## 5 Conclusions and Future Work
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **DPSO** & **PSO** & **GA** & **DE** \\ \hline \(f_{5}\) Ave & **4.560\(\times\)10\({}^{11}\)** & 8.607\(\times\)10\({}^{36}\) & **2.788\(\times\)10\({}^{10}\)** & 3.949\(\times\)10\({}^{19}\) \\ \hline Std & 0.000 & 2.361\(\times\)10\({}^{21}\) & 7.629\(\times\)10\({}^{-6}\) & 3.277\(\times\)10\({}^{4}\) \\ \hline \(f_{6}\) Ave & **7.538\(\times\)10\({}^{1}\)** & 8.185\(\times\)10\({}^{1}\) & **6.799\(\times\)10\({}^{1}\)** & 7.900\(\times\)10\({}^{1}\) \\ \hline Std & 0.000 & 4.263\(\times\)10\({}^{-14}\) & 4.263\(\times\)10\({}^{-14}\) & 1.421\(\times\)10\({}^{-14}\) \\ \hline \(f_{7}\) Ave & 8.224\(\times\)10\({}^{1}\) & _8.116\(\times\)10\({}^{1}\)_ & **6.852\(\times\)10\({}^{1}\)** & **6.462\(\times\)10\({}^{1}\)** \\ \hline Std & 0.000 & 0.000 & 4.263\(\times\)10\({}^{-14}\) & 0.000 \\ \hline \(f_{8}\) Ave & 8.228\(\times\)10\({}^{1}\) & _8.116\(\times\)10\({}^{1}\)_ & **6.852\(\times\)10\({}^{1}\)** & **6.462\(\times\)10\({}^{1}\)** \\ \hline Std & 4.263\(\times\)10\({}^{-14}\) & 0.000 & 4.263\(\times\)10\({}^{-14}\) & 0.000 \\ \hline \(f_{9}\) Ave & **1.963\(\times\)10\({}^{4}\)** & 5.403\(\times\)10\({}^{5}\) & 2.195\(\times\)10\({}^{5}\) & **3.387\(\times\)10\({}^{2}\)** \\ \hline Std & 7.276\(\times\)10\({}^{-12}\) & 3.492\(\times\)10\({}^{-10}\) & 2.910\(\times\)10\({}^{-11}\) & 5.684\(\times\)10\({}^{-14}\) \\ \hline \(f_{10}\) Ave & **3.243\(\times\)10\({}^{4}\)** & 1.549\(\times\)10\({}^{6}\) & 2.234\(\times\)10\({}^{6}\) & **6.067\(\times\)10\({}^{3}\)** \\ \hline Std & 2.183\(\times\)10\({}^{-11}\) & 1.164\(\times\)10\({}^{-9}\) & 1.397\(\times\)10\({}^{-9}\) & 4.547\(\times\)10\({}^{-12}\) \\ \hline \(f_{11}\) Ave & **7.982\(\times\)10\({}^{4}\)** & 2.034\(\times\)10\({}^{7}\) & 2.279\(\times\)10\({}^{5}\) & **8.108\(\times\)10\({}^{4}\)** \\ \hline Std & 5.821\(\times\)10\({}^{-11}\) & 7.451\(\times\)10\({}^{-9}\) & 2.910\(\times\)10\({}^{-11}\) & 4.366\(\times\)10\({}^{-11}\) \\ \hline \(f_{12}\) Ave & **6.891\(\times\)10\({}^{1}\)** & 1.443\(\times\)10\({}^{3}\) & **2.789\(\times\)10\({}^{2}\)** & 3.996\(\times\)10\({}^{2}\) \\ \hline Std & 1.421\(\times\)10\({}^{-14}\) & 6.821\(\times\)10\({}^{-13}\) & 1.705\(\times\)10\({}^{-13}\) & 2.842\(\times\)10\({}^{-13}\) \\ \hline \(f_{13}\) Ave & **7.068\(\times\)10\({}^{1}\)** & 7.193\(\times\)10\({}^{2}\) & 2.165\(\times\)10\({}^{2}\) & **1.123\(\times\)10\({}^{2}\)** \\ \hline Std & 2.842\(\times\)10\({}^{-14}\) & 4.547\(\times\)10\({}^{-13}\) & 0.000 & 5.684\(\times\)10\({}^{-14}\) \\ \hline \(f_{14}\) Ave & **1.647\(\times\)10\({}^{2}\)** & 2.841\(\times\)10\({}^{2}\) & 2.389\(\times\)10\({}^{2}\) & **7.673\(\times\)10\({}^{1}\)** \\ \hline Std & 8.527\(\times\)10\({}^{-14}\) & 1.705\(\times\)10\({}^{-13}\) & 1.137\(\times\)10\({}^{-13}\) & 4.263\(\times\)10\({}^{-14}\) \\ \hline \(f_{15}\) Ave & **2.256** & 3.689 & **2.932** & _3.500_ \\ \hline Std & 8.882\(\times\)10\({}^{-16}\) & 