2.3.1. Particle Swarm Optimization (PSO)

Particle swarm optimization is a nature inspired stochastic optimization technique proposed by J. Kennedy and R. C. Eberhard in 1995. It is a population-based computationally inexpensive technique that is inspired by the social behaviour of fish schooling and bird flocking. The methodology of the algorithm is that the swarm of particles fly in the search space and finds the optimal solution by updating their own best solution and the best solution obtained by the swarms. The swarm is randomly initialized as particles in N-dimensional search space with position *x<sup>i</sup>* and velocity *v<sup>i</sup>* . The position of the particles represents the probable solution, and the velocity represents the rate of change of position of the particle concerning the current position. The particles change their positions with respect to the positions of the best particle. The velocity update equations are given by:

$$v\_i^d(t+1) = w \times v\_i^d(t) + c\_1 \times r\_1 \times \left( pbest\_i^d(t) - x\_i^d(t) \right) + c\_2 \times r\_2 \times \left( gbest^d - x\_i^d \right) \tag{6}$$

$$
\boldsymbol{\alpha}\_i^d(t+1) = \boldsymbol{\alpha}\_i^d(t) + v\_i^d(t+1) \tag{7}
$$

where *v d i* (*t*) and *x d i* (*t*) represents the velocity and position of *i*th particle in *d*th dimension at *t*th iteration, *v d i* (*t* + 1) and *x d i* (*t* + 1) is the velocity and position of the ith particle in dth dimension at (*t* + 1)th iteration. *pbest<sup>d</sup> i* represents the current best position of the particles and *gbest<sup>d</sup>* represents the best position among all the particles in *d*th dimension, *c*<sup>1</sup> and *c*<sup>2</sup> are the acceleration parameter, *r*<sup>1</sup> and *r*<sup>2</sup> are the random number in the range [0, 1] and *w* is the inertial weight vector which maintains balance between exploration and exploitation.
