Scheduling unit processing time jobs on an m-machine flow-shop

The sequential production of identical jobs and the flow-shop machine setting are extremely common in real-life applications. We study a scheduling problem that combines these two elements: jobs of identical processing time, with job-dependent weights, and a given common due date processed on an m-machine flow-shop. The (just-in-time) objective is to minimize the maximum earlier/tardiness cost. We introduce a polynomial time solution in both cases of (i) a non-restrictive (i.e., sufficiently large) due date, and (ii) a restrictive due date (which restricts the number of early jobs).