A Practical Method for Timetable Rescheduling in Subway Networks During the End-Of-Service Period

Hindawi Journal of Advanced Transportation Volume 2018, Article ID 5914276, 9 pages https://doi.org/10.1155/2018/5914276

Research Article A Practical Method for Timetable Rescheduling in Subway Networks during the End-of-Service Period

Wenkai Xu , Peng Zhao , and Liqiao Ning

School of Trafc and Transportation, Beijing Jiaotong University, Beijing 100044, China

Correspondence should be addressed to Peng Zhao; bjtu [email protected]

Received 11 January 2018; Revised 30 April 2018; Accepted 8 May 2018; Published 27 June 2018

Academic Editor: Zhi-Chun Li

Copyright © 2018 Wenkai Xu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tis 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. Te �-constraint method and linearization techniques are utilized to obtain the approximate Pareto optimal solutions within limited seconds, allowing for fguring out the trade-of between the two objectives. Te method is validated by numerical experiments for diferent delay scenarios based on a real-world case: the Beijing subway network.

1. Introduction before the end-of-service period, there is always a next connecting train which the transfer passenger can board. Recently, there are several contributions to the last-train But during the end-of-service period, the missed connecting timetabling problem of a subway system, which focused only train may be the last train of the connecting line; if so, the on the last train of each line in the network and expected transfer passenger cannot fnish his/ her trip via the subway to generate a more efciently scheduled timetable for all last system, which will bring a lot of inconvenience to the transfer trains [1–4]. However, it is very common that the scheduled passenger. timetable cannot be implemented because of unavoidable As a result, in order to deal with the disturbances occur- delays that occur at an operational level [5]. As a result, ring during the end-of-service period, the frst contribution this study is devoted to the timetable rescheduling problem of this study is that a timetable rescheduling model is during a specifc period: the end-of-service period. proposed from a stakeholder-oriented perspective with the Typically, most subway systems will be closed to the consideration of benefts of both passengers and operating public at midnight or thereabouts for maintenance. Owing to agencies. On the one hand, we seek to minimize the total the diferences in passenger fow characteristics between dif- transfer waiting time (TTWT) of all transfer passengers, and ferent lines in a subway network, the operational time frames a penalty time is adopted if transfer passengers miss their vary considerably among diferent lines. To be specifc, the last connecting trains, which benefts improving the level of end-of-service period in this study is defned as a period of service (LOS) afer disturbances. On the other hand, we try time from the scheduled departure time of the earliest last to minimize the deviation between the rescheduled timetable train from its originating station (among all last trains of all and the scheduled timetable, which also benefts passengers lines) to the time when all trains fnish their jobs. who do not need to transfer. Because of unavoidable disturbances in the daily opera- In addition, in contrast to previous studies that focused tion, a lot of contributions have been made to the timetable only on the last train of each line, there are multiple trains rescheduling problem during other periods (e.g., peak hours) running on each line during the end-of-service period, which [6]. But there is a considerable diference between the end-of- means that the train connection relationship in transfer service period and other periods. For example, if a transfer stations becomes much more complicated. But, the timetable passenger misses a connecting train due to a disturbance rescheduling is carried out through real-time adjustment of 2 Journal of Advanced Transportation an existing schedule, with a consequent need for fast compu- Schachtebeck and Schobel¨ [13], and Dollevoet et al. [14, 15] tation. In order to solve the practical problem of a large-scale by considering limited capacity of tracks, priority decisions, and complex network efciently, we utilize the �-constraint rerouting passengers, and limited capacity of stations. Binder method and some linearization techniques to convert the et al. [16] proposed an ILP model with three objectives: proposed model into an integer linear programming (ILP) the passenger satisfaction, the operational costs, and the model that can be solved by Cplex speedily, which constitutes deviation from the scheduled timetable. Strategies include the second contribution of this study. canceling, delaying, rerouting the trains, and scheduling Te third contribution of this study is that a real-world emergency trains. case study of the Beijing subway network is presented to More recently, there are several publications focusing validate the efectiveness of the proposed method. Historical on timetable rescheduling of a subway system. However, automaticfarecollection(AFC)dataoftheBeijingsubway