This study considers single processor scheduling problems incorporating a rate-modifying activity (RMA) as well as processing time deterioration rates. The RMA fully restores the processing speed of a processor. As discussed in recent literature, the optimal position of an RMA for a single processor scheduling problem with the objective of makespan minimization, $1|p_{\left[j\right]}^A={\alpha }_{\left[j\right]}{S}_{\left[j\right]},rm|C_{max}$, where the processing time of a job ($p_{\left[j\right]}^A:\left[j\right]\text{denotes}\text{the}j\text{th}\text{position})$is determined by its start time ($S_{\left[j\right]})$and position-dependent rate (${\alpha }_{\left[j\right]})$, can be found easily. This study considers six variants of the problem: (P1) $1|p_{\left[j\right]}^A=\overline{p}+{\alpha }_{\left[j\right]}{S}_{\left[j\right]},rm|C_{max}$, $\left(\mathrm{P}{1}^{+}\right)1\left|p_{\left[j\right]}^A=\overline{p}+{\alpha }_{\left[j\right]}{S}_{\left[j\right]},drm\right|C_{max}$, (P2) 1$|p_{\left[i,j\right]}^A=p_j+{\alpha }_{\left[i\right]}{S}_j,rm|C_{max}$, $\left(\mathrm{P}{2}^{+}\right)1\left|p_{\left[i,j\right]}^A=p_j+{\alpha }_{\left[i\right]}{S}_j,drm\right|C_{max}$, (P3) $1|p_j^A={\alpha }_j{S}_j,rm|C_{max}$, and $\left(\mathrm{P}{3}^{+}\right)1\left|p_j^A={\alpha }_j{S}_j,drm\right|C_{max}$, where $\overline{p}$ is the common basic processing time, $p_j$is the basic processing time for job j, [i,j] is the job j scheduled at the ith position, and $drm$ is the deteriorating RMA. We show that (P1) and (P1⁺), where the common basic processing time is considered, can be converted into the problem $1|p_{\left[j\right]}^A={\alpha }_{\left[j\right]}{S}_{\left[j\right]},rm|C_{max}$, so that the optimal position of the RMA can be found easily. In the other variants, the optimal sequences of jobs and positions of the RMA must be determined. In (P2) and (P2⁺), the basic processing time for each job ($p_j)$is considered, whereas, in (P3) and (P3⁺), the deterioration rate depends not on the position of the job (${\alpha }_{\left[j\right]})$ but on the job itself (${\alpha }_j$). We show that the optimal schedules for (P2), (P2⁺) and (P3), (P3⁺) can be found by solving the assignment problems and the knapsack problems, respectively.

本研究考虑了包含速率修改活动 (RMA) 以及处理时间恶化率的单处理器调度问题。RMA 完全还原处理器的处理速度。正如在最近的文献中所讨论的那样，以最小化完工时间为目标的单处理器调度问题的 RMA 的最佳位置，$1|{磷}_{\left[j\right]}^{\mathrm{一种}}={\alpha }_{\left[j\right]}{秒}_{\left[j\right]},r米|C_{米\mathrm{一种}X}$, 其中作业的处理时间 (${磷}_{\left[j\right]}^{\mathrm{一种}}：\left[j\right]\text{表示}\text{这}j\text{日}\text{位置})$由其开始时间决定（${秒}_{\left[j\right]})$和位置相关率（${\alpha }_{\left[j\right]})$，很容易找到。本研究考虑了该问题的六个变体：(P1)$1|{磷}_{\left[j\right]}^{\mathrm{一种}}=\overline{磷}+{\alpha }_{\left[j\right]}{秒}_{\left[j\right]},r米|C_{米\mathrm{一种}X}$, $\left(\mathrm{磷}{1}^{+}\right)1\left|{磷}_{\left[j\right]}^{\mathrm{一种}}=\overline{磷}+{\alpha }_{\left[j\right]}{秒}_{\left[j\right]},dr米\right|C_{米\mathrm{一种}X}$, (P2) 1$|{磷}_{\left[\mathrm{一世},j\right]}^{\mathrm{一种}}={磷}_j+{\alpha }_{\left[\mathrm{一世}\right]}{秒}_j,r米|C_{米\mathrm{一种}X}$, $\left(\mathrm{磷}{2}^{+}\right)1\left|{磷}_{\left[\mathrm{一世},j\right]}^{\mathrm{一种}}={磷}_j+{\alpha }_{\left[\mathrm{一世}\right]}{秒}_j,dr米\right|C_{米\mathrm{一种}X}$, (P3) $1|{磷}_j^{\mathrm{一种}}={\alpha }_j{秒}_j,r米|C_{米\mathrm{一种}X}$， 和 $\left(\mathrm{磷}{3}^{+}\right)1\left|{磷}_j^{\mathrm{一种}}={\alpha }_j{秒}_j,dr米\right|C_{米\mathrm{一种}X}$， 在哪里 $\overline{磷}$ 是常见的基本处理时间， ${磷}_j$是作业j的基本处理时间，[ i,j ] 是调度在第i个位置的作业j，并且$dr米$是不断恶化的 RMA。我们证明了 (P1) 和 (P1 ⁺ )，其中考虑了共同的基本处理时间，可以转化为问题$1|{磷}_{\left[j\right]}^{\mathrm{一种}}={\alpha }_{\left[j\right]}{秒}_{\left[j\right]},r米|C_{米\mathrm{一种}X}$，以便可以轻松找到 RMA 的最佳位置。在其他变体中，必须确定 RMA 的最佳作业顺序和位置。在 (P2) 和 (P2 ⁺ ) 中，每个作业的基本处理时间 (${磷}_j)$被考虑，而在（P3）和（P3 ⁺）中，恶化率不取决于工作的位置（${\alpha }_{\left[j\right]})$ 但在工作本身（${\alpha }_j$）。我们表明，(P2), (P2 ⁺ ) 和 (P3), (P3 ⁺ )的最佳调度可以通过分别解决分配问题和背包问题来找到。