Intuitionistic principal value Z-linguistic hybrid geometric operator and their applications for multi-attribute group decision-making

Abstract

Problems often encountered in the modeling of uncertain linguistic multi-attribute group decision-making in targeted poverty alleviation are how to describe and aggregate uncertain and imprecise information more effectively. In this paper, in order to effectively describe the fuzziness and reliability of linguistic evaluation information, and emphasize the role of linguistic evaluation value, we firstly propose this concept of the intuitionistic principal value Z-linguistic set. Then, we propose intuitionistic principal value Z-linguistic hybrid geometric operator regarding the position and ordered weight to aggregate linguistic evaluation information more effectively. Finally, an example of targeted poverty alleviation is given to illustrate effectiveness and superiority based on the operator with different position weights.

Appendix

In this part, we mainly give the proof of some theorems in this paper.

1.1 The proof of Theorem 2

Proof

• (1) According to Definition 12, rule (1) and (2) are clearly correct.

• (2) For rule (3):

$\begin{array}{l} \because \lambda {(}{A_1} \oplus {A_2}{)}= \lambda \left\langle \begin{array}{l} [{s_{\theta ({x_1}) + \theta ({x_2})}},{s_{\tau ({x_1}) + \tau ({x_2})}}],(1 - (1 - \mu ({x_1})(1 - \mu ({x_2}))),\nu ({x_1})\nu ({x_2})); \frac{{{p^2}({x_1}) + {p^2}({x_2})}}{{p({x_1}) + p({x_2})}} \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{\lambda {(} \theta ({x_1}) + \theta ({x_2}))}},{s_{\lambda (\tau ({x_1}) + \tau ({x_2}))}}],(1 - {(1 - \mu ({x_1}))^\lambda }{(1 - \mu ({x_2}))^\lambda },{(\nu ({x_1}))^\lambda }{(\nu ({x_2}))^\lambda }); \frac{{{p^2}({x_1}) + {p^2}({x_2})}}{{p({x_1}) + p({x_2})}} \end{array} \right\rangle , \end{array}$ and

$\begin{array}{l} \lambda {(}{A_1}{)} \oplus \lambda {(}{A_2}{)}=\left\langle {[{s_{\lambda \theta ({x_1})}},{s_{\lambda \tau ({x_1})}}], (1 - {{(1 - \mu ({x_1}))}^\lambda },{{(\nu ({x_1}))}^\lambda });p({x_1})} \right\rangle \oplus \\ \left\langle {[{s_{\lambda \theta ({x_2})}},{s_{\lambda \tau ({x_2})}}], (1 - {{(1 - \mu ({x_2}))}^\lambda },{{(\nu ({x_2}))}^\lambda });p({x_2})} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\lambda {(}\theta ({x_1}) + \theta ({x_2}))}},{s_{\lambda (\tau ({x_1}) + \tau ({x_2}))}}],(1 - {(1 - \mu ({x_1}))^\lambda }{(1 - \mu ({x_2}))^\lambda },{(\nu ({x_1}))^\lambda }{(\nu ({x_2}))^\lambda }); \frac{{{p^2}({x_1}) + {p^2}({x_2})}}{{p({x_1}) + p({x_2})}} \end{array} \right\rangle , \end{array}$

$\therefore \lambda {(}{A_1} \oplus {A_2}{)}=\lambda {A_2} \oplus \lambda {A_1}$.

The proof of rule (3) is complete.

(3) For rule (4)

