Incorporating multimodal coordination into timetabling optimization of the last trains in an urban railway network

https://doi.org/10.1016/j.trc.2020.102889Get rights and content

Highlights

  • Three models in a progressive fashion to optimize LTTP incorporating multimodal coordination.

  • Three LTTP models are mixed-integer linear programming by linearization techniques.

  • The complexity of the model will not exceed traditional ST-LTTP.

  • Classifying the stations and lines, the transferability between different transport modes is optimized.

  • The space-time distribution of the arrivals and departures of the connecting modes are considered.

Abstract

Urban rail transit (URT) provides efficient and low-cost services for passengers. It is a common issue for operators to coordinate the last trains of a URT network. This paper discusses three models in a progressive fashion to optimize the last train timetable incorporating multimodal coordination. The first model maximizes the transferability at transfer stations without the distinction of the stations. The second model, based on a refined classification of stations and lines, optimizes the transferability at transfer stations between different transport modes. The third model maximizes the multimodal coordination taking into account the space-time distribution of the arrivals and departures of the connecting modes. These models are formulated as mixed-integer linear programming by linearization techniques for finding the optimal timetable solutions. The proposed models are tested in the Beijing URT network connecting three railway stations and two airport terminals. The numerical results indicate that the proposed models can effectively improve the coordination among the last trains within the URT network and between the URT and the connecting modes.

Introduction

Urban rail transit (URT) provides secure and punctual passenger mobility services in large volume and plays an increasingly important role in alleviating road congestion and reducing energy consumption. The total length of the URT worldwide has been rising rapidly recently. For instance, in China, 35 cities had URT with a total length over 4750 km in 2017 and the total length will be 6000 km in 2020 (Huang et al., 2019). Passenger mobility and travel demand, especially in mega-cities, rely heavily on URT. For example, the Beijing URT affords 3.85 billion ridership in 2018 with a growth rate of 1.9% annually according to the 2019 Beijing Transport Annual Report.1

Much research has been focusing on improving the services of the URT (Wong et al., 2008, Niu and Zhou, 2013, Wu et al., 2015, Niu et al., 2015, Canca and Zarzo, 2017, Guo et al., 2018, Sun et al., 2018, Canca and Barrena, 2018, Shang et al., 2019, Lv et al., 2019, Zhang et al., 2019, Guo et al., 2020, Yang et al., 2020). Several main research subjects include the network design (Jin et al., 2013, An and Lo, 2016, Gutiérrez-Jarpa et al., 2018), line planning (Goossens et al., 2006, Fu et al., 2015) and timetabling (Hassannayebi et al., 2016, Hassannayebi and Zegordi, 2017, Shang et al., 2018, Yang et al., 2019a, Hassannayebi et al., 2019, Yang et al., 2019b), rescheduling (Gao et al., 2016, Binder et al., 2017, Ortega et al., 2018, Zhu and Goverde, 2019), and combinations of the above for the planning and operations (Canca et al., 2017, Yue et al., 2017, Yan and Goverde, 2019, Canca et al., 2019). For a developed URT system, the timetabling optimization becomes the most efficient and cost-saving way to maximize their services (such as minimizing transfer time and maximizing transfer accessibility) or to minimize the generalized costs (Guo et al., 2016).

The URT seldom provides full-day (24-h) services due to the low ridership and the need for maintenance at night. Therefore, there is a special and practical timetabling problem of the last trains in the URT system in the late evening. If the connection at the transfer station fails, the passengers who need transfers in the URT system cannot reach their destinations. Therefore, the last train timetabling problem (LTTP) has drawn much attention (Zhou et al., 2013, Dou et al., 2015, Dou and Guo, 2017, Xu et al., 2018, Zhou et al., 2019).

The LTTP in the past research focused on the coordination at transfer stations (referred to as ST-LTTP by Zhou et al. (2019) for station-transferability based LTTP). Kang et al. (2015a) proposed a model to reveal the relationships between passenger transfer connection time and waiting time with the aim of maximizing the transfer connection headways, and developed a solution algorithm based on genetic algorithm (GA). Kang et al. (2015b) further suggested a method to find a timetable minimizing the deviation from the original timetable and maximizing the average transfer redundant time and the network accessibility. To minimize the standard deviation of transfer redundant times and balance the last train transfers in subway networks, a non-linear optimization model and a heuristic algorithm were developed by Kang and Zhu (2017). To deal with large-scale URT networks, Kang and Meng (2017) put forward a global optimization model in a mixed-integer linear (MIL) formulation. A two-phase decomposition method was proposed to solve the large-scale problems globally by decomposing the original MIL model into two MIL models. Yang et al. (2017) considered the decision-makers’ risk preferences under uncertainty and formulated an optimization model for the last train timetabling that explicitly considers the number of successful transfers and the running times of last trains, for which a tabu search algorithm was employed to find the solutions. From the viewpoint of balancing different objectives of the transport operators, a bi-level programming model was proposed by Yin et al. (2019). The upper level aimed to improve the social service efficiency and the lower level to cut down the revenue loss for the operators. Since the URT is not only the transport mode for city trips, Kang et al. (2019) paid attention to the synchronization between URT and other modes (e.g., bus). They presented an optimization-based approach that deposed the LTTP by developing a mixed-integer linear programming (MILP) model to bridge coordination between the last trains and buses. To maintain model tractability, they decomposed the model into two smaller MILP models: maximizing the last train connections and minimizing the waiting times of rail-to-bus passengers.

