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Research ArticleScheduling Additional Train Unit Services on Rail Transit Lines
Zhibin Jiang1 Yuyan Tan2 and Oumlzguumlr YalccedilJnkaya3
1 School of Transportation Engineering Key Laboratory of Road and Traffic Engineering of theMinistry of Education Tongji University4800 Caorsquoan Road Shanghai 201804 China
2 Institute of Railway Systems Engineering and Traffic Safety Technical University of Braunschweig Pockelsstraszlige 338106 Braunschweig Germany
3Department of Industrial Engineering Dokuz Eylul University 35160 Buca-Izmir Turkey
Correspondence should be addressed to Yuyan Tan ytantu-bsde
Received 4 June 2014 Accepted 31 July 2014 Published 25 September 2014
Academic Editor Wuhong Wang
Copyright copy 2014 Zhibin Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with the problem of scheduling additional train unit (TU) services in a double parallel rail transit line and amixedinteger programming (MIP) model is formulated for integration strategies of new trains connected by TUs with the objective ofobtaining higher frequencies in some special sections and special time periods due to mass passenger volumes We took timetablescheduling and TUs scheduling as an integrated optimization model with two objectives minimizing travel times of additionaltrains and minimizing shifts of initial trains We illustrated our model using computational experiments drawn from the real railtransit line 16 in Shanghai and reached results which show that rail transit agencies can obtain a reasonable new timetable fordifferent managerial goals in a matter of seconds so the model is well suited to be used in daily operations
1 Introduction
Transit scheduling is the processes of computing the fre-quency of services the number of required vehicles thetiming of their travel and other related operating elementsThe outcomes of scheduling include graphical and numericalschedules for operators and supervisors timetables for thepublic and operating data for a line [1] The rail transittimetable is aimed to meet the passenger demand whichvaries during the hours of a day the day of a week from oneseason to another and so forth [2] On rail transit lines due tothe high frequencies and strict stock capacities in terminalsthe timetable scheduling and the TUs scheduling should beconsidered simultaneously Inserting some new train servicesinto an initial timetable is one of the important methods inthe process of redeveloping a timetable
The primary motivation of this research based on addi-tional demands occurrence in the rail transit lines of Shang-hai These additional demands causing timetabling prob-lems have been determined by the Shanghai ShentongMetroOperation Company which is the responsible authority forthe daily operations The authority thinks it is an important
problem and needed to be solved more efficiently accuratelyand fast Up toMarch 2014 there have been 14 rail transit lines(with an operating route length of 538 kilometers and 329stations) operated in Shanghai On a normal weekday morethan 8 million people use the Shanghai rail transit networkPlanning of the rail transit operations primarily concerns thetimetable and two other main resources the rolling stocksand the crews Planning of these resources undergoes twomain phases (tactical and short-term planning) before theactual operationThe planning horizon in tactical planning isfrom one month up to one year The steps conducted duringthis planning phase are constructing several initial timetables(for working days weekend days holidays etc) which satisfydifferent service demands and allocating the rolling stocksand the crews to the initial timetables On the other hand theshort-term planning phase refers to planning taskswith a timehorizon of a fewdays up to onemonth In this phase the initialplans are adapted to the demands of the corresponding daysSpecial holidays and events that attract a lot of people suchas exhibitions concerts and major sports events generallyrequire an offered capacity in different times and positionsConsequently some train services are required to be inserted
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 954356 13 pageshttpdxdoiorg1011552014954356
2 Mathematical Problems in Engineering
to improve the capacity of some special sections with timewindowsThemost commonway is inserting additional trainservices into the initial timetables
The problem which is called scheduling additional trainunit services (SATUS) problem is a problem in which newtrains connected by a number of TUs start their trip from adepot or reversing tracks after collectively visiting a numberof routes they return to the starting points The SATUSproblem is a complicated one because the efficient circulationof TUs is an important consideration for operators of railtransit trains additionally the large number of trips linksand paths to be considered rapidly increases the numberof variables and constraints in any model developed TheSATUS problem is not a real-time rescheduling problemsince themain difference between it and timetable reschedul-ing in short-term planning or in disruption managementis the absence of uncertainty and the fact that the latter ismuch less time-critical while the first is often thought as atemporary redevelopment strategy of an initial timetable
This paper deals with the problem of SATUS in a doubleparallel rail transit line and a MIP model is formulated forintegration strategies of new trains connected by TUs withthe objective of obtaining higher frequencies in some specialsections and special time periods due to mass passengervolumes
The study contributes a number of new features tocapture the influence of specific elements that have not beenconsidered in studies on the SATUS problem in the relatedliterature First of all the approach decides on timetablesand TUs schedules using an integrated optimization modelaccording to sections and turnback capacities Second amaximum deviation for arrival or departure times of trainsin an initial timetable all-station-stopping policy and expressservice strategy linking orders and time windows of newinserted trains are also considered Finally thismodel has twoobjectives minimizing travel times of additional trains andminimizing shifts of initial trains
The paper is organized as follows In Section 2 a shortreview related to the SATUS problem is provided Afterthat Section 3 introduces a brief summary of the relevantconcepts in the model description A MIP model includingsets parameters decision variables and objective functionsis presented in Section 4 Section 5 illustrates the proposedmodel with an example The conclusions and future studiesare summarized in Section 6
2 Literature Review
The SATUS problem is related to a variety of topics in theliterature The first and foremost is railway transportationTrain scheduling rescheduling and routing problems havehad a great deal of attention in recent years There are twomain timetable variants One of the variants is the periodic(or cyclic) timetable that is repeated every given time periodfor example every hour with only slight differences betweenpeak hours and off-peak hours The other variant is thenonperiodic timetable which allows following the passengerdemands with the frequencies of the trains In both cases
timetables are usually repeated every day although theremaybe differences between weekdays and weekend
Cacchiani et al [3 4] gave a detailed review of theliterature on timetable scheduling The timetable schedulingproblem in a rail transit system in which TUs crews andpassengers are incorporated into a single planning frame-work science is complex various constraints and objectivesshould be considered simultaneously Due to its importanceand complexity which have been acknowledged in variouspublications topics related to this issue have attracted consid-erable attention in the literature A multiphase semiregulartimetable which divides a day into several time periods andeven applies the vehicle-departing interval for each periodmay somehow help to accommodate peak-hour demandwhile maintaining a certain level of service for passengersboarding at nonpeak hours Guihaire and Hao [5] presenteda global review of the crucial strategic and tactical steps oftransit planning and also discussed the scheduling problemwith phase regular for a transit corridor Ceder [6] provided acomprehensive modeling framework for determining vehicledeparture time with either even headways or even averageloads with a special focus on smoothing the transitionsbetween time periods These studies provide useful methodsfor optimizing frequency for a particular time period while aunified framework is critically needed for scheduling meth-ods that can consider uneven headways and time-dependentdemand patterns Jiang et al [7 8] presented a computationaltimetable scheduling method in rail transit line with mul-tiroutes or circle route and a timetable designing softwarenamed Train Plan Maker (TPM) was developed and appliedbymanymetro operation companies in China Niu and Zhou[9] focused on optimizing a passenger train timetable ina heavily congested urban rail corridor A binary integerprogramming model incorporated with passenger loadingand departure events was constructed to provide a theoreticdescription for the problem under consideration Freyss et al[10] focused on the skip-stop operation for rail transit linesusing a single one-way track and the system was modeled byusing a continuous approximation approach
