Appendix 1
1.1 The Proof of Theorem 10.
Theorem 10
Assume \(\mathscr {\tilde{Q}}_{k}=\left( [\alpha _k, \beta _k], [\gamma _k, \delta _k]\right) (k=1, 2, \ldots , n)\) be a family of IVIFNs, and \(g, h \ge 0\). Then the fusion result from Eq (13) is also an IVIFN, and it can be acquired from
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}, \mathscr {\tilde{Q}}, \ldots , \mathscr {\tilde{Q}} \right) \\&\quad =\left( \begin{array}{cccc}\left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\gamma _k\right) \\ +h\xi \left( 1-\gamma _j\right) \end{array}\right) \right) \right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\delta _k\right) \\ +h\xi \left( 1-\delta _j\right) \end{array}\right) \right) \right) \right) \right) \right) \right] \end{array}\right) . \end{aligned}$$
Proof
Since
$$\begin{aligned}&\mathscr {\tilde{Q}}_{i}^{g}= \left( \begin{array}{cccc} \left[ \xi ^{-1}\left( g\xi \left( \alpha _k\right) \right) , \xi ^{-1}\left( g\xi \left( \beta _k\right) \right) \right] , \\ \left[ 1-\xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) \right) , 1-\xi ^{-1}\left( g\xi \left( 1-\delta _k\right) \right) \right] \end{array}\right) ,\\&\mathscr {\tilde{Q}}_{j}^{h}= \left( \begin{array}{cccc} \left[ \xi ^{-1}\left( h\xi \left( \alpha _j\right) \right) , \xi ^{-1}\left( h\xi \left( \beta _j\right) \right) \right] , \\ \left[ 1-\xi ^{-1}\left( h\xi \left( 1-\gamma _j\right) \right) , 1-\xi ^{-1}\left( h\xi \left( 1-\delta _j\right) \right) \right] \end{array}\right) . \end{aligned}$$
Then
$$\begin{aligned} \mathscr {\tilde{Q}}_{i}^{g} \otimes \mathscr {\tilde{Q}}_{j}^{h}&\,=\,\left( \begin{array}{cccc}\left[ \xi ^{-1}\left( \xi \left( \xi ^{-1}\left( g\xi \left( \alpha _k\right) \right) \right) +\xi \left( \xi ^{-1}\left( h\xi \left( \alpha _j\right) \right) \right) \right) ,\right. \\ \left. \xi ^{-1}\left( \xi \left( \xi ^{-1}\left( g\xi \left( \beta _k\right) \right) \right) +\xi \left( \xi ^{-1}\left( g\xi \left( \beta _j\right) \right) \right) \right) \right] , \\ \left[ 1-\xi ^{-1}\left( \xi \left( \xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) \right) \right) +\xi \left( \xi ^{-1}\left( h\xi \left( 1-\gamma _j\right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \xi \left( \xi ^{-1}\left( g\xi \left( 1-\delta _k\right) \right) \right) +\xi \left( \xi ^{-1}\left( h\xi \left( 1-\delta _j\right) \right) \right) \right) \right] \end{array}\right) ;\\&\,=\,\left( \begin{array}{cccc}\left[ \xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) , \xi ^{-1}\left( g\xi \left( \beta _k\right) +\xi \left( \beta _{j}\right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) +h\xi \left( 1-\gamma _j\right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( g\xi \left( 1-\delta _k\right) +h\xi \left( 1-\delta _j\right) \right) \right] \end{array}\right) , \end{aligned}$$
and
$$\begin{aligned}&\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\mathscr {\tilde{Q}}_{k}^{g} \otimes \mathscr {\tilde{Q}}_{j}^{h} \\&\quad =\left( \begin{array}{cccc} \left[ 1-\xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right] ,\\ \left[ \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) +h\xi \left( 1-\gamma _j\right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( g\xi \left( 1-\delta _k \right) +h\xi \left( 1-\delta _j\right) \right) \right) \right) \right] \end{array}\right) . \end{aligned}$$
Furthermore
