Equity-oriented integrated optimization of train timetable and stop plans for suburban railways system

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Highlights

  • Quantitative evaluation the equity issue in train scheduling is developed.

  • An integrated model to optimize the train stop plan and timetable is formulated.

  • An improved extended adaptive large neighborhood search approach is developed.

  • The model is tested for case studies of Chinese Shanghai-Jinshan suburban railway.

  • The relationship between fairness, efficiency, and cost is analyzed.

Abstract

In the suburban railway system, when oversaturation occurs in the railway line or the train skips the same station continuously, passengers waiting at different stations may receive a varying share of train resources. As a result, some passengers have to endure a very long waiting time, causing issues of inequity. Thus, this paper focuses on improving the system-wide equity performance through integrated optimization of the train timetable and stop plan while ensuring travel efficiency and cost rationality. As a part of the study, a novel equation to estimate equity performance by considering the difference in passenger travel utility is developed. Then, an integrated optimization model combining the train timetable and stop plan is proposed to minimize total travel time, train running time, and equity performance. Unlike other studies in the literature, this paper does not define any initial set of stopping plans, and synchronously optimizes the train timetable and stops the plan. This facilitates obtaining better solutions by expanding the solution space. Furthermore, it overcomes the NP-hardness of the investigated problem by using the adaptive large-scale neighborhood search (ALNS) algorithm to solve the multi-objective mixed-integer linear programming model. Finally, this paper discusses the original and optimized scheduling indicators of a real-world case based on the Shanghai Jinshan railway and performs a sensitivity analysis on equity weights. The results show that the proposed approach can improve the overall benefits of efficiency, cost, and equity by 23%. Specifically, in terms of equity, the passenger travel utility variance was reduced by 42.8%. This analysis provides essential information to railway managers in formulating the train operation plan.

Introduction

Nowadays, the travel demand of residents between cities and suburbs has increased dramatically due to the rapid expansion of the urban scale. Compared with private cars or other railways (e.g., metro and high-speed railways), the suburban railway system offers high speed and good accessibility that attracts many commuter passengers daily. Different from the operating characteristics of the metro (e.g., the distance between adjacent stations is relatively short), suburban railways mainly serve medium and long-distance commuters, and the temporal and spatial distribution of passenger flow is unbalanced (Tang and Xu, 2022). Hence, optimizing train operation planning problems for suburban railways has attracted much interest over the past years.

The passenger travel suburban railway system has prominent tidal characteristics such that the peak hours occur in the morning and evening (Bin et al., 2014). When the train capacity cannot meet the passenger demand during peak hours, some passengers cannot board the first train, so they must wait for subsequent trains with remaining capacity. Especially when the capacity of each train continues to get full at the upstream station, the passengers at the downstream station must wait for multiple trains to get on successfully. From the perspective of social equity, public resources should be fairly allocated to individuals and groups (Litman, 2002). While this fact is essential, it is often neglected in actual operations. For instance, if multiple trains skip the same station continuously or the capacity of the train arrival station is full, some passengers have to wait for a long time (T. Zhang et al., 2018a), resulting in exasperation among passengers due to the inequity problem. Indeed, this phenomenon is widely spread and is not conducive to the sustainable development of public transport.

Increasing train capacity or building more infrastructure is a straightforward solution to the problem of service inequity. However, this plan requires large capital investment and urban space resources, it is no longer the main means to solve the problem. On the other hand, the aforementioned inequity problem can be mitigated by proper timetable adjustment and train stop scheduling. The flexible train stop plan means that each train can stop at or skip a station along its line based on the dynamic passenger demand. The flexible stop plan can better match the passenger flow demand of different stations or different OD pairs with the train capacity, so as to effectively balance efficiency and accessibility. This approach to operation can significantly affect the train timetable quality.

