# Hesitant Picture 2-Tuple Linguistic Aggregation Operators Based on Archimedean T-Norm and T-Conorm and Their Use in Decision-Making

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- (1)
- if SC(b) < SC(e), then b < e,
- (2)
- if SC(b) > SC(e), then b > e,
- (3)
- when SC(b) = SC(e), and
- (a)
- AC(b) = AC(e), then b = e,
- (b)
- AC(b) < AC(e), then b < e,
- (c)
- AC(b) > AC(e), then b > e.

## 3. Operational Laws of Hesitant Picture 2-Tuple Linguistic Sets

**Definition**

**6.**

- $\mathsf{\Psi}(1,z)=z$.
- $\mathsf{\Psi}(z,u)=\mathsf{\Psi}(u,z)$.
- $\mathsf{\Psi}(z,\mathsf{\Psi}(u,v\left)\right)=\mathsf{\Psi}\left(\mathsf{\Psi}\right(z,u),v)$.
- If $z<{z}^{\prime}$, $u\le v$, then $\mathsf{\Psi}(z,u)\le \mathsf{\Psi}({z}^{\prime},v)$.

**Definition**

**7.**

- $\mathsf{\Phi}(0,z)=z$.
- $\mathsf{\Phi}(z,u)=\mathsf{\Phi}(u,z)$.
- $\mathsf{\Phi}(z,\mathsf{\Phi}(u,v\left)\right)=\mathsf{\Phi}\left(\mathsf{\Phi}\right(z,u),v)$.
- If $z<{z}^{\prime}$, $u\le v$, then $\mathsf{\Phi}(z,u)\le \mathsf{\Phi}({z}^{\prime},v)$.

**Definition**

**8.**

**Definition**

**9.**

- Addition operation$$\begin{array}{ccc}\hfill b\oplus {b}^{{}^{\prime}}& =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\delta [d({\u25b5}_{b}^{-1})]+\delta [d({\u25b5}_{{b}^{\prime}}^{-1})])),{\delta}^{-1}(\delta ({\mu}_{b})+\delta ({\mu}_{{b}^{{}^{\prime}}})),{\vartheta}^{-1}(\vartheta \hfill \\ & & ({\eta}_{b})+\vartheta ({\eta}_{{b}^{{}^{\prime}}})),{\vartheta}^{-1}(\vartheta ({\nu}_{b})+\vartheta ({\nu}_{{b}^{{}^{\prime}}}))\rangle |\langle ({s}_{\theta (b)},{\alpha}_{b}),{\mu}_{b},{\eta}_{b},{\nu}_{b}\rangle \in b,\hfill \\ & & \langle ({s}_{\theta ({b}^{{}^{\prime}})},{\alpha}_{{b}^{{}^{\prime}}}),{\mu}_{{b}^{{}^{\prime}}},{\eta}_{{b}^{{}^{\prime}}},{\nu}_{{b}^{{}^{\prime}}}\rangle \in {b}^{{}^{\prime}}\};\hfill \end{array}$$
- Multiplication operation$$\begin{array}{ccc}\hfill b\otimes {b}^{{}^{\prime}}& =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\vartheta [d({\u25b5}_{b}^{-1})]+\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})])),{\vartheta}^{-1}(\vartheta ({\mu}_{b})+\vartheta ({\mu}_{{b}^{{}^{\prime}}})),{\delta}^{-1}\hfill \\ & & (\delta ({\eta}_{b})+\delta ({\eta}_{{b}^{{}^{\prime}}})),{\delta}^{-1}(\delta ({\nu}_{b})+\delta ({\nu}_{{b}^{{}^{\prime}}}))\rangle |\langle ({s}_{\theta (b)},{\alpha}_{b}),{\mu}_{b},{\eta}_{b},{\nu}_{b}\rangle \in b,\hfill \\ & & \langle ({s}_{\theta ({b}^{{}^{\prime}})},{\alpha}_{{b}^{{}^{\prime}}}),{\mu}_{{b}^{{}^{\prime}}},{\eta}_{{b}^{{}^{\prime}}},{\nu}_{{b}^{{}^{\prime}}}\rangle \in {b}^{{}^{\prime}}\};\hfill \end{array}$$
- Scalar-multiplication operation$$\begin{array}{c}\hfill \begin{array}{c}\kappa b=\{\langle \u25b5(T\xb7{\delta}^{-1}(\kappa \delta \left[d({\u25b5}_{b}^{-1})\right])),{\delta}^{-1}(\kappa \delta ({\mu}_{b})),{\vartheta}^{-1}(\kappa \vartheta ({\eta}_{b})),{\vartheta}^{-1}(\kappa \vartheta ({\nu}_{b}))\rangle |\hfill \\ \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\langle ({s}_{\theta (b)},{\alpha}_{b}),{\mu}_{b},{\eta}_{b},{\nu}_{b}\rangle \in b\},\kappa >0;\hfill \end{array}\end{array}$$
- Exponential operation$$\begin{array}{c}\hfill \begin{array}{c}{b}^{\kappa}=\{\langle \u25b5(T\xb7{\vartheta}^{-1}(\kappa \vartheta \left[d({\u25b5}_{b}^{-1})\right])),{\vartheta}^{-1}(\kappa \vartheta ({\mu}_{b})),{\delta}^{-1}(\kappa \delta ({\eta}_{b})),{\delta}^{-1}(\kappa \delta ({\nu}_{b}))\rangle |\hfill \\ \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\langle ({s}_{\theta (b)},{\alpha}_{b}),{\mu}_{b},{\eta}_{b},{\nu}_{b}\rangle \in b\},\kappa >0.\hfill \end{array}\end{array}$$