methods were mostly proposed at a single-line level. Xu system is available to obtain the number of transfer passen- et al. [17] modeled the problem as a discrete event model gers of each connection, as an important input of our model. considering service balance performance of both directions Te approximate Pareto frontier is obtained by calculating on a double-track subway line. Te model is expected to the approximate Pareto optimal solutions, which helps us minimize the total delay time of all trains rather than all understand the trade-of of the two objectives. passengers. Gao et al. [18] integrated the skip-stop pattern Te rest of this study is organized as follows. Section 2 into the rescheduling model for a double-track subway line. reviews some recent studies about timetable rescheduling An iterative algorithm was proposed to solve the model and last-train timetabling. Te stakeholder-oriented model based on the model decomposition. Xu et al. [19] proposed for timetable rescheduling during the end-of-service period a passenger-oriented model for rescheduling on a subway is proposed in Section 3. Section 4 presents the model line considering the limited train capacity. Te delay time conversion and the solution strategy. Some experiments of alighting passengers and the penalty time of stranded based on a real-world case, the Beijing subway network, are passengers constitute the generalized delay time, which is carried out in Section 5. Section 6 draws some conclusions expected to be minimized. andfuturedirectionsinbrief. 2.2. Last-Train Timetabling. An enormous amount of litera- 2. Literature Review tures contribute to the timetabling problem, like Caprara et al.[20],ZhouandZhong[21],LeeandChen[22],Cacchiani Teliteraturereviewpresentedinthissectionfocusesontwo and Toth [5], and Yang et al. [23]. However, the last-train aspects: timetable rescheduling and last-train timetabling. timetabling problem of a subway system is an emerging issue Some recent publications are reviewed below in detail. in recent years. However, all related publications focused only on the last train of each line. Zhou et al. [1] developed an 2.1. Timetable Rescheduling. Terearealotofstudiesfocus- optimization model to reduce passengers’ transfer waiting ing on timetable rescheduling, which can be classifed by time for last trains and inaccessible passenger volume of disturbance or disruption, microscopic or macroscopic, and all origin-destination pairs. Coordinated departure times for passenger-oriented or train-oriented [6]. Various approaches all last trains are determined by the model. Kang et al. [2] have been developed in these previous studies. established a programing model with adjustable running From a train-oriented perspective, D’Ariano et al. [7] times and dwell times to obtain coordinated arrival and regarded the timetable rescheduling problem as a huge departure times of last trains at transfer stations. A genetic job shop scheduling problem with no-store constraints and algorithm was designed to solve the model. Kang et al. modeled the problem with an alternative graph formulation [3] modeled the problem as a mean-variance model to to minimize the deviation from the scheduled timetable. Tey improve the efciency of transfer passengers. Te model proposed a branch and bound algorithm with implication was solved by a genetic simulated annealing algorithm. rules enabling the speed up of the computation. Tornquist¨ Kang and Zhu [4] studied the same problem in Kang et and Persson [8] presented a MIP model to minimize a cost al. [2] and designed a new heuristic algorithm outper- function based on train delays considering reordering and forming both genetic algorithms and simulated annealing rerouting of trains. But for certain scenarios, it is difcult to algorithms. fnd good solutions within seconds. Terefore, Krasemann Based on all publications reviewed above, we present [9] designed a greedy algorithm to quickly fnd a good the focus of this study here. In case there is a disturbance solution by performing a depth-frst search. Dundar¨ and occurring in a subway network during the end-of-service S, ahin [10] developed a genetic algorithm to minimize the period, this study is working to ofer a practical method total weighted delay. Te algorithm could reduce total delay for timetable rescheduling from the stakeholder-oriented time by around half in comparison to an artifcial neural perspective, at a macroscopic level. Disturbances (i.e., delays network method developed to mimic the decision behavior of 3 to 10 minutes) will not make passengers change their of dispatchers. predetermined origin and destination stations and paths. To From a passenger-oriented perspective, since Schobel¨ [11] the best of our knowledge, this study is the frst attempt on proposed the frst MIP model for the delay management timetable rescheduling during the end-of-service period in a problem to minimize the total delay time of all passen- subway network and real data from the AFC system is used as gers, the model has been further extended in Schobel¨ [12], the model input. We want to fgure out the trade-of between Journal of Advanced Transportation 3

