TheythenderivefromthisboundtheasymptoticoptimalityoftheWSPRalgorithmforthesinglemachineanduniformparallelma-chineproblems.Chou001alsoextendsthisresulttothestochasticversionofthesinglemachineproblem1jxjstoch;rjjE[PwjCj],wherethemetricistomin-imizetheexpectedtotalweightedcompletiontimes,E[PwjCj].TheyprovethattheWeightedShortestExpectedProcessingTimeWSEPTruleisasymptotically

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52 optimalfor1jxjstoch;rjjE[PwjCj]aslongasthejobweightsandprocessingrequirementsareboundedandtheprocessingrequirementsareindependentlydis-tributedwithknownmeanvalues.Forshopproblems,KaminskyandSimchi-Levi01bstudytheowshopprob-lemFmjjPCjandshowthattheSPTruleisasymptoticallyoptimalaslongasthejobprocessingrequirementsarecontinuously,independently,andidenticallydis-tributedi.i.d..KaminskyandSimchi-Levi99andXiaetal.00studythemoregeneralowshopproblemFmjjPwjCj.TheyuseprobabilisticanalysistocharacterizetheeectivenessoftheWSPTruleandshowthattheWSPTruleisasymptoticallyoptimalforFmjjPwjCjundersomemildprobabilisticassumptionsonthedistributionsofjobprocessingtimesandweights.BuildingontheresultsofChouetal.01,Liu2001presenttwoonlineheuristicsandonesemi-onlineheuristicwhichareasymptoticallyoptimalfortheowshopproblemFmjrjjPwjCjwithboundedjobprocessingrequirementsandboundedweights.Alloftheseheuris-ticsarecloselyrelatedtotheWSPRAalgorithm.Forthetotalweightedowtimemetric,noasymptoticresultisknownuptothisdate.Afewresultsareavailableinthedomainofcompetitiveanalysis.TheinterestedreadersarereferredtoChekuriandKhanna002andChekurietal.001.Forthetotalweightedstretchmetric,noanalyticalresultisknownsofarandthespecialcaseoftotalunweightedstretchhavebeenanalyzed,inthecompetitiveanalysisdomain,byMuthukrishnanetal.999andBenderetal.0023.3TotalWeightedCompletionTimeMetricAlthoughitisknownthatanynondelayalgorithmisasymptoticallyoptimalfor1jxjstoch;rjjPwjCj,Qmjxjstoch;rjjPwjCjandFmjxjistoch;rjjPwjCjaslongasmachinecapacityisadequateandsomemildconditionsontheweightsandprocessingrequirementshold,therateofconvergenceofarandomlyselectednondelayalgorithmtotheoptimalsolutionisstillanopenquestion.Inthissection,weperform

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53 extensivesimulationstudiesontheperformancesoftwosimplenondelayheuristicsdenotedbyrandomandFCFS,respectively.Inrandom,wheneveramachineisavailablewerandomlyselectoneofthejobsthatarewaitingtobeprocessednext.InFCFS,jobsareprocessedintheorderoftheirarrivals.Ideally,wewouldliketoreporttheperformanceofrandomandFCFSrelativetotheoptimaloinesolutions.However,sinceallthesethreeproblemsareNP-hardeveninthedeterministicsetting,ndingtheoptimaloinesolutionsisprohibitivelyexpensive.Toovercomethisdiculty,wecomparetheperformancesofrandomandFCFStosomelowerboundsseesection3.1.1oftheoptimaloinesolutionsoftheseproblems.Furthermore,wenotethatanynondelayalgorithmforastochasticonlineschedulingproblemisalsoapplicabletoitsdeterministiccounterpartbutnotviceversa.ThusbysimulationwecanevaluatetheperformancesofrandomandFCFSbycomparingitwiththeWSPRAbasedalgorithms,whichhavebeenshown,byChouetal.2001andLiu01,resp.,tobeasymptoticallyoptimalforthedeterministiconlineproblems1jrjjPwjCj,QmjrjjPwjCjandFmjrjjPwjCj.3.3.1LowerBoundsTheUniformParallelMachineProblem.First,wepresentapreemptivesinglemachinerelaxationfortheuniformparallelmachineproblemQmjrjjPwjCj.Thepreemptivesinglemachineschedulingproblem,referredtoasproblemP1m,iscon-structedasfollows.ToeveryinstanceIoftheproblemQmjrjjPwjCj,weassociateaninstanceI1mofthesinglemachineproblem1jrj;pmtnjPwjCj,withthesamenumberofjobsandsamejobcharacteristicsasIandamachinespeedsm=Pmi=1si.Chouetal.2001showsthattheoptimalsolutionofI1m,denotedbyZ1m,isalowerboundofoptimalsolutionofI,denotedbyZm.AlthoughZ1mprovidesalowerboundforQmjrjjPwjCj,itremainsprohibitivelyexpensivetosolvesincetheproblem1jrj;pmtnjPwjCjisalsoNP-hard.Toresolve

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54 thisdiculty,weusethemeanbusydaterelaxationofgeneralschedulingprob-lemsintroducedbyGoemans97togeneratelowerboundsforZ1m.Givenafeasibleschedule,themeanbusydateMjofajobjisdenedtobetheaverageofthetimeinstantsatwhichjobjisbeingprocessing.Itcanbecalculatedbytheformula:Mj=1 xjZT0sjttdt,wheresjtisthespeedatwhichjobjisbe-ingprocessedattimetandTisthetimehorizonof.Whenthespeedfunctionssjtj2f1;2;;ngarepiecewiseconstantont,themeanbusydateMjcanbeexpressedastheweightedaverageofthemidpointsofthetimeintervalsduringwhichjobjisprocessedatconstantspeed.Goemans97alsoseeChouetal.2001showsthatforanygivenscheduleof1jrj;pmtnjPwjCj,Pnj=1wjMj+xj=2isalowerboundofPnjwjCj.Hence,let1jrj;pmtnjPwjMjdenotethemeanbusydaterelaxationoftheproblem1jrj;pmtnjPwjCjandletMjj21;2;;ndenotetheoptimalsolutionto1jrj;pmtnjPwjMj,thenPnj=1wjMj+1 2xjprovidesalowerboundof1jrj;pmtnjPwjCjand,consequently,ofQmjrjjPwjCj.Furthermore,Goemans97showsthatthefollowinggreedyalgorithm,denotedbyLP,opti-mallysolvesproblem1jrj;pmtnjPwjMj:Atthecompletiontimeandthereleasetimeofanyjob,considerallthejobscurrentlyinthesystemandtheonewiththelargestratioofwj=xjisselectedtostartorresumeprocessingimmediately,evenifthisforcesthepreemptionofacurrentlyin-processjob.Ifnojobisavailable,themachinestayidleuntilatleastonejobarrives.AlgorithmLPiseasilyimplementableandwillbeusedtocomputelowerboundsfortheuniformparallelmachineproblemQmjxjstoch;rjjPwjCj.TheShopProblemsandSingleMachineProblem.Wepresentasinglema-chinerelaxationforthedeterministicowshopproblemFmjrjjPwjCj,introducedbyLiu001.Thenonpreemptivesinglemachineproblem,referredtoasP1F,iscon-structedasfollows.ToeveryinstanceIoftheproblemFmjrjjPwjCj,weassociateaninstanceI1Fofthesinglemachineproblem1jrjjPwjCj,whichhasamachine

