## 5 Numerical Examples
### The 15-Bar Planar Truss  
The first benchmark example is a 15-bar planar truss structure involving continuous variables (Figure 4). The modulus of elasticity is \(10^{4}\) ksi (1 ksi = 6.8948 MPa) and the material density is 0.1 lb/in\({}^{3}\). An applied concentrated load of P exerts a 10-kip downward force. This example has a total of twenty-three design variables, consisting of fifteen sizing and eight shape variables, as given below:  
\[ \begin{array}{ccccc}\text{\emph{Sizing variables}}&A=[A_{1}\ A_{2}\ \ A_{3}\cdots\ A_{15}]\\ \\ \text{\emph{Layout variables}}&X=[x_{2}=x_{6},\ x_{3}=x_{7},\ y_{2},y_{3},y_{4},y_{6},y_{7},y_{8}]\\ \\ \end{array} \]  
The allowable cross-sections for sizing variables are given in the discrete profile set outlined as {0.111, 0.141, 0.174, 0.220, 0.270, 0.287, 0.347, 0.440, 0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.800, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.300, 10.850, 13.330, 14.290, 17.170, 19.180) (in\({}^{2}\) (1 in\({}^{2}\) = 6.4516 cm\({}^{2}\)). The maximum  
Figure 4: The 15-bar planar truss (1 in. = 2.54 cm, 1 kip = 4.448 kN). Figure 3: Statistical findings from the sensitivity analysis conducted for different _MP_ values. stress for all members must not exceed \(\pm 25\) ksi. The layout constraints are considered as given below:  
\[ \begin{array}{ll}100\leq x_{1}\leq 140\ \ (in);\ 220\leq x_{3}\leq 260\ \ (in);\ \ 100\leq y_{2}\leq 140\ \ (in);\\ 100\leq y_{3}\leq 140\ \ (in);\ 50\leq y_{4}\leq 90\ \ (in);\ \ \ -20\leq y_{6}\leq 20\ \ (in);\\ -20\leq y_{7}\leq 20\ \ (in);20\leq y_{8}\leq 60\ \ (in);\end{array} \]  
Table 5 presents the sizing and layout variables of NCSBO and provides a comprehensive comparison with results from previous methods, including GA, ARSAG, PSO, GCPSO, IGA, FA, TLBO, MHS, ABC, and MLA. An optimal weight of 72.224 lb was achieved, requiring 5430 structural analyses in NCSBO, whereas CSBO yielded an optimal weight of 72.66 lb, requiring 7620 structural analyses. NCSBO demonstrates the most effective weight minimization among all of the methods reported in Table 5. It is noteworthy that NCSBO is the most powerful method in terms of convergence speed. SCPSO and MHS required 4500 and 5000 structural analyses, respectively, to achieve their optimum designs. NCSBO, however, reached the optimized results for SCPSO and MHS after only 4260 and 3990 structural analyses, respectively. This comparison demonstrates the superior convergence capability of NCSBO. In the table, ODNSA represents the number of structural analyses that obtained the optimum design, while TNSA is the total number of structural analyses to terminate the optimization process. Moreover, STD refers to the standard deviation of independent runs, and CVP denotes the constraint violation percentage values for optimum designs. Figure 5 displays the stress distributions within the members of the 15-bar planar truss, as determined by CSBO and NCSBO. The maximum stresses are recorded as 24.998597 ksi for CSBO and 24.999438 ksi for NCSBO, while the minimum stresses are 24.999924 ksi and  
\begin{table}
\begin{tabular}{l l l l l l l l l l l l l} \hline \hline \multirow{2}{*}{**No. **} & \multirow{2}{*}{**Design**} & \multirow{2}{*}{**GA**} & \multirow{2}{*}{**ARSAGA**} & \multirow{2}{*}{**SCPSO**} & \multirow{2}{*}{**IGA**} & \multirow{2}{*}{**FA**} & \multirow{2}{*}{**TLBO**} & \multirow{2}{*}{**MHS**} & \multirow{2}{*}{**D-**} & \multirow{2}{*}{**LBC**} & \multirow{2}{*}{**MDA**} & \multirow{2}{*}{**LBC**} & \multirow{2}{*}{**MDA**} & \multirow{2}{*}{**LBC**} & \multirow{2}{*}{**MLA**} & \multirow{2}{*}{**CSBO**} & \multirow{2}{*}{**NCSBO**} \\  & & [54] & & [55] & & [56] & & [57] & & [58] & & [59] & & [60] & [61] & [62] & [63] \\ \hline  
(-164.484\) & \(-20\) & \(-51.799\) & \(-10.067\) & \(-17.564\) & \(3.1411\) & \(-59.472\) & \(-10.6651\) & \(-19.598\) & \(-5.9798\) \\
22 & Y7 & \(-16.4484\) & \(-20\) & \(4.2193\) & \(-1.339\) & \(-5.821\) & \(-6.6997\) & \\(-24.993868\) ksi, respectively.