Disruption Management with Rescheduling of Trips and Vehicle Circulations

0.15 MB·English

by  Sandor P. Fekete

Disruption Management with Rescheduling of Trips and Vehicle
Circulations
Sa´ndorP. Fekete∗ AlexanderKro¨ller∗ Martin Lorek∗† Marc Pfetsch ‡
1
1
0
2 Abstract

n This paper introduces a combined approach for the recovery of a timetable by rescheduling trips and vehicle
a circulations for a rail-based transportation system subject to disruptions. We propose a novel event-based integer
J programming(IP)model.Featuresincludeshiftingandcancelingoftripsaswellasmodifyingthevehicleschedules

1 bychanging or truncating the circulations. Theobjective maximizes thenumber of recovered trips, possibly with
1 delay,whileguaranteeingaconflict-freenewtimetablefortheestimatedtimewindowofthedisruption. Wedemon-
stratetheusefulnessofourapproachthroughexperimentsforreal-lifetestinstancesofrelevantsize,arisingfromthe

] subwaysystemofVienna. Wefocusonscenariosinwhichonedirectionofonetrackisblocked,andtrainshaveto
C
bescheduledthroughthisbottleneck.Solvingtheseinstancesismadepossiblebycontractingpartsoftheunderlying
O
event-activitygraph;thisallowsasignificantsizereductionoftheIP.Usually,thesolutionsfoundwithinoneminute
. areofgoodqualityandcanbeusedasgoodestimatesofrecoveryplansinanonlinecontext.
h
t
a
m 1 Introduction
[

1.1 Disruption ManagementinPassengerTraffic
1
v
Mobilityofpeopleisofgrowingimportanceinmodernsocieties. Forinstance,especiallyindenselypopulatedEuro-
3
2 peancities,publictransportationsystemsformarelevanteconomicfactor. However,possibledisruptionofoperations
1 insuchsystemswillalwaysremainunavoidable. Increasedpressuretowardseconomicaloperationhasamplifiedthe
2 impactandthefrequencyofsuchdisruptions.Beingoftenoptimizedtothelimit,transportationsystemshavebecome
.
1 moreinterconnected,andthereforemorevulnerabletodisruptions.
0 Thismakesit indispensableto developmoreadvancedmethodsfor disruptionmanagementof passengertraffic.
1 RailwaydisruptionmanagementisdefinedinJespersen-Grothetal. [1]astheprocessoffindinganewtimetableby
1
rerouting, delaying, or canceling trains and reschedulingthe resources like rolling stock and the crew such that the
:
v newtimetableisfeasiblewithrespecttothenewschedules. Thishastobeperformedwhiletheeffectsofdisruptions
i arestillunfolding,thatis,disruptionmanagementisanonlineproblem.
X
The recovery of the timetable, the vehicle schedule, and the crew schedule is usually performed sequentially.
r
a Whilepreviousworkhasfocusedoneitherdelaymanagement(i.e.,makingsurethatthedelayofpassengersandthe
inconveniencecausedbymissedtrainsiskeptlow)oronderivingnewcirculationsonagivendispatchingtimetable,
aparticularaspectthathasyettobeconsideredfromascientificsideistocombinereschedulingoftripsandvehicle
circulationstoobtainafeasibledispatchingtimetable. Thegoalistoperformasmanytripsoftheoriginaltimetable
aspossible.Clearly,thisisanimportantnextstepwhenfurtheradvancingoptimizationtowardsreal-timemethodsfor
stabilizingrail-basedtransportationsystems.
∗DepartmentofComputerScience,TUBraunschweig,D-38106Braunschweig,Germany,Email: {s.fekete,m.lorek,a.kroeller,m.pfetsch}@tu-
bs.de.http://www.cs.ibr.tu-bs.de/alg
†SupportedbyafellowshipfromSiemensAG.
‡DepartmentofMathematics,TUBraunschweig,D-38106Braunschweig,Germany,Email:[email protected].
1
1.2 Related Work
The challengesand problemsconnectedwith passengerrailway transportationhave beenintensivelystudied for the
last decades, and operationsresearch methods have been successfully applied. For example, Liebchen investigated
the optimization of periodic timetables [2]. However, in daily operation, unforeseen events occur and may lead to
disruptions of the timetable. Different aspects of this problem have been considered, and various approaches and
modelsexist. A generalframeworkfor the problemsand solution methodsin disruptionmanagementcan be found
in[1]. Theproblemweinvestigateinthispaperiscloselyrelatedtothedelaymanagementproblem. Mixed-integer
approachesforitareintroducedbyScho¨bel[3,4]andextendedbycapacityconstraintsforthetracks[5]. Schachte-
beck[6]addscirculationconstraintstothemodelandperformsamatchingatthedepotstopreventalargedelayfrom
spreadingintothenextcirculationactivity. Thecomputationalcomplexityoftheoriginaldelaymanagementproblem
isinvestigatedin[7]: itturnsouttobeNP-hard,eveninspecialcases,seealso[8]. Simulationstudiescanbefound
in[9,10,11]. To¨rnquist[12]givesanoverviewovertheexistingmodelsanddecisionsupportsystems. Healsoinves-
tigatesthedisturbancemanagementonn-tracks[13]. GattoandWidmayerconsiderrelatedproblemsasanonlinejob
shopmodel[14]. Adecisionsupportsystemforrollingstockreschedulingduringdisruption,wheretherollingstock
isbalancedaccordingtoagivendispatchingtimetable,isprovidedbyNielsen[15].
