## 3 The Proposed System
### Energy Model
#### 3.2.5 [PERSON]'s Hawks Optimization Algorithm (HHO)  
HHO was offered by [31] as a new meta-heuristic optimization method inspired by [PERSON]'s hawks' cooperative foraging behaviour. [PERSON]'s hawks can show a range of pursuit patterns that are compatible with the dynamic nature of the environment and the prey's fleeing tendencies. [PERSON]'s hawks are hunters who hunt in groups. Several hawks strike the victim from different random positions at a predetermined interval in this interesting hunting tactic. The quest might take a single dive or numerous fast dives over several minutes. A brief description of a mathematical model is as follows [32]. **Firstly:** Exploration Phase (H): [PERSON]'s hawks frequently sit in unusual places, wait, and scan the desert for prey. Two perching strategies are used, namely, one related to the locations of other family members and the prey, and the other on the random value \(r\). \[H(t+1)= \begin{cases}H_{r}(t)-r_{1}|H_{r}(t)-2r_{2}H(t)&r\geq 0.5\\ (H_{pray}(t)-H_{r}(t))-r_{3}(lb-r_{4}(ub-lb))&r<0.5\end{cases}  \tag{16}\]  
\[H_{M}(t+1)=\frac{1}{N}\sum_{N}^{1}H_{i}(t)\]  
where:  
* (\(t\)): the current position. * HM: the average value of the current position of all hawks. * (\(t\)): the random hawk position. * (\(t\)): represents the best position of the prey. * \(ub\) and \(lb\): the upper and lower bound of the problem. **Secondly:** The Phase Change from Exploration to Exploitation. The HH algorithm uses a transition mechanism based on the prey's fleeing energy to move from exploration to exploitation when \(|e|<1\), and then alters the exploitative behaviours. The prey's energy is modelled as follows, with a drop in energy as the prey flees. \[e=e_{0}\bigg{(}1-\frac{t}{T}\bigg{)} \tag{17}\]  
where:  
* \(e_{0}\) represents the initial state and the escaping energy of the prey, respectively. * \(t\) and \(T\), the current and maximum number of iterations, respectively. Thirdly: The exploitation phase depends on the fleeing energy of the prey and the chasing methods of [PERSON]'s hawks, four distinct chasing and attacking strategies are offered. The pursuit strategy is determined by parameter r, which determines whether the prey has a probability of successfully escaping (\(r<0.5\)) or not (\(r\geq 0.5\)) before being attacked. 1. Simple besiege (\(\mathbf{H(t+1)}\)) When \(r\geq 0.5\) and \(\mid\!e\!\mid\geq 0.5\), the prey still has enough energy to try to flee, thus [PERSON]'s hawks surround it softly to tire it before attacking it. The positions are updated as the following: \[H(t+1)=\Delta H(t)-\big{\lvert}\!\mid\!*\ H_{prey}(t)-H(t)\big{\lvert}\Delta H(t )=H_{prey}(t)-H(t)\] (18) \[J=2(1-r_{5})\]
2. Harsh besiege (\(\mathbf{H(t+1)}\)) When \(r\geq 0.5\) and \(\mid\!e\!\mid\!<0.5\), the prey has low fleeing energy, allowing [PERSON]'s hawks to easily encircle and attack it. The positions are updated as the following: \[H(t+1)=H_{prey}(t)-e\big{\lvert}\!\mid\!a??H(t)\big{\lvert}\] (19)
3. Besiege softly with a series of quick dives (\(\mathbf{H(t+1)}\)) When r < 0.5 and \(\mid\!\mid\!E\!\mid\geq 0.5\), the prey has enough energy to flee, therefore [PERSON]'s hawks engage in a gentle besiege by performing multiple fast dives around the prey, trying to gradually adjust its position and direction. Next is a model of this behaviour: \[P_{1}=H_{prey}(t)-e\big{\lvert}\!\mid\!J\ *\ H_{prey}(t)-H(t)\big{\lvert}\] (20) \[P_{2}=P_{1}+v\ *\ LEV\ Y(D)\] \[H(t+1)= \begin{cases}P_{1}&F(P_{1})<F(H(t))\\ P_{2}&F(P_{1})<F(H(t))\end{cases} \] (21)
4. Harsh besiege with gradual rapid dives When \(\mid\!e\!\mid\!<0.5\) and \(r<0.5\), because the prey lacks the energy to flee, the hawks launch a severe besiege to reduce the distance between their average position and the prey, and then attack and kill it. The next is a mathematical depiction of this behaviour: \[P_{1}=H_{prey}(t)-e\big{\lvert}\!\mid\!J\ *\ H_{prey}(t)-H(t)\big{\lvert}\] (22) \[P_{2}=P_{1}+v\ *\ LEV\ Y(D)\] \[H(t+1)= \begin{cases}P_{1}&F(P_{1})<F(H(t))\\ P_{2}&F(P_{1})<F(H(t))\end{cases}