# Distribution-Based Approaches to Deriving Weights from Dual Hesitant Fuzzy Information

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (i)
- A dual hesitant fuzzy weighted averaging (DHFWA) operator of the dimension $n$ is a mapping DHFWA: ${\Omega}^{n}\to \Omega $, which has an associated $n-$ dimensional vector $w=({w}_{1},{w}_{2},\cdots ,{w}_{n})$, with ${w}_{i}>0,$ $i=1,2,\cdots ,n$ and ${\sum}_{i=1}^{n}{w}_{i}}=1$, such that$$\mathrm{DHFW}{\mathrm{A}}_{w}\left({d}_{1},{d}_{2},\cdots ,{d}_{n}\right)=\underset{i=1}{\overset{n}{\oplus}}\left({w}_{i}{d}_{i}\right)={\displaystyle {\cup}_{{\gamma}_{i}\in {h}_{i},{\eta}_{j}\in {g}_{j}}\left\{\left\{1-{\displaystyle \prod _{i=1}^{n}{\left(1-{\gamma}_{i}\right)}^{{w}_{i}}}\right\},\left\{{\displaystyle \prod _{i=1}^{n}{\left({\eta}_{i}\right)}^{{w}_{i}}}\right\}\right\}}$$
- (ii)
- A dual hesitant fuzzy weighted geometric (DHFWG) operator of the dimension $n$ is a mapping DHFWG: ${\Omega}^{n}\to \Omega $, which has an associated $n-$ dimensional vector $w=({w}_{1},{w}_{2},\cdots ,{w}_{n})$, with ${w}_{i}>0,$ $i=1,2,\cdots ,n$ and ${\sum}_{i=1}^{n}{w}_{i}}=1$, such that$$\mathrm{DHFW}{\mathrm{G}}_{w}\left({d}_{1},{d}_{2},\cdots ,{d}_{n}\right)=\underset{i=1}{\overset{n}{\otimes}}{\left({d}_{i}\right)}^{{w}_{i}}={\displaystyle {\cup}_{{\gamma}_{i}\in {h}_{i},{\eta}_{j}\in {g}_{j}}\left\{\left\{{\displaystyle \prod _{i=1}^{n}{\left({\gamma}_{i}\right)}^{{w}_{i}}}\right\},\left\{1-{\displaystyle \prod _{i=1}^{n}{\left(1-{\eta}_{i}\right)}^{{w}_{i}}}\right\}\right\}}$$

- (i)
- if ${s}_{{d}_{1}}>{s}_{{d}_{2}}$, then ${d}_{1}$ is superior to ${d}_{2}$, denoted by ${d}_{1}\succ {d}_{2}$;
- (ii)
- if ${s}_{{d}_{1}}={s}_{{d}_{2}}$, then
- (a)
- if ${p}_{{d}_{1}}={p}_{{d}_{2}}$, then ${d}_{1}$ is equivalent to ${d}_{2}$, denoted by ${d}_{1}\sim {d}_{2}$;
- (b)
- if ${p}_{{d}_{1}}>{p}_{{d}_{2}}$, then ${d}_{1}$ is superior than ${d}_{2}$, denoted by ${d}_{1}\succ {d}_{2}$.

## 3. Weighting Methods Based on Dual Hesitant Fuzzy Elements (DHFEs)

#### 3.1. The Distance Measures for DHFEs

**Definition**

**1.**

- (1)
- $0\le r\left({d}_{1},{d}_{2}\right)\le 1$;
- (2)
- $r\left({d}_{1},{d}_{2}\right)=0$if and only if${d}_{1}$=${d}_{2}$;
- (3)
- $r\left({d}_{1},{d}_{2}\right)=r\left({d}_{2},{d}_{1}\right)$.

**Definition**

**2.**

- (1)
- $0\le s\left({d}_{1},{d}_{2}\right)\le 1$;
- (2)
- $s\left({d}_{1},{d}_{2}\right)=1$if and only if${d}_{1}$=${d}_{2}$;
- (3)
- $s\left({d}_{1},{d}_{2}\right)=s\left({d}_{2},{d}_{1}\right)$.

