Developing Average Run Length for EWMA Control Chart for Stationary Auto-regressive Moving Average Model with Factor Model

The efficacy of control charts is evaluated using the primary indicator, Average Run Length (ARL), which assesses chart performance and process behavior. This research aims to determine the formula that can be utilized with the Fredholm integral method to compute the ARL base on exponentially weighted moving average (EWMA) chart for auto-regressive moving average process with factor variables (ARMAX (p,q,r)). The fixed-point principle confirmed the solution's being and irreproducibility, validating our proposed formula. Also, numerical methods for integral equations using Gauss-Legendre quadrature were used to estimate ARL so we could compare them with the output from the proposed explicit formula. The validity of the proposed formula was evaluated based on two criteria: accuracy and computational efficiency. We discover that ARL values obtained from the explicit formula are identical to those approximated via the numerical method, yielding an accuracy of 100%. Furthermore, the processing time required to calculate the ARL using the given formula is less than that of the numerical method. Hence, the proposed formula for the ARMAX(p,q,r) process can serve as a substitute for computing ARL values in EWMA chart. An alternative method for estimating ARL is the explicit formula provided for the ARMAX(p,q,r) process, as demonstrated using gasoline price data.