# A Multi-Objective Evolutionary Algorithm Model for Product Form Design Based on Improved SPEA2
## 4 Discussion
### Performance of ISPEA2  
The performance of MOEA algorithm is the most critical factor in determining the comprehensive performance of PFDMOEAM. After the comparison of second generation classical MOEAs, we chose to improve SPEA2 into the MOEA required by the PFDMOEAM built in this paper. To verify the performance of ISPEA2, ISPEA2 and SPEA2 were independently run 30 times under the same parameter settings (see Table 11) to compare the performance of two algorithms by using convergence metric, spacing metric and coverage metric [42; 43; 44]. Since the true Pareto front in this study is unknown, we adopted a common comparative method suitable for this situation, which is to construct a reference Pareto front to represent the true Pareto front by a pair-wise comparison after collecting all Pareto solutions from all independent runs of the two algorithms [7; 45]. Let \(A=(A_{1},A_{2},\ldots,A_{|A|})\) be a solution set obtained by ISPEA2 or SPEA2, and \(P^{*}=(P^{*}_{1},P^{*}_{2},\ldots,P^{*}_{|P^{*}|})\) be a solution set on the true Pareto front, the three metrics can be calculated as follows. Convergence metric \(C\) represents the average distance from the solution set obtained to the true Pareto front, which can be calculated as [42; 46; 47]:  
\[C_{(|A|)}=\frac{\sum\limits_{i=1}^{|A|}d_{i}}{|A|} \tag{21}\]  
\[d_{i}=\min\limits_{j=1}^{|P^{*}|}(\sqrt{\sum\limits_{z=1}^{Z}(\frac{f_{z}(i)- f_{z}(j)}{f_{z}^{\max}-f_{z}^{\min}})}^{2}) \tag{22}\]  
where \(d_{i}\) is the minimum normalized Euclidean distance from each solution in \(A\) to \(P^{*}\); \(f_{z}^{\max}\) and \(f_{z}^{\min}\) are the maximum and minimum function values of the \(z\)th objective function in the true Pareto solution set \(P^{*}\), respectively. The smaller the \(C\) value is, the better the convergence performance will be. When \(C=0\), it means that all solutions in \(A\) are on the true Pareto front. Spacing metric \(S\) denotes the distance between two consecutive solutions and measures the diversity performance of algorithm, which is computed as [43; 47]:  
\[S=\sqrt{\frac{1}{|A|-1}{\sum\limits_{i=1}^{|A|}{(d_{i}-\overline{d})}}^{2}} \tag{23}\]  
where \(\overline{d}\) is the mean of \(d_{i}\). A smaller \(S\) value indicates that the obtained solutions are distributed more evenly throughout the objective space. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Solution No.**} \\ \cline{2-5}  & **7** & **17** & **9** & **26** \\ \hline Luxurious & 1.6205 & 1.1264 & 2.1569 & 2.2264 \\ Steady & 2.6535 & 1.6408 & 1.9342 & 1.8708 \\ Modern & 2.1379 & 2.4006 & 1.3423 & 1.4606 \\ C\({}_{\text{i}}\) score & 0.6587 & 0.3861 & 0.4879 & 0.5118 \\ Rankings & 1 & 4 & 3 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Expert evaluation result for four selected Pareto solutions. Coverage metric \(C(A^{1},A^{2})\) denotes that the obtained solutions of one algorithm dominate the obtained solutions of the other algorithm, which is defined as [44; 48]:  
\[C(A^{1},A^{2})=\frac{\left\{\left\langle X\right|Y\in A^{2},\exists X\in A^{1}: F(X)>F(Y)\right\rangle\right\}}{\left|A^{2}\right|} \tag{24}\]  
where \(C(A^{1},A^{2})\) means the ratio of the number of solutions dominated by \(A^{1}\) in \(A^{2}\) to all solutions in \(A^{2}\), and \(C(A^{2},A^{1})\) indicates the opposite case. If \(C(A^{1},A^{2})\geq C(A^{2},A^{1})\), then \(A^{1}\) are better than \(A^{2}\). To compare \(C(A^{1},A^{2})\) values of the two algorithms from a statistical perspective, \(A^{1}\) and \(A^{2}\) solution sets are obtained by further calculation of Pareto solution generation after collecting all solutions of all runs of the two algorithms, respectively. After calculating the metrics mentioned above, we computed the means and standard deviations of \(C\) and \(S\) metrics and the \(C(A^{1},A^{2})\) metric values of two algorithms (Table 17) and plotted the trend graph of \(C\) and \(S\) metric values of 30 independent runs (Figure 11).