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Optimization Models and Methods for Network Flow Problems Arising in Railroad Industry

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Title:
Optimization Models and Methods for Network Flow Problems Arising in Railroad Industry
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1 online resource (139 p.)
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english
Creator:
Icyuz, Ilksen Ece
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University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Richard, Jean-Philippe P
Committee Members:
Guan, Yongpei
Geunes, Joseph Patrick
Yin, Yafeng
Acharya, Dharma

Subjects

Subjects / Keywords:
coal -- heuristics -- largescalemips -- polyhedralstudy -- railroads -- unittrains
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In this thesis, we develop models and methods for solving certain network flow problems that occur in the railroad industry. First, we study the monthly coal train reservations planning problem that involves a variety of operational and tactical decisions such as train formation, routing and scheduling. We propose a two-model heuristic solution methodology that is time-efficient and produces good quality solutions. On instances coming directly from railroads, we show numerically that our approach outperforms current practice. Second, we introduce an optimization model that incorporates all operational restrictions and specifications of an American Class I railroad company's actual planning process. Due to the physical size of the network and the level of details required for forecast planning, we present a heuristic algorithm to solve the problem. This heuristic allows operators to obtain reliable forecasts quickly as they receive train requests. Third, we study a specific class of network flow maximization problems that we call unsplittable maximum flow problems (UFP). In these problems, some of the vertices are required to satisfy no-split, no-merge restrictions. These problems provide a streamlined abstraction of the models that we developed for unit train scheduling. We show that an ideal formulation of the unsplittable requirement can be obtained in a higher-dimensional space.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Statement of Responsibility:
by Ilksen Ece Icyuz.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Richard, Jean-Philippe P.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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Applicable rights reserved.
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lcc - LD1780 2013
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UFE0045667:00001

MISSING IMAGE

Material Information

Title:
Optimization Models and Methods for Network Flow Problems Arising in Railroad Industry
Physical Description:
1 online resource (139 p.)
Language:
english
Creator:
Icyuz, Ilksen Ece
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Richard, Jean-Philippe P
Committee Members:
Guan, Yongpei
Geunes, Joseph Patrick
Yin, Yafeng
Acharya, Dharma

Subjects

Subjects / Keywords:
coal -- heuristics -- largescalemips -- polyhedralstudy -- railroads -- unittrains
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre:
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
In this thesis, we develop models and methods for solving certain network flow problems that occur in the railroad industry. First, we study the monthly coal train reservations planning problem that involves a variety of operational and tactical decisions such as train formation, routing and scheduling. We propose a two-model heuristic solution methodology that is time-efficient and produces good quality solutions. On instances coming directly from railroads, we show numerically that our approach outperforms current practice. Second, we introduce an optimization model that incorporates all operational restrictions and specifications of an American Class I railroad company's actual planning process. Due to the physical size of the network and the level of details required for forecast planning, we present a heuristic algorithm to solve the problem. This heuristic allows operators to obtain reliable forecasts quickly as they receive train requests. Third, we study a specific class of network flow maximization problems that we call unsplittable maximum flow problems (UFP). In these problems, some of the vertices are required to satisfy no-split, no-merge restrictions. These problems provide a streamlined abstraction of the models that we developed for unit train scheduling. We show that an ideal formulation of the unsplittable requirement can be obtained in a higher-dimensional space.
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.
Statement of Responsibility:
by Ilksen Ece Icyuz.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Richard, Jean-Philippe P.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045667:00001


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OPTIMIZATIONMODELSANDMETHODSFORNETWORKFLOWPROBLEMS ARISINGINRAILROADINDUSTRY By ILKSENECEICYUZ ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c 2013IlksenEceIcyuz 2

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Idedicatethisworktomyparentsandmydeargrandfather,MehmetIcyuz. 3

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ACKNOWLEDGMENTS Firstandforemost,IowemydeepestandmostsinceregratitudetomyPh.D. advisor,Dr.Jean-PhilippeP.Richard.IcannotpossiblyexpresshowmuchIadmired hismentorship,knowledgeandwisdomandhowluckyIfeelforhavingsuchakind, patientandunderstandingmentorduringmygraduatestudies.Withouthiscontinuous encouragementandpersistenthelp,thisthesiswouldnotbepossible. Iwouldliketoacknowledgethemembersofmythesiscommittee,Dr.Joseph Geunes,Dr.YongpeiGuan,Dr.YafengYinandDr.DharmaAcharya,fortheircontinuous supportandconsiderationthroughoutmyPh.D.journey.Iamalsothankfultoallmy professorsandcolleaguesintheIndustrialandSystemsEngineeringdepartment aswellasintheComputerandInformationScienceandEngineeringdepartment.I greatlyappreciatedandbenetedfromtheirbroadvision,deepknowledge,insightsand selessnessinsharing.Ienjoyedallourdiscussionsthatledmetoconstantlychallenge myselfintellectuallyinandoutsidetheclassroom. Inaddition,startingwithDr.ErdemEskigun,Iwouldliketothankallmycoworkers intheOperationsResearchunitatCSXTransportationforprovidingmeassistancein broadeningmyunderstandingandknowledgeintheindustryIconductedresearchon, fortheirwelcomingattitudewhileIwassettlingdowninanewworkenvironment,andfor theircontinuedfriendship, Lastbutnotleast,IamindebtedtoallmyfriendsinGainesvillewhohavebecomea familytomeduringalltheseyearsthatwewereapartfromourbelovedones.Theyhave alwaysbeenthereformetosharelaughsandtears.Mostimportantly,Iwanttothank mydearparentsSemaandYusufandmylovelysisterIlkeforalltheloveandsupport theyhavegivenmeovertheyearsdespitethelongdistancesthatseparatedus.Finally, Iwouldliketothankmyanc e,Dr.FerhatAy,withwhomIwillbeunitingmylifeinJuly 2013. Thankstoeveryonewhohaseverhadtheslightestbitofroleinmyjourney. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................9 CHAPTER 1INTRODUCTION...................................10 2ATWO-MODELSOLUTIONAPPROACHFORTHEMONTHLYCOALTRAIN RESERVATIONSPLANNINGPROBLEM.....................13 2.1ProblemCharacteristicsandDescription...................14 2.1.1ReservationProcess..........................15 2.1.2CoalNetwork..............................15 2.1.3CarsandBuildLocations.......................16 2.1.4FactorsAffectingtheQualityofaReservationPlan.........17 2.1.5ProblemSizeandComplexity.....................18 2.2LiteratureReview................................19 2.3SolutionMethodology.............................23 2.3.1AggregatedCapacityAssignmentProblemACAP.........25 2.3.1.1Problemformulation.....................25 2.3.1.2Solutionprocedure......................28 2.3.2DetailedPlanningProcedureDPP..................31 2.3.2.1Constructiveheuristic....................33 2.3.2.2Improvementheuristic....................45 2.3.3LoopingProcedure...........................49 2.4ComputationalResults.............................51 3MONTHLYCOALRESERVATIONSPLANNING..................63 3.1PlanningMonthlyCoalReservations.....................64 3.1.1CollectingReservations........................65 3.1.2CoalMines...............................68 3.1.3LoadDestinations...........................68 3.1.4RailTracks...............................69 3.1.5RailCars................................69 3.1.6Locomotives...............................72 3.1.7TravelTimes...............................73 3.1.8TimeRestrictions............................73 3.1.9CurrentMonthvs.PlanningMonth..................74 3.1.10CharacteristicsofBetterReservationPlans.............74 5

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3.2SolutionApproach...............................75 3.2.1HandlingPrivateSetsandSystemCars...............75 3.2.2BuildingaForecastModel.......................78 3.2.3MIPModel................................81 3.2.3.1Data..............................81 3.2.3.2Variables...........................83 3.2.3.3Constraints..........................84 3.2.3.4Objectivefunction......................87 3.2.4SolutionProcedure:IssuesandSolution...............88 4UNSPLITTABLENETWORKFLOWPROBLEMSINTRANSPORTATIONMODELS ANDAPPLICATIONS................................91 4.1PolyhedralAnalysis..............................95 4.1.1APolynomialExtendedFormulationof S k;l ..............98 4.1.2StrongFormulationsintheSpaceofOriginalVariables.......104 4.1.2.1Convexhulldescriptionsforspecialcases.........107 4.1.2.2Facet-deninginequalitiesfor PS k;l .............112 4.2ComputationalExperiments..........................127 4.3Conclusion...................................129 5CONCLUSION....................................131 APPENDIX ACONVEXHULLOFSNUFPEXAMPLE......................133 REFERENCES.......................................136 BIOGRAPHICALSKETCH................................139 6

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LISTOFTABLES Table page 2-1ComputationalresultsforMCTRPP.........................60 2-2Actualvs2-stepsolutionprocedureperformancecomparison..........61 2-3ACAPImprovementheuristicperformances.....................61 2-4DPPImprovementheuristicperformances.....................62 4-1ComparativeanalysisofdifferentformulationsforsolvingtheUFProot relaxation........................................130 7

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LISTOFFIGURES Figure page 2-1CtRAPnetwork....................................37 2-2ETSPnetwork.....................................42 4-1Singlenodeunsplittableowproblem........................96 4-2Anetworkwithmultipleno-splitnodes.......................97 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy OPTIMIZATIONMODELSANDMETHODSFORNETWORKFLOWPROBLEMS ARISINGINRAILROADINDUSTRY By IlksenEceIcyuz August2013 Chair:Jean-PhilippeP.Richard Major:IndustrialandSystemsEngineering Inthisthesis,wedevelopmodelsandmethodsforsolvingcertainnetworkow problemsthatoccurintherailroadindustry.First,westudythe monthlycoaltrainreservationsplanningproblem thatinvolvesavarietyofoperationalandtacticaldecisions suchastrainformation,routingandscheduling.Weproposeatwo-modelheuristic solutionmethodologythatistime-efcientandproducesgoodqualitysolutions.On instancescomingdirectlyfromrailroads,weshownumericallythatourapproach outperformscurrentpractice.Second,weintroduceanoptimizationmodelthat incorporatesalloperationalrestrictionsandspecicationsofanAmericanClassI railroadcompany'sactualplanningprocess.Duetothephysicalsizeofthenetworkand thelevelofdetailsrequiredforforecastplanning,wepresentaheuristicalgorithmto solvetheproblem.Thisheuristicallowsoperatorstoobtainreliableforecastsquicklyas theyreceivetrainrequests.Third,westudyaspecicclassofnetworkowmaximization problemsthatwecall unsplittablemaximumowproblems UFP.Intheseproblems, someoftheverticesarerequiredtosatisfyno-split,no-mergerestrictions.These problemsprovideastreamlinedabstractionofthemodelsthatwedevelopedforunit trainscheduling.Weshowthatanidealformulationoftheunsplittablerequirementcan beobtainedinahigher-dimensionalspace. 9

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CHAPTER1 INTRODUCTION TheUSrailroadinfrastructureoperatesovermorethan 200 ; 000 kmofrailtrackson whichtrainsmovebetweenterminalswithvariousspecialisationssuchasloading, unloading,crewchange,maintenanceandinspection.Trains,whicharetherail transportationunits,areextremelydependentoneachoneofthesecomponents. BecauseofthemassivesizeofthisinfrastructureintheUS,planningmodelscan becomeverydifculttosolve.Further,iftheseproblemsarenotsolvedappropriately, railroadsmayexperiencevariabilityinservicetimesanddelays.Sucheventsimpair thecompetitivenessofrailroadtransportationoverothermodesoftransportationifthey becomeprevalent.Therefore,railroadsareinconstantneedofefcientdecisionsupport toolstoincreasereliability,competitivenessandprotability. Railroadsoperationsinvolvedifferentlevelsofplanningdecisionsstrategic, tactical,operationalthatareofteninterrelated.Thecomplexityofthosedecisionsis largelyduetotherailroads'capital-intensiveinfrastructure,aswellastoitsmassive sizeintheUnitedStatesterritories.Forthesereasons,optimizationhasfound manyfruitfulapplicationsinrailroadplanningproblems.Optimizationisalsooneof reasonsthatrailroadtransportationhasemergedasanenergy-efcient,cost-effective, andenvironmentallyfriendlyalternativetootherrelativelymoreexiblemodesof transportation. Overthepastdecades,OperationsResearchORtoolsandmethodologieshave foundawideeldofapplicationintherailtransportationoffreightastheyoftenresultin substantialcostsavings.ProblemsaddressedinthepastusingOR-basedmethodscan beroughlyclassiedintothefollowingthreecategories: Problemsinvolvingstrategicdecisions:Theseproblemsinvolveinfrastructure designandredesign.Theyincluderelocationofyards/sidings,capacityincreaseof tracksegments,railcarandlocomotiveeetssizingandstructuring. 10

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Problemsinvolvingtacticaldecisions:Theseproblemstypicallyinvolvethe schedulingofsubsystems.Tacticalproblemsincludeblocking,trainscheduling, locomotiveschedulingandmaintenancescheduling. Problemsinvolvingoperationaldecisions:Theseproblemsincludedecision problemsrelatedtocrewrostering,traindispatching,andconsistbustingand formation. Theproblemswestudyinthisthesishavebothtacticalandoperationalfeatures. Wenextsummarizebrieysomeoftheproblemsthatshowsimilaritiestotheapplications westudyinthisthesis. Trainschedulingisawell-studiedproblemintheliterature.Itisoftenintertwined withtheblockingproblem,whichconsistsinconsolidatingindividualshipmentsinto blockssoastoreduceintermediatehandlingefforts.Giventheblockingnetwork,train schedulingproblemcreatestrainsi.e.,determinestheorigin,destination,intermediary stationsandstarttimeofatraintogetherwiththefrequencyofcreationandassigns createdtrainstotheblocks.Theconsistbustingandformationproblemaimstoassign theappropriatecountandtypeofequipmenttotrainsbasedonthetimeandlocation ofavailabilityoftherailcars.Thelocomotiveschedulingproblemseekstoassign appropriatesetoflocomotivestoalreadyscheduledtrains.Itprovidesadetailed routingmapoflocomotivesoverthenetworkconsideringrestrictionssuchasstructural limitationsoflocomotivetypes,fuelingandmaintenanceshopschedules.Thetrain dispatchingproblemseekstodeterminedetailedtrainmovementmapsaimingto minimizeoverallsystemdelaywhilesatisfyingnumerousoperationalconstraints enforcedbytherailroadinfrastructure.Crewrosteringistheproblemofassigning crewstotrainswiththeaimofminimizingsystemsdelayduetopoorplanningwhile satisfyingvarioussafetyandunionrules.Formoredetailsontheseandotherproblem typesarisinginfreighttransportationthereaderisreferredtoAssad1980,Cordeau etal.1998andCrainicandLaporte1997.Thesearticlespresentcomprehensive reviewswhereauthorsincorporatedifferentproblemcategorizationsandsummarizethe typesofoptimizationmethodsappropriatefordifferentplanningdecisions. 11

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InChapter2,wepresentanefcientsolutionmethodologytoapracticalproblem arisingintherailroadsindustry.Ourprobleminvolvestacticalandoperationaldecisions inbothforecastandplanninglevels.Forthesakeofsimplicity,wepresentinChapter2 astreamlinedversionoftheproblemthatcontainsonlyitssalientfeatures.InChapter3, wedescribetheoperationalconcernsthatourindustrycollaboratorhastotakeinto accountwhileproducinganimplementableplaninforecast.Wethenshowthesener characteristicscanbeincorporatedintoourmodels.Thesemodelsarethebasisfor theUnitTrainPlanningSystemUTPS,adecisionsupporttoolthatwasdeveloped incollaborationwiththeOperationsResearchteamofourindustrycollaborator,an AmericanClassIrailroadcompany.InChapter4,westudyastreamlinedversionofthe unittrainschedulingproblem.Inparticular,weconsideranetworkowproblemwitha combinatorialrestrictiononcertainnodes.Thiscombinatorialrestrictionrequiresthat theowenteringanodebedirectedtoanoutgoingarcunchanged.Thiscombinatorial requirementisomni-presentinthepracticalmodelsdescribedinbothChapter2and Chapter3.Inparticular,weperformapolyhedralanalysisofthesinglenodeunsplittable owproblem. InChapter5,wesummarizethecontributionsofthisthesisandpresentpossible directionsforfutureresearch. 12

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CHAPTER2 ATWO-MODELSOLUTIONAPPROACHFORTHEMONTHLYCOALTRAIN RESERVATIONSPLANNINGPROBLEM Railroadsfacediverseplanningproblems.Manyofthemaresolvedmanually duetothelackofefcientdecisionsupporttools.Althoughrecentstudiesshow thatsomeoftheseproblemscanbesuccessfullytackledwithOperationsResearch ORmodels,andalthoughsolutionalgorithmshavebeenproposed,avastarrayof business-specicproblemshaveyettobestudied.Modelingthemrequiresathorough understandingofthesysteminwhichtheyarise,sothatbusiness-specicstrategic, tacticalandoperationalgoalsareaddressedaccurately.Further,itiscrucialthatsolution methodologiesaredevelopedthatprovidegood-qualitysolutionstothesemodels inasufcientlysmallamountoftime.Becauseofthelargescaleofrailroadsinthe UnitedStates,improvementsinoperationcanhaveasignicantpracticalimpactdueto increasedcustomersatisfaction,moreefcientassetmanagementandreducedsystem uncertainty. Inthischapter,wedevelopanefcientsolutionmethodologyforonesuch practicalproblem.Theproblemweconsiderarisesintheunittrainoperationi.e., railtransportationofasinglehigh-volumemerchandisefromasinglepre-speciedload origintoasinglepre-speciedloaddestinationwithoutloading/unloadingoperation betweenoriginanddestination.Grain,bulk,rockandcoalareamongthevarious commoditieshandledthroughunittrains.Althoughinthischapter,wefocusspecically oncoaltransportation,theproposedmodelsandsolutionmethodologiescanbe extendedtootherunittraincommoditieswithsuitablemodications. IntheUnitedStates,70%ofcoaldeliveriesareperformedbyrailroads.The competitiveadvantageofrailroadsoverothermodesoftransportationcomesfromtheir lowcostoverlongdistances;seeGormanandHarrod2011fordetails.According toAAR,AssociationofAmericanRailroads2010,coalisimportanttorailroadsasit accountsforover47%oftheirtotalfreighttonnageandover25%oftheirtotalfreight 13

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revenue.FurtherAASHTO,AmericanAssociationofStateHighwayandTransportation Ofcial2012statesthatservingthecoalindustryisvitaltorailroadsasthisbusinessis thefoundationofcarrierprots.Hence,railroadshaveakeeninterestinimprovingtheir processesandensuringthatcoaltransportationremainscost-effective. Theproblemwestudyinthischapteristhemonthlyschedulingofcoalunittrains. Becauseorderrequestsarereferredtoasreservationsinthecoaltransportation business,werefertoourproblemasthemonthlycoaltrainreservationsplanning problemMCTRPP. Theremainderofthischapterisorganizedasfollows.In x 2,wegiveadetailed descriptionoftheproblemandintroducerelevantterminology.Wealsoarguethat straightforwardMIPmodelsandapproachesarenotadequatetosolveit.In x 3,we provideabriefsurveyofexistingrailroadtransportationmodelsthataresimilarto theonewedevelop.Inparticular,wecompareandcontrastourproblemandsolution methodologieswiththosealreadystudiedintheliterature.Wethenproposein x 4an approachtosolveit.Thismethodologyconstructsimplementable,time-efcientand near-optimalsolutionstotheproblemthroughthesolutionofseveralMIPmodels.We giveageneraldescriptionofeachmodelandoutlinethealgorithmsweusetosolve them.Wepresentcomputationalresultsin x 5andgiveconcludingremarksin x 6. 2.1ProblemCharacteristicsandDescription MCTRPPistheproblemofdeterminingthelargestadequatesubsetofcoal reservationsthatarailroadshouldacceptinamonthtogetherwithamonthlyschedule thatsupportsthissubsetofreservations.Thisproblemiscomplexbecauserailroads haveavarietyofinternalcapacitylimitationsthatrestrictthenumberofreservationsthat canbesatised.Itrequireslong-termplanningasrailroadshaveavestedinterestin deliveringreservationstheycommittedtoatthebeginningofthemonth. Inthissection,wedescribeingreaterdetailthesalientfeaturesoftheproblem. Thereexistsavarietyofotherlessimportantpracticalfeaturesthatrailroadshaveto 14

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consider.Forthesakeofsimplicityinpresentation,wedonotdescribetheminthis chapter.WereferthereadertoChapter3foradiscussion. 2.1.1ReservationProcess Attheendofeachmonth,powercompanies,coalexportfacilityownersand othercustomersthatwerefertoasconsigneesplacerequestsforcoaltrainsi.e., reservationstorailroads.Whilecommunicatingtheirreservationrequests,consignees specifytheminealsoreferredtoasloadoriginwherethecoalwillbeloadedfrom, thecustomersitei.e.,destinationthatthecoalwillbedeliveredto,thetypeand ownershipofpreferredequipmenti.e.,railcarsthatwillcarrytheload,andthenumber andtonnageofcarsneeded.Becauseofsystemcapacitylimitationsandbecauseunit trainshavelowpriorityonrailroadnetworks,itisoftenimpossibleforrailroadstosatisfy allreservationrequestsduringthemonth.Therefore,coalmanagershavetodecide, sometimesinamatterofminutes,whethertoacceptorrejectareservation.When areservationisaccepted,therailroadcommitstothedeliveryofcoalwithrequested specicationsanddesiredequipment.Itiscrucialthatthesecommitmentsaremetmost ofthetime. 2.1.2CoalNetwork Coalnetworksaretheassociationofrailroadsegmentsandofthreetypesof terminalsthatcorrespondtovertices:loadorigins,destinationsandcrewchangeyards. Asmentionedbefore,minesarereferredtoasloadorigins.Loadoriginstendtobe concentratedinspecicregionsofthecountry.Destinationshoweverdifferwidelyin locationsandsizes.Theyrangefromsmallpowerutilityplantstolargecoalexporters. Asaresult,destinationsexhibitmorevariabilityintheircharacteristicsthanloadorigins. Fromoneterminalofthecoalnetworktoanother,weassumethattrainstravel alongshortestdistancepathsonthesubgraphformedbycrewchangeyards.Inother words,loadoriginsanddestinationsarenotvisitedbytrainsunlessaload-inora load-outeventisscheduled.Alongtheseknownpaths,thetotaltraveltimeisassumed 15

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tobethesumoftransittimesthroughnetworktracksegmentsanddwelltimesi.e., thetimeatrainspendsataterminalawaitingforterminaloperationssuchasloading, unloading,inspectionandcrewchangetobeperformedincurredatallterminalsalong thepath.Dwelltimesatterminalsdependonthetypeofworkbeingperformedonthe visitingtrain.Thedurationofthisworkisspecictoeachterminaldependingrstonits functionalityinthecoalnetworki.e.,whetheritisacrew-changeyard,aloadoriginora destination,whetherroutineinspection/maintenanceisperformedatthisterminaletc. andsecond,onwhetherthetrainisloadedornot. Becauseloadoriginsanddestinationshavedifferentsizes,speedsandcapacities forloading/unloadingequipment,thenumberoftrainstheycanserveeverydaydiffers. Werefertothemaximumnumberoftrainsthatcanbeprocessedataloadoriginor destinationduringasingledayasitsdailytraincapacity. 2.1.3CarsandBuildLocations Coalistransportedfromloadoriginstodestinationsinrailcars.Railcarsareeither ownedbyrailroadsorbyconsignees.Werefertothecarsthatrailroadsownassystem carsandtothecarsconsigneesownasprivatecars.Eachreservationrequestcomes withanownershipspecicationforthecarstobeusedwhileformingthetrainthatwill fullltherequest.Inadditiontotheownership,carsalsodifferinthewaytheyarebuilt andinsomeoftheirphysicalcharacteristics.Werefertosetsofstructurallydifferent carsascartypes.GenericcartypesincludeGondolasandHoppercarsofvarious makeandcapacities.Inthischapter,weusethetermcarkindtouniquelydescribethe combinationofacartype,carownershipandownerforprivatecarsi.e.,consignee. Further,werefertothecombinationofcarkindsusedtosatisfyareservationasitscar consist. Foreachdestinationterminal,onecarkindisidentiedaspreferredwhileseveral othersmightbeexcludedbytheconsigneebasedontheterminalequipmenthandling capabilities.Consigneesexpecttheirreservationstobesatisedwiththeirpreferred 16

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carkind.Thisisnotalwayspossiblehoweverasthenumberofcoalcarsofeachkind inthesystemislimited.Therefore,consigneesalsohavealistofadmissiblecarkinds associatedwitheachdestination.Wheneverthereisnotasufcientamountofcarsof thepreferredkindtosatisfyareservation,itispossible,althoughnotdesirable,toassign amixofcarsofdifferentkindsfromtheadmissiblelist.Carsofakindexcludedata destinationhowevershouldneverbeassignedtoreservationshavingthisdestination. Asreservationsmaydifferinthenumberofcarstheyrequire,carshaveto bedetachedfromorattachedtotrainsfromtimetotimeduringthemonth.Trains howevercanonlybeformedorbustedatcertaincrewchangeyardswherebothskilled manpowerandappropriateequipmentareavailable.Werefertotheseyardsasbuild locations.Thus,whileforminglongertrainsfromshorterones,emptycarsneedto beretrievedfromtheirpoollocationsorfromtheterminalassociatedwiththeirlast assignmentandbroughttobuildlocationstobeattachedtoatrain.Similarly,carsare removedfromlongtrainstotreservationswithsmallerdemandsatbuildlocations. Sinceresizingoperationsarecostlyandtime-consuming,theyshouldbeavoided wheneverpossible.Forthisreason,whenagroupofreservationsrequiresexactlythe samenumberofcarswithidenticalcarpreferences,itisdesirabletoassignthesame carconsisttoallreservationsinthegroupinordertopreventunnecessaryvisitstobuild locations.Further,whendecidingoncarstoassigntoareservation,itisimportantto considertheinitialemplacementofcargroupsinordertoavoidlongtraveldistances fromtheseinitialpositionstobuildlocationsaswellasfrombuildlocationstoload originsoncethetrainisformed. 2.1.4FactorsAffectingtheQualityofaReservationPlan Intheprevioussections,wedescribedsomeofthehardrequirementsthat mustbetakenintoaccountwhendesigningamonthlyplanforcoaltransportation. However,whendecidingwhetherornottoacceptareservation,severalothersofter considerationsmustbeconsidered.First,itisimportanttoacceptasmanyreservations 17

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aspossiblesincerejectingareservationresultsinlostopportunityandcustomer dissatisfaction.Ontheotherhand,itisimportantnottoaccepttoomanyreservations sincethismightleadtoreservationsnotbeingsatisedattheendofthemonthand fortherailroadtonotbeabletodeliveronitscommitments.Second,itisimportantto developaschedulewherethemajorityofreservationsarecompletelysatisedduring themonth.Reservationsinitiatedbutnotcompletedduringthemonthshouldnotbe countedequallysincetheywillconsumeequipmentandterminalcapacitiesduringthe followingmonth.Weshouldalsoavoidproducingscheduleswheremanyreservations areinitiatedduringthelastfewdaysofthemonth.Finally,itisimportanttosatisfyas manyreservationsaspossiblewiththeirpreferredcartypessoastomaximizecustomer satisfaction. Theaboveaspectsformsofterconcernsthatarebestconsideredinsideofobjective functions.Therelativeweightsofeachofthesefactorscanthenbeadjustedbasedon railroadpriorities. 2.1.5ProblemSizeandComplexity InapreliminaryattempttosolveMCTRPP,weformulatedtheproblemasa mixedintegerprogramMIPoveratime-spacenetwork.Assuming m loadorigins, d destinations, y yards, t =30 planningdaysand k carkindsavailableforcoal transportation,thecorrespondingtime-spacenetworkhas m + y + d t 21 ; 000 potentialverticesand m d + y m + y + d t t +1 = 2 k 11.5billionpotential arcs.Wedonotrevealtheactualnumbersheresincethesizeofthecustomerbaseof ourindustrycollaboratorisconsideredtobesensitiveinformation.Evenwithoutwriting downthedetailsoftheMIPformulation,itisclearthatthenumberofintegerdecision variablesrepresentingcarowsisofthesameorderofmagnitudeasthenumberof arcs.Eventhoughthetime-spacenetworkissparse,thenumberofdecisionvariables ofsuchMIPeasilyreacheshundredsofmillions.Programsofthisscaleexceedthe 18

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capacityofcurrentoff-the-shelfcommercialsolvers.Further,someofthefeaturesofthe modelwedescribelatermaketheuseofdecompositionmethodsdifcult. 2.2LiteratureReview Criticalplanningdecisionsarisingintherailindustryhavehistoricallybeen formulatedascomplexmathematicalprograms.Thiscomplexitycanbeattributedto thephysicalinfrastructureofrailroads.Thisinfrastructure,althoughcapital-intensive andoperationallyconstraining,isalsoacompetitiveadvantageoverothermodesof transportationsinceitallowsrailtransportationtobeenergy-efcient,cost-effective andenvironment-friendly.Inordertopreservethiscompetitiveedge,awidevariety ofOperationsResearchtoolsandmethodologieshavebeenproposed.Fortherail transportationoffreight,inparticular,comprehensivereviewsbyAssad1980,Cordeau etal.1998andCrainicandLaporte1997canbefoundintheliteraturethataddress differentlevelsofplanningdecisions.TworecentsurveysbyHarrodandGorman 2011andNemaniandAhuja2011summarizerecentadvancesinthearea.The formerdifferentiatesthreetypesoftrainsoperatedbyAmericanRailroadsasunit trains,dedicatedtrainsandloosecartrains.Theauthorsfocusmostlyonproblems arisingintheoperationofloosecartrains.Theyclassifyproblemsoftrainfrequency settingandcarroutingasstrategicdecisions,problemsofsettingdepartureandarrival timesoftrains,re-routingofcarstoagreewithtrainschedulesastacticaldecisions andproblemsofcarassignments,traindispatching,andmodicationofen-routetrain pathsinresponsetonetworkdelaysasoperationaldecisions.NemaniandAhuja 2011classifyexistingworkunderthreemaincategories:trainplanning,locomotive schedulingandcrewscheduling.Fortrainplanning,theygroupexistingworkunderthe categoriesofrailroadblocking.e.,aclassicationproblemaimingatreducingcosts byminimizingintermediatehandlingofrailcars,trainschedulingi.e.,givenablocking mapandnetworkcapacities,derivingadetaileddescriptionoftrainroutes,frequencies andtimetablesandtraindispatchingi.e.,derivingdetailedtimetablesateachtrack 19