Recently, some research concerned the origin–destination (OD) accessibility in the URT system (referred to as DR-LTTP by Zhou et al. (2019) for destination-reachability based LTTP). Yao et al. (2019) presented a bi-level model framework to maximize the number of passengers served by the last trains and minimize their transfer waiting time at the upper lever, considering passenger path choice with a detour routing strategy at the lower level. The solution of the upper-level model was found by a GA, and the lower level model was solved by a semi-assignment algorithm. Chen et al. (2019b) suggested three models in a progressive relationship. First, a basic model fine-tuned the last train timetable given the bounds of dwell time. Then, allowing mutual transfers with the prolonged dwell time to maximize transfer accessibility, a bi-objective model was proposed to seek the trade-off between the transfer accessibility and the dwell time extension. Finally, they treated the heterogeneity of transfer walking time as a random variable and employed a discrete approximation of the nonlinear model. To maximize the accessible OD pairs for last train services, a MIP model was proposed with the objective of maximizing the percentage of passengers who successfully reach their destinations (Chen et al., 2019a). Considering passenger assignment in LTTP, Zhou et al. (2019) formally introduced the DR-LTTP in a MILP and solved the problem by an existing optimization solver.

The above studies have mainly addressed transferability at the stations and destination-reachability for passenger demand. However, several limitations exist in the above problem formulations. First, ST-LTTP focuses on the passengers’ transferability only at transfer stations, but the successful transfers at stations do not necessarily imply higher reachability, especially in a large-scale URT network (Zhou et al., 2019). Second, although the DR-LTTP takes into account passengers’ path choice in the URT network and is more dedicated to improving individuals’ accessibility, it is hard to find the optimal solution in a large-scale network and supposedly difficult than the ST-LTTP due to the vast path permutations. Third, neither of the ST-LTTP and DR-LTTP has incorporated multimodal coordination between URT and other connecting modes. Finally, the space–time distribution of the connecting modes is largely overlooked in the LTTP.

To address the above limitations and pay more attention to the coordination between URT and other modes, we propose a model framework for the LTTP that considers passenger path choice with less complexity and ensures valid transferability and high reachability at transfer stations.

To improve transferability, the space–time distribution of the arrivals and departures of other connecting modes is also incorporated. For instance, as the hubs for railway and airplane, the Beijing railway stations (i.e., Beijing South Railway Station, Beijing West Railway Station, Beijing Railway Station) and the Beijing Capital International Airport involve heavy passenger flows. As exemplified in Fig. 1(a, b), there are quite a few trains and flights arriving and departing in the late evening, which fall within the service time windows of the last trains in the URT system. To present the model framework clearly, three models are discussed in a progressive relationship. First, the basic model maximizes the transferability at transfer stations without station distinction. Then, the second model maximizes the transferability at transfer stations involving other modes. To ensure valid transferability, we classify the stations and lines at different levels and maximize transferability at the key transfer stations. A bi-objective (maximizing transferability at transfer stations without station distinction and maximizing transferability at the key transfer stations) model is transformed into a single objective model by modifying the first objective function as a constraint. The classification of the lines and stations obviate the enumeration of passenger path sets suggested by Zhou et al. (2019), which is almost impossible in large URT networks. Finally, considering the space–time distributions of other connecting modes (i.e., arrival and departure times of different public transit services), the third model maximizes the coordination at the transfer stations for connecting modes, reflecting the valid connections between the last trains of URT to passengers’ destination zones. Particularly, the models are different from Chen et al., 2019a, Zhou et al., 2019 in deposing the passenger path choice process and different from Kang et al. (2019) in guaranteeing valid transferability. The three models are all formulated in a MIL formulation, which can be solved by the existing MIL solvers. Note that the last-second or even the last-third trains are not considered in our models. Despite relevant to the proposed model framework, incorporating these trains would significantly increase the model complexity but achieve limited improvement in the objectives as found in Zhou et al. (2019).

The remainder of the paper is organized as follows. Section 2 formulates the three progressive models considering the coordination between the last trains of the URT and the connecting modes. Section 3 presents a comprehensive case study considering the connection between the URT system and the railway and air transport in Beijing (China). Finally, Section 4 concludes the main contributions and provides suggestions for future research.

Section snippets

Model

In this section, we introduce the notations and assumptions and present three model formulations progressively. For each model, the objective, constraints, and the relation to the existing studies are discussed.

Case study

In this section, the URT network in Beijing is employed to validate the effectiveness of the proposed models. Note that, each line is divided into two different sub-lines by the up- and down-directions (for instance, subway Line1- “sline1” includes “sline1up” and “sline1down”). As shown in Fig. 5, each URT line is marked with two distinctive running directions. The stations are numbered in Table A1. Line and station characteristics are given in Table A2. The travel time from the starting

Conclusions and future work

The timetabling of the last trains in a URT network is a challenging topic. Considering the coordination with the connecting modes, this paper suggests three models in progressive relationship to optimize the last train timetable. Based on a basic model (Model 1), Models 2 and 3 take into account the valid transferability at transfer stations, which can not only achieve the accessibility of the DR-LTTP proposed by Zhou et al., (2019), but also consider passenger paths with reduced complexity.

CRediT authorship contribution statement

Kang Huang: Conceptualization, Methodology, Validation, Writing - original draft. Jianjun Wu: Conceptualization, Methodology, Writing - review & editing, Supervision. Feixiong Liao: Conceptualization, Methodology, Writing - review & editing, Supervision. Huijun Sun: Methodology, Writing - review & editing. Fang He: Methodology, Writing - review & editing. Ziyou Gao: Resources, Methodology, Writing - review & editing.

Acknowledgements

This work is supported by the National Key R&D Program of China (2019YFB1600200), National Natural Science Foundation of China (71771018, 71890972/71890970, 71621001), the China National Funds for Distinguished Young Scientists (71525002), the State Key Laboratory of Rail Traffic Control and Safety (No. RCS2020ZZ001).

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