CKH

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Chaotic Krill Herd algorithm
Gai-Ge Wang
a
, Lihong Guo
b,
⇑
, Amir H. Gandomi
c,1
, Guo-Sheng Hao
a
, Heqi Wang
b
a
School of Computer Science and Technology, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China
b
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China
c
Department of Civil Engineering, The University of Akron, Akron, OH 44325, USA
article info
Article history:
Received 20 August 2012
Received in revised form 26 September
2013
Accepted 10 February 2014
Available online 3 March 2014
Keywords:
Global optimization problem
Krill Herd
Chaotic maps
Multimodal function
abstract
Recently, Gandomi and Alavi proposed a meta-heuristic optimization algorithm, called Krill
Herd (KH). This paper introduces the chaos theory into the KH optimization process with
the aim of accelerating its global convergence speed. Various chaotic maps are considered
in the proposed chaotic KH (CKH) method to adjust the three main movements of the krill
in the optimization process. Several test problems are utilized to evaluate the performance
of CKH. The results show that the performance of CKH, with an appropriate chaotic map, is
better than or comparable with the KH and other robust optimization approaches.
Ó2014 Elsevier Inc. All rights reserved.
1. Introduction
The process of optimization is essentially the choice of a vector within a search space. The selected vector can maximize/
minimize an objective function to provide the best solution. Generally, modern intelligent approaches are used to deal with
these types of optimization problems. Such optimization approaches can be categorized into two groups in view of their nat-
ures: (1) deterministic, and (2) random intelligent approaches. The deterministic approaches using gradient have a strict
step. They produce the identical solution if its initial starting values are the same with each other when solving the same
problem. Contrary to the deterministic approaches, gradient-free stochastic algorithms are based on random walks. There-
fore, the optimization process cannot be repeated under any conditions. However, in most cases, both of them are capable of
finding the same final optimal solutions [44]. Recently, nature-inspired meta-heuristic algorithms show a powerful and effi-
cient performance for dealing with high-dimension nonlinear optimization problems [12,53].
To some extent, all meta-heuristic approaches manage to make a trade-off between intensification (local search) and ran-
domization (global search) [50]. These robust nature-inspired meta-heuristic approaches are utilized to tackle NP-hard prob-
lems such as task-resource assignment. They can fully exploit the useful information of the whole population to find optimal
solutions. Up to now, significant research has been done on evolution theory. The process of evolution is idealized as a kind of
gradient-free method, called genetic algorithms (GAs) [6,18,55]. Since then various nature-inspired meta-heuristic ap-
proaches have been proposed such as evolutionary strategy (ES) [1,2], particle swarm optimization (PSO) [27,28,40,56],
ant colony optimization (ACO) [4], differential evolution (DE) [10,19,38], firefly algorithm (FA) [7], biogeography-based
optimization (BBO) [29,37], cuckoo search (CS) [9,49], probability-based incremental learning (PBIL) [36], big bang–big
http://dx.doi.org/10.1016/j.ins.2014.02.123
0020-0255/Ó2014 Elsevier Inc. All rights reserved.
⇑
Corresponding author. Tel.: +1 3689827126.
E-mail addresses: gaigewang@163.com,gaigewang@gmail.com (G.-G. Wang), guolh@ciomp.ac.cn (L. Guo), a.h.gandomi@gmail.com,ag72@uakron.edu
(A.H. Gandomi), guoshenghaoxz@tom.com (G.-S. Hao), whq200808@gmail.com (H. Wang).
1
URL: http://www.gozips.uakron.edu/~ag72.
Information Sciences 274 (2014) 17–34
Contents lists available at ScienceDirect
Information Sciences
journal homepage: www.elsevier.com/locate/ins

crunch algorithm [5,23,25,26], harmony search (HS) [15,17,47], charged system search (CSS) [24], animal migration optimi-
zation (AMO) [31], teaching–learning-based optimization (TLBO) [3,35], bat algorithm (BA) [11,52], and the KH method [8].
In fact, KH is a new kind of swarm intelligence optimization approach inspired by the herding behavior of krill [8]. In KH, the
objective function for the krill movement is decided by the distances of each krill from density of the krill swarm and food.
The position for each krill is made up of three parts: (i) movement induced by other individuals (ii) foraging motion, and (iii)
physical diffusion. The crucial advantage of KH algorithm is its simplicity, making KH implement easily and lending itself to
parallel computation [44].
In general, KH lends itself strongly to exploitation. However, it cannot always implement global search well. Thus, in some
cases, KH fails to find global optimal solution. The search strategy used in basic KH is mainly based on random walks. Thus, it
cannot always deal with the problem successfully [45]. Different strategies have been added to the basic KH method [43,45]
with the aim of improving its performance [45].
With the development of the nonlinear dynamics, chaos concept has been widely considered in various applications [34].
In this context, one of the most famous applications is the introduction of chaos theory into the optimization methods [48].
Up to now, the chaos theory has been successfully combined with several meta-heuristic optimization methods [13]. Some
major efforts in this area includes hybridizing chaotic sequences with memetic differential evolution algorithm [22],FA[13],
gravitational search algorithm [20], imperialist competitive algorithm [41], charged system search [33,42], PSO [14,46], and
GAs [21].
In the present study, chaotic KH-based (CKH) methods are introduced for the purpose of accelerating the convergence of
KH. Various one-dimensional chaotic maps are employed in place of the parameters used in KH. The performance of the pro-
posed approach is tested on fourteen benchmark problems. Experimental results indicate that CKH performance is superior
to KH, ACO, BA, CS, DE, ES, GA, PBIL, and PSO. This is mainly because deterministic chaotic signals are used to replace linearly
declined values.
The organization of this paper is as follows. Firstly, a brief overview of the basic KH algorithm and 12 chaotic maps are
given in Section 2. The detailed presentation of the proposed CKH approach is provided in Section 3. Subsequently, the tuning
of the inertia weights and selecting the optimal CKH are described in Section 4. In addition, the performance of the CKH ap-
proach is verified using fourteen benchmarks. Finally, a summary of the present work is represented in Section 5.
2. Overview of the KH and chaotic maps
2.1. KH method
KH [8] is a new type of meta-heuristic method for solving optimization problems. This method is inspired by the herding
of krill swarms when searching for food in nature. For each krill, its position in search space is influenced by three compo-
nents described below [45]:
i. movement induced by other krill;
ii. foraging action;
iii. random diffusion.
For simplicity, the above three motions in KH can be idealized to the following Lagrangian model [8] as shown in Eq. (1).
dX
i
dt ¼N
i
þF
i
þD
i
ð1Þ
where N
i
,F
i
, and D
i
are, respectively, corresponding to the above three motions for the ith krill [8]. The krill number is rep-
resented by i, and tis considered as generation.
2.1.1. Motion induced by other krill
The direction of the first motion,
a
i
, is approximately calculated according to the following three factors: target effect,
local effect, and repulsive effect. For krill i, this movement can be modeled below:
N
new
i
¼N
max
a
i
þ
x
n
N
old
i
ð2Þ
where
a
i
¼
a
local
i
þ
a
targ et
i
ð3Þ
and N
max
is the maximum speed,
x
n
is the inertia weight in [0,1], N
old
i
is the previous motion,
a
local
i
and
a
targ et
i
are the local
effect and the target effect, respectively [45]. According to the literature [8], we set N
max
to 0.01 (m s
1
) in our study.
2.1.2. Foraging motion
The second motion is determined by the two main factors: the food location and the previous experience with respect to
the food position. For the ith krill, the expression of this motion can be provided below [45]:
18 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

