Last train scheduling for maximizing passenger destination reachability in urban rail transit networks

Transportation Research Part B 129 (2019) 79–95

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Last train scheduling for maximizing passenger destination reachability in urban rail transit networks Yu Zhou a,b, Yun Wang a,b, Hai Yang b, Xuedong Yan a,∗ a

MOT Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China b Department of Civil and Environmental Engineering, The Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 1 February 2019 Revised 21 July 2019 Accepted 3 September 2019

Keywords: Urban rail transit Last train scheduling Passenger assignment Destination reachability Mixed integer linear programming

a b s t r a c t As urban rail transit (URT) systems usually do not operate for the whole day, the last train service offers the last daily chance for late-night passengers to utilize URT services to reach their target destination stations. This paper formally introduces and models the destination-reachability based last train timetabling problem (DR-LTTP in abbreviation) in URT networks, which involves both the last train timetabling and the passenger assignment. The DR-LTTP is formulated as a mixed integer linear programming and can be solved by existing commercial optimization software. The model is illustrated with a simple numerical example on a minimum spanning tree network, and comparison experiments are conducted between DR-LTTP model and station-transferability based last train timetabling problem (ST-LTTP in abbreviation). Finally, a real case study with Beijing URT network is conducted to test the performance of our model. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Passenger mobility, especially in large cities worldwide, relies heavily on urban rail transit (URT). For example, in Hong Kong, over 36% of passengers’ daily trips involved its MTR (mass transit railway) system in 2016; provision of convenient, efficient, comfortable and reliable services is of great importance. Research on URT over the past serval decades has overwhelmingly focused on planning and operations, such as network design, line planning, timetabling and rescheduling (Farahani et al., 2013; Schöbel, 2012; Cacchiani et al., 2014). Recently, scheduling of last train services in URT networks has received attentions, which is always a practical issue because URT systems usually do not operate for the whole day. The last train service offers the last daily chance for late-night passengers to utilize URT services. The studies conducted so far on the last train service issue focus on the last train service coordination for passenger transfer at stations, which is termed as the last train timetabling problem (LTTP) (Kang et al., 2015a). LTTP can be regarded as a special type of transit timetabling problem (TTP). Generally, the traditional TTP aims at determining an optimal timetable for each transit line by establishing train departure and arrival times at each station, subject to capacity constraints as well as some necessary operational constraints (Laporte et al., 2017). Different objectives are considered in the TTP, such as passenger waiting time (Niu and Zhou, 2013; Barrena et al., 2014), robustness (Cacchiani and Toth, 2012; Liu and Dessouky, 2019; Robenek et al., 2018), energy consumption (Wang and Goverde, 2019), travel time ∗

Corresponding author. E-mail address: [email protected] (X. Yan).

https://doi.org/10.1016/j.trb.2019.09.006 0191-2615/© 2019 Elsevier Ltd. All rights reserved.

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(Warg et al., 2019) and operator profits (Cacchiani et al., 2016). In general, these studies on TTP are concerned with normal time of day operations without considering the impacts of last train services and thus reachability is not an issue. The major difference of the LTTP on URT networks from the traditional TTP lies in the passengers’ transferability at stations, which is governed by the arrival and departure times of last trains at all transfer stations. Kang et al. (2014, 2015a) treated the LTTP as a last train transfer coordination problem (hereinafter, it is referred to as station-transferability based LTTP or ST-LTTP in abbreviation); a non-linear programming model is developed to maximize the transfer connection headways, which are integrated by the transfer connection time and transfer waiting time at transfer stations; and the model is then solved by a genetic algorithm. Kang et al. (2015b) proposed a rescheduling model for last trains by considering train delays caused by incidents that occurred in train operations. The model seeks a rescheduled timetable that is close to the original timetable, to minimize the running time and the dwell time and to maximize the average transfer redundant time and transferability. Furthermore, Kang and Meng (2017) formulated the LTTP as a mixed-integer linear programming problem and proposed an effective two-phase decomposition method, which decomposes the original problem into two sub-problems of smaller sizes and is able to find global optimum for large-scale problems. Xu et al. (2018) proposed a bi-objective method for the LTTP, which aims to minimize the total passenger waiting time as well as the deviation to the original timetable, then an ε -constraint method was put forward to obtain the approximate Pareto optimal solutions. Kang et al. (2019) addressed the LTTP and bus bridging service problem, which aims to provide stranded passengers in the URT system with bus service to reach their destinations. Nonetheless, the above methods proposed so far for the LTTP aim at reducing transfer waiting time and increasing successful transfers at stations. These methods do not guarantee reachability of passengers to their target destination stations; especially for those passengers who need multiple transfers, which should be a primary concern of the LTTP (hereinafter, the problem is referred to as destination-reachability based LTTP or DR-LTTP in abbreviation). Evidently, reachability requires transferability at stations, but more successful transfers at stations does not always imply a higher reachability especially in large-scale URT networks. More importantly, in contrast with the ST-LTTP, which is independent of passenger movement patterns, the DR- LTTP must take into account passengers’ path choice or assignment on the URT networks. Clearly, the DR-LTTP is different from traditional concept of accessibility considered in urban network design problem (Tong et al., 2015; Di et al., 2018). The accessibility-based network design problem is to improve individuals’ accessibility or the ease of reaching desired activities centers or nodes in terms of travel time and cost. The reachability in the URT system near the closure time directly pertains to the last train timetable or availability of services; some passengers may not reach their destination station and have to change to other transportation modes. Nevertheless, both accessibility between activity locations and reachability to destinations in a URT network requires consideration of passenger assignment or path choice, which is usually included in the traditional transit network design problem (Liu and Zhou, 2016), operational line planning problem (Canca et al., 2017; Guan et al., 2006) and passengers flow state estimation (Shang et al., 2019). It should be emphasized destination reachability in the LTTP is ensured as long as one of the reasonable passenger path is feasible. We note that a recent study was carried out independently by Chen et al. (2019) on the timetable synchronization of last trains for destination reachability on URT networks. While both dealing with the DR-LTTP, the formulations of our model and the model by Chen et al. (2019) follow two totally different lines of thoughts. Instead of directly tracing arrival times of passengers at each transfer station in Chen et al. (2019), we consider last train timetabling and passenger path choice/assignment simultaneously and use a number of logical relations to represent directional transferability and destination-station reachability in a general URT network. Then such logical relations are converted into equivalent linear constraints. Consequently, the DR-LTTP is skillfully formulated as a mix integer linear programming (MILP) model, which is solvable for large-size problems with standard solvers such as CPLEX and GAMS. The remainder of the paper is organized as follows. In the next section, an example is presented to illustrate the STLTTP examined before and the new DR-LTTP of interest. Section 3 formulates the DR-LTTP as a MILP. Section 4 elucidates the model with a simple numerical example on a minimum spanning tree network. Section 5 further tests the model with Beijing URT network together with three network simplification methods. Section 6 concludes the paper and highlights avenue for future research.

