# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### 2D Path Planning
Hybrid HS, hybrid BBBC, HFSA, and MsinDE found very similar path lengths between 40.983 and 40.994 mm; these trajectories are practically symmetric about the line connecting the point near obstacle 1, where trajectories diverge, and the end point B. Since the trajectory designed with standard JAYA between obstacles 1 and 7 is more curved than those designed with hybrid HS and HFSA, the length of standard JAYA's optimal path increased to 41.050 mm. The detailed analysis of the optimized trajectory revealed that, in order to minimize length, trajectory should be rectilinear between obstacles 1 and 7 as well as in the final branch from obstacle 7 to end point B. Interestingly, FHGWJA was the only optimizer to satisfy this requirement. In fact, the optimal path found with FHGWJA between obstacles 1 and 7 may be fitted in the XY plane via a linear regression with R\({}^{2}\) = 0.999 vs. only R\({}^{2}\) = 0.992 and R\({}^{2}\) = 0.994, respectively, for hybrid HS (HFSA's trajectory is very close to the one designed with hybrid HS) and hybrid BBBC (MsinDE's trajectory is very close to the one designed with hybrid BBBC). Furthermore, the optimal path of FHGWJA may be linearly fitted with R\({}^{2}\) = 0.998 vs. only R\({}^{2}\) = 0.991 and R\({}^{2}\) = 0.994 for hybrid HS and hybrid BBBC, respectively. Besides the higher curvature of the trajectories designed with the other algorithms between obstacles 1 and 7, the final branches of the optimal trajectories designed via hybrid HS, HFSA, hybrid BBBC, and MsinDE show changes in concavity that make trajectory deviate from a straight line. Figure 7b compares the optimized trajectories obtained with FHGWJA and its competitors in the 2D path planning problem variant including 20 optimization variables. The figure shows the trajectories of the best and worst optimization runs of FHGWJA as well as the other five best solutions quoted in Table 2. The new obstacles 8 and 10 introduced in the problem became active for all algorithms while the optimized trajectory approached the new obstacle 9 only for mAHS. This explains why the optimized path lengths of hybrid HS, HFSA, MsinDE, and hybrid BBBC varied at most by 0.037 mm (i.e., between 40.997 mm  
Figure 7: Comparison of optimized trajectories found with FHGWJA and its competitors in the 2D path planning problem: (**a**) 7 base points and 14 variables; (**b**) 10 base points and 20 variables. and 41.034 mm) while path length increased by 0.041 mm just passing from hybrid BBBC to mAHS (i.e., from 41.034 mm to 41.075 mm). Again, FHGWJA designed the straightest trajectory in the path branches between obstacles 1 and 7 and between obstacle 7 and end point B. Such a behavior occurred for both the best and worst FHGWJA optimization runs. Conversely, trajectories designed with hybrid HS/BBBC/SA and MsinDE algorithms exhibited significant changes in slope and curvature especially in the final branch of the path or between obstacles 10 and 7. For example, the final branch of the FHGWJA's optimal paths may be linearly fitted with R\({}^{2}\) = 0.997 vs. 0R\({}^{2}\) = 0.965, R\({}^{2}\) = 0.940, and R\({}^{2}\) = 0.978 for hybrid HS, hybrid BBBC, and HFSA, respectively. In summary, the FHGWJA algorithm developed in this study is an efficient and robust tool for solving highly nonlinear 2D path planning problems in robotics. The proposed algorithm could always design shorter paths requiring less analyses to complete optimization runs than its competitors.