Due to the high latitude and complexity of the integrated optimization of multiple decision variables, most current researches focus on the train skip-stop plan and timetable methods separately (Xu, Li, & Li, 2016, Zhu et al., 2017, Zhu et al., 2016). However, the timetable is closely related to the train stop plan, and the design of the train stop plan significantly influences the formulation of the timetable. Additionally, separately handling the train timetabling and stop planning cannot cope with the service imbalance, which is not conducive to maximizing operational benefits and improving the system equity performance.

Hence, this paper intends to explore the integrated optimization of timetable and train stop plan for redistributing train capacities and arrival times at each station and improve the overall equity performance. First, the service inequity problem caused by the contradiction between train supply and passenger demand is analyzed. The equity performance is measured by the variance of passenger travel utility for all OD pairs. Second, the efficiency, cost, and equity are comprehensively considered, and an integrated optimization model of train stop planning and timetable based on dynamic passenger flow demand is established. The model’s objective herein is to minimize the passenger travel time, train cost, and the variance of passenger travel utility for all OD pairs. Furthermore, an extended adaptive large-scale neighborhood search algorithm (ALNS) is designed to solve this model. Finally, a real-world case is conducted to demonstrate the performance and reliability of the proposed approach.

In recent years, the equity of operations management has gradually received attention in rail services. Train stop plans and timetables were proved can improve overall service quality and equity performance to some extent. Lots of scholars have made excellent contributions to various optimizations in the field of train stop plans and timetables.

(1) Train stops plan optimization

In the suburban railway system, the travel demand of passengers is unevenly distributed in time and space, and the train skip-stop plan has been proven effective in balancing efficiency and passenger demand (Du and Yang, 2018, Wan et al., 2020). Sun et al. (Sun et al., 2018) developed a train stop optimization model for express/local by selecting a combination of train stops from a given set of stop patterns to minimize the operating cost and the passenger’s total travel time. The results show that for the commuter railway serving long-distance, the express/local trains operation plan can effectively shorten passengers’ travel time. Yang et al. (Yang et al., 2019) and Fan et al. (Fan & Ran, 2021) constructed an optimization model with the trains stopped at AB stations and verified that the AB stop station plan effectively reduced the passenger travel time compared with the all-stop station. Tang et al. (Tang & Xu, 2022) set the principle of the express train stop at the first-level station by classifying the station level and constructing a bi-level programming model for the train optimization scheme under the express/local stop mode. The results showed that compared with all stopping modes, the express/local stopping mode can effectively reduce agency operation costs and passenger traveling costs. The above train stop plan optimization problem requires preset a set of train stop plans, in which the stop plans set become the key to a high-quality stop plan. For the train’s flexible skip-stop mode, Rajabighamchi et al. (Rajabighamchi et al., 2019) proposed a robust skip-stop method based on a two-stage scenario to address the problem of uncertain arrival rates during peak hours. In their study, the objective was to minimize passengers’ travel time and waiting time at stations, and the results showed that the skip-stop method saves about 5% of the total travel time. The above studies have proved that the train skip-stop plan can improve passenger efficiency and reduce the training cost.

(2) Train timetable optimization

Several scholars have provided excellent studies about the train timetable problem. Two optimization perspectives are distinguished in the study of timetables. The first one is based on the perspective of passenger demand, and the second one is based on the railway operator (Canca et al., 2014, Petering et al., 2016). Canca et al. (Canca et al., 2014) and Niu et al. (Niu et al., 2015) established a nonlinear integer programming model that fits the arrival and departure train times to a dynamic demand to minimize the average waiting time of passengers. In practice, the cost is also a critical factor. Many researchers tend to balance passengers’ and operational costs. Robenek et al. (Robenek et al., 2016) constructed a mixed-integer linear programming (MILP) timetable optimization model to maximize the operation company’s profit while maintaining passenger satisfaction. Yin (Yin et al., 2017) proposed an integrated approach for the train timetable problem on a bi-direction urban metro line to minimize the cost (i.e., energy consumption) and passenger waiting time. Shakibayifar et al. (Shakibayifar et al., 2019) believe that railway management is a multi-objective optimization problem involving the operators’ cost and passenger service level, so a multi-objective model-based optimization framework was constructed.

