*4.3. The Proposed Hybrid Multi-Population GTO and BWO*

The proposed HMGTO-BWO is introduced to solve complex and nonlinear optimization issues. The following steps describe the procedure of the proposed HMGTO-BWO:

**Step 1**: Define the parameters of the proposed HMGTO-BWO as well as the constraints of the problem.

**Step 2:** Generate a set of populations randomly.

**Step 3:** Divide the populations into three subpopulations (N1, N2, N3), where N1 = N2 = N/3 and N3 = N − (N1 + N2), where N, N1, N2, and N3 are numbers of the population, the first subpopulation, the second subpopulation, and the third subpopulation, respectively.

**Step 4:** Update the populations in each subpopulation group based on the GTO, as illustrated in Section 4.1.

**Step 5:** Accept the new updated subpopulations if their values are better than those of the old populations.

**Step 6:** Combine the three subpopulations as one vector; it represents the initial populations of the BWO technique.

**Step 7:** Update the populations based on BWO, including the swimming motion, the Levy flight motion, and the fall of BWs.

**Step 8:** Repeat Step 3 to Step 7 until the stopping criterion is satisfied.

The step procedures of the suggested algorithm are depicted in Figure 4.

**Figure 4.** Flowchart of the proposed HMGTO-BWO.

The HGTO-BWO computational complexity is based on the initialization, fitness assessment, and updating of the silverbacks and BW, and it can be described as follows:

$$\text{O}(\text{HGTO} - \text{BNO}) = (\text{Sub}.\text{Population1} + \text{Sub}.\text{Population2} + \text{Sub}.\text{Population3})\_{\text{GTO}} + \text{BNO} \tag{35}$$

$$\begin{aligned} O(HGDO-BWO) &= \begin{bmatrix} O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_1\right) + O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_1 \times D\right) \times 2 \end{bmatrix} + \begin{bmatrix} O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_2\right) + O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_2 \times D\right) \times 2 \end{bmatrix} \\ &+ \begin{bmatrix} O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_3\right) + O\left(t\_{\text{max}} \times \frac{1}{3}\text{N}\_3 \times D\right) \times 2 \end{bmatrix} + O(N \times (1 + 1.1 \times t\_{\text{max}})) \\ &= O(\text{N} \times (1 + t\_{\text{max}} + TD) \times 2 + (1 + 1.1 \times t\_{\text{max}})) \end{aligned} \tag{36}$$
 
$$\text{where } D \text{ is the dimension of the problem.}$$

#### **5. Testing of Benchmark Function**

For a fair comparison between the suggested HGTO-BWO and the other algorithmic approaches, the maximum iterations number was set to 500; the population size was assigned to 30; and 30 runs were conducted for each considered optimizer. The proposed HGTO-BWO was investigated via the traditional benchmark functions and CEC 2019 functions. The fetched results were compared to TSA, GWO, WOA, SCA, HS, BWO, and GTO. The algorithms' parameters are presented in Table A1 in Appendix A.
