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共 158 条

Advances in Engineering Software 95 (2016) 51–67

Contents lists available at ScienceDirect

Advances in Engineering Software

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

The Whale Optimization Algorithm

Seyedali Mirjalili

∗

, Andrew Lewis

School of Information and Communication Technology, Griﬃth University, Nathan Campus, Brisbane, QLD 4111, Australia

Griﬃth College, Mt Gravatt, Brisbane, QLD 4122, Australia

a r t i c l e i n f o

Article history:

Received 7 August 2015

Revised 8 January 2016

Accepted 15 January 2016

Keywords:

Optimization

Benchmark

Constrained optimization

Particle swarm optimization

Algorithm

Heuristic algorithm

Genetic algorithm

Structural optimization

a b s t r a c t

This paper proposes a novel nature-inspired meta-heuristic optimization algorithm, called Whale Opti-

mization Algorithm (WOA), which mimics the social behavior of humpback whales. The algorithm is in-

spired by the bubble-net hunting strategy. WOA is tested with 29 mathematical optimization problems

and 6 structural design problems. Optimization results prove that the WOA algorithm is very competi-

tive compared to the state-of-art meta-heuristic algorithms as well as conventional methods. The source

codes of the WOA algorithm are publicly available at http://www.alimirjalili.com/WOA.html

1. Introduction

Meta-heuristic optimization algorithms are becoming more and

more popular in engineering applications because they: (i) rely

on rather simple concepts and are easy to implement; (ii) do not

require gradient information; (iii) can bypass local optima; (iv)

can be utilized in a wide range of problems covering different

disciplines.

Nature-inspired meta-heuristic algorithms solve optimization

problems by mimicking biological or physical phenomena. They

can be grouped in three main categories (see Fig. 1 ): evolution-

based, physics-based, and swarm-based methods. Evolution-based

methods are inspired by the laws of natural evolution. The search

process starts with a randomly generated population which is

evolved over subsequent generations. The strength point of these

methods is that the best individuals are always combined together

to form the next generation of individuals. This allows the popu-

lation to be optimized over the course of generations. The most

popular evolution-inspired technique is Genetic Algorithms (GA)

[1] that simulates the Darwinian evolution. Other popular algo-

rithms are Evolution Strategy (ES) [110] , Probability-Based Incre-

mental Learning (PBIL) [ 111] , Genetic Programming (GP) [2] , and

Biogeography-Based Optimizer (BBO) [3] .

∗

Corresponding author. Tel.: + 61 434555738.

E-mail address: seyedali.mirjalili@griﬃthuni.edu.au , ali.mirjalili@gmail.com

(S. Mirjalili).

Physics-based methods imitate the physical rules in the uni-

verse. The most popular algorithms are Simulated Annealing (SA)

[4,5] , Gravitational Local Search (GLSA) [6] , Big-Bang Big-Crunch

(BBBC) [7] , Gravitational Search Algorithm (GSA) [8] , Charged

System Search (CSS) [9] , Central Force Optimization (CFO) [10] ,

Artiﬁcial Chemical Reaction Optimization Algorithm (ACROA)

[11] , Black Hole (BH) [12] algorithm, Ray Optimization (RO) [13]

algorithm, Small-World Optimization Algorithm (SWOA) [14] ,

Galaxy-based Search Algorithm (GbSA) [15] , and Curved Space

Optimization (CSO) [16] .

The third group of nature-inspired methods includes swarm-

based techniques that mimic the social behavior of groups of ani-

mals. The most popular algorithm is Particle Swarm Optimization,

originally developed by Kennedy and Eberhart [17] . PSO is inspired

by the social behavior of bird ﬂocking. It uses a number of par-

ticles (candidate solutions) which ﬂy around in the search space

to ﬁnd the best solution ( i.e. the optimal position). Meanwhile,

they all trace the best location (best solution) in their paths. In

other words, particles consider their own best solutions as well as

the best solution the swarm has obtained so far. Another popular

swarm-based algorithm is Ant Colony Optimization, ﬁrst proposed

by Dorigo et al. [18] . This algorithm is inspired by the social be-

havior of ants in an ant colony. In fact, the social intelligence of

ants in ﬁnding the closest path from the nest and a source of food

is the main inspiration of this algorithm. A pheromone matrix is

evolved over the course of iteration by the candidate solutions.