arr Feeder Line  Tl,s, Train Arrival Time Walking Waiting

x, =1 Tdep  Connecting Line l,s, Train Departure Time

Figure 1: Te successful transfer connection. diferent objectives and provide a method of decision support connections determined in the scheduled timetable might to dispatchers. change. To describe connections between trains of diferent lines, the binary variable �V,V� is introduced. �V,V� =1indicates that transfer passengers on feeder train V can transfer from 3. Model Formulation � � station � on line � to station � on line � and board the � 3.1. Notations. Some necessary parameters are defned as connecting train V successfully. Figures 1 and 2 depict the follows: successful and failed transfer connections, respectively. � �={�|�= : the set of subway lines in a network, As a result, �V,V� can be determined by the following 1, 2, ..., �},where� is the total number of lines. Specifcally, formula: a double-track subway line in practice is considered as two one-way lines. {1, �� −�� −�� ≥0 � � � = {� | � = 1, 2, ..., � } if ��,��,V� �,�,V �,�� :thesetofallstationsonline , � � , �V,V� = { (1) � � where � is the total number of stations on line . {0, else ��: the set of trains still in operation on line � during the end-of-service period, �� ={V | V = 1, 2, ..., ��},where�� is the last train of line �. 3.3. Transfer Waiting Time and Penalty. In normal daytime operation, travelers waiting at a given station may be unable ℎ�:theminimumheadwayofline� during the end-of- service period. to take the frst available train, e.g., if that train has no spare � capacity for additional passengers (Schmockeretal.,2011).¨ ��,�: the minimum running time for trains running from station � to �+1on line �, including additional time of train However, it seems reasonable to suppose that demand during starting and braking at stations. the end-of-service period is generally low enough to permit � the assumption that capacity is always available. As a result, ��,�: the minimum dwell time for trains stopping at station � � all passengers are assumed to board the frst arriving train on line�� . � V � afer they reach the platform in this study. When there is a �,�,V: the planned arrival time for train at station on �� line �. successful transfer connection, V,V� can be determined by the �� following formula: ��,�,V: the planned departure time for train V from station � on line �. �V V �� ,�� : the number of transfer passengers on train ,who �V,V� � � �� need to transfer from station on line to station on line �� −� −� , V =1 � � =1 � � � { � � � �,�,V �,�� if & V,V (2) .Inreality,station and station are the same station with = � ,� ,V { �� diferent serial numbers on diferent lines. � −� −� , V >1,� � =0 � � =1 �� { ��,��,V� �,�,V �,�� if V,V −1 & V,V ��,�� : the average time for transfer passengers walking from the platform of station � on line � to the platform of � � station � on line � .Itisobviousthatdiferentpeoplehave But for transfer passengers, the last train of the con- diferent walking speed. For model simplifcation, the trans- necting line is the last chance to fnish their trips. If the fer walking time is assumed to be constant for passengers of connection to the last connecting train is broken, it will the same transfer direction in this study. bring a lot of inconvenience to transfer passengers. To avoid Te decision variables are presented as follows: this undesired phenomenon as much as possible, a penalty �� ,�,V: the actual arrival time for train V at station � on line time � is introduced in this study. If the missed connecting �. train is the last train of the connecting line, the transfer �� ,�,V : the actual departure time for train V from station � waiting time of these transfer passengers (i.e., failed transfer on line �. passengers, FTP) equals the penalty time; see the following �� V,V� : the waiting time of transfer passengers who are from formula: � � train V and transfer to station � on line � . Tis is a period of � �� time from passengers reaching the platform of station � to � � =� , V = � � � � =0 � � V,V � if � & V,V (3) the actual departure time of train V from station � .

�� 3.2.TrainConnectionRelationship.During the end-of-ser- In summary, the complete �V,V� can be determined by the vice period, once a delay occurs in the network, train following formula: 4 Journal of Advanced Transportation

arr Feeder Line  Tl,s, Train Arrival Time Walking

x, =0 Tdep  Connecting Line l,s,  Train Departure Time

Figure 2: Te failed transfer connection.

