The effect of machine availability on the worst-case performance of LPT

We consider the makespan minimization parallel machine scheduling problem where each machine may be unavailable for a known time interval. For this problem, we investigate how the worst-case behavior of the longest processing time first algorithm (LPT) is affected by the availability of machines. In particular, for given m machines, we analyze the cases where arbitrary number, λ, ranging from one to m-1, machines are unavailable simultaneously. Then, we show that the makespan of the schedule generated by LPT is never more than the tight worst-case bound of 1+12⌈m/(m-λ)⌉ times the optimum makespan.