a multiobjective optimization for train routing at the high...
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Research ArticleA Multiobjective Optimization for Train Routing at theHigh-Speed Railway Station Based on Tabu Search Algorithm
Ziyan Feng , Chengxuan Cao , Yutong Liu, and Yaling Zhou
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China
Correspondence should be addressed to Chengxuan Cao; [email protected]
Received 10 June 2018; Accepted 16 September 2018; Published 3 October 2018
Academic Editor: Mahmoud Mesbah
Copyright © 2018 Ziyan Feng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper focuses on the train routing problem at a high-speed railway station to improve the railway station capacity andoperational efficiency. We first describe a node-based railway network by defining the turnout node and the arrival-departure linenode for the mathematical formulation. Both considering potential collisions of trains and convenience for passengersâ transferin the station, the train routing problem at a high-speed railway station is formulated as a multiobjective mixed integer nonlinearprogramming model, which aims to minimize trainsâ departure time deviations and total occupation time of all tracks and keepthe most balanced utilization of arrival-departure lines. Since massive decision variables for the large-scale real-life train routingproblem exist, a fast heuristic algorithm is proposed based on the tabu search to solve it. Two sets of numerical experiments areimplemented to demonstrate the rationality and effectiveness of proposed method: the small-scale case confirms the accuracy ofthe algorithm; the resulting heuristic proved able to obtain excellent solution quality within 254 seconds of computing time on astandard personal computer for the large-scale station involving up to 17 arrival-departure lines and 46 trains.
1. Introduction
Generally, the station is composed of bottleneck and arrival-departure lines which is vital for trains arriving, departing,running, shunting operations, and so on. The efficiency ofrailway transport faces challenges due to increasing passen-gers and freight transportation demands. In order to enhancethe capacity of the railway station, operators can be dedicatedto adding the number of track lines, which requires plentyof manpower, materials, and financial resources obviously.By contrast, making proper arrangements of trainsâ routesat bottleneck and arrival-departure lines based on existingtrack lines can save lots of resources and achieve the sameeffect. Hence, attention is paid to arrange the trainsâ routesreasonably in the station to improve the operational efficiencyand reduce the operation costs [1, 2] (DâAriano et al., 2008).
In general, the railway planning process is divided intostrategic level, tactic level, and operational level. In this paper,we focus on the tactic level, especially on the timetableoptimization and railway track allocation/train routing ina complex high-speed railway station. At present, the opti-mization of the train routing problem (TRP) in the railway
station is mainly based on the mathematical optimizationmodel. Some studies aimed at the complicated bottleneckof the railway station, as DâAriano et al. [3] considered thatreducing trainsâ collisions at the bottleneck can improvepunctuality without decreasing the capacity usage of the linesand a detailed model for conflict resolution and differentalgorithmswas illustrated. Based on the robustness of routingassignment, Jia et al. [4] proposed the optimization of routingutilization at bottleneck. Kang et al. [2] presented a bottleneckoptimization model thereafter to enhance the capacity byreasonably arranging routes and turnouts. Other parts of theresearches concentrated on the problems of arrival-departurelines occupancy. For example, Billionnet [5] consideredassigning trains to the available tracks at a railway stationinstead of taking into account the collision at the bottleneck.Caprara et al. [1] proposed a 0-1 integer programming modelto describe the routing problem of trains at the stationplatform; moreover, the quadratic objective function waslinearized and solved by integer linear programming. Besides,an optimizationmodel is presented byQiao et al. [6] based onthe train schedules to arrange suitable arrival-departure linesfor passenger trains. Furthermore, the research of the TRP
HindawiMathematical Problems in EngineeringVolume 2018, Article ID 8394397, 22 pageshttps://doi.org/10.1155/2018/8394397
2 Mathematical Problems in Engineering
with joint optimization of bottleneck and arrival-departurelines is also found in the literature. For instance, Zwaneveld etal. [7] considered both bottleneck and arrival-departure linesand proposed a 0-1 programming model to arrange trainspassing the railway station. But the problem only at small-scale railway stations can be solved by branch-cut method.Based on the graph theory, Corman et al. [8] rearranged trainrouting in real-time unpredictable events and found the bestsolution using truncated branch-and-bound and tabu searchalgorithms.
In the past few decades, there have been limitedresearches on TRP in high-speed railway stations. Moststudies were associated with the train timetable problems [9â11]. TRP is tantamount to selecting a sequence of tracks fora train from its origin to destination, with the objective ofminimizing the sum of travel time, the total operating cost,and/or increasing the capacity of railway network. Xu et al.[11] defined the objective function to minimize deviationsbetween trainsâ arrival time at the destination and originaltimetable. The optimization model proposed by Li et al.[12] aimed to minimize the total delay of all trains in therailway network. Liu et al. [13] developed a mathematicalmodel whose aim is minimizing the total occupation time ofstation bottleneck sections to avoid train delays. In addition,some research focused on the optimization by keeping themost balanced utilization arrival-departure lines like Qiaoet al. [6]. One of the objective functions considered byZhou et al. [14] and Zhou et al. [10] was to minimize thetotal travel time on the track. Apart from this, the mostimportant constraint should be taken into account was thespatiotemporal interactions between each train operationroute in the TRP (DâAriano et al., 2008) [2, 4, 12]. Further-more, it should be considered that trains which occupy thesame arrival-departure lines should satisfy a headway timeconstraint [1, 6]. Zhou et al. [14] and Corman et al. [8]proposed that trains should stop for enough time to ensurethe transfer time of passengers and crews. What is more,Fang et al. [15] analyzed a comprehensive survey on differentmodels by a clear classification based on the different scale,infrastructures, objectives, and constraints.
It is well known that the TRP is the NP-hard problem[16] and unlikely to get an exact optimal solution in ashort computational time under the large-scale and complexsituation. In order to get an approximate optimal solutionrapidly, many studies proposed different heuristic algorithmsbased on different strategies. Corman and Meng [17] intro-duced the online dynamic models and algorithms for the railtraffic management in order to provide punctual and reliableservices. Specifically, Ahuja et al. [18] considered the issue ofrailway scheduling and presented a heuristic algorithm to getthe approximate optimal solution in short time. Carey et al.[19] studied the large-scale problem that involved lots of trainschedules and routes and proposed heuristic algorithms tosolve it. Liu et al. [13] designed a hybrid algorithm betweengenetic algorithm and the simulated annealing algorithm.In recent years, for the sake of further improving thecomputational speed and quality of solution, some studieshave proposed improving heuristic algorithms. He et al.[20] proposed an improve branch-and-price algorithm to
deal with the large scale integer programming. Zhou etal. [10] used an efficient train-based lagrangian relaxationdecomposition to the simultaneous passenger train routingand timetabling problem. Additionally, some studies usedcommercial software like Qi et al. [21] who obtained for anapproximate optimal solution within 30s by a local searchheuristic algorithmusing CPLEX solver andXu et al. [11] whosolved train routing and timetabling problemwith switchablerules by CPLEX solver with OPL language.
Besides that, some studies dealt with the integer pro-gramming problem using tabu search algorithm, as Isaaiand Singh combined the heuristic with tabu search andsimulated annealing search control strategies to deal withthe train timetabling problem. Similar, Corman et al. [8]rearranged trainsâ routes based on the graph theory andgot the approximate optimal solution of the combination oftruncated branch-and-bound and tabu search algorithms. Liet al. [12] proposed a tabu search algorithm, and they pointedout that the algorithm relies on a better initial solution;otherwise the result obtained is not stable. In addition tothis, Goh et al. proposed a tabu search with sampling andperturbation to generate feasible solutions. The tabu searchalgorithms and variable neighborhood are applied by Sama etal. [22] to improve the solution for the real-timemanagementproblem of scheduling and routing trains in complex andbusy railway networks.
As can be found in numerous studies, there were fewelaborate mathematical models to describe TRP and furtherstudy on TRP in the network of the high-speed railwaystation. In this research, with the motivation of greatlyimproving the solving efficiency of TRP in the network of thehigh-speed railway station,we intend to provide the followingcontributions:
(i) Describe a railway network by defining the turnoutnode and the arrival-departure line node on accountof the traditional layout of the railway station, which isregarded as a directed graph. The nodes are regardedas vertices, and the actual connection of the lines isregarded as arcs. The railway station is divided intothree parts and the connection sets are built to satisfythe connection relationship of them.
(ii) According to the given nominal timetable, the cal-culation methods of trains occupying each track areelaborated. Then formulate the TRP in the networkof the high-speed railway station as a multiobjectivemixed integer nonlinear programming model, inwhich both are considered the potential collisions oftrains and the convenience for passengersâ transferin the high-speed railway station. In the proposedmodel, it is not only minimizing train departuretime deviations with the most balanced utilization ofarrival-departure lines but also minimizing the totaloccupation times of all tracks.
(iii) The TRP is NP-hard problem. Therefore, it is veryunlikely to devise a polynomial-time (exact) algo-rithm for it. In order to get an approximate optimalsolution rapidly, a heuristic algorithm is proposed
Mathematical Problems in Engineering 3
based on the tabu search to solve the large-scale TRPin the network of the high-speed railway station.
(iv) Numerical examples are implemented to demonstratethe effectiveness and efficiency of proposed method.By taking advantage of an efficient tabu search algo-rithm, we can solve the model rapidly for a smallcase. The results we obtained from the algorithm arethe same as obtained directly from CPLEX solver.In the large-scale case involving 17 arrival-departurelines and 46 trains between 16:00âŒ19:00, the resultingheuristic proved able to obtain excellent solutionquality within 254 seconds of computing time on astandard personal computer.
The remainder of this paper is organized as follows.Section 2 presents a detailed description of TRP problem.Section 3 provides the mathematical formulation for the TRPin the network of the high-speed railway station. Section 4deals with the development of a heuristic algorithm based onthe tabu search. Section 5 describes the instances used andprovides computational results. Finally, some conclusionsand further research directions are presented in Section 6.
2. Train Routing Problem Description
2.1. Layout of Railway Station. We consider a railway stationas illustrated in Figure 1. This railway station consists ofbottlenecks and some arrival-departure lines whose lengthis ð¿ ï¿œí . The left side of the station in Figure 1 is defined asleft bottleneck whose length is ð¿ ï¿œí. The route set of it isð¿ = {ð1, ð2, â â â , ðï¿œí1}. Similarly, the right side is defined asright bottleneck whose length is ð¿ï¿œí. The route set of it isð = {ð1, ð2, â â â , ðï¿œí2}. The bottleneck involves tracks wheretrains arrive and depart. For instance, an outbound trainarrives from node ð1 and stops at arrival-departure line. InFigure 2(a), there are 4 possible routes for the outbound trainthat arrives from node ð1, whose set is ð¿1 = {ð1, ð2, ð3, ð4} inthe left bottleneck. Similar, an inbound train departs fromarrival-departure line to the node ð2.There are also 4 possibleroutes for the inbound train departs fromnodeð2 as shown inFigure 2(b) whose set is ð¿2 = {ð5, ð6, ð7, ð8}. So the route set ofthe left bottleneck is ð¿ = ð¿1âªð¿2 = {ð1, ð2, . . . , ð8}. In particular,at large stations, the number of routes in the bottleneck maybe larger than the number of arrival-departure lines due to theexistence ofmultiple crossovers and turnouts.The route set ofarrival-departure lines is ð = {ð 1, ð 2, . . . , ð ï¿œí}, which containsI, II, 3, and 4 as shown in Figure 1.
Normally, track lines are divided by insulation joints. Inthis paper, the turnout node and the arrival-departure linenode are defined as follows to describe the railway station.
