Combined timetabling of periodic and aperiodic trains using an efficient train-based Lagrangian relaxation decomposition

To better meet passenger travel demands and meanwhile improve regularity and flexibility of train operation, this study focuses on solving the optimization problem of periodic and aperiodic train timetabling, and establishes an integer linear programming model with many binary variables and non-negative integer variables. This model systematically considers the constraints of equal-interval operation of periodic trains, dwell times, arrival and departure safety time intervals, no overtaking requirements in the section and so on. Due to the NP-hard property of the established model, a space-time discretized network is constructed to reformulate and transform the periodic and aperiodic train timetabling problem into two types of shortest path-choice problems related to periodic and aperiodic trains, respectively. And an efficient train-based Lagrangian relaxation decomposition algorithm is designed to solve the reformulated model. Finally, the feasibilities of the proposed model and algorithm are verified by a set of numerical experiments, involving a small-scale case and a real-world case derived from Beijing-Shanghai high-speed railway.