$$\begin{aligned}&\frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\mathscr {\tilde{Q}}_{k}^{g} \otimes \mathscr {\tilde{Q}}_{j}^{h} \\&\quad =\left( \begin{array}{cccccc} \left[ 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\xi \left( \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \right) ,\right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\xi \left( \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \right) \right] ,\\ \left[ \xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\xi \left( \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\gamma _k\right) \\ +h\xi \left( 1-\gamma _j\right) \end{array}\right) \right) \right) \right) \right) ,\right. \\ \left. \xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\xi \left( \xi ^{-1}\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\delta _k\right) \\ +h\xi \left( 1-\delta _j\right) \end{array}\right) \right) \right) \right) \right) \right] \end{array}\right) .\\&\quad =\left( \begin{array}{ccc} \left[ 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) ,\right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \right] ,\\ \left[ \xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) +h\xi \left( 1-\gamma _j\right) \right) \right) \right) \right) ,\right. \\ \left. \xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( g\xi \left( 1-\delta _k\right) +h\xi \left( 1-\delta _j\right) \right) \right) \right) \right) \right] \end{array}\right) . \end{aligned}$$
Accordingly
$$\begin{aligned}&\left( { {\frac{1}{n\left( { {n-1} }\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} {\mathscr {\tilde{Q}}_{k}^{g} \otimes \mathscr {\tilde{Q}}_{j}^{h}}}}\right) ^{\frac{1}{g+h}} \\&\quad =\left( \begin{array}{cccccc} \left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( \mathfrak {a} \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { i,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( \mathfrak {b} \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( \mathfrak {c} \right) \right) \right) \right) \right) \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1 }\\ {k\ne j} \end{array}}^{n}\xi \left( \mathfrak {d} \right) \right) \right) \right) \right) \right] \end{array}\right) . \end{aligned}$$
Where
$$\begin{aligned}&\mathfrak {a}= 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) ;\\&\mathfrak {b}=1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) ;\\&\mathfrak {c}=1-\xi ^{-1}\left( g\xi \left( 1-\gamma _k\right) +h\xi \left( 1-\gamma _j\right) \right) ;\\&\mathfrak {d}=1-\xi ^{-1}\left( g\xi \left( 1-\delta _k\right) +h\xi \left( 1-\delta _j\right) \right) . \end{aligned}$$
Accordingly, we can attain that Eq (13)is correct. \(\square\)
Appendix 2
1.1 The Proof of Theorem 11.
Theorem 11
(Idempotency) Assume \(\{\mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n}\}\) be a family of IVIFNs, and \(g, h \ge 0\). If all \(\mathscr {\tilde{Q}}_{k}\) are equal \(\mathscr {\tilde{Q}}\), i.e., \(\mathscr {\tilde{Q}}_{k}=\mathscr {\tilde{Q}}=\langle [\alpha , \beta ], [\gamma , \delta ]\rangle (k=1, 2, \ldots , n)\), then
$$\begin{aligned} IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) =\mathscr {\tilde{Q}}. \end{aligned}$$
(47)
Proof
Since \(\mathscr {\tilde{Q}}_{k}=\mathscr {\tilde{Q}}=\langle [\alpha , \beta ], [\gamma , \delta ]\rangle\)
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}, \mathscr {\tilde{Q}}, \ldots , \mathscr {\tilde{Q}} \right) \\&\quad =\left( \begin{array}{cccc}\left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1 }\\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\gamma _k\right) \\ +h\xi \left( 1-\gamma _j\right) \end{array}\right) \right) \right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \begin{array}{ccc}g\xi \left( 1-\delta _k\right) \\ +h\xi \left( 1-\delta _j\right) \end{array}\right) \right) \right) \right) \right) \right) \right] \end{array}\right) .\\&\quad =\left( \begin{array}{cccc} \left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \alpha \right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \beta \right) \right) \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\gamma \right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\delta \right) \right) \right) \right) \right) \right) \right) \right] \end{array}\right) .