This study proposes a biobjective optimization method for timetable rescheduling during the end-of-service period of a subway network, taking all stakeholders’ interests into consideration. We seek to minimize the total transfer waiting time for all transfer passengers, meanwhile minimizing the deviation to the scheduled timetable. The
Recently, there are several contributions to the last-train timetabling problem of a subway system, which focused only on the last train of each line in the network and expected to generate a more efficiently scheduled timetable for all last trains [
Typically, most subway systems will be closed to the public at midnight or thereabouts for maintenance. Owing to the differences in passenger flow characteristics between different lines in a subway network, the operational time frames vary considerably among different lines. To be specific, the end-of-service period in this study is defined as a period of time from the scheduled departure time of the earliest last train from its originating station (among all last trains of all lines) to the time when all trains finish their jobs.
Because of unavoidable disturbances in the daily operation, a lot of contributions have been made to the timetable rescheduling problem during other periods (e.g., peak hours) [
As a result, in order to deal with the disturbances occurring during the end-of-service period, the first contribution of this study is that a timetable rescheduling model is proposed from a stakeholder-oriented perspective with the consideration of benefits of both passengers and operating agencies. On the one hand, we seek to minimize the total transfer waiting time (TTWT) of all transfer passengers, and a penalty time is adopted if transfer passengers miss their last connecting trains, which benefits improving the level of service (LOS) after disturbances. On the other hand, we try to minimize the deviation between the rescheduled timetable and the scheduled timetable, which also benefits passengers who do not need to transfer.
In addition, in contrast to previous studies that focused only on the last train of each line, there are multiple trains running on each line during the end-of-service period, which means that the train connection relationship in transfer stations becomes much more complicated. But, the timetable rescheduling is carried out through real-time adjustment of an existing schedule, with a consequent need for fast computation. In order to solve the practical problem of a large-scale and complex network efficiently, we utilize the
The third contribution of this study is that a real-world case study of the Beijing subway network is presented to validate the effectiveness of the proposed method. Historical automatic fare collection (AFC) data of the Beijing subway system is available to obtain the number of transfer passengers of each connection, as an important input of our model. The approximate Pareto frontier is obtained by calculating the approximate Pareto optimal solutions, which helps us understand the trade-off of the two objectives.
The rest of this study is organized as follows. Section
The literature review presented in this section focuses on two aspects: timetable rescheduling and last-train timetabling. Some recent publications are reviewed below in detail.
There are a lot of studies focusing on timetable rescheduling, which can be classified by disturbance or disruption, microscopic or macroscopic, and passenger-oriented or train-oriented [
From a train-oriented perspective, D’Ariano et al. [
From a passenger-oriented perspective, since Schöbel [
More recently, there are several publications focusing on timetable rescheduling of a subway system. However, methods were mostly proposed at a single-line level. Xu et al. [
An enormous amount of literatures contribute to the timetabling problem, like Caprara et al. [
Based on all publications reviewed above, we present the focus of this study here. In case there is a disturbance occurring in a subway network during the end-of-service period, this study is working to offer a practical method for timetable rescheduling from the stakeholder-oriented perspective, at a macroscopic level. Disturbances (i.e., delays of 3 to 10 minutes) will not make passengers change their predetermined origin and destination stations and paths. To the best of our knowledge, this study is the first attempt on timetable rescheduling during the end-of-service period in a subway network and real data from the AFC system is used as the model input. We want to figure out the trade-off between different objectives and provide a method of decision support to dispatchers.
Some necessary parameters are defined as follows:
The decision variables are presented as follows:
During the end-of-service period, once a delay occurs in the network, train connections determined in the scheduled timetable might change. To describe connections between trains of different lines, the binary variable
The successful transfer connection.
The failed transfer connection.
As a result,
In normal daytime operation, travelers waiting at a given station may be unable to take the first available train, e.g., if that train has no spare capacity for additional passengers (Schmöcker et al., 2011). However, it seems reasonable to suppose that demand during the end-of-service period is generally low enough to permit the assumption that capacity is always available. As a result, all passengers are assumed to board the first arriving train after they reach the platform in this study. When there is a successful transfer connection,
But for transfer passengers, the last train of the connecting line is the last chance to finish their trips. If the connection to the last connecting train is broken, it will bring a lot of inconvenience to transfer passengers. To avoid this undesired phenomenon as much as possible, a penalty time
In summary, the complete
The rescheduling model is mainly subject to some operational requirements to ensure the safety of the operation and the feasibility of the rescheduled timetable.
