## 4 Case Study
### Sensitivity and Comparison Analysis
#### 4.4.2 Comparison with Rough-TOPSIS Method
If the combination of weights is altered, the ideal utility reference (Table 13) will also change, thereby causing the optimal decision to change. Upon further analysis, it was determined that TOPSIS focuses on the distance between a solution and the ideal reference, rather than balancing conflicts in interest. Therefore, the quality of the final decision depends on the weighting of the objectives. In the rough TOPSIS-3 and TOPSIS-4 results, the two best solutions are very similar to each other, as they have relative closeness values of 0.813/0.759 and 0.811/0.880, respectively. This will make it difficult for the designer to differentiate the priority of the solutions, which will trigger fluctuations in the decision-making process. Based on the analysis above, it can be deduced that the antagonism between CV and economic utilities makes it difficult to accurately determine their relative importance, especially if one tries to maximize one type of utility or the other without considering its effects on the overall design. Consequently, the proposed decision analysis model adopts Shapley values to balance the interests of both players in this cooperative game, to select an optimal solution that satisfies both players. Hence, TOPSIS can transform a multi-objective decision problem into a single-objective evaluation using relative closeness index and obtain the distance-optimal solution. However, this method cannot consider the interaction between CV and economic objectives in the design process, especially, when there is a conflicting relationship between the two objectives. Then, the influence of their own interests on the overall design expectation needs to be fully considered in order to avoid unstable decision results. This study constructs a comprehensive weight model integrating entropy weight and BWM method, which can ensure the objectivity of the program value acquisition process. Simultaneously, Shapley is used to fairly weigh the contribution of CV and economic objectives to obtain a fair objective benefit ratio. Furthermore, the proposed decision method focuses only on objectives' game utilities derived from the initial scheme value data without other auxiliary mathematical models, which can reduce the objectivity of the decision results for design concept under uncertainty. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Decision Method**} & \multirow{2}{*}{**Optimal Scheme**} & \multicolumn{2}{c}{**Game Utility**} \\ \cline{3-4}  & & \(U_{E}(S_{E},S_{T})\) & \(U_{T}(S_{E},S_{T})\) \\ \hline Rough TOPSIS-1 (\(W_{p}\),\(W_{E}\)) = (0.1,0.9) & CS\({}_{5}\) & 1.070 & 1.029 \\ Rough TOPSIS-2 (\(W_{p}\),\(W_{E}\)) = (0.3,0.7) & CS\({}_{5}\) & 1.070 & 1.029 \\ Rough TOPSIS-3 (\(W_{p}\),\(W_{E}\)) = (0.5,0.5) & CS\({}_{5}\) & 1.070 & 1.029 \\ Rough TOPSIS-4 (\(W_{p}\),\(W_{E}\)) = (0.7,0.3) & CS\({}_{6}\) & 1.110 & 0.977 \\ Rough TOPSIS-5 (\(W_{p}\),\(W_{E}\)) = (0.9,0.1) & CS\({}_{6}\) & 1.110 & 0.977 \\ Weighted Shapley value method & CS\({}_{1}\) & 0946 & 1.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The optimal scheme for two decision methods.