Abstract: This study presents the application of the Numerical Integral Equation (NIE) method to estimate the Average Run Length (ARL) of the Double Exponentially Weighted Moving Average (DEWMA) control chart under the Autoregressive with Exogenous Variables model, ARX(p,r). Four numerical approaches—Midpoint Rule, Trapezoidal Rule, Gauss Rule, and Simpson’s Rule are implemented to approximate ARL and evaluated in terms of estimation accuracy and computational efficiency (CPU time). The findings reveal that all four methods yield highly consistent ARL estimates across parameter variations, indicating their interchangeability. Among them, the Midpoint and Trapezoidal Rules achieve the shortest computation times, whereas Simpson’s Rule requires the longest. A comparative analysis of DEWMA and EWMA control charts, assessed through Expected Average Run Length (EARL), shows that DEWMA consistently outperforms EWMA. Specifically, DEWMA provides substantially lower EARL values, demonstrating its superior capability in rapid detection and process monitoring, particularly for mean shifts influenced by exogenous variables.

Keywords: Numerical Integral Equation Method, Average Run Length, Autoregressive model with exogenous variables, Double Exponentially Weighted Moving Average

Pages: 90-105