An approximate algorithm for a high-multiplicity parallel machine scheduling problem

We consider a high-multiplicity parallel machine scheduling problem where the objective is to minimize the weighted sum of completion times. We suggest an approximate algorithm and we prove that it is asymptotically exact. The algorithm exploits a convex quadratic relaxation of the problem to fix a partial schedule, consisting of most jobs, and then assigns the residual jobs following a simple and general rule. The quality of the obtained solution is evidenced by some numerical tests.