Single machine scheduling with a restricted rate-modifying activity to minimize the weighted makespan

We investigate the single machine scheduling problem with a restricted rate-modifying activity (RMA) aimed at minimizing the weighted makespan. The RMA is an activity that modifies the machine’s production rate while occupying it for a specified duration. Importantly, its starting time is constrained to fall within a predetermined interval $[a,b]$. Our results demonstrate that the special case where parameter $a=0$ admits a polynomial-time solution, while the cases with $b=\infty$ or $b=a$ are proven to be binary NP-hard. For these NP-hard problems, we develop pseudo-polynomial time dynamic programming solutions. Notably, in the scenario where splitting is permitted, we establish a polynomial-time algorithm for the $b=a$ case, which in turn enables the derivation of a 2-approximation algorithm for the non-split version. Additionally, by combining our dynamic programming approach with the vector trimming technique, we achieve fully polynomial-time approximation schemes (FPTAS) for all NP-hard variants under consideration.