Scheduling on proportionate flow shop with job rejection and common due date assignment

We study proportionate flow shop scheduling problem allowing job rejection. The objective function is to minimize the earliness, tardiness, due date cost, and rejection cost. We consider the collaboration between the manufacturer and supplier. The common due date is measured in two-ways, which is only related to accepted jobs (total rejection case) or all jobs share a common due date (semi-rejection case). We show the relation between proportionate flow shop and single-machine problems, use the properties of single-machine to solve proportionate flow shop problems with job rejection, and propose the corresponding dynamic programming algorithm and algorithm complexity for the problems.