Online refueling policy for liner ships with offline learning

Liner shipping vessels consume substantial bunker fuel to maintain reliable service on fixed routes, yet volatile and spatially-correlated fuel prices across ports pose critical challenges for refueling optimization. This study addresses the online refueling problem where fuel prices are revealed sequentially, requiring real-time decisions on whether and how much to refuel at each port based on current inventory and observed prices. We formulate the problem as a Markov decision process and characterize the optimal policy through the Bellman equation. To capture the complex dynamics of fuel prices, we develop a multivariate Gaussian mixture (GM) model that approximates the joint price distribution across ports, coupled with GM regression for dynamic price prediction at unvisited locations. We propose an approximate dynamic programming framework that leverages Gaussian processes to model value functions and introduces a novel sampling procedure to generate informative training samples for policy learning. Through extensive computational experiments on synthetic datasets as well as empirical data from an Asia-Mediterranean shipping route, we demonstrate that our proposed policy achieves superior performance relative to established benchmark policies. Our findings reveal the GM model’s exceptional capability in capturing complex fuel price dependencies and volatility patterns, while the integration of predictive learning within the optimization framework yields significant cost reductions. This work contributes a theoretically grounded, data-driven methodology for dynamic refueling optimization under uncertainty, offering significant implications for operational efficiency and strategic fuel management in maritime logistics.