The stopschedules for passenger trains are important to the operation planning of highspeed trains, and they decide the quality of passenger service and the transportation efficiency. This paper analyzes the specific manifestation of passenger travel convenience and proposes the concepts of interstation accessibility and degree of accessibility. In consideration of both the economic benefits of railway corporations and the travel convenience of passengers, a multitarget optimization model is established. The model aims at minimizing stop cost and maximizing passenger travel convenience. Several constraints are applied to the model establishment, including the number of stops made by individual trains, the frequency of train service received by each station, the operation section, and the 01 variable. A hybrid genetic algorithm is designed to solve the model. Both the model and the algorithm are validated through case study.
Train schedules program is a very important task railway transportation organization operation plan [
Line planning is one of the most crucial and complex problems in transportation organization and has been explored by many researchers. However, it is essentially a multitarget combination optimization problem, which awaits an effective and universal solution algorithm. At present, the method is mainly focused on mathematical programming [
Most studies treat stopschedule planning as a subproblem of line planning. Based on their objectives, those studies generally employ the following three types of models.
A few studies investigated stopschedule planning problem. Such studies generally assume that line plans and various stop modes are already known. Goossens [
China’s line planning and stopschedule planning are significantly different from those in West Europe and Japan. In China, train line planning is based on passenger traffic flow. Based on passenger flow properties, characteristics and patterns, the operation priority, origin and destination, quantity, route, marshaling, stopschedule, passenger seat utilization, and passenger carriage utilization are reasonably arranged, with organization schemes covering from passenger flow to train flow. A highspeed railway stopschedule plan should, after deciding train routes, train types, marshaling, and operation pairs, reasonably arrange the orders of stopping for each train based on passenger flow requirements and train collaboration. The line planning and stopschedule planning of China are different from those of West Europe and Japan for two reasons. On the one hand, China’s transportation organization mode is significantly different from that of West Europe and Japan; the former is organized transportation, and the latter is planned transportation. On the other hand, the average length of OD sections in China is much longer than those in West Europe and Japan. The much more complicated railway networks of China also disenable the train operation diagrams to be cyclic, adding difficulties to the compilation of line plans and stopschedule plans.
In this paper, we propose two concepts, interstation accessibility and degree of accessibility. On the basis of considering the economic benefits of railway corporations and the travel convenience of passengers, a multitarget optimization model for stopschedule planning is established, aiming at minimizing train stop costs and maximizing passenger travel convenience. A corresponding hybrid genetic algorithm is also designed.
The contribution of this paper is twofold. First, in order to better describe the number of options for passengers to travel and express the travel convenience for passengers, the concepts of accessibility and degree of accessibility are proposed. Compared with directly using travel time and wait time as indicators of passenger benefit, these two concepts are more reasonable. Meanwhile, this paper presents a hybrid genetic algorithm to solve the multitarget model. Since this paper focuses on the highspeed railway stopschedule planning problem in China, whose railway networks are much larger and more complicated than those of West Europe and Japan, the problem cannot be effectively solved by precise algorithms such as column generation algorithm, lagrangian relaxation method, or branch and bound method. The proposed heuristic algorithm, although cannot produce the optimal solution, provides satisfying solutions, which is of practical importance.
The remainder of this paper is organized as follows. Section
Passenger travel convenience is closely associated with departure time of the chosen train and degree of freedom when passengers choose trains. Through questionnaire survey, He et al. [
Passengers’ travel convenience in different time periods for departure (unit: hour).
In a reasonable range of departure time of highspeed trains, the flexibility of passengers’ travel time and the number of travel schemes increase with the number of trains in the line plan that satisfies passengers’ travel requirements. More travel schemes means greater flexibility, that is, higher travel convenience, as shown in Figure
Variation of passenger travel convenience with the number of available travel schemes.
