# Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Introduction for Picture Fuzzy Sets

#### 1.2. The Development State for Picture Fuzzy Aggregation Operators

#### 1.3. Main Contributions for This Paper

## 2. Basic Concepts and Properties for Picture Fuzzy Sets

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Definition**

**4**

**Proposition**

**1.**

**Definition**

**5.**

**Definition**

**6**

**Definition**

**7**

## 3. New Transformation Approach for Picture Fuzzy Sets

**Proposition**

**2.**

**Definition**

**8.**

**Theorem**

**1.**

## 4. New Geometric Aggregation Operators for Picture Fuzzy Sets

**Theorem**

**2.**

**Remark**

**3.**

**Theorem**

**3.**

**Theorem**

**4.**

**Remark**

**4.**

## 5. Application to Multi-Criteria Decision Making

#### 5.1. Numerical Example

**Example**

**1.**

#### 5.2. Application to Multi-Criteria Decision Making

#### 5.2.1. Algorithm for Multi-Criteria Decision Making

**Step 1:**Normalized decision matrix:

**Step 2:**Calculating the aggregation values of each alternative ${A}_{i}(i=1,2,\dots ,m)$ by using the aggregation operator $PFWG$(or $PFOWG$, or $PFHG$).

**Step 3:**Calculating the score function values $S({A}_{i})$ of aggregation alternative by Definition 6.

**Step 4:**Ranking the alternatives by Definition 7, the greatest score value is the best alternative.

#### 5.2.2. Application to Multi-Criteria Decision Making

**Example**

**2.**

**Step 1:**Since the five attributes are all benefit attributes, the picture fuzzy decision matrix is normalized decision matrix.

**Step 2:**Calculating the aggregation values of each alternative by using the proposed $PFWG$ aggregation operator, we have

**Step 3:**Calculating the score function values of each aggregation alternative by Definition 6, we have

**Step 4:**Ranking the alternatives by Definition 7, the greatest score value is the best alternative. We have $S({A}_{1})>S({A}_{3})>S({A}_{2})$. Therefore

**Step 1:**Since the five attributes are all benefit attributes, the picture fuzzy decision matrix is the normalized decision matrix.

**Step 2:**Calculating the aggregation values by using the proposed $PFOWG$ aggregation operator, where the weighting vector of $PFOWG$ aggregation operator is $w=(0.2,0.3,0.1,0.1,0.3)$, we have

**Step 3:**Calculating the score function values of each aggregation alternative by Definition 6, we have

**Step 4:**Ranking the alternatives by Definition 7, the greatest score value is the best alternative. We have $S({A}_{3})>S({A}_{2})>S({A}_{1})$. Therefore

**Step 1:**Since the five attributes are all benefit attributes, the picture fuzzy decision matrix is the normalized decision matrix.

**Step 2:**Calculating the aggregation values of each alternative by using the proposed $PFHG$ aggregation operator. Where the weighting vector of $PFHG$ aggregation operator is $W=(0.112,0.236,0.304,0.236,0.112)$ based on the normal distribution in [39], we have

**Step 3:**Calculating the score function values of each aggregation alternative according to the Definition 6, we can get

**Step 4:**Ranking the alternatives according to the Definition 7, the greatest score value is the best alternative. We have $S({A}_{1})>S({A}_{3})>S({A}_{2})$. Therefore

**Example**

**3**

**Step 1:**Since the four attributes are all benefit attributes, the decision matrix is the normalized decision matrix.

**Step 2:**Calculating the aggregation values of each alternative by using the $PFWG,PFOWG$ and $PFHG$ operators, respectively. The results in Table 8.

**Step 3:**Calculating the score function values of each aggregation alternative according to the Definition 6, the results in Table 9.

**Step 4:**Ranking the alternatives according to the Definition 7, the greatest score value is the best alternative. The results in Table 10.

