The major objective of this work is to present a train rescheduling model with train capacity constraint from a passengeroriented standpoint for a subway line. The model expects to minimize the average generalized delay time (AGDT) of passengers. The generalized delay time is taken into consideration with two aspects: the delay time of alighting passengers and the penalty time of stranded passengers. Based on the abundant automatic fare collection (AFC) system records, the passenger arrival rate and the passenger alighting ratio are introduced to depict the shortterm characteristics of passenger flow at each station, which can greatly reduce the computation complexity. In addition, an efficient genetic algorithm with adaptive mutation rate and elite strategy is used to solve the largescale problem. Finally, Beijing Subway Line 13 is taken as a case study to validate the method. The results show that the proposed model does help neutralize the effect of train delay, with a 9.47% drop in the AGDT in comparison with the trainoriented model.
The train rescheduling problem is one of the most crucial problems in rail transit operation and management. During the course of daily operation, trains are inevitably affected by unexpected accidents or technical problems, which leads to the deviations from the original timetable as well as delays. If dispatchers can not handle it immediately, the delay may propagate to other trains, which will do great harm to the normal operation and disturb passengers’ trips seriously. Many researchers devoted themselves to studying the train rescheduling problem, which has become a research focus currently.
In order to achieve the realtime and intelligent train rescheduling, a lot of studies have been carried out in proposing regulation rules, presenting rescheduling models, and designing solution algorithms. Among so many studies, most proposed models were built from a trainoriented point of view with minimizing the delay time of trains, the number of delayed trains or deviations from the original timetable, and so on [
In addition, many researchers focus on heuristic algorithms to accelerate the speed of computation. Meng et al. [
To sum up, most researchers conceived the train rescheduling problem from a trainoriented viewpoint, and few works paid attention to passengers’ interests. As for this problem in an urban rail transit system, considering the actual characteristics of urban rail transit lines: being shorter in length, high passenger flow volume, and high service frequency, a train rescheduling model for an urban rail transit line should be presented from a passengeroriented perspective rather than a trainoriented point of view. Currently, during the actual operation process, train rescheduling mainly depends on dispatchers’ dispatching orders, which are based on their experience and craftsmanship without intelligent decision support. But, with passengers’ rising requirements for the level of service (LOS) of a rail transit system, train rescheduling should be more precise and scientific, which is what this work is expected to do. The main contributions of this work are summarized as follows:
A train rescheduling model is proposed from a passengeroriented viewpoint. In this model, the train capacity and stranded passengers are taken into consideration, which make the model more practicable. In addition, the prediction of stranded passengers will remind the corresponding stations to take timely measures of passenger flow control.
The passenger arrival rate and the passenger alighting ratio of each station are introduced to capture the different shortterm passenger flow characteristics of each station [
An efficient genetic algorithm with adaptive mutation rate and elite strategy is designed to obtain a goodenough solution of a practical problem within acceptable duration, which is a key factor for realtime application.
A realworld case study of Beijing Subway Line 13 is carried out to test the method proposed in this work. The results show that the performance of the passengeroriented model is much better than the trainoriented model’s.
A passengeroriented model for train rescheduling is presented in this part. For presentation simplicity, the necessary symbols and notations are listed as follows:
In an urban rail transit system, the passenger flow characteristics of a station can be captured by abundant historical AFC records. Each AFC record includes the accurate time of a passenger entering and leaving a station. As for transfer passengers, the accurate time of entering and leaving their transfer stations can be obtained by an assignment model [
In order to depict the passenger flow characteristics of a station, the arrival rate
In this work, passengers fall into five categories: passenger on board (
Passenger on board (
Alighting passenger (
Boarding passenger (
Arrival passenger (
Stranded passenger (
The model is mainly subject to some operational requirements to ensure the safety of the operation and the feasibility of the timetable optimized by the proposed model.
Under the limitation of traction and brake performance of trains, the length of each section, safety requirements, and so on, the actual running time of trains in each section must be longer than the minimum running time [
Similar to section running time, the actual dwell time of trains at each section must be longer than the minimum dwell time [
Regarding all trains running on the subway line, they should meet the requirements of the minimum arrival and departure headway of the line, as shown in (
Obviously, the rescheduled timetable cannot be earlier than the planned timetable and all variables in this practical problem must be integers, as shown in formulas (
For most previous studies about train rescheduling problem, their optimal objectives tend to be designed from a trainoriented point of view. For instance, a trainoriented objective can be calculated by (
However, in this work, the train rescheduling problem is considered from a passengeroriented perspective with two aspects: the delay time of alighting passengers and the penalty time of stranded passengers. The delay time of each alighting passenger equals the delay time of the train arriving at his or her destination station. The total delay time of alighting passengers can be calculated by
As for stranded passengers, they have to spend extra time, at least a headway, waiting for the next train. The penalty factor
Consequently, the passengeroriented objective of minimizing the AGDT of passengers is presented by (
The train rescheduling problem is considered as one of the most intractable problems in the operation and management of rail transit system [
A chromosome represents a solution in the genetic algorithm. Each train’s actual arrival time
Chromosome representation.
Each number in a rectangle in Figure
The fitness of an individual in the population represents that the individual is good or bad. Meanwhile, it determines the possibility that the individual can be selected to generate the new individual. The passengeroriented model is a model with a minimizing objective, so the objective function with some relatively minor modifications is the fitness function; see (
The other main operations in the genetic algorithm are summarized as follows. The method of roulette is adopted in selecting operation and the singlepoint crossover is used in crossover operation. As for mutation operation, the value of each gene on the chromosome can change within the determined lower and upper bound according to the adaptive mutation rate, which can be determined by (
The detailed algorithmic steps are depicted as follows.
This step is as follows:
Set the initial parameters: population size
Input the initial data:
Input the serial number of the delayed train, the delay position, and the delay time.
Generate the initial population
Calculate the fitness
This step is as follows:
Calculate the selecting probability
Make crossover operation in
Make mutation operation in
Calculate the fitness
Select individuals in
Calculate the objective value of (
Elite strategy: replace the worst individual with the best individual in
This step is as follows:
Update
If
The performance of the passengeroriented model for train rescheduling and the genetic algorithm is tested by a realworld case of Beijing Subway Line 13. Beijing Subway Line 13 is a semiloop line with 16 stations in total, as shown in Figure
The length and the minimum train running time of each section.
Section  Length/m 

