## 5 An Illustrative Example
### Conducting the Proposed Method
**  
The comprehensive weights vector can be obtained by combining the objective weights vector and the subjective weights vector according to Equation (16). It was \(\omega^{0}=[0.2103,\ 0.4069,\ 0.2119,\ 0.2404,\ 0.2405]\). **Step 4: Identify \(\lambda\)-fuzzy-measures. **  
The parameter \(\lambda\) is identified according to Equation (17), which was \(-0.5262\). The weights of all subsets of the power set of the criterion set can be calculated according to Equation (18), as shown in Table 5.  
Figure 6: The 3-dimensional histograms of \(C\). **Step 5: Calculate the grey fuzzy correlation degree. **  
**Step 5:1: Calculate correlation coefficients. **  
Select \(\{C^{+}\}=\ \{0.5763,0.9854,1.3439,1.3681,1.4790\}\) as the positive ideal solution scheme and \(\{C^{-}\}=\ \{-1.9837,-1.7970,-1.2968,-1.5178,-0.9849\}\) as the negative ideal solution scheme. According to Equations (21) and (22), the positive correlation matrix \(\xi^{+}\) and the negative correlation matrix \(\xi^{-}\) were found as follows. \[\xi^{+}=\left[ \begin{array}{cccc}0.9579&0.8924&0.6907&1.0000&1.0000\\ 0.3605&0.3415&0.4232&0.6035&0.6784\\ 0.9192&0.5424&0.3533&0.3333&0.3693\\ 0.6914&0.5803&1.0000&0.6383&0.5353\\ 1.0000&1.0000&0.4232&0.4391&0.3842\\ 0.9579&0.6747&0.6310&0.4572&0.4004\end{array} \right],\]  
\[\xi^{-}=\left[ \begin{array}{cccc}0.3663&0.3562&0.4198&0.3333&0.3693\\ 1.0000&1.0000&0.6815&0.4268&0.4477\\ 0.3723&0.4797&1.0000&1.0000&1.0000\\ 0.4296&0.4535&0.3533&0.4110&0.5437\\ 0.3605&0.3415&0.6815&0.5806&0.9050\\ 0.3663&0.4088&0.4454&0.5517&0.8265\end{array} \right].\]  
**Step 5.2: Calculated grey fuzzy Choquet integral. **  
The positive and negative grey fuzzy integral vectors can be obtained by applying Equations (23) and (24) on the correlation matrix and the comprehensive weights vector which, respectively, were \(R^{+}=[1.0000,\,0.5314,\,0.5824,\,0.7486,\,0.8085,\,0.7103]\) and \(R^{-}=[0.4665,\,1.0000,\,0.9881,\,0.5708,\,0.7381,\,0.6792]\). **Step 6: Calculate fuzzy distance to ideal solution. **  
The positive and negative ideal solution distance vectors can be calculated by applying Equations (25), (26) on the standardized evaluation matrix and the comprehensive weights vector, which, respectively, were \(D^{+}=[0.1119,\,0.7536,\,0.7771,\,0.3408,\,1.0000,\,0.9583]\) and \(D^{-}=[0.4690,\,0.3669,\,0.4180,\,0.4585,\,1.0000,\,0.9597]\). **Step 7: Calculate the comprehensive proximity degree. **  
Normalize \(R^{+},R^{-},D^{+},D^{-}\) according to Equation (27). Set the parameter p as 0.5, which means the importance of shape and distance is the same. The positive and negative comprehensive proximity degree vectors can be calculated by applying Equation (28) on the grey fuzzy integral vector and the ideal solution distance vector, which, respectively, were \(S^{+}=[0.7345,\,0.4491,\,0.5002,\,0.6036,\,0.9043,\,0.8350]\) and \(S^{-}=[0.2892,\,0.8768,\,0.8826,\,0.4558,\,0.8691,\,0.8188]\). The final comprehensive evaluation vector can be obtained by aggregating the positive and negative comprehensive proximity degree vector according to Equation (29) which was \(CS=[0.7175,\,0.3387,\,0.3617,\,0.5697,\,0.5099,\,0.5049]\). The sequence of objects was \(1>4>5>6>3>2\) \begin{table}
\begin{tabular}{c c c c c c} \hline
**Sets** & **Measures** & **Sets** & **Measures** & **Sets** & **Measures** \\ \hline
1 & 0.2103 & 2 & 0.4069 & 3 & 0.2119 \\
4 & 0.2404 & 5 & 0.2405 & 12 & 0.5722 \\
13 & 0.3988 & 14 & 0.4241 & 15 & 0.4242 \\
23 & 0.5734 & 24 & 0.5958 & 25 & 0.5959 \\
34 & 0.4255 & 35 & 0.4256 & 45 & 0.4504 \\
123 & 0.7203 & 124 & 0.7402 & 125 & 0.7402 \\
134 & 0.5888 & 135 & 0.5888 & 145 & 0.6109 \\
234 & 0.7413 & 235 & 0.7413 & 245 & 0.7609 \\
345 & 0.6121 & 1234 & 0.8696 & 1235 & 0.8696 \\
1245 & 0.8870 & 1345 & 0.7547 & 2345 & 0.8880 \\
12345 & 1.0000 & & & & \\ \end{tabular} where 1, 2, 3, 4 represent the sets of \(x_{11},x_{12},x_{13}\) and \(x_{14}\), respectively (Similarly hereinafter).  
\end{table} 
Table 5: Fuzzy measures.