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蝠鲼优化算法（Manta Ray Foraging Optimization, MRFO）是一种新型的智能优化算法，源自于蝠鲼在海洋中觅食的行为模式。蝠鲼在海洋中寻找食物时展现出的高效和灵活策略，为解决复杂优化问题提供了灵感。MRFO算法在解决多模态、非线性、高维度的问题上具有较高的潜力，广泛应用于工程、经济、生物医学和社会科学等领域。 MRFO的核心思想是模拟蝠鲼在海洋中的两种主要行为：随机漂移和捕食行为。随机漂移代表了蝠鲼在广阔区域内的探索能力，而捕食行为则体现了对最优解的精细化搜索。算法初始阶段，蝠鲼种群随机分布在解决方案空间中，随着迭代进行，种群不断调整其位置，以接近最优解。在蝠鲼优化算法中，每个个体蝠鲼代表一个潜在的解决方案，它们通过与环境交互来更新自己的位置。蝠鲼的移动由两个主要因素决定：一是随机漂移，即在一定范围内随机改变位置；二是捕食行为，蝠鲼会根据当前解的质量和其他蝠鲼的位置来调整自身位置，接近最优解。这个过程可以通过数学公式表示，包括对当前位置和邻近最优解的动态调整。 "j.engappai.2019.103300.pdf" 文件可能是蝠鲼优化算法的原始研究论文，它详细介绍了算法的理论基础、实现步骤以及与其他优化算法的比较。通过阅读这篇论文，读者可以深入理解MRFO的工作原理，包括如何设定参数、如何初始化种群、如何定义适应度函数以及如何进行迭代更新。 "MRFO.zip" 文件则包含了蝠鲼优化算法的源代码，可能是用Python或 MATLAB 等编程语言实现的。这些代码对于实践者来说极其宝贵，可以直接应用到实际问题中，或者作为学习和改进的基础。通过分析和运行这些代码，开发者可以更好地理解算法的执行流程，并可能对其进行修改和优化以适应特定的优化任务。蝙鲼优化算法作为人工智能和机器学习领域的一种智能优化算法，与其他如遗传算法、粒子群优化、蚁群算法等并列，它们共同构成了解决复杂优化问题的重要工具箱。学习和掌握这种算法有助于提升我们在解决实际问题时的计算效率和精度，特别是在处理大规模优化问题时，能够展现出良好的性能和鲁棒性。

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Engineering Applications of Artificial Intelligence 87 (2020) 103300

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence

journal homepage: www.elsevier.com/locate/engappai

Manta ray foraging optimization: An effective bio-inspired optimizer for

engineering applications

Weiguo Zhao

a,b

, Zhenxing Zhang

, Liying Wang

∗

School of Water Conservancy and Hydropower, Hebei University of Engineering, Handan, Hebei, 056021, China

Illinois State Water Survey, Prairie Research Institute, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA

A R T I C L E I N F O

Keywords:

Global optimization

Engineering design

Manta ray foraging optimization

Metaheuristic

Constrained problems

Optimization algorithm

Heuristic algorithm

A B S T R A C T

A new bio-inspired optimization technique, named Manta Ray Foraging Optimization (MRFO) algorithm,

is proposed and presented, aiming to providing a novel algorithm that provides an alternate optimization

approach for addressing real-world engineering issues. The inspiration of this algorithm is based on intelligent

behaviors of manta rays. This work mimics three unique foraging strategies of manta rays, including chain

foraging, cyclone foraging, and somersault foraging, to develop an efficient optimization paradigm for solving

different optimization problems. The performance of MRFO is evaluated, through comparisons with other state-

of-the-art optimizers, on benchmark optimization functions and eight real-world engineering design cases. The

comparison results on the benchmark functions suggest that MRFO is far superior to its competitors. In addition,

the real-world engineering applications show the merits of this algorithm in tackling challenging problems in

terms of computational cost and solution precision. The MATLAB codes of the MRFO algorithm are available

at https://www.mathworks.com/matlabcentral/fileexchange/73130-manta-ray-foraging-optimization-mrfo.

