Single-machine scheduling with production and rejection costs to minimize the maximum earliness

In this paper, we consider the single-machine scheduling problem with production and rejection costs to minimize the maximum earliness. If a job is accepted, then this job must be processed on the machine and a corresponding production cost needs be paid. If the job is rejected, then a corresponding rejection cost has to be paid. The objective is to minimize the sum of the maximum earliness of the accepted jobs, the total production cost of the accepted jobs and the total rejection cost of the rejected jobs. We show that this problem is equivalent to a single-machine scheduling problem to minimize the maximum earliness with two distinct rejection modes. In the latter problem, rejection cost might be negative in the rejection-award mode which is different from the traditional rejection-penalty mode in the previous literatures. We show that both of two problems are NP-hard in the ordinary sense and then provide two pseudo-polynomial-time algorithms to solve them. Finally, we also show that three special cases can be solved in polynomial time.