1. Turnout node: the intersection of the lines in thestation. It includes turnouts (as shown in Figure 1,ð5 and ð6, etc.), crossovers (as shown in Figure 1, ð7,etc.), and the position of signal which located in theentrance of the bottleneck (i.e., the boundary point ofthe station).
2. Arrival-departure line node: the connection pointsof bottleneck and arrival-departure lines in the
railway station, namely, the signal position of theentrance of the arrival-departure lines (i.e.. the nodesð8âŒð10 in Figure 1).
During the operation, the outbound trains can only arrivefrom turnout node ð1 and depart from turnout node ð3as shown in Figure 1. Similar, the inbound trains can onlyarrive from turnout node ð4 and depart from turnout nodeð2. However, no matter in which direction trains are, any ofarrival-departure lines can be occupied. Hence the physicalnetwork of the railway is regarded as a directed graph. Thenodes are regarded as vertices, and the actual connection ofthe lines is regarded as arcs. Letð· denote the set of all verticesand ðŽ denote the set of all arcs in the network.
2.2. Trains. We consider a set of trains in both directions.The set of trains is denoted by ð¶ = {ð1, ð2, â â â , ðï¿œí}, in whichð¶1 = {ð1, ð3, ð5, â â â } is the set of outbound trains and ð¶2 ={ð2, ð4, ð6, â â â } is the set of inbound trains. The heterogeneoustrains are taken into account. We assume that these trainsare categorized into two types: slow trains and fast trains. Inaddition, there are shunting operation beside train receptionand departure operations. The speeds of shunting trains areslower than the ordinary trains.
In addition, some of the fast trains pass the stationwithout stopping while some trains stop at platform forpassengers to alight or transfer. Different trainsâ stop timemay be different, whether they are fast or slow trains. Aswe all know, the speed of trains is not a constant, while itchangeswith the driverâs brakingwhen the trainwould stop atthe platform. Similarly, the speed of trains does not increasesuddenly when departing from the platform. Therefore,the additional time that provides trainsâ acceleration anddeceleration is considered. At the same time, the length oftrains is also taken into account to meet the actual situation.
2.3. Conflicts and Potential Conflicts. The conflicts of trackswhich trains occupy are the most imperative problem to besolved in TRP. In other words, trains cannot occupy the sametrack at the same time. A conflict occurs whenever trainstraverse the same track and do not respect theminimum timeinterval at the bottleneck and arrival-departure lines. What ismore, it would result in potential conflicts. These situationsare discussed separately as follows.
2.3.1. Train Routing Conflicts at Bottleneck. The bottleneckconsists of channels with some turnouts, crossovers, andother facilities, which may cause spatial intersections oftrainsâ alternative sets of routes, especially when some unpre-dictable events or interruptions occurred. So it is an impor-tant task for train dispatchers to avoid the train collision andto assure the trains passing the tracks orderly. The conflictsbetween two trains in the bottleneck mainly occurred at theintersection of the track segments (the segment ð1âŒð2 asshown in Figure 3) and the intersection of crossovers (thepoint ð3 as shown in Figure 3).
In Figures 3(a) and 3(b), trains ð1/ð2 travel in thesame/reverse direction, and the collision occurs at the tracksegment between nodes ð1 and ð2. Similarly, the collision
4 Mathematical Problems in Engineering
I
II
d1
d2
d3
d4
d5 d6
d7
d8
d9
LrLsLl
d10
inbound
outbound3
4
Figure 1: An illustration of a railway station.
d1
d2
3
4
outbound
inbound
I
II
(a)
d1
d2
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4inbound
outbound
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Figure 2: The route set of left bottleneck in the railway station.
d1 d2
c1c2c3
d3
(a)
d1 d2
d3
(b)
Figure 3: The same/reverse direction conflicts of trains routing.
occurs at the track segment at the node ð3 when the trains ð2and ð3 travel in the same and reverse direction, respectively.The methods to dissipate conflicts will be depicted in detailin Section 3.
2.3.2. Train Routing Conflicts at Arrival-Departure Lines.When two consecutive trains plan to occupy the same arrival-departure line in the railway station, they should follow aheadway time to suit the limited infrastructure capacity.Thatis, one train occupies this arrival-departure line a specificheadway time later than the other one. As shown in Figure 4,there are two trains ð1 and ð2. The origin and destinationof train ð1 are A and C, respectively, and the origin anddestination of train ð2 are B and C. Obviously, if the route ð1is assigned to ð1 and ð2 occupies route ð2 in a short interval,this would cause a conflict at platform 1. In this case, the lattertrain should stop andwait until the previous train leaves fromthe platform 1 for a reasonable headway time. It may cause thelatter train to be behind the schedule time and even to havesecond delay in severe cases. But if the train ð2 occupies route
ð3, the conflicts would be avoided and the train operationalefficiency would be ensured in the railway station.
Therefore, if the two trainsâ departure interval is smallerthan the headway time ð1ï¿œí, it is better to arrange them atdifferent lines. Just as illustrated in Figure 4, if the differenceof two trainsâ departure time is bigger than the headway timeð1ï¿œí, the same arrival-departure line can be arranged.
2.3.3. Potential Conflicts of Train Routing. It is noteworthythat only considering trainsâ routing at bottleneck and head-way time at arrival-departure lines cannot completely meetthe requirement of safe operation.There are further potentialconflicts.
For instance, the outbound train ð1 would stop at arrival-departure line II from outboundmain line while the inboundtrain ð2 would depart from line 3 to the inbound mainline (as shown in Figure 5(a), the yellow dotted line andthe blue dotted line represent the route of trains ð1 and ð2,respectively).There is no time provided to avoid the potentialconflict area (the red circle area in Figure 5), since it is not in
Mathematical Problems in Engineering 5
A
B
C
p1p2p3
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2
Figure 4: The conflict at arrival-departure lines.
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Potential conflicting area
outbound
inboundII
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I
Potential conflicting area
c1c2
outbound
inboundII
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Figure 5: The illustrative of outbound and inbound trainsâ potential conflicts.
the context of the two conflicts described above. Therefore,a time separation should be adopted even though two trainsdo not occupy the same track; i.e., ð1 first arrives and then ð2departs later or ð2 first leaves and then ð1 arrives later, the sameas shown in Figure 5(b). It is necessary to set constraints sothat two consecutive trains arrive the potential conflict areain a time separation.
In addition, when two trains are in the same direction,the conflicts between them are not potential conflicts. Just asshown in Figure 6, ð1 and ð2 are outbound trains. It is apparentthat there exists a conflict in the red circle area. However, dueto the conflicts at the bottleneck between the two trains, theconstraints are set to avoid the collision; namely, conflicts inthe rad circle area are also dissolved. Therefore, the potentialconstraints we discussed are only among the reverse trains.
2.4. The Calculation Methods of Track Occupation Time. Inthis paper, not only is the train routing chosen as the decisionvariables but also optimized the schedule of trains to startentering the station according to the planned schedule whentrains depart from the station. Thus the time for trainspassing the conflict tracks needs be calculated. In addition,the convenience factors of passengers transferring at stationare taken into account, and the associated trainsâ dwell-timeintersection should be long enough. As a consequence, thetrainsâ travel time in different track sections should be clearlycalculated.
To our knowledge, as long as a track segment is occupied,the lines, the fouling posts, and the signals are also occupiedat the same time.We elaborate the calculation of the occupiedtime and the end time of occupied of trains passing throughthe left bottleneck, the arrival-departure lines, and the rightbottleneck in turn. The occupied trains at the bottleneckare divided into two situations: train reception and train
departure, whose calculatedmethod of train occupied time oftracks is same. So only take the outbound trains as an exampleto explain in detail.
2.4.1. The Calculation of Trains Travel Time and the End Timeof Occupied at the Bottleneck. For outbound trains ð â ð¶1 inthe left bottleneck ð â ð¿, they start to enter the station at ð¡ï¿œí +ð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí and pass each node with speed Vï¿œí.The track is unlockedafter their tails pass the end node of the track section. So thetrainsâ occupied time of each track segment is calculated asfollows:
ð¡ï¿œí = ð¿ï¿œíVï¿œí
, (1)
where ð¡ï¿œí is occupancy time of a certain track segment ð andð¿ï¿œí is length of the track segment ð. Thus, the travel time ofoutbound trains ð â ð¶1 from the left bottleneck signals to thearrival-departure lines is calculated with the following:
ð¡ï¿œíï¿œíï¿œí = (ð¿ ï¿œí + ð¿ï¿œí)Vï¿œí
, (2)
where ð¿ ï¿œí is length of routes which is sum of each tracksegment in the left bottleneck and ð¿ï¿œí is the length of the trainð. In Figure 7, the length of ð¿ï¿œí1âŒï¿œí10 is the sum of lengths ofð1âŒð6 and ð6âŒð10; the length of ð¿ï¿œí1âŒï¿œí11 is the sum of lengthsof ð1âŒð6 and ð6âŒð11.That is, the length of the different routesin the left bottleneck is different.The end of the left bottleneckoccupied time of outbound trains is
ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí = ð¡ï¿œí + ð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí . (3)
Take the railway station in Figure 1 as an example tocalculate the occupancy time of one route of outbound train
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Non-potential conflicting areac1c2
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Figure 6: The illustrative of the same direction trains occupied different lines.
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Figure 7: The calculation of trainsâ travel time in station.
in the left bottleneck. As shown in Figure 7, it is assumed thatthe train arrives from the node ð1 and stops at line I (i.e., thegreen dashed line in Figure 7). Then the travel time of trainð from the left bottleneck signal to the line ðŒ is calculatedas ð¡ï¿œí1ï¿œíï¿œí ðŒ = (ð¿ï¿œí1âŒï¿œí10 + ð¿ï¿œí)/Vï¿œí. The end of the left bottleneckoccupied time is ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 = ð¡ï¿œí + ð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí + ð¡ï¿œí1ï¿œíï¿œí ðŒ .
The occupied time of outbound trains departing from thestation through the right bottleneck is calculated as
ð¡ï¿œíï¿œíï¿œíï¿œí = (ð¿ï¿œí + ð¿ï¿œí)Vï¿œí
, (4)
Similarly, the travel time of trains departing from the lineðŒ and leaving to node ð3 through the right bottleneck (i.e., theblue dashed line in Figure 7) is calculated as ð¡ï¿œí1ï¿œíï¿œíï¿œí ðŒ = (ð¿ï¿œí9âŒï¿œí3 +ð¿ï¿œí)/Vï¿œí.
For the inbound trains ð â ð¶2, the calculation of trainstravel time and the end time of occupied at right bottleneckare as follows, respectively:
ð¡ï¿œíï¿œíï¿œí = (ð¿ï¿œí + ð¿ï¿œí)Vï¿œí
. (5)
ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí = ð¡ï¿œí + ð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí . (6)
The travel time in the left bottleneck when the traindeparts from the station is calculated by
ð¡ï¿œíï¿œíï¿œíï¿œí = (ð¿ ï¿œí + ð¿ï¿œí)Vï¿œí
. (7)
2.4.2. The Calculation of Trains Travel Time and the End ofOccupied Time at Arrival-Departure Lines. The occupationof arrival-departure lines can be partitioned into two kinds.One is that some fast trains pass the station directly withoutstopping at the platform, and the other one is that trains stopfor a time interval before leaving the station.
In the first case, the trainâs travel time through arrival-departure lines can be obtained directly according to thelength of the arrival-departure lines and the speed of thetrains: ð¡ï¿œíï¿œí = (ð¿ ï¿œí + ð¿ï¿œí)/Vï¿œí.
