## 4 Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **DPSO** & **PSO** & **GA** & **DE** \\ \hline \(f_{0}\) Norm & 2.246\(\times 10^{-9}\) & 1.000 & **1.006\(\times 10^{-16}\)** & _0.000_ \\ \hline \(f_{1}\) Norm & **8.388\(\times 10^{-3}\)** & 1.147\(\times 10^{-1}\) & 1.000 & _0.000_ \\ \hline \(f_{2}\) Norm & _0.000_ & 2.710\(\times 10^{-5}\) & **6.438\(\times 10^{-11}\)** & 1.000 \\ \hline \(f_{3}\) Norm & _3.810\(\times 10^{-6}\)_ & 1.000 & _0.000_ & **1.593\(\times 10^{-8}\)** \\ \hline \(f_{4}\) Norm & _0.000_ & **4.428\(\times 10^{-15}\)** & _2.682\(\times 10^{-1}\)_ & 1.000 \\ \hline \(f_{5}\) Norm & **4.154\(\times 10^{-5}\)** & _0.000_ & 1.000 & 1.447\(\times 10^{-3}\) \\ \hline \(f_{6}\) Norm & _7.895\(\times 10^{-1}\)_ & 1.000 & **7.057\(\times 10^{-1}\)** & _0.000_ \\ \hline \(f_{7}\) Norm & _0.000_ & 1.000 & **8.062\(\times 10^{-1}\)** & _8.557\(\times 10^{-1}\)_ \\ \hline \(f_{8}\) Norm & **9.287\(\times 10^{-1}\)** & 1.000 & _0.000_ & 9.443\(\times 10^{-1}\) \\ \hline \(f_{9}\) Norm & _2.543\(\times 10^{-2}\)_ & 1.000 & **7.551\(\times 10^{-3}\)** & _0.000_ \\ \hline \(f_{10}\) Norm & _0.000_ & 1.000 & **6.413\(\times 10^{-2}\)** & _5.706\(\times 10^{-1}\)_ \\ \hline \(f_{11}\) Norm & _0.000_ & 1.000 & _4.721\(\times 10^{-2}\)_ & **6.814\(\times 10^{-3}\)** \\ \hline \(f_{12}\) Norm & _0.000_ & 1.000 & **1.595\(\times 10^{-1}\)** & _5.604\(\times 10^{-1}\)_ \\ \hline \(f_{13}\) Norm & _0.000_ & _9.106\(\times 10^{-1}\)_ & **1.182\(\times 10^{-1}\)** & 1.000 \\ \hline \(f_{14}\) Norm & **3.266\(\times 10^{-2}\)** & 1.000 & _6.224\(\times 10^{-1}\)_ & _0.000_ \\ \hline \(f_{15}\) Norm & _8.635\(\times 10^{-1}\)_ & 1.000 & **5.252\(\times 10^{-1}\)** & _0.000_ \\ \hline \(f_{16}\) Norm & **3.222\(\times 10^{-1}\)** & 1.000 & _0.000_ & 5.351\(\times 10^{-1}\) \\ \hline \(f_{17}\) Norm & **1.238\(\times 10^{-1}\)** & _9.600\(\times 10^{-1}\)_ & 1.000 & _0.000_ \\ \hline \(f_{18}\) Norm & **4.544\(\times 10^{-1}\)** & 1.000 & _5.639\(\times 10^{-1}\)_ & _0.000_ \\ \hline \(f_{19}\) Norm & 4.795\(\times 10^{-1}\) & 1.000 & _0.000_ & **3.025\(\times 10^{-1}\)** \\ \hline \(f_{20}\) Norm & **1.288\(\times 10^{-2}\)** & _1.848\(\times 10^{-1}\)_ & 1.000 & _0.000_ \\ \hline \(f_{21}\) Norm & _0.000_ & 1.000 & _2.130\(\times 10^{-1}\)_ & **1.616\(\times 10^{-1}\)** \\ \hline \(f_{22}\) Norm & **2.277\(\times 10^{-2}\)** & _0.000_ & 1.000 & 3.147\(\times 10^{-1}\) \\ \hline \(f_{23}\) Norm & _5.228\(\times 10^{-1}\)_ & _0.000_ & 1.000 & **1.202\(\times 10^{-2}\)** \\ \hline Rank Factor & 1.000 & 3.744 & 2.202 & 1.584 \\ \hline Rank Sum & 47 & 80 & 61 & 52 \\ \hline Overall Rank & 1 & 4 & 3 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Normalized fitness w.r.t. each function problem with 21 dimensions and the resulting ranking: _rank 1 (best)_, **rank 2**, _rank 3_, and rank 4.  
problem does not exist for DE and GA. The likely factor that makes some of these problems more difficult for DE and GA is that the population size, which they rely on for diversity, is relatively small compared to the number of dimensions. This way, they are entirely reliant on warping and mutation to increase the variety of candidate solutions. Without randomly generating new genomes, they can only make do with trying to improve on the dimensions for which there is a sufficient variety of overall fit individuals to experiment on. From Figures A1-A12, it is apparent that most algorithms tend to converge or approach convergence within the first 500 iterations, with relatively small improvements thereafter. The last improvement in the global optimum is given at \(x\)-axis value 1.0 in the plots--shown in Tables A4-A6--but data with next to no improvement at the end were removed to improve clarity where possible. The sudden improvement given by DPSO may be due to the fact that the dimension-wise activation of attractors is similar in principle to the crossover method found in GA and DE.