Train Schedule Optimization in Public Rail Transport · Preface This thesis deals with train scheduling problems with an emphasis on public rail transport. In partic-ular, we assume

Vom Fachbereichfur MathematikundInformatik

derTechnischenUniversitat Braunschweig

genehmigteDissertation

zur ErlangungdesGradeseines

DoktorsderNaturwissenschaften(Dr. rer. nat.)

ThomasLindner

Train ScheduleOptimizationin Public Rail Transport

30.Juni 2000

1. Referent:Prof.Dr. UweT. Zimmermann

2. Referent:Priv.-Doz.Dr. MichaelL. Dowling

eingereichtam: 19.April 2000

Acknowledgements

Many peoplehave contributedto this thesisin oneway or theother. I would like to thankUwe Zim-mermannandtheothermembersof theDepartmentof MathematicalOptimizationfor their support.Themotivatingatmospherein thedepartmentprovidedanexcellentframework for scientificworking.

I would also like to expressmy gratitudeto Leo Kroon, AlexanderSchrijver and MatthiasKristafor makingtestdataandotherrailroad-relatedinformationavailable.It hasbeenveryhelpful to knowRobertBixby. Withouthim, I wouldneverhavefoundthe‘hiddenparameters’in theCPLEXsoftwarewhichacceleratedtheMIP solutionprocessby a factorof 10 in somecases.

Specialthanksgoto Karl Nachtigall.It hasbeenapleasureworking togetherwith him onchapter3. Ihopethatwewill beableto solve thePESPinstance14someday. This instanceis officially known as‘Jul18’, becauseit wasgeneratedJuly18 someyearsagoandnobodycanremembertheexactyear!

My lastspecialthanksgoto MichaelBussieck,notonly for thosemany scientificdiscussions,but alsofor personalsupportand– probablythemostimportant– for encouragingmeto startwriting a thesison trainscheduling.

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Preface

This thesisdealswith train schedulingproblemswith anemphasison public rail transport. In partic-ular, we assumea periodicscheduleanda fixedrailroadtracknetwork, which is commonfor publicrail transport.

Thefundamentalmathematicalmodeldiscussedhereis thePeriodicEventSchedulingProblem(PESP)introducedby SerafiniandUkovich in 1989. In a few words,the PESPis theproblemof finding afeasibleschedulefor someperiodicallyrecurringeventssubjectto certainconstraints.ThePESPisknown to beNP-completeandthereforebelongsto aclassof problemsassumedto beveryhard.

Wewill analyzedifferentexistingalgorithmsfor solvingPESPinstances.Basedonthisinvestigations,wemodify thesealgorithmsto achieveamuchbetterperformancefor probleminstancesfrom practice.Furthermore,we discusspolyhedral aspectsof a mixedinteger programming(MIP) formulationofthePESP, therebyderiving valid inequalitiesandproving somepropertiesof theseinequalities.Wecombineexisting algorithmic ideaswith new ideasfrom thesepolyhedralinvestigationsin ordertoobtainanew algorithmthatcanbesuccessfullyappliedto PESPinstances.

Therearemany criteriafor evaluatingschedules.ThePESPitself is afeasibilityproblem.Weextenditby anobjective functionrepresentingtheoperational costsof realizingaschedule.Thecostapproachis basedon a modelsuggestedby Claessens.Theresultingmodelis calledminimumcostschedulingproblem(MCSP).

The decisionversionof the MCSPis shown to be NP-complete.We presenta MIP formulationoftheproblem.With thehelpof polyhedral methodslike preprocessingtechniques,valid inequalities,a specificrelaxation, a branch-and-boundanda cutting planeprocedure,we areableto solve real-world instancesof theMCSP, which is not possiblewithin a reasonableamountof time whenusingthedirectMIP formulationandacommercialMIP solver.

The mathematicalmodelsandalgorithmsintroducedin this thesisare testedon practicalinstancesobtainedfrom therailroadcompaniesof Germany (DeutscheBahnAG) andtheNetherlands(Neder-landseSpoorwegen).

Thecostapproachof theMCTP belongsto thestrategic planningmethods,i.e. it is usedto evaluatepossiblescenarios5–15yearsaheadin thefuture.Ourexperiencesshow thatit is possibleto producesolutionsof theMCSPfor practicallyrelevantproblemsizesin afew minutes,which is acceptableforstrategic planning. Moreover, our algorithmdetermineslower boundson thecostsandthusenablesus to give boundson the quality of the solutions(if we arenot able to solve the probleminstanceexactly).

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An importantpoint is the transferof mathematicalmodels/ ideasinto practice.Mathematicalideastendto beabstract andnon-intuitiveandarethereforedisregardedby practitionersif they arenotcare-fully introduced.In orderto overcometheseobstacles,theGermanFederal Ministry of EducationandResearch fundeda seriesof projectson MathematicalMethodsfor SolvingProblemsin IndustryandBusiness. In thesejoint projects,mathematicians,engineersandsoftwaredeveloperswork together,transferringmathematicalideasinto practicalsoftware. Application fields are, for example,traffic,logistics,medicineor finance.This thesisemergedfrom theprojectTrain ScheduleOptimizationinPublicTransportation.

Thethesisis organizedasfollows: In chapter1, we give anintroductionto traffic planningin generalandwith respectto schedules.Chapter2 providesanoverview of existingmodelsfor trainschedulingandincludessomeextensionsof the models. In particular, the PESPandthe MCSParedescribed.Furthermore,we discusscomputationalcomplexity aspectsof thePESPandtheMCSP. In chapter3,thePESPis investigatedin detail.Wepresentexisting algorithmsfor solvingPESPinstancesandde-velopmodificationsandnew algorithmsthatallow a muchfastersolutionof suchinstances.We alsogive theoreticalresultson thepolyhedralstructureof thePESP. In chapter4, we introducealgorithmsfor solvingMCSPinstances.With thehelpof adecompositionidea,wedevelopa relaxationiterationanda branch-and-boundapproachfor the MCSP. Both methodsrequirethe solutionof certainsub-problems,which arealsoexamined.Chapter5 containscomputationalresultsfor our real-world testinstances,andthelastchapterdealswith conclusionsandsuggestionsfor furtherresearch.

Contents

1 Public Rail Transport Planning 1

1.1 HierarchicalRailroadPlanningLevels . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 TrainSchedulePlanning:An Overview . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Models for Train Scheduling 9

2.1 RailroadNetworksandTrainSchedules . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 ThePeriodicEventSchedulingProblem(PESP). . . . . . . . . . . . . . . . . . . . 13

2.3 EventGraphModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 LinearModel with IntegerVariables . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Extensionsof thePESP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 ScheduleOptimizationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 CostModel for Line Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 CostModel for TrainScheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9.1 Complexity Resultson thePESP. . . . . . . . . . . . . . . . . . . . . . . . 26

2.9.2 Complexity ResultsonCostOptimalScheduling . . . . . . . . . . . . . . . 26

3 FeasibleSchedules 33

3.1 Preprocessing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 BasicPropertiesof thePESP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 MixedIntegerProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Odijk’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 ConstraintPropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Algorithm of SerafiniandUkovich . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Arc Choicefor theGeneralizedSerafini-Ukovich Algorithm . . . . . . . . . . . . . 47

3.8 PolyhedralStructureof thePESP. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8.1 TheUnboundedTimetablePolyhedron . . . . . . . . . . . . . . . . . . . . 52

3.8.2 CycleCuttingPlanes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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3.8.3 ChainCuttingPlanes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.8.4 SimpleLifting Proceduresfor Flow Inequalities. . . . . . . . . . . . . . . . 57

3.8.5 SingleBoundImprovement . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8.6 Flow InequalitiesandSingleBoundImprovement. . . . . . . . . . . . . . . 59

3.9 Branch-and-CutMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Cost Optimal Schedules 65

4.1 MixedIntegerProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 ProblemDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 RelaxationIterationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Branch-and-BoundMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 SolvingMCTP instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6 SolvingFSPinstances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.7 ExactSolutionof theNonlinearProblem. . . . . . . . . . . . . . . . . . . . . . . . 80

5 Computational Results 85

5.1 TestInstances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 HardwareandSoftware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 PESPResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 OptimizationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Conclusionsand Suggestionsfor Further Research 99

A Computational Complexity 101

A.1 TheProblemClassesPandNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 NP-completeProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

B Mixed Integer Linear Programs 103

B.1 LinearandMixedIntegerLinearPrograms. . . . . . . . . . . . . . . . . . . . . . . 103

B.2 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.3 SolutionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C ShortestPath Problems 111

C.1 ClassicalShortestPathProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

C.2 Gauss-JordanMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.3 FeasibleDifferentialProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Bibliography 123

Index 127

Chapter 1

Public Rail Transport Planning

Nowadays,public rail transportplanningis a highly complex task. Too many objectsinteractwitheachotherto bemanageablesimultaneously(cf. table1.1, detailsarefound in theGeschaftsberichtder DeutschenBahn[20]). Varioussubproblemsof differentnaturelike network design,schedulingor routingoccur, andthesolutionsof mostof thosesubproblemsdependon thesolutionsof theothersubproblems.Dueto severecompetitionfrom othertransportationmodes,therail industryis eagertoimprove its operationalefficiency andrationalizeits planningdecisions.Analytical modelsgetmoreandmoreimportantin supportingmanagerialdecision-making.Theprocessof privatizationof publictransportationcompaniesenforcestheefficientutilization of resources.

38000 km of network

40000 trainsperday

66 billion travelerkilometers

15 billion grossinvestment(in DM)

250000 employees

130 licenseagreementswith otherrailroadcompanies

Table1.1: Referencenumbersof theGermanrailroadcompany DeutscheBahnAG (1998)

Differentdemandsonthetransportservicecomefrom thedifferentdepartmentsof arailroadcompany.Themarketingdepartmentsrequesttakingcareof thepassengers’wisheslike minimizationof traveltime,pleasantchangesfrom onetrain to another(shortwaiting time,oppositeplatforms).Thelogisticdepartmentspayattentionto thecostaspects.They areresponsiblefor theefficient usageof rollingmaterialandpersonnel.Available rolling stockhasto be consideredaswell ascrew rulings. Thedepartmentsmaintainingthe network take careof operational constraints occurring,for example,for concurrentuseof critical points (single tracks,stations,switches,signals). All these,usuallyconflicting,demandsareshown in figure1.1.

Apart from economicalaspects,political decisionsandprestigiousinvestmentprojectsinfluencetheplanningprocess.In 1995,Baron[3] describesthesituationof public transportationin Germany, con-

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2 CHAPTER1. PUBLIC RAIL TRANSPORT PLANNING

OperationalConstraints

Marketing Cost

?

Figure1.1: Conflictingdemands

cluding that transportpolicy andplanningwill remaina playingfield of scientists,lobbyists,politi-cians,gurus,fanaticsandconcernedcitizensfor manyyears to come, andit will keepgenerationsofjournalistsbusy.

Dueto thetremendoussizeof thepublicrail traffic system,theplanningprocessis dividedinto severalsteps(alsosee[10]). A diagramof thishierarchicaldecompositionis givenin figure1.2.

Crew Management

Planningof Rolling Stock

Train SchedulePlanning

Line Planning

Analysisof Demand

Figure1.2: Hierarchicalplanningprocess

In afirst step,thepassenger demandhasto beanalyzed.As aresult,theamountof travelerswishingtogo from certainoriginsto certaindestinationsis known. As a subsequenttask,linesaredetermined,i.e. routeswheretrainsrun. Also, the frequenciesfor the lines aredetermined.Afterwards,in thetrain scheduleplanningstep,all arrival anddeparturetimesof thelinesarefixed.Thishasto bedonesubjectto theperiodicityof thesystem(theGermanrailroadtrain scheduleoperateswith a periodofonehour, for example).Now enginesandcoacheshave to beassembledto trains,whichareassignedto lines.This is calledplanningof rolling stock. A similar taskis thecrew management, whichmeans

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thedistribution of personnelin orderto guaranteethateachtrain is equippedwith thenecessarystaff.

Every singlestepin this processis a difficult task. We will discussthesestepsfurther in section1.1.A problemof thedecompositionis that theoptimalsolutionfor onepartservesasfixedinput for thesubsequentproblem. Onecannotexpectan overall optimal solutionin the end. It is even possiblethat at somepoint, former decisionshave to be changed,anda part or the completeprocesshastobe repeated.Nevertheless,this hierarchyprovides a partition of the traffic planningproblemintomanageabletasks.

Anotherclassicalpoint of view [2,33] is thepartitionof theplanningprocessinto strategic, tacticalandoperational planninglevel, table1.2.

Planninglevel Timehorizon Goal

Strategic 5–15years Resourceacquisition

Tactical 1–5years Resourceallocation

Operational 24 h – 1 year Day-by-daydecisions

Table1.2: Planninglevels

Onthestrategic planninglevel,possibleinfrastructureinvestmentsareexamined.Thegoalis to decideaboutresourceacquisition(i.e. building new traffic links etc.). Suchprojectsmayhave a durationof5–15years,andthustheview of thefutureplaysanimportantrole. Theanalysisof passengerdemandandthedesignof line plansbelongto thisplanninglevel. It is alsopossibleto examinetrainschedulesatthispointof time,e.g.in orderto examinetheeffectof acertaininfrastructureproposalonthetraveltime.

The tactical planninglevel focuseson resourceallocationin the mediumterm. Here, the generalpatternof traffic flow is derived from invariable infrastructureand customerdemanddata. Moredetailedline plansand train schedulesare developed,as well as generalpatternsfor rolling stockcirculationandcrews.

Day-by-daydecisionsconstitutetheoperationalplanninglevel. Here,dueto unexpectedeventslikebreakdowns, specialtrainsor short term changesin the infrastructurecausedby constructionsites,partsof ascheduleor rolling stockandcrew assignmentpatternshave to berearranged.

During the lastdecade,theuseof mathematicaloptimizationmodelsfor rail transportplanningandthustheautomaticcomputationof line plans,schedules,crew patternsetc.hasincreasedsignificantly(for anoverview we refer to [18]). In theeighties,theapplicationanddevelopmentof mathematicalmodelswashinderedby insufficient computationalcapabilitiesand the problemsof collectingandorganizingtherelevantdata,whichmany railroadcompaniescouldnotafford.

Thesituationhaschangedremarkablyduringthelastyears.Increasingcomputerspeedandprogressin mathematicalmethodsenabledthedevelopmentandsolutionof probleminstancesof morerealisticmodels(for lineplanningproblems,Bussieck[10] discussedthesedevelopments).As wehavealreadymentioned,competition,privatizationandderegulationrequirethe efficient useof resourcesfor thecompanies.Thishasaffectedair transportationcompaniesto anespeciallylargeextent.


In Germany, the winter train schedule1998/1999was the first one to be developedwith the helpof computerscompletely(cf. [20]). However, this doesnot meanthat the schedulewasgeneratedautomatically, but with thehelpof decisionsupportsystemsandgraphicaluserinterfaces.

A next stepwill bethesimultaneousplanningof severalhierarchicallevels, in thehopeof achievingbetteroverall solutions.In theNetherlands,thedecisionsupportsystemDONS(Designerof NetworkSchedules)assiststheplannersin routingandscheduling(cf. [38]). TheCADANS moduleof DONSgeneratesschedules,consideringtherailwayinfrastructureonly from aglobalpointof view. A secondmodule,STATIONS, is responsiblefor checkingwhethera scheduleis feasiblewith respectto theroutingof trainsthroughtherailway stations,i.e. with the track layout. A comprehensive survey ofdiscreteoptimizationtechniquesin public rail transportcanbefoundin [9].

Besidesoptimizationmodels,simulationtoolsfor traffic planningarewidelyusedtocomparedifferentscenariosfor complex problemswithin shortcomputingtimes.

1.1 Hierar chical Railr oadPlanning Levels

We will shortly focuson thedifferenthierarchicalplanningstepsfrom figure1.2. Sincethesestepsinfluenceeachother, it is of interestto discussthem and their connectionto train schedulingto acertainextent.Ourpresentationfollows [10] at thispoint.

PassengerDemand

In orderto establisha customer-orientedtransportationservice,thepassenger demandor traffic vol-umemustbegivenor estimated.Theconventionalform of passengerdemanddatais aso-calledorigindestinationmatrix (OD-matrix). An entry

i j of this matrix givesthenumberof peoplewishingto

travel from locationi to location j.

Sophisticatedmodelsandmethodshave beendevelopedin orderto determineOD-matrices.A num-berof cost-intensive interviews of customers mayform a basisfor statisticalmethodsestimatingtheoverall demand.Anotherapproachare traffic censuseson the networklinks (like railroadtracksorstreets).Statistical[40] andmathematicalprogramming[58] methodsthatgenerateOD-matricesfromsuchlink traffic censusesareavailable. A disadvantageof this approachis that from the traffic vol-umeon the links, OD-matriceswith differently structuredentriescanbe obtained.An exampleforthis situationis given in figure1.3. Anotherproblemis that the routestaken by thetravelersremainunknown. Nevertheless,OD-matricesarewidely usedin traffic planningmodels.

Thereis anothergeneralproblemconcerningthe predictionof the passengers’behavior or wishes:The demandestimationbasedon methodslike interviews or traffic censusonly reflectsthe currenttransportationservicesituation.If theline planor train scheduleis changed,passengersmaychangetheirbehavior in anunpredictableway.

The main link betweenpassengerdemandandtrain schedulingis the problemof establishingtrainconnectionswith adequatewaiting times for travelerswithout a direct train from the origin to thedestinationof their trip. Thesetravelerswould like to have enoughtime to changethetrain (even incaseof smalldelay),but of coursedonotwish to wait for a long time.

Whenestablishingconnectionsfrom OD-matrixdata,onefacestwo mainproblems:

1.1. HIERARCHICAL RAILROAD PLANNING LEVELS 5

C D E

A B

7

2

5

1

6

1

3 0 2 4

A B C D E

A 6

B 4

C 1 7

D 2 1 1

E 1

A B C D E

A 4

B 1

C 4 5

D 1 2

E 1

Figure1.3: DifferentOD-matricesfor thesamelink traffic volume Choiceof routes: As we have alreadymentioned,theroutesof thetravelersarenotdeterminedby thematrix. Wemayrely on theassumptionthattravelersmostlychoosea shortest-path-likeroutefor their trip. However, a shortrouteconcerninglengthin km maybeservedby a slowertrain or requireanuncomfortabletrainchange. Choiceof locationfor train change: If passengersneedto changebetweenlinesrunningon thesamerailroadtrackfor sometime, they maydo soatseveralstations.

In thesesituations,personalpreferencesof thepassengersplay an importantrole, andobjective de-cision criteria cannotbe given. For train scheduleplanning,oneshouldtry to establishat leastone“good” connectionin thiscase.

Line Planning

A line is givenby a routeanda correspondingfrequency. Therouteis givenby a pathin therailroadtracknetwork. Thefrequency determineshow oftenthis line is served in accordanceto thescheduleperiod. Line planningmeansto selectlines from a setof feasiblelinessubjectto certainconstraintsandpursuingcertainobjectives.

Somepossibleconstraintsare that theremust be enoughlines (or trains respectively) to carry allpassengers,thecapacityof tracksmustnot beexceededor that therequiredtrainsmustbeavailable.Common(and,asalways,conflicting)objectivesareminimizationof costsor maximizingthenumberof travelerswith a directconnection.Bussieckdiscussesline planningproblemsextensively in [10].Wewill useandextendsomeconceptsfoundin [10] in orderto establishanew modelfor costoptimaltrainscheduling.

Theperiodicityof theschedulehasto bekept in mind whendesigningline plans.In general,severaltrains(or so-calledtrain compositions) arerequiredin orderto serve a line, becausenormallya trainhasnot traveledthecompleterouteandbackin onescheduleperiod.

Railroadcompaniesusuallyoffer differenttrain serviceto their customers.In Germany for example,InterCity andInterCityExpresstrainsconnectprincipal centersof the country. Thesetrainsarefastandequippedcomfortablywith dining car, phoneor boardservices.InterRegio trainsareslower andconnectprincipal centersaswell asdistrict towns. Additionally, thereare regional trains (like theAggloRegio trainsin theNetherlands).All thesetrainshave to sharetheglobalnetwork.


For theplanningprocess,thenetwork is oftendecomposedinto supplynetworkscorrespondingto thedifferenttrain services(InterCity network, InterRegio network etc.). If a line planfor a singlesupplynetwork hasto bedeveloped,theglobaldemandinformationsuchasgiven by an OD-matrixhastobe adaptedfor the supplynetworks. For example,thereareapproachesto split an OD-matrix intodifferentmatricesfor thesupplynetworks(systemsplit procedure,cf. [50]).

Theline planservesasdirectinput for thetrainschedulingproblem,wherearrival anddeparturetimesfor thelineshave to befound.Furthermore,theline plandetermineswhich travelershave to changeatrainduringtheir trip andthusneedacceptableconnectiontimes.

Train SchedulePlanning

Thetrain scheduleconstitutesthebackboneof public rail transportplanning.Thegenerationof trainschedulesis the coresubjectof this thesis. A detailedintroductionto train schedulingproblemsisgivenin section1.2.

A trainscheduleconsistsof thearrival anddeparturetimesof thelinesatcertainpointsof thenetwork.Dependingon therequiredresolution, thesepointsarestations(low resolution)or evenswitchesandimportantsignalpoints(high resolution). For the railroadnetwork of Germany, in the former caseapproximately8000suchpointsareconsidered,in thelattercaseabout27000points.

In general,schedulesfor public transportareperiodical,i.e. thescheduleis repeatedafterabasictimeperiodor, for short,period.

Theschedulefixesarrival anddeparturetimesfor linesandthusfor all trainsof theline. An individualtrain correspondsto a trip of theline. Theassignmentof enginesandcoachesto thesetrips (or trainsrespectively) is donein asubsequentstep.

Planning of Rolling Stockand Crews

Thetripsestablishedby thetrainschedulemustbeperformedby somevehicles(motorunit, coaches)and crews (like enginedrivers, conductorsetc.). Optimizationmethodsfor vehicle schedulinginpublic transportationaredescribedin [26,39]. Sincethedispatchof rolling stockandpersonnelhasthe main influenceon the overall transportservicecosts,optimizationmethodsareessentialat thisstep,andotherpartsof theplanningmayhave to berevised.

Crew managementnot only consistsof dispatchingtrain crews, but alsolocal staff like cleaningstaffor ticket office staff. Often, therearecomplex constraintsystemsfor suchduties,e.g.dueto unioncontractsfor breakregulationsor workingtimes.Railwaycrew managementexperiencesarereportedin [11].

1.2 Train SchedulePlanning: An Overview

We will startwith a shorthistorical introductionon train schedules(detailscanbe found in [45]).In 1871, the first train scheduleconferencein Germany facedthe difficult taskof coordinatingtheschedulesof the80railroadcompaniesexisting in Germany at thattime(cf. [22]). Thefirst schedules

1.2. TRAIN SCHEDULEPLANNING: AN OVERVIEW 7

introducedfor long distancetrains in the world werenon-periodic. The reasonmight be that longdistancetrainswerescheduledrarely(usuallyonly onetrain perday),thereforea periodwould havebeensenseless.In highly congestedurbanareas,periodicor fixedintervalscheduleswereusedalmostfromthebeginning(e.g.undergroundtrainsin London1863,Budapest1896,Paris1900,Berlin1902).

Themainadvantageof periodicschedulesfrom customers’pointof view is thatthey areeasyto keepin mind. An examplefrom [45] clearlyshows this fact,seefigure1.4.

Schedule1991/92 Schedule1995/96

departurefor directionKaiserslautern, Neustadt,Saarbrucken

hour5678910

hour5678910

35 5233 43

21 33 43 5838 52

3002 43

25 36 4606 36 4606 25 36 4606 36 4606 25 36 4606 36 46

Figure1.4: Non-periodicandfixedinterval scheduleatPirmasens

By introducingperiodic schedulesfor long distancetraffic in 1939, the railroad company of theNetherlands,NederlandseSpoorwegen, marked a new epoch. Other Europeancountriesfollowedmuch later: Denmarkin 1974,Switzerlandin 1984,Belgium andAustria in 1991. In Germany, afixedinterval schedulefor InterCity trainswith a periodof onehourwasintroducedin 1979.Begin-ning in 1985/86,InterRegio trainsstepby stepweregivena periodof two hours.From1992/93,alsoregionaltrainswerescheduledin afixedinterval.

A furtherdevelopmentis the(perfect)integratedfixedinterval schedule(see[27]). This is a periodicschedulewith specificjunctionpoints,whereall trainsservingthatpoint arrive anddepartnearlyatthesametime. Thus,at the junctions,transferis possiblebetweenany pair of lines. If thereis onlyonejunction,suchascheduleobviously alwaysexists.

Traditionally, schedulesarevisualizedby timespacediagramslike in figure1.5. In suchdiagrams,fora particularrouteof thenetwork, all trainsservingthe routearerepresented.Onecandetectcriticalpointsor conflictssimply by looking at sucha diagram: The trains speedsare representedby therespective gradients,andcrossingsindicatethattrainsovertake or encountereachother.

Besidesthealreadymentionedtrainchangetimestherearemany otherconstraintsfor aschedule:Themostimportantonesaresafetyconstraints. Trainson thesametrackhave to keepa certainheadwaydistance.Onnetwork linkswith only asingletrack,trainsmustnotstartfromdifferentdirectionsatthesametime. Frequentlyusedobjectivesfor trainscheduleplanningaretheminimizationof travel time,whichmainlycorrespondsto aminimizationof waitingtimefor trainchanges,minimizationof certaincostsor maximizationof certainprofits. A comprehensive introductionto constraints,objectivesandmodelsfor trainschedulesis foundin chapter2.

In thelast few years,computersoftwarehasbeendevelopedthat is capableof effectively supportingtheconstructionof schedules.Softwareproductslike ROMAN (ROuteMANagement,this is usedin


Time

Groningen

Assen

Zwolle

Amersfort

Utrecht

Gouda

Rotterdam

8:00 8:15 8:30 8:45

Figure1.5: Timespacediagramfor ascheduleon therouteGroningen-Rotterdam

Germany andAustria)storeinformationon track topology, engineandcoachpropertiesor availablecrews in databases.Thus, the runningtime of trainscanbe calculatedin advance. Graphicaluserinterfacesenablescheduleplannersto constructor edit schedulesinteractively basedon time spacediagramslikein figure1.5.Conflicts(likemissingheadway)areautomaticallyindicatedonthescreen.After thegenerationof aschedule,simulationscanbeperformed.

However, with a few exceptionslike theDONSsystem(which is mainlyusedfor strategic planning),an automaticgenerationor even optimizationof schedulesis practically impossibleat the moment.Most of theknown algorithmsaresimply too slow for networksof practicalsize.Evenworse,math-ematicalmodelsfor someaspects(like environmentaleffects,which will bea key aspectin thenextfew years,cf. [21]) still have to bedeveloped.

Chapter 2

Models for Train Scheduling

In this chapter, mathematicalmodelsfor theproblemsof generatingandoptimizing train scheduleswill bepresented.Theperiodiceventschedulingproblem(PESP), which is theproblemof finding afeasibleschedulesubjectto aparticularclassof constraints,formsacentralpartof thechapter. Severaloptimizationcriteriafor train schedulesareintroduced,anda new modelfor costoptimalschedulingis presented.Themodelcanbeformulatedasamixedintegerprogram.At theendof thechapter, thecomputationalcomplexity of theproblemof costoptimalschedulingis analyzed.

2.1 Railr oadNetworks and Train Schedules

A railroad network is usually representedby an undirectedgraphG V E , whereV is the set

of nodesandE is the setof edges.Dependingon the requiredresolution,the nodesmay representstationsor evenswitchesandimportantsignalpoints.Theedgesrepresentrailroadtracks.

A line is modeledasa vectorof nodesv1 vn with vi V for every i 1 n , vi v j for

i j, andvi vi 1 E for every i 1 n 1 . We always assumethat all lines are served

periodicallywith thesameperiodT , i.e. we do not considerline frequencies.If therearelineswith differentperiods,we mayusetheleastcommonmultiple of all periodsasoursingleperiod,thiswill bediscussedfurtherbelow. Thesetof lineswill bedenotedby . Notethat theelementsof arevectorsof differentdimensions.

Let r v1 vn be a line. In our models,we assumethat trains of this line run from v1

to vn (via v2 ) andback to v1 via vn 1 (this is not true in somereal world cases:thereexistlines usingcyclesinsteadof onepathin both directions). We will usethe notationv r if thereisa numberi 1 n suchthatv vi andthenotation

v v r (andalso

v v r) if thereis a

numberi 1 n 1 suchthatv vi andv vi 1.

In general,theeventsthathave to bescheduledarethearrivalsor departuresof linesatsomelocationsrepresentedby nodesv V. We considerperiodic events, i.e. arrivals or departuresof a line, andindividual events, i.e. arrivalsor departuresof aparticulartrain of a line. A formal definitionis givennow:

9

10 CHAPTER2. MODELSFORTRAIN SCHEDULING

Definition: A scheduleπ for a setof events ˆ is a mappingπ : ˆ! . For an event e ˆ , π

e

is called the event time of e. A periodic event e is a countablesetof (so-calledindividual) events e" i #%$ i '& suchthattheeventtime πe" i # is givenby π

e" 0# )( T * i.

By definingtheeventtimefor anindividualeventof aperiodicevent,theeventtimesof all individualeventsof theperiodiceventaredefined.For a set

of periodicevents,let

0 : e" 0#+$ e . Byassigningtimesto eachelementof

0, all timesof individualeventsof theelementsof

areassigneda time.

Definition: A scheduleπ for asetof periodicevents

isamappingπ :,-

definedby amappingπfor thecorrespondingindividual events:π

e% x : . π

e" 0# / x for eache .

In ourmodels,we will usethefollowing notationfor ourperiodicevents:

avr 0 µ arrival of line r, directionµ, at stationv

dvr 0 µ departureof line r, directionµ, at stationv

For simplicity, we will alwaysassumethat our graphnodesrepresentstations.The directionindexmaybe0 or 1 andis interpretedasfollows: If r v1 vn , direction0 means“on thewayfrom v1

to vn”, while direction1 meanson theway back.Theindex will beomittedif therecanbeno misun-derstanding.Theindividual eventsof theseperiodiceventscorrespondto thearrivalsanddeparturesof individual trainsservinga line, i.e. the trips.

Many scheduleconstraintsof practicalinterestcanbeformulatedassocalledperiodic interval con-straints for theperiodicevents( [37,48,57] etc.).They have thefollowing form:

πe π

e)(21 l u3 T : . 4

z576 πe" 0# )( l 8 π

e9" 0# : z * T 8 π

e" 0# )( u (2.1)

with e e , l u . We will alsousethe notation“e e l u is a periodic interval constraint”

in order to express(2.1). Unionsof periodic intervals canbe modeledby intersectionsof periodicintervals(e.g. 1 10 203 60 ; 1 30 403 60 1 10 403 60 < 1 30 803 60).

Someexamplesfor scheduleconstraintsthatcanbemodeledasperiodicinterval constraintsaregivenhere(cf. [37,48,57]): Traveltimes: Supposethat

v v r andthatl is theminimumandu themaximumallowedtime

for trainsof line r for theway from v to v . Thiscanbeexpressedby thefollowing constraint:

πav=

r 0 0 πdv

r 0 0 )(>1 l u3 T (2.2)

Notethat,dependingon thechoiceof z from (2.1), individual eventswith differentindicesforarrival anddeparturemaybelongto thesametrain. A similar constraintfor theotherdirectioncanbegiveneasily. If thetravel timesareconstant,we cansetl u. Waiting times: If the waiting time for line r at stationv hasto be in the interval 1 l u3 , thefollowing constrainthasto besatisfied:

πdv

r 0 0 πav

r 0 0 )(>1 l u3 T (2.3)

2.1. RAILROAD NETWORKSAND TRAIN SCHEDULES 11 Turnaroundtimes: If r v1 vn , we needa constraintof this form:

πavn

r 0 1 πdvn

r 0 0 )(>1 l u3 T (2.4)

A turnaroundtime constraintfor theotherdirectionis notnecessarybecauseit is givenimplic-itly by usingperiodicinterval constraints. Time for train changes: As we have alreadydiscussedin section1.1, thosepassengerswitha changefrom onetrain to anotheronewould like to have a certainconnectiontime. This isprovidedby a constraintof this type:

πdv

r = 0 µ= πav

r 0 µ )(21 l u3 T (2.5)

We have alreadyseenthat it is very difficult to determinesuchstationsv and lines r r . Insection5.4,wegive aheuristicalgorithmfor determininglinesandstationsfor trainchanges. Headwaytimes: If

v v r1 and

v v r2 for r1 r2 andthereis only onerailroadtrack

leadingfrom v to v , the trainsof the lines r1 andr2 have to run on this sametrack. In orderto avoid crashes,they shouldkeepa certainheadway distance(which is equivalentto a certainheadway time). If the train speedsareconstant(which is normally assumedfor strategic andtacticalplanningmodels),theheadway timesonly needto beguaranteedat thestations,leadingto oneperiodicinterval constraintfor departuretimesandoneconstraintfor arrival times:

πdv

r2 0 µ πdv

r1 0 µ )(>1 l u3 Tπav=

r2 0 µ πav=

r1 0 µ )(>1 l u3 T (2.6)

An upperboundfor theheadway time is alsonecessary, becausetherehasto beaheadway timefor precedingandfor following trains.

Therearecasesin which theheadway constraintsdo not have thedesiredeffect. This will bediscussedin detail in section2.5.

If thereare lines r1 rm with periodsT1 Tm, one can choosethe leastcommonmultiple Tof T1 Tm and replaceeachline r i by a set of virtual lines r i 0 1 r i 0 T ? Ti whosedepartureandarrival timesareconnectedby periodicinterval constraintslike

dvr i @ j A 1 0 µ av

r i @ j 0 µ (21 Ti Ti 3 T (2.7)

This procedurepresentsanotherproblemfor thetreatmentof train changes.It is not known whichofthevirtual linesareusedfor thechange.A constraintof theform4

i 5 1 0 B B BC0 T ? T1

4j 5 10 B B BC0 T ? T2

πdv

r2 @ j 0 µ2 π

av

r1 @ i 0 µ1D(21 l u3 T (2.8)

needsto be satisfied,but this is not an interval constraint. Only in somespecialcasesthe con-straint(2.8)canbetransformedinto asetof interval constraints.


Proposition 2.1 If travelers needto change from line r1 with period T1 to line r2 with period T2 atstation v with time interval 1 l u3 (with u l E T2) and T1 c * T2 with c F , then the followingconditionis equivalentto (2.8)G

q 57H 1 0 B B BC0 cI πdv

r2 @ 1 0 µ2 π

av

r1 @ 1 0 µ1)(21 l ( q * T2 u ( T1 ( q 1:* T2 3 T1 (2.9)

Proof: In orderto simplify thenotation,theindicesfor directionsandstationsareomitted.Supposethat (2.8) is true, i.e. thereare i F 1 T J T1 , j F 1 T J T2 , z1 z2 z3 K& andt 1 l u3 suchthatthefollowing conditionshold:

πa" 0#r1 @ 1 )( i 1L* T1 π

a" 0#r1 @ i : z1 * T

πd " 0#r2 @ 1 D( j 1L* T2 π

d " 0#r2 @ j L z2 * T

πa " 0#r1 @ i )( t π

d " 0#r2 @ j L z3 * T

With z : " z2 z3 z1 #M TT1

( i 1 (notethatz & ) this canbetransformedto

πa " 0#r1 @ 1 )( t j 1:* T2 π

d " 0#r2 @ 1 : z * T1

Now determinek N : min k O& $ k * c j 1+P q . It follows thatkNQ* c j 1+P q, butkNR 1)*

c j 1S8 q 1. Becauseof

πa " 0#r1 @ 1 )( l ( q * T2 8 π

a " 0#r1 @ 1 D( t ( kN * c j 1:* T2 π

d " 0#r2 @ 1 : z kN :* T1 π

a " 0#r1 @ 1 D( t ( T1 ( k NT 1:* c j 1L* T2 8 π

a " 0#r1 @ 1 D( u ( T1 ( q 1:* T2

theconstraint(2.9)canbesatisfiedfor every q 1 c .Conversely, let (2.9)betrue. In this case,we have thefollowing conditions:

πa " 0#r1 @ 1 )( t1 π

d " 0#r2 @ 1 : z1 * T1 t1 1 l ( T2 u ( c * T2 3

πa " 0#r1 @ 1 )( t2 π

d " 0#r2 @ 1 : z2 * T1 t2 1 l ( 2 * T2 u ( c ( 1:* T2 3

...