Section  Length/m 


Xizhimen–Dazhongsi  2839  215  Huoying–Lishuiqiao  4785  275 
Dazhongsi–Zhichunlu  1206  95  Lishuiqiao–Beiyuan  2272  135 
Zhichunlu–Wudaokou  1829  125  Beiyuan–Wangjingxi  6720  385 
Wudaokou–Shangdi  4866  285  Wangjingxi–Shaoyaoju  2152  135 
Shangdi–Xierqi  2538  185  Shaoyaoju–Guangximen  1110  85 
Xierqi–Longze  3623  265  Guangximen–Liufang  1135  85 
Longze–Huilongguan  1423  95  Liufang–Dongzhimen  1769  125 
Huilongguan–Huoying  2110  135 
Beijing Subway Line 13.
According to the planned timetable of Line 13 downdirection, the planned train diagram for trains whose departure times are between 8:30 am and 9:30 am is obtained in Figure
The planned train diagram of Beijing Subway Line 13 (8:30 am to 9:30 am).
Based on abundant historical AFC records of Beijing Subway Line 13, the passenger arrival rate
Station 


Station 



Xizhimen  166  0  Huoying  38  0.25 
Dazhongsi  93  0.01  Lishuiqiao  32  0.23 
Zhichunlu  95  0.03  Beiyuan  12  0.08 
Wudaokou  132  0.06  Wangjingxi  9  0.27 
Shangdi  38  0.13  Shaoyaoju  11  0.13 
Xierqi  130  0.19  Guangximen  6  0.13 
Longze  50  0.24  Liufang  3  0.3 
Huilongguan  50  0.29  Dongzhimen  0  1 
Using the passengeroriented model and the genetic algorithm proposed in this work, the practical problem is solved within 30 seconds by programing in MATLAB R2014b on an Intel Pentium dualcore CPU 3.1 GHz and 8 GB RAM desktop computer. Meanwhile, the problem is also solved by the trainoriented model mentioned above, using Lingo. The necessary parameters are given in Table
Necessary parameters.
Parameter  Value 


40 

800 

0.8 

0.05 

0.01 

20 

1356 

160 

500 
Solution results.
Model  AGDT/s 

The trainoriented model  65.07 
The passengeroriented model  58.91 
Improvement  −9.47% 
The convergence curve of the genetic algorithm.
The highlight of this work is that the train capacity is taken into consideration. With this constraint, the number of passengers on board (
The number of passengers on board of the 6th train.
In addition, with the constraint of train capacity, there is a possibility that some passengers can not board the arriving train. In this experiment, it is found that there are stranded passengers (
The number of stranded passengers at Wudaokou station and Xierqi station.
The changes in the number of stranded passengers at Xierqi station.
The train rescheduling problem is always a hot problem in rail transit operation and management. With passengers’ rising requirements for the LOS of an urban rail transit system, a passengeroriented model is much better than a trainoriented model. In this work, a passengeroriented model with train capacity constraint is presented to minimize the AGDT of passengers, which consists of the delay time of alighting passengers and the penalty time of stranded passengers. In order to meet the realtime requirement, an efficient genetic algorithm is proposed to solve the practical and complex problem. Finally, the case study of Beijing Subway Line 13 is carried out to verify the method proposed in this work. The results show the following:
Compared to the trainoriented model, the passengeroriented model has obviously optimal effects on the generalized delay time of passengers, with a 9.47% decrease in the AGDT.
In comparison with the passengeroriented model without the constraint of train capacity, the model with train capacity constraint is more corresponding to the reality and the number of passengers on board cannot be greater than the train capacity.
With the constraint of train capacity, the number of stranded passengers can be counted by the proposed model so that the stations with increasing number of stranded passengers can be detected. Then, the corresponding stations are able to take preventive measures in time.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is financially supported by the National Natural Science Foundation of China (no. 51478036).