# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### 2D Path Planning
The optimal positions of base points found with FHGWJA were (3.6617;1.3944), (6.5927;2.7934), (16.562;11.705), (21.231;17.081), (24.186;19.826), (27.855;26.254), and (28.626;28.196) mm. The other algorithms ranked as follows: hybrid HS [81], MsinDE (derived from [96]), hybrid BBBC [81], HFSA [81], mAHS (derived from [94]), mBBBC-UBS (derived from [95]), and improved JAYA [81]. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Algorithm** & **Best (mm)** & **Average (mm)** & **Worst (mm)** & **ST Dev (mm)** & **Number of Analyses** \\ \hline FHGWJA (present) & 40.811 & 40.859 & 40.890 & 3.331 \(\times\) 10\({}^{-2}\) & 4581 \(\pm\) 1127 \(\blacktriangle\) \\ \hline Standard GWO & 41.104 & 41.112 & 41.129 & 1.340 \(\times\) 10\({}^{-2}\) & 50,000 \\ Standard JAYA & 41.050 & 41.092 & 41.162 & 6.080 \(\times\) 10\({}^{-2}\) & 17,204 \\ Improved JAYA [81] & 41.082 & 41.104 & 41.119 & 1.850 \(\times\) 10\({}^{-2}\) & 6523 \\ \hline Hybrid HS [81] & 40.983 & 41.040 & 41.105 & 6.133 \(\times\) 10\({}^{-2}\) & 4150 \\ Hybrid BBBC [81] & 40.986 & 41.078 & 41.101 & 6.110 \(\times\) 10\({}^{-2}\) & 3611 \\ HFSA [81] & 40.994 & 41.048 & 41.113 & 6.035 \(\times\) 10\({}^{-2}\) & 4127 \\ mAHS (der. from [94]) & 41.059 & 41.067 & 41.095 & 1.890 \(\times\) 10\({}^{-2}\) & 7237 \\ mBBBC–UBS (der. from [95]) & 41.081 & 41.099 & 41.112 & 1.136 \(\times\) 10\({}^{-2}\) & 5724 \\ MsinDE (der. from [96]) & 40.985 & 41.002 & 41.026 & 2.122 \(\times\) 10\({}^{-2}\) & 6291 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimization results for the 2D path planning problem solved with 14 design variables. The rate of success of FHGWJA was 100% because all independent optimization runs designed shorter trajectories than the best solutions obtained with its competitors. In particular, the worst solution of FHGWJA achieved a path length of 40.890 mm while the shortest paths designed via all other optimizers ranged between 40.983 and 41.104 mm. Such a behavior occurred because FHGWJA inherently balances exploration and exploitation phases enhancing the exploration capability of \(\alpha\), \(\beta\), and \(\delta\) wolves without risk of stagnation and forcing the optimizer to search very high-quality solutions regardless of performing exploration or exploitation. Interestingly, the \(W(X_{itr}^{\rightarrow})\leq 1.1W(X_{best}^{\rightarrow})\) elitist strategy implemented in FHGWJA was effective also for mAHS, mBBBC-UBS, and MsinDE that improved their optimized solutions with respect to the original AHS [94], BBBC-UBS [95], and sinDE [96] algorithms utilized in [81]: in particular, MsinDE enhanced its overall rank up to the 3 rd position. FHGWJA was the fastest optimizer overall in the 2D path planning problem with 14 variables, requiring on average only 4581 analyses vs. 50,000 and 17,204 analyses, respectively, required for standard GWO and JAYA to converge to their optimized solutions. Furthermore, the present algorithm was about 42.3% faster than the improved JAYA. The computational speed for the fastest optimization run of FHGWJA was practically the same as those of the HS/BBBC/SA hybrid algorithms: only 3837 analyses vs. 4150, 3611, and 4127 analyses, respectively. However, the present algorithm generated a feasible intermediate design with a path length of 40.961 mm (hence, shorter than optimized path lengths of hybrid HS/BBBC/SA ranging between 40.983 and 40.994 mm) within only 2875 analyses. FHGWJA ranked 6 th in terms of standard deviation on optimized path length after mBBBC-UBS, standard GWO, improved JAYA, mAHS, and MsinDE, which, however, completed their optimizations within more analyses than the present algorithm and converged to worse designs. The present algorithm was also robust enough with respect to computational cost showing only 24.6% dispersion on the number of analyses required in the 20 independent optimization runs. Figure 5 confirms the superiority of FHGWJA over its competitors in terms of convergence behavior.