# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Planar 200-Bar Truss Structure  
The planar 200-bar truss structure shown in Figure 2 was optimized with 200 sizing variables corresponding to the cross-sectional areas of each element. The structure is subject to five independent loading conditions:  
1. 4449.741 N (i.e., 1000 lbf) in the positive X-direction at nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, 71;
2. 44.497 kN (i.e., 10,000 lbf) in the negative Y-direction at nodes 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 70, 71, 72, 73, 74, 75;
3. Loading conditions a) and b) acting together. 4. 4449.741 N (i.e., 1000 lbf) in the negative X-direction at nodes 5, 14, 19, 28, 33, 42, 47, 56, 61, 70, 75;
5. Loading conditions (b) and (d) acting together. This test problem has 3500 non-linear constraints on nodal displacements and element stresses. The displacements of all free nodes in both coordinate directions X and Y must be less than \(\pm\)1.27 cm (i.e., \(\pm\)0.5 in). The allowable stress (the same in tension and compression) is 206.91 MPa (i.e., 30,000 psi). All non-linear constraints were independently evaluated, and no constraint grouping was adopted. The same was done for all test problems considered in this study. Cross-sectional areas vary between 0.64516 cm\({}^{2}\) (i.e., 0.1 in\({}^{2}\)) and 645.16 cm\({}^{2}\) (i.e., 100 in\({}^{2}\)). In order to make a direct comparison with the hybrid HS algorithm of Ref. [51], LSSO-HHSIA optimizations were also executed for different population sizes: respectively, \(N_{\text{POP}}\) = 20, 50, 100, 200, 500 and 1000. The same was done for the other hybrid HS algorithm of Ref. [53] compared with LSSO-HHSIA. Table 1 shows that all HS variants were practically insensitive to population size. However, only the present algorithm always converged to feasible designs while the other hybrid HS variants found optimal solutions that slightly violate displacement constraints. The robustness of LSSO-HHSIA is confirmed by the fact that it is not possible to establish any direct relationship between population size and computational cost of the optimization process in terms of the required number of structural analyses. Figure 2: Schematic of the planar 200-bar truss structure. Table 2 shows the optimization results obtained by LSSO-HHJSJA and its competitors in the 200-bar truss problem. The results of the independent optimization runs that provided the lowest and the highest structural weights for each algorithm are respectively denoted by \"Best\" and \"Worst\" in the table. Furthermore, results of the optimization runs completed within the smallest and largest number of structural analyses are respectively denoted by \"Fastest\" and \"Slowest\" in the table. The same nomenclature has been utilized for all results tables presented in the article. The present algorithm clearly outperformed the other metaheuristic methods because it achieved the lowest structural weight and always converged to feasible solutions. LSSO-HHJSJA was the best optimizer overall because it designed the lightest structure (weighing 12,483.339 kg) within only 5637 analyses. The other algorithms found heavier designs in their best optimization runs and completed those runs within at least 5679 analyses. CMLPSA also converged to feasible designs in all optimization runs but its best weight was almost 10 kg heavier than the one found by LSSO-HHJSJA. The worst optimization runs of JAYA, adaptive HS variants, BBBC-UBS, sinDE and SQP-MATLAB converged to feasible solutions, but the corresponding structural weights were between 18-19 kg (for BBBC-UBS, JAYA and SQP-MATLAB) and 57-293 kg (for sinDE and adaptive HS variants) heavier than the worst design found by the present algorithm.