# An Approach for the Analysis of Energy Resource Selection Based on Attributes by Using Dombi T-Norm Based Aggregation Operators

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Let $X$ be the set. An IFS$B$ is:

**Definition**

**2**

**.**Let $X$ be the set. An IVIFS $B$ is:

**Definition**

**3**

**.**Let $M=\left(\left[{\mu}_{M}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)\right],\left[{v}_{M}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)\right]\right)$ and $N=\left(\left[{\mu}_{N}^{l}\left(\u0377\right),{\mu}_{N}^{u}\left(\u0377\right)\right],\left[{v}_{N}^{l}\left(\u0377\right),{v}_{N}^{u}\left(\u0377\right)\right]\right)$ be two IVIFNs and $\lambda >0$ be any arbitrary real number. Then:

- $M\subseteq N$ iff ${\mu}_{M}^{l}\left(\u0377\right)\le {\mu}_{N}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)\le {\mu}_{N}^{u}\left(\u0377\right)$ and ${v}_{M}^{l}\left(\u0377\right)\ge {v}_{N}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)\ge {v}_{N}^{u}\left(\u0377\right)$.
- $M=N$ iff ${\mu}_{M}^{l}\left(\u0377\right)={\mu}_{N}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)={\mu}_{N}^{u}\left(\u0377\right)$ and ${v}_{M}^{l}\left(\u0377\right)={v}_{N}^{l}\left(\u0377\right)={v}_{M}^{u}\left(\u0377\right)={v}_{N}^{u}\left(\u0377\right)$
- $M\u0377N=\left(\left[{\mu}_{M}^{l}\left(\u0377\right)\vee {\mu}_{N}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)\vee {\mu}_{N}^{u}\left(\u0377\right)\right],\left[{v}_{M}^{l}\left(\u0377\right)\wedge {v}_{N}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)\wedge {v}_{N}^{u}\left(\u0377\right)\right]\right)$
- $M\cap N=(\left[{\mu}_{M}^{l}\left(\u0377\right)\wedge {\mu}_{N}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)\wedge {\mu}_{N}^{u}\left(\u0377\right)\right],\left[{v}_{M}^{l}\left(\u0377\right)\vee {v}_{N}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)\vee {v}_{N}^{u}\left(\u0377\right)\right]$.
- ${M}^{c}=\left(\left[{v}_{M}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)\right],\left[{\mu}_{M}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)\right]\right)$
- $M\oplus N=\left(\begin{array}{c}\left[{\mu}_{M}^{l}\left(\u0377\right)+{\mu}_{N}^{l}\left(\u0377\right)-{\mu}_{M}^{l}\left(\u0377\right){\mu}_{N}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right)+{\mu}_{N}^{u}\left(\u0377\right)-{\mu}_{M}^{u}\left(\u0377\right){\mu}_{N}^{u}\left(\u0377\right)\right],\\ \left[{v}_{M}^{l}\left(\u0377\right){v}_{M}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right){v}_{N}^{u}\left(\u0377\right)\right]\end{array}\right)$.
- $M\otimes N=\left(\begin{array}{c}\left[{\mu}_{M}^{l}\left(\u0377\right){\mu}_{M}^{l}\left(\u0377\right),{\mu}_{M}^{u}\left(\u0377\right){\mu}_{N}^{u}\left(\u0377\right)\right],\\ \left[{v}_{M}^{l}\left(\u0377\right)+{v}_{N}^{l}\left(\u0377\right)-{v}_{M}^{l}\left(\u0377\right){v}_{N}^{l}\left(\u0377\right),{v}_{M}^{u}\left(\u0377\right)+{v}_{N}^{u}\left(\u0377\right)-{v}_{M}^{u}\left(\u0377\right){v}_{N}^{u}\left(\u0377\right)\right]\end{array}\right)$.
- $\lambda M=\left[1-{\left(1-{\mu}_{M}^{l}\left(\u0377\right)\right)}^{\lambda}1-{\left(1-{\mu}_{M}^{u}\left(\u0377\right)\right)}^{\lambda}\right],\left[{\left({v}_{M}^{l}\left(\u0377\right)\right)}^{\lambda}{\left({v}_{M}^{u}\left(\u0377\right)\right)}^{\lambda}\right]$.
- ${M}^{\lambda}=\left[{\left({\mu}_{M}^{l}\left(\u0377\right)\right)}^{\lambda}{\left({\mu}_{M}^{u}\left(\u0377\right)\right)}^{\lambda}\right],\left[1-{\left(1-{v}_{M}^{l}\left(\u0377\right)\right)}^{\lambda}1-{\left(1-{v}_{M}^{u}\left(\u0377\right)\right)}^{\lambda}\right]$.

