Tramp ship routing and scheduling with integrated bunker optimization

A tramp ship operator often has contracted cargoes that must be carried and seeks to maximize profit by carrying optional cargoes. Hence, tramp ships operate much like taxies following available cargo and not according to fixed route networks and itineraries as liner ships. Marine fuel is referred to as bunker and bunker costs constitute a significant part of daily operating costs. There can be great variations in bunker prices across bunker ports so it is important to carefully plan bunkering for each ship. As ships operate 24 h a day, they must refuel during operations. Therefore, route and schedule decisions affect the options for bunkering. Current practice is, however, to separate the two planning problems by first constructing fleet schedules and then plan bunkering for these fixed schedules. In this paper we explore the effects of integrating bunker planning in the routing and scheduling phase for a tramp operator sailing full shiploads, i.e. carrying at most one cargo onboard each ship at a time. We present a mixed integer programming formulation for the integrated problem of optimally routing, scheduling and bunkering a tramp fleet carrying full shiploads. Aside from bunker integration, this model also extends standard formulations by using load dependent costs, speed and bunker consumption. We devise a solution method based on column generation with a dynamic programming algorithm to generate columns. The method is heuristic mainly due to discretization of the continuous bunker purchase variables. We show that the integrated planning approach can increase profits and that the decision of which cargoes to carry and on which ships is affected by the bunker integration and by changes in the bunker prices.