F
i
¼V
f
b
i
þ
x
f
F
old
i
ð4Þ
where
b
i
¼b
food
i
þb
best
i
ð5Þ
and V
f
is the foraging speed,
x
f
is the inertia weight between 0 and 1, F
old
i
is the previous foraging motion, b
food
i
is the food
attraction and b
best
i
is the effect of the best fitness. In our study, we set V
f
to 0.02 [8].
2.1.3. Physical diffusion
The third motion is essentially a random process. The model of this motion can be expressed according to two factors: a
maximum diffusion speed and a random vector. This model can be expressed below [8]:
D
i
¼D
max
dð6Þ
where D
max
is the diffusion speed, and dis the random vector in [1, 1] [45].
2.1.4. Main procedure of the KH method
The second and third motion involve two global and two local schemes. These schemes can work simultaneously which
makes KH a robust and efficient method [45]. The position of krill ifrom tto t+
D
tcan be formulated by Eq. (7) [45].
X
i
ðtþ
D
tÞ¼X
i
ðtÞþ
D
tdX
i
dt ð7Þ
Note that
D
tis a key constant and should be fine-tuned in terms of the specific problem. The reason is that
D
tcan be
regarded as a scale factor of the speed vector. Moreover, in KH, the inertia weights (
x
n
,
x
f
) are set to 0.9 at the start of
the KH to highlight exploration. They are later linearly decreased to 0.1 in order to stimulate exploitation [8].
Furthermore, in order to enhance the performance of the KH, genetic reproduction mechanisms have been introduced
into the KH algorithm [8].
In general, the KH method can be described as shown in Fig. 1, and its responding flowchart is illustrated in Fig. 2. Further
information about the above movements and KH method can be found in [8].
2.2. Chaotic maps
There is a special kind of random-based optimization algorithms, namely chaotic optimization algorithm (COA) [13]. COA
employs chaotic variables rather than random variables [13]. The chaos has the property of the non-repetition and ergodic-
ity. Therefore, it can perform downright searches at higher speeds compared to the stochastic searches that mainly rely on
probabilities [13]. In the present study, 1-D, non-invertible maps are used to produce chaotic sets. Twelve distinguished 1-D
Krill herd algorithm
Begin
Step 1: Initialization. Initialize the generation counter G, the population P of NP krill
randomly, V f, Dmax, and N max.
Step 2: Fitness calculation . Calculate fitness for each krill according to its initial position.
Step 3: While G < MaxGeneration do
Sort the population according to their fitness.
for i=1:NP (all krill) do
Perform the following motion calculation.
Motion induced by other individuals
Foraging motion
Physical diffusion
Implement the genetic operators.
Update the krill position in the search space.
Calculate fitness for each krill according to its new position.
end for i
G = G+1.
Step 4: end while
End.
Fig. 1. Krill Herd algorithm.
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 19

maps [13] are described in Table 1. In this table, kmeans the index of the chaotic sequence, and x
k
represents the kth number
in the chaotic sequence.
3. The proposed CKH approach
As it is presented in Eqs. (2) and (4), the main parameters of KH are the inertia weights (
x
n
,
x
f
) that represent the vari-
ations of the global optimal attraction. The values of the weights are vitally important in deciding the convergence speed and
how to update the position of the krill in KH.
Though the basic KH is very powerful, the solutions are still deviating from each other at the end of the search. In KH, the
inertia weights (
x
n
,
x
f
) are set to 0.9 at the early search stage to highlight exploration. Finally, they are linearly decreased to
0.1 to stimulate exploitation [8].
In the standard KH, there is no need to keep linearly decrements. In fact, a chaotic varying inertia weights (
x
n
,
x
f
) may be
more favorable for the search, which may also make the method converge to the best solution with a fast speed as will be
demonstrated in the following section. After normalization, the range of a chaotic map is always between 0 and 1. So, in this
paper, we use chaotic maps to adjust inertia weights (
x
n
,
x
f
), and such chaos-improved KH can be named as the chaotic KH.
The basic framework of the CKH method and its responding flowchart are shown in Figs. 3 and 4, respectively. In Figs. 3
and 4,16i6NP represents the krill number, and NP is the total number of the krill in the population, i.e. population size.
The tuning of inertia weights (
x
n
,
x
f
) using chaotic maps also improve the ability of KH to avoid local optima when solv-
ing a multimodal function. Comparing with other optimization methods, this could be a merit for this method.
Is termination
condition met?
Output the bestsolution
Motion induced by other individuals
Foraging motion
Physical diffusion
Implement the genetic operator(s)
Initialization
Update the krill individual position
Fitness evaluation
Start
End
Fig. 2. Flowchart of KH algorithm.
20 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

