Train rescheduling model with train delay and passenger impatience time in urban subway network

Transcription

1 JOURNAL OF ADVANCED TRANSPORTATION J. Adv. Transp. 2016; 50: Published online 9 February 2017 in Wiley Online Library (wileyonlinelibrary.com) Train rescheduling model with train delay and passenger impatience time in urban subway network Qu Zhen 1 and Shi Jing 2 * 1 State Tobacco Monopoly Administration, Beijing , China 2 Department of Civil Engineering, Tsinghua University, Beijing , China SUMMARY This paper considers the train rescheduling problem with train delay in urban subway network. With the objective of minimizing the negative effect of train delay to passengers, which is quantified with a weighted combination of travel time cost and the cost of giving up the planned trips, train rescheduling model is proposed to jointly synchronize both train delay operation constraints and passenger behavior choices. Space time network is proposed to describe passenger schedule-based path choices and obtain the shortest travel times. Impatience time is defined to describe the intolerance of passengers to train delay. By comparing the increased travel time due to train delay with the passenger impatience time, a binary variable is defined to represent whether the passenger will give up their planned trips or not. The proposed train rescheduling model is implemented using genetic algorithm, and the model effectiveness is further examined through numerical experiments of real-world urban subway train timetabling test. Duration effects of the train delay to the optimization results are analyzed. Copyright 2017 John Wiley & Sons, Ltd. KEY WORDS: rescheduling; train delay; impatience time; urban subway; genetic algorithm 1. INTRODUCTION Train rescheduling problem is the daily task faced by operators when train delay makes timetable infeasible and increases unnecessary waiting times of passengers in urban subway network. During train operation process, the planned timetable is sometimes disrupted by emergencies such as train delays, extraordinary passenger volumes or some other accidents. Some of these disruptions are severe enough to prevent the urban subway system from planned operation. When these situations occur, the planned timetable may become infeasible, and train rescheduling is approached as a real-time schedule recovery problem in urban subway. The focus of this paper is on train rescheduling problem in urban subway network, where the aim is to reduce the effect of train delay and to enhance passenger satisfaction. With the urban subway network scale expanding, there are many alternative path choices from origin to destination, so passenger behavior choice is more autonomous and random. The diversity of passenger behavior choices and network complexity added the difficulty of train rescheduling problem. Passengers may wait for the delayed train, change their path choices or give up their planned trips according to real-time timetable and train delay information. In condition of disruption cases, passengers are looking forward to a timetable with good coordination between different lines so that they can reduce the extra waiting time due to the effect of train delay. Train rescheduling should adjust the timetable in an effective way to make the delayed passengers return to the planned state as much as possible. However, in actual urban subway operation, rescheduling problem is exclusively conducted by human dispatchers based on their experience, and they lack an overall understanding of the subway network and scientific theory guidance. *Correspondence to: Shi Jing, Department of Civil Engineering, Tsinghua University, Beijing , China. jingshi@tsinghua.edu.cn Copyright 2017 John Wiley & Sons, Ltd.

2 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 1991 The experiences of railway planning process have a vital significance for urban subway operations and management. Cacchiani et al. [1] present an overview of recovery models and algorithms for realtime railway disruption management, including real-time rescheduling of timetable, rolling stock and crew duties. In most urban subway network, there are no across-line trains because lines are separate from each other and each direction of a line has a separate rail track. Thus, the main propose of rescheduling process in urban subway network is to adjust the current timetable of each line to ensure the rational use of available trains. In recent years, many researchers focus on train rescheduling problem with the consideration of passenger complaints when there is train delay. With the goal of minimizing the total passenger delay, Schöbel [2] studies the train rescheduling problem which consider if the connecting trains should wait in a station for delayed feeder trains or not. Heilporn et al. [3] presented two equivalent mixed integer linear programming models for delay management problem with the objective of minimizing the total delay. Zhao and Zeng [4] developed a heuristic method for optimizing transit network planning consisting of route network design, vehicle headway setting and timetable creation to minimize passenger costs. Krumke et al. [5] have discussed extra dwell time of a single train to minimize the increased travel time of passengers or money refunded due to a large delay. Kanai et al. [6] proposed the method to measure passenger dissatisfaction and use it as objective functions of their simulation model in condition of a strategy for maintaining or dropping connections. Corman et al. [7] presented a bi-objective model with the goal of minimizing the delays and maximizing the total value of satisfied connections. Corman et al. [8] have evaluated the rescheduled timetables in relation to the total delay, average delay and weighted summary of traveling and waiting times. D Ariano [9] evaluated the benefits of local rerouting strategies to minimize the delays between consecutive trains. A comparison with advanced rescheduling solutions shows the high potential of iterative rerouting and rescheduling strategies to minimize train delays and to improve the use of the infrastructure capacity. D Ariano [10] studied a train scheduling problem in railway, which is modeled by an alternative graph formulation. A branch and bound algorithm is proposed to solve a case study based on a bottleneck area of the Dutch rail network. Dynamic passenger demand has been considered to improve the performance of train rescheduling problem in recent years. Niu and Zhou [11] proposed a binary integer programming model incorporated with passenger loading and departure events and analyzed the characteristic of time-dependent demands under oversaturated conditions. Cadarso et al. [12] explicitly deal with the effects of the disruption on the passenger demand and proposed a two-step approach that combines an integrated optimization model with a model for the passenger behavior. Canca et al. [13] built a nonlinear integer programming model that fits the arrival and departure times of trains to a dynamic behavior of passenger demand. Barrena et al. [14] presented three formulations for the problem of design and optimization of train rescheduling problem in condition of passenger dynamic demand. Barrena et al. [15] also studied the design and optimization of train timetables for a rail rapid transit line in a dynamic demand environment, with the objective of minimizing the average passenger waiting time at stations from the perspective of passengers. Niu and Zhou [16] focused on how to minimize the total passenger waiting time at stations by computing and adjusting train timetables for a rail corridor with given timedependent origin to destination passenger demand matrices. In highly interconnected subway network, the passenger path choices are important parts of train rescheduling problem because passenger would adjust path choices when they face train delay. In recent studies, schedule-based path searching algorithms have been studied with the consideration of dynamic train service network. Wong and Tong [17] developed one of the earliest schedule-based path searching algorithms which was further enhanced in Tong et al. [18] [19]. In their optimal path searching process, waiting time, in-vehicle time and transfer time were considered. This schedulebased optimal path searching algorithm is an example adapted from Dijkstra s algorithm for the transit system. Dollevoet [20] assumed that the passengers know which connections between trains will be maintained in the near future. They may decide to take an alternative route to their destination. Dollevoet [21] showed that the possibility for passengers to choose an alternative route through the network leads to a significant reduction of their delays. Although there is a comprehensive body of literatures on train rescheduling problem with the consideration of passenger complaints, dynamic passenger demand and passenger path choices, what has been neglected in most papers so far are listed in the following three aspects:

