## 3 The Proposed Model
### Steps Involved in HMSFO Algorithm
The average weighted displacement factor \(D\in R\) can be defined using the variance of fish fitness \((\Delta f_{k})\) by involving the fish that have the highest fitness in the solitary movement and is given as follows:  
\[\overset{\rightarrow}{D}(t)=\frac{\sum_{k=1}^{N}\Delta\overset{\rightarrow} {\tau}\Delta f_{k}}{\sum_{k=1}^{N}\Delta f_{k}}, \tag{9}\]  
In Equation (9), \(N\) represents the maximal capacity of the cluster and the \(\Delta\tau_{k}^{msf}\) vector denotes the variance among the novel and present locations of fish. Consequently, the present position of all the fish can be upgraded by:  
\[\overset{\rightarrow}{\tau}(t+1)=\overset{\rightarrow}{\tau}_{k}^{msf}(t)+ \overset{\rightarrow}{D}(t). \tag{10}\]  
This operator becomes instrumental to deal with the exploration and exploitation capability interconnected with the swarm. In these contexts, the rise in the total weight of the swarm indicates the remarkable efficiency of the search function and the additional increase in swarm size decreases the exploration of another region. Therefore, the size of the swarm can be controlled by means of centroid over its compression and expansion techniques. Furthermore, centroid \(G\) is calculated on the basis of the current location of the fish and its corresponding weight can be given as follows:  
\[\overset{\rightarrow}{G}(t)=\frac{\sum_{k=1}^{N}\overset{\rightarrow}{\tau }M_{k}(t)}{\sum_{k=1}^{N}M_{k}(t)}. \tag{11}\]  
Once the swarm size is constantly rising in the present iteration, then each fish updates the current position. Likewise, the existing position of every fish is upgraded once swarm size reduces in the current iteration:  
\[\overset{\rightarrow}{\tau}^{msf}(t+1)=\overset{\rightarrow}{\tau}^{msf}(t )-\gamma(0,1)*\sigma_{m}\left[\frac{\overset{\rightarrow}{\tau}^{msf}(t)- \overset{\rightarrow}{G}(t)}{d\left(\overset{\rightarrow}{\tau}_{k}^{msf}(t),\overset{\rightarrow}{G}(t)\right)}\right] \tag{12}\]  
\[\Delta f_{k}=f\Big{(}\tau_{k}^{msf}(t+1)\Big{)}-f\Big{(}\tau_{k}^{msf}(t)\Big{)}, \tag{5}\]0  
In this study, the presented HMSFO-EARS technique involves the integration of the MSFO algorithm with the ABHC concept [26]. It is a new local search-based technique that is an adapted version of \(\beta HC\). This study shows that \(A\beta HC\) gives an improved performance when compared to others. To increase the exploitation capability and quality of the concluding solution, \(A\beta HC\) is incorporated into the basic MSFO to help find the neighbourhood of the optimum solution. For the existing solution \(X_{i}=\left(x_{i,1},x_{i,2},\ldots,x_{i,D}\right)\), \(A\beta HC\) iteratively produces the improved solution \(X_{i}^{\prime\prime}=\left(x_{i,1}^{\prime\prime},x_{i,2}^{\prime\prime}, \ldots,x_{i,D}^{\prime\prime}\right)\) according to the \(\mathcal{N}\)-operator and \(\beta\)-operator. First, the \(N\)-operator transfers \(X_{i}\) to the new neighborhood solution \(X_{i}^{\prime}=\left(x_{i,1}^{\prime},x_{i,2}^{\prime},\ldots,x_{i,D}^{\prime \prime}\right)\) that is determined by the following expression:  
\[\Delta f_{k}=f\Big{(}\tau_{k}^{msf}(t+1)\Big{)}-f\Big{(}\tau_{k}^{msf}(t)\Big{)}, \tag{5}\]1  
\[\Delta f_{k}=f\Big{(}\tau_{k}^{msf}(t+1)\Big{)}-f\Big{(}\tau_{k}^{msf}(t)\Big{)}, \tag{5}\]2  
Whereas \(U(O,1)\) indicates a random value between zero and one, \(x_{ij}\) represents the value of the decision parameter in \(j\) - th dimensions, \(t\) indicates the existing iteration, _Maxiter_ indicates the maximal amount of iterations, \(N\) suggests the bandwidth distance amongst the existing solution and the neighbour, \(D\) represents the dimension vector, and \(K\) represents a constant. Then, the decision variable of novel solution \(X_{i}^{\prime\prime}\) is allocated from the present solution or arbitrarily from the presented range of the \(\beta\)-operator:  
\[x_{i,j}^{\prime}\leftarrow \begin{cases}x_{i,r},&if\ r_{8}<\beta\\ x_{i,r}^{\prime},&else\end{cases}