Moreover, the second significant improvement is the addition of elitism scheme to the CKH. As with other meta-heuristic
methods, a kind of elitism is typically introduced in order to hold the optimal solutions. This puts a stop to the best solutions
from being corrupted by motion calculation operators. Here, an elitism strategy is utilized to memorize the property of the
best krill in CKH. Consequently, even if motion calculation operation ruins its corresponding krill, it has been held and can be
recovered if required.
Table 1
Twelve different chaotic maps.
No. Name Definition
M1 Chebyshev map x
k+1
= cos(kcos
1
(x
k
))
M2 Circle map
a
x
kþ1
¼x
k
þbða=2
p
Þsinð2
p
kÞmodð1Þ
M3 Gaussian map x
kþ1
¼0x
k
¼0
1=x
k
modð1Þotherwise
;1=x
k
modð1Þ¼
1
x
k

1
x
k
hi
M4 Intermittency map x
kþ1
¼
e
þx
k
þcx
n
k
0<x
k
6P
x
k
P
1P
P<x
k
<1

M5 Iterative map x
kþ1
¼sin
a
p
x
k

;a
e
(0, 1)
M6 Liebovitch map
x
kþ1
¼
ax
k
0<x
k
6P
Px
k
P
2
P
1
P
1
<x
k
6P
2
1bð1x
k
ÞP
2
<x
k
61
8
<
:
M7 Logistic map x
k+1
=ax
k
(1 x
k
)
M8 Piecewise map
x
kþ1
¼
x
k
P
06x
k
<P
x
k
P
0:5P
P6x
k
<
1
2
1Px
k
0:5P
1
2
6x
k
<1P
1x
k
P
1P6x
k
<1
8
>
>
>
<
>
>
>
:
M9 Sine map x
kþ1
¼
a
4
sinð
p
x
k
Þ;0<a64
M10 Singer map x
kþ1
¼
l
7:86x
k
23:31x
2
k
þ28:75x
3
k
13:302875x
4
k

M11 Sinusoidal map x
kþ1
¼ax
2
k
sinð
p
x
k
Þ
M12 Tent map x
kþ1
¼
x
k
0:7
x
k
<0:7
10
3
ð1x
k
Þx
k
P0:7

a
With a= 0.5 and b= 0.2, it geneguos rates chaotic sequence in (0, 1).
CKH algorithm
Begin
Step 1: Initialization. Initialize the generation counter G, the population P of NP krill
randomly, V
f
, D
max
, and N
max
and KEEP; set initial value of the chaotic map c
0
randomly, and inertia weights
Step 2: Fitness calculation. Calculate fitness for each krill according to its initial position.
Step 3: While G < MaxGeneration do
Sort the population according to their fitness.
Memorize the KEEP best krill.
Update inertia weights using chaotic maps
for i=1: NP(all krill) do
Perform the following motion calculation.
Motion induced by the presence of other individuals
Foraging motion
Physical diffusion
Update the krill position in the search space.
Calculate fitness for each krill according to its new position.
end for i
Replace the KEEP worst krill with the KEEP best krill.
G = G+1.
Step 4: end while
End.
Fig. 3. CKH algorithm.
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 21

4. Experimental results
In this section, various experiments on optimization benchmark problems are implemented to verify the performance of
the proposed meta-heuristic CKH method.
In order to get an unbiased comparison of CPU times, all the experiments are performed using the same PC with the de-
tailed settings as shown in Table 2.
Is termination
condition met?
Output the best solution
Motion induced by other individuals
Foraging motion
Physical diffusion
Initialization
Set initial value of the chaotic map
c0randomly, and inertia weights
Update the krill individual position
Fitness evaluation
Start
End
Update inertia weights using chaotic
maps
Memorize the KEEP best krill
Replace the KEEP worst krill with the
KEEP best krill
Fitness evaluation
ω
= (
ω
n
,
ω
f
) = c
0
.
(
ω
= c
t+1
).
Fig. 4. Flowchart of CKH algorithm.
22 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

Fourteen different benchmark functions are used to evaluate our proposed meta-heuristic CKH method. The expressions
and properties of these benchmarks can be presented in Tables 3 and 4, respectively [44]. More information with reference to
all the benchmarks can be found in [54].
For all the algorithms used in this work, population size NP, an elitism parameter Keep, and maximum generation Maxgen
are assigned to 50, 2 and 50, respectively. The optimal results obtained by each algorithm for each benchmark are explained
in the following sections. It should be pointed out that different scales are used to normalize the results in the tables. There-
fore, values cannot be comparative between them. The dimension of each function used in our work is 20 (i.e., d= 20).
4.1. The performance of CKH with different chaotic maps
CKH is capable of significantly improving the solution quality by using the chaotic maps. Here, the work of adjusting the
inertia weights
x
=(
x
n
,
x
f
) are implemented. And the value of
x
is substituted with various chaotic maps presented in Sec-
tion 2.2. We normalized all the chaotic maps in [0.1,0.9] so as to achieve this.
100 implementations of the CKH algorithm are carried out on fourteen benchmarks to obtain typical statistical results.
The results of the simulations are recorded in Tables 5 and 6.Table 5 represents the average values obtained by the CKH
method, averaged over 100 simulations. Table 6 also illustrates the best values obtained by CKH method from 100 simula-
tions. We must point out, the marks M1, M2, ...,M12 in the Tables 5 and 6 are short for the twelve responding chaotic maps
in the Section 2.2.
Table 2
The detailed settings.
Name Detailed settings
Hardware
CPU Pentium IV processor
Frequency 2.0 GHz
RAM 512 MB
Hard drive 160 GB
Software
Operating system Windows XP3
Language MATLAB R2012a (7.14)
Table 3
Benchmark functions.
No. Name Definition
F01 Ackley fð~
xÞ¼20 þe20 e
0:2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n
P
n
i¼1
x
2
i
pe
1
n
P
n
i¼1
cosð2
p
x
i
Þ
F02 Fletcher–Powell fð~
xÞ¼P
n
i¼1
ðA
i
B
i
Þ
2
;A
i
¼P
n
j¼1
ða
ij
sin a
j
þb
ij
cos a
j
ÞB
i
¼P
n
j¼1
ða
ij
sin x
j
þb
ij
cos x
j
Þ
F03 Griewank fð~
xÞ¼P
n
i¼1
x
2
i
4000
Q
n
i¼1
cos
x
i
ffi
i
p

