The unbounded parallel batch machine scheduling with release dates and rejection to minimize makespan

In this paper, we consider the unbounded parallel batch machine scheduling with release dates and rejection. A job is either rejected with a certain penalty having to be paid, or accepted and processed in batches on the parallel batch machine. The processing time of a batch is defined as the longest processing time of the jobs contained in it. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. We show that this problem is binary NP-hard and provide a pseudo-polynomial-time algorithm. When the jobs have the same rejection penalty, the problem can be solved in polynomial time. Finally, a 2-approximation algorithm and a fully polynomial-time approximation scheme are given for the problem.