# A Novel Similarity Measure for Interval-Valued Intuitionistic Fuzzy Sets and Its Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

**Definition**

**1**

**.**A fuzzy set A in the unverse of discourse $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$ is defined as follows:

**Definition**

**2**

**.**An intuitionistic fuzzy set A in a universe of discourse $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$ is defined as follows:

**Definition**

**3**

**.**An interval-valued intuitionistic fuzzy set A in a universe of discourse $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$ is defined as follows:

**Definition**

**4**

**.**For every two IVIFSs A and B in the universe of discourse X, we have the following relations:

- (1):
- $A\subseteq B$ iff $(\forall x\in X){\mu}_{A}^{-}\left(x\right)\le {\mu}_{B}^{-}\left(x\right)$ and ${\mu}_{A}^{+}\left(x\right)\le {\mu}_{B}^{+}\left(x\right)$ and ${\nu}_{A}^{-}\left(x\right)\ge {\nu}_{B}^{-}\left(x\right)$ and ${\nu}_{A}^{+}\left(x\right)\ge {\nu}_{B}^{+}\left(x\right).$
- (2):
- $A\cup B=\u2329x,[max({\mu}_{A}^{-}\left(x\right),{\mu}_{B}^{-}\left(x\right)),max({\mu}_{A}^{+}\left(x\right),{\mu}_{B}^{+}\left(x\right))],[min({\nu}_{A}^{-}\left(x\right),{\nu}_{B}^{-}\left(x\right)),min({\nu}_{A}^{+}\left(x\right),{\nu}_{B}^{+}\left(x\right))]\u232a.$
- (3):
- $A\cap B=\u2329x,[min({\mu}_{A}^{-}\left(x\right),{\mu}_{B}^{-}\left(x\right)),min({\mu}_{A}^{+}\left(x\right),{\mu}_{B}^{+}\left(x\right))],[max({\nu}_{A}^{-}\left(x\right),{\nu}_{B}^{-}\left(x\right)),max({\nu}_{A}^{+}\left(x\right),{\nu}_{B}^{+}\left(x\right))]\u232a.$
- (4):
- $A=B$ iff $(\forall x\in X){\mu}_{A}^{-}\left(x\right)={\mu}_{B}^{-}\left(x\right)$ and ${\mu}_{A}^{+}\left(x\right)={\mu}_{B}^{+}\left(x\right)$ and ${\nu}_{A}^{-}\left(x\right)={\nu}_{B}^{-}\left(x\right)$ and ${\nu}_{A}^{+}\left(x\right)={\nu}_{B}^{+}\left(x\right)$.
- (5):
- ${A}^{c}=\u2329x,[{\nu}_{A}^{-}\left(x\right),{\nu}_{A}^{+}\left(x\right)],[{\mu}_{A}^{-}\left(x\right),{\mu}_{A}^{+}\left(x\right)]\u232a$

**Definition**

**5**

**.**Let A and B be interval-valued intuitionistic fuzzy sets in the unverse of discourse $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$, a mapping $S:IVIFS\left(X\right)\times IVIFS\left(X\right)\to [0,1]$, $S(A,B)$ is called to be a similarity measure between A and B, if $S(A,B)$ satisfies the following properties:

- (S1):
- $0\le S(A,B)\le 1,$
- (S2):
- $S(A,B)=1$ if and only if $A=B,$
- (S3):
- $S(A,B)=S(B,A),$
- (S4):
- If $A\subseteq B\subseteq C$, then $S(A,C)\le S(A,B)$, and $S(A,C)\le S(B,C)$.

## 3. Some Existing Similarity Measures

## 4. A New Similarity Measure between Interval-Valued Intuitionistic Fuzzy Sets

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Example**

**1.**

## 5. Geometric Interpretation of the Novel Similarity Measure

## 6. Applications

#### 6.1. Pattern Recognition

#### 6.1.1. Algorithms for Pattern Recognition

**Step 1**. Calculate the similarity measure $S(B,{A}_{j})$ between B and ${A}_{j}$ $(j=1,\dots ,m)$.

**Step 2**. Choose the maximum one $S(B,{A}_{{j}_{0}})$ from $S(B,{A}_{j})$ $(j=1,2,\dots ,m)$, i.e., $S(B,{A}_{{j}_{0}})=\underset{1\le j\le m}{max}S(B,{A}_{j})$. Then, the test sample B is classified the pattern ${A}_{{j}_{0}}$.

