# Attribute Control Chart Construction Based on Fuzzy Score Number

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Plotted Data of a Control Chart Based on Fuzzy Number

## 3. Control Limits of a Fuzzy Number-Based Control Chart and Its Nonconformity Judgment

#### 3.1. Possibility and Necessity Measures

#### 3.2. Control Chart Nonconformity Judgment Rules

- (1)
- The possibility measure of $\stackrel{~}{S}$ under known $\overline{S}$ must be no less than the preset $\alpha (0<\alpha \le 1)$ (i.e., $Pos(\overline{S}|\stackrel{~}{S})\ge \alpha $ (See [20]));
- (2)
- The necessity measure of $\stackrel{~}{S}$ under known $\overline{S}$ must be no less than the preset $\beta (0<\beta \le 1)$ (i.e., $Nec(\overline{S}|\stackrel{~}{S})\ge \beta $).

#### 3.3. Parameters of Threshold for Nonconformity Judgment

## 4. Basic Form of a Fuzzy Control Chart

## 5. Case Study

#### 5.1. Construction of a Fuzzy Control Chart Based on Triangular Fuzzy Number

#### 5.2. Analysis of Control Chart

## 6. Influence of the of Membership Function Type on a Fuzzy Control Chart

- (1)
- As for the possibility measure, triangular membership function has the narrowest control limit (i.e., the highest control level), trapezoid type takes second place, and π-type function is last.
- (2)
- As for the necessity measure, π-type function has the narrowest control limit (i.e., the highest control level), trapezoid type takes second place, and triangular-type function is last; there is no distinct difference between the trapezoid-type and Gauss-type function, and their control limits are almost overlapping. Furthermore, the control interval of each sample has little difference.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of two fuzzy measures. (

**a**) Possibility measure (Pos); (

**b**) necessity measure (Nec).

**Figure 6.**Determination of parameters. (

**a**) Determination of parameter $\alpha $; (

**b**) determination of parameter $\beta $.

**Figure 7.**Supremum (Sup(S

_{i})

_{a}) and infimum (Inf(S

_{i})

_{a}) of fuzzy number $\alpha \mathrm{-cut}$.

Sample No. | Pos | $\mathit{S}\mathit{u}\mathit{p}{(\tilde{{\mathit{S}}_{\mathit{i}}})}_{\mathit{\alpha}}$ | $\mathit{I}\mathit{n}\mathit{f}{(\tilde{{\mathit{S}}_{\mathit{i}}})}_{\mathit{\alpha}}$ | Nec | $\mathit{S}\mathit{u}\mathit{p}{(\tilde{{\mathit{S}}_{\mathit{i}}})}_{1-\mathit{\beta}}$ | $\mathit{I}\mathit{n}\mathit{f}{(\tilde{{\mathit{S}}_{\mathit{i}}})}_{1-\mathit{\beta}}$ |
---|---|---|---|---|---|---|

1 | 0.83 | 6.4 | 6.14 | 0.47 | 6.45 | 6.06 |

2 | 0.8 | 5.8 | 5.4 | 0.3 | 5.9 | 5.3 |

3 | 1 | 6.6 | 5.4 | 0.25 | 6.9 | 5.1 |

4 | 0.8 | 7.3 | 6.2 | 0 | 7.45 | 5.8 |

5 | 0.25 | 7.7 | 7.3 | 0 | 7.8 | 7.2 |

Type | Sample Fuzzy Number | Pos | $\mathit{S}\mathit{u}\mathit{p}\mathit{(}\tilde{{\mathit{S}}_{\mathit{i}}}{\mathit{)}}_{\mathit{a}}$ | $\mathit{I}\mathit{n}\mathit{f}\mathit{(}\tilde{{\mathit{S}}_{\mathit{i}}}{\mathit{)}}_{\mathit{\alpha}}$ | Nec | $\mathit{S}\mathit{u}\mathit{p}\mathit{(}\tilde{{\mathit{S}}_{\mathit{i}}}{\mathit{)}}_{1-\mathit{\beta}}$ | $\mathit{I}\mathit{n}\mathit{f}\mathit{(}\tilde{{\mathit{S}}_{\mathit{i}}}{\mathit{)}}_{1-\mathit{\beta}}$ |
---|---|---|---|---|---|---|---|

