Polynomial-time solutions for minimizing total load on unrelated machines with position-dependent processing times and rate-modifying activities

We study the problem of minimizing total load on parallel unrelated machines. Job processing times are assumed to be machine- and position-dependent in the most general way. The scheduler may perform a rate-modifying maintenance activity on each machine. The processing times of the jobs scheduled after the maintenance are reduced. The maintenance time and the impact on the following jobs are machine-dependent. We introduce a solution algorithm which is polynomial when a constant bounds the number of machines. The special case of (general) job deterioration, the more general case in which job rejection is allowed, and the extension to the setting of job processing times, which are controllable through allocating a limited resource, are also studied. All these scheduling problems are shown to be solved in polynomial time.