## 4 Consistency Repaired Methods for an HMPR  
Motivated by the logarithmic least squares model [31], the priority weight vector can be derived by using the following optimization model:  
\[ \begin{array}{l}(\mathbf{M-1})\ \min\!J_{1}=\sum_{i,j=1}^{n}\left(\ln f(p_{ ij})-\ln w_{i}+\ln w_{j}\right)^{2}\\ s.t.\ \ \sum_{i=1}^{n}w_{i}=1,w_{i}>0,i\in N.\end{array}  \tag{4}\]  
In fact, since \(f(p_{ij})\cdot f(p_{ji})=1,i,j\in N\), then we have  
\[ \begin{array}{l}J_{1}=\sum_{i<j}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j} \right)^{2}+\sum_{i>j}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)^{2}\\ =\sum_{i<j}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)^{2}+\sum_{i>j}\left( -\left(-\ln f(p_{ij})+\ln w_{i}-\ln w_{j}\right)\right)^{2}\\ =\sum_{i<j}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)^{2}+\sum_{j<i}\left( \ln f(p_{ji})-\ln w_{j}+\ln w_{i}\right)^{2}\\ =2\sum_{i<j}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)^{2},\end{array} \]  
thus, the developed optimization model (M-1) can be converted into the following optimization model:  
\[ \begin{array}{l}(\mathbf{M-2})\ \min\!J_{1}=\sum_{i<j}\left(\ln f(p_{ ij})-\ln w_{i}+\ln w_{j}\right)^{2}\\ s.t.\ \ \sum_{i=1}^{n}w_{i}=1,w_{i}>0,i\in N.\end{array}  \tag{5}\]  
According to Definition 6, we know that an HMPR \(P=\left(p_{ij}\right)_{n\times n}\) is consistent, then there exists a normalized crisp weight vector \(w=\left(w_{1},w_{2},\cdots,w_{n}\right)^{T}\), such that \(f(p_{ij})=w_{i}/w_{j},\forall i,j\in N\), i.e., \(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}=0,i,j\in N\), then we have  
\[\sum_{j=1}^{n}\left(\ln f(p_{ij})-\ln w_{i}+\ln w_{j}\right)=0,\]  
thus  
\[\sum_{j=1}^{n}\ln f(p_{ij})-n\ln w_{i}+\sum_{j=1}^{n}\ln w_{j}=0.\]  
That is  
\[\ln w_{i}=\frac{1}{n}{\sum_{j=1}^{n}\ln w_{j}}+\frac{1}{n}{\sum_{j=1}^{n}\ln f (p_{ij})},\]  
i.e.,  
\[w_{i}=k\cdot e^{\frac{1}{n}{\sum_{j=1}^{n}\ln f(p_{ij})}}=k\cdot\left(\prod_{ j=1}^{n}f(p_{ij})\right)^{1/n},\]where \(k=\left(\prod_{j=1}^{n}w_{j}\right)^{1/n}\). Since \(\sum_{i=1}^{n}w_{i}=1\), then we have \(k=1/\sum_{i=1}^{n}\left(\prod_{j=1}^{n}f(p_{ij})\right)^{1/n}\), one can get that  
\[w_{i}=\left(\prod\
olimits_{j=1}^{n}f(p_{ij})\right)^{1/n}/\sum_{i=1}^{n} \left(\prod\
olimits_{j=1}^{n}f(p_{ij})\right)^{1/n} \tag{6}\]  
Therefore, the following Algorithm 1 is designed to be adjusted the consistency of the HMPR \(P\):  
```
Step 1. Suppose that \(P^{(t)}=\left(p_{ij}^{(t)}\right)_{n\times n}=P=\left(p_{ij}\right)_{n\times n}\) and \(t=0\), and pre-set the threshold \(\delta_{0}\), the controlling parameter \(\theta\) and the maximum number of iterations \(t_{\max}\); Step 2. Derive the priority vector \(\widehat{w}^{(t)}=\left(\widehat{w}_{1}^{(t)},\widehat{w}_{2}^{(t)},\cdots, \widehat{w}_{n}^{(t)}\right)^{\mathrm{T}}\) by Equation (6); Step 3. Determine the consistency index \(CI(P^{(t)})\) by using Equation (3); Step 4. If \(CI(P^{(t)})\leq\delta_{0}\) or \(t>t_{\max}\), then go to Step 7. Otherwise, go to Step 5; Step 5. Let \(P^{(t+1)}=\left(p_{ij}^{(t+1)}\right)_{n\times n}\), where \(p_{ij}^{(t+1)}=\left\{\gamma_{ij,k}^{(t+1)}\middle|k=1,2,\cdots,\left|p_{ij}^{ (t+1)}\middle|\right\},\gamma_{ij,k}^{(t+1)}=\left(\gamma_{ijk}^{(t)}\right)^{1 -\theta}\cdot\left(\widetilde{w}_{i}^{(t)}/\widetilde{w}_{j}^{(t)}\right)^{\theta}\) Step 6. Let \(t=t+1\) and return to Step 2; Step 7. Output \(P^{(t)}\), \(\widehat{w}^{(t)},CI(P^{(t)})\) and \(t\); Step 8. End. ```  
**Algorithm 1**The consistency adjusting process of HMPR based on logarithm least squares model  
In the following, we prove that the developed Algorithm 1 is convergent. **Theorem 1**. : _Let \(P=\left(p_{ij}\right)_{n\times n}\) be an HMPR, \(\theta(0<\theta<1)\) be the adjusted parameter, \(\left\{P^{(t)}\right\}\) be a collection of HMPRs in Algorithm 1. If \(CI(P^{(t)})\) is the consistency index of \(P^{(t)}\), then we have_  
\[CI(P^{(t+1)})<CI(P^{(t)})\text{ for each }t,\text{ and }\lim_{t\to\infty}CI(P^{(t)})=0 \tag{7}\]  
**Proof.