The hub transportation problem with chance constrained due dates

Railway transportation offers an efficient and reliable service between stations. However, in remote areas, reduced accessibility due to large distances between stations and passengers' origin/destination points presents a significant challenge for railway companies. In low-density areas and in the countryside in particular, the supply of complementary transport to the train is often limited, or even nonexistent during off-peak hours, and does not always allow for an intermodal journey. It is essential to be able to offer public transport authorities a complementary offer to provide this feeder transport service to or from stations to make the journey as seamless as possible. A key aspect to operate this service successfully is to ensure that shuttles arrive at the train station before the time that passengers are due to take their train. Consequently, finding an optimal routing plan for shuttles is a highly time-sensitive problem. Besides, an inaccurate representation of travel times can easily lead to infeasible routing decisions with respect to meeting the due date that passengers are required to be at the train station. An algorithm based on column generation is proposed to address the problem. The pricing problem incorporates a chance constrained approach, deeming a route feasible if the shuttle can reach the train station before the passengers' due date. Travel time uncertainty is modeled using a discrete and finite uncertainty set to represent various travel time scenarios. The chance constrained approach ensures that the final decision remains feasible by meeting the passengers' due date in at least a specified proportion of the travel time scenarios