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<div align="justify"><b>Problem statement:</b> Although economic Xbar-R chart designs do guarantee the minimal&#13; operating cost, they typically have poor levels of statistical performance measures. The obvious&#13; limitation of the economic design is that the Type I error rate seems to be very high for many situations&#13; and will cause a large number of false alarms. This situation leads to an investigation of appropriate&#13; levels of control chart variables which consist of a sample size, an interval between samples or&#13; sampling frequency and the control chart limits. <b>Approach:</b> Evolutionary operations via the weighed&#13; centroid modified simplex, WCMSM and linear constrained response surface optimization, LCRSOM,&#13; methods are applied to optimize the Xbar-R chart variables in the analytical model called as the&#13; operating cost function. WCMSM allows the simplex to converge more rapidly towards an optimum&#13; via the weighted centroid of the hyperface by expansion and multiple ways of simplex contraction&#13; along the line of conventional reflection in order to speed up the convergence. LCRSOM is a&#13; combination of the principles of experimental designs, least squares and related mathematical&#13; programming models to approach the optimum. <b>Results:</b> The computational results for economic&#13; Xbar-R charts via both methods reveal that the process quality level affects the total cost more than&#13; relevant errors from selecting and inspecting samples including identifying assignable causes.&#13; However, total costs obtained from the LCRSOM seem to be better for all process scenarios.&#13; <b>Conclusions: </b>On Xbar-R charts, if a process is at the high quality level the total cost is better than a&#13; process with a low quality level without a consideration of errors from selecting/inspecting samples&#13; and identifying assignable causes.</div

Weighted Centroid Modified Simplex and Linear Constrained Response Surface Optimization Methods for the Xbar-R Chart Variable Determination