Scheduling multi‐operation jobs on a single machine

Abstract

We consider the problem of scheduling n multi‐operationjobs on a single machine. Eachjob comprises up to F operations that belong to different families.Changeovers of production from one family to another have associated set‐up times.A job completes when all of its operations have been processed. This type of problem arisesin the manufacture of food products. It combines the batching aspect of the well‐knownfamily scheduling models with an assembly element (where the job's operationsare assembled to make the final product).Our analysis covers three classic optimality criteria: the maximum lateness, theweighted number of late jobs, and the sum of job completion times. We show that the problemof minimizing the maximum lateness is equivalent to its counterpart without assembly.This enables us to derive extensions of known complexity results and to indicate appropriatealgorithms. The number of late jobs problem is shown to be binary NP‐hard when thereare two families, and unary NP‐hard when there are an arbitrary number of families,even when all set‐up times are identical. For a fixed number of families, we give adynamic programming algorithm to minimize the weighted number of late jobs, which requirespseudo‐polynomial running time. A similar algorithm solves the sum of completion timesproblem in polynomial time, under the additional assumption that the processing times ofoperations between families are agreeable.