Approximation algorithm for uniform quay crane scheduling at container ports

At a container port, container vessels are served by quay cranes for loading and unloading containers. Each vessel is typically split into bays from head to tail where containers are stored. Parallel quay cranes can process different bays simultaneously, and their processing efficiency significantly affects the turn-around time of a container vessel. Sharing a single traveling rail, the quay cranes cannot crossover each other, and this phenomenon is referred as the non-crossing constraint. In addition, the quay cranes may have different processing speeds due to gradual equipment updates. Inspired by updating activities of cranes in modern container terminals, this paper studies a scheduling problem with two uniform quay cranes, aiming at minimizing the turn-around time of a vessel, i.e., the makespan. We mainly develop an integrated approximation algorithm which is min{(s + 1)/s, (s + 1)2/(s + 2)}-approximation, where the two quay cranes are of processing speeds 1 and s(≥ 1), respectively.