The buffer sizing problem in unreliable production lines is a complex combinatorial optimization problem. In the formulation of the problem, the system consists of n machines and n fixed-size buffers in series. These machines produce parts with two different quality levels: conforming and non-conforming parts. The production line can contain inspection stations whose job is to reject the non-conforming parts from the line. Thus, the design goal is to select the optimal combination of buffer sizes and inspection stations to meet the demand for conforming parts while minimizing total investment cost. In this paper, we propose an exact algorithm that not only finds an optimal location of the inspection stations but also optimizes the number of needed inspection stations. We present new theoretical results on buffer sizes bounds and stationarity and cost function convexity. These results help reducing the complexity of the algorithm. With this algorithm, we are able to solve to optimality large problems with 20 machines.