This research considers the problem of scheduling jobs on parallel machines with non-common due dates and additional resource constraints. The objective is to minimize the total absolute deviation of job completion times about the due dates. Job processing times are assumed to be the same. This problem is motivated by restrictions that occur in the handling and processing of jobs in certain phases of semiconductor manufacturing and other production systems. We examine two problems. For the first of these, the number of different types of additional: resources and resource requirements per job are arbitrary. The problem is formulated as a zero-one integer linear programming and the Lagrangian relaxation approach is used. For the second case, there exists one single type of additional resource and the resource requirements per job are zero or one. We show how to formulate the problem as an assignment problem.