In the second case, we consider the deceleration timeð¡ï¿œíï¿œíï¿œí when the speed of train ð decelerates to 0 to stop at theplatform and the acceleration time ð¡ï¿œíï¿œíï¿œí when the trainâs speedincreases to Vï¿œí from 0 to leave the station.Then the travel timeon the arrival-departure lines is ð¡ï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí .
For better expression in both cases, we adopt a binaryvariable ðï¿œí ï¿œíï¿œí to imply if train ð would stop at arrival-departureline ð . Then the travel time on the arrival-departure lines canbe calculated as follows:
ð¡ï¿œíï¿œí = ðï¿œí ï¿œíï¿œí (ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí ) + (1 â ðï¿œí ï¿œíï¿œí ) (ð¿ ï¿œí + ð¿ï¿œí)Vï¿œí
. (8)
The departure time from the arrival-departure line ð ofthe stopped trains is:
for the outbound trains ð â ð¶1: ð¡ï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí , (9)
for the inbound trains ð â ð¶2: ð¡ï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí . (10)
The end of occupied time at the arrival-departure lines isgiven at the same time:
for the outbound trains ð â ð¶1: ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œí , (11)
for the inbound trains ð â ð¶2: ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œí . (12)
In general, the relationship of trainâs occupied time onthe track can be illustrated clearly in Figure 8. Figures 8(a)and 8(b) indicate the process of outbound trains and inboundtrains passing through the station, respectively. The trendsof curves in two graphs are similar since the principle ofoutbound trains and inbound trains is similar. The ordinate
Mathematical Problems in Engineering 7
T
S
tc t ftc
right-bottleneck
arrival-departure lines
tcst d tcst a
Outbound trains
t fendtctclt end
tcst sleft-bottleneck
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right-bottleneck
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Inbound trains
t fendtctcrt end
tcs
(b)
Figure 8: A space-time network of outbound and inbound trains passing through the rail station.
indicates trainâs position when passing through the station,and the abscissa shows the time correspondingly.The red linerepresents trains that do not stop at the station and the blueone represents trains that would stop at the station.
2.5. Problem Statement and Notation
2.5.1. Problem Description and Assumption. TRP is regardedas a pivotal component in providing high efficiency of oper-ation, which can greatly affect the quality of train schedulesand passengers service levels. In practice, TRP is often basedon train planning problem, often predesigned by predicteddemands and service capability. As mentioned above, thedescription of the physical railway station is restated inthis paper. Moreover, the character of trains, the conflictswhich may occur, and the calculation of trainsâ travel timeare introduced. In general, the problems we studied arethe train routing optimization problems which involve lotsof heterogeneous trains and many safety constraints in acomplex high-speed railway station.
Mathematically, we use binary decision variables to rep-resent whether a train is allocated on a track segment or not.When a set of feasible decision variables are determined, therouting for each train can be specified concurrently. Thenan integer decision variable is quoted to ensure punctualityof trains based on planned timetable. One of the objectivesdiscussed in this paper is to minimize trainsâ total traveltime on the tracks in the station to ensure trains passing thestation in a short time. It reduces the energy consumptionand ensures the punctuality rate. It is worth noting that theroute is shortest if train is allotted on the sidings close tomain lines. Otherwise, the distance would increase if thetrain is arranged to stop on others, especially the furthestarrival-departure line from the main line. Nevertheless, itwould cause serious wear and tear on the tracks if one arrival-departure line is often occupied repeatedly. Therefore, it is ofimportance tomaintain the occupancy balance of the arrival-departure lines, which is reflected in the number of trains on
the arrival-departure lines and the duration time. Finally, thepunctuality of trains is considered so that it would not createtrains delay to reduce the operation efficiency.
Therefore, a mixed integer programming model is builtbased on the constraints of safety and station ability. Thedetails and algorithm are discussed in Section 3.The assump-tions throughout this paper are listed as follows.
Assumption 1. The trains pass the station at a constant speedVï¿œí if they do not stop at the platform. For stopped trains,the additional acceleration and deceleration time of hetero-geneous trains are same, respectively, for simplification.
Assumption 2. Due to the safety requirements of the foulingpost, trains must stay within the fouling post. So we assumethat all trains satisfy the safety requirements of the foulingpost to ensure that the tails of trains would not collide.
Assumption 3. Each train has a planned departure time andcannot depart from the station earlier than the predetermineddeparture time.
2.5.2. Notations. Regarding the trains, input data includesset/index, velocity, length, and other property parameters.In addition, it also covers relationship between trains, thetime of trainsâ setup, travel, and dwelling. Based on this basicinput data of trains, we can further determine the relationshipbetween trains and routes and trains occupation time andoccupation end time, also predetermined as inputs of ourmodel. As for the railway network, its input data involvesthe set/index of nodes and arcs, the set of connection oftrack segments, the length of different areas, and safety timeinterval and headway time. The details are summarized inTable 1.
The outputs of TRP compose the traverse route sets oftrains when they arrive at and depart form the station andthe starting time of trains to pull into the station, as well astheir precedence relation of two trains at the same track. Thedetails are introduced in Table 2.
8 Mathematical Problems in Engineering
Table 1: The indices, parameters, and sets.
Notations Descriptionð¶ Set of trainsð¶1, ð¶2 Set of outbound trains and inbound trains respectivelyc Index of trainsð Set of arrival-departure liness Index of arrival-departure linesð¿, ð Set of routes at left bottleneck and right bottleneck respectivelyl, r Index of routes at left bottleneck and right bottleneck respectivelyð Set of trainsâ routesD,A Set of nodes and arcs of railway network respectivelyð ï¿œíï¿œí , ð ï¿œíï¿œí Set of left and right bottleneck routes connected to the arrival-departure line s respectivelyðŒï¿œí Weight of the heterogeneous trainsðœ Weight of different objective functionsð¿ ï¿œí, ð¿ ï¿œí Length of routes in the left and the right bottleneck respectivelyð¿ ï¿œí Length of the arrival-departure linesð¿ ï¿œí Length of trainsð¿ï¿œí1ï¿œí2 Length of the track segment between ð1 and ð2ð¡âï¿œí Planned starting time of train c to pull into the stationð¡ï¿œíï¿œí Actual departure time of train c from the stationVï¿œí Speed of train cð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí Setup time of train cð¡ï¿œíï¿œíï¿œíï¿œíï¿œí Occupation end time at the left bottleneck of outbound trainsð¡ï¿œíï¿œíï¿œíï¿œíï¿œí Occupation end time at the right bottleneck of inbound trainsð¡ï¿œíï¿œíï¿œíï¿œíï¿œí Occupation end time at the arrival-departure linesð¡ï¿œíï¿œí Travel time at the arrival-departure line s of train cð¡ï¿œí ï¿œíï¿œí Dwell time at the arrival-departure line s of train cð¡ï¿œíï¿œíï¿œí , ð¡ï¿œíï¿œíï¿œí Additional times corresponding to acceleration and deceleration of trains, respectivelyð¡ï¿œíï¿œíï¿œí , ð¡ï¿œíï¿œíï¿œíï¿œí Travel time at left bottleneck to arrive and at right bottleneck to depart of outbound train cð¡ï¿œíï¿œíï¿œí , ð¡ï¿œíï¿œíï¿œíï¿œí Travel time at right bottleneck to arrive and at left bottleneck to depart of inbound train cð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 Time interval between trains ð1 and ð2 entering the conflict track segmentsðï¿œí ï¿œíï¿œí Binary parameters, if train c would stop at arrival-departure line s, ðï¿œí ï¿œíï¿œí = 1, else, ðï¿œí ï¿œíï¿œí = 0ðŸï¿œí1ï¿œí2 Binary parameters, if potential conflicts exist, ðŸï¿œí1ï¿œí2 = 1, else, ðŸï¿œí1ï¿œí2 = 0ðï¿œí1ï¿œí2 Binary parameters, if there are conflicts between routes ð1 and ð2, ðï¿œí1ï¿œí2 = 1, else, ðï¿œí1ï¿œí2 = 0, ð, ð , ð â ðð¿ï¿œí 1ï¿œí 2 Binary parameters, if arrival-departure lines ð 1 and ð 2 are close to the same platform, ð¿ï¿œí 1 ,ï¿œí 2 = 1, else, ð¿ï¿œí 1 ,ï¿œí 2 = 0ðï¿œí1ï¿œí2 Binary parameters, if there is transfer relationship between trains ð1 and ð2, ðï¿œí1 ,ï¿œí2 = 1, else, ðï¿œí1 ,ï¿œí2 = 0ð1ï¿œí Headway time of trains which occupy the same arrival-departure linesð2ï¿œí Minimum time interval of two trains at the potential conflict areað3ï¿œí Minimum transfer time of passengers on the platform
3. Mathematical Model of TRP in High-SpeedRailway Station
In this section, we put forward a mathematical model forTRP in a high-speed rail station. To depict this problemmore clearly, the following discussion will concentrate onspecifying each part of the models, including the objectivefunctions and systematic constraints.
3.1. Objective Function
3.1.1. Utilization Balanced of the Arrival-Departure Lines.During railway operations, the challenge is to decrease the
overall cost by means of a more efficient use of availableresources as mentioned in Giacco et al. [23]. It is worthnoting that the route is shortest if train is allotted on thesidings close to main lines. Otherwise, the distance wouldincrease if the train is arranged to stop on others, especiallythe furthest arrival-departure line from the main line. If theobjective is only to minimize the trainsâ total travel time,a large number of trains would like to stop at the arrival-departure lines near themain line, while the arrival-departurelines far from the main line are not occupied. If things go onlike this it would cause serious wear and tear on the tracks ifone arrival-departure line is often occupied repeatedly, whichmay bring extra costs owing to the frequent maintenance
Mathematical Problems in Engineering 9
Table 2: Decision variables.
Notations Descriptionð¥ï¿œíï¿œí Binary variables, if the arrival-departure line s is occupied by the train c, ð¥ï¿œíï¿œí = 1. Otherwise, ð¥ï¿œíï¿œí = 0ð¥ï¿œíï¿œíï¿œí Binary variables, if the route l in the left bottleneck is occupied by the outbound train c when pulls in, ð¥ï¿œíï¿œíï¿œí = 1.
Otherwise, ð¥ï¿œíï¿œíï¿œí = 1ð¥ï¿œíï¿œíï¿œí Binary variables, if the route l in the left bottleneck is occupied by the inbound train c when departs from the
station, ð¥ï¿œíï¿œíï¿œí = 1. Otherwise, ð¥ï¿œíï¿œíï¿œí = 1ð¥ï¿œíï¿œíï¿œí Binary variables, if the route r in the right bottleneck is occupied by the inbound train c when pulls in, ð¥ï¿œíï¿œíï¿œí = 1.
