We address single-machine scheduling problems for which the actual processing times of jobs are subject to various effects, including a positional effect, a cumulative effect and their combination. We review the known results on the problems to minimize the makespan, the sum of the completion times and their combinations and identify the problems for which an optimal sequence cannot be found by simple priority rules such as Shortest Processing Time (SPT) and/or Longest Processing Time (LPT). Typically, these are problems to minimize the sum of the completion times under a deterioration effect, and we verify under which conditions for these problems an optimal permutation is V-shaped (an LPT subsequence followed by an SPT subsequence). We demonstrate that previously used techniques for proving that an optimal sequence is V-shaped are not properly justified. We use the corrected method to describe a wide range of problems with a pure positional effect and a combination of a cumulative effect with a positional effect for which an optimal sequence is V-shaped. On the other hand, we show that even the refined approach has its limitations.

中文翻译：

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在单台机器上优化 V 形优化排序的条件，以最大限度地减少综合效果下的总完成时间

我们解决了单机调度问题，作业的实际处理时间受到各种影响，包括位置影响、累积影响及其组合。我们审查问题的已知结果以最小化完工时间、完成时间的总和及其组合，并确定无法通过简单的优先级规则（例如最短处理时间 (SPT) 和/或最长）找到最佳序列的问题处理时间 (LPT)。通常，这些是在恶化效应下最小化完成时间总和的问题，我们验证在哪些条件下这些问题的最佳排列是 V 形的（LPT 子序列后跟 SPT 子序列）。我们证明了以前用于证明最佳序列是 V 形的技术没有得到适当的证明。我们使用校正方法来描述具有纯位置效应和累积效应与位置效应的组合的广泛问题，其中最优序列为 V 形。另一方面，我们表明即使是经过改进的方法也有其局限性。