Single operation earliness-tardiness scheduling with machine activation costs

We consider a static, single operation, non-pre-emptive, deterministic scheduling problem in which a set of n jobs is to be processed on k identical machines. Jobs assigned to each machine have a common due date. The number of machines (k) is unknown. Activating a machine will require additional costs to be incurred. The objective is to find an optimal sequence, the optimal number of machines (k), and the respective due dates to minimize the weighted sum of earliness, tardiness, and machine activation costs. We propose a polynomial time algorithm to solve the problem.