# Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively
## 4 The Basic PSO for Solving the TP
\begin{table}
\begin{tabular}{l l l l l l l} \hline  & & **PSO** & **DWPSO** & **TVAC** & **TrigAC-PSO** & **FSVAC** \\ \hline  & Mean & 187.4 & 183.9 & 184.5 & 183.6 & 183 \\  & Var & 3.687818 & 1.449138 & 1.581139 & 1.264911 & 0 \\
**Pr12** & Min & 183 & 183 & 183 & 183 & 183 \\  & Max & 193 & 186 & 186 & 186 & 183 \\  & cv\% & 1.967886 & 0.788003 & 0.856986 & 0.688949 & 0 \\ \hline  & Mean & 804.6 & 799 & 799 & 799 & 799 \\  & Var & 11.80584 & 0 & 0 & 0 & 0 \\
**Pr13** & Min & 799 & 799 & 799 & 799 & 799 \\  & Max & 827 & 799 & 799 & 799 & 799 \\  & cv\% & 1.467293 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 302.8 & 284.2 & 289.2 & 278.1 & 273 \\  & Var & 15.38975 & 7.871185 & 15.59772 & 8.491172 & 0 \\
**Pr14** & Min & 290 & 273 & 273 & 273 & 273 \\  & Max & 328 & 291 & 328 & 295 & 273 \\  & cv\% & 5.082481 & 2.769594 & 5.393403 & 3.05328 & 0 \\ \hline  & Mean & 1184.3 & 1160 & 1160 & 1160 & 1160 \\  & Var & 76.14321 & 0 & 0 & 0 & 0 \\
**Pr15** & Min & 1160 & 1160 & 1160 & 1160 & 1160 \\  & Max & 1401 & 1160 & 1160 & 1160 & 1160 \\  & cv\% & 6.429386 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 212.9 & 200 & 200 & 200 & 200 \\  & Var & 8.69802 & 0 & 0 & 0 & 0 \\
**Pr16** & Min & 200 & 200 & 200 & 200 & 200 \\  & Max & 221 & 200 & 200 & 200 & 200 \\  & cv\% & 4.085496 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 246.9 & 241.6 & 240 & 240 & 240 \\  & Var & 6.773314 & 2.796824 & 0 & 0 & 0 \\
**Pr17** & Min & 240 & 240 & 240 & 240 & 240 \\  & Max & 256 & 248 & 240 & 240 & 240 \\  & cv\% & 2.743343 & 1.157626 & 0 & 0 & 0 \\ \hline  & Mean & 820 & 820 & 820 & 820 & 820 \\  & Var & 0 & 0 & 0 & 0 & 0 \\
**Pr18** & Min & 820 & 820 & 820 & 820 & 820 \\  & Max & 820 & 820 & 820 & 820 & 820 \\  & cv\% & 0 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 191.6 & 190.1 & 190.3 & 190.1 & 190 \\  & Var & 0.843274 & 0.316228 & 0.674949 & 0.316228 & 0 \\
**Pr19** & Min & 190 & 190 & 190 & 190 & 190 \\  & Max & 192 & 191 & 192 & 191 & 190 \\  & cv\% & 0.440122 & 0.166348 & 0.354676 & 0.166348 & 0 \\ \hline  & Mean & 83.9 & 83 & 83 & 83 & 83 \\  & Var & 1.449138 & 0 & 0 & 0 & 0 \\
**Pr20** & Min & 83 & 83 & 83 & 83 & 83 \\  & Max & 86 & 83 & 83 & 83 & 83 \\  & cv\% & 1.72722 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 3493.6 & 3468.4 & 3477.6 & 3460 & 3470.8 \\  & Var & 57.87765 & 13.1673 & 24.84262 & 0 & 12.76105 \\
**Pr21** & Min & 3460 & 3456 & 3460 & 3460 & 3460 \\  & Max & 3625 & 3492 & 3519 & 3460 & 3495 \\  & cv\% & 1.656676 & 0.379636 & 0.714361 & 0 & 0.367669 \\ \hline  & Mean & 914.7 & 910 & 910 & 910 & 910 \\  & Var & 10.133 & 0 & 0 & 0 & 0 \\
**Pr22** & Min & 910 & 910 & 910 & 910 & 910 \\  & Max & 938 & 910 & 910 & 910 & 910 \\  & cv\% & 1.107795 & 0 & 0 & 0 & 0 \\ \hline  & Mean & 1675.2 & 1670.2 & 1672 & 1670 & 1670.6 \\  & Var & 7.743097 & 0.421637 & 3.018462 & 0 & 1.577621 \\
**Pr23** & Min & 1670 & 1670 & 1670 & 1670 & 1670 \\  & Max & 1694 & 1671 & 1679 & 1670 & 1675 \\  & cv\% & 0.462219 & 0.025245 & 0.18053 & 0 & 0.094434 \\ \hline \end{tabular}
\end{table} 
Table 11: _Cont._The overall results demonstrate two inferences of decisive importance: first, the PSO algorithm and its variations have successfully solved the TP with maximum accuracy and efficiency; second, TrigAC-PSO, beyond any doubt, is the leading option for solving the TP in terms of both stability and the solution's quality. ## 7 Conclusions

As technology is developing, the need for product improvement and trading is of high priority in obtaining a more economical solution. The PSO algorithm was applied with success in order for the TP to be solved. Furthermore, two new variations were introduced and compared to already-known variations. These variations induced exceptional results and indicated their superiority against the existing variations and the well-known exact methods in the literature. The proposed PSO variations have been tested in a variety of test instances with different combined values of inertia weight as well as social and personal