# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 1 Introduction
In their research, they utilized ranking methods for numbers to evaluate the fuzzy objective values of the objective function and determine the optimal alternative [11]. [PERSON] and [PERSON] introduced a novel algorithm known as the fuzzy zero point method for finding a fuzzy optimal solution for the FTP, where the transportation cost, supply and demand are represented by trapezoidal fuzzy numbers [12]. [PERSON] and [PERSON] formulated a two-stage-cost-minimizing FTP, where supply and demand are trapezoidal fuzzy numbers, while the costs remain crisp [13]. [PERSON] and [PERSON] introduced a method for solving the FTP by utilizing octagonal fuzzy numbers [14]. [PERSON] and [PERSON] based their research on [PERSON]'s robust ranking method [15]. They proposed an alternative algorithm for finding an initial basic solution to the fuzzy triangular and trapezoidal transportation problem [15]. In the literature, there are many other surveys consisted of problems which are solved by various methods via generalized trapezoidal fuzzy numbers. Specifically, generalized fuzzy numbers extend the concept of classical fuzzy numbers by introducing additional parameters to provide a more versatile representation of uncertainty. [PERSON] and [PERSON] proposed a new approach for solving the FTP using generalized fuzzy numbers, which had not been applied till that point [16]. Building upon the concept of [PERSON] and [PERSON], [PERSON] strived to diminish the computational complexity of the existing method. The technique applied in his study was simpler, computationally efficient and novel compared to the proposed method by [PERSON] and [PERSON] [17]. [PERSON] contributed to the previously mentioned research by proposing the concept of generalized fuzzy numbers when there are no prescribed limits on the membership function to the normal form [18]. FTP approximation is a crucial aspect in the realm of computational intelligence algorithms. Various optimization techniques including metaheuristic algorithms and mathematical programming approaches have been adapted or developed to handle fuzzy transportation problems effectively. [PERSON] solved the transportation problem with fuzzy coefficients using genetic algorithms (GA) [19]. [PERSON] and [PERSON] emphasized on four-dimensional multi-item TP through GA and particle swarm optimization (PSO) [20]. New approaches in metaheuristics to solve the fixed TP in a fuzzy environment were developed by [PERSON] et al. [21]. [PERSON] et al. design a hybrid GA-PSO algorithm to solve the traveling salesman problem (TSP) [22]. Last but not least, [PERSON] and [PERSON], in 2021, suggested an extension of PSO for solving the FTP [23]. Their research, though, focused on solutions presented in a crisp form from fuzzy sets. All the abovementioned techniques, achieved satisfactory results in optimizing some instances of the fuzzy transportation problem. However, they were applied to a rather small set of instances with more less-similar characteristics. So, it is obvious that a more generic optimization algorithm should be designed and applied to optimize all the above fuzzy transportation instances, achieving results comparable or even better to the ones found so far. Motivated by the aforementioned applications of metaheuristic algorithms, this study focuses on the effective application of PSO to solve the FTP. Twenty-eight distinct instances are tacked using ten well-established methods found in the existing literature. The spotlight will be on the trigonometric acceleration coefficients-PSO (TrigAC-PSO) algorithm, a variation of the classical PSO algorithm, introduced by [PERSON] and [PERSON], in 2023 [24]. Extensive experimentation was carried out to assess the stability and the performance of the proposed algorithm in navigating transportation problems with fuzzy costs. Furthermore, various particle configurations were examined to evaluate TrigAC-PSO's capacity to achieve optimal results, demonstrating its superior performance.