AMS Subj. Classiﬁcation: 90C57; 90C10;Rail transportation is very rich in terms of problems that can be modelled and solved using mathematical optimization techniques. The train scheduling problem as the most important part of a rail operating policy has a very signiﬁcant impact on a rail company proﬁt considering the fact that from the quality of a train timetable depends a ﬂow of three most important resources on rail network: cars, locomotives and crews. The train timetabling problem aims at determining a periodic timetable for a set of trains that does not violate track capacities and satisﬁes some operational constraints. In this paper, we developed an integer programming approach for determining an optimal train schedule for a single, one-way track linking two major stations, with a number of intermediate stations between. The application has been tested on a realistic example suggested by the PE “Serbian Railways”. Obtained results show a potential for a practical application of proposed approach

Bojović, Nebojša

Kratica, Jozef

Marić, Miroslav

Milenković, Miloš

Stanojević, Predrag

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————New Series Vol. 24, 2010, Fasc.3-4Mathematical Optimization for theTrain Timetabling ProblemPredrag Stanojevic´ 1, Miroslav Maric´ 1, Jozef Kratica 2,Nebojˇsa Bojovic´ 3, Milosˇ Milenkovic´ 3Rail transportation is very rich in terms of problems that can be modelled and solvedusing mathematical optimization techniques. The train scheduling problem as the most im-portant part of a rail operating policy has a very significant impact on a rail company profitconsidering the fact that from the quality of a train timetable depends a flow of three mostimportant resources on rail network: cars, locomotives and crews. The train timetabling prob-lem aims at determining a periodic timetable for a set of trains that does not violate trackcapacities and satisfies some operational constraints. In this paper, we developed an integerprogramming approach for determining an optimal train schedule for a single, one-way tracklinking two major stations, with a number of intermediate stations between. The applicationhas been tested on a realistic example suggested by the PE “Serbian Railways”. Obtainedresults show a potential for a practical application of proposed approach.AMS Subj. Classification: 90C57; 90C10;Key Words: rail transportation, scheduling, timetabling, integer programming1. IntroductionIn most countries the railway traffic system represents a very importantpart of the backbone transport system. Traffic and transport policies are striv-ing towards decreasing road traffic pollution by increasing railway usage whenappropriate. At the same time, the available railway systems are partly over-saturated creating bottlenecks on major links. An important issue is thus howto best use the existing capacity while ensuring sustainability and attractive-ness. The train scheduling problem arises in several contexts [1],[2]. In real-timescheduling [3], it helps the dispatchers who are managing the traffic to make op-timal decision about which trains to stop and where, as updated data about trainpositions becomes available [6],[8]. In tactical planning, it consists of finding thebest master schedule or timetable on a regular basis (weekly, monthly, yearly)304 P. Stanojevic´, M. Maric´, J. Kratica, N. Bojovic´, M. Milenkovic´[5], [7]. In strategic planning, it relates to investment decisions in rail infrastruc-ture such as building new stations, extending current station track lengths, andupgrading single-line segments [9], [4] to double lines. The timetable planningproblem aims at determining a timetable for a set of trains which does not vi-olate track capacities and satisfies some operational constraints. The timetableis a schedule of trains on a given rail infrastructure. It contains the arrival anddeparture times of the trains at each intermediate point of their route. Due toits complex nature, the construction of the actual timetable as it is operatedin practice, is still mainly a human planning process. Traditionally, plannersuse two types of graphs as the main tools for constructing a timetable. The so-called time-space diagram graphically represents the train movements that takeplace in between stations, on the tracks. One axis displays time, and the otheraxis depicts space. Each line in the time-space diagram depicts a train, whichis indicated by the number next to the line. Lines with a positive inclinationcorrespond to trains from Station E to Station D, and lines with a negative in-clination to trains in opposite direction. Flat lines indicate fast trains, becausethose trains cover a large distance in a short time, and steep lines indicate slowtrains. When a train dwells for some time at a station, this gives a vertical lineat that station, because time moves on while the location of the train