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Optimization Models and Algorithms for Solving Large-Scale Network Design, Routing and Scheduling Problems

Permanent Link: http://ufdc.ufl.edu/UFE0041977/00001

Material Information

Title: Optimization Models and Algorithms for Solving Large-Scale Network Design, Routing and Scheduling Problems
Physical Description: 1 online resource (100 p.)
Language: english
Creator: Bog, Suat
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: consolidation, design, freight, network, optimization, routing, scheduling, service, transportation
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: Freight transportation is an important component of economy and constitutes largest portion of the logistics cost. Decisions regarding freight transportation have a high impact on customer service levels, economic efficiency and competitiveness of the firms. In this dissertation, we study network design, routing and scheduling problems arising in freight transportation industry. We focus on optimization problems faced mainly by consolidation-based freight carriers. The emphasis is given on developing computationally efficient optimization models and algorithms which capture all the real life complexities of studied problems and result into near-optimal solutions in reasonable time limits. We first consider transportation network disruption problem encountered by carriers for maintenance of their tangible assets. In this context, we study the Curfew Planning Problem (CPP) encountered by rail carriers for the maintenance of railway tracks. Maintenance regions which are under absolute curfew cause a complete blockage of the rail traffic. While providing high customer service levels, railroads need to perform maintenance on their tracks with minimum possible disruptions in transportation network. We propose four iterative algorithms that decompose the problem into efficient smaller integer programming models working on shorter time horizons. These models are flexible to run with dynamically changing data sets throughout iterations. Hence, one can enter a partial maintenance schedule and the algorithm can complete the rest of the schedule. Proposed algorithms are applied on the whole rail network, for the complete yearly planning horizon and can capture all the real life complexities affecting the implementability of the resulting solution. They provide a successful way for handling complicating decisions and constraints implicitly. We tested our algorithms on real life instances from a major North American Railroad company and obtained very good solutions in practical time limits. Next, we consider integrated transportation planning and propose a novel model to solve combined network design and commodity routing problem. In this problem, our emphasis is on non-bifurcated flow of commodities where each commodity should flow on a single path from its origin to its destination. Non-bifurcated flow arises frequently in freight transportation for consolidation-based carriers such as less-than-truckload trucking, express package delivery and railway freight routing. Earlier methods mostly solved network design and commodity routing problems separately in a sequential manner for bifurcated network design problems. We propose a holistic approach to solve these highly interrelated problems in an integrated manner. The proposed model involves binary path based design and path based flow variables. Traditional network design models in the literature use arc based design variables, and hence they are not suitable for incorporating asset management related constraints into the model. We adapted our generic model on an instance obtained from a major railroad company to solve combined train routing and block-to-train assignment problem. The model could handle many asset management related constraints required by the railroad and it resulted in good quality solutions within reasonable time limits. Finally, we consider service network design problem on hub-and-spoke networks. We propose a novel network shrinking based decomposition which allows the generation of a smaller time space network of hub to hub connections. In this three phased decomposition, we apply all three types of consolidation. In the first phase of the decomposition, we perform facility based consolidation on a space network where we sort and consolidate shipments for their next location. In the second phase, we perform temporal consolidation by holding shipments in hubs to be able to generate larger shipments and in the third phase, we perform multi-stop consolidation to improve direct non-hub to hub and hub to non-hub connections. This generic approach can be utilized by all consolidation-based carriers operating on hub-and-spoke networks, such as express package delivery, less-than-truckload service providers, freight rail carriers, etc. We could solve a fairly large scale practical problem using the proposed decomposition scheme. We applied our algorithm on a real life instance for a less-than-truckload motor carrier and obtained considerable improvements in transportation costs and load capacity utilizations in a reasonable time limit.
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 Suat Bog.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ahuja, Ravindra K.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041977:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041977/00001

Material Information

Title: Optimization Models and Algorithms for Solving Large-Scale Network Design, Routing and Scheduling Problems
Physical Description: 1 online resource (100 p.)
Language: english
Creator: Bog, Suat
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: consolidation, design, freight, network, optimization, routing, scheduling, service, transportation
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: Freight transportation is an important component of economy and constitutes largest portion of the logistics cost. Decisions regarding freight transportation have a high impact on customer service levels, economic efficiency and competitiveness of the firms. In this dissertation, we study network design, routing and scheduling problems arising in freight transportation industry. We focus on optimization problems faced mainly by consolidation-based freight carriers. The emphasis is given on developing computationally efficient optimization models and algorithms which capture all the real life complexities of studied problems and result into near-optimal solutions in reasonable time limits. We first consider transportation network disruption problem encountered by carriers for maintenance of their tangible assets. In this context, we study the Curfew Planning Problem (CPP) encountered by rail carriers for the maintenance of railway tracks. Maintenance regions which are under absolute curfew cause a complete blockage of the rail traffic. While providing high customer service levels, railroads need to perform maintenance on their tracks with minimum possible disruptions in transportation network. We propose four iterative algorithms that decompose the problem into efficient smaller integer programming models working on shorter time horizons. These models are flexible to run with dynamically changing data sets throughout iterations. Hence, one can enter a partial maintenance schedule and the algorithm can complete the rest of the schedule. Proposed algorithms are applied on the whole rail network, for the complete yearly planning horizon and can capture all the real life complexities affecting the implementability of the resulting solution. They provide a successful way for handling complicating decisions and constraints implicitly. We tested our algorithms on real life instances from a major North American Railroad company and obtained very good solutions in practical time limits. Next, we consider integrated transportation planning and propose a novel model to solve combined network design and commodity routing problem. In this problem, our emphasis is on non-bifurcated flow of commodities where each commodity should flow on a single path from its origin to its destination. Non-bifurcated flow arises frequently in freight transportation for consolidation-based carriers such as less-than-truckload trucking, express package delivery and railway freight routing. Earlier methods mostly solved network design and commodity routing problems separately in a sequential manner for bifurcated network design problems. We propose a holistic approach to solve these highly interrelated problems in an integrated manner. The proposed model involves binary path based design and path based flow variables. Traditional network design models in the literature use arc based design variables, and hence they are not suitable for incorporating asset management related constraints into the model. We adapted our generic model on an instance obtained from a major railroad company to solve combined train routing and block-to-train assignment problem. The model could handle many asset management related constraints required by the railroad and it resulted in good quality solutions within reasonable time limits. Finally, we consider service network design problem on hub-and-spoke networks. We propose a novel network shrinking based decomposition which allows the generation of a smaller time space network of hub to hub connections. In this three phased decomposition, we apply all three types of consolidation. In the first phase of the decomposition, we perform facility based consolidation on a space network where we sort and consolidate shipments for their next location. In the second phase, we perform temporal consolidation by holding shipments in hubs to be able to generate larger shipments and in the third phase, we perform multi-stop consolidation to improve direct non-hub to hub and hub to non-hub connections. This generic approach can be utilized by all consolidation-based carriers operating on hub-and-spoke networks, such as express package delivery, less-than-truckload service providers, freight rail carriers, etc. We could solve a fairly large scale practical problem using the proposed decomposition scheme. We applied our algorithm on a real life instance for a less-than-truckload motor carrier and obtained considerable improvements in transportation costs and load capacity utilizations in a reasonable time limit.
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 Suat Bog.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ahuja, Ravindra K.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041977:00001


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OPTIMIZATIONMODELSANDALGORITHMSFORSOLVINGLARGE-SCALE NETWORKDESIGN,ROUTINGANDSCHEDULINGPROBLEMS By SUATBO G ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010SuatBog 2

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Dedicatedtomyparents(BingulandYakup), mywife(Gozde), andmybrother(Murat) 3

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ACKNOWLEDGMENTS Firstofall,IwouldliketothankmyadvisorDr.RavindraK.Ah ujaforhissupport andguidance.Hewasagreatmentor.Hegavemeenoughfreedomto workonmy researchbuthewasalsoavailabletoguidemewhenIneededhe lp.Ialsowanttothank himforopportunitiesheprovided. IwouldliketothankDr.J.ColeSmith,Dr.PanosM.Pardalosa ndDr.Sanjay Rankafortheirwillingnesstoserveonmycommittee,forthe irtimeandvaluable feedback.IalsowanttothankDr.JosephGeunesforhissuppo rt,smilingfaceand hisendlesshumorsinouroce. IwanttothankmyfriendYusufwithwhomIstartedthisjourne ywith.Hehadbeen agreatroommate.WithhimthersttwoyearsofmyPh.D.hadbe enveryenjoyable. IwouldliketoacknowledgemyfriendandcolleagueGoncafor hersupportandfor numerousdelightfulconversationswehadonalmosteveryth ing. Iamgratefultomyparentsandmybrotherwhoalwayssupporte dme.Finally,I wanttothankmybelovedwifeGozde.Shealwayssupportedan dencouragedme.Shewas awakeandalwayswithmewhenIwasworkinglate.Hertrust,con denceandsupport gavemegreatmotivationandenergytocompletethiswork. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 12 1.1BackgroundandMotivation .......................... 12 1.1.1Consolidation-basedFreightTransportation .............. 13 1.1.2NetworkDesignModels ......................... 13 1.1.3TransportationPlanningLevels .................... 14 1.2ContributionsandOverview .......................... 14 1.2.1TransportationNetworkDisruption .................. 15 1.2.2IntegratedTransportationPlanning .................. 16 1.2.3ServiceNetworkDesign ......................... 17 2ITERATIVEALGORITHMSFORTHECURFEWPLANNINGPROBLEM 20 2.1Introduction ................................... 20 2.2ProblemDescription .............................. 22 2.2.1ProblemInputs ............................. 24 2.2.2ProblemConstraints .......................... 25 2.2.2.1Performanceconstraints ................... 25 2.2.2.2Feasibilityconstraints .................... 26 2.3SolutionApproaches .............................. 26 2.3.1PartitioningtheProjects ........................ 27 2.3.2SchedulingJamboreeProjects ..................... 27 2.3.31-WeeklyAlgorithm ........................... 28 2.3.4 k -WeeklyAlgorithm ........................... 36 2.3.5Backtracking ............................... 40 2.3.5.11-weeklyalgorithmwithbacktracking ............ 41 2.3.5.2 k -weeklyalgorithmwithbacktracking ............ 42 2.4ComputationalTests .............................. 42 2.4.1TestingwithRealLifeInstances .................... 43 2.4.2ComparisonofAlgorithms ....................... 44 2.4.3TestingwithSmallInstances ...................... 46 2.5SummaryandConclusions ........................... 50 5

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3COMBINEDNETWORKDESIGNANDCOMMODITYROUTINGPROBLEM 52 3.1Introduction ................................... 52 3.2ProblemDescription .............................. 55 3.3TraditionalBifurcatedMulticommodityNetworkDesignM odels ...... 56 3.4Non-bifurcatedNetworkDesignProblem ................... 58 3.5ProposedNon-bifurcatedNetworkDesignModel ............... 59 3.6SummaryandConclusions ........................... 61 4TRAINROUTINGANDBLOCK-TO-TRAINASSIGNMENTPROBLEMS .. 62 4.1Introduction ................................... 62 4.2ProblemDescription .............................. 63 4.3SolutionApproach ............................... 66 4.3.1GenericModelTailoredfortheIntegratedProblem ......... 66 4.3.2IterativeTrainConstructionApproach ................ 71 4.4ComputationalExperience ........................... 72 4.5SummaryandConclusions ........................... 72 5SERVICENETWORKDESIGNPROBLEMONHUB-AND-SPOKENETWORK 74 5.1Introduction ................................... 74 5.2ProblemDescription .............................. 77 5.2.1ProblemInputs ............................. 78 5.2.2ProblemConstraints .......................... 78 5.2.3ProblemOutputs ............................ 79 5.3Decomposition-BasedSolutionApproach ................... 79 5.3.1Phase1:FacilityConsolidation .................... 79 5.3.2Phase2:TemporalConsolidation ................... 82 5.3.2.1Networkshrinkingapproach ................. 83 5.3.2.2Spacetimemodelfortheshrunknetwork .......... 84 5.3.3Phase3:Multi-stopConsolidation ................... 86 5.4ComputationalExperience ........................... 89 5.5SummaryandConclusions ........................... 91 REFERENCES ....................................... 93 BIOGRAPHICALSKETCH ................................ 100 6

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LISTOFTABLES Table page 2-11-weeklyiterativesteps ................................ 32 2-2Sizeofthe2007and2008instances ......................... 43 2-3Numberofviolationsdetectedinthesolutionsimplement edbytherailroad ... 43 2-4Comparisonofviolations:nobacktracking,1versus k weekly(2007) ....... 44 2-5Comparisonofviolations:nobacktracking,1versus k weekly(2008) ....... 44 2-6Comparisonofviolations:withbacktracking,1versus k weekly(2007) ..... 45 2-7Comparisonofviolations:withbacktracking,1versus k weekly(2008) ..... 45 2-8Averageruntimesofthetopvesolutionswithrespecttot otalviolations .... 46 2-9Computationalresultswithsmallinstances:planningh orizon5weeks ...... 49 2-10Progressofexactoptimizationmodel ........................ 50 4-1Percentageimprovementsofproposedmodelanditerativ eapproach ....... 73 4-2Comparisonofruntimesforproposedmodelanditerative approach ....... 73 5-1Comparisonofincrementalandzero-basedsolutionswit hLTLcarrier'ssolution 91 5-2Optimalitygapandtotalrunningtimesforeachphaseofp roposedalgorithm .. 91 7

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LISTOFFIGURES Figure page 2-11-weeklyalgorithm .................................. 36 2-2 k -weeklyalgorithm .................................. 40 2-31-weeklyalgorithmwithbacktracking ........................ 42 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy OPTIMIZATIONMODELSANDALGORITHMSFORSOLVINGLARGE-SCALE NETWORKDESIGN,ROUTINGANDSCHEDULINGPROBLEMS By SuatBog August2010 Chair:RavindraK.AhujaMajor:IndustrialandSystemsEngineering Freighttransportationisanimportantcomponentofeconom yandconstituteslargest portionofthelogisticscost.Decisionsregardingfreight transportationhaveahigh impactoncustomerservicelevels,economiceciencyandco mpetitivenessoftherms. Inthisdissertation,westudynetworkdesign,routingands chedulingproblemsarising infreighttransportationindustry.Wefocusonoptimizati onproblemsfacedmainlyby consolidation-basedfreightcarriers.Theemphasisisgiv enondevelopingcomputationally ecientoptimizationmodelsandalgorithmswhichcapturea llthereallifecomplexitiesof studiedproblemsandresultintonear-optimalsolutionsin reasonabletimelimits. Werstconsidertransportationnetworkdisruptionproble mencounteredbycarriers formaintenanceoftheirtangibleassets.Inthiscontext,w estudytheCurfewPlanning Problem(CPP)encounteredbyrailcarriersforthemaintena nceofrailwaytracks. Maintenanceregionswhichareunderabsolutecurfewcausea completeblockageof therailtrac.Whileprovidinghighcustomerservicelevel s,railroadsneedtoperform maintenanceontheirtrackswithminimumpossibledisrupti onsintransportationnetwork. Weproposefouriterativealgorithmsthatdecomposethepro blemintoecientsmaller integerprogrammingmodelsworkingonshortertimehorizon s.Thesemodelsarerexible torunwithdynamicallychangingdatasetsthroughoutitera tions.Hence,onecan enterapartialmaintenancescheduleandthealgorithmcanc ompletetherestofthe schedule.Proposedalgorithmsareappliedonthewholerail network,forthecomplete 9

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yearlyplanninghorizonandcancaptureallthereallifecom plexitiesaectingthe implementabilityoftheresultingsolution.Theyprovidea successfulwayforhandling complicatingdecisionsandconstraintsimplicitly.Wetes tedouralgorithmsonreallife instancesfromamajorNorthAmericanRailroadcompanyandobt ainedverygood solutionsinpracticaltimelimits. Next,weconsiderintegratedtransportationplanningandpr oposeanovelmodelto solvecombinednetworkdesignandcommodityroutingproble m.Inthisproblem,our emphasisisonnon-bifurcatedrowofcommoditieswhereeach commodityshouldrowon asinglepathfromitsorigintoitsdestination.Non-bifurca tedrowarisesfrequentlyin freighttransportationforconsolidation-basedcarriers suchasless-than-truckloadtrucking, expresspackagedeliveryandrailwayfreightrouting.Earl iermethodsmostlysolved networkdesignandcommodityroutingproblemsseparatelyi nasequentialmannerfor bifurcatednetworkdesignproblems.Weproposeaholistica pproachtosolvethesehighly interrelatedproblemsinanintegratedmanner.Thepropose dmodelinvolvesbinarypath baseddesignandpathbasedrowvariables.Traditionalnetw orkdesignmodelsinthe literatureusearcbaseddesignvariables,andhencetheyar enotsuitableforincorporating assetmanagementrelatedconstraintsintothemodel.Weada ptedourgenericmodelon aninstanceobtainedfromamajorrailroadcompanytosolvec ombinedtrainroutingand block-to-trainassignmentproblem.Themodelcouldhandle manyassetmanagement relatedconstraintsrequiredbytherailroadanditresulte dingoodqualitysolutionswithin reasonabletimelimits. Finally,weconsiderservicenetworkdesignproblemonhuband-spokenetworks.We proposeanovelnetworkshrinkingbaseddecompositionwhic hallowsthegeneration ofasmallertimespacenetworkofhubtohubconnections.Int histhreephased decomposition,weapplyallthreetypesofconsolidation.I ntherstphaseofthe decomposition,weperformfacilitybasedconsolidationon aspacenetworkwhere wesortandconsolidateshipmentsfortheirnextlocation.I nthesecondphase,we 10

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performtemporalconsolidationbyholdingshipmentsinhub stobeabletogeneratelarger shipmentsandinthethirdphase,weperformmulti-stopcons olidationtoimprovedirect non-hubtohubandhubtonon-hubconnections.Thisgenerica pproachcanbeutilized byallconsolidation-basedcarriersoperatingonhub-andspokenetworks,suchasexpress packagedelivery,less-than-truckloadserviceproviders ,freightrailcarriers,etc.Wecould solveafairlylargescalepracticalproblemusingthepropo seddecompositionscheme.We appliedouralgorithmonareallifeinstanceforaless-than -truckloadmotorcarrierand obtainedconsiderableimprovementsintransportationcos tsandloadcapacityutilizations inareasonabletimelimit. 11

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CHAPTER1 INTRODUCTION 1.1BackgroundandMotivation Freighttransportationisoneofthekeyactivitiesofalogi sticssystemandamajor sectorcontributingtoeconomy.Decisionsregardingfreig httransportationaectall supplychainplayers,costofnalproducts,networkdesign ,locationoffacilities, resourceallocationsandutilizations,customerservicel evels,economiceciencyand competitivenessinthemarket.Freighttransportationcos tsconstitutethelargestportion oflogisticscostsinUSoverthelasttwodecadesaccordingto AnnualStateofLogistics ReportsreleasedbytheCouncilofSupplyChainManagementP rofessionals(CSCMP). 20thAnnualReportreleasedin2009indicatesthattotalUSlog isticscostswas$1.3 trillionin2008andfreighttransportationcostsrepresen t64%oftotallogisticscosts whichisequivalentto6.1%ofGrossDomesticProduct(GDP)( CSCMP 2009 ).In2008, transportationactivitiescontributedto9.5%ofGDP( USDOTRITABTS 2010d )and totalemploymentintransportationsectorwas9.7%oftotal USlaborforce( USDOT RITABTS 2010c ).Freighttransportationisahighlydynamiceld.Transpo rtation activitiesareaectedbyvariousfactorssuchasgrowthord eclineineconomicactivity, globalization,energyprices,infrastructurecapacity,e nvironmentalconcerns,changes inregulations,advanceoftechnology,internetshopping, andjust-in-timeinventory management( Crainic 2003 ; USDOTRITABTS 2010a ).Typicallyfreighttransportation industryrequiresalargeinvestmentincapitalassets.Tra nsportationcompaniesneed tomakecomplexsetofinterrelateddecisionswhichhavetra de-osamongeachother. Tomanagetheirassetseectively,inacostecientwayandt omaintainhighcustomer servicelevels,theyneedtoplanandoptimizetheirtranspo rtationactivities.Otherwise,it mightnotbepossibletostaycompetitiveandmakeprotinth isdynamic,complexand demandingindustry. 12

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Therearevemajortransportationmodesusedtomovegoodsb etweensupplyand demandpoints:trucking,watertransportation,railtrans portation,airtransportation, andpipelines.In2007,railtransportationandtruckingac countedfor39.5%and28.6% oftotalton-milesofdomesticfreightrespectively.Theyc onstitutethemostcommonly usedtransportationmodesandarefollowedbypipelines,wa tertransportationandair transportation.Comparedto1990,ton-milesofdomesticfr eightforrailandtrucking increasedby71%and55.2%respectively( USDOTRITABTS 2010b ). 1.1.1Consolidation-basedFreightTransportation Consolidationisacommonmethodusedtoobtainconsiderabl ecostsavingsin freighttransportation.Theprocesstakesadvantageofeco nomiesofscaleprincipleby consolidatingsmallshipmentsintolargerones.Consolida tion-basedfreighttransportation isappliedbyvariousserviceproviderssuchasless-than-t ruckload(LTL)motorcarriers, postalservices,railways,shippinglines,etc.Therearet hreemajortypesofconsolidation: Facility,temporalandmulti-stop.Infacilityconsolidat ion,inboundshipmentsaresorted andconsolidatedtoformoutboundshipmentsthataremovedj ointlytoanotherhub.In temporalconsolidation,shipmentsareholdandaggregated overtimetobeabletoship largeshipments.Multi-stopconsolidationisusedforpick -upanddeliveryroutes.Several customersareservedtogetheronapick-upordeliveryroute insteadofusingoneshipment foreachcustomer.1.1.2NetworkDesignModels Planningproblemsforfreighttransportationareoftenexp ressedusingnetwork designmodels.Thesemodelsareusuallydierentextension sofgenericmulticommodity capacitatednetworkdesign(MCND)formulationthathasmany applicationsintransportation, telecommunication,energy,computer,andproduction-dis tributionsystems( Balakrishnan etal. 1997 ; Magnanti&Wong 1984 ; Minoux 1989 ).Infreighttransportationcontext, networkdesignmodelsarefrequentlyusedtoconstructandi mprovenetworks,build serviceroutesandschedules,andallocateresourcestojob s.Collectionoftheseinterrelated 13