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55 speedofone,thesamenumberofjobswiththesamejobcharacteristicsasinstanceIexceptthateachjobjinI1Fhasaprocessingrequirementxj=xj1,namelytheprocessingrequirementofjobjontherstmachineininstanceI.LetZFandZ1Fdenote,resp.,theoptimalsolutionvalueofinstanceIandtheoptimalsolutionvalueofitssinglemachinerelaxationI1F.Liu2001showsthatZ1FisalowerboundofZFSince1jrjjPwjCjisalsoNP-hard,Z1Fremainsprohibitivelyexpensivetosolve.WethenusetherelaxationintroducedbyDyerandWolsey1990for1jrjjPwjCjtogeneratethelowerboundsforZ1F.DyerandWolsey90developedthefollowingintegerprogrammingformulationfor1jrjjPwjCj:MinnXj=1wjCjD0s:t:nXj=1yjt1;8t=1;2;;T;TXt=1yjt=xj;8j=1;2;;n;xj 2+1 xjTXt=1t)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 2yjt=Cj;8j=1;2;;n;y2Y;yjt2f0;1g;8j=1;2;;n;t=rj+1;;T;whereTistheschedulinghorizon,yjt=1ifjobjisbeingprocessedinthetimeperiod[t)]TJ/F15 11.955 Tf 12.001 0 Td[(1;tandYareconstraintsimposingthateachjobjisprocessedduringxjconsecutiveperiods.Althoughinthisformulationitisassumedthatjobshaveintegralprocessingrequirementsandreleasetimes,itiseasytoseethatformulationD0isapplicableaslongastheprocessingrequirementsandreleasetimestakerationalvalues,sincethetimeunitintheformulationcanbechosentorepresentonlyafractionoftheactualunittimeperiod,e.g.,1=3minute.NowletproblemDdenotethelinearprogrammingproblemobtainedfromD0bydroppingtheconstraintsy2Y

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56 andtheintegralityconstraints,thentheoptimalobjectivevalueofproblemDisalowerboundofboththeproblem1jrjjPwjCjandtheproblemFmjrjjPwjCj.DyerandWolsey1990seealsoHall,etal.1997provethatthealgorithmLP,describedinsection3.1.1.1,optimallysolvestheproblemD.Hence,wecanusethealgorithmLPtocalculatelowerboundsfortheoineoptimalsolutionsofboththesinglemachineproblem1jxjstoch;rjjPwjCjandtheowshopproblemFmjxjistoch;rjjPwjCj.Furthermore,itiseasytoverifythatproblemP1FalsoprovidesvalidlowerboundsfortheoineoptimalsolutionsofOmjxjistoch;rjjPwjCjandJmjxjistoch;rjjPwjCj.ThusthealgorithmLPcanalsobeusedtogeneratelowerboundsforthesetwoproblems.WecanfurthermodifyP1Ftoobtainatighterlowerboundforthejobshopproblem,Jmjxjistoch;rjjPwjCj,asfollows:toeveryinstanceIoftheproblemJmjrjjPwjCj,weassociateaninstanceI1Jofthesinglemachineproblem1jrjjPwjCj,whichhasamachinespeedofoneandthesamenumberofjobsasinstanceI,witheachjobjhasaprocessingrequirementx0j=xj1andreleasetimer0j=rj+Pi2'jxji,where'jisthesetofmachinesonwhichjobjhasbepro-cessedbeforebeingprocessedontherstmachineininstanceI.InlaterdiscussionswewillapplythealgorithmLPtoI1JtoobtainalowerboundofeachinstanceofJmjxjistoch;rjjPwjCj.3.3.2ExperimentDesignandSimulationResultsTorecapitulate,wetestthefollowing4dierentalgorithmsforeachoftheveproblemsweconsider: 1. random:Inthisheuristic,wheneveramachineiisavailable,considerallthejobsthatarewaitingformachineiandrandomlyselectonejobtobeprocessednext.Ifnojobiswaitingforit,machineistaysidleuntilatleastonejobbecomesavailable.

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57 2. FCFS:Inthisheuristic,wheneveramachineiisavailable,considerallthejobsthatarewaitingformachineiandtheonethathasbeenwaitingforthelongestisselectedtobeprocessednext.Ifnojobiswaitingforit,machineistaysidleuntilatleastonejobbecomesavailable.Note:thewaitingtimeofjobjformachineiisdenedtobethedierencebetweenthetimewhenjobjbecomesavailableformachineiandthecurrenttime 3. WSPRA:Inthisheuristic,wheneveramachineiisavailable,considerallthejobsthatarewaitingformachineiandselectajobwiththelargestratioofwj=xjtobeprocessednext.Ifthereisnojobwaitingforif,machineistaysidleuntilatleastonejobbecomesavailable.Note:fortheshopproblemsxj=Pmi=1xji 4. LP:Tosimplifythenotation,fortheuniformparallelmachineproblemwedenesm=Pmi=1siandfortheowshopproblemandsinglemachineproblemwedenesm=1.Wealsodenexji=xji2f1;2;;mgfortheuniformparallelmachineproblemandxj1=xjforthesinglemachineproblem.Nowconsiderapreemptivesinglemachineproblemwithmachinespeedsm,weightwj,jobprocessingrequirementsxj1andreleasetimer0j=rj+Pi2'jxji,where'jisthesetofmachineswhichjobjmustvisitbeforevisitingtherstma-chine.Wheneverajobisreleasedorcompleted,considerallthejobsthatarecurrentlyinthesystemandthejobwiththelargestratioofwj=xj1isselectedtostartorresumeprocessingimmediately,evenifthisforcesthepreemp-tionofacurrentlyin-processjob.Ifnojobisavailable,themachinestaysidleuntilatleastonejobarrives.Whenschedulingisnishedthetotalweightedmeanbusydate,Pnj=1wjMj+xj=2,iscalculatedifitistheuniformparallelmachineproblemandthetotalweightedcompletiontime,Pnj=1wjCj,iscalcu-latedotherwise.Note:inthesinglemachineproblem,uniformparallelmachine

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58 problem,owshopproblemandopenshopproblemtheset'jisalwaysempty,8j2f1;2;;ng.Asmentionedbefore,WSPRAwasproved,byChouetal.01andLiu01,resp.,tobeasymptoticallyoptimalforthedeterministiconlineproblems1jrjjPwjCj,QmjrjjPwjCjandFmjrjjPwjCjwithboundedjobprocessingrequirementsandweights.AlgorithmLPprovidesalowerboundfortheoptimaloinesolutionofeachinstanceofthevestochasticonlineschedulingproblemswestudy.Wedenotethetotalweightedcompletiontimesobtainedbyapplyingrandom,FCFS,WSPRAandLPby,resp.,wcr,wcF,wcwandwclWeusethreedierentdistributionstogeneratedatasetsfortesting.Werstgeneratealltheparameters,includingprocessingrequirements,inter-arrivaltimesandweights,usinguniformdistributions.Thesimulationisthenrepeatedusingalltheparametersgeneratedfromexponentialdistributionsandthenfromempiricaldistributions.Theparametersofthethreedistributionsarerandomlyselectedsuchthattheyhavethesamemeanvalues.Specically,ifletdenotethemeanvalue,thenthecorrespondinguniform,exponentialandempiricaldistributionsare,resp.,Uniform0;2],Exponential,andEmpirical0:2;0:6;0:5;0:92;0:3;1:4,whereintheempiricaldistribution0.2,0.5and0.3aretheprobabilitiesthattherandomvariablewilltakeavalueof,resp.,0:6,0:92and1:4.Thuswehavethemeanvalueequalto0:20:6+0:50:92+0:31:4=and,again,theseparametersarerandomlyselected.WeusetheArenasimulationpackageseeKeltonetal.2002toimplementallofoursimulationstudies.ThevisualinterfaceandmodellingexibilityofArenaallowustomodeltheveproblems,thefourdierentalgorithmsandthethreedierentperformancemeasureseasily.Thenumberofsimulationrunsischosensuchthatthehalf-widthofeachcondenceintervaloftheweightedcompletiontimeislessthan10%oftheaveragevalue.