#### 6.1.2. Applications for Pattern Recognition

**Example**

**2.**

**Example**

**3**

**.**In this example, a pattern recognition example about classification of building materials is used to illustrate the proposed similarity measure. Suppose that there are four classes of building material, which are denoted by the IVIFSs ${A}_{j}=\{<{x}_{1},[{\mu}_{{A}_{j}}^{-}\left({x}_{1}\right),{\mu}_{{A}_{j}}^{+}\left({x}_{1}\right)],[{\nu}_{{A}_{j}}^{-}\left({x}_{1}\right),{\nu}_{{A}_{j}}^{+}\left({x}_{1}\right)]>,\dots ,<{x}_{12},[{\mu}_{{A}_{j}}^{-}\left({x}_{12}\right),{\mu}_{{A}_{j}}^{+}\left({x}_{12}\right)],[{\nu}_{{A}_{j}}^{-}\left({x}_{12}\right),{\nu}_{{A}_{j}}^{+}\left({x}_{12}\right)]>|{x}_{1},\dots ,{x}_{12}\in X\}$ $(j=1,\dots ,4)$ in the feature space $X=\{{x}_{1},{x}_{2},\dots ,{x}_{12}\}$, and there is an unknown pattern B:

#### 6.2. Applications for Medical Diagnosis

**Example**

**4.**

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Comparison of similarity measures in the environment of IVIFSs (interval-valued intuitionistic fuzzy set) (counter-intuitive cases are in bold type; $p=1$ in ${S}_{1}$ and ${S}_{2}$; $p=1,{t}_{1}=2,{t}_{2}=3$ in ${S}^{p}$).

1 | 2 | 3 | 4 | |
---|---|---|---|---|

${A}_{i}$ | $<[0.20,0.30],[0.40,0.60]>$ | $<[0.20,0.30],[0.40,0.60]>$ | $<[0.20,0.30],[0.30,0.50]>$ | $<[0.20,0.30],[0.30,0.50]>$ |

${B}_{i}$ | $<[0.30,0.40],[0.40,0.60]>$ | $<[0.30,0.40],[0.30,0.50]>$ | $<[0.30,0.40],[0.40,0.60]>$ | $<[0.30,0.40],[0.30,0.50]>$ |

${S}_{1}$ [24] | 0.90 | 0.90 | 0.90 | 0.95 |

${S}_{2}$ [24] | 0.90 | 0.90 | 0.90 | 0.90 |

${S}_{D}$ [28] | 1.00 | 0.98 | 0.95 | 0.94 |

${S}^{p}$ | 0.95 | 0.90 | 0.80 | 0.94 |

Feature1 | Feature2 | Feature3 | Feature4 | |
---|---|---|---|---|

${A}_{1}$ | $<[0.10,0.50],[0.20,0.30]>$ | $<[0.10,0.30],[0.00,0.20]>$ | $<[0.30,0.50],[0.20,0.40]>$ | $<[0.20,0.50],[0.10,0.30]>$ |

${A}_{2}$ | $<[0.20,0.40],[0.15,0.35]>$ | $<[0.20,0.20],[0.05,0.15]>$ | $<[0.20,0.60],[0.30,0.30]>$ | $<[0.30,0.40],[0.15,0.25]>$ |

${A}_{3}$ | $<[0.15,0.30],[0.30,0.40]>$ | $<[0.20,0.40],[0.50,0.60]>$ | $<[0.50,0.60],[0.15,0.35]>$ | $<[0.25,0.45],[0.30,0.40]>$ |

${A}_{4}$ | $<[0.20,0.35],[0.10,0.65]>$ | $<[0.35,0.60],[0.05,0.30]>$ | $<[0.15,0.30],[0.40,0.55]>$ | $<[0.15,0.25],[0.45,0.55]>$ |

B | $<[0.30,0.40],[0.10,0.50]>$ | $<[0.10,0.40],[0.25,0.40]>$ | $<[0.20,0.30],[0.10,0.35]>$ | $<[0.15,0.40],[0.20,0.50]>$ |

**Table 3.**Pattern recognition result under different similarity measures (counter-intuitive cases are in bold type; $p=1$ in ${S}_{1}$ and ${S}_{2}$; $p=1,{t}_{1}=2,{t}_{2}=3$ in ${S}^{p}$;

**N.A.**means method is not applicable).

$\mathit{S}({\mathit{A}}_{1},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{2},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{3},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{4},\mathit{B})$ | Classification Results | |
---|---|---|---|---|---|

${S}_{1}$ [24] | 0.87 | 0.87 | 0.86 | 0.87 | N.A. |

${S}_{2}$ [24] | 0.75 | 0.76 | 0.79 | 0.76 | ${A}_{3}$ |

${S}_{W}$ [25] | 0.78 | 0.79 | 0.78 | 0.79 | N.A. |

${S}_{D}$ [28] | 0.82 | 0.86 | 0.88 | 0.88 | N.A. |

${S}^{p}$ | 0.82 | 0.81 | 0.88 | 0.75 | ${A}_{3}$ |

**Table 4.**Pattern recognition results under different similarity measures (counter-intuitive cases are in bold type; $p=1$ in ${S}_{1}$ and ${S}_{2}$, $p=1,{t}_{1}=2,{t}_{2}=3$ in ${S}^{p}$).

