**1. Introduction**

Many real-world optimization problems are increasingly becoming challenging [1]. Therefore a new bio-inspired optimization technique, named Manta Ray Foraging Optimization (MRFO) algorithm, is presented, aiming to providing a novel algorithm that provides an alternate optimization approach for addressing real-world engineering issues [2]. The contribution of this algorithm is the foraging behaviors of manta rays and the characteristic of their foraging behaviors [3]. The optimization results demonstrate the MRFO optimizer can lead to promising improvement on solution precision with less computation cost compared with other well-established optimizers.

**2. Inspiration of MRFO**

The inspiration of MRFO is based on intelligent foraging behaviors of manta rays. There are three unique foraging strategies of manta rays, for the purpose of hunt for food [4]. They are as described as follows;

**Chain foraging:**The chain foraging strategy is which, when 50 or more manta rays start foraging, they lineup, one behind another, forming an orderly line. Smaller male manta rays are backed upon female ones and swim on top of their back stomach the beats of the female’s pectoral fins [5]. Consequently, the plankton which is missed by previous manta rays will be scooped up by ones behind them. Cooperating with each other in this way, they can funnel the most amount of plankton into their gills and improve their food rewards.**Cyclone foraging**: The cyclone foraging strategy is which, when the concentration of plankton is very high, dozens of manta rays gather together [6]. Their tail ends link up with heads in a spiral to generate a spiraling vertex in the eye of the cyclone and the filtered water moves up towards the surface. This pulls the plankton into their open mouths.**Somersault foraging:**The final foraging strategy is somersault foraging. This is one of the most splendid sceneries in nature. When manta rays find a food source, they will do a series of backwards somersaults, circling around the plankton to draw it towards manta rays [7]. Somersault is a random, frequent, local and cyclical movement, which helps manta rays to optimize food intake. Although these foraging behaviors are rare in nature.

**3. Manta ray Foraging Optimization Algorithm (MRFO)**

Manta rays are fancy and one of the largest marine creatures although they appear to be terrible. Manta rays have a flat body from top to bottom and a pair of pectoral fins, with which they elegantly swim as freely fly [8]. They also have a pair of cephalic lobes that extend in front of their giant, terminal mouths. Without sharp teeth, manta rays feed on plankton made mostly of microscopic animals from the water. When foraging, they funnel water and prey into their mouths using horn-shaped cephalic lobes. Manta rays eat a large amount of plankton daily wise. An adult manta ray can eat 5kg of plankton on daily basis. The most interesting thing about manta rays is their foraging behaviors, and they may travel alone or in groups of more than 50 and their foraging strategy is observed in groups.

**4. Numerical Implementation of MRFO**

MRFO is implemented by three foraging behaviors including chain foraging, cyclone foraging and somersault foraging. The mathematical models are described below.

**4.1. Chain Foraging**

In MRFO, manta rays can observe the position of plankton and swim towards it. The higher the concentration of plankton in a position is, the better the position is. That is, in every iteration, each individual is updated by the best solution found so far and the solution in front of it. This mathematical model of chain foraging is represented as follows;

Where, is the position of xth individual at time n in 𝑑im dimension, rand is a random vector within the range of [0, 1], φ is a weight coefficient, is the plankton with high concentration.

**4.2. Cyclone Foraging**

The cyclone foraging strategy of manta is to spirally move towards the food, each manta ray swims towards the one in front of it. An individual not only follows the one in front of it but only moves towards the food along a spiral path. The mathematical equation modeling the spiral-shaped movement of manta rays is defined as follows;

This motion behavior may be extended to d space. For this mathematical model of cyclone foraging can be defined as;

Where α is the weight coefficient, T is the maximum number of iterations, and rand1 is the rand number in [0, 1].Each individual to search for a new position far from the current best one by assigning a new random position in the entire search space position. This mechanism focuses MRFO to achieve an extensive global search; its mathematical equation is presented below;

Where is a random position randomly produced in the search space, LB and UB are the lower and upper limits of the dimension, respectively.

