## 3 Proposed Method
### Improved SMA
The proposed method is compared with the state-of-the-art algorithms: OBLSMAL, ESMA, MEO, JADE, and RWGWO. The population size (N) and the maximum number of iterations were fixed at 30 and 500 for all comparison algorithms. ### Welded Beam Design Problem

The design diagram for the structural problem of a welded beam [55] is shown in Figure 3. The objective of structural design optimization of welded beams is to minimize the total cost, subject to certain constraints, which are the shear stress \(\tau\), the bending stress \(\sigma\) on the beam, the buckling load \(P_{c}\), and the deflection \(\delta\) of beam. Four variables are considered in this problem, welded thickness (\(h\)), the bar length (\(l\)), bar height (\(t\)), and the thickness of the bar (\(b\)). The mathematical equations of this problem are shown below:

Consider:

\[\mathbf{x}=[x_{1}\ x_{2}\ x_{3}\ x_{4}]=[h\ l\ t\ b];\]

minimize:

\[f(\mathbf{x})=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2});\]

Figure 2: Convergence figures on test functions F1–F23. subject to:

\[g_{1}(\mathbf{x}) =\sqrt{(\tau^{\prime})^{2}+2\tau^{\prime}\tau^{\prime\prime}\frac{x_ {2}}{2R}+(\tau^{\prime\prime})^{2}}-\tau_{max}\leq 0;\] \[g_{2}(\mathbf{x}) =\frac{6 PL}{x_{3}^{2}x_{4}}-\sigma_{max}\leq 0;\] \[g_{3}(\mathbf{x}) =x_{1}-x_{4}\leq 0;\] \[g_{4}(\mathbf{x}) =0.10471x_{1}^{2}+0.04811x_{3}x_{4}(14+x_{2})-5\leq 0;\] Figure 2:0 Figure 2:1 Figure 2:2

where:

Figure 2:3 Figure 2:4 Figure 2:5 Figure 2:6 Figure 2:7 Figure 2:8

range of variables:

Figure 2:9

Figure 3.0 Welded beam design problem. In Table 11, the results of the proposed DQOBLSMA and other well-known comparative optimization algorithms are given. It is clear from Table 11 that the proposed DQOBLSMA provides promising results for the optimal variables compared to other well-known optimization algorithms. The DQOBLSMA obtained a minimum cost of 1.695436 when \(h\) = 0.205598, \(l\) = 3.255605, \(t\) = 9.036367, and \(b\) = 0.205741. ### Tension/Compression Spring Design

The design goal for extension/compression springs [56] is to obtain the minimum optimum weight under four constraints: deviation (\(g_{1}\)), shear stress (\(g_{2}\)), surge frequency (\(g_{3}\)), and deflection (\(g_{4}\)). As shown in Figure 4, three variables need to be considered. They are the wire diameter (\(d\)), the mean coil diameter (\(D\)), and the number of active coils (\(N\)). The mathematical description of this problem is given below:

Consider:

Figure 3.1

minimize:

Figure 3.2

subject to:

Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6

range of variables:

Figure 3.7

\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Optimal Values for Variables**} & \multirow{2}{*}{**Optimal Cost**} \\ \cline{2-2} \cline{4-6}  & \(h\) & \(l\) & \(t\) & \(b\) & **Optimal Cost** \\ \hline DQOBLSMA & 0.205598 & 3.255605 & 9.036367 & 0.205741 & 1.695436 \\ OBLSMAL & 0.253062 & 1.842203 & 8.270240 & 0.253229 & 1.726511 \\ ESMA & 0.201567 & 3.357515 & 8.983361 &



The results of the DQOBLSMA and other comparative algorithms are presented in Figure 3.8 The proposed DQOBLSMA achieved the best solution to the problem. The DQOBLSMA obtained a minimum cost of 0.012719 when \(d\) = 0.050000, \(D\) = 0.317425, and \(N\) = 14.028013. ### Pressure Vessel Design

The pressure vessel design problem is a four-variable, four-constraint problem in the industry field that aims to reduce the total cost of a given cylindrical pressure vessel [57].