# Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively
## 4 The Basic PSO for Solving the TP
\begin{table}
\begin{tabular}{l r r r r r} \hline  & **PSO** & **DWPSO** & **TVAC** & **TrigAC-PSO** & **FSVAC** \\ \hline Pr.01 & 0.0333 & 0.2 & 0.4666 & 0.5667 & 0.2333 \\ Pr.02 & 0.7667 & 1 & 1 & 1 & 1 \\ Pr.03 & 1 & 1 & 1 & 1 & 1 \\ Pr.04 & 1 & 1 & 1 & 1 & 1 \\ Pr.05 & 0.2333 & 0.6667 & 0.8333 & 0.8667 & 0.8333 \\ Pr.06 & 0.3667 & 1 & 0.9667 & 1 & 1 \\ Pr.07 & 0.7667 & 0.9 & 1 & 1 & 0.9 \\ Pr.08 & 0 & 0 & 0.2667 & 0.1667 & 0.1334 \\ Pr.09 & 0.0333 & 0.3 & 0.2 & 0.2667 & 0.1 \\ Pr.10 & 0 & 0 & 0.0333 & 0 & 0.0333 \\ \hline \end{tabular}
\end{table} 
Table 4: Accuracy of PSO, DWPSO, TVAC, TrigAC-PSO and FSVAC for 10 particles. Figure 2: Average percentage deviation for each method. The accuracy rate results for 15 particles are presented in Table 5. DWPSO achieved 65.31% accuracy, whereas TVAC reached 66.99%. It is of particular interest that TrigAC-PSO achieved the highest accuracy rate once again by reaching 69.8%. Finally, FSVAC obtained an accuracy rate equal to 66.56%. \begin{table}
\begin{tabular}{l r r r r r} \hline \hline  & **PSO** & **DWPSO** & **TVAC** & **TrigAC-PSO** & **FSVAC** \\ \hline



verage** & 0.383338 & 0.595834 & 0.611459 & 0.6281313 & 0.594603 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._

\begin{table}
\begin{tabular}{l r r r r r} \hline \hline  & **PSO** & **DWPSO** & **TVAC** & **TrigAC-PSO** & **FSVAC** \\ \hline
**Pr01** & 0.0667 & 0.6 & 0.6333 & 0.6333 & 0.4333 \\
**Pr02** & 0.9667 & 1 & 1 & 1 & 1 \\
**Pr03** & 1 & 1 & 1 & 1 & 1 \\
**Pr04** & 1 & 1 & 1 & 1 & 1 \\
**Pr05** & 0.7 & 0.9 & 1 & 1 & 1 \\
**Pr06** & 0.7 & 1 & 0.9667





In summary, the proposed method FSVAC-PSO, although it did not demonstrate the highest average success rate, was very accurate in calculating the optimal solution in cases where the aforementioned variations were unable to approach the optimal solution. In more detail, this research experimented on population sizes of 10, 15, 20 particles over 32 well-known test instances used in the respective literature. For each problem, as already mentioned, 30 independent experimental runs were conducted. In the case of 10 particles, the classical PSO found the optimal solution in only in 3 out of 32 test instances in all 30 runs (9.4%); DWPSO found the optimal solution in 10 out of 32 test instances in all 30 runs (31.25%); while TVAC and Trig-PSO managed to find the optimal solution in 9 out of 32 test instances in all 30 runs (28.13%); finally, FSVAC was shown to be the best PSO variation, finding the optimal solution in 11 out of 32 test instances in all 30 runs (34.4%). In the case of 15 particles, FSVAC also showed the best performance by finding in the optimal solution in 18 out of 32 test instances in all 30 runs (56.25%); the classic PSO found the optimal solution in 4 out of 32 (12.5%) test instances, and TVAC in 11 out of 32, in all 30 runs; last but not least, both DWPSO and TrigAC-PSO found the optimal value in 13 out of 32 test instances in all 30 runs (40.63%). In the case of 20 particles, the variations TrigAC-PSO and FSVAC are still more accurate than the other PSO variations since they succeeded in finding the optimal solution in 18 out of 32 and in 17 out of 32 test instances in all 30 runs, respectively. The other PSO variations attained relatively lower success rates in finding the optimal solution in all of their runs. In the following table (Table 7), the most important statistical measures in the cases of 20 particles for 30 independent runs are represented for all PSO variations. These experimental results demonstrate the very good performance and stability of the proposed PSO variations in solving the TP. As presented, in all cases, the mean value is very close to the best one, showing that all these variations are not only efficient but also quite stable.