An Approximation of ARL of the DHWMA Control Chart for MA(q) Process using the Numerical Integral Equation Method

Quality control of products and services is essential across various industries and sectors, including manufacturing, engineering, public health, economics, finance, and medicine. It is critical to ensure that product characteristics align with customer requirements while maintaining consistent quality and standards. Control charts are essential tools in Statistical Process Control (SPC), used to detect changes in process means or variability, and play a critical role in monitoring, controlling, and enhancing quality within production processes. This study evaluates the Average Run Length (ARL) of the Double Homogeneously Weighted Moving Average (DHWMA) control chart for Moving Average processes (MA(q)) with exponential white noise, using Numerical Integral Equation (NIE) methods. These methods include Simpson’s Rule, the Midpoint Rule, the Trapezoidal Rule, and the Gauss–Legendre Quadrature. Additionally, the study compares the performance of the DHWMA and HWMA control charts. The results indicate that all four NIE methods yield similar ARL values. However, CPU time analysis shows that the Midpoint Rule generally requires less computation time than the other methods. Furthermore, when evaluating change-point detection effectiveness, the DHWMA control chart consistently outperforms the HWMA chart across all shift sizes, as demonstrated by its lower Expected Average Run Length (EARL).