Approximation Algorithm for Minimizing the Weighted Number of Tardy Jobs on a Batch Machine

We consider the problem of minimizing the weighted number of tardy jobs ($\sum_{j=1}^{n}w_jU_j$) on an unbounded batch processing machine. The batch processing machine can process up to B (B *** n ) jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. For a batch of jobs, the processing time of the batch is equal to the largest processing time among the jobs in this batch. In this paper, we design a fully polynomial time approximation scheme (FPTAS) to solve the unbounded batch scheduling problem $1|B\geq n|\sum_{j=1}^{n}w_jU_j.$ This is the strongest possible polynomial time approximation result that we can derive for an NP-hard problem (unless P = NP holds).