remainsunchanged. Intersecting lines indicate that the two corresponding trains meetat the point of intersection. This is clearly only allowed at stations, or if thetrains are using different tracks. The advantage of this diagram is that it makesit much more simple and intuitive to read the timetable and to detect conflicts.In Fig.1 is an example of a train diagram with 21 trains and 19 stations.The X axis represents time of day, the Y axis the sequence of stations indistance scale. Lines indicate the movement of trains, with the slope indicatingdirection and speed, horizontal meaning stand-still. Usually, the outbound di-rection is defined upwards. Fig 2. displays a possible movement of four trainson a single track line connecting the stations A-E. Intermediate meet-points arelocated at all stations from B to D. The horizontal and vertical axes representtime and space, respectively.Even in this simple example there are many possible combinations of sta-tions and times for trains to be pulled over to allow meets and passes. Therefore,the train meet-pass problem is a very large-scale combinatorial optimizationproblem. When several trains in both directions are scheduled, many conflictsmay arise. Each conflict involves trains moving either in opposite directions orin the same direction. Depending on the chosen solution for a conflict involv-ing two trains, the location and the time of later conflicts may change, newconflicts at different locations and times may arise, and existing conflicts mayMathematical Optimization for the Train Timetabling Problem 305Figure 1. Train diagram exampleFigure 2. Graph of train movements306 P. Stanojevic´, M. Maric´, J. Kratica, N. Bojovic´, M. Milenkovic´disappeared. Thus, the number of feasible solutions to train conflicts can bevery large. Also, it is very important to notice that train scheduling decisionmay have different economic effects. Sequences that reduce travel times may de-crease investments in cars and engines as well as crew and fuel costs. Regularityand transportation velocity, which also depend on the schedule, are importantfactors to the satisfaction of customers needs and thus are related to railwaycompany revenues.In practice, a timetable is constructed by specifying, a time-space path foreach train through a railway network, which is drawn in the time-space diagrams,and by specifying the platform tracks that the train occupies, which are drawnin the platform-occupation charts. Generally, a timetable is not constructedfrom scratch. Rather, adjustments are made to an existing timetable, typically,to the timetable of the previous year. Trains are added to the existing timetable,deleted from it, or the schedule of an already existing train is adjusted. Duringthe process of adding, deleting, and adjusting the schedules of trains, one usuallyruns into problems at a certain moment in time. It may not be possible toschedule a new train as desired, because there is no capacity on the track orin the station, or because capacity is only available when a connection can notbe realized. In such a case, some of the already scheduled trains have to berescheduled. This can be achieved by shifting an already scheduled train tosome earlier or later point in time, by adjusting the planned trip time of a train,or by relocating a train to a different platform. Cycling through this processof scheduling trains, and backtracking on previously made choices in case of a’dead end’, one may eventually arrive at a complete timetable. The timetableconstruction process is quite complex, and it may take a team of planners severalmonths to create a complete timetable. Modification of the last year timetableimplies that the new timetable inherits properties that may be unnecessary andcostly. The construction process is very time consuming. Thus, for time reasons,there is no possibility to optimize the timetable, and the planner is often satisfiedto find a feasible timetable.In this paper we present an optimization model for the problem of time-table optimization. This algorithm treats some kind of trade-off between depar-ture times for each of a given set of trains and the total waiting time needed fortrain conflict resolution. Considering the fact that an optimal timetable gives agood base for making the optimal decision for routing of locomotives, cars andcrews, this strategy of making timetables based on a flexible train departurescan be a very efficient strategy for improving the timetable construction. Wetest this algorithm on a realistic example. The computation times are moderateconsidering the sizes of the optimization problems. Of course, there are manyMathematical Optimization for the Train Timetabling Problem 307aspects we do not consider, but these may be added later. We hope that thesuggested procedure will be a first step towards intelligent computer assistedprofit maximizing allocation of track