Let qN : max q ' 1 c $ t1 P l ( q * T2 . Obviously t1 P l ( qNR* T2 holds.Wewill now show thatt1 8 u ( qNT* T2 is alsotrue:

If qN+ c, theclaimis correctfrom thediscussionabove. Let qNUE c. Sincetheintervalsfor theti havea length E T1, thechoiceof theti andzi is uniquelydetermined.It caneasilybeseenthatzq z1 forq 8 qN andthatzq z1 1 for q V qN . Now considercondition(2.9) for qNQ( 1. Wehave

πa" 0#r1 @ 1 )( tqWX 1 π

d " 0#r2 @ 1 : zqWY 1 * T1 tqWZ 1 1 l ( qN ( 1L* T2 u ( c ( qN :* T2 3Z

SincetqWY 1 t1 ( z1 * T1 zqWY 1 * T1 t1 ( T1, it follows thatt1 1 l ( qN ( 1:* T2 T1 u ( qN * T2 3 .Now we have shown that

πa " 0#r1 @ 1 )( t ( qN * T2 π

d " 0#r2 @ 1 : z1 * T1 for a t 1 l u3Z

2.2. THE PERIODICEVENT SCHEDULINGPROBLEM (PESP) 13

0 6 12 21 27 36 42 51 5760

Valid timest

t 1 6 573 60

t 1 21 723 60

t 1 36 873 60

t 1 51 1023 60

Figure2.1: Valid changingtimesfor T T1 60,T2 15, l 6, u 12

Fromthis,onecandirectly find thecorrectvaluesto satisfy(2.8). [An exampleillustrationis givenin figure2.1.

Therearemany otheraspectsof railway schedulingwhich cannotbe expressedasperiodicintervalconstraints(for exampleconstraintsreferringto individual trains). Someof themwill be discussedlaterin thischapter.

2.2 The Periodic Event SchedulingProblem(PESP)

Theproblemof findingaschedulefor periodiceventssubjectto periodicinterval constraintshasbeenexaminedby several authors. In [59], SerafiniandUkovich definedthe Periodic EventSchedulingProblem(PESP)similar to figure2.2.

PeriodicEventSchedulingProblem(PESP):

Given: T timeperiodsetof periodicevents\setof periodicinterval constraintsfor

Find: π :

]^ schedulesatisfyingall constraintsfrom

\or stateinfeasibility

Figure2.2: PeriodicEventSchedulingProblem

SerafiniandUkovich provedthatthePESPis NP-complete[59]. Odijk [49] provedthat theproblemis NP-completeevenfor fixedT V 2. More detailson thiscanbefoundin section2.9.

ThePESPhasbeenextensively examinedby severalauthors(for example[42,49,57,59]). Many oftheiralgorithmicapproachesto solve PESPinstanceswill bediscussedin chapter3. Apart from trainscheduling,thePESPhasbeenappliedto traffic light scheduling[60] andairlinescheduling[30].

Moreover, thePESPis abasisfor many scheduleoptimizingmodels.Someof themwill bepresentedin section2.6.


2.3 Event Graph Model

Often, the PESPis interpretedasa problemon the correspondingPESPeventgraph. For a PESPinstance,thedirectedeventgraph _` Vab AaS is definedasfollows: For eache , thereis anodeve Va . For each

e e l u \ , there is an arc from ve to ve= with a correspondingperiodic inter-

val 1 l u3 T .

An examplefor a (part of a) network and the event graphto the correspondingPESPinstanceisgivenin figure2.3. In theexamplecase,only onedirectionof thelinesis considered,sotherearenodirectionindices.

Network

line 1line 2line 3

stationA

stationB

stationC

Eventgraph

travel / waittrain changeheadway

aA1 dA

1 aC1 dC

1

aA2 dA

2 aB2 dB

2

aA3 dA

3 aB3 dB

3

Figure2.3: Network andeventgraphto thecorrespondingPESPinstance

In theterminologyof [59], a mappingϕ : Va c is calledpotential. Every scheduleπ for

repre-

sentsa potential(andvice versa).For a potentialϕ, thecorrespondingmappingδ : Aa d defined

by δ

v v/ ϕve: ϕ

v for eacharcfrom v to v is calledtension.

A potentialϕ (andthecorrespondingtension)is calledfeasible, if for eacharc from ve to ve= in Aarepresentingthe interval constraintc

e eX l u \ , thereexists a zc K& suchthat δ

ve ve= ft zc * T for at 1 l u3 , i.e. therespectiveschedulesatisfiesall periodicinterval constraints.zc is calledthemoduloparameterfor thatparticulararc.

For a b , we definethefollowing notation:

a g b modT : . 4z576 b a z * T

2.4 Linear Model with Integer Variables

ThePESPcanalsobe interpretedastheproblemof finding a solutionto a setof linear inequalitieswheresomevariableshave to take an integer value. From figure 2.2 and(2.1), onecanseethat a

2.5. EXTENSIONSOFTHE PESP 15

solution forXh z (where

his the vector of π

e , e , and z is the vectorof zc, c \ ) for the

problem ijjk jjl l 8 πe : π

e: zc * T 8 u for eachc e e l u \

πe

for eache zc & for eachc \ m jjnjjo (2.10)

is asolutionfor thePESPinstancegivenby T \ .

Usingtheeventgraphformulationof thePESPwith pqa asthenodearc incidencematrix (seechap-ter3 for details),l asthevectorof lowerandu asthevectorof upperinterval bounds,thelinearsystemcanbewritten as ijjk jjl l 8 p Ta h Tz 8 uh OrVs r

z & rA s r mjjnjjo (2.11)

Therearealgorithmsbasedon this formulation(anexampleis Odijk’s algorithm[47], which will bediscussedin chapter3).

Severalconstraintswhich cannotbeexpressedasperiodicinterval constraintscanbegivenaslinearconstraintsandthuscanbeaddedto theformulation(2.10)or (2.11).Somearegivenin section2.5.

2.5 Extensionsof the PESP

In this sectionwe will discusssomeextensionsto theperiodicinterval constraintmodel,which willenableusto considerotherpracticalscheduleconstraints.

SingleTrack Connections

On single track connections(i.e. wherethe samesingletrack is usedfor trainsof both directions),trainsmustnot startfrom differentdirectionsat thesametime (for obvious reasons).To beexact, ifthereis a line l1 runningfrom v to v anda line l2 runningfrom v to v onthesametrack,thefollowingconstraintsmustbeobeyed(directionindicesareomitted):t

πdv= " 0#

l2uP π

av= " 0#

l1: z * T z '&

πav " 0#

l2u8 π

dv " 0#

l1: z * T ( T for thesamez v (2.12)

In otherwords,a trainof line l2 canonly startafterthearrival of a trainof line l1 in v andmustarrivein v beforethenext train of line l1 departsthere.

Theseconstraintsarenot periodicinterval constraints.Only if thetravel timesareconstant,they canbetransformedinto interval constraints:

πav= " 0#

l1% π

dv " 0#

l1)( t1

πav " 0#

l2/ π

dv= " 0#

l2)( t2 vw π

dv " 0#

l1D( t1 8 π

dv= " 0#

l2L z * T 8 π

dv " 0#

l1)( T t2

Theintegerlinearformulations(2.10)and(2.11)enableusto addsingletrackconnectionconstraintslike (2.12)to themodelby requestingz z for therespective pairsof inequalities.


Representative Trains

Sincewe have usedperiodicinterval constraintsfor the travel, waiting andturnaroundtime of eachline l , thecorrespondingindividual eventsdv " 0#

l 0 µ or av " 0#l 0 µ do not necessarilybelongto thesametrain.

Thesameholdsfor differentvaluesof v.

Obviously, if thereis a solutionof a train schedulingPESP, thenthereis alsoa solutionwhereall theindividual eventswith index

0 correspondto thesametrain (onemaysimply addsuitablemultiples

of T to the event times). Therefore,we will now alwaysassumethat in a PESPsolution, individ-ual eventswith index

0 correspondto the sametrains for eachline. Thesetrains will be called

representativetrains.

Using representative trainscanbe modeledby requestingz 0 for theperiodicinterval constraintsfor traveling,waiting andturnaroundtime.

Another Constraint Type for the Headway Problem

As we have mentioned,periodic interval constraintsare not sufficient to provide correctheadwaytimes.An exampleis givennow:

Let two lines l1 and l2 run on the sametrack vom v to v (line direction indiceswill be omitted).Let T 60 andtheheadway time h 2 for trainsof line l2 following thoseof line l1 andvice versa.Furthermore,let thetravel timesbegivenby π

av=

l1 π

dv

l1x(1 20 223 T andπ

av=

l2 π

dv

l2x(1 16 183 T .

We assumethat the trainshave constantspeed(which we do not know). Following section2.1, weshouldintroducetheseconstraints:

πdv

l2 πdv

l1 )(21 2 583 T and πav=

l2 πav=

l1 )(21 2 583 TThisdoesnot leadto thedesiredresult:A feasiblescheduleis givenby

πdv " 0#

l1y 0 π

dv " 0#

l2/ 2 π

av= " 0#

l1/ 20 π

av= " 0#

l2y 18

whichmeansthata trainof line l2 hasovertakena trainof line l1 (rememberthatrepresentative trainsareconsidered),whichmaybeimpossibleonthetrackfrom v to v . Figure2.4showsacorrespondingtimespacediagram.

line l1

line l2v

v10 20

Figure2.4: Timespacediagram:a trainpassestheotherone

With thehelpof thefollowing proposition,wewill deriveanew typeof constraintthatwill behelpfulfor handlingtheheadway time problem.

2.5. EXTENSIONSOFTHE PESP 17

Proposition 2.2 Let l1 and l2 be lines running on a railroad track from v to v . Assumethat trainspeedsare constantandthefollowing conditionshold (directionindicesareomitted):

πdv

l2 π

dv

l1D(21 l u3 T with 0 E l 8 u E T

πav=

l2 π

av=

l1D(21 l uz3 T with 0 E l D8 u)E T

Thentrainsfromdifferentlinesovertake each otherif andonly if for theseconstraints,condition(2.1)is satisfiedfor differentvaluesof z (with representativetrains).

Proof: Weassumethatit is possibleto find amappingfrom therailroadtrackfrom v to v to theinter-val 1 0 13 preservingcontinuity(this is alwaysdonewhendrawing timespacediagrams,for example).

Let σ1x bethetimeatwhich thetrain relatedto to dv " 0#

l1passesthepointcorrespondingto x 1 0 13 .

Let σ2x bedefinedanalogouslyfor line l2. Setσ

x% σ2

xL σ1

x . Becauseof theconstanttrain

speeds,σ is amonotonefunctionon theinterval 1 0 13 . It is alsocontinous.

Trainsfrom different lines overtake eachother if andonly if thereis an x 0 1 anda k & forwhichσ

xy k * T is true(this is obvious,becausein thiscase,trainsof differentlinesareat thesame

positionon thetrackat thesametime).

Now supposethat(2.1) is satisfiedfor bothconstraintswith thesamevalueof z, i.e.

πdv " 0#

l2/ π

dv " 0#

l1)( zT ( t with t 1 l u3 and π

av= " 0#

l2y π

av= " 0#

l1)( zT ( t with t 1 l u 3

for somez & . Thenσ0U zT ( t andσ

1U zT ( t . Sinceσ is monotone,therecannotbe an

x 0 1 for which σ is amultipleof T.

Otherwisesupposethattheconditionis satisfiedfor differentvalueszandz , i.e.

πdv " 0#

l2y π

dv " 0#

l1)( zT ( t with t 1 l u3 and π

av= " 0#

l2y π

av= " 0#

l1)( z T ( t with t 1 l u 3

Supposez E z (z V z canbedealtwith analogously).Now σ0T zT t, σ

1Q z T t , andbecause

of thecontinuityof σ, therehasto beanx 0 1 with σx% z * T. [

Wemaynow avoid trainovertakingconflictsby demandingthatfor somepairsof interval constraints,the valuesof z in condition (2.1) are equal(and representative trains are used). This leadsto anextensionof thePESPcalledJPESP(PESPwith joinedconstraints), seefigure2.5.

Disadvantagesof Feasibility Models

Theschedulingmodelsdiscussedsofaronlyconsiderfeasibility. Thisleadsto twomaindisadvantagesfor thepracticaluseof algorithmsbasedon thosemodels: Practicalinstancesmaybeinfeasible.Froma theoreticalpoint of view, this doesnot presenta

problem,but in practicea schedulehasto begenerated.In orderto make theinstancefeasible,someconstraintshaveto berelaxed.But it is notatall clearwhichconstraintsshouldberelaxedor how they shouldberelaxed.A practicalalgorithmwill have to decidethat. If a schedulehasbeenfound, thereis no informationwhetherthereare“better” schedules.Inpractice,therearemany criteria for evaluatingschedules(someof themwill be mentionedinsection2.6). Wewill developanew costoptimizationmodelfor train schedulingin section2.8(whichwill bebasedon acostoptimizationmodelfor line planning).


PeriodicEventSchedulingProblemwith joinedconstraints(JPESP):

Given: T timeperiodsetof periodicevents\setof periodicinterval constraintsfor

| \Fq\setof joining conditions

Find: π :]^

schedulesuchthat travel, waiting, turnaroundconstraintsaresatisfiedwith z 0 all otherconstraintsfrom\

aresatisfiedwith arbitraryz & c1 c2 w zc1 zc2

or stateinfeasibility

Figure2.5: PeriodicEventSchedulingProblemwith joinedconstraints

2.6 ScheduleOptimization Models

Railroadcompanieshave many different(andconflicting)optimizationcriteriafor schedules,includ-ing: Minimizationof total travel time for passengers: An importantaspectdeterminingthe attrac-

tivenessof a scheduleis to keepthe trip timesfor passengersshort. Sincetrain speedsoftencannotbe variedmuch,the largestoptimizationpotentialherecomesfrom the waiting timesfor passengerswhoneedto changefrom onetrain to another. In caseof variablewaiting times,thesetimesshouldalsobekeptassmallaspossible.

Let ¯\ | \ bethesetof train changetime constraintsfrom section2.1. Supposethat for everyc ¯\ , thenumberof passengersωc who needtherespective connectionis known (aswe havepointedout in section1.1, it is difficult to determinethesenumbers).Thenthesumof waitingtimesfor all passengersis givenby

∑~av

l1 @ µ10 dv

l2 @ µ20 l 0 ux c5 ¯ ωc *x π dv

l2 0 µ2: π

av

l2 0 µ2L zc * T / (2.13)

This is a linearexpressionin thevaluesof πe andthuscanbeaddedto thePESPformulation

in orderto geta mixed integer linearprogram(MIP). In [37], Krista solved this MIP (with anadditionalcosttermfor trainwaiting timeverysimilar to (2.13))for severalrealworld networkinstanceswith acommercialMIP solver. Nachtigall[41,42] developedabranch-and-cutmethodto solve theproblem.

An alternative approachfor minimizing thetrain changetime is given in [19]. There,DadunaandVoßusedaquadraticsemiassignmentmodelandatabu searchheuristic.Kolonko etal. lookfor paretooptimalsolutionsconcerningminimumtrip time andinvestmentcostfor upgradingthenetwork tracks[36]. In this case,a greedyheuristicanda geneticalgorithmareused.

2.6. SCHEDULEOPTIMIZATION MODELS 19 Maximizationof robustnessin caseof train delays: In practice,traindelaysoccurveryoften. Inthissituation,othertrainsusingthesametrackmayhaveto wait,andsotheoverallsystemdelayincreasesin a cascade-like process.Furthermore,if passengersarrive at their train changingstationlate,they maylosetheir connection.Alternatively, othertrainshave to wait, andagainthe total delay increases.To avoid this, onecan try to maximizethe minimum headway oftrainsarriving or departingat thesamepoint in thenetwork. As a consequence,all trainshaveaheadway thatis largerthanactuallyrequired.In caseof delays,thecorrespondingconstraintsmaybedisobeyed,aslong astheactuallyrequiredheadway is guaranteed.

This approachhasbeenfollowedby Heuschet al. in [31], wherea generalizedgraphcoloringmodelandacorrespondingbacktrackingalgorithmis presented. Maximizationof profit / minimizationof costs: Thereareseveral ideasfor estimatingtheprofit/ costof a trainschedule,resultingin differentmodelsfor scheduleoptimization:

Brannlundet al. developedamodelfor aprofit maximizingschedulein [6]. In their model,theprofit dependsonthetimethatcertaintrainspasscertainpartsof thenetwork. They formulateabinaryvariablelinearprogramandgive heuristicsolutionsby aLagrangianrelaxationmethod.

In [12,13], Carey considersa minimal costschedulingmodel,wheretherearetrip time costs,dwell time costsor costsfor specialarrival anddeparturetimes.Themodelresultsin a binaryvariablelinearprogram,which is treatedheuristically.

Anothermodel introducedby Higgins et al. in [32] usesa weightedsum of delay and trainoperatingcosts.On their binaryprogram,a specialbranch-and-boundmethodwith nonlinearsubproblemsis used.Themodelis only usedonsingleline rail corridors.

In section2.8,wewill introduceanew modelfor minimalcosttrainscheduling,which is basedon acostmodelfor line planning. Minimizationof theperiod: Theremaybesituationsin whichtheminimalpossibleperiodfor atraffic systemis of interest.This approachis somewhatdifferentfrom theothers,sincefor thisproblem,theperiodis variable.In [7,52], theproblemis formulatedandsolvedasaneigenvalueproblemfor themaxplus-algebra.

An overview onoptimizationmodelsfor train routingandschedulingcanbefoundin [18].

Anotherideafor scheduleoptimizationis the“minimization of theinfeasibility” of a PESPinstance.Let π beaschedule(whichmaybeinfeasible)for aPESPinstance.Thenfor eachc e e l u \ ,theconstraint violation εc is definedby

εc : minzc 56 max 0 π e L π

e: zc * T u l π e L π

e: zc * T (2.14)

Somepossibilitiesfor “infeasibility minimization”usingεc are: Find ascheduleπ suchthatthenumberof constraintsc with εc V 0 is minimized. Find ascheduleπ suchthat∑c5 εc is minimized. Find ascheduleπ suchthatmaxc5 εc is minimized.


2.7 CostModel for Line Planning

In thissection,wewill describeamodelfor cost-orientedline planning.Basedonthismodel,wewilldevelopa cost-orientedmodelfor trainschedulingin section2.8. Theline planningmodelwasintro-ducedby Claessensin [16] in cooperationwith theDutchrailroadcompany NederlandseSpoorwegen(NS) andRailned(a Dutchstateorganizationresponsiblefor capacityplanning,managementof theinfrastructureandfor railroadsafety).It wasfurtherexaminedandtestedonpracticalnetworksof theNetherlandsin [17] and[10].

Given a network G V E , a set of possiblelines P and a set of possiblefrequencies r for

eachr P, theline optimizationis to find asubset | P andfrequenciesfr r for eachr suchthatcertainconstraintsaresatisfiedandacertainobjective is minimized.

In the modelproposedin [16], not only lines andfrequenciesaredetermined,but alsonumbersofcoachesfor the trains(i.e. trainsdo not have a fixed lengtha priori). The following costaspectsareconsideredby themodel: Fixedcostper scheduleperiodper motorunit andper coach: This includesdepreciationcost,

capitalcost,fixedmaintenancecostor costfor overnightparking. Costperkmpermotorunit andpercoach: Examplesareenergy andmaintenancecost.

In orderto determinethecostof a scheduleperiodof a particularline r P, we needto know thenumberof traincompositionsrequiredfor operatingtheline andthedistancethetrainshave to run.

Let r P, let fr r be a frequency for r and let tr be the time requiredby a train to fulfill acompletecirculation. Sincethis time may dependon the actualscheduleand thus is not knownexactly in advance,anestimationtr is used.Thenumberof trainsrequiredfor the line is thengivenby

γr : fr * trT (2.15)

Let dr bethelengthof a circulationof line r. During a scheduleperiod,thesumof thedistancesrunby all trainsof line r is dr * fr (which is independentof γr , asonecaneasilyverify!).

An examplefor thecalculationof γr is givenin figure2.6.

Thefollowing typesof constraintsareconsideredin [16]: Numbers of coaches: For eachline r P, thereis a lowerandanupperboundfor thenumberof coaches. Line frequencyfor edges: For eachnetwork edge,thereis a lower andanupperboundfor thesumof thefrequenciesof linesrunningon thatedge. Travelercapacity: On eachnetwork edge,thereis a lower boundfor the sumof the travelercapacitiesof thetrainsrunningon thatedgein onescheduleperiod.

2.7. COSTMODEL FORLINE PLANNING 21

stationA stationB stationC stationD

stationA stationB stationC stationD

20 km 25km 25 km

20 km 25km 25 km

fr 1, trainspeed60km/h,waiting andturnaroundtime ignored γr 3

fr 2, trainspeed60km/h,waiting andturnaroundtime ignored γr 5

Figure2.6: Circulationof trainsfor differentfrequencies

In [16], a nonlinearinteger programis constructedto solve the problem. We will give a slightlymodifiedversionof theformulationhere.Thevariablesare:

xr r frequency of line r P

wr & numberof coachesfor thetrainsof line r P

With thenotationof table2.1 for theinput data,themodelis givenin figure2.7.

Theresultsobtainedwith thismodelandaheuristicsolutionprocedurearereportedto bequiteunsat-isfactory(cf. [16]). Betterresultswereproducedwith two kindsof linearizationsof theCOSTNLPmodel.They will bediscussednow.

Insteadof usingintegervariablesfor thefrequency, in [10] binaryvariablesareintroducedindicatingthata certainfrequency is usedor not used.Furthermore,for eachfeasiblefrequency for a line, anintegervariablefor thenumberof coachesis used:

xr 0 f 0 1 line r P is usedwith frequency f r

wr 0 f q& numberof coachesin trainsof frequency f r for line r P

This substitutionleadsto a linear integer programmingmodel,figure 2.8. There,someconstraintshave to beaddedto ensurethatonly onefrequency is usedfor a line andthatno coachesareusedifthecorrespondingfrequency is not selected.


Cfix fixedcostpermotorunit CfixC fixedcostpercoach

Ckm km costpermotorunit CkmC km costpercoach

dr circulationlengthof line r tr estimatedcirculationtime of line r

W min. # coachespertrain W max.# coachespertrain

l f re

min. line frequency for edgee l f re max.line frequency for edgee

Ne # travelersonedgee coachcapacity

T timeperiod

Table2.1: Parametersfor cost-relatedline optimization

Nonlinearintegerprogramfor cost-relatedline optimization(COSTNLP):

min ∑r 5 P xr * tr J T +* Cfix ( wr * CfixC D( xr * dr * Ckm ( wr * CkmC l f r

e8 ∑

r 57 P 0 r e

xr 8 l f re for eache E* ∑r 5 P 0 r e

xr * wr P Ne for eache E

W 8 wr 8 W for eachr P

xr r for eachr P

wr & for eachr P

Figure2.7: Nonlinearintegerprogramfor cost-relatedline optimization

Integerlinearprogramfor cost-relatedline optimization(COSTILP):

min ∑r 57 P

∑f 57 r f * tr J T S* xr 0 f * Cfix ( wr 0 f * CfixC )( f * dr * xr 0 f * Ckm ( wr 0 f * CkmC

l f re

8 ∑r 57 P 0 r e

∑f 57 r

f * xr 0 f 8 l f re for eache E* ∑r 57 P 0 r e

∑f 57 r

f * wr 0 f P Ne for eache E

W * xr 0 f 8 wr 0 f 8 W * xr 0 f for eachr P and f r

∑f 5 r

xr 0 f 8 1 for eachr P

xr 0 f 0 1 for eachr P and f r

wr 0 f & for eachr P and f r

Figure2.8: Integerlinearprogramfor cost-relatedline optimization

For this model,severalpreprocessingtechniquesanda cuttingplanealgorithmhave beendevelopedin [10]. For several practicaloptimizationinstancesof NederlandseSpoorwegen,high quality so-

2.8. COSTMODEL FORTRAIN SCHEDULING 23

lutions (i.e. solutionswith small MIP gapsor even optimal solutions)could be found by usingtheCOSTILPmodel(with preprocessing,cuttingplanesandtheuseof thecommercialmodelingsystemGAMS (cf. [23] and[24]) andthecommercialMIP solver CPLEX [34]).

Anotherlinearizationapproachfor COSTNLPis foundin [17]. In thiscase,notonly binaryvariablesareusedasfrequency indicatorsbut alsofor thenumbersof coaches:

wr 0 f 0 c 0 1 line r P is usedwith frequency f r andc coaches

Thismethodresultsin a linearprogramwith (a lot of) binaryvariables,seefigure2.9.

Binaryvariablelinearprogramfor cost-relatedline optimization(COSTBLP):

min ∑r 57 P

∑f 57 r

W

∑c W

f * tr J T S* Cfix ( c * CfixC )( f * dr * Ckm ( c * CkmC S* wf 0 r 0 cl f r

e8 ∑

r 57 P 0 r e∑f 57 r

W

∑c W

f * wr 0 f 0 c 8 l f re for eache E* ∑r 57 P 0 r e

∑f 57 r

W

∑c W

f * c * wr 0 f 0 c P Ne for eache E

∑f 57 r

W

∑c W

wr 0 f 0 c 8 1 for eachr P

wr 0 f 0 c 0 1 for eachr P, f r andc W W Figure2.9: Binaryvariablelinearprogramfor cost-relatedline optimization

Thesolutionsgeneratedby this modelwerenot asgoodasthosefor COSTILP. As reportedin [10],on the onehandthe binary variablesprovided a betterLP relaxation,while on the otherhand,thebranch-and-boundprocessfor MIP solvingwassloweddown by theenormoussizeof theproblem.

In [10], COSTILPwasextendedto cover several supplynetworks (for exampleInterCity andInter-Regio network) simultaneously. Apart from largerproblemsizes,otherpracticalproblemsmayoccurfor suchmodels: The modelmay selectcheaper, but slower and thereforelessattractive train typesfor many

lines. Interactionsbetweenthetrain typesaredifficult to control(for exampletrainspeeds).

2.8 CostModel for Train Scheduling

We will now assemblethe ideasfrom feasibility modelsfor schedulesandfor costoptimizationforlinesto getanew modelfor costoptimalscheduling. Supposethataline planhasbeenfound(i.e. | P hasbeenselected).In our model,we assumethat the railroadcompany still is facedthe taskofassigningtrain typesto thelines.A train typeis characterizedby:

24 CHAPTER2. MODELSFORTRAIN SCHEDULING cost,capacityof coaches,boundsonnumberof coaches(asin theline planningmodel) speed

Thepossibilityof choosingfrom asetof train typesmayresultfrom thefactthatthesupplynetworksfor the lines arenot fixed in advance(althoughthis may causedifficulties asmentionedat the endof section2.7) or that thereactually are several motor units and coacheswith different propertiesavailablefor thesamesupplynetwork. Let denotethesetof train types.

Thechoiceof train typesinfluencestheschedulevia thespeed.Let r| bethesetof feasibletrain

typesfor line r. Thenthetravel timeconstraintsfor line r arerelaxedfrom

πav=

r 0 µ πdv

r 0 µ )(21 l u3 T to 4τ 57 r

πav=

r 0 µ πdv

r 0 µ D(21 lτ uτ 3 T Only for somecombinationsof train types,therewill bea feasibleschedule.Our modelwill deter-mine theminimumcosttrain typecombination(includingnumbersof coaches)for which a feasiblescheduleexists.

SincetheCOSTILPmodelgave thebestresultsfor theline optimizationproblem,we will developasimilar integer linearprogrammingmodelfor theschedulingproblem.Our variablesare(for a shortnotation,we now usea andd for eventtimesinsteadof events):

xr 0 τ train typeτ r is usedfor line r wr 0 τ numberof coachesof typeτ for trainsof line r av

r 0 µ arrival time of individual train0 , directionµ of line r in v

dvr 0 µ departuretimeof individual train

0 , directionµ of line r in v

z vectorof moduloparametersfor JPESPconstraints

The constantsform the line optimizationmodelsaregiven an additionalindex for the train type ifnecessary. Thetravel timeboundsnow dependon thetrain types:

travvv=τ minimumtravel time for trainsof typeτ from v to v

travvv=τ maximumtravel time for trainsof typeτ from v to v

wait wait minimumandmaximumwaiting time

turn turn minimumandmaximumturnaroundtime

Of course,the time boundsmaydependon othercriteriaaswell. The completeMIP modelfor theminimumcostschedulingproblem(MIP-MCSP)is givenin figure2.10.Notethat: in orderto avoid a nonlinearmodel,we still useestimations(dependingon the train type) for

thecirculationtime, weassumethat is thesetof linesafterhaving introducedacommonperiod(i.e.all lineshavethesamefrequency), in thetravel timeconstraint,µ is thedirectionin whichnodev is directly followedby v ,

2.9. COMPUTATIONAL COMPLEXITY 25

Mixedintegerlinearprogramfor minimumcostscheduling(MIP-MCSP):

min ∑r 57 ∑

τ 5 r tr 0 τ J T +* xr 0 τ * Cfixτ ( wr 0 τ * CfixC

τ D( dr * xr 0 τ * Ckmτ ( wr 0 τ * CkmC

τ ∑

r 57U0 r e∑

τ 57 r

τ * wr 0 τ P Ne for eache E

Wτ * xr 0 τ 8 wr 0 τ 8 Wτ * xr 0 τ for eachr andτ r

∑τ 57 r

xr 0 τ 1 for eachr ∑

τ 5 r

travvv=τ xr 0 τ 8 av=

r 0 µ dvr 0 µ 8 ∑

τ 57 r travvv=τ xr 0 τ for eachr ,

v v r

wait 8 dvr 0 µ av

r 0 µ 8 wait for eachr , v r, µ

turn 8 avnr 0 1 dvn

r 0 0 8 turn for eachr v1 vn otherJPESPconstraints

xr 0 τ 0 1 for eachr andτ r

wr 0 τ & for eachτ andτ r

avr 0 µ

for eachr , v r, µ

dvr 0 µ

for eachr , v r, µ

z integervectorfor correspondingJPESPdimension

Figure2.10:Mixedintegerlinearprogramfor minimumcostscheduling becauseof theperiodicity, only oneturnaroundconstraintperline is needed.

Themodelconsistsof two blocksof constraintswhichareonly connectedby thetrain typevariables.Thefirst threeclassesof constraintsareverysimilar to theline optimizationconstraints.Theremain-ing classesform a JPESPif the train typesarefixed. We will usetheseblocksfor a decompositionmethodin section4.2.

If thetravel timeestimationtr 0 τ is replacedby dv1r 0 1 av1

r 0 0 ( turn for eachr v1 vn , themodelbecomesexact,but nonlinear. In section4.7,we will developanalgorithmfor solvingthis nonlinearproblem(whichwill be,however, muchtooslow for probleminstancesizesof practicalinterests).

2.9 Computational Complexity

In this section,wewill giveanoverview on computationalcomplexity resultson thePESP. Addition-ally, we will examinethecomplexity of thecostoptimalschedulingproblemfrom section2.8. Someremarksoncomplexity canbefoundin appendixA, moredetailson thissubjectaregivenin [25].


2.9.1 Complexity Resultson the PESP

For thefollowing theorems,weassumethatwearegivenintegralvaluesasinterval boundsandperiod.Also theschedulesareexpectedto beintegervalued.

Theorem 2.1 ThePESPis NP-complete.

A proof is givenin [59]. TheHamiltoncycle problem(HCP),which is NP-complete,canbepolyno-mially transformedto thePESP. TheHCPis theproblemto determinewhethera non-directedgraphcontainsa cycle covering all verticesexactly once. The problemof the proof is that the periodTdependson thesizeof theHCPinstance.

For practical instances,there is always a fixed period (e.g. 60 for hourly trains). Therefore,it isinterestingto examinethecomplexity of thePESPfor fixedT.

Theorem 2.2 ThePESPis in P for T 2.

In [45], Nachtigallprovedthis theorem.Thereareonly two reasonableinterval boundsin caseof T 2: 1 0 03 and 1 1 13 . If thereis a constraintc

e e 0 0 \ it follows that πeg π

e mod2,

otherwiseπee g π

e mod2. Theexistenceof aschedulefor suchaninstancecanobviouslychecked

by asimplelabelingprocedureworkingwith complexity O $ \ $ .

Theorem 2.3 ThePESPis NP-completefor fixedT V 2.

Odijk [47] shows that instancesof the K-colorability problemfor fixed K, which is known to beNP-complete[25], canbe polynomially transformedto instancesof the PESPwith periodK. TheK-colorability problemis the problemof determining,whetherfor a given graphG

V E andanumberK , K E $V $ , thereis a mappingc : V

1 K suchthat cv c

ve whenever

i j E.

ToagivenK-colorabilityprobleminstanceG V E , constructaPESPinstance_, V E , whereE is obtainedfrom E by choosingarbitrarydirectionsfor theelementse E. SetT K andtake inter-val boundsl 1, u T 1 for every arc. Obviously, thereis a one-to-onecorrespondencebetweenfeasiblepotentialsfor _ andfeasiblecoloringsof G.

2.9.2 Complexity Resultson Cost Optimal Scheduling

Theminimumcostschedulingproblemdiscussedin section2.8containsseveraldifficult aspects.Aswe have seenin section2.9.1,finding a feasibleschedulefor a set of periodic interval constraintsis alreadyNP-complete.We will now focuson the selectionof train types,which will alsoleadtoNP-completeproblems.

We will now considertheminimumcostschedulingproblemwithout theschedulingconstraint(i.e.we areonly concernedwith thefirst block of constraintsin figure2.10). This problemwill becalledminimumcost typeproblem(MCTP). Furthermore,we focuson a decisionversionof the problem,namelydeterminingwhetherthereis a choiceof train typesandnumbersof carssuchthat all con-straintsaresatisfiedandtheobjective functiondoesnot exceeda certainvalue.This problemwill becalledDecision-MCTPandis givenin figure2.11independentlyof amodel(suchasMIP-MCSP).


Decisionversionof theminimumcosttypeproblem(Decision-MCTP):

Given: G V E network graph setof lines setof train types r| setof feasibletrain typesfor eachr

Cfix CfixC Ckm CkmC : 0 costfunctions

W W : boundsfor numbersof coaches

d : lengthof line circulation

γ : estimatednumberof traincompositions : coachcapacity

N : E numberof travelers

K O costlimit

Find: x : andw : suchthat

1. xr r for eachr

2.Wxr U8 w

r +8 W

xr for eachr

3. ∑r 57 : r e

x r L* w r UP Ne for eache E

4. ∑r 57 γ

r x r :*x Cfix x r )( w

r :* CfixC x r ( d

r :*x Ckm x r )( w

r :* CkmC x r 8 K


Figure2.11:Decisionversionof theminimumcosttypeproblem

Theorem 2.4 TheDecision-MCTPis NP-complete.

Proof: Theproblemis in NP: A solutionconsistsof thevaluesof x andw andis thereforepolynomi-ally boundedin theinput size.Theproperties1–4canof coursebecheckedin polynomialtime.

We will now show that instancesof theknapsackproblemof figure2.12,which is known to beNP-complete[25], canbe polynomially transformedto instancesof the Decision-MCTP, therebycom-pletingtheproof.

Consideraninstanceof theknapsackproblemwith U U1 Un (i.e. $U $ n). Wewill constructan equivalent Decision-MCTPinstanceon the network graphdepictedin figure 2.13. The graphconsistsof onenodeXi for eachui U , i ' 1 n andtwo othernodesY andZ. Thereis oneedgefrom eachXi toY andanotheredgefromY to Z. Let thelengthof all theseedgesbe1, N

Xi Y y 1

for every i 1 n andN

Y Z y F ( n.

Let r1 rn , line r i beinga line runningfrom Xi viaY to Z andback.Further, let i 0 1 and i 0 2be thefeasibleline typesfor line r i . Set 10 1 10 2 n 0 1 n 0 2 . For eachtrain type, let 1 betheonly feasiblenumberof coaches.Moreover, let γ

r τ % 1 for eachpairwith r , τ .


Knapsackproblem:

Given: U set

h : U sizefunction

f : U valuefunction

H sizelimit

F valuebound

Find: U | U suchthat

∑u5 U = f ufP F and ∑

u 5 U = h uf8 H


Figure2.12:Knapsackproblem

Z

Y

X1 X2 Xn 1 Xn

Figure2.13:Network from theproof of theorem2.4

Definethecoststructurefor theinstanceasfollows: Cfix τ Ckm τ T CkmC τ y 0 for everyτ ,CfixC i 0 1 T 1 for every i q 1 n , CfixC i 0 1 R h

ui ( 1 for every i O 1 n . Let thecapacity

be Ti 0 1 % 1 and Ti 0 2 y fui )( 1 for every i 1 n . Finally, defineK : H ( n.

Obviously, this Decision-MCTPinstancecanbeconstructedfrom theknapsackprobleminstanceinpolynomialtime. It remainsto show thattheknapsackinstanceis feasible,if andonly if theDecision-MCTP instanceis feasible.

Let theknapsackinstancebefeasible,andlet U | U bea setsatisfyingtheknapsackproblemcon-straints.In thiscase,we obtainasolutionfor theDecision-MCTPinstanceby setting:

xr i / t i 0 1 if ui U i 0 2 if ui U and w

r i y 1

for eachi 1 n . Of course,thefirst two conditionsaresatisfiedby this choice,thesameholdsfor thetravelercapacityconstraintsontheedges

Xi Y . Now considerthecapacityontheedge

Y Z :

n

∑i 1

wr i :* x r i y ∑

i: ui 5 U = f ui )( 1D( ∑i: ui 5 U = 1 P F ( n N

Y Z

Thecostconstraintis alsofulfilled:


n

∑i 1

wr i L* CfixC x r i y ∑

i: ui 5 U = h ui )( 1)( ∑i: ui 5 U = 1 8 H ( n K

Conversely, let theDecision-MCTPinstancebefeasibleandlet x andw begivensuchthat thecorre-spondingconditionshold. By choosing

U ui$ x r i /¡ i 0 2 i 1 n

thevalueconstraintof theknapsackproblemis truebecauseof

∑i: x " r i #eQ i @ 1 1 ( ∑

i: x " r i #eR i @ 2 f ui )( 1¢P F ( n

∑i: x " r i #Q i @ 2 f

ui ¢P F

∑i: ui 5 U = f ui ¢P F

Analogously, thesizeconstraintcanbeverified. [At this point, one could conjecturethat algorithmsfor knapsackproblemscould be usedto solvepracticalinstancesof theMCTP, but thereis anotherdifficulty:

Theorem 2.5 TheDecision-MCTPis NP-completeevenif there is onlyonetrain type.

