# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 3586-Bar Truss Tower
[51], which illustrated only the case \(N_{\text{POP}}\) = 500; (ii) to make optimization runs of adaptive HS variants [38; 39], improved/parameterless JAYA [54; 55; 56], BBBC-UBS [57] and sinDE [58] computationally affordable as the computational cost of those algorithms is proportional to the population size \(N_{\text{POP}}\). In regard to issue (i), the hybrid HS, hybrid BBBC and HFSA algorithms of [53] were used to solve this test problem as none of the metaheuristic algorithms described in Ref. [51] could find a feasible solution. In regard to issue (ii), by setting \(N_{\text{POP}}\) = 20 and a computational budget of 20,000 structural analyses, it was possible to complete up to 1000 optimization iterations for adaptive HS and sinDE, and about 2000 iterations for the improved/parameterless JAYA and BBBC-UBS algorithms. The selected 20,000 analyses computational budget is consistent with the data listed in Table 4 for the 1938-bar tower, a similar structure yet with about one-half nodes/elements than the 3586-bar tower. The largest fraction of computation time entailed by a single structural analysis of a 3D truss structure is associated with the inversion of the global stiffness matrix \([K]\) to solve the linear system \(\{F\}\) = \([K][u]\): this operation was found to be four times more expensive for the 3586-bar tower. No data are listed in Table 6 for adaptive HS variants [38; 39] and sinDE [58] because the best candidate designs included in the population after 20,000 structural analyses still \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Optimized Weight (ton)** & **Number of Structural Analyses** & **Constraint Violation (\%)** \\ \hline \multirow{3}{*}{**LSSO-JAYA Present**} & Best: 323.175 & Best: 10,997 & \\ \cline{2-4}  & Worst: 323.722 & Worst: 11,753 & Feasible \\ \cline{2-4}  & Mean/STD: 323.287 \(\pm\) 0.158 & Mean/STD: 11,262 \(\pm\) 534 & \\ \hline \multirow{2}{*}{**Hybrid HS with LS [51]**} & 325.381 & 11,312 & 0.06110 \\ \cline{2-4}  & Best: 323.611 & Best: 11,504 & Best: 0.03679 \\ \cline{2-4}  & Worst: 324.794 & Worst: 12,046 & Worst: 0.02170 \\ \cline{2-4}  & Mean/STD: 324.202 \(\pm\) 0.4358 & Mean/STD: 11,904 \(\pm\) 628 & Mean/STD: 0.0317 \(\pm\) 0.00986 \\ \hline \multirow{2}{*}{**Hybrid BBBC with LS [51]**} & 325.980 & 14,616 & 0.06180 \\ \cline{2-4}  & Best: 324.246 & Best: 12,356 & Best: 0.02376 \\ \cline{2-4}  & Worst: 325.752 & Worst: 13,403 & Worst: 0.03750 \\ \cline{2-4}  & Mean/STD: 325.299 \(\pm\) 0.5607 & Mean/STD: 13,295 \(\pm\) 789 & Mean/STD: 0.0269 \(\pm\) 0.0104 \\ \hline \multirow{2}{*}{**CMLPSA [51]**} & 326.185 & 16,240 & 0.105 \\ \cline{2-4}  & Best: 323.567 & & \\ \cline{2-4}  & Worst: 325.329 & 14,466 \(\pm\) 565 & \\ \cline{2-4}  & Mean/STD: 324.385 \(\pm\) 0.4283 & & Feasible \\ \hline \multirow{3}{*}{**Improved/parameterless JA**} & Best: 323.977 & & \\ \cline{2-4}  & Worst: 324.431 & Computational budget: 20,000 & \\ \cline{1-1} \cline{2-4}  & Mean/STD: 324.130 \(\pm\) 0.287 & & \\ \hline \multirow{3}{*}{**BBBC-UBS [57]**} & Best: 325.097 & & \\ \cline{1-1} \cline{2-4}  & Worst: 327.681 & Computational budget: 20,000 & \\ \cline{1-1} \cline{2-4}  & Mean/STD: 325.541 \(\pm\) 1.083 & & \\ \hline \multirow{2}{*}{**SLP-MATLAB \& DOT**} & 326.278 & 16,480 & Feasible \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimization results obtained for the 3586-bar tower design example. were between 10% and 15% heavier than the optimum design found by LSSO-HHSJA and violated displacement constraints by an amount between 3% and 5%. Once again, using parameter adaptation without any line search strategy does not allow to efficiently solve large structural optimization problems. LSSO-HHSJA was once again the most efficient optimizer overall converging to the lowest structural weight of 323.175 ton, within the smallest number of structural analyses, 10,997. The hybrid HS and hybrid BBBC variants including line search strategies and 1-D probabilistic search developed in [53] found lighter designs than those reported in [51] also reducing constraint violations.