Single machine scheduling with small operator-non-availability periods

Through an industrial application, we were confronted with the planning of experiments where human intervention of a chemists is required to handle the starting and termination. This gives rise to a new type of scheduling problems, 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 experiments lasting several days whereas the operator unavailability periods are the 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, 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. We finally focus on the practical case of periodic and equal small unavailability periods. We prove that this problem is not in FPTAS for more than 3 periods and we derive an FPTAS for the two-period case. keywords: One machine scheduling, unavailability constraints, forbidden start and end, approximation