Minimizing Total Weighted Late Work in a Proportionate Flow Shop with Job Rejection

In this paper, we investigate a scheduling problem with optional job rejection in a proportionate flow shop setting, where the job processing times are machine independent. A solution to our problem is characterized by (i) partitioning the set of jobs into a set of accepted jobs and a set of rejected jobs, and (ii) scheduling the accepted jobs in a proportionate flow shop setting. The aim is to find a solution to minimize the sum of total weighted late work of the accepted jobs and total rejection cost of the rejected jobs. When all jobs share a common due date, we show that the single-machine case is [Formula: see text]-hard by reduction from the Subset Sum problem. When the operations of all jobs have equal processing times, we solve the case in [Formula: see text] time by reducing it into a linear assignment problem. For the general problem, we first provide a pseudo-polynomial-time algorithm via the dynamic programming method, then we convert it into a fully polynomial time approximation scheme. As a byproduct, we also resolve an open question in the literature.