1. Introduction

Many real-world optimization problems are increasingly becoming

challenging as they often concern a big number of decision variables,

complex nonlinear constraints and objective functions. The global opti-

mization using traditional approaches like numerical methods becomes

less powerful especially when objective functions or constraints have

multiple peaks. Metaheuristic algorithms, powerful tools for handling

challenging optimization problems, are increasingly becoming popular.

The popularity drives from the following aspects.

First, the most outstanding characteristic of metaheuristic algo-

rithms is their simplicity. These metaheuristic methods possess basic

theories or mathematical models which derive from nature. These

methods are generally simple and easy to implement. The ease-to-

use allows one to apply metaheuristics to solve real-world problems.

Moreover, it is also easy to develop their variants according to existing

methods. Second, these optimization technologies can be viewed as

a black box, meaning that it is able to offer a set of outputs for a

given problem for a set of inputs. Furthermore, scholars may eas-

ily modify the structures and parameters of these methods in order

to obtain satisfactory solutions. Third, randomness is one the most

important characteristics of metaheuristic algorithms. This allows meta-

heuristic algorithms to explore the entire search space and prevent

them from trapping into local optima effectively. More specially, it

makes many metaheuristics successful to solve problems with unknown

∗

Corresponding author.

E-mail addresses: zhaoweiguo@hebeu.edu.cn (W. Zhao), zhang538@illinois.edu (Z. Zhang), wangliying@hebeu.edu.cn (L. Wang).

search space or multiple local optima. Finally, these metaheuristics are

highly versatile and flexible, implying their practicability to various

different types of optimization problems including non-linear prob-

lems, non-differentiable problems, or complex numerical problems with

plentiful of local minima. Many metaheuristic algorithms have been

presented and successfully applied to different areas. These algorithms

are mainly categorized into three classes (Hare et al., 2013): evolution-

based (Mühlenbein et al., 1988), physics-based (Geem et al., 2001), and

swarm-based (Krause et al., 2013).

Evolution-based algorithms simulate natural evolution like chemo-

taxis, reproduction elimination and dispersal, and migration (Passino,

2002; De Falco et al., 2012). Genetic algorithm (GA), proposed by

Holland (1975), is a famous and widely used evolutionary algorithm

(EA). One of the main features of GA is that it does not require deriva-

tive in the search space that is existing in mathematical optimization

approaches. GA evolves a population by emulating survival of the

fittest in nature, it may provide efficient solutions and avoid local

optima. Since its emergence, a range of variants have been developed to

improve GA. With its popularity, many other evolutionary algorithms,

including differential evolution (DE) (Rocca et al., 2011), evolutionary

programming (EP) (Juste et al., 1999), evolutionary strategies (ES)

(Beyer and Schwefel, 2002), memetic algorithm (MA) (Moscato et al.,

2007), and so on, have been proposed. In addition, scores of novel

https://doi.org/10.1016/j.engappai.2019.103300

Received 21 December 2018; Received in revised form 17 September 2019; Accepted 14 October 2019

Available online xxxx

W. Zhao, Z. Zhang and L. Wang Engineering Applications of Artificial Intelligence 87 (2020) 103300

EAs have been presented recently, including biogeography-based op-

timization (BBO) (Simon, 2009), bacterial foraging optimization (BFO)

(Passino, 2002), artificial algae algorithm (AAA) (Uymaz et al., 2015),

bat algorithm (BA) (Yang and Hossein Gandomi, 2012), monkey king

evolutionary (MKE) (Meng and Pan, 2016), et al. (Punnathanam and

Kotecha, 2016; Pan, 2012; Mehrabian and Lucas, 2006; Civicioglu,

2013).

Physics-based algorithms mimic physical laws in universe. Simu-

lated annealing (SA) (Kirkpatrick et al., 1983) is one of the most

well-known physics-based techniques. SA is analogy with thermody-

namics in physical material. Annealing is to minimize energy use,

specifically with the way that heated metals cool and crystallize. Re-

cently, multiple new physics-inspired techniques are developed, in-

cluding gravitational search algorithm (GSA) (Rashedi et al., 2009),

electromagnetism-like mechanism (EM) algorithm (Birbil and Fang,

2003), particle collision algorithm (PCA) (Sacco and De Oliveira, 2005),

vortex search algorithm (VSA) (Doğan and Ölmez, 2015), water evap-

oration optimization (WEO) (Kaveh and Bakhshpoori, 2016), atom

search optimization (ASO) (Zhao et al., 2019a), big bang–big crunch

algorithm (BB-BC) (Genç et al., 2010), et al. (Shah-Hosseini, 2011;