Once the timetable scheduling has been defined therolling stock and TUs assignments must be done An integerprogramming model was considered by Alfieri et al [11] todetermine the rolling stock circulation for multiple rollingstock types on a single line and on a single day and thismodelwas extended by Fioole et al [12] by including combining andsplitting trains as it happens at several locations in the Dutchtimetables Cadarso et al [13] studied the disruptionmanage-ment problem of rapid transit rail networks Besides optimiz-ing timetable and rolling stock schedules they explicitly dealtwith the effects of disruption on the passenger demandsTheyproposed a two-step approach that combines an integratedoptimization model (for the timetable and the rolling stock)with a model for the behaviors of passengers Lin and Kwan[14] proposed a two-phase approach for the TUs schedulingproblemThefirst phase assigned and sequenced train trips toTUs temporarily ignoring some station infrastructure detailswhich was modeled as an integer fixed-charge multicom-modity flow (FCMF) problem The second phase focused onsatisfying the remaining station detailed requirements which
Mathematical Problems in Engineering 3
was modeled as a multidimensional matching problem witha mixed integer linear programming (MILP) formulationEberlein et al [15] studied a real-time deadheading problemin transit operations control Haghani and Banihashemi [16]proposed an innovative multiple depot vehicle schedulingwith route time constraints (MDVSRTC) model to solve bustransit vehicle scheduling problems After that they derived asingle depot vehicle scheduling with route time constraints(SDVSRTC) model to solve the same problem [17] Yu etal [18] presented a partway deadheading strategy for transitoperations to improve transit service of the peak directions oftransit routes
Inserting additional train into an existing timetable isa common technique used in railway systems Burdett andKozan [19] considered techniques for scheduling additionaltrain services integrated into current timetables and involvinggeneral time window constraints fixed operations mainte-nance activities and periods of section unavailability Flieret al [20] addressed the recurring problem of adding a trainpath that is a schedule for a single train in terms of trackallocation in space and time to a given dense timetable on acorridor which is an important subnetwork in form of a pathbetween two major stations
The SATUS problem includes the timetable schedulingand the TUs circulation problems therefore it is usuallymuchmore complex and difficult to solve than the models dealingwith a single phase Cadarso and Marın [2] proposed anintegrated MIP model to adapt the frequencies in a timetabletogether with rolling stock circulation in order to deal withincreased passenger demands and traffic congestion in a rapidtransit network They also took into account the shuntingof rolling stocks in depots Canca et al [21] proposed atactical model to determine optimal policies of short-turningand nonstopping at certain stations considering differentobjectives such as minimizing the passenger overload andpreserving certain level of quality of service
Our study contributes a number of new features tocapture the influence of specific elements which have notbeen studied in the related literature as given in previoussection
3 Problem Description
In this section the SATUS problem in rail transit linesis described in detail Firstly the rail transit line and theroutes are introduced After that we describe the timetableand the TUs circulation problems Then headway and traintraveling times are introduced and finally how the capacityof turnback operation is modeled has been explained
31 Rail Transit Network and Route The rail transit line withbranch linking depots is considered to be a simple networkwith a collection of stations and sections as illustrated inFigure 1 A rail transit network119866 is defined by a set of stations119878 that are connected to each other by a set of sections 119861 Therail transit line in the model consists of parallel double lineswhere trains follow a loop running from a certain station
b5
b1 b2 b3 b4
Intermediate stationDepot link station
TurnbacksDepot
RouteRoute
s1 s2 s3
s6
s4 s5
rs5s1rs1s5
Figure 1 Rail transit line infrastructure definition
s1 s2 s3 s4 s5
s6 1s1s5tr1
tr4tr3
tr1
tr1
tr1 tr1 tr1
tr2 tr2 tr2 tr2
tr2
tr2
tp
Figure 2 Train route infrastructure definition
denoted as a starting point to an end station with right-handrunning rule
A train route is a group of trains that run bidirectionalbetween two stations on the rail transit line All trains inthe same route have the same size capacity and operatingcharacteristics and additionally they always visit the samesequence of stations We define 119903
119904119894 119904119895as a route linked by the
stations 119904119894and 119904119895 Rail transit line can be characterized by two
main train route styles (1) normal cyclic routes and (2) depotlinking routes The first one comprises the daily operationsof fixed train cyclic running paths with trains stopping andproviding passenger loading services (119903
1199041 1199045 1199031199045 1199041
1199031199041 1199044
and1199031199044 1199041
in Figure 1) The latter refers to the route linking depotwith a main turnback station in which trains sometimes donot stop and cannot provide passenger services (119903
1199046 1199041 1199031199041 1199046
1199031199046 1199042
and 1199031199042 1199046
in Figure 2)A train track path is defined as the detailed train running
path from an original station to a destination station includ-ing the specified tracks in all stations Let 119904
119894(tr119895) describe the
track tr119895of the station 119904
119894 then the train track path from 119904
1(tr4)
to 1199045(tr1) in Figure 2 can be expressed by
tp11199041 1199045
= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)
(1)
32 Timetable and TUs Circulation In rail transit lines atime-distance diagram has the line (distance) plotted onthe vertical and time on the horizontal axes As shown inFigure 3 the line is divided in sections with uniform speedsThe plot of every run of a train and TUs indicated by anumber shows all scheduled elements (travel time speedetc) of the train on each section and at each terminalThe horizontal axis also shows headways as time distancesbetween subsequent train runs and cycle time (119879
119888) as time
distances between two successive departures related to the
4 Mathematical Problems in Engineering
Layover time u2 u2
u2 u2
u2
6 7
1
u1
u1
u1 u1
u1Pull-out train
4 9
Tc
582 3
-depot
s1
s2
s3
s4
s5-
Figure 3 Time-distance diagram for a rail transit line
same TUs from a terminal The whole diagram shows trainarrivalsdepartures at each reference point along the linelayover time as well as locations and times where trainsmeetTime-distance diagram can also show pull-outs and pull-ins of trains from depots for operations on some sectionsdifferent stopping times and so forth
In our model the set of trains considered is given by 119879 =
119879ini
cup119879add where119879ini denotes the set of initial trains that have
a prescribed timetable and 119879add denotes the set of additional
trains that need to be inserted to the original timetableFor each train 119894 isin 119879
ini a timetable is specified consistingof the following
(i) an ordered sequence of trains 119905119894
(ii) an ordered sequence of TUs 119880119895
(iii) an ordered sequence of trains linked byTU 119895 and119906119895=
119905119894 119905
119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
(v) the departure time from 119891119894 the arrival time to 119897
119894 and
the arrival and departure times for the intermediatestations in 119878
119894
119891119894 119897119894 of the train 119894
(vi) the exact track path 119896119894that is allocated to the train 119894
on each station(vii) themaximumdeviation for arrival or departure times
of trains(viii) the minimum and the maximum dwell times at each
station in 119878119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
For each train 119894 isin 119879add a timetable is specified consisting
of the following
(i) a sequence of TUs 119906add119895
(ii) an ordered sequence of new trains 119905add119894
(iii) an ordered sequence of trains linked by TU 119895 and
119906add119895
= 119905add119894
119905add119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
sj+1
bk
sj
haa
rpp
hdd rsp
rss
rps
tde
tac
Figure 4 Illustration of headways and train traveling times
(v) the exact track path 119896119894that is allocated to the train 119894
on each station(vi) the desired departure time window from 119891
119894 the
minimum and the maximum dwell times at eachstation in 119878
119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
33 Headways and Train Traveling Times The minimumheadway on a line is determined by the physical charac-teristics of the system (technology methods of driving andcontrol and required degree of safety) and station operations(rate of boardingalighting departure control etc) In ourmodel we consider the express service strategy so theheadways need to be defined separately for departing andarriving Set ℎ
119889119889to be the minimum headway of two suc-
cessive trains departing from stations and ℎ119886119886
