Tramp ship routing and speed optimization with tidal berth time windows

Tramp shipping companies need to schedule shipping routes and decide on appropriate speeds based on short-term demand. This task differs from traditional vehicle routing problems (VRPs) in that the ship may wait for the tide, which changes with time. The wait time is a nonlinear function of the load, and in this paper, we describe this kind of wait as the ship following a tidal berth time windows. Additionally, the speed of the ship affects both the wait time and the sailing cost. This paper proposes a mixed-integer nonlinear programming model to tackle this problem. A branchand-price framework is applied to solve the model efficiently, decomposing the model into a set partitioning master problem and an elementary shortest path subproblem. A labeling algorithm incorporating ship speed is developed to handle the subproblem, and further enhancements are made by optimizing the speed separately. Computational experiments show the effectiveness and accuracy of the proposed solution approach for large-scale instances. Moreover, considering the tidal time windows allows for exploiting the practical benefits of raising tides, which benefits the tramp shipping industry.