#### 5.3. Comparative Analysis the Conditions of Using Some Aggregation Operators

**Condition**

**1.**

**Condition**

**2.**

**Condition**

**3.**

**Condition**

**4.**

## 6. Application to Pattern Recognition

#### 6.1. Algorithm for Pattern Recognition

**Step 1:**Calculating the aggregation values $\langle {\mu}_{{P}_{j}},{\eta}_{{P}_{j}},{\nu}_{{P}_{j}}\rangle $ of each known pattern ${P}_{j}=\left\{\langle {x}_{i},{\mu}_{{p}_{j}}({x}_{i}),{\eta}_{{p}_{j}}({x}_{i}),{\nu}_{{p}_{j}}({x}_{i})\rangle |{x}_{i}\in X\right\}(j=1,2,\dots ,m)$. Calculating the aggregation values $\langle {\mu}_{S},{\eta}_{S},{\nu}_{S}\rangle $ of unknown pattern $S=\left\{\langle {x}_{i},{\mu}_{s}({x}_{i}),{\eta}_{s}({x}_{i}),{\nu}_{s}({x}_{i})\rangle |{x}_{i}\in X\right\}$ by using the proposed aggregation operator $PFWG$ (or $PFOWG$, or $PFHG$), where $\omega =(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n})$.

**Step 2:**Calculating the distance $d(\langle {\mu}_{{P}_{j}},{\eta}_{{P}_{j}},{\nu}_{{P}_{j}}\rangle ,\langle {\mu}_{S},{\eta}_{S},{\nu}_{S}\rangle )(j=1,2,\dots ,m)$ between ${P}_{j}$ and S after aggregation by using the distance [15]:

**Step 3:**Select the minimum one $d(\langle {\mu}_{{P}_{{j}_{0}}},{\eta}_{{P}_{{j}_{0}}},{\nu}_{{P}_{{j}_{0}}}\rangle ,\langle {\mu}_{S},{\eta}_{S},{\nu}_{S}\rangle )$ from $d(\langle {\mu}_{{P}_{j}},{\eta}_{{P}_{j}},{\nu}_{{P}_{j}}\rangle ,\langle {\mu}_{S},{\eta}_{S},{\nu}_{S}\rangle )(j=1,2,\dots ,m)$. Then the unknown pattern S belongs to the known pattern ${P}_{{j}_{0}}.$

#### 6.2. Application to Pattern Recognition

**Example**

**4**

**Step 1:**Calculating the aggregation values of each pattern by using the proposed aggregation operator $PFWG$. Where $\omega =(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. The aggregation values in Table 14.

**Step 2:**Calculating the distances of the aggregate values: $d({P}_{1},S)=0.1364,$ $d({P}_{2},S)=0.0560.$

**Step 3:**Select the minimum distance: since $d({P}_{1},S)>d({P}_{2},S),$ so sample S belongs to known pattern ${P}_{2}$.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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References | Aggregation Operators |
---|---|

Jana et al. [21] | $PFW{G}_{1}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{1}{1+{\{{\sum}_{j=1}^{n}{\omega}_{j}{(\frac{1-{\mu}_{{\alpha}_{j}}}{{\mu}_{{\alpha}_{j}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{\omega}_{j}{(\frac{{\eta}_{{\alpha}_{j}}}{1-{\eta}_{{\alpha}_{j}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{\omega}_{j}{(\frac{{\nu}_{{\alpha}_{j}}}{1-{\nu}_{{\alpha}_{j}}})}^{\xi}\}}^{\frac{1}{\xi}}}\rangle .$ |

$PFOW{G}_{1}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{1}{1+{\{{\sum}_{j=1}^{n}{w}_{j}{(\frac{1-{\mu}_{{\alpha}_{\sigma (j)}}}{{\mu}_{{\alpha}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{w}_{j}{(\frac{{\eta}_{{\alpha}_{\sigma (j)}}}{1-{\eta}_{{\alpha}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{w}_{j}{(\frac{{\nu}_{{\alpha}_{\sigma (j)}}}{1-{\nu}_{{\alpha}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}}\rangle .$ | |

$PFH{G}_{1}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{1}{1+{\{{\sum}_{j=1}^{n}{W}_{j}{(\frac{1-{\mu}_{{\tilde{\alpha}}_{\sigma (j)}}}{{\mu}_{{\tilde{\alpha}}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{W}_{j}{(\frac{{\eta}_{{\tilde{\alpha}}_{\sigma (j)}}}{1-{\eta}_{{\tilde{\alpha}}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}},1-\frac{1}{1+{\{{\sum}_{j=1}^{n}{W}_{j}{(\frac{{\nu}_{{\tilde{\alpha}}_{\sigma (j)}}}{1-{\nu}_{{\tilde{\alpha}}_{\sigma (j)}}})}^{\xi}\}}^{\frac{1}{\xi}}}\rangle .$ | |