**Definition**

**4**

**.**Let $\alpha =\left(\left[{\mu}_{\alpha}^{l},{\mu}_{\alpha}^{u}\right],\left[{v}_{\alpha}^{l},{v}_{\alpha}^{u}\right]\right)$ be any IVIFN. Then, the SF is:

**Definition**

**5**

**.**Let $\alpha =\left(\left[{\mu}_{\alpha}^{l},{\mu}_{\alpha}^{u}\right],\left[{v}_{\alpha}^{l},{v}_{\alpha}^{u}\right]\right)$ and $\beta =\left(\left[{\mu}_{\beta}^{l},{\mu}_{\beta}^{u}\right],\left[{v}_{\beta}^{l},{v}_{\beta}^{u}\right]\right)$ be two IVIFNs. Then:

- If $\Phi \left(\alpha \right)>\Phi \left(\beta \right)$, then $\alpha >\beta $.
- If $\Phi \left(\alpha \right)<\Phi \left(\beta \right)$, then $\alpha <\beta $.

**Definition**

**6**

**.**Let the real numbers be $a$ and $b$ . Then, DTN is:

**Definition**

**7.**

**Definition**

**8**

**.**Let ${\alpha}_{i}=\left({\mu}_{i},{v}_{i}\right)\left(i=1,2,\dots ,n\right)$ be a number of IFNs. Then, intuitionistic fuzzy Dombi weighted averaging (geometric) operators are defined by:

## 3. IVIF Dombi Operational Laws

**Definition**

**9**

**.**Let $\alpha =\left(\left[{\mu}_{\alpha}^{l},{\mu}_{\alpha}^{u}\right],\left[{v}_{\alpha}^{l},{v}_{\alpha}^{u}\right]\right)$, $\beta =\left(\left[{\mu}_{\beta}^{l},{\mu}_{\beta}^{u}\right],\left[{v}_{\beta}^{l},{v}_{\beta}^{u}\right]\right)$ be two IVIFNs, and $\mathsf{\u0444}\ge 1and\lambda 0$ be any real number. Then, DTN and TCN operations of IVIFNs are:

