## 5 Illustrative Example Results and Discussion  
### Numerical Example  
There is a city that was affected by fog-haze for a long time, and the scientists found that there are four main influence factors \(x_{1},x_{2},x_{3},x_{4}\) for this city's fog-haze. In order to determine the most important influence factor and rank these factors for fog-haze, a group scientist compares these four factors with each other and then provides the following preference information, HMPR \(P=\left(p_{ij}\right)_{4\times 4}\)[23]:  
\[P=\left( \begin{array}{cccc}\{1\}&\{1/7,1/6,1/5\}&\{1,3\}&\{1/8,2/7\}\\ \{5,6,7\}&\{1\}&\{1,2\}&\{7,9\}\\ \{1/3,1\}&\{1/2,1\}&\{1\}&\{1/4\}\\ \{7/2,8\}&\{1/9,1/7\}&\{4\}&\{1\}\end{array} \right).\]  
Now, we apply this paper's Algorithms 3 and 4 respectively to select the most important factor for fog-haze. ```
Step 1. Let \(t=0\),\(P^{(t)}=P\), \(\delta_{0}=0.3\) and \(\theta=0.1\) Step 2. By Equation (6), we obtain the priority vector: \[\widetilde{w}^{(0)}=\left(0.0934,0.5512,0.1090,0.2464\right)^{\text{T}}.\]
```  
**Algorithm 3**The consistency adjusting process of HMPR based on logarithm least squares model ```
Step 3. By Equation (3), we determine the consistency index \(CI(P^{(0)})=0.7573\) Step 4. As \(CI(P^{(0)})=0.7573>\delta_{0}\), then we utilize Step 5 in Algorithm 1 to repair the consistency of the HMPR \(P^{(0)}\) and derive the new HMPR \(P^{(1)}\): \(P^{(1)}=\left( \begin{array}{cccc}\{1\}&\{0.1453,0.1669,0.1967\}&\{0.9847,2.64 67\}&\{0.1397,0.2939\}\\ \{5.0839,5.9916,6.8823\}&\{1\}&\{1.1759,2.1944\}&\{6.2452,7.8302\}\\ \{0.3778,1.0155\}&\{0.4557,0.8504\}&\{1\}&\{0.2647\}\\ \{3.4025,7.1581\}&\{0.1277,0.1601\}&\{3.7779\}&\{1\}\end{array} \right)\) Step 5. By Equation (3), we determine the consistency index \(CI(P^{(0)})=0.7573\). Because \(CI(P^{(1)})=0.6133>\delta_{0}\), then by using Step 5 in Algorithm 1, we have \(P^{(2)}=\left( \begin{array}{cccc}\{1\}&\{0.1476,0.1672,0.1938\}&\{0.9711,2.3 645\}&\{0.1544,0.3015\}\\ \{5.1601,5.9824,6.7772\}&\{1\}&\{1.3606,2.3855\}&\{5.6358,6.9081\}\\ \{0.4229,1.0297\}&\{0.4192,0.7350\}&\{1\}&\{0.2786\}\\ \{3.3170,6.4781\}&\{0.1448,0.1774\}&\{3.5888\}&\{1\}\end{array} \right)\) Step 6. By Equation (3), we obtain the consistency index \(CI(P^{(2)})=0.49688>\delta_{0}\), then according to Step 5 in Algorithm 1, we have \(P^{(3)}=\left( \begin{array}{cccc}\{1\}&\{0.1497,0.1674,0.1912\}&\{0.9591,2.1 363\}&\{0.1689,0.3089\}\\ \{5.2296,5.9727,6.6820\}&\{1\}&\{1.5515,2.5716\}&\{5.1383,6.1714\}\\ \{0.4681,1.0427\}&\{0.3889,0.6445\}&\{1\}&\{0.2918\}\\ \{3.2416,5.9202\}&\{0.1620,0.1946\}&\{3.4272\}&\{1\}\end{array} \right)\)  Applying Equation (6), the priority vector of HMPR \(P^{(3)}\) can be determined : \(\tilde{w}^{(3)}=(0.0883,0.4437,\)\(0.1908,0.2772)^{\mathrm{T}}\). Step 7. We calculate the consistency index \(CI(P^{(3)})=0.2923\). Step 8. Since \(CI(P^{(3)})<\delta_{0}\), then the iteration stops, and \(P^{(3)}\) is acceptable consistent HMPR. Step 9. Output \(\tilde{w}^{(3)}=\left(0.0883,0.4437,0.1908,0.2772\right)^{\mathrm{T}}\). As \(w^{(3)}_{2}>w^{(3)}_{4}>w^{(3)}_{1}>w^{(3)}_{3}\), then we have \(x_{2}\succ x_{4}\succ x_{1}\succ x_{3}\). Therefore, the most important factor for \(\mathrm{fog-haze}\) is \(x_{2}\). ```

**Algorithm 4**The consistency adjusting process of HMPR based on linear optimization model

``` Step 1\({}^{\prime}\). Let \(t=0,\tilde{P}^{(t)}=P,\delta_{0}=0.3\) and \(\theta=0.1\). Step 2\({}^{\prime}\). Using a model (M-4), we get the optimal deviation values \(\widehat{d}^{+(0)}_{23}=0.8124,\widehat{d}^{-(0)}_{23}=1.0245,\widehat{d}^{+ (0)}_{24}=1.4958,\widehat{d}^{-(0)}_{12}=\widehat{d}^{-(0)}_{12}=\widehat{d}^{- (0)}_{14}=\widehat{d}^{+(0)}_{14}=\widehat{d}^{-(0)}_{23}=\widehat{d}^{-(0)}_{ 24}=\widehat{d}^{-(0)}_{34}=\widehat{d}^{+(0)}_{34}=0\), and the priority weight vector can be obtained as follows: \(\tilde{w}^{(0)}=\left(0.0745,0.4695,0.0994,0.3566\right)^{\mathrm{T}}\). Step 3\({}^{\prime}\). Utilizing Equation (3) to get the consistency index \(CI(\tilde{P}^{(0)})=0.7033\). Step 4\({}^{\prime}\).