3 1992 Q. ZHEN AND S. JING (1) Most papers assume a fixed path choice of passengers in railway system where the timetable is published to society. In these papers, passengers will take exactly the path they have planned and have to wait until the same train connection takes place again if there is train delay. However, this assumption is usually not valid in urban subway system, where the passenger path choices are more autonomous due to the shorter departure frequency and lower time cost. Passengers will adjust their path choices according to the rescheduling timetable. (2) In most papers, the passenger demands from origins to destinations will not be affected by train delay. Transportation research has shown that impatience and time pressure are determining factors for traffic rule violation and behavior by Naveteur et al. [22]. In practice, the passenger demands are not only time dependent along the day, but also extremely sensitive to train delay because passengers have their own impatience time. If urban subway service is not provided quickly enough, sometimes the passengers may change or give up their planned trips. Thus, there will be a decline in passenger demands under the effect of train delay. However, few literatures have paid attention to the effect of train delay to passenger demands. (3) Most papers minimize the total travel time, average waiting time or delay time as the objective function of their train rescheduling models. In urban subway system, when passengers are waiting at the platform for the next coming train, which is exactly delayed, there is a maximum time boundary which the passengers can accept; otherwise, the planned trip cancelations will happen. Thus, the objective function of train rescheduling model should consider the cost of the planned trips cancelations, which is ignored by most literatures. Compared to the existing literatures, this study offers the three main contributions as below: First, we focus on the dynamic choice of passengers when they face train delay, and three scenarios are considered as follows: (i) keep waiting at platform until they can board the next train; (ii) choose other alternative path in subway network to avoid train delay; and (iii) give up their planned trips. Given a rescheduled timetable under train delay, each passenger will adjust his path choice dynamically. Second, we consider the real-time demand of potential passengers, who are extremely sensitive to train delay and have their own impatience time. Passengers will consider whether it is worth for them to wait or give up their planned trips. Impatience time is defined in terms of the intolerance of passengers to train delay, which may affect their planned trips. Each passenger will compare the increased travel time due to delay with their own impatience time, and the time-dependent demand from origin to destination will be changed. Third, the aim of the train rescheduling model in this study is to reduce negative effects of train delay to passengers, which is quantified by a weighted combination of the total travel time and the number of planned trips cancelations. Train rescheduling model is established from the view of both operators to reduce negative effects of train delay and passenger behavior choices to train delay situation. The rest of this paper is organized as follows. In section 2, a detailed problem statement, assumptions, optimization process and symbol notations are presented. In section 3, train rescheduling model is put forward to design a rescheduled timetable using time-dependent passenger demands. Based on the space time subway network, passenger dynamic path choice model under train delay is proposed in section 4. In section 5, a genetic algorithm is proposed to solve optimization model. In section 6, the passenger impatience time under train delay is presented based on questionnaire survey s empirical analysis. Quality of train rescheduling model and the genetic algorithm are illustrated by case study, which is composed of six busy lines from the Beijing subway network. Moreover, duration analysis of train delay is proposed. Conclusions and views of future works are shown in section PROBLEM STATEMENT This study considers an urban subway network with m lines, where lines are numbered as 1 +,1,2 +,2,...,m +, m, sequentially. Note that a physical line l (l =1,2,..,m) with two opposite directions shown in Figure 1, is defined as line l + (up-direction) and line l (down-direction) to denote the bi-directional characteristic, respectively. The stations on each line l (l =1,2,..,m) with two

4 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 1993 Figure 1. Illustration of an urban subway line. opposite directions are numbered as 1, 2,..,n l, sequentially. Train services on line l (l =1,2,..,m) move from station 1 to station n l. Suppose that there are some emergency events which cause delay of train j at station u on line l (l + or l )and disorganize the planned timetable as shown in Figure 2. Train delay period [t 1, t 2 ]is considered where t 1 is the initial instant when the emergency begins and t 2 is the time instant set by the operator indicating when the emergency is estimated to be finished. There is no other train delay on other lines l {1, 2,..., m},l l (l + or l ). Given delayed line l (l + or l ), delayed train j and delayed station u, rescheduling process in this paper is to adjust timetable variables (stopping time at each station, arrival time at first station) of line 1(1 + or 1 ), line 2 (2 + or 2 ),, line m(m + or m ) during study time period [0, T] to recover smooth operation of urban subway Fundamental assumption To build a more reasonable optimization model, the following assumptions are further introduced: Assumption 1. We assume that transfer time is equal to the average value for different passengers. In practice, transfer times from one platform to another platform are uncertain for different passengers. Moreover, it depends on many factors such as congestion in transfer station, boarding and alighting duration of passengers. Assumption 2. We assume that train overtaking and crossing are not allowed during the subway operations. A platform can only accommodate one train at a time, and there is no overtaking at any station of the line. Assumption 3. During the rescheduling process, train stopping times at each station and train arrival times at first station are adjustable, and the running times between two successive stations are given and measured according to actual operation. Passenger travel time of the planned path with no delay is given and can be measured according to the subway operation. Assumption 4. We assume that the initial states of subway network, passenger time-dependent demand, line physical data and train operation data are known or can be estimated via the available information. The real-time train delay information has been announced to passengers who are rational Figure 2. Situation of delayed line.