Proof: We formulatethe Decision-MCTPwith only onetrain type (Decision-MCTP1) asshown infigure2.14. Becausethereis only onetrain type, thecoachcapacitycanbescaledsuchthat ' 1,andthusthecapacityfunctionis omittedfor theDecision-MCTP1.Sincethecostfor motorunitsareconstantif thereis only onetrain type,thosecostsarenot containedin theDecision-MCTP1.

Of course,the Decision-MCTP1is in NP. We may modify every Decision-MCTP1instanceto anequivalentinstancewith feasiblenumberof coachesbetween0 andW W by setting(in thisorder:)

Ne : max

tK£NeL ∑

r 5U0 r e

W ¤ 0 v for eache E

K : K ∑r 5 W * γ r L* CfixC ( d

r :* CkmC

W : W W

W : 0

If thisprocedureleadsto K E 0, we immediatelyknow it is infeasible.

In the following, we show thatevery instanceof theproblemof finding feasibleline planswith fre-quencybound1, which is introducedand shown to be NP-completein [10], can be polynomiallytransformedto suchamodifiedDecision-MCTP1instance.Theproblemof findingfeasibleline plansis theproblemof choosingsomelines from a givensetof linessuchthat for eachnetwork edgethesumof thefrequency of the linesrunningover it is boundedby certainnumbers.In [10] it is shownthatthisproblemis NP-completeevenif for all edges,thelower andupperboundsareequalto 1.


Decisionversionof theMCTP with onetrain type(Decision-MCTP1):

Given: G V E network graph setof lines

CfixC CkmC O 0 costcoefficients

W W O boundsfor numbersof coaches

d : lengthof line circulation

γ : estimatednumberof traincompositions

N : E numberof travelers

K ¥ costlimit

Find: w : suchthat

1.W 8 wr f8 W for eachr

2. ∑r 57 : r e

wr +P N

e for eache E

3. ∑r 57 γ

r :* w r :* CfixC ( d

r L* w r :* CkmC 8 K


Figure2.14:Decisionversionof theMCTPwith onetrain type

In this case,theonly possibleline frequency is 1. Thereforeit is sufficient to show the polynomialtransformationof instancesof findingfeasibleline planswith frequencybound1 (FLP1), figure2.15,to Decision-MCTP1instancesin orderto prove thattheDecision-MCTP1is NP-complete.

For anFLP1instance,weconstructacorrespondingDecision-MCTP1instanceby choosingCfixC 0,CkmC 1,W 0,W 1, N

e/ 1 for eache E, d

r % thenumberof edgesin r for eachr ,

γr % 1 for eachr , K $E $ . This is obviously a polynomialtransformation.It remainsto show

thatthis instanceis feasibleif andonly if theFLP1instanceis feasible.

Let theFLP1instancebefeasiblewith solution ¦ | . Define

wr / t

1 if r 0 if r

Thecapacityconstraintis satisfiedtrivially. Thecostconstraintis alsofulfilled:

∑r 57 d

r :* w r S ∑

r 57 = d r / ∑r 57 = ∑e5 r

1 ∑e5 E

∑r 57 = 0 r e

1 $E $ 8 K

Conversely, let theDecision-MCTP1instancebefeasiblewith a solutionw. Choose : r $wr / 1 .

Consideranedgee E. Fromthecapacityconstraintof theDecision-MCTP1instancewe get

∑r 57 = 0 r e

1 ∑r 5U0 r e

wr +P 1


Feasibleline planproblemwith frequency bound1 (FLP1):

Given: G V E network graph setof lines

Find: | suchthat

∑r 57U0 r e

1 1 for eache E


Figure2.15:Feasibleline planproblemwith frequency bound1

Now supposethat thereis anedgee E with ∑r 57U0 r e1 α V 1. This would bea contradictiontothecostconstraintof theDecision-MCTP1instance,because

∑r 57 d

r :* w r / ∑

r 57 = ∑e5 r1 ∑

e5 E∑r §¨ =r © e 1 ∑

r §X¨ =r © e= 1 ( ∑

e§ Ee ª« e= ∑

r §¨ =r © e 1 P α ( $E $ 1%V $E $ K

Thiscompletestheproof. [By the theoremsof this section,we have seenthatall principlepartsof thecostoptimalschedulingproblem,i.e. the selectionof train types(cf. theorem2.4), selectionof numbersof coaches(theorem2.5), determinationof a schedulefor given train types,i.e. known intervals for travel time (theo-

rem2.3)

belongto aclassof problemssupposedto bedifficult to solve. Thismotivatestheuseof aMIP modelfor theMCSP.

A direct solution of the MIPs of figure 2.10 for practical instancesis impossiblein a reasonableamountof time (i.e. even small practicalinstancesrequiredsolution timesof several days). For astrategic planningtool, thesolutiontimesshouldnotexceeda few minutes.

In chapter4, we will develop a decompositionmethodwhich acceleratesthe solutionprocesssig-nificantly. With the method,it is possibleto solve instances(or at leastto get feasiblesolutionsofacceptablequality) of practicalinterestin a few minutes.


Chapter 3

FeasibleSchedules

In this chapterwe discussknown algorithmsanddesignnew algorithmsfor solvingPESPinstances.Thesolutionof PESPinstanceswill form acrucialpartof ourscheduleoptimizationalgorithmswhichwill beintroducedin chapter4.

Notation and Concepts

Many PESPalgorithmsarebasedon ideasrelatedto the event graphformulationof PESP(cf. sec-tion 2.3). We will shortly introducesomefurther notationsand conceptsbeforeexamining PESPalgorithms.

Theeventgraph _¡ Vab AaS wasintroducedin section2.3. It canhave parallelarcs.Let n : $Va $andm : $Aa $ . In orderto getashortnotation,let thesetsVa andAa beorderedandlet theelementsof Va becalled1 n. We will usethenotationa Aa , a : i

j to describethata is anarc from

nodei Va to node j Va . i and j arecalledendpointsof a.

A chaina1 ar is a sequenceof arcssuchthatai andai 1 (1 8 i 8 r 1) areadjacent,i.e. they

have a commonnode. That endpointof a1 which is not an endpointof a2 andthat endpointof ar

which is notanendpointof ar 1 arecalledendpointsof thechain.A chainis saidto beelementaryif,for eachnodewhich is anendpointof anarcof thechain,thereis atmostonearcstartingfrom andatmostonearcendingin thenode.A cycleis a chainwhoseendpointscoincide.If thereis a mappingν : 1 r Va suchthatai : ν

i ν

i ( 1 for eachi 1 r 1 , thechainis alsocalled

path. If, in additionto thiscondition,ar : νr ν

1 for acycle, thecycle is alsocalledcircuit.

Leta1 ar bea chainwith ai a j for each1 8 i E j 8 r. Let therebeno loop in a1 ar .

An arc aρ, ρ 1 r 1 is saidto have positiveorientation in the chain, if aρ : i

j and j isan endpointof aρ 1 (otherwiseit is said to have negative orientation). ar is said to have positiveorientation,if ar : i

j andi is anendpointof ar 1 (analogouslynegativeorientationis defined).The

incidencevectorp m representingthechainis definedby

pa : ijjk jjl 1 if a aρ for a ρ 1 r with positive orientation 1 if a aρ for a ρ 1 r with negative orientation

0 if a is not containedin thechain.

for eacha Aa33

34 CHAPTER3. FEASIBLESCHEDULES

Let p : max 0 pa 7 a5 A s andp : max 0 pa 7 a 5 A s . Notethatp p p .

If, for every pairof nodesof _ , thereis achainwith thesenodesasendpoints,_ is calledconnected.Every connectedgraphcontainsa spanningtree , which is a cycle-free,connectedsubgraphcon-tainingall nodesof _ . Let thearcsof bedenotedby A . Let µ : Aa

beacostfunctionfor thearcsof _ . is calledminimumspanningtreeif theweight∑a5¬) µ

a is minimalamongtheweights

of all spanningtrees.Calculatinga minimumspanningtreecanbeefficiently done,for example,bythe well known algorithmsof Kruskal andPrim (see[1]). Arcs a with a A arecallednon-tree,co-treearcsor chords.

Thenodearc incidencematrix p® °¯ ia hasonerow for eachnodei andonecolumnfor eacharca.Theentriesof thematrix aredefinedby¯

ia : ijjk jjl 1 if a : j

i for somenode j 1 if a : i

j for somenode j

0 otherwise.

An examplefor a graphandthecorrespondingnodearcincidencematrixaregivenin figure3.1. Let±k denotethecolumnof the transposednodearcincidencematrix correspondingto nodek.

Adding a co-treearca to a treegeneratesa uniqueelementarycycle. The incidencevector ² of thiscycle with γa 1 (notethat if ² is an incidencevectorof a cycle, ³² is the incidencevectorof thesamecycle,but with thedirectionreversed)formsa row of thenetworkmatrix Γ of thegraph.Sinceevery spanningtree hasexactly n 1 arcs,therearem n ( 1 co-treearcs,andthereforeΓ hasm n ( 1 rows (andm columns).The numberm n ( 1 is calledcyclomaticnumberof thegraph.Whenusingasuitablenumberingof thearcs,thenetwork matrixcanbesplit into theform Γ ´1N E 3 ,whereE denotesthe unit matrix associatedwith all co-treearcs. An examplefor a sucha networkmatrix is givenin figure3.1,wherearcsrepresentedby thick arrows form thespanningtree .

1 2

3 4a3

a1

a4

a2 a5 µ¶¶¶¶·`¸ 1 ¸ 1 0 ¸ 1 0

1 0 0 0 ¸ 1

0 1 ¸ 1 0 0

0 0 1 1 1

¹Zºººº»Nodearcincidence

matrix ¼ ½0 ¸ 1 ¸ 1 1 0

1 ¸ 1 ¸ 1 0 1 ¾Network matrix Γ

Figure3.1: Graphandcorrespondingnodearcincidenceandnetwork matrix

Let π : Va c bea potential(correspondingto a schedule)for _ . Let πi : π

i for a nodei Va ,

andleth

bethevectorof πi , i 1 n . Thecorrespondingtension,representedasavectorx, canbe calculatedasx ´p T h . Every tensionis characterizedby the fact that the sumalonga cycle iszero.This meansx is a tension(associatedwith apotential

h) if andonly if Γx 0.

For an arc a Aa , let la be the lower andua be the upperboundfor the interval of the constraintcorrespondingto a. Theinterval 1 la ua 3 is calledspan, andtheexpressionua la is calledspanlength.

3.1. PREPROCESSING 35

In contrastto thespan, 1 la ua 3 T : x z * T $ x z '& la 8 x 8 ua denotestheso-calledperiodicextensionof 1 la ua 3 . By thisdefinition,(2.1)candirectlybeinterpretedasbeingmemberof thecorrespondingset.

Wewill usethenotation *z modT :

t 1 0 T x ¿

x modT : min x z * T $ z & x z * T P 0 (it follows thatπi g π j modT is equivalentto

πi modT

π j modT). This notationis extendedto vectorsx andsetsX by

x modT : xi modT andX modT : x modT $ x X .

3.1 Preprocessing

The running timesof the algorithmspresentedin this chapterexponentiallydependon the sizeofthePESPeventgraph.Often thesolutioncanbeacceleratedremarkablyby reducingthegraphsizein a preprocessingstepbeforeactuallystartingthe solutionalgorithm. The preprocessingmethodsdiscussedin thissectioncanbedividedinto two categories: Reducingthenumberof nodesandarcsof thegraph Reducingtheinterval width of theperiodicinterval constraints

Reducingthe Number of Nodesand Ar csof the Graph

Therearesomesituationswherenodesor arcscanbedeletedfrom an instancewithout changingitsfeasibility status: Trivially feasibleor infeasiblearcs: If thegraphcontainsanarcawith ua la P T, theconstraint

correspondingto a canobviously alwaysbe satisfied. Therefore,a canbe deletedfrom thegraph.

Similarly, if ua la E 0, theconstraintcannotbesatisfiedat all, andtheinstanceis infeasible.

If thereis a loop a : i

i with the interval 1 la ua 3 containinga multiple of T, the arc canbedeleted.If theinterval doesnot containsuchamultiple, theproblemis infeasible. Arcswith singlepoint interval: If thereis anarca : i

j with la ua, thenthearcandoneof

thenodescanbedeleted:Replaceeveryarca : i j with interval 1 la= ua= 3 by anarca : i iwith interval 1 la= la ua= la 3 . Analogously, arcsa : j

i for nodesi canbereplaced.Now

node j andarca canbedeleted. Nodeswith onlyoneincidentarc: If thereis anode j whichis only incidentto onearc(a : j

ior a : i

j), theconstraintcorrespondingto a canalwaysbesatisfied,andnode j andarca can

bedeleted.

36 CHAPTER3. FEASIBLESCHEDULES Nodeswith only two incidentarcs: If thereis a node j with only two incidentarcsa : i

janda : j

k, the nodeandthe two arcscanbe replacedby onearc a : i

k with interval1 la ( la= ua ( ua= 3 . Components: If thegraphconsistsof severalcomponents(i.e. theunderlyingundirectedgraph

is not connected),thePESPinstancesfor thecomponentscanbesolvedseparately. If andonlyif all theseinstancesarefeasible,thewholeinstanceis feasible.

If the graphconsistsof two partsthat areonly connectedby onearc, the partscanbe solvedseparately, andthe potentialsof the solutionof onepart have to be increased/decreased by aconstantin orderto geta feasibletensionon theconnectingarc.

If thegraphconsistsof two partsthatareonly connectedby two arcsa : i

j anda : i j (without lossof generalityit is assumedthat i and i arein thesamepart),we canchooseonepart(without lossof generalitythepartwith i andi ) andsolve T PESPinstancesarisingfromaddinganarca : i

i with interval 1 k k3 , k K 0 T 1 to thepart (assumingthatonly

integer datais considered).By doing this, all feasibletensionsπi = πi and thusall feasibletensionsπ j = π j canbedetermined.Arcscorrespondingto theconstraintsfor π j = π j cannowbeaddedto thepartof thegraphcontainingj and j . Now thispartcanbesolved.Thismethodis only usefulif onepartis “small” comparedto theotherone(sinceT PESPinstancesfor thisparthave to besolved),andif thenumberof constraintsfor π j = π j doesnotgrow toomuch.

A feasiblesolutionof theremainingproblem(s)canobviouslybeextendedto asolutionof theoriginalproblem.

ReducingInter val Widths

Let a Aa with feasibleinterval 1 la ua 3 . Oftenit happensthatfor every feasiblesolution,thetensionof arca is anelementof asubsetS ÀÁ1 la ua 3 . Sometimes,theinterval maythenbereplacedby anotherinterval S : 1 l a ua 3 with S

|SLÀ1 la ua 3 . Sometimes,it is possibleto detectsucha possibility in a

preprocessingstep.

In orderto developsuchapreprocessingmethod,wewill useaconstraintpropagationapproach:Lookat theexampleof figure3.2.There,thetimespan1 20 303 60 is givenfor thearcfrom node1 to node2.However, π2 π1 g 20 mod60 is notpossiblebecauseof theothertwo arcs.Wecanactuallyreplacethespan1 20 303 60 by 1 21 303 60. Theconstraintpropagationmethodinvestigatesall triplesof nodesina recursive manner, until no intervals canbereduced.We will now give a formal descriptionof thisapproach.

Let T denotea fixed period. A setU|

is saidto be T-periodic, if for all u U andz K& alsou ( zT U . Suchasetcanbewrittenas

U : u ( zT $ u U| 1 0 T z &

Let ST : s ( zT $ s S z ¥& . ST is aperiodicset.If a : i

j is anarc,then 1 la ua 3 T is theperiodicsetcontainingthefeasibletensionvaluesπ j πi. Let thesetof all T-periodicsetsbedenotedby Peranddefinetheoperations

U Â V : U < V and U Ã V : u ( v $ u U v V

3.1. PREPROCESSING 37

1 2

3T 60

Ä20Å 30Æ : ∆1 Ç 2

Ä11Å 16Æ Ä

10Å 15ÆFigure3.2: Thespan∆1 0 2 canbereplacedby 1 21 303

for U V Per.

Algorithm 3.1calculatesa periodicsetMi j for eachpairof nodesi j . Eachfeasiblepotential

hhas

to satisfyπ j πi Mi j for every arca : i

j. If 0 Mii for a nodei, thentheinstanceis infeasible,becauseno potentialcanfulfill πi πi g 0 modT.

In contrastto this,0 Mii for all nodesdoesnot imply that thePESPinstanceis feasible,asonecanseefrom thecounterexamplein figure3.3.

3 4

1 2

T 10

È1 Å 3ÉÈ1 Å 3ÉÈ

0 Å 1É È0 Å 1ÉÈ

2 Å 3ÉÈ1 Å 2É

Figure3.3: InfeasiblePESPinstancewith 0 Mii for eachnodei

In the example,non-convex spansare given. However, thesecan be modeledby intersectionsofseveral interval constraints(cf. section2.1).

On theonehand,thePESPinstanceis infeasible:Assumeπ1 g 0 mod10, thentherearetwo cases:π4 g 2 mod10 w π2 g 1 mod10 w π3 g 0 mod10 w π4 π3 1 3 T , π4 g 3 mod10 w π2 g3 mod10 w π3 g 1 mod10 w π4 π3 1 3 T .

On theotherhand,for eacharca : i

j, thevalueof Mi 0 j is givenby theoriginal spanof figure3.3,and0 Mi 0 i for eachnodei.

Our preprocessingmethodwill now work asfollows: Let a : i

j be an arc with interval 1 la ua 3 T .For every feasiblesolutionof thePESPinstance,π j πi 1 la ua 3 T < Mi j . Wecanconstructastrongerinitial constraintsystemin thisway: Wecanreplacethearca by severalarcsin suchawaythatπ j πi 1 la ua 3 T < Mi j is demanded

explicitely (recallthatMi j maybeaunionof periodicintervals).By thisprocedure,thenumberof arcsmaybeincreasedto suchanextentthatthePESPalgorithmbecomesvery slow.


Algorithm 3.1ConstraintPropagationfor Preprocessingfor each

i j Va Va do

if i j thenMi j : T * &

elseif thereis anarca : i

j or a : j

i thenMi j : ËÊ a:i Ì j 1 la ua 3 T < ÍÊ a: j Ì i 1Î ua la 3 T

elseMi j : 1 0 T T

end ifend formodification: truewhile modificationdo

modification: falsefor all

i j k Va Va Va with k i j do

A : Mik Ã Mkj

if i j and 0 Mi j thenStop.Instanceis infeasible

end ifif A Ï Mi j then

Mi j : Mi j Â Amodification: true

end ifend for

endwhileStop.Mi j hasbeencalculatedfor all pairsof nodes. Weonly modify theinterval boundsla andua for thearcin suchawaythatua la is minimized,

but still Mi j| 1 la ua 3 T is fulfilled. An exampleis givenin figure3.4.

intervalÄla Å ua Æ

Mi j

new interval0

0

0

60

60

60

10 45

10 50

10 20 30 45

Figure3.4: Reducingtheinterval width

3.2. BASIC PROPERTIESOF THE PESP 39

3.2 BasicPropertiesof the PESP

RecallthataPESPinstanceis givenby a time periodT, a graph _] Vab Aa+ , n : $Va $ , m : $Aa $ ,asetof spans 1 la ua 3 $ a Aa+ , andto solve thePESPinstancemeansfindingapotential

h n andavectorof moduloparametersz q& m with

la 8 π j πi za * T 8 ua for eacha : i

j AaU (3.1)

Wewill shortenournotationbyV : Va andA : Aa .

Let thesetof feasiblesolutionsfor aPESPinstancebedenotedbyÐ: Xh z $ h n z '& m la 8 π j πi za * T 8 ua for eacha : i

j A

Theconvex hull convÐ

of thissetis calledthe(unbounded)timetablepolyhedron. For afixedvectorof moduloparametersz '& m, define

Πz : h $ la 8 π j πi za * T 8 ua for eacha : i

j A

andlet Ñ: z q& m $ Π z /0

Theproblemof deciding

Πz ? /0

for a given z is a feasibledifferentialproblem(seeappendixC.3) with spans1 l ( z * T u ( z * T 3 andcanbesolvedby a shortestpathproblemin a modifiedgraph _Ò with _ÒD

VaU A; A , A : A,A : thesetof counterarcsfor eacha A (cf. appendixC.3). Thearclengthsfor _Ò aregivenby

µa : tua ( za * T a A la za * T a A

According to (C.2), the feasibledifferential problemis soluble if and only if for eachcycle withincidencevector ² , Ó T ² (² SP 0. FromÓ T ² (² y u ( z * T T ² l ( z * T T ² uT ² lT ² ( TzT ²it follows thatΠ

z /0 is equivalentto

uT ² lT ² ( TzT ²P 0 . zT ²P 1T* lT ² uT ² (3.2)

Sincez is integral,we have thefollowing proposition:

Proposition 3.1 Πz /0 if andonly if for everyelementarycycle² Tz P 1

T lT ² uT ² (3.3)


This resulthasalsobeenshown by polyhedralargumentsin [47]. Theinequalitesof this propositioncanbe usedascutting planesandwill play an importantrole in this chapter. They arecalledcyclecuttingplanes.

The cycle cutting planescannotbe usedin order to remove all non-integer solutionsfrom the in-equality system(3.1). Figure 3.5 shows an example. The constraintsystemconsistingof the in-equalities(3.1) and all the cycle cutting planeshas the fractional solution

h 0 7 1 2 T , z

0 0 0 12 1

2 12 T . The PESPinstanceis infeasible: Let π1 g 0 mod10. Thenwe have to con-

siderthe following two cases:π3 g 0 mod10 andπ3 g 1 mod10. Proposition3.4 will show that ifthereis no integral solutionto a PESPinstancewith integerdatal, u andT, thenthereis no solutionatall.

π3 g 0 mod10 w π2 g 1 mod10 becauseof arc a4. From arc a1 andarc a5 it follows that π4 g2 mod10, which is a contradictionto arc a6. Onecanobtainan analogouscontradictionfor π3 g1 mod10.

T 10

3

1 2

4

Cyclecuttingplanes:¸ 1 Ô z2 ¸ z3 ¸ z4 Ô 0

0 Ô z1 ¸ z2 ¸ z5 Ô 1

0 Ô z4 Õ z5 ¸ z6 Ô 1

0 Ô z1 ¸ z3 ¸ z6 Ô 1¸ 1 Ô ¸ z1 Õ z2 ¸ z4 Õ z6 Ô ¸ 1

0 Ô z2 ¸ z3 Õ z5 ¸ z6 Ô 0

0 Ô z1 ¸ z3 ¸ z4 ¸ z5 Ô 0a6Ä

3 Å 11Æ

Ä3 Å 11Æa2Ä

0 Å 1Æ a3 a5Ä0 Å 1ÆÄ

1 Å 2Æa4Ä

2 Å 3Æ a1

Figure3.5: Cyclecuttingplanesfor aPESPinstance

Thefollowing resulthasbeenprovenin [49] and[59]:

Proposition 3.2 If a PESPinstanceis feasible, thenfor each vectorof moduloparameters z andforeach fixedspanningtreethere existsa vectorof moduloparameters z with za 0 for all treearcsand

Πz modT Π

z modT /0

Proof: Considera fixedspanningtreeof theconnectedgraph _ . Fix anarbitrarynode,saynode1,asthetreeroot node.Thenfor eachothernodei, thetreecontainsa uniquelydeterminedchainwithincidencevectorpi from node1 to nodei. Let z '& m and

h Πz .

Now defineh

andz by

πk : π k pk Tz * T and za : za p j pi Tz for eachnodek andeacharca : i

j. Wewill now show thatza 0 for treearcsand

h Πz .

Considera treearca : i

j. At first observe thatp j pi ea, whereea denotestheunit vectorwithea

a 1 andeab for eacharcb a. It follows thatza 0 and

π j πi za * T π j p j Tz * T π i ( pi Tz * T π j π i za * T 1 la ua 3Z

3.3. MIXED INTEGERPROGRAMMING 41

For aco-treearca : i

j, we have

π j πi za * T π j p j Tz Ö* T π i ( pi Tz * T ( za ( p j Tz Ö pi Tz eL* T π j π i za * T 1 la ua 3ZIt follows that

h Πz . Since

h g h modT, theproof is complete. [Now assumethatwe have a fixedspanningtree with associatednetwork matrix Γ. Two schedulesgiven by

hand

h withh g h modT areequivalent. Proposition3.2 allows us to consideronly

scheduleswith moduloparameterof 0 on all treearcs.DefineÑ : z Ñ $ za 0 for a ¥Note thateachrow of Γ is the (transposed)incidencevector ² T

a of a cycle which containsonly treearcsandexactlyonechorda. Theinducedcyclecuttingplanesfor thecycleandits countercyclegiveboundson themoduloparameterof all chordsby thefollowing proposition:

Proposition 3.3 Let ² Ta bea row of thenetworkmatrix Γ andlet z Ñ . Then

za : ^ 1T lT ² a uT ² a 8 za 8× 1

T uT ² a lT ² a Ø : za

It follows thatÑ is finite.

Thenext propositiondealswith integralsolutionsof aPESPinstance.It has,for example,beenprovenin [45].

Proposition 3.4 Leta feasiblePESPinstancebegivenbyVa , Aa , l u q& rA s r andT O& . Thenthereexistsa feasiblepotential

h '& rVs r .Proof: SincethePESPinstanceis feasible,thereis a vectorz & rA s r suchthat l 82p T h zT 8 u.Thisconstraintsystemcanalsobewrittenas£ p Tp T

¤ h 8 £u ( zT l zT

¤ Thecoefficient matrix of this systemis totally unimodular, andtheright handsideis integral. There-foreanintegersolution

hexists. [

3.3 Mixed Integer Programming

A straightforward approachto solve PESPinstancesis the useof the mixed integer programmingformulationgiven by (2.11). We canstrengthenthe formulation (i.e. addconstraintssuchthat theoriginal LP solutiongetsinfeasible,seealsoappendixB) in thefollowing ways:

42 CHAPTER3. FEASIBLESCHEDULES Theintegervariablescorrespondingto thearcsof aspanningtreecanbefixedto zero(proposi-tion 3.2). Boundsfor theotherintegervariablesareobtainedfrom proposition3.3. Cyclecuttingplanes(3.3)maybeused.

Additionally, wemayfix thepotentialof onenodeto anarbitraryvalue(sayπ1 : 0): Ifh

is afeasibleschedule,thenalso

h ( c * 1 with c is a feasibleschedule.1 denotesthevectorcontainingonly1-entries.

Our practicalexperience(seechapter5) shows that the MIP solutionprocessis possiblefor someinstances.Neverthelessthereis aproblemwith thisapproach:As wehavealreadymentioned,in caseof aninfeasibleinstance,wewould like to detecta“reason”for theinfeasibility, or in otherwords,wewould like to have anideahow to relaxtheinstancein sucha way that it becomesfeasible.Thereisno obviousway for gettingsuchinformationfrom theMIP branch-and-boundtree.

3.4 Odijk’ s Algorithm

In [47,48], Odijk suggestsa PESPalgorithm basedon the MIP formulationof PESP(cf. (2.10)).However, thealgorithmdoesnotsolve theMIP instancedirectly, but in a two-stepiterative procedurethatprofitsfrom theeffect of thecyclecuttingplanes(3.3). Wewill now describesomeideasleadingto thealgorithm.For moreinformation,[47,48] canbeconsulted.

Notethatsolving(2.10)for fixedz, or equivalentlyfinding ah Π

z , canbeformulatedasa linear

programmingproblemand thus can be solved efficiently. Let z & m and define l " z # : l ( z * T,u " z # : u ( z * T andlet

LPz : ijjk jjl max 0

subjectto l " z # 8¦p T h 8 u " z #h P 0

(3.4)h P 0 doesnot reallypresentaconstraintfor theschedule.Thedualproblemof LPz is givenby

DPz : t

minu " z # Ty l " z # Ty

subjectto p y y +P 0(3.5)

This problemeitherhasan optimal solutionwith objective value0 (e.g.y 0 andy 0) or isunboundedfrom below. In thefirst case,LP

z alsohasfeasiblesolutions,anda schedule

hcanbe

determined.Otherwise,anextremerayy y with

u " z # Ty l " z # Ty E 0 canbefound.In [47],

Odijk shows thatfrom this ray, acyclecuttingplanecanbeconstructedwhich is violatedby z.

Theseideasareintegratedin the following iterative procedure:During eachiteration,a polytopePcontainingcandidatesfor vectorsof modulo parametersz is kept. By the help of a backtrackingprocedure,a z P <Ù& m is selected(this is themaintimeconsumingpartof thealgorithm).If nosuchvectorexists, the PESPinstanceis infeasible. Otherwise,DP

z is solved. If it is unboundedfrom

below, a cycle cutting planeviolatedby z is constructed,andP is replacedby the intersectionof P

3.5. CONSTRAINTPROPAGATION 43

with thecut. If DPz hasanoptimalsolutionwith objective value0, the iterationprocedurecanbe

stopped,sinceasolutionπ of thePESPinstancecanbefound.

Initially, P is thepolytopedescribedby theboundconstraintsfrom proposition3 3 andza 0 for treearcsfor afixedspanningtree .

Thecompletemethodis givenby algorithm3.2.Practicalexperiencesshow thatthismethodcanonlyhandlesmallPESPinstancesin a reasonableamountof time (cf. [45], [47]).

Algorithm 3.2Odijk’s AlgorithmChooseaspanningtree .P : z m $ za 0 for eacha A

Ù andza 8 za 8 za for eacha A Ú

loopif P <Ù& m /0 then

Stop.ThePESPinstanceis infeasible.end ifChoosez P <Ú& m.if DP

z hasanoptimalsolution

y y with objective value0 then

Constructoptimalsolutionh

for LPz .

Stop.Xh z is asolutionfor thePESPinstance.

end ifFromanextremeray

y y with negative objective valuefor DP

z , constructa cycle cutting

planeÛ Tz 8 α0 which is violatedby z.P : P <¥ z m $ Û Tz 8 α0

end loop

3.5 Constraint Propagation

Voorhoeve hasdevelopedanothersolutionmethodfor PESPinstancesin [62]. His algorithmextendstheconstraintpropagationmethodintroducedin section3.1.

Assumethat thepotentialof anarbitrarynodehasbeenfixedto anarbitraryvalue(e.g.π1 : 0) andthatMi j hasbeencalculatedfor eachpair

i j of nodes.Considerthesetwo cases: 0 Mii for somenodei. ThenthePESPinstanceis infeasible. 0 Mii for all nodesi. In this case,thePESPinstancemaybefeasible.

Thecalculationof Mi j is integratedinto thefollowing constraintpropagationprocedure:If 0 Mii forall nodesi, anotherpotential,sayπ2, is fixedin suchawaythatπ2 π1 M12. ThisprobablyreducesthesetsMi j for otherpairsof nodes

i j . Theprocedureis repeated.If 0 Mii for a nodei at some

step,onehasto backtrack,andthepotentialof thevariablethatwasfixed in thestepbeforeis givenanothervalue.

Thealgorithmterminateseitherwith afeasiblefixingh

of all potentialsor with aproofof infeasibility(all potentialsleadto abacktrackingstep).


Voorhoeve’s methodis given as algorithm 3.3. Computationalresultswith this methodare ratherdeterring[4]. A main reasonfor the behavior is that therearetoo many possibilitiesfor the fixingof potentialswhenT is large (like T 60). In this case,the searchtreesoongetstoo large to bemanageable.

Algorithm 3.3Voorhoeve’s ConstraintPropagationAlgorithmChooseanarbitrarynodei V andsetπi : 0.loop

CalculateMi j for eachi j V

V.

if 0 Mii for somenodei thenif no backtrackingis possiblethen

Stop.ThePESPinstanceis infeasible.end ifPerforma backtrackingstep,i.e. changethe potentialof the variablethat wasfixed before.If necessary, do further backtracking. If no suchbacktrackingis possible,stop. The PESPinstanceis infeasible.

elseChooseanodei V suchthatπi is not fixedyet.Fix πi in suchaway thatπ j πi Mi j for all nodesj.

end ifend loop

3.6 Algorithm of Serafini and Ukovich

SerafiniandUkovich introducea backtrackingmethodfor solving PESPinstancesin [59]. In thismethod,thepossiblevectorsof moduloparametersareinvestigated.

Thealgorithmstartswith determiningaminimumspanningtree concerningthespanlengthsua lafor eacha A anda feasiblepotential

hfor thegraphignoring thechords.

his obtainedby fixing

thepotentialof onenodeandchoosinganarbitraryfeasibletensionon thetreearcs.Afterwards,allchordsaresortedin orderof increasingspanlength.Let thisorderbea1 am n 1.

Now assumethatthealgorithmis searching at level k, whichmeansthat Themoduloparametersfor thetreearcsaresetto 0, andthemoduloparametersfor all chordsar with r 1 k 1 have beenfixed. A potential

his known which is feasibleup to level k 1, i.e. the periodic interval con-

straints(3.1)aresatisfiedfor thetreearcsandfor a1 ak 1. Thealgorithmis looking for an integerzak for arcak : i

j suchthath

canbemadefeasibleup to level k withoutchangingfixedmoduloparameters.

Thesearchproducesoneof thesetwo results:

3.6. ALGORITHM OF SERAFINI AND UKOVICH 45 Thereexistsa zak q& with π j πi zak 1 lak uak 3 . Thereforeh

is feasibleup to level k. Thereis az '& with

π j πi z * T E lak 8 uak E π j πi z 1L* T In this case,the algorithmtries to raiseor lower the tensionvalueπ j πi in sucha way thatthe tensiongetsfeasiblefor arc ak (cf. figure 3.6) andthe feasibility for the othertensionsismaintained(without changingany of thefixedmoduloparameters).This is a feasibledifferen-tial problemandcanbesolved by a modifiedDijkstra shortestpathprocedurethat is given inappendixC.

Supposethatone(or both)attempts,i.e. to lower or to raisethetension,fails. In this situation,theshortestpathalgorithmfindsa circuit of negative length(cf. appendixC). Thearcsof thiscircuit (thesecircuits)arecalledblocking arcsandplay animportantrole for thebacktrackingstepof thealgorithm:Thetensionvalueof arcak canonly belowered(or raised)for therequiredamount,if at leastthemoduloparameterof oneof theblockingarcsis changed.

tensionfeasible feasible feasible

lak ¸ T uak ¸ T lakuak lak Õ T uak Õ T

π j ¸ πi ¸ zT π j ¸ πi ¸ÝÜ z ¸ 1Þ T

raisetensionlower tension

Figure3.6: Raisingor loweringthetension

The result of this proceedingis either a feasiblemodulo parameterzak and a (possiblymodified)potentialwhich is feasibleup to level k or infeasibility. Stating infeasibility meansthat it is notpossibleto extendthepartial solution za1 zak ß 1 to a feasiblez-vectorfor thecompleteinstance(i.e. Π

z f /0 for all z & m with za1 zak ß 1 fixed asdoneby the algorithm). In this case,the

moduloparameterof a previously investigatedlevel hasto be changed.To be moreexact, let à betheunionof thesetsof blockingarcsfor loweringor raisingthetensionof arcak. Thenthealgorithmbacktracksto level k with k : max κ $ aκ àá .For eachlevel k, informationhasto be storedon thevaluesfor zak thathave alreadybeentested.Ifall valueszak zak

zak have leadto infeasibility, the algorithmhasto backtrackto level k 1.In [59], SerafiniandUkovich suggesta methodfor storingall relevant informationin a list structure(ratherthanatreestructure).However, theirpseudocodecontainsanerror. There,onpage565,line 7,theassignmentà2â /0 maycausethatpartsof thesearchspacearenot investigatedandinfeasibilityis statedalthougha feasiblesolutionexists. Nevertheless,this errorcanbecorrectedassuggestedbyNachtigallin [43].

Thesizeof thebacktrackingsearchtreemaybeof order∏m n 1k 1 1 ( zak zak

, i.e. exponentialin thenumberof arcs. This explainswhy SerafiniandUkovich suggestto take a minimum spanningtreeconcerningspanlengthsanto orderthearcsin orderof increasingspanlengths.Onecanheuristically


assumethatthenumberof possiblemoduloparametersfor level k is smaller, as“more restrictive” theconstraintsof thestarttreeandthelevels1 k 1 are.

