# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Shape Optimization of a Concrete Gravity Dam Under Multiple Earthquake Loads
The last inequality constraint \(G_{7}\) in Equation (21) is relative to geometric proportions in the vertical direction, while the equality constraint \(H_{8}\) requires the dam height to be 150 m.  
The concrete volume per unit width of 10,502.1 m\({}^{3}\) of the real dam was significantly reduced to 7448.774 m\({}^{3}\) via the multi-level cross entropy optimizer (MCEO) [97] and the flying squirrel optimizer (FSO) [98]. However, Refs. [97; 98] neither considered the equality constraint \(H_{8}\) nor the presence of the horizontal earthquake force 2F\({}_{\rm E}\). These issues were instead included in [81], which presented the results of hybrid HS, hybrid BBBC, hybrid fast simulated annealing (HFSA), adaptive harmony search (AHS) [94], big bang-big crunch with upper bound strategy (BBBC-UBS) [95], and sinusoidal differential evolution [96]. For this reason, the FHGWJA algorithm developed in this study was compared with hybrid HS/BBBC/SA as well as with modified adaptive harmony search (mAHS), modified big bang-big crunch with upper bound strategy (mBBBC-UBS), and modified sinusoidal differential evolution (MsinDE). Similar to the 2D path planning problem, mAHS, mBBBC-UBS, and MsinDE included the \(W(X_{i,tr}^{\rightarrow})\leq 1.1W(X_{best}^{\rightarrow})\) elitist strategy implemented in FHGWJA to enhance the performance of the original algorithms. Similar to the 2D path planning problem, 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 \(N_{\rm POP}=10\) in this study. Table 3 compares the optimization results of FHGWJA and its competitors for the dam problem. The number of structural analyses corresponds to the number of evaluations of constraints \(G_{1-7}(\overrightarrow{X})\) and \(H_{g}(\overrightarrow{X})\). The average number of structural analyses and corre  
FPGWJA was the best optimizer overall also in this test problem. In fact, it converged to the lowest concrete volume for unit width of 7079.558 m\({}^{3}\), which is 0.411% smaller than the 7108.772 m\({}^{3}\) for the best design quoted in literature so far for hybrid BBBC [81]. All algorithms performed significantly better than MCEO [97] and FSO [98]: in fact, optimized values of unit volume ranged from 7059.558 m\({}^{3}\) to 7121.134 m\({}^{3}\) vs. 7448.774 m\({}^{3}\) of Refs. [97; 98]. The solutions quoted in Table 3 fully satisfied design constraints \(G_{1}(\overset{\rightarrow}{X})\) through \(G_{7}(\overset{\rightarrow}{X})\) and practically the geometric equality constraint \(H_{8}(\overset{\rightarrow}{X})\) on dam height (in fact, violation never exceeded 6.067 \(\times\) 10\({}^{-3}\)%). The algorithms ranked as follows in terms of optimized concrete volume values: FHGWJA, hybrid BBBC, mBBBC-UBS, mAHS, hybrid HS, MsinDE, HFSA, improved JAYA, standard GWO, and standard JAYA. FHGWJA was also the fastest and most robust optimizer. In fact, it converged to the global optimum of 7059.558 m\({}^{3}\) in all independent optimization runs with zero standard deviation. Conversely, none of its competitors could achieve zero standard deviation on optimized concrete volume values. The present algorithm was on average between 4.7- and 6.5-times faster than its competitors; the fastest optimization run converging to the global optimum was completed by FHGWJA within only 2207 structural analyses vs. 16,924 structural analyses required for mBBBC-UBS that, however, obtained a higher value for the dam's concrete volume. The hybrid formulation implemented in FHGWJA was significantly more efficient than its component algorithms GWO and JAYA that converged in their standard formulation to the best solutions of 7117.441 m\({}^{3}\) and 7121.134 m\({}^{3}\), respectively. The improved JAYA algorithm of Ref.