The number of available travel schemes can be direct accessible scheme value or transfer accessible scheme value. Direct accessible scheme value (
Generally, when the number of available travel schemes is large, the passengers’ choices on departure and arrival times are more dispersed (given that train operation diagrams are usually plotted in a balanced manner). Passengers tend to travel at times that are more suitable to their preferences and travel purposes. Therefore, the number of available travel schemes, compared with travel time, has greater influence on passenger’s travel convenience. To a certain extent, the influence of departure time on transfer convenience can be ignored, which means passenger’s travel convenience is solely decided by the number of available travel schemes.
In order to better describe the value of available travel schemes for passengers to choose and express the degree of passengers’ travel convenience, we propose in this section the concepts of interstation accessibility and degree of accessibility.
Interstation accessibility refers to whether the passenger flow within an OD pair can travel by direct train or through one transfer. In order to improve passengers’ travel convenience and easiness, reduce their fatigue, and avoid losing passenger flow due to frequent transfers, we allow in this study at most one transfer during a passenger’s travel, which means every passenger can get to their destination by at most one transfer. Accessibility is a state describing whether an OD pair is accessible. Two states, direct accessible and transfer accessible, are used based on the need for transfer. As shown in Figure
A line plan and its interstation accessibility.
The degree of accessibility refers to the values of schemes to travel within any OD pair through direct trains or one transfer. Like accessibility, the degree of accessibility can be degree of direct accessibility or degree of transfer accessibility. The degree of direct accessibility means the total value of the schemes to take direct trains to reach destinations for passengers within an OD pair, whereas the degree of transfer accessibility means the total value of the schemes to reach destination by transferring once for passengers within an OD pair. For the latter, the trains taken by passengers should not be any of the direct trains within the OD pair. If a direct train is available to reach the destination, then taking this train and transferring to another halfway to reach the destination are not counted as a transfer accessibility scheme. Assuming that in Figure
The degree of accessibility not only reflects the accessibility of OD pairs in the highspeed railway network, but also describes the amount of travel options for passengers. Higher degree of accessibility indicates better accessibility of the railway network and more available travel schemes. Therefore, the degree of accessibility, to a certain extent, describes the passenger’s travel convenience.
Figure
A highspeed railway line and the trains operating on it.
A variable
The train set is
For an OD pair (origin and destination are
The total degree of accessibility of this OD pair (
For some OD pairs, due to some intermediate stations’ inability to fulfill technical requirements of passenger transfers or other technical reasons, passengers cannot get to their destinations through transfers even though the route passes through stations that connecting trains can stop. This portion of passenger flow can only travel through direct trains. To reduce unnecessary stops at such intermediate stations so as to avoid increasing transportation cost and prolonging travel time, the degrees of transfer accessibility between such OD pairs should be regulated. Transfer accessibility schemes that do not fulfill the transfer requirements should be eliminated, thus optimizing the degree of transfer accessibility.
For the situation shown in Figure
While establishing a highspeed train stopschedule optimization model, the operation benefit of railway transportation corporations should be considered, and the principle is to maximize the operation benefit. On the other hand, the passenger’s travel convenience should also be considered, which means that the degree of OD accessibility in the railway network should be expressed. Therefore, this section establishes a multitarget optimization model for stopschedule planning with the objectives of minimizing stop cost and maximizing passenger travel convenience.
In order to simplify the complexity of the model and to make the model more accurate, we make some basic assumptions before establishing the model:
Highspeed railways in China are usually doubletrack; the up direction trains and the down direction trains are independent of each other. We generally considered the same number of trains operated in the same section from opposite direction. We only study a fixed direction of the train stop schedule optimization problem in this paper.
We assume that the line plan is known condition when we consider stop scheme optimization problem. Namely, passenger’s flow of every OD pair, train operation section, train types, the number of trains, and train marshalling are all known conditions.