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problemsiscalledasservicenetworkdesignintheliteratu re.Foralltypeofcarriers, designingareliable,timeandcostecientservicenetwork iscrucialinordertooperate protablyandmaintainhighcustomerservicelevels. Crainic ( 2000 )makesagood reviewofnetworkdesignmodels,relevantsolutionapproac hesandservicenetworkdesign formulationsinfreighttransportation.Inarecentsurvey Wieberneit ( 2008 )reviews dierentformulationsandsolutionframeworksforservice networkdesignproblems. 1.1.3TransportationPlanningLevels Planningandoptimizationproblemsthatneedtobeaddresse dbyfreighttransportation managerscanbeclassiedaccordingtothreeplanninglevel s:Strategical,tacticaland operationallevelplanning( Crainic 2000 ; Crainic&Laporte 1997 ).Strategicalplanning includeslongtermdecisionsandrequireslargecapitalinv estments.Decisionsrelatedto ownershipofresourcesandlocationsofthefacilitiesaret ypicalexamplesofstrategical planning.Tacticalplanningdecisionsarerelatedtooptim alutilizationofresourcesover amediumtermhorizon.Servicenetworkdesignisanexampleo ftacticalplanninglevel andincludesdecisionsregardingserviceselection,shipm entrouting,repositioningof emptyvehiclesandconsolidationworksatterminals.Opera tionalplanningdecisions aregivenforashorttermhorizoninadynamicsetting.Onthi slevel,localmanagers anddispatchersperformadjustmentsonthetacticalplansb ycontrollingserviceand maintenanceschedules,routinganddispatchingofvehicle sandcrews. 1.2ContributionsandOverview Transportationofresourcesisthedrivingcomponentofthe problemswestudy.These problemsaremostlyNP-hardproblems,andforlarge-scaleap plicationsitisdicultto ndoptimalsolutionsorevenfeasiblesolutions.Forthese typesofinstances,wecombine mixedintegerprogrammingwithnetworkoptimizationandhe uristictechniquesinnovel ways.Suchhybridalgorithmstakeadvantageofbothexactan dheuristicmethodologies. Inthisdissertation,wecontributetoliteraturebydesign ingcomputationallyecient algorithmsthatprovidenear-optimalsolutionsinreasona bletimelimits.Mainapplication 14

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areaofourresearchhasbeenfreighttransportationnetwor ks.Wehaveconducted researchinthreeimportantareasrelatedtofreighttransp ortation:transportationnetwork disruption,integratedtransportationplanningfornon-b ifurcatednetworkdesignproblems andservicenetworkdesignonhub-and-spokenetworks.Belo wwehighlightsomeofour maincontributionsineachoftheseeldsofstudy.1.2.1TransportationNetworkDisruption Freighttransportationindustryrequiresalargeinvestme ntincapitalassets.While operatinginareliableandcostecientway,carriershavet operformregularmaintenance activitiesfortheirexpensiveresources.Theyneedtosche dulerequiredmaintenance activitiessuchthattransportationnetworkdisruptionsa reatminimumpossiblelevels. Inthiscontext,westudytheCurfewPlanningProblem(CPP)e ncounteredbyrail carriersforthemaintenanceofrailwaytracks.Inrailroad terminology,aregionisunder absolutecurfewifamaintenanceprojectcausesacompleteb lockageoftherailtrac. Whileprovidingsatisfactoryservicelevels,railroadsmu stperformmaintenanceontheir trackswithoutcausingdisruptionsintrainschedules.The CPPistodesignanannual timetabletocompleteagivensetofrepairsandreplacement jobs(rail-workandtie-work) ontherailwaytracksforasetofteamsspecializedinrail-w ork(railteam)ortie-work (tieteam).Wedeveloptheworkscheduleforeachteamsuchth atthedisruptionsintrain routesareminimized.Qualityandimplementabilityofasol utiondependshighlyonthe curfewrelatedperformanceconstraints.Wepublishedtwop apersonthisproblem.In ourrstpaper Bogetal. ( 2010 ),wedevelopednoveliterativedecompositionalgorithms tominimizethenumberofviolationsinperformanceconstra ints.Previousmethods developedforthisproblemareappliedonasingletrackofth ewholerailnetwork,fora shorttermhorizon(aweek)andaremostlyusefultomodifyan existingtimetable(seee.g. Budaietal. 2006 ; Higginsetal. 1999 ; Lake&Ferreira 2002 ).In Bogetal. ( 2010 ),we proposedfouriterativealgorithmstominimizepossiblera iltracdisruptions.Wetested ouralgorithmsonreallifeinstancesfromamajorNorthAmeric anRailroadcompany 15

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andobtainedverygoodsolutionsinpracticaltimelimits.O urmethodwasappliedto thewholetrainnetworkforalongtermhorizon(ayear).Weco nsideredalmostallof thereal-lifeconstraintsaectingtheimplementabilityo ftheresultingsolution.The proposediterativealgorithmsarerexibleinthesensethat userscanprovideapartial scheduleandalgorithmcanassigntherestoftheschedule.I noursecondpaper Nemani etal. ( 2010 ),wepresentedfourdierentsolutionapproachesfortheCP P:(i)time-space networkmodel,(ii)duty-generationmodel,(iii)column-g enerationmodel,and(iv) decomposition-baseddutygenerationheuristics.Thispap errepresentsourthoughtprocess andallthedevelopedmethodsinanorderedwayduringthesol utionoftheCPP.Thelast solutionapproach(decomposition-baseddutygenerationh euristics)takesinsightsfrom theiterativealgorithmspresentedintherstpaper.Itist hebestapproachforgenerating solutionsfromscratch.Iterativeapproachisthebestfori ncrementaloptimization.Inthis paper,wereducedperformanceconstraintviolationsby75% andincreasedcrewworklife qualitybyreducingtraveldistancealongtheyearby15%.Wi ththesetwostudies,wewon asecondplaceawardinINFORMSstudentpapercompetitiononM anagementSciencein RailroadApplicationsin2009.1.2.2IntegratedTransportationPlanning LiteratureonMulticommodityCapacitatedNetworkDesign(M CND)problems mostlyfocusonbifurcatedrowofcommodities(seee.g. Crainicetal. 2000 ; Ghamlouche etal. 2003 ; Holmberg&Yuan 2000 ; Magnantietal. 1993 ).Inbifurcatedcase,a commoditycanrowonseveralpathsfromitsorigintoitsdest ination.Inthisstudy, wefocusonnon-bifurcatedrowofcommoditieswhereacommod itycanrowonasingle pathfromitsorigintoitsdestination.Wecancategorizeou rproblemasNon-bifurcated MulticommodityCapacitatedNetworkDesign(NMCND)problemwh ichbelongstothe generalclassofMCNDproblems.NMCNDproblemarisesinmanyrea llifesystems suchascomputernetworks,telecommunicationnetworks,fr eighttransportationin consolidation-basedcarriers(less-than-truckloadtruc king,expresspackagedeliveryand 16

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railwayfreightrouting).Theprobleminvolvestwoclosely interrelateddecisions:network designandcommodityrouting.Manystudiesintheliteratur edividethisprobleminto twoseparatestageswhicharesolvedseparatelyinasequent ialmanner.Inthisstudy, weproposeanovelmathematicalmodelwhichecientlysolve sthesetwoproblemsin anintegratedmanner.Theproposedmodelinvolvesbinarypa thbaseddesignandpath basedrowvariables.Traditionalnetworkdesignmodelsint heliteratureusearcbased designvariables,andhencetheyarenotsuitableforincorp oratingassetmanagement relatedconstraintsintothemodel.Wetestedourmodelonan instanceobtainedfroma majorrailroadcompanytosolvecombinedtrainroutingandb lock-to-trainassignment problem.Themodelcouldhandlemanyassetmanagementrelat edconstraintsrequiredby therailroadanditresultedingoodqualitysolutionswithi nreasonabletimelimits. 1.2.3ServiceNetworkDesign Bringingthefreightattherighttime,totherightplaceisv erycriticalforcarriers. Mostcarriersannouncetheirservicecommitmentsandprovi destrictdeliverytimes fortheircustomers.Inthisstudy,wefocusonservicenetwo rkdesignproblemof consolidation-basedfreightcarrierswhichoperateonhub -and-spokenetworks.Inputs oftheproblemareservicecommitmentsandnetworkoftermin allocationswhichinclude hublocationsandend-of-lineterminals.Hub-and-spokenet worksarefrequentlyutilized tosolveconsolidationproblemswhere,insteadofsendinge achshipmentdirectlytoits destination,shipmentsarecombinedintoloadsandroutedt hroughhubs.Thegoalisto installloadsonthelinksofthegiventhenetworkanddecide onroutesforeachshipment suchthattotaltransportationcosts(totalmileagecosts) areminimized.Eachrouteis asequenceofloadsashipmentshouldtaketotravelfromitso rigintodestination.We assumethatshipments'demandquantitiesarenotlargeenou ghtollloadcapacities. Onceashipmentisloadedfromitsorigin,ittravelshubloca tionsandnallyreachesits destinationterminal.Forthisproblem,weproposeadecomp ositionapproachbasedonan innovativenetworkshrinkingidea.Proposedapproachisne ededbyallconsolidationbased 17

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carriers,suchasexpresspackagedelivery,less-than-tru ckload(LTL)serviceproviders, freightrailcarriers,etc.Usingtheproposeddecompositio nscheme,wesolvedfairlylarge scalepracticalproblemofanLTLcarrierandwereabletodec reaseweeklycostsofthe carrierby15%andincreasetheirloadcapacityutilization by5%. Theoutlineofthisdissertationisasfollows.InChapter 2 ,wedevelopfouriterative algorithmstosolvethecurfewplanningproblem(CPP)encou nteredbyrailroads.The goaloftheproblemistoschedulemaintenanceofrailwaytra ckssuchthatpossible transportationnetworkdisruptionsareminimized.Wesimp lifytheoriginalproblem byrstsolving1-weeklymodelsiteratively.Wethenextend thisapproachbysolving k -weeklymodelsandusingbacktrackingidea.Backtrackingh elpstohavemorefar-sighted approachbysolvingformultiple k -weeklyor1-weeklyperiodsandmodifyingthecurrent period'ssolutionifapossiblefutureviolationisforesee n. InChapter 3 ,wedescribethenon-bifurcatednetworkdesignmodelwehav e developedforsolvingnetworkdesignandcommodityrouting problemssimultaneously. Non-bifurcatedrowofcommoditieswhereeachcommodityrows onasinglepathis frequentlyobservedinfreighttransportationnetworks.W erstgiveanoverviewof bifurcatednetworkdesignmodelsintheliterature,thenwe explaindetailsofourproposed non-bifurcatednetworkdesignmodel. InChapter 4 ,westudycombinedtrainroutingandblock-to-trainassign ment problemswhichconstituteanimportantpartindevelopinga railroad'soperatingplan. Weillustratehowweadaptourgenericnon-bifurcatednetwo rkdesignmodelforthesetwo highlyinterrelatedproblems.Wealsoextendthebasicmode lbyincorporatingmanyreal lifeconstraintstogenerateanimplementablesolution.Co mputationaltestsonareallife instanceshowtheeectivenessofourapproachoveranitera tivemethodandrailroad's solution. InChapter 5 ,wefocusonservicenetworkdesignproblemonhub-and-spok enetworks. Weproposeanoveldecompositionapproachbasedonanetwork shrinkingidea.We 18

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providedetailsofourthreephaseddecompositionschemewh ichiscomposedoffacility consolidation,temporalconsolidationandmulti-stopcon solidationphases.Weillustrate ourcomputationalexperienceonareallifeinstanceforale ss-than-truckloadmotor carrier. 19

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CHAPTER2 ITERATIVEALGORITHMSFORTHECURFEWPLANNINGPROBLEM 2.1Introduction RailroadsarevitaltoAmerica'seconomicalpowerandcompet itiveness,moving40 percentofthenation'sfreight(inton-miles).Freightrai lroadsspendmorethan20billion dollarseachyearfortrackandequipmentmaintenance,rene walandexpansion.In2006, freightrailroadsintheUnitedStateshad54billiondollars oftotalrevenue( Association ofAmericanRailroads 2008a ).Theinvestmentandrevenuesareexpectedtogrowsince itisprojectedthatdemandforfreightrailroadwillrise87 .6percentby2035comparedto 2002levels( U.S.DepartmentofTransportation 2007 ).Althoughcapitalinvestmentand revenuesareveryhigh,railindustrylagsbehindmostindus triesintermsofprotability ( AssociationofAmericanRailroads 2008b ).Consideringthepotentialsavingsand performanceimprovements,allocatingandutilizingresou rcesinanecientandtimely mannercanhelpimprovetheprotability.Possiblenetwork disruptionsduetoinecient maintenanceschedulingofrailwaytracksmayresultintens ofmillionsofdollarsinlost revenues.MaintenanceschedulingplannersinamajorNorthAm ericanrailroadcompany describethecurfewplanningproblem(CPP)asoneofthemost importantanddicult problems.TheyalsomentionthattheCPPiscurrentlybeings olvedeachyearmanually byagroupofvetosixmaintenanceschedulingplanners.Bui ldingaschedulefrom scratchtakesplannersabouttwoweeks.Railroadsneedadec isionsupportsystemthat facilitatesthecreationofanannualmaintenancetimetabl efortheirteams.Maintenance schedulingproblemalsoarisesinavarietyofindustries.T ypicalapplicationareasinclude aircraft,vehiclereet,powergeneration,pavement,highw ay,reneryandproduction facilities.Asurveyonmaintenanceschedulingliterature canbefoundin Oke ( 2004 ). Thereareanumberofstudiesthatconsidertheimportanceof minimizingrailtrac disruptions.Theconrictbetweenrailoperationsandraili nfrastructuremaintenance isemphasizedin Lake&Ferreira ( 2002 ).Theyformulateashort-termmaintenance 20

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schedulingproblemasabinaryintegernonlinearprogrammi ngproblemandapplya twostepheuristictechniquetosolvetheformulation.Afea siblesolutionisfoundinthe rststepandaheuristic(simulatedannealing,localsearc h,multiplelocalsearchortabu search)isusedtoimprovethefeasiblesolutioninthesecon dstep.Thebestresultsare obtainedwithsimulatedannealing. Higgins ( 1998 )providesanintegerprogrammingmodel thataimstominimizethedisruptionstotrainservicesandr educemaintenancecosts. Theirmodelisnotappliedonawholenetwork,itisappliedon a300kmtrackcorridor withafourdayplanninghorizon.Theobjectivefunctionisc onstructedtominimizethe interferencedelaysandprioritizednishingtimesofmain tenanceactivities.Nonlinearity oftheconstraintsinthemodelandsizeoftheproblemforcet heuseofheuristics.They rstndafeasiblesolutionandimproveitusingtabusearch Higginsetal. ( 1999 )use thesameapproachona89kmtrackcorridorandobtainsimilar reductionsinobjective functionvalue(about7%)ascomparedtotheschedulecreate dmanually.Another studythatconsidersoneraillinkisby Budaietal. ( 2006 ).Theydiscussthepreventive maintenanceschedulingproblemwiththeintentofminimizi ngthetrackpossessioncosts andmaintenancecostsforonelinkinarailnetwork.Possess ioncostsaredeterminedby thetimeatrackisoccupiedformaintenanceandcannotbeuse dforrailtrac.Their paperconsidersthegroupingofpreventivemaintenanceact ivitiesandgivesfourdierent heuristicsforsolvingtheproblem. Adynamicschedulegenerationtechniquefortherollinghor izonisdiscussedby Cheungetal. ( 1999 ).TheirstudyisbasedonrealtimedatafromHongKongsubwayr ail system.Foroccasionallyusedtracks,whichisthecaseinAus traliaandsomeEuropean countries, Budai&Dekker ( 2004 )showthatthetrackpossessionismodeledinbetween operations. Budaietal. ( 2004 )introduceaslightlydierentversionoftheproblemwhere theobjectiveiscompletingtheprojectwithinthetrack'sf reetime.Theycreateadynamic scheduleforcarryingoutpreventivemaintenanceactiviti esandproposethreeheuristics fortacklingtheproblemunderseverallimitations.Ofthes ethree,theMax-to-Min 21

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heuristicwasfoundtobethemostsuccessful. Grimes&Barkan ( 2006 )performastudy formeasuringthecost-eectivenessofrailwayinfrastruc turerenewalmaintenance.Their resultsshowthatifrailroadsconstraintherenewalmainte nancetoreducetheoverall capitalexpenditures,increasingmaintenanceexpensesth atfollowwillmorethanosetthe initialtemporaryreductionsincapitalspending. Mostoftheexistingstudiesworkforasingle-trackonashor t-termhorizonand thereforecanbeusedtomodifyageneratedtimetableaftera networkdisruption occurs.Manyoftheseapproachesresultinhighcomputerrun ningtimesandthusare notwell-suitedtohandlethereal-lifecomplexitiesofacu rfewplanningproblemwhichis denedonthewholenetworkwithalong-termhorizon. Inthisstudy,wefocusonthemaintenanceschedulingofrail waytracksandaim towardsconstructingamaintenancetimetablethatminimiz espossiblerailtrac disruptions.Weproposefouriterativealgorithmsthatpro duceverygoodsolutions withinpracticaltimelimits.Proposedalgorithmshavethe abilityofhandlingsome criticaldecisionsimplicitly.Thealgorithmsareapplied totheentirerailnetworkand overalong-termhorizon.Weproviderexibleintegerprogra mmingformulationsthat aredesignedtorunwithdynamicallychangingdatasetsthro ughoutiterations.The algorithmsdevelopedconsiderallofthereal-lifeconstra intsaectingtheperformanceof theresultingmaintenanceplan.Inadditiontotheintroduc tion,thischapterisorganized intofourothersections:Section 2.2 providestheproblemdescriptionandexplainsthe probleminputsandconstraints.Section 2.3 presentsoursolutionapproaches.Section 2.4 presentsthecomputationaltestsandSection 2.5 containsourconcludingremarks. 2.2ProblemDescription Railwaytrackmaintenanceschedulingproblemisbroughtto ournoticebyamajor USrailroadcompany.Thisproblemiscalledasthecurfewplan ningproblem(CPP)by therailroadcompany.Notethattherestoftheterminologyus edthroughoutthischapter alsocomesfromtherailroad. 22

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Foramajorrailroadcompany,eachyeararound2500repairjo bshavetobe completedonrailwaytracknetwork.Seventypesofmaintena ncejobsareperformed onthetracks:Capacityjobs,curvepatching,concreterepa ir,gauging,out-of-face new,out-of-facerepairandtie-surfacing.Thesejobtypes areclassiedundertwomain categoriesasrailjobsandtie-surfacingjobsinwhichthe rstsixtypesfallintorailjob category.Thejobsarepartitionedintoabout300projectsb ytheirtypeandgeographical proximity,i.e.railjobsinaspecicregionaregroupedint oarailprojectandsimilarly tie-surfacingjobsinthisregionaregroupedintoatie-sur facingproject.Thedurationof arailprojectdependsonthetrackdistancebeingrepairedo rreplaced,whilethatofa tie-surfacingprojectiscalculatedbyaddingthenumberof tiesbeingprocessed.Duration alsodependsonthesizeofcrewworkingontheproject.Around 800crewsaregrouped into18to19teams,whicharesupposedtocompletetheprojec tsduringtheyearandat thesametimemeetvariousbusinessrequirements.Theworki sscheduledattheproject level.Thetimetablehastospecifywhenandbywhichteameac hprojectshouldbe started. Aprojectis\active"inaweekifitisscheduledinthatweek. Someprojectsare locatedontherailroadyardsandsomeareonthemainline.If aprojectisontherailroad yard,itdoesnotcauseadisruptioninrailtrac.Mainlines canbesingle-trackedor double-tracked.Ifaprojectislocatedonasingle-tracked mainlinethenitcausea completeblockageoftherailtracandtheregionwherethep rojectisscheduledis calledtobe\underabsolutecurfew".Theprojectswhichare notonsingle-tracked mainlinerequire\normalcurfew".Fortherailroadsabsolu tecurfewismuchmorecrucial andvitalcomparedtothenormalcurfew.Theprojectsrequir ingabsolutecurfewshould bescheduledcarefullyinordertominimizethepossiblenet workdisruptionsinrailtrac. Therailroadnetworkisdividedinto\subdivisions",eachc ontainingasetofprojects. Subdivisionsmayinvolvemorethanonetracksegmentandmay requirebothrailwork andtie-surfacingwork.Asubdivisionisconsideredtobeun derabsolutecurfewinaweek 23

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ifatleastoneoftheabsolutecurfewrequiringprojectsins idethesubdivisionisactivein thatweek.Inordertocontrolpossiblenetworkdisruptions ,somebusinessrequirements maybeenforcedonthenumberofsubdivisionsthatareundera bsolutecurfewatany week.Insiderailroadnetwork,thereareusually 10 to 12 \servicecorridors"whichare composedofseveralsubdivisions.Eachservicecorridorco nsistsofspeciedtracksegments intherailroad,butsomeservicecorridorshaveoverlappin gtracksegments.Therefore, thesamesubdivisioncanbeinmorethanoneservicecorridor .Alargeportionoffreight transportationiscarriedoverservicecorridors.Inorder topreventdisruptionsthatmay aectmovementofhighvolumesoffreight,eachservicecorr idormayhaveatmostoneof itssubdivisionsunderabsolutecurfewatanyweek. Therearefourtypesofteams:smallandlargerailteams(SRa ndLR);andsmalland largetie-surfacingteams(STandLT).Theprojectcompleti ontimesaregiveninterms ofweeks.Alargeteamcannishaprojectroughlytwiceasfas tasasmallteam.After aprojectisnished,teamsrelocateovertheweekends.Comb iningtwoteamstonisha projectinashortertimeiscalled\splitting".Splittingm ayalsobeappliedtosatisfysome businessrequirementsthatmustbeenforcedineachweek. Eachyeartherearesomespecicweekswheretracksarenotus edduetocoalmine closuresorspecialvacations.Thesefreeweeksarecalled\ jamboreeweeks"(andusually consistoftwoconsecutiveweekseachyear).Thesetofproje ctsthathastobecompleted duringjamboreeweeksarecalled\jamboreeprojects".Them ainpriorityofthesespecial weeksiscompletingthejamboreeprojects,somostofthebus inessrequirementsare relaxed.Whenjamboreeweeksstart,teamsmayhavetointerr upttheirongoingprojects andtravellongdistancestostartworkingonthejamboreepr ojects. 2.2.1ProblemInputs Projects: Thelistofprojectnames,thetype(railortie-surfacing)a ndthe subdivisionforeachproject,whetheraprojectrequiresab solutecurfew,andthe numberofweeksrequiredforcompletingtheprojectbyasmal lteamandbyalarge team. 24

