We study the single-machine preemptive scheduling problem with multiple maintenance activities to minimize the total late work, in which the jobs must be processed in the time space not occupied by the maintenance intervals. For this problem, we present a polynomial algorithm to determine the optimal schedule and establish a formula expression to the optimal value. Moreover, our result is used to correct some minor errors in the literature related to the single-machine (preemptive or non-preemptive) scheduling with one maintenance activity to minimize the total late work. 

We study the single-machine preemptive scheduling problem with multiple maintenance activities to minimize the total late work, in which the jobs must be processed in the time space not occupied by the maintenance intervals. For this problem, we present a polynomial algorithm to determine the optimal schedule and establish a formula expression to the optimal value. Moreover, our result is used to correct some minor errors in the literature related to the single-machine (preemptive or non-preemptive) scheduling with one maintenance activity to minimize the total late work.