the minimumheadway of two successive trains arriving to stations asshown in Figure 4 Each time when an intermediate station ispassed by a train the spent times in decelerating stoppingand accelerating of the vehicle are saved at the successivestation So this model considers acceleration time (119905ac) anddeceleration time (119905de) as shown in Figure 4 There are fourexecution modes for train traveling at section 119887
119896 namely (1)
bypassing stations 119904119895and 119904119895+1
(119903119901119901) (2) bypassing station 119904
119895
but stopping at station 119904119895+1
(119903119901119904) (3) stopping at station 119904
119895but
bypassing station 119904119895+1
(119903119904119901) and (4) stopping at both stations
119904119895and 119904119895+1
(119903119904119904) So 119903
119901119904 119903119904119901 and 119903
119904119904can be calculated by the
following respectively
119903119901119904
= 119903119901119901
+ 119905de
119903119904119901
= 119903119901119901
+ 119905ac
119903119904119904
= 119903119901119901
+ 119905ac + 119905de
(2)
34 Layover Time and Turnback Operation Layover time isthe time between the scheduled arrival and departure of avehicle at a transit terminal Minimum layover time includesthe dwell time for alighting and boarding of passengersthe time for changing the train operator and conductingany necessary inspections and brake tests and the time formoving and locking the crossover switches and the time forrecovery of the schedule if it is needed Maximum layover
Mathematical Problems in Engineering 5
T2
T1
Platform
tbTAS12
u1 u2u3
654321
tRst
Number of crossing points le 1
(a) Turnback operation with crossover located in advance of astation (TAS)
T2
T1
Platform
u1
u2u3
65 43 21
T3
T4
tbTBS12
Number of crossing points le 3
(b) Turnback operation with crossover located in back of astation (TBS)
Figure 5 Track occupation of turnback operation process at a terminal
time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate
There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V
3in Figure 5(a)) arrival to the station must insure
that the first departing train (V2in Figure 5(a)) which linked
with the first arrival train (V1in Figure 5(a)) has left from
the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V
119894 V119895) in which the train V
119894and the
consecutive train V119895share the same TUs at a terminal can be
calculated by
tbTASV119894 V119895 isin [119905119886
V119894 119905119889
V119895 + 119905119877
st] (3)
where 119905119886
V119894 119905119889
V119895 are the arrival time of train V119894and the departure
time of train V119895at the terminal respectively So the capacity
constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one
On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot
be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V
119894 V119895) at the terminal can be calculated by
tbTBSV119894 V119895 isin [119905119886
V119894 119905119889
V119895] (4)
4 Model Description
The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions
41 Sets The sets below contain the basic information for ourmathematical model
119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904
119894 119904119895) in
the rail transit line with 119904119894 119904119895isin 119878
119878119879 set of turnback stations
119879 = 119879ini
cup 119879add set of all trains consisting of
additional trains 119879add and initial trains 119879ini119880 = 119880
inicup119880
add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894
1015840
) with 119894 lt 1198941015840 when the
train 119894 and the consecutive train 1198941015840 share the same TUs
at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879
119878119894
isin 119878 set of stations that the train 119894 visits119861119894
isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
to improve the capacity of some special sections with timewindowsThemost commonway is inserting additional trainservices into the initial timetables
The problem which is called scheduling additional trainunit services (SATUS) problem is a problem in which newtrains connected by a number of TUs start their trip from adepot or reversing tracks after collectively visiting a numberof routes they return to the starting points The SATUSproblem is a complicated one because the efficient circulationof TUs is an important consideration for operators of railtransit trains additionally the large number of trips linksand paths to be considered rapidly increases the numberof variables and constraints in any model developed TheSATUS problem is not a real-time rescheduling problemsince themain difference between it and timetable reschedul-ing in short-term planning or in disruption managementis the absence of uncertainty and the fact that the latter ismuch less time-critical while the first is often thought as atemporary redevelopment strategy of an initial timetable
This paper deals with the problem of SATUS in a doubleparallel rail transit line and a MIP model is formulated forintegration strategies of new trains connected by TUs withthe objective of obtaining higher frequencies in some specialsections and special time periods due to mass passengervolumes
The study contributes a number of new features tocapture the influence of specific elements that have not beenconsidered in studies on the SATUS problem in the relatedliterature First of all the approach decides on timetablesand TUs schedules using an integrated optimization modelaccording to sections and turnback capacities Second amaximum deviation for arrival or departure times of trainsin an initial timetable all-station-stopping policy and expressservice strategy linking orders and time windows of newinserted trains are also considered Finally thismodel has twoobjectives minimizing travel times of additional trains andminimizing shifts of initial trains
The paper is organized as follows In Section 2 a shortreview related to the SATUS problem is provided Afterthat Section 3 introduces a brief summary of the relevantconcepts in the model description A MIP model includingsets parameters decision variables and objective functionsis presented in Section 4 Section 5 illustrates the proposedmodel with an example The conclusions and future studiesare summarized in Section 6
2 Literature Review
The SATUS problem is related to a variety of topics in theliterature The first and foremost is railway transportationTrain scheduling rescheduling and routing problems havehad a great deal of attention in recent years There are twomain timetable variants One of the variants is the periodic(or cyclic) timetable that is repeated every given time periodfor example every hour with only slight differences betweenpeak hours and off-peak hours The other variant is thenonperiodic timetable which allows following the passengerdemands with the frequencies of the trains In both cases
timetables are usually repeated every day although theremaybe differences between weekdays and weekend
Cacchiani et al [3 4] gave a detailed review of theliterature on timetable scheduling The timetable schedulingproblem in a rail transit system in which TUs crews andpassengers are incorporated into a single planning frame-work science is complex various constraints and objectivesshould be considered simultaneously Due to its importanceand complexity which have been acknowledged in variouspublications topics related to this issue have attracted consid-erable attention in the literature A multiphase semiregulartimetable which divides a day into several time periods andeven applies the vehicle-departing interval for each periodmay somehow help to accommodate peak-hour demandwhile maintaining a certain level of service for passengersboarding at nonpeak hours Guihaire and Hao [5] presenteda global review of the crucial strategic and tactical steps oftransit planning and also discussed the scheduling problemwith phase regular for a transit corridor Ceder [6] provided acomprehensive modeling framework for determining vehicledeparture time with either even headways or even averageloads with a special focus on smoothing the transitionsbetween time periods These studies provide useful methodsfor optimizing frequency for a particular time period while aunified framework is critically needed for scheduling meth-ods that can consider uneven headways and time-dependentdemand patterns Jiang et al [7 8] presented a computationaltimetable scheduling method in rail transit line with mul-tiroutes or circle route and a timetable designing softwarenamed Train Plan Maker (TPM) was developed and appliedbymanymetro operation companies in China Niu and Zhou[9] focused on optimizing a passenger train timetable ina heavily congested urban rail corridor A binary integerprogramming model incorporated with passenger loadingand departure events was constructed to provide a theoreticdescription for the problem under consideration Freyss et al[10] focused on the skip-stop operation for rail transit linesusing a single one-way track and the system was modeled byusing a continuous approximation approach
