Research Article | Open Access
Ziyan Feng, Chengxuan Cao, Yutong Liu, Yaling Zhou, "A Multiobjective Optimization for Train Routing at the High-Speed Railway Station Based on Tabu Search Algorithm", Mathematical Problems in Engineering, vol. 2018, Article ID 8394397, 22 pages, 2018. https://doi.org/10.1155/2018/8394397
A Multiobjective Optimization for Train Routing at the High-Speed Railway Station Based on Tabu Search Algorithm
This paper focuses on the train routing problem at a high-speed railway station to improve the railway station capacity and operational efficiency. We first describe a node-based railway network by defining the turnout node and the arrival-departure line node for the mathematical formulation. Both considering potential collisions of trains and convenience for passengers’ transfer in the station, the train routing problem at a high-speed railway station is formulated as a multiobjective mixed integer nonlinear programming model, which aims to minimize trains’ departure time deviations and total occupation time of all tracks and keep the most balanced utilization of arrival-departure lines. Since massive decision variables for the large-scale real-life train routing problem exist, a fast heuristic algorithm is proposed based on the tabu search to solve it. Two sets of numerical experiments are implemented to demonstrate the rationality and effectiveness of proposed method: the small-scale case confirms the accuracy of the algorithm; the resulting heuristic proved able to obtain excellent solution quality within 254 seconds of computing time on a standard personal computer for the large-scale station involving up to 17 arrival-departure lines and 46 trains.
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 of railway transport faces challenges due to increasing passengers and freight transportation demands. In order to enhance the capacity of the railway station, operators can be dedicated to adding the number of track lines, which requires plenty of manpower, materials, and financial resources obviously. By contrast, making proper arrangements of trains’ routes at bottleneck and arrival-departure lines based on existing track lines can save lots of resources and achieve the same effect. Hence, attention is paid to arrange the trains’ routes reasonably in the station to improve the operational efficiency and reduce the operation costs [1, 2] (D’Ariano et al., 2008).
In general, the railway planning process is divided into strategic level, tactic level, and operational level. In this paper, we focus on the tactic level, especially on the timetable optimization and railway track allocation/train routing in a complex high-speed railway station. At present, the optimization of the train routing problem (TRP) in the railway station is mainly based on the mathematical optimization model. Some studies aimed at the complicated bottleneck of the railway station, as D’Ariano et al.  considered that reducing trains’ collisions at the bottleneck can improve punctuality without decreasing the capacity usage of the lines and a detailed model for conflict resolution and different algorithms was illustrated. Based on the robustness of routing assignment, Jia et al.  proposed the optimization of routing utilization at bottleneck. Kang et al.  presented a bottleneck optimization model thereafter to enhance the capacity by reasonably arranging routes and turnouts. Other parts of the researches concentrated on the problems of arrival-departure lines occupancy. For example, Billionnet  considered assigning trains to the available tracks at a railway station instead of taking into account the collision at the bottleneck. Caprara et al.  proposed a 0-1 integer programming model to describe the routing problem of trains at the station platform; moreover, the quadratic objective function was linearized and solved by integer linear programming. Besides, an optimization model is presented by Qiao et al.  based on the train schedules to arrange suitable arrival-departure lines for passenger trains. Furthermore, the research of the TRP with joint optimization of bottleneck and arrival-departure lines is also found in the literature. For instance, Zwaneveld et al.  considered both bottleneck and arrival-departure lines and proposed a 0-1 programming model to arrange trains passing 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.  rearranged train routing in real-time unpredictable events and found the best solution using truncated branch-and-bound and tabu search algorithms.
In the past few decades, there have been limited researches on TRP in high-speed railway stations. Most studies were associated with the train timetable problems [9–11]. TRP is tantamount to selecting a sequence of tracks for a train from its origin to destination, with the objective of minimizing the sum of travel time, the total operating cost, and/or increasing the capacity of railway network. Xu et al.  defined the objective function to minimize deviations between trains’ arrival time at the destination and original timetable. The optimization model proposed by Li et al.  aimed to minimize the total delay of all trains in the railway network. Liu et al.  developed a mathematical model whose aim is minimizing the total occupation time of station bottleneck sections to avoid train delays. In addition, some research focused on the optimization by keeping the most balanced utilization arrival-departure lines like Qiao et al. . One of the objective functions considered by Zhou et al.  and Zhou et al.  was to minimize the total travel time on the track. Apart from this, the most important constraint should be taken into account was the spatiotemporal interactions between each train operation route in the TRP (D’Ariano et al., 2008) [2, 4, 12]. Furthermore, it should be considered that trains which occupy the same arrival-departure lines should satisfy a headway time constraint [1, 6]. Zhou et al.  and Corman et al.  proposed that trains should stop for enough time to ensure the transfer time of passengers and crews. What is more, Fang et al.  analyzed a comprehensive survey on different models 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  and unlikely to get an exact optimal solution in a short computational time under the large-scale and complex situation. In order to get an approximate optimal solution rapidly, many studies proposed different heuristic algorithms based on different strategies. Corman and Meng  introduced the online dynamic models and algorithms for the rail traffic management in order to provide punctual and reliable services. Specifically, Ahuja et al.  considered the issue of railway scheduling and presented a heuristic algorithm to get the approximate optimal solution in short time. Carey et al.  studied the large-scale problem that involved lots of train schedules and routes and proposed heuristic algorithms to solve it. Liu et al.  designed a hybrid algorithm between genetic algorithm and the simulated annealing algorithm. In recent years, for the sake of further improving the computational speed and quality of solution, some studies have proposed improving heuristic algorithms. He et al.  proposed an improve branch-and-price algorithm to deal with the large scale integer programming. Zhou et al.  used an efficient train-based lagrangian relaxation decomposition to the simultaneous passenger train routing and timetabling problem. Additionally, some studies used commercial software like Qi et al.  who obtained for an approximate optimal solution within 30s by a local search heuristic algorithm using CPLEX solver and Xu et al.  who solved train routing and timetabling problem with switchable rules by CPLEX solver with OPL language.