Other swarm-based techniques are listed in Table 1 . This class

of meta-heuristic methods started to be attractive since PSO was

http://dx.doi.org/10.1016/j.advengsoft.2016.01.008

52 S. Mirjalili, A. Lewis / Advances in Engineering Software 95 (2016) 51–67

Meta-heuristic algorithms

Evolutionary algorithms Physics-based algorithms

Swarm-basssed algorithms

GA ES

GP DE

GSA CSS

CFO BBBC

PSO ACO

ABS CS

Human-based algorithms

TLBO HS

TS GSO

Fig. 1. Classiﬁcation of meta-heuristic algorithms.

Table 1

Swarm-based optimization algorithms developed in literature.

Algorithm Inspiration Year of

proposal

PSO [17] Bird ﬂock 1995

Marriage in Honey Bees Optimization

Algorithm (MBO) [19]

Honey Bees 2001

Artiﬁcial Fish-Swarm Algorithm (AFSA) [20] Fish swarm 2003

Termite Algorithm [21] Termite colony 2005

ACO [18] Ant colony 2006

ABC [22] Honey Bee 2006

Wasp Swarm Algorithm [23] Parasitic wasp 2007

Monkey Search [24] Monkey

2007

Wolf pack search algorithm [25] Wolf herd 2007

Bee Collecting Pollen Algorithm (BCPA) [26] Bees 2008

Cuckoo Search (CS) [27] Cuckoo 2009

Dolphin Partner Optimization (DPO) [28] Dolphin 2009

Bat-inspired Algorithm (BA) [29] Bat herd 2010

Fireﬂy Algorithm (FA) [30] Fireﬂy 2010

Hunting Search (HS) [31] Group of animals

2010

Bird Mating Optimizer (BMO) [32] Bird mating 2012

Krill Herd (KH) [33] Krill herd 2012

Fruit ﬂy Optimization Algorithm (FOA) [34] Fruit ﬂy 2012

Dolphin Echolocation (DE) [35] Dolphin 2013

proven to be very competitive with evolution-based and physical-

based algorithms. Generally speaking, swarm-based algorithms

have some advantages over evolution-based algorithms. For ex-

ample, swarm-based algorithms preserve search space informa-

tion over subsequent iterations while evolution-based algorithms

discard any information as soon as a new population is formed.

They usually include less operators compared to evolutionary ap-

proaches (selection, crossover, mutation, elitism, etc. ) and hence

are easier to implement.

It is worth mentioning here that there are also other meta-

heuristic methods inspired by human behaviors in the litera-

ture. Some of the most popular algorithms are Teaching Learning

Based Optimization(TLBO) [36,37] , Harmony Search (HS) [38] , Tabu

(Taboo) Search (TS) [39–41] , Group Search Optimizer (GSO) [42,43] ,

Imperialist Competitive Algorithm (ICA) [44] , League Championship

Algorithm (LCA) [45,46] , Firework Algorithm [47] , Colliding Bod-

ies Optimization (CBO) [4 8,4 9] , Interior Search Algorithm (ISA)

[50] , Mine Blast Algorithm (MBA) [51] , Soccer League Competi-

tion (SLC) algorithm [52,53] , Seeker Optimization Algorithm (SOA)

[54] , Social-Based Algorithm (SBA) [55] , Exchange Market Algo-

rithm (EMA) [56] , and Group Counseling Optimization (GCO) al-

gorithm [57,58] .

Population-based meta-heuristic optimization algorithms share

a common feature regardless of their nature. The search process is

divided into two phases: exploration and exploitation [59–61] . The

optimizer must include operators to globally explore the search

space: in this phase, movements ( i.e. perturbation of design vari-

ables) should be randomized as most as possible. The exploitation

phase follows the exploration phase and can be deﬁned as the pro-

cess of investigating in detail the promising area(s) of the search

space. Exploitation hence pertains to the local search capability

in the promising regions of design space found in the exploration

phase. Finding a proper balance between exploration and exploita-

tion is the most challenging task in the development of any meta-

heuristic algorithm due to the stochastic nature of the optimization

process.

