# Optimization of Multi-Day Flexible EMU Routing Plan for High-Speed Rail Networks
## 1 Introduction
### Research on Flexible Train Formation  
In recent years, the increasing convenience of high-speed rail has led to growing passenger demand, with significant variations in flow across stations along high-speed lines. In response, flexible EMU composition strategies are being explored to dynamically match these demand fluctuations. Consequently, a number of researchers have begun to investigate how passenger demand patterns and train formation configurations influence EMU routing plans.  
[PERSON] et al. [17] investigated the energy cost implications of flexible train formations on high-speed rail. They developed an optimization model to minimize energy consumption, considering passenger demand and train composition constraints. Case studies revealed that flexible formations significantly reduce energy costs under low-demand conditions, providing theoretical support for the broader adoption of flexible EMU operation strategies. [PERSON] et al. [18] developed an integer programming model to minimize operational costs, allowing train coupling and decoupling at specific stations based on Dutch railway practices. The model incorporated composition constraints and was solved using CPLEX. [PERSON] et al. [19] proposed a mixed-integer programming model under a fixed timetable and set seat demand, also accounting for coupling and decoupling operations. They designed an exact branch-and-bound algorithm and validated their approach with Dutch railway data. [PERSON] et al. [20] extended this line of research by formulating an integer program that integrates train composition and passenger demand, aiming to minimize total operational costs. The model was solved using a Benders decomposition method. [PERSON] et al. [21] proposed a trip sequence graph to model EMU movements and coupling/splitting operations in high-speed rail networks, incorporating maintenance constraints and the periodicity of trip sequences. Based on this graph, they formulated both path-based and arc-based integer linear programming models. A tailored branch-and-price algorithm was developed for the path-based model. Application to the eastern China high-speed rail network demonstrated the models' effectiveness and practical relevance through numerical experiments. [PERSON] et al. [22] developed an optimization model for high-speed rail operations with variable train formations, aiming to maximize ticket revenue while accounting for passenger demand and flexible EMU configurations. Their results showed that, under sufficient rolling stock availability, the optimized variable formation strategy significantly outperformed fixed formation plans in terms of revenue generation.  
Studies integrating passenger demand, train formation, and EMU circulation planning are also common in urban rail transit. [PERSON] et al. [23] jointly optimized train formation and rolling stock scheduling via a multi-objective mixed-integer nonlinear programming (MINLP) model based on a time-space network, which was linearized into single-objective MILP and solved using CPLEX. [PERSON] et al. [24] proposed a \"shadow train\" model to optimize coupling-based circulation plans, aiming to minimize both total connection costs and the variance in rolling stock utilization, solved via a multi-objective chaotic particle swarm algorithm. [PERSON] et al. [25] developed a two-stage stochastic programming model to address the dynamic and stochastic nature of commuting demand, incorporating both flexible train formations and robust passenger flow control. The model was reformulated into MILP for solving using CPLEX. [PERSON] et al. [26] tackled uneven urban passenger flow using a path-based MILP model with flexible train formations, solved using a branch-and-price algorithm. However, due to differences in demand patterns and operational characteristics, these urban transit models--typically ignoring empty runs--require significant adaptation before being applied to high-speed rail.