This constraint is to input the delay information (e.g., the delayed train, delay time, and position) to the model; see the following formula:
During the process of rescheduling, the actual arrival and departure times of trains at stations cannot be earlier than the scheduled times; see the following formulas:
Under the limitations of the traction and brake performance of trains, the length of each section, safety requirements, and the actual running times of trains in sections must be longer than the minimum running times [
Adjusting the dwell time is an important measure for dispatchers to control subway trains. Similar to the section running time, the actual dwell times of trains at stations must be longer than the minimum dwell times [
As we mentioned above, there is more than one train still running on each line during the end-of-service period. Thus, all trains running on each line should meet the requirements of minimum headway during the end-of-service period; see the following formulas:
We present two objectives to be optimized here. First, we try to minimize the total transfer waiting time (TTWT) for all transfer passengers, which helps to improve the LOS of the system after disturbances; see formula (
The two objectives and constraints (
Owing to the huge complexity of the timetable rescheduling problem, especially when solving a real-world case of a large-scale and complex network, many heuristic algorithms have been proposed to speed solving this problem. Examples include greedy algorithm [
The
By the
Among all constraints, constraints (
On the premise of the objective to minimize the total transfer waiting time of all transfer passengers, formula (
Finally, the single-objective model obtained in Section
To validate the method proposed in this study, the Beijing subway network is used as a real-world case study. By the end of 2016, the Beijing subway network consisted of 18 double-track lines (i.e., 36 one-way lines), 53 transfer stations, and 225 ordinary stations with an average daily ridership of 9.998 million passengers. A sketch map of the Beijing subway network without Airport Express is shown in Figure
Sketch map of the Beijing subway network without Airport Express.
Owing to the difference in passenger flow characteristics, different lines have different operational time frames. Among all last trains of all lines in Beijing subway network, the earliest one is the last train of Fangshan Line from SZ to GGZ, starting at 22:00. According to the definition in this study, the end-of-service period of the Beijing subway case is from 22:00 to the time when all trains finish their jobs, a period of time about 2.5 hours. In addition, the starting time is (22:00) reset to 0 and then all times are changed according to the time lag and the minimum time unit is second.
Table
A sample of the AFC data with key information.
Smart card ID | Station_In | Time_In | Station_Out | Time_Out |
---|---|---|---|---|
20714652 | Beijingxizhan | 22:00:13 | Xizhimen | 22:49:24 |
79292837 | Lishuiqiao | 22:04:35 | Beijingzhan | 22:53:42 |
50124710 | Dawanglu | 22:25:29 | Chaoyangmen | 22:47:07 |
78241891 | Jinsong | 23:02:28 | Wukesong | 23:55:02 |
22069171 | Zhichunlu | 23:30:51 | Longze | 23:57:49 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
In order to prove that the proposed method is effective, various delay scenarios are generated randomly in terms of delayed train, delay position, and delay time. Numerical experiments based on these delay scenarios are carried out. Detailed information about these delay scenarios is listed in Table
Delay scenarios in detail.
ID | Line | Scenario | Delay time |
---|---|---|---|
1 | Line 1 from SHD to PGY | The | 9 min |
2 | Line 2 outer loop | The | 10 min |
3 | Line 4 from TGY to AB | The | 8 min |
4 | Line 5 from TB to SJZ | The | 6 min |
5 | Line 6 from LC to HW | The | 8 min |
6 | Line 7 from BX to JHC | The | 7 min |
7 | Line 8 from NG to ZXZ | The | 9 min |
8 | Line 9 from GT to GGZ | The | 10 min |
9 | Line 10 outer loop | The | 5 min |
10 | Line 15 from QX to BB | The | 7 min |
We test these delay scenarios with
Solution results of different scenarios.
Scenario | TTWT/s | Decrement | Number of FTP | Decrement | ||
---|---|---|---|---|---|---|
| | | | |||
1 | 11418266 | 8879746 | 22.23% | 1304 | 734 | 43.71% |
2 | 11623197 | 9064789 | 22.01% | 1372 | 796 | 41.98% |
3 | 11312422 | 8956396 | 20.83% | 1307 | 745 | 43.00% |
4 | 11720791 | 9337384 | 20.33% | 1408 | 868 | 38.35% |
5 | 11632670 | 9345290 | 19.66% | 1402 | 849 | 39.44% |
6 | 11400663 | 9220033 | 19.13% | 1317 | 811 | 38.42% |
7 | 11387058 | 9532154 | 16.29% | 1340 | 888 | 33.73% |
8 | 11474331 | 8422855 | 26.59% | 1331 | 604 | 54.62% |
9 | 11374509 | 9196693 | 19.15% | 1325 | 810 | 38.87% |
10 | 11398591 | 9237736 | 18.96% | 1346 | 816 | 39.38% |
The TTWT includes the transfer waiting time of all successful transfer passengers and the penalty time of failed transfer passengers. For most scenarios, there is a considerable decrease of about 20% in the TTWT as well as a big decline in the number of FTP, about 40% compared to those of the rescheduled timetable with
During the daily operation, different disturbances may lead to different delay times. In this experiment, we focus on Scenario 1 and set delay time changing from 5 to 10 minutes to test the effect of the method on different delay times. Similarly, all corresponding problems are solved within 2 seconds by Cplex 12.6.2 on the same computer. Table
Solution results of Scenario 1 with different delay times.
