# Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators and Their Application to Multi-Attribute Decision Making

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Interval-Valued Hesitant Fuzzy Sets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.2. Probabilistic Interval-Valued Hesitant Fuzzy Sets

**Example**

**1.**

**Note**

**1.**

**Note**

**2.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

- If $S({h}_{1})>S({h}_{2})$, then ${h}_{1}>{h}_{2}$, that is, ${h}_{1}$ is greater than ${h}_{2}$.
- If $S({h}_{1})=S({h}_{2})$, then
- (a)
- If $\varphi ({h}_{1})>\varphi ({h}_{2})$, then ${h}_{1}<{h}_{2}$. That is, ${h}_{1}$ is less than ${h}_{2}$.
- (b)
- If $\varphi ({h}_{1})<\varphi ({h}_{2})$, then ${h}_{1}>{h}_{2}$. That is, ${h}_{1}$ is greater than ${h}_{2}$.
- (c)
- If $\varphi ({h}_{1})=\varphi ({h}_{2})$, then ${h}_{1}={h}_{2}$. That is, ${h}_{1}$ equals ${h}_{2}$.

#### 2.3. Archimedean Norm

#### 2.4. Probabilistic Interval-Valued Hesitant Fuzzy Operation Rules

**Definition**

**7.**

- (1)
- ${h}^{c}=\{([1-{\gamma}^{(k)-},1-{\gamma}^{(k)+}],{p}^{(k)})\}$
- (2)
- ${h}_{1}\oplus {h}_{2}=\{(S([{\gamma}_{1}^{(k)-},{\gamma}_{1}^{(k)+}],[{\gamma}_{2}^{(k)-},{\gamma}_{2}^{(k)+}]),\overline{{p}_{1}^{(k)}+{p}_{2}^{(k)}})\}=\{([{f}^{-1}(f({\gamma}_{1}^{(k)-})+f({\gamma}_{2}^{(k)-})),{f}^{-1}(f({\gamma}_{1}^{(k)+})+f({\gamma}_{2}^{(k)+}))],\overline{{p}_{1}^{(k)}+{p}_{2}^{(k)}})\}$
- (3)
- ${h}_{1}\otimes {h}_{2}=\{(T([{\gamma}_{1}^{(k)-},{\gamma}_{1}^{(k)+}],[{\gamma}_{2}^{(k)-},{\gamma}_{2}^{(k)+}]),\overline{{p}_{1}^{(k)}+{p}_{2}^{(k)}})\}=\{([{g}^{-1}(g({\gamma}_{1}^{(k)-})+g({\gamma}_{2}^{(k)-})),{g}^{-1}(g({\gamma}_{1}^{(k)+})+g({\gamma}_{2}^{(k)+}))],\overline{{p}_{1}^{(k)}+{p}_{2}^{(k)}})\}$
- (4)
- $\lambda h=\{([{f}^{-1}(\lambda f({\gamma}^{(k)-})),{f}^{-1}(\lambda f({\gamma}^{(k)+}))],{p}^{(k)})\}$
- (5)
- ${h}^{\lambda}=\{([{g}^{-1}(\lambda g({\gamma}^{(k)-})),{g}^{-1}(\lambda g({\gamma}^{(k)+}))],{p}^{(k)})\}$

**Theorem**

**1.**

- (1)
- ${h}_{1}\oplus {h}_{2}={h}_{2}\oplus {h}_{1}$;
- (2)
- ${h}_{1}\otimes {h}_{2}={h}_{2}\otimes {h}_{1}$;
- (3)
- $\lambda ({h}_{1}\oplus {h}_{2})=\lambda {h}_{1}\oplus \lambda {h}_{2},\lambda >0$;
- (4)
- ${({h}_{1}\otimes {h}_{2})}^{\lambda}={h}_{1}^{\lambda}\otimes {h}_{2}^{\lambda},\lambda >0$;
- (5)
- ${\lambda}_{1}h\oplus {\lambda}_{2}h=({\lambda}_{1}+{\lambda}_{2})h,{\lambda}_{1},{\lambda}_{2}>0$;
- (6)
- ${h}^{{\lambda}_{1}}\otimes {h}^{{\lambda}_{2}}={h}^{{\lambda}_{1}+{\lambda}_{2}},{\lambda}_{1},{\lambda}_{2}>0$.

