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This paper considers a container liner shipping problem with speed optimisation (CLSP-SO) to minimise the total costs of the fleet, which includes operating costs, capital costs and voyage costs. A mixed-integer nonlinear programming model is first formulated to illustrate the problem, in which the oil consumption of ships is treated as a cubic function of speeds. Then, the computational complexity of the problem is analysed and a lower bound is given based on the theoretical optimised speed of ships. To solve the problem, a probabilistic tabu search (PTS)-based algorithm is developed considering the NP-hardness of the problem. Extensive computational experiments on randomly generated data and a real-world case are conducted and the performance of the proposed method is compared with the lower bound and that of the basic tabu search (TS) algorithm. The results show that the proposed PTS-based algorithm obtains satisfactory solutions with respect to lower bounds in reasonable computation time and it outperforms the basic TS-based algorithm.

Probabilistic tabu search algorithm for container liner shipping problem with speed optimisation