On guaranteed in-control performance for Shewhart X and X⁻ control charts

Recently, two methods have been published in this journal to determine adjusted control limits for the Shewhart control chart in order to guarantee a pre-specified in-control performance. One is based on the bootstrap approach (Saleh et al. (2015)), and the other is an analytical approach (Goedhart, Schoonhoven, and Does (2017)). Although both methods lead to the desired control chart performance, they are still difficult to implement by the practitioner. The bootstrap is rather computationally intensive, while the analytical approach consists of multiple integrals and derivatives. In this letter to the editor we simplify the analytical expressions provided in Goedhart, Schoonhoven, and Does (2017) by using the theory on tolerance intervals for individual observations as given in Krishnamoorthy and Mathew (2009).