Inthispaper,wefocusonscenariosinwhichonedirectionofatracknetworkhastobeshutdown,suchthattrains
have to be scheduled through a bottleneck. This leads to the track allocation problem studied by Borndo¨rfer and
Schlechte [16], whichis shown to be NP-hard in [8]. A columngenerationapproachto timetablingtrains thougha
corridorispresentedbyCacchianietal.[17]. Huismanetal.developedcolumngenerationapproachestoreschedule
railcrew[18]andsolvedthecombinedcrewreschedulingproblemwithretimingoftripsbyallowingsmallchanges
tothetraintimetables[19]. Anoverviewofmodelsandmethodsforsimilarproblemsthatariseintheairlinecontext
canbefoundinClausen[20].
Noneoftheabovepapersconsidersreschedulingoftripsandcirculationsasanintegralpartofderivingafeasible
dispatchingtimetable. Clearly, we canimaginereal-lifescenariosthatrequireanintegratedapproachfordisruption
management,e.g.,inanonlinesituationwithtrainsalreadycirculatingintherailnetworkandaninfeasibletimetable
duetoheavydisruptions.
1.3 OurContribution
Inthispaper,weintroduceacombinedapproachforreschedulingtripsandvehiclecirculationsinordertodealwith
unplanned disruptions. We derive a new re-optimized dispatching timetable with respect to a feasible new vehicle
schedule that performs as much of the original trips as possible. Especially in subway systems, the utilization of
the system is high. The high frequency of trains circulating in the system, the limited possibilities for overtaking,
swapping tracks, and temporarily parking, as well as the bottleneck sections caused by disruptions force us to add
strongerresourcerestrictionsandintegratefurtherpossibilitiesintothemodelinordertoobtainafeasibledispatching
timetable.Ourmainmathematicalcontributionsareanappropriatemodelbasedonintegerprogrammingthatcombines
reschedulingoftripsandcirculations,aswellasamathematicalcontractiontechniqueforsimplifyingthemodel. On
thepracticalside,weareabletodemonstratetheusefulnessofthismathematicaloptimizationapproachbyproviding
optimalandnear-optimalsolutionsforasystemofreal-lifeinstancesthatarebasedonthesubwaysystemofVienna.
Therestofthispaperisorganizedasfollows.Inthefollowingsection,wedescribedetailedaspectsofdisruptions
intheoperationofrail-basedtransportation.Thisleadstoanoptimizationmodelbasedonintegerlinearprogramming.
Mathematicalaspectsofsolvingreal-lifeinstancesarediscussed, andcomputationalresultsarepresentedbeforewe
concludewithadiscussionoffuturework.
2 Disruption in Public Transport
Disturbancesinthedailyoperationofpublicpassengertransportcannotbeavoided. Thiscanincludedelayscaused
by longerwaiting times, becausean unexpectednumberofpassengersenteror leavethe train, a driverarrivinglate
at a relief point, or a technical problem with the mechanism of an automatic door. These disruptionslead to small
delays, whichusuallycan beabsorbedbythebuffertimesintegratedinto thetimetable. Ifbiggerdisruptionsoccur,
2
e.g.,avehiclebreaksdownorapartofthetrackrequiresunplannedwork,thiscausesatemporaryshutdownonatrack
section. Suchdisruptionsleadtosevereviolationsofthetimetable,whichhavetoberesolvedbythedispatchers.
As an illustrating example, consider a blocked track section on a 2-track network. A possible option includes
reroutingtrips,whichoriginallyusetheblockedtrack,viatheoppositetrack.Thisleadstoalargenumberofconflicts
onthattracksection,becausetrainsdrivingintooppositedirectionshavetosharethesametrackresourcesandcompete
forfreetimeslotstopasstroughthisbottleneck.Considertwotrainsthatarescheduledtoenterthesharedtracksection
inoppositedirectionsatthesametime. Assumethattheminimumtransittimeis5minutesandtheminimumsafety
timebetweentwotrainsusingthesametrackinoppositedirectionisoneminute. Ifthecycletimeis5minutesand
weallowamaximumdelayof5minutes,itisnotpossibletoschedulebothtrainsthroughthebottleneck.
