# Group Decision-Making for Hesitant Fuzzy Sets Based on Characteristic Objects Method

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- 1.
- $kh=\bigcup _{\gamma \in h}\{1-{(1-\gamma )}^{k}\};$
- 2.
- ${h}_{1}\oplus {h}_{2}=\bigcup _{{\gamma}_{1}\in {h}_{1},{\gamma}_{2}\in {h}_{2}}\{{\gamma}_{1}+{\gamma}_{2}-{\gamma}_{1}{\gamma}_{2}\};$
- 3.
- ${h}_{1}\otimes {h}_{2}=\bigcup _{{\gamma}_{1}\in {h}_{1},{\gamma}_{2}\in {h}_{2}}\{{\gamma}_{1}{\gamma}_{2}\}.$

**Definition**

**4.**

**Definition**

**5.**

- 1.
- The support of $\tilde{A}$ is $S(\tilde{A})=\left(\right)open="\{"\; close="\}">x:{\mu}_{\tilde{A}}(x)0$
- 2.
- The core of $\tilde{A}$ is $C(\tilde{A})=\left(\right)open="\{"\; close="\}">x:{\mu}_{\tilde{A}}(x)=1$

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

- 1.
- $T(x,y)=T(y,x)$ (commutativity),
- 2.
- $T(x,y)\le T(x,z)$, if $y\le z\phantom{\rule{4pt}{0ex}}$(monotonicity),
- 3.
- $T(x,T(y,z))=T(T(x,y),z)$ (associativity),
- 4.
- $T(x,1)=x$ (neutrality of one).

## 3. COMET for MCGDM Using HFS

${C}_{1}=\left(\right)open="\{"\; close="\}">{F}_{11}^{b\delta},{F}_{12}^{b\delta},...,{F}_{1{c}_{1}}^{b\delta}$ |

${C}_{2}=\left(\right)open="\{"\; close="\}">{F}_{21}^{b\delta},{F}_{22}^{b\delta},...,{F}_{2{c}_{2}}^{b\delta}$ |

$\vdots $ |

${C}_{n}=\left(\right)open="\{"\; close="\}">{F}_{n1}^{b\delta},{F}_{n2}^{b\delta},...,{F}_{n{c}_{n}}^{b\delta}$ |

$C({C}_{1})=\left(\right)open="\{"\; close="\}">C({F}_{11}^{t}),C({F}_{12}^{t}),...,C({F}_{1{c}_{1}}^{t})$ |

$C({C}_{2})=\left(\right)open="\{"\; close="\}">C({F}_{21}^{t}),C({F}_{22}^{t}),...,C({F}_{2{c}_{2}}^{t})$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

$C({C}_{n})=\left(\right)open="\{"\; close="\}">C({F}_{n1}^{t}),C({F}_{n2}^{t}),...,C({F}_{n{c}_{n}}^{t})$ |

$C{O}_{1}=\left(\right)open="\{"\; close="\}">C({F}_{11}^{t}),C({F}_{21}^{t}),...,C({F}_{n1}^{t})$ |

$C{O}_{2}=\left(\right)open="\{"\; close="\}">C({F}_{11}^{t}),C({F}_{21}^{t}),...,C({F}_{n2}^{t})$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

$C{O}_{s}=\left(\right)open="\{"\; close="\}">C({F}_{1{c}_{1}}^{t}),C({F}_{2{c}_{2}}^{t}),...,C({F}_{n{c}_{n}}^{t})$ |

$\mathbf{ME}J=\left(\right)open="["\; close="]">\begin{array}{cccc}{\tilde{h}}_{11}& {\tilde{h}}_{12}& \cdots & {\tilde{h}}_{1s}\\ {\tilde{h}}_{21}& {\tilde{h}}_{22}& \cdots & {\tilde{h}}_{2s}\\ \vdots & \vdots & \ddots & \vdots \\ {\tilde{h}}_{s1}& {\tilde{h}}_{s2}& \cdots & {\tilde{h}}_{ss}\end{array}$ |

- ${\tilde{h}}_{\alpha \beta}^{\sigma (\omega )}+{\tilde{h}}_{\beta \alpha}^{\sigma ({l}_{{\tilde{h}}_{\alpha \beta}}-\omega +1)}=1,\alpha ,\beta =l,2,...,s;$
- ${\tilde{h}}_{\alpha \alpha}=\{0.5\},\alpha =l,2,...,s;$
- ${l}_{{\tilde{h}}_{\alpha \beta}}={l}_{{\tilde{h}}_{\beta \alpha}},\alpha ,\beta =l,2,...,s.$

