# An Improved Cockroach Swarm Optimization

Authors: I. C. Obagbuwa; A. O. Adewumi
Journal: The Scientific World Journal
 (2014)

---

## Abstract

Hunger component is introduced to the existing cockroach swarm optimization (CSO) algorithm to improve its searching ability and population diversity. The original CSO was modelled with three components: chase-swarming, dispersion, and ruthless; additional hunger component which is modelled using partial differential equation (PDE) method is included in this paper. An improved cockroach swarm optimization (ICSO) is proposed in this paper. The performance of the proposed algorithm is tested on well known benchmarks and compared with the existing CSO, modified cockroach swarm optimization (MCSO), roach infestation optimization RIO, and hungry roach infestation optimization (HRIO). The comparison results show clearly that the proposed algorithm outperforms the existing algorithms.


---

## Body

## 1. Introduction

Swarm intelligence (SI) is a method of computing whereby simple decentralized agents get information by interacting locally with one another and their environment [1]. The local information received is not controlled centrally; local interaction of agents results in amazing and emergent global patterns which can be adopted for solving problems [1].SI algorithms draw inspiration from insects and animals social behaviour and have been proven in literature to be efficient in solving global optimization problems. Examples of existing SI algorithms include particle swarm optimization (PSO), ant colony optimization (ACO), and bee colony optimization (BCO). PSO based on bird social behaviour, introduced by Kennedy and Eberhart [2], has been applied to several problems, including power and management processes [3, 4] and combinatorial optimization problem in [5]. ACO based on ant social behaviour, introduced by Dorigo [6], has been applied to problems such as vehicle routing problem [7] and network routing problem [8]. BCO based on bees social behaviour, introduced by Pham et al. [9], has been applied to real world problems by Karaboga and his research group [10–12].One of the recent developments in SI is cockroach optimization [13–16]. Cockroach belongs to Insecta Blattodea, abodes in warm, dark, and moist shelters, and exhibits habits which include chasing, swarming, dispersing, being ruthless and omnivorous, and food searching. Cockroaches interact with peers and respond to their immediate environment and make decisions based on their interaction such as selecting shelter, searching for food sources and friends, dispersing when danger is noticed, and eating one another when food is scarce.The original cockroach swarm optimization (CSO) algorithm, introduced by Zhaohui and Haiyan [14], was modified by ZhaoHui with the introduction of inertial weight [15]. CSO algorithms [14, 15] mimic chase swarming, dispersion, and ruthless social behaviour of cockroaches.Global optimization problems are considered as very hard problems, ever increasing in complexity. It became necessary to design better optimization algorithms; this necessitated the design of a better cockroach algorithm. This paper extends MCSO with the introduction of another social behaviour called hunger behaviour. Hunger behaviour prevents local optimum and enhances diversity of population. An improved cockroach swarm optimization (ICSO) is presented in this paper.The organization of this paper is as follows: Section2 presents CSO, MCSO, and ICSO models with algorithmic steps; Section 3 shows the experiments carried out and results obtained; the paper is summarised in Section 4.


## 2. Cockroach Swarm Optimization

CSO algorithm is a population based global optimization algorithm which has been applied to problems in literature including [17–19]. CSO [14] models are given as follows.(1) Chase-Swarming Behaviour. 						(1)xi={xi+step·rand·(pi-xi),xi≠pixi+step·rand·(pg-xi),xi=pi,where xi is the cockroach position, step is a fixed value, rand is a random number within [0,1], pi is the personal best position, and pg is the global best position. Consider
(2)pi=Optj{xj,|xi-xj|≤visual},
where perception distance visual is a constant, j=1,2,…,N, i=1,2,…,N. Consider(3)pg=Opti{xi}(2) Dispersion Behaviour.
(4)
x
i
=
x
i
+
rand
(
1
,
D
)
,
i
=
1,2
,
…
,
N
,
where rand(1,D) is a D-dimensional random vector that can be set within a certain range.(3) Ruthless Behaviour.
(5)
x
k
=
p
g
,
where k is a random integer within [1,N] and pg is the global best position.


### 2.1. Modified Cockroach Swarm Optimization

ZhaoHui presented a modified cockroach swarm optimization (MCSO) [15] with the introduction of inertial weight to chase swarming component of original CSO as shown below. Other models remain as in original CSO.Chase-swarming behaviour is as follows:(6)xi={w·xi+step·rand·(pi-xi),xi≠piw·xi+step·rand·(pg-xi),xi=pi,
where w is an inertial weight which is a constant.


### 2.2. Improved Cockroach Swarm Optimization

In this paper, MCSO is extended with additional component called hunger behaviour.


#### 2.2.1. Hunger Behaviour

At interval of time, when cockroach is hungry, it migrates from its comfortable shelter and friends company to look for food [13, 20]. Hunger behaviour is modelled using partial differential equation (PDE) migration techniques [21]. Cockroach migrates from its shelter to any available food source xfood within the search space. A threshold hunger is defined, when cockroach is hungry and threshold hunger is reached; it migrates to food source. Hunger behaviour prevents local optimum and enhances diversity of population.PDE migration equation is described by Kerckhove [21]:
(7)∂u∂t=-c∂u∂x
with u(0,x)=u0(x).Parameterc is the controlling speed of the migration. u is the population size, t is time, and x is location or position. u(t,x) is the population size at time t in location x with u(0,x)=u0(x) being the initial population distribution. Consider
(8)∂u∂t=-c∂u∂x,∂u∂t+c∂u∂x=0.The characteristic equations are(9)dt1=dxc=du0,dx-cdt=0.By integration, we have(10)x-ct=α,u=u(α),u=u(x-ct),u[t,x]=u0[-ct+x].Consider displacement = speed × time.Inu0(x-ct), u0(x) displaces ct.u
0
(
x
-
c
t
) satisfies migration equation at any initial population distribution u0(x) [21].Hunger behaviour is modelled as follows:If(hunger==thunger)(11)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number [0,1].