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sectionthatminimizestotaldeviationfromtheoriginalplanwhilerespectingoperational andsafetyconstraints.Fortheblockingproblem,theauthorsdescribeaneffective neighborhoodsearchalgorithm.Forthetrainschedulingproblem,theyproposea decompositionapproachthatroutestrainsandassignsblockstotrainsintherstphase andthatderivesexacttraintimesandfrequenciesinthesecond.Finally,forthetrain dispatchingproblem,theauthorsformulateatime-spacenetworkmodelandsuggestan IP-basedheuristicthatdividestheplanninghorizonintooneshortandonelongportion. Fewarticlesintheliteratureaddresscoaltrainschedulingspecically.Liuand Kozan2011modelacoaltrainschedulingproblemthatconsiderstrackcapacitiesasa job-shopschedulingproblemwheretrainroutesaretreatedasjobs,singletracksections aretreatedasmachines,multipletracksectionsasparallelmachines,theeventofa trainpassingthroughasectionasanoperationandsectiontransittimesasoperational processingtimes.Tosolvetheproblem,theauthorsproposeaconstructiveandan improvementheuristicthatcombinesaneighborhoodsearchalgorithmwithtabusearch. Changetal.1981rstaddressthecoalallocationproblemwhichseekstominimize thecostofreallocatingcoaldemandandsupplyinthephysicalcoalnetworksubjectto pollutionconstraints.Theyenforcedemandrestrictionsoncustomersbasedontheir needsaswellassupplyrestrictionsonminingareasbasedontheircapacities.Second, theybuildaownetworkmodelthataimstooptimallyassignthecoaltrafcderivedin therstmodel,fromsupplytodemandvertices.Biellietal.1984considertheproblem ofschedulingmultipleunittraintypes.Theauthorsconsidereachtypeasadifferent commodity.Intheproposedmodel,transittimesofunittrainsareminimizedwhilethe totalvolumetransportedismaximizedsubjecttotrackcapacitiesandelasticsupplyand demandpatterns.Theauthorsuseresourcedirectivedecompositionmethodstosolve theproblem.Theydesignthemasterproblemtoallocatecapacitytoeachcommodity, andthesubproblemstoactassinglecommodityproblemsoncethecapacityallocations aredetermined. 20

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BecauseUSrailroadscommonlyuseblockingasamethodtoincreasethe efciencyoftheiroperations,themajorityofexistingworkexaminesthecarconsist busting-and-formationproblemindependentlyfromthetrainschedulingandrouting problems.SheraliandSuharko1998addresstherepositioningofemptycarsforthe railtransportationofautomobiles.Threetypesofrailcarsareavailableinthisproblem. Theobjectiveistominimizecostandunmetdemandgivenuncertaintiesintransittimes andtime-dependentprioritiesdependingonlatenesstolerancesforthecustomers. Theauthorsproposetwomodelsforthisproblem.Thesecondincludesblockingwhile therstdoesnot.TheyformulatebothproblemsasMIPsontime-spacenetworksand considerwaystoimprovetheassociatedformulations.Theplanninghorizonisassumed tobebetweenveandeightdays. ThecarassignmentproblemhasbeenaddressedmostlyformajorEuropean railroadssincetheydonotoperateinblockingmodeforfreightduetoshorttravel distancesandtrainlengths.Cesellietal.2008studytheproblemofdetermining trainschedulesundercapacityrestrictionsandpick-upanddeliverytimewindows. TheproblemarisesatthetwoyardsthattheCargoExpressserviceoftheSwiss FederalRailwaysoperatesonanovernightdeliverycycle.Intheabsenceofblocking, classicationofcarsoccursattheseyards.Inparticular,carsaredetachedfrom incomingtrains,classied,andattachedbacktooutgoingtrainsunlesstheyhave reachedtheirnaldestination.Theauthorsproposethreemethodstosolvethisproblem thatwaspreviouslysolvedmanuallyonceayear.Theycomparetheresultsobtained usingacommercialsolvertothesolutionstheyobtainbyapplyingdecomposition techniquesandbranch-and-cut.Theyalsoformulatetheapplicationasaside-constrained setpartitioningproblem.Theysolvethismodelusingcolumngeneration.Theythen comparetheperformanceofthissolutionapproachwiththersttwoandarguethat thelatterperformssignicantlybetteronlargepracticalinstances.Campetellaetal. 2006alsostudythemovementofemptycarsofvarioustypesinanItalianrailroad. 21

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Theauthorsmodeltheproblemasamulti-commoditynetworkdesignproblemwitha nonlinearobjectivefunction.Inordertodeterminetrainfrequencies,theyuseatabu searchenhancedwithadditionalheuristicmechanisms. Laietal.2010distinguishdifferenttraintypesi.e.,intermodal,manifest,unit, local,passenger,commuterandformulateamixed-integernetworkdesignmodel withtrackcapacitiestominimizetransportationandmaintenancecostsonanetwork inwhichalltrainssharethesametracks.Differenttraintypesaretreatedasdifferent commodities.Gorman1998considersatrainroutingproblemwithcapacityconstraints thatseekstominimizecostwhilerequiringthatdemandismet.Theauthorsolvesthe applicationbydecomposingitintotwosubproblems:trainschedulinganddemandow. Geneticandtabu-enhancedgeneticmetaheuristicsareusedtoobtainsolutions. Finally,Lawleyetal.2008bformulateatime-spaceschedulingmodelforbulk unittrains.Theyseektomaximizethesatiseddemandexpressedintermsofunit trainsdeliveredtotheirloaddestinationswhileminimizingwaittimesatyardsunder yardandtrackcapacitylimitations.Theyassumethatcareetsarehomogeneousand thatalltrainshaveexactlythesamesize.Theauthorsdividetheplanninghorizoninto sub-periodsandsolvetheassociatedproblemssequentiallyusingacommercialMIP solver. Tothebestofourknowledge,thereisnopreviousstudyinthefreightrailroad literaturethatincorporatesallthecharacteristicsofMCTRPP.Inparticular,noprevious workdealswithheterogenouscareetassignmentstounittrainsallowingcar substitutionsunderdestination-dependentrestrictionsalongwithrequirementsrelated totrainroutingandscheduling.AlthoughLawleyetal.2008baddressspecicallyunit trainschedulingwheredemandisexpressedintermsoftrainsratherthancars,car eetsareassumedtobehomogeneousandonlysmalltrainsofaxedsizeofforty carsareconsidered.Wementionthatourattemptsatformulatingpracticalinstancesof MCTRPPasatime-spacenetworkowproblemfailedduetotheextremelylargesizeof 22

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theresultingformulationoncecarsubstitutionisconsidered.Forpracticalinstances,it wasnotevenpossibletoloadtheproblemintooff-the-shelfcommercialsolverssuchas GUROBIandCPLEX. Forthisreason,weintroduceatwo-phasesequentialsolutionapproachthat providesgoodqualitysolutionsforlargenetworkswithhighvolumesofcoaldemandthat areencounteredinpractice.Theformulationweproposefortherstphaseconsiders theproblematanaggregatelevelandissolvedinafewminutesforpracticalinstances. Thereforeitmightbeusedasatooltoperformfastsupply-versus-demandanalysisin adynamicdemandenvironmentwhereapromptresponsemechanismisrequiredto interactwithcustomers.Themodelwedevelopforthesecondphaseprovidesaprecise schedulefortrainstogetherwiththeexactlistofassignmentsthatcarsreceiveandtheir associatedrouting.Thisscheduleprovidesadetaileddescriptionofcarassignments totrainstogetherwithoperationaldetailsabouttherepositioningofemptycars.Italso performsprecisecapacityvericationacrossthenetworkandthroughoutthemonth. Thesecondphasemodelissolvedusingaheuristic.Onpracticalinstances,this heuristicobtainshigh-qualitysolutionsforMCTRPP.Itsrunningtimeissufcientlysmall thatitissuitableforpracticalusebyourindustrycollaborator. 2.3SolutionMethodology Giventheextremelylargesizeoftheprobleminstancesencounteredinpractice, weconcentrateondevelopingheuristicsolutionmethods.Inparticular,weintroducea two-modelsequentialsolutionproceduretoconstructgood-qualityfeasiblesolutionsina reasonableamountoftime. Therstmodelcomparessupplyanddemandandexaminestheirrelationswith networkcapacitiestomakeinformeddecisionsaboutwhatreservationstoacceptduring theplanningmonth.Toenablethisdetermination,wemakesimplifyingassumptions thatwediscusslaterinthissection.WerefertothisrstmodelasAggregatedCapacity AssignmentProblemACAP.ACAPprovidesagoodestimateofthemaximumnumber 23

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ofreservationsthatisreasonabletoacceptduringtheplanningmonth.Moreover,once solved,ACAPassignstentativedatestoacceptedreservationsincludingarrivaland departurefrombothloadoriginsanddestinations.Weusethesetentativedatesto deriveprecedencerelationsbetweenreservations.Theserelations,togetherwiththecar consistsassignedtoeachreservationbyACAP,arethenusedasinputforthesecond modelthatwerefertoasDetailedPlanningProcedureDPP.TheaimofDPPisto deriveafeasiblesolutiontoMCTRPPthatsatisestherestrictionsthatwereignoredin ACAP. Asmentionedabove,wemakesimplicationstoformulateACAPinordertoobtain arst-cutsolutionthatcanbeobtainedinasmallamountoftime.Specically,we assumethatthemovementofcarsassociatedwithareservationisinitiatedfromthe destinationterminal.Thetrainthendirectlyproceedstotheminewhereemptycarsare loadedwhichconcludestherstlegofthecycle.Itisthensentbacktothedestination wherecarsareunloadedwhichconcludesthesecondlegofthecycle.Thetotaltime thatbothlegstaketosatisfyareservationisthereforereferredtoasitscycletime.The timerequiredtocollecttherequestedamountofemptycarsofthedesiredkind,toform thetrainandtosendthetraintotheloadoriginisincorporatedwithintherstlegof thecycle.Further,weassumethatcarsassociatedwiththistripareusedonlyduring thelengthofthecycle.Oncereleasedfromthecurrentreservation,weassumethat theyareimmediatelyavailableatanypointofthenetworkforotherreservationsthat requirethem.Thissimplicationkeepsthemodelsimplesincetheexactpositionand movementsofcarsacrossthenetworkandthroughoutthemonthdoesnotneedtobe tracked.Italsoallowsustoconsidercarsinpools,ratherthanasindividualentities. BecausethemovementofcarsisapproximatedinACAP,capacitiesareimposed atanaggregatelevel.ItfollowsthatthescheduleproposedbyACAPtypicallydoes notdirectlyyieldafeasiblesolutiontoMCTRPP.OurexperiencehoweveristhatACAP providesgood-qualityrst-cutsolutionsthatcanbeadequatelyrenedthroughDPP.In 24

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DPP,werepresentpreciselythesequenceofreservationsthatcarsareassignedto.In otherwords,wederivethespecicjourneythateachcartakesduringthemonthstarting fromtheterminalwhereitbecomesavailable.Thismorefaithfulrepresentationofthe tripenablesexactcapacityvericationatallcapacity-restrictedterminalsofthenetwork. Asaresult,DPPproducesanimplementablereservationschedulethatissupported byspeciccarassignments.Precedencerelationshipsbetweenreservationsandcar consistassignmentdecisionsarepassedfromACAPtoDPPandprovidethemain vectorsofinteractionbetweenthetwomodels. 2.3.1AggregatedCapacityAssignmentProblemACAP ThemainassumptionwemakeinformulatingACAPisthatthecarsusedtosatisfy reservationsoriginatetheirtripattheirloaddestination,maketheirwaytotheloadorigin andthenbacktothedestination.Aftercoalhasbeenunloaded,carsarereleasedback tothepoolstowhichtheybelong.Atthistime,weassumethattheyareimmediately availableatanypointofthenetworkthatrequiresthem.Itfollowsthat,inACAP,the initialpositionofcarsatthebeginningofthemonthdoesnotneedtobeconsidered.We musthoweverrecordthenumberofcarsthatisavailableduringeachdayofthemonth. Withtheseassumptionsinmind,wenowdescribeACAP. 2.3.1.1Problemformulation Wedenotethesetsofvertices n ,loadoriginsLDOR m ,destinationsDEST d consignees c andreservations r by N M D C ,and R respectively.For n 2 N = M [ D wealsouse R n torepresentthesubsetofreservationshavingterminal n asloadorigin ordestination. Railcarsareindexedby k 2 K .AsexplainedinSection2.1.3,carkind k isa uniquecombinationoftype,ownershipandowningconsignee.Werepresentthesubset ofcarkindsthatareadmissibleforreservation r by K r anddenotethepreferredcar kindamongthissubsetby k 2 K r .Wedenotethesubsetofcarkindsexcludedat destination d by e K d 25

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Eachreservation r hasaspecicloadorigin, m r ,destination, d r ,consignee placingtherequest, c r ,aswellasthenumberofcars r thatmustbedelivered. Theplanninghorizonischosentobeamonthwheredaysareindexedby t 2 T Foreachreservation r ,wecreatealistofcandidateschedules l 2 L r thatdifferbytheir startday l .Thescheduleassociatedwith l =0 correspondstotheoptionofrejecting reservation r .Wethereforehaveapproximately 30 schedulesforeachreservation. Givenareservation r ,werefertothesubsetofschedulesforwhichthereservationis completedi.e.,theloadedtrainreachesitsdestinationandisunloadedbeforethelast dayofthemonthby L 0 r Wereferto r;l;t hours asthenumberofcarhoursthatreservation r requiresduring day t ifitissatisedaccordingtoschedule l ,i.e.,ifitisscheduledtostartonday l Parameter r;l;t n issetto 1 wheneverterminal n isbeingvisitedbythetrainassociated withreservation r duringday t ifitisscheduledtostartonday l .Itissetto 0 otherwise. Next,weintroducenotationforthedatathatisusedtoimposethehardconstraints ofthemodel.Everyloadoriginanddestinationhasadailytraincapacitythatwe denoteby n;t .Duetopreviousorcurrentassignments,dailycaravailabilitymaynotbe constantovertheplanninghorizon.Werepresentby k t thenumberofcarhoursofkind k availableduringday t Finally,inordertoaddressthevarioussoftconcernslistedinSection2.1.4,we introducepositivepenaltyweightsthatwillappearintheobjectivefunctionofourmodel. Parameter 1 penalizesrejectedreservations, 2 rewardsreservationsbeingcompleted withinthemonthand 3 favorsassignmentsofpreferredcarkindstoreservations. Wenextdenethethreesetsofdecisionvariablesofthemodel.Firstwedene thebinaryvariable y r;l toequal 1 ifreservation r issatisedaccordingtoschedule l and 0 otherwise.Secondwedenetheintegervariable x k r;l toindicatethenumberof carsofkind k thatareassignedtoreservation r satisedaccordingtoschedule l .Third, wedenetheintegervariable s r;l torepresentthenumberofcarsthatarenotofthe 26

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preferredcarkind,yetareusedtosatisfyreservation r accordingtoschedule l .We formulateACAPas: minimize 1 X r 2 R y r; 0 )]TJ/F24 11.9552 Tf 15.278 0 Td [( 2 X r 2 R X l 2 L 0 r nf 0 g y r;l + 3 X r 2 R X l 2 L r nf 0 g s r;l subjectto: X l 2 L r y r;l =1 ; 8 r 2 R X r 2 R X l 2 L r nf 0 g r;l;t hours x k r;l k t ; 8 k 2 K; 8 t 2 T X k 2 K r x k r;l = r y r;l ; 8 r 2 R; 8 l 2 L r nf 0 g x k r;l + s r;l = r y r;l ; 8 r 2 R; 8 l 2 L r nf 0 g X r 2 R n X l 2 L r nf 0 g r;l;t n y r;l n;t ; 8 n 2 N; 8 t 2 T y r;l 2f 0 ; 1 g ; 8 r 2 R; 8 l 2 L r 0 x k r;l r ;x k r;l 2 Z ; 8 r 2 R; 8 l 2 L r ; 8 k 2 K r 0 s r;l r ;s r;l 2 Z ; 8 r 2 R; 8 l 2 L r : Intheaboveformulation,theobjectivefunction2seekstominimizethenumber ofrejectedreservations,tomaximizethenumberofreservationscompletedwithin themonthandtominimizethenumberofcarsofnonpreferredkindsassignedto reservations.Constraint2inconjunctionwithConstraint2ensuresthateach reservationiseithergivenaspecicscheduleorrejected.Constraint2together withConstraint2ensuresthatthetotalnumberofcarhoursusedbyaccepted reservationsdoesnotexceedthedailycarhourcapacityforeachcarkind.Constraint 2imposesthateachreservationbeassignedcarsthatbelongtoitsadmissibleset ofcarkindswhileConstraints2and2ensurethatthenumberofcarsofthe preferredkindusedforeachreservationisaccuratelycomputed.NotethatConstraint 27

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2isintroducedtopenalizepositivevaluesof s r;l intheobjective.Finally,Constraint 2limitsthedailynumberoftrainsprocessedatloadoriginsanddestinations. 2.3.1.2Solutionprocedure AlthoughACAPisacommonintegerprogram,itisdifculttosolvewithregular off-the-shelfsolversforpracticalinstances.Inparticular,thecommercialsoftwarewe experimentedwithdidnotconsistentlyproducegood-qualitysolutionsforthisproblem withinanacceptableamountoftime.Webelievethatoneoftheissuesthatcontributes tothisdifcultyisthelargesizeofthemodelforpracticalinstances.Inparticular, althoughitispossibletoloadmodelsofpracticalinstancesinmemorywithGUROBI 5.0.2.,solvingtheLPrelaxationoftheproblemtypicallyrequiresseveralhoursof computation.Forthisreason,wediscussnextheuristicprocedurestosolveACAP. Whileexperimentingwiththemodel,weobservedthatanimportantdriverofits difcultyisthefeaturethatallowstheassignmentofheterogeneousmixesofcarsto eachreservation.Inparticular,weobservedthat,whencarassignmentsareallowed onlytothepreferredcartype,good-qualitysolutionstothemodelareobtainedrelatively quicklyusingGUROBI5.0.2.indefaultmode.Wealsoobservedthat,whenmixed assignmentsareallowed,themodelcanstillbesolvedrelativelyfastwithoff-the-shelf solversifthenumberofreservationsisabouthalfoftheaveragenumberofreservations encounteredinpractice.Thesetwoobservationspromptedustoimplementthe followingheuristictwo-stepsolutionproceduretoproducegood-qualitysolutions toACAPinashortamountoftimeforprobleminstanceshavingalargenumberof reservations.Althoughforsmallprobleminstances,wecansolveACAPdirectlywith GUROBI5.0.2.,weapplyourheuristicproceduretoallinstancesizes.Thisdecisionis motivatedbythefactthatourindustrialcollaboratorconsidersthesizeofitscustomer basetobesensitiveinformation. Intherststep,ACAPissolvedallowingonlytheassignmentofthepreferred carkindtoeachreservation.Inthesecondstep,thesolutionobtainedisimproved 28

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usingalarge-scalelocalsearchprocedure.Inthisprocedure,wextheschedules andcarassignmentsofalargefractionofthereservationstobeidenticaltothose oftheincumbentsolution.WethenformulateanewACAPmodelfortheremaining reservationsallowingcarsubstitution,andre-optimizewiththehopeofndingan improvedscheduleforthispartofthesolution.Wechoosereservationstore-optimize amongthoseinitiatedwithinagiventimewindoworwithinthosethatsharecommon resourceswiththereservationsofanunder-servedconsignee.Forthelatter,weselect consigneesaccordingtotheirdroppercentagesi.e.,percentageofreservationsthatare rejected.Thesetwovariantsofthelocalimprovementalgorithmarethenappliedtothe incumbentsolution.Ifanimprovedsolutionisfound,itiskept.Otherwise,theincumbent solutioniskeptandanotherneighborhoodissearched.Wenextdescribeinmoredetail theconstructivepartofoursolutionprocedureinSection2.3.1.2.1whilewedescribethe twolocalsearchheuristicsinSection2.3.1.2.2. 2.3.1.2.1Constructivephase .Thisphase,althoughsimple,iscrucialinterms oftheoverallimplementabilityofthealgorithm.Itgeneratesagood-quality,feasible solutionwithinareasonableamountoftime.ItconsistsofsolvingACAPwiththe additionalrestrictionthatreservationsareonlyassignedtheirpreferredcarkind. Thiscanbeachievedbyremovingthevariablescorrespondingtocarsubstitution fromACAP.Inthesolutionofthismodel,acceptanceratiosaresometimeslowfor specicdestinationsorconsigneessincereservationscompeteforlimitedresources. Theimprovementphaseoftheheuristicisdesignedtohelpwithmissedsubstitution opportunitiesbyre-consideringrejectedreservations. 2.3.1.2.2Improvementphase .Weconsidertwodifferentwaysofimprovingan existingsolution:localsearchbytimewindowandlocalsearchbyconsignee. 2.3.1.2.2.1Localsearchbytimewindow Inthisheuristic,weconsideranexisting feasiblesolutionthatwasobtainedfromtheconstructivephaseorthroughafewrounds ofthelocalimprovementprocedure.Weconsideratimewindow [ T;T + T ] where T is 29

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chosenrandomlyand T isaparameterofthelocalsearch.Ourgoalistoimprovethe solutioninsideofthistimewindowwithafocusonthereservationsthatarecurrentlynot satisedandarecurrentlysatisedandhavetheirstartdateinsideofthetimewindow. Tothisend,wecreateavariantofACAPwherevariablesrelatedtoreservationsbeing satisedintheincumbentsolutionandstartingatatime t thatisnotinthetimewindow [ T;T + T ] arexedtotheircurrentvalues.Inotherwords,welookforsolutionsin whichthesereservationsaresatisedatexactlythesametime,inexactlythesame way.Thevariablesassociatedwithallotherreservationsareallowedtotakeanyoftheir admissiblevalues,leadingtoanewinstanceofACAPthathastoscheduleasmaller subsetofreservations.InthisnewinstanceofACAP,weallowcarsubstitutionswhile stillimposingcapacityconstraintsovertheplanninghorizon.Thismodelissolvedwith apre-settimelimitusingGUROBI5.0.2.Whenthetimelimitisreachedoranoptimal solutionisobtained,weinspectthesolutionproducedifany.Ifthissolutionsatises morereservationsthantheincumbentsolution,itiskept.Otherwiseitisdiscardedand anotherlocalimprovementmoveisinitiated. Severalaspectsmustbeconsideredinorderforthisproceduretobeeffectiveand efcient.First,theparameter T mustbechosenrelativelylargesoastoallowthe MIPsearchtouncovernew,bettersolutions.However,theparameter T shouldnotbe chosensolargethatitmakestheMIPsearchtime-consuming. 2.3.1.2.2.2Localsearchbyconsignee Inthisheuristic,weconsideragainan existingfeasiblesolutionthatwasobtainedfromtheconstructivephaseorthrougha fewroundsofthelocalimprovementprocedure.Ourgoalistoimprovethesolution forthoseconsigneeshavingadroppercentageaboveapre-determinedthreshold.To thisend,wecreateavariantofACAPinwhichcarsubstitutionisallowed.Werank theconsigneesbydroppercentagesandrandomlypickonefromthelistwhosedrop percentageisabovethepre-setthreshold.Weidentifyallcurrentlysatisedreservations thatsharesomeoftheresourcesthattheselectedconsignee'sreservationsuse.More 30

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preciselyweselect,inorderofpriority,theconsignee'sownreservations,andthen thosereservationsthatsharetheirloadorigins,destinationsandcartypes.Westop addingreservationstoourlistwhenitssizebecomeslargerthanapre-determined target.Ifthelistproducedistooshort,wecontinuebyrandomlypickinganothereligible consigneeandrepeatingthestepsdescribedabove.Oncethetargetproblemsizeis reached,wexallreservationsnotinthelistthatarecurrentlysatisedtotheirvalues intheincumbentsolutionandallowallothervariablestotakeanyoftheiradmissible values.Theresultingmodelisthensolvedwithapre-settimelimitusingacommercial solver.Theresultingsolutioniseitherkeptordiscardedinthesamemannerasthatis describedinSection2.3.1.2.2.1. 2.3.2DetailedPlanningProcedureDPP ThemainsimplicationofACAPisthatitignoreshowcarsaretransportedfrom thedestinationoftheircurrentassignmenttotheoriginoftheirnextassignment. Thisassumptionismadefortractabilitypurposes.Inordertoderiveamoreprecise schedule,weintroduceadetailedplanningprocedureDPPthattracksindividual carsanddeterminespecicpathsthattheyfollowthroughthemonth.Asaresult,it allowsforexactdistancesbeingtraveledtobecomputedandenablesprecisecapacity vericationsatloadoriginsanddestinations. SinceattemptstoformulateMCTRPPasasingleoptimizationproblemfaileddueto itslargesize,weimplementedthefollowingsequentialapproachtoobtaingood-quality feasiblesolutions.Thisalgorithmiscomposedofaconstructiveandanimprovement phasethataredescribedinSections2.3.2.1and2.3.2.2respectively.Theconstructive phaseisitselfcomposedoftwophases: Car-to-ReservationAssignmentProblemCtRAP: Inthisphase,wedetermine whatcarsareassignedtowhatreservationsandhowcarsowfromtheirinitiallocation througheachofthereservationstheyareassignedtoduringthemonth.Inparticular, thisphaseproducesapartialorderbetweenreservations.Wementionhoweverthatno 31

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specicstarttimesareassignedtoeventsatthisstage.Inotherwords,wecreatepaths thatspecicgroupsofcarsfollowthroughoutthemonthwithoutpreciselydescribing wheneacheventi.e.,loading,unloadingortransitonthispathisperformed. EventTimeStampingProblemETSP: Inthisphase,wedetermineastarting timeforeachoftheeventsonthepathsthatwerederivedinthesolutionofCtRAP. Thesetimesareassignedinawaythatrespectsdailynetworkcapacitiesatloadorigins anddestinations.Becauseallprecedenceconstraintsbetweenthedifferenteventson apatharealreadyset,thisproblemissimilartojob-shopschedulingBlazewiczetal., 1996;JainandMeeran,1999.Weuseavariantoftheshiftingbottleneckheuristic proposedinAdamsetal.1988thatwedescribeinmoredetailinSection2.3.2.1.2 toobtainasolution.Oncetimesareassignedinsuchawaythatcapacitiesatload originsanddestinationsaresatised,wehaveobtainedadetailedschedulethatis implementableinpractice.Reservationsthatareassignedaload-indatefallingoutside oftheplanninghorizonarerejectedpurgedfromthesolution. Becausethisconstructivealgorithmisheuristic,wedonotexpectittoproduce anoptimalsolutiontotheproblem.Therefore,inordertoimprovethesolution,we useaneighborhoodsearchheuristic.Thisheuristicconsidersanexistingdetailed solutiontogetherwithreservationsacceptedinACAPbutpurgedinDPP.Itthenseeksto determineabetterschedulethatsatisesalargernumberofacceptedreservations. Improvementheuristic: IntheimprovementstepofDPP,weperformagreedy large-scaleneighborhoodsearch.Atahigh-level,thealgorithmreleasestheschedule ofasubsetofreservationsthatusethesamecarkindsbycancelingspeciccarpaths. Themaincriterionweusetoselectpathstocancelisthenumberofpurgedreservations theycontain.Usingresidualcarandfacilitycapacitiesinthenetwork,wethengreedily attempttoreschedulethecanceledreservations,i.e.,toreschedulereservationsonthe canceledpathincludingthosethatwerepurged.Toproduceanarrayofpossiblenew solutions,weincorporaterandomnesstosomekeydecisionsoftheprocedure,such 32

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astheorderinwhichwepickthereservationstoberescheduledandtheselectionof carsupplysources.Thereforewhenrunningthealgorithm,wemightobtaindifferent newsolutionsforagivensetofcanceledreservations.Forthisreason,werepeatthe greedyassignmentprocedureoverapre-setnumberofiterationsinordertoidentify betterschedules.Whenasolutionisdiscoveredthatsatisesanincreasednumber ofreservationsormaintainsthenumberofreservationssatisedwhiledecreasingthe overallmakespan,itisselectedtoreplacetheincumbentdetailedsolution. 2.3.2.1Constructiveheuristic 2.3.2.1.1Car-to-reservationassignmentproblemCtRAP 2.3.2.1.1.1Description Thegoaloftherstphaseoftheconstructiveheuristicis, foreachcarkind,todetermineaowofcarsthroughthemonthbetweenreservations thatminimizesthenumberofsplit-and-mergeoperationsrequiredfortheformationof trains.Asplitofacargroupoccurswhenthetrainthatistobeformedrequiresfewer carsthanwhatiscurrentlyinthecargroup.Amerge,ontheotherhand,happenswhen multiplecargroupsneedtobemergedintoasingletrain.Ourgoalistondsuchow thatisconsistentwiththeprecedencerelationsbetweenreservationsderivedfromthe solutionofACAP,theinitialpositionofcargroupsandtheirassociatedcarcounts.The precedencerelationsmentionedaboverequirethatcarsassignedtoreservation r i can beusedasapossiblesourceofcarsforreservation r j onlyforthosereservations r j whosestarttimeinthesolutionofACAPcomeslaterthantheendtimeofreservation r i TheformulationofCtRAPseekstocoverallthereservationsassociatedwith agivencarkindbymovingcarsofthiskindfromtheirinitiallocationsinawaythat minimizesthenumberofsplit-and-mergeoperationsthatarecarriedacrossthenetwork andthroughoutthemonthandthetotaldistancetraveledbytrains.Thisproblemcanbe modeledandsolvedasanintegerprogramthatwedescribeinSection2.3.2.1.1.2.This modelalwayshasatleastonefeasiblesolutionsincewearenotrequiringthatthepaths createdbesufcientlyshorttotwithintheplanninghorizon. 33