Besidescomputationtime, thereis anotherreasonfor keepingthesearchtree“as smallaspossible”motivatedfrom ourcostoptimizationalgorithmsin chapter4: In caseof aninfeasiblePESPinstance,we will have to analyzethe “reason”for the infeasibility. Formally, we will needto determinea setof arcswhoseinterval constraintscannotbesatisfiedsimultaneously. It will beof advantageto find asmallsetof arcswith this property, andthuswe would like to detectinfeasibility of a PESPinstanceassoonaspossible. Obviously, a setof arcswhoseconstraintscannotbe satisfiedsimultaneouslyis containedin thesetof arcsfrom thespanningtreeandthosechordsthathave beenexaminedformoduloparameterfixing.

Developmentsof Schrijver and Steenbeek

Schrijver andSteenbeekobserve that thePESPalgorithmof SerafiniandUkovich investigatessomepartsof thesearchspaceagainandagain[56]. This is causedby thefactthatafterabacktrackingstepfrom level k down to level kDE k, thefeasiblemoduloparametersfor all levelsk( 1 k( 2 k 1areforgottenby thealgorithmandhave to berecalculatedby shortestpathalgorithms,whichmaybetime consumingfor larger graphs. The ideain [56] is to dynamicallyreorganizethe searchtreeinsucha way thatthis informationcanbekeptaftereachbacktrackingstep.In detail (seefigure3.7),abacktrackingstepfrom level k to level k is performedby exchangingthearcsof level k andk 1 andthencontinuingto investigatethat level which is associatedwith arcak= . After changingthemoduloparameterof ak= , thecurrentpotentialis still feasiblefor all chordsa1 ak= 1 ak= 1 ak 1. Thus,are-computation,asdoneby thePESPalgorithm,of thevaluesassociatedwith thearcsak= 1 ak 1

is notnecessaryany more.

level exchange

...

ak

ak ã 1

...

akä å 1

akä...

...

ak

akäak ã 1

...

akä å 1

...

Figure3.7: Exchangeof arcsduringabacktrackingstep

GeneralizedAlgorithm

Thealgorithmof SerafiniandUkovich canbegeneralizedin someways:

3.7. ARC CHOICEFORTHE GENERALIZEDSERAFINI-UKOVICH ALGORITHM 47 Choiceof start tree : Insteadof choosingaminimumspanningtreeconcerningspanlengths,an arbitrarystart treecanbe used. An examplefor a PESPinstancewherethe SerafiniandUkovich starttreeleadsto a largesearchtreeis givenin figure3.8.

3 4

1 2T 60Ä

10Å 25Æ Ä10Å 25ÆÄ10Å 25Æ æXæXæ Ä

30Å 55ÆÄ5 Å 30Æ Ä

6 Å 31ÆFigure3.8: On thisgraph,theSerafini-Ukovich starttreeshouldnotbeused.

If thestarttreeis chosenastheminimumspanningtreeconsideringspanlengths,noneof theparallelarcsof thisgraphis chosen.As onecanseefrom theparallelarcs,π4 π2 g 30 modTmustbe fulfilled. For someof theseparallelarcs,several moduloparameterscanbe selectedin sucha way that a feasiblepotentialcanbe found which cannotbe extendedto a feasiblepotentialfor thecompletegraph.In contrastto this, if oneof theparallelarcsis chosenfor thestarttree,thereis only onepossibilityfor themoduloparameterof all parallelarcsandevenforall arcsof thegraph.No backtrackingis needed.

In practice,thestarttreesuggestedby SerafiniandUkovich providescomparablygoodresults.The exampleof figure 3.8 seemsto be very artificial. Moreover, our preprocessingmethodseliminateall parallelarcsof thegraph,asonecaneasilyverify. Choiceof an arc to be examinedat level k: At level k, one can choosean arbitrary arc awhosemoduloparameterhasnot beenfixedandlook for a moduloparameterza. As we havealreadymentioned,SerafiniandUkovich suggestanexaminationorderdeterminedin advance.Schrijver andSteenbeekmodify this orderduring the algorithm. We will discussseveral arcchoicerules in section3.7 that lead to a considerableaccelerationof the algorithm for ourpracticalinstances.

A generalizedversionof thealgorithmof SerafiniandUkovich is givenasalgorithm3.4.

3.7 Ar c Choicefor the GeneralizedSerafini-Ukovich Algorithm

Wehavealreadymentionedthattheorderby whichthechordsarechosenfor moduloparameterfixinghasmuchinfluenceon thesearchtreeandthuson thesolutiontime of thealgorithmof SerafiniandUkovich. In this section,we will give many new suggestionsfor this choiceof chords,which oftenleadto bettersolutiontimesor evento thesolutionof instancesthatcouldnotbesolvedby theoriginalalgorithmbecauseof lackof time.


Algorithm 3.4GeneralizedSerafini-Ukovich AlgorithmChooseaspanningtree . SetS: A

Ú .za : 0 for all a SDeterminea feasiblepotential

hfor thegraph

V S with thefixedmoduloparameters.

loopChooseachorda Aç S.Za : za

$ za is a feasiblemoduloparameterfor a if only arcsa S areconsidered,but withtheir fixedmoduloparametersif Za /0 then

Chooseaz Za.za : zS: S ;è aDeterminea feasiblepotential

hfor thegraph

V S with thefixedmoduloparameters.

if S A thenStop.

his a feasiblepotentialfor thePESPinstance.

end ifelse

Performbacktracking:Chooseanothervaluefrom Za= for thearca whosemoduloparameterwasfixedin the iterationbefore. If this is not possible,performfurtherbacktracking.Deletethe correspondingarcsfrom S. If thereareno moremoduloparametersfor the arc whosemoduloparameterwasfixedin thefirst iteration,stop.ThePESPinstanceis infeasible.

end ifend loop

Ordering by the Boundsfor the Modulo Parametersin Advance

In orderto heuristicallykeepthesearchtree“small”, it hasalreadybeenpointedout that the “mostrestrictive” chordsshouldbe selectedascandidatesfor the fixing of moduloparametersfirst. Thisleadsto the following idea: From proposition3.3, we canderive an upperboundon the numberoffeasiblemoduloparametersfor eachchord.Insteadof orderingthechordsby increasingspanlength,onecanorderthemby this upperbound.This leadsto a remarkableaccelerationof thealgorithmingeneral(cf. thecomputationalresultsin chapter5). For example,whenusingthis rule, the instancefrom figure3.8 is solvedwithoutbacktracking,evenif theSerafini-Ukovich starttreeis chosen.

A generaldisadvantageof a fixed orderingof chordsin advanceis that during the algorithm,onlyinformationconcerningtreearcsandchordswith fixedmoduloparametersis used(exceptfor thefactthat the chordshave beenordered).Even in theSchrijver/Steenbeekversion,the orderingfrom theinitializationhasthemaininfluenceon thebehavior of thealgorithm.

Ordering by the Number of Modulo Parametersin eachSearch TreeNode

With the fixing of somemoduloparameters,the boundsfor the non-fixed moduloparametersmaychange.Onecanrecalculatetheseboundsexactly after every fixing of a moduloparameter, i.e. inevery nodeof thesearchtree,andchoosethechordwith the“most restrictive” bounds.

3.7. ARC CHOICEFORTHE GENERALIZEDSERAFINI-UKOVICH ALGORITHM 49

A naive implementationof this methodwould apply thestandardDijkstra algorithmtwice to deter-mine the boundsfor the moduloparameterof a chorda : i

j: In a first step,the algorithmstarts

with root nodei. If the algorithmterminateswith a label λj , the tensionπ j πi canbe raisedat

mostby λj without changingalreadyfixedmoduloparameters(cf. appendixC). Startingwith root

node j, theresultinglabelλj is theamountby that thetensionπ j πi canbelowered.Fromthese

values,thepossiblemoduloparametersza caneasilybefound.

However, assoonasa chorda with a moduloparameterboundwidth za za 0 or an infeasiblechord a is found, the bounddeterminationin this searchtree nodecan be stopped. By choosingchorda for moduloparameterfixing (or backtracking),onecanavoid creatingadditionalbranchesin the searchtree. Moreover, if the bestboundwidth found so far is w, the Dij raise and Dij lower

procedures,suppliedwith a propervalueof δ, canbeusedto decidewhetherthecurrentlyexaminedchordhasa boundwidth of at leastw (theadvantageof the Dij raise and Dij lower proceduresis thatinthiscase,theproceduresprobablyterminatewithout having generatedacompleteshortestpathtree).

By usingthis technique,oftenmany moduloparameterscanbefixedbeforeanotherbranchingoccursin thesearchtree.

Our experimentshave shown that it is often usefulto examinethechordsin a cyclic order: Let thechordswith non-fixedmoduloparametersbeorderedasa1 ar . If we stopthesearchat chordaρbecausethereis only onefeasiblemoduloparameter, thenwe canstartthechordexaminationin thenext iterationof algorithm3.4with chordaρ 1 andcontinuewith chorda1 afterexaminingchordar .Thecompletealgorithmfor choosingachordfor moduloparameterfixing is givenby algorithm3.5.

Several Chords with the sameNumber of FeasibleModulo Parameters

If theminimumnumberof feasiblemoduloparameterswN ( 1 is V 1, it is usefulto examinethose

chordswith boundwidth wN moreclosely. In general,therearemany chordswith boundwidth wN .Assumethat the chordsag, g 1 k have boundwidth wN (with wN V 0). Let z0

g zwWg bethe feasiblemoduloparametersfor chordag. Now, all subtreescorrespondingto zh

g, g 1 k ,h ¦ 0 wN areexamined. If therearechordswith moduloparametersleadingto infeasibilitybeforea branchingoccursin the subtree,then choosea chord with the maximal numberof suchmoduloparameters.Otherwise,let dg0 h bethemaximalnumberof nodesin thesubtreecorrespondingto themoduloparameterzh

g beforea furtherbranchingoccurs(i.e.wN V 0 again).Now let

dg : minw57H 00 B B BC0wW I dg0w andchoosechordagW with dgW max

g 57H 1 0 B B BC0 k I dg Thiscorrespondsto “looking ahead”andthenchoosingthechordlocally leadingto a largestdelayoffurtherbranchingsanthesubtree.dg is calledlook-aheadvalueof chordag. An exampleis shown infigure3.9.

Maintaining a CandidateList for Look-Ahead

In general,therearetoo many chordswith boundwidth wN for examiningthemall in a reasonableamountof time. Instead,a candidatelist for chordinvestigationshouldbeused.


Algorithm 3.5ChoosingaChordin theGeneralizedSerafini-Ukovich AlgorithmLet thechordswith non-fixedmoduloparametersbea1 ar .Let as be the chord that was examinedbeforea chord was chosenin the previous iteration ofalgorithm3.4. If thefirst examinedchordin thepreviousiterationwaschosen,sets : 0.ρ : s ( 1wN : ∞ Jêé w is thecurrentboundwidth, wN thebestfoundboundwidth é7Jwhile ρ sdo

if ρ V r thenρ : 1

end ifLet aρ : i

j. Determinez : max z '& $ π j πi z * T P lρ .

if π j πi z * T 8 uρ thenw : 0

elsew : ë 1

end ifδ : lρ ( wN w:* T π j πi z * T Use Dij raise with parameterδ to determinethe maximal amountλ with λ 8 δ by which thetensionπ j πi canberaisedwithoutchangingfixedmoduloparameters.

w : w ( × λ lρ ( π j πi z * TT

Øif w E wN then

Analogously, increasew for loweringthetension.if w E wN then

ρ N : ρ; wN : wif w E 1 then

Stop.Choosechordaρ. Jêé w ( 1 $Zaρ$ 8 1 é7J

end ifend if

end ifρ : ρ ( 1

endwhileStop.Choosechordaρ W .

A simplestrategy is to stopthe look-aheadprocessaftera limit of k chords,kìE k andchoosethatchordagW with the bestfound valuedgW so far. Often, betterresultscanbe obtainedfor a dynamiclook-aheadlimit: The processis stoppedafter the productof the numberκ of alreadyinvestigatedarcsandthebestfoundvaluedgW exceedsacertainboundD.

Anotherheuristicapproachis thefollowing: For eachchordag that is to beexamined,anadjacencyvaluecg is determined.Thevaluecg is definedas

cg : ∑a § S

a incident with ag

T ( la ua

3.7. ARC CHOICEFORTHE GENERALIZEDSERAFINI-UKOVICH ALGORITHM 51

wí 1chorda1 chorda2

z01 z1

1 z02 z1

2

d1 Ç 0 2 d1 Ç 1 1 d2 Ç 0 0 d2 Ç 1 3î ïÖð ñd1 1

î ïòð ñd2 0

choosechorda1

Figure3.9: Determiningachordfor moduloparameterfixing by “looking ahead”

whereS is the setof thosearcswhosemoduloparametersarealreadyfixed (algorithm3.4). cg islarge if ag is incidentwith many arcsfrom Sandespeciallyif thosearcshave a small span.We canheuristicallyhopethata largevalueof cg correspondsto a “highly restrictive” chordag.

The adjacency value can be computedvery fast, comparedto look-aheadvalues. We can alwayschoosethe arc with highestadjacency valuefor moduloparameterfixing or even combinethe twoapproaches:Startingwith thechordwith highestadjacency value,weonly calculatelook-aheadvaluesfor chordswith relatively highadjacency value(sayα * cg, whereag is thechordwith thehighestlook-aheadvaluesofar, α E 1). This strategy is givenby algorithm3.6.

Algorithm 3.6ChoosingaChordwhenwN V 0Let a1 ak be the setof chordswith non-fixed moduloparameters.Let c1 P cκ for all κ 2 k .q : 0; dN : ë 1; cN : c1; g : 1;loop

if g V k or q * dN P D thenStop.ChoosechordaN .

end ifif $Zag

$ wN ( 1 thenq : q ( 1if cg P α * cN then

if dg V dN or (dg dN and cg V cN ) thendN : dg; cN : cg; aN : ag

end ifend if

end ifg : g ( 1

end loop


We have appliedall of themethodsfrom this section(andcombinations)to a setof PESPinstancesfrom practice.Fromtheresultsgivenin chapter5, onecanseethatthereareinstancesthatcouldonlybesolved(or provento beinfeasible)with thehelpof thesemethods.Ontheotherhand,theadditionalcalculationstake muchtimeevenfor instancesthatcanbesolvedwith theoriginal algorithm.

3.8 Polyhedral Structure of the PESP

In this section,we will examinethe polyhedralstructureof PESP. Recall the definitionsfrom sec-tion 3.2: Ð Xh z n & m $ l 8,p T h Tz 8 u Ñ z q& m $ thereis a

h n suchthatl 8,p T h Tz 8 u Note that

Ñis the projectionof

Ðon the moduloparameters.The convex hull of

Ðis calledun-

boundedtimetablepolyhedron.

We alreadyknow that themoduloparametercanbefixedon thearcsof a fixedspanningtree andtherebyboundsz andz for themoduloparameterscanbeobtained(with za za 0 for treearcs).Asaconsequence,we will alsoexamineboundedversionsof

Ðand

Ñ:Ð z z : Xh z n & m $ l 8p T h Tz 8 u z 8 z 8 z Ñ z z : z '& m $ thereis a

h n suchthatl 8,p T h Tz 8 u z 8 z 8 z conv

Ð z z is calledboundedtimetablepolyhedron.

In this section,we will derivenew resultsfor cuttingplanesof PESPinstancesandderive anew classof cuttingplanesfor suchinstances.

3.8.1 The UnboundedTimetable Polyhedron

Let ó T h (ô Tz P ϕ0 beavalid inequalityfor convÐ

. Without lossof generalityassumethatall spans1 la ua 3 fulfill ua la E T (otherwisethisspancanbeignored).LetXh z Ð . Let k beanodeandlet

µ q& . Then

π i : tπi ( µT if i k

πi otherwiseand za : ijjk jjl za ( µ if a : i

k for anodei

za µ if a : k

i for anodei

za otherwise

leadto a feasiblesolutionXh z Ð . It is easyto seethatz z ( µ

±k. Notetható T h (ô Tz 2ó T h (ô Tz ( µ

ξk * T (ô T ±

k (3.6)

Proposition 3.5 If ó T h (Fô Tz P ϕ0 is a valid inequalityfor convÐ

, then

ξk * T (Fô T ±k 0 for all nodesk

3.8. POLYHEDRAL STRUCTUREOF THE PESP 53

Proof: LetXh z Ð . Assumethat δ : ξkT (ô T ±

k E 0 for a nodek. Setρ : ´ó T h (ô Tz andµ : 1 ( ϕ0 ρ J δ.

By theprocedureabove,Xh z Ð canbegenerated.Now by (3.6)ó T h (ô Tz 2ó T h (ô Tz (öõ 1 ( ϕ0 ρ

δ ÷ * δ Áó T h¥ø ô Tzø

δø

ϕ0 ρ ϕ0ø

δ E ϕ0 ThiscontradictsùXúû°ü z ûeýSþÿ . An analogouscontradictioncanbefoundfor δ 0. Defineaprojection

f :

n m mùXúÒü z ý x : n

∑k 1

ù πk k ý T zandlet X : f ùùXúÒü z ýýxùXú ü z ý+þÿü 0 >ú T 1 .Theorem 3.1 T ú ø Tz ϕ0 is a valid inequalityfor conv ÿ if andonly if ξk 1

T

T k for allnodesk and

Tx T ϕ0 is a valid inequalityfor convX.

Proof: Let T ú ø Tz ϕ0 be a valid inequality for conv ÿ . Thenξk ! 1T

T k follows fromproposition3.5.Theotherconditionis alsofulfilled: T ú ø" Tz ϕ0

n

∑k 1

T k

Tπkø" Tz ϕ0

ϕT # n

∑k 1

kπkø

z T $ T ϕ0 Tx T ϕ0

Conversely, let ξk 1T

T k for all nodesk andlet Tx % T ϕ0 for X. Now consideranelementùXú ü z ýSþÿ . Wehave to show T ú ø& Tz ϕ0.

For each ùXúÒü z ý þ¦ÿ , thereexist uniquelydeterminedintegersµi , i þ 1 ü('('('ü n suchthat 0 πiø

µi T T. Settingπ ûi : πi µi T and z û : z ∑i µi i leadsto a feasiblepoint ùXú û ü x û ý þ¦ÿ andx : f ùXúbûü z ûýSþ X. It follows that

ϕ0 1T

T # n

∑k 1

ú ûk k z û T $) T ú û ø" Tz û T ú ø" Tz T T * T n

∑k 1

µk k T ú ø& Tz üsinceT ξk + T k. Theproof is complete. Regard

asa flow on thearcsof thegraph , . Theamount

T k canbeinterpretedasinflow minusoutflow atnodek, seefigure3.10.Now consideravalid inequality

Tz ϕ0 for conv - , whichcanbeunderstoodasinducedby avalid inequality T ú ø. Tz ϕ0 with / 0. Fromtheorem3.1it followsthat T k 0 for all nodesk. This conditionis known asflow conservationlaw. This givesthenext

theorem.

54 CHAPTER3. FEASIBLESCHEDULEST k 0 0

T k 1 0

T k 2 0

demand143 Tξ1

demand143 Tξ2

demand143 Tξ3

demand143 Tξ4

Figure3.10:Flow conservation: ξ1 5 ξ2 5 ξ3 5 ξ4 0

Theorem 3.2 Tz ϕ0 is a valid inequalityfor conv - , if andonly if

is a flow fulfilling 6 T 0

andϕ0 min 7 Tz z þ8-:9'

3.8.2 Cycle Cutting Planes

An importantclassof cuttingplanesfor PESPinstancesis givenby thecyclecuttingplanesintroducedas(3.3): ;

Tz =< 1T > lT ;@? uT

;BADCFEfor eachelementarycyclewith incidencevector

;.

3.8.3 Chain Cutting Planes

We will now introducea new classof cuttingplanes.Considera systemof m disjoint arcsbetweentwo nodes1 and2, whereonly lower boundsaregiven:

S: 7 ù π1 ü π2 ü z1 ü('('('ü zm ý+þ 2 HG m la π2 π1 za T for all a 1 ü('('('ü m9For eacharca define

ka : 1Tùù la ý modT la ý+þ G and l ûa : ù la ý modT '

Thus,l ûa ka T 5 la. Assumethat0 l û1 l û2 )'('(' l ûm T. For technicalreasons,definel û0 : l ûm T.Set

αi : l ûi l ûi ? 1 for i 1 ü('('('ü m'It is easyto seethattheα-valueshave thefollowing properties:

(1) 0 α j T for eachj 1 ü('('('ü m(2) ∑m

j 1α j T

(3) ∑mj i

A1α j l ûm l ûi


Proposition 3.6 For each ù π1 ü π2 ü z1 ü('('('ü zm ý+þ S, thefollowing inequalityis valid:

π2 π1 l ûm 5 m

∑j 1

α j ù zj k j ý (3.7)

Proof: Let ù π1 ü π2 ü z1 ü('('('ü zm ý/þ Swith tensionx : π2 π1. Definexû : ´ù xý modT. ThenxûI x 5 kTfor anintegerk. Furthermore,thereis auniquelydeterminedindex i þJ 0 ü('('('ü m fulfilling

0 l û1 l û2 '('('K l ûi xû l ûi A 1 %'('('L l ûm T üwherei 0 meansxûM l û1.

From l j x zjT we know that l ûj l j 5 k jT x zjT 5 k jT xû 5 ù k j k zj ý T. This impliesù k j k zj ý@ 0 for all j 1 ü('('('ü i and ù k j k zj ý@ 1 for j i 5 1 ü('('('ü m. Therefore,

π2 π1 x x 5 m

∑j 1

α jzj i

∑j 1

α jzj m

∑j i

A1

α jzj xû kT 5 m

∑j 1

α jzj 5 i

∑j 1

α j ù k k j ý 5 m

∑j i

A1

α j ù k k j 5 1ý xû kT 5 m

∑j 1

α jzj 5 km

∑j 1

α j m

∑j 1

α jk j 5 m

∑j i

A1

α j xû 5 m

∑j 1

α j ù zj k j ý 5 l ûm l ûi l ûm 5 m

∑j 1

α j ù zj k j ýThiscompletestheproof. Now, we will discussthe caseof an arbitrarygraph , and a feasiblepoint ùXúü z ý þÿ . Considertwo nodes1 and2 of , andm pathsfrom node1 to node2 with incidencevectorsp1 ü('('('ü pm. Foreachi þ8 1 ü('('('ü m , define

zi : ù p Ai ý Tz ù p ?i ý Tz and l i : ù p Ai ý T l ù p ?i ý Tu 'Obviously, we have l i π2 π1 Tzi for eachi þN 1 ü('('('ü m . Usingthebox constraints(3.3)on themoduloparametersleadsto zi ù p Ai ý Tz ,ù p ?i ý Tz : zi . From proposition3.6, we obtainthe validinequality

π2 π1 l ûm 5 m

∑j 1

α j ù zj k j ý (3.8)

for ÿ . This inequalityis calledchaincuttingplane.

For graphsizesgiven by practicalinstances,it is importantto identify “small” pathsetsleadingto“effective” chaincuttingplanes.Thisproblemwill beaddressedin thefollowing.

Supposethatwearegivenasystemof mpathsfrom node1 to node2 andthechaincuttingplane(3.8).Thepathsaredenotedby p1 ü('('('ü pm andcorrespondto incidencevectorsp1 ü('('('ü pm. Now, exchangeonepathof thissystem,sayp1, by apathq with incidencevectorq andl ù q ý : ù q A ý T l Fù q ? ý Tu such


that l1 O l ù q ý modT. Togetherwith themodifiedmoduloparameterboundzû1 : öù q A ý Tz ,ù q ? ý Tu,themodifiedintegerkû1 with ù l ù q ýý modT kû1T 5 l ù q ý yieldstheinequality

π2 π1 l ûm 5 α1 ù zû1 kû1 ý 5 m

∑j 2

α j ù zj k j ýP'Hence,thechaincuttingplanegetstighterby thepathexchange,if

α1 ù zû1 kû1 ýB α1 ù z1 k1 ý or equivalently zû1 kû1 z1 k1 'Sincek1 1

T ùù l1 ý modT l1 ý , kû1 1T ùù l ù q ýý modT l ù q ýý and ù l1 ý modT `ù l ù q ýý modT, thepath

exchangeimprovesthecuttingplane,if andonly if

zû1 5 l ù q ýT z1 5 l1

Tü

which meanszû1 T 5 l ù q ý@ z1 T 5 l1. Thebestimprovementof this exchangetypecanbefoundbysolvinga longestpathproblem:For a given valueτi ù l i ý modT, we have to find a longestpathqfrom node1 to node2 with ù l ù q ýý modT τi , wherethearclengthsaregivenby

µ

Aa : za T 5 la and µ

?a : za T ua '

Themodulopathproblem

max µ ù p ý. p incidencevectorfor apathfrom node1 to node2 andùù p A ý T l ù p ? ý Tu ý modT τ (3.9)

can,for fixedperiodT, besolvedin polynomialtime by thefollowing dynamicprogrammingformu-lation:

For eachnodei andeachmodulovalueτ 0 ü('('('ü T 1, define

Fk ù τ ü i ý : max µ ù p ý. p incidencevectorof a pathfrom 1 to i with k arcsandùù p A ý T l ù p ? ý Tu ý modT τ Startingwith

F0 ù τ ü i ý : !Q 0 if τ 0 andi 1

∞ otherwiseüwe obtain

Fk

A1 ù τ ü i ýD max max Fk ù τ û°ü j ý 5 µ

Aa a : j i andla 5 τ û O τ modT ü

max Fk ù τ û ü j ý 5 µ

?a a : i j and ua 5 τ û O τ modT LK'


3.8.4 SimpleLifting Proceduresfor Flow Inequalities

Theflow inequalities Tz ϕ0 maycontaincoefficientsϕi RþJK 1 ü 0 ü 1 . In many of thosecases,the

definedboundson themoduloparametersallow a simplecoefficient reduction(or lifting) procedure.A shortview on coefficient reductionandits effecton thesolutionof MIPs is givenin appendixB.

Without lossof generalityassumethat 0 za za and ϕa 0 (otherwise,usethe transformationzûa : za za for arcswith za R 0 andϕa 0, andthetransformationzûa : za za for arcswith za R 0andϕa 0; this leadsto an inequalityof thedesiredtype,which canbe transformedbackafter thelifting).

Let ϕ1 ü('('('ü ϕk 0 andϕi 0 for i k. Moreover, assumeρ : gcd ϕ1 ü('('('ü ϕk S 1. Otherwise,theinequalitymaybetightenedby

1ρ Tz < ϕ0

ρ

E 'For k 2 andϕ0 0, theinequalityϕ1z1 5 ϕ2z2 ϕ0 canbelifted to< ϕ0

ϕ2

Ez1 5 < ϕ0

ϕ1

Ez2 < ϕ2

0

ϕ1 ϕ2

E 'As an example,the inequality z1 5 2z2 1 can be transformedto z1 5 z2 1 by this method. Afractionalsolutionlike ù 0 ü 1

2 ý is infeasiblefor thetransformedinequality.

Thelifting procedurecanbeappliedsuccessively to eachvariablein thefollowing way: Defineϕ û2 : gcd ϕ2 ü('('('ü ϕk and

zû2 : 1ϕ û2 k

∑a 2

ϕaza 'Thisyieldstheinequalityϕ1z1 5 ϕ û2zû2 ϕ0. Now, onecanapplythelifting procedureto thisequationandcontinueby selectingall othervariablesz2 ü('('('ü zk.

3.8.5 SingleBound Impr ovement

For many combinatorialPESPalgorithms,it is easyto useadditional information on the boundsof the moduloparameters,while it may be difficult to useinformationfrom generalcutting planes.Therefore,we will now focuson thegenerationof moduloparameterboundsby cuttingplanesfor - .

Assumethat T Tz β0 is a cutting planefor - . Consideran arc a with βa 0 anddefine T a : T8 βa ea. Thenβaza β0 ùUT Aa ý Tz 5 ùUT ?a ý Tz '

In caseof < 1βa > β0 ¦ùUT Aa ý Tz 5 ùUT ?a ý Tz

C E za (3.10)

we obtainanimprovedlowerboundfor themoduloparameterof arca.

A singleboundseparation algorithm for a class V of cuttingplanesis a methodto find, for anarca,eitheracuttingplane ùUT Tz β0 ý/þ4V which improvesthecurrentboundsfor themoduloparameterza

accordingto (3.10)or to prove thattheboundsfor za cannotbeimprovedby acuttingplanefrom V .


Define V b astheclassof all singleboundinequalitiesresultingfrom (3.10). Adding all inequalitiesfrom V b to the polytopeconv ù ÿ@W ù z ü z ýý (possibly)yields strengthenedboundsz û z andz û z forthemoduloparameters.Thisgivesa tighterrelaxationpolytope

conv ù ÿ W ù z û ü z û ýýX : V b ù conv ù ÿ W ù z ü z ýýýP'If conv ù ÿ W ù z û°ü z û ýý R /0, we canagainapplythesingleboundcuttingplanesto theimprovedbounds.Proceedingin a recursive manner, we finally obtaineitheranemptypolytopeor a polytopewherenoboundcanbeimproved.Thispolytopeis calledthe V b-kernel. An examplefor thecalculationof sucha V b-kernelis shown in figure3.11.

z1

z2 z1 3 z2 Y 3 1

z1 Z z2 Y 3

Initial bounds: 0 Y z1 Y 3, 0 Y z2 Y 4[ 1H\ z1 Z z2 Y 3 ] z1 3 z2 Y 3 1

Improve boundsfor z2:

z2 Y 3 z1 Z 3 Y 3 z1 Z 3 1 3

z2 _ z1 Z 1 _ z1 Z 1 1 1

Improve boundsfor z1:

z1 Y z2 3 1 Y z1 3 1 1 2

No furtherimprovementis possible.

The[ b-kernelhasbeenfound.

Figure3.11:Calculationof the V b-kernel

Theorem 3.3 Let thetimeperiodT befixed.If thesingleboundseparation problemfor a class V ofcuttingplanesis polynomiallysoluble, thenthe V b-kernelcanbecalculatedin polynomialtime.

Proof: Thenumberof possiblemoduloparametersza za 5 1 for achorda is boundedby n 5 1. Thisfollows from thefactthateachchordgeneratesauniquelydeterminedcycle in thespanningtree.Thiscyclehasatmostn arcs.Sincewehaveua la T for eachspan la ü ua a , thebounddifferenceza zacannotbelargerthann.

During the kernelcalculationwe only canimprove m n bounds(otherwisewe get an emptypoly-tope).If theboundseparationproblemcanbesolvedin polynomialtimewith complexity f ù mü ný , thecalculationof thekernelcanbedonewithin complexity m n f ù mü ný .


Proposition 3.7 Thesingleboundseparation problemfor theclassof cycleinequalitiesfor anarc a :i j canbesolvedby calculatinga shortestpathfromnode j to nodei with thefollowing objective(p is theincidencevectorof thepath):ù p A ý T ù u 5 Tz ýbù p ? ý T ù l 5 Tz ýP'Proof: Considera cycle inequality

;Tz γ0. Let a : i j beanarcof thecorrespondingcycle, i.e.

γa R 0. Then

; : p 5 ea, wherep is theincidencevectorof apathfrom node j to nodei. From(3.2),we have

0 ù ; A ý T ù u 5 Tz ýù ; ? ý T ù l 5 Tz ý Tza 5 ua 5 ù p A ý T ù u 5 Tz ýù p ? ý T ù l 5 Tz ý Tza 5 ua 5 ù p A ý T ù u 5 Tz ýù p ? ý T ù l 5 Tz ýThisyieldstheinequality

za < 1T > ua ¦ù p A ý T ù u 5 Tz ý 5 ù p ? ý T ù l 5 Tz ý CcE 'The right handsideof this inequality only dependson the lengthof the pathbelongingto p withrespectto thearclengthsfrom theproposition.In orderto geta tight boundfor za, theright handsidehasto bemaximized,which correspondsto finding a shortestpathaccordingto thearcweightsfromtheproposition. 3.8.6 Flow Inequalities and SingleBound Impr ovement

We will now examinesingleboundimprovementby cuttingplanes Tz ϕ0 where

is a flow with6 T 0 (cf. section3.8.1). Suchan inequality is calledflow inequality. Considerthe exampleof

figure3.12.

T 1 10

1 2

a0

a1

a2

d1 ] 3ed3 ] 10ed8 ] 13e

1 Y π2 3 π1 3 10z0 Y 3

3 Y π2 3 π1 3 10z1 Y 10

8 Y π2 3 π1 3 10z2 Y 13

Figure3.12:Exampleinstance

Thecyclecuttingplanesfor theexampleleadto 9 10ù z1 z0 ýf 0 12 10ù z2 z0 ýf 5 2 10ù z1 z2 ýf 10

or equivalently

z1 z0 0 ùhg7ýz2 z0 1 ùhgLg7ý

0 z1 z2 1 '


Theresultinglowerboundinequalitiesfor thevariablez1 are:

z1 z0 and z1 z2

Combining ùhg7ý and ùhgLg7ý leadsto z1 z2 1, whichgivesthebetterbound

z1 z2 5 1 üwhichcannotbegeneratedby thesingleboundseparationalgorithmfor cycle inequalities.Thisboundcanonly befoundby combiningcycles.This ideawill begeneralizedin thefollowing.

Considera set I of cyclesandthecorrespondingincidencevectors

;i, i þ I . Let the resultingcycle

inequalitiesbedenotedby αi ; Ti z αi . Let a 1 beafixedarc,and

Tz ϕ0 beaflow inequalitywith ϕ1 0.

Define

ϕ

Aa : Q ϕa if ϕa 0

0 otherwiseand ϕ

?a : Q ϕa if ϕa 0

0 otherwise'Theinducedsingleboundflow inequalityfor variablez1 is givenby

z1 ji 1ϕ1

# ϕ0 ∑a k 1

ϕ

Aa za 5 ∑

a k 1

ϕ

?a za $ml '

For simplicity we assumeboundsza : 0 za za for all arcs(a correspondingtransformationwasgivenin section3.8.4).Now, assumethat theflow

is generatedby theabove introducedsystemof

elementarycycles,whereoneachcyclewith incidencevector

;i , theamountεi is circulating.Then ∑

i n I εi

;i ü

andby combiningthecycle inequalitieswe obtain Tz ϕ0 : ùpo A ý T q ùpo ? ý T q 'Without lossof generalityassumethatϕ1 1. Theresultingsingleboundinequalityis thengivenby

z1 ¡ùpo A ý T q ùpo ? ý T q ∑a k 1

ϕ

Aa za 5 ∑

a k 1

ϕ

?a za '

3.9 Branch-and-Cut Method

In this sectionwe describea PESPalgorithmwhich usestheideasof theSerafini-Ukovich algorithmandcombinesit with thepolyhedralresultsof section3.8. Thebasicideais to usea branch-and-cut(seeappendixB) methodappliedto thetimetablepolyhedronconv ù ÿ W ý .Themethodstartswith z andz fromproposition3.3.Then,thefollowing principleisapplied:Consider

a relaxationset ˜ÿ W ù z ü z ýr¡ÿ W ù z ü z ý for which thedecisionproblem ˜ÿ W ù z ü z ý ? /0 shouldbe easy. Incaseof ˜ÿ W ù z ü z ý R /0, weobtainanelementùXúÒü z ýSþ ˜ÿ W ù z ü z ý . If z is integral,we have foundasolution

3.9. BRANCH-AND-CUT METHOD 61

for thePESPinstance.Otherwise,we pick a fractionalmoduloparameterza Rþ G andgeneratetwoproblemswith disjunctive solutionspacesby demandingza ts za u or za wv za x .The subproblemsonly differ from the original problemby new bounds. A binary searchtree likein figure 3.13 is obtained. At eachnodeof the searchtree,singleboundimprovementcutscanbeapplied.

˜yKzX z ] z |˜y z z1 ] z |

˜yKz z2 ] z | ˜yKzD z1 ] z1 |˜y z z ] z2 |

˜yKzX z3 ] z2 | ˜yKzD z ] z3 |Figure3.13:Binarysearchtreefor thebranch-and-cutmethod

The performanceof thealgorithm(i.e. thesize/ shapeof thesearchtree)dependson many points,includingthese:~ In general,thereis morethanonefractionalmoduloparameter. Wehave to decideon whichof

thoseto branch.~ If ˜ÿ W ù z ü z ý R /0, the determinationof a point ùXú ü z ý þ ˜ÿ W ù z ü z ý canbe donein differentways.Anotherideais to addheuristicsfor minimizingthenumberof fractionalmoduloparameterstothisdeterminationalgorithm.

For ourbranch-and-cutmethod,wewill usetheconvex hull of thefollowing enlargedfeasibleset:

˜ÿ@W ù z ü z ý : DùXúü z ýSþ n m l 6 T ú Tz u ü z z z Proposition 3.8 For 0 ua la T andintegral boundsza za considerthedecisionproblem

conv ù ˜ÿ@W ù z ü z ýý ? /0 ' (3.11)

If z R z, thesetis empty. Otherwise, (3.11)is equivalentto thefeasibledifferential problem

Π ù z ü z ý : ú) l 5 Tz 6 T ú u 5 Tz ? /0 'Proof: Wewill show that Q ú z ùXú ü z ýfþ conv ù ˜ÿ W ù z ü z ýý Π ù zü zýP'Theimplication ùXúü z ýSþ conv ù ˜ÿ W ù z ü z ýýD l 5 Tz 6 T ú u 5 Tz is obvious.


