Integrated timetable rescheduling and passenger reassignment during railway disruptions

Integrated timetable rescheduling and passenger reassignment during railway disruptions

Zhu, Y.; Goverde, R.M.P.

DOI

10.1016/j.trb.2020.09.001

Publication date

2020

Document Version

Final published version

Published in

Transportation Research. Part B: Methodological

Citation (APA)

Zhu, Y., & Goverde, R. M. P. (2020). Integrated timetable rescheduling and passenger reassignment during

railway disruptions. Transportation Research. Part B: Methodological, 140, 282-314.

https://doi.org/10.1016/j.trb.2020.09.001

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ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Integrated

timetable

rescheduling

and

passenger

reassignment

during

railway

disruptions

Yongqiu

Zhu

∗

,

Rob M.P.

Goverde

Department of Transport and Planning, Delft University of Technology, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 17 July 2019 Revised 18 August 2020 Accepted 1 September 2020 Keywords: Railways Disruption management Timetable rescheduling Passenger reassignment Fix-and-optimize algorithm

a

b

s

t

r

a

c

t

During railwaydisruptions,mostpassengers may not beableto findpreferred alterna-tivetrainservicesduetothecurrentwayofhandlingdisruptionsthatdoesnottake pas-sengerresponsesintoaccount.Toofferbetteralternativestopassengers,thispaper pro-posesanovelpassenger-orientedtimetablereschedulingmodel,whichintegratestimetable reschedulingandpassengerreassignmentintoaMixedIntegerLinearProgrammingmodel withtheobjectiveofminimizinggeneralizedtraveltimes:in-vehicletimes,waitingtimes atorigin/transferstationsandthenumberoftransfers.Themodelappliesthedispatching measuresofre-timing,re-ordering,cancelling,flexiblestoppingandflexibleshort-turning trains, handles rolling stockcirculations at bothshort-turning and terminal stations of trains,andtakesstationcapacityintoaccount.Tosolvethemodelefficiently,anAdapted Fix-and-Optimize(AFaO)algorithmisdeveloped.Numericalexperimentswerecarriedout toapart oftheDutchrailways.The resultsshow that theproposed passenger-oriented timetablereschedulingmodelisabletoshortengeneralizedtraveltimessignificantly com-pared toanoperator-orientedtimetablereschedulingmodel thatdoesnotconsider pas-sengerresponses. Byallowing only10minmore traindelaythanan optimal operator-orientedreschedulingsolution,thepassenger-orientedmodel isabletoshortenthe gen-eralized traveltimesoverallpassengers bythousands ofminutesinall considered dis-ruptionscenarios.Withapassenger-orientedrescheduledtimetable,morepassengers con-tinuetheirtraintravelsafter adisruptionstarted, comparedtoarescheduledtimetable fromthe operator-orientedmodel.The AFaOalgorithmobtains high-qualitysolutions to thepassenger-orientedmodelinupto300s.

© 2020TheAuthor(s).PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Railwaysystemsplayanimportantroleinpeople’sdailytravellingsothattheoperationsarerequiredasreliableas pos-sibletoensurepassengerpunctuality.Unfortunately,unexpecteddisruptionsoccurintherailwaysonadailybasis(Zhuand Goverde,2017),duringwhichmanytrainservicesaredelayedandcancelledthatdisturbpassengerplannedjourneys signifi-cantly.Whenreschedulingatimetableincaseofadisruption,trafficcontrollersdecidewhichserviceshavetobedelayedor cancelledintermsofpre-designedcontingencyplans,wheretheimpactonpassengersisconsideredtoaverylimitedextent

∗ _{Corresponding author.}

E-mail addresses: y.zhu-5@tudelft.nl (Y. Zhu), r.m.p.goverde@tudelft.nl (R.M.P. Goverde).

https://doi.org/10.1016/j.trb.2020.09.001

0191-2615/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

(Ghaemietal., 2017b).As a result,therescheduled trainservices maynot be passenger-friendly.Forexample,passengers mayhardly find alternative train services to reach the expecteddestinations in reasonabletravel times. Toprovide pas-sengerswithbetteralternatives duringdisruptions, itisnecessarytorescheduleatimetableinamorepassenger-oriented way.

1.1. Thestate-of-the-artontimetablerescheduling

Passenger-oriented timetable rescheduling started from the field of delay management that decides whether a train should wait for a delayed feeder train to guarantee the transfer connection of some passengers. Schöbel (2001) is the first one dealing with this problem based on the assumption that if passengers missed the transfer connections, they would wait for a complete cycletime to catch the next connection considering that the planned timetable is periodic.

Dollevoetetal.(2012)make anextension byintroducingthe possibilityofrerouting passengers whoareassumedtotake theshortestpathsfortheirfollowingtravelsincaseofmissedtransfers.Bothpapersdescribetheinfrastructureata macro-scopiclevelneglecting signalsandblock sections.Toimprovesolutionfeasibilityinpractice,Cormanetal.(2017) propose adelaymanagement modelin whichtheinfrastructure isdescribedata microscopiclevel.Albert etal.(2017) formulate passengerbehavioursinstations(e.g.queueinginboarding trains)atamicroscopiclevelto describepassengerinfluences ontraindelaysratherthanconsideringtheimpactoftraindelaysonpassengerbehavioursonly.

Delay management deals with the interaction between timetable and passengers, but not the interaction between timetableandreduced infrastructureavailability,which howevermust be takeninto account by disruptionmanagement. Operator-orienteddisruptionmanagementconsidersonlythelatterkindofinteraction,whilepassenger-orienteddisruption managementconsidersbothkindsofinteractions.Inpractice,disruptionmanagementconsistsofthreephasesstartingfrom thedisruptiveevent(failure)(Ghaemietal.,2017b).Thefirstphaseconsistsofgettinginformationaboutthedisruptionand itslocation,guaranteeingsafety,estimatingtheexpecteddurationanddecidingonthereschedulingmeasures.Inthesecond phasetherescheduledtimetableisappliedandinthethirdphasethetrafficrecoverstotheoriginaltimetable.Atpresent, thefirstphasecantakeupquitesome timedependingonhowexistingcontingencyplansneedtobe adjusted,howmany changeshavetobemadetothedispatchingplans,andhowdriverscanbeinformedofdisruptionsahead.Speedingupthis processisrequiredtoavoidqueuingofstrandedtrains.Atimelimitof5minutetocomputeareschedulingsolutionwillbe sensibleformainlinerailwaynetworksandwouldimplyabigimprovementon thecurrentpractice.Notethatthispaper handlesseriousdisruptionsofblockedtracksthatgobeyondsimplere-timingorre-orderingdecisions.

Mostliterature ondisruption management is operator-oriented,including(Ghaemietal., 2017, 2018;Meng andZhou, 2011;Veelenturf etal., 2015;Zhan etal.,2015, 2016;Zhu andGoverde,2019). Thedifferences amongthesepaperslie in theconsidered railwaylines(single-track linesordouble-track lines),the adopteddispatchingmeasures, whether consid-eringthe transition from the planned timetable to the disruption timetable andvice versa, the extent of infrastructure description(macroscopicormicroscopiclevel),thenumberofconsidereddisruptions(singledisruptionormultiple disrup-tions),and/orthecharacteristicofdisruptionlength(deterministicoruncertain).Thesimilarityamongthesepapersisthat theyalluseoperator-orientedobjectives:e.g.,minimizingtraindelaysand/orcancellations,inwhichaconstantcancellation penalty isused to representthedelay ofcancellingeach train. There area few papers that considerboth operators and passengers.Bettinelli etal.(2017)associatedispatching decisionswithdifferentpenaltiesconsidering theextents oftheir impactsonpassengers.Forexample,amajorchangeinatrainpathisassociatedwithabiggerpenality.Louwerseand Huis-man(2014)includeatermintheobjectivetobalancethenumbersofcancelledtrainsinbothdirectionstodistributethe disruptionimpactevenlyoverthedifferentpassengergroupsincaseofpartialtrackblockage.