Otherwise, ð¥ï¿œíï¿œíï¿œí = 1ð¥ï¿œíï¿œíï¿œí Binary variables, if the route r in the right bottleneck is occupied by the outbound train c when departs from the
station, ð¥ï¿œíï¿œíï¿œí = 1. Otherwise, ð¥ï¿œíï¿œíï¿œí = 1ð¡ï¿œí Integer variables, indicates the starting time of train c to pull into the station
ð¿ï¿œí1 ,ï¿œí2 Binary variables, indicates the trains ð1 and ð2 precedence relation at the same track. If the train ð1 precedes ð2,ð¿ï¿œí1 ,ï¿œí2 = 1. Otherwise, ð¿ï¿œí1 ,ï¿œí2 = 0
and repaired, the utilization proportionality of the arrival-departure lines should be kept when arranging trainsâ routes.The served trains provided by the arrival-departure linesinclude passing trains and stopping trains. As can be seen, theequilibrium is not only reflected in the number of trains, butalso in the trainâs travel time that involves dwell time on thearrival-departure lines. It is converted into the mathematicalexpression and represented by the sum of variances of thenumber of trains and their dwell times:
min ð§1= 1ðâï¿œí âï¿œí
(âï¿œíâï¿œí¶
ð¥ï¿œíï¿œí â ðð)2
+ 1ðâï¿œí âï¿œí
(âï¿œíâï¿œí¶
ð¡ï¿œíï¿œí ð¥ï¿œíï¿œí â âï¿œíâï¿œí¶ ð¡ï¿œíï¿œí ð )2
.(13)
In the objective function (13),m is the number of arrival-departure lines and ð is the number of trains (i.e., the numberof arrival-departure lines that are occupied). âï¿œíâï¿œí¶ ð¡ï¿œíï¿œí ð¥ï¿œíï¿œí denotes the sum of all trainsâ travel time on the arrival-departure lines. The first half of the function (13) shows thenumber of trainsâ occupancy balance and the second halfmeans that the trainsâ travel time are relatively balanced onthe arrival-departure lines.
3.1.2. Minimize the Trainsâ Total Travel Time. Trainsâ traveltime on the tracks is another important factor in theoptimization of the train routing problem. So the secondoptimization objective function isminimizing the trainsâ totaltravel time in the station:
min ð§2= âï¿œí âï¿œí
âï¿œíâï¿œí¿
âï¿œíâï¿œí
âï¿œíâï¿œí¶1
ðŒï¿œí (ð¥ï¿œíï¿œíï¿œíð¡ï¿œíï¿œíï¿œí + ð¥ï¿œíï¿œí ð¡ï¿œíï¿œí + ð¥ï¿œíï¿œíï¿œíð¡ï¿œíï¿œíï¿œíï¿œí )+âï¿œí âï¿œí
âï¿œíâï¿œí¿
âï¿œíâï¿œí
âï¿œíâï¿œí¶2
ðŒï¿œí (ð¥ï¿œíï¿œíï¿œíð¡ï¿œíï¿œíï¿œí + ð¥ï¿œíï¿œí ð¡ï¿œíï¿œí + ð¥ï¿œíï¿œíï¿œíð¡ï¿œíï¿œíï¿œíï¿œí ) .(14)
In (14), the front part indicates the total travel time ofoutbound trains ð â ð¶1; the last half expresses inbound trainsð â ð¶2 total travel time in the railway station. In order toensure the satisfaction of passengers, the train reception anddeparture operations are punctuality and cannot be adjusteddrastically. Compared to this, the shunting operations canbe adjusted to be relatively flexible. At the same time, thepunctuality requirements are different for different types oftrains (passenger trains and freight trains). Therefore, theweight coefficient ðŒï¿œí is introduced to indicate the significanceof each train.
3.1.3. Minimize the Trainsâ Total Departure Time Deviations.For railway operators, the punctuality of trains is necessaryfor the order and efficient operation of the station. Mean-while, it is also the most concern of passengers. Therefore,the third optimization objective considered in this paper isminimizing the trainsâ total deviations between the trainsâstarting time to enter the station and the planned timetablesof trains pulling in
min ð§3 = âï¿œíâï¿œí¶
ðŒï¿œí (ð¡ï¿œí â ð¡âï¿œí ) . (15)
In (15), the same ðŒï¿œí weight coefficient is quoted. ð¡ï¿œí isthe starting time of train ð pulling into the station, and ð¡âï¿œírepresents the planned entering time of trains which cancalculate through pregiven timetable.
In summary, the linearly weighted compromise approachis adopted to handle the objective functions, and the objectivefunction studied in this paper is
min ð = ðœ1ð§1 + ðœ2ð§2 + ðœ3ð§3. (16)
In the objective function (16), ðœï¿œí (ð = 1, 2, 3) are theweight coefficients with 0 †ðœï¿œí †1 (ð = 1, 2, 3) and meet theequation requirement: ðœ1 +ðœ2 +ðœ3 = 1. It is set to distinguishthe importance of different objectives. The specific values aregiven in the cases study.
10 Mathematical Problems in Engineering
d1 d2 b1
b2
b3
c1c2
(a)
d1 d2b3
b2
b1
c1c2
(b)
Figure 9: Conflict resolution when (a) two same direction trainsâ routing conflict, where (b) is a conflict of two reverse direction trainsâroutes.
3.2. Constraints
3.2.1. Constraints of Conflicts and Dispersion. There are twotypes of routing conflicts: on the overlap track segments oftrainsâ routes and the potential conflicts at turnout nodesbeside arrival-departure line. More detailed formulation ofeach set of constraints is provided as follows.
(1) Conflicts and Dispersion of Trainsâ Routes on Track Seg-ments. The conflicts on the overlap track segments of trainsâroutes are discussed into the conflicts at left bottleneck,arrival-departure lines, and right bottleneck when trains gothrough the railway station. Here we build mathematicalformulas to illustrate how to mitigate the conflicts.
If there is a conflict owing to the delay or interruption,that is, there would be hostile routes of trains, railwayoperators should take certain measures to ensure trains passthe hostile track segments orderly and efficiently. Figure 9 isa schematic diagram of routing conflicts of the same/reversedirection.The conflict occurs at track segmentð1âŒð2 of trainsð1 and ð2, and track insulation joints ð1, ð2 and ð3. Whenthe same direction conflicts take place (i.e., as shown inFigure 9(a)), if ð1 is arranged first, train ð2 can enter afterthe tail of train ð1 passing the insulation joint ð2 to suitthe safety requirement. Similar, when the reverse directionconflict occurs as shown in Figure 9(b), train ð2 can enter afterthe tail of train ð1 passes the insulation joint ð2 if train ð1 isarranged first; train ð1 can enter after the tail of train ð2 passesthe insulation joint ð2 if train ð2 is arranged first.
In accordance with the progress of the trains, eachpossible conflict is considered and stated as follows.
(i) Routes Conflicts at the Left Bottleneck. As describedpreviously, we first discuss two outbound trainsâ routes whichhave overlapped tracks when arriving at the station from theleft bottleneck. It should be formulated as ð¡ï¿œí2 âð¡ï¿œí1 âð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 ⥠0, orð¡ï¿œí1 â ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 ⥠0, where ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 is the time from train ð1 startingpulling into the station to the tail of it passing through theinsulation joint ð2, i.e., the time interval of trains to enter therailway station. The value can be obtained by (1). Similarly,ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 is the time from train ð2 starting entering to the tail of itpassing through the insulation joint ð1.
Considering the order and relationship between trainsand routes which are chosen, the constraints can be expressedas
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí2 â ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +ð(1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1, ð2 â ð¶1, ð1, ð2 â ð¿, (17)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí1 â ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶1, ð1, ð2 â ð¿. (18)
In the constraints (17) and (18), ð¿ï¿œí1 ,ï¿œí2 is a binary variablethat indicates the trainsâ precedence relationship passingthrough the overlapping tracks. If train ð1 precedes trainð2, ð¿ï¿œí1 ,ï¿œí2 = 1. Otherwise, ð¿ï¿œí1 ,ï¿œí2 = 0. M is a sufficientlylarge number. Not only the safety time interval but also theroutesâ selection relationship are taken into account. So inconstraints (17) and (18), if a route ð1 is chosen in the leftbottleneck by outbound train ð1 (ð¥ï¿œíï¿œí1ï¿œí1 = 1, else, ð¥ï¿œíï¿œí1ï¿œí1 = 0) anda route ð2 is occupied by outbound train ð2 to enter the station(ð¥ï¿œíï¿œí2ï¿œí2 = 1, else, ð¥ï¿œíï¿œí2ï¿œí2 = 0) and, in the meantime, there arespatial conflicts between routes ð1 and ð2 (if there are spatialconflicts between routes, ðï¿œí1ï¿œí2 = 1. Otherwise, ðï¿œí1ï¿œí2 = 0.), thenthe safe time interval ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 should be held between entrancetime of trains ð1 and ð2.
Secondly, the situation for two inbound trains that departfrom the station through the left bottleneck is similar. Thatis, after the front train leaving the station, the latter traincan depart from the arrival-departure lines through the leftbottleneck after the safe time interval ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 . ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 âð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 âð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 â¥0 or ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 ⥠0must hold which are equivalent tothe following constraints:
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +ð(1 â ð¿ï¿œí1 ,ï¿œí2))⥠0, âð1, ð2 â ð¶2, ð1, ð2 â ð¿, (19)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶2, ð1, ð2 â ð¿. (20)
Mathematical Problems in Engineering 11
The occupation ending time on the arrival-departurelines can be calculated by (6) and (12), which is ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí = ð¡ï¿œí +ð¡ï¿œí ï¿œíï¿œí¡ï¿œí¢ï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œí .
Thirdly, for two opposite direction trains, an outboundtrain ð1 would arrive at the station through the left bottleneckwhile an inbound train ð2 would depart from the arrival-departure lines; then theymay cause route conflictswhich canbe avoided by the following constraints:
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +ð(1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð1, ð2 â ð¿, (21)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð1, ð2 â ð¿. (22)
(ii) Routes Conflicts at the Right Bottleneck. Similar to theconflicts in the left bottleneck, the trains routing conflicts fortwo inbound trains, two outbound trains, and two oppositedirection trains should satisfy the following constraints in theright bottleneck.
When two inbound trains arrive at the station throughthe right bottleneck, the route conflicts may occur.Therefore,we set following constraints to avoid collision:
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí2 â ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +M (1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1, ð2 â ð¶2, ð1, ð2 â ð , (23)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí1 â ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶2, ð1, ð2 â ð . (24)
For two outbound trains which would depart from thestation through the right bottleneck
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +M (1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1, ð2 â ð¶1, ð1, ð2 â ð , (25)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œí1 â ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶1, ð1, ð2 â ð . (26)
For two opposite direction trains, considering the orderand relationship between trains and routes which are chosen,the constraint can be
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 +M (1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð1, ð2 â ð , (27)
ð¥ï¿œíï¿œí1ï¿œí1ð¥ï¿œíï¿œí2ï¿œí2ðï¿œí1ï¿œí2 (ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð1, ð2 â ð . (28)
Among these six constraints, the safe time interval ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2should be kept. The same with the discussion above, theoccupation end time on the arrival-departure lines can becalculated by (6) and (12).
(iii) Routes Conflicts on the Arrival-Departure Lines. The trainrouting conflicts which may occur on the arrival-departurelines has been described in Section 2. Here we propose thealgebraic formulas to resolve the conflicts. If the difference oftwo trainsâ departure time is smaller than the headway timeð1ï¿œí, in other words, if train ð1 occupies the arrival-departureline first, train ð2 can enter from bottleneck after the end ofoccupied time at arrival-departure lines of train ð1 and after aheadway time: (ð¡ï¿œíï¿œí2 â ð¡ï¿œí ï¿œíï¿œí â ð¡ï¿œíï¿œíï¿œí ) â (ð¡ï¿œíï¿œí1 + ð¡ï¿œíï¿œíï¿œí ) â ð1ï¿œí ⥠0, whereð¡ï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí for the outbound trains and for theinbound trains ð¡ï¿œíï¿œí = ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí + ð¡ï¿œíï¿œíï¿œí + ð¡ï¿œí ï¿œíï¿œí which can be calculatedaccording to (9) and (10) described in Section 2. Accordingto Assumption 1, the additional acceleration and decelerationtime of trains that run in different speed are same, and theformula is converted to ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œí1 â ð¡ï¿œí2ï¿œí â ð1ï¿œí ⥠0. Similar, iftrain ð2 enters first, it should be satisfied: ð¡ï¿œíï¿œí1 â ð¡ï¿œíï¿œí2 â ð¡ï¿œí1ï¿œí âð1ï¿œí â¥0.
