## 3 Background
### Whale Optimization Algorithm  
_WOA_ is a recent _SI_ method, proposed by [PERSON] et al. [17]. The method mimics the social behavior of the humpback whale to ensure a fast exploration of search space. In addition, _WOA_ mimics the bubble-net technique to ensure an effective exploitation of promising regions. The heuristic has been tested over multiple optimization and engineering problems and has proved itself as a global optimizer [17].  
Let us describe how the basic algorithm operates. In the intensification phase, the _WOA_ sets the prey to be the best solution overall the search space \(\overrightarrow{X}^{*}(t)\) and swarm members applies the bubble-nettechnique to surround the prey. Mathematically, the helix-shaped path taken by each whale \(\overrightarrow{X}(t)\) to the prey can be computed as:  
\[\overrightarrow{D} =|\overrightarrow{X}^{*}(t)-\overrightarrow{X}(t)|, \tag{6}\] \[\overrightarrow{X}(t+1) =\overrightarrow{D}\cdot e^{bl}\cdot\cos(2\pi l)+ \overrightarrow{X}^{k}(t), \tag{7}\]  
where \(b\) stands for a constant that represents the logarithmic spiral shape, \(l\) is a random number in the range of \([-1,1]\), and \(\overrightarrow{X}^{k}(t)\) stands for the best solution. Considering the diversification phase, the _WOA_ uses two exploration methods to cover search space. These methods simulate the social behavior of the humpback whale. The first diversification method is based on selecting a random whale \(\overrightarrow{X}_{rand}(t)\) from the current swarm to be the prey. Then, the swarm members reposition themselves according to:  
\[a=2-t*(2/Max\_Iteration), \tag{8}\] \[A=2*a*r-a,\] (9) \[C=2*r,\] (10) \[\overrightarrow{D}=|C\cdot\overrightarrow{X}_{rand}(t)- \overrightarrow{X}(t)|.\] (11) \[\overrightarrow{X}(t+1)=\overrightarrow{X}_{rand}(t)-A\cdot \overrightarrow{D}, \tag{12}\]  
where \(r\) represents a random number, \(C\) and \(A\) are controlling coefficients, \(t\) represents the current iteration number, _Max_Iteration_ represents the maximum number of iterations, and \((\cdot)\) represents element by element multiplication. The second diversification method was setting the prey to be the best solution \(\overrightarrow{X}^{*}(t)\). Swarm members update their new positions according to:  
\[\overrightarrow{D}=|C\cdot\overrightarrow{X}^{*}(t)- \overrightarrow{X}(t)|. \tag{13}\] \[\overrightarrow{X}(t+1)=\overrightarrow{X}^{*}(t)-A\cdot \overrightarrow{D}. \tag{14}\]  
The choice between the two diversification methods is controlled by the value of \(A\) parameter. In starting iterations (\(|A|<1\)), so _WOA_ focuses on discovering the whole search space by computing the new positions of swarm members according to Equation (12). The value of \(|A|\) decreases with the increase in number of iterations, which makes _WOA_ computes new positions of swarm members according to Equation (14). This mechanism improves the search efficiency of _WOA_ by exploring the whole search space in starting iterations and exploiting promising regions in the final ones.  
Choosing between the exploration and the exploitation phases is done randomly according to the random parameter \(p\). _WOA_ uses a 50% value to provide the balance between intensification and diversification methods. The following equation describes the selection criteria of _WOA_[13]:  
\[\overrightarrow{X}(t+1)=\left\{\begin{array}{l l}\overrightarrow{X}^{k}(t)- A\cdot\overrightarrow{D}&if\quad p<0.5,\\ \overrightarrow{D}\cdot e^{bl}\cdot\cos(2\pi l)+\overrightarrow{X}^{k}(t)&if \quad p\geq 0.5\end{array}\right. \tag{15}\]  
where \(p\) is a random number in range of \([0,1]\). Next section discusses the computational representation of hybridizing _WOA_ with _TS_.