# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 5 Conclusions  
In order to solve the many-objective optimization problems well, an improved MOEA/D with optimal DE schemes, called MOEA/D-oDE, is proposed in this paper. MOEA/D-oDE under the framework of MOEA/D decomposes a many-objective problem into a set of single objective subproblems, and each subproblem is handled with evolutionary operators. All of these subproblems can be solved in parallel. MOEA/D-oDE has two distinguishing points from MOEA/D. First, a new combined decomposition method named the WST approach is introduced and employed in MOEA/D-oDE. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objectives**} & \multirow{2}{*}{**Problem**} & \multicolumn{3}{c}{**MOEA/D-oDE**} & \multicolumn{3}{c}{**MOEA/DD**} & \multicolumn{3}{c}{**MOEA/D-AWA**} \\ \cline{3-10}  & & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** \\ \hline \multirow{5}{*}{4} & DTLZ1 & 0.0422 & 0.0361 & 0.0172 & 1.2946 & 0.4973 & 0.4400 & 2.1900 & 2.1200 & 0.0363 \\  & DTLZ2 & 0.0538 & 0.0530 & 0.0003 & 0.0515 & 0.0510 & 0.0003 & 6.0200 & 5.8600 & 0.0764 \\  & DTLZ3 & 0.0586 & 0.0540 & 0.0094 & 4.7469 & 2.0666 & 1.7146 & 6.1600 & 5.9200 & 0.1360 \\  & DTLZ4 & 0.0019 & 0.0017 & 0.0038 & 0.0549 & 0.0142 & 0.0236 & 5.9900 & 5.8900 & 0.0536 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistics of the IGD metric obtained by MOEA/D-oDE, MOEA/DD and MOEA/D-AWA on DTLZ1–DTLZ4 with 4 objectives. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Objectives**} & \multirow{2}{*}{**Problem**} & \multicolumn{3}{c}{**MOEA/D-oDE**} & \multicolumn{3}{c}{**MOEA/DD**} & \multicolumn{3}{c}{**MOEA/D-AWA**} \\ \cline{3-10}  & & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** & **Mean** & **Best** & **Std** \\ \hline \multirow{5}{*}{4} & DTLZ1 & 0.0017 & 0.0016 & 0.0032 & 0.4917 & 0.3544 & 0.1311 & 0.0150 & 0.0027 & 0.0276 \\  & DTLZ2 & 0.0025 & 0.0022 & 0.0003 & 0.0007 & 0.0007 & 0 & 0.0074 & 0.0071 & 0.0002 \\ \cline{1-1}  & DTLZ3 & 0.0028 & 0.0023 & 0.0049 & 2.2195 & 1.3006 & 0.5813 & 0.0272 & 0.0073 & 0.0497 \\ \cline{1-1}  & DTLZ4 & 0.0110 & 0.0107 & 0.0001 & 0.0007 & 0.0006 & 0 & 0.0073 & 0.0068 & 0.0002 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistics of the GD metric obtained by MOEA/D-oDE, MOEA/DD and MOEA/D-AWA on DTLZ1–DTLZ4 with 4 objectives. The WST approach combines the advantages of the weighted sum and the weighted Tchebycheff approach. It focuses not only on one single aspect, but also the entire direction of objectives. Second, we design an optimal DE scheme, which combines two DE crossover operators and two scaling factors \(F\) to generate and select the best child solution to do the a posteriori computing in each subproblem. In the experimental studies, MOEA/D with the WST method is compared with MOEA/D with the weighted sum approach and the Tchebycheff approach in DTLZ1-DTLZ4 with 4-6 objectives firstly, which reflects the effectiveness of the new decomposition method. Then, MOEA/D-oDE is tested on three sets of six continuous test instances with 4-6 objectives, and the results obtained by MOEA/D-oDE are compared with those obtained by MOEA/D and NSGA-II. Experimental results indicate that MOEA/D-oDE in solving the problems with a higher dimension of objectives outperforms MOEA/D and NSGA-II in most situations. In the end, the comparison with MOEA/DD and MOEA/D-AWA on DTLZ1-DTLZ4 with four objectives is made to reflect the performance of MOEA/D-oDE from another point of view. Additionally, several sets of parameter sensitivity experiments were carried out. In a word, to some extent, MOEA/D-oDE can solve the many-objective optimization problems well. In the future, we plan to research how to make the DE's trial vectors obtain the best child solutions dynamically. Additionally, the multi-objective discrete problems, such as the multi-objective TSP (travelling salesman problem) problems, are also our research targets.