## 3 Multi-Objective Optimization
### Multi-Objective Hybrid Butterfly Optimization and Particle Swarm Optimization Algorithm  
Due to the inherent conflicting nature of objective functions in multi-objective optimization problems, there is no single solution that simultaneously minimizes all considered objective functions. Therefore, in solving MOO problems, it is crucial to obtain a set of solutions, called Pareto-optimal solutions (Pareto front), which are considered optimal because they represent a trade-off between the objectives, meaning that improving one objective comes at the cost of worsening another. The acquired Pareto set offers decision makers a variety of possibilities from which to pick based on their preferences and priorities. Let us consider the MOO problem in Equation (1); then, the feasible domain \(\mathbb{F}\) is a set of solutions that satisfy all the constraints imposed on the considered problem, which is written as  
\[\mathbb{F}=\left\{ \begin{aligned} \mathbf{x}\mid g_{k}( \mathbf{x}_{i})\geq 0\wedge h_{l}(\mathbf{x}_{i})=0\wedge x_{ij}\in\left[x_{j}^{Lower },x_{j}^{Upper}\right]\right\}\end{aligned} \right. \tag{37}\]  
Then, for each solution \(\mathbf{x}\in\mathbb{F}\), we say that it is Pareto-optimal if and only if there does not exist any point \(\mathbf{y}\in\mathbb{F}\) for which \(f(\mathbf{y})\leq f(\mathbf{x})\), such that \(f(\mathbf{y})\
eq f(\mathbf{x})\). **Definition 1**. : _Non-dominated solution: The solution \(\mathbf{x}\) is said to be non-dominated (or Pareto-optimal) if and only if there does not exist another solution \(\mathbf{y}\) such that \(f_{i}(\mathbf{x})\leq f_{i}(\mathbf{y})\), \(\forall i\in\{1,2,\ldots,M\}\) with at least one strict inequality, i.e., \(\exists j\in\{1,2,\ldots,M\}:f_{j}(\mathbf{x})<f_{j}(\mathbf{y})\)_  
**Definition 2**. : _Pareto front: The set, which is composed of non-dominated Pareto-optimal solutions, is called the Pareto front (PF), denoted as \(X=\left\{ \begin{aligned} \mathbf{x}\in\mathbb{F}\end{aligned}  \middle|\,\exists\mathbf{y}\in\mathbb{F}\,,\text{ such that }f(\mathbf{y})>f(\mathbf{x})\,\right\}\)._  
The BOA and PSO algorithms, however, were both initially proposed for single-objective optimization problems. Later, different versions of the PSO algorithm for MOO problems, as well as, in recent years, the BOA algorithm, were developed [18]. Furthermore, to successfully solve MOO problems, we need to provide an adequate balance between exploration and exploitation and keep population diversity. As a result, in order to successfully handle complex MOO problems, the proposed multi-objective Hybrid Butterfly Optimization and Particle Swarm Optimization Algorithm, named HMOBPSO, aims to discover the Pareto optimal of the considered problem. This is accomplished by introducing a selection operator based on crowding distance, as well as a separate archive in which all non-dominated solutions found in each iteration are saved. Furthermore, with the introduction of the appropriate hybridization between BOA and PSO algorithms, which utilizes the adaptive parameter in determining which strategy to apply to the current optimization stage, the algorithm effectively maintains the population diversity. By incorporating a hybrid approach, the algorithm is able to overcome the limitations of each individual method and achieve better performance in solving complex multi-objective optimization problems. Firstly, the selection strategy, based on the crowding distance, is used to select the solutions that will be kept in the archive \(\mathbb{P}_{s}\). The crowding distance is a measure of how crowded a solution is in the objective space. Solutions that are too close to each other are considered redundant and only one of them needs to be kept in the archive. The selection strategy ensures that the archive contains a diverse set of non-dominated solutions that cover the entire objective space. In each iteration, we compare all solutions written to the archive \(\mathbb{P}_{s}\) in order to determine whether the newly obtained solution should enter the archive.