**3. GWO**

GWO is a recent soft computing approach that mimics the social activities of gray wolves. This algorithm depicts leadership, tracking, surrounding and attacking prey

[7] activities of the species. In this algorithm a specific number of gray wolves in a group travel through a multi-dimensional search space in search of prey. The position of gray wolves are considered as different position variables and the distances of the prey from the gray wolves determine the fitness value of the objective function. The individual gray wolf adjusts its position and moves to the better position. The GWO saves the best solutions obtained through the course of iterations. The goal of this algorithm is to reach to the prey by the shortest possible route. The movement of each individual is influenced by four processes. Their hunting mechanism is as follows:


These steps are shown in **Figure 1** [7].

The GWO algorithm was anticipated by Mirjalili et al. [7]. Gray wolves are related to the Canidae family and are zenith predator. A pack approximately consists of a group of 5to12 wolves. Their society is divided on the basis of hierarchy. The leader is a couple called the 'Alphas'. They take all the decisions for the pack and these decisions are then communicated to the pack. All the members of the pack respect the leader with keeping their tails down. The alpha is the best member who can manage the pack in a better way. The second level in this hierarchy is the 'Beta' wolves. It is an assistant wolf next to alpha after the current wolf gone. It assists alpha and keeps discipline in the pack. The third level in this hierarchy is the 'Delta' wolves. The lowest ranked gray wolf is 'Omega'. They are the scapegoat or the babysitters. Amidst all the social hierarchy, there is an exciting social activity of the gray wolf is group hunting (optimization).

#### **Figure 1.**

*Hunting steps of Gray Wolf: (A) chasing, approaching and tracking prey (B–D) pursuing, harassing, and encircling (E) Stationary situation and attacking [7].*

The encircling behavior of gray wolves may be modeled mathematically as per following Eqs. (11) and (12).

$$\vec{D} = \left| \vec{C} \,\vec{X}\_p(t) - \vec{X}(t) \right| \tag{10}$$

$$
\overrightarrow{X}(t+1) = \overrightarrow{X}\_p(t) - \overrightarrow{A}\,\overrightarrow{D} \tag{11}
$$

Where *X* ! *<sup>p</sup>* and *X* ! are the respective vectors corresponding to the position of the prey and the gray wolf, and *t* designates the present iteration.

The coefficient vectors *A* ! and *C* ! can befound out as given in (13) and (14).

$$
\overrightarrow{\vec{A}} = 2\overrightarrow{a}.\overrightarrow{r}\_1 - \overrightarrow{a} \tag{12}
$$

$$
\overrightarrow{\mathbf{C}} = \mathbf{2}.\overrightarrow{r\_2} \tag{13}
$$

Where !*r* <sup>1</sup> and *r* ! <sup>2</sup> vectors are randomly chosen in the range [0, 1], the values of *a* ! are gradually varied from 2 to 0 with the increase of iteration so as to put emphasis on exploration and exploitation, respectively.

All the wolves of the pack keep updating their locations as per the location of the senior wolves in the pack. Moreover, the location of the prey would be a random place encircled by the alpha, beta and delta during their search because they are more experienced in hunting. The following Eqs. (15)–(17) are proposed to revise the position of all wolves as per the locations obtained so far by the best candidates as the alpha, beta and delta.

$$
\overrightarrow{D}\_a = \left| \overrightarrow{\mathbf{C}}\_1 \overrightarrow{X}\_a - \overrightarrow{X} \right| \\
\overrightarrow{D}\_\beta = \left| \overrightarrow{\mathbf{C}}\_2 \overrightarrow{X}\_\beta - \overrightarrow{X} \right|, \\
\overrightarrow{D}\_\delta = \left| \overrightarrow{\mathbf{C}}\_3 \overrightarrow{X}\_\delta - \overrightarrow{X} \right| \tag{14}
$$

$$
\overrightarrow{X}\_1 = \overrightarrow{X}\_a - \overrightarrow{A}\_1.\left(\overrightarrow{D}\_a\right), \overrightarrow{X}\_2 = \overrightarrow{X}\_\beta - \overrightarrow{A}\_2.\left(\overrightarrow{D}\_\beta\right), \ \overrightarrow{X}\_3 = \overrightarrow{X}\_\delta - \overrightarrow{A}\_3.\left(\overrightarrow{D}\_\delta\right) \tag{15}
$$

$$
\overrightarrow{X}(t+1) = \frac{\overrightarrow{X}\_1 + \overrightarrow{X}\_2 + \overrightarrow{X}\_3}{3} \tag{16}
$$

Diverging and converging towards the prey in order to search and attack the prey is what the gray wolves follow. In the mathematical modeling of divergence, we use the value of j j *A* > 1 or j j *A* < 1 for the searching wolf to deviate from the prey to emphasize exploration.

At the beginning of search process, a pack of gray wolves is randomly initialized in the GWO algorithm. Each wolf in the searching place updates its gap from the prey. Finally, the algorithm ends the optimization when termination limit is attained.