**Remark**

**1.**

- (1)
- $b\oplus {b}^{{}^{\prime}}=\left\{\langle \u25b5\left({\u25b5}_{b}^{-1}+{\u25b5}_{{b}^{\prime}}^{-1}-\frac{{\u25b5}_{b}^{-1}\xb7{\u25b5}_{{b}^{\prime}}^{-1}}{T}\right),{\mu}_{b}+{\mu}_{{b}^{{}^{\prime}}}-{\mu}_{b}{\mu}_{{b}^{{}^{\prime}}},{\eta}_{b}{\eta}_{{b}^{{}^{\prime}}}.{\nu}_{b}{\nu}_{{b}^{{}^{\prime}}}\rangle \right\}.$
- (2)
- $b\otimes {b}^{{}^{\prime}}=\left\{\langle \u25b5\left(\frac{{\u25b5}_{b}^{-1}\xb7{\u25b5}_{{b}^{\prime}}^{-1}}{T}\right),{\mu}_{b}{\mu}_{{b}^{{}^{\prime}}},{\eta}_{b}+{\eta}_{{b}^{{}^{\prime}}}-{\eta}_{b}{\eta}_{{b}^{{}^{\prime}}},{\nu}_{b}+{\nu}_{{b}^{{}^{\prime}}}-{\nu}_{b}{\nu}_{{b}^{{}^{\prime}}}\rangle \right\}.$
- (3)
- $\kappa b=\left\{\langle \u25b5(T\xb7(1-(1-\frac{{\u25b5}_{b}^{-1}}{T}){}^{\kappa})),1-{(1-{\mu}_{b})}^{\kappa},{\left({\eta}_{b}\right)}^{\kappa},{\left({\nu}_{b}\right)}^{\kappa}\rangle \right\},\kappa >0.$
- (4)
- ${b}^{\kappa}=\left\{\langle \u25b5({({\u25b5}_{b}^{-1})}^{\kappa}\xb7{T}^{1-\kappa}),{\mu}_{b}^{\kappa},1-{(1-{\eta}_{b})}^{\kappa},1-{(1-{\nu}_{b})}^{\kappa}\rangle \right\},\kappa >0.$

**Remark**

**2.**

- (1)
- $b\oplus {b}^{{}^{\prime}}=\left\{\langle \u25b5\left({T}^{2}\xb7\frac{{\u25b5}_{b}^{-1}+{\u25b5}_{{b}^{\prime}}^{-1}}{{T}^{2}+{\u25b5}_{b}^{-1}\xb7{\u25b5}_{{b}^{\prime}}^{-1}}\right),\frac{{\mu}_{b}+{\mu}_{{b}^{{}^{\prime}}}}{1+{\mu}_{b}{\mu}_{{b}^{{}^{\prime}}}},\frac{{\eta}_{b}{\eta}_{{b}^{{}^{\prime}}}}{1+(1-{\eta}_{b})(1-{\eta}_{{b}^{{}^{\prime}}})},\frac{{\nu}_{b}{\nu}_{{b}^{{}^{\prime}}}}{1+(1-{\nu}_{b})(1-{\nu}_{{b}^{{}^{\prime}}})}\rangle \right\}.$
- (2)
- $b\otimes {b}^{{}^{\prime}}=\left\{\langle \u25b5\left(T\xb7\frac{{\u25b5}_{b}^{-1}\xb7{\u25b5}_{{b}^{\prime}}^{-1}}{{T}^{2}+(T-{\u25b5}_{b}^{-1})(T-{\u25b5}_{{b}^{\prime}}^{-1})}\right),\frac{{\mu}_{b}{\mu}_{{b}^{{}^{\prime}}}}{1+(1-{\mu}_{b})(1-{\mu}_{{b}^{{}^{\prime}}})},\frac{{\eta}_{b}+{\eta}_{{b}^{{}^{\prime}}}}{1+{\eta}_{b}{\eta}_{{b}^{{}^{\prime}}}},\frac{{\nu}_{b}+{\nu}_{{b}^{{}^{\prime}}}}{1+{\nu}_{b}{\nu}_{{b}^{{}^{\prime}}}}\rangle \right\}.$
- (3)
- $\kappa b=\left\{\langle \u25b5(T\xb7\frac{{(T+{\u25b5}_{b}^{-1})}^{\kappa}-{(T-{\u25b5}_{b}^{-1})}^{\kappa}}{{(T+{\u25b5}_{b}^{-1})}^{\kappa}+{(T-{\u25b5}_{b}^{-1})}^{\kappa}}),\frac{{(1+{\mu}_{b})}^{\kappa}-{(1-{\mu}_{b})}^{\kappa}}{{(1+{\mu}_{b})}^{\kappa}+{(1-{\mu}_{b})}^{\kappa}},\frac{2{\eta}_{b}^{\kappa}}{{(2-{\eta}_{b})}^{\kappa}+{\eta}_{b}^{\kappa}},\frac{2{\nu}_{b}^{\kappa}}{{(2-{\nu}_{b})}^{\kappa}+{\nu}_{b}^{\kappa}}\rangle \right\},\kappa >0.$
- (4)
- ${b}^{\kappa}=\left\{\langle \u25b5(T\xb7\frac{2{\left({\u25b5}_{b}^{-1}\right)}^{\kappa}}{{(2T-{\u25b5}_{b}^{-1})}^{\kappa}+{\left({\u25b5}_{b}^{-1}\right)}^{\kappa}}),\frac{2{\mu}_{b}^{\kappa}}{{(2-{\mu}_{b})}^{\kappa}+{\mu}_{b}^{\kappa}},\frac{{(1+{\eta}_{b})}^{\kappa}-{(1-{\eta}_{b})}^{\kappa}}{{(1+{\eta}_{b})}^{\kappa}+{(1-{\eta}_{b})}^{\kappa}},\frac{{(1+{\nu}_{b})}^{\kappa}-{(1-{\nu}_{b})}^{\kappa}}{{(1+{\nu}_{b})}^{\kappa}+{(1-{\nu}_{b})}^{\kappa}}\rangle \right\},\kappa >0.$

**Theorem**

**1.**

- (1)
- $b\oplus {b}^{\prime}={b}^{\prime}\oplus b,$
- (2)
- $b\otimes {b}^{\prime}={b}^{\prime}\otimes b,$
- (3)
- $\kappa (b\oplus {b}^{\prime})=\left(\kappa b\right)\oplus \left(\kappa {b}^{\prime}\right),$
- (4)
- ${(b\otimes {b}^{\prime})}^{\kappa}={b}^{\kappa}\otimes {\left({b}^{\prime}\right)}^{\kappa},$
- (5)
- $({\kappa}_{1}+{\kappa}_{2})b=\left({\kappa}_{1}b\right)\oplus \left({\kappa}_{2}{b}^{\prime}\right),$
- (6)
- ${b}^{{\kappa}_{1}}\otimes {b}^{{\kappa}_{2}}={b}^{{\kappa}_{1}+{\kappa}_{2}}.$

