Appendix
Proof
Since \(0\le \tilde{\gamma }_j^L \le \tilde{\gamma }_j^U \le 1\) for all j, then we have
$$\begin{aligned}&0\le \frac{\prod \nolimits _{j=1}^n {(1+\tilde{\gamma }_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le \frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\le 1, \end{aligned}$$
(31)
which indicates that the aggregated value by the IVIHEPWA operator is an IVHFE. In addition, by mathematical induction on n:
-
1.
For \(n=2\): Since
$$\begin{aligned} \frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }\tilde{h}_1= & {} \left\{ \left[ \frac{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}, \right. \right. \nonumber \\&\left. \left. \left. \frac{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{(1+\tilde{\gamma }_1^U )^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}} \right] \right| \right. \nonumber \\&\left. \tilde{\gamma }_1 \in \tilde{h}_1 \right\} , \end{aligned}$$
(32)
$$\begin{aligned} \frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }\tilde{h}_2= & {} \left\{ \left[ \frac{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}, \right. \right. \nonumber \\&\left. \left. \left. \frac{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}} \right] \right| \right. \nonumber \\&\left. \tilde{\gamma }_2 \in \tilde{h}_2 \right\} . \end{aligned}$$
(33)
Then, we have
$$\begin{aligned}&\mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2 \right) =\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }\tilde{h}_1 \oplus _\varepsilon \frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }\tilde{h}_2\nonumber \\&\quad =\left\{ \left[ \frac{\frac{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}+\frac{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}}{1+\frac{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}\cdot \frac{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}}, \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{\frac{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}+\frac{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}}{1+\frac{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}}\cdot \frac{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}} \right] \right| \right. \nonumber \\&\quad \left. \tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 \right\} \nonumber \\&\quad =\left\{ {\left[ {\frac{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1+\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^L \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1-\tilde{\gamma }_2^L \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}},} \right. } \right. \nonumber \\&\quad \left. \left. \left. \frac{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1+\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_1^U \right) ^{\frac{T_1 }{\sum \nolimits _{i=1}^n {T_i } }}\left( 1-\tilde{\gamma }_2^U \right) ^{\frac{T_2 }{\sum \nolimits _{i=1}^n {T_i } }}} \right] \right| \right. \nonumber \\&\quad \left. \tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 \right\} . \end{aligned}$$
(34)
(2) Suppose Eq. (8) holds for \(n=k\), that is:
$$\begin{aligned}&\mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_k \right) \nonumber \\&\quad =\left\{ \left[ \frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }, \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{\prod \nolimits _{j=1}^k {(1+\tilde{\gamma }_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} } \right] \right| \right. \nonumber \\&\quad \left. {\begin{array}{*{20}c} {\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 ,} \\ {\ldots ,\tilde{\gamma }_k \in \tilde{h}_k } \\ \end{array} } \right\} . \end{aligned}$$
(35)
Then, when \(n=k+1\), by the operational laws described in Sect. 3, we have
$$\begin{aligned}&\mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_{k+1}\right) \\&\quad =\left\{ \left[ \frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }, \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} } \right] \right| \right. \nonumber \\&\quad \left. \begin{array}{*{20}c} {\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 ,} \\ {\ldots ,\tilde{\gamma }_k \in \tilde{h}_k } \\ \end{array} \right\} \end{aligned}$$
$$\begin{aligned}&\quad \oplus _\varepsilon \left\{ \left. \left[ \frac{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}, \right. \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{\left( 1+\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}} \right] \right| \right. \nonumber \\&\quad \left. \tilde{\gamma }_{k+1} \in \tilde{h}_{k+1} \right\} \end{aligned}$$
