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The cumulative sum (CUSUM) control chart is a well-known process monitoring tool that is sensitive to small-to-moderate changes in process parameters. In this paper, we propose an approximated average run length (ARL) method based on the numerical integral equation (NIE) method for monitoring the mean of a long-memory autoregressive fractionally integrated process with an exogenous variable (ARFIX) running on a CUSUM control chart. The approximated ARL based on the NIE method is realized by solving a system of linear equations and integration based on the partitioning and summation of the area under the curve of a function derived by using the Gauss-Legendre quadrature. In a comparative study with the ARL based on analytical formulas, the proposed approximated ARL method could detect shifts of various sizes in the process mean of an ARFIX process running on a CUSUM control chart. In addition, the proposed method was compared with their analytical formulas in terms of the relative percentage change (r%) to verify the accuracy of the ARL results. The results revealed that the ARL results obtained from the NIE method are an approach to analytical formulas with r% of less than 0.25. Hence, the NIE method is very accurate and in excellent agreement with the analytical formulas approach. Apparently, the NIE method is an alternative as efficiently as the analytical method for this situation. It also performed well in comparison with the approximated ARL for the same process running on an exponentially weighted moving average control chart. In addition, real datasets are also used to demonstrate the efficacy of the proposed method.
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