# Constrained Multi-Objective Optimization of Simulated Tree Pruning with Heterogeneous Criteria
## 1 Introduction  
Pruning is one of the most important horticultural intervention techniques with which the vegetative and reproductive growth of fruit trees can be balanced. This balance is important for producing a stable and high quality product in consecutive yield seasons [1, 2]. Proper tree pruning also helps to prevent tree degradation, and reduces the chance of disease by providing sufficient light and air flow through the tree canopy [3]. The knowledge of tree pruning is, thus, an essential skill for both professional and amateur fruit growers. In recent years, a number of software tools have emerged for computer-aided horticultural education [4, 5, 6, 7]. They complement the field training by allowing the user to perform various actions on simulated tree models within a 3D virtual environment. The effects of such interactive tree manipulation on its growth are simulated to provide an informative feedback to the user. The use of software tools can be extended beyond education, since they allow analysis and comparison of various tree training techniques. Another functionality enabled by software is algorithmic determination and evaluation of pruning performed on virtual trees with respect to specific goals. Existing research in this direction includes goal-oriented assessment of pruning effects [8], selection of pruning points in 3D tree reconstructions from images or point clouds as part of automated pruning [9, 10, 11], and algorithmic pruning optimization for support in computer-aided education [12]. In a previous study, the immediate and delayed effects of selective pruning on fruit tree development were assessed using the estimated amount of light received by the flower buds in the current and the next season [13]. In this case, the light intake was a proxy measure for the achieved ratio of vegetative and reproductive growth resulting from pruning, which was the object of optimization. It was shown that pre-growth andpost-growth evaluation of a single metric can result in a complex bi-objective optimization problem with conflicting goals. The limitation of existing approaches to tree pruning optimization is that they are focused on a single pruning objective, which is the improvement of lighting conditions in the crown. Only [PERSON] et al. [11] used the crown volume as an additional component in the calculation of the pruned tree's final score. The problem of light intake maximization was approached from the perspective of both single-objective [12] and multi-objective optimization [13]. However, in the latter case, the objectives were homogeneous and their values were computed by the same function in different value spaces (after pruning and after regrowth). In practice, however, the objectives of pruning can reflect various additional goals, such as maintaining a desirable tree shape, volume, or balance [14]. In this paper, we propose and evaluate the multi-objective pruning optimization with such heterogeneous criteria, which are evaluated on the same pruned model. This extends to the existing methodology by including diverse goals considered in manual pruning, and provides new educational and analytical functionality for the optimization framework, which is implemented within the EduAPLE virtual tree simulation tool [15]. Multi-objective optimization (MO) problems arise in many real-life and scientific tasks. Research of MO methods has produced a large number of efficient algorithms, which were used to solve practical problems ranging from groundwater remediation [16] to workload balancing [17]. Multi-objective problems can be approached using scalarization, Pareto-based algorithms, or hybrid approaches [18]. Scalarization to single-objective problems can employ mature and competitive optimization methods for specialized tasks [19]. However, state-of-the art results in multi-objective optimization problems were achieved by Pareto-based algorithms, which construct an approximation of a Pareto-optimal set of solutions [20].