Minimizing the total tardiness and job rejection cost in a proportionate flow shop with generalized due dates

We study a scheduling problem involving both partitioning and scheduling decisions. A solution for our problem is defined by (i) partitioning the set of jobs into a set of accepted and a set of rejected jobs and (ii) scheduling the set of accepted jobs in a proportionate flow shop scheduling system. For a given solution, the jth largest due date is assigned to the job with the jth largest completion time. The quality of a solution is measured by two criteria, one for each set of jobs. The first is the total tardiness of the set of accepted jobs, and the second is the total rejection cost. We study two problems. The goal in the first is to find a solution minimizing the sum of the total tardiness and the rejection cost, while the goal in the second is to find a solution minimizing the total rejection cost, given a bound on the total tardiness. As both problems are \({\mathcal {N}}{\mathcal {P}}\)-hard, we focus on the design of both exact algorithms and approximation schemes.