�� V,V ��,�,V −��,�,V ≥��,� (9) �� −� −� , V =1 � � =1 { ��,��,V� �,�,V �,�� if & V,V { 3.4.5. Headway. As we mentioned above, there is more than { �� {� � � � −� −� � , if V >1,�V,V�−1 =0& �V,V� =1 (4) = � ,� ,V �,�,V �,� one train still running on each line during the end-of-service { � {� , V = � � � � =0 period. Tus, all trains running on each line should meet { � if � & V,V { the requirements of minimum headway during the end-of- {0, else service period; see the following formulas: �� −�� ≥ℎ 3.4. Model Constraints. Te rescheduling model is mainly �,�,V+1 �,�,V � (10) subject to some operational requirements to ensure the �� −�� ≥ℎ safety of the operation and the feasibility of the rescheduled �,�,V+1 �,�,V � (11) timetable. 3.5. Optimization Objectives. We present two objectives to be 3.4.1. Initial Delay. Tis constraint is to input the delay optimized here. First, we try to minimize the total transfer information (e.g., the delayed train, delay time, and position) waiting time (TTWT) for all transfer passengers, which helps to the model; see the following formula: to improve the LOS of the system afer disturbances; see �� formula (12). Te frst objective also benefts increasing the �� ≥ � +� �∗,�∗,V∗ �∗,�∗,V∗ � number of successful transfer passengers afer disturbances (5) because of the penalty time set for failed transfer passengers �� or ��∗,�∗,V∗ ≥ ��∗,�∗,V∗ +�� (FTP) who miss their last connecting trains. Second, we seek to minimize the deviation between the rescheduled timetable V∗ �∗ �∗ where indicates the delayed train running on line . and the scheduled timetable, which is usually dispatchers’ V∗ represents the station, where train is located when the frst goal in practice; see formula (13). Passengers who do disturbance occurs or the frst station that is going to be not need to transfer afer disturbances also beneft from the V∗ � visited by train afer the disturbance occurs. � represents second objective. the delay time. V �� min ∑ ∑ ∑ ∑ ∑ ∑ ��,�� ×�V,V� �∈� ��∈� �∈� ��∈� V∈� V�∈� (12) 3.4.2. Actual Arrival and Departure Time. During the process � �� of rescheduling, the actual arrival and departure times of �� trains at stations cannot be earlier than the scheduled times; ∑ ∑ ∑ (�� − � +�� − � ) min �,�,V �,�,V �,�,V �,�,V (13) see the following formulas: �∈� �∈�� V∈�� ≥ � �,�,V �,�,V (6) Te two objectives and constraints (1) and ((4)-(11)) �� above consist of the complete rescheduling model during �� ≥ � �,�,V �,�,V (7) the end-of-service period. Te model is called a stakeholder- oriented model because interests of both operation and 3.4.3. Section Running Time. Under the limitations of the passengers (transfer and nontransfer) are considered in the traction and brake performance of trains, the length of each two objectives. Te trade-of between the two objectives may section, safety requirements, and the actual running times of help dispatchers make decisions. trains in sections must be longer than the minimum running times [24]; see the following formula: 4. Model Solution �� −�� ≥�� ,�+1,V �,�,V �,� (8) Owing to the huge complexity of the timetable rescheduling problem, especially when solving a real-world case of a large- 3.4.4. Dwell Time. Adjusting the dwell time is an important scale and complex network, many heuristic algorithms have measure for dispatchers to control subway trains. Similar to been proposed to speed solving this problem. Examples the section running time, the actual dwell times of trains at include greedy algorithm [9], particle swarm algorithm [25], stations must be longer than the minimum dwell times [24]; and genetic algorithm [10]. However, in this study, we utilize see the following formula: the �-constraint method [26] to convert the model into a Journal of Advanced Transportation 5