2. An illustrative example and problem definition Consider a simplified URT network shown in Fig. 1, consisting of four lines and eleven stations, with B, C, E and G being transfer stations. Each column of Table 1 displays the arrival and departure time of last train at/from the transfer stations in each line. We first consider a last passenger who aims to travel from Station A to Station F. This passenger needs to make a transfer at Station B from the southbound RL to the eastbound GL. But, with the given timetable, this transfer is infeasible or this passenger cannot reach his destination station, because the last train of the eastbound GL already departed from Station B at 22:38 before the arrival of the last train of southbound RL at 22:40. We now consider another passenger who aims to travel from Station A to Station K. This passenger needs to make two transfers at Stations B and E to reach his destination station K. Obviously, Station A to Station K is reachable if and only if the two transfers can be met. But, with the given last train timetable, the second transfer is infeasible or this passenger fails to reach his destination station. This case shows that station-transferability does not imply destination-reachability.

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Table 1 Timetable of last train service. Line

RL-U

RL-D

GL-U

GL-D

BL-U

BL-D

YL-U

YL-D

Dep. Arr. Dep. Arr. Dep. Arr.

A 22:35 B 22:40 B 22:42 C 22:45 C 22:47 D 22:55

D 22:48 C 22:55 C 22:57 B 23:00 B 23:03 A 23:08

E 22:31 B 22:35 B 22:38 G 22:43 G 22:45 H 22:53

H 22:30 G 22:36 G 22:38 B 22:43 B 22:45 E 22:47

I 22:45 C 22:49 C 22:52 G 22:58 G 23:00 J 23:04

J 22:36 G 22:40 G 22:42 C 22:48 C 22:51 I 22:55

K 22:32 E 22:38

E 22:46 K 22:51

Fig. 1. URT network example.

Destination station reachability may also need consideration of the train(s) preceding to the last train of a line. For example, we again consider the last passenger who failed to travel from Station A to Station K. If, however, the second last (penultimate) train of GL leaves station B at 22:42 and arrive at station E at 22:45, then this passenger can take the last train of RL from A to B (arrive at station B at 22:40) and catch up with the penultimate train of GL, and then make a transfer at station E to catch up with the last train of YL, and finally reach his destination station. This case shows that we should take the last several trains of a line into consideration in the LTTP. Finally, we consider a passenger who aims to travel from Station A to Station G. There are normally two feasible paths for this passenger: A→B→F→G with a total travel time of 14 (min) and A→B→C→G with a total travel time of 19 (min). But with the given last train time table, the first shorter path is not feasible and this passenger has to take the second alternative detour path. This case shows that passenger assignment with a maximum allowable detour time must be considered in dealing with the DR-LTTP. With these observations, the DR-LTTP can be defined as follows: For a given set of passenger groups by origin-destination (OD) pairs, a set of candidate paths for each passenger group, and a timetable of non-last trains in a URT network, the DRLTTP is to determine the last train timetabling (departure time from terminal stations, section running time and station dwelling time within certain ranges) and passenger path choice to maximize the number of passengers who can reach their target destination stations by taking the last train services from their entry stations (passengers’ destination-reachability). 3. The destination-reachability based LTTP In this section, we give a detailed description of the elements involved in the DR-LTTP. We first introduce the directional station-transferability and passenger assignment, which are then used to measure destination-reachability, and finally we formulate the DR-LTTP as a single mixed integer linear programming problem. Before the model formulation, some assumptions are made as follows: Assumption 1. we assume that the OD demand of different passenger group is given. This data can be obtained by the historic passenger demand through Automatic Fare Collection (AFC) systems (Chen et al., 2019). Assumption 2. the transfer time is known and fixed for all passengers (Kang et al., 2015a). The transfer time is the minimum required time interval for a passenger to get off an arrival train and walk to get on next line departure train at a transfer station, which can be gained through field survey.

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Assumption 3. the capacity of last train service is sufficient to accommodate all passengers. This assumption is reasonable because the late night passenger demand is relatively low. 3.1. Directional station-transferability Let L be the set of all directional train services in the URT network. Each train li ∈ Ldenotes the i-th last train service of line l, and train services are consecutively numbered in reverse order (i.e. the train where i = 1 represents the last train service). li ∈ L is characterized by a tuple (ρ + (l),ρ − (l)), where ρ + (l) indicates the line name or number and ρ − (l) denotes the line direction (e.g., Beijing URT line 1 and eastbound or westbound direction). A station can be either a transfer station or a non-transfer station. The DR-LTTP of interest is mainly concerned with the transfer stations where passenger transfers take place. Let S be the set of all transfer stations in the URT network, and let L(s) be the set of trains serving station s ∈ S. In order to represent directional transfer, for each transfer station s ∈ S, each train li ∈ L(s) is divided into two trains: an inbound train li+ (s ), on which passengers arrive at transfer station s and an outbound train li− (s ), on which passengers departure from the transfer station s, respectively. Let L+ (s) and L− (s)be the set of all inbound and outbound trains at transfer station s ∈ S, respectively. Obviously, a transfer can be defined as follows: a passenger arrives at a transfer station s ∈ S by taking an inbound train li + (s) of line l and leaves the station by taking an outbound train l − j (s) of another line l . Note that a passenger traverses a transfer station from the inbound to the outbound train of the same line is not regarded as a transfer. Thus, the set of all possible directional transfers at transfer station s ∈ S is given by D(s ) = {(li , l  j )(s )|li ∈ L+ (s ), l  j ∈ L− (s )&ρ + (l ) = ρ + (l  )}, where (li ,l j )(s)denotes the directional transfer from the i-th last inbound train of line l to the j-th last outbound train of line l at transfer station s. In the URT last train service, some directional transfers in the set of D(s) at a transfer station s may be infeasible. Feasibility of a directional transfer (li ,l j )(s) ∈ D(s) depends on the arrival time of inbound train li+ (s ) and departure time of outbound train l j− (s ) at the transfer station s ∈ S. The arrival time of the last train at a transfer station is determined by its departure time from the terminal station, running time from the terminal station to the transfer station, and the total station dwell time before reaching the transfer station; and the departure time is further determined by the dwell time at the transfer station. According to the analysis above, we need to determine the departure time of the last train, the running time and the dwell time at transfer station. So decision variable tl1 is introduced to denote the departure time of the last train on line l from its terminal station, decision variable tl1 ,s represents the running time of the last train from the transfer station s-1 to transfer station s on line l, and decision variable l1 ,s is the dwell time of the last train at station s on line l. Another dep , 1 ,s

