Exact Average Run Length Evaluation on One-Sided and Two-Sided Extended EWMA Control Chart with Correlated Data

Abstract

The Extended Exponentially Weighted Moving Average (Extended EWMA) control chart is one of the control charts that can rapidly detect a minor shift. Using the average run length (ARL), the control charts' effectiveness can be evaluated. This research aims to derive the explicit formulations for the ARL on one-sided and two-sided Extend EWMA control charts for the MA(1) model with exponential white noise, as they have not been previously presented. The analytical solution accuracy was determined and compared to the numerical integral equation (NIE) method. The results indicate that the ARL calculated using the explicit formula and the NIE method are nearly identical, demonstrating the validity of the explicit formula. In addition, this study is extended to compare the performance with the Exponentially Weighted Moving Average (EWMA) control chart. The results show that the Extended EWMA control chart has superior efficacy to the EWMA control chart. To demonstrate the efficacy of the proposed method, the analytical solution of ARL is finally applied to real-world data on Thailand's monthly fuel price.