Optimal vessel speed and fleet size for industrial shipping services under the emission control area regulation

This study aims to determine the optimal vessel speed and ship fleet size for an industrial shipping service operating through the emission control areas (ECAs). To comply with the ECA regulation, the industrial shipping service operator can adjust its vessel speeds and fleet size to minimize the annual fleet cost, which includes the total fleet operating cost and in-transit cargo inventory cost. A mixed-integer convex cost-minimization model is developed for the determination of optimal vessel speeds and fleet size, in which the vessel speeds are continuous decision variables and the fleet size is an integer decision variable. The analytical optimal solution of the developed model is identified by relaxing the integer restriction of the fleet size variable. It is shown that how the in-transit cargo inventory cost can play a role in changing the way that the optimal vessel speed is identified. We also demonstrate that the minimal annual fleet cost could be attained by simply comparing two candidate analytical solutions. Numerical studies are carried out to investigate the economic and environmental impacts of the ECA regulation on several representative industrial shipping services. Recommendations are also provided on the proper modeling choice of fleet size in different contexts.