Two‐stage production scheduling with an outsourcing option

This paper considers a two‐stage production scheduling problem in which each activity requires two operations to be processed in stages 1 and 2, respectively. There are two options for processing each operation: the first is to produce it by utilizing in‐house resources, while the second is to outsource it to a subcontractor. For in‐house operations, a schedule is constructed and its performance is measured by the makespan, that is, the latest completion time of operations processed in‐house. Operations by subcontractors are instantaneous but require outsourcing cost. The objective is to minimize the weighted sum of the makespan and the total outsourcing cost. This paper analyzes how the model’s computational complexity changes according to unit outsourcing costs in both stages and describes the boundary between NP‐hard and polynomially solvable cases. Finally, this paper presents an approximation algorithm for one NP‐hard case.