#### 2.2.2. Improved Cockroach Swarm Optimization Models

(1) Chase-Swarming Behaviour. 										(12)xi={w·xi+step·rand·(pi-xi),xi≠pi,w·xi+step·rand·(pg-xi),xi=pi,
where w is an inertial weight which is a constant, step is a fixed value, rand is a random number within [0,1], pi is the personal best position, and pg is the global best position. Consider
(13)pi=Optj{xj,|xi-xj|≤visual},
where perception distance visual is a constant, j=1,2,…,N, i=1,2,…,N. Consider
(14)pg=Opti{xi}(2) Hunger Behaviour. If hunger==thunger,
(15)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number within [0,1].(3) Dispersion Behaviour. 										(16)xi=xi+rand(1,D),i=1,2,…,N,
where rand(1,D) is a D-dimensional random vector that can be set within a certain range.(4) Ruthless Behaviour. 										(17)xk=pg,
where k is a random integer within [1,N] and pg is the global best position.The algorithm for ICSO is illustrated in Algorithm1 and its computational steps given as follows.(1)
Initialise cockroach swarm with uniform distributed random numbers and set all parameters with values.(2)
Findpi and pg using (12) and (13).(3)
Perform chase-swarming using (11).(4)
Perform hunger behaviour using  (14)(5)
Perform dispersion behaviour using  (15).(6)
Perform ruthless behaviour using  (16).(7)
Repeat the loop until stopping criterion is reached.Series of experiments are conducted in Section 3 using established global optimization problems to test ICSO performance. The performance of ICSO is compared with that of existing algorithms RIO, HRIO, CSO, and MCSO.Algorithm 1:An improved cockroach swarm optimization algorithm.
INPUT: Fitness function:f(x),x∈RD
set parameters and generate an initial population of cockroach
setpg=x1
for  i=2toN  do
if  f(xi)<f(pg)  then
pg=xi
end if
end for
for  t=1toTmax⁡  do
for  i=1toN  do
for  j=1toN  do
if  abs(xi-xj)<visual;f(xj)<f(xi)  then
pi=xi
end if
end for
if  pi==xi  then
xi=w·xi+step·rand·(pg-xi)
else
xi=w·xi+step·rand·(pi-xi)
end if
if  f(xi)<f(pg)  then
pg=xi
end if
end for
if  Hunger==thunger  then
xi=xi+(xi-ct)+xfd)
hungeri=0
Incrementhungeri counters
end if
for  i=1toN  do
xi=xi+rand(1,D)
if  f(xi)<f(pg)  then
pg=xi
end if
end for
k=randint([1,N])
xk=pg;
end for
Check termination condition


## 2.1. Modified Cockroach Swarm Optimization

ZhaoHui presented a modified cockroach swarm optimization (MCSO) [15] with the introduction of inertial weight to chase swarming component of original CSO as shown below. Other models remain as in original CSO.Chase-swarming behaviour is as follows:(6)xi={w·xi+step·rand·(pi-xi),xi≠piw·xi+step·rand·(pg-xi),xi=pi,
where w is an inertial weight which is a constant.


## 2.2. Improved Cockroach Swarm Optimization

In this paper, MCSO is extended with additional component called hunger behaviour.


### 2.2.1. Hunger Behaviour

At interval of time, when cockroach is hungry, it migrates from its comfortable shelter and friends company to look for food [13, 20]. Hunger behaviour is modelled using partial differential equation (PDE) migration techniques [21]. Cockroach migrates from its shelter to any available food source xfood within the search space. A threshold hunger is defined, when cockroach is hungry and threshold hunger is reached; it migrates to food source. Hunger behaviour prevents local optimum and enhances diversity of population.PDE migration equation is described by Kerckhove [21]:
(7)∂u∂t=-c∂u∂x
with u(0,x)=u0(x).Parameterc is the controlling speed of the migration. u is the population size, t is time, and x is location or position. u(t,x) is the population size at time t in location x with u(0,x)=u0(x) being the initial population distribution. Consider
(8)∂u∂t=-c∂u∂x,∂u∂t+c∂u∂x=0.The characteristic equations are(9)dt1=dxc=du0,dx-cdt=0.By integration, we have(10)x-ct=α,u=u(α),u=u(x-ct),u[t,x]=u0[-ct+x].Consider displacement = speed × time.Inu0(x-ct), u0(x) displaces ct.u
0
(
x
-
c
t
) satisfies migration equation at any initial population distribution u0(x) [21].Hunger behaviour is modelled as follows:If(hunger==thunger)(11)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number [0,1].


### 2.2.2. Improved Cockroach Swarm Optimization Models

(1) Chase-Swarming Behaviour. 										(12)xi={w·xi+step·rand·(pi-xi),xi≠pi,w·xi+step·rand·(pg-xi),xi=pi,
where w is an inertial weight which is a constant, step is a fixed value, rand is a random number within [0,1], pi is the personal best position, and pg is the global best position. Consider
(13)pi=Optj{xj,|xi-xj|≤visual},
where perception distance visual is a constant, j=1,2,…,N, i=1,2,…,N. Consider
(14)pg=Opti{xi}(2) Hunger Behaviour. If hunger==thunger,
(15)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number within [0,1].(3) Dispersion Behaviour. 										(16)xi=xi+rand(1,D),i=1,2,…,N,
where rand(1,D) is a D-dimensional random vector that can be set within a certain range.(4) Ruthless Behaviour. 										(17)xk=pg,
where k is a random integer within [1,N] and pg is the global best position.The algorithm for ICSO is illustrated in Algorithm1 and its computational steps given as follows.(1)
Initialise cockroach swarm with uniform distributed random numbers and set all parameters with values.(2)
Findpi and pg using (12) and (13).(3)
Perform chase-swarming using (11).(4)
Perform hunger behaviour using  (14)(5)
Perform dispersion behaviour using  (15).(6)
Perform ruthless behaviour using  (16).(7)
Repeat the loop until stopping criterion is reached.Series of experiments are conducted in Section 3 using established global optimization problems to test ICSO performance. The performance of ICSO is compared with that of existing algorithms RIO, HRIO, CSO, and MCSO.Algorithm 1:An improved cockroach swarm optimization algorithm.
INPUT: Fitness function:f(x),x∈RD
set parameters and generate an initial population of cockroach
setpg=x1
for  i=2toN  do
if  f(xi)<f(pg)  then
pg=xi
end if
end for
for  t=1toTmax⁡  do
for  i=1toN  do
for  j=1toN  do
if  abs(xi-xj)<visual;f(xj)<f(xi)  then
pi=xi
end if
end for
if  pi==xi  then
xi=w·xi+step·rand·(pg-xi)
else
xi=w·xi+step·rand·(pi-xi)
end if
if  f(xi)<f(pg)  then
pg=xi
end if
end for
if  Hunger==thunger  then
xi=xi+(xi-ct)+xfd)
hungeri=0
Incrementhungeri counters
end if
for  i=1toN  do
xi=xi+rand(1,D)
if  f(xi)<f(pg)  then
pg=xi
end if
end for
k=randint([1,N])
xk=pg;
end for
Check termination condition


## 2.2.1. Hunger Behaviour

At interval of time, when cockroach is hungry, it migrates from its comfortable shelter and friends company to look for food [13, 20]. Hunger behaviour is modelled using partial differential equation (PDE) migration techniques [21]. Cockroach migrates from its shelter to any available food source xfood within the search space. A threshold hunger is defined, when cockroach is hungry and threshold hunger is reached; it migrates to food source. Hunger behaviour prevents local optimum and enhances diversity of population.PDE migration equation is described by Kerckhove [21]:
(7)∂u∂t=-c∂u∂x
with u(0,x)=u0(x).Parameterc is the controlling speed of the migration. u is the population size, t is time, and x is location or position. u(t,x) is the population size at time t in location x with u(0,x)=u0(x) being the initial population distribution. Consider
(8)∂u∂t=-c∂u∂x,∂u∂t+c∂u∂x=0.The characteristic equations are(9)dt1=dxc=du0,dx-cdt=0.By integration, we have(10)x-ct=α,u=u(α),u=u(x-ct),u[t,x]=u0[-ct+x].Consider displacement = speed × time.Inu0(x-ct), u0(x) displaces ct.u
0
(
x
-
c
t
) satisfies migration equation at any initial population distribution u0(x) [21].Hunger behaviour is modelled as follows:If(hunger==thunger)(11)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number [0,1].