Afew works focuson passenger-oriented disruption management. Cadarsoetal. (2013) propose a two-step approach inwhich a frequency-based passenger assignment model isperformed first toestimate thepassenger demand andthen areschedulingmodel (fortimetableandrollingstock) issolved toaccommodate thepassenger demandasmuch as pos-sible.The adopteddispatching measures are limitedto cancellingoriginal trains andinserting additionaltrains. Zhu and Goverde(2019c)adopta schedule-basedpassengerassignmentmodeltoobtainthetravel pathofeachpassengerinterms ofthe plannedtimetable.Withthisinformation, thepotential impact ofeach dispatchingdecision onpassenger planned travelsisestimated,whichisusedasweightintheobjectivetominimizingpassengerdelays.Theadopteddispatching mea-suresincludere-timing,re-ordering,cancelling,flexiblestopping(i.e.addingextrastopsandskippingscheduledstops),and flexibleshort-turning.Short-turninga trainmeansthat atrain stopsatthelast possiblestationbeforethe blockedtracks andthecorrespondingrollingstockturnsatthatstationtoservetheoppositeoperation.Flexibleshort-turningmeansthat each trainis givena full choiceof short-turningstation candidates,andthe modeldecides the optimalstationandtime ofshort-turningatrain.BothCadarsoetal.(2013) andZhuandGoverde(2019c)considerstaticpassengerdemand,which neglectthatpassengersmaychooseother travelpathsratherthantheplannedonesduetotherescheduledtrainservices. Toformulatepassengerbehaviourinamorerealisticway,itisnecessarytotakeintoaccountpassengerresponses towards therescheduled trainservices. Veelenturfet al.(2017)propose an iterativeapproach that embedsa timetable reschedul-ingmodel andapassenger assignment modelintoan iterativeframework whereateach iteration an adjustmentwillbe appliedonthetimetableifit reducesthe totalpassengerinconvenienceasevaluatedby thepassengerassignmentmodel. The adjustments are restricted to adding stops. Binder et al. (2017) propose an integrated approach of formulating the timetablereschedulingandthepassengerassignmentintoone singlemodelthat computesarescheduledtimetablebyan optimizationsolverdirectly. Theapplied dispatchingmeasures includere-timing,re-ordering,cancelling, globalre-routing

and insertingadditional trains. The rolling stockcirculations at the short-turningand terminal stations oftrains are ne-glected.Gaoetal.(2016)alsoproposeatimetablereschedulingmodelconsideringdynamicpassengerflows,whilefocusing ontherecovery phase ofa disruption.Asthe targetcaseisametro corridor,allpassengers areassumedtochoosedirect trains(i.e.notransfers).Thedispatchingmeasuresofstop-skippingandre-timingareusedtoadjustthetimetabletoreduce passengerwaitingtimesatstations.Duetothecomputationalcomplexity,themasterproblemofgeneratingarescheduled timetableisdecomposedintoaseriesofsub-problemsthat eachreschedulesone trainonly.When solvingasub-problem foronetrain,thestoppingpatternsandtimeschedulesofthepreviousconsideredtrainsareallfixed.

1.2. Thescientificgapsonpassenger-orientedtimetablerescheduling

Formulatingpassengerre-routingasamulti-commodityflowproblemisamethodcommonlyusedintheliterature.For exampleinBinder etal.(2017)andCormanetal.(2017),atimetableisformulated intoadirectedacyclicgraph(DAG)to describepassengerpathchoices.Then,thepassengerre-routingismodelledasamulti-commodityflowproblem,inwhich passengers flow through the arcs of the DAG that is updated according to the rescheduled timetable. The challengesof modellingpassengerre-routingthiswaymainlylieintwoaspects:(1)howtoformulateaDAGfromatimetabletodescribe morepathattributeswithlimitednodes/arcs,and(2)howtoreformulateaDAGdynamicallyduringtimetablerescheduling whenpassenger re-routingisintegrated. Theexisting literatureeitheruses asimplemethodofformulating aDAG,which cannot reflectcertain path attributes(e.g. the number oftransfers), oradopts a formulation method that will lead to a large-size ofDAG iffocusing on arailway networkwithhigh-frequency services. Also,limited dispatchingmeasures (e.g. no flexible stopping) are used inthe literature, which need tobe extended to explore more alternative pathchoices for passengersduringdisruptions.However,includingmoredispatchingmeasureswillincreasethecomplexityofreformulating aDAGduringtimetablerescheduling.Anotherchallengeisdesigninganefficientalgorithmtosolvetheintegratedtimetable reschedulingandpassengerre-routingmodelwithhigh-qualitysolutionsinanacceptabletime.Thishasbeenreportedasa challengingtaskintheliteraturesofar(Cormanetal.,2017;Binderetal.,2017).

1.3. Thecontributionsofthispaper

Thispapercontributestotheliterature byimprovedmethodsofDAGformulationandreformulationtoenableabetter integratedtimetablereschedulingandpassengerre-routingmodelintermsoftheconsiderationsofmultiplepathattributes andmultiple dispatchingmeasures. Thispaperalsocontributes withan efficientalgorithm tosolve theintegratedmodel withoptimalornear-optimalsolutions.Thekeycontributionsofthispaperaresummarizedasfollows.

• AnimprovedmethodofformulatingaDAG(calledanevent-activitynetworkinthispaper)fromatimetableisproposed, byexplicitlydistinguishingpassengeractivities atoriginstations,transferstations (ifany)andtrainswithouttime dis-cretization.

• Anewconcept,the transitionnetwork,isproposed toenablethedynamicformulationofevent-activitynetworks con-sideringtheimpactsofmultipledispatchingmeasures,thecharacteristicsofthedisruption,theoperationalrequirements oftrains,andthetravelrequirementsofpassengers.

• Forthefirsttime,thedispatchingmeasureofflexiblestopping(addingandskippingstops)isformulatedwithpassenger re-routinginarailwaynetwork(insteadofonecorridor)wheretransfersareallowed.

• Anadaptedfix-and-optimize(AFaO)algorithmisdesignedtoiterativelysolvetheproposedpassenger-orientedtimetable reschedulingmodel.The algorithmallowsto balancethesolutionqualityandcomputationtimeby changingtheinput parameter.

• Thepassenger-orientedtimetablereschedulingmodelisabletogeneratereschedulingsolutionswithshortergeneralized travel times than an operator-oriented model according to results of real-life instances in part of the Dutch railway network.

This paper considers single disruption that blocks tracks between stations completely assuming that the duration of the disruptionis knownatthebeginning ofthe disruption,andwill notchange overtime. We describeinfrastructure at a macroscopiclevel andhandle railwaynetworkswith bothsingle-track anddouble-track railway lines.We use the dis-patchingmeasures ofre-timing, re-ordering,cancelling, flexibleshort-turning,andflexible stoppingto computea feasible rescheduled timetablefromthe start ofa disruption untilit is fullyrecovered. Atrain isassumed tohave unlimited ca-pacity,which meansthat a passengeris ableto boardanytrain ifhe/shedecides toboardthistrain. Thisis becausewe focus onproviding better alternative trainservices to passengers so that thepossible impact ofvehicle capacityon pas-sengersisneglected. Inthisway,wecan gettheoptimalrescheduledtimetableintermsofgeneralizedtravel times.This optimalrescheduled timetablecan thenbe used asaninput to rollingstockreschedulingthat aims toaccommodate the passenger demand asmuch aspossible.Forexample, Kroon etal.(2014) andVan derHurk etal.(2018) both deal with passenger-orientedrollingstockreschedulingwitharescheduledtimetablegivenasinput.