Once the timetable scheduling has been defined therolling stock and TUs assignments must be done An integerprogramming model was considered by Alfieri et al [11] todetermine the rolling stock circulation for multiple rollingstock types on a single line and on a single day and thismodelwas extended by Fioole et al [12] by including combining andsplitting trains as it happens at several locations in the Dutchtimetables Cadarso et al [13] studied the disruptionmanage-ment problem of rapid transit rail networks Besides optimiz-ing timetable and rolling stock schedules they explicitly dealtwith the effects of disruption on the passenger demandsTheyproposed a two-step approach that combines an integratedoptimization model (for the timetable and the rolling stock)with a model for the behaviors of passengers Lin and Kwan[14] proposed a two-phase approach for the TUs schedulingproblemThefirst phase assigned and sequenced train trips toTUs temporarily ignoring some station infrastructure detailswhich was modeled as an integer fixed-charge multicom-modity flow (FCMF) problem The second phase focused onsatisfying the remaining station detailed requirements which
Mathematical Problems in Engineering 3
was modeled as a multidimensional matching problem witha mixed integer linear programming (MILP) formulationEberlein et al [15] studied a real-time deadheading problemin transit operations control Haghani and Banihashemi [16]proposed an innovative multiple depot vehicle schedulingwith route time constraints (MDVSRTC) model to solve bustransit vehicle scheduling problems After that they derived asingle depot vehicle scheduling with route time constraints(SDVSRTC) model to solve the same problem [17] Yu etal [18] presented a partway deadheading strategy for transitoperations to improve transit service of the peak directions oftransit routes
Inserting additional train into an existing timetable isa common technique used in railway systems Burdett andKozan [19] considered techniques for scheduling additionaltrain services integrated into current timetables and involvinggeneral time window constraints fixed operations mainte-nance activities and periods of section unavailability Flieret al [20] addressed the recurring problem of adding a trainpath that is a schedule for a single train in terms of trackallocation in space and time to a given dense timetable on acorridor which is an important subnetwork in form of a pathbetween two major stations
The SATUS problem includes the timetable schedulingand the TUs circulation problems therefore it is usuallymuchmore complex and difficult to solve than the models dealingwith a single phase Cadarso and Marın [2] proposed anintegrated MIP model to adapt the frequencies in a timetabletogether with rolling stock circulation in order to deal withincreased passenger demands and traffic congestion in a rapidtransit network They also took into account the shuntingof rolling stocks in depots Canca et al [21] proposed atactical model to determine optimal policies of short-turningand nonstopping at certain stations considering differentobjectives such as minimizing the passenger overload andpreserving certain level of quality of service
Our study contributes a number of new features tocapture the influence of specific elements which have notbeen studied in the related literature as given in previoussection
3 Problem Description
In this section the SATUS problem in rail transit linesis described in detail Firstly the rail transit line and theroutes are introduced After that we describe the timetableand the TUs circulation problems Then headway and traintraveling times are introduced and finally how the capacityof turnback operation is modeled has been explained
31 Rail Transit Network and Route The rail transit line withbranch linking depots is considered to be a simple networkwith a collection of stations and sections as illustrated inFigure 1 A rail transit network119866 is defined by a set of stations119878 that are connected to each other by a set of sections 119861 Therail transit line in the model consists of parallel double lineswhere trains follow a loop running from a certain station
b5
b1 b2 b3 b4
Intermediate stationDepot link station
TurnbacksDepot
RouteRoute
s1 s2 s3
s6
s4 s5
rs5s1rs1s5
Figure 1 Rail transit line infrastructure definition
s1 s2 s3 s4 s5
s6 1s1s5tr1
tr4tr3
tr1
tr1
tr1 tr1 tr1
tr2 tr2 tr2 tr2
tr2
tr2
tp
Figure 2 Train route infrastructure definition
denoted as a starting point to an end station with right-handrunning rule
A train route is a group of trains that run bidirectionalbetween two stations on the rail transit line All trains inthe same route have the same size capacity and operatingcharacteristics and additionally they always visit the samesequence of stations We define 119903
119904119894 119904119895as a route linked by the
stations 119904119894and 119904119895 Rail transit line can be characterized by two
main train route styles (1) normal cyclic routes and (2) depotlinking routes The first one comprises the daily operationsof fixed train cyclic running paths with trains stopping andproviding passenger loading services (119903
1199041 1199045 1199031199045 1199041
1199031199041 1199044
and1199031199044 1199041
in Figure 1) The latter refers to the route linking depotwith a main turnback station in which trains sometimes donot stop and cannot provide passenger services (119903
1199046 1199041 1199031199041 1199046
1199031199046 1199042
and 1199031199042 1199046
in Figure 2)A train track path is defined as the detailed train running
path from an original station to a destination station includ-ing the specified tracks in all stations Let 119904
119894(tr119895) describe the
track tr119895of the station 119904
119894 then the train track path from 119904
1(tr4)
to 1199045(tr1) in Figure 2 can be expressed by
tp11199041 1199045
= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)
(1)
32 Timetable and TUs Circulation In rail transit lines atime-distance diagram has the line (distance) plotted onthe vertical and time on the horizontal axes As shown inFigure 3 the line is divided in sections with uniform speedsThe plot of every run of a train and TUs indicated by anumber shows all scheduled elements (travel time speedetc) of the train on each section and at each terminalThe horizontal axis also shows headways as time distancesbetween subsequent train runs and cycle time (119879
119888) as time
distances between two successive departures related to the
4 Mathematical Problems in Engineering
Layover time u2 u2
u2 u2
u2
6 7
1
u1
u1
u1 u1
u1Pull-out train
4 9
Tc
582 3
-depot
s1
s2
s3
s4
s5-
Figure 3 Time-distance diagram for a rail transit line
same TUs from a terminal The whole diagram shows trainarrivalsdepartures at each reference point along the linelayover time as well as locations and times where trainsmeetTime-distance diagram can also show pull-outs and pull-ins of trains from depots for operations on some sectionsdifferent stopping times and so forth
In our model the set of trains considered is given by 119879 =
119879ini
cup119879add where119879ini denotes the set of initial trains that have
a prescribed timetable and 119879add denotes the set of additional
trains that need to be inserted to the original timetableFor each train 119894 isin 119879
ini a timetable is specified consistingof the following
(i) an ordered sequence of trains 119905119894
(ii) an ordered sequence of TUs 119880119895
(iii) an ordered sequence of trains linked byTU 119895 and119906119895=
119905119894 119905
119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
(v) the departure time from 119891119894 the arrival time to 119897
119894 and
the arrival and departure times for the intermediatestations in 119878
119894
119891119894 119897119894 of the train 119894
(vi) the exact track path 119896119894that is allocated to the train 119894
on each station(vii) themaximumdeviation for arrival or departure times
of trains(viii) the minimum and the maximum dwell times at each
station in 119878119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
For each train 119894 isin 119879add a timetable is specified consisting
of the following
(i) a sequence of TUs 119906add119895
(ii) an ordered sequence of new trains 119905add119894
(iii) an ordered sequence of trains linked by TU 119895 and
119906add119895
= 119905add119894
119905add119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
sj+1
bk
sj
haa
rpp
hdd rsp
rss
rps
tde
tac
Figure 4 Illustration of headways and train traveling times
(v) the exact track path 119896119894that is allocated to the train 119894
on each station(vi) the desired departure time window from 119891
119894 the
minimum and the maximum dwell times at eachstation in 119878
119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
33 Headways and Train Traveling Times The minimumheadway on a line is determined by the physical charac-teristics of the system (technology methods of driving andcontrol and required degree of safety) and station operations(rate of boardingalighting departure control etc) In ourmodel we consider the express service strategy so theheadways need to be defined separately for departing andarriving Set ℎ
119889119889to be the minimum headway of two suc-
cessive trains departing from stations and ℎ119886119886
the minimumheadway of two successive trains arriving to stations asshown in Figure 4 Each time when an intermediate station ispassed by a train the spent times in decelerating stoppingand accelerating of the vehicle are saved at the successivestation So this model considers acceleration time (119905ac) anddeceleration time (119905de) as shown in Figure 4 There are fourexecution modes for train traveling at section 119887
119896 namely (1)
bypassing stations 119904119895and 119904119895+1
(119903119901119901) (2) bypassing station 119904