This study describes a new meta-heuristic optimization algo-

rithm (namely, Whale Optimization Algorithm, WOA) mimicking

the hunting behavior of humpback whales. To the knowledge of

the present authors, there is no previous study on this subject

in the optimization literature. The main difference between the

current work and the recently published papers by the authors

(particularly GWO [62] ) is the simulated hunting behavior with

random or the best search agent to chase the prey and the use of

a spiral to simulate bubble-net attacking mechanism of humpback

whales. The eﬃciency of the WOA algorithm developed in this

research is evaluated by solving 29 mathematical optimization

problems and six structural optimization problems. Optimization

results demonstrate that WOA is very competitive compared to

the state-of-the-art optimization methods.

The rest of the paper is structured as follows. Section 2 de-

scribes the Whale Optimization Algorithm developed in this study.

Test problems and optimization results are presented and dis-

cussed in Sections 3 and 4 , respectively, for mathematical func-

tions and structural optimization problems. Section 5 summarizes

the main ﬁndings of this study and suggests directions for future

research.

S. Mirjalili, A. Lewis / Advances in Engineering Software 95 (2016) 51–67 53

Fig. 2. Bubble-net feeding behavior of humpback whales.

2. Whale Optimization Algorithm (WOA)

In this section the inspiration of the proposed method is ﬁrst

discussed. Then, the mathematical model is provided.

2.1. Inspiration

Whales are fancy creatures. They are considered as the biggest

mammals in the world. An adult whale can grow up to 30 m

long and 180 t weight. There are 7 different main species of this

giant mammal such killer, Minke, Sei, humpback, right, ﬁnback,

and blue. Whales are mostly considered as predators. They never

sleep because they have to breathe from the surface of oceans. In

fact, half of the brain only sleeps. The interesting thing about the

whales is that they are considered as highly intelligent animals

with emotion.

According to Hof and Van Der Gucht [63] , whales have com-

mon cells in certain areas of their brains similar to those of hu-

man called spindle cells. These cells are responsible for judgment,

emotions, and social behaviors in humans. In other words the spin-

dle cells make us distinct from other creatures. Whales have twice

number of these cells than an adult human which is the main

cause of their smartness. It has been proven that whale can think,

learn, judge, communicate, and become even emotional as a hu-

man does, but obviously with a much lower level of smartness. It

has been observed that whales (mostly killer whales) are able to

develop their own dialect as well.

Another interesting point is the social behavior of whales. They

live alone or in groups. However, they are mostly observed in

groups. Some of their species (killer whales for instance) can

live in a family over their entire life period. One of the biggest

baleen whales is humpback whales ( Megaptera novaeangliae ). An

adult humpback whale is almost as size of a school bus. Their

favorite prey are krill and small ﬁsh herds. Fig. 2 shows this

mammal.

The most interesting thing about the humpback whales is their

special hunting method. This foraging behavior is called bubble-net

feeding method [64] . Humpback whales prefer to hunt school of

krill or small ﬁshes close to the surface. It has been observed that

this foraging is done by creating distinctive bubbles along a circle

or ‘9’-shaped path as shown in Fig. 2 . Before 2011, this behavior

was only investigated based on the observation from surface. How-

ever, Goldbogen et al. [65] investigated this behavior utilizing tag

sensors. They captured 300 tag-derived bubble-net feeding events

of 9 individual humpback whales. They found two maneuvers asso-

ciated with bubble and named them ‘upward-spirals’ and ‘double-

loops’. In the former maneuver, humpback whales dive around

12 m down and then start to create bubble in a spiral shape around

the prey and swim up toward the surface. The later maneuver in-

cludes three different stages: coral loop, lobtail, and capture loop.

Detailed information about these behaviors can be found in [65] .

It is worth mentioning here that bubble-net feeding is a unique

behavior that can only be observed in humpback whales. In this

work the spiral bubble-net feeding maneuver is mathematically

modeled in order to perform optimization.