- $\alpha \oplus \beta =\left(\begin{array}{c}\left[1-\frac{1}{1+{\left\{{\left(\frac{{\mu}_{\alpha}^{l}}{1-{\mu}_{\alpha}^{l}}\right)}^{\u0444}+{\left(\frac{{\mu}_{\beta}^{l}}{1-{\mu}_{\beta}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},,1-\frac{1}{1+{\left\{{\left(\frac{{\mu}_{\alpha}^{u}}{1-{\mu}_{\alpha}^{u}}\right)}^{\u0444}+{\left(\frac{{\mu}_{\beta}^{u}}{1-{\mu}_{\beta}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right],\\ \left[\frac{1}{1+{\left\{{\left(\frac{1-{v}_{\alpha}^{l}}{{v}_{\alpha}^{l}}\right)}^{\u0444}+{\left(\frac{1-{v}_{\beta}^{l}}{{v}_{\beta}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},\frac{1}{1+{\left\{{\left(\frac{1-{v}_{\alpha}^{u}}{{v}_{\alpha}^{u}}\right)}^{\u0444}+{\left(\frac{1-{v}_{\beta}^{u}}{{v}_{\beta}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right]\end{array}\right)$
- $\alpha \otimes \beta =\left(\begin{array}{c}\left[\frac{1}{1+{\left\{{\left(\frac{1-{\mu}_{\alpha}^{l}}{{\mu}_{\alpha}^{l}}\right)}^{\u0444}+{\left(\frac{1-{\mu}_{\beta}^{l}}{{\mu}_{\beta}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},\frac{1}{1+{\left\{{\left(\frac{1-{\mu}_{\alpha}^{u}}{{\mu}_{\alpha}^{u}}\right)}^{\u0444}+{\left(\frac{1-{v}_{\beta}^{u}}{{\mu}_{\beta}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right],\\ \left[1-\frac{1}{1+{\left\{{\left(\frac{{\mu}_{\alpha}^{l}}{1-{v}_{\alpha}^{l}}\right)}^{\u0444}+{\left(\frac{{v}_{\beta}^{l}}{1-{v}_{\beta}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},,1-\frac{1}{1+{\left\{{\left(\frac{{v}_{\alpha}^{u}}{1-{v}_{\alpha}^{u}}\right)}^{\u0444}+{\left(\frac{{v}_{\beta}^{u}}{1-{v}_{\beta}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right]\end{array}\right)$
- $\lambda \alpha =\left(\begin{array}{c}\left[1-\frac{1}{1+{\left\{\lambda {\left(\frac{{\mu}_{\alpha}^{l}}{1-{\mu}_{\alpha}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},1-\frac{1}{1+{\left\{\lambda {\left(\frac{{\mu}_{\alpha}^{u}}{1-{\mu}_{\alpha}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right],\\ \left[\frac{1}{1+{\left\{\lambda {\left(\frac{1-{v}_{\alpha}^{l}}{{v}_{\alpha}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},\frac{1}{1+{\left\{\lambda {\left(\frac{1-{v}_{\alpha}^{u}}{{v}_{\alpha}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right]\end{array}\right)$
- ${\alpha}^{\lambda}=\left(\begin{array}{c}\left[\frac{1}{1+{\left\{\lambda {\left(\frac{1-{\mu}_{\alpha}^{l}}{{\mu}_{\alpha}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},\frac{1}{1+{\left\{\lambda {\left(\frac{1-{\mu}_{\alpha}^{u}}{{\mu}_{\alpha}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right],\\ \left[1-\frac{1}{1+{\left\{\lambda {\left(\frac{{v}_{\alpha}^{l}}{1-{v}_{\alpha}^{l}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}},1-\frac{1}{1+{\left\{\lambda {\left(\frac{{v}_{\alpha}^{u}}{1-{v}_{\alpha}^{u}}\right)}^{\u0444}\right\}}^{\frac{1}{\u0444}}}\right]\end{array}\right)$

**Theorem**

**1.**

- 1.
- $\alpha \oplus \beta =\beta \oplus \alpha $
- 2.
- $\alpha \otimes \beta =\beta \otimes \alpha $
- 3.
- $\lambda \left(\alpha \oplus \beta \right)=\lambda \alpha \oplus \lambda \beta $
- 4.
- $\left({\lambda}_{1}+{\lambda}_{2}\right)\alpha ={\lambda}_{1}\alpha \oplus {\lambda}_{2}\alpha $
- 5.
- ${\left(\alpha \otimes \beta \right)}^{\lambda}={\alpha}^{\lambda}\otimes {\beta}^{\lambda}$
- 6.
- ${\alpha}_{1}^{\lambda}\otimes {\alpha}_{2}^{\lambda}={\alpha}^{{\lambda}_{1}+{\lambda}_{2}}$

## 4. Dombi AOs

**Definition**

**10**

**.**The IVIFDWA operator for some IVIFNs ${\alpha}_{i}=\left(\left[{\mu}_{i}^{l},{\mu}_{i}^{u}\right],\left[{v}_{i}^{l},{v}_{i}^{u}\right]\right)$ is a function ${\alpha}^{n}\to \alpha ,$ such that

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Definition**

**11.**

**Theorem**

**6.**

**Remark**

**1.**

**Definition**

**12.**

^{th}biggest weighted IVIF number of ${\alpha}_{i}({\dot{\alpha}}_{i}=n{\xi}_{i}{\alpha}_{i}$and $n$is the balancing coefficient.