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Teams: Thelistofteamsandtheirtypes(SR,LR,STorLT). Subdivisions: Thelistofsubdivisionnamesandthelistofalladjoiningsu bdivision pairs. ServiceCorridors: Thelistofservicecorridornames. Subdivision-ServiceCorridorMapping: Thelistofthesubdivisionsineach servicecorridor. TimeWindow: Eachprojectmayhavespecicstartandendweeksthatshowth e intervalinwhichtheprojectisallowedtobeactive.Theser estrictionsarecausedby variousreasons.Forinstance,maintenanceteamsmayavoid workinginthenorth duringwinterortheymayavoidworkingataregionifamajors portseventtakes placeatthattime.Timewindowsisusefultoreducetheprobl emsizebuttheyalso reducethefeasibleregion. 2.2.2ProblemConstraints Twogroupsofconstraintsmustbesatised: PerformanceConstraints arerequiredbytherailroad.Theydeterminethe implementabilityoftheschedulegenerated. FeasibilityConstraints areduetoproblemcharacteristics. 2.2.2.1Performanceconstraints 1. AbsoluteCurfewConstraints: Theremaybeatmost subdivisionsthat areunderabsolutecurfewperweek.Thevalueofparameter isspeciedbythe railroad.Aswestatedearlier,allabsolutecurfewrequirin gprojectsinasubdivision areregardedasonlyoneabsolutecurfewiftheyareactivein thesameweek. 2. ServiceCorridorAt-MostConstraints: Atmostonesubdivisionwithina servicecorridormaybeunderabsolutecurfewinanyweek. 3. MutuallyExclusiveSubdivisionConstraints: Anypairofadjoiningsubdivisions inthelistofsuchpairsshouldnotbeunderabsolutecurfews imultaneously. 4. TimeWindowConstraints: Timewindowsofjamboreeprojectsmustbehonored ashardconstraints.Thetimewindowsofotherprojectsareh onoredasmuchas possible. 5. DistanceConstraints: Onceaprojectisnished,alloftheresources(heavy equipments,teammembers,etc.)shouldbemovedtothenextp rojectoverthe weekend.Thetraveldistancelimitbetweenprojectsisspec iedas400miles. 25

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2.2.2.2Feasibilityconstraints 1. Eachteammayworkononlyoneprojectatatime. 2. Eachprojectshouldbecompleted. 3. Railprojectsmustbedonebytherailteams,andtie-surfaci ngprojectsbythe tie-surfacingteams. 4. Noteamshouldbeidleduringtheyear.Breaksinthescheduleo fateamcanbeat theendoftheyear,ifrequired. 5. Therstweekoftheyearisaholiday. Thesecondfeasibilityconstraintsmaysuggestthatthepro blemofarranging schedulesforrailteamsandtie-surfacingteamsisseparab le.However,thedecomposition oftheproblemisnotpossible,becauseperformanceconstra intssuchasabsolutecurfew, servicecorridorat-most,andmutuallyexclusivesubdivis ionconstraintshavetobeapplied tobothtypesofteams.Forinstance,asubdivisioncanbeund erabsolutecurfewdueto bothrailprojectsandtie-surfacingprojects. WedesignedfouriterativealgorithmstosolvetheCPP.Inor dertopresentour thoughtprocessbetter,wepresentthealgorithmsintheord ertheyaredeveloped. Inthersttwoapproaches,wesolve1-weeklyand k -weeklyintegerformulations iteratively,andfortheothersweextendtheseapproachesb yusingbacktrackingidea.The objectivefunctionoftheCPPistominimizetheamountofvio lationsintheperformance constraints. 2.3SolutionApproaches Wepresentfouralgorithmswehavedevelopedinordertosolv etheCPP.Inall algorithms,therearetwocommonstrategiesapplied.Ther stinvolvespartitioningthe setofrailprojectsandtie-surfacingprojectsamongsmall andlargeteamsandsolving weeklyor k -weeklymodelsgiventhepartitionathand.Thesecondstrat egystartsby solvingforjamboreeweeksandassigningthejamboreeproje ctsandnishesbycompleting therestofthemaintenancescheduleinaregularway,starti ngfromthesecondweekofthe 26

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year.Wenowpresentthesecommonpointsindetailandthenex plainthefouralgorithms developed.2.3.1PartitioningtheProjects Forpartitioningtheprojectsanintegerprogramming(IP)f ormulationissolved separatelyforbothrailandtie-surfacingprojects.Itisu sedtopartitiontheprojects amongsmallandlargeteams.Bythepartitioningformulatio n,weensurethatthetotal timeofprojectsthatareassignedtoateamtypedoesnotexce edtheavailablenumberof workingweeksforthatteamtype.Theavailablenumberofwor kingweeksforateamtype canbefoundbymultiplyingthenumberofteamsofthattypeby thenumberofweeksin ayear.Itistheupperboundoftotaltimeofprojectsthatcan beassignedtoateamtype. Wealsoenforcealowerboundlimitonthetotaltimeofprojec tsassigned.Thelimits appliedtorailandtie-surfacingteamsarethesame.Bythis way,theaveragenumberof workingweeksforsmallorlargerailandtie-surfacingteam siskeptcloser.Wecreatea coupleofpartitionsbychangingthelimitsofworkingweeks assignedtoateamtype.For thesepurposesweusethefollowingstraightforwardconstr aints: 1. TotaltimeofrailprojectsthatareassignedtoSRteamsmust belessthanthe availablenumberofworkingweeksforSRteamsandgreaterth anthelowerbound limitdeterminedontheworkingweeks. 2. TotaltimeofrailprojectsthatareassignedtoLRteamsmust belessthanthe availablenumberofworkingweeksforLRteamsandgreaterth anthelowerbound limitdeterminedontheworkingweeks. Theseconstraintsareappliedsimilarlytothetie-surfaci ngprojects. 2.3.2SchedulingJamboreeProjects Beforeassigninganyotherproject,wescheduletheproject sthathavetobedone withinjamboreeweeks.First,jamboreeprojectsarepartit ionedamongsmallandlarge teamssothatthesumoftheprojectdurationsassignedtoapa rticularteamtypedoesnot exceedtheavailableworkingweeks.Then,wesolveanassign mentmodelthatassignseach jamboreeprojecttooneortwoteamsdependingontheirdurat ion.Notethatjamboree 27

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projectshavepriorityduringjamboreeweeks.Whileapproa chingjamboreeweeks,ifa project'sdurationcrossesajamboreeprojectthatisalrea dyassignedthentheprojectis interrupteduntilthejamboreeprojectiscompleted.2.3.31-WeeklyAlgorithm Inordertoformanannualmaintenanceschedule,wesolveali nearintegerprogramming formulationforeachweekiteratively.Sincetherstweeko feachyearisaholiday,we startbysolvingforthesecondweek.Oncethemodelissolved forweek w ,someslotsfor thefollowingweeksarealsolledsinceeachprojectmaytak emorethanaweek.Hence whenthemodelissolvedforweek w +1 ,someoftheslotsmayalreadybeoccupieddue totheassignmentsmadeonpreviousweeks.Theschedulethat isgeneratediterativelyis keptinthe M matrix.The M wt entryinthismatrixshowstheprojectassignedtoteam t inweek w .Eachtimebeforesolvingthemodelforaweek w ,werecordtwosetsusingthe matrix M :Therstisthesetofunoccupiedteamslots( E ).Theset E isutilizedwhile formingtheconstraintsofthe1-weeklyformulation.These ts E SR E LR E ST and E LT are thedisjointsubsetsoftheset E andtheydenetheunoccupiedteamslotsavailablefor aparticularteamtypeinweek w .Thesecondsetrecordedis P split whichincludesthe projectsthataretobesplit.Aprojectisnecessarilyadded tothissetifthefollowing conditionsaresatised: 1. Theproject'sdurationismorethantheremainingweeksofth eplanningperiod(a year). 2. Thereareatleasttwoavailableslotsfortheteamtypetowhi chtheprojectis assigned. Ifaproject'sdurationismorethantheremainingweeksbuti fthesecondconditionis notsatised,thenweaddtheprojecttotheset P na whichshowsthattheprojectshould notbeassignedinthisweek.Notethat P na isatemporarylistforthecurrentweek's iterationinthe1-weeklyalgorithm.Wealsomaintainthese tofavailableprojects( P ) throughoutthealgorithm,andthissetkeepsdecreasingasw eiteratethroughtheweeks. 28

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Initiallytheset P consistsofalltheprojectstobeassigned.Astheprojectsar eassigned totheteams,theyareremovedfromtheset P Beforepresentingthe1-weeklyformulationandthealgorit hmindetail,weshow theschematicrepresentationoftherstfewweekstohelpwi thunderstandinghowthe algorithmrows.InTable 2-1 ,noneoftheprojectsinitiallyareassignedtoateamandall oftheslotsareempty.After1-weeklymodelissolvedforthes econdweek,theemptyslots thatwillbelledinthethirdweekbecomedenite.Theproce sscontinuesinasimilar fashiononaweeklybasisuntilnoprojectsremainforassign ment.Notethatthesolution ofthe1-weeklymodelinaweek w dependsonthesets P and E whicharebothdependent uponthedecisionsmadeinthepreviousweeks. 1-WeeklyIntegerProgrammingFormulation. Theformulationissolvedfor eachweek w ,startingfromthesecondweektothelastweekoftheyear.We nowpresent thelinearintegerformulation.Consideringthatthesetso favailableprojectsandslots ( P and E )changedynamicallyforeachweekduringtherowofthealgor ithm,themodel givenassumesthatweareinweek w Indices: w :Week p :Project u :Servicecorridor s :Subdivision a :Adjoiningsubdivisionpair e :Teamtype t :Teamnumber Sets: W :Setofallweeksintheplanninghorizon W J :Setofjamboreeweeks P R :Setofrailprojects 29

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P TS :Setoftie-surfacingprojects P :Setofallprojects, P = P R [ P TS P :Setofavailableprojects P split :Setofprojectsthathavetobesplit K :Setofteamtypes: K = f SR LR ST LT g E e :Setofunoccupiedteamslotsforteamtype e forthecurrentweek w E :Setofallunoccupiedteamslotsforthecurrentweek w E = S e 2 K E e P s :Setofabsolutecurfewprojectsundersubdivision s J :Setofjamboreeprojects, J P T e :Setofteamsoftype e 2 K T :Setofallteams, T = S e 2 K T e S :Setofsubdivisions U :Setofservicecorridors S adj :Setofpairsofadjoiningsubdivisions P e :Setofprojectsassignedtoteamtype e incurrentpartition Parameters: e :Numberofavailableteamsoftype e r p :1ifproject p requiresabsolutecurfew,0otherwise :Maximumnumberofabsolutecurfewsallowedperweek m s :Totalnumberofprojectsthatrequireabsolutecurfewinsu bdivision s su :1ifsubdivision s isinservicecorridor u n u :Numberofsubdivisionswith m s su > 0 inservicecorridor u d ij :Traveldistancebetweenprojects i and j ; i j 2 P TW pw :Numberoftimewindowsviolatedweeksif p isassignedtoanyteamsinweek w dlim :Traveldistancelimitbetweentwoconsecutiveprojects r pt :1ifadistanceviolationoccursincaseprojectpisassigne dtoteamt duetopreviousweek'sassignment( d p ` p > d lim M w 1 t = p ` ) 30

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c va :Perunitcostofabsolutecurfewconstraintviolation c vm :Perunitcostofmutuallyexclusivesubdivisionconstrain tviolation c vsc :Perunitcostofservicecorridorat-mostconstraintviola tion c tw :Perunitcostoftimewindowsconstraintviolation c vd :Perunitcostofdistanceconstraintviolation DecisionVariables: x pt :1ifproject p 2 P isstartedbyteam t 2 E (inthecurrent1-weeklyperiod), 0otherwise y s :1ifsubdivision s 2 S isunderabsolutecurfewincurrentweek w ,0otherwise va :Variabletoconvertcurfewconstrainttoasoftconstraint vm a :Variabletoconvertmutuallyexclusiveconstrainttoasof tconstraint; a =( s 1 s 2 ) 2 S adj vsc u :Variabletoconvertservicecorridorat-mostconstraintt oasoftconstraint; u 2 U Theobjectiveistominimizetheamountofviolationsinperf ormanceconstraints (absolutecurfew,mutuallyexclusivesubdivisions,servi cecorridorat-most,timewindows anddistancelimitconstraints).Forexample,iftotalnumb erofsubdivisionsunder absolutecurfewinaweekis18,whilethelimit onabsolutecurfewis15,thenwecount theamountofviolationforabsolutecurfewconstraintinth isweekas3.Thequalityofthe providedsolutionisdeterminedbytheamountofviolations inperformanceconstraints. min c va va + c vm X a 2 S adj vm a + c vsc X u 2 U vsc u + c tw TW pw X p 2 P X t 2 E x pt + c vd X p 2 P X t 2 E r pt x pt (2{1) Apartfromthebasicobjectivefunction( 2{1 ),wemakethefollowingmodicationto theobjectiveduringtherowofthealgorithm:Wetrytogette amsclosertothelocations ofthejamboreeprojectswhilemovingtowardthejamboreewe eks.Sixweekspriorto thejamboreeweeks,westarttoputdistanceviolationcosts forsomeoftheavailable projectsbyconsideringtheirdistancetothejamboreeproj ectahead.Bytheaidofthis 31

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Table2-1.1-weeklyiterativesteps InitialTableau Team1 Team2 Team3 ... ... ... TeamK Week1 Week2 Week3 ... ... Week52 Beforesolvingforweek3 Team1 Team2 Team3 ... ... ... TeamK Week1 Week2 137 218 157 275 121 75 160 Week3 157 275 121 160 Week4 157 275 Week5 275 Week6 ... ... Week52 Beforesolvingforweek4 Team1 Team2 Team3 ... ... ... TeamK Week1 Week2 137 218 157 275 121 75 160 Week3 264 269 157 275 121 282 160 Week4 269 157 275 282 Week5 275 Week6 ... ... Week52 Beforesolvingforweek5 Team1 Team2 Team3 ... ... ... TeamK Week1 Week2 137 218 157 275 121 75 160 Week3 264 269 157 275 121 282 160 Week4 258 269 157 275 91 282 278 Week5 275 91 278 Week6 91 278 Week7 91 Week8 91 Week9 ... ... Week52 32

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modication,themodelisinclinedtoselectprojectsthata reclosertothejamboree projects'locations. Constraints: 1. Determineifsubdivision s isunderabsolutecurfewornot.Ifproject p isin subdivision s andrequiresabsolutecurfew,then y s x pt constraintassuresthat y s equals 1 ifproject p isstartedbyteam t inthecurrentweek. y s x pt 8 s 2 S p 2 P s t 2 E (2{2) 2. Duetotheassignmentsinpreviousweeks,someteamsmayalre adybeoccupied. Ifthereisanactiveprojectinsubdivision s whichrequiresabsolutecurfewthen subdivision s issettobeunderabsolutecurfew. y s =1 8f s 2 S : M wt = p p 2 P s t = 2 E g (2{3) 3. Ifthereisnoprojectthatrequiresabsolutecurfewinsubdi vision s ,thensubdivision s cannotbeunderabsolutecurfew. y s =0 8f s 2 S : m s =0 g (2{4) 4. Eachprojectcanonlybeassignedtoateamthatisqualiedto dothatproject x pt =0 8 p 2 P e t = 2 E e e 2 K (2{5) 5. Jamboreeprojectscannotbeassignedifthecurrentweek w isnotajamboreeweek. x pt =0 8f p 2 J t 2 T w = 2 W J g (2{6) 6. Atmostoneprojectcanbeassignedtoeachavailableteamine achweek.This meansthatateamcanhandleoneprojectatatime. X p 2 P x pt 1 8 t 2 E (2{7) 7. Ifaprojectistobesplit( p 2 P split ),thenitisassignedtotwoteamsinweek w X t 2 E e x pt =2 8 p 2 P split p 2 P e e 2 K (2{8) 8. Iftheproject'sdurationislargerthantheremainingweeks butitcannotbesplitin thecurrentweek( p 2 P na ),wedonotassigntheprojectinthegivenweek. x pt =0 8f p 2 P na t 2 E g (2{9) 33

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9. Eachprojectcanbestartedbyatmosttwoteamsofthesametyp e. X t 2 E e x pt 2 8 p 2 P e (2{10) 10. Forthecurrentweek,ifnumberofavailableprojectsoftype e isgreaterthanor equaltounoccupiedteamsofthesametype,aprojectwillbea ssignedtoeachof theseunoccupiedteams. X p 2 P e x pt =1 8 t 2 E e : P e E e 8 e 2 K (2{11) 11. Forthecurrentweek,ifthenumberofavailableunoccupiedt eamsoftype e islarger thantheavailableprojectsofthesametype( P e = 2 P split ),assigneachofthese projectstoatleastoneoftheavailableteams.Notethatproj ectsthatareinset P split arenotpartofthisconstraintsincetheyreservetwounoccu piedteamsas statedinconstraint( 2{8 ). X t 2 E e x pt 1 8 p 2 P e = 2 P split : P e = P split < j E e j 2 j P split j 8 e 2 K (2{12) 12. Sumofsubdivisionsthatareunderabsolutecurfewmustbele ssthanorequaltothe givenabsolutecurfewlimit(Thisconstraintisappliedasa hardorsoftconstraintin specicplacesofthealgorithm). X s 2 S y s + va (2{13) 13. Adjoiningsubdivisionpairsaremutuallyexclusive.Theyca nnotbeunderabsolute curfewinthesameweeksimultaneously(appliedasahardors oftconstraint interchangeably). y s 1 + y s 2 1+ vm a 8 a =( s 1 s 2 ) 2 S adj (2{14) 14. ServiceCorridorAt-MostConstraints:Atmostonesubdivis ionwithinaservice corridormaybeunderabsolutecurfewinanyweek. X s 2 S y s su 1+ vsc u 8 u 2 U (2{15) 15. Boundsonintegervariables. x pt 2f 0,1 g8 p 2 P 8 t 2 E (2{16) y s 2f 0,1 g8 s 2 S (2{17) 34

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16. Boundsonviolationvariables.Notethatenforcingintegral ityforthesevariablesis notrequiredduetotheconstructionofdenedconstraints. Theamountofviolation thatcanoccureachweekdierforeachtypeofviolationvari ables.Thatiswhy,the upperboundsofthesevariablesaredierent.Forinstance, thelowerboundforva variablesis0andupperboundisnumberofteamsminusthewee klylimitonthe numberofsubdivisionsunderabsolutecurfew.Forinstance ,ifthereare19teams andiftheweeklylimitonmaximumnumberofsubdivisionsund erabsolutecurfew is15,thenviolationamountinanyweekcanbeatmost4.Thisc anhappeninthe worstcasewhen19dierentprojectsareassignedto19teams inthisweekandall these19projectsarelocatedindierentsubdivisionsandi ftheyallrequireabsolute curfew. va 2 [ 0, j T j ] (2{18) vm a 2 [ 0,1 ] 8 a 2 S adj (2{19) vsc u 2 [ 0, n u 1 ] 8 u 2 U (2{20) AlgorithmDetails. Theoutlineofour1-weeklyalgorithmisshowninFigure 2-1 .Eachiterationofthealgorithmconstitutesndingthepro ject-teammatchings foragivenweek w .Whileassigningprojectstotheteams,wesolvealinearint eger formulationseveraltimes.Firstwetrytohonoralloftheab solutecurfew,mutually exclusivesubdivisions,andservicecorridorat-mostcons traintssimultaneously.Thisis relativelyeasyfortherstseveralweeks,becausetherear emanyavailableprojectsto choosefromandthusmorerexibility.However,aswemovetowa rdthefuture,itmay notbealwayspossibletosatisfyalloftheperformancerela tedconstraintsatthesame timeashardconstraints.Insuchcases,weintroducetheper formanceconstraintsas softconstraintsandpenalizetheviolationsintheobjecti vefunction.Relaxationofhard constraintsmaybedoneinamultistepapproachbyrelaxingt hemonebyoneandintwos untilwendafeasiblesolution.Thisapproachisusefulwhe ntherelativeimportanceof constraintsdiersandinasituation,forinstance,wherey oudonotwantaviolationofa specicconstraintforaslongaspossibleandcansacriceh avingmoreviolationsinother constraints. The1-weeklyformulationthatwehaveprovidedhastheabili tytomakesplitting decisions.Themodelmakesthesplitifitndsitdesirable. Notethatthisshouldnot 35

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beconfusedwiththeusageoftheinputset P split whichincludestheprojectsthatare splitnecessarily.Aprojectisinset P split ifthereisnotenoughtimetodothisprojectin theremainingweeksoftheyearlyplanningperiodbyconside ringtheproject'sduration. Splittingaprojectbetweentwoteamscanhelpinthefollowi ngway:Imaginethattwo availableteamsstartworkingontwodierentprojectsanda ssumethattheseprojects bothrequireabsolutecurfewandarelocatedindierentsub divisions.Inthiscase,eachof theseteamswillcauseanincreaseinthenumberofsubdivisi onswhichareunderabsolute curfew.Inweekswhereabsolutecurfewconstraintcannotbe satised,lettingtwoteams handlethesameprojectatthesametimemaydecreasethenumb erofsubdivisionsthat areunderabsolutecurfew,andthusmayallowtheconstraint tobeheld.Thisaction willalsoshortenthehandlingtimeoftheprojectbyhalf.Th ereforesplittingalengthy projectmayalsodecreasethechanceofacontinuingviolati onforlongperiodsoftime. Ourobservationsshowthatthesplittingabilitymakesitea siertondfeasiblesolutions. Stepsofthealgorithm Input: Projects,teams,timewindowsofprojects,subdivisions,a ndservicecorridors Output: Maintenancetimetablefortheteams Step1. Initialization: 1.1 Gettheinputsandformthenecessarysetsandarrays 1.2 Partitiontheprojects Partitionthesetofrailprojectsamongsmallandlargerail teams Partitionthesetoftie-surfacingprojectsamongsmalland largetie-surfacingteams 1.3 ConstructandsolveanIPforjamboreeweekstoassignthejam boreeprojects Step2. Foreachpartitionobtained,ndayearlymaintenancesched ule: Setw=2 2.1 Constructandsolve1-weeklyformulationforweek w withabsolutecurfew, mutuallyexclusiveandservicecorridorconstraintsashar dconstraints Ifmodelisinfeasible,solvetheformulationbyrelaxingal lperformanceconstraints 2.2 Implementweek w 'sschedule.Recordthenumberofperformanceviolationsin week w Set w = w +1 andgoto2.1 Figure2-1.1-weeklyalgorithm 2.3.4 k -WeeklyAlgorithm Thepromisingresultsobtainedbyusingthe1-weeklyalgori thmledustodevelop theideafurther.Foreachiteration,insteadofsolvingfor oneweek'sassignment,wesolve k -weeklyIPmodels.Thealgorithmsolvesfora k -weeklyperiodandimplementsonlythe 36

