Abstract
Purpose
This study aims to improve the passenger accessibility of passenger demands in the end-of-operation period.
Design/methodology/approach
A mixed integer nonlinear programming model for last train timetable optimization of the metro was proposed considering the constraints such as the maximum headway, the minimum headway and the latest end-of-operation time. The objective of the model is to maximize the number of reachable passengers in the end-of-operation period. A solution method based on a preset train service is proposed, which significantly reduces the variables of deciding train services in the original model and reformulates it into a mixed integer linear programming model.
Findings
The results of the case study of Wuhan Metro show that the solution method can obtain high-quality solutions in a shorter time; and the shorter the time interval of passenger flow data, the more obvious the advantage of solution speed; after optimization, the number of passengers reaching the destination among the passengers who need to take the last train during the end-of-operation period can be increased by 10%.
Originality/value
Existing research results only consider the passengers who take the last train. Compared with previous research, considering the overall passenger demand during the end-of-operation period can make more passengers arrive at their destination. Appropriately delaying the end-of-operation time can increase the proportion of passengers who can reach the destination in the metro network, but due to the decrease in passenger demand, postponing the end-of-operation time has a bottleneck in increasing the proportion of passengers who can reach the destination.
Keywords
Citation
Wen, F., Bai, Y., Zhang, X., Chen, Y. and Li, N. (2023), "Last train timetable optimization for metro network to maximize the passenger accessibility over the end-of-service period", Railway Sciences, Vol. 2 No. 2, pp. 273-288. https://doi.org/10.1108/RS-03-2023-0012
Publisher
:Emerald Publishing Limited
Copyright © 2023, Fang Wen, Yun Bai, Xin Zhang, Yao Chen and Ninghai Li
License
Published in Railway Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Optimization of metro timetable refers to determining the arrival and departure times of trains on different lines in the network at transfer stations, so as to improve the transfer efficiency of passengers. In off-peak and peak hours, the optimization of the timetable focuses on reducing the waiting time for passenger transfer to improve the service level of the network (Yu, Han, Dong, Li, & Yao, 2015). Whereas at the end-of-operation period, a train transfer failure between lines will cause a decrease in passenger accessibility of the network service, thus passengers would not be able to reach their destination by taking the metro (Guo, Jia, & Qin, 2015). Therefore, it is necessary to study the optimization of the last train timetable from the perspective of service passenger accessibility, so that boosting passenger accessibility as far as possible of the metro.
Existing studies on the optimization for the timetable of the last train of the metro can be mainly divided into two categories: the first category focuses on improving the number of successful transfer connections at the transfer station and the second category optimizes the passenger accessibility of passengers in the network. For the first category, references (Xu, Zhang, & Jiang, 2008; Kang, Wu, Sun, Zhu, & Gao, 2015; Kang & Meng, 2017; Chen, Bai, Feng, & Li, 2017, Chen, Mao, Bai, Ho, & Li, 2019a, Chen, Mao, Bai, Ho, & Li, 2019b; Guo et al., 2020) optimized the timetable of the last train, to improve the number of passengers transferring successfully among the last train and reduce the waiting time for the transfer; Ning, Zhao, Xu, Qiao, and Yao (2016) formulated an optimization model of train timetable in the end-of-operation period to minimize the waiting time for transfer and the number of failed transfer passengers, considering all trains in the period. However, in the metro network, optimizing train transfers merely does not guarantee the largest number of passengers reaching their destinations.
