Citation
Algorithms and Complexity Analysis for Integer Multicommodity Network Flow and Robust Single-Machine Scheduling Problems

Material Information

Title:
Algorithms and Complexity Analysis for Integer Multicommodity Network Flow and Robust Single-Machine Scheduling Problems
Creator:
Tadayon, Bita
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (22 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
SMITH,JONATHAN COLE
Committee Co-Chair:
GEUNES,JOSEPH PATRICK
Committee Members:
GUAN,YONGPEI
RANKA,SANJAY
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Commodities ( jstor )
Hubs ( jstor )
Linear programming ( jstor )
Mathematical robustness ( jstor )
Mathematical sequences ( jstor )
Mink ( jstor )
Objective functions ( jstor )
Robust optimization ( jstor )
Scheduling ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
budgeted-uncertainty -- complexity-analysis -- congestion -- cutting-planes -- dynamic-programming -- integer-programming -- linearization -- multicommodity-flow -- network-optimization -- node-reliabilities -- robust-optimization -- scheduling -- single-machine -- uncertainty
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
We address several optimization problems in which data elements are uncertain. For each problem we apply an appropriate method for uncertainty representation, propose efficient mathematical formulations, and present solution methods. We first consider the problem of sending a set of multiple commodities from their origin to destination nodes via intermediate hubs (multicommodity network flow problem). We assume that each hub node is associated with a reliability function, which depends on the total flow that crosses that hub. The probability that each commodity is successfully relayed from its origin to its destination is given by the product of hub reliabilities on the commodity's path. The problem we consider seeks to find minimum-cost commodity paths such that each commodity reaches its destination with a sufficiently large probability. We first formulate the problem as a nonlinear multicommodity network flow problem and prove that it is strongly NP-hard. We then present two linearization techniques for this formulation, and propose a pair of lower- and upper-bounding formulations, which can then be used within an exact cutting-plane algorithm to solve the problem. Finally, we analyze the computational effectiveness of our proposed strategies on a set of randomly generated instances. As our second line of research we consider the group of single-machine scheduling problems with uncertainty in their parameter values. In particular we focus on robust optimization as an appropriate method of dealing with uncertainty in several scheduling environments. We fist present a comprehensive survey of robust single-machine scheduling problems, classify the literature, and introduce open problems in this area. As the output of this survey we propose the possibility of improving existing robust scheduling models by applying recent developments in robust optimization in this area. Accordingly, as the next step of our research we study a robust single-machine scheduling problem where job processing times are subject to uncertainty with their values belonging to independent continuous intervals. We consider four alternative optimization criteria and apply state-of-the-art robust optimization methods to define three different uncertainty sets. We then explore the problem of determining the worst-case scenario (set of job processing times) corresponding to a given job schedule and objective. We then analyze the problem of scheduling jobs to minimize the worst-case objective, given each combination of objective and uncertainty set. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: SMITH,JONATHAN COLE.
Local:
Co-adviser: GEUNES,JOSEPH PATRICK.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-11-30
Statement of Responsibility:
by Bita Tadayon.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
11/30/2014
Resource Identifier:
907295067 ( OCLC )
Classification:
LD1780 2014 ( lcc )

Downloads

This item has the following downloads:


Full Text

PAGE 1

ALGORITHMSANDCOMPLEXITYANALYSISFORINTEGERMULTICOMMODITYNETWORKFLOWANDROBUSTSINGLE-MACHINESCHEDULINGPROBLEMSByBITATADAYONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

PAGE 2

c2014BitaTadayon 2

PAGE 3

Tomyparents,NaderandNooshin,fortheirendlesslove,support,andencouragement 3

PAGE 4

ACKNOWLEDGMENTS Firstandforemost,Iwouldliketothankmyadvisor,Dr.J.ColeSmith,forhisvaluablesupport,guidance,andencouragement,throughoutthecourseofmystudiesatUniversityofFlorida.Withouthishelpandguidance,thisdissertationwouldnotbepossible.HecontinuallyconveyedaspiritofadventureinregardtoresearchandmadethisjourneyanexcitingpathtowardbecomingtheresearcherIamtoday.Iambeyondgratefulforhavingtheopportunitytoworkwithhimandcouldnothavewishedforabetteradvisor.IwouldalsoliketothankDr.JosephGeunes,Dr.YongpeiGuan,andDr.SanjayRankaforservingonmyPh.D.committeeandfortheirinsightfulcommentsandsuggestions.DuringmyyearsatUniversityofFlorida,Ihadthepleasureofmeetingmanyamazinggraduatestudentcolleagueswhobecamegreatfriendsofmine.Iwouldliketothankmyofcemates,SoheilHemmati,MikePrince,andDeonBurchettforallthehelpandsupporttheyhavegivenmeduringthelastthreeyears.IwouldalsoliketothankmydearfriendsShantihSpanton,CinthiaPerez,AyseNurArslan,ZehraMelisTeksan,SvetlanaMoiseeva,SaeedGhadimi,AndrewRomich,JingMa,VijayPappu,ChrysasVogiatzis,JoseWalteros,JorgeA.Sefair,DmytroKorenkevych,YenTang,ChaoyueZhao,GregoryPastukhov,andOlegShirokikhforcreatingsomanywonderfulmemoriesforme.Lastbutcertainlynotleast,mydeepestgratitudegoestomydearestparents,NaderandNooshin,whotaughtmehowtofollowmydreamsandencouragedmeineverydecisionImadeinmylife.Iwanttothankmydearuncleandaunt,NasserandFautee,foralwaysbeingthereformeandmakingmefeelhomewhileawayfromhome.FinallyIwanttothankmyboyfriend,Roosbeh,forhisloveandsupportandallthelaughterhegavemethroughouttheseyears. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2ALGORITHMSFORANINTEGERMULTICOMMODITYNETWORKFLOWPROBLEMWITHNODERELIABILITYCONSIDERATIONS ........... 14 2.1MotivationandLiteratureReview ....................... 14 2.2ProblemStatementandComplexity ..................... 17 2.2.1NotationandDescription ........................ 18 2.2.2NonlinearFormulationofMCFNR ................... 21 2.2.3ComplexityAnalysis .......................... 24 2.3LinearizationoftheMathematicalModel ................... 29 2.3.1Approach1 ............................... 30 2.3.2Approach2 ............................... 32 2.4Lower-andUpper-BoundingScheme .................... 34 2.5Cutting-PlaneAlgorithm ............................ 36 2.6ComputationalExperiments .......................... 38 2.6.1TestProblemGeneration ........................ 38 2.6.2Results ................................. 40 3ASURVEYOFROBUSTSINGLE-MACHINESCHEDULINGPROBLEM .... 48 3.1Motivation .................................... 48 3.2RobustnessandUncertaintyDenitions ................... 51 3.2.1RobustnessMeasures ......................... 51 3.2.2UncertaintyRepresentation ...................... 54 3.3RobustSingle-MachineSchedulingLiteratureClassication ........ 56 3.4RobustOptimizationinOtherSchedulingProblems ............. 61 4ALGORITHMSANDCOMPLEXITYANALYSISFORROBUSTSINGLE-MACHINESCHEDULINGPROBLEMS ............................. 65 4.1Motivation .................................... 65 4.2ProblemDenitionandNotation ....................... 67 4.3ComplexityResultsandAlgorithms ...................... 69 4.3.1MinimizingTotalCompletionTime ................... 70 5

PAGE 6

4.3.1.1ScenarioGenerationProblem ............... 70 4.3.1.2RobustOptimizationProblem ................ 74 4.3.2MinimizingTotalWeightedCompletionTime ............. 78 4.3.2.1ScenarioGenerationProblem ............... 78 4.3.2.2RobustOptimizationProblem ................ 79 4.3.3MinimizingMaximumLateness/Tardiness .............. 84 4.3.3.1ScenarioGenerationProblem ............... 84 4.3.3.2RobustOptimizationProblem ................ 86 4.3.4MinimizingNumberofLateJobs ................... 87 4.3.4.1ScenarioGenerationProblem ............... 87 4.3.4.2RobustOptimizationProblem ................ 91 5COMPUTATIONALANALYSISFORASPECIALCASEOFROBUSTSINGLE-MACHINESCHEDULINGPROBLEM ....................... 106 5.1ApproximateSolutions ............................. 106 5.1.1UpperBound .............................. 106 5.1.2LowerBounds .............................. 108 5.1.2.1Single-StageApproaches .................. 108 5.1.2.2Multi-StageApproaches ................... 110 5.1.3TestProblemGeneration ........................ 111 5.1.4Results ................................. 112 5.2ExactSolution ................................. 112 6CONCLUSIONSANDFUTURERESEARCH ................... 120 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 130 6

PAGE 7

LISTOFTABLES Table page 2-1Comparisonofcut-generationstrategies1,2,and3 ............... 42 2-2ComparisonofGloMIQOandtheS1cutting-planealgorithm .......... 44 2-3ComparisonofthethreeUi-andRi-valueselectionstrategies .......... 45 2-4Comparisonoflower-andupper-boundingmodels ................ 47 3-1ComplexityresultsobtainedintheliteratureforrobustSMSP .......... 60 4-1ComplexityresultsforrobustSMSPunderbudgeteduncertainty ........ 70 5-1Evaluationoflower-andupper-boundingheuristicalgorithms .......... 113 5-2Evaluationofexactsolutionmethod(2n10) ................. 116 5-3Evaluationofexactsolutionmethod(n=10) ................... 117 7

PAGE 8

LISTOFFIGURES Figure page 2-1Problemillustration .................................. 19 2-2Convertingthereliabilityonarctothereliabilityonnode ............. 21 2-3Anexampletransformationfrom3DM ....................... 27 2-4Solutiontimeofinstancesusingthreecutting-planegenerationstrategies ... 43 4-1Creatingsequence00from0 ............................ 74 4-2Dynamic-programmingSGPnetworkwithPj2JUjcriterionandUS2 ...... 90 4-3SGPnetworkforthePj2JUjcriterionunderUS2 ................. 101 8

PAGE 9

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyALGORITHMSANDCOMPLEXITYANALYSISFORINTEGERMULTICOMMODITYNETWORKFLOWANDROBUSTSINGLE-MACHINESCHEDULINGPROBLEMSByBitaTadayonMay2014Chair:J.ColeSmithMajor:IndustrialandSystemsEngineeringInthisdissertationweaddressseveraloptimizationproblemswithuncertaindataelements.Foreachproblemweapplyanappropriatemethodforuncertaintyrepresentation,proposemathematicalformulations,andpresentsolutionmethods.Werstconsideramulticommoditynetworkowprobleminwhichintermediatenodes(hubs)mayfailtosuccessfullyrelaytheow.Wemodelthisuncertaintybyassociatingeachhubnodewithareliabilityfunction,whichdependsonthetotalowthatcrossesthathub.Theprobabilitythateachcommodityissuccessfullytransferredfromitsorigintoitsdestinationisgivenbytheproductofhubreliabilitiesonthecommodity'spath.Weseektondminimum-costcommoditypathssuchthateachcommodityreachesitsdestinationwithasufcientlylargeprobability.WerstformulatetheproblemasanonlinearmulticommoditynetworkowproblemandprovethatitisstronglyNP-hard.Wethenpresenttwolinearizationtechniques,andproposeapairoflower-andupper-boundingformulations,whichcanthenbeusedwithinanexactcutting-planealgorithmtosolvetheproblem.Finally,weanalyzethecomputationaleffectivenessofourproposedstrategiesonasetofrandomlygeneratedinstances.Asoursecondlineofresearchweconsidersingle-machineschedulingproblemswithuncertaintyintheirparametervalues.Wefocusonrobustoptimizationasanappropriatemethodofdealingwithuncertaintyinseveralschedulingenvironments.We 9

PAGE 10

rstpresentacomprehensivesurveyofrobustsingle-machineschedulingproblems,classifytheliterature,andintroduceopenproblemsinthisarea.Thissurveyproposesthepossibilityofimprovingexistingrobustschedulingmodelsbyapplyingrecentdevelopmentsinrobustoptimizationinthisarea.Accordingly,asthenextstepofourresearchwestudyarobustsingle-machineschedulingproblemwherejobprocessingtimesaresubjecttouncertaintywiththeirvaluesbelongingtoindependentcontinuousintervals.Weconsiderfouralternativeoptimizationcriteriaandapplystate-of-the-artrobustoptimizationmethodstodenethreedifferentuncertaintysets.Then,giveneachcombinationofobjectivefunctionanduncertaintyset,weexploretheproblemofdeterminingtheworst-casescenario(jobprocessing-timevalues)correspondingtoagivenjobschedule,andanalyzetheproblemofschedulingjobstominimizetheworst-caseobjective. 10

PAGE 11

CHAPTER1INTRODUCTIONIntheprocessofdevelopingmathematicalrepresentationsofreal-worldoptimizationproblems,theultimategoalistoobtainthesimplestmodelthatexpressestheproblemasaccuratelyaspossible.Researchersfaceawidevarietyofchallengesinproblemrepresentation,especiallyinthepresenceofuncertainty.Duetovariationsinprocessandenvironmentaldata,uncertaintyiscommoninmanypracticalproblems.Basedonavailableinformationaboutthebehaviorofuncertainparametersandthedesiredperformancelevel,onemustselectanuncertaintyrepresentationmethodandprescribeanapproachforsolvingtheproblem.Inourresearchwestudytheeffectofuncertaintyintwowell-knownintegerprogrammingproblems:Themulticommoditynetworkowproblemandthesingle-machineschedulingproblem.Foreachproblem,weselectanappropriatemethodtorepresentuncertainty,andexplorealternativeformulationstrategiesfortheproblem.Wetheninvestigatetheproblemcomplexityandprescribealgorithmsforsolvingeachproblem.Thisdissertationbeginsbyconsideringtheproblemofsendingasetofmultiplecommoditiesfromtheirorigintodestinationnodesonanetworkinwhichintermediatenodes(hubs)mayfailtocorrectlydeliverows.Inseveralreal-worldapplications,suchascommunicationortransportationnetworks,theriskoffailureinowtransmissionthroughahubnodeincreasesasthetrafcpassingthroughthehubincreases.Accordingly,inChapter 2 wemodeltheuncertaintyinowtransmissionbydeningthereliabilityofeachhubnode(i.e.,theprobabilitythatitcorrectlyrelayseachcommodityowthatpassesthroughit)asanonincreasingfunctionofthetotalowthatcrossesthathub.Additionally,inmanyapplications,suchastransferringindividualmessagesinacommunicationnetworkortransportinggoodsinarailwaynetwork,theowofeachcommoditycannotbesplitamongdifferentpaths.Therefore,inChapter 2 werequiretheowbetweenacommodity'soriginanddestinationtofollowasinglepath 11

PAGE 12

(amulticommodityowproblemwiththisassumptionisreferredtoasanintegermulticommodityowproblem).Theprobabilitythateachcommodityissuccessfullyrelayedfromitsorigintoitsdestinationisthengivenbytheproductofhubreliabilitiesonthecommodity'spath.Theproblemseekstondminimum-costcommoditypathssuchthateachcommodityreachesitsdestinationwithasufcientlylargeprobability.Weformulatethisproblemasanonlinearmixed-integerprogrammingproblemandprovethatitisstronglyNP-hard.Wethenpresenttwolinearizationtechniquesforthisformulation,andproposeapairoflower-andupper-boundingformulations,whichcanthenbeusedwithinanexactcutting-planealgorithmtosolvetheproblem.Thesecondresearchfocusinthisdissertationstudiessingle-machineschedulingproblemsinthepresenceofuncertainty.InChapter 3 ,afterpresentinganoverviewofdifferentapproachestohedgeagainstdatauncertaintyinschedulingproblems,wefocusonrobustoptimizationasanappropriatemethodofdealingwithuncertaintyinseveralschedulingenvironments.Inthischapterwepresentacomprehensivesurveyofrobustoptimizationapplicationsinsolvingsingle-machineschedulingproblems.Weintroducethreedifferentrobustnessmeasuresandfourdifferentuncertaintysetsthathavebeenappliedintherobustschedulingliterature.Wethenclassifytheresultsobtainedforthespeciccaseofrobustsingle-machineschedulingproblemswithdifferentobjectivefunctions,undereachcombinationofuncertaintysetandrobustnessmeasure,andintroduceopenproblemsinthisarea.Ourultimategoalinthischapteristoaddressthegrowinggapbetweentheliteratureofrobustoptimizationandrobustscheduling,andencouragetheclosureofthisgap.Totakeasteptowardsimprovingexistingrobustschedulingmodels,weapplystate-of-the-artrobustoptimizationtechniquesinChapter 4 todeneandsolveasingle-machineschedulingprobleminwhichjobprocessingtimesareuncertain.Weassumethatjobprocessing-timevaluescanberepresentedasindependentcontinuousintervalsandweseektoguaranteeaminimumqualityfortheobjectivevalueinthe 12

PAGE 13

worst-casescenario.Wedenetheproblemasarobustoptimizationproblemandintroducethreealternativeuncertaintysetstocontrolthelevelofconservatismandmoderatetheworst-casescenariointheproblem.Wethenstudytheproblemunderfouralternativeoptimizationcriteria,specically,minimizingtotalcompletiontime,minimizingtotalweightedcompletiontime,minimizingmaximumlateness,andminimizingnumberoflatejobs.InChapter 5 wefocusononespeciccaseoftherobustsingle-machineschedulingproblemintroducedinChapter 4 ,whereweseektominimizenumberoflatejobsintheworst-casescenarioinwhichnomorethanacertainnumberofjobscantakeontheirworst-caseprocessing-timevalues.Weimplementthemixed-integerprogramming(MIP)formulationpresentedforthisprobleminChapter 4 andsolveasetofrandomly-generatedtestproblemsusingCPLEXdefaultMIPsolvers.Wealsointroduceanupperboundandalternativelowerboundsfortheproblemandtesttheirperformancebyapplyingthemonasetofrandomly-generatedinstances. 13

PAGE 14

CHAPTER2ALGORITHMSFORANINTEGERMULTICOMMODITYNETWORKFLOWPROBLEMWITHNODERELIABILITYCONSIDERATIONS 2.1MotivationandLiteratureReviewThemulticommodityow(MCF)problemseekstosatisfydemandsamongasetofcommoditiesatminimumcost,acrossadirectednetworkhavingcapacitatedarcs.Thecommoditiesareassociatedwithanoriginanddestinationnode,andwithademandquantity.Additionally,manyapplicationsrequireowbetweenacommodity'soriginanddestinationtofollowasinglepath.Givenowcostsforeacharconthenetwork,theproblemofsimultaneouslyshippingallcommoditydemandsonthenetworkataminimumcostwhereeachcommodity'sowfollowsasinglepathisreferredtoastheintegermulticommodityowproblem[ 10 11 ].ThischapterexaminesintegerMCFsonnetworks,inwhichintermediatenodesonanorigin-destinationpathmayfailtocorrectlydeliverows.Forthisproblem,weassumethatwhenanodefailstoproperlyrelayacommodityow,theowitselfispropagatedthroughthenetworkasdesired,butthecontentsoftheowhavesomehowbeendamaged.Thismaybethecaseinshippingfragilecontentsorinrelayinginformationinacommunicationnetwork.Thereliabilityofeachnode(i.e.,theprobabilitythatitcorrectlyrelayseachcommodityowthatpassesthroughit)ismodeledasanonincreasingfunctionoftheloadassignedtoit,whereloadisgivenbythetotalamountofowthatcrossesthenode.Givenanorigin-destinationrouteforeachcommodity,thereexistsaBooleanrandomvariablecorrespondingtoeachcommodity/nodepair,whichspecieswhetherornotthenodewillsuccessfullyrelaythecommodityow.(Foranynodethatdoesnotserveasanintermediatenodeonthecommodity'spath,therandomvariableisirrelevant.)Weassumethattheserandomvariablesaremutuallyindependent,andsotheprobabilityofsuccessfullytransmittingowonapathiscalculatedastheproductofnodereliabilitieslyingonthepath.(Forinstance,iftwocommoditiesbothsendowthroughacommonintermediatenode,thentheprobability 14

PAGE 15

thatthisnodesuccessfullyrelaysowfromtherstcommodityisindependentoftheprobabilitythatitsuccessfullyrelaysowfromthesecondcommodity.Also,iftwonodesbothlieonsomecommodity'spath,thentheprobabilitythattherstnodesuccessfullyrelaysthecommodity'sowisindependentoftheprobabilitythatthesecondnodesuccessfullyrelaystheow.)TheproblemweexamineistheintegerMCF,subjecttotherestrictionthateachcommoditymustbesuccessfullydeliveredwithasufcientlylarge(specied)probability.Wecallthisthemulticommodityowproblemwithnodereliabilityconstraints(MCFNR).Ahujaetal.[ 1 ]provideageneraldiscussionofMCFmodelsandalgorithms.Seealso[ 5 47 ]forcomprehensivesurveysonclassicalMCFresearch,andtherecentsurveybyOuorouetal.[ 67 ]whichfocusesonalgorithmsforsolvingnonlinearconvexMCFproblems.Arc-basedandpath-basedformulationsaretwocommonlyusedstrategiesformodelingMCFs.Arc-basedformulationsincludedecisionvariablesthatdeterminehowmuchowforeachcommodityisshippedoneacharc,andresultinapolynomialnumberofconstraintsandvariables.Bycontrast,path-basedformulationsrequirefewerconstraints,butanexponentialnumberofvariables,oneforeverypathconnectingacommodityoriginanddestination.TosolvetheintegerMCFusingapath-basedformulation,Barnhartetal.[ 10 11 ]presentacolumn-generationmodelforsolvinglinearprogrammingrelaxationsfortheintegerMCF,andprescribeabranch-and-price-and-cutapproachforsolvingtheproblem.Brunettaetal.[ 20 ]studyseveralclassesofvalidinequalitiesobtainedfromalternativeformulationsoftheproblem,andproposeabranch-and-cutalgorithmforsolvingtheproblem.TheyalsostudythepolyhedralstructureoftheMCFpolytopeforthespecialcaseinwhichallowsareinteger,allcommoditieshaveunitdemand,andallarcshaveunitcapacities.Similartoourproblem,manyMCFapplicationsinvolvesideconstraintsthatmustbesatisedinadditiontothestandardMCFconstraints.HolmbergandYuan[ 37 ]examine 15

PAGE 16

time-delayandreliabilitysideconstraintsforacommunicationnetwork,inwhichthereliabilityofeachpathiscalculatedbasedonthearcfailureratesthatlieonthatpath.Distinctfromourproblem,thearcfailureratesdonotdependontheamountoftrafccrossingthearc;moreover,theirproblempermitstheuseofmultiplepathstosendowsbetweeneachorigin-destinationpairandrequiresallpathsto(independently)satisfytheminimumreliabilityrequirement.Tosolvethisproblem,theauthorsprescribeacolumn-generationalgorithmforsolvingapath-basedmodel.Networkowreliabilityproblemshavereceivedmuchattentioninthenetworkowoptimizationliterature,particularlywhenarcornodereliabilitiesarenotfunctionsoftheowtheytransmit,asisthecaseinourstudy.Inthesestudies,reliabilityisoftendenedastheprobabilityofsatisfyingallcommoditydemands,givenuncertainarccapacities.TheproblemofevaluatingnetworkreliabilityinthiscontextisNP-hard[ 21 ].TwoprimaryapproachesusedtoevaluatenetworkreliabilityintheliteratureemploytheconceptsofMinimalPaths(MPs)andMinimalCuts(MCs).FoundationalapproachesusingMPscanbefoundin[ 6 38 55 57 60 68 82 85 ].Yehgeneralizesthenetworkreliabilityproblemtoconsidercaseswithunreliablenodes[ 86 ]andwithabudgetconstraint[ 88 ].Lin[ 58 59 ]extendsthereliabilityproblemtoaccommodatemulticommoditycases.TheMCapproachhasbeenusedin[ 42 74 86 87 ],andiscloselyrelatedtoMPalgorithmsbythemax-owmin-cuttheorem[ 30 ].Anotherlineofresearchregardsnetworkdesignproblemsinwhichanetwork'stopologyandarccapacitiesaredeterminedinawaythatdemandscanbesatisedviaacost-efcientroutingscheme[ 9 19 22 32 34 42 ].Forinstance,Gavishetal.[ 33 ]addresstheproblemofdesigningareliablenetworkwithminimumpossiblecost.Theyassumethatarcsandnodesinthenetworkhavespecicfailureratesandthenetworkstateuctuatesbasedonthestatusofthearcsandnodes.Thegoalistodesignanetworksuchthatthecapacityassignedtoeacharcislargeenoughtosatisfythedemandsinallpossiblenetworkstates. 16

PAGE 17

Incontrasttothepapersdiscussedabove,MCFNRconsidersthesituationinwhichnodereliabilitiesdependontheamountofowtheytransmit.Tothebestofourknowledge,theeffectofcongestiononthereliabilityofMCFproblemsasdenedinourstudyhasnotbeenconsideredintheliterature.However,intransportationapplicationstheeffectofcongestionontraveltime(whichisanalogoustoourreliabilityanalysis)hasbeenstudiedcomprehensively.Jahnetal.[ 40 ]considertheproblemofroutingowinatransportationnetworkinwhichallvehicleshavingacommonoriginanddestinationareassumedtobeonecommodity.Intheirmodel,traveltimeoneacharcisassumedtobeadifferentiablenondecreasingfunctionoftherateoftrafconthatarc.Accordingly,trafcloadoneacharcaffectstheminimumtotaltraveltime.Moreworkintheareaoftrafccongestionintransportationnetworkscanbefoundin[ 79 83 89 ].Theremainderofthischapterisorganizedasfollows.InSection 2.2 ,wedevelopamathematicalprogrammingformulationforMCFNRandanalyzethecomplexityoftheproblem.InSection 2.3 ,weintroducetwoapproachestolinearizethemathematicalformulationproposedinSection 2.2 .Wethenpresentlower-andupper-boundingschemesfortheprobleminSection 2.4 ,andapplythoseboundswithinacutting-planealgorithmtosolvetheprobleminSection 2.5 .InSection 2.6 ,weemployourcutting-planealgorithmtosolveasetofrandomlygeneratedinstancesofdifferentsizes.Wealsocomparetheobjectivefunctionvaluesobtainedfromsolvingthelower-andupper-boundingmodelsonourtestproblems. 2.2ProblemStatementandComplexityWeprovideaformaldescriptionofMCFNRinSection 2.2.1 ,andpresentamathematicalprogrammingformulationinSection 2.2.2 .WethenshowthatMCFNRisstronglyNP-hardinSection 2.2.3 ,evenunderdataassumptionsthatsimplifytheproblem. 17

PAGE 18

2.2.1NotationandDescriptionWebeginbyprovidingnotationandassumptionsusedforMCFNR.DeneKtobethesetofcommodities.Foreachcommodityk2K,deneO(k)asitsoriginnode,D(k)asitsdestinationnode,anddkasitsdemandquantity.Commodityk2Kmustbesuccessfullytransmittedtoitsdestinationwithprobabilityatleastk,whichisaparameterboundedby0
PAGE 19

commoditysuccessfullyis0.56,whichsatisestheminimumprobabilityrequirement.However,if1=2=0.6,thentheprevioussolutionisnolongerfeasible.Forthisnewinstance,commodity1usesintermediatenodes1and4,butcommodity2usesintermediatenodes3and4atoptimality.Theloadsofnodes1,3,and4arethen3,2,and5,respectively,yieldingcorrespondingnodereliabilitiesof0.91,0.96,and0.75.Thereliabilityofthecommodity1pathis0.68,andthereliabilityofthecommodity2pathis0.72.Bothprobabilitiessatisfytheminimumrequirement,andthetotalcostis17. Figure2-1. Problemillustration Remark2.1. Withoutlossofgenerality,weassumethattheterminalsareperfectlyreliable,whereasthereliabilityofeachhubisanonincreasingfunctionoftheamountofowpassingthroughthehub.Observethatthecaseofunreliableterminalscanbehandledbyadjustingthek-valuesbasedonthetotal(xed)oworiginatingfromorarrivingtoeachterminal.Thisgeneralizationisvalidbecausetheloadofeachterminalisconstantinanyfeasiblesolution,andisdeterminedbythetotaldemandofcommoditiesthatoriginatefromorarriveatthatterminal.Forinstance,supposeinExample 2.1 thatthereliabilityofallnodes(hubsandterminals)aredenedusingthesamereliabilityfunction)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[((loadofnodei)2=100.Inanyfeasiblesolution,eachcommoditypassesthroughitsoriginanddestinationterminals,whosereliabilitieswouldbe0.91forO(1)andD(1)and0.96forO(2)andD(2).Nowsupposethatwedesire 19

PAGE 20

eachcommoditytobesuccessfullytransmittedtoitsdestinationwithprobability0.5.Wecantreattheterminalsasbeingperfectlyreliable,andscalethereliabilitythresholdsas1=0.5=(0.912)and2=0.5=(0.962).Inthismanner,allterminalscanequivalentlybetreatedashavingperfectreliability.ThearcsetinMCFNRispartitionedintoarcsAf(i,j)ji,j2HghavingbothincidentnodesinH,andarcs~Af(i,p)[(p,i)ji2H,p2Tgthatincludealleligibleterminal-hubassignments.Notethatassignmentsareundirected,i.e.,(i,p)2~Aifandonlyif(p,i)2~A,8i2H,p2T.Recallthatthecostoftransferringeachunitofcommodityk2Konarc(i,j)2A[~AisgivenbyCkij.Theproblemisthereforetondanassignmentofterminalstohubs,andasetofpathstotransmitallcommodities,ataminimumtotalcost,whileguaranteeingthateachcommodityk2Kissuccessfullytransmittedtoitsdestinationwithatleastaprobabilityofk. Remark2.2. TheMCFNRassumesthatnodesmayfailtocorrectlyrelayow,butarcsareperfectlyreliable.Thisassumptionisreasonable,forinstance,whenexamininganetworkofpeoplecommunicatingelectronicallyinanorganization.Nodesrepresentthepeople,andarcsdeterminepossiblecommunicationsbetweenthosepeople.Theprobabilityofmistakesmadebyeachpersonwhoisresponsiblefordirectingmessagesthroughthesystemincreasesastheworkloadonthatpersonincreases.Comparedtohumanerrorrate,electroniccommunication(onthearcs)isvirtuallyawless.Inotherapplications,however,itisnotreasonabletoassumethatthearcsareperfectlyreliable.Ourmodelcaneasilybeadjustedtocoverthoseinstancesaswell,byreplacingeachunreliablearc(i,j)withtwoperfectlyreliablearcs(i,ij)and(ij,j),whereijisadummynodewhosereliabilityisdenedusingtheinitialarc'sreliabilityfunction.Afterthismodication,theowpassingthedummynodeisexactlythesameastheinitialarcow,whichyieldsanequivalentproblemhavingperfectlyreliablearcs.ThismodicationisillustratedinFigure 2-2 20