Besides that, some studies dealt with the integer programming problem using tabu search algorithm, as Isaai and Singh combined the heuristic with tabu search and simulated annealing search control strategies to deal with the train timetabling problem. Similar, Corman et al.  rearranged trains’ routes based on the graph theory and got the approximate optimal solution of the combination of truncated branch-and-bound and tabu search algorithms. Li et al.  proposed a tabu search algorithm, and they pointed out that the algorithm relies on a better initial solution; otherwise the result obtained is not stable. In addition to this, Goh et al. proposed a tabu search with sampling and perturbation to generate feasible solutions. The tabu search algorithms and variable neighborhood are applied by Samà et al.  to improve the solution for the real-time management problem of scheduling and routing trains in complex and busy railway networks.
As can be found in numerous studies, there were few elaborate mathematical models to describe TRP and further study on TRP in the network of the high-speed railway station. In this research, with the motivation of greatly improving the solving efficiency of TRP in the network of the high-speed railway station, we intend to provide the following contributions:(i)Describe a railway network by defining the turnout node and the arrival-departure line node on account of the traditional layout of the railway station, which is regarded as a directed graph. The nodes are regarded as vertices, and the actual connection of the lines is regarded as arcs. The railway station is divided into three parts and the connection sets are built to satisfy the connection relationship of them.(ii)According to the given nominal timetable, the calculation methods of trains occupying each track are elaborated. Then formulate the TRP in the network of the high-speed railway station as a multiobjective mixed integer nonlinear programming model, in which both are considered the potential collisions of trains and the convenience for passengers’ transfer in the high-speed railway station. In the proposed model, it is not only minimizing train departure time deviations with the most balanced utilization of arrival-departure lines but also minimizing the total occupation times of all tracks.(iii)The TRP is NP-hard problem. Therefore, it is very unlikely to devise a polynomial-time (exact) algorithm for it. In order to get an approximate optimal solution rapidly, a heuristic algorithm is proposed based on the tabu search to solve the large-scale TRP in the network of the high-speed railway station.(iv)Numerical examples are implemented to demonstrate the effectiveness and efficiency of proposed method. By taking advantage of an efficient tabu search algorithm, we can solve the model rapidly for a small case. The results we obtained from the algorithm are the same as obtained directly from CPLEX solver. In the large-scale case involving 17 arrival-departure lines and 46 trains between 16:00~19:00, the resulting heuristic proved able to obtain excellent solution quality within 254 seconds of computing time on a standard 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 TRP in the network of the high-speed railway station. Section 4 deals with the development of a heuristic algorithm based on the tabu search. Section 5 describes the instances used and provides computational results. Finally, some conclusions and further research directions are presented in Section 6.
2. Train Routing Problem Description
2.1. Layout of Railway Station
We consider a railway station as illustrated in Figure 1. This railway station consists of bottlenecks and some arrival-departure lines whose length is . The left side of the station in Figure 1 is defined as left bottleneck whose length is . The route set of it is . Similarly, the right side is defined as right bottleneck whose length is . The route set of it is . The bottleneck involves tracks where trains arrive and depart. For instance, an outbound train arrives from node and stops at arrival-departure line. In Figure 2(a), there are 4 possible routes for the outbound train that arrives from node , whose set is in the left bottleneck. Similar, an inbound train departs from arrival-departure line to the node . There are also 4 possible routes for the inbound train departs from node as shown in Figure 2(b) whose set is . So the route set of the left bottleneck is . In particular, at large stations, the number of routes in the bottleneck may be larger than the number of arrival-departure lines due to the existence of multiple crossovers and turnouts. The route set of arrival-departure lines is , which contains I, II, 3, and 4 as shown in Figure 1.