2.2. Mathematical model and optimization algorithm

In this section the mathematical model of encircling prey, spiral

bubble-net feeding maneuver, and search for prey is ﬁrst provided.

The WOA algorithm is then proposed.

2.2.1. Encircling prey

Humpback whales can recognize the location of prey and en-

circle them. Since the position of the optimal design in the search

space is not known a priori, the WOA algorithm assumes that the

current best candidate solution is the target prey or is close to

the optimum. After the best search agent is deﬁned, the other

search agents will hence try to update their positions towards the

best search agent. This behavior is represented by the following

equations:



D = |



C .

−→

∗

(t) −



X (t ) | (2.1)



(

t + 1

)

−→

∗

(

)

−



A ·



D (2.2)

where t indicates the current iteration,



A and



C are coeﬃcient vec-

tors, X

∗

is the position vector of the best solution obtained so

far,



X is the position vector, | | is the absolute value, and · is

an element-by-element multiplication. It is worth mentioning here

that X

∗

should be updated in each iteration if there is a better so-

lution.

The vectors



A and



C are calculated as follows:



A = 2



a ·



r −



a (2.3)



C = 2 ·



r (2.4)

where



a is linearly decreased from 2 to 0 over the course of it-

erations (in both exploration and exploitation phases) and



r is a

random vector in [0,1].

Fig. 3 (a) illustrates the rationale behind Eq. (2.2) for a 2D prob-

lem. The position ( X , Y ) of a search agent can be updated according

to the position of the current best record ( X

∗

, Y

∗

). Different places

around the best agent can be achieved with respect to the current

position by adjusting the value of



A and



C vectors. The possible

updating position of a search agent in 3D space is also depicted

in Fig. 3 (b). It should be noted that by deﬁning the random vector

(



r ) it is possible to reach any position in the search space located

between the key-points shown in Fig. 3 . Therefore, Eq. (2.2) allows

any search agent to update its position in the neighborhood of the

current best solution and simulates encircling the prey.

The same concept can be extended to a search space with

n dimensions, and the search agents will move in hyper-cubes

around the best solution obtained so far. As mentioned in the pre-

vious section, the humpback whales also attack the prey with the

bubble-net strategy. This method is mathematically formulated as

follows:

2.2.2. Bubble-net attacking method (exploitation phase)

In order to mathematically model the bubble-net behavior of

humpback whales, two approaches are designed as follows:

1 Shrinking encircling mechanism: This behavior is achieved by de-

creasing the value of



a in the Eq. (2.3) . Note that the ﬂuctua-

tion range of



A is also decreased by



a . In other words



A is a

54 S. Mirjalili, A. Lewis / Advances in Engineering Software 95 (2016) 51–67

X*-X

-Y

(X,Y)

(X*,Y*)

(X*,Y)

(X,Y*)

(X,Y*-Y)

(X*-X,Y)

(X*,Y*-Y)

(X*-X,Y*-Y)

(X*-X,Y*)

(X,Y,Z)

(X*,Y*,Z*)

(X,Y*-Y,Z*-Z)

(X*-X,Y,Z*-Z)

(X*,Y*-Y,Z*-Z)(X*-X,Y*-Y,Z-Z*)

(X*-X,Y*,Z*-Z)

(X,Y*,Z)

(X,Y*-Y,Z)

(X,Y*,Z*)

(X,Y,Z*)

(X*,Y*,Z*-Z) (X,Y*,Z*-Z)

(X*,Y,Z*-Z) (X,Y,Z*-Z)

(X*-X,Y,Z*)

(X*-X,Y,Z) (X*,Y,Z)

(X,Y,Z*)

(X*,Y,Z*)

Fig. 3. 2D and 3D position vectors and their possible next locations ( X

∗

is the best solution obtained so far).

Fig. 4. Bubble-net search mechanism implemented in WOA ( X

∗

is the best solution obtained so far): (a) shrinking encircling mechanism and (b) spiral updating position.

random value in the interval [ −a , a ] where a is decreased from

2 to 0 over the course of iterations. Setting random values for



A in [ −1,1], the new position of a search agent can be deﬁned

anywhere in between the original position of the agent and the

position of the current best agent. Fig. 4 (a) shows the possi-

ble positions from ( X , Y ) towards ( X

∗

, Y

∗

) that can be achieved

by 0 ≤A ≤1 in a 2D space.