capacity in timetabling.2. Model formulationWe consider a single, one-way track linking two major stations with anumber of intermediate stations. This case is particularly interesting becauserailway networks are mostly contained from single track sections. On thesesections made of a track carrying traffic in opposite directions, track resource islimited by great traffic densities.The following data represent problem parameters:T - set of trainsS - set of stationsQ - set of discrete time intervalsdt - departure station of train tat - arrival station of train tPath(t) - set of stations between dt and atedt - estimated departure of train tmdt - maximum allowed delay of train tldt = edt +mdtf(t, s) - travel time of train t alongside the track segment sPrev(t, s) - previous station in the path of train tNext(t, s) - next station in the path of train teat,s - earliest arrival of train t to station s (calculated from edt and f(t, l))eat,s = edt +s∑l=dtf(t, l)mws - maximum number of trains waiting simultaneously at station smis - maximum number of trains simultaneously waiting at, or exiting fromstation sxdept,dt,k- an arc which denotes train t departing at time kxarrt,at,k - an arc which denotes train t arriving at time kxwaitt,s,k - an arc denoting train t waiting in station s at time kxoutt,s,k - an arc denoting that train t is leaving station s at time kArcs - set of all arcs xdept,dt,k, xarrt,at,k, xwaitt,s,kandxoutt,s,k,∀t,∀k,∀sArcs = {xdept,dt,k|∀t,∀k} ∪ {xarrt,at,k|∀t,∀k} ∪ {xwaitt,s,k |∀t,∀k,∀s} ∪ {xoutt,s,k|∀t,∀k,∀s}Exit(t, s, k) - set of arcs denoting train t leaving node (s,k) except for departurearcs, because there is no enter arc for themExit(t, s, k) ={xoutt,s,k, xwaitt,s,k,}∪{xarrt,at,k|s = at}308 P. Stanojevic´, M. Maric´, J. Kratica, N. Bojovic´, M. Milenkovic´Enter(t, s, k) - set of arcs denoting train t entering node (s,k) except for arrivalarcs, because there is no exit arc for themEnter(t, s, k) ={xoutt,Pr ev(s),k−f(t,s), xwaitt,s,k−1,}∪{xdept,at,k|s = dt}The problem is formulated as an integer programming model with deci-sion variables x ∈ Arcs representing the flow of a train on an arc. The formu-lation is as follows:minimize z =∑t∈T(ldt∑k=edt(k − edt)xdept,dt,k +∑s∈Path(t)eat,s+mdt∑k=eat,sxwaitt,s,k) (1)Subject to:ldt∑k=edtxdept,dt,k= 1,∀t ∈ T (2)lat∑k=satxarrt,at,k = 1,∀t ∈ T (3)∑x∈Enter(t,s,k)x− ∑x∈Exit(t,s,k)x = 0,∀t ∈ T,∀s ∈ Path(t),∀k ∈ Q, ea(t, s) ≤ k ≤ ea(t, s) +md(t) (4)∑t∈Txwaitt,s,k ≤ mws,∀s ∈ S,∀k ∈ Q (5)∑t∈T(xwaitt,s,k + xoutt,s,k) ≤ mis,∀s ∈ S,∀k ∈ Q (6)∑t∈T out∑l≤k−1,l+f(t,s)≥kxoutt,s,l+∑t∈T in∑l≤k−1,l+f(t,s+1)≥kxoutt,s+1,l ≤ 1,∀s ∈ S,∀k ∈ Q (7)x ∈ {0, 1},∀x ∈ ArcsThe objective function (1) minimizes the sum of train delays since thearc costs represent the delays in discretized time units. Equation (2) ensuresthat for each train there is a unit outflow from its departure node to one ofthe nodes that is the copy of its origin station at a time within its possibledeparture window. The constraint of equation (3) ensures that for each trainthere is a unit flow from one of the nodes that is the copy of its destination nodeto its arrival node. Constraint set (4) provides the flow conservation constraints.Constraint set (5) ensures that the number of trains waiting simultaneously instation be equal or less than the maximum number of tracks serving for trainoperations. Constraint set (6) ensures that the total number of trains stayingin station and trains which at the same time pass through the station be lessthan the maximum number of available tracks. Constraint (7) is dedicated tocapacity of sections between the stations. Therefore, the total number of trainscan not be greater than one, if we are considering inter-station system of trafficregulation.Mathematical Optimization for the Train Timetabling Problem 309Figure 3. Optimal solution for a train timetabling problem3. Computational resultsIn this section, we present computational results for a real world exampleof the train dispatching problem. The example is based on actual data for a 24-hour planning horizon on a part of the major line of Serbian railways. Withinthis time interval we have 25 trains of opposite directions and 12 potentialmeet/pass points for each of them. According to data from the graphical traintimetable, for solving all conflicts raising on this set of trains a total waitingtime of 42 minutes is planned. Next table contains all necessary data.Of course, we also considered the possibility that is not for any train al-lowed to wait in any station due to incompatibility of its length and the lengthof station tracks or for some other reasons. Fig. 3 contains an optimal schedulewith minimal train delays for given example. Table 2. presents the computa-tional results for a given set of trains.The minimum total waiting time, as a sum of the time spent in departurestations and the time needed for meeting/overtaking operations for this