The remainder of the paper is organized asfollows. Section 2 introduces the general framework of establishing the passenger-oriented timetablerescheduling model.Section 3 explainshow to formulate a timetableinto an event-activity network,whichisadirectedacyclicgraphwitheventsasnodesandactivitiesasarcstodescribepassengerpathchoices.A

Fig. 1. An overview of the passenger-oriented timetable rescheduling model.

pathisconstitutedbyaseriesofconnectedeventsandactivities.Theplannedtimetablecan beformulatedintoan event-activitynetwork



_plan,whichisthenextendedtoatransitionnetwork



∗thatenablesthedynamicformulationof event-activity networks duringtimetable rescheduling. A transition network is a combination of all events and activities that couldbeinanyevent-activitynetworksformulatedfromfeasiblerescheduledtimetablestowardsthedisruptionconcerned.

Section4introducesthemethodofconstructingatransitionnetwork.Basedonatransitionnetwork,thepassenger-oriented timetablereschedulingmodelisproposedinSection 5followedbySection6that introducesthemethodsofreducing the computational complexityofthe model.InSection 7, extensivenumerical experimentswere carried out toa partof the Dutchrailways.Finally,Section8concludesthepaperandpointsoutfutureresearchdirections.

2. Generalframework

Thispaperintegratestimetablereschedulingwithpassengerre-routingintoanMILPmodel,forwhichtwopreprocessing stepsareneeded.Fig.1givesanoverviewofthemodel.

The first preprocessing step transforms the planned timetable into an event-activity network



plan, which is a

di-rectedacyclicgraphusedtodescribe passengerpath choices.The methodofconstructingan event-activity networkfrom atimetableisintroduced inSection 3.Incaseofa disruption,theplannedtimetablewill becomeinfeasible,andsodoes thecorrespondingevent-activity network



plan thatnowisunable toreflectthepaths currentlyavailable intherailways.

Underthiscircumstance,thetimetablehastoberescheduled,andduringreschedulingthecorrespondingevent-activity net-workshavetobeupdatedaswelltoconsidertimetable-dependentpassengerbehaviours.Toenableadynamicevent-activity networkformulationduringtimetablerescheduling,weperformthesecondpreprocessingsteptoconstructatransition net-work



∗. Atransitionnetworkisextended fromtheevent-activity network



_plan by addingall eventsandactivitiesthat could exist in any event-activity network



dis corresponding to a feasible rescheduled timetable obtainedfor a specific

disruption.Inother words,



∗=i



idis∪



plan,where



idis refers tothe event-activitynetwork corresponding tothe ith

feasiblerescheduledtimetable.Foronespecific disruptionthereareusually multiplefeasiblerescheduledtimetables.Note that



∗varieswiththedisruptioncharacteristics (i.e.location andstarting/endingtime)andthedispatchingmeasures al-lowed.Atransitionnetwork



∗ isnotadirectedacyclicgraphasitincludesthepossibilityofchangingtheorderoftrains. ThemethodofconstructingatransitionnetworkisintroducedinSection4.

Table 1 Event attributes.

Symbol Description

st e The corresponding station of event e ∈ E \ E penal

tr e The corresponding train of event e ∈ E ar ∪ E de ∪ E dde

tl e The corresponding train line of event e ∈ E ar ∪ E de ∪ E dde

λe The corresponding departure event of e ∈ E dde

o e The scheduled time of event e ∈ E ar ∪ E de ∪ E dde

The constructed transitionnetwork,the plannedtimetable, the disruption characteristics,andthe allowed dispatching measures are all necessary inputsto establishthe passenger-orientedtimetable reschedulingmodel,which is formulated as an MILP in this paper. This model consists of the constraints for three purposes: 1) timetable rescheduling, 2) dy-namic event-activity network formulation,and3) passenger reassignment. The timetable reschedulingconstraints ensure a rescheduled timetable does not violate anyinfrastructure andoperational restrictions. The constraints relevant to the dynamicevent-activity network formulationdecide whichactivities andeventsof



∗ should be selected toconstruct an event-activitynetwork



dis intermsofarescheduledtimetable.Thepassengerreassignmentconstraintsdecidetheweight

ofeachactivityof



disfromtheperspectivesofpassengers,andassigneachpassengertoonepathonly.Apathisdescribed

by a sequenceof connectedactivities. The totalactivity weightof apath is thegeneralizedtravel time ofthis path.The objectiveofthemodelisminimizingthegeneralizedtraveltimesofallpassengers.Bythismodel,arescheduledtimetable that leadstothe shortestgeneralizedtravel timesofall passengers can beobtained, aswell asthe pathchosen by each passengerundertherescheduledtimetable.

3. Event-activitynetwork

Thissectiondefinesanevent-activitynetwork,whichisarepresentationofatimetableandallowspassengerpathchoices to be described. Anevent-activity network needs to be reconstructedifthe corresponding timetableis rescheduled.This sectionintroduceshowtoformulateanevent-activitynetworkgivenafixedtimetable.

3.1. Events

Sixtypesofeventsarecreatedinan event-activitynetwork.Theyare arrivalevents,departure events,duplicate depar-tureevents, entryevents,exitevents anda penalty event, whichconstitute the sets Ear, Ede,Edde,Eentry,Eexitand Epenal,

respectively. Inparticular,Ear=Ealightar ∪Earpass, andEde=Edeboard∪E pass

de , whereE alight

ar isthe setofarrival eventsthat

corre-spondtopassenger alighting,andEboard

de isthe setofdeparture eventsthatcorrespond topassengerboarding.The arrival

(departure)eventsassociatedtoathroughtrainthatdonotcorrespondtopassengeralighting(boarding)constitutetheset ofEarpass(Epassde ).

The attributes of events are indicated in Table 1. Note that an event e _∈ E_dde is the duplicate of a departure event

e∈Eboard

de withexactly the sameattributes whiche has, andwithan extra attribute

λ

e toindicate the departure event

ecorrespondingtoe:E_dde=

{

e

|

λ

e=e,e∈E_deboard

}

.Oneandonlyoneduplicateiscreatedforadepartureevente∈E_deboard.

Duplicate departure eventsare used forconstructing wait, boarding andtransfer activities, which are explained inmore detailinSection 3.2.Notethat thispaperdefinestheseactivitiesdifferentlythanZhuandGoverde(2019a).AsforEpenal,it

containsonlyonepenalty eventforconstructingthepenalty arcsthatenablepassengerswho cannotfindpreferredpaths toleavetherailways.

3.2. Activities

Anactivityisadirectedarcbetweentwodifferentevents.Tentypesofactivitiesarecreatedinanevent-activitynetwork, whichareconstructedasfollows.

Aentry=

{

(

e,e

)

|

e∈Eentry,e∈Edde,ste=ste

}

.Entryactivitiesenablepassengerstoentertherailwayswhenarrivingatthe

origins.