$\begin{array}{l} \because {\lambda _1}{A_1} \oplus {\lambda _2}{A_1}={\lambda _1}\left\langle \begin{array}{l} [{s_{\theta ({x_1})}},{s_{\tau ({x_1})}}],(\mu ({x_1}),\nu ({x_1})); p({x_1}) \end{array} \right\rangle \oplus {\lambda _2}\left\langle \begin{array}{l} [{s_{\theta ({x_1})}},{s_{\tau ({x_1})}}],(\mu ({x_1}),\nu ({x_1})); p({x_1}) \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{\lambda _1}\theta ({x_1})}},{s_{{\lambda _1}\tau ({x_1})}}],(1 - {(1 - \mu ({x_1}))^{{\lambda _1}}},{(\nu ({x_1}))^{{\lambda _1}}}); p({x_1}) \end{array} \right\rangle \oplus \\ \left\langle \begin{array}{l} [{s_{{\lambda _2}\theta ({x_1})}},{s_{{\lambda _2}\tau ({x_1})}}],(1 - {(1 - \mu ({x_1}))^{{\lambda _2}}},{(\nu ({x_1}))^{{\lambda _2}}}); p({x_1}) \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{\lambda _1}\theta ({x_1}) + {\lambda _2}\theta ({x_1}))}},{s_{{\lambda _1}\tau ({x_1}) + {\lambda _2}\tau ({x_1}))}}],(1 - {(1 - \mu ({x_1}))^{{\lambda _1}}}{(1 - \mu ({x_1}))^{{\lambda _2}}},{(\nu ({x_1}))^{{\lambda _1}}}{(\nu ({x_1}))^{{\lambda _2}}}); \frac{{{p^2}({x_1}) + {p^2}({x_1})}}{{p({x_1}) + p({x_1})}} \end{array} \right\rangle , \end{array}$ and

$\begin{array}{l} ({\lambda _1}+{\lambda _2}) \otimes {A_1} = \left\langle {[{s_{({\lambda _1}+{\lambda _2})\theta ({x_1})}}, {s_{({\lambda _1}+{\lambda _2})\tau ({x_1})}}],(1 - {{(1 - \mu ({x_1}))}^{{\lambda _1}+{\lambda _2}}}, {{(\nu ({x_1}))}^{{\lambda _1}+{\lambda _2}}});p({x_1})} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{\lambda _1}\theta ({x_1}) + {\lambda _2}\theta ({x_1}))}},{s_{{\lambda _1}\tau ({x_1}) + {\lambda _2}\tau ({x_1}))}}],(1 - {(1 - \mu ({x_1}))^{{\lambda _1}}}{(1 - \mu ({x_1}))^{{\lambda _2}}},{(\nu ({x_1}))^{{\lambda _1}}}{(\nu ({x_1}))^{{\lambda _2}}}); \frac{{{p^2}({x_1}) + {p^2}({x_1})}}{{p({x_1}) + p({x_1})}} \end{array} \right\rangle , \end{array}$

$\therefore {\lambda _1}{A_1} \oplus {\lambda _2}{A_1} = ({\lambda _1} + {\lambda _2}) \otimes {A_1}$.

The proof of rule (4) is complete.

(4) For rule (5)

$\begin{array}{l} \because {A_1}^{{\lambda _1}} \otimes {A_1}^{{\lambda _2}}= \left\langle {[{s_{{{(\theta ({x_1}))}^{{\lambda _1}}}}}, {s_{{{(\tau ({x_1}))}^{{\lambda _1}}}}}],({{(\mu ({x_1}))}^{{\lambda _1}}}), (1 - {{(1 - \nu ({x_1}))}^{{\lambda _1}}});{p^{{\lambda _1}}}({x_1})} \right\rangle \\ \otimes \left\langle {[{s_{{{(\theta ({x_1}))}^{{\lambda _2}}}}}, {s_{{{(\tau ({x_1}))}^{{\lambda _2}}}}}],({{(\mu ({x_1}))}^{{\lambda _2}}}), (1 - {{(1 - \nu ({x_1}))}^{{\lambda _2}}});{p^{{\lambda _2}}}({x_1})} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_1}))}^{{\lambda _1}}} \times {{(\theta ({x_1}))}^{{\lambda _2}}}}},{s_{{{(\tau ({x_1}))}^{{\lambda _1}}} \times {{(\tau ({x_1}))}^{{\lambda _2}}}}}],({(\mu ({x_1}))^{{\lambda _1}}}{(\mu ({x_1}))^{{\lambda _2}}},((1 - {(1 - \nu ({x_1}))^{{\lambda _1}}}) + \\ {(1 - {(1 - \nu ({x_1}))^{{\lambda _2}}}) - (1 - {(1 - \nu ({x_1}))^{{\lambda _1}}})(1 - {(1 - \nu ({x_1}))^{{\lambda _2}}})}{)}; {p^{{\lambda _1}}}({x_1}) \times {p^{{\lambda _2}}}({x_1}) \end{array} \right\rangle , \end{array}$ and