The second category of study optimizes the last train timetable from the perspective of OD (origin-destination) reachability. Chen et al. (2019a, b) further considered the transfer of passengers between the last train and the non-last train on different metro lines and optimized the timetable of the last train with the objective of maximizing the number of passengers who can reach their destination among the passengers taking the last train at the origin station (passengers departing by the last train). Considering the demand of passengers departing by the last train, Zhou, Wang, Yang, and Yan (2019) proposed an MILP (mixed integer linear program) model for the last train timetable optimization, which can be solved by Cplex. In the end-of-operation period, not only the passengers departing by the last train may not be able to reach their destination but passengers taking the non-last train at the origin station of the travel may also be unreachable to their destination. Yao, Liu, Liu, and Yang (2018) optimized OD reachability of the network considering passengers taking the last train in the entire travel. Yang, Di, Dessouky, Gao, and Shi (2020) proposed an MILP model based on a space-time network with the consideration of the passenger demand departing by the non-last train, and designed a Lagrange relaxation algorithm to solve the model. This study considers all passenger demands at the end-of-operation period, which may lead to a large scale of problems and difficulty in achieving efficient solutions. Wen et al. (2019) proposed a mathematical model aiming at the maximum total number of reachable OD pairs at each time during the end-of-operation period, but the model did not consider the difference in passenger demand with different departure times. In the existing studies, only part of the passenger demand (such as the passengers departing by the last train) in the end-of-operation period is taken into consideration to optimize the last train timetable, while all OD passenger demands in the period are not considered in detail (passengers departing by the non-last train are overlooked), so it is difficult to ensure the maximum reachability of the network service.
Tackling the issues raised above, this paper, oriented toward all passenger demands in the end-of-operation period, proposes an optimization model of the last train timetable to maximize the number of reachable passengers of the metro network in the end-of-operation period; in view of the characteristics of various departure times and great difficulty in model solution during the period, a solution method based on preset train services is designed to solve the optimized last train timetables.
2. Illustrating existing problems
2.1 Definition of parameters
The metro network consists of lines and stations, and the up and down directions are considered as two lines. Definition of parameters:
t0 is defined as the start time of the end-of-operation period, and the end-of-operation period is divided into several time points with an interval of
For the passenger group
In the end-of-operation period, the train services on the line
2.2 Analysis of passenger accessibility
Passenger accessibility varies along with departure time and is directly related to the connectivity of effective paths between ODs. At a certain moment, if there is at least one connected path between ODs, OD is considered as reachable (Xu, Zhang, Guo, & Du, 2014; Chen, Mao, Bai, Li, & Tang, 2020). Taking the passenger group
Taking the path
3. Optimization model of last train timetable
3.1 Model assumptions
It is assumed that the passenger heterogeneity is low and the fluctuation of the time of transfer by foot is small at the end-of-operation period, which indicates the passengers in the same transfer direction at the transfer station have the same time of transfer by foot.
The effective path between OD takes the K short physical path before loop-free.
Assuming that the train capacity is sufficient and there is no passenger stranded, passengers always choose to take the train arriving first.
3.2 Modeling
3.2.1 Decision variables
In this paper, to improve the OD reachability of the network during the end-of-operation period, the arrival and departure times (
3.2.2 Constraints
Constraints on the adjustment range of section operation and dwell times of the last train
Let tΔ1 and tΔ2 be the maximum and minimum values of section operation time adjustment, tΔ3 and tΔ4 the maximum and minimum values of dwell time adjustment, and the time adjustment should meet the actual operation requirements, so the arrival and departure times of the last train at each station should be within the adjustment range, i.e.
Constraint of maximum headway
To ensure the level of passenger service, the headway between the last train and the penultimate train at the first station should not be greater than the maximum headway
Constraint of minimum headway
To ensure train operation safety in the section, the headway between the last train and the penultimate train should not be less than the minimum headway
To ensure the safety of station operation, at the same station, the headway between the arrival time of the last train and the departure time of the penultimate train should not be less than the minimum headway
Constraint on the latest end-of-operation time
To avoid interference with night maintenance, the end-of-operation time of the terminal station
3.2.3 Objective function
Defined
3.2.4 Evaluation of passenger accessibility
Evaluation of passenger accessibility
Defined
Evaluation of path connectivity
Defined
Evaluate whether the passenger can take the train in the ride section
Defined
Evaluation of the train service in the ride section
The value of the variable
Evaluation of the arrival time of the passenger group at the origin station of the ride section
In Equation (12), if
Evaluation of the arrival time of the passenger group at the destination station of the ride section
In Equation (13), the arrival time
4. Solution method
The optimization model of the last train timetable is a mixed integer nonlinear programming model. Its number of decision variables and constraints are mainly affected by the number of passenger groups and the size of the train set
To reduce the number of variables, a solution method based on a preset train service is proposed. The preset train service refers to the earliest train service that the passenger group can board in the travel path according to the departure time of the passenger group, the timetable information of the non-last train and the adjustment range of the timetable of the last train. If the passenger group does not need to take the last train in the travel path, the passenger group can still reach the destination after optimization, and this part of the passenger group is eliminated by the algorithm; if the optional train of the passenger group is the last train, the OD reachability of the passenger group can be determined by estimating whether the passenger group can get on the last train. Thus, the model is linearized to an MILP model and the timetable optimization model can be quickly solved by the commercial solver.