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59 WeuseE[crl],E[cFl]andE[cwl]todenote,resp.,theaverageratioswcr=wcl,wcF=wclandwcw=wclandusep,wandLtodenote,resp.,themeanvaluesoftheprocessingrequirements,jobweightsandinter-arrivaltimes.Duetolimitedspace,weareunabletoincludeallthesimulationresultsinthispaper.SelectedsimulationresultsofthevedierentschedulingproblemsarepresentedinTables3.1throughTable3.6.Specically,thesimulationresultsforthestochasticonlinesinglemachine,uniformparallelmachine,owshop,openshopandjobshopproblemswithalltheparametersgeneratedfromuniformdistributionsarepresented,resp.,inTable3.1,Table3.2,Table3.4,Table3.5andTable3.6.Table3.3showsthesimulationresultsforthestochasticonlineuniformparallelmachineproblemwithalltheparametersgeneratedfromexponentialandempiricaldistributions.Figure 3{1 andFigure 3{2 illustratetherelativeperformancesofrandom,FCFSandWSPRAforthestochasticonlineuniformparallelmachineproblemandowshopproblem,resp.,underthetotalweightedcompletiontimemetric.Fromthesimulationresultsweobservethatwhenthetotalweightedcomple-tiontimemetricisused,bothrandomandFCFSperformreasonabllywellandarecomparabletotheWSPRAbasedalgorithms.Specically,weobservethat 1. bothrandomandFCFSconvergefasttotheoptimalsolutionasnincreases.TheconvergencebecomesfasterasthevalueofsmL pincreasesorasthenumberofmachinesdecreases; 2. thetotalweightedcompletiontimesobtainedbyapplyingrandomandFCFSareveryclosetothetotalweightedcompletiontimeobtainedbyapplyingWSPRA.Thisistrueforalltheveproblemswetestandevenforprob-lemsofsmallsizeslessthan6%ofdierence; 3. ThetotalweightedcompletiontimesobtainedbyapplyingWSPRA,randomandFCFSgetevencloserastheratiosmL pincreasesorthenumberofmachinesincreases;

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60 Fromthesimulationstudy,itappearsthatbothrandomandFCFSarealsoasymptoticallyoptimalforthestochasticonlineopenshopandjobshopproblemsseeTable3.5andTable3.6,whichsuggestthattheasymptoticoptimalityresultsprovedinChapter1forthestochasticsinglemachineproblem,uniformparallelma-chineproblemandowshopproblemmaybeextendabletothesetwoproblems.Furthermore,theperformancesofrandomandFCFSarealmostidenticalforalloftheveproblemsstudied.Thetotalweightedcompletiontimemetricseemstobeveryinsensitivetodierentschedulingalgorithmsanddoesnotnecessarilyrevealvaluableinformationaboutthegoodnessorbadnessofthesealgorithms. Table3{1:Simulationresultsfor1jxjstoch;rjjPwjCj. n p=10,w=10,UniformDistri. L=10.5 L=12 E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.0333 1.0325 1.0024 1.0132 1.0131 1.0017 200 1.0226 1.0227 1.0014 1.0065 1.0065 1.0009 500 1.0128 1.0130 1.0007 1.0029 1.0029 1.0004 1000 1.0080 1.0080 1.0004 1.0014 1.0014 1.0002 3000 1.0032 1.0031 1.0001 1.0005 1.0005 1.0001 3.4TotalWeightedFlowTimeandTotalWeightedStretchMetricIdeally,forthetotalweightedowtimemetricandthetotalweightedstretchmetricweshouldalsocomparetheperformancesofrandom,FCFSandWSPRAtothelowerboundsoftheoineoptimalsolutionsofthevestochasticonlineschedulingproblems.Thesametechniquesthatweusetogeneratelowerboundsforthetotalweightedcompletiontimemetriccanalsobeusedtogeneratevalidlowerboundsforthetotalweightedowtimemetric,sincethetwoobjectivefunctionsdieronlybyanadditiveterm,i.e.,thetotalweightedreleasetime,whichisconstantandindependent

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61 Table3{2:SimulationresultsforQmjxjstoch;rjjPwjCj. s1=1:5;s2=0:5s1=0:2;s2=0:3;s3=0:4;s4=0:5;s5=0:6 n p=10,w=10,UniformDistri. L=5.5,m=2* L=5.5,m=5** E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.0469 1.0470 1.0257 1.0935 1.0946 1.0793 200 1.0275 1.0272 1.0132 1.0517 1.0515 1.0405 500 1.0117 1.0117 1.0054 1.0218 1.0219 1.0163 1000 1.0060 1.0060 1.0027 1.0109 1.0110 1.0082 3000 1.0020 1.0020 1.0009 1.0037 1.0037 1.0027 Table3{3:SimulationresultsforQmjxjstoch;rjjPwjCj. n p=10,w=10,L=5.5,m=2,s1=1.5,s2=0.5 ExponentialDistri. EmpiricalDistri. E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.0864 1.0854 1.0317 1.0241 1.0240 1.0219 200 1.0525 1.0529 1.0170 1.0124 1.0124 1.0111 500 1.0291 1.0292 1.0070 1.0050 1.0050 1.0045 1000 1.0163 1.0164 1.0036 1.0025 1.0025 1.0022 3000 1.0056 1.0056 1.0012 1.0008 1.0008 1.0007

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62 Table3{4:SimulationresultsforFmjxjistoch;rjjPwjCj. n p=10,w=10,UniformDistr. L=11,m=3 L=11,m=5 E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.1163 1.1163 1.0935 1.1991 1.1999 1.1771 200 1.0684 1.0682 1.0542 1.1208 1.1217 1.1071 500 1.0341 1.0342 1.0269 1.0599 1.0598 1.0519 1000 1.0180 1.0181 1.0144 1.0318 1.0318 1.0279 3000 1.0060 1.0060 1.0048 1.0110 1.0110 1.0097 Table3{5:SimulationresultsforOmjxjistoch;rjjPwjCj. n p=10,w=10,UniformDistri. L=11,m=3 L=11,m=5 E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.0837 1.0881 1.0585 1.1299 1.1361 1.1037 200 1.0473 1.0508 1.0307 1.0735 1.0754 1.0551 500 1.0213 1.0225 1.0126 1.0318 1.0325 1.0225 1000 1.0116 1.0123 1.0065 1.0159 1.0163 1.0113 3000 1.0042 1.0044 1.0022 1.0058 1.0059 1.0038 Table3{6:SimulationresultsforJmjxjistoch;rjjPwjCj. n p=10,w=10,UniformDistr. L=11,m=3 L=11,m=5 E[crl] E[cFl] E[cwl] E[crl] E[cFl] E[cwl] 100 1.1187 1.1259 1.0613 1.2019 1.2039 1.1176 200 1.0749 1.0799 1.0343 1.1252 1.1295 1.0658 500 1.0329 1.0358 1.0141 1.0590 1.0634 1.0283 1000 1.0169 1.0186 1.0072 1.0339 1.0370 1.0150 3000 1.0060 1.0065 1.0025 1.0117 1.0130 1.0051

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63 Figure3{1:Relativeperformancesofrandom,FCFS,andWSPRAtoLPforQmjxjstoch;rjjPwjCj Figure3{2:Relativeperformancesofrandom,FCFS,andWSPRAtoLPforFmjxjistoch;rjjPwjCj

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64 oftheschedule.However,thesetechniquesdonotprovideasymptoticallytightlowerboundsforthetotalweightedowtimemetric.Forinstance,thealgorithmLPcanbeusedtoprovidevalidlowerboundforthetotalweightedowtimemetricbutthegapbetweenthelowerboundthatitprovidesandtheoptimaloinesolutionincreasesastheproblemsizeincreases.Forthetotalweightedstretchmetric,developingasymptoticallytightboundisevenharder.Infact,therehasbeenanongoingresearchintheliteraturetodevelopeasilycomputabletightboundsforthetotalweightedowtimemetricandtotalweightedstretchmetricbutsofarnosuchalgorithmshasbeenreported.HencewewillcomparetheperformancesofrandomandFCFSagainsttheperformancesofWSPRAunderthesetwoperformancemeasures.Weusethesamethreedistributions,i.e.,theuniform,exponentialandempiricaldistributions,togeneratetheprocessingrequirements,inter-arrivaltimesandweights.Thenumberofsimulationrunsischosensuchthatthehalf-widthofthecondenceintervaloftheweightedowtimeislessthan10%oftheaveragevalue.Wedenotethetotalweightedowtimesobtainedbyapplyingrandom,FCFSandWSPRAby,resp.,wfr,wfFandwfwandthetotalweightedstretchesobtainedbyapplyingrandom,FCFSandWSPRAby,resp.,wer,weFandwew.WeuseE[frw],E[fFw],E[erw]andE[eFw]todenote,resp.,theaverageratioswfr=wfw,wfF=wfw,wer=wew,andweF=wew.SelectedsimulationresultsofthevedierentschedulingproblemsarepresentedinTables3.7throughTable3.9.Specically,thesimulationresultsforthestochasticonlineuniformparallelmachine,owshop,andjobshopproblemswithalltheparametersgeneratedfromuniformdistributionsarepresented,resp.,inTable3.7,Table3.8,andTable3.9.Weobservethatwhenthetotalweightedowtimemetricorthetotalweightedstretchmetricisused,bothrandomandFCFSperformbadlycomparedwithWSPRA.Thegaptendstoincreasedramaticallyasthenumberofjobsincreasesorasthenumberofmachinesdecreases.Itisevidentthatthetotalweightedowtime

