# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 3 Results
Because the canonical NSGA-II does not use an external archive, the initial diversity with a small population can be exhausted before a large enough pool of feasible non-dominated solutions can be constructed. MOEA/D-EAM prefers smaller mutation rates than NSGA-II and SPEA2 because the changes are propagated faster across the population via the neighborhood mechanism. The probability mass function for different mutation types was taken from the study by [PERSON] et al. [13], where \(\mathbf{P}=\langle P_{\text{move}},P_{\text{add}},P_{\text{remove}}\rangle= \langle 0.3,0.35,0.35\rangle\) performed well. Using equal probabilities for extending and shortening the solution vectors also encourages searching with mean pruning size. An empirical lower bound value 0.9 was used for the constraint objective \(f_{4}\), i.e., no more than 10% of the tree's biomass should be removed by pruning, which is a pragmatical limit. In order to reduce the search space and prevent negligible cuts, the same values for heuristic constraints \(A_{\text{max}}=4\) and \(m_{\text{min}}=10\) as in [13] were used. The number of cuts in a pruning solution was also constrained empirically to the range \([5,25]\). The number of objective evaluations in the final experiments was set to \(M=10,000\). The NSGA-II, SPEA2, and MOEA/D-EAM optimization methods were executed \(q=11\) times for each tree model. Within each method, the obtained \(q\) non-dominated sets were compared in order to determine the number of overall non-dominated solutions in them. The fronts were then sorted by the decreasing number of non-dominated solutions. The best, median, and worst run of both optimization methods were finally used in the analysis of results and method comparison. Figure 7 shows 3D approximations of Pareto fronts, obtained by the best runs of NSGA-II and SPEA2 for the tree models from Figure 6. Figure 8 shows the same comparison for MOEA/D-EAM and SPEA2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Configuration** & \multicolumn{3}{c}{**Parameter Value /**} \\  & \(\mathbf{Ps/Mr/Cr}\) & \multicolumn{3}{c}{**RNI**} \\ \hline \multirow{5}{*}{NSGA-II} & \(P_{S}\) & 20 & 30 & 50 & 100 \\  & \(P_{S}/0.05/0.8\) & 0.579 & 0.667 & 0.971 & 0.875 \\ \cline{2-6}  & \(Mr\) & 0.01 & 0.03 & 0.05 & 0.1 \\  & \(50/Mr/0.8\) & 0.636 & 0.679 & 0.857 & 0.676 \\ \cline{2-6}  & \(Cr\) & 0.7 & 0.8 & 0.9 & 1.0 \\  & \(50/0.05/Cr\) & 0.750 & 0.771 & 0.692 & 0.696 \\ \hline \multirow{5}{*}{SPEA2} & \(P_{S}\) & 20 & 30 & 50 & 100 \\  & \(P_{S}/0.05/0.8\) & 0.644 & 0.806 & 0.808 & 0.684 \\ \cline{2-6}  & \(Mr\) & 0.01 & 0.03 & 0.05 & 0.1 \\  & \(50/Mr/0.8\) & 0.689 & 0.769 & 0.885 & 0.812 \\ \cline{2-6}  & \(Cr\) & 0.7 & 0.8 & 0.9 & 1.0 \\  & \(50/0.05/Cr\) & 0.658 & 0.962 & 0.880 & 0.724 \\ \hline \multirow{5}{*}{MOEA/D-EAM} & \(P_{S}\) & 20 & 30 & 50 & 100 \\  & \(P_{S}/0.03/1.0\) & 0.833 & 0.870 & 0.750 & 0.619 \\ \cline{2-6}  & \(Mr\) & 0.01 & 0.03 & 0.05 & 0.1 \\ \cline{1-1}  & \(30/Mr/1.0\) & 0.944 & 1.000 & 0.800 & 0.636 \\ \cline{1-1} \cline{2-6}  & \(Cr\) & 0.7 & 0.8 & 0.9 & 1.0 \\ \cline{1-1}  & \(30/0.03/Cr\) & 0.720 & 0.706 & 0.875 & 0.879 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis of meta-parameter effects on quality of non-dominated solution sets, measured by the RNI metric [32]. For each optimization method, the parameters \(P_{S}\), \(Mr\) and \(Cr\) were individually modified. The RNI for a configuration with specific value of the observed parameter is computed with respect to other configurations using different values of the same parameter. In order to convey a better sense of shape and relation to reference solutions, the 2D projections of the Pareto front approximation for the tree in Figure 5(a) are shown in Figure 9. The situation is similar for other tree models.