$\begin{array}{l} {A_1}{^{{\lambda _1} + {\lambda _2}}}=\left\langle {[{s_{{{(\theta ({x_1}))}^{{\lambda _1} + {\lambda _2}}}}}, {s_{{{(\tau ({x_1}))}^{{\lambda _1} + {\lambda _2}}}}}], ({{(\mu ({x_1}))}^{{\lambda _1} + {\lambda _2}}}),(1 - {{(1 - \nu ({x_1}))}^{{\lambda _1} + {\lambda _2}}});{p^{{\lambda _1} + {\lambda _2}}}({x_1})} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{{(\theta ({x_1}))}^{{\lambda _1}}} \times {{(\theta ({x_1}))}^{{\lambda _2}}}}},{s_{{{(\tau ({x_1}))}^{{\lambda _1}}} \times {{(\tau ({x_1}))}^{{\lambda _2}}}}}],({(\mu ({x_1}))^{{\lambda _1}}} {(\mu ({x_1}))^{{\lambda _2}}},((1 - {(1 - \nu ({x_1}))^{{\lambda _1}}}) + (1 - {(1 - \nu ({x_1}))^{{\lambda _2}}})\\ - (1 - {(1 - \nu ({x_1}))^{{\lambda _1}}})(1 - {(1 - \nu ({x_1}))^{{\lambda _2}}});{p^{{\lambda _1}}}({x_1}) \times {p^{{\lambda _2}}}({x_1}) \end{array} \right\rangle , \end{array}$

$\therefore {(}{A_1}{)}^{{\lambda _1}} \otimes {(}{A_1}{)}^{{\lambda _2}}={(}{A_1}{)}{^{{\lambda _1} + {\lambda _2}}}$.

The proof of rule (5) is complete.

(5) For rule (6)

$\begin{array}{l} \because {A_1}^\lambda \otimes {A_2}^\lambda =\left\langle {[{s_{{{(\theta ({x_1}))}^\lambda }}}, {s_{{{(\tau ({x_1}))}^\lambda }}}],({{(\mu ({x_1}))}^\lambda }), (1 - {{(1 - \nu ({x_1}))}^\lambda });{p^\lambda }({x_1})} \right\rangle \\ \otimes \left\langle {[{s_{{{(\theta ({x_2}))}^\lambda }}}, {s_{{{(\tau ({x_2}))}^\lambda }}}],({{(\mu ({x_2}))}^\lambda }), (1 - {{(1 - \nu ({x_2}))}^\lambda });{p^\lambda }({x_2})} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_1}))}^\lambda } \times {{(\theta ({x_2}))}^\lambda }}}, {s_{{{(\tau ({x_1}))}^\lambda } \times {{(\tau ({x_2}))}^\lambda }}}], {(\mu ({x_1}))^\lambda }{(\mu ({x_2}))^\lambda },((1 - {(1 - \nu ({x_1}))^\lambda }) + (1 - {(1 - \nu ({x_2}))^\lambda })\\ - (1 - {(1 - \nu ({x_1}))^\lambda })(1 - {(1 - \nu ({x_2}))^\lambda }));{p^\lambda }({x_1}) \times {p^\lambda }({x_2}) \end{array} \right\rangle , \end{array}$ and

$\begin{array}{l} {A_1} \otimes {A_2}{^\lambda }= \left\langle \begin{array}{l} [{s_{{{(\theta ({x_1}) \times \theta ({x_2}))}^\lambda }}}, {s_{{{(\tau ({x_1}) \times \tau ({x_2}))}^\lambda }}}],({(\mu ({x_1}) \mu ({x_2}))^\lambda },1 - (1 - {((\nu ({x_1}) + \nu ({x_2}) - \nu ({x_1})\nu ({x_2})))^\lambda }));\\ {(p({x_1}) \times p({x_2}))^\lambda } \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{{(\theta ({x_1}))}^\lambda } \times {{(\theta ({x_2}))}^\lambda }}}, {s_{{{(\tau ({x_1}))}^\lambda } \times {{(\tau ({x_2}))}^\lambda }}}], {(\mu ({x_1}))^\lambda }{(\mu ({x_2}))^\lambda },((1 - {(1 - \nu ({x_1}))^\lambda }) + (1 - {(1 - \nu ({x_2}))^\lambda })\\ - (1 - {(1 - \nu ({x_1}))^\lambda })(1 - {(1 - \nu ({x_2}))^\lambda }));{p^\lambda }({x_1}) \times {p^\lambda }({x_2}) \end{array} \right\rangle , \end{array}$

$\therefore {(}{A_1}{)}^\lambda \otimes {(}{A_2}{)}^\lambda = {(}{A_1} \otimes {A_2}{)}{^\lambda }$.

