## 3 Methodology
Based on the comparisons of \(h_{win}\) for _best-\(f_{best}\)_ and _mean-\(f_{best}\)_, FIPSaDE has the highest values, followed by FIPS and SaDE. Therefore, FIPSaDE consistently produces a group of better-quality solutions than its original variants when problem dimensionality and population size increase. For most tested functions, FIPSaDE proves to produce better _best-\(f_{best}\)_ and _mean-\(f_{best}\)_ in the swarm space compared to the other algorithms as the problem space increases. FIPSaDE, a hybrid of both FIPS and SaDE, can maneuver and manage the swarm solutions more effectively. Therefore, FIPSaDE improves performance in finding the best fitness point as the problem space increases. The variable _mean-\(f_{best}\)_ for FIPSaDE, FIPS and SaDE is denoted as \(\mu_{{}_{FIPSaDE}}\), \(\mu_{{}_{FIPS}}\) and \(\mu_{{}_{SaDE}}\), respectively. A \(t\)-test is performed with a significance level, \(\alpha\) of 0.05 to evaluate the hypothesis whether \(\mu_{{}_{FIPSaDE}}\) of the proposed algorithm is better than \(\mu_{{}_{FIPS}}\) and \(\mu_{{}_{SaDE}}\) or not, as shown below. \[H_{0}:\mu_{{}_{FIPSaDE}}\geq\mu_{{}_{FIPS}}\] \[H_{1}:\mu_{{}_{FIPSaDE}}<\mu_{{}_{EIPS}}\]

\[H_{0}:\mu_{{}_{FIPSaDE}}\geq\mu_{{}_{SaDE}}\] \[H_{1}:\mu_{{}_{FIPSaDE}}<\mu_{{}_{SaDE}}\]

\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{3}{c}{_Best-\(f_{best}\)_} & \multicolumn{3}{c}{_Mean-\(f_{best}\)_} & \multicolumn{3}{c}{_Std-\(f_{best}\)_} \\ \cline{2-10}  & **FIPS** & **SaDE** & **FIPSaDE** & **FIPS** & **SaDE** & **FIPSaDE** & **FIPS** & **SaDE** & **FIPSaDE** \\ \hline F1 & 1153.25 & 450.00 & \(\mathbf{-450.00}\) & 2462.33 & 450.00 & \(\mathbf{-450.00}\) & 1044.25 & **0.00** & **0.00** \\ F2 & 12,312.68 & 967.61 & \(\mathbf{-450.00}\) & 19,296.90 & 3624.73 & \(\mathbf{-450.00}\) & 2864.28 & 1944.03 & **0.00** \\ F3 & 19,742,966.18 & \(\mathbf{1.55}\) & 26,213.65 & 46,306,427.65 & 64,427.16 & \(\mathbf{939.247}\) & 18,623,890.22 & 2,656,381.58 & **54,148.58** \\ F4 & 18,353.52 & 117,796.54 & \(\mathbf{-448.2}\) & 29,351.56 & 26,706.41 & \(\mathbf{-364.74}\) & 7366.24 & 7776.96 & **94.60** \\ F5 & 8344.21 & 7515.12 & \(\mathbf{3823.55}\) & 12,



The results of hypothesis tests for the paired FIPSaDE-FIPS and FIPS-SaDE based on different settings of problem dimensionality and population size are shown in Table 10. If the \(p\)-values are less \(\alpha\), they are bolded, indicating that the associated \(H_{0}\) is rejected. As a result, there is enough evidence to accept \(H_{1}\). The \(p\)-values associated with rejecting \(H_{0}\) are bolded and its frequency is denoted as \(h_{(-H_{0})}\). At the bottom of the table, \(h_{(-H_{0})}\) shows the frequency \(H_{0}\) being rejected for different settings. For the setting \(10D25N\), \(\mu_{{}_{FIPSaDE}}\) is significantly better than \(\mu_{{}_{FIPS}}\) and \(\mu_{{}_{SaDE}}\) in 10 and 9 functions, respectively. When the setting increases from \(10D25N\) to \(50D50N\), the values \(h_{(-H_{0})}\) for FIPSaDE-FIPS increase from 10 to 18 and 22. Therefore, the frequency that FIPSaDE is significantly better than FIPS increases. When similar comparisons are made for FIPSaDE-SaDE, an increasing trend is observed, but it is less obvious. The findings about \(h_{(-H_{0})}\) show that the performances of FIPSaDE are significantly better than FIPS and SaDE when the problem dimensionality and population size increase. ## 6 Conclusions

In this study, a hybrid of FIPS and SaDE called FIPSaDE is proposed, and its performance is validated by running the algorithm against the benchmark functions while also being compared to their respective original version, the SaDE and FIPS as well as it variants, such as DE and DE-PSO. The self-adaptation strategy of SaDE is adapted and maneuvered by the FIPS particle swarm, preventing the solutions from being trapped in the local region. Each algorithm can adaptively adjust the parameter values while the swarm is searching for the best solution needed.