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2.3.2.1.1.2CtRAPnetworkandmodelformulation First,wedescribethedata neededtoset-upCtRAP.Wethenexplainhowthisdatacanbeprocessedintoa networkthatisusedastheactualinputtoourmodel.Wenextdescribethevariableswe deneonthisnetworkandnallypresenttheconstraintsandtheobjectivefunctionof themodel. AswediscussedinSection2.1.3,carsareattachedtoordetachedfromtrains atbuildlocations.Weassumethateachreservationhastwoknownandxedbuild locations:onetoformanincomingtrainandtheothertobustanoutgoingtrain.We refertothebuildlocationassociatedwiththeloadoriginof r asthein-buildlocationof reservation r r in .Similarly,werefertothebuildlocationassociatedwiththedestination of r astheout-buildlocationofreservation r r out .Wedenotethecarcountrequestof reservation r by n r .ThestartandendtimesassignedbyACAPtoreservation r ifany aredenotedby t start r and t end r ToconstructtheCtRAPnetworkweusethesetofbuildlocationsdenotedby L thesetofreservationsacceptedforthemonthbyACAPdenotedby R andthesetof groupsofcarsavailableforassignmentdenotedby G .Assignment-readyrailcarsare splitintogroupsbasedonthelocationandthetimeatwhichtheybecomeavailable. Forinstance,thesamelocationmightbeassociatedwithtwodifferentgroupvertices inthenetworkifthereisagroupofcarsreadytobeassignedatthebeginningofthe monthandthereisanothergroupthatbecomesavailableafewdayslater.Givenacar group g 2G ,wedenotethetimeatwhich g becomesavailableby t g ,thelocationwhere g becomesavailableby l g andthenumberofcarsin g by n g .Grouplocationsmightnot alwaysbebuildlocations.Therefore,wheneverasplitormergehastobeperformedina carconsistbeforesendingittoareservation,thecorrespondingcargroupmustrstbe transportedtothein-buildlocationofitsrstreservationassignment. WearenowreadytointroducetheCtRAPnetwork G = V ; E ,where V and E are thesetofvertices v andarcs e of G respectively.Set V consistsofastartvertexthat 34

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wedenoteas v start ,groupvertices g 2G ,reservationvertices r 2R togetherwiththeir associatedin-andout-buildlocationsvertices r in 2L in and r out 2L out respectively, andanendvertexthatwedenoteas v end i.e., V = f v start ;v end g[L[G[R Set E containsmultipletypesofarcs.First,weintroducearcsbetweenstart andgroupvertices.Wealsoconnectallin-buildlocationverticestotheirassociated reservationvertices,andconnectallreservationverticestotheirassociatedout-build locationsvertices.AsillustratedinFigure2-1-a,weplaceanarcbetweenagroup vertex g andareservationvertex r onlyifthereisanexactcarcountmatchandifthe condition t g t start r issatised.Incasecarcountsdonotmatch,weintroducethearc g;r in if t g t start r toallowsplit-and-mergeoperations.Weintroducearcsbetweentwo reservationvertices r i and r j onlyifthestarttimeof r j t start r j ,comesaftertheendtimeof r i t end r i ,andthecarcountrequestofthetworeservationsisidentical.When t start r j >t end r i andcarcountsdonotmatch,wecreatearcsfrom r i to r in j r out i to r in j and r out i to r j as showninFigure2-1-b.Finally,weconnectallverticestotheendvertex.For v 2 V ,we denotethesetofarcsemanatingfromavertex v by + v .Similarlywedenotethesetof arcsbringingowintovertex v by )]TJ/F25 7.9701 Tf -0.449 -7.294 Td [(v .Wedenotethelengthofarc e by d e WenextpresentanMIPformulationforCtRAP.Foreacharc e 2 E ,weintroduce integervariable y e torepresenttheowofcarsonarc e andbinaryvariable x e toindicate whetherarc e carriesowornot. 35

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TheMIPformulationforCtRAPis minimize X l 2L out X e 2 + l x e + X l 2L in X e 2 )]TJ/F26 5.9776 Tf -0.299 -6.416 Td [(l x e + X e 2 E d e x e subjectto X e 2 + v y e = X e 2 )]TJ/F26 5.9776 Tf -0.299 -4.838 Td [(v y e ; 8 v 2 V nf v start ;v end g X e 2 + g y e n g ; 8 g 2G X e 2 + r y e = n r ; 8 r 2R y e Mx e ; 8 e 2 E X e 2 )]TJ/F26 5.9776 Tf -0.299 -4.839 Td [(r x e =1 ; 8 r 2R X e 2 + r x e =1 ; 8 r 2R x e 2f 0 ; 1 g ; 8 e 2 E y e 2 Z + ; 8 e 2 E : Intheaboveformulation,theobjectivefunction2seekstominimizesplits whichoccurattheout-buildlocationofreservationvertices,mergeswhichoccurat thein-buildlocationofreservationverticesandthetotaldistancetraveledbycoaltrains. Constraint2enforcesowbalanceateachvertexofthenetworkexceptstartand endvertices.Constraint2ensuresthatthenumberofcarsusedineachcargroup doesnotexceedthesizeofthegroupandConstraint2statesthatthenumberof carsassignedtoeachreservationexactlymatchesitscarcountrequest.Constraint 2allowsowtobecarriedonanarconlyifthisarcisopen.Finally,Constraints 2and2ensurethatcarssatisfyingareservationarriveatandleavethis reservationasasingletrain. 2.3.2.1.1.3Solutionprocedure Theabovemodelisamixed-integerprogram thatcanbesolvedusingcommercialsolver.Itmustbesolvedmultipletimes,once 36

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A B Figure2-1.CtRAPnetwork.aArcsemanatingfromagroupvertex.bArcsemanating fromareservationvertexanditsout-buildlocation. foreachcarkind.Dependingonthenumberofcarsinacarkindandthenumberof reservationsthatmustbesatisedusingthem,theformulationmightbecomelargeand time-consumingtosolve.Althoughinourpracticalexperience,themodelissolvedto optimalityveryquicklyformostcarkinds,weobservedthatsolutiontimescanbevery longincertaincasesinvolvinglargepoolsofsystemcars.Inthiscase,evennding afeasiblesolutioncanprovetobedifcult,whichcreatesasignicanthurdleforthe overallsolutionprocedure.Wethereforeintroduceaheuristicforobtainingasolution tothisproblem.Inthisheuristic,werstfeedtheproblemtoacommercialsolverwith thehopethatitndsanoptimalsolutionquickly.Ifso,thesolutioniskeptandthe 37

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followingcarkindisconsidered.Whenagood-qualityfeasiblesolutionisnotobtainedin areasonableamountoftime,asmeasuredbysomepre-settimelimit,theoptimization isstopped.Wethenswitchtoanoptimizationproblemwherethegoalissimplytoassign carstoreservationsirrespectiveofwhetherornotthisisdonewithalargenumberof split-and-mergeoperations.Theonlymodicationthatwemaketotheaboveformulation tocreatethisnewmodelistoreplacetheobjectivefunction2with minimize X g 2G X e 2 + g y e : Anyfeasiblesolutiontothisnewmodelisfeasiblefortheinitialmodelastheyshare thesamefeasibleregion.Thenewproblemissolvedusingacommercialsolver.For practicalinstancesandforthelargestofourcreatedprobleminstances,ourextensive experimentsshowthatcommercialsolversobtaingoodqualitysolutionstothenew modelwithinthetimelimitthatweimposetocontroltheoverallsolutiontimeofthe constructiveprocedure.Afterasolutionisobtained,itisfedbacktotheinitialCtRAP modelandusedbythesoftwareasastartingpointtoobtainbettersolutions.The originalformulationisthenrununtileitheranoptimalsolutionisfoundorthetimelimitis reached.Inthisrun,wecanguaranteetheexistenceofafeasiblesolutionwhich,inthe worstcase,isthestartingsolution.Werepeatthisprocedureforallcarkinds. Theaboveprocedureyieldsacarowfromthestartvertextotheendvertexthat passesthroughcargroupandthatcoversallreservationvertices.Aswewishtotrack speciccarsthroughoutthemonth,wemustdecomposethisowintoacollectionof pathsforeachcarkind.UsingatraditionalowdecompositionapproachAhujaetal., 1993,weconstructonesuchpossiblepathdecompositiontodescribethemonthly routesofcoalcarsinthenetwork.Whenevermultipleowsmergeatavertexandare splitintodifferentowsfromthatvertex,thedecompositionofowintopathsobtained isoneofmany.BecausetheCtRAPnetworkisacyclicsincearcsareconsistentwith thetentativescheduleobtainedinACAP,thisdecompositioncanbeachievedwithout 38

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theuseofdirectedcycles.Moreprecisely,weinitializethesetofpaths P tobeformed solelyofpaths p ,composedofasinglearcfromthestartvertextoacargroupvertex. Allarcsusedtoinitializethesepathsareremovedfromthesetofarcs E oftheCtRAP network.Next,weiterateoveralltheremainingarcs e 2 E .Ifvertex i ofarc e = i;j appearsasthelastvertexofapath p therstoneencounteredduringthesearchin P weadd e topath p afterperformingoneofthefollowingoperationsandafteraltering P and E accordingly:Iftheowonarc e f e ,andtheowonpath p f p ,areequal,thenwe simplyremovearc e from E ,if f e >f p then e iskeptin E anditsowisresetto f e )]TJ/F24 11.9552 Tf 12.086 0 Td [(f p ,if f p >f e ,thenweremovearc e from E andcreateanewpath p 0 identicalto p thatweadd to P .Weset f p 0 tobe f p )]TJ/F24 11.9552 Tf 11.96 0 Td [(f e and f p tobe f e .Withinanitenumberofiterations,theow isdecomposedintopaths. 2.3.2.1.2EventtimestampingproblemETSP 2.3.2.1.2.1Description AftersolvingCtRAP,wehaveaprecisedescriptionofthe sequenceofassignmentsthatcarsreceivethroughthemonth.However,westillneedto determinetheexactload-inandload-outtimeofalltheacceptedreservations.Thistask isnottrivialsincethereareloadoriginanddestinationcapacitiesthatmustberespected acrossthecoalnetworkandthroughthemonth.Toensurethatthesecapacitiesare respected,weuseaheuristicthatisavariationoftheshiftingbottleneckprocedure Adamsetal.,1988forjob-shopscheduling. 2.3.2.1.2.2ETSPNetwork Inthetraditionaljob-shopschedulingproblem,weare givenasetofjobseachcomposedofseveraloperationsandasetofmachineson whichtheseoperationsmustbeperformedoneatatime.Wearealsogivenasetof precedencerelationsbetweenoperationsthataredictatedbythejobdescriptions. Werefertotheserelationsasjob-precedencerelations.Forinstance,weknowthat operation i ofajobmustbeperformedbeforeoperation i+1 ofthisjob.However,weare notgiventheprecedencerelationsbetweenoperationsofdifferentjobsonmachines. Werefertotheserelationsthatmustbedeterminedasmachine-precedencerelations. 39

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Asolutiontothejob-shopschedulingproblemcanthereforeberepresentedasa network.Inthisnetwork,eachvertexrepresentsanoperationexceptforthestartand endverticeswhichrepresentthebeginningandendoftheschedulingperiod.Ajob arc i,j isincludedbetweenvertex i andvertex j ifoperation i mustbeperformed beforeoperation j asdictatedbytheassociatedjob-precedencerelationbetweenthese operations.Weconnectthestartvertextoalloperationverticeshavingnopredecessors intheirjobdescription.Likewise,weconnectallverticesthathavenosuccessorin theirjobdescriptiontotheendvertex.Weassociatewitheachjobarc i,j where i is anoperationvertex,thetimeittakestocompleteoperation i .Algorithmsforjob-shop schedulingcanbeseenasmethodswhich,startingfromthenetworkcontainingonly job-precedencearcs,addnecessarymachine-precedencearcsuntiltheresulting scheduledoesnotviolatethecapacityconstraintsofthemachines.Givenasetof job-andmachine-precedencerelationships,thestarttimesofanoperationcanbe computedasthelengthofthelongestpathfromthestartvertextothecorresponding operationvertex.Onceallnecessarymachine-arcshavebeenaddedtothenetwork,the makespanofthescheduleissimplyobtainedasthelongestpathinthenetworkfromthe startvertextotheendvertex. Inourapplication,thecapacitycentersaretheloadoriginsanddestinations. Forthisreason,werefertothemasmachinesinthefollowingdescription.Because machineshavelimitedcapacities,itisnecessarytosequenceoperationsonthem adequatelyi.e.,determinetheorderinwhichoperationsareperformedonthe machinesothatcapacityisnotexceeded.Thedescriptionofalljobsthatneedto beperformedcanbeinferredfromthesolutionofCtRAP.Infact,thesolutionofCtRAP yieldsacollectionofpathsbetweencarsupply,buildlocationandreservationvertices thatspecicgroupsofcarsfollowthroughthemonth.Thesepathshoweverdonot containspecicoperationsatloadoriginsanddestinations.Toconformtothejob-shop schedulingframework,wethereforespliteveryreservationnodeintothreeconsecutive 40

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sub-operationsnodestheloadingofthetrainatthemine,thetransportationofcoalto thedestinationandthedumpingoftheloadatthedestination.Therefore,intheETSP network,thesetofoperationsiscomposedofthesenewETSPverticestogetherwith theCtRAPverticesthatdonotcorrespondtoreservationsi.e.,carsupplyvertices,inandout-buildlocationvertices.Thismodicationisnecessarybecausetherstand thelastoftheseoperationsusecapacityattheloadoriginanddestinationrespectively. ItisillustratedinFigure2-2.Toreinforcetheanalogytojob-shopscheduling,werefer tothesepaths,aftersplitting,asjobsandtoeachoftheverticestheyvisitintheESTP networkasoperations. Theprocessingtimesthatweassociatewithjobarcsinthenetworkaresettoload origindwelltimesforarcsemanatingfromloadoriginverticesandarrivingatreservation transitvertices,traveltimesfromloadoriginstodestinationsforarcsemanatingfrom reservationtransitverticesandarrivingatdestinationverticesandnally,thesum ofdestinationdwelltimesandtraveltimestothenextlocationforarcsemanating fromdestinationvertices.Theprocessingtimethatweassociatewithamachine-arc isthedwelltimeofthecorrespondingloadoriginordestinationifitismorethan 24 hours.Otherwise,wesetitto 24 hoursinordertomakesurethatthedailycapacityof eachmachineisrespectedonanygivenday.Wedescribethedetailsofhowwetreat machineswithdailycapacitylargerthan 1 inSection2.3.2.1.2.3. 2.3.2.1.2.3SolutionProcedure:AdaptedShiftingBottleneckAlgorithm Having nowobtainedthedescriptionofjobs,operationsandmachines,wearereadyto describeourvariantofthejobshop-schedulingheuristicknownasshiftingbottleneck algorithmAdamsetal.,1988.Usingthisalgorithm,wesetmachine-precedence relationsbetweenreservationsforeachmachineloadoriginanddestinationwiththe goalofminimizingthetotalmakespanassociatedwiththecompletionofalloperations. Atahigh-level,theadaptedshiftingbottleneckalgorithmproceedsasfollows.In therststep,eachmachineisconsideredindividuallyandislabeledasnonxed.Each 41

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A B Figure2-2.ETSPnetwork.aSingle-vertexrepresentationofreservations.b Three-verticesrepresentationofreservations. operationonanonxedmachineisgivenastartingtimethatiscompatiblewiththe capacityofthemachineandtheprecedenceconstraintscontainedinthejobdescription job-arcs.Thistime-labelingisobtainedbyspecifyinganorderbetweenoperationson themachine.Thissequencingstepcorrespondstosettingprecedencearcsbetween operationsonthegivenmachinemachine-arcs.Thesequencingprocedureweusefor eachmachineisanalogoustothatofAdamsetal.1988andwillbedescribedinmore detailinAlgorithm2. Onceeachnonxedmachineissequencedfollowingtheproceduredescribedin Algorithm2,anyonethatdeterminesthemakespani.e.,thelongestpathfromstartto endwhenitsmachine-precedencearcsareaddedtotheETSPnetworkisconsidered tobethebottleneckmachine.Themachine-precedencearcscorrespondingtothe scheduleofthebottleneckmachinearethenaddedtotheETSPnetworkandthis machineisconsideredtobexed.Inthefollowingstepsoftheprocedure,weconsider 42

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eachofthenonxedmachinesandsequencethemsimilarlyinaniterativemanner.The sequencingisagainperformedbyconsideringthecapacityofthemachines,together withthejob-precedenceconstraintsbutalsoallthemachine-precedenceconstraintsthat werecreatedforthexedmachines.Ateachiteration,thebottlenecknonxedmachine isidentiedanditsmachine-precedencearcsareaddedtotheETSPnetwork.The procedureiscontinueduntilallmachinesarexed. Toimprovethequalityofthesolutionobtained,eachtimeamachineisxed,the setofalreadyxedmachinesisreconsidered.Inparticular,ifamachine-precedence arcofaxedmachineisonthelongestpathfromstarttoend,theassociatedmachine isidentiedascritical.Anymakespanbetterthantheoneassociatedwiththecurrent sequenceonacriticalmachine,usesanarcselectioninwhichatleastonearcon thelongestpathisreversedAdamsetal.,1988.Inanattempttoidentifysuch ashortermakespan,eachcriticalmachineisre-sequencedgiventhenewarc structureoftheETSPnetwork.Usingthesingle-machinesequencingprocedureof Algorithm2.Wethenreplacetheoldsetofmachine-precedencearcswiththenewset ofmachine-precedencearcs. WesummarizeourvariantoftheshiftingbottleneckalgorithminAlgorithm1.In thedescription,werefertosetofmachinesas M andtothesetsofxedandnonxed machinesas M xed and M nonxed respectively. InAlgorithm2,wedescribetheprocedureweutilizetosequenceasinglemachine inAlgorithm1.Weintroducethenecessarynotationnext.WedeneLongestPath i;j tobethelengthofthelongestpathfromvertex i tovertex j inthecurrentETSP network.Wedenotetheprocessingtimeofoperation n by n:pt .Werefertothelength ofthelongestpathfromthestartvertextooperation n andfromoperation n totheend vertexas n:head and n:tail respectively.Wedenotethemaximumlengthamongtails as tail max andtheprocessingtimeofthechosenoperationwithmaximumtaillength as pt max .Let m:slot size bethedailytraincapacityofmachine m .Weassociatewith 43

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machine mm:slot size slotsthatprocessatmostoneoperationatatime.Inorder tosequenceoperationsonmachine m ,werstcalculateheadandtaillengthsofall operationson m .Wethenidentifyalloperationshavingnopredecessorrelationshipwith otheroperationsthatareyettobesequencedon m .Startingfromtheoperationhaving theshortesthead,wesequentiallyassignthemtothemachineslotthatisbecoming availableearliest.Webreaktiesbygivingprioritytooperationshavinglongesttailand processingtimelengths. Therearetwosubstantialmodicationstotheoriginalshiftingbottleneckalgorithm thatwemaketoaccommodatespecicproblemcharacteristics.First,asinglejob i.e.,sequenceofreservationsmightrequiretheuseofthesamemachinei.e., loadoriginordestinationseveraltimes.Forinstance,thismayhappenwhenaset ofcarsisassignedrepeatedlytoreservationssharingthesameloadoriginand destination.Whenwesequencethisloadoriginordestination,itisimportantnotto createmachine-precedencearcsthatcontradictthepre-establishedjob-precedence relations.Further,asweproceedwithmachinesequencing,machine-precedencearcs ofdifferentmachinesmightalsocontradicteachother.Forthisreason,throughoutthe algorithm,wekeeptrackofallancestryrelationshipsbetweenoperationsusingavariant oftopologicalsortAhujaetal.,1993.Thisancestryinformationbetweenoperations changeseverytimemachine-precedencearcsareaddedordeleted.Itthereforemust beconstantlyupdated.Inparticular,eachtimewexamachinei.e.,xthesequenceof operations v tobeprocessedonthemachineweupdatethelistofoperationsthatmust occurbeforeandaftereveryoperation v Thesecondmodicationthatwemaketotheoriginalshiftingbottleneckalgorithm istoaddressthefactthat,inourproblem,loadoriginsanddestinationscanserve morethanonetrainatatime.InAdamsetal.1988however,theauthorsassume thatmachinescanonlyprocessoneoperationatatime.Whenadaptingouralgorithm toaccommodatethisdifference,weintroducetheconceptofslotsformachines.In 44

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particular,slotsmodelmultiplesimultaneoussequencesofoperationsonamachine thathasadailycapacityofmorethanonetrain.Asmentionedearlier,wedene thenumberofslotsofagivenmachine m tobeitsdailytraincapacity.Eachtimea newoperation n issequencedonmachine m ,wecheckfortheslot s whosecurrent queueofoperationsterminatesrst,add n tothequeueof s notethatoperation n willhavenomachine-precedencerelationswithoperationsthatareassignedto otherslotsandupdatetheavailabilityofthisslotbasedontheprocessingtimeof operation n onmachine m .Oncealltheoperationsaresequencedonmachine m weaddtheassociatedmachine-precedencearcstotheETSPnetworkusingthe operation-sequencesoftheslotsi.e.,listoforderedoperationsthatareassignedto eachslot. 2.3.2.2Improvementheuristic 2.3.2.2.1Description .TheprocedurewedescribedinSection2.3.2.1.2 associatesload-inandload-outdatestoalloperationsonthepathsderivedinSection 2.3.2.1.1.Sincethecollectionofthesepathscoversallthereservationsthatwere satisedinthesolutionofACAP,eachreservationacceptedinACAPisassigneda scheduleafterETSPissolved.Theload-indatesofsomeoftheacceptedreservations howevermightexceedtheendoftheplanninghorizon,resultinginthepurgingof thesereservations.Intheimprovementprocedure,weaimtodeterminewhether somere-arrangementsofthepathscanleadtoasmallernumberofreservations beingpurged.Inordertodeterminewhethersuchre-arrangementsarepossible, weselectafewpathsthatarepartofthesolutionoftheconstructiveheuristicand cancelthem.Cancelingapathreleasesthetimeassignmentsandtheprecedence relationsofreservationsonthepath,allowingustoreschedulethem.Cancelinga pathalsofreesthecorrespondingcarstogetherwiththecapacitytheyusedatload originsanddestinations.Wecanthenconsiderallcanceledreservationsandtry togreedilyaddthembacktotheschedule,inawaythatmaximizesthenumberof 45

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reservationssatisedthroughthemonth.Becauseourgoalistoincreasethenumber ofreservationssatisedduringthemonth,wepredominantlycancelpathsthatcontain purgedreservations.Further,toremainconsistentwiththesolutionofACAP,wedo notchangethecarconsistselectedforareservation.Therefore,duringaniteration oftheimprovementheuristic,welimitourselvestocancelingpathsthatusethesame carkind.Evenunderthisassumption,allreleasedreservationsarenotstructurally identical.Inparticular,splitreservationsi.e.,reservationshavingamixtureofcarkinds intheirconsistmightappearincertainpathsthatwedonotcancelwhileappearingin somepathswecancelinagiveniteration.Toovercomethisdifculty,werequirethat newschedulesproducedbesuchthatthecarconsistrequestedbyasplitreservation becomesavailableatthelocationwheretheemptytrainisformedbeforeitscurrent pre-setstartdayi.e.,thedayonwhichtheemptyunittrainstartsitstriptowardsthe loadoriginasassignedintheincumbentdetailedsolution.Thisrestrictionensuresthat theassociatedpathsthatwerenotcanceledwillnothavetobedelayeduntilthepart ofthesplitreservationbeinggreedilyrescheduledarrives.Infact,delayingthesepaths mayleadtocapacityissuesandtoalargernumberofreservationsbeingpurged.To implementthisrequirement,weintroducethenotionofdemandvertextorepresentthe requirementthataparticularnumberofcarsofthekindcurrentlyconsideredisdelivered toaparticularlocationbyacertaindateinordertosatisfyasplitreservation.Oncecars enterademandvertex,theyareconsideredtobeinuseuntilthecorrespondingsplit reservationiscompleted.Atthispoint,theyarereleasedbackforassignmenttothe network.Therefore,weintroduceanassociatednotionofreleasevertextorepresentthe factthataparticularnumberofcarsbecomeavailableatacertainlocationatacertain timeafterthecompletionofasplitreservation. Carsassociatedwithapathrstpassthroughcargroupvertices.Asmentioned inSection2.3.2.1.1,eachcargroupvertexcorrespondstoauniquelocationandtime combinationatwhichthecargroupbecomesavailableforassignment.Inthecontext 46

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oftheimprovementheuristic,werefertothemassupplyvertices.Itispossiblethatthe networkhascargroupsthatarenotusedinthesolutionoftheconstructiveheuristic. Wealsorefertothemassupplyvertices.Hence,togetherwiththereleasevertices, wehaveasetofcarresourceswithspeciccarcount,locationandtimeofavailability specications.Werefertothesespecicationsasattributeswhiledescribingthe followingalgorithms.Wealsohaveasetofreleasedreservationanddemandvertices withcarcount,locationand,fordemandvertices,duedateattributes. 2.3.2.2.2Solutionalgorithm .Inourlocalsearchsolutionalgorithm,werun manyiterationsinwhichweselectdifferentsetsofpathstocancel.Givenasetofpaths tocancel,wegeneratethelistsofsupply,demand,releaseandreservationvertices describedinSection2.3.2.2.1.Theselistsarepassedtotheimprovementprocedure thatwillseektoreschedulecanceledandpurgedreservations.Sinceweincorporate somelevelofrandomizationinsideofthere-schedulingprocedure,werunthealgorithm severaltimesforthesamesetofcanceledpaths.Werefertotheserunsastrials.Next, wedescribeinAlgorithm3thestepsinvolvedinatrialthatseekstoproduceanew assignmentforthereservationsthatwerecanceled. Toproduceagreedyassignment,werstcomputetheresidualcapacitiesofthe networkafterallselectedpathshavebeencanceledwhiletheothersarexedexactly asintheincumbentsolution.Wethensortallsupplyandreleaseverticeswerefer tothesetofsupplyandreleaseverticesas e S bydatesofavailability.Thenforeach reservationordemandvertex e r thatisnotscheduledyet,weinvestigatesupplyor releaseverticesthatcanbeusedpossiblyincombinationtosatisfy e r .Tothisend, westartbylookingattherst h supplyandreleaseverticeswithearliestavailabilityfor h =1 ;:::; k e Sk where k e Sk representsthenumberofelementsin e S .Wedenotetheset ofsortedsupplyandreleaseverticescontainingtherst h elementsof S as e S h .Among them,wepickthosewhosesumislargestandassigntheircarconsistto e r .Wethen computethetimeittakestosendcarsfromthesesupplyandreleaseverticesto e r .By 47

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incrementing h ateachiteration,weobtaindifferentwaysofsupplying e r withcars.We associateascorewitheachassignmentalowerscoreisabetterscorebasedona weightedcombinationofthetimeatwhichcarsarrivewedenotethecorresponding weightby ,thenumberofsplit-and-mergeoperationsinvolvedforwhichwexa maximumlimitof k wedenotethecorrespondingweightby ,andanincentivefor notmissingtheduedateif e r isademandvertex.Rememberthatwhenaduedateis associatedwithavertex,wearerequiredtodelivertherequestedcarsbythispre-set duedate.Therefore,if e r isexaminedlateinprocess,allthesuppliesthatcansatisfythis requestontimehaveprobablybeenalreadyassignedtootherreservationsordemand vertices.Ifthisundesirablesituationhappens,weincreasetheincentiveassociatedwith thecorrespondingdemandverticestohelpthembeconsideredearlierinsubsequent trials. Foreachreservationordemandvertex e r ,wekeepalistofthe 1 differentsupply assignmentswithminimumscores.Oncethislistisestablished,werandomlypick onesupplyassignmentinthelistandmakeitourcandidatesupplycombinationfor e r Werepeatthisprocedureforallnonscheduledreservationanddemandverticesand createalistof 2 reservationordemandverticeswithbestscores.Oncethislistis constructed,werandomlyselectareservationordemandvertex e r inthelist.Sincethe supplyassignmentassociatedwith e r isknown,weapplyittotheproblem.Wethen remove e r fromthelistofreservationanddemandverticestobescheduled.If e r isa reservationvertex,wesuitablyreducethecapacitiesofitsloadoriginanddestination and,basedon e r 'scompletiontime,wecreateanewsupplyvertex n supply .Wesetthe carcountof n supply tobethecarcountrequestedby e r ,wesetthelocationofavailability of n supply tobethedestinationof e r andnally,wesetthetimeofavailabilityof n supply to bethecompletiontimeof e r .If e r isademandvertex,thensimilarly,wecreatearelease vertex n release andpopulateitsattributes.Asthereservationassociatedwithademand vertexisnotconsideredtobereleasedi.e.,itsscheduleandthereforeloadoriginand 48

