# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 6 Case Studies and Experimental Results
It provides a numerical representation of how much individual data values differ from the average or desired outcome and comprises a metric of each algorithm's stability. Deviation is given by the following formula:  
\[Dev=\frac{x_{ij}-optimal}{optimal} \tag{14}\]  
where \(x_{ij}\) is the current solution. Based on the data presented in Table 4, it is apparent that VAM, MOMC and TrigAC-PSO appear to be more efficient than the other methods. Therefore, the solutions achieved by NWC differ from the optimal solution by 27.3%, the results of LCM by 10.12% and MOMC by 7.5%. [PERSON]'s approximation method presented higher levels of efficiency since the average of the export solutions differ from the optimal solutions by 2.7%. The TrigAC-PSO method is now highlighted as the most efficient method, surpassing all other methods, with a percentage close to zero. This indicates its remarkable stability and accuracy in predicting the optimal value with near-perfect precision. In Boolean (classical) logic, transportation costs are represented as precise, fixed values. In fuzzy logic, transportation costs, as supply and demand quantities, are represented as approximate values, allowing for gradual transitions between costs and shipments. The pivotal aspect of our study was the visual representation of the final solution using fuzzy numbers. In classical logic, a solution is deemed optimal only if its value aligns precisely with that of the optimal solution. However, the scientific quandary arises: How equitable is it to discard other solutions even when their deviation from the optimal one is infinitesimal? This quandary sparked our original concept, i.e., to represent the costs incurred by exact methods using fuzzy sets and to ascertain the degree of truth inherent in each algorithm's solution belonging to this set. Consequently, every solution is now endowed with its unique value, irrespective of its optimality status. The binary classification of classical logic, where 0 signifies non-optimality and 1 denotes optimality, is supplanted by a continuum with the interval \([0,\ 1]\), wherein each number signifies the solution's degree of belongingness to the fuzzy set. In this study, the Gaussian membership function will be employed to ascertain the truthfulness of each solution. Figure 3: Number of optimal solutions achieved by each method. The Gaussian membership function is defined by a bell-shaped curve characterized by its mean \((\mu)\) and standard deviation \((\sigma)\). The function assigns a degree of membership to each element in a fuzzy set, based on its proximity to the mean. Mathematically, the Gaussian membership function is given by the formula:  
\[\mu(x)=e^{\frac{-(x-y)^{2}}{2\sigma^{2}}} \tag{15}\]  
where  
* \(x\) is the input value;
* \(\mu\) is the mean of the fuzzy numbers;
* \(\sigma\) is the standard deviation of the fuzzy numbers. The following table (Table 5) shows the degree of truth for each solution of the 28 problems. We notice that among the methods that were unable to precisely identify the optimal solution, they achieved values that were remarkably close to the optimal cost. The degree of truth studies how closely the solution obtained by each method approaches our optimal solution offering a clearer perspective. When the degree of truth is 1, the NWM solutions exhibited optimality with a degree of truth of 0.6061 (equivalent to 60.61% optimality). In contrast, both LCM and MOMC attained optimality with a substantially higher degree, scoring 0.8261 (82.61% optimality) and 0.8546 (85.46% optimality), respectively, surpassing the NWM. While VAM succeeded in discovering the optimal solution in half of the instances, its remaining solutions were notably proximal, as VAM achieved the optimal with degree of truth as high as 0.9638 (96.38%). Yet again, TrigAC-PSO stands out significantly, having achieved the optimal solution in 24 out of 28 problems. However, in instances where TrigAC-PSO did not achieve the desired results, its attained soluti