First train timetabling and bus service bridging in intermodal bus-and-train transit networks

Highlights

• A bus bridging approach to cooperate with the first train operations in a subway network.

• An explicitly integrated MILP model for the first train and bus bridging timetabling problems.

• A tailored algorithm for optimally solving the first train timetabling and bus service bridging problems.

• The model and the approach are applied to the Beijing subway network.

Abstract

Subway system is the main mode of transportation for city dwellers and is a quite significant backbone to a city's operations. One of the challenges of subway network operation is the scheduling of the first trains each morning and its impact on transfers. To deal with this challenge, some cities (e.g. Beijing) use bus ‘bridging’ services, temporarily substituting segments of the subway network. The present paper optimally identifies when to start each train and bus bridging service in an intermodal transit network. Starting from a mixed integer nonlinear programming model for the first train timetabling problem, we linearize and reformulate the model using the auxiliary binary variables. Following that, the bus bridging model is developed to cooperate with the first train operation for reducing long transfer waiting times. After realizing the low computational efficiency of solving the integrated model, a tailored algorithm is designed to optimally solve the first train timetabling and bus service bridging problems. The exact models and algorithms are applied to the Beijing subway network to test their effectiveness and computational efficiency. Numerical results show that our approaches decrease the total passenger waiting time by 53.4% by a combined effect of adjusting the first train departure times and operating 27 bridging buses on 7 routes.

Introduction

Since the 1970s, with the increasingly heavy problems of the urbanized energy crisis, resource shortage, traffic congestion and traffic accidents, etc., subway systems have attracted much attention around the world. The level of urban traffic supply has been improved due to the rapid expansion of subway networks; the traffic congestion has been reduced; the urban territories have been expanded. However, subway systems in metropolises have increasingly become complex in recent years. It is, therefore, more difficult to manage the operations of subway systems under concentrated networking.

There is a famous saying that “well begun is half done”. The first train transfer problem is crucial to morning passengers. If the transfer connectivity from the first train to other trains in a subway network is well addressed, morning commuters can save plenty of time and it further helps to reduce congestion during morning rush hours. The first train transfer problem is mainly due to the expansion of the subway network. As Fig. 1(a) shows, passengers riding the first trains on line A and line B transfer to the first train on line C. They have to wait (t3 + R3 + D1 − t1 − R1) min and (t3 + R3 + D1 + R4 + D2 − t2 − R2) min, respectively. The transfer waiting times usually increase much more at some stations as the network becomes large (R3 + D1 ≫ R1;  R3 + D1 + R4 + D2 ≫ R2), and/or the inappropriate setting of the first trains’ departure times (t3 ≫ t1;  t3 ≫ t2). Fig. 1(b) shows a sample of the first train diagram from the Beijing subway in 2016, which depicts a serious problem of long waiting time for its first train connection. For instance, when a passenger riding the first train transfers from the up-train direction on line 6 to the down-train direction on line 10 at Ci-Shou-Si station, he/she has to wait for as long as 59 min. Moreover, according to our statistics, the total transfer waiting time of the first train passengers in the full Beijing subway system (more than 240 transfer directions) was over 22,790 min daily in 2016, considering the transfer passenger volumes. It is an unsolved problem of subway network operations that the first train transfers should be well synchronized. Moreover, Mohring et al. (1987) found that passengers often viewed their waiting time twice the actual time. Therefore, this study will address the first train transfer problem to reduce transfer waiting times for the early-morning passengers, aiming to allow for well-timed connection transfers in large subway systems.

Note that the previous study Kang et al. (2016) reduced the total transfer waiting time for passengers from 22,790 min to 16,380 min daily by optimizing the early train departure times for the Beijing subway case (9 bi-directional lines and 31 transfer stations with the first train departure times between 4:50 and 5:10, a range of 20 min adjustment). Although the optimized timetables reduced the total waiting time by 28.1%, a figure of 16,380 min is still a huge waiting period and the longest waiting time of a transfer direction reaches as much as 70 min. An efficient approach to avoid long waiting times for the morning passengers in practice is to coordinate well-designed train operations with bus bridging services, which are provided in the event of inefficient first train transfers. The design of temporary bus services and their routes during the first train periods can improve connectivity in large-size subway networks. This type of temporary network is often referred to as the bus bridging service (Jin et al., 2015), and will be introduced in detail in Section 2.3. When the first train transfer is not efficient in the rail network, bridging buses would ferry affected commuters waiting for the connecting trains (these bridging buses run from one station to the other stations with different locations on the subway lines) and minimize the impact of a single mode of transportation on early morning commuters.

