白骨顶鸡优化算法COOT_optimization

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Expert Systems With Applications 183 (2021) 115352

Available online 9 June 2021

A new optimization method based on COOT bird natural life model

Iraj Naruei

, Farshid Keynia

Department Engineering, Kerman Branch, Islamic Azad University, Kerman, Iran

Department of Energy Management and Optimization, Institute of Science and High Technology and Environmental Sciences, Graduate University of Advanced

Technology, Kerman, Iran

ARTICLE INFO

Keywords:

Optimization techniques

Meta-heuristic algorithm

Coot Birds

Coot optimization

ABSTRACT

Recently, many intelligent algorithms have been proposed to nd the best solution for complex engineering

problems. These algorithms can search volatile and multi-dimensional solution spaces and nd optimal answers

timely. In this paper, a new meta-heuristic method is proposed that inspires the behavior of the swarm of birds

called Coot. The Coot algorithm imitates two different modes of movement of birds on the water surface: in the

rst phase, the movement of birds is irregular, and in the second phase, the movements are regular. The swarm

moves towards a group of leading leaders to reach a food supply; the movement of the end of the swarm is in the

form of a chain of coots, each of coot which moves behind its front coots. The algorithm then runs on a number of

test functions, and the results are compared with well-known optimization algorithms. In addition, to solve

several real problems, such as Tension/Compression spring, Pressure vessel design, Welded Beam Design, Multi-

plate disc clutch brake, Step-cone pulley problem, Cantilever beam design, reducer design problem, and Rolling

element bearing problem this algorithm is used to conrm the applicability of this algorithm. The results show

that this algorithm is capable to outperform most of the other optimization methods. The source code is currently

available for public from: https://www.mathworks.com/matlabcentral/leexchange/89102-coot-optimization-

algorithm.

1. Introduction

Optimization is the process of nding the best answer or the global

optimal point for a problem. In optimizing the problems, the optimal

global point is the minimum or maximum value of a function. Optimi-

zation issues can be found in all elds of study, which makes optimi-

zation techniques an essential and important direction of study for

researchers. The meta-heuristic algorithms are a kind of random algo-

rithms which used to nd an optimal response. Optimization methods

and algorithms are categorized into two groups of exact algorithms and

approximate algorithms. Exact algorithms are capable of nding the

optimal answer in a precise manner, but they are not efcient enough for

strict optimization problems and their execution time expands expo-

nentially with the dimensions of the problems. Approximate algorithms

are capable of nding good (near optimal) solutions at a short time for

strict optimization problems. Approximate algorithms are divided into

three categories: heuristic algorithms, meta-heuristic and hyper heuristic

algorithms. The two main problems are the innovative algorithms,

catching them at the local optimum points and early convergence into

these points. Meta-heuristic algorithms to solve the decit heuristic

algorithms have been proposed (Spall, 2005). In fact, meta-heuristic

algorithms are one of a kind of approximate optimization algorithms

that have solutions for Escape from Local Optimum points and can be

applied to a wide range of problems. Various categories of these algo-

rithms have been developed in recent decades (Mahdavi et al., 2015).

The capabilities of the meta-heuristic processes can be a simple, exible,

non-inference mechanism and avoid local optimums. Meta-heuristic

processes are inspired by physical phenomena, animal behavior,

evolutionary concepts, and human phenomena. A category of known

algorithms is presented in Fig. 1. Many articles have tried to classify

optimization algorithms based on their inspiration (Ertenlice & Kalayci,

2018; Hussain et al., 2018; Sotoudeh-Anvari & Hafezalkotob, 2018).

An important question arises here, when there are famous algorithms

such as those mentioned above, what is needed to offer and present new

algorithms. According to the NFL theorem (Blum & Roli, 2003; Wolpert

& Macready, 1997), there is no optimization algorithm that can solve all

optimization problems. In fact, the average performance of the opti-

mizers is almost the same. So there are a lot of problems that are not still

well solved notwithstanding the popular optimization algorithms, and

offering new algorithms can solve such problems. This is the motivation

* Corresponding author.

E-mail addresses: irajnaruei@iauk.ac.ir (I. Naruei), f.keynia@kgut.ac.ir (F. Keynia).

Contents lists available at ScienceDirect

Expert Systems With Applications

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

https://doi.org/10.1016/j.eswa.2021.115352

Received 19 February 2020; Received in revised form 10 April 2021; Accepted 3 June 2021

Expert Systems With Applications 183 (2021) 115352

of the creation of a new bird collective-behavioral based optimization

algorithm called Coot to compete with current algorithms in solving

problems. In this paper, a new meta-heuristic algorithm is introduced

that imitates the movements of the COOT on the water.