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destinationcapacityconsumptionsarexedasintheincumbentsolution,thereisno needtoupdatenetworkcapacitiesinthiscase. Thisprocedureisiterateduntilallreservationanddemandverticesofthecanceled setofpathsaregivenaschedule.Wemighthoweverbreaktheloopearlierifthe makespanofthegreedyschedulebecomestoolargeorifthetimeatwhichademand vertexissatisedisoverdue.Insuchasituation,weincreasetheincentiveusedin thescoreformulaforthisdemandvertex.Ifattheendofapre-setnumber I oftrials asolutionisobtainedinwhichalargersetofreservationsissatised,orashorter makespanisobtainedforthesamenumberofsatisedreservationswhencompared totheincumbentsolution,wereplacetheincumbentsolutionwiththisnewimproved solution.Otherwise,weconsideranotherneighborhood. 2.3.3LoopingProcedure Asawaytoimprovethequalityofcar-to-reservationassignmentsforsystemcars, weintroducethenotionoflooping.Themotivationforthisprocedureisthat,when multiplereservationssharethesameloadorigin,destination,carkindandrequestedcar count,thenitisquiteefcientandnaturalinpracticetosatisfythemoneaftertheother, onaloop,usingthesameequipment.Suchassignments,besidesbeingpractical,also helptoreducethenumberofdecisionsthatmustbemadebythesubsequentmodels, therebyprovidingadecreaseinsolutiontimes. TheloopingprocedureinteractswithbothACAPandDPP.First,itneedstoaccess anACAPsolutionsincecarconsistassignmentsforreservationsarerequiredinorder toidentifyloopablegroupsofreservations.Rememberthatconsistassignments aredeterminedduringthesolutionofACAPandarebasedoncarpreferencesof customersandonoverallnetworkcapacities.Therefore,wecanonlyperformlooping afterACAPissolved.Itissimpletoguaranteethatsufcientcarandterminalcapacities areavailableinthenetworkforreservationsinvolvedintheloopingproceduresince wecanestablishtheresourceconsumptionofreservationsfromtheirplacementinthe 49

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loop.Theschedulesofloopedreservationshowever,arenotalwayscompatiblewith thoseassignedintheinitialACAPsolutionandmayresultincapacityconicts.Toavoid suchconicts,andensurethefeasibilityoftheglobalsolution,weadjustcar,loadorigin anddestinationcapacitiestotakeintoaccounttheconsumptionofloopedreservations, afterloopingisperformed.WethensolveavariantofACAPinwhichweconsideronly non-loopedreservationstodeterminetheiradjustedschedulesaswellastheiradjusted carassignments. Second,loopingdecisionsmustalsobeconsideredduringDPP.Inparticular, CtRAPisthestepofourconstructiveprocedurethatassignscarstothepathsthat theyfollowduringthemonth.Wemustensurethatthesequenceofreservationsin theloopedpathsisnotbemodiedbyCtRAP.Tothisend,weintroducethenotion ofconcatenatedreservations.WhencreatingverticesforCtRAPinthepresenceof looping,werepresentallreservationsinaloopasasingleconcatenatedreservation vertex.Thissinglevertexactsasaregularreservationvertexrequestingthecarcount commontoallloopedreservations,havingtheloadoriginandstarttimeofitsrst reservationandtheloaddestinationandendtimeofitslastreservation.Acarow throughthisvertexsigniescarstravelingthroughthecorrespondingsequenceof reservationsfromloadorigintodestination.Withtheuseofconcatenatedreservation nodes,wecansolveCtRAPasdescribedinSection2.3.2.1.1.3andusethesamepath decompositionalgorithmtoderivepathsthatcarsfollowthroughthemonth.Itthen remainstoexpandtheconcatenatedverticesbackintopathsbyreplacingthemwith thesequenceofreservationsofthecorrespondingloops.Theexpandedpathsthatwe obtaincanthenbeusedtoconstructtheETSPnetworkwithoutfurthermodications. Weconcludethissectionbydescribingtheprocedurethatweusetogenerate loops.Werstcreategroupsforreservationsthatshareloadorigin,destination, requestedcarcountandassignedcarkind.Usingthereservationsofeachofthese groups,westartconstructingloopsbyschedulingreservationsoneafteranother.We 50

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tentativelysetthestartdayoftherstreservationonthelooptobetherstdayof themonthinwhichasufcientnumberofcarsisavailable.Wesetthestartdayof thefollowingreservationstobetheenddayofthepreviousreservationontheloop. Oncewehavesetatentativestartdayforareservation,wedeterminewhenthetrain reachestheloadoriginanddestinationaccordingtothecycledurationcommontoall reservationsofthegroup.Wethenchecktheassociatedresidualcapacitiesofload originsanddestinationsonthesearrivaldays.Ifbothloadoriginanddestinationhave sufcientcapacityonthedesignatedday,weacceptthetentativestartdayandcontinue theprocesswiththenextreservation.Otherwise,weincrementthestartdayofthe reservationbyoneandcomputewhetherloadoriginanddestinationcapacitiesare compatiblewiththisschedule.Werepeatthisprocedureuntilwendastartdayforthe reservationthatdoesnotviolateresidualcapacities.Wekeepaddingreservationsin thisfashionuntilnoreservationfromthecurrentgroupcantintheremainingdaysof theplanningperiod.Further,wedonotwanttocreateshortloopsbeingcompletedfar beforetheendofmonthaswebelievethatthesereservationsarebetterhandledusing theregularsolutionprocedure.Therefore,wedenethenotionofafulllooptobeone thatissuchthat,ifonemorereservationwithsamecycletimewastobeaddedtothe loop,thetotaltimerequiredtocompleteallreservationsontheloopwouldexceedthe durationoftheplanningperiod.Weonlyconsiderfullloopswhenapplyingthelooping procedure. 2.4ComputationalResults Weperformedaseriesofcomputationalexperimentsbasedonthecoalnetworkof ourcollaborator,anAmericanClassIrailroadcompanyCSXCorporationInc.,2013. Thenetworkconsistsofapproximately700terminalsand2300tracksegments.We implementedtheproposedsequentialprocedureincludingloopinginVisualC#under MicrosoftVisualStudio2010andtestedtheexperimentsonadesktopwith2.27GHz XeonIntelProcessorand6GBRAM. 51

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Forthepurposeofthischapterandbecausesomeofthepracticaldataissparse andproprietary,wegenerateacollectionofrealisticprobleminstancesofincremental sizes.Size,inthecontextofMCTRPP,referstothemonthlynumberofrequested reservations.Foreachselectedsize,wecreate 10 differentinstances.Eachfamilyof instanceislabeledwiththeassociatedproblemsize.Inordertocreatechallengingyet realisticinstancesevenwithsmallproblemsizes,weselecttheavailablecarresources tobeproportionaltotheproblemsizewhilekeepingvertexcapacities,networkstructure, emptyandloadeddwelltimesatverticesandtransittimesontracksegmentsintactand independentoftheproblemsize. Wederiverealisticcarcountsbycarkindandreasonablenumberofreservations byconsigneeandownershipforallproblemsizesusingthepracticalinstancesofthe problemthatwereprovidedbyourindustrycollaborator.Inparticular,weinferthe averageratioofsystemreservations s andprivatereservations p =1 )]TJ/F24 11.9552 Tf 12.134 0 Td [( s fromthe distributionofreservationsinave-monthperiod.Foreachprobleminstanceofsize n wesimplysetthedesiredcountsofsystemandprivatereservationstobe s n and p n respectively.Wedeterminethedistributionofprivatereservationsamongconsignees similarlybyinferringreservationpercentagesperconsigneefromourcollaborator's data.Werescalethereservationpercentagesperconsigneeaveragedovervemonths inorderforthesummationofallreservationpercentagestoreach 100% andround fractionalreservationcountstoexactlymatchthedesiredproblemsize.Wethen randomlypopulatesystemandprivatereservationsetsfromacollectionofreservations thatwereobservedinpracticeduringavemonthperiod.Likewise,wedetermine systemcarcountsbycarkindusingtheaverageratioofsystemcarkindsoversystem reservations.Todeterminethecountofeachprivatecarkind,weusepercentageratios calculatedbycarkind. Inourgeneratedinstances,allthecarsareassumedtobeavailableontherstday oftheplanningmonth.Forsystemcars,arandombuildlocationisselectedasthestart 52

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locationofthecars.Weassumethatprivatecarsareavailableatthepoollocationof theconsigneei.e.,ayardusuallyownedbyorinservicefortheconsigneewherethe privatelyownedcarsarekeptwhentheyareidle.InTable2-1,wepresenttheresults obtainedusingouralgorithmforvariousproblemsizes.Thistablecontainstheinstance names,ACAPacceptedreservationcounts,averageACAPCPUtimesinminutes, DPPacceptednonpurgedreservationcounts,averageDPPCPUtimesinminutes andpercentgapsbetweenACAPandDPPsolutionsforeachproblemsize.Withinthe columnsofACAPandDPPacceptedreservationscounts,weprovidetheminimum, averageandmaximumcountsoverthe10instancesgeneratedforeachproblemsize. Likewise,minimum,averageandmaximumpercentgapsareprovidedunderthePercent Gapcolumn.WeobservethattheresultsofACAPprovideagoodrough-cutestimatefor thenumberofreservationsthatcanbesatisedthroughthemonth. Tovalidatethepracticalrelevanceofouralgorithm,wecompareactualreservation acceptanceratioswiththeratiosprovidedbyACAPandDPPonpracticalinstancesof theproblemcollectedover 5 monthsbyourcollaborator.Theseresultsarepresented inTable2-2.Resultsimplythatanaverageof 8 : 88% improvementcanbeachieved. Further,ontheseproblems,theresultsproposedbyDPPliewithinanaveragegapof 3 : 26% fromACAPresults. Wealsoconductedastudytoexaminetheperformanceofbothofourimprovement proceduresfortheproblemspresentedinTable2-1.Wesetthemaximumnumber ofreleasedreservationstobe50and75forinstancesofsizeupto750,100and 150fortherestintheimprovementstepsofACAPandDPPrespectively.Inboth procedures,werunthealgorithmfor5iterations.InDPP,wesetthenumberoftrials tobe100foreachiteration.Theaverageimprovementwithrespecttotheconstructive stepsolutionisgiveninTable2-3forACAPandinTable2-4forDPP.Forinstances ofsmallsizes,weobservedthattheimprovementphaseofACAPisveryeffective. Thiscanbeattributedtothefactthatwheninstancesaresmallcarsubstitutionis 53

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crucialtoobtaingoodsolutionsandisnotavailableduringtheconstructivephase. OurcomputationalexperiencealsosuggeststhattheACAPimprovementheuristicis usefulmostlyduringtherstfewiterations.TheresultsofTable2-3arenotsubstantially changedifthenumberofiterationsisincreased.TheaverageimprovementinDPPis smallercomparedtoACAPhoweverincreasingthenumberofsatisedreservationsby evenanaverageof 1% issignicantforourindustrycollaborator. 54

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Algorithm1 Eventtimestampingalgorithm Input: M Output: Machine-precedencearcsforall m i in M ,starttimesofalloperations n in ETSPnetwork. Method: Set M nonxed M M xed ; while M nonxed 6 = ; do Let max =0 forall m i 2 M nonxed do Sequence m i byapplyingAlgorithm2. Additsmachine-precedencearcstemporarilytoETSPnetwork. Updateprecedence-relationsbetweenalloperationstemporarily. Calculate m i ,themakespanof m i if m i > max then max m i endif endfor Declaremachine m i yielding max asbottleneckmachine m b Addmachine-precedencearcsof m b toETSPnetwork. Updateprecedence-relationsbetweenalloperations. Move m b from M nonxed to M xed forall m i 2 M xed nf m b g do if anymachine-precedencearcof m i isonthelongestpathofcurrentETSP network then Declare m i ascriticalmachine m c Re-sequence m c Updatemachine-precedencearcsof m c Updateprecedence-relationsbetweenoperations. endif endfor endwhile CalculatestarttimesforalloperationsinETSPnetwork. 55

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Algorithm2 Single-machinesequencingalgorithm Input: Nonxedmachine m ,setofoperations N m tobesequencedon m Output: Machine-precedencearcsof m Method: Set t = 1 forall operations n tobesequencedonmachine m do n:head LongestPath Start n n:tail LongestPath n End )]TJ/F24 11.9552 Tf 12.622 0 Td [(n:pt if n:headtail max then tail max n:tail pt max n:pt n n elseif n:tail = tail max then if n:pt pt max then pt max n:pt n n endif endif endif endfor n : start time t Add n totheoperation-sequenceofslot s s .next availability time max f s .next availability time ;n : start time g + max f n:pt; 24 g Remove n from Q t max f m .next earliest available slot time ; min f n:head j n 2 Q and n hasno predecessorin Q gg endwhile 56

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Algorithm3 Improvementprocedure Input: SetofCtRAPpaths P .Setofcanceledpaths P 0 Output: Anewscheduleforthereleasedreservationsofthepathsin P 0 Method: Createcollection C ofreleasedreservationsonthepathsof P 0 forall paths p 2P 0 do Addsupplyvertex n supply of p tothesetofsupplyvertices N supply forall reservations r appearingin p do if 9 p 0 2PnP 0 suchthat r appearsin p 0 then Createdemandvertex n demand ,populateitsattributes,additto N demand Createreleasevertex n release ,populateitsattributes,additto N release else Createreservationvertex n res ,populateitsattribute,additto N res endif endfor endfor Updatecapacitiesofloadoriginsanddestinations. 57

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Algorithm3 Improvementprocedurecontinued for trial i =1 to I do Initializesetofscheduledvertices R sch ; Initializesetofverticestobescheduled R non-sch N res [N demand while R non-sch 6 = ; do Sortsetofsupplyvertices e SN release [N supply bytimeofavailability. forall e r 2R non-sch do if e r 2N demand then Set incentive e r =0 else Set incentive e r =bigM endif for h =1 to k e Sk do forall e s 2 e S h do if sumofmax k f car count e S 1 ;:::; car count e S h g > car count e r then Select k k suchthat max k )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 f car count e S 1 ;:::; car count e S h g < car count e r and max k f car count e S 1 ;:::; car count e S h g car count e r Setmax k f car count e S 1 ;:::; car count e S h g astheassignmentof car count e r forall cargroups e s inmax k f car count e S 1 ;:::; car count e S h g do Computetimeatwhich e s arrivesatbuildlocationof e r ifneeded. endfor Computethenumberofsplit-and-mergeoperations y sm inthis assignment. Computecompletiontime t e r end of e r giventothisassignment. Computethescoreofthisassignmentto e r as t e r end + y sm )]TJ/F15 11.9552 Tf 11.173 0 Td [(incentive e r endif endfor endfor Keep 1 minimum-scoredsupplyassignmentsfor e r .Pickonerandomly. if e r 2N demand and t e r end > e r: due date then Increase incentive e r breakwhile; endif endfor Keep 2 minimum-scoredreservations.Selectonerandomly. Applythesupplyassignmentfortheselectedreservation e r if t e r end current-makespan then breakwhile; endif ...continuesonnextsection endwhile endfor 58

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Algorithm3 Improvementprocedurecontinued for trial i =1 to I do while R non-sch 6 = ; do ...continuingfromprevioussection Updateloadoriginanddestinationresidualcapacities. Remove e r from R non-sch if e r 2N demand then Createareleasevertexfor e r elseif e r 2N res then Createasupplyvertexfor e r endif Addcreatedvertexto e S Add e r to R sch endwhile Computethemakespanoftrial t andthenumberofreservationssatised. if thesolutionobtainedattheendoftrial t isbetterthantheincumbentsolution then Setthissolutionasthenewincumbentsolution. endif if thesolutionobtainedattheendoftrial t satisesallthereservationsbeforethe endofthemonth then Updatetheincumbentsolution. Createsetofpathsassociatedwiththeincumbentsolution. breakfor; endif endfor Updatetheincumbentsolution. Createsetofpathsassociatedwiththeincumbentsolution. 59

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Table2-1.ComputationalresultsforMCTRPP. Collection ACAPAcceptedACAPDPPAcceptedDPPPercentGap ReservationCountCPUReservationCountCPU TimeTime minavgmaxminavgmaxminavgmax MCTRPP1008992956.68689921.00.002.845.43 MCTRPP2001841881927.51731811863.91.094.027.81 MCTRPP5004684734804.945246147031.70.842.685.01 MCTRPP7506366807147.061664968193.61.164.566.50 MCTRPP10008448799177.0771817844176.52.336.9510.05 MCTRPP1250965103510849.3902950990265.86.228.1811.87 MCTRPP150011581190121213.5106810911129375.05.018.3210.73 MCTRPP175012841327139714.8115612001249474.77.519.0912.22 MCTRPP200012611430150019.9119913121366594.64.929.5910.78 60

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Table2-2.Actualvs2-stepsolutionprocedureperformancecomparison. PlanningActualACAPDPPPercent MonthAcceptanceRatioAcceptanceRatioAcceptanceRatioImprovement Month183.37%95.08%91.63%8.27% Month281.19%94.90%92.60%11.41% Month382.30%91.84%88.99%6.69% Month474.67%88.19%83.93%9.26% Month578.45%90.69%87.24%8.79% Table2-3.ACAPImprovementheuristicperformances. CollectionPercent%SuccessfulCPUTime ImprovementIterationsinmin. minavgmax MCTRPP1002.2052.4580.39441.4 MCTRPP20031.9460.9576.64361.2 MCTRPP50027.3934.5041.99301.4 MCTRPP75018.3624.0031.53462.6 MCTRPP100018.0420.9924.66402.7 MCTRPP125010.2514.1317.44343.3 MCTRPP15008.1712.3115.11304.0 MCTRPP17509.3411.4912.68385.3 MCTRPP2000012.4613.99308.8 61

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Table2-4.DPPImprovementheuristicperformances. CollectionPercent%SuccessfulCPUTime ImprovementIterationsinmin. minavgmax MCTRPP10001.502.2761.0 MCTRPP20000.892.25101.7 MCTRPP50001.262.872410.7 MCTRPP75000.423.012429.7 MCTRPP100001.063.331855.4 MCTRPP125000.541.431286.9 MCTRPP150000.301.5714134.9 MCTRPP175000.310.6014124.9 MCTRPP200000.611.0116144.4 62

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CHAPTER3 MONTHLYCOALRESERVATIONSPLANNING OurindustrycollaboratorisoneoflargestrailtransportationsuppliersintheUnited States.Thecompany'snetworkhasapproximately21,000milesoftracksin23states andservessomeofthelargestpopulationcentersinthecountry.Nearlytwo-thirdof Americanslivewithinthecompany'sserviceterritory.Withitsdominatingrailpresence overtheUSeasternterritories,thecompanyhasaccesstoover70ocean,riverand lake-portterminals. Theunittrainserviceisoneofthemostprotablebusinessesofourcollaborator Thalji,2012.Inrailroadsterminology,aunittrainisatrainoperatinggenerallyintact betweenpointoforiginandnaldestination,typicallyhaulingasinglebulkcommodity, composedoflikecars,andequippedwithhigh-tensilecouplers.Boththecompany anditscustomersconsiderunittrainstobeeconomical,reliableandefcientforthe shipmentoflargequantitiesofasingleproduct.Commoditiestransportedviaunittrains musthavesufcientlyhighvolumetowarrantthededicationofasingletrain.Grain,bulk, mineralsandwastearesomeofthecommoditiesoftentransportedthroughunittrains. Thetonnageofexportcoalthatistransportedalmosttripledinvolumefrom2006 to2011.Withthisincrease,coalhasbecomeamajorpartoftheunittrainserviceof thecompany.Tosupporttheaforementionedbusinessgrowth,coaloperationshave continuallyadjustedtheircapacityandimprovedtheirpractices.Besidesitscontinued investmentandupgradesinequipment,tracksandinfrastructure,thecompanyhas alsoseekedtoimproveitsreservationandinformationsystemstomaximizecustomer satisfaction,andtoincreaseprotabilityofthecoalbusiness. Aspartofanefforttoimprovethecoalreservationssystem,acollaborationwas startedbetweentheUniversityofFloridaandOperationsResearchteamofourindustry collaboratorthatresultedinthecreationoftheUnitTrainPlanningSystemUTPS. Thistoolhelpstodeterminealevelofreservationstoacceptforboththeongoingand 63

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upcomingmonthsthatissufcientlycognizantofthesystemcapacitiesthatsupporting trainschedulescanbelaterbuilt.Oncefullyimplemented,thistoolisestimatedto provideconsiderablereductionsinrailcarcycletimesandtobringsubstantialannual savingsascycletimereductionsbringopportunitiestocarrymoreloads.Thetoolisalso expectedtoimprovetheproductivityofcoaldirectorsandcoordinatorsbyallowingthem toswitchfromthecurrentmanualplanningandschedulingdecisionstoanoptimization tooltherebystreamliningtheirdailyroutines. InSection3.1,wegivethedetailsofthemonthlycoalreservationsplanning problem.WethendiscussinSection3.2thechallengeswefacedinobtainingagood forecastplanthatiscompatiblewithactualplanning.Wealsodescribehowwehandled differentpracticalaspectsoftheproblemandpresenttheheuristicsolutionmethodwe developedtoconstructtheforecastplan. 3.1PlanningMonthlyCoalReservations UTPSaimstostreamlineandimprovethecoalreservationprocessofourindustry collaborator.Reservationsarethecoalrequestsfortheupcomingmonththatcustomers placetothecompany.Reservationsspecifythenumberoftrainsandcarsacustomer requeststopickupcoalatspeciccoalminesandtodeliverittospecicdestinations. Becausethenumberofreservationrequestsislarge,itmightnotbepossibleforthe companytosatisfythemallduringtheupcomingmonth.Infact,mineshavelimitedtrack capacitiesandloadingmachinerythatrestrictthenumberoftrainsthatcanvisitthem duringaday.Similarly,customerlocationshavelimitedtrackcapacitiesandunloading machinerythatlimitsthenumberoftrainsandcarsthattheycanaccommodateeach day.Further,sinceunittrainshavelowpriorityamongthevarioustypesoftrains circulatingintherailnetwork,somerailtrackslocatedbetweenbusyterminalsmight alsobelimiting.Forthosetracksegments,identiedasbottleneck,adailycoaltrain capacityisalsoincurredtopreventcongestion.Moreover,theeetofcarsthatthe companyusesforcoaltransportationislimitedinsizeandcontainscarsthatdifferfrom 64

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eachotherduetosomedistinctivecharacteristicsthatreducethenumberofpossible reservation-to-carassignmentstremendously.Finally,railcarsmustbemovedusinga locomotiveeetwhichisitselfverylimitedandiscomposedofveryvaluableresource thatrequireshighutilization. Animportantproblemforourindustrycollaboratoristhereforetodeterminethe largestsuitablesubsetofreservationsthatcanbeacceptedsothatthereissufcient internalcapacityduringthemonthtosatisfythesecommitments.Althoughthemost importantdrivingfactorofthesolutionisthenumberofreservationsaccepted,other factorsalsocomeintoconsiderationwhendeterminingwhetherasolutionisofgood quality.Weprovidenextadescriptionofthesefactorstogetherwiththedetailsofthe overallcoalreservationssystem. 3.1.1CollectingReservations Coalcustomersareaskedtocommunicatetheirreservationrequeststothe companyfortheplanningmonthoneweekpriortoitsstart.Thisweekisreferredto astheplanningweek.Asinglecustomermightplaceseveralrequestsforthemonth. Theserequestsmaydifferintheminesatwhichcoalistobeloadedbutalsointhe destinationswherecoalmustbedelivered.Foreachreservationorder,thenumberof carsrequested,theminealsoreferredtoasloadoriginandtheloaddestinationare speciedbythecustomer. Ourindustrycollaboratorrequiresthatreservationrequestsbeplacedearlysoasto givecoalcoordinatorssufcienttimetobestplantheuseofresourcesfortheupcoming month.Thistimealsoallowscustomersandminestocompletetheiragreements beforetrainsaresenttomines.Thisrequirementishowevernotalwaysrespectedby customersandreservationrequeststendtotrickleinafterthebeginningofthemonth duringwhichtheymustbesatised.Historicaldataindicatesthatonlyanaverageof 35%ofreservationrequestsarebeingenteredinthesystembeforethestartofthe planningweek. 65

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Reservationrequestsdonotneedtobeacceptedorrejectedimmediatelyafterthey arereceived.Duringtheplanningweek,coaloperationsprioritizereservationsentered inthesystem.Theythenmarkreservationswithhigherpriorityaspublished.Published reservationscanbeseenasacommitmentthatthecompanymakestoitscustomersto satisfytheirreservations.Onceareservationispublished,everyeffortshouldbemade tosatisfyitinthewayitispublished.Inparticular,thepublishedload-inandload-out datesshouldberespectedasmuchaspossible.Althoughitispreferabletosatisfy apublishedreservationwiththepublishedequipment,thisisnotviewedascritical. Droppingapublishedreservationisonlyconsideredunderexcruciatingcircumstances. Whenithappens,railroadcompanyremovesthecorrespondingreservationfromthe system.Thecustomermightthenconsiderplacinganewreservationrequestforthe followingmonthasareplacementforthiscancelledreservation. Inmanycases,thepublishedload-inandload-outdatesaremostlydecidedbythe railroadcompanyasreservationsdonotcomewitharmduedate.Somereservations howeverhavespecicduedatesandshouldbesatisedontime.Werefertosuch reservationsaspermitreservations.Mostreservationsarenotpermitreservationsand thecompanyhasmoreexibilityinschedulingthem.Examplesofsuchreservations includethosehavingaportasloaddestinationwhereavesselwithstricttimetableneed tobeloadedwithcoalbeforeitleaves.Satisfyingsuchareservationlateorearlycan causeproblem.Whenatrainarrivesearlyatitsloadorigin,thecoalorderedmightnot havebeenminedyet.Insuchasituation,theequipmentandtheenginesmighthave tobekeptidleforsometimewhiletheycouldhavebeenusedforotherassignments. Similarly,satisfyingarequestlatecancauseseveredissatisfactiononthepartofthe customer. Coaloperationsdonotpublishallrequestedreservationsandmightpublish reservationsinastaggeredfashion.Severalfactorscontributetothedecisionof publishingorrejectingareservation.Thedrivingfactorinthisdecisionisrelatedto 66

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thecapacitiesofthenetwork.Asecondimportantfactorariseswhencustomersenter morereservationrequeststhantheyactuallyneedinordertoavoidbeingshortedlater duringthemonth.Athirdfactoristheoverallrequestedvolumeofeachcustomerunder consideration.Forinstance,rejectingasinglereservationofasmallcustomerhavinga totalmonthlyrequestofthreetrainsislessdesirablethanrejectingareservationfroma biggercustomerthatrequestsdozensoftrains.Forthesereasons,coaloperationshas heavilyreliedonhistoricalaveragestoacceptorrejectreservations. Inadditiontoprovidingcarsandtrainsforitscoalunit,therailroadcompanyalso providesservicefortrainsoriginatingfromloadoriginsorheadingtodestinationsthat arenotwithinitsterritories.ThesereservationsarereferredtoasN-reservations. IncomingN-reservationtrainsarriveatrailroadjunctionterminalswithequipmentthat belongstothetransferringrailroad.Themainresponsibilityofthecompanyinthecase ofincomingN-reservationsistoattachsuitablelocomotivecombinationstothesetrains andtodelivertheloadtotherequireddestinationservicedbytherailroadcompany.In caseofoutgoingN-reservations,theemptyequipmentisdeliveredtotherequiredmine withintheterritoryoftherailroadcompanyand,onceloaded,thetrainistransferred backtothedesiredjunctionterminalusingthelocomotivesownedbythecompany. TrainsassociatedwithN-reservationsdonottypicallyconsumecarresourcesbutneed tobeconsideredinplanningmodelsbecausetheyutilizemines,customerterminalsand bottleneckrailtracks. Thelargevolumemakesitdifcultforahumanoperatortoconsiderallsystem restrictionswhendecidingwhichreservationsshouldbesatisedduringtheupcoming planningmonth.Further,theprocessisdynamicandmakingmanualadjustments throughoutthemonthtoaccommodatenewreservationsandunexpecteddelays isatremendouschallenge.Whensuchunexpectedsituationsoccur,ourindustry collaboratortriestoreschedulereservationswithoutperturbingtoomuchpreviously madecommitments. 67

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3.1.2CoalMines Associatedwitheachreservationisaspecicminewherecoalistobeloaded. Minesareconcentratedinspecicareasofthecountry.Whenconsideredindividually,a minecantypicallyloadasingletrainatthetime.Dependingonitsspatialconguration andloadingequipment,itmightonlybeabletoaccommodatealimitednumberofcars atatime.Further,loadingmechanismsusuallydeterminethespeedoftheloading process.Asaresult,loadingtimesforeachminedifferandaffectthedailynumberof reservationsthattheycanhandle.Fromthesenumbers,itispossibletoinferdailytrain capacitiesandcarcapacitiesatmines.Sometimes,minesarelocatedinsuchproximity thattheysharecommonrailtracks.Whenseveralminesarelocatedinsuchaclose area,theyaresaidtoformabranch.Whenthishappens,atrainbeingloadedata mineofthebranchmightblockthetrackthatanothertrainwouldneedtousetoreach aclosebymine.Thesetoftrackscommontoallminesinabranchimposecapacity restrictionsonthemaximumnumberoftrainsandcarsthatcanbeloadedonasingle dayattheminesinthatbranch. Variousotherrequirementscanbeconsideredformines.Forinstance,eventhough certainminescanaccommodateonetrainperday,itmightnotbepossibleforthemto loadatrainonconsecutivedays.Thereforemoreelaboratecapacityrestrictionsmight existthatareimposedoveracoupleofdays.Finallyminesperiodicallyclose.Trains cannotbesenttoaclosedmine. 3.1.3LoadDestinations Oncetheircarsareloadedwithcoal,trainstravelfromtheminetothedestination speciedbytheassociatedcustomer,whichisreferredtoasconsignee.Some destinations,suchasexportlocationsandpiersaresharedbymultipleconsignees. Itisalsopossibleforasingleconsigneetoownmultipledestinationsandtohave destination-specicrailcarpreferences. 68

