# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 2 The FHGWJA Algorithm
However, this task entails a new constraint evaluation to compute the penalized cost function value for each new trial design. In order to reduce the number of constraint evaluations required in the optimization search, FHGWJA utilizes an elitist strategy retaining only the trial solutions that are very likely to improve design. Hence, FHGWJA initially compares only the cost function \(W(\overrightarrow{X_{i,tr}})\) computed for the new trial design \(\overrightarrow{X_{i,tr}}\) with the penalized cost function \(W_{p}(\overrightarrow{X_{best}})\) computed for the current best record. If \(W(\overrightarrow{X_{i,tr}})>W_{p}(\overrightarrow{X_{best}})\), the new trial design \(\overrightarrow{X_{i,tr}}\) is directly rejected because it certainly cannot improve the current best record as the cost function value \(\overrightarrow{X_{i,tr}}\) is by itself larger than the penalized cost of \(\overrightarrow{X_{best}}\) which also accounted for any constraint violation. If \(W(\overrightarrow{X_{i,tr}})<W_{p}(\overrightarrow{X_{best}})\), the cost function value \(W(\overrightarrow{X_{i,tr}})\) computed for the new trial solution \(\overrightarrow{X_{i,tr}}\) is compared with 1.1 times the cost function value \(W(\overrightarrow{X_{best}})\) computed for the current best record. Ifit holds \(W(\overrightarrow{X_{i,tr}})\leq 1.1W(\overrightarrow{X_{best}})\), the new trial solution \(\overrightarrow{X_{i,tr}}\) is provisionally included in the new population. The \(1.1W(\overrightarrow{X_{best}})\) threshold was selected upon the following rationale. Exploration is characterized by a high probability of improving design: hence, the \(W(\overrightarrow{X_{i,tr}})>1.1W(\overrightarrow{X_{best}})\) condition is not likely to occur. In exploitation, local minima should be bypassed by the optimizer to converge to the global optimum. Similar to SA, FHGWJA is allowed to provisionally accept slightly worse candidate solutionsthan \(\vec{X_{best}}\). The threshold level of acceptance probability in state-of-the-art SA formulations such as [81; 90] is 0.9 if the ratio between the cost function increment \([W(\vec{X_{i,tr}})-W(\vec{X_{best}})]\) and the annealing temperature \(T\) is 0.1. This means that there is a 90% probability to provisionally accept a worse design than \(\vec{X_{best}}\) and make it improve in the next iterations. Since \(T\) is initially set equal to the expected optimum cost (or roughly corresponds to the cost function value \(W(\vec{X_{best}})\)), it is reasonable to assume that solutions up to 10% worse than \(\vec{X_{best}}\) may improve \(\vec{X_{best}}\) itself in the next few iterations. Since each new trial solution \(\vec{X_{i,tr}}\) provisionally included in the new population was generated by FHGWJA using the classical GWO scheme based on the positions of wolves \(\alpha\), \(\beta\), and \(\delta\), FHGWJA adopts a JAYA-based scheme to avoid stagnation. For that purpose, when \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{best}})\), a new trial solution \((\vec{X_{i,tr}})^{\prime}\) is defined as follows:  
\[(\vec{X_{i,tr}})^{\prime}=\vec{X_{i,tr}}+\vec{\lambda_{1}}\cdot(\vec{X_{best} }-\vec{X_{i,tr}})-\vec{\lambda_{2}}\cdot(\vec{X_{\delta}}-\vec{X_{i,tr}}) \tag{13}\]  
where \(\vec{\lambda_{1}}\) and \(\vec{\lambda_{2}}\) are two vectors of \(NDV\) random numbers generated in the [0,1] interval. Equation (13) is relative to exploitation as it perturbs the good design \(\vec{X_{i,tr}}\) that was provisionally included in the new population being close to the current best record or likely better than it. The \(\delta\) wolf is temporarily selected as the worst individual of the population, and FHGWJA is forced to locally search in a region of design space containing only high-quality solutions. The new trial solution \((\vec{X_{i,tr}})^{\prime}\) is compared with \(\vec{X_{i,tr}}\) to select the solution that finally updates population.