${a}_{1j}\in [C({F}_{1{k}_{1}}^{t}),C({F}_{1({k}_{1}+1)}^{t})];$ |

${a}_{2j}\in [C({F}_{2{k}_{2}}^{t}),C({F}_{2({k}_{2}+1)}^{t})];$ |

$\vdots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ |

${a}_{nj}\in [C({F}_{n{k}_{n}}^{t}),C({F}_{n({k}_{n}+1)}^{t})];$ |

## 4. An Illustrative Example

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Alternatives | ${\mathit{C}}_{1}$ (LR) | ${\mathit{C}}_{2}$ (R/U) | Bill Amount | Original Rank |
---|---|---|---|---|

${A}_{1}$ | 150 | 1500 | 1650 | 2 |

${A}_{2}$ | 50 | 2000 | 2050 | 3 |

${A}_{3}$ | 250 | 1250 | 1500 | 1 |

${A}_{4}$ | 30 | 2150 | 2180 | 4 |

**Table 2.**LR-type Group Fuzzy Numbers (GFNs) selected by the Decision-Makers (DMs) for criteria ${C}_{1}$.

DM1 | $\{(30,30,200),(30,200,300),(200,300,300)\}$ |

$\{(30,30,30,170),(30,170,220,300),(220,300,300,300)\}$ | |

DM2 | $\{(30,30,200),(30,200,300),(200,300,300)\}$ |

$\{(30,30,30,180),(30,180,230,300),(230,300,300,300)\}$ | |

DM3 | $\{(30,30,200),(30,200,300),(200,300,300)\}$ |

$\{(30,30,30,160),(30,160,215,300),(215,300,300,300)\}$ |

DM1 | $\{(1200,1200,1800),(1200,1800,2500),(1800,2500,2500)\}$ |

$\{(1200,1200,1200,1600),(1200,1600,1900,2500),(1900,2500,2500,2500)\}$ | |

DM2 | $\{(1200,1200,1800),(1200,1800,2500),(1800,2500,2500)\}$ |

$\{(1200,1200,1200,1700),(1200,1700,1900,2500),(1900,2500,2500,2500)\}$ | |

DM3 | $\{(1200,1200,1800),(1200,1800,2500),(1800,2500,2500)\}$ |

$\{(1200,1200,1200,1650),(1200,1650,1950,2500),(1950,2500,2500,2500)\}$ |

Average of the Membership Values Obtained from LR-Type GFNs for Criterion ${\mathit{C}}_{2}$ | |||
---|---|---|---|

30 | 50 | 150 | 250 |

$(1,0,0)$ | $(0.8824,0.1176,0)$ | $(0.2941,0.7059,0)$ | $(0,0.5000,0.5000)$ |

$(1,0,0)$ | $(0.8567,0.1433,0)$ | $(0.8567,0.1433,0)$ | $(0,0.6425,0.3575)$ |

1250 | 1500 | 2000 | 2150 |

$(0.9167,0.0833,0)$ | $(0.5000,0.5000,0)$ | $(0,0.7143,0.2857)$ | $(0,0.5000,0.5000)$ |

$(0.8880,0.1120,0)$ | $(0.3278,0.6722,0)$ | $(0,0.8586,0.1414)$ | $(0,0.6010,0.3990)$ |

