# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 6 Case Studies and Experimental Results  
In this section, the TrigAC-PSO algorithm will be utilized to solve the fuzzy transportation problem. A total of twenty-eight numerical examples will be resolved with results derived from four other established methods. To ensure a thorough investigation, the numerical examples encompass both triangular and trapezoidal fuzzy numbers. Moreover, the inclusion of generalized fuzzy numbers in our domain adds a distinctive dimension, making the application of our variation particularly intriguing in this context. Initially, the problems will be evaluated in comparison to methods whose reliability has remained consistent over the years. In this context, the TrigAC-PSO algorithm will be assessed alongside the northwest method, the least-cost method, [PERSON]'s approximation method and the maximum supply with minimum cost method (MOMC) which was introduced in 2015 by [PERSON] et al. (2015). Their classification technique showed a spectacular computation advantage characterized by a higher processing speed and less use of memory. Both MOMC and the previously mentioned methods have been employed in attempting to solve the fuzzy transportation problem. This poses a challenge for us, as we strive to optimize the values of the given problems using TrigAC-PSO. At this stage, it would be beneficial to make a reference to the initialization process. In the initial steps of Algorithm 1, two vectors, namely \(Supply\) and \(Demand\), are created as input (line 1 and 2). Subsequently, the variables \(m\) and \(n\) are calculated, representing the values of the parameters \(Supply\) and \(Demand\), respectively. Then, a matrix is generated containing random Monte Carlo real numbers (line 7). Line 10 rounds the elements of the candidate solution matrix to the nearest integer, as the commodity quantities must be non-negative integers. The subsequent lines of the algorithm commence a process of realignment and redistribution of matrix \(L\) to align with the specified supply and demand amounts. In lines 11 and 12, the sum of all elements in each row of matrix L is stored in vector \(Sumrow\), and similarly, the sum of all elements in each column is stored in vector \(Sumcol\). Then, two new vectors, \(s\) and \(d\), are created by subtracting \(Sumrow\) from \(Supply\) and \(Sumcol\) from \(Demand\), respectively. Following this, for each cell in the final matrix, any discrepancies are identified and adjusted accordingly, ensuring that the excess amounts in vectors \(s\) and \(d\) are zeroed out. The output of Algorithm 1 is a matrix comprising the initial solution, known as an initial basic feasible solution (IBFS). Subsequently, the PSO algorithm is employed to find the particle achieving the optimal position and its corresponding optimal transportation cost. The supply and demand values are crisp values, whereas transportation cost is depicted using fuzzy sets. However, the expanded positions of the particles, while adhering to demand and supply constraints, exhibited negative or fractional values. Such values are incompatible with the nature of the solution, which necessitates quantities represented solely by positive integers. Consequently, necessary adjustments were implemented to ensure the integrity of the particle's positions. These sub-algorithms were presented in 2023 by [PERSON] and [PERSON] (2024). To derive the final solution, fuzzy operationsoutlined in Table 1 will be applied, culminating in the representation of the final iteration cost as a fuzzy number. ```
1. Define Supply = [\(s_{1},s_{2},...,s_{m}\)]
2. Define Demand = [\(d_{1},d_{2},...,d_{n}\)]
3. Define m = length (Supply)
4. Define n = length (Demand)
5. Initialize a solution matrix I = zeros (m, n)
6. Initialize a supporting table B = zeros (\(m\times n\))
7. Generate a new random number x, x = -log (rand(m \(\times\) n, 1)
8. Set x = x/Sum(x)
9. Take a matrix L = reshape (B x, [m, n])
10.