Robust optimization of train scheduling with consideration of response actions to primary and secondary risks.

Nowadays, with the rapid development of rail transportation systems, passenger demand and the possibility of the risks occurring in this industry have increased. These conditions cause uncertainty in passenger demand and the development of adverse impacts as a result of risks, which put the assurance of precise planning in jeopardy. To deal with uncertainty and lessen negative impacts, robust optimization of the train scheduling problem in the presence of risks is crucial. A two-stage mixed integer programming model is suggested in this study. In the first stage, the objective of the nominal train scheduling problem is to minimize the total travel time function and optimally determine the decision variables of the train timetables and the number of train stops. A robust optimization model is developed in the second stage with the aim of minimizing unsatisfied demand and reducing passenger dissatisfaction. Additionally, programming is carried out and the set of optimal risk response actions is identified in the proposed approach for the presence of primary and secondary risks in the train scheduling problem. A real-world example is provided to demonstrate the model's effectiveness and to compare the developed models. The results demonstrate that secondary risk plays a significant role in the process of optimal response actions selection. Furthermore, in the face of uncertainty, robust solutions can significantly and effectively minimize unsatisfied demand by a slightly rise in the travel time and the number of stops obtained from the nominal problem.