The joint economic-statistical design of X and R charts for nonnormal data

We consider the joint economic-statistical design of X and R control charts under the assumption that the quality measurement and the in-control time have Johnson and Weibull distributions. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The four parameters–the sample size n, time h between successive samples, and the control factors k1 and k2 for the X and R charts–are determined so that the mean hourly loss-cost is minimized under constraints on the Type I and II error probabilities. We have generalized the Costa model to accommodate the Johnson and Weibull distributions. Sensitivity to nonnormality, shift, and Weibull scale parameter is considered in our analysis. Our sensitivity analysis shows that the optimal design parameters are sensitive to nonnormality. Comparisons of the fully economic and economic-statistical designs are given.