Aexit=

{

(

e,e

)

|

e∈Earalight,e∈Eexit,ste=ste

}

. Exitactivities enablepassengers toleave the railwayswhenarriving atthe

destinations, Aenpenal=



e,e



e∈Eentry,e∈Epenal



,andAexpenal=



ee



e∈Epenal,e∈Eexit



.Entrypenaltyactivitiesandexitpenalty activitiestogetherenablepassengerstodroptherailwaysincasenopreferredpathscanbefound,

A_board=



e_,e



e_∈E_dde,e_∈Eboard de ,e=

λ

e



.Boardingactivitiesenablepassengerstoboardatrain.Eachduplicate depar-tureeventislinkedtoitscorrespondingdepartureevent.

Arun=



e,e



e∈Ede,e∈Ear,tre=tre,ste istheupstreamstationadjacenttoste



.Runningactivitiesenablepassengers totravelfromonestationtoanotherinatrain.

Fig. 2. A planned timetable with the constructed transition network.

A_dwell=





e_,e



_

e_∈E_aralight_,e_∈Eboard

de ,tre=tre,ste=ste,oe− oe>0

.Dwellactivitiesenablepassengerstowaitata sta-tioninatrain.

Apass=



e,e



e∈Earpass,e∈Epassde ,tre=tre,ste=ste,oe− oe=0



. Pass-through activities enable passengers to pass

throughastationina train.Notethat itisnecessarytodistinguish theplannedpass-throughanddwellactivitiessothat wecanrecognizetheskipped(extra)stopsinarescheduledtimetablebecausethedispatchingmeasureofflexiblestopping isappliedinthispaper.

Await=



_

e,e





e∈Edde,e=argmin



oe

|

oe ≥ oe,e∈Edde,tre=tre,ste=ste



.Waitactivitiesenablepassengerstowait

atastation.Eachduplicatedepartureeventislinkedtothenexttime-closestduplicatedepartureeventthatisatthesame stationbutcorrespondstoanothertrain.

Atrans=



_

e_,e



_

e_∈E_aralight_,e₌argmin



o_e

|

o_e≥ oe+trans_e,e ,e∈Edde,tre =tre,ste=ste



. Transfer activities enable pas-sengerstotransferfromonetraintoanother.Eacharrivaleventislinkedtothenexttime-closestduplicatedepartureevent thatoccursatleasttrans

e,e later atthesamestation butcorrespondsto anothertrain.Here, transe,e represents theminimum

transfertimerequiredfromthearrival traintre to anotherdeparturetraintre, whicharealongsidethesameplatformor

differentplatformsaffectingthevalueof_trans e,e .

Anevent-activitynetworkis



=

(

E,A

)

,whichisadirectedacyclicgraph(DAG).IntheblueboxofFig.2,allnodesand arcscoloredinblackconstituteanevent-activitynetworkformulatedfromtheplannedtimetableshownintheleft.

3.3.Weightsofactivities

Theweightsofactivitiesaredeterminedasfollows:

wa=

β

vehicle

(

oe− oe

)

, a=

(

e,e

)

∈Arun∪Adwell∪Apass,

wa=

β

wait

(

oe− oe

)

, a=

(

e,e

)

∈Await,

wa=

β

wait

(

oe− oe

)

+

β

trans, a=

(

e,e

)

∈Atrans,

wga=oe− tgori, a=

(

e,e

)

∈Aentry:oe≥ torig

wg_a=Tmax

g , a=

(

e,e

)

∈Aenpenal

wa=0, a=

(

e,e

)

∈Aboard∪Aexit∪Aexpenal,

where

β

_vehicle and

β

_wait representrespectivelypassengerpreferenceonin-vehicletimesandwaitingtimesatstations,and

β

trans refers to the time penalty of one transfer. Note that the weight of an entry activity or entry penality activity is

passengerdependent. tori

g isthe time when passenger group garrives at theorigin station,andthis paperassumesthat

eachpassengergroupghasanacceptablemaximumgeneralizedtraveltimeTmax

Table 2

Sets relevant to a transition/event-activity network. Notation Description

∗ _{Transition network: }∗_{= ( E} ∗_{, A} ∗₎

plan Event-activity network formulated from the planned timetable: plan = 

E plan , A plan_{and } plan ⊂∗

dis Event-activity network formulated from any possible disruption timetable by adjusting the planned timetable: dis ⊂∗

E ∗ _{Set of events in }∗

E plan _{Set of events in }

plan : E plan ⊂ E ∗

E plan

i Set of i events in plan , i ∈ {ar, de, dde, entry, exit, penal}: E _{i ⊂ E} plan plan

E alight, plan

ar Set of arrival events that correspond to passenger alighting in plan : E alightar , plan⊆ E arplan

E pass, plan

ar Set of arrival events that do not correspond to passenger alighting in plan : E passar , plan= E arplan\ E alightar , plan

E board, plan

de Set of departure events that correspond to passenger boarding in plan : E deboard, plan⊆ E plan de

E pass, plan

de Set of departure events that do not correspond to passenger boarding in plan : E depass, plan= E plan de \ E board, plan de A ∗ _{Set of activities in }∗ A ∗

i Set of i activities in ∗: A ∗i ⊂ A ∗, i ∈ {wait, trans, board, entry, exit} A plan _{Set of activities in }

plan : A plan ⊂ A ∗

A plan

i Set of i activities in plan : A plan_{i ⊂ A} plan , i ∈ {run, dwell, pass, wait, trans, board, entry, exit, enpenal, expenal}

A undis

k Set of undisrupted k activities in ∗: A undisk ⊂ A plank , k ∈ {run, dwell, pass, wait, trans, board, entry, exit} A dis

k 1 Set of disrupted k 1 activities in 

∗_{: A} dis

k 1 = A

plan

k 1 \ A

undis

k 1 , k 1 ∈ {run, dwell, pass} A dis

k 2 Set of disrupted k 2 activities in 

∗_{: A} dis

k 2 = A

∗

k 2\ A

undis

k 2 , k 2 ∈ {wait, trans, board, entry, exit}

4. Transitionnetwork

Thissectiondefinesa transitionnetwork,whichallows adynamicevent-activitynetworkformulationduringtimetable rescheduling.Thetransitionnetwork



∗ isanextension oftheevent-activity network



plan formulatedfromtheplanned

timetablebyaddingalleventsandactivitiesthatcouldexistinanyrescheduledtimetables.Inotherwords,



∗representsall possibletimetableadjustments,whichcanbeusedtodescribethealternativepathsavailabletopassengersduringtimetable rescheduling.Beforegivingthedetailsofconstructingatransitionnetwork,anexampleonasimplecaseisgivenbelowto explainthebasicidea.

ExampleFig.2showsaplannedtimetablewiththreestations A,BandC,andtwotrainstr1 andtr2.Bothtrainsstart

fromAandendatBwithtraintr1 additionallystoppingatB.Inthebluebox,theeventsandactivities inblackconstitute

theevent-activitynetwork



plan fromtheplannedtimetable, whiletheeventsandactivities inblackandorangetogether

constitutethetransitionnetwork



∗.Inthiscase,



∗isextendedfrom



planby addinganeweventandeightnew

activ-ities(coloredinorange)thatdonot existintheplannedtimetablebutcouldexistinarescheduled timetable.Duetothe dispatchingmeasureofre-ordering,traintr1coulddepartlaterthantraintr2 atstationA,althoughtraintr1 wasoriginally

plannedtodepart earlierthan traintr2.Consideringthispossibletrain orderchange,an extrawait activityisaddedfrom

event(dde,tr2,A)toevent(dde,tr1,A),whichcreatesacyclebetweenbothevents.Thisdisablesatransitionnetworktobe

aDAG.Duetothedispatchingmeasureofflexiblestopping,anextrastop couldbeaddedtotraintr2 atstationB.Thus,a

newevent(dde,tr2,B)isaddedaswellasanentryactivity,aboardingactivity,awaitactivity,twotransferactivities,and

anexitactivity.Ascanbeseenentry/exitpenaltyactivitiesalwaysremainthesamein



∗asin



_plan.