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65 Table3{7:SimulationresultsforQmjxjstoch;rjjPwjfjandQmjxjstoch;rjjPwjRj. s1=1:5;s2=0:5s1=0:2;s2=0:3;s3=0:4;s4=0:5;s5=0:6 n p=10,w=10,UniformDistri. L=5.5,m=2* L=5.5,m=5** E[frw] E[fFw] E[erw] E[eFw] E[frw] E[fFw] E[erw] E[eFw] 100 1.3790 1.3847 2.3972 2.4183 1.1291 1.1418 1.5329 2.0728 200 1.5175 1.5094 2.7128 2.5583 1.2008 1.2022 2.2423 2.2214 500 1.5448 1.5424 3.4875 2.8507 1.2375 1.2395 2.9026 2.3433 1000 1.5740 1.5708 2.1860 3.6154 1.2419 1.2471 2.5478 3.5289 3000 1.6027 1.6069 3.1682 3.4985 1.2537 1.2575 2.6455 2.9903 Table3{8:SimulationresultsforFmjxjistoch;rjjPwjfjandFmjxjistoch;rjjPwjRj. n p=10,w=10,UniformDistri. L=11,m=3 L=11,m=5 E[frw] E[fFw] E[erw] E[eFw] E[frw] E[fFw] E[erw] E[eFw] 100 1.1805 1.1804 1.2091 1.2074 1.1038 1.1058 1.1119 1.1151 200 1.1966 1.2000 1.2208 1.2284 1.1122 1.1162 1.1192 1.1243 500 1.2092 1.2179 1.2330 1.2450 1.1265 1.1253 1.1348 1.1330 1000 1.1987 1.2086 1.2217 1.2334 1.1180 1.1172 1.1262 1.1244 3000 1.2067 1.2062 1.2333 1.2307 1.1152 1.1147 1.1221 1.1213

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66 Table3{9:SimulationresultsforJmjxjistoch;rjjPwjfjandJmjxjistoch;rjjPwjRj. n p=10,w=10,UniformDistri. L=11,m=3 L=11,m=5 E[frw] E[fFw] E[erw] E[eFw] E[frw] E[fFw] E[erw] E[eFw] 100 1.5032 1.5661 1.6547 1.7259 1.4543 1.4654 1.5314 1.5464 200 1.6605 1.7414 1.8304 1.9443 1.5882 1.6308 1.6779 1.7325 500 1.7487 1.8640 1.9372 2.0911 1.7220 1.8253 1.8257 1.9453 1000 1.7717 1.9015 1.9661 2.1362 1.8407 1.9803 1.9568 2.1160 3000 1.8198 1.9298 2.0349 2.1729 1.8666 2.0417 1.9941 2.1847 metricandthetotalweightedstretchmetricaremuchmoresensitivetoevenslightchangesinthesystemparametersanddierentalgorithmsthanthetotalweightedcompletiontimemetricintheonlineschedulingenvironment.Thismightsuggestthatthetotalweightedowtimemetricandthetotalweightedstretchmetricaremoreappropriatemetricsformanyonlineschedulingproblems.Despitetheextremelylimitedeortthathasbeendevotedsofartothesetwometricsintheonlineenvi-ronment,theinterestandpotentialareimmense.Researchonasymptoticresultsofalgorithmsunderthetotalweightedowtimemetricorthetotalweightedstretchmetricintheonlineenvironmentisimportantandyetchallenging.Oneofthemajorchallengesistodeveloptightlowerboundsforthetotalweightedcompletiontimeandtotalweightedstretch.3.5ConcludingRemarksInthischapterwestudyvedierentstochasticonlineschedulingproblemsunderthreedierentperformancemeasures.Withtheobjectiveofminimizingthetotalweightedcompletiontime,weshowthattwogenericnondelayalgorithmsperformverywellandconvergetotheoptimalsolutionsveryfastforalloftheseveproblems.ItappearsthattheasymptoticoptimalityresultspresentedinChapter1forthe

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67 stochasticsinglemachineproblem,uniformparallelmachineproblemandowshopproblemmaybeextendabletothestochasticonlineopenshopandjobshopproblems.Thesimulationresultsalsosuggestthat,comparedwiththetotalweightedcompletiontimemetric,thetotalweightedowtimemetricandtotalweightedstretchmetricaremoresensitiveandmaybebetterperformancemeasuresforschedulingintheonlineschedulingenvironment.Averyinterestingdirectionoffutureresearchwouldbetodevisealgorithmsthatareasymptoticallyoptimalforthetotalweightedowtimemetricorthetotalweightedstretchmetrics.Oneofthemajorchallengesistodeveloptightlowerboundsforthetotalweightedcompletiontimeandtotalweightedstretch.

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CHAPTER4NEWMODELFORSTRATEGICSUPPLYCHAINLOCATION4.1IntroductionSupplychainnetworkdesigndecisionsaregenerallycostlyanddiculttoreverse,andtheirimpactspansalongtimehorizon.Duringthetimewhendesigndecisionsareineect,manydecisionparameters-demands,costs-maychangedramatically.Unfortunately,mostfacilitylocationmodelsinliteraturearestaticanddeterministic.Inthepastfewdecades,researchershavedealtwiththeuncertaintiesbydeninganumberofpossiblefuturescenarios.Facilitysitesthateitheroptimizetheexpectedperformanceoroptimizetheworst-caseperformanceoverallthescenariosarethenchosen.However,inpracticefacilitiesarenottypicallydesignedforeithertheaveragecaseortheworst-casescenario.Forexample,airportsareneversizedforeitheranaverageday,sincedoingsowouldresultinsignicantunder-capacitymuchofthetime.Ontheotherhand,airportareneversizedforthepeaktravelday,e.g.,theSundayofThanksgivingweekendintheUS,sincedoingsowouldbeprohibitivelyexpensive.Morerecently,Daskin,HesseandRevelle97developedamodelcalled-reliablep-medianminimaxregretmodel.Thismodelidentiesthelocationpatternthatminimizesthemaximumregretwithrespecttoanendogenouslyselectedsubsetofscenarioswhosecollectiveprobabilityofoccurrenceisatleastsomeuser-denedvaluea.Inthisway,theplannercanbe100%surethattheregretwillbenomorethanthatfoundbythemodel.Althoughthisworkrepresentsanimportantbreakthrough,ithasseriousdrawbacks.Incontrast,wehavedevelopedanewmodelwhichminimizestheexpectedregretwithrespecttoanendogenouslyselectedsubset 68

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69 ofworst-casescenarioswhosecollectiveprobabilityofoccurrenceisexactly1-.Thisnewmodel,whichwecall-reliablemeanExcessregretmodel,ormeanExcessmodelforshort,hasdemonstratedsignicantimprovementsoverDaskinetal'smodelinournumericaltests.Moreover,aheuristichasbeendevelopedtoecientlysolvethe-reliablep-medianminimaxregretmodelbysolvingaseriesofmeanExcesssub-problems.Therestofthischapterisorganizedasfollows.InSection4.2webrieyreviewsomeoftheliteratureonscenariomodelinginthecontextofstochasticfacilitylo-cation.InSection4.3weformulateournewmodelandcompareitwithDaskinetal's-reliablep-medianminimaxregretmodel.ComputationalresultsarepresentedinSection4.4.InSection4.5wepresentaheuristic,whichinvolvessolvingaseriesof-reliablemeanExcessregretsub-problems,forthe-reliablep-medianminimaxregretproblem.Finally,weconcludeandproposefuturedirectionsofresearchinSection4.6.4.2LiteratureReviewInthepastfewdecades,researchershaveusedscenarioplanningtodealwiththeuncertaintiesinstrategicfacilitylocation.Inscenarioplanning,thedecisionmakeridentiesanumberoffuturepossiblescenariosandestimatethelikelihoodofeachscenariooccurring.Scenarioplanningwaschosenprimarilybecause,aspointedoutinSnyder,DaskinandTeo02,itallowsthedecisionmakerstomodeldependenceamongrandomparameters.Forexample,futuredemandsarelikelytobecorrelated.Soascosts.Ifcontinuousapproachisusedtomodelsuchcorrelations,thentheproblemtendstobecomeintractable.Sheppard74isamongtherstwhousescenarioplanningtomodeluncer-taintiesinfacilitylocation.Hismodelgivesasittingplanthatminimizestheexpectedcostoverallscenarios.Daskin,HoppandMedina92demonstratethatuseofthe

