We study a scheduling problem on parallel identical machines, where the objective function is maximizing the weighted number of jobs completed exactly at their due-dates. The scheduler may reject a subset of the jobs, and the total permitted rejection cost is assumed to be bounded. Thus, when a decision has to be made regarding a certain job, the following options need to be considered: either (\(\mathrm{i}\)) identify the set of machines on which the job can be scheduled on time, and assign the job to one of these machines, or (\(\mathrm{ii}\)) reject the job (if possible), or (\(\mathrm{iii}\)) delay the job. A pseudo-polynomial dynamic programming algorithm is introduced for this NP-hard problem. Medium size problems are solved in reasonable running times. We then study a different version of the problem, in which job-dependent due-windows are considered, and the job processing times are assumed to be identical. This version is proved to be NP-hard, and a pseudo-polynomial dynamic programming algorithm is introduced as well. Our numerical study indicates that medium size instances can be handled for this version. In addition, for both problems, an alternative solution procedure based on integer-linear-programming formulation is introduced.

我们研究了并行相同机器上的调度问题，其中目标函数是最大化在到期日准确完成的作业的加权数量。调度程序可能会拒绝作业的子集，并且假定总的允许拒绝成本是有界的。因此，当必须对某个作业做出决定时，需要考虑以下选项： ( \(\mathrm{i}\) ) 标识可以按时安排作业的机器集，以及将作业分配给其中一台机器，或 ( \(\mathrm{ii}\) ) 拒绝该作业（如果可能），或 ( \(\mathrm{iii}\)) 延迟工作。针对这个NP-hard问题引入了一种伪多项式动态规划算法。中等规模的问题在合理的运行时间内得到解决。然后，我们研究了该问题的不同版本，其中考虑了与作业相关的到期窗口，并且假定作业处理时间相同。该版本被证明是 NP-hard 的，并且还引入了伪多项式动态规划算法。我们的数值研究表明这个版本可以处理中等大小的实例。此外，针对这两个问题，介绍了一种基于整数线性规划公式的替代求解过程。