Normally, track lines are divided by insulation joints. In this paper, the turnout node and the arrival-departure line node are defined as follows to describe the railway station.1.Turnout node: the intersection of the lines in the station. It includes turnouts (as shown in Figure 1, and , etc.), crossovers (as shown in Figure 1, , etc.), and the position of signal which located in the entrance of the bottleneck (i.e., the boundary point of the station).2.Arrival-departure line node: the connection points of bottleneck and arrival-departure lines in the railway station, namely, the signal position of the entrance of the arrival-departure lines (i.e.. the nodes in Figure 1).
During the operation, the outbound trains can only arrive from turnout node and depart from turnout node as shown in Figure 1. Similar, the inbound trains can only arrive from turnout node and depart from turnout node . However, no matter in which direction trains are, any of arrival-departure lines can be occupied. Hence the physical network of the railway is regarded as a directed graph. The nodes are regarded as vertices, and the actual connection of the lines is regarded as arcs. Let denote the set of all vertices and denote the set of all arcs in the network.
We consider a set of trains in both directions. The set of trains is denoted by , in which is the set of outbound trains and is the set of inbound trains. The heterogeneous trains are taken into account. We assume that these trains are categorized into two types: slow trains and fast trains. In addition, there are shunting operation beside train reception and departure operations. The speeds of shunting trains are slower than the ordinary trains.
In addition, some of the fast trains pass the station without stopping while some trains stop at platform for passengers to alight or transfer. Different trains’ stop time may be different, whether they are fast or slow trains. As we all know, the speed of trains is not a constant, while it changes with the driver’s braking when the train would stop at the platform. Similarly, the speed of trains does not increase suddenly when departing from the platform. Therefore, the additional time that provides trains’ acceleration and deceleration is considered. At the same time, the length of trains is also taken into account to meet the actual situation.
2.3. Conflicts and Potential Conflicts
The conflicts of tracks which trains occupy are the most imperative problem to be solved in TRP. In other words, trains cannot occupy the same track at the same time. A conflict occurs whenever trains traverse the same track and do not respect the minimum time interval at the bottleneck and arrival-departure lines. What is more, it would result in potential conflicts. These situations are discussed separately as follows.
2.3.1. Train Routing Conflicts at Bottleneck
The bottleneck consists of channels with some turnouts, crossovers, and other facilities, which may cause spatial intersections of trains’ alternative sets of routes, especially when some unpredictable events or interruptions occurred. So it is an important task for train dispatchers to avoid the train collision and to assure the trains passing the tracks orderly. The conflicts between two trains in the bottleneck mainly occurred at the intersection of the track segments (the segment as shown in Figure 3) and the intersection of crossovers (the point as shown in Figure 3).
In Figures 3(a) and 3(b), trains travel in the same/reverse direction, and the collision occurs at the track segment between nodes and . Similarly, the collision occurs at the track segment at the node when the trains and travel in the same and reverse direction, respectively. The methods to dissipate conflicts will be depicted in detail in 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 a headway time to suit the limited infrastructure capacity. That is, one train occupies this arrival-departure line a specific headway time later than the other one. As shown in Figure 4, there are two trains and . The origin and destination of train are A and C, respectively, and the origin and destination of train are B and C. Obviously, if the route is assigned to and occupies route in a short interval, this would cause a conflict at platform 1. In this case, the latter train should stop and wait until the previous train leaves from the platform 1 for a reasonable headway time. It may cause the latter train to be behind the schedule time and even to have second delay in severe cases. But if the train occupies route , the conflicts would be avoided and the train operational efficiency would be ensured in the railway station.
Therefore, if the two trains’ departure interval is smaller than the headway time , it is better to arrange them at different lines. Just as illustrated in Figure 4, if the difference of two trains’ departure time is bigger than the headway time , the same arrival-departure line can be arranged.
2.3.3. Potential Conflicts of Train Routing
It is noteworthy that only considering trains’ routing at bottleneck and headway time at arrival-departure lines cannot completely meet the requirement of safe operation. There are further potential conflicts.
For instance, the outbound train would stop at arrival-departure line II from outbound main line while the inbound train would depart from line 3 to the inbound main line (as shown in Figure 5(a), the yellow dotted line and the blue dotted line represent the route of trains and , respectively). There is no time provided to avoid the potential conflict area (the red circle area in Figure 5), since it is not in the context of the two conflicts described above. Therefore, a time separation should be adopted even though two trains do not occupy the same track; i.e., first arrives and then departs later or first leaves and then arrives later, the same as shown in Figure 5(b). It is necessary to set constraints so that two consecutive trains arrive the potential conflict area in a time separation.
In addition, when two trains are in the same direction, the conflicts between them are not potential conflicts. Just as shown in Figure 6, and are outbound trains. It is apparent that there exists a conflict in the red circle area. However, due to the conflicts at the bottleneck between the two trains, the constraints are set to avoid the collision; namely, conflicts in the rad circle area are also dissolved. Therefore, the potential constraints we discussed are only among the reverse trains.