Thus the constraints should be satisfied as follows:
ð¥ï¿œí1ï¿œí ð¥ï¿œí2ï¿œí (ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œí1 â ð¡ï¿œí2ï¿œí â ð1ï¿œí +M (1 â ð¿ï¿œí1 ,ï¿œí2)) ⥠0,âð1, ð2 â ð¶, ð1, ð2 â ð¿, ð â ð (29)
ð¥ï¿œí1ï¿œí ð¥ï¿œí2ï¿œí (ð¡ï¿œíï¿œí1 â ð¡ï¿œíï¿œí2 â ð¡ï¿œí1ï¿œí â ð1ï¿œí +Mð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶, ð1, ð2 â ð¿, ð â ð. (30)
The value of ð¡ï¿œíï¿œí can be calculated from (3), (6), (9), and(10).
(2) Potential Conflicts and Avoidance. In small scale stations,there are a small number of arrival-departure lines andplatforms that any of them can be occupied by trains. Asshown in Figure 5, the outbound train ð1 may occupy thearrival-departure line 3 to stop when the operation is busy.In this case, it may cause potential conflicts between oppositedirection trains if they choose the arrival-departure lineson the same side. Thus measures should be adopted toavoid the two trains arriving the potential conflicting areasimultaneously.The trainsâ order should be determined by thedecision variable ð¿ï¿œí1 ,ï¿œí2 . In Figure 5(a), if train ð1 arrives at thestation through the left bottleneck, train ð2 can depart fromthe arrival-departure line 3 after the tail of train ð1 passes thepotential conflicting area after a safety time interval ð2ï¿œí (i.e.,the inequality ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí â ð2ï¿œí ⥠0 should be satisfied). Similar,if train ð2 leaves the station first, after a safety time interval,train ð1 can enter the station through the left bottleneck afterthe tail of train ð2 passes the potential conflicting area (i.e.,satisfying ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð2ï¿œí ⥠0).
12 Mathematical Problems in Engineering
To sum up, we use the following formula to avoidpotential conflicts:
ð¥ï¿œíï¿œí1ï¿œíð¥ï¿œí2ï¿œí ðŸï¿œí1ï¿œí2 (ð¡ï¿œíï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí â ð2ï¿œí +ð(1 â ð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð â ð¿, ð â ð, (31)
ð¥ï¿œíï¿œí1ï¿œíð¥ï¿œí2ï¿œí ðŸï¿œí1ï¿œí2 (ð¡ï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí2 â ð2ï¿œí +ðð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð â ð¿, ð â ð. (32)
It is suitable for solving the potential conflicts as shownin Figure 5(a). Constraints (33) and (34) are used to solve theproblems of the situation in Figure 5(b):
ð¥ï¿œíï¿œí1ï¿œíð¥ï¿œí2ï¿œí ðŸï¿œí1ï¿œí2 (ð¡ï¿œí2 â ð¡ï¿œíï¿œíï¿œíï¿œíï¿œí1 â ð2ï¿œí +ð(1 â ð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð â ð , ð â ð, (33)
ð¥ï¿œíï¿œí1ï¿œíð¥ï¿œí2ï¿œí ðŸï¿œí1ï¿œí2 (ð¡ï¿œíï¿œí1 â ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí â ð2ï¿œí +ðð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1 â ð¶1, ð2 â ð¶2, ð â ð , ð â ð. (34)
3.2.2. The Limited Capacity Restrictions. This constraint setconsiders the limited infrastructure capacity. Obviously, atrain can only utilize one route at bottleneck and arrival-departure lines. To guarantee the safe operation of trainstraveling in the railway station, the constraints are listed asfollows:
âï¿œí âï¿œí
ð¥ï¿œíï¿œí = 1 âð â ð¶, (35)
âï¿œíâï¿œí¿
ð¥ï¿œíï¿œíï¿œí = 1 âð â ð¶1, (36)
âï¿œíâï¿œí
ð¥ï¿œíï¿œíï¿œí = 1 âð â ð¶1, (37)
âï¿œíâï¿œí
ð¥ï¿œíï¿œíï¿œí = 1 âð â ð¶2, (38)
âï¿œíâï¿œí¿
ð¥ï¿œíï¿œíï¿œí = 1 âð â ð¶2. (39)
Constraint (35) ensures that only one arrival-departureline can be occupied by one train. Constraints (36) and (37)guarantee that each train must utilize one and only one routeto traverse the left and right bottleneck for each outboundtrain, respectively. Similarly, only one route in the right andleft bottleneck can be chosen by each inbound train throughconstraints (38) and (39), respectively.
3.2.3. Connectivity of Train Routing Constraints. Trainsâ rout-ing connectivity should be ensured. That is, for an outbound
train, the route occupied at the left bottleneck should connectto the arrival-departure line which must connect to the routein the right bottleneck. Otherwise, it is conceivable that thetrain cannot pass the discontinuous route through the station.Therefore, these restrictions can guarantee the feasibility oftrainsâ routes.
ð¥ï¿œíï¿œí = âï¿œíâï¿œí ðð
ð¥ï¿œíï¿œíï¿œí, âð â ð¶1, ð â ð, ð â ð¿, (40)
ð¥ï¿œíï¿œí = âï¿œíâï¿œí ðð
ð¥ï¿œíï¿œíï¿œí, âð â ð¶2, ð â ð, ð â ð¿, (41)
ð¥ï¿œíï¿œí = âï¿œíâï¿œí ðð
ð¥ï¿œíï¿œíï¿œí, âð â ð¶2, ð â ð, ð â ð , (42)
ð¥ï¿œíï¿œí = âï¿œíâï¿œí ðð
ð¥ï¿œíï¿œíï¿œí, âð â ð¶1, ð â ð, ð â ð , (43)
whereð ï¿œíï¿œí andð ï¿œíï¿œí are sets of left and right bottleneck routesconnected to the arrival-departure line ð respectively.3.2.4. Restrictions of Passengersâ Transfer at Station. It repre-sents one of the concerns of railway operators to facilitatepassengers transfer to improve the service level. Nowadays,various large scale high-speed rail stations have set upshortcut channels in the railway station to offer conveniencefor passengersâ transfer in station. This enables passengers toachieve their traveling purpose in a shorter time and cost.There are transfer elevators at each platform in some largeand well-equipped facilities stations. And passengers canquickly find the corresponding ticket gate directly throughthe information from the ticket or the electronic screen.
Two trains that may have transfer relationship are dis-cussed in this paper. A constraint is set to keep the two trainsstopped at two near platforms to ensure that passengers orcrews transfer in a comfortable time:
ð¥ï¿œí1ï¿œí 1ð¥ï¿œí2ï¿œí 2ð¿ï¿œí 1ï¿œí 2ðï¿œí1ï¿œí2 ⥠1, âð1, ð2 â ð¶, ð 1, ð 2 â ð, (44)
At the same time, the target train should stop at theplatform for a time interval ð3ï¿œí to ensure that all passengerscomplete transfer comfortably, which is required to allowpassengers alight from one train, move to the correspondingplatform track, and board the other train. So we obtain thefollowing constraints:
ðï¿œí1ï¿œí2ð¥ï¿œí1ï¿œí 1ð¥ï¿œí2ï¿œí 2 (ð¡ï¿œíï¿œí2 â (ð¡ï¿œíï¿œí1 â ð¡ï¿œí ï¿œí1ï¿œí ) â ð3ï¿œí +ð(1 â ð¿ï¿œí1 ,ï¿œí2))⥠0, âð1, ð2 â ð¶, ð 1, ð 2 â ð, (45)
ðï¿œí1ï¿œí2ð¥ï¿œí1ï¿œí 1ð¥ï¿œí2ï¿œí 2 (ð¡ï¿œíï¿œí1 â (ð¡ï¿œíï¿œí2 â ð¡ï¿œí ï¿œí2ï¿œí ) â ð3ï¿œí +ðð¿ï¿œí1 ,ï¿œí2) ⥠0,âð1, ð2 â ð¶, ð 1, ð 2 â ð. (46)
Mathematical Problems in Engineering 13
3.2.5. Variable Feasible Ranges. The following constraintsindicate the feasible ranges of the variables:
ð¡ï¿œí â ð¡âï¿œí ⥠0, âð â ð¶, (47)
ð¡ï¿œí â ð, âð â ð¶, (48)
ð¥ï¿œíï¿œí , ð¥ï¿œíï¿œíï¿œí, ð¥ï¿œíï¿œíï¿œí, ð¥ï¿œíï¿œíï¿œí, ð¥ï¿œíï¿œíï¿œí â {0, 1} ,ð â ð, ð â ð¶, ð â ð¿, ð â ð , (49)
ð¿ï¿œí1 ,ï¿œí2 â {0, 1} , ð â ð¶. (50)
Constraint (47) indicates that all trains cannot departfrom the station before the planned timetable, where integerdecision variable ð¡ï¿œí indicates the start time measured inminutes to enter the station of train ð. Constraints (49) and(50) ensure that those decision variables are 0-1 variables.
Herein, the train routing problem for heterogeneoustrains can be formulated as the following model, which isessentially a mixed integer nonlinear programming model:
min ð = ðœ1ð§1 + ðœ2ð§2 + ðœ3ð§3,s.t. constraints (17)â(50). (51)
4. Solution Methodology Based onTabu Search
The number of decision variables of the model will increasegreatly as the increasing number of optional routes of trainswhich is NP-hard problemmentioned in Ahuja et al. [18] andCarey et al. [19]. In the proposed model, the situation scaleof the values of variables ð¥ is huge as the number of them isgreat. For instance, when ð¶ = {ð1, ð2} and ð¿ = {ð1, ð2, ð3}, thereare 26 kinds of values of ð¥ = [ð¥ï¿œíï¿œí1ï¿œí1 , ð¥ï¿œíï¿œí1ï¿œí2 , ð¥ï¿œíï¿œí1ï¿œí3 , ð¥ï¿œíï¿œí2ï¿œí1 , ð¥ï¿œíï¿œí2ï¿œí2 , ð¥ï¿œíï¿œí2ï¿œí3],where each element of ð¥ may take 0 or 1. It would cost asignificant amount of time to solve the problemunder a largercase andmaynot even get the optimal solution in a reasonablecomputation time. Apart from this, the mathematic modelis nonlinear and there is no algorithm which can solve suchproblems accurately at present. Therefore in this section, wedesign a tabu search algorithm stepwise to obtain the near-optimal solution based on the nonlinear characteristic of themodel for the TRP with a large scale railway station.
The tabu search algorithm is a deterministic meta-heuristic on account of local search [24], which makesextensive use of memory for guiding the search. From theincumbent solution, non-tabumoves define a set of solutionswhich called the neighborhood of the incumbent solution.The best solution is selected as the new incumbent solutionat each step and stored in the tabu list to avoid being trappedin local optima and re-visiting the same solutions. When thenumber of solutions achieves the length of the tabu list, theearliest one which entered the list is released. The algorithmstops until reaching the termination conditions.