In the following, we introduce how to construct a transition network by extending the event-activity network



plan

correspondingtoa plannedtimetable.Thesetnotationwiththesuperscriptofplanrepresentstheevents/activitiessetsin



plan.Thesetnotationwiththesuperscriptof∗representstheextendedevents/activitiesin



∗.Table2showsthenotation

ofsetsrelevanttoatransition/event-activitynetwork.

4.1. Extendedevents

Alleventsofevent-activitynetwork



_planareincludedinthetransitionnetwork



∗,inwhichonlythesetofduplicate departureeventsisextended

E_dde∗ =



e



λ

e=e,e∈E_deplan

, whereE_deplan=E_deboard,plan∪E_depass,plan.Here, E_deboard,plan andE_depass,plan representrespectively thesetofdepartureeventsthatcorrespondanddonotcorrespondtopassengerboardingintheplannedtimetable.Recall thatinanevent-activitynetwork,duplicatesareonlycreatedfordepartureeventsthatcorrespondtopassengerboarding.

4.2. Extendedactivities

All activities ofevent-activity network



plan are included inthe transitionnetwork



∗, inwhich five typesof

activi-tiesareextendedincludingA∗_wait,A∗_trans,A∗_board,A∗_entry andA∗_exit.Exceptentry/exit penaltyactivities,each typeofactivitiesis classifiedintotwosubsets,undisruptedanddisrupted:

Aplan_i =Aundis

i ∪Adisi , i∈

{

run,dwell,pass

}

,

A∗_k=Aundis

Wedefinean activityanundisrupted activityifbothofthetwoeventsinthisactivitywereoriginallyplannedto occur beforetstart or aftertend+R, inwhich R isthe time length required forthe normalschedule to be fullyrecovered after

thedisruption ends. Inthat sense, an undisrupted activity isan activity that willnever be differentthan plannedin the rescheduledtimetable.In thispaper,we ensurean arrival(departure) eventthat wasoriginally scheduledtooccur before thedisruption start tstart oratleast Rminutes laterthan the disruptionendtend will not bedelayed/cancelled. Thisalso

appliestoduplicatedeparture events,which arealwayswiththesameoccurrencetimesastheir correspondingdeparture events.We define an activitya disrupted activityif atleast one of the two eventsin thisactivity could be cancelledor delayed.Inthatsense,adisruptedactivityisan activitythatcouldbedifferentthanplannedintherescheduledtimetable. Thispaper requiresthat only the events,which were originally plannedto occur duringthe period [tstart,tend+R] could

becancelledordelayed.Theseeventscan correspondtoanystations,whicharenot distinguishedbetweendisruptedand undisruptedinthepaper.Basedonthese,wedecidewhetheranactivityisundisruptedordisruptedasfollows.

4.2.1. Running,dwell,andpass-throughactivities

Thedisruptedandundisruptedrunning,dwell,andpass-throughactivitiesarerespectivelydefinedas

Adis

i =



_

e_,e



_∈Aplan_i

|

tstart≤ oe<tend+Rortstart≤ oe<tend+R

, i_∈

{

run_,dwell_,pass

}

_,

Aundis

i =A

plan

i

\

Adisi ,i∈

{

run,dwell,pass

}

,

whereoe refers to the original scheduledtime of e, tstart (tend) represents the start (end) time of the disruption,and R

representsthedurationrequiredforthedisruptiontimetableresumingtotheplannedtimetableafterthedisruptionends.

4.2.2. Entry,exit,andboardingactivities

ThedisruptedentryactivitiesaredefinedasAdis

entry=Adisentry,1 ∪Adisentry,2,where

Adis_entry,1 =





e,e



∈Aplan_entry

|

tstart≤ oe<tend+R

,

A_entrydis,2 =





e,e





e∈E_entryplan,e∈E_dde∗

\

E_ddeplan,ste=ste,tstart≤ oe <tend+R

.

ThedisruptedexitactivitiesaredefinedasAdis exit=A

dis,1 exit ∪A

dis,2 exit,where

Adis_exit,1₌





e_,e



_∈Aplan_exit

|

tstart≤ oe<tend+R

,

Adis_exit,2=





e,e





e∈Earpass,plan,e∈Eexitplan,ste=ste,tstart≤ oe<tend+R

. ThedisruptedboardingactivitiesaredefinedasAdis

board=A

dis,1 board∪A

dis,2

board,where

Adis_board,1 =





e,e



∈Aplan_board

|

tstart≤ oe<tend+R

,

Adis_board,2 =





e,e





e∈E_dde∗

\

Eplan_dde,e∈E_depass,plan,e=

λ

e,tstart≤ oe<tend+R

.

A_entrydis,1_,Adis_exit,1_,andA_boarddis,1 representrespectivelytheentry,exit,andboardingactivitiesthatcouldbecancelledduetothe disruption.Adis_entry,2_,Adis_exit,2_, andA_boarddis,2 representrespectivelytheentry,exit,andboardingactivities thatare notin



_plan but mightbe neededduetoextrastopsaddedina rescheduledtimetable.Theundisrupted entry,exit,andboardingactivities arerespectivelydefinedasAundis

entry=A

plan

entry

\

Adisentry,1,Aundisexit =A plan exit

\

A

dis,1

exit,andAundisboard=A plan board

\

A

dis,1 board.

4.2.3. Waitactivities

To construct disrupted wait activities, we first define three event sets, Emax

dde =

{

argmax

{

oe

|

e∈E plan

dde,oe<tstart,ste=

st

}}

st∈ST, Eddemin=

{

argmin

{

oe

|

e∈E plan

dde,oe≥ tend+R,ste=st

}}

st∈ST, and Eddedis =

{

e∈Edde∗

|

tstart≤ oe<tend+R

}

, in which ST is

thesetofstations.Set Emax

dde includesateachstation st∈STthelatestduplicate departureeventbeforetstart. SetEddemin

in-cludesateach stationst ∈STtheearliestduplicatedeparture eventaftertend+R.The eventsinEddemax andEmindde willnot be

affectedbythedisruption,whileEdis

dde includesallduplicatedepartureeventsthatcouldbeaffectedbythedisruption.

BasedonEmax dde,E

min

dde andE

dis

dde,thesetofdisruptedwaitactivitiesisdefinedasA dis wait=



j∈{1,...,4}Adiswait,j,inwhich

Adis_wait,1=



e,e



e∈Emax

dde,e∈Edisdde,ste=ste,oe− oe≤ max_wait



,

Adis_wait,2₌



e_,e



e_∈Edis dde,e∈E

min

dde,ste=ste,oe− oe≤ max_wait+D



, Adis_wait,3=



e,e



e,e∈Edis

dde,e=e,ste=ste,0≤ oe− oe≤ max_wait+ D



,

Adis_wait,4=



e,e



e,e∈Edis

Table 3 Decision variables.

Symbol Description Module

x e Continuous variable deciding the rescheduled time of an event e ∈ E plan

ar ∪ E deplan ∪ E dde∗ . 1, 2, 3

c e Binary variable with value 1 deciding event e ∈ E plan

ar ∪ E plande ∪ E ∗dde is cancelled, and 0 otherwise. 1, 2

s a Binary variable deciding whether a scheduled stop a ∈ A plan

dwell is skipped or 1,

2 whether an extra stop is added to a ∈ A plan

pass .