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70 scenarioplanningapproachcanleadtotheadoptionoftheworstpossibleinitialde-cisionunderconditionsoffutureuncertainty.Theyproposeaforecasthorizon-basedapproachtofacilityplanningovertime.GhoshandMcLaerty82proposeamodelinwhicheitherthesumoftheregretsorthesumofsquaredregretsoverallscenarios.Itshouldbenotedthattheobjectiveofminimizingthesumoftheregretsisequivalenttominimizingtheexpectedregretwithallscenarioshavingthesameprobability.SerraandMarianov1998lookataprobleminwhichtheparametersofthenetwork,includingtraveltimes,demandanddistances,changeoverthecourseofadayandtheymodeleachperiodofthedayasscenario.Theyidentifysolutionsthateitherminimizethemaximumtraveltimeoverthescenariosorsolutionsthatminimizethetotalregret.Theydenetheregretofeachscenariotobethedierencebetweentheobjectivefunctionvaluesgivenbytheoverallsolutionandtheoptimalsolutionforthatsinglescenario.Serra,RatickandReVelle996studyamaximumcaptureproblemwheretheobjectiveistoselectthelocationsofserversforanenteringrmwhichwishestomaximizeitsmarketshareinamarketwherecompetitorsarealreadyinposition.Theirmodelseithermaximizetheminimumcaptureassociatedwithanyscenarioorminimizetheexpectedregretoverallthescenarios.Current,RatickandReVelle1998studyproblemswherethetotalnumberoffacilitiestobelocatedisuncertainandtheobjectiveiseithertominimizetheexpectedopportunitylossortominimizethemaximumopportunityloss.Theopportunitylossisdenedasthedierencebetweentheobjectivefunctionvaluewhentheinitialfacilitylocationsmustbeincludedinthenalsitingplan,andtheobjectivefunctionvaluewhenthereisnosuchaconstraint.Theminimumexpectedopportunitylosscriteriandstheinitialsetoffacilitylocationsthatminimizetheexpectedopportunitylossesacrossallscenarios.Theminimaxopportunitylosscriteriandstheinitialfacilitylocationssuchthatthemaximumlossisminimizedoverallscenarios.

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71 Morerecently,Snyder,DaskinandTeostudythestochasticlocationproblemwithriskpoolingwhichseekstolocatedistributioncenterstominimizethetotalxedlocationcosts,transportationcosts,andinventorycosts.TheyproposeamodelthatminimizetheexpectedcostofthesystemacrossallscenariosanddevelopaLagrangian-relaxation-basedexactalgorithmtosolvethemodel.TheworkbyDaskin,HesseandRevelle1997iscloselyrelatedtothispaper.Theydevelopedamodelcalled-reliablep-medianminimaxregret,whichndsthesitingplanthatminimizesthemaximumregretwithrespecttoanendogenouslyselectedsubsetofscenarioswhosecollectiveprobabilityofoccurrenceisatleast.Inthisway,theplannercanbe100%surethattheregretwillbenomorethanthatfoundbythemodel.ThereaderisreferredtoOwenandDaskin998foramorecomprehensivereviewofrecentdynamicandstochasticfacilitylocationproblems.4.3ModelWeconsiderthestochasiticp-medianprobleminthecontextofscenarioplan-ning,giventhefollowingnotation:i=1,,m:indexofdemandnodesj=1,,n:indexofcandidatelocationsk=1,,K:indexofpossiblescenarioshik:thedemandatnodeiunderscenariokdijk:distancefromnodeitocandidatesitejunderscenariokp:numberoffacilitiestolocate^Vk:best-medianvaluethatcanbeobtainedunderscenariokqk:probabilitythatscenariokwillbetheonethatisrealizedDecisionvariables:xj=8><>:1:ifwelocateatcandidatenodej0:otherwise

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72 yijk=8><>:1:ifdemandnodeiisassignedtoafacilityatjunderscenariok0:otherwiseWedenetheregretassociatedwithscenariokandaselectionoflocationsasthedierencebetweentheobjectivefunctionvaluethedemand-weightedtotaldistancevaluethatresultsfromthislocationselectionandthebestobjectivefunctionvaluethatwecouldattainifwelocateatthebestpossiblesitesforscenariokalone.Namely,letVkdenotethedemand-weightedtotaldistanceunderthecompromiselocation,i.e.,Vk=XiXjhikdijkYijk,thentheregretassociatedwithscenariokisdenedtobeRk=Vk)]TJ/F15 11.955 Tf 13.742 3.022 Td[(^Vk.First,letshaveacloserlookatthe-reliablep-minimaxregretmodelproposedbyDaskin,HesseandRevelle1997.Topresenttheirmodel,weneedtodenethefollowing:mk=alargeconstantspecictoscenarioksuchthatmkRkzk=8><>:1:ifscenariokisincludedinthesetoverwhichthemaximizationistaken0:otherwiseThe-reliablep-minimaxregretproblemcanbeformulatedasfollows:MinimizeW .1 subjectto:nXj=1xj=p .2 nXj=1yijk=1;8i;k .3 yijk)]TJ/F22 11.955 Tf 11.956 0 Td[(xj08i;j;k .4 Rk)]TJ/F15 11.955 Tf 11.955 0 Td[(mXi=1nXj=1hikdijkyijk)]TJ/F15 11.955 Tf 13.741 3.022 Td[(^Vk=0;8k .5

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73 KXk=1qkzk .6 W)]TJ/F22 11.955 Tf 11.956 0 Td[(Rk+mk)]TJ/F22 11.955 Tf 11.956 0 Td[(zk0;8k .7 xj2f0;1g;8j .8 yijk2f0;1g;8i;j;k .9 zk2f0;1g;8k .10 Theobjectivefunction 4.1 minimizesthe-reliablep-minimaxregret.Constraint 4.2 stipulatesthatexactlypfacilitiesaretobelocated.Constraint 4.3 statesthateachdemandnodemustassignedtoexactlyonefacilityineachscenario.Constraint 4.4 statesthatdemandsaticannotbeassignedtoafacilityjunderscenariokunlessafacilityislocatedatnodej.Constraint 4.5 denestheregretassociatedwithscenariok,asdiscussedpreviously.Finally,constraint 4.6 and 4.7 stipulatethattheprobabilityassociatedwiththesetofscenariosoverwhichthemaximumregretiscomputedmustbeatleast.Thus,the-reliablep-minimaxregretproblemminimizesthemaximumregretoverasubsetofthepossiblescenarios,withtheaddedstipulationthattheprobabilityofrealizingascenariothatisnotincludedmustbeatmost1-.Inaddition,byvaryingoveranappropriaterange,thedecisionmakercanidentifyaportfolioofsitingplan.Althoughthe-reliablep-minimaxregretcapturesimportantreal-lifemanagerialconcernsthatarenotreectedinmodelsthatminimizeexpectedregretormaximumregret,ithasseveralmajordrawbacks.Firstofall,itprovidesnowaytoassessthemagnitudeoftheregretsassociatedwiththescenariosthatarenotincludedinthe-reliableset.Thus,decisionmakersthatusesitingplansprovidedbythismodelfacetheriskofrealizingascenarioassociatedwithaveryhighregret.Inaddition,thismodelisincapableofdistinguishingbetweensituationswherethepotentialexcessregretsareonlyalittlebitworse,andthosewherethepotentialexcessregretsareoverwhelming.Moreover,conductedinthispapernumericalexperimentsshowthat