PAGE 21

Figure2-2. Convertingthereliabilityonarctothereliabilityonnode Generally,theassumptionofhavingreliabilityfunctionsonnodesresultsinamodelwhichismorecomprehensivecomparedtotheonewitharcreliabilityfunctions.Thisistruesincethediscussedmodicationcanbeappliedtoincorporatearcswithreliabilityfunctionsintothemodel,evenwhenthenetworkisundirected.Ontheotherhand,ifarcreliabilityfunctionsareusedtobuildthemodelasimilarmodicationcanbeusedtoincorporatethenodeswithreliabilityfunctionsintothemodelbysplittingthenodeintotwoandputtingadummyarcwiththesamereliabilityfunctioninbetweenfordirectednetworks.However,themodicationtechniquefailswhenthenetworkisundirected. 2.2.2NonlinearFormulationofMCFNRWenowformulateMCFNRasamixed-integernonlinearprogramming(MINLP)problem.Webeginbyaddressingthehubreliabilityfunctions,whicharenonincreasingfunctionsofthehub'sload.Weallowthehubreliabilityfunctiontotakeonanyform(assumingthatreliabilityisanonincreasingfunctionofload).However,onerecurringexampleinthischapterexaminesthecaseinwhichthereliabilityfunctionofhubi2Hisgivenby1)]TJ /F8 11.955 Tf 12 9.69 Td[((loadofnodei)2=m2i,whereparametermiisthemaximumloadthathubicantakebeforeitsreliabilitydropstozero.Thisparticularfunctioncapturesthecaseinwhichreliabilitydegradesatanincreasingrateasitsloadincreases(i.e.,asaconcavefunctionofload),upuntilthepointthatthehubreliabilitydropstozero.(Notethatthisfunctionnecessitatesadditionalconstraintsthatrestricttheloadofnodeitobenomorethanmi.)ToformulateMCFNR,werstdenethefollowingsetsthatappearinourformulation.Foreachi2H,denetheforwardstar,FS(i)=fj2H:(i,j)2Agandthereverse 21

PAGE 22

star,RS(i)=fh2H:(h,i)2Ag.Similarly,letfFS(p)=fi2H:(p,i)2~Agrepresentthesetofhubsthatcanbeassignedtoterminalp2T,andalsodenefRS(i)=fp2T:(p,i)2~Agasthesetofterminalsthatcanbeassignedtohubi2H.Foreveryp2T,dene(p)=fk2K:O(k)=porD(k)=pgasthesetofcommoditieswhoseoriginordestinationterminalisp.Ourformulationutilizesthefollowingdecisionvariables.Ypi=8>><>>:1,ifhubiisassignedtoterminalp0,otherwise,8(p,i)2~AXkij=8><>:1,ifarc(i,j)isusedtotransfercommodityk0,otherwise,8(i,j)2A,k2KUi=totalamountofowpassingthroughhubi,8i2HRi=reliabilityofhubi,8i2HWeproposethefollowingMINLPformulationforMCFNR,giventhecapacityfunctionsdescribedabove,wherewedeneYpi0,8p2T,i2H:(p,i)=2~A.MinX(i,j)2AXk2KCkijdkXkij+X(p,i)2~AXk2(p)CkpidkYpi (2)subjectto:Xi2fFS(p)Ypi=1,8p2T (2)Xp2fRS(i)Ypii,8i2H (2)Xj2FS(i)Xkij)]TJ /F8 11.955 Tf 18.87 11.35 Td[(Xh2RS(i)Xkhi=YO(k),i)]TJ /F3 11.955 Tf 11.96 0 Td[(YD(k),i,8k2K,i2H (2)Ui=Xk2Kdk0@Xj2FS(i)Xkij+YD(k),i1A,8i2H (2)Ri=1)]TJ /F3 11.955 Tf 13.54 8.09 Td[(U2i m2i,8i2H (2) 22

PAGE 23

Uimi,8i2H (2)0@Xi2fFS(O(k))RiYO(k),i1AY(i,j)2ARXkijjk,8k2K (2)Xkij2f0,1g,8(i,j)2A,k2K (2)Ypi2f0,1g,8(p,i)2~A (2)Theobjectivefunction( 2 )minimizesthecostoftransferringallcommodities'owsamongthehubs(therstterm)andbetweeneachassignedterminal-hubpair(thesecondterm).Constraints( 2 )ensurethatexactlyonehubisassignedtoeachterminalandconstraints( 2 )imposeanupperbound,i,onthenumberofterminalsthatcanbeassignedtohubi2H.Theow-balanceconstraintscorrespondingtoeachcommodityk2Kateachhubi2Harestatedbyconstraints( 2 ).Foreveryhubi2H,constraint( 2 )denesthehubload,Ui,asthesumofthedemandvaluesoverallcommoditiesthatpassthroughhubi.Thereliabilityofeachhubi2Histhendenedinconstraint( 2 ).Constraint( 2 )guaranteesthatthehubloadisnomorethanmiforeachhubi2H.Constraints( 2 )stateareliabilitythresholdinequalityforeverycommodity.Intheseconstraints,foreachcommodityk2K,thestatementinparenthesesrepresentsthereliabilityofthersthubvisitedinthepathofthatcommodity.Thisvalueisthenmultipliedbyreliabilityofallotherhubsonthepathforktocalculatetheprobabilityofsuccessfullytransferringcommodityktoitsdestination.Finally,constraints( 2 )and( 2 )representlogicalrestrictionsonX-andY-variables,respectively.NotethatthenonnegativityoftheU-andR-variablesisimpliedby( 2 ),( 2 ),and( 2 ),alongwiththenonnegativityofthed-values,X-variables,andY-variables.WenowemployasimilarapproachusedbyAndreasandSmith[ 4 ]toreformulateconstraints( 2 ).Deneskitobetheprobabilitythatcommodityksuccessfullyreacheshubifromitsoriginterminal,fork2K,giventhatitspathvisitshubi.Constraints( 2 )canbesubstituted 23

PAGE 24

bythefollowinginequalities:skjRjski+)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Xkij,8k2K,(i,j)2A (2)ski1)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(Ri)YO(k),i,8k2Ki2fFS(O(k)) (2)skik,8k2K,i2H. (2)Constraints( 2 )and( 2 )canbetightenedbynotingthatifhubiliesonthe(unique)pathforcommodityk,thenatleastdkunitsofowpassesthroughthisnode,andsoskimaynotexceedk=1)]TJ /F3 11.955 Tf 10.7 0 Td[(d2k=m2i.Moreover,ifweknowthatiisnotthersthubvisitedinthepathforcommodityk,thenwealsoknowthatthemessagehasalreadypassedatleastoneotherhub.Therefore,forthesehubstheupperboundonskiwillbe(k)2.Asaresult,constraints( 2 )and( 2 )canberevisedas:skjRjski+(k)21)]TJ /F3 11.955 Tf 11.96 0 Td[(Xkij)]TJ /F8 11.955 Tf 11.95 16.86 Td[(1)]TJ /F5 11.955 Tf 16.43 8.09 Td[(1 kYO(k),j,8k2K,(i,j)2A (2)ski(k)2)]TJ /F5 11.955 Tf 11.95 0 Td[(((k)2)]TJ /F3 11.955 Tf 11.96 0 Td[(Ri)YO(k),i,8k2K,i2fFS(O(k)). (2)NotethatinourMINLPformulation,variablesUiandRi,aswellasconstraints( 2 )and( 2 )areincludedinthemodelforthesakeofsimplicityandcanberemovedbysubstitutingthecorrespondingvaluesintheremainingconstraints.OurMINLPformulationthuscontainsO(j~Aj+jHjjKj+jAjjKj)variablesandO(jTj+jHjjKj+jAjjKj)constraints. 2.2.3ComplexityAnalysisBecausetheintegerMCFisstronglyNP-hard,itisnotsurprisingthatMCFNRisalsostronglyNP-hard.Infact,weshowinthissectionthatMCFNRremainsstronglyNP-hardevenforthesimpliedcaseinwhichallcommoditieshaveunitdemands,allhubsaresubjecttoacommonafnereliabilityfunction,theminimumrequiredprobabilityofsuccessfullytransmittingeachcommodityisthesamevalueforallcommodities,andthereisnocostfortransferringowinthenetwork. 24

PAGE 25

Theorem2.1. ThespecialcaseofMCFNRinwhichdk=1andk=,8k2K;Ckij=0,8(i,j)2A[~A,k2K;andRi=f(Ui),8i2H,isNP-hard,wherefisanafnefunctionand0<<1isaconstantparameter. Proof. Webeginbydeningthedecisionproblem,MCFNRD,correspondingtoMCFNRasfollows:DoesthereexistafeasibleMCFNRsolution,regardlessofcost?MCFNRDbelongstoNPbecausewecanverifyifaroutingschemeisfeasibleinpolynomialtimebyenumeratingthesetofcommoditiespassingthrougheachhub,calculatingtheloadofeachhub,andthendeterminingtheprobabilityofsuccessfullydeliveringeachcommodity.ThisprocedurerequiresO(jHjjKj)steps,andsoMCFNRDbelongstoNP.Next,weshowthatMCFNRDisNP-complete.Ourproofemploysapolynomialtransformationfromthe3-DIMENSIONALMATCHING(3DM)problem(knowntobestronglyNP-complete[ 31 ])toanequivalentinstanceofMCFNRD. 3DM:LetW,X,andYbenite,disjointsetswithjWj=jXj=jYj=,anddeneZWXYasasetoftriples(w,x,y)suchthatw2W,x2X,andy2Y.(Weassume2.)A3DMsolutionconsistsofasubsetMZsuchthatjMj=andforanytwodistincttriples(w1,x1,y1)2Mand(w2,x2,y2)2M,wehavew16=w2,x16=x2,andy16=y2.Totransformanarbitrary3DMinstancetoanequivalentMCFNRDinstance,wecreateanetworkconsistingofgroupsofregularhubs,plusanadditionalgroupofdummyhubs.Theithregulargroupincludes2regularhubs:hubsfx1i,...,xigcorrespondingtoxi2Xandhubsfy1i,...,yigcorrespondingtoyi2Y.Therearealso2dummyhubs,givenbyfx1d,...,xdgandfy1d,...,ydg.Wenextintroduceasetof4commodities,eachhavingaunitdemand,whichforeaseinexpositionweclassifyintothreecategories:primary(commodities),secondary(2commodities),anddummy(commodities).SetTconsistsof2+6terminals.Primarycommodityi2f1,...,ghasanoriginterminalwiandadestinationterminalti.Forthesecondarycommodities,commoditieshaveacommonoriginOxand 25

PAGE 26

destinationDx,andtheothercommoditieshaveacommonoriginOy,anddestinationDy.Finally,alldummycommoditieshaveacommonoriginwdanddestinationtd.Next,weconstructtheset~AinthetransformedMCFNRDinstancebyspecifyingfFS(p)foreveryp2T.First,fFS(wi),8i=1,...,,isthesetofallxjisuchthat(wi,xj,yk)2Zforsomeyk2Y.Also,fFS(ti)=fy1i,...,yig,8i=1,...,.Forthesecondaryterminals,wehave:fFS(Ox)=fx11,...,x1g,fFS(Oy)=fy11,...,y1g,fFS(Dx)=fx1d,...,xdg,andfFS(Dy)=fy1d,...,ydg.Finally,fFS(wd)=fx1d,...,xdg,andfFS(td)=fy1d,...,ydg.WenowlistallarcsthatbelongtoA. foreachi2f1,...,)]TJ /F5 11.955 Tf 11.96 0 Td[(1ganarcexistsfromxjitoxji+1andfromyjitoyji+1, foreach(wi,xj,yk)2M,thereexistsanarcfromxjitoyki, foreachi2f1,...,g,arcs(xi,xid),(xid,yid),and(yi,yid)exist.Notethatonlypaths,eachoftheformOxxi1xixidDx,8i2f1,...,g,connectOxandDx;onlypaths,Oyyi1yiyidDy,8i2f1,...,g,connectOyandDy;andonlypaths,wdxidyidtd,8i2f1,...,g,connectthedummyterminals.WechoosethehubreliabilityfunctionandthresholdreliabilityvaluessothateachpathinafeasibleMCFNRDsolutionvisitsatmosttwohubshavingaloadoftwo,withallothernodesvisitedbythepathhavingaloadofone.WeusealinearhubreliabilityfunctiongivenbyRi=1)]TJ /F6 11.955 Tf 12.21 0 Td[("(Ui)]TJ /F5 11.955 Tf 12.2 0 Td[(1),8i2H,forsome0<"1=2.Havingaminimumrequiredprobabilityofi=(1)]TJ /F6 11.955 Tf 12.01 0 Td[(")2,8i2H,willthenensurethatthespeciedconditionsaresatised:Because1)]TJ /F5 11.955 Tf 12.52 0 Td[(2"<(1)]TJ /F6 11.955 Tf 12.51 0 Td[(")2,nohubcanhavealoadexceedingtwoina 26

PAGE 27

feasiblesolution,andbecauseRi=(1)]TJ /F6 11.955 Tf 12.16 0 Td[(")whenUi=2,eachpathvisitsnomorethantwohubshavingaloadoftwo.Figure 2-3 illustratesthetransformationfroma3DMinstancewith=3andZ=f(w1,x2,y3),(w1,x2,y2),(w1,x1,y3),(w2,x1,y3),(w2,x3,y2),(w3,x3,y3),(w3,x1,y1)g.Squaresrepresenttheprimaryanddummyterminals,roundedsquaresdepictthesecondaryterminals,andcirclessymbolizethehubs.Primaryanddummyorigin-destinationpairsaredrawnvertically,andthesecondarypairsaredepictedhorizontally.The3DMsolutionisgivenbyM=f(w1,x2,y3),(w2,x3,y2),(w3,x1,y1)g.Eachofthe2uniquepathsbetweenthesecondaryterminals(connectingOxandDx,andconnectingOyandDy)isusedtotransferonesecondarycommodityow,andeachdummycommodityistransferredviaoneofthepathsfromwdtotd.TheselectedprimarypathsaremarkedbydashedarcsinFigure 2-3 Figure2-3. Anexampletransformationfrom3DM 27

PAGE 28

ToprovethatthetransformedMCFNRDinstanceisequivalenttothe3DMinstance,weshowthatthereexistsa3DMsolutionifandonlyifthereexistsanMCFNRDsolution.First,supposethatasolutionMtothe3DMinstanceexists.Afeasibleroutingschemecanbeobtainedbysendingtheowofeachprimarycommodityithroughthepathwixjiykiticorrespondingtothetriple(wi,xj,yk)2M.Eachdummycommodityi2f1,...,gisroutedonthepathwdxidyidtd.Foreachj2f1,...,g,onesecondarycommoditycanberoutedalongthepathOxxj1xjxjdDxandanotheralongthepathOyyj1yjyjdDy.ToseethatthisroutingschemeisfeasiblefortheMCFNRDinstance,rstnotethatalldummyhubshavealoadoftwo(becauseonesecondaryandonedummycommodityuseeachdummynode).Eachregularhubhasaloadofatleastonebecauseofthesecondarycommodityows,andforeach(wi,xj,yk)2M,regularhubsxjiandykihavealoadoftwobecauseoftheprimarycommodityowonthepathwixjiykiti.Wenowshowthatatmosttwohubswithaloadoftwoexistoneachpath,andsothethresholdreliabilityconstraintsaresatised.Theonlypathsthatvisitmorethantwohubsaretheonescorrespondingtothesecondarycommodities;thesepathsincludetwohubswithaloadoftwo:Eitherxjiandxjd,oryjiandyjd,forsome1i.Ifbothxjiandxjk(yjiandyjk)havealoadoftwofor1i
PAGE 29

commodityowthroughxjiandyjiisone,8iandjinf1,...,g,andthroughxjdandyjdistwo,8j2f1,...,g.Becauseeachsecondarycommodityusesadummynode(withaloadoftwo),onlyoneregularhubonthesecondarypathcanrelayaprimarycommodity.Thatis,atmostonenodeinfxi1,...,xig,andonenodeinfyi1,...,yig,canbeusedinaprimarycommoditypath.Togeneratea3DMsolution,wethusletMconsistofall(wi,xj,yk)triplesthatcorrespondtoaprimarypathforcommodityioftheformwixjiykiti.Asaresult,eachelementofw,x,andyappearsexactlyinonetriplecontainedinM,andsoMprovidesafeasible3DMsolution.Finally,notethatthetransformednetworkhasapolynomialnumberofnodesandarcs,andapolynomialnumberofcommoditiesneedtobetransferredviathenetwork.Moreover,thenumericaldataforthetransformedproblemislimitedtotheonlytworelevantvaluesofRithatneedtobestored(1forahubhavingaloadofone,and(1)]TJ /F6 11.955 Tf 11.96 0 Td[(")forthosehavingaloadoftwo)alongwith=(1)]TJ /F6 11.955 Tf 12.1 0 Td[(")2.Bysetting,e.g.,"=1=2,alldatacanberepresentedusingaconstantnumberofbits.ThetransformationthereforeshowsthatMCFNRDisstronglyNP-complete. Beforeconcludingthissection,wenotethatMCFNRDremainsstronglyNP-completespecicallyforthehubreliabilityfunctionsgivenin( 2 ).Wecanusesuchafunctionintheproofbychoosingmi=8,8i2H,whichistwicethetotaldemandvalue.Inordertosatisfytheconditions,theminimumprobabilityinthiscasewouldbe=(1)]TJ /F5 11.955 Tf 12.69 0 Td[(22=(8)2)2(1)]TJ /F5 11.955 Tf 12.69 0 Td[(1=(8)2)()]TJ /F10 7.97 Tf 6.59 0 Td[(1),becausethelengthofthelongestpathinthetransformedinstanceis+1,andatmosttwohubsonthispathmaycarrytwounitsofow.WeomitfurtherdetailsofhowthischoiceofforcesMCFNRDsolutionstocorrespondto3DMinstancesforbrevity. 2.3LinearizationoftheMathematicalModelInthemathematicalmodelpresentedintheprevioussection,constraints( 2 )and( 2 )includebilineartermsthatmaketheproblemnon-convex.Therefore,to 29

PAGE 30

assistusinndingaglobaloptimalsolution,wereformulatetheproblemusinganequivalentmixed-integerlinearprogram(MILP).ThedifcultyinlinearizingthismodelstemsfromthefactthatneitherRjnorskiinthetermRjskiisbinary-valued,whichthenprohibitstheuseofstandardlinearizationtechniquesforquadraticprograms.However,basedontheassumptionthateachcommodityisroutedonasinglepath,thereexistnitelymanypossiblevaluesforUi,andbyextensionforRi.Usingthisfactandthelinearizationmethodgivenin[ 64 ],weprovidetwodifferentapproachesforobtainingMILPformulationsinSections 2.3.1 and 2.3.2 .Furthermore,therstapproachthatweprovidespecicallyassumesthereliabilityfunctiongivenin( 2 ),whilethesecondapproachisvalidforgeneral(non-increasing)reliabilityfunctions. 2.3.1Approach1DenebinaryvariablesQki,8i2H,k2K,equaltooneifhubiisusedtotransfercommoditykandzerootherwise.Thesevariablescanbedenedbythefollowingequalities:Qki=Xj2FS(i)Xkij+YD(k),i,8i2H,k2K. (2)Also,notingthatUi=Pk2KdkQki,constraints( 2 )simplifyto:Ri=1)]TJ /F5 11.955 Tf 17.46 8.09 Td[(1 m2i24Xk2K(dk)2Qki+2X1k1
PAGE 31

wk1k2iQk1i+Qk2i)]TJ /F5 11.955 Tf 11.95 0 Td[(1,8i2H,1k1
PAGE 32

atmostoneofwhichiscontinuous.Therefore,theycanbelinearizedusingthesametechniquediscussedbefore.LinearizationusingthisapproachrequiresatotalofO(jAjjKj3)variablesandthesameorderofconstraints,assumingthatjAjjHj.Asaresult,thislinearizationmethod(potentially)increasesthenumberofvariablesbyafactorofO(jKj2)overthenonlinearformulation. 2.3.2Approach2Wenowrevisittheoriginalformulationgivenby( 2 )( 2 ).BecauseUicantakeonlyanitenumberofvalues,8i2H,wecandetermineallpotentialvaluesforUiviathefollowingtwo-stageapproach. Therststagedetermineswhichcommoditiescansendowsthroughhubi.Thisstagerstexecutesadepth-rstsearchstartingathubi,usingarcsinthereversedirection,todeterminewhichoriginnodescansendowtonodei.Similarly,wethenexecutedepth-rstsearchstartingathubi,usingarcsintheforwarddirection,todeterminewhichdestinationnodescanbereachedfromhubi.ThesetofcommoditieskwhoseoriginscanreachiandwhosedestinationsarereachablefromiisdenotedbyK0.ThecomplexityofthisstepisO(jAj+j~Aj+jKj). Inthesecondstage,weemploydynamicprogrammingtodetermineallpossiblesumsoftheformPk2K00dk,foreachK00K0,whichinturnyieldallpotentialvaluesforUi.Toachievethis,indexthecommoditiesinK0as1,...,jK0j.Forj=0,...,jK0jands=0,...,minfmi,PjK0jk=1dkg,denejsasabinaryvariablethatequals1ifandonlyifthereexistsasubsetoffd1,...,djgsuchthatthesumofelementsinequalss.Initially,setall-variablesto0,except00=1.(Also,denejs0ifs<0,foreveryj.)Then,foreachj=1,...,jK0j,thealgorithmconsiderseveryvalues=0,...,minfmi,PjK0jk=1dkginorder,andsetsjs=maxfj)]TJ /F10 7.97 Tf 6.58 0 Td[(1,s,j)]TJ /F10 7.97 Tf 6.58 0 Td[(1,s)]TJ /F4 7.97 Tf 6.59 0 Td[(djg.Attheendofthisprocess,allpotentialUi-valuescoincidewiththosesforwhichjK0js=1.ThecomplexityofthisstepisO(jKjminfmi,PjKjk=1dkg).IfjKjjAj+j~Aj,thenthesecondstepdominatesthecomplexity.Else,ifjAj+j~Aj>jKj,theneitherstepcoulddominate;hence,theoverallalgorithmcomplexityisO(jAj+j~Aj+jKjminfmi,Pk2Kdkg). 32

PAGE 33

AssumingthatGi+1possibleUi-valuesaregenerated,thesetofallpossiblevaluesforUicanberepresentedasfV0,...,VGig.Accordingly,foreachpossiblevalueofUi,thecorrespondingvalueforRicanbecalculatedusingequation( 2 ).Observethatinthiscase,ourapproachdoesnotdependonthequadraticformof( 2 ),andhence,anyvalidreliabilityfunctioncanbeusedwithinApproach2.LetfF0,...,FGigbethesetofpossiblevaluesforRi,8i2H.TosubstituteUiandRiinthemodelbydiscretevariables,wedenenewbinaryvariablesugi,8i2H,g2f0,...,Gig,whichequaloneifUi=Vg(andRi=Fg),andequalzerootherwise.WecannowformulateanMILPbasedonourMINLPformulationinwhichconstraints( 2 )and( 2 )arerevisedas:GiXg=0Vgugi=Xk2Kdk0@Xj2FS(i)Xkij+YD(k),i1A,8i2H (2)Ri=GiXg=0Fgugi,8i2H (2)GiXg=0ugi=1,8i2H (2)ugi2f0,1g,8i2H,g2f0,...,Gig. (2)Theconstraintsofthisrevisedformulationinclude(X,Y)2X,constraints( 2 ),( 2 ),( 2 ),and( 2 )( 2 ).Withtheabovemodications,bothUiandRicanbesubstitutedbylinearfunctionsofdiscretevariables,andsoalltermsin( 2 )and( 2 )arenoweitherlinearorquadratic.Also,becauseeachquadratictermistheproductoftwoboundedvariables,atmostoneofwhichiscontinuous,theycanbelinearizedbydeninganewvariableforeachproductandusingthesamemethodusedinSection 2.3.1 .TolinearizeourMINLPusingthisapproach,weneedatotalofO(j~AjG+jHj(jKj+G)+jAjjKjG)variablesandO(j~AjG+jHjjKj+jAjjKjG)constraints,whereG=maxfGi,8i2Hg.BecauseGisexponentiallylargeingeneral,thisformulationmay 33

PAGE 34

betoolargetobesolvedwithinpracticalcomputationallimits.Wethusexploreanalternativesolutionmethodologybasedonthisformulationinthenexttwosections. 2.4Lower-andUpper-BoundingSchemeSinceMCFNRisNP-hard,andourexactformulationstendtobeintractableduetotheirsize,weinsteadinvestigatelower-andupper-boundingschemesforMCFNRbysolvingpolynomial-sizeMILPformulations.Werstdiscussourlower-boundingmodel,whichissimilartoApproach2presentedinSection 2.3.2 .However,hereweonlyconsiderasubsetofpossibleUi-values(andcorrespondingRi-values).Ourlower-boundingmodelthensetsUitoequalthelargestvalueinthissubsetthatdoesnotexceedtheloadofnodei(i.e.,UiisroundeddowntothenearestvalueinthesubsetofpossibleUi-values).ThevalueforRithencorrespondstotheestimatedUi-value,andisthuspossiblyanoverestimationofthetruereliabilityofhubi.Usingthisstrategyyieldsalowerboundontheoptimalobjectivefunctionvalue,becauseifthereliabilityconstraints(constraints( 2 ),( 2 ),and( 2 ))aresatisedbythehubreliabilityvalues,thentheywillbesatisedusingtheoverestimatedvaluesforRiaswell.Asaresult,thelower-boundingproblem(inwhichweoverestimateRi-values)isarelaxationoftheoriginalproblem.Toimplementthisidea,wepick+1possiblevaluesforUifromthesetfV0,...,VGig,andindexthesevaluesasfVf0g,...,Vfgg,whereVfjg
PAGE 35

newbinaryvariables~ugiasfollows:~ugi=8>><>>:1,ifUiVfgg0,otherwise8i2H,g2f1,...,g.Wethenneedtoaddthefollowingconstraintstotheproblem:~ugiUi)]TJ /F3 11.955 Tf 11.95 0 Td[(Vfgg+1 Pk2Kdk)]TJ /F3 11.955 Tf 11.95 0 Td[(Vfgg+1,8i2H,g2f1,...,g. (2)TheestimatedvalueforRiinthelower-boundingproblemisgivenas:RLi=1+Xg=1~ugi(Ffgg)]TJ /F3 11.955 Tf 11.96 0 Td[(Ffg)]TJ /F10 7.97 Tf 6.59 0 Td[(1g),8i2H. (2)Similarly,wecanobtainareduced-sizemodelinwhichweunderestimatetheRi-values(andoverestimatetheUi-values),whichthenyieldsanupperboundfortheproblem.Inthiscase,wemodifythedenitionof~ugisothatitequalsoneifandonlyifUi>Vfgg(asastrictinequality),foreachi2Handg=0,...,)]TJ /F5 11.955 Tf 12.1 0 Td[(1.Weenforcethisrelationshipbytheconstraints:~ugiUi)]TJ /F3 11.955 Tf 11.95 0 Td[(Vfgg Pk2Kdk)]TJ /F3 11.955 Tf 11.95 0 Td[(Vfgg,8i2H,g2f0,...,)]TJ /F5 11.955 Tf 11.95 0 Td[(1g, (2)andconstraintheestimatedRi-valuesintheupper-boundingproblemas:RUi=1+)]TJ /F10 7.97 Tf 6.58 0 Td[(1Xg=0~ugi(Ffg+1g)]TJ /F3 11.955 Tf 11.96 0 Td[(Ffgg). (2)Wecannowformulatethelower-boundingandupper-boundingproblemsbyrevisingmodel( 2 )( 2 ),inwhich( 2 )isreplacedwith( 2 ),( 2 ),and( 2 ),asfollows.First,substitutevariableRiinmodel( 2 )( 2 )byRLi(orRUifortheupper-boundingmodel)andreplaceconstraints( 2 )bytheexpressiondeningRLiin( 2 )(orRUiin( 2 )).Next,addconstraints( 2 )(or( 2 ))tothemodelalongwithbinarinessrestrictionsonthe~u-variables.ThesamemethodsdiscussedinSection 2.3 canthenbeimplementedtolinearizethelower-andupper-boundingmodels. 35

PAGE 36

2.5Cutting-PlaneAlgorithmInthissection,wepresentamethodforobtaininganoptimalsolutionforMCFNRbyappendingcuttingplanestothelower-boundingformulationinSection 2.4 .First,westartwithsubsetsfVf0g,...,VfggandfFf0g,...,Ffgg,andformulatethelower-boundingproblemdescribedinSection 2.4 .Wethenexecutebranch-and-boundonthelower-boundingproblemuntilaninteger-feasiblesolution,(^X,^Y),isfound.Giventhemulticommodityowroutesprescribedby(^X,^Y),wecalculatetheactualUi-andRi-valuesusingequations( 2 )and( 2 ),respectively,andcomputethereliabilityofeachpath.BecausetheMINLPfeasibleregionisasubsetofthelower-boundingproblem'sfeasibleregion,theactualreliabilityofapathforsomecommodityk2Kmightbelessthankinthesolutiongivenby(^X,^Y).Ifso,thiscommodityissaidtobeaviolatedcommodity.Dene~KKasthesetofallviolatedcommodities.Ifnoviolatedcommoditiesexist(~K=;),thentheactualreliabilityvaluesgivenbythesolution(^X,^Y)areallatleastk,8k2K,and(^X,^Y)isfeasibletoMCFNR.Otherwise,foreachlower-boundingsolutionhavingviolatedcommodities,wecutoffthecurrentsolutionviaacuttingplane,asdescribedbelow.Togenerateacuttingplane,consideraset~Kofviolatedcommodities.Foreachviolatedcommoditypath,weneedtoeitherrevisethepath,orreducetheloadofahubvisitedbythepath.Fork2~K,letPkbethesetofhubsthatlieoncommodityk'spath.Moreover,deneIkasthesetofallincomingarcstothehubsinPkwhosecorrespondingX-variablesequal1inthecurrentsolution.Similarly,let~Ikbethesetofallterminal-hubpairsassignedinthecurrentlower-boundingsolution,suchthatthehubbelongstoPk.Formally,givenacurrentsolutionwithowandassignmentvalues(^X,^Y),andviolatedcommodityk2~K:Ik=f(i,j,k0)j(i,j)2A,j2Pk,k02K,^Xk0ij=1g,~Ik=f(p,i)2~Aji2Pk,^Ypi=1g. 36

