# Metaheuristic Optimized Multi-Level Classification Learning System for Engineering Management
## 4 Metaheuristic-Optimized Multi-Level Classification System  
### Benchmarking of the Enhanced Metaheuristic Optimization Algorithm  
This section evaluates the efficiency of the enhanced FA by testing benchmark functions to elucidate the characteristics of optimization algorithms. Ten complex benchmark functions with different characteristics and dimensions [74; 75] were used herein to evaluate the performance of the enhanced FA. This investigation used 200 for the number of fireflies and 1000 for the maximum number of iterations.  
Table 3 presents numerical benchmark functions and their optimal values that are obtained, using the enhanced FA. The results indicate that the enhanced FA yielded all of the optimal values, which were very close to the analytically obtained values. Therefore, the proposed enhanced FA is promising.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No.** & **Benchmark Functions** & **Dimension** & \begin{tabular}{c} **Minimum** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Maximum** \\ **Value** \\ \end{tabular} & \begin{tabular}{c} **Mean of** \\ **Optimum** \\ \end{tabular} & \
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} &
\begin{tabular}{c} **Total Time** \\ **(s)** \\ \end{tabular} \\ \hline
1 & Griewank & 10 & \(3.03\times 10^{-11}\) & \(3.75\times 10^{-10}\) & \(1.36\times 10^{-10}\) & \(8.44\times 10^{-11}\) & \(2.10\times 10^{4}\) \\  & \(f(x)=\sum\limits_{i=1}^{d}\frac{x_{i}^{2}}{\log 200}-\prod\limits_{i=1}^{d}cos \left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & 30 & \(7.84\times 10^{-8}\) & \(2.36\times 10^{-7}\) & \(1.51\times 10^{-7}\) & \(4.49\times 10^{-7}\) & \(1.99\times 10^{4}\) \\  & \(x_{i}=(-600,600)\) & & & & & & \\ Minimum f(0,...) = 0 & 50 & \(5.40\times 10^{-7}\) & \(1.74\times 10^{-6}\) & \(1.17\times 10^{-6}\) & \(3.01\times 10^{-7}\) & \(2.34\times 10^{4}\) \\
2 & Deb 01 & \(10\) & \(-1\) & \(-1\) & \(-1\) & \(4.98\times 10^{-12}\) & \(1.54\times 10^{4}\) \\  & \(f(x)=-\frac{1}{2}\sum_{i=1}^{d}sin^{6}(5^{\circ}\text{p}^{\circ}\text{p}^{ \circ}\text{x})\) & 30 & \(-1\) & \(-8.34\times 10^{-1}\) & \(-9.93\times 10^{-1}\) & \(3.12\times 10^{-2}\) & \(1.85\times 10^{4}\) \\  & Minimum f(0,...) = \(-1\) & 50 & \(-1\) & \(-5.24\times 10^{-1}\) & \(-9.31\times 10^{-1}\) & \(1.39\times 10^{-1}\) & \(2.26\times 10^{4}\) \\
3 & Csendes & 10 & \(7.04\times 10^{-11}\) & \(1.06\times 10^{-5}\) & \(9.57\times 10^{-7}\) & \(2.09\times 10^{-6}\) & \(3.55\times 10^{4}\) \\  & \(f(x)=\sum\limits_{i=1}^{d}x_{i}^{6}\Big{(}2+\sin\frac{1}{x_{i}}\Big{)}\) & 30 & \(4.39\times 10^{-6}\) & \(2.39\times 10^{-3}\) & \(5.07\times 10^{-4}\) & \(5.66\times 10^{-4}\) & \(4.27\times 10^{4}\) \\  & \(x_{i}=[-1;1]\) & & & & & & \\ Minimum f(0,...) = 0 & 50 & \(3.78\times 10^{-4}\) & \(6.53\times 10^{-3}\) & \(1.49\times 10^{-3}\) & \(1.22\times 10^{-3}\) & \(4.91\times 10^{4}\) \\
4 & [PERSON] & 10 & \(2.80\times 10^{-12}\) & \(8.65\times 10^{-12}\) & \(4.82\times 10^{-12}\) & \(1.59\times 10^{-12}\) & \(1.50\times 10^{4}\) \\ \(f(x)=\sum_{i=1}^{d}x_{i}^{2}\); \(x_{i}=[-5.12;5.12]\) & 30 & \(7.40\times 10^{-11}\) & \(3.33\times 10^{-4}\) & \(1.11\times 10^{-5}\) & \(6.08\times 10^{-5}\) & \(1.97\times 10^{4}\) \\ Minimum f(0,...) = 0 & 50 & \(1.39\times 10^{-4}\) & \(4.45\times 10^{-2}\) & \(8.07\times 10^{-3}\) & \(9.66\times 10^{-3}\) & \(2.37\times 10^{4}\) \\
5 & Alpine 1 & 10 & \(6.69\times 10^{-7}\) & \(5.49\times 10^{-4}\) & \(2.03\times 10^{-5}\) & \(9.99\times 10^{-5}\) & \(1.50\times 10^{4}\) \\ \(f(x)=\sum\limits_{i=1}^{d}|x_{i}\sin(x_{i})+0.1\,x_{i}\,|\) & 30 & \(6.80\times 10^{-6}\) & \(7.43\times 10^{-3}\) & \(5.21\times 10^{-4}\) & \(1.65\times 10^{-3}\) & \(2.07\times 10^{4}\) \\ \(x_{i}=[-10;10]\) & & & & & & \\ Minimum f(0,...) = 0 & 50 & \(2.43\times 10^{-5}\) & \(4.95\times 10^{-3}\) & \(9.43\times 10^{-4}\) & \(1.42\times 10^{-3}\) & \(2.34\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical benchmark functions.