# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Planar 200-Bar Truss Structure
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Optimized Weight (kg)** & **Number of Structural Analyses** & **Constraint Violation (\%)** \\ \hline \multirow{4}{*}{**LSSO-HHSIJA Present**} & Best: 12,483.339 & Best: 5637 & \multirow{4}{*}{Feasible} \\  & Worst: 12,484.135 & Fastest: 5436 & \multirow{4}{*}{Feasible} \\  & Mean: 12,483.791 & Slowest: 6373 & \\  & STD: 0.3144 & Mean/STD: 5806 \(\pm\) 357 & \\ \hline \multirow{4}{*}{**Hybrid HS with LS [51]**} & Best: 12,490.332 & Best: 5679 & Best: 0.00730 \\  & Worst: 12,490.542 & Fastest: 5668 & Worst: 0.00750 \\  & Mean: 12,490.430 & Slowest: 6604 & Mean: 0.00695 \\  & STD: 0.07470 & Mean/STD: 6031 \(\pm\) 377 & STD: 0.000772 \\ \hline \multirow{4}{*}{**Hybrid HS with LS [53]**} & Best: 12,489.457 & Best: 6040 & Best: 0.003250 \\  & Worst: 12,489.778 & Fastest: 5621 & Worst: 0.002802 \\  & Mean: 12,489.596 & Slowest: 6372 & Mean: 0.003254 \\  & STD: 0.1291 & Mean/STD: 5966 \(\pm\) 324 & STD: 0.0002565 \\ \hline \multirow{4}{*}{**Hybrid BBBC with LS [51]**} & Best: 12,490.439 & Best: 7745 & Best: 0.00559 \\  & Worst: 12,490.932 & Fastest: 1924 & Worst: 0.00625 \\  & Mean: 12,490.680 & Slowest: 9460 & Mean: 0.00626 \\  & STD: 0.1773 & Mean/STD: 5652 \(\pm\) 2912 & STD: 0.000347 \\ \hline \multirow{4}{*}{**CMLPSA [51]**} & Best: 12,492.888 & Best: 11,726 & \multirow{4}{*}{Feasible} \\  & Worst: 12,493.290 & Fastest: 10,338 & \\  & Mean: 12,493.081 & Slowest: 12,118 & \\  & STD: 0.2014 & Mean/STD: 11,394 \(\pm\) 935 & \\ \hline \multirow{4}{*}{**Improved/parameterless JAYA [54; 55; 56]**} & Best: 12,490.603 & Best: 12,869 & Best: 0.03490 \\  & Worst: 12,502.175 & Fastest: 12,316 & Worst: Feasible \\  & Mean: 12,494.460 & Slowest: 14,326 & Mean: 0.01888 \\  & STD: 3.056 & Mean/STD: 12,818 \(\pm\) 601 & STD: 0.01644 \\ \hline \multirow{4}{*}{**AHS [38]**} & Best: 12,497.475 & Best: 16,981 & Best: 0.1570 \\  & Worst: 12,777.339 & Fastest: 15,063 & Worst: Feasible \\  & Mean: 12,567.441 & Slowest: 21,412 & Mean: 0.08363 \\  & STD: 174.716 & Mean/STD: 17,130 \(\pm\) 2996 & STD: 0.06836 \\ \hline \multirow{4}{*}{**SAHS [39]**} & Best: 12,495.939 & Best: 15,812 & Best: 0.1824 \\  & Worst: 12,669.384 & Fastest: 13,384 & Worst: Feasible \\  & Mean: 12,537.54 & Slowest: 17,464 & Mean: 0.09718 \\  & STD: 144.817 & Mean/STD: 15,618 \(\pm\) 1681 & STD: 0.07942 \\ \hline \multirow{4}{*}{**BBBC-UBS [57]**} & Best: 12,490.035 & Best: 15,250 & Best: 0.05540 \\  & Worst: 12,502.346 & Fastest: 13,865 & Worst: Feasible \\  & Mean: 12,497.562 & Slowest: 16,198 & Mean: 0.03048 \\  & STD: 4.170 & Mean/STD: 14,795 \(\pm\) 1141 & STD: 0.02837 \\ \hline \multirow{4}{*}{**sinDE [58]**} & Best: 12,502.536 & Best: 21,653 & Best: 0.05880 \\  & Worst: 12,541.640 & Fastest: 17,124 & Worst: Feasible \\  & Mean: 12,520.999 & Slowest: 22,635 & Mean: 0.03030 \\  & STD: 16.369 & Mean/STD: 20,766 \(\pm\) 2472 & STD: 0.03017 \\ \hline \multirow{4}{*}{**SQP-MATLAB**} & Best: 12,491.400 & Best: 28,198 & Best: 0.05131 \\  & Worst: 12,503.300 & Fastest: 24,619 & Worst: Feasible \\  & Mean: 12,498.034 & Slowest: 38,410 & Mean: 0.03298 \\ \cline{1-1}  & STD: 5.661 & Mean/STD: 30,990 \(\pm\) 5957 & STD: 0.03969 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization results obtained for the 200-bar truss design example. SAHS [39] exhibited the largest constraint violations amongst the different methods as well as the largest structural weights for the worst optimization runs. A very important point is to limit the number of descent directions utilized to form high quality trial designs and update population in the current iteration. The JAYA-based Equations (14), (16) and (19) combined by LLSO-HHSJA with the line search-based HS architecture greatly help to accomplish these tasks. Furthermore, JAYA naturally tries to avoid low quality designs, and this reduces the probability of dealing with infeasible regions of search space. Finally, Equations (14), (16) and (19) make it unnecessary to perform 1-D probabilistic searches based on SA if trial designs are worse than the current best record.