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2016-09-19
AUTHORSSalah Haridy, Ahmed Maged, Saleh Kaytbay, Sherif Araby
ABSTRACTThe control chart is one of the most powerful techniques in statistical process control (SPC) to monitor processes and ensure quality. The sample size n plays a critical role in the overall performance of any control chart. This article studies the effect of n on the performance of Shewhart control charts, which have traditionally been used for monitoring both the mean and variance of a variable (e.g., the diameter of a shaft and the temperature of a surface). The study is conducted under different combinations of false alarm rate and process shift. The detection speed of the Shewhart charts is evaluated in terms of average extra quadratic loss (AEQL) which is a measure of the overall performance. It is found that n = 2 is the best sample size of the Shewhart X_&R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$\end{document} and X_&S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$\end{document} charts. The comparative study reveals that the X_&R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$\end{document} and X_&S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$\end{document} charts with n = 2 outperform the X_&R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$\end{document} and X_&S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$\end{document} charts with n ≥ 4 by at least 9 and 7 %, respectively, in terms of AEQL. These results contradict the common knowledge in SPC niche that n between 4 and 6 is usually recommended for the X_&R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$\end{document} and X_&S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$\end{document} charts. More... »
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