An improved algorithm for a two-stage production scheduling problem with an outsourcing option

Abstract We consider a two-stage production scheduling problem where each operation can be outsourced or processed in-house. For each operation in the same machine, the ratio of its outsourcing cost to its processing time is constant. The objective is to minimize the sum of the makespan and the total outsourcing cost. It is known that this problem is either polynomial time solvable or NP-hard according to the conditions of the ratios. Even though approximation algorithms for NP-hard cases had been developed, their tight worst-case performance ratios are still open. In this paper, we carefully analyze the approximation algorithms to identify their tight worst-case performance ratios for cases. In one case, we propose a new approximation algorithm with a better and tight worst-case performance ratio. In the process of analyzing the algorithm, we propose a technique utilizing nonlinear optimization.