## 3 Computational Experiments Comparing MOMmCAA to Other Algorithms  
MOLAPO, GS, MOPSO, NSGA-II, MOMVO, and MNMA were compared to MOMmCAA in order to identify the best performance in calculating Pareto-optimal solutions. The initial parameters of all described algorithms are summarized in Table 1. Each experiment employed 50 PS solutions and a maximum of 1000 iterations. The proposed algorithm was tested in 29 diverse case studies, including 27 unconstrained and constrained mathematical problems and two real-world engineering design problems. The original Matlab implementations of these algorithms were taken directly from the web addresses indicated in the reference articles. The Matlab code of the MOMmCAA can be downloaded from Github using the link [[https://github.com/juanseck/MOMmCAA](https://github.com/juanseck/MOMmCAA)]([https://github.com/juanseck/MOMmCAA](https://github.com/juanseck/MOMmCAA)) (accessed on 2 September 2024). The MOMmCAA and other algorithms were executed in Matlab 2015a on a PC with a 3.1 GHz Intel Xeon CPU with 64 GB of RAM using the macOS Sonoma operating system. Thirty independent runs were made for each algorithm on every benchmark function. A number of different metrics were used to compare the results of the algorithms, as described below. **Hypervolume (HV)**: The diverseness in the search space through the hypervolume metric was first introduced by [PERSON] et al. to escalate the diversity in both decision space and objective space [55]. The \(HV\) of a set of solutions measures the size of the portion of the objective space dominated by those solutions as a group. In general, \(HV\) is favored because it captures both the closeness of the solutions to the optimal set and (to some extent) the distribution of solutions across the objective space in a single scalar. The \(HV\) value measures both convergence and diversity, and can be calculated using the equation  
\[HV=\cup_{i}Z(s)\mid s\in PF, \tag{2}\]  
where \(Z(s)\) refers to the hypercube bounded by a solution \(s\) in the obtained \(PF\). A larger \(HV\) value indicates a better approximation of the \(PF\). **Contribution (C)**: The Contribution metric counts the number of PS points used in the combined solution of all algorithms. This metric is an extension of the Purity metric [56] \begin{table}
\begin{tabular}{c l} \hline \hline
**Algorithm** & **Parameters** \\ \hline MOLAPO & \(FE=2,cap=50\) \\ GS & \(numparts=20,cap=50\) \\ MOPSO & \(C1=C2=2\), \(w=numax-t\times(wmax-wmin)/(tmax)\), \\  & \(wmax=0,0,wmin=04,cap=50\) \\ NSGA-II & \(procs=0.7,\,cross=2*round(procs*100/2),\,pmut=0.4\), \\  & \(mut=round(pmut*100)\), \(\mu=0.02,\,Sigma=0.1*(wmax-vmin)\) \\ MNMA & \(\delta^{\prime}=2,\,\delta^{\prime\infty}=0.5,\delta^{\prime\lambda}=-0.5,\gamma=5\) \\ MOMmCAA & \(cap=50,\,n_{S}=11,\,n_{me}=3,\,n_{int}=5,\,prop_{w}=5,2\leq n_{r}\leq 5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the algorithms used for comparison: MOLAPO, GS, MOPSO, NSGA-II, MNMA, and MOMmCAA. For two approximation Pareto sets \(A\) and \(B\), where \(B\subset A\), the \(C\) metric assigns \(A\) a higher measure than \(B\). For \(O\geq 2\) MOEAs applied to a problem, let \(R_{i}\) be the non-dominated solutions obtained by the \(i\)-th MOEA for \(1\leq i\leq O\). The union of all these sets is \(R=\cup_{i=1}^{O}R_{i}\). The set \(R^{*}\) of non-dominated solutions is calculated from \(R\). Let \(r_{i}^{*}\) be the number of non-dominated sets in \(R^{*}\) obtained by the \(i\)-th MOEA:  
\[r_{i}^{*}=\{s\mid s\in R_{i}\text{ and }s\in R^{*}\}. \tag{3}\]  
Thus, the \(C_{i}\) metric of the \(i\)-th MOEA is defined as  
\[C_{i}=\frac{|r_{i}^{*}|}{|R_{i}|}. \tag{4}\]  
The \(C_{i}\) value may lie between \([0,1]\), with a value nearer to \(1\) indicating better performance. **Epsilon Indicator (EI)**:  
The Epsilon Indicator was defined in [57]. It measures the minimum value of the scalar \(e\) required to make the Pareto front (\(PF\)) dominated by the approximation set \(S\). Epsilon values fall within the range of \([1,\infty)\).