## 3 Application of Industrial Styling Decision-Making Methods
### Result Validation and Comparison of Various Weighting Methods  
At present, entropy weight-TOPSIS is a traditional decision-making method that takes into account both subjective and objective factors among comprehensive weighting methods [35; 36]. In order to validate the feasibility of the decision-making approach proposed in this paper, a comparative calculation was conducted through the entropy weight method coupled with the matrix G obtained from the previous expert evaluation method. The specific calculation process is exhibited in Figure 4, and MATLAB R2022a software was introduced for validation calculations. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \
\begin{tabular}{c} **Primary** \\ **Indicator** \\ \end{tabular} &
\begin{tabular}{c} **Secondary** \\ **Indicator** \\ \end{tabular} & \multicolumn{2}{c}{**Comparison Matrix**} & \multicolumn{2}{c}{**Subjective Weight**} & \multicolumn{2}{c}{**Comprehensive Weight**} & \multicolumn{1}{c}{**Ranking**} \\ \hline \multirow{4}{*}{X1} & X11 & 1 & 2 & 3 & 4 & 0.467 & 0.267 & 1 \\  & X12 & 1/2 & 1 & 1/3 & 2 & 0.160 & 0.108 & 3 \\  & X13 & 1/3 & 3 & 1 & 3 & 0.278 & 0.169 & 2 \\  & X14 & 1/4 & 1/2 & 1/2 & 1 & 0.095 & 0.076 & 4 \\ \hline \multirow{4}{*}{X2} & X21 & 1 & 1 & 3 & 1/2 & 0.169 & 0.118 & 3 \\  & X22 & 1 & 1 & 3 & 1/3 & 0.191 & 0.127 & 2 \\  & X23 & 1/3 & 1/3 & 1 & 1/2 & 0.074 & 0.066 & 4 \\  & X24 & 5 & 3 & 5 & 1 & 0.565 & 0.316 & 1 \\ \hline \multirow{4}{*}{X3} & X31 & 1 & 1/4 & 5 & 4 & 0.275 & 0.171 & 2 \\  & X32 & 4 & 1 & 5 & 4 & 0.550 & 0.310 & 1 \\  & X33 & 1/5 & 1/5 & 1 & 1 & 0.082 & 0.074 & 3 \\  & X34 & 1/4 & 1/4 & 1 & 1 & 0.092 & 0.079 & 4 \\ \hline \multirow{4}{*}{X4} & X41 & 1 & 1 & 3 & 1/3 & 0.200 & 0.132 & 2 \\  & X42 & 1 & 1 & 4 & 1/4 & 0.200 & 0.128 & 3 \\ \cline{1-1}  & X43 & 1/3 & 1/4 & 1 & 1/4 & 0.075 & 0.065 & 4 \\ \cline{1-1}  & X44 & 3 & 4 & 4 & 1 & 0.523 & 0.293 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical table of the comprehensive weighting results for indicators. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicator** & **Correlation Degree** & **Ranking** \\ \hline X4 & 0.109 & 1 \\ X3 & 0.102 & 2 \\ X1 & 0.095 & 3 \\ X2 & 0.009 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Grey weighted correlation degree results and ranking. The outcomes of design element importance, including the positive and negative ideal solutions and Euclidean distances (namely, relative closeness values), are presented in Table 9.  
In line with the checking process depicted in Figure 4, the positive ideal solution U+ = [0.260, 0.389, 0.529, 0.459] and the negative ideal solution U\(-\) = [0.0345, 0.0139, 0.01920, 0.02330] were obtained. According to the standardized importance decision matrix and the positive and negative ideal solutions, the ranking of importance in the four domains was unraveled with the use of the entropy weight-TOPSIS method: X4 \(>\) X1 \(>\) X2 \(>\) X3. Overall, both calculation methods yielded the same conclusion regarding the optimal solution. However, there were slight differences in the ranking of domain importance for subsequent analysis, validating the basic feasibility of the design decision-making method proposed in this investigation. In terms of the difficulty of decision-making operations, both methods require two rounds of quantitative evaluation, making the difficulty roughly equivalent. Comparatively, the entropy weight-TOPSIS method, which completely relies on  
Figure 4: Comparison of program feasibility based on the entropy weight-TOPSIS method.