# A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**6.**

## 3. COMET Method with NIVTFNs

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

**Step 2:**Generate the COs

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

**Step 4:**Preference values of COs

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

## 4. An Illustrative Example

**Step 1:**Suppose that ${N}_{1},$${N}_{2},$ and ${N}_{3}$ represent the three families of subsets of $\mathcal{C}$ selected for the criteria ${C}_{1}$, ${C}_{2}$ and ${C}_{3}$ respectively, where

**Step 2:**The solution of COMET is obtained for different numbers of COs which can be obtained by taking the Cartesian product of the sets $C\left({N}_{1}\right),\phantom{\rule{4pt}{0ex}}C\left({N}_{2}\right)\phantom{\rule{4pt}{0ex}}$ and $C\left({N}_{3}\right)$. The list of all the COs with their set values are given as under:

**Step 3:**For the comparison of COs, suppose that the expert provides his/her pairwise judgments in the form of pre-defined linguistic scales in the form of NIVTFNs as expressed in Table 2. The most preferred CO will get linguistic variable “absolutely important”, the largest weaker CO will get linguistic variable “weakly important” and the COs with same comparison will get the linguistic variable “equally important”.

**Step 4:**Now, we calculate the vector $SJ=[{v}_{1},{v}_{2},\dots ,{v}_{18}]$ based on $MEJ$ as mentioned in Step 4. The first component ${v}_{1}$ can be computed by using Equation (1) as follows:

**Step 5:**The preference interval indicating the approximate preference value of the first alternative ${A}_{1}=\{84,4,3\}$ is computed by using Formula (6), which is given as follows:

## 5. Conclusions

- interval-valued intuitionistic fuzzy sets,
- hesitant fuzzy linguistic term sets,
- hesitant intuitionistic fuzzy linguistic term sets,
- etc.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Alternatives | ${\mathit{C}}_{1}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{C}}_{2}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{C}}_{3}$ | Reference Ranking |
---|---|---|---|---|

FP | BM | EP | ||

${A}_{1}$ | 84 | 8 | 3 | 2 |

${A}_{2}$ | 65 | 7 | $3.5$ | 8 |

${A}_{3}$ | 73 | 6 | $3.7$ | 7 |

${A}_{4}$ | 76 | 8 | $4.2$ | 6 |

${A}_{5}$ | 80 | 7 | $3.5$ | 4 |

${A}_{6}$ | 61 | 6 | $3.8$ | 9 |

${A}_{7}$ | 80 | $6.5$ | $3.7$ | 5 |

${A}_{8}$ | 85 | 8 | $4.5$ | 1 |

${A}_{9}$ | 59 | 6 | $3.5$ | 10 |

${A}_{10}$ | 79 | 8 | 4 | 3 |

Sr. No | Variable | Value |
---|---|---|

1 | $\mathrm{Weekly}\phantom{\rule{4.pt}{0ex}}\mathrm{Important}\phantom{\rule{4.pt}{0ex}}\left(WI\right)$ | $(0,0,0,0.1,0.2)$ |

2 | $\mathrm{Equally}\phantom{\rule{4.pt}{0ex}}\mathrm{Important}\phantom{\rule{4.pt}{0ex}}\left(EI\right)$ | $(0.2,0.2,0.2,0.3,0.4)$ |

3 | $\mathrm{Fairly}\phantom{\rule{4.pt}{0ex}}\mathrm{Important}\phantom{\rule{4.pt}{0ex}}\left(FI\right)$ | $(0.3,0.3,0.3,0.4,0.5)$ |

4 | $\mathrm{Strongly}\phantom{\rule{4.pt}{0ex}}\mathrm{Important}\phantom{\rule{4.pt}{0ex}}\left(SI\right)$ | $(0.7,0.7,0.7,0.8,0.9)$ |

5 | $\mathrm{Absolutely}\phantom{\rule{4.pt}{0ex}}\mathrm{Important}\phantom{\rule{4.pt}{0ex}}\left(AI\right)$ | $(0.8,0.8,0.8,0.9,1)$ |

${\mathit{CO}}_{1}$ | ${\mathit{CO}}_{2}$ | ${\mathit{CO}}_{3}$ | ${\mathit{CO}}_{4}$ | ${\mathit{CO}}_{5}$ | ${\mathit{CO}}_{6}$ | ${\mathit{CO}}_{7}$ | ${\mathit{CO}}_{8}$ | ${\mathit{CO}}_{9}$ | |
---|---|---|---|---|---|---|---|---|---|

$C{O}_{1}$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{2}$ | $FI$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ |

$C{O}_{3}$ | $FI$ | $FI$ | $EI$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ |

$C{O}_{4}$ | $FI$ | $FI$ | $WI$ | $EI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ |

$C{O}_{5}$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $FI$ | $FI$ | $FI$ |

$C{O}_{6}$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $EI$ | $SI$ | $FI$ | $FI$ |

$C{O}_{7}$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $EI$ | $WI$ | $WI$ |

$C{O}_{8}$ | $FI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ | $FI$ | $EI$ | $WI$ |

$C{O}_{9}$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $EI$ |

$C{O}_{10}$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ |

$C{O}_{11}$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $SI$ | $FI$ | $FI$ |

$C{O}_{12}$ | $AI$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $SI$ | $FI$ | $FI$ |