PAGE 37

OnenecessaryconditionforfeasibilityistorequireatleastoneXk0ij-variabletoequalzero,for(i,j,k0)2Ik,oratleastoneYpi-variabletoequalzero,for(p,i)2~Ik.Thisconditionisenforcedbythefollowinginequality:X(i,j,k0)2Ik(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Xk0ij)+X(p,i)2~Ik(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Ypi)1. (2)Inequality( 2 )isvalid,becauseiftheleft-hand-sideiszero,thencommoditykcontinuestousethesamepathasbefore,andtheloadoneveryhubinPkisatleastaslargeasitwasintheprevioussolution(whichledtocommoditykbeingaviolatingcommodity).Thus,anysolutioninwhichtheleft-hand-sideof( 2 )equalszeromustbeinfeasible.Moreover,( 2 )isacuttingplanebecauseitsleft-hand-sideevaluatestozerointhecurrentinfeasiblesolution(^X,^Y).Wenowshowthatthecutting-planealgorithmconvergesnitely.Notethateachfeasiblesolutiontothelower-boundingproblemcorrespondstoadistinctsetofX-andY-variables.Asaresult,thenumberofdifferentsolutionscannotbemorethan2jAjjKj+j~Aj.Moreover,cutting-planealgorithmvisitseachintegersolutionatmostonceinthebranch-and-boundtree,andsothegivenalgorithmconvergesnitely. Remark2.3. Therearevariousimplementationoptionsforthecutting-planealgorithmbasedonthenumberofcutsweaddateachnodeofthebranch-and-boundtreeforwhichinequalitiesoftheform( 2 )aregenerated.Here,weimplementthreedifferentstrategiestocutoffasolutionhavingviolatedcommodities~K.Strategy1.Generatej~Kjcutting-planeinequalitiesoftheform( 2 ),oneforeachviolatedcommodity.Strategy2.Generateasinglecutbyaggregatingcutting-planeconstraints( 2 )forallviolatedcommodities.Strategy3.Deneamostviolatedcommodity(km)asaviolatedcommodityforwhichthedifferencebetweenthereliabilitythresholdvalueandtheactualprobabilityof 37

PAGE 38

successfuldeliveryisthelargestamongallviolatedcommodities.Generateasinglecut( 2 )correspondingonlytocommoditykm.Notethatthelasttwostrategiesgeneratefewercuttingplanesthantherststrategy.However,strategy1tendstocutoffmoreinfeasiblesolutionsthantheothertwostrategies,thusreducingthesizeofthebranch-and-boundtreeexploredbythealgorithm. 2.6ComputationalExperimentsInthissectionweevaluatethecomputationalefciencyofourcutting-planealgorithmsforsolvingMCFNRbytestingthemon40randomlygeneratedinstances.WerstpresentourtestproblemsanddescribeourrandomgenerationroutineinSection 2.6.1 .Then,inSection 2.6.2 ,weimplementallthreecutting-planestrategiesusingCPLEX12.2viaILOGConcertTechnology,alongwithabriefcomparisontoapremierquadraticoptimizationsolver.AllcomputationswereperformedonanIntelCorei5witha2.40GHzprocessorand4.0GBRAM. 2.6.1TestProblemGenerationWerstdescribetheschemethatweusetorandomlygenerateMCFNRinstances.BecauseMCFNRisanewproblemintheliterature,wehavealsomadeourtestinstancesavailableat http://www.ise.ufl.edu/cole .TheinstancesareclassiedintoeightcategoriesbasedonthecombinationofparametervaluesjTj2f5,10g,jHj2f10,20g,andjKj2f5,10g,withoneexception.WhenjTj=10andjKj=5,allcommunicationswouldbeunidirectional,i.e.,eachterminalservesaseitheranoriginoradestinationforexactlyonecommodity.Toavoidthissituation(whichwouldseemtoberareinpractice),wemodifyjTj2f5,7gwhenjKj=5.ForeachcombinationofjTj,jHj,andjKj,werandomlygenerateveinstances.ToformthesetA,foreachpairofhubsi,j2H,i6=j,wegeneratearc(i,j)withprobability0.5.Theset~Aisgeneratedbyapplyingthesamemethodforeachundirectedarc(p,i)suchthatp2Tandi2H.Wethengenerateanarcfromeachisolatedhub(ahub 38

PAGE 39

withnoincomingoroutgoingarcs)toarandomhub.Anintegerrandomnumberwithdiscreteuniformdistributionbetween1and5isgeneratedasthecostofeacharcinA.Thesamemethodisusedtogeneratethecostofarcsin~A,exceptthattheintegervaluesaregenerateduniformlybetween1and15.Foreachcommodity,theoriginterminalisrandomlyselectedfromthesetT.Thedestinationterminalisthengeneratedviathesamemethod,exceptthattheprocessisrepeateduntilthecommodity'sdestinationisdifferentfromitsorigin,andthegeneratedorigin-destinationpairisnotthesameasanyothercommodity'sorigin-destinationpair.Next,i-values(foralli2H)aregeneratedrandomlybetween1andjTj.Wethenensurethateveryterminalisanoriginordestinationforatleastonecommodity,andthatallterminalscanbeassignedtothehubs,i.e.,Pi2HijTj.Thedemandvalueforeachcommodityisproducedbygeneratingarandomnumberwithuniformdistributionbetween1and5.Wesetmi=2Pk2Kdk,8i2H,whichthusobviatestheneedforconstraintslimitingUimi,foreachi2H.Wethengeneratek-valuesbyrstdeninganewparameterasthelargestterminalload,i.e.,=maxp2TPk2(p)dk.Thenwedenekasfollows:k=1)]TJ /F6 11.955 Tf 14.46 8.08 Td[(2 m22,8k2K, (2)wheremisthecommonvalueassignedtoallmi-valuesintheinstance.Inthismanner,allcommodityreliabilitythresholdsequalacommonvalue,.Settingassuchtendstogenerateinstancesthatarechallengingtosolve,anddonotgenerallyadmitoptimalsolutionsinwhichallcommoditypairssimplyuseashortestpathconnectingtheiroriginsanddestinations.Morespecically,ourchoiceofguaranteesthefeasibilityofsolutionsforwhicheachterminalisassignedtoaseparatehub,andeachcommoditypathpassesthroughatmosttwohubs(eachofwhichwouldthereforehavealoadofatmost).Severalofourinstancesdonotsatisfythiscriteria,andsotheseinstances 39

PAGE 40

mayormaynotbefeasible.Eachinfeasibleinstancegeneratedinthismannerwasdiscarded,sothatallinstancesinourtestsethaveoptimalsolutions.TogeneratethesetofpossiblevaluesforUi,andbyextensionforRi,recallthatwerequireV0=0(F0=1)andV=Pk2Kdk(F=0.75,notingthateachmi=2Pk2Kdk).OurstrategyforselectingtheremainingV-andF-valuesisinspiredfromtheshapeofthereliabilityfunctiondenedin( 2 ),whichisconcaveanddecreasing.First,weset=jKj.Toobtainatighterrelaxationofthisfunction,wemaywishtoselectmoreV-valuescorrespondingtohigherhubloads.Ontheotherhand,becauseweareunlikelytorelayalargeportionofthetotalcommodities'loadsthroughasinglehub,selectingseverallargevaluesforUiisnotreasonable.Therefore,wetendtodistributethevaluesmoredenselyaroundthecenterofthereliabilityfunction'srange.Toaccomplishthisgoal,werstsortthecommoditiesinnondecreasingorderoftheirdemandvalues,andobtaintheorderedsetfdf1g,...,dfjKjgg.Then,wedeneVi+1=8>><>>:Vi+dbjKj=2c)]TJ /F4 7.97 Tf 10.35 0 Td[(i+1,8i2f0,...,bjKj=2cgVi+di+1,8i2fbjKj=2c+1,...,jKj)]TJ /F5 11.955 Tf 17.94 0 Td[(1g.NotethatthenumberofselectedvaluesinthismethodisjKj+1. 2.6.2ResultsWepresentcomputationalresultsregardingthecomparisonofcut-generationstrategiesinTable 2-1 ,wherealltimesarereportedinCPUseconds.Weallowaone-hour(3600seconds)timelimit,andfortheinstancesthatexceedthistimelimit,wereporttherelativeoptimalitygapthatwasachieved.InTable 2-1 ,S1,S2,andS3refertocolumn-generationstrategies1,2,and3,respectively.ThecolumnlabeledInstancedescribestheinstancesasXYZ-NinwhichX,Y,andZdenotethelevel(Lforlarge,andSforsmall)ofparametersjTj,jHj,andjKj,respectively,andNistheinstancenumberwithinthecategoryspeciedbyXYZ.Inthesecondcolumn(Time(Gap%)),wereportthesolutiontimeorrelative 40

PAGE 41

optimalitygapforeachofthestrategies,dependingonwhethertheinstancehasbeensolvedtooptimalitywithinthetimelimitornot,respectively.Foreachstrategy,thenumberofcutsgeneratedandnumberofnodesinthebranch-and-boundtreeforeachstrategyarepresentedincolumnsCutsandNodes,respectively.Table 2-1 showsthatthenumberofinstancesthatcanbesolvedwithinonehourusingstrategies1,2,and3are38,36,and35,respectively.Also,theaveragesolutiontimefortheinstancesthathavebeensolvedusingallthreestrategies(35instances)are167,184,and226usingstrategies1,2,and3,respectively.Figure 2-4 depictsthenumberofsolvedinstancesbyincreasingthetimelimit,usingeachstrategy.Thehorizontalaxisinthisgurerepresentsthelimitweallowforthesolutiontimeandthegraphrepresentsthenumberofinstancesthathavebeensolvedtooptimalitywithinthattimelimit,usingthecorrespondingstrategy.Basedonthisgraph,strategy1appearstobethemosteffectivestrategyoverall,althoughthedifferencesintheperformancesofthesestrategiesappeartobesmall.Asaresult,weusestrategy1togeneratecutsinourremainingexperiments.Recallthatbecauseweselectedaquadraticcapacityfunctionintheseexperiments,wecouldalternativelysolvetheseMCNFinstancesviaamixed-integerquadraticoptimizer.Accordingly,wemodeledtheMCNFinGAMS24.1.1andsolvedtheMINLPformulationdirectlyusingtheGloMIQO2solver.Weadjustedtherelativeoptimalitygapparameter(optcr)to0.001inordertoobtainoptimalsolutionsfortheseinstances.Table 2-2 presentstheresultsofusingthissolver(inthecolumnmarkedGloMIQO)tosolvetheforegoinginstancesasbefore(againwiththeonehourtimelimit),andcomparesthemwiththeresultsobtainedfromourcut-generationstrategy1(S1).Thecolumn(Time(Gap%))givestheexecutiontimesforinstancessolvedtooptimalitybythemethods,ortherelativeoptimalitygapsforinstancesthatreachedthetimelimit.Thethird(LB)andfourth(UB)columnsrespectivelydepictthelowerandupperboundvaluesobtainedusingeachmethod.Thecellscorrespondingtotheinstancesinwhich 41

PAGE 42

Table2-1. Comparisonofcut-generationstrategies1,2,and3 Time(Gap%)CutsNodesInstanceS1S2S3S1S2S3S1S2S3 LLL-1139149137101010312312312LLL-2132812000010LLL-32918(5.42%)(1.48%)7511496261302036124490LLL-480161843103116323116LLL-59219000010LLS-1121000000LLS-2454000121212LLS-3483371000LLS-44411010351102888LLS-55187102090299LSL-1113150901548520472143944LSL-2180112742723963444621404817075249LSL-316151660(0.84%)4811942391874091971336LSL-43048317433848137826678233682889LSL-5413405389392128122521249812268LSS-1217916448625231631566727607LSS-2111000000LSS-3313030401938630255265432LSS-4000000000LSS-5111000000SLL-118921927221780SLL-2(10.69%)(11.54%)(11.38%)86223834422971933215347SLL-3112811000010SLL-4163316000010SLL-5821659025156126270164SLS-1222000000SLS-2232000000SLS-3112000000SLS-4(7.14%)(2.27%)(17.11%)164826251312446612502646154SLS-5881(37.58%)(3.62%)7109041573244845849532870SSL-1845456599863535294971555220958SSL-2111111000323232SSL-3476490401512108521064366SSL-48177142430308330SSL-517191510121174217183SSS-1000000000SSS-2001000000SSS-3111000000SSS-411107103413951241636SSS-5111000000 42

PAGE 43

Figure2-4. Solutiontimeofinstancesusingthreecutting-planegenerationstrategies GloMIQOfailstogenerateafeasiblesolutionwithinthetimelimitaremarkedby.WeconcludethatGloMIQOappearstobethemoreefcientsolveroninstancesthatarerelativelyeasytosolve.ItspotentialadvantagesareunderscoredmostdramaticallyintheLSLinstances,exceptforLSL-5.However,forthemostdifcultinstances,S1ispreferable.NotethatGloMIQOfailedtosolvesixinstancestooptimality,whileS1failedtosolvetwoinstances.Furthermore,GloMIQOfailstoevenndafeasiblesolutiontothreeinstanceswithinthetimelimit.Again,itisworthnotingthatS1canbeappliedforgeneralcapacityfunctions,anddoesnotrelyonquadraticstructures.WenextstudyandcomparetwoalternativemethodsinselectingV-andF-valuestoinitializeourcutting-planealgorithm.Theoriginalmethod(inwhichUi-valuesarechosennearthecenterofthereliabilityfunctionrange)istermedmethod1.Thesecondandthirdmethodsdividetheinterval[0,Pk2Kdk]intoseveralsubintervalsoflength 43

PAGE 44

Table2-2. ComparisonofGloMIQOandtheS1cutting-planealgorithm Time(Gap%)LBUBInstanceGloMIQOS1GloMIQOS1GloMIQOS1 LLL-1139312312LLL-2613283283283283LLL-32918337337LLL-4(7.42%)80287293310293LLL-579339339339339LLS-121118118118118LLS-234121121121121LLS-33492929292LLS-43463636363LLS-52595959595LSL-119113366366366366LSL-21131801269269269269LSL-31491615396396396396LSL-4833048374374374374LSL-52188413344344344344LSS-11421204204204204LSS-201133133133133LSS-31131136136136136LSS-400101101101101LSS-501136136136136SLL-1718300300300300SLL-2(2.42%)(10.69%)415.68381.35426427SLL-3611382382382382SLL-4616416416416416SLL-52182377377377377SLS-122154154154154SLS-222234234234234SLS-321123123123123SLS-4(7.14%)175.5189SLS-5777881171171171171SSL-1978845592592592592SSL-2111360360360360SSL-3(1.65%)47418420425420SSL-4118395395395395SSL-5217460460460460SSS-100191191191191SSS-200230230230230SSS-311176176176176SSS-4811273273273273SSS-511123123123123 44

PAGE 45

dmed(medianofdemandvalues)anddmin(minimumofdemandvalues),respectively.Method2thussets=dPk2Kdk=dmede.NotethatthelastsubintervalhaslengthPk2Kdk)]TJ /F3 11.955 Tf 12.16 0 Td[(dmed()]TJ /F5 11.955 Tf 12.16 0 Td[(1)dmed,withstrictinequalityholdingifPk2Kdkisnotdivisiblebydmed.Theseobservationsapplytomethod3aswell,withdmedreplacedbydmin.Totestthethreemethodspresentedabove,werandomlygenerated20instancesforeachoftheeightcategoriesdiscussedinSection 2.6.1 .However,thedemandvaluesforcommoditiesarenowgeneratedrandomlybetween1and100,inordertoobservethedifferencesbetweenthemethods.Table 2-3 presentsthenumberofinstancessolvedwithinonehourlimit(#solved),andtheaveragesolutiontime(AverageTime)forinstancesgeneratedbyeachmethod(M1,M2,M3).NotethatthenumberofselectedvaluesforUiimpactsthenumberofvariablesinthelower-boundingmodel,andtendstocreatelargeformulationsforthethirdmethodinparticular.Thecomputationaltimelimitremainsonehourfortheseinstances. Table2-3. ComparisonofthethreeUi-andRi-valueselectionstrategies #solvedAverageTimeCategoryM1M2M3M1M2M3 LLL1818127096001843LLS1920205872LSL1719157524991537LSS202019522212SLL141510114910321779SLS171717543543610SSL16161011488352043SSS20201947210 BasedonthedatapresentedinTable 2-3 ,manyoftheinstancesgeneratedbythethirdmethodcannotbesolvedduetomemoryortimelimits.Thoseinstancesthatweresolvedwithinthetimelimitbythethirdmethodrequireconsiderablymoretimetosolvethantheothertwomethods.Theperformanceofthealgorithmontheinstancesgeneratedbymethods1and2dependsonthesizeoftheinstancesandinparticular,thesizeofparameterjKj.ForsmallervaluesofjKj,therstmethodappearstobe 45

PAGE 46

favorable.However,forthemorechallenginginstancesinwhichjKjtakesitsupperboundvalue,thesecondmethodispreferable.Next,weexaminethestrengthofthelowerandupperboundsforMCFNR,aspresentedinSection 2.4 .Wesolvethelower-andupper-boundingmodelsfortheinstancesinTable 2-1 andcomparetheboundswiththeiroptimalobjectivevaluesinTable 2-4 .Weallowaone-hourtimelimitandpresentthecalculatedboundfortheproblem(LBforLowerBoundandUBforUpperBound),thetotalsolutiontime(Time)inseconds,andthegapbetweentheboundandtheoptimalobjectivefunctionvalue(Gap%)inTable 2-4 .ThelastcolumnofTable 2-4 liststheoptimalobjectivefunctionvalueoftheinstancesthathavebeensolvedtooptimalitywithintheone-hourtimelimitusingourcutting-planealgorithm.Forinstancesinwhichwewereunabletocomputeanupperboundwithinonehour,wereportinthetable.Thecellsmarkedwith*correspondtotheinstancesforwhichthecutting-planealgorithmfailstoterminatewithinonehour.Additionally,theupper-boundingmodelisinfeasibleforinstanceLLS-4,andismarkedwithINF.AccordingtoTable 2-4 thequalityoflowerandupperboundsfortheinstanceswhosecorrespondingupper-boundingmodelhavebeensolvedwithinanhourarecomparable.However,theupper-boundingmodeltakeslongertoterminatethanthelower-boundingmodel,becausendingfeasiblesolutionsfortheupper-boundingmodelismoredifcultthanforthelower-boundingmodel. 46

PAGE 47

Table2-4. Comparisonoflower-andupper-boundingmodels Lower-boundingmodelUpper-boundingmodelCuttingplaneInstanceLBTimeGap%UBTimeGap%OptimalValue LLL-1309220.96312470.00312LLL-2283130.00283110.00283LLL-33272002.9734022030.89337LLL-4287182.05293450.00293LLL-5339140.0033980.00339LLS-111820.0011820.00118LLS-212150.0012130.00121LLS-38428.70360092LLS-46123.17INF1363LLS-59025.26106611.5895LSL-1366980.003661350.00366LSL-22647201.8626913850.00269LSL-3388632.023600396LSL-43741110.003600374LSL-5343590.293603794.65344LSS-1165019.122282511.76204LSS-213310.003600133LSS-312865.8813610.00136LSS-410110.00118216.83101LSS-513610.0013610.00136SLL-1300100.003600300SLL-238133*426554**SLL-3382140.0038280.00382SLL-4416180.00416200.00416SLL-5373121.063600377SLS-115420.003600154SLS-223430.003600234SLS-312320.0012320.00123SLS-41701*3600*SLS-515837.603600171SSL-1586561.016077472.53592SSL-236040.003600360SSL-3418110.48420170.00420SSL-439051.27416195.32395SSL-546060.0046440.87460SSS-119110.003600191SSS-223010.003600230SSS-317610.0018525.11176SSS-426921.4730912513.19273SSS-512310.00146218.70123 47

PAGE 48

CHAPTER3ASURVEYOFROBUSTOFFLINESINGLE-MACHINESCHEDULINGPROBLEM 3.1MotivationTheclassofsingle-machineschedulingproblemsweexamineinthissurveychapterseektoschedulenonpreemptivejobsonasinglemachine,whichiscapableofperformingonlyonetaskatatime.DeningJasthesetofalljobs,eachjobj2Jistypicallyassociatedwithdataattributessuchasprocessingtime(pj),weight(wj),andduedate(dj).Theobjectiveofascheduledependsonthecompletiontime(Cj)ofeachjobj2J.DeneLj(lateness)asCj)]TJ /F3 11.955 Tf 12.44 0 Td[(dj,Tj(tardiness)asmaxf0,Cj)]TJ /F3 11.955 Tf 12.44 0 Td[(djg,andUjasabinaryindicatorvalueequalto1ifandonlyifCj)]TJ /F3 11.955 Tf 12.46 0 Td[(dj>0.Commonsingle-machineschedulingobjectivesincludeminimizingtotalcompletiontime(Pj2JCj),totalweightedcompletiontime(Pj2JwjCj),andthenumberoflatejobs(Pj2JUj).Also,deningLmax=maxj2JfLjg(andTmaxandCmaxanalogously),othercommonobjectivesincludeminimizingLmax,Tmax,andCmax.(Thelatterobjectiveisalsoreferredtoasmakespan.)Deterministicschedulingproblemsassumethattheexactvaluesofallparametersareknown.Asaresult,itisstraightforwardtocalculatethejobcompletiontimesandobjectivefunctionvaluecorrespondingtoeachsequence.Moreover,eachproblemlistedaboveispolynomiallysolvableusingdeterministicdata[ 50 66 73 ].Duetovariabilityofprocessandenvironmentaldata,uncertaintyiscommoninmanypracticalschedulingproblems.Therefore,severalresearchershavedevelopedmethodstohedgeagainstdatauncertaintyinthisarea.Reactive(online)schedulingdealswithadjustingjobschedulesasdataisrealizedinordertoreducetheeffectofdisruptionsandunpredicteddelays.Usingthisapproach,theschedulergeneratesinitialscheduleswithoutconsideringuncertainty,andthenrevisesthescheduleasdisruptionsoccur.Onlineschedulingisspecicallyusefulforsituationsinwhichlimitedinformationaboutuncertaindataisavailableinadvance,anditispracticalfortheschedulertodynamicallyadjusttheschedules.Thisreactiveapproachcanaccommodateawide 48

PAGE 49

varietyofdisruptions(suchasmachinebreakdowns,processinterruptions,orparametervaluevariations)andissuitableforproblemsinwhichdisruptionsaredifculttopredict.Forexample,processschedulinginanoperatingsystemcanbemodeledasanonlineschedulingproblemsincetheoperatingsystemtypicallydoesnotknowtheexecutiontimeofaprocessbeforeitscompletion,anditispossiblefortheoperatingsystemtointerleavetheexecutionofdifferenttasksbytemporarilyinterruptingataskandresumingitsexecutionatalatertime(preemption).Anotherexampleregardstheschedulercomponentofawebserver.Thenumberofusersandthelengthofeachuser'stasksisunknownandverydifculttopredict.Therefore,onlineschedulingisanappropriatemethodformodelingtheproblemofschedulingusers'webtransactions.See[ 70 ]foranoverviewofonlineschedulingandasurveyofresultsobtainedinthisarea,and[ 61 ]foranexampleofonlinesingle-machinescheduling.However,itisnotalwaysrealistictoassumethatschedulescanberevisedafterdisruptions.Inthesesituations,apredictive(ofine)schedulingapproachprescribessolutionsthatarerelativelyinsensitivetochangesininputdata.Animplicitassumptionmadeinofineschedulingisthatalluncertaindatavaluesarerealizedafterthedecisionshavebeenmade.Stochasticprogrammingandrobustoptimizationaretwoimportantofinemodelingschemesthathavebeenappliedforsolvingseveralschedulingproblemsunderuncertainty.Stochasticoptimization,whichwasrstintroducedbyDantzig[ 26 ],assumesknowledgeofdataprobabilitydistributions.Thismethodseekstooptimizetheexpectedobjectivefunctionvalue,whileensuringthatconstraintsaresatisedatleastwithsufcientlyhighprobability.Werefertheinterestedreadertotextbooks[ 18 36 43 ]andthereferencestherein.PinedoandSchrage[ 69 ]presentasurveyofstochasticoptimizationapplicationsinsolvingschedulingproblems.Robustoptimizationisanalternativeapproachfordealingwithuncertaindata[ 12 72 ].Inrobustoptimization,datauncertaintyisusuallyrepresentedbycontinuousordiscreteuncertaintysetsandthefeasibilityofthesolutionisguaranteedwith 49

PAGE 50

respecttoanypossibledataoutcomewithintheuncertaintyset.Twomainfactorsmotivatetheuseofrobustoptimization.First,itisnotalwayspossibletoestimatedataprobabilitydistributionswithdesiredprecision.Second,instochasticprogramming,theproblemsizeincreasesdrasticallywiththenumberofuncertainparameters,whichinducessubstantialcomputationalchallenges.Anotheressentialdifferencebetweenthestochasticprogrammingandrobustoptimizationapproachesisthatstochasticprogrammingmodelstypicallyaimatoptimizingexpectedsystemperformance,whilerobustoptimizationmodelsfocusonguaranteeingaminimumqualityforthesolutionintheworstcase.Inthedeterministicschedulingliterature,problemsareclassiedintodifferentcategoriesbasedonthemachineenvironment(),jobcharacteristics(),andtheoptimizationcriterion(),andweassumethattheexactvaluesofallparametersareknown.Singlemachine(1),m-machineowshop(Fm),andm-machinejobshop(Jm)areexamplesofcommonmachineenvironments(),whilehavingprecedencerelationsbetweenjobs(prec),orhavingfamiliesofjobswithsimilarcharacteristics(fml)areexamplesofjobcharacteristics().AnyminimizationcriterionsuchasCmax,PCj,andPUj,isanexampleof.Accordingly,wecanspecifyeverydeterministicschedulingproblemusingGraham'snotationjjsuggestedin[ 35 ].Inrobustoptimization,however,weassumethatdifferentjob-relatedparameterssuchaspj,dj,andwjareuncertainandweoptimizewithrespecttotheworst-casedatarealizationusingamin-maxobjective.Therefore,wealsospecifytheuncertainparametersinthenotation,denotedby.Finally,wespecifytherobustnessmeasurefortheproblem,,whichweexplainindetailinSection 3.2.1 .TheoverallinstancedescriptionforarobustschedulingproblemisthengivenbyMinMax(jj,).Inthischapterwereviewtherobustoptimizationtechniquesthathavebeenusedforsolvingstandardschedulingproblemsunderuncertainty.Notethatthemajorityofresearchinthisareaisfocusedonsingle-machineschedulingproblems(SMSP).Thus, 50

PAGE 51

weexploreandclassifytheliteratureofrobustSMSP,whileaddressingsomeotherstandardschedulingproblemsthathavebeeninvestigated.Theremainderofthechapterisorganizedasfollows.InSection 3.2 ,werepresentdifferentrobustnesscriteriaanduncertaintyrepresentationsthathavebeenintroducedinrobustoptimizationliteratureandaddresstheuseofeachmethodinscheduling.TheninSection 3.3 ,wediscussthedetailsofexistingresearchintheareaofrobustSMSP,classifythemaccordingtothecategoriespresentedinSection 3.2 ,andexploretheexistinggapsandopenproblems.WethenaddressrelatedrobustschedulingresultsinSection 3.4 3.2RobustnessandUncertaintyDenitionsTodenearobustschedulingproblem,weneedtorepresentthepotentialvaluesofuncertainparametersintheproblemandspecifyameasurebywhichweevaluatetherobustnessofaparticularsolution.Inthissection,wepresentthemostcommonrobustnessmeasuresanduncertaintyrepresentationsforrobustoptimizationingeneralandforrobustschedulinginparticular.Wealsodiscussthebenets,limitations,andpotentialapplicationsforeachscheme.Deneasequence,,asapermutationofjobsanddenotethesetofallpossiblepermutationsby.Ascenario,,isaparticularrealizationofuncertainparameters,whererepresentsthe(possiblyinnite-cardinality)setofallpossiblescenarios.LetZbetheobjectivefunctionvalueofjobsequenceunderdatarealization.DeneZasthebestachievable(optimal)objectivefunctionvaluewhendatascenariohappens.Weusethisnotationtopresentrobustnessanduncertaintydenitionsinthefollowingtwosubsections. 3.2.1RobustnessMeasuresWediscussthethreemostcommonrobustnessmeasuresthatariseintheschedulingeld:Absoluterobustness,robustdeviation,andrelativerobustdeviation[ 49 ].(SeealsoSabuncuogluandGoren[ 71 ]foramoredetailedclassicationfor 51

PAGE 52

possiblerobustnessandstabilitymeasures,alongwithapplicationsofsomemeasuresinsolvingrobustSMSPs.)Absoluterobustnessseekstominimizethemaximumobjectivefunctionvalueoverallscenarios.Thatis,whendealingwithabsoluterobustnessmeasure,weconsidertheworst-casescenariocorrespondingtoeachsequenceofjobs,andselectthesequencewhoseworst-caseobjectivevalueisminimum,comparedtoallotherfeasiblesequences.Wecanmathematicallystateabsoluterobustnessasmin2max2Z.Thismeasureisparticularlyusefulforsituationsinwhichoneseekstoguaranteeatleastacertainqualityforthesolutionoverallpossiblescenarios.Tounderstandtherobustdeviationmeasure,werstintroducetheconceptofregret.Inthecontextofrobustscheduling,adecisionmakerchoosesaschedulebeforeobservingthedatavalues.Afterthedataisobserved,thedecisionmaker'sregretisgivenbythedifferencebetweenhis/herchosenschedule'sobjectiveandtheobjectiveoftheretrospectiveoptimalsolution,i.e.,theoptimalsolutiongivenknowledgeofthedataoutcome.Robustdeviationseekstominimizethelargestpossibleregret,andcanbestatedasmin2max2(Z)]TJ /F3 11.955 Tf 12.1 0 Td[(Z).Minimizingrobustdeviationmaybeappropriatewhendeterminingaschedulewhoseperformance,comparedtothecorrespondingoptimalperformance,isrelativelyinsensitivetodatarealization.Solutionsthatminimizerobustdeviationcanalsobeinterpretedasuniformlysuboptimalsolutions,i.e.,-optimalsolutionsforalldatarealizations,withassmallaspossible.Relativerobustdeviationminimizesthemaximumrelative(orpercentage)deviationfromoptimality,i.e.,min2max2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[((Z)]TJ /F3 11.955 Tf 11.96 0 Td[(Z)=Z.Infact,relativerobustdeviationisanormalizedrobustdeviationmeasurethatseekstominimizetherelativeregretintheproblem.Sincethevalueofrelativeregretonlydependsontheratioofobjectivevalues,itcanbeusedtocreatebenchmarkstocomparethequalityofdifferentproblems'solutions. 52