�� ≥� −� −� −�×(1−� � ), V =1 single-objective model, and some linearization techniques V,V� ��,��,V� �,�,V �,�� V,V if (18) are adopted to reformulate all nonlinear constraints into �� ≥� −� −� −�×� � , V >1 linear constraints. Ten, the original model is converted to V,V� ��,��,V� �,�,V �,�� V,V −1 if (19) a single-objective ILP model, which can be solved by Cplex �� ≥� ×(1−� � ), V = � � rapidly. By solving the problem for diferent values of �,the V,V� � V,V if � (20) approximateParetofrontiercanbeshowntounderstandthe �� ≥0, ∀V ∈�, V ∈�� trade-of between the two objectives. V,V� � � (21) Finally, the single-objective model obtained in Section 4.1 4.1. Te �-Constraint Method. Te �-constraint method is is converted into a single-objective ILP model with objective good at solving multiobjective models to obtain a set of function (12) and linear constraints ((5)-(11)), ((14)-(16)), and approximate Pareto optimal solutions. During the end-of- ((18)-(21)), which can be solved by Cplex within limited service period, we should put transfer passengers on the frst seconds. place because they may miss their last connecting trains due to delayed feeder trains. As a result, the objective function 5. Case Study (13) is chosen as the �-constraint and reformulated by the Tovalidatethemethodproposedinthisstudy,theBeijing following formula: 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, �,�,V �,�,V �,�,V �,�,V and 225 ordinary stations with an average daily ridership of �∈� �∈�� V∈�� (14) 9.998 million passengers. A sketch map of the Beijing subway ≤�∗ × (1+�) network without Airport Express is shown in Figure 3. Words with all letters in uppercase are acronyms of stations’ names. ∗ Owing to the diference in passenger fow characteristics, where � indicates the optimal objective value when only diferent lines have diferent operational time frames. Among objective function (13) is considered in the model. � is a coef- all last trains of all lines in Beijing subway network, the earliest fcient representing dispatchers’ tolerability to the deviation one is the last train of Fangshan Line from SZ to GGZ, starting between the rescheduled and scheduled timetables, �≥0. at 22:00. According to the defnition in this study, the end-of- By the �-constraint method, the original model is con- service period of the Beijing subway case is from 22:00 to the verted into a single-objective model with the objective func- time when all trains fnish their jobs, a period of time about tion (12) and constraints (1), ((4)-(11)), and (14). 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 4.2. Linearization. Among all constraints, constraints (1) the minimum time unit is second. and (4) are nonlinear constraints. In order to speed the Table 1 shows a sample of the real AFC data with key process of model solution, constraints (1) and (4) need to be information of the Beijing subway system. Te number reformulated into linear constraints. of transfer passengers is the key to deciding whether a � is introduced to represent a big enough positive connecting train should wait for a delayed feeder train or integer; then formula (1) can be easily replaced by the depart on time. From the real data recorded in the AFC following linear formulas: system, we can obtain the approximate number of transfer passengers during the end-of-service period by a “Passenger- to-Train” assignment method [27]. Te AFC data, the data of �� −� −� <�×� � average transfer walking time in all transfer stations, and the ��,��,V� �,�,V �,�� V,V (15) data of the scheduled timetable are all provided by the Beijing �� −� −� ≥�×(� � −1) Rail Transit Control Center. ��,��,V� �,�,V �,�� V,V (16) 5.1. Scenario-Based Experiments. In order to prove that the On the premise of the objective to minimize the total proposed method is efective, various delay scenarios are transfer waiting time of all transfer passengers, formula (4) generated randomly in terms of delayed train, delay position, can be relaxed into formula (17). Ten formula (17) can be and delay time. Numerical experiments based on these delay replaced by linear formulas ((18)-(21)). scenarios are carried out. Detailed information about these delay scenarios is listed in Table 2. We test these delay scenarios with � = 0.25, ℎ� = �� V,V� 2 minutes, and �� =1hour (i.e., 3600 seconds). All �� corresponding problems are solved within 2 seconds by Cplex � −� −� , V =1 � � =1 { ��,��,V� �,�,V �,�� if & V,V { 12.6.2 on a laptop computer with Intel Core i7-7700HQ CPU { �� {� � � � −� −� � , if V >1,�V,V�−1 =0& �V,V� =1 (17) @ 2.8 GHz, 8 GB RAM. For benchmarking, we also test ≥ � ,� ,V �,�,V �,� { � these delay scenarios with �=0,whichissimilartoactual {� , V = � � � � =0 { � if � & V,V { behaviors of dispatchers to minimize the deviation. Table 3 {0, �� reports the solution results in detail. 6 Journal of Advanced Transportation

CX TB SGZ Changping Line Line 8 5 Line ZXZ BB XEQ Line 13 AB Line 4 Line 15 QX

GT DZM XZM Line 6 LC HW NG Line 1 SH SHD TQ

PGY Line 9 Batong Line Line 2

Line 7 JHC BX Line 14 West Line 14 East XJ BN ZGZ Line 10 GGZ SJZ SZ Fangshan Line Yizhuang Line TGY CQ Figure 3: Sketch map of the Beijing subway network without Airport Express.

Table 1: 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 ......

