Approximation schemes for minimizing total (weighted) completion time with release dates on a batch machine

A batch machine is a machine that can process a number of jobs simultaneously as a batch, and the processing time of a batch is equal to the longest processing time of the jobs assigned to it. In this paper we present a polynomial time approximation scheme (PTAS) for scheduling a batch machine to minimize the total completion time with job release dates. Also, we present a fully polynomial time approximation scheme (FPTAS) for scheduling an unbounded batch machine, which can process an arbitrary number of jobs simultaneously, to minimize the total weighted completion time with job release dates.