Depending on the length of the bottleneck and the frequency of the railway system, updating the timetable by
delayingtrips will notsuffice. If trainsqueueup in frontof the bottleneck, passengertravel time will increase dra-
matically. More precisely, the dispatcher has differentpossibilities for dealing with this situation. The timetabling
perspectiveincludesthat
1. tripscanbedelayedwithinacertaintimewindowand
2. tripscanbecanceled.
Inadditionreschedulingofthevehiclecirculationsallowsthat
1. trainscanbeinstructedtotruncatetheircirculationandturnearly,inordertoservetripsinitiallyplannedfora
differenttrain,
2. trainscanbeusedtoshuttleinsidethesingle-tracksections,
3. trainscanreturnearlytoadepot,and
4. replacementvehiclescanbeused.
Usually, timetabling and vehicle scheduling are planned sequentially. Recently, research has focused more and
moreonintegratingdifferentstepsoftheplanningprocess,asproposedin[21]. Ifheavydisruptionsoccur,thesetwo
aspectsarestronglyinterweavedandneedtobehandledatthesametime.
Inthefollowing,wepresentamodelbasedonintegerlinearprogrammingthatfindsafeasibledispositiontimetable
andnewvehicleschedules. Itrespectsthecapacityconstraintsofthetracksandallowsdroppingtripsandearlyturns.
Thegoalistoreturntotheinitialtimetablewithinacertaintimehorizon.
3 Our Model
3.1 A GraphRepresentation
InthefollowingweconsiderarailnetworkG=(S,J)withSandJrepresentingthesetofstationsandtracksanda
givendispatchingtimetablep thatincludesthescheduleforeachvehicleinV. Ourmodelfordisruptionmanagement
withreschedulingoftripsandvehiclecirculationsisbasedonthewidelyusedconceptofanevent-activitynetwork.
ThisstructurewassuggestedbySerafiniandUkovich[22]andalsousedbyNachtigall[23]forperiodictimetabling
problems,seealso[24]. Inthecontextofdelaymanagementproblemsevent-activitynetworksareusedin[4]. Weuse
amodelsimilarto[6]andextendtheevent-activitynetworktoaccountforearlyturnaroundsandcapacityconstraints
atthestationplatforms.
Anevent-activityN=(E,A) isa directedgraph,wherethenodesinEdenotetheeventsandtheedgesinA are
calledactivities.Inoursettingthetimetablep consistsofscheduledtrips. Atripconnectstwostationswithaspecified
trainviaacertaintrack. Asequenceofconsecutivetripsbetweentwoterminalstationsiscalledaline. Asequenceof
linesiscalledacirculation. Eachtripgeneratestwoevents: adepartureeventv∈E fromastation,andanarrival
dep
eventw∈E atanadjacentstation. Thesetwoeventsareconnectedbyadrivingactivitya=(v,w). Together,they
arr
modelaconstraintbetweenthoseevents,meaningthateventwcannothappenbeforeeventvhastakenplace,plusthe
minimumdurationL assignedtoactivitya.
a
3
Similarto[4],wedistinguishbetweendifferenttypesofactivities.
Trainactivities(A )representthedriving,waitingandturningoperationsofatrain.
train
1. Drivingactivities(A ⊂E ×E )representscheduledtripsbetweentwostations.
drive dep arr
2. Waitingactivities(A ⊂E ×E )representthescheduledwaitingtimesatastationtoletpassengersenter
wait arr dep
andleavethetrain.
3. Turningactivities(A ⊂E ×E )connectanarrivaleventwithadepartureevent,butinadditiontowaiting
turn arr dep
arcs,theeventsconnectedbyaturningactivitydonotbelongtotripsonthesameline;instead,theyrepresent
thepossibilityfortrainstotruncatetheircurrentcirculationandcontinueonandifferentline.
Headwayactivitiescanbesplitintotwosubsetsofactivitiesthatdealwiththelimitedcapacityoftracksbetween
twostations(A )oronaplatform(A ):
track station
1. Theheadwayactivities(A ⊂E ×E )modeltheheadwayconditionbetweentwoeventsthatsharethe
track dep dep
same track resourcebetween two stations. For each a=(v,w)∈A , there is a correspondingactivity a˜=
track
(w,v)∈A sothattogethertheymodelaprecedenceconstraint,i.e., indicatewhichofthedepartureevents
track
willtakeplacefirst. Thus,onlyoneofeachpairofactivitiescanbeactive.
2. Ifwehavetoroutetrainsthroughacorridorconsistingofseveralstationsonasingletrack,wehavetointroduce
asecondkindofheadwayactivity(A ⊂E ×E ).Thesehavetomakesurethattrainsdrivinginopposite
station arr arr
directionsdonotenterthesameplatformandblockeachother;inotherwords, theyensurethatthe departure
event of some train takes place before another train arrives at the same platform. Unfortunately, two arrival
eventscouldimplymorethanonepairofprecedenceconstraints.Infact,thiskindofheadwayactivitiesinvolves
four events, i.e. each possible departure event correspondingto the arrival events. In our model we add the
possibilities for trains to turn at certain points, so the point of time a train may enter a station is strongly
connectedtothedirectionofthesuccessivetripofthetrain.
