# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Calibration of Nonlinear Hydrologic Models
[81] showed a similar convergence speed to FHGWJA after about 750 function evaluations. Table 6 presents the results obtained with FHGWJA and its competitors in the problem variant with 25 variables; the same nomenclature of Table 5 is utilized. For the sake of clarity, the table reports only the optimal values of channel storage \(S_{t}\) (included as optimization variables 4 through 25) found with FHGWJA. The present algorithm was the best optimizer also in this test problem variant. In fact, it converged to the lowest value overall of SSQ = 36.761, followed by the HFSA and hybrid HS/BBBC algorithms of Ref. [81] that obtained SSQ = 36.762 and 36.763, respectively. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Algorithm** & \(K\) & \(h\) & \(M\) & Best & Average & Worst & ST Dev & **NFE** \\ \hline FHGWJA (Present) \({}^{\star}\) & 0.08611 & 0.2869 & 1.8681 & 36.761 & 36.761 & 36.761 & \(2.919\times 10^{-12}\) & \(4438\pm 544\) \\  & & & & & & & 4122 \\ \hline Standard GWO & 0.08576 & 0.2870 & 1.8694 & 36.818 & 36.870 & 36.934 & \(1.395\times 10^{-2}\) & 50,000 \\ Standard JAYA & 0.08620 & 0.2868 & 1.8677 & 36.765 & 36.766 & 36.769 & \(1.435\times 10^{-3}\) & 22,663 \(\pm\) 8359 \\  & & & & & & & 13,820 \\ Improved JAYA [81] & 0.08591 & 0.2866 & 1.8689 & 36.806 & 36.806 & 36.807 & \(1.078\times 10^{-7}\) & 7276 \(\pm\) 2367 \\  & & & & & & & 5476 \\ \hline Hybrid HS [81] & 0.08614 & 0.2869 & 1.8705 & 36.763 & 36.763 & 36.763 & \(3.333\times 10^{-11}\) & 4744 \(\pm\) 1954 \\  & & & & & & & 3277 \\ Hybrid BBBC [81] & 0.08567 & 0.2868 & 1.8718 & 36.763 & 36.763 & 36.763 & \(8.416\times 10^{-11}\) & 4863 \(\pm\) 1607 \\  & & & & & & & 3656 \\ HFSA [81] & 0.08599 & 0.2867 & 1.8670 & 36.762 & 36.762 & 36.762 & \(1.032\times 10^{-10}\) & 5144 \(\pm\) 1733 \\  & & & & & & & 3887 \\ \hline mAHS (der. [94]) & 0.08624 & 0.2862 & 1.8678 & 36.774 & 36.774 & 36.775 & \(1.263\times 10^{-8}\) & 5385 \(\pm\) 760 \\  & & & & & & & 4610 \\ mBBBC–UBS (der. [95]) & 0.08621 & 0.2872 & 1.8693 & 36.786 & 36.786 & 36.787 & \(1.149\times 10^{-8}\) & 5146 \(\pm\) 797 \\  & & & & & & & 4488 \\ MsinDE (der. [96]) & 0.08604 & 0.2864 & 1.8676 & 36.786 & 36.786 & 36.787 & \(1.456\times 10^{-8}\) & 5848 \(\pm\) 1016 \\ \hline  & & & & & & & & 5261 \\ \hline \hline \end{tabular} \({}^{\star}\) The optimal channel storage values \(S_{t}\) found with FHGWJA are [27.723;26.729;32.45;48.04.41;47.329; 199.195;244.112;229.25;296.016;300.904;291.677;26.154;229.724;187.670;146.693;107.247;75.785;84.842;41.014;30.810; 25.732;20.491) m\({}^{3}\).  
\end{table} 
Table 6: Optimization results of the hydrologic model calibration problem solved with 25 variables. Figure 11: Comparison of convergence curves of FHGWJA and its competitors for the 3-variable hydrologic model calibration problem. Standard JAYA ranked 5 th overall in terms of SSQ but required on average 5.1-times more function evaluations than FHGWJA to complete optimization runs. Improved JAYA and standard GWO found the worst solutions amongst all optimizers (respectively, SSQ = 36.806 and 36.818), requiring on average, respectively, about 1.64- and 11.3-times more function evaluations than the present algorithm. These results confirm the validity of the hybridization scheme implemented in FHGWJA. The \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{best}})\) elitist strategy of FHGWJA significantly improved the performance of the AHS/BBBC-UBS/sinDE algorithms of Refs. [94; 95; 96] also in the 25-variables problem variant. In particular, in their best optimization runs, mAHS reduced the optimized cost from 37.078 of AHS [94] to 36.774 (i.e., the 6 th best result amongst those quoted in Table 6), mBBBC-UBS from 36.826 of BBBC-UBS [95] to 36.786, and MsinDE from 37.101 of sinDE [96] to 36.786. Furthermore, mAHS, mBBBC-UBS, and MsinDE reduced average computational cost by 32.7%, 34.4%, and 28.4% with respect to the original algorithms of Refs. [94; 95; 96], thus becoming competitive enough with FHGWJA in terms of average computational speed.