## 3 Hybridization Method
### Krill Herd (KH)
The dynamic behavior of a swarm of bats is controlled using a frequency-tuning approach, and the balance between exploration and exploitation can be managed by tweaking algorithm-dependent parameters in the bat algorithm. The steps in BA are as follows:

* All bats employ echolocation to sense distance, and a bat's location \(x_{i}\) is regarded as a solution to a problem. * Bats look for prey by flying randomly at location \(x_{i}\) with variable frequency (from the smallest frequency \(f_{min}\) to the largest value \(f_{max}\)) with varying wavelengths _lambda_ and loudness \(A\). They can automatically alter the wavelengths (or frequencies) of their emitted pulses as well as the rate of pulse emission \(r\) based on the target's distance. * The value of loudness ranges from a large positive number \(A_{0}\) to the smallest value \(A_{min}\). Each bat \(i\) should have its position (\(x_{i}\)) and velocity (\(v_{i}\)) defined in a \(d\)-dimensional solution space. Position \(x_{i}\) should be updated as the iterations progress. \[f_{k}=f_{min}+(f_{max}-f_{min})rand \tag{8}\]

\[v_{k}^{t+1}=v_{k}^{t}+(x^{*}-x_{k}^{t})f_{k} \tag{9}\]

\[x_{k}^{t+1}=x_{k}^{t}+v_{K}^{t+1} \tag{10}\]

where \(rand\in[0,1]\) is a stochastic vector drawn from a uniform distribution. In this case, \(x^{*}\) is the current global best position (solution) found after evaluating all solutions across all \(n\) bats. A new solution is produced locally for each bat using the random walk, defined as

\[X_{new}=X_{old}+\epsilon\langle A^{k+1}\rangle \tag{11}\]

where \(\epsilon\in[-1,1]\) is a stochastic number, while \(\langle A_{k}^{t+1}\rangle\) is the mean loudness of bats at the current time step. As iterations proceed, the loudness and rate of pulse emission are modified. As the bat comes closer to its meal, the volume decreases, and the pulse rate increases. The following equations are the formulas for loudness and puls e rate update:

\[N_{i}^{new}=N^{max}\alpha_{i}+\omega_{n}N_{i}^{old} \tag{3}\]0

\[N_{i}^{new}=N^{max}\alpha_{i}+\omega_{n}N_{i}^{old} \tag{3}\]1

where \(1<\omega<1\) and \(\eta<1\) are constants. As \(t\rightarrow\infty\), we have \(A_{k}^{t}\to 0\) and \(r_{k}^{t}\to r_{k}^{0}\). The initial parameters are defined as follows: \(A_{0}\in[1,2]\) and emission rate \(r_{0}\in[0,1]\). ```
1: Define objective function
2: Initialize the flock of bats
3: Define frequency of pulse
4: Set loudness and pulse rates
5:while\(t\leq t_{max}\)do
6: Adjust value of frequency
7: Update velocities
8: Update locations/solutions
9:if\((rand>r_{k})\)then
10: Select a solution from a set of best solutions
11: Create a local solution nearby the selected best solution
12:endif
13: Create a new solution for a bat flying randomly
14:if\((rand<A_{k}\) & \(F(x_{k})<F(x^{*}))\)then
15: Adopt new solutions
16: Increase \(r_{k}\)
17: Decrease \(A_{k}\)
18:endif
19: Evaluate all bats and pick the current best bat
20:endwhile
```

**Algorithm 3** Bat algorithm. ### Summary

The proposed hyper-heuristic hybridization approach, which is applied to the hybridization of the BA and KH algorithms, is visually presented in \[N_{i}^{new}=N^{max}\alpha_{i}+\omega_{n}N_{i}^{old} \tag{3}\]2 Whereas low-performing agents strive to copy the behavior of well-performing agents by flipping to their style of behavior, a fascinating collective behavior emerges as a result of the basic process of probabilistic state flipping. If, however, a single state applies to the bulk of agents, worse-performing agents begin to migrate to another state. Such opportunistic behavior is a trade-off between exploration and exploitation. The exploration process aids in getting near the global minimum or a local minimum, whereas the exploitation process aids in more precisely locating the global minimum. As a result, the hyper-heuristic strikes a balance between local and global qualities since the baseline algorithm may have a biased behavior towards global or local searching, thus allowing the diversification of the search to prevent becoming locked in a local optimum.