Inaddition,theheadwayconstraintstransmitstheprioritydecisiontakenfromtheprecedenceconstraintsonthe
tracksofonesideofthestationtotheotherside.
Asanexample,considertwotrainsk,l∈VandthreestationA,B,C∈S. TrainkmovesfromstationAthrough
stationB tostationC andtrainl drivesintotheoppositedirectionsfromstationC throughstationBto station
A. SoifthedepartureeventfromstationAtoBoftrainkisscheduledtotakeplacebeforethedepartureevent
oftrainl atstationB toA, thenalso traink hasto departfromstationB toC beforetrainl leavesstationC in
oppositedirectionstostationB.
Inourmodeltrainscouldbeorderedtoreturntoadepotandreservetrainsmaybeinserted. Therefore,weextend
the event-activity network by adding depots Di ∈D and replacement capacities RDi for each depot. For each trip
that ends at a station connected to a depot we add a trip to the depot, i.e., a departure event at the station and an
arrivaleventatthedepotconnectedbyadrivingactivity. ThesetsE containsthearrivaleventsatthedepotsand
depot
A thecorrespondingdrivingactivities. Trainsstoredinthedepotsmaybecomereinsertedafteraminimumidle
return
time, so analogously, the arrival events at a depot become connected to the departure events of possible follow on
trips.Similarly,replacementcapacitiesareinserted,byaddinganeventr∈E foreachdepotsupplyingreplacement
repl
vehiclesandconnectingthemtothenetwork.
Despitereschedulingthetimetableeventsourmodelshouldbeabletomodifythevehicleschedulestofindfeasible
newcirculations,performunplannedturnsortoreturntoadepotatcertainstation. Therefore,aclassicalflowmodel
isintegrated.Eachactivitya∈A cantransportaflowofatmostoneunit.Foreacheventi∈E,theoutflowequals
train
theinflow. Thus,everyeventwithaningoingactivityarcthattransportsoneunitofflowhasapredecessoreventand
duetoflowconservationisalsoconnectedtoasuccessorevent. Thegoalistofindfeasibleflowsinthenetwork,i.e.,
trainsalwayshaveasequenceoffollowingtripsanddonotstandidleandblockthetracks.
Afeasibletimetablep ∈NEassignsatimep toeachtimetableeventv∈Eandafeasibleflowthroughthenetwork
0 v
representsthecirculationoftrains.
4
3.2 Integer Programming Formulation
Asolutionofthefollowingintegerprogrammingmodelyieldsanewfeasibledispatchingtimetablep ∗∈ZE andvehi-
+
clescheduleincaseofadisruptionbasedontheoriginaltimetablep ∈ZE andthecorrespondingvehiclecirculations.
+
(cid:229) (cid:229)
max caya (1)
v∈Edepa∈d +(v)
subjectto
M(1−ya)+xw+p w≥xv+p v+Lmain ∀a=(v,w)∈Atrain, (2)
M(1−ya)+xw+p w≤xv+p v+Lmaax ∀a=(v,w)∈Adrive, (3)
(cid:229) (cid:229)
M(3− ya− ya−gvw)
a∈d +(w) a∈d +(v)
+xw+p w≥xv+p v+Lvw ∀(v,w)∈Atrack, (4)
gvw+gwv=1 ∀(v,w)∈Atrack, (5)
−M(3−yvw−yv′w′−hvv′)
+xw+p w+Svv′ ≤xv′+p v′ ∀(v,w),(v′,w′)∈Atrain:
(v,v′)∈Astation, (6)
hvv′+hv′v=1 ∀(v,v′)∈Astation, (7)
(cid:229) (cid:229)
ya− ya=0 ∀v∈Edep∪Earr, (8)
a∈d −(v) a∈d +(v)
(cid:229) (cid:229)
ya− ya≤1 ∀v∈Edepot, (9)
a∈d −(v) a∈d +(v)
(cid:229) ya≤RDi ∀v∈Erepl,Di∈D, (10)
a∈d +(v)
ya∈{0,1} ∀a∈A, (11)
xv∈Z+ ∀v∈E, (12)
ga∈{0,1} ∀a∈Atrack, (13)
ha∈{0,1} ∀a∈Astation (14)
Thevariablesofthemodelareasfollows: yisthecirculationoftrains,x determinesthedelayofeventv,g and
v a
h representprecedenceconstraints(seebelow).Inthefollowing,wesuccessivelydiscusstheconstraintsoftheabove
a
model.
