## 4 Experiment, Results & Discussion
### Optimization of Benchmark Problems
#### 4.1.1 Number of Agents  
There are only two parameters to be selected for PSA. The first is the number of iterations. This is related to the computational ability of the computing platform and the complexity of the problems. Hence, for this experiment the number of iteration is set to 1000. The second parameter is the number of agents. In this section the effect of the number of agents is investigated. The number of agents tested are 10, 20, 30, 40, and 50. All the settings are run 30 times; the best results of the run are tabulated in Table 2. Since the benchmark functions are minimization problems, the best results are the minimum value from the 30 runs of each setting. It can be seen that there is no one number of agents that works best for all test functions. However, the number of agents equal to 50 obtained a higher number of best results compared to the others. The Friedman signed rank test of 1 \(\times\) N is then conducted to identify the best number of agents using this set of benchmark functions. The Friedman signed rank test is a statistical method commonly used to provide an unbiased analysis of metaheuristics' performance [26; 27; 28]. The lower the Friedman rank of an algorithm, the better the algorithm is. A 1 \(\times\) N test compares the best-ranked algorithm with the other algorithms. If the Friedman statistic value-which is distributed according to \(\chi^{2}\) with \(k-1\) degrees of freedom, where \(k\) is the number of test (here \(k=5\), i.e.,: 5 number of agents tested)-has \(p\)-value of lesser than the significance level adopted, \(\alpha\), then the null hypothesis that states that all settings tested are on par with each other is rejected. In order to identify the significant difference, a post hoc procedure known as the Holm post hoc procedure is adopted. In this work, \(\alpha=0.05\) is adopted. Other than 0.05, 0.1 \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Function Name** & **Ideal Fitness** \\ \hline f4 & Shifted and Rotated & 400 \\  & [PERSON]’s Function & 400 \\ f5 & Shifted and Rotated [PERSON]’s Function & 500 \\ f6 & Shifted and Rotated & 600 \\  & Weierstrass Function & 600 \\ f7 & Shifted and Rotated & 700 \\  & [PERSON]’s Function & 700 \\ f8 & Shifted [PERSON]’s Function & 800 \\ f9 & Shifted and Rotated & 900 \\  & [PERSON]’s Function & 1000 \\ f10 & Shifted Schwefel’s Function & 1000 \\ f11 & Shifted and Rotated & 1100 \\  & [PERSON]’s Function & 1200 \\ f12 & Shifted and Rotated Katsuura & 1200 \\  & Function & \\  & Shifted and Rotated & 1300 \\ f13 & HappyCat Function & 1300 \\  & Shifted and Rotated HGBat & 1400 \\  & Function & \\  & Shifted and Rotated Expanded & \\ f15 & [PERSON]’s plus [PERSON]’s & 1500 \\  & Function & \\ f16 & Shifted and Rotated Expanded & 1600 \\  & Scaffer’s F6 Function & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of Benchmark Functions. is also commonly used by researchers. The smaller the value, the stricter the statistical test in accepting the null hypothesis. In this work, the statistical test is carried using Knowledge Extraction based on Evolutionary Learning (KEEL), which is an open-source tool with statistical analysis modules [26]. KEEL is a user friendly software developed by a group of researchers for research and academic purposes [26; 27; 28]. The Friedman ranks are listed in the last row of Table 2. The Friedman statistical value (distributed according to \(\chi^{2}\) with 4 degrees of freedom) is 28.030769 and its \(p\)-value is 0.000012, which is less than a significance level of 0.05. Therefore, a significant difference exists among the five settings. The Holm post-hoc is used to determine which setting is on par or worse than the best-ranked number of agents. Table 3 shows the \(p\)-value and Holm value of the post hoc test. The Holm's procedure rejects those hypotheses that have a \(p\)-value < Holm. Hence, a number of agents equal to 10 is significantly worse than 50, and the others are on par with each other. Thus, it can be concluded that a number of agents above 20 is recommended for PSA.