## 3 Hesitant Multiplicative Preference Relations and Consistency Index  
In what follows, inspired by MPR and score function of HME, we define a new version of HMPR, and then the consistency of HMPR and consistency index of HMPR are presented.  
Definition 5: An HMPR \(P\) on \(X\) can be defined as reciprocal matrix \(P=\left(p_{ij}\right)_{n\times n}\subset X\times X\), where \(p_{ij}\) is an HME, which indicates the possible preference degrees of alternative \(x_{i}\) over \(x_{j}\), and it satisfies  
\[f(p_{ij})\cdot f(p_{ji})=1,p_{ii}=\{1\},\left|p_{ij}\right|=\left|p_{ji}\right|,i,j\in N \tag{1}\]  
where \(f(p_{ij})\) and \(\left|p_{ij}\right|\) are the score function of \(p_{ij}\) and the number of values in \(p_{ij}\), respectively.  
For a HMPR \(P=\left(p_{ij}\right)_{n\times n}\), since \(f(p_{ij})=\left(\prod_{k=1}^{|p_{ij}|}\gamma_{ij,k}\right)^{\
icefrac{{1}}{{ p_{ij}}}}\geq\left((1/9)^{|p_{ij}|}\right)^{\
icefrac{{1}}{{p_{ij}}}}=1/9\) and \(f(p_{ij})=\left(\prod_{k=1}^{|p_{ij}|}\gamma_{ij,k}\right)^{\
icefrac{{1}}{{ p_{ij}}}}\leq\left(9^{|p_{ij}|}\right)^{\
icefrac{{1}}{{p_{ij}}}}=9\), then \(f(p_{ij})\in[1/9,9],i,j\in N\). On the other hand, from Definition 5, one can obtain that \(f(p_{ij})\cdot f(p_{ji})=1,i,j\in N\). Therefore, by using score function, we can transform the HMPR \(P\) into an MPR \(F=\left(f_{ij}\right)_{n\times n}\), where \(f_{ij}=f(p_{ij}),\forall i,j\in N\).  
Therefore, the following consistency of HMPR is introduced.  
Definition 6: Assume that \(P=\left(p_{ij}\right)_{n\times n}\) is an HMPR, where \(p_{ij}\) is HME, then \(P=\left(p_{ij}\right)_{n\times n}\) is called consistent HMPR, if there exists a normalized crisp weight vector \(w=\left(w_{1},w_{2},\cdots,w_{n}\right)^{\mathsf{T}}\), such that  
\[f(p_{ij})=w_{i}/w_{j},\forall i,j\in N \tag{2}\]  
where \(w_{i}>0,\forall i\in N,\sum_{i=1}^{n}w_{i}=1\).  
From Equation (2), we have \(\ln f(p_{ij})=\ln w_{i}-\ln w_{j},\forall i,j\in N\). However, for the HMPR provided by DMs, it is difficult to satisfy the consistency, and then Equation (2) cannot hold, it means that there exist \(i,j\in N\), such that \(\ln f(p_{ij})\
eq\ln w_{i}-\ln w_{j}\), then we can use \(\left(\ln f(p_{ij})-\left(\ln w_{i}-\ln w_{j}\right)\right)^{2}=\left(\ln f(p_ {ij})-\ln w_{i}+\ln w_{j}\right)^{2}\) to measure the deviation between \(\ln f(p_{ij})\)and \(\ln w_{i}-\ln w_{j}\). Therefore, the values of \(\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)^{2}(i,j\in N)\) can be used to measure the consistency level of the HMPR \(P=\left(p_{ij}\right)_{n\times n}\).  
**Definition 7**.: _Assume that \(P=\left(p_{ij}\right)_{n\times n}\) is an HMPR, \(w=\left(w_{1},w_{2},\cdots,w_{n}\right)^{T}\) is the priority weight vector derived from \(P\) satisfying \(w_{i}>0,\forall i\in N,\sum_{i=1}^{n}w_{i}=1\), then the consistency index of \(P\) is defined as_  
\[CI(P)=\frac{2}{n(n-1)}{\sum_{i<j}}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j} \right)^{2} \tag{3}\]  
The smaller is the value of \(CI(P)\), the better is the consistency of HMPR \(P\). If \(CI(P)=0\), then \(P\) is consistent. If we provide the threshold \(\delta_{0}\) and \(CI(P)\leq\delta_{0}\), then \(P\) is called of acceptable consistency.