# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 6 Case Studies and Experimental Results
\begin{table}
\begin{tabular}{c c c c} \hline
**Ranking Function for Two Classical Fuzzy** & \multicolumn{3}{c}{**Ranking Function for Generalized**} \\
**Numbers [7]** & & **Trapezoidal Fuzzy Numbers [17]** \\ \hline \hline \multirow{4}{*}{\(\bullet\)} & \multirow{2}{*}{Let \(\widetilde{A}=(a,b,c)\) a triangular fuzzy} & \multirow{2}{*}{\(\bullet\)} & \multirow{2}{*}{Let} \\  & & & \(\widetilde{A}_{1}=(a_{1},b_{1},c_{1},d_{1};w_{1})\) \\  & & & and \\  & & & \(\widetilde{A}_{2}=(a_{2},b_{2},c_{2},d_{2};w_{2})\) \\  & & & be two generalized trapezoidal fuzzy \\  & & & numbers and \(w=min(w_{1},w_{2})\). \\  & & & Then, \\  & & & then, \\  & & & \(\mathfrak{R}\left(\widetilde{A}_{1}\right)=\frac{w\cdot(a_{1}+b_{1}+c_{1}+d_{1})}{ 4}\) \\  & & & and \(\mathfrak{R}\left(\widetilde{A}_{2}\right)=\frac{w\cdot(a_{2}+b_{2}+c_{2}+d_{2})}{ 4}\) \\ \hline \end{tabular}
\end{table} 
Table 6: The formulae of ranking functions. \begin{table}
\begin{tabular}{c c c c c} \hline
**Pr.** & **Ekanayake** & **TrigAC-PSO** & **Ekanayake** & **TrigAC-PSO** \\  & **Optimal Solution** & **Optimal Solution** & **Membership Value** & **Membership Value** \\ \hline Pr. 02 & (400, 845, 1315) & (400, 845, 1315) & 1.0000 & 1.0000 \\ Pr. 11 & (144, 285, 453) & (144, 285, 453) & 1.0000 & 1.0000 \\ Pr. 12 & (20, 89, 89) & (20, 89, 89) & 1.0000 & 1.0000 \\ Pr. 13 & (5960, 7620, 9630) & (5960, 7620, 9630) & 1.0000 & 1.0000 \\ Pr. 14 & (64, 124, 206) & (64, 124, 206) & 1.0000 & 1.0000 \\ Pr. 16 & (370, 735, 1145, 1595) & (370, 715, 1085, 1635) & 0.9951 & 1.0000 \\ Pr. 17 & (400, 640, 875, 1370) & (400, 640, 875, 1370) & 1.0000 & 1.0000 \\ Pr. 18 & (105, 150, 195) & (104, 149, 194) & 0.9999 & 1.0000 \\ Pr. 22 & (148, 322, 418) & (148, 322, 418) & 1.0000 & 1.0000 \\ Pr. 23 & (347, 566, 760) & (347, 554, 752) & 0.9950 & 1.0000 \\ Pr. 25 & (12, 55, 88, 117) & (12, 55, 88, 117) & 1.0000 & 1.0000 \\ Pr. 26 & (52, 106, 176, 230) & (52, 106, 176, 230) & 1.0000 & 1.0000 \\ \hline \end{tabular}
\end{table} 
Table 7: Comparison between [PERSON] and Ekanayake and the TrigAC-PSO method. Upon reviewing the table above, the following conclusions can be drawn:  
* [PERSON]'s approach as well as [PERSON] and [PERSON]'s approach exhibited zero deviation from the optimal solution, establishing them as the preferred methods for solving the TP involving fuzzy generalized numbers;
* TrigAC-PSO's performance in this scenario was exceptional, yet again. This method almost reached an ultimate solution with a deviation rate of 1.71%;
* [PERSON] and [PERSON]'s method demonstrated commendable efficiency with a deviation from the optimum standing at 2.89%;
* Conversely, the outcomes derived from [PERSON]'s method exhibited a substantial deviation from the optimal solution, amounting to 20.54%. Based on these assessments, it is apparent that our method, the TrigAC-PSO algorithm, ensures maximum accuracy, completeness and stability, adeptly handling both the classical TP and the more intricate FTP with remarkable proficiency. Metaheuristic algorithms consistently emerge as optimal solutions for optimization problems with the PSO algorithm standing out as particularly advantageous. In our innovative variation, TrigAC-PSO, we have methodically designed the cognitive and social component to align with our philosophy, prioritizing the particle's assimilation of swarm behavior and collective experience to follow its own decision-making process. Additionally, the inertia weight \(w\) intricately intertwines with the number of iterations outlined in our algorithm. The remaining consideration is to assess whether the number of particles, comprising our swarm, influences the evolution and quality of our solutions. The solutions for 28 problems were found using 20, 35 and 50 particles, generated by Algorithm 1. These solutions are summarized in the table below (Table 10). The highlighted indications show the cases where the method failed to identify the optimal solution but achieved a value very close to it.