Tramp Ship Routing and Scheduling with Voyage Separation Requirements

In this paper we explore tramp ship routing and scheduling. Tramp ships operate much like taxies following the available demand as opposed to liner ships that operate more like busses on a fixed route network according to a published timetable. Tramp operators can determine some of their demand in advance by entering into long term contracts and then try to maximise profits from optional voyages found in the spot market. Routing and scheduling a tramp fleet to best utilise fleet capacity according to current demand is therefore an ongoing and complicated problem. Here we add further complexity to the routing and scheduling problem by incorporating voyage separation requirements that enforce a minimum time spread between some voyages. The incorporation of these separation requirements helps balance the conflicting objectives of maximising profit for the tramp operator and minimising inventory costs for the charterer, since these costs increase if similar voyages are not performed with some separation in time. We have developed a new and exact Branch-and-Price procedure for this problem. We use a dynamic programming algorithm to generate columns and use a modified time window branching scheme to enforce the voyage separation requirements which we relax in the master problem. Computational results show that our algorithm finds very good if not optimal solutions extremely fast though one instance requires longer time. We also compare our method to an earlier a priori path generation method which we show is not optimal. Computational results confirm this as our algorithm consistently finds solutions that are equally good or better than those from the a priori generation method. Furthermore, for all but one instance, our solutions are found in the same or shorter time than those from the a priori generation method.