2.4. The Calculation Methods of Track Occupation Time
In this paper, not only is the train routing chosen as the decision variables but also optimized the schedule of trains to start entering the station according to the planned schedule when trains depart from the station. Thus the time for trains passing the conflict tracks needs be calculated. In addition, the convenience factors of passengers transferring at station are taken into account, and the associated trains’ dwell-time intersection should be long enough. As a consequence, the trains’ travel time in different track sections should be clearly calculated.
To our knowledge, as long as a track segment is occupied, the lines, the fouling posts, and the signals are also occupied at the same time. We elaborate the calculation of the occupied time and the end time of occupied of trains passing through the left bottleneck, the arrival-departure lines, and the right bottleneck in turn. The occupied trains at the bottleneck are divided into two situations: train reception and train departure, whose calculated method of train occupied time of tracks is same. So only take the outbound trains as an example to explain in detail.
2.4.1. The Calculation of Trains Travel Time and the End Time of Occupied at the Bottleneck
For outbound trains in the left bottleneck , they start to enter the station at and pass each node with speed . The track is unlocked after their tails pass the end node of the track section. So the trains’ occupied time of each track segment is calculated as follows:where is occupancy time of a certain track segment and is length of the track segment . Thus, the travel time of outbound trains from the left bottleneck signals to the arrival-departure lines is calculated with the following:where is length of routes which is sum of each track segment in the left bottleneck and is the length of the train . In Figure 7, the length of is the sum of lengths of and ; the length of is the sum of lengths of and . That is, the length of the different routes in the left bottleneck is different. The end of the left bottleneck occupied time of outbound trains is
Take the railway station in Figure 1 as an example to calculate the occupancy time of one route of outbound train in the left bottleneck. As shown in Figure 7, it is assumed that the train arrives from the node and stops at line I (i.e., the green dashed line in Figure 7). Then the travel time of train from the left bottleneck signal to the line is calculated as . The end of the left bottleneck occupied time is .
The occupied time of outbound trains departing from the station through the right bottleneck is calculated as
Similarly, the travel time of trains departing from the line and leaving to node through the right bottleneck (i.e., the blue dashed line in Figure 7) is calculated as .
For the inbound trains , the calculation of trains travel time and the end time of occupied at right bottleneck are as follows, respectively:
The travel time in the left bottleneck when the train departs from the station is calculated by
2.4.2. The Calculation of Trains Travel Time and the End of Occupied Time at Arrival-Departure Lines
The occupation of arrival-departure lines can be partitioned into two kinds. One is that some fast trains pass the station directly without stopping at the platform, and the other one is that trains stop for a time interval before leaving the station.
In the first case, the train’s travel time through arrival-departure lines can be obtained directly according to the length of the arrival-departure lines and the speed of the trains: .
In the second case, we consider the deceleration time when the speed of train decelerates to 0 to stop at the platform and the acceleration time when the train’s speed increases to from 0 to leave the station. Then the travel time on the arrival-departure lines is .
For better expression in both cases, we adopt a binary variable to imply if train would stop at arrival-departure line . Then the travel time on the arrival-departure lines can be calculated as follows:
The departure time from the arrival-departure line of the stopped trains is:
The end of occupied time at the arrival-departure lines is given at the same time:
In general, the relationship of train’s occupied time on the track can be illustrated clearly in Figure 8. Figures 8(a) and 8(b) indicate the process of outbound trains and inbound trains passing through the station, respectively. The trends of curves in two graphs are similar since the principle of outbound trains and inbound trains is similar. The ordinate indicates train’s position when passing through the station, and the abscissa shows the time correspondingly. The red line represents trains that do not stop at the station and the blue one represents trains that would stop at the station.
2.5. Problem Statement and Notation
2.5.1. Problem Description and Assumption
TRP is regarded as a pivotal component in providing high efficiency of operation, which can greatly affect the quality of train schedules and passengers service levels. In practice, TRP is often based on train planning problem, often predesigned by predicted demands and service capability. As mentioned above, the description of the physical railway station is restated in this paper. Moreover, the character of trains, the conflicts which may occur, and the calculation of trains’ travel time are introduced. In general, the problems we studied are the train routing optimization problems which involve lots of heterogeneous trains and many safety constraints in a complex high-speed railway station.
Mathematically, we use binary decision variables to represent whether a train is allocated on a track segment or not. When a set of feasible decision variables are determined, the routing for each train can be specified concurrently. Then an integer decision variable is quoted to ensure punctuality of trains based on planned timetable. One of the objectives discussed in this paper is to minimize trains’ total travel time on the tracks in the station to ensure trains passing the station in a short time. It reduces the energy consumption and ensures the punctuality rate. It is worth noting that the route is shortest if train is allotted on the sidings close to main lines. Otherwise, the distance would increase if the train is arranged to stop on others, especially the furthest arrival-departure line from the main line. Nevertheless, it would cause serious wear and tear on the tracks if one arrival-departure line is often occupied repeatedly. Therefore, it is of importance to maintain 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, the punctuality of trains is considered so that it would not create trains delay to reduce the operation efficiency.
