## 3 Manipulation of Swarms Positions and Space Solution Using Multi-Threaded Techniques
### Proposition II
The influence of the increase in coefficients values causes the multitude of a given step or displacement of individuals. Therefore, in our considerations we do not analyze the impact of these coefficients on the method and accuracy of the obtained solutions, and each parameter was chosen in a random way in the range \(\langle 0.1,0.4\rangle\). The obtained values of coefficients were respectively

* \(\xi=0.23\), \(m=30\), \(q_{0}=0.4\),
* \(\phi_{p}=0.15\), \(\phi_{s}=0.32\), \(best\_ratio=20\),
* \(\zeta=0.31\), \(\beta_{pop}=0.28\), \(I_{pop=0.18}\)
* \(\lambda=0.37\), \(\delta=0.12\), \(\varphi=0.33\), \(\rho=0.21\), \(bestratio=20\),
* \(k=0.15\), \(\beta_{0}=0.21\), \(\alpha=0.23\). ### Performance Metrics

For the purpose of evaluating algorithms, several basic metrics have been used. The accuracy of the optimization algorithms is evaluated by the average value of the solution obtained from the tests carried out what can be presented as

\[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]1

and error calculated as an absolute value between the ideal and obtained solution which is

\[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]2The second aspect is parallelization evaluated by two metrics - acceleration Y and efficiency \(\Psi\). Acceleration is the ratio of sequential execution time of the algorithm defined as

\[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]3

where \(\xi\) is execution time measured for one processor, and \(\varphi\) is execution time measured for \(pc\) processors. The second assessment is made by the following formula

\[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]4

In addition, scalability with the number of cores is measured in accordance with [PERSON]'s law

\[a=\langle a_{1},a_{2}\rangle=\left\langle a_{1},\frac{a_{2}}{pc}\right\rangle \cup\bigcup_{k=1}^{pc-1}\left\langle\frac{ka_{2}}{pc},\frac{a_{2}}{pc}(k+1) \right\rangle. \tag{27}\]5

where \(\Theta\) is the proportion of execution time of the proposal to the original versions. For our measurements, \(\Theta\) was determined as the quotient of the average time for all algorithms for \(pc\) cores and the sum of time needed for \(pc\) and one processor. ### Results

Firstly, we analyzed the impact of different coefficient values on the algorithms. We noticed that the coefficient values depend on the function itself--the more local extremes, the higher the values should be. This is due to the fact that individuals have to get out in such a minimum location, hence the large values of coefficients can prolong movement in one iteration and allow escape. Such reasoning forced us to depend on the value of coefficients from the pseudorandom generator. This action, combined with averaging the obtained results, enabled to obtain averaged solutions. It was performed for all versions of the algorithms--the original and three proposed modifications in this paper. The obtained solution are presented in Tables 2-5 and errors values are in Tables 6-9. In all cases, the first proposition--the use of devoted local search--reduced the error values in almost every case. Of course, there were cases when the selected algorithms had a minimal difference between the results (see CSA results), although it may be due to bad initial position of individuals. In contrast, the second proposal related to the division of the solution space brought quite a big drop in the value of errors for each case.