\\&\quad =\left( \begin{array}{cccc} \left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( n\left( n-1\right) \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \alpha \right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( n\left( n-1\right) \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \beta \right) \right) \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( n\left( n-1\right) \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\gamma \right) \right) \right) \right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\left( n\left( n-1\right) \xi \left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\delta \right) \right) \right) \right) \right) \right) \right) \right] \end{array}\right) \\&\quad = \left( \begin{array}{cccc} \left[ \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \alpha \right) \right) \right) \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( \beta \right) \right) \right) \right) \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\gamma \right) \right) \right) \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\left( 1-\xi ^{-1}\left( \left( g+h\right) \xi \left( 1-\delta \right) \right) \right) \right) \right) \right] \end{array}\right) .\\&\quad = \left( \begin{array}{cccc} \left[ \xi ^{-1}\left( \frac{1}{g+h}\left( g+h\right) \xi \left( \alpha \right) \right) , \right. \\ \left. \xi ^{-1}\left( \frac{1}{g+h}\left( g+h\right) \xi \left( \beta \right) \right) \right] , \\ \left[ 1-\xi ^{-1}\left( \frac{1}{g+h}\left( g+h\right) \xi \left( 1-\gamma \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \frac{1}{g+h}\left( g+h\right) \xi \left( 1-\delta \right) \right) \right] \end{array}\right) . =\left( \begin{array}{cccc} \left[ \xi ^{-1}\xi \left( \alpha \right) , \right. \\ \left. \xi ^{-1}\xi \left( \beta \right) \right] ,\\ \left[ 1-\xi ^{-1}\left( \xi \left( 1-\gamma \right) \right) , \right. \\ \left. 1-\xi ^{-1}\left( \xi \left( 1-\delta \right) \right) \right] \end{array}\right) \\&\quad =\left( [\alpha , \beta ], [\gamma , \delta ]\right) \end{aligned}$$
Accordingly, the idempotency of IVIFCBM operator is proved. \(\square\)
Appendix 3
The Proof of Theorem 12.
Theorem 12
(Monotonicity) Assume \(\{\mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n}\}\) \(\{\mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'}\}\) be two families of IVIFNs and \(h, g \ge 0\). If \(\{\mathscr {\tilde{Q}}_{k}=\langle [\alpha _{k}, \beta _{k}], [\gamma _{k}, \delta _{k}]\rangle\), \(\{\mathscr {\tilde{Q}}_{k}^{'}=\langle [\alpha _{k}^{'}, \beta _{k}^{'}], [\gamma _{k}^{'}, \delta _{k}^{'}]\rangle\), \(\alpha _{k} \ge \alpha _{k}^{'}, \beta _{k}\ge \beta _{k}^{'}, \gamma _{k}\le \gamma _{k}^{'}\) and \(\delta _{k} \le \delta _{k}^{'} (k=1, 2, \ldots , n)\),
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \nonumber \\&\quad \ge IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'}\right) \end{aligned}$$
(48)
Proof
Let \(IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) =\langle [\alpha , \beta ], [\gamma , \delta ]\rangle\), \(IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'} \right) =\langle [\alpha ^{'}, \beta ^{'}], [\gamma ^{'}, \delta ^{'}]\rangle\), where
$$\begin{aligned}&\alpha =\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) }{n\left( n-1\right) }\right) \right) \right) ,\\&\alpha ^{'}=\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) \right) }{n\left( n-1\right) }\right) \right) \right) ,\\&\beta =\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) }{n\left( n-1\right) }\right) \right) \right) ,\\&\beta ^{'}=\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{\left( \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) \right) }{n\left( n-1\right) }\right) \right) \right) . \end{aligned}$$
Since \(\alpha _{k} \ge \alpha _{k}^{'}, \beta _{k}\ge \beta _{k}^{'}\), \(\xi (t)\) and \(\xi ^{-1}(t)\) are decreasing functions. \(1-\xi (t)\) and \(1-\xi ^{-1}(t)\) are increasing functions. Then we have
$$\begin{aligned} g\xi \left( \alpha _i\right) +h\xi \left( \alpha _j\right) \le g\xi \left( \alpha _{i}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) ,\\ g\xi \left( \beta _i\right) +h\xi \left( \beta _j\right) \le g\xi \left( \beta _{i}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) , \end{aligned}$$