## 2.2.2. Improved Cockroach Swarm Optimization Models

(1) Chase-Swarming Behaviour. 										(12)xi={w·xi+step·rand·(pi-xi),xi≠pi,w·xi+step·rand·(pg-xi),xi=pi,
where w is an inertial weight which is a constant, step is a fixed value, rand is a random number within [0,1], pi is the personal best position, and pg is the global best position. Consider
(13)pi=Optj{xj,|xi-xj|≤visual},
where perception distance visual is a constant, j=1,2,…,N, i=1,2,…,N. Consider
(14)pg=Opti{xi}(2) Hunger Behaviour. If hunger==thunger,
(15)xi=xi+(xi-ct)+xfood,
where xi denotes cockroach position, (xi-ct) denotes cockroach migration from its present position, c is a constant which controls migration speed at time t, xfood denotes food location, thunger denotes hunger threshold, and hunger is a random number within [0,1].(3) Dispersion Behaviour. 										(16)xi=xi+rand(1,D),i=1,2,…,N,
where rand(1,D) is a D-dimensional random vector that can be set within a certain range.(4) Ruthless Behaviour. 										(17)xk=pg,
where k is a random integer within [1,N] and pg is the global best position.The algorithm for ICSO is illustrated in Algorithm1 and its computational steps given as follows.(1)
Initialise cockroach swarm with uniform distributed random numbers and set all parameters with values.(2)
Findpi and pg using (12) and (13).(3)
Perform chase-swarming using (11).(4)
Perform hunger behaviour using  (14)(5)
Perform dispersion behaviour using  (15).(6)
Perform ruthless behaviour using  (16).(7)
Repeat the loop until stopping criterion is reached.Series of experiments are conducted in Section 3 using established global optimization problems to test ICSO performance. The performance of ICSO is compared with that of existing algorithms RIO, HRIO, CSO, and MCSO.Algorithm 1:An improved cockroach swarm optimization algorithm.
INPUT: Fitness function:f(x),x∈RD
set parameters and generate an initial population of cockroach
setpg=x1
for  i=2toN  do
if  f(xi)<f(pg)  then
pg=xi
end if
end for
for  t=1toTmax⁡  do
for  i=1toN  do
for  j=1toN  do
if  abs(xi-xj)<visual;f(xj)<f(xi)  then
pi=xi
end if
end for
if  pi==xi  then
xi=w·xi+step·rand·(pg-xi)
else
xi=w·xi+step·rand·(pi-xi)
end if
if  f(xi)<f(pg)  then
pg=xi
end if
end for
if  Hunger==thunger  then
xi=xi+(xi-ct)+xfd)
hungeri=0
Incrementhungeri counters
end if
for  i=1toN  do
xi=xi+rand(1,D)
if  f(xi)<f(pg)  then
pg=xi
end if
end for
k=randint([1,N])
xk=pg;
end for
Check termination condition


## 3. Simulation Studies

The speed, accuracy, robustness, stability, and searching capabilities of ICSO are evaluated in this section with 23 benchmark test functions. The test functions were adopted from [22–24]; any further information about the test functions can be found in these references. The test functions are of different characteristics such as unimodal (U), multimodal (M), separable (S), and nonseparable (N). Table 1 of this paper shows the test functions used, whose problem ranges from 2 to 30 in dimension as in [22–24].Table 1
Benchmark test functions.
Number
Range
D
C
Functions
Description
1
[−100, 100]
30
US
Step
f
(
x
)
=
∑
i
=
1
n
(
⌊
x
i
+
0.5
⌋
)
2
2
[−100, 100]
30
US
Sphere
f
(
x
)
=
∑
i
=
1
n
x
i
2
3
[−10, 10]
30
US
Sumsquares
f
(
x
)
=
∑
i
=
1
n
i
x
i
2
4
[−100, 100]
2
MS
Bohachevsky1
f
(
x
)
=
x
1
2
+
2
x
2
2
-
0.3
cos
⁡
(
3
π
x
1
)
-
0.4
cos
⁡
(
4
π
x
2
)
+
0.7
5
[−100, 100]
2
MN
Bohachevsky2
f
(
x
)
=
x
1
2
+
2
x
2
2
-
0.3
cos
⁡
(
3
π
x
1
)
(
4
π
x
2
)
+
0.3
6
[−100, 100]
2
MN
Bohachevsky3
f
(
x
)
=
x
1
2
+
2
x
2
2
-
0.3
cos
⁡
(
3
π
x
1
+
4
π
x
2
)
+
0.3
7
[0, 180]
20
UN
Sinusoidal20
f
(
x
)
=
-
[
A
∏
i
=
1
n
sin
⁡
⁡
(
x
i
-
z
)
+
∏
i
=
1
n
sin
⁡
⁡
(
B
(
x
i
-
z
)
)
]
A
=
2.5
,
B
=
5
,
z
=
30
8
[−100, 100]
30
UN
Quadric
f
(
x
)
=
∑
i
=
1
n
(
∑
i
=
1
n
x
j
)
2
9
[−100, 100]
2
UN
Easom
f
(
x
)
=
-
cos
⁡
x
1
cos
⁡
x
2
·
exp
⁡
(
-
(
x
1
-
π
)
2
)
exp
⁡
(
-
(
x
2
-
π
)
2
)
10
[−10, 10]
2
UN
Matyas
f
(
x
)
=
0.26
(
x
1
+
x
2
)
-
0.48
x
1
x
2
11
[−5, 10]
10
UN
Zakharov
f
(
x
)
=
∑
i
=
1
n
(
x
i
)
2
+
(
∑
i
=
1
n
0.5
i
x
i
)
2
+
(
∑
i
=
1
n
0.5
i
x
i
)
4
12
[−10, 10]
24
UN
Powell
f
(
x
)
=
∑
i
=
1
n
/
k
(
x
4
i
-
3
+
10
x
4
i
-
2
)
2
+
5
(
x
4
i
-
1
-
x
4
i
)
2
+
(
x
4
i
-
2
-
x
4
i
-
1
)
4
+
10
(
x
4
i
-
3
-
x
4
i
)
4
13
[−10, 10]
30
UN
Schwefel2.22
f
(
x
)
=
∑
i
=
1
n
|
x
i
|
+
∏
i
=
1
n
|
x
i
|
14
[−30, 30]
30
UN
Rosenbrock
f
(
x
)
=
∑
i
=
1
n
-
1
[
100
(
x
i
+
1
-
x
i
2
)
2
+
(
x
i
-
1
)
2
]
15
[−5.12, 5.12]
30
MS
Rastrigin
f
(
x
)
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
(
2
π
x
i
)
+
10
16
[−100, 100]
2
MN
Schaffer1
f
(
x
)
=
0.5
+
sin
⁡
2
⁡
x
1
2
+
x
2
2
2
-
0.5
[
1
+
0.001
(
x
1
2
+
x
2
2
)
]
2
17
[−100, 100]
30
MN
Schaffer2
f
(
x
)
=
(
x
1
2
+
x
2
2
)
0.25
(
sin
⁡
2
⁡
(
50
(
x
1
2
+
x
2
2
)
0.1
)
+
1
)
18
[−600, 600]
30
MN
Griewangk
f
(
x
)
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
(
x
i
i
)
+
1
19
[−32, 32]
30
MN
Ackley
f
(
x
)
=
-
20
exp
(
-
0.2
∑
i
=
1
n
‍
(
x
i
2
/
n
)
)
⁡
-
⁡
exp
(
∑
i
=
1
n
‍
cos
⁡
(
2
π
x
i
/
n
)
)
⁡
+
20
+
e
20
[−5, 5]
2
MN
Three hump camel back
f
(
x
)
=
2
x
1
2
-
1.05
x
1
4
+
1
6
x
1
6
+
x
1
x
2
+
x
2
2
21
[−5, 5]
2
MN
Six hump camel back
f
(
x
)
=
4
x
1
2
-
2.1
x
1
4
+
1
3
x
1
6
+
x
1
x
2
-
4
x
2
2
+
4
x
2
4
22
[
-
128, 128]n
9
UN
Storn's Tchebychev
f
(
x
)
=
p
1
+
p
2
+
p
3,
23
[
-
32768, 32768]n
17
Storn's Tchebychev
where
p
1
=
{
(
u
-
d
)
2
if
u
<
d
0
if
u
≥
d
u
=
∑
i
=
1
n
(
1.2
)
n
-
i
x
i
p
2
=
{
(
v
-
d
)
2
if
v
<
d
0
if
v
≥
d
v
=
∑
i
=
1
n
(
-
1.2
)
n
-
i
x
i


