Is the weekly service frequency constraint tight when optimizing ship speeds and fleet size for a liner shipping service?

Abstract

Weekly service frequency is often maintained in liner shipping. The previous studies present two kinds of the weekly service frequency constraints. For a single liner shipping service, the two kinds of the weekly service frequency constraints can be described by using an equation and inequality, respectively. In order to explore whether the weekly service frequency constraint is tight or not in different cases, this paper addresses the optimization of ship speeds and fleet size for a single liner shipping service. According to realistic ship speeds at different legs in practice, we introduce different minimum and maximum allowable sailing speeds at various legs into our studied problem. Our problem can be formulated as a mixed-integer nonlinear programming model. By using the Karush-Kuhn-Tucker conditions, our analytical results show that, when considering the optimization of ship speeds, the weekly service frequency constraint is tight in most cases. Without considering the effect of the minimum and maximum allowable sailing speeds, we find that, the weekly service frequency constraint is always tight and ships would sail at the average speed. When considering the effect of the minimum and maximum allowable sailing speeds at various legs, the optimal ship speed at each leg should be selected from three options, based on which we further propose a ship speed choice problem. The proposed ship speed choice problem is also formulated as a mixed-integer nonlinear programming model, which can be efficiently solved by using the outer approximation method. Finally, a number of numerical experiments are carried out to account for the effectiveness of our analytical results, models and methods.