A timespace network based optimization method is designed for highspeed rail train timetabling problem to improve the service level of the highspeed rail. The general timespace path cost is presented which considers both the train travel time and the highspeed rail operation requirements: (1) service frequency requirement; (2) stopping plan adjustment; and (3) priority of train types. Train timetabling problem based on timespace path aims to minimize the total general timespace path cost of all trains. An improved branchandprice algorithm is applied to solve the large scale integer programming problem. When dealing with the algorithm, a rapid branching and node selection for branchandprice tree and a heuristic train timespace path generation for column generation are adopted to speed up the algorithm computation time. The computational results of a set of experiments on China’s highspeed rail system are presented with the discussions about the model validation, the effectiveness of the general timespace path cost, and the improved branchandprice algorithm.
The train timetabling problem (TTP) plays a critical role in the organization and operation of the railway system. TTP for the highspeed rail (HSR), which is different from conventional lines on operational requirements, brings a tremendous challenge for the operation company. For example, the frequency of highspeed train from Wuhan to Guangzhou in China is one hundred pairs for weekdays, compared to only ten pairs before the WuhanGuangzhou HSR was operated [
TTP schedules the movement of trains on tracks in the railway system, and our research focuses on the offline timetable optimization problem. It should be noted that the relevant optimization problems of TTP like the train scheduling problem, the train dispatching problem, and the track allocation problem will also be reviewed in the following. TTP has attracted many researches in these decades, which are usually classified into periodic train timetabling problem and nonperiodic timetabling problem according to the operation mode.
Periodic timetable is applied in some European countries like Germany and The Netherlands, due to the fact that it is easy to be used, understood, and operated. After Serafini and Ukovich introduced the concept of the Periodic Event Scheduling Problem [
A wide range of studies has been devoted to the nonperiodic train timetable problem. The heuristic methods were addressed in a number of researches because of a short time for acquiring a feasible solution. However, solution quality of the heuristic approach cannot be guaranteed. Cai et al. proposed a local optimality criterion to resolve the meet and pass event in a singletrack railway line. The conflict resolving measures assigned trains according to the traffic prediction [
A sequence optimization method inspired by the job shop problem is a commonly used method to TTP. The trains are considered as the tasks and the tracks are considered as the machines. This type of method can solve the train timetabling problem with the metaheuristics algorithms [
Integer programming based formulation can guarantee the solution quality with less speed than the heuristic approaches. As the offline timetabling problem is less sensitive to the time requirement, integer programming based formulations are a more popular approach for nonperiodic train timetabling. These approaches usually adopted the timespace network or the likely network to describe the trains’ movement on the tracks, and designed an exact algorithm to obtain a more optimal solution. Brännlund et al. suggested an integer programming approach to determine a profit maximizing schedule. The profit was given by the departure time minus a per minute cost for unnecessary waiting along the track. The problem was solved with a Lagrangian relaxation approach in which track capacity constraints are relaxed. They assigned a price to each track and separated the problem into dynamic programs [
By comparing the above methods to the train timetabling problem, our study applies the timespace network to formulate the mathematical model. Note that minimizing the total train travel time and maximizing the total train profits are the commonly used objective functions in the previous train timetabling researches. However, it cannot handle the HSR requirements like the service frequency in different hours, stopping plan adjustment, and train priority. Furthermore, although branchandprice is a promising method for the train timetabling problem [
Our study makes the following specific contributions. First, a new timespace network generation method is presented to describe trains’ movement in stations in more detail and facilitate to adjust the train stopping plan and formulate the constraints in the model. Second, the general timespace path cost is proposed to consider the new situation of HSR. Minimizing the total train general timespace path cost aims to make the system more efficient by abiding the requirement of service frequency in different hours, stopping plan adjustments and train priority. Third, an improved branchandprice (IBAP) algorithm integrating with column generation, heuristic timespace path generation, and rapid branch strategy is designed, which is suitable for handling large scale problems and speeds up the solving processes.
This paper is organized as follows. After presenting the necessary notation and describing the problems in detail of TTP and timespace network in Section
Set of time interval, indexed by
Set of stations, indexed by
Set of sections, indexed by
Set of section outflow from station
Set of section inflow to station
Set of timespace nodes, indexed by n,
Set of timespace nodes at station
Set of timespace nodes at station
Set of timespace nodes of which
Set of timespace arcs, indexed by a,
Set of timespace arcs of train r, indexed by
