2020 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP 2020)
Honolulu, Hawaii, USA, February 28 - March 2, 2020

Whale Optimization Algorithm with Increased Number of Agents for Global Search
Hidehiro Sugioka, Ken Kamiyotsumoto, Yoko Uwate, and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {sugioka, kamiyotsumoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

2. Whale optimization algorithm

In this study, we focuse on global search of Whale Optimization Algorithm (WOA). In the conventional WOA, the number of agents for global search is reduced by the probability determined by the number of iterations. In the proposed
method, there are always certain number of agents for global
search until number of iteration exceeds half of the maximum
number of iterations. The proposed method and the conventional WOA are compared with searching for optimal valueof the benchmark functions. Their minimum values, averagevalues and standard deviations are compared. The proposed
method shows better results than the conventional WOA.

Whale Optimization Algorithm (WOA) is proposed by
Mirjalili and Lewis in 2016. WOA is nature-inspired metaheuristic optimization algorithm which simulate the feeding
behavior of humpback whales. The feeding behavior is performed by bubble net method. Whales take three types ofactions: Encircling prey, Bubble-net attacking method andSearch for prey in WOA.

1. Introduction
Swarm intelligence (SI) is an artificial intelligence thatsimulates the behavior of creatures that form swarms. Examples of animals simulated are ants, bees, fishes, etc. Thefeature of SI is each individual can only take simple movements, and the whole group shows complicated movements.
SI is important because simple controls are more efficientthan complex controls. SI is used for driving automation andfor analysis of human movement during evacuation and traffic
jams.
Recently, optimization problems have attracted attention.
Optimization derive the best result under the condition or deriving the method. Examples of optimization problems aretraveling salesman problems and knapsack problems. Optimization problems are often nonlinear, it is difficult to derive
an optimal value. Metaheuristic algorithms are metaheuristic
and are used to seeking the optimal solution. Optimization
by SI is an effective optimization method. Examples are Ant
Colony Optimization (ACO) [1], Artificial Bee Colony Algorithm (ABC) [2], Particle Swarm Optimization (PSO) [3],and Whale Optimization Algorithm (WOA) [4].
In this study WOA is used. WOA have insufficient globalsearch. We propose WOA with a certain number of whales
that perform global search, when number of iteration is lessthan half of the maximum iteration. The search ability of theproposed method is better than WOA.

(a) Encircling prey
Humpback whales find prey and surround prey. There is aprey at the position of the best agent in WOA, and that prey isthe optimal solution. After the best agent is determined, theother agents update their position towards the best agent. The
movement is represented by Eqs. (1) and (2).
−→−→ −→−→
D = | C .X ∗ (t) − X (t)|(1)−→∗−→−→−→X (t + 1) = X (t) − A . D,(2)−→−→where t indicates the current iteration, A and C are coeffi−→∗−→cient vectors, X is the position vector, X is the position vector of the best solution obtained, . is an element-by-element−→−→
multiplication. A and C are determined by Eqs. (3) and (4).
−→A = 2⃗a.r⃗1 − ⃗a−→
C = 2r⃗2 ,(3)(4)

where ⃗a is linearly decreased from 2 to 0 over the course ofiterations, r⃗1 and r⃗2 are random vector in [0, 1].
(b) Bubble-net attacking method
In the bubble-net attacking method, the humpback whalemoves along the spiral path toward the prey. Equations (5)and (6) represent the relationship between whale position andprey position for simulating spiral movement.

179−→−→−→
D′ = |X ∗ (t) − X (t)|(5)−→−→−→X (t + 1) = D′ ebl cos (2πl ) + X ∗ (t),(6)−→
where D′ is vector of the distance between the whale andthe best agent, b is a constant for defining the shape of thelogarithmic helix, l is a random number in [−1, 1]. The probability of choosing either Eq. (2) or (6) to update the whaleposition is 50%. The equation of movement is selected byEq. (7).

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

Step 8.

Calculate fi of each search agent.

Step 9.

Update X* if there is a better solution.

Step 10. Repeat Step 3 to Step 9 tmax times.

{ −→−→−→−→X (t + 1) = X (t) − A . D(if p < 0.5)−→−→−→X (t + 1) = D′ ebl cos (2πl ) + X ∗ (t), (if p ≥ 0.5)

(7)
3. Proposed method

where p is random number in [0, 1].
(c) Search for prey
Besides the bubble-net attacking method, humpback
whales randomly search for prey. The position of the whale−→is updated with value of A greater than 1 or less than −1 tomove the agent away from the chosen whale. The update oflocation plays a global search role in WOA. The movement
during search for prey is represented by Eqs. (8) and (9).

