In practice, vehicle scheduling is planned on a variable timetable so that the departure times of trips can be shifted in tolerable ranges, rather than on a fixed timetable, to decrease the required fleet size. This paper investigates the vehicle scheduling problem on a variable timetable with the constraint that each vehicle can perform limited trips. Since the connection-based model is difficult to solve by optimization software for a medium-scale or large-scale instance, a designed path-based model is developed. A Benders-and-Price algorithm by combining the Benders decomposition and column generation is proposed to solve the LP-relaxation of the path-based model, and a bespoke Branch-and-Price is used to obtain the integer solution. Numerical experiments indicate that a variable timetable approach can reduce the required fleet size with a tolerable timetable deviation in comparison with a fixed timetable approach. Moreover, the proposed algorithm is greatly superior to GUROBI in terms of computational efficiency and guarantees the quality of the solution.
Rescheduling is often needed when trains stay in segments or stations longer than specified in the timetable due to disturbances. Under crowded situations, it is more challenging to return to normal with heavy passenger flow. Considering making a trade-off between passenger loss and operating costs, we present a train regulation combined with a passenger control model by analyzing the interactive relationship between passenger behaviors and train operation. In this paper, we convert the problem into a Markov decision process and then propose the management strategy of regulating the running time and controlling the number of boarding passengers. Owing to the high dimensions of the large-scale problem, we applied the Approximate Dynamic Programming (ADP) approach, which approximates the value function with state features to improve computational efficiency. Finally, we designed three experimental scenarios to verify the effectiveness of our proposed model and approach. The results show that both the proposed model and the approach have a good performance in the cases with different passenger flows and different disturbances.
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