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Destinationsdifferwidelyandrangefromsmallpowerutilityplantstolargecoal exporters.Destinationshavedailytrainandcarcapacitiesthataresimilartothoseof mines.Actualdailycapacitiestendhowevertobemuchlarger.Forinstance,yards associatedwithcoalexportterminalshavemultipleliningswithsufcientunloading machinerytosupporttheunloadingofupto15trainssimultaneously. 3.1.4RailTracks Thecoalnetworkofourindustrycollaboratorcoversaround700terminalsand 2300railtracksthatconnectallterminals.Totravelfromoneterminaltoanother,coal trainsuseapathofshortestdistancethatavoidsunnecessaryvisitstoloadorigins anddestinations.Asaresult,somerailtracksappearinnumerouspathsleading tohigh-volumeareasthathostsignicantlymoretrainsthanothers.Thecompany identiesthesebusyrailtracksasbottleneckandmaintainsadynamiclistoftheserail tracksasthevolumeofcoaltransportedshiftstovariouspartsofthenetworkovertime duetochangingdemandpatterns.Inordertopreventcongestionalongthebottleneck railtracks,adailymaximumnumberofallowedtrainsisdeterminedandenforcedduring planning.Thedailytraincapacitiesalongrailtracksisspecictothedirectiontrainsare travelingontherailtracks.Forinstance,arailtrackthatliesbetweenterminalAand terminalBmightbeconsideredtobebottleneckandmighthaveastringentdailytrain capacityrestriction,while,inthereversedirection,itmighthaveadifferentdailytrain capacityoritmightnotevenbeconsideredtobeabottleneck. 3.1.5RailCars Coalistransportedfromorigintodestinationinrailcars.Therailroadcompany ownsvarioustypesofcarsthatdifferintheirbuilt,capacityandphysicalcharacteristics. Furthermore,someconsigneesownpoolsofcarsthatthecompanyusestotransport thecoalassociatedwiththeirreservations.Coalcarsarethereforedifferentiatedby ownership.Werefertocarsthatthecompanyownsassystemcars.Werefertothe othersasprivatecars.Weusethevocablecarkindtodescribethecombinationofacar 69

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typeandacarownership.Ifthecarisprivate,carkindalsoencompassestheowning consignee. Thenumberofcarsrequiredforareservationvariesbutisneverlessthan 75 in orderforthereservationtowarranttheformationofaunittrain.Thenumberofcars onacoaltrainismostoftenlessthan 150 butcanbemoreinsomerareoccurrences. Virtuallyallmultiplesof 5 carsbetweenthisupperandlowerboundareobservedin practice. Whenchoosingtheactualcarstobeassignedtoareservation,itisimportant toadheretosomeguidelines.First,thecarspecicationsofcustomerswithrespect totheirreservationsmustbeaddressedwithcare.Thedemandofcustomerswho requestedtheirreservationstobesatisedwithsystemcarswerefertothese reservationsassystemreservationsmustbesatisedfromtheeetofsystemcars andcannotbesatisedfromtheirownprivatecars,norfromcarsbelongingtoother customers/consignees.Thedemandofcustomerswhorequestedtheirreservations tobesatisedusingtheirownprivatecareetswerefertothesereservationsas privatereservationsmustbesatisedwithcarsownedbythesecustomers.Theonly raredeviationtothisrule,whichinvolvestheassignmentofacombinationofprivate andsystemcarstoasinglereservation,occurswhencustomersexplicitlyenterinan agreementwiththerailroadcompanytoprovidethemwithsystemcarswheneversome portionoftheireetisinuse,unavailableorinsufcientinsizetocarrytheamountof coaltheyrequireduringthemonth.Further,somecustomerswhodonotownprivate carsmightstillrequestprivatecarhandlingwhentheyleasecarsfromotherprivatecar owners.Insuchsituations,weconsidertheleaseetobetheownerofthecars. Foreachsystemreservation,customersdeclareapreferredcarkind.The consigneeexpectsthereservationtobesatisedwiththiscarkindifitisavailable. Preferredcarkindsareafunctionofloaddestinations.Customersalsospecifyalist ofadmissiblecarkindsforeachreservation.Thesecarkindscanbeusedwhenthe 70

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preferredcarkindisnotavailable.Finally,somecarkindsarestrictlyforbiddenfrom certaindestinations.Werefertothemasexcludedcarkinds.Acarkindisexcludedfrom adestinationwhen,forinstance,thatdestinationisnotequippedtounloadcarsofthis makeorbuilt.Itispossible,althoughnotdesirable,tomeetthedemandofacustomer requestingsystemcarswithamixofdifferentadmissible,includingpreferred,carkinds. Theprivatecarsofaconsigneearebundledingroupsthatarecalledsets.Some setsthataconsigneeoperatesmightbeexcludedfromsomedestinationsofthis consignee.Excludeddestinationsoftenarisefromthefactthatconsigneesprefer keepingtheirsetscyclingbetweenthesamemineanddestinationpairs.Itfollows thatprivatereservationsareassignedtoasetratherthantoacollectionofdisparate cars.Althoughcustomerspreferkeepingthesesetsasintactaspossiblethroughthe month,thenumberofcarsrequestedinareservationmightnotalwaysmatchasetsize, therebytriggeringchangesinsetcompositions.Thecarconsistofaset,i.e.,thecounts andtypesofthecarsformingtheset,isthereforeexibletothedegreethatresizing shouldinvolveonlyafewcars.Consigneesowningsetstypicallyalsohavesparecar pools.Therefore,wheneverasetisassignedtoareservationrequiringmorecarsthan thenumberofcarscurrentlyintheset,thesetconsistisexpandedusingcarsfrom thesparepool.Similarly,whenthesethasmorecarsthanrequiredbythereservation, someofitscarsarereturnedtothesparepooltomatchtherequestedreservationsize. Whenreservationsdifferinthenumberofcarstheyrequire,carshavetobeeither detachedfromorattachedtotrains.Resizingthecarconsistofatrainisaprocessthat canonlybeperformedatcertainspecicyardsthatwerefertoasbuildlocations.In fact,resizingoperationsrequireskilledmanpowernotavailableateveryterminalofthe networktoperformnecessarysafetychecks.Resizingoperationsshouldthereforebe avoidedwheneverpossible.Asaresult,itiscommonpracticetousetheexactsame carstoconsecutivelyserveseveralreservationsrequestingthesamenumberofcars. 71

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Besidestheserequirementsandpreferences,itisalsoimportanttoconsiderthe currentlocationofcarswhendecidingonassignmentssothatcarsleavingaload destinationcanbedeliveredtothenextloadoriginwithouthavingtoundertakealong journeyacrossthelargeterritoryoverwhichtherailroadcompanyoperates. 3.1.6Locomotives Coalcarsneedenginepowertobemovedfromonelocationtoanother.The railroadcompanyownsvarioustypesoflocomotivedifferinginpullingcapabilities. Althoughthereisnodedicatedpooloflocomotivesforcoaltransportation,thecompany usestwospecictypesoflocomotivestopullcoaltrains,withthepossibleuseofa kickerthathelpswithheavierloads.Dependingonthenumberofcarsonthetrain andthetopologyofthepathfollowedfromloadorigintoloaddestination,itmightbe necessarytoassigntwoorthreelocomotivestoacoaltrain.Coaltransportationofour industrycollaboratorreliesmostlyonthreedifferentlocomotivecombinations.Amongall admissiblelocomotivecombinationsthatmeetthepullingpowerrequirementofapath, thecompanytypicallyusestheonewithsmallestpullingpowertopreservevaluable resources.Werefertothiscombinationasmostdesired. Sinceitisnotclearatthebeginningofthemonthwhatspeciclocomotivescan beusedtopullcoaltrains,anoverallmaximumlimittogetherwithmaximumlimits bylocomotivetypeareprovidedtocoaloperationsforplanningpurposes.Typically, locomotivesthatarealreadypullingcoaltrainsatthestartoftheplanningmonthcan againbeconsideredfornewcoalreservationassignmentslaterduringthemonth. Additionally,providedthattheaforementionedmaximumlimitsarerespected,other locomotivesnishingtheirassignmentswithotherbulkcommoditiescanalsobe consideredforfuturecoalassignments. Becauselocomotiveschedulingisnotunderthecontrolofcoaloperations,our modelshouldnotschedulelocomotives.Rather,givencoaldemand,itshouldaimat 72

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estimatingthenumberoflocomotivesthatwillbeneededthroughoutthemonthandat guaranteeingthatthisthenumberstayswithinreasonablelimits. 3.1.7TravelTimes Traveltimesanddistancesbetweenmines,loaddestinationsandbuildlocations needtobeestimatedaccuratelyinordertoobtainanimplementableforecastplan. Althoughdistancesarerelativelysimpletoobtainsincetrainstendtofollowthesame pathsbetweenoriginsanddestinations,traveltimesarehardertoevaluate.Infact, thesetimesareacombinationofthetimeneededtotraverseeachrailtrackreferred toastransittimeandthedwelltimethatroutinerailroadoperationstakeatterminals alongthepath.Theseroutineoperationsarefundamentaltorailroadsforthehandling offreightandforthebreakingup,makingup,forwarding,servicingoftrainsandcrew changesCSXCorporationInc.,2013.Dependingontheterminaltype,suchasbuild location,crewchangeyard,loaddestinationormineservingyard,anddependingon whetherthecarsattachedtothetrainareloadedorempty,theaveragedwelltime varies.Variationsarealsoobservedintransittimessinceunittrainshavethelowest priorityonrailroads.Infact,coaltrainsoftenhavetobesloweddownordelayedto leavewayforotherloadsorpassengertrainswithhigherpriorities.Becausehistorical averagesarefairlyreliableandvariationsarelimited,weassumethatdwelltimesand transittimesaredeterministic. 3.1.8TimeRestrictions Therearetwoimportantdatesassociatedwiththescheduleofareservation. Therstoneistheload-indateatwhichthetrainreachesthemine.Thesecondis theload-outdateatwhichthetrainreachesitsdestination.Therearetypicallymore restrictionsonload-indatesastrainscannotbeloadedifcoalhasnotbeenminedand cannotbeloadedifanothertrainisalreadybeingloaded.Further,consigneesspecify onlytheload-indateforpermitreservations.Theload-outdatecanbecomputedfrom theload-indatebyaddingloadeddwelltimesatthemineandintermediaryyards,and 73

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transittimesfromloadorigintoloaddestination.Althoughthesedwellandtransittimes arenotconstantacrossalltrips,theyarefairlystableandtherefore,thedifference betweenload-outandload-indatesisroughlyconstant. 3.1.9CurrentMonthvs.PlanningMonth Oncethereservationrequestsfortheupcomingmonththatwerefertoasplanning monthstarttobeplacedduringtheongoingmonththatwerefertoascurrentmonth, theplanningforecasttoolisexpectedtobeusedforboththeremainingdaysof thecurrentmonthandtheplanningmonth.Therefore,capacityutilizationsdueto reservationsinitiatedandnotcompletedduringthecurrentmonthshouldbeusedto correctlycalculatetheresidualdailycapacitiesoftheplanningmonth. Reservationsofthecurrentmontharetypicallyassignedwithastatusthattellsthe mostup-to-dateinformationabouttheexecutionstatusofthereservation.Inparticular, forthecurrentmonthforecast,weneedtotreatreservationswithstatusactivatedor scheduledaspublishedreservationsfortheplanningmonthsincetheyarealready associatedwithstrictduedates. 3.1.10CharacteristicsofBetterReservationPlans Intheprevioussections,wedescribedsomeofthehardrequirementsthatmust beconsideredwhendesigninganimplementablemonthlyplanforthecoalreservation ordersplacedattherailroadcompany.Whendecidingwhetherornottoaccepta reservation,severalothersofterconsiderationsareimportant. Ontheonehand,itiscrucialtoacceptasmanyreservationsaspossiblesince rejectingareservationresultsinsignicantcustomerdissatisfaction.Ontheother hand,itisalsoimportantnottoovercommitsincethiswillresultintrainsnotbeingable toreachtheirdestinationsbytheendofthemonth.Werefertothesereservations, initiatedbutnotcompletedwithinthemonthaspurged.Anoverabundanceofpurged reservationsimpliesthattherailroadcompanyisnotabletodeliverloadsasscheduled andtherebythatsignicantcustomerdissatisfactionistriggered. 74

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Whilemakingeveryefforttoincreasethenumberofacceptedreservations,the railroadcompanyshouldalsodistributetheamountofrejectedreservationsfairlyacross loadoriginsanddestinations.Inparticular,itisimportantthatreservationsarenot predominantlyrejectedfromsmallercustomerssincerejectingasignicantportionof suchareservationmightwipethiscustomer'sprotmarginentirely. Tosafeguardagainstrejectedreservations,somecustomersentermorereservation requeststhantheyneed.Fromtheperspectiveoftherailroadcompany,itmight thereforebeappropriatetodevelopareservationplanthatdoesnotdiffertoomuch fromhistoricalaveragessothatitavoidsplanningforreservationsthatwillultimately becancelledbythecustomer.Inthepast,thishasbeendonebyanalyzingordering patternsandseasonalityandbynotplanningforreservationlevelsthataresignicantly differentfromthoseobservedinpreviousmonths. Asweseektomaximizethenumberofsatisedreservations,optimizedschedules willtendtohavealargenumberofrequestsinitiatedonthelastdaybecausethese reservationsconsumeonlycapacityofthefollowingmonth,whichisnotconsidered whileplanningthecurrentmonth.Inordertoavoidsuchdeceptivesolutions,weneedto ensurethatthemajorityofreservationsacceptedarecompletedduringthemonth. Finally,aswementionedearlier,whenschedulingactivatedorscheduledreservations ofthecurrentmonth,publishedreservationsoftheplanningmonthorpermitreservations, deviationsfromalreadysetduedatesshouldbeavoidedasmuchaspossible.Further, reservationsshouldbesatisedwiththeirpreferredcareetsasmuchaspossiblesoas tomaximizecustomersatisfactionandcoaltrainsshouldbeassignedtheirmostdesired locomotivecombinationssoastopreventinefcientuseofenginepower. 3.2SolutionApproach 3.2.1HandlingPrivateSetsandSystemCars Theassignmentofcarstoreservationsisanimportantpartoftheforecastplanning model.Thelocationandavailabilityofcargroupsandsetsmustbeconsideredwhen 75

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makingtheseassignmentdecisions.Inthecaseofsystemreservations,preferred, admissibleandexcludedcartypesareintroducedtomodelcustomerpreferences. Associatedvariablesareintroducedinthemodeltodeterminethecompositionofthecar consistassignedtoeachacceptedreservation.Anadditionalvariableisintroducedto countthenumberofcarsofthepreferredkindassignedtoeachacceptedreservation. Theassignmentofpreferredcartypescanthenbeencouragedbypenalizingadmissible carutilizationintheobjectivefunction. Inthecaseofprivatereservations,however,carkindscannotbetreatedinthe samewaysinceprivatereservationrequestsrequiretheassignmentofsetsinstead ofindividualcars.Thecarconsistsofthesesetsincorporatediversecartypeswith differentcounts.Further,owningconsigneesdonotwantthecarconsistsofsets tobealtereddramatically.Themajorityofconsigneesalsoenforceexcludedsets fromparticulardestinationstoguaranteethatthesesetskeeponservingtheload destinationsthatconsigneesassignedtothem. Inordertoincorporatethenotionofsettothemodel,weintroducethemasspecial carkinds.Ifso,eachreservationnowhasmultiplepreferredcarkinds.Further,ifwe introduceallpossiblesetsasadmissiblecarkinds,thenthemodelwilltreatthemas sparecarkinds,whicharesupposedtobeusedonlytoadjustsetconsistcompositions tomatchtherequestedcarnumbers.Therefore,thesetupusedtoassigncarstosystem reservationscannotbeuseddirectly.Instead,wedevelopaprocedurethatassignssets toreservationsinagreedyfashionbeforeanyoptimizationisperformed.Theprocedure preassignssetstoreservationsbasedontheireligibilityandavailability. Asetiseligibletobeassignedtoareservationifitbelongstothesameconsignee asthereservationrequest.Further,thenumberofcarsinthecarconsistoftheset candeviatefromtherequestedcarcountofthereservationonlybyatolerancethatis presetbythecoaloperationsteamoftherailroadcompany.Thistoleranceisimposedto preventradicalchangesinthecarconsistofsetsthroughoutthemonth.Finally,inorder 76

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forasettobeconsideredeligibleforareservation,itshouldnotappearintheexcluded setslistofthedestinationofthecorrespondingreservation. Itisoftenthecasethatsetsareeligibletobeassignedtoalargenumberof reservations.Itisthereforeimportantthatduringthepre-assignmentofsetsto reservations,asetisnotassignedtotoomanyreservations.Forthisreason,we introducethenotionofoccupancyoftheset.Whenever,asetisassignedwitha reservationrequest,thecycletimeofthisreservationisaddeduptotheoccupancyof theset.Wethenusethesetoccupancytocheckwhetherthissetisstillavailablefor newassignments.Wementionthat,althoughsetsarepre-assignedtoreservations, noprecedencerelationbetweenreservationsissetduringthisphase.Infact,these relationswillbedeterminedwithinthesolutionalgorithmthatwedescribelaterin thischapter.Inotherwords,weusethecycletimesasameantonotoverbook anyparticularset.Ifasetdoesnothaveenoughroominitsmonthlyscheduleto accommodateanewreservation,eventhoughitiseligible,theprocedurewillassign anothersettothisreservation. Asmentionedearlier,published,activatedandscheduledreservationsare assignedasetatthetimetheyarepublished.Therefore,forthesereservations,the setassignmentproceduremustrstcheckwhatcanbetheearliesttimeatwhichthe preassignedsetcanstartservingthereservation.Ifthattimefallsoutsidethetolerable timeintervalthatisdeterminedbasedonthealreadyassignedload-indateforthat reservation,thenitseekstoassignanothersetthatmeetsthetimerestrictionsofthat reservation.Ifnosuchsetcanbefoundthenthepre-assignedsetiskeptandtardiness isreported. Thesetsassignedtoagroupofreservationsbytheproceduredescribedabove donotnecessarilymatchperfectlythenumberofcarsrequestedbythereservations. Inordertogivethemodeltheexibilitytoattachanddetachsparecarstoandfrom sets,weintroduceeachsetasadifferentcarkindtothemodel.Further,weintroduce 77

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thepre-assignedsetsaspreferredcarkindtotheirassociatedreservations.Thespare carsarethenintroducedasadmissiblecarkindsiftheirtypesarenotexcludedby thedestinationofthereservation.Moreover,weaddanewconstrainttoourmodel thatensuresthatpreferredsetsofprivatereservationsoutnumberanygroupofthe admissiblecars.Weaddthisconstrainttopreventsituationwherereservationsare satisedusingcarsbelongingentirelytothesparepool. 3.2.2BuildingaForecastModel Tomakeinformeddecisionsduringtheprocessofpublishingreservations,our industrycollaboratorneedsafastplanningforecasttool.Thistoolwouldbeusedseveral timesadayasnewreservationsrequestsareenteredinthesystem.Itistherefore importantthattherunningtimeofthistoolbelimitedtoafewminutesbutthatitalso producesreliable,goodqualityforecastsolutionsthatcanresultinimplementableplans. Toconstructasolutioninsuchasmallamountoftime,somesimplicationsin theformulationoftheproblemarenecessaryaspreciselytrackingthemovementof carsthroughthemonthincreasesthecomplexityofthemodel.Inparticular,inan actualmonthlyreservationsplan,carsandlocomotivesmustbefollowedthroughtheir assignments,whichcausesthenumberofvariablesintheformulationoftheproblem todramaticallyincrease.Therefore,modelingcarandlocomotivemovementsasthey occurinpracticeandobtaininggood-qualitysolutionsintheallowedtimeframeis unlikelywithcurrentoff-the-shelfsolvers.Wethereforesimplifycarandlocomotive movementsinawaythatisconsistentwiththeactualpractice. Moreprecisely,weassumethatthemovementofcarsandlocomotivesassociated withareservationisinitiatedfromtheloaddestination.Locomotivesandcarsthen directlyproceedtothemine,carsareloadedandthensentbacktotheloaddestination. Thetimeittakestosatisfyareservationisthereforeassumedtobeequaltotheduration ofafullcyclestartingandendingattheloaddestination.Further,weassumethat carsandlocomotiveswillbeusedonlyduringthelengthofthecycleandwillbecome 78

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availableatanypointofthenetworkforotherreservationsimmediatelyaftertheyare releasedfromthecurrentreservation.Withthissimplication,themodelcanbekept simpleastheexactpositionandmovementsofcarsthroughoutthemonthandacross thenetworkdoesnotneedtobecarefullymonitored.Thissimplicationalsoallowsusto considerthecarsandlocomotivesaspools,ratherthanindividualresources.Therefore, capacitiescanbeimposedatanaggregatedailylevel. Throughexcludeddestinations,customersrequestingprivatereservationsoften ensurethattheirsetsareassignedtoreservationshavingthesameloadoriginand destinationinasequence.Inthecaseofsystemcars,itisalsoquiteefcientinterms ofequipmentutilizationtoassignthesamegroupofcarstoconsecutivereservations requiringthesameamountofcarsandhavingthesameloadoriginanddestination. Therefore,adesirablecharacteristicofreservationplanisthattheyassignsystemcar groupsandprivatesetsincycleswheneverpossible.Oursimplifyingassumptioncanbe viewedasageneralizationofthispracticeoverallreservations. Intheformulationwedevelop,themainvariablesarebinarydecisionsthatdescribe whethertosatisfyorrejectareservation,togetherwiththedayofthemonththe schedulewhenthereservationcycleisstarted,incaseitissatised.Wedenetwo moresetsofdecisionvariablesthatassociateasuitablecargrouporsetaswellasa locomotivecombinationwithsatisedreservations. WeformulatethevariouscapacityrequirementsdescribedinSection3.1as hardconstraintsinourMIPmodel.Inparticular,dailycar-hourrestrictionsforeach systemcarkind,privatesetandpoolcartypeisstrictlyenforced.Similarly,daily locomotive-hourlimitationsbylocomotivetypearealsoenforced.Dailycarandtrain capacitiesatmine,branchesanddestinationsareimposed.Finally,traincapacitieson railtracksareimposed.Itispossibletoimposesuchconstraintssincethevariables describingschedulesimplicitlycontaintheinformationneededtodeterminewhen capacitiesareused. 79

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Inadditiontocapacityconstraints,werequirethatareservationiseitherrejected orsatisedaccordingtoasinglescheduleamongthepossiblealternatives.For privatereservations,weimposethatthereservationisnotsatisedsolelywithcars fromthesparepooloftheconsignee.Instead,werequirethatitissatisedusingthe pre-assignedsetthatcanbealteredinsizetoacertainextenti.e.,addingorremoving afewcarswherethenumberisspeciedbytheconsigneeusingcarsfromthespare pool. TotakeintoaccounttheconcernsdescribedinSection3.1.10,weintroducevarious incentivesandpenaltiesintheobjective.Theseincentivesandpenaltiesareappliedto slackvariables.Thesevariablesensurethattheconstraintsthattheyappearinarenot enforcedashardconstraints.Instead,theyarepenalizedorrewardedintheobjective functionthroughweightsassignedtoeachsetofthesevariablesaccordingtothe prioritiesofthecoaloperationsteam. Inourimplementation,themostimportantoftheaboveaspectsisthatasmooth distributionofacceptedreservationsisestablishedamongminesanddestinations. Moreprecisely,rejectiondecisionscausingdramaticbusinesslossesatminesand destinationsmustbepenalized.Thesecondmostimportantpenaltyisassociated withrejectedreservations.Thethirdisrelatedtoduedaterequirementsi.e.,permit, published,activatedorscheduledreservationsofthecurrentmonth.Deviationsfrom thespeciedduedatesarepenalized.Fourth,selectingschedulesaccordingtowhich reservationsarecompletedwithintheplanningmonthisrewardedforalltheremaining reservations.Additionally,inordertopreventanabundanceofreservationsaccepted withaschedulestartingtowardstheendofthemonth,weimposecapacityconstraints onanadditionaltimeframeofaweek.Bydoingso,wediscouragetheassignment oflast-day-schedulestoreservations,sincethecapacitiesthattheseschedulesuse outsideofplanningperiodarealsorestricted.Fifth,anysystemcarassignmentother thanthepreferredcarkindoranyprivatesetassignmentotherthanthepre-assignedset 80

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ispenalized.Sixth,anylocomotivecombinationassignmentthatisnotthemostdesired ispenalized.Finally,foranygivenmineanddestination,stayingbetween 70% and 130% ofthehistoricalaveragenumberofacceptedreservationsisrewarded.Detailsofthe modelarepresentedinSection3.2.3. 3.2.3MIPModel 3.2.3.1Data Themodelisbasedonavarietyofsetsthatcanbebroadlycharacterizedas associatedwithnetworklocations,reservations,cars,locomotives,railtracksandtime andloadingschedules.Adetaileddescriptionisgivennext: Networklocations: N :setofnodes n M :setofmines m M N D :setofdestinations d D N Reservations: C :setofconsignees c R :setofreservations r R n :subsetofreservationstouchingnode n R n R ,where n 2 N R m :subsetofreservationshavingnode m asmine, R m R ,where m 2 M R d :subsetofreservationshavingnode d asdestination, R d R ,where d 2 D R s :subsetofreservationsrequestingsystemcars, R s R R p :subsetofreservationsrequestingprivatecars, R p R R dr :subsetofreservationshavingadeadlinerequirement, R dr R R nr :subsetofreservationsthatareN-reservations, R nr R Cars: K :setofrailcars k .Acarkind k isauniquecombinationoftype,ownership,and ownerconsigneeiftheownerisprivate. 81

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K r :subsetofcarkindsthatareadmissibleforreservation r K r K .Thepreferred carkindinthissubsetisdenotedby k 2 K r f K d :subsetofcarkindsexcludedfromdestination d Locomotives: M :setoflocomotivetypes m thatcanbeassignedtocoaltrains. r :setoflocomotivecombinations thatcanbeassignedtocoaltrains.Themost desirablelocomotivecombinationamongtheadmissiblecombinationslistisdenotedby 2 r Railtracks: E :setofbottleneckrailtracksegments e Timeandloadingschedules: T :setofdays t intheplanningmonth. L r :setofcandidateschedulesforreservation r whichdifferbystartday l Scheduleassociatedwith l =0 correspondstothecasewherethereservation r is notsatised. L 0 r :subsetofschedulesforreservation r tobecompletedbeforethelastdayofthe month. Nextweintroducenotationforthedatathatisusedtoimposethehardconstraints ofthemodel: r;l;t hours :numberofcarhoursthatreservation r requiresduringday t ifitissatised accordingtoschedule l k t :numberofcarhoursthatcarkind k isavailableduringday t r :totalnumberofcarsrequestedbyreservation r ;m :numberoflocomotivesoftype m appearinginlocomotivecombination m t :numberoflocomotivehoursoftype m availableduringday t r;l;t n :binarydatathatequals 1 ifterminal n isvisitedbythetrainassociatedwith reservation r duringday t giventhatitissatisedaccordingtoschedule l 82

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! n;t :numberofcarsthatcanbeprocessedatnode n duringday t b;t :numberofcarsthatcanbeloadedatbranch b duringday t n;t :numberoftrainsthatcanbeprocessedatnode n duringday t b;t :numberoftrainsthatcanbeloadedatbranch b duringday t r;l;t e :binarydatathatequals 1 ifbottlenecktracksegment e isvisitedbythetrain associatedwithreservation r duringday t giventhatitissatisedaccordingtoschedule l e;t :numberoftrainsthatcanpassbytracksegment e duringday t lb n :minimummonthlythresholdforthenumberofreservationsassociatedwith node n 2 M [ D thatshouldbesatised. ub n :maximummonthlythresholdforthenumberofreservationsassociatedwith node n 2 M [ D thatshouldbesatised. 3.2.3.2Variables Wenextdescribetheeightsetofvariablesweuseinthemodel.Therstsetof variablesisusedtodeterminewhichscheduleifanyisassignedtoeachreservation. Binaryvariable y r; 0 takesthevalue 1 ifreservation r isrejectedand 0 otherwise. Similarly,binaryvariable y r;l takesthevalue 1 ifreservation r issatisedaccording toschedule l 2 L r nf 0 g and 0 otherwise.Thesecondsetofvariablesisusedto determinethecarconsistthatisusedtosatisfyreservation r .Inparticular,integer variable x k r;l representsthenumberofcarsofkind k 2 K r usedtosatisfyreservation r accordingtoschedule l 2 L r nf 0 g .Thisvariableisintegerformostreservationsexcept forthosethataresubjecttoprivatetreatment.Inthiscase,thevariableisdenedto besemi-integerforthepreferredcarkind.Ifthecarcountoftheset 0 islessthanthe carcountrequest 00 ofthereservation,thenthisvariablewilltakeeitherthevalue 0 oranintegervaluebetween 0 and 00 .Ifthecarcountoftheset 0 isgreaterthanor equaltothecarcountrequest 00 ofthereservation,thenthisvariablewilltakeeither thevalue 0 ,orthevalue 00 .Thethirdsetofvariablesisusedtodeterminelocomotive 83

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combinationassignmentsofreservations.Binaryvariable w r;l takesthevalue 1 ifthe locomotivecombination isassignedtoreservation r whichissatisedaccordingto schedule l 2 L r nf 0 g and 0 otherwise.Thefourthsetofvariablesisusedtorepresent thenumberofcarsassignedtoreservation r thatarenotofitspreferredcarkind.In particular, sk r;l isthetotalnumberofcarsofnon-preferredkindsusedinsatisfying r Thefthsetofvariablesisusedtorepresentwhetherareservationissatisedwithits mostdesiredlocomotivecombination.Inparticular, s r;l takesthevalue 1 ifthemost desiredlocomotivecombinationisnotusedforreservation r and 0 otherwise.The sixthsetofvariablesisusedtodeterminewhetherthenumberofreservationssatised foramineordestinationfallsbelow lb n if lb n j R n j .Inparticular,wedeneinteger variable a lb n tobethedifferencebetweenthetotalnumberofacceptedreservations havingnode n asmineordestinationand lb n .Weimposethat )]TJ/F24 11.9552 Tf 9.299 0 Td [(lb n a lb n j R n j forall n 2 M [ D with lb n j R n j .Theseventhsetofvariablesisusedtodeterminewhether thenumberofreservationssatisedforamineordestinationisabove ub n if j R n j ub n Inparticular,wedeneintegervariable a ub n tobethedifferencebetween ub n andthetotal numberofacceptedreservationshavingnode n asmineordestination.Weimposethat ub n )-234(j R n j a ub n ub n forall n 2 M [ D with j R n j ub n .Theeighthsetofvariables m representstheacceptedreservationsratioofallreservationsoriginatingfrommine m for m 2 M .Theninthsetofvariables d representstheacceptedreservationsratioofall thereservationsheadedtodestination d for d 2 D 3.2.3.3Constraints Wenextdescribetheconstraintsofthemodel. Eachreservation r shouldeitherbesatisedordropped. X l 2 L r y r;l =1 8 r 2 R: Thetotalnumberofcarhoursutilizedbyacceptedreservationsusingspeciccar kind k onagivenday t shouldnotexceedthedailycar-hourcapacityofkind k onthat 84