Chuang and Jiang, 2007; Shah-Hosseini, 2009; Kaveh and Dadras,

2017; Kaveh and Talatahari, 2010; Zheng et al., 2010; Javidy et al.,

2015; Mirjalili and Hashim, 2012; Zheng, 2015; Flores et al., 2011;

Tamura and Yasuda, 2011; Rao et al., 2012; Zarand et al., 2002;

Shen and Li, 2009; Kripka and Kripka, 2008; Patel and Savsani, 2015;

Eskandar et al., 2012; Moghaddam et al., 2012).

Swarm-based algorithms simulate social behaviors of species like

self-organization and division of labor (Beni and Wang, 1993; Ab Wa-

hab et al., 2015). Two outstanding examples are particle swarm op-

timization (PSO) (Kennedy and Eberhart, 1995) and ant colony opti-

mization (ACO) (Dorigo et al., 1996). PSO inspired by bird flocking

behaviors updates each agent in a population by its best individual

agent and the best global agent. ACO is inspired by the foraging

behaviors of ant swarms, ants search for the most effective route

from their nest to the food source by the intensity of pheromones

which reduces over time. Other examples of swarm-inspired algorithms

include glowworm swarm optimization (GSO) (Krihnanand and Ghose,

2009), grey wolf optimization (GWO) (Mirjalili et al., 2014), artifi-

cial ecosystem-based optimization (AEO) (Zhao et al., 2019b), shark

smell optimization (SSO) (Abedinia et al., 2014), firefly algorithm (FA)

(Yang, 2010), supply–demand-based optimization (SDO) (Zhao et al.,

2019c), spotted hyena optimization (SHO) (Gaurav and Vijay, 2017),

and so on (Yang and Deb, 2009; Oftadeh et al., 2010; Kiran, 2015;

Mohamed et al., 2017; Cuevas et al., 2013; Askarzadeh, 2014; Saremi

et al., 2017; Akay and Karaboga, 2012a,b; Mirjalili, 2016; Kaveh and

Farhoudi, 2013; Yang, 2010; Mucherino and Seref, 2007; Gandomi and

Alavi, 2012).

It is worth mentioning that there are other recently developed meta-

heuristics motivated from social behaviors and ideology in humans.

Some of the most well-known ones include parliamentary optimization

algorithm (POA) (Borji, 2007), artificial human optimization (AHO)

(Gajawada, 2016), continuous opinion dynamics optimization (CODO)

(Kaur et al., 2013), league championship algorithm (LCA) (Kashan,

2009), social group optimization (SGO) (Satapathy and Naik, 2016),

ideology algorithm (IA) (Huan et al., 2016), and so on (Kuo and Lin,

2013; Moosavian and Roodsari, 2014; Kumar et al., 2018; Xu et al.,

2010; Ray and Liew, 2003).

Swarm-based algorithms have some unique features. Some histor-

ical information about the swarm is restored to provide a basis for

every agent to update individual positions by using interaction rules

over subsequent iterations. In general, swarm-based techniques share

two specific behaviors, exploration and exploitation (Alba and Dor-

ronsoro, 2005; Lynn and Suganthan, 2015). Exploration is to search

a wide variable space for promising solutions which are not neighbor

to the current solution, and this search should be as extensive and

random as possible. This behavior generally contributes to escaping

local optima. Exploitation is to confine the search to a small region

found in the exploration process to refine the solution. Essentially

this behavior implements local search in a promising space. Based on

these two behaviors, swarm-based algorithms have superiority over

other two types of algorithms.

Some might question why new optimization algorithms are still

developed despite of so many existing algorithms. The answer can be

found in the No Free Lunch Theorem of Optimization (Wolpert and

Macready, 1997), which describes that there is no optimizer perform-

ing the best for all optimization problems. So developing an effec-

tive swarm-inspired optimizer to solve specific real-world problems

motivates this work.

This paper proposes a new metaheuristic algorithm, named manta

ray foraging optimization (MRFO), which simulates the foraging be-

haviors of manta rays. This algorithm has three foraging operators,

including chain foraging, cyclone foraging, and somersault foraging.

