# Minimizing Total Weighted Completion Time with Unexpected Machine Unavailability

## Abstract

In the past two decades, scheduling with machine availability constraints has received more and more attention. Until now most research has focused on the setting where all machine unavailability information is known at the beginning of scheduling horizon. In real world, this is impractical in some cases.

In this article, we consider the situation where the scheduler has to make scheduling decisions without any knowledge of the machine unavailable intervals. In particular, we study the problem of minimizing the 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 breakdown model and emergent job model. We first show that no \(\tfrac{\sqrt{5}+1}{2}\)-competitive online algorithm exists for breakdown model, and no \(\tfrac{11-\sqrt{2}}{7}\)-competitive online algorithm exists for emergent job model. Then we show that the simple LPT (Largest Processing Time first) rule can give a 2-competitive ratio and 9/5-competitive ratio for breakdown model and emergent job model, respectively. We show the ratios are tight by examples. For offline case, we show that First Fit LPT (FF-LPT) rule can give a tight approximation ratio of 2 and 4/3 for breakdown model and 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* increases. When *n* ≥ 50, the worst error ratio of LPT is about 8.7 %, and the worst error ratio of FF-LPT is about 0.7%. So in both cases, the error ratio is quite far from the theoretical bound.

## Keywords

Completion Time Single Machine Competitive Ratio Total Weight Completion Time Idle Interval## Preview

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