Last-train timetable synchronization for service compatibility maximization in urban rail transit networks with arrival uncertainties

•Service compatibility maximization is proposed for last train timetabling.•A probabilistic scenario set is proposed to represent the arrival uncertainties.•A novel mixed-integer nonlinear programming model is developed and linearlized.•A robust processing method is improved to meet the probabilistic scenario set.•The effectiveness of our approaches is verified using real-world case experiments. The last train services of urban rail transit offer all passengers the final chance to reach their destinations. In the context of multimodal transport, considering the random delayed arrivals of late-night vehicles of other transport modes at urban hub stations and their adverse effects on multimodal passengers transferring to urban rail transit, a probabilistic scenario set is established to determine the arrival uncertainties of multimodal passengers. In each scenario, the timetable synchronization problem is formulated as a mixed-integer nonlinear programming model that requires a performance trade-off between destination reachability and the remaining path distance of both non-multimodal and multimodal passengers, integrally termed last train service compatibility. Through linearization techniques, the model is transformed into an equivalent mixed-integer linear programming form, which can be efficiently solved using the Gurobi solver to obtain the optimal last train timetable for each scenario and form an alternative scheme set. An improved probabilistic scenario set-based regret value theory is then developed, in which a novel regret value calculation method is proposed. The scheme with the minimum total weighted opportunity loss in all scenarios is selected as the optimal robust. Real case experiments based on the Chengdu–Chongqing high-speed railway line and Chengdu metro network are conducted to test the performance of our model. The results show that compared with the original timetable, the optimized timetable reduces the number of unreachable passengers by 34.29 % and the sum of the average remaining path distance of non-multimodal and multimodal passengers by 57.43 % with the help of path planning for all passengers. The proposed approach is proven not only to balance the demands of both reachable and unreachable passengers, but also to significantly improve the robustness of the last train timetable to arrival uncertainties of multimodal passengers and reduce their service inequity.