In this paper, we considers the outsourcing decision problem in a single machine scheduling problem. The decision problem is to determine for each job whether to be processed on an in-house manufacturing or external facilities(outsourcing). Moreover, this paper considers a situation where each job has a due date. The objective of the problem is to minimize the outsourcing cost, subject to the due date constraints. The considered problem is proved to be NP-hard. Some solution properties and valid inequalities are derived, and an effective lower bound is derived based on the LP-relaxation. The results of experimental tests are presented to evaluate the performance of the suggested lower bound.