# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Optimal Crashworthiness Design of a Vehicle Subject to Side Impact
However, the optimized values of variables \(\mathrm{x_{g}}\) and \(\mathrm{x_{g}}\) listed in Table 7 for the IGWO and chaotic GWO do not coincide with the available discrete values {0.192;0.345} originally set for this test problem: the optimized designs of the IGWO and GWO would become infeasible as soon as \(\mathrm{x_{8}}\) is set equal to 0.192 or 0.345, thus violating constraint functions \(G_{6}\), \(G_{7}\), \(G_{8}\), and \(G_{9}\) by up to 13.9 and 23.7%, respectively. Interestingly, the average optimized weight achieved with FHGWJA over the 20 independent runs was lower than the best solutions of IGWO and chaotic GWO: only 21.3905 kg vs. 21.39473 and 21.46164 kg, respectively. Furthermore, the present algorithm required on average only 1309 structural analyses vs. the 30,000 analyses indicated in [69] for the IGWO and chaotic GWO. The hybrid optimizer proposed here clearly outperformed the standard GWO and standard JAYA that obtained 22.84755 and 23.19155 kg, respectively. Furthermore, FHGWJA required on average only 1309 structural analyses vs. 50,000 of standard GWO and 7560 of standard JAYA for completing the search process. The adaptive dynamic self-learning grey wolf optimization algorithm (ASGWO) [70] ranked below the standard GWO in terms of optimized weight (22.87188 kg vs. 22.84755 kg) but it required only 15,000 structural analyses vs. 50,000 analyses of the standard GWO. In summary, using rather simple elitist strategies and properly refining/repairing trial solutions (so that they always lie on descent directions) allowed FHGWJA to outperform complex GWO schemes combining special perturbations of design variables or several strategies taken from various metaheuristic algorithms. This happened because only the present algorithm can always generate high-quality trial solutions throughout the optimization process. The starling murmuration optimizer (SMO) and the other algorithms compared in Ref. [70] also were less efficient than FHGWJA. In fact, they required 30,000 structural analyses vs. only 1309 of the present algorithm to find heavier optimized weights than FHGWJA (i.e., from 22.84298 to 23.12717 kg vs. only 21.38340 of FHGWJA). The present algorithm was very robust: the standard deviation on optimized weight was only 0.0333% of the average optimized weight while the ratio between the standard deviation on number of structural analyses and the average number of structural analyses was 26.2%. Statistical data relative to the other algorithms compared with FHGWJA usually were not available in the literature. This can be explained in view of the formulation of the optimal crashworthiness design problem. The cost function \(W(\overset{\rightarrow}{X})\) stated by Equation (26) depends only on variables \(\mathrm{x_{1}}\), \(\mathrm{x_{2}}\), \(\mathrm{x_{3}}\), \(\mathrm{x_{4}}\), \(\mathrm{x_{5}}\), and \(\mathrm{x_{7}}\), while \(\mathrm{x_{10}}\) is the only variable entering in the expressions of all constraint functions (see Equation (27)). Hence, the search space of this problem hosts many competitive solutions corresponding to significantly different optimized values of design variables with obvious implications in terms of statistical dispersion of solutions. For example, the optimized values of the first seven variables for the solutions listed in Table 7 varied by at most 27.8% while optimized values of variables \(\mathrm{x_{8}}\) through \(\mathrm{x_{11}}\) varied from 36% to 259.5% where the largest variation occurred for variable \(\mathrm{x_{10}}\) that must be adjusted to satisfy all constraints. Remarkably, FHGWJA was always able to precisely approach and efficiently explore and exploit the best region of search space in all optimization runs in view of its inherent ability to generate high-quality trial solutions over the whole search process. This reduced by a great deal the standard deviation of the optimized solution.