two auxiliary decision variables, tlarr,s and tl 1

transfer station s ∈ S. Clearly, tlarr,s and 1

s 

tlarr = tl1 + 1 ,s

tl1 ,k +

k=1

tldep = tl1 + ,s 1

s  k=1

s−1 

dep tl ,s 1

are introduced to respectively represent its arrival/departure time at/from the

are given by

l1 ,k ,s ∈ S, l1 ∈ L(s )

(1)

k=1

tl1 ,k +

s 

l1 ,k ,s ∈ S, l1 ∈ L(s )

(2)

k=1

In order to provide passengers with enough time to get off/on the train, a minimum dwell time should be guaranteed. On the other hand, excessive dwell time shouldn’t be allowed with the consideration of service quality. So, constraint (3) should be meet.

¯ s ∈ S, l1 ∈ L(s ) l ,s ≤ l1 ,s ≤  l1 ,s

(3)

1

Due to the infrastructure limitation, there is an upper bound for the running speed of train, so the running time should larger than a minimum value. Also, the running time can’t be too long or the service quality will be decreased. Therefore, constraint (4) should be observed.

tl

1 ,s

≤ tl1 ,s ≤ t¯l ,s s ∈ S, l1 ∈ L(s )

(4)

1

Obviously, to ensure a feasible directional transfer, the departure time of the outbound train should be later than the arrival time of the inbound train by a minimum amount of time required for passengers to transfer. Namely, tlarr + t(trans ≤ ,s l ,l  )(s ) i

dep , j ,s

tl 

i

j

where t(trans denotes the minimum passenger transfer time from train li to train l j at transfer station s. The following l ,l  )(s ) i

j

transferability matrix [x(li ,l  j )(s ) ] is introduced to represent the feasibility of a directional transfer (li ,l j )(s) ∈ D(s), which denotes the transfer from the i-th last train of line l to the j-th last train of line l at transfer station s, where the auxiliary binary variable x(li ,l  j )(s ) is defined below.



x(li ,l  j )(s ) =

1

if tlarr + t(trans ≤ t dep  ,s l ,l  )(s )

0

otherwise

i

i

j

l j ,s ,

s ∈ S, (li , l j )(s ) ∈ D(s )

(5)

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83

Fig. 2. Path generation process.

where x(li ,l  j )(s ) = 1 means a feasible directional transfer and 0 otherwise. The above logical relationship can be equivalently expressed by the following inequalities:







arr trans M · x(li ,l  j )(s ) − 1 ≤ tldep  ,s − tli ,s − t (li ,l  j )(s ) ≤ M · x (li ,l  j )(s ) − ε , s ∈ S, (li , l j )(s ) ∈ D (s ) j

(6)

where M is a sufficiently large positive number and ε is a sufficiently small positive number. 3.2. Passenger assignment Now we move to consider passenger path assignment. We begin with the usual passenger path assignment on the URT networks where directional transfer is always feasible. Let K be the set of passenger origin-destination pairs or passenger groups, and Pk be the set of pre-identified reasonable paths available for passenger group k, k ∈ K. Here, the path is denoted by a sequence of origin station, directional transfers and destination station. We assume the feasible paths are predefined, which can be firstly ex ante determined using following method: Firstly, the paths can be determined using the kth shortest path method (Guan et al., 2006) and only consider the stations sequence which the passengers travel with a due consideration of a maximum allowable detour time. Then every path can be divided into several paths based on which train the passenger will actually take and which transfer the passenger will make. Fig. 2 illustrates this process, suppose that a group of passengers have chosen a path (s→A→B→e) according to the kth shortest path method. Then the passengers have i • i paths according to the directional transfers which they will make. Note that we don’t consider whether the directional transfer is feasible in every path, infeasible path will be eliminated in our later model. A binary decision variable yp is introduced to denote whether passengers in group k, k ∈ K, are assigned to path p, p ∈ Pk :



yp =

1 0

if passengers in group k choose path p , otherwise

p ∈ Pk , k ∈ K

(7)

A passenger is free to choose any one of feasible and reasonable paths to reach the destination, and so passengers in the same group may take different paths to get to their destinations. For simplicity, we further assume that passengers in the same group choose the same path, which means that



y p = 1, k ∈ K

(8)

p∈Pk

3.3. Destination-reachability For the DR-LTTP, a usual reasonable path may become infeasible or a target destination station may not be reachable due to service unavailability after last trains or some infeasible station transfers. To describe the destination-reachability p p problem we introduce a path-transfer incidence matrix [α(l ,l  )(s ) ] where α(l ,l  )(s ) = 1 if path p, p ∈ Pk ,k ∈ K, contains i

j

directional transfer (li ,l j )(s) ∈ D(s) at transfer station s ∈ S, and 0 otherwise.