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74 minimizingthe-reliableminimaxregretmayleadtoanundesirableincreaseintheworst-caseregret.Thisisintuitivesinceinthe-reliableminimaxregretmodelthemagnitudeoftheregretsassociatedwiththe100)]TJ/F22 11.955 Tf 12.715 0 Td[(%worstscenariosarenotconsidered.Finally,mathematicallythe-reliableminimaxregretmodelhassomeseriouslimitations.The-reliableminimaxmodelhasearlierbeenstudiedinthestochasticprogrammingliterature,althoughnotinastrategicfacilitylocationcontext.Ithasbeenshownsee,e.g.,MausserandRosen1998that-reliableminimaxregretisanonsmooth,nonconvex,andmultiextremefunctionwithrespecttothedecisionvariablesx;yandthereforeiscomputationallyverydiculttosolve.Duetothelimitationsofthe-reliablep-minimaxregretmodel,inthispaperwedevelopanewmodelwhichminimizestheexpectedregrettheprobability-weightedregretwithrespecttoanendogenouslyselectedsubsetofworst-casescenarioswhosecollectiveprobabilityofoccurrenceisexactly1-.Topresentournewmodel,weneedthefollowingadditionaldenitionsandnotations:X2:thep-medianfeasibilityconstraintset,i.e.,constraints 4.2 4.3 4.4 4.5 4.8 and 4.9 X:valueofdecisionvariableX=x;yRX;k:regretasafunctionofXandscenarioindexkfX;=PfkjRX;kg:withXxedatX,thecollectiveprobabilityofthosescenariosinwhichtheregretdoesnotexceedX=minf2R:fX;g:withXxedatX,theminimumvaluesuchthatfX;,i.e.,the-quantileoftheregretsofthe

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75 KscenariosX=Xk:RX;k>XRX;kqk1 Pkqk:withXxedatX,theconditionalprobability-weightedaverageoftheregretsstrictlyexceedingXWiththesenewnotations,itiseasytoseethatthe-reliablep-minimaxregretproblemcanberewrittenasfollows:MinimizeX .11 subjectto:X2Incontrast,ournewproblem,whichwerefertoasthe-reliablemeanExcessregretproblem,canbeformulatedasfollows:MinimizeX+)]TJ/F22 11.955 Tf 11.955 0 Td[(X .12 subjectto:X2where=[fX;X)]TJ/F22 11.955 Tf 11.955 0 Td[(] 1)]TJ/F22 11.955 Tf 11.956 0 Td[(2[0;1].Notethattheobjectivefunctionistheweightedaverageofthethe-quantileoftheregretsoftheKscenarios,X,andthecon-ditionalexpectationoftheregretsstrictlyexceedingX,X,insteadofonlyX.Thisisbecause,inthescenarioplanningcontext,thecollectiveprobabilityofthosescenarioswhichhaveregretsstrictlyexceedingXmaybestrictlylessthan.Therefore,theobjectivefunctioninformulation 4.12 minimizestheexpectedregretwithrespecttoanendogenouslyselectedsubsetofworst-casescenarioswhosecollectiveprobabilityofoccurrenceisexactly1-.SinceboththeregretfunctionRX;kandthefeasibilitysetareconvexinX,itcanbeshownthatseeRockafellarandUryasev2000and2001formulation 4.12

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76 canbereducedintothefollowingform:MinimizeFX;=+1 1)]TJ/F22 11.955 Tf 11.956 0 Td[(KXk=1qkMaxf[RX;k)]TJ/F22 11.955 Tf 11.955 0 Td[(];0g .13 subjectto:X2whereisafreevariable.Hencethe-reliablemeanExcessregretmeanExcessproblemcanbeformulatedasthefollowingmixedintegerproblem:meanExcessMinimizeFx;y;=+1 1)]TJ/F22 11.955 Tf 11.955 0 Td[(KXk=1qkUk .14 subjectto:nXj=1xj=pnXj=1yijk=1;8i;kyijk)]TJ/F22 11.955 Tf 11.955 0 Td[(xj0;8i;j;kRk)]TJ/F15 11.955 Tf 11.955 0 Td[(mXi=1nXj=1hikdijkyijk)]TJ/F15 11.955 Tf 13.742 3.022 Td[(^Vk=0;8kUkRk)]TJ/F22 11.955 Tf 11.955 0 Td[(;8k .15 xj2f0;1g;8jyijk2f0;1g;8i;j;kUk0;8k .16 ItcanbeveriedthatseeRockafellarandUryasev2000and2002Fx;y;isconvexwithrespecttobothandx;y.Inaddition,minimizingFx;y;givesboththeoptimal-reliablemeanExcessregretandthecorrespondingnon-optimal-reliableminimaxregret.Specically,afterthe-reliablemeanExcessregretproblemissolvedoptimally,theobjectivefunctionvaluegivestheminimum-reliablemeanExcessregret,thesolutionx;ygivestheoptimalsitingplanandthesolutiongivesthethecorrespondingnon-optimal-reliableminimaxregret.

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77 The-reliablemeanExcessregrethasseveralsignicantadvantagesoverthe-reliableminimaxregretmodel.Firstofall,bydenition,the-reliablemeanExcessregretisanupperboundofthecorresponding-reliableminimaxregret.Therefore,minimizing-reliablemeanExcessregretwillalsoleadtoalow-reliableminimaxregret.Second,minimizationof-reliablemeanExcessregretavoidsthedanger-ousincreaseintheworst-caseregret.Third,becausetheobjectivefunctionofthe-reliableminimaxregretmodelisconvexwithrespecttobothandx;y,itiscomputationallymucheasiertosolve.4.4ComputationalResultsInthissection,wesummarizeourcomputationalresultswithboththe-reliableminimaxregretproblemandthe-reliablemeanExcessregretproblemoutlinedabove.AllofourcomputationalexperimentsarebasedonthedatafoundinDaskin995andDaskin,HesseandRevelle97on88majorUScities.Specically,weusetheninescenariosofthe88-cityproblemfoundinDaskin,HesseandRevelle1997togeneratemorescenariostobeusedinourcomputationalexperiments.Forexample,inordertohaveKK9alternativescenarios,wegeneratebK=9cscenariosfromeachscenario,say,scenarios,oftheninescenariosinDaskin,HesseandRevelle997,usinganormaldistributionwhichhasameandemandequaltothedemandofscenariosandastandarddeviationequalto1=10ofthedemandofscenarios.TheremainingK)]TJ/F15 11.955 Tf 10.565 0 Td[(9bK 9cscenariosarethengeneratedinthesamewayfromscenarioNo.veinDaskin,HesseandRevelle97.Allofourtestsinvolvesitingvefacilitiesinveofthe88cities,i.e.,p=5.Inallruns,itisassumedthatall88demandnodesarealsoeligiblecandidatesites.Inaddition,theonlydierencebetweenanytwoscenariosgeneratedisthedemands.Namely,dierentscenarioshavedierentdemandvolumesatthe88cities.Butthe

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78 distancebetweenanytwocitiesisscenario-independent.Therefore,inboththe-reliableminimaxregretproblemandthe-reliablemeanExcessregretproblemwereplaceyijkbyyij,therebysignicantlyreducingthenumberofdecisionvariables.Notethat,inboththe-reliableminimaxregretproblemandthe-reliablemeanExcessregretproblem,thevalues^Vk's,i.e.,theminimumdemand-weightedtotaldistancevaluesthatwecouldattainifwelocateatthebestpossiblesitesforscenariokk=1;2;;Kalone,arerequiredasinputsparameters.Weobtainthesevaluesbyoptimallysolvingthep-medianproblemforeachoftheKscenarios.Inaddtion,notethatinthe-reliableminimaxregretproblem,thevaluesmk's,i.e.,theupperboundofthelargestpossibleregretforeachscenariokk=f1;2;;Kg,arealsorequiredasinputparameters.Weusethreedierentproce-durestoattainthesevalues.IntherstprocedureseeDaskin,HesseandRevelle1997,foreachscenariok,wecomputetheregretassociatedwithlocatingattheoptimallocationsfoundforeachoftheotherK)]TJ/F15 11.955 Tf 12.302 0 Td[(1scenarios.mkisthentakenasthemaximumofthesequantities.ThisprocedurerequiressolvingKK)]TJ/F15 11.955 Tf 11.183 0 Td[(1+1sub-problems.Inthesecondprocedure,eachmkk=f1;2;;Kgissettotheconstantvalueof3:01011,whichislargeenoughtobeusedastheupperboundoftheRk's.Inthethirdprocedure,wecomputethemaximumpossibleregretassociatedwitheachscenariokbysolvingthep-medianproblemassociatedwithit,withtheobjectivefunctionchangedtomaximizeRk.ThisprocedurerequiressolvingKsub-problems.Tofacilitateourdiscussion,wedenotethe-reliablep-medianminimaxregretprob-lemswiththerst,secondandthirdprocedureforsolvingthemk'sbyRPMMI,MinimaxIIandMinimaxIII,respectively.Bothproblemsaswellastheirsub-problemsarecodedwithMicrosoftVisualStudio.netandsolvedusingCPLEXversion8.1onaDellpersonalcomputer.TheoperatingsystemisMicrosoftWindowsXPProfessionalEdition.Allthecomputationtimespresentedinthissectionareinsecondsunlessotherwisestated.