**Definition**

**13**

**.**Let ${\alpha}_{i}=\left(\left[{\mu}_{i}^{l},{\mu}_{i}^{u}\right],\left[{v}_{i}^{l},{v}_{i}^{u}\right]\right)$ be a number of IVIFNs. Then, IVIF Dombi weighted geometric (IVIFDWG) operator is a function ${\alpha}^{n}\to \alpha $ such that:

**Proof.**

**Theorem**

**10.**

**Proof**

**.**The proof follows the same pattern as Theorem 3. □

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Definition**

**14**

**.**Let ${\alpha}_{i}=\left(\left[{\mu}_{i}^{l},{\mu}_{i}^{u}\right],\left[{v}_{i}^{l},{v}_{i}^{u}\right]\right)$be number of IVIFNs. Then, IVIF Dombi order weighted geometric (IVIFDOWG) operator of dimension n is a function${\alpha}^{n}\to \alpha $, such that:

**Theorem**

**13.**

**Remark**

**2.**

**Theorem**

**14.**

**Theorem**

**15.**

**Theorem**

**16.**

**Definition**

**15**

**.**Let ${\alpha}_{i}$ and ${\u0377}_{i}$be two sets of IVIFNs. Then, IVIF Dombi hybrid geometric (IVIFDHG) operator of dimension n is a function ${\alpha}^{n}\to \alpha $ , such that:

^{th}biggest weighted intuitionistic fuzzy value of ${\u0377}_{i}\left({\u0377}_{i}=n{\xi}_{i}{\u0377}_{i},i=1,2,\dots ,n\right).$n is the balancing coefficient.

## 5. Model for MADM Using IVIF Information

#### 5.1. Algorithm

**Step**

**1:**

**Step**

**2:**

**Step**

**3:**

**Step**

**4:**

**Step**

**5:**

#### 5.2. Case Study

#### 5.3. Example

- ${Q}_{1}$: Cost.
- ${Q}_{2}$: Quantity.
- ${Q}_{3}$: Reliability.
- ${Q}_{4}$: Sustainability.

#### 5.4. Impact of $m$ on Ranking Results

## 6. Comparative Study

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DTN | Dombi t-norm |

TN | T-norm |

TCN | T-conorm |

MG | Membership grade |

NMG | Non-MG |

AO | Aggregation operator |

IVIF | Interval-valued intuitionistic fuzzy |

IVIFN | IVIF number |

IVIFDWA | IVIF Dombi weighted averaging |

IVIFDWG | IVIF Dombi weighted geometric |

FS | Fuzzy set |

IFS | Intuitionistic FS |

HD | Hesitancy degree |

MADM | Multi-attribute decision-making |

IVIFDOWA | IVIF Dombi ordered weighted averaging |

IVIFDHA | IVIF Dombi hybrid averaging |

IVIFDWG | IVIF Dombi ordered weighted geometric |

IVIFDHG | IVIF Dombi hybrid geometric |

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**Figure 1.**A graphical depiction of the ranking results is given in Table 4.

**Figure 2.**A graphical representation of the comparative study is given in Table 7.

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

MD | NMD | MD | NMD | MD | NMD | MD | NMD | |||||||||

L | U | L | U | L | U | L | U | L | U | L | U | L | U | L | U | |

${J}_{1}$ | $0.25$ | $0.7$ | $0.07$ | $0.23$ | $0.03$ | $0.05$ | $0.7$ | $0.2$ | $0.1$ | $0.3$ | $0.5$ | $0.4$ | $0.08$ | $0.01$ | $0.01$ | $0.05$ |

${J}_{2}$ | $0.57$ | $0.39$ | $0.03$ | $0.04$ | $0.3$ | $0.2$ | $0.1$ | $0.4$ | $0.06$ | $0.05$ | $0.11$ | $0.31$ | $0.69$ | $0.3$ | $0.05$ | $0.03$ |

${J}_{3}$ | $0.85$ | $0.1$ | $0.04$ | $0.02$ | $0.06$ | $0.02$ | $0.08$ | $0.01$ | $0.05$ | $0.03$ | $0.07$ | $0.02$ | $0.65$ | $0.2$ | $0.08$ | $0.01$ |

${J}_{4}$ | $0.7$ | $0.2$ | $0.03$ | $0.04$ | $0.2$ | $0.5$ | $0.62$ | $0.23$ | $0.07$ | $0.23$ | $0.08$ | $0.45$ | $0.15$ | $0.5$ | $0.13$ | $0.4$ |

$IVIFDWAOperator$ | $IVIFDWGOperator$ | |
---|---|---|

${J}_{1}$ | $\left(\left[0.1212,0.3768\right],\left[0.2500,0.3969\right]\right)$ | $\left(\left[0.1327,0.2713\right],\left[0.3558,0.2342\right]\right)$ |