p
3
=
∑
j
=
0
m
{
(
w
j
-
1
)
2
if
w
j
>
1
(
w
j
+
1
)
2
if
w
j
<
-
1
0
if
-
1
≤
w
j
≤
1
w
j
=
∑
i
=
1
n
(
2
j
m
-
1
)
n
-
i
x
i
, for n=9: d=72.661, and m=60 for n=17: d=10558.145, and m=100.
D
: dimension; C: characteristic; U: unimodal; S: seperable; N: non-separable.All algorithms were implemented in MATLAB 7.14 (R2012a) and run on a computer with 2.30 GHz processor with 4.00 GB of RAM. Experimental setting of [13–15] is used for the experiments of this paper; experiment runs 20 times with maximum iteration 1000, perception distance visual=5, the largest step was step=2, and inertia weight was w=0.618; we defined hunger threshold thunger=0.5 and hunger as a randomly generated number [0,1] in each iteration for ICSO. Cockroach parameters [13] are used for RIO and HRIO; c0=0.7 and cmax⁡=1.43, hunger threshold thunger=100, and hunger as randomly generated number [0,(thunger-1)]. Cockroach population size N=50 is used in this paper for all the algorithms. Further details about RIO, HRIO, CSO, and MSCO can be found in [13–15].ICSO along with similar algorithms, that is, CSO, MSCO, RIO, and HRIO, was implemented with several simulation experiments conducted and reported. Success rate, average and best fitness, standard deviation (STD), and execution time in seconds are used as performance measure for comparative purpose (see Tables2, 3, and 4 of this paper).Table 2
Simulation results of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Fn.
Dim.
Opt.
RIO
HRIO
CSO
MCSO
ICSO
1
Boha1
2
0
Ave.
3.4405
E
-
05
3.2877
E
-
04
2.9893
E
02
3.5153
E
-
09
0.0000
STD
2.5963
E
-
05
3.0334
E
-
04
5.0332
E
02
1.4392
E
-
08
0.0000
Best
1.3520
E
-
07
5.2651
E
-
06
2.0651
E
-
05
0.0000
0.0000
Success
20/20
20/20
5/20
20/20
20/20
Time
1.137525
0.886356
23.913237
0.075212
0.097187
2
Boha2
2
0
Ave.
4.2829
E
-
05
4.6703
E
-
04
9.0941
E
02
8.4459
E
-
12
0.0000
STD
3.0070
E
-
05
3.4047
E
-
04
1.7794
E
03
2.9240
E
-
11
0.0000
Best
2.2910
E
-
06
9.374
E
-
06
1.3775
E
-
05
0.0000
0.0000
Success
20/20
20/20
4/20
20/20
20/20
Time
0.998178
0.946887
26.492095
0.072021
0.074106
3
Boha3
2
0
Ave.
5.3479
E
-
05
4.7575
E
-
04
7.4284
E
02
2.1388
E
-
14
0.0000
STD
2.9141
E
-
05
2.3273
E
-
04
1.6739
E
03
4.8670
E
-
14
0.0000
Best
3.1200
E
-
06
4.6981
E
-
05
2.3093
E
-
07
0.0000
0.0000
Success
20/20
20/20
3/20
20/20
20/20
Time
1.089920
0.885252
25.028054
0.080908
0.068189
4
3camel
2
0
Ave.
1.4962
E
-
02
4.3021
E
-
04
5.003
E
09
7.098
E
-
11
5.9853
E
-
31
STD
6.6769
E
-
02
2.8371
E
-
04
1.7137
E
10
3.0201
E
-
10
2.5457
E
-
30
Best
1.1739
E
-
06
2.2449
E
-
05
1.7642
E
-
05
3.1395
E
-
19
2.2320
E
-
53
Success
19/20
20/20
12/20
20/20
20/20
Time
4.231533
0.794983
18.281683
0.104132
0.078845
5
6camel
2
-
1.03163
Ave.
-
4.3522
E
-
01
-
4.7652
E
-
01
1.5763
E
05
-
1.0263
E
-
08
-
2.9798
E
-
25
STD
3.3322
E
-
01
3.1284
E
-
01
7.0503
E
05
4.4391
E
-
08
1.3325
E
-
24
Best
-
1.0215
-
1.0034
-
9.4052
E
-
01
-
1.9879
E
-
07
-
5.9589
E
-
24
Success
20
/
20
20
/
20
19
/
20
20
/
20
20
/
20
Time
0.406355
0.330198
5.723039
0.0945856
0.086637
6
Easom
2
-
1
Ave.
-
1
-
1
-
4.3165
E
-
01
-
1
-
1
STD
3.7518
E
-
02
2.1031
E
-
02
3.4470
E
-
01
1.4897
E
-
08
4.4116
E
-
17
Best
-
1
-
1
-
1
-
1
-
1
Success
20/20
20/20
20/20
20/20
20/20
Time
0.124022
0.107303
0.106738
0.077179
0.092393
7
Matyax
2
0
Ave.
4.9470
E
-
05
3.2297
E
-
04
7.5712
2.6876
E
-
13
4.0732
E
-
35
STD
3.0244
E
-
05
2.6018
E
-
04
1.1247
E
01
8.9347
E
-
13
1.8125
E
-
34
Best
6.2897
E
-
06
1.2684
E
-
05
8.8777
E
-
06
6.6695
E
-
21
1.1292
E
-
55
Success
20/20
20/20
11/20
20/20
20/20
Time
0.973322
0.711734
13.559576
0.88536
0.076693
8
Schaffer1
2
-
1
Ave.
-
1.9069
-
1.6211
-
2.9174
E
-
01
-
1
-
1
STD
7.0381
E
-
01
5.9214
E
-
01
7.5142
E
-
01
5.9575
E
-
07
4.1325
E
-
15
Best
-
2.7458
-
2.7164
-
2.7438
-
1
-
1
Success
20/20
20/20
20/20
20/20
20/20
Time
0.109048
0.086433
0.119076
0.072400
0.081599
9
Schaffer2
2
0
Ave.
2.0179
E
-
03
1.6566
E
-
03
7.1618
3.3168
E
-
04
2.2149
E
-
09
STD
2.6407
E
-
03
1.4451
E
-
03
5.3095
3.0328
E
-
04
2.9483
E
-
09
Best
6.2423
E
-
05
4.1422
E
-
04
2.8354
E
-
01
1.5810
E
-
05
1.9383
E
-
14
Success
2
/
20
13
/
20
0
/
20
20
/
20
20
/
20
Time
62.567654
31.415836
29.194283
0.084127