PAGE 53

Robustdeviationandrelativerobustdeviationmeasuresareappropriateforenvironmentsinwhichthequalityofsolutionsareevaluatedafterthedataisrealized.Insuchcasesthe(relative)deviationortherelativedeviationoftheselecteddecisionfromtheoptimaldecisionfortherealizedscenarioisaplausiblequalitymeasure.Forhighlycompetitivemarkets,wherearmneedstohaveasatisfactoryperformancecomparedtoitscompetitors,underanypotentiallyrealizablescenario,theuseofthesemeasuresisalsoappropriate[ 49 ].Tounderstandthedifferencebetweenrobustnessmeasures,consideranexampleinwhichweseektondatwo-jobschedulethatminimizestotalcompletiontime,wherep12[4,5]andp22[1,6].Ifweapplytheabsoluterobustnessmeasure,theworst-casescenarioforanysequenceoccurswhenp1=5andp2=6.Therefore,therobustoptimalsolutionisobtainedbyschedulingjob1beforejob2(sequence1,2).However,whenrobustdeviationisapplied,thelargestregretvalueequals4forsequence1,2(whenp1=5andp2=1)andequals2forsequence2,1(whenp1=4andp2=6).Therefore,schedulingjob2beforejob1isfavorablewhenrobustdeviationmeasureisused.Notethatusingrelativerobustdeviationforthisinstancealsoresultsinsequence2,1beingoptimal(largestrelativeregretvalueequals4/7forsequence1,2and2/14forsequence2,1).However,robustdeviationandrelativerobustdeviationmeasuresmayresultindifferentoptimalsolutionsinotherinstances.Forexample,supposethatwemodifytheaboveexamplebylettingp12[4,5]andp22[2,8].Givensequence1,2,thelargestabsoluteregretvalue(3)isobtainedwhenp1=5andp2=2:Theobjectiveofsequence1,2is12,whiletheoptimalobjectivevalueforthisscenariois9.Sequence2,1,ontheotherhand,resultsinthelargestregretvalueof4,andthereforesequence1,2ispreferablewithrespecttorobustdeviationmeasure.However,therelativeregretvalueis3/9forsequence1,2and4/16forsequence2,1,andsosequence2,1isoptimalunderrelativerobustdeviationmeasure. 53

PAGE 54

3.2.2UncertaintyRepresentationIntherobustoptimizationliterature,severalmethodsofexpressinguncertainparametervalueshavebeenproposed.Inthissection,wediscussfourmainmethodsofuncertaintyrepresentation,presentbenetsanddrawbacksofeachmethod,andaddresssomeapplicationsofeachmethodinrobustscheduling.ThemosttraditionalmethodofpresentinguncertaintyinrobustoptimizationproblemswasdevelopedbySoyster[ 76 ].Soysterassumesthateachuncertaininputdata,independentofalltheothers,cantakeonanyvalueswithinacontinuousinterval(forthesakeofbrevity,wecallthisrepresentationintervaluncertainty)andproposesalinearoptimizationmodeltosolvetheproblem.Thesimplicityofthismethodanditsresultingformulationsmotivatesitsextensiveapplicationinrobustschedulingproblems.See[ 7 44 62 65 ]asexamplesofusingintervaluncertaintytorepresentthevaluesofdifferentjob-specicparametersofSMSPhavingdifferentoptimizationcriteria.NotethatifdatauncertaintyisrepresentedusingSoyster'smethodandweseektominimizetheabsoluterobustness,settingallparameterstotheirworst-casevalueswillgenerateaworst-casescenariocorrespondingtoanysequenceofjobs.Therefore,robustSMSPshavingtheabsoluterobustnesscriterionunderintervaluncertaintyisequivalenttoadeterministicSMSPinwhichalldataelementstakeontheirworst-casevalues.However,generatingaworst-casescenariowhenrobustdeviationorrelativerobustdeviationmeasuresareappliedisnotobvious.Forexample,supposeweseektominimizetotalcompletiontimeinanSMSPwithtwojobswherep12[1,5]andp22[3,4]andweuserobustdeviationmeasure.Theworst-casescenarioforthesequenceinwhichjob1isprocessedbeforejob2iswhenp1takesonitslargestpossiblevalue(5)andp2takesonitssmallestpossiblevalue(3),andsotheregretisgivenby2.Similarly,forthereversesequence(job2beforejob1),intheworst-casescenariowehavep1=1andp2=4,whichresultsinrobustdeviationvalueof3.Therefore,sequence 54

PAGE 55

1,2minimizesrobustdeviation.Notethattheoptimalsequenceforthisproblemwithabsoluterobustnessmeasureistheopposite(job2rstandjob1second).Animportantdisadvantageofthismethodisthatcorrelationsbetweenthevaluesofparametersisnotaddressed.Onewaytocapturethesecorrelationsistoenumerateallpossiblescenariosandrepresentuncertainparametersasasetofdiscretescenariosfortheirnumericalvalues.Werefertothismethodasscenario-baseduncertainty.IntherobustSMSPliterature,usingscenario-baseduncertaintyiscommon[ 3 25 27 63 ].Thismethodallowsthedecisionmakertoconsidertherelationshipbetweenalluncertainfactorsintheschedulingenvironmentandincludeallpossiblecasesinthemodel.Italsomaintainscontroloverthelevelofconservatismintheproblem.However,thisapproachpotentiallyrequiresenumerationofalargenumberofdataoutcomes,whichiscomputationallyintractableinsomecases.Althoughintervaluncertaintyresultsinsimplerobustformulationsforseveralproblems,ittendstogenerateschedulesthatareover-conservativeinthecaseofabsoluterobustness.Ben-TalandNemirovski[ 13 15 ]andEl-Ghaouietal.[ 28 29 ]addressthisissueandintroduceamethodtoreducethelevelofconservatismbyconningthedatatobelongtouncertaintysetsintheformofellipsoids.Theyproposeefcientalgorithmstosolvetheresultingconvexoptimizationproblems.BertsimasandSim[ 16 17 ]proposebudgeteduncertaintyasanalternativeapproachtocontrolthelevelofconservatismforgeneralrobustoptimizationproblems,whichproducesalinearrobustformulation.Thismethodassumesthatalluncertainparametersindependentlytakeonvaluesaccordingtoasymmetricdistributionwithknownmeanvalues(theiridealvalues)andperturbationbounds.Sinceitisunlikelythatallparametersfailtotakeontheirpredictedvalues,theylimitthenumberofperturbedparameters(theoneswithvaluesotherthantheiridealvalues)ineachconstraintandintheobjectivefunction,separately.Theyalsolimittheamountofperturbationinthevalueofeachindividualparameter.Usingthismethodtorepresentuncertaincoefcients 55

PAGE 56

oftheobjectivefunctionandconstraintsofanoptimizationproblem,theyproposearobustprogrammingproblemofmoderatelylargersize.Theyprovethatusingbudgeteduncertainty,arobustlinearprogrammingproblemcanstillbesolvedasalinearprogram,andtherobustcounterpartofapolynomiallysolvable0-1linearoptimizationproblemremainspolynomiallysolvable.Tothebestofourknowledge,budgeteduncertaintymethodisappliedtorobustSMSPinonlyonepaper[ 77 ],whichwediscussinSection 3.3 .Thismethodhasalsobeendirectlyappliedinseveralotherschedulingproblemsintheareaofchemicalprocessscheduling,aswediscussinSection 3.4 3.3RobustSingle-MachineSchedulingLiteratureClassicationInthissection,wereviewtherobustsingle-machineschedulingliteratureandclassifyexistingresearchinthisareaaccordingtotheirrobustnessmeasuredenitionanduncertaintyrepresentation.Kasperski[ 45 ]summarizessomeoftheresultsobtainedintherobustSMSPliteraturefordifferentcases(speciedbyarobustnessmeasureandanuncertaintyrepresentation)andintroducesopencasesinthisarea.Ashortsurveyoftheresultsisalsopresentedin[ 2 ].Here,weprovideanupdatedsurveyoftheresultsobtainedineachcategoryofSMSPs.AsstatedinSection 3.2.2 ,thecaseofabsoluterobustnesswhendealingwithcontinuousintervalsofuncertaintycanbeequivalentlysolvedasadeterministicproblemandthereforeisoutofthescopeofthischapter.However,whenparametervaluesareconnedtobelongtosomeuncertaintysets,asinbudgeteduncertaintymethod,thereisnoobviousequivalentdeterministicversionoftheproblem.TadayonandSmith[ 77 ]denethreealternativeuncertaintysetsfortheabsoluterobustSMSPwhenprocessing-timevaluesarepresentedasindependentcontinuousintervals.Inuncertaintysets1,2,and3,theyrespectivelyrequirethetotaldelay,thenumberofdelayedjobs,andthetotalratiobywhichtheprocessingtimesareincreasedtobenomorethanaconstantvalue.TheystudythecomplexityoftheproblemMinMax(1jjZ,pj), 56

PAGE 57

whereZ2fPCj,PwjCj,Lmax,Tmax,PUjgundereachuncertaintyset,proposeexactalgorithmsforpolynomiallysolvableproblems,andpresentmixed-integerprogrammingformulationsfortheNP-hardproblemsandtheoneswithunknowncomplexity.WepresentthedetailsandtheresultsofthisresearchinChapter 4 .AbsoluterobustSMSPinpresenceofscenario-baseduncertaintyhasbeenstudiedinseveralpapers.YangandYu[ 84 ]provethatproblemMinMax(1jjPCj,pj)withscenario-baseduncertaintyisNP-hardforallthreerobustnessmeasures(eveninthespecialcaseofhavingonlytwoscenarios).TheyalsopresentanexactdynamicprogrammingalgorithmwithexponentialcomplexityandtwopolynomialtimeheuristicstosolvethisrobustSMSP.AloulouandDellaCroce[ 3 ]studyseveralotherabsoluterobustSMSPswithscenario-baseduncertaintyindifferentjob-relatedparameters.Whentheschedulerseekstominimizethenumberoflatejobs,theyprovethatproblemsMinMax(1jjPUj,pj)andMinMax(1jjPUj,pj,dj)areNP-hard,butthecaseofMinMax(1jjPUj,dj)isstillopen.Also,whentheobjectivefunctionoftheproblemisminimizingfmax2fCmax,Lmax,Tmaxgandprecedencerelationsbetweenjobsinthesequenceareallowed,theyproposepolynomialalgorithmsforsolvingtheprobleminwhichpj,dj,orbothareexpressedasasetofdiscretescenarios.Inaddition,theyprovethatproblemMinMax(1jjPwjCj,wj)withscenario-baseduncertaintyisNP-hard.(NoticethatNP-hardnessofMinMax(1jjPwjCj,pj)alsofollowsfromNP-hardnessofMinMax(1jjPCj,pj),whichwasprovedin[ 84 ]).Morerecently,Mastrolillietal.[ 63 ]seektogenerateapproximationschemesforproblemMinMax(1jjPwjCj,fwj,pjg).TheyprovethattheproblemcannotbeapproximatedwithinO(log1)]TJ /F11 7.97 Tf 6.58 0 Td[(n)forany>0,unlessquasi-polynomialalgorithmsexistforNP.Forthespecialcaseofunweightedjobs,theyproposea2-approximationalgorithmforsolvingMinMax(1jjPCj,pj)andprovethattheproblemisNP-hardtoapproximatewithinafactorlessthan6/5. 57

PAGE 58

Despitetheextensiveapplicationofabsoluterobustnessintherobustoptimizationliterature,robustdeviationhasbeeninvestigatedmorewidelyinSMSPliterature.Whenindependentcontinuousintervalsareusedtorepresentuncertainprocessingtimes,LebedevandAverbakh[ 51 ]provethatthegeneralcaseofproblemMinMaxdev(1jjPCj,pj)isNP-hard;however,theyshowthatwhenallintervalsofuncertaintyhavethesamecenter,theproblemcanbesolvedpolynomially(inO(nlogn)time)ifthenumberofjobsisodd,andisNP-hardotherwise.Montemanni[ 65 ]presentstherstmixed-integerlinearprogrammingformulationforproblemMinMaxdev(1jjPCj,pj)withintervaluncertaintyandtranslatessomepreprocessingrulesintovalidinequalities.KasperskiandZielinski[ 46 ]provethatanyminmaxrobustdeviationSMSPwithtotalcompletiontimecriterionandintervaluncertainty,whoseequivalentdeterministicproblemispolynomiallysolvable,canbeapproximatedwithinafactorof2.Amoregeneralcaseofthisproblemwithweightedsumofcompletiontimecriterion(MinMaxdev(1jjPwjCj,pj))hasbeenstudiedin[ 75 ]inwhichsomedominancerelationsareintroducedandaheuristicalgorithmisproposedtondagoodnondominatedsequence.Robustdeviationinthecaseofintervaluncertaintyisalsoconsideredin[ 7 ],[ 44 ],and[ 62 ]fordifferentSMSPs.Averbakh[ 7 ]provesthatMinMaxdev(1jjmaxfwjTjg,wj)ispolynomiallysolvablebypresentinganO(n3)algorithmfortheproblem.Kasperski[ 44 ]proposesapolynomialalgorithm(withO(n4)complexity)foroptimallysolvingMinMaxdev(1jprecjLmax,fpj,djg).Luetal.[ 62 ]deneaspecicSMSPwithtotalcompletiontimeobjectivefunctioninwhichjobsaregroupedintofamiliesandasequence-dependentfamilysetuptimeexists.Theyprovethatwhenbothjobprocessingtimesandfamilysetuptimesarepresentedasintervalsofuncertainty,minimizingrobustdeviationinthecorrespondingproblemisNP-hard.Theythenproposeasimulatedannealing-basedalgorithmtondgoodqualitysolutionsfortheproblem. 58

PAGE 59

SeveralresearchersstudyrobustdeviationinSMSPsinwhichparametersarepresentedasdiscretescenarios.DanielsandKouvelis[ 25 ]examineproblemsMinMaxdev(1jjPCj,pj)andMinMaxrel(1jjPCj,pj)whentheuncertaintysetconsistsofdiscretescenariosforthepj-values.TheyprovethatbothproblemsareNP-hardandpresentsomedominancerelationsthatcanbeusedtodeterminetherelativejobpositionsinarobustschedule.Then,theyusetheseresultstodevelopanexactbranch-and-boundalgorithmforsolvingtherobustproblem.Theyalsopresentintuitiveheuristicapproachesandevaluatetheirefciencyandaccuracythroughasetofrandomly-generatedinstances.Additionally,DanielsandKouvelisprovethatevenwhenprocessingtimesarepresentedasindependentcontinuousintervals,theworst-casescenarioforeachsequencebelongstothenitesetofextreme-pointscenarios(scenariosinwhicheachparametertakesonitslower-orupper-boundvalues).Accordingtothisresult,theyclaimthattheformulationspresentedforscenario-baseduncertaintyarealsovalidforintervaluncertainty.Asstatedbefore,YangandYu[ 84 ]approachthesameproblemwithrobustdeviationasoneofthethreerobustnessmeasuresandpresentadifferentNP-hardnessprooffortheproblem.Amoregeneralproblemwiththeweightedsumofcompletiontimecriterion(usingrobustdeviationmeasureandscenario-baseduncertaintyinjobprocessingtimes)isstudiedin[ 27 ],whereasetofvalidinequalitiesfortheconvexhullofitsfeasibleregionispresentedandisthenutilizedtodesignacutting-planealgorithmforsolvingtheproblem.Tothebestofourknowledge,relativerobustdeviationhasbeenexplicitlyconsideredinonlythreeSMSPpaperssofar.Averbakh[ 8 ]studiesthismeasureofrobustnessforrobustcombinatorialoptimizationproblemsingeneralandforSMSPasaspecialcase,andprovesthattheproblemMinMaxrel(1jjPCj,pj)withintervaluncertaintyisNP-hard.DanielsandKouvelisstatethattheanalysis,results,andsolutionmethodspresentedin[ 25 ]forthecaseofrobustdeviationcanbeappliedfortherelativerobustdeviation 59

PAGE 60

measurewithaslightmodication.YangandYu[ 84 ]provetheNP-hardnessofproblemMinMax(1jjPCj,pj)withscenario-baseduncertaintyforallthreerobustnessmeasures(includingrelativerobustdeviation).WesummarizetheresultsobtainedintheliteratureofrobustSMSPinTable 3-1 .Inthistable,wespecifythecomplexityofrobustSMSPproblemsundereachrobustnessmeasure(absoluterobustnessabs.rob.,robustdeviationrob.dev.,andrelativerobustdeviationrel.rob.dev.)andeachuncertaintyrepresentation(budgeteduncertaintybudg.uncert.,intervaluncertaintyinterval,andscenario-baseduncertaintyscenario),aswellasthereferencefromwhichtheresultisextracted.Eachproblemisspeciedbyitsobjectivefunctionanduncertainparameterintherstcolumnlabeledas(obj.,param.).ThecomplexityoftheproblemsthatareprovedtobeNP-hardarelistedinthetableasNPH.Notethatbudgeteduncertaintyhasbeenconsideredonlyforabsoluterobustnesscriterion.Amongtheresultsobtainedin[ 77 ],weonlyincludetheonesabouttherstuncertaintysetinthistable,forthesakeofbrevity.Forinformationaboutthecomplexityofeachproblemundertheothertwouncertaintysets,refertoChapter 4 .Thecellswhosecorrespondingproblemshavenotbeenyetstudiedintheliteraturearemarkedby-. Table3-1. ComplexityresultsobtainedintheliteratureforrobustSMSP abs.rob.rob.dev.rel.rob.dev.(obj.,param.)budg.uncert.scenariointervalscenariointervalscenario (PCj,pj)O(nlogn)[ 77 ]NPH[ 84 ]NPH[ 51 ]NPH[ 25 84 ]NPH[ 8 ]NPH[ 25 84 ](PwjCj,pj)Open[ 77 ]NPH[ 84 ]NPH[ 51 ]NPH[ 25 84 ]NPH[ 8 ]NPH[ 25 84 ](PwjCj,wj)-NPH[ 3 ]----(PwjCj,fpj,wjg)-NPH[ 84 ]NPH[ 51 ]NPH[ 25 84 ]NPH[ 8 ]NPH[ 25 84 ](PUj,pj)O(nlogn)[ 77 ]NPH[ 3 ]----(PUj,dj)-Open[ 3 ]----(PUj,fpj,djg)-NPH[ 3 ]----(Lmax,pj)O(nlogn)[ 77 ]O(n2jSj)[ 3 ]O(n4)[ 44 ]---(Lmax,dj)-O(n2+njSj)[ 3 ]O(n4)[ 44 ]---(Lmax,fpj,djg)-O(n2jSj)[ 3 ]O(n4)[ 44 ]---(Tmax,pj)O(nlogn)[ 77 ]O(n2jSj)[ 3 ]----(Tmax,dj)-O(n2+njSj)[ 3 ]----(Tmax,fpj,djg)-O(n2jSj)[ 3 ]----(Cmax,pj)-O(n2jSj)[ 3 ]----(maxwjTj,wj)--O(n3)[ 7 ]--60

PAGE 61

3.4RobustOptimizationinOtherSchedulingProblemsAlthoughrobustoptimization,asdenedinthischapter,hasbeenmostfrequentlystudiedforthesingle-machineschedulingenvironment,otherschedulingproblemshavebeenalsoaddressedintheliterature.Moreover,theconceptofrobustnessinsomestudiesextendsbeyondthestandarddenitionspresentedearlierinthischapter,andinclude(a)therestrictionthatascheduleremainsfeasiblewithacertainprobability[ 41 ],(b)thatascheduleguaranteesacertainperformancelevel[ 24 81 ]),or(c)theincorporationofdifferentuncertaintiesinschedulingenvironments(e.g.,unpredictableproductioninterruptions[ 52 ]).Inthissection,weintroducesomeoftheotherareasofschedulinginwhichdifferentvariationsofrobustoptimizationhasbeenappliedtohedgeagainstuncertaintyinproductionenvironments.Kouvelisetal.[ 48 ]applyarobustnessdenitionsimilartotheonepresentedinthischaptertocomplywithuncertaintyinatwo-machineowshopenvironment.TheyprovethatproblemMinMaxdev(F2jjCmax,pj)isNP-hard.Theyalsodiscussthepropertiesofrobustschedulesanddevelopexactandheuristicsolutionapproachesfortheproblemunderbothintervaluncertaintyandscenario-baseduncertaintyinjobsprocessingtimes.Althoughellipsoidalandbudgeteduncertainty(presentedbyBen-TalandNemirovski[ 13 15 ],El-Ghaouietal.[ 28 29 ],andBertsimasandSim[ 16 17 ])havenotbeenwidelyaddressedinrobustsingle-machineschedulingliteratureyet,ithasbeenappliedtootherschedulingproblems,mainlyintheareaofchemicalprocessscheduling.Productionscheduling,asoneofthemostimportantproblemsinindustrialplantoperations,hasbeenstudiedinseveralresearchpapers.Short-termproductionschedulingseekstodetermineanoptimalassignmentofavailableresourcestoproductiontasksovertimewhilesatisfyingtheproductionrequirementsatduedates.Dependingontheplantlayoutandthesequenceoftasks,processschedulingisusuallyintheformofastandardowshoporjobshopschedulingproblem.IerapetritouandFloudas[ 39 ]proposeamixed-integerlinearprogrammingformulationforthisproblem. 61

PAGE 62

LiandIerapetritou[ 54 ]consideruncertaintyinparametervaluesintheformulationpresentedin[ 39 ].Theyimplementthethreemaincontinuousuncertaintyrepresentationapproachesforrobustoptimization,i.e.,Soyster'smethod,Ben-TalandNemirovski'smethod,andBertsimasandSim'smethod(budgeteduncertainty),toformulatetherobustprocessschedulingproblemgivenuncertaintyregardingproductunitprice,taskprocessingtimes,andproductdemandvalues.TheycomparethethreeformulationsbysolvingtestinstancesfromlinearformulationsusingCPLEXandtheonesfromnonlinearformulationsusingtheDICOPTsolverviaGAMS.TheresultsoftheircomputationsshowthatBertsimasandSim'smethodyieldsthemostappropriateformulationfortherobustschedulingproblem,accordingtothefactthatthismethoddoesnotincreasetheproblemsizesubstantiallyandmaintainsthelinearityofthemodel.Linetal.[ 56 ]andJanaketal.[ 41 ]consideruncertaintyindifferentparametersoftheprocessschedulingproblemasformulatedin[ 39 ].Uncertainparameters(price,processingtimes,anddemands)arepresentedasindependentcontinuousintervalsin[ 56 ]andasrandomvariableswithknownprobabilitydistributionsin[ 41 ].Janaketal.[ 41 ]denearobustsolutiontobeonethatisfeasibleinallconstraintswithacertainprobability.MathematicalformulationsfortherobustproblemsarepresentedinbothpapersandtheefciencyofmodelsareinvestigatedbysolvingtestinstancesfromlinearformulationsusingCPLEXandfromnonlinearmodelsusingDICOPT.LiandLerapetritou[ 53 ]reviewtheliteratureofprocessschedulingunderuncertainty.Theyintroducerobustoptimizationasoneofthealternativeapproachesindealingwithuncertaintyinprocessschedulingandaddressthemainadvancesandchallengesinthisareaofresearch.Verderameetal.[ 80 ]provideabroaderoverviewofplanningandschedulingproblemsunderuncertainty,acrossmultiplesectors.Althoughtheobjectivefunctionandconstraintsofthemodelvariesgreatlyamongdifferentsectors, 62

PAGE 63

theyaddressthecommonattributeofrequiringastandardrobustnessdenitionanduncertaintyrepresentationinallareasandencourageinterdisciplinaryresearch.Leonetal.[ 52 ]deneadifferenttypeofuncertaintyinastandardjobshopenvironmentinwhichdisruptionsoccurrandomlyduringtheprocess.Theyseektondasequenceofjobsthatisrobustunderdifferentscenariosofdisruptions,assumingthatpreemptionisnotallowed(whenajob'sprocessisinterrupted,ithastoberestarted),theperformancemeasureismakespan,andtheorderofjobscannotberevisedafterinterruptionsoccur.Theydenetherobustnessmeasureasarandomlyweightedsumoftheexpectedmakespanandtheexpecteddifferencebetweentheactualmakespansandthedeterministicmakespan(withoutinterruptions).Becausetheeffectofeachdisruptiondependsontheoutcomeofallpreviousdisruptions,theycomputetherobustnessmeasureonlyinthecaseofasingleinterruption.Forthecaseofseveraldisruptions,theydevelopasurrogatemethodandembeditinageneticalgorithmtogeneraterelativelyrobustschedules.Arobustsolutioninsomeapplicationsisdenedasasolutionthatguaranteesacertainlevelofperformanceunderallpossibledatarealizations.DanielsandCarrillo[ 24 ]denea-robustsolutionasajobsequencethatmaximizesthelikelihoodofachievingatotalcompletiontimevaluenogreaterthanaconstantvalueinasingle-machineschedulingproblemwithuncertainprocessingtimes.Theyassumethatjobsprocessingtimesareindependentrandomvariableswithknownmeansandvariancesthatcanberepresentedasdiscretescenarioswheneachscenariooccurswithacertainprobability.TheyprovethattheproblemisNP-hardanddevelopanexactbranch-and-boundalgorithmandapolynomialheuristicforsolvingtheproblem.Wuetal.[ 81 ]considerasimilarproblemwithindependentnormallydistributedprocessing-timevaluesandpresentthreemodels:primal(maximizingtheprobabilityofobtainingacertainlevelofperformance),dual(minimizingthelevelthatcanbeachievedwithaxedprobability),andhybrid.Theydeterminethefeasibilityofeachmodelforasetof 63

PAGE 64

randomlygeneratedproblemsandstudytheeffectofsomedominancerulesinsolvingtheproblems. 64

PAGE 65

CHAPTER4ALGORITHMSANDCOMPLEXITYANALYSISFORROBUSTSINGLE-MACHINESCHEDULINGPROBLEMS 4.1MotivationWeexamineinthischapteraschedulingprobleminwhichasetofjobs,J,mustbeprocessedonasinglemachine,oneatatime,withoutpreemption.Everyjobj2Jrequiresaspecicamountoftime,pj,tobeprocessedonthemachine,andinsomeapplications,isassociatedwithaduedate,dj.Also,forsituationsinwhichthejobsarenotequallyimportant,jobjisassociatedwithaweight,wj.Wefocusinthischapteronseveralalternativeobjectivefunctions.DeningCjtobethecompletiontimeofjobj,weconsiderthedue-dateindependentobjectivesofminimizingtotalcompletiontimeofjobs(PCj)andminimizingtotalweightedcompletiontimeofjobs(PwjCj).Forcasesinwhichduedatesarerelevant,weconsidertheproblemofminimizingthenumberoflatejobs(PUj,whereUj=1ifCj>djandis0otherwise),minimizingthemaximumlateness(Lmax=maxjfLjg,whereLj=Cj)]TJ /F3 11.955 Tf 12.36 0 Td[(dj),orminimizingthemaximumtardiness(Tmax=maxjfTjg,whereTj=maxf0,Cj)]TJ /F3 11.955 Tf 11.96 0 Td[(djg).Schedulingproblemsunderuncertaintyhavereceivedanextensiveamountofattentionduringthelastfewdecades.Schedulesthatareoptimalwithrespecttodeterministicdatacanbesuboptimalinpracticeduetouncertaintiesinparameterssuchasprocessingtimes,duedates,andweights.Twoapproachesthatcanbeusedtomodeluncertaintyinoptimizationproblemsincludestochasticprogrammingandrobustoptimization.Stochasticprogrammingtypicallyseekstooptimizeasolution'sexpectedobjectivevalue.Thisapproachrequiressomeknowledgeoftheprobabilitydistributionforallnondeterministicparameters,whichisoftenhandledbysamplingstrategies(see,e.g.,[ 18 43 ]).Robustoptimizationisanalternativeapproachfordealingwithuncertaindata[ 12 72 ],whichassumesthatalluncertaindatavaluesarerealizedafterthedecisionshavebeenselected.Inrobustoptimizationproblems,thedecisionvariablesmustremain 65

PAGE 66

feasibleunderanydataoutcome.Theobjective(forminimizationproblems)seekstominimizethemaximumpossibleobjectivefunctionvaluethatcouldoccurfortheselecteddecisionvariables.Inthecontextofourschedulingproblems,asolutionreferstoajobpermutation,whichremainsfeasibleforanydatarealization.Thechallengesthatwefaceinthesestudiesistocharacterizewhichdataoutcomesresultinworst-caseobjectivefunctionvalues(asafunctionofthechosenschedules),andtodeterminehowtominimizethoseworst-casevalues.Severalcriteriacanbeappliedtomeasuretherobustnessofaparticularsolution.KouvelisandYu[ 49 ]introducethreegeneralrobustnessmeasurescalledabsoluterobustness,robustdeviation,andrelativerobustdeviation.Absoluterobustnessisusedwhenthegoalistominimizetheobjectivefunctionoftheworst-casescenario,aswedointhischapter.Robustdeviation(orabsoluteregret)seekstominimizethelargestpossibledifferencebetweentheobservedobjectivefunctionvalueandtheoptimalobjectivefunctionvalue.Relativerobustdeviation(orrelativeregret)minimizesthelargestpossibleratioofrobustdeviationtotheoptimalobjectivefunctionvalue.Kaspersky[ 45 ]summarizessomeresultsforrobustschedulingproblems(speciedbyarobustnessmeasurementandanuncertaintyrepresentation)andintroducesopencasesinthisarea.Aissietal.[ 2 ]presentasurveyofregret-basedcombinatorialoptimizationproblems.Seealso[ 71 ]foradifferentcategorizationforrobustnessandstabilitymeasures,alongwithareviewofsingle-machineschedulingproblems(SMSP)ineachcategory.Foramorerecentsurveyoftheresultsobtainedintheliteratureofrobustsingle-machinescheduling,seeChapter 3 orrefertothetechnicalreport[ 78 ].Inthischapterweconsiderabudgeteduncertaintymodelinwhichprocessingtimesareuncertainandareconnedtosomespeciedinterval,andwherewelimitthetotalmagnitudeofdeviationfromtheiridealvalues.Thisapproach,inamoregeneralsense,constrainsuncertaindatatoliewithinsomepolyhedron.Asopposedtointervaluncertainty,ouranalysisprohibitsdatafromsimultaneouslytakingonworst-casevalues, 66