4.1 Preset method of train service in ride section
The preset train service of the ride section is related to the timetable of the line corresponding to the ride section and the time when the passenger group arrives at the origin station of the ride section. For the passenger group
Step 1
Given the ride section
Step 2
Based on the preset train service
Step 3
Determine whether
4.2 Reformulated model based on preset train service
With the preset train service of each ride section, it is assumed that the passenger group only selects the preset train in the travel path, and a reformulated model of the coordination optimization model of the last train timetable (hereinafter referred to as the original model) is proposed.
Since the reformulated model only considers whether the passenger group
For the constraint formula Equation (12), the constraint of the surplus time of the passenger group
Similar to the original model, for the first ride section in the path
The rest constraints remain unchanged, and the reformulated model is obtained as follows:
Eqs. (1)–(6), Eqs. (8)–(9), Eqs. (15)–(18)
Compared with the original model, the decision variables and constraint scale of train service evaluation in the reformulated model are significantly reduced; and the reformulated model is a linear model, which can be quickly solved by Cplex software.
As the model only optimizes the timetable of the last train on each line, passenger groups who can arrive at the destination station by taking a non-last train during the trip will still reach the destination station after optimization. For the passenger groups that need to take the last train during their trip, their passenger accessibility is affected by the last train timetable. In order to improve the solution efficiency, the preset method is first used to estimate the preset train service of each path of all passenger groups in the ride section during the end-of-operation period. According to the preset train service, the passenger groups that must take the last train during the trip are eliminated from passenger demand B during the end-of-operation period, and their set is represented by
5. Case study
Taking Wuhan Metro as the background, the up-direction and down-direction lines in the metro network are regarded as two lines. There are 48 key stations, including transfer stations, origin and terminal stations of the line, and stations with the largest passenger flow between two adjacent transfer stations. The simplified network diagram is shown in Figure 3. The green and blue arrows indicate the up direction and down direction of the line, respectively; dots indicate key stations in the network. Two key stations constitute the key OD (2,256 pairs in total). Three valid paths are considered between each OD. Parameter setting: the end-of-operation period is 21:30 to 24:00; the dwell time of each station in the initial timetable is 30–60 s. The running time in sections is within 1–4 minutes; the minimum headway and the minimum departure–arrival headway at stations are both 2 minutes. The maximum headway between the last train and the penultimate train at the origin station of the line is 10 minutes. The end-of-operation time of the line can be delayed by 10 minutes.
5.1 Comparison of solving efficiency between reformulated model and original model
The end-of-operation period is divided by intervals
Table 1 shows that in the reformulated model, the difference between the preset and the accurate number of people who can reach the destination is only 44. The reason is that the preset method assumes that passengers only choose the preset train in their travel path, but passengers may not choose the preset train. The small difference indicates that the preset method has high accuracy in estimating the train taken by passengers in the ride section and that the evaluation error of the model is small. With the decrease of interval, the problem scale and the computation challenge increase, the relative difference between the two models increases and the computation time becomes longer. The smaller the interval of passenger demand is, the more obvious the advantage of the fast-solving speed of the reformulated model is. Therefore, the reformulated model can get the approximate optimal solution in a relatively short time compared with the original model.
5.2 Optimization results
With the set
It can be seen from Figures 4 and 5 that compared with the original, the optimized reachable OD percentage and the number of destination-reachable passengers in the end-of-operation period are increased to a certain extent; the reachable OD percentage increases significantly during the period 22:31–22:56 and increases by 15.6% at 22:32.
It can be seen from Table 2 that within the passenger flow
Although the optimized timetable slightly increases the travel time of some passengers who can reach their destination before optimization, it effectively improves the percentage of reachable OD and the number of destination-reachable passengers of the whole network during the end-of-operation period and improves the overall service level of the network.