- (1)
- The dual hesitant normalized Hamming distance between two DHFEs ${d}_{1}$ and ${d}_{2}$:$${r}_{dhnh}({d}_{1},{d}_{2})=\frac{1}{2{l}_{h}}{\displaystyle \sum _{j=1}^{{l}_{h}}\left|{h}_{1}^{\sigma \left(j\right)}\left(x\right)-{h}_{2}^{\sigma \left(j\right)}\left(x\right)\right|}+\frac{1}{2{l}_{g}}{\displaystyle \sum _{j=1}^{{l}_{g}}\left|{g}_{1}^{\sigma \left(j\right)}\left(x\right)-{g}_{2}^{\sigma \left(j\right)}\left(x\right)\right|}$$
- (2)
- The dual hesitant normalized Euclidean distance between two DHFEs ${d}_{1}$ and ${d}_{2}$:$${r}_{dhne}({d}_{1},{d}_{2})={\left[\frac{1}{2{l}_{h}}{\displaystyle \sum _{j=1}^{{l}_{h}}{\left|{h}_{1}^{\sigma \left(j\right)}\left(x\right)-{h}_{2}^{\sigma \left(j\right)}\left(x\right)\right|}^{2}}+\frac{1}{2{l}_{g}}{\displaystyle \sum _{j=1}^{{l}_{g}}{\left|{g}_{1}^{\sigma \left(j\right)}\left(x\right)-{g}_{2}^{\sigma \left(j\right)}\left(x\right)\right|}^{2}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$
- (3)
- The dual hesitant normalized Hamming–Hausdorff distance between two DHFEs ${d}_{1}$ and ${d}_{2}$:$${r}_{dhnhh}({d}_{1},{d}_{2})=\mathrm{max}\left\{\underset{j}{\mathrm{max}}\left|{h}_{1}^{\sigma \left(j\right)}\left(x\right)-{h}_{2}^{\sigma \left(j\right)}\left(x\right)\right|,\underset{k}{\mathrm{max}}\left|{g}_{1}^{\sigma \left(k\right)}\left(x\right)-{g}_{2}^{\sigma \left(k\right)}\left(x\right)\right|\right\}$$

**Definition**

**3.**

**Remark**

**1.**

**Remark**

**2.**

**Proof.**

**Definition**

**4.**

#### 3.2. Three Weighting Methods Based on DHFEs

**Theorem**

**1.**

**Proof.**

#### 3.3. The Weighting Methods Based on HFEs

**Definition**

**5.**

**Definition**

**6.**

- (1)
- The hesitant normalized Hamming distance between two HFEs:$${r}_{hnh}(A,B)=\frac{1}{{l}_{x}}{\displaystyle \sum _{j=1}^{{l}_{x}}{\left|{h}_{A}^{\sigma \left(j\right)}\left(x\right)-{h}_{B}^{\sigma \left(j\right)}\left(x\right)\right|}^{}}$$
- (2)
- The hesitant normalized Euclidean distance between two HFEs:$${r}_{hne}(A,B)={\left[\frac{1}{{l}_{x}}{\displaystyle \sum _{j=1}^{{l}_{x}}{\left|{h}_{A}^{\sigma \left(j\right)}\left(x\right)-{h}_{B}^{\sigma \left(j\right)}\left(x\right)\right|}^{2}}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$
- (3)
- The hesitant normalized Hamming–Hausdorff distance between two HFEs:$${r}_{hnhh}(A,B)=\underset{j}{\mathrm{max}}\left\{\left|{h}_{A}^{\sigma \left(j\right)}\left(x\right)-{h}_{B}^{\sigma \left(j\right)}\left(x\right)\right|\right\}$$

## 4. Illustrative Examples

#### 4.1. Illustrative Examples for DHFEs

**Example**

**1.**

**Step 1.**Calculate the mean $\overline{d}$ and the standard deviation $\sigma $ according to the Euclidean distance Equation(4), respectively: $\overline{d}$ = {{0.544,0.411,0.389}, {0.411,0.389,0.367}}, and $\sigma $ = 0.189.

**Step 2.**Determine the experts’ weights by using Equations (8), (10) and (13) respectively. Thus, the gained weights for ${d}_{i}\left(i=1,2,\cdots ,9\right)$ are listed in Table 1.

**Case 1.**Comparisons among the three methods with respect to the distances $r\left({d}_{i},\overline{d}\right)\left(i=1,2,\cdots ,9\right)$. To provide a clear analysis of them, we put the weights of the DHFEs ${d}_{i}\left(i=1,2,\cdots ,9\right)$ obtained from these three proposed methods, respectively, into Figure 1.

**Case 2.**Comparisons among the three methods and Xu’s method with respect to the ranking of scores. Since Xu’s method is designed for the OWA operator, then in order to conduct some comparisons, we first rank the DHFEs ${d}_{i}\left(i=1,2,\cdots ,9\right)$ based on the technique [4] as follows: ${d}_{7}\succ {d}_{9}\succ {d}_{4}\succ {d}_{8}\succ {d}_{3}\succ {d}_{1}\succ {d}_{6}\succ {d}_{2}\succ {d}_{5}$.