The objective function (1) aims at maximizing the number of trips that still will be served in the re-optimized
dispatchingtimetablep ∗,possiblywithdelay.Becausethefrequencyofmostsubwaysystemsisquitehigh,canceling
few trips will lead to only a minor delay and inconveniencefor the passengers. Therefore, minimizing the overall
delay,asitisdoneinmostpublicationsaboutdelaymanagement,isonlyasecondarygoal.
Atripisservedifthecorrespondingdepartureeventv∈E hasanoutflowofoneunitofflow,i.e.,a=(v,w)∈
dep
A ispartofatraincirculation.Wecanattachanadditionalcostcoefficientc toeachdrivingactivitya;thisallows
drive a
aweightingofthetrips.
Given an original timetable p and a corresponding vector x∈ZE, representing the delays x of each event v,
+ v
Constraints(2)requirethatifanactivitya=(v,w)isinthesolution,theearliesttimeforeventwtostartisthestarting
timeofthepredecessoreventp ,plustheoccurreddelayx andtheminimumdurationLmin ofactivitya. Withrespect
v v a
tothebuffertimes,adelayx ofeventvmightcauseadelayx ofthesuccessoreventw. Weassumethatthedriving
v w
times are symmetric. Constraints (3) make sure that a driving activity a is bounded by a maximal duration Lmax,
a
whichincludespossiblebuffertimesandsafetymargins,butshouldnotbetoolargetopreventtrainsbeingidlefora
longperiodoftime. Theprecedenceconstraintsformulatedby(4)and(5) forsharedtrackresources,comein pairs
for(v,w),(w,v)∈A andensurethatconflictingeventsv,w∈E thatuse the same trackresourceare correctly
track dep
5
Station A Station B Station C
u v a w z
driving waiting driving
guw′ hvv′ gwu′
gw′u hv′v gu′w
driving waiting driving
z′ w′ a˜ v′ u′
Figure1: Theevent-activitynetworkforaconflictonasingletrack. OnetrainisdrivingfromstationAtostationC
whileanothertrainisdrivingin theoppositedirection. Thenodesrepresentthecorrespondingdepartureandarrival
eventsanda,a˜thewaitingactivities.Furthermorethedisjunctiveheadwayactivitiesareshown,denotedbythevariable
nameofcorrespondingconstraintoftheIP.
scheduledtopreventdeadlocksandkeepsafetymargins.Themodelimplies
1 : ifeventvtakesplacebeforew,
g =
vw (cid:26) 0 : otherwise,
whereL representstheminimalwaitingtimeforeventwtostartifwisscheduledafterv. Ifthetripscorresponding
vw
tothedepartureeventsv,w∈E usethetrackinthesamedirection,L isthesafetymarginbetweenbothdeparture
dep vw
events; if the track is used in opposite direction, L contains the maximum allowed transit time for the trip and a
vw
safetymarginS .
vw
Similarlytheprecedenceconstraints(6),(7)forthetracksinsideastationensurethatnotwotrainsusethesame
platformatthesametime,wherethemodelimplies
1 : ifeventwtakesplacebeforeevent,v′
hvv′ =(cid:26) 0 : otherwise.
Regardingtwo conflictingarrivaleventsv,v′ ∈E at somestation andtheir correspondingdepartureeventsw,w′ ∈
arr
E witha=(v,w),a˜=(v′,w′)∈A denotingtheactivities,ifthesolutioncontainsbothactivities(y =y =1)
dep train a a˜
andhvv′ =1,departureeventwtakesplacebeforethearrivaleventv′ attheplatformwithrespecttothesafetymargin
S (seeFigure1).
vv′
ThesafetymarginsS andS shouldbeequalanddependonthedirectionofthecorrespondingtrips:
vw vv′
Lemma 1. The safetymarginS fortwo trainsdrivinginoppositedirectionsthatenterandleavea stationonthe
vv′
sametrackshouldsatisfy
Svv′ > 12 max{(Lmwzax−Lmwzin),(Lmuvax−Lmuvin)}.
Proof. Theeventactivitynetworkfortwotrains,drivingthroughastationinoppositedirectionsonasingletrackis
shownin Figure1. Inthis case, we havey =y =1, i.e., the solutioncontainsthese conflictingtrips. To simplify
a a˜
notation,wesetp ∗=p +x forallv∈E,andweusethenewtimesofadispatchingtimetable. Inordertoobtaina
v v v
feasibletimetable,theheadwayvariablesg ,h ,g shouldbeequal.Weshowthat
uw′ vv′ wu′
Svv′ > 12 max{(Lmwzax−Lmwzin),(Lmuvax−Lmuvin)}
impliesguw′ =hvv′ =gwu′ =1.