**Proof.**

- (1)
- $$\begin{array}{ccc}\hfill b\oplus {b}^{\prime}& =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\delta [d({\u25b5}_{b}^{-1})]+\delta [d({\u25b5}_{{b}^{\prime}}^{-1})])),{\delta}^{-1}(\delta ({\mu}_{b})+\delta ({\mu}_{{b}^{\prime}})),\hfill \\ & & {\vartheta}^{-1}(\vartheta ({\eta}_{b})+\vartheta ({\eta}_{{b}^{\prime}})),{\vartheta}^{-1}(\vartheta ({\nu}_{b})+\vartheta ({\nu}_{{b}^{\prime}}))\rangle \},\hfill \\ & =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\delta [d({\u25b5}_{{b}^{\prime}}^{-1})]+\delta [d({\u25b5}_{b}^{-1})])),{\delta}^{-1}(\delta ({\mu}_{{b}^{\prime}})+\delta ({\mu}_{b})),\hfill \\ & & {\vartheta}^{-1}(\vartheta ({\eta}_{{b}^{\prime}})+\vartheta ({\eta}_{b})),{\vartheta}^{-1}(\vartheta ({\nu}_{{b}^{\prime}})+\vartheta ({\nu}_{b}))\rangle \},\hfill \\ & =& {b}^{\prime}\oplus b.\hfill \end{array}$$
- (2)
- $$\begin{array}{ccc}\hfill b\otimes {b}^{\prime}& =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\vartheta [d({\u25b5}_{b}^{-1})]+\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})])),{\vartheta}^{-1}(\vartheta ({\mu}_{b})+\vartheta ({\mu}_{{b}^{\prime}})),\hfill \\ & & {\delta}^{-1}(\delta ({\eta}_{b})+\delta ({\eta}_{{b}^{\prime}})),{\delta}^{-1}(\delta ({\nu}_{b})+\delta ({\nu}_{{b}^{\prime}}))\rangle \},\hfill \\ & =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})]+\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}(\vartheta ({\mu}_{{b}^{\prime}})+\vartheta ({\mu}_{b})),\hfill \\ & & {\delta}^{-1}(\delta ({\eta}_{{b}^{\prime}})+\delta ({\eta}_{b})),{\delta}^{-1}(\delta ({\nu}_{{b}^{\prime}})+\delta ({\nu}_{b}))\rangle \},\hfill \\ & =& {b}^{\prime}\otimes b.\hfill \end{array}$$
- (3)
- $$\begin{array}{cc}& \kappa (b\oplus {b}^{\prime})\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\kappa \delta [d(T\xb7{\delta}^{-1}(\delta (d({\u25b5}_{b}^{-1}))+\delta (d({\u25b5}_{{b}^{\prime}}^{-1}))))])),{\delta}^{-1}(\kappa \delta ({\delta}^{-1}(\delta ({\mu}_{b})\hfill \\ & +\delta ({\mu}_{{b}^{\prime}})))),{\vartheta}^{-1}(\kappa \vartheta ({\vartheta}^{-1}(\vartheta ({\eta}_{b})+\vartheta ({\eta}_{{b}^{\prime}})))),{\vartheta}^{-1}(\kappa \vartheta ({\vartheta}^{-1}(\vartheta ({\nu}_{b})+\vartheta ({\nu}_{{b}^{\prime}}))))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\kappa (\delta [d({\u25b5}_{b}^{-1})]+\delta [d({\u25b5}_{{b}^{\prime}}^{-1})]))),{\delta}^{-1}(\kappa (\delta ({\mu}_{b})+\delta ({\mu}_{{b}^{\prime}}))),{\vartheta}^{-1}(\kappa \hfill \\ & (\vartheta ({\eta}_{b})+\vartheta ({\eta}_{{b}^{\prime}}))),{\vartheta}^{-1}(\kappa (\vartheta ({\nu}_{b})+\vartheta ({\nu}_{{b}^{\prime}})))\rangle \}.\hfill \end{array}$$$$\begin{array}{cc}& (\kappa b)\oplus (\kappa {b}^{\prime})\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\delta [d(T\xb7{\delta}^{-1}(\kappa \delta [d({\u25b5}_{b}^{-1})]))]+\delta [d(T\xb7{\delta}^{-1}(\kappa \delta [d({\u25b5}_{{b}^{\prime}}^{-1})]))])),{\delta}^{-1}(\delta \hfill \\ & ({\delta}^{-1}(\kappa \delta ({\mu}_{b})))+\delta ({\delta}^{-1}(\kappa \delta ({\mu}_{{b}^{\prime}})))),{\vartheta}^{-1}(\vartheta ({\vartheta}^{-1}(\kappa \vartheta ({\eta}_{b})))+\vartheta ({\vartheta}^{-1}(\kappa \vartheta ({\eta}_{{b}^{\prime}})))),\hfill \\ & {\vartheta}^{-1}(\vartheta ({\vartheta}^{-1}(\kappa \vartheta ({\nu}_{b})))+\vartheta ({\vartheta}^{-1}(\kappa \vartheta ({\nu}_{{b}^{{}^{\prime}}}))))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}(\kappa (\delta [d({\u25b5}_{b}^{-1})]+\delta [d({\u25b5}_{{b}^{\prime}}^{-1})]))),{\delta}^{-1}(\kappa (\delta ({\mu}_{b})+\delta ({\mu}_{{b}^{\prime}}))),{\vartheta}^{-1}(\kappa (\vartheta \hfill \\ & ({\eta}_{b})+\vartheta ({\eta}_{{b}^{\prime}}))),{\vartheta}^{-1}(\kappa (\vartheta ({\nu}_{b})+\vartheta ({\nu}_{{b}^{\prime}})))\rangle \}.\hfill \end{array}$$