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rstweekofthisperiod.Thenitcontinuessolvingforthene xt k -weeklyperiodwhich startswiththeweekaftertheimplementedweek.Thepurpose ofthe k -weeklyalgorithm istoimprovetheoptimizationbysolvingtheIPmodeloveral ongertimehorizon. Focusingonassigningprojectsforonlyoneweekallowsthem odeltoassignprojectsto teamswithoutconsideringtheupcomingweeks.Hence,themod elmaynotforeseepossible futureviolations.Asthevalueofthe k getsbigger,thecomputationaltimeofthemodel getshigherduetoanincreasingcombinatorialeect.There fore,wesetthevalueof k as highaspossiblewhiletryingtokeepaniteration'ssolutio ntimewithinpracticallimits. k -WeeklyIntegerProgrammingFormulation. Ourlinearintegerprograming formulationusedinndingasolutionfora k -weeklyperiodisshownbelow.Westartby presentingonlytheadditionalsetsandparametersusedfor thisformulation. Sets: W ` :Setofweeksincludedinthecurrent k -weeklyperiod E w e :Setofunoccupiedteamslotsforteamtype e inweek w E w :Setofallunoccupiedteamslotsinweek w W p w :Setofpotentialstartingweeksforproject p thatpassesthroughweek w : f w ` 2 W p w : w w ` w d t pe = 2 e +1, p 2 P e w ` 2 W ` g W p w :Setofpotentialstartingweeksforproject p thatpassesthroughweek w ifitis notsplit: w ` 2 W p w : w d t pe = 2 e w ` w t pe +1, p 2 P e w ` 2 W ` Parameters: # :Totaldurationofavailableprojects w 1 :Firstweekofthecurrent k -weeklyperiod w L :Lastweekofthecurrent k -weeklyperiod t pe :Numberofweekstocompleteproject p giventhatitisassignedtoteamtype e r ` pt :1ifadistanceviolationoccursincaseprojectpisassigne dtoteamt inweek w 1 duetopreviousweek'sassignment( d p ` p > d lim M w 1 1 t = p ` ) 37

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DecisionVariables: x ptw :1ifproject p isstartedbyteam t inweek w ,0otherwise y sw :1ifsubdivision s isunderabsolutecurfewinweek w ,0otherwise va w :Variabletoconvertanabsolutecurfewconstrainttoasoft constraint vm aw :Variabletoconvertamutuallyexclusiveconstrainttoaso ftconstraint vsc uw :Variabletoconvertaservicecorridorat-mostconstraint toasoftconstraint min w L X w 1 ( c va va w + c vm X a 2 S adj vm aw + c vsc X u 2 U vsc uw + c tw TW pw X p 2 P X t 2 T x ptw ) (2{21) + c vd X p 2 P X t 2 T r ` pt x ptw 1 subjectto y sw x pt w 8 s 2 S w 2 W `, p 2 P s t 2 E w w 2 W p w (2{22) y sw x pt w X t ` 6 = t x pt w 8 s 2 S w 2 W `, p 2 P s t 2 E w w 2 W p w (2{23) y sw =1 8f s 2 S w 2 W `: M wt = p p 2 P s t = 2 E w g (2{24) y sw =0 8f s 2 S w 2 W `: m s =0 g (2{25) x ptw =0 8f p 2 P e t = 2 E w e w 2 W `, e 2 K g (2{26) x ptw =0 8f p 2 J w = 2 W J t 2 E w g (2{27) x ptw =0 8 p 2 P e t 2 E w e w 2 W `: w + d t pe = 2 e 1 > j W j (2{28) X t ` 6 = t x pt ` w x ptw 8 p 2 P e t 2 E w e w 2 W `: w + t pe 1 > j W j (2{29) X p 2 P x ptw 1 8f t 2 E w w 2 W ` g (2{30) X t 2 E w e x ptw 2 8f p 2 P e w 2 W `, e 2 K g (2{31) X t 2 E w e X p 2 P e X w 2 W p w x ptw = j E w e j ( 8 w 2 W `: # e w X w 1 j E w e j 8 e 2 K ) (2{32) 38

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X p 2 P T 1 X w = F 1 x pt w (1 x ptw )( T 1 F 1 +1), 8 p 2 P e w 2 W ` n w L t 2 E w (2{33) X p 2 P T 2 X w = F 2 x pt w (1 x ptw + X t ` 6 = t x pt ` w )( T 2 F 2 +1), 8 p 2 P e w 2 W ` n w L t 2 E w (2{34) X s 2 S y sw + va w 8 w 2 W ` (2{35) y s 1 w + y s 2 w 1+ vm aw 8 a =( s 1 s 2 ) 2 S adj w 2 W ` (2{36) X s 2 S y sw su 1+ vsc uw 8 u 2 U w 2 W ` (2{37) x ptw 2f 0,1 g8 p 2 P 8 t 2 E w 8 w 2 W ` (2{38) y sw 2f 0,1 g8 s 2 S 8 w 2 W ` (2{39) va w 2 [ 0, j T j ] 8 w 2 W ` (2{40) vm aw 2 [ 0,1 ] 8 a 2 S adj 8 w 2 W ` (2{41) vsc uw 2 [ 0, n u 1 ] 8 u 2 U 8 w 2 W ` (2{42) where F 1 = w +1 T 1 = min f w L w + d t pe = 2 e 1 g T 2 = min f w L w + t pe 1 g F 2 = w + d t pe = 2 e Mostoftheconstraintsresemblethe1-weeklyformulatione xceptthattheyaredenedfor allweeksinthe k -weeklyperiod.Constraints( 2{33 )and( 2{34 )arethemostimportant constraintsinthe k -weeklymodel.Ifanavailableproject p isstartedatweek w bya team t ,thentheseconstraintsensurethatnootherprojectscanbe assignedtoteam t untilthisprojectends.Notethatthe k -weeklymodelllsaportion( k j T j )ofthe M matrix,theninsidetherowofthealgorithmtheschedulemat rix M iscompleted takingintoaccounttheprojectswhosedurationexceedsthe periodlength.Numberof variablesandconstraintsinthe k -weeklymodelareO( k [ j P jj T j + j S j + S adj + j U j ] )and 39

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O( k [ j S jj T j + j T j + j P j + S adj + j U j ] )respectively.Theyare k timesthenumberofvariables andconstraintsofthe1-weeklymodel. AlgorithmDetails. Theoutlineofour k -weeklyalgorithmisshowninFigure 2-2 Stepsofthealgorithm Input: Projects,teams,timewindowsofprojects,subdivisions,a ndservicecorridors Output: Maintenancetimetablefortheteams Step1. Initialization: 1.1 Gettheinputsandformthenecessarysetsandarrays 1.2 Partitiontheprojects Partitionthesetofrailprojectsamongsmallandlargerail teams Partitionthesetoftie-surfacingprojectsamongsmalland largetie-surfacingteams 1.3 ConstructandsolveanIPforjamboreeweekstoassignthejam boreeprojects Step2. Foreachpartitionobtained,ndayearlymaintenancesched ule: Startwiththerst k -weeklyperiod 2.1 Constructandsolvethe k -weeklyformulationbysettingperformanceconstraintsas hard constraints Ifmodelisinfeasible,solvebyrelaxingallperformanceco nstraints 2.2 Implementweek w 1 'sschedule.Recordthenumberofperformanceviolationsin week w 1 Continuetothenext k -weeklyperiodthatstartswith w 1 +1 ,thengoto2.1 Figure2-2. k -weeklyalgorithm Byapplyinga k -weeklyformulationineachiteration,weminimizepossibl eviolations thatmayoccurinalongertimehorizon.Thegoalistobetterp redictthefuture.Forthis reason,wesolvefor k weeksandimplementtherstweekofthese k weeksiterativelyto theendoftheyear.2.3.5Backtracking Inordertobettermanagepossiblefutureviolations,wedel ayedimplementingthe solutionof1-weeklyor k -weeklyformulationsandcontinuedtosolveforaspeciedp ortion oftheremainingtimehorizon.Ifaviolationisobservedthe nthesolutionofthecurrent periodischangedbyintroducinganewconstraint.Afterobta ininganalternativesolution, weagainsolvefortheupcomingperiodsandcheckwhetherany violationsappear.The processofchangingthecurrentperiod'sassignmentsconti nuesuntilnofutureviolations areseen.Iffutureviolationscannotbepreventedinanyoft heattemptsthenwesolve byrelaxingtheperformanceconstraintsandcontinuewitht hefollowingperiod.Wehave namedtheprocessofcheckingpossiblefutureviolationsan dthengoingbackandchanging 40

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thecurrentassignmentincaseofforeseenviolationsas\ba cktracking".Wewillnow explaintheideaforthe1-weeklyand k -weeklyalgorithms. 2.3.5.11-weeklyalgorithmwithbacktracking Eachiterationofthealgorithmincludesbacktracking.Afte rwendasolutionforthe currentweek w ,wedon'timplementthescheduleimmediately.First,thepr ojectsthatare assignedtotheemptyslotsaresaved.Then,wesolveforanum berofweeksinthefuture andeachtimewendaviolationwealtertheassignmentofwee k w .Forthispurpose,a constraintthatacceptsatmost ofthepreviouslyassignedprojectsinweek w isadded totheoriginalformulation.Theconstraintisgivenin( 2{43 ).Theset P \ isthesetof projectspreviouslyassignedtotheemptyslots.Thevalueo f equals j E j 1 initially anddecreasesby 1 eachtimeafutureviolationisdetectedforthecurrentseto fassigned projects. X p 2 P \ X t 2 E x pt (2{43) AlgorithmDetails. Theoutlineofour1-weeklyalgorithmwithbacktrackingis displayedinFigure 2-3 Instep2.1a,wemoveforwardatleasttwoweeksdependingont hecurrentweek w Ifwecannotndaviolationintheupcomingweeks,weacceptt hecurrentassignments ofprojectstotheteams.Otherwise,wetrytochangethecurr entsolutionandrecheck forpossiblefutureviolations.Intheworstcase,weforcet hemodeltochangeallof theprojectsassignedandndacompletelydierentsolutio n.Wethenrecheckforthe upcomingweeks.Afterthispointthevalueof decreasesbelowzero.Ifwecannotnda solutionthatletsusavoidfutureviolationsduringthelas tattempt,wesimplyacceptthe solutionathandandcontinuewiththenextweek. 41

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Stepsofthealgorithm Input: Projects,teams,timewindowsofprojects,subdivisions,a ndservicecorridors Output: Maintenancetimetablefortheteams Step1. Initialization: 1.1 Gettheinputsandformthenecessarysetsandarrays 1.2 Partitiontheprojects Partitionthesetofrailprojectsamongsmallandlargerail teams Partitionthesetoftie-surfacingprojectsamongsmalland largetie-surfacingteams 1.3 ConstructandsolveanIPforjamboreeweekstoassignjambor eeprojects Step2. Foreachpartitionobtained,ndayearlymaintenancesched ule: Set w =2 2.1 Constructandsolve1-weeklyformulationforweek w withabsolutecurfew, mutuallyexclusiveandservicecorridorconstraintsashar dconstraints 2.1a Ifmodelissatised,solvefromweek w +1 to max f w +( j W j w ) = 5, w +2 g Ifnoneoftheweekscauseaviolation,gotostep2.2 Elseif 0 ,addconstraint( 2{43 )thatacceptsatmost ofthe previouslyassignedprojectsinweek w .Set = 1 andreturntostep2.1 Else( < 0 ),gotostep2.2 2.1b Ifmodelisinfeasible,solvebyrelaxingallperformanceco nstraints 2.2 Implementweek w 'sschedule.Recordthenumberofperformanceviolationsin week w .Set w = w +1 = j E j 1 andgoto2.1 Figure2-3.1-weeklyalgorithmwithbacktracking 2.3.5.2 k -weeklyalgorithmwithbacktracking The k -weeklyalgorithmbyitselfconsiders k weeksandimplementstherstweek. Inthe k -weeklyalgorithmwithbacktracking,weextendtheideabys olvingthreemore k -weeklyperiodstodecidewhethertoacceptthesolutionfor week w AlgorithmDetails. Theprocessissimilarto1-weeklyalgorithmwithbacktrack ing. Notethatineachiterationofthealgorithm,modeldetermine sthescheduleofa k -weekly period,hencewerecordtheunoccupiedteamslotsforalloft he k -weekstoformthesets E w .Onceamodelissolved,wesavetheprojectsassignedtothes eslots.Inorderto checkfutureviolations,weprefertosolveforthree k -weeklyperiods.Ifwedonotnda violationinthesethree k -weeklyperiods,weacceptthesolutionfor w 1 ,otherwiseweadd aconstraintoftype( 2{43 )tochangethesetofassignedprojectsassignedtotheempty slots. 2.4ComputationalTests WeimplementedouralgorithmsinJavausingCPLEX11.Comput ationalexperiments wererunona2.5GHzPCwith2GBofmemory. 42

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2.4.1TestingwithRealLifeInstances Wehavetestedouralgorithmsontworeallifeinstancesprov idedbytherailroad company.Thesetwoinstancesbelongtoyears2007and2008re spectively.Thesizeof theseinstancesisgiveninTable 2-2 Table2-2.Sizeofthe2007and2008instances ParametersYear2007Year2008 NumberofProjects(Rail/Tie)254(139/115)265(150/115)NumberofTeams1919NumberofRailTeams(Small/Large)10(9/1)10(7/3)NumberofTie-surfacingTeams(Small/Large)9(2/7)9(2/7)NumberofWeeks5151NumberofServiceCorridors1010NumberofSubdivisions282282 For2007instance,minimumprojectdurationis1weekandmax imumdurationis36 weeks.Averageprojectdurationis5.19weekswithastandard deviationof5.04weeks. For2008instance,minimumandmaximumprojectdurationsar e1weekand28weeks respectively.Averageandstandarddeviationofprojectdur ationsare5.07and4.78weeks. Forbothoftheseinstances,proportionoftotaldurationof projectstotheavailable workingdurationisover0.95.Weassumeequalprioritybetw eenperformanceconstraints; however,railroadplannerscanusedierentcostparameter sandhencegivedierent weightstotheperformanceconstraintviolationsbasedupo ntheirbusinesspriority.The violationsinthesolutionsimplementedbytherailroadcom panyin2007and2008are showninTable 2-3 Table2-3.Numberofviolationsdetectedinthesolutionsimp lementedbytherailroad Violations2007Instance2008Instance Time-Window1547DistanceConstraints5930AbsoluteCurfew20MutuallyExclusiveSubdivisions2943ServiceCorridor2452 TotalViolations268132 43

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2.4.2ComparisonofAlgorithms Nobacktracking,1versus k weekly. Thenumberofviolationsinperformance constraintsfor1-weeklyand k -weeklyalgorithmsarepresentedinTables 2-4 and 2-5 Asweexpected,the k -weeklyalgorithmperformsbetterthanthe1-weeklyalgori thm. Notethatinthe k -weeklyalgorithmwesolvefor k weeksandimplementtherstofthese weeksineachiteration.Inthe k -weeklyalgorithm,atthestagewherewemayhaveto relaxallperformanceconstraints,asolutionwithmorevio lationsintherstweekmay beaccepted,butwhentotalnumberofviolationsinall k weeksareconsidered k -weekly approachismoreholisticandhastheadvantageofmakingbet terprojectassignments. Hence,itmayresultinalowernumberofoverallviolations.W etested k -weeklyalgorithm bysettingvalueof k as3,4,5and6.Usingthesevalues,weobservedthatiterative model sizesandrunningtimeofthealgorithmwasinthepracticall imits. Table2-4.Comparisonofviolations:nobacktracking,1ver sus k weekly(2007) Violations1-weekly k -weekly (2007instance)w/obacktrackingw/obacktracking Time-Window7192DistanceConstraints6042AbsoluteCurfew01MutuallyExclusiveSubdivisions55ServiceCorridor2111 TotalViolations157151 Table2-5.Comparisonofviolations:nobacktracking,1ver sus k weekly(2008) Violations1-weekly k -weekly (2008instance)w/obacktrackingw/obacktracking Time-Window00DistanceConstraints3032AbsoluteCurfew00MutuallyExclusiveSubdivisions2016ServiceCorridor4525 TotalViolations9573 Withbacktracking,1versus k weekly. Weobservethatbacktrackingimproves thesolutionsofboth1-weeklyand k -weeklyalgorithmswhencomparedtothecase 44

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wherenobacktrackingexists. k -Weeklywithbacktrackingperformsbetterthan1-weekly withbacktrackingasweexpect.Comparisonsbetween1-week lywithbacktrackingand k -weeklywithbacktrackingalgorithmsarepresentedinTabl es 2-6 and 2-7 Table2-6.Comparisonofviolations:withbacktracking,1v ersus k weekly(2007) Violations1-weekly k -weekly (2007instance)w/backtrackingw/backtracking Time-Window9678DistanceConstraints5242AbsoluteCurfew00MutuallyExclusiveSubdivisions01ServiceCorridor511 TotalViolations153132 Table2-7.Comparisonofviolations:withbacktracking,1v ersus k weekly(2008) Violations1-weekly k -weekly (2008instance)w/backtrackingw/backtracking Time-Window00DistanceConstraints4222AbsoluteCurfew00MutuallyExclusiveSubdivisions46ServiceCorridor1526 TotalViolations6154 Comparisonofallalgorithms. Allofthefouralgorithmsdevelopedprovides consistentlybettersolutionsinalmostalltypesofperfor manceconstraintscompared tothesolutionsimplementedbytherailroad(Table 2-3 ).Feedbackfromtherailroad companyabouttheobtainedsolutionswasverypositive.The yareconsideringour iterativealgorithmsfordeployingintheupcomingyears.As perourobservations, thesuccessoftheiterativealgorithmsdependsontheirabi litytohandlesomecritical decisionsinvolvedintheoverallproblemimplicitly.Thes ecriticaldecisionsaredistance constraintswhichaecttheorderofprojectshandledbyate amandthedecisionsof assigningprojectstoalargeorasmallteam.Computationti mesfortheprovided solutionsaregiveninTable 2-8 .Weobservethatbestsolutionsareobtainedbyusing k -weeklywithbacktrackingalgorithms.Asweexpect,duetoth esizeoftheinteger 45

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formulationssolvedineachiteration1-weeklyalgorithmw ithoutbacktrackingrunsfaster comparedtothe k -weeklyalgorithmwithoutbacktracking.Whenwecompare1weekly withoutbacktrackingand1-weeklywithbacktrackingalgor ithms,thereisanexpected increaseinsolutiontimesduetothebacktrackingprocess. Thispatternisthesameif k -weeklywithoutbacktrackingand k -weeklywithbacktrackingalgorithmsarecompared. Notethatgeneratingthissolutionmanuallytakestwoweeksf ortherailroadplanners. Table2-8.Averageruntimesofthetopvesolutionswithresp ecttototalviolations Runtime(secs) 1-weeklyw/obacktracking65.13k -weeklyw/obacktracking512.11 1-weeklyw/backtracking304.30k -weeklyw/backtracking920.57 2.4.3TestingwithSmallInstances Wehavemodiedthe k -weeklyformulationtoobtainanexactformulationforthe curfewplanningproblem.Inparticular,weremovedpartiti oningofprojectsamongsmall andlargeteamsandaddeddistancevariablesexplicitlytot he k -weeklymodel.Notethat byusingiterativealgorithmswehavetheadvantageofenfor cingthedistanceconstraints implicitly.Iterativealgorithmscheckthepreviousslot' sorweek's(inthe1-weeklycase) assignmentsinordertodeterminetheassignmentsoffuture slotsorweeks.Intheanalysis madefortheoriginalinstance,wehaveseenthatthisapproa chhelpsidentifygoodquality solutions.Furthermorethefollowingresultsdemonstrate thattheexactmodelswith explicitdistancevariablesarenoteasytosolvewithinpra cticaltimelimits.Wehave denedthefollowingadditionalvariablesandconstraints toformtheexactformulation: AdditionalDecisionVariablesandConstraints. ls p :1ifproject p isassignedtoalargeteam,0otherwise; p 2 P vd pw :1ifproject p violatesdistanceconstraintinweek w ,0otherwise; p 2 P w 2 W 46

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min w L X w 1 ( c va va w + c vm X a 2 S adj vm aw + c vsc X u 2 U vsc uw + c tw TW pw X p 2 P X t 2 T x ptw ) (2{44) + c vd X p 2 P X w 2 W vd pw subjectto x ptw X p ` 6 = p d pp ` d lim x p ` t ( w + t pe ) vd pw + X t ` 6 = t x ptw 8f p 2 P e t 2 E w e w 2 W e 2 K g (2{45) x ptw X p ` 6 = p d pp ` d lim x p ` t ( w + d t pe = 2 e ) vd pw +1 X t ` 6 = t x ptw 8f p 2 P e t 2 E w e w 2 W e 2 K g (2{46) ls p x ptw 8 p 2 P R 8 t 2 T 2 8 w 2 W (2{47) X w 2 W X t 2 T 1 x ptw (1 ls p )2, 8 p 2 P R (2{48) X w 2 W X t 2 T 1 x ptw 1 ls p 8 p 2 P R (2{49) X w 2 W X t 2 T 2 x ptw ls p 8 p 2 P R (2{50) X w 2 W X t 2 T 2 x ptw 2 ls p 8 p 2 P R (2{51) ls p x ptw 8 p 2 P TS 8 t 2 T 4 8 w 2 W (2{52) X w 2 W X t 2 T 3 x ptw (1 ls p )2, 8 p 2 P TS (2{53) X w 2 W X t 2 T 3 x ptw 1 ls p 8 p 2 P TS (2{54) X w 2 W X t 2 T 4 x ptw ls p 8 p 2 P TS (2{55) X w 2 W X t 2 T 4 x ptw 2 ls p 8 p 2 P TS (2{56) 47

