## 3 Acceptability Measurement of an IVIFPR
### An Error-Analysis-Based Extension Method for Determining the IVIF Priority Weights  
Inspired by [PERSON] [26], this subsection proposes an error-analysis-based extension method to determine the IVIF priority weights from two special extractive IVFPRs. Assume that IVFPR \(R^{\mu}=\left(r_{ij}^{\mu}\right)_{n\times n}\) with \(r_{ij}^{\mu}=\left[\underline{\mu}_{ij},\overline{\mu}_{ij}\right]\) for all \(i<j\) is extracted from IVIFPR \(\widetilde{R}\), which is the lowest preferred matrix. Let  
\[m_{ij}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij}\right),\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{21}\]  
We have \(m_{ij}\geq 0\) and  
\[m_{ij}+m_{ji}=\frac{1}{2}\left(\underline{\mu}_{ij}+\overline{\mu}_{ij} \right)+\frac{1}{2}\left(1-\overline{\mu}_{ij}+1-\underline{\mu}_{ij}\right) =1,\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{22}\]  
\[m_{ii}=\frac{1}{2}(0.5+0.5)=0.5,\ \text{for all}\ i,j=1,2,\ldots,n \tag{23}\]  
Then, we obtain an expected fuzzy preference relation \(M_{ij}=\left(m_{ij}\right)_{n\times n}\). Additionally, let  
\[h_{ij}=\frac{1}{2}\left(\overline{\mu}_{ij}-\underline{\mu}_{ij}\right),\ \text{for all}\ i,j=1,2,\ldots,n,i<j \tag{24}\]  
Then we get an error matrix \(H=\left(h_{ij}\right)_{n\times n}\), which has the following properties:  
\[h_{ij}\in\left[0,1\right],\ h_{ij}=h_{ji},\ h_{ii}=0,\ \text{for all}\ i,j=1,2, \ldots,n \tag{25}\]For any fuzzy preference relation \(B=\left(b_{ij}\right)_{n\times n}\) with \(b_{ij}\geq 0,b_{ij}+b_{ji}=1\), \(b_{ii}=0.5,i,j=1,2,\ldots,n\). To derive the priority vector \(w=\left(w_{1},w_{2},\ldots,w_{n}\right)^{T}\) of \(B\), a simple formula proposed in Xu [33] as following:  
\[w_{i}=\frac{1}{n(n-1)}\left(\sum\
olimits_{j=1}^{n}b_{ij}+\frac{n}{2}-1\right) \text{, for }i=1,2,\ldots,n \tag{26}\]  
where \(w_{i}\geq 0\), \(i=1,2,\ldots,n\) and \(\sum_{i=1}^{n}w_{i}=1\). Then, the priority vector \(w^{\mu}=\left(w_{1}^{\mu},w_{2}^{\mu},\ldots,w_{n}^{\mu}\right)^{T}\) of the expected fuzzy preference relation \(M=\left(m_{ij}\right)_{n\times n}\) can be obtained using Equations (26) and (21), where  
\[w_{i}^{\mu}=\tfrac{1}{n(n-1)}\left(\sum\
olimits_{j=1}^{n}m_{ij}+\tfrac{n}{2} -1\right)=\tfrac{1}{n(n-1)}\left(\tfrac{1}{2}\sum\
olimits_{j=1}^{n}\left( \underline{\mu}_{ij}+\overline{\mu}_{ij}\right)+\tfrac{n}{2}-1\right)\text{, for }i=1,2,\ldots,n \tag{27}\]  
Using Equations (10) and (25), the error priority vector \(\Delta w^{\mu}=\left(\Delta w_{1}^{\mu},\Delta w_{2}^{\mu},\ldots,\Delta w_{n }^{\mu}\right)^{T}\) can be obtained as  
\[\left(\Delta w_{i}^{\mu}\right)^{2}=\left(\tfrac{1}{n(n-1)}\right)^{2}\sum \
olimits_{j=1}^{n}\left(\Delta h_{ij}\right)^{2}=\left(\tfrac{1}{n(n-1)} \right)^{2}\sum\
olimits_{j=1}^{n}\left(\tfrac{1}{2}\left(\overline{\mu}_{ij }-\underline{\mu}_{ij}\right)\right)^{2}\text{, for }i=1,2,\ldots,n \tag{28}\]  
Solving Equation (28), we have  
\[\Delta w_{i}^{\mu}=\frac{1}{2n(n-1)}\sqrt{\sum\
olimits_{j=1}^{n}\left( \overline{\mu}_{ij}-\underline{\mu}_{ij}\right)^{2}}\text{, for }i=1,2,\ldots,n \tag{29}\]  
Thus, the priority vector \(w^{\mu}=\left(w_{1}^{\mu},w_{2}^{\mu},\ldots,w_{n}^{\mu}\right)^{T}\) with \(w_{i}^{\mu}=\left[w_{i}^{\mu}-\Delta w_{i}^{\mu},w_{i}^{\mu}+\Delta w_{i}^{\mu }\right]\) for the IVFPR \(R^{\mu}=\left(r_{ij}^{\mu}\right)_{n\times n}\) can be determined, which denotes the importance degree of alternative \(x_{i}\). Similarly, using Equations (27) and (29), the priority vector \(w^{v}=\left(w_{1}^{v},w_{2}^{v},\ldots,w_{n}^{v}\right)^{T}\) with \(w_{i}^{v}=\left[w_{i}^{v}-\Delta w_{i}^{v},w_{i}^{v}+\Delta w_{i}^{v}\right]\) for the IVFPR \(R_{v}=\left(r_{ij}^{p}\right)_{n\times n}\) can also be obtained, which also reflects the importance degree of alternative \(x_{i}\).