The performance of MRFO is tested using 31 test functions and 8 engi-

neering problems. The test results discover that the proposed method

significantly outperforms those well-known metaheuristics.

This study is structured as follows. Section 2 detailedly introduces

MRFO algorithm and the concepts behind it. 31 mathematical optimiza-

tion problems and 8 real-world engineering problems are employed

to check the validity of MRFO in Sections 3 and 4, respectively.

Finally, Section 5 gives some conclusions and suggests future research

directions.

2. Manta Ray Foraging Optimization (MRFO)

2.1. Inspiration

Manta rays are fancy creatures although they appear to be terrible.

They are one of the largest known marine creatures. Manta rays have

a flat body from top to bottom and a pair of pectoral fins, with which

they elegantly swim as birds freely fly. They also have a pair of cephalic

lobes that extend in front of their giant, terminal mouths. Fig. 1(A)

provided by Swanson Chan on Unsplash depicts a foraging manta ray,

and Fig. 1(B) shows the structure of a manta ray. 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. Then the prey is filtered from the

water by modified gill rakers (Dewar et al., 2008). Manta rays can be

divided into two distinct species. One is reef manta rays (manta alfredi)

living in Indian Ocean and western and south Pacific, which can reach

5.5 m in width. The other is giant manta rays (manta birostris) found

throughout tropical, subtropical and warm temperate oceans, which

can reach 7 m in width. They have existed for about 5 million years.

The average life span is 20 years but many never reach this age because

they are hunted by fishers (Miller and Klimovich, 2016).

Manta rays eat a large amount of plankton every day. An adult

manta ray can eat 5 kg of plankton on daily basis. Oceans are believed

to be the richest source of plankton. However, plankton is not evenly

dispersed or regularly concentrated in some specific areas, which are

formed with ebb and flow of tides or change of seasons. Interestingly,

manta rays are always good at finding abundant plankton. The most

intriguing thing about manta rays is their foraging behaviors, and they

may travel alone or in groups of up to 50 but foraging is often observed

in groups. These creatures have evolved a variety of fantastic and

intelligent foraging strategies.

The first foraging strategy is chain foraging (Johnna, 2016). When

50 or more manta rays start foraging, they line up, one behind another,

forming an orderly line. Smaller male manta rays are piggybacked upon

female ones and swim on top of their backs to match the beats of

the female’s pectoral fins. 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.

W. Zhao, Z. Zhang and L. Wang Engineering Applications of Artificial Intelligence 87 (2020) 103300

Fig. 1. (A) A foraging manta ray, and (B) structure of a manta ray.

The second foraging strategy is cyclone foraging (Gene and George,

2014). When the concentration of plankton is very high, dozens of

manta rays gather together. 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.

The final foraging strategy is somersault foraging (Rebecca and

Bobbie, 2015). 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. 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, they are

highly effective. We mathematically model these foraging behaviors

and develop a new metaheuristic algorithm named Manta Ray Foraging

Optimization (MRFO) to perform a global optimization.

2.2. Mathematical model

MRFO is inspired by three foraging behaviors including chain for-

aging, cyclone foraging and somersault foraging. The mathematical

models are described below.

2.2.1. 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. Although the best solution is not known,

MRFO assumes the best solution found so far is the plankton with high

concentration manta rays want to approach and eat. Manta rays line

up head-to-tail and form a foraging chain. Individuals except the first

move towards not only the food but also the one in front of it. 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

𝑥

𝑑

𝑖

(𝑡 + 1) =



𝑥

𝑑

𝑖

(𝑡) + 𝑟 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) + 𝛼 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 1

𝑥

𝑑

𝑖

(𝑡) + 𝑟 ⋅ (𝑥

𝑑

𝑖−1

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) + 𝛼 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 2, … , 𝑁

(1)

𝛼 = 2 ⋅ 𝑟 ⋅



𝑙𝑜𝑔(𝑟) (2)

where, 𝑥

𝑑

𝑖

(𝑡) is the position of 𝑖th individual at time t in 𝑑th dimension,

r is a random vector within the range of [0, 1], a is a weight coef-

ficient, 𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) is the plankton with high concentration. Fig. 2 depicts

this foraging behavior in a 2-D space. The position update of the 𝑖th

individual is determined by the position 𝑥

𝑖−1

(t) of the (i-1)th current

individual and the position 𝑥

𝑏𝑒𝑠𝑡

(t) of the food.

