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

Abstract

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