Parallel machine scheduling with almost periodic maintenance and non-preemptive jobs to minimize makespan

We introduce and study a parallel machine scheduling problem with almost periodic maintenance activities. We say that the maintenance of a machine is -almost periodic if the difference of the time between any two consecutive maintenance activities of the machine is within . The objective is to minimize the makespan Cmax, i.e., the completion time of the last finished maintenance. Suppose the minimum and maximum maintenance spacing are T and T′=T+ , respectively, then our problem can be described as Pm,MS[T,T′]Cmax. We show that this problem is NP-hard, and unless P=NP, there is no polynomial time ρ-approximation algorithm for this problem for any ρ<2. Then we propose a polynomial time 2T′/T-approximation algorithm named BFD-LPT to solve the problem. Thus, if T′=T, BFD-LPT algorithm is the best possible approximation algorithm. Furthermore, if the total processing time of the jobs is larger than 2m(T′+TM) and min{pi} T, where TM is the amount of time needed to perform one maintenance activity, then the makespan derived from BFD-LPT algorithm is no more than that of the optimal makespan. Finally, we show that the BFD-LPT algorithm has an asymptotic worst-case bound of 1+σ/(1+2σ) if min{pi} T, where σ=TM/T′.