0.082320
Dim. denotes dimension. Opt. denotes optimum value. Boha1 denotes Bohachevsky1. Boha2 denotes Bohachevsky2. Boha3 denotes Bohachevsky3. 3camel denotes three hump camel back. 6camel denotes six hump camel back.Table 3
Simulation results of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Fn.
Dim.
Opt.
RIO
HRIO
CSO
MCSO
ICSO
10
Sphere
30
0
Ave.
2.2168
E
-
05
1.6676
E
-
04
1.8123
E
02
1.5201
E
-
12
3.3448
E
-
34
STD
2.4528
E
-
05
2.4018
E
-
04
8.1048
E
02
6.7224
E
-
12
1.3324
E
-
33
Best
5.7627
E
-
09
5.5635
E
-
08
4.9195
E
-
07
2.9978
E
-
24
2.8205
E
-
54
Success
20/20
20/20
19/20
20/20
20/20
Time
0.617544
0.557871
25.378161
0.82512
0.199373
11
Rastrigin
30
0
Ave.
3.8135
E
-
05
3.2150
E
-
04
3.6022
E
03
9.1994
E
-
11
0.0000
STD
3.4436
E
-
05
3.0003
E
-
04
5.5728
E
03
3.9456
E
-
10
0.0000
Best
2.7098
E
-
07
2.1450
E
-
07
3.1340
E
-
04
0.0000
0.0000
Success
20/20
20/20
5/20
20/20
20/20
Time
0.956329
0.826770
71.811170
0.175563
0.369987
12
Rosenbrock
30
0
Ave.
2.5281
E
06
3.3571
E
06
9.5067
E
11
2.9000
E
01
2.9000
E
01
STD
4.0528
E
06
7.1150
E
06
2.2713
E
12
0.0000
0.0000
Best
1.6773
E
04
3.7562
E
04
4.4068
E
01
2.9000
E
01
2.9000
E
01
Success
0/20
0/20
0/20
0/20
0/20
Time
126.618734
127.469638
81.361663
76.084929
78.572185
13
Ackley
30
0
Ave.
2.0001
E
01
2.0005
E
01
1.9222
E
01
5.1593
E
-
06
1.0651
E
-
15
STD
3.0455
E
-
03
1.5671
E
-
02
5.8258
1.9149
E
-
05
7.9441
E
-
16
Best
2.0001
E
01
1.9998
E
01
2.0133
E
01
6.4623
E
-
09
8.1818
E
-
16
Success
0/20
0/20
0/20
20/20
20/20
Time
122.216187
117.635854
82.227210
0.235012
0.192339
14
Quadric
30
0
Ave.
2.4498
E
-
05
2.2711
E
-
04
3.4991
E
-
04
4.4754
E
-
13
7.2183
E
-
28
STD
2.7957
E
-
05
2.3635
E
-
04
3.3725
E
-
04
1.9751
E
-
12
3.2218
E
-
27
Best
1.1360
E
-
08
5.8230
E
-
07
4.1551
E
-
08
5.6309
E
-
23
5.910
E
-
52
Success
20/20
20/20
20/20
20/20
20/20
Time
0.718785
0.512242
31.075809
0.247456
0.227244
15
Schwefel2.22
30
0
Ave.
2.3131
E
02
2.4395
E
02
2.9013
E
54
6.3587
E
-
06
6.0407
E
-
16
STD
1.3193
E
02
1.2341
E
02
1.2971
E
55
1.1936
E
-
05
1.2203
E
-
15
Best
6.7400
E
01
1.7354
E
01
3.6854
E
01
5.9410
E
-
08
5.1670
E
-
24
Success
0/20
0/20
0/20
20/20
20/20
Time
128.445013
127.084387
79.924516
0.217104
0.219296
16
Griewangk
30
0
Ave.
7.9510
E
-
01
7.7746
E
-
01
2.6148
E
01
3.3151
E
-
11
0.0000
STD
3.7583
E
-
01
2.5454
E
-
01
3.6626
E
01
1.4672
E
-
10
0.0000
Best
2.9324
E
-
01
3.2031
E
-
01
6.3912
E
-
05
0.0000
0.0000
Success
0/20
0/20
5/20
20/20
20/20
Time
126.872461
126.210153
70.852376
0.211351
0.210934
17
Sumsquare
30
0
Ave.
1.9818
E
03
4.6771
E
03
9.0499
E
05
4.2446
E
-
11
1.5600
E
-
24
STD
2.8370
E
03
6.7104
E
03
1.0253
E
06
1.2930
E
-
10
6.9785
E
-
24
Best
1.6463
E
01
2.0516
E
02
1.8730
E
02
1.49990
E
-
16
1.3765
E
-
47
Success
0/20
0/20
0/20
20/20
20/20
Time
122.748646
125.154349
78.809270
0.273780
0.236129
18
Sinusoidal
30
-
3.5
Ave.
-
4.2587
E
-
01
-
3.7898
E
-
01
-
2.449
-
3.1030
-
3.1030
STD
2.6632
E
-
01
1.9791
E
-
01
1.0203
5.0473
E
-
05
1.9436
E
-
14
Best
-
1.1922
-
8.3111
E
-
01
-
3.3087
-
3.1032
-
3.1030
Success
20/20
20/20
20/20
20/20
20/20
Time
0.204559
0.240200
0.234205
0.200361
0.217635
Dim. denotes dimension. Opt. denotes optimum value.Table 4
Simulation results of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
Dim.
Opt.
RIO
HRIO
CSO
MCSO
ICSO
19
Zakharov
30
0
Ave.
1.0167
E
04
1.0216
E
04
6.3663
E
18
2.3878
E
-
09
4.1579
E
-
26
STD
3.8643
E
03
5.1012
E
03
2.2732
E
19
8.8529
E
-
09
1.8549
E
-
25
Best
2.6634
E
03
2.3151
E
03
1.3578
E
09
2.0954
E
-
15
6.3965
E
-
57
Success
0/20
0/20
0/20
20/20
20/20
Time
115.192226
114.691827
79.926232
0.205280
0.259202
20
Step
30
0
Ave.
0.0000
0.0000
2.0004
E
04
0.0000
0.0000
STD
0.0000
0.0000
8.4815
E
04
0.0000
0.0000
Best
0.0000
0.0000
0.0000
0.0000
0.0000
Success
20/20
20/20
16/20
20/20
20/20
Time
0.686403
0.633264
39.136696
0.239525
0.225102
21
Powell
24
0
Ave.
1.8348
E
-
03
3.7434
E
-
03
1.0840
E
08
2.6031
E
-
12
1.8207
E
-
24
STD
1.6248
E
-
03
6.1711
E
-
03
4.1180
E
08
6.9959
E
-
12
5.6824
E
-
24
Best
9.6693
E
-
05
6.8033
E
-
04
5.2392
E
01
1.2287
E
-
19
1.2265
E
-
54
Success
2/20
12/20
0/20
20/20
20/20
Time
122.796991
92.876086
74.794730
1.527170
0.853751
22
ST
9
0
Ave.
