# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### 2D Path Planning  
Test cases (i-ii) regarded a classical robotics problem: to determine the optimal 2D path connecting two points A and B with a piecewise cubic spline trajectory, defined by a certain number of control points (\(NCP\)), that passes through points A, B, and a series of base points (\(NBP\)) while avoiding a set of obstacles (\(NOB\)). The problem objective is to minimize the total length of the trajectory. In an obstacle-free environment, the shortest path between A and B obviously is a straight line. Cubic splines allow for building a smooth and continuous curved path avoiding obstacles. Path planning complexity increases with the number of obstacles and the number of base points; coordinates of base points are selected as optimization variables. The 2D path planning problem searching for the minimum length of a 2D trajectory, passing through \(NBP\) base points so as to avoid \(NOB\) obstacles, and defined by \(NCP\) control points connected by (\(NCP-1\)) spline segments, can be formulated as follows:  
\[\textit{Minimize}\ W(\overset{\rightarrow}{X})=\sum_{r=1}^{NCP}\sqrt{(xc_{r+1} -xc_{r})^{2}+(yc_{r+1}-yc_{r})^{2}}+PENALTY \tag{18}\]  
\[\textit{Subject to}\ \  
centers shifted by 0.001 mm in order to avoid constraint singularity. This is the worst-case scenario when the trajectory intersects all obstacles. The corresponding path lengths were 231.2 mm for 7 base points and 238.5 mm for 10 base points, while initial constraint violation raised to 159,816% and 204,741%, respectively. For both problem variants, the FHGWJA hybrid optimizer developed here was compared with the best optimizers quoted in the literature: namely, hybrid harmony search (hybrid HS), hybrid big bang-big crunch (hybrid BBBC), and hybrid fast simulated annealing (HFSA) [81]. Such a comparison should be considered highly indicative because the hybrid optimizers developed in [81] combined efficient metaheuristic search engines with approximate line search strategies and gradient information while FHGWJA repeatedly utilizes elitist strategies based on the concept of descent direction. The classical grey wolf optimizer (GWO) and the classical and improved JAYA formulations [81] also were compared with FHGWJA to prove the advantages of the hybrid optimizer. Finally, modified harmony search optimization with adaptive parameter updating (mAHS) (derived from the AHS algorithm of Ref. [94]), modified big bang-big crunch with upper bound strategy (mBBBC-UBS) (derived from the BBBC-UBS algorithm of Ref. [95]), and modified sinusoidal differential evolution (MsinDE) (derived from the sinDE algorithm of Ref. [96]) were enhanced with the \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{opt}})\) elitist strategy of FHGWJA to gather specific information on the efficiency of this strategy. The selected population size in Ref. [81] for the hybrid HS, hybrid BBBC, and improved JAYA algorithms was 20, while it was set equal to 10 in this study for the new optimization runs carried out for mAHS, mBBBC-UBS, and MsinDE to have a homogeneous basis of comparison with FHGWJA and its component algorithms, the standard GWO and standard JAYA. All other internal parameters required for SHGWJA's competitors were left the same as those indicated in the original references [81; 94; 95; 96]. The same approach was adopted for the two problem variants with 14 and 20 optimization variables. Table 1 presents the optimization results for the 2D path planning problem variant with 14 design variables (seven base points and seven obstacles). All data listed in the table refer to feasible solutions; the number of analyses refers to the number of times that nonlinear constraints were evaluated. FHGWJA was the best optimizer overall and designed the shortest path of 40.811 mm, outperforming the component algorithms, the standard GWO and standard JAYA, that designed longer trajectories of 41.104 mm and 41.050 mm, respectively.