energy efficient parallel machine scheduling problem with deterioration effect to minimize makespan and total energy consumption

With the growing concerns about the environment and global warming, energy consumption has become a priority for manufacturers. however, parallel machine systems consume a large amount of energy per unit of time, considering the running of several machines in parallel. meanwhile, these manufacturing systems are subject to deterioration effects, which may increase unnecessarily the total energy consumption. thus, we treat a parallel machines scheduling problem with energy consumption and deterioration effect to minimize makespan and the total consumed energy. we consider that job energy consumption depends on its actual processing time, and for each position, a quantity of energy is wasted due to deterioration. first, we present a mathematical programming m model. this model is not able to give solutions in reasonable computational time. however we adapt to our problem a multi-objective simulated annealing algorithm. to evaluate the efficiency of the algorithm, we test it for small, medium and large instances.