# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Shape Optimization of a Concrete Gravity Dam Under Multiple Earthquake Loads
[81], implementing a simplified elitist strategy where each new trial solution \(\overset{\rightarrow}{X_{i,tr}}\) is included in the new population only if it improves its counterpart individual \(\overset{\rightarrow}{X_{i}}\) (i.e., if it holds \(W(\overset{\rightarrow}{X_{i,tr}})\leq W(\overset{\rightarrow}{X_{i}})\)), also could not reduce the unit volume of the dam below 7117.44 m\({}^{3}\). The classical GWO was on average about 13-times more computationally \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Best (m\({}^{3}\))** & **Average (m\({}^{3}\))** & **Worst (m\({}^{3}\))** & **ST Dev (m\({}^{3}\))** & **Number Structural** & **Constraint** \\  & **(m\({}^{3}\))** & **(m\({}^{3}\))** & **(m\({}^{3}\))** & **(m\({}^{3}\))** & **Analyses** & **Violation (\%)** \\ \hline FHGWJA (Present) & 7079.558 & 7079.558 & 7079.558 & 0 & \(\dfrac{3834\pm 1597}{2207}\) & 1.441 \(\times 10^{-3}\) \\ \hline Standard GWO & 7117.441 & 7119.288 & 7125.568 & 3.4995 & 50,000 & \(2.423\times 10^{-3}\) \\ Standard JAYA & 7121.134 & 7125.213 & 7129.822 & 5.3946 & 25,000 & \(1.338\times 10^{-3}\) \\ Improved JAYA & 7117.437 & 7119.141 & 7120.890 & 1.7269 & 22,537 \(\pm\) 1996 & \(4.076\times 10^{-4}\) \\
[81] & 7109.608 & 7112.340 & 7113.474 & 2.5882 & 18,487 \(\pm\) 3226 & \(3.641\times 10^{-5}\) \\ Hybrid BBBC [81] & 7108.772 & 7111.175 & 7112.275 & 1.8606 & 18,074 \(\pm\) 2692 & \(3.029\times 10^{-5}\) \\ HFSA [81] & 7113.456 & 7114.225 & 7114.533 & 1.5385 & 19,390 \(\pm\) 1324 & \(1.343\times 10^{-3}\) \\ mAHS & & & & & & & \\ (derived from [94]) & 7109.419 & 7111.476 & 7113.292 & 2.0719 & 24,687 \(\pm\) 2916 & \(3.333\times 10^{-3}\) \\ mBBBC-UBS & & & & & & & \\ (derived from [95]) & 7109.122 & 7110.306 & 7111.004 & 0.7912 & 16,924 & \(6.067\times 10^{-3}\) \\ (MsimDE & & & & & & & \\ (deerived from [96]) & 7113.123 & 7114.350 & 7115.267 & 0.8983 & 24,196 \(\pm\) 2048 & \(4.145\times 10^{-3}\) \\ \hline MCEO [97] & 7448.74 & N/A & N/A & \(3.274\times 10^{-2}\) & 35,000 & 32 \\ FSO [98] & 7448.74 & N/A & N/A & N/A & N/A & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimization results of the concrete gravity dam design problem. expensive than the present algorithm, while classical JAYA and improved JAYA were on average about 6.5- and 5.9-times slower than FHGWJA. The \(W(X_{i,tr}^{\rightarrow})\leq 1.1W(X_{best}^{\rightarrow})\) elitist strategy implemented in FHGWJA again improved significantly the performance of the mAHS, mBBBC-UBS, and MsinDE algorithms with respect to the original AHS, BBBC-UBS, and sinDE formulations documented in [94; 95; 96]: in particular, the best solutions of mAHS, mBBBC-UBS, and MsinDE became very close to those of hybrid HS/BBBC/SA while computational cost decreased on average by 16.5%, 6.1%, and 35.2%, respectively. Figure 9 clearly shows the superior convergence behavior of FHGWJA with respect to its competitors. The figure compares the optimization histories of the best runs for the algorithms listed in Table 3; the average FHGWJA convergence curve also is shown in the figure. The plot is limited to the first 21,000 structural analyses of the optimization history and to the 7000-14,000 m\({}^{3}\) concrete volume interval for the sake of clarity. It can be seen from the figure that the best and average optimization runs' convergence curves of FHGWJA always lie below those of the other algorithms. FHGWJA started its best optimization run from the very large cost of 18,372.761 m\({}^{3}\) (i.e., about 2.6 times the globally optimum length of 7079.558 m\({}^{3}\) found with FHGWJA) while the initial cost for all other optimizers ranged between 13,265.001 and 14,209.219 m\({}^{3}\). However, the present algorithm immediately recovered the initial gap in cost function with respect to its competitors. The hybrid simulated annealing algorithm (HFSA) of Ref. [81] was the only optimizer to compete in convergence speed with FHGWJA for the first 100 structural analyses.