þ1
F04 Penalty #1 fð~
xÞ¼
p
30
10 sin
2
ð
p
y
1
ÞþP
n1
i¼1
ðy
i
1Þ
2
½1þ10 sin
2
ð
p
y
iþ1
Þ þ ðy
n
1Þ
2
no
þP
n
i¼1
uðx
i
;10;100;4Þ;y
i
¼1þ0:25ðx
i
þ1Þ
F05 Penalty #2 fð~
xÞ¼0:1 sin
2
ð3
p
x
1
ÞþP
n1
i¼1
ðx
i
1Þ
2
½1þsin
2
ð3
p
x
iþ1
Þ þ ðx
n
1Þ
2
½1þsin
2
ð2
p
x
n
Þ
n oþP
n
i¼1
uðx
i
;5;100;4Þ
F06 Quartic with noise fð~
xÞ¼P
n
i¼1
ðix
4
i
þUð0;1ÞÞ
F07 Rastrigin fð~
xÞ¼10 nþP
n
i¼1
ðx
2
i
10 cosð2
p
x
i
ÞÞ
F08 Rosenbrock fð~
xÞ¼P
n1
i¼1
½100ðx
iþ1
x
2
i
Þ
2
þðx
i
1Þ
2

F09 Schwefel 2.26 fð~
xÞ¼418:9829 DP
D
i¼1
x
i
sinðjx
i
j
1=2
Þ
F10 Schwefel 1.2 fð~
xÞ¼P
n
i¼1
P
i
j¼1
x
j

2
F11 Schwefel 2.22 fð~
xÞ¼P
n
i¼1
jx
i
jþQ
n
i¼1
jx
i
j
F12 Schwefel 2.21 fð~
xÞ¼max
i
fjx
i
j;16i6ng
F13 Sphere fð~
xÞ¼P
n
i¼1
x
2
i
F14 Step fð~
xÞ¼6nþP
n
i¼1
bx
i
c

In benchmark function F02, the matrix elements a
nn
,b
nn
e
(100,100),
a
n1
e
(
p
,
p
) are draw from uniform distribution.

In benchmark functions F04 and F05, the definition of the function u(x
i
,a,k,m) is as follows:
uðxi;a;k;mÞ¼
kðxiaÞm;xi>a
0;a6xi6a
kðxiaÞm;xi<a
8
>
<
>
:
:
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 23

From Tables 5 and 6, it can be seen that the basic KH combined with the M9 (Sine map) or M10 (Singer map) perform
better than others. For M9 (Sine map) and M10 (Singer map), the M10 (Singer map) significantly outperforms the M9 (Sine
map) when multiple runs are made. Accordingly, we choose it as the proper map to form the best chaotic KH (CKH) that will
be studied in more detail in Section 4.2.
4.2. CKH vs. other optimization algorithms
The performance of CKH was compared on benchmark problems with nine other optimization algorithms [45]. The meth-
ods included in the comparative study are ACO [4],BA[51,52],CS[49],DE[30,38],ES[1,2],GA[18],KH[8], PBIL [36], and PSO
Table 4
Properties of benchmark functions, lb denotes lower bound, ub denotes upper bound, opt denotes optimum point.
No. Function lb ub opt Continuity Modality
F01 Ackley 32.768 32.768 0 Continuous Multimodal
F02 Fletcher–Powell 
pp
0 Continuous Multimodal
F03 Griewangk 600 600 0 Continuous Multimodal
F04 Penalty #1 50 50 0 Continuous Multimodal
F05 Penalty #2 50 50 0 Continuous Multimodal
F06 Quartic with noise 1.28 1.28 1 Continuous Multimodal
F07 Rastrigin 5.12 5.12 0 Continuous Multimodal
F08 Rosenbrock 2.048 2.048 0 Continuous Unimodal
F09 Schwefel 2.26 512 512 0 Continuous Multimodal
F10 Schwefel 1.2 100 100 0 Continuous Unimodal
F11 Schwefel 2.22 10 10 0 Continuous Unimodal
F12 Schwefel 2.21 100 100 0 Continuous Unimodal
F13 Sphere 5.12 5.12 0 Continuous Unimodal
F14 Step 5.12 5.12 0 Discontinuous Unimodal
Table 5
Mean normalized values with twelve different chaotic maps.
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12
F01 4.03 1.32 1.26 4.44 1.33 4.29 1.18 1.27 1.24 1.00 1.24 1.25
F02 1.00 1.22 1.25 2.92 1.19 2.82 1.22 1.20 1.24 1.13 1.24 1.32
F03 7.38 1.17 1.11 29.79 1.71 35.59 1.07 1.15 1.02 1.00 1.07 1.13
F04 37.50 1.48 1.01 6.7E3 6.58 7.6E3 1.24 1.67 1.01 1.00 1.38 1.14
F05 17.81 1.05 1.19 898.85 2.91 957.45 1.25 1.12 1.00 1.06 1.30 1.10
F06 4.62 1.15 1.12 132.64 1.95 131.40 1.20 1.21 1.03 1.00 1.21 1.13
F07 1.68 1.00 1.04 2.59 1.00 2.51 1.02 1.03 1.01 1.03 1.04 1.02
F08 1.99 1.24 1.17 15.49 1.45 15.03 1.16 1.19 1.09 1.00 1.20 1.10
F09 1.16 1.00 1.01 1.34 1.01 1.10 1.01 1.01 1.03 1.05 1.01 1.03
F10 1.05 1.06 1.08 2.73 1.11 2.61 1.05 1.05 1.03 1.00 1.04 1.02
F11 1.00 1.04 1.05 2.42 1.09 2.20 1.04 1.04 1.06 1.06 1.07 1.10
F12 4.17 1.06 1.09 6.65 1.29 6.57 1.11 1.05 1.00 1.04 1.06 1.02
F13 7.61 1.15 1.09 36.83 1.73 40.34 1.11 1.18 1.07 1.00 1.06 1.08
F14 8.69 1.29 1.17 39.13 1.82 42.42 1.08 1.11 1.03 1.00 1.16 1.12
Table 6
Best normalized values with twelve different chaotic maps.
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12
F01 8.03 1.98 1.69 4.54 1.57 3.88 1.14 1.83 1.76 1.00 1.31 1.64
F02 1.44 1.50 1.98 3.59 1.57 2.60 1.93 1.10 1.41 1.41 2.19 1.00
F03 9.31 1.40 1.44 6.53 2.41 16.75 1.42 1.55 1.24 1.68 1.00 1.43
F04 3.12 2.23 1.91 4.0E4 2.34 2.1E5 3.81 4.17 2.07 1.00 3.09 3.28
F05 57.53 5.50 1.00 1.0E3 6.82 3.1E3 6.33 7.75 8.06 4.43 19.54 4.27
F06 5.20 1.19 2.34 48.35 3.67 75.48 1.00 1.19 1.80 1.16 2.58 2.06
F07 1.74 1.00 1.25 2.74 1.05 2.65 1.19 1.14 1.20 1.18 1.17 1.04
F08 2.00 1.14 1.26 4.84 1.86 7.92 1.26 1.16 1.07 1.00 1.13 1.01
F09 1.27 1.05 1.09 1.29 1.25 1.09 1.00 1.06 1.00 1.13 1.14 1.07
F10 1.20 1.21 1.31 1.58 1.12 2.95 1.35 1.13 1.24 1.07 1.11 1.00
F11 1.07 1.34 1.26 1.93 1.00 1.86 1.14 1.17 1.21 1.13 1.21 1.31
F12 10.33 1.58 1.41 10.63 2.12 6.48 1.64 1.86 1.35 1.51 1.00 1.39
F13 17.54 1.27 1.75 26.25 4.00 40.23 2.21 1.72 1.00 1.20 2.33 1.41
F14 18.66 1.87 1.69 50.27 3.62 41.49 1.42 1.48 1.15 1.00 1.93 1.60
24 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