**Example**

**2.**

**Part 1. Evaluation process to choose the appropriate product manager.**

**Step 1.**Calculating weights. Using the hamming distance in Equation (3), we can get the weights of the experts ${p}_{j}\left(j=1,2,3,4\right)$ derived from Equations (8), (10) and (13) respectively as follows:

**Step 2.**Evaluations. We use Equations (1) and (2) to calculate the final scores of the candidates ${A}_{i}\left(i=1,2,3,4,5\right)$. For convenience, we assume that $\mathrm{DHFW}{\mathrm{A}}_{1}$, $\mathrm{DHFW}{\mathrm{A}}_{2}$ and $\mathrm{DHFW}{\mathrm{A}}_{3}$ represent the aggregated values obtained by the DHFWA operator using ${w}_{j}^{(1)}\left(j=1,2,3,4\right)$, ${w}_{j}^{(2)}\left(j=1,2,3,4\right)$ and ${w}_{j}^{(3)}\left(j=1,2,3,4\right)$ respectively, and $\mathrm{DHFW}{\mathrm{G}}_{1}$, $\mathrm{DHFW}{\mathrm{G}}_{2}$ and $\mathrm{DHFW}{\mathrm{G}}_{3}$ are obtained by the DHFWG operator using ${w}_{j}^{(1)}\left(j=1,2,3,4\right)$, ${w}_{j}^{(2)}\left(j=1,2,3,4\right)$ and ${w}_{j}^{(3)}\left(j=1,2,3,4\right)$ correspondingly. The results are shown in Table 6:

**Part 2. Discussion**

#### 4.2. Illustrative Examples for HFEs

**Example**

**3.**

**Step 1.**Calculate the mean $\overline{d}$ and the variance $\sigma $ of these HFEs:

**Step 2.**With using Equations (19), (21) and (22), the appropriate weights of ${h}_{i}\left(i=1,2,\cdots ,9\right)$ are determined, which are listed in Table 7.

**Example**

**4.**

## 5. Concluding Remarks

- (1)
- The mean and the standard deviation of a collection of DHFEs have been first defined to describe the mid one(s) and the divergence degrees of a collection of DHFEs.
- (2)
- Some distances for DHFEs have been introduced to depict the relationships between the mean and DHFEs.
- (3)
- Based on the natures of the linear function, the inverse proportion function, and the normal distribution function, three weighting methods for DHFEs have been developed, respectively.
- (4)
- These weighting methods have been extended to accommodate hesitant fuzzy information.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The weights of the three methods with respect to the distances $r\left({d}_{i},\overline{d}\right)\left(i=1,2,\cdots ,9\right)$.

**Table 1.**The weights ${w}_{i}^{\left(j\right)}\left(i=1,2,\cdots ,9;j=1,2,3\right)$ for the DHFEs ${d}_{i}\left(i=1,2,\cdots ,9\right)$.

${\mathit{d}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}^{\left(1\right)}(\mathbf{Ranking})$ | ${\mathit{w}}_{\mathit{i}}^{\left(2\right)}(\mathbf{Ranking})$ | ${\mathit{w}}_{\mathit{i}}^{\left(3\right)}(\mathbf{Ranking})$ |
---|---|---|---|

