## 5 Illustrative Example Results and Discussion
### Discussions  
In what follows, we utilize the Algorithm I proposed by [PERSON] and [PERSON] [28] to cope with the aforementioned problem, and then the following steps are involved:  
**Step 1\({}^{\prime\prime}\).** Let optimized parameter \(\xi=1\), then we obtain the normalized HMPR \(\overline{p}=\left(\overline{p}_{ij}\right)_{4\times 4}\) as follows:  
\[\overline{p}=\left(\begin{array}{cccc}\{1\}&\{1/7,1/6,1/5\}&\{1,3,3\}&\{1/8,2 /7,2/7\}\\ \{5,6,7\}&\{1\}&\{1,2,2\}&\{7,9,9\}\\ \{1/3,1/3,1\}&\{1/2,1/2,1\}&\{1\}&\{1/4,1/4,1/4\}\\ \{7/2,7/2,8\}&\{1/9,1/9,1/7\}&\{4,4,4\}&\{1\}\end{array}\right).\]  
**Step 2\({}^{\prime\prime}\).** Utilize Equation (21) in Zhang and Wu [28] to construct the MPRs \(A_{\rm s}(s=1,2,3)\) from \(\overline{p}=\left(\overline{p}_{ij}\right)_{4\times 4}\):  
\[A_{1}=\left(\begin{array}{cccc}1&1/7&1&1/8\\ 7&1&1&7\\ 1&1&1&1/4\\ 8&1/7&4&1\end{array}\right),A_{2}=\left(\begin{array}{cccc}1&1/6&3&2/7\\ 6&1&2&9\\ 1/3&1/2&1&1/4\\ 7/2&1/9&4&1\end{array}\right),A_{3}=\left(\begin{array}{cccc}1&1/5&3&2/7\\ 5&1&2&9\\ 1/3&1/2&1&1/4\\ 7/2&1/9&4&1\end{array}\right).\]  
**Step 3\({}^{\prime\prime}\).** Acceptable consistency of \(A_{\rm s}(s=1,2,3)\) is checked by Algorithm I in Xu and Wei [17]. Due to vast amount of computation, we would not list the iterative calculation process of adjusting the consistency of \(A_{\rm s}(s=1,2,3)\). After six iterations of Algorithm I in Xu and Wei [17], any of \(A_{\rm s}^{(6)}(s=1,2,3)\) is acceptably consistent.  
**Step 4\({}^{\prime\prime}\).** Employ Equation (28) in Zhang and Wu [28] to obtain the weight vectors of \(A_{\rm s}^{(6)}(s=1,2,3)\):  
\[W(A_{1}^{(6)}) = \left(0.4557,2.7600,0.4677,1.4346\right)^{\rm T},\] \[W(A_{2}^{(6)}) = \left(0.3404,3.2789,0.6613,1.0911\right)^{\rm T},\] \[W(A_{3}^{(6)}) = \left(0.6185,2.9184,0.7418,1.3343\right)^{\rm T}.\]  
**Step 5\({}^{\prime\prime}\).** By Equation (31) in Zhang and Wu [28], we obtain the interval weight vector of \(P=\left(p_{ij}\right)_{4\times 4}\) as follows:  
\[w_{1}=[0.3404,0.6185],w_{2}=[2.7600,3.2489],w_{3}=[0.4677,0.7418],w_{4}=[1.091 1,1.4346].\]  
**Step 6\({}^{\prime\prime}\).** Compute the degree of possibility of \(w_{i}\geq w_{j}(i,j=1,2,3,4)\) by Equation (32) in Zhang and Wu [28], we have \(w_{2}>w_{4}>w_{3}>w_{1}\), and then the ranking of the four main influence factors is \(x_{2}\succ x_{4}\succ x_{1}\succ x_{3}\). Therefore, the most important factor for fog-haze is \(x_{2}\).  
From the above numerical example and comparison with Algorithm I in [PERSON] and [PERSON] [28], the proposed decision-making algorithms have the following characteristics:  
(1) According to the above decision-making process, it is observed that our algorithms and [PERSON] and [PERSON]'s [28] approach produce the same ranking of the four influence factors for fog-haze, which means that our algorithms are reasonable.  
(2) It is clear that the decision-making process in our approaches would be more straight-forward and efficient than the Algorithm 1 proposed by [PERSON] and [PERSON] [28]. In fact, in the process of consistency-improving, our approaches utilize the original HMPR information provided by DMs and all the calculations directly using the HMEs to produce results, which can preserve the original information of DMs. However, with the [PERSON] and [PERSON]'s [28] method, one must transfer the original HMPR given by the DMs into its corresponding MPRs, therefore it seems to be an indirect computation process. Meanwhile, in the process of obtaining the interval weight vector, it may be derived the same interval weights for different alternatives when the number of alternatives is too large, which leads to the original information losses.  
(3) Our approaches investigate effective methods to improve the consistency of HMPRs, so that the improved HMPRs are acceptably consistent. Furthermore, in order to obtain the acceptably consistent HMPRs (or MPRs), the required number of iterations by our algorithms are less than [PERSON] and [PERSON]'s [28] approach.