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In the subway system, passenger crowding in peak hours is likely to cause train delays that easily propagate to following trains, resulting in a lower efficiency of the system. Consequently, this paper focuses on determining a robust timetable for the trains on the one hand, i.e., finding a better timetable to avoid delay propagation as much as possible in case of a crowded subway system. On the other hand, this paper considers the energy efficiency, i.e., reducing the total energy consumption during operations by selecting appropriate speed profiles and maximizing the utilization of regenerative braking energy. A related mathematical optimization model is formulated with the objective of maximizing the robustness and minimizing the total energy consumption. In order to solve this model, an efficient algorithm, i.e., simulation-based variable neighborhood search algorithm, is presented to obtain a good timetable in reasonable amount of time. Finally, experiments are implemented to show the performance of the proposed algorithm.

Considering the dynamics and randomness of passenger demand, this paper investigates a train timetabling problem in the stochastic environment for an urban rail transit system. With the scenario-based representation of passenger distribution, an integer nonlinear programming (INLP) model is first formulated to simultaneously optimize the total number of train services, headway settings and speed profile selection decision during the planning time horizon, in which the expected total service cost is treated as the objective function. Through an analysis of the features of the nonlinear constraints, a reformulation method is proposed to develop an equivalent integer linear programming (ILP) model that can be easily solved by commercial software. Moreover, a variable neighborhood search algorithm is developed to find the approximate optimal solutions for large-scale problems within the tolerable computing time. Finally, two sets of numerical experiments, with the operation environments of a simple urban rail transit line and Fuzhou Metro Line 1, are implemented to verify the solution quality and effectiveness of the proposed methods.