5.3 Verification of the necessity of considering the overall passenger demand in the end-of-operation period
In existing studies, only the passenger groups departing by the last train are considered to optimize the last train timetable. In order to verify the necessity of considering the overall passenger demand during the end-of-operation period, a comparative case is set up, where set
The time interval of passenger flow data is 1 minute. The optimization results are shown in Table 3. It can be seen that in the compared case and the case in this study, the percentages of reachable passengers in passenger flow
5.4 Sensitivity analysis of the latest end-of-operation time
The latest end-of-operation time of each line restricts the adjustment range of the arrival and departure times of the last train at each station, thus having a great impact on the service passenger accessibility in the network. In this paper, the latest end-of-operation time
As can be seen from Figure 6, without the delay of the end-of-operation time of each line, the percentage of destination-reachable passengers increases by 6.5%; with the delay of the latest end-of-operation time, the percentage of destination-reachable passengers in the passenger flow
6. Conclusions
From the perspective of improving the passenger accessibility during the end-of-operation period, a last train timetable optimization model is proposed. Considering the model is difficult to be solved due to the large number of passenger groups and train services during the end-of-operation period, this study proposes a solution method based on preset trains, which reformulates the original model into a mixed integer linear programming model with fewer decision variables and can achieve a fast solution.
Compared with the original model, the reformulated model and the solution method can get a high-quality solution in a relatively short time with a small error. The smaller the interval of passenger flow data, the more significant the advantage of the fast solution. After optimization, the percentage of passengers who can reach their destination successfully by last train during the end-of-operation period increases by 10%, which verifies the effectiveness of this model. Compared with the previous studies that only consider the passengers who take the last train, the number of passengers who take the last train and can reach the destination in the end-of-operation period increased by 3.0%, which indicates that it is necessary to consider the overall passenger demand in the end-of-operation period for the study on the optimization of the last train schedule.
In this paper, the passenger accessibility of the network is optimized by adjusting the last train timetable. In future research, it can be considered to coordinate and optimize all train timetables in the end-of-operation period, expand the model solution space and achieve a better optimization effect.
Figures
Solution results of the original model and reformulated model under passenger flow data with different intervals
| Interval/min | Model | The preset number of people who can reach the destination/pax | The accurate number of people who can reach the destination/pax | Upper limit/pax | Gap/% | Computation time/min |
|---|---|---|---|---|---|---|
| 10 | Original | 23,745 | 23,789 | 23,792 | 0.01 | 11 |
| Reformulated | 23,789 | 0.01 | 3 | |||
| 5 | Original | 27,816 | 27,458 | 27,923 | 1.69 | 60 |
| Reformulated | 27 860 | 0.23 | 11 | |||
| 1 | Original | 29,108 | 28,579 | 29,391 | 2.84 | 780 |
| Reformulated | 29,152 | 0.82 | 65 |
Source(s): Authors’ own work
Percentage of destination-reachable passengers and average travel time before and after optimization
| Timetable | Percentage of destination-reachable passengers in passenger flow, | Average travel time of destination-reachable passengers/min | Average travel time of all passengers/min |
|---|---|---|---|
| Original | 32.3 | 39.8 | 79.2 |
| Optimized | 42.3 | 45.0 | 76.7 |
Source(s): Authors’ own work
Comparison of the number of destination-reachable passengers and increase after optimization
| Timetable | Passenger flow departing by the last train | Passenger flow needing the last train during travel | |||
|---|---|---|---|---|---|
| Number of destination-reachable passengers/pax | Increase in destination-reachable passengers/% | Number of destination-reachable passengers/pax | Increase in destination-reachable passengers/% | ||
| Original | 9,959 | 22,249 | |||
| Optimized | Comparative case | 12,443 | 24.9 | 28,477 | 28.0 |
| Case of this study | 12,360 | 24.1 | 29,152 | 31.0 | |
Source(s): Authors’ own work
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Acknowledgements
This research was supported by Talents Funds for Basic Scientific Research Business Expenses of Central Colleges and Universities (Grant No. 2021RC228) and Special Funds for Basic Scientific Research Business Expenses of Central Colleges and Universities (Grant No. 2021YJS103).