2 Spiral updating position: As can be seen in Fig. 4 (b), this ap-

proach ﬁrst calculates the distance between the whale located

at ( X , Y ) and prey located at ( X

∗

, Y

∗

). A spiral equation is then

created between the position of whale and prey to mimic the

helix-shaped movement of humpback whales as follows:



(

t + 1

)

−→



· e

· cos

(

2 π l

)

−→

∗

(

)

(2.5)

where

−→



= |

−→

∗

(t) −



X (t ) | and indicates the distance of the i th

whale to the prey (best solution obtained so far), b is a constant

for deﬁning the shape of the logarithmic spiral, l is a random

number in [ −1,1], and . is an element-by-element multiplica-

tion.

Note that humpback whales swim around the prey within a

shrinking circle and along a spiral-shaped path simultaneously. To

model this simultaneous behavior, we assume that there is a prob-

ability of 50% to choose between either the shrinking encircling

mechanism or the spiral model to update the position of whales

during optimization. The mathematical model is as follows:



X ( t + 1 ) =



−→

∗

(

)

−



A ·



D if p < 0 . 5

−→



· e

· cos

(

2 π l

)

−→

∗

(

)

if p ≥ 0 . 5

(2.6)

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

In addition to the bubble-net method, the humpback whales

search for prey randomly. The mathematical model of the search

is as follows.

2.2.3. Search for prey (exploration phase)

The same approach based on the variation of the



A vector

can be utilized to search for prey (exploration). In fact, humpback

whales search randomly according to the position of each other.

Therefore, we use



A with the random values greater than 1 or less

than −1 to force search agent to move far away from a reference

whale. In contrast to the exploitation phase, we update the posi-

tion of a search agent in the exploration phase according to a ran-

domly chosen search agent instead of the best search agent found

so far. This mechanism and |



A | > 1 emphasize exploration and al-

low the WOA algorithm to perform a global search. The mathemat-

ical model is as follows:



D = |



C ·

−−→

rand

−



X | (2.7)



(

t + 1

)

−−→

rand

−



A .



D (2.8)

S. Mirjalili, A. Lewis / Advances in Engineering Software 95 (2016) 51–67 55

(X,Y)

(X*,Y*)

(X*,Y)

(X,Y*)

(X,Y*-AY)

(X*-AX,Y)

(X*,Y*-AY)

(X*-AX,Y*-AY)

(X*-AX,Y*)

A=2

A=1.8

A=1.5

A=1.4

A=1

Fig. 5. Exploration mechanism implemented in WOA ( X

∗

is a randomly chosen

search agent).

Initialize the whales population X

(i = 1, 2, ..., n)

Calculate the fitness of each search agent

X*=the best search agent

while (t < maximum number of iterations)

for each search agent

Update a, A, C, l, and p

if1 (p<0.5)

if2 (|A| < 1)

Update the position of the current search agent by the Eq. (2.1)

else if2 (|A|

Select a random search agent ( )

Update the position of the current search agent by the Eq. (2.8)

end if2

else if1 (p

0.5)

Update the position of the current search by the Eq. (2.5)

end if1

end for

Check if any search agent goes beyond the search space and amend it

Calculate the fitness of each search agent

Update X* if there is a better solution

t=t+1

end while

return X*

Fig. 6. Pseudo-code of the WOA algorithm.

where

−−→

rand

is a random position vector (a random whale) chosen

from the current population.

Some of the possible positions around a particular solution with



A > 1 are depicted in Fig. 5.

The WOA algorithm starts with a set of random solutions. At

each iteration, search agents update their positions with respect

to either a randomly chosen search agent or the best solution ob-

tained so far. The a parameter is decreased from 2 to 0 in order to

provide exploration and exploitation, respectively. A random search

agent is chosen when |



A | > 1, while the best solution is selected

when |



A | < 1 for updating the position of the search agents. De-

pending on the value of p , WOA is able to switch between either a

spiral or circular movement. Finally, the WOA algorithm is termi-

nated by the satisfaction of a termination criterion.