In our model, we notice that constraints (17)-(34) andconstraints (45) and (46) are nonlinear. It is obvious that ifthe variable ð¥ is determined, the model (51) would becomean integer linear programming model that determines the
departure time of each train. For such mathematical pro-gramming model, there are already mature algorithms, suchas branch and bound and cutting plane, etc. Taking con-straints (17) and (18) as an example, the decision variablesð¥ï¿œíï¿œí1ï¿œí1 and ð¥ï¿œíï¿œí2ï¿œí2 are binary variables, and ðï¿œí1ï¿œí2 is known binaryparameters according to wether there are conflicts betweenroutes ð1 and ð2. There are two cases due to the value ofdecision variables. The first is that the value of any variablesor parameter is 0 among the decision variables ð¥ï¿œíï¿œí1ï¿œí1 , ð¥ï¿œíï¿œí2ï¿œí2and parameter ðï¿œí1ï¿œí2 . Then the left side of the formula is 0,and the constraint is effective apparently. The second caseis that the values of them are 1, then the constraints weresatisfied if the formulas are satisfied inside the brackets.Hence the inequality constraints are obviously satisfied inthe first case. And in the second case, the constraints turninto an integer linear constraints, not only for constraints(17) and (18), but for other constraints in the model (51).As a result, we set the decision variable ð¥ as the tabuobject. Based on that, through generating neighborhoodof incumbent solution and searching a good solution ateach step, we can gradually approach the near-optimalsolution.
To deal with the problems mentioned above in themodel presented in this paper, we discuss the setting of theneighborhood of incumbent solution, the selection of initialsolution, and the scheme of algorithm in detail below.
4.1. Neighborhood of Incumbent Solution. The establishmentof incumbent solution neighborhood is crucial for a betterdirection of the search, which affects the quality of solution ateach step. In our model, we hope that trains would have theshortest travel time, which performances the shortest routes.As mentioned, the travel distance would increase if trainsare arranged on the sidings far from the main line. In ourproblem, we need to determine a good quality initial solutionand tabu move to reach the near-optimal solution. Thereforewe generate two different neighborhoods of the incumbentsolution, respectively:
ðï¿œíï¿œíï¿œíï¿œí¡ï¿œíï¿œíï¿œí: based on the incumbent routing set, whichcontains routes in bottleneck and arrival-departureline, the two adjacent sidings and correspondingroutes in bottleneck are accommodated into it. Thatis, the neighborhood contains not only the incumbentarrival-departure lines of trains and correspondingroutes in left and right bottleneck, but also the twoadjacent sidings and corresponding routes in thebottleneck.ðï¿œí¡ï¿œíï¿œíï¿œí¢: the neighborhood of incumbent routing set fortabu move containsðï¿œíï¿œíï¿œíï¿œí¡ï¿œíï¿œíï¿œí, which involves four adja-cent siding and corresponding routes in bottleneck.Besides, remove the situation that trains with closedeparture times and occupy the same sidings.
In general, ðï¿œíï¿œíï¿œíï¿œí¡ï¿œíï¿œíï¿œí is used to determine a good qualityinitial solution, whose scale is smaller. And we need tosearch the solution roughly in a short time. In contrast tothis, ðï¿œí¡ï¿œíï¿œíï¿œí¢ is a wide neighborhood in order to avoid empty
14 Mathematical Problems in Engineering
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
j2
j3
g1
g2
d13
d14
d17
d12
d15
d16
d19
d20
d21
d22
d23
d18
d31
d24
d25
d26
d27
d28
d29
d30 d32
d33
d34z1
z2
j1
I
II
3
45
6
outbound
Figure 10: The structure of a small-scale station.
neighborhoods as far as possible while avoiding to search inall possible situations. It is thus worthwhile distinguishing thedifferent neighborhood structures.
4.2. Determined of Initial Solution. As describe above, thequality of initial solution is crucial for the performanceof tabu search. A good initial feasible solution can greatlyimprove the speed of searching for the optimal solution.If we randomly assign a set of routes as the incumbentsolution, it would take a lot of time to search, and thequality of the near-optimal solution cannot be guaranteed.The objective value is much smaller obviously than theprevious strategy if starting from a good incumbent solu-tion. So in this paper, we can get a feasible solution oftrain routing set through our prior knowledge of trainsâcharacter and construction of the railway station whichis just a preliminary program. In view of this, the cor-responding routing set solution is selected which satisfiesconstraints and at the same time reaches the minimumobjective function in the solution neighborhoods at nextstep. Then regard this solution as a new incumbent solution.By cycling this step until the iterative termination condi-tion is satisfied, a batter initial solution of the tabu searchalgorithm is determined. The procedure is summarized inTable 3.
4.3. The Algorithm Scheme. In this paper, we first obtainthe initial routes based on method describe in Table 3.Then we generate the neighborhood ðï¿œí¡ï¿œíï¿œíï¿œí¢ of the incumbentsolution to reduce the range of the search. And we solve thecorresponding MLP in theðï¿œí¡ï¿œíï¿œíï¿œí¢ to choose the best solution.Finally, the best solution is chosen as the new incumbentsolution and stored in the tabu list. The steps shown inTable 3 are repeated in sequence recursively. We summarizethe procedure of this heuristic algorithm in Table 4.
In this case the tabu list contains the situation of alltrainsâ routes. The aspiration criterion is set to reinforce thelocal search to avoid losing an admirable state. That is, if asolution in the tabu list is largely superior to others, thenreconsider and search it as a new incumbent solution. Theother solutions are forbidden in the tabu list to avoid beingtrapped in local optima or infinite loops.
5. Numerical Experiments
In this section, two sets of numerical experiments are imple-mented to show the performance of our proposed model.Specifically, for the models formulated, a small case is imple-mented to demonstrate the application and performance, inwhich CPLEX solver is used to obtain exact optimal solutionwhich compares with the near-optimal solution obtained bythe proposed heuristic algorithm based on the tabu search,while in large scale case experiment, we apply the proposedheuristic algorithm to the Jinan West high-speed railwaystation, in which a tabu search algorithm is designed inPYTHON 2.7.13 to obtain the near-optimal solutions of trainsrouting on a Windows 10 platform with Intel(R) Core(TM)i7-8550U CPU and 8G RAM.
5.1. A Small-Scale Case Study. In this case, we consider a twomain ðŒ and II lines railway station as shown in Figure 10whichis outbound and inbound direction, respectively. In the leftbottleneck, there are a reception line ð§1 of outbound trains, adeparture line ð§2 of inbound trains, and a locomotive waitingtrack ð1. At the same time, there are 6 arrival-departure lines,of which arrival-departure lines II, 3 and 4 next to a platform.In the right bottleneck, there are 4 lines including a departureline ð1 of outbound trains, a reception line ð2 of inboundtrains. and two locomotive waiting tracks ð2 and ð3. Thedistances between connection points in this railway stationare shown in Table 5.
In order to test the effectiveness of the proposed modelsand solution algorithms, we derive a set of instances in thissmall-scale rail station with different numbers of trains (thetype of trains is exhaustive to demonstrate the versatilityand correctness of the model). Table 6 shows the originand destination of each train, as well as the characteristicparameters of them. Trains ð1 and ð4 are outbound trains,while ð3 is inbound train. At the same time, there is a transferrelationship between ð1 and ð3. Trains ð2 and ð5 are shuntingoperations, wherein ð2 travels from the locomotive waitingtracks ð2 to ð1 and ð5 travels from ð1 to be an originating train(i.e., train ð5 would choose an arrival-departure line to stopand then depart from the station through the departure nodeð1). In order to distinguish the importance of operations,the higher punctuality required for train reception, and
Mathematical Problems in Engineering 15
Table 3: The scheme of determining initial solution.
Step 1. Input initial information, including trainsâ character, railway station parameters, train speed profiles, trains withtransfer relationship, planned timetable, etc., set ð = 0.Step 2. Choose a feasible routes set as incumbent solution ð¥ððð = ð¥ð through prior knowledge of trains and parameters of therailway station, get objective value ð ð¥ððð.Step 3. Generate neighborhoodððððð¡ððð(ð¥ððð) of ð¥ððð, and corresponding neighborhood solution: Sððððð¡ððð(ð¥ððð).Step 4. Search the best solution ð ð¥â in Sððððð¡ððð(ð¥ððð) and corresponding routing set ð¥â
Step 5. If the objective value of ð ð¥âis better than that of ð ð¥ððð, ð ð¥ððð = ð ð¥â, ð¥ððð = ð¥â; otherwise, let ð¥ððð as current solutioncontinuously.Step 6. If the value of ð¥ððð does not change, ð+ = 1; otherwise, go to step 7.Step 7. If ð == 1, output ð ð¥ððð and the corresponding routing set ð¥ððð, stop. Else, go to step 3.
Table 4: Pseudocode of the tabu search algorithm.
Input: ð¥ï¿œíï¿œíï¿œí as an initial solution, parameter ðï¿œí1ï¿œí2 , ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 , ð¡ï¿œíï¿œíï¿œíï¿œí2ï¿œí1 , etcðð¿ = 0assign the incumbent solution: ð¥ï¿œíï¿œíï¿œí€ = ð¥ï¿œíï¿œíï¿œíwhile the frequency of one solution is not reached:
generate neighborhoodðï¿œí¡ï¿œíï¿œíï¿œí¢(ð¥ï¿œíï¿œíï¿œí€)updateðï¿œí¡ï¿œíï¿œíï¿œí¢(ð¥ï¿œíï¿œíï¿œí€) by removing duplicate and solutions that do not satisfy constraintsfor i inðï¿œí¡ï¿œíï¿œíï¿œí¢(ð¥ï¿œíï¿œíï¿œí€):
get solutions: S(i) satisfied the constraints in the mathematical modelððï¿œí¡ï¿œíï¿œíï¿œí¢(ð¥ï¿œíï¿œíï¿œí€).append(S(i)) which is neighborhood solutionget the decision variable ð¡ï¿œí and ð¿ï¿œí1 ,ï¿œí2 correspondingly
choose the non-tabu optimal solution ð ð¥ï¿œè in ððï¿œí¡ï¿œíï¿œíï¿œí¢(ð¥ï¿œíï¿œíï¿œí€)search the corresponding routing set: ð¥ï¿œè ð¥ï¿œíï¿œíï¿œí€ = ð¥ï¿œè ðð¿.append(ð¥ï¿œíï¿œíï¿œí€)if len(ðð¿)>5:
del ðð¿[0]if the number of iterations is an integer multiple of 5:
if exist a solution ð¥ï¿œíï¿œí ï¿œí satisfy aspiration criterion:ð¥ï¿œíï¿œíï¿œí€ = ð¥ï¿œíï¿œí ï¿œíð¡ððð¢ððð ð¡.append(ð¥ï¿œíï¿œíï¿œí€)if len(ðð¿)>5:
del ðð¿[0]Output: optimal solution: ð¥ï¿œíï¿œíï¿œí€, ð¡ï¿œí and ð¿ï¿œí1 ,ï¿œí2
departure operations compared to the shunting operations,we set the weights to 0.9 and 0.1, respectively. As for themultiobjective of our model, we attach the most importanceto the punctuality, then the total travel time and utilizationbalanced of the arrival-departure lines are considered evenly.Thus set the parameters ðœ1 = ðœ2 = 0.3 and ðœ3 = 0.4. Inaddition to this, the preparation time of each train is 1min,and the additional time that provides trainsâ acceleration anddeceleration is 2min and 1min, respectively.
We assume the headway time ð1ï¿œí = 2min, the minimumtime interval of potential area of two trains ð2ï¿œí = 1min,and the minimum transfer time offered to passengers in thearrival-departure lines ð3ï¿œí = 10min. It is easy to neglect inthis small example that if two opposite direction trains pass
through node ð9 or ð27 one after the other, there may bepotential conflicts.