Therefore, a mixed integer programming model is built based on the constraints of safety and station ability. The details and algorithm are discussed in Section 3. The assumptions throughout this paper are listed as follows.
Assumption 1. The trains pass the station at a constant speed if they do not stop at the platform. For stopped trains, the additional acceleration and deceleration time of heterogeneous trains are same, respectively, for simplification.
Assumption 2. Due to the safety requirements of the fouling post, trains must stay within the fouling post. So we assume that all trains satisfy the safety requirements of the fouling post to ensure that the tails of trains would not collide.
Assumption 3. Each train has a planned departure time and cannot depart from the station earlier than the predetermined departure time.
Regarding the trains, input data includes set/index, velocity, length, and other property parameters. In addition, it also covers relationship between trains, the time of trains’ setup, travel, and dwelling. Based on this basic input data of trains, we can further determine the relationship between trains and routes and trains occupation time and occupation end time, also predetermined as inputs of our model. As for the railway network, its input data involves the set/index of nodes and arcs, the set of connection of track segments, the length of different areas, and safety time interval and headway time. The details are summarized in Table 1.
The outputs of TRP compose the traverse route sets of trains when they arrive at and depart form the station and the starting time of trains to pull into the station, as well as their precedence relation of two trains at the same track. The details are introduced in Table 2.
3. Mathematical Model of TRP in High-Speed Railway Station
In this section, we put forward a mathematical model for TRP in a high-speed rail station. To depict this problem more clearly, the following discussion will concentrate on specifying each part of the models, including the objective functions 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 available resources as mentioned in Giacco et al. . It is worth noting that the route is shortest if train is allotted on the sidings close to main lines. Otherwise, the distance would increase if the train is arranged to stop on others, especially the furthest arrival-departure line from the main line. If the objective is only to minimize the trains’ total travel time, a large number of trains would like to stop at the arrival-departure lines near the main line, while the arrival-departure lines far from the main line are not occupied. If things go on like this it would cause serious wear and tear on the tracks if one arrival-departure line is often occupied repeatedly, which may bring extra costs owing to the frequent maintenance and repaired, the utilization proportionality of the arrival-departure lines should be kept when arranging trains’ routes. The served trains provided by the arrival-departure lines include passing trains and stopping trains. As can be seen, the equilibrium is not only reflected in the number of trains, but also in the train’s travel time that involves dwell time on the arrival-departure lines. It is converted into the mathematical expression and represented by the sum of variances of the number of trains and their dwell times:
In the objective function (13), m is the number of arrival-departure lines and is the number of trains (i.e., the number of arrival-departure lines that are occupied). denotes the sum of all trains’ travel time on the arrival-departure lines. The first half of the function (13) shows the number of trains’ occupancy balance and the second half means that the trains’ travel time are relatively balanced on the arrival-departure lines.
3.1.2. Minimize the Trains’ Total Travel Time
Trains’ travel time on the tracks is another important factor in the optimization of the train routing problem. So the second optimization objective function is minimizing the trains’ total travel time in the station:
In (14), the front part indicates the total travel time of outbound trains ; the last half expresses inbound trains total travel time in the railway station. In order to ensure the satisfaction of passengers, the train reception and departure operations are punctuality and cannot be adjusted drastically. Compared to this, the shunting operations can be adjusted to be relatively flexible. At the same time, the punctuality requirements are different for different types of trains (passenger trains and freight trains). Therefore, the weight coefficient is introduced to indicate the significance of each train.
3.1.3. Minimize the Trains’ Total Departure Time Deviations
For railway operators, the punctuality of trains is necessary for the order and efficient operation of the station. Meanwhile, it is also the most concern of passengers. Therefore, the third optimization objective considered in this paper is minimizing the trains’ total deviations between the trains’ starting time to enter the station and the planned timetables of trains pulling in
In (15), the same weight coefficient is quoted. is the starting time of train pulling into the station, and represents the planned entering time of trains which can calculate through pregiven timetable.
In summary, the linearly weighted compromise approach is adopted to handle the objective functions, and the objective function studied in this paper is
In the objective function (16), are the weight coefficients with and meet the equation requirement: . It is set to distinguish the importance of different objectives. The specific values are given in the cases study.
3.2.1. Constraints of Conflicts and Dispersion
There are two types of routing conflicts: on the overlap track segments of trains’ routes and the potential conflicts at turnout nodes beside arrival-departure line. More detailed formulation of each set of constraints is provided as follows.
(1) Conflicts and Dispersion of Trains’ Routes on Track Segments. The conflicts on the overlap track segments of trains’ routes are discussed into the conflicts at left bottleneck, arrival-departure lines, and right bottleneck when trains go through the railway station. Here we build mathematical formulas 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, railway operators should take certain measures to ensure trains pass the hostile track segments orderly and efficiently. Figure 9 is a schematic diagram of routing conflicts of the same/reverse direction. The conflict occurs at track segment of trains and , and track insulation joints and . When the same direction conflicts take place (i.e., as shown in Figure 9(a)), if is arranged first, train can enter after the tail of train passing the insulation joint to suit the safety requirement. Similar, when the reverse direction conflict occurs as shown in Figure 9(b), train can enter after the tail of train passes the insulation joint if train is arranged first; train can enter after the tail of train passes the insulation joint if train is arranged first.
