## 5 Numerical Examples
### The 18-Bar Planar Truss
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Design** & **SCPSO** & **VGA** & **GA** & **GSO** & **IGSO** & **D-ICDE** & **ARC** & **MLA** [63] & **CSBO** & **NCSBO** \\  & **Variables** & [56] & [64] & [65] & [66] & [66] & [61] & [62] & **MLA** [63] & **CSBO** & **NCSBO** \\ \hline
1 & G\({}_{1}\) & 12.5 & 12.5 & 12.25 & 12.25 & 12.25 & 13 & 12.5 & 9.75 & 12.50 & **12.5** \\
2 & G\({}_{2}\) & 17.5 & 16.25 & 18 & 18.25 & 18.25 & 17.5 & 17.75 & 18.75 & 17.50 & **17.5** \\
3 & G\({}_{3}\) & 5.75 & 8 & 5.25 & 4.75 & 4.75 & 6.5 & 5.75 & 5.6 & 60.5 & **5.75** \\
4 & G\({}_{4}\) & 3.75 & 4 & 4.25 & 4.25 & 4.25 & 3 & 3.75 & 3.5 & 3.75 & **3.75** \\
5 & X\({}_{3}\) & 907.2491 & 891.9 & 913 & 916.9 & 920.812 & 914.06 & 912.997 & 991.4057 & 906.747 & 907.7788 \\
6 & Y\({}_{3}\) & 179.8671 & 145.3 & 186.8 & 191.971 & 170.912 & 183.46 & 183.681 & 187.5341 & 179.528 & 180.4711759 \\
7 & X\({}_{5}\) & 646.7873 & 610.6 & 650 & 654.224 & 604.506 & 640.53 & 642.714 & 662.7803 & 653.496 & 637.368089 \\
8 & Y\({}_{5}\) & 141.8271 & 118.2 & 150.5 & 156.1 & 139.87 & 133.74 & 143.892 & 162.039 & 139.864 & 141.332995 \\
9 & Y\({}_{7}\) & 407.9442 & 385.4 & 418.8 & 423.5 & 409.416 & 406.12 & 411.692 & 427.350 & 406.841 & **408.0822344** \\
10 & Y\({}_{7}\) & 94.0599 & 72.5 & 97.4 & 102.571 & 917.74 & 92.63 & 97.1476 & 94.025 & 95.461 & 439.8800027 \\
11 & X\({}_{9}\) & 198.7897 & 184.4 & 204.8 & 207.519 & 198.775 & 196.69 & 200.909 & 210.1569 & 198.719 & **198.7942297** \\
12 & Y\({}_{9}\) & 29.5157 & 23.4 & 26.7 & 28.579 & 29.504 & 37.06 & 30.2191 & 28.3101 & 29.457 & 29.51760031 \\ Best weight (lb) & 4512.365 & 4616.8 & 4547.9 & 4538.7676 & 4553.12 & 4554.29 & 4537.06 & 427.157 & 4530.13 & **4521.2144** \\ Mean weight (lb) & 4551.709 & N/A & N/A & N/A & N/A & N/A & 4585.110 & 4317.984 & 4539.63 & 4534.3 \\ Worst weight (lb) & 4661.268 & N/A & N/A & N/A & N/A & N/A & 4627.524 & 4348.598 & 4553.22 & 4548.19 \\ ODNSA & 4500 & N/A & N/A & 50,000 & 50,000 & 8025 & 270 & 30,000 & 7585 & 5900 \\ TNSA & 4500 & N/A & N/A & 50,000 & 50,000 & 8025 & 18,000 & 30,000 & 18,000 & 18,000 \\ STD (lb) & 37.69 & N/A & N/A & N/A & N/A & N/A & N/A & 9,977 & 24.769 & 5.73 & 5.38 \\ CVP (\%) & 0 & 0.0079 & None & None & 94.28 & None & None & 24.87 & None & None \\ \hline \hline \end{tabular}  
* NCSBO achieved a weight of 4537.06 lb after 5150 structural analyses (the result of ABC [62]).  
\end{table} 
Table 6: Comparison of optimized designs for the 18-member truss. Figure 9: Comparison of member stresses with the allowable limit at the optimized design of the 18-bar truss: **(a)** CSBO and **(b)** NCSBO. Figure 10 depicts the optimized configurations derived from the initial layout of the 18-bar truss, accompanied by a stress color map indicating each member's utilization relative to its stress limit. In the optimal design obtained with NCSBO, member 15 reaches near-maximum stress capacity, as represented by the orange coloration. Conversely, in the CSBO optimal design, member 15 exhibits a lower stress utilization, indicated by a yellow coloration, signifying that the stress capacity remains underutilized. The remaining members exhibit similar stress capacities in both optimal designs. The reduction in weight relative to the number of analyses performed using both NCSBO and CSBO methods is shown in Figure 11. A comparison of the convergence behavior shows that NCSBO converges earlier; however, both methods exhibit similar reduction patterns with the number of analyses. For NCSBO, the final optimal solution is achieved after 5900 analyses, whereas for CSBO, the optimal result is reached after 7585 analyses. Both solutions remain stable until they reach the termination criterion of 18,000 analyses. Figure 11: Comparison of convergence curves for the 18-bar truss. Figure 10: The optimized shape of the 18-bar truss with a color map of stresses in members by (**a**) CSBO and (**b**) NCSBO.