5 1994 Q. ZHEN AND S. JING and will make choices dynamically. Passengers will minimize the travel time when they choose the path in subway network Optimization process In this paper, train rescheduling is modeled as a hybrid process with both discrete and continuous events. Trains are scheduled according to discrete timetables and passenger demand from origins to destinations is continuously time dependent. Train rescheduling is also a hybrid process with the interaction effect of passengers and operators. During the rescheduling process, each passenger will choose the optimal path dynamically based on real-time timetable information, which is called schedule-based path choice. When timetable is adjusted by operators, the passengers will adjust their path choices, too. Thus, dynamic path choice model is proposed. Given the assumption 1 4 and the following conditions: (a) timetable and line physical data of urban subway network; (b) number of trains to be rescheduled on each line; (c) urban subway operation constraints according to train operation data; (d) impatience time function of passengers under train delay; (e) time-dependent origin to destination passenger demand; The proposed optimization process of train rescheduling problem is shown in Figure 3 and further explained in the following: Train rescheduling model aims to minimize the weighted combination of (i) travel time costs and (ii) costs of planned trips cancelations. We define binary variables to represent whether the passengers will give up their planned trips or not, which are determined and updated by comparing the increased travel time due to delay with the impatience time of passengers. If the increased travel time is longer than the impatience time, the binary variable is equal to 1, or 0 vice versa. Genetic algorithm is used to obtain the optimal rescheduled timetables. Dynamic path choice aims to obtain the optimal path choice and shortest travel time for each passenger under the condition of train delay. According to assumption 4, the real-time timetable of the urban subway network is known for all passengers, and they will choose the paths dynamically. Given delayed timetable and time-dependent demand from origin to destination, Floyd Warshall algorithm based on space time network is used to solve path choice and corresponding travel time for each passenger. Figure 3. Optimization process.

6 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 1995 Decision variables are train stopping times and arrival times. Given these decision variables, timetable and space time subway network can be determined, which is the basis of passenger path choice. Constrains of decision variables are proposed to ensure the feasibility of train operation in condition of train delay Notation Parameters of optimization process are as follows: T study time boundary t index of times, t =1,2,...,T t 1 time instant when train delay begins t 2 time instant when train delay is estimated to be finished t delay duration of train delay N number of trains on each line over the study period j index of trains, j =1,2,...,N m number of bi-directional lines in urban subway network l index of lines, l =1,2,..,m,l + is up direction and l is down direction n l number of stations on line l u index of stations, u =1,2,...,n l O set of origins D set of destinations o l, u origin at station u on line l d l, u destination at station u on line l S set of nodes in graph of subway network L set of links in graph of subway network j; min TAl;1 earliest arrival time of train j at the first station on line l (l + or l ) j; max TAl;1 latest arrival time of train j at the first station on line l (l + or l ) TS min minimum stopping time of trains at stations TS max maximum stopping time of trains at stations TH min minimum time headway between two successive trains at the same station TH max maximum time headway between two successive trains at the same station Variables of optimization process are as follows: TA j arrival time of train j at station u on line l (l + or l ) TD j departure time of train j at station u on line l (l + or l ) TS j stopping time of train j at station u on line l (l + or l ) TR j running time of train j at segment u u + 1 on line l (l + or l ) TH j headway between train j and train j + 1 at station u on line l (l + or l ) T l ;u transfer time between station u on line l and station u on line l f t o ;d l ;u number of passengers from origin to destination with a certain arrival time t at origin T t o ;d l ;u shortest travel time of OD pairs f t o ;d l ;u under train delay T pla o ;d l ;u travel time of the planned path from o l, u to d l, u with no delay T alt o ;d l ;u travel time of the alternative path from o l, u to d l, u under train delay η t o ;d l ;u binary variable represents whether OD pairs f t o ;d l ;u give up their planned trips or not T increased increased travel time due to train delay impatience time which passengers can accept due to train delay T impatience 3. TRAIN RESCHEDULING MODEL During the rescheduling process under train delay, each passenger will consider whether to give up his or her planned trip or not. The aim of train rescheduling model is to minimize the negative effects of train delay to passengers shown as the objective function in section 3.1. Decision variables are train