The pseudo code of the WOA algorithm is presented in Fig. 6.

From theoretical stand point, WOA can be considered a global

optimizer because it includes exploration/exploitation ability. Fur-

thermore, the proposed hyper-cube mechanism deﬁnes a search

space in the neighborhood of the best solution and allows other

search agents to exploit the current best record inside that domain.

Adaptive variation of the search vector A allows the WOA algo-

Table 2

Description of unimodal benchmark functions.

Function V_no Range f

min

(x ) =



i =1

30 [ −100,100] 0

(x ) =



i =1

| x

| +



i =1

| x

| 30 [ −10,10] 0

(x ) =



i =1

(



j−1

)

30 [ −100,100] 0

(x ) = max

{ | x

| , 1 ≤ i ≤ n } 30 [ −100,100] 0

(x ) =



n −1

i =1

[ 100

( x

i +1

− x

)

( x

− 1

)

] 30 [ −30,30] 0

(x ) =



i =1

( [ x

+ 0 . 5 ]

)

30 [ −100,100] 0

(x ) =



i =1

+ random [ 0 , 1

) 30 [ −1.28,1.28] 0

rithm to smoothly transit between exploration and exploitation: by

decreasing A , some iterations are devoted to exploration (| A | ≥1)

and the the rest is dedicated to exploitation (| A | < 1). Remarkably,

WOA includes only two main internal parameters to be adjusted

( A and C ).

Although mutation and other evolutionary operations might

have been included in the WOA formulation to fully reproduce

the behavior of humpback whales, we decided to minimize the

amount of heuristics and the number of internal parameters thus

implementing a very basic version of the WOA algorithm. How-

ever, hybridization with evolutionary search schemes may be the

subject of future studies

3. Results and discussion

The numerical eﬃciency of the WOA algorithm developed in

this study was tested by solving 29 mathematical optimization

problems. The ﬁrst 23 problems are classical benchmark functions

utilized in the optimization literature [66–69] . WOA was com-

pared with state-of-the-art swarm-based optimization algorithms.

Tables 2–4 summarize the test problems reporting the cost func-

tion, range of variation of optimization variables and the optimal

value f

min

quoted in literature. The other 6 test problems consid-

ered in this study (see Table 5 ) regard composite benchmark func-

tions considered in the CEC 2005 special session (see Ref. [70] for

the detailed description of the composite functions). These bench-

mark functions are shifted, rotated, expanded, and combined vari-

ants of the most complicated mathematical optimization problems

presented in literature [71] . Note that V_no indicates the number

of design variables in Tables 2 –5 . For all the algorithms, a popula-

tion size and maximum iteration equal to 30 and 500 have been

utilized.

Generally speaking, benchmark functions can be divided into

four groups: unimodal, multimodal, ﬁxed-dimension multimodal,

and composite functions. Fig. 7 shows the typical 2D plots of

the cost function for some test cases considered in this study. It

should be noted that the difference between ﬁxed-dimensional

multi-modal functions in Table 4 and multi-modal functions in

Table 3 is the ability of deﬁning the desired number of design

variables. The mathematical formulations of ﬁxed-dimensional

test functions do not allow us to tune the number of design

variables, but they provide different search space compared to

multi-modal test functions in Table 3 . It is also worth mentioning

here that the composite test functions provide challenging test

functions by shifting the global optimum to random positions

before each run, locating the global optimum on the boundaries

of the search spaces occasionally, and rotating the functions

using F (x ) = f ( R ∗ x ) formula where R is an orthogonal rotation

matrices calculated using Salmon’s method [72] . As Table 5 shows,

composite test functions also combine different search spaces of

n other primitive test functions by stretching and manipulating

their coverage range. It should be noted that σ deﬁnes the cov-

erage range of the primitive test functions and λ indicates the

stretch/compression level of each of the primitive test functions.

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2021-06-02

骗子，根本不是解决TSP问题的代码，只是WOA算法源码加TSP问题的数据集，没有解决的源码，不要下载。

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