$\mathit{S}({\mathit{A}}_{1},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{2},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{3},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{4},\mathit{B})$ | Recognition Results | |
---|---|---|---|---|---|

${S}_{1}$ [24] | 0.59 | 0.58 | 0.81 | 0.97 | ${A}_{4}$ |

${S}_{2}$ [24] | 0.53 | 0.53 | 0.79 | 0.94 | ${A}_{4}$ |

${S}_{W}$ [25] | 0.48 | 0.47 | 0.74 | 0.94 | ${A}_{4}$ |

${S}_{D}$ [28] | 0.64 | 0.56 | 0.83 | 0.98 | ${A}_{4}$ |

${S}^{p}$ | 0.60 | 0.58 | 0.85 | 0.97 | ${A}_{4}$ |

${\mathit{x}}_{1}$ (Temperature) | ${\mathit{x}}_{2}$ (Cough) | ${\mathit{x}}_{3}$ (Headache) | ${\mathit{x}}_{4}$ (Stomach Pain) | |
---|---|---|---|---|

${A}_{1}$ (Viral fever) | $<[0.8,0.9],[0.0,0.1]>$ | $<[0.7,0.8],[0.1,0.2]>$ | $<[0.5,0.6],[0.2,0.3]>$ | $<[0.6,0.8],[0.1,0.2]>$ |

${A}_{2}$ (Typhoid) | $<[0.5,0.6],[0.1,0.3]>$ | $<[0.8,0.9],[0.0,0.1]>$ | $<[0.6,0.8],[0.1,0.2]>$ | $<[0.4,0.6],[0.1,0.2]>$ |

${A}_{3}$ (Pneumonia) | $<[0.7,0.8],[0.1,0.2]>$ | $<[0.7,0.9],[0.0,0.1]>$ | $<[0.4,0.6],[0.2,0.4]>$ | $<[0.3,0.5],[0.2,0.4]>$ |

${A}_{4}$ (Stomach problem) | $<[0.8,0.9],[0.0,0.1]>$ | $<[0.7,0.8],[0.1,0.2]>$ | $<[0.7,0.9],[0.0,0.1]>$ | $<[0.8,0.9],[0.0,0.1]>$ |

**Table 6.**Computed results under different similarity measures (counter-intuitive cases are in bold type; $p=1$ in ${S}_{1}$ and ${S}_{2}$; $p=1,{t}_{1}=2,{t}_{2}=3$ in ${S}^{p}$).

$\mathit{S}({\mathit{A}}_{1},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{2},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{3},\mathit{B})$ | $\mathit{S}({\mathit{A}}_{4},\mathit{B})$ | Recognition Result | |
---|---|---|---|---|---|

${S}_{1}$ [24] | 0.81 | 0.89 | 0.86 | 0.84 | ${A}_{2}$ |

${S}_{2}$ [24] | 0.73 | 0.80 | 0.78 | 0.73 | ${A}_{2}$ |

${S}_{W}$ [25] | 0.82 | 0.80 | 0.79 | 0.77 | ${A}_{2}$ |

${S}_{D}$ [28] | 0.82 | 0.91 | 0.86 | 0.84 | ${A}_{2}$ |

${S}^{p}$ | 0.83 | 0.89 | 0.87 | 0.85 | ${A}_{2}$ |

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**MDPI and ACS Style**

Luo, M.; Liang, J.
A Novel Similarity Measure for Interval-Valued Intuitionistic Fuzzy Sets and Its Applications. *Symmetry* **2018**, *10*, 441.
https://doi.org/10.3390/sym10100441

**AMA Style**

Luo M, Liang J.
A Novel Similarity Measure for Interval-Valued Intuitionistic Fuzzy Sets and Its Applications. *Symmetry*. 2018; 10(10):441.
https://doi.org/10.3390/sym10100441

**Chicago/Turabian Style**

Luo, Minxia, and Jingjing Liang.
2018. "A Novel Similarity Measure for Interval-Valued Intuitionistic Fuzzy Sets and Its Applications" *Symmetry* 10, no. 10: 441.
https://doi.org/10.3390/sym10100441