# 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
Table 4 lists the optimized designs obtained with FHGWJA and its main competitors; coordinate values are expressed in meters. The data relative to the standard GWO and JAYA and the improved JAYA are omitted for the sake of brevity as these algorithms converged to the worst solutions overall. The 7079.558 m\({}^{3}\) dam's concrete volume design obtained with FHGWJA is very similar to the optimal solutions obtained with the other algorithms: all design variables changed by at most 5.6% except X\({}_{3}\) that was fixed to its lower bound of 1 mm by FHGWJA and between 4.0001 m and 4.1358 m for the other algorithms. Figure 9: Comparison of convergence curves of FHGWJA and its competitors for the concrete dam optimization problem. Figure 10 compares the optimized shapes of the dam (coordinates are expressed in meters) obtained with FHGWJA and its competitors with the real structure and the optimum design of Refs. [97,98]. For the sake of clarity, the average of hybrid HS, hybrid BBBC, and HFSA configurations and the average of mAHS, mBBBC-UBS, and MsinDE configurations are shown in the figure. It can be seen that all designs resemble the aspect ratio of the real dam: in particular, the ratio between optimized base length and height is very close to 1. The optimized profiles are slightly shifted in the positive X-direction with respect to the real dam and the dam configurations optimized in [97,98] to counteract the effect of the additional horizontal earthquake force 2\(\mathrm{F_{E}}\) included in the design problem. The smaller inclination of the bottom segment of the dam's upstream side increases the dam's storage capacity with respect to the real structure. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Variables** \\ **(m)** \\ \end{tabular} } & \multirow{2}{*}{**FHGWJA**} & \multirow{2}{*}{\begin{tabular}{c} **Hybrid** \\ **HS** [81] \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Hybrid** \\ **BBBC** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **HFSA** \\ **[81]** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **mAHS** \\ **(der. [94])** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **mBBC-UBS** \\ **(der. [95])** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **MsinDE** \\ **(der. [96])** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **MCEO/FSO** \\ **[97,98]** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Existing** \\ **Dam** \\ \end{tabular} } \\ \hline X\({}_{1}\) & 5,0000 & 5,0000 & 5,0006 & 5,0004 & 5,0394 & 5,0175 & 5,0256 & 5,1678 & 4 \\ X\({}_{2}\) & 50.000 & 49.670 & 49.554 & 49.924 & 49.211 & 49.333 & 49.497 & 25.1978 & 42 \\ X\({}_{3}\) & 1.0000 & 4.0132 & 4.0043 & 4.0001 & 4.1358 & 4.0846 & 4.0866 & 4.9556 & 4.2 \\ X\({}_{4}\) & 69.784 & 70.084 & 70.309 & 68.244 & 70.006 & 70.053 & 69.483 & 55.7811 & 50 \\ X\({}_{5}\) & 24.835 & 24.499 & 24.539 & 23.955 & 23.973 & 24.175 & 24.066 & 16.0874 & 12.5 \\ X\({}_{6}\) & 30.000 & 30.239 & 30.131 & 31.673 & 30.696 & 30.924 & 31.090 & 21.0009 & 58 \\ X\({}_{7}\) & 40.000 & 39.997 & 40.025 & 39.984 & 39.974 & 39.968 & 39.919 & 29.9036 & 23.2 \\ X\({}_{8}\) & 99.770 & 100.39 & 100.49 & 99.241 & 100.51 & 100.75 & 100.26 & 120.169 & 140 \\ X\({}_{9}\) & 80.000 & 80.261 & 80.164 & 81.874 & 80.566 & 81.348 & 80.947 & 85.382 & 105 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of optimized designs obtained with FHGWJA and its competitors in the concrete gravity dam problem. Figure 10: Comparison of concrete dam optimized shapes obtained with FHGWJA and its competitors. The safety factors against sliding and overturning evaluated for the optimal solution of FHGWJA became 5.738 and 3.470, respectively, practically the same as those quoted in [81]. Hence, reducing the concrete volume did not affect at all the level of structural safety of the dam. This proves the suitability of the proposed FHGWJA algorithm for civil engineering design problems.