${J}_{2}$ | $\left(\left[0.5490,0.2538\right],\left[0.0926,0.1511\right]\right)$ | $\left(\left[0.1644,0.1349\right],\left[0.0701,0.1818\right]\right)$ |

${J}_{3}$ | $\left(\left[0.6550,0.1178\right],\left[0.0914,0.0282\right]\right)$ | $\left(\left[0.1205,0.0575\right],\left[0.0692,0.0150\right]\right)$ |

${J}_{4}$ | $\left(\left[0.3690,0.3901\right],\left[0.0911,0.1538\right]\right)$ | $\left(\left[0.1828,0.3443\right],\left[0.2033,0.3550\right]\right)$ |

Scores | $IVIFDWAOperator$ | $IVIFDWGOperator$ |
---|---|---|

${J}_{1}$ | $0.1591$ | $0.1466$ |

${J}_{2}$ | $0.3568$ | $0.1316$ |

${J}_{3}$ | $0.3548$ | $0.0844$ |

${J}_{4}$ | $0.3328$ | $0.1838$ |

$RankingAnalysis$ | |
---|---|

$IVIFDWAoperator$ | ${J}_{2}>{J}_{3}>{J}_{4}>{J}_{1}$ |

$IVIFDWGoperator$ | ${J}_{4}>{J}_{1}>{J}_{2}>{J}_{3}$ |

M | Ranking Order | Optimal Alternative |
---|---|---|

$\mathbf{1}$ | ${J}_{2}>{J}_{3}>{J}_{4}>{J}_{1}$ | ${J}_{2}$ |

$\mathbf{2}$ | ${J}_{4}>{J}_{3}>{J}_{2}>{J}_{1}$ | ${J}_{4}$ |

$\mathbf{3}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

$\mathbf{7}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

$\mathbf{10}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

$\mathbf{30}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

M | Ranking Order | Optimal Alternative |
---|---|---|

$\mathbf{1}$ | ${J}_{4}>{J}_{1}>{J}_{2}>{J}_{3}$ | ${J}_{4}$ |

$\mathbf{4}$ | ${J}_{4}>{J}_{1}>{J}_{2}>{J}_{3}$ | ${J}_{4}$ |

$\mathbf{10}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

$\mathbf{20}$ | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ | ${J}_{4}$ |

Operator | Results |
---|---|

IVIFDWA operator | ${J}_{4}>{J}_{3}>{J}_{2}>{J}_{1}$ |

IVIFDWG operator | ${J}_{4}>{J}_{2}>{J}_{3}>{J}_{1}$ |

IVIFWA operator [48] | ${J}_{2}>{J}_{3}>{J}_{4}>{J}_{1}$ |

IVIFWG operator [49] | ${J}_{2}>{J}_{4}>{J}_{3}>{J}_{1}$ |

IVIFHM operator [50] | ${J}_{2}>{J}_{4}>{J}_{3}>{J}_{1}$ |

IVIFWGBM operator [51] | ${J}_{2}>{J}_{4}>{J}_{3}>{J}_{1}$ |

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**MDPI and ACS Style**

Waqar, M.; Ullah, K.; Pamucar, D.; Jovanov, G.; Vranješ, Ð.
An Approach for the Analysis of Energy Resource Selection Based on Attributes by Using Dombi T-Norm Based Aggregation Operators. *Energies* **2022**, *15*, 3939.
https://doi.org/10.3390/en15113939

**AMA Style**

Waqar M, Ullah K, Pamucar D, Jovanov G, Vranješ Ð.
An Approach for the Analysis of Energy Resource Selection Based on Attributes by Using Dombi T-Norm Based Aggregation Operators. *Energies*. 2022; 15(11):3939.
https://doi.org/10.3390/en15113939

**Chicago/Turabian Style**

Waqar, Mujab, Kifayat Ullah, Dragan Pamucar, Goran Jovanov, and Ðordje Vranješ.
2022. "An Approach for the Analysis of Energy Resource Selection Based on Attributes by Using Dombi T-Norm Based Aggregation Operators" *Energies* 15, no. 11: 3939.
https://doi.org/10.3390/en15113939