# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 8 Conclusions  
This study investigates the train scheduling problem involving express-local trains under a cross-line operation framework between suburban railways and urban rail transit. By adopting a space-time network modeling approach, an integrated physical network of the suburban railway and urban rail systems is constructed. Based on this, various categories of passenger space-time arcs and train operation arcs are further defined. A mathematical optimization model is formulated to minimize the total system cost, including both the total passenger travel cost and operator cost, while accounting for constraints such as passenger flow distribution and minimum departure intervals. To solve the model, an extended adaptive large neighborhood search (E-ALNS) algorithm is developed, incorporating customized destroy and repair operators tailored for express-local train combinations and cross-line timetable design. The model is reformulated into a linearized form to enable comparison with exact optimization results from Gurobi. Numerical experiments demonstrate that the proposed algorithm produces high-quality solutions with a relative gap below 4% in all tested instances, and 3.02% in the base demand scenario. Further comparisons among independent, full cross-line, and hybrid cross-line modes show that the hybrid mode achieves the best performance, reducing passenger travel costs by 6% and improving overall cost efficiency despite a slight increase in operating cost. These results indicate that integrating long and short cross-line trains enables a more balanced trade-off between supply and demand. Sensitivity analysis on key parameters reveals that a 5- min transfer walking time yields the most efficient outcome, while excessively short or long walking times degrade system performance. Higher penalty and waiting costs encourage operational adjustments but increase total cost due to value accumulation. A 10- min passenger maximum waiting time achieves a good balance between service coverage and waiting time, minimizing unserved passengers without excessive delay. Future research can be extended in several directions. First, the passenger demand in cross-line operations may exhibit temporal fluctuations; incorporating dynamic demand into the timetable optimization framework would enhance model applicability. Second, this study focuses on mixed operation with express-local trains capable of flexible stop-skipping; further analysis can be conducted by comparing scenarios where only express or local trains are deployed. Third, although this study adopts a single-objective cost-based  
Figure 10: Comparison of metrics for different maximum passenger waiting time. formulation to ensure computational tractability and consistency in evaluating operational and passenger-related costs, future research could explore multi-objective formulations to better balance trade-offs among service quality, equity, and cost-efficiency, especially in contexts involving conflicting interests between operators and passengers. Finally, the current model assumes fixed train formation sizes. Future work may explore flexible train formations in conjunction with demand-responsive passenger assignments to achieve more refined optimization results. Conceptualization, [PERSON] and X.G. ; methodology, J.Z. ; software, [PERSON] and J.P.; validation, [PERSON] and [PERSON]; formal analysis, [PERSON]; investigation, [PERSON] and X.G. ; resources, [PERSON] and [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON]; visualization, [PERSON]; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, X.G. All authors have read and agreed to the published version of the manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2025 JBZX001), the National Natural Science Foundation of China (U2469201), and Technology Research and Development Program Project of China State Railway Group Co., Ltd. (grant number P2024X007).