## 3. Generalized Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators

#### 3.1. GPIVHFOWA Operator

**Definition**

**8.**

**Theorem**

**2.**

**Proof.**

#### 3.2. The GPIVHFOWG Operator

**Definition**

**9.**

**Theorem**

**3.**

#### 3.3. The Properties of the GPIVHFOWA Operator and the GPIVHFOWG Operator

**Property**

**1**

**(idempotency).**

**Proof.**

**Property**

**2**

**(monotonicity).**

**Property**

**3**

**(boundedness).**

**Property**

**4**

**(commutativity).**

## 4. Relationship among the Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators

**Case 1:**When $g(t)=-\mathrm{ln}(t)$, the GPIVHFOWA operator and the GPIVHFOWG operator are transformed into probabilistic interval-valued hesitant fuzzy ordered weighted averaging (PIVHFOWA) operators and probabilistic interval-valued hesitant fuzzy ordered weighted geometric (PIVHFOWG) operators respectively, as follows:

**Case 2:**When $g(t)=-\mathrm{ln}\left[\left(2-t\right)/t\right]$, the GPIVHFOWA operator and GPIVHFOWG operator are transformed into probabilistic interval-valued hesitant fuzzy Einstein ordered weighted averaging (PIVHFEOWA) operator and the probabilistic interval-valued hesitant fuzzy Einstein ordered weighted geometric (PIVHFEOWG) operator, as follows:

**Lemma**

**1.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

## 5. A Group Decision Making Model Based on the PIVHF Information Aggregation Method

## 6. Illustrative Example

- Step 1’: Same as step 1.
- Step 2’: Based on the original decision matrix $H={({h}_{ij})}_{5\times 4}$, the comprehensive attribute values ${h}_{i}(i=1,2,3,4,5)$ of each supplier are calculated by using Equations (27) and (28). Considering the limitation of the length of the paper and the elements in ${h}_{i}$ can reach up to 81, then we are not list the comprehensive attribute values ${h}_{1}$ in details.
- Step 3’: Compute the score functions (Definition 3) $S({h}_{i})(i=1,2,3,4,5)$ of the comprehensive IVHFEs, which are shown in Table 4.
- Step 4’: According to the value of the score function, five suppliers are sorted, and the results are shown in Table 5. The calculation results show that the supplier with the best comprehensive performance is ${X}_{5}$.

- (1)
- Probabilistic interval-valued hesitant fuzzy elements can not only represent all possible decision information, but also reflect the importance of each decision information.
- (2)
- Comparing the numbers of interval-valued hesitant fuzzy elements, it is found that the decision making process of the information aggregation proposed by our method is more simplified and convenient than that of Chen et al. [28].
- (3)
- According the overall results of PIVHFEs derived by the PIVHFOWA operator, the PIVHFOWG operator, the PIVHFEOWA operator and the PIVHFEOWG operator, it is found that the relationship among these probabilistic interval-valued hesitant fuzzy aggregation operators are consistent with the Theorem 7.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | |

${X}_{1}$ | {([0.6,0.8],0.5),([0.4,0.6],0.2),([0.0,0.5],0.3)} | {([0.6,0.8],0.4),([0.5,0.7],0.3),([0.2,0.4],0.3)} |

${X}_{2}$ | {([0.5,0.7],0.6),([0.4,0.6],0.1),([0.2,0.4],0.3)} | {([0.4,0.6],0.5),([0.3,0.5],0.1),([0.1,0.3],0.4)} |

${X}_{3}$ | {([0.7,0.9],0.2),([0.4,0.6],0.4),([0.0,0.2],0.4)} | {([0.6,0.8],0.3),([0.3,0.5],0.5),([0.2,0.4],0.2)} |

${X}_{4}$ | {([0.4,0.6],0.6),([0.3,0.5],0.2),([0.2,0.4],0.2)} | {([0.4,0.6],0.4),([0.2,0.4],0.4),([0.0,0.2],0.2)} |

${X}_{5}$ | {([0.6,0.8],0.3),([0.2,0.4],0.4),([0.1,0.3],0.3)} | {([0.7,0.9],0.6),([0.6,0.8],0.2),([0.3,0.5],0.2)} |

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${X}_{1}$ | {([0.2,0.4],0.5),([0.1,0.3],0.4),([0.0,0.2],0.1)} | {([0.7,0.9],0.3),([0.4,0.6],0.3),([0.2,0.4],0.4)} |

${X}_{2}$ | {([0.5,0.7],0.3),([0.4,0.6],0.5),([0.0,0.2],0.2)} | {([0.7,0.9],0.6),([0.5,0.7],0.3),([0.0,0.2],0.1)} |

${X}_{3}$ | {([0.4,0.6],0.7),([0.1,0.3],0.2),([0.0,0.2],0.1)} | {([0.7,0.9],0.4),([0.2,0.4],0.5),([0.1,0.3],0.1)} |

${X}_{4}$ | {([0.6,0.8],0.5),([0.5,0.7],0.2),([0.3,0.5],0.3)} | {([0.5,0.7],0.5),([0.3,0.5],0.3),([0.2,0.4],0.2)} |

${X}_{5}$ | {([0.5,0.7],0.4),([0.4,0.6],0.4),([0.1,0.3],0.2)} | {([0.7,0.9],0.5),([0.4,0.6],0.1),([0.2,0.4],0.4)} |

**Table 2.**Comprehensive attribute values of each supplier are calculated with the probabilistic interval-valued hesitant fuzzy ordered weighted averaging (PIVHFOWA) operator.