Set of passing timespace arcs occurring at station s, indexed by
Set of dwell timespace arcs occurring at station
Set of additional waiting timespace arcs occurring at station s, indexed by a, and
Set of timespace arcs occurring at section e, indexed by a, and
Set of dummy timespace arcs, indexed by a,
Set of maintenance windows, indexed by
Set of timespace paths, indexed by
Set of timespace paths of train
Set of timespace paths through station s, indexed by
Set of timespace paths through station
Set of timespace paths through station
Set of timespace paths through station
Beginning time of maintenance widow mw,
Ending time of maintenance widow
Source timespace node of timespace arc
Sink timespace node of timespace arc
Departure time of timespace arc
Departure station of timespace arc
Arrive time of timespace arc
Arrive station of timespace arc
Running time of timespace arc
General cost of timespace arc
General cost of timespace path
Arrive time of section
Leave time of section
Arrive time of station
Departure time of station
The train which timespace path
Direction of train
Direction of timespace path
Source dummy node,
Sink dummy node,
01 parameters with
a 01 variable with
A train timetable is a document setting out information on service time to assist passengers to plan a trip and listing the time when a train is scheduled to arrive at and depart from specified locations. The train timetabling problem for HSR is an offline optimization problem in our study. It does not need to schedule trains on each track. Therefore, physical railway network
An illustration for the macroscopic railway network and the timespace network.
Timespace network is a familiar method for the slot allocation problem considering the spatial and temporal requirements. Timespace network shows the position of individual train in time and in space. It is very useful for understanding and describing the operation of trains. In timespace network models, time horizon
It does not permit trains to stop on sections; that is, there are only timespace running arcs on sections which are adopted to describe the movement of trains. Train activities are more complicated in stations than on sections. Stations are split into two types of timespace node to specify the trains activities in more detail and are depicted in the Figure
Furthermore, if the stopping plan can be adjusted, especially in the peak hour, the total system may be more efficient. Adding and deleting stopping stations are the two adjustment measures. If train
A timespace network for a train.
As the speed, stop plan, origin station, and destination station of each train are different, it should generate the timespace network
Many railway train timetabling methods addressed that the optimization objective in their approaches was to minimize the total train travel time. Nonetheless, it may be insufficient to ensure that each train is scheduled in a reasonable timespace path. For example, in Figures
Twotrain scheduling plan with same travel time and the illustration for timespace areas.
Plan 1
Plan 2
Moreover, there are different operational conditions and requirements between HSR and the conventional lines. Note that the train frequency in different hour should be set as diverse values to satisfy the passenger travel fluctuations in a day. Despite the fact that the stopping plan can be adjusted, that situation should occur as less as possible. In addition, trains are classified into different types with different priorities. The higher level trains have the higher priory than the lower ones. The high level trains are suggested not to be overtaken by the lower level trains. Given these reasons, a general timespace path cost is presented to acquire a high service level of timetable. Note that all the above requirements are like the soft constraints in the mathematical models. Hence, these requirements are transformed as the penalty functions. If the conditions on the variables are not satisfied, these are penalized in the objective function.
The running area in timespace network is introduced firstly. Supposing that a train departs at time
Then, the general timespace path cost of the above requirements is introduced below.
The general cost of timespace arc
Train timetabling problem based on timespace path (TTPTP) combined with optimization objectives and constraints is designed. TTPTP is a pathbased network flow formulation.
Except part of the requirements like the service frequency requirement, stopping plan adjustment, and priority of train types, the constrains or business requirements for HSR train timetabling also include the safety requirements, maintenance time windows, station capacity requirement, minimum dwell time at station, and section running time. The timespace running arc guarantees section running time requirement. The safety requirements are realized by means of minimum headway time; refer to the research [
The illustration for the three types of headway constraints in timespace network.
Note that both minimum headway constraints on arrival and departure time at stations ensure that one train can be overtaken by other trains at a station with the safety requirements.
The illustration for the maintenance window constraints and minimum dwell time at stations constraints in timespace network.
Obviously, there are huge timespace arcs and nodes when timespace network is applied to TTP. Thus, the number of timespace path, which consists of timespace arc and node, is far beyond what we can enumerate. In the next section, a diligently designed algorithm is presented to solve the largescale problem.
The train timetabling optimization is a largescale linear programming problem. At present, the commonly used applicable algorithm for that problem is branchandprice (BAP) algorithm [
IBAP is a hybrid algorithm, which includes many producers. The flow chart of IBAP shown in Figure
The parameters for IBAP.




