# Safe Three-Dimensional Assembly Line Design for Robots Based on Combined Multiobjective Approach
## 4 Experimental Results and Analysis
### Performance Comparison with Approach  
The population size is 100, the maximum number of iterations is 10,000, the crossover probability is 0.8, and the mutation probability is 0.1. Two objective functions and six decision variables areestablished. When the DE strategy parameters are F = 0.5 and CR = 0.3, the global search algorithm yields the best convergence speed and search ability.  
The benchmark functions ZDT1 and DTLZ1 are applied to evaluate the performance of NSGA-II [8], MOCell [38]. and SE-NSGA2 in order to verify the performance of SE-NSGA2. NSGA-II and MOCell are two mainstream multi-objective optimization methods, which have a good performance of the convergence effect. The NSGA-II [8] is that GAs evolutionary algorithm has a strong global optimization capability. The proposed approach SE-NSGA2 is an improved approach of NSGA-II method. The MOCell algorithm [38] has received wide attention due to its diversity performance. Therefore, the benchmark functions ZDT1 and DTLZ1 are applied in order to evaluate the performance of NSGA-II [8], MOCell [38], and SE-NSGA2.  
The inverted generational distance (IGD) is an evaluation index that reflects the overall performance of the algorithm. The smaller the value, the better the overall convergence and distribution performance. The IGD expression (10) is  
\[\mathrm{IGD}(\mathrm{P},\mathrm{Q})=\sum_{\mathrm{v}\in\mathrm{P}}\mathrm{d}( \mathrm{v},\mathrm{Q})/|\mathrm{P}| \tag{10}\]  
P is the set of the Pareto front uniformly distributed, \(|\mathrm{P}|\) is the number of the solution set, Q is the optimal solution set in the Pareto front, and d(v, Q) is the smallest Euclidean distance from v to Q in the Pareto front. The smaller the value of d(v, Q) is, the better the convergence of the algorithm is. If this value is large, the distribution is scattered. The MOCell, NSGA-II, and SE-NSGA2 algorithms were compared based on the test function. Table 3 presents the statistical quantitative results of the IGD values for 30 independent runs. The smaller the value, the better the performance of the corresponding method. The data in Table 3 directly indicate that SE-NSGA2 performs better than the other methods, and the distribution and convergence of the solution set are better.  
Three objective functions are established in order to observe and verify the algorithm performance based on convergence and diversity. Figure 5 shows a 3D visualization of the Pareto front sets. The statistical results of the Pareto front are compared. The figure shows that the distribution and convergence of MOCell are worse than those of NSGA-II and SE-NSGA2. The Pareto sets of NSGA-II and SE-NSGA2 have dense points in the plot and, as indicated in Table 3, the IGD value of SE-NSGA2 is smaller than the value of NSGA-II, which is, SE-NSGA2 yields convergence and distribution results. Moreover, when compared with the original algorithm, the proposed method increases the diversity and produces better distribution and convergence results with fewer overlapping solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Test Function** & **NSGA-II** & **SE-NAGA2** & **MOCell** \\ \hline ZDT1 & \(1.8758\times 10^{-2}\) & \(1.6865\times 10^{-2}\) & \(1.7515\times 10^{-2}\) \\ DTZ1 & \(4.9333\times 10^{-2}\) & \(4.1731\times 10^{-2}\) & \(6.9575\times 10^{-2}\) \\ \hline \end{tabular}
\end{table}
Table 3: Statistical results of the IGD.