Minimizing total load on parallel machines with linear deterioration

We study a scheduling problem with linear deterioration of job processing times. We consider parallel identical machines. The objective is minimum total load, i.e., minimum total processing times on all the machines. We propose a simple heuristic of a greedy type for this NP-hard problem. A lower bound based on the geometric mean of the job deterioration factors is also introduced. The greedy heuristic is shown to guarantee an optimal solution for small size problems, and our numerical tests indicate that in general it produces very close-to-optimal schedules.