## 5 Numerical Examples  
This paper assesses the efficiency of the proposed algorithm for optimizing the sizing and layout of trusses utilizing well-known benchmark examples from the literature. The benchmark examples include a 15-bar planar truss, an 18-bar planar truss, a 47-bar planar truss, and a 25-bar spatial truss for which discrete size and continuous shape optimization are performed. The algorithm's performance is assessed over 20 independent runs for each test example, with a population size of 45. The maximum iteration is set to 400 iterations for four examples. The distribution of stresses in the members of optimum trusses obtained by CSBO and NCSBO is illustrated using color maps. The colors in the color map vary between values of 0, corresponding to light blue for the minimum stress, and 1, corresponding to dark red for the maximum stress. Sensitivity analyses were carried out for standard CSBO and the NCSBO. The goal was to identify the optimal values for the population size (_Npop_) and the initial number of weak individuals (_NR_) in standard CSBO. In NCSBO, since the initial number of weak individuals (_NR_) is adaptively determined, it was not necessary to identify this parameter. Instead, in addition to the population size, a sensitivity analysis for the mutation probability (_MP_) was performed to ascertain the most appropriate values. The benchmark problem used for the sensitivity analysis was a 15-bar truss structure. This problem was optimized separately for nine different population sizes ranging from 15 to 300 for both CSBO and NCSBO. The results, presented in Table 2 and visualized in Figure 1, indicate that the most suitable population size is 45. After establishing \(N_{pop}=45\), a sensitivity analysis was performed for the _NR_ parameter in CSBO. Since _NR_ represents the weak solutions in the population, its possible values range from 0 to \(N_{pop}\). To systematically investigate its impact, optimizations were conducted for _NR_ values varying from 0 to 45 in increments of 5. The results are reported in Table 3 and Figure 2. The optimal _NR_ value was identified as 15. For the NCSBO method, the mutation probability (_MP_) was analyzed within the range of 0 to 1, with increments of 0.1. The optimization results demonstrated that an _MP_ value of 0.3 yields the best performance in terms of solution quality, as reported in Table 4 and Figure 3. The results of sensitivity analyses are considered applicable to the subsequent examples utilized in this paper as they provide insights into the parameter settings that are expected to be effective across different truss structures. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \hline \multicolumn{1}{c}{\(N_{pop}\)} & **15** & **30** & **45** & **60** & **75** & **90** & **120** & **150** & **300** \\ \hline Best weight (lb *) & 74.388 & 72.946 & **72.688** & 73.742 & 75.491 & 76.713 & 81.030 & 83.963 & 106.672 \\ Mean (lb) & 78.745 & 76.891 & **75.92** & 76.555 & 78.890 & 80.002 & 84.043 & 88.110 & 118.454 \\ Std (lb) & 2.197 & 2.061 & **1.966** & 2.131 & 1.516 & 1.854 & 1.686 & 1.899 & 5.744 \\ Mean + Std (lb) & 80.942 & 78.952 & **77.886** & 78.686 & 80.407 & 81.856 & 85.730 & 90.008 & 124.199 \\ \hline \multicolumn{1}{c}{\(\mathbf{CSBO}\)} & & & & & & & & & \\ \hline \multicolumn{1}{c}{\(N_{pop}\)} & **15** & **30** & **45** & **60** & **75** & **90** & **120** & **150** & **300** \\ \hline Best weight (lb) & 73.177 & 73.048 & **72.843** & 75.051 & 75.038 & 77.52 & 81.789 & 83.953 & 110.311 \\ Mean (lb) & 78.61 & 76.275 & **76.452** & 78.102 & 78.577 & 80.207 & 84.488 & 88.349 & 120.024 \\ Std (lb) & 2.845 & 2 & **2.116** & 1.917 & 2.188 & 1.882 & 1.894 & 2.187 & 5.863 \\ Mean + Std (lb) & 81.455 & 78.275 & **78.568** & 80.019 & 80.765 & 82.089 & 86.382 & 90.536 & 125.887 \\ \hline \hline \end{tabular}
* lb = 4.4482 N.  
\end{table} 
Table 2: Sensitivity analyses to find the optimal \(N_{pop}\) value in CSBO and NCSBO for a 15-bar truss.