[16,27]. In DE, the DE/rand/1/bin scheme is used. It is worth mentioning that Gandomi and Alavi [8] have demonstrated that
the KH II shows the best performance compared to the existing algorithms. Thus, in this work, KH II is considered as the basic
KH method. The parameter settings in all experiments for each algorithm are shown in Table 7 [44].
Tables 8 and 9 record the average and the best results from 100 runs, respectively. Table 8 shows that, on average, CKH
performs better than other methods on eleven of the fourteen benchmarks (F01–F08, and F12–F14) when searching for func-
tion minimum. CS, DE and GA are the second most effective, performing best on one of the fourteen benchmarks (F10, F11
and F09, respectively). From Table 9, it can be seen that, CKH performs better than other methods on eight of the fourteen
benchmarks (F01, F03, F06–F08, and F12–F14). GA and ACO are the second and third most effective, respectively, performing
the best on the benchmarks F02, F09, F10 and F04–F05 when multiple runs are made. CS and DE are the fourth most effec-
Table 7
Parameters settings.
Algorithms Parameters
ACO Initial pheromone value
s
0
=1E6, pheromone update constant Q= 20, exploration constant q
0
= 1, global pheromone decay rate
q
g
= 0.9, local pheromone decay rate
q
l
= 0.5, pheromone sensitivity s= 1, and visibility sensitivity b=5
BA Loudness A= 0.95, pulse rate r= 0.5, and scaling factor
e
= 0.1
CKH The foraging speed V
f
= 0.02, the maximum diffusion speed D
max
= 0.005, the maximum induced speed N
max
= 0.01
CS A discovery rate p
a
= 0.25; for DE, a weighting factor F= 0.95 and a crossover constant CR = 0.4
DE A weighting factor F= 0.95 and a crossover constant CR = 0.4
ES The number of offspring k= 10 produced in each generation, and standard deviation
r
= 1 for changing solutions
GA Roulette wheel selection, single point crossover with a crossover probability of 1, and a mutation probability of 0.01
KH The foraging speed V
f
= 0.02, the maximum diffusion speed D
max
= 0.005, the maximum induced speed N
max
= 0.01
PBIL A learning rate of 0.05, 1 good population member and 0 bad population members to use to update the probability vector each
generation, an elitism parameter of 1, and a 0 probability vector mutation rate
PSO An inertial constant = 0.3, a cognitive constant = 1, and a social constant for swarm interaction = 1
Table 8
Mean normalized values.
ACO BA CKH CS DE ES GA KH PBIL PSO
F01 3.51 4.39 1.00 3.90 2.74 4.28 3.74 1.20 4.46 3.70
F02 2.65 3.84 1.00 1.87 1.04 2.68 1.08 1.06 2.36 2.27
F03 2.25 37.64 1.00 14.06 3.74 17.94 6.91 6.10 39.12 14.28
F04 3.8E3 7.6E3 1.00 343.53 24.30 2.9E3 45.91 435.21 6.0E3 545.34
F05 1.1E3 1.1E3 1.00 116.70 13.70 515.59 24.94 101.82 1.1E3 131.40
F06 8.68 126.33 1.00 20.39 3.13 106.70 6.57 24.48 121.83 24.12
F07 1.78 2.72 1.00 2.11 1.57 2.46 1.67 2.02 2.50 1.85
F08 14.74 15.34 1.00 4.08 2.17 19.11 3.62 5.02 15.56 4.28
F09 1.21 4.06 2.21 3.08 2.32 2.85 1.00 4.80 3.60 3.52
F10 1.57 3.75 1.02 1.00 2.14 2.42 1.63 5.14 2.40 1.64
F11 2.36 3.88 1.34 2.28 1.00 3.62 1.78 8.4E10 2.95 2.23
F12 4.85 8.51 1.00 5.72 6.35 7.66 6.66 1.73 8.16 6.55
F13 21.70 46.81 1.00 16.18 4.14 46.71 14.12 7.04 46.44 16.90
F14 3.54 42.51 1.00 15.68 3.90 26.29 8.22 6.43 44.57 15.78
Time 3.18 1.00 5.06 2.06 2.00 2.12 2.32 4.94 1.14 2.54
Table 9
Best normalized values.
ACO BA CKH CS DE ES GA KH PBIL PSO
F01 4.35 6.32 1.00 5.14 3.95 6.21 4.43 1.10 7.18 5.66
F02 10.52 12.96 2.86 6.07 3.42 9.50 1.00 18.56 6.78 7.94
F03 2.52 39.53 1.00 14.13 4.03 22.21 4.06 9.93 71.07 24.54
F04 1.00 7.0E7 163.92 3.1E5 1.2E4 4.8E7 155.30 1.6E7 2.9E8 3.6E6
F05 1.00 1.5E8 8.6E3 1.0E7 1.5E6 8.4E7 1.2E5 3.3E7 3.1E8 1.0E7
F06 6.78 111.99 1.00 14.21 3.02 191.23 2.23 51.11 169.98 27.53
F07 2.01 2.79 1.00 2.41 1.91 2.84 1.40 2.46 2.93 2.09
F08 13.56 9.19 1.00 2.79 2.12 16.53 1.65 4.47 16.31 1.97
F09 2.68 12.56 7.07 9.37 8.10 9.58 1.00 19.10 14.09 10.85
F10 1.15 2.98 1.01 1.00 2.60 2.63 1.00 2.29 2.80 1.72
F11 2.18 2.97 1.23 1.99 1.00 2.17 1.26 2.5E5 3.32 1.81
F12 7.94 14.43 1.00 9.41 12.12 17.02 5.38 2.33 18.18 10.41
F13 52.12 109.09 1.00 39.42 11.07 175.14 14.74 19.58 180.26 57.42
F14 2.83 48.10 1.00 18.54 5.39 43.68 3.58 8.93 86.33 23.91
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 25

