A branch-and-cut algorithm for scheduling train platoons in urban rail networks

With the emerging of virtual coupling technologies, the concept of train platoon, where different vehicles can be flexibly and dynamically grouped or decoupled, has become a hot research topic. In this study, we investigate the scheduling of train platoons for urban rail networks with time-dependent demand to mitigate passenger inconvenience. We propose a mixed-integer linear programming (MILP) model that simultaneously optimizes the train-platoon (de)coupling strategies, arrival/departure times at each station, and the running orders of trains, while considering limited rolling stock resources at the depots and the safety of trains at cross-line zones. To tackle computational challenges in real-world instances, we develop a customized branch-and-cut solution algorithm, based on the analysis of mathematical properties of our MILP model, to generate (near-)optimal solutions more efficiently. In particular, we propose three sets of valid inequalities that are dynamically added to the model to strengthen the linear relaxation bounds at each node. We also design a customized branching rule in the search tree by imposing to branch on the key decision variables regarding the train orders at the cross-line zones. Real-world case studies based on the operational data of Beijing metro network are conducted to verify the effectiveness of our approach. The results demonstrate that our branch-and-cut-based approach evidently outperforms commercial solvers in terms of solution quality and computational efficiency. Compared to the current train schedule with fixed compositions in practice, our approach with flexible coupling strategies can reduce the passenger dissatisfaction by over 15%.