Parallel machines scheduling with rate-modifying activities and total absolute differences penalties

In this paper we consider identical parallel machines scheduling problems with rate modifying, activities. in This model each machine has a rate-modifying activity that can change the processing rate of machines under consideration. Hence the actual processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs minimizing core to objective function. We concentrate on two goals We show that the problems remain polynomially solvable under the proprod model.