Appendix A
To prove Eq. (18), we used the principle of mathematical induction.
Step 1. For \(n=2\), we have.
$${\varphi }_{1}.{\beta }_{1}=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right),$$
$${\varphi }_{2}.{\beta }_{2}=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right),$$
$$\mathrm{CPFDWA}({\beta }_{1},{\beta }_{2})={\varphi }_{k}{\beta }_{k}\oplus{\varphi }_{k}{\beta }_{k}$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$\oplus\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+\sum_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
which is true for \(n=2\).
Step 2. Suppose that Eq. (18) is true for \(n=l\) then,
$$CPFDWA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{l}\right)=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+\sum_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
Step 3. In this step, we will prove that Eq. (18) is true for \(n=l+1\).
By Eq. (17), we have
$$CPFDWA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{l+1}\right)=\sum_{k=1}^{l}{\varphi }_{k}{\beta }_{k}+{\varphi }_{l+1}{\beta }_{l+1}$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+\sum_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$\oplus\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+\sum_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right).$$
As a result, Eq. (18) is true for \(n=l+1\), and thus, it is satisfied for all \(n\). Therefore,
$$\mathrm{CPFDWA}\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{n}\right)=\left(\begin{array}{l}\left(\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+\sum_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right),$$
which completed the proof.
Appendix B
To prove Eq. (22), we used the principle of mathematical induction,
Step 1. For \(n=2\), according to Definition 22, we can obtain.
$${\beta }_{1}^{{\varphi }_{1}}=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right),$$
$${\beta }_{2}^{{\varphi }_{2}}=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right),$$
$$\mathrm{CPFDWA}({\beta }_{1},{\beta }_{2})={\beta }_{1}^{{\varphi }_{1}}\otimes {\beta }_{1}^{{\varphi }_{1}}=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$\otimes \left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\frac{1}{\sqrt{1+{\left\{{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{L}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}^{U}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }+{\varphi }_{2}{\left(\frac{1}{{\left({\upsilon }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+{\left\{{{\varphi }_{1}\left(\frac{1}{{\left({\vartheta }_{{R}_{1}}\right)}^{2}}-1\right)}^{2\eta }+{{\varphi }_{2}\left(\frac{1}{{\left({\vartheta }_{{R}_{2}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{2}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+\sum_{k=1}^{2}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
which is true for \(n=2\).
Step 2. Suppose that Eq. (22) is true for \(n=l\) then,
$$CPFDWA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{l}\right)=\left(\begin{array}{l}\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\\ \left(\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+\sum_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
Step 3. In this step, we will prove that Eq. (22) is true for all \(n=l+1\).
By Eq. (17), we have
$$CPFDWG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{l+1}\right)=\sum_{k=1}^{l}{\varphi }_{k}{\beta }_{k}\otimes {\varphi }_{l+1}{\beta }_{l+1}$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+\sum_{k=1}^{l}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$\otimes \left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\varphi }_{l+1}{\left(\frac{1}{{\left({\upsilon }_{{R}_{l+1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+{\left\{{{\varphi }_{l+1}\left(\frac{1}{{\left({\vartheta }_{{R}_{l+1}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right)$$
$$=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{l+1}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+\sum_{k=1}^{l+1}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right).$$
As a result, Eq. (22) is true for \(n=l+1\), and thus, it satisfies for all \(n\). Therefore,
$$CPFDWA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{n}\right)=\left(\begin{array}{l}\left(\left[\begin{array}{l}\frac{1}{\sqrt{1+{\sum }_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \frac{1}{\sqrt{1+{\sum }_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right],\left[\begin{array}{l}\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{L}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\\ \sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}^{U}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\end{array}\right]\right),\\ \left(\sqrt{1-\frac{1}{1+{\sum }_{k=1}^{n}{\left\{{\varphi }_{k}{\left(\frac{1}{{\left({\upsilon }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}},\frac{1}{\sqrt{1+\sum_{k=1}^{n}{\left\{{{\varphi }_{k}\left(\frac{1}{{\left({\vartheta }_{{R}_{k}}\right)}^{2}}-1\right)}^{2\eta }\right\}}^{\frac{1}{\eta }}}}\right)\end{array}\right).$$
which completed the proof.