The proof of rule (6) is complete.□

1.2 The proofs of Theorems 4 to 7

The proof of Theorem 4:

Let $IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_j}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_j}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_j}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_j}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_j}))}^{{\omega _j}}}} \end{array} \right\rangle,$

$IPVZLWG{A_\omega }(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta (x_j^{'}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau (x_j^{'}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu (x_j^{'}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu (x_j^{'}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p(x_j^{'}))}^{{\omega _j}}}} \end{array} \right\rangle$,

Since $A_j^{'} \le {A_j}$ for all $(j = 1,2,\ldots ,n)$, we have $IPVZLWG{A_\omega }(A_1^{'},A_2^{'},\ldots ,A_n^{'}) \le IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n})$.

The proof of Theorem 5:

Since ${A_j} = A (j = 1,2,\ldots ,n)$, $\sum \nolimits _{j = 1}^n {{\omega _j}} = 1$,

$\begin{array}{l} IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_j}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_j}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_j}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_j}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_j}))}^{{\omega _j}}}} \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{{(\theta ({x_j}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}},{s_{{{(\tau ({x_j}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}}],({(\mu ({x_j}))^{\sum \limits _{j = 1}^n {{\omega _j}} }},1 - {(1 - \nu ({x_j}))^{\sum \limits _{j = 1}^n {{\omega _j}} }}); {(p({x_j}))^{\sum \limits _{j = 1}^n {{\omega _j}} }} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\theta ({x_j})}},{s_{\tau ({x_j})}}],(\mu ({x_j}),1 - (1 - \nu ({x_j})); p({x_j}) \end{array} \right\rangle =A, \end{array}$

Then, we have $IPVZLOW{G_\omega }({A_1},{A_2},\ldots ,{A_n}) = A$.

The proof of Theorem 6:

Since ${x_{\min }} \le {x_j} \le {x_{\max }}$ for all $j(j = 1,2,\ldots ,n)$, and ${\omega _j} \in [0,1]$, $\sum \nolimits _{j = 1}^n {{\omega _j}} = 1$, according to Theorems 5 and 6, then,

$\begin{array}{l} IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n})\\ = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_j}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_j}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_j}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_j}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_j}))}^{{\omega _j}}}} \end{array} \right\rangle \\ \ge \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\mathrm{min}}}}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\mathrm{min}}}}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\mathrm{min}}}}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\mathrm{min}}}}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_{min}}))}^{{\omega _j}}}} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_{{\mathrm{min}}}}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}},{s_{{{(\tau ({x_{{\mathrm{min}}}}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}}],({(\mu ({x_{{\mathrm{min}}}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }},1 - {(1 - \nu ({x_{{\mathrm{min}}}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }}); {(p({x_{min}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\theta ({x_{{\mathrm{min}}}})}},{s_{\tau ({x_{{\mathrm{min}}}})}}],(\mu ({x_{{\mathrm{min}}}}),1 - (1 - \nu ({x_{{\mathrm{min}}}}))); p({x_{{\mathrm{min}}}}) \end{array} \right\rangle ={A_{\min }}, \end{array}$

$\begin{array}{l} IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n}) \\ = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_j}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_j}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_j}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_j}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_{max}}))}^{{\omega _j}}}} \end{array} \right\rangle \\ \le \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\mathrm{max}}}}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\mathrm{max}}}}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\mathrm{max}}}}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\mathrm{max}}}}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p({x_{max}}))}^{{\omega _j}}}} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_{{\mathrm{max}}}}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}},{s_{{{(\tau ({x_{{\mathrm{max}}}}))}^{\sum \limits _{j = 1}^n {{\omega _j}} }}}}],({(\mu ({x_{{\mathrm{max}}}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }},1 - {(1 - \nu ({x_{{\mathrm{max}}}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }}); {(p({x_{max}}))^{\sum \limits _{j = 1}^n {{\omega _j}} }} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\theta ({x_{{\mathrm{max}}}})}},{s_{\tau ({x_{{\mathrm{max}}}})}}],(\mu ({x_{{\mathrm{max}}}}),1 - (1 - \nu ({x_{{\mathrm{max}}}}))); p({x_{{\mathrm{max}}}}) \end{array} \right\rangle ={A_{\max }}, \end{array}$