PAGE 67

andinsteadconcentratesonaless-conservativeanalysisofdatarealizations.See[ 17 ]forathoroughdiscussionofbudgeteduncertaintymodels.Inordertospecifytheprobleminthischapter,weapplythesamenotationintroducedinChapter 3 ,i.e.,(jj,).Thedenitionofeachparameter(,,,,and)inthisnotationispresentedinSection 3.1 .Theremainderofthischapterisorganizedasfollows.InSection 4.2 weprovidetheproblemdenitionanddiscussthreedifferentuncertaintysetsthatconstraintotaldeviationintheidealparametervalues.InSection 4.3 weinvestigateabsoluterobustnessintheSMSPundereachuncertaintysetwithfourcommonly-usedminimizationcriteria:totalcompletiontime(PCj),totalweightedcompletiontime(PwjCj),maximumlateness/tardiness(LmaxorTmax),andnumberoflatejobs(PUj). 4.2ProblemDenitionandNotationLetJ=f1,...,ngbethesetofjobstobeprocessed.Theidealprocessingtimeofjobj2J,givenbypj,isdenedasthe(best-case)timerequiredbythemachinetoprocessjobj,assumingthatnootherfactorsaffecttheprocess.Theactualprocessingtimeofjobjcanbelongerthantheidealprocessingtime,inwhichcasewesaythatjobjisdelayed.Denotethequantityofdelayforjobjbyj.Sinceinseveralapplications,jobshavinglongerprocessingtimesaremorelikelytobedelayedbyagreatervalue,weassumethatjisaproportionofpj(i.e.,j=kjpjforsomenonnegativemultiplierkj)andlimitthetotaldelayforeachjobj2JbyrequiringthatkjK(jKpj).Inadditiontolimitingthedelayofeachjob,wealsorestrictthetotalamountofdelaytocontrolthelevelofuncertaintyintheproblem,i.e.,(1,...,n)2S,whereSisthesetofallpossible-values.Inparticular,westudythreeclassesofuncertaintysets.Uncertaintyset1(US1)requiresthetotalamountofdelaytobenomorethanaconstant,,i.e.,Pj2Jj.Forthenextuncertaintyset,wedenemjasabinaryvariablethatequalsoneifjobjisdelayed,andzerootherwise.Uncertaintyset2(US2)limitsthenumberofdelayedjobsbyaninteger,M,i.e.,Pj2JmjM.Finally, 67

PAGE 68

uncertaintyset3(US3)ensuresthatthetotalratiobywhichtheprocessingtimesareincreasedcannotbemorethanaconstant,,i.e.,Pj2Jkj.Here,withoutlossofgenerality,weassumethatPj2JKpj,Mn,andKn.Wedeneasequence,,asapermutationofjobsanddenotethesetofallpossiblepermutationsby.Thejthjobinisdenotedbyj.Ascenario,,inthischapterisaparticularrealizationofjobprocessingtimes,whererepresentsthe(innite-cardinality)setofallscenariosinouruncertaintyset.Givenajobsequenceanddatascenario,deneCjtobethecompletiontimeofthejthjob,andZtobetheobjectivefunctionvalueofthesequence.Theactualprocessingtimeofthejthjobinunderscenarioisdenotedbypj.Fortheidealscenarioinwhichalljobstaketheiridealprocessingtimes,wedenotethecompletiontimeofthejthjobby~Cj,andtheobjectivevaluecorrespondingtothissequenceasZ.Correspondingtoeachsequence,aworst-casescenario(())isdenedtobeascenariothatmaximizestheobjectivefunctiongiventhejobsequence(wherefornotationsimplicity,()isreplacedby,unlessitresultsinconfusion).Tobetterunderstandtheproblem,weinvestigatetheworst-casescenarioandtheoptimalsequenceforallthreeuncertaintysetsinaprobleminstancepresentedinExample 4.1 Example4.1. Consideratwo-jobinstancewherep1=8,w1=10,p2=1,andw2=1.LetK=0.5,anduncertaintybudgetscorrespondingtoUS1,US2,andUS3aregivenby=4,M=1,and=0.5,respectively.Weseektoidentifyasequenceofjobsthatminimizestotalweightedcompletiontimeundertheabsoluterobustnessmeasure.Therearetwopossiblesequencesforthisinstance.Therstsequence(1)schedulesjob1beforejob2.Byinspection,theworst-casescenarioforallthreeuncertaintysetsisachievedbyincreasingtheprocessingtimeofjob1byitsmaximumpossiblevalue(4).Therefore,p11=p1=12,p12=p2=1,andZ1=10(12)+1(12+1)=133forallthreeuncertaintysets. 68

PAGE 69

Thesecondsequence(2)schedulesjob2rstandjob1second.ForUS1,theworst-casescenarioincreasestheprocessingtimeofjob2by0.5(minfKp2,g),andtheprocessingtimeofjob1by3.5(minfKp1,)]TJ /F5 11.955 Tf 13.25 0 Td[(0.5g).Therefore,forUS1,p21=p2=1.5,p22=p1=11.5,andZ2=1(1.5)+10(1.5+11.5)=131.5.ForUS2andUS3,theworst-casescenarioincreasestheprocessingtimeofjob1by4.Asaresult,inthosecases,p21=p2=1,p22=p1=12,andZ2=1(1)+10(1+12)=131.Hence,thesecondscheduleisoptimal,regardlessofwhichuncertaintysetwechoose. 4.3ComplexityResultsandAlgorithmsInthissectionweexamineourrobustschedulingproblemforeachcombinationofobjectivefunctionsdescribedinSection 4.1 anduncertaintysetsdenedinSection 4.2 .Ineachcase,werstcharacterizetheproblemofgeneratingtheworst-casescenarioforagivensequence(whichwecallthescenario-generationproblem(SGP)),andthenusetheobtainedresultstodeterminetherelativepositionofjobsinarobustschedule.TheSGPcanbeformulatedas:MaxR(1,...,n) (4)subjectto:F(1,...,n)b (4)(1,...,n)2S, (4)whereR(1,...,n)isthetotalincreaseintheidealobjectivevalue(towhichwereferasthetotalpenalty)bydelayingjobsandF(1,...,n)istheamountofuncertaintybudgetthathasbeenusedduetojobdelays.WesummarizeinTable 4-1 theresultstobepresentedinthissection.Thistablegivesthealgorithmcomplexitythatweobtainedforeachproblem(SGPandrobustoptimizationproblem)undereachobjectivefunction(PCj,PwjCj,LmaxorTmax,andPUj)andeachuncertaintyset(US1,US2,andUS3).Forthoseproblemswherewecouldnotidentifyapolynomial-timealgorithm,wedenotethecomplexityresult 69

PAGE 70

byOpen(MIP)ifwespecifyamixed-integerprogramming(MIP)formulationfortheproblemandbyOpenotherwise.Notethatfortheseproblems,wehavealsonotidentiedanNP-hardnessproof,whichiswhywespecifythattheircomplexityremainsopen. Table4-1. ComplexityresultsforrobustSMSPunderbudgeteduncertainty SGPRobustoptimizationproblemObjectiveUS1US2US3US1US2US3 Pj2JCjO(n)O(nlogn)O(nlogn)O(nlogn)O(nlogn)O(nlogn)Pj2JwjCjO(n)O(nlogn)O(nlogn)Open(MIP)Open(MIP)Open(MIP)LmaxorTmaxO(n)O(nlogM)O(nlogd=Ke)O(nlogn)O(nlogn)O(nlogn)Pj2JUjO(n)O(Mn2)Open(MIP)O(n2)Open(MIP)Open 4.3.1MinimizingTotalCompletionTimeWeseektondasequencethatminimizesthemaximumtotalcompletiontimeofjobs(Pj2JCj)underallpossiblescenarios.Thedeterministicversionofthisproblemissolvablebysequencingthejobsinnondecreasingorderoftheirp-values,i.e.,inshortestprocessingtime(SPT)order[ 73 ].WeshowinSection 4.3.1.1 howtheSGPissolvedinthiscase,andthenprovethatanSPTjoborderingoptimizestherobustschedulingproblemunderallthreeuncertaintysets. 4.3.1.1ScenarioGenerationProblemTheSGPformulationcorrespondingtominimizingtotalcompletiontimecanbemathematicallystatedby( 4 )( 4 )inwhich( 4 )canbeequivalentlyrestatedasPnj=1Fj(j)b,wherethespecicformofeachFj-functionandthevalueofbdependonwhichuncertaintysetisused.WenextprovethatwecanalsoexpressR(1,...,n)asalinearfunctionofregardlessoftheuncertaintyset. Theorem4.1. Fortheproblemofminimizingtotalcompletiontime,theSGPobjectivefunction,R(1,...,n),canbeexpressedasPnj=1Rj(j),whereRj(j)=(n)]TJ /F3 11.955 Tf 11.88 0 Td[(j+1)j,8j=1,...,n. Proof. Weprovethetheorembyinductiononthenumberofjobs.Firstsupposethatn=1.Assumingthatwedelaythisjobby1,thetotalcompletiontimeisZ=~C1+1, 70

PAGE 71

andsothetotalpenaltyisR(1)=R1(1)=Z)]TJ /F5 11.955 Tf 14.37 2.66 Td[(~C1=1,whichisequivalentto(n)]TJ /F5 11.955 Tf 12.64 0 Td[(1+1)1whenn=1.Next,supposebyinductionthatthetotalpenaltyforasequenceofmjobsisgivenbyPmj=1(m)]TJ /F3 11.955 Tf 12.48 0 Td[(j+1)j.Wewillshowthatforasequenceofm+1jobs,thetotalpenaltyisgivenbyR(1,...,m+1)=Pm+1j=1(m)]TJ /F3 11.955 Tf 12.39 0 Td[(j+2)j.Let0bethesequenceoftherstmjobsof.Thetotalcompletiontimeof0isPmj=1~Cj+Pmj=1(m)]TJ /F3 11.955 Tf 13.04 0 Td[(j+1)jbytheinductionassumption.Wenowappendjobm+1totheendof0tocreate.Thetotalcompletiontimeofunderscenario,Z,isgivenbyPmj=1~Cj+Pmj=1(m)]TJ /F3 11.955 Tf 11.75 0 Td[(j+1)j+Cm+1,becauseaddingjobm+1totheendofdoesnotaffectthecompletiontimesoftheprecedingjobs.NotethatCm+1=Cm+pm+1+m+1andCm=~Cm+Pmj=1j.Therefore,Z=mXj=1~Cj+mXj=1(m)]TJ /F3 11.955 Tf 11.95 0 Td[(j+1)j+~Cm+mXj=1j+pm+1+m+1.Since~Cm+pm+1=~Cm+1,wehaveZ=m+1Xj=1~Cj+mXj=1(m)]TJ /F3 11.955 Tf 11.95 0 Td[(j+1)j+m+1Xj=1j,soZ=Pm+1j=1~Cj+Pm+1j=1(m)]TJ /F3 11.955 Tf 11.66 0 Td[(j+2)j.ThetotalpenaltyisthereforeR(1,...,m+1)=Z)]TJ /F8 11.955 Tf 11.95 8.96 Td[(Pm+1j=1~Cj=Pm+1j=1(m)]TJ /F3 11.955 Tf 11.95 0 Td[(j+2)j,andtheproofiscomplete. Next,weaddresstheoptimizationoftheSGPcorrespondingtoeachuncertaintyset.Foreachuncertaintyset,weprescribeanordering,O,ofjobstodelay,suchthatthefollowinggreedydelayruleoptimallysolvestheSGP: Foreachj=1,...,n,inthisorder,delayjobOjbythemaximumamountallowedby( 4 )and( 4 ). Lemma1. IfOj=j,8j=1,...,n,thenthegreedydelayruleyieldsanoptimalsolutionfortheSGPunderUS1. Proof. ForUS1,delayingthejthjobbyjusesjoftheuncertaintybudget,.Therefore,Fj(j)=j,8j=1,...,n,andb=in( 4 ).Moreover,jcantake 71

PAGE 72

onanyvaluebetween0andKpj.Hence,byTheorem 4.1 ,wecanformulatetheSGPunderUS1asfollows.MaxnXj=1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(j+1)j (4)subjectto:nXj=1j (4)0jKpj,8j=1,...,n. (4)Becausetheobjectivecoefcientsof1,...,nformadecreasingsequence,thegreedydelayruleundertheorderingspeciedbyLemma( 1 )optimizesthisproblem. SolvingtheSGPunderUS1requiresO(1)operationsforeachj=1,...,n,andtherefore,itstotalcomplexityisO(n).ForouranalysisregardingUS2andUS3,wedeneRjasthelargestpossiblevaluethatRj(j)couldtakeifunconstrainedby( 4 ),i.e.,Rj=(n)]TJ /F3 11.955 Tf 12.09 0 Td[(j+1)Kpj.ThefollowinglemmasestablishoptimalSGPsolutionsunderUS2andUS3. Lemma2. ConsideranorderingOobtainedbysortingjobsj=1,...,ninnonincreasingorderoftheirR-values.ThegreedydelayrulewiththisorderingyieldsanoptimalsolutionunderUS2. Proof. InUS2,eachjobthatisdelayedbyapositiveamountconsumesoneunitofthetotaluncertaintybudget,M.Asaresult,in( 4 ),b=MandFj(j)=mj,8j=1,...,n,wheremjisonewhenjispositiveandzerootherwise.Therefore,theSGPinthiscasecanbestatedasfollows.MaxnXj=1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(j+1)j (4)subjectto: 72

PAGE 73

nXj=1mjM (4)0jKpjmj,8j=1,...,n (4)mj2f0,1g,8j=1,...,n. (4)AnoptimalsolutionforthisproblemselectsMvariablesjtotakeontheirupperbounds(Kpj).Inparticular,theMvariableshavingthelargestR-valueswillbechosentoequaltheirupperbounds,thuscompletingtheproof. Lemma3. ConsidertheorderingOobtainedbysortingjobsj=1,...,ninnonincreasingorderoftheirR-values.Giventhisordering,thegreedydelayruleyieldsanoptimalsolutionunderUS3. Proof. InUS3,delayingthejthjobbyjuseskj=j=pjoftheuncertaintybudget,.Therefore,substitutingj=pjkjinTheorem 4.1 ,theSGPisgivenbyMaxnXj=1(n)]TJ /F3 11.955 Tf 11.96 0 Td[(j+1)pjkj (4)subjectto:nXj=1kj (4)0kjK,8j=1,...,n. (4)Observethatthisproblemisequivalenttomaximizing(1=K)Pnj=1Rjkj.Therefore,wecangenerateanoptimalsolutionforthisproblemviathegreedydelayrule.Thiscompletestheproof. AccordingtoLemmas 2 and 3 ,tosolvetheSGPunderUS2andUS3,wesortthejobsbytheirR-values(O(nlogn)),andperformthegreedydelayrule(O(n)).Thus,thetotalcomplexityofsolvingtheSGPunderUS2andUS3isO(nlogn). 73

PAGE 74

4.3.1.2RobustOptimizationProblemLemmas 1 3 permitustocharacterizeoptimalsolutionstoourrobustschedulingproblem. Theorem4.2. AnySPTscheduleisoptimalforproblemMinMax(1jjPCj,pj),underallthreeuncertaintysets. Proof. Supposethat0isanoptimalsequenceofjobsthatdoesnotfollowtheSPTorder,i.e.,thereexistssomej2f1,...,ngsuchthatp0j>p0j+1.Weshowthatwecanimprovethesequencebycreatinganalternativesequence00,inwhichweswaptheorderofthejthandthe(j+1)stjobsin0andretaintheorderingofallotherjobs.Figure 4-1 illustratesthetwosequences.Forbrevityinnotation,wesubstitute(0)0 Figure4-1. Swappingtheorderofthejthandthe(j+1)stjobsin0tocreate00 ((00)00)andZ(0)0(Z(00)00)by0(00)andZ0(Z00)torepresentjobdelaysandtotalcompletiontimeintheworst-casescenarioscorrespondingto0(00),respectively.WethusseektoshowthatZ0)]TJ /F3 11.955 Tf 11.96 0 Td[(Z00=nXq=1(n)]TJ /F3 11.955 Tf 11.96 0 Td[(q+1)h(p0q)]TJ /F3 11.955 Tf 11.95 0 Td[(p00q)+(0q)]TJ /F6 11.955 Tf 11.95 0 Td[(00q)i>0. (4)Because0and00areidenticalexceptforthejthand(j+1)thjobs,( 4 )reducestothefollowing:Z0)]TJ /F3 11.955 Tf 11.96 0 Td[(Z00=(p0j)]TJ /F3 11.955 Tf 11.96 0 Td[(p0j+1)+nXq=1(n)]TJ /F3 11.955 Tf 11.96 0 Td[(q+1)(0q)]TJ /F6 11.955 Tf 11.95 0 Td[(00q)>0. 74

PAGE 75

Hence,becausep0j>p0j+1,itsufcestoshowthatnXq=1(n)]TJ /F3 11.955 Tf 11.95 0 Td[(q+1)(0q)]TJ /F6 11.955 Tf 11.96 0 Td[(00q)0. (4)FirstconsiderUS1,whereLemma 1 guaranteesthatthejobsaregreedilydelayedintheorder1,...,n.Asaresult,0q=00q,8q2~J,where~J=f1,...,ngnfj,j+1g.Therefore,itsufcestoshowthat(n)]TJ /F3 11.955 Tf 11.95 0 Td[(j+1)(0j)]TJ /F6 11.955 Tf 11.95 0 Td[(00j)+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(j)(0j+1)]TJ /F6 11.955 Tf 11.96 0 Td[(00j+1)0. (4)Next,notethat0j+0j+1=00j+00j+1,because0q=00qforallq2~J,andPnq=10q=Pnq=100q.Therefore,theleft-hand-sideof( 4 )reducesto0j)]TJ /F6 11.955 Tf 11.89 0 Td[(00j.Also,wehavethat0j00j,becausethelargestdelayforeachjobisaproportionofitsprocessingtimeandp0j>p0j+1.Thus,( 4 )holdstrue.ForUS2andUS3,Lemmas 2 and 3 guaranteethattheworst-casescenariodelaysjobsbytheirlargestpossibleamount,innonincreasingorderoftheirR-values.WefocusonUS3here,withanalysisforUS2followingasadirectresult.ByLemma3,theconditiongivenby( 4 )isequivalenttothefollowing:nXi=1R0ik0i)]TJ /F4 7.97 Tf 18.31 14.95 Td[(nXi=1R00ik00i0. (4)Toprovethetheoremfortheseuncertaintysets,werstestablishthefollowingfacts.Fact1.R0q=R00q,8q2~J,because0q=00q,8q2~J.Fact2.R0j+1p0j+1.Fact3.R0j+R0j+1)]TJ /F5 11.955 Tf 13.63 2.66 Td[(R00j)]TJ /F5 11.955 Tf 13.63 2.66 Td[(R00j+1>0,becausebysubstitution,wehaveR0j+R0j+1)]TJ /F5 11.955 Tf -446.36 -21.25 Td[(R00j)]TJ /F5 11.955 Tf 13.31 2.66 Td[(R00j+1=K(p0j)]TJ /F3 11.955 Tf 11.95 0 Td[(p0j+1)>0(sincep0j>p0j+1). 75

PAGE 76

LetD0(D00)bethesetofjobpositionsthataredelayedinschedule0(00),anddenel02argmini2D0fR0ig(l002argmini2D00fR00ig).Wecanthenestablishthefollowingfact.Fact4.Inanoptimalsolution,k0i=K,8i2D0nfl0g(k00i=K,8i2D00nfl00g),k0i=0,8i2f1,...,ngnD0(k00i=0,8i2f1,...,ngnD00),andk0l0=k00l00=k,forsome0maxfR00j,R00j+1g,8q2D0.Hence,j62D00andj+162D00,andbyFact4,LHS=0. Case2.j2D0,j+162D0,j62D00,j+162D00Forthiscase,notethatbecauseR0q=R00q,8q2~J,wehavethatD0differsfromD00byonlyoneelement;inparticular,j2D0,j62D00,l0062D0,andl002D00. Case2-1.l0=jAccordingtoFacts1and4,LHS=kR0j)]TJ /F3 11.955 Tf 12.78 0 Td[(kR00l00.Notethatl002~J,andsoR0l00=R00l00(byFact1).Becausej2D0andl0062D0,wehaveR0jR00l00andsoLHS0. Case2-2.l06=jLHS=KR0j+kR0l0)]TJ /F3 11.955 Tf 12.18 0 Td[(KR00l0)]TJ /F3 11.955 Tf 12.18 0 Td[(kR00l00.SinceR0l0=R00l0(byFact1),wehaveLHS=KR0j)]TJ /F5 11.955 Tf 12.47 0 Td[((K)]TJ /F3 11.955 Tf 12.47 0 Td[(k)R00l0)]TJ /F3 11.955 Tf 12.47 0 Td[(kR00l00=(K)]TJ /F3 11.955 Tf 12.47 0 Td[(k)(R0j)]TJ /F5 11.955 Tf 13.82 2.66 Td[(R00l0)+k(R0j)]TJ /F5 11.955 Tf 13.83 2.66 Td[(R00l00),whichisnonnegativesinceKk,R0jR00l0,andR0jR00l00bythesamereasoningpresentedinCase2-1. Case3.j2D0,j+162D0,j2D00,j+162D00 Case3-1.l0=j,l00=jLHS=kR0j)]TJ /F3 11.955 Tf 11.96 0 Td[(kR00j,whichispositivebyFact2. Case3-2.l06=j,l00=j 76

PAGE 77

LHS=KR0j+kR0l0)]TJ /F3 11.955 Tf 12.18 0 Td[(KR00l0)]TJ /F3 11.955 Tf 12.18 0 Td[(kR00j.SinceR0l0=R00l0(byFact1),LHSsimpliestoKR0j)]TJ /F5 11.955 Tf 12.38 0 Td[((K)]TJ /F3 11.955 Tf 12.39 0 Td[(k)R00l0)]TJ /F3 11.955 Tf 12.38 0 Td[(kR00j,whichispositivesinceR0j>R00jbyFact2,andR0jR00l0,orelsel0wouldnotbethelowest-priorityjobtobedelayedinD0. Case3-3.l006=jFacts1and2implythatR0j>R00l0.Hence,ifl006=j,thenl06=jaswell.Inthiscase,wehavel0=l00byLemma 3 andFact1.Therefore,LHS=KR0j)]TJ /F3 11.955 Tf 11.95 0 Td[(KR00j>0. Case4.j2D0,j+162D0,j62D00,j+12D00Theproofforthiscaseissymmetrictothatforcase3. Case5.j2D0,j+162D0,j2D00,j+12D00Notethatforthiscase,wehavel06=j,orelse,byLemma 3 itwouldbeimpossibletodelayboth00jand00j+1in00asassumedinthiscase. Case5-1.l00=jLHS=KR0j+kR0l0)]TJ /F3 11.955 Tf 11.52 0 Td[(KR00j+1)]TJ /F3 11.955 Tf 11.52 0 Td[(kR00j.BecauseR0l0R0j+1(orelsewewouldhavedelayed0j+1insteadof0l0),weconcludethatLHSKR0j+kR0j+1)]TJ /F3 11.955 Tf 12.09 0 Td[(KR00j+1)]TJ /F3 11.955 Tf -399.23 -14.45 Td[(kR00j.SinceR0j+1
PAGE 78

LHS=KR0j+KR0j+1)]TJ /F3 11.955 Tf 11.95 0 Td[(KR00j+1)]TJ /F3 11.955 Tf 11.95 0 Td[(KR00j,whichispositivebyFact3. Theorem 4.2 impliesthatonecansolvetherobustoptimizationproblemunderallthreeuncertaintysetsbysortingthejobsbytheirprocessingtimes,whichimpliestheworst-casecomplexityofO(nlogn)fortheproblem. 4.3.2MinimizingTotalWeightedCompletionTimeInthissection,weconsidertheproblemMinMax(1jjPwjCj,pj).Thedeterministicversionofthisproblemcanbesolvedbysequencingthejobsinnondecreasingorderoftheratiopj=wj,whichformsaweightedshortestprocessingtime(WSPT)order[ 73 ].However,wewillshowinthissectionthattheWSPTruledoesnotalwayscreaterobustoptimalschedulesinthepresenceofuncertainty. 4.3.2.1ScenarioGenerationProblemWerstshowthatthecorrespondingSGPcanbemathematicallystatedby( 4 )( 4 ),wherebothR(1,...,n)andF(1,...,n)canbeexpressedasthesumofseparablefunctions. Theorem4.3. ThetotalobjectivevalueincreaseR(1,...,n)=Pnj=1Rj(j),whereRj(j)=(Pnq=jwq)j,8j=1,...,n. Proof. SimilartotheproofofTheorem 4.1 Asdiscussedbefore,F(1,...,n)canberestatedasPnj=1Fj(j)accordingtotheuncertaintyset.NotethatthefunctionsFj(j),8j=1,...,n,donotdependontheoptimizationcriteriaandtherefore,thefeasibleregionoftheproblemforUS1,US2,andUS3arethesameastheonespresentedinformulations( 4 )( 4 ),( 4 )( 4 ),and( 4 )( 4 ),respectively.WedenetheobjectivefunctionofthethreeproblemsaccordingtoTheorem 4.3 (wherewesubstitutejbypjkjforthecaseofUS3in( 4 )aswedidin( 4 )).Therefore,theSGPcanbestatedas( 4 ),( 4 ),and 78

PAGE 79

( 4 )forUS1,US2,andUS3,respectively.MaxnXj=1 nXq=jwq!j,subjecttoconstraints( 4 )and( 4 ) (4)MaxnXj=1 nXq=jwq!j,subjecttoconstraints( 4 )( 4 ) (4)MaxnXj=1 nXq=jwq!pjkj,subjecttoconstraints( 4 )and( 4 ) (4)AdjustingourdenitionofRjasRj=(Pnq=jwq)Kpj,8j=1,...,n,similarproofsdemonstratethatLemmas 1 3 holdfortheproblemofminimizingweightedcompletiontime.Therefore,thecomplexityofsolvingSGPundereachuncertaintysetissimilartotheonespresentedinSection 4.3.1.1 4.3.2.2RobustOptimizationProblemAlthoughsolvingtheSGPiseasy,creatingarobustoptimalsequenceforthisproblemisnotstraightforward.Example 4.1 inSection 4.2 demonstratesacaseinwhichtheWSPTruledoesnotprovidearobustoptimalsolutionfortheproblemofminimizingtotalweightedcompletiontime.Becausep1=w1><>>:1,ifjobjisscheduledastheqthjobinthesequence0,otherwise,8j2J,q=1,...,nIij=8>><>>:1,ifjobjisscheduledafterjobi0,otherwise,8i,j2J~Cj=completiontimeofjobjwhereallprocessingtimestakeontheiridealvalues,8j2J 79

PAGE 80

yj=thepercentageoflargestallowabledelayofjobjused(yj=kj=K),8j2Jmj=8>><>>:1,ifjobjisdelayed0,otherwise,8j2JThemin-maxmathematicalformulationoftheproblemispresentedbelow,where(I)representsthetotalincreaseintheobjectivefunctionvaluecausedbydelayedjobsinpresenceofuncertainty.Thedenitionof(I)foreachofthethreeuncertaintysets,US1,US2,andUS3,willbepresentedinformulations( 4 )( 4 ),( 4 )( 4 ),and( 4 )( 4 ),respectively.MinXj2Jwj~Cj+(I) (4)subjectto:nXq=1xjq=1,8j2J (4)Xj2Jxjq=1,8q=1,...,n (4)Iijxiq+xjs)]TJ /F5 11.955 Tf 11.96 0 Td[(1,8i,j2J,1q
PAGE 81

value,whichiseither0or1accordingtoConstraints( 4 ).Therefore,wecanrelaxtheassumptionofIijbeingbinary.Next,wedene(I)foreachuncertaintysetbyadjustingmodels( 4 ),( 4 ),and( 4 )usingthedecisionvariablesofthemodelpresentedin( 4 )( 4 ).Notethatthetermsjin( 4 )and( 4 ),andpjkjin( 4 ),areequivalenttoKpjyj.Bysubstitutingfory,weobtainthefollowingthreeformulationsfortheSGPcorrespondingtoeachuncertaintyset.ForUS1,wehave:(I)=MaxXj2J wj+Xi2JwiIji!Kpj!yj (4)subjectto:Xj2JKpjyj (4)0yj1,8j2J. (4)ForUS2,themodelisdenedasfollows:(I)=MaxXj2J wj+Xi2JwiIji!Kpj!yj (4)subjectto:Xj2JmjM (4)0yjmj,8j2J (4)mj2f0,1g,8j2J. (4)Finally,forUS3,wedene:(I)=MaxXj2J wj+Xi2JwiIji!Kpj!yj (4)subjectto: 81

PAGE 82

Xj2JKyj (4)0yj1,8j2J. (4)Next,wewillshowhowtoconvertthemodelpresentedin( 4 )( 4 )toanMIP.ForUS1andUS3,notethatbecause( 4 )( 4 )and( 4 )( 4 )arelinearprograms,wecanreplace(I)bytheoptimalobjectivefunctiontotheirdualformulations(duetothestrongdualitytheorem).ForUS2,though,theformulationpresentedin( 4 )( 4 )isanMIP.Theorem 4.4 presentsalinearprogramthatisequivalentto( 4 )( 4 ),thusallowingustoemploythestrongdualitytheoremtoformulate( 4 )( 4 )asanMIPforUS2. Theorem4.4. Theoptimalvalueofthefollowinglinearprogramequalstheoptimalvalueoftheproblemformulatedin( 4 )( 4 ).(I)=MaxXj2J wj+Xi2JwiIji!Kpj!yj (4)subjectto:Xj2JyjM (4)0yj1,8j2J. (4) Proof. Thecoefcientmatrixdeningtheconstraintsetfortheproblemformulatedin( 4 )( 4 )istotallyunimodular,whichthusimpliesthatanoptimalsolutionto( 4 )( 4 )mustexistinwhichally-variablesarebinaryvalued(notingthatthefeasibleregionoftheproblempresentedin( 4 )( 4 )isnonemptyandbounded).Also,becausethem-variablesonlyappearin( 4 )and( 4 ),thestructureofthoseconstraintsguaranteesthatanoptimalsolutionexistsinwhichmj=1onlyifyj=1,8j2J.Thisfact,combinedwith( 4 ),ensuresthatyj=mj.Substitutingoutthem-values,theformulationspresentedin( 4 )( 4 )andin( 4 )( 4 )becomeidentical. 82