119895
but stopping at station 119904119895+1
(119903119901119904) (3) stopping at station 119904
119895but
bypassing station 119904119895+1
(119903119904119901) and (4) stopping at both stations
119904119895and 119904119895+1
(119903119904119904) So 119903
119901119904 119903119904119901 and 119903
119904119904can be calculated by the
following respectively
119903119901119904
= 119903119901119901
+ 119905de
119903119904119901
= 119903119901119901
+ 119905ac
119903119904119904
= 119903119901119901
+ 119905ac + 119905de
(2)
34 Layover Time and Turnback Operation Layover time isthe time between the scheduled arrival and departure of avehicle at a transit terminal Minimum layover time includesthe dwell time for alighting and boarding of passengersthe time for changing the train operator and conductingany necessary inspections and brake tests and the time formoving and locking the crossover switches and the time forrecovery of the schedule if it is needed Maximum layover
Mathematical Problems in Engineering 5
T2
T1
Platform
tbTAS12
u1 u2u3
654321
tRst
Number of crossing points le 1
(a) Turnback operation with crossover located in advance of astation (TAS)
T2
T1
Platform
u1
u2u3
65 43 21
T3
T4
tbTBS12
Number of crossing points le 3
(b) Turnback operation with crossover located in back of astation (TBS)
Figure 5 Track occupation of turnback operation process at a terminal
time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate
There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V
3in Figure 5(a)) arrival to the station must insure
that the first departing train (V2in Figure 5(a)) which linked
with the first arrival train (V1in Figure 5(a)) has left from
the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V
119894 V119895) in which the train V
119894and the
consecutive train V119895share the same TUs at a terminal can be
calculated by
tbTASV119894 V119895 isin [119905119886
V119894 119905119889
V119895 + 119905119877
st] (3)
where 119905119886
V119894 119905119889
V119895 are the arrival time of train V119894and the departure
time of train V119895at the terminal respectively So the capacity
constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one
On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot
be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V
119894 V119895) at the terminal can be calculated by
tbTBSV119894 V119895 isin [119905119886
V119894 119905119889
V119895] (4)
4 Model Description
The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions
41 Sets The sets below contain the basic information for ourmathematical model
119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904
119894 119904119895) in
the rail transit line with 119904119894 119904119895isin 119878
119878119879 set of turnback stations
119879 = 119879ini
cup 119879add set of all trains consisting of
additional trains 119879add and initial trains 119879ini119880 = 119880
inicup119880
add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894
1015840
) with 119894 lt 1198941015840 when the
train 119894 and the consecutive train 1198941015840 share the same TUs
at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879
119878119894
isin 119878 set of stations that the train 119894 visits119861119894
isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
was modeled as a multidimensional matching problem witha mixed integer linear programming (MILP) formulationEberlein et al [15] studied a real-time deadheading problemin transit operations control Haghani and Banihashemi [16]proposed an innovative multiple depot vehicle schedulingwith route time constraints (MDVSRTC) model to solve bustransit vehicle scheduling problems After that they derived asingle depot vehicle scheduling with route time constraints(SDVSRTC) model to solve the same problem [17] Yu etal [18] presented a partway deadheading strategy for transitoperations to improve transit service of the peak directions oftransit routes
Inserting additional train into an existing timetable isa common technique used in railway systems Burdett andKozan [19] considered techniques for scheduling additionaltrain services integrated into current timetables and involvinggeneral time window constraints fixed operations mainte-nance activities and periods of section unavailability Flieret al [20] addressed the recurring problem of adding a trainpath that is a schedule for a single train in terms of trackallocation in space and time to a given dense timetable on acorridor which is an important subnetwork in form of a pathbetween two major stations
The SATUS problem includes the timetable schedulingand the TUs circulation problems therefore it is usuallymuchmore complex and difficult to solve than the models dealingwith a single phase Cadarso and Marın [2] proposed anintegrated MIP model to adapt the frequencies in a timetabletogether with rolling stock circulation in order to deal withincreased passenger demands and traffic congestion in a rapidtransit network They also took into account the shuntingof rolling stocks in depots Canca et al [21] proposed atactical model to determine optimal policies of short-turningand nonstopping at certain stations considering differentobjectives such as minimizing the passenger overload andpreserving certain level of quality of service
Our study contributes a number of new features tocapture the influence of specific elements which have notbeen studied in the related literature as given in previoussection
3 Problem Description
In this section the SATUS problem in rail transit linesis described in detail Firstly the rail transit line and theroutes are introduced After that we describe the timetableand the TUs circulation problems Then headway and traintraveling times are introduced and finally how the capacityof turnback operation is modeled has been explained
31 Rail Transit Network and Route The rail transit line withbranch linking depots is considered to be a simple networkwith a collection of stations and sections as illustrated inFigure 1 A rail transit network119866 is defined by a set of stations119878 that are connected to each other by a set of sections 119861 Therail transit line in the model consists of parallel double lineswhere trains follow a loop running from a certain station
b5
b1 b2 b3 b4
Intermediate stationDepot link station
TurnbacksDepot
RouteRoute
s1 s2 s3
s6
s4 s5
rs5s1rs1s5
Figure 1 Rail transit line infrastructure definition
s1 s2 s3 s4 s5
s6 1s1s5tr1
tr4tr3
tr1
tr1
tr1 tr1 tr1
tr2 tr2 tr2 tr2
tr2
tr2
tp
Figure 2 Train route infrastructure definition
denoted as a starting point to an end station with right-handrunning rule
A train route is a group of trains that run bidirectionalbetween two stations on the rail transit line All trains inthe same route have the same size capacity and operatingcharacteristics and additionally they always visit the samesequence of stations We define 119903
119904119894 119904119895as a route linked by the
stations 119904119894and 119904119895 Rail transit line can be characterized by two
main train route styles (1) normal cyclic routes and (2) depotlinking routes The first one comprises the daily operationsof fixed train cyclic running paths with trains stopping andproviding passenger loading services (119903
1199041 1199045 1199031199045 1199041
1199031199041 1199044
and1199031199044 1199041
in Figure 1) The latter refers to the route linking depotwith a main turnback station in which trains sometimes donot stop and cannot provide passenger services (119903
1199046 1199041 1199031199041 1199046
1199031199046 1199042
and 1199031199042 1199046
in Figure 2)A train track path is defined as the detailed train running
path from an original station to a destination station includ-ing the specified tracks in all stations Let 119904
119894(tr119895) describe the
track tr119895of the station 119904
119894 then the train track path from 119904
1(tr4)
to 1199045(tr1) in Figure 2 can be expressed by
tp11199041 1199045
= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)
(1)
32 Timetable and TUs Circulation In rail transit lines atime-distance diagram has the line (distance) plotted onthe vertical and time on the horizontal axes As shown inFigure 3 the line is divided in sections with uniform speedsThe plot of every run of a train and TUs indicated by anumber shows all scheduled elements (travel time speedetc) of the train on each section and at each terminalThe horizontal axis also shows headways as time distancesbetween subsequent train runs and cycle time (119879
119888) as time
distances between two successive departures related to the
4 Mathematical Problems in Engineering
Layover time u2 u2
u2 u2
u2
6 7
1
u1
u1
u1 u1
u1Pull-out train
4 9
Tc
582 3
-depot
s1
s2
s3
s4
s5-
Figure 3 Time-distance diagram for a rail transit line
same TUs from a terminal The whole diagram shows trainarrivalsdepartures at each reference point along the linelayover time as well as locations and times where trainsmeetTime-distance diagram can also show pull-outs and pull-ins of trains from depots for operations on some sectionsdifferent stopping times and so forth
In our model the set of trains considered is given by 119879 =
119879ini
cup119879add where119879ini denotes the set of initial trains that have
a prescribed timetable and 119879add denotes the set of additional
trains that need to be inserted to the original timetableFor each train 119894 isin 119879