- (4)
- $$\begin{array}{cc}& {(b\otimes {b}^{\prime})}^{\kappa}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\kappa \vartheta [d(T\xb7{\vartheta}^{-1}(\vartheta [d({\u25b5}_{b}^{-1})]+\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})]))])),{\vartheta}^{-1}(\kappa \vartheta ({\vartheta}^{-1}(\vartheta \hfill \\ & ({\mu}_{b})+\vartheta ({\mu}_{{b}^{\prime}})))),{\delta}^{-1}(\kappa \delta ({\delta}^{-1}(\delta ({\eta}_{b})+\delta ({\eta}_{{b}^{\prime}})))),{\delta}^{-1}(\kappa \delta ({\delta}^{-1}(\delta ({\nu}_{b})+\delta ({\nu}_{{b}^{\prime}}))))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\kappa (\vartheta [d({\u25b5}_{b}^{-1})]+\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})]))),{\vartheta}^{-1}(\kappa (\vartheta ({\mu}_{b})+\vartheta ({\mu}_{{b}^{\prime}}))),{\delta}^{-1}\hfill \\ & (\kappa (\delta ({\eta}_{b})+\delta ({\eta}_{{b}^{\prime}}))),{\delta}^{-1}(\kappa (\delta ({\nu}_{b})+\delta ({\nu}_{{b}^{\prime}})))\rangle \}.\hfill \\ \\ & {b}^{\kappa}\otimes {({b}^{\prime})}^{\kappa}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\kappa \vartheta [d({\u25b5}_{b}^{-1})]+\kappa \vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})])),{\vartheta}^{-1}(\kappa \vartheta ({\mu}_{b})+\kappa \vartheta ({\mu}_{{b}^{\prime}})),{\delta}^{-1}\hfill \\ & (\kappa \delta ({\eta}_{b})+\kappa \delta ({\eta}_{{b}^{\prime}})),{\delta}^{-1}(\kappa \delta ({\nu}_{b})+\kappa \delta ({\nu}_{{b}^{\prime}}))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\kappa (\vartheta [d({\u25b5}_{b}^{-1})]+\vartheta [d({\u25b5}_{{b}^{\prime}}^{-1})]))),{\vartheta}^{-1}(\kappa (\vartheta ({\mu}_{b})+\vartheta ({\mu}_{{b}^{\prime}}))),{\delta}^{-1}\hfill \\ & (\kappa (\delta ({\eta}_{b})+\delta ({\eta}_{{b}^{\prime}}))),{\delta}^{-1}(\kappa (\delta ({\nu}_{b})+\delta ({\nu}_{{b}^{\prime}})))\rangle \},\hfill \\ =& {(b\otimes {b}^{\prime})}^{\kappa}.\hfill \end{array}$$
- (5)
- Notice that$$\begin{array}{c}\hfill \begin{array}{c}({\kappa}_{1}+{\kappa}_{2})b=\{\langle \u25b5(T\xb7{\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta [d({\u25b5}_{b}^{-1})])),{\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta ({\mu}_{b})),{\vartheta}^{-1}\hfill \\ (({\kappa}_{1}+{\kappa}_{2})\vartheta ({\eta}_{b})),{\vartheta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\vartheta ({\nu}_{b}))\rangle \}.\hfill \end{array}\end{array}$$$$\begin{array}{cc}& ({\kappa}_{1}b)\oplus ({\kappa}_{2}b)\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}({\kappa}_{1}\delta [d({\u25b5}_{b}^{-1})]+{\kappa}_{2}\delta [d({\u25b5}_{b}^{-1})])),{\delta}^{-1}({\kappa}_{1}\delta ({\mu}_{b})+{\kappa}_{2}\delta ({\mu}_{b})),{\vartheta}^{-1}({\kappa}_{1}\hfill \\ & \vartheta ({\eta}_{b})+{\kappa}_{2}\vartheta ({\eta}_{b})),{\vartheta}^{-1}({\kappa}_{1}\vartheta ({\nu}_{b})+{\kappa}_{2}\vartheta ({\nu}_{b}))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta [d({\u25b5}_{b}^{-1})])),{\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta ({\mu}_{b})),{\vartheta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\vartheta ({\eta}_{b})),\hfill \\ & {\vartheta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\vartheta ({\nu}_{b}))\rangle \}.\hfill \end{array}$$