0.0000
0.0000
0.0000
0.0000
0.0000
STD
0.0000
0.0000
0.0000
0.0000
0.0000
Best
0.0000
0.0000
0.0000
0.0000
0.0000
Success
20/20
20/20
20/20
20/20
20/20
Time
0.435911
0.426320
0.437944
0.431122
0.436741
23
ST
17
0
Ave.
0.0000
0.0000
0.0000
0.0000
0.0000
STD
0.0000
0.0000
0.0000
0.0000
0.0000
Best
0.0000
0.0000
0.0000
0.0000
0.0000
Success
20/20
20/20
20/20
20/20
20/20
Time
1.066161
1.052169
1.159830
1.089657
1.147114
Dim. denotes dimension. Opt. denotes optimum value.ICSO locates minimum values for the tested benchmark problems such as Bohachevsky, Rastrigin, Easom, Schaffer, Step, and Storn’s Tchebychev problems as shown in Tables2, 3, and 4. The comparison of the average performance of ICSO with that of RIO, HRIO, CSO, and MCSO is shown in Table 5; the comparison result clearly shows that ICSO outperforms other algorithms. Similarly, the best performance of ICSO with that of RIO, HRIO, CSO, and MCSO is shown in Table 6; ICSO has better performance than others.Table 5
Comparison of average performance of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
RIO
HRIO
CSO
MCSO
ICSO
Optimum
1
Bohachevsky1
3.4405
E
-
05
3.2877
E
-
04
2.9893
E
02
3.5153
E
-
09
0.0000
0
2
Bohachevsky2
4.2829
E
-
05
4.6703
E
-
04
9.0941
E
02
8.4459
E
-
12
0.0000
0
3
Bohachevsky3
5.3479
E
-
05
4.7575
E
-
04
7.4284
E
02
2.1388
E
-
14
0.0000
0
4
3 Hump camel back
1.4962
E
-
02
4.3021
E
-
04
5.003
E
09
7.098
E
-
11
5.9853
E
-
31
0
5
6 Hump camel back
-
4.3522
E
-
01
-
4.7652
E
-
01
1.5763
E
05
-
1.0263
E
-
08
-
2.9798
E
-
25
-
1.03163
6
Easom
-
1
-
1
-
4.3165
E
-
01
-
1
-
1
-
1
7
Matyax
4.9470
E
-
05
3.2297
E
-
04
7.5712
2.6876
E
-
13
4.0732
E
-
35
0
8
Schaffer1
-
1.9069
-
1.6211
-
2.9174
E
-
01
-
1
-
1
-
1
9
Schaffer2
2.0179
E
-
03
1.6566
E
-
03
7.1618
3.3168
E
-
04
2.2149
E
-
09
0
10
Sphere
2.2168
E
-
05
1.6676
E
-
04
1.8123
E
02
1.5201
E
-
12
3.3448
E
-
34
0
11
Rastrigin
3.8135
E
-
05
3.2150
E
-
04
3.6022
E
03
9.1994
E
-
11
0.0000
0
12
Rosenbrock
2.5281
E
06
3.3571
E
06
9.5067
E
11
2.9000
E
01
2.9000
E
01
0
13
Ackley
2.0001
E
01
2.0005
E
01
1.9222
E
01
5.1593
E
-
06
1.0651
E
-
15
0
14
Quadric
2.4498
E
-
05
2.2711
E
-
04
3.4991
E
-
04
4.4754
E
-
13
7.2183
E
-
28
0
15
Schwefel2.22
2.3131
E
02
2.4395
E
02
2.9013
E
54
6.3587
E
-
06
6.0407
E
-
16
0
16
Griewangk
7.9510
E
-
01
7.7746
E
-
01
2.6148
E
01
3.3151
E
-
11
0.0000
0
17
Sumsquare
1.9818
E
03
4.6771
E
03
9.0499
E
05
4.2446
E
-
11
1.5600
E
-
24
0
18
Sinusoidal
-
4.2587
E
-
01
-
3.7898
E
-
01
-
2.449
-
3.1030
-
3.1030
-
3.5
19
Zakharov
1.0167
E
04
1.0216
E
04
6.3663
E
18
2.3878
E
-
09
4.1579
E
-
26
0
20
Step
0.0000
0.0000
2.0004
E
04
0.0000
0.0000
0
21
Powell
1.8348
E
-
03
3.7434
E
-
03
1.0840
E
08
2.6031
E
-
12
1.8207
E
-
24
0
22
ST9
0.0000
0.0000
0.0000
0.0000
0.0000
0
23
ST17
0.0000
0.0000
0.0000
0.0000
0.0000
0
Number of good optimums
4
4
2
7
23
ST9 denotes Storn's Tchebychev 9. ST17 denotes Storn's Tchebychev 17.Table 6
Comparison of best performance of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
RIO
HRIO
CSO
MCSO
ICSO
Optimum
1
Bohachevsky1
1.3520
E
-
07
5.2651
E
-
06
2.0651
E
-
05
0.0000
0.0000
0
2
Bohachevsky2
2.2910
E
-
06
9.374
E
-
06
1.3775
E
-
05
0.0000
0.0000
0
3
Bohachevsky3
3.1200
E
-
06
4.6981
E
-
05
2.3093
E
-
07
0.0000
0.0000
0
4
3 hump camel back
1.1739
E
-
06
2.2449
E
-
05
1.7642
E
-
05
3.1395
E
-
19
2.2320
E
-
53
0
5
6 hump camel back
-
1.0215
-
1.0034
-
9.4052
E
-
01
-
1.9879
E
-
07
5.9589
E
-
24
-
1.03163
6
Easom
-
1
-
1
-
1
-
1
-
1
-
1
7
Matyax
6.2897
E
-
06
1.2684
E
-
05
8.8777
E
-
06
6.6695
E
-
21
1.1292
E
-
55
0
8
Schaffer1
-
2.7458
-
2.7164
-
2.7438
-
1
-
1
-
1
9
Schaffer2
6.2423
E
-
05
4.1422
E
-
04
2.8354
E
-
01
1.5810
E
-
05
1.9383
E
-
14
0
10
Sphere
5.7627
E
-
09
5.5635
E
-
08
4.9195
E
-
07
2.9978
E
-
24
2.8205
E
-
54
0
12
Rosenbrock
1.6773
E
04
3.7562
E
04
4.4068
E
01
2.9000
E
01
2.9000
E
01
0
14
Quadric
1.1360
E
-
08
5.8230
E
-
07
4.1551
E
-
08
5.6309
E
-
23
5.910
E
-
52
0
15
Schwefel2.22
6.7400
E
01
1.7354
E
01
3.6854
E
01
5.9410
E
-
08
5.1670
E
-
24
0
16
Griewangk
2.9324
E
-
01
3.2031
E
-
01
6.3912
E
-
05
0.0000
0.0000
0
17
Sumsquare
1.6463
E
01
2.0516
E
02
1.8730
E
02
1.49990
E
-
16
1.3765
E
-
47
0
18
Sinusoidal
-
1.1922
-
