Single machine scheduling with small operator-non-availability periods

Abstract

For an industrial application in the chemical industry, we were confronted with the planning of experiments, where human intervention of a chemist is required to handle the starting and termination of each of the experiments. This gives rise to a new type of scheduling problem, namely problems of finding schedules with time periods when the tasks can neither start nor finish. We consider in this paper the natural case of small periods where the duration of the periods is smaller than any processing time. This assumption corresponds to our chemical experiments lasting several days, whereas the operator unavailability periods are typically single days or week-ends. These problems are analyzed on a single machine with the makespan as criterion.

We first prove that, contrary to the case of machine unavailability periods, the problem with one small operator non-availability period can be solved in polynomial time. We then derive approximation and inapproximability results for the general case of k small unavailability periods, where k may be part of the input or k may be fixed. We consider in particular the practical case of periodic and equal small unavailability periods. We prove that all these problems become NP-hard if one has k≥2 small unavailability periods and the problems do not allow fully polynomial time approximation schemes (FPTAS) for k≥3. We then analyze list scheduling algorithms and establish their performance guarantee.