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day. X r 2 R X l 2 L r nf 0 g r;l;t hours x k r;l k t 8 k 2 K; 8 t 2 T: Agivenreservation r 2 R canonlybeassignedtoavailablecartypesfromits admissiblecarkindslist. X k 2 K r x k r;l = r y r;l 8 r 2 R; 8 l 2 L r nf 0 g : Everyreservation r 2 R hasapreferredcarkind k thatisfavoredovertherestof acceptablecarkinds.Thenumberofcarsofthepreferredcarkindusedinsatisfying r is computedthrough: x k r;l + sk r;l = r y r;l 8 r 2 R; 8 l 2 L r nf 0 g : Further,forprivatereservations,thenumberofcarsassignedtoaprivatesetmust bemadeofmorecarsfromtheinitialsetthanspares.Rememberthatcarssubjectto privatesettreatmentareassignedinprioritytothecarkindrepresentingtheirassigned setbutcanalsousespares. x k r;l x k r;l 8 r 2 R p ; 8 k 2 K r nf k g : Thetotalnumberoflocomotivehoursutilizedbyacceptedreservationsinagivenday t shouldnotexceedthedailylocomotive-hourcapacityforanylocomotivetype m 2 M X r 2 R X l 2 L r nf 0 g r;l;t hours X 2 r ;m w r;l m t 8 m 2 M; 8 t 2 T: Givenareservation r 2 R ,asinglelocomotivecombination 2 r canbeassigned to r X 2 r w r;l = y r;l 8 r 2 R; 8 l 2 L r nf 0 g : 85

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Mostdesiredlocomotivecombination isfavoredovertherestofacceptable locomotivecombinations.Whetherthemostdesiredlocomotivecombinationisusedfor reservation r isdeterminedthrough: w r;l + s r;l = y r;l 8 r 2 R; 8 l 2 L r nf 0 g : Thetotalnumberofcarshandledatnode n 2 N duringday t shouldnotexceedthe maximumnumberthat n canload/dump/accomodateduringday t X r 2 R n X l 2 L r nf 0 g r r;l;t n y r;l n;t 8 n 2 N; 8 t 2 T: Thetotalnumberofcarshandledatminebranch b 2 B duringday t shouldnot exceedthemaximumnumberthat b canloadduringday t X m 2 M b X r 2 R m X l 2 L r nf 0 g r r;l;t m y r;l b;t 8 b 2 B; 8 t 2 T: Thetotalnumberofreservationshandledatnode n duringday t shouldnotexceed themaximumnumberthat n canload/dump/accomodateduringday t X r 2 R n X l 2 L r nf 0 g r;l;t n y r;l n;t 8 n 2 N; 8 t 2 T: Thetotalnumberofreservationshandledatminebranch b 2 B duringday t should notexceedthemaximumnumberthat b canloadduringday t X m 2 M b X r 2 R m X l 2 L r nf 0 g r;l;t m y r;l b;t 8 b 2 B; 8 t 2 T: Totalnumberofreservationspassingbyabottleneckrailtrack e 2 E duringday t shouldnotexceedthemaximumnumberthat e canaccommodateduringthatday t X r 2 R e X l 2 L r nf 0 g r;l;t e y r;l e;t 8 e 2 E; 8 t 2 T: 86

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Thetotalmonthlynumberofreservationshandledatnode n 2 M [ D shouldto begreaterthan lb n if lb n islessthanorequaltothetotalrequestedamount.Similarly,it shouldbelessthan ub n if ub n islessthanorequaltothetotalrequestedamount. P r 2 R d P l 2 L r nf 0 g y r;l )]TJ/F24 11.9552 Tf 11.955 0 Td [(a lb d lb d 8 d 2 D with lb d j R d j : P r 2 R d P l 2 L r nf 0 g y r;l + a ub d min f ub d g8 d 2 D with j R d j ub d : Thereservationacceptanceratioofeachmineanddestinationshouldbeatleastas bigasthecorrespondingvariable j R m j)]TJ/F29 11.9552 Tf 23.181 8.967 Td [(P r 2 R m y r; 0 j R m j m 8 m 2 M: j R d j)]TJ/F29 11.9552 Tf 21.832 8.967 Td [(P r 2 R d y r; 0 j R d j d 8 d 2 D: 3.2.3.4Objectivefunction Theobjectiveofthemodelistominimizeallpenaltiesassociatedwiththesetof chosenschedules: min )]TJ/F24 11.9552 Tf 11.955 0 Td [( 1 X m 2 M m + X d 2 D d + 2 X r 2 R y r; 0 + 3 X r 2 R dr X l 2 L r nf 0 g r;l y r;l )]TJ/F24 11.9552 Tf 15.278 0 Td [( 4 X r 2 R n R dr X l 2 L 0 r nf 0 g y r;l + 5 X r 2 R X l 2 L r nf 0 g sk r;l + 6 X r 2 R X l 2 L r nf 0 g s r;l )]TJ/F24 11.9552 Tf 15.279 0 Td [( 7 X m 2 M a lb m + a ub m + X d 2 D a lb d + a ub d : Thevariousobjectivetermscanbeexplainedasfollows.Therstobjectiveterm aimsatmaximizingtheminimumreservationacceptanceratioofeachmineand destination.Inotherwords,itseekstoestablishasmoothdistributionofaccepted reservationsamongminesanddestinations.Thesecondobjectivetermappliesa penaltyforeachreservation r thatisnotsatised.Thethirdtermisimposedonly forreservationswithdeadlinerequirementsandpenalizesanydeviationfromthe targetdeadlineontheload-indate.Thefourthobjectivetermisimposedonlyfor 87

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reservationswithnodeadlinerequirements,andaimstorewardreservationsthatare completedwithinthemonth.Thefthobjectivetermseekstominimizereservations thatarenotassignedtheirpreferredcarkinds.Likewise,thesixthtermtriestominimize reservationsthatarenotassignedtheirmostdesirablelocomotivecombination.The seventhobjectivetermpenalizesdeviationsfromhistoricalaverages. Thecoaloperationsteamofourindustrycollaboratorprioritizedtheseseven objectivecomponentssothat 1 2 3 4 5 ; 6 7 > 0 3.2.4SolutionProcedure:IssuesandSolution Themodeldescribedaboveisamixedintegerproblem.However,duetothe highmonthlyvolumeofthecoalrequeststhattherailroadcompanyseekstoconsider withinthismodel,duetothefactthattheplanningperiodisamonth,andduetothe levelofdetailssuchasspeciccarandlocomotiveassignmentsassociatedwitheach reservation,off-the-shelfsolversdidnotprovidegoodsolutionsinthedesiredamount oftime.Inparticular,eventheLPrelaxationofthemodelcouldnotbesolvedwithin theallowedtimelimits.Thefailureofcommercialsoftwaresinproducinggoodquality solutionsinashortamountoftimeledustoanalyzetheproblemcharacteristicsfurther andtodevelopasequentialmethodologythatallowsustoreducethesizeofproblems thatarefedtocommercialsoftware. Thealgorithmweproposeconsistsofaconstructiveandanimprovementphase. Intheconstructivephase,wefeedthemodeltoacommercialsolverrestrictingthecar assignmentstobeperformedonlyusingpreferredcarkinds.Inthisstep,themodel producesafeasiblesolutionveryquicklyduetosmallernumberofvariablesusedto formulatetheproblem.Thefeasiblesolutionobtainedattheendofthisconstructive phaseisthenfedtotheimprovementphaseasaninput.Intheimprovementphase, werandomlychooseoneoftwolocalsearchprocedures.Bothproceduresconsidera sub-problemofsmallersizethatseekstooptimallyscheduleasubsetofreservations whiletheschedules,andcarandlocomotiveassignmentsoftherestofthereservations 88

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arexedtotheirvaluesintheincumbentsolution.Inallthemodelsweconsiderin theimprovementphase,boththepreferredcarassignmentsandtheadmissiblecar assignmentsareallowed.Therefore,unlikeconstructivephase,theproblemsizeis smallnotbecausecar-to-reservationdecisionsarelimitedbutbecausethenumberof reservationsconsideredissmall. Therstlocalsearchprocedurethatwedevelopmakesuseofrandomlyselected timewindowstoimprovetheincumbentsolution.Werandomlygeneratethestartdate ofthetimewindowateachiterationoftheprocedure.Theselectionoftheintervallength ismoredelicatesincealargerintervalmightallowbettersolutionstobeuncovered howeveritmightalsoimpairthespeedofthesearch.Onceatimewindowisselected, wecreateoursub-problemtorescheduleonlythereservationsthataresatisedinthe incumbentsolutionwithastartingdatethatfallsinsideoftheselectedtimeinterval. Wealsoconsiderthereservationsthatarerejectedintheincumbentsolutionaspart ofoursub-problem.Wethensolvetheresultingmodelusingcommercialsoftwarewith apre-settimelimit.Ifabettersolutionisfoundwithintheprovidedtime,theincumbent solutionisupdated. Inthesecondlocalsearchprocedure,weseektoreschedulethereservations ofasubsetofconsignees.Wealsoconsiderthereservationssharingaloadorigin, destinationorcarresourceswiththesepreselectedreservationsintheincumbent solution.Similartotheprocedurethatusestimewindows,wealsotrytoschedulethe reservationsthatwererejectedintheincumbentsolution.Toobtaintheaforementioned subsetofconsignees,weselectthemrandomlyamongtheconsigneeshavinga rejectedreservationspercentagegreaterthanapresetthresholdvalue.Thisselection routinealsohelpsinensuringthatrejectionsarenotpredominantlytargetingsmaller customers.Similartotherstprocedure,wextheschedulesandassignmentsofall otherreservationsthatarenotconsideredwithinthesub-problem.Wethensolvethe 89

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modelusingcommercialsoftwarewithapre-settimelimit.Theincumbentsolutionis updatedifabettersolutionisobtained. 90

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CHAPTER4 UNSPLITTABLENETWORKFLOWPROBLEMSINTRANSPORTATIONMODELSAND APPLICATIONS Inthischapter,westudynetworkowproblemsinwhichsomeoftheverticesare requiredtosatisfyno-split,no-mergerestrictions.Theserestrictionsimposethatthe owenteringanunsplittablevertexofthenetworkleavesthisvertexunchanged.In particular,anenteringowshouldnotbesplitacrossmultipleoutgoingarcsormerged withotherincomingows.Unsplittableowproblemshavevariouspracticalapplications includingunittrainschedulingforrailroads,seeChapter2andChapter3andbandwidth allocationinheterogeneousnetworksarisingininformationinfrastructurecontextHu etal.,2009. Inrailroadsforinstance,theproblemarisesspecicallyintheunittrainbusiness asunittrainsarekeptintactbetweentheiroriginanddestination,i.e.,nocarcanbe attachedtoordetachedfromthosetrainsexceptwhenformingthem.Trainformation canonlybeaccomplishedincertainterminalshavingtherequiredequipment, machinery,andmanpower.Therefore,unittrainnetworkshaveanheterogeneous structurethatconsistofamajorityofterminals/nodeswithno-split,no-mergerestrictions whileafewothersareequippedfortrainformationandcanthereforehandlesplit-and-merge operations.Assumingthatallunittrainsareidenticalandcarrythesametypesofcars seeLawleyetal.2008aforanapplication,theproblemofdeterminingthemaximum numberoftrainrequeststhatcanbesatisedthenreducestosolvinganunsplittable maximumowproblemUFPinwhichowleavingthesupplynodesofanetworkis maximizedgivenno-split,no-mergerestrictionsonselectedvertices. Next,wedescribefurthertheparticularstructureofUFPproblemsthatwewillsolve inthischapterastheyrelatetounittrainscheduling.Let G V [f s;t g ;E beadirected multigraphwhere V isthesetofvertices, s isthesourcevertex, t isthesinkvertex,and E isthemultisetofarcs.Wedenoteby S V thesetofsupplyvertices.Similarly,we denoteby D V thesetofdemandvertices.Thereisanarcfromthesourcevertex 91

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s toallsupplyverticesandfromalldemandverticestothesinkvertex t V represents thesetofverticeswithno-split,no-mergerestrictions.Wedenotethesetofverticesthat haveadirectedarcintovertex j by V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j .Similarlywerepresentby V + j thesetofvertices towardswhichvertex j hasadirectedarc.Eacharc i;j 2 E whichrepresentstherail tracksofarailnetworkhasadailymaximumlimitonthenumberoftrainsthatcanpass by i;j .Wemodelthedailytraincapacityofanarc i;j bycreatingasmanycopiesof thisarcasthecapacitylimit.Wedenotethesetofcopiesofanarc i;j by K i;j .Further, eacharc i;j;k hasarestrictiononthemaximumnumberofcarsassociatedwitheach trainthatwedenoteby u i;j .Weintroducelowerbounds l d;j = c d = u d;j onthearcs leavingademandnode d tomakesurethatonlyatrainwiththerequestednumberof carsleaves d .Weset l i;j =0 forallarcsfrom i 2 V;i 6 = d to j 2 V .Wealsomention thateacharc i;j alsohasaglobalcapacitythatisbindingforthetotalnumberof carsassociatedwithallthetrainspassingthroughthisarc.Inthescopeofthepractical applicationsweconsider,theglobalarccapacitiesareconsideredtobesufciently large.Finally,weuse c s torepresentthenumberofcarsavailableatsupplyvertex s 2 S Wecreateadummysupplynode s 0 withinnitecapacitytomakesurethatthemodelis feasible.Maximizingthenumberofcarsprovidedtocustomersthencorrespondstond amaximumowfromsupplynodestodemandnodesin G V [f s;t g ;E WenextpresentapossiblemathematicalprogrammingformulationforUFP.This modelhastwosetsofdecisionvariables.First,weintroduceintegervariables y k i;j to representtheowfromnode i tonode j throughthecapacityslot k 2 K i;j whereboth i and j 2 V .Second,weintroducebinaryvariables z j i 1 ;k 1 ;i 2 ;k 2 toequal 1 iftheowentering tovertex j fromvertex i 1 throughcapacityslot k 1 ofarc i 1 ;j leavesvertex j forvertex i 2 throughcapacityslot k 2 ofarc j;i 2 .Usingthesevariables,UFPcanbeformulatedas thefollowingmixedintegerprogram: 92

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max X s 2 S nf s 0 g ;j 2 V n S X k 2 K s;j y k s;j s.t. y k 1 i 1 ;j )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i 1 ;j )]TJ/F24 11.9552 Tf 11.955 0 Td [(z j i 1 ;k 1 ;i 2 ;k 2 y k 2 j;i 2 ; 8 j 2 V; 8 i 1 2 V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j ; 8 k 1 2 K i 1 ;j ; 8 i 2 2 V + j ; 8 k 2 2 K i 2 ;j y k 2 j;i 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u j;i 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(z j i 1 ;k 1 ;i 2 ;k 2 y k 1 i 1 ;j ; 8 j 2 V; 8 i 1 2 V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j ; 8 k 1 2 K i 1 ;j ; 8 i 2 2 V + j ; 8 k 2 2 K i 2 ;j X i 2 2 V + j X k 2 2 K j;i 2 z j i 1 ;k 1 ;i 2 ;k 2 1 ; 8 j 2 V; 8 i 1 2 V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j ; 8 k 1 2 K i 1 ;j X i 1 2 V )]TJ/F26 5.9776 Tf -1.882 -6.225 Td [(j X k 1 2 K i 1 ;j z j i 1 ;k 1 ;i 2 ;k 2 1 ; 8 j 2 V; 8 i 2 2 V + j ; 8 k 2 2 K j;i 2 X j 2 V + s X k 2 K s;j y k s;j c s ; 8 s 2 S nf s 0 g X i 2 V )]TJ/F26 5.9776 Tf -1.882 -6.225 Td [(j X k 2 K i;j y k i;j )]TJ/F29 11.9552 Tf 14.036 11.358 Td [(X i 2 V + j X k 2 K j;i y k j;i =0 ; 8 j 2 V n V l i 1 ;j y k 1 i 1 ;j X i 2 2 V + j X k 2 2 K j;i 2 z j i 1 ;k 1 ;i 2 ;k 2 u i 1 ;j ;y k 1 i 1 ;j 2 Z + ; 8 j 2 V;i 1 2 V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j ; 8 k 1 2 K i 1 ;j l j;i 2 y k 2 j;i 2 X i 1 2 V )]TJ/F26 5.9776 Tf -1.882 -6.225 Td [(j X k 1 2 K i 1 ;j z j i 1 ;k 1 ;i 2 ;k 2 u j;i 2 ;y k 2 j;i 2 2 Z + ; 8 j 2 V;i 2 2 V + j ; 8 k 2 2 K j;i 2 z j i 1 ;k 1 ;i 2 ;k 2 2f 0 ; 1 g ; 8 j 2 V; 8 i 1 2 V )]TJ/F25 7.9701 Tf -2.601 -8.012 Td [(j ; 8 k 1 2 K i 1 ;j ; 8 i 2 2 V + j ; 8 k 2 2 K i 2 ;j : 93

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Intheaboveformulation,theobjectivefunction4seekstomaximizethecarow leavingsupplyverticesnotincludingthedummysupplynode s 0 .Constraint4in conjunctionwithConstraint4ensuresthatwhen z j i 1 ;k 1 ;i 2 ;k 2 equals 1 ,theincoming owfromvertex i 1 tovertex j throughcapacityslot k 1 equalstheoutgoingowfrom j to i 2 throughcapacityslot k 2 .Constraint4imposesthatifthereisaowentering vertex j throughcapacityslot k 1 ofarc i 1 ;j ,thenitisassignedtoasinglecapacityslot ofoneoftheoutgoingarcsofvertex j .Constraint4ensuresthattheowleaving asupplyvertexdoesnotexceedtheinitialamountofsupplyofthatvertex.Constraint 4limitsthenumberofcarsthatcanbesentthrougheacharc i;j 2 E .Finally,we introducetheowbalanceconstraintsforverticesthatarenotin V inConstraint4. Proposition4.0.4.1. UFPisNP-hard. Proof. WereducetheunsplittableowproblemwithsetofterminalpairsUFP-STP KolmanandScheideler,2002toUFP. UFP-STP INSTANCE: Anetwork G = V;E with n 1 verticesand m arcs,arccapacities c i;j : E R + ,asetof k terminalpairs T = f s i ;t i g for i 2 K withdemand d i 2 Z QUESTION: Isthereasubsetof S T suchthatthedemandofalltherequests i 2 S aremetandthearccapacitiesarerespected. WenextshowthatthesolutiontoUFP-STPisyesifandonlyifwecannda solutiontothefollowinginstanceofUFP. GiventheaboveinstanceofUFP-STPingraph G ,weconstructaninstance G 0 = V 0 ;E 0 ofUFPasfollows:Werstadd s i and t i verticesto V 0 .Weassigntheir demandtobe ~ d i =2 n 2 d i )]TJ/F24 11.9552 Tf 12.561 0 Td [(i .Wethenconstructandassociateunsplittablenodes s i 0 and t i 0 withtheoriginalsupplyanddemandnodes s i and t i respectively.Weplacearcs s i ;s i 0 and t i 0 ;t i thathave ~ d i asthelowerboundontheowthatcanpassthrough thesearcs.Wethenaddallnodesin V nf s i ;t i g forall i 2 K into V 0 .Foreach i;j 2 E where i;j 2 V nf s i ;t i g forall i 2 K ,wecreate k copiesof i;j 2 E 0 ,eachcopy 94

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havingacorrespondingindividualarccapacityof ~ d i forall i 2 K .Wereplacethearcs s i ;j and i;t j 2 E with k copiesof s i 0 ;j 2 E 0 and i;t i 0 2 E 0 ,eachcopyhavinga correspondingindividualarccapacityof ~ d i forall i 2 K .Finally,weassignaglobalarc capacityonallarcs i;j 2 E 0 thatisequalto ~ c i;j =2 n 2 c i;j TheabovereductionwouldguaranteethatanysolutiontothisinstanceofUFP respectsthepairmatchingsinUFP-STP.Observethatnotwodemandvalues ~ d i are equal.Thisisensuredby P i 2 K ~ d i = P i 2 K )]TJ/F24 11.9552 Tf 5.48 -9.684 Td [(d i )]TJ/F25 7.9701 Tf 18.47 4.707 Td [(i 2 n 2 > P i 2 K d i )]TJ/F15 11.9552 Tf 12.336 0 Td [(1 .Giventhat,togetherwith thelowerboundrestrictionsonarcs s i ;s i 0 and t i 0 ;t i andthefactthat s i 0 sand t i 0 sare unsplittableensurethattheowiscarriedbetweentheoriginalpairs.Moreover,the scaledglobalarccapacitiesensuresthatanyfeasibleowpathon G remainsfeasiblein G 0 .Theyalsoguaranteethatnoarc i;j 2 E 0 isusedonmultiplepaths.Therefore,by solvingthisinstanceofUFP,wecansolvetheinitialinstanceofUFP-STP.Since ~ d i sand ~ c i;j sareofpolynomiallengthin n andweperformpolynomialnumberofoperationsin reducing G to G 0 ,weconcludethatUFPisNP-hard. GiventhattheproblemweareinterestedinsolvingisNP-hard,wewillseekto obtaingloballyoptimalsolutionthroughbranch-and-cuttechniques.Inparticular,in Section4.1,weseektodeterminewhethertheintegerprogrammingmodelofthe no-splitrequirementweusedaboveisstrong,andwhetherthisrequirementcanbe alternativelyhandledthroughcuttingplanes.InSection4.2,wereportcomputational experimentsthatcomparethenumericalstrengthofthedifferentformulationsproposed aswellasthespeedwithwhichthecorrespondingrelaxationsaresolved. 4.1PolyhedralAnalysis Inthissection,westudythesinglenodeunsplittableowproblem SNUFP with k enteringarcsand l leavingarcsasdepictedinFigure4-1.Inthisproblem,weconsider anetworkwithasinglenode,whereabalanceequationisimposedontheincoming andoutgoingarcs.Wedenotetheowonincomingarc i by x i for i 2 K := f 1 ;:::;k g 95

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Figure4-1.Singlenodeunsplittableowproblem. Similarly,wedenotetheowonoutgoingarc j by y j for j 2 L := f 1 ;:::;l g .Wedenote thecapacityofincomingarc i by u i for i 2 K andthecapacityofoutgoingarc j by v j for j 2 L .Asitiscommoninnetworkowmodels,werequirethat u i 2 Z and v j 2 Z .For i 2 K and j 2 L ,wedene w i;j :=min f u i ;v j g .Further,werequirethateachnonzero incomingow x i bepairedwithasingleoutgoingow y j ofsamevaluei.e., x i = y j andthateachnonzerooutgoingow y j bepairedwithasingleincomingow x i ofsame value.Werefertosuchows x;y asunsplittable.Inotherwords,wesaythataow x;y isunsplittableifthereisamatching M inthecompletebipartitegraphwithnode sets V 1 = K and V 2 = L suchthat x i = y j forall i;j 2 M x i =0 ifnode i isexposed in M ,and y j =0 if j isexposedin M .Weusethenotation nosplit x;y torepresentthe requirementthataow x;y isunsplittable. Usingthenotationintroducedabove,thesetwestudycanbeformulatedas S k;l = 8 > > > > > < > > > > > : P i 2 K x i = P j 2 L y j x;y 2 R k + l 0 x i u i ; 0 y j v j nosplit x;y 9 > > > > > = > > > > > ; : Intheremainderofthischapter,weimpose Assumption4.1.0.1. 0
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Figure4-2.Anetworkwithmultipleno-splitnodes ThereisnolossofgeneralityinAssumption4.1.0.1sincearcscanbereordered. Assumption4.1.0.2isalsowithoutlossofgenerality.Infact,when u k >v l ,nooutgoing arccanaccommodateaowofvalue u k .Thecapacityofarc k canthereforebereduced tothemaximumoutgoingcapacity,whichis v l becauseofAssumption4.1.0.1.Similarly, when v l >u k ,noincomingowwillexceed u k andtherefore v l canbereducedto u k .In theremainderofthischapter,weuse w =min f u k ;v l g = u k = v l Weareinterestedinderivingstrongformulationsfor S k;l bothintheoriginalspace ofvariables x;y andinhigher-dimensionalspaces.Strongformulationsof S k;l can beusedtogeneratebetterMIPmodelsforproblemshavingno-splitrequirementsin general,andforUFPinparticular.Thereasonthatwefocusonthesingle-nodecase isthree-fold.First,itisthenaturalforminwhichthisrequirementoccurs.Second, single-noderelaxationsforcloselyrelatedxed-chargednetworkproblemshavebeen showntoyieldveryusefulcutsforgeneralnetworksGuetal.,1999;Padbergetal., 1985;RoyandWolsey,1986.Inparticular,theresultingow-covercutsarenowpart ofallcommercialsolvers.Third,thesingle-nodemodelcanbeusedasanontrivial relaxationofnetworkproblemshavingmultipleno-splitnodes.Weelaborateonthis commentnext. 97

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ConsiderthenetworkofFigure4-2wherenodesthatareshadedhaveano-split requirementwhiletheothersonlyneedtosatisfybalanceequations.Itiseasytosee thatthereexistsano-splitrequirementbetweentheredarcsvariables x 2 x 3 x 5 x 6 x 9 andthegreenarcsvariables x 8 x 10 and x 11 inFigure4-2.Moregenerally,forasubset ofvertices S ofanetwork,dene )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [( S tobethesetofarcswhoseheadbelongto S andwhosetaildoesnotbelongto S .Similarly,dene + S tobethesetofarcswhose tailbelongsto S andwhoseheaddoesnotbelongto S .If S isasubsetofnodeswith no-splitrequirement,thenthesinglenodeunsplittableowmodelwithincomingarcs )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [( S andoutgoingarcs + S isarelaxationoftheinitialproblem.Therefore,strong validinequalitiesforthissinglenodeunsplittableowrelaxationprovidecutsforthe moregeneralmodel. 4.1.1APolynomialExtendedFormulationof S k;l Beforefocusingonstrongformulationsof S k;l intheoriginalspaceofvariables x i and y j ,werstpresentapolynomially-sizeextendedformulationthatisinspiredbythe naturalIPformulationoftheno-splitrequirement.Firstweintroducebinaryvariables z i;j for i 2 K and j 2 L .Thesevariablesequal 1 iftheowonincomingarc i isroutedonto 98

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outgoingarc j .TheresultingMIPformulationis: X i 2 K z i;j 1 ; 8 j 2 L; X j 2 L z i;j 1 ; 8 i 2 K; x i )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F24 11.9552 Tf 11.955 0 Td [(z i;j y j ; 8 i 2 K; 8 j 2 L; y j )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j )]TJ/F24 11.9552 Tf 11.956 0 Td [(z i;j x i ; 8 i 2 K; 8 j 2 L; 0 x i X j 2 L u i z i;j ; 8 i 2 K; 0 y j X i 2 K v j z i;j ; 8 j 2 L; 0 z i;j 1 ; 8 i 2 K; 8 j 2 L; z i;j 2 Z ; 8 i 2 K; 8 j 2 L: Wedenoteby T k;l thesetoffeasiblesolutionsto4-4.Welet PT k;l = conv T k;l and LPT k;l betheLPrelaxationof T k;l ,i.e., LPT k;l = f x;y;z 2 R k + l + kl j x;y;z satises4-4 g .Werstarguethat T k;l isindeedavalidformulationof S k;l Proposition4.1.1.1. proj x;y T k;l = S k;l Proof. First,weshowthat S k;l proj x;y T k;l .Consideranunsplittableow x;y 2 S k;l Bydenition,thereexistsamatching M inthecompletebipartitegraphwithvertices K;L suchthat x i = y j 8 i;j 2 M ,and x i =0 and y j =0 forallvertices i 2 K and j 2 L thatareexposedin M .Dene z i;j toequal 1 if i;j 2 M and 0 otherwise.The solution x;y;z satisesconstraints4,4,4and4because M isa matching.Considernowconstraints4and4.If z i;j =1 ,theseconstraintsare satisedsince x i = y j .If z i;j =0 ,theseconstraintsarealsosatisedsince x i 2 [0 ;u i ] and y j 2 [0 ;v j ] .Constraint4issatisedsince x i =0 and P j 2 L z i;j =0 when i isexposedin M and 0 x i u i and P j 2 L z i;j =1 when i iscoveredby M .Asimilar argumentcanbeappliedtoshowthat4issatised. 99

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Second,weshowthat proj x;y T k;l S k;l .Consideranysolution x;y;z 2 T k;l ,we mustshowthat x;y 2 S k;l .Dene M = f i;j 2 K L j z i;j =1 g M isamatching inthecompletebipartitegraphwithnodes V 1 = K and V 2 = L .When i;j 2 M 4and4implythat x i = y j ,while4and4requirethat 0 x i u i and 0 y j v j .If i doesnotbelongtoanedgeof M ,thenconstraint4implies that x i =0 .Similarly, y j =0 forall j thatdonotbelongtoanedgeof M .Finally,the balanceequationissatisedsinceeachvariable x i withnon-zerovalueisinone-to-one correspondencewithavariable y j ofthesamevalue. Weobservethat LPT k;l isnotastrongformulationoftheno-splitrestriction.In fact,thispolyhedronhasextremepointsthatdonotevensatisfythebalanceequation constraint. Example4.1.1.1. Consideraninstanceof LPT 2 ; 2 where u = ; 7 and v = ; 7 .Note thatthesolutionwhere x = ; 7 y = ; 7 z 1 ; 1 = 4 7 z 1 ; 2 = 3 7 and z 2 ; 1 = 3 7 and z 2 ; 2 = 4 7 is anextremepointof LPT 2 ; 2 Althoughthebalanceequationisnaturallysatisedwhenthevariablesareinteger, itisnotsointheLPrelaxation.Oneobviouswaytoimprovetheformulationistherefore toaddtheconstraint P i 2 K x i = P j 2 L y j tothedenitionof T k;l ,therebyyieldingaset thatwerefertoas T k;l .Wethencandene P T k;l and LP T k;l asabove.Unfortunately, theenhancedLPrelaxation LP T k;l stilldoesnothaveverticeswhere z i;j arebinaryas illustratedinthefollowingexample. Example4.1.1.2. ConsidertheproblemofExample4.1.1.1.Thesolutionwhere x = ; 5 y = ; 7 z 1 ; 1 = 3 5 z 1 ; 2 = 2 5 and z 2 ; 1 = 2 5 and z 2 ; 2 = 3 5 isanextremepointof LP T 2 ; 2 Inparticular,weconcludefromExample4.1.1.2that LP T k;l hasextremepointsthat correspondtoowsthatviolatetheno-splitrequirementandhaveassociatedvariables z i;j thatarenotbinary.Toeliminatetheseundesirablepointsfromtherelaxation,we 100