tive. They have the best performance on the benchmarks F10 and F11, respectively. It can be seen that replacing inertia
weights
x
with chaotic maps can greatly enhance the performance of the KH.
Moreover, the calculated time demands of the ten optimization algorithms were similar. The average running times for
these algorithms are collected and presented in Table 8. BA was the most efficient algorithm, while CKH was the tenth fastest
of the ten methods.
Furthermore, optimization process of each algorithm is given in Figs. 5–18. The values shown in these figures are the
average function optimum achieved from 100 runs. Here, all the values are not normalized, and they are true function values.
In addition, it should be noted that the global optima of the benchmarks (F04, F05, and F11) are illustrated in the form of the
semi-logarithmic convergence plots. Also, KH II is short for KH in Figs. 5–18.
Fig. 5 shows the values obtained by the ten methods on the F01 function, which is a multimodal function with a narrow
global minimum basin (F01
min
= 0) and many minor local optima. From Fig. 5, CKH performance is superior to the nine meth-
ods in optimization process, while KH II performs the second best in this benchmark. Moreover, all the methods have the
almost same initial values, while CKH overtakes the other 9 methods. All methods clearly outperform the PBIL algorithm.
Fig. 6 shows the optimization values for F02 function. For this case, the figure clearly shows that CKH, DE and GA, KH II
performances differ from each other. From Table 8 and Fig. 6, CKH has the best performance for this multimodal function.
ACO, BA, CS, ES, PBIL, and PSO fail to find the best solution in this benchmark function.
0 5 10 15 20 25 30 35 40 45 50
4
6
8
10
12
14
16
18
20
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 5. Performance comparison on the F01 Ackley function.
0 5 10 15 20 25 30 35 40 45 50
2
3
4
5
6
7
8
9
10
11
12 x 105
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 6. Performance comparison on the F02 Fletcher–Powell function.
26 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

0 5 10 15 20 25 30 35 40 45 50
0
50
100
150
200
250
300
350
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 7. Performance comparison on the F03 Griewank function.
0 5 10 15 20 25 30 35 40 45 50
104
105
106
107
108
109
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 8. Performance comparison on the F04 Penalty #1 function.
0 5 10 15 20 25 30 35 40 45 50
105
106
107
108
109
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 9. Performance comparison on the F05 Penalty #2 function.
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 27

0 5 10 15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
40
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 10. Performance comparison on the F06 Quartic (with noise) function.
0 5 10 15 20 25 30 35 40 45 50
80
100
120
140
160
180
200
220
240
260
280
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 11. Performance comparison on the F07 Rastrigin function.
0 5 10 15 20 25 30 35 40 45 50
500
1000
1500
2000
2500
3000
3500
4000
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 12. Performance comparison on the F08 Rosenbrock function.
28 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

Fig. 7 shows the optimization values for F03 function. F03 has a strange property, as it is much easier to solve for higher
dimensions than lower dimensions [32]. From Fig. 7, it can be seen that CKH has the best performance for this benchmark.
For other algorithms, ACO, BA, GA, and PSO have the fastest convergence rate initially towards the global optimum. However,
they are outperformed by CKH after some generations. Further, ACO and DE work very well because they rank 2 and 3,
respectively, among ten algorithms.
Fig. 8 illustrates the values for F04 Penalty #1 function. From Fig. 8, apparently, CKH overtakes all other approaches. For
this case, PSO and GA show a faster convergence rate initially, however they seem to be trapped into sub-optimal values as
the procedure proceeds, especially PSO. Although slower, DE performs the second best in the optimization process.
Fig. 9 illustrates the values achieved on F05 function. For this benchmark, very similar to F04 in Fig. 8, CKH significantly
overtakes all other approaches in the optimization process.
Fig. 10 illustrates the values achieved for the ten methods when using F06. With reference to this benchmark, very similar
to F02 Fletcher–Powell function as shown in Fig. 6, the figure reveals that ACO, CKH, DE and GA perform slightly different
from each other. However, Table 8 and Fig. 10 show that CKH provides better results for this problem. BA, CS, ES, KH II, PBIL,
and PSO do not succeed in this benchmark function. At last, DE and GA converge to the value that is very close to CKH’s.
Fig. 11 illustrates the functions values for the F07 Rastrigin function that is a difficult multimodal function with a unique
global minimum of F07
min
= 0 and several local optima. When attempting to solve F07, methods may be converged into a
local value. Therefore, a method that can keep a larger diversity is well capable of producing better values. At first glance
0 5 10 15 20 25 30 35 40 45 50
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 13. Performance comparison on the F09 Schwefel 2.26 function.
0 5 10 15 20 25 30 35 40 45 50
0.5
1
1.5
2
2.5
x 10 4
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 14. Performance comparison on the F10 Schwefel 1.2 function.
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 29

