Variable sample size and variable sampling interval Shewhart control chart with estimated parameters

The theoretical results about variable sample size and sampling intervals (VSSI) Shewhart control chart are concerned with the case where the mean and standard deviation of the given process are known. However, in practice the mean and standard deviation are unknown. In this case, the common treatment is to estimate the unknown parameters using historical data. After determining the parameters we treat the control limits as though fixed. Up to now we have no idea about the average time to signal (ATS) of VSSI X¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{X}}$$\end{document} charts when the control limits are estimated. This paper will calculate the mean of the ATS for VSSI X¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{X}}$$\end{document} charts when the control limits are estimated in three different ways. The results show that parameter estimation has significant effect on the in-control ATS, but has little effect on the out-of-control ATS. These results could show us some good reference idea for choosing adequate number of sample groups m and the sample size n of historical data. Finally, this paper obtains the optimal design of the maximum sample size n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_2$$\end{document} when other parameters are fixed.