Wei [22] | $PFW{G}_{2}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{\gamma {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{j}})}^{{\omega}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1)(1-{\mu}_{{\alpha}_{j}})]}^{{\omega}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{j}})}^{{\omega}_{j}}},$ |

$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\alpha}_{j}}]}^{{\omega}_{j}}-{\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{j}})}^{{\omega}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\alpha}_{j}}]}^{{\omega}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{j}})}^{{\omega}_{j}}},$$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\alpha}_{j}}]}^{{\omega}_{j}}-{\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{j}})}^{{\omega}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\alpha}_{j}}]}^{{\omega}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{j}})}^{{\omega}_{j}}}\rangle .$ | |

$PFOW{G}_{2}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{\gamma {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1)(1-{\mu}_{{\alpha}_{\sigma (j)}})]}^{{w}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}},$ | |

$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\alpha}_{\sigma (j)}}]}^{{w}_{j}}-{\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\alpha}_{\sigma (j)}}]}^{{w}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}},$$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\alpha}_{\sigma (j)}}]}^{{w}_{j}}-{\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\alpha}_{\sigma (j)}}]}^{{w}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}}\rangle .$ | |

$PFH{G}_{2}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle \frac{\gamma {\prod}_{j=1}^{n}{({\mu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1)(1-{\mu}_{{\tilde{\alpha}}_{\sigma (j)}})]}^{{W}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{({\mu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}},$ | |

$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\tilde{\alpha}}_{\sigma (j)}}]}^{{W}_{j}}-{\prod}_{j=1}^{n}{(1-{\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\eta}_{{\tilde{\alpha}}_{\sigma (j)}}]}^{{W}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}},$$\frac{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\tilde{\alpha}}_{\sigma (j)}}]}^{{W}_{j}}-{\prod}_{j=1}^{n}{(1-{\nu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}}{{\prod}_{j=1}^{n}{[1+(\gamma -1){\nu}_{{\tilde{\alpha}}_{\sigma (j)}}]}^{{W}_{j}}+(\gamma -1){\prod}_{j=1}^{n}{(1-{\nu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}}\rangle .$ | |

Ashraf et al. [25] | $PFW{G}_{3}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{j}})}^{{\omega}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{j}})}^{{\omega}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\alpha}_{j}})}^{{\omega}_{j}}\rangle .$ |

$PFOW{G}_{3}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}\rangle .$ | |

$PFH{G}_{3}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}\rangle .$ | |

Wang et al. [26] | $PFW{G}_{4}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{j}}+{\eta}_{{\alpha}_{j}})}^{{\omega}_{j}}-{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{j}})}^{{\omega}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{j}})}^{{\omega}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\alpha}_{j}})}^{{\omega}_{j}}\rangle .$ |

$PFOW{G}_{4}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{\sigma (j)}}+{\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}-{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}\rangle .$ | |

$PFH{G}_{4}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\tilde{\alpha}}_{\sigma (j)}}+{\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}-{\prod}_{j=1}^{n}{({\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},{\prod}_{j=1}^{n}{({\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},1-{\prod}_{j=1}^{n}(1-{{\nu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}\rangle .$ | |

Wei [31] | $PFW{G}_{5}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{j}})}^{{\omega}_{j}},1-{\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{j}})}^{{\omega}_{j}},1-{\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{j}})}^{{\omega}_{j}}\rangle .$ |

$PFOW{G}_{5}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},1-{\prod}_{j=1}^{n}{(1-{\eta}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}},1-{\prod}_{j=1}^{n}{(1-{\nu}_{{\alpha}_{\sigma (j)}})}^{{w}_{j}}\rangle .$ | |

$PFH{G}_{5}({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})=\langle {\prod}_{j=1}^{n}{({\mu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},1-{\prod}_{j=1}^{n}{(1-{\eta}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}},1-{\prod}_{j=1}^{n}{(1-{\nu}_{{\tilde{\alpha}}_{\sigma (j)}})}^{{W}_{j}}\rangle .$ |

PFWG | PFOWG | PFHG | |
---|---|---|---|

Jana et al. [21] ($\xi =1$) | Cannot be calculated | Cannot be calculated | Cannot be calculated |

Wei [22] ($\gamma =2$) | $\langle \mathbf{0.0000},0.1269,0.1258\rangle $ | $\langle \mathbf{0.0000},0.1269,0.1258\rangle $ | $\langle \mathbf{0.0000},0.1269,0.1258\rangle $ |