- (6)
- Based on$$\begin{array}{ccc}\hfill {b}^{{\kappa}_{1}}& =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}({\kappa}_{1}\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}({\kappa}_{1}\vartheta ({\mu}_{b})),{\delta}^{-1}({\kappa}_{1}\delta ({\eta}_{b})),{\delta}^{-1}({\kappa}_{1}\hfill \\ & & \delta ({\nu}_{b}))\rangle \}\phantom{\rule{3.33333pt}{0ex}}and\hfill \\ \hfill {b}^{{\kappa}_{2}}& =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}({\kappa}_{2}\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}({\kappa}_{2}\vartheta ({\mu}_{b})),{\delta}^{-1}({\kappa}_{2}\delta ({\eta}_{b})),{\delta}^{-1}({\kappa}_{2}\hfill \\ & & \delta ({\nu}_{b}))\rangle \},\hfill \end{array}$$$$\begin{array}{cc}& {b}^{{\kappa}_{1}}\otimes {b}^{{\kappa}_{2}}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(\vartheta [d(T\xb7{\vartheta}^{-1}({\kappa}_{1}\vartheta [d({\u25b5}_{b}^{-1})]))]+\vartheta [d(T\xb7{\vartheta}^{-1}({\kappa}_{2}\vartheta [d({\u25b5}_{b}^{-1})]))])),{\vartheta}^{-1}\hfill \\ & (\vartheta ({\vartheta}^{-1}({\kappa}_{1}\vartheta ({\mu}_{b})))+\vartheta ({\vartheta}^{-1}({\kappa}_{2}\vartheta ({\mu}_{b})))),{\delta}^{-1}(\delta ({\delta}^{-1}({\kappa}_{1}\delta ({\eta}_{b})))+\delta ({\delta}^{-1}({\kappa}_{2}\delta ({\eta}_{b})))),\hfill \\ & {\delta}^{-1}(\delta ({\delta}^{-1}({\kappa}_{1}\delta ({\nu}_{b})))+\delta ({\delta}^{-1}({\kappa}_{2}\delta ({\nu}_{b}))))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}({\kappa}_{1}\vartheta [d({\u25b5}_{b}^{-1})]+{\kappa}_{2}\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}({\kappa}_{1}\vartheta ({\mu}_{b})+{\kappa}_{2}\vartheta ({\mu}_{b})),{\delta}^{-1}({\kappa}_{1}\hfill \\ & \delta ({\eta}_{b})+{\kappa}_{2}\delta ({\eta}_{b})),{\delta}^{-1}({\kappa}_{1}\delta ({\nu}_{b})+{\kappa}_{2}\delta ({\nu}_{b}))\rangle \}.\hfill \\ \\ & {b}^{{\kappa}_{1}+{\kappa}_{2}}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\vartheta ({\mu}_{b})),{\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta ({\eta}_{b})),\hfill \\ & {\delta}^{-1}(({\kappa}_{1}+{\kappa}_{2})\delta ({\nu}_{b}))\rangle \}\hfill \\ =& \{\langle \u25b5(T\xb7{\vartheta}^{-1}({\kappa}_{1}\vartheta [d({\u25b5}_{b}^{-1})]+{\kappa}_{2}\vartheta [d({\u25b5}_{b}^{-1})])),{\vartheta}^{-1}({\kappa}_{1}\vartheta ({\mu}_{b})+{\kappa}_{2}\vartheta ({\mu}_{b})),{\delta}^{-1}({\kappa}_{1}\hfill \\ & \delta ({\eta}_{b})+{\kappa}_{2}\delta ({\eta}_{b})),{\delta}^{-1}({\kappa}_{1}\delta ({\nu}_{b})+{\kappa}_{2}\delta ({\nu}_{b}))\rangle \}.\hfill \end{array}$$By using the above equations, we can obtain ${b}^{{\kappa}_{1}}\otimes {b}^{{\kappa}_{2}}={b}^{{\kappa}_{1}+{\kappa}_{2}}$.

## 4. Some Aggregation Operators of HP2TLSs Based on Archimedean T-norm and T-conorm

**Definition**

**10.**

**Theorem**

**2.**

**Theorem**

**3.**

- (1)
- (Idempotency): Let ${b}_{i}$ be some HP2TLSs, if all ${b}_{i}(i=1,2,\dots ,n)$ are identical, i.e., ${b}_{i}=b$, for all i, then$$\begin{array}{c}\hfill \mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})=b.\end{array}$$
**Proof.**Let ${b}_{i}=b$ and $\langle x,\mu ,\eta ,\nu \rangle \in b$, where $x=({s}_{\theta (b)},{\alpha}_{b})$, for all $(i=1,2,\dots ,n)$, then we have$$\begin{array}{c}\hfill \begin{array}{c}\mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})\hfill \\ \Rightarrow \mathit{\text{ATS-HP}}2TLWA(b,b,\dots ,b)\hfill \\ ={\displaystyle \bigcup _{b}}\{\u25b5(T\xb7{\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta [d({\u25b5}_{b}^{-1})])),{\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta (\mu )),{\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta (\eta )),\hfill \\ {\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta (\nu ))\}\hfill \\ ={\displaystyle \bigcup _{b}}\{\langle ({s}_{\theta (b)},{\alpha}_{b}),\mu ,\eta ,\nu \rangle \}\hfill \\ =b.\hfill \end{array}\end{array}$$ - (2)
- (Boundedness): Let ${b}_{i}$ $(i=1,2,\dots ,n)$ be some HP2TLSs, if ${b}^{-}=min\{{b}_{i}\}$ and ${b}^{+}=max\{{b}_{i}\}$, then$$\begin{array}{c}\hfill {b}^{-}\le \mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})\le {b}^{+},\end{array}$$$$\begin{array}{ccc}\hfill {b}^{-}& =& min\{{b}_{i}\}=\{\langle min({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),min({\mu}_{i}),max({\eta}_{i}),max({\nu}_{i})\rangle |\langle ({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),\hfill \\ & & {\mu}_{i},{\eta}_{i},{\nu}_{i}\rangle \in b\},\hfill \\ \hfill {b}^{+}& =& max\{{b}_{i}\}=\{\langle max({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),max({\mu}_{i}),min({\eta}_{i}),min({\nu}_{i})\rangle |\langle ({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),\hfill \\ & & {\mu}_{i},{\eta}_{i},{\nu}_{i}\rangle \in b\}.\hfill \end{array}$$
**Proof.**Since $d(u)=\frac{u}{T}$, $\delta (u)$ and ${\delta}^{-1}(u)$ are two increasing functions, $\vartheta (u)$ and ${\vartheta}^{-1}(u)$ are two decreasing functions, then we have:$$\begin{array}{c}\hfill \begin{array}{c}d({\u25b5}_{{b}^{-}}^{-1})\le d({\u25b5}_{{b}_{i}}^{-1}).\hfill \\ \Rightarrow \delta [d({\u25b5}_{{b}^{-}}^{-1})]\le \delta [d({\u25b5}_{{b}_{i}}^{-1})]\hfill \\ \Rightarrow {\sum}_{i=1}^{n}{w}_{i}\delta [d({\u25b5}_{{b}^{-}}^{-1})]\le {\sum}_{i=1}^{n}{w}_{i}\delta [d({\u25b5}_{{b}_{i}}^{-1})]\hfill \\ \Rightarrow \u25b5(T\xb7{\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta [d({\u25b5}_{{b}^{-}}^{-1})]))\le \u25b5(T\xb7{\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta [d({\u25b5}_{{b}_{i}}^{-1})])).\hfill \end{array}\end{array}$$In the same way, we can obtain ${\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta (min({\mu}_{i})))\le {\delta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\delta ({\mu}_{i}))$, ${\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta (min({\eta}_{i})))\ge {\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta ({\eta}_{i})),$ ${\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta (min({\nu}_{i})))\ge {\vartheta}^{-1}({\sum}_{i=1}^{n}{w}_{i}\vartheta ({\nu}_{i}))$. According to Definition 3, 4, and 5, we can verify that ${b}^{-}\le \mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})$. Similarity, $\mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})\le {b}^{+}$ can be proved. Thus, the following inequality holds:$$\begin{array}{c}\hfill {b}^{-}\le \mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})\le {b}^{+}.\end{array}$$ - (3)
- (Monotonicity): Let ${b}_{i}$ and ${b}_{i}^{{}^{\prime}}$ $(i=1,2,\dots ,n)$ be two HP2TLSs, and $\langle ({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),{\mu}_{i},{\eta}_{i},{\nu}_{i}\rangle \in {b}_{i}$, $\langle ({s}_{\theta ({b}_{i}^{{}^{\prime}})},{\alpha}_{{b}_{i}^{{}^{\prime}}}),{\mu}_{i}^{{}^{\prime}},{\eta}_{i}^{{}^{\prime}},{\nu}_{i}^{{}^{\prime}}\rangle \in {b}_{i}^{{}^{\prime}}$. If any ${b}_{i}$ and ${b}_{i}^{{}^{\prime}}$ satisfy $\langle ({s}_{\theta ({b}_{i})},{\alpha}_{{b}_{i}}),{\mu}_{i},{\eta}_{i},{\nu}_{i}\rangle \le \langle ({s}_{\theta ({b}_{i}^{{}^{\prime}})},{\alpha}_{{b}_{i}^{{}^{\prime}}}),{\mu}_{i}^{{}^{\prime}},{\eta}_{i}^{{}^{\prime}},{\nu}_{i}^{{}^{\prime}}\rangle $, then$$\begin{array}{c}\hfill \mathit{\text{ATS-HP}}2TLWA({b}_{1},{b}_{2},\dots ,{b}_{n})\le \mathit{\text{ATS-HP}}2TLWA({b}_{1}^{{}^{\prime}},{b}_{2}^{{}^{\prime}},\dots ,{b}_{n}^{{}^{\prime}}).\end{array}$$

