# Metaheuristic Optimized Multi-Level Classification Learning System for Engineering Management
## 3 Methodology
### Optimization in Machine Learning
#### 3.2.2 Enhanced Firefly Algorithm
\[\alpha^{t}=\alpha_{0}\theta^{t} \tag{7}\]  
where \(\alpha_{0}\) is the initial randomization parameter; \(\alpha^{t}\) is the randomization parameter in the \(t^{\text{th}}\) generation; \(\theta\) is the randomness reduction constant (\(0<\theta<1\)), and \(t\) is the number of the iteration. The selected value of \(\theta\) in this implementation is 0.9 based on the literature, and \(t\in[0\), \(t_{max}\)), where \(t_{max}\) is the maximum number of generations. Levy Flight: Increasing Movement and Mimicking Insects  
A random walk is the outstanding characteristic of Levy flight in which the step length follows a Levy distribution [70]. Equation (8) provides the step length \(s\) in [PERSON]'s algorithm. \[\text{Levy }\sim s=\frac{u}{\left|v\right|^{1/\tau}} \tag{8}\]  
where Levy is a Levy distribution with an index \(\tau;s\) denotes a power-law distribution; and \(u\) and \(v\) are drawn from normal distributions, as follows. New solutions are obtained around the optimal solution using a Levy walk, which expedites the local search. \[u\sim N(0,\sigma_{u}^{2}),v\sim N(0,\sigma_{v}^{2}) \tag{9}\]  
\[\text{where }\sigma_{u}=\left\{\frac{\Gamma(1+\tau)\sin(\pi\tau/2)}{\Gamma[(1+ \tau)/2]\tau 2^{(\tau-1)/2}}\right\}^{1/\tau}\sigma_{v}=1 \tag{10}\]  
Here, \(\Gamma(\text{t})\) is the Gamma function. \[\Gamma(\text{t})=\int\limits_{0}^{\infty}z^{t-1}e^{-z}dz \tag{11}\]  
Notably, the aforementioned metaheuristic components supplement the basic FA to improve the effectiveness and efficiency of optimization process. The movement of the \(i^{\text{th}}\) firefly that is attracted to a brighter \(j^{\text{th}}\) firefly is thus modified as follows:  
\[x_{i}^{t+1}=x_{i}^{t}+\beta(x_{j}^{t}-x_{i}^{t})+\alpha^{t}\text{sign}[\text{ rand }-0.5]\otimes\text{Levy} \tag{12}\]  
Table 1 presents the default settings of the parameters used in the enhanced FA. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Parameter** & **Setting** & **Purpose** \\ \hline Swarm and metaheuristic settings & Number of fireflies & User defined; default value: 80 & Population number \\ \cline{2-4}  & Max generation & User defined; default value: & Constrain implementation of \\  & & \(tmax=40\) & algorithm \\ \cline{2-4}  & Chaotic logistic map & Random generation; biotic potential \(\eta=4\) & Generate initial population with high diversity \\ \hline Brightness & Objective function & Accuracy & Calculate firefly brightness \\ \hline Attractiveness & \(\beta\)min & Default value \(\beta_{0}=0.1\) & Minimum value of attractive \\  & & & parameter \(\beta\) \\ \cline{2-4}  & Chaotic Gauss/mouse & & \\  & map & Random generation & Automatically tune \(\beta\) parameter \\ \cline{2-4}  & \(\gamma\) & Default value \(\gamma=1\) & Absorption coefficient \\ \hline  & \(\alpha\) & Default value \(\alpha_{0}=0.2\) & Randomness of firefly movement \\ \cline{2-4}  & Adaptive inertia weight & Default value \(\theta=0.9\) & Control the local and global search capabilities of swarm algorithm \\ \cline{2-4}  & & & Accelerate the local search by \\  & Lévy flight & Default value \(\tau=1.5\) & generating new optimal \\  & & & neighborhoods around the obtained \\  & & & best solution \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Default settings of parameters of enhanced FA.