Ashraf et al. [25] | $\langle \mathbf{0.0000},\mathbf{0.0000},0.1288\rangle $ | $\langle \mathbf{0.0000},\mathbf{0.0000},0.1288\rangle $ | $\langle \mathbf{0.0000},\mathbf{0.0000},0.1288\rangle $ |

Wang et al. [26] | $\langle 0.2060,\mathbf{0.0000},0.1288\rangle $ | $\langle 0.2060,\mathbf{0.0000},0.1288\rangle $ | $\langle 0.2060,\mathbf{0.0000},0.1288\rangle $ |

Wei [31] | $\langle \mathbf{0.0000},0.1322,0.1288\rangle $ | $\langle \mathbf{0.0000},0.1322,0.1288\rangle $ | $\langle \mathbf{0.0000},0.1322,0.1288\rangle $ |

The proposed | $\langle 0.1574,0.1525,0.1237\rangle $ | $\langle 0.1574,0.1525,0.1237\rangle $ | $\langle 0.1574,0.1525,0.1237\rangle $ |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${A}_{1}$ | $\langle 0.1,0.2,0.1\rangle $ | $\langle 0.4,0.3,0.2\rangle $ | $\langle 0.0,0.0,1.0\rangle $ | $\langle 0.2,0.2,0.1\rangle $ | $\langle 0.2,0.1,0.4\rangle $ |

${A}_{2}$ | $\langle 0.2,0.0,0.7\rangle $ | $\langle 0.4,0.1,0.1\rangle $ | $\langle 0.1,0.5,0.0\rangle $ | $\langle 0.3,0.2,0.4\rangle $ | $\langle 0.0,0.0,1.0\rangle $ |

${A}_{3}$ | $\langle 0.3,0.1,0.6\rangle $ | $\langle 0.0,0.0,1.0\rangle $ | $\langle 0.2,0.4,0.0\rangle $ | $\langle 0.1,0.4,0.1\rangle $ | $\langle 0.4,0.1,0.4\rangle $ |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ | |
---|---|---|---|---|---|

${A}_{{\sigma}_{1}}$ | $\langle 0.4,0.3,0.2\rangle $ | $\langle 0.2,0.2,0.1\rangle $ | $\langle 0.1,0.2,0.1\rangle $ | $\langle 0.2,0.1,0.5\rangle $ | $\langle 0.0,0.0,1.0\rangle $ |

${A}_{{\sigma}_{2}}$ | $\langle 0.4,0.1,0.1\rangle $ | $\langle 0.1,0.5,0.0\rangle $ | $\langle 0.3,0.2,0.4\rangle $ | $\langle 0.2,0.0,0.7\rangle $ | $\langle 0.0,0.0,1.0\rangle $ |

${A}_{{\sigma}_{3}}$ | $\langle 0.2,0.4,0.0\rangle $ | $\langle 0.4,0.1,0.4\rangle $ | $\langle 0.1,0.4,0.1\rangle $ | $\langle 0.3,0.1,0.6\rangle $ | $\langle 0.0,0.0,1.0\rangle $ |

${\tilde{\mathit{A}}}_{{\mathit{\sigma}}_{1}}$ | ${\tilde{\mathit{A}}}_{{\mathit{\sigma}}_{2}}$ | ${\tilde{\mathit{A}}}_{{\mathit{\sigma}}_{3}}$ | |
---|---|---|---|

${C}_{1}$ | $\langle 0.5352,0.3005,0.1327\rangle $ | $\langle 0.5352,0.1112,0.1006\rangle $ | $\langle 0.5352,0.1112,0.3219\rangle $ |

${C}_{2}$ | $\langle 0.1056,0.1198,0.0675\rangle $ | $\langle 0.0513,0.3162,0.0000\rangle $ | $\langle 0.1056,0.2620,0.0000\rangle $ |

${C}_{3}$ | $\langle 0.1000,0.2000,0.1000\rangle $ | $\langle 0.1633,0.1296,0.3909\rangle $ | $\langle 0.0513,0.2416,0.0747\rangle $ |

${C}_{4}$ | $\langle 0.2845,0.1298,0.4214\rangle $ | $\langle 0.2000,0.0000,0.7000\rangle $ | $\langle 0.3000,0.1000,0.6000\rangle $ |