Conversely, let ú>þ n with l 5 Tz 6 T ú u 5 Tz. Defineq : 1Tù u 6 T ú³ý and T : 1

Tù l 86 T ú³ýP'

Now q z, T z and q T . Togetherwith z z this shows

max q ü z min Tü z K'For eachz with max q ü z z min Tü z , we obtain z z z and q z %T , or equivalentlyl 6 T ú4 T z u. Hence,ùXú ü z ý+þ conv ù ˜ÿ W ù z ü z ýý . Fromthisproof it follows that ùXú ü z ýSþ conv ù ˜ÿ W ù z ü z ýý if andonly if ú,þ Π ù z ü z ý and

zla ùXú ý : 1

Tù ua 5 Tza ù π j πi ýý za 1

Tù la 5 Tza ù π j πi ýýX : zu

a ùXú³ýP'This meansthata potential úþ Π ù z ü z ý is feasiblewith arca : i j, if theinterval ` zl

a ü zua a containsat

leastoneintegervalueza þ G . Otherwiseú is infeasiblefor thisarc.Wewill usethis factto applytheheuristicmethodof algorithm3.7to minimizethenumberof fractionalmoduloparameters.

Thecompletebranch-and-cutmethodis describedasalgorithm3.8. As we have alreadymentioned,purelinear(mixedinteger)programmingapproachesseemto beinadequatefor thesolutionof PESPinstances.For this reasonwe usethe feasibledifferentialproblemrelaxation. In orderto maintainthisproblemstructureateverynodeof thesearchtree,weuseonly cuttingplanescompatiblewith thestructure.Thesingleboundcutsfrom section3.8.5have thisproperty.

For apracticalimplementationof thebranch-and-cutmethod,thedatastructuresusedfor representingthetreehave to bechosencarefully. Thereis noobviouswayfor asufficient list structurelike in [59].

Someconceptsfrom section3.7canbeadaptedto thebranch-and-cutmethod.As anexample,a look-aheadvaluefor eacharcwith afractionalmoduloparameteris givenby theamountof boundreductionduring the subsequentV b-kernelcalculation. Sincethis calculationis a time consumingprocess,itis importantto reducethe numberof arcswith fractional modulo parametersfor this approachinparticular.

3.9. BRANCH-AND-CUT METHOD 63

Algorithm 3.7Minimizing FractionalValuesLet ú bea (notnecessarilyfeasible)potential.F : a þ A ú is feasiblewith a R : A Ffor eacha þ F do

Determinetheuniqueintegerza with

la 5 Tza la 5 Tza π j πi ua 5 Tza ua 5 Tza 'end forfor eacha þ A do

da : !Q la 5 Tza if a þ F

la 5 Tza otherwiseda : Q ua 5 Tza if a þ F

ua 5 Tza otherwise

end forwhile R R /0 do

Choosea þ R.R : R a Let δl ü δu betheamountfor which thetensionhasto beloweredor raisedto make thepotentialfeasible.if Dij lower ù δl ü d ü d üZúü ; ý or Dij raise ù δr ü d ü d üZúü ; ý succeedsthen

Updatethepotentialaccordingto thesolution.Updatetheboundsfor a by da : la 5 Tza andda : ua 5 Tza.

end ifendwhileStop.


Algorithm 3.8Branch-and-CutMethodwith FDPRelaxation: ÿ W ù z ü z ý

loopif /0 thenStop.Theinstanceis infeasible.

end ifChooseÿ þ . Let themoduloparameterboundsof ÿ bedenotedby z ü z.

: ÿJAdd singleboundimproving cutsto ÿ . g calculatethe V b-kernel gif conv ù ˜ÿ'ý /0 then

continue g FDPrelaxationinfeasiblegend ifChooseùXú û ü z û ý+þ conv ù ˜ÿqý .Try to reducethenumberof fractionalvaluesof z û . Let theupdatedsolutionbegivenby ùXúü z ý .if z þ G m then

Stop. ùXúÒü z ý is a feasiblesolutionfor theinstance.end ifChooseanarca þ A with za Rþ G to branchon.Generatetwo new problemsÿ 1 and ÿ 2 from ÿ with adjustedbounds

z1a : s za u and z2

a : v za x ': ÿ 1 üÿ 2

end loop

Chapter 4

CostOptimal Schedules

In this chapter, solutionmethodsfor theMIP formulationof theminimumcostschedulingproblemfrom section2.8, MIP-MCSP, arepresented.Sincea direct MIP solutionwith a commercialsolverturnsout to beimpossiblefor practicalprobleminstances,adecompositionapproachis developed.

The decompositionis integratedinto a relaxationiterationalgorithmand into a branch-and-boundalgorithm. Fastmethodsfor solving thesubproblemsarisingfrom thedecompositionaregiven. Attheendof this chapter, thealgorithmsareextendedin sucha way thata certainnonlinearversionofthemodelfor minimumcostschedulingcanbehandled.

4.1 Mixed Integer Programming

A straightforwardattemptto solve theMIPs for minimumcosttrainschedulingof section2.8(cf. fig-ure2.10)is thedirectuseof commercialMIP solversontheprobleminstances.For practicalinstances(suchastheInterCitynetwork of Germany), thesolutionof theMIPsmaytake severaldays.In manycases,it is not even possibleto find optimal solutionswith our computerhardware and software(cf. section5.2).

As we have alreadymentioned,our intention is to develop a model that can be usedfor strategicplanning. In order to analyzeor comparedifferentscenarios,the solutiontimesshouldnot exceeda few minutesfor practicalinstances.In the following sections,we will develop strategiesthat willenableusto solve instancesor at leastto find solutionsof practicalinterestwithin suchatimehorizon.

4.2 ProblemDecomposition

Two widely usedclassesof solutionmethodsfor solvingMIPs (seeappendixB) are:~ branch-and-boundmethods~ iterative relaxationmethods(e.g.cuttingplanemethods)

65

66 CHAPTER4. COSTOPTIMAL SCHEDULES

Bothclassesrequiresolvingarelaxationof therespective MIP. Thetypically appliedrelaxationis theLP relaxation(cf. appendixB). This is in particularthecasefor thecommercialMIP solver CPLEX,whichwe have used.

For theminimumcostschedulingproblem,wewill developanothertypeof relaxationleadingto muchbetterresultsconcerningsolutiontime.

Considerthecoefficient matrix of theobjective functionandtheconstraintsfor theMIP-MCSP, fig-ure4.1.Thegraycolor indicatesthattherearenonzerocoefficientsfor therespective variableandthecorrespondingclassof constraints.

Variables

w x z a d

objective function

travelercapacity

numberof coaches

exactlyonetrain type

travel time

otherperiodicinterval constraints

moduloparameterconstraints(JPESP)

MCTP (for fixedtrain types)FSP

Figure4.1: Structureof objective functionandconstraintsof theMCSP

With the exceptionof the x-variablesin the travel time constraints,the matrix canbe divided intoa two-block diagonalmatrix. Oneblock representstheproblemof minimizing thecostconsideringonly theconstraintsfor capacity, numberof coachesandselectionof onetrain type.Thisproblemwascalledminimumcosttypeproblem(MCTP) in section2.9. Theintegerprogrammingformulationofthis problem,which is given by this block, will be calledIP-MCTP in the following. The problemof satisfyingthe constraintsof the otherblock for fixedtrain typeswill be called feasiblescheduleproblem(FSP). TheFSPis aJPESP(cf. section2.5).

If thetrain typesarefixed,theMCTP andtheFSPcanbesolvedseparately. Thesolutionof theFSPdoesnot influencethe objective value,becausethe correspondingvariablesarenot in the objectivefunction. If the FSPis feasible,the optimal solutionof the MCTP andthe feasiblesolutionfor theFSPcanbecombinedto anoptimalsolutionfor thecompleteproblem.

Wewill usetheMCTPasarelaxationfor theMCSP. Basedonthisrelaxation,weproposearelaxationiterationalgorithmin section4.3anda branch-and-boundalgorithmin section4.4. In bothcases,wewill have to solve many instancesof theMCTP andtheFSP. Therefore,fastsolutiontechniquesfortheseproblemsaredevelopedin section4.5(for theMCTP) andsection4.6(for theFSP).

4.3 Relaxation Iteration Method

For ashortintroductiononmixedintegerprogramsandsolutionmethodswereferto appendixB. Wewill now develop a relaxationiterationmethodfor solving MCSPinstances.From appendixB weknow severalcrucialpointsfor thedesignof suchanalgorithm:

4.4. BRANCH-AND-BOUND METHOD 67~ Choiceof therelaxation: As initial relaxationfor theiterationmethodfor aMIP-MCSPinstancewe usethecorrespondingMCTP instance.This can,for example,be interpretedasan integerprogramof considerablysmallersize(IP-MCTP).By usingtheideasof section4.5,we will beableto solve theMCTP instancefor practicalnetworkswith acommercialMIP solver in a fewseconds.

The main disadvantageof the IP-MCTP relaxationis the fact that after reducingthe solutionspacein an iteration,we will have to restarttheMIP solutionprocesscompletely(thereis nosuchobviousidealike thedualsimplex algorithmfor LP relaxationiteration).Therefore,it willbeimportantto keepthenumberof iterationssmall.~ Feasibility check: Let the optimal solution of the relaxationbe given by the vectorx of thetrain typevariablesandthevectorw of variablesfor numbersof coaches.We now needto findout whetherthe FSPconstraintscanbe satisfiedwith x. This problemis a JPESP. A simpleapproachto solve JPESPinstancesis given by mixed integer programming(with an arbitraryobjective function— only feasibility is important).In section4.6,wewill developanalgorithmfor JPESPinstancesbasedonthePESPalgorithmbySerafiniandUkovich (cf. chapter3),whichwill have certainadvantages.~ Reductionof thesolutionspace: If theFSPinstanceof someiterationis infeasible,at leastoneof thetrain typeshasto bechangedin orderto geta feasiblesolutionof theMCSPinstance.Letτr bethetrain typefor line r þ in theoptimalsolutionof therelaxation.Then,thefollowinglinearconstraintfor thetrain typevariablesis introduced:

∑r n xr τr ( 1 (4.1)

This inequalitycanbeaddedto theIP-MCTPinstancein thesubsequentiteration.

Sincetherearemany possiblecombinationsof train typesin practicalinstances,we mayhave to adda lot of inequalitiesto theoriginal MIP-MCTP instance,eventuallyslowing down theMIP solutionprocess.In orderto avoid this it wouldbehelpful if wecouldexcludeseveralinfeasiblecombinationsof train typeswith thesameinequality.

Onepromisingideais to detecta “comparablysmall” setof lines ˆ which alreadycausestheinfeasibility of the presentFSPinstance. An approachto find sucha set is given in section4.6.Using ˆ , theconstraint(4.1)canbereplacedby

∑r n ˆ xr τr ˆ 1 ' (4.2)

It mayevenbeallowedto excludeseveraltrain typesfor oneline at thesametime if thesetypeshavethesamespeed.

Theideasof this sectionarecombinedin algorithm4.1.

4.4 Branch-and-BoundMethod

A shortoverview on branch-and-boundmethodsandin particularfor suchmethodsfor thesolutionof MIPs is given in appendixB. We will now focus on the importantpoints for the designof a


Algorithm 4.1RelaxationIterationAlgorithm for theMCSPloop

if theIP-MCTPis infeasiblethenStop.TheMCSPis infeasible.

end ifLet thevectorof optimaltypesfor theIP-MCTPbegivenby x andthevectorof optimalnumbersof coachesgivenby w.if theFSPfor x is feasiblewith solutionvectorsa ü d ü z then

Stop.An optimalsolutionis givenby x ü w ü a ü d ü z.end ifLet ˆ beasetof linesleadingto theinfeasibilityof FSPfor x. Add inequality(4.2)for ˆ to theIP-MCTP.

end loop

branch-and-boundalgorithmfor thesolutionof theMCSP.~ Choiceof relaxation: Like in section4.3, the IP-MCTPrelaxationis used.Theremarksgivenin thatsectionalsoapplyto thebranch-and-boundmethod.~ Feasibilitycheck: Again this is donelike in section4.3.~ Choiceof division/ partition: If theFSPinstanceis infeasible,we needto changeat leastoneof the train types. Let τr be the train type for line r þ¡ , andlet ρ : w . An obvious wayto divide theset ¢ 1 '('(' ¢ ρ of possible(but not necessarilyfeasible)combinationsfor traintypesis givenby¢ 1

'('(' ¢ ρ ùh¢ 1 τ1 7ý ¢ 2 '('(' ¢ ρ

'('(' ¢ 1 '('(' ¢ ρ

?1 ùh¢ ρ τρ 7ýP' (4.3)

Accordingto this scheme, new problemshave to begeneratedin this case.In section4.3we have suggestedreplacing by a “small” set ˆ£ which is alreadycausingtheconflict.This methodcanalsobeappliedhere.

By usingthesemethods,a first versionof a branch-and-boundalgorithmfor theMCSPis obtained,seealgorithm4.2.Again,weassumethat j r1 ü('('('ü rρ . Of course,if i 1 thenotation ¢ 1 '('(' ¢ i

?1 ¢ i τi ¢ i

A1 '('(' ¢ ρ means¢ 1 τ1 ¢ 2

'('(' ¢ ρ andis understoodsimilarly for theotherspecialcases.

We will now presentseveral methodswhich acceleratedalgorithm4.2 for our practicalproblemin-stancesconsiderably:~ LP relaxationfor theIP-MCTP: Insteadof directly solvingtheIP-MCTPfor ¢ û þ , we only

solvetheLP relaxationof theIP-MCTPin afirst step.If it is infeasibleor hasanobjectivevalue c¤ , i.e.worsethanthevalueof thebestknown solution,thenext branch-and-boundnodecanbeexaminedimmediately(without consideringtheIP).

4.4. BRANCH-AND-BOUND METHOD 69

Algorithm 4.2SimpleBranch-and-BoundAlgorithm for theMCSPc¤ : ∞;

: LL¢ 1

'('(' ¢ ρ L ; l W1 ¥¦ ¦ ¦§¥ W ρ : + ∞

loopif /0 thenStop. If c¤ ∞, theproblemis infeasible.Otherwise,anoptimalsolutionis givenby x ¤ , w ¤ ,a ¤ , d ¤ , z ¤ .

end ifChoose¢ û þ .

: L¢Ùû¨if theMCTPfor ¢û is infeasiblethen

continueend ifLet anoptimalsolutionof theIP-MCTPbedefinedby thevectorsx andw with optimalvaluec.if c c¤ then

continueend ifif theFSPfor x is feasiblewith solutionvectorsa, d, z then

c¤ : c; x ¤ : x; w ¤ : w; a ¤ : a; d ¤ : d; z ¤ : z: ˆ¢ l ˆW c¤

continueend ifLet ˆ beasetof linesleadingto theinfeasibilityof FSPfor x andlet τi bethetrain typefor liner i þ in theMCTPsolutiondefinedby x.for i 1 to ρ do

if r i þ ˆ thenLet ¢ ¤ denote¢ 1

'('(' ¢ i

?1 ¢ i τi ¢ i A 1

'('(' ¢ ρ.l W© : c

: L¢ ¤ end if

end forend loop~ Bounddominance: Let ¢û : ¢ û1 '('(' ¢ûρ and ¢û û : +¢û û1

'('(' ¢û ûρ betwo possible(but notnecessarilyfeasible)combinationsof train types.If¢ û r¢ û û and l W«ª l W«ª ª üwe cansetl W¬ª ª : l W«ª , sincetheproblemfor ¢ û is a relaxationof theproblemfor ¢ û û .Thereare two situationsin the algorithmwherethis dominancecanbe exploited. After thesolutionof the IP-MCTP (or even after the solutionof the correspondingLP relaxation),theoptimalvaluec is anew lower boundfor theproblemdefinedby ¢û . Assumethatthereis aset¢ û û)þ with ¢û û¢ û . Thenthis lowerboundis alsovalid for ¢ û û . If alreadyc c¤ holds, ¢û ûcanberemovedfrom

withouteverbeingexaminedfurther. Thesameappliesif theIP-MCTP

for ¢û or thecorrespondingLP relaxationis infeasible.

Thesecondsituationis thedivision of ¢û . Sometimes,betterboundsfor thenew problemscanbeobtainedfrom elementsalreadyin

.


Theseimprovementsleadto algorithm4.3.

Algorithm 4.3 ImprovedBranch-and-BoundAlgorithm for theMCSPc¤ : ∞;

: LL¢ 1

'('(' ¢ ρ L ; l W 1 ¥¦ ¦ ¦§¥ W ρ : + ∞loop

if /0 thenStop. If c¤® ∞, theproblemis infeasible.Otherwise,anoptimalsolutionis givenby x ¤ , w ¤ ,a ¤ , d ¤ , z ¤ .

end ifChoose¢ û þ .

: L¢Ùû¨if LP relaxationof IP-MCTPfor ¢ û is infeasiblethen

: L¢ û ûDþ ¯¢û û°¢ û¨ ;continue

end ifif LP relaxationof IP-MCTPhasanoptimalvalue c¤ then

: L¢ û û þ ¯¢ û û °¢ û ;continue

end ifif theIP-MCTPfor ¢û is infeasiblethen

: L¢ û ûDþ ¯¢û û°¢ û¨ ;continue

end ifLet anoptimalsolutionof theIP-MCTPbedefinedby thevectorsx andw with optimalvaluec.if c c¤ then

: L¢ û û þ ¯¢ û û °¢ û ;continue

end ifif FSPfor x is feasiblewith solutionvectorsa, d, z then

c¤ : c; x ¤ : x; w ¤ : w; a ¤ : a; d ¤ : d; z ¤ : z: ˆ¢ l ˆW c¤

continueend ifLet ˆ beasetof linesleadingto theinfeasibilityof FSPfor x andlet τi bethetrain typefor liner i þ in theMCTPsolutiondefinedby x.for i 1 to ρ do

if r i þ ˆ thenLet ¢ ¤ denote¢ 1

'('(' ¢ i

?1 ¢ i τi ¢ i A 1

'('(' ¢ ρ.l W© : max c ü max l W«ª ª I¢û û)þ j± ¢ û û²¢ ¤ L

: L¢ ¤ end if

end forend loop

As describedin appendixB, in a branch-and-boundprocessone can usedifferent nodeselectionrules. For our implementation,we have usedthe following one: Alwaysthe nodewith the lowest

4.5. SOLVING MCTPINSTANCES 71

lower boundfor theobjective valueis chosen.If thereareseveralnodeswith thesamelowestbound,theonewith thelargestset ¢ ¤ is used.By thisselection,wehopethatweobtain“good” lowerboundsandourboundingstrategiescanbeappliedoften.

4.5 Solving MCTP instances

In this section,solutionmethodsfor IP-MCTPinstancesarepresented.A directsolutionof the “IP-MCTPpart” of theMIP-MCSPmodelfrom section2.8is possiblefor someinstanceswith acommer-cial MIP solver, but it still takestoo muchtime (recall that thedecompositionalgorithmsmayneedto solve many suchinstances).In this section,we introducea binaryvariablemodelfor theMCTP,preprocessingtechniquesandcuttingplanesleadingto a remarkablespeedupof thesolutionprocess.

Binary Variable Model

For thecostoptimal line planningmodel,two integer linearformulations(COSTILPandCOSTBLP,cf. section2.7) have beenexamined. Our IP-MCTP model was developedfrom COSTILP. In ananalogouswayto COSTBLP, wecanformulateabinarymodelfor theMCTP. Therefore,weintroducethefollowing variables:

wr τ c line r usestrain typeτ with c coaches

ThebinaryvariablemodelBP-MCTPfor theMCTP is given in figure4.2. As onecanseefrom ta-ble 4.1, theconstraintmatrix from theBP-MCTPhasfewer rows, but morecolumnsthanthematrixfor IP-MCTP. For our practicalinstances,theBP-MCTPprovidesbetterLP relaxationsand shortersolutiontimes.This experienceis differentfrom costoptimal line planning,wherethebinaryformu-lation gave betterLP relaxationsandthegeneralinteger formulationgave bettersolutiontimes. WemaythereforereplaceIP-MCTPby BP-MCTPin ourdecompositionalgorithms.

Binaryvariablemodelfor MCTP (BP-MCTP):

min ∑r n ∑

τ n W r Wτ

∑c Wτ

> v tr τ T x ù Cfixτ 5 c CfixC

τ ý 5 dr ù Ckmτ 5 c CkmC

τ ý C wr τ c∑

r n r ³ e ∑τ n W r Wτ

∑c Wτ ´ τ c wr τ c Ne for eache þ E

∑τ n W r Wτ

∑c Wτ

wr τ c 1 for eachr þwr τ c þ 0 ü 1 for eachr þ , τ þJ¢ r , c þJ Wτ ü('('('ü Wτ

Figure4.2: Binary variablemodelfor theMCTP


Model Numberof rows Numberof columns Numberof nonzeroentries

IP-MCTP 5 ¶µ· 5 2 ∑r n ¶¢ r 2 ∑

r nI ¶¢ r ∑en E ∑

r ¸p¹r º e ¶¢ r 5 5 ∑

r n ¶¢ r BP-MCTP 5 ¶µ· ∑

r n ∑τ n W r 1 5 Wτ Wτ ∑

en E ∑r ¸»¹r º e ∑

τ n W r 1 5 Wτ Wτ5 ∑r n ∑

τ n W r 1 5 Wτ Wτ

Table4.1: Comparisonof themodelsIP-MCTPandBP-MCTP

The following propositionshows that theoptimal solutionof the LP-relaxationfor a BP-MCTPin-stancecannotbe worsethanthe optimal solutionof the relaxationfor the correspondingIP-MCTPinstance.

Proposition 4.1 Let zI be the optimal solutionvalueof the LP-relaxationof an IP-MCTP instanceandlet zB betheoptimalsolutionvalueof theLP-relaxationof thecorrespondingBP-MCTPinstance.Then

zI zB 'Proof: We show that for eachfeasiblesolutionwB of theBP-MCTPinstance,thereexistsa feasiblesolution ù xI ü wI ý of theIP-MCTPinstancewith thesameobjectivevalue.Let wB beafeasiblesolutionof theBP-MCTPinstance.Define

xIr τ : Wτ

∑c Wτ

wBr τ c and wI

r τ : Wτ

∑c Wτ

c wBr τ c '

Onecaneasilyverify that the obtainedsolution ù xI ü wI ý satisfiesall constraintsof IP-MCTP. As anexample,we considertheconstraintsfor connectingx- andw-variables:

Wτ xr τ Wτ Wτ

∑c Wτ

wr τ c Wτ

∑c Wτ

c wr τ c wr τ Wτ Wτ

∑c Wτ

wr τ c Wτ xr τTheobjective functionvaluesof wB and ù xI ü wI ý areidentical:

zI ∑r nI ∑

τ n W r v tr τ T x ù xr τ Cfixτ 5 wr τ CfixC

τ ý 5 dr ù xr τ Ckmτ 5 wr τ CkmC

τ ý∑

r n ∑τ n W r # v tr τ T x #m# Wτ

∑c Wτ

wr τ c $ Cfixτ 5 # Wτ

∑c Wτ

c wr τ c $ CfixCτ $

5 dr ## Wτ

∑c Wτ

wr τ c $ Ckmτ 5 # Wτ

∑c Wτ

c wr τ c $ CkmCτ $m$+ zB

Thiscompletestheproof.


Preprocessing

By preprocessingtheprobleminstances,wetry to reducetheirsizesor to improvetheirLP relaxationsin orderto acceleratethesolutionby ouralgorithms(cf. sectionB.3). For IP-MCTPinstances(or BP-MCTP instancesrespectively), we will develop suchpreprocessingtechniquesnow. The ideasaremainlybasedon combinatorialpropertiesof theproblem.

Often,onecanfind out in advancethaton somenetwork edgee, every feasiblesolutionof theMCTPleadsto a travelercapacityNe 5 ν, ν 0. In this case,Ne canbeincreasedby ν without changingtheoptimalsolution,but possiblytherebyobtaininga betterLP relaxationvalue.Somesituationswherethetravelercapacitycanbeincreasedarediscussednow:~ Greatestcommondivisor increase: Let e þ E andlet Γe bethegreatestcommondivisor of all

feasiblecoachcapacitiesfor trainsservingtheedgee, i.e.

Γe gcd

´ τ r ¸p¹r º e τ þJ¢ r ¼¾½¿ 'Sinceevery train runningover e hasa capacitywhich is a multiple of Γe, thetravelercapacityon every network edgeecanbemodifiedin thefollowing way:

N ù eý : < N ù eýΓe

E Γe

This choicedoesnotaffect thefeasibilityof asolutionof MCTP.~ Line capacityboundincrease: Let ´ r bea lower boundfor thenumberof travelerscarriedbyline r in any feasiblesolutionof MCTP. Thenthefollowing increaseis valid:

N ù eý : max

N ù eýü ∑r ¸»¹r º e ´ r ¼¾½¿

A simpleboundis ´ r min Wτ ´ τ τ þ.¢ r . If thereis a setof network edgesE ûM E onlyservedby r, possiblyabetterboundis givenby

´ r min À c ´ τ τ þ8¢ r ü c þ8 Wτ ü('('('ü Wτ ü c ´ τ maxen E ª N ù eýÂÁ·'

Cutting Planes

Wewill now introducecuttingplanesfor BP-MCTPinstances.

Proposition 4.2 Let e þ E be a networkedge and let r1 ü('('('ü rn þ4 be the lines containinge. Letn 2. Moreover, let d1 ü('('('ü dn

?1 þJ 1 ü('('('ü Ne 1 with d : ∑n

?1

i 1 di Ne. ThenÃÄÅn

?1

∑i 1

∑τ n W ri ∑

c ¸hÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì di

wr i τ c ÍÎÏ 5 ∑τ n W rn ∑

c ¸ÐÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì Ne Ñ d wrn τ c 1 (4.4)

is a valid inequalityfor theBP-MCTP.


Proof: We assumethe contrary. Becauseof the integrality, all variablesappearingin (4.4) have totake avalueof 0. It follows that

∑r ¸»¹r º e ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c n

∑i 1

∑τ n W ri Wτ

∑c Wτ

c ´ τ wr i τ c n

?1

∑i 1

di 5 Ne d Ne üwhich is acontradictionto thetravelercapacityinequalityof theBP-MCTP. For our practicalinstances,many of thesecutsareviolatedby therespective LP relaxation,althoughthe edgesaremostly served only by a few lines. Even if n 2, thereare too many suchviolatedinequalitiesto addthemall. As the following propositionshows, for n 2 it is only necessarytoconsider(4.4) for a few valuesof d1.

Proposition 4.3 Let e þ E be a network edge which is only servedby the lines r1 and r2. Let´ 11 ü('('('ü ´ k1

1 be all possibletrain capacitiesfor trains of line r1. Let ´ 11 ´ 2

1 t'('('¬ ´ k11 , and let

therespectivevaluesfor r2 bedefinedanalogously.

Furthermore, let a solution for the LP relaxationof the BP-MCTPbe given by the vector w. Letd1 þJ 1 ü('('('ü Ne 1 such that condition(4.4) is violatedfor w.

Thenthere is alsoa d1 þ8 ´ 11 5 1 ü('('('ü ´ k1

1 5 1 such that theconditionis violatedfor w.

Proof: Notethatd1 ´ 11 holds(otherwise(4.4)cannotbeviolated).

We considertwo cases:Eitherd1 ´ k11 5 1 or thereis in index i þÒ 1 ü('('('ü k1 1 suchthat ´ i

1 5 1 d1 ´ i

A1

1 . In thefirst case,choosed1 : ´ k11 5 1. We thenobtain

∑τ n W r1 ∑

c ¸hÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì d1

wr1 τ c 5 ∑τ n W r2 ∑

c ¸hÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì Ne Ñ d1

wr2 τ c ∑τ n W r2 ∑



c ¸ÐÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì Ne Ñ d1

wr2 τ c 1 'In otherwords,(4.4) is violated.

Otherwise,choosed1 : ´ i1 5 1. Thenit follows that

∑τ n W r1 ∑

c ¸hÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì d1

wr1 τ c 5 ∑τ n W r2 ∑



c ¸ÐÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì d1

wr1 τ c 5 ∑τ n W r2 ∑

c ¸ÐÆ Wτ Ç È È È ÇWτ Éc Ê Ë τ Ì Ne Ñ d1

wr2 τ c ∑

τ n W r1 ∑c ¸ÐÆ Wτ Ç È È È ÇWτ É

c Ê Ë τ Ì d1

wr1 τ c 5 ∑τ n W r2 ∑


wr2 τ c 1 üandagain(4.4) is violated.This completestheproof. As a consequence,in the caseof n 2 only few inequalitieshave to be checked for violation. Forlargervaluesof n, thechoicefrom proposition4.3canbeusedasaheuristic.

Wewill now analyzethequality of thecuts(4.4). Considera graphconsistingonly of two nodesandaconnectingedgeewith two linesr1 andr2 runningovere. Let thepossibletraincapacities(resultingfrom thecombinationsof train typesandnumbersof coaches)for line r1 begivenby ´ 1

1 ü('('('ü ´ k11 with´ 1

1 +'('('b ´ k11 . In contrastto proposition4.3, ´ i

1 ´ i

A1

1 is possiblefor somei þ& 1 ü('('('ü k1 1 ifthesamecapacitycanbeobtainedby selectingdifferentcombinationsof train typesandnumbersofcoaches.Let therespective valuesbedefinedfor line r2.


Proposition 4.4 With thesedefinitions,thepolyhedron P describedby theconstraints

∑τ n W r1 Wτ

∑c Wτ

wr1 τ c 1 ü ∑τ n W r2 Wτ

∑c Wτ

wr2 τ c 1 ü w 0

andtheconstraints(4.4) for all valuesof d1 fromproposition4.3 is integral.

Proof: Weshow thattheconstraintmatrix is aninterval matrixandthustotally unimodular(cf. [46]).Sincetheright handsidesof theconstraintsareintegral, thepropositionfollows.

Let usorderthecolumnsof theconstraintmatrixasfollows. Orderthevariablesfor line r1 by increas-ing capacity´ i

1 andthevariablesfor line r2 by decreasingcapacity. Now theconstraintmatrix lookslike this (boundconstraintsareomitted):ÃÄÄÄÄÄÄÄÅ ´ 1

1 ´ 21 '('(' ´ k1

1 ´ k22 '('(' ´ 2

2 ´ 12

1 1 1 '('(' 1 1 0 0 '('(' 0 0 0

0 0 0 '('(' 0 0 1 1 '('(' 1 1 1

0 1 '('('j'('('Ó'('('Ô'('(' '('('Ô'('(' 1 0 '('(' 0

0 0 1 '('('Ó'('('Ô'('(' '('('Ô'('('j'('(' 1 0 0

etc.

ÍÎÎÎÎÎÎÎÏ∑τ ∑cwr1 τ c 1

∑τ ∑cwr2 τ c 1

from proposition4.3

from proposition4.3

from proposition4.3

Obviously, it is aninterval matrix. Now compareP andthe polyhedronPû given by the LP relaxationof the correspondingBP-MCTPinstance.Every integerpoint of P satisfiestheBP-MCTPtravelercapacityconstraintsandthusis anelementof Pû . Conversely, all integerpointsfrom Pû arealsoin P: Theconstraintsfrom proposition4.4areeitherthesameasin theBP-MCTPdescriptionor arefulfilled becauseof proposition4.2.

SinceP is integral, the addition of all cuts from proposition4.3 is sufficient to obtain an integralpolyhedronfor thespecialgraphwe have examined.For generalinstances,we canconcludethat ifthereis an edgewith two lines runningover it, thereareno “better” cutsfor BP-MCTPwhich areusingonly the informationof thetraveler volumeon thatedgeandtheavailablecapacitiesfor trainsof thetwo linesrunningover thatedge.

In [10], Bussieckintroducesseveralclassesof cuttingplanesfor theline optimizationmodelwe havepresentedin section2.7. Oneof theseclasses((5.19)/(5.20)in [10]) canbeadaptedto theBP-MCTP.Thefollowing propositiondealswith thisclass:

Proposition 4.5 Let E û«° E, NE ª : ∑en E ª Ne. Let αE ªr denotethe numberof edges of E û that are

containedin line r. Let αE ª : maxr n αE ªr . Thenthefollowing inequalityis valid:

∑r ¸»¹

αE ªr Ì 1 ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c i NE ªαE ª l (4.5)


Proof: For eache þ E û we have

∑r ¸p¹r º e ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c Ne üandtherefore

∑en E ª ∑r ¸»¹

r º e ∑τ n W r Wτ

∑c Wτ

c ´ τ wr τ c NE ª 'Now

αE ª ∑r ¸p¹

αE ªr Ì 1 ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c ∑r ¸p¹

αE ªr Ì 1 ∑

e E ªr º e ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c ∑

en E ª ∑r ¸p¹r º e ∑

τ n W r Wτ

∑c Wτ

c ´ τ wr τ c NE ª 'Divide this inequalityby αE ª . Sincethe left handsideremainsintegral, the right handsidemay beroundedup. We may even divide by αE ª Γ, whereΓ is the greatestcommondivisor of all traincapacities. Theeffectof thecuttingplanes(4.5)is visualizedin figure4.3.Assumethatfor eachof thethreelinesgivenin thepicture,thereis only onetrain type,andlet thecapacityof onecoachof this typebe10.Let thefeasiblenumbersof coachesbe1 or 2.

Network Lines Solution

Ne ∑c

10c Õ wr Ö c33

33 33

16× 516× 5 16× 5

Figure4.3: Solutionwithout inequalities(4.5)

With (4.5)we obtain∑r c10c wr c 50,which is violatedby thesolution.

For ourpracticalinstances,thesecutshadonly avery smalleffecton theLP relaxationvalue.In fact,they slowedtheIP solutionprocessdown. Therefore,we finally have notusedthem.

4.6 Solving FSPinstances

Theproblemof finding a feasibleschedulefor fixedtrain typescanbeformulatedassatisfyinga setof linearconstraintswith integervariables.A simplesolutionapproachis addinganarbitrarylinear

4.6. SOLVING FSPINSTANCES 77

objective functionandusinga commercialMIP solver. For practicalinstances,this methodtakestoomuchtime.

Sincethestructureof theFSPandthePESPis similar, anotherideais to usea PESPalgorithmandadaptit to theFSP. In chapter3, severalalgorithmsfor solvingPESPinstanceshave beenintroduced.Most of themcanbeextendedin orderto handleFSPinstances.We have chosento adaptthePESPalgorithmof SerafiniandUkovich (seesection3.6) for theFSP. This algorithmis fastenoughfor ourinstancesandconsumesonly asmallamountof memory.

Wewill alsodiscussamethodfor findinga“small” setof linescausingtheconflict in caseof infeasibleinstances.

Modification of the Algorithm of Serafini and Ukovich

ThePESPalgorithmof SerafiniandUkovich hasbeenintroducedin section3.6. Variantsof this al-gorithmwhich leadto anaccelerationfor many practicalinstanceshavebeendiscussedin section3.7.In orderto usethe algorithm(or its variants)for FSPinstanceswe will have to modify it in suchaway thatfor every solution,thefollowing two conditionsarefulfilled (cf. section2.5):~ Themoduloparametersfor travel time,waiting time andturningtimeconstraintsare0.~ Themoduloparametersfor certainpairsof headway constraintsareidentical.

If thereis no PESPsolutionsatisfyingtheseadditionalconstraints,theFSPinstanceis infeasible.Ofcourse,if thePESPinstancealreadyis infeasible,thensois theFSPinstance.

Wecanensurethatthemoduloparametersof thetravel time,waitingtimeandturningtimeconstraintsare0 by takingthecorrespondingarcsinto thestarttreeof thealgorithmof SerafiniandUkovich. Thisis possiblebecausethosearcsform aspanningforestof theeventgraph.

A naive idea to provide the equality of somemodulo parametersin the algorithm of SerafiniandUkovich is to executea backtrackingstepif this equality is violatedsomewherein the searchtree.This doesnot work correctly in general. An exampleis given in figure 4.4. For start tree 1, thealgorithmstatesthatno feasiblesolutionexists,but for starttree2, a feasiblesolutionis found. Theproblemis thatit is notpossibleto setthemoduloparameterto 0 onanarbitraryspanningtreewithoutchangingthesolutionspace(nostatementanalogousto proposition3.2exists).

For our FSPinstances,we will show in proposition4.6 that it is possibleto selecta start treesuchthat the algorithmgives the desiredresult: If thereexists a solutionfor an instancewith a moduloparameterof 0 on the travel time, waiting time andturning time arcs,togetherwith equalmoduloparameterson certainpairsof headway arcs,thenthereexistsa solutionwith theadditionalpropertythat on the chosenstart tree,all moduloparametersare0. This is possiblebecauseof the specialstructureof theFSP.

An examplefor an FSPevent graphis given in figure4.5. Here,theeventsaregiven upperindices1 ü 2 ü('('(' in theorderof appearancein theline circulations.Ri is thesetof all eventsbelongingto line r i .Thisnotationis usedin proposition4.6.


Instance(T 1 10) Starttree1 Starttree2

1 2

3

1 2

3

1 2

3

samemod.par.required

d0 ] 1ed

3 ] 5e d7 ] 9e

z 1 0 Ø z 1¡3 1

z 1 0

infeasible

z 1 0 z 1 0

feasible: ϕ1| 1 0,

ϕ2| 1 10,ϕ

3| 1 3

Figure4.4: Thenaive algorithmmaystatefeasibilityonly for somestarttrees.