The rest of the paper is organized as follows: Section 2 presents a

literature review of previous meta-heuristic algorithms. Section 3 de-

scribes the proposed COOT algorithm. The results and discussion of the

proposed algorithm, benchmark functions, and real problems are pre-

sented in Sections 4–5. Finally, Section 6 concludes this article and offers

suggestions for future work.

2. Related works

In recent years, optimization has become a popular research eld

and an economical way to nd an optimal solution to complex problems.

According to Fig. 1, we have divided the optimization algorithms into

four categories based on the type of inspiration: Swarm Based Algo-

rithms, which include Particle Swarm Optimization (PSO) (Eberhart &

Kennedy, 2002), Firey Algorithm (FFA) (X.-S. Yang, 2010), Ant Colony

Optimization (ACO) (Colorni et al., 1991), Articial Bee Colony (ABC)

(Basturk & Karaboga, 2006), Krill Herd (KH) (Gandomi & Alavi, 2012),

multi-leader whale optimization algorithm (MLWOA) (Abd Elaziz et al.,

2021), and Enhanced Crow Search Algorithm (ECSA) (Ouadfel & Abd

Elaziz, 2020). The second group is evolutionary algorithms such as

Genetic Algorithm (GA) (Holland, 1967), Evolution Strategy (ES)

(Rechenberg, 1973), Genetic Programming (GP) (Angeline, 1994),

Biogeography Based Optimizer (BBO) (Simon, 2008), Evolutionary

Programming (EP) (Yao et al., 1999), Differential Evolution (DE) (Storn

& Price, 1995), Virulence Optimization Algorithm (VOA) (Jaderyan &

Khotanlou, 2016). The third category is physics-based algorithms such

as Simulated Annealing (SA) (Kirkpatrick, 2014), Black Hole Algorithm

(BH) (Hatamlou, 2013), Curved Space Optimization Algorithm (CSO)

(Moghaddam et al., 2012), and Ray Optimization (RO) (Kaveh &

Khayatazad, 2012). The fourth category is human-based algorithms such

as Harmony Search (HS) (Manjarres et al., 2013), Teaching Learning

Based Optimization (TLBO) (Rao et al., 2011), Imperialist Competitive

Algorithm (ICA) (Atashpaz-Gargari & Lucas, 2007), Exchanged Market

Algorithm (EMA) (Ghorbani & Babaei, 2014), Thermal Exchange Opti-

mization (TEO) (Kaveh & Dadras, 2017) and Tabu Search (TS) (Glover,

1989). There are other algorithms that are a combination of algorithms

in these four categories, such as Cooperative Meta-heuristic Algorithms

(Elaziz et al., 2021), volleyball premier league algorithm based on sine

cosine Algorithm (VPLSCA) (Moghdani et al., 2020), and competitive

chain-based Harris Hawks Optimizer (Elaziz et al., 2020). Some of the

most famous are described below. Genetic Algorithm (GA) is the most

popular evolutionary inspirational technique that mimics the principles

of Charles Darwin’s Compatibility Survival Theory. This method in-

volves the selection process, the crossover, and the mutation process to

replace the worst solution in each generation. In this algorithm, solu-

tions are improved according to the best solutions obtained by each

particle so far and the best found solution of the overall swarm. The ACO

algorithm imitates the collective behavior of ants in nding the shortest

path from the nest to the food source. One of the most important and

most interesting behavior of ants is their behavior in nding food, and in

particular how to nd the shortest path between food and nest. This kind

of behavior of the ants has a kind of swarm intelligence that has recently

attracted the attention of scientists. In the natural world, ants of some

species (initially) wander randomly, and upon nding food return to

their colony while laying down pheromone trails. If other ants nd such

a path, they are likely not to keep travelling at random, but instead to

follow the trail, returning and reinforcing it if they eventually nd food.

The algorithm (DE) is presented to overcome the main defect of the

Fig. 1. Categorization of meta-heuristic algorithms.

