# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 2 Literature Review
Numerical experiments on the Beijing Metro demonstrated the method's effectiveness in accelerating recovery by strategically skipping stations and reducing passenger accumulation. Furthermore, other works have explored cross-line scheduling under system interoperability. [PERSON] et al. [PERSON] et al. (2018) optimized the intersections and frequency of cross-line trains from the perspectives of saving travel time and enterprise cost. [PERSON] et al. [PERSON] et al. (2018) proposed a rescheduling model integrating flexible stopping and short-turning to minimize passenger delays, using time-dependent demand weights. Case studies on Dutch railways showed that flexible stopping significantly reduced delays and improved operational adaptability during disruptions. [PERSON] et al. [PERSON] et al. (2018) constructed an operation scheme based on infrastructure constraints. From a comparative perspective, [PERSON] (2018) highlighted the institutional differences between Japan and Europe in organizing through-train services, noting Japan's integrated management and operator collaboration versus Europe's vertically separated and open-access framework, which presents important implications for the design of cross-line rail systems. [PERSON] et al. [17] evaluated the impact of cross-line trains on line throughput capacity, and constructed a cross-line capacity allocation model with the objective of minimizing the cost of vehicle use, train operation cost, and passenger travel cost based on the analysis of the consumption of station capacity by cross-line train operation, which provided a quantitative optimization method for cross-line operation. [PERSON] et al. [18] proposed an optimization model to balance passenger cost and train load during peak cross-line operations. Introducing section load rate balance, the model used turn-back stations and train frequencies as decision variables to minimize travel cost, load imbalance, and operating cost. Case studies on Beijing's Changing Line and Line 8 showed improved capacity balance and reduced overcrowding. However, most existing studies treat cross-line operation and express-local train scheduling as independent systems. Skip-stop operation, a well-known acceleration method in urban rail transit, assigns each train a subset of stations to stop at. [PERSON] et al. [19] propose a nonlinear integer optimization model--later relaxed and linearized--to minimize passengers' average travel time. The model accounts for five station arrangement scenarios, a safety collision constraint, and accommodates transfers and backtracking. Calculations show that this mode saves about 10 percent in passenger travel time compared to the all-stop strategy. [PERSON] et al. [20] investigated the fast and slow train scheduling problem under the fixed overrun station mode and quantitatively analyzed the relationship between energy consumption, travel time, and timetable to reduce the operational energy consumption by optimizing the train overrun points and at the same time reduce the passenger travel time. [PERSON] et al. [21] formulated skip-stop pattern optimization as a bi-level model to minimize passengers' generalized travel cost without increasing operational cost. The upper level is a mixed-integer skip-stop optimization model, while the lower level is a schedule-based transit assignment that captures passenger route choices. A heuristic solution method was developed to tackle the large-scale urban setting. Tested by example, the method reduced passenger travel time and lowered generalized cost. [PERSON] et al. [22] introduced a path scale Logit model with fare factors to analyze the distribution of passenger flows under the cross-line hopping stations of national railroads and suburban railroads. A unified modeling approach that simultaneously considers both aspects remains largely unexplored. An enhanced adaptive large neighborhood search (ALNS) algorithm [23] is developed to efficiently solve the proposed model. [PERSON] et al.