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Definition**

**11.**

**Theorem**

**4.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Remark**

**10.**

## 5. A Novel Method of Solving MADM Problems

**Step 1**: Obtain the judgment information matrix:

**Step 2**: Fuse all values ${r}_{ij}$$(j=1,2\dots ,n)$ of each alternative ${D}_{i}$ by using the ATS-HP2TLWA operator or the ATS-HP2TLWG operator. Then, we can obtain the collective results:

**Step 3**: Rank ${r}_{i}$$(i=1,\dots ,5)$ according to descending order with respect to the values derived from Definitions (3)–(5), and the ideal alternative can be derived.

## 6. Application of the Designed Method and Discussion

**Step 1**: Obtain the judgment information matrix, which is delineated in Table 1.

**Step 2**: Employ the ATS-HP2TLWA operator or the ATS-HP2TLWG operator to aggregate all attributes of ${D}_{i}$$(i=1,2,3,4,5)$. Then, collective results of the two operators are given in Table 2 and Table 3, separately. Through Table 2, we can find that EHP2TLWA operator is a special form of HHP2TLWA operator. When $\gamma =2$, HHP2TLWA operator degenerates into an EHP2TLWA operator.

**Step 3**: Obtain score values and sort the alternatives in accordance with Definitions (3)–(5), and the results of two operators are presented in Table 4 and Table 5, separately.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | |

${D}_{1}$ | $\left\{\langle ({s}_{4},0),0.53,0.33,0.09\rangle \right\}$ | $\{\langle ({s}_{2},0),0.89,0.08,0.03\rangle ,$ |

$\langle ({s}_{3},0),0.75,0.05,0.06\rangle \}$ | ||

${D}_{2}$ | $\left\{\langle ({s}_{1},0),0.73,0.12,0.08\rangle \right\}$ | $\left\{\langle ({s}_{4},0),0.13,0.64,0.21\rangle \right\}$ |

${D}_{3}$ | $\left\{\langle ({s}_{5},0),0.91,0.03,0.02\rangle \right\}$ | $\left\{\langle ({s}_{1},0),0.07,0.09,0.05\rangle \right\}$ |

${D}_{4}$ | $\{\langle ({s}_{5},0),0.85,0.09,0.05\rangle ,$ | $\left\{\langle ({s}_{5},0),0.74,0.16,0.10\rangle \right\}$ |

$\langle ({s}_{4},0),0.54,0.35,0.11\rangle $} | ||

${D}_{5}$ | $\{\langle ({s}_{5},0),0.90,0.05,0.02\rangle ,$ | $\left\{\langle ({s}_{1},0),0.68,0.08,0.21\rangle \right\}$ |

$\langle ({s}_{6},0),0.91,0.03,0.02\rangle \}$ | ||

${\mathit{C}}_{\mathbf{3}}$ | ${\mathit{C}}_{\mathbf{4}}$ | |

${D}_{1}$ | $\left\{\langle ({s}_{1},0),0.73,0.12,0.08\rangle \right\}$ | $\left\{\langle ({s}_{4},0),0.13,0.64,0.21\rangle \right\}$ |

${D}_{2}$ | $\{\langle ({s}_{3},0),0.03,0.82,0.13\rangle ,$ | $\left\{\langle ({s}_{4},0),0.73,0.15,0.08\rangle \right\}$ |

$\langle ({s}_{4},0),0.74,0.15,0.11\rangle \}$ | ||

${D}_{3}$ | $\left\{\langle ({s}_{4},0),0.04,0.85,0.1\rangle \right\}$ | $\{\langle ({s}_{2},0),0.68,0.26,0.06\rangle ,$ |

$\langle ({s}_{3},0),0.75,0.08,0.06\rangle \}$ | ||

${D}_{4}$ | $\left\{\langle ({s}_{3},0),0.02,0.89,0.05\rangle \right\}$ | $\left\{\langle ({s}_{1},0),0.08,0.84,0.06\rangle \right\}$ |

${D}_{5}$ | $\{\langle ({s}_{3},0),0.05,0.87,0.06\rangle ,$ | $\left\{\langle ({s}_{4},0),0.13,0.75,0.09\rangle \right\}$ |

