Abstract—Biogeography-Based Optimization (BBO) is a recently developed global optimization algorithm and has shown its ability to solve complex optimization problem. In order to speed the optimization process and get better results, an improved opposition-based biogeography optimization (IOBBO) method is proposed. By dividing the range of values into several areas, the proposed method produce more evenly distributed initial population. Considering some of the optimal values are not in the center of the problem domain, in this paper, the current optimal solution is introduced. In the process of taking the opposite populations, the current optimal solution is instead with the range of the upper and lower limits. Experiments results clearly showed that the IOBBO o u t p e r f o r ms t h e o p p o s i t i o n - b a s e d b i o g e o g r a p h y optimization(OBBO) and improvement of Opposition-based Differential Evolution(IODE) on five benchmark test functions.

I. INTRODUCTION IOGEOGRAPHY based optimization (BBO) is a new evolutionary algorithm which was proposed by D. Simon[1] . The basic idea is based on the migration of

species between habitats to complete the flow of information, the migration process by adjusting the immigration rate and emigration rate, migration topology, migration interval and migration strategies to achieve information sharing, improve the habitat suitability to obtain the optimal solution.

In a group of neighboring islands, the islands represent the solutions and they are ranked by their habitat suitability index (HSI), where a higher HSI presents a superior �tness value. The islands are comprised of solution features named suitability index variables (SIV), equivalent to GA’s genes.

BBO is one of the newest evolutionary algorithms, but it has been proven to be a useful optimization algorithms. Markov analysis in [2] proved that BBO outperformed GA on simple unimodal, multimodal and deceptive benchmark functions when used with low mutation rates. D. Simon provides experimental studies comparing BBO’s performance to many other EA’s on a wide set of benchmarks[3].

However, BBO algorithm still leaves some room for improvement, as many other techniques have been developed to enhance other EAs. D. Simon and M. Ergezer proposed a Manuscript received June 30, 2011. This work was supported in part by the Program for Liaoning Excellent Talents in University (2008RC32) and Program for Liaoning Science and Technology Innovative Research Team in University (2009T062, LT2010058). Xin YANG is with the School of Information and Control Engineering, Liaoning Shihua University (e-mail: yangxin6678010@163.com). Jiangtao CAO is with the School of Information and Control Engineering, Liaoning Shihua University, (e-mail: jtcao@lnpu.edu.cn). Kairu LI is with the School of Control Science and Engineering, Dalian University of Technology. Ping LI is with the School of Information and Control Engineering, Liaoning Shihua University, Professor (e-mail: liping@lnpu.edu.cn).

strategy based on opposition-based learning (OBL), Oppositional Biogeography-Based Optimization (OBBO) [4].Which not only calculates the current search point OBBO also calculate its opposite point. At the same time by assessing the current search point and the opposite point, it can get a better approximation of the global optimum.

In this paper, an Improved Opposition-based Biogeography Optimization (IOBBO) was proposed which by dividing the range of values into several areas, the proposed method produce more evenly distributed initial population, and moreover, the current optimal solution is introduced. In the process of taking the opposite populations, the current optimal solution is instead with the range of the upper and lower limits. In order to verify IOBBO performance, we test it on 5 well-known benchmark functions. The results show that IOBBO achieves better results than OBBO and IODE on the majority of test problems.

II. THE BIOGEOGRAPHY-BASED OPTIMIZATION ALGORITHM

Biology Geography is a discipline which researches the law of species about distribution, migration and extinction in the habitat. Species of plants and animals living in different habitats, different habitats are isolated from each. some habitats may tend to accumulate more species than others because they posses certain environmental features that are more suitable to sustaining those species than habitats with fewer species. Geographical areas that are well suited as residences for biological species are said to have a high habitat suitability index (HSI). Features that correlate with HSI include factors such as rainfall, diversity of vegetation, diversity of topographic features, land area, and temperature. The variables that characterize habitability are called suitability index variables (SIVs). SIVs can be considered the independent variables of the habitat, and HSI can be executed using these variables. Habitats with a high HSI tend to have a large number of species, while those with a low HSI have a small number of species. The worst solution has the highest immigration rate; hence, it has a very high chance of borrowing features from other solutions, helping it to improve for the next generation. The best solution has a very low immigration rate, indicating that it is less likely to be altered by the other solutions. The emigration rate works similarly.

Mathematically the concept of emigration and immigration can be represented by a probabilistic model. Let us, consider the probability SP that the habitat contains exactly S species at time t . SP changes from time t to time t t+ Δ as follows:

( ) ( ) ( )( )( ) ( )1 1 1 1

1S S S S

S S S S

P t t P t t

P t t P t t

λ μ

λ μ− − + +

+ Δ = − + Δ

+ Δ + Δ (1)

Improved Opposition-based Biogeography Optimization Xin Yang, Jiangtao Cao, Kairu Li, and Ping Li

B

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE 642

Where sλ and sμ are the immigration and emigration rates when there are S species in the habitat. This equation holds because in order to have S species at time t t+ Δ , one of the following conditions must hold: 1) there were S species at time t , and no immigration or emigration occurred between t to t t+ Δ ; 2) there were 1S − species at time t , and one species immigrated