# Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively
###### Abstract
The PSO algorithm was first introduced by Dr. [PERSON] and Dr. [PERSON] in 1995 and was known as a novel population-based stochastic algorithm, working out complex non-linear optimization problems [8]. The basic idea was originally inspired by simulations of the social behavior of animals such as bird flocking, fish schooling, etc. Possessing their own intelligence, birds of the group connect with each other, sharing their experiences, and follow and trust the mass in order to reach their food or migrate safely without knowing in advance the optimal way to achieve it. The proposed research is expected to enhance the abilities of both the social behavior and personal behavior of the birds. It is observed that the original PSO has deficits in premature convergence, especially for problems with multiple local optimums [9]. The swarm's ability to function with social experience as well as personal experience is determined in the algorithm through two stochastic acceleration components, known as the cognitive and social components [10]. These components have the aptitude to guide the particles in the original PSO method to the optimum point as the correct selection of their values is the key influence on the success and efficiency of the algorithm. Much research has been carried out with a focus on finding out the best combination of these components [10]. First, this paper examines approaches that have already been applied with great success to solve the TP. Adding to the above, two new PSO variations are presented and applied to solve the TP, operating proper transformations of the main PSO parameters. Experimental results show that these new PSO variations have very good performance and efficiency in solving the TP compared to the former methods. In order to confirm the technical merit and the applied value of our study, 32 instances of the TP with different sizes have been solved to evaluate and demonstrate the performance of the proposed PSO variations. Their experimental results are compared with those of well-known exact methods, proving their superiority over them. One major innovation of the proposed variations is the appropriate combination of acceleration coefficients (parameters \(c_{1},c_{2}\)) and inertia weight (parameter \(w\)) [11] (see Section 3) in order to come up with better computational results compared to existing approaches. Exhaustive experimental results demonstrate that the performance of the new PSO variations noted significantly higher performance not only compared to the exact methods already applied to solve the TP but also compared to the other PSO variations already introduced in the respective literature. Furthermore, in order to check the stability of the proposed PSO variations, many different combinations of the main PSO parameters were tested and validated. The contribution of the paper is as follows:  
* According to our knowledge, PSO has already been applied for solving the fixed-charged TP, and a heuristic approach was used in order to find the shortest path in a network of routes with a standard number of points connected to each other. For the first time, the PSO-based algorithms are applied to solve the basic TP in a large amount of test instances effectively, not only finding the optimal means of items distribution but also discovering the optimal value. * Moreover, two new PSO variations are introduced, which sustain balance between exploration and exploitation of the search space. These variations proved to be very efficient in solving the TP, achieving better results compared not only to deterministic but also to other already-known PSO-based methods. * A thorough experimental analysis has been performed on the PSO variations applied to solve the TP to prove their efficiency and stability. The remainder of the paper is organized as follows: Section 2 presents the mathematical formulation of the TP. The PSO algorithm is briefly described in Section 3. Section 4 presents the initialization procedure of the basic feasible solutions and the steps of the PSO algorithm for the TP.