0 5 10 15 20 25 30 35 40 45 50
102
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 15. Performance comparison on the F11 Schwefel 2.22 function.
0 5 10 15 20 25 30 35 40 45 50
10
20
30
40
50
60
70
80
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 16. Performance comparison on the F12 Schwefel 2.21 function.
0 5 10 15 20 25 30 35 40 45 50
0
10
20
30
40
50
60
70
80
90
100
Number of generations
benchmark function value
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
Fig. 17. Performance comparison on the F13 Sphere function.
30 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

it is clear that CKH greatly overtakes all other approaches. Moreover, PSO has the fastest convergence rate initially towards
global value. However, it is outperformed by CKH after 16 generations. Further, DE and GA compares well compared with
other algorithms.
Fig. 12 shows the functions values for F08 function, which can also be regarded as a multimodal function. For this case,
analogous to the F07 in Fig. 11, clearly, CKH greatly overtakes all other approaches. In addition, PSO has the fastest conver-
gence rate initially towards the global solution. However, it is outperformed by CKH after 22 generations. Further, the per-
formance of DE and GA is very good for this case.
Fig. 13 reveals the equivalent values for the F09 Schwefel 2.26 function. From Fig. 13, GA performs better than other nine
methods involving CKH in the optimization process, while ACO performs the second best in this multimodal benchmark.
Moreover, KH II is superior to CKH.
Fig. 14 displays the values for F10 function. From Fig. 14, for this case, analogous to the F02 Fletcher–Powell function as
shown in Fig. 6, it can be concluded that: (i) the performance of CKH and CS is similar. (ii) KH II is only inferior to CKH and CS,
and performs the third best in this benchmark function. However, referring to Table 8 and Fig. 14, CS slightly outperforms
CKH and the other approaches. PSO has the fastest convergence rate initially at finding the global minimum. However, it is
overtaken by CKH and CS after 26 iterations.
Fig. 15 reveals the function values for F11 Schwefel 2.22 function. From Fig. 15, DE has the stable convergence rate to-
wards the global solution and overtakes all other approaches in this unimodal benchmark function. CKH is only inferior
to DE, and performs the second best in this unimodal function.
Fig. 16 reveals the function values for F12 Schwefel 2.21 function. At first glance, it is obvious that CKH has the fastest
convergence rate towards the global solution in the whole optimization process. CKH reaches the optimal solution signifi-
cantly earlier than other algorithms. KH II is only inferior to CKH, and performs the second best in this unimodal function.
Fig. 17 shows the function values for F13 Sphere function, which is also known as de Jong’s function, and has a single glo-
bal value F13
min
= 0, therefore, it is easy to solve. From Fig. 17, CKH has the fastest convergence rate towards the global solu-
tion and overtakes all other methods. Fig. 17 shows that GA performs better than DE at the beginning of the search, but DE
eventually finds the value of CKH, while GA cannot. KH II fails to find the global value within the maximum number of
iterations.
Fig. 18 displays the function values for F14 Step function. CKH shows the fastest convergence rate towards the global
solution and overtakes all other methods. Referring to Fig. 18, ACO shows a little faster convergence than DE. Both ACO
and DE are only inferior to CKH and reach a minimum that is very close to CKH’s.
Considering the results shown in Figs. 5–18, it can be concluded that the CKH’s performance is superior to or at least quite
competitive with the other nine algorithms. Further, benchmarks F02, F04–F08, and F10 illustrate that PSO performs much
better than the other methods initially, while later it is overtaken by other approaches.
4.3. Comparisons using t-test
Based on the final search results of 100 independent trials on every function, Table 10 presents the tvalues on every func-
tion of the two-tailed test with the 5% level of significance between the CKH and other optimization methods. In the table,
the value of twith 198 degrees of freedom is significant at
a
= 0.05 by a two-tailed test. Boldface indicates that CKH performs
ACO
BA
CKH
CS
DE
ES
GA
KH
PBIL
PSO
0
0.5
1
1.5
2
2.5
3
3.5
4x 104
benchmark function value
0 5 10 15 20 25 30 35 40 45 50
Number of generations
Fig. 18. Performance comparison on the F14 Step function.
G.-G. Wang et al. / Information Sciences 274 (2014) 17–34 31