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

$C{O}_{1}$ | $\{0.5\}$ | $\{0.8,1\}$ | $\{0.8,0.9\}$ | $\{0.7,0.8\}$ | $\{0.8,0.9,1\}$ |

$C{O}_{2}$ | $\{0,0.2\}$ | $\{0.5\}$ | $\{0.8,1\}$ | $\{0,0.1,0.2\}$ | $\{0.9,1\}$ |

$C{O}_{3}$ | $\{0.1,0.2\}$ | $\{0,0.2\}$ | $\{0.5\}$ | $\{0,0.2,0.3\}$ | $\{0.1,0.2\}$ |

$C{O}_{4}$ | $\{0.2,0.3\}$ | $\{0.8,0.9,1\}$ | $\{0.7,0.8,1\}$ | $\{0.5\}$ | $\{0.8,0.9,1\}$ |

$C{O}_{5}$ | $\{0,0.1,0.2\}$ | $\{0,0.1\}$ | $\{0.8,0.9\}$ | $\{0,0.1,0.2\}$ | $\{0.5\}$ |

$C{O}_{6}$ | $\{0,0.2\}$ | $\{0,0.2\}$ | $\{0,0.1\}$ | $\{0,0.2\}$ | $\{0,0.1,0.2\}$ |

$C{O}_{7}$ | $\{0,0.2\}$ | $\{0.8,1\}$ | $\{0,0.8\}$ | $\{0,0.2\}$ | $\{0.8,0.9,1\}$ |

$C{O}_{8}$ | $\{0,0.1,0.2\}$ | $\{0.1,0.2\}$ | $\{0.7,0.8,1\}$ | $\{0.1,0.2\}$ | $\{0,0.1,0.2\}$ |

$C{O}_{9}$ | $\{0,0.2\}$ | $\{0,0.1\}$ | $\{0.2,0.3\}$ | $\{0,0.1,0.2\}$ | $\{0,0.2\}$ |

$\mathit{C}{\mathit{O}}_{6}$ | $\mathit{C}{\mathit{O}}_{7}$ | $\mathit{C}{\mathit{O}}_{8}$ | $\mathit{C}{\mathit{O}}_{9}$ | $\mathit{S}\mathit{J}$ | |
---|---|---|---|---|---|

$C{O}_{1}$ | $\{0.8,1\}$ | $\{0.8,1\}$ | $\{0.8,0.9,1\}$ | $\{0.8,1\}$ | $0.999999$ |

$C{O}_{2}$ | $\{0.8,1\}$ | $\{0,0.2\}$ | $\{0.8,0.9\}$ | $\{0.9,1\}$ | $0.999980$ |

$C{O}_{3}$ | $\{0.9,1\}$ | $\{0,0.2\}$ | $\{0,0.2,0.3\}$ | $\{0.7,0.8\}$ | $0.995033$ |

$C{O}_{4}$ | $\{0.8,1\}$ | $\{0.8,1\}$ | $\{0.8,0.9\}$ | $\{0.8,0.9,1\}$ | $0.999998$ |

$C{O}_{5}$ | $\{0.8,0.9,1\}$ | $\{0,0.1,0.2\}$ | $\{0.8,0.9,1\}$ | $\{0.8,1\}$ | $0.999751$ |

$C{O}_{6}$ | $\{0.5\}$ | $\{0,0.1,0.2\}$ | $\{0.2,0.3\}$ | $\{0.8,0.9,1\}$ | $0.968745$ |

$C{O}_{7}$ | $\{0.8,0.9,1\}$ | $\{0.5\}$ | $\{0.8,1\}$ | $\{0.8,0.9\}$ | $0.999970$ |

$C{O}_{8}$ | $\{0.7,0.8\}$ | $\{0,0.2\}$ | $\{0.5\}$ | $\{0.7,0.8,0.9\}$ | $0.996636$ |

$C{O}_{9}$ | $\{0,0.1,0.2\}$ | $\{0.1,0.2\}$ | $\{0.1,0.2,0.3\}$ | $\{0.5\}$ | $0.841614$ |

Alternatives | ${\mathit{C}}_{1}$ (LR) | ${\mathit{C}}_{2}$ (R/U) | Original Ranking | Preference Values | New Ranking |
---|---|---|---|---|---|

${A}_{1}$ | 150 | 1500 | 2 | $0.5725$ | 3 |

${A}_{2}$ | 50 | 2000 | 3 | $0.6236$ | 2 |

${A}_{3}$ | 250 | 1250 | 1 | $0.6272$ | 1 |

${A}_{4}$ | 30 | 2150 | 4 | $0.5281$ | 4 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S.
Group Decision-Making for Hesitant Fuzzy Sets Based on Characteristic Objects Method. *Symmetry* **2017**, *9*, 136.
https://doi.org/10.3390/sym9080136

**AMA Style**

Faizi S, Sałabun W, Rashid T, Wątróbski J, Zafar S.
Group Decision-Making for Hesitant Fuzzy Sets Based on Characteristic Objects Method. *Symmetry*. 2017; 9(8):136.
https://doi.org/10.3390/sym9080136

**Chicago/Turabian Style**

Faizi, Shahzad, Wojciech Sałabun, Tabasam Rashid, Jarosław Wątróbski, and Sohail Zafar.
2017. "Group Decision-Making for Hesitant Fuzzy Sets Based on Characteristic Objects Method" *Symmetry* 9, no. 8: 136.
https://doi.org/10.3390/sym9080136