−−−→−−−→ −→Drand = |Xrand − X (t)|−→−−−→ −→ −−−→X (t + 1) = Xrand − A .DrandStep 7.

(8)
(9)

WOA performs a global search using Eqs. (8) and (9).
However, Eqs. (8) and (9) represents a linear movement, andEqs. (8) and (9) cannot perform sufficient global search. Each
number of whales performing local search and global search
is determined by a. The value of a decreases, then the number of whales in the global search decreases. If a is less than
1, there are no whales in the global search. The number ofiterations in this case is more than half of tmax . We propose
WOA with a certain number of whales that perform global
search when a is 1 or more. The number of whales constantly
searching globally is three, four or five.
4. Simulation result−−−→
Xrand is position vector of whales randomly selected fromthe population.

We find the minimum values of the benchmark functions
using the proposed method and WOA [5]-[8]. Five of thebenchmark functions are unimodal and seven are multimodal.
(d) Procedure of WOA
Table 1 shows the formula, range, and optimal value (OV) of
The procedure for WOA to search for solution is shown unimodal benchmark functions. x is variable in the function.
xi denotes i-dimensional x.
below.
Step 1.

Step 2.

Initialize the whales population n and the maximum number of iterations tmax .

Table 1: unimodal benchmark functions.
Name−→
Set the position of the whale Xi . i is the number−→of the whale. Xi is determined randomly within
the range.

Formula30
∑
x2i

Range

OV

[-100,100]

0[-10,10]0{}|xi |, 1 ≤ i ≤ 30[-100,100]

0[100(xi+1 − x2i )2 + (xi − 1)2 ][-30,30]0[-100,100]0f1i=130∑f2|xi | +∏

|xi |

i=1

Step 3.

Step 4.

Calculate the fitness value fi and update the best−→
whale position X ∗ .

f3f4−→ −→
Update ⃗a, r⃗1 , r⃗2 , A , C , l and p.

Step 5.

Update the position of the whale by Eq. (2),(6)−→or (9). Equations are selected by | A | and p.

Step 6.

Repeat Step 3 to Step 5 until all search agents aredone.

max30
∑
i=1

f530
∑
i=1

180

(xi + 0.5)2

Table 2 shows the formula, range, and optimal value ofmultimodal benchmark functions.

Table 3: Results in unimodal functions.
namef1

Name30∑f6

Formula√−xi sin( |xi |)AVE−84STD

6.9753 × 10−846.0423 × 10

7.1808 × 10−623

4.7185 × 10−278

1.6108 × 10−61

1.6433 × 10−600(WOA)

Table 2: multimodal benchmark functions.

MIN

Range

OV

4

1.4505 × 10−79

2.0660 × 10−60

3.7556 × 10−59[-500,500]-12569

51.4185 × 10−57

7.3986 × 10−60

9.2180 × 10−590(WOA)

1.3178 × 10−50

3.4737 × 10−40

6.1151 × 10−393

1.8813 × 10−52

2.1338 × 10−401.5445 × 10−3943.2534 × 10−491.3163 × 10−391.2485 × 10−3853.4654 × 10−492.1338 × 10−391.5445 × 10−380(WOA)

7.9001 × 10−13

1.0573 × 10−54.5413 × 10−539.1663 × 10−145

8.9641 × 10−6

3.0281 × 10−548.1488 × 10−191.0425 × 10−53.3772 × 10−553.4718 × 10−139

1.9876 × 10−59.8165 × 10−50(WOA)

9.7889 × 10−8

1.0943 × 101

1.2743 × 101

32.1159 × 10−71.0439 × 101

1.2394 × 101

43.4053 × 10−7

9.8581 × 100

1.2165 × 101

51.2294 × 10−7

1.0047 × 101

1.2445 × 1010(WOA)

1.2743 × 10−11

4.8566 × 10−3

7.1133 × 10−3

3

5.6239 × 10−144.6502 × 10−3

5.9021 × 10−3

45.6076 × 10−13

4.4180 × 10−3

5.8768 × 10−3

51.9573 × 10−15

4.2435 × 10−3

6.3817 × 10−3

i=1f730∑[x2i − 10 cos(2πxi ) + 10][-5.12,5.12]

