# Solving the Fuzzy Transportation Problem by a Novel Particle Swarm Optimization Approach
## 1 Introduction  
In the realm of logistics and supply chain management, the transportation problem (TP) stands as a fundamental challenge, aiming to satisfy the distribution of goods from suppliers to consumers while minimizing overall transportation costs. Over the years, extensive research has been conducted to develop efficient algorithms and methods for solving various transportation problems with precise and deterministic values of supply and demand units such as transportation costs. The northwest corner method (NWC) is one of the methods that discovers a basic feasible solution to various transportation problems [1]. The method is named after the fact that it stands from the northwest corner cell of the transportation table and proceeds in a systematic manner. Despite the fact that is an easy and applicable method, it does not always guarantee an optimal solution [1]. The least-cost method (LCM) is an alternative method, which focuses on selecting the cell with the minimum cost during the allocation process [2]. [PERSON]'s approximation method (VAM) is known for yielding values relatively close to the optimum, often aligning precisely with it [3]. Although VAM is more complex than the LCM, since it imposes penalties on cells characterized by high transportation costs, it tends to achieve more favorable initial solutions than LCM [3]. However, in real-world scenarios, it is difficult to predict supply and demand quantities \"\(a\) priori\" as they are inextricably linked by the season and by the trends, prevailing by consumers and the financial circumstances of each corporation. Equivalent situations unfold with the prices of transportation which usually depend on factors such as fluctuating fuel prices, the weather, dynamic market conditions and varying transportation routes. To handle with these uncertainties, researchers have turned to fuzzy logic, a mathematical framework that allows for the representation of vagueness and ambiguity in decision-making processes. Going into further detail, fuzzy set theory provides a robust foundation for modeling and analyzing problems with imprecise information, emerging as a promising avenue to optimize transportation problems. A fuzzy transportation problem (FTP) is a variation of the traditional transportation problem in which the transportation cost, supply and demand are fuzzy quantities [4]. There is some research that has relied on fuzzy environments, proving that the fuzzy decision-making method is becoming paramount. The inspirers of the idea of decision-making in fuzzy environments were [PERSON] and [PERSON] in 1970 [5]. Following their study, new findings emerged, with the most recent studies being noteworthy of mention. [PERSON] et al., in 1996, introduced the concept of an optimal solution for the TP with fuzzy coefficients represented as fuzzy numbers and developed an algorithm to attain an optimal solution [6]. [PERSON], [PERSON] and [PERSON] presented a new algorithm for finding an initial basic feasible solution of the TP when the transportation matrix contains both fuzzy and crisp numbers [7]. Another intriguing respective was offered by [PERSON] and [PERSON] in 2009 [8]. Their method was based on the minimization of transportation cost as well as time of transportation when the demand, supply and shipping cost per unit of the quantities are fuzzy. [PERSON], two years later, introduced an approach for solving a wide range of transportation problems by transforming fuzzy quantities into crisp values using their new method [4]. During the same year, [PERSON] et al. suggested their individual version for solving the fuzzy transportation problem, utilizing the simplex algorithm [9]. [PERSON] also worked on fuzzy exact algorithms. He successfully developed the fuzzy version of [PERSON]'s and MODI methods for obtaining a fuzzy initial basic solution and fuzzy optimal solution, respectively [10]. [PERSON] and [PERSON] formulated an alternative method for dealing with a special type of FTP, in which the cost is depicted as triangular fuzzy number.