6
Station A Station B Station C
u v w z
driving driving
guw′ hvv′ gwu′
turning turning
gw′u hv′v gu′w
driving driving
z′ w′ v′ u′
Figure2: Example:TrainsareallowedtoturnatB
(i) hvv′ =1 ⇒ gwu′ =1
Ifhvv′ =1thenp ∗w+Svv′ <p ∗v′,becauseof(6). WehaveLu′w=Su′w+Lmu′av′x=Svv′+Lmu′avx′. Ifgwu′ =0,itfollows
that
p ∗w≥p ∗u′+Lmu′ivn′+Svv′
p ∗w+Svv′ ≥p ∗u′+Lmu′ivn′+2Svv′
p ∗v′ ≥p ∗u′+Lmu′ivn′+2Svv′
p ∗v′+Lmu′av′x≥p ∗u′+Lmu′av′x+Lmu′ivn′+2Svv′ ≥p ∗v′+Lmu′ivn′+2Svv′
Lmu′avx′ ≥Lmu′ivn′+2Svv′
Svv′ ≤ 12(Lmu′avx′ −Lmu′ivn′),
whichisincontradictiontotheconditiononS .
vv′
(ii) gwu′ =1 ⇒ hvv′ =1
Assumegwu′ =1andhvv′ =0. Becauseof(4),wehavep ∗u′ ≥p ∗w+Lmwzin+Svv′. Becauseof(6)andhvv′ =0,it
followsthatp ∗u′ ≥p ∗v′+Svv′+Lmwzin+Svv′,whichisincontradictiontotheconditionofSvv′ >0.
(iii) hvv′ =1 ⇒ guw′ =1and
(iv) guw′ =1 ⇒ hvv′ =1canbeshownanalogously. 2
Rememberthatwe cannotalways expressthe headwayconstraintsvia a single variable, asthe conditionguw′ =
hvv′ =gwu′ doesnothold,iftrainsareallowedtoturn(seeFigure2).
Aswedescribedabove,aflowmodelisintegratedtomodelthecirculationoftrains. Constraint(8)ensuresflow
conservationontheevents,i.e.,theoutflowshouldbeequaltotheinflow.Aflowinthenetworkencodesthesequence
of tripsfor eachtrain duringthe observedtime interval. Note thattrainsrepresenta single commodity. Depotscan
consumeflow(9)andinitiateflow(10)withrespecttothemaximumnumberofreplacementtrainsRDiofdepotDi∈D.
Thebig-Mhastobechosensufficientlylargeinordertoyieldacorrectmodel;analogousto[6],weset
M:=Y+ max (p −p +L +S ),
w v vw vw
(v,w)∈A
whereY denotesthemaximumalloweddelayforeacheventandweletLvw =Svv′ =0,onarcswheretheyhavenot
beendefined.InourcaseY isboundedbythecycletime.
Wesummarize:
Observation1. Feasibledispatchingtimetablesp ∗=p +xcorrespondpreciselytothefeasiblesolutionsoftheinteger
programwithconstraints(2)–(14).
7
3.3 Reducing the Sizeofthe Integer Program
The computationalrunningtime forsolvingthe IPis mainlyinfluencedby the numberof precedenceconstraintsin
the system. These constraintsinvolvebig-M constantsandthuslead to a weak LPrelaxation: herewe canset each
g andh witha∈A to 1. Thismakesitdesirabletoreducethenumberofprecedenceconstraintsbyfixingthem
a a head 2
whereverpossible. Thiscanbeachievedbyconsideringtheschedulinglogicinherentin thesystem. Ifa disruption
occurs, decisions for parts of the network imply decisionsfor other parts, e.g., if there are alternating sequencesof
drivingandwaitingactivitieswithnoturningpossibilities. Furthermore,trainsdrivingintothesamedirectioncannot
passeachother. Thus,variablescanbefixedbyusingthenaturalprecedencerelations. Inasecondstep,wecanbuild
blocks of such sequences and contract the driving and waiting activities into just one driving activity with the new
minimumdrivingtime. Becausethedelayofaneventispassedtosuccessiveevents,thetraveltimeisboundedbythe
maximumalloweddelaywithrespecttothebuffertimes. Safetyintervalsareinheritedfromtheoriginalevent-activity
network. The cost of the departureevents is set to the sum of the correspondingcontracteddeparture events. This
processyieldsareducedevent-activitynetworkNr=(Er,Ar).
Lemma2. AninstanceofthereducednetworkNr=(Er,Ar)hasafeasiblesolution,ifandonlyifthecorresponding
instanceoftheoriginalevent-activitynetworkN=(E,A)hasafeasiblesolution,andtheirobjectivesareequal.
Proof. Letp rbeafeasibledispatchingtimetableforthereducedevent-activitynetworkNr=(Er,Ar).Leta=(v,w)∈
Arbeadrivingactivity,andletEvw ⊂E ∪E bethecorrespondingsequenceofeventsfromthestarteventv∈E
seq dep arr dep
totheendeventw∈E .Ifx ≥x inNr,wereducethedelayoftheeventsalongthesequenceaslateaspossible;this
arr v w
isnecessarytopreventblockingoffollowingtrains. Ifx ≤x ,weincreasethedelayassoonaspossibletopreventa
v w
collisionwiththetraininfront.Theotherdirectionfollowsanalogously. 2
NotethattheoveralldelayofadispatchinginthetheoriginalnetworkN thatresultsfromtheapplicationofthe
strategyintheaboveproofmightbehigherthantheminimalpossibledelayinN. Thisimposesnoseriousrestriction,
sinceourobjectiveisinsensitivetodelays.
