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 ɛ -almost periodic if the difference of the time between any two consecutive maintenance activities of the machine is within ɛ . The objective is to minimize the makespan C max , i.e., the completion time of the last finished maintenance. Suppose the minimum and maximum maintenance spacing are T and T ′ = T + ɛ , respectively, then our problem can be described as Pm , MS [ T , T ′ ] | | C max . 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 2 T ′ / 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 2 m ( T ′ + T M ) and min { p i } ⩾ T , where T M is the amount of time needed to perform one maintenance activity, then the makespan derived from BFD-LPT algorithm is no more than 3 2 that of the optimal makespan. Finally, we show that the BFD-LPT algorithm has an asymptotic worst-case bound of 1 + σ / ( 1 + 2 σ ) if min { p i } ⩾ T , where σ = T M / T ′ .