# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### 2D Path Planning
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{1}{c}{**Algorithm**} & \begin{tabular}{c} **Best** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Average (mm)** \\ \end{tabular} & \begin{tabular}{c} **Worst** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **ST Dev (mm)** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Analyses** \\ \end{tabular} \\ \hline FHGWJA (present) & 40.898 & 40.901 & 40.924 & \(1.161\times 10^{-2}\) & 5153 \(\pm\) 248 \(\blacktriangle\) \\ \hline Standard GWO & 41.134 & 41.146 & 41.165 & \(1.276\times 10^{-2}\) & 50,000 \\ Standard JAYA & 41.112 & 41.160 & 41.187 & \(3.102\times 10^{-2}\) & 11,778 \\ Improved JAYA [81] & 41.141 & 41.184 & 41.209 & \(2.808\times 10^{-2}\) & 11,393 \\ \hline Hybrid HS [81] & 40.997 & 41.053 & 41.116 & \(4.861\times 10^{-2}\) & 7408 \\ Hybrid BBBC [81] & 41.034 & 41.091 & 41.108 & \(3.165\times 10^{-2}\) & 7542 \\ HFSA [81] & 41.004 & 41.060 & 41.127 & \(5.028\times 10^{-3}\) & 8245 \\ mAHS (der. from [94]) & 41.075 & 41.092 & 41.116 & \(1.682\times 10^{-2}\) & 8541 \\ mBBBC-UBS (der. from [95]) & 41.091 & 41.123 & 41.150 & \(2.412\times 10^{-2}\) & 7492 \\ MsinDE (der. from [96]) & 41.030 & 41.077 & 41.092 & \(2.641\times 10^{-2}\) & 8762 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization results for the 2D path planning problem solved with 20 design variables. Figure 5: Comparison of convergence curves of FHGWJA and its competitors for the 2D path planning problem solved with 14 design variables. In the vertical axis, the ”,” notation indicates the decimal signs. in particular, computational cost decreased on average by 36%. Interestingly, the high level of nonlinearity of 2D path planning problem variants including 14 and 20 optimization variables was better handled with FHGWJA than with the hybrid HS/BBBC/SA algorithms of Ref. [81]. This occurred in spite of the fact that hybrid HS/BBBC/SA utilized gradient information and multiple line searches to minimize the number of analyses. However, this limited the approximation quality and hence the ability of the optimizers to generate trial solutions effectively located in regions of the search space where \(X_{best}^{\rightarrow}\) is likely to improve. Figure 6 compares the optimization histories of the best runs for FHGWJA and its competitors in the 2D path planning problem solved with 20 design variables; the average FHGWJA convergence curve is also shown in the figure. The plot is limited to the first 9000 analyses of optimization history for the sake of clarity. Similar to the 14-variable problem variant, the best and average optimization runs' convergence curves of FHGWJA practically always lie below those of the other algorithms. All algorithms started their best optimization runs from the very large cost of 238.54 mm (i.e., about 5.83 times the optimum trajectory length of 40.898 mm found with FHGWJA). However, the present algorithm recovered the gap in cost function with respect to target optimum in a much faster way than the other algorithms. The hybrid simulated annealing algorithm (HFSA) of Ref. [81] was the only optimizer to compete in convergence speed with FHGWJA but only after 2000 analyses, that is when the distance from target optimum was less than 5%. Figure 7a compares the optimized trajectories obtained with FHGWJA and its competitors in the 2D path planning problem variant including 14 design variables. For the sake of clarity, the figure shows only the best six solutions quoted in Table 1. It can be seen that optimized paths involve only obstacles 1 and 7. Furthermore, the trajectory branch limited by the start point A and obstacle 1 is practically the same for all optimizers. Optimal  
Figure 6: Comparison of convergence curves of FHGWJA and its competitors for the 2D path planning problem solved with 20 design variables. trajectories diverge between obstacles 1 and 7: the slope is larger for hybrid HS, HFSA, and standard JAYA than for the present algorithm, hybrid BBBC, and MsinDE.