and
$$\begin{aligned} 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \ge 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) , \\ 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \ge 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) . \end{aligned}$$
Then
$$\begin{aligned}&\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \\&\quad \le \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) ,\\&\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \\&\quad \le \sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) , \end{aligned}$$
and
$$\begin{aligned}&\frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \\&\quad \le \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) ,\\&\frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \\&\quad \le \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} {k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) . \end{aligned}$$
Furthermore
$$\begin{aligned}&1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1 }\\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \\&\quad \le 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) \right) ,\\&1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \\&\quad \le 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) \right) . \end{aligned}$$
and
$$\begin{aligned}&\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \\ \ge&\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) \right) \right) \\&\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \\ \ge&\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) \right) \right) . \end{aligned}$$
Moreover
$$\begin{aligned}&\frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \\&\quad \ge \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) \right) \right) \\&\frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \\&\quad \ge \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) \right) \right) . \end{aligned}$$
Hence
$$\begin{aligned}&\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _k\right) +h\xi \left( \alpha _j\right) \right) \right) \right) \right) \right) \\&\quad \ge \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \alpha _{k}^{'}\right) +h\xi \left( \alpha _{j}^{'}\right) \right) \right) \right) \right) \right) \\&\xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _k\right) +h\xi \left( \beta _j\right) \right) \right) \right) \right) \right) \\&\quad \ge \xi ^{-1}\left( \frac{1}{g+h}\xi \left( 1-\xi ^{-1}\left( \frac{1}{n\left( n-1\right) }\sum \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n} \xi \left( 1-\xi ^{-1}\left( g\xi \left( \beta _{k}^{'}\right) +h\xi \left( \beta _{j}^{'}\right) \right) \right) \right) \right)
\right) . \end{aligned}$$
i.e., \(\alpha \ge \alpha ^{'}, \beta \ge \beta ^{'}\).
Analogously, we also acquire \(\gamma \le \gamma ^{'}, \delta \le \delta ^{'}\).
Besides, the following three particular situations are explored:
-
(1)
If \(\alpha \ge \alpha ^{'}, \beta \ge \beta ^{'}\) and \(\gamma \le \gamma ^{'}, \delta \le \delta ^{'}\), then
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \\&\quad > IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'}\right) ; \end{aligned}$$
-
(2)
If \(\alpha = \alpha ^{'}, \beta = \beta ^{'}\) and \(\gamma \le \gamma ^{'}, \delta \le \delta ^{'}\), then
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \\&\quad > IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'}\right) ; \end{aligned}$$
-
(3)
If \(\alpha = \alpha ^{'}, \beta = \beta ^{'}\) and \(\gamma = \gamma ^{'}, \delta = \delta ^{'}\), then
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \\&\quad = IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}^{'}, \mathscr {\tilde{Q}}_{2}^{'}, \ldots , \mathscr {\tilde{Q}}_{n}^{'}\right) . \end{aligned}$$
Accordingly, the monotonicity of IVIFCBM operator is proved.
