# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 4 Experimental Study
### Experimental Results and Analysis
#### 4.4.2 MOEA/D-oDE Compared with MOEA/D and NSGA-II  
All compared algorithms run 20-times independently for each test problem in the process of our experiments. Table 4 shows the IGD metric values obtained by three algorithms, including the mean, the best and the standard values of the 20 IGD metric results. From the table, we can see that the performance of MOEA/D-oDE is better than the other two algorithms except for the DTLZ4 problem with six objectives. For this problem, NSGA-II performs the best. Because of this, we cannot say that \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objectives**} & \multirow{2}{*}{**Problem**} & \multicolumn{4}{c}{**MOEA/D-WST**} & \multicolumn{4}{c}{**MOEA/D-WS**} & \multicolumn{4}{c}{**MOEA/D-TCH**} \\ \cline{3-11}  & & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** \\ \hline \multirow{4}{*}{4} & DTLZ1 & 0.2590 & 0.2580 & 0.0110 & 0.2640 & 0.2630 & 0.1620 & 0.2630 & 0.2630 & 0.0003 \\  & DTLZ2 & 0.2040 & 0.2030 & 0.0007 & 0.2740 & 0.2720 & 0.0010 & 0.2710 & 0.2680 & 0.0010 \\  & DTLZ3 & 0.2050 & 0.2040 & 0.0004 & 0.2740 & 0.2480 & 0.0067 & 0.2720 & 0.2710 & 0.0007 \\  & DTLZ4 & 0.2840 & 0.2030 & 0.1380 & 0.5710 & 0.2730 & 0.1900 & 0.4540 & 0.2710 & 0.1780 \\ \hline \multirow{4}{*}{5} & DTLZ1 & 0.2430 & 0.2420 & 0.0008 & 0.2830 & 0.2740 & 0.0183 & 0.2760 & 0.2740 & 0.0025 \\  & DTLZ2 & 0.3140 & 0.3120 & 0.0013 & 0.5500 & 0.5440 & 0.0047 & 0.4700 & 0.4680 & 0.0014 \\  & DTLZ3 & 0.3110 & 0.3100 & 0.0007 & 0.4710 & 0.4690 & 0.0008 & 0.5500 & 0.5180 & 0.0120 \\  & DTLZ4 & 0.4060 & 0.3100 & 0.1140 & 0.7300 & 0.5610 & 0.1280 & 0.5980 & 0.4690 & 0.1400 \\ \hline \hline \multirow{4}{*}{6} & DTLZ1 & 0.2680 & 0.2680 & 0 & 0.2420 & 0.2060 & 0.0173 & 0.2680 & 0.2680 & 0 \\  & DTLZ2 & 0.3470 & 0.3420 & 0.0026 & 0.7350 & 0.7230 & 0.0099 & 0.4810 & 0.4750 & 0.0121 \\ \cline{1-1}  & DTLZ3 & 0.3440 & 0.3410 & 0.0021 & 0.7340 & 0.7090 & 0.0109 & 0.4890 & 0.4750 & 0.0146 \\ \cline{1-1}  & DTLZ4 & 0.5880 & 0.3410 & 0.1590 & 0.8100 & 0.7250 & 0.0826 & 0.6910 & 0.4740 & 0.1340 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics of the IGD metric obtained by MOEA/D-WST, MOEA/D-WS and MOEA/D-TCH on DTLZ1–DTLZ4. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objectives**} & \multirow{2}{*}{**Problem**} & \multicolumn{4}{c}{**MOEA/D-WST**} & \multicolumn{4}{c}{**MOEA/D-WS**} & \multicolumn{4}{c}{**MOEA/D-TCH**} \\ \cline{3-11}  & & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** \\ \hline \multirow{4}{*}{4} & DTLZ1 & 0.0220 & 0.0219 & 0.0224 & 10.300 & 2.7300 & 17.300 & 0.0307 & 0.0306 & 0.0308 \\  & DTLZ2 & 0.0061 & 0.0061 & 0.0061 & 0.0062 & 0.0064 & 0.0063 & 0.0066 & 0.0072 & 0.0072 \\  & DTLZ3 & 0.0061 & 0.0061 & 0.0062 & 1.8300 & 0.0063 & 9.8600 & 0.0072 & 0.0072 & 0.0072 \\  & DTLZ4 & 0.0062 & 0.0061 & 0.0062 & 0.0070 & 0.0062 & 0.0079 & 0.0072 & 0.0069 & 0.0075 \\ \hline \multirow{4}{*}{5} & DTLZ1 & 0.0228 & 0.0227 & 0.0230 & 10.300 & 4.8800 & 14.600 & 0.0302 & 0.0302 & 0.0303 \\  & DTLZ2 & 0.0099 & 0.0098 & 0.0101 & 0.0100 & 0.0096 & 0.0107 & 0.0146 & 0.0145 & 0.0146 \\ \cline{1-1}  & DTLZ3 & 0.0098 & 0.0096 & 0.0098 & 2.4400 & 0.0098 & 5.1600 & 0.0146 & 0.0145 & 0.0146 \\ \cline{1-1}  & DTLZ4 & 0.0103 & 0.0096 & 0.0098 & 0.0121 & 0.0102 & 0.0197 & 0.0140 & 0.0129 & 0.0147 \\ \hline \multirow{4}{*}{6} & DTLZ1 & 0.0131 & 0.0131 & 0.0131 & 22.000 & 8.1700 & 47.200 & 0.0131 & 0.0131 & 0.0131 \\ \cline{1-1}  & DTLZ2 & 0.0117 & 0.0116 & 0.0119 & 0.0099 & 0.0095 & 0.0104 & 0.0169 & 0.0160 & 0.0172 \\ \cline{1-1}  & DTLZ3 & 0.0117 & 0.0114 & 0.0119 & 1.7600 & 0.0397 & 3.3000 & 0.0168 & 0.0159 & 0.0171 \\ \cline{1-1}  & DTLZ4 & 0.0131 & 0.0114 & 0.0119 & 0.0209 & 0.0134 & 0.0349 & 0.0171 & 0.0154 & 0.0182 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistics of the GD metric obtained by MOEA/D-WST, MOEA/D-WS and MOEA/D-TCH on DTLZ1–DTLZ4. the decomposition-based methods must be better than dominance-based methods.