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Notethat w L correspondstothelastweekofplanninghorizonfortheexac tformulation. Constraints( 2{45 )and( 2{46 )enforcethedistanceconstraints.Ifproject p isassignedto twoteamsthentheterm P t ` 6 = t x ptw equalsoneandconstraint( 2{45 )becomesredundant sinceitwouldtake d t pe = 2 e weekstocompletetheproject,otherwisetheterm P t `6 = t x ptw equalszeroandconstraint( 2{46 )isredundantasexpected.Constraints( 2{47 )to( 2{51 ) determinewhetherarailprojectisassignedtoasmallorlar gerailteamandtakethe necessaryaction.Ifarailprojectisassignedtoasmall(la rge)railteamthenconstraints ( 2{48 )to( 2{51 )assurethattheprojectisnotdonebytheotherteamtypeand itis assignedtoatleastoneandatmosttwosmall(large)railtea ms.Constraints( 2{52 )to ( 2{56 )performthesametasksfortieprojects. Weformed20randominstances.Inordertoforminstanceshav ingsimilarcharacteristics astherealones,wekeepfollowingstatisticssameastherea llifeinstances:theratioof thenumberofweeksrequiredtocompletethegivenprojectst othenumberofavailable workingweeks,thepercentageofabsolutecurfewrequiring projectsandthepercentage ofmutuallyexclusivesubdivisionpairs.Fortherst10ins tances,theplanninghorizon's lengthis5weeks.Fortheremaining10instances,thelength oftheplanninghorizonis 10weeks.Therunningtimelimitfortheexactformulationis setto8hours.Wecheck thebestintegersolutionsdetectedbytheexactmodelafter 3,5,10,and15minutes andafter1hourand8hours.Thegoalistoobservehowthequal ityofthesolutions foundbytheexactoptimizationmodelevolvesovertime.InT able 2-9 ,thesecondand thirdcolumnsshowthepercentagegapandabsolutegapbetwe enthebestsolutionfrom iterativealgorithmsandthebestsolutionfromtheexactmo del.Thefollowing(H-Time andE-timecolumns)showthetimesrequiredtondthesesolu tionsbyheuristiciterative algorithmsandbyexactoptimizationmodelrespectively. Given8hoursrunningtime,only6ofthesmallinstances(ind icatedwithanasterisk) aresolvedtooptimalitybytheexactmodel.Theaveragerunn ingtimeforthese6 48

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Table2-9.Computationalresultswithsmallinstances:pla nninghorizon5weeks Instances%GapAbs.GapH-TimeE-Time (secs)(secs) 10.00%052.4128,80025.77%315.5328,80033.70%241.6928,800 4*11.54%313.91587.815*1.96%115.382,964.026*0.00%05.024,083.747*5.13%221.785,397.778*4.55%269.9911,682.17 98.51%463.1328,800 10*2.86%130.361,455.69 Average4.40%1.832.92 instancesis4,361.86secs.Forthesamesetofinstancesthe averagerunningtimesof iterativealgorithmsis26.07secs,withanaverageabsolut egapof1.5violations. Table 2-10 presentstheprogressofexactoptimizationmodel.Theseco ndandthird columnsshowthetimeandgapofrstfeasiblesolutionsfoun dbyexactmodel.The remainingsixcolumnspresenthowthesolutionsfoundbyexa ctformulationevolvedover time.Percentagegapsgivenaretheoptimalitygapsreporte dbyCPLEXafterspecied amountoftime.For4ofthe10instances,exactmodelendedwi thonlyfeasiblesolutions after8hours.Itisobservedthatexactmodelcouldndanini tialstartingsolutionwith anaveragegapof60.40%inaverageof422.77secsover10inst ances. Thisobservationcoupledwiththeevaluationofsolutionsf oundindiscretetime intervalsshowsthatexactmodelcannotndgoodsolutionsw ithinpracticaltimelimits eveninthecaseofsmallinstances.Thisbehaviorisclearer whenwesolveforinstances withaplanninghorizonof10weeks.Inall10oftheseinstanc es,exactmodelcould notreportafeasiblesolutionwithin8hoursofrunningtime .Thedicultyinsolving exactformulationisduetotwomainreasons:Exactformulat ionrequiresaddingdecision variablestodeterminewhetheraprojectshouldbeassigned toasmalloralargeteam. Secondly,italsorequiresaddingdistancelimitconstrain tstotheformulationexplicitly. Ontheotherhand,iterativealgorithmsdealwithdistancec onstraintsimplicitlydueto 49

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Table2-10.Progressofexactoptimizationmodel Instances Time 1 % Gap 1 %GapinExactOptimizationModel (secs)3mins5mins10mins15mins1hr8hrs 1536.5855.36%--21.43%11.29%5.08%3.45%2506.6055.02%--38.14%38.14%18.92%7.69%3293.2856.17%34.41%34.41%12.86%12.86%6.15%3.17%4251.0259.08%-12.82%0.00%5161.9575.89%67.55%36.46%36.46%19.74%0.00%6576.8432.01%--23.88%23.88%1.92%0.00%7208.6265.95%-57.32%44.57%17.74%1.92%0.00%8709.6093.52%---87.54%13.60%0.00%9536.5068.60%--51.42%51.42%17.81%5.51% 10446.7042.42%--23.89%16.50%0.00% Average422.7760.40% theirconstruction.Furthermore,iterativealgorithmspa rtitiontheprojectsamongsmall andlargeteamsbeforetheiterativemodelsaresolved.Iter ativemodelsarebasedonsets ofprojectswhicharepre-partitionedamongsmallandlarge teams. 2.5SummaryandConclusions Thecurfewplanningproblem(CPP)isanimportantreal-life problemfacedby railroads.Theproblemdealswithschedulingrailtrackmai ntenanceforthewholetrain networkoveralong-termhorizon.Ourstudypresentsfourdi erentalgorithmsfornding theproblem'ssolutionwiththeobjectiveofminimizingthe totalnumberofviolationsin theperformanceconstraints.Thealgorithmsaretestedusi ngreal-lifeinstancesprovided byamajorNorthAmericanrailroadcompany.Comparisonsamong algorithmsare provided.Thebestperformingalgorithmisthe k -weeklyalgorithmwithbacktracking.In general,backtrackinghelpsimproveboththe1-weeklyand k -weeklyalgorithms.There aretwomainreasonswhybacktrackingisbenecial:Firstly ,thesolutionofthecurrent weekaectstheemptyslotsofthefollowingweeks.Duetothe alreadyoccupiedteams andongoingprojects,itmaybemorediculttondaschedule whichdoesnotviolate performanceconstraintsinthefollowingweeks.Secondly, inanyiteration,manypossible waysexistforassigningprojectstotheteams.Particularl ywhentherearemanyavailable projects,alternativewaysformakingtheassignmentsincr ease.However,usingaproject 50

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inalaterstagecansometimesbemorebenecialbecauseitma ycreateapossibilityto honortheperformanceconstraints.Backtrackingmayhelpd elaytheassignmentofa projectbycheckingforpossiblefutureviolations. Wealsocomparetheperformanceviolationsofthesolutioni mplementedbythe railroadcompanywithoursolutions.Thesolutionqualitya sdeterminedbythenumberof violationsintheperformanceconstraintsisimprovedbyar ound50%comparedtothatof therailroadcompany.Moreover,therunningtimesofthealg orithmsfallwithinpractical levelsandtheappliedmethodsarerexible.Theintegerprog rammingformulations providedaredesignedtorunwithdynamicallychangingdata sets.Hence,onecan enterapartialscheduleandthealgorithmcancompletether estoftheschedule.For instance,afterhalfoftheyearpassesandsomeprojectsare completed,railroadplanners canfeedtheremainingprojectsintothealgorithmandcondu ctanewrunbyusing dierentweightsfordierentperformancecriteria.Moreo ver,theplannercanpreassign arail/tie-surfacingprojecttoasmalloralargerail/tiesurfacingteam,modifyagiven partitionorelseprovideanewpartitionthatconsidersthe irusualbusinesspractices. Inatypicalexactformulation,distanceconstraintsandde cisionsofassigningany projecttoasmalloralargeteammaketheproblemevenmoreco mplex.Inthisstudy, wehaveprovidedasuccessfulwayofhandlingcomplicatedde cisionsandconstraints. Ouriterativealgorithmsimplicitlydealwiththesecomple xdecisionsandconstraints bydeningrexibleformulationsonshortplanninghorizons andthensolvingthem successively. 51

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CHAPTER3 ANOVELGENERICMODELTOSOLVECOMBINEDNETWORKDESIGNAND COMMODITYROUTINGPROBLEMS 3.1Introduction TheMulticommodityCapacitatedNetworkDesign(MCND)proble mhasmany applicationsintransportation,telecommunication,ener gy,computer,andproduction systems( Balakrishnanetal. 1997 ; Magnanti&Wong 1984 ; Minoux 1989 ).Thisclassof problemsisfrequentlyusedtoconstructandimprovenetwor ks,buildserviceroutesand schedules. InMCND,severaldistinctcommoditieswithgivendemandaret oberoutedona givennetwork.Eachcommodityisdenedbyitsuniqueorigin -destinationnodepairs. Dependingontheapplicationacommoditymightbedata,prod ucts,electricity,etc.Flow ofthecommoditiesisachievedbyinstallingfacilitiesont hearcsofthenetwork.Facilities aretransportationmediumsthatarerequiredtocarrytheco rrespondingcommodities. Eachfacilityhasacapacityandfacilitiesareinstalledin discreteamounts.Thereare twotypesofcostsforeacharc:Fixedcostforeachfacilityi nstalledandvariablecost forroutingoneunitofeachcommodityonafacilityinstalle d.Theobjectiveisdesigning networkandarrangingtherowofcommoditiessuchthattotal costisminimizedwhile satisfyingcapacityconstraintsanddemandrequirements. Inthisgenericversionofthe problem,rowofcommoditiescanbefractionalmeaningthats everalpathscanbeused tosatisfydemandofacommodityfromitsorigintoitsdestin ation.Hence,MCND problemcanbecalledasbifurcatedorsplittableMCNDproble m.Mostofthestudies intheliteraturefocusonbifurcatednetworkdesignproble ms.However,non-bifurcated problems,whereeachcommodityshouldrowonasinglepath,a riseinmanyapplications suchascomputernetworks( Gavish&Altinkemer 1990 ),telecommunicationnetworks ( vanHoeseletal. 2002 2003 ),expresspackagedelivery,andfreighttransportationin consolidation-basedcarriers. 52

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MCNDisawell-knownNP-hardproblem( Magnanti&Wong 1984 ; Minoux 1989 ). Hence,heuristicapproachesaremostlyusedforthesolution oftheproblemcomparedto exactapproaches.UncapacitatedversionsoftheMCNDarestud iedextensivelyinthe literatureandseveralecientspecializedalgorithmshad beendeveloped( Balakrishnan etal. 1989 ; Holmberg&Hellstrand 1998 ; Magnantietal. 1986 ; Magnanti&Wong 1984 ).UncapacitatednetworkdesignproblemisalsoNP-Hard( Balakrishnanetal. 1989 ; Magnanti&Wong 1984 )asthecapacitatedones.However,MCNDproblemis comparativelymoredicultanditarisesmoreofteninreallifecases. ThereareseveralvariantsofthegenericbifurcatedMCNDpro blem.Ifatmostone unitoffacilitycanbeinstalledoneacharc,thentheproble miscalledasxed-charge multicommoditycapacitatednetworkdesignproblem(seee. g. Crainicetal. 2000 ; Ghamloucheetal. 2003 ; Holmberg&Yuan 2000 ).Inthiscase,binarydesignvariables areusedforeacharcinsteadofgeneralintegervariables. Forsomenetworkdesignapplications,routingcostscanbei gnoredandonlyxed costsareimportant.Intheseapplications,wearetryingto installmultiplefacilitiesof samecapacityontheedgesofthenetwork.Thisvariantofthe MCNDisintroducedas networkloadingproblem(NLP)by Magnantietal. ( 1993 )and Magnantietal. ( 1995 ). Withrespecttocapacityusage Magnantietal. ( 1993 )and Magnantietal. ( 1995 )solve undirectedversionoftheNLPwherecapacityofafacilityins talledonanedgeisshared bythecommoditiesrowinginbothdirections.Hence,inundir ectedmodelssumofrow inbothdirectionsshouldnotexceedthecapacityofthefaci lity.Inthebidirected(also calledasdirected)NLP,onceafacilityisinstalled,sameca pacitycanbeusedseparately bycommoditiesrowingindierentdirections(seee.g. vanHoeseletal. 2002 2003 ).In directedmodels,upperboundofrowforcommoditiesrowingi ndierentdirectionsare constrainedseparatelyusingtwoconstraintsetsbythecap acityoftheinstalledfacility. Severaldierentmathematicalformulationshadbeenpropo sedintheliteraturein ordertomodelgenericbifurcatedMCNDproblemanditsvarian ts(suchasxed-charge 53

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MCND,networkloadingproblem,non-bifurcatedMCND).Thesef ormulationshavesome commoncharacteristicswhenweconsidertypesofvariables used.Theymostlyutilize arc-basedorpath-basedrowvariablesandarc-based(edgebased)designvariables.For instance, Crainicetal. ( 2000 )and Katayamaetal. ( 2009 )usepath-basedrowvariables andarc-baseddesignvariables. Ghamloucheetal. ( 2003 ), Holmberg&Yuan ( 2000 ), Frangioni&Gendron ( 2009 ), Crainicetal. ( 2001 ), Crainicetal. ( 2004 ),and Ghamlouche etal. ( 2004 )usearc-basedrowvariablesandarc-baseddesignvariable s. Bergeretal. ( 2000 )and Gendronetal. ( 2002 )solvenetworkloadingproblemsandusepath-based rowvariablesandedge-baseddesignvariables. Magnantietal. ( 1995 )and Bartolini& Mingozzi ( 2009 )usearc-basedrowvariablesandedge-baseddesignvariabl es.Notethat edge-baseddesignvariablesaremostlyusedfornetworkloa dingproblemswhichismore commonintelecommunicationsandcomputernetworks.Theon lyexceptionweknowis thepaperby Bartolini&Mingozzi ( 2009 )whichusesedge-baseddesignvariablestosolve anon-bifurcatednetworkdesignproblem. Onecommoncharacteristicofalltheformulationsistousea rc-based(oredge-based) designvariables.Anarc-baseddesignvariableisusedtodet erminenumberoffacilities tobebuiltonanarc.Inthisstudy,weutilizebinarypath-ba seddesignvariablestosolve anon-bifurcatednetworkdesignproblem.Inthisproblem,f acilitiesareinstalledonthe arcsandprovideacapacityonlyinthedirectionofthearcst heyareinstalledon.The usualpracticetosolvethiskindofproblemsisrstdecidin gonthefacilitiestobebuilt thendetermininghowthecommoditieswillrowoverthefacil itiesinasequentialmanner. Wecandierentiatebetweencompletelyisolatedandintera ctiveiterativealgorithms.In acompletelyisolatediterativealgorithm,underlyingsub problemsaresolvedincomplete isolationandresultofonesubproblemisusedasaninputfor theother;whereas,inan interactiveiterativealgorithmtwoproblemsaresolvedby iteratingbetweeneachanother, hence,resultsofbothproblemsaecteachother.Inatypica linteractiveiterativesolution procedure,yourstbuildasetofpotentialfacilitiesonth elinksofthenetwork,then 54

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selectacommoditybasedonsomecriteriaandidentifyaseto fpotentialpathswhichcan carrythiscommodityoverthenetworkofinstalledfaciliti es.Foreachpotentialpathwhich isasequenceofinstalledfacilities,youassigncandidate commodityandotheravailable commoditieswhichcangofromtheorigintothedestinationo fthepotentialpathand calculatecostofpathswithassignedblocks.Thenyouselec tthebestpathwithadded commoditiesonit,updateunassignedcommoditiesandlisto fpotentialpaths,andrepeat samestepsuntilnocommoditiesremainunassigned. Gorman ( 1998 )and Barnhartetal. ( 2002 )provideinteractiveiterativealgorithmsforrailroadop eratingplanandexpress shipmentdelivery,respectively. Networkdesignandcommodityroutingdependsoneachotheran dsolvingthese problemssequentiallymayresultintosuboptimalsolution s.Ourmotivationisto createaholisticapproachthatprovidesanintegratedsolu tiontonetworkdesign andcommodityroutingproblems.Forthisproblem,weintrod uceanewformulation basedonpath-basedrowvariablesandpath-baseddesignvar iables.Sinceweutilize path-baseddesignvariables,therearenoxedchargeconst raintsintheintroduced model.Proposedmodelenablesustosolvenetworkconstruct ionandcommodityrow problemssimultaneouslyinamoreeectivewaycomparedtot hesequentialapproachfor apracticalsizeinstanceofamajorrailroadcompany.Furth ermore,theresultantholistic approachutilizingpath-baseddesignvariablesisrexible enoughtohandlemanybusiness constraintsspecictotheparticularapplicationcontext .Theproposedgenericmodelcan beparticularlybenecialforassetmanagementconsiderat ioninvariousapplicationsand canbeadaptedtosolvenon-bifurcatednetworkdesignprobl emsofotherfreightcarriers suchasless-than-truckloadserviceprovidersandexpress packagedeliverycompanies. 3.2ProblemDescription Inourproblem,eachcommodityhastofollowasinglepathfro mitsorigintoits destination.Thisversionoftheproblemisknownasnon-bif urcated(orunsplittable,or binary)problem.Commoditieshavetosharethecapacityoft heinstalledfacilitiesonthe 55

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arcsofthenetwork.Initially,therearenofacilitiesinst alledonthearcs.Weshoulddecide atwhicharcsfacilitiesaretobeinstalledandhowthecommo ditiesrowontheinstalled facilitieswithoutexceedingtheircapacity.Thegoalisto minimizexedcostofinstalled facilitiesandvariableroutingcostswhilesatisfyingthe transportationdemand.Wecan categorizeourproblemasNon-bifurcatedMulticommodityCa pacitatedNetworkDesign (NMCND)problemwhichbelongstothegeneralclassofMCNDprobl ems.NMCND problemarisesinmanyreallifesystemssuchasless-than-t ruckloadtrucking,express packagedeliveryandrailwayfreightrouting. MaininputsfortheNMCNDproblemarethenetworksandcommodit ies.Commodities canbeanytypeoffreight(blocks,asetofproducts,letters ,messages,etc.)andare denedbydistinctorigin-destinationpairs.Weneedtoins tallfacilitiesonarcssuchthat allcommoditiescanbefeasiblycarriedoverthenetwork.Fa cilitiescanbeanytypeof transportationmodes(cables,trucks,trains,planes,shi ps)dependingontheapplication area(telecommunication,railways,trucking,etc.). Let G =( N A ) bethedirectednetworkwhere N isthesetofnodesand A istheset ofarcs.Let K denotethesetofcommodities.Eachcommodity k 2 K hasdemandof d k whichneedstobetransportedfromitsoriginnode O ( k ) toitsdestinationnode D ( k ) Usingthisnotation,wewilldemonstratepreviousmodelsint heliteratureandthenew model. 3.3TraditionalBifurcatedMulticommodityNetworkDesignM odels ClassicalformulationforbifurcatedMCNDusesarc-basedro wvariablesand arc-baseddesignvariables.Thisformulationispresented in Frangioni&Gendron ( 2009 ) asfollows; min X k 2 K X ( i j ) 2 A d k c k ij x k ij + X ( i j ) 2 A f ij y ij (3{1) 56

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subjectto X j 2 N + i x k ij X j 2 N i x k ji = k i 8 i 2 N k 2 K (3{2) X k 2 K d k x k ij u ij y ij 8 ( i j ) 2 A (3{3) 0 x k ij 1 8 ( i j ) 2 A k 2 K (3{4) y ij 2 Z + 8 ( i j ) 2 A (3{5) where N + i = f j 2 N j ( i j ) 2 A g N i = f j 2 N j ( j i ) 2 A g and k i =1 if i = O ( k ) k i = 1 if i = D ( k ) and k i =0 otherwise.Continuous x k ij variablesarethearc-basedrowvariables whichindicatethefractionofcommodity k rowingonarc ( i j ) 2 A andinteger y ij variablesarethearc-baseddesignvariables. c k ij isthevariablecostofmovingoneunitof commodity k onarc ( i j ) and f ij isthexedcostofinstallingoneunitoffacilityonarc ( i j ) d k isthedemandofcommodity k and u ij istherowcapacityprovidedbyafacility installedonarc ( i j ) AnotherfrequentlyusedmodeltoformulateMCNDistheonewith path-basedrow variablesandarc-baseddesignvariables.Thisformulatio niscanbepresentedasfollows (seee.g. Crainic 2000 ,forasimilarrepresentationofthismodel); min X p 2 P d p c p x p + X ( i j ) 2 A f ij y ij (3{6) subjectto X p 2 P k x p =1 8 k 2 K (3{7) X p 2 P ( i j ) d p x p u ij y ij 8 ( i j ) 2 A (3{8) 0 x p 1 8 p 2 P (3{9) y ij 2 Z + 8 ( i j ) 2 A (3{10) 57