\(\square\)
Appendix 4
Case 7: When \(\xi (t)=\ln \left( \frac{e^{-\varsigma t}-1}{e^{-\varsigma }-1}\right) (\varsigma \ne 0)\), we have
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \nonumber \\&\quad =\left( \begin{array}{cccc} \left[ -\frac{1}{\varsigma }\ln \left( u_{4}\left( e^{-\varsigma }-1\right) +1\right) , -\frac{1}{\varsigma }\ln \left( v_{3}\left( e^{-\varsigma }-1\right) +1\right) \right] , \\ \left[ 1+\frac{1}{\varsigma }\ln \left( p_{3}\left( e^{-\varsigma }-1\right) +1\right) , 1+\frac{1}{\varsigma }\ln \left( q_{3}\left( e^{-\varsigma }-1\right) +1\right) \right] \end{array}\right) , \end{aligned}$$
where
$$\begin{aligned}&u_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&v_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( v_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&p_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( p_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&q_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( q_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&u_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&v_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( v_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&p_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( p_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&q_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( q_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&u_{3}{''}=\left( \frac{e^{-\varsigma \alpha _{k}}-1}{e^{-\varsigma }-1}\right) ^{g}\left( \frac{e^{-\varsigma \alpha _{j}}-1}{e^{-\varsigma }-1}\right) ^{h},\\&p_{3}{''}=\left( \frac{e^{-\varsigma \left( 1-\gamma _{k}\right) }-1}{e^{-\varsigma }-1}\right) ^{g}\left( \frac{e^{-\varsigma \left( 1-\gamma _{j}\right) }-1}{e^{-\varsigma }-1}\right) ^{h},\\&v_{3}{''}=\left( \frac{e^{-\varsigma \beta _{k}}-1}{e^{-\varsigma }-1}\right) ^{g}\left( \frac{e^{-\varsigma \beta _{j}}-1}{e^{-\varsigma }-1}\right) ^{h},\\&q_{3}{''}=\left( \frac{e^{-\varsigma \left( 1-\gamma _{k}\right) }-1}{e^{-\varsigma }-1}\right) ^{g}\left( \frac{e^{-\varsigma \left( 1-\gamma _{j}\right) }-1}{e^{-\varsigma }-1}\right) ^{h}. \end{aligned}$$
Appendix 5
Case 8: When \(\xi (t)=\ln \left( \frac{1-\varsigma (1-t)}{t}\right)\) with \(\varsigma \in [-1, 1)\), we have
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \nonumber \\&\quad =\left( \begin{array}{ccc} \left[ \frac{1-\varsigma }{u_{4}-\varsigma }, \frac{1-\varsigma }{v_{4}-\varsigma } \right] , \left[ 1-\frac{1-\varsigma }{p_{4}-\varsigma }, 1-\frac{1-\varsigma }{q_{4}-\varsigma }\right] \end{array}\right) , \end{aligned}$$
where
$$\begin{aligned}&u_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}}, v_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}},\\&p_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}}, q_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}};\\&u_{4}^{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4}{''}-\varsigma }}\right) ^{\frac{1}{n\left( n-1\right) }}, v_{4}^{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{v_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }},\\&p_{4}^{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{p_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }}, q_{4}^{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{q_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }}\\&u_{4}{''}=\left( \frac{1-\varsigma \left( 1-\alpha _{k}\right) }{\alpha _{k}}\right) ^{g}\left( \frac{1-\varsigma \left( 1-\alpha _{j}\right) }{\alpha _{j}}\right) ^{h},\\&v_{4}{''}=\left( \frac{1-\varsigma \left( 1-\beta _{k}\right) }{\beta _{k}}\right) ^{g}\left( \frac{1-\varsigma \left( 1-\beta _{j}\right) }{\beta _{j}}\right) ^{h},\\&p_{4}{''}=\left( \frac{1-\varsigma \gamma _{k}}{1-\gamma _{i}}\right) ^{g}\left( \frac{1-\varsigma \gamma _{j}}{1-\gamma _{j}}\right) ^{h}, q_{4}{''}=\left( \frac{1-\varsigma \delta _{k}}{1-\delta _{i}}\right) ^{g}\left( \frac{1-\varsigma \delta _{j}}{1-\delta _{j}}\right) ^{h}. \end{aligned}$$