T | ${\tilde{S}}_{1}$ $\left(5.58,6.22,6.35,6.75\right)$ | 0.9162 | 6.43 | 6.092 | 0.4588 | 6.47 | 6.03 |

${\tilde{S}}_{2}$ $\left(4.70,5.50,5.70,6.50\right)$ | 0.8750 | 5.86 | 5.34 | 0.2500 | 5.94 | 5.26 | |

${\tilde{S}}_{3}$ $\left(3.30,5.70,6.30,8.70\right)$ | 1.0000 | 6.78 | 5.22 | 0.1875 | 7.02 | 4.98 | |

${\tilde{S}}_{4}$ $\left(3.40,6.60,7.15,8.35\right)$ | 0.8750 | 7.39 | 5.96 | 0 | 7.51 | 5.64 | |

${\tilde{S}}_{5}$ $\left(6.6,7.4,7.6,8.4\right)$ | 0.1875 | 7.76 | 7.24 | 0 | 7.84 | 7.16 | |

$\overline{S}$ $\left(5.10,5.90,6.10,6.90\right)$ | - | 6.26 | 5.74 | - | 6.66 | 5.34 | |

G | ${\tilde{S}}_{1}$ $\left(0.2251,6.285\right)$ | 0.9078 | 6.44 | 6.13 | 0.4550 | 6.48 | 6.09 |

${\tilde{S}}_{2}$ $\left(0.4247,5.6\right)$ | 0.895 | 5.88 | 5.32 | 0.2418 | 5.96 | 5.24 | |

${\tilde{S}}_{3}$ $\left(1.274,6\right)$ | 1 | 6.85 | 5.15 | 0.1757 | 7.08 | 4.93 | |

${\tilde{S}}_{4}$ $\left(0.7432,6.875\right)$ | 0.7551 | 7.37 | 6.38 | 0.0387 | 7.50 | 6.25 | |

${\tilde{S}}_{5}$ $\left(0.4247,7.5\right)$ | 0.2103 | 7.78 | 7.22 | 0.0016 | 7.86 | 7.14 | |

$\overline{S}$ $\left(5.10,5.90,6.10,6.90\right)$ | - | 6.28 | 5.72 | - | 6.66 | 5.34 | |

P | ${\tilde{S}}_{1}$ $\left(5.58,6.22,6.35,6.75\right)$ | 0.986 | 6.47 | 6.02 | 0.4209 | 6.51 | 5.97 |

${\tilde{S}}_{2}$ $\left(4.7,5.5,5.7,6.5\right)$ | 0.9687 | 5.96 | 5.25 | 0.1250 | 6.01 | 5.19 | |

${\tilde{S}}_{3}$ $\left(3.3,5.7,6.3,8.7\right)$ | 1 | 7.05 | 4.94 | 0.0703 | 7.23 | 4.77 | |

${\tilde{S}}_{4}$ $\left(3.4,6.6,7.15,8.35\right)$ | 0.9688 | 7.53 | 5.59 | 0 | 7.62 | 5.36 | |

${\tilde{S}}_{5}$ $\left(6.6,7.4,7.6,8.4\right)$ | 0.0703 | 7.85 | 7.15 | 0 | 7.91 | 7.09 | |

$\overline{S}$ $\left(5.10,5.90,6.10,6.90\right)$ | - | 6.36 | 5.65 | - | 6.59 | 5.41 |

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**MDPI and ACS Style**

Hou, S.; Wang, H.; Feng, S.
Attribute Control Chart Construction Based on Fuzzy Score Number. *Symmetry* **2016**, *8*, 139.
https://doi.org/10.3390/sym8120139

**AMA Style**

Hou S, Wang H, Feng S.
Attribute Control Chart Construction Based on Fuzzy Score Number. *Symmetry*. 2016; 8(12):139.
https://doi.org/10.3390/sym8120139

**Chicago/Turabian Style**

Hou, Shiwang, Hui Wang, and Shunxiao Feng.
2016. "Attribute Control Chart Construction Based on Fuzzy Score Number" *Symmetry* 8, no. 12: 139.
https://doi.org/10.3390/sym8120139