${d}_{1}$ | 0.123(3) | 0.124(3) | 0.161(3) |

${d}_{2}$ | 0.097(8) | 0.035(8) | 0.057(8) |

${d}_{3}$ | 0.130(1) | 0.461(1) | 0.173(1) |

${d}_{4}$ | 0.108(5) | 0.050(5) | 0.101(5) |

${d}_{5}$ | 0.096(9) | 0.034(9) | 0.053(9) |

${d}_{6}$ | 0.118(4) | 0.082(4) | 0.142(4) |

${d}_{7}$ | 0.099(7) | 0.037(7) | 0.063(7) |

${d}_{8}$ | 0.127(2) | 0.208(2) | 0.169(2) |

${d}_{9}$ | 0.103(6) | 0.042(6) | 0.080(6) |

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

${A}_{1}$ | {{0.4,0.3}, {0.5}} | {{0.5,0.4}, {0.4,0.3}} | {{0.3,0.2}, {0.6}} | {{0.5,0.4}, {0.5}} |

${A}_{2}$ | {{0.6}, {0.4}} | {{0.5,0.2,0.1}, {0.4}} | {{0.2}, {0.8,0.7,0.5}} | {{0.5}, {0.5,0.4}} |

${A}_{3}$ | {{0.8,0.6}, {0.2}} | {{0.7}, {0.2,0.1}} | {{0.6,0.5,0.4}, {0.4}} | {{0.7,0.6,0.5}, {0.3}} |

${A}_{4}$ | {{0.8}, {0.1}} | {{0.3,0.2,0.1}, {0.2}} | {{0.6,0.5}, {0.4}} | {{0.6}, {0.4,0.3,0.2}} |

${A}_{5}$ | {{0.6,0.5}, {0.4}} | {{0.4,0.3,0.2}, {0.5}} | {{0.5,0.4}, {0.2}} | {{0.4,0.3,0.2}, {0.5}} |

**Table 3.**The weights ${w}_{j}^{(1)}\left(j=1,2,3,4\right)$ for the experts ${p}_{j}\left(j=1,2,3,4\right)$.

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

${A}_{1}$ | 0.264 | 0.241 | 0.237 | 0.258 |

${A}_{2}$ | 0.247 | 0.254 | 0.233 | 0.266 |

${A}_{3}$ | 0.256 | 0.244 | 0.236 | 0.264 |

${A}_{4}$ | 0.229 | 0.234 | 0.266 | 0.271 |

${A}_{5}$ | 0.257 | 0.250 | 0.243 | 0.250 |

**Table 4.**The weights ${w}_{j}^{(2)}\left(j=1,2,3,4\right)$ for the experts ${p}_{j}\left(j=1,2,3,4\right)$.

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

${A}_{1}$ | 0.567 | 0.100 | 0.090 | 0.243 |

${A}_{2}$ | 0.207 | 0.251 | 0.153 | 0.390 |

${A}_{3}$ | 0.244 | 0.140 | 0.108 | 0.507 |

${A}_{4}$ | 0.130 | 0.141 | 0.324 | 0.405 |

${A}_{5}$ | 0.326 | 0.241 | 0.191 | 0.241 |

**Table 5.**The weights ${w}_{j}^{(3)}\left(j=1,2,3,4\right)$ for the experts ${p}_{j}\left(j=1,2,3,4\right)$.

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

${A}_{1}$ | 0.368 | 0.167 | 0.136 | 0.329 |

${A}_{2}$ | 0.233 | 0.276 | 0.150 | 0.341 |

${A}_{3}$ | 0.308 | 0.200 | 0.128 | 0.363 |

${A}_{4}$ | 0.149 | 0.172 | 0.330 | 0.349 |

${A}_{5}$ | 0.312 | 0.250 | 0.188 | 0.250 |

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

$\mathrm{DHFW}{\mathrm{A}}_{1}$ | −0.095 | −0.041 | 0.399 | 0.360 | 0.035 |

$\mathrm{DHFW}{\mathrm{A}}_{2}$ | −0.110 | −0.013 | 0.387 | 0.315 | 0.034 |

$\mathrm{DHFW}{\mathrm{A}}_{3}$ | −0.091 | −0.011 | 0.414 | 0.319 | 0.027 |

$\mathrm{DHFW}{\mathrm{G}}_{1}$ | −0.132 | −0.149 | 0.349 | 0.209 | −0.032 |

$\mathrm{DHFW}{\mathrm{G}}_{2}$ | −0.129 | −0.105 | 0.352 | 0.212 | −0.031 |

$\mathrm{DHFW}{\mathrm{G}}_{3}$ | −0.117 | −0.106 | 0.373 | 0.202 | −0.037 |

**Table 7.**Rankings of the aggregation results for DHFEs using ${w}_{j}^{(i)}\left(i=1,2,3;j=1,2,3,4\right)$.

$\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{1}$ | $\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{2}$ | $\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{3}$ | |
---|---|---|---|

Ranking | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}$ |

$\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{1}$ | $\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{2}$ | $\mathbf{D}\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{3}$ | |

Ranking | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}\succ {A}_{2}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{2}\succ {A}_{1}$ |

**Table 8.**The weights ${w}_{j}^{(4)}\left(j=1,2,3,4\right)$ for the experts ${p}_{j}\left(j=1,2,3,4\right)$.

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

${A}_{1}$ | 0.244 | 0.219 | 0.298 | 0.239 |

${A}_{2}$ | 0.265 | 0.209 | 0.305 | 0.222 |

${A}_{3}$ | 0.280 | 0.288 | 0.195 | 0.237 |

${A}_{4}$ | 0.370 | 0.112 | 0.245 | 0.274 |

${A}_{5}$ | 0.264 | 0.250 | 0.235 | 0.250 |

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

$\mathrm{DHFW}{\mathrm{A}}_{4}$ | −0.114 | −0.064 | 0.422 | 0.453 | 0.034 |

**Table 10.**The weights ${w}_{i}^{\left(j\right)}\left(i=1,2,\cdots ,9;j=1,2,3\right)$ for the HFEs ${h}_{i}\left(i=1,2,\cdots ,9\right)$.

