Explicit ARL Computational for a Modified EWMA Control Chart in Autocorrelated Statistical Process Control Models

This study presents an innovative development of the exponentially weighted moving average (EWMA) control chart, explicitly adapted for the examination of time series data distinguished by seasonal autoregressive moving average behavior—SARMA(1,1)_L under exponential white noise. Unlike previous works that rely on simplified models such as AR(1) or assume independence, this research derives for the first time an exact two-sided Average Run Length (ARL) formula for the Modified EWMA chart under SARMA(1,1)_L conditions, using a mathematically rigorous Fredholm integral approach. The derived formulas are validated against numerical integral equation (NIE) solutions, showing strong agreement and significantly reduced computational burden. Additionally, a performance comparison index (PCI) is introduced to assess the chart’s detection capability. Results demonstrate that the proposed method exhibits superior sensitivity to mean shifts in autocorrelated environments, outperforming existing approaches. The findings offer a new, efficient framework for real-time quality control in complex seasonal processes, with potential applications in environmental monitoring and intelligent manufacturing systems.