Custom Timetable Maker in PHP

Unpredictable disturbances that occur when the train is running often make the actual timetable deviate from the planned timetable and deteriorate service quality for passengers. To provide a robust planned timetable that is reliable in operation, this paper presents a two-stage distributionally robust railway timetabling (DRRT) model, where the decisions in the first and second stages are the planned and actual timetables, respectively. To deal with the uncertain travel time, the proposed DRRT model assumes the true distribution belongs to a Wasserstein ambiguity set centered at the empirical distribution constructed from finite samples and aims to minimize the sum of the total idle time, travel time, and worst-case expected delay over the ambiguity set. To solve this model, we design a decomposition algorithm with provable convergence, which iteratively solves a master problem and a set of subproblems. To enhance the algorithm efficiency, a blend of two popular decomposition methods is proposed to integrate their advantages, and the solution of the subproblems is accelerated by using an uncapacitated minimum-cost flow interpretation of the second-stage optimization problem. Experiments on real-world instances demonstrate the effectiveness of these enhancements. The results also show that our model prominently outperforms the sample average approximation model in terms of the out-of-sample performance, especially when only a small sample set is available. Moreover, at the expense of equal traffic efficiency, our model is more robust than those models that do not utilize distribution information. Note to Practitioners—This paper was motivated by the problem of using finite historical data to provide a robust planned railway timetable, aiming at minimizing the traffic efficiency losses and expected delays under the worst-case distribution. Existing studies on dealing with the uncertain travel time usually assumed that the distribution is exactly known, or the distribution is completely unknown but the travel time belongs to an uncertainty set. However, in practice, the exact distribution is hard to know but can only be coarsely estimated from data. This paper proposes a data-driven two-stage model by assuming that the true distribution belongs to a 1-Wasserstein ball centered at the empirical distribution with a bounded support set, which is a middle ground between the above two distribution assumptions. To solve this problem efficiently, we propose a customized algorithm with provable convergence and then enhance it by exploiting problem-specific properties. This distributionally robust model can be easily shifted to other timetabling problems, and the algorithm enhancement techniques can also be applied to efficiently solve other two-stage recourse models with similar properties.