α(pl ,l  )(s) = i

j

1 0

i

j

if path p contains directional transfer (li , l  j )(s ) , p ∈ Pk , k ∈ K, s ∈ S otherwise

(9)

As a result, the destination-reachability in the DR-LTTP can be measured by the transferability matrix together with the path-transfer incidence matrix and passenger assignment. Let zk be an indicator of the destination-reachability of passenger group k, k ∈ K, by the last train service, which is defined as follows:



k

z =

1 0

if x(li ,l  j )(s ) − otherwise



p∈Pk

α(pli ,l  j )(s) y p ≥ 0, ∀(li , l  j )(s ) ∈ D(s ), s ∈ S

,k ∈ K

(10)

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In the above equation, zk = 1 means passenger group k can reach their target destination station. Namely, a passenger can reach his destination station if and only if all directional transfers contained in his chosen path are feasible. To represent the above logical relationship in a linear manner, we first introduce a new auxiliary binary variable zk(l ,l  )(s ) i

defined as follows:



if x(li ,l  j )(s ) −

1

zk(li ,l  j )(s ) =

0

 p∈Pk

α(pli ,l  j )(s) y p ≥ 0

otherwise

, k ∈ K, (li , l j )(s ) ∈ D(s ), s ∈ S

j

(11)

Then, the logic relationship (10) for any passenger group k ∈ K, is equivalent to



k

z =





1

if

0

otherwise

s∈S (li ,l  j )(s )∈D(s )

zk(l ,l  )(s ) = ND i j

, k∈K

(12)

where ND is the total number of directional transfers in the URT network, or the total number of inequalities in (10). In other word, zk = 1 if all inequalities in (10) hold and zk = 0 otherwise. Consequently, the above logical relationships (11) and (12) can be respectively expressed by the following inequalities:

zk(li ,l  j )(s ) − 1 ≤ x(li ,l  j )(s ) −







M · zk − 1 ≤



 p∈Pk

α(pl ,l  )(s) y p ≤ M · zk(li ,l  j )(s) − ε , k ∈ K, (li , l j )(s ) ∈ D(s ), s ∈ S

(13)

zk(li ,l  j )(s ) − ND ≤ zk − ε , k ∈ K

(14)

i

s∈S (li ,l  j )(s )∈D(s )

j

where M is again a sufficiently large positive number and ε is a sufficiently small positive number. 3.4. The optimization model We are now ready to present our model for the DR-LTTP, which is formulated as the following mixed-integer linear programming problem: max



qk z k

(15)

k∈K

subject to s 

tlarr = tl1 + 1 ,s

tl1 ,k +

k=1

tldep = tl1 + ,s 1

s 

s−1 

l1 ,k ,s ∈ S, l1 ∈ L(s )

(16)

k=1

tl1 ,k +

k=1

s 

l1 ,k ,s ∈ S, l1 ∈ L(s )

(17)

k=1

l ,s ≤ l1 ,s ≤ l1 ,s s ∈ S, l1 ∈ L(s )

(18)

≤ tl1 ,s ≤ t¯l ,s s ∈ S, l1 ∈ L(s )

(19)

1

tl

1 ,s

1





arr trans  M · x(li ,l  j )(s ) − 1 ≤ tldep  ,s − tli ,s − t (li ,l  j )(s ) ≤ M · x (li ,l  j )(s ) − ε , (li , l j )(s ) ∈ D (s ), s ∈ S j



y p = 1, k ∈ K

(21)

p∈Pk

zk(li ,l  j )(s ) − 1 ≤ x(li ,l  j )(s ) −





M · zk − 1 ≤



(20)





α(pli ,l  j )(s) y p ≤ M · zk(li ,l  j )(s) − ε , k ∈ K, (li , l j )(s ) ∈ D(s ), s ∈ S

(22)

zk(li ,l  j )(s ) − ND ≤ zk − ε , k ∈ K

(23)

p∈Pk

s∈S (li ,l  j )(s )∈D(s )

tlearly ≤ tl1 ≤ tllate ,l ∈ L 1

(24)

1

where qk is the number of the last train passengers in group k ∈ K,

early tl 1

and tllate are the earliest and latest departure time 1

of the last train of line l ∈ L from the terminal station. The objective function (15) is to maximize the total number of passengers who can reach their target destination stations by the last train service; as illustrated above, constraints (16) to (20) form the directional station-transferability matrix; constraint (21) represents passenger assignment; constraints (22) and (23) form the destination-reachability matrix; finally, the last constraint (24) is an operational constraint newly added here.

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3.5. DR-LTTP and ST-LTTP The ST-LTTP proposed by Kang et al. (2015a, 2017) aims to enhance the station-transferability of the last train service network. The available transfer connection time is used to judge the station-transferability which can be expressed by Eq. (25). arr trans T(l1 ,l  1 )(s ) = tldep  ,s − tl1 ,s − t (l1 ,l  1 )(s )

(25)

1

If the above buffer time is greater than zero, then passengers can make a transfer through the last train service or the directional transfer is feasible. The ST-LTTP model is then to seek the solution to maximize the number of successful directional transfers. While the DR-LTTP aims to determine a last train timetable to make as many passengers as possible reach their target stations. The DR-LTTP has the following advantages over the ST-LTTP. (1) Station-transferability is independent of passenger demand. Whether a directional transfer at a station is feasible solely depends on the arrival time of the last train on a line and the departure time of the last train on another line. (2) For a passenger whose path contains more than one transfer, reachability to his/her target station depends on the feasibility of all directional transfers along the path. One infeasible directional transfer will lead to the unreachability of a whole path, and thus more feasible directional transfers do not necessarily imply more passengers can reach their destination stations. (3) The DR-LTTP involves passengers’ path choice. Multiple paths are generally available in a large URT network and a passenger may take an alternative path to reach his destination station even the shortest path is infeasible. (4) The DR-LTTP takes the i-th last train into consideration. If a passenger’s path contains more than one transfer, the first train passenger take must be the last train, while the next train after transfer may not be the last train, so the last several train should be considered. 4. Numerical analysis on a test network In this section, numerical analysis is carried out to test the feasibility of the proposed method based on a simple test network. In the case of a minimum spanning tree network, each passenger has only one defined path to get to the destination. Thus, the path assignment variable yp is held constant and solving the model becomes much simpler. Taking the advantage of the simple structure of minimum spanning tree networks, we illustrate how the i-th (i ≥ 2) last train affect the outcome of the DR-LTTP. Finally, we make a comparison between DR-LTTP with ST-LTTP. 4.1. A simple example with a minimum spanning tree network As shown in Fig. 3, the test URT network consists of three physical lines and 15 stations. Station M and Station N are two transfer stations: Line 1 crosses Line 2 and Line 3 at Station M and Station N, respectively. We take last three trains into consideration, so there are totally 144 directional transfers (2 (stations)∗ 8 (directions) ∗ (3∗ 3 train combination)), and 210 passenger groups (origin-destination pairs). We need to coordinate the last train timetable for the six URT lines so as to maximize the number of passengers successfully reaching their destinations. Other parameters are set as follows: the number of passengers in each group is generated randomly within the range from 1 to 80; Table 2 lists the bound of running time, dwell time as well as the minimal and maximal last train departure time from the terminal stations; the walking time for all transfers is set to 2 min.

