## 6 Results
### Real-World Constrained Design Problems
* For the SpRe problem, \(b\in\ [2.6,\ 3.6],\ m\in\ [0.7,\ 0.8],\ z\) an integer \(\in\ [17,\ 28],\ l_{1}\in\ [7.3,\ 8.3],\ l_{2}\in\ [7.3,\ 8.3],\ l_{1}\in\ [2.9,\ 3.9],\ d_{1}\in\ [5.0,\ 5.5]\) \[\qquad\qquad C_{PV-1}\longrightarrow 0.0193\frac{R}{R_{s}}-1.0\leq 0\] \[\qquad\qquad C_{PV-2}\longrightarrow 0.00954\frac{R}{l_{1}}-1.0\leq 0\] \[\qquad\qquad C_{PV-3}\longrightarrow\frac{4}{3}\frac{R}{L}+\frac{1 296000}{tL^{3}R}-1.0\leq 0\] \[\qquad\qquad C_{PV-4}\longrightarrow\frac{L}{240}-1.0\leq 0\] \[\qquad\qquad C_{SP-1}\longrightarrow\frac{27}{\frac{392}{82}}-1.0\leq 0\] \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]0 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]1 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]2 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]3 (30b) \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]4 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]5 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]6 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]7 \[f_{PV}(T_{h},T_{s},L,R)=0.6224T_{s}RL+1.7781T_{h}R^{2}+19.84 LT_{h}^{2}+3.1661 LT_{ s}^{2} \tag{29a}\]8  
In addition to the optimization algorithm, the solution quality obtained for constrained problems also depends on the constraint handling method [40]. The authors of [33] offer a good review of the available strategies. They also propose variants of the feasibility rules introduced in [40]. The formulations proposed in [33] use a penalty factor (\(p_{v}\)) to disadvantage particles in position of constraint violation, see Equation (31). In this expression \(F_{\textit{Pbest}}\) represents the best fitness value identified among the particles respecting all constraints. Equation (32a) gives the expression for the \(p_{v}\) variant considered in the present study. In this equation \(\overline{C}_{j}\) represents the normalized version of a constraint \(j\) belonging to a group of \(m\) constraints. The constraint equations presented in Equations (30a) and (30b) are normalized expressions. More precisely, in order to avoid converging toward particle positions nearly respecting the imposed constraints, the present simulations used the modified version of Equation (32a) given by Equation (32b). This formulation ensures that the \(Fv^{*}\) evaluation for particles in position of constraint violation is never smaller than twice \(Fv_{best}\). It therefore eliminates all those particles from the final exploitation period. \[Fv^{*}= \begin{cases}Fv\text{ for particles respecting all constraints}\\ \qquad p_{v}\times\max\{Fv_{best},Fv\}\text{ otherwise}\end{cases}  \tag{31}\]  
\[p_{v}{=}\max\bigl{\{}\max\bigl{\{}0,\overline{C}_{j}\bigr{\}},\;j\;\in\;[1,m] \bigr{\}} \tag{32a}\] \[p_{v}=\max\bigl{\{}\bigl{(}\max\{\overline{C}_{j}\},\;j\;\in\;[1,m]\;\bigr{)},2 \bigr{\}} \tag{32b}\]  
Both the PV and the SpRe problems have been analyzed in numerous publications. Some studies present solutions nearly respecting the imposed constraints or tolerating a given level of constraint violation. Some studies also ignore the integer limitation on \(x_{h}\) and \(x_{s}\) (PV problem) or on \(z\) (SpRe problem). Most of the publications accepting constraint errors present lower weight values. Refs. [32; 34; 35; 36; 37] give some examples of those results for the PV, while Refs. [37; 39] give examples for the SpRe. A strict review of the current literature indicates that the optimal solutions fully respecting the constraints defined in Equations (30a) and (30b) are as given below.