Analytical design of ARL based on a modified EWMA control chart under ARIMA processes

In statistical process control, the performance of control charts is commonly evaluated using the average run length (ARL), which represents the expected number of observations required to signal a change in the monitored process. In many practical applications, quality characteristics are observed as time series data, motivating the incorporation of time series models into control chart design. The principal innovation of this study is the development of a new explicit analytical formula for the ARL of the modified exponentially weighted moving average (EWMA) control chart for autoregressive integrated moving average (ARIMA) processes subject to exponentially distributed white noise. The proposed explicit solution enables direct and instantaneous computation of ARL. Its accuracy and consistency are validated against the numerical integral equation (NIE) method using a conformity measure, demonstrating nearly identical results. Comparative analysis demonstrates that the explicit formulation provides equivalent accuracy while substantially reducing computational time. The detection performance of the modified EWMA chart is further investigated under various design parameters and shift sizes revealing improved sensitivity to small shifts compared with the standard EWMA chart. The practical applicability of the proposed approach is illustrated using real disaster datasets, where the modified EWMA chart detects structural changes more rapidly than the EWMA chart. These findings confirm the accuracy, computational efficiency, and practical effectiveness of the proposed explicit ARL formulation for monitoring autocorrelated processes.