**4.3. Somersault Foraging**

Each individual tends to swim to and somersault to a new position. Therefore, they always update their positions around the best position found so far. The mathematical model can be described as follows

Where som is the somersault factor that decides the somersault range of manta rays and 𝑆om= rand2 and rand3 are two random numbers in [0, 1].

Therefore, the overall time complexity of this MRFO is given as

O (MRFO) = O (T (O chain foraging+ O cyclone foraging+ O Somersault foraging) (10)

Where T is the maximum number of iterations.

**5. Pseudo code of MRFO**

**6. Flowchart of MRFO**

**7. Advantages of MRFO**

**8. Applications of MRFO**

The MRFO algorithm demonstrates the practicability and superiority in handling challenging engineering problems. The below following engineering design problems are employed to evaluate MRFO. They are;

- Compression string design [9].
- Pressure vessel design [10].
- Welded beam design [11].
- Speed reducer design [12].
- Rolling element bearing design [13].
- Multiple disc design [14].

**Reference**

1. Zhao W, Zhang Z, Wang L (2020) Manta ray foraging optimization: An effective bio-inspired optimizer for engineering applications. Engineering Applications of Artificial Intelligence 87:103300. doi: 10.1016/j.engappai.2019.103300

2. Ab Wahab M, Nefti-Meziani S, Atyabi A (2015) A Comprehensive Review of Swarm Optimization Algorithms. PLOS ONE 10:e0122827. doi: 10.1371/journal.pone.0122827

3. Akay B, Karaboga D (2012) A modified Artificial Bee Colony algorithm for real-parameter optimization. Information Sciences 192:120-142. doi: 10.1016/j.ins.2010.07.015

4. Askarzadeh A (2014) Bird mating optimizer: An optimization algorithm inspired by bird mating strategies. Communications in Nonlinear Science and Numerical Simulation 19:1213-1228. doi: 10.1016/j.cnsns.2013.08.027

5. Bayraktar Z, Komurcu M, Bossard J, Werner D (2013) The Wind Driven Optimization Technique and its Application in Electromagnetism. IEEE Transactions on Antennas and Propagation 61:2745-2757. Doi: 10.1109/tap.2013.2238654

6. (2014) Das richtige Licht für die Gesundheit. Der Orthopäde 43:703-704. doi: 10.1007/s00132-014-2336-9

7. Birbil Ş, Fang S (2003) Journal search results – Cite This For Me. Journal of Global Optimization 25:263-282. doi: 10.1023/a:1022452626305

8. Zhang X, Gong C (2013) Dual-Buck Half-Bridge Voltage Balancer. IEEE Transactions on Industrial Electronics 60:3157-3164. doi: 10.1109/tie.2012.2202363

9. Civicioglu P (2013) Backtracking Search Optimization Algorithm for numerical optimization problems. Applied Mathematics and Computation 219:8121-8144. doi: 10.1016/j.amc.2013.02.017

10. Cuevas E, Cienfuegos M, Zaldívar D, Pérez-Cisneros M (2013) A swarm optimization algorithm inspired in the behavior of the social-spider. Expert Systems with Applications 40:6374-6384. doi: 10.1016/j.eswa.2013.05.041

11. Gandomi A, Alavi A (2012) Krill herd: A new bio-inspired optimization algorithm. Communications in Nonlinear Science and Numerical Simulation 17:4831-4845. doi: 10.1016/j.cnsns.2012.05.010

12. Zong Woo Geem, Joong Hoon Kim, Loganathan G (2001) A New Heuristic Optimization Algorithm: Harmony Search. SIMULATION 76:60-68. doi: 10.1177/003754970107600201

13. Hare W, Nutini J, Tesfamariam S (2013) A survey of non-gradient optimization methods in structural engineering. Advances in Engineering Software 59:19-28. doi: 10.1016/j.advengsoft.2013.03.001

14. Kannan B, Kramer S (1994) An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and Its Applications to Mechanical Design. Journal of Mechanical Design 116:405-411. doi: 10.1115/1.2919393

Can You Upload Source Code….

mail us decoderzmt@gmail.com