significantly better than the compared algorithms. For instance comparing the CKH and DE, the former significantly outper-
formed the latter on twelve functions (F01, F03–F08, and F10–F14), does as good as the latter on two functions (F02 and F09).
This indicates that the CKH generally performs better than DE on the solution accuracy. Though the CKH performed slightly
weaker on some functions, Table 10 also reveals that it outperforms the other nine methods for most functions.
4.4. The study of the number of fitness evaluations
In order to fully investigate the advantage of the CKH method, the number of fitness evaluations is further studied. We
look at the number of fitness evaluations for every function for each method to the fixed function values. In the experiments,
the fixed function values are set to opt +1.opt indicates the optimum for every function that can be found in fifth column of
Table 4. The maximum of fitness evaluations is set to 50,000. That is to say, if a method cannot converge to the values opt +1
before the maximum number of fitness evaluations is reached (50,000), it will stop, too, and return only up to 50,000 fitness
evaluations. All the results are recorded in Table 11.Table 11 shows that, for the six functions F04–F05, F10, F12–F14, CKH
can find the satisfactory results with the least fitness evaluations. KH is somewhat inferior to the CKH method in terms of
number of fitness evaluations, and it can converge to the fixed values with the least fitness evaluations for the five functions
F02–F03 and F07–F09. Not only do DE and GA also perform well, but also they use the least evaluations when solving the test
problems F01, F11 and F06.
5. Discussion
From the experiments conducted in Sections 4.1–4.4 on the benchmarks, it is demonstrated that CKH performs better
than or comparable to the basic KH and other optimization approaches. Moreover, the implementation of CKH is simple
and easy, and no effect is made to fine-tune the parameters. The work carried out in this paper demonstrates the CKH to
be robust over all kinds of benchmark functions.
There are many ways to verify the performance of the meta-heuristic optimization algorithms. Benchmark evaluation is a
simple way that is widely used. However, it is not a perfect approach. Firstly, the parameters used in these methods cannot
Table 10
Comparisons between CKH and other methods at
a
= 0.05 on a two-tailed t-tests.
ACO BA CS DE ES GA KH PBIL PSO
F01 65.34 121.01 84.83 62.15 118.07 70.03 12.03 131.81 96.11
F02 27.74 24.46 17.31 0.41 23.41 0.97 25.90 22.50 24.36
F03 18.19 34.53 34.66 32.93 45.49 17.60 34.59 74.05 63.74
F04 6.10 14.50 12.72 8.37 19.39 5.57 20.14 24.72 13.82
F05 7.07 15.99 13.08 16.05 25.77 6.59 24.55 39.43 15.60
F06 14.58 18.21 17.46 17.26 35.63 11.62 32.35 43.30 23.09
F07 31.77 52.85 48.61 28.53 62.35 23.33 39.06 70.16 40.54
F08 37.70 25.27 24.03 30.04 37.33 17.06 33.43 42.38 24.08
F09 26.39 36.54 18.13 1.46 12.61 28.36 39.12 29.90 21.10
F10 10.16 18.63 2.41 22.72 25.45 10.23 15.24 26.84 12.46
F11 22.33 1.01 15.17 11.29 29.63 7.60 1.00 43.07 9.08
F12 43.68 67.68 57.24 79.07 95.40 43.09 12.68 109.18 45.62
F13 34.57 36.20 41.53 35.45 67.82 25.75 37.26 66.18 59.59
F14 19.71 40.08 35.81 32.87 60.56 18.21 31.53 66.39 52.55
Table 11
The number of fitness evaluations for different methods.
ACO BA CKH CS DE ES GA KH PBIL PSO
F01 50000 50000 19028 26820 18380 50000 50000 28478 50000 50000
F02 50000 50000 49995 50000 50000 50000 50000 49981 50000 50000
F03 50000 50000 49995 50000 50000 50000 50000 49981 50000 50000
F04 50000 50000 15109 23040 16600 50000 21410 30683 50000 50000
F05 50000 50000 7797 25660 18480 50000 42160 21713 50000 50000
F06 2160 50000 2282 2260 3140 50000 1450 5949 50000 27890
F07 50000 50000 49995 50000 50000 50000 50000 49981 50000 50000
F08 50000 50000 49995 50000 50000 50000 50000 49981 50000 50000
F09 50000 50000 49995 50000 50000 50000 50000 49981 50000 50000
F10 50000 50000 43813 50000 50000 50000 50000 49981 50000 50000
F11 50000 50000 22260 19780 15100 50000 50000 47836 50000 50000
F12 50000 50000 10261 50000 47280 50000 50000 17969 50000 50000
F13 50000 50000 3939 8120 8760 50000 18220 8425 50000 50000
F14 50000 50000 8645 30380 19980 50000 50000 21804 50000 50000
32 G.-G. Wang et al. / Information Sciences 274 (2014) 17–34

be adjusted carefully. It is well-known that for these optimization methods, they always have some randomness. Therefore,
different tuning parameters might make a big difference in their performance. Secondly, most practical application problems
are not necessarily related with benchmark functions. Finally, benchmark evaluation might arrive at diverse conclusions if
we change population size or maximum number of generations [44].
When a meta-heuristic method is used to solve a certain problem, the running time is a key factor. If a method wastes a
long time searching for the best or second best solution, it must be impractical [45]. CKH does not seem to demand an
impractical amount of computational effort. In comparison with other algorithms considered in this study, CKH was the
tenth fastest. In CKH, calculating the chaotic sequence takes a relatively long time. How to make the CKH’s converge faster
still deserves further scrutiny.
6. Conclusions
The chaos theory and KH method are hybridized in order to design a novel improved meta-heuristic chaotic KH method
(CKH) for solving optimization problems. Various chaotic maps are used to regulate the inertia weights,
x
, of KH. The Singer
map is selected as its
x
through comparing various chaotic KH variants to form the best CKH. The simulations show that the
usage of deterministic chaotic signals instead of linearly decreasing values is an important modification of the KH method.
The experimental results show that the tuned KH significantly enhances the reliability of the global optimality and the qual-
ity of the solutions. In addition, the results reveal that CKH considerably enhances the KH’s performances on most multi-
modal and unimodal problems. Moreover, CKH is simple and easy to implement.
Another advantage of CKH is that there are fewer parameters to adjust compared with other population-based optimiza-
tion methods. The performance of a method that relies on parameters is mainly determined by those parameters. In such
cases, it is hard to select the optimal parameters for an optimization method [14]. The present study proposes a new tech-
nique to deal with this difficult issue by using chaotic maps in place of these parameters. In CKH, the chaos theory is used to
adjust the key parameter,
x
. However, the introduction of chaos theory leads to another problem, which makes the approach
take a long time to select the best chaotic map in place of the parameter,
x
.
A limitation of this study is that only fourteen benchmarks are used to verify the performance of CKH. Thus, in order to
evaluate the proposed method more in-depth, more comprehensive problems should be tested such as the high-dimensional
(dP20) CEC 2005 test suite [39]. Furthermore, CKH may be compared with other state-of-the-art optimization methods. In
the future, the CKH method can be extended to solve constrained optimization problems such as the constrained optimiza-
tion CEC 2005 test suite [39]. Herein, two mechanisms are used to improve the KH method. An interesting topic for future
research would be investigating the real impact of elitism and the use of chaos in order to analyze their contributions to the
algorithm performance. For more verification, the CKH method might be applied to solve practical engineering problems
such as benchmark structural optimization problems. We will also perform mathematical analysis of the proposed method
using dynamic system such as Markov chain to prove and explain its convergence.
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