## 3 Manipulation of Swarms Positions and Space Solution Using Multi-Threaded Techniques
### Proposition II
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Function** & **GA** & **PSOA** & **FA** & **CSA** & **WSA** & **ACO** \\ \hline \(f_{1}\) & 1603.4 & 1584 & 1622.72 & 1583.12 & 1573.44 & 1602 \\ \(f_{2}\) & 2399.4 & 2280 & 2583 & 2439 & 2482 & 2403 \\ \(f_{3}\) & 789.36 & 801.68 & 786.72 & 762 & 782.32 & 773 \\ \(f_{4}\) & 1247.84 & 1291.5 & 1260.16 & 1245.2 & 1249.84 & 1267 \\ \(f_{5}\) & 4785.44 & 4772.24 & 4843.52 & 4762.56 & 4670.16 & 4694.95 \\ \(f_{6}\) & 571.12 & 589.6 & 581.4 & 583.44 & 586.08 & 582.2 \\ \(f_{7}\) & 648 & 583.44 & 603 & 586.08 & 602.8 & 589.2 \\ \(f_{8}\) & 2491 & 2429 & 2539 & 2349 & 2483 & 2424 \\ \(f_{9}\) & 601.5 & 728.64 & 704.88 & 620.25 & 724.24 & 636 \\ \(f_{10}\) & 2124 & 2148 & 2348 & 2189 & 2290 & 2201 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Running time values achieved by all algorithms for each test functions for proposition II. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **GA** & **PSOA** & **FA** & **CSA** & **WSA** & **ACO** \\ \hline \(f_{1}\) & 1603.6 & 1349.25 & 1345.5 & 1367.25 & 1335.75 & 1401 \\ \(f_{2}\) & 2418.24 & 2286.75 & 2449.8 & 2399.76 & 2690.25 & 2650 \\ \(f_{3}\) & 788.48 & 756 & 673.5 & 678 & 672 & 673 \\ \(f_{4}\) & 1258.4 & 1060.5 & 1062.75 & 1061.25 & 1060.5 & 1064 \\ \(f_{5}\) & 4094.25 & 4091.25 & 4122.75 & 4001 & 4104 & 4005 \\ \(f_{6}\) & 483 & 567.6 & 488.25 & 491.25 & 486 & 473 \\ \(f_{7}\) & 584.32 & 495.75 & 545.75 & 497.25 & 496.5 & 491 \\ \(f_{8}\) & 2202 & 2121 & 2292 & 2160.75 & 2176.5 & 2189 \\ \(f_{9}\) & 718.96 & 601.5 & 703 & 597 & 635 & 606 \\ \(f_{10}\) & 2103 & 2014 & 2261 & 2127 & 2221 & 2134 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Running time values achieved by all algorithms for each test functions for proposition III. chosen proposal, another algorithm proved to be the best. It is easy to notice that depending on the chosen proposal, another algorithm proved to be the best. Without any modification the best algorithm was classic version of PSOA and ACO. Adding first proposition the best one were PSOA and FA, CSA, ACO are in the second place equally, and with second proposals, there are the same scores. The third modification allowed CSA and ACO to be the best algorithms. Of course, obtained results depend on the equations of motion, their length and other factors affecting such a large palette of metaheuristic methods. Not only, the accuracy was measured, but the duration of action with using multithreading techniques. Measured time values are presented in Table 4, 10, 11, 12, 13 and on Figure 3. The use of any modification shortens the operating time for almost every case compared to the original versions. What is interesting, first and second proposition shortened time of approximately the same value, when the third obtained the best result in this aspect. To accurately assess the operation time, we used the formulas described in Equations (37) and (38), the obtained results are presented in Table 14. The worst results were achieved for the first modification, than the third one and the second one as the best one in terms of acceleration. Scalability for each proposition (having 6 cores) were approximately successively 1.79, 1.79 and 1.86. To analyze these values, we also calculated the scalability for the 2 and 4 cores which results are presented in Figure 4. Ideal solution would be linear curve, but the more cores are used, the worst scalability is. In the case of the first two proposals, it decreases quite rapidly. However, the scalability of proposition III only minimally decreases after using more than 4 cores. Another aspect is the number of iteration needed for the sequential version to get similar results (approximately) to the presented proposals. The obtained data are presented in Table 15. In the case of 6 cores (for proposition I and II), the number of iteration must be increased by almost 22-26%. Such a large discrepancy is caused by the randomness of the algorithms (for example, the initial distribution of the population).