Single-machine scheduling with total late work and job rejection

We study single-machine scheduling problems with rejection, where we can refuse to process jobs in the shop at a certain cost. Solutions to the problems investigated in this study consist of (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 on the single machine. The quality of a solution is measured by two criteria. The first is the total late work 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 late work and the total rejection cost, while the goal in the second is to find a solution minimizing the total rejection cost, given a bound on the total late work. As both problems are NP -hard, we focus on the design of pseudo-polynomial time algorithms and approximation schemes.