The algorithm designed in Section 4 is used to solve themodel of this small case. First, an incumbent solution oftrainsâ routing is chosen based on the scheme described inTable 3,whose quality has a great influence on the efficiency ofsearching the near-optimal solution.The prior knowledge weconsidered here includes our analysis of the station structure,the nature of each train, and the route by which the train ismost likely to occupy, etc. For instance, there is a transferrelationship between ð1 and ð3, while the arrival-departurelines II and 4 are located on either side of the platform.Therefore, we can arrange arrival-departure lines II and 4,respectively, as initial routes of trains ð1 and ð3. In terms of
16 Mathematical Problems in Engineering
Table 5: Distances between points in the railway network in Figure 10.
Node:ðï¿œí Node:ðï¿œí Distance(m) Node:ðï¿œí Node:ðï¿œí Distance(m) Node:ðï¿œí Node:ðï¿œí Distance(m)1 2 90 9 11 60 22 28 851 9 305 10 12 80 23 28 902 4 60 11 16 85 24 25 903 5 48 11 17 90 25 34 3003 6 60 12 18 750 25 26 604 5 60 13 19 750 26 29 1004 6 48 14 20 750 27 30 1485 8 175 15 21 750 28 27 606 7 100 16 22 750 29 31 607 14 170 17 23 750 29 32 907 9 90 18 24 80 30 31 908 10 90 19 25 125 30 32 608 13 125 20 26 170 31 33 509 15 125 21 27 125 33 34 60
Table 6: The parameters of trains in the small case.
Train Origin Destination Average velocity(km/h) Operations Planned
timetableStop time(min)
ð1 ð§1 ð1 30 Passing 11:25 17ð2 ð2 ð1 15 Shunting 11:28 0ð3 ð2 ð§2 30 Passing 11:30 10ð4 ð§1 ð1 30 Passing 11:34 6ð5 ð1 ð1 15 Shunting 11:37 30
Table 7: The start time of trains to enter the station.
Train ð1 ð2 ð3 ð4 ð5Start time 11:25 11:28 11:30 11:34 11:37Delay time(min) 0 0 0 0 0
the origin and destination of train ð2, there is only one route.And for the remaining two trains, they can be organized tostop at the arrival-departure lines close to the entrance line,as ð4 occupies arrival-departure line ðŒ and ð5 would stop atarrival-departure line 4.
At the same time, there are 5 trains stopping and passingthrough the railway station. In the case that origin anddestination are prespecified, the number of each trainâs routeat left bottleneck, arrival-departure line, and right bottleneckis 6, respectively. Therefore, there are 65 possibilities for therouting arrangement of 5 trains. Of course, it may increaseexponentially sharply with the number of trains and arrival-departure lines. Therefore, in order to reduce computationtime, we take the trainsâ routes arrangement as tabu decisionvariable based on the scheme described in Table 4. Then weget the near-optimal routing arrangement of each train (asshown in Figure 11) and the start time when trains enter thestation (as shown in Table 7) based on the designed algorithmin a short time.
The routing arrangements can be seen clearly in Figure 11.From our intuitive, the objective that utilization balanced of
the arrival-departure lines is not only satisfied but also meetsthe transfer relationship between ð1 and ð3. What is more, alltrains are of punctuality. It is noteworthy that the results weobtained from the algorithm are the same as obtained directlyfrom CPLEX solver, and the result suits the requirements oftrain operations absolutely.Therefore, it can be demonstratedthat the proposed model can optimize the routes of trainsand the algorithm can get a reasonable solution to ensure thepunctuality of train and the rationality of routing.
5.2. Large-Scale Case Experiment. To test the effectivenessand efficiency of our proposed train routing problem modeland approach, this section applies the proposed model to areal-world case study on the Jinan West high-speed railwaystation in China, which involves 17 arrival-departure linesincluding 4 main lines and 8 platforms to the operations ofreception and departure. The network of railway is describedby themethodmentioned in Section 2.1 as shown in Figure 12.
In Figure 12, arrival-departure lines ðŒ and III are out-bound main lines and II and IV are inbound main lines,respective. Arrival-departure lines ðŒ and II are used to receiveand depart trains between Beijing and Shanghai, and thedirection from Shanghai toward Beijing is defined as theinbound direction, while it is outbound direction. Similarly,arrival-departure lines III and IV are used to receiving anddeparting trains between Beijing and Qingdao, while thedirection from Qingdao toward Beijing is defined as theinbound direction, whereas it is outbound direction. The
Mathematical Problems in Engineering 17
d1
d2
d3
d4
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d6
d7
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d28
d29
d30 d32
d33
d34z1
z2
j 1
c1c2c3c4c5
outbound
Figure 11: The route arrangements of the small case.
11
III
III
IV
9
87
65
1314
16
17
outbound
j1
j2
j3
j4
z1
z2
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e2inbound
d1
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d8d10
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d25 d27
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d37 d54
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d74 d81d83 d86 d91
d88d90
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d94 d95
d96 d97
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d76 d82 d84
d85d75d65
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outbound
inbound
12
15
10
Figure 12: Construction of Jinan West Railway Station in China.
arrival-departure lines 5âŒ10 are located on the side of theoutbound, while arrival-departure lines 11âŒ17 are located onthe side of the inbound. The distribution of the platform isshown in Figure 12. In addition, there are four locomotivewaiting tracks ð1âŒð4.
Owing to the scale of the station, the number of turnoutsand crossovers is large, so it is difficult to collect theactual length between all track segments. Thus, the distancesbetween various connection points are not listed here. Itis described that if a train stops on the arrival-departureline which is far away from the main line, this would costmore travel time. In the real-world operation, trains needto align with the platform signs when they arrive at thestation which need a little more time, and this need not beconsidered when trains depart from the station. In addition,in terms of our common sense, the trainsâ speed will decreasewhen arrive at the station due to the limit of lateral speedat turnouts. So there is insignificant difference in travelingtime in the station between trains running at 300 km/h and
250 km/h. Combined with the calculation methods of trackoccupied time described in Section 2.4 and the large numberof observation results in the control center, the occupancytime of each route is shown in Table 8.The parameters ðŒï¿œí = 1since the shunting operation is not regarded.
In the real-world station trains reception and departureoperations, the nonstopped trains directly pass through thestation on main lines, while the stopped trains can stop onany arrival-departure lines except main lines. However, ifinbound trains enter the station from ð2 and stop on thearrival-departure line 5 in the station as shown in Figure 12,this not only causes route conflicts with the outbound trainsbut also brings inconvenience to the outbound trains, whichoccupies more equipment and increases the difficulty ofoperations. Therefore, the inbound train should occupy thearrival-departure line on the side of inbound main linecorrespondingly (i.e., inbound trains enter the station fromð2 or ð2 and stop at arrival-departure lines 11âŒ17). Like theoutbound trains, namely, outbound trains enter the station
18 Mathematical Problems in Engineering
j4
g2
e2d96
d97
d89
d87
d98
d99
d100
d101
d76 d82 d84
d85d75
d65
d77
d78
d68
d79
d80
d62
d63
d64
d66
d67
d69
d70
Figure 13: The parallel routes of two trains at bottleneck.
Table 8: Travel time of trains on the left bottleneck, the arrival-departure lines, and the right bottleneck.
Stoppedarrival-departureline of outboundtrains
Travel time ofreception at theright bottleneckð¡ï¿œíï¿œíï¿œí (s)
Travel time ofdeparture at the left
bottleneckð¡ï¿œíï¿œíï¿œíï¿œí (s)Stopped
arrival-departure lineof inbound trains
Travel time ofreception at the left
bottleneckð¡ï¿œíï¿œíï¿œí (s)Travel time of
departure at the rightbottleneckð¡ï¿œíï¿œíï¿œíï¿œí (s)
11 100 80 5 125 10512 105 85 6 120 10013 110 90 7 115 9514 115 95 8 110 9015 120 100 9 105 8516 125 105 10 100 8017 130 110
from ð§1 or â1 and stop at arrival-departure lines 5âŒ10. As aresult, the trainsâ travel time on the left or right bottleneckis listed in Table 8. We select all trains passing through theJinanWest railway station between 16:00 and 19:00, that is, 46trains (including 27 outbound trains and 19 inbound trains).The parameters are shown in Table 9.
Since only a period of time is selected, we assume that therailway stations are all vacant.Thedata of every track segmentcannot be obtained accurately; thus the value of ð¡ï¿œíï¿œíï¿œíï¿œí1ï¿œí2 is 1min.If the large instance is solved accurately, only the situationof a train occupying the arrival-departure lines needs tocalculate 1746 times. Such a large scale problem belongs tothe NP-hard as mentioned above. We adopt the heuristicalgorithm designed, starting from a set of routes of 46 trainswhich is the best solution chosen through the iterations ofmethod described in Table 3 to obtain the superior initialsolution. Based on that, the tabu search algorithm is implied.We choose the routes with the lowest objective value asthe target of the next search. Consequently the efficiencyof searching a better solution is ensured. After that, thenear-optimal solution with train routing is obtained within254 seconds, and the results of different direction trains areshown in Tables 10 and 11, respectively (only a part of nodesclearly expresses the routes occupied by train duo to the largenumber of nodes).
As shown in Tables 10 and 11, it can be seen that thefrequency of arrival-departure lines occupied are almostequal, which satisfies the objective of utilization balanced of
them.Thenonconflict parallels are occupied in the bottleneckas far as possible for two trains whose arrival time are closeto each other. As for the inbound trains G134 and G4218,the arrival time of them are close and may cause a conflictin the right bottleneck. Therefore, routes are arranged on theparallel respectively as shown in Figure 13 (the red and greendotted lines represent the routes of the two trains, instead ofthe red and the yellow dotted line) to prevent the possibilityof collision.
In addition, the planned start timetable to enter thestation is obtained by the arrival time minus the travel timeof the reception in the bottleneck according to Table 9,which is same with the actual start time calculated by thealgorithm. That is, the model we propose and designedalgorithm can get the trains routing arrangement in a highquality and short time based on the punctuality of trains.From the analysis results of the examples, the proposedmodel and the designed algorithm can solve the TRP effi-ciently.
It is worth mentioning that the problem we discussed is aNP-hard problem. Owing to the nonlinearity and numerousdecision variables of the proposedmathematicalmodel, thereis neither proper commercial software available nor an algo-rithmwhich can solve such problems accurately at present. Atthe same time, the heuristic algorithm we designed can solvethe problem accurately and efficiently as mentioned above.So we did not compare the designed algorithm with othermethods in the large-scale case.
Mathematical Problems in Engineering 19
Table 9: Parameters of trains passing through the Jinan West railway station.
