# Novel Parameterized Distance Measures on Hesitant Fuzzy Sets with Credibility Degree and Their Application in Decision-Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Property**

**1.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Main Results

#### 3.1. Analysis on Hesitance Degree

#### 3.2. Novel Distance Measures with Three Factors

**Definition**

**5.**

**Definition**

**6.**

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

- In classical distance measures, the hesitance degree index of HFE is often calculated in addition to operations with the values of HFEs in classical distance measures. In contrast, the distance measures proposed in this study transfer the hesitance degree index to a credibility factor. Specifically, the credibility factor of HFE is calculated in multiplication operations with the values of HFEs in newly proposed distance measures, which handles the relationship between the cardinal number and the element values of hesitant fuzzy set well.
- In the newly proposed distance measures, there are three parameters. These parameters can be adjusted by decision makers according to the specific decision-making environment, which is beneficial for combining subjective and objective decision-making information, making the decision-making results more objective.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HFS | Hesitant fuzzy set |

HFE | Hesitant fuzzy element |

## References

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

${A}_{1}$ | $\left\{0.5,0.4,0.3\right\}$ | $\left\{0.9,0.8,0.7,0.1\right\}$ | $\left\{0.5,0.4,0.2\right\}$ | $\left\{0.9,0.6,0.5,0.3\right\}$ |

${A}_{2}$ | $\left\{0.5,0.3\right\}$ | $\left\{0.9,0.7,0.6,0.5,0.2\right\}$ | $\left\{0.8,0.6,0.5,0.1\right\}$ | $\left\{0.7,0.4,0.3\right\}$ |

${A}_{3}$ | $\left\{0.7,0.6\right\}$ | $\left\{0.9,0.6\right\}$ | $\left\{0.7,0.5,0.3\right\}$ | $\left\{0.6,0.4\right\}$ |

${A}_{4}$ | $\left\{0.8,0.7,0.4,0.3\right\}$ | $\left\{0.7,0.4,0.2\right\}$ | $\left\{0.8,0.1\right\}$ | $\left\{0.9,0.8,0.6\right\}$ |

${A}_{5}$ | $\left\{0.9,0.7,0.6,0.3,0.1\right\}$ | $\left\{0.8,0.7,0.6,0.4\right\}$ | $\left\{0.9,0.8,0.7\right\}$ | $\left\{0.9,0.7,0.6,0.3\right\}$ |

**Table 2.**Deviations between each alternative and the ideal alternative where $(\alpha ,\beta )=(0.9,0.1)$.

$\mathit{\lambda}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | Ranking |
---|---|---|---|---|---|---|

$\lambda =1$ | 0.2612926 | 0.2612927 | 0.2631 | 0.2628 | 0.2582 | ${A}_{5}\succ {A}_{1}\succ {A}_{2}\succ {A}_{4}\succ {A}_{3}$ |

$\lambda =2$ | 0.2622 | 0.2645 | 0.2636 | 0.2646 | 0.2585 | ${A}_{5}\succ {A}_{1}\succ {A}_{2}\succ {A}_{3}\succ {A}_{4}$ |

$\lambda =6$ | 0.2661 | 0.2774 | 0.2656 | 0.2723 | 0.2593 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

$\lambda =10$ | 0.2699 | 0.2891 | 0.2672 | 0.2801 | 0.2602 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

**Table 3.**Deviations between each alternative and the ideal alternative where $(\alpha ,\beta )=(0.7,0.3)$.

$\mathit{\lambda}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | Ranking |
---|---|---|---|---|---|---|

$\lambda =1$ | 0.2983 | 0.3021 | 0.2869 | 0.2904 | 0.2751 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =2$ | 0.3003 | 0.3054 | 0.2880 | 0.2940 | 0.2752 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =6$ | 0.3083 | 0.3186 | 0.2921 | 0.3073 | 0.2758 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =10$ | 0.3156 | 0.3304 | 0.2960 | 0.3178 | 0.2763 | ${A}_{5}\succ {A}_{3}\succ {A}_{1}\succ {A}_{4}\succ {A}_{2}$ |

**Table 4.**Deviations between each alternative and the ideal alternative where $(\alpha ,\beta )=(0.5,0.5)$.

$\mathit{\lambda}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | Ranking |
---|---|---|---|---|---|---|

$\lambda =1$ | 0.3409 | 0.3494 | 0.3143 | 0.3227 | 0.2944 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =2$ | 0.3445 | 0.3528 | 0.3173 | 0.3298 | 0.2958 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =6$ | 0.3588 | 0.3662 | 0.3274 | 0.3515 | 0.3002 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =10$ | 0.3707 | 0.3780 | 0.3347 | 0.3641 | 0.3032 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

**Table 5.**Deviations between each alternative and the ideal alternative where $(\alpha ,\beta )=(0.3,0.7)$.

$\mathit{\lambda}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | Ranking |
---|---|---|---|---|---|---|

$\lambda =1$ | 0.3900 | 0.4041 | 0.3459 | 0.3604 | 0.3166 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =2$ | 0.3959 | 0.4076 | 0.3524 | 0.3729 | 0.3205 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =6$ | 0.4192 | 0.4211 | 0.3715 | 0.4051 | 0.3316 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =10$ | 0.4368 | 0.4329 | 0.3822 | 0.4197 | 0.3385 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{1}$ |

**Table 6.**Deviations between each alternative and the ideal alternative where $(\alpha ,\beta )=(0.1,0.9)$.

$\mathit{\lambda}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | ${\mathit{A}}_{5}$ | Ranking |
---|---|---|---|---|---|---|

$\lambda =1$ | 0.4465 | 0.4674 | 0.3823 | 0.4044 | 0.3418 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =2$ | 0.4559 | 0.4711 | 0.3941 | 0.4247 | 0.3497 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{2}$ |

$\lambda =6$ | 0.4914 | 0.4848 | 0.4250 | 0.4694 | 0.3702 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{1}$ |

$\lambda =10$ | 0.5157 | 0.4963 | 0.4390 | 0.4864 | 0.3833 | ${A}_{5}\succ {A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{1}$ |

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**MDPI and ACS Style**

Li, J.; Zhang, F.; Li, Q.; Sun, J.; Yee, J.; Wang, S.; Xiao, S.
Novel Parameterized Distance Measures on Hesitant Fuzzy Sets with Credibility Degree and Their Application in Decision-Making. *Symmetry* **2018**, *10*, 557.
https://doi.org/10.3390/sym10110557

**AMA Style**

Li J, Zhang F, Li Q, Sun J, Yee J, Wang S, Xiao S.
Novel Parameterized Distance Measures on Hesitant Fuzzy Sets with Credibility Degree and Their Application in Decision-Making. *Symmetry*. 2018; 10(11):557.
https://doi.org/10.3390/sym10110557

**Chicago/Turabian Style**

Li, Jiaru, Fangwei Zhang, Qiang Li, Jing Sun, Janney Yee, Shuhong Wang, and Shujun Xiao.
2018. "Novel Parameterized Distance Measures on Hesitant Fuzzy Sets with Credibility Degree and Their Application in Decision-Making" *Symmetry* 10, no. 11: 557.
https://doi.org/10.3390/sym10110557