Therefore, we have ${A_{\min }}=\min ({A_1},{A_2},\ldots ,{A_n}) \le IPVZLOW{G_\omega }({A_1},{A_2},\ldots ,{A_n}) \le \max ({A_1},{A_2},\ldots ,{A_n}) = {A_{\max }}$.

The proof of Theorem 7:

Let

$IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_j}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_j}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_j}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_j}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p(x_j))}^{{\omega _j}}}} \end{array} \right\rangle$,

$IPVZLWG{A_\omega }(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta (x_j^{'}))}^{{\omega _j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau (x_j^{'}))}^{{\omega _j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu (x_j^{'}))}^{{\omega _j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu (x_j^{'}))}^{{\omega _j}}}} ); \prod \nolimits _{j - 1}^n {{{(p(x_j^{'}))}^{{\omega _j}}}} \end{array} \right\rangle$,

Since $(A_1^{'},A_2^{'},\ldots ,A_n^{'})$ be any permutation of $({A_1},{A_2},\ldots ,{A_n})$ and ${A_i} \otimes {A_j}={A_j} \otimes {A_i}$, so

$IPVZLWG{A_\omega }(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = IPVZLWG{A_\omega }({A_1},{A_2},\ldots ,{A_n}).$

1.3 The proofs of Theorems 9 to 12

The proof of Theorem 9:

Let $IPVZLOW{G_w}(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta (x_{{\sigma _j}}^{'}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau (x_{{\sigma _j}}^{'}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu (x_{{\sigma _j}}^{'}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu (x_{{\sigma _j}}^{'}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}^{'}}))}^{{w_j}}}} \end{array} \right\rangle$,

Since $A_j^{'} \le {A_j}$ for all $(j = 1,2,\ldots ,n)$, it follows that ${({A_j})^{n{w_j}}} \le {(A_j^{'})^{n{w_j}}}$. Hence, ${A_{{\sigma _j}}} \le A_{{\sigma _j}}^{'}$ for all $j (j = 1,2,\ldots ,n)$ . Then, we have $IPVZLOW{G_w}(A_1^{'},A_2^{'},\ldots ,A_n^{'}) \le IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n})$.

The proof of Theorem 10:

According to ${({A_j})^{n{w_j}}}=A$ for all $j (j = 1,2,\ldots ,n)$, since ${A_j} = A (j = 1,2,\ldots ,n)$, $\sum \nolimits _{j = 1}^n {{w_j}} = 1$ and Theorem 8, we have

$\begin{array}{l} IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}) \\ = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\sigma _j}}}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\sigma _j}}}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\sigma _j}}}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\sigma _j}}}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}}))}^{{w_j}}}} \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{{{(\theta ({x_{{\sigma _j}}}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}},{s_{{{(\tau ({x_{{\sigma _j}}}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}}],({(\mu ({x_{{\sigma _j}}}))^{\sum \limits _{j = 1}^n {{w_j}} }},1 - {(1 - \nu ({x_{{\sigma _j}}}))^{\sum \limits _{j = 1}^n {{w_j}} }}); {(p({x_{{\sigma _j}}}))^{\sum \limits _{j = 1}^n {{w_j}} }} \end{array} \right\rangle \\ =\left\langle \begin{array}{l} [{s_{\theta ({x_{{\sigma _j}}})}},{s_{\tau ({x_{{\sigma _j}}})}}],(\mu ({x_{{\sigma _j}}}),1 - (1 - \nu ({x_{{\sigma _j}}}))); p({x_{{\sigma _j}}}) \end{array} \right\rangle =A, \end{array}$

Then, we have $IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}) = A$.

