Coordinating transfers between bus and metro train in real time: A logic-based branch-and-bound method

The concept of Mobility-as-a-Service and on-demand public transit increasingly depends on integrating multiple transportation modes to offer seamless door-to-door travels. However, coordinating these modes in real time to address fluctuating demand and disturbances remains a major technical challenge. We tackle this issue by formulating a mixed-integer quadratic programming model to generate coordinated bus and train adjustment strategies. The model captures key factors such as traffic dynamics, passenger loads, and vehicle overtaking. To address the computational challenge of the mixed-integer property, we incorporate logic-based concepts into a branch-and-bound framework, analyzing the logical relationships among variables to improve solution efficiency. This logic-based branch-and-bound method exploits the distinct strengths of discrete and continuous components. It employs logical inference to guide the search, incorporates domain reduction to accelerate computation, and constructs reduced continuous optimization problems to efficiently update bounds and estimate logical values. Computational results demonstrate that the proposed coordinated adjustment strategy effectively improves vehicle punctuality and headway regularity, while reducing the number of stranded non-transfer and transfer passengers. The solution method demonstrates desirable computational efficiency in real-world settings, which is suitable for real-time applications.