If a ∈ A plan

dwell , then s a = 1 indicates a is skipped.

If a ∈ A plan

pass , then s a = 1 indicates a is added with a stop.

y a Binary variable with value 1 deciding activity a ∈ ∗_{is effective in }

dis , and 0 otherwise. 2, 3

u g a Binary variable with value 1 deciding activity a ∈ ∗_{is chosen by passenger group g , and 0 otherwise. 3}

w g a Continuous variable deciding the weight of activity a ∈ ∗_{perceived by each passenger in group g 3}

Module 1: timetable rescheduling; Module 2: dynamic event-activity network formulation; Module 3: passenger reassignment

Here, D represents the maximum allowed delay per event, and max

wait represents the maximum waiting time that a

passenger would like to spend at a station. We assume that max

wait≥ D. Then, undisrupted wait activities are defined as

Aundis wait =A plan wait

\

(

A plan wait∩A dis wait

)

. 4.2.4. Transferactivities

To construct disrupted transfer activities, we first establish two event sets, Edis

ar =

{

e

|

e∈Earplan,tstart≤ oe<tend+R

}

,

and Etrans

ar =

{

e

|

e∈E

plan

ar ,oe<tstart,

(

e,e

)

∈Aplan_trans,tstart≤ oe<tend+R

}

. Eardis includes the arrival events that could be

de-layed/cancelledduetothedisruption.Etrans

ar containsthearrivaleventsthatwillnotbedelayed/cancelledbythedisruption

butthecorrespondingplannedtransferactivitiescouldbecancelledduetothedisruption. BasedonEdis

ar andEartrans,thedisruptedtransferactivitiesaredefinedasAdistrans=



j∈{1,...,5}Adistrans,j,where

Adis_trans,1=



e,e



e∈Etrans

ar ,e∈Emindde,tre=tre,ste =ste,_e,etrans_ ≤ oe− oe≤ maxtrans



, Adis_trans,2=



e,e



e∈Etrans

ar ,e∈Edisdde,tre=tre,ste =ste,oe− oe≤ maxtrans



, Adis_trans,3=



e,e



e∈Edis

ar ,e∈Eddemin,tre=tre,ste =ste,trans_e,e ≤ oe− oe≤ maxtrans+D



, Adis_trans,4=



e,e



e∈Edis

ar ,e∈Edisdde,tre=tre,ste =ste,0≤ oe− oe≤ maxtrans+D



, Adis_trans,5=



e,e



e∈Edis

ar,e∈Eddedis,tre=tre,ste=ste,trans_e,e − D≤ oe− oe<0



,

Here, trans

e,e representsthe minimum transfertime, andmaxtrans representsthe maximumtransfer time that a passenger

wouldliketospendatastation.Weassumethatmax

trans≥ D>trans_e,e .Adistrans,1 andA dis,2

trans arebothrelatedtoEtransar ,whileAdistrans,3,

A_transdis,4 andA_transdis,5 areallrelatedtoEdis

ar.UndisruptedtransferactivitiesarethendefinedasAundistrans =A plan trans

\

(

A

plan

trans∩Adistrans

)

.

5. Passenger-orientedtimetablereschedulingmodel

Inthissection,weformulatethepassenger-orientedtimetablereschedulingproblemasanMILPmodel,withthe objec-tiveofminimizinggeneralizedtraveltimes.TheMILPmodelconsistsofthreeconstraintmodules:1)timetablerescheduling, 2)dynamicevent-activitynetworkformulation,and3)passengerreassignment.

Thetimetablereschedulingmodulecomputesafeasiblerescheduledtimetable.Thedynamicevent-activitynetwork for-mulationmoduleformulatesanevent-activitynetwork



_discorresponding totherescheduledtimetablebasedonthe pre-constructedtransitionnetwork



∗.Thepassengerreassignmentmoduledecidestheweightofeachactivitya∈



∗_perceived

byeachpassenger,andassignseachpassengertothepathwiththeshortestgeneralizedtraveltimeperceivedbythis pas-senger.

The constraints used in the timetable rescheduling module are all from Zhu and Goverde (2019c) so that we do not presentthem in thispaper, neither the decision variables that are only used in thismodule. We refer to Zhu and Goverde(2019c)fordetails.Inthispaper,wepresenttheconstraintsinthemodulesofthedynamicevent-activitynetwork formulationandthe passenger reassignment,aswell asthe corresponding decisionvariables. Table3 liststhesedecision variables andthe modules inwhich they are used.The notationof parameters/setscan be found inTable 19in the Ap-pendix.Notethattherescheduledtimexeofanyeventethatwasoriginallyscheduledtooccurbeforetstartoraftertend+R

isforcedtobethesameasitsoriginalscheduledtimeoebyconstraintsfromZhuandGoverde(2019c).Inotherwords,our

modelrespectswhathasalreadyhappenedbeforethebeginningofthedisruption,andrecoversthedisruptionbacktothe normalscheduleatlatestRtimeaftertheendofthedisruption.

Duetoflexible stopping,scheduledstopscouldbe skippedandextrastopscould beadded.The scheduledstops (non-stops) can alsobe cancelled,dueto short-turningor completetrain cancellation. Table show all possiblestop types ina rescheduledtimetable,andthecorrespondingvaluesoftherelevantdecisionvariables.Therearespecificconstraintsinthe

timetablereschedulingmoduletolimit thevalue combinationsofce,ce andsa.We referto ZhuandGoverde(2019c)for

details.

5.1. Dynamicevent-activitynetworkformulation

The dynamic event-activity network formulation module decides which events and activities of the transition net-work



∗ areeffectiveinan event-activitynetwork



dis corresponding toa rescheduledtimetableby respectingtherules

of constructing an event-activity network introduced in Section 3. Recall that



∗=

(

E∗,A∗

)

, where E∗=Earplan∪Eplande ∪

E_dde∗ ∪E_entryplan ∪E_exitplan∪E_penalplan, andA∗=Aplanrun ∪Aplandwell∪A plan

pass∪A∗wait∪A∗trans∪A∗board∪A∗entry∪A∗exit∪A plan

enpenal∪A

plan

expenal. In

partic-ular,Aplan_{i =}Aundis

i ∪A

dis

i ,i∈

{

run,dwell,pass

}

,andA∗j=Aundisj ∪A dis

j ,j∈

{

wait,trans,board,entry,exit

}

,whichmeansthatin

thetransitionnetwork



∗, eachkind ofactivityset consistsoftwo subsets:an undisrupted activity set,anda disrupted activityset.Foranundisrupted activity,bothofthecorrespondingeventswillnot bedelayed/cancelledbythedisruption; whileforadisruptionactivity,atleastoneofthecorrespondingeventscouldbedelayed/cancelledbythedisruption.

5.1.1. Decidingwhicheventsareeffectivein



dis

The binarycancellation decision ce of an event e∈Earplan∪Edeplan∪Edde∗ is equivalent to deciding whether thisevent is

effectivein



dis.Anevente∈Eplanar ∪Edeplan∪Edde∗ iseffectivein



disifitisnotcancelled,ce=0.Thecancellationdecisionce

andtherescheduledtimexeofanarrival(departure)evente∈Earplan

(

e∈Edeplan

)

aredeterminedinthetimetablerescheduling

module.Aduplicatedepartureevente∈E_dde∗ isrequiredtobecancelled/keptsimultaneouslyasitscorrespondingdeparture evente∈E_deplan,andtherescheduledtimesofbotheventsareforcedtobethesame:

ce=ce, e∈Edde∗ ,e∈E plan de ,

λ

e=e, (1) xe=xe, e∈Edde∗ ,e∈E plan de ,

λ

e=e, (2)

where

λ

_e isagivenattributeindicatingthedepartureeventcorrespondingtoduplicatedepartureevente. Anevente∈Eplan_entry∪E_exitplan∪E_penalplan isalwayseffectiveinany



dis.

