# A Multi-Objective Evolutionary Algorithm Model for Product Form Design Based on Improved SPEA2
## 3 Empirical Study
### Construction of Multi-Objective Prediction Model (MOPM)  
In this section, we constructed three BPNNs optimized by GA to predict calculated scores of three target adjectives. Each BPNN selected the classic three-layer structure, and the number of neurons in the input layer, hidden layer and output layer were 7, 10 and 1, respectively. GA was adopted to generate the optimal initial weights and thresholds for three BPNNs. After being trained, these BPNNs were further used as the fitness function to predict the calculated scores of three target adjectives in ISPEA2. Table 8 reflects the specific parameter values of GABP, some of which are derived from [PERSON] et al. [40]. The details of the algorithm parameters are manifested in Table 8, The values of some parameters in Table 8 and Table 11 are referenced to the studies. Figure 9 demonstrates the prediction errors of three BPNNs when the number of iterations is 20,000.  
As can be seen from Figure 9, there are small errors between the predicted and true values of all samples for three target adjectives. In addition, we further used the root mean square error (RMSE) to evaluate the performances of these BPNNs, which is defined as  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{ns}{{{({\psi_{i}^{*}-{\psi_{i}})}^{2}}}}} {ns}}(ns=45) \tag{20}\]  
where _ns_ is the number of samples; \(\psi_{i}^{*}\) and \(\psi_{i}\) are the BPNN predicted and true values of the _i_th sample, respectively.  
Table 9 reflects that RMSE of the three BPNNs decrease rapidly as the number of iterations increases. After 20,000 epochs, the final RMSE values are 0.0455, 0.0534 and 0.0526, respectively, all within the acceptable tolerance of less than 0.1, which demonstrates that the convergence performance of each network is satisfactory. It can be seen that GABP has built a MOPM with high prediction accuracy, which provides a strong guarantee for the effective implementation of ISPEA2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Rankings** & **Representative Adjectives** & **FIOCS** \\ \hline
1 & Luxurious (Target adjective) & 2.936 \\
2 & Steady (Target adjective) & 2.702 \\
3 & Modern (Target adjective) & 1.670 \\
4 & Male & 1.304 \\
5 & Avant-garde & 1.162 \\
6 & Elegant & 0.996 \\
7 & Simple & 0.850 \\
8 & Dynamic & 0.641 \\
9 & Smooth & 0.340 \\
10 & Generous & 0.332 \\
11 & Formal & 0.322 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Final image objective comprehensive score (FIOCS) and rankings of 11 representative adjectives.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Parameters of BP**} \\ \hline
**Hidden Layer** & **Output Layer** & **Number of** & **Number of** & **Number of** & **Training** & **Learning** & **Training** \\
**Function** & **Function** & **Samples** & **Samples** & **Epochs** & **Rate** & **Goal** \\ \hline
**Sigmoid** & Purelin & 40 & 5 & 1000 & 0.1 & 0.00001 \\ \hline \hline \multicolumn{8}{c}{**Parameters of GA**} \\ \hline
**Population Size** & **Crossover Rate** & **Mutation Rate** & **Crossover Operator** & **Mutation Operator** & **Selection Mechanism** & **Evolutionary Epochs** \\ \hline
20 & 0.4 & 0.2 & Single-point crossover & Gaussian mutation & Roulette wheel & 20,000 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Algorithm parameters of genetic algorithm and back propagation neural network (GABP).