# A Multi-Objective Evolutionary Algorithm Model for Product Form Design Based on Improved SPEA2
## 3 Empirical Study
### Optimal Solution Selection and Verification  
In order to obtain the optimal solution, we calculated the weights of three target adjectives and the comprehensive evaluation index \(\Omega_{i}\) of all Pareto solutions by using entropy-TOPSIS, Table 13 manifests the weights of three target adjectives. Table 14 demonstrates the detailed \(\Omega\) values and rankings of 30 Pareto solutions. Since \(\Omega_{7}\) ranks first, it was determined to be the optimal solution.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Target Adjectives** & **Weights** \\ \hline Luxurious & 0.3756 \\ Steady & 0.3155 \\ Modern & 0.3089 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Weights of three target adjectives determined by entropy weight method.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**ISPEA2**} & Population Size & Archive Size & Crossover Rate & \(\alpha\) & Mutation Rate & \(ms\) & \(md\) & \(mn\) & \(ub\) & \(lb\) \\ \cline{2-11}  & 30 & 30 & 0.6 & [\(-\)0.5,1.5] & 0.4 & 0.2 & 0.01 & 100 & 1 & \(-\)1 \\ \hline \multirow{2}{*}{**SPEA2**} & Population size & Archive size & Crossover rate & \(\alpha\) & Mutation rate & \multicolumn{3}{c}{ Mutation parameter} & \\ \cline{2-11}  & 30 & 30 & 0.6 & [0,1] & 0.4 & & & & default & \\ \hline \hline \end{tabular}
\end{table}
Table 11: Algorithm parameters of ISPEA2 and SPEA2.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Target Adjectives**} & \multicolumn{8}{c}{**PC Number**} \\ \cline{2-11}  & **PC 1** & **PC 2** & **PC 3** & **PC 4** & **PC 5** & **PC 6** & **PC 7** \\ \hline Luxurious & 0.880 ** & \(-\)0.588 ** & 0.025 & \(-\)0.056 & 0.022 & 0.113 & \(-\)0.059 \\ Steady & 0.669 ** & \(-\)0.682 ** & 0.071 & \(-\)0.229 & 0.000 & \(-\)0.004 & \(-\)0.009 \\ Modern & 0.606 ** & \(-\)0.350 * & \(-\)0.-0.156 & 0.102 & 0.014 & 0.216 & 0.008 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Correlation coefficients between PC scores and IEMSs of the three target adjectives.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Target Adjectives** & **Weights** \\ \hline Luxurious & 0.3756 \\ Steady & 0.3155 \\ Modern & 0.3089 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Weights of three target adjectives determined by entropy weight method.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Solution No.**} & \multirow{2}{*}{**Rankings**} & \multicolumn{6}{c}{**PC Scores**} & \multicolumn{6}{c}{**Calculated Scores**} & \multirow{2}{*}{\(\Omega_{i}\)**Values**} & \multirow{2}{*}{**Solution Profiles**} \\ \cline{3-3} \cline{5-11}  & & **PC1** & **PC2** & **PC3** & **PC4** & **PC5** & **PC6** & **PC7** & **Luxurious** & **Steady** & **Modern** \\ \hline
7 & 1 & 0.9949 & \(-\)0.5920 & 0.4877 & 0.0764 & \(-\)0.5240 & \(-\)0.1886 & 0.1235 & 1.6796 & 3.1099 & 2.0175 & 0.5961 \\
26 & 15 & 0.9988 & \(-\)0.1619 & 0.3512 & \(-\)0.5988 & \(-\)0.4142 & \(-\)0.3302 & 0.4411 & 2.2418 & 1.5539 & 1.5578 & 0.5356 \\
9 & 16 & 0.9825 & \(-\)0.1751 & 0.2984 & \(-\)0.1685 & \(-\)0.4301 & \(-\)0.3580 & \(-\)0.6725 & 1.7722 & 1.5778 & 2.2506 & 0.5321 \\
17 & 30 & 0.8956 & \(-\)0.0676 & 0.241 & \(-\)0.2338 & \(-\)0.2140 & \(-\)0.3084 & 0.2815 & 1.2124 & 1.5272 & 2.6007 & 0.4463 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Automatic calculation result for four selected Pareto solutions.