Fig. 3. A simple test network.

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Y. Zhou, Y. Wang and H. Yang et al. / Transportation Research Part B 129 (2019) 79–95 Table 2 Running time, dwell time and departure time bound (in minute). Line

Line1-up Line1-down Line2-up Line2-down Line3-up Line3-down

Running time

Dwell time

Departure time

min

max

min

max

min

max

5 5 5 5 5 5

20 20 20 20 20 20

1 1 1 1 1 1

5 5 5 5 5 5

1382 1384 1386 1388 1390 1392

1500 1500 1500 1500 1500 1500

Table 3 Coordinated last train timetable of DR-LTTP with consideration of last three trains. Line

Line1-U

Line1-D

Line2-U

Line2-D

Line3-U

Line3-D

Dep. Arr. Dep. Arr. Dep. Arr.

A 22:40 M 22:45 M 22:48 N 23:08 N 23:11 D 23:18

D 22:48 N 23:08 N 23:11 M 23:16 M 23:17 A 23:25

E 23:10 M 23:15 M 23:18 I 23:28

I 22:30 M 22:46 M 22:49 E 22:55

J 22:49 N 23:09 N 23:12 O 23:30

O 22:47 N 23:07 N 23:10 J 23:28

Table 4 Impacts of the i-th last train on destination reachability. The i-th last train

1

2

3

4

5

Passengers reachable to destination Total number of passengers Percentage DR

7319 8554 85.6%

7395 8554 86.5%

7489 8554 87.5%

7489 8554 87.5%

7489 8554 87.5%

Percentage DR=Number of passengers reachable to destination/Total number of passengers.

Our model with this test example data is solved using CPLEX 12.6.2 in a personal computer with an Intel Core i5 3.30 GHz CPU and 8 GB RAM. It takes only 8.5 s to get the optimal solution by the branch and bound method. Table 3 lists the coordinated last train timetable including the last train departure time from terminal stations and the arrival and departure time at/from transfer stations. Based on the above timetable and the given last 2nd and 3rd train timetable, we can further identify feasible directional transfers and destination-reachability. The results show that 78 of the 144 directional transfers are feasible, accounting for 54.2%. Of all 8554 passengers, 1065 passengers’ all paths include infeasible directional transfers. It means that 7489 passengers can get to their destinations, with a percentage reachability of 87.5%.

4.2. Impacts of the i-th last train In the DR-LTTP, “LT” (last train) refers to the train that passengers take at their entry station, while the train of the next line they transfer to may not be the last train of that line. As already mentioned, passengers may transfer and catch up the i-th last train of next line. This indicates that the several last trains should be taken into consideration in the DR-LTTP. Sensitivity test on the i-th last train is conducted to look into its impacts on the destination reachability. We consider the above example in Fig. 3. Table 4 displays the results associated with consideration of the i-th last train for i=1, 2, 3, 4, 5. From the results in Table 4, the number of passengers reachable to destination stations increases marginally with i. Clearly, this increase mainly comes from the passengers need to make multiple transfers. For example, the last train passengers from entry stations (I, H, G) to destination stations (L, O) need to make two transfers at stations (M, N). If only the last train of the transfer line (Line 1-U) is considered, these passengers successfully make a directional transfer at station M but failed to transfer at station N. When the 2nd last train is considered (the departure time of the 2nd last train of Line1-U from station M was given to be 22:12), then these passengers can catch up with the penultimate train of Line1-U and then catch up with the last train of Line3-U and reach their destination stations. When the number of i-th last train increasing to 4 and 5, the DR-Rate keeps unchanged, and no extra passengers can successfully get to their destination through the i-th last train. Note that the number of destination-reachable passengers ceases to increase when i=4, 5 in this simple example. In general, several last trains should be considered for large networks but it is conceivable that it is enough to consider the 2nd or the 3rd last train particularly when the departure times of last trains of all lines do not differ very much.

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Table 5 Coordinated last train timetable of DR-LTTP with only consideration of last train. Line

Line1-U

Line1-D

Line2-U

Line2-D

Line3-U

Line3-D

Dep. Arr. Dep. Arr. Dep. Arr.

A 23:10 M 23:20 M 23:22 N 23:28 N 23:30 D 23:38

D 23:22 N 23:29 N 23:31 M 23:39 M 23:41 A 23:50

E 23:29 M 23:39 M 23:42 I 23:50

I 23:08 M 23:20 M 23:22 E 23:32

J 23:19 N 23:29 N 23:30 O 23:40

O 23:17 N 23:28 N 23:31 J 23:41

Table 6 Coordinated last train timetable of ST-LTTP with only consideration of last train. Line

Line1-U

Line1-D

Line2-U

Line2-D

Line3-U

Line3-D

Dep. Arr. Dep. Arr. Dep. Arr.

A 23:54 M 0:04 M 0:06 N 0:12 N 0:14 D 0:22

D 23:47 N 23:54 N 23:56 M 0:04 M 0:06 A 0:16

E 23:54 M 0:04 M 0:07 I 0:20

I 23:52 M 0:04 M 0:06 E 0:18

J 23:10 N 23:20 N 23:21 O 23:35

O 0:00 N 0:11 N 0:14 J 0:28

Table 7 Results comparison between DR-LTTP and ST-LTTP.