${\mathit{h}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}^{\left(1\right)}(\mathbf{Ranking})$ | ${\mathit{w}}_{\mathit{i}}^{\left(2\right)}(\mathbf{Ranking})$ | ${\mathit{w}}_{\mathit{i}}^{\left(3\right)}(\mathbf{Ranking})$ |
---|---|---|---|

${h}_{1}$ | 0.118(3) | 0.121(3) | 0.152(3) |

${h}_{2}$ | 0.106(6) | 0.048(6) | 0.082(6) |

${h}_{3}$ | 0.123(1) | 0.463(1) | 0.168(1) |

${h}_{4}$ | 0.109(5) | 0.056(5) | 0.099(5) |

${h}_{5}$ | 0.102(9) | 0.041(9) | 0.060(9) |

${h}_{6}$ | 0.115(4) | 0.089(4) | 0.137(4) |

${h}_{7}$ | 0.106(7) | 0.047(7) | 0.079(7) |

${h}_{8}$ | 0.120(2) | 0.172(2) | 0.161(2) |

${h}_{9}$ | 0.103(8) | 0.041(8) | 0.062(8) |

${G}_{1}$ | ${G}_{2}$ | ${G}_{3}$ | ${G}_{4}$ | |

${A}_{1}$ | {0.4,0.3} | {0.5,0.4} | {0.3,0.2} | {0.5,0.4} |

${A}_{2}$ | {0.6} | {0.5,0.2,0.1} | {0.2} | {0.5} |

${A}_{3}$ | {0.8,0.6} | {0.7} | {0.6,0.5,0.4} | {0.7,0.6,0.5} |

${A}_{4}$ | {0.8} | {0.3,0.2,0.1} | {0.6,0.5} | {0.6} |

${A}_{5}$ | {0.6,0.5} | {0.4,0.3,0.2} | {0.5,0.4} | {0.4,0.3,0.2} |

**Table 12.**The aggregation results of the candidates ${A}_{i}\left(i=1,2,\cdots ,5\right)$ based on the HFEs.

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

$\mathrm{HFW}{\mathrm{A}}_{1}$ | 0.383 | 0.418 | 0.641 | 0.592 | 0.411 |

$\mathrm{HFW}{\mathrm{A}}_{2}$ | 0.380 | 0.423 | 0.634 | 0.567 | 0.411 |

$\mathrm{HFW}{\mathrm{A}}_{3}$ | 0.394 | 0.417 | 0.652 | 0.600 | 0.398 |

$\mathrm{HFW}{\mathrm{G}}_{1}$ | 0.364 | 0.343 | 0.617 | 0.492 | 0.379 |

$\mathrm{HFW}{\mathrm{G}}_{2}$ | 0.367 | 0.353 | 0.617 | 0.546 | 0.385 |

$\mathrm{HFW}{\mathrm{G}}_{3}$ | 0.379 | 0.346 | 0.632 | 0.534 | 0.371 |

**Table 13.**Rankings of the aggregation results for the HFEs using ${w}_{j}^{(i)}\left(i=1,2,3;j=1,2,3,4\right)$.

$\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{1}$ | $\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{2}$ | $\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{A}}_{3}$ | |
---|---|---|---|

Ranking | ${A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}$ |

$\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{1}$ | $\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{2}$ | $\mathbf{H}\mathbf{F}\mathbf{W}{\mathbf{G}}_{3}$ | |

Ranking | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}\succ {A}_{2}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{5}\succ {A}_{1}\succ {A}_{2}$ | ${A}_{3}\succ {A}_{4}\succ {A}_{1}\succ {A}_{5}\succ {A}_{2}$ |

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## Share and Cite

**MDPI and ACS Style**

Su, Z.; Xu, Z.; Zhao, H.; Liu, S.
Distribution-Based Approaches to Deriving Weights from Dual Hesitant Fuzzy Information. *Symmetry* **2019**, *11*, 85.
https://doi.org/10.3390/sym11010085

**AMA Style**

Su Z, Xu Z, Zhao H, Liu S.
Distribution-Based Approaches to Deriving Weights from Dual Hesitant Fuzzy Information. *Symmetry*. 2019; 11(1):85.
https://doi.org/10.3390/sym11010085

**Chicago/Turabian Style**

Su, Zhan, Zeshui Xu, Hua Zhao, and Shousheng Liu.
2019. "Distribution-Based Approaches to Deriving Weights from Dual Hesitant Fuzzy Information" *Symmetry* 11, no. 1: 85.
https://doi.org/10.3390/sym11010085