Train Terminal Arrivaltime
Departuretime
Stop time(min) Direction Origin/Destination in
station Operations
G30 Beijing South 16:00 16:02 2 Inbound ð2/ð§2 PassingG215 Shanghai 16:05 16:08 3 Outbound ð§1/ð1 PassingG191 Qingdao 16:00 16:09 9 Outbound â1/ð1 PassingG143 Shanghai 16:10 16:12 2 Outbound ð§1/ð1 PassingG168 Beijing South 16:12 16:16 4 Inbound ð2/ð§2 PassingG17 Shanghai 16:22 16:24 2 Outbound ð§1/ð1 PassingG132 Beijing South 16:21 16:28 7 Inbound ð2/ð§2 PassingG145 Shanghai 16:15 16:34 19 Outbound ð§1/ð1 PassingG322 Beijing South 16:31 16:39 8 Inbound ð2/ð§2 PassingG1203 Shanghai 16:37 16:43 6 Outbound ð§1/ð1 PassingG412 Beijing South 16:37 16:44 7 Inbound ð2/ð§2 PassingG193 Qingdao 16:42 16:47 5 Outbound â1/ð1 PassingG35 Hangzhou East 16:51 16:53 2 Outbound ð§1/ð1 PassingG56 Beijing South 16:51 16:53 2 Inbound ð2/ð§2 PassingG161 Anqing 16:57 16:58 2 Outbound ð§1/ð1 PassingG4 Beijing South 16:59 17:01 2 Inbound ð2/ð§2 PassingG53 Hangzhou East 17:00 17:03 3 Outbound ð§1/ð1 PassingG134 Beijing South 17:09 17:11 2 Inbound ð2/ð§2 PassingG4218 Beijing South 17:01 17:15 14 Inbound ð2/â2 Passing
G351 HuangshanNorth 17:17 17:19 2 Outbound ð§1/ð1 Passing
G45 Jiangshan 17:22 17:24 2 Outbound ð§1/ð1 PassingG475 Rongcheng 17:29 17:31 2 Outbound â1/ð1 PassingG194 Beijing South 17:28 17:32 4 Inbound ð2/â2 PassingG138 Beijing South 17:33 17:36 3 Inbound ð2/ð§2 PassingG1267 Qingdao East 17:34 17:40 6 Outbound â1/ð1 PassingG330 Tianjin West 17:37 17:40 3 Inbound ð2/ð§2 PassingG21 Shanghai 17:39 17:41 2 Outbound ð§1/ð1 PassingG147 Shanghai 17:43 17:45 2 Outbound ð§1/ð1 PassingG140 Beijing South 17:41 17:45 4 Inbound ð2/ð§2 PassingD6077 Rongcheng - 17:48 - Outbound ð1/ð1 OriginatingG195 Qingdao 17:57 18:00 3 Outbound â1/ð1 PassingG18 Beijing South 17:59 18:01 2 Inbound ð2/ð§2 PassingG37 Hangzhou East 18:02 18:04 2 Outbound ð§1/ð1 PassingG164 Beijing South 17:53 18:06 13 Inbound ð2/ð§2 PassingG1235 Shanghai 18:08 18:14 6 Outbound ð§1/ð1 PassingG149 Shanghai 18:13 18:18 5 Outbound ð§1/ð1 PassingG142 Beijing South 18:18 18:21 3 Inbound ð2/ð§2 PassingG474 Beijing South 18:27 18:29 2 Inbound ð2/â2 PassingG1231 Shanghai 18:17 18:29 12 Outbound ð§1/ð1 PassingG197 Qingdao 18:28 18:31 3 Outbound â1/ð1 PassingG324 Beijing South 18:25 18:33 8 Inbound ð2/ð§2 PassingG39 Hangzhou East 18:33 18:35 2 Outbound ð§1/ð1 PassingG23 Shanghai 18:39 18:41 2 Outbound ð§1/ð1 PassingG52 Tianjin West 18:46 18:49 3 Inbound ð2/ð§2 PassingG1257 Shanghai 18:50 18:53 3 Outbound ð§1/ð1 PassingG153 Shanghai 18:56 18:58 2 Outbound ð§1/ð1 PassingThe â-â in the table indicates that the corresponding train is originating train. The entry time and stop time depend on the actual situation. It is assumed herethat D6077 train enters from ï¿œí1 and departs after 30 min.
20 Mathematical Problems in Engineering
Table 10: The routes occupied by outbound trains through the Jinan West railway station between 16:00 and 19:00.
Train Nodes of trains passing through at the leftbottleneck when arrive at the station.
Arrival-departureline
Nodes of trains passing through at the rightbottleneck when depart from the station.
G215 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1G191 â1, ð18, ð32, ð40 8 ð57, ð72, ð81, ð86, ð88, ð1G143 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð88, ð90, ð93, ð1G17 ð§1, ð22, ð23, ð26, ð33, ð42 10 ð59, ð73, ð74, ð90, ð93, ð1G145 ð§1, ð6, ð8, ð10, ð13, ð17, ð37 5 ð54, ð83, ð86, ð88, ð90, ð93, ð1G1203 ð§1, ð22, ð23, ð26, ð33, ð42 10 ð59, ð73, ð74, ð90, ð93, ð1G193 â1, ð18, ð39 7 ð56, ð81, ð91, ð92, ð1G35 ð§1, ð22, ð23, ð26, ð41 9 ð58, ð74, ð88, ð90, ð93, ð1G161 ð§1, ð6, ð8, ð18, ð32, ð40 8 ð57, ð72, ð81, ð86, ð88, ð90, ð93, ð1G53 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1G351 ð§1, ð6, ð8, ð10, ð13, ð17, ð37 5 ð54, ð83, ð86, ð88, ð90, ð93, ð1G45 ð§1, ð22, ð23, ð26, ð33, ð42 10 ð59, ð73, ð74, ð90, ð93, ð1G475 â1, ð18, ð39 7 ð56, ð81, ð91, ð92, ð1G1267 â1, ð26, ð33, ð42 10 ð59, ð73, ð74, ð1G21 ð§1, ð6, ð8, ð18, ð32, ð39 7 ð56, ð81, ð86, ð88, ð90, ð93, ð1G147 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1D6077 ð1, ð17, ð31, ð38 6 ð35, ð71, ð83, ð86, ð88, ð1G195 â1, ð26, ð33, ð42 10 ð59, ð73, ð74, ð1G37 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1G1235 ð§1, ð6, ð8, ð18, ð32, ð40 8 ð57, ð72, ð81, ð86, ð88, ð90, ð93, ð1G149 ð§1, ð6, ð8, ð10, ð13, ð17, ð31, ð38 6 ð55, ð71, ð83, ð86, ð88, ð90, ð93, ð1G1231 ð§1, ð6, ð8, ð18, ð32, ð39 7 ð56, ð81, ð86, ð88, ð90, ð93, ð1G197 â1, ð18, ð32, ð40 8 ð57, ð72, ð81, ð86, ð88, ð1G39 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1G23 ð§1, ð22, ð23, ð26, ð33, ð42 10 ð59, ð73, ð74, ð90, ð93, ð1G1257 ð§1, ð6, ð8, ð26, ð41 9 ð58, ð74, ð90, ð93, ð1G153 ð§1, ð6, ð8, ð18, ð32, ð40 8 ð57, ð72, ð81, ð86, ð88, ð90, ð93, ð1
6. Conclusions
This paper focuses on modeling and solving the TRP basedon considering heterogeneous trains and detailed structureof the rail station. The main research work and conclusionsare summarized as follows:
(1) A detailed mathematical formulation for TRPdescribes the routing arrangement in a large andcomplex high-speed railway station. The turnoutnode and the arrival-departure line node are definedto describe the layout of the railway station instead ofthe traditional railway network. The heterogeneoustrains are taken into account; and the potentialcollisions of trains and convenience for passengerstransferring at station are considered as constraints.Then we propose a method to calculate the occupiedtime of each track and describe the TRP problemmore completely and realistically.
(2) A high-efficient algorithm based on tabu search isproposed based on the proposed model. We set
two different neighborhoods for searching incumbentsolution and tabu move to search the near-optimalsolution. The scale of first neighborhood is smaller tosearch in a short time, and the second neighborhoodis wilder in order to avoid empty neighborhoods asfar as possible. Then we set two strategies of them,respectively.
(3) The correctness of the proposed model is verifiedwith a small example that includes all types of trainsand operations. The results obtained by the proposedtabu search algorithm are the same as those obtaineddirectly from CPLEX solver. The result is fully inline with the requirements of train operations. In thelarge scale case, we chose the actual trains within3 hours passing through the Jinan West railwaystation which involves 17 arrival-departure lines. Theexcellent solutions are obtainedwithin 254 seconds bythe designed algorithm.
It is realistic and easy to understand our proposed model.However, it has the characteristics of non-linearity and has
Mathematical Problems in Engineering 21
Table 11: The routes occupied by inbound trains through the Jinan West railway station between 16:00 and 19:00.
Train Nodes of trains passing through at the rightbottleneck when arrive at the station.
Arrival-departureline
Nodes of trains passing through at the leftbottleneck when depart from the station.
G30 ð2, ð98, ð99, ð76, ð75, ð64 11 ð47, ð34, ð28, ð27, ð24, ð§2G168 ð2, ð89, ð87, ð76, ð65 12 ð48, ð28, ð15, ð11, ð§2G132 ð2, ð98, ð99, ð82, ð78, ð67 14 ð50, ð36, ð25, ð15, ð11, ð§2G322 ð2, ð98, ð99, ð76, ð75, ð64 11 ð47, ð34, ð28, ð27, ð24, ð§2G412 ð2, ð98, ð99, ð82, ð77, ð66 13 ð49, ð35, ð25, ð15, ð11, ð§2G56 ð2, ð89, ð87, ð82, ð68 15 ð51, ð25, ð15, ð11, ð§2G4 ð2, ð98, ð99, ð82, ð77, ð66 13 ð49, ð35, ð25, ð15, ð11, ð§2G134 ð2, ð98, ð99, ð82, ð78, ð67 14 ð50, ð36, ð25, ð15, ð11, ð§2G4218 ð2, ð100, ð101, ð85, ð80, ð70 17 ð53, ð30, ð19, â2G194 ð2, ð76, ð75, ð64 11 ð47, ð34, ð28, â2G138 ð2, ð89, ð87, ð84, ð79, ð69 16 ð52, ð29, ð19, ð15, ð11, ð§2G330 ð2, ð98, ð99, ð82, ð77, ð66 13 ð49, ð35, ð25, ð15, ð11, ð§2G140 ð2, ð98, ð99, ð82, ð78, ð67 14 ð50, ð36, ð25, ð15, ð11, ð§2G18 ð2, ð89, ð87, ð76, ð65 12 ð48, ð28, ð15, ð11, ð§2G164 ð2, ð89, ð87, ð82, ð68 15 ð51, ð25, ð15, ð11, ð§2G142 ð2, ð98, ð99, ð82, ð77, ð66 13 ð49, ð35, ð25, ð15, ð11, ð§2G474 ð2, ð76, ð65 12 ð48, ð28, â2G324 ð2, ð89, ð87, ð84, ð79, ð69 16 ð52, ð29, ð19, ð15, ð11, ð§2G52 ð2, ð89, ð87, ð76, ð65 12 ð48, ð28, ð15, ð11, ð§2
difficulties to solve themathematical formulas.Therefore, onepossible future direction is to improve themodel. Second, thetabu search algorithm needs to set effective rules to generatethe incumbent solutionâs neighborhood and rules for tabumove. In this respect, further improvements are needed toobtain a better near-optimal solution more efficiently. Moreimportantly, future research does not restrict to a railwaystations. With the rapid development of high-speed railway,there are more train routing and scheduling problems,train rerouting and rescheduling optimization problems, etc.,which need to be further studied.
Data Availability
Previously reported [construction and related data of JinanWest Railway Station in China] data were used to supportthis study and are available at [https://www.researchgate.net/publication/276080928 Using Simulated Annealing in aBottleneck Optimization Model at Railway Stations].These prior studies (and datasets) are cited at relevant placeswithin the text as [20] (i.e., [2]). The [trains informationpassing through the Jinan West railway station] data usedto support the findings of this study have been deposited inthe [Railway Customer Service Center of China] repository([https://kyfw.12306.cn/otn/leftTicket/init]).
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper.
Acknowledgments
This research was supported by the Research Foundationof State Key Laboratory of Rail Traffic Control and Safety,Beijing Jiaotong University, China (nos. RCS2017ZT012,RCS2018ZZ001, and RCS2018ZZ003), and the FundamentalResearch Funds for theCentralUniversities (no. 2018YJS193).
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