The above research provided the basis for preparing train operation plans, but the optimization of the train timetable and stop plans was researched independently, and the benefit of the optimized improvement plan is limited. To further maximize the transportation benefits, some scholars began to combine the optimization of the train stop plan and timetable.

(3) Integrated optimization of train timetable and stop plans

The integrated optimization problem primarily focuses on passenger efficiency and train cost (Barrena et al., 2014; X. Li & Lo, 2014). Wang et al. (Y. Wang et al., 2014) developed a double-layer optimization model of stop plan and timetable to minimize the total passenger travel time and train energy consumption, and the train stop plans were selected from a set of pre-determined stop sets. Meng et al. (Meng & Zhou, 2019) developed an integrated optimization model of the train stop plan and timetable to minimize total passenger travel time by preset a set of stop plans. Moreover, passenger demand is simplified as a function but is not a dynamic time variable. The above research simplified the optimization problem by providing a preset set of train stops, which will limit the optimization space of the model. Boroun et al. (Boroun et al., 2020) proposed a joint optimization model for timetabling and train stop planning with dynamic time constraints and used the heuristic algorithm to generate approximate solutions. Their results proved that the heuristic algorithm helps obtain high-quality solutions when solving the integrated optimization problem. Zhang et al. (Y. Zhang et al., 2022) adopted a multi-commodity network flow model to simultaneously optimize the periodic train timetable and stop plan, and minimize the number of trains used and the average passenger waiting time. Research closely related to our paper is, Dong et al. (Dong et al., 2020) conducted the integrated optimization of the train timetable and the stop plans based on the dynamic OD demand of passenger flow to minimize passengers’ total waiting and train running times. In their study, the model was solved by the ALNS algorithm and the results showed that the ALNS approach obtains high-quality solutions in a short duration.

The above research about optimizing objectives for minimizing total passenger waiting time and train cost can result in equity issues. For example, in the optimization process, the train skips the same station continuously, or the transport capacity is oversaturated, resulting in significant differences in the waiting time of passengers at different stations.

(4) Passenger travel equity

To the best of the authors’ knowledge, studies about service equity are scarce. Wang et al. (Z. Wang et al., 2012) pointed out the problem of only pursuing efficiency and ignoring equity in traffic control and proposed an optimization method for the on-ramp control by using the queue delay Gini coefficient to measure the equity. Wu et al. (Wu et al., 2015) constructed a timetable optimization model for urban subway networks. The object was to minimize the maximum waiting time at any transfer station. This study showed that the equity performance of transfer passengers in the network is significantly improved by optimizing the timetable. Shang et al. (Shang et al., 2018) focus on improving system-wide equity performance in an oversaturated rail transit network by minimizing the number of passengers with the maximum number of missed trains in the train skip-stop optimization model. The result showed that the optimized skip-stop pattern can change the equity states for nearly 40% of passengers. Gong et al. (Gong et al., 2020) proposed an integrated optimization model for passenger flow control and train schedules at saturation conditions by minimizing the number of missed trains of passengers.

The above studies aim to minimize the transfer waiting time or the number of missed trains respectively but have not thoroughly studied the equity of passenger waiting time for timetable. Moreover, the research only focused on equity and rarely discussed the relationship between equity and efficiency. Li et al. (D. Li et al., 2019) proposed the min-max equity and α-equity indicators and constructed a train timetable optimization model to balance efficiency and equity. The result showed that considering equity can decrease the efficiency of the timetable. However, this paper only focuses on optimizing the train timetable under the preset stop plan. A summary of traffic equity research and a comparison of model characteristics between the proposed solution in this study and the existing ones in the literature is shown in Table 1.

The inequity problem is essential for passengers but is usually neglected by operators. Thus, this study aims to find an innovative approach to solve the integrated optimization problem of stop plans and timetables and simultaneously achieve the equitable distribution of all passengers. Based on the characteristics of the proposed model, a large-scale neighborhood search algorithm (ALNS) is designed for the solution. The main contributions of this article are as follows.

