# Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively
###### Abstract  
The Transportation Problem (TP) is a special type of linear programming problem, where the objective is to minimize the cost of distributing a product from a number of sources to a number of destinations. Many methods for solving the TP have been studied over time. However, exact methods do not always succeed in finding the optimal solution or a solution that effectively approximates the optimal one. This paper introduces two new variations of the well-established Particle Swarm Optimization (PSO) algorithm named the Trigonometric Acceleration Coefficients-PSO (TrigAc-PSO) and the Four Sectors Varying Acceleration Coefficients PSO (FSVAC-PSO) and applies them to solve the TP. The performances of the proposed variations are examined and validated by carrying out extensive experimental tests. In order to demonstrate the efficiency of the proposed PSO variations, thirty two problems with different sizes have been solved to evaluate and demonstrate their performance. Moreover, the proposed PSO variations were compared with exact methods such as Vogel's Approximation Method (VAM), the Total Differences Method 1 (TDMI), the Total Opportunity Cost Matrix-Minimal Total (TOCM-MT), the Juman and Hoque Method (JHM) and the Biliqis Chastine Erma method (BCE). Last but not least, the proposed variations were also compared with other PSO variations that are well known for their completeness and efficiency, such as Decreasing Weight Particle Swarm Optimization (DWPSO) and Time Varying Acceleration Coefficients (TVAC). Experimental results show that the proposed variations achieve very satisfactory results in terms of their efficiency and effectiveness compared to existing either exact or heuristic methods. t 202320233010  
proven that in problems with a small range of values and a relatively small number of variables, the above exact methods are quite efficient. In some cases, TP has a complex structure, multifaceted parameters and a huge amount of data to be studied. Therefore, exact methods do not succeed in finding a suitable solution in an acceptable time period; a result, it is unpractical to use them. Taking into consideration the above, apart from conventional solution techniques, various heuristic and metaheuristic methods have been designed to capitalize on their potential capabilities. Specifically, metaheuristic algorithms attempt to find the best feasible solution, surpassing the other technique as much in terms of quality as in computational time [4]. [PERSON], [PERSON] and [PERSON] applied a Genetic Algorithm (GA) for a two-stage TP using priority-based encoding, showing that the GA has been receiving great attention and can be successfully applied for combinational optimization problems [5]. Ant Colony Optimization (ACO) algorithms have already proven their efficiency in many complex problems; they constitute a very useful optimization tool for many transportation problems in cases where it is impossible to find an algorithm that finds the optimal solution or in cases where the time interval does not make it possible to approve this solution [6]. The applications of hybrid methods with the combination of two or more heuristic, metaheuristic or even exact methods are also widespread. Interesting research was undertaken in 2019 by [PERSON], [PERSON] and [PERSON] [7]. In their study, four metaheuristic algorithms, including Red deer Algorithm (RDA), Stochastic Fractal Search (SFS), Genetic Algorithm (GA) and Simulated Annealing (SA), as well as two hybrid algorithms, the RDA and GA (HRDGA) algorithm and the Hybrid SFS and SA (HRDGA) algorithm, were utilized to solve the TP, demonstrating significant effectiveness [7]. Motivated by the above-mentioned applications of metaheuristic algorithms to cope with the TP, this work deals with the application of Particle Swarm Optimization (PSO) to solve the TP effectively.