7 1996 Q. ZHEN AND S. JING stopping times and arrival times shown in section 3.2. Constrains of decision variables are proposed to ensure the feasibility of train operation in condition of train delay in section Objective function The objective function of train rescheduling model is to minimize the weighted combination of total travel time and the number of planned trip cancelations during the study time period, which can be calculated with Equation (1). minf ¼ α t ½0;TŠ o O d l ;u D η t o ;d l ;u f t o ;d l ;u!þ β t ½0;TŠ o O d l ;u D 1 η t o ;d l ;u f t o ;d l ;u T t o ;d l ;u! (1) η t o ;d l ;u f t o ;d l ;u t ½0;TŠ o O d l ;u D number of planned trips cancelations; t ½0;TŠ o O d l ;u D 1 η t o ;d l ;u f t o ;d l ;u T t o ;d l ;u the total travel time of passengers; coefficient α and β are proposed to balance the magnitude of two kinds in the objective function; f t o ;d l ;u number of passengers from origins o l, u to destinations d l, u with arrival time t; η t o ;d l ;u binary variable is shown in Equation (2), if η t o ;d l ;u ¼ 1, the passengers f t o ;d l ;u will give up their planned trips; if η t o ;d l ;u ¼ 0, the passengers f t o ;d l ;u will not give up their planned trips; ( η t o ;d l ;u ¼ 1; T impatience T increased 0; T impatience > T increased (2) T increased increased travel time in Equation (3) is defined to represent the difference value between the shortest travel time under delay and planned travel time with no delay. n o T increased ¼ max T t o ;d l ;u T pla o ;d l ;u ; 0 (3) T t o ;d l ;u shortest travel time calculated by space time based algorithm in section 4.4; T pla o ;d l ;u T impatience planned travel time measured according to subway practical operation; impatience time represent the maximum time boundary which passengers can accept in order to finish their planned trips. There is a function between the impatience time T impatience and planned travel time T pla o ;d l ;u and the curve fitting will be discussed in section 6.1, Equation (31) Decision variables Using the arrival time TA j l;1 at the first station, the stopping time TSj at each station u and the running time TR j of each segment u u + 1, we can determine the arrival time TAj and departure time TDj at each station step by step. The following Equations (4) and (5) should be satisfied for any train j {1, 2,..., N} on any line l {1, 2,..., m}. TA j ¼ TAj l;1 þ u 1 i¼1 TS j l;i þ u 1 TR j l;i ; u ¼ 2; 3; :::; n l (4) i¼1 TD j ¼ TAj þ TSj ; u ¼ 1; 2; :::; n l (5) According to assumption 3, train stopping times and train arrival times are adjustable, and the running times are can be measured according to subway operation. Decision variables of train rescheduling model are shown as TA j l;1 ; TSj l;1 ; TSj l;2 ; :::,j {1, 2,..., N},l {1, 2,..., m}.

8 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY Constraints Given a time period [t 1, t 2 ] of delayed train j occurred at station u on line l (l + or l ), the above decision variables should adapt to the following constrains (6) (19) to ensure the feasibility of a rescheduled timetable Stopping time constraints For lines with no delay l (l + or l ) l, l {1, 2,..., m}, we obtain constraint (6) to ensure that the stopping time has a feasible range for any station u {1, 2,..., n l } and any train j {1, 2,..., N}. TS min TS j l ;u TSmax l; ðl þ or l Þ l; l f1; 2; mg (6) The minimum boundary of stopping time ensures that all passengers can board and alight the train at station during the stopping time period. Besides, the maximum boundary of stopping time ensures that the train would not stay at station for too long. The values of minimum and maximum boundary can be set by the experiences of the operators and operational practices. For delayed line l (l + or l ), the stopping time constraints are reset below: (a) First, for stopping time TS j of train j at station u, the minimum and maximum boundary of TSj under train delay should be equal to delay time in Equation (7). t delay TS j t delay (7) (b) Second, for stopping time TS j of trainjat the rest of stations u u, u {1, 2,..., n l}, we obtain constraint (8) the same as constraint (6) to ensure that TS j has a feasible range. TS min TS j TSmax (8) (c) Third, for each successive trainj {j + 1,..., N}, the stopping time of these trains should be reset due to the effect of delayed train j. As we can see in Figure 4, the effect of delayed trainj to the successive train j {j + 1,..., N} is listed in three cases below: n o Case 1. if the successive trainj has departed from the first station j j jtd j l;1 t 1, and it is located in the interval between successive station u 1 and u or is exactly stopping at station u, TA j t 1 > TD j 1. Then, there is an extra increased stopping time of this trainj at station u to ensure the Figure 4. Illustration of successive trains on delayed line.

9 1998 Q. ZHEN AND S. JING feasibility of a rescheduled timetable. The minimum and maximum boundary of TS j should be reset in Equation (9). The stopping time of train j at other stations u u, u {1, 2,..., n l } should satisfy the constraint (10) the same as constraint (6). n o n o TS min þ t delay TS j TSmax þ t delay ; j j jtd j l;1 t 1 ; u ujta j t 1 > TD j 1 (9) n o TS min TS j TSmax ; j j jtd j l;1 t 1 ; u u (10) Case 2. if the successive trainj has not departed from the first station, and it is exactly stopping at the first station TD j l;1 > t 1 > TA j l;1, then we suppose that there is an extra increased stopping time of the this train at the station 1 to ensure the feasibility of a rescheduled timetable. The minimum and maximum boundary of TS j l;1 should be reset in (11). The stopping time of trainj at other stations 2, 3,, n l should satisfy the constraint (12) the same as constraint (6). n o TS min þ t delay TS j l;1 TSmax þ t delay ; j j jtd j l;1 > t 1 > TA j l;1 (11) n o TS min TS j TSmax ; j j jtd j l;1 > t 1 > TA j l;1 ; u ¼ 2; 3; ; n l (12) Case 3. if the successive trainj has not arrived at the first station TA j l;1 t 1, it means trainj is stopping at dummy station in this case. Delayed trainjwill not affect the stopping rime of trainj but affect the arrival time of train j 0 at the first station in section Thus, the stopping time of trainj at each station should satisfy the constraint (13) the same as constraint (6). n o TS min TS j TSmax ; j j jta j l;1 t 1 ; u ¼ 1; 2; 3; ; n l (13) In conclusion, for delayed line l (l + or l ), the adjustment of stopping time constraint (7), constraint (9) and constraint (11) due to the effect of train delay are shown in Figure 5. Figure 5. Illustration of stopping time constraints.