Line r1 Line r2 '('(' Line rρ

Arrival A

DepartureA

Arrival B

DepartureB

ArrivalC

DepartureC

Arrival D

DepartureD

...

travel/wait/turn

trainchange

headway

R1 R2 Rρr11

r21

r31

r41

r51

r61

r71

r81

r12

r22

r32

r42

r52

r62

r1ρ

r2ρ

r3ρ

r4ρ

Figure4.5: Examplefor anFSPeventgraph

Proposition 4.6 Considera feasibleFSPinstancewith eventgraph , . Then,for each spanningtreeSof , that containsthetraveltime, waiting timeandturning timearcs,there existsa feasiblepotentialwith a moduloparameterof 0 on thetreearcs.

Proof: Let an FSPinstanceanda feasiblepotentialϕ be given. Let ! r1 ü('('('ü rρ and let theeventsof line r i , i þ 1 ü('('('ü ρ be denotedby r1

i ü r2i ü('('('ü rki

i . r1i is the arrival at the first stationin

directionµ 0, r2i the correspondingdepartureetc., rki

i is the departureat the first station(i.e. thestationof r1

i ) in directionµ 1, which is the last event of the line that hasto be considered.LetRi : r1

i ü('('('ü rkii for eachi þÒ 1 ü('('('ü ρ .

Now we will constructa potentialϕ û which is also feasible,but leadsto a moduloparameterof 0on all tree arcs. Note that by addinga multiple of the period to the potentialsof all nodesin a

4.6. SOLVING FSPINSTANCES 79

setRi, i þ8 1 ü('('('ü ρ , themoduloparametersof pairsof arcsneedingthesamemoduloparametersarechangedin thesameway. Hence,onecanadda multiple of theperiodto thepotentialsof sucha setwithout violating thecorrespondingcondition.

Now consideran arc ù r ji ü r j ª

i ª ý with i ü i ûþ" 1 ü('('('ü ρ , j þ" 1 ü('('('ü ki , j ûRþ" 1 ü('('('ü ki ª with a nonzeromoduloparameterz with respectto ϕ. Subtractz T from thepotentialsof all nodesv þ VÙ with a

path(arcdirectionis ignoredhere)from r j ªi ª to v containingonly arcsfrom Sìù r j

i ü r j ªi ª ý . This procedure

changesthemoduloparameteronly on onetreearc,namely ù r ji ü r j ª

i ª ý . In fact, thenodesv areexactlythemembersof aunionof someRi, i þJ 1 ü('('('ü ρ .By applying this method,the modulo parameterof the arc ù r j

i ü r j ªi ª ý is set to 0, while the modulo

parametersof the otherarcsof S remainunchanged.Further, only moduloparametersof completesetsRi, i þJ 1 ü('('('ü ρ arechanged,andsothemoduloparametersof arcsneedingthesameparameterarealwayschangedin thesameway. Thus,we caniteratively constructa potentialϕ with a moduloparameterof 0 onall treearcs. This propositionallows us to usethealgorithmof SerafiniandUkovich to solve FSPinstances.Weonly needto selectall travel time, waiting time andturning time constraintsfor thestarttree. If themoduloparametersfor arcsneedingthesamevaluearechosendifferently, abacktrackingstepcanbeexecuted.

Finding a Setof Lines CausingInfeasibility

As wehavealreadypointedout, it wouldbehelpful for ouralgorithmsif wecouldnotonly detectthatanFSPinstanceis infeasible,but alsocoulddeterminewhich linesactuallycausetheconflict.

Look at the exampleof figure 4.6. Supposethat after constructingthe start tree, the algorithmofSerafiniandUkovich triesto satisfytheinterval conditionfor thedashedarcandfails (infeasibility).If sometravel timeintervalsarechangedin R2 for example,andthealgorithmis restarted,thenthestarttreeis constructedfrom thesamearcsagain(whichwill bethecasewhenthealgorithmis executedassuggested),thesamealgorithmicstepswill beperformed.Thedashedarcagaincannotbesatisfied.Therefore,changinga train typeon line r2 will not resolve theconflict. This argumentationwill beformalizedin thefollowing proposition.'('(' '('('

'('('R1 R2

R3

starttreearcconnectingRi andRj

non-treearctobeinserted

Figure4.6: FSPstarttreeandanew arc


Proposition 4.7 Let the start tree S for the FSP version of the algorithm of Serafini and Ukovichalwaysbechosenindependentlyof thetrain types.Let theorder in which non-treearcsareexaminedbeindependentof thetrain types.Let there bean infeasibleFSPinstancewhere theinfeasibilityhasbeendetectedafter examiningthenon-treearcsa1 ü('('('ü aq þ AÙ . Let Qi be thenodesetof thecyclecontainingai andthepathfromtheendnodeof ai to thestartnodeusingonly arcsfromS.

Then the FSP instanceremainsinfeasibleif only train typesfor lines with nodesnot containedin Ú q

i 1 Qi are changed.

Proof: In this case,thealgorithmperformsthesamesteps. With thisproposition,wecandefine

ˆ : r þ i nÛ 1 ¦ ¦ ¦ qÜ r containsanodeof Qi ¼ ½¿for our relaxationiterationor branch-and-boundalgorithm.

4.7 Exact Solution of the Nonlinear Problem

In section2.7, we introducedtheestimationtr for thecirculationtime of line r in orderto calculatethenumberof trainsrequiredfor theoperationof theline in theline planningmodels.Theestimationwasalsousedin thescheduleoptimizationmodel.

Theactualcirculationtime tr for a line r ù v1 ü('('('ü vn ýUþH dependson theschedule.If we assumethattheminimumtime for turningfrom directionµ 1 to directionµ 0 is used,tr is givenby

tr dv1r 1 av1

r 0 5 turn ' (4.6)

Let N-MCSPbe thenonlinearmodelobtainedfrom MIP-MCSPby replacingtr τ by tr τ andaddingtheconstraints(4.6) for eachline. We will now constructanalgorithmfor solvingN-MCSPexactly.Therefore,we will modify the FSPalgorithmso that we canfind out the numberof trainsusedina feasiblesolutionandthecorrespondingcost. After this modification,we will adaptthe relaxationiterationandthebranch-and-boundmethod.

Determining Costwith the FSPAlgorithm

The numberof trainsrequiredfor a line and thusthe costof a solutionof an FSPinstancecanbederived from certainmoduloparametersof anextendedeventgraph.We will now againdistinguishbetweeneventsandeventtimes.Thenumberof trainsneededfor a line r ù v1 ü('('('ü vn ýfþ is givenby

γr : i π ù dv1r 1 ýb π ù av1

r 0 ý turn

TlÙü

wherewe usetheminimumturningtime,becauseourobjective is to minimizethenumberof trains.

Now inserttwo arcsfor eachline r ù v1 ü('('('ü vn ýSþÝ :

4.7. EXACT SOLUTION OFTHE NONLINEAR PROBLEM 81~ cendr : ù dv1

r 1 ü aendr ü turn 5 T 1 ü turn 5 T 1ý (alwaysusethisarcin thestarttree)~ cloop

r : ù av1r 0 ü aend

r ü 0 ü T 1ýThearcscloop

r for eachr þÞ arealwaysfeasible.Now themoduloparameteronanarccloopr is exactly

thenumberof trainsneededfor line r. In orderto verify this,weconsidertwo cases:~ π ù dv1r 1 ý π ù av1

r 0 ý 5 turn k T for k þ"ß . In this case,γr k. Now considerthe modulo

parameterfor thearccloopr :

0 π ù aendr ýb π ù av1

r 0 ý z T T 1 z þ G0 π ù dv1

r 1 ýb π ù av1r 0 ý 5 turn 5 T 1 z T T 1

0 k T 5 T 1 z T T 1 z k

Recallthatrepresentative trainsareusedandthustheπ-variablescorrespondto thesametrain.~ Now let π ù dv1r 1 ýM π ù av1

r 0 ý 5 turn k T 5 t with k þÝß , t þ 1 ü('('('ü T 1 . Weobtainγr k 5 1.In ananalogousway

0 k T 5 t 5 T 1 z T T 1 z þ G üwhichcanonly betruefor z k 5 1.

Let τr bethefixedtrain typeof trainsof line r andlet wr bethefixednumberof coachesof trainsofline r. Thenthefixedcostpartof anFSPsolutionwith moduloparametervectorz is givenby

∑r n zcloop

r > Cfix

τr 5 wr CfixCτr

C ' (4.7)

Thekm-orientedcostpart is still independentof theschedule.Notethatnot only thetrain types,butalsothenumberof coachesinfluencesthecostof theschedule.

TheFSPversionof thealgorithmof SerafiniandUkovich stopsassoonasafeasiblesolutionis found.Instead,wenow evaluate(4.7)andexecuteabacktrackingstep.By thisprocedure,thecompletesearchtreeis examined.If thereareno morepossiblemoduloparameters,thealgorithmterminates.If therearefeasiblesolutions,anoptimalsolutionis returned.Otherwise,infeasibility is stated.

Wecanreducethenumberof searchtreenodesthathave to bevisitedby severaltechniques:~ In every node,we cancalculatea lower boundon eachmoduloparameter(cf. chapter3) andthereforea lower boundon thecostof theschedule.If thisboundexceedsthevalueof thebestknown solution,thesubtreefor thenodedoesnotneedto beexamined.~ As aheuristic,whenbranchingonanarccloop

r wecanalwayschoosethesubtreewith thelowermoduloparameterfor cloop

r first, hopingto find a low costsolutionearly. With sucha solution,it maybeeasierto find subtreeswith a lowerboundexceedingthebestknown costbound.~ We mayheuristicallybranchon arcscloop

r early in thesearchtreein orderto get stronglowerbounds.


Exact Relaxation Iteration Method

Wewill now extendalgorithm4.1from section4.3in orderto getanexactsolutionmethodfor theN-MCSP. As arelaxation,BP-MCTPisusedinsteadof IP-MCTP. Onreasonfor doingthisaretheshortersolutiontimes.Anotherreasonis givenbelow. Therearetwo maindifferencesbetweenalgorithm4.1andtheexactmethodgivenin thissection:~ The new algorithmdoesnot stopassoonasan FSPinstanceis feasible,but cutsoff the one

combinationof train typesandnumbersof coachesfrom theBP-MCTPthatled to theinstance:

Let τr be thetrain typeandcr thenumberof coachesof line r þH in theBP-MCTPsolutionthatled to theFSPinstance.Thentheinequality

∑r n wr τr cr ( 1 (4.8)

givesthedesiredresultfor themodelBP-MCTP. For thegeneralintegermodel,thereis nosuchsimpleinequality. This is thesecondadvantageof usingBP-MCTPinsteadof IP-MCTP.

Inequality(4.2) canbeeasilytransferredto theBP-MCTP. Let τr be the train typeof theBP-MCTP solution.Thecorrespondingbinarymodelinequalityis givenby

∑r n ˆ Wτr

∑c Wτr

wr τr c ˆ( 1 ' (4.9)~ For the relaxationBP-MCTP, lower boundson the line circulationtime tr areusedinsteadofestimationstr . Thelowerboundsarecalculatedby alwayschoosingthelowerboundfor travel,waiting andturningtime.

Algorithm 4.4 is anexact relaxationiterationmethodfor solvingN-MCSP. It is basedon thebinaryformulationBP-MCTP, thusthereis no train typevectorx.

Practicalexperiencesshow thatthismethodis veryslow. In theworstcase,∏r n ∏τ n W r ù 1 5 Wτ Wτ ýiterationsarenecessary. We have implementeda versionof thealgorithmonly consideringthetraintypesfor thecuts,i.e. inequality(4.8) is replacedby

∑r n Wτr

∑c Wτr

wr τr c + 1 'This inexactversionis still muchtooslow (cf. chapter5).

Exact Branch-and-BoundMethod

It is alsopossibleto designanexactbranch-and-boundmethodfor theN-MCTP. As in thecaseof theexactrelaxationiterationalgorithm,weuseBP-MCTPwith lowerboundson thecirculationtimeasarelaxation.In orderto considerthenumberof coachesfor eachFSPinstance,we extendthedataforasubproblem.

4.7. EXACT SOLUTION OFTHE NONLINEAR PROBLEM 83

Algorithm 4.4ExactRelaxationIterationAlgorithm for theN-MCSPc¤ ∞loop

if theBP-MCTPis infeasiblethenStop. If c¤à ∞, the problemis infeasible. Otherwise,an optimal solutionwith valuec¤ isgivenby w ¤ , a ¤ , d ¤ , z ¤ .

end ifLet w beanoptimalsolutionvectorof theBP-MCTP(usingtr ) with objective valuec.if c c¤ then

Stop. If c¤à ∞, the problemis infeasible. Otherwise,an optimal solutionwith valuec¤ isgivenby w ¤ , a ¤ , d ¤ , z ¤ .

end ifif theFSPfor w is feasiblethen

Let an optimal solution for the FSPconcerning(4.7) be given by a, d, z. Let the objectivevaluebec.if c c¤ then

c¤ : c; w ¤ : w; a ¤ : a; d ¤ : d; z ¤ : zend ifAdd cut (4.8) to theBP-MCTP.

elseLet ˆ beasetof linesleadingto theinfeasibility of theFSPfor w.Add cut (4.9) to theBP-MCTP.

end ifend loop

Let thefeasiblecombinationsof train typesandnumbersof coachesfor a line r þ bedefinedby¢ cr : 7 ù τ ü cý τ þJ¢ r ü c þJ Wτ ü('('('ü Wτ 9 '

If j r1 ü('('('ü rρ , thenasubproblemis definedby ¢ c1 '('(' ¢ c

ρ .

Theexactbranch-and-boundmethodfor theN-MCTP is givenby algorithm4.5. Again, this methodis tooslow for practicalinstances,andevenaversionconsideringonly train typesfor generatingnewsubproblemsdoesnot seemto bepromising(seecomputationalresultsin chapter5).


Algorithm 4.5ExactBranch-and-BoundAlgorithm for theN-MCSPc¤ : ∞;

: LL¢ c

1 '('(' ¢ c

ρ L ; l W c1 ¥¦ ¦ ¦§¥ W c

ρ : ∞loop

if /0 thenStop.If c¤@ ∞, theproblemis infeasible.Otherwise,(w ¤ , a ¤ , d ¤ , z ¤ ) is optimal.

end ifChoose¢ ûDþ .

: L¢ û if LP relaxationof theBP-MCTPfor ¢û is infeasiblethen

: L¢ û ûDþ ¯¢û û°¢ û¨ ; continueend ifif LP relaxationof BP-MCTPhasanoptimalvalue c¤ then

: L¢ û û þ ¯¢ û û °¢ û ; continueend ifif theBP-MCTPfor ¢û is infeasiblethen

: L¢ û û þ ¯¢ û û °¢ û ; continueend ifLet anoptimalsolutionof BP-MCTPbedefinedby thevectorw with optimalvaluec.if c c¤ then

: L¢ û û þ ¯¢ û û °¢ û ; continueend ifif FSPfor w is feasiblethen

Let anoptimalsolutionfor theFSPconcerning(4.7) begivenby a, d, z with objective valuecû .if cû c¤ then

c¤ : cû ; w ¤ : w; a ¤ : a; d ¤ : d; z ¤ : z;

: ˆ¢ l ˆW c¤áend ifLet τi bethetrain typeandqi thenumberof coachesfor line r i þ in theBP-MCTPsolutiondefinedby w.for i 1 to ρ do

Let ¢ ¤ denote¢ c1 '('(' ¢ c

i

?1 ¢ c

i Dù τi ü qi ý ¢ ci

A1 '('(' ¢ c

ρ .l W© : max c ü max l W«ª ª I¢ û û þ j± ¢ û û ²¢ ¤ L

: L¢ ¤ end forcontinue

end ifLet ˆ beasetof linesleadingto theinfeasibilityof theFSPfor w andlet τi bethetrain typeforline r i þ in theBP-MCTPsolutiondefinedby w.for i 1 to ρ do

if r i þ ˆ thenLet ¢ ¤ denote¢ c

1 '('(' ¢ c

i

?1 ¢ c

i Dù τi ü qý q þJ Wτiü('('('ü Wτi L ¢ c

i

A1 '('(' ¢ c

ρ .l W© : max c ü max l W«ª ª I¢ û û þ j± ¢ û û ²¢ ¤ L

: L¢ ¤ end if

end forend loop

Chapter 5

Computational Results

In this chapter, we reportoncomputationalexperienceswith themodelsPESPandMCSPintroducedin theprevious chapters.We have testedseveralalgorithmsfor datafrom therailroadcompaniesofGermany andtheNetherlands.In section5.1,weshortlydescribethetestinstances.Theothersectionscontaincomputationalresultsfor PESPinstances(section5.3)andMCSPinstances(section5.4).

5.1 TestInstances

We have testedour algorithmson realnetworks from theGermanrailroadcompany Deutsche Bahn(DB) andthe railroadcompany of the NetherlandsNederlandseSpoorwegen (NS). For the Germanrailroad,we usednetwork data,line plans,origin destinationmatricesand costdatafor the Inter-City (IC) andInterRegio (IR) supplynetworks. For therailroadof theNetherlands,we obtainedtherespective datafor theInterRegio (IR), InterCity (IC) andAggloRegio (AR) supplynetworks.

Furthermore,the PESPalgorithmswere testedon 15 specialinstanceswe obtainedfrom NS. Theconstraintsetsof theseinstancescontaina subsetof socalledmarketingconstraints V M %V . Theseconstraintsarenotabsolutelynecessary, but try to makethetimetableattractive for passengers.Exam-plesfor thoseconstraintsaretrainchangetimeconstraintsor constraintsensuringthatlinesrunningonthesametrackhave a very largeheadway in orderto geta shortwaiting time for passengerswishingto travel with anarbitraryof thoselines. Theinstancesobtainedfrom instances1–15by ignoringthemarketingconstraintsarecalled1a–15a.

We startwith a shortcharacterizationof the30 resultingPESPinstances.Thenumbersof nodesandarcsof theinstancesaregivenin table5.1.

Somecharacteristicsof theoptimizationinstancescanbefoundin table5.2. For all instances,4 dif-ferenttrain typeshavebeenconsidered.A heuristicmethodfor thedeterminationof stationsandlinesusedby passengersfor changingtrains is discussedshortly during the analysisof the optimizationresults.

As anexample,theInterCitynetwork of theNetherlandsis givenin figure5.1. Costoptimallinesforthisnetwork andfor theothernetworksof theNetherlandshave beendeterminedin [10].

85

86 CHAPTER5. COMPUTATIONAL RESULTS

Inst. # Nodes # Arcs Inst. # Nodes # Arcs Inst. # Nodes # Arcs

1 1866 14205 11 536 4705 6a 1344 7477

2 1672 14707 12 265 1491 7a 2338 12725

3 1672 11331 13 2233 14183 8a 2338 12725

4 125 925 14 2395 14446 9a 2338 12725

5 197 1118 15 2621 13175 10a 2338 12725

6 1345 9443 1a 1866 12967 11a 536 4318

7 2339 13906 2a 1596 11010 12a 264 1259

8 2339 13924 3a 1596 9752 13a 2224 11925

9 2339 14264 4a 124 721 14a 2338 12717

10 2339 14102 5a 196 920 15a 2621 12953

Table5.1: Numbersof nodesandarcsfor thePESPinstances

DB-IC DB-IR NS-IC NS-IR NS-AR

# Nodes 90 297 36 38 122

# Edges 107 384 48 40 134

# Lines 31 89 25 21 117

Average# edgesin a line 7.5 5.9 5.0 5.8 4.2

Table5.2: Problemcharacteristicsfor the5 optimizationinstances

5.2 Hardware and Software

Our computationalexperimentshave beenperformedon a 400 MHz PentiumII PC with 256 MBmainmemoryandoperatingsystemLinux.

The algorithmshave beencodedin C. For the solutionsof MIPs andLPs,we usedthe commercialsolver CPLEX,version6.5.Detailson thissolver canbefoundin [34].

5.3 PESPResults

In this sectionwe presentanddiscussexperienceswith severalPESPalgorithms,i.e. algorithmsforfinding a feasibleschedule.We have implementedthePESPpreprocessingmethodsfrom section3.1(without the decompositionmethods,which could not be appliedto our instances).Applying thepreprocessingleadsto a remarkablereductionof the event graphsizesof our setof instances,seetable5.3.

Thepreprocessingof eachinstancerequiredalwayslessthanoneminute.

In chapter3,differentalgorithmsfor solvingPESPinstanceswerediscussed.Wehavetestedseveraloftheseon our instances.Theobtainedresultscanbefoundin table5.4,wherethecolumnscorrespondto differentalgorithms:~ MIP: We have solved instancesby usingCPLEX on the MIP formulationof the PESP. The

5.3. PESPRESULTS 87

Lw

Hr

Gn

Asn

Zl

Dv

Aml

ApdAmf

Ah

Hgl

Es

Nm

Ed

Ht

Ut

Asd

Ledn

Gd

Shl

Hlm

GvGvc

Rtd

Ddr

Rsd Bd

Ehv

Wt

Vl

Rm

Std

Mt Hrl

Dvd

Tb

Figure5.1: InterCitynetwork of theNetherlands

solution timesdependon the parametersettingsof CPLEX. We obtainedthe bestresultsbychangingonly afew parametersfrom theirdefault values:Weusedstrongbranching, automaticgenerationof fractionalcutsandanaggregator toleranceof 10

?6. For a detailedexplanation

of theseparameters,wereferto [5,34].~ SU: This is the original algorithmof SerafiniandUkovich with the correctionof Nachtigall(cf. section3.6).~ SUâ : This is thealgorithmof SerafiniandUkovich, but with anarcpreorderby thenumberoffeasiblemoduloparameters(cf. section3.7)~ SUã : WehaveexaminedtheSerafini-Ukovich algorithmwith adynamicstrategy for thechoiceof arcs. In section3.7, we have discussedseveral suchstrategies. In table 5.4, the fastestsolutiontimeobtainedby acertaincombinationof thesestrategiesis given.A detailedanalysisof thestrategieswill bepresentedbelow.


I Nodes/Arcs Nodes/Arcs I Nodes/Arcs Nodes/Arcs I Nodes/Arcs Nodes/Arcs

original preproc. original preproc. original preproc.

1 1866/14205 450/2975 11 536/4705 234/1430 6a 1344/7477 1327/7157

2 1672/14707 1619/14404 12 265/1491 197/1056 7a 2338/12725 1340/7823

3 1672/11331 515/7924 13 2233/14813 1098/7540 8a 2338/12725 1340/7823

4 125/925 118/715 14 2395/14446 1287/8310 9a 2338/12725 1340/7821

5 197/1118 182/951 15 2621/13175 446/1915 10a 2338/12725 1340/7819

6 1345/9443 1096/6665 1a 1866/12967 828/4829 11a 536/4318 422/2132

7 2339/13906 1247/7923 2a 1596/11010 706/3790 12a 264/1259 218/1147

8 2339/13924 1247/7937 3a 1596/9752 706/3655 13a 2224/11925 1285/7254

9 2339/14264 1247/8141 4a 124/721 123/694 14a 2338/12717 1340/7811

10 2339/14102 1247/8050 5a 196/920 174/862 15a 2621/12953 446/1842

Table5.3: Reductionof theeventgraphsizeby preprocessing~ BC: Resultsfor thebranch-and-cutalgorithmfrom section3.9)aregivenhere.

The g -symbol in table 5.4 in the following tablesindicatesthat therewasno result after the limitof 10hoursof computationtime. For someinstances,it is actuallyunknown whetherthey arefeasibleor infeasible. This is indicatedby a questionmark in the correspondingcolumn. Therearesomeinstanceswhich couldbesolved (or provento be infeasible),but with a solutiontime of muchmorethan10 hours(e.g.severaldays).

For someinstances,the algorithmof SerafiniandUkovich detectsinfeasibility beforebuilding thesearchtree(0 nodesrequired).In thiscase,theinstanceis trivially infeasible(cf. section3.1).

A detailedexaminationof theeffectsof arcchoicestrategiesandheuristicsof section3.7 is givenintable5.5,wherethefollowing arcchoiceruleshave beenconsidered:~ A: arcwith minimalnumberof feasiblemoduloparameters;arcsearchis terminatedassoonas

anarcwith lessthan2 feasiblemoduloparametersis found~ B: asA, but thearcsareexaminedin a cyclic way~ C: asB, but if thereareseveralarcswith theminimumnumberof feasiblemoduloparameters2, thearcwith maximallook-aheadvalueis chosen(cf. section3.7); 5 arcswith theminimumnumberof feasiblemoduloparametersareexamined~ D: asC, but morethan5 arcsareexaminedaslong astheproductof thenumberof examinedarcsandthebestlook-aheadvalueis 100~ E: asD, but arcsareexaminedaslongastheproductis 200~ F: asE, but arcswith anadjacency valuelessthan 1

3 of theadjacency valueof the“bestarcsofar” areignored;thearcwith thehighestadjacency valueis examinedfirst

Fromtheresultswe canseethatby usingthenew arcchoicestrategiesfor theSerafini-Ukovich algo-rithm, somePESPinstancescouldbesolvedthatwereimpossibleto solve by theoriginal algorithm.

5.3. PESPRESULTS 89

I Feas. MIP SU SUä SUå BC

time time nodes time nodes time nodes time

1 no 1 s 1 s 0 1 s 0 1 s 0 1 s

2 no 1 s 1 s 0 1 s 0 1 s 0 1 s

3 no 1 s 1 s 0 1 s 0 1 s 0 1 s

4 no 1:34h 1:31h 12311697 0:45h 7080808 797s 488559 47s

5 yes 48 s 183s 258255 5 s 9760 3 s 770 4 s

6 ? æ æ æ æ æ7 ? æ æ æ æ æ8 ? æ æ æ æ æ9 ? æ æ æ æ æ10 ? æ æ æ æ æ11 yes æ æ æ 1783s 1262 116s

12 yes 1:09h 1 s 1812 1 s 891 8 s 858 6 s

13 ? æ æ æ æ æ14 ? æ æ æ æ æ15 yes æ æ æ 196s 3437 13s

1a ? æ æ æ æ æ2a yes æ æ æ æ æ3a yes æ æ æ æ æ4a yes 348s æ 273s 583071 5 s 572 3 s

5a yes æ 1 s 689 1 s 689 8 s 689 7 s

6a ? æ æ æ æ æ7a yes æ æ æ æ æ8a ? æ æ æ æ æ9a ? æ æ æ æ æ10a ? æ æ æ æ æ11a yes æ æ æ æ 300s

12a yes æ æ 1:11h 3600108 30s 930 30s

13a ? æ æ æ æ æ14a ? æ æ æ æ æ15a yes æ æ æ 1121s 1440 60s

Table5.4: PESPsolutiontimesandnumbersof searchtreenodes( g : no resultafter10 h)

Thebranch-and-cutmethodgivesevenbettersolutiontimes.With thatmethod,it waspossibleto findsolutionsfor the instances3a and7a within a time limit of oneday (which wasimpossiblefor theSerafini-Ukovich algorithmor its variants).A solutionfor instance2awasfoundaftera few days.

90 CHAPTER5. COMPUTATIONAL RESULTSI

AB

CD

EF

time

node

stim

eno

des

time

node

stim

eno

des

time

node

stim

eno

des

11

s0

1s

01

s0

1s

01

s0

1s

0

21

s0

1s

01

s0

1s

01

s0

1s

0

31

s0

1s

01

s0

1s

01

s0

1s

0

418

81s

5007

5879

7s

4885

590:

35h

3373

620:

35h

3320

560:

41h

3727

0314

15s

2128

60

517

s77

03

s77

035

s82

644

s90

155

s77

045

s79

1

6

çççççç

7

çççççç

8

çççççç

9

çççççç

10

çççççç

11

ççç

1783

s12

620:

53h

1150

0:49

h12

64

1214

s86

08

s85

829

s85

834

s85

851

s85

847

s92

7

13

çççççç

14

çççççç

1527

2s

1621

196

s34

3790

2s

1460

679

s18

3013

59s

3099

859

s14

77

1a

çççççç

2a

çççççç

3a

çççççç

4a6

s65

55

s57

239

s85

115

2s

647

176

s61

718

9s

1050

5a12

s68

98

s68

961

s68

919

7s

719

362

s10

8239

9s

718

6a

çççççç

7a

çççççç

8a

çççççç

9a

çççççç

10a

çççççç

11a

çççççç

12a

30s

930

35s

930

765

s10

917

1020

s10

2617

41s

4442

1494

s18

21

13a

çççççç

14a

çççççç

15a

çç

1121

s14

400:

41h

1437

1:20

h15

170:

54h

1404

Table5.5: Resultsfor variantsof theSerafini-Ukovich algorithm( g : no resultafter10 h)

5.4. OPTIMIZATION RESULTS 91

5.4 Optimization Results

In this sectionwe presentresultson theminimumcostschedulingproblemof section2.8 for our realworld testinstances.

Wewill atfirst describeaheuristicmethodfor determininglinesandstationswhereatrainchangetimeconstraintshouldbeestablished.For thisheuristic,OD-matricesareused.In asecondstep,wereporton experienceswith the relaxationiterationalgorithm4.1 andthe branch-and-boundalgorithm4.3,assumingthattheaccelerationmethodsfor thesubproblemsMCTPandFSP(cf. sections4.5and4.6)areapplied.We will thenanalyzetheeffectsof theaccelerationmethodsin detail. In thelastpartofthissection,wegiveresultsfor thealgorithms4.4and4.5for thenonlinearproblemwherethenumberof usedtrainsdependson theschedule.

Determining Lines and Stationsfor Train ChangeTime Constraints

Let G ¡ù V ü E ý bethenetwork graphof arailroadnetwork. Let ωi j bethenumberof travelerswishingto travel from stationvi þ V to stationv j þ V in a certaintime (e.g.oneyear; aswe have alreadymentioned,it is very difficult to determinesuchnumbersin practice).Thematrix Ω with entriesù Ω ý i j : ωi jis calledorigin-destination-matrix or OD-matrix.

A greedyheuristicfor determininglinesandstationswheretrainchangetimeconstraintsareusefulformany travelersis givenby algorithm5.1.There,asetè of trainchangetimeconstraintsis constructed.An elementof è consistsof a sourceline r þ4 , a destinationline r ûQþ4 , a sourceline directionµ þJ 0 ü 1 , adestinationline directionµûìþJ 0 ü 1 andastationv þ V wherethetrain changetime hasto beprovided.

Thealgorithmrequiresmany MCSPinstancesto besolved. We have useda variantwherethealgo-rithm terminatesassoonas èÒ hasreachedacertainvalue.

In figure5.2, therelative amountof travelersin theNS-IR network with a directconnectionor withonetrain changewith time constraint(i.e. with a “good connection”)dependingon the numberofintroducedtrainchangetimeconstraintsis depicted.

Dir ect MIP Solution

Wehavetriedto solveMIP-MCSPinstanceswith thecommercialsolverCPLEXdirectly. In table5.6,thesolutiontime or optimality gapafter10 h computationtime is given. An ∞-entrymeansthatnotevena feasiblesolutionwasfoundin thetime limit.

As one can see,only instanceswith very few train changetime constraintscan be solved by thismethod. To be moreexact: Our experimentsshowed that only if the optimal combinationof traintypesandnumbersof coachesallowed a feasiblesolution, the correspondingMIP-MCSPinstancecouldbesolved.


Algorithm 5.1DeterminingLinesandStationsfor TrainChangeTimeConstraintsé: V V Dù vü výà v þ V è : /0

loopifé /0 thenStop. è hasbeengenerated.

end ifChooseù vi ü v j ý+þ é suchthatωi j max ωk l xù vk ü vl ýfþ é é

: é Dù vi ü v j ýif thereis a line r þ connectingvi andv j then

continueend ifif traveling from vi to v j is possibleonly with 2 trainchangesthen

continueend ifˆè : êDù rk ü µk ü r l ü µl ü vm ýë rk þ¡]ü µk þ& 0 ü 1 ü r l þ¡`ü µl þ& 0 ü 1 ü vm þ V suchthat it is possibleto travel from vi to vm usingline rk in directionµk andto travel from vm to v j usingline r l indirectionµl while C R /0 do

Choosec þ ˆèˆè : ˆè¡ c if MCSPwith train changetime constraintsfrom è c is solvablewithout exceedingtimelimit, iterationlimit (for MCSPalgorithm4.1)or nodelimit (for MCSPalgorithm4.3) thenè : è c

breakend if

endwhileend loop

Inst. ì íì Time Inst. ì íì Time Inst. ì íì Time

(or gap) (or gap) (or gap)

DB-IC 0 1:45h DB-IR 20 1549s NS-IR 0 1067s

DB-IC 40 ∞ NS-IC 0 1:24h NS-IR 20 ∞DB-IR 0 233s NS-IC 40 ∞ NS-AR 0 2.7%

Table5.6: Resultsfor adirectsolutionof MIP-MCSPinstances

GeneralPerformanceof Relaxation Iteration Algorithm 4.1

In table5.7,resultswith therelaxationiterationalgorithm4.1aregiven.Wehave testedour instanceswith severalnumbers èÒ of train changetime constraints.For eachcombination,theoverall compu-tationtime, thenumberof requirediterationsandthecost(in monetaryunits)of theoptimalschedule(or thebestknown solutionif thetime limit wasexceeded)aregiven.

It is assumedthatfor eachiterationof thealgorithm,a “small” set ˆ of linescausingtheinfeasibility


60

65

70

75

80

85

90

95

100

0 5 10 15 20 25 30 35 40

travelerswith directconnection

travelerswith at least2 trainchanges

Figure5.2: Relative amountof travelerswith “goodconnection”(NS-IR)

is generated(cf. chapter4). If this is not done,many moreiterationsarerequired.For example,forthe NS-IR instancewith èNL 20, even after 10 h (andover 250 iterations)the lower boundof theseconditerationof thealgorithmwith ˆ wasnotachieved.

Inst. ì íì Time # Iter. Cost Inst. ì íì Time # Iter. Cost

DB-IC 0 219s 1 1.3722 NS-IC 30 28s 1 4.0548

DB-IC 10 183s 1 1.3722 NS-IC 40 æ 235 _ 4 × 0690åDB-IC 20 279s 1 1.3722 NS-IR 0 33s 1 2.6984

DB-IC 30 104s 1 1.3722 NS-IR 10 27s 1 2.6984

DB-IC 40 æ 925 _ 1 × 3858ä NS-IR 20 989s 89 2.7792

DB-IR 0 4 s 1 1.7534 NS-IR 30 1259s 91 2.7792

DB-IR 10 7 s 3 1.7588 NS-IR 40 1083s 91 2.7792

DB-IR 20 11 s 4 1.7592 NS-AR 0 0:47h 1 7.4852

DB-IR 30 15 s 5 1.7594 NS-AR 10 1:04h 1 7.4852

DB-IR 40 64 s 22 1.7689 NS-AR 20 1:11h 1 7.4852

NS-IC 0 30 s 1 4.0548 NS-AR 30 1:23h 1 7.4852

NS-IC 10 35 s 1 4.0548 NS-AR 40 1:24h 1 7.4852

NS-IC 20 35 s 1 4.0548 ä optimal: 1.3863 å optimal: 4.0709

Table5.7: Resultsfor therelaxationiterationalgorithm4.1

The increasingsolutiontime per iteration,causedby the additionalconstraintsfor the IP-MCTP, isshown in figure5.3for theexampleof theNS-IRnetwork, èÒá 20.

As onecansee,the resultsof thedecomposition-basedrelaxationiterationalgorithmleadsto muchbetterresults.Only two of thetestinstancescouldnotbesolvedwith thismethod.


0

5

10

15

20

25

10 20 30 40 50 60 70 80

Figure5.3: Time (in s) requiredfor eachiteration(NS-IR, èNá 20)

GeneralPerformanceof Branch-and-boundAlgorithm 4.3

Theresultsfor thebranch-and-boundalgorithm4.3areshown in table5.8. There,thesolutiontime,thenumberof nodesin thesearchtreeor theremainingoptimalitygapafterthetimelimit andthetimerequiredto find the optimal solution(if theproblemwassolved to optimality within the time limit)aregiven.

Inst. ì íì Time # Nodes Time Inst. ì íì Time # Nodes Time

(or gap) opt. sol. (or gap) opt. sol.

DB-IC 0 219s 1 219s NS-IC 30 28s 1 28s

DB-IC 10 183s 1 183s NS-IC 40 æ 0.07% 264s

DB-IC 20 279s 1 279s NS-IR 0 33s 1 33s

DB-IC 30 104s 1 104s NS-IR 10 27s 1 27s

DB-IC 40 9:31h 37746 480s NS-IR 20 1:00h 1486 110s

DB-IR 0 4 s 1 4 s NS-IR 30 1:19h 2017 135s

DB-IR 10 14s 5 14 s NS-IR 40 1:20h 2009 132s

DB-IR 20 15s 8 8 s NS-AR 0 0:47h 1 0:47h

DB-IR 30 27s 18 12 s NS-AR 10 1:04h 1 1:04h

DB-IR 40 122s 103 51 s NS-AR 20 1:11h 1 1:11h

NS-IC 0 30s 1 30 s NS-AR 30 1:23h 1 1:23h

NS-IC 10 35s 1 35 s NS-AR 40 1:24h 1 1:24h

NS-IC 20 35s 1 35 s

Table5.8: Resultsfor thebranch-and-boundalgorithm4.3


In orderto getanideaof theeffect of theaccelerationmethodsleadingfrom thesimplebranch-and-boundalgorithm4.2 to algorithm4.3, thesolutiontimesandnumberof nodesfor someinstancesforthesimplealgorithm4.2areshown in table5.9.

