In this paper we consider single machine scheduling problems with a common due-date for all jobs, arbitrary monotone earliness and tardiness costs and arbitrary breakdown and repair processes. We show that the problem is equivalent to a deterministic one without breakdowns and repairs and with an equivalent cost function of a job's completion time. A V-shaped schedule without idle times is shown to be optimal, if this equivalent cost function is quasi-convex.
Conversely, we show that a V-shaped schedule may fail to be optimal if the property does not apply. We derive general conditions for the earliness and tardiness cost structure and repair and breakdown processes under which the equivalent cost function is quasi-convex. When a V-shaped schedule is optimal, an efficient (though pseudo-polynomial) algorithm can be used to compute an optimal schedule.
Federgruen, Awi, and Gur Mosheiov. "Single machine scheduling problems with general breakdowns, earliness and tardiness costs." Operations Research 45, no. 1 (1997): 66-71.
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