# A Modified Artificial Bee Colony Algorithm Based on the Self-Learning Mechanism
## 4 Experiments and Results
### Experimental Results  
This paper uses five indexes to analysis the experimental results: the best solution (Best), median solution (Median), worst solution (Worst), average solution (Mean), and standard deviation (Std). Tables 2-4 show the experimental results obtained by each algorithm in the 20 independent runs. The results suggest that SLABC offers the higher solution accuracy on almost all the functions except function \(f_{4}\) with \(D=30,60,100\). It can be seen from the formulation of Rosenbrock function (\(f_{4}\)) that the first term \(100(x_{i+1}-x_{i})^{2}\) mainly influence the function value, which means that only the values among different dimensions in the candidate solutions are almost the same, the function value is smaller. Using the mean and standard deviation of the selected two solution, the distABC algorithm obtained the new candidate solutions, in which the values among different dimensions are of uniform size. Therefore, based on the distributed solution update rule, distABC algorithm gets the best results on Rosenbrock function (\(f_{4}\)). Moreover, in the case of functions \(f_{5}\),\(f_{6}\),\(f_{8}\) with \(D=30\), the solution accuracy of SLABC are equal with the best of other algorithms. The Rastrigin function (\(f_{5}\)), [PERSON] function (\(f_{6}\)) and [PERSON]'s Problem 2.26 function (\(f_{8}\)) are multimodal functions and are easy to obtain the optimal solutions with \(D=30\). As the dimension gets higher, the difficulty of obtaining the optimal solution increases gradually. However, the solution accuracy of SLABC are better than the functions \(f_{5}\),\(f_{6}\),\(f_{8}\) with \(D=60\) and \(D=100\), which proves that SLABC algorithm outperforms the other algorithms. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Type** & **Test Function** & **Formulation** & **Search Range** & **Minimum Value** \\ \hline \multirow{4}{*}{Unimodal} & Sphere & \(f_{1}(\overrightarrow{x})=\sum\limits_{i=1}^{D}x_{i}^{2}\) & \(x_{i}\in[-100,100]\) & \(f_{1}(\overrightarrow{q})=0\) \\  & Quartic & \(f_{2}(\overrightarrow{x})=\sum\limits_{i=1}^{D}x_{i}^{4}+random(0,1)\) & \(x_{i}\in[-1.28,1.28]\) & \(f_{2}(\overrightarrow{q})=0\) \\  & Schwefel’s Problem 2.22 & \(f_{3}(\overrightarrow{x})=\sum\limits_{i=1}^{D}|x_{i}|+\prod\limits_{i=1}^{D}|x _{i}|\) & \(x_{i}\in[-10,10]\) & \(f_{3}(\overrightarrow{q})=0\) \\  & Rosenbrock & \(f_{4}(\overrightarrow{x})=\sum\limits_{i=1}^{D}|100(x_{i+1}-x_{i})^{2}(1-x_{i} )^{2}|\) & \(x_{i}\in[-30,30]\) & \(f_{4}(\overrightarrow{1})=0\) \\ \hline \multirow{4}{*}{Multimodal} & [PERSON] & \(f_{5}(\overrightarrow{x})=\sum\limits_{i=1}^{D}|x_{i}^{2}-10\cos(2\pi x_{i})+ 10|\) & \(x_{i}\in[-5.12,5.12]\) & \(f_{5}(\overrightarrow{q})=0\) \\  & [PERSON] & \(f_{6}(\overrightarrow{x})=\sum\limits_{i=1}^{D}x_{i}^{2}-\prod\limits_{i=1}^{D} \cos(\frac{\pi x_{i}}{q^{2}})+1\) & \(x_{i}\in[-600,600]\) & \(f_{6}(\overrightarrow{q})=0\) \\  & [PERSON] & \(f_{7}(\overrightarrow{x})=-20\exp(-0.2\sqrt{\sum\limits_{i=1}^{D}x_{i}^{2}/D)+ 20-\exp{(\sum\limits_{i=1}^{D}\cos{(2\pi x_{i})}/D)}+e}\) & \(x_{i}\in[-30,30]\) & \(f_{7}(\overrightarrow{q})=0\) \\  & [PERSON]’s Problem 2.26 & \(f_{8}(\overrightarrow{x})=-418.9829D-\sum\limits_{i=1}^{D}\left(x_{i}sin(\sqrt {|x_{i}|})\right)\) & \(x_{i}\in[-500,500]\) & \(f_{8}(\overline{420,9687})=0\) \\  & Penalized & \(f_{9}(\overrightarrow{x})=0.1\left\{\sin^{2}(3\pi x_{1})+\sum\limits_{i=1}^{D} \left(x_{i}-1\right)^{2}[1+\sin^{2}(3\pi x_{i+1})]+\left(x_{D}-1\right)^{2} \right\}\) & \(x_{i}\in[-50,50]\) & \(f_{9}(\overrightarrow{1})=0\) \\  & & \(+\sum\limits_{i=1}^{D}u(x_{i},5,100,4)\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical benchmark functions.