Mathematical model for Quay Crane Scheduling Problem with spatial constraints

In the last decades, competition between port container terminals, especially between geographically close one, is rapidly increasing. To improve this competitiveness, terminal managers try to achieve rapid container vessel loading and unloading, that corresponds to a reduction of the time in port for vessels. In this paper, we focus our attention on the operational decision problem related to the seaside area of maritime container terminals. In particular, we study The Quay Crane Scheduling Problem (QCSP) which is considered as a core task of managing maritime container terminals and the optimization of these operations affects significantly the time spent by vessels at berth. The main goal behind this planning problem is to find the optimized sequence of loading and unloading tasks on a set of deployed quay cranes in order to exploit the full performances of port's resources while reducing the berth's total time occupation by vessels. In this paper, we provide a rich model for quay crane scheduling problem that covers important parameters such as ready time and due dates of Quay cranes (QCs), safety margin in order to avoid congestion between QCs and precedence relations among tasks. The proposed model seeks for a more compact mathematical formulation that can be easily solved by a standard optimization solver. Thus, we formulated the Quay Crane Scheduling Problem as a mixed-integer linear model that minimizes the sum of the QCs holding cost and tardiness penalty cost.