# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Planar 200-Bar Truss Structure
Indeed, these searches were significantly reduced passing from the hybrid HS formulation [51] to its enhanced version [53]. However, they still played some role in the optimization process, thus complicating the practical implementation of the algorithm. Interestingly, improved/parameterless JAYA [54; 55; 56] ranked right after LSSO-HHSJA, hybrid HS and BBBC variants [51; 53] and CMLPSA [51]. In fact, while most of the algorithms listed in Table 2 converged to structural weights between 12,483.3 and 12,492.9 kg, JAYA's average constraint violation was only 0.01888%, much less than for BBBC-UBS [57] (0.03048%), sinDE [58] (0.03030%) and adaptive HS [38; 39] (between 0.08363 and 0.09718%). This confirms the inherent ability of JAYA to move towards good regions of search space but also that a sufficient number of descent directions must be considered in order to form new trial designs of very high quality in each iteration. LSSO-HHSJA ranked 2 nd overall after hybrid BBBC [51] in terms of the average number of required structural analyses but BBBC's computational cost increased by almost five times passing from \(N_{\text{POP}}\) = 20 to 1000: in particular, the slowest optimization run of hybrid BBBC, corresponding to \(N_{\text{POP}}\) = 1000, took 9460 structural analyses vs. only 6373 analyses of LSSO-HHSJA (just one more analysis than the hybrid HS variant of Ref. [53]). CMLPSA [51], JAYA [54; 55; 56] and BBBC-UBS [57] ranked after the hybrid HS and hybrid BBBC algorithms and were on average two times slower than LSSO-HHSJA. Adaptive HS [38; 39] variants and sinDE [58] were the slowest metaheuristic methods and required on average from 15,618 to 20,766 structural analyses vs. only 5806 analyses of the present algorithm. SQP-MATLAB was considerably slower than the other optimizers and required five times more analyses than LSSO-HHSJA. Such behavior was seen regardless of starting SQP optimizations from very conservative or unconservative designs and can be explained with the inherent complexity of the 200-bar truss problem that makes it difficult for SQP to solve the approximate sub-problems built in each iteration. The present algorithm was slightly less robust in terms of optimized weight than the other HS, BBBC and SA methods analyzed in Refs. [51; 53] but considerably less sensitive to initial population/design than all other optimizers including SQP-MATLAB. However, the standard deviation on optimized weight was always less than 1.4% of the best design for all algorithms compared in this study. LSSO-HHSJA obtained the second lowest standard deviation on the required number of structural analyses after the hybrid HS variant of Ref. [53] but the referenced algorithm obtained slightly heavier designs that violate displacement constraints on average by 0.003254%. Hence, the present algorithm is also the most robust optimizer overall for the 200-bar truss design problem. Figure 3 compares the convergence curves recorded for the best optimization runs of LSSO-JAYA and its competitors. For the sake of clarity, the plot is limited to the first 13,000 structural analyses, and the 8000-128,000 kg structural weight range is selected for the Y-axis. Second, the curves relative to hybrid HS variants [51; 53] are averaged. Last, the figure includes only the plot relative to the best adaptive HS method [38; 39]. It can be seen from the figure that all methods started their optimizations from initial populations, including best individual/designs about 16 times heavier than the target optimization weight of about 12,490 kg. However, structural weight was drastically reduced (by about 150,000 kg) within the first 400 structural analyses by all optimizers except hybrid BBBC with line search strategy [51]. CMLPSA [51] also utilizes line search to perturb design but the set of descent directions considered for this purpose delivers only one trial design at a time while LSSO-JAYA, hybrid HS and hybrid BBBC [51; 53] operate on a population of candidate solutions.