TTWT/s | Delay time/s | |||||
---|---|---|---|---|---|---|
300 | 360 | 420 | 480 | 540 | 600 | |
| 11417881 | 11388673 | 11389273 | 11379193 | 11418266 | 11421626 |
| 9373458 | 9227160 | 9132110 | 8998431 | 8879746 | 8743882 |
Decrement | 17.91% | 18.98% | 19.82% | 20.92% | 22.23% | 23.44% |
With the delay time increasing from 5 to 10 minutes, our method can reduce the TTWT by 17.91% to 23.44% compared with that of the rescheduled timetable with
Our proposed biobjective model for timetable rescheduling during the end-of-service period aims to minimize the TTWT for all transfer passengers, meanwhile minimizing the deviation to the scheduled timetable. However, in the actual process of decision-making, it is difficult for dispatchers to obtain the optimal solution for multiple criteria. As a result, we are interested in the trade-off between objectives and adopt the
Scenario 1 is still an example to obtain approximate Pareto optimal solutions by changing the value of
Numerical results with changing
In addition, for each approximate Pareto optimal solution, we calculate the TTWT, the number of FTP, and the total travel time (TT) of all trains involved in the end-of-service period by
A comparison between total TT, TTWT, and the number of FTP.
Rescheduled | Total TT/s | Increment | TTWT/s | Decrement | FTP | Decrement |
---|---|---|---|---|---|---|
| 661345 | - | 11418266 | - | 1304 | - |
| 661345 | 0 | 9800056 | 14.17% | 932 | 28.53% |
| 661345 | 0 | 9452463 | 17.22% | 857 | 34.28% |
| 661381 | 36 | 9223486 | 19.22% | 810 | 37.88% |
| 661382 | 37 | 9034198 | 20.88% | 751 | 42.41% |
| 661351 | 6 | 8879746 | 22.23% | 734 | 43.71% |
| 661477 | 132 | 8726061 | 23.58% | 686 | 47.39% |
| 661447 | 102 | 8589964 | 24.77% | 669 | 48.70% |
| 661351 | 6 | 8396420 | 26.47% | 594 | 54.45% |
| 661477 | 132 | 8241846 | 27.82% | 546 | 58.13% |
| 661447 | 102 | 8102096 | 29.04% | 529 | 59.43% |
During the end-of-service period, a passenger who does not need to transfer can catch his or her train definitely even if the train is late, but a transfer passenger may miss the last connecting train because of the late feeder train. As a result, based on all our numerical results in Section
The timetable rescheduling problem can be optimized by many objectives because of its inherently multicriterion nature. It is difficult to tell which solution is the optimal solution. But in terms of some specific criteria, we can figure out that a solution is better or worse than others. As a result, one of the major contributions of this study is that a biobjective optimization method is proposed from the stakeholder-oriented perspective to tackle the timetable rescheduling problem during the end-of-service period of a subway network, which allows us to figure out the trade-off between the LOS (in terms of the TTWT and the number of FTP) and the operation (in terms of the deviation to the scheduled timetable). We utilize the
In addition, given the actual characteristics of the end-of-service period as well as the fact that a high tolerability to the deviation will not lead to a big extension in the total travel time, we think that dispatchers should put transfer passengers’ interests in the first place when rescheduling during the end-of-service period, which will benefit the overall LOS after disturbances.
For future extension, the timetable rescheduling problem during the end-of-service period can take passengers rerouting into consideration, especially when the delay time is long. However, train rerouting seems an unreasonable option because in most subway systems, trains belonging to a line cannot run on other lines. Anyway, contributions should be devoted to improving the LOS of subway systems after disturbances.
All related data are included within this article.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (51478036) and the Fundamental Research Funds for the Central Universities (2018YJS074).