# A Multi-Objective Bio-Inspired Optimization for Voice Disorders Detection: A Comparative Study
## 3 Material and Methods
### Proposed Method
#### 3.2.1 Algorithms Compared  
Multi-objective problems have two or three simultaneous conflicting objectives. Although improving one objective might lead to a degradation in the other one, multi-objective optimization algorithms search to have the optimal set of solutions that are called the Pareto set solutions. The Pareto solutions are trade-off solutions where no solution is better than the others. Mathematically, a multi-objective optimization problem with constraints is defined by Equations (1)-(3) [30]. In Equation (1), \(F(X)\) is a minimization function. \(X\) is an n-dimensional vector of decision variables, which is defined by \(X=x_{1},\ldots,x_{n}\)The \(k\) variable is the number of objective functions. Equations (2) and (3) present the inequality (\(g(x)\)) and equality (\(h(x)\)) constraints, respectively. \[Minimise\;\;F(X)=[f_{1}(X),\;f_{2}(X),\ldots,f_{k}(X)] \tag{1}\]  
\[g_{m}(x)\leq 0,\;m=1,\;2,\ldots M \tag{2}\]  
\[h_{j}(x)=0,\;j=1,\;2,\ldots J \tag{3}\]  
Comparing the obtained solutions of multi-objective optimization is done via the Pareto-Dominance. Consider a two-objective minimization problem with two solutions, represented as \(\vec{x_{1}}\) and \(\vec{x_{2}}\). We say that \(\vec{x_{1}}\) dominates \(\vec{x_{2}}\), denoted as (\(\vec{x_{1}}\preceq\vec{x_{2}}\)), if the condition described in (Equation (4)) is fulfilled. \[\forall i:f_{i}(\vec{x_{1}})\leq f_{i}(\vec{x_{2}})\quad\&\quad\exists j:f_{j} (\vec{x_{1}})<f_{j}(\vec{x_{2}}),\;where\;\;i\;\&\;j\in\{1,2,\ldots k\} \tag{4}\]  
This means that all elements of \(f(\vec{x_{1}})\) are no worse than their corresponding elements in \(f(\vec{x_{2}})\), with at least one element being strictly better. Figure 1 illustrates the concept of dominance, highlighting the green and yellow colored solutions. \(\vec{x_{b}}\) is superior in objective (1) but inferior in objective (2), while \(\vec{x_{a}}\) excels in objective (2) and performs worse in objective (1). Consequently, both \(\vec{x_{a}}\) and \(\vec{x_{b}}\) are considered non-dominated solutions, forming part of the Pareto optimal set, denoted as \(P^{*}=\vec{x^{*}}\in\Omega\). In contrast, solution c is dominated by both solutions a and b, which are superior. Four well-regarded evolutionary multi-objective optimization algorithms were used and compared for the optimization of voice disorder detection (The NSGA-II, SPEA-II, and MOEA/D). The NSGA-II is a modified version of the NSGA algorithm and improves over the first version by being faster and incorporating a better sorting and selection strategy. Its objective is to find a set of solutions that are non-dominated with respect to each other. The NSGA-II algorithm relies on the genetic algorithm while employing non-dominated sorting and Crowding-distance fitness. It works by producing an initial population of solutions and then evaluating them based on multiple objectives. It ranks solutions into different Pareto-fronts using non-dominated sorting, where solutions in the first front are non-dominated by any others. To preserve diversity, it calculates a crowding distance, ensuring solutions are spread across the objective space. Through selection, crossover, and mutation, NSGA-II creates new generations, combining parent  
Figure 1: A depiction of dominance, where both \(f_{1}\) and \(f_{2}\) are minimization functions. The grey points represent dominated solutions, while the blue dots indicate non-dominated solutions. and offspring populations to maintain the best solutions. This process is repeated until a stopping criterion is met, ultimately yielding a set of Pareto-optimal solutions [6]. The SPEA-II is an advanced multi-objective optimization algorithm developed to find and maintain a diverse set of Pareto-optimal solutions. It works by keeping an external archive of non-dominated solutions, allocating fitness values based on both the number of solutions a candidate dominates (strength) and a density estimation to ensure diversity.