$\langle ({s}_{4},0),0.68,0.26,0.06\rangle \}$ |

HP2TLWA Operator | EHP2TLWA Operator | |

${D}_{1}$ | $\{\langle ({s}_{3},0.1783),0.5597,0.2756,0.1092\rangle ,$ | $\{\langle ({s}_{3},0.0832),0.5307,0.2913,0.1104\rangle ,$ |

$\langle ({s}_{3},0.2583),0.5220,0.2629,0.1171\rangle \}$ | $\langle ({s}_{3},0.1717),0.4977,0.2794,0.1180\rangle \}$ | |

${D}_{2}$ | $\{\langle ({s}_{3},0.2870),0.5545,0.2761,0.1019\rangle ,$ | $\{\langle ({s}_{3},0.2257),0.5222,0.2987,0.1023\rangle ,$ |

$\langle ({s}_{4},-0.4022),0.6999,0.1659,0.0969\rangle \}$ | $\langle ({s}_{4},-0.4610),0.6939,0.1688,0.0972\rangle \}$ | |

${D}_{3}$ | $\{\langle ({s}_{3},0.4821),0.6159,0.2166,0.0551\rangle ,$ | $\{\langle ({s}_{3},0.3757),0.5758,0.2398,0.0554\rangle ,$ |

$\langle ({s}_{4},-0.2442),0.6520,0.1352,0.0551\rangle \}$ | $\langle ({s}_{4},-0.3069),0.6130,0.1511,0.0554\rangle \}$ | |

${D}_{4}$ | $\{\langle ({s}_{3},-0.0404),0.2807,0.6078,0.0675\rangle ,$ | $\{\langle ({s}_{3},-0.1667),0.2486,0.6418,0.0676\rangle ,$ |

$\langle ({s}_{3},0.3532),0.4251,0.4632,0.0576\rangle \}$ | $\langle ({s}_{3},0.1870),0.3665,0.5182,0.0577\rangle \}$ | |

${D}_{5}$ | $\{\langle ({s}_{4},-0.1550),0.4756,0.3647,0.0642\rangle ,$ | $\{\langle ({s}_{4},-0.2132),0.4175,0.4172,0.0647\rangle ,$ |

$\langle ({s}_{4},0.0918),0.6217,0.2539,0.0642\rangle ,$ | $\langle ({s}_{4},0.0501),0.5905,0.2776,0.0647\rangle ,$ | |

$\langle ({s}_{6},0),0.4866,0.3293,0.0642\rangle ,$ | $\langle ({s}_{6},0),0.4266,0.3839,0.0647\rangle ,$ | |

$\langle ({s}_{6},0),0.6296,0.2292,0.0642\rangle \}$ | $\langle ({s}_{6},0),0.5976,0.2536,0.0647\rangle \}$ | |

HHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | FHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | |

${D}_{1}$ | $\{\langle ({s}_{3},0.0832),0.5307,0.2913,0.1104\rangle ,$ | $\{\langle ({s}_{3},0.1346),0.5458,0.2841,0.1100\rangle ,$ |

$\langle ({s}_{3},0.1717),0.4977,0.2794,0.1180\rangle \}$ | $\langle ({s}_{3},0.2187),0.5108,0.2720,0.1177\rangle \}$ | |

${D}_{2}$ | $\{\langle ({s}_{3},0.2257),0.5222,0.2987,0.1023\rangle ,$ | $\{\langle ({s}_{3},0.2587),0.5402,0.2874,0.1021\rangle ,$ |

$\langle ({s}_{4},-0.4610),0.6939,0.1688,0.0972\rangle \}$ | $\langle ({s}_{4},-0.4292),0.6971,0.1674,0.0971\rangle \}$ | |

${D}_{3}$ | $\{\langle ({s}_{3},0.3757),0.5758,0.2398,0.0554\rangle ,$ | $\{\langle ({s}_{3},0.4286),0.5971,0.2287,0.0553\rangle ,$ |

$\langle ({s}_{4},-0.3069),0.6130,0.1511,0.0554\rangle \}$ | $\langle ({s}_{4},-0.2763),0.6338,0.1434,0.0553\rangle \}$ | |

${D}_{4}$ | $\{\langle ({s}_{3},-0.1667),0.2486,0.6418,0.0676\rangle ,$ | $\{\langle ({s}_{3},-0.1003),0.2669,0.6237,0.0676\rangle ,$ |

$\langle ({s}_{3},0.1870),0.3665,0.5182,0.0577\rangle \}$ | $\langle ({s}_{3},0.2717),0.3982,0.4902,0.0577\rangle \}$ | |

${D}_{5}$ | $\{\langle ({s}_{4},-0.2132),0.4175,0.4172,0.0647\rangle ,$ | $\{\langle ({s}_{4},-0.1847),0.4478,0.3916,0.0645\rangle ,$ |

$\langle ({s}_{4},0.0501),0.5905,0.2776,0.0647\rangle ,$ | $\langle ({s}_{4},0.0710),0.6065,0.2665,0.0645\rangle ,$ | |

$\langle ({s}_{6},0),0.4266,0.3839,0.0647\rangle ,$ | $\langle ({s}_{6},0),0.4578,0.3574,0.0645\rangle ,$ | |

$\langle ({s}_{6},0),0.5976,0.2536,0.0647\rangle \}$ | $\langle ({s}_{6},0),0.6139,0.2423,0.0645\rangle \}$ |

HP2TLWG Operator | EHP2TLWG Operator | |

${D}_{1}$ | $\{\langle ({s}_{2},0.4623),0.3502,0.4146,0.1317\rangle ,$ | $\{\langle ({s}_{3},-0.4223),0.3773,0.3936,0.1296\rangle ,$ |

$\langle ({s}_{3},-0.4358),0.3443,0.4127,0.1344\rangle \}$ | $\langle ({s}_{3},-0.3174),0.3685,0.3910,0.1326\rangle \}$ | |

${D}_{2}$ | $\{\langle ({s}_{3},-0.2192),0.2358,0.5070,0.1090\rangle ,$ | $\{\langle ({s}_{3},-0.1267),0.2708,0.4679,0.1082\rangle ,$ |