The reducednetwork Nr usually contains significantly fewer nodes and arcs, resulting in a smaller IP instance.
Thisenablesustoobtainsolutionsforsignificantlylargernetworks;inourexperiments,thenumberofbinaryvariables
wasreducedbyroughlyafactorofthree.
4 Experiments
In this section we report on computational experiments, which we conducted to evaluate the applicability of our
models.Theyarebasedonthereal-lifetimetableofthesubwaylineU6inVienna(Figure3). WeemployedtheFalko
toolofSiemensAG[25]. Timetablesaregivenwithagranularityofonesecond.
Theconsideredsubwaylinehas24stations,withterminalstationsSiebenhirtenandFloridsdorf;theregulartravel
timebetweentheseterminalsis34minutes. Weusetypicalfrequenciesof5or10minutesinourexperiments.Trains
are circulatingbetweenthe terminals. ThestationsFloridsdorf,Michelbeuern,andAlterlaa are connectedto depots
thateachcontainareservetraininourscenarios. SidingsexistatSiebenhirten. Ourinitialtimetablecontainsbuffer
timesofabout10%ofthescheduleddrivingtimesbetweenstationsandattheendoftheline.
We consider a variety of different disruptions with focus on the blockage of one side of the two tracks. Such
disruptions often yield very hard instances and lead to severe problems in the daily operation of a line. During a
disruption, a section of tracks on one side is blocked for a time interval between 5 minutes and 2 hours; until the
sectionisreopened,notrainisallowedtoentertheblockedsection.Thetracktopologyallowsswitchingthetrackand
performingaturnatseveralstations. Tripsaffectedbytheblockedsectioncanusetheunblockedtrackintheopposite
direction. More specifically, we allow trains to pass the switches that are immediately beforeand after the blocked
section. Thecirculationofatraincanbetruncatedbyintroducingaturning.
For each scenario below, we use our model to generate a feasible dispatching timetable. We have to return to
the originaltimetable within 60 minutesafter reopeningthe blocked sections. Choices for trains include turning at
specifiedstations,increasingtheirdelays,orreturningtoadepot;themaximumalloweddelayofatripisequaltothe
cycletime.
8
U6
ke
Siebenhirten ErlaaerSAtlrt.erlaa Philadelphiabr¨uc WestbahnBhurofggassTehaliastrJ.osefsta¨dAtlesrerStSrt.r.MichelbeuernNussdorSfpeirttSetlra.u Floridsdorf
00 02 04 09 15 17 18 20 22 23 26 28 34
Figure3: TheViennasubwaylineU6. Boxesrepresentthestations: awhitebox,ifthestationisconnectedtoadepot
orsiding.ThenumbersbelowarethedrivingtimesfromSiebenhirtentothestations.
4.1 Results
WetestedfourscenariosontheViennaU6thatdifferinlocation,transittimes,thetopologyoftheswitchesandturn
possibilitiestoevaluatetheimpactofdisruptions:
Scenario1: A disruption occurs between Michelbeuern and Westbahnhof. These stations are located close to the
centerofthelineandareheavilyused. Tripsbetweenthesestationsareassignedtotheoppositetrack. Trains
running from Michelbeuern to Westbahnhof have to switch the track after Michelbeuern and return to their
originally assigned track between Burggasse and Westbahnhof. We allow trains to turn at Michelbeuern and
Westbahnhof. Thedrivingtimebetweenthesestationsisabout7minutesandthetrainshavetosharethesame
trackin4stations.
Scenario2: A disruption occurs between Philadelphiabru¨ckeand Erlaaer Straße that are located near the terminal
stationSiebenhirten. TrainshavetoswitchthetrackinfrontofthestationPhiladelphiabru¨ckeandbackagain
beforearrivingatErlaaerStraße. Weallowtrainstoturnatthesestations. Thedrivingtimeis7minutesandthe
trainssharethesametrackin4stations. ThedepotAlterlaaislocatedbetweenthosestations.
Scenario3: A disruption occurs between Nußdorfer Straße and Michelbeuern. Trains have to switch shortly after
NußdorferStraßeandreturntotheiroriginaltrackbeforearrivingatMichelbeuern.Optionsforturningaregiven
atMichelbeuernand(fortrainsgoingtoFloridsdorf)afterSpittelau. Thedrivingtimethroughthebottleneckis
3minutesbutpointsforturningarenotclosetothebottlenecksection.
