# 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 c c} \hline  & **NWM** & **LCM** & **VAM** & **MOMC** & **TrigAC-PSO** \\ \hline Pr. 01 & 0.000000000 & 0.050505050 & 0.050505050 & 0.0505050 & 0.011655011 \\ Pr. 02 & 0.226542450 & 0.019569930 & 0.00000000 & 0.001933556 & 0.00000000 \\ Pr. 03 & 0.054284910 & 0.094827269 & 0.000000000 & 0.224555478 & 0.000000000 \\ Pr. 04 & 0.188775510 & 0.359693877 & 0.367346940 & 0.359693877 & 0.00000000 \\ Pr. 05 & 0.00000000 & 0.014457831 & 0.036144580 & 0.036144578 & 0.000000000 \\ Pr. 06 & 0.237623762 & 0.193069306 & 0.00000000 & 0.039603960 & 0.00000000 \\ Pr. 07 & 0.144067796 & 0.011299435 & 0.0000000000 & 0.016949152 & 0.000000000 \\ Pr. 08 & 0.170854271 & 0.120603015 & 0.02010050 & 0.000000000 & 0.00000000 \\ Pr. 09 & 0.000000000 & 0.148148148 & 0.02574900 & 0.059581320 & 0.000000000 \\ Pr. 10 & 0.14841269 & 0.11111111 & 0.03174603 & 0.000000000 & 0.000000000 \\ Pr. 11 & 0.278911564 & 0.000000000 & 0.00000000 & 0.275510204 & 0.00000000 \\ Pr. 12 & 0.681914893 & 0.000000000 & 0.000000000 & 0.003039514 & 0.00000000 \\ Pr. 13 & 0.000151251 & 0.000000000 & 0.000000000 & 0.000151251 & 0.00000000 \\ Pr. 14 & 0.499846954 & 0.000000000 & 0.000000000 & 0.00000000 & 0.000000000 \\ Pr. 15 & 0.974654377 & 0.131336405 & 0.060606060 & 0.207373271 & 0.041935483 \\ Pr. 16 & 0.08042049 & 0.021919442 & 0.000000000 & 0.00000000 & 0.000000000 \\ Pr. 17 & 0.170802192 & 0.080669711 & 0.000000000 & 0.006088280 & 0.000000000 \\ Pr. 18 & 0.181208054 & 0.020134228 & 0.000000000 & 0.006711409 & 0.000000000 \\ Pr. 19 & 0.382899628 & 0.00000000 & 0.00000000 & 0.14981413 & 0.000000000 \\ Pr. 20 & 0.392733535 & 0.027523238 & 0.001104230 & 0.083443207 & 0.000000000 \\ Pr. 21 & 0.085327784 & 0.085327784 & 0.064255980 & 0.022892820 & 0.000000000 \\ Pr. 22 & 0.644711051 & 0.148766475 & 0.00000000000 & 0.149036837 & 0.0000000000 \\ Pr. 23 & 0.0755657573 & 0.01212752 & 0.012122750 & 0.054389053 & 0.000000000 \\ Pr. 24 & 0.501813869 & 0.738523790 & 0.005336960 & 0.035509976 & 0.02057207 \\ Pr. 25 & 0.367647059 & 0.073529412 & 0.029411760 & 0.000000000 & 0.000000000 \\ Pr. 26 & 0.177304965 & 0.049645390 & 0.000000000 & 0.0000000000 \\ Pr. 27 & 0.189721159 & 0.066156370 & 0.06509620 & 0.066156370 & 0.000000000 \\ Pr. 28 & 0.774801587 & 0.256613757 & 0.0000000000 & 0.256613757 & 0.091269841 \\ Average & 0.272940340 & 0.101237633 & 0.027501980 & 0.075030869 & 0.005889912 \\ \hline \end{tabular}
\end{table} 
Table 4: The deviation (dev) of the methods for the 28 test instances. from the optimal one, as indicated by the Gaussian membership function. TrigAC-PSO's precision is evident, reaching the optimal solution with a remarkable truth score of 0.9991 (99.91% optimality). The selection of these methods was deliberate, guided by two principal factors. Firstly, they were chosen because they have been established for years, verifying their reliability. Secondly, our intention was to demonstrate a compelling narrative: Despite their perceived obsolescence and occasional failure to attain optimality, these methods possess the potential for resurgence and redemption through fuzzy sets, transcending the boundaries of classical logic to reclaim their erstwhile prominence. Upon culmination of the exhaustive comparative analysis involving the aforementioned methods, it became apparent that TrigAC-PSO appeared as the best choice, distinguishing itself as the quintessential option. In a quest to examine further its rustiness and precision, TrigAC-PSO underwent additional tests against contemporaneous methods introduced concurrently. Firstly, the proposed algorithm is tested on twelve instances, from Table 2, against the algorithm introduced by [PERSON] and [PERSON], in 2023 [15]. This algorithm was based on [PERSON]'s robust ranking method [16]. But how is a ranking function defined?