We assume that the station capacity and line capacity have no restrictive effect on the train operation. That is to say, station capacity, facilities and equipment capacity, and other factors have no effect on the design of the train stop scheme.
We assume that the size and nature of passenger’s flow of the railway line between the departure station and arrival station do not vary from the train stop scheme changed. That is to say, the change of the stop schedule will not cause the fluctuations of the passenger flow. Besides, we do not consider the passenger flow loss due to the transfer.
We assume that the frequency of the train service shown in this paper is considered on a daily basis.
A train and station set
In addition,
Different levels of trains cost differently for stops. Assuming that, on the highspeed railway line, a train costs
The degree of OD accessibility expresses the accessibility within an OD pair, yet cannot accurately reflect the passenger flow’s demand for accessibility. Thus, we introduce a passenger flow degree coefficient matrix and use the product of passenger flow degree coefficient and degree of accessibility to indicate the convenience provided to passengers.
The degree of OD accessibility is
For the
Based on the passenger flow matrix, a passenger flow degree coefficient matrix
For train
On a highspeed railway line, the line plan must ensure a certain amount of stops to ensure OD accessibility, satisfy passengers’ demand for train stops, and ease passengers’ traveling and transferring. Meanwhile, excessive stops should be reduced in order to avoid overly long travel time and high stop cost. Therefore, certain constraints are put on the maximum number and proportion of stops. Trains of different types travel at different speeds and thus have different preferences for stops, leading to different ranges for the number of stops.
Assuming that the highest and lowest numbers of stops made by trains on a highspeed railway line are
The number of train stops is associated with the number of stations within the train’s operation section. For different railway line sections, the required numbers of stops are different.
Different levels of node stations have different volumes of passenger flow. Higherlevel stations often have higher social and economic status; thus their geographic locations play more important roles in the road network. This means higher volumes of passenger flow and that higher frequency of service is required. In order to meet the demands of different stations, minimum train stop times that meet passenger flow requirements are set for different levels of train stations.
This constraint means that for a train station, the number of times that a train stops at it (including departure and arrival) should not be lower than the demand
The set of operation sections on a railway line is denoted by
The specific operation section of each train is determined and constrained. Train
The constraints on operation section are expressed as follows:
The expressions above indicate that a train never stops at a station outside its operation section but always stops at the departure and arrival stations of the section.
The optimization of highspeed train stopschedule planning is an NPhard problem. As the number of train stations and the number of trains increase, the solution combinations increase explosively, and the difficulty of obtaining the global optimal solution increases in geometric progression. When solving largescale practical problems, conventional optimization algorithms often fail to obtain the ideal optimal solution. Considering the fact that this study uses a nonlinear hybrid integer programming model with a 01 variable as a constraint, a hybrid genetic algorithm that integrates simulated annealing strategy is proposed.
The procedure of using the hybrid genetic algorithm to solve highspeed train stopschedule optimization problems is as follows.
Initialize algorithm parameters, including population size popsize, crossover possibility regulation parameters
Generate initial population pop through chromosome coding, with the current generation number
Calculate the adaptability of each chromosome in current population, select the optimum individuals for next generation, and perform roulette wheel selection to the remaining individuals.
Calculate the crossover possibility and perform uniform crossover operations to the population.
Calculate the mutation possibility and perform mutation operations to the population.
Randomly choose a chromosome from the progeny population and generate its neighborhood solutions. Choose one from the two solutions and update the current generation number to
If
Figure
Flow diagram of the proposed algorithm.
In the hybrid genetic algorithm, adaptive genetic algorithm is used to control the direction of the global optimization, simulated annealing strategy is used to improve the ability of neighborhood search, and a combination of adaptive genetic algorithm and simulated annealing strategy can improve the algorithm performance.
At each iteration, a current individual
Based on the preparation to various data of the WuhanGuangzhou highspeed railway train stopschedule schemes in [
The minimum service frequencies required by train stations.