2.2.2. Cyclone foraging

When a school of manta rays recognize a patch of plankton in deep

water, they will form a long foraging chain and swim towards the food

by a spiral. This similar spiral foraging strategy can be found in WOA

(Mirjalili and Lewis, 2016). However, for the cyclone foraging strategy

of manta ray swarms, in addition to spirally move towards the food,

each manta ray swims towards the one in front of it. That is, manta ray

swarms in line developing a spiral perform foraging. Fig. 3 illustrates

the cyclone foraging behavior in a 2-D space. 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 in a 2-D space can be defined as



𝑋

𝑖

(𝑡 + 1) = 𝑋

𝑏𝑒𝑠𝑡

+ 𝑟 ⋅ (𝑋

𝑖−1

(𝑡) − 𝑋

𝑖

(𝑡)) + 𝑒

𝑏𝑤

⋅ cos(2𝜋𝑤) ⋅ (𝑋

𝑏𝑒𝑠𝑡

− 𝑋

𝑖

(𝑡))

𝑌

𝑖

(𝑡 + 1) = 𝑌

𝑏𝑒𝑠𝑡

+ 𝑟 ⋅ (𝑌

𝑖−1

(𝑡) − 𝑌

𝑖

(𝑡)) + 𝑒

𝑏𝑤

⋅ sin(2𝜋𝑤) ⋅ (𝑌

𝑏𝑒𝑠𝑡

− 𝑌

𝑖

(𝑡))

(3)

where w is a random number in [0, 1].

This motion behavior may be extended to a n-D space. For sim-

plicity, this mathematical model of cyclone foraging can be defined

𝑥

𝑑

𝑖

(𝑡 + 1) =



𝑥

𝑑

𝑏𝑒𝑠𝑡

+ 𝑟 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) + 𝛽 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 1

𝑥

𝑑

𝑏𝑒𝑠𝑡

+ 𝑟 ⋅ (𝑥

𝑑

𝑖−1

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) + 𝛽 ⋅ (𝑥

𝑑

𝑏𝑒𝑠𝑡

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 2, … , 𝑁

(4)

𝛽 = 2𝑒

𝑟

𝑇 −𝑡+1

𝑇

⋅ sin(2𝜋𝑟

) (5)

where 𝛽 is the weight coefficient, T is the maximum number of itera-

tions, and 𝑟

is the rand number in [0, 1].

All individuals randomly perform the search with respect to the

food as their reference position, so the cyclone foraging has a good

exploitation for the region with the best solution found so far. This

behavior is also used to substantially improve the exploration. We can

force 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 as their reference position. This mechanism focuses mainly on the

exploration and enables MRFO to achieve an extensive global search,

its mathematical equation is presented below

𝑥

𝑑

𝑟𝑎𝑛𝑑

= 𝐿𝑏

𝑑

+ 𝑟 ⋅ (𝑈 𝑏

𝑑

− 𝐿𝑏

𝑑

) (6)

𝑥

𝑑

𝑖

(𝑡 + 1) =



𝑥

𝑑

𝑟𝑎𝑛𝑑

+ 𝑟 ⋅ (𝑥

𝑑

𝑟𝑎𝑛𝑑

− 𝑥

𝑑

𝑖

(𝑡)) + 𝛽 ⋅ (𝑥

𝑑

𝑟𝑎𝑛𝑑

− 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 1

𝑥

𝑑

𝑟𝑎𝑛𝑑

+ 𝑟 ⋅ (𝑥

𝑑

𝑖−1

(𝑡) − 𝑥

𝑑

𝑖

(𝑡)) + 𝛽 ⋅ (𝑥

𝑑

𝑟𝑎𝑛𝑑

− 𝑥

𝑑

𝑖

(𝑡)) 𝑖 = 2, … , 𝑁

(7)

where 𝑥

𝑑

𝑟𝑎𝑛𝑑

is a random position randomly produced in the search

space, 𝐿𝑏

𝑑

and 𝑈𝑏

𝑑

are the lower and upper limits of the 𝑑th dimension,

respectively.