Te TTWT includes the transfer waiting time of all requirements for LOS from passengers, our method is much successful transfer passengers and the penalty time of failed better than dispatchers when tackling a disturbance which transfer passengers. For most scenarios, there is a consider- leads to a long delay time. able decrease of about 20% in the TTWT as well as a big decline in the number of FTP, about 40% compared to those 5.3. Trade-Of between Objectives. Our proposed biobjective of the rescheduled timetable with �=0,whichisusedto model for timetable rescheduling during the end-of-service mimic dispatchers’ behaviors. Tese numerical results show period aims to minimize the TTWT for all transfer passen- that the LOS of subway systems afer disturbances can be gers, meanwhile minimizing the deviation to the scheduled improved obviously by our method. timetable. However, in the actual process of decision-making, it is difcult for dispatchers to obtain the optimal solution for 5.2. Delay Time Analysis. During the daily operation, dif- multiple criteria. As a result, we are interested in the trade- ferent disturbances may lead to diferent delay times. In of between objectives and adopt the �-constraint method to this experiment, we focus on Scenario 1 and set delay time obtain a set of approximate Pareto optimal solutions, allowing changing from 5 to 10 minutes to test the efect of the dispatchers to choose one solution simply. method on diferent delay times. Similarly, all corresponding Scenario 1 is still an example to obtain approximate Pareto problems are solved within 2 seconds by Cplex 12.6.2 on the optimal solutions by changing the value of � from 0 to 0.5. same computer. Table 4 shows the solution results in detail. Te numerical results are shown in the lef part of Figure 4. With the delay time increasing from 5 to 10 minutes, our With � rising from 0 to 0.5, the TTWT decreases rapidly; method can reduce the TTWT by 17.91% to 23.44% compared however, the deviation increases steadily. Te right part of with that of the rescheduled timetable with �=0.Inother Figure 4 shows us the approximate Pareto frontier, which can words, the gap in TTWT between the rescheduled timetables demonstrate the trade-of between the two objectives. Tere by dispatchers and by our method is becoming bigger with is an obvious trend between the two objectives: a decrease the increasing delay time. As a result, with the improving in the TTWT corresponding to an increase in the deviation. Journal of Advanced Transportation 7

Table 2: Delay scenarios in detail.

ID Line Scenario Delay time st 1Line1fromSHDtoPGYTe1 train departs late at Guomao. 9 min st 2 Line 2 outer loop Te 1 train arrives late at Yhgong. 10 min nd 3 Line 4 from TGY to AB Te 2 train departs late at Xizhimen. 8 min nd 4 Line 5 from TB to SJZ Te 2 train departs late at Dongdan. 6 min st 5Line6fromLCtoHWTe1 train arrives late at Bsqnan. 8 min nd 6 Line 7 from BX to JHC Te 2 train arrives late at Ciqikou. 7 min nd 7 Line 8 from NG to ZXZ Te 2 train departs late at Huoying. 9 min rd 8 Line 9 from GT to GGZ Te 3 train arrives late at Bsqnan. 10 min rd 9Line10outerloopTe3 train departs late at Shaoyaoju. 5 min st 10 Line 15 from QX to BB Te 1 train arrives late at Wangjingxi. 7 min

Table 3: Solution results of diferent scenarios. TTWT/s Number of FTP Scenario Decrement Decrement � = 0 � = 0.25 � = 0 � = 0.25 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%

Dispatchers should weigh up interests of all stakeholders 6. Conclusions andmakedecisionswiththebestsolutionintheirminds according to the actual situation. Te timetable rescheduling problem can be optimized by In addition, for each approximate Pareto optimal solu- many objectives because of its inherently multicriterion tion, we calculate the TTWT, the number of FTP, and the nature. It is difcult to tell which solution is the optimal total travel time (TT) of all trains involved in the end-of- solution. But in terms of some specifc criteria, we can ∑ ∑ (�� −��) fgure out that a solution is better or worse than others. service period by �∈� V∈� �,� ,V �,1,V ,asshowninTable5. � � As a result, one of the major contributions of this study According to the numerical results we can conclude that a is that a biobjective optimization method is proposed from high tolerability to the deviation only causes a rather small the stakeholder-oriented perspective to tackle the timetable extension in the total TT of all involved trains. For example, rescheduling problem during the end-of-service period of a among all rescheduled timetables, the biggest increment in subway network, which allows us to fgure out the trade-of � = 0.45 totalTTisonly132seconds( ), but there is a between the LOS (in terms of the TTWT and the number 27.82% decrement in the TTWT and a 58.13% decrement in of FTP) and the operation (in terms of the deviation to the the number of FTP compared to those of the rescheduled scheduled timetable). We utilize the �-constraint method to �=0 timetable with . obtain approximate Pareto optimal solutions within limited During the end-of-service period, a passenger who does seconds for diferent delay scenarios based on a real-world not need to transfer can catch his or her train defnitely case, the Beijing subway network, which can help dispatchers even if the train is late, but a transfer passenger may to make decisions during the process of rescheduling. miss the last connecting train because of the late feeder In addition, given the actual characteristics of the end- train. As a result, based on all our numerical results of-service period as well as the fact that a high tolerability in Section 5.3, Figure 4, and Table 5 in particular, we to the deviation will not lead to a big extension in the total strongly suggest that dispatchers should take more interests travel time, we think that dispatchers should put transfer from transfer passengers into consideration when reschedul- passengers’ interests in the frst place when rescheduling ing timetable afer disturbances during the end-of-service during the end-of-service period, which will beneft the period. overall LOS afer disturbances. 8 Journal of Advanced Transportation