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where P k isthesetofpathsforcommodity k and P = [ Kk =1 P k P ( i j ) isthesetof pathsthatrowsthrougharc ( i j ) .Wesetdemandofapathequaltothedemandof itscorrespondingcommodity( d p = d k forall p 2 P k ).Continuous x p variablesarethe path-basedrowvariableswhichindicatethefractionofcom modity k thatrowsonpath p 2 P k 3.4Non-bifurcatedNetworkDesignProblem InordertoconvertgenericbifurcatedMCNDformulationtoth enon-bifurcated MCNDproblem,wecansimplydenerowvariablesasbinaryvari ablesintheabove twoformulations.Notethatrequiringrowvariablestotakeb inaryvaluesconsiderably increasesthedicultyofthenon-bifurcatednetworkdesig nproblemcomparedtothe bifurcatedcase.TheresultingNMCNDformulationwitharc-ba sedrowvariablesand arc-baseddesignvariablesisasfollows: min X k 2 K X ( i j ) 2 A d k c k ij x k ij + X ( i j ) 2 A f ij y ij (3{11) subjectto X j 2 N + i x k ij X j 2 N i x k ji = k i 8 i 2 N k 2 K (3{12) X k 2 K d k x k ij u ij y ij 8 ( i j ) 2 A (3{13) x k ij 2f 0,1 g8 ( i j ) 2 A k 2 K (3{14) y ij 2 Z + 8 ( i j ) 2 A (3{15) NMCNDwithpath-basedrowvariablesandarc-baseddesignvari ablesisformulatedas follows: min X p 2 P d p c p x p + X ( i j ) 2 A f ij y ij (3{16) 58

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subjectto X p 2 P k x p =1 8 k 2 K (3{17) X p 2 P ( i j ) d p x p u ij y ij 8 ( i j ) 2 A (3{18) x p 2f 0,1 g8 p 2 P (3{19) y ij 2 Z + 8 ( i j ) 2 A (3{20) 3.5ProposedNon-bifurcatedNetworkDesignModel Inallofthefourformulationsdenedforbifurcatedandnon -bifurcatednetwork designproblems,amajorproblemistheexistenceofxedcos tsofinstallingfacilities. Inallformulations,whiledeningcapacityconstraints,w emultiplyarc-baseddesign variables(oredge-baseddesignvariablesespeciallyinca seofnetworkloadingproblems) withthecapacityoftheinstalledfacilitiesintheright-h andsideoftheconstraint.In ordertoremovethesexed-chargeconstraints,wepresenta novelformulationwhichuse path-baseddesignvariablesandpath-basedrowvariables. Inthisnewmodel,selectinga path-baseddesignvariablecorrespondsselectingallthea rcsofthispathinthetraditional arc-baseddesignformulation.Oncepathsfordesignvariab lesareenumerated,weuse whatwecallfacility-arcsinsteadoftraditionalarcsinth eformulation.Atraditional arcisaregulararcofthenetworkandinarc-baseddesignfor mulationsdecisiontobe takenisthenumberoffacilitiestoinstallonthearc.Afaci lity-arcresemblesafacility installedonaspecicarc,henceitisan(arc,facility)pai r.Proposedmodeldecidesto usethisfacilityifatleastonecommoditypaththatrowsont hepathofthisfacility-arc isselected.Afacility-archasthecapacityofthefacility thatisplannedtoinstallonthat arc.Inthenewformulation,wecanconstructourcapacityco nstraintsonthefacility-arcs insteadoftraditionalarcs.Withthehelpoffacilityarcs, xed-chargeconstraintsused intheformulationsutilizingarc-baseddesignvariablesa reremoved.Insteadofdening anintegervariablefordesignvariables,weareabletouseb inaryvariablestodetermine 59

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whetherweuseadesignpathornot.Hence,inthemodel,binary variablesareusedfor bothdesignandrowdecisions.Theresultingformulationha sastructurethatallowsless fractionalvaluesinitsLPrelaxation.Theproposedformul ationisasfollows: min X p 2 P d p c p x p + X p 2 P f p y p (3{21) subjectto X p 2 P k x p =1 8 k 2 K (3{22) X p 2 P p ( i j ) d p x p u p ij 8 p 2 P ,( i j ) 2 A p (3{23) y p x p 8 p 2 P p 2 P p (3{24) x p 2f 0,1 g8 p 2 P (3{25) y p 2f 0,1 g8 p 2 P (3{26) where P isthesetofalldesignpaths.Notethateachcommoditypath p 2 P isdenedby asequenceoffacility-arcs. P p representsthesetofallcommoditypathsthatrowsthrough designpath p .Hence, P p includesanycommoditypath p thatrowsonatleastoneofthe facility-arcsofthisdesignpath p P p ( i j ) showsthesetofcommoditypathsthatrowon designpath p whichispassingthrougharc ( i j ) Proposedformulationcanbeappliedinmanynon-bifurcated networkdesign applications.Itmightbeespeciallyusefulwhenthefacili tiesinstalledaremovingobjects liketrucks,trains,aircrafts,etc.Intheseapplications ,path-baseddesignvariablescan easilyhelptodeneassetmanagementrelatedbusinesscons traints.Forinstance,a path-baseddesignvariablecanbethoughtasatruck'sroute .Arowpathvariablecan representacommoditytravelingonasegmentofagiventruck 'spath.Notethata commoditycanalsouseseveraldesignpaths(e.g.severaltr uckpaths)onitswayfromits origintoitsdestination. 60

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3.6SummaryandConclusions Wefocusedonanon-bifurcatednetworkdesignproblemwhere eachcommodityrows onasinglepathfromitsorigintoitsdestination.Inthispr oblem,facilitiesareinstalled onthearcsandeachfacilityprovidesacapacityinthedirec tionofthearconwhichit isinstalled.Initially,therearenofacilitiesinstalled onthearcs.Suchnon-bifurcated networkdesignproblemsmayariseforvariousfreightcarri erssuchasless-than-truckload serviceproviders,railfreightcarriers,expresspackage deliverycompanies,etc.Wepropose anovelmodelwiththegoalofsolvingnetworkdesignandcomm odityroutingproblems simultaneously.Forthispurpose,weusepath-basedrowvar iablesandpath-baseddesign variableswhicharebothbinaryvariables.Asfarasweknow,p ath-basedbinarydesign variableshadnotbeenusedintheliteratureforthesolutio nofnon-bifurcatednetwork designproblems.Furthermore,thesetwohighlyrelatedpro blemsweremostlysolved usingsequentialapproaches.Proposedmodelprovidesahol isticapproachandisrexible forincorporatingassetmanagementconsiderations.Inthe nextchapter,weadaptthe proposedmodeltosolveapracticalsizeinstanceofamajorr ailroadcompany. 61

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CHAPTER4 INTEGRATIONOFTRAINROUTINGANDBLOCK-TO-TRAINASSIGNMENT PROBLEMS:ANAPPLICATIONOFTHEPROPOSEDNETWORKDESIGNMODEL 4.1Introduction Operatingplanofarailroaddeterminesmovementofrailcar loads,locomotives andcrewsoverrailnetwork.Developinganoperatingplanre quirestosolveblocking, trainrouting,block-to-trainassignment,emptycardistr ibutionandtraintimetabling problems.Theseproblemsareoftensolvedseparatelydueto sizeanddicultyofthe overallproblem.Asanexample,amajorNorthAmericanrailroad companytypically operatesaround400trainsdailytomove1300blocksonanetw orkof12000nodesand 2000nodes. Theobjectiveinsolvingtrainroutingandblock-to-traina ssignmentproblemsisto constructasetoftrainssuchthatallblockscanbecarriedf easiblyoverthetrainnetwork. Hence,themaindecisionstobegivenaretrains(origin,dest inationandrouteofeach train)andblock-to-trainassignments.Thesehighlyinter relatedproblemsareusually solvedinasequentialmanner.Candidatetrainsareconstru ctedrstandthenblocksare assignedtocandidatetrainsusingacostfunctionandbestc andidatetrainisselectedto bebuilt.Thisprocesscontinuesinaniterativewayuntilno blocksremaintoberouted. Railroadsneedamoreholisticapproachforcreatingmultip letrainsatoncewhilefeasibly routingblocksoverthesetoftrainsformed. Cordeauetal. ( 1998 )makeaverygoodreviewofoptimizationmodelsforthe trainroutingandschedulingproblems.Therstpartofthei rreviewsurveystrain routingproblemsinthecontextofrailfreighttransportat ionandthesecondpartreviews optimizationmodelsfortrainschedulinginbothfreightan dpassengertransportation. Jha etal. ( 2008 )solveblock-to-trainassignmentproblemandassumethatb lockingplan,train routingsandtheirschedulearegiven. Dierentapproacheshavebeenintroducedtosolveinterrel atedrailroadoperating planproblemssimultaneously. Gorman ( 1998 )and Ahujaetal. ( 2005 )conrmthat 62

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mostpapersintheliteraturesolvetrainroutingprobleman dblock-to-trainassignment problemseparately.Aniterativeschemeisusedtosolvethes etwoproblemssuccessively (seee.g. Crainicetal. 1984 ; Crainic&Rousseau 1986 ; Haghani 1989 ; Keaton 1989 1991 ). Gorman ( 1998 )alsodecomposestheproblemsimilarly.Theauthorrstgen erates candidatetrainschedulesbyusingtabu-enhancedgenetics earchandthencheckscostof eachschedulebyroutingblocksoverthescheduleinanitera tiveway. Gorman ( 1998 )'s maincontributionistoproduceadetailedweeklyoperating planwhichprovideswhich trainrunswhichday. Inthisstudy,weaddressthetrainroutingandblock-to-tra inassignmentproblems andadaptourgenericnon-bifurcatedmodeltosolvethesepr oblemssimultaneously.We assumethatblockingplanisavailable.Weincorporateloco motiveandcrewconsiderations intothissinglemodel.Theresultingmodelissuccessfulin ndingagoodsolutionin reasonabletimelimitsforarealworldinstanceofamajorNor thAmericanrailroad company.Theproposedmodelisalsosuitableforobtaininga nincrementalsolutionin casesometrainpathsorblockpathsarefavorableandselect edinadvancebytherailroad. 4.2ProblemDescription Maininputsoftheproblemarerailnetworkandblocks.While buildingtrainsand assigningblockstotrains,weneedtosatisfysomefeasibil ityconstraintsandalsomany businessconstraints.Feasibilityconstraintsaretheone sthatalreadyexistinthegeneric model.Thedenitionsoftheseconstraintsusingthetermin ologyadaptedfromthis problemareasfollows: 1. Exactlyoneblockpathisselectedforeachblock. 2. Relationbetweenblockpathandtrainpathvariables:Ifabl ockpaththatrowsona trainpathtisselectedthentrainpathtshouldalsobeselec ted. Businessconstraintsincorporatedintotheproposedgener icoptimizationmodelareas follows: 63

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1. TrainloadcapacitiesatlinksThisconstraintrestrictsthemaximumvolume(measuredbyn umberofcars)ona trainwhilepassingthroughanarc. 2. TraincapacitiesatnodesNumberoftrainsstartingandterminatingatanodeislimited 3. TraincapacitiesatlinksNumberoftrainsthatcanpassthroughalinkislimited. 4. BlockswapsatnodesNumberofblockswapsatanodeislimited. 5. LocomotiveimbalanceatnodesInordertodeterminelocomotiveimbalance,werstneedtod eterminelocomotive requirementsforatrain.Forthispurpose,weintroduceloc omotiverequirement variablesthatshowtheminimumlocomotiverequirementfor eachtrain.Our assumptionisthatminimumnumberoflocomotivesrequiredt opullatraindepends onthemaximumlocomotiverequirementoverallthesegments ofthetrainroute.In ordertondtheimbalance,weneedtheactualrowoflocomoti vesoriginatingand terminatingateachstation.Forthispurpose,wedeneanot hersetoflocomotive rowvariables,whichisequaltothenumberofactivelocomot ivesifthereisno locomotivesdeadheadedonthetrain.Wethenpenalizetheim balancebetweenthe numberoflocomotivesattachedtooriginatingtrainsandte rminatingtrainsateach station. 6. CrewimbalanceoncrewdistrictsWedenecrewimbalancevariablestodeterminetheimbalanc eoftrainsrunning inoppositedirectionsonacrewdistrictandthenpenalizec rewimbalanceinthe objectivefunction. 7. Work-eventcapacityThenumberoftrainsthatcanstopatanodeislimited.Aselec tedblockpath generateswork-eventsatallitsswitchingnodesandaselec tedtrainpathgenerates work-eventsatallitscrew-changingnodes. 8. CrewworktimelimitMaximumworktime(ontracktimeplusstoptime)inacrewdist rictislimited. Apartfromtheusualxedandvariablecostsmentionedintheg enericmodel,wealso neededtoincorporatemanyotherobjectiveswhicharespeci ctothisproblem.These objectivetermscanbelistedasfollows: 64

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TrainCosts 1. TrainStartCostFixedcostofrunningapotentialtrain 2. TrainMilesCostTraindistancecostforeachpotentialtrain 3. TrainWorkEventCostsCostincurredduetoworkeventswhenatrainstopsataninter mediatestationon itsroute.Theintermediateworkeventsoccurduetopickups andsetosofblocksor crewchanges. CarCosts 1. BlockSwapsCostBlockswapcostsforeachblockpath.Individualcostsdepen donthenumberof blockswapsperformedinthecorrespondingblockpath. 2. CarMovementCostCarmovementcostiscalculatedforeachblockpathbymultip lyingcorresponding block'svolumewithblockpath'sdistance. 3. CarHoursCostCartraveltimecostdependsontraveltimeofcarsonselecte dblockpaths. Locomotivecosts 1. LocomotiveLightTravelCostLighttravelisduetoimbalanceoflocomotiverowsoverthen etwork.Ifthereisan imbalance,locomotiveshavetowithoutpullingatrainwhic hiscalledaslighttravel. 2. LocomotiveOwnershipCostThisistheduetoperhourcostofowningalocomotive. 3. LocomotiveActivePullingCostThisistheperhouroperatingcostofalocomotiveactivelyp ullingatrain. 4. LocomotiveDeadheadingCostThisisthedeadheadingcostofalocomotiveperhour Crewcosts 1. ActiveCrewCostForeachcrewlinkonthepathofapotentialtrain,wecalcula tecrewstartcost,crew wagesandtripcost.Crewwagesdependonthetraveltimeoftr ainonthegiven crewlink. 65

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2. CrewImbalanceCostImbalanceinnumberoftrainsrunninginoppositedirection sonacrewdistrictis multipliedwiththecostofacrewdeadheadinginthatcrewdi strict. 4.3SolutionApproach 4.3.1GenericModelTailoredfortheIntegratedProblem Pathbasedrowanddesignvariablesformthecoreofthepropo sednon-bifurcated capacitatednetworkdesignmodel.Inthisspecicproblem, commoditiesarethegiven blocksandeachcommodityistobecarriedbytrainsfromthei rorigintotheirdestination. Inamodelwitharcbaseddesignvariables,onecanassumeins tallingtransportation mediumswithsucientcapacityonthearcsofthenetwork.How ever,actualrouteofa traincannotbedeterminedbythisapproach.Sincetrainsar emovingoverthenetwork, usingapath-basedmodelwouldbebenecialtodeterminetra inroutesandincorporate manyassetmanagementrelatedobjectivesintothemodelwhi chwouldotherwisenotbe possibleunlessasequentialapproachisused.Blockpathva riablescorrespondtopath basedrowvariablesandtrainpathvariablescorrespondtop athbaseddesignvariables intheproposedmodel.TheproposedNMCNDformulationcanbead aptedtoformulate integratedtrainroutingandblock-to-trainassignmentby deningthefollowingsets, parametersanddecisionvariables. Sets: N :Setofnodes A :Setofarcs A t :Setofarcsofpotentialtrainpath t T :Setofallpotentialtrainpaths T + i :Setoftrainpathsoriginatingatnode i T i :Setoftrainpathsterminatingatnode i B :Setofallblocks :Setofallpotentialblockpaths b :Setofpotentialblockpathsforblock b 66

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t :Setofpotentialblockpathsthatrowontrainpath t t ( i j ) :Setofpotentialblockpathsthatusearc ( i j ) androwontrainpath t T ( i j ) :Setoftrainsthatpassthrougharc ( i j ) i :Setofpotentialblockpathsthatmakesablockswapatnode i W ti :Setofblockpathsthatcauseaworkeventfortrain t atnode i W U ti :Setofblockpathsthatcauseapick-upworkeventfortrain t atnode i W O ti :Setofblockpathsthatcauseaset-oworkeventfortrain t atnode i Z i :Setoftrainpathsthatstopsatcrew-changenode i L :Setofallcrewlinks L t :Setofcrewlinksoftrainpath t T l T l :Setoftrainpathsthatgoesinoppositedirectionsoncrewl ink l N t :Setofnodesoftrainpath t N l t :Setofnodesofcrewlink l whichareusedbytrain t Parameters: d p :Numberofcarsintheblockassociatedwithblockpath p ,(notethat d p = d b 8 p 2 b ) u t ( i j ) :Capacityoftrain t onlink ( i j ) intermsofnumberofcarsitcancarry O i :Maximumnumberoftrainsthatcanoriginateatnode i E i :Maximumnumberoftrainsthatcanterminateatnode i F :Maximumnumberoflocomotives(activeanddeadheading)th atcanbe attachedtoatrain @ :Minimumnumberofactivelocomotivesrequiredtopullatra in } :Maximumnumberofactivelocomotivesthatcanbeusedtopul latrain H ( i j ) :Horsepowerrequirementperunitweightonarc ( i j ) H std :Horsepowerofastandardlocomotive w p :Weightoftheblockwhichiscarriedbyblockpath p (Notethat w p = w b 8 p 2 b ) 67

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` ( i j ) :Maximumnumberoftrainsthatcanpassthrougharc ( i j ) r i :Blockswapcapacityatnode i C i :Work-eventcapacityofnode i l t :Traveltimeoftrain t oncrewlink l u i :Stoptimeforapick-upworkeventatnode i o i :Stoptimeforaset-oworkeventatnode i < :Maximumcrewworktimeinacrewlink f t :Fixedcostofrunningpotentialtrainpath t m t :Trainmilesforpotentialtrain t c 1 :Costofmovingatrainpermile c 2 :Intermediateworkeventscostperstop n p :Numberofblockswapsforblockpath p p :Blockswapcostdependingonthevolumeofthecorrespondin gblockfor blockpath p v p :Costofmovingcarsofblockpath p & p :Cartraveltimecostforblockpath p dependingonvolumeofthe correspondingblockandtraveltimeofthecarsalongtheblo ckpath l :Costpaidtoactivecrewsoncrewlink l l :Crewdeadheadingcostforthecrewlink l r :Lighttravelpenaltyforalocomotive c 3 :Costofactivelypullingatrainperhour c 4 :Costofdeadheadingalocomotiveperhour c 5 :Costofowningalocomotiveperhour t :Traveltimeofpotentialtrainpath t inhours DecisionVariables: x p :1ifcandidateblockpath p 2 isselected,0otherwise y t :1ifpotentialtrainpath t 2 T isselected,0otherwise 68

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~ v i :Locomotiverowimbalanceatnode i v l :Crewimbalanceatcrewlink l = i :Amountofviolationofwork-eventcapacityatnode i w ti :1iftrain t stopsatnode i forawork-event,0otherwise r t :Activelocomotiverequirementfortrainpath t z t :Numberoflocomotivesattachedtotrain t (activeplusdeadheadinglocomotives) Notethatmodelletsustodenedierentcapacitiesoneachli nkforagiventrain. Thisismorerealisticsincerailroadsmayrequiretheloadl imitstobechangedaccording togeographicalconditionsonthatarcoftherailnetwork.Ho wever,onecanalsousea constantcapacity,inthatcase u t ( i j ) = u t forallarcs ( i j ) .Usingthesenotationsthemodel isformulatedasfollows: min X t 2 T f t y t + c 1 X t 2 T m t y t + c 2 X t 2 T X i 2 N t w ti + X p 2 p n p x p + X p 2 v p x p + X p 2 & p x p (4{1) + X t 2 T X l 2 L t l y t + X l 2 L l v l + r X i 2 N ~ v i + c 3 X t 2 T t r t + c 4 X t 2 T t ( z t r t )+ c 5 X t 2 T t z t subjectto X p 2 b x p =1 8 b 2 B (4{2) X p 2 t ( i j ) d p x p u t ij 8 t 2 T ,( i j ) 2 A t (4{3) y t x p 8 t 2 T p 2 t (4{4) X t 2 T + i y t O i 8 i 2 N (4{5) X t 2 T i y t E i 8 i 2 N (4{6) X p 2 i x p r i 8 i 2 N (4{7) 69

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X t 2 T ( i j ) y t ` ( i j ) 8 ( i j ) 2 A (4{8) w ti x p 8 t 2 T i 2 N t p 2 W ti (4{9) w ti y t 8 t 2 T i 2 N t : t 2 Z i (4{10) X t 2 T w ti C i + = i 8 i 2 N (4{11) v l X t 2 T l y t X t 2 T l y t 8 l 2 L (4{12) v l X t 2 T l y t X t 2 T l y t 8 l 2 L (4{13) y t l t + X i 2 N l t ( X p 2 W U ti x p u i + X p 2 W O ti x p o i ) <8 t 2 T l 2 L t i 2 N l t (4{14) r t X t ( i j ) x p w p H ( i j ) = H std 8 t 2 T ,( i j ) 2 A t (4{15) @ y t r t } y t 8 t 2 T (4{16) r t z t Fy t 8 t 2 T (4{17) ~ v i X t 2 T + i z t X t 2 T i z t 8 i 2 N (4{18) ~ v i X t 2 T i z t X t 2 T + i z t 8 i 2 N (4{19) x p 2f 0,1 g8 p 2 (4{20) y t 2f 0,1 g8 t 2 T (4{21) w ti 2f 0,1 g8 t 2 T i 2 N t (4{22) v l v i 0 8 l 2 L i 2 N (4{23) Constraints( 4{2 )ensurethatexactlyoneblockpathisselectedforeachbloc k. Constraints( 4{3 )restrictthesumofblockvolumesassignedtoatrainonanar c. Constraints( 4{4 )formtherelationbetweenrowanddesignvariables.Itrequ iresthat atrainpathisselectedifablockpathrowsonitandthatbloc kpathisselectedin 70