10 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY Arrival time constraints For lines l (l + or l ) l, l {1, 2,..., m} with no delay, we obtain the constraint (14) to enforce the arrival time of any train j {1, 2,..., N} atfirst station has a reasonable boundary. Constraint (14) can tighten the feasible space of the resulting optimization models and reduce solution searching time. The j; min j; max values of TAl;1 and TAl;1 can be set by the experiences of the operators or the operational practices of an existing timetable. TA j; min l ;1 TA j l ;1 TA j; max l ;1 ; l ðl þ or l Þ l; l f1; 2; :::; mg (14) For delayed line l (l + or l ), the arrival time constraints of trains at the first station are changed due to train delay and reset below: First, for successive train j {j + 1,..., N} which has arrived at the first station TA j l;1 < t 1, the arrival time constraints at first station are not changed and the same as Equation (15). n o j; min TAl;1 TA j j; max l;1 TAl;1 ; j j jta j l;1 < t 1 (15) Second, for successive trainj {j + 1,..., N} which has not arrived at the first station TA j l;1 t 1, the arrival time constraints at first station are changed and reset in Equation (16), shown in Figure 5. n o j ; min TAl;1 þ t delay TA max l;1 TAj ; l;1 þ t delay ; j j jta j l;1 t 1 (16) Safety headway constraints The station-based safety headway TH j is defined to ensure at most one train can stop at the same station during a small time period. For line l (l + or l ) l, l {1, 2,..., m}with no delay, both the arrival time and the departure time of two successive trainj and train j + 1 at the same station u should satisfy constraint (17) for any trainj {1, 2,..., N 1}, any station u {1, 2,..., n l }. The minimum and maximum boundary of safety headway can be set by the experiences of the operators and operational practices. n o TH min min TA jþ1 j l ;u TAl ;u ; TDjþ1 l ;u TDj l ;u TH max (17) For delayed line l(l + or l ), the maximum boundary of safety headway will be reset in Equation (18) for any train j {1, 2,..., N 1}, any station u {1, 2,..., n} l because the train arrival time and departure time is changed. n TH min min TA jþ1 TA j jþ1 ; TD j TDl u o TH max þ t delay (18) Finally, according to assumption 2, a platform can only accommodate one train at a time and there is no overtaking. Thus, the arrival time TA jþ1 and the departure time TDj of two successive train j and train j + 1 at station u should satisfy constraint (19) for any train j {1, 2,..., N 1}, any station u {1, 2,..., n l }and any line l(l + or l ). TA jþ1 > TDj ; l f 1; 2; m g; u f 1; 2; n lg; j f1; 2; N 1g (19) 3.4. Efficiency analysis In this paper, train rescheduling model under train delay is suitable for different delay situations, no matter what the train delay occurs at an intermediate station or transfer station. Given the delayed situation, the decision variables on lines with no delay should adapt to constrain (6), constrain (14), constrain (17) and constrain (19) shown in section 3.3. For delayed lines, the decision variables should adapt to constraint (7) (13), constraint (15) (16) and constraint (18) (19) shown in section 3.3.

11 2000 Q. ZHEN AND S. JING If train delay occurs at transfer station modeled in Figure 6, more than one line is affected. For delayed line l and l, each direction of each line can be modeled as one separate line and corresponding decision variables is still adapted to the above constraints. In this case, train rescheduling problem can also be described and solved by our model. 4. DYNAMIC PATH CHOICE In urban subway network where lines are highly interconnected, there are many feasible paths from origin to destination, which is composed of a simple listing of nodes along the itinerary with departure time windows. What is more, for nodes which belong to a delayed line, the departure time windows will be updated and adapt to the current delay situation. During the rescheduling process under train delay, passenger path choice and the shortest travel time may be changed because they can choose the path dynamically. Thus, a space time subway network under train delay is proposed in this section. Floyd Warshall algorithm based on space time network is used to obtain the optimal path choice and shortest travel time for each passenger Space time network Space time network has been widely used in transportation network models, which aims to integrate physical transportation networks with travelers time-dependent movements (Tong and Zhou [23]). In this paper, to model the passenger dynamic path choice, a space time subway network is represented by a directed graph G ={S, L}. Physical station u is modeled as space time station s(u, t), u =1,2,, n l for each line l (l + or l ) with time interval σ shown in Figure 7. L ={s(u, t) s(u, t ) s(u, t), s(u, t ) S} is the set of space time links; S ={s(u, t) u = 1, 2,..., n l, t = 0, 2,..., T} is the set of space time station nodes; Passenger path choice decision can be described as a sequence of links traveling at this space time subway network, namely the space time traveling link. Each link represents a process of subway travel, and these links tracing the movement of passenger form a space time path. Here two paths are listed below: Path 1 consider a specific OD from station u to station u + 2 with a certain arrival time t at station u. Path 2 consider a specific OD from station u + 2 to station u with a certain arrival time t +5σ at station u Space time link Boarding link Boarding link is proposed to represent the process that passengers wait to board the next train. The weight of boarding link su; ð tþ s u; TD j represents the minimum waiting time of passengers to board the next train, which can be calculated in (20) by two factors: (a) System time t of passengers arrive at station s(u, t) on line l; Figure 6. Illustration of delay occurs at transfer station.

