3.2.2. Hierarchy (H)

Horses are dependent. They follow a leader throughout their life, which is a behavior that people frequently exhibit; based on the principle of domination [38], a mature stallion or a filly is also in charge of overseeing a herd of wild horses. Coefficient *hm* in HHO is defined as the propensity of a group of horses to follow the direction of the most skilled and powerful horse. Horses follow the law of hierarchy when they are between the ages of 5 and 15 years according to research.Equations (30) and (31) are useful for establishing this.

$$H\_m^{iter, age} = h\_m^{iter, age} \left( P\_{lbh}^{iter-1} - P\_m^{iter-1} \right) \tag{30}$$

$$h\_m^{iter,age} = h\_m^{iter-1,age} \times w\_{\mathcal{S}} \tag{31}$$

*<sup>H</sup>iter*,*age <sup>m</sup>* shows the optimal horse's location using the velocity variable. *<sup>P</sup>iter*−<sup>1</sup> *lbh* identifies where the finest horse is standing.

#### 3.2.3. Sociability (Soc)

Horses may cohabit with various animal species and require social contact. Living in a herd protects wild horses from predators that may pursue them. Pluralism increases their chances of surviving and makes escaping simpler. Due to their social nature and the fact that they are so unique, horses regularly fight with one another.

Some horses seem to prefer being around other animals like sheep and cattle, but they despise being by themselves [38].

The following calculations demonstrate that horses between the ages of 5 and 15 years are most interested in being with a herd.

$$\operatorname{Soc}\_{\mathfrak{M}}^{\operatorname{iter, age}} = \operatorname{soc}\_{\mathfrak{M}}^{\operatorname{iter, age}} \left[ \left( \frac{1}{N} \sum\_{j=1}^{N} P\_j^{\operatorname{iter} - 1} \right) - P\_{\mathfrak{M}}^{\operatorname{iter} - 1} \right] \operatorname{age} = \beta\_\prime \gamma \tag{32}$$

$$\text{soc}\_{\text{ml}}^{\text{iter,age}} = \text{soc}\_{\text{ml}}^{\text{iter-1,age}} \times \omega\_{\text{Soc}} \tag{33}$$

The above-mentioned equations reveal the following:

*Sociter*,*age <sup>m</sup>* explains the social motion vector that the ith horse presents.

*soc iter*,*age <sup>m</sup>* demonstrates how the horse is facing the direction of group ith.

iter, which has a parameter of *ωs*, reduces the iteration with each cycle.

The total number of horses is expressed by N.

Age is a representation of each horse's age range.

By evaluating these factors, the derivations of coefficients γ and β are carried out.
