# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Calibration of Nonlinear Hydrologic Models
It can be seen from Table 5 that the present algorithm always converged to the lowest value of SSQ = 36.760 in all independent optimization runs thus reaching 0 standard deviation on optimized cost. Hence, FHGWJA was the most robust optimizer overall. Standard and improved JAYA, modified AHS/BBBC-UBS/sinDE, COBSA, and NMS also found the globally optimum solution. However, none of these algorithms converged to the same design in all optimization runs: standard deviations on optimized SSQ values ranged between 1.442 \(\times\) 10\({}^{-4}\) (improved JAYA) and 0.01 (COBSA). All other methods found optimized solutions ranging between SQQ = 36.761 (standard GWO and hybrid HS/BBBC/SA) and 36.77 (HS, IICSA, GA, DE, and PSO). The hybrid FHGWJA algorithm was significantly more efficient than its components, the GWO and JAYA. In particular, the standard GWO converged to SSQ = 36.761 after 50,000 function evaluations (i.e., 25-times slower than FHGWJA) while standard JAYA converged to SSQ = 36.760 but it was on average 2.7-times slower than the present algorithm. Improved JAYA was on average 20.3% slower than FHGWJA while their fastest optimization runs practically required the same number of function evaluations (i.e., 1637 for FHGWJA vs. 1766 for improved JAYA). However, the fastest optimization run of im \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) & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.760 & 36.760 & 0 & \(\begin{array}{c}2005\pm 386\\ 1637\end{array}\) \\ \hline Standard GWO & 0.08621 & 0.2869 & 1.8681 & 36.761 & 36.762 & 36.764 & \(1.258\times 10^{-3}\) & 50,000 \\ Standard JAYA & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.761 & 36.763 & \(1.500\times 10^{-3}\) & \(\begin{array}{c}5438\pm 756\\ 4331\end{array}\) \\ Improved JAYA [81] & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.761 & 36.762 & \(1.442\times 10^{-4}\) & \(\begin{array}{c}2412\pm 684\\ 1766\end{array}\) \\ \hline Hybrid HS [81] & 0.08633 & 0.2869 & 1.8677 & 36.761 & 36.762 & \(36.762\) & \(4.892\times 10^{-4}\) & \(\begin{array}{c}2547\pm 1035\\ 1331\end{array}\) \\ Hybrid BBBC [81] & 0.08618 & 0.2869 & 1.8682 & 36.761 & 36.762 & \(1.769\times 10^{-5}\) & \(\begin{array}{c}2468\pm 1102\\ 1281\end{array}\) \\ HFSA [81] & 0.08632 & 0.2869 & 1.8678 & 36.761 & 36.762 & \(36.762\) & \(3.891\times 10^{-4}\) & \(\begin{array}{c}2755\pm 1344\\ 1326\end{array}\) \\ mAHS (der. [94]) & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.761 & \(36.762\) & \(1.543\times 10^{-4}\) & \(\begin{array}{c}2368\pm 967\\ 1670\end{array}\) \\ mBBBC-UBS (der. [95]) & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.761 & \(36.762\) & \(2.044\times 10^{-4}\) & \(\begin{array}{c}2344\pm 835\\ 1711\end{array}\) \\ MsinDE (der. [96]) & 0.08620 & 0.2869 & 1.8681 & 36.760 & 36.762 & \(36.763\) & \(2.119\times 10^{-3}\) & \(\begin{array}{c}2966\pm 1293\\ 2213\end{array}\) \\ \hline HS [102] & 0.0870 & 0.2870 & 1.8661 & 36.77 & N/A & N/A & N/A & 20,000 \\ IICSA [103] & 0.0865 & 0.2870 & 1.8675 & 36.77 & N/A & N/A & N/A & N/A \\ COBSA [104] & 0.0864 & 0.2869 & 1.8678 & 36.76 & 36.77 & 36.77 & 0.01 & 100,000 \\ GA [104] & 0.0865 & 0.2869 & 1.8674 & 36.77 & 36.85 & 37.24 & 0.13 & N/A \\ DE [104] & 0.0863 & 0.2869 & 1.8680 & 36.77 & 37.09 & 38.04 & 0.36 & N/A \\ PSO [104] & 0.0867 & 0.2867 & 1.8668 & 36.77 & 36.84 & 37.25 & 0.14 & N/A \\ BFGS [105] & 0.0863 & 0.2869 & 1.8679 & 36.768 & N/A & N/A & N/A & N/A \\ NMS [106] & 0.0862 & 0.2869 & 1.8681 & 36.76 & N/A & N/A & N/A & 221 \({}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimization results of the hydrologic model calibration problem solved with 3 unknown parameters. proved JAYA converged to SSQ = 36.762 vs. the global optimum of 36.760 reached with the present algorithm in all runs.