# Optimization of Multi-Day Flexible EMU Routing Plan for High-Speed Rail Networks
## 1 Introduction
### Research on Operation Optimization in EMU Routing Plans  
In existing research, EMU routing plans are usually abstracted as vehicle routing problems with capacity constraints. The goal of optimization is to use fewer EMUs and build models based on multi-commodity networks. [PERSON] et al. [2] developed an arc-based network flow model, using graph theory to represent connection decisions and constraints, and proposed heuristic methods including local search and Lagrangian relaxation, validated with real-world data. [PERSON] et al. [3] proposed a fast heuristic for the train unit assignment problem, which seeks to allocate rolling stock to scheduled trips with known seat demands. The method is based on the Lagrangian relaxation of a natural integer programming formulation and solves a sequence of assignment problems. Compared to existing approaches, their heuristic offers significantly faster computation while maintaining high-quality solutions, making it well-suited for real-time applications and integration with other planning phases. [PERSON] et al. [4] addressed EMU operations on Korea's high-speed rail by ensuring the coverage of all services and minimizing empty movements and then reassigning EMUs under repositioning constraints to better utilize network-wide loops, solved via a two-stage approach. [PERSON] et al. [5] compared two exact methods for rolling stock (re)scheduling--one flow-based and one path-based--demonstrating that explicitly modeling unit order improves computational efficiency and practical applicability under real-world conditions. [PERSON] et al. [6] addressed the high-speed train unit routing problem by incorporating both time- and mileage-based maintenance constraints. They developed a general connection-based routing model and adapted it to China's two-day circulation structure. To improve tractability, a network reduction strategy was introduced. The study also extended the model to multi-depot scenarios and validated its effectiveness using real-world data from the Chinese high-speed railway. [PERSON] et al. [7] investigated EMU routing optimization under practical operational requirements by considering three distinct objectives: minimizing the number of circulations, minimizing connection time between trips, and minimizing the number of train-sets used. They analyzed the performance and trade-offs under each objective to evaluate their impact on circulation planning outcomes. With increasingly refined EMU management standards, operators impose stage-specific maintenance requirements to ensure functional safety. As a result, EMU routing optimization research has expanded to incorporate maintenance planning. [PERSON] et al. [8; 9; 10] addressed this by developing integer programming models that minimize the total cost of maintenance-related turnarounds. They designed heuristic algorithms combined with the CPLEX solver and validated their approach using real-world data from the Dutch railway network. [PERSON] et al. [11] developed a mixed-integer programming model to minimize both the number of train-sets used and the number of empty movements, while incorporating maintenance requirements and depot locations. A branch-and-bound algorithm was designed to solve the model, which was validated using case studies from Serbian railways. [PERSON] et al. [12] introduced maintenance job duration constraints to ensure on-time inspections and proposed an integer programming model aimed at minimizing operating costs. They solved it using a column generation approach combined with Lagrangian relax ation, achieving tight bounds within reasonable computation time. [PERSON] et al. [13] formulated a highly scalable mixed-integer programming model based on hyper-graphs that considers train compositions, maintenance needs, and depot capacity limits. A local search heuristic was implemented and applied to practical cases on German railways. In the context of disruptions, [PERSON] et al. [14] proposed a rescheduling model to minimize connection costs, depot storage, and empty mileage.