${C}_{5}$ | $\langle 0.0000,0.0000,1.0000\rangle $ | $\langle 0.0000,0.0000,1.0000\rangle $ | $\langle 0.0000,0.0000,1.0000\rangle $ |

References | PFWG | PFOWG | PFHG |
---|---|---|---|

Jana et al. [21] | Cannot be calculated | Cannot be calculated | Cannot be calculated |

Wei [22] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ |

Ashraf et al. [25] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ |

Wang et al. [26] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ |

Wei [31] | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ | ${A}_{1}={A}_{2}={A}_{3}$ |

The proposed operator | ${A}_{1}>{A}_{3}>{A}_{2}$ | ${A}_{3}>{A}_{2}>{A}_{1}$ | ${A}_{1}>{A}_{3}>{A}_{2}$ |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | |
---|---|---|---|---|

${A}_{1}$ | $\langle 0.53,0.33,0.09\rangle $ | $\langle 0.89,0.08,0.03\rangle $ | $\langle 0.42,0.35,0.18\rangle $ | $\langle 0.08,0.89,0.02\rangle $ |

${A}_{2}$ | $\langle 0.73,0.12,0.08\rangle $ | $\langle 0.13,0.64,0.21\rangle $ | $\langle 0.03,0.82,0.13\rangle $ | $\langle 0.73,0.15,0.08\rangle $ |

${A}_{3}$ | $\langle 0.91,0.03,0.02\rangle $ | $\langle 0.07,0.09,0.05\rangle $ | $\langle 0.04,0.85,0.10\rangle $ | $\langle 0.68,0.26,0.06\rangle $ |

${A}_{4}$ | $\langle 0.85,0.09,0.05\rangle $ | $\langle 0.74,0.16,0.10\rangle $ | $\langle 0.02,0.89,0.05\rangle $ | $\langle 0.08,0.84,0.06\rangle $ |

${A}_{5}$ | $\langle 0.90,0.05,0.02\rangle $ | $\langle 0.68,0.08,0.21\rangle $ | $\langle 0.05,0.87,0.06\rangle $ | $\langle 0.13,0.75,0.09\rangle $ |

$\mathit{PFWG}$ | $\mathit{PFOWG}$ | $\mathit{PFHG}$ | |
---|---|---|---|

${A}_{1}$ | $\langle 0.4336,0.4912,0.0752\rangle $ | $\langle 0.5102,0.4253,0.0645\rangle $ | $\langle 0.4145,0.5166,0.0689\rangle $ |

${A}_{2}$ | $\langle 0.5545,0.3024,0.1093\rangle $ | $\langle 0.5102,0.4253,0.0645\rangle $ | $\langle 0.4494,0.3832,0.1289\rangle $ |

${A}_{3}$ | $\langle 0.6159,0.2904,0.0937\rangle $ | $\langle 0.4693,0.3619,0.1688\rangle $ | $\langle 0.5445,0.3128,0.1427\rangle $ |

${A}_{4}$ | $\langle 0.4251,0.4949,0.0800\rangle $ | $\langle 0.4214,0.4976,0.0810\rangle $ | $\langle 0.3515,0.5622,0.0863\rangle $ |

${A}_{5}$ | $\langle 0.4756,0.4288,0.0690\rangle $ | $\langle 0.4710,0.4372,0.0663\rangle $ | $\langle 0.3838,0.5076,0.0806\rangle $ |

$\mathit{PFWG}$ | $\mathit{PFOWG}$ | $\mathit{PFHG}$ | |
---|---|---|---|

$S({A}_{1})$ | 0.3584 | 0.4457 | 0.3456 |

$S({A}_{2})$ | 0.4452 | 0.2232 | 0.3205 |

$S({A}_{3})$ | 0.5222 | 0.3005 | 0.4018 |

$S({A}_{4})$ | 0.3451 | 0.3404 | 0.2652 |

$S({A}_{5})$ | 0.4066 | 0.4047 | 0.3032 |

Ranking | |
---|---|

$PFWG$ | ${A}_{3}>{A}_{2}>{A}_{5}>{A}_{1}>{A}_{4}$ |

$PFOWG$ | ${A}_{1}>{A}_{5}>{A}_{4}>{A}_{3}>{A}_{2}$ |

$PFHG$ | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{5}>{A}_{4}$ |

References | $\mathit{PFWG}$ | $\mathit{PFOWG}$ | $\mathit{PFHG}$ |
---|---|---|---|

