## 4 Case Study
### Calculate the Dominance Degree  
According to Equation (13), with the loss coefficient \(\theta\) being 1, the dominance degree of RRS \(O_{\rm i}\) relative to RRS \(O_{\rm k}\) under criterion \(Z_{\rm i}\) can be calculated:  
\[\varphi_{\rm ik}^{1}= \begin{array}{llll}O_{1}&\begin{array}{llll}O_{2} \\ O_{3}\end{array}&\begin{array}{llll}0&-1.650&-0.813&-1.700\\ 0.640&0&0.443&-0.533\\ 0.315&-1.141&0&-1.195\\ 0.660&0.207&0.464&0\end{array}\end{array}  \tag{24}\]  
\[\varphi_{ik}^{2}= \begin{array}{llll}O_{1}&\begin{array}{llll}0&0.351&0.104 &0.351\\ -3.250&0&-3.161&0\\ O_{3}&-0.965&0.341&0&0.341\\ O_{4}&-3.250&0&-3.161&0\end{array}\end{array}  \tag{25}\] \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**RRS**} & \multicolumn{4}{c}{**Criterion \(Z_{j}\)**} \\ \cline{2-5}  & \(Z_{\bf 1}\) & \(Z_{\bf 2}\) & \(Z_{\bf 3}\) & \(Z_{\bf 4}\) \\ \hline \(O_{1}\) & (0.333, 0.083, 0.001) & (1, 0.083, 0.001) & (0.75, 0.083, 0.001) & (0.125, 0.083, 0.001) \\ \(O_{2}\) & (0.958, 0.083, 0.001) & (0, 0.083, 0.001) & (0.208, 0.083, 0.001) & (0.875, 0.083, 0.001) \\ \(O_{3}\) & (0.583, 0.083, 0.001) & (0.958, 0.083, 0.001) & (0.625, 0.083, 0.001) & (0.125, 0.083, 0.001) \\ \(O_{4}\) & (1, 0.083, 0.001) & (0, 0.083, 0.001) & (0.125, 0.083, 0.001) & (0.875, 0.083, 0.001) \\ \hline \end{tabular}
\end{table} 
Table 6: AAC of the risk reduction scheme. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**RRS**} & \multicolumn{4}{c}{**Criterion \(Z_{j}\)**} \\ \cline{2-5}  & \(Z_{\bf 1}\) & \(Z_{\bf 2}\) & \(Z_{\bf 3}\) & \(Z_{\bf 4}\) \\ \hline \(O_{1}\) & (0.324, 0.083, 0.001) & (1, 0.083, 0.001) & (0.748, 0.083, 0.001) & (0.123, 0.083, 0.001) \\ \(O_{2}\) & (0.973, 0.083, 0.001) & (0, 0.083, 0.001) & (0.207, 0.083, 0.001) & (0.878, 0.083, 0.001) \\ \(O_{3}\) & (0.572, 0.083, 0.001) & (0, 0.973, 0.083, 0.001) & (0.625, 0.083, 0.001) & (0.123, 0.083, 0.001) \\ \(O_{4}\) & (1, 0.083, 0.001) & (0, 0.083, 0.001) & (0.125, 0.083, 0.001) & (0.878, 0.083, 0.001) \\ \hline \end{tabular}
\end{table} 
Table 7: Weighted cloud decision matrix \(CR\). \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**RRS**} & \multicolumn{4}{c}{**Criterion \(Z_{j}\)**} \\ \cline{2-5}  & \(Z_{\bf 1}\) & \(Z_{\bf 2}\) & \(Z_{\bf 3}\) & \(Z_{\bf 4}\) \\ \hline \(O_{1}\) & (0, 0.083, 0.001) & (1, 0.083, 0.001) & (1, 0.083, 0.001) & (0, 0.083, 0.001) \\ \(O_{2}\) & (0.960, 0.083, 0.001) & (0, 0.083, 0.001) & (0.132, 0.083, 0.001) & (1, 0.083, 0.001) \\ \(O_{3}\) & (0.367, 0.083, 0.001) & (0.973, 0.083, 0.001) & (0.803, 0.083, 0.001) & (0, 0.083, 0.001) \\ \(O_{4}\) & (1, 0.083, 0.001) & (0, 0.083, 0.001) & (0, 0.083, 0.001) & (1, 0.083, 0.001) \\ \hline \end{tabular}
\end{table} 
Table 8: Normalized weighted cloud decision matrix. \[\varphi_{ik}^{3}= \begin{array}{l}O_{1}\\ O_{2}\\ O_{3}\\ O_{4}\end{array} \left[ \begin{array}{l}0\\ -1.688\\ -0.674\\ -1.891\\ -0.610\\ \end{array}\begin{array}{l}0.533\\ 0.213\\ -1.399\\ -0.499\\ -1.580\\ \end{array}\begin{array}{l}0.597\\ -1.399\\ 0.193\\ 0.499\\ \end{array} \right] \tag{26}\]  
\[\varphi_{ik}^{4}= \begin{array}{l}O_{1}\\ O_{2}\\ O_{3}\\ O_{4}\end{array} \left[ \begin{array}{l}0\\ -2.441\\ 0.459\\ 0.459\\ -2.441\\ 0.459\\ \end{array}\begin{array}{l}-2.441\\ 0.459\\ 0.459\\ 0.459\\ \end{array}\begin{array}{l}0.597\\ -1.399\\ 0.193\\ 0.499\\ \end{array}\begin{array}{l}0.597\\ -1.399\\ 0.499\\ -1.580\\ \end{array}\begin{array}{l}0.597\\ -1.399\\ 0.499\\ \end{array}\begin{array}{l}0.193\\ -1.580\\ \end{array}\begin{array}{l}0.193\\ -1.598\\ 0.459\\ \end{array}\begin{array}{l}0.193\\ -1.580\\ \end{array} \begin{array}{l}0. \(O_{4}\) are reversed when the coefficient is 0.1 and 0.2. It can be explained that the ranking of schemes may be changed according to the DMs' various psychological behaviors. In addition, the ranking result \(O_{4}>O_{2}>O_{3}>O_{1}\) shows a stable trend when DMs prefer risk avoidance. As \(\theta\) increases, the gap between the overall dominance degrees of schemes \(O_{2}\) and \(O_{3}\) also increases.