## 5 Bio-Inspired Methods with Parameter Adaptation  
The BCO algorithm has proven to be a good technique to optimize parameters [35; 63], which is why we perform an automated search that allows better performance of the BCO algorithm. For the studied case of the fuzzy controller, the fitness function in the BCO algorithm is represented with the Mean Square Error (MSE) shown in Equation (12). For each Follower Bee for \(N\) Cycles, the Type-1 FLS design for the BCO algorithm is evaluated and the objective is to minimize the error. Therefore, for designing the fuzzy systems, which dynamically adjust the _alpha_ and _beta_ parameters, the iteration and diversity metrics are considered as inputs. The distribution of membership functions in the inputs and outputs is realized in a symmetrical way. The design of input variables can be appreciated in Figure 1 for the type-1 fuzzy logic system. Figure 1: Type-1 fuzzy system for dynamic parameter adaptation in BCO (Bee Colony Optimization). The fuzzy system illustrated in Figure 1 is type-1, has iteration and diversity as inputs and has the _Beta_ and _Alpha_ parameters as outputs. In this case, the inputs were granulated into three trapezoidal membership functions and outputs into five trapezoidal membership functions. It contains nine rules that were designed to control the behavior of BCO. The fuzzy system in Figure 2 is an interval type-2 system, has the same number of membership functions per input and output as the type-1 fuzzy system, but now, as type-2 trapezoidal membership functions, the fuzzy rule base is the same as in type-1 because the knowledge is not changing only the type of membership functions. Table 1 contains the fuzzy rules used for both fuzzy systems in Figures 1 and 2; these rules were designed based on several experiments to create knowledge from parameters of BCO and how to control its behavior or model a proper behavior. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Inputs**} & \multicolumn{2}{c}{**Outputs**} \\ \cline{2-5}  & **Iteration** & **Diversity** & **Beta** & **Alpha** \\ \hline
1 & Low & Low & High & Low \\
2 & Low & Medium & MediumHigh & Medium \\
3 & Low & High & MediumHigh & MediumLow \\
4 & Medium & Low & MediumHigh & MediumLow \\
5 & Medium & Medium & Medium & Medium \\
6 & Medium & High & MediumLow & MediumHigh \\
7 & High & Low & Medium & High \\
8 & High & Medium & MediumLow & MediumHigh \\
9 & High & High & Low & High \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy rules for dynamic Beta and Alpha. Figure 2: Interval type-2 fuzzy system for dynamic parameter adaptation in BCO. The distribution of membership functions from the type-1 FLS is taken as reference to design the interval type-2 FLS. In addition, membership functions from interval type-2 FLS were designed symmetrically. T1 FLS and IT2 FLS have the same rules and are of Mamdani style. Using the described methodology for parameter adaptation presented in Section 4, some parameters in PSO are dynamically adjusted using a type-1 and interval type-2 fuzzy systems. Based on several experiments, the best parameters were selected to be adjusted over the iterations, in this case, \(C_{1}\), \(C_{2}\) and the constriction factor \(C\) from Equation (6), which are the most important and have a big impact in the behavior of the algorithm, so controlling these parameters allows to control the entire algorithm. Figures 3 and 4 present the type-1 fuzzy system and the interval type-2 fuzzy system used to dynamic parameter adaptation of the parameters of PSO, using as inputs metrics iteration and diversity. The fuzzy system illustrated in Figure 3 is type-1, has iteration and diversity as inputs and has the parameters \(C_{1}\), \(C_{2}\) and _constriction factor_ as outputs. In this case, the inputs were granulated into three triangular membership functions, and the outputs into five triangular membership functions. It contains nine rules that were designed to control the behavior of PSO.