## 3 Acceptability Measurement of an IVIFPR
### An Optimization Model to Obtain the Acceptable Multiplicative Consistent IVIFPR
\]  
Then Model (17) is converted into:  
\[\min\sum_{i=1}^{n}\sum_{j=i+1}^{n}\Big{(}s_{ij}^{-}+t_{ij}^{-}+s_{ij}^{+}+t_{ ij}^{+}+p_{ij}^{-}+q_{ij}^{-}+p_{ij}^{+}+q_{ij}^{+}\Big{)}\]  
\[\begin{array}{l}\left\{\begin{array}{l}\underline{\varphi}\\ \frac{\varphi}{4\epsilon_{ij}^{n}}\sum_{i=1}^{n}\sum_{k=i+1}^{n}\sum_{j=k+1}^{n} \left( \begin{array}{c}\alpha_{ikj}^{-}+\alpha_{ikj}^{+}+\beta_{ikj}^{-}+\beta_ {ikj}^{+}\\ +\gamma_{ikj}^{-}+\gamma_{ikj}^{+}+\delta_{ikj}^{-}+\delta_{ikj}^{+}\\ \end{array} \right)+(1-\varphi)\varepsilon\leq\text{MCI}_{0}\\ \varepsilon\geq\alpha_{ikj}^{-}+\alpha_{ikj^{\prime}}^{+}\varepsilon\geq\beta_ {ikj}^{-}+\beta_{ikj^{\prime}}^{+}\varepsilon\geq\gamma_{ikj}^{-}+\gamma_{ikj^ {\prime}}^{+}\varepsilon\geq\delta_{ikj}^{-}+\delta_{ikj}^{+}(i,j,k=1,2,\ldots, n,i<k<j).\\ s_{ij}^{-}-t_{ij}^{-}=\underline{\mu}_{ij}-\underline{\mu}_{jj^{\prime}}\)\(s_{ij}^{+}-t_{ij}^{+}=\overline{\mu}_{ij}-\overline{\mu}_{ij}(i,j=1,2,\ldots,n,i<j)\\ p_{ij}^{-}-q_{ij}^{-}=\underline{\
u}_{ij}-\underline{\phi}_{ij}\)\(p_{ij}^{+}-q_{ij}^{+}=\overline{\
u}_{ij}-\overline{\
u}_{ij}(i,j=1,2,\ldots,n,i<j)\\ \alpha_{ikj}^{-}-\alpha_{ikj}^{+}=ln\underline{\theta}_{ij}+ln\Big{(}1- \underline{\theta}_{ik}\Big{)}+ln\Big{(}1-\underline{\theta}_{kj}\Big{)}-ln \Big{(}1-\underline{\theta}_{ij}\Big{)}-ln\Big{(}1-\underline{\theta}_{ij} \Big{)}-ln\underline{\theta}_{ik}-ln\underline{\theta}_{kj}\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,\ i<k<j)\\ \beta_{ikj}^{-}-\beta_{ikj}^{+}=ln\overline{\pi}_{ij}+ln\big{(}1-\overline{ \pi}_{ik}\big{)}+ln\Big{(}1-\overline{\pi}_{kj}\Big{)}-ln\Big{(}1-\overline{ \pi}_{ij}\big{)}-ln\overline{\pi}_{ik}-ln\overline{\pi}_{kj}\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,\ i<k<j)\\ \gamma_{ikj}^{-}-\gamma_{ikj}^{+}=ln\underline{\theta}_{ij}+ln\big{(}1- \underline{\phi}_{ik}\big{)}+ln\Big{(}1-\underline{\phi}_{kj}\Big{)}-ln\Big{(}1 -\underline{\phi}_{ij}\Big{)}-ln\underline{\phi}_{ik}-ln\underline{\phi}_{kj} \\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,\ i<k<j)\\ \delta_{ikj}^{-}-\delta_{ikj}^{+}=ln\overline{\phi}_{ij}+ln\big{(}1-\overline{ \phi}_{ik}\big{)}+ln\Big{(}1-\overline{\phi}_{kj}\Big{)}-ln\big{(}1-\overline{ \phi}_{ij}\big{)}-ln\overline{\phi}_{ik}-ln\overline{\phi}_{kj}\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k>j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k>j)\\ \hskip 142.26378 pt(i,j,k=1,2,\ldots,n,i<k<j)\\ \hskip 142. * Using Equations (14) and (15), obtain \(HD\Big{(}\widetilde{Q}\Big{)}=1.5890\) and \(CD\Big{(}\widetilde{Q}\Big{)}=2.7726\). Then, by Equation (16), calculate the multiplicative consistency index \(\text{MCI}\Big{(}\widetilde{Q}\Big{)}=2.1801\). Since \(\text{MCI}\Big{(}\widetilde{Q}\Big{)}>\text{MCI}_{0}\), the IVIFPR \(\widetilde{Q}\) is unacceptable multiplicative consistency index and go to next step. * Using Model (18), an optimization model is constructed as follows: \[\min\Bigg{(}\begin{array}{c}s_{12}^{-}+t_{12}^{-}+s_{12}^{+}+t_{12}^{+}+p_{1 2}^{-}+q_{12}^  
\[\underline{\mu}_{ij}\leq\underline{r}_{ij}\leq 1-\overline{v}_{ij},\ \overline{\mu}_{ij}\leq \overline{r}_{ij}\leq 1-\underline{v}_{ij},\ \underline{r}_{ij}+\overline{r}_{ji}=1\ \text{and}\ \underline{r}_{ii}= \overline{r}_{ii}=0.5\ \text{for all}\ i,j=1,2,\ldots,n.\]  
To simplify calculation, in what follows, two special extractive IVFPRs are participated in determining priority weights of alternatives for an IVIFPR. Let \(\left[\underline{\mu}_{ij},\overline{\mu}_{ij}\right]\leq\left[\underline{t} _{ij},\overline{r}_{ij}\right]\leq\left[1-\overline{v}_{ij},1-\underline{v}_{ ij}\right]\), two special IVFPRs \(R^{\mu}=\left(r_{ij}^{\mu}\right)_{n\times n}\) and \(R^{v}=\left(r_{ij}^{v}\right)_{n\times n}\) are extracted from the IVFPR \(\widetilde{R}=\left(\overline{r}_{ij}\right)_{n\times n}\), respectively, where  
\[r_{ij}^{\mu}=\left\{