We use the same example in Fig. 1(a) to better illustrate the idea of intermodal bus-and-train cooperation. Assume that the first trains on line A, line B, and line C depart from their depots at time 0. The segment travel time is given on each line of Fig. 2(a). The dwell time at each station is set to 5. The transfer walking time at each transfer station is 0. As shown in Fig. 2(a), the transfer waiting time from line A to line C is 0 + 70 + 5 − 0 − 20 = 55 min. The transfer waiting time from line B to line C is 0 + 70 + 5 + 30 + 5 − 0 − 50 = 60 min. If we deploy a bridging bus departing from S1 at time 15 and heading to S2 and line C (bus running time from S1 to S2 is 50), the waiting time from line A to the bus is 15 + 5 − 20 = 0 min and from line B to the bus is 15 + 5 + 50 + 5 − 50 = 25 min. As can be observed, transfer waiting times and journey times are reduced effectively by riding the bridging buses. Moreover, one may suggest deploying deadheading trains (serving passengers from S1 on Line C) before the first trains to connect transfer passengers. This strategy is effective in reducing waiting times and journey times. However, it requires subway operators to run two extra deadheading trains from the depot of Line C with an extra running time of 140 min and extra operational costs. Hence, the train-and-bus mode is the best for both passengers and companies. Without losing generality, Fig. 2(b) integrates the feeder train, connecting train and bus bridging timetables into one diagram, which shows the advantage of bridging buses in reducing first train transfer waiting times. As shown, passengers in the feeder train T1 transfer to the connecting train T1’ at the exchange station with a three-headway waiting time. This scenario occurs because of the differences in first train departure times and network structure. Long transfer waiting time problems will become more prominent and serious with the extending of the subway networks. With bridging buses (denoted by B1 and B2), early morning passengers could enjoy a fast transfer and smooth connection services. The transfer waiting times can be reduced significantly, as shown in Fig. 2(b). A practical case of the cooperation between the first trains and bridging buses is detailed in Appendix A. Indeed, there are additional costs in operating the bridging buses such as fuel costs and labor costs. However, if we do not dispatch bridging buses, passengers may be forced to ride taxis as commuting tools at transfer stations. In the early morning, there are not many taxis cruising in the streets, making it difficult for passengers to take taxis. On the other hand, it is expensive to take a taxi compared to subway trains and bridging buses. In view of the effectiveness of the above approach, this paper will study the problem of coordinating bus bridging services with the first train timetables in intermodal bus-and-train transport networks. In the following, some related literature is reviewed.

The Railway passenger transportation problem is a systematic problem that is usually planned in three levels of hierarchy: strategic, tactical, and operational control, as illustrated in Fig. 3. The tactical and operational problems are at a high dynamic level (Almodóvar and García-Ródenas, 2013). When the planning horizon becomes short, the buffer responding time reduces significantly, which makes the above problems need dynamic approaches (Narayanaswami and Rangaraj, 2011). Table 1 summarizes the motivation and objectives of the recently published literature on scheduling and rescheduling of the railway operations. This table has three columns: Column 1 lists the main railway scheduling and rescheduling operations problems that are discussed in the literature related to this paper. Columns 2 and 3 list scheduling and rescheduling objectives, respectively. In this section, a brief literature review of railway operations will be discussed.

Scheduling is a time allocation of resources to meet demands in completing a task (Abdolmaleki et al., 2020). The train timetable scheduling problem aims to achieve a conflict-free railway timetable consisting of all the train arrival and departure times at each station. Cacchiani and Toth (2012) performed an intensive study on the train timetabling problems and underlined the differences between methods for dealing with the nominal and robust versions. When designing railway train timetables, objectives developed by the operating companies are profit-oriented, such as reducing operating costs (Ibarra-Rojas et al., 2014), minimizing train travel times (Chevrier et al., 2013), and improving energy efficiency (Wang and Goverde, 2019). From the passengers’ perspective, operating companies will always employ measures regularly to minimize waiting times (Niu and Zhou, 2013; Lin and Ku, 2014; Niu et al., 2015; Kang et al., 2015a; Zhou et al., 2019). For example, Dou et al. (2015) optimized network-based bus timetables to coordinate the last train services in a subway network, by using tailor-made coordination principles for a bus network. Other methods to solve the timetable coordination problem can be found in Wong et al. (2008), Shafhi and Khani (2010), and Niu et al. (2015).

Railway disruption management can be divided into two categories: train timetabling and real-time rolling stock rescheduling. For deterministic or stochastic reasons, it sometimes requires the reallocation of resources for task completion, which is referred to as rescheduling. For example, Jin et al. (2015) presented a mixed integer linear programming (MILP) model of deploying bus services for the disruptions of urban railway transit networks. Train timetable rescheduling aims to coordinate the arrival-departure times of trains at stations to improve the level of services. Acuna-Agost et al. (2011b) defined the train rescheduling problem after incidents as a reparation problem, for which a MIP was formulated to minimize the difference between the original timetable and the rescheduling timetable. In such problems, MIP models were usually developed to optimize train delays and reduce the number of canceled trains. There is a wealth of literature on timetable rescheduling, e.g., Carey and Lockwood (1995), Narayanaswami and Rangaraj (2013), Corman et al. (2014), Veelenturf et al. (2015), and many more.

Although there exist plenty of studies on train scheduling or rescheduling problems, few works on the integrated optimization of bus bridging services and first train services in an intermodal public transport network can be found. Two previous studies, Dou et al. (2015) and Kang et al. (2019) investigated how to design bus routes for midnight passengers who failed to catch the last trains. They aimed to improve subway network accessibility. In contrast, this paper focuses on the first train and bus bridging coordination problem to improve the first train connections and synchronizations in large subway networks. The innovative problem setting considered is essentially different from the above-mentioned two studies. The main contributions in this paper are as follows. First, we design a bus bridging approach to cooperate with the first train operations. This is a novel method to cope with the first train transfer problem in a large network. Second, we build an explicitly integrated MILP model for the first train and bus bridging timetabling problems to minimize transfer waiting times for the early-morning passengers. Third, after realizing the low computational efficiency of solving the integrated model, a tailored algorithm is designed for optimally solving the first train timetabling and bus service bridging problems. We apply the model and the approach to the Beijing subway network. The results indicate that the optimal first train and bus bridging timetables decrease the total waiting time from a base of 22,790 min to 10,628.5 min daily by a combined effect of adjusting the first train departure times and operating 27 bridging buses on 7 routes.