PAGE 83

Next,wepresentarobustMIPformulationcorrespondingtoeachuncertaintyset.Deneuasthedualvariableassociatedwiththeweightconstraint(Constraint( 4 ),( 4 ),and( 4 )forUS1,US2,andUS3,respectively),andvjasthedualvariableassociatedwithboundingconstraintscorrespondingtoj2J(Constraints( 4 ),( 4 ),and( 4 )forUS1,US2,andUS3,respectively).TheMIPformulationforUS1ispresentedbelow.MinXj2Jwj~Cj+u+Xj2Jvj (4)subjectto:Constraints( 4 )( 4 ),Kpju+vj wj+Xi2JwiIji!Kpj,8j2J (4)u0,vj0,8j2J. (4)ForUS2,themodelispresentedbelow.MinXj2Jwj~Cj+Mu+Xj2Jvj (4)subjectto:Constraints( 4 )( 4 ),u+vj wj+Xi2JwiIji!Kpj,8j2J (4)u0,vj0,8j2J. (4)Finally,forUS3,wehavethefollowingMIPformulation.MinXj2Jwj~Cj+u+Xj2Jvj (4)subjectto:Constraints( 4 )( 4 ), 83

PAGE 84

Ku+vj wj+Xi2JwiIji!Kpj,8j2J (4)u0,vj0,8j2J. (4)Linearityofformulationspresentedin( 4 )( 4 ),( 4 )( 4 ),and( 4 )( 4 )impliesthatwecansolveeachproblemusingastandardMIPsolver. 4.3.3MinimizingMaximumLateness/TardinessWhenminimizingthemaximumlatenessortardinessamongalljobsinaschedule,givendeterministicdata,theproblemcanbesolvedoptimallybysequencingthejobsinnondecreasingorderoftheirduedates(dj).Theseschedulesareknownasearliestduedate(EDD)schedules,duetoLawler[ 50 ].WeshowinthissectionthatEDDschedulesremainoptimalunderallthreeuncertaintysets. 4.3.3.1ScenarioGenerationProblemWebeginbystatingasimpleO(n)algorithmforsolvingSGPunderUS1. Theorem4.5. ForUS1,delayingjobsbytheirlargestpossiblevalues,intheorderthattheyappearin,solvestheSGPcorrespondingto. Proof. WeprovethatthestrategydescribedinTheorem 4.5 maximizesthecompletiontimeforeachjobinthesequence,andthereforemaximizesthelargestlateness/tardinessoccurringinthesequence.Bycontradiction,supposethatintheworst-casescenariocorrespondingtoasequence,thejthjobisdelayed(pjpj+,forsome>0)whilethereisajobinpositioni
PAGE 85

ForUS2,anoptimalsolutiontotheSGPcanbeobtainedusingthefollowingpolynomialalgorithm.Foreachjobpositionj=1,...,nin,deneGjasfollows.IfjM,thenGj=f1,...,jg,andifj>M,thenGjcontainsMjobshavingthelongestprocessingtimesamongalljobsinpositions1,...,j.(Thatis,forthecaseinwhichj>Mwehavethati2Gjfor1ijimpliesthatpipqforeveryqsuchthat1qjandq=2Gj.)Recallthat~Cjistheidealcompletiontimeofthejthjobin(whenalljobstakeontheiridealprocessingtime).DeneR0j=KPi2Gjpi+~Cj)]TJ /F3 11.955 Tf 12.46 0 Td[(djasthemaximumlatenessforjobjinschedule;thislatenessisachievedbydelayingalljobsinGjbytheirlargestpossibleamount.Hence,enumeratingthemaximumR0j-valueoverallj=1,...,nidentiestheworst-caseobjectivefunctionvaluecorrespondingto.Lettingq2argmaxj2f1,...,ngfR0jg,theworst-caseobjectiveisR0q.AnidenticalanalysisholdsforthecaseofmaximumtardinessbysettingR0j=maxf0,KPi2Gjpi+~Cj)]TJ /F3 11.955 Tf 11.95 0 Td[(djg.ForUS3,wemodifytheapproachusedforUS2.DeneGj,8j=1,...,n,asbefore,exceptthatthissetnowcontainstheminfd=Ke,jglongest-processing-timejobsinf1,...,jg.Also,denoteasajobwiththeshortestprocessingtimeinGj(2argmini2Gjfpig).Foreachj=1,...,n,computeR0j=KXi2Gjfgpi+()]TJ /F3 11.955 Tf 11.95 0 Td[(Kb=Kc)p+~Cj)]TJ /F3 11.955 Tf 11.95 0 Td[(djforthemaximumlatenesscase,andR0j=max8<:0,KXi2Gjfgpi+()]TJ /F3 11.955 Tf 11.96 0 Td[(Kb=Kc)p+~Cj)]TJ /F3 11.955 Tf 11.96 0 Td[(dj9=;forthemaximumtardinesscase.BythesamelogicgivenforthecaseofUS2,theworst-caseobjectiveisgivenbyR0qforsomeq2argmaxj2f1,...,ngfR0jg.ThisresultisachievedbydelayingjobsibyKpi,8i2Gq)-333(fg,anddelayingjobby()]TJ /F3 11.955 Tf 11.95 0 Td[(Kb=Kc)p. 85

PAGE 86

WenowshowthatthecomplexityofthealgorithmisO(nlogM)forUS2andO(nlog(d=Ke))forUS3.ThemainoperationinthealgorithmisformingthesetsGj,8j=1,...,n,andcalculatingthesumofprocessingtimesineachset.WestoreeachsetGjasa(sorted)binarytreeandrecursivelycalculateitusingGj)]TJ /F10 7.97 Tf 6.59 0 Td[(1.ForUS2,wecreateGjfromGj)]TJ /F10 7.97 Tf 6.59 0 Td[(1byinsertingjobjinthesortedbinarytreeGj)]TJ /F10 7.97 Tf 6.58 0 Td[(1(O(logM)operations)andadditsprocessingtimetoPi2Gj)]TJ /F17 5.978 Tf 5.75 0 Td[(1pi(O(1)operations).Whenj>M,wealsoremoveajobhavingtheshortestprocessingtimeinGj(tokeepthecardinalityofGjequaltoM).ThisrequiresO(1)operationstolocateandremovethisjob,andtosubtractitsprocessingtimefromPi2Gjpi.Findingthemaximumlateness/tardinessrequiresO(n)operationsinapostprocessingstep.Thus,thetotalcomplexityofthealgorithmforUS2isO(nlogM).AsimilardiscussionforUS3establishesacomplexityofO(nlogd=Ke)forthealgorithm. 4.3.3.2RobustOptimizationProblemWenowshowthatEDDschedulingresultsinanoptimalalgorithmfortherobustoptimizationproblemdiscussedinthissubsection,underallthreeuncertaintysets. Theorem4.6. AscheduleformedbytheEDDruleisoptimalfortherobustoptimizationproblemofminimizingmaximumlateness/tardinessinSMSP,underallthreeuncertaintysets. Proof. ConsideranEDDsequence,andsupposethatjhasthemaximumlatenessortardinessintheworst-casescenarioofsequence(Lmax=Lj=Cj)]TJ /F3 11.955 Tf 12.93 0 Td[(djorTmax=Tj=maxf0,Cj)]TJ /F3 11.955 Tf 12.9 0 Td[(djg).Supposethatisnotoptimal,andthat0isanoptimalsequence.First,notethatatleastonejobappearingbeforejinschedulemustappearafterjin0(orelse,applyingthesamedelaystojobsinschedule0asappliedintheworstcaseforscheduleyieldsatleastasmuchlateness(ortardiness)forjobjinschedule0).Letjob0qbethelatestscheduledjobin0amongthosejobsscheduledbeforejin.Inschedule0,jobj,alongwithalljobsscheduledpriortojin,arescheduledpriorto0qin0.Therefore,usingthesamedelaysin0asin 86

PAGE 87

,job0qcompletesin0afterjobjcompletesin.Moreover,d0qdjsinceisanEDDsequenceand0qisscheduledbeforejin.Asaresult,thevalueofLmax(Tmax)correspondingto0intheworst-casescenarioisatleastaslargeastheonecorrespondingto.Thiscontradictstheassumptionthatissuboptimal(if0isoptimal)andcompletestheproof. Theorem 4.6 impliesthatsolvingtherobustoptimizationproblem,underallthreeuncertaintysets,requiresO(nlogn)operations. 4.3.4MinimizingNumberofLateJobsMinimizingthenumberoflatejobswithdeterministicdataispolynomiallysolvablebyMoore'salgorithm[ 66 ],whichworksasfollows.WerstschedulethejobsinEDDordertoform.Then,weinvestigatethejobsintheorderthattheyappearinuntilwendtherstlatejob,sayj.Wethenremove(orreject)ajobinf1,...,jghavingthelongestprocessingtime,andcontinuetothenextlatejobuntilalljobshavebeenscheduledinorremoved.Wethenscheduleanyremovedjobsattheendof,inanyarbitraryorder,toformanoptimalsolution.Thisalgorithmmustbemodiedinthepresenceofuncertaintytoyieldanoptimalschedule.Forexample,consideratwo-jobinstancewherep1=p2=1,d1=1,andd2=2.LetK=1,andletuncertaintybudgetsbegivenby=M==1forthethreeuncertaintymodels.Schedulingjob1beforejob2producesnolatejobsassumingdeterministicdata.However,thissequenceresultsintwolatejobsinpresenceofuncertainty(forallthreeuncertaintysets)bydelayingjob1.Thereversesequence(job2beforejob1)yieldsonlyonelatejobintheworstcaseinallthreeuncertaintymodels. 4.3.4.1ScenarioGenerationProblemForUS1,theSGPissolvedbythesamemethodastheonepresentedinSection 4.3.3.1 forUS1(delayingjobsbythelargestpossibleamountintheorderthattheyappearin).AsdiscussedintheproofofTheorem 4.5 ,thisstrategyresultsinthelargestpossiblecompletiontimeforeveryjobinthesequence.Accordingly,it 87

PAGE 88

createsthemaximumnumberofdelayedjobsinthesequence.ItthenfollowsthatthecomplexityoftheSGPproblemunderUS1isO(n).ForUS2,weconstructadynamic-programmingalgorithmfortheSGP.Denefj(l,r)asthemaximumcompletiontimeofthejthjobin,whenwedelayroftherstjjobsbytheirlargestpossiblevalueandcreatellatejobs(amongtherstjjobs)bythisaction.Ifitisimpossibletocreatellatejobsbydelayingrjobs(amongtherstjjobsin),thenfj(l,r)=0.Aworst-casescenariocorrespondstothelargestpossiblevalueoflforwhichfn(l,M)ispositive.Westartbyinitializingfj(l,r)=0,8j,l=0,...,n,andr=0,...,M.Todescribeourrecursion,wedeneanindicatorfunction,I,suchthatIfg=1ifistrue,andIfg=0otherwise.Thefollowingrecursionconsidersfourpossiblecases,onecorrespondingtoeachcombinationofwhetherornotjwillbelate,andwhetherornotjisdelayed.Foreachj,l=1,...,nandr=0,...,M,wehave:fj(l,r)=max8>>>>>>>>>><>>>>>>>>>>:)]TJ /F24 9.963 Tf 4.57 -8.07 Td[(fj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l,r)+pjIffj)]TJ /F27 6.974 Tf 6.22 0 Td[(1(l,r)+pjdjg)]TJ /F24 9.963 Tf 4.57 -8.07 Td[(fj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l,r)]TJ /F22 9.963 Tf 9.96 0 Td[(1)+pj(1+K)Iffj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l,r)]TJ /F22 9.963 Tf 9.96 0 Td[(1)+pj(1+K)djg)]TJ /F24 9.963 Tf 4.57 -8.07 Td[(fj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l)]TJ /F22 9.963 Tf 9.96 0 Td[(1,r)+pjIffj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l)]TJ /F22 9.963 Tf 9.96 0 Td[(1,r)+pj>djg)]TJ /F24 9.963 Tf 4.57 -8.07 Td[(fj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l)]TJ /F22 9.963 Tf 9.96 0 Td[(1,r)]TJ /F22 9.963 Tf 9.97 0 Td[(1)+pj(1+K)Iffj)]TJ /F27 6.974 Tf 6.23 0 Td[(1(l)]TJ /F22 9.963 Tf 9.96 0 Td[(1,r)]TJ /F22 9.963 Tf 9.96 0 Td[(1)+pj(1+K)>djg. (4)Thersttwocasescorrespondtotheeventinwhichjisnotlate,andthelasttwocasescorrespondtotheeventinwhichjislate.Therstandthird(secondandfourth)casescorrespondtotheeventinwhichjisnot(is)delayed.Observethatifoneofthecasesdoesnotapply,thecorrespondingindicatorfunctionequalszeroandthecaseisignoredinthecomputationoffj(l,r).Thealgorithmstartsbysettingf0(0,0)=0.Theprocedurecomputesf1(l,r)forallcombinationsoflandr,thenf2(l,r)forallcombinationsoflandr,andsoon,uptofn(l,r)forallcombinationsoflandr.Afteridentifyinganoptimalobjectivefunctionvalue(largestlforwhichfn(l,r)>0),thesolutionleadingtothisvaluecanbefoundbybacktracking.ThecomplexityofthisalgorithmisO(Mn2),sinceO(Mn2)f-valuesmustbecomputed,and( 4 )requiresO(1)effortforeachf-value.Thealgorithmisillustratedbythefollowingexample. 88

PAGE 89

Example4.2. Considerafour-jobsequence=(1,2,3,4)wherep1=4,p2=6,p3=2,p4=10,d1=5,d2=12,d3=15,d4=30,K=0.5,andM=3.Weseektoidentifyjobprocessingtimevaluesthatmaximizethenumberoflatejobsin.Figure 4-2 demonstratesthecalculationofthef-valuesin( 4 ),inaforwardpropagationmanner.Inthisgure,ifweencountermultiplecandidatesforsomefj(l,r),thenonlyonenodecorrespondingtolargestvaluesoffj(l,r)isretained(deletednodesareshadedgray).Also,thereisnodelay4branchemergingfromnodef3(3,3)=18,becausethreejobshavealreadybeendelayedatthisnode,whichisthemaximumnumberofdelayedjobs.Theworst-casescenarioisgivenbydelayingjobs1,2,and4bytheirlargestpossiblevalue(p1=6,p2=9,p3=2,p4=15),whichresultsinfourlatejobs.Thenodecontainingfn(l,r)havingthelargestvalueofl,andthepathfromf0(0,0)=0tothisnode,aredisplayedusingthickarrows.NotethatwecannotdirectlyextendtheproposeddynamicprogrammingalgorithminordertosolveSGPunderUS3:risnolongerintegralinthatcase,anditisnotclearhowtoobtainanitestatespaceoverwhichourrecursiontakesplace.Therefore,weleaveopenthequestionofwhetherSGPispolynomiallysolvableforthisproblemunderUS3.Asanalternative,weproposeanMIPformulationforthecorrespondingSGPunderUS3.WerstdenetheSGPdecisionvariablescorrespondingtosequence.Uj=8>><>>:1,ifjobjislate0,otherwise,8j=1,...,nCj=completiontimeofjobj,8j=1,...,n 89

PAGE 90

Figure4-2. Dynamic-programmingSGPnetworkwithPj2JUjcriterionandUS2 kj=theproportionofpjbywhichwedelayjobj,8j=1,...,nOurMIPformulationfortheSGPunderUS3isgivenasfollows.MaxXj2JUj (4)subjectto:Cj=jXi=1(pi(1+ki)),8j=1,...,n, (4)Uj1+Cj)]TJ /F3 11.955 Tf 11.96 0 Td[(dj)]TJ /F6 11.955 Tf 11.95 0 Td[(j dj+j)]TJ /F5 11.955 Tf 13.54 2.66 Td[(~Cj,8j=1,...,n, (4)0kjK,8j=1,...,n, (4) 90

PAGE 91

nXj=1kj (4)Uj2f0,1g,8j=1,...,n, (4)wherejisthesmallestvaluebywhichjobjcanbelate.Notethatwecanassume,withoutlossofgenerality,thatdj+j)]TJ /F5 11.955 Tf 14.71 2.66 Td[(~Cj>0(Ifdi+i)]TJ /F5 11.955 Tf 14.71 2.66 Td[(~Ci0forsomei2J,theniislate,regardlessofthedelayscenario.Inthatcase,wexUi=0andremoveConstraint( 4 )wherej=i.)Itthenfollowsthatifjobjisnotlate,wehave)]TJ /F5 11.955 Tf 9.3 0 Td[(1(Cj)]TJ /F3 11.955 Tf 12.27 0 Td[(dj)]TJ /F6 11.955 Tf 12.27 0 Td[(j)=(dj+j)]TJ /F5 11.955 Tf 13.85 2.65 Td[(~Cj)<0andsoConstraint( 4 )forcesUj=0.Ontheotherhand,whenjislate,theright-hand-sideofConstraint( 4 )isgreaterthan1andsoUj=1atoptimality.Inpractice,becausethek-variablesarecontinuous,itisnecessarytouseverysmallvaluesfor-constantsinthismodel.Practicallyspeaking,onemightsetjasthesmallestdetectablevaluethatcausesajobtobelate.Forinstance,ifprocessingtimesaremeasuredinminutes,andjobjisnotpracticallylateuntilitisveminutespastdue,thenj=5. 4.3.4.2RobustOptimizationProblemWerstpresentamodicationofMoore'salgorithmforsolvingtheproblemofminimizingthenumberoflatejobsinaSMSPunderuncertainty.Then,inTheorem 4.7 ,weprovethattheproposedalgorithmgeneratesanoptimalrobustsequenceunderUS1.ModiedMoore's(MM)Algorithm. Step0. Initialize=astheremaininguncertaintybudget.LetbeanEDDscheduleofjobsand(initiallyempty)betheschedulethatweconstructusingthisalgorithm.Also,letR(initiallyempty)bethesetofrejectedjobsandrbethelastrejectedjob.DeneIitobethesubsetofjobsthatcompletebeforetheirdeadlineswhenweusetheMMalgorithmtoschedulejobsf1,...,ig(withI0=;).Initializei=j=1,whereiisthejobpositioncurrentlyunderexaminationinandjisthejobpositionbeingscheduledin. 91

PAGE 92

Step1. Tentativelyscheduleiinthejthpositionof(j=i)andsetpj=pj+minf,Kpjg.Updatethevalueoftoequal)]TJ /F5 11.955 Tf 12.54 0 Td[(minf,Kpjg.Ifjislatein,thengotoStep2;otherwise,gotoStep3. Step2. Adjustthescheduleofjobsinasfollows: Step2-1. Findq2argmaxs2f1,...,jgfpsg,andchooser=qtobethenextrejectedjob.AddqtoR,updatetoequal+pq)]TJ /F3 11.955 Tf 12.39 0 Td[(pq,andsetpq=pq.GotoStep2-2. Step2-2. Ifq=j,thengotoStep4;otherwise,gotoStep2-3. Step2-3. Setq=q+1.If>0andpq<(1+K)pq,thengotoStep2-4;otherwise,gotoStep2-5. Step2-4. Updatethevalueofpqtoequalpq+minf,(1+K)pq)]TJ /F3 11.955 Tf 12.56 0 Td[(pqgandupdatethevalueoftoequal)]TJ /F5 11.955 Tf 11.95 0 Td[(minf,(1+K)pq)]TJ /F3 11.955 Tf 11.96 0 Td[(pqg.GotoStep2-5. Step2-5. IncrementthevalueofqbyoneandgotoStep2-2. Step3. SetIi=Ii)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fjg.IncrementiandjbyoneandgotoStep5. Step4. SetIi=Ii)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fjgnfrg.IncrementibyoneandgotoStep5. Step5. Ifin,thengotoStep1;otherwise,schedulethejobsinRinpositionsj,...,nof,inanyorder,andterminate.InordertoprovetheoptimalityoftheMMalgorithm,werstpresentsomedenitionsandlemmastofacilitatetheproof.GivenanysetofjobsS,wedenoteanEDDsequenceofjobsinSasEDD(S)anddenethelengthofS(denotedbyCmax(S))astheworst-casemakespanvalueofjobsinSunderUS1(i.e.,Cmax(S)=Pj2Spj).WhenalljobsinStakeontheiridealprocessingtimes,makespanisdenotedby~Cmax(S)=Pj2Spj. Lemma4. Cmax(S1)}Cmax(S2)ifandonlyif~Cmax(S1)}~Cmax(S2),where}referstoanyofthefollowingoperations:<,,and=. Proof. Weprovethelemmaforthecaseinwhich}isthe
PAGE 93

ofjobsintheworst-casescenariocorrespondingtoanarbitrarysequenceofjobsinSi.Therefore,wehaveCmax(Si)=~Cmax(Si)+~Si,fori=1,2.NotethattheinequalitiesjKpj,8j2S1[S2,and~Si,fori=1,2,implythat~Si=minf,K~Cmax(Si)g,fori=1,2.Wecanthereforeconcludethatif~Cmax(S1)<~Cmax(S2),wehave~S1~S2andthereforeCmax(S1)
PAGE 94

(b)jIijjSj,8feasibleSf1,...,ig, (4)(c)Cmax(Ii)Cmax(S),8feasibleSf1,...,igsuchthatjSj=jIij.First,notethatforasinglejob(1),theMMalgorithmgeneratesI1=f1gifp1d1andI1=;otherwise.Hence( 4 )holdsfori=1.Fori2,weprovethat( 4 )holdsbycontradiction.Letq(2)bethesmallestvalueofiforwhich( 4 )doesnothold.LetDqbeaminimum-lengthfeasiblesubsetoff1,...,qgamongallmaximum-cardinalityfeasiblesubsetsofthesejobs.NotingthatIqisfeasiblebyconstruction,thenoneofthefollowingtwoconditionsholdsifIqdoesnotsatisfy( 4 ):Case1.jIqjCmax(Dq).Toanalyzethesecases,werstobservethatineverystepoftheMMalgorithm,atmostonemorejobwillbeaddedtothesetI,i.e.,jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1jjIqjjIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j+1.Furthermore,ifjIqj=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j+1,thenIq=Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1[q.ConsiderCase1rst.Wemusthavethatq2DqandjDqj=jIqj+1.Toseethis,notethatifq62Dq,thenDqisafeasiblesubsetoff1,...,q)]TJ /F10 7.97 Tf 6.59 0 Td[(1g,whichcontradictstheassumptionthat( 4 )holdsforIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1sincejDqj>jIqjjIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j.Also,ifjDqjjIqj+2,thenitfollowsthatjDqnfqgj>jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j,whichagaincontradictstheassumptionthat( 4 )holdsforIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1.DeneDq)]TJ /F10 7.97 Tf 6.59 0 Td[(1=Dqnfqg,andnotethatDq)]TJ /F10 7.97 Tf 6.59 0 Td[(1isafeasiblesubsetwithjIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j=jDq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j(jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1jjDq)]TJ /F10 7.97 Tf 6.59 0 Td[(1jisimpossiblebecausejDqj=jDq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j+1andjDqj>jIqjjIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j).Therefore,jDqj=jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j+1andsojIqj=jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j.Byinduction,Cmax(Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1)Cmax(Dq)]TJ /F10 7.97 Tf 6.59 0 Td[(1),whichimpliesthat~Cmax(Iq)]TJ /F10 7.97 Tf 6.58 0 Td[(1)~Cmax(Dq)]TJ /F10 7.97 Tf 6.58 0 Td[(1)byLemma 4 .Addingpqtobothsidesofthelatterinequalityyields~Cmax(Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fqg)~Cmax(Dq)andbyLemma 4 ,Cmax(Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fqg)Cmax(Dq).Thus,addingqtothesetIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1createsafeasiblesetI0qwithjI0qj=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j+1,whichcontradictstheassumptionthatIqwasgeneratedbytheMMalgorithmsinceIq=Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1. 94

PAGE 95

Hence,fortheremainderofthisproof,assumethatCase2holds,butCase1doesnot.Oneofthefollowingtwocasesmustarise:(2a)jIqj=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j+1;or(2b)jIqj=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j.InCase(2a),q2Iqandq2Dq(orelsetheinductionassumptioniscontradicted).DeningDq)]TJ /F10 7.97 Tf 6.58 0 Td[(1asabove,wehavethatIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1=IqnfqgandjDq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1j.ButthenhavingCmax(Iq)>Cmax(Dq)impliesthat~Cmax(Iq)>~Cmax(Dq)(byLemma 4 ).Subtractingpqfrombothsidesofthelatterinequalityyields~Cmax(Iq)]TJ /F10 7.97 Tf 6.58 0 Td[(1)>~Cmax(Dq)]TJ /F10 7.97 Tf 6.58 0 Td[(1).Lemma 4 thenguaranteesthatCmax(Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1)>Cmax(Dq)]TJ /F10 7.97 Tf 6.59 0 Td[(1),contradictingtheassumptionthat( 4 )holdsforIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1.InCase(2b),jIqj=jIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1jimpliesthatIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fqgisinfeasible,andsotheMMalgorithmaddsqtoIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1andremovesajobsIhavingthelongestprocessingtime(sI2argmaxfpsjs2Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fqgg)fromIq)]TJ /F10 7.97 Tf 6.59 0 Td[(1toformIq.Therefore,Cmax(Iq)Cmax(Iq)]TJ /F10 7.97 Tf 6.59 0 Td[(1).SinceCmax(Dq)`gandnotethatSDq(accordingtothedenitionof`).Let(D)beafeasiblesequenceofjobsinDqandformanewsequence0(D)byremovingjobqandthejobsinSfrom(D),processingtheremainingjobsasearly 95

PAGE 96

aspossible(inthesameorderrelativetoeachotherasbefore),andthenprocessingjob`,followedbythejobsinSinEDDorder.Notethat0(D)isapossiblescheduleforthejobsinD0q)]TJ /F10 7.97 Tf 6.59 0 Td[(1.Weclaimthat0(D)containsnolatejobs.Toprovethisclaim,notethatC(0(D))iC((D))i,8i2Dqn(S[fqg),sinceweschedulethesejobsin0(D)nolaterthantheywerescheduledin(D),whichfeasiblyscheduledthesejobs.Letm=jIq)]TJ /F10 7.97 Tf 6.58 0 Td[(1j=jD0q)]TJ /F10 7.97 Tf 6.58 0 Td[(1jandrecallthat0(D)m=EDD(Iq)]TJ /F10 7.97 Tf 6.58 0 Td[(1)m2S[f`g.AlsonotethatC0(D)m><>>:1,ifjobjisscheduledasthekthjobinthesequence0,otherwise,8j2J,k=1,...,n.Next,wedenedecisionvariablesthatmimictheSGPdynamicprogrammingprocessexplainedinSection 4.3.4.1 .Thefollowingvariablesaredenedwithrespectto 96

PAGE 97

combinationsofk-,l-,andr-valuesandsignifythatloftherstkjobswillbelate,giventhatrofthosekjobshavebeenmaximallydelayed.Forbrevity,wesimplyrefertoadelaytriple(k,l,r)inthedenitionsbelow.fklr=largestcompletiontimeofthekthjobin,givendelaytriple(k,l,r)8k=0,...,n,l=0,...,k,r=0,...,minfk,Mguklr1=8>>>>>><>>>>>>:1,ifthedelayedversionofthe(k+1)stjobinwouldnishafteritsdeadline,givendelaytriple(k,l,r)0,otherwise,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1guklr0=8>>>>>><>>>>>>:1,ifthenon-delayedversionofthe(k+1)stjobinwouldnishafteritsdeadline,givendelaytriple(k,l,r)0,otherwise,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg.Intuitively,ourformulationspeciesajobsequenceusingx-variablesanddeterminesthef-andu-variablescorrespondingtothesequence.Thesevaluesinduceanetworkwhosenodescorrespondto(k,l,r)delaytriples,asdepictedinFigure 4-3 .Arcsinthisnetworkexistfrom(n,l,r)toT,foralll=0,...,nandr=0,...,minfk,Mg,allwiththelengthof0.Everyotherarcinthenetworkisfromanode(k,l,r)to(k+1,l0,r0)forsomek=0,...,n)]TJ /F5 11.955 Tf 12.2 0 Td[(1,l=0,...,k,l0=l,l+1,r=0,...,minfk,Mg,andr0=r,r+1,anditslengthis0ifl0=land1ifl0=l+1.Thus,thelengthofeachpathfromS(correspondingto(0,0,0))toTdeterminesthenumberoflatejobsresultingfromthedelaystrategycorrespondingtothepath.Accordingly,thelongestpathinthisnetworkyieldsthemaximumnumberoflatejobs,giventhevaluesofu-variablesforajobsequence.Tomodelthelongest-pathproblemoverthis(acyclic)network,wedenethefollowingbinaryvariables: 97

PAGE 98

yklrl0r0=8>>>>>><>>>>>>:1,ifthelongestpathusesanarcconnectingdelaytriple(k,l,r)to(k+1,l0,r0)inthenetwork0,otherwise,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,l0=l,l+1,r=0,...,minfk,Mg,r0=rifr=M,r0=r,r+1ifr><>>:1,ifthelongestpathusesanarcconnectingdelaytriple(n,l,r)tonodeTinthenetwork0,otherwise,8l=0,...,n,r=0,...,M.Deneandasarbitrarilysmallandlargenumbers,respectively,andlet(u)bethemaximumpossiblenumberoflatejobs,givenu.Furthermore,denedmin=minj2Jfdjg,dmax=maxj2Jfdjg,andpmin=minj2Jfpjg.Themin-maxformulationispresentedbelow.Min(u) (4)subjectto:nXk=1xjk=1,8j2J (4)Xj2Jxjk=1,8k=1,...,n (4)fklr+Pj2J[pj(1+K))]TJ /F24 9.963 Tf 9.96 0 Td[(dj]xj(k+1) Pj2Jpj(1+K))]TJ /F24 9.963 Tf 9.96 0 Td[(dminuklr11+fklr+Pj2J[pj(1+K))]TJ /F24 9.963 Tf 9.97 0 Td[(dj]xj(k+1))]TJ /F37 9.963 Tf 9.97 0 Td[( dmax)]TJ /F24 9.963 Tf 9.96 0 Td[(pmin(1+K)+,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g (4)fklr+Pj2J[pj)]TJ /F24 9.963 Tf 9.97 0 Td[(dj]xj(k+1) Pj2Jpj(1+K))]TJ /F24 9.963 Tf 9.96 0 Td[(dminuklr01+fklr+Pj2J[pj)]TJ /F24 9.963 Tf 9.96 0 Td[(dj]xj(k+1))]TJ /F37 9.963 Tf 9.96 0 Td[( dmax)]TJ /F24 9.963 Tf 9.96 0 Td[(pmin+,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)f(k+1)(l+1)(r+1)24fklr+(1+K)Xj2Jpjxj(k+1)35)]TJ /F37 9.963 Tf 9.96 0 Td[((1)]TJ /F24 9.963 Tf 9.97 0 Td[(uklr1), 98

