# Intuitionistic Fuzzy Sets in Multi-Criteria Group Decision Making Problems Using the Characteristic Objects Method

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^{2}

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^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

- if $Sc\left(\tilde{A}\right)<Sc\left(\tilde{B}\right),$ then $\tilde{A}<\tilde{B};$
- if $Sc\left(\tilde{A}\right)=Sc\left(\tilde{B}\right),$ and

- (i)
- $H\left(\tilde{A}\right)<H\left(\tilde{B}\right),$ then $\tilde{A}<\tilde{B};$
- (ii)
- $H\left(\tilde{A}\right)=H\left(\tilde{B}\right),$ then $\tilde{A}=\tilde{B}.$

**Definition**

**2.**

- $k\tilde{A}=(1-{(1-{\mu}_{\tilde{A}})}^{k},{\left({\nu}_{\tilde{A}}\right)}^{k})$, $k\in [0,1];$
- $\tilde{A}\oplus \tilde{B}=({\mu}_{\tilde{A}}+{\mu}_{\tilde{B}}-{\mu}_{\tilde{A}}{\mu}_{\tilde{B}},{\nu}_{\tilde{A}}{\nu}_{\tilde{B}});$
- $\tilde{A}\otimes \tilde{B}=({\nu}_{\tilde{A}}{\nu}_{\tilde{B}},{\mu}_{\tilde{A}}+{\mu}_{\tilde{B}}-{\mu}_{\tilde{A}}{\mu}_{\tilde{B}}).$

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 3. MCDM with COMET Method Using IFSs

**Step 1:**Define the space of the problem as follows:

**Step 2:**Generate the COs:

**Step 3:**Rank and evaluate the COs:

**Step 4:**Consistency measure:

`k=length(unique(SJ));`

`P=zeros(t,1);`

`for i=1:k`

`ind=find(SJ == max(SJ))`

`P(ind)=(k-i)/(k-1);`

`SJ(ind)= min(SJ)-1;`

`end`

**Step 5:**Inference in a fuzzy model and final ranking:

## 4. Illustrative Example

## 5. Conclusions

^{th}alternative ${A}_{5}=\{225,1750\}$ in the given decision problem, then, the original ranking of five alternatives is obtained as ${A}_{3}\succ {A}_{1}\succ {A}_{5}\succ {A}_{2}\succ {A}_{4}$. The preference value of ${A}_{5}$ using the proposed method is obtained as $0.9271$, which makes the new ranking order as ${A}_{3}\succ {A}_{2}\succ {A}_{5}\succ {A}_{1}\succ {A}_{4}$. From both ranking orders as calculated above, it can be easily observed that the inclusion of the new alternative ${A}_{5}$ does not affect the ranking order of the remaining alternatives. This observation justifies the basis of our claim. The prominent characteristic of the proposed approach is to provide a valuable and flexible way to efficiently assist the DMs under an uncertain environment. Furthermore, the proposed approach can be applied for both TIFNs and IFNs, which reflects the uncertainty appropriately. In the future, we hope that the COMET method can be applied to MCDM/MCGDM problems under more uncertain environments such as interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, hesitant fuzzy linguistic term sets, and so on.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MCGDM | Multi-Criteria Group Decision Making |

MCDM | Multi-Criteria Decision Making |

GDM | Group Decision Making |

DM | Decision Maker |

IF | Intuitionistic Fuzzy |

IFS | Intuitionistic Fuzzy Set |

IFN | Intuitionistic Fuzzy Number |

TIFN | Triangular Intuitionistic Fuzzy Number |

IPR | Intuitionistic Preference Relations |

HFS | Hesitant Fuzzy Set |

COMET | Characteristic Objects METhod |

MEJ | Matrix of Expert Judgments |

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**Figure 1.**The flowchart of the proposed approach combining the advantages of the Characteristic Object Method (COMET) and Triangular Intuitionistic Fuzzy Numbers (TIFNs).

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

${A}_{1}$ | 150 | $1.5$ | 1650 | 2 |

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

${A}_{3}$ | 250 | $1.25$ | 1500 | 1 |

${A}_{4}$ | 30 | $2.15$ | 2180 | 4 |

DM1 | $\left\{\right(0,0,180;0,0,190),(0,200,350;0,200,360),(200,300,380;200,300,400\left)\right\}$ |

DM2 | $\left\{\right(0,0,190;0,0,250),(0,200,380;0,200,390),(200,300,400;200,300,400\left)\right\}$ |

DM3 | $\left\{\right(0,0,170;0,0,210),(0,200,370;0,200,380),(200,300,340;200,300,390\left)\right\}$ |

DM1 | $\left\{\right(1100,1200,1600;1000,1200,1700),(1200,1800,2500;1100,1800,2600),$ |

$(1800,2500,2800;1700,2500,3000)\}$ | |

DM2 | $\left\{\right(1050,1200,1500;1000,1200,1600),(1100,1800,2700;1000,1800,2900),$ |

$(1800,2500,3000;1800,2500,3000)\}$ | |

DM3 | $\left\{\right(1150,1200,1400;1000,1200,1600),(1300,1800,2900;1100,1800,3000),$ |

$(1800,2500,2850;1800,2500,2900)\}$ |

**Table 4.**Comparison of the ranking obtained using intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs) with the original ranking.

Alternatives | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | Original | Ranking | Preference | Ranking |
---|---|---|---|---|---|---|

(LR) | (R/U) | Ranking | Using | Values Using | Using | |

HFSs | IFSs | IFSs | ||||

${A}_{1}$ | 150 | $1.5$ | 2 | 3 | $0.8502$ | 3 |

${A}_{2}$ | 50 | 2 | 3 | 2 | $0.9069$ | 2 |

${A}_{3}$ | 250 | $1.25$ | 1 | 1 | $0.9849$ | 1 |

${A}_{4}$ | 30 | $2.15$ | 4 | 4 | $0.8479$ | 4 |

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**MDPI and ACS Style**

Faizi, S.; Sałabun, W.; Rashid, T.; Zafar, S.; Wątróbski, J.
Intuitionistic Fuzzy Sets in Multi-Criteria Group Decision Making Problems Using the Characteristic Objects Method. *Symmetry* **2020**, *12*, 1382.
https://doi.org/10.3390/sym12091382

**AMA Style**

Faizi S, Sałabun W, Rashid T, Zafar S, Wątróbski J.
Intuitionistic Fuzzy Sets in Multi-Criteria Group Decision Making Problems Using the Characteristic Objects Method. *Symmetry*. 2020; 12(9):1382.
https://doi.org/10.3390/sym12091382

**Chicago/Turabian Style**

Faizi, Shahzad, Wojciech Sałabun, Tabasam Rashid, Sohail Zafar, and Jarosław Wątróbski.
2020. "Intuitionistic Fuzzy Sets in Multi-Criteria Group Decision Making Problems Using the Characteristic Objects Method" *Symmetry* 12, no. 9: 1382.
https://doi.org/10.3390/sym12091382