# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 6 Case Studies and Experimental Results
An efficient approach for comparing fuzzy numbers is via the use of the racking function \(\mathfrak{R}:F(R)\rightarrow\mathbb{R}\) where \(F(\mathfrak{R})\) is a set of fuzzy numbers defined on a set of real numbers, which convents a fuzzy number into a crisp value, under specific circumstances [16]. The following table (Table 6) gives the formulae of the ranking functions, which have been applied in this manuscript. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No** & **NWM** & **LCM** & **VAM** & **MOMC** & **TrigAC-PSO** \\ \hline Pr. 01 & 1.0000 & 0.9767 & 0.9767 & 0.9767 & 0.9950 \\ Pr. 02 & 0.7115 & 0.9985 & 1.0000 & 0.9999 & 1.0000 \\ Pr. 03 & 0.6022 & 0.3488 & 1.0000 & 0.0304 & 1.0000 \\ Pr. 04 & 0.7844 & 0.4699 & 0.4553 & 0.4699 & 1.0000 \\ Pr. 05 & 1.0000 & 0.9137 & 0.9944 & 0.9944 & 1.0000 \\ Pr. 06 & 0.6140 & 0.7183 & 1.0000 & 0.9862 & 1.0000 \\ Pr. 07 & 0.7561 & 0.9983 & 1.0000 & 0.9961 & 1.0000 \\ Pr. 08 & 0.8319 & 0.8319 & 0.9949 & 1.0000 & 1.0000 \\ Pr. 09 & 1.0000 & 0.9981 & 0.9697 & 0.9868 & 1.0000 \\ Pr. 10 & 0.7483 & 0.8445 & 0.9862 & 1.0000 & 1.0000 \\ Pr. 11 & 0.6045 & 1.0000 & 1.0000 & 0.6139 & 1.0000 \\ Pr. 12 & 0.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ Pr. 13 & 0.9980 & 1.0000 & 1.0000 & 0.9980 & 1.0000 \\ Pr. 14 & 0.2271 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ Pr. 15 & 0.0000 & 0.1311 & 0.6147 & 0.0060 & 0.9870 \\ Pr. 16 & 0.9947 & 0.9951 & 1.0000 & 1.0000 & 1.0000 \\ Pr. 17 & 0.9974 & 0.9709 & 1.0000 & 0.9995 & 1.0000 \\ Pr. 18 & 0.4868 & 0.9911 & 1.0000 & 0.9999 & 1.0000 \\ Pr. 19 & 0.7559 & 1.0000 & 1.0000 & 0.9692 & 1.0000 \\ Pr. 20 & 0.4762 & 0.9924 & 0.9999 & 0.9665 & 1.0000 \\ Pr. 21 & 0.9999 & 0.9999 & 0.9999 & 0.9999 & 1.0000 \\ Pr. 22 & 0.9718 & 0.9921 & 1.0000 & 0.9920 & 1.0000 \\ Pr. 23 & 0.9093 & 0.9967 & 0.9967 & 0.9514 & 1.0000 \\ Pr. 24 & 0.0000 & 0.0000 & 0.9995 & 0.9930 & 0.9999 \\ Pr. 25 & 0.2235 & 0.9893 & 1.0000 & 1.0000 & 1.0000 \\ Pr. 26 & 0.2590 & 0.9750 & 1.0000 & 1.0000 & 1.0000 \\ Pr. 27 & 0.0190 & 0.9998 & 0.9998 & 0.9997 & 1.0000 \\ Pr. 28 & 0.0000 & 0.0000 & 1.0000 & 0.0000 & 0.9930 \\ Average & 0.6061 & 0.8262 & 0.9639 & 0.8546 & 0.9991 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The Gaussian membership function for the 28 test instances. Once more, TrigAC-PSO algorithm stands out by attaining the optimal solution with a membership degree of 1, demonstrating its superiority. Similarly, the method of Ekanayake and Ekanayake yields impressive outcomes, with an aggregated membership degree reaching 0.9995. At this juncture, it is intriguing to delve into the realm of generalized fuzzy numbers, where the application of a metaheuristic approach to the FTP marks a pioneering endeavor. Five problems, from Table 2, will be subjected to comparison against TrigAC-PSO, shedding light on the algorithm's adaptability and efficacy in undertaking challenges modeled by fuzzy generalized numbers [16; 17; 30; 39]. The ensuing table (Table 8) illustrates the outcomes generated by four distinct methods applied to five of the aforementioned problem sets. These solutions are depicted as fuzzy numbers, subsequently subjected to defuzzification through the ranking function (Table 6). This process facilitates an in-depth analysis of the deviation from the optimal values, as shown in Table 9, thus enhancing comprehension for the reader.