PAGE 99

8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g (4)f(k+1)l(r+1)24fklr+(1+K)Xj2Jpjxj(k+1)35)]TJ /F37 9.963 Tf 9.96 0 Td[(uklr1,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g (4)f(k+1)(l+1)r24fklr+Xj2Jpjxj(k+1)35)]TJ /F37 9.963 Tf 9.97 0 Td[((1)]TJ /F24 9.963 Tf 9.96 0 Td[(uklr0),8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)f(k+1)lr24fklr+Xj2Jpjxj(k+1)35)]TJ /F37 9.963 Tf 9.97 0 Td[(uklr0,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)fklr0,8k=0,...,n,l=0,...,k,r=0,...,minfk,Mg (4)uklr02f0,1g,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)uklr12f0,1g,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g. (4)Theobjectivefunction( 4 )minimizes(u),whichcorrespondstothenumberoflatejobsintheworst-casescenario.(TheoptimalvalueofSGPcorrespondsto(u),whichwillbecalculatedviathemodelpresentedin( 4 )( 4 )next.)Constraints( 4 )and( 4 )enforceapermutationscheduleforthejobs.InConstraints( 4 )and( 4 )wecanassume,withoutlossofgenerality,thatthedenominatorsarepositive.Toseethis,notethatPj2Jpj(1+K))]TJ /F3 11.955 Tf 12.34 0 Td[(dmin0impliesthatnolatejobexists,regardlessofthesequenceandthedelayscenario.Ontheotherhand,dmax)]TJ /F3 11.955 Tf 12.21 0 Td[(pmin(1+K)+0(dmax)]TJ /F3 11.955 Tf 12.34 0 Td[(pmin+0)impliesthatthedelayed(non-delayed)versionofeveryjobinJislate,regardlessofthesequence.Alloftheaforementionedcasesresultinoptimization 99

PAGE 100

problemswithtrivialsolutionsandareexcludedfromourstudies.Accordingly,theupperbound(lowerbound)ofbothConstraints( 4 )and( 4 )takeonvalueswithin[0,1)(within(,0])whenthe(k+1)stjobnishesontime,andwithin[1,1)(within(0,1])whenitnishesafteritsdeadline.Therefore,Constraints( 4 )(Constraints( 4 ))setuklr1(uklr0)tooneifthedelayed(non-delayed)versionofthe(k+1)stjobinnishesafteritsdeadline,andtozerootherwise.NotethattheseconstraintsaresimilartoConstraints( 4 ),withadditionallower-boundingconstraints(thelower-boundingconstraintsinthiscasearenotautomaticallyenforcedasisthecaseintheformulationpresentedin( 4 )( 4 )andmustbeexplicitlyaddedtotheproblem).Thenextfourconstraintsforcethef-variablestotakeontheircorrectvalues.Giventhatthekthjobinnishesatfklr,Constraints( 4 )(Constraints( 4 ))calculatethecompletiontimeofdelayedversionofthe(k+1)stjobwhenuklr1=1(uklr1=0).Similarly,Constraints( 4 )(Constraints( 4 ))calculatethecompletiontimeofnon-delayedversionofthe(k+1)stjob,whenuklr0=1(uklr0=0).Finally,Constraints( 4 )( 4 )statelogicalrestrictionsonthemodelvariables.NotethatthevalueofcanbedenedusingthesamelogicdiscussedinSection 4.3.4.1 fordeningjinConstraints( 4 ).Moreover,thelargestpossiblecompletiontimeofjobsinJestablishesapracticalvalueforinConstraints( 4 )( 4 ),i.e.,=(1+K)Pj2Jpj.Theproblemofndingalongestpathisstatedasfollows.(u)=Maxn)]TJ /F27 6.974 Tf 6.23 0 Td[(1Xk=0kXl=024minfk,MgXr=0yklr(l+1)r+minfk,M)]TJ /F27 6.974 Tf 6.23 0 Td[(1gXr=0yklr(l+1)(r+1)35 (4)subjectto:yklr(l+1)(r+1)uklr1,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g (4)yklrl(r+1)1)]TJ /F24 9.963 Tf 9.96 0 Td[(uklr1,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.97 0 Td[(1g (4)yklr(l+1)ruklr0,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4) 100

PAGE 101

Figure4-3. SGPnetworkforthePj2JUjcriterionunderUS2.Thevalue[a,b]oneacharcindicatesthatthearchasacostofaandacapacityofb. yklrlr1)]TJ /F24 9.963 Tf 9.96 0 Td[(uklr0,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)y00011+y00001+y00010+y00000=1, (4)yklr(l+1)(r+1)+yklrl(r+1)+yklr(l+1)r+yklrlr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(k)]TJ /F27 6.974 Tf 6.23 0 Td[(1)(l)]TJ /F27 6.974 Tf 6.23 0 Td[(1)(r)]TJ /F27 6.974 Tf 6.23 0 Td[(1)lr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(k)]TJ /F27 6.974 Tf 6.22 0 Td[(1)l(r)]TJ /F27 6.974 Tf 6.22 0 Td[(1)lr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(k)]TJ /F27 6.974 Tf 6.23 0 Td[(1)(l)]TJ /F27 6.974 Tf 6.22 0 Td[(1)rlr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(k)]TJ /F27 6.974 Tf 6.22 0 Td[(1)lrlr=0,8k=1,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4)ynlr,T)]TJ /F24 9.963 Tf 9.96 0 Td[(y(n)]TJ /F27 6.974 Tf 6.22 0 Td[(1)(l)]TJ /F27 6.974 Tf 6.23 0 Td[(1)(r)]TJ /F27 6.974 Tf 6.23 0 Td[(1)lr)]TJ /F24 9.963 Tf 9.97 0 Td[(y(n)]TJ /F27 6.974 Tf 6.23 0 Td[(1)l(r)]TJ /F27 6.974 Tf 6.22 0 Td[(1)lr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(n)]TJ /F27 6.974 Tf 6.23 0 Td[(1)(l)]TJ /F27 6.974 Tf 6.22 0 Td[(1)rlr)]TJ /F24 9.963 Tf 9.96 0 Td[(y(n)]TJ /F27 6.974 Tf 6.23 0 Td[(1)lrlr=0,8l=0,...,n,r=0,...,minfn,Mg (4)yklrl0r00,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg (4) 101

PAGE 102

l0=l,l+1,r0=rifr=M,r0=r,r+1ifr
PAGE 103

r0=rifr=M,r0=r,r+1ifr
PAGE 104

everyarcfromanode(k,l,r)pointstoanode((k+1),l0,r0)inthenetwork(l0=l,l+1,andr0=r,r+1),anditscostiseither0or1,wehavew(k+1)l0r0)]TJ /F3 11.955 Tf 12.19 0 Td[(wklr2f0,)]TJ /F5 11.955 Tf 9.3 0 Td[(1g,whichthenimpliesthatyklrlr02f0,1g.Theabovediscussionshowsthatwecanimplementstandardlinearizationtechniquesforquadraticprogramstolinearize( 4 ),similartothemethodimplementedinSection 2.3 tolinearizethequadratictermsin( 2 ).Deneklrl0(r+1)1=uklr1vklrl0(r+1)andklrl0r0=uklr0vklrl0rforeachdelaytriple(k,l,r),andl0=l,l+1.TheMILPformulationoftherobustproblemispresentedbelow.Minn)]TJ /F27 6.974 Tf 6.22 0 Td[(1Xk=0kXl=0minfk,M)]TJ /F27 6.974 Tf 6.23 0 Td[(1gXr=0klr(l+1)(r+1)1+vklrl(r+1))]TJ /F37 9.963 Tf 9.96 0 Td[(klrl(r+1)1+n)]TJ /F27 6.974 Tf 6.23 0 Td[(1Xk=0kXl=0minfk,MgXr=0klr(l+1)r0+vklrlr)]TJ /F37 9.963 Tf 9.96 0 Td[(klrlr0+w000 (4)subjectto:klrl0r0uklr0,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg,l0=l,l+1 (4)klrl0r0vklrl0r,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg,l0=l,l+1 (4)klrl0r0uklr0+vklrl0r)]TJ /F22 9.963 Tf 9.96 0 Td[(1,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg,l0=l,l+1 (4)klrl0r00,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,Mg,l0=l,l+1 (4)klrl0(r+1)1uklr1,8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g,l0=l,l+1 (4)klrl0(r+1)1vklrl0(r+1),8k=0,...,n)]TJ /F22 9.963 Tf 9.97 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g,l0=l,l+1 (4)klrl0(r+1)1uklr1+vklrl0(r+1))]TJ /F22 9.963 Tf 9.96 0 Td[(1,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g,l0=l,l+1 (4)klrl0(r+1)10,8k=0,...,n)]TJ /F22 9.963 Tf 9.96 0 Td[(1,l=0,...,k,r=0,...,minfk,M)]TJ /F22 9.963 Tf 9.96 0 Td[(1g,l0=l,l+1 (4)( 4 )( 4 )and( 4 )( 4 ) (4)AlthoughtheabovediscussionledustoanMILPformulationfortheproblemdiscussedinthissectionunderUS2,theproblemcomplexityremainsopen.InChapter 5 ,weinvestigatethelevelofdifcultyofthisproblembyimplementingthemodelpresentedin( 4 )( 4 )andsolvingasetofrandomly-generatedinstancesusing 104

PAGE 105

CPLEX.Wealsoexploreseveralheuristicideastogeneratelowerandupperboundsfortheproblemandimprovetheoptimalitygap.Asthelastcaseintroducedinourresearch,weconsidertheproblemofrobustSMSPwiththeobjectiveofminimizingthenumberoflatejobsunderUS3.RecallthatwewereunabletogenerateapolynomialalgorithmtosolvetheSGPcorrespondingtothisproblem.Also,notethatbinaryconstraintsintheMILPformulationpresentedin( 4 )( 4 )impliesnonconvexityoftheSGP.Thus,weomitfurtherdiscussioninthisregardandleavethisproblemasanopenquestion. 105

PAGE 106

CHAPTER5COMPUTATIONALANALYSISFORASPECIALCASEOFROBUSTSINGLE-MACHINESCHEDULINGPROBLEMInthischapterweprovideamoredetailedstudyforaspecialcaseofrobustsingle-machineschedulingproblemswheretheobjectiveistominimizethenumberoflatejobsintheworst-casescenariounderuncertaintyset2(US2),asdenedinSection 4.3.4 .WeusethesamenotationanddenitionsasthosepresentedinChapter 4 .Theremainderofthischapterisorganizedasfollows.InSection 5.1 wepresentheuristicupper-andlower-boundingalgorithmstogenerateandevaluateapproximatesolutionsforasetofrandominstances.TheninSection 5.2 wetesttheMILPformulationpresentedinSection 4.3.4 (see( 4 )( 4 ))bysolvingasetofrandomly-generatedinstancesusingCPLEX,wherewespecifytheupper-boundingsolutionpresentedinSection 5.1 asanincumbentsolutiontotheproblem.AllcomputationsinthischapterareperformedonanIntelCorei7witha1.80GHzprocessorand8.0GBRAM. 5.1ApproximateSolutionsInthissectionwepresentpolynomialalgorithmstondupperandlowerboundsfortheproblemdiscussedinthischapter.Wethensolveasetofrandomlygeneratedinstancesusingthesealgorithmsandspecifytheoptimalitygap. 5.1.1UpperBoundInordertondanupperboundfortheproblem,weuseaheuristicupper-bounding(HUB)algorithmtogenerateasequenceofjobs.Then,wesolvetheSGPcorrespondingtothecreatedsequenceusingthedynamic-programmingalgorithmpresentedinSection 4.3.4.1 .Astheoutputofthisprocess,weobtainthemaximumnumberoflatejobscorrespondingtoanarbitrarysequenceofjobs,whichyieldsanupperboundfortheproblem.TheHUBalgorithmisdescribedasfollows. Step0. LetbeanEDDsequenceofjobsand(initiallyempty)bethesequencethatweconstructusingthisalgorithm.Also,letR(initiallyempty)bethesetofrejectedjobsandrbethelastrejectedjob.DeneIitobethesubsetofjobsthatcomplete 106

PAGE 107

beforetheirdeadlineswhenweusethisalgorithmtoschedulejobsf1,...,ig(withI0=;).Also,deneGjtobethesetofminfM,jgjobshavingthelongestprocessingtimesamongalljobsinpositions1,...,j.(Thatis,forthecaseinwhichj>M,wehavethats2Gjfor1sjimpliesthatpspqforeveryqsuchthat1qjandq=2Gj.)Initializei=j=1,whereiisthejobpositioncurrentlyunderexaminationinandjisthejobpositionbeingscheduledin. Step1. Tentativelyscheduleiinthejthpositionof(j=i).Now,ifPjs=1ps+K(Ps2Gjps)>dj,thengotoStep2;otherwise,gotoStep3. Step2. Adjustthescheduleofjobsinasfollows: Step2-1. Findq2argmaxs2f1,...,jgfpsg,andchooser=qtobethenextrejectedjob.AddqtoRandgotoStep2-2. Step2-2. Ifq=j,thengotoStep4;otherwise,gotoStep2-3. Step2-3. Setq=q+1.GotoStep2-4. Step2-4. IncrementthevalueofqbyoneandgotoStep2-2. Step3. SetIi=Ii)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fjg.IncrementiandjbyoneandgotoStep5. Step4. SetIi=Ii)]TJ /F10 7.97 Tf 6.59 0 Td[(1[fjgnfrg.IncrementibyoneandgotoStep5. Step5. Ifin,thengotoStep1;otherwise,schedulethejobsinRinpositionsj,...,nof,inanyorder,andterminate.AftergeneratingusingtheHUBalgorithm,wecalculatethemaximumnumberoflatejobsinbysolvingthecorrespondingSGPusingthedynamic-programmingalgorithmpresentedinSection 4.3.4.1 .Wenextprovethattheworst-casecomplexityofthepresentedupper-boundingprocedureisO(Mn2).WerstclaimthatthecomplexityofHUBalgorithmisnomorethanO(Mn2).NotethatStep0ofHUBisexecutedonlyonceandrequiresgeneratingbysortingthejobsinEDDorder;sothecomplexityofthisstepisO(nlogn).Step1isexecutedntimes,andforeachj=M+1,...,n,itidentiesMjobshavingthelongestprocessingtimesinf1,...,jgtoformthesetGj(O(Mj)).Therefore,the 107

PAGE 108

worst-casecomplexityofthisstepisO(Mn2).Step2isperformedamaximumofO(n)times;moreover,foreachj=1,...,n,thissteprequiresndingajobhavingthelongestprocessingtime(O(j)).Thus,theworst-casecomplexityofthisstepisO(n2).Steps3and4eachrequireO(1)operationsandisperformednomorethanO(n)times.Finally,Step5requiresO(n)operationsoverthecourseofthealgorithm.Asaresult,theworst-casecomplexityofHUBalgorithmisO(Mn2),andsotheoverallcomplexityoftheupper-boundingprocedureisO(Mn2)sincethecomplexityofourdynamic-programmingalgorithmisalsoO(Mn2)(seeSection 4.3.4.1 ). 5.1.2LowerBoundsInthissectionweconsidersingle-stageandmulti-stageapproachestogeneratelowerboundsfortheproblem.Wepresentthedetailsoftheseapproachesinthefollowingsubsections. 5.1.2.1Single-StageApproachesInordertogeneratelowerboundsfortheproblem,weassumethatthesetofdelayedjobsisxed,regardlessofthesequenceofjobs.Weconsiderthreedifferentstrategiesforselectingthesetofdelayedjobs.Foreachcasewerstdeterminethejobprocessing-timevaluesviathecorrespondingstrategy;thenwendasequenceofjobswithminimumnumberoflatejobs,giventhatprocessingtimesstayunchanged.Becausetheprocessingtimesarepre-determinedforeachcase,ndingasequencethatminimizesthenumberoflatejobsisadeterministicproblemandcanbesolvedoptimallyusingMoore'salgorithm[ 66 ],presentedinSection 4.3.4 .Notethatthismethodobtainsalowerboundfortheproblembecausewereplacetheoptimalobjectivevalueoftheinner(maximization)probleminthemin-maxrobustoptimizationformulationoftheproblemwiththeobjectivevalueofafeasiblesolution.Next,weintroducethreedifferentstrategiesforselectingdelayedjobs.Then,inSection 5.1.4 ,wecalculatethelowerboundcorrespondingtoeachstrategyasdiscussedabove,forasetofrandomly-generatedinstances. 108

PAGE 109

Lowerbound1(LB1): Togeneratetherstlowerbound,weassumethatMjobshavinglongestprocessing-timevaluesaredelayed.Incaseseveraljobshavingthesameprocessingtimeexist,thejobswithearliestduedatesaredelayed. Lowerbound2(LB2): Foroursecondlowerbound,werstexecutetheHUBalgorithmpresentedinSection 5.1.1 anddeterminethesetofjobsthataredelayedwhenthealgorithmterminates.Wethenchoosethesamesetofjobstobedelayed. Lowerbound3(LB3): Tocreateourlastsingle-stagelowerbound,weuseaheuristiclower-bounding(HLB)algorithmtogenerateasetofdelayedjobs(D)asfollows. Step0. LetbeanEDDscheduleofjobsand(initiallyempty)betheschedulethatweconstructthroughthecourseofthisalgorithm.Also,letD(initiallyempty)bethesetofdelayedjobs.Initializei=j=1,whereiisthejobpositioncurrentlyunderexaminationinandjisthejobpositionbeingscheduledin. Step1. Tentativelyscheduleiinthejthpositionof(j=i).Ifjislateinwithoutbeingdelayed,thensetpj=pjandgotoStep2;otherwise,ifeitherjisnotlateinafterbeingdelayedorjDj=M,setpj=pjandgotoStep3.Else(jislateinonlyafterbeingdelayedandjDj
PAGE 110

Step6. IfjDj
PAGE 111

thenextiteration,wextheprocessingtimestobetheonesgeneratedinthepreviousiteration.Wecontinuetheprocedureinasimilarway,andattheendofeachiterationweupdatethelowerboundifwendagreatervalue.Westoptheprocedurewhenthelowerbounddoesnotimprovein10consecutiveiterations.Notethatalthoughourmulti-stageproceduresrequiremoreeffortcomparedtothesingle-stageprocedures,theyyieldbetterbounds.Inaddition,theaveragesolutiontimeforthe100instanceshaving100jobseachislessthanonesecondforbothLB4andLB5.ThedetailsofourtestinstancesandtheresultsofourcomputationsarepresentedinSections 5.1.3 and 5.1.4 ,respectively. 5.1.3TestProblemGenerationWegenerate100randominstancesoftherobustSMSPunderUS2,eachhaving100jobs(n=100),asfollows.Foreachinstance,werandomlygenerateanintegerbetween1andn)]TJ /F5 11.955 Tf 13.18 0 Td[(1asthemaximumnumberofdelayedjobs(M).ThevalueofparameterKandthevaluesofjobprocessingtimes(pj,8j2J)arerandomlydrawnfromtheintervalof[0,2]and[0,10],respectively.Inordertogeneratejobduedates,weintroduceaconstant,,asarandomlygeneratedvaluebetween0and5.Then,foreachjobj2J,wedene)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(jasfollows,)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(j=+jXi=1pi(1+K),8j2J, (5)andsetthevalueofdjtoarandomnumberintheintervalof[pj(1+K),)]TJ /F4 7.97 Tf 17.14 -1.8 Td[(j].Usingthismethodtogeneratedue-datevaluestendstoavoidinstanceshavingtrivialsolutions,i.e.,instancesinwhichdj-valuesareeithertoosmall(e.g.dj
PAGE 112

5.1.4ResultsTocompareandevaluateourupperandlowerbounds,wesolvetherandominstancespresentedinSection 5.1.3 usingeachalgorithm.Table 5-1 presentstheresultsofourcomputations.Inthistablewespecifyrandominstancesbynumbersbetween1and100inColumnInstance.Foreachinstancewepresenttheobjectivevalueobtainedbythelower-boundingAlgorithmi(LBi)presentedinSection 5.1.2 inColumnLBi(i=1,...,5),andthetheoneobtainedbytheupper-boundingalgorithm(UB)presentedinSection 5.1.1 inColumnUB.WethenevaluatetheoptimalitygapforeachinstancebypresentingthedifferencebetweentheupperboundandthegreatestlowerboundinColumnUB-Max(LBs).BasedontheresultspresentedinTable 5-1 ,multi-stagelowerbounds4and5(LB4andLB5)providethebestboundsinallofourinstances,whichispredictableduetotheirinteractivenature.In24instances,weareabletoclosethegapbetweenthelowerandupperboundsandprovetheoptimalityofthesequencegeneratedbyourupper-boundingalgorithm.WefurtherinvestigatethequalityofourupperboundbycomparingittotheoptimalobjectivevalueofsmallerinstancesinSection 5.2 5.2ExactSolutionInthissectionweimplementtheMILPformulationpresentedinSection 4.3.4 forsolvingtherobustSMSPwiththeobjectiveofminimizingthenumberoflatejobsunderUS2,i.e.,( 4 )( 4 ),usingCPLEX12.3viaILOGConcertTechnology.Werstseektoestimatethesizeofinstancesthatcanbesolvedtooptimalityinareasonabletimeusingthisformulation.Tothatend,werandomlygenerate20instancesviathesamemethodpresentedinSection 5.1.3 ,withaslightmodication:wenowassumethatthenumberofjobs(n)ineachinstanceisanintegerrandomnumberwithdiscreteuniformdistributionbetween2and10.(ThevalueofinourMILPformulationissetto0.1forallinstances.)WethensolveeachinstanceusingCPLEXasfollows. 112

PAGE 113

Table5-1. Evaluationoflower-andupper-boundingheuristicalgorithms InstanceLB1LB2LB3LB4LB5UBUB-Max(LBs) 127202727272812668891783627981894181318181826852314232424251622222222222207667771698232523252627192019202324295102020202222253118491010221212202020222226413847910211114252625262626015231823232329616261926262630417241624242425118141214151524919181818222228620262626272727021141514161522622231523242426223231423242424024101110111219725262226262626026172017201924427171719202029928262626262627129192120222223130201520212121031107101111231232252425262627133252625272727034617882315351041011122311362425252727281372222222424284381313141515216392117212121254401010101010122 113

PAGE 114

Table 5-1 .Continued InstanceLB1LB2LB3LB4LB5UBUB-Max(LBs) 412422242424240422825282828280432312232323230442121212223263451514161716236462518252525250471818181818191482016202020255492313232323285501819182020222511922202323252521291214142395319171920212545411121213142175526242625262605614151416161935716151619192455823222323232415922212222232856021212122222426124172426262606264799167633017302930300643435353535350651919192020301066251925252527267261926262626068252525252525069241324242425170232323232323071222322242424072888881137316101717172697419151920202667533445947688999167772120212222253781512151616215791912191919234802020202121210 114

PAGE 115

Table 5-1 .Continued InstanceLB1LB2LB3LB4LB5UBUB-Max(LBs) 8121172122212978225182525252508311101114132288491091010177852425252626260861617162020244871516151615193882011202020233893247716990232323232323091959131320792101010111217593171818202025594961012122412951710171817246962522252525294975456617119815151515151729924152424242841003223323232320 Asapreprocessingstep,weusethealgorithmpresentedinSection 5.1.1 togenerateanupper-boundingsequenceandadelayscenarioforeachinstance.WethenaddthissolutionasanincumbentsolutiontothecorrespondinginstanceinCPLEX.Thispreprocessingstepislikelytoexpeditetheprocessofoptimallysolvingtheproblembyreducingthesizeofthebranch-and-boundtree(fathomingbrancheswhosecorrespondingobjectivevaluesaregreaterthanourupper-boundvalue)earlyon,whichmaydramaticallyreducethesolutiontime,especiallywhentheupperboundistight.WealsosettheoptimalitytoleranceinCPLEXto0.999,becausetheobjectivefunctionoftheproblem(numberoflatejobs)isintegerandanydifferencelessthanonebetweenthelowerandupperboundprovestheoptimalityofourbestincumbentsolution.WethensolveeachinstanceusingCPLEXuntileithertheoptimalitytolerance,ortheone-hour(3600seconds)timelimitisreached. 115

PAGE 116

Table5-2. Evaluationofexactsolutionmethod(2n10) InstancenMKUBTimeNodesCPLEXLBCPLEXUB 1960.96033600156511.500321060.28533600327561.04733840.411340484792.00134611.4001001.00015650.811253031.01326611.9891000.02617871.4161200.00218521.2562102.00029751.1693144192.005310211.2820000.000011311.5701000.013112640.2292301.010213950.7141200.002114730.6393125912.017315981.4883219780232.001316531.8922102.000217761.15333415082.004318511.5641000.007119741.843320978682.005320750.88321615461.0012 TheresultsoftheexperimentsdiscussedinthissectionarepresentedinTable 5-2 .Inthistableforeachinstancenumber(1,...,20)listedinColumnInstance,thecorrespondingparametervalues,n,M,andK,arepresentedinColumnsn,M,andK,respectively.Theupper-boundvaluegeneratedinthepreprocessingstepislistedinColumnUB.Afterimplementingthediscussedsolutionmethod,werecordthesolutiontime(Time)inseconds,numberofnodesinthebranch-and-boundtree(Nodes),andthecalculatedboundsfortheprobleminCPLEX(CPLEXLBforLowerBoundandCPLEXUBforUpperBound).NotethatsolutiontimeoftheinstancesinwhichCPLEXfailstosolvetheproblemtooptimalitywithinthetimelimitarepresentedas.AccordingtotheresultspresentedinTable 5-2 ,mostoftheinstanceswithupto10jobsaresolvableinlessthanonehour;however,thesolutiontimegrowssubstantiallybyincreasingthesizeoftheproblem.Therefore,wepredictthatourmethodwillfailtosolvemostoftheinstanceswith10jobsormoreinaone-hourtimeframe.Another 116

PAGE 117

pointworthnotinghereisthatinalltheinstancesthathavebeensolvedtooptimality,theupperboundgeneratedinthepreprocessingstepisoptimal.Thisresultimpliesthattheupper-boundingalgorithmpresentedinSection 5.1.1 islikelytoprovideatightupperbound,evenforlargerinstances.Tofurtherevaluatethequalityofourupper-boundingalgorithmwerandomlygenerate20otherinstances,eachhaving10jobs,usingthemethodpresentedinSection 5.1.3 .Asmentionedbefore,severaloftheseinstancesmightnotbesolvedtooptimalitywithinonehourusingourexactsolutionmethod,butweexpecttosolvesomeofthem.Comparingtheoptimalobjectiveoftheserelativelylargeinstancestoourupperboundwillthenprovideabetterevaluationoftheupperboundquality.Table 5-3 presentstheresultsofthisstudy. Table5-3. Evaluationofexactsolutionmethod(n=10) InstanceMKUBLB1TotaltimeNodesCPLEXLBCPLEXUB 191.17041869360089751.5004240.11428214333031.0022340.185220821442261.0042451.60645723600174221.0434581.4443670360080431.5263661.78034843600216241.2133771.4422622223550961.0032841.13834243600156451.6253961.21933563600459281.07031050.760225036366641.00121170.6826876360067962.07861281.23041221360078101.11741330.00931143600577421.77831481.55535583600181821.05531570.6874747360068321.27741650.181267371856341.00121760.8802780132260611.00221821.969284955995631.00121961.83735233600185921.05132050.44334373600228841.3333 ThecolumnsinTable 5-3 aresimilartotheonesinTable 5-2 ,exceptthatColumnnisomittedforbrevity(n=10forallinstances)andColumnTimeissubstitutedby 117

PAGE 118

twocolumns:LB1,whichreportsthetime(seconds)ittakesforthesolvertondtherstlowerboundvalueofgreaterorequalto1,andTotaltime,whichcontainsthetotalsolutiontimeinseconds,similartoTimeinTable 5-2 .AccordingtotheresultsobtainedinTable 5-3 ,theonlyinstancesthataresolvedtooptimalitywithinonehouraretheoneshavingtheoptimalvalueof2.Thisimpliesthatlowerboundimprovementhappensveryslowlyandinallinstancesexceptone,itsvalueliesintheintervalof(1,2)evenafteronehour;therefore,wereachoptimalityonlyininstanceshavingupper-boundvalueof2.Itisalsoworthnotingthatourinitialupperboundhasnotbeenimprovedinanyoftheinstances,andthatitisequaltotheoptimalvalueinallinstancesthathavebeensolvedtooptimality.Thisresultonceagainimpliesthatourupper-boundingalgorithmcreatesstrongboundsfortheproblemevenforlargerinstances.Asthelaststepofourresearchweinvestigatewhetherthevaluesgeneratedbyourupper-boundingalgorithmarelocallyoptimalintheinstancesinwhichCPLEXfailstondtheoptimalsolutionwithinthetimelimit.Tothatend,weconduct2-optlocalsearch[ 23 ]forallsuchinstancesbyswappingeverypairofjobsinthesequencegeneratedbyourupper-boundingalgorithmandcalculatingthemaximumnumberoflatejobscorrespondingtoeachresultingsequenceusingthedynamicprogrammingalgorithmpresentedinSection 4.3.4.1 .Weobservethatforeachinstancethemaximumnumberoflatejobscorrespondingtoeveryswappedsequenceisatleastaslargeastheupper-boundvalueforthatinstance.Theresultofthisexperimentimpliesthatnoimprovementcanbemadebyswappingjobpairsinanyoftheinstancesandthereforeourupper-boundingalgorithmtendstobelocallyoptimal. Remark5.1. Althoughtheupper-boundingalgorithmpresentedinSection 5.1.1 succeedsinndinganoptimalsolutionineveryinstancethatissolvedtooptimalityinourexperiments,andislocallyoptimuminotherinstances,wecannotconcludelocaloptimalityofthisalgorithm.Indeedthisalgorithmfailstondtheoptimalsolutionin 118