I. Naruei and F. Keynia

Expert Systems With Applications 183 (2021) 115352

genetic algorithm, namely the lack of local search in this algorithm. The

main difference between genetic algorithms and algorithms (DE) is in

the selection operators. In the ABC model, the colony consists of three

groups of bees: employed bees, onlookers and scouts. It is assumed that

there is only one articial employed bee for each food source. In other

words, the number of employed bees in the colony is equal to the

number of food sources around the hive. Employed bees go to their food

source and come back to hive and dance on this area. The employed bee

whose food source has been abandoned becomes a scout and starts to

search for nding a new food source. Onlookers watch the dances of

employed bees and choose food sources depending on dances. The rey

algorithm (FA) is a meta-heuristic algorithm, inspired by the ashing

behavior of reies. The primary purpose of a rey’s ash is to act as a

signal system to attract other reies. The Firey Algorithm is based on

the life of the rey worms. In the rey algorithm, worms tend to be

attracted to high-attraction light. The worm with less light can be

absorbed into the worm with more light. In this situation, the swarm

moves like a particle swarm algorithm (PSO). Of course, the degree of

motion is also taken randomly to prevent the early convergence of the

rey algorithm. Bat algorithm is a heuristic algorithm proposed by

Yang in 2010 and has been inspired by a property, named as echolo-

cation, which guides the bats’ movements during their ight and

hunting even in complete darkness. The Salp swarm algorithm is

inspired by the movement of a chain of marine species called Salp that

has a clear body and its movement is like jellysh, and water is pumped

through the body and causes the salp to move. Salps form a chain to

achieve fast and harmonious movement to nd food. GSA is a nature-

inspired algorithm based on Newton’s famous law of gravity and the

law of motion. In this algorithm, search agents are a set of objects that

can be thought of as planets of a system. The optimum region, like a

black hole, sucks the planets toward you. Information about the tness

of each object is stored in the form of gravitational masses and inertia.

The exchange of information and the impact of objects on each other are

under the force of gravitation.

Optimization algorithms have advantages and disadvantages.For

example, the pso algorithm solves problems with small and simple di-

mensions well, but has poor performance in high-dimensional, complex,

and hybrid problems, and suffers from premature convergence. Success-

History based Adaptive Differential Evolution (SHADE) algorithm, an

improved version of the DE algorithm, solves hybrid and complex

problems well and does not suffer from premature convergence, but it

does not solve high-dimensional problems well. Genetic algorithms and

gravitational search have the same disadvantages. In this article, we

tried to solve these disadvantages.

3. Coot optimization algorithms

3.1. Description

the Coots are small water birds that are members of the rail family,

Rallidae. They constitute the genus Fulica, the name being the Latin for

“coot” (Paillisson & Marion, 2001). Coots have prominent frontal shields

or other decoration on the forehead, with red to dark red eyes and

colored bills. Many, but not all, have white on the under the tail. The

literature on American coot behavior includes extensive work on

breeding, habitat and migratory behavior (Randler, 2005; Varo & Amat,

2008; Zhang et al., 2011). There are many behaviors and movements of

coots on the water. In this article, we used the behavior of this bird on

the water surface to provide a new optimization method. Coots travel at

angles to their direction of motion and appear to come well within what

is, for surf scoters, a zone of repulsion. The behavior of coot’s swarm on

water consists of Three movements. A disordered movement of activity,

as shown in Fig. 2a, and a synchronized movement, shown in Fig. 2b.

The coots also make a chain move on the surface of the water, and each

coot moves behind its front coot, as shown in Fig. 3 (Trenchard, 2012).

Coots have different collective behaviors, our goal in this paper is to

simulate collective movements (regular and irregular movements on the

surface of the water. The whole group is directed toward the target

(food)by a few coots in front of the group, which we consider group

leaders.

Therefore, we consider the four different moves of coots on the water

surface, which are as follows.

1- Random movement to this side and that side

2- Chain movement

3- Adjusting the position based on the group leaders

4- Leading the group by the leaders towards the optimal area (leader

movement)

3.2. Mathematical model and algorithm

the basic framework of all optimization algorithms is the same. The

algorithm starts with ( x

→

) = { x

→

, x

→

, ... x

→

} an initial random popula-

tion. This random population is repeatedly evaluated by the target

Fig. 2. (a) Disordered phase, (b) Synchronized phase, (Trenchard, 2012).

Fig. 3. Moving the chain of coots on the water (Trenchard, 2012).

I. Naruei and F. Keynia

Expert Systems With Applications 183 (2021) 115352

function and a target value is determined (O

→

) = {O

,...O

}. It is also

improved by a set of rules that are the core of an optimization technique.

