Robust ship scheduling with multiple time windows

We present a ship scheduling problem concerned with the pickup and delivery of bulk cargoes within given time windows. As the ports are closed for service at night and during weekends, the wide time windows can be regarded as multiple time windows. Another issue is that the loading/discharging times of cargoes may take several days. This means that a ship will stay idle much of the time in port, and the total time at port will depend on the ship's arrival time. Ship scheduling is associated with uncertainty due to bad weather at sea and unpredictable service times in ports. Our objective is to make robust schedules that are less likely to result in ships staying idle in ports during the weekend, and impose penalty costs for arrivals at risky times (i.e., close to weekends). A set partitioning approach is proposed to solve the problem. The columns correspond to feasible ship schedules that are found a priori. They are generated taking the uncertainty and multiple time windows into account. The computational results show that we can increase the robustness of the schedules at the sacrifice of increased transportation costs.