2.665\(\times\)10\({}^{-15}\) & 1.332\(\times\)10\({}^{-15}\) & 1.332\(\times\)10\({}^{-15}\) \\ \hline \(f_{16}\) Ave & **2.663** & 3.510 & _2.897_ & **2.871** \\ \hline Std & 0.000 & 1.332\(\times\)10\({}^{-15}\) & 1.332\(\times\)10\({}^{-15}\) & 4.441\(\times\)10\({}^{-16}\) \\ \hline \(f_{17}\) Ave & **2.540** & 2.970 & **2.960** & 3.017 \\ \hline Std & 4.441\(\times\)10\({}^{-16}\) & 2.220\(\times\)10\({}^{-15}\) & 4.441\(\times\)10\({}^{-16}\) & 8.882\(\times\)10\({}^{-16}\) \\ \hline \(f_{18}\) Ave & **6.945\(\times\)10\({}^{2}\)** & 1.831\(\times\)10\({}^{3}\) & 1.122\(\times\)10\({}^{3}\) & **1.005\(\times\)10\({}^{3}\)** \\ \hline Std & 1.137\(\times\)10\({}^{-13}\) & 9.095\(\times\)10\({}^{-13}\) & 2.274\(\times\)10\({}^{-13}\) & 6.821\(\times\)10\({}^{-13}\) \\ \hline \(f_{19}\) Ave & **1.708\(\times\)10\({}^{2}\)** & 2.040\(\times\)10\({}^{3}\) & _6.648\(\times\)10\({}^{2}\)_ & **5.270** \\ \hline Std & 0.000 & 4.547\(\times\)10\({}^{-13}\) & 3.411\(\times\)10\({}^{-13}\) & 8.882\(\times\)10\({}^{-16}\) \\ \hline \(f_{20}\) Ave & **1.545\(\times\)10\({}^{2}\)** & 1.397\(\times\)10\({}^{3}\) & _6.831\(\times\)10\({}^{2}\)_ & **3.054\(\times\)10\({}^{2}\)** \\ \hline Std & 8.527\(\times\)10\({}^{-14}\) & 0.000 & 0.000 & 0.000 \\ \hline \(f_{21}\) Ave & **2.142\(\times\)10\({}^{2}\)** & 7.904\(\times\)10\({}^{3}\) & **3.584\(\times\)10\({}^{3}\  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **DPSO** & **PSO** & **GA** & **DE** \\ \hline _f\({}_{0}\)_ & _1.375\(\times 10^{39}\)_ & 6.121\(\times 10^{27}\) & **6.249\(\times 10^{11}\)** & **9.395\(\times 10^{9}\)** \\ \hline Std & 0.000 & 0.000 & 0.000 & 3.815\(\times 10^{-6}\) \\ \hline _f\({}_{1}\)_ & **2.600\(\times 10^{3}\)** & _2.824\(\times 10^{8}\)_ & 2.417\(\times 10^{6}\) & **5.773\(\times 10^{6}\)** \\ \hline Std & 1.490\(\times 10^{-8}\) & 0.000 & 9.537\(\times 10^{-7}\) & 2.794\(\times 10^{-9}\) \\ \hline _f\({}_{2}\)_ & **2.401\(\times 10^{6}\)** & 4.489\(\times 10^{12}\) & **1.091\(\times 10^{7}\)** & 1.657\(\times 10^{27}\) \\ \hline Std & 8.731\(\times 10^{-11}\) & 9.766\(\times 10^{-4}\) & 3.725\(\times 10^{-9}\) & 6.400\(\times 10^{1}\) \\ \hline _f\({}_{5}\)_ & 2.028\(\times 10^{10}\) & 5.324\(\times 10^{38}\) & 5.551\(\times 10^{13}\) & **8.484\(\times 10^{27}\)** \\ \hline Std & 1.407\(\times 10^{15}\) & 2.951\(\times 10^{20}\) & 2.013\(\times 10^{6}\) & 2.199\(\times 10^{12}\) \\ \hline _f\({}_{4}\)_ & **1.012\(\times 10^{20}\)** & **5.906\(\times 10^{11}\)** & 3.516\(\times 10^{25}\) & 1.311\(\times 10^{26}\) \\ \hline Std & 3.815\(\times 10^{-6}\) & 2.441\(\times 10^{-4}\) & 2.147\(\times 10^{10}\) & 6.872\(\times 10^{10}\) \\ \hline _f\({}_{5}\)_ & **2.981\(\times 10^{20}\)** & **2.988\(\times 10^{21}\)** & 7.175\(\times 10^{27}\) & 1.038\(\times 10^{25}\) \\ \hline Std & 1.342\(\times 10^{8}\) & 0.000 & 4.398\(\times 10^{12}\) & 0.000 \\ \hline _f\({}_{6}\)_ & 8.179\(\times 10^{1}\) & 8.342\(\times 10^{1}\) & **8.114\(\times 