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reformulate T k;l inanevenhigher-dimensionalspaceasfollows. X i 2 K z i;j 1 ; 8 j 2 L; X j 2 L z i;j 1 ; 8 i 2 K; x i = X j 2 L w i;j z i;j )]TJ/F24 11.9552 Tf 11.955 0 Td [(s i;j ; 8 i 2 K; y j = X i 2 K w i;j z i;j )]TJ/F24 11.9552 Tf 11.955 0 Td [(s i;j ; 8 j 2 L; 0 s i;j z i;j 1 ; 8 i 2 K; 8 j 2 L; z i;j 2 Z ; 8 i 2 K; 8 j 2 L; Theintuitiveideabehindtheaboveformulationisthat,whenaconnectionisestablished betweenincomingarc i andoutgoingarc j i.e., z ij =1 ,then,inextremepointsofthe problem,either x i = y j =0 or x i = y j = w ij .When z ij =1 ,thevariable s ij thentakesthe value 0 if x i = y j = w ij andtakesthevalue 1 if x i = y j =0 .When z ij =0 ,variable s ij onlytakesvalue 0 Wedenotethesetoffeasiblesolutionsto4?? by U k;l .Similartobefore, weusethenotation PU k;l =conv U k;l anddenotetheLPrelaxationof U k;l by LPU k;l = f x;y;z;s 2 R k + l +2 kl j x;y;z;s satises4-4 g .Wenextargue inProposition4.1.1.2that U k;l isavalidformulationfortheproblem. Proposition4.1.1.2. PT k;l =proj x;y;z PU k;l Proof. Werstprovethat PT k;l proj x;y;z PU k;l .Since T k;l isbounded,itsconvex hullisalsotheconvexhullofitsextremepoints.Itthereforesufcestoshowthateach extremepointbelongsto proj x;y;z U k;l toobtainthat PT k;l convproj x;y;z U k;l = proj x;y;z PU k;l .First,wearguethatinanyextremesolution x;y;z of T k;l with z i;j =1 x i = y j = w i;j or x i = y j =0 .Assumenot,thenthereexistsanextremepoint x;y;z with z i;j =1 and 0
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x;y;z = 1 2 x; y; z + 1 2 ~ x; ~ y; ~ z x;y;z isnotanextremepointof T k;l .Now,consideran extremepoint x;y;z of T k;l .Bytheaboveresult, x i = y j 2f 0 ;w i;j g forall i;j such that z i;j =1 .Nextdenethesolution x; y; z; s asfollows.If x i = y j = w i;j and z i;j =1 thenweset x i = x i y j = y j z i;j = z i;j and s i;j =0 .If x i = y j =0 and z i;j =1 ,thenwe set x i = x i y j = y j z i;j = z i;j and s i;j =1 .Finally,wesetallothervariablesto 0 .This solution x; y; z; s 2 U k;l ,asdesired. Forthereverseinclusion,weshowrstthat,foreachfeasiblepoint x;y;z;s to U k;l x;y;z 2 T k;l .Weobservethatinanysolution x;y;z;s of U k;l ,if z i;j =1 and s i;j =1 ,then x i = y j =0 .Similarly,if z i;j =1 and s i;j =0 ,then x i = y j = w i;j Constraints4and4arethereforesatisedsince,if z i;j =1 ,then x i = y j andotherwise x i 2 [0 ;u i ] and y j 2 [0 ;v j ] .Constraint4isalsosatisedsince x i P j 2 L z i;j w i;j P j 2 L z i;j u i .Asimilarargumentcanbeusedtoshowthat4is satised.Therefore proj x;y;z U k;l T k;l .Itfollowsthat proj x;y;z PU k;l PT k;l TheresultofProposition4.1.1.2statesthattheconvexhullsofthetwosetsare equal.Thestrongerstatementthanthesetsthemselvesareequalisnottruesince U k;l onlycontainsextremesolutionsof T k;l .Asaresult, U k;l cannotbeusedasaformulation oftheno-splitrequirement,unlessvariables s ij arerelaxedtobecontinuous.The advantageofformulation U k;l over T k;l isthatformulation LPU k;l isnaturallyintegralin thatvariables z i;j and s i;j take0-1valuesinextremepointsof U k;l .Weprovethisresult next. Proposition4.1.1.3. PU k;l = LPU k;l 102

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Proof. Theconstraintmatrixassociatedwith U k;l isoftheform 2 6 6 6 6 6 6 6 6 6 6 4 U 1 000 I 00 I 0000 I 0 )]TJ/F24 11.9552 Tf 9.299 0 Td [(II 0000 I A 1 A 2 I 0000 A 3 A 4 0 I 000 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 z s x y t u v 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 1 1 0 0 0 3 7 7 7 7 7 7 7 7 7 7 5 ; afterslackvariables t u and v areintroduced.Eachextremepointof LPU k;l corresponds toatleastonebasicfeasiblesolutionoftheabovesystem.Further,inabasicfeasible solution,variables x i and y j arebasicfor i 2 K and j 2 L .Considernowthebasic columnscorrespondingto z s t u ,and v .Thevaluesofthesevariablesareobtainedby solvingasquaresubsystemof 2 6 6 6 6 4 U 1 0 I 00 I 00 I 0 )]TJ/F24 11.9552 Tf 9.299 0 Td [(II 00 I 3 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 z s t u v 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 4 1 1 0 3 7 7 7 7 5 thatcontainsalltherowsoftheinitialsystem.Because U 1 istotallyunimodularTU sinceitistheconstraintmatrixofanassignmentproblem,itiseasilyveriedthatthe abovematrixisTU.Itfollowsthatthesolutionofthesquaresubsystemthatyieldsthe positivevaluesof z s t u ,and v isintegral.Inparticular,vectors z and s areinteger. Finally,since x = )]TJ/F24 11.9552 Tf 9.299 0 Td [(A 1 z )]TJ/F24 11.9552 Tf 12.286 0 Td [(A 2 s and y = )]TJ/F24 11.9552 Tf 9.299 0 Td [(A 3 z )]TJ/F24 11.9552 Tf 12.286 0 Td [(A 4 s ,weconcludethat x and y arealso integer. 103

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WeconcludefromtheproofofPropositions4.1.1.2and4.1.1.3thatthebalance equationisnotnecessaryintheformulationof U k;l asitissatisedatallextremepoints oftheformulation.Since PU k;l = LPU k;l and LPU k;l ispolynomially-sizedinthedata inputs,itispossibletooptimizealinearfunctionover PU k;l inpolynomial-timeusing forexampleKachiyan'sellipsoidalgorithmforLP.Theproblemofoptimizingalinear function P i 2 K i x i + P j 2 L j y j over PS k;l canalsobereformulatedasanassignment problemoverthebipartitenetworkwithvertices K;L wherethecostofarc i;j isset tobe max f i + j w ij ; 0 g Althoughtheformulationoftheno-splitrequirementweproposeinthissectionis ideal,itisrelativelylarge.Forthepracticalproblemsrelatedtounittrainschedulingwe studied,inwhichmonthlytransportationproblemsareexpressedoveratime-space network,thisformulationisoftentoolargetobehandleddirectlythroughcommercial solver.Onepossiblewaytocircumventthissizeproblemistohandletheextended reformulationthroughcolumngeneration.Webelievethatsuchapproachisvery appropriateandmostlikelyquiteeffectiveforsolvingproblemsinwhichtheonly combinatorialconstraintsinadditiontothetransportationstructureoftheproblemare no-splitconstraints.Theproblemsweultimatelywouldliketosolve,however,havea varietyofothercombinatorialrequirementsontheowvariables.Forthisreason,we believethatobtainingastrongformulationoftheno-splitrequirementinthespaceof originalvariablesislikelytobemoreusefulforpracticalproblemsascuttingplanescan behandledwithlittleeffortincurrentcommercialcodes. 4.1.2StrongFormulationsintheSpaceofOriginalVariables Inthissection,westudytheconvexhullof S k;l inthenaturalspaceofvariables x i and y j .Inparticular,wedene PS k;l =conv S k;l .Inthefollowingproposition,weshow that PS k;l isnotfull-dimensional. Proposition4.1.2.1. dim PS k;l = k + l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 104

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Proof. Let M = betheequalitysetof S k;l .Clearly,rank M = 1 sinceallfeasible solutionsto S k;l satisfy P i 2 K x i = P j 2 L y j .Therefore,dim PS k;l = k + l )]TJ/F20 11.9552 Tf 9.299 0 Td [(rank M = k + l )]TJ/F15 11.9552 Tf 10.194 0 Td [(1 Ontheotherhand,thevectors x 0 ;y 0 = ; 0 ~ x i ; ~ y i = e i ; e 1 for i 2 K and x j ; y j = e 1 ; e j for j 2 K nf 1 g areafnelyindependentsolutionsof S k;l andtherefore showthat dim PS k;l k + l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Because PS k;l isnotfull-dimensional,facet-deninginequalitiesfor PS k;l have multiplerepresentationsthatareobtainedfromeachotherthroughscalarmultiplication andthroughadditionoftheequation P m 2 K x m = P j 2 L y j .Intheremainderofthissection, wheneverconsideringavalidinequalityfor PS k;l thatcontainvariablesotherthan x 1 only, weusethisequalitytoeliminatevariable x 1 Observation4.1.2.1. If P i 2 K i x i + P j 2 L j y j denesface F of PS k;l ,then F is alsodenedbytheinequality P m 2 K nf 1 g m )]TJ/F24 11.9552 Tf 12.765 0 Td [( 1 x m + P j 2 L j + 1 y j since x 1 = P j 2 L y j )]TJ/F29 11.9552 Tf 24.534 8.967 Td [(P m 2 K nf 1 g x m Itispossibletoobtainadescriptionof PS k;l inthespaceoforiginalvariables bynumericallyprojecting LPU k;l ontothespaceof x i and y j variablesthrough, say,Fourier-Motzkinelimination;seeZiegler1995foradescription.Similarly,itis well-knownthatfacet-deninginequalitiesoftheprojectioncorrespondtoraysofthe projectionconeoftheformulation.Wementionhoweverthatnotallextremeraysofthe projectionconecorrespondtofacet-deninginequalitiesof PS k;l ;seeforinstanceBalas 2005.Todescribethedesiredprojectioncone,werstintroducemultipliers foreach 105

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oftheconstraintsoftheformulationasfollows: x i )]TJ/F29 11.9552 Tf 11.955 11.357 Td [(X j 2 L w i;j z i;j + X j 2 L w i;j s i;j =0 ; 8 i 2 K; 4 i y j )]TJ/F29 11.9552 Tf 11.955 11.358 Td [(X i 2 K w i;j z i;j + X i 2 K w i;j s i;j =0 ; 8 j 2 L; 4 j X i 2 K z i;j 1 ; 8 j 2 L; 4 j X j 2 L z i;j 1 ; 8 i 2 K; 4 i )]TJ/F24 11.9552 Tf 9.298 0 Td [(s i;j 0 ; 8 i 2 K; 8 j 2 L; 4 i;j )]TJ/F24 11.9552 Tf 11.955 0 Td [(z i;j + s i;j 0 ; 8 i 2 K; 8 j 2 L; 4 i;j z i;j 1 ; 8 i 2 K; 8 j 2 L: 4 i;j Theprojectionconeisthenobtainedbyrequiringtheinequalityobtainedbyaggregating inequalities4-4withweights haszerocoefcientsforvariables z ij and s ij Thefollowingresultensues. Proposition4.1.2.2. Let beanextremerayof C = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 2 R t )]TJ/F24 11.9552 Tf 9.298 0 Td [( 4 i w i;j )]TJ/F24 11.9552 Tf 11.956 0 Td [( 4 j w i;j + 4 j + 4 i )]TJ/F24 11.9552 Tf 11.955 0 Td [( 4 i;j + 4 i;j =0 ; 8 i 2 K; 8 j 2 L; 4 i w i;j + 4 j w i;j )]TJ/F24 11.9552 Tf 11.955 0 Td [( 4 i;j + 4 i;j =0 ; 8 i 2 K; 8 j 2 L; 4 j ; 4 i ; 4 i;j ; 4 i;j ; 4 i;j 0 ; 8 i 2 K; 8 j 2 L 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; where t =2 k +2 l +3 kl Then, X i 2 K 4 i x i + X j 2 L 4 j y j X j 2 L 4 j + X i 2 K 4 i + X i 2 K X j 2 L 4 i;j 106

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isvalidfor PS k;l .Further,allfacet-deningof PS k;l areoftheform4forsome extremerayof C Proposition4.1.2.2hasseveralpossibleuses.First,itcanbeusedtoderive closed-formexpressionforvalidinequalitiesfor PS k;l .Further,foranysolution x ;y = 2 PS k;l ,Proposition4.1.2.2showsthatthereisapolynomial-timealgorithmtoidentifythe validinequalityfor PS k;l thatismost-violatedat x ;y .Infact,thisinequalitycanbe derivedbysolvingthefollowingcut-generatinglinearprogramCGLP z =max X i 2 K 4 i x i + X j 2 L 4 j y j )]TJ/F29 11.9552 Tf 11.955 11.357 Td [(X j 2 L 4 j )]TJ/F29 11.9552 Tf 11.291 11.358 Td [(X i 2 K 4 i )]TJ/F29 11.9552 Tf 11.955 11.358 Td [(X i 2 K X j 2 L 4 i;j s:t: 2 C X j 2 L 4 j + X i 2 K 4 i + X i 2 K X j 2 L 4 i;j =1 ; wherethenormalizationequality4isaddedtoguaranteethattheproblemremains boundedwhenaviolatedinequalitycanbefound.Inparticular,iftheoptimalvalue z ofCGLPispositive,thereisaviolatedcut.Otherwise, x ;y 2 PS k;l .Forpractical purposes,theaboveCGLPisavaluablecomputationaltoolthatcanbeusedin cut-generationroutines.CGLPishoweverrelativelylargeanditsoptimalsolutions arenotguaranteedtoyieldfacet-deningfor PS k;l .Forthisreason,wenextseekto identifyfamiliesoffacet-deninginequalitiesthatcanbegeneratedquicklyandare guaranteedtobestrongfor PS k;l 4.1.2.1Convexhulldescriptionsforspecialcases Alineardescriptionof PS k;l inthespaceoforiginalvariablesdoesnotseem straightforwardtoobtain.Toillustratethediversityofstructureinitsinequalities,we presentintheAppendix5acompletelineardescriptionoftheconvexhullofasimple unsplittablenodeowproblemswiththreeinowsandthreeoutows.Next,wedisplaya fewoftheseinequalities. 107

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Example4.1.2.1. Consider ^ S k;l = 8 > > > > > > > < > > > > > > > : x;y 2 R 3 R 3 x 1 + x 2 + x 3 = y 1 + y 2 + y 3 x 1 3 ;x 2 7 ;x 3 13 y 1 5 ;y 2 11 ;y 3 13 nosplit x;y 9 > > > > > > > = > > > > > > > ; Thelineardescriptionof P ^ S k;l has45equalities/inequalities.Amongthem,wend A3x 3 0 A9 y 1 5 A112 x 2 2 x 3 + 5 y 1 15 A14 5 x 2 2 x 3 + 5 y 1 50 A176 x 3 + 13 y 3 91 A228 x 2 + 5 x 3 + 20 y 1 125 A23 13 x 3 2 y 3 143 A3252 x 2 70 x 3 + 91 y 3 273 A41 143 x 3 78 y 2 88 y 3 715 Althoughwedonotknowacompletelineardescriptionof PS k;l inthegeneralcase, suchdescriptioniseasilytoobtainwhen k =1 or l =1 .Disjunctiveprogramming techniquesBalas,1979canalsobeusedtoobtainsuchadescriptionwhen k = l =2 First,weconsiderthecasewherethereisasinglearcleavingthenode. Proposition4.1.2.3. Alineardescriptionoftheconvexhullof S k; 1 isgivenby PS k; 1 = 8 > > > > > < > > > > > : P i 2 K x i = y x;y 2 R k +1 P i 2 K x i w i; 1 1 x i 0 8 i 2 K 9 > > > > > = > > > > > ; : 108

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Proof. Theextremepointsof S k; 1 canbeeasilyidentiedtobe x i ;y i = w i; 1 e i ;w i; 1 for i 2 K togetherwith x 0 ;y 0 = 0 ; 0 .Projecting PS k; 1 ontothespaceof x variables thevariable y isnotneededasitisdenedthroughtherelation y = P i 2 K x i ,weobtain apolytopedenedastheconvexhullof x i = w i; 1 e i for i 2 K and x 0 =0 .Since thesepointsareafnelyindependent,theresultingpolytopeisasimplexoftheform S k = f x 2 R k j x 0 ; T x g ,where i w i; 1 = for i 2 K .Weconcludethat T x = is infactoftheform X i 2 K x i w i; 1 1 : Togetherwiththeequality y = P i 2 K x i ,weobtaintheresult. Becausetheroleofvariables x and y issymmetricinourmodel,itissimpleto verifythatProposition4.1.2.3alsoprovidesalineardescriptionfortheconvexhullof PS 1 ;l .Next,weobtainalineardescriptionoftheconvexhullof S 2 ; 2 throughdisjunctive programming. Proposition4.1.2.4. Alineardescriptionoftheconvexhullof S 2 ; 2 isgivenby PS 2 ; 2 = 8 > > > > > > > > > > > < > > > > > > > > > > > : x 1 + x 2 = y 1 + y 2 x 1 0 ;x 2 0 x;y 2 R 4 x i + v 1 min f u i ;v 2 g )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F24 11.9552 Tf 9.077 -10.593 Td [(y 2 min f u i ;v 1 g8 i 2f 1 ; 2 g y 1 0 ;y 2 0 u 1 min f u 2 ;v j g )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F24 11.9552 Tf 9.077 -10.593 Td [(x 2 + y j min f u 1 ;v j g8 j 2f 1 ; 2 g 9 > > > > > > > > > > > = > > > > > > > > > > > ; : Proof. WeprovethisresultusingdisjunctiveprogrammingBalas,1979.Infact S 2 ; 2 = P 1 [P 2 where P 1 = f x;y 2 R 4 j x 1 = y 1 ;x 2 = y 2 ; 0 x u ; 0 y v g and P 2 = f x;y 2 R 4 j x 1 = y 2 ;x 2 = y 1 ; 0 x u ; 0 y v g .Usingdisjunctive 109

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programming, conv P 1 [P 2 canbeexpressedinahigherdimensionalspaceas X = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : x = x +~ x;y = y +~ y x 1 = y 1 ; x 2 = y 2 x;y; x; y; ~ x; ~ y; 2 R 13 ~ x 1 =~ y 2 ; ~ x 2 =~ y 1 0 x u ; 0 y v 0 ~ x )]TJ/F24 11.9552 Tf 11.955 0 Td [( u ; 0 ~ y )]TJ/F24 11.9552 Tf 11.955 0 Td [( v 0 1 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; since P 1 and P 2 arebounded.Since proj x;y X =clconv P 1 [P 2 ,wenextproject X ontothespaceof x;y variablesusingFourier-Motzkinelimination;seeTheorem1.4in Ziegler1995.Usingequalities ~ x = x )]TJ/F15 11.9552 Tf 12.86 0 Td [( x ~ y = y )]TJ/F15 11.9552 Tf 12.928 0 Td [( y ~ x 1 =~ y 2 ~ x 2 =~ y 1 x 1 = y 1 x 2 = y 2 weeliminate ~ x; ~ y; y fromtheformulationof X toobtain proj x;y; x; X = 8 > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > : x 1 = x 1 + y 2 )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 2 x 2 = x 2 + y 1 )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 1 0 x 1 w 11 0 x 2 w 22 0 y 2 )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 12 0 y 1 )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 21 0 1 9 > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > ; : 110

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Afterprojecting x; y ,weobtain proj x;y; X = 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > : x 1 + x 2 = y 1 + y 2 0 y 1 + y 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(x 2 w 11 + )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 12 0 x 2 w 22 + )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 21 0 y 1 w 11 + )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 21 0 y 2 w 22 + )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 12 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(y 1 w 22 y 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(x 2 w 11 x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(y 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 21 y 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( w 12 0 1 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > ; : Finally,weprojectoutvariable .Becausevariables x and y playsymmetricalroles inthedenitionof S 2 ; 2 ,itissufcienttoconsiderthecasewhere u 1 v 1 .Wetherefore 111

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havethat u 1 v 1 v 2 = u 2 proj x;y X 1 = 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : x 1 + x 2 = y 1 + y 2 0 y 1 + y 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(x 2 u 1 0 x 2 u 2 0 y 1 v 1 0 y 2 u 2 u 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(u 2 x 2 + u 2 y 2 u 1 u 2 u 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 x 2 + v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 + u 1 y 2 u 2 v 1 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 + v 1 x 2 + u 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 1 y 2 u 1 u 2 u 2 x 2 + v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 y 2 u 2 v 1 u 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 y 1 + v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 y 2 v 1 u 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(u 2 1 v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 x 2 + u 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(v 1 y 1 v 1 u 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 1 x 2 + u 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [(v 1 + u 1 y 1 v 1 u 2 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 1 x 2 + v 1 y 1 v 1 u 1 9 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > ; = 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : x 1 + x 2 = y 1 + y 2 0 y 1 + y 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(x 2 u 1 0 x 2 0 y 1 0 y 2 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 x 2 + u 2 y 2 u 1 u 2 u 2 x 2 + v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 y 2 u 2 v 1 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 1 x 2 + v 1 y 1 v 1 u 1 9 > > > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > > > ; : 4.1.2.2Facet-deninginequalitiesfor PS k;l InSection4.1.2.1,wecharacterizetheconvexhullof PS k;l when k =1 or l =1 and k = l =2 .Inthissectionwedescribeseveralfamiliesoffacet-deninginequalities for PS k;l when k> 1 and l> 1 .Manyfamiliesofinequalitieswederivehavemirror equivalentsthatfollowfromthefactthatvariables x and y playsymmetricalrolesinthe denitionof PS k;l Remark4.1.2.1. Observethatvariables x and y playasymmetricroleinthedenitionof S k;l .Itfollowsthatifinequality X m 2 K m u;v x m + X n 2 L n u;v y n u;v 112

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isfacet-deningfor PS k;l undersomeconditions C u;v onthecapacities u and v ofthe incomingandoutgoingarcs,respectively,then X m 2 L m v;u x m + X n 2 K n v;u y n v;u isfacet-deningfor PS l;k underconditions C v;u First,westudytrivialinequalitiesof PS k;l ,i.e.,inequalitiesthatarevalidfor LPS k;l Wenextpresentconditionsunderwhichtheseinequalitiesarefacet-deningfor PS k;l Proposition4.1.2.5. Inequalities x i 0 for i 2 K and y j 0 for j 2 L arefacet-dening for PS k;l Proof. Inequality x i 0 isvalidfor PS k;l sinceitisvalidfor LPS k;l .Let 2 K nf i g Considernowthepoints x n ; y n = e ;e n for n 2 L and ^ x m ; ^ y m = e m ;e 1 for m 2 K nf i g where isasufcientlysmallpositiverealnumber.These k + l )]TJ/F15 11.9552 Tf 12.272 0 Td [(1 points belongto S k;l ,areafnelyindependentandsatisfy x i 0 atequality.Thisshowsthat x i 0 isfacet-deningfor PS k;l .ItfollowsfromRemark4.1.2.1that y j 0 isalso facet-deningfor PS k;l ItfollowsfromProposition4.1.2.5thatalllowerboundinequalitiesarefacet-dening forthesetdescribedinExample4.1.2.1.IntheAppendix5,weobservethatthese inequalitiesappearinthedescriptionof PS k;l underthelabels A 1 A 7 .Next,we determinewhentheupperboundconstraintsonowvariablesarefacet-deningfor PS k;l Proposition4.1.2.6. Let i 2 K nf k g and j 2 L nf l g besuchthat v j u i .Then y j v j isfacet-deningfor PS k;l Proof. Inequality4isvalidfor PS k;l sinceitisvalidfor LPS k;l .Toproveitdenes afacetof PS k;l ,weconstructthepoints x m ;_ y m = v j e m ; v j e j for m = i;:::;k 113

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x m ; y m =_ x i ;_ y i + e m ; e l for m =1 ;:::;i )]TJ/F15 11.9552 Tf 12.688 0 Td [(1 ~ x n ;~ y n =_ x i ;_ y i + e 1 ; e n for n =1 ;:::;l )]TJ/F15 11.9552 Tf 12.397 0 Td [(1 ,with n 6 = j and ^ x ;^ y =_ x i +1 ;_ y i +1 + e i ; e l where isasufciently smallpositiverealnumber.These k + l )]TJ/F15 11.9552 Tf 11.565 0 Td [(1 afnelyindependentpointsbelongto S k;l and satisfy4atequality. ItfollowsbyRemark4.1.2.1that x i u i isfacet-deningfor PS k;l for i 2 K nf k g and j 2 L nf l g with u i v j .WhenappliedtoExample4.1.2.1,Proposition4.1.2.6andthe ensuingremarkshowthatinequalities x 1 3 x 2 7 and y 1 5 arefacet-deningfor P ^ S 3 ; 3 .Theseinequalitiesarelabeled A 8 A 10 intheAppendix5. InProposition4.1.2.4,weshowedthattheconvexhullof S 2 ; 2 isdescribedbythe trivialboundinequalitiesstudiedinPropositions4.1.2.5and4.1.2.6togetherwithsome inequalitiesthatinvolveasinglevariable x i andasinglevariable y j .Next,wegeneralize theseinequalitiesandpresentconditionsunderwhichtheyarefacet-deningfor PS k;l Proposition4.1.2.7. Let j 2 L besuchthat v j = v l .Let p 2 K suchthat u p = u k and u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1
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assumethat y j = x m for m 2f 1 ;:::;p )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 g .Then y j = x m u m u p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 .Wewritethat LHS u k y j u k u p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 wheretherstinequalityholdsbecausethecoefcientsof x i arenonpositiveandthe secondinequalityholdsbecausecoefcientof y j u p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 .Third,assumethat y j =0 ThenLHSisnonpositivesincethecoefcientsof x i arenonpositiveand y j =0 .Itfollows that4issatisedsince u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u k isnonnegative. Next,weshowthat4isfacet-deningfor PS k;l .Let F bethefaceof PS k;l denedby4.Let k X m =1 m x m + l X n =1 n y n beanyinequalitythatdenes F .Weshowthat4and4arescalarmultiplesof eachotheruptoadditionoftheequality P m 2 K x m = P n 2 L y n .Considerthepoint ~ x m ; ~ y m = u k e m ; u k e j where m 2f p;:::;k g in F .Wewritethat m = u k )]TJ/F24 11.9552 Tf 11.955 0 Td [( j for m = p;:::;k: Second,considerthepoints x n ;_ y n =~ x k ; ~ y k + e 1 ; e n for n =1 ;:::;l ,with n 6 = j .We writethat k u k + 1 + j u k + n = : Substituting4for m = k in4,weobtainthat 1 = )]TJ/F24 11.9552 Tf 9.299 0 Td [( n for n =1 ;:::;l; with n 6 = j: Third,considerthepoints x m ; y m =~ x k ; ~ y k + e m ; e t for m =2 ;:::;p )]TJ/F15 11.9552 Tf 11.975 0 Td [(1 ,where t =1 when j = l and t = j +1 otherwise.Wewritethat k u k + m + j u k + t = for m =2 ;:::;p )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 115

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Subtracting4for m = k from4weconcludethat m = )]TJ/F24 11.9552 Tf 9.299 0 Td [( t for m =2 ;:::;p )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Finally,weconsider x; y = u p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 e p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ; u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 e j in F .Wewritethat p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + j u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = : Combiningtheresultsin4,weobtainthat p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( j = m = )]TJ/F24 11.9552 Tf 9.299 0 Td [( n for m =1 ;:::;p )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ;n =1 ;:::;l; with n 6 = j: Therefore,wewritethat = X m 2 K m x m + X n 2 L n y n = X m 1 2f 1 ;:::;p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 g u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( j x m 1 + X m 2 2f p;:::;k g u k )]TJ/F24 11.9552 Tf 11.955 0 Td [( j x m 2 + X n 2 L nf j g )]TJ/F29 11.9552 Tf 11.291 16.857 Td [( u k )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.956 0 Td [( j y n + j y j = X m 1 2f 1 ;:::;p )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 g u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x m 1 + X m 2 2f p;:::;k g u k x m 2 + X n 2 L nf j g )]TJ/F29 11.9552 Tf 11.291 16.857 Td [( u k )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y n = X m 2 2f p;:::;k g u k )]TJ/F24 11.9552 Tf 20.912 8.088 Td [( u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x m 2 + u p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y j ; wherethethirdequalityholdssince P m 2 K x m = P n 2 L y n andthefourthequalityissatised becauseofObservation4.1.2.1. ThenextresultisadirectconsequenceofProposition4.1.2.7andRemark4.1.2.1. Proposition4.1.2.8. Let i 2 K besuchthat u i = u k .Let p 2 L suchthat v p = v l and v p )]TJ/F22 7.9701 Tf 6.587 0 Td [(1
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distinct.Wethenwrite u 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 3 x 3 + u 3 y 3 u 2 u 3 ; whichreducesto )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 x 3 +13 y 3 91 : ToapplyProposition4.1.2.8,wemustselect i =3 and p =3 sincethecapacitiesofall outgoingarcsaredistinct.Wethenwrite v 3 x 3 + v 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 3 y 3 v 2 v 3 ; whichreducesto 13 x 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 y 3 143 : Therstinequalityislabeledas A 17 intheAppendix5whilethesecondislabeled A 23 Allinequalitiesdescribedsofarcontainnomorethantwovariables.Thenextfamily containsasequenceofconsecutivenonzerocoefcientsforthe x variables. Proposition4.1.2.9. Let i 2 K nf 1 g and j 2 L besuchthat u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1
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First,weassumethat y j = x p .Then x p = y j v j .Wewritethat LHS v j v j )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x p + v j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(v j y j v 2 j v j )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j = v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 wheretherstinequalityholdsbecausethecoefcientsof x m arenonpositiveandthe secondinequalityholdsbecausecoefcientof x p and y j arenonnegative. Second,weassumethat y j = x q for q 2f i;:::;k gnf p g .Then y j = x q v j and x p u p .Wewritethat LHS u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(v j u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j x q + v j v j )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x p + v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j x q = u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x q + v j v j )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + v j u p v j )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 wheretherstinequalityholdsbecause y j = x q andbecausethecoefcientsof x m are nonpositive,andthesecondinequalityholdsbecausethecoefcientof x q and x p are nonnegative. Third,weassumethat y j = x q for q 2f 1 ;:::;i )]TJ/F15 11.9552 Tf 10.566 0 Td [(1 g .Then y j = x q u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 and x p u p Wethenwritethat LHS v j v j )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x p + v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j y j v j v j )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u p + v j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(v j u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 wheretherstinequalityholdsbecausethecoefcientsof x m arenonpositive,andthe secondinequalityholdsbecausethecoefcientof x p and y j arenonnegative. 118

