## 3 Methodology
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **FIPS** & **DE** & **SaDE** & **DE-PSO** & **FIPSaDE** \\ \hline F1 & 1.6913E+00 & 5.6843E-14 & **1.8948E-15** & 5.6843E-14 & 7.7548E-11 \\ F2 & 1.7168E+00 & 2.5199E-01 & **5.6843E-14** & 1.8182E+01 & 9.2022E-11 \\ F3 & 4.8917E+03 & 6.3587E+05 & 3.3801E+04 & 5.8865E+04 & **3.0103E+01** \\ F4 & 1.0672E+01 & 2.0546E+02 & **7.0025E-11** & 2.1796E+01 & 9.2033E-11 \\ F5 & **0.0000E+00** & 9.1538E+02 & 6.6696E-13 & 2.3041E-12 & 8.9070E-11 \\ F6 & 5.0970E+01 & 2.6962E+00 & 1.2781E+00 & 1.3425E+05 & **5.3154E-01** \\ F7 & 1.2671E+03 & **7.6374E-02** & 1.2670E+03 & 1.8818E-01 & 1.2670E+03 \\ F8 & 2.0217E+01 & 2.0227E+01 & 2.0384E+01 & **2.0081E+01** & 2.0110E+01 \\ F9 & 8.0601E+00 & **5.6843E-14** & 2.6532E-01 & **5.6843E-14** & 4.9748E-01 \\ F10 & 8.6778E+00 & 1.6527E+01 & 6.9076E+00 & 1.0315E+01 & **5.8371E+00** \\ F11 & **1.7263E+00** & 6.9768E+00 & 3.0350E+00 & 1.7695E+00 & 2.3054E+00 \\ F12 & 2.8017E+02 & **2.4992E+01** & 5.6407E+02 & 1.5613E+03 & 3.0828E+02 \\ F13 & 5.6892E+01 & **2.6653E-02** & 8.8579E-01 & 4.2952E-01 & 3.2816E-01 \\ F14 & 2.1172E+00 & 3.0997E+00 & 3.1387E+00 & 2.2248E+00 & **1.9601E+00** \\ F15 & **1.8014E+02** & - & 2.8100E+02 & 1.9472E+02 & 3.1617E+02 \\ F16 & **1.0019E+02** & - & 1.0218E+02 & 1.1402E+02 & 1.0514E+02 \\ F17 & 1.0719E+02 & 1.0722E+03 & 1.1966E+02 & 1.4561E+02 & **1.0136E+02** \\ F18 & 7.6687E+02 & 1.6996E+03 & **7.0154E+02** & 7.8021E+02 & 7.8756E+02 \\ F19 & 7.7130E+02 & 1.8139E+03 & **5.5847E+02** & 7.5198E+02 & 7.2585E+02 \\ F20 & 7.4034E+02 & 1.7606E+03 & 6.8165E+02 & 7.6896E+02 & **6.4081E+02** \\ F21 & 6.9422E+02 & 1.2334E+03 & 7.0185E+02 & **5.4388E+02** & 7.3634E+02 \\ F22 & 7.1363E+02 & 1.2253E+03 & 7.7433E+02 & **7.4833E+02** & 7.5392E+02 \\ F23 & 7.8007E+02 & 1.2661E+03 & 8.5139E+02 & **7.9163E+02** & 7.9806E+02 \\ F24 & 2.8126E+02 & 7.4243E+02 & **2.0000E+02** & 5.3977E+02 & 2.2654E+02 \\ F25 & 1.7506E+03 & 7.4292E+02 & 1.7507E+03 & **5.4948E+02** & 1.7495E+03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of average error values for 10\(D\) test problem and \(N=25\) (10D25N). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **FIPS** & **DE** & **SaDE** & **DE-PSO** & **FIPSaDE** \\ \hline F1 & 1.0003E+03 & **1.3264E-14** & 1.0232E-13 & 1.3873E+02 & 6.6317E-14 \\ F2 & 7.4895E+03 & 8.1760E+03 & 5.9137E+02 & 3.3581E+02 & **2.3457E-12** \\ F3 & 9.6860E+06 & 1.2073E+07 & 3.0102E+06 & 1.4827E+06 & **8.9114E+04** \\ F4 & 1.3059E+04 & 3.2880E+04 & 8.8330E+03 & 4.2988E+02 & **3.2211E+01** \\ F5 & 5.0737E+03 & 1.006 FIPSaDE with its original variants, that is, FIPS and SaDE are further analyzed based on the best, the mean and standard deviation of \(f_{best}\) for different configurations of problem dimensionality and population size for all functions. The \(f_{best}\) obtained by FIPSaDE, FIPS and SaDE for all functions are shown in Tables 7-9, summarizing the best, the mean and standard deviation for \(f_{best}\) over a number of runs, which can be either 20 runs or 30 runs. The best, mean and standard deviation results of \(f_{best}\) are denoted as _best-\(f_{best}\)_, _mean-\(f_{best}\)_ and _std-\(f_{best}\)_, respectively. For each table, the algorithm producing the best results for the same function is bolded and its frequency, \(h_{win}\) is calculated at the bottom. For the setting of \(10D25N\), FIPSaDE has the highest \(h_{win}\) for _best-\(f_{best}\)_, _mean-\(f_{best}\)_ and _std-\(f_{best}\)_. FIPS and SaDE have similar \(h_{win}\) values _best-\(f_{best}\)_, _mean-\(f_{best}\)_ for the same setting. When the setting increases to \(30D30N\) and \(50D50N\), FIPSaDE is still associated with the highest \(h_{win}\) for _best-\(f_{best}\)_, _mean-\(f_{best}\)_ and _std-\(f_{best}\)_. Therefore, FIPSaDE shows better results for _best-\(f_{best}\)_, _mean-\(f_{best}\)_ and _std-\(f_{best}\)_ with the increase of problem dimensionality and population size. producing lowest variations of solutions is slightly lower at high problem dimensionality and population size.