Successful directional transfer Total directional transfer Percentage ST Passengers get to destination Total passenger Percentage DR

DR-LTTP

ST-LTTP

12 16 75.0% 7319 8554 85.6%

13 16 81.3% 7017 8554 82.0%

4.3. Comparison between DR-LTTP and ST-LTTP In order to make a comparison between DR-LTTP and ST-LTTP, we test the same case with only consideration of the last train service between DR-LTTP model and ST-LTTP model. The last train timetable figured out by these two models is shown in Tables 5 and 6. The result of successful directional transfer can be represented by the reticular diagram (Tzieropoulos et al., 2010) shown in Fig. 4 (a line with arrow represents one directional transfer with departure time and arrive time at transfer station) and comparison result is shown in Table 7. As shown in Fig. 4 and Table 7, the timetables computed by DR-LTTP and ST-LTTP models achieve 12 and 13 successful directional transfers respectively. Although one more directional transfer realized by ST-LTTP, but the number of passengers who can reach their destinations with DR-LTTP is nearly 4% more than the one with ST-LTTP. The reason is that the DR-LTTP model aims to maximize the passengers who can successfully reach their destination station, the directional transfers (“1U-3U-N” and “3D-1D-N”) achieved by DR-LTTP meet greater travel demand than the directional transfers (“2U-1U-M”, “1D-2D-M” and “3D-1U-M”) achieved by ST-LTTP. Specifically, for a path with more than one transfer, the passenger can successfully get to the destination only when all directional transfers are feasible, so the destination-reachability is directly related to station-transferability but a good station-transferability does not ensure a good destination-reachability. The major concern of last train service is to make as much as possible passengers to reach their destination. The comparison reveals that the DR-LTTP model can provide a better service than the ST-LTTP model, and the directional transfers with a higher travel demand are maintained. The DR-LTTP model focuses on assigning passengers to reasonable paths and adjusting timetable to improve destination-reachability.

5. Real case application: Beijing URT network We now move on to the application of the proposed model to a real URT network: Beijing URT network. It is a vital mass transit system serving more than 100 million passengers every day. Also, it is one of the most complicated URT networks worldwide and passenger transfers are rather common. Thus, last train passenger destination-reachability is indeed one of the important operational issues in Beijing URT systems. With the Beijing URT network, we first analyze the complexity of the optimization model and then problem size reduction is implemented to make the model easier to solve. Finally, we test the timetable computed by DR-LTTP model compared to the published Beijing URT last train timetable.

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Fig. 4. Reticular diagrams for DR-LTTP and ST-LTTP.

5.1. Data preparation Fig. 5 depicts the map of Beijing URT network as in December, 2018. It consists of 20 physical lines (40 URT lines) and 295 stations. The up-direction is defined from north to south for vertical line and from west to east for horizontal line. As for the loop line, such as line 2 and line 10, the clockwise direction is set as the up-direction. Among the 295 stations, there are totally 55 transfer stations. The Beijing URT network contains 86730 passenger groups. For analysis purpose, we randomly generate all group (origindestination) demands as random numbers from 1 to 20. In the dense Beijing URT network, there may be various paths for some passenger groups. As already stated in Section 2, only a few reasonable paths are considered and the other paths of excessive detours are ignored, for simplification, we assume that only the three shortest paths are reasonable candidate paths when there are more than three paths for a group of passengers. Other information, including walking time, the i-th (i ≥ 2) last train timetable, etc., are collected from the Beijing URT corporation webpage. Note that the earliest last train departure times from terminal stations are set based on the published

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Fig. 5. Map of the Beijing URT network.

Table 8 Published last train departure time and the obtained bounds in Beijing URT network. Line

Published time

Earliest

Latest

Line

Published time

Earliest

Latest

1-up 1-down 2-up 2-down 4-up 4-down 5-up 5-down 6-up 6-down 7-up 7-down 8-up 8-down 9-up 9-down 10-up 10-down 13-up 13-down

23:30 23:30 23:03 23:00 22:45 22:38 22:48 23:11 22:40 22:49 23:15 22:25 22:05 23:05 23:19 22:40 22:49 22:41 22:42 22:42

23:20 23:20 22:55 22:55 22:40 22:25 22:44 23:05 22:45 22:45 23:13 22:00 22:00 22:55 23:15 22:40 20:55 20:55 22:38 22:38

1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00

14E-up 14E-down 14W-up 14W-down 15-up 15-down 16-up 16-down BT-up BT-down CP-up CP-down FS-up FS-down JC-up JC-down XJ-up XJ-down YZ-up YZ-down

22:30 22:40 22:10 22:30 23:15 22:11 22:30 22:55 23:40 23:00 23:35 23:35 22:20 23:25 22:52 22:30 22:20 22:58 23:20 22:40

22:28 22:38 22:10 22:30 23:10 21:58 22:25 22:51 23:35 22:55 22:55 23:35 22:15 23:15 22:45 22:25 22:15 22:55 23:15 22:35

1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1:00

early

of line l ∈ L is penultimate train’s departure time in the corporation webpage. Specifically, the latest departure time tl set to be 2 min later than the published penultimate train’s departure time to keep the minimum headway constraint, and the earliest departure time tllate is set to be 1500 (refer to 1:00 of next day). Table 8 lists the published last train departure time and the obtained and latest last train departure times for all lines in Beijing URT network.

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Y. Zhou, Y. Wang and H. Yang et al. / Transportation Research Part B 129 (2019) 79–95 Table 9 Number of variables and constraints in the DR-LTTP optimization model.

Variable Variable Variable Variable

tl1 tl1 ,s

l1 ,s yp(k)

Variable zk Variable x(li ,l j  )(s ) Variable zk

(li ,l j  )(s )

Cons. (16)–(19) Cons. (20)

Number at most

Beijing case

After simplification

|L| |L| • |S| |L| • |S|  |P (k )|

40 2200 2200 156118

40 2200 2200 26389

|K| 

86730 398

16255 398

62134964

10502822

8800 398

8800 398

86730 62134964

30733 10502822

40

40

k∈K

s ∈S

|D(s )|

k∈K

|P (k )| ·

 s∈S

|D(s )|

4 • |L| • |S|  |D(s )|

s∈S

Cons. (21), (23) Cons. (22)

|K|  k∈K

Cons. (24)

|L|

|P (k )| ·

 s∈S

|D(s )|

5.2. Computational complexity Before carrying out the experiment, we analyze the complexity of our formulation with the Beijing URT network, including the total numbers of variables and critical constrains in the problem. Typically, there are two types of variables in the proposed DR-LTTP optimization model. Binary decision variables include feasible directional transfer indicator (x(l ,l  )(s ) i j

and zk

(li ,l j  )(s )

), passenger path selection indicator (yp(k) ), and destination-reachability indicator (zk ); continuous decision vari-

ables include the last train departure time, tl1 , from terminal station in each URT line, running time from station s-1 to station s, tl1 ,s , and dwell time at station s, l1 ,s . Obviously, the complexity of the model depends on the number of URT lines (|L|), number of transfer stations (|S|) and number of directional transfers (|D(s)|) at each transfer station, number of passenger groups (|K|) and number of reasonable paths (|P(k)|) for each passenger group. Note that the directional transfers and reasonable paths are related to the number of i-th last train, the more i-th last train we consider, the larger scale of the problem. Table 9 lists the number of variables and constrains in the mixed-integer linear programming model. In the Beijing URT  network example with only consideration of last train (refer as scenario 1), |L| = 40, |S| = 55, |D(s )| = 398, |K| = 86730, s∈S  p(k) , zk and zk | P ( k ) | = 156118 . Thus, there are 62378210 binary variables with respect to x , y , and total   (l ,l )(s ) k∈K i j