$$\begin{aligned}&\quad =\left\{ \left[ \frac{\frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }+\frac{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-(1-\tilde{\gamma }_{k+1}^L )^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}}{1+\frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\cdot \frac{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^L \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}}, \right. \right. \nonumber \\&\quad \left. \left. \left. \frac{\frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }+\frac{\left( 1+\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-\left( 1-\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}}{1+\frac{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^k {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^k {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\cdot \frac{(1+\tilde{\gamma }_{k+1}^U )^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}-(1-\tilde{\gamma }_{k+1}^U )^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}{\left( 1+\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}+\left( 1-\tilde{\gamma }_{k+1}^U \right) ^{\frac{T_{k+1} }{\sum \nolimits _{i=1}^n {T_i } }}}} \right] \right| \right. \nonumber \\&\quad \left. \begin{array}{*{20}c} {\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 ,} \\ {\ldots ,\tilde{\gamma }_{k+1} \in \tilde{h}_{k+1} } \\ \end{array} \right\} \end{aligned}$$
$$\begin{aligned}&\quad =\left\{ \left[ \frac{\prod \nolimits _{j=1}^{k+1} {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^{k+1} {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^{k+1} {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^{k+1} {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }, \right. \right. \nonumber \\&\quad \left. \left. \frac{\prod \nolimits _{j=1}^{k+1} {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^{k+1} {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^{k+1} {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^{k+1} {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} } \right] \right| \nonumber \\&\quad \left. \begin{array}{*{20}c} {\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2 ,} \\ {\ldots ,\tilde{\gamma }_{k+1} \in \tilde{h}_{k+1} } \end{array} \right\} , \end{aligned}$$
(36)
i.e., Eq. (8) holds for \(n=k+1\). Thus, Eq. (8) holds for all n. So, we complete the proof of Theorem 3. \(\square \)
Proof
Since \(\tilde{a}_j^L =\mathop {\min }\limits _k \tilde{\gamma }_{jk}^L \le \tilde{\gamma }_j^L \le \mathop {\max }\limits _k \tilde{\gamma }_{jk}^L =\tilde{b}_j^L \), \(\tilde{a}_j^U =\mathop {\min }\limits _k \tilde{\gamma }_{jk}^U \le \tilde{\gamma }_j^U \le \mathop {\max }\limits _k \tilde{\gamma }_{jk}^U =\tilde{b}_j^U \), for all j, then
$$\begin{aligned}&\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{a}_j^L }{1-\tilde{a}_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} \nonumber \\&\quad \le \prod \limits _{j=1}^n {\left( {\frac{1+\tilde{\gamma }_j^L }{1-\tilde{\gamma }_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} \nonumber \\&\quad \le \prod \limits _{j=1}^n {\left( {\frac{1+\tilde{b}_j^L }{1-\tilde{b}_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}},\end{aligned}$$
(37)
$$\begin{aligned}&\quad \quad \prod \limits _{j=1}^n {\left( {\frac{1+\tilde{a}_j^U }{1-\tilde{a}_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} \le \prod \limits _{j=1}^n {\left( {\frac{1+\tilde{\gamma }_j^U }{1-\tilde{\gamma }_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}\nonumber \\&\quad \le \prod \limits _{j=1}^n {\left( {\frac{1+\tilde{b}_j^U }{1-\tilde{b}_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}, \end{aligned}$$
(38)
and then
$$\begin{aligned}&1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{a}_j^L }{1-\tilde{a}_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1}\nonumber \\&\quad \le 1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{\gamma }_j^L }{1-\tilde{\gamma }_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1}\nonumber \\&\quad \le 1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{b}_j^L }{1-\tilde{b}_j^L }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1},\end{aligned}$$
(39)
$$\begin{aligned}&1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{a}_j^U }{1-\tilde{a}_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1}\nonumber \\&\quad \le 1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{\gamma }_j^U }{1-\tilde{\gamma }_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1}\nonumber \\&\quad \le 1-\frac{2}{\prod \limits _{j=1}^n {\left( {\frac{1+\tilde{b}_j^U }{1-\tilde{b}_j^U }}\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +1}, \end{aligned}$$
