# Optimization of Stop Plan for Skip-Stop Operation on Suburban Railway Line
## 1 Introduction  
With its development of metropolises and urban agglomerations, China is now planning and constructing regional rapid rail transit networks within urban areas and between urban agglomerations on a large scale. Suburban rail connects the central area of a metropolis to its surrounding satellite cities, operates on long routes with long distances between stations, and primarily serves commuters during peak periods. To minimize commuter travel times and improve train utilization efficiency, suburban rail services are generally operated with a combination of express and local trains. This mode of operation referred to as E/L mode, express trains can satisfy the needs of passengers who travel over long distances, with rapid and direct service, whereas local trains accommodate the needs of those traveling over short distances along the same route [1]. E/L mode refers to the concurrent operation of both local trains that stop at every station and express trains that skip some or even all stations between two points along the same route, with the goal of meeting the needs of different passengers. E/L mode is shown in Figure 1.  
Researchers around the world have studied E/L mode in urban rail transit systems. [PERSON] et al. [2], [PERSON] et al. [3], and [PERSON] et al. [4] studied stop plans of express trains from the perspective of passenger travel efficiency and found the following: (1) Allowing  
Figure 1: E/L mode. express trains to skip stations can effectively reduce both the train turnaround times and the travel times of passengers on long journeys; and (2) using reasonable schedules that allow express trains to overtake local trains can maximize route capacity. [PERSON] et al. [5] developed a model incorporating several factors (e.g., inter-station distances, changes in passenger flow, and departure intervals) to optimize the stop plan for an E/L mode of the Seoul subway system and found that while operating both express and local trains during peak periods increased the passenger wait time \(t_{twait}\), this mode of operation reduced the total passenger travel time \(t_{total}\). [PERSON] et al. [6] examined four influencing factors for the attractiveness of express train services, namely, the direction of the route, the average passenger trip length, the scale of the passenger flow, and the level of imbalance, of which the average passenger trip length was identified as the principal factor. The optimization problem of stop plan for skip-stop operation is usually defined as a mathematical optimization problem, and researchers have designed the mathematical models and algorithms. [PERSON] et al. [7], [PERSON] et al. [8] and [PERSON] and [PERSON] [9] developed the optimization models to minimize energy consumption and passenger travel costs for urban and suburban rail transit, respectively. [PERSON] [10], [PERSON] et al. [11], and [PERSON] et al. [12] established the nonlinear integer programming models to minimize passenger travel times, with several parameters (e.g., the passenger flow demand, departure intervals, and the number of operating trains) as constraints, and they designed algorithms to solve these models. [PERSON] et al. [13] presented the formulation of the skip-stop problem as a bi-level optimization problem, and a heuristic method for large-scale urban networks is presented to solve this extremely complex bi-level problem, where the skip-stop optimization is a mixed-integer problem. [PERSON] et al. [14] developed a model where the objectives consider the minimization of probability of train delays, energy consumption, and travel time of the trains, a parallelism selection genetic algorithm was designed to solve the model. Moreover, some scholars have studied the synchronization optimization of the train stopping plan and timetable under E/L mode. [PERSON] et al. [15] optimized the train timetable for a route based on a certain stop plan in E/L mode. [PERSON] et al.