ini a timetable is specified consistingof the following
(i) an ordered sequence of trains 119905119894
(ii) an ordered sequence of TUs 119880119895
(iii) an ordered sequence of trains linked byTU 119895 and119906119895=
119905119894 119905
119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
(v) the departure time from 119891119894 the arrival time to 119897
119894 and
the arrival and departure times for the intermediatestations in 119878
119894
119891119894 119897119894 of the train 119894
(vi) the exact track path 119896119894that is allocated to the train 119894
on each station(vii) themaximumdeviation for arrival or departure times
of trains(viii) the minimum and the maximum dwell times at each
station in 119878119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
For each train 119894 isin 119879add a timetable is specified consisting
of the following
(i) a sequence of TUs 119906add119895
(ii) an ordered sequence of new trains 119905add119894
(iii) an ordered sequence of trains linked by TU 119895 and
119906add119895
= 119905add119894
119905add119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
sj+1
bk
sj
haa
rpp
hdd rsp
rss
rps
tde
tac
Figure 4 Illustration of headways and train traveling times
(v) the exact track path 119896119894that is allocated to the train 119894
on each station(vi) the desired departure time window from 119891
119894 the
minimum and the maximum dwell times at eachstation in 119878
119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
33 Headways and Train Traveling Times The minimumheadway on a line is determined by the physical charac-teristics of the system (technology methods of driving andcontrol and required degree of safety) and station operations(rate of boardingalighting departure control etc) In ourmodel we consider the express service strategy so theheadways need to be defined separately for departing andarriving Set ℎ
119889119889to be the minimum headway of two suc-
cessive trains departing from stations and ℎ119886119886
the minimumheadway of two successive trains arriving to stations asshown in Figure 4 Each time when an intermediate station ispassed by a train the spent times in decelerating stoppingand accelerating of the vehicle are saved at the successivestation So this model considers acceleration time (119905ac) anddeceleration time (119905de) as shown in Figure 4 There are fourexecution modes for train traveling at section 119887
119896 namely (1)
bypassing stations 119904119895and 119904119895+1
(119903119901119901) (2) bypassing station 119904
119895
but stopping at station 119904119895+1
(119903119901119904) (3) stopping at station 119904
119895but
bypassing station 119904119895+1
(119903119904119901) and (4) stopping at both stations
119904119895and 119904119895+1
(119903119904119904) So 119903
119901119904 119903119904119901 and 119903
119904119904can be calculated by the
following respectively
119903119901119904
= 119903119901119901
+ 119905de
119903119904119901
= 119903119901119901
+ 119905ac
119903119904119904
= 119903119901119901
+ 119905ac + 119905de
(2)
34 Layover Time and Turnback Operation Layover time isthe time between the scheduled arrival and departure of avehicle at a transit terminal Minimum layover time includesthe dwell time for alighting and boarding of passengersthe time for changing the train operator and conductingany necessary inspections and brake tests and the time formoving and locking the crossover switches and the time forrecovery of the schedule if it is needed Maximum layover
Mathematical Problems in Engineering 5
T2
T1
Platform
tbTAS12
u1 u2u3
654321
tRst
Number of crossing points le 1
(a) Turnback operation with crossover located in advance of astation (TAS)
T2
T1
Platform
u1
u2u3
65 43 21
T3
T4
tbTBS12
Number of crossing points le 3
(b) Turnback operation with crossover located in back of astation (TBS)
Figure 5 Track occupation of turnback operation process at a terminal
time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate
There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V
3in Figure 5(a)) arrival to the station must insure
that the first departing train (V2in Figure 5(a)) which linked
with the first arrival train (V1in Figure 5(a)) has left from
the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V
119894 V119895) in which the train V
119894and the
consecutive train V119895share the same TUs at a terminal can be
calculated by
tbTASV119894 V119895 isin [119905119886
V119894 119905119889
V119895 + 119905119877
st] (3)
where 119905119886
V119894 119905119889
V119895 are the arrival time of train V119894and the departure
time of train V119895at the terminal respectively So the capacity
constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one
On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot
be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V
119894 V119895) at the terminal can be calculated by
tbTBSV119894 V119895 isin [119905119886
V119894 119905119889
V119895] (4)
4 Model Description
The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions
41 Sets The sets below contain the basic information for ourmathematical model
119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904
119894 119904119895) in
the rail transit line with 119904119894 119904119895isin 119878
119878119879 set of turnback stations
119879 = 119879ini
cup 119879add set of all trains consisting of
additional trains 119879add and initial trains 119879ini119880 = 119880
inicup119880
add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894
1015840
) with 119894 lt 1198941015840 when the
train 119894 and the consecutive train 1198941015840 share the same TUs
at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879
119878119894
isin 119878 set of stations that the train 119894 visits119861119894
isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Layover time u2 u2
u2 u2
u2
6 7
1
u1
u1
u1 u1
u1Pull-out train
4 9
Tc
582 3
-depot
s1
s2
s3
s4
s5-
Figure 3 Time-distance diagram for a rail transit line
same TUs from a terminal The whole diagram shows trainarrivalsdepartures at each reference point along the linelayover time as well as locations and times where trainsmeetTime-distance diagram can also show pull-outs and pull-ins of trains from depots for operations on some sectionsdifferent stopping times and so forth
In our model the set of trains considered is given by 119879 =
119879ini
cup119879add where119879ini denotes the set of initial trains that have
a prescribed timetable and 119879add denotes the set of additional
trains that need to be inserted to the original timetableFor each train 119894 isin 119879
ini a timetable is specified consistingof the following
(i) an ordered sequence of trains 119905119894
(ii) an ordered sequence of TUs 119880119895
(iii) an ordered sequence of trains linked byTU 119895 and119906119895=
119905119894 119905
119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
(v) the departure time from 119891119894 the arrival time to 119897
119894 and
the arrival and departure times for the intermediatestations in 119878
119894
119891119894 119897119894 of the train 119894
(vi) the exact track path 119896119894that is allocated to the train 119894
on each station(vii) themaximumdeviation for arrival or departure times
of trains(viii) the minimum and the maximum dwell times at each
station in 119878119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
For each train 119894 isin 119879add a timetable is specified consisting
of the following
(i) a sequence of TUs 119906add119895
(ii) an ordered sequence of new trains 119905add119894
(iii) an ordered sequence of trains linked by TU 119895 and
119906add119895
= 119905add119894
119905add119901
(iv) an ordered sequence of stations 119878119894
= 119891119894 119897119894 isin 119878
that the train 119894 visits where 119891119894is the first (origin)
station and 119897119894is the last (destination) station
sj+1
bk
sj
haa
rpp
hdd rsp
rss
rps
tde
tac
Figure 4 Illustration of headways and train traveling times
(v) the exact track path 119896119894that is allocated to the train 119894
on each station(vi) the desired departure time window from 119891
119894 the
minimum and the maximum dwell times at eachstation in 119878
119894
119891119894 119897119894 and the trip time at each section
119887 = 119895 1198951015840
with 119895 1198951015840
isin 119878119894
33 Headways and Train Traveling Times The minimumheadway on a line is determined by the physical charac-teristics of the system (technology methods of driving andcontrol and required degree of safety) and station operations(rate of boardingalighting departure control etc) In ourmodel we consider the express service strategy so theheadways need to be defined separately for departing andarriving Set ℎ
119889119889to be the minimum headway of two suc-
cessive trains departing from stations and ℎ119886119886
the minimumheadway of two successive trains arriving to stations asshown in Figure 4 Each time when an intermediate station ispassed by a train the spent times in decelerating stoppingand accelerating of the vehicle are saved at the successivestation So this model considers acceleration time (119905ac) anddeceleration time (119905de) as shown in Figure 4 There are fourexecution modes for train traveling at section 119887
119896 namely (1)
bypassing stations 119904119895and 119904119895+1
(119903119901119901) (2) bypassing station 119904
119895
but stopping at station 119904119895+1