8.3111
E
-
01
-
3.3087
-
3.1032
-
3.1030
-
3.5
19
Zakharov
2.6634
E
03
2.3151
E
03
1.3578
E
09
2.0954
E
-
15
6.3965
E
-
57
0
20
Step
0.0000
0.0000
0.0000
0.0000
0.0000
0
21
Powell
9.6693
E
-
05
6.8033
E
-
04
5.2392
E
01
1.2287
E
-
19
1.2265
E
-
54
0
22
ST9
0.0000
0.0000
0.0000
0.0000
0.0000
0
23
ST17
0.0000
0.0000
0.0000
0.0000
0.0000
0
Number of good optimums
4
4
5
11
22
ST9 denotes Storn's Tchebychev 9. ST17 denotes Storn's Tchebychev 17.ICSO algorithm has consistent performance in each iteration. This is proved by very low standard deviation of the average optimal recoded during experiments. The ICSO average optimal STD is compared with the STD of RIO, HRIO, CSO, and MCSO in Table7. ICSO has better minimum STD than others.Table 7
Comparison of standard deviation of mean global optimum values of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
RIO
HRIO
CSO
MCSO
ICSO
1
Bohachevsky1
2.5963
E
-
05
3.0334
E
-
04
5.0332
E
02
1.4392
E
-
08
0.0000
2
Bohachevsky2
3.0070
E
-
05
3.4047
E
-
04
1.7794
E
03
2.9240
E
-
11
0.0000
3
Bohachevsky3
2.9141
E
-
05
2.3273
E
-
04
1.6739
E
03
4.8670
E
-
14
0.0000
4
3 hump camel back
6.6769
E
-
02
2.8371
E
-
04
1.7137
E
10
3.0201
E
-
10
2.5457
E
-
30
5
6 hump camel back
3.3322
E
-
01
3.1284
E
-
01
7.0503
E
05
4.4391
E
-
08
1.3325
E
-
24
6
Easom
3.7518
E
-
02
2.1031
E
-
02
3.4470
E
-
01
1.4897
E
-
08
4.4116
E
-
17
7
Matyax
3.0244
E
-
05
2.6018
E
-
04
1.1247
E
01
8.9347
E
-
13
1.8125
E
-
34
8
Schaffer1
7.0381
E
-
01
5.9214
E
-
01
7.5142
E
-
01
5.9575
E
-
07
4.1325
E
-
15
9
Schaffer12
2.6407
E
-
03
1.4451
E
-
03
5.3095
3.0328
E
-
04
2.9483
E
-
09
10
Sphere
2.4528
E
-
05
2.4018
E
-
04
8.1048
E
02
6.7224
E
-
12
1.3324
E
-
33
11
Rastrigin
3.4436
E
-
05
3.0003
E
-
04
5.5728
E
03
3.9456
E
-
10
0.0000
12
Rosenbrock
4.0528
E
06
7.1150
E
06
2.2713
E
12
0.0000
0.0000
13
Ackley
3.0455
E
-
03
1.5671
E
-
02
5.8258
1.9149
E
-
05
7.9441
E
-
16
14
Quadric
2.7957
E
-
05
2.3635
E
-
04
3.3725
E
-
04
1.9751
E
-
12
3.2218
E
-
27
15
Schwefel2.22
1.3193
E
02
1.2341
E
02
1.2971
E
55
1.1936
E
-
05
1.2203
E
-
15
16
Griewangk
3.7583
E
-
01
2.5454
E
-
01
3.6626
E
01
1.4672
E
-
10
0.0000
17
Sumsquare
2.8370
E
03
6.7104
E
03
1.0253
E
06
1.2930
E
-
10
6.9785
E
-
24
18
Sinusoidal
2.6632
E
-
01
1.9791
E
-
01
1.0203
5.0473
E
-
05
1.9436
E
-
14
19
Zakharov
3.8643
E
03
5.1012
E
03
2.2732
E
19
8.8529
E
-
09
1.8549
E
-
25
20
Step
0.0000
0.0000
8.4815
E
04
0.0000
0.0000
21
Powell
1.6248
E
-
03
6.1711
E
-
03
4.1180
E
08
6.9959
E
-
12
5.6824
E
-
24
22
ST9
0.0000
0.0000
0.0000
0.0000
0.0000
23
ST17
0.0000
0.0000
0.0000
0.0000
0.0000
Number of good STD
2
2
2
4
23
ST9 denotes Storn's Tchebychev 9. ST17 denotes Storn's Tchebychev 17.ICSO locates good solutions in each experiment; this is proved by the success rate of the algorithm. Table8 shows the comparison of the success rate of the proposed algorithm with the existing algorithms RIO, HRIO, CSO, and MCSO. ICSO has 100% success rate in all test functions except Rosenbrock.Table 8
Comparison of success performance of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
RIO
HRIO
CSO
MCSO
ICSO
1
Bohachevsky1
1
1
2.5
1
1
2
Bohachevsky2
1
1
0.2
1
1
3
Bohachevsky3
1
1
0.15
1
1
4
3 hump camel back
0.95
1
0.6
1
1
5
6 hump camel back
1
1
0.95
1
1
6
Easom
1
1
1
1
1
7
Matyax
1
1
0.55
1
1
8
Schaffer1
1
1
1
1
1
9
Schaffer2
0.1
0.65
0
1
1
10
Sphere
1
1
0.95
1
1
11
Rastrigin
1
1
0.25
1
1
12
Rosenbrock
0
0
0
0
0
13
Ackley
0
0
0
1
1
14
Quadric
1
1
1
1
1
15
Schwefel2.22
0
0
0
1
1
16
Griewangk
0
0
0.25
1
1
17
Sumsquare
0
0
0
1
1
18
Sinusoidal
1
1
1
1
1
19
Zakharov
0
0
0
1
1
20
Step
1
1
0.8
1
1
21
Powell
0.1
0.6
0
1
1
22
ST9
1
1
1
1
1
23
ST17
1
1
1
1
1
Number of 100% success rates
14
15
6
22
22
ST9 denotes Storn's Tchebychev 9. ST17 denotes Storn's Tchebychev 17.ICSO utilizes minimum time in executing the selected test function. Table9 shows the comparison of the execution time of ICSO and that of RIO, HRIO, CSO, and MCSO; ICSO is shown to have utilized minimum time.Table 9
Comparison of exec1ution time of RIO, HRIO, CSO, MCSO, and ICSO.
SN
Function
RIO
HRIO
CSO
MCSO
ICSO
1
Bohachevsky1
1.137525
0.886356
23.913237
0.075212
0.097187
2
Bohachevsky2
0.998178
0.946887
26.492095
