# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 1938-Bar Tower
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Optimized Weight (ton)** & **Number of Structural Analyses** & **Constraint Violation (\%)** \\ \hline \multirow{3}{*}{**LSSO-JAYA**} & Best: 98.822 & Best: 7529 & \\  & Worst: 98.860 & Worst: 7780 & Feasible \\  & Mean/STD: 98.832 \(\pm\) 0.01980 & Mean/STD: 7680 \(\pm\) 284 & \\ \hline \multirow{2}{*}{**Hybrid HS with LS**} & Best: 100.008 & Best: 7931 & \\  & Worst: 100.240 & Worst: 8694 & Feasible \\  & Mean/STD: 100.147 \(\pm\) 0.467 & Mean/STD: 8395 \(\pm\) 657 & \\ \hline \multirow{2}{*}{**Hybrid BBBC with LS**} & Best: 99.164 & Best: 8167 & \\  & Worst: 99.225 & Worst: 9655 & Feasible \\  & Mean/STD: 99.179 \(\pm\) 0.02687 & Mean/STD: 8861 \(\pm\) 526 & \\ \hline \multirow{2}{*}{**HFSA**} & Best: 99.794 & \multirow{2}{*}{13201 \(\pm\) 598} & \multirow{2}{*}{Feasible} \\  & Worst: 101.251 & & \\  & Mean/STD: 100.523 \(\pm\) 0.516 & & \\ \hline \multirow{2}{*}{**Improved JA**} & Best: 99.255 & Best: 20,051 & \\  & Worst: 99.265 & Worst: 21,980 & \\  & Mean/STD: 99.263 \(\pm\) 0.003536 & Mean/STD: 21,136 \(\pm\) 843 & \\ \hline \multirow{2}{*}{**AHS**} & Best: 100.750 & Best: 19,139 & \\  & Worst: 103.421 & Worst: 17,184 & \\  & Mean/STD: 101.919 \(\pm\) 1.653 & Mean/STD: 18,394 \(\pm\) 1134 & \\ \hline \multirow{2}{*}{**SAHS**} & Best: 100.120 & Best: 15,437 & \\  & Worst: 104.368 & Worst: 14,297 & Feasible \\  & Mean/STD: 102.623 \(\pm\) 2.188 & Mean/STD: 15,201 \(\pm\) 1383 & \\ \hline \multirow{2}{*}{**BBBC-UBS**} & Best: 101.120 & Best: 17,461 & \\  & Worst: 102.628 & Worst: 19,980 & Feasible \\  & Mean/STD: 101.335 \(\pm\) 0.3033 & Mean/STD: 18,930 \(\pm\) 666 & \\ \hline \multirow{2}{*}{**SLP-DOT**} & \multicolumn{1}{c}{102.789} & \multicolumn{1}{c}{12,310} & \multicolumn{1}{c}{Feasible} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimization results obtained for the 1938-bar tower design example. and SAHS [39] are inherently more sensitive than LSSO-HHSIA to the sequence of generated random numbers in the search process. In fact, while SAHS and AHS, respectively, need \(NDV\) or (\(NDV\) + 2) random numbers to form a new trial design in the current iteration, LSSO-HHSIA may need up to (\(NDV\) + 2) + 2 \(\times\)\(NDV\)\(\times\) (\(N_{\text{POP}}-2\)) random numbers because the new trial design must also be evaluated according to the four cases described in Section 3.2. Hence, LSSO-HHSIA may dispose of a much larger number of optimal combinations of random numbers than SAHS and AHS and this makes the present algorithm less sensitive to independent optimization runs. Convergence curves of the best optimization runs executed for the algorithms listed in Table 4 are compared in Figure 5. Again, the plot is limited to the first 13,000 structural analyses for the sake of clarity while the 50-650 ton weight range is represented for the \(Y\)-axis; the figure includes only the plot relative to the best adaptive HS method [38; 39]. Interestingly, LSSO-HHSIA generated better intermediate designs than the hybrid HS with line search [53] throughout optimization process. Adaptive HS was initially faster than the present algorithm, but such a fast weight reduction resulted in the presence of several steps in the convergence curve with marginal improvements in design. LSSO-HHSIA recovered the weight gap with respect to adaptive HS within only 3000 structural analyses. Hybrid fast SA [53] and improved JAYA [54] were the most competitive algorithms with LSSO-HHSIA but they soon reduced their structural weight reduction rates crossing the LSSO-HHSIA's best run convergence curve after about 4850 and 5150 analyses, respectively. SLP-DOT soon generated infeasible designs and had to penalize weight in order to recover constraint violation.