Development of average run length formulas for EWMA control chart under the AR(1) with quadratic trend model for detecting and monitoring process variability

Abstract

This paper proposes an explicit formula for calculating the average run length of an exponentially weighted moving average (EWMA) control chart based on the first-order autoregressive model with quadratic trends. Furthermore, the study presents a technique for estimating the average run length using the numerical integral equation (NIE) method. This enables a comparison between the results of the explicit formula and the numerical integral equation method. The two ARL solutions obtained from the explicit formula and numerical integral equation method are similar and identical with an absolute percentage difference of less than 0.001. Thereby, the explicit formula accurately corresponds to the NIE method. Additionally, the explicit formulas are more computationally efficient as they require fewer computations compared to the NIE approach