A Branch and Bound Solution Method for the Crane Scheduling Problem

Typical cargo ships spend 60% of their time in port, costing their owners about $1000 per hour. In this paper, we attack such costs with a method to speed loading and unloading. We model the need for container handling as generic “work,” which cranes can do at a constant rate. Each hold of each ship has a given amount of this work and cranes can interrupt their work on individual holds without any loss of efficiency. In the parlance of scheduling theory, this constitutes an “open shop” with parallel, identical machines, where jobs consist of independent, single-stage, preemptable tasks. Practical problems often involve only a few ships but many holds; the complication of preemptable tasks makes them very complex. The paper presents a branch and bound method which, for this model, minimizes delay costs (weighted tardiness). As part of the method, we extend previous solutions to the feasibility problem of preemptive machine scheduling (to cases where multiple machines can work simultaneously on a single task). Computational results and extensions to more complicated problems are offered. Certain concepts developed here may also be applicable to other problems, both in scheduling and elsewhere. In particular, they may lead to optimal solutions of problems for which feasibility determination methods already exist.