(40)
which means that
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {(1+\tilde{a}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {(1-\tilde{a}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {(1+\tilde{a}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {(1-\tilde{a}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le \frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le \frac{\prod \nolimits _{j=1}^n {(1+\tilde{b}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {(1-\tilde{b}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {(1+\tilde{b}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {(1-\tilde{b}_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }, \end{aligned}$$
(41)
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {(1+\tilde{a}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {(1-\tilde{a}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {(1+\tilde{a}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {(1-\tilde{a}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le \frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le \frac{\prod \nolimits _{j=1}^n {(1+\tilde{b}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {(1-\tilde{b}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {(1+\tilde{b}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {(1-\tilde{b}_j^U )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }. \end{aligned}$$
(42)
According to the Definitions 4 and 5, we obtain that \(s\le \mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \le t\), which completes the proof. \(\square \)
Proof
For any \(\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2,\ldots ,\tilde{\gamma }_n \in \tilde{h}_n \), using the Lemma 1, we have
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad = 1-\frac{2\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le 1-\frac{2\prod \nolimits _{j=1}^n {(1-\tilde{\gamma }_j^L )^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\sum \nolimits _{j=1}^n {\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }\left( 1+\tilde{\gamma }_j^L \right) } +\sum \nolimits _{j=1}^n {\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }\left( 1-\tilde{\gamma }_j^L \right) } }\nonumber \\&\quad = 1-\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}, \end{aligned}$$
(43)
where the equality holds if and only if \(\tilde{\gamma }_1^L =\tilde{\gamma }_2^L =\cdots =\tilde{\gamma }_n^L \).
Similarly, we have
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \le 1-\prod \limits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}, \end{aligned}$$
(44)
where the equality holds if and only if \(\tilde{\gamma }_1^U =\tilde{\gamma }_2^U =\cdots =\tilde{\gamma }_n^U \).
Then by Definition 4 and Definition 5, we know that the score function of \(\mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n\right) \) is less than the score function of \(\mathrm{HIVFPWA}(\tilde{h}_1 ,\tilde{h}_2,\ldots ,\tilde{h}_n )\), i.e.,
$$\begin{aligned}&s\left( \mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \right) \nonumber \\&\le s\left( \mathrm{HIVFPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \right) , \end{aligned}$$
it follows that
$$\begin{aligned} \mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \le \mathrm{HIVFPWA}(\tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n ), \end{aligned}$$
where the equality holds if and only if \(\tilde{h}_1 =\tilde{h}_2 =\cdots =\tilde{h}_n \), which completes the proof. \(\square \)
Proof
For any \(\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2,\ldots ,\tilde{\gamma }_n \in \tilde{h}_n \), using the Lemma 1, we have
$$\begin{aligned}&\frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \ge \frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\sum \nolimits _{j=1}^n {\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }\left( 2-\tilde{\gamma }_j^L \right) } +\sum \nolimits _{j=1}^n {\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }\tilde{\gamma }_j^L } }\nonumber \\&\quad =\prod \limits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}, \end{aligned}$$
(45)
where the equality holds if and only if \(\tilde{\gamma }_1^L =\tilde{\gamma }_2^L =\cdots =\tilde{\gamma }_n^L \).
Similarly, we have
$$\begin{aligned}&\frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad \ge \prod \limits _{j=1}^n {\left( \tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}, \end{aligned}$$
(46)
where the equality holds if and only if \(\tilde{\gamma }_1^U =\tilde{\gamma }_2^U =\cdots =\tilde{\gamma }_n^U \).
