# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 1938-Bar Tower
In particular, sinDE's best record after 104,000 structural analyses still weighted about 111.4 ton (i.e., more than 10 ton heavier than the optimized designs achieved by its competitors) and violated displacement constraints by about 1%. SQP-MATLAB could not complete a single loop optimization successfully and was hence alternated with SQP-DOT. However, after about 25,000 structural analyses, the SQP's current best record was still heavier than the feasible solutions obtained by all metaheuristic algorithms compared in Table 4 (i.e., about 101.5 kg vs. at most 101.120 kg) and violated constraints by 0.258%. \begin{table}
\begin{tabular}{c l} \hline \hline \multicolumn{1}{c}{**Loading Condition 1**} \\ \hline \hline
**X** & None \\ \hline
**Y** & None \\ \hline \hline
**Z** & None \\ \hline \hline \multicolumn{1}{c}{**Loading condition 3**} \\ \hline \hline
**X** & None \\ \hline \hline \multirow{5}{*}{**Y**} & **-4.448** **kN@** nodes 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 110, 118,It can be seen from Table 4 that LSSO-HHSIA was the best algorithm also in this design example because it converged to the global minimum weight of 98.822 kg within only 7529 structural analyses. All metaheuristic algorithms found a feasible solution ranking in the following order in terms of optimized weight: LSSO-HHSIA, hybrid BBBC with line search [53], improved JAYA [54], HFSA [53], hybrid HS with line search [53], SAHS [39], AHS [38] and BBBC-UBS [57]. In particular, structural weight increased by about 2.33% passing from the present algorithm to BBBC-UBS. The computational cost of optimization process varied much more significantly than structural weight: from the 7529 structural analyses required by the present algorithm to the 20,051 analyses required by improved JAYA [54]. In general, the optimizers that implemented more sophisticated line search strategies such as LSSO-HHSIA, hybrid HS and hybrid BBBC [53], HFSA [53] and improved JAYA [54] outperformed their competitors. Interestingly, improved JAYA was about 2.7 times slower than LSSO-HHSIA, while hybrid HS and hybrid BBBC were at most 15% slower than the present algorithm. This is a direct consequence of the fact that improved JAYA actually worked with a simplified line search strategy comparing each trial design \(\mathbf{X_{k}}^{new}\) only with its counterpart design \(\mathbf{X_{k}}^{pre}\) of the current population. Conversely, hybrid HS and hybrid BBBC always considered a rather large set of descent directions to form a new trial design \(\mathbf{X_{\text{TR}}}\). Parameter adaptation confirmed itself definitely less effective than line search strategy. In fact, the AHS [38] and SAHS [39] algorithms exhibited the largest average weights and the heaviest worst designs over the independent optimization runs. LSSO-HHSIA was also the most robust optimizer overall with the lowest standard deviations on optimized weight and required number of structural analyses. As expected, adaptive HS variants [38; 39] were characterized by the largest statistical dispersions on structural weight and computational cost. This is fully consistent with the fact that AHS [38]