Effects of the optimal n-step toll scheme on bulk carriers queuing for multiple berths at a busy port

Compared with the liner container ships, the tramp bulk carriers often have to queue for vacant berths at a busy port. In order to solve this problem, this paper develops the optimal n-step toll schemes (n=1, 2, 3, …) to bulk carriers in a queue for multiple berths. Bulk carriers' queuing times at the anchorage will be rationally decreased after pricing the toll schemes. This paper also shows the regularities in all equilibrium results under the toll schemes. The above outcomes are useful if the flexible pricing policy to a queue of bulk carriers is considered by the port authorities.