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thesolution.Constraints( 4{5 )and( 4{6 )enforcetheoriginatingandterminatingtrain capacitiesatnodes.Constraints( 4{7 )limitthenumberofblockswapsoccurringateach node.Constraint( 4{8 )restrictsthetotalnumberoftrainsthatarepassingthrou gheach arc.Constraints( 4{9 ),( 4{10 )and( 4{11 )areusedtolimitthenumberoftrainsthatcan stopatanode(workeventcapacity).Constraints( 4{12 )and( 4{13 )areusedtocalculate thedierenceoftrainsrunninginoppositedirectionsonac rewdistrict.Constraints ( 4{14 )limitthemaximumcrewworktime(traveltimeplusstoptime )inacrewdistrict. Constraints( 4{15 )and( 4{16 )determinetheactivelocomotivesrequiredtopulleach train.Constraints( 4{17 )areusedtocalculateactualnumberoflocomotives(active and deadheading)attachedtoatrain.Constraints( 4{18 )and( 4{19 )determinetheimbalance oflocomotivesateachnode.Objectivefunction( 4{1 )minimizesthecostsrelatedtotrains (trainstarts,trainmiles,trainworkevents),cars(block swaps,carmovement,carhours), locomotives(lighttravel,activepulling,deadheading,l ocomotiveownership)andcrews (activecrewcost,crewimbalance).4.3.2IterativeTrainConstructionApproach Iterativeapproachbuildstrainsonebyoneuntilforroutin gallunassignedblocks. Thepseudocodeofthissequentialmethodisasfollows: Enumeratepotentialtrainpaths Constructtrainsiterativelyasfollows: { Selectacandidateblockforwhichatrainisbuilt.Thecandi dateblockis selectedbasedondailyvolumeandnumberofpotentialtrain s.Ascalingvalue isusedtodetermineminimumblockvolume.Asmodeliterates, thescaling factorisreducedbythefactoroftwo. { Forthecandidateblock,identifyasetofpotentialtrainsw hichcancarrythis block. { Assignblockstoeachpotentialtrainusingagreedyheuristi c. Foreachpotentialtrain,assigncandidateblockandotherb locksgoing fromtheorigintothedestinationofthetrain. 71

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Computecostimpact:Inaniterativeloop,assesstheobject ivefunction impactofaddingablocktothetrainandcheckthefeasibilit yofaddingthe block.Addthebestfeasibleblocktothepotentialtrainandi terateuntilno moreblockscanbeadded. { Constructonetraintobeaddedinthesolution:Amongthepote ntialtrains withassignedblocks,selecttheonewithbestobjectivefun ctionvalueifitis feasiblefrom crewrouteandtrainloadperspective nodesandlinkscapacityperspective { Updateunassignedblocks,listofpotentialtrainsandallth eotherstatistics. Repeatsamesteps,byreducingtheminimumblockvolumewhic hcanbe candidateinnewtrainconstructionbyhalfuntilallblocks arerouted. 4.4ComputationalExperience WetryoursolutionmethodonarealinstanceofamajorNorthAme ricanRailroad. Inthisinstance,1200blocksaretoberoutedfromtheirorig intotheirdestination.We compareresultswegetusingourintegratedmodelwiththeso lutionimplementedbythe railroadandalsowiththeresultswegetusingtheiterative approach. InTable 4-1 ,wepresentpercentimprovementsofiterativeapproachand proposed holisticapproachoverrailroadsolution.Overallcostimp rovementsfortheholisticand iterativeapproachesare15.84%and4.46%successively.Ex ceptforthenumberofblock swaps,holisticapproachconsistentlyhasbetterimprovem entsforallcostfactorscompared totheiterativeapproach.Increaseinnumberofblockswaps isacceptablesincetotal numberoftrainsandaveragelengthofthetrainsdecreasesi ntheholisticapproach.This leadstomoreblockswapsandbetterutilizationoftraincap acities.Averagenumber ofcarsonatrainarcis82.84fortheholisticapproach.Fort herailroadsolutionand iterativeapproach,averagenumberofcarsinatrainarcis7 2.01and75.16consecutively. Computerruntimesoftheiterativeapproachandproposedmo delisgiveninTable 4-2 4.5SummaryandConclusions Weadaptedourgenericnon-bifurcatedmodelforthesolutio noftrainroutingand block-to-trainassignmentproblems.Usingthegenericnonbifurcatedmodelwhich 72

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Table4-1.Percentageimprovementsofproposedmodelandit erativeapproachoverthe railroadsolution IterativeApproachHolisticApproach TrainCostSavings TrainStarts1.909.48 TrainMiles-0.8312.59 TrainWorkEvents0.497.54 CarCostSavings TotalCar-Hours1.444.62 CarMovementCost0.230.28 TotalBlockSwaps-4.82-13.25 LocomotiveCostSavings Totallocomotiverequirement13.6023.84 OwnershipCost2.6810.01 ActivePullingCost1.103.94 DeadheadingCost2.1610.32 LightTravelCost9.4621.43 CrewCostSavings TotalCrewRequirements-0.6113.11 ActiveCrewCost1.2310.76 CrewImbalanceCost31.6764.76 TotalOperatingPlanCost 4.4615.84 Table4-2.Comparisonofruntimesforproposedmodelandite rativeapproach IterativeApproachHolisticApproach Computationaltime(secs) 302.35395.56 utilizesbinarypathbaseddesignandrowvariableshasledu sincorporatemanyasset relatedbusinessconstraints.Notethatthisformulationis alsosuitableforapplying anincrementalapproachincasesometrainpathsorblockpat hsarefavorableforthe railroad.Weappliedoursolutionapproachonareallifeins tanceofamajorNorth Americanrailroadcompanyandobtainedagoodqualitysoluti onwhichimprovesthe solutionoftherailroadanditerativeapproachinafewminu tes. 73

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CHAPTER5 ADECOMPOSITIONAPPROACHFORSERVICENETWORKDESIGNPROBLEMS ONHUB-AND-SPOKENETWORKS 5.1Introduction Traditionalnetworkdesignproblemsaretakenintoanewlev elwiththeadditionof timedimensionintothedesign.Carriershavetoprovidestr ictdeliverytimes(usually calledasservicecommitments)fortheirservicestheyanno uncebetweendierent origin-destinationterminalpairsovertheirnetwork.Des igningareliableandcost ecientservicenetworkiscriticalintheoperationofalls erviceproviders.Atimeand costecientplanconsistsofanoptimalloadscheduleandsh ipmentroutesthatbringall shipmentstotheirdestinationsontimeatminimaloperatin gcost.Aloadcorresponds toacapacitatedfacilityinstalledonalinkifweconsiderc apacitatedmulticommodity networkdesignproblems.Inservicenetworkdesigncontext ,aloadcanbebetterdened asaconsolidationofdierentshipmentstravelingtogethe ronalinkfromoneterminalto another. Crainic ( 2000 )reviewsnetworkdesignmodels,relevantsolutionapproac hesand servicenetworkdesignformulationsinfreighttransporta tion.Inamorerecentpaper Wieberneit ( 2008 )reviewsdierentformulationsandsolutionframeworksfo rservice networkdesignproblems.Authorpresentsvedierentpract icalservicenetworkdesign problemsinliteraturearisinginexpressshipmentdeliver y,lettermaildeliveryonright networks,andless-than-truckloadoperationsinEuropean dNorthAmerica.Service networkdesignreviewswhicharespecializedonaspecictr ansportationmode,long-haul orintermodaloperationsinclude Christiansenetal. ( 2007 )formaritimetransportation, Assad ( 1980 )and Cordeauetal. ( 1998 )forrailtransportation, Crainic ( 2003 )for long-haultransportation,and Crainic&Kim ( 2007 )forintermodaltransportation. Formulationsforservicenetworkdesignproblemscanbecla ssiedasstatic (frequency)ortime-dependent(dynamic).Thisclassicat ionismostlydueto Crainic ( 2000 )and Crainic&Kim ( 2007 ).Instaticmodels,timedimensionisimplicitly 74

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consideredandservicesandtheirfrequenciesaredetermin ed.Itisassumedthatdemand doesnotvaryduringtheplanninghorizon.Someexamplesofs taticformulationshave beenpresentedby Crainic&Rousseau ( 1986 )and Crainic&Roy ( 1988 )formultimodal transportation, Powell&She ( 1983 ), Powell ( 1986 )and Powell&She ( 1989 )forLTL motorcarriers, Keaton ( 1992 )and Newtonetal. ( 1998 )forrailtransportation, Barnhart &Schneur ( 1996 ), Armacostetal. ( 2002 ), Kimetal. ( 1999 )forexpressshipmentservices, Christiansenetal. ( 2004 )formaritimetransportation.Intime-dependentmodels, movementofresourcesisrepresentedintimeanddecisionsi nvolvingservicesandtheir detailedschedulesaremade.Sincetimedimensionisexplic itlyconsidered,formulations arebasedontime-spacenetworksandresultantnetworksand formulationsarelarger insize.Time-dependentformulationexamplesarestudiedb y Andersenetal. ( 2009a ), Andersenetal. ( 2009b ),and Pedersenetal. ( 2009 )forrailintermodaloperations, Haghani ( 1989 )and Gorman ( 1998 )forrailtransportation, Smilowitzetal. ( 2003 )forexpress shipmentservices, Farvolden&Powell ( 1994 )forLTLcarrierservices. Andersenetal. ( 2009a )integrateassetmanagementconsiderationsintoservice networkdesignmodelsforconsolidation-basedfreightcar riersandcomparefourdierent formulationsforservicenetworkdesignwithassetmanagem ent.Thefourformulations combinearcandcycledesignvariableswitharcandpathrowv ariables.Authorsgenerate 21instancesandcomparetheperformanceofformulationsus ingtheseinstances.They alsotestimpactofassetmanagementconstraintsonthesolu tionqualityandsolutiontime bygraduallyremovingassetrelatedconstraintsfromtheor iginalarc-arcformulationof servicenetworkdesignwithassetmanagement. Andersenetal. ( 2009b )presentanew modelforservicenetworkdesignwithassetmanagementandm ultiplereetcoordination. Themodeladdressesintermodaltransportationoperations andaimstodetermineservice departuretimessuchthatdemandthroughputtimeandxedco stofoperatingthereets areminimized.Authorstestthemodelonanactualrailinterm odalapplication.Other 75

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recentcontributionsincorporatingassetmanagementinto servicenetworkdesigninclude Andersen&Christiansen ( 2009 ), Pedersenetal. ( 2009 ),and Teypazetal. ( 2010 ). Liumetal. ( 2009 )and Hoetal. ( 2009 )addressservicenetworkdesignproblems withstochasticdemand. Liumetal. ( 2009 )studytheimportanceofintroducing stochasticityinservicenetworkdesignproblemsusingasm allsize,basicproblem. Ho etal. ( 2009 )proposesamodelinspiredby Liumetal. ( 2009 )anduseametaheuristic methodtosolvelargeinstancesofservicenetworkdesignpr oblemwithstochasticdemand. Hub-and-spokenetworksarefrequentlyutilizedtosolvecon solidationproblems, where,insteadofsendingeachshipmentdirectlytoitsdest ination,shipmentsare combinedintoloadsandroutedthroughhubs.Hub-and-spokeb asednetworksare composedofendoflineterminalsandbreak-bulks(hublocat ions).Break-bulklocations areusedtoconsolidatedemandcomingfromdierentdestina tions. Jarrahetal. ( 2009 ) solvesareal-life,large-scaleinstanceoftheservicenet workdesignprobleminthecontext oftheLTLindustryonahub-and-spokesystem.Theypresenta novelnetworkdesign modelanddecomposethismassivemodelintoasetofecienti ntegerprogramming modelsforeachdestinationterminalalongwithacoordinat ingmasternetworkdesign problem.Ineachsubproblemtheygeneratealoadplanningtr eewhichdenesthe feasiblefreightrowtoadestinationterminalfromallothe rterminalsinthenetwork. Loadplanningtreesisintroducedasdecisionvariablesint hemasternetworkdesign formulation.Linearprogramming(LP)relaxationofthemas ternetworkdesignmodel issolvedusingaslope-scalingheuristicwhichwasrstint roducedby Kim&Pardalos ( 1999 ).LPrelaxationsineachiterationoftheslope-scalingheu risticaresolvedusing columngenerationwitheachcolumncorrespondingtoafeasi bleload-planningtree. Authorsuseamodiedslopescalingheuristicthatusesagrad ualcostingstrategytoslow downtheconvergenceoftheheuristic.Theygeneratemajorp otentialcostsavingsforthe targetLTLcarrierinabouttwohoursforeachrun. 76

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Westudyaservicenetworkdesign(SND)problemonahub-and-s pokenetwork andproposenetworkshrinkingbaseddecompositionschemef oritssolution.Network shrinkingideaallowsthegenerationofsmallertime-space networksthatcanbesolved withinreasonabletimelimits.Theapproachcanbeutilized byallconsolidationbased carriers,suchasexpresspackagedelivery,less-than-tru ckloadserviceproviders,freight railcarriers,etc.Intherstphaseofthisalgorithmwesol veanintegerprogramming modelbyusingslopescalingheuristicandproposeimprovem entstepswhichsignicantly decreasedthecostsofthesolutionobtainedbyslope-scali ngheuristic.Thesecondphase solvesamodelonatime-spacenetworkwhichiscomposedofon lyhublocations.The thirdphaseimprovesthenon-hubtohubandhubtonon-hubcon nectionsoftherst phase.Weappliedouralgorithmonareallifeinstanceforal ess-than-truckloadmotor carrierandobtainedconsiderableimprovementsintranspo rtationcostsandloadcapacity utilizationsinareasonabletimelimit. 5.2ProblemDescription Thegoaloftheproblemistoinstallloadsonthelinksoftheg iventhenetworkand decideonroutesforeachshipmentsuchthattotaltransport ationcosts(totalmileage costs)areminimized.Eachrouteisasequenceofloadsaship mentshouldtaketotravel fromitsorigintodestination.Weassumethatshipments'de mandquantitiesarenotbig enoughtollloadcapacities.Onceashipmentisloadedfrom itsorigin,ittravelshub locationsandnallyreachesitsdestinationterminal.Note thatusuallytwohubsare visitedontherouteoftheshipment.However,weallowmoreth antwohubstobevisited. Weassumethatintermediatehandling(sortingandregroupi ng)ofshipmentscantake placeonlyathubs.Atnon-hublocations,onlypick-upandse t-oareallowed.Inorder tomakebetterconsolidationathubs,shipmentsareallowed tobeholdathubs.Holding isusefultoaggregatedierentshipmentsintoasingleload andutilizeloads'capacities better.Holdingofshipmentsisvalidaslongasshipmentssat isfytheircorresponding servicecommitments.Alltheshipmentsfollowasinglepathf romtheirorigintotheir 77

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destination.Hence,wesolveanon-bifurcatedservicenetwo rkdesignproblem.Inputs, constraintsandoutputsofourservicenetworkdesignprobl emaregivenasfollows: 5.2.1ProblemInputs Locations: Therearemultiplelocationseithershipmentsoriginateor terminate. Therearetwotypesoflocationshubsandnon-hubs(alsocall edasend-of-line terminals).Wearegivensetofnon-hublocationsandsetofh ublocations. Linkswiththeirdistanceandtraveltime: Travelingtimeisobtainedbyusing averagespeedoftransportationmedium.Weassumethatalmo stalllocationsare connectedtoeachotherdirectly.Thisisespeciallytruefo rroadnetworks. Service: Eachserviceisdenedbyanorigin,destinationlocationpa ir. Servicecommitments: Foreachserviceandforeachspecicdayoftheweek, carrierprovidescorrespondingservicecommitments.Each servicecommitmentis denedbyitscorrespondingserviceoriginlocation,servi cedestinationlocation, cutoday,cutotime,recoveryday,recoverytimeandplann edweight.Planned weightsareusuallyconstructedbythecarrierlookingatth eirhistoricaldemand databetweenorigin-destinationterminalpairs.Forexamp le,ifashipmentisready atlocationA(originlocation)onMonday(cutoday)by9pm( cutotime)than thecostumercanpickitupfromlocationB(destinationloca tion)at8am(recovery time)onTuesday(recoveryday).Servicesandcorrespondin gservicecommitments areprovidedfromalmostalllocationstoallotherlocation s. Loadcapacity: Capacityoftransportationmedium. Transportationcostpermile 5.2.2ProblemConstraints Servicecommitmenttimelimits: Durationsofshipmentroutesshouldsatisfy theircorrespondingservicecommitmenttimelimit,i.e.,a llshipmentsmustarriveto theirdestinationsbeforetheirrecoverydayandtime. Loadcapacitylimit: Totalshipmentweighttravelingonaloadshouldnotexceed itscapacity. ShipmentFlowrule: Flowsdictatenextvialocationfromaparticularlocationo n aparticularday.Ateachlocationonagivenday,allshipmen tswiththesameactual destinationshouldberoutedtothesamenextvialocation. 78

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5.2.3ProblemOutputs ShipmentRoutingPlan: Foreachpossibledestinationfromanorigin,wendthe routetofollow.Notethatshipmentroutesshouldobeytherow rules. LoadsandLoadschedule: Identifyloadstobeusedanddeterminedeparture timeandarrivaltimeoftheload.Therecanbemultipleloads departingfroma locationonthesameday. Theobjectiveoftheproblemistocreatealoadplanandshipm entroutingplanwhich minimizescost.Weassumethatmajorcostistransportation costpermile.Handling costattheterminalsisrelativelysmallcomparedtothecos tofmovingpackagesbetween terminals.Maingoalisreducingthemileagewhilemaintain ingtheservice. 5.3Decomposition-BasedSolutionApproach Wedecomposetheproblemintothreeparts.Intherstphase, wedeterminerowsof shipmentsusingaspacenetworkcomposedofalllocations.T hisphaseperformsfacility consolidationbydecidingonhowinboundshipmentsaresort edandconsolidatedfor theirnextvialocations.Inthesecondphase,wedeterminel oadsandloaddeparture timesonashrunkspace-timenetworkwhichisformedusingon lyhublocations.This phaseperformsatemporalconsolidationbyholdingshipmen tsovertimeinordertoform largershipments.Thirdphaseisusedforpick-upanddelive ryroutes.Weimprovedirect non-hubtohubandhubtonon-hubconnectionsbyperformingm ulti-stopconsolidation. Inthisphaseseveralnon-hublocationsassignedtoapartic ularhubareservedtogetheron asingleroute.5.3.1Phase1:FacilityConsolidation Intherstphase,usingtheoverallnetworkandshipmentwei ghts,wearetryingto identifyshipmentroutesthatsatisfyrowrules.Whilerout ingallshipments,themodel alsotakesintoaccountloadcapacities.Volumeofshipment routespassingthrougha speciclinkshouldnotexceedthetotalcapacityofloadsin stalledonthelinkandroute traveltimesshouldsatisfycommitmenttimelimits.Wesolv ephase1problemonaspace network. 79

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Let N denotethesetofnodesinthenetwork.Eachnodecanbeorigin ,destination orhublocation. A denotethesetofarcsinthenetwork. c ij isthecostincurredbyeach loadusedonarc ( i j ) u ij isthecapacityofeachpotentialloadonarc ( i j ) .Decision variable y ij equals 0 ifnoloadismadeonarc ( i j ) ,otherwise y ij denotesthenumber ofloadsmadeonarc ( i j ) .Plannedserviceweightsneedtoberoutedforeachservice ( S = f ( u v ): u 2 N v 2 N g ,weindexeachserviceby s ).Plannedserviceweights ( w s s 2 S )equaltothetotalplannedweightforthecorrespondingser viceduringthe wholeplanningweek.Notethatsomeportionofthisplannedwe eklyweightcanbe realizeddemandwhichispreorderedbeforetheweekstartsa ndsomeportioncanbedue totheestimateofthecarrierusinghistoricaldata. s denotesthemaximumnumberof hourstheservicefromnode u tonode v cantake.Weenumeratedirectedshipmentroutes foreachservice.Theset P denotesthesetofallpotentialshipmentroutesandtheset P s P denotethesetofpotentialshipmentroutesforservice s 2 S .Weuseindex p todenoteashipmentroute.Weset w p equalto w s forall p 2 P s anddeneabinary decisionvariable x p foreachpotentialshipmentroute p 2 P x p equals 1 whenservice s usesshipmentroute p 2 P s ,anditiszerootherwise.Tobeabletosetupthemodel,for eacharc ( i j ) 2 A wedetermineallthepotentialshipmentroutesthatcontain thearc anddenotethissetby P ij .Wealsopreparethesets Q ijks and W ik consideringthesetof potentialshipmentroutesenumerated.Thesetwosetsareus edtodeneshipmentrow ruleconstraints. Q ijks isthesetofallshipmentroutesforservice s withdestination k whichrowonarc ( i j ) W ik isthesetofallnextvialocationsvisitedafterleavingnod e i bythepotentialshipmentrouteshavingdestination k .Variable z ijk takesthevalueof 1 if atleastoneservicewithdestination k usesarc ( i j ) .Weadda z ijk variabletothemodel fornodes f i j k g onlyifcorrespondingset W ik hasmorethan 1 element. min X ( i j ) 2 A c ij y ij (5{1) 80

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subjectto X p 2 P s x p =1 8 s 2 S (5{2) X p 2 P ij w p x p u ij y ij 8 ( i j ) 2 A (5{3) z ijk X p 2 Q ijks x p 8 i k j s : j Q ijks j > 0 (5{4) X j 2 W ik z ijk 1 8 i k : j W ik j > 1 (5{5) x p 2f 0,1 g8 p 2 P (5{6) y ij 2 Z + 8 ( i j ) 2 A (5{7) 0 z ijk 1 8 i j k 2 N (5{8) Constraints( 5{2 )statethatexactlyoneshipmentrouteisselectedforeachs ervice. Capacityconstraints( 5{3 )requirethatmaximumweightrowingonanarcdoesnot exceedthecapacityinstalledonthearc.Constraints( 5{4 )and( 5{5 )ensurethatall shipmentswiththesamedestinationareroutedtothesamene xtvialocationafterpassing throughagivenlocation. Inordertosolvethisproblem,weadoptslopescalingheuris tictechniquedueto Kim &Pardalos ( 1999 )andimproveditssolutionbypostprocessingsteps.Inthis iterative solutionapproach,wesolvetheLPrelaxationofthemodel,a ndweupdatexedcharge costsforlinksandthensolvetheLPrelaxationagainuntiln oimprovementisfoundfora certainnumberofiterations.Fixedchargecostsforeachli nkisupdatedconsideringthe loadcapacityutilizationfactors,whichisfoundbydividi ngfractionalnumberofloads fromtheLPrelaxationsolutionwiththeroundednumberoflo adsonthislink.Hence, loadcapacityutilizationfactoronarc ( i j ) iscalculatedby ij = y ij = d y ij e .Fixedcharge costofalinkisupdatedbydividingwithloadcapacityutili zationfactorofthelink. Hence,ifloadcapacityutilizationfactorislow,xedcharg ecostofthelinkincreases,this inturnmakesitmoreprobableforthecorresponding y ij valuetogetclosertoitsrounded 81