Jana et al. [21] | ${A}_{3}>{A}_{5}>{A}_{4}>{A}_{2}>{A}_{1}$ | ${A}_{1}>{A}_{5}>{A}_{3}>{A}_{4}>{A}_{2}$ | ${A}_{1}>{A}_{3}>{A}_{5}>{A}_{4}>{A}_{2}$ |

Wei [22] | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{5}>{A}_{4}$ | ${A}_{1}>{A}_{5}>{A}_{3}>{A}_{4}>{A}_{2}$ | ${A}_{3}>{A}_{2}>{A}_{1}>{A}_{5}>{A}_{4}$ |

Ashraf et al. [25] | ${A}_{2}>{A}_{1}>{A}_{3}>{A}_{5}>{A}_{4}$ | ${A}_{1}>{A}_{5}>{A}_{3}>{A}_{4}>{A}_{2}$ | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{4}>{A}_{5}$ |

Wang et al. [26] | ${A}_{2}>{A}_{3}>{A}_{5}>{A}_{1}>{A}_{4}$ | ${A}_{5}>{A}_{1}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{3}>{A}_{1}>{A}_{4}>{A}_{2}>{A}_{5}$ |

Wei [31] | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{5}>{A}_{4}$ | ${A}_{1}>{A}_{5}>{A}_{3}>{A}_{4}>{A}_{2}$ | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{4}>{A}_{5}$ |

The proposed | ${A}_{3}>{A}_{2}>{A}_{5}>{A}_{1}>{A}_{4}$ | ${A}_{1}>{A}_{5}>{A}_{4}>{A}_{3}>{A}_{2}$ | ${A}_{3}>{A}_{1}>{A}_{2}>{A}_{5}>{A}_{4}$ |

References | Condition 1 | Condition 2 | Condition 3 | Condition 4 |
---|---|---|---|---|

${\mathit{\mu}}_{\mathit{j}}\ne \mathbf{0},{\mathit{\eta}}_{\mathit{j}}\ne \mathbf{0}$ | ${\mathit{\mu}}_{\mathit{i}}=\mathbf{0}$${\mathit{\eta}}_{\mathit{i}}=\mathbf{0}$ | $\{\mathit{\mu},\mathit{\eta}\}$$\{\mathit{\mu},\mathit{\nu}\}$$\{\mathit{\eta},\mathit{\nu}\}$ | $\{\mathit{\mu},\mathit{\eta},\mathit{\nu}\}$ | |

Jana et al. [21] | Yes | No Yes | No No No | No |

Wei [22] | Yes | No Yes | No No No | No |

Ashraf et al. [25] | Yes | No No | No No No | No |

Wang et al. [26] | Yes | Yes No | Yes No No | No |

Wei [31] | Yes | No Yes | No No No | No |

The proposed operators | Yes | Yes Yes | Yes Yes Yes | Yes |

Patterns | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ |
---|---|---|---|

${P}_{1}$ | $\langle 0.3,0.2,0.1\rangle $ | $\langle 0.5,0.1,0.2\rangle $ | $\langle 0.6,0.1,0.3\rangle $ |

${P}_{2}$ | $\langle 0.6,0.1,0.3\rangle $ | $\langle 0.1,0.2,0.5\rangle $ | $\langle 0.6,0.3,0.1\rangle $ |

S | $\langle 0.5,0.3,0.2\rangle $ | $\langle 0.3,0.4,0.2\rangle $ | $\langle 0.4,0.3,0.2\rangle $ |

Patterns | Aggregation Values |
---|---|

${P}_{1}$ | $\langle 0.4808,0.1278,0.3915\rangle $ |

${P}_{2}$ | $\langle 0.4759,0.2483,0.2759\rangle $ |

S | $\langle 0.4056,0.3323,0.2621\rangle $ |

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## Share and Cite

**MDPI and ACS Style**

Luo, M.; Long, H.
Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application. *Symmetry* **2021**, *13*, 119.
https://doi.org/10.3390/sym13010119

**AMA Style**

Luo M, Long H.
Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application. *Symmetry*. 2021; 13(1):119.
https://doi.org/10.3390/sym13010119

**Chicago/Turabian Style**

Luo, Minxia, and Huifeng Long.
2021. "Picture Fuzzy Geometric Aggregation Operators Based on a Trapezoidal Fuzzy Number and Its Application" *Symmetry* 13, no. 1: 119.
https://doi.org/10.3390/sym13010119