5.1.2. Decidingwhichactivitiesarealwayseffectiveinany



dis

Entry/exitpenaltyactivities,andundisruptedactivitiesareeffectiveinany



dis:

ya=1, a∈Aplan_enpenal∪Aplan_expenal, (3)

ya=1, a∈



Aundis

k



k∈K,K=

{

run,dwell,pass,wait,trans,board,entry,exit

}

, (4)

whereyaisabinaryvariablewithvalue1indicatingthatactivityaiseffectivein



dis,and0otherwise.Recallthatbothof

theeventscorrespondingtoanundisruptedactivitywillnotbedelayed/cancelledduetothedisruption.

5.1.3. Decidingwhichdisruptedrunactivitiesareeffectivein



dis

Recall that a running activityis froma departure event eto an arrival event e, which correspondto the same train atneighbouring stations. Adisrupted runningactivity in thetransition network



∗ will be effective inan event-activity network



disifneitherofthecorrespondingeventsiscancelled:

ya=1− ce, a=

(

e,e

)

∈Adisrun, (5)

ya=1− ce, a=

(

e,e

)

∈Adisrun. (6)

Notethat inthetimetablereschedulingmodule(ZhuandGoverde,2019c), thedeparture eventeandthearrival evente

inthesamerunningactivityare forcedtobe cancelled/keptsimultaneously: ce = ce,whichiswhywe useequalitiesfor

(5)and(6).

5.1.4. Decidingwhichdisrupteddwell/pass-throughactivitiesareeffectivein



dis

Recallthat a dwell(pass-through)activityis fromanarrival eventeto adeparture evente, whichcorrespondto the sametrainatthesamestation.Wedecidewhetheradisrupteddwell(pass-through)activityof



∗willbeeffectivein



dis

by:

ya≤ 1− ce, a=

(

e,e

)

∈Adis_dwell∪Adispass, (7)

ya≤ 1− ce, a=

(

e,e

)

∈Adisdwell∪Adispass, (8)

ya≥ 1− ce− ce, a=

(

e,e

)

∈Adis_dwell∪Adis_pass. (9)

Constraints(7)and(8)meanthatadisrupteddwell(pass-through)activitywillnotbeeffectivein



disifatleastoneofthe

correspondingeventsiscancelled;otherwise,itmustbeeffective(9).RecallthatAdis

dwell⊆ A

plan

Table 4

The stop type of activity a = (e, e _{) ∈ A} plan

dwell in a rescheduled timetable according to

c e , c e  and s a . c e c e  s a Stop type 0 0 0 Stop 0 0 1 Skipped stop 1 0 0 Cancelled stop 0 1 0 Cancelled stop 1 1 0 Cancelled stop Table 5

The stop type of activity a = (e, e _{) ∈ A} plan

pass in a rescheduled timetable according to

c e , c e and s _{a .} c e c e  s a Stop type 0 0 0 Extra stop 0 0 1 Non-stop 1 0 1 Cancelled non-stop 0 1 1 Cancelled non-stop 1 1 1 Cancelled non-stop

5.1.5. Decidingwhichdisruptedentryactivitiesareeffectivein



dis

Recall that an entryactivity isfroman entryeventetoa duplicatedeparture evente,which bothcorrespond tothe samestation.Weuseabinaryparameterr_e withvalue1toindicatethata(duplicate)departureeventecorrespondstoa trainorigindeparture,and0otherwise.Fora disruptedentryactivitya=



e,e



ofwhichtheduplicatedepartureevente

correspondstoatrainorigindeparture,awillbeeffectivein



_dis ifeisnotcancelled:

ya=1− ce, a=

(

e,e

)

∈Adisentry,re=1. (10)

Fora disrupted entryactivity a=



e,e



ofwhich the duplicatedeparture event e doesnot correspond to a train origin departure,weestablishedthefollowingconstraintstodecidewhetheraiseffectivein



_dis:

ya≤ 1− ce, a=

(

e,e

)

∈Adisentry,re=0, (11)

ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Adisentry,re=0,e=

λ

e,a=

(

e,e

)

∈Adisdwell∪Adispass, (12)

ya≥ 1− sa− ce− ce, a=

(

e,e

)

∈Adisentry,re=0,e=

λ

e,a=

(

e,e

)

∈Adisdwell∪Adispass. (13)

Constraint(11) meansthat adisrupted entryactivitya=

(

e,e

)

will notbe effectivein



dis ifits correspondingduplicate

departureeventeiscancelled.Otherwise,awillbeeffectiveonlyifitscorrespondingduplicatedepartureeventeis asso-ciatedwitharealstopa=

(

e,e

)

∈Adis

dwell∪Adispass thathasce=0,ce=0andsa =0(seeTable4andTable5),inwhich

e



isthedepartureeventcorrespondingtoe:e=

λ

e.Thisisrepresentedby(12)and(13).

5.1.6. Decidingwhichdisruptedboardingactivitiesareeffectivein



dis

Recall that a boarding activity is from a duplicate departure event e to the corresponding departure event e. For a disruptedboarding activitya=



e,e



ofwhich theduplicate departureeventecorresponds toa trainorigindeparture, a

willbeeffectivein



disifeisnotcancelled.:

ya=1− ce, a=

(

e,e

)

∈Adisboard,re=1. (14)

Foradisruptedboardingactivitya=



e,e



ofwhichtheduplicatedepartureeventedoesnot correspondtoatrainorigin departure,wedecidewhetheraiseffectivein



disby

ya≤ 1− ce, a=

(

e,e

)

∈Adisboard,re=0, (15) ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Adisboard,re=0,a=

(

e,e

)

∈A dis dwell∪A dis pass, (16)

ya≥ 1− sa− ce− ce, a=

(

e,e

)

∈Adis_board,re=0,a=

(

e,e

)

∈Adisdwell∪Adispass. (17)

Constraint(15)meansthatadisruptedboardingactivityawillnotbeeffectivein



_disifitscorrespondingduplicate depar-tureeventeiscancelled.Otherwise,awillbeeffectiveonlyifitscorrespondingdepartureeventeisassociatedwithareal

stopa=

(

e,e

)

∈Adis

5.1.7. Decidingwhichdisruptedexitactivitiesareeffectivein



_dis

Recallthatanexitactivityisfromanarrivaleventetoanexitevente,whichbothcorrespondtothesamestation.We useabinaryparameterfe withvalue1toindicatethat anarrival eventecorrespondsto atraindestinationarrival,and0

otherwise.Fora disruptedexitactivitya₌



e_,e



ofwhich thearrivaleventecorrespondstoa traindestinationarrival,a

willbeeffectivein



dis ifeisnotcancelled:

ya=1− ce, a=

(

e,e

)

∈Adisexit,fe=1. (18)

Fora disruptedexitactivitya=



e,e



ofwhichthearrival evente doesnotcorrespond toatrain destinationarrival, we establishedthefollowingconstraintstodecidewhetheraiseffectivein



dis:

ya≤ 1− ce, a=

(

e,e

)

∈Adisexit,fe=0, (19)

ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Aexitdis,fe=0,a=

(

e,e

)

∈Adisdwell∪Adispass, (20)

ya≥ 1− sa− ce− ce, a=

(

e,e

)

∈Adisexit,fe=0,a=

(

e,e

)

∈Adisdwell∪Adispass, (21)

Constraint(19)meansthatadisruptedexitactivitya=

(

e,e

)

willnotbeeffectivein



disifitscorrespondingarrivalevente

iscancelled.Otherwise,awillbeeffectiveonlyifitscorrespondingarrivaleventeisassociatedwitharealstopa=

(

e,e

)

∈

Adis dwell∪A

dis

pass thathasce=0,ce=0andsa =0.Thisisstatedby(20)and(21).

