An efficient optimal solution of a two-crane scheduling problem

This paper studies a two-crane scheduling problem in a port terminal. These two cranes are deployed in a block of a port terminal. They can move along a bi-directional traveling lane and must maintain a safe distance between them to avoid collision. A group of export containers in a block is required to be transported by cranes from their storage location to an access point of the block. Each container is associated with a due date. The problem is to find a schedule such that all containers are carried to the access point before their due dates. If such schedule does not exist, then the problem becomes to find schedules to minimize the maximum tardiness, the number of tardy jobs, respectively. In this paper, we first identify the necessary and sufficient conditions for the existence of a feasible schedule, i.e., a schedule transports all containers to the access point before their due dates and does not violate the requirement to maintain a safe distance between two cranes. We prove that if there exists at least one feasible schedule, then there must be a permutation schedule with containers sequenced in earliest due date (EDD) order to be feasible. An efficient algorithm is developed to find a feasible schedule given its existence. Furthermore, we provide efficient algorithms for minimizing the maximum tardiness, the number of tardy jobs, the makespan and the total completion time, respectively.