Scheduling on proportionate flowshop with total late work and job rejection

In this paper, we consider proportionate flowshop scheduling problems allowing job rejection. The goal is to determine the set of accepted jobs, the set of rejected jobs, and the sequence of accepted jobs so that the sum of total late work and total rejection costs is minimized. When the completion time of a job is greater than its due date, the late work problem occurs. For late work, we study the total late work, which refers to the last operation of the job. Regarding the due date, we investigate two cases: (i) the due date is related to the job, and (ii) the due date can be assigned (assignable due date). Both cases are NP-hard problems; we analyze the properties of the problems and design pseudo-polynomial dynamic programming and alternative pseudo-polynomial time algorithms.