# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 2 Materials and Methods
### Objectives  
In a previous study by [PERSON] et al. [13], the bi-objective value of a pruning solution \(\mathbf{x}\) for a given tree model \(\mathcal{T}\) was defined using the pre-growth objective value \(f_{\text{pre}}\) and the post-growth objective value \(f_{\text{post}}\):  
\[\mathbf{f}(\mathbf{x})=(f_{\text{pre}}(\mathcal{T}_{\mathbf{x}}),f_{\text{post}}(\mathcal{ T}_{\mathbf{x}})). \tag{3}\]  
Figure 2: Realization of a pruning vector (i.e., genotype) \(\mathbf{x}=(3,5,8)\) is a pruned tree model (i.e., phenotype). In this example, the cut on internode 5 is deactivated by the cut on internode 3 up the tree hierarchy. The quantity \(f(\mathcal{T}_{/x})\) is the estimated light intake of a tree \(\mathcal{T}\) after being pruned according to \(x\). The pre-growth objective \(f_{\text{pre}}\) reflects the short-term (i.e., immediate) effects of pruning, which can be assessed directly on the pruned tree model. On the other hand, the post-growth objective \(f_{\text{post}}\) serves as a proxy measure for long-term (i.e., delayed) pruning effects. These are estimated by averaging the light intake over the results of multiple stochastic growth simulations of the pruned tree. Improving the light conditions within the crown is one of the principal goals of tree pruning. However, there are other important aspects that practitioners need to consider, especially when performing the pruning as a corrective measure in neglected or damaged trees. For example, the amount of removed biomass may need to be constrained in order to prevent inflicting too much stress on the tree, or the disrupted tree balance needs to be restored by improving biomass distribution. Such heterogeneous criteria can constitute conflicting objectives for the pruning optimization task. In this paper, we analyze the results of such multi-objective optimization by introducing the following pruning objectives:  
* The average light exposure of flower buds after pruning, denoted by \(f_{1}\), and calculated as: \[f_{1}(\mathbf{x})=\frac{1}{N_{f}}\sum_{i\in B_{\text{f}}(\mathcal{T}_{/x})}q_{i}^ {2}.\] (4) Here, \(\mathcal{B}_{\text{f}}(\mathcal{T}_{/x})\subset\mathcal{B}(\mathcal{T}_{/x})\) denotes the set of the pruned tree's flower buds. Note that this objective promotes aggressive pruning, since the maximum value can be achieved by a few fully exposed buds. * Conformance of crown shape to the desired training form, which is denoted by \(f_{2}\). Its value is estimated using the inverse Hausdorff distance between the convex hull of the pruned tree and the hull's bounding volume of target symmetric shape: \[f_{2}(\mathbf{x})=h^{-1}(\mathcal{H}(\mathcal{T}_{/x}),\mathcal{C}(\mathcal{T}_{/ x})).\] (5) The convex hull \(\mathcal{H}\) of the pruned tree is constructed using its branch tips. Once the hull is obtained, its bounding volume \(\mathcal{C}\) with target shape is determined (Figure 3). In our experiments, a cylindrical target shape was pursued, but other common training forms can be used (e.g., conical). The Hausdorff distance between two shapes is defined as: \[h(X,Y)=\max\{\sup_{x\in X}\inf_{y\in Y}d(x,y),\sup_{y\in Y}\inf_{x\in X}d(x,y)\},\] (6) where \(d\) denotes the standard Euclidean distance between two points. * Tree balance, denoted by \(f_{3}\), and represented by the inverse horizontal distance \(d_{\mathbf{c}}\) between the above ground biomass center of gravity \(\mathbf{c}\) and the vertical axis going through the stem origin: \[f_{3}(\mathbf{x})=\frac{1}{1+d_{\mathbf{c}}}.\] (7) The center of gravity location \(\mathbf{c}\) for a tree \(\mathcal{T}_{/x}\) with internodes \(\mathcal{I}(\mathcal{T}_{/x})\) is computed as: \[\mathbf{c}=\frac{\sum\limits_{i\in\mathcal{I}(\mathcal{T}_{/x})}m_{i}\mathbf{s}_{i}} {\sum\limits_{i\in\mathcal{I}(\mathcal{T}_{/x})}m_{i}}.\] (8) Here, \(\mathbf{s}_{i}\) is the midpoint of internode \(i\), and \(m_{i}\) is its mass.