Because population-based optimization techniques look for the optimal

amount of optimization problems, there is no guarantee of nding a

solution in one run. However, with sufcient numbers of random solu-

tions and optimization steps (iteration), the probability of nding the

global optimal increases. The population is randomly generated in the

sightly space using the formula (1):

CootPos(i) = rand(1, d).*(ub − lb)+lb (1)

Where CootPos(i) is the coot position, d the number of variables or

problem dimensions, lb is the lower bound of the search space and ub is

the upper bound of the search space that is dened as the formula (2).

Each variable may have a different lower bound and upper bound

problem.

lb = [lb

, lb

, ..., lb

], ub = [ub

, ub

, ..., ub

] (2)

After generating the initial population and determining the position

of each agent, the tness of each solution must be calculated using O

f( x

→

) the objective function. We select the NL number of coots as group

leaders. The choice of leaders is random.

Now, the four movements of coots on the surface of the water, which

were mentioned in the previous section, are implemented.

3.2.1. Random movement to this side and that side

To implement this movement, we consider a random position ac-

cording to formula (3) in the search space and move the coot towards

this random position.

Q = rand(1, d).*(ub − lb)+lb (3)

This coot movement explores different parts of the search space. If

the algorithm gets stuck in the local optimal, this movement will cause

the algorithm to escape from the local optimal. coot’s new position is

calculated according to formula (4).

CootPos(i) = CootPos(i)+A ×R2 ×(Q − CootPos(i)) (4)

Where R2 is a random number in the interval [0, 1], A is calculated

according to formula (5). Fig. 4 depicts the random movement of the

coot in different directions.

A = 1 − L × (

Iter

) (5)

Where L is current iteration, Iter is maximum iteration.

3.2.2. Chain movement

The average position of two coots can be used to implement chain

movement (Mirjalili et al., 2017). Another way to implement a chain

movement is that we rst calculate the distance vector between the two

coots and then move the coot toward the other coot about half the dis-

tance vector. We used the rst method and the new position of the coot is

calculated according to formula (6).

CootPos(i) = 0.5 × (CootPos(i − 1)+CootPos(i))(6)

Where CootPos(i-1) is second coot. Fig. 5 shows the chain movement of

the coots.

3.2.3. Adjusting the position based on the group leaders

Usually the group is led by a few coots in front of the group and the

rest of the coots have to adjust their position based on the group’s

leaders and move toward them. A question that might be asked is that

each coot will adjust its position based on which leader. The average

position of the leaders can be considered, and the coots can update their

position based on this average position. Considering the average posi-

tion causes premature convergence. To implement this movement, we

use a mechanism according to formula (7) to select the leader.

K = 1 +(i MOD NL) (7)

Where i is the index number of the current coot, NL is the number of

leaders, and K is the leader’s index number. Fig. 6 shows the mechanism

for choosing a leader by coot.

the coot(i) must update its position based on the leader’s k. Formula

(8) calculates the next position of the coot based on the selected leader.

CootPos(i) = LeaderPos(k)+2 × R1 ×cos(2R

)

× (LeaderPos(k)− CootPos(i)) (8)

Where CootPos(i) is the current position of coot, LeaderPos(k) is selected

leader position, R1 is a random number in the interval [0, 1],

is the

same pi value as 3.14, and R is a random number in the interval [− 1, 1].

3.2.4. Leading the group by the leaders towards the optimal area (leader

movement)

The group must be directed towards a goal (optimal area), so leaders

Fig. 4. Random movement to this side and that side. Fig. 5. Chain movement of the coots.

I. Naruei and F. Keynia

Expert Systems With Applications 183 (2021) 115352

need to update their position toward the goal. Formula 9 is recom-

mended to update the position of leaders. This formula looks for better

positions around this current optimal point. Sometimes leaders have to

move away from the current optimal position to nd better positions.

This formula provides a good way of getting closer to optimal location

and getting away from it.

LeaderPos(i) =











B ×R3 ×cos(2R

)× R4 < 0.5

(gBest − LeaderPos(i)) + gBest

B ×R3 ×cos(2R

)× R4 >= 0.5

(gBest − LeaderPos(i)) − gBest











(9)

Where gBest is the best position ever found, R3 and R4 are random

number in the interval [0, 1],

is the same pi value as 3.14, R is a

random number in the interval [− 1, 1], and B is calculated according to

formula (10).

Fig. 6. mechanism for choosing a leader by coot.

Fig. 7. update of the leaders position relative to the best position.

I. Naruei and F. Keynia

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