Best possible approximation algorithms for single machine scheduling with increasing linear maintenance durations

We consider a single machine scheduling problem with multiple maintenance activities, where the maintenance duration function is of the linear form f(t) = a+bt with a ≥ 0 and b > 1. We propose an approximation algorithm named FFD-LS2I with a worst-case bound of 2 for problem. We also show that there is no polynomial time approximation algorithm with a worst-case bound less than 2 for the problem with b ≥ 0 unless P = NP, which implies that the FFD-LS2I algorithm is the best possible algorithm for the case b > 1 and that the FFD-LS algorithm, which is proposed in the literature, is the best possible algorithm for the case b ≤ 1 both from the worst-case bound point of view.