## 4 Case Study
### Sensitivity and Comparison Analysis
#### 4.4.2 Comparison with Rough-TOPSIS Method  
It is widely known that TOPSIS is a classical multi-objective ranking method [49; 54], which can be used to obtain the optimal scheme closest to the ideal solution. However, before determining the ideal utility reference, the weights of the CV and the economy objectives need to be set, followed by the analysis of the decision results. To this end, \(\text{P}_{\text{E}}\) and \(\text{P}_{\text{T}}\) are given five weight value groups: (\(W_{P}\), \(W_{E}\)) \(\in\) {(0.1, 0.9), (0.3, 0.7), (0.5, 0.5), (0.7, 0.3), (0.9, 0.1)]. By weighting the game utility in Table 10, the positive utility reference (\(WUL_{c}\)) and negative-ideal utility reference (\(WUL_{c}\)) for the CV and economic objectives can be determined, respectively, as shown in Table 12.  
After that, based on different weight distributions of CV and economic objectives, the distance between each positive ideal (\(dis_{i}^{*}\)) and negative ideal reference (\(dis_{i}^{0}\)) is determined based on Equations (25) and (26). \[dis_{i}^{*}=\sqrt{\sum_{c=1}^{2}\left(W_{c}U_{i}^{c}-WUL_{c}^{*}\right)^{2}}, \ c\in P,\ E \tag{25}\]  
\[dis_{i}^{0}=\sqrt{\sum_{c=1}^{2}\left(W_{c}U_{i}^{c}-WUL_{c}^{0}\right)^{2}}, \ c\in P,\ E \tag{26}\]  
where \(dis_{i}^{0}\) is the distance between \(U_{c}\) and the negative ideal solution \(WUL^{0}\) in \(\text{CS}_{i}\); \(dis_{j}^{*}\) is the distance between \(U_{c}\)(\(\text{Sch}_{i}\)) and the positive ideal solution \(WUL^{t}\) in \(\text{CS}_{i}\); \(U_{j}^{c}\) = (\(U_{P}\)(\(S_{P}\)(\(\text{CS}_{1}\)), \(S_{E}\)(\(\text{CS}_{1}\))), \(U_{E}\)(\(S_{P}\)(\(\text{CS}_{1}\)), \(S_{E}\)(\(\text{CS}_{1}\)))). Finally, based on Equation (27), the scheme with the minimum distance between the game utility and the positive ideal solution is obtained, which is the comprehensive optimal scheme. \[C_{i}^{*}=dis_{i}^{0}/\left(dis_{i}^{0}+dis_{i}^{*}\right),\ i\in 1,2, \ldots,K \tag{27}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & Utility Function \(U_{c}\)(\(S_{E}\), \(S_{T}\)) & & & \\ \cline{2-10}
**Utility** & \(U_{P}\) & \(U_{E}\) & \(U_{P}\) & \(U_{E}\) & \(U_{P}\) & \(U_{E}\) & \(U_{P}\) & \(U_{E}\) & \(U_{P}\) & \(U_{E}\) \\ \cline{2-10}
**Reference** & **(\(W_{P}\), \(W_{E}\)) = (0.1, 0.9)** & **(\(W_{P}\), \(W_{E}\)) = (0.3, 0.7)** & **(\(W_{P}\), \(W_{E}\)) = (0.5, 0.5)** & **(\(W_{P}\), \(W_{E}\)) = (0.7, 0.3)** & **(\(W_{P}\), \(W_{E}\)) = (0.9, 0.1)** \\ \hline \(WUL_{c}^{*}\) & 0 where \(C_{j}^{*}\) represents the relative closeness index between the scheme \(i\) and positive ideal reference, and the greater the \(C_{j}^{*}\) value, the better the comprehensive value of the scheme. Under different weighting groups, the relative closeness values of each scheme can be calculated by Equations (25)-(27) and the optimal decision option can be analyzed. Then, the decision results obtained by the two methods are given in Table 13.  
To explore the influence law of game utility under each weight combination for the CV and economic objectives, game utility is analyzed based on the comparison of decision results in Table 13, as illustrated in Figure 6. It is demonstrated that if the weight of the CV objectives is equal to or smaller than 0.5 (\(w_{p}\) = 0.1, 0.3, 0.5), the optimal solution will be CS\({}_{5}\), which prioritizes CV. If the weight of the CV objectives is greater than 0.5 (\(w_{p}\) = 0.7, 0.9), the optimal solution will be CS\({}_{6}\), which prioritizes design expectations associated with economic objectives. If the rough TOPSIS-2 decision analysis method is adopted, the ideal utility references for (\(w_{p}\), \(w_{E}\)) = (0.3, 0.7) are (\(WUL_{p}^{*}\), \(WUL_{E}^{*}\)) = (0.333, 0.730) and (\(WUL_{p}^{0}\), \(WUL_{E}^{0}\)) = (0.270, 0.671), respectively. The solution closest to the ideal utility reference is CS\({}_{5}\), and its distribution of utilities is (\(U_{P}\), \(U_{E}\)) = (1.070, 1.029).