Minimizing fuel emissions by optimizing vessel schedules in liner shipping with uncertain port times

We consider the problem of designing an optimal vessel schedule in the liner shipping route to minimize the total expected fuel consumption (and emissions) considering uncertain port times and frequency requirements on the liner schedule. The general optimal scheduling problem is formulated and tackled by simulation-based stochastic approximation methods. For special cases subject to the constraint of 100% service level, we prove the convexity and continuous differentiability of the objective function. Structural properties of the optimal schedule under certain conditions are obtained with useful managerial insights regarding the impact of port uncertainties. Case studies are given to illustrate the results.