## 3 Acceptability Measurement of an IVIFPR
### Multiplicative Consistency Index of an IVIFPR
: _Given an IVIFPR \(\widetilde{R}=\left(\widetilde{r}_{ij}\right)_{n\times n}\) with \(\widetilde{r}_{ij}=\left(\widetilde{u}_{ij},\widetilde{v}_{ij}\right)\) where \(\widetilde{\mu}_{ij}=\left[\underline{u}_{ij^{\prime}},\overline{\mu}_{ij}\right]\), \(\widetilde{v}_{ij}=\left[\underline{v}_{ij},\overline{v}_{ij}\right]\), it concludes that_  
\[\underline{\Upsilon}_{ijk,\mu}=\underline{\Upsilon}_{ijk,\mu}=\underline{ \Upsilon}_{jk,\mu}=\underline{\Upsilon}_{kj,\mu}=\underline{\Upsilon}_{kij, \mu}=\underline{\Upsilon}_{ikj,\mu},\]  
\[\overline{\tau}_{ijk,\mu}=\overline{\tau}_{jk,\mu}=\overline{\tau}_{jk,\mu}= \overline{\tau}_{kj,\mu}=\overline{\tau}_{kj,\mu}=\overline{\tau}_{kj,\mu},\]  
\[\underline{\Upsilon}_{ijk,\
u}=\underline{\Upsilon}_{jk,\
u}=\underline{ \Upsilon}_{jki,\
u}=\underline{\Upsilon}_{kij,\
u}=\underline{\Upsilon}_{kj, \
u}=\underline{\Upsilon}_{kj,\
u},\]  
\[\overline{\tau}_{ijk,\
u}=\overline{\tau}_{jik,\
u}=\overline{\tau}_{jk,\
u}= \overline{\tau}_{kij,\
u}=\overline{\tau}_{kij,\
u}.\]  
From Theorem 1, it can be seen that the deviations related to subscripts \(i,j\) and \(k\) are not influenced by the orderings of their subscripts. Therefore, Equation (12) is transformed into the formula as follows:  
\[ \begin{split} D\left(\widetilde{R}\right)=\sum_{i=1}^{n}\sum_{k=i +1}^{n}\sum_{j=k+1}^{n}\left\{\left|\underline{\Upsilon}_{ikj,\mu}\right|^{p} +\left|\overline{\tau}_{ikj,\mu}\right|^{p}+\left|\underline{\Upsilon}_{ikj, \
u}\right|^{p}+\left|\overline{\tau}_{ikj,\
u}\right|^{p}\right\}^{1/p}\\ \text{for all }i,j,k=1,2,\ldots,n,\ i<k<j\end{split}  \tag{13}\]  
where the deviation \(D\left(\widetilde{R}\right)\) depends on the parameter \(p(p\geq 0)\). Accordingly, as \(p(1\leq p<\infty)\) increases, more importance is granted to the largest deviation. If \(p=1\), Equation (13) degenerates into Hamming distance. If \(p=+\infty\), Equation (13) degenerates into Chebyshev distance. Therefore, integrating two special cases (\(p=1\) and \(p=+\infty\)), we have the following definition of the multiplicative consistency index for an IVIFPR. **Definition 6**. : _For an IVIFPR \(\widetilde{R}=\left(\widetilde{r}_{ij}\right)_{n\times n}\), when \(p=1\), the multiplicative consistency degree of an IVIFPR can be measured by Hamming distance, which is defined as_  
\[HD\left(\widetilde{R}\right)=\frac{1}{4\alpha_{n}^{3}}\sum_{i=1}^{n}\sum_{k=i+1 }^{n}\sum_{j=k+1}^{n}\left[\left|\underline{\Upsilon}_{ikj,\mu}\right|+\left| \overline{\tau}_{ikj,\mu}\right|+\left|\underline{\Upsilon}_{ikj,\
u}\right|+ \left|\overline{\tau}_{ikj,\
u}\right|\right] \tag{14}\]  
**Definition 7**. : _For an IVIFPR \(\widetilde{R}=\left(\widetilde{r}_{ij}\right)_{n\times n}\), when \(p=+\infty\), the multiplicative consistency degree of an IVIFPR can be measured by Chebyshev distance, which is defined as_  
\[CD\left(\widetilde{R}\right)=\max_{1<i<k<j}\left\{\left|\underline{\Upsilon} _{i,j,\mu}\right|,\left|\overline{\tau}_{ikj,\mu}\right|,\left|\underline{ \Upsilon}_{ikj,\
u}\right|,\left|\overline{\tau}_{ikj,\
u}\right|\right\} \tag{15}\]  
_Considering the majority principle, the Hamming distance employed in Equation (14) is to check the multiplicative consistency degree of an IVIFPR \(\widetilde{R}\). In contrast, considering the minority principle, the Chebyshev distance adopted in Equation (15) is to verify the multiplicative consistency degree of an IVIFPR \(\widetilde{R}\)._  
**Definition 8**. : _Combining the \(HD\left(\widetilde{R}\right)\) with \(CD\left(\widetilde{R}\right)\) the multiplicative consistency index (MCI) of an IVIFPR \(\widetilde{R}\) is defined as_  
\[\mathrm{MCI}\left(\widetilde{R}\right)=\varphi HD\left(\widetilde{R}\right)+ \left(1-\varphi\right)CD\left(\widetilde{R}\right) \tag{16}\]  
_where parameter \(\varphi\)\((0\leq\varphi\leq 1)\) denotes DMs' risk attitude.