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79 Table 4{1 presentsthecomputationtimesforoptimallysolvingmeanExcess,MinimaxI,MinimaxIIandMinimaxIII,withthereliabilitylevelxedat95%.AllthecomputationtimesinTable 4{1 excludetheinput/outputtimes,aswellasthetimeneededtosolveforthe^Vk'sandmk's.Notethat,MinimaxIIcouldsolvenomorethan99scenarioswhileMinimaxIIIcouldsolvenomorethan45scenarios.Figure 4{1 illustratesthecomputationtimesneededtooptimallysolvemeanExTable4{1:SolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. No.ofscenariosTotalSolutionTime MinimaxIMinimaxIIMinimaxIIImeanExcess 972909585278292114484534961972213772288140979343*130997098738188111*1941261356198298*129305*2031621317**119023*2121984706****23923410225****21827912135****276324******385 *amountoftimespentbeforeCPLEXstoppedwithoutndinganysolution**nottested cess,MinimaxI,MinimaxIIandMinimaxIIIasthenumberofscenariosincreases.Notethat,asthenumberofscenariosincreases,thecomputationtimesofMinimaxI,MinimaxIIandMinimaxIIIincreaseexponentiallywhilethecomputationaltimeofmeanExcessincreasesonlylinearly.Table 4{2 presentsthetotalcomputationtimesneededforoptimallysolvingmeanExcess,MinimaxI,MinimaxIIandMinimaxIII,withthereliabilitylevelxedat95%.Specically,thetotalcomputationtimesofmeanExcessincludethetimeneededtosolveforthe^Vk'sandthetimeneededtosolvethe-reliablemeanExcessregretproblem.ThetotalcomputationtimeofMinimaxI,MinimaxIIandMinimaxIII

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80 Figure4{1:SolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcesswith=0.95. includethetimeneededtosolveforthe^Vk'sandmk'saswellasthetimeneededtosolvethe-reliablep-medianminimaxregretproblem.Alltheinput/outputtimesareexcluded.ThetotalcomputationtimesinTable 4{2 areillustratedinFigure 4{2 .Weobservethat,asthenumberofscenariosincreases,thetotalsolutiontimesofMini-maxI,MinimaxIIandMinimaxIIIincreaseexponentiallywhilethetotalsolutiontimeofmeanExcessincreasesinalinearmanner.Wealsonotethat,whenthenumberofscenariosincreasesto279,ittakesmorethan88hourstooptimallysolveMinimaxI.ConsidertheexponentiallyincreasingcomputationtimeofMinimaxI,itseemsim-practicaltosolveMinimaxIoptimallyformorethan279scenarios.ForMinimaxIIandMinimaxIII,CPLEXwasunabletondanysolutionformorethan99scenariosand45scenarios,respectively.Itisobviousthattheinputparametersmk'shaveansignicantimpactonthesolutiontimeofthe-reliablep-medianminimaxregret

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81 Table4{2:TotalSolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. No.ofscenariosTotalSolutionTime MinimaxIMinimaxIIMinimaxIIImeanExcess 997101110962743513117087451841695841213725912155979586*28099156578990188528*44512630716198671*129928*57716264317**119988*773198118394****1019234193056****1235279318242****1650324******2191 *amountoftimespentbeforeCPLEXstoppedwithoutndinganysolution**nottested Figure4{2:TotalSolutionTimesforMinimaxI,MinimaxII,MinimaxIIIandmeanEx-cesswith=0.95.

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82 problem.Inparticular,tighterupperboundsofRk'sdramaticallyreducethesolutiontimeofthe-reliablep-medianminimaxregretproblembuttakemuchlongertimetoobtain.Table3presentsthesolutionsforthe-reliablemeanExcessregretproblemandthe-reliablep-medianminimaxregretproblem.Notethat,thesolutiontothe-reliablep-medianminimaxregretproblemisindependentofwhichprocedureisusedtosolveforthemk's.Therefore,inTable3weonlypresentthesolutionsofMinimaxI,sinceitwasabletosolveupto279scenarios.Inaddition,inTable3wealsopresenttheworst-casescenarioregretsassociatedwiththetwoproblems,andthe-reliablep-medianminimaxregretassociatedwithminimizingthe-reliablemeanExcessregret.FromTable 4{3 weobservethat,in9outofthe10instancesthatwetested, Table4{3:SolutionsofMinimaxI,MinimaxII,MinimaxIIIandmeanExcess. No.of )]TJ/F17 10.909 Tf 8.485 0 Td[(reliablemeanExcessregret )]TJ/F17 10.909 Tf 8.484 0 Td[(reliablep-medainminimax scenarios )]TJ/F17 10.909 Tf 8.485 0 Td[(reliablemean-reliableworst-case )]TJ/F17 10.909 Tf 8.485 0 Td[(reliableworst-case accessregretminimaxregretregret minimaxregretregret 9 224742881122474288112247428811 22474288112247428811 27 225873831120519962842321909291 20519962842321909291 45 230241775222003213392468654038 22003213392468654038 72 240558207322464491862532201475 22464491862532201475 99 243259611122702604832644634954 22702604832644634954 126 234678602322288053882648265518 22288053882648265518 162 235353920922140903642496477548 21993486732552036761 198 249500711922783544672827041127 22783544672827041127 234 240913909322436779462980885404 22436779462980885404 279 245551295722513284822869632915 22513284822869632915 minimizingthe-reliablemeanExcessregretalsoledtotheminimizationofthe-reliablep-medianminimaxregret.Intheremainingoneinstance,i.e.,theonewith162scenarios,minimizingthe-reliablemeanExcessregretgivesan-reliablep-medianminimaxregretwhichis0.67%higherthantheoptimal-reliablep-median

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83 minimaxregret.Ontheotherhand,ineachofthe10instancestested,the-reliablemeanExcessregretmodelgivesaworst-casescenarioregretnobiggerthanthatisgivenbythe-reliablep-medianminimaxregretmodel.Particularly,inoneinstance,i.e.,theinstancewith162scenarios,theworst-casescenarioregretgivenbythesolutiontothe-reliablemeanExcessregretmodelis2.23%lowerthantheworst-casescenarioregretgivenbythesolutiontothe-reliablep-medianminimaxregretmodel.4.5AMeanExcessBasedHeuristicforSolvingtheMinimaxModelInthissectionwepresentaheuristicthatcanbeusedtoecientlysolvethemini-maxmodel.ThisheuristicismodiedfromtheheuristicproposedbyLarsen,MausserandUryasev002andinvolvessolvingaseriesofmeanExcesssub-problems.Theideabehindthisheuristicissimple:lowvaluesofminimaxregretcanbeobtainedbysolvingameanExcessmodelwithsomenewidenedsothatthevaluesofthetwomeasurescoincideorgetascloseaspossible.InthisheuristicwestartwithoptimallysolvingthemeanExcessmodelassociatedwiththeoriginalminimaxproblem.Then,ateachiteration,werenderaportionoftheworstscenariosinactiveandsolveameanExcesssub-problemwiththefollowingcharacteristics:1.onlytheremainingactivescenariosareconsideredinthesub-problem;2.thecondenceleveliofthesub-problemisdenedsuchthatthemeanExcessregretofthesub-problemcoincide,oriscloseto,theminimaxregretoftheoriginalproblem.Thus,byconstructingandsolvingaseriesofmeanExcesssub-problemsthatcloselyapproximatetheminimaxregretoftheoriginalproblem,wecansystematicallyreducetheminimaxregretoftheoriginalproblem.Wenowpresentaformaldescriptionofthisheuristic:Step0.Initialization 1. Set0=,i=0,activescenariosetH0=fk:k=1;3;;Kg,taboolistT=;,currentminimaxregret0=1,andbestminimaxregret0=1.

