Average Run Length of Double Modified Exponentially Weighted Moving Average Control Chart by Numerical Integral Equation

Abstract

Through the use of Numerical Integral Equation (NIE) techniques—specifically, the Gaussian, Midpoint, Trapezoidal, and Simpson's rules—and the Double Modified Exponentially Weighted Moving Average (DMEWMA) control chart, this study aims to investigate the Average Run Length (ARL) approximation.  Assuming that process data follow continuous probability distributions, specifically the exponential and Weibull distributions, the analytical framework is developed.  Additionally, a thorough performance comparison is conducted between the DMEWMA control chart and two well-known substitutes: the Exponentially Weighted Moving Average (EWMA) control chart and the Modified Exponentially Weighted Moving Average (MEWMA) control chart. This comparative analysis is based on two critical performance indicators: the out-of-control Average Run Length (ARL₁) and computational efficiency, as quantified by CPU processing time. The empirical results demonstrate that all NIE-based methods produce ARL estimates that are statistically indistinguishable, affirming their mutual accuracy. Nonetheless, with respect to computational performance, the Midpoint and Trapezoidal rules exhibit superior efficiency, achieving reduced processing times. Furthermore, at all levels of shift magnitude, the DMEWMA control chart continuously outperforms the MEWMA and EWMA charts in terms of sensitivity to changes in the process mean.  All things considered, the results of this study highlight the effectiveness and usefulness of using NIE techniques for ARL estimation in sophisticated statistical process control. The proposed methodology not only yields precise and computationally efficient results but also exhibits strong applicability to a wide array of real-world datasets, thereby reinforcing its potential as a robust and versatile tool for contemporary process monitoring.