## 3 The Proposed Model
### Steps Involved in HMSFO Algorithm  
The presented HMSFO-EARS technique involves the integration of the MSFO algorithm with the Adaptive \(\beta\)-Hill Climbing (ABHC) concept. In this study, a novel MSFO algorithm is developed that is impacted by the search (food) analogy of cooperative/societal response or the swarming rudiment of fishes living in a turbid or muddy water habitat [25]. Generally, every single fish impacts the surroundings through its own neighbourhood efforts. Mainly, the MSFO technique is intended for solving the presented energy management problems by efficiently handling the energy flow through the networks toward decreasing the losses and production cost along with system and operation constraints amongst meeting the load demands. Every fish that exists in the muddy water implements a localized search in a random fashion through solitary elements to search for a possible solution and it is evaluated in the following Equation (4):  
\[\tau_{k}^{msf}(t+1)=\tau_{k}^{msf}(t)+r\times\sigma,r\in[-1,1]. \tag{4}\]  
The mobility of fish can be represented by the novel and existing locations \(\tau_{k}^{msf}(t+1)\) and \(\tau_{k}^{msf}(t)\), correspondingly. In addition, random number \(r\) is formed by using a uniformly distributed range within \([-1,1]\) and the variable liable for the maximumstep is denoted by \(\sigma\). Consequently, all the fishes allow movement towards the optimum position if the fitness function at the neighbourhood position is higher than the existing position. \(f(\tau_{k}^{msf}(t+1))>f(\tau_{k}^{msf}(t))\) positions become equivalent and are determined by \(\tau_{k}^{msf}(t+1)=\tau_{k}^{msf}(t);\Delta\tau_{k}^{msf}=0\) is defined by the following equation:  
\[\Delta f_{k}=f\Big{(}\tau_{k}^{msf}(t+1)\Big{)}-f\Big{(}\tau_{k}^{msf}(t)\Big{)}, \tag{5}\]  
\[\Delta\tau_{k}^{msf}=\tau_{k}^{msf}(t+1)-\tau_{k}^{msf}(t), \tag{6}\]  
Furthermore, the development of exploitation capability in succeeding iterations has led to reducing the amplitude of the step \(\sigma\) variable and it is estimated as follows:  
\[\sigma(t+1)=\sigma(t)-\Big{[}\frac{\sigma_{i}-\sigma_{f}}{\textit{Total Iterations}}\Big{]}, \tag{7}\]  
In Equation (7), the first and last steps are represented by \(\sigma_{i}\) and \(\sigma_{f}\). The first step must be higher than the last step. Figure 2 depicts the flowchart of the MSFO technique. The algorithm begins with population initialization. Then, the fitness function is determined with foraging--impulsive and non-impulsive. Next, the current state of the soil fish can be evaluated and the optimal one is chosen as the target solution. The process gets repeated until the stopping criteria are satisfied. The foraging operators are closely connected to all the fish's weight updating model. It can be defined by the success rate  
Figure 2: Flowchart of MSFO technique. accomplished in the present iteration. The rise in fish weight maximizes the probability of acquiring an optimal region. Moreover, the weight function can be expressed by:  
\[M_{k}(t+1)=M_{k}(t)+\left[\frac{\Delta f_{k}}{\max(|\Delta f_{k}|)}\right], \tag{8}\]  
In Equation (8), \(M_{k}(t+1)\) and \(M_{k}(t)\) denote the mass of fish at novel and recent positions, correspondingly. The change occurs in the target functions concerning present and novel locations regarding the fish \(k\) that is denoted by \(\Delta f_{k}\) and \(\Delta f_{k}\) is assumed as zero once the fish remain in a similar position. In addition, some additional efforts are included in the fitness assessment for guaranteeing improved convergence towards the optimal area over design space through the mass scale \((M_{s})\). * Each fish is born with a mass closer to \(\frac{M_{s}}{2}\). * Parameters are likely to limit the fish mass through \(M_{s}\) within \([1,\;M_{s}]\). Such cooperative operators reveal the displacement pattern established by the fish, i.e., effective together with solitary movement and \(f_{k}\
eq 0\).