Scheduling overnight trains for improving both last and first train services in an urban subway network

In most urban subway networks, passengers may make a transfer at night or in the early morning to arrive at their destinations. Given that passengers may confronted with transfer failure at night or long waiting in the morning, scheduling overnight trains is beneficial to both the last and first train services in the urban metro network, in particular, to minimize the number of failed transfer passengers in the last train service and the transfer waiting time for the first train service. Then, we formulate the overnight train scheduling problem as a mixed integer linear programming model based on a well-designed time-discretized space–time network. To handle large-scale applications, we decompose the proposed model into two space–time path-searching sub-problems and a semi-assignment problem through Lagrangian relaxation–based decomposition and solution framework. Finally, we adopt Beijing Subway Network as a practical case study. The results show that both the number of failed transfer passengers for last train service and the passenger transfer waiting time for first train service decrease significantly if the operators can schedule the overnight trains in Beijing Subway Network according to the optimization results of the proposed model.