# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 3586-Bar Truss Tower
HFSA [53] totally outperformed CMLPSA [51] converging to a lighter feasible design, very close to the global optimum found by the present algorithm (i.e., 323.567 vs. 323.175 ton). Hence, HFSA should be considered the 2 nd best optimizer overall after LSSO-HHSJA. Improved/parameterless JAYA [54; 55; 56] and BBBC-UBS [57] also found feasible solutions, lighter than those quoted in [51] for hybrid HS, hybrid BBBC and CMPLSA, but their optimization runs were always stopped before reaching a fully converged solution in the sense of Equation (20). However, improved/parameterless JAYA's design was only 0.248% heavier than the global optimum found by LSSO-HHSJA. This confirms the utility of integrating JAYA-based operators in a harmony search architecture. BBBC-UBS [57] also reached a rather satisfactory performance (i.e., feasible solution with only 0.595% weight penalty with respect to LSSO-HHSJA) but was outperformed by improved/parameterless JAYA [54; 55; 56], which adopts a very similar elitist strategy as BBBC-UBS to accept/reject new trial designs but has the inherent capability to move towards the best design of the population and move away from the worst design. SQP also converged to a feasible solution that is about 1% heavier than the best design found by LSSO-HHSJA. However, the rate of success of this gradient-based algorithm was rather low and less than 10% of the different optimization runs started from each of the 500 designs included in the LSSO-HHSJA's initial population converged to feasible designs lighter than 330 ton. The average constraint violation was of the order of 0.15%, similar to that reported in [51] for SQP. Similar to that seen for the 200-bar truss and 1938-bar tower problems, the number of structural analyses varied more significantly than the optimized weight passing from one algorithm to another. HFSA [53] was on average about 30% slower than LSSO-HHSJA while improved/parameterless JAYA [54; 55; 56] and BBBC-UBS [57] were about two times slower than the present algorithm. Multiple line searches put in the context of a population-based search appear to be the most efficient approach to handle the complex design spaces with many design variables and nonlinear constraints that characterize large-scale structural optimization problems. LSSO-HHSJA presented the lowest standard deviations on optimized weight and required structural analyses. All of the algorithms compared in Table 6 were actually robust and the ratio between standard deviation on optimized weight and average optimized weight never exceeded 0.34%. This confirms the validity of the selected competitors of LSSO-HHSJA that provide effective information on the real performance of the proposed algorithm. Optimal values of sizing variables determined for the three design examples were not listed in the results tables for the sake of brevity. In the spatial towers examples, cross-sectional areas of bars increased from top to bottom of each segment thus realizing uniform stiffness designs. Each segment contributed to the total structural weight by the same extent regardless of having 3586 or 1938 elements in the tower. The bottom segment (i.e., respectively, \"5\" and \"4\" for the 3586-bar and 1938-bar towers) counted by almost 50% of the total weight. For example, Figure 7 compares the optimized cross-sections by LSSO-HHSJA and its competitors for the 3586-bar tower problem. Distributions relative to hybrid HS/BBBC variants with line search of [51; 53] are averaged for the sake of clarity. It can be seen from Figure 7 that cross-sectional areas of each segment are distributed in fashion of \"oscillating waves\" between very small and very large areas. Values of large areas tend to increase from the top to the bottom of the segment while values of small areas tend to be constant. In particular, the SQP's design is characterized by linearly increasing element areas towards the bottom of each segment while the other members are sized at their minimum gage.