Cruise service planning considering berth availability and decreasing marginal profit

The service is a sequence of ports (harbor cities) that the cruise ship visits. In this decision problem, the constraint about the availability of berths at each port is taken into account. In reality, if a cruise service is executed by the ship repeatedly for several times, the profit earned by the cruise service in each time decreases gradually. This effect of decreasing marginal profit is also considered in this study. We propose a nonlinear integer programming model to cater to the concavity of the function for the profit of operating a cruise service repeatedly. To solve the nonlinear model, two linearization methods are developed, one of which takes advantage of the concavity for a tailored linearization. Some properties of the problem are also investigated and proved by using the dynamic programming (DP) and two commonly used heuristics. In particular, we prove that if there is only one candidate cruise service, a greedy algorithm can derive the optimal solution. Numerical experiments are conducted to validate the effectiveness of the proposed models and the efficiency of the proposed linearization methods. In case some parameters needed by the model are estimated inexactly, the proposed decision model demonstrates its robustness and can still obtain a near-optimal plan, which is verified by experiments based on extensive real cases.