Then by Definition 4 and Definition 5, we know that the score function of \(\mathrm{HIVFPWG}(\tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n )\) is less than the score function of \(\mathrm{HIVFPWG}(\tilde{h}_1 ,\tilde{h}_2,\ldots ,\tilde{h}_n )\), i.e.,
$$\begin{aligned}&s\left( \mathrm{HIVFPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n\right) \right) \nonumber \\&\le s\left( \mathrm{IVHFEPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n\right) \right) , \end{aligned}$$
it follows that
$$\begin{aligned}&\mathrm{HIVFPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \\&\le \mathrm{IVHFEPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) , \end{aligned}$$
where the equality holds if and only if \(\tilde{h}_1 =\tilde{h}_2 =\cdots =\tilde{h}_n \). This completes the proof of Theorem 8. \(\square \)
Proof
For any \(\tilde{\gamma }_1 \in \tilde{h}_1,\tilde{\gamma }_2 \in \tilde{h}_2,\ldots ,\tilde{\gamma }_n \in \tilde{h}_n \), then
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }- \frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^L\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\\&\quad =\frac{\prod \nolimits _{j=1}^n {\left( 2+\tilde{\gamma }_j^L -\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 2-3\tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( \root n \of {3}\tilde{\gamma }_j^L -\root n \of {3}\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\left( {\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\right) \cdot \left( {\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\right) }. \end{aligned}$$
Since \(0\le \tilde{\gamma }_j^L \le \tilde{\gamma }_j^U \le 1\) for all j, then we have
$$\begin{aligned}&2+\tilde{\gamma }_j^L -\left( \tilde{\gamma }_j^L \right) ^2\ge 0,2-3\tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\nonumber \\&\quad \ge 0,\tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\ge 0,\root n \of {3}\tilde{\gamma }_j^L -\root n \of {3}\left( \tilde{\gamma }_j^L \right) ^2\ge 0,\\&\quad j=1,2,\ldots ,n. \end{aligned}$$
And note that
$$\begin{aligned}&\left( 2+\tilde{\gamma }_j^L -\left( \tilde{\gamma }_j^L \right) ^2\right) -\left( 2-3\tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) \\&\quad \;-\left( \tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) - \left( \root n \of {3}\tilde{\gamma }_j^L -\root n \of {3}\left( \tilde{\gamma }_j^L \right) ^2\right) \\&\quad =\left( 3-\root n \of {3}\right) \tilde{\gamma }_j^L +\left( \root n \of {3}-3\right) \left( \tilde{\gamma }_j^L \right) ^2\\&\quad =\tilde{\gamma }_j^L \left( 3-\root n \of {3}\right) \left( 1-\tilde{\gamma }_j^L \right) \ge 0,\\&\quad j=1,2,\ldots ,n. \end{aligned}$$
Using the Lemma 2, we have
$$\begin{aligned}&\prod \nolimits _{j=1}^n {\left( 2+\tilde{\gamma }_j^L -\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} \nonumber \\&\quad -\prod \nolimits _{j=1}^n {\left( 2-3\tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}\nonumber \\&\quad -\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L +\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}}\nonumber \\&\quad -\prod \nolimits _{j=1}^n {\left( \root n \of {3}\tilde{\gamma }_j^L -\root n \of {3}\left( \tilde{\gamma }_j^L \right) ^2\right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} \ge 0,\quad (70) \end{aligned}$$
and then
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad - \frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^L \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\ge 0.\nonumber \\ \end{aligned}$$
(47)
Similarly, we have
$$\begin{aligned}&\frac{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} -\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 1+\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( 1-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\nonumber \\&\quad -\frac{2\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }{\prod \nolimits _{j=1}^n {\left( 2-\tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} +\prod \nolimits _{j=1}^n {\left( \tilde{\gamma }_j^U \right) ^{\frac{T_j }{\sum \nolimits _{i=1}^n {T_i } }}} }\ge 0.\nonumber \\ \end{aligned}$$
(48)
According to Definition 4 and Definition 5, we know that the score value of \(\mathrm{IVHFEPWA}(\tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n )\) is less than the score value of \(\mathrm{IVHFEPWG}(\tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n )\), i.e.,
$$\begin{aligned}&s\left( \mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \right) \\&\ge s\left( \mathrm{IVHFEPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n\right) \right) , \end{aligned}$$
then, it follows that
$$\begin{aligned}&\mathrm{IVHFEPWA}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) \nonumber \\&\ge \mathrm{IVHFEPWG}\left( \tilde{h}_1,\tilde{h}_2,\ldots ,\tilde{h}_n \right) . \end{aligned}$$
So, we complete the proof of Theorem 9. \(\square \)
