An application of fuzzy random variables to control charts.

*(English)*Zbl 1206.93062Summary: The two most significant sources of uncertainty are randomness and incomplete information. In real systems, we wish to monitor processes in the presence of these two kinds of uncertainty. This paper aims to construct a fuzzy statistical control chart that can explain existing fuzziness in data while considering the essential variability between observations. The proposed control chart is an extension of Shewhart’s \(\overline X - S^2\) control charts in fuzzy space. The proposed control chart avoids defuzzification methods such as fuzzy mean, fuzzy mode, fuzzy midrange, and fuzzy median. It is well known that using different representative values may cause different conclusions to be drawn about the process and vague observations to be reduced to exact numbers, thereby reducing the informational content of the original fuzzy sets. The out-of-control states are determined based on a fuzzy in-control region and a simple and precise graded exclusion measure that determines the degree to which fuzzy subgroups are excluded from the fuzzy in-control region. The proposed chart is illustrated with a numerical example.

##### MSC:

93C42 | Fuzzy control/observation systems |

93E03 | Stochastic systems in control theory (general) |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

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\textit{A. Faraz} and \textit{A. F. Shapiro}, Fuzzy Sets Syst. 161, No. 20, 2684--2694 (2010; Zbl 1206.93062)

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