x 100000 x 100000 120 4500 120

110 4000 110 Deviation/s

100 3500 100 TTWT/s TTWT/s

90 3000 90

80 2500 80 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2500 3000 3500 4000 4500  Deviation/s TTWT Deviation Figure 4: Numerical results with changing � and the approximate Pareto frontier.

Table 4: Solution results of Scenario 1 with diferent delay times.

Delay time/s TTWT/s 300 360 420 480 540 600 �=0 11417881 11388673 11389273 11379193 11418266 11421626 � = 0.25 9373458 9227160 9132110 8998431 8879746 8743882 Decrement 17.91% 18.98% 19.82% 20.92% 22.23% 23.44%

Table 5: A comparison between total TT, TTWT, and the number of FTP.

Rescheduled Timetable Total TT/s Increment TTWT/s Decrement FTP Decrement � = 0.00 661345 ---- 11418266 ---- 1304 ---- � = 0.05 661345 0 9800056 14.17% 932 28.53% � = 0.10 661345 0 9452463 17.22% 857 34.28% � = 0.15 661381 36 9223486 19.22% 810 37.88% � = 0.20 661382 37 9034198 20.88% 751 42.41% � = 0.25 661351 6 8879746 22.23% 734 43.71% � = 0.30 661477 132 8726061 23.58% 686 47.39% � = 0.35 661447 102 8589964 24.77% 669 48.70% � = 0.40 661351 6 8396420 26.47% 594 54.45% � = 0.45 661477 132 8241846 27.82% 546 58.13% � = 0.50 661447 102 8102096 29.04% 529 59.43%

For future extension, the timetable rescheduling prob- Acknowledgments lem during the end-of-service period can take passengers rerouting into consideration, especially when the delay time is Tis work was supported by the National Natural Sci- long. However, train rerouting seems an unreasonable option ence Foundation of China (51478036) and the Fundamental because in most subway systems, trains belonging to a line Research Funds for the Central Universities (2018YJS074). cannot run on other lines. Anyway, contributions should be devoted to improving the LOS of subway systems afer References disturbances. [1] W. Zhou, L. Deng, M. Xie, and X. Yang, “Coordination opti- Data Availability mization of the frst and last trains’ departure time on urban rail transit network,” Advances in Mechanical Engineering,vol. Allrelateddataareincludedwithinthisarticle. 2013, Article ID 848292, 12 pages, 2013. [2]L.Kang,J.Wu,H.Sun,X.Zhu,andZ.Gao,“Acasestudyon Conflicts of Interest the coordination of last trains for the Beijing subway network,” Transportation Research Part B: Methodological,vol.72,pp.112– Te authors declare that they have no conficts of interest. 127, 2015. Journal of Advanced Transportation 9

