Mathematical Model

The population basically follows the pathfinder for various activities. However, in order to do so, we need to specify the position of both the pathfinder and any arbitrary population member. The position of a member says X_i is defined as:

Where Xi is the position vector of ith follower at k iteration, r₁ and r₂ are random variables uniformly generated in the range of [0,1], α is the coefficient for interaction and β is the coefficient of attraction. The value of α, ß are set in such a way that there is the perfect balance between interaction (i.e. the magnitude of movement of any member together with its neighbour) and attraction (i.e. the random distance for keeping the herd roughly with the leader). The optimum values for α, ß should be around 1. ε is the vector of vibration and it is defined as follows:

where, , u₁ is random variable in range [-1,1]

The position of pathfinder is defined as:

X_p is the position vector of the pathfinder, K is the current iteration, r₃ if a random variable in the range of [0,1], A is the vector of fluctuation rate. A is defined as follows:

where u2 is a random variable in the range [-1,1], K_max is the maximum iteration.

ε, A can provide random movement (walk) for all members. Therefore setting different values for them ensures exploration and exploitation. The term  in ε, A ensures exploration and exploitation phases of a metaheuristic technique. At first   is very small, thereby resulting in rapid change constituting to exploration. Then at later stages, its value becomes eventually small or even zero, constituting exploitation.

Nature is full of social behaviours for performing different tasks. Although the ultimate goal of all individuals and collective behaviours is survival, creatures cooperate and interact in groups, herds, schools, colonies, and flocks for several reasons: hunting, defending, navigating, and foraging. In order to mimic these characteristics of animals, swarm-intelligence based optimization algorithms are introduced.

For example – Ant Colony optimization, Cat Swarm Optimization, Particle Swarm optimization