Appendix 6
Case 16: When \(\xi (t)=\ln \left( \frac{e^{-\varsigma t}-1}{e^{-\varsigma }-1}\right) (\varsigma \ne 0)\), we have
$$\begin{aligned}&IVIFCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \nonumber \\&\quad =\left( \begin{array}{ccc}\left[ -\frac{1}{\varsigma }\ln \left( u_{3}\left( e^{-\varsigma }-1\right) +1\right) , -\frac{1}{\varsigma }\ln \left( v_{3}\left( e^{-\varsigma }-1\right) +1\right) \right] , \\ \left[ 1+\frac{1}{\varsigma }\ln \left( p_{3}\left( e^{-\varsigma }-1\right) +1\right) , 1+\frac{1}{\varsigma }\ln \left( q_{3}\left( e^{-\varsigma }-1\right) +1\right) \right] \end{array}\right) \end{aligned}$$
where
$$\begin{aligned}&u_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&v_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&p_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}},\\&q_{3}=\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{'}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) ^{\frac{1}{g+h}}; \\&u_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( u_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&v_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( v_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&p_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( p_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }},\\&q_{3}{'}=\left( \prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{e^{-\varsigma \left( 1+\frac{1}{\varsigma }\ln \left( q_{3}{''}\left( e^{-\varsigma }-1\right) +1\right) \right) }-1}{e^{-\varsigma }-1}\right) \right) ^{\frac{1}{n\left( n-1\right) }};\\&u_{3}{''}=\left( \frac{e^{-\varsigma \left( u_{3''}^{i} \right) }-1}{e^{-\varsigma }-1}\right) ^{g} \left( \frac{e^{-\varsigma \left( u_{3''}^{j} \right) }-1}{e^{-\varsigma }-1}\right) ^{h},\\&v_{3}{''}=\left( \frac{e^{-\varsigma \left( v_{3''}^{i} \right) }-1}{e^{-\varsigma }-1}\right) ^{g} \left( \frac{e^{-\varsigma \left( v_{3''}^{j} \right) }-1}{e^{-\varsigma }-1}\right) ^{h},\\&p_{3}{''}=\left( \frac{e^{-\varsigma \left( p_{3''}^{i} \right) }-1}{e^{-\varsigma }-1}\right) ^{g} \left( \frac{e^{-\varsigma \left( p_{3''}^{j} \right) }-1}{e^{-\varsigma }-1}\right) ^{h},\\&p_{3}{''}=\left( \frac{e^{-\varsigma \left( q_{3''}^{i} \right) }-1}{e^{-\varsigma }-1}\right) ^{g} \left( \frac{e^{-\varsigma \left( q_{3''}^{j} \right) }-1}{e^{-\varsigma }-1}\right) ^{h}\\&u_{3''}^{i}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \left( 1-\alpha _{k}\right) }-1}{e^{-\varsigma }-1}\right) ^{n\omega _{k}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&u_{3''}^{j}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \left( 1-\alpha _{j}\right) }-1}{e^{-\varsigma }-1}\right) ^{n\omega _{j}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&v_{3''}^{i}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \left( 1-\beta _{k}\right) }-1}{e^{-\varsigma }-1}\right) ^{n\omega _{k}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&v_{3''}^{j}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \left( 1-\beta _{j}\right) }-1}{e^{-\varsigma }-1}\right) ^{n\omega _{j}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&p_{3''}^{i}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \gamma _{k}}-1}{e^{-\varsigma }-1}\right) ^{n\omega _{k}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&p_{3''}^{j}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \gamma _{j}}-1}{e^{-\varsigma }-1}\right) ^{n\omega _{j}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&q_{3''}^{i}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \delta _{k}}-1}{e^{-\varsigma }-1}\right) ^{n\omega _{k}}\left( e^{-\varsigma }-1\right) +1\right) ,\\&q_{3''}^{j}=1+\frac{1}{\varsigma }\ln \left( \left( \frac{e^{-\varsigma \delta _{j}}-1}{e^{-\varsigma }-1}\right) ^{n\omega _{j}}\left( e^{-\varsigma }-1\right) +1\right) . \end{aligned}$$
