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As an important means of transportation, urban rail transit provides effective mobility, sufficient punctuality, strong security, and environment-friendliness in large cities. However, this transportation mode cannot offer a 24-h service to passengers with the consideration of operation cost and the necessity of maintenance, that is, a final time should be set. Therefore, operators need to design a last train timetable in consideration of the number of successful travel passengers and the total passenger transfer waiting time. This paper proposes a bi-level last train timetable optimization model. Its upper level model aims to maximize the number of passengers who travel by the last train service successful and minimize their transfer waiting time, and its lower level model aims to determine passenger route choice considering the detour routing strategy based on the last train timetable. A genetic algorithm is proposed to solve the upper level model, and the lower level model is solved by a semi-assignment algorithm. The implementation of the proposed model in the Beijing urban rail transit network proves that the model can optimize not only the number of successful transfer directions and successful travel passengers but also the passenger transfer waiting time of successful transfer directions. The optimization results can provide operators detailed information about the stations inaccessible to passengers from all origin stations and uncommon path guides for passengers of all origin–destination pairs. These types of information facilitate the operation of real-world urban rail transit systems. [ABSTRACT FROM AUTHOR]

• Present a new cyclic train timetabling model based on extended time-space network. • Decompose the cyclic train timetabling model through Lagrangian relaxation and ADMM. • Solve each train-specific sub-problem efficiently with forward dynamic programming. • Compare with the PESP-based model on a set of real-world instances. The cyclic train timetabling problem aims to synchronize limited operational resources toward a master periodic schedule of transport services. By introducing an extended time-space network construct, this paper proposes a new type of integer programming model reformulation for the cyclic train timetabling problem on a double-track railway corridor at the macroscopic level. This reformulation method also holds the promises to be applied in a broader set of routing and scheduling problems with periodic activity requirements. We also hope that this space-time network extension technique, as a special version of variable splitting methods in the dual decomposition literature, could potentially bridge the modeling gaps between cyclic and non-cyclic timetables. Specifically, the existing mathematical programming model for the periodic event scheduling problem (PESP) is transformed into a multi-commodity network flow model with two coupled schedule networks and side track capacity constraints. In addition, two dual decomposition methods including Lagrangian relaxation and Alternating Direction Method of Multipliers (ADMM), are adopted to dualize the side track capacity constraints. For each train-specific sub-problem in an iterative primal and dual optimization framework, we develop an enhanced version of forward dynamic programming to find the time-dependent least cost master schedule across the time-space network over multiple periods. ADMM-motivated heuristic methods with adjusted penalty parameters are also developed to obtain good upper bound solutions. Based on real-world instances from the Beijing-Shanghai high-speed railway corridor, we compare the numerical performance between the proposed reformulation and the PESP model that involves the standard optimization solver. [ABSTRACT FROM AUTHOR]