PAGE 119

someotherinstances.Forexample,considerathree-jobinstancewheretheprocessingtimesarep1=2,p2=10,andp3=3,andtheduedatesared1=3,d2=19,andd3=19.Also,letK=M=1.Theupper-boundingalgorithmpresentedinSection 5.1.1 generatesthesequence3,1,2(or3,2,1)sincebothjobs1and2arerejectedinStep2ofthealgorithmandarescheduledattheendofthesequence.Thissequenceresultsintwolatejobs,ifwedelayjob2.However,theoptimalsequenceis1,3,2whichcontainsnomorethanonelatejob,underanydelayscenario.Moreover,sequence3,1,2isnotlocallyoptimal,sinceswappingthersttwojobsresultsinanimproved(andgloballyoptimal)solution. 119

PAGE 120

CHAPTER6CONCLUSIONSANDFUTURERESEARCHInChapter 2 weconsideredamulticommoditynetworkowprobleminwhichtheprobabilityofsuccessfulowtransmissionthrougheachhubnodedecreasesastheowpassingthroughthehubincreases.Ourcomputationalresultsledustoconcludethatapowerfulgeneral-purposesolver,suchasGloMIQO,isanappropriatetoolforsolvingprobleminstancesthatarerelativelysmall,ifthenodalreliabilityfunctionsarequadratic.However,formoredifcultinstances,orforinstancesthatcannotbeformulatedasmixed-integerquadraticprograms,werecommendedtheuseofourcutting-planeprocedure.Futureresearchinthisareamayfocusongeneratingheuristicmethodstoimprovetheoptimalitygapfortheinstancesthatarenotsolvableinareasonabletimeframeusingourexactcutting-planealgorithm.Additionally,akeyassumptioninourmodelisthatowsthroughoutthenetworkarepossiblyruinedintransit,butarestillpassedalongtothedestinationaftertheyhavebeenruined.Analternativemodelmayexaminethecaseinwhichinspectionofthecargobeingshippedtakesplaceateachnode.Ifthecargoisfoundtobedamaged,itcanbeimmediatelydiscardedinsteadofbeingforwardedontoitsdestination.Asaresult,theprobabilitiesofsuccessfullyshippingcommodityowschangesinthiscase,sincetheunsuccessfulshipmentofonecommoditymaynowimprovetheoddsthatanothercommodityissuccessfullyshipped.Othervariationsmayconsiderthecaseinwhichowscanbemisdirected,orwhensplitowsareallowed.Eachofthesevariationswouldrequiredifferentapproachesthantheonesprescribedhere.InChapter 3 westudiedthetaxonomyoftherobustsingle-machineschedulingproblemsandpresentedacompletesurveyinthisarea.Thisstudyledustoconcludethatdespitetheextensiveuseofdiscretescenariosandindependentcontinuousintervalsinrepresentinguncertaindataintheliteratureofrobustsingle-machine 120

PAGE 121

scheduling,certainshortcomingsexistinbothrepresentations.Ononehand,itisnotalwayseasytodeterminetheexactpossiblescenariosforuncertainparametersintheproblem,especiallywhenthenumberofpossiblevaluesisverylarge.Ontheotherhand,incaseofrepresentingdataasindependentintervalsofuncertainty(Soyster'smethod),correlationsamongparametervaluesarenotaddressedandtheobtainedworst-casescenariosareveryunlikelytohappeninpractice.Therefore,usingmethodssimilartotheonessuggestedin[ 16 77 ]tocontrolthelevelofconservatisminschedulingproblemswithuncertaindataisapromisingareaoffutureresearch.InChapter 4 weappliedstate-of-the-artrobustoptimizationmethodstodeneandsolveasingle-machineschedulingproblemwithuncertainjobprocessingtimes,underfouralternativeoptimizationcriteria.Foreachproblem,thegoalwastondaschedulethatminimizestheworst-caseobjectivefunctionvalue.Weassumedthatjobprocessingtimescanberepresentedusingindependentcontinuousintervals.Wethenmoderatedthelevelofconservatismintheproblembyconningprocessingtimevaluestobelongtothreealternativeuncertaintysets.Foreachproblem(distinguishedbyitsoptimizationcriterionandtheuncertaintyset),westudiedthedifcultyofthescenario-generationproblemandtheresultingrobustoptimizationproblem.ResultspresentedinTable 4-1 implythatalthoughtheproblemofndingaworst-casescenarioispolynomiallysolvableforalmosteveryproblem(expectfortheproblemofminimizingnumberoflatejobsunderuncertaintyset3,whosecomplexityremainsunknown),theirrobustcounter-partsarenotobviouslysolvableinpolynomialtime(e.g.,fortheproblemofminimizingweightedsumofcompletiontimesunderallthreeuncertaintysetsandfortheproblemofminimizingnumberoflatejobs,underuncertaintysets2and3).Forsolvingtherobustproblemswithaknownworst-casescenariogenerationcomplexity,weeitherdevelopedoptimalalgorithms,orderivedamixed-integerprogrammingmodel.Futureresearchinthisareacanbedevotedtoexploringthecomplexityofopenproblemsintroducedinthischapter. 121

PAGE 122

InChapter 5 westudytheproblemofminimizingnumberoflatejobsinrobustsingle-machineschedulingproblemunderuncertaintyset2,asdenedinChapter 4 .Werstpresentupper-andlower-boundingheuristicalgorithmsfortheproblemandevaluatethembysolvingasetofrandomly-generatedinstancesusingeachheuristicandcomparingthebounds.Wethenimplementthesingle-stagemixed-integerprogrammingformulationpresentedforthisprobleminChapter 4 bysolvingasetofrandominstancesusingCPLEX.AccordingtotheresultsobtainedinChapter 5 ,instanceswithlessthan10jobsarelikelytobesolvedusingourmixed-integerprogrammingformulation( 4 )( 4 )inlessthananhour.Moreover,theupper-boundingschemepresentedinthischaptertendstogenerategood-qualitysolutionsforallinstancesandcanbeusedasagoodheuristicalgorithmfortheproblem.Asafutureresearchinthisarea,anefforttodevelopefcientexactalgorithmsforsolvinglargerinstancesoftheproblemwouldbeofinterest.Also,tighterlowerboundscanhelpreducethegapand/orproveoptimalityinsomeinstances. 122

PAGE 123

REFERENCES [1] Ahuja,R.K.,T.L.Magnanti,J.B.Orlin.1993.NetworkFlows:Theory,Algorithms,andApplications.PrenticeHall,UpperSaddleRiver,NJ. [2] Aissi,H.,C.Bazgan,D.Vanderpooten.2009.Min-maxandmin-maxregretversionsofcombinatorialoptimizationproblems:Asurvey.EuropeanJournalofOperationalResearch197(2)427. [3] Aloulou,M.A.,F.DellaCroce.2008.Complexityofsinglemachineschedulingproblemsunderscenario-baseduncertainty.OperationsResearchLetters36(3)338. [4] Andreas,A.K.,J.C.Smith.2008.Mathematicalprogrammingalgorithmsfortwo-pathroutingproblemswithreliabilityconstraints.INFORMSJournalonComputing20(4)553. [5] Assad,A.A.1978.Multicommoditynetworkowsasurvey.Networks8(1)37. [6] Aven,T.1985.Reliabilityevaluationofmultistatesystemswithmultistatecomponents.IEEETransactionsonReliability34473. [7] Averbakh,I.2000.Minmaxregretsolutionsforminimaxoptimizationproblemswithuncertainty.OperationsResearchLetters27(2)57. [8] Averbakh,I.2005.Computingandminimizingtherelativeregretincombinatorialoptimizationwithintervaldata.DiscreteOptimization2(4)273. [9] Barahona,F.1996.Networkdesignusingcutinequalities.SIAMJournalonOptimization6(3)823. [10] Barnhart,C.,C.A.Hane,P.H.Vance.1996.Integermulticommodityowproblems.W.H.Cunningham,S.T.McCormick,M.Queyranne,eds.,IntegerProgrammingandCombinatorialOptimization,LectureNotesinComputerScience,vol.1084.Springer-Verlag,Berlin,58. [11] Barnhart,C.,C.A.Hane,P.H.Vance.2000.Usingbranch-and-price-and-cuttosolveorigin-destinationintegermulticommodityowproblems.OperationsResearch48(2)318. [12] Ben-Tal,A.,L.El-Ghaoui,A.Nemirovski.2009.RobustOptimization.PrincetonUniversityPress,Princeton,NJ. [13] Ben-Tal,A.,A.Nemirovski.1998.Robustconvexoptimization.MathematicsofOperationsResearch23(4)769. [14] Ben-Tal,A.,A.Nemirovski.1999.Robustsolutionstouncertainlinearprograms.OperationsResearchLetters251. 123

PAGE 124

[15] Ben-Tal,A.,A.Nemirovski.2000.Robustsolutionsoflinearprogrammingproblemscontaminatedwithuncertaindata.MathematicalProgramming88(3)411. [16] Bertsimas,D.,M.Sim.2003.Robustdiscreteoptimizationandnetworkows.MathematicalProgramming9849. [17] Bertsimas,D.,M.Sim.2004.Thepriceofrobustness.OperationsResearch52(1)35. [18] Birge,J.R.,F.V.Louveaux.1997.IntroductiontoStochasticProgramming.Springer-Verlag,NewYork. [19] Bley,A.2003.ALagrangianapproachforintegratednetworkdesignandroutinginIPnetworks.ProceedingsofInternationalNetworkOptimizationConference(INOC2003).107. [20] Brunetta,L.,M.Conforti,M.Fischetti.2000.Apolyhedralapproachtoanintegermulticommodityowproblem.DiscreteAppliedMathematics101(1)13. [21] Colbourn,C.J.1987.TheCombinatoricsofNetworkReliability.OxfordUniversityPress,NewYork,NY. [22] Crainic,T.G.,M.Gendreau,J.Farvolden.2000.Simplex-basedTabuSearchforthemulticommoditycapacitatedxedchargenetworkdesignproblem.INFORMSJournalonComputing12(3)223. [23] Croes,G.A.1958.Amethodforsolvingtraveling-salesmanproblems.OperationsResearch6(6)791. [24] Daniels,R.L.,J.E.Carrillo.1997.-Robustschedulingforsingle-machinesystemswithuncertainprocessingtimes.IIETransactions29(11)977. [25] Daniels,R.L.,P.Kouvelis.1995.Robustschedulingtohedgeagainstprocessingtimeuncertaintyinsingle-stageproduction.ManagementScience41(2)363. [26] Dantzig,G.1955.Linearprogrammingunderuncertainty.ManagementScience1(3)197. [27] deFariasJr.,I.R.,H.Zhao,M.Zhao.2010.Afamilyofinequalitiesvalidfortherobustsinglemachineschedulingpolyhedron.ComputersandOperationsResearch37(9)1610. [28] El-Ghaoui,L.,H.Lebret.1997.Robustsolutionstoleast-squareproblemswithuncertaindatamatrices.SIAMJournalofMatrixAnalysisandApplications181035. [29] El-Ghaoui,L.,F.Oustry,H.Lebret.1998.Robustsolutionstouncertainsemideniteprograms.SIAMJournalonOptimization933. 124

PAGE 125

[30] Ford,L.R.,D.R.Fulkerson.1962.FlowsinNetworks.PrincetonUniversityPress,Princeton,NJ. [31] Garey,M.R.,D.S.Johnson.1979.ComputersandIntractability:AGuidetotheTheoryofNP-completeness.W.H.Freeman&Co.,Princeton,NJ. [32] Gavish,B.,K.Altinkemer.1990.Backbonenetworkdesigntoolswitheconomictradeoffs.ORSAJournalonComputing2(3)236. [33] Gavish,B.,P.Trudeau,M.Dror,M.Gendreau,L.Mason.1989.Fiberopticcircuitnetworkdesignunderreliabilityconstraints.IEEEJournalonSelectedAreasinCommunications7(8)1181. [34] Gendron,B.,T.G.Crainic,A.Frangioni.1998.Multicommoditycapacitatednetworkdesign.B.Sanso,P.Soriano,eds.,TelecommunicationsNetworkPlanning.KluwerAcademicsPublishers,Norwell,MA,1. [35] Graham,R.L.,E.L.Lawler,J.K.Lenstra,A.H.G.RinnooyKan.1979.Optimizationandapproximationindeterministicmachinescheduling:Asurvey.AnnalsofDiscreteMathematics,vol.5.287. [36] Heyman,D.P.,M.J.Sobel.2004.StochasticModelsinOperationsResearch:VolumeII,StochasticOptimization.DoverPublications,Mineola,NY. [37] Holmberg,K.,D.Yuan.2003.Amulticommoditynetwork-owproblemwithsideconstraintsonpathssolvedbycolumngeneration.INFORMSJournalonComputing15(1)42. [38] Hudson,J.C.,K.C.Kapur.1985.Reliabilityboundsformultistatesystemswithmultistatecomponents.OperationsResearch33153. [39] Ierapetritou,M.G.,C.A.Floudas.1998.Effectivecontinuous-timeformulationforshort-termscheduling.1.Multipurposebatchprocesses.Industrial&EngineeringChemistryResearch37(11)4341. [40] Jahn,O.,R.H.Mohring,A.S.Schulz,N.E.Stier-Moses.2005.System-optimalroutingoftrafcowswithuserconstraintsinnetworkswithcongestion.OperationsResearch53(4)600. [41] Janak,S.L.,X.Lin,C.A.Floudas.2007.Anewrobustoptimizationapproachforschedulingunderuncertainty:II.Uncertaintywithknownprobabilitydistribution.Computers&ChemicalEngineering31(3)171. [42] Jane,C.C.,J.S.Lin,J.Yuan.1993.Onreliabilityevaluationofalimited-ownetworkintermsofminimalcutsets.IEEETransactionsonReliability42354. [43] Kall,P.,S.W.Wallace.1994.StochasticProgramming.JohnWileyandSons,Chichester,UK. 125

PAGE 126

[44] Kasperski,A.2005.Minimizingmaximalregretinsinglemachinesequencingproblemwithmaximumlatenesscriterion.OperationsResearchLetters33(4)431. [45] Kasperski,A.2008.DiscreteOptimizationwithIntervalData:MinmaxRe-gretandFuzzyApproach(StudiesinFuzzinessandSoftComputing),vol.228.Springer-Verlag,Berlin. [46] Kasperski,A.,P.Zielinski.2006.Anapproximationalgorithmforintervaldataminmaxregretcombinatorialoptimizationproblems.InformationProcessingLetters97(5)177. [47] Kennington,J.L.1978.Asurveyoflinearcostmulticommoditynetworkows.OperationsResearch26(2)209. [48] Kouvelis,P.,R.L.Daniels,G.Vairaktarakis.2000.Robustschedulingofatwo-machineowshopwithuncertainprocessingtimes.IIETransactions32(5)421. [49] Kouvelis,P.,G.Yu.1997.RobustDiscreteOptimizationandItsApplications.KluwerAcademicPublishers,Norwell,MA. [50] Lawler,E.L.1973.Optimalsequencingofasinglemachinesubjecttoprecedenceconstraints.ManagementScience19(5)544. [51] Lebedev,V.,I.Averbakh.2006.Complexityofminimizingthetotalowtimewithintervaldataandminmaxregretcriterion.DiscreteAppliedMathematics154(15)2167. [52] Leon,V.J.,S.D.Wu,R.H.Storer.1994.Robustnessmeasuresandrobustschedulingforjobshops.IIETransactions26(5)32. [53] Li,Z.,M.G.Ierapetritou.2008.Processschedulingunderuncertainty:Reviewandchallenges.Computers&ChemicalEngineering32(4)715. [54] Li,Z.,M.G.Ierapetritou.2008.Robustoptimizationforprocessschedulingunderuncertainty.Industrial&EngineeringChemistryResearch47(12)4148. [55] Lin,J.S.,C.C.Jane,J.Yuan.1995.Onreliabilityevaluationofacapacitatedownetworkintermsofminimalpathsets.Networks25(3)131. [56] Lin,X.,S.L.Janak,C.A.Floudas.2004.Anewrobustoptimizationapproachforschedulingunderuncertainty:I.Boundeduncertainty.Computers&ChemicalEngineering28(6)1069. [57] Lin,Y.K.2001.Asimplealgorithmforreliabilityevaluationofastochastic-ownetworkwithnodefailure.ComputersandOperationsResearch28(13)1277. 126

PAGE 127

[58] Lin,Y.K.2002.Two-commodityreliabilityevaluationforastochasticownetworkwithnodefailure.ComputersandOperationsResearch291927. [59] Lin,Y.K.2007.Reliabilityevaluationforaninformationnetworkwithnodefailureundercostconstraint.IEEETransactionsonSystems,ManandCyberneticsPartA:SystemsandHumans37180. [60] Lin,Y.K.,J.Yuan.1998.Anewalgorithmtogenerated-minimalpathsinamultistateownetworkwithnon-integerarccapacities.InternationalJournalofReliability,QualityandSafetyEngineering5(3)269. [61] Liu,M.,C.Chu,Y.Xu,J.Huo.2012.Anoptimalonlinealgorithmforsinglemachineschedulingtominimizetotalgeneralcompletiontime.JournalofCombinatorialOptimization23(2)189. [62] Lu,C.C.,S.W.Lin,K.C.Ying.2012.Robustschedulingonasinglemachinetominimizetotalowtime.ComputersandOperationsResearch39(7)1682. [63] Mastrolilli,M.,N.Mutsanas,O.Svensson.2013.Singlemachineschedulingwithscenarios.TheoreticalComputerScience47757. [64] McCormick,G.P.1976.Computabilityofglobalsolutionstofactorablenonconvexprograms:PartIconvexunderestimatingproblems.MathematicalProgramming10(1)147. [65] Montemanni,R.2007.Amixedintegerprogrammingformulationforthetotalowtimesinglemachinerobustschedulingproblemwithintervaldata.JournalofMathematicalModellingandAlgorithms6(2)287. [66] Moore,J.M.1968.Annjob,onemachinesequencingalgorithmforminimizingthenumberoflatejobs.ManagementScience15(1)102. [67] Ouorou,A.,P.Mahey,J.-Ph.Vial.2000.Asurveyofalgorithmsforconvexmulticommodityowproblems.ManagementScience46(1)126. [68] Peixin,Z.,Z.Xin.2009.Asurveyonreliabilityevaluationofstochastic-ownetworksintermsofminimalpaths.InternationalConferenceonInformationEngineeringandComputerScience2009(ICIECS2009).1. [69] Pinedo,M.L.,L.Schrage.1982.Stochasticshopscheduling:Asurvey.M.A.H.Dempster,J.K.Lenstra,A.H.G.RinnooyKan,eds.,DeterministicandStochasticScheduling.Reidel,Dordrecht. [70] Pruhs,K.,J.Sgall,E.Torng.2004.Onlinescheduling.J.Leung,ed.,HandbookofScheduling:Algorithms,Models,andPerformanceAnalysis,chap.15.CRCPress,BocaRaton,FL. [71] Sabuncuoglu,I.,S.Goren.2009.Hedgingproductionschedulesagainstuncertaintyinmanufacturingenvironmentwithareviewofrobustnessandstability 127

PAGE 128

research.InternationalJournalofComputerIntegratedManufacturing22(2)138. [72] Smith,J.C.,S.Ahmed.2010.Introductiontorobustoptimization.J.Cochran,ed.,WileyEncyclopediaofOperationsResearchandManagementScience.Wiley,Hoboken,NJ,2574. [73] Smith,W.E.1956.Variousoptimizersforsingle-stageproduction.NavalResearchLogisticsQuarterly3(1)59. [74] Soh,S.,S.Rai.2005.Anefcientcutsetapproachforevaluatingcommunication-networkreliabilitywithheterogeneouslink-capacities.IEEETransactionsonReliability54133. [75] Sotskov,Y.,N.Egorova.2008.Sequencingjobswithuncertainprocessingtimesandminimizingtheweightedtotalowtime.Tech.rep.,InstituteofInformationTheoriesandApplicationsFOIITHEA.URL http://hdl.handle.net/10525/1009 [76] Soyster,A.L.1973.Convexprogrammingwithset-inclusiveconstraintsandapplicationstoinexactlinearprogramming.OperationsResearch211154. [77] Tadayon,B.,J.C.Smith.2014.Algorithmsandcomplexityanalysisforrobustsingle-machineschedulingproblems.Tech.rep. [78] Tadayon,B.,J.C.Smith.2014.Asurveyofrobustofinesingle-machineschedulingproblem.Tech.rep. [79] Taylor,M.A.P.,J.E.Woolley,R.Zito.2000.Integrationoftheglobalpositioningsystemandgeographicalinformationsystemsfortrafccongestionstudies.TransportationResearchPartC:EmergingTechnologies8(1)257. [80] Verderame,P.M.,J.A.Elia,J.Li,C.A.Floudas.2010.Planningandschedulingunderuncertainty:Areviewacrossmultiplesectors.Industrial&EngineeringChemistryResearch49(9)3993. [81] Wu,C.W.,K.N.Brown,J.C.Beck.2009.Schedulingwithuncertaindurations:Modeling-robustschedulingwithconstraints.ComputersandOperationsResearch36(8)2348. [82] Xue,J.1985.Onmultistatesystemanalysis.IEEETransactionsonReliability34329. [83] Yang,Hai,WeiXu,BingShengHe,QiangMeng.2010.Roadpricingforcongestioncontrolwithunknowndemandandcostfunctions.TransportationResearchPartC:EmergingTechnologies18(2)157. [84] Yang,J.,G.Yu.2002.Ontherobustsinglemachineschedulingproblem.JournalofCombinatorialOptimization6(1)17. 128

PAGE 129

[85] Yeh,W.C.1998.Arevisedlayered-networkalgorithmtosearchforalld-Minpathsofalimited-owacyclicnetwork.IEEETransactionsonReliability47(4)436. [86] Yeh,W.C.2001.Asimplealgorithmtosearchforalld-MPswithunreliablenodes.ReliabilityEngineeringandSystemSafety73(1)49. [87] Yeh,W.C.2004.AsimpleMC-basedalgorithmforevaluatingreliabilityofstochastic-ownetworkwithunreliablenodes.ReliabilityEngineeringandSys-temSafety83(1)47. [88] Yeh,W.C.2005.Anewapproachtoevaluatereliabilityofmultistatenetworksunderthecostconstraint.Omega33(3)203. [89] Zhao,L.,Y.C.Lai,K.Park,N.Ye.2005.Onsetoftrafccongestionincomplexnetworks.PhysicalReviewE71(2)026125(1). 129

PAGE 130

BIOGRAPHICALSKETCH BitaTadayonwasbornin1986inIran.Shereceivedherbachelor'sdegreein2008andhermaster'sdegreein2010fromtheDepartmentofIndustrialEngineeringatSharifUniversityofTechnology,Tehran,Iran.InAugust2010,shejoinedthePh.D.programintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFlorida.Herresearchinterestslieindifferentareasofoperationsresearch,includingintegerprogramming,networkoptimization,decompositionapproachestolarge-scaleoptimizationproblems,androbustoptimization.ShereceivedherDoctorofPhilosophydegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainthespringof2014andstartedherjobasaSeniorOperationsResearchAnalystatSabreAirlineSolutionsinthesummerof2014. 130


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E2CY226VK_BXDTER INGEST_TIME 2014-10-03T21:28:12Z PACKAGE UFE0046568_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES



PAGE 1

This article was downloaded by: [128.227.215.210] On: 06 February 2014, At: 17:49Publisher: Institute for Operations Research and the Management Sciences (INFORMS)INFORMS is located in Maryland, USA Management SciencePublication details, including instructions for authors and subscription information:http://pubsonline.informs.org Expectation and Chance-Constrained Models andAlgorithms for Insuring Critical PathsSiqian Shen, J. Cole Smith, Shabbir Ahmed, To cite this article:Siqian Shen, J. Cole Smith, Shabbir Ahmed, (2010) Expectation and Chance-Constrained Models and Algorithms for InsuringCritical Paths. Management Science 56(10):1794-1814. http://dx.doi.org/10.1287/mnsc.1100.1208 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial useor systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisherapproval. For more information, contact permissions@informs.org.The Publisher does not warrant or guarantee the articles accuracy, completeness, merchantability, fitnessfor a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, orinclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, orsupport of claims made of that product, publication, or service.Copyright 2010, INFORMSPlease scroll down for articleit is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, managementscience, and analytics.For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org

PAGE 2

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

PAGE 3

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

PAGE 4

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

PAGE 5

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

PAGE 6

!%%+"&FG !" & )&* &% 1-% -@D + ( %-+ & ( %. + & ( %. + + ( %/% 0 ( %1 % ( % ( % (&'%&%&%-&)%4&+@ &4,-&4%%'&& ) -) )% ) +&4% % F&+)% ) G1ALM 2$ %' 2 %&%' <&)&4% <) 2$ & 3 ) & %' 2 & 3 ) & %=)%& 2$ &),)%, &-4%-)4%-,)%,-& ),&4%4%-,+ %' 2 &&& %2'A',)&+)&4%4%-,+ ) & <-)4& 2 %' 2$ )% 2$ %' 2 %%<%2+%&4 %-%)%,-&%'%&+ 1A %%%&&+%%&%&&%%) %' 4' ' %' ". J" J= N % N / % N% / %$ 2$ 2 %' ,% % / & 2$ 2 / . / E) <& . / N% 4 0 & % / %',% 4 0 %$ / N% %&,% 4 1 / 0%'&, 4 1 N 40 "*'&$C &%&'24&+)A%+% -))% %C' -%+%'% 40 40 FG 4)%<%-)& %'-%4%-&.&&%')&&%%)+),)+&4%&%'& '&+%;<+%C' -&%)%&4&%)-&,&&%%) %'%4&')'&-&,".J" J==)&,A&-+%&)&&&%& '-&+-%&%,,%4&'&) -%&%-,K*)%)'-% 0,% 4%&'%%-'& %,)%%<)&))%,-&<-' &%'--&-&+% <)% 0, 4&%4&%,)%,% '-&'-&-&+% ,%)$&4%%%&4&%'-% -&".J" J=<)&)%2&''&,, -&+%,!..B%)<&.&& 5 & ".J" J= N% % / %' . N*&%&2')% 4%'&'$ )% 6 4 *&%&+'-% 07<&)% 7 4 07 & 7 7" 5 7 # 0# 7 6 %' 1 / 0 1 % =),&&&%),,. &&F*'&$(G B%-&+%FG+%'%' )'-%,.&&%'&&+)%).&&%'&%2&% '%, 8$& '-&+ 4<) 8$&)&&-,%% +)%-'&% & ) %'-) % ) ) ;<4'-%+%',. &&<&&% 2 8'-&+ 4<) 8&)%$&,%%+)%-'&% ) & % %'-) % ) ) C2%.&& &% 2 '-&+ 4-2,)%&%& %%$&-, 5 *,)&&&&%)') % ,&%& &+%&24%&' 1-%%'&+.&&&&)%

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

!%%+"&FG !" ,-&FKG%'4$&<&&%< &&--%,-&FK6G;< 2)"**)'FH2<+% !%?%?&%%'*'&$#G %$&%&+)''&+'4&' -')%&+%%)%< 4%2)%&> C $%&)%' ,,&%<&+)+%<-'%&& %-%'<-'%'&%&+-& &2-&+%&42%+4%-, C B %&%<&)%&>, C %' %&B-)%'A4%,-& +''%L)+M%'LA4 +M&42,)4$%' 4$%%B%%&%% %/ <) % ('&+4$%' 4$%&42%' / %),,-&+-,'%'&% -&42&%66&% 4$A4+&%-&+))&'6' ,),% & C %&*'&& %+&)%&'-&+(.J#OFJ5 G4&%J5(=)+2%'% -%&%,'%"33%"%<&) %!;>%'5C *!(-% &%&%'&(.3'B%<% )-F6'G&&& ,!)#-! )4$%<--FG %',)4$%<--FG -&+&%'-%4%-%'&+.&& %')-%&%-&=% )% %$ &%' %4+)%$ &-&&-%'%4%+(.3',%) &%4% D %&42, -%+&),%&4%<&)&) &&& C ,%&;<4 &%&+ C %%'%&%&(.3& %'&-&+%+%&&%-&+ C %&<'-)&%&+% %&%'<&)%&%)%P&% -%&%$&'2=% =)%4%+-,-+%'2%)% &%$&%2%&%,-&,%& %')%4%+(.3&&%2)% A4&,&%%')%, C ''%,&%)%4&+ %''%4'<&)&)&&& ,)A4+4$%% C%-<2'4%& &%%)%&)C'-&+ %/-&',-%+&)<%4&+&, &%-%&%,,%'%%) )%F%G'&24)''&$' &+'FG')'$&&24-".J" J=2& %+%&+%&)'-%F%' -'-%42'-+&4&.& &G=&-%)-%&%&% ,-%%)<4'&%66<&) %&2%2&+(.J#O)' '&$'&+'#%),) D % ?%%&$(.3)-4-&+)& %%)%'<&)%%4%+, '-&+-'&&)'+23&+ )%&%<%'')

PAGE 11

!%%+"&FG !" !" # !"#$% &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& &&& %'4') J=)%'-".J" J=2&%+%&+%))%2-'-% 42)&/-%',)%< 4'<&)&))-&&& ,# )&%<4&'&&+)%&&+& %-&%&'2-$%&%'"=. 'J %' &42'%4%+ -&-'%'&-'%?'-%&,% 4%%&E))%4 %+'-%&'-&4%-,% '-& &-%;<$%&)$<)&) %%&,%%I'-%&'-& %& 3 &9-<)))% <&&-'&)&%-&<%& %+&4&%<'%% & '%&+',)& 3 4%-%'< $%&),/-2&<)&)%%'&,, &,)&-%&-'&)%&'& %-& B'-)&$&% %01/ %' %001/ &%&+-6)%%%&&' &+--))%)P,%''% %4+)A+-F%'LPMG, <'2)$P,%&)'+F%'LPMG%'=)+-%'&'

PAGE 12

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

PAGE 13

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
PAGE 14

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

PAGE 15

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

PAGE 16

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

PAGE 17

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

PAGE 18

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

PAGE 19

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

PAGE 20

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

PAGE 21

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
PAGE 22

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