$\langle ({s}_{3},0.0314),0.6168,0.2145,0.1029\rangle \}$ | $\langle ({s}_{3},0.1475),0.6379,0.2029,0.1022\rangle \}$ | |

${D}_{3}$ | $\{\langle ({s}_{3},-0.2405),0.2454,0.5059,0.0634\rangle ,$ | $\{\langle ({s}_{3},-0.1334),0.2835,0.4608,0.0631\rangle ,$ |

$\langle ({s}_{3},0.2453),0.2552,0.4610,0.0634\rangle \}$ | $\langle ({s}_{3},0.3358),0.2987,0.4002,0.0631\rangle \}$ | |

${D}_{4}$ | $\{\langle ({s}_{2},0.2533),0.1058,0.7611,0.0591\rangle ,$ | $\{\langle ({s}_{2},0.3935),0.1187,0.7394,0.0590\rangle ,$ |

$\langle ({s}_{2},0.1550),0.0966,0.7766,0.0713\rangle \}$ | $\langle ({s}_{2},0.2597),0.1041,0.7634,0.0710\rangle \}$ | |

${D}_{5}$ | $\{\langle ({s}_{3},0.3401),0.1696,0.6943,0.0805\rangle ,$ | $\{\langle ({s}_{3},0.4344),0.1869,0.6680,0.0793\rangle ,$ |

$\langle ({s}_{3},0.4641),0.1700,0.6930,0.0805\rangle ,$ | $\langle ({s}_{4},-0.3983),0.1876,0.6658,0.0793\rangle ,$ | |

$\langle ({s}_{4},-0.3589),0.3710,0.4849,0.0805\rangle ,$ | $\langle ({s}_{4},-0.2599),0.4046,0.4519,0.0793\rangle ,$ | |

$\langle ({s}_{4},-0.2236),0.3719,0.4828,0.0805\rangle \}$ | $\langle ({s}_{4},-0.0845),0.4059,0.4487,0.0793\rangle \}$ | |

HHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | FHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | |

${D}_{1}$ | $\{\langle ({s}_{3},-0.4547),0.3773,0.3936,0.1296\rangle ,$ | $\{\langle ({s}_{3},-0.4779),0.3641,0.4053,0.1310\rangle ,$ |

$\langle ({s}_{3},-0.3549),0.3685,0.3910,0.1326\rangle \}$ | $\langle ({s}_{3},-0.3741),0.3569,0.4032,0.1338\rangle \}$ | |

${D}_{2}$ | $\{\langle ({s}_{3},-0.1564),0.2708,0.4679,0.1082\rangle ,$ | $\{\langle ({s}_{3},-0.1708),0.2545,0.4885,0.1087\rangle ,$ |

$\langle ({s}_{3},0.1055),0.6379,0.2029,0.1022\rangle \}$ | $\langle ({s}_{3},0.0918),0.6277,0.2093,0.1026\rangle \}$ | |

${D}_{3}$ | $\{\langle ({s}_{3},-0.1239),0.2835,0.4608,0.0631\rangle ,$ | $\{\langle ({s}_{3},-0.1886),0.2652,0.4843,0.0633\rangle ,$ |

$\langle ({s}_{3},0.3361),0.2987,0.4002,0.0631\rangle \}$ | $\langle ({s}_{3},0.2898),0.2775,0.4327,0.0633\rangle \}$ | |

${D}_{4}$ | $\{\langle ({s}_{2},0.3948),0.1187,0.7394,0.0590\rangle ,$ | $\{\langle ({s}_{2},0.3230),0.1126,0.7505,0.0591\rangle ,$ |

$\langle ({s}_{2},0.2445),0.1041,0.7634,0.0710\rangle \}$ | $\langle ({s}_{2},0.2090),0.1009,0.7697,0.0712\rangle \}$ | |

${D}_{5}$ | $\{\langle ({s}_{3},0.4303),0.1869,0.6680,0.0793\rangle ,$ | $\{\langle ({s}_{3},0.3866),0.1784,0.6820,0.0801\rangle ,$ |

$\langle ({s}_{4},0.2870),0.1876,0.6658,0.0793\rangle ,$ | $\langle ({s}_{4},-0.4731),0.1789,0.6803,0.0801\rangle ,$ | |

$\langle ({s}_{4},-0.2793),0.4046,0.4519,0.0793\rangle ,$ | $\langle ({s}_{4},-0.3092),0.3879,0.4698,0.0801\rangle ,$ | |

$\langle ({s}_{5},-0.4720),0.4059,0.4487,0.0793\rangle \}$ | $\langle ({s}_{4},-0.1588),0.3889,0.4672,0.0801\rangle \}$ |

HP2TLWA Operator | EHP2TLWA Operator | HHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | FHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | |
---|---|---|---|---|

${D}_{1}$ | 2.2969 | 2.1888 | 2.1888 | 2.2462 |

${D}_{2}$ | 2.6355 | 2.5578 | 2.5574 | 2.5999 |

${D}_{3}$ | 2.8581 | 2.7213 | 2.7213 | 2.7910 |

${D}_{4}$ | 2.0440 | 1.8793 | 1.8793 | 1.9658 |

${D}_{5}$ | 3.7158 | 3.5839 | 3.5839 | 3.6514 |

Ranking | ${r}_{5}>{r}_{3}>{r}_{2}>{r}_{1}>{r}_{4}$ | ${r}_{5}>{r}_{3}>{r}_{2}>{r}_{1}>{r}_{4}$ | ${r}_{5}>{r}_{3}>{r}_{2}>{r}_{1}>{r}_{4}$ | ${r}_{5}>{r}_{3}>{r}_{2}>{r}_{1}>{r}_{4}$ |

HP2TLWA Operator | EHP2TLWA Operator | HHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | FHP2TLWA Operator ($\mathit{\gamma}\mathbf{=}\mathbf{2}$) | |
---|---|---|---|---|

${D}_{1}$ | 1.5257 | 1.6329 | 1.6112 | 1.5805 |

${D}_{2}$ | 1.9307 | 2.0435 | 2.0188 | 1.9892 |

${D}_{3}$ | 1.7824 |