Abstract: This research establishes a definitive closed-form equation for the average run length (ARL) of the double modified exponentially weighted moving average (DMEWMA) control chart employed for the ARMA(p,q) process. The proposed formula is derived analytically and validated through comparison with the numerical integral-equation (NIE) method. To assess the agreement between the two methods, we examine the percentage of accuracy. To evaluate the monitoring capabilities of the DMEWMA chart, additional performance metrics—SDRL and MRL—are computed across various chart design configurations. We use RMI, AEQL, and PCI to examine how well the DMEWMA chart detects mean-process shifts when applied to the modified EWMA (MEWMA) and the EWMA charts. A real-world example of monthly unemployment-rate production-index data from Bangkok and southern Thailand demonstrates the effectiveness of the explicit ARL-based DMEWMA chart in detecting changes in process means. The findings demonstrate that the DMEWMA chart outperforms the MEWMA and EWMA charts in identifying process shifts.

Keywords: Autoregressive and Moving Average, Average Run Length, Double Modified Exponentially Weighted Moving Average, zero-state, process- shift detection, Numerical Integral equation

Pages: 693-710