Optimal Design of a CUSUM Chart for a Mean Shift of Unknown Size

A cumulative sum (CUSUM) control chart is one of the most popular methods used to detect a process mean shift. When one specific size of the mean shift is assumed, the CUSUM chart can be optimally designed in terms of average run length (ARL). In practice, however, the size of the mean shift is usually unknown, and the CUSUM chart can perform poorly when the actual size of the mean shift is significantly different from the assumed size. In this paper, we assign a probability distribution to the size of the mean shift to represent its lack of knowledge. We use an ARL-based performance measure, called expected weighted run length (EWRL), and propose a method to optimally design a CUSUM chart based on EWRL. This method can be easily extended to other CUSUM-based control charts, such as weighted CUSUM and multi-CUSUM charts proposed in the literature. The numerical results show that the CUSUM or the CUSUM-based chart can be improved by our proposed method in terms of EWRL.