Inst. ì íì Time Nodes Inst. ì íì Time Nodes Inst. ì í:ì Time Nodes

(gap) (gap) (gap)

DB-IC 40 æ 0.01% DB-IR 40 132s 132 NS-IR 20 1:20h 2105

Table5.9: Resultsfor thesimplebranch-and-boundalgorithm4.4

The branch-and-boundalgorithm gives feasiblesolutionsin a few secondsor minutesfor our testinstances.Moreover, thequality of thesesolutionsis quiteacceptable(aftera few minutes,theopti-mality gapwaslessthan1% in our testcases).However, thealgorithmis slower thantherelaxationiterationalgorithm(if it doesnot terminatewith anoptimalsolutionat therootnode).

Solving MCTP instances

We will now discussthe differentmethodsfor solving MCTP instances.As exampleinstances,wehave chosenthe MCTP instancesfrom the first iteration of algorithm 4.1 (or the root nodefromalgorithm4.3respectively).

In table5.10,thenumberof rows, columnsandnon-zerosof theMIPs, therelative gapbetweentheoptimalLP solutionandtheoptimalMIP solutionandthesolutiontime for thegeneralintegermodelIP-MCTPandthebinarymodelBP-MCTParegiven.

Thesolver CPLEX containsa MIP preprocessor(see[34]) for reducingtheMIP size. Thenumbersfrom table 5.10 were obtainedafter using the preprocessor. We have also testedseveral variablebranchingstrategies(cf. appendixB). Thesolutiontimesarealwaysgivenfor thefasteststrategy fortheparticularinstance.

IntegermodelIP-MCSP BinarymodelBP-MCSP

Inst. #Con. #Var. # î1 0 Root Time Inst. #Con. #Var. # î1 0 Root Time

gap gap

DB-IC 204 244 1304 2.4% 4 s DB-IC 74 735 3435 0.9% 1 s

DB-IR 319 413 1436 2.4% 1 s DB-IR 69 471 1376 0.3% 1 s

NS-IC 148 179 826 2.5% 129s NS-IC 57 793 3237 2.5% 20s

NS-IR 86 107 570 1.5% 5 s NS-IR 41 545 2741 1.5% 2 s

NS-AR 471 617 2618 2.8% ä NS-AR 178 2221 9328 2.1% 865sä : terminateswith memoryfailure

Table5.10:Resultsfor differentMCTPmodels

Theeffectof usingthecuttingplanesfrom section4.5canbeseenin table5.11.There,thenumberofcuttingplaneiterationsbeforestartingtheMIP branch-and-boundprocess,thenumberof usedcuts,therelative gapbetweentheLP solutionandtheMIP solutionandthesolutiontimearegiven.


I # Iter. # Cuts Root Time I # Iter. # Cuts Root Time

gap gap

DB-IC 3 7 0.6% 1 s DB-IR 2 18 0.05% 1 s

NS-IC 10 134 0.6% 9 s NS-IR 10 94 0.9% 4 s

NS-AR 9 123 1.1% 934s

Table5.11:Resultsfor theBP-MCTPwith cuttingplanealgorithm

We canseethat for our instances,theBP-MCTPformulationgivesbetterresults(i.e. solutiontimes)thantheIP-MCTPformulation.Theuseof cuttingplanesprovidesanadditionalaccelerationin somecases.

Solving FSPinstances

In orderto comparethedifferentFSPalgorithmsresultingfrom thevariantsof thealgorithmof Ser-afini andUkovich, wehave testedthemby integratingtheminto therelaxationiterationalgorithm4.1andthe branch-and-boundalgorithm4.3. Our MCSPinstancesleadto the FSPinstancesizesfromtable 5.12. Note that for all instancesarising from the sameMCSP instances,thesenumbersareidentical.

Inst. ì íì ìVïì ìAïì ì ð®ì Inst. ì íì ìVïì ìA ï«ì ì ð®ìDB-IC 40 924 1529 298 NS-IR 20 488 941 224

DB-IR 20 2112 2295 116 NS-AR 0 1968 3956 1012

NS-IC 40 496 783 134

Table5.12:FSPinstancesizes

We will now discusstheresultsfor differentvariantsof FSPalgorithmsderived from variantsof theSerafini-Ukovich algorithm.We have appliedthesevariantsto thefirst 200nodesof thebranch-and-boundtreeof algorithm4.3for our testinstances(if thereweresomany nodes).

Table5.13shows detailedresultsfor thedifferentalgorithms.We have separatedfeasibleandinfea-sible instancesandgiven minimal, maximal,averagesolutiontime andthe mediansof the solutiontimes.Ouralgorithmicvariantsare:~ SU:original algorithmof SerafiniandUkovich~ SUâ : SerafiniandUkovich algorithmwith arcpreorderby numberof feasiblemoduloparame-

ters~ SUã : SerafiniandUkovich algorithmwith arcchoicestrategy A from table5.5~ SU*: SerafiniandUkovich algorithmwith arcchoicestrategy B from table5.5


MCSP FeasibleInstances InfeasibleInstances

Inst. ì í:ì Algo. # FSP SolutionTime # FSP SolutionTime

Inst. Min. Max. Avg. Med. Inst. Min. Max. Avg. Med.

DB-IC 40 SU 1 æ æ æ æ 1 1 s 1 s 1 s 1 s

DB-IC 40 SUä 0 4 1 s æ æ 1 s

DB-IC 40 SUå 1 253s 253s 253s 253s 40 1 s 14s 5 s 5 s

DB-IC 40 SU* 1 æ æ æ æ 13 1 s 3 s 2 s 2 s

DB-IR 10 SU 2 2 s 2 s 2 s 2 s 2 1 s 1 s 1 s 1 s

DB-IR 10 SUä 2 2 s 2 s 2 s 2 s 2 1 s 1 s 1 s 1 s

DB-IR 10 SUå 2 4 s 5 s 4 s 4 s 2 1 s 1 s 1 s 1 s

DB-IR 10 SU* 2 4 s 4 s 4 s 4 s 2 1 s 1 s 1 s 1 s

NS-IC 40 SU 1 1 s 1 s 1 s 1 s 55 1 s 1 s 1 s 1 s

NS-IC 40 SUä 1 9 s 9 s 9 s 9 s 75 1 s 42s 8 s 1 s

NS-IC 40 SUå 1 7 s 7 s 7 s 7 s 77 1 s 5 s 2 s 1 s

NS-IC 40 SU* 1 2 s 2 s 2 s 2 s 78 1 s 2 s 1 s 1 s

NS-IR 20 SU 0 200 1 s 1 s 1 s 1 s

NS-IR 20 SUä 1 1 s 1 s 1 s 1 s 93 1 s 602s 3 s 1 s

NS-IR 20 SUå 1 48s 48s 48s 48 s 95 1 s 3 s 1 s 1 s

NS-IR 20 SU* 1 23s 23s 23s 23 s 95 1 s 3 s 1 s 1 s

NS-AR 0 SU 1 æ æ æ æ 0

NS-AR 0 SUä 1 92s 92s 92s 92 s 0

NS-AR 0 SUå 1 1:10h 1:10h 1:10h 1:10h 0

NS-AR 0 SU* 1 0:34h 0:34h 0:34h 0:34h 0

Table5.13:Resultsfor FSPinstanceswith differentalgorithms

Algorithms for the Nonlinear Problem

We have also tried to solve the N-MCSP with methodslike algorithm 4.4 and 4.5. As we havealreadymentionedin section4.7,thesealgorithmsarevery slow. Resultswith our implementationofasimplifiedversionof thesealgorithmsareshown in table5.14andtable5.15.

Therearemany instancesfor whicheventhefirst FSPinstancewith optimizationcouldnotbesolved(thefirst BP-MCTPinstancewasalwayssolvedin a few secondsor a few minutes).

As onecanseefrom the tables,theexactcalculationrevealsthat thenumbersof trainsareoveresti-matedto acertainextentby usingtr τ insteadof tr τ.


Inst. ì íì Time # Iter. Cost Inst. ì íì Time # Iter. Cost

or gap or gap

DB-IC 0 1161s 2 1.3396 NS-IC 30 æ 17 0.3%

DB-IC 10 æ 1 3.6% NS-IC 40 æ 270 _ 3 × 9268

DB-IC 20 æ 18 1.3% NS-IR 0 0:55h 2 2.5116

DB-IC 30 æ 11 0.4% NS-IR 10 1:00h 2 2.5116

DB-IC 40 æ 896 _ 1 × 3544 NS-IR 20 æ 307 0.4%

DB-IR 0 277s 2 1.6968 NS-IR 30 æ 507 _ 2 × 6190

DB-IR 10 æ 124 0.04% NS-IR 40 æ 499 _ 2 × 6192

DB-IR 20 æ 630 0.04% NS-AR 0 æ 1 _ 6 × 7253

DB-IR 30 æ 27 0.04% NS-AR 10 æ 1 _ 6 × 7253

DB-IR 40 æ 166 0.2% NS-AR 20 æ 1 _ 6 × 7253

NS-IC 0 928s 2 3.9075 NS-AR 30 æ 1 _ 6 × 7253

NS-IC 10 æ 1 0.6% NS-AR 40 æ 1 _ 6 × 7253

NS-IC 20 1084s 2 3.9075

Table5.14:Resultsfor therelaxationiterationalgorithm4.4for thenonlinearproblem

Inst. ì íì Time # Nodes Cost Inst. ì íì Time # Nodes Cost

(or gap) (or gap)

DB-IC 0 1161 2 1.3396 NS-IC 30 æ 4 0.8%

DB-IC 10 æ 1 3.6% NS-IC 40 æ 4065 0.7%

DB-IC 20 æ 6 0.1% NS-IR 0 0:55h 1 2.5116

DB-IC 30 æ 6 0.4% NS-IR 10 1:00h 1 2.5116

DB-IC 40 æ 3680 _ 1 × 3544 NS-IR 20 æ 513 1.1%

DB-IR 0 277s 1 1.6968 NS-IR 30 æ 7134 1.4%

DB-IR 10 æ 124 0.04% NS-IR 40 æ 4785 _ 2 × 6174

DB-IR 20 æ 51 0.02% NS-AR 0 æ 1 _ 6 × 7253

DB-IR 30 æ 51 0.02% NS-AR 10 æ 1 _ 6 × 7253

DB-IR 40 æ 430 0.2% NS-AR 20 æ 1 _ 6 × 7253

NS-IC 0 928s 1 3.9075 NS-AR 30 æ 1 _ 6 × 7253

NS-IC 10 æ 1 0.6% NS-AR 40 æ 1 _ 6 × 7253

NS-IC 20 1084s 1 3.9075

Table5.15:Resultsfor thebranch-and-boundalgorithm4.5 for thenonlinearproblem

Chapter 6

Conclusionsand Suggestionsfor FurtherResearch

In this thesis,wehave presentedanddevelopedmodelsandalgorithmsfor generatingandoptimizingtrain schedules.From a theoretical point of view, theseproblemsbelongto a classof very hardproblems.

Despitethis fact, for practical instances,our algorithmsperformquite satisfactory, i.e. we canfindoptimal solutionsfor small or mediumsizednetworks like the InterCity or InterRegio networks ofGermany or theNetherlands.Ourrelaxation-basedalgorithmsproducesolutionswith provablequalityin a few minutes.It is worthmentioningthattheoretical considerationsontheproblemstructurewereof greathelpwhendesigningpractical algorithms.For largernetworks,a decompositioninto regional networksseemsto beadequate.For thesesubnet-works,solutionscanbe producedthat have to be combinedto an overall solution. This will not bepossibledirectly in somecases,but requiresmall adaptationsthat have to be performedmanually.This is alsothetraditionalwayof generatingschedules.The sameholdsfor anotherobvious goal, namelythe combinationof supplynetworks. Sincelinesfrom differentsupplynetworksoftenusethesamephysicalrailroadtracks,their schedulescannotbeconsideredseparately. Again,onecantry to solve thecombinedproblemdirectly or by adecomposi-tion method.

A practicalrequirementthathasto betakeninto accountin thefutureis theconsiderationof multipleobjectives. We have seenseveral evaluationcriteria of practicalrelevancefor schedules,includingminimizationof traveltime, maximizationof robustnessor minimizationof cost.

Ourmodelconsiderstheminimizationof operational costdirectly. Aspectslike minimizationof totaltravel time are only taken into considerationindirectly by settingan upperboundon the time fortrain changes.The fact that long waiting timesat stationsmay requirean additional(costly) traincompositionalsoleadsto anindirectreductionof travel time.

Thereareseveralprincipleapproachesfor consideringmultiple objectivessimultaneously:~ Weightedsumof objectives: We can constructcombinedmodelswith an objective functionbeingtheweightedsumof theoriginalobjective functions.In practice,thisapproachoftenleadsto unsatisfactoryresults.Thatobjective with thehighestweight is consideredonly, regardlessof theotherobjectives.

99

100 CHAPTER6. CONCLUSIONSAND SUGGESTIONSFORFURTHERRESEARCH~ Constraintsfor somecriteria: Anothercommonmethodis theoptimizationof onlyonecriterionsubjectto othercriteriarequiringacertain“acceptancelevel”. In thiscase,constraintsaregivenfor thoseobjectivesthatareignoredby theoptimization.

TheMCSPcanbeinterpretedassuchanapproach.Theoperationalcostsareminimizedwhilethetrainchangetimemustnotexceedacertainlimit. A similar approachis thefollowing.~ Pareto optimal solutions: A feasiblesolution of a multi-objective problemis called paretooptimal if all other feasiblesolutionwith a betterobjective value for a singlecriterion haveworseobjective valuesfor at leastoneothercriterion.

Ourexperimentswith severalsetsof trainchangetimeconstraintsè show paretooptimalsolu-tionsconcerningtheobjectivestravelers with “good” connectionandcost, seefigure6.1 (thisis notabsolutelycorrectbecausewehave usedaheuristicto determinetheset è ).

1.75

1.755

1.76

1.765

1.77

4400 4600 4800 5000 5200 5400 5600 5800 6000

Cost

Travelerswith “good” connection

ññ ñ ñ

ñ

Figure6.1: Paretooptimalsolutionsfor theGermanInterRegio network

Anotherproblemwehavealreadymentionedis thatachange in thetransportservice, likeanotherlineplanorschedule,affectsthetravelers’ behavior. Thepassengerdemanddatausedfor thegenerationofaschedulemayactuallybecomeworthlessby theintroductionof thatschedule.In orderto overcomethis problem,practitionerstry to simulatethe travelers’ behavior andthustry to estimatethe actualeffectof aschedule(or line planetc.).

Often, an iterative approachis used: After a scheduleis obtained,the demanddatais updatedac-cordingto a simulation. Afterwardsanotherscheduleis generated.Onemay hopethat this methodconverges, althoughthereis, of course,nomathematicaljustificationfor suchabehavior.

Naturally, thefinal goal is transportplanningwithout thehierarchical decompositionfrom figure1.2on page2. However, this goalseemsto beout of reachat themoment,dueto eachstepbeingstill ahardproblemfor real-world-sizedinstances.

Appendix A

Computational Complexity

A topic of maininterestfor designingalgorithmsis thequestion“How long doesit take in theworstcaseto solveaprobleminstanceof acertainsize?”.Anotherpointmaybetheamountof memorythatis needed.An approachto answersuchquestionsis givenby thetheoryof computationalcomplexityof algorithms.Thereis a lot of literatureconcerningthissubject,for example[1,25,51]. Wewill giveashortdescriptionof someideason computationalcomplexity here.

Thesizeof a probleminstancecanbeunderstoodasthenumberof bits that is neededto describeallthedatathatdefinestheinstance.For example,thesizeof aninteger i R 0 is sizeù i ý@ 1 5 v log2 i xin this case.The runningtimeof analgorithmmaybemeasuredasthenumberof “basicsteps”likeassignmentsteps,addition,subtraction,multiplication,division,comparisonof two numbers,thatarerequiredto solve aprobleminstance.In general,therunningtimedependson thesizeof theinstance.

In orderto definetheworstcasecomplexity of analgorithm,the“big O” notationis used.Let f : ß bea function. An algorithmis saidto have worstcaserunningtime (or complexity) of O ù f ù nýý ifthereareconstantsc 0 andn0 þNß suchthat the running time doesnot exceedc f ù ný for eachinstanceof sizen n0.

A.1 The ProblemClassesP and NP

An algorithmis calledpolynomialtimealgorithmif thereexistsapolynomial f suchthatthealgorithmhasa runningtime of O ò f ò nó(ó . An algorithmis calledexponentialtimealgorithm if its runningtimecanonly beboundedby anexponentialfunction,but notby a polynomial.

We will focusonly on decisionproblemsin the following. A problemis calleddecisionproblemiftheanswerconsistsonly of ananswer“yes” or “no”. A minimizationproblemcanbesolvedby usingdecisionproblemalgorithmsvia binarysearchtechniques(“is thereasolutionwith anobjective valueô

c”). Therefore,if thereis a polynomialtime algorithmto solve a decisionproblem,thenthereisalsoapolynomialtime algorithmfor suchacorrespondingoptimizationproblem.

The setof all decisionproblemsfor which a polynomial time solutionalgorithmexists, is denotedby P.

Many importantdecisionproblems(and thereforemany importantoptimizationproblems)are notknown to have polynomial time solutionalgorithms. However, for many problems,a situationlike

101

102 APPENDIX A. COMPUTATIONAL COMPLEXITY

in thefollowing exampleis given: Considera mixedintegerprogrammingproblem(cf. appendixB)andthequestionif thereis a solutionx with anobjective value

ôc. If we aregivena solutionx with

objective valueô

c, wecancheckthattheanswerto ourproblemis “yes” in polynomialtime(simplyby evaluatingtheobjective function for x). Note thaton theotherhand,if we aregivena solutionxwith objective value õ c, we cannotverify that the answeris “no” that easily. This motivatesthefollowing definitionof theproblemclassNP.

A decisionproblemis saidto bein theclassNPif andonly if, for every instancefor which theansweris “yes”, thereis a certificate, namelya binary string whoselengthis polynomially boundedby thesizeof the input data,anda polynomialtime algorithmwhich, whensuppliedwith the input dataoftheprobleminstanceandthecertificate,confirmsthattheansweris indeed“yes” in polynomialtime.

In ourexample,this certificatewouldconsistof thebinaryencodingof x. Notethat,if theanswerforaninstanceis “no”, thereis nothingsaidaboutcertificatesor polynomialtimealgorithms.

A.2 NP-completeProblems

Considertwo problemsΠ andΠ ö . Π is said to be polynomiallytransformableto Π ö , if thereis analgorithmwhich, for every instanceI of Π, constructsaninstanceI ö of Π ö (i.e. it takestheinput datafrom I andconstructsthe input datafor I ö ) in polynomialtime suchthat theanswerto I ö is “yes” ifandonly theanswerfor I is “yes”.

A problemis said to be NP-completeif and only if it is in NP and every problemin NP can bepolynomiallytransformedto it.

Many importantdecisionproblemsfor which no polynomial time solutionalgorithmis known (forexamplethedecisionversionof solvingmixedintegerprograms)havebeenshown to beNP-complete.In [25], thereis a list of problemswhichwereknown to beNP-completealreadyin 1979.

If thereis a polynomial time solutionalgorithmfor a singleNP-completeproblem,then therearepolynomial time solution algorithmsfor all NP-completeproblems. In this case,we would haveP ÷ NP, which is hardlybelieved.Therefore,if wecanshow thataproblemis NP-complete,wehavereasonto believe thatthereis no polynomialtime for solvingit.

In orderto show thataproblemΠ is NP-completeit is sufficient to show thatit is in NPandthatthereis anNP-completeproblemthatcanbepolynomiallytransformedto Π.

Appendix B

Mixed Integer Linear Programs

B.1 Linear and Mixed Integer Linear Programs

Theproblemof theform

given A ø¡ù m ú n û b ø4ù m û c øHù n

minimize cTx

subjectto Axô

b

x ø¡ù n

(B.1)

is calledlinearprogrammingproblemor, for short,linear program (LP). Theset ü x ø8ù n ý Axô

b þis called feasiblesetor feasibleregion. The function f : ù n ÿ ù definedby f ò x ó®÷ cTx is calledobjectivefunction. For thetheoryof linearprogramsandsolutionmethods,we referto [15] and[55].

If somecomponentsof x arerequestedto have integer values,the problemis calledmixedintegerlinear program (MIP). If all componentshave to beinteger, it is an integer linear program (IP). Thecorrespondingfeasiblesetsaregivenby theintersectionof ü x ý Ax

ôb þ andtheintegrality constraints.

Someliteraturethatdealswith suchproblemsis [46,55].

B.2 Polyhedra

In this sectionsomewell known resultsand notionsof polyhedral theory are presented.A morecomprehensive discussioncanbefoundin [46].

A polyhedron P "ù n is thesetof pointssatisfyingafinite numberof linearinequalities,i.e.asetthatcanbe describedas ü x øJù n ý Ax

ôb û A øJù m ú n û b øJù m þ . If a polyhedronP is bounded(that is,

if thereis an ω ø8ù suchthat P ü x øù n ý ω ô xiô ω for eachi ø&ü 1 ûû n þLþ wherexi arethe

componentsof x), it is alsocalledpolytope. A polyhedronP is of dimensionk, denotedby dimP ÷ k,if themaximumnumberof affinely independentpointsin P is k 1.

An inequality Txô α0, ø¡ù n, α0 ø¡ù is calledvalid inequalityfor P if P )ü x ø4ù n ý Tx

ô α0.If Tx

ô α0 is avalid inequalityfor P, thenF ÷%ü x ø P ý Tx ÷ α0 þ is calleda faceof P. In thiscase Txô α0 is saidto generateF.

103

104 APPENDIX B. MIXED INTEGERLINEAR PROGRAMS

A faceF is said to be proper if F ÷ /0 andF ÷ P. A faceF is called facet if dimF ÷ dimP 1.Thesinglepoint of a zerodimensionalfaceF ÷ü x cþ of a polytopeP is calledextremepoint of P. Apoint x ø P is anextremepoint of a polytopeP if andonly if theredo not exist x ö û x ö ö ø P suchthatx ÷ 1

2x ö 12x ö ö .

Thefeasiblesetof LP is a polyhedron.In general,the feasiblesetof MIP or IP is not a polyhedron.Assumenow that the feasibleset is a polytope. Many MIP solutionalgorithmsstartby solving thecorrespondingLP (thesocalledLP relaxation, cf. sectionB.3) andfinding anoptimalextremepoint(notethatif thereis anoptimalpoint in apolytope,thenthereis alsoanoptimalextremepoint). If thispoint doesnot satisfythe integrality constraints,which is thenormalcase,thealgorithmsstartsomeother, usuallytimeconsumingprocedure.

Let P ÷ü x øHù n ý Axô

b þ bea polytope.FromWeyl’s theorem,we know thattheconvex hull of thefeasibleset conv òhü x ø n ý Ax

ôb þó@÷ : C for thecorrespondingIP canbedescribedasa polytopeü x øù n ý Aö x ô b ö þ with Aö ø4ù m ú n, b ö ø4ù m . Theextremepointsof this convex hull all satisfythe

integrality constraints.If Aö andb ö wereknown, onecouldstartthesolutionalgorithmwith Aö andbö ,andit wouldonly needto solve theLP relaxationto giveanoptimalsolutionfor IP, cf. [46]. A similarstatementcanbegivenfor MIP.

Unfortunatelyit is very difficult to find Aö andb ö for agivenset ü x ø n ý Axô

b þ if only A andb areknown, which is theusualcase.Accordingto [46], for eachfacetof thepolytopeC, avalid inequalityis necessaryin a description ü x øÒù n ý Aö x ô b ö þ for C, andthereis no polynomialφ suchthat thenumberof facetsof C is boundedby φ ò sizeò A û b ó(ó , cf. [55].

Therefore,it is practicallyimpossibleto find all facetsof C. However, onecantry to find somefacetsor at leastvalid inequalitiesfor C heuristically. Considerthe exampleof figure B.1. In the middle,the feasibleregion P for the LP relaxationis given. On the right, onecanseetheconvex hull C ofthefeasiblesetof IP. After addingtheconstraint2x1 x2

ô3 (i.e. thedashedconstraint),theoptimal

solutionof LP is integral althoughthis inequalityis notevena facet.

min x2

s.t. 12x1 x2 1

2 14x1 x2 1

4

4x1 x2 6

x1 x2 1

x2

1 x1

P

1

x2

1 x1

C

FigureB.1: IP, feasibleregion without integrality constraints,convex hull of feasibleregion

In asocalledcuttingplanealgorithm(cf. sectionB.3) for IP solution,thefirst stepconsistsof solvingtheLP relaxationof theprobleminstance.Let theoptimalpoint for theLP relaxationbex . If it isintegral, the IP solutionhasbeenfound. Otherwiseonetries to find a valid inequality Tx

ô α0 fortheconvex hull of thefeasiblesetsuchthat Tx õ α0. This inequalityis addedto theLP relaxationandthenew linear problemis solved. Theprocessis continuediteratively, until an integer solution

B.3. SOLUTION METHODS 105

hasbeenfound(seesectionB.3).

For this type of algorithm it is importantthat, given a set of points S and an additionalpoint x ,one candecidewhetherx ø convS and in caseof x ø convS give a valid inequality for convSwhich is violatedby x. Thisproblemis calledseparation problemandcanin generalbepolynomiallytransformedinto theoriginal optimizationproblem(andvice versa),see[29]. Nevertheless,in manyspecialcasessomeclassesof facetsor valid inequalitiescanbefoundin polynomialtime.

B.3 Solution Methods

In thefollowing, weassumehaving anIP instancemin ü cTx ý Axô

b û x ø n þ . Thepresentedmethodscanbe easilyextendedfor the solutionof MIP instances.Let S : ÷ü x ø n ý Ax

ôb þ (i.e. S is the

feasiblesetfor theinstance).

Relaxation Iteration Algorithms

Oneclassof solutionmethodsare relaxationiteration algorithms, seealgorithmB.1. Thesealgo-rithmstry to minimizetheobjective functionon a setR S. If anoptimalsolutionx is foundwithx Sø S, thenx is an optimal solutionfor the IP instance.OtherwiseR is replacedby a setRö withR Rö S and the procedureis restarted. In mostcasesthesealgorithmsaredesignedin suchawaythatduringthesolutionof min ü cTx ý x ø Rþ , informationgeneratedby previousiterationscanbereused.

Algorithm B.1 RelaxationIterationAlgorithmChooseR S.loop

if R ÷ /0 thenStop.Theproblemis infeasible.

end ifCalculatez ÷ min ü cTx ý x ø Rþ andacorrespondingsolutionx .if x ø S then

Stop.x is anoptimalsolution.end ifChooseRö with R Rö S.R : ÷ Rö .

end loop

An importantexamplefor this typeof algorithmsarefractionalcuttingplanealgorithms(or, for short,cutting planealgorithms). Here,R ÷tü x øNù n ý Ax

ôb þ is choseninitially, i.e. the corresponding

LP instance(the so calledLP relaxation) is solved. If x ø S, which meansthat x hasa fractionalcomponent,Rö is chosenas the intersectionof R with an additional linear inequality (the cuttingplane) which is violatedby x , but valid for every x ø S. Onecanshow that thereis alwayssuchaninequalityandthat theseinequalitiescanbechosenin sucha way that thealgorithmterminatesafterafinite numberof iterations,see[28] or [46].


Therearecutting planealgorithmsworking with a particularsetof cutting planessuchthat thereisnotalwaysaninequalityviolatedby x , but valid for eachx ø S. They proceedwith othertechniquesassoonasin someiteration,thereis nocuttingplaneviolatedby x , but valid for eachx ø S.

Onemotivation for usingcuttingplanealgorithmsis thefact thatafteraddinganinequalityto R, thesolutionx is still dually feasible,and the minimization in the next iterationcanbe donefrom anadvancedbasisfor thedualsimplex algorithm(cf. [46]). Anotheradvantageis thatin every iteration,z is a lowerboundon theoptimalsolutionof theoriginal IP instance.

Enumerative Algorithms

Anotherclassof algorithmsoftenusedfor thesolutionof MIPs is givenby enumerativealgorithms.Let S ÷ r

ρ 1Sr with Sρ1 Sρ2 ÷ /0 if ρ1 ÷ ρ2, ρ1 ø4ü 1 ûû r þ , ρ2 ø¡ü 1 ûPû r þ . Then ü Sρý ρ ÷ 1 ûû r þ

is saidto beapartition of S. Thefactthat

min ü cTx ý x ø SþS÷ minρ "! 1 # $ $ $%# r & ü cTx ý x ø Sρ þ

suggestsusinga divide-and-conqueralgorithm. ThesetSρ maybepartitionedagain,anda partitiontreestructureis obtained.Thewayof partitioningSis of coursethecrucialpoint for thealgorithm.If Swaspartitionedinto setsthatcontainonly oneelement(theextremecase),we wouldhave acompleteenumeration.Thisprocedureusuallyexhaustsall computationalresources,if it is appliedto practicalprobleminstances.

Insteadof usingpartitions,onemay alsousedivisions. ü Sρý ρ ÷ 1 ûû r þ is calleddivision of S if

S ÷ rρ 1 Sr (no furtherconditionis needed).

Thereareseveral simplecriteria for stoppingfurther partition somewherein the partition treeat asetSρ:' Sρ ÷ /0' theoptimalsolutionfor min ü cTx ý x ø Sρ þ is known' it canbeshown thatmin ü cTx ý x ø Sρ þ)( z , wherez is thesolutionvalueof anelementx , for

whichwe alreadyknow x ø S

Let Rρ bea setwith Rρ Sρ, andlet zà÷ min ü cTx ý x ø Sρ þ , i.e. theoptimalvalueof a relaxation.Let x bea correspondingvaluefor x. Thenthefollowing situationsallow usto usethecriteriafromabove:' Rρ ÷ /0' x ø Sρ' z*( z , wherez is thesolutionvalueof aknown solution

Algorithmsusingthesepartition,relaxationandstoppingcriteria ideasarefrequentlycalledbranch-and-boundalgorithms.A generalbranch-and-boundalgorithmfor solving integer programmingin-stancesis givenby algorithmB.2.


Algorithm B.2 Branch-and-BoundAlgorithmz : ÷ ∞; +÷ü Sþ ; lS : ÷ ∞loop

if +÷ /0 thenStop.If z@÷ ∞, thentheproblemis infeasible.Otherwise,x is optimalwith valuez .

end ifChooseSöø,+ andasetRö- Sö .+ : ÷.+0/ü SöUþif RöL÷ /0 then

continueend ifz : ÷ min ü cTx ý x ø Rö þ with optimalsolutionx.if z ( z then

continueend ifif x ø Sö then

z : ÷ z; x : ÷ x+ : ÷.+1/ü S ý lS ( záþcontinue

end ifChooseapartition r

ρ 1 Sr of Sö .lSρ : ÷ z for eachρ øJü 1 ûû r þ+÷.+32 S1 2 S2 2 2 Sr

end loop

In the algorithm, a set + of setsSρ S is maintainedfor which the objective function hasto beminimized. Often, Sρ is identifiedwith the correspondingminimizationproblemandthus itself iscalledproblem. Associatedwith eachSρ ø1+ is a lowerboundlSρ suchthatcTx ( lSρ for eachx ø Sρ.It is possibleto have lSρ ÷ ∞. The bestknown solutionvaluethat the algorithmhasfound so faris z . zX÷ ∞ meansthatnosolutionhasbeenfoundyet. If z54 ∞, thecorrespondingsolutionis givenby x .Again, the mostpopulartype of relaxationis theLP relaxation.Many commercialcomputercodes(like CPLEX, which hasbeenusedin our experiments)usethis type of relaxation. In this case,asolutionx ø Sö hasat leastonefractionalcomponent,sayt 4 xi 4 t 1 for anindex i ø8ü 1 ûû n þ andanintegert. A possiblepartitionthenconsistsof thesetsS1 : ÷ Sö ü x ø S ý xi

ôt þ andS2 : ÷ Sö ü x ø

S ý xi ( t 1 þ . This canbeexpressedasanadditionallinearinequalityfor eachsetandthusthedualsimplex algorithmseemsto bea promisingmethodfor thesolutionof theproblemsarisingfrom S1

andS2.

In anLP-basedbranch-and-boundprocess,severaldecisionshave to bemade.Essentiallythesedeci-sionsare:' choiceof Sö ø1+

108 APPENDIX B. MIXED INTEGERLINEAR PROGRAMS' choiceof a fractionalcomponentof x ø Rö , if it has

Thefirst decisionis oftencallednodeselection, theseconddecisionchoiceof thebranchingvariable.We presentsomesuggestionsfor thesedecisionshere. For further informationwe refer to [46], fordetailsconcerningavailability of suchstrategiesin thecommercialsoftwarewe have used,see[23]and[34].

A widely usednodeselectionrule is thedepthfirst search rule. Thenodesof thebranch-and-boundtree(althougha problemset + is maintained,it canbeinterpretedasa treestructure)arevisitedin adepthfirst searchorder. Anotherrule is thebestboundrule. In thiscase,theelementSö ø,+ with

lS ÷ minS 76 lS

is selected.By doingthis,we try to improve thelower boundon thesolutionfor S. Recallthat in thebranchandboundalgorithm,z is anupperboundfor thesolution,minS 76 lS a lowerbound.

Thechoiceof thebranchingvariableis frequentlydoneby a maximuminfeasibilityor by a minimuminfeasibility rule. In the first casethe fractional componentxi wherexi

98 xi : is “closestto 12” is

selected,in the lattercasethecomponentwhich is closestto an integervalue. Anotherrule thathasbeensuccessfullyappliedto practicalproblemsis thestrongbranching rule (a descriptionof this isgivenin [61], for example).

An advantageof branch-and-boundalgorithms,comparedwith relaxationiterationalgorithms,is thepossiblegenerationof feasible(but not necessarilyoptimal) solutionswhile examining sometreenodes.Fromthelower boundsof all remainingtreenodesandthebestknown solution,anoptimalitygapcanbecalculated(i.e. oneknows aninterval containingtheoptimalsolutionvalue).

Theideasof cuttingplanesandbranch-and-boundcanbecombinedeffectively:' Cut-and-branch: Thesealgorithmsstartwith cutting planes,until somestoppingcriterion isfulfilled. Thenabranch-and-boundprocessis startedontheproblemwith theaddedconstraints.' Branch-and-cut: In this case,at every nodeof thebranch-and-boundtree,acuttingplanealgo-rithm is started.As soonasa stoppingcriterionis fulfilled, thebranchingis continuedandthegeneratedcutsareappliedin thecompleterespective subtree.Note that in suchanalgorithm,cuttingplanesonly valid for asubtreecanbeapplied.

Preprocessing

Sometimesthe sizeof a MIP canbe reducedbeforeactuallystartingto solve it. By looking at thespecificproblemstructure,onecanoftenfind variableswhosevaluesin anoptimalsolution(or evenin a feasiblesolution)canbe easilydeterminedin advance.Thus,they canbe replacedby constantvalues.Thisprocessis calledvariablefixing. Similarly, onemaydetectredundantconstraints.

Often,coefficientsof theconstraintmatrixcanbemodifiedin suchawaythatthefeasiblesetremainsunchanged,but non-integerextremepointsof thecorrespondingLP areavoided.Suchaprocedureiscalledcoefficientreductionandis especiallyhelpful for cuttingplanealgorithms(seesectionB.2). Anexampleis givenin figureB.2, wherechangingtheconstraint 2

3x1 x2ô

0 to 12x1 x2

ô0 gives

thesamefeasibleset,but leadsto anintegeroptimalsolutionalreadyfor theLP relaxation.


For someproblemswith aparticularstructure,coefficient reductionschemesareknown.

A moredetailedinvestigationof preprocessingtechniquescanbefoundin [35] and[54].

min x2

s.t. 23x1 x2 0

x1 2 x2 0

x1 x2 1

x2

1 x1

23x1 x2 0 12x1 x2 0

FigureB.2: Changingacoefficient leadsto anintegeroptimalsolutionfor LP here

Let P ÷ ü x øù n ý Axô

b þ andC ÷ conv òhü x ø; ý Axô

b þó . We know thatC P. By coefficientreductionaswell asby the additionof cutting planeswe try to find polyhedraPö with C Pö=< P,hoping that a relaxationfrom ü x ø> n ý Ax

ôb þ to Pö givesbetterboundsfor the solutionor less

non-integral extremepointsthana relaxationto P.


Appendix C

ShortestPath Problems

Let G ÷ ò V û Aó beadirectedgraph,andsupposethateacharca is associatedwith a lengthor costµa.Let ýV ý ÷ : n, ýA ý ÷ : m andlet thenodesbedenotedby 1 ûû n. Theshortestpathproblemis to finda minimumcostpathfrom a sourcenodeto a destinationnode.In orderto formalizethis ideaandtodiscussalgorithmsfor solvingshortestpathproblems,we shortly focuson somealgebraicstructuresrelatedto suchproblems.For details,wereferto [63].