10^{1}\)** & **7.567\(\times 10^{1}\)** \\ \hline Std & 1.421\(\times 10^{-14}\) & 0.000 & 1.421\(\times 10^{-14}\) & 2.842\(\times 10^{-14}\) \\ \hline _f\({}_{7}\)_ & **7.943\(\times 10^{6}\)** & 8.394\(\times 10^{1}\) & **8.307\(\times 10^{1}\)** & _8.329\(\times 10^{2}\)_ \\ \hline Std & 1.421\(\times 10^{-14}\) & 7.105\(\times 10^{-14}\) & 2.842\(\times 10^{-14}\) & 4.263\(\times 10^{-14}\) \\ \hline _f\({}_{8}\)_ & **8.311\(\times 10^{2}\)** & 8.394\(\times 10^{1}\) & **7.225\(\times 10^{2}\)** & _8.329\(\times 10^{2}\)_ \\ \hline Std & 0.000 & 7.105\(\times 10^{-14}\) & 4.263\(\times 10^{-14}\) & 4.263\(\times 10^{-14}\) \\ \hline _f\({}_{9}\)_ & 1.220\(\times 10^{6}\) & 3.937\(\times 10^{2}\) & **5.202\(\times 10^{2}\)** & **2.246\(\times 10^{2}\)** \\ \hline Std & 6.985\(\times 10^{-10}\) & 7.451\(\times 10^{-9}\) & 1.746\(\times 10^{-10}\) & 8.731\(\times 10^{-11}\) \\ \hline _f\({}_{10}\)_ & **1.817\(\times 10^{6}\)** & 2.345\(\times 10^{7}\) & **1.674\(\times 10^{6}\)** & _1.345\(\times 10^{2}\)_ \\ \hline Std & 5.821\(\times 10^{-11}\) & 1.118\(\times 10^{-8}\) & 6.985\(\times 10^{-10}\) & 0.000 \\ \hline _f\({}_{11}\)_ & **2.353\(\times 10^{6}\)** & 1.266\(\times 10^{2}\) & _8.217\(\times 10^{6}\)_ & **3.199\(\times 10^{6}\)** \\ \hline Std & 0.000 & 1.863\(\times 10^{-9}\) & 3.492\(\times 10^{-10}\) & 5.821\(\times 10^{-11}\) \\ \hline _f\({}_{12}\)_ & **4.166\(\times 10^{2}\)** & 1.787\(\times 10^{3}\) & **6.351\(\times 10^{2}\)** & _1.185\(\times 10^{3}\)_ \\ \hline Std & 1.137\(\times 10^{-13}\) & 2.274\(\times 10^{-13}\) & 1.137\(\times 10^{-13}\) & 4.547\(\times 10^{-13}\) \\ \hline _f\({}_{13}\)_ & **5.709\(\times 10^{2}\)** & 7.462\(\times 10^{2}\) & **5.936\(\times 10^{2}\)** & 7.634\(\times 10^{2}\) \\ \hline Std & 3.411\(\times 10^{-13}\) & 1.137\(\times 10^{-13}\) & 4.547\(\times 10^{-13}\) & 3.411\(\times 10^{-13}\) \\ \hline _f\({}_{14}\)_ & **3.199\(\times 10^{2}\)** & 6.809\(\times 10^{2}\) & _5.400\(\times 10^{2}\)_ & **3.077\(\times 10^{2}\)** \\ \hline Std & 5.684\(\times 10^{-14}\) & 3.411\(\times 10^{-13}\) & 4.547\(\times 10^{-13}\) & 0.000 \\ \hline _f\({}_{15}\)_ & 3.883 & 3.941 & **3.740** & **3.517** \\ \hline Std & 0.000 & 1.332\(\times 10^{-15}\) & 4.441\(\times 10^{-16}\) & 4.441\(\times 10^{-16}\) \\ \hline _f\({}_{8}\)_ & **3.693** & 3.958 & **3.566** & 3.776 \\ \hline Std & 1.332\(\times 10^{-15}\) & 4.441\(\times 10^{-16}\) & 0.000 & 4.441\(\times 10^{-16}\) \\ \hline _f\({}_{27}\)_ & **3.045** & 3.731 & 3.764 & **2.943** \\ \hline Std & 1.332\(\times 10^{-15}\) & 8.882\(\times 10^{-16}\) & 0.000 & 4.441\(\times 10^{-16}\) \\ \hline Std & 1.364\(\times 10^{-13}\) & 3.932\(\times 10^{3}\) & 2.477\(\times 10^{2}\) & **5.943\(\times 10^{2}\)** \\ \hline Std & 1.364\(\times 10^{-12}\) & 1.819\(\times 10^{-12}\) & 1.364\(\times 10^{-12}\)Figure 17: The average In-Sample (\(f_{0}\), \(f_{1}\), \(f_{2}\)) and Elliptical (\(f_{3}\), \(f_{4}\), \(f_{5}\)) global best results over time for 7 dimensions.