This paper considers outsourcing decisions in a scheduling problem. The objective of the problem is to minimize the outsourcing costs under the constraints of the capacity of outsourcing facilities and due date of each job. The problem is composed of three kinds of decisions. The first decision is the selection of jobs to be processed in-house, the second is to schedule the in-house processing jobs with due date constraints, and the last is to select an outsourcing provider for each outsourced job where the outsourcing provider has a capacity constraint. Some optimality conditions and solution properties for the problem are presented. A solution algorithm with pseudo-polynomial complexity is suggested to find the optimal solution of the problem. The main contributions of this paper are as follows: (1) the mathematical model for the problem is proposed, (2) the pseudo-polynomial algorithm, Depth First Search (DFS) is developed to find the optimal solution and (3) several optimality properties for the problem are addressed. Numerical experiments show that the DFS algorithm has better results than Dynamic Programming (DP) in efficiency even for larger numbers of jobs and outsourcing providers.

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