# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 4 Experimental Study
### Comparison Algorithms and Performance Metrics  
In this paper, we compare MOEA/D with the WST approach to MOEA/D with WS and the Tchebycheff approach, named MOEA/D-WST, MOEA/D-WS and MOEA/D-TCH, respectively, to test the performance of the WST approach firstly. For this part, we test the DTLZ1-DTLZ4 with 4-6 objectives representatively. After that, we compare MOEA/D-oDE with two other well-known algorithms, MOEA/D [18] and NSGA-II [14], on all six test instances. Lastly, MOEA/DD [28] with DE operators and MOEA/D-AWA [26] have been tested to compare MOEA/D-oDE on DTLZ1-DTLZ4 with four objectives representatively.  
As shown above, MOEA/D is a famous MOEA based on decomposition, and MOEA/D-oDE is developed from the MOEA/D, so we choose the MOEA/D to test all six test instances at the same time, and we should point out that the Tchebycheff approach is used in MOEA/D for comparing. In addition, NSGA-II is the most widely-cited algorithm in the MOEA field, which is based on the Pareto-dominance principle. Although NSGA-II is relatively long, its influence is still profound. On the other hand, MOEA/DD and MOEA/D-AWA are all of the recently-proposed promising algorithms. In other words, all four algorithms are influential, so it is worth comparing them. In addition, we should point out that the results of MOEA/DD are run by \"PlatEMO\" [34], and we acknowledge the team that made this. The source codes of MOEA/D-AWA can be found on the personal web sites of the author Qi.  
During the experiments, the inverted generational distance (IGD) [35; 36; 37] metric and the generational distance (GD) [38] are used to determine the quality of an algorithm. The IGD metric value can reflect the convergence and diversity of the results obtained by algorithms. Let \(P^{*}\) be a group of uniformly-distributed Pareto-optimal points in PF, as well as \(P\) be a group of points for PF approximation; then, the IGD metric can be defined as follows [10; 37]:  
\[\text{IGD}(P^{*},P)=\frac{\Sigma_{v\in P^{*}}d(v,p)}{|P^{*}|} \tag{12}\]  
where \(d(v,p)\) is the minimum Euclidean distance between \(v\) and any point in \(P\) and \(|P^{*}|\) is the cardinal number of \(P^{*}\). If \(P^{*}\) is a set of large points that could represent the PF well, in a way, it means that the IGD\((P^{*},P)\) could measure the diversity and convergence of \(P\). Therefore, \(P\) must be very close to the true PF and does not miss any part of the PF in order to have a smaller IGD\((P^{*},P)\) value. Thus, we can conclude that the smaller the IGD metric value is, the better the algorithm performs.  
For generational distance (GD) [38]: Let \(P^{*}\) be a set of uniformly-distributed points in the objective space along the PF and \(P\) be an approximation set to the PF; the generational distance from \(P\) to \(P^{*}\) is defined as:  
\[\begin{split}\text{GD}(P,P^{*})=\frac{\sqrt{\sum_{v\in P}(d(v,P^{* }))^{2}}}{|P|}\\ d(v,P^{*})=\min_{x^{*}\in P^{*}}\left\{\sqrt{\sum_{i=1}^{m}(f_{i} (x)-f_{i}(x^{*}))^{2}}\right\}\end{split} \tag{13}\]  
where \(m\) is the number of objectives and \(d(v,P^{*})\) is the minimum Euclidean distance between \(v\) and the points in \(P^{*}\). The generational distance could indicate the distance between the approximate Pareto-optimal front obtained by the algorithms and the ideal Pareto-optimal front distances. Consequently, the value of \(\mathrm{GD}(P,P^{*})\) is smaller, which means the the convergence of the approximate Pareto-optimal front obtained by the algorithms is better; meanwhile, it is closer to the ideal Pareto-optimal front distances. The best situation is that while \(\mathrm{GD}(P,P^{*})=0\), it means that all solutions in \(P\) are the Pareto-optimal solutions.  
In our experiments, the number of the solutions in setting \(P^{*}\) selected to calculate the value of the IGD metric is 1000 uniformly-distributed points for four-objective problems, 2000 uniformly-distributed points for five-objective problems and 3003 uniformly-distributed points for six-objective problems.