(li ,l j )(s )

4440 continuous variables with respective to tl1 , tl1 ,s and l1 ,s . The formulation is a sufficiently large-scale mixed-integer linear programming problem. 5.3. Network simplification Solving the above extremely large-scale mixed-integer linear programming problem is computationally demanding. Here we propose network simplification methods to reduce the problem size and accelerate the solving process. From Table 9, it can be found that the huge number of variables and constrains are primarily due to the large number of passenger groups. The number of passenger groups directly governs the sizes of variables yp(k) , zk , and as a result, governs the size of variable zk  . Moreover, the numbers of these variables govern the number of corresponding constraints. So, if the (li ,l j )(s )

number of passenger groups can be reduced, the number of variables as well as constraints would drop significantly so that the program could be solved within a reasonable time. There are three ways to reduce the number of passenger groups in the DR-LTTP. Firstly, we can eliminate passenger groups if there is a candidate path for passengers to reach destination without transfers. It is obvious that passengers in such groups can always get to their destination independent of the timetable coordination. Such passenger groups related variables yp(k) , zk , and zk  can be preset to one or can be eliminated to reduce the size of the problem. The rule for (li ,l j )(s )

eliminating such variables is given as follows.

⎧  ⎫ ⎨   ⎬ eliminate passenger group k for k α(pl(,kl ) )(s) = 0 ∃ p(k ) ∈ P (k ) ⎩ (l ,l  )(s)∈D(s) i j ⎭ i j

Secondly, we can integrate some passenger groups into one passenger group when their reachability is always the same with each other. From the perspective of reachability, attentions should be paid to which directional transfer passengers would take. In an URT network, passengers in some groups may always need to take the same directional transfers even though they have different paths. For such passenger groups, reachability will also be the same. For example, as shown in Fig. 6, passengers from TunDian station to ZhiChunLi station and passengers from YongFeng station to ZhiChunLi station

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91

Fig. 6. Passenger group integration example.

both have to take line 16 to reach XiYuan station and then transfer to line 4 to reach HaiDianHuangZhuang station and then transfer to line 10 to reach the destination. The reachability of both passenger groups depends on the feasibility of the same two directional transfers. Actually, if departure stations of passenger groups are between BeiAnHe station and NongDaNanLu station and destination stations are among the rest of stations, these passenger groups can be integrated into the passenger groups with the same departure station, i.e., NongDaNanLu station. In other word, we can merge BeiAnHe station to NongDaNanLu station into one single station. For the same reason, the stations between two consecutive transfer station also can be treated as one station. Thirdly, if a path of a group of passengers contains an infeasible directional transfer, then the passengers can’t reach his/her destination through this path, such path can be eliminated from the path set. Let tr(k)be the set of all directional transfer in path k, The eliminated process is shown as follow: Step 1: Select and remove a directional transfer (li ,l j )(s) from path tr(k), if i = 1, j = 1ori = 1, j = 1, then go to step 2; else if i = 1, j = 1, then go to step 3; else if i = 1, j = 1, then go to step 4. Step 2: If tr(k) is not empty, go to step 1; else path k can’t be eliminated. dep dep Step 3: Let min = tlarr,s + minheadway, if tl  ,s − min − t(trans ≥ 0 and tr(k) is not empty, go to step 1; else if tl  ,s − min − l ,l  )(s ) 2

i

j

j

j

dep j ,s

≥ 0 and tr(k) is empty, the path k can’t be eliminated; else if tl  t(trans l ,l  )(s ) i

j

− min − t(trans < 0, the path k can be l ,l  )(s ) i

j

eliminated. dep dep Step 4: If tl  ,s − tlarr − t(trans ≥ 0 and tr(k) is not empty, go to step 1; else if tl  ,s − tlarr − t(trans ≥ 0 and tr(k) is empty, ,s ,s l ,l  )(s ) l ,l  )(s ) j

i

i

j

j

dep j ,s

the path k can’t be eliminated; else iftl 

i

i

j

− tlarr − t(trans < 0, the path k can be eliminated. ,s l ,l  )(s ) i

i

j

Based on the above observation, we conduct such integration in the whole Beijing URT network. Fig. 7 illustrates the simplified Beijing URT network while stations in the black rectangle represent integrated stations. After simplification, the total number of stations reduces from 295 to 138. On the other hand, the number of passenger groups is reduced from 86730 to 18906. Furthermore, 2651 passenger groups without taking transfers can be eliminated. Finally, there are totally 16255 passenger groups in the simplified Beijing URT network. The number of variables and constraints after simplification is listed in Table 9. Obviously, the problem size can be reduced by nearly five-sixths compared to that of the original problem. In order to get a better last train timetable, we also simplify the scenario of Beijing URT case with the 2nd last train and the 3rd last train based on the same rules. 5.4. Results and discussion The three simplified Beijing URT network scenarios are solved by CPLEX 12.6.2 in a workstation equipped with Intel Core i7-3700 3.40 GHz CPU and 32GB RAM. Scenario 1 (with the 1st last train) and Scenario 2 (with the 2nd last train) can obtain optimal solution, the results are reported in Table 10, and the solution processes are shown in Fig. 8. As for scenario 1, CPLEX finds the optimal solution within 1 h, total 293137 passengers can successfully get to their destination with a

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Fig. 7. Simplified Beijing URT network. Table 10 Results for different scenarios.

