Non-normality effects on the economic–statistical design of X¯ charts with Weibull in-control time

We consider the economic–statistical design of X¯-control charts for non-normal quality measurements. Specifically, we assume that the sample average X¯ has a Johnson distribution. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The cost model, proposed by McWilliams, is used to determine the optimal design parameters—the sample size, time between successive samples, and number of standard deviations away from the center line. This work is a generalization of Rahim’s models; for example, it combines mainly three of Rahim’s models: (i) economic design of X¯ chart under non-normality [Rahim, M.A., 1985. Economic model of X¯-chart under non-normality and measurement errors. Computer and Operations Research 12, 291–299], (ii) economic design of X¯ chart under Weibull shock models [Banerjee, P.K., Rahim, M.A., 1988. Economic design of X¯-control charts under Weibull shock models. Technometrics 30, 407–414], and (iii) economic–statistical design of X¯ charts with non-Markovian in-control times [Al-Oraini, H.A., Rahim, M.A., 2003. Economic statistical design of X¯ control charts for systems with gamma (λ, 2) in-control times. Journal of Applied Statistics 30, 397–409].Our sensitivity analysis shows that non-normality has a significant effect on the design parameters and hence should not be ignored. Sensitivity to the Weibull shape and the process-mean shift are also considered. We also compare the economic–statistical and fully economic designs for non-normal data.