5.1.8. Decidingwhichdisruptedwaitactivitiesareeffectivein



_dis

Recallthatawaitactivityisfromaduplicatedepartureeventetothenexttime-closestduplicatedepartureeventethat occursatthesamestationbutcorresponds toa differenttrain.Wedecide whetheradisruptedwait activitya=

(

e,e

)

is effectivein



disby ya≤ 1− ce, a=

(

e,e

)

∈Adis_wait, (22) ya≤ 1− ce, a=

(

e,e

)

∈Adiswait, (23) ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Adiswait,re=0,e=

λ

e,a=

(

e,e

)

∈A dis dwell∪A dis pass, (24)

ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Adiswait,re=0,e=

λ

e,a=

(

e,e

)

∈Adisdwell∪A dis pass, (25) ya+ya≤ 1, a=

(

e,e

)

∈Adiswait,a=

(

e,e

)

∈A dis wait (26) a∈Adis wait, tail(a)=e ya≤ 1, e∈E_ddedis, (27) a∈Adis wait, head(a)=e ya≤ 1, e∈Edisdde, (28)

xe− xe≤ M

(

1− ya

)

, a=

(

e,e

)

∈Adiswait,a=

(

e,e

)

∈Adiswait, (29)

xe− xe≤ M

(

1− ya

)

, a=

(

e,e

)

∈Adis_wait,a=

(

e,e

)

∈Adis_wait, (30) xe− xe≥ −M

(

1− ya

)

, a=

(

e,e

)

∈Adiswait, (31)

wheretail(a)referstothetaileventofanactivity:theeventwhichanactivitystartsfrom,head(a)referstotheheadevent ofanactivity:theeventwhichanactivitydirectsto,andMisasufficientlylargenumberofwhichthevalueissetto2880. Constraints(22)and(23)meanthatadisruptedwaitactivitywillnotbeeffectivein



_disifatleastoneofthecorresponding eventsiscancelled.Constraint (24)(25)requiresa disruptedwaitactivity a=

(

e,e

)

tobe ineffectiveifthecorresponding duplicatedeparture evente(e) doesnot correspondto atrain origindeparture andisnot associatedwitharealstop. A duplicatedepartureeventcouldberelevanttomultipledisruptedwaitactivitiesinatransitionnetwork,whileatmostone oftheseactivities canbeeffectiveinanevent-activitynetwork



_dis(26)–(28).Constraints(29)–(31)togetherensurethata duplicatedepartureeventecanonlybelinkedtothenexttime-closestduplicatedepartureeventtoconstructaneffective waitactivityin



dis.

5.1.9. Decidingwhichdisruptedtransferactivitiesareeffectivein



_dis

Recallthatatransferactivityisfromanarrivaleventetothenexttime-closestduplicatedepartureevente thatoccurs atthesamestationasebutcorresponds toadifferenttrain.Wedecidewhetheradisruptedtransferactivitya=

(

e,e

)

is effectivein



_dis by

ya≤ 1− ce, a=

(

e,e

)

∈Adistrans, (32)

ya≤ 1− ce, a=

(

e,e

)

∈Adistrans, (33)

ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Atransdis ,fe=0,a=

(

e,e

)

∈Adis_dwell∪Adispass, (34)

ya≤ 1− sa+ce+ce, a=

(

e,e

)

∈Adistrans,re=0,e=

λ

e,a=

(

e,e

)

∈Adisdwell∪Adispass, (35)

a∈Adis trans,

tail(a)=e

ya≤ 1, e∈Eartrans∪Edisar, (36)

xe− xe≤ M

(

1− ya

)

, a=

(

e,e

)

∈Adistrans,a=

(

e,e

)

∈Adistrans, (37)

xe− xe≤ M

(

1− ya

)

, a=

(

e,e

)

∈Adistrans,a=

(

e,e

)

∈Adistrans, (38)

xe− xe≥ −M

(

1− ya

)

+transe,e , a=

(

e,e

)

∈Adistrans, (39)

where _trans

e,e refers tothe minimumtransfer time. Constraints (32)and(33)means that a disruptedtransfer activitywill

notbeeffectivein



disifatleastoneofthecorrespondingeventsiscancelled.Constraint(34)requiresadisruptedtransfer

activity a₌

(

e_,e

)

to be ineffectiveif thecorresponding arrival eventedoes not correspondto a train destination arrival andisnotassociatedwitharealstop.Constraint(35)requiresadisruptedtransferactivitya=

(

e,e

)

tobeineffectiveifthe correspondingduplicatedepartureeventedoesnotcorrespondtoatrainorigindepartureandisnotassociatedwithareal stop.Constraint(36)meansthatforanarrivalevente∈Etrans

ar ∪Eardis,whichhasmultipledisruptedtransferactivitiesstarting

fromit, atmostone oftheseactivities will be effectiveinan event-activity network



_dis.Constraints (37)–(39) together ensurethatanarrivaleventecanonlybelinkedtothenexttime-closestduplicatedepartureeventtoconstructaneffective transferactivityofwhichtheminimumtransfertimemustberespected.

5.2. Passengerreassignment

Therecould bemultiplepassengers whoshareexactlythesamejourneysintermsoftheplannedtimetable:thesame originstation,thesamearrival timeattheoriginstation,thesamedestination,andthesameexpectedgeneralizedtravel time fromtheoriginto thedestination. Thesepassengers formasame groupg _∈G,whichis assumedtobe inseparable incaseofa disruption,andwillnot changethedestination.Grepresents thesetofpassengergroups, possibly consisting of a singlepassenger. Recall that a path is a sequence ofconnected activities. Decidingwhich path will be chosen by a passenger groupis equivalentto decidingwhich activities willbe chosen bythisgroup, whileeach groupg isassociated withthesameactivitychoicesetA∗.Thepassengerreassignmentmoduledecideswhichactivitya∈A∗willbechosenbya passengergroupgandtheweightofeachactivitya∈A∗perceivedbyg.

5.2.1. Assigningeachpassengergrouptoonepathonly

Anactivitya∈A∗cannotbechosenbyapassengergroupifaisnoteffectivein



dis(ya=0):

uga≤ ya, a∈A∗,g∈G, (40)

where ug_a is a binarydecision with value 1 indicating that activity a _∈A∗ is chosen by passenger group g _∈ G, and 0 otherwise.

A path that could be chosen by a passenger group g must start from an entry (entry penalty) event corresponding to his/heroriginOg, endinan exit(exitpenalty) eventcorrespondingto his/herdestination Dg,andincludeatleastone

intermediateeventtoconnectthem:

a∈Aundis entry∪Adisentry∪A

plan enpenal, sttail(a)=Og uga=1, g∈G, (41) a∈Aundis entry∪Adisentry∪A

plan enpenal,

sttail(a)=Og