Two-stage distributionally robust optimization for maritime inventory routing

This work addresses uncertain sailing times and uncertain waiting times at ports in a maritime inventory routing problem (MIRP). As the probability distribution of these uncertain parameters is difficult to estimate and hence not known exactly, we propose a two-stage distributionally robust optimization (DRO) approach in which the uncertainty is described by a Wasserstein ambiguity set. Our model is based on a continuous-time arc-flow mixed integer linear programming (MILP) formulation of the MIRP, and an equivalent robust counterpart of the two-stage DRO problem is derived under the 1-norm Wasserstein metric. We also develop a tailored Benders decomposition algorithm that combines the strengths of Pareto-optimal and high-density cuts to solve large-scale model instances. Computational case studies, including a real-world industrial case considering the maritime transportation of refined diesel along the east coast of China, demonstrate the benefits of the DRO model and the effectiveness of the proposed Benders decomposition algorithm. In general, compared to a traditional stochastic programming approach, the DRO model yields routing solutions that are significantly less sensitive to variations in sailing and port waiting times, and exhibit improved out-of-sample performance.