 

Station level  I  IV  IV  II  IV  I  III  IV  II  IV  II  III  IV  III  I 
Minimum frequency of train service (times)  40  8  8  30  8  40  20  8  30  8  30  20  8  20  40 
The number of stops by each type of trains.
Section number 
Operation 
Number of stops 
Number of trains  Number of stops made by the trains  

TypeA highspeed trains  TypeB highspeed trains  TypeA highspeed trains  TypeB highspeed trains  


15  50  2 




6  0  2 




10  18  8 


Illustration of train operation sections and operation lines.
Assume that typeA and typeB highspeed trains operate at the speeds of 300 km/h and 200 km/h, respectively, and all trains are 8car marshaled. In addition, it is assumed that each stop made by a typeA train costs
A hybrid genetic algorithm that combines simulated annealing strategy was adopted, and MATLAB7.11.0 software program was used to solve the example case. Considering the solving time and result precision, the parameters of the hybrid genetic algorithm were set to the following: population size
Frequency of train service received by each station.














 

Gtrain service times  50  7  6  28  6  43  23  5  21  5  28  19  7  19  68 
Dtrain service times  4  3  3  2  3  12  6  5  9  7  7  8  6  7  10 
Total service times  54  10  9  30  9  55  29  10  30  12  35  27  13  26  78 
Degree of direct accessibility within each OD pair.
OD 

















—  10  9  30  9  29  18  4  17  6  20  16  6  16  52 

—  —  2  7  3  8  1  2  3  2  5  4  2  3  10 

—  —  —  4  2  7  2  1  2  3  2  6  2  4  9 

—  —  —  —  5  13  11  4  8  3  7  10  4  8  30 

—  —  —  —  —  5  1  1  5  1  2  4  1  3  9 

—  —  —  —  —  —  7  9  20  10  24  22  9  22  55 

—  —  —  —  —  —  —  3  12  8  15  9  6  12  29 

—  —  —  —  —  —  —  —  4  6  7  5  4  4  10 

—  —  —  —  —  —  —  —  —  6  13  13  10  12  30 

—  —  —  —  —  —  —  —  —  —  6  6  4  7  12 

—  —  —  —  —  —  —  —  —  —  —  9  6  10  35 

—  —  —  —  —  —  —  —  —  —  —  —  7  15  27 

—  —  —  —  —  —  —  —  —  —  —  —  —  6  13 

—  —  —  —  —  —  —  —  —  —  —  —  —  —  26 

—  —  —  —  —  —  —  —  —  —  —  —  —  —  — 
Iterative solution process via hybrid genetic algorithm.
Based on the passenger flow between OD pairs that have transfer stations between them, the degrees of transfer accessibility on different transfer nodes were calculated for OD pairs with relatively large passenger flow. In this study, the degree of transfer accessibility is determined by the value of transfer schemes. However, not all trains that meet the requirements of transfer accessible provide convenient travel schemes to passengers. This computation method is to a certain extent defective in practice. It is more suitable to use the minimum numbers of connecting trains fulfilling transfer accessible requirements as the chosen degrees of transfer accessibility.
In Table
Degrees of accessibility for OD pairs with large volumes of passenger flow.
OD  Volume of passenger 
Degree of direct 
Degrees of transfer accessibility with different transfer nodes  Degree of accessibility  





Sum  

3128  29 

—  —  —  0/0  29 

362  18 


—  —  242/11  29 

281  17 


—  —  286/13  30 

239  16 




307/21  37 

3239  52 




52/2  54 

776  30  — 



26/1  31 

645  10  —  — 

—  0/0  10 

2701  24  —  — 

—  78/6  30 

272  22  —  — 


8/1  23 

587  22  —  — 


9/1  23 

5015  55  —  — 


0/0  55 

316  15  —  — 

—  42/6  21 

1190  29  —  — 


0/0  29 

399  10  —  — 


0/0  10 

340  12  —  —  — 

21/3  15 

1638  30  —  —  — 

0/0  30 

457  12  —  —  — 

0/0  12 
By analyzing the degrees of direct and transfer accessibilities for OD pairs with higher volumes of passenger flow, it can be seen that, with the stopschedule plan produced by the model, the accessibilities between all OD pairs are good. For such OD pairs, the traveling of passengers is primarily made by direct trains. Transfers are generally made at station
According to the operation sections, number of trains, and service frequency for each station (Table
This paper proposes a stopschedule optimization model that aims at minimizing the train stop cost for railway transportation corporations and maximizing passenger travel convenience. The two objectives constrain each other, but also interconnect; they are integrated as a whole. The model considers both the economic benefits of railway transportation corporations and the travel convenience of passengers. Such a considerate strategy helps the transportation corporations to attract customers. When considering the travel convenience of passengers between OD pairs, instead of simply maximizing the degree of OD accessibility, the degree of accessibility is optimized based on actual volumes of passenger flow, making the model more suitable for practical use. In addition, configuring the operation section as constraints caters the multioperation section characteristic of China’s highspeed railway system, making the model able to adapt to passenger flow requirements of different lines or different stages of the same line and widening the model’s application.
The concept of the degree of transfer accessibility refers to the number of schemes for a passenger to travel between an OD pair through one train transfer, and its value is the product of the numbers of trains before and after the transfer. This definition, however, is not optimal. In the future, a better and quantified calculation method may be adopted in order to obtain better stopschedule plans.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (Projects nos. 61273242 and 61403317), the Fundamental Research Funds for the Central Universities (Project no. 2682015CX043), Science and Technology Plan of China Railway Corporation (Projects nos. 2014X004D and 2015X008B), Soft Science Foundation of Sichuan Province STA of China (Project no. 2015ZR0141), and the Foundation of Sichuan Provincial Education Department (Project no. 14ZB0258).