## 5 The Case Study Results and Analysis
### Constructing a Multi-Attribute Evaluation System for Alternative Schemes
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**P\({}_{1}\)** & **F\({}_{1}\)** & **F\({}_{2}\)** & **F\({}_{3}\)** & **F\({}_{4}\)** & **F\({}_{5}\)** \\ \hline D\({}_{1}\) & (0.5,0.6,0.7) & (0.5,0.6) & (0.5) & (0.6,0.7) & (0.5,0.7) \\  & (0.3,0.4) & (0.3,0.4) & (0.5) & (0.4) & (0.4,0.5) \\ D\({}_{2}\) & (0.7,0.8) & (0.7,0.8) & (0.7,0.8) & (0.4,0.5) & (0.4,0.5) \\  & (0.2,0.3) & (0.2,0.3) & (0.2,0.3) & (0.5,0.6) & (0.5,0.6) \\  & (0.5,0.6) & (0.4,0.5) & (0.5,0.6) & (0.4,0.5) & (0.3,0.4) \\ D\({}_{3}\) & (0.4,0.5) & (0.5,0.6) & (0.4,0.5) & (0.5,0.6) & (0.6,0.7) \\  & (0.6,0.7) & (0.7,0.8) & (0.8,0.9) & (0.5,0.6) & (0.6) \\ D\({}_{4}\) & (0.4,0.5) & (0.2,0.3) & (0.2,0.3) & (0.4,0.5) & (0.4,0.5) \\  & (0.5,0.6) & (0.4,0.5) & (0.5,0.6) & (0.5,0.7) & (0.5,0.7) \\  & (0.4,0.5) & (0.5,0.6) & (0.4,0.5) & (0.3,0.4) & (0.4,0.5) \\  & (0.5,0.6) & (0.8,0.9) & (0.7,0.8) & (0.6,0.7) & (0.6,0.7) \\  & (0.4,0.5) & (0.1,0.2) & (0.3,0.4) & (0.4,0.5) & (0.4,0.5) \\  & (0.5,0.6) & (0.5,0.6) & (0.5,0.6) & (0.7,0.8) & (0.6,0.8) \\  & (0.5,0.6) & (0.5,0.6) & (0.5,0.6) & (0.7,0.8) & (0.6,0.8) \\  & (0.4,0.5) & (0.3,0.4) & (0.3,0.5) & (0.2,0.3) & (0.2,0.4) \\  & (0.5,0.6,0.7) & (0.7,0.8) & (0.7,0.9) & (0.7,0.9) & (0.6,0.7) \\  & (0.2,0.3) & (0.2,0.4) & (0.2,0.3) & (0.2,0.3) & (0.4,0.5) \\  & (0.5,0.6) & (0.5,0.6) & (0.5,0.6) & (0.6,0.7) & (0.5,0.6) \\  & (0.4,0.5) & (0.3,0.4) & (0.2,0.4) & (0.4,0.5) & (0.3,0.4) \\  & (0.6,0.7) & (0.6,0.7) & (0.7,0.8) & (0.6,0.7) & (0.6,0.8) \\  & (0.4,0.6) & (0.2,0.3) & (0.3,0.5) & (0.3,0.5) & (0.2,0.3) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pythagorean hesitant fuzzy expert evaluation of the initial matrix. Figure 3: Coffee machine form design alternatives. Step 4: In accordance with Equations (7)-(9), we calculated the gray relational coefficients between each attribute in the expert decision matrix and the ideal scheme. This process resulted in the formation of the gray relational coefficient decision matrix. \[\xi= \begin{bmatrix}0.353761&0.471005&0.528207&0.482712&0.407610\\ 0.613569&0.641101&0.574858&0.628577&0.543885\\ 0.333333&0.355429&0.583090&0.639851&1.000000\\ 0.416773&0.653372&0.496996&0.624123&0.511832\\ 0.432265&0.605221&0.426778&0.736923&0.923160\\ 0.818977&0.531262&0.580595&0.691064&0.838431\\ 1.000000&1.000000&0.835902&0.603181&0.510097\\ 0.576758&0.409787&1.000000&1.000000&0.593669\end{bmatrix} \]  
Step 5: We calculated the correlation weights for the attributes of each scheme using Equations (10) and (11), as shown in Table 4.  
Step 6: Using Equations (12)-(15), we calculated the group utility value \(\mathbf{S_{i}}\), individual regret value \(\mathbf{R}_{i}\), benefit ratio \(\mathbf{Q}_{i}\), and closeness value \(\mathbf{OV}_{i}\) for each scheme. Finally, we derived the comprehensive score for each scheme based on Equation (16), as shown in Table 5.  
Step 7: Consistent with the expert group in Step 1, the user evaluation information was aggregated and standardized. After obtaining the standardized evaluation matrix, the similarity between the evaluation matrices of each user was calculated using Equations (3), (4), (17) and (18) to obtain the user similarity matrix. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) \\ \hline
0.177493 & 0.183588 & 0.201573 & 0.220597 & 0.216750 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Associated weights of attributes for each scheme.