In accordance with the progress of the trains, each possible conflict is considered and stated as follows.
(i) Routes Conflicts at the Left Bottleneck. As described previously, we first discuss two outbound trains’ routes which have overlapped tracks when arriving at the station from the left bottleneck. It should be formulated as , or , where is the time from train starting pulling into the station to the tail of it passing through the insulation joint , i.e., the time interval of trains to enter the railway station. The value can be obtained by (1). Similarly, is the time from train starting entering to the tail of it passing through the insulation joint .
Considering the order and relationship between trains and routes which are chosen, the constraints can be expressed as
In the constraints (17) and (18), is a binary variable that indicates the trains’ precedence relationship passing through the overlapping tracks. If train precedes train , . Otherwise, . M is a sufficiently large number. Not only the safety time interval but also the routes’ selection relationship are taken into account. So in constraints (17) and (18), if a route is chosen in the left bottleneck by outbound train (, else, ) and a route is occupied by outbound train to enter the station (, else, ) and, in the meantime, there are spatial conflicts between routes and (if there are spatial conflicts between routes, . Otherwise, .), then the safe time interval should be held between entrance time of trains and .
Secondly, the situation for two inbound trains that depart from the station through the left bottleneck is similar. That is, after the front train leaving the station, the latter train can depart from the arrival-departure lines through the left bottleneck after the safe time interval . or must hold which are equivalent to the following constraints:
Thirdly, for two opposite direction trains, an outbound train would arrive at the station through the left bottleneck while an inbound train would depart from the arrival-departure lines; then they may cause route conflicts which can be avoided by the following constraints:
(ii) Routes Conflicts at the Right Bottleneck. Similar to the conflicts in the left bottleneck, the trains routing conflicts for two inbound trains, two outbound trains, and two opposite direction trains should satisfy the following constraints in the right bottleneck.
When two inbound trains arrive at the station through the right bottleneck, the route conflicts may occur. Therefore, we set following constraints to avoid collision:
For two outbound trains which would depart from the station through the right bottleneck
For two opposite direction trains, considering the order and relationship between trains and routes which are chosen, the constraint can be
(iii) Routes Conflicts on the Arrival-Departure Lines. The train routing conflicts which may occur on the arrival-departure lines has been described in Section 2. Here we propose the algebraic formulas to resolve the conflicts. If the difference of two trains’ departure time is smaller than the headway time , in other words, if train occupies the arrival-departure line first, train can enter from bottleneck after the end of occupied time at arrival-departure lines of train and after a headway time: , where for the outbound trains and for the inbound trains which can be calculated according to (9) and (10) described in Section 2. According to Assumption 1, the additional acceleration and deceleration time of trains that run in different speed are same, and the formula is converted to . Similar, if train enters first, it should be satisfied: .
Thus the constraints should be satisfied as follows:
(2) Potential Conflicts and Avoidance. In small scale stations, there are a small number of arrival-departure lines and platforms that any of them can be occupied by trains. As shown in Figure 5, the outbound train may occupy the arrival-departure line 3 to stop when the operation is busy. In this case, it may cause potential conflicts between opposite direction trains if they choose the arrival-departure lines on the same side. Thus measures should be adopted to avoid the two trains arriving the potential conflicting area simultaneously. The trains’ order should be determined by the decision variable . In Figure 5(a), if train arrives at the station through the left bottleneck, train can depart from the arrival-departure line 3 after the tail of train passes the potential conflicting area after a safety time interval (i.e., the inequality should be satisfied). Similar, if train leaves the station first, after a safety time interval, train can enter the station through the left bottleneck after the tail of train passes the potential conflicting area (i.e., satisfying ).
To sum up, we use the following formula to avoid potential conflicts:
3.2.2. The Limited Capacity Restrictions
This constraint set considers the limited infrastructure capacity. Obviously, a train can only utilize one route at bottleneck and arrival-departure lines. To guarantee the safe operation of trains traveling in the railway station, the constraints are listed as follows:
Constraint (35) ensures that only one arrival-departure line can be occupied by one train. Constraints (36) and (37) guarantee that each train must utilize one and only one route to traverse the left and right bottleneck for each outbound train, respectively. Similarly, only one route in the right and left bottleneck can be chosen by each inbound train through constraints (38) and (39), respectively.
3.2.3. Connectivity of Train Routing Constraints
Trains’ routing connectivity should be ensured. That is, for an outbound train, the route occupied at the left bottleneck should connect to the arrival-departure line which must connect to the route in the right bottleneck. Otherwise, it is conceivable that the train cannot pass the discontinuous route through the station. Therefore, these restrictions can guarantee the feasibility of trains’ routes.
where and are sets of left and right bottleneck routes connected to the arrival-departure line respectively.