W. Zhao, Z. Zhang and L. Wang Engineering Applications of Artificial Intelligence 87 (2020) 103300

Fig. 2. Chain foraging behavior in a 2-D space.

Fig. 3. Cyclone foraging behavior in a 2-D space.

2.2.3. Somersault foraging

In this behavior, the position of the food is viewed as a pivot. Each

individual tends to swim to and fro around the pivot and somersault to

a new position. Therefore, they always update their positions around

the best position found so far. The mathematical model can be created

as follows

𝑥

𝑑

𝑖

(𝑡 + 1) = 𝑥

𝑑

𝑖

(𝑡) + 𝑆 ⋅ (𝑟

⋅ 𝑥

𝑑

𝑏𝑒𝑠𝑡

− 𝑟

⋅ 𝑥

𝑑

𝑖

(𝑡)), 𝑖 = 1, … , 𝑁 (8)

where S is the somersault factor that decides the somersault range of

manta rays and 𝑆 = 2, 𝑟

and 𝑟

are two random numbers in [0, 1].

As seen from Eq. (8), with definition of the somersault range, it

is possible for each individual to move to any position in a new

search domain located between the current position and its symmetrical

position around the best position found so far. As the distance between

the individual position and the best position found so far reduces, the

perturbation on the current position gets reduced, too. All individuals

approximate gradually to the optimal solution in the search space.

Accordingly, the range of somersault foraging is adaptively reduced

as iterations increase. Fig. 4 shows the sketch of somersault foraging

behavior in MRFO.

Fig. 5 shows that three individuals evolved 100 times in the search

space according to Eq. (8). The sampled points randomly distribute

between the current positions and their symmetrical positions around

𝑥

𝑏𝑒𝑠𝑡

, and the sampled points become sparse as the distance reduces.

The dense points around 𝑥

𝑏𝑒𝑠𝑡

can contribute substantially to exploita-

tion and the sparse ones can contribute positively to exploration.

Similar to other metaheuristic optimizers, MRFO starts by generat-

ing a random population in the domain of problem. At each iteration,

each individual updates its position with respect to both the one in

front of it and the reference position. The value of t/T decreases

from 1/T to 1 to respectively perform exploratory and exploitative

search. The current best solution is chosen as a reference position for

Fig. 4. Somersault foraging behavior in MRFO.

Fig. 5. Somersault foraging behavior of three individuals in a 2-D space.

the exploitation when t/T < rand, while a random position randomly

generated in the search space is chosen as a reference position for the

exploration when t/T > rand. Meanwhile, according to the random

number, MRFO can switch between the chain foraging behavior and the

cyclone foraging behavior. Then individuals update their own positions

with respect to the best position found so far by somersault foraging.

All the update and calculation are interactively preformed until the

stop criterion is met. Eventually, the position and the fitness value of

the best individual are returned. The pseudocode of MRFO algorithm

is given in Fig. 6.

With the description of MRFO algorithm, some remarks are noted.

(a) MRFO is motivated from three foraging strategies of manta rays,

including chain foraging, cyclone foraging, and somersault foraging,

which can effectively improve the optimization ability of MRFO from

different aspects.

(b) The proposed chain foraging strategy makes each individual

update its position with respect to the one in front of it and the current

global best solution.

W. Zhao, Z. Zhang and L. Wang Engineering Applications of Artificial Intelligence 87 (2020) 103300

Fig. 6. Pseudocode of MRFO algorithm.

update its position with respect to both the one in front of it and

the reference position. Whether the best position obtained so far or

a random position produced in the search space is chosen as the

reference position depends on the value of t /T. The former contributes

to exploitation and the latter to exploration.

(d) The gradual increase of the value of t/T encourages MRFO to

smoothly transit from exploratory search to exploitative search.

(e) With the value of rand, MRFO can switch between the chain

foraging and the cyclone foraging.

(f) The somersault foraging allows individuals to adaptively search

in a changing search range.

(g) MRFO is very easy to implement and require few parameters to

be adjusted.

2.3. Time complexity of the MRFO

The time complexity of MRFO depends on the number of variables,

the number of individuals, and the maximum of iterations. Somersault

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