## 4 Consistency Repaired Methods for an HMPR
** Suppose that \(\widetilde{w}^{(t)}=\left(\widetilde{w}_{1}^{(t)},\widetilde{w}_{2}^{(t)}, \cdots,\widetilde{w}_{n}^{(t)}\right)^{\mathrm{T}}\) is the priority weight vector of \(P^{(t)}\) for each \(t\), from the above analysis, we know that \(\widetilde{w}^{(t)}=\left(\widetilde{w}_{1}^{(t)},\widetilde{w}_{2}^{(t)}, \cdots,\widetilde{w}_{n}^{(t)}\right)^{\mathrm{T}}\) also is the optimal weight vector by solving model (M-2) for \(P^{(t)}\). Thus, we have  
\[\sum_{i<j}\left(\ln f(p_{ij}^{(t+1)})-\ln\widetilde{w}_{i}^{(t+1)}+\ln \widetilde{w}_{j}^{(t+1)}\right)^{2}\leq\sum_{i<j}\left(\ln f(p_{ij}^{(t+1)})- \ln w_{i}^{(t+1)}+\ln w_{j}^{(t+1)}\right)^{2} \tag{8}\]  
Let \(w_{i}^{(t+1)}=\widetilde{w}_{i}^{(t)}\), \(i\in N\), then we have  
\[\sum_{i<j}\left(\ln f(p_{ij}^{(t+1)})-\ln\widetilde{w}_{i}^{(t+1)}+\ln \widetilde{w}_{j}^{(t+1)}\right)^{2}\leq\sum_{i<j}\left(\ln f(p_{ij}^{(t+1)})- \ln\widetilde{w}_{i}^{(t)}+\ln\widetilde{w}_{j}^{(t)}\right)^{2} \tag{9}\]  
In addition, according to Step 5 in Algorithm 1, we have  
\[\ln f(p_{ij}^{(t+1)})=\left(\prod\
olimits_{k=1}^{\left|p_{ij}^{ (t+1)}\right|}\gamma_{ij,k}^{(t+1)}\right)^{1/\left|p_{ij}^{(t+1)}\right|}= \ln\left(\prod\
olimits_{k=1}^{\left|p_{ij}^{(t+1)}\right|}\left(\left(\gamma_ {ij,k}^{(t)}\right)^{1-\theta}\cdot\left(\widetilde{w}_{i}^{(t)}/\widetilde{w }_{j}^{(t)}\right)^{\theta}\right)\right)^{1/\left|p_{ij}^{(t+1)}\right|}\] \[=\ln\left(\left(\left(\prod\
olimits_{k=1}^{\left|p_{ij}^{(t)} \right|}\gamma_{ij,k}^{(t)}\right)^{1/\left|p_{ij}^{(t)}\right|}\right)^{1- \theta}\cdot\left(\widetilde{w}_{i}^{(t)}/\widetilde{w}_{j}^{(t)}\right)^{ \theta}\right)=(1-\theta)\ln f(p_{ij}^{(t)})+\theta(\ln\widetilde{w}_{i}^{(t )}-\ln\widetilde{w}_{j}^{(t)}).\]Therefore,  
\[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]0 \[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]1 \[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]2 \[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]3 \[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]4  
i.e., \(CI(P^{(t+1)})=(1-\theta)CI(P^{(t)})<CI(P^{(t)})\) for each \(t\). Furthermore, on the one hand, according to Equation (10), we get that  
\[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]5  
On the other hand, it is obvious that \(CI(P^{(t)})\geq 0\), hence \(\underset{t\rightarrow\infty}{\lim}CI(P^{(t)})=0\).\(\square\)  
From Definition 6, if HMPR \(P=\left(p_{ij}\right)_{n\times n}\) is consistent, then Equation (2) holds. Hence, it can be rewritten as \(\ln f(p_{ij})=\ln w_{i}-\ln w_{j},\forall i,j\in N\). However, in many real situations, due to fuzziness and uncertainty, the HMPR provided by DMs is usually inconsistent, thus Equation (2) cannot hold, i.e., there exist \((i,j)\in N\times N\), such that \(\ln f(p_{ij})\
eq\ln w_{i}-\ln w_{j}\). In this case, some non-negative deviation variables \(d_{ij}^{-}\) and \(d_{ij}^{+},d_{ij}^{-}\cdot d_{ij}^{+}=0,i,j\in N\) are introduced, such that  
\[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]6  
The smaller is the value of deviation variables \(d_{ij}^{-}\) and \(d_{ij}^{+}\), the better is the consistency of HMPR. Therefore, we develop a linear optimization model to derive the smallest deviation variables and priority weight vector as follows:  
\[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]7 \[s.t.\left\{ \begin{array}{c}\ln f(p_{ij})+d_{ij}^{-}-d_{ij}^{+}= \ln w_{i}-\ln w_{j},i,j\in N,\\ \sum_{i=1}^{n}w_{i}=1,w_{i}>0,i\in N.\end{array} \right.\]  
From Definition 5 and Equation (11), one can obtain that  
\[d_{ij}^{-}-d_{ij}^{+} =\ln w_{i}-\ln w_{j}-\ln f(p_{ij})=-(\ln w_{j}-\ln w_{i}-\ln(1/f(p _{ij})))\] \[=-(\ln w_{j}-\ln w_{i}-\ln f(p_{ji}))=-(d_{ji}^{-}-d_{ji}^{+})=d_ {ji}^{+}-d_{ji}^{-},\]  
i.e., \(d_{ij}^{-}+d_{ji}^{-}=d_{ij}^{+}+d_{ji}^{+}\). As \(d_{ij}^{-},d_{ij}^{+}\geq 0\) and \(d_{ij}^{-}\cdot d_{ij}^{+}=0,i,j\in N\), it follows that \(d_{ij}^{-}=d_{ji}^{+},d_{ij}^{+}=d_{ji}^{-},i,j\in N\).