3.2.4. Restrictions of Passengers’ Transfer at Station
It represents one of the concerns of railway operators to facilitate passengers transfer to improve the service level. Nowadays, various large scale high-speed rail stations have set up shortcut channels in the railway station to offer convenience for passengers’ transfer in station. This enables passengers to achieve their traveling purpose in a shorter time and cost. There are transfer elevators at each platform in some large and well-equipped facilities stations. And passengers can quickly find the corresponding ticket gate directly through the information from the ticket or the electronic screen.
Two trains that may have transfer relationship are discussed in this paper. A constraint is set to keep the two trains stopped at two near platforms to ensure that passengers or crews transfer in a comfortable time:
At the same time, the target train should stop at the platform for a time interval to ensure that all passengers complete transfer comfortably, which is required to allow passengers alight from one train, move to the corresponding platform track, and board the other train. So we obtain the following constraints:
3.2.5. Variable Feasible Ranges
The following constraints indicate the feasible ranges of the variables:
Constraint (47) indicates that all trains cannot depart from the station before the planned timetable, where integer decision variable indicates the start time measured in minutes to enter the station of train . Constraints (49) and (50) ensure that those decision variables are 0-1 variables.
Herein, the train routing problem for heterogeneous trains can be formulated as the following model, which is essentially a mixed integer nonlinear programming model:
4. Solution Methodology Based on Tabu Search
The number of decision variables of the model will increase greatly as the increasing number of optional routes of trains which is NP-hard problem mentioned in Ahuja et al.  and Carey et al. . In the proposed model, the situation scale of the values of variables is huge as the number of them is great. For instance, when and , there are 26 kinds of values of , where each element of may take 0 or 1. It would cost a significant amount of time to solve the problem under a larger case and may not even get the optimal solution in a reasonable computation time. Apart from this, the mathematic model is nonlinear and there is no algorithm which can solve such problems accurately at present. Therefore in this section, we design a tabu search algorithm stepwise to obtain the near-optimal solution based on the nonlinear characteristic of the model for the TRP with a large scale railway station.
The tabu search algorithm is a deterministic meta-heuristic on account of local search , which makes extensive use of memory for guiding the search. From the incumbent solution, non-tabu moves define a set of solutions which called the neighborhood of the incumbent solution. The best solution is selected as the new incumbent solution at each step and stored in the tabu list to avoid being trapped in local optima and re-visiting the same solutions. When the number of solutions achieves the length of the tabu list, the earliest one which entered the list is released. The algorithm stops until reaching the termination conditions.
In our model, we notice that constraints (17)-(34) and constraints (45) and (46) are nonlinear. It is obvious that if the variable is determined, the model (51) would become an integer linear programming model that determines the departure time of each train. For such mathematical programming model, there are already mature algorithms, such as branch and bound and cutting plane, etc. Taking constraints (17) and (18) as an example, the decision variables and are binary variables, and is known binary parameters according to wether there are conflicts between routes and . There are two cases due to the value of decision variables. The first is that the value of any variables or parameter is 0 among the decision variables , and parameter . Then the left side of the formula is 0, and the constraint is effective apparently. The second case is that the values of them are 1, then the constraints were satisfied if the formulas are satisfied inside the brackets. Hence the inequality constraints are obviously satisfied in the first case. And in the second case, the constraints turn into 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 as the tabu object. Based on that, through generating neighborhood of incumbent solution and searching a good solution at each step, we can gradually approach the near-optimal solution.
To deal with the problems mentioned above in the model presented in this paper, we discuss the setting of the neighborhood of incumbent solution, the selection of initial solution, and the scheme of algorithm in detail below.
4.1. Neighborhood of Incumbent Solution
The establishment of incumbent solution neighborhood is crucial for a better direction of the search, which affects the quality of solution at each step. In our model, we hope that trains would have the shortest travel time, which performances the shortest routes. As mentioned, the travel distance would increase if trains are arranged on the sidings far from the main line. In our problem, we need to determine a good quality initial solution and tabu move to reach the near-optimal solution. Therefore we generate two different neighborhoods of the incumbent solution, respectively: : based on the incumbent routing set, which contains routes in bottleneck and arrival-departure line, the two adjacent sidings and corresponding routes in bottleneck are accommodated into it. That is, the neighborhood contains not only the incumbent arrival-departure lines of trains and corresponding routes in left and right bottleneck, but also the two adjacent sidings and corresponding routes in the bottleneck. : the neighborhood of incumbent routing set for tabu move contains , which involves four adjacent siding and corresponding routes in bottleneck. Besides, remove the situation that trains with close departure times and occupy the same sidings.
In general, is used to determine a good quality initial solution, whose scale is smaller. And we need to search the solution roughly in a short time. In contrast to this, is a wide neighborhood in order to avoid empty neighborhoods as far as possible while avoiding to search in all possible situations. It is thus worthwhile distinguishing the different neighborhood structures.
