Manta ray foraging optimization algorithm

Zhao et al. [43] designed the MRFO algorithm, a bioinspired optimizer. MRFO assumes the actions of manta rays in catching the prey. MRFO utilizes three foraging procedures of manta rays: chain foraging, cyclone foraging, and somersault foraging. Chain foraging mimics the process of necessary food searching. Foraging manta rays systematically grasp up to capture the disappeared or undetected prey in the chain by the last manta ray. Cyclone foraging occurs when the collection amount of the prey is significant. The head is paired with the manta ray’s tail, making a spiral to create an edge in a cyclone’s eye. In somersault foraging, manta rays perform backward rotation and circle movements throughout as the prey planktons move them into their open lips [78]. Like various meta-heuristic algorithms, MRFO’s initialization start is defined randomly to generate random positions for a set of agents X, followed by getting the most desirable agent Xbest with the best fitness value. The agents are updated according to the aforementioned three strategies. The following are the mathematical models of these strategies.

Chain foraging In this process, the agent Xk is updated at iteration k using Eq. (16)

where xkt is the kth position in time k, r is a random vector in [0, 1], α is the weight coefficient, and xbestt is the high-concentration plankton [43].

Cyclone foraging In this process, the agent Xk is updated following Eq. (18)

where β is the weight coefficient. Furthermore, based on a random position provided in the search space, the cyclone foraging step agents can change their position to improve the MRFO’s exploration. Thus, the current agent location is updated using the following equation:

where Low and Up are the lower and upper limits of the search space.

Somersault foraging The mathematical formulation used to update the agent in this phase is defined in the following equation:

where s is the somersault factor equal to 2, and r1 and r2 are the random numbers in [0, 1].

To conclude, the MRFO algorithm is initialized by generating a random population within specified permissible boundaries. The position-update procedure depends on the individual manta ray at the front of the recent one and the estimated pivot position. Changing from exploration to exploitation phase depends on the (itr/maxitr) ratio value. The exploitation phase is determined when (itr/maxitr<r), in which the current most suitable position is recognized as a pivot position. The algorithm moves to the exploration phase when (itr/maxitr>r). Furthermore, the algorithm can shift between chain foraging and cyclone foraging based on a randomly generated number. Somersault foraging then takes steps to update the individuals’ current status through the current best approach. These three distinct foraging mechanisms are conducted interchangeably to simultaneously achieve the optimization problem’s optimal global solution and meet the predefined end criterion.