## 6 Results
### Real-World Constrained Design Problems  
The original design of the algorithm targeted 5D problems, essentially because real-world engineering jobs normally involve simulations associated with demanding computations roughly proportional to the problem dimensionality. Nevertheless, the proposed algorithm structure is not limited to this frame of reference. The following considers two constrained design problems defined as \"real-world problems\" in the literature [32; 33; 34; 35; 36; 37; 38; 39]: the weight optimization of a pressure vessel and the weight optimization of a speed reducer. Regardless of their \"real-world\" tag, these problem definitions lead to simple objective functions involving only low-cost computations. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**S-EPSO: \(N\)-1000**} & \multicolumn{2}{c}{**S-EPSO: \(N\)-1000**} & \multicolumn{2}{c}{**S-EPSO: \(N\)-1500**} \\  & & \multicolumn{2}{c}{**and \(\gamma^{*}\)-10\({}^{4}\)**} & \multicolumn{2}{c}{**and \(\gamma^{*}\)-2\(\times 10^{4}\)**} & \multicolumn{2}{c}{**and \(\gamma^{*}\)-2\(\times 10^{4}\)**} \\ \hline \multicolumn{2}{c}{**Accuracy**} & \multirow{2}{*}{\(PR\)} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} \\  & **Level \(\epsilon\)** & & & & & & \\ \hline \multirow{6}{*}{F\({}_{7}\)} & \(10^{-1}\) & 0.947 & 0.100 & 0.942 & 0.060 & **0.987** & **0.580** \\  & \(10^{-2}\) & 0.882 & 0.020 & 0.867 & 0.000 & **0.972** & **0.360** \\ \cline{1-1}  & \(10^{-3}\) & 0.754 & 0.000 & 0.758 & 0.000 & **0.863** & 0.000 \\ \cline{1-1}  & \(10^{-4}\) & 0.651 & 0.000 & 0.690 & 0.000 & **0.754** & 0.000 \\ \cline{1-1}  & \(10^{-5}\) & 0.607 & 0.000 & **0.654** & 0.000 & 0.603 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: S-EPSO results for F\({}_{7}\) with \(\gamma^{*}\) and particle number \(N\) increases. The following equations give the objective function for the pressure vessel (PV) and the speed reducer (SpRe) problems, respectively, while Equations (30a) and (30b) show the associated constraints. The PV case includes four optimization variables and four constraints (Equation (30a)), whereas the SpRe problem describes a 7D domain and includes eleven constraints given by Equation (30b). In Equation (29a) \(T_{h},T_{s},L\) and \(R\) are the thickness of the cylinder wall, the thickness of the head wall, the cylinder length and the cylinder inner radius, respectively [32]. In Equation (29b), \(b,m,z,l_{1},l_{2},d_{1}\) and \(d_{2}\) are the gear face width, the tooth module, the tooth number, the length of the input shaft, the length of the output shaft, the diameter of the input shaft and the diameter of the output shaft [37]. \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\] \[f_{SpRe}(b,m,z,l_{1},l_{2},d_{1},d_{2})\] \[\qquad\qquad=0.7854 bm^{2}\big{(}3.33332^{2}+14.933z-43.0934\big{)}+0.7 854\big{(}l_{1}d_{1}^{2}+l_{2}d_{2}^{2}\big{)}-1.508\big{(}d_{1}^{2}+d_{2}^{2} \big{)}\] (29b) \[\qquad\qquad\qquad+7.477\big{(}d_{1}^{3}+d_{2}^{3}\big{)}\] The optimization variables also respect the following ranges: * For the PV problem, \(T_{h}\) and \(T_{s}\) are given by \(x_{h}0.0625\) and \(x_{s}0.0625\), respectively, where \(x_{h}\) and \(x_{s}\) represent integer values \(\in\ [1,\ 99],\ L\ \in\ [10,\ 200]\) and \(R\ \in\ [10,\ 200]\) (See [32]).