# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Weight Minimization of a Planar 200-Bar Truss Structure  
The last test case solved in this study regarded the weight minimization of the planar 200-bar truss structure shown in Figure 15. The structure, made of steel (Young's modulus of 206.91 GPa; mass density of 7833.413 kg/m\({}^{3}\)), is composed of 200 elements connected by 77 nodes. Because of structural symmetry, the 200 elements are categorized in 29 groups as indicated in Table 8: the elements of each group have the same cross-sectional area, which is taken as a sizing optimization variable. Hence, this test problem has 29 design variables. The structure is subjected to three independent loading conditions:  
1. Concentrated forces of 4.45 kN (i.e., 1000 lbf) acting in the positive X-direction at nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, and 71 (denoted by the green horizontal arrows in Figure 15);
2. Concentrated forces of 44.497 kN (i.e., 10,000 lbf) acting in the negative Y-direction at nodes 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 42, 43, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 59, 60,61,62,64, 66, 68, 70, 71, 72, 73, 74, and 75 (denoted by the vertical horizontal arrows in Figure 15);  
Figure 14: Comparison of convergence curves of FHGWJA and its competitors for the optimal crashworthiness problem. The truss weight must be minimized under 1200 nonlinear constraints on element stresses that should not exceed \(\pm\)68.97 MPa (i.e., \(\pm\)10,000 psi, the same limit stress under tension and compression). The cross-sectional areas taken as sizing variables can vary between 0.64516 cm\({}^{2}\) (i.e., 0.1 in\({}^{2}\)) and 645.16 cm\({}^{2}\) (i.e., 100 in\({}^{2}\)). Figure 15: Schematic of the planar 200-bar truss including applied loads and kinematic constraints. (MCOA) that uses chaotic sequences instead of random sequences for perturbing solutions (i.e., 11,544.007 kg, [112]); (iv) the cyclic neighborhood network topology particle swarm optimizer (CNNT-PSO) that handles population diversity based on the interactions between each particle and its neighboring individuals (i.e., 11,545.330 kg, [113]); (v) the hybrid geometric mean optimizer (hGMO) that integrates GMO with variable neighborhood to enhance exploitation (i.e., 11,545.568, [114]). Other advanced MHAs converged to lighter designs than the 11,542.4 kg target weight, but their optimized solutions were infeasible. For example, the corrected multi-level and multi-point simulated annealing (CMLPSA) algorithm of Ref. [90], that generates the new trial design using a population of candidate designs lying on descent directions rather than developing a single point as is conducted in classical SA, found an optimized weight of 11,542.137 kg (practically the same value of target optimum weight) violating stress constraints by only 0.0709%. The hybrid HPSACO algorithm [115], combining harmony search, particle swarm, and ant colony optimization, found an optimum weight of 11,410.737 kg but stress constraints were violated by 9.97%. An improved Harris hawks optimization algorithm was used in [116]: the optimized design weighted 11,533.983 kg (i.e., 0.073% less than target optimum weight) with 9.55% violation on stress constraints. PSOHHO [117], combining the particle swarm optimization and Harris hawks optimization methods, obtained 11,374.477 kg structural weight (i.e., 1.46% reduction) but the corresponding solution violated stress constraints by about 27.6%. Finally, the biogeography-based optimization [118] reduced structural weight to 11,320.758 kg (i.e., 1.92% reduction) but the corresponding design violated stress constraints by 11.33%. In view of the above arguments, the optimization results obtained with the FHGWJA algorithm developed in this study were compared with those of the hybrid HS/BBBC/SA [81], ECM [110], LSHADE-Epsin [111], MCOA [112], CNNT-PSO [113], and hGMO [114] algorithms providing the best feasible solutions available in the literature.