Definition: A nonemptysetH with internalcomposition? : H @ H ÿ H is calledsemigroup, if

a ? ò b ? có÷ ò a ? bó-? c for all a û b û c ø H A semigroupis calledmonoidif it containsanelementewith

e ? a ÷ a ? e ÷ a for all a ø H In this case,e is saidto bea neutralelementof H. If a neutralelementexists, it is alwaysuniquelydetermined.A semigroupis calledcommutative, if

a ? b ÷ b ? a for all a û b ø H Definition: A commutative semigroupis calledordered, if

aô

b A a ? cô

b ? c for all a û b û c ø H An elementa of anorderedsemigroupis saidto bepositiveif

bô

a ? b for all b ø H Definition: Let ò Rû ?mó be a commutative monoid with neutralelement0 and let ò RûCB ó be a (notnecessarilycommutative) monoidwith neutralelement1, where0 ÷ 1. ò Rû ? ûCB ó is calledasemiringwith unity 1 andzero 0, if

a B ò b ? có ÷ ò a B bó-? ò a B cóò b ? có B a ÷ ò b B aó-? ò c B aó0 ÷ a B 0 ÷ 0 B a

DFEEGEEH for all a û b û c ø R111

112 APPENDIXC. SHORTESTPATH PROBLEMS

We will shortlyspeakof thesemiringR. Thefirst two conditionsarecalledlaws of distributivity. Ifall elementsof R areidempotentwith respectto ? , R is called idempotentsemiring. If ò RûCB ó is acommutative monoid,R is calledcommutativesemiring.

Let p : ÷+ò a1ûû ar ó beapathin thegraphG andlet thearccostvaluesbeelementsof acommutative

semiring.Theweightw ò pó of thepathis thendefinedas

w ò pó : ÷ rIk 1

µak

Let Pi j denotethesetof all pathsfrom i to j in G. Theproblemof determining

µ ò i û j ó : ÷KJ

p Pi j

w ò pófor a pair of nodesò i û j ó anda correspondingpathis calledalgebraic pathproblem. It is commontodefineµ ò i û i ó : ÷ 1. If R ÷òpùL2¡ü ∞ þ û min û mó with zero∞ andunity 0, thealgebraicpathproblemistheclassicalshortestpathproblem. In this case,it is commonto defineµ ò i û j óD÷ ∞ if ò i û j óMø A.

Definition: A semiringis calledcompleteif thefollowing conditionsarefulfilled:' Ji I

ai ø R is well definedfor countablesetsI , ai ø R for all i ø I

' Ji I

ai ÷NJj J

OP Ji I j

ai QR for partitions ò I jû j ø J ó of I

' b BTS Ji I ai U ÷VJ

i I

ò b B ai ó and S Ji I

ai U B b ÷VJi I

ò aiB bó for all b ø R

C.1 ClassicalShortestPath Problem

Wewill now considertheclassicalshortestpathproblemwith possiblenegativearccostsin thesemir-ing òpùN24ü ∞ þ û min û mó . If Pi j ÷ /0 andif G doesnot containcircuits of negative weight, thenthereexists a shortestpathfrom i to j. Shortestpathproblems(with possiblynegative arc costs)canbesolvedby oneof theclassicallabelcorrectingmethodsdescribed,for example,in [1]. Thosemethodseithersolve theproblemor detecta circuit with negative weight in polynomialtime (notethat in thiscasethereis no solutionfor theproblem).

Notethatfor shortestpathweightsthetriangle inequality

µ ò i û j ó B µ

ò j û k óW( µ ò i û k ó for all i û j û k ø N

is valid. If onewishesto have all shortestpathsfrom onesourcenodeu to all othernodesv of thegraph,this canbedescribedby a shortestpathtree, which is a treewith root u whereevery uniquelydeterminedpathfrom u to anothernodev is ashortestpathfrom u to v.

C.1. CLASSICAL SHORTESTPATH PROBLEM 113

ShortestPath Algorithms

We will briefly describelabelsettingor labelcorrectingalgorithmsto calculateshortestpathsfrom afixednode1 to all othernodesof agraph.Thesealgorithmsarebasedon iterationschemesandassigna labelλ ò i ó to eachnodei. During an iteration,λ ò i ó givesthe lengthof thebestpathfrom node1 tonodei foundsofar. Initially, thelabelsaredefinedby λ ò 1óD÷ 0 andλ ò i óD÷ ∞ for all nodesi ÷ 1.

In eachiterationof the algorithms,a setof nodes(the so calledcandidatelist L) is scanned,whichmeansthatfor eachnodei in this list, all nodesj with anarca : i ÿ j areinvestigated.If

λ ò j óõ λ ò i ó B µaû

thepathlengthfrom node1 to node j canbeimprovedby combiningthepathto nodei with lengthλ ò i ówith thearca : i ÿ j of lengthµa (wherecombiningmeanstheuseof thesemiringoperation“ B ”).In this case,λ ò j ó is setto λ ò i ó B µa. After an iteration,a new candidatelist is obtainedby thesetofimprovednodes.Initially thecandidatelist only containstherootnode1.

GenerallabelcorrectingmethodschooseasublistL ö- L to bescanned.TheiterationprocessstopsifL ÷ /0.

If therearenegative arc costs,the Bellman-Ford algorithm shouldbe used(algorithm C.1, whichcan be easily adaptedin order to calculatenot only the shortestpath weights,but also the pathsthemselves), whereat eachiteration, the completelist L ö : ÷ L is scanned.If after n iterationsthecandidatelist is notempty, theshortestpathproblemis notsoluble,i.e. thegraphcontainsacircuit ofnegative length.

Algorithm C.1 Bellman-Ford Algorithmλ ò 1ó : ÷ µ1 j for each j øJü 1 û n þL : ÷ü 1 þfor k ÷ 1 to n do

L : ÷ /0for eachl ø L do

for eacharca : l ÿ j doif λ ò j ó@õ λ ò l ó B µa then

λ ò j ó : ÷ λ ò l ó B µa

L : ÷ L 2Hü j þend if

end forend forL : ÷ L

end forif L ÷ /0 then

Stop.Shortestpathweightsfrom node1 to all othernodeshave beencalculated.end ifStop.Thegraphcontainsacircuit with negative weight.


If all arcshave a non-negative costµa ( 0, theshortestpathproblemis soluble,andthelabelsettingmethodproposedby Dijkstra (algorithm C.2, which againcan be adaptedto determinethe corre-spondingpaths)is a very efficient algorithm. During eachstep,thescanlist L ö : ÷+ü i áþX L containsexactly thenodei ø L with minimumlabelλ ò i có : ÷ min ü λ ò i ó ý i ø L þ . Theassumptionthatall arccostsarenon-negativeguaranteesthattheshortestpathfrom node1 to nodei hasalreadybeenfound,i.e. λ ò i có@÷ µ ò 1 û i có . Thus,if we areonly interestedin theshortestpathfrom node1 to a fixedgoalnode,we donothave to run thealgorithmuntil L ÷ /0, but maystopwhenever thegoalnodehasbeenselectedasscannode.

Algorithm C.2 Dijkstra’s Algorithmλ ò j ó : ÷ µ1 j for eachj øJü 1 ûû nþN : ÷ V /ü 1 þloop

Determinek ø N with λ ò k óD÷ min ü λ ò j ó ý j ø N þ .N : ÷ N /ü k þ / Y L ö : ÷ü k þZY /if N ÷ /0 then

Stop.Shortestpathweightsfrom node1 to all othernodeshave beencalculatedend ifλ ò j ó : ÷ min ü λ ò j ó û λ ò k ó B µkj þ for all j ø N

end loop

C.2 Gauss-Jordan Method

For a completesemiring,the generalizedGauss-Jordan method(algorithmC.3) canbe usedto de-termineshortestpathweightsfor all pairsof nodessimultaneously(cf. [63]). For anelementa of asemiring,define

a

: ÷ 1 ? a ? ò a B aó[? ò a B a B aó[? Algorithm C.3 GeneralizedGauss-JordanMethod

for each ò i û j ó@ø V @ V doSetMi j : ÷ 1 if i ÷ j andMi j : ÷\J

a:i ] j

µa otherwise

end forfor k ÷ 1 to n do

Mkk : ÷ M kk

Mik : ÷ MikB Mkk for all i ÷ k

Mkj : ÷ MkkB Mkj for all j ÷ k

Mi j : ÷ Mi j ? ò MikB Mkj ó for all i û j ÷ k

end forStop.Shortestpathweightsaregivenby µ ò i û j óB÷ Mi j

C.3. FEASIBLEDIFFERENTIAL PROBLEM 115

C.3 FeasibleDiffer ential Problem

Let G ÷êò V û Aó bea directedgraph,whereeacharca ø A is associatedwith a span ^ la û ua _ (in a non-periodicsense).Thefeasibledifferential problem(FDP) is to find apotentialπ suchthat

π j πi ø^ la û ua _ for eacha : i ÿ j

Thisproblemhasbeenexaminedin [53].

An FDPinstancecanbesolvedasa shortestpathprobleminstanceon a specialgraphGö÷tò V ö û Aö ó ,whichis constructedin thefollowing way:V ö¯÷ V, Aöá÷ A 2:ü aö : j ÿ i ý a : i ÿ j ø A þ . If a : i ÿ j ø A,thearcaö : j ÿ i is calledcounterarc of a. Eacharcis assigneda lengthµa with

µa : ÷a` ua if a ø A la if a is acounterarc.

Now theFDPinstanceis feasibleif andonly if thereexistsapotentialb for Gö with

π j πi

ôµa for eacha : i ÿ j ø Aö (C.1)

Let ashortestpathfrom anarbitrarynode,saynode1, to all othernodesof Gö exist (with µa1B µa2 : ÷

µa1 µa2 for a1û a2 ø Aö ). In this case,Gö doesnot containnegative circuit. Theinequality

µ ò 1 û j ó ô µ

ò 1 û i ó[ µaû

which is valid for every arc a : i ÿ j of Gö shows that the potentialdefinedby πi : ÷ µ ò 1 û i ó fulfillsinequality(C.1). Conversely, if thereis a negative circuit in Gö , inequality(C.1) cannotbe satisfiedfor thearcsof thatcircuit.

Consideracircuit with incidencevector cö of Gö . By traversingeachcounterarcin negativedirection,this circuit canbe uniquelydescribedby a cycle in G. Let ced (cgf ) denotethe incidencevectorsofthesetof arcsof this cycle which aretraversedin positive (negative) direction.Thenthepathlengthof thecircuit canbeexpressedby h

T c ö ÷ uT c d lT c f Therefore,theshortestpathproblemin Gö is solvableif andonly if for all cyclesin G (with incidencevectorsced and cgf asdescribedabove),

uT c d lT c f ( 0 (C.2)

Whensolving PESPinstances,the following subproblemplaysan importantrole: Supposethat wehave a solution b of anFDPinstancewith spansda

û da _ for eacharca andexactly oneof theboundsfor onearc hasto be modified. The tasknow is to find a feasiblepotentialfor the new spansor toprove infeasibility.

Assumethatthelowerboundof q : u ÿ v wasraisedfrom dq to d öq õ dq. If πv πu ( d öq, thepotential

is still feasiblefor this tightenedproblem.Now supposeπv πu 4 d öq. In this case,we have to raise

thetensionxq ÷ πv πu at leastfor theamountδ : ÷ d öq πv πu.


Consideragainthe bi-directedgraphGö anddefineanotherarc lengthµöa for eacharc a : i ÿ j byµöa : ÷ da

π j πi if a ø A andµöa : ÷ da π j πi if a is a counterarc(with µöa1

B µöa2: ÷ µöa1

µöa2).

For all arcsa ø A û a ÷ q thearclengthfor a andits counterarcarepositive,andthereforetheDijkstraalgorithm(algorithmC.2)canbeappliedto find ashortestpathfrom u to v in Gö .Supposethatduringtheiterationprocessthegoalnodev hasnot yet beenselectedasscannode,andlet the currentscannodebe i . If λ ò i ói( δ, thenµö ò u û vó)( µö ò u û i ó ÷ λ ò i óM( δ. A simplecasediscussionshows, thatthenthemodifiedpotential

π öi : ÷a` πi λ ò i ó δ if λ ò i óg4 δπi otherwise

is a feasiblepotentialfor the modifiedFDP instance.If nodev is labeledwith λ ò vój4 δ, thenthereexistsa negative circuit for weightµ in for Gö , andthemodifiedFDPinstanceis infeasible:Considerthe circuit consistingof the shortestpath ò a1

ûû ar ó for weight µö from nodeu to nodev and thecounterarcfor q. Weknow that

r

∑k k 1

ak:i l j m A µöak r

∑k k 1

ak:i l j counterarc

µöak4 δ

r

∑k k 1

ak:i l j m A dak π j πi r

∑k k 1

ak counterarc to a:i l j

da π j πi 4 dq

πv πu

πu πv r

∑k k 1

ak:i l j m A dak r

∑k k 1

ak counterarc to a

da 4 dq πv πu

ûwhich is a contradictionto (C.2) In this case,thearcsof thecircuit (or thecorrespondingcycle arcsin G) arecalledblocking arcs.

The modified Dijkstra algorithm (which will stop as soonas a negative circuit hasbeenfound orλ ò i óg( δ for a nodei ) will bedenotedby Dij lower ò δ û d û d û b û c@ó for thetensionloweringversionandby Dij raise ò δ û d û d û b û c@ó for thetensionraisingversion.

List of Algorithms

3.1 ConstraintPropagationfor Preprocessing . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Odijk’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Voorhoeve’s ConstraintPropagationAlgorithm . . . . . . . . . . . . . . . . . . . . 44

3.4 GeneralizedSerafini-Ukovich Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 ChoosingaChordin theGeneralizedSerafini-Ukovich Algorithm . . . . . . . . . . 50

3.6 ChoosingaChordwhenw õ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7 Minimizing FractionalValues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8 Branch-and-CutMethodwith FDPRelaxation. . . . . . . . . . . . . . . . . . . . . 64

4.1 RelaxationIterationAlgorithm for theMCSP . . . . . . . . . . . . . . . . . . . . . 68

4.2 SimpleBranch-and-BoundAlgorithm for theMCSP . . . . . . . . . . . . . . . . . 69

4.3 ImprovedBranch-and-BoundAlgorithm for theMCSP . . . . . . . . . . . . . . . . 70

4.4 ExactRelaxationIterationAlgorithm for theN-MCSP . . . . . . . . . . . . . . . . 83

4.5 ExactBranch-and-BoundAlgorithm for theN-MCSP . . . . . . . . . . . . . . . . . 84

5.1 DeterminingLinesandStationsfor TrainChangeTimeConstraints. . . . . . . . . . 92

B.1 RelaxationIterationAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.2 Branch-and-BoundAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

C.1 Bellman-Ford Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

C.2 Dijkstra’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.3 GeneralizedGauss-JordanMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

117

118 LIST OF ALGORITHMS

List of Symbols

NP Complexity class 102

P Complexity class 101

An Arc setof aneventgraph 14

Cfix Fixedcostpermotorunit 22

CfixC Fixedcostpercoach 22

Ckm Costperkm permotorunit 22

CkmC Costperkm percoach 22

Cfixτ Fixedcostpermotorunit of typeτ 24

CfixCτ Fixedcostpercoachof typeτ 24

Ckmτ Costperkm permotorunit of typeτ 24

CkmCτ Costperkm percoachof typeτ 24

E Setof edges 9

G Graph 9

Ne Numberof travelers(passengerdemand)on edgee 22

Per Periodicsets 36

T Basictimeperiod 9

V Setof nodes 9

Vn Nodesetof aneventgraph 14

W Minimal numberof coachespertrain 22

W Maximalnumberof coachespertrain 22

Wτ Minimal numberof coachespertrain of typeτ 24

Wτ Maximalnumberof coachespertrain of typeτ 24oSetof periodicinterval constraints 13pSetof periodicevents 10p 0 Setof correspondingindividual eventswith index 0 for

p10

ˆp Setof individual events 10qr Setof possiblefrequenciesfor line r 20r

Eventgraphof aPESPinstance 14

119

120 LIST OF SYMBOLSsSetof joining constraints 18tUnboundedtimetablepolyhedron 39,52tguBoundedtimetablepolyhedron 52vSetof lines 9

ˆv Setof linescausinginfeasibilityof anFSPinstance 67v P Setof possiblelines 20wSetof train types 24

or: Spanningtreefor aPESPinstance 34wr Setof feasibletrain typesfor line r 24x

Setof train changetimeconstraints 91ySetof feasiblemoduloparameters 39,52y uSetof feasiblemoduloparameterswith zeroon aspanningtree 41,52zCapacityof onecoach 22z

τ Capacityof onecoachof train typeτ 24 n Nodearcincidencematrix for PESPeventgraphr

34

avr # µ Arrival of line r, directionµ, atnodev 10

dr Lengthof circulationof line r 20

dvr # µ Departureof line r, directionµ, at nodev 10

l f re

Minimal line frequency onedgee 22

l f re Maximal line frequency on edgee 22

tr Circulationtime for a trainof line r 20

tr Estimatedcirculationtime for a trainof line r 20

tr # τ Estimatedcirculationtime for line r with train typeτ 24

travvvτ Minimum travel time for trainsof typeτ from v to vö 24

travvvτ Maximumtravel time for trainsof typeτ from v to vö 24

turn Minimum turnaroundtime 24

turn Maximumturnaroundtime 24

wr Numberof coachesof trainsof line r 21

wr # f Numberof coachesof trainsof frequency f for line r 21

wr # τ Numberof coachesof typeτ for trainsof line r 24

wr # f # c Line r is usedwith frequency f andc coaches(indicatorvariable) 23

wait Minimum waiting time 24

wait Maximumwaiting time 24

xr Frequency of line r 21

xr # f Line r is usedwith frequency f (indicatorvariable) 21

xr # τ Line r is usedwith train typeτ 24

121

Γ Network matrixof agraph 34

Π ò z ó Setof feasiblepotentialsfor fixedmoduloparameters 39

γr Estimatednumberof traincompositionsrequiredfor line r 20|k Columnbelongingto nodek in thetransposedincidencematrix 34

µ ò i û j ó Shortestpathweightfrom nodei to node j 112

π Schedulefor periodicevents 10

π Schedulefor individual events 9^ a û b_ T Periodicextensionof theinterval ^ a û b_ with periodT 10,34

p d , p f Positive andnegative partof avector 33

a b modT Modulo operation 14ò~ ó mod t Modulo projection 35

a : i ÿ j a is anarcfrom i to j 33

Dij lower , Dij raise ModifiedDijkstra procedures 116

O ò f ò nó(ó Complexity class 101

122 LIST OF SYMBOLS

Bibliography

[1] R.K. Ahuja, T.L. MagnantiandJ.B.Orlin. NetworkFlows— Theory, Algorithms,andAppli-cations. PrenticeHall, 1993.

[2] A.A. Assad.Models for Rail Transportation.TransportationResearch Part A 14, 205–220,1980.

[3] P. Baron. Transportationin Germany: A Historical Overview. Transportation ResearchPart A 29,9–20,1995.

[4] J.H.A. Van den Berg and M.A. Odijk. DONS: ComputerAided Designof Regular ServiceTimetables.In Proceedingsof CompRail94, Madrid,1994.

[5] R.E.Bixby. Privatecommunication,1999.

[6] U. Brannlund,P.O. Lindberg, A. Nou, J.-E.Nilsson.Railway TimetablingusingLagrangianRelaxation.TransportationScience32 (4), 358–369,1998.

[7] H. Braker. AlgorithmsandApplicationsin TimedDiscreteEventSystems. PhDthesis,Univer-sity Delft. 1993.

[8] M.R. Bussieck,P. KreuzerandU.T. Zimmermann.Optimallinesfor railwaysystems.EuropeanJ. Oper. Res., 96,54–63,1996.

[9] M.R. Bussieck,T. WinterandU.T. Zimmermann.Discreteoptimizationin publicrail transport.Math.Programming, 79 (1–3),415–444,1997.

[10] M.R. Bussieck.Optimal Lines in Public Rail Transport.PhD thesis,TechnischeUniversitatBraunschweig,1998.

[11] A. Caprara,M. Fischetti,P. Toth, D. Vigo andP.L. Guida.Algorithms for railway crew man-agement.Math.Programming, 79 (1–3),125–141,1997.

[12] M. Carey. A modelandstrategy for train pathingwith choiceof lines,platforms,androutes.TransportationResearch B 28 (5), 333–353,1994.

[13] M. Carey. Extendinga train pathingmodel from one-way to two-way track. TransportationResearch B 28 (5), 395–400,1994.

[14] T. Christof.Ein VerfahrenzurTransformationzwischenPolyederdarstellungen. Master’s thesis.Universitat Augsburg, 1991.

123

124 BIBLIOGRAPHY

[15] V. Chvatal.Linearprogramming. FreemanandCompany, 1983.

[16] M.T. Claessens.De kost-lijnvoering.Master’s thesis,Universityof Amsterdam,1994.

[17] M.T. Claessens,N.M. vanDijk andP.J.Zwaneveld. Costoptimalallocationof rail passengerlines.EuropeanJ. Oper. Res., 110(3) 474–489,1998.

[18] J.-F. Cordeau,P. Toth andD. Vigo. A Survey of OptimizationModelsfor Train RoutingandScheduling.TransportationScience, 32 (4), 380–404,1998.

[19] J.R. Dadunaand S. Voß. Practical Experiencesin ScheduleSynchronization.Computer-AidedTransitScheduling:Proceedingsof the6th InternationalWorkshopon Computer-AidedSchedulingof Public Transport, LectureNotesin EconomicsandMathematicalSystems,vol-ume430,1995.

[20] DeutscheBahn.Geschaftsbericht1998.

[21] DeutscheBahn.Umweltbericht1998.

[22] W. Filz. Erste Fahrplankonferenzder deutschenEisenbahnen(20.4.1871).Zeitzeichen 35-960420(manuscriptfor radiotransmission),WestdeutscherRundfunk,1996.

[23] GAMS DevelopmentCorp.GAMS— A User’s Guide, 1998.

[24] GAMS DevelopmentCorp.GAMS— TheSolverManuals, 1999.

[25] M.R. Garey andD.S.Johnson.Computers andIntractability — A Guideto theTheoryof NP-Completeness. FreemanandCompany, 1979.

[26] M. Grotschel,A. Lobel and M. Volker. OptimierungdesFahrzeugumlaufsim offentlichenNahverkehr. In K.-H. Hoffmann,W. Jager, T. LohmannandH. Schnuck(editors),Mathematik– Schlusseltechnologie fur die Zukunft, Springer, 1997.

[27] R. Gobertshahn.Der integraleTaktfahrplan– FundamentderneuenNahverkehrsstrategievonDeutscherBundesbahnundDeutscherReichsbahn.Die DeutscheBahn5/93,357–362,1993.

[28] R.E. Gomory. Outline of an algorithm for integer solutionsof linear programs.Bull. Amer.Math.Soc., 64,275–278,1958.

[29] M. Grotschel,L. LovaszandA. Schrijver. GeometricAlgorithmsandCombinatorialOptimiza-tion. Springer, Berlin, 1988.

[30] I. GertsbakhandP. Serafini.PeriodicTransportationScheduleswith Flexible DepartureTimes.EuropeanJ. Oper. Res., 50,298–309,1991.

[31] P. Heusch,F. Meisgenand E. Speckenmeyer. CATS — ComputerAided Tram Scheduling.ReportNo. 97-262,Universitat zu Koln, 1997.

[32] A. Higgins,E. KozanandL. Ferreira.Optimal Schedulingof Trainson a SingleLine Track.TransportationResearch B, 30 (2), 147–161,1996.

BIBLIOGRAPHY 125

[33] J.S.Hooghiemstra.Designof regularinterval timetablesfor strategic andtacticalrailwayplan-ning.In J.Allan, C.A. Brebbia,R.J.Hill, G. Sciutto,andS.Sone,editors.RailwaySystemsandManagement, volume1 of Computers in RailwaysV. ComputationalMechanicsPublications,1996.

[34] ILOG. ILOG CPLEX6.5ReferenceManual. 1999.

[35] E.L. Johnson,M.M. KostrevaandU.H. Suhl.Solving0-1-IntegerProgrammingProblemsAris-ing from LargeScalePlanningModels.Oper. Res., 33,803-819,1985.

[36] M. Kolonko, K. NachtigallandS.Voget.ExponatderUniversitatHildesheimaufderCeBit96:Optimierung von Taktfahrplanen mit genetischenAlgorithmen. HildesheimerInformatik-Berichte, 8/96,1996.

[37] M. Krista.VerfahrenzurFahrplanoptimierungdargestelltamBeispielderSynchronzeiten.PhDthesis,TechnischeUniversitat Braunschweig,1996.

[38] L.G. Kroon and P.J. Zwaneveld. Routing trains throughrailway stationsincluding shuntingdecisions.In: P. Kleinschmidt,A. Bachem,U. Derigs,D. Fischer, U. Leopold-Wildburger andR. Mohring,editors:OperationsResearch Proceedings1995, 438–444,Springer, Berlin, 1995.

[39] A. Lobel.Optimalvehicleschedulingin public transit. PhDthesis,TU Berlin, 1997.

[40] M.J. Maher. Inferenceson trip matricesfrom observationson link volumes:A Bayesianstatis-tical approach.TransportationResearch B, 17,435–447,1983.

[41] K. Nachtigall.A BranchandCut Approachfor PeriodicNetwork Programming.HildesheimerInformatik-Berichte, 29/94,1994.

[42] K. Nachtigall.CuttingPlanesfor aPolyhedronAssociatedwith aPeriodicNetwork. TechnicalReportIB 112-96/17,DLR, Braunschweig,1996.

[43] K. Nachtigall.HasPESPanError?1997.

[44] K. NachtigallandS.Voget.Minimizing WaitingTimesin IntegratedFixedInterval Timetablesby upgradingRailway Tracks.EuropeanJ. Oper. Res., 103,610–627,1997.

[45] K. Nachtigall.PeriodicNetwork OptimizationandFixed Interval Timetables.Habilitationss-chrift. Universitat Hildesheim,1998.

[46] G.L. Nemhauserand L.A. Wolsey. Integer and Combinatorial Optimization. John Wi-ley & Sons,1988.

[47] M.A. Odijk. Constructionof PeriodicTimetables— Part I: A CuttingPlaneAlgorithm. Tech-nical Report94-61,Departmentof MathematicsandComputerScience.Delft University ofTechnology, 1994.

[48] M.A. Odijk. Constructionof PeriodicTimetables— Part II: An Application.TechnicalReport94-71, Departmentof Mathematicsand ComputerScience.Delft University of Technology,1994.

126 BIBLIOGRAPHY

[49] M.A. Odijk. A constraintgenerationalgorithmfor theconstructionof periodicrailway timeta-bles.TransportationResearch B, 30 (6), 1996.

[50] C. Oltrogge. Linienplanungfur mehrstufige Bedienungssystemeim offentlichen Personen-verkehr. PhDthesis,TechnischeUniversitat Braunschweig,1994.

[51] C.H.PapadimitriouandK. Steiglitz.CombinatorialOptimization—AlgorithmsandComplex-ity. PrenticeHall, 1982.

[52] C. Poppe.Fahrplan-Algebra.SpektrumderWissenschaft 11,18–21,1995.

[53] R.T. Rockafellar. NetworkFlowsandMonotropic Optimization. JohnWiley & Sons,1984.

[54] M.W.P. Savelsbergh. Preprocessingand probing techniquesfor mixed integer programmingproblems.ORSAJ. Comput., 6 (4), 445–454,1994.

[55] A. Schrijver. Theoryof LinearandInteger Programming. JohnWiley & Sons,1986.

[56] A. Schrijver andA. Steenbeek.Dienstregelingontwikkeling voor Railned.TechnicalReport,Centrumvoor WiskundeenInformatica,1994.

[57] A. Schrijver. Routing and Timetabling by Topological Search.ProceedingsInternationalCongressof Mathematicians, Berlin, 1998.

[58] H.D. Sherali,R. Sivanandan,A.G. Hobeika.A linearprogrammingapproachfor synthesizingorigin-destinationtrip tablesfrom link traffic volumes.TransportationResearch B, 28, 213–233,1994.

[59] P. SerafiniandW. Ukovich. A mathematicalmodel for periodicschedulingproblems.SIAMJ. Disc.Math.2 (4), 1989.

[60] P. SerafiniandW. Ukovich. A mathematicalmodelfor thefixed-timetraffic controlproblem.EuropeanJ. Oper. Res.42,152–165,1989.

[61] S.Thienel.ABACUS—A Branch-And-CUtSystem. PhDThesis,Universitat Koln.

[62] M. Voorhoeve. Rail Schedulingwith DiscreteSets.Unpublishedreport,EindhovenUniversityof Technology, TheNetherlands,1993.

[63] U.T. Zimmermann.Linear andCombinatorialOptimizationin OrderedAlgebraic Structures.Number10 in Annalsof DiscreteMathematics,1981.

Index

adjacency value,50algebraicpathproblem,112arrival time,6

basictimeperiod,6blockingarc,45,116branch-and-boundalgorithm,106

certificate,102chain,33

elementary, 33chaincuttingplane,55changeof trains,4chord,seenon-treearccircuit, 33co-treearc,seenon-treearccoefficient reduction,108complexity, 101cost

of a linefixed,20perkm, 20

of anarc,111counterarc,115crew planning,6cuttingplane,105cuttingplanealgorithm,104cycle,33cyclecuttingplane,40cyclomaticnumber, 34

decisionproblem,101departuretime,6dimension

of a polyhedron,103division,106

endpoint

of achain,33of anarc,33

enumerative algorithm,106event,9eventgraph,14eventtime,9exponentialtime algorithm,101extremepoint,103

faceof apolyhedron,103

facet,103feasibledifferentialproblem,115feasiblepotential,14feasibleregion,seefeasiblesetfeasiblescheduleproblem,66feasibleset,103feasibletension,14fixedinterval schedule,7fractionalcuttingplanealgorithm,105FSP, seefeasiblescheduleproblem

graphconnected,34

headway, 7

incidencematrix,34incidencevector, 33individual event,9integerlinearprogram,103integratedfixedinterval schedule,7

joinedconstraints,17JPESP, seePESPwith joinedconstraintsjunction,7

kernelo b-kernel,58

127

128 INDEX

lengthof anarc,seecostof anarc

line, 9line planning,5linearprogram,103look-aheadvalue,49

MCSP, seeminimumcostschedulingproblemMCTP, seeminimumcosttypeproblemminimumcostschedulingproblem,24minimumcosttypeproblem,26mixedintegerlinearprogram,103moduloparameter, 14monoid,111

non-treearc,34NP-complete,102

objective function,103OD-matrix,seeorigin destinationmatrixoperationalplanning,3orientation

in a chain,33origin destinationmatrix,4, 91

partition,106passengerdemand,4path,33period,seebasictimeperiodperiodicevent,9periodiceventschedulingproblem,13periodicextension,34periodicinterval constraint,10periodicschedule,6, 7periodicset,36PESP, seeperiodiceventschedulingproblemPESPwith joinedconstraints,17polyhedron,103polynomialtimealgorithm,101polytope,103potential,14

relaxation,105relaxationiterationalgorithm,105representative trains,16resolution,6

rolling stock,6runningtime

of analgorithm,101

schedulefor individual events,9for periodicevents,10

scheduleplanning,6semigroup,111semiring,111separationproblem,104shortestpathproblem,111shortestpathtree,112size

of aprobleminstance,101span,34spanlength,34spanningtree,34

minimum,34strongbranching,108supplynetwork, 6

tacticalplanning,3tension,14timespacediagram,7timetablepolyhedron

bounded,52unbounded,39,52

traffic volume,seepassengerdemandtraincomposition,5train type,23triangleinequality, 112trip, 6

valid inequality, 103variablefixing, 108

DeutscheZusammenfassung

Die vorliegendeArbeit befaßtsichmit FahrplanoptimierungunterBerucksichtigungderVerhaltnissebeim spurgefuhrten,offentlichenPersonenverkehr. Insbesonderewird davon ausgegangen,daßderFahrplansichnacheinerbestimmtenZeitperiode(z.B. eineStunde)wiederholensoll.

EinFahrplanbestehtausdenAnkunfts-undAbfahrtzeitendereinzelnenVerkehrslinienanbestimmtenPunktenim Verkehrsnetz,etwa denBahnhofen beim Eisenbahn-Fernverkehr. FahrplanelassensichnachunterschiedlichenKriterien bewerten. Im Mittelpunkt dieserArbeit stehtdie Minimierung derdurcheinenFahrplanentstehendenBetriebskostenfur die Fahrzeuge.

In Kapitel1 wird dieFahrplanerstellungalsTeil derVerkehrsplanungdargestellt.DiesePlanungwirdnormalerweiseals hierarchischerProzeßbetrachtet. Die einzelnenTeilaufgabenwie etwa Linien-planung,FahrplanungoderPersonaleinsatzplanung, werdenin demKapitel vorgestellt,undeswirdaufgezeigt,wie siesichgegenseitigbeeinflussen.

Kapitel 2 stellt mathematischeModelle zur Fahrplanerstellungvor. Eine zentraleBedeutunginner-halbdieserArbeit kommtdabeidemsogenanntenPeriodicEventSchedulingProblem(PESP)zu,dasim Jahr1989von Serafiniund Ukovich eingefuhrt wurde. DasPESPist ein Zulassigkeitsproblem,berucksichtigtalsokeineOptimierungsaspekte.Weiterhinwerdenin demKapitel ausder LiteraturbekannteFahrplanbewertungsansatzeerlautert.EinneuesModell zurkostenoptimalenFahrplangestal-tung, dassogenannteMinimumCostSchedulingProblem(MCSP), wird entwickelt. Es kombiniertIdeendesPESPsmit einemvon Claessensim Jahr1994vorgeschlagenenKostenkonzeptzur Lin-ienoptimierung.DasMCSPlaßtsich als gemischt-ganzzahligeslinearesProgrammdarstellen.Da-ruberhinausenthalt Kapitel2 Ergebnissezur Komplexitat desPESPsunddesMCSPs.

Das PESPwird in Kapitel 3 genaueruntersucht. Es werdenausder Literatur bekannteLosungs-algorithmenvorgestellt.DurcheinigeModifikationenandenVerfahrenlaßtsichdie LosungszeitfurausPraxissichtrelevanteProbleminstanzgroßendeutlichverkurzen.Desweiterenenthalt dasKapitelneueResultatein BezugaufdiepolyedrischeStrukturdesPESPs.Mit Hilfe dieserErgebnissewird einneuesBranch-and-Cut-Verfahrenzur Bearbeitungvon PESP-Instanzenentwickelt, dasnocheinmaleinewesentlicheBeschleunigungdesLosungsvorgangsermoglicht.

Eine direkteLosungder gemischt-ganzzahligen linearenProgrammefur interessanteVerkehrsnetz-großenmittels kommerziellerSoftwareerwiessichaufgrundzu langerRechenzeitenundzu hohemSpeicherbedarf– selbstbei massivem Hardwareeinsatz– als nicht moglich. In Kapitel 4 wird eineDekompositionsideebeschriebenundsowohl in ein Schnittebenenverfahrenals auchin ein Branch-and-Bound-Verfahrenintegriert. Als Teilproblemetretenin jeder Iteration bzw. in jedemKnotenPESP-ahnlicheProblemeauf. Mit den VerfahrenausdiesemKapitel konnenin akzeptablerZeit

Losungenvonhoher, beweisbarerQualitatgeneriertwerden.Fur kleinereVerkehrsnetzeist sogareineexakteOptimierungmoglich. DasKapitel endetmit der Betrachtungeinesnichtlinearengemischt-ganzzahligenModells,dasdieFahrplankostennochetwasgenauerberechnet.Fur diesesModell wirdein exakterLosungsalgorithmusangegeben,derallerdingsfur praktischeProblemgroßenzu langsamist.

In Kapitel 5 werdenRechenergebnissefur die in dieserArbeit vorgestelltenneuenVerfahrenprasen-tiert. Dazuwurdenvon denEisenbahnbetreibernDeutsche BahnAG und NederlandseSpoorwegenPraxisdatenzur Verfugunggestellt.

DasletzteKapitelderArbeit enthalt Anregungenfur diemathematischeBearbeitungsehrgroßerProb-leminstanzen.Weiterhinwird ein Ausblick auf zukunftigeModelleundMethodenzur Fahrplanopti-mierunggegeben.

Lebenslauf

Name: ThomasLindner

Geburtsdatum: 14.Oktober1972

Geburtsort: Wolfenbuttel

Familienstand: verheiratet

Staatsangehorigkeit: deutsch

Bildungsgang: 1979–1992 BesuchderGrundschule,Orientierungsstufeund

desGymnasiums

1992–1993 Zivildienst

1993–1997 StudiumderMathematik(Dipl.) ander

TU Braunschweig

Beschaftigungszeiten: 1995–1997 studentischeHilfskraft in derAbteilung

MathematischeOptimierungderTU Braunschweig

1997–2000 wissenschaftlicherMitarbeiterin derAbteilung

MathematischeOptimierungderTU Braunschweig

Documents

Train Schedule Optimization in Public Rail Transport · Preface This thesis deals with train scheduling problems with an emphasis on public rail transport. In partic-ular, we assume