Exact Run Length Calculation on EWMA Control Charts with Seasonal and Exogenous Factors using SARMAX Models

This research evaluates the efficacy of EWMA control charts based on SARMAX models, contrasting explicit formulae and Numerical Integral Equation (NIE) techniques for determining Average Run Length (ARL). The research examines six SARMAX models under varying smoothing factors (= 0.15, 0.20, 0.25) with ARL0=370. Analysis shows that SARMAX(2,1,2)12 has the largest control limits while SARMAX(1,2,1)12 has the lowest. It's clear that increasing the parameter increases both the control limits and the ARL, but this comes at the cost of making the process less sensitive to changes. The findings validate explicit formulas as a reliable, efficient, and practical alternative for ARL computation in EWMA control charts applied to SARMAX processes, particularly beneficial for real-time monitoring applications.