12 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 2001 Figure 7. Illustration of space time station. (b) Train departure times TD j ; j 1; 2; :::; N at station s u; TDj on line l; n o su; ð tþ s u; TD j ¼ min TD j >0 tjj 1; 2; :::; N (20) Running link Running link is proposed to represent that train run between two successive stations. The weight of running link s u; TD j s uþ 1; TD j þ1 represents train running time between two successive station s u; TD j and station s uþ 1; TD j þ1 on line l, which can be calculated with (22). By comparing departure time TD j with system time t of passengers arrive at station s(u, t), train sequence number n_s(u, t) which passengers board at station s u; TD j can be calculated with (21). n o n su; ð tþ ¼ min jjtd j t (21) s u; TD j s uþ 1; TD j þ1 ¼ TD n su;t ð Þ þ1 TD n su;t ð Þ (22) Dummy link Dummy link is proposed to connect two stations s(u, t),s(n l u +1,t) with opposite directions at the same line l, shown in Figure 8. The process of passenger walking between platforms with opposite directions at the same station can be practical measured. Thus, the weight of dummy link s(u, t) s(n l u +1,t) can be determined Transfer link Transfer link is proposed to represent that passengers walk between different lines. The weight of transfer link represents the passengers walking time between different stations on different lines. According to our assumption 1, the weight of transfer link is a static value for different passengers, and it is determined according to practical measure in each transfer station. For passengers who walk from the station u on line l to the station u on line l, Figure 9 illustrates schematically a process of transfer link, the weight of which listed as follows:

13 2002 Q. ZHEN AND S. JING Figure 8. Illustration of dummy link. (a) walking time T l ;u (b) walking time T l ;u (c) walking time T l ;u (d) walking time T l ;u Figure 9. Illustration of transfer link. from station s(u, t) on line l to station s u ; t þ T l ;u on line l ; su; ð tþ s u ; t þ T l ;u ¼ T l ;u (23) from station s(u, t) on line l to station s n l u þ 1; t þ T l ;u on line l ; su; ð tþ s n l u þ 1; t þ T l ;u ¼ T l ;u (24) from station s(n l u, t) on line l to station s u ; t þ T l ;u on line l ; sn ð l u; tþ s u ; t þ T l ;u ¼ T l ;u (25) from station s(n l u, t) on line l to station s n l u þ 1; t þ T l ;u on line l ; sn ð l u; tþ s n l u þ 1; t þ T l ;u ¼ T l ;u (26) A case process of boarding link, running link and transfer link in urban subway space time network is shown in Figure 10. Passenger starts its path from station s(u, t) on line l with the arrival time t. Passenger waiting time to board the next train j from station s(u, t) to stations u; TD j is TD j t. Train running time between two successive stations u; TD j and s uþ 1; TD j þ1 is TD j þ1 TDj. Passenger walking time from station s(u, t) on line l to station s u ; t þ T l ;u on line l is T l ;u Train delay effect on shortest travel time The shortest travel time for passenger is an important variable in the objective function of the train rescheduling model, and it can be presented with the minimum sum of successive links which connect a listing of nodes from origin to destination.

14 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 2003 Figure 10. Illustration of space time urban subway network. Given the origin and destination, the planned path of passengers with no delay is determined according to space time subway network. If there are no intersections between the planned path and delayed line such as case 1 in Figure 11, it means that train delay will not change the path choices of passengers. On the contrary, if there are overlapping parts between the planned path and delayed line, such as case 2 and case 3 in Figure 11, it means that the passenger choices might be changed due to the effect of train delay. Waiting at the delayed line such as case 2 may not be the optimal choice because there is alternative path connecting origin with destination such as case 3. Figure 11. Train delay effect on path choice.

15 2004 Q. ZHEN AND S. JING Case 1. Planned path has no connectivity with delayed line. Passengers are not affected by train delay and will not change the planned path with no delay from origin to destination. In this case, the shortest travel time will not be changed and equal to the planned value. T t o ;d l ;u ¼ T pla o ;d l ;u (27) Case 2. Planned path has connectivity with delayed line choose the planned path. Passengers are affected by train delay and choose the planned path and wait at the platform on delayed line for the next arrival train. In this case, the shortest travel time under train delay will be longer than the value of planned path. T t o ;d l ;u ¼ T pla o ;d l ;u þt delay (28) Case 3. Planned path has connectivity with delayed line choose the alternative path. Passengers are affected by train delay and choose the alternative path with no delay. In this case, the shortest travel time under train delay is equal to the travel time of the alternative path. T t o ;d l ;u ¼ T alt o ;d l ;u (29) Case 4. Planned path has connectivity with delayed line-only one choice. Passengers are affected by train delay, and there is no other alternative path and the passengers have to wait at the origin station for the next arrival train. In this case, the shortest travel time of planned path due to train delay will be longer than the planned value. T t o ;d l ;u ¼ T pla o ;d l ;u þt delay (30) In conclusion, for each passenger, train delay effect on the shortest travel time depends on the specific distribution of the planned path and the connectivity of subway network. The actual increased travel time of passengers due to train delay might be smaller than delayed time Space time-based algorithm As we can see (Dreyfus [24]), the shortest path algorithm based on fixed weight of links have already been proved to be solved by traditional algorithm such as Dijkstra effectively. However, in a schedulebased urban subway network, the weight of links is time dependent, and the passengers will choose the path dynamically according to the real-time timetable. Thus, traditional algorithm is not adapted in this study. In computer science, the Floyd Warshall algorithm is an algorithm for finding shortest paths in a weighted graph with positive or negative edge weights (Hougardy [25]). A single execution of the algorithm will find the summed weights of the shortest paths between all pairs of vertices. Based on space time network, Floyd Warshall algorithm is used to calculate the shortest paths which can update the time-dependent weight of links, shown in Table I. 5. GENETIC ALGORITHM Train rescheduling problem in urban subway network has been proved belonging to the NP-hard class (Ibarra-Rojas et al. [26]). In this paper, train rescheduling model is a non-convex programming problem with objective function (1) and constraints (6) (19), where the non-convexity is due to the passenger travel time in objective function which is calculated by the process of passenger dynamic path choice based on space time algorithm.

