## 3 Acceptability Measurement of an IVIFPR
### An Error-Analysis-Based Extension Method for Determining the IVIF Priority Weights
\[w_{i}^{v}=\frac{1}{n(n-1)}\left(\frac{1}{2}\sum\
olimits_{j=1}^{n}\left(2- \overline{v}_{ij}-\underline{v}_{ij}\right)+\frac{n}{2}-1\right)\text{, for }i=1,2,\ldots,n \tag{30}\]  
\[m_{ij}+m_{ji}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij} \right)+\frac{1}{2}\left(1-\overline{\mu}_{ij}+1-\underline{\mu}_{ij}\right) =1,\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{22}\]0  
Due to the interval-valued priority vector \(w^{\mu}\) and \(w^{v}\) refer to the initial IVIFPR. An IVIFPR should give an IVIFPR weight estimate [9]. Hence, assume that \(w=\left(w_{1},w_{2},\ldots,w_{n}\right)^{T}\) with \(w_{i}=\left(\left[w_{i}^{\mu-},w_{i}^{\mu+}\right],\left[w_{i}^{v-},w_{i}^{v+} \right]\right)\) for \(i=1,2,\ldots,n\) is an IVIF priority weight vector, \(\left[w_{i}^{\mu-},w_{i}^{\mu+}\right]\) and \(\left[w_{i}^{v-},w_{i}^{v+}\right]\) can be interpreted as degree ranges of the membership and the non-membership for an IVIFPR, respectively. Similar to [PERSON] et al. [9], the interval-valued priority vector \(w^{\mu}\) and \(w^{v}\) can be fused to build the normalized one as follows:  
\[m_{ij}+m_{ji}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij} \right)+\frac{1}{2}\left(1-\overline{\mu}_{ij}+1-\underline{\mu}_{ij}\right) =1,\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{22}\]1  
with  
\[m_{ij}+m_{ji}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij} \right)+\frac{1}{2}\left(1-\overline{\mu}_{ij}+1-\underline{\mu}_{ij}\right) =1,\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{22}\]2  
\[m_{ij}+m_{ji}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij} \right)+\frac{1}{2}\left(1-\overline{\mu}_{ij}+1-\underline{\mu}_{ij}\right) =1,\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{22}\]3where \(\lambda=max\Big{\{}max_{i=1,2,\ldots,n}\Big{\{}max\Big{\{}\underline{w}_{i}^{\mu}, \underline{w}_{i}^{v}\Big{\}}-min\Big{\{}\overline{w}_{i}^{\mu},\overline{w}_{ i}^{v}\Big{\}}+1\Big{\}},1\Big{\}}\), which can guarantee that the IVIF priority weight \(w_{i}(i=1,2,\ldots,n)\) satisfies \(w_{i}^{\mu+}+w_{i}^{v+}\leq 1\). Consequently, the IVIF priority weights \(w_{i}=\Big{(}\Big{[}w_{i}^{\mu-},w_{i}^{\mu+}\Big{]},\big{[}w_{i}^{v-},w_{i}^{v +}\big{]}\Big{)}(i=1,2,\ldots,n)\) are obtained from Equation (32). In the following, a numerical example is applied to illustrate how to obtain weight vector in terms of interval-valued intuitionistic fuzzy values. **Example 2**: _Continuing on from Example 1, an acceptable multiplicative consistent IVIFPR \(\hat{Q}\) can be obtained in Example 1, and then the priority weights with regard to IVIFV can be determined by the proposed error-analysis-based extension method._  
**Step 1**: Using Equation (20), two special IVFPRs \(Q^{\mu}\) and \(Q^{v}\) can be extracted from the acceptable multiplicative consistent IVIFPR \(\hat{Q}\), which are shown in the following:  
\[Q^{\mu}=\left( \begin{array}{ccc}[0.5000,0.5000]&[0.4000,0.6000]&[0.2030,0.364 4]\\ [0.4000,0.6000]&[0.5000,0.5000]&[0.3000,0.3000]\\ [0.6356,0.7970]&[0.7000,0.7000]&[0.5000,0.5000]\end{array} \right)\]  
\[Q^{v}=\left( \begin{array}{ccc}[0.5000,0.5000]&[0.6000,0.8000]&[0.5723,0.9412 ]\\ [0.2000,0.4000]&[0.5000,0.5000]&[0.5000,0.8000]\\ [0.0588,0.4277]&[0.2000,0.5000]&[0.5000,0.5000]\end{array} \right)\]
**Step 2**: Using Equations (27) and (29), the priority vector \(w^{\mu}=\left(w_{1},w_{2},w_{3}\right)^{T}\) with \(w_{i}^{\mu}=\Big{[}w_{i}^{\mu}-\Delta w_{i}^{\mu},w_{i}^{\mu}+\Delta w_{i}^{ \mu}\Big{]}(i=1,2,3)\) for the IVFPR \(Q^{\mu}\) can be determined as follows:  
\[w^{\mu}=\left([0.2759,0.3187],[0.2833,0.3167],[0.3893,0.4162]\right)^{T}\]  
Similarly, using Equations (30) and (31), the priority vector \(w^{v}=\left(w_{1},w_{2},w_{3}\right)^{T}\) with \(w_{i}^{v}=\left[w_{i}^{v}-\Delta w_{i}^{v},w_{i}^{v}+\Delta w_{i}^{v}\right](i= 1,2,3)\) for the IVFPR \(Q^{v}\) can be determined as follows:  
\[w^{v}=\left([0.3745,0.4444],[0.2950,0.3550],[0.2259,0.3052]\right)^{T}\]  
Thus, using Equation (32), the priority vector in terms of IVIFVs can be obtained as follows:  
\[