# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Calibration of Nonlinear Hydrologic Models  
Test problems (iv-v) regarded the calibration of nonlinear hydrologic models including 3 or 25 unknown parameters. This is a typical inverse problem in the hydrology field [99; 100; 101]. The Muskingum model was selected in this study. The difference between observed river outflows and their counterpart computed with the Muskingum model over a period of 126 h, split in 21 intervals \(\Delta t=6\) h, was minimized. The inverse problem was formulated as an unconstrained optimization problem:  
\[Minimize\;SSQ=\sum_{t=1}^{NCT}(O_{\sigma o,t}-O_{\sigma o,t})^{2} \tag{22}\]  
where \(O_{\sigma o,t}\) and \(O_{\sigma o,t}\), respectively, are the observed and computed river outflows at time \(t\); \(NCT=22\) is the number of control points defined by the 21 time intervals. The computed river outflow \(O_{\sigma o,t}\) can be determined using the classical three-parameter nonlinear Muskingum model:  
\[O_{\sigma o,t}=\frac{1}{1-h}\bigg{(}\frac{S_{t}}{K}\bigg{)}^{\frac{1}{t}}(1-h )\;I_{t} \tag{23}\]  
In Equation (23), \(K\), \(h\), and \(r\) are the parameters of Muskingum model to be calibrated, respectively, taken as optimization variables \(x_{1}\), \(x_{2}\), and \(x_{3}\); \(S_{t}\) and \(I_{t}\), respectively, are the channel storage and measured river inlet at time \(t\). The channel storage \(S_{t}\) at time \(t\) is updated as follows:  
\[S_{t+1}=S_{t}+\dot{S_{t}}\Delta t \tag{24}\]  
The storage rate \(\dot{S_{t}}\) at time \(t\) is as follows:  
\[\dot{S_{t}}=\frac{1}{1-h}\cdot\bigg{(}\frac{S_{t}}{K}\bigg{)}^{\frac{1}{t}}+ \frac{1}{1-h}\cdot I_{t} \tag{25}\]  
Two variants of the hydrologic model calibration problem were solved in this study: (i) the classical problem including only 3 optimization variables, the unknown parameters \(K\), \(h\), and \(r\)[102; 103; 104; 105; 106]; (ii) the extended problem solved in [81] including also the 22 storage values \(S_{t}\) as additional design variables for a total of 25 optimization variables. The side constraints on Muskingum model parameters and storage values are as follows: \(0.01\leq k\leq 0.2\); \(0.2\leq h\leq 0.3\); \(1.5\leq r\leq 2.5\), and \(1\leq S_{t}\leq 1000\) m\({}^{3}\). The FHGWJA algorithm developed in this study was compared with the best solutions of the literature obtained with following: (i) metaheuristic methods such as the hybrid HS/BBBC/SA algorithms of Ref. [81], other HS variants [102], improved immune clonal selection (IICSA) [103], backtracking search with orthogonal initialization and chaotic map (COBSA) [104], genetic algorithms (GA) [104], differential evolution (DE) [104], and particle swarm optimization (PSO) [104]; (ii) modified adaptive harmony search (mAHS), modified big bang-big crunch with upper bound strategy (mBBBC-UBS), and modified sinusoidal differential evolution (MsinDE) including the FHGWJA's elitist strategy \(W(\overset{\rightarrow}{X_{i,tr}})\leq 1.1W(\overset{\rightarrow}{X_{best}})\) to improve the original AHS/BBBC-UBS/sinDE algorithms of Refs. [94; 95; 96]; (iii) gradient-based methods such as the Broyden-Fletcher-Goldfarb-Shanno algorithm (BFGS) [105]; (iv) zero-order heuristic methods such as the Nelder-Mead Simplex algorithm (NMS) [106]. FHGWJA was also compared with the standard GWO as well as standard and improved JAYA variants [81]. Similar to the previous test cases, population size was set equal to 20 in Ref. [81] for hybrid HS, hybrid BBBC, and improved JAYA, while mAHS, mBBBC-UBS, and MsinDE were run with \(N_{\text{POP}}=10\) in the present study. Table 5 presents the optimization results of FHGWJA and its competitors for the three-variables problem variant. The table reports the average number of evaluations of the SSQ cost function (\(NFE\)) and corresponding standard deviation, along with the number of function evaluations required by the fastest optimization run when available.