## 2 Materials and Methods
### Insbbo  
Biogeography-based optimization (BBO) is a nature-inspired meta-heuristic algorithm first proposed by [PERSON] in 2008 [49]. The algorithm draws inspiration from the science of biogeography, which studies the distribution of biological species across different geographical locations over time. In natural ecosystems, habitats that are highly suitable for life tend to have a high number of species, while less hospitable environments host fewer species. The migration of species from high-suitability habitats to low-suitability ones, along with occasional random mutations, forms the fundamental biological processes that BBO mimics in its search strategy. This elegant concept allows BBO to efficiently explore and exploit complex search spaces by combining solution sharing (through migration) and diversity maintenance (through mutation), making it highly effective for solving optimization problems where both global exploration and local refinement are crucial. In BBO, each candidate solution to the optimization problem is analogized as a habitat. The quality of a solution is quantified using a metric called the habitat suitability index (HSI), where a higher HSI indicates a better solution. Habitats with higher HSIs share their features with others through a process known as migration, and random modifications called mutations introduce variability to avoid premature convergence to local optima [50; 51; 52]. Unlike traditional evolutionary algorithms where populations evolve through selection, crossover, and mutation, BBO emphasizes the migration of features among solutions while maintaining each individual habitat in its original form after each iteration. The migration process is governed by two main rates: the emigration rate (\(\mu\)) and the immigration rate (\(\lambda\)). High-HSI habitats have high emigration rates and low immigration rates, meaning they tend to share their features with others rather than accepting features from them. Conversely, low-HSI habitats have high immigration rates, allowing them to improve by adopting features from better habitats. The relationship between immigration, emigration, and the number of species in a habitat is typically modeled linearly. Specifically, the emigration rate \(\mu_{i}\) and immigration rate \(\lambda_{i}\) for \(i\)-th habitat are defined as shown in Equations (9) and (10) [53]:  
\[\mu_{i}=E\times\left(\frac{i}{N}\right) \tag{10}\]  
\[\lambda_{i}=I\times\left(1-\frac{i}{N}\right) \tag{11}\]  
where \(i\) represents the rank of the habitat in terms of suitability, \(N\) is the total number of habitats. During the migration phase of BBO, solutions improve by exchanging information between habitats based on their respective immigration and emigration rates. Specifically, each habitat is considered as a candidate for accepting new features, and for each feature, a decision is made whether to migrate information from a donor habitat based on the host's immigration rate. The mathematical formulation governing the migration process is presented in Equation (12) [51]:  
\[H_{j}\left(SIVs\right)\leftarrow\ H_{j}\left(SIVs\right)+H_{i}\left(SIVs\right) \tag{12}\]  
where \(H_{j}\) represents the host habitat, \(H_{i}\) represents the guest habitat, and \(SIVs\) denotes the set of suitability index variables (SIVs), which are the key features or decision variables of the solution. This operation implies that the feature values of the host habitat are updated by incorporating corresponding feature values from the guest habitat. Essentially, the host habitat adopts part of the characteristics from the guest habitat, thereby improving or diversifying its solution structure. In this migration process, the selection of \(H_{i}\) as the guest is probabilistic and weighted by the emigration rate, favoring habitats with higherHSI values as sources of useful features. Meanwhile, the likelihood of a host \(H_{j}\) accepting incoming information is determined by its immigration rate, which is inversely related to its own HSI.