(119903119901119904) (3) stopping at station 119904
119895but
bypassing station 119904119895+1
(119903119904119901) and (4) stopping at both stations
119904119895and 119904119895+1
(119903119904119904) So 119903
119901119904 119903119904119901 and 119903
119904119904can be calculated by the
following respectively
119903119901119904
= 119903119901119901
+ 119905de
119903119904119901
= 119903119901119901
+ 119905ac
119903119904119904
= 119903119901119901
+ 119905ac + 119905de
(2)
34 Layover Time and Turnback Operation Layover time isthe time between the scheduled arrival and departure of avehicle at a transit terminal Minimum layover time includesthe dwell time for alighting and boarding of passengersthe time for changing the train operator and conductingany necessary inspections and brake tests and the time formoving and locking the crossover switches and the time forrecovery of the schedule if it is needed Maximum layover
Mathematical Problems in Engineering 5
T2
T1
Platform
tbTAS12
u1 u2u3
654321
tRst
Number of crossing points le 1
(a) Turnback operation with crossover located in advance of astation (TAS)
T2
T1
Platform
u1
u2u3
65 43 21
T3
T4
tbTBS12
Number of crossing points le 3
(b) Turnback operation with crossover located in back of astation (TBS)
Figure 5 Track occupation of turnback operation process at a terminal
time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate
There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V
3in Figure 5(a)) arrival to the station must insure
that the first departing train (V2in Figure 5(a)) which linked
with the first arrival train (V1in Figure 5(a)) has left from
the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V
119894 V119895) in which the train V
119894and the
consecutive train V119895share the same TUs at a terminal can be
calculated by
tbTASV119894 V119895 isin [119905119886
V119894 119905119889
V119895 + 119905119877
st] (3)
where 119905119886
V119894 119905119889
V119895 are the arrival time of train V119894and the departure
time of train V119895at the terminal respectively So the capacity
constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one
On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot
be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V
119894 V119895) at the terminal can be calculated by
tbTBSV119894 V119895 isin [119905119886
V119894 119905119889
V119895] (4)
4 Model Description
The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions
41 Sets The sets below contain the basic information for ourmathematical model
119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904
119894 119904119895) in
the rail transit line with 119904119894 119904119895isin 119878
119878119879 set of turnback stations
119879 = 119879ini
cup 119879add set of all trains consisting of
additional trains 119879add and initial trains 119879ini119880 = 119880
inicup119880
add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894
1015840
) with 119894 lt 1198941015840 when the
train 119894 and the consecutive train 1198941015840 share the same TUs
at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879
119878119894
isin 119878 set of stations that the train 119894 visits119861119894
isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
T2
T1
Platform
tbTAS12
u1 u2u3
654321
tRst
Number of crossing points le 1
(a) Turnback operation with crossover located in advance of astation (TAS)
T2
T1
Platform
u1
u2u3
65 43 21
T3
T4
tbTBS12
Number of crossing points le 3
(b) Turnback operation with crossover located in back of astation (TBS)
Figure 5 Track occupation of turnback operation process at a terminal
time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate
There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V
3in Figure 5(a)) arrival to the station must insure
that the first departing train (V2in Figure 5(a)) which linked
with the first arrival train (V1in Figure 5(a)) has left from
the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V
119894 V119895) in which the train V
119894and the
consecutive train V119895share the same TUs at a terminal can be
calculated by
tbTASV119894 V119895 isin [119905119886
V119894 119905119889
V119895 + 119905119877
st] (3)
where 119905119886
V119894 119905119889
V119895 are the arrival time of train V119894and the departure
time of train V119895at the terminal respectively So the capacity
constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one
On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot
be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V
119894 V119895) at the terminal can be calculated by
tbTBSV119894 V119895 isin [119905119886
V119894 119905119889
V119895] (4)
4 Model Description
The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions
41 Sets The sets below contain the basic information for ourmathematical model
119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904
119894 119904119895) in
the rail transit line with 119904119894 119904119895isin 119878
119878119879 set of turnback stations
119879 = 119879ini
cup 119879add set of all trains consisting of
additional trains 119879add and initial trains 119879ini119880 = 119880
inicup119880
add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894
1015840
) with 119894 lt 1198941015840 when the
train 119894 and the consecutive train 1198941015840 share the same TUs
at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879
119878119894
isin 119878 set of stations that the train 119894 visits119861119894
isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
119891119894 set of first (starting) travelling station of the train
119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894
119894 isin 119879
42 Parameters The model uses the following parameterswhich are all assumed to be integer valued
119905min119867
the minimum time of the planning horizon119905max119867
the maximum time of the planning horizon
119909119886ini119894119895
the departure time of the train 119894 from the station119895 119894 isin 119879
ini 119895 isin 119878119894
119909119889ini119894119895
the arrival time of the train 119894 at the station 119895119894 isin 119879
ini 119895 isin 119878119894
ℎ119889119889 the minimum headway time between two con-
secutive departuresℎ119886119886 theminimumheadway time between two consec-
utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at
stations 119904119894and 119904119895 119887 = (119904
119894 119904119895) isin 119861
dwmin119894119895
the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895
the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878
119894119862min119895
the minimum layover time at the terminal 119895 119895 isin
119878119879
119862max119895
themaximum layover time at the terminal 119895 119895 isin
119878119879
119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894
1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878
119879 (119894 1198941015840) isin
119877119895 = 0 otherwise
119905119888119895 the maximum number of TUs at the same time at
the terminal 119895 119895 isin 119878119879
119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879
ini
43 Decision Variables The following variables are used inthe model
119909119886
119894119895 the departure time of the train 119894 at the station 119895
119894 isin 119879 119895 isin 119878119894
119909119889
119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879
119895 isin 119878119894
120593119894119895 binary variable = 1 if the train 119894 stops at the station
119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise
120587119889
1198941198941015840119895 binary variable = 1 if the train 119894 departures
before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894= 0 otherwise120587119886
1198941198941015840119895 binary variable = 1 if the train 119894 arrives before
the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878
119894 = 0otherwise120579119901119903119895
binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878
119879 119903 = (119894 1198941015840
) isin 119877119895 119894 1198941015840 isin 119879
= 0 otherwise119899TU119901119895
the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878
119879 119901 isin 119875
44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects
(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains
min119865119905
119865119905= sum
119894isin119879add
(119909119886
119894119897119894
minus 119909119889
119894119891119894
) (5)
(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains
min119865119904
119865119904= sum
119894isin119879ini119895isin119878119894
[10038161003816100381610038161003816(119909119886
119894119895minus 119909119886ini119894119895
)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889
119894119895minus 119909119889ini119894119895
)10038161003816100381610038161003816]
(6)
45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows
451 Timetable Constraints Consider the following
119909119886
1198941198951015840 = 119909119889
119894119895+ 119903119887+ 119905119886119886
sdot 120593119894119895
+ 119905119886119889
sdot 1205931198941198951015840
119887 = (119895 1198951015840
) isin 119861119894
119894 isin 119879
(7)
119909119889
119894119895minus 119909119886
119894119895ge dwmin119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(8)
119909119889
119894119895minus 119909119886
119894119895le dwmax119894119895
sdot 120593119894119895 119894 isin 119879 119895 isin 119878
119894
(9)
