We study single-machine scheduling problems with step-learning, where an improvement in processing time is experienced if a job is started at, or after, a job-dependent learning-date. We consider minimizing two functions: the number of late jobs and the total late work, and we show that when at least a common due-date or common learning-date is assumed, the problem is NP-hard in the ordinary sense; however, when both are arbitrary, the problem becomes strongly NP-hard. For each of the problems where at least one of the dates is assumed to be common, we analyze the structure of an optimal job schedule with and without idle time and propose pseudo-polynomial time dynamic programming algorithms. We also show that the problem of minimizing the weighted number of late jobs with step-learning can be solved with a minor change to the algorithms for the unweighted case. In addition to this, we show that when a common due-date is assumed and no idle time is allowed, the problem of minimizing the total late work is equivalent to that of minimizing the makespan. Furthermore, we provide a more efficient algorithm to solve the problem of minimizing makespan under the assumption of a common learning-date than the one in the existing literature. Lastly, we show that our analysis can also be applied to the case of step-deterioration, where instead, the processing times of jobs increase at a given date.

中文翻译：

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通过步骤学习最大限度地减少延迟作业的数量和总延迟作业


我们用步进学习来研究单机调度问题，如果一个作业在依赖于作业的学习日期或之后开始，那么处理时间就会有所缩短。我们考虑最小化两个函数：延迟工作的数量和总的延迟工作，我们表明，当至少假设一个共同的到期日期或共同的学习日期时，问题是通常意义上的 NP-hard;但是，当两者都是任意的时，问题就变得非常 NP-hard。对于每个假设至少有一个日期是共同的问题，我们分析了有和没有空闲时间的最佳作业计划的结构，并提出了伪多项式时间动态规划算法。我们还表明，通过步进学习最小化延迟作业的加权数量的问题可以通过对未加权情况的算法进行微小的更改来解决。除此之外，我们还表明，当假设一个共同的到期日期并且不允许空闲时间时，最小化总延迟工作的问题等同于最小化制作时间的问题。此外，我们提供了一种比现有文献中更有效的算法来解决在共同学习日期假设下最小化 makespan 的问题。最后，我们表明我们的分析也可以应用于阶梯恶化的情况，相反，作业的处理时间在给定日期增加。