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84 2. SelectstepsizetobeS;Step1.SolvemeanExcesssub-problem 1. OptimallysolvethemodelmeanExcessdenedontheactivesetHo,withthecondencelevelequaltoiandthefollowingadditionalconstraints:thesolutionmustbedierentfromeverysolutioninthetaboolistT. 2. Basedonthesolutiontothesub-problem: a SorttheKscenariosinthenon-decreasingorderoftheirregrets. b Denotetheorderedscenariosbykl;l=1;2;;K. c DenotetheSscenarioswiththelargestregretsintheactivesetby~i. 3. AddthecurrentsolutiontothetaboolistT.Step2.Estimatetheminimaxregret 1. Calculatethecurrentminimaxregret:i=Rkl,wherel=minfl:lPt=1qktg 2. Calculatethebestminimaxregretsofar:0=minfi;0g.Step3.Re-initializationIfPt2Hi=~iqt
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85 andthattheunderliningproblemwasalinearprogrammingproblem.However,inournewheuristicwenolongermaketheseassumptions.Thereforeourimplementationissignicantlydierent.Second,inournewheuristic,ateachiterationweaddthenewlyobtainedsolutiontothetaboolistsothatnosolutionisallowedtobere-visitedinthesubsequentiterations.Thisenhancestheeciencyoftheheuristicsincetheminimaxregretcannotbereducedunlessanewsolutionisgeneratedateachiteration.Third,inLarsenelal'sheuristic,ateachiteration,twoadditionalsetsofconstraintswereaddedwhichrequirethatonceascenariobecomesinactive,itremainsworsethananyofthescenariosintheactivesetinallsubsequentiterations.Suchconstraintsleadtoasmallerfeasibleregion.Incontrast,inournewheuristicwehaveremovedtheseconstraintsand,astheresult,ateachiterationweareactuallyminimizinganupperboundofthecurrentminimaxregretoftheoriginalproblemandthereforepossibllyreducingtheminimaxregretoftheoriginalproblem.Toseethis,recallthatateachiterationinourheuristic,thesub-problemisformulatedinsuchawaythatthe0-minimaxregretofthesub-problem,where0==Pt2Hiqt,isequaltoorveryclosetothe-minimaxregretoftheoriginalproblem,basedonthesolutionobtainedinthepreviousiteration.Whenthesub-problemisoptimallysolved,therankingofaninactivescenario,say,k0,intheKscenariosintermsofregrets,maychangebecauseanewsolutionhasbeengeneratedandthetwosetsofconstraintsinLarsenetal'sheuristichavebeenremoved.Now,iftheregretofscenariok0givenbythenewsolutionisgreaterthanorequaltothe0-minimaxregretofthesub-problemgivenbythenewsolution,thenthe0-minimaxregretofthesub-problemremainsequaltoorveryclosetothe-minimaxregretoftheoriginalproblem.Otherwise,iftheregretofscenariok0givenbythenewsolutionissmallerthanthe0-minimaxregretofthesub-problemgivenbythenewsolution,thenthe0-minimaxregretofthesub-problemisanupperboundtothe-minimax

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86 regretoftheoriginalproblem.Ineithercase,weareminimizinganupperboundofthecurrentminimaxregretoftheoriginalproblemateachiteration.Itmayalsobeworthnotingthat,inthisheuristic,onceascenarioisrenderedinactive,itsregretisnolongertakenintoaccountinthesubsequentsub-problems.Therefore,theactual-meanExcessregretoftheoriginalproblemwillprobablyin-creaseaswefurtherreducetheminimaxregret.Wetesttheeectivenessofourheuristicusingtheoutlier"-instanceinthe10probleminstancesthatwestudiedinthelastsection,i.e.,theinstancewith162scenarios.RecallfromTable 4{3 that,foreachoftheother9instances,minimizingthe-meanExcessregretalreadyledtotheminimizationofthe-minimaxregret.Theinstancewith162scenariosistheonlyoneforwhichminimizingthe-meanExcessregretgavean-minimaxregretthatislargerthantheoptimal-minimaxregret.Ourheuristic,withthestepsizesetto1,terminatedafter9iterationswhenitwasusedtosolvethisinstance.However,theoptimal-minimaxregretwasfoundattheseconditeration.Thetwoiterationstookatotalof391secondstonish,comparingwiththe1317secondsthattheMinimaxImodelspentbeforeitfoundtheoptimalsolution.Ifwecomparethetotaltimes,whichalsoincludethethetimesneededtosolveforthe^Vk'sandmk's,thenittookourheuristic965secondsandtooktheMinimaxImodel64317secondstondtheoptimal-minimaxregret.4.6DiscussionsandConclusionInthischapterwehaveoutlinedanewapproachtostrategicfacilitylocationplanning.Inthisapproach,decisionmakersidentifyanumberoffuturescenariosandestimatethelikelihoodofeachscenariooccurring.Themodelthenndsasolutionwhichminimizestheexpectedregretwithrespecttoanendogenouslyselectedsubsetofworst-casescenarioswhosecollectiveprobabilityofoccurrenceisexactly1-.Ournewmodel,the-reliablemeanExcessregret,hasanumberofadvantagesoverthe-reliablep-medianminimaxmodel.Firstofall,bydenition,-reliablemeanExcess

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87 regretisanupperboundofthecorresponding-reliablep-medianminimaxregret.Therefore,minimizing-reliablemeanExcessregretwillalsoleadtoalow-reliablep-medianminimaxregret.Second,minimizationof-reliablemeanExcessregretavoidsthedangerousincreaseintheworst-caseregret.Third,the-reliablemeanExcessmodeliscomputationallymucheasiertosolve.Alloftheseadvantageshavebeendemonstratedbyournumericalexperiments.Inaddition,wepresentaheuristicfortheminimaxmodelwhichinvolvessolvingaseriesofmeanExcesssub-problems.Ourcomputationalresultsshowthatthisheuristiccanbeusedtoecientlysolvetheminimaxmodel.The-reliablemeanExcessregretmodelhastremendouspotentialsinawiderangeofapplicationsinsupplychainmanagement,capacityplanning,andnancialengineering,etc..Forinstance,itcanbeusedtodesignrobustsupplychainstohedgeagainstuncertaintiesindemand,costs,orotherparameters.Intheareaofhydra-electricpowergeneration,themodelcanbeusedtominimizethecostassociatedwiththecostlystartupsandshutdownsofback-upthermalgenerating-units.Inthenearfuture,weplantoextendthismodeltomulti-stageandmulti-dimensiontosolvemorecomplexreallifeproblems.

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BIOGRAPHICALSKETCHGangChenwasborninthecityofHanZhong,Shaanxiprovince,ChinaonOctober28th,1977.HenishedhighschoolattheFirstMiddleSchoolofMianCountyin1994andstartedhisundergraduatestudyinoneofthemostprestigiousuniversitiesinChina,XianJiaotongUniversity,inthecityofXian,China.ThereheearnedaBachelorofEngineeringdegreeinmechanicalengineeringandautomation,andaBachelorofManagementdegreeinindustrialengineeringwithaminorinbusinessin1999.HebecameagraduatestudentintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridainAugust1999andearnedaMasterofScienceinoperationsresearchinJuly2002.HeiscurrentlyaPh.D.candidateinthedepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridaandplanstonishhisPh.D.inoperationsresearchinDecember2003.Hisareasofresearchincludestochasticmodelingandanalysis,combinatorialandheuristicoptimization,andsimulationmodelingandanalysis.Areasofapplicationincludeschedulingandresourceallocationwithuncertainty,riskmanagement,supplychainandlogisticssystemsanalysis,andproductionandoperationsmanagement.