The average run length for continuous distribution process mean shift detection on a modified EWMA control chart
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Abstract
Herein, we provide an approximated average run length (ARL) solved by applying integral equations for detecting shifts in the process mean on a modified exponentially weighted moving average (EWMA) control chart when the observations are from continuous distributions such as gamma or Weibull. We compared numerical approximations of the ARL using four quadrature rules: the composite midpoint, trapezoidal, and Simpson’s rules and the Gauss-Legendre rule. The shape and scale parameters of four continuous distributions of the observations: Gamma (2, 1), Gamma (3, 1), Weibull (2, 1), and Weibull (3, 1) were determined according to their skewness. The criterion for evaluating the performances of the four quadrature rules and control charts was the out-of-control ARL and CPU Time. Our analysis reveals that the accuracies of the four quadrature rules to approximate the ARL on a modified EWMA control chart with observations from either gamma or Weibull distributions were similar. However, the Gauss-Legendre rule provided the simplest ARL calculation and achieved the highest accuracy for the given number of nodes. Meanwhile, the results reveal the superiority of the modified EWMA control chart over the standard one in terms of detecting a shift in the process mean. In addition, the efficacies of the control charts using the approximated ARL solutions were also demonstrated using a continuous distribution of real observations.
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