[3]L.Kang,X.Zhu,J.Wu,H.Sun,S.Siriya,andT.Kanokvate, [20] A. Caprara, M. Fischetti, and P.Toth, “Modeling and solving the “Departure time optimization of last trains in subway networks: train timetabling problem,” Operations Research,vol.50,no.5, mean-variance model and GSA algorithm,” Journal of Comput- pp.851–861,2002. ing in Civil Engineering,vol.29,no.6,ArticleID04014081,2014. [21] X. Zhou and M. Zhong, “Single-track train timetabling with [4] L. Kang and X. Zhu, “Strategic timetable scheduling for last guaranteed optimality: branch-and-bound algorithms with trains in urban railway transit networks,” Applied Mathematical enhanced lower bounds,” Transportation Research Part B: Modelling: Simulation and Computation for Engineering and Methodological,vol.41,no.3,pp.320–341,2007. Environmental Systems,vol.45,pp.209–225,2017. [22] Y. Lee and C. Chen, “A heuristic for the train pathing and [5] V.Cacchiani and P.Toth, “Nominal and robust train timetabling timetabling problem,” Transportation Research Part B: Method- problems,” European Journal of Operational Research,vol.219, ological,vol.43,no.8-9,pp.837–851,2009. no. 3, pp. 727–737, 2012. [23] X. Yang, X. Li, Z. Gao, H. Wang, and T. Tang, “A cooperative [6] V.Cacchiani, D. Huisman, M. Kidd et al., “An overview of recov- scheduling model for timetable optimization in subway sys- ery models and algorithms for real-time railway rescheduling,” tems,” IEEE Transactions on Intelligent Transportation Systems, Transportation Research Part B: Methodological,vol.63,pp.15– vol.14,no.1,pp.438–447,2013. 37, 2014. [24]R.C.W.Wong,T.W.Y.Yuen,K.W.Fung,andJ.M.Y.Leung, [7] A. D’Ariano, D. Pacciarelli, and M. Pranzo, “Abranch and bound “Optimizing timetable synchronization for rail mass transit,” algorithm for scheduling trains in a railway network,” European Transportation Science,vol.42,no.1,pp.57–69,2008. JournalofOperationalResearch,vol.183,no.2,pp.643–657, [25] X. Meng, L. Jia, and Y. Qin, “Train timetable optimizing and 2007. rescheduling based on improved particle swarm algorithm,” [8] J. TornquistandJ.A.Persson,“N-trackedrailwaytrafcre-¨ Transportation Research Record, no. 2197, pp. 71–79, 2010. scheduling during disturbances,” Transportation Research Part [26] O.J.Ibarra-Rojas,R.Giesen,andY.A.Rios-Solis,“Anintegrated B: Methodological,vol.41,no.3,pp.342–362,2007. approach for timetabling and vehicle scheduling problems to [9] J. Tornquist¨ Krasemann, “Design of an efective algorithm for analyze the trade-of between level of service and operating fast response to the re-scheduling of railway trafc during costs of transit networks,” Transportation Research Part B: disturbances,” Transportation Research Part C: Emerging Tech- Methodological,vol.70,pp.35–46,2014. nologies,vol.20,no.1,pp.62–78,2012. [27] W. Zhu, W. Wang, and Z. Huang, “Estimating Train Choices of RailTransitPassengerswithRealTimetableandAutomaticFare [10] S. Dundar¨ and I. S¸ahin, “Train re-scheduling with genetic algo- Collection Data,” JournalofAdvancedTransportation,vol.2017, rithms and artifcial neural networks for single-track railways,” ArticleID5824051,12pages,2017. Transportation Research Part C: Emerging Technologies,vol.27, no.2,pp.1–15,2013. [11] A. Schobel,¨ “A model for the delay management problem based on mixed-integer-programming,” Electronic Notes in Teoretical Computer Science,vol.50,no.1,pp.1–10,2001. [12] A. Schobel,¨ “Capacity constraints in delay management,” Public Transport,vol.1,no.2,pp.135–154,2009. [13] M. Schachtebeck and A. Schobel,¨ “To wait or not to wait-and who goes frst? Delay management with priority decisions,” Transportation Science,vol.44,no.3,pp.307–321,2010. [14] T. Dollevoet, D. Huisman, M. Schmidt, and A. Schobel,¨ “Delay management with rerouting of passengers,” Transportation Science,vol.46,no.1,pp.74–89,2012. [15] T. Dollevoet, D. Huisman, L. Kroon, M. Schmidt, and A. Schobel,¨ “Delay management including capacities of stations,” Transportation Science,pp.1–19,2014. [16] S. Binder, Y. Maknoon, and M. Bierlaire, “Te multi-objective railway timetable rescheduling problem,” Transportation Research Part C: Emerging Technologies,vol.78,pp.78–94,2017. [17]X.Xu,K.Li,andL.Yang,“Reschedulingsubwaytrainsbya discrete event model considering service balance performance,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems,vol.40,no.2,pp. 1446–1466, 2016. [18] Y. Gao, L. Kroon, M. Schmidt, and L. Yang, “Rescheduling a metro line in an over-crowded situation afer disruptions,” Transportation Research Part B: Methodological,vol.93,pp.425– 449, 2016. [19] W. Xu, P. Zhao, and L. Ning, “A passenger-oriented model for train rescheduling on an urban rail transit line considering train capacity constraint,” Mathematical Problems in Engineering,vol. 2017,ArticleID1010745,9pages,2017. International Journal of Rotating Advances in Machinery Multimedia

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