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downintegervalue.Inordertopreventslopescalingheuris ticgettingstuckinlocal optima,weapplygradualcostingofthelinksbyupdatingxe dcostofonlyaspecied numberoflinks. Improvementapproachfortheslope-scalingsolution. Inordertoimprove thesolutionobtainedbyslopescalingheuristic,weapplyf ollowingsteps.Let y int bethe integersolutionwegetafterroundingupthecontinuousyva riablesatthebestiterationof theslopescalingheuristic. Step1: Wesolveanintegerprogrammingmodel(IP-1)wherecostcoe cientsinthe objectivefunctionaretheupdatedxedcostsatthebestite rationoftheheuristic.We give y int asthestartingsolutiontoIP-1model.Thismodelissolvedv eryquicklysincewe usetheupdatedcostsatthebestiterationoftheheuristic; ittakesabout 2 minutesto solveittooptimality.SolutionofIP-1improvesthesoluti onoftheslopescalingheuristic around 17% Step2: Wegettheintegersolutionfromstep1.Wecalculatetotalwe ightrowing oneacharcbycheckingtheselectedshipmentroutesandthen ndactualloadutilization ofeacharc'scapacitybydividingtotalrowonthearcbycapa cityofaload.Wesolve anewintegerprogrammingmodel(IP-2)wherewereplaceyvar iableswithintegerpart offractionalloadutilizationfactorplusabinaryvariabl e.Notethatloadutilizationcan beviewedasthefractionalnumberofloadsrequiredoneacha rc.Inordertondinteger part,werounddownfractionalloadutilizationofeacharc; i.e.,ifloadutilizationis 5.6 foranarc,theninIP-2model,wereplaceywith 5 plusabinaryvariable.InIP-2,weuse actualxedcostsforeacharcandgivethesolutioninstep1a sthestartingsolutionfor thenewIPmodel.SolutionofIP-2improvesthesolutionofIP -1around 7% 5.3.2Phase2:TemporalConsolidation Inthesecondphase,timecomponentisaddedtothespacenetw ork,whichhelpsto determineloadsandtheirarrivalanddeparturetimes.Fore achserviceandeachdayof week,plannedservicecommitmentweightsareroutedoverth espace-timenetwork. 82

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Thesolutionofthephase1isusedasaninputandshipmenthol dingstructureand resultingloadtimingsarefoundusingaspace-timenetwork .Weutilizetheoutputsof rstphaseinthesecondphase.Thearcsselectedandrowsfou ndforeachserviceareused asaninput.Sincerowrulesaregiven,enumerationofpossib lepathswillberestricted andlesstimeconsuming.Forthesameservicedepartingondi erentdaysoftheweek, sequenceoflocationsoftheselectedroutesisgoingtobeth esamebutshipmentholding structuremaydier.5.3.2.1Networkshrinkingapproach Sincerowsarealreadygiven,weexpecttosolveaneasierint egerprogrammingmodel inthesecondphasehowever,sizeofthenetworkisconsidera blylargercomparedtothe spacenetwork.Tobeabletosolvetheproblemmodelonthespa ce-timenetwork,weuse networkshrinkingidea.Shrunknetworkisformedbyusingon lyhublocations.Inthe shrunknetwork,allhublocationsarefullyconnected.Shru nknetworkresultsintoaspace timemodelwhichhasconsiderablysmallersizecomparedtot heoriginalnetwork. Followingapproachisusedtoformhubtohubshrunknetwork: Weusethelocation sequencesformedforallservicesintherstphase.Mostoft helocationsequences haveasimilarstructureofstartingwithanon-hublocation andtravelingseveralhub locationsandterminatingatanon-hubdestinationlocatio n.Forinstance,onepossible shipmentroutemighttraversefollowinglocations:non-hu borigin,hub 1 ,hub 2 ,non-hub destination.Weshrinkthispotentialshipmentroutesotha titstartsathub 1 andends withhub 2 .Wethensetservicecommitmentearliestdeparturetime(cu t-otime)from hub 1 astheearliesttimeshipmentcanreachhub 1 locationfromitsnon-huborigin location.Similarly,wedeterminelatestarrivaltime(rec overytime)tohub 2 location. Followingthisroutine,wemodifycut-oandrecoverytimes ofallservicecommitments suchthattheyallgofromhubtohublocations.Wethencreate datedarcsbetweenhubs. Forallarcssuggestedbythespacenetwork,wedeterminethe oneswhosetailandhead 83

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nodesarehubnodesandcreatetwelvedatedarcsforeachdayo fweek.Departuretimesof datedarcsaredistributedinequalintervals. For 7 -dayshrunkspacetimenetwork,weslightlychangethenotat iontorepresent datedcomponents. ^ N denotesthesetofnodesinthenetwork.Eachnoderepresents a speciclocationonaspecicdayofweek.Inthesecondphase ,weusethearcsselected intherstphase,however,wereplaceeacharcbyasetofdate darcsbetweensamepair oftailandheadnodeswithdierentdeparturetimes.Theset ofdatedarcsisshownby ^ A .EachdatedarcisdenedbyanarcIDandaspecicdeparturet ime;hence,adated arcrepresentsasingleloaddepartingatacertaintime.Pla nnedfreightsforeachservice commitmentneedtoberoutedoverthespace-timenetwork.We denotethesetofservice commitmentsby C PathenumerationforSpace-TimeNetwork. Foreachservicecommitment c 2 C ,weenumeratesetofpotentialshipmentroutes( P c ).Thepathenumerationis exactlythesameasforspacenetworkwithfollowingminorch anges:Onlythosepathsare consideredwhichhavethesamelocationsequenceassuggest edbyspacenetworksolution. Atanylocationshipmentisallowedtostaytodepartonupcom ingdaysiftheservice commitmentrecoverytimepermits.Thecostofapathisconsi deredtobethetimetaken toreachthedestination.5.3.2.2Spacetimemodelfortheshrunknetwork Notethatweareusingdatedarcsinthespacetimenetworkande achdated arcrepresentsasingleloadwithaspecicdeparturetime.D atedarcsresemblethe facility-arcsintroducedinthechapter2.Thatiswhy,weut ilizethenon-bifurcated networkdesignmodelproposedtosolvecombinednetworkdes ignandcommodityrouting problem.Hence,whiledeningcapacityconstraints,wedon' thavetousexed-charge constraints.Thisresultsintothefollowingmodel: min X ( i j ) 2 ^ A c ij y ij (5{9) 84

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subjectto X p 2 P c x p =1 8 c 2 C (5{10) X p 2 P ij w p x p u ij 8 ( i j ) 2 ^ A (5{11) y ij X p 2 P c ij x p 8 ( i j ) 2 ^ A c 2 C (5{12) x p 2f 0,1 g8 p 2 P (5{13) y ij 2f 0,1 g8 ( i j ) 2 ^ A (5{14) Objectiveofthespace-timemodelisthesameasthespacenet workmodel.Constraints ( 5{10 )areverysimilartotheoneusedinthespacenetworkmodel.I nthespacemodel, wewereselectingexactlyoneshipmentrouteforeachservic e.Inthesecondphase, exactlyoneshipmentrouteisselectedforeachserviceandf oreachdayoftheweekservice isprovided(henceforeachservicecommitment).Constrain ts( 5{11 )aredierent,in thespacenetwork;weweredeterminingnumberofloadstobeu sedoneacharc.For space-timenetwork,betweentwolocationsweconstructsev eraldatedarcsandeachof themrepresentsaload-arcdeninganarcidandaloaddepart ingataspeciedtime. Tobeabletosetupthemodel,weagainformtheset P ij where ( i j ) isadatedarc;that is,wekeepthesetofpotentialroutesrowingondatedarc ( i j ) .Notethat w p denotes theplannedservicecommitmentweightforthecorrespondin gserviceandday.Weset w p equalto w c forall p 2 P c .Secondconstraintensuresthattotalweightofshipments assignedtoadatedarcdoesnotexceedloadcapacity.Constr aints( 5{12 )areusedtoset uptherelationbetweendesignvariablesandrowvariables. Theyensurethatdatedarc ( i j ) isselectedifatleastonepotentialshipmentrouterowingo nthisarcisselected. Notethatadatedarcissimplyaloadwithgivendepartureanda rrivaltimesonitstail andheadnodesrespectively.Whiledeningconstraint( 5{12 ),weagainusedthetightened 85

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versionoftheconstraintbyformingtheset P c ij whichincludethesetofpotentialroutes forservicecommitment c rowingondatedarc ( i j ) 5.3.3Phase3:Multi-stopConsolidation Afterdetermininghubtohubrows,non-hubtohubandhubtonon -hubassignments comingfromtherstphaseisimprovedbyconnectingnonhubl ocationsbetween themselves.Inthisway,wecandecreasetransportationcos ts,makebetterconsolidation anduselessnumberofvehiclestopickuploadsfromnon-hubl ocationsandbringthemto hublocations,similarlyweuselessvehiclestodelivershi pmentsfromahublocationto thenon-hublocationsitisconnected. Asaresultofphase1,weknowwhichnon-hublocationsareconn ectedtoagivenhub andfromphase2,wealsoknowtheloaddeparturetimesofeach servicecommitmentfrom hublocations.Objectiveinthethirdphaseistoconstructl oadsandtheirdeparturetimes suchthatseveralnon-hublocationscanbevisitedconsecut ivelybeforearrivingtotherst hublocationorafterdepartingfromthelasthublocationon theshipmentroute.Inthis phase,weallowpick-upsfromothernon-hublocationsforno n-hubtohubportionofthe route.Similarly,set-oispermittedatothernon-hubloca tionsforhubtonon-hubportion oftheroute.Thisresultsintobetterconsolidationanddec reasedoverallmileage. SolutionApproachforPhase3. Wesolvethirdphaseforeachhublocation separately.Eachhublocationalongwithallitsassociated non-hublocationsforms asub-problem.Non-hubtohubandhubtonon-hubconnectionsa reimprovedin separatesteps.First,wesolveallsub-problemstoimprove non-hubtohubconnectionsof associatedshipmentroutes.Secondly,wesolvesub-proble msforimprovinghubtonon-hub connections.Wewillrstdescribetheprocessfornon-hubt ohubconnections.Same approachisappliedforimprovinghubtonon-hubconnection s. Forthenon-hubtohubconnections,inasub-problemforagiv enhub,weconsider setofservicecommitmentswhoseshipmentroutespassthrou ghthehublocationafter leavingtheirnon-huboriginlocations.Foreachsub-probl em,weformaspace-time 86

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networkinvolvingasinglehublocationandseveralnon-hub locations.Asinphase2,we constructmultipledatedarcsconnectingsamepairofnodes .Foreachnon-hublocation, weenumeratedirectedpathsoriginatingatnon-hublocatio nsandterminatingatthe hublocation.Notethatthereisonlyonehubineachsub-probl em.Eachdirectedpath representsasequenceofloadsanddescribesapick-upseque nce.Amongthesetofdirected paths,wealwayshavethepathgoingfromnon-huborigintoth ehublocationdirectly. Therearealsopathsvisitingmultiplenon-hubintermediat elocationsbeforearrivingto thehublocation. Foreachsub-problem,weusethefollowingnotationtoconst ructtheinteger programmingmodelfornon-hubtohubconnections: ~ C :Setofservicecommitmentsoriginatingfromnon-hublocat ionsandgoingthrough samehublocation. a c :Availabletimeoftheservicecommitment c 2 ~ C atitsoriginlocation. l c :Latesttimebywhichshipmentrouteforservicecommitment c shouldreachthe hublocation.Noticethatweknowthedeparturetimeoftherou teforthisservice commitmentfromthehublocationusingsecondphase. P :Setofdirectedpathsenumerated R c :Setofrowpathsforaservicecommitment c whichcanfeasiblycarryplanned weightforthecommitmenttoitshublocation.Flowpathsare subpathsofdirected paths.Weincludearowpathtoset R c ifthecorrespondingwholedirectedpath picksupplannedweightforservicecommitment c aftertime a c andcancarryit tohublocationbeforetime l c R p :Setofrowpathsrowingonpath p Q ijkc :Setofallshipmentroutesforservicecommitment c withdestination k whichrow onarc ( i j ) .Noticethatweusedatednodestoformthisset.Hence,node i representaspeciclocationandadepartureday. 87

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W ik :Setofallnextvialocationsvisitedafterleavingnode i bythepotentialshipment routeshavingdestination k Let x r beabinaryvariabletakingvalueof 1 ifplannedweightforservicecommitment c isassignedtopartialdirectedpath r 2 R c andlet y p beabinaryvariabletakingvalueof 1 ifadirectedpathisselected.Then,theproblemcanbeformu latedasfollows: min X p 2 P c p y p (5{15) subjectto X r 2 R c x r =1 8 c 2 C (5{16) X r 2 R p w r x r u p 8 p 2 P (5{17) y p x r 8 p 2 P r 2 R p (5{18) z ijk X r 2 Q ijkc x r 8 i k j c : j Q ijkc j > 0 (5{19) X j 2 W ik z ijk 1 8 i k : j W ik j > 1 (5{20) x r 2f 0,1 g8 r 2 R (5{21) y p 2f 0,1 g8 p 2 P (5{22) 0 z ijk 1 8 i j k 2 N (5{23) Constraints( 5{16 )statethatthatforeachservicecommitment;exactlyonepa rtial directedpathisselected.Constraints( 5{17 )ensurethattotalweightassignedtoa directedpathcannotexceedloadcapacity.Constraint( 5{18 )setstherelationbetween designpathsandrowpaths.Ifarowpath(asubpathofcorresp ondingdirectedpath) isselectedforaservicecommitmentthenthecorresponding directedpath(designpath) shouldbechosenbythemodel.Sincewetrytochangenon-hubt ohubandhubto non-hubpartsofshipmentroutesinthethirdphase,weneedt ocheckifshipmentrow 88

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rulesaresatised.Hence,constraints( 5{19 )and( 5{20 )areincludedinthemodelasin therstphase.Noticethattherearenoxedchargeconstrain tssinceweformulatedthe problembyusingtheproposednon-bifurcatednetworkdesig nmodel. 5.4ComputationalExperience Theproposeddecompositionapproachforsolvingservicene tworkdesignproblem onhub-and-spokenetworksisneededbyallconsolidationba sedcarriers,suchasexpress packagedelivery,less-than-truckloadserviceproviders ,freightrailcarriers,etc.Weapplied ourapproachtosolveareallifeinstanceofaless-than-tru ckloadcarrier. ThecaseforaLess-than-TruckloadMotorCarrier. ANorthAmerican less-than-truckloadcarriermaintainsaround 60 terminals(end-of-lineterminalsand hubs).Thecompanyhastimesensitiveshipmentswithstrict deliverydates.Shipments arehandledat 10 hubs.Intheselocationsarrivingshipmentsaresorted.The nsome packagesaredeliveredtocustomersandothersareagainreg roupedtosendtoother hubs.Thehandlingcostofthecompanyattheterminalsisrel ativelysmallcompared tothecostofmovingpackagesbetweenterminals;therefore ,themaingoalistoreduce themileagecostwhilemaintainingtheservice.Carrierpay stodriverspermile.Ifaload distanceislessthanacertainlimit,theypayfortherateof singledriverelsetheypay fortherateofteamdriverforthisload.Servicecommitment saresetbythecarrieras aninputconsideringtheirhistoricaldata.Ineachweek,th ecarrierhasaround 21000 servicecommitments.Aprovidedsolutionshouldgiveloads cheduleoverthenetwork andshipmentroutingplanforeachservicedenedbetweenor igindestinationterminals. Thecarrierrequiresanalgorithmthatcanimprovetheircur rentsolution(incremental algorithm).Theyarealsolookingforatoolwhichhastheabi litytondabettersolution fromscratch(zero-basedalgorithm)comparedtotheircurr entoperatingplancost. Developedalgorithmshouldselectbestroutestohandleshi pmentsanddecideonusing certainsetofhubsamonggivencandidateset.Thecompanydo esnothavesophisticated 89

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planningsystem.Managersroutepackagesbasedonthehisto ricaldataandtheirprevious experience. Inordertoobtainanincrementalsolution,wegetcarrier's shipmentroutesforeach servicecommitmentandusesetoflocationsvisitedbyeachs hipmentrouteasaninput. Hence,phase1ofouralgorithmisnotneededtobeusedinthisc ase.Weusephase2and changeonlytheholdingstructureofshipmentsandkeepthes etoflocationsvisitedby eachshipmentroutesameasthegiveninput.Inphase3,wehav ethechanceofimproving agivensolutionbyvisitingasetofnon-hublocationsinste adofusingdirectnon-hub tohubordirecthubtonon-hubconnections.Inourexperimen ts,weobservethatboth phase2and3improvedthesolutionsforhubtohubandnon-hub tohub(orviceversa) connectionssignicantly.InTable 5-1 ,wepresentresultsweobtainedusingourzero-based algorithmandincrementalapproachandcomparethemwithcu rrentlyimplemented solutionbythecarrier.Thecolumn\Proposed-1"showsther esultswhenweusecarrier's solutionasastartingsolution(incrementalapproach),th ecolumntitled\Proposed-2" presentstheresultswhenwendasolutionfromscratch(zer o-basedsolution).\LTL Carrier"columnrepresentsthesolutionimplementedbythe carrier.Weobservethatthe algorithmpresentedjustifyitselfbyimprovingagivensta rtingsolutionbythecarrier, alsobyformingasolutionfromscratchwhichresultsintobe stsolutionamongthesethree solutionspresented.InTable 5-2 ,wepresentoptimalitygaps(%)andtotalcomputerrunningt imeinseconds foreachphaseofourproposeddecompositon-basedalgorith m.Noticethatinphase3an integerprogrammingmodelissolvedforeachhubseparately .Forthisphase,optimality gapreportedistheaverageofoptimalitygapsobtainedover allhubs.Weobservethat wecansolveoptimizationmodelsformostofthehubs(90%)to optimalityinamatterof seconds.Forasmallportionofthehubswhereshipmenttrac isveryhigh,optimization modeltriestoclosethegaptilltheruntimelimitof15minut es.Hence,totalsolution timereportedis2254.06seconds.Inphase1,slopescalingh euristicandimprovement 90

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Table5-1.Comparisonofincrementalandzero-basedsoluti onsobtainedby proposedalgorithmwithLTLcarrier'ssolution Proposed-1Proposed-2LTLCarrier NumberofLoads Non-HubtoHub301310323HubtoNon-Hub385410462 HubtoHub600556710 Total 1,2861,2761,495 Cost Non-HubtoHub$91,574.96$84,148.08$94,242.14HubtoNon-Hub$102,477.84$109,573.98$118,466.58 HubtoHub$395,739.98$367,977.98$448,626.04 Total $589,792.78$561,700.04$661,334.76 Avg.LoadUtilization Non-HubtoHub50.90%46.37%48.79%HubtoNon-Hub41.42%35.47%36.06% HubtoHub52.70%54.27%44.63% Total 48.34%45.37%43.16% stepsruninlessthanthirtyminutes.Inordertoreporttheo ptimalitygapoftheresulting solution,weinputtheresultingsolutionasastartingsolu tiontotheexactoptimization modelandreportthegapafterthreeminutes.Forphase2,weu searunningtimelimitof 1hourforthespacetimemodel.Theoptimalitygapreportedi sthegapweobtainedafter therunningtimelimitisexceeded. Table5-2.Optimalitygapandtotalrunningtimesforeachph aseofproposedalgorithm OptimalityGapRunningTime(secs) Phase15.971684.58Phase23.883600.00Phase30.792254.06 5.5SummaryandConclusions Weproposedadecomposition-basedapproachforthesolutio nofaservicenetwork designproblemonahub-and-spokenetwork.Weappliedourap proachonareallife instanceforaless-than-truckloadcarrierandobtainedag oodqualitysolutionina practicalrunningtime.Proposeddecompositionincludesa ninnovativenetworkshrinking approachthatseparatesthenetworkintothreepartitions: non-hubtohub,hubtohub 91

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andhubtonon-hubnetwork.Networkshrinkingresultsintosm allerspacetimenetworks andcorrespondingspacetimemodelscanbesolvedsatisfact orilywithinrunningtime limits.Intherstphaseofouralgorithm,weadaptslopesca lingheuristictosolve correspondingintegerprogrammingmodelonaspacenetwork andproposeimprovement stepsfortheresultingsolution.Weobservethatimproveme ntstepssignicantlydecreased thecosts.Heuristicsolutionofphase1isimprovedapproxim atelyby23%.Inphase3,we improvedthesolutionsfordirectnon-hubtohubandhubtono n-hubconnections.This isachievedbylettingseveralintermediatenon-hublocati onstobevisitedconsecutively beforearrivingthersthublocationorafterdepartingfro mthelasthublocationon theshipmentroute.Wecoulddecreasetransportationcosts by8.64%ontheaverage fornon-hubtohubandhubtonon-hubconnections.Similarly ,wecouldmakebetter consolidationovertheservicenetworkbyincreasingloadc apacityutilizationby3.15%on theaveragefornon-hubtohubandhubtonon-hubconnections 92

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BIOGRAPHICALSKETCH SuatBogwasborninSamsun,Turkeyin1981.HereceivedhisB. S.degreein industrialengineeringfromBogaziciUniversity,Istanb ulin2004andhisM.S.degree inindustrialengineeringfromKocUniversity,Istanbulin 2006.SinceAugust2006,he hasbeenpursuinghisdoctoraldegreeintheDepartmentofIn dustrialandSystems EngineeringattheUniversityofFlorida.Suat'sacademicre searchisfocusedon solvinglarge-scalediscreteoptimizationproblems.Hisre searchinterestsincludeinteger programming,networkoptimizationandheuristics.InhisP h.D.study,heworkedwith Dr.RavindraK.Ahujaandconductedresearchonnetworkdesig n,routingandscheduling problemsemerginginfreighttransportationindustry.Spe cically,hefocusedonthree importantareasrelatedtofreighttransportation:integr atedtransportationplanning fornon-bifurcatednetworkdesignproblems,servicenetwo rkdesignonhub-and-spoke networksandtransportationnetworkdisruption. 100