16 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 2005 Table I. Space time Floyd Warshall algorithm. Input: Directed graph G ={S, L} S(G)={s(u, t) u = 1, 2,..., n l, t = 0, 2,..., T} is the set of space time station nodes; S(G) = {1, 2,..., n}, n = u t is the number of space time station nodes in network; L(G)={s(u, t) s(u, t ) s(u, t), s(u, t ) S} is the set of space time links; s(u, t) s(u, t )is calculated with the weight function (21) (27); Output: An n n matrix d(:) such that d[s(u, t), s(u, t )] donates the shortest travel time of the path from space time station s(u, t) to space time station s(u, t ); 1 d[s(u, t), s(u, t )] : =, s(u, t) s(u, t ) 2 d[s(u, t), s(u, t)] : = 0, s(u, t) 3 d[s(u, t), s(u, t )] : = s(u, t) s(u, t ), s(u, t) s(u, t ) L 4 for s(u, t): =1 to n do 5 for s(u, t ): =1to n do 6 for s(u, t ): =1 to n do 7 if d[s(u, t ), s(u, t )] > d[s(u, t ), s(u, t)] + d[s(u, t), s(u t )] then d[s(u, t ), s(u, t )] = d[s(u, t ), s(u, t)] + d[s(u, t), s(u, t )] 8 for s(u, t): =1 to n do 9 ifd[s(u, t), s(u, t)] < 0 then return ( graph contains a negative cycle ) Several approaches, e.g., pattern search (Hooke and Jeeves [27]), sequential quadratic programming3 (SQP) (Boggs and Tolle [28]), mixed-integer (non) linear programming (Wang et al. [29]), iterative convex programming (Wang et al. [30]) and evolutionary algorithms (Bocharnikov et al. [31]; Ding et al. [32]; Yang et al. [33]) can be applied to solve train rescheduling problem. In recent years, intelligence algorithms such as genetic algorithm, simulated annealing algorithm and artificial neural network algorithm are commonly used to solve the problem. Among these algorithms, the genetic algorithm (GA) is mostly used and has been used by Shafahi and Khani [34], Hassannayebi et al. [35], Arbex and Cunha [36], Kang et al. [37], Kang et al. [38], Wu et al. [39] and Wang and Tang [40]. The idea of GA is to simulate the mechanism of living beings evolving and natural selection. In general, the procedure for GA includes the following: (i) generate a certain number of chromosomes randomly; (ii) evaluate the quality of those chromosomes by the fitness function; (iii) obtain fine chromosomes via selection, crossover and mutation operations; and (iv) terminate when the difference in the best solutions between two generations is less than a given parameter or a maximum number of iterations are reached. In this paper, GA is used to solve the optimization model. Decision variables (arrival times and stopping times) which satisfy the constraint (6) (19) under train delay in section 3 forms the chromosome of GA. The first population is created according to the initial timetables under train delay. A sample chromosome of the urban subway network timetable is shown in Figure 12. The objective function in Equation (1) is chosen as the fitness function of GA. The process of the GA to solve the rescheduling model is described as follows. Step 1 Initialization. 1.1 Input the initial GA parameters: population size I, initial generation n = 0, the maximum number of generations N. 1.2 Input the basic parameters of urban subway network G ={S, L},T,t 1,t 2, minimum and maximum boundaries of decision variables TAl;1,TAl;1, TS min,ts max,th min,th max. j; min j; max 1.3 Input the passenger time-dependent demand f t o ;d l ;u and passenger impatience function. Figure 12. A sample chromosome of the network timetable.

17 2006 Q. ZHEN AND S. JING 1.4 Initialize a parent chromosome TA j l;1 ; TS j l;1 ; TSj l;2 ; :::; j {1, 2,..., N}, l {1, 2,..., m}. 1.5 Take a mutation operation to the parent chromosome and generate another chromosome within constraints (6) (16) of the decision variables. Adopt crossover operation to these two chromosomes and generate two children chromosomes. 1.6 Check the chromosomes. If they satisfy constraint (17) (19), then turn to Step 1.7. Otherwise, return to Step Repeat 1.5 until the population size is equal to I. Step 2 Creating and selecting a new population. 2.1 Calculate the shortest travel time T t o ;d l ;u for each pair of OD f t o ;d l ;u by space time-based algorithm in section Calculate binary variable η t o ;d l ;u according to Equation (2) in section Calculate each fitness function value F(i) of chromosomes in the current population.! Fi ðþ¼ α η t o ;d l ;u f t o ;d l ;u!þ β 1 η t o ;d l ;u f t o ;d l ;u T t o ;d l ;u t ½0;TŠ o O d l ;u D t ½0;TŠ o O d l ;u D 2.4 Calculate probability using fðþ¼fi i ðþ= I Fi ðþand adopt roulette wheel regulation to i¼1 choose the chromosomes to form the parent population. 2.5 Adopt roulette wheel regulation, and reproduce a new population by copying the chromosomes according to f(i). 2.6 Obtain another I new chromosomes by crossover operation. 2.7 With these 2 I chromosomes, calculate the fitness function value of the feasible chromosomes and select Ichromosomes as an elite population. 2.8 Select the chromosomes to mutate. 2.9 If the chromosome is feasible, turn to next; otherwise, delete it Calculate each fitness function value in the current population. Step 3 Update and judge. 3.1 Update n = n If n = N, stop. Otherwise, return to Step CASE STUDY Urban subway network in Beijing is developing with a rapid increasing number of lines these years. At the same time, the planned subway timetable is sometimes disrupted by train delays. Thus, Beijing subway is chosen as target in this case study Impatience time analysis under train delay To obtain adequate suitable data from passengers, a subsequent web-based Stated Preference (SP) survey was carried out within the same group of passengers who often take subway in Beijing. In this SP survey, passengers were classified by their trip purposes when they took subway. In condition of train delay, each passenger would face different scenarios with a sequence values of travel times (20 min, 30 min, 40 min, 50 min and 60 min) from origin to destination and choose their own impatience time. A total of 317 valid questionnaires were collected in this SP survey. The frequency distribution histogram between the impatience time and travel time is shown in Figure 13, the form of which would translate with different travel time values. Power function is used to express relationship between the planned travel time T pla o ;d l ;u and the impatience time T impatience according to distribution histogram, shown in Equation (31).

