# Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively
## 4 The Basic PSO for Solving the TP
As a result, the particle is mostly influenced by its own knowledge, while the influence of the swarm on it is limited; in the second sector, the value of \(c_{1}\) is decreasing while the value of \(c_{2}\) is increasing to an equilibrium between the knowledge of the particle gained at the previous sector and the experience of the swarm; in the third sector, the value of \(c_{1}\) is decreasing while the value of \(c_{2}\) is increasing--explicitly, the particles are allowed to move towards the global best position, following the swarm's movements; as a result, information about the global best is reallocated to all the particles for more exploration before the swarm finally converges [11]; in the fourth sector, the particles head toward both their own personal best and global best observed by the whole swarm--the concept of this variation is based on the combination of all types of different searching behaviors, as they arise for different values of the coefficient acceleration parameters, culminating in equilibrium between exploitation and exploration of the search space; finally, in the last iteration, the two coefficient parameters are equated. The formulation is represented in detail below:

* In the first Iteration, as already mentioned:\[c_{2}=2;\]
* In the first sector: \[c_{1}=\frac{\left(2\cdot c_{1f}-c_{1i}\right)\cdot t}{t_{max}}-1\] (13) where \(c_{1i}=2\), \(c_{1f}=3\),\(c_{2i}=2\) and \(c_{2f}=1\);
* In the second sector: \[c_{1}=\frac{c_{1f}\cdot c_{1i}\cdot t}{2-t_{max}}-1\] (14) \[c_{2}=0.5+\frac{c_{2f}+c_{2j}}{2}\cdot\frac{t}{t_{max}}\] (15) where \(c_{1i}=2\), \(c_{1f}=0.5\),\(c_{2i}=2\) and \(c_{2f}=2\);
* In the fourth sector: \[c_{1}=c_{1i}-\left(c_{1i}-c_{1f}\right)\cdot\frac{t}{t_{max}}; \tag{8}\]0 (16) where \(c_{1i}=0.5\),\(c_{1f}=2\),\(c_{2i}=2.5\) and \(c_{2f}=2\);
* In the last iteration: \[c_{1}=c_{1i}-\left(c_{1i}-c_{1f}\right)\cdot\frac{t}{t_{max}}; \tag{8}\]1 \[c_{1}=c_{1i}-\left(c_{1i}-c_{1f}\right)\cdot\frac{t}{t_{max}}; \tag{8}\]2

In the above formulations, \(c_{1i},\ c_{1f},\ c_{2i}\) and \(c_{2f}\) are initial and final values of cognitive and social components acceleration factors, respectively. To improve the solution quality, these coefficients are updated in such a way that the values increase and decrease at a steady pace. According to this approach, the solution avoids being trapped into a local optimum, as shown by the experimental results presented in Section 6. As for the inertia weight \(w\), Equation (12) is used to provide the necessary momentum for particles to roam across the search space. ## 6 Case Studies and Experimental Results

In this section, the proposed variations of the PSO algorithm are applied in thirty two well-known numerical examples of the TP, as shown in \[c_{1}=c_{1i}-\left(c_{1i}-c_{1f}\right)\cdot\frac{t}{t_{max}}; \tag{8}\]3 The numerical examples of this study come from the research of [PERSON], who compared five different methods, which will be presented briefly below, regarding their performance in solving the TP [22]. Vogel's Approximation Method (VAM) is an iterative procedure such that in each step, proper penalties for each available row and column are taken into account through the least cost and the second-least cost of the transportation matrix [22]. The Total Differences Method 1 (TDM1) was introduced by [PERSON] in 2017. The method is based on VAM's innovation to use penalties for all rows and columns of the transportation matrix. The TDM1 was developed by calculating penalty values only for rows of the transportation matrix [23]. [PERSON] et al., in 2019, represented their new method, known as the Total Opportunity Cost Matrix Minimal Cost (TOCM-MT). This method has a mechanism with which to check the value of the least-cost cell before allocating the maximum units \(x_{ij}\); this is in contradiction to the TDM1, which directly allocates the maximum units \(x_{ij}\) to the least cost [24]. [PERSON] and [PERSON], in 2015, developed a formulation method called the Juman and Hoque method (JHM).