Co( 

UB( 

LB( 


IBAP algorithm flow chart.
Branchandprice search tree is similar to BranchandBound tree. It is the framework of the algorithm and is used to get the integer solution by adding the new cuts iteratively. Each node
A rapid branching strategy just adds a specific type of cut into the nodes. A selected node is generally divided into left and right nodes with the cuts of
There are a lot of
Furthermore, an optimal gap
TTPTP is a pathbased network flow problem. As known that when a pathbased network flow problem is relaxed as a linear programming, it can be solved by column generation effective referring to the research [
The integer variable is relaxed to continuous variables firstly; then a
Using the duality of linear problem, the dual restricted master problem (DRMP) can be acquired as follows:
According to Constraint (
Constraint (
A heuristic timespace path generation is designed to avoid the degeneration in column generation algorithm, like the phenomenon of trapping in local optima in the neighborhood search algorithms. The heuristic timespace path generation can enlarge the neighborhood of the current solution. Then, it can improve the current integer solution with the more generated timespace paths.
The timetable can be described as a matrix
Given a trains’ matrix, a feasible timetable can be acquired by changing the sequence with a simple neighborhood search algorithm. Then, the decoding of the trains’ matrix is used by a headway time path for each section and station.
The headway time path with artificial arc is defined to obtain a situation that there are no conflicts on each section. As shown in Figure
(1) Assign a feasible timespace path for each train by trains’ matrix with timespace arcs in the sections and stations of the train.
(2)
(4) Get a headway path on each section and reassign timespace arc for each train
(5) Copy all stations into the adjustment station set
(5)
(6) Sorte the adjustment stations by the number of passing trains. Select the maximum number station.
If more than one station’s number of passing trains is the maximum number, the downbound station is selected firstly.
Moreover, if all the stations have the same direction, the station is selected by downbound direction.
(7) For the selected station, find a headway path. Reassign the timespace arc inside the station.
(8) Adjust the connect section timespace arc and remove the station from adjustment station set.
(11)
An example for heuristic timespace path generation.
Note that the number of parallel tracks at the station is not considered in the heuristic producer.
Three features are investigated through a series of computation experiments: (1) the timespace network validity for TTP, (2) the effectiveness of the general timespace path cost to meet the HSR operation requirements, and (3) the effectiveness of the IBAP algorithm in realsize instances. The train types are described above. The algorithm is implemented in Visual Studio 2008 platform with C# language and ILOG Cplex12.4 software for solving DRMP problem in column generation. The experiments are conducted on a PC with a 2.10 Ghz CPU, 2 GB RAM.
A simple railway network shown in Figure
The timetable for the instances.
In Figure
The effectiveness of general timespace path cost is testified by a 17 h period train timetable, which schedules 143 trains on BeijingShanghai HSR shown in Figure
Firstly, the actual departure train number and the design departure train number for each hour in Figure
The actual departure train number and the designed departure train number for each hour.
Then, the total train travel time is 49977 min solved by IBAP, while the actual timetable of BeijingShanghai HSR of 2012 is 51133 min. Obviously, service frequency requirement and stopping plan adjustment make the railway system more efficient.
Finally, according to the timetable in Figure
The examples for the overtaking.
There are 9 instances with different time horizon, train numbers, and networks in three HSR networks in China, which are ZhengzhouXian HSR, BeijingShanghai HSR, and BeijingGuangzhou HSR. The column “Network” is expressed as “(section number, station number)”. The periods include 5, 10, and 17 hours, and number of trains is 40, 80, and 140, respectively. The “CPU times” and “solution quality” list the performance of each algorithm, and the values are used to compare the effectiveness of the algorithms.
Three algorithms including the rolling horizon algorithm (RHA) in [
Solving Statistics for 9 instances.
#  Network  Period  Train numbers  CPU time (s)  Solution  

Cplex  RHA  GBAP  IBAP  Cplex  RHA  GBAP  IBAP  
1  (10, 9)  5  40  331  175  302  241  62300  5923  5802  5451 
2  (10, 9)  10  80  657  235  398  354  122501  11323  10689  9878 
3  (10, 9)  17  140  —  422  525  467  —  24432  23490  23412 
4  (23, 22)  5  40  —  446  464  347  —  14981  13821  13323 
5  (23, 22)  10  80  —  876  678  543  —  25234  24723  24591 
6  (23, 22)  17  140  —  1113  796  686  —  51234  50321  49943 
7  (40, 39)  5  40  —  1042  987  826  —  25189  24890  23414 
8  (40, 39)  10  80  —  2283  1531  953  —  46903  46667  45594 
9  (40, 39)  17  140  —  3225  1825  1221  —  86443  85564  82432 
It is obvious that IBAP outperforms three algorithm described before. Due to that TTPTP is decomposed into a more small size problem literately, both the IBAP and GBAP are able to solve instance 1–9. IBAP, which adds heuristic timespace path generation, has a more powerful solving ability compared with GBAP. Table
In this paper, a train timetabling optimization method based on timespace network for HSR is proposed. The timespace network is constructed by the way that the time is discretized into small time interval to describe trains’ movement in stations and sections. A general timespace path cost is applied to obtain the HSR operational requirement, and can acquire a higher quality timetable. With a set of computational experiments, the method is verified to be effective to acquire qualified timetables, which abided the HSR requirements. The instances with different sizes show that the designed algorithm outperforms the candidate algorithms.
Our ongoing research focuses on two major aspects. First, the movement of trains inside the station is more complex in the real practice and should be implemented into a more precise simulation way. Then, the robustness of the timetable should also be considered in the future.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was sponsored by National Basic Research Program of China (no. 2012CB725403), National Natural Science Foundation of China (no. 61374202), and National Key Technology R&D Program of the Ministry of Science and Technology (2011BAG01B0103 and 2011BAG01B021). The authors thank the anonymous referees for their valuable suggestions.