Scenario4: Here,adisruptionoccursbetweenAlserStraßeandThaliastraße,sothattrainshavetoswitchthetrack
afterMichelbeuernandreturnafterJosephsta¨dterStraße. Thedrivingtimebetweenthesestationsis5minutes
and the optionsfor turningare given at Michelbeuernand Westbahnhof, which are not close to the disrupted
tracksection.
Forourexperiments,weusedtheIPsolverCPLEX12.1onaPCwitha3.0GHzIntelCore2DuoCPUand2Gbytes
ofRAM.
Thetables1–6areorganizedbyscenariosandcyclefrequencies;rowscorrespondtodisruptiontimesinminutes,
indicatedinthefirstcolumn. Columns2showsthenumberofbinariesthatcorrespondtotheprecedenceconstraints
andthe flowontheactivityarcs. Allcomputationswereperformedusingthereducednetwork. Column3givesthe
numberofgeneralvariablesforthedelays. Thefourthcolumnshowstheoriginalnumberoftripsintheundisrupted
timetableduringthe observedtime window. Column5 showsthenumberof tripsin the newdispatchingtimetable,
derivedwithCPLEXinamaximumruntimeof1800seconds,whilethesixthcolumnprovidesthebestboundprovided
byCPLEX.Column7showsthenecessarysolutiontimeinseconds,or(incaseofdelaysof45minutesormore)1800
ifCPLEXcouldnotestablishoptimalityofasolutionwithing1800seconds. Column8indicateswhetheroptimality
could be proved, or the gap with respect to an upper bound. The final column 9 shows the number of trips in the
dispositiontimetable,ifwestopCPLEXfindingasolutioninonly60secondsandsucceedinfindingatleastafeasible
solution.Forthiscomputation,theMIPnodeselectionstrategyofCPLEXissettobestestimatesearch.
9
Table1:SCENARIO1: DISRUPTIONBETWEENMICHELBEUERNANDWESTBAHNHOF,CYCLETIMEOF5MINUTES
dur bin int |Adep| sol u.bound time status 60s
5 628 306 171 165 165.00 0.25 opt 165
10 745 331 177 171 171.00 3.41 opt 171
15 861 356 198 183 183.00 0.87 opt 183
20 978 381 207 189 189.00 9.42 opt 189
30 1215 431 231 207 207.00 24.88 opt 207
45 1578 506 267 231 231.00 141.56 opt 228
60 950 581 303 252 269.63 1800 7.0% 252
90 2721 731 375 276 351.14 1800 27.2% –
120 3528 881 447 – 424.62 1800 – –
Table2: SCENARIO 1: DISRUPTION BETWEEN MICHELBEUERN AND WESTBAHNHOF, CYCLE TIME OF 10 MIN-
UTES
dur bin int |Adep| sol u.bound time status 60s
5 254 150 85 85 85.00 0.08 opt 85
10 307 164 92 92 92.00 0.09 opt 92
15 365 175 97 97 97.00 0.89 opt 97
20 416 189 104 104 104.00 0.13 opt 104
30 524 214 116 110 110.00 4.04 opt 110
45 694 250 133 127 127.00 231.72 opt 127
60 854 289 152 146 146.00 559.87 opt 140
90 1193 364 188 170 188.00 1800 10.5% 170
120 1453 439 224 200 224.00 1800 12.0% 197
Table 3: SCENARIO 2: DISRUPTION BETWEEN PHILADELPHIABRU¨CKE AND ERLAAER STR., CYCLE TIME OF 5
MINUTES
dur bin int |Adep| sol u.bound time status 60s
5 819 327 172 162 169.00 1.51 opt 162
10 913 353 184 184 184.00 0.28 opt 184
15 1092 379 196 183 183.00 5.30 opt 183
20 1231 405 208 192 192.00 7.64 opt 192
30 1515 457 232 209 209.00 59.11 opt 209
45 1956 535 268 236 243.00 1800 2.9% 235
60 2415 613 685 258 284.31 1800 10.2% 252
90 2721 731 375 276 351.14 1800 27.2% 272
120 3528 881 447 – 424.62 1800 – –
Table4: SCENARIO 2: DISRUPTION BETWEEN PHILADELPHIABRU¨CKE AND ERLAAER STR., CYCLE TIME OF 10
MINUTES
dur bin int |Adep| sol u.bound time status 60s
5 348 158 86 86 86.00 0.08 opt 86
10 411 177 92 92 92.00 0.09 opt 92
15 473 184 98 98 98.00 0.14 opt 98
20 511 203 104 104 104.00 0.90 opt 104
30 623 229 116 116 116.00 0.12 opt 116
45 863 262 134 130 130.00 24.65 opt 130
60 1065 307 152 146 146.00 591.15 opt 145
90 1482 385 188 180 184.87 1800 2.7% 174
120 1917 463 224 209 221.00 1800 5.7% 209
10