Constraints (7) define the arrival time to the station 1198951015840
from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and
the deceleration time (if a train stops at the station 1198951015840) At each
station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
452 Headway Constraints Consider the following
119909119889
119894119895minus 119909119889
1198941015840119895ge ℎ119889119889
sdot 120587119889
1198941198941015840119895minus 119872 sdot (1 minus 120587
119889
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(10)
119909119889
1198941015840119895minus 119909119889
119894119895
ge ℎ119889119889
sdot (1 minus 120587119889
1198941198941015840119895) minus 119872 sdot 120587
119889
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(11)
119909119886
119894119895minus 119909119886
1198941015840119895ge ℎ119886119886
sdot 120587119886
1198941198941015840119895minus 119872 sdot (1 minus 120587
119886
1198941198941015840119895)
119894 1198941015840
isin 119879 119895 isin 119878119894
(12)
119909119886
1198941015840119895minus 119909119886
119894119895
ge ℎ119886119886
sdot (1 minus 120587119886
1198941198941015840119895) minus 119872 sdot 120587
119886
1198941198941015840119895
119894 1198941015840
isin 119879 119895 isin 119878119894
(13)
120587119886
11989411989410158401198951015840 = 120587119889
1198941198941015840119895 119887 = (119895 119895
1015840
) isin 119861119894
119894 1198941015840
isin 119879 (14)
120587119889
1198941198941015840119895= 120587119886
1198941198941015840119895 119895 isin 119878
119894
119894 1198941015840
isin 119879 (15)
The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train
453 Time Deviation Constraints Consider the following
119909119886
119894119895minus 119909119886ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
119909119889
119894119895minus 119909119889ini119894119895
isin [minus119905inimax 119878 119905
inimax 119878]
119887 = (119895 1198951015840
) isin 119861119894
119895 isin 119878119894
119894 isin 119879
(16)
Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable
454 Layover Time and Turnback Operation ConstraintsConsider the following
119909119889
1198941015840119895minus 119909119886
119894119895isin [119862
min119895
119862max119895
]
(119894 1198941015840
) isin 119877119895
119895 isin 119878119894
119894 1198941015840
isin 119879
(17)
119899TU119901119895
= sum
119903=(1198941198941015840)isin119877119895
120579119901119903119895
119901 isin 119875 119895 isin 119878119879
119894 1198941015840
isin 119879
(18)
119899TU119901119895
le 119905119888119895 119901 isin 119875 119895 isin 119878
119879
(19)
Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked
by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901
is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style
5 Computational Experiments
51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations
We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec
The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS
The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4
52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Figure 6 Infrastructure of Shanghai rail transit line 16
Table 1 Train routes and track path information in line 16
Route ID Route information Route track path ID Detail track path information
R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations
R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations
R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations
R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations
R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations
R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations
R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations
Table 2 Computation parameters in line 16
Parameter Value119905min119867
500119905max119867
1000ℎ119889119889
180 secℎ119886119886
180 sec119905ac 30 sec119905de 30 secdwmin119894119895
30 secdwmax119894119895
60 sec119862
min119895
DSL (180 sec) LSR (270 sec) EHN (60 sec)119862
max119895
600 sec119905119888119895
2 at DSL 1 at LSR and EHN
the time window of starting time for the new trains Table 5summarizes the studied scenarios
53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute
As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015 017
007006
021
013 023
027
201
203
001002
003
004
005
006
007
005
U2
U2008
009
007
010011
009
012
011
012
010
014
018
011
008
038
012
040
202 204
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U2008 010 003 005 007 009 011 012
U2001 002 004 006 008 010 003
U2
Figure 7 Time-distance diagram obtained by the scenario 1
Table 3 Parameters of the new trains with one TU linking
New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1
Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)
U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1
Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)
U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1
Table 4 Time windows of the new trains
Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000
timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)
have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)
The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
U2
U2 U2001 003 004002
003
201
005 009
015017
007006
021
013
023
027
203
001002
003
004
005
006
007
005
008
009
007
010011
009
012
011
012
010
014
018
011
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
051
045
031
055
041
047
029
053
043
033
057
035
059DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
008
038
202
040
012
204026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
001 002 004 006U2
008 010 003 005 007 009 011 012U2
001 002 004 006 008 010 003
Figure 8 Time-distance diagram obtained by the scenario 2
Table 5 Different scenarios
Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905
119894) New TU ID Time window scheme ID
1 min 119865119905
0 sec U2 TW32 min 119865
1199050 sec U2 TW4
3 min 119865119904
300 sec U1 TW14 min 119865
119904300 sec U1 TW2
5 min 119865119904
270 sec U1 TW26 min 119865
119904240 sec U1 TW2
7 min 119865119904
210 sec U1 TW28 min 119865
119904180 sec U1 TW2
9 min 119865119904
150 sec U1 TW210 min 119865
119904120 sec U1 TW2
Table 6 Computation results of the scenarios
Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution
Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec
It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025103
051105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021
013
023
027
101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028022
046
032
016
042
024 048
030
020
044
034
036
001 002 004 006
U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)
DSL
LGA
SY
EHN
HN
WAP
XC
EHTHSH
EZP
LSR
DEP
500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900
500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800
003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001
037
061
039
019
049
025
103
051
105
045
031
055
041
047
029
053
043
033
057
035
059
001003
004002
003
005009
015017
007006
021013
023
027101
106
001
002
003
004
005
006
007
005
U1
U1
008
009
007
010011
009
012
011
012
010
014
018
102
011
008
038
012
040
104
026
050
028
022
046
032
016
042
024
048
030
020
044
034
036
001 002 004 006U1
008 010 003 005 007 009 011 012 001 002 004 006 008 010 003
U1
U1
U1 U1
Figure 10 Time-distance diagram obtained by the scenario 4 (the dotted lines are the initial trains)
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
0540 550 600 610 620
DSL
LGA
SY
003 005 007U1
019
002
004102
103
025
029
(a)
DSL0
540 550 600 610 620
LGA
SY
019
003 005 007U1
002
004
102
103
025
029
(b)
Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)
DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL
6 Conclusions and Future Work
In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations
However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865
119905and119865119904(119865119905+119904
= 119865119905sdot1205721+119865119904sdot1205722)
where the coefficients of 1205721and 120572
2are hard to evaluate On
the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this article
Acknowledgments
This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support
References
[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005
[2] L Cadarso and A Marın ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012
[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007
[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014
[5] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008
[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009
[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010
[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010
[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013
[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013
[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006
[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006
[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013
[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013
[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997
[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002
[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003
[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012
[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009
[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009
[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of