Parallel-machine scheduling with deteriorating jobs and rate-modifying activities to minimize total completion time

This article explores the parallel-machine scheduling problems in which the processing time of a job is a linear function of its starting time and the job-independent deterioration rates are identical for all jobs. We implemented a rate-modifying activity to change the processing time of the jobs that followed this activity on each machine. Therefore, the actual processing time of a job varied given that it depended not only on its starting time but also on whether it was scheduled before or after the rate-modifying activity. Hence, the objective of this study was to determine the job sequence on each machine and the position of rate-modifying activity on each machine so that the total completion time was minimized. We showed that the problem could be solved in polynomial time when the number of machines was fixed.