Jointly optimizing ship sailing speed and bunker purchase in liner shipping with distribution-free stochastic bunker prices

This paper jointly designs the optimal ship sailing speeds on shipping voyages and the optimal amount of bunker fuel to purchase at each port of a shipping network operated by a container liner shipping company. Bunker prices at these ports are assumed to be correlated random variables. Considering the difficulties in calibrating these prices into specific joint probability distribution in practice, this study merely requires some fundamental descriptive statistics information of these bunker prices, including lower and upper bounds, means and covariances, which can be tangibly estimated from historical data. To solve this problem, a mixed integer programming model is first formulated for deterministic bunker prices to minimize the sum of ship operating cost and bunker consumption cost. This model is subsequently extended to incorporate stochastic bunker prices by developing a series of approximation techniques using the bunker price descriptive statistics information. A numerical example based on real-case price data of a liner shipping network from an international shipping company shows that the proposed model is able to simultaneously control the average bunker purchase cost as well as the risk resulting from the extremely high bunker prices.