Minimizing the weighted number of early and tardy jobs with bounded batch

Processing jobs individually is extended to the situation where the jobs can be formed in batches to minimize the weighted number of early and tardy jobs. Bounded version of batch scheduling is discussed here and all jobs share a common due window. When the early and tardy penalty coefficients depend on the jobs and the due window location is a decision variable, the problem is proved to be strongly NP-Complete and several optimal properties are proposed. Then several special cases are solved.