# A Novel Hybrid Harris Hawk-Arithmetic Optimization Algorithm for Industrial Wireless Mesh Networks
## 2 Proposed Hybrid Harris Hawks-Arithmetic Optimization Algorithm
### Exploration  
This section proposes an exploration mechanism for hybridizing the HHO and AOA algorithms. The algorithm draws inspiration from the hunting behavior of Harris hawks, capitalizing on their exceptional eyesight to effectively track and recognize prey. However, the prey may not be immediately visible in some cases, prompting the hawks to patiently wait, observe, and monitor the objective location for several hours until detecting a potential target.  
The algorithm expertly utilizes Harris hawks to represent potential solutions, with the ultimate goal of comparing the best solution to the ideal outcome. The algorithm flawlessly imitates the hawks' actions by strategically placing them in different locations and using various approaches until the desired solution is found. Each strategy is meticulously employed in equal measure to ensure maximum efficiency. The hawks rest closer to the prey (rabbit) based on the positions of other family members during \(q<0.5\). When the condition is met with \(q\geq 0.5\), the hawks will randomly perch on tall trees within the range of their family members. Instead of selecting the random tall trees, the best position nearest to the rabbit will be identified using the AOA algorithm, which will further enhance the chances of reaching and identifying the target much faster. Therefore, the exploration strategy of the proposed HHAOA is obtained using the equation given below.  
\[Y_{j+1}^{i}=\begin{cases}y_{\text{rand}}-r_{1}\mid y_{\text{rand}}-2r_{2}(y_{j} \div(\text{MOP}+\epsilon)\times((UB_{j}\\ -LB_{j})\times\ \mu+LB_{j})\mid,q\geq 0.5\ \&\ \epsilon\geq 0.5\\ y_{\text{rand}}-r_{1}\mid y_{\text{rand}}-2r_{2}(y_{j}\times\text{MOP}\times((UB _{j}-LB_{j})\\ \times\ \mu+LB_{j})\mid,q\geq 0.5\ \&\ \epsilon<0.5\\ y_{\text{B}}-y_{m}-r_{3}\big{[}LB_{j}+r_{4}\big{(}UB_{j}-LB_{j}\big{)}\big{]},q <0.5,\end{cases} \tag{1}\]  
where \(Y_{i}^{j+1}\) describes the location of \(t\)-th solution in the top layer (HHO) corresponding to the \(j\)-th search solution in the AOA layer. \(j\) represents the current iteration of the algorithm. \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\), \(c\), and \(q\) are the random numbers that lies in the span of [0,1]. \(y_{rand}\), \(y_{B}\), \(y_{m}\), \(UB\), \(LB\), MOP are the random hawk selected, the best location obtained so far in the present iteration, average mean of the hawk's position, upper and lower bound ranges of the variables, and the math-optimizer probability coefficient. The MOP and average mean of the hawk's location values are calculated based on the equation given below.  
\[\text{MOP}(j)=1-\left(\frac{t}{T}\right)^{1/\alpha} \tag{2}\]  
\[y_{m}(j)=\frac{1}{N}\sum_{j=1}^{N}y_{i}(j), \tag{3}\]  
where \(\alpha\) is a crucial parameter factor that determines the precision of the exploitation phase, \(t\), and \(T\) represent the current and the highest possible number of iterations, respectively. The position of every hawk in iteration j is represented by \(y_{i}(t)\), and the total number of hawks is denoted by \(N\).