4.2. Determined of Initial Solution
As describe above, the quality of initial solution is crucial for the performance of tabu search. A good initial feasible solution can greatly improve the speed of searching for the optimal solution. If we randomly assign a set of routes as the incumbent solution, it would take a lot of time to search, and the quality of the near-optimal solution cannot be guaranteed. The objective value is much smaller obviously than the previous strategy if starting from a good incumbent solution. So in this paper, we can get a feasible solution of train routing set through our prior knowledge of trains’ character and construction of the railway station which is just a preliminary program. In view of this, the corresponding routing set solution is selected which satisfies constraints and at the same time reaches the minimum objective function in the solution neighborhoods at next step. Then regard this solution as a new incumbent solution. By cycling this step until the iterative termination condition is satisfied, a batter initial solution of the tabu search algorithm is determined. The procedure is summarized in Table 3.
4.3. The Algorithm Scheme
In this paper, we first obtain the initial routes based on method describe in Table 3. Then we generate the neighborhood of the incumbent solution to reduce the range of the search. And we solve the corresponding MLP in the to choose the best solution. Finally, the best solution is chosen as the new incumbent solution and stored in the tabu list. The steps shown in Table 3 are repeated in sequence recursively. We summarize the procedure of this heuristic algorithm in Table 4.
In this case the tabu list contains the situation of all trains’ routes. The aspiration criterion is set to reinforce the local search to avoid losing an admirable state. That is, if a solution in the tabu list is largely superior to others, then reconsider and search it as a new incumbent solution. The other solutions are forbidden in the tabu list to avoid being trapped in local optima or infinite loops.
5. Numerical Experiments
In this section, two sets of numerical experiments are implemented to show the performance of our proposed model. Specifically, for the models formulated, a small case is implemented to demonstrate the application and performance, in which CPLEX solver is used to obtain exact optimal solution which compares with the near-optimal solution obtained by the proposed heuristic algorithm based on the tabu search, while in large scale case experiment, we apply the proposed heuristic algorithm to the Jinan West high-speed railway station, in which a tabu search algorithm is designed in PYTHON 2.7.13 to obtain the near-optimal solutions of trains routing 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 two main and II lines railway station as shown in Figure 10 which is outbound and inbound direction, respectively. In the left bottleneck, there are a reception line of outbound trains, a departure line of inbound trains, and a locomotive waiting track . 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 departure line of outbound trains, a reception line of inbound trains. and two locomotive waiting tracks and . The distances between connection points in this railway station are shown in Table 5.
In order to test the effectiveness of the proposed models and solution algorithms, we derive a set of instances in this small-scale rail station with different numbers of trains (the type of trains is exhaustive to demonstrate the versatility and correctness of the model). Table 6 shows the origin and destination of each train, as well as the characteristic parameters of them. Trains and are outbound trains, while is inbound train. At the same time, there is a transfer relationship between and . Trains and are shunting operations, wherein travels from the locomotive waiting tracks to and travels from to be an originating train (i.e., train would choose an arrival-departure line to stop and then depart from the station through the departure node ). In order to distinguish the importance of operations, the higher punctuality required for train reception, and departure operations compared to the shunting operations, we set the weights to 0.9 and 0.1, respectively. As for the multiobjective of our model, we attach the most importance to the punctuality, then the total travel time and utilization balanced of the arrival-departure lines are considered evenly. Thus set the parameters and . In addition to this, the preparation time of each train is 1 min, and the additional time that provides trains’ acceleration and deceleration is 2 min and 1 min, respectively.
We assume the headway time min, the minimum time interval of potential area of two trains min, and the minimum transfer time offered to passengers in the arrival-departure lines min. It is easy to neglect in this small example that if two opposite direction trains pass through node or one after the other, there may be potential conflicts.
The algorithm designed in Section 4 is used to solve the model of this small case. First, an incumbent solution of trains’ routing is chosen based on the scheme described in Table 3, whose quality has a great influence on the efficiency of searching the near-optimal solution. The prior knowledge we considered here includes our analysis of the station structure, the nature of each train, and the route by which the train is most likely to occupy, etc. For instance, there is a transfer relationship between and , while the arrival-departure lines 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 and . In terms of the origin and destination of train , there is only one route. And for the remaining two trains, they can be organized to stop at the arrival-departure lines close to the entrance line, as occupies arrival-departure line and would stop at arrival-departure line 4.
At the same time, there are 5 trains stopping and passing through the railway station. In the case that origin and destination are prespecified, the number of each train’s route at left bottleneck, arrival-departure line, and right bottleneck is 6, respectively. Therefore, there are 65 possibilities for the routing arrangement of 5 trains. Of course, it may increase exponentially sharply with the number of trains and arrival-departure lines. Therefore, in order to reduce computation time, we take the trains’ routes arrangement as tabu decision variable based on the scheme described in Table 4. Then we get the near-optimal routing arrangement of each train (as shown in Figure 11) and the start time when trains enter the station (as shown in Table 7) based on the designed algorithm in a short time.