(1) In order to better measure equity, based on the theory (proportional distribution according to utility theory) proposed by Li et al. (D. Li et al., 2019), a new equity framework is constructed using the idea of variance to minimize the differences of passenger travel utility for all OD pairs. This paper accurately obtains the passenger travel utility for all OD pairs by matching trains capacity and passenger flows demand, which improves the equity on the whole. In addition, the equity weight influence on efficiency and cost is analyzed to test the existence of a trade-off between efficiency, cost, and equity.

(2) A mixed-integer programming model is developed to obtain an optimized timetable and train skip-stop plan that compromises the performance of efficiency, cost, and equity. Unlike previous research studies, the proposed model has two features. First, the total passengers’ travel time, train cost, and the variance of passenger travel utility for all OD pairs are taken as the optimization objectives. Second, the model comprehensively optimizes the train timetable and stop plan without defining any initial timetable or stop plan set. This is beneficial to expand the solution space and obtain a train operation plan with better overall performance.

(3) An improved extended ALNS and simulated annealing (SA) are designed to obtain a near-optimal solution. Two repair rules of train insertion and train stop from the perspective of stranded passengers are put forward to expand the search space.

Section snippets

Problem description

Indeed, optimizing the train timetable and stop plans is a resource allocation problem that can cause inequity issues when only minimizing the total passenger waiting time. For example, when the rail line is oversaturated, passengers waiting at the station may not get on the train until they wait for a train with residual transportation capacity, especially the passengers at the downstream stations. Furthermore, there will be a problem that some passengers at certain stations or OD pairs wait

Methodology

Consider a one-way suburban railway line with S stations, and mainly focus on finding a solution where trains stop or skip and when trains depart and arrive at stations along with the railway line in this paper to minimize the total travel time (including WT and IVT), operating costs R, and equity coefficient fα-equity. With this goal, the research proposes an integrated optimization method for train timetables and stop plans without a predefined skip-stop set or timetable. The output optimized

Algorithm introduction

The presented integrated model is a mixed-integer nonlinear model and can be proven to be an NP-hard problem. In order to achieve the solution to the NP-hard problems, Barrena et al. (4) proposed an ALNS algorithm to solve large-scale integrated optimization problems. The ALNS algorithm was applied to optimize the train timetable and stop plans (Canca et al., 2017; T. Zhang et al., 2018b). The solution quality and computational efficiency of the ALNS algorithm have been proved (Dong et al., 2020

Case introduction

This study selects the Jinshan railway of Shanghai, China as a case for analysis, and route direction as shown in Figure 7. Jinshan Railway is the regional suburban railway that connects the urban center of Shanghai to the Jinshan suburbs. By 2021, the total railway length is 56.5 km with eight stations, the design speed is 160 km/h, the number of trains per day is 37 pairs, and the average daily passenger flow is 32,000 passengers (Chen, 2021). The direction from Shanghai South to Jinshan Wei

Conclusion

This study focused on developing a new modeling framework to improve the system-wide equity performance by optimizing trains’ timetables and stop plans. Within the study context, an equity index associated with the variance of travel utility for all OD pairs passengers was proposed to measure the overall service equity. A multi-objective mixed-integer linear programming model was formulated to minimize passenger travel time, train cost, and travel utility for all OD pairs. Furthermore, an

CRediT authorship contribution statement

Juan Shao: Conceptualization, Methodology, Software, Writing – original draft. Yan Xu: Data curation, Supervision, Writing – review & editing. Lishan Sun: Conceptualization, Funding acquisition, Resources, Supervision, Writing – review & editing. Dewen Kong: Supervision. Huabo Lu: .

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors would like to acknowledge the financial support for this study provided by the National Natural Science Foundation of China (Nos.71901008), the Beijing Municipal Education Commission Science and Technology Program General Project (KM202010005001), and Beijing Municipal Education Commission Science and Technology Program General Project (KM202110005002).

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