Real-Time Rolling Stock and Timetable Rescheduling in Urban Rail Transit Systems

Unexpected disruptions in urban rail transit systems cause the infeasibility of the initial train schedule and delays or cancelations of a lot of trains. Even though some recent studies begun to address the rolling stock and timetable optimization problem (RSTO), there is still a large gap between theoretical models and practical applications due to the real-time requirements of train rescheduling decisions. In this work, we first model RSTO using a path-based formulation, in which each path refers to a spatial-temporal trajectory of a rescheduled train in the considered network. The optimal set of paths can minimize the expected cost of train cancelation and train delay time. Our formulation also considers a series of operational constraints, such as train headway constraints, shortturning constraints and rolling stock constraints. We develop an efficient branch-and-price framework that decomposes the problem into a restricted master problem and a set of pricing subproblems, where we iteratively generate promising paths with negative reduce costs. We show that each subproblem is a resource-constrained shortest path problem and can be solved efficiently by an improved label setting algorithm by proving its optimality conditions. We compare the tightness of our new path-based formulation with state-of-art formulations and test our branch-and-price approach on real-world instances from Beijing rail transit. The results show that our approach can generate near-optimal solutions in less than three minutes with small duality gap, which evidently outperforms existing formulations and fulfills the requirement of rail managers in practical applications.