A necessary condition for the elimination of crane interference

Abstract

Cooper smelters and steel mills utilize large overhead cranes to transport material from one location to another within the plant. The instructions "move material from source A to sink B" are termed "jobs". Because all cranes share the same track and cannot pass each other, the set of jobs may impose a set of conflicting demands on the cranes. Two cranes assigned to specified jobs may be mutually blocked in their transit from source to sink. This is known as crane interference. The completion of one of the jobs must of necessity delay the completion of the other. Delays due to crane interference may be minimized by an appropriate assignment of cranes to jobs.

In this paper, a model is presented in which the scheduling problem reduces to that of constructing batches of jobs which can be assigned to cranes so that crane interference is eliminated. A necessary but not sufficient condition for such job batches can be determined by deriving the partition number. The partition number is defined to be the minimum cardinality of an ordered partition of an n-tuple. An efficient algorithm (O(n²)) is described to determine a minimum ordered partition.

This is followed by a graph-theoretic representation of the problem in which it is shown that the determination of the partition number is equivalent to finding the stability number of a transitive 1-graph. The complement graph is identified as a permutation graph.

Job batches are constructed using a heuristic procedure. If the necessary condition is not satisfied, then the batching strategy will not completely eliminate crane interference. There may exist another strategy which will eliminate interference. Therefore, other strategies should be examined before the use of the heuristic procedure.

Supported by the Natural Sciences and Engineering Research Council of Canada