# A Multi-Objective Bio-Inspired Optimization for Voice Disorders Detection: A Comparative Study
## 3 Material and Methods
### Experimental Setup  
The experimental setup for optimizing the detection of voice disorders using NSGA-II, SPEA-II, and MOEA/D was conducted using the PyGMO library [32], applying default parameters for all the algorithms and machine learning models involved. The machine learning algorithms are implemented using the Scikit-learn library [33]. The default parameters are presented in Table 2.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Algorithm** & **Parameter** & **Default Value** \\ \hline \multirow{4}{*}{KNN} & (n\_neighbors) & 5 \\  & Weights & Uniform \\  & Algorithm & Auto \\  & Metric & Minkowski \\ \hline \multirow{4}{*}{RF} & n\_estimators & 100 \\  & Criterion & Gini \\ \cline{1-1}  & max\_depth & None \\ \cline{1-1}  & min\_samples\_split & 2 \\ \cline{1-1}  & min\_samples\_leaf & 1 \\ \cline{1-1}  & Bootstrap & True \\ \hline \hline \end{tabular}
\end{table}
Table 2: Default values of KNN, RF, MLP, and SVM in Scikit-learn.  
Regarding the SVM algorithm, several experiments were conducted to find the best values of the \(\gamma\) and \(C\) hyper-parameters per dataset [34]. In this regard, cost and gamma for each dataset are set as in Table 3.  
Furthermore, Table 4 summarizes the used hyper-parameters of NSGA-II, SPEA-II, and MOEA/D as the default values defined in PyGMO library. Those hyper-parameters control key aspects of the optimization behaviour, such as genetic operators, neighborhood structures. Regarding the MOEA/D, the neighborhood-based selection and aggregation function (Tchebycheff) are critical to its decomposition strategy.  
The performance evaluation was performed via five-fold cross-validation of the training subset to ensure robust performance assessment. The population and offspring sizes were set to 30, with 50 generations per run. To account for variability, each experiment was repeated across 15 independent runs. The HV was computed using the worst-performing solution among all Pareto-fronts, and the reference point for the IGD was derived from all solutions across all runs per dataset. The experiments were executed on a dedicated server to handle the computational demands of the process. The server has a processor of 12 th Gen Intel(R) Core(TM) i9-12900 KF, and a memory of 125 GB.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **NSGA-II** & **SPEA2** & **MOEA/D** \\ \hline Population Size & 30 & 30 & 30 \\ Crossover Prob. & 0.95 & 0.95 & 1.0 \\ Mutation Prob. & 0.01 & 0.01 & - \\ Distribution Index (SBX) & 10 & 10 & 20 \\ Distribution Index (Mutation) & 10 & 50 & 20 \\ Number of Generations & 50 & 50 & 50 \\ Neighborhood Size & N/A & N/A & 20 \\ Decomposition Method & N/A & N/A & Tchebycheff \\ \hline \hline \end{tabular}
\end{table}
Table 4: Default hyper-parameters for multi-objective algorithms in PyGMO.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Algorithm** & **Parameter** & **Default Value** \\ \hline \multirow{6}{*}{MLP} & hidden\_layer\_sizes & (100,) \\  & Activation & Relu \\  & Solver & Adam \\  & Alpha & 0.0001 \\  & Learning Rate & Constant \\  & max\_iter & 200 \\  & Tolerance (tol) & \(1\times 10^{-4}\) \\ \hline \multirow{6}{*}{SVM} & Regularization Parameter (C) & See Table 3 \\  & Kernel & RBF \\ \cline{1-1}  & Degree & 3 \\ \cline{1-1}  & Gamma & See Table 3 \\ \cline{1-1}  & Coef0 & 0.0 \\ \cline{1-1}  & Tolerance (tol) & \(1\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Dataset** & **Cost (C)** & **Gamma (\(\gamma\))** \\ \hline SLI & 33.20983613 & \(9.700\times 10^{-6}\) \\ SVD & 5.37391496 & \(1.880\times 10^{-6}\) \\ VOICED & 16.89381839 & \(4.590\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Best SVM parameters (Cost and Gamma) for the SLI, SVD, and VOICED datasets.