The proof of Theorem 11:

Since ${x_{\min }} \le {x_{{\sigma _j}}} \le {x_{\max }}$, ${A_{\min }} \le A_j^{n{w_j}} \le {A_{\max }}$ for all $j (j = 1,2,\ldots ,n)$ and ${w_j} \in [0,1]$, $\sum \nolimits _{j = 1}^n {{w_j}} = 1$, according to Theorems 5 and 6, then,

$\begin{array}{l} IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n})\\ = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\sigma _j}}}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\sigma _j}}}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\sigma _j}}}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\sigma _j}}}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}}))}^{{w_j}}}} \end{array} \right\rangle \\ \ge \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{\min }}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{\min }}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{\min }}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{\min }}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{\min }}))}^{{w_j}}}} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_{\min }}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}},{s_{{{(\tau ({x_{\min }}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}}],({(\mu ({x_{\min }}))^{\sum \limits _{j = 1}^n {{w_j}} }},1 - {(1 - \nu ({x_{\min }}))^{\sum \limits _{j = 1}^n {{w_j}} }}); {(p({x_{min}}))^{\sum \limits _{j = 1}^n {{w_j}} }} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\theta ({x_{\min }})}},{s_{\tau ({x_{\min }})}}],(\mu ({x_{\min }}),1 - (1 - \nu ({x_{\min }}))); p({x_{\min }}) \end{array} \right\rangle ={A_{\min }}, \end{array}$

$\begin{array}{l} IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}) \\ = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\sigma _j}}}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\sigma _j}}}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\sigma _j}}}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\sigma _j}}}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}}))}^{{w_j}}}} \end{array} \right\rangle \\ \le \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{\max }}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{\max }}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{\max }}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{\max }}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{\max }}))}^{{w_j}}}} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{{{(\theta ({x_{\max }}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}},{s_{{{(\tau ({x_{\max }}))}^{\sum \limits _{j = 1}^n {{w_j}} }}}}],({(\mu ({x_{\max }}))^{\sum \limits _{j = 1}^n {{w_j}} }},1 - {(1 - \nu ({x_{\max }}))^{\sum \limits _{j = 1}^n {{w_j}} }}); {(p({x_{max}}))^{\sum \limits _{j = 1}^n {{w_j}} }} \end{array} \right\rangle \\ = \left\langle \begin{array}{l} [{s_{\theta ({x_{\max }})}},{s_{\tau ({x_{\max }})}}],(\mu ({x_{\max }}),1 - (1 - \nu ({x_{\max }}))); p({x_{\max }}) \end{array} \right\rangle ={A_{\max }}, \end{array}$

Therefore, we have ${A_{\min }}=\min ({A_1},{A_2},\ldots ,{A_n}) \le IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}) \le \max ({A_1},{A_2},\ldots ,{A_n})={A_{\max }}$.

The proof of Theorem 12:

Let $IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j - 1}^n {({{(\theta ({x_{{\sigma _j}}}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j - 1}^n {({{(\tau ({x_{{\sigma _j}}}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu ({x_{{\sigma _j}}}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu ({x_{{\sigma _j}}}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}}))}^{{w_j}}}} \end{array} \right\rangle$,

$IPVZLOW{G_w}(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = \left\langle \begin{array}{l} [{s_{\prod \nolimits _{j = 1}^n {({{(\theta (x_{{\sigma _j}}^{'}))}^{{w_j}}})} }},{s_{\prod \nolimits _{j = 1}^n {({{(\tau (x_{{\sigma _j}}^{'}))}^{{w_j}}})} }}],\prod \limits _{j = 1}^n {{{(\mu (x_{{\sigma _j}}^{'}))}^{{w_j}}}} ,(1 - \prod \limits _{j = 1}^n {{{(1 - \nu (x_{{\sigma _j}}^{'}))}^{{w_j}}}} ); \prod \nolimits _{j - 1}^n {({{(p({x_{{\sigma _j}}{'}}))}^{{w_j}}}} \end{array} \right\rangle$,

Since $(A_1^{'},A_2^{'},\ldots ,A_n^{'})$ be any permutation of $({A_1},{A_2},\ldots ,{A_n})$ and ${A_i} \otimes {A_j}={A_j} \otimes {A_i}$, so

$IPVZLOW{G_w}(A_1^{'},A_2^{'},\ldots ,A_n^{'}) = IPVZLOW{G_w}({A_1},{A_2},\ldots ,{A_n}).$