# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Calibration of Nonlinear Hydrologic Models
The superiority of FHGWJA over its competitors is confirmed by the fact that the present algorithm achieved the lowest standard deviation on optimized cost (only 2.919 \(\times\) 10\({}^{-12}\) vs. 3.333 \(\times\) 10\({}^{-11}\) to 1.395 \(\times\) 10\({}^{-2}\) of the other algorithms). Remarkably, the worst solution of FHGWJA practically coincided with its best solution, and it was always better than the best solutions found with its competitors. Furthermore, FHGWJA was faster than the other optimizers completing on average the 20 independent optimization runs within only 4438 function evaluations vs. 4744 to 5848 evaluations of hybrid HS/BBBC/SA and modified AHS/BBBC-UBS/sinDE. Interestingly, the hybrid HS/BBBC/SA algorithms of Ref. [81] completed their fastest optimization runs within less function evaluations than FHGWJA (respectively, only 3277, 3656, and 3887 vs. 4122). However, FHGWJA found intermediate solutions corresponding to SSQ = 36.763 (i.e., the best solutions of hybrid HS and hybrid BBBC) always within only 3200 function evaluations, and SSQ = 36.762 (i.e., the best solution of HFSA) always within only 3700 function evaluations. Figure 12 compares the optimization histories of the best runs for the algorithms listed in Table 6; the average FHGWJA convergence curve also is shown in the figure. The plot is limited to the first 4500 function evaluations of optimization history and to the 35-80 cost function interval for the sake of clarity. The convergence behavior depicted in the figure resembles that observed for the three-variable problem variant. The best and average optimization runs' convergence curves for FHGWJA again lie below those of the other algorithms practically over the whole search history. FHGWJA started its best optimization run from the very large cost of 3547.602 (i.e., about 96.5 times the target optimum of 36.761 quoted in Table 6) while the initial cost for all other optimizers ranged between 461.502 and 2063.123, except the improved JAYA variant of Ref. [81] that started from 75.965. However, the present algorithm immediately recovered the initial gap in cost function with respect to its competitors reducing cost to 37.2 (just 1.2% more than the target optimum) within only 320 function evaluations while all other algorithms required at least 995 function evaluations to reach the same intermediate cost function value. The hybrid big bang-big crunch algorithm of Ref. [81] was competitive with FHGWJA only for the first 190 function evaluations. The \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{opt}})\) elitist strategy of FHGWJA allowed mAHS and MsinDE to approach the cost function reduction rate of FHGWJA from 220 to 310 function evaluations, while mBBBC-UBS generated the closest intermediate designs to those of FHGWJA between 650 and 1000 function evaluations. The excellent performance of FHGWJA in the two variants of the hydrologic model calibration problems confirms the suitability of the proposed algorithm for highly nonlinear optimization problems. Interestingly, the higher level of design freedom introduced in the optimization search by the additional 22 variables did not affect at all the robustness of FHGWJA that was able to converge practically to the same values (with at most 0.105% difference) of Muskingum model parameters (i.e., \(K\), \(h\), and \(m\)) and channel storage values \(S_{t}\) found in the three-variables problem variant.