## 3 Manipulation of Swarms Positions and Space Solution Using Multi-Threaded Techniques
### Proposition II
This points to the fact that the size of the space is very important for metaheuristics--a search of the same area in less time and without necessarily increasing computing needs is a very important issue. The proposed division of space is one of the many cases that can be corrected, but it is one that significantly improves solution for each test function indicates the direction of future research. Moreover, the combination of these two proposition improved the obtained results for many cases, but not for all. GA and PSOA improved solutions for more than 5 cases, when FA improved the score for 9 from 10 benchmark functions. For better visualization the error values, The average error obtained for each version of the algorithm is shown in \[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]6 The graph shows that the error value is the smallest when applying proposition 2 or 3, and 1 has an approximately constant error. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **GA** & **PSOA** & **FA** & **CSA** & **WSA** & **ACO** \\ \hline \(f_{1}\) & 0.03072 & 0.00027 & 0.08178 & 0.00217 & 0.0039 & 0.00032 \\ \(f_{2}\) & 0.09337 & 0.07208 & 0.05691 & 0.07437 & 0.10287 & 0.06821 \\ \(f_{3}\) & 0.00001 & 0 & 0.00006 & 0.00005 & 0.00002 \\ \(f_{4}\) & 0.00079 & 0.00083 & 0.00166 & 0.00015 & 0.00011 & 0.00029 \\ \(f_{5}\) & \(-186.438\) & \(-186.597\) & \(-186.563\) & \(-186.633\) & \(-186.694\) \\ \(f_{6}\) & 0.0001 & 0 & 0.0001 & 0.00003 & 0 & 0 \\ \(f_{7}\) & 0.00124 & 0.00023 & 0.00005 & 0.00002 & 0.00013 & 0.00012 \\ \(f_{8}\) & \(-391.968\) & \(-391.893\) & \(-391.9878\) & \(-391.983\) & \(-391.926\) & \(-391.978\) \\ \(f_{9}\) & 0.01592 & 0.03343 & 0.0199 & 0.00995 & 0.01 & 0.00899 \\ \(f_{10}\) & 0.00314 & 0 & 0.00971 & 0.00117 & 0.00073 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Averaged solution values achieved by all algorithms for each test functions for proposition II. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **GA** & **PSOA** & **FA** & **CSA** & **WSA** & **ACO** \\ \hline \(f_{1}\) & 0.07593 & 0.08617 & 0.26872 & 0.16432 & 0.00257 & 0.00319 \\ \(f_{2}\) & 0.12283 & 0.16489 & 0.18691 & 0.1729 & 0.1275 & 0.13129 \\ \(f_{3}\) & 0.00172 & 0.27991 & 0.05206 & 0.00948 & 0.00029 & 0.00192 \\ \(f_{4}\) & 0.00506 & 0.00963 & 0.00174 & 0.01354 & 0.00981 & 0.00166 \\ \(f_{5}\) & \(-186.014\) & \(-185.831\) & \(-185.843\) & \(-185.824\) & \(-185.805\) & \(-185.815\) \\ \(f_{6}\) & 0.00001 & 0 & 0.00001 & 0.0002 & 0.00001 & 0.00001 \\ \(f_{7}\) & 0.00105 & 0.00062 & 0.66179 & 0.00035 & 0.00025 & 0.00037 \\ \(f_{8}\) & \(-391.329\) & \(-391.58\) & \(-391.344\) & \(-391.746\) & \(-391.598\) & \(-391.594\) \\ \(f_{9}\) & 0.19899 & 0.07731 & 0.13266 & 0.07822 & 0.1328 & 0.09898 \\ \(f_{10}\) & 0.00103 & 0.00172 & 0.04368 & 0.33444 & 0.00103 & 0.00098 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Averaged solution values achieved by all original algorithms for each test functions.