0√ ∑3012−20 exp(−0.2 30i=1 Xi )∑30
1− exp( 30cos(2πx))+20+eii=1[-32,32]0f930301 ∑ 2 ∏xi
x −cos( √ ) + 1
4000 i=1 i i=1i[-600,600]

f10

14x21 − 2.1x41 + x61 + x1 x2 − 4x22 + 4x42[-5,5]

1.0316[-5,5]

0.398

f2

i=1

f8

f11

35.1 2 5x + x1 − 6)24π 2 11 π+10(1 − 8π ) cos x1 + 10(x2 −[1 + (x1 + x2 + 1) (19 − 14x1 + 3x21
−14x2 + 6x1 x2 + 3x22 )] × [30 + (2x1 + x2 + 1)2×(18 − 32x1 + 12x21 + 48x2 − 36x1 x2 + 27x22 )]

0f3

2

f12

[-2,2]

f4

3

In each test function, we define tmax = 500, n = 30. We
repeated the search 500 times and found the minimum values,
average values and standard deviations of the search resultswith the conventional WOA and the proposed methods. The
minimum values, average values and standard deviations areshown in Table 3. The numbers in the second column are thepopulation of whales for global search constantly. If it is 0,
the conventional WOA is used.
According to Table 3, the minimum value obtained by
the proposed method is much smaller than the conventional
WOA in f1 and f3 . The best values of average value andstandard deviations are found by the proposed method in all
unimodal functions except f1 . The proposed method withthree agents for global search found the best average value
and standard deviations in two types of unimodal functions,and found better average value and standard deviations thanWOA in all unimodal functions except f1 .
In multimodal function, the minimum values, average values and standard deviations are shown in Table 4. According
to Table 4, the conventional WOA and the proposed methodfind the same minimum value in all multimodal functions.
All best values of average values and standard deviations arefound by the proposed methods. In particular, when there arealways three agents for global search, average value and standard deviations are smaller than the conventional WOA in allfunctions. From the above, WOA with three agents for global
search has the best search ability in the proposed methods.

f55. Conclusion
WOA cannot perform a sufficient global search. In the proposed method, there are always certain number of agents for
global search until half of the maximum number of iterations.
The number of the agents are three, four or five. We compared
the search performance of the proposed method and conventional WOA with five kinds of unimodal benchmark functions
and seven kinds of multimodal benchmark functions. In unimodal function, the proposed method find best value of average and standard deviations. The minimum value obtained by
the proposed method is much smaller than the conventional
WOA in two types of unimodal functions. The proposed
method performed best in all multimodal functions. WOA
with three whale for global search has the best search capabilities in the proposed methods.

181

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Proceedings of the 1995 IEEE international conferenceon neural networks, p.19428, 1995.

Table 4: Results in multimodal functions.
name

f6

f7

f8

MIN0(WOA)−1.2569 × 10

−1.2531 × 10

1.8893 × 102

3−1.2569 × 104−1.2559 × 104

6.5279 × 101

4−1.2569 × 104−1.2553×104

1.1135 × 102

5−1.2569 × 104−1.2564 × 104

4.7501 × 101

f11

f12−1706.0396 × 103.4097 × 10

305.6843 × 10−173.1263 × 10−16403.5527 × 10−172.9511 × 10−16506.7502 × 10−173.3963 × 10−160(WOA)

4.4409 × 10−16

4.5018 × 10−15

2.6610 × 10−15

34.4409 × 10−16

4.4160 × 10−152.5423 × 10−15

4.4409 × 10−16−15−152.6831 × 104.4409 × 10−16−152.6987 × 10−15−3−2

4.5623 × 10
4.5581 × 10

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−160(WOA)

5f10

STD44f9

AVE40(WOA)

0

6.200 × 10

6.9374 × 1030

1.4273 × 10−3

2.2831 × 10−2401.2602 × 10−3

2.1420 × 10−250

2.6824 × 10−3

3.5339 × 10−20(WOA)−1.0316−1.02894.7233 × 10−2

3−1.0316−1.0295

4.0684 × 10−2

4−1.0316−1.0293

4.6685 × 10−2

5−1.0316−1.0294

3.9914 × 10−20(WOA)

3.9789 × 10−1

4.0553 × 10−11.6005 × 10−2

3

3.9789 × 10−1

4.0475 × 10−1

1.2734 × 10−2

43.9789 × 10−1

4.0475 × 10−1

1.3225 × 10−253.9789 × 10−1

4.0330 × 10−1

1.2482 × 10−20(WOA)

3.00003.12840.2382733.0000

3.10520.1914843.0000

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