Appendix 7
Case 17: When \(\xi (t)=\ln \left( \frac{1-\varsigma (1-t)}{t}\right)\) with \(\varsigma \in [-1, 1)\), we have
$$\begin{aligned}&IVIFWCBM^{g, h} \left( \mathscr {\tilde{Q}}_{1}, \mathscr {\tilde{Q}}_{2}, \ldots , \mathscr {\tilde{Q}}_{n} \right) \nonumber \\&\quad =\left( \left[ \frac{1-\varsigma }{u_{4}-\varsigma }, \frac{1-\varsigma }{v_{4}-\varsigma } \right] , \left[ 1-\frac{1-\varsigma }{p_{4}-\varsigma }, 1-\frac{1-\varsigma }{q_{4}-\varsigma }\right] \right) , \end{aligned}$$
where
$$\begin{aligned}&u_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}}, v_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{v_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}},\\&p_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{p_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}}, q_{4}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{q_{4}{'}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4}{'}-\varsigma }}\right) ^{\frac{1}{g+h}};\\&u_{4}{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k \ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }}, v_{4}{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{v_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }},\\&p_{4}{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{p_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }}, q_{4}{'}=\prod \limits _{\begin{array}{c} { k,j=1} \\ {k\ne j} \end{array}}^{n}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{q_{4}{''}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4}{''}-\varsigma }}\right) ^{ \frac{1}{n\left( n-1\right) }};\\&u_{4}{''}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4''}^{i}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4''}^{i}-\varsigma }}\right) ^{g}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{u_{4''}^{j}-\varsigma }\right) }{1-\frac{1-\varsigma }{u_{4''}^{j}-\varsigma }}\right) ^{h},\\&v_{4}{''}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{p_{4''}^{i}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4''}^{i}-\varsigma }}\right) ^{g}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{p_{4''}^{j}-\varsigma }\right) }{1-\frac{1-\varsigma }{p_{4''}^{j}-\varsigma }}\right) ^{h},\\&p_{4}{''}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{v_{4''}^{i}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4''}^{i}-\varsigma }}\right) ^{g}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{v_{4''}^{j}-\varsigma }\right) }{1-\frac{1-\varsigma }{v_{4''}^{j}-\varsigma }}\right) ^{h},\\&q_{4}{''}=\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{q_{4''}^{i}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4''}^{i}-\varsigma }}\right) ^{g}\left( \frac{1-\varsigma \left( \frac{1-\varsigma }{q_{4''}^{j}-\varsigma }\right) }{1-\frac{1-\varsigma }{q_{4''}^{j}-\varsigma }}\right) ^{h};\\&u_{4''}^{i}=\left( \frac{1-\varsigma \alpha _{k}}{1-\alpha _{k}}\right) ^{n\omega _{k}}, u_{4}^{j}=\left( \frac{1-\varsigma \alpha _{j}}{1-\alpha _{j}}\right) ^{n\omega _{j}},\\&v_{4''}^{i}=\left( \frac{1-\varsigma \beta _{k}}{1-\beta _{k}}\right) ^{n\omega _{k}}, v_{4''}^{j}=\left( \frac{1-\varsigma \beta _{j}}{1-\beta _{j}}\right) ^{n\omega _{j}},\\&p_{4''}^{i}=\left( \frac{1-\varsigma \left( 1-\gamma _{k}\right) }{\gamma _{k}}\right) ^{n\omega _{i}}, p_{4''}^{j}=\left( \frac{1-\varsigma \left( 1-\gamma _{j}\right) }{\gamma _{j}}\right) ^{n\omega _{i}},\\&q_{4''}^{i}=\left( \frac{1-\varsigma \left( 1-\delta _{k}\right) }{\delta _{k}}\right) ^{n\omega _{k}}, q_{4''}^{j}=\left( \frac{1-\varsigma \left( 1-\delta _{j}\right) }{\delta _{j}}\right) ^{n\omega _{i}}. \end{aligned}$$
