**2. Particle swarm optimization**

Particle swarm optimization (PSO) is accepted as the second population-based algorithm inspired from animals. Since James Kennedy (a social psychologist) and Russell C. Eberhart simulated the bird flocking and fish schooling foraging behaviors, they have used this simulation to the solution of an optimization problem and published their idea in a conference in 1995 [3] for the optimization of continuous nonlinear functions. There are two main concepts in the algorithm: velocity and coordinate for each particle. Each particle has a coordinate and an initial velocity in a solution space. As the algorithm progresses, the particles converge toward the best solution coordinates. Since PSO is quite simple to implement, it requires less memory and has no operator. Due to this simplicity, PSO is also a fast algorithm. Different versions of PSO have been developed, using some operators since the first version of PSO was published.

In the first versions of PSO, the velocity was calculated with a basic formula using current velocity, personal best and local best values in the formula, multiplying stochastic variables. The current particle updates its previous velocity, not only its previous best but also the global best. The total probability was distributed between local and global best using stochastic variables.

In the next versions, in order to control the velocity, an inertia weight was introduced by Shi and Eberhart in 1998 [4]. Inertia weight balances the local and global search ability of algorithm. Inertia weight specifies the rate of contribution of previous velocity to its current velocity. Researchers made different contributions to the inertia weight concept. Linearly, exponential or randomly decreasing or adaptive inertia weight was introduced by different researchers [5]. In the next version of PSO, a new parameter called constriction factor was introduced by Clerc and Kenedy [6, 7]. Constriction factor (K) was introduced in the studies on stability and convergence of PSO. Clerc indicates that the use of a constriction factor insured convergence of the PSO. A comparison between inertia weight and constriction factor was published by Shi and Eberhart [8].

Nearly all engineering discipline and science problems have been solved with PSO. Some of the most studied problems solved with PSO are from Electrical Engineering, Computer Sciences, Industrial Engineering, Biomedical Engineering, Mechanical Engineering and Robotics. In Electrical Engineering, power distribution problem [9] is solved with PSO. Another most studied problem in Electrical Engineering is economic dispatch problem [10, 11]. In Computer Sciences, face localization [12], edge detection [13], image segmentation [14], image denoising [15], image filtering [16] problems are solved with PSO. In Industrial Engineering, examination timetabling problems [17], traveling salesman problem [18], and job-shop scheduling problems [19] are solved with PSO. In Robotics, particle swarm optimization in coconut tree plucking robot is introduced [20], and path planning problem is solved with PSO [21]. In these studies, it has been proven PSO success in point of both performance and speed in most of the studies.

After the main PSO algorithm was studied and evolved with some parameters, hybrid algorithms were designed and developed by researchers. The most prominent ones are hybrid PSO with genetic algorithm (GA) [22] and particle swarm optimization with chaos [23–25] and quantum chaotic PSO [26]. In the next section, some of the recent hybrid PSO algorithms are presented.
