# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 6 Case Studies and Experimental Results
A brief examination reveals minimal deviation among solutions, often rendering them almost identical. However, a more thorough examination was deemed necessary. Each of the 28 problems was subjected to runs with varying numbers of particles and 20 tests conducted for each instance. Table 11 expands the accuracy rate of each algorithm for 20, 35 and 50 particles. The accuracy rate is commonly defined based on the following relationship:  
\[Accuracy=\frac{TOR}{TR}\]  
where TOR is the total number of runs where an optimal solution was found and TR is the number of runs in general. In our scenario, the experiments were carried out for 20 runs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pr.** & **[PERSON] [17]** & **[PERSON] and [PERSON] [39]** & **[PERSON] and [PERSON] [16]** & **[PERSON] et al. [30]** & **TrigAC-PSO** \\ \hline Pr. 01 & (117, 205, 352, 613, 0.2) & (117, 205, 35, 613, 0.2) & (147, 220), 382, 603, 0.2) & (197, 240), 382, 643), 0.2) & (147, 220), 382, 583), 0.2) \\ Pr. 04 & (315, 810, 1220, 1575, 0.2) & (318, 813, 1220, 1582, 0.2) & (315, 810, 1220, 1575, 0.2) & (415, 970), 1460, 1865; 0.2) & (315, 810, 1220, 1575; 0.2) \\ Pr. 24 & (5148, 6475, 7802, 9244; 0.6) & (5148, 6475, 7802, 9244; 0.6) & (5307, 6294, 7, 872, 9460; 0.6) & (5307, 6294, 7, 872, 9460; 0.6) & (5608, 0.6) \\ Pr. 27 & (376, 436, 474, 543; 0.2) & (376, 436, 474, 543; 0.2) & (413, 459, 506, 572, 0.2) & (411, 455, 509, 563; 0.2) & (404, 460, 502, 559; 0.2) \\ Pr. 28 & (294, 348, 408, 462; 0.2) & (294, 348, 408, 462; 0.2) & (294, 348, 408, 462; 0.2) & (294, 348, 408, 462; 0.2) & (294, 348, 408, 462; 0.2) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison TrigAC-PSO among four established methods. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pr.** & **[PERSON] [17]** & **[PERSON] and [PERSON] [39]** & **[PERSON] and [PERSON] [16]** & **[PERSON] et al. [30]** & **TrigAC-PSO** \\ \hline Pr. 01 & 0.0000 & 0.0000 & 0.0505 & 0.1360 & 0.0117 \\ Pr. 04 & 0.0000 & 0.0028 & 0.0000 & 0.2015 & 0.0000 \\ Pr. 24 & 0.0000 & 0.0000 & 0.0278 & 0.0278 & 0.0213 \\ Pr. 27 & 0.0000 & 0.0000 & 0.0662 & 0.6617 & 0.0524 \\ Pr. 28 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Average & 0.0000 & 0.0006 & 0.0289 & 0.2054 & 0.0171 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The deviation (dev) of the methods for 5 numerical examples. The data represented in Table 11 indicate that the algorithm achieves an accuracy rate of 66.07% for 20 particles. For 35 particles, the algorithm's accuracy rate improves to 74.03%. Furthermore, solutions obtained using 50 particles exhibit even higher accuracy, reaching 75.54%. This pattern indicates that as the number of particles increases, there is enhanced interaction and information exchange within the swarm, resulting in higher solution accuracy rates. Figure 4 also reveals the high levels of accuracy. \begin{table}
\begin{tabular}{c c c c c} \hline
**Pr.** & **20 Particles** & **35 Particles** & **50 Particles** & **Optimal** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 10: Solutions for 20, 35 and 50 particles. Figure 4: The accuracy of TrigAC-PSO for 20, 35 and 50 particles. In the following table (Table 12), the most important statistical measures for 20 independent runs are represented for 20, 35 and 50 particles, respectively. The experimental findings highlight the remarkable stability of our proposed algorithm in solving the FTP. Across all scenarios, the mean values closely approximate the optimal solution, regardless of whether 20, 35 or 50 particles are utilized. This emphasizes not only the efficiency but also the robustness of the TrigAC-PSO method. Notably, the CV value, a crucial metric for measuring an algorithm's stability, remains consistently low. Specifically, for 20 particles, the mean CV is 1.82%, reducing to 1.53% for 35 particles and further decreasing to 0.91% for 50 particles. This tendency suggests that increasing the number of particles improves algorithm's stability.