Suppliers | Comprehensive Decision Attribute Values |
---|---|

${X}_{1}$ | {([0.5845,0.8045],0.4250),([0.3840,0.5896],0.3000),([0.1350,0.3816],0.2750)} |

${X}_{2}$ | {([0.5584,0.7774],0.5000),([0.4104,0.6133],0.2500),([0.0630,0.2314],0.2500)} |

${X}_{3}$ | {([0.6243,0.8375],0.4000),([0.2463,0.4487],0.4000),([0.0986,0.2997],0.2000)} |

${X}_{4}$ | {([0.4809,0.6851],0.5000),([0.3188,0.5232],0.2750),([0.1672,0.3693],0.2250)} |

${X}_{5}$ | {([0.6531,0.8618],0.4500),([0.4453,0.6547],0.2750),([0.1991,0.4004],0.2750)} |

**Table 3.**Comprehensive attribute values of each supplier are calculated with the probabilistic interval-valued hesitant fuzzy ordered weighted geometric (PIVHFOWG) operator.

Suppliers | Comprehensive Decision Attribute Values |
---|---|

${X}_{1}$ | {([0.5083,0.7258],0.4250),([0.3241,0.5471],0.3000),([0.0000,0.3601],0.2750)} |

${X}_{2}$ | {([0.5261,0.7298],0.5000),([0.3967,0.5996],0.2500),([0.0000,0.2259],0.2500)} |

${X}_{3}$ | {([0.5976,0.8011],0.4000),([0.2182,0.4291],0.4000),([0.0000,0.2838],0.2000)} |

${X}_{4}$ | {([0.4690,0.6708],0.5000),([0.2942,0.5002],0.2750),([0.0000,0.3397],0.2250)} |

${X}_{5}$ | {([0.6395,0.8409],0.4500),([0.4071,0.6155],0.2750),([0.1772,0.3867],0.2750)} |

Operators | ${\mathit{X}}_{\mathbf{1}}$ | ${\mathit{X}}_{\mathbf{2}}$ | ${\mathit{X}}_{\mathbf{3}}$ | ${\mathit{X}}_{\mathbf{4}}$ | ${\mathit{X}}_{\mathbf{5}}$ |
---|---|---|---|---|---|

PIVHFOWA | [0.1336,0.2079] | [0.1325,0.1999] | [0.1227,0.1915] | [0.1219,0.1898] | [0.1570,0.2260] |

PIVHFOWG | [0.1044,0.1905] | [0.1207,0.1904] | [0.1088,0.1829] | [0.1051,0.1831] | [0.1495,0.2180] |

PIVHFEOWA | [0.1313,0.2062] | [0.1312,0.1989] | [0.1215,0.1905] | [0.1208,0.1888] | [0.1560,0.2251] |

PIVHFEOWG | [0.1074,0.1938] | [0.1218,0.1917] | [0.1098,0.1842] | [0.1057,0.1841] | [0.1506,0.2192] |

IVHFOWA | [0.3678,0.5919] | [0.3439,0.5407] | [0.3231,0.5286] | [0.3223,0.5259] | [0.4325,0.6390] |

IVHFOWG | [0.2775,0.5443] | [0.3076,0.5184] | [0.2719,0.5047] | [0.2544,0.5036] | [0.4079,0.6144] |

Operators | Ranking Results |
---|---|

PIVHFOWA operator | ${X}_{5}\succ {X}_{1}\succ {X}_{2}\succ {X}_{3}\succ {X}_{4}$ |

PIVHFOWG operator | ${X}_{5}\succ {X}_{2}\succ {X}_{1}\succ {X}_{3}\succ {X}_{4}$ |

PIVHFEOWA operator | ${X}_{5}\succ {X}_{1}\succ {X}_{2}\succ {X}_{3}\succ {X}_{4}$ |

PIVHFEOWG operator | ${X}_{5}\succ {X}_{2}\succ {X}_{1}\succ {X}_{3}\succ {X}_{4}$ |

IVHFOWA operator | ${X}_{5}\succ {X}_{1}\succ {X}_{2}\succ {X}_{3}\succ {X}_{4}$ |

IVHFOWG operator | ${X}_{5}\succ {X}_{2}\succ {X}_{1}\succ {X}_{3}\succ {X}_{4}$ |

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**MDPI and ACS Style**

Wu, W.; Li, Y.; Ni, Z.; Jin, F.; Zhu, X.
Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators and Their Application to Multi-Attribute Decision Making. *Algorithms* **2018**, *11*, 120.
https://doi.org/10.3390/a11080120

**AMA Style**

Wu W, Li Y, Ni Z, Jin F, Zhu X.
Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators and Their Application to Multi-Attribute Decision Making. *Algorithms*. 2018; 11(8):120.
https://doi.org/10.3390/a11080120

**Chicago/Turabian Style**

Wu, Wenying, Ying Li, Zhiwei Ni, Feifei Jin, and Xuhui Zhu.
2018. "Probabilistic Interval-Valued Hesitant Fuzzy Information Aggregation Operators and Their Application to Multi-Attribute Decision Making" *Algorithms* 11, no. 8: 120.
https://doi.org/10.3390/a11080120