Train timetable optimization for urban railway systems under the virtual formation mode combined with the rolling stock utilization strategy

The distribution of passenger demands on certain urban railway lines exhibits obvious spatiotemporal imbalances, posing challenges for the traditional fixed formation mode. This paper presents the optimization of the virtual formation train timetable and rolling stock utilization strategy, which aims to maximize the quantity of connections and minimize the number of detained passengers. A mixed-integer nonlinear programming model (MINLP) is formulated to characterize this problem, in which the coupling/decoupling operations between different types of rolling stock are considered. By applying linearization techniques, the aforementioned MINLP model can be transformed into a mixed-integer linear programming (MILP) model. To effectively address the model, a two-stage (TS) optimization approach is designed to decompose the original problem into two sequential steps for the solution. In the first stage, a reduced-scale optimization problem is solved, focusing solely on a subset of services; then, the partial binary variables obtained from the first stage are incorporated into the original problem for further resolution in the second stage. Furthermore, we design an accelerated technique of bound contraction based on logical inference to enhance the solving efficiency of the second stage. Five sets of numerical experiments based on the Beijing metro Yizhuang line are conducted to verify the effectiveness and practicability of the model and algorithm. The experimental results illustrate that the virtual formation mode can effectively address the spatiotemporal imbalances of passenger demands on the line. The proposed TS approach is also proven to exhibit greater efficiency than traditional heuristic algorithms, such as genetic algorithm (GA), for large-scale problems.