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Fourth,weassumethat y j =0 .Then x p u p andwewritethat LHS v j v j )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 x p v j v j )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 u p = v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 u p v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 2 j u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 wheretherstinequalityholdsbecausethecoefcientsof x m arenonpositiveand y j =0 ,thesecondinequalityholdsbecausethecoefcientof x p isnonnegative,andthe thirdinequalityholdsbecause u p v j Wenextarguethat4isfacet-deningfor PS k;l .Let F bethefaceof PS k;l denedby4.Let X m 2 K m x m + X n 2 L n y n beanyvalidinequalitythatalsodenes F .Wenextshowthat4isascalarmultiple of4uptoadditionoftheequality P m 2 K x m = P n 2 L y n .Becausethepoint x 0 ;y 0 = v j e p ; v j e j 2F ,wemusthavethat p v j + j v j = : Similarly,thepoints x m ; y m = x 0 ;y 0 + e m ; e l for m =1 ;:::;i )]TJ/F15 11.9552 Tf 12.255 0 Td [(1 belongto F and showthat p v j + m + j v j + l = : Subtracting4from4yieldsthat m = )]TJ/F24 11.9552 Tf 9.299 0 Td [( l for m =1 ;:::;i )]TJ/F15 11.9552 Tf 12.145 0 Td [(1 .Now,consider thepoints ^ x n ; ^ y n = x 0 ;y 0 + e 1 ; e n for n =1 ;:::;l )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ,with n 6 = j in F .Wehavethat p v j + 1 + j v j + n = : Subtracting4from4yieldsthat 1 = )]TJ/F24 11.9552 Tf 9.298 0 Td [( n for n =1 ;:::;l )]TJ/F15 11.9552 Tf 12.165 0 Td [(1 ,with n 6 = j .We concludethat m = )]TJ/F24 11.9552 Tf 9.298 0 Td [( n for m =1 ;:::;i )]TJ/F15 11.9552 Tf 12.254 0 Td [(1 and n =1 ;:::;l )]TJ/F15 11.9552 Tf 12.255 0 Td [(1 ,with n 6 = j .Consider 119

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nextthepoint x; y = u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 e i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ; u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 e j + u p e p ; u p e l in F whichisfeasiblesincewe assumed j 6 = l .Wehavethat i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + p u p + j u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + l u p = : Werewrite4as )]TJ/F24 11.9552 Tf 9.299 0 Td [( l u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + v j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j u p + j u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + l u p = : whereweuse i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F24 11.9552 Tf 9.299 0 Td [( l and p = v j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j from4.Simplifyingfurther,weobtain l = 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 + j = )]TJ/F24 11.9552 Tf 9.298 0 Td [( m for m =1 ;:::;i )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 as u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 > 0 Finally,considerthepointsof F ~ x m ;~ y m = v j e m ; v j e j + u p e p ; u p e l for m = i;:::;k with m 6 = p .Wewritethat m v j + p u p + j v j + l u p = : Simplifying,weobtain m = 1 v j )]TJ/F24 11.9552 Tf 13.151 8.088 Td [(u p v 2 j )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + u 2 p v 2 j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( j 120

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Usingthefactthat p = v j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j from4,4and4,wenowcanwritethat = k X m =1 m x m + l X n =1 n y n = )]TJ/F25 7.9701 Tf 13.595 14.944 Td [(i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X m =1 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + j x m + k X m = i;i 6 = p 1 v j )]TJ/F24 11.9552 Tf 13.151 8.088 Td [(u p v 2 j )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + u 2 p v 2 j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( j x m + v j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j x p + l X n =1 ;n 6 = j 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.087 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + j y n + j y j = )]TJ/F25 7.9701 Tf 13.595 14.944 Td [(i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 X m =1 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 40.461 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 x m + k X m = i;i 6 = p 1 v j )]TJ/F24 11.9552 Tf 13.15 8.087 Td [(u p v 2 j )]TJ/F24 11.9552 Tf 40.462 8.087 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.956 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + u 2 p v 2 j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 x m + v j x p + l X n =1 ;n 6 = j 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y n : Finally,usingobservation4.1.2.1,weconcludethat = k X m = i;i 6 = p 1 v j )]TJ/F24 11.9552 Tf 13.151 8.087 Td [(u p v 2 j )]TJ/F24 11.9552 Tf 40.462 8.087 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 + 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.087 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x m + 1 v j + 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.088 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 x p )]TJ/F24 11.9552 Tf 11.956 0 Td [( 1 u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 )]TJ/F24 11.9552 Tf 40.462 8.087 Td [(u p v j u p )]TJ/F24 11.9552 Tf 11.955 0 Td [(u i )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y j : Thisinequalityisascalarmultipleof4. WenextillustratetheresultofProposition4.1.2.9onanexample. Example4.1.2.3. Considertheset P ^ S 3 ; 3 denedinExample4.1.2.1.Wehavethat u 1 =3
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whichreduceto p =2:5 x 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 x 3 +20 y 1 50 p =3: )]TJ/F15 11.9552 Tf 9.298 0 Td [(8 x 2 +5 x 3 +20 y 1 125 : Therstinequalityislabeledas A 14 intheAppendix5whilethesecondislabeled A 22 Anothersimilarfamilyofbandinequalitiesispresentednext. Proposition4.1.2.10. Let j 2 L nf l g and i 2 K nf k g besuchthat u i
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Next,weshowthat4isfacet-deningfor PS k;l .Let F bethefaceof PS k;l dened by4.Let k X m =1 m x m + l X n =1 n y n beanyinequalitythatdenes F .Weshowthat4and4arescalarmultiples ofeachotheruptoadditionoftheequality P m 2 K x m = P n 2 L y n .Considerthepoints x m ;_ y m = v j e m ; v j e j for m = i +1 ;:::;k in F .Wewritethat m v j + j v j = : If i +1
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Finally,considerthepoint ^ x ;^ y = u i e i ; u i e j .Wewritethat i u i + j u i = ; whichcanberewrittenas i = u i )]TJ/F24 11.9552 Tf 11.955 0 Td [( j : Using4,4impliesthat l = )]TJ/F24 11.9552 Tf 9.299 0 Td [( 1 = )]TJ/F29 11.9552 Tf 11.291 13.271 Td [( u i )]TJ/F24 11.9552 Tf 11.955 0 Td [( j .Considering4,4, 4and4,wewritethat = k X m =1 m x m + l X n =1 n y n = i X m =1 u i )]TJ/F24 11.9552 Tf 11.956 0 Td [( j x m + k X m = i +1 v j )]TJ/F24 11.9552 Tf 11.955 0 Td [( j x m + l X n =1 ;n 6 = j )]TJ/F24 11.9552 Tf 12.155 8.088 Td [( u i + j y n + j y j = i X m =1 u i x m + k X m = i +1 v j x m + l X n =1 ;n 6 = j )]TJ/F24 11.9552 Tf 12.155 8.088 Td [( u i y n = k X m = i +1 v j )]TJ/F24 11.9552 Tf 14.811 8.088 Td [( u i x m + u i y j = k X m = i +1 u i )]TJ/F24 11.9552 Tf 11.955 0 Td [(v j u i v j x m + v j u i v j y j wherethethirdequalityholdssince P m 2 K x m = P n 2 L y n ,andthefourthequalityholds becauseofobservation4.1.2.1.Thisshowsthat4isascalarmultipleof4 andcompletestheproof. WenextillustratetheresultofProposition4.1.2.10onanexample. Example4.1.2.4. Considertheset P ^ S 3 ; 3 denedinExample4.1.2.1.Wehavethat u 1 =3
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whichreducesto )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 x 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 x 3 +5 y 1 15 : Thisinequalityislabeledas A 11 intheAppendix5. Weconcludethissectionbypresentingafamilyoffacet-deninginequalitiesfor PS k;l havingthecharacteristicthatcoefcientsofvariables x i and y j canbecomevery large. Proposition4.1.2.11. Let j 2 L besuchthat v j = v l .Then k X m =2 k Y n =2 ;n 6 = m u n u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u m # x m + k Y n =2 u n y j k Y n =1 u n isfacet-deningfor PS k;l Proof. Werstshowthat4isvalidfor PS k;l .Therearethreecases.Intheensuing discussion,weuseLHSasashorthandnotationfortheleft-hand-sideof4. First,assumethat y j = x m for m 2f 2 ;:::;k g .Then y j = x m u m .Wewritethat LHS k Y n =2 ;n 6 = m u n u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u m # x m + k Y n =2 u n x m k Y n =2 ;n 6 = m u n u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u m # u m + k Y n =2 u n u m = k Y n =2 u n [ u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u m + u m ] = k Y n =1 u n ; whererstinequalityholdssincethecoefcientsof x m arenonpositiveand x m = y j whilethesecondinequalityholdssince x m u m .Second,assumethat y j = x 1 .We writethat LHS k Y m =2 u m x 1 k Y m =2 u m u 1 = k Y m =1 u m 125

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whererstinequalityholdssincethecoefcientsof x m arenonpositiveand x 1 = y j andthesecondinequalityholdssince x 1 u 1 .Third,assumethat y j =0 .ThenLHSis nonpositivesincethecoefcientsof x m arenonpositiveand y j =0 .Itfollowsthat4 issatisedsince Q k m =1 u m isnonnegative. Next,weshowthattheinequalityisfacet-deningfor PS k;l .Inparticular,wedene x m ;_ y m = u m e m ; u m e j for m =1 ;:::;k and x n ; y n =_ x k ;_ y k + e 1 ; e n for n =1 ;:::;l with n 6 = j .These k l )]TJ/F15 11.9552 Tf 12.247 0 Td [(1 pointsareafnelyindependentandbelongtothefaceof PS k;l denedby4. WenextillustratetheresultofProposition4.1.2.11onanexample. Example4.1.2.5. Considertheset P ^ S 3 ; 3 denedinExample4.1.2.1.ToapplyProposition4.1.2.11,wemustselect j =3 sincethecapacitiesofalloutgoingarcsaredistinct. Weobtaintheinequality u 3 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 2 x 2 + u 2 u 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(u 3 x 3 + u 2 u 3 y 3 u 1 u 2 u 3 whichreducesto )]TJ/F15 11.9552 Tf 9.299 0 Td [(52 x 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(70 x 3 +91 y 1 273 : Thisinequalityislabeledas A 32 intheAppendix5.Itisclearthatasymmetric inequalitycanbeobtainedusingRemark4.1.2.1.Inparticular,weobtain v 2 v 3 x 3 + v 3 v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 2 y 2 + v 2 v 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(v 3 y 3 v 1 v 2 v 3 whichreducesto 143 x 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(78 y 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(88 y 3 715 : Thisinequalityislabeledas A 41 intheAppendix5. 126

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4.2ComputationalExperiments Wenextprovideacomparativeempiricalstudyofthedifferentformulationswe developedinprevioussections.WerandomlygenerateUFPinstanceshaving500,1000 and2000nodes,respectively.Inparticular,wecreate10differentinstancesofeach size.Eachfamilyislabeledwithitsassociatednumberofnodes. Inordertomaintainadifcultylevelthatiscommoninpracticalapplications,we generatetheinstancespecicsinawaythatisreminiscentoftheunittrainnetwork structureofrailroads.Weselectthesupplynodecounttobebetween5%and10%of theproblemsize.Supplynodesareassumedtobesplittablebecausetheycorrespond tocarpoollocationswheretherequiredmachineryandmanpowertoformatrainis available.Demandnodesareassumedtobeunsplittableastrainsusuallyleavetheir destinationsintheformtheyarrived.Withrespecttoreservations,wesetthenumber ofdemandnodestobehalfoftheproblemsize.Wethenrandomlygeneratesplittable nodesthatcanbeseenasbuildlocationsintheunittraincontext.Werestrictthe numberofsplittablenodestobebetween1%and3%oftheproblemsize.Wesetall remainingnodesasunsplittablenodes. Next,werandomlygeneratethenumberofoutgoingarcsforeachnodebetween1 and10.Foreacharcthatwecreate,weassignadailytraincapacityrandomlybetween 1and7.Thiscorrespondstocreatingseveralcopiesofthearc,whichwillthenbe requiredtoaccommodatenomorethanasingletrain.Weassigncarcapacitiestoeach ofthesearccopiessothattheyreectactualtrainsizes.Inparticular,eacharcisgiven acarcapacitythatisanintegervaluerandomlyselectedbetween3and7times25. Foreachinstance,wesolvetheLPrelaxationoftheproblemusingformulations LPU k;l LPT k;l and LPS k;l .Fortheformulationbasedon LPS k;l ,weusetwomethods. Intherst,wegeneratecutsusingCGLP.Inthesecond,weiterativelygeneratecuts fromthefamiliesoffacet-deninginequalitieswedevelopedinSection4.1.2.2.We performedallexperimentsonalaptopcomputerwith2.50Ghz.IntelCorei5-2450 127

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processorand8GBRAMona64-bitoperatingsystem.Wecodedtheassociated routinesusingC#inMicrosoftVisualStudio2012.Wesolvedallthelinearprograms usingGurobi5.5with.NETenvironment. InTable4-1,wepresenttheLPboundsweobtainwitheachofthedifferent formulations.Eachcolumncorrespondstoaninstancefamilyofthesizeprovidedin theheader.Rowsaregroupedaccordingtotheformulationused.Foreachofthese formulations,weprovidethepercentimprovementoverthecompletelysplittableow problemandassociatedCPUtimesinseconds.TherowlabeledUFormulation presentsresultsfor LPU k;l ,therowlabeledTformulationpresentsresultsfor LPT k;l TherowlabeledCGLPpresentsresultsfor LPS k;l wherecutsareaddedthrougha separationroutinethatusesCGLP,andtherowlabeledKnownCutspresentsresults for LPS k;l wherecutsareaddedthroughaseparationroutinethatmakesuseoffamilies offacet-deninginequalitieswedevelopedinSection4.1.2.2,respectively.Finally, therowlabeledSFormulationpresentstheCPUtimesthat LPS k;l takes.Foreach problemsize,weprovidetheminimum,averageandmaximumpercentimprovements fortheaforementioned10differentinstancesofthesamesize. Theempiricalresultsindicatethatthereisvalueinstudyingtheno-splitrequirement further.Inparticular,thetrivialTformulation,althoughitissmallerandsolvesfast, doesnotprovideboundsasgoodasthoseprovidedbytheUformulation.Further, thestrengthoftheUformulationcanbeharnessedthroughprojectionasweobtain theexactsameboundsastheUformulationintheoriginalxy-space.Atthispoint,the timerequiredtoobtaintheseboundsinthexy-spaceisstilllarge.However,webelieve thatthiscanberesolvedthroughadequateuseoftheCGLPandthroughseparation ofclosed-formcuts.Infact,knowncutscanbeusedtoseparatesolutionsviolating theno-splitconditionsinamoreefcientmanner.Preliminarycomputationindicates howeverthatwemightnothavediscoveredyetthemostimportantoftheseinequalities. 128

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Thereforefurtherfamiliesoffacet-deninginequalitiesshouldbederivedinorderto obtainbetterboundsinatime-efcientmannerintheoriginalspaceofxy-variables. 4.3Conclusion Inthischapter,westudiedacombinatorialrequirementthatoccursintheformulation ofnetworkowproblemsarisinginrailroads.Westudiedhowtobestformulatethis requirement.Inparticular,weshowedthatthestrengthofextendedformulationscanbe harnessedinthespaceoforiginalvariablesthroughthesolutionofacut-generationLP. Althoughrunningtimesatthispointarestillslower,webelievethatthisproblemcanbe resolvedthroughthederivationofnewfamiliesoffacet-deninginequalities.Further, webelievethattheadvantageofthexyformulationwillbecomemoreapparentduring branching.OurCGLPapproachcanstillbeappliedinthenodesofthetreeifithasbeen decidedalreadythatsomeow x i willnotberoutedtosomecollectionofoutgoingarcs y j .Infact,itsufcestosetthecorrespondingvariablesto z i;j to 0 intheUformulation. MaintainingsmallerLPsthatdonotcontainuselessvariablesforpreventingpotential splitviolationsshouldhavesignicantbenetsintheresultingbranch-and-boundtree. 129

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Table4-1.ComparativeanalysisofdifferentformulationsforsolvingtheUFProot relaxation. Collection UFP500UFP1000UFP2000 UFormulation %Improvement min0.00%1.03%15.38% avg15.18%12.82%19.12% max34.12%25.11%23.30% CPUTimeinsec.1.752.9411.74 TFormulation %Improvement min0.00%1.03%11.70% avg13.19%10.67%16.51% max31.82%21.02%21.21% CPUTimeinsec.0.280.661.87 KnownCuts %Improvement min0.00%0.00%0.00% avg0.82%0.77%0.73% max2.62%2.30%1.45% CPUTimeinsec.3.236.6831.83 CGLP %Improvement min0.00%1.03%15.38% avg15.18%12.82%19.12% max34.12%25.11%23.30% CPUTimeinsec.33.3276.71227.17 SFormulationCPUTimeinsec.0.010.030.04 130

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CHAPTER5 CONCLUSION Wesummarizethecontributionsofthisthesisnext. First,wepresentedamethodtoobtaingood-qualitysolutionsinareasonable amountoftimeforthemonthlycoaltrainreservationsplanningproblem.Tothebest ofourknowledge,thisproblemhasnotbeenaddressedinthislevelofdetailinearlier studies.Consideringcarresources,dailytraincapacitiesatloadoriginsanddestinations andcarpreferencesofcustomers,themethodweproposeprovidesasetofcoal reservationsthattherailroadshouldcommittosatisfy.Moreover,theproducedsolution isassociatedwithadetailedassignmentofcoalcarsofvariouskindstoreservations, detailedtrainschedulesandroutesthroughoutaplanningmonth.Thedetailedmonthly planincorporatesbothoperationalandtacticalconcernsofthedecisionmakers. Ournumericalresultssuggestthatasignicantbusinessimprovementmightbe gainedbyusingtheproposedsequentialapproachinsteadofcurrentpractice.Because ouralgorithmsresultinamoreefcientallocationofresources,webelievethatthey willcontributetomaintainthecompetitiveadvantageofrailroadsoverothermodesof transportationformajorrevenue-bringingfreightcommoditiessuchascoal.Theideas describedherecanbeadaptedtootherhighvolumeunittraincommodities.Thescope ofMCTRPPitselfmightbeextendedtoincorporateadditionalproblemcharacteristics suchasunittrainlocomotiveschedulingoradditionalcapacityrestrictionsenforced regionally,oncertainbusiness-signicanttimeintervals i.e. ,bidiurnalorweeklyload origincapacitiesoralongtracksegments.Webelievethatthemethodologyproposed inChapter2issufcientlyexibletobecustomizedbasedontheplanningneedsof differentrailroads.Wealsobelievethatthemodelsandsolutionideasthatwepresent hereareapplicabletootherrailroadproblemsandthereforecouldbringefciency, reliabilityandsubstantialcostsavingstovariousbusinessesofUSrailroads.This researchillustratesthattherailroadindustrystillhasmuchtobenetfromincorporating 131

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ORtoolsintothecomplexdecisionsystemsthatsupporttheirplanningandscheduling processes. Second,wehaveshowninChapter3thatthemodelsdescribedaboveare sufcientlyexibletoaccommodateavarietyofother,ner,practicalrequirements thatspecicrailroadsmayface.Inparticular,ourmodelwasimplementedbyour industrycollaboratorinadecisionsupporttoolcalledUTPS. Third,wehaveprovidednewinsightsonthestructureofthecertainowproblems thatarisefrequentlyduringthemodelingofrailroadapplications.Wehaveshownthat thetraditionalintegerprogrammingformulationoftheproblemisnotidealbutthatitcan bestrengthenedthroughtheadditionofsuitablevariablesorconstraints.Asystematic studyofsuchrequirementscouldleadtonewmethodologiesfortransportationproblems withcombinatorialrequirementsthatwouldallowtheexactsolutionoftheseproblems throughbranch-and-boundalgorithms.Currently,thesemodelsarenotsolvablethrough state-of-the-artcommercialsolversandneedtobesolvedwithheuristics. Thereareavarietyofresearchdirectionsthatstemfromtheworkpresentedinthis thesis.Onthepracticalside,aninterestingavenueofresearchistogeneralizeMCTRPP tootherunittraincommodities.Suchcommoditieshavedifferentreservationprocesses thatwouldrequirethatourcurrentmodelsbeadapted.Onthetheoreticalside,itwould beinterestingtostudyothercombinatorialrequirementsthatareoftenimposedon practicalowmodels.Forinstance,trainsareoftenrequiredtostaysufcientlylongat ayard.Inatime-spacenetwork,thiscorrespondstofollowapresetpathinthenetwork uponenteringspecicnodesofthenetwork.Determiningthebestwayofmodelingthis combinatorialrequirement,whichisageneralizationoftheno-splitconstruct,wouldbe ofbothpracticalandtheoreticalinterest. 132

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APPENDIXA CONVEXHULLOFSNUFPEXAMPLE Alineardescriptionoftheconvexhullofthefollowingsinglenodeunsplittableow problem ^ S k;l = 8 > > > > > > > < > > > > > > > : x;y 2 R 3 R 3 x 1 + x 2 + x 3 = y 1 + y 2 + y 3 x 1 3 ;x 2 7 ;x 3 13 y 1 5 ;y 2 11 ;y 3 13 nosplit x;y 9 > > > > > > > = > > > > > > > ; isgivenby A1 x 1 + x 2 + x 3 y 1 y 2 y 3 =0 A2x 2 0 A3x 3 0 A4y 1 0 A5y 2 0 A6y 3 0 A7 x 2 + x 3 y 1 y 2 y 3 0 A8x 2 x 3 + y 1 + y 2 + y 3 3 A9 y 1 5 A10 x 2 7 A112 x 2 2 x 3 + 5 y 1 15 A12 2 x 3 + y 2 33 A13 7 x 2 2 y 2 2 y 3 35 A14 5 x 2 2 x 3 + 5 y 1 50 A15 4 x 2 x 3 + 4 y 1 53 A164 x 3 + 11 y 2 77 A176 x 3 + 13 y 3 91 A188 x 2 8 x 3 + 20 y 1 + 11 y 2 93 133

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A198 x 2 2 x 3 + 8 y 1 + y 2 113 A2010 x 2 10 x 3 + 25 y 1 + 13 y 3 114 A212 x 2 8 x 3 + 20 y 1 11 y 2 123 A228 x 2 + 5 x 3 + 20 y 1 125 A23 13 x 3 2 y 3 143 A24 14 x 2 + 7 y 2 4 y 3 147 A25x 2 10 x 3 + 25 y 1 + 13 y 3 159 A26 10 x 2 4 x 3 + 10 y 1 + 11 y 2 177 A27 21 x 2 6 y 2 + 7 y 3 196 A28 28 x 2 + 13 x 3 8 y 2 8 y 3 205 A29 14 x 2 + 4 x 3 + 16 y 1 + y 2 205 A3044 x 2 56 x 3 + 77 y 2 231 A31 15 x 2 6 x 3 + 15 y 1 + 13 y 3 241 A3252 x 2 70 x 3 + 91 y 3 273 A334 x 2 + 28 x 3 + 7 y 2 385 A344 x 2 + 28 x 3 + 10 y 1 + 7 y 2 415 A35 56 x 2 + 26 x 3 11 y 2 16 y 3 445 A3632 x 2 + 20 x 3 + 80 y 1 + 5 y 2 515 A37 39 x 2 + 13 x 3 + 39 y 1 2 y 3 533 A3852 x 2 70 x 3 + 91 y 2 + 91 y 3 546 A3944 x 2 56 x 3 + 110 y 1 + 77 y 2 561 A4052 x 2 70 x 3 + 130 y 1 + 91 y 3 663 A41 143 x 3 78 y 2 88 y 3 715 A42104 x 2 140 x 3 + 203 y 2 + 182 y 3 1239 A4378 x 2 + 65 x 3 + 195 y 1 10 y 3 1300 A44 273 x 2 + 143 x 3 78 y 2 88 y 3 2080 A45220 x 2 280 x 3 + 385 y 2 + 364 y 3 2247 134

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Gorman,M.F.1998.Anapplicationofgeneticandtabusearchestothefreightrailroad operatingplanproblem. AnnalsofOperationsResearch 78 51. Gorman,M.F.,S.Harrod.2011.Operationsresearchapproachestoassetmanagement infreightrail. WileyEncyclopediaofOperationsResearchandManagementScience Gu,Z.,G.L.Nemhauser,M.W.P.Savelsbergh.1999.Liftedowcoverinequalitiesfor mixed0-1integerprograms. MathematicalProgramming 85 439. Harrod,S.,M.F.Gorman.2011.Operationsresearchforfreighttrainroutingand scheduling. WileyEncyclopediaofOperationsResearchandManagementScience Hu,Y.,J.Lan,C.Wan.2009.Analgorithmforunsplittableowprobleminexible recongurablenetwork. FrontierofComputerScienceandTechnology,2009. FCST'09.FourthInternationalConferenceon .IEEE,543. Jain,A.S.,S.Meeran.1999.Deterministicjob-shopscheduling:Past,presentand future. EuropeanJournalofOperationalResearch 113 390. Kolman,P.,C.Scheideler.2002.Improvedboundsfortheunsplittableowproblem. proceedingsofthethirteenthannualACM-SIAMsymposiumondiscretealgorithms SocietyforIndustrialandAppliedMathematics,184. Lai,Y.C.R.,M.H.Dingler,C.E.Hsu,P.C.Chiang.2010.Optimizingtrainnetwork routingwithheterogeneoustrafc. TransportationResearchRecord:Journalofthe TransportationResearchBoard 2159 69. Lawley,M.,V.Parmeshwaran,J.-P.P.Richard,A.Turkcan,A.Dalal,D.Ramcharan. 2008a.Atime-spaceschedulingmodelforoptimizingrecurringbulkrailcardeliveries. TransportationResearchPartB:Methodological 42 438. Lawley,M.,V.Parmeshwaran,J.-P.P.Richard,A.Turkcan,M.Dalal,D.Ramcharan. 2008b.Atimespaceschedulingmodelforoptimizingrecurringbulkrailcardeliveries. TransportationResearchPartB:Methodological 42 438. Liu,S.Q.,E.Kozan.2011.Optimisingacoalrailnetworkundercapacityconstraints. FlexibleServicesandManufacturingJournal 23 90. Nemani,A.K.,R.K.Ahuja.2011.Ormodelsinfreightrailroadindustry. WileyEncyclopediaofOperationsResearchandManagementScience Padberg,M.W.,T.J.VanRoy,L.A.Wolsey.1985.Validlinearinequalitiesforxedcharge problems. OperationsResearch 33 842. Roy,T.J.Van,L.A.Wolsey.1986.Validinequalitiesformixed0-1programs. Discrete AppliedMathematics 14 199. Sherali,H.D.,A.B.Suharko.1998.Atacticaldecisionsupportsystemforemptyrailcar management. TransportationScience 32 306. 137

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Thalji,J.2012.Portoftampapreparestounveilnewrailroadlinetoboost cargo,ethanolmarkets. http://www.tampabay.com/news/business/ port-of-tampa-prepares-to-unveil-new-railroad-line-to-boost-cargo-ethanol/ 1253265 .Accessed:03/11/2013. Ziegler,G.M.1995. LectureonPolytopes ,vol.152.Springer. 138

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BIOGRAPHICALSKETCH IlksenEceIcyuzreceivedherPh.D.fromtheDepartmentofIndustrialandSystems EngineeringattheUniversityofFlorida.SheworkedunderthesupervisionofDr. Jean-PhilippeP.Richardontransportationproblemsarisinginrailroadsindustry.Her researchfocusesondevelopingefcientheuristicmethodsforlarge-scaleplanning andschedulingproblems,andonthepolyhedralanalysisofnetworkowproblems withpracticalcombinatorialrestrictions.DuringherPh.D.studies,shealsoperformed aninternshipatCSXTransportation,inJacksonville,FL,whereshecontributedinthe developmentofadecisionsupporttoolformonthlyunittrainscheduling. ShereceivedherB.S.degreefromtheDepartmentofManufacturingSystems EngineeringatSabanciUniversity,Turkeyin2008.TwoyearsafterenteringthePh.D. programattheUniversityofFlorida,shereceivedherM.Sc.degreeinindustrialand systemsengineeringin2010. 139