set oftrains on a part of Serbian rail network is 185 minutes. We implemented ouralgorithm in C programming language and CPLEX 10.0 solver. All computa-tional tests were conducted on a 3.0-GHz Intel Pentium 4 processor with 2GbRAM. For this example CPLEX loading time was 8 minutes long and optimalsolution has been found for 25 seconds of CPU time.4. ConclusionsDue to the complexity of rail operations, the expected growth of trafficand the limited possibilities of enhancing the infrastructure, effective timetabledesign strategy play a key role in improving the level of railway service. Inthis paper we presented a model for tactical train scheduling on a single linetrack. Unlike most models, we tried here to make a trade-off between the timeof train departing and the train schedule. For each train, depending on itsimportance and future train connections we defined an interval of departure asa rolling horizon within which we can depart our train. Obtained results arevery promising. Considering the fact that time for deriving the optimal solutionis the crucial parameter if we want to use this model as a part of decision310 P. Stanojevic´, M. Maric´, J. Kratica, N. Bojovic´, M. Milenkovic´Train No.DeparturestationArrivalstationDeparturetimeArrival time1 Novi Sad Subotica 22:52 24:502 Subotica Novi Sad 02:40 03:373 Novi Sad Subotica 09:04 11:004 Subotica Novi Sad 17:00 18:545 Novi Sad Subotica 05:09 07:166 Subotica Novi Sad 18:40 20:457 Novi Sad Subotica 11:35 13:328 Subotica Novi Sad 05:45 07:429 Novi Sad Subotica 20:19 22:1810 Subotica Novi Sad 14:22 16:1911 Novi Sad Subotica 10:10 12:1612 Subotica Novi Sad 11:17 13:1713 Novi Sad Subotica 13:05 15:1714 Subotica Novi Sad 07:20 09:3015 Novi Sad Subotica 15:25 17:5516 Subotica Novi Sad 04:33 06:5417 Novi Sad Subotica 04:33 06:4218 Subotica Novi Sad 10:29 12:5319 Novi Sad Subotica 07:38 09:4920 Subotica Novi Sad 15:48 18:0421 Novi Sad Subotica 19:15 21:2622 Subotica Novi Sad 13:00 15:1823 Novi Sad Subotica 22:20 24:2524 Subotica Novi Sad 19:21 21:3825 Novi Sad Subotica 22:22 24:34Table 1. Timetable for a given set of trainssupport system for on-line train dispatching, this model can be used as a toolfor real-time train dispatching.Mathematical Optimization for the Train Timetabling Problem 311Train No. Optimal departure timeTime deviation in minutesrespect to earliestdeparture time1 22:54 22 02:49 93 09:05 14 17:01 15 05:09 06 18:50 107 11:43 98 06:12 279 20:30 1110 14:34 1211 10:10 012 11:27 1013 13:15 1014 07:27 715 15:25 016 04:43 1017 04:33 018 10:31 219 07:39 120 15:48 021 19:32 1722 13:04 423 22:39 1924 19:21 025 22:23 1Table 2. Optimal solution for a given train timetabling problemReferences[1] C a p r a r a , M . F i s c h e t t i , P . T o t h , Modeling and Solving the TrainTimetabling Problem.Operations Research 50, 2002, 851-861.[2] J . C o r d e a u , P .T o t h , D .V i g o , A Survey of Optimization Models forTrain Routing and Scheduling.Transportation Science , 32, 1998, 380-404.[3] A . D ’A r i a n o , D . P a c c i a r e l l i , M . P r a n z o , Effects of FlexibleTimetables in Real-Time Scheduling of Rail Operations.Preprints of the 6th312 P. Stanojevic´, M. Maric´, J. Kratica, N. Bojovic´, M. Milenkovic´Triennial Symposium on Transportation Analysis, TRISTAN 2007, 1-7.[4] A . H i g g i n s , E . K o z a n , L . F e r r e i r a ,Optimal Scheduling of Trainson a Single Line Track. Transportation Research, Part B, 30(2), 1996, 147-161.[5] D . J o v a n o v i c, Improving Railroad In-time Performance: Models, Algo-rithms and Applications. PhD thesis, Decision Science Department, WhartonSchool, University of Pennsylvania, 1989.[6] E . P e t e r s e n , A . T a y l o r , C . Ma r t l a n d ,An Introduction toComputer-Assisted Train Dispatch. Journal of Advanced Transportation, 20,1986, 63-72.[7] G . S a h i n , R . Ah u j a , C . C u nh a , New approaches for the train dis-patching problem. submitted to Transportation Research Part B., 2005.[8] R . S a u d e r , W . We s t e rma n , Computer Aided Train Dispatching:Decision Support Through Optimization. Interfaces, 13, 1983, 24-37.[9] B . S z p i g e l , Optimal Train Scheduling on a Single Track Railway. InProceedings of the IFORS Conference on Operational Research’72, 1973, 343-361.1 Faculty of MathematicsUniversity of Belgrade,Studentski Trg 1611000 Belgrade, SerbiaE-Mails: djape@sbb.rs, maric.m@sbb.rs2 Mathematical Instituteof Serbian Academy of Science and ArtsKneza Mihaila 3611001 Belgrade, SerbiaE-Mail: jkratica@mi.sanu.ac.rs3 Faculty of Transport and Traffic EngineeringUniversity of BelgradeVoivode Stepe 30511000 Belgrade, SerbiaE-Mails: nb.bojovic@sf.bg.ac.rs, m.milenkovic@sf.bg.ac.rs

Mathematical Optimization for the  Train Timetabling Problem

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Mathematical Optimization for the Train Timetabling Problem

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