A note on minimizing total weighted completion time with an unexpected machine unavailable interval

Abstract

Recently, Huo et al. (J Sched 17(2):161–172, 2014) addressed single-machine scheduling problems with an unexpected machine unavailable interval. In their study, both the start time and the length of the unavailable interval are unknown beforehand. Two models were considered according to the way that the machine becomes unavailable, the breakdown model and the emergent job model. In this note, we further study several single-machine scheduling problems with an unexpected machine unavailable interval. For the breakdown model to minimize the total weighted completion time, we give a better lower bound which shows that the simple LPT rule can give the best possible competitive ratio. For the emergent job model to minimize the total weighted completion time, we give a new lower bound and design a best possible algorithm with a competitive ratio of $1+4\alpha /(4+\alpha ^2)$, where $\alpha \approx 0.6109$ is the root in (0, 1) of the equation $23\alpha ^4+24\alpha ^3+72\alpha ^2-32\alpha -16=0$. This improves upon the worst-case bound $(11-\sqrt{2})/7$ of the heuristic presented by Huo et al. (J Sched 17(2):161–172, 2014). Moreover, for minimizing the total completion time, we prove no 9/7- and 5/4-competitive online algorithm exist for the breakdown model and emergent job model, respectively. Then, we propose a best possible algorithm for the latter model.