0.072021
0.074106
3
Bohachevsky3
1.089920
0.885252
25.028054
0.080908
0.068189
4
3 hump camel back
4.231533
0.794983
18.281683
0.104132
0.078845
5
6 hump camel back
0.406355
0.330198
5.723039
0.0945856
0.086637
6
Easom
0.124022
0.107303
0.106738
0.077179
0.092393
7
Matyax
0.973322
0.711734
13.559576
0.88536
0.076693
8
Schaffer1
0.109048
0.086433
0.119076
0.072400
0.081599
9
Schaffer2
62.567654
31.415836
29.194283
0.084127
0.082320
10
Sphere
0.617544
0.557871
25.378161
0.82512
0.199373
11
Rastrigin
0.956329
0.826770
71.811170
0.175563
0.369987
12
Rosenbrock
126.618734
127.469638
81.361663
76.084929
78.572185
13
Ackley
122.216187
117.635854
82.227210
0.235012
0.192339
14
Quadric
0.718785
0.512242
31.075809
0.247456
0.227244
15
Schwefel2.22
128.445013
127.084387
79.924516
0.217104
0.219296
16
Griewangk
126.872461
126.210153
70.852376
0.211351
0.210934
17
Sumsquare
122.748646
125.154349
78.809270
0.273780
0.236129
18
Sinusoidal
0.204559
0.240200
0.234205
0.200361
0.217635
19
Zakharov
115.192226
114.691827
79.926232
0.205280
0.259202
20
Step
0.686403
0.633264
39.136696
0.239525
0.225102
21
Powell
122.796991
92.876086
74.794730
1.527170
0.853751
22
ST9
0.435911
0.426320
0.437944
0.431122
0.436741
23
ST17
1.066161
1.052169
1.159830
1.089657
1.147114
Number of minimum execution times
—
2
—
9
12
ST9 denotes Storn's Tchebychev 9. ST17 denotes Storn's Tchebychev 17.To determine the significant difference between the performance of the proposed algorithm and the existing algorithms, test statistic of Jonckheere-Terpstra (J-T) test was conducted using the statistical package for the social science (SPSS). The Null hypothesis test for J-T test is that there is no difference among several independent groups. As the usual practice in most literature,P value threshold value for hypothesis test was set to 0.05. If P value is less than 0.05, the Null is rejected which means there is significant difference between the groups. Otherwise the Null hypothesis is accepted. Table 10 shows the result of J-T test; P value (Asymp. Sig.) was computed to be 0.001. The P value is less than the threshold value 0.05; therefore, there is significant difference in performance of ICSO and that of RIO, HRIO, CSO, and MCSO for benchmarks evaluated.Table 10
Jonckheere-Terpstra test statisticsa.
Fitness
Number of levels in algorithm
5
N
114
Observed J-T statistic
1952.000
Mean J-T statistic
2599.500
STD of J-T statistic
199.355
Standard data of J-T statistic
-
3.245
Asymp. Sig. (2-tailed)
0.001
aGrouping variable: algorithm.Effect size of the significant difference is the measure of the magnitude of the observed effect. The effect sizer, (1>r<0) of the significant difference of J-T test, was calculated as
(18)r=ZN,
where Z is the standard data of J-T statistic as shown in Table 10, N is the total number of samples, and N=114. Consider
(19)Z=x-μσ,
where x denotes observed J-T statistic, μ denotes the mean J-T statistic, and σ denoted the standard deviation of J-T statistic. Consider
(20)Z=1952-2599199.355=-3.245,r=-3.245114=-0.3.The distance between the observed data and the mean in units of standard deviation is absolute value of|Z| (Z is negative when observed data is below the mean and positive when above). The effect size 0.3 is of medium size, using Cohen’s guideline on effect size [25, 26]. The statistics of 0.3 effect size shows that there is significant difference of medium magnitude between proposed algorithm and existing algorithms.


## 4. Conclusion

Cockroach swarm optimization algorithm is extended in this paper with a new component called hunger component. Hunger component enhances the algorithm diversity and searching capability. An improved cockroach swarm optimization algorithm is proposed. The efficiency of the proposed algorithm is shown through empirical studies where its performance was compared with that of existing algorithms, that is, CSO, MSCO, RIO, and HRIO. Results show its outstanding performance compared to the existing algorithms. Application of the algorithm to real life problems can be considered in further studies.



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*Source: 375358-2014-05-14.xml*