Scenario 1 Scenario 2 Scenario 3

CPU time

BestInteger

Optimal solution

Total passenger

Passenger without transfer

Percentage DR

46 m 41s 13 h 52 m 8s 23 h 12 m 20s

256173 279028 277326

256173 279028 /

389953 389953 389953

36964 36964 36964

75.1% 81.0% 80.6%

Percentage DR = (Best integer+Passenger without transfer)/Total passenger. Table 11 Cplex engine log for Beijing URT real case with last 3rd train. Nodes

Best Integer

Best Bound

Gap

Computational Time (Without data reading)

0 1684∗ 18326 35485∗ 45776 52356∗ 78233

/ 254702 259524 265679 272639 277326 277326

350215 / 311297 / 322797 / 301584

/ 33.22% 27.25% 23.00% 19.27% 16.32% 11.03%

6 min 42 s 38 min 40 s 6 h 6 min 17 s 9 h 51 min 52 s 14 h 22 min 18 s 16 h 00 min 23 s 23 h 12 min 20 s

∗

represents that the integer solution is found at this node.

percentage DR of 75.1%. In scenario 2, the 2nd last train is taken into consideration, the problem size become larger due to more directional transfers and passenger paths. After nearly 14 hours’ computation, the optimal solution is found. Because passengers have more choices to make a transfer, more people can reach their target station by the last train service, and the percentage DR rise by 6% compared to scenario 1. For the scenario 3, we consider the 3rd last train. The number of passenger path and directional transfer grow much larger than scenario 1 and scenario 2. The CPLEX solver terminated due to out of memory after nearly 1 day’s computation. The optimization processes are reported in Table 11. Within 24 hours’ solution process, the gaps are reduced gradually from 33.22%, to 23.00%, to 16.32%, and to 11.03% finally with a termination. The best integer objective function value is 277326, which is obtained after about 16 hours’ solver running. It means that the destination-reachability is over 80.6% with the coordinated timetable as well as the last 2nd and last 3rd timetable.

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Fig. 8. Solution process of scenario 1 and scenario 2. Table 12 Performances of optimal and published timetables. Timetable

Destination-reachability Passenger groups

Optimal (only last train) Published (only last train) Optimal (2nd last train) Published (2nd last train)

Passengers

Number

Total

Percentage

Number

Total

Percentage

13002 10254 14533 12025

18906 18906 18906 18906

68.7% 54.2% 76.9% 63.6%

293137 245052 315992 268358

389953 389953 389953 389953

75.1% 62.8% 81.0% 68.8%

Furthermore, we make a comparison with the current real timetable used by the Beijing URT Corporation. Table 12 compares the performances between the published timetable and the computed timetable by our model. The results show that the performance of the last train service is enhanced substantially when the timetable is coordinated. Specifically, the destination-reachability in terms of the number of both passenger groups and the number of passengers increases compared to that of the current published timetable. When only the last train is considered, the percentage DR increases from 62.8% with the published timetable to 75.1% with the computed timetable. When the 2nd last train is considered, the percentage DRs are 68.8% and 81.0% for the published and computed timetables, respectively. 6. Conclusion This paper presents a fundamental optimization model for the DR-LTTP in URT networks. The model is formulated as a mixed-integer linear programming and can be solved by existing commercial optimization software such as the standard branch and bound method used in the numerical examples. The model produces a coordinated timetable for the last train services that maximizes the destination-reachability, i.e., the number of passengers successfully reaching their destination stations. A simple minimum spanning tree network is used to test the feasibility of the proposed model. Then a detailed analysis is carried out with the same test network, in which sensitivity analysis on the number of the i-th last train is conducted. We also make a detailed comparison between DR-LTTP and ST-LTTP and the results shows that the DR-LTTP can provide passengers with better services. Furthermore, the Beijing URT network, a large-scale URT network, is applied to test the performance of our model. We put forward three methods to simplify the problem size. The results show that the computation time and solution quality are satisfactory. Finally, we make a comparison of the coordinated timetable with the current timetable in operation, more passengers can reach their destination stations with the optimized timetable. Further research work is recommended in three avenues: one is to develop more efficient algorithms for solving larger URT network problems by exploring the model structure and decomposition and relaxation methods. The second one is to

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consider some more specific factors related to the last train timetabling problem, such as passengers’ waiting time, running speed optimization, etc. The last one is to consider the first train timetabling problem (Kang et al., 2016a, 2016b) with the LTTP simultaneously, which aims to minimize the waiting time of early passengers and maximize the destinationreachability of late-night passengers. Acknowledgements This research was supported in part by a grant from the Hong Kong’s Research Grants Council (HKUST16211218) and a grant from National Natural Science Foundation of China (71890974/71890970). Appendix A The following notations are used throughout the paper. Sets L: S: L(s): L+ (s): L− (s): D(s): K: Pk :

set set set set set set Set set

of of of of of of of of

all URT lines. all URT transfer stations. trains serving transfer station s ∈ S. all inbound trains associated with transfer station s ∈ S. all outbound trains associated with transfer station s ∈ S. all directional transfers associated with transfer station s ∈ S. passenger groups, grouped by their origins and destinations. pre-identified feasible paths in the URT network for last train passengers in group k.

Parameters

ρ + (l): ρ − (l): tl1 ,s :

l1 ,s : (li ,l j )(s) : t(trans l ,l  )(s ) i

j

tlearly (tllate ): 1 1 α(pl ,l  )(s) :

name of line l. direction of line l. travel time of the last train from the terminal station to the transfer station s ∈ S on line l. dwell time of the last train at transfer station s ∈ S in line l ∈ L(s). directional transfer from the i-th last inbound train of line l to the j-th last outbound train of linel at transfer station s. minimum passenger transfer time from line l to line l at transfer station s ∈ S.

qk :

earliest (latest) departure time of the last train of line l from the terminal station. path-transfer incident matrix indicating whether path p, p ∈ Pk ,k ∈ K, contains directional transfer (li ,l j )(s) ∈ D(s) at transfer station s ∈ S. number of passengers in group k ∈ K.

Variables tl1 : tlarr,s :

departure time of the last train in line l. arrival time of last train at transfer station s ∈ S in line l.

i

j

1

tldep : 1 ,s x(li ,l j  )(s ) :

yp : zk : zk

(li ,l j  )(s )

:

departure time of last train from transfer station s ∈ S in line l. 1 if transfer from the i-th last train of line l to the j-th last train of line l at the transfer station s is feasible, and 0 otherwise, (li ,l j )(s) ∈ D(s), s ∈ S. 1 if passengers in group k choose path p, and 0 otherwise, p ∈ Pk , k ∈ K. 1 if passengers in group k can get to their destination by the last train service, and 0 otherwise, k ∈ K. 1 if the corresponding inequality can be satisfied, and 0 otherwise, k ∈ K, (li , l j )(s ) ∈ D(s ).

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