$C{O}_{13}$ | $FI$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ |

$C{O}_{14}$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ |

$C{O}_{15}$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $FI$ | $FI$ | $FI$ |

$C{O}_{16}$ | $SI$ | $FI$ | $FI$ | $FI$ | $EI$ | $WI$ | $FI$ | $FI$ | $FI$ |

$C{O}_{17}$ | $AI$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $AI$ | $FI$ | $FI$ |

$C{O}_{18}$ | $AI$ | $AI$ | $AI$ | $SI$ | $FI$ | $FI$ | $AI$ | $SI$ | $FI$ |

${\mathit{CO}}_{10}$ | ${\mathit{CO}}_{11}$ | ${\mathit{CO}}_{12}$ | ${\mathit{CO}}_{13}$ | ${\mathit{CO}}_{14}$ | ${\mathit{CO}}_{15}$ | ${\mathit{CO}}_{16}$ | ${\mathit{CO}}_{17}$ | ${\mathit{CO}}_{18}$ | |
---|---|---|---|---|---|---|---|---|---|

$C{O}_{1}$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{2}$ | $WI$ | $WI$ | $WI$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{3}$ | $WI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{4}$ | $WI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{5}$ | $FI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ | $EI$ | $WI$ | $WI$ |

$C{O}_{6}$ | $FI$ | $FI$ | $WI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $WI$ |

$C{O}_{7}$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{8}$ | $WI$ | $WI$ | $WI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{9}$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{10}$ | $EI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{11}$ | $FI$ | $EI$ | $WI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ | $WI$ |

$C{O}_{12}$ | $FI$ | $FI$ | $EI$ | $SI$ | $FI$ | $FI$ | $FI$ | $FI$ | $WI$ |

$C{O}_{13}$ | $WI$ | $WI$ | $WI$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{14}$ | $WI$ | $WI$ | $WI$ | $FI$ | $EI$ | $WI$ | $WI$ | $WI$ | $WI$ |

$C{O}_{15}$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $EI$ | $FI$ | $WI$ | $WI$ |

$C{O}_{16}$ | $FI$ | $WI$ | $WI$ | $FI$ | $FI$ | $WI$ | $EI$ | $WI$ | $WI$ |

$C{O}_{17}$ | $FI$ | $FI$ | $WI$ | $SI$ | $FI$ | $FI$ | $FI$ | $EI$ | $WI$ |

$C{O}_{18}$ | $FI$ | $FI$ | $FI$ | $FI$ | $SI$ | $FI$ | $FI$ | $FI$ | $EI$ |

**Table 5.**Ranking of Alternatives with COMET using NIVTFNs (Rank), TOPSIS ($TOPSI{S}_{R}$), and reference ranking ($Referenc{e}_{R}$), where $R{C}_{i}$ relative closeness.

Alternatives | Preference Intervals | ${\mathit{P}}_{\mathit{r}}\left({\mathit{A}}_{\mathit{i}}\right)$ | Rank | ${\mathit{RC}}_{\mathit{i}}$ | ${\mathit{TOPSIS}}_{\mathit{R}}$ | ${\mathit{Reference}}_{\mathit{R}}$ |
---|---|---|---|---|---|---|

${A}_{1}$ | $[0.0193,0.0906]$ | $0.05495$ | 2 | 0.8469 | 2 | 2 |

${A}_{2}$ | $[0.0126,0.0242]$ | $0.01840$ | 8 | 0.6521 | 8 | 8 |

${A}_{3}$ | $[0.0120,0.0274]$ | $0.01970$ | 7 | 0.7321 | 7 | 7 |

${A}_{4}$ | $[0.0144,0.0720]$ | $0.04320$ | 4 | 0.7655 | 6 | 6 |

${A}_{5}$ | $[0.0084,0.0741]$ | $0.04125$ | 5 | 0.8052 | 3 | 4 |

${A}_{6}$ | $[0.0065,0.0189]$ | $0.01270$ | 9 | 0.6106 | 9 | 9 |

${A}_{7}$ | $[0.0071,0.0725]$ | $0.03980$ | 6 | 0.8043 | 4 | 5 |

${A}_{8}$ | $[0.0809,0.1894]$ | $0.13515$ | 1 | 0.8584 | 1 | 1 |

${A}_{9}$ | $[0.0054,0.0186]$ | $0.01200$ | 10 | 0.5904 | 10 | 10 |

${A}_{10}$ | $[0.0114,0.0982]$ | $0.05480$ | 3 | 0.7963 | 5 | 3 |

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**MDPI and ACS Style**

Faizi, S.; Sałabun, W.; Ullah, S.; Rashid, T.; Więckowski, J.
A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique. *Symmetry* **2020**, *12*, 516.
https://doi.org/10.3390/sym12040516

**AMA Style**

Faizi S, Sałabun W, Ullah S, Rashid T, Więckowski J.
A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique. *Symmetry*. 2020; 12(4):516.
https://doi.org/10.3390/sym12040516

**Chicago/Turabian Style**

Faizi, Shahzad, Wojciech Sałabun, Samee Ullah, Tabasam Rashid, and Jakub Więckowski.
2020. "A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique" *Symmetry* 12, no. 4: 516.
https://doi.org/10.3390/sym12040516