Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables

The exponentially weighted moving average (EWMA) control chart is a popular tool used to monitor and identify slight unnatural variations in the manufacturing, industrial, service processes. In general, charts operate under assumption of normality observation attention quality feature, but it not easy maintain this practice. such situations, data random processes are correlated data, as stock price economic field or air pollution environment field. characteristics performance measured by run length (ARL). article, we present new explicit formula ARL for EWMA based on MAX(q,r) process. proposed process proved using Fredholm integral equation technique. Moreover, values also assessed numerical equations method Gaussian, midpoint, trapezoidal rules. Banach's fixed point theorem guarantees existence uniqueness solution. Furthermore, accuracy absolute percentage relative error compared with method. results found that formula's similar those obtained method; errors less than 0.0001 percent. As result, essential conclusion outperforms computational time. Consequently, have been alternative approaches computing chart. They would be applied various fields, including economics, environment, biology, engineering, others.