# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 3 Results  
The goals of the experiments presented in this section are to:  
1. Determine the properties of Pareto front approximations obtained from pruning optimization with multiple heterogeneous objectives. In particular, we are interested in the relation of generated pruning solutions to three reference proposals, corresponding to non-pruning, non-selective pruning to cylindrical shape, and distance-based pruning, where the secondary branches are removed if their distance to the primary ones is below the threshold. 2. Investigate the pruning patterns produced by MO, and how they reflect the trade-offs between conflicting pruning goals. 3. Evaluate the effect of the proposed constraint objectives on the resulting Pareto front approximations and solution characteristics. 4. Compare the performance of NSGA-II, SPEA2, and MOEA/D-EAM on this problem, which is important from the perspective of future framework development. In the continuation, we first describe the configuration and workflow of experiments, and then address the above research questions in order. The hardware used for the experiments was a desktop computer with Intel i7 CPU, NVIDIA GeForce GTX 1060 GPU, and 16 GB of RAM. The software environment included the Linux operating system (kernel version 5.13.11) and the GCC compiler version 11.1. The tree models for experiments were generated by using the simulator with growth parameters \(C_{1}=80,C_{2}=40,C_{3}=80,C_{4}=20\), and \(p_{f}=0.08\). The models are shown in Figure 6, and present different levels of complexity for pruning optimization. Their main structural properties and the corresponding combinatorial search space sizes are reported in Table 1.  
Tuning of methods' meta-parameters was performed with a grid search using the pools \(Ps\in\{20,30,50,100\}\), \(Mr\in\{0.01,0.03,0.05,1.0\}\), and \(Cr\in\{0.7,0.8,0.9,1.0\}\). The best out of 5 optimization runs with 5000 objective evaluations was selected for each configuration. The configurations were ordered using the hypervolume indicator \(I_{H}\)[31]. For NSGA-II and SPEA2, the configuration \(Ps/Mr/Cr=50/0.05/0.8\) was the configuration with the highest \(I_{H}\) value. Its performance was also stable across a selection of tree models, and was therefore used in the final experiments. For MOEA/D-EAM, a smaller \(Ps=30\) and \(Mr=0.03\) in combination with \(Cr=1\) proved to be optimal. The MOEA/D-EAM uses additional neighborhood size \(Ns\) parameter, which was set to the recommended value \(Ps/10\). A wider neighborhood setting \(Ns=Ps/5\) was also experimented with, but performed worse than the default. In order to analyze the effects of individual meta-parameters in more detail, we used the ratio of non-dominated individuals (RNI) [32], which is computed between solution sets of the selected reference configuration and the modified configurations. The best solutions, in terms of RNI, were selected out of 5 optimization runs and used for configuration comparison. The effects of individual meta-parameters are reported in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tree Model** & \begin{tabular}{c} **\# of** \\ **Internodes** \\ \end{tabular} & \
\begin{tabular}{c} **\# of Potential** \\ **Cut Locations** \\ \end{tabular} &
\begin{tabular}{c} **Search** \\ **Space Size** \\ \end{tabular} \\ \hline Figure 5(a) & 1121 & 161 & \(1.63\times 10^{29}\) \\ Figure 5(b) & 1234 & 145 & \(9.75\times 10^{27}\) \\ Figure 5(c) & 1919 & 253 & \(2.54\times 10^{34}\) \\ Figure 5(d) & 2506 & 341 & \(5.92\times 10^{37}\) \\ Figure 5(e) & 2647 & 354 & \(1.55\times 10^{38}\) \\ Figure 5(f) & 3740 & 429 & \(2.18\times 10^{40}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of experimental tree models. Figure 6: Tree models used in the experiments. It can be observed that, for NSGA-II and MOEA/D-EAM, the selection of \(Cr\) is less sensitive than for SPEA2. For NSGA-II, larger population sizes can be used than for the other two methods, while SPEA2 performs comparatively for population sizes 30 and 50.