## 3 Hybridization Method
### Krill Herd (KH)  
Krill herd (KH) is a swarm intelligence optimization approach influenced by Antarctic krill herding behavior and takes into account krill's Lagrangian and evolutionary activities in their native habitat [6]. Krill can form enormous herds that can span hundreds of meters in length. Whenever predators (such as penguins or other birds) attack krill herds, they remove a krill, resulting in a lower krill density. Following a predator assault, krill formation is a multi-objective process aimed at increasing the krill density and accessing food. The krill swarm density, krill distance, and food density all influence the goal function in KH. The krill position is determined by the following factors: (i) movement caused by other krill, (ii) foraging movement, and (iii) physical diffusion. When looking for the maximum density and food, every individual krill migrates towards the best feasible solution. The smallest distances between individual krill and food, as well as the largest density of the herd, are regarded as the objective function for krill movement, which eventually leads to krill herding around the global minima. The KH method uses the Lagrangian model in a \(d\)-dimensional solution space as follows:  
\[\frac{dX_{i}}{dt}=N_{i}+F_{i}+D_{i} \tag{2}\]  
where \(N_{i},F_{i}\), and \(D_{i}\) are the movement caused by other krill, foraging movement, and physical scattering of the \(i\)th krill. The direction of motion, \(alpha_{i}\), of the movement caused by other krill is approximated by the target swarm density, local swarm density, and repulsive swarm density. This movement may be characterized for a krill individual as shown below, where \(N_{m}ax\) is the greatest induced speed, \(omega_{n}\) is the motion inertia weight in \([0,1]\), and \(N_{i}^{old}\) is the final movement. \[N_{i}^{new}=N^{max}\alpha_{i}+\omega_{n}N_{i}^{old} \tag{3}\]  
The two basic components estimate the foraging motion. The first is the food position, and the second is prior knowledge of the food position. This movement may be approximated for the \(i\)th krill as:  
\[F_{i}=V_{f}\beta_{i}+\omega_{f}F_{i}^{old} \tag{4}\]  
where  
\[\beta_{i}=\beta_{i}^{food}+\beta_{i}^{best} \tag{5}\]  
and \(V_{f}\) is foraging velocity, \(\omega_{f}\) is the inertia of foraging motion, and \(F^{o}ld_{i}\) is the final foraging movement. In general, the random dispersal of krill individuals might be considered a random process. A maximum scattering velocity and a random directed vector can be used to define this motion. It is defined as:  
\[D_{i}=D^{max}\delta \tag{6}\]  
where \(D^{m}ax\) is the greatest scattering velocity, and \(\delta\) is the stochastic vector with values from \([-1,1]\). The spatial vector of a krill throughout the interval \(t\) to \(t+\Delta t\) is calculated based on the three above-mentioned motions and varying factors of the motion during this time, as follows:  
\[X_{i}(t+\Delta t)=X_{i}(t)+\Delta t\frac{dX_{i}}{dt} \tag{7}\]  
The KH algorithm is summarized in Algorithm 2. ```

```
1: Define the bounds, parameter(s), and so forth. 2: Randomly create the starting population in search space. 3: Assess each krill based on its position. 4: Perform movement calculation. 5: Calculate movement influenced by other krill: foraging movement and physical scattering. 6: Update the krill location in the search space. 7: Go to Line 3 until the stopping criteria are reached. ```
**Algorithm 2** Pseudo-code for the KH Algorithm. ### Bat Algorithm (BA)

The bat algorithm was proposed by [PERSON] in [53] and motivated by the echolocation behavior of bats. Microbat echolocation can be characterized as follows: each virtual bat flies at a random velocity \(v_{i}\) at position (solution) \(x_{i}\) with varying frequency or wavelength and loudness \(A_{i}\). Its frequency, loudness, and pulse emission rate \(r\) fluctuate as it seeks for and finds its victim. A local random walk is used to enhance the search. The best ones are chosen until the specified stop requirements are reached.