We explore the problem of scheduling n jobs on a single machine in which there are m fixed machine non-availability intervals. The target is to seek out a feasible solution that minimizes total weighted late work. Three variants of the problem are investigated. The first is the preemptive version, the second is the resumable version, and the third is the non-resumable version. For the first one, we present an \(O((m+n) \log n)\)-time algorithm to solve it. For the second one, we develop an exact dynamic programming algorithm and a fully polynomial time approximation scheme. For the third one, we first demonstrate that it is strongly \(\mathcal{NP}\mathcal{}\)-hard for the case where all jobs have the unit weight and common due date, and then we develop a pseudo-polynomial time algorithm for the unit weight case where the number of non-availability intervals is fixed, finally we propose a pseudo-polynomial time algorithm for the case where there is only one non-availability interval.

我们探讨了在有m个固定的机器不可用间隔的单台机器上调度n个作业的问题。目标是寻找一个可行的解决方案，以最大限度地减少总加权后期工作。研究了该问题的三个变体。第一个是抢占版，第二个是可恢复版，第三个是不可恢复版。对于第一个，我们提出了一个\(O((m+n) \log n)\)时间算法来解决它。对于第二个，我们开发了一个精确的动态规划算法和一个完全多项式时间近似方案。对于第三个，我们首先证明它是强\(\mathcal{NP}\mathcal{}\)-hard 对于所有工作都有单位权重和共同截止日期的情况，然后我们开发了一个伪多项式时间算法，用于单位权重的情况，其中不可用间隔的数量是固定的，最后我们提出了一个伪多项式对于只有一个不可用区间的情况的时间算法。