18 TRAIN RESCHEDULING MODEL WITH TRAIN DELAY 2007 Figure 13. Frequency distribution histogram between the impatience time and travel time. General model: b T impatience ¼ a T pla o ;d l ;u þ c (31) Coefficients (with 95% confidence bounds): a = 32.36, b = , c = Goodness of fit: SSE: 347.7, R-square: , Adjusted R-square: , RMSE: As we can see from the results of fitted function: (a) increasing cumulative effect: the longer travel time from origin to destination, the longer impatience time passengers are willing to wait. (b) marginal decreasing effect: with the increase of travel time, the growth of corresponding impatience time is slowing Initial input data and parameter settings The proposed train rescheduling model under train delay and GA is illustrated with simplified subway network in the center of Beijing shown in Figure 14, which includes six lines (bi-direction), 14 transfer stations and 11 intermediate stations. The real number of stations in these lines is 14 transfer stations and 27 intermediate stations. Compared with real Beijing subway network, some intermediate stations of simplified version are omitted and the passenger demands of which are merged to their adjacent transfer stations. In condition of train delay, passenger path choice from origin to destination will only change at transfer station. Thus, merging passenger demands of intermediate stations to their adjacent transfer stations will not affect the effectiveness and validity of train rescheduling model. Figure 14 provides the up-train direction and down-train direction of each line, which indicates the train operation. Kang et al. [34] proposed that the subway network capacity to neutralize the effects of train delay is in the range of 15 min 20 min. In this paper, we assume that there is 20-min train delay time on a specific station shown in Figure 14. Transfer times of passengers between different lines are determined according to the practical measure. In this case, parameters TH min,th max,ts min and TS max are taken as 2 min, 10 min, 1 min and 3 min, respectively. For the same segment on each line, running times for each train have the same values on both up-direction and down-direction, we measure running

19 2008 Q. ZHEN AND S. JING Figure 14. A case of Beijing urban subway network. times based on actual subway timetable shown in Table II. Under the effect of 20-min delay time on up-train direction of line 1, the stopping times and arrival times of successive trains are adjusted to ensure the constraint (6) (16) station by station. Minimum and maximum train stopping times and arrival times on up-train direction of line 1 are shown in Tables III V. Table II. Running times between two successive stations. Segment Segment Segment Segment Segment line 1 (1 2) 2 min line 2 (2 3) 5 min line 2 (9 10) 3 min line 4 (4 5) 3 min line 6 (1 2) 2 min line 1 (2 3) 3 min line 2 (3 4) 4 min line 2 (10 1) 3 min line 4 (5 6) 2 min line 6 (2 3) 3 min line 1 (3 4) 4 min line 2 (4 5) 3 min line 3 (1 2) 2 min line 5 (1 2) 2 min line 6 (3 4) 3 min line 1 (4 5) 5 min line 2 (5 6) 3 min line 3 (2 3) 3 min line 5 (2 3) 4 min line 6 (4 5) 4 min line 1 (5 6) 3 min line 2 (6 7) 4 min line 4 (1 2) 2 min line 5 (3 4) 3 min line 6 (5 6) 3 min line 1 (6 7) 2 min line 2 (7 8) 3 min line 4 (2 3) 3 min line 5 (4 5) 3 min line 6 (6 7) 2 min line 2 (1 2) 5 min line 2 (8 9) 3 min line 4 (3 4) 3 min line 5 (5 6) 2 min Table III. Minimum and maximum stopping times on the up-direction of line 1. Train Station number Station 1 Station 2 Station 3 Station 4 Station 5 Station 6 Station7 1 2 min/2 min 2 min/2 min 2 min/2 min 15 min/15 min 2 min/2 min 2 min/2 min 2 min/2 min 2 1 min/3 min 1 min/3 min 16 min/18 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 3 1 min/3 min 16 min/18 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 4 16 min/18 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 5 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min 1 min/3 min Table IV. Earliest and latest arrival times at the first station (up-train direction). Train Line 1 line 2 Line 3 Line 4 Line 5 Line :00/12:00 12:01/12:01 11:59/11:59 12:01/12:01 11:59/11:59 12:01/12: :02/12:06 12:03/12:07 12:03/12:03 12:03/12:07 12:01/12:05 12:05/12: :06/12:10 12:07/12:11 12:07/12:07 12:07/12:11 12:05/12:09 12:09/12: :10/12:14 12:11/12:15 12:11/12:11 12:11/12:15 12:09/12:13 12:13/12: :34/12:38 12:15/12:19 12:15/12:15 12:15/12:19 12:13/12:17 12:17/12: :19/12:23 12:19/12:19 12:19/12:23 12:17/12:21 12:21/12: :23/12:27 12:23/12:23 12:23/12:27 12:21/12:25 12:25/12: :27/12:31 12:27/12:27 12:27/12:31 12:25/12:29 12:29/12: :31/12:35 12:31/12:31 12:31/12:35 12:29/12:33 12:33/12:33