Minimizing total weighted completion time with an unexpected machine unavailable interval

Abstract

In the past two decades, scheduling with machine availability constraints has received considerable attention. Until recently most research has focused on the setting where all machine unavailability information is known at the beginning of the scheduling horizon. In reality, this is not practical in some cases. The machine may become unavailable to process jobs due to a machine breakdown or an occurrence of an emergent job that has to be processed immediately. In both cases, the start time of the unavailable interval is unknown beforehand, and the length of the interval may not be known until the end of the interval. In this article, we consider the situation in which the scheduler has to make scheduling decisions without any knowledge of the machine unavailable intervals. Of particular interest is the problem of minimizing total weighted completion time. When there are two or more unavailable intervals on a single machine, Fu et al. (2009) have shown that the problem is exponentially inapproximable even when jobs' weights are equal to their processing times and one has full knowledge of unavailability. So in this paper we consider the scheduling problem on a single machine with a single unavailable period. And, we assume that every job has a weight proportional to its processing time. Based on whether the unavailable interval is due to a breakdown or an emergent job, we have the breakdown model and the emergent job model. First we show that no $$\tfrac{\sqrt{5}+1}{2}$$ 5 + 1 2 -competitive online algorithm exists for the breakdown model, and no $$\tfrac{11-\sqrt{2}}{7}$$ 11 ? 2 7 -competitive online algorithm exists for the emergent job model. Next, we show that the simple LPT rule can give a 2- and a $$\tfrac{9}{5}$$ 9 5 -competitive ratio for the breakdown model and the emergent job model, respectively. Further, we show that the ratios are tight by examples. For the offline case, we show that the First Fit LPT (FF-LPT) rule can give a tight approximation ratio of 2 and 4/3 for the breakdown model and the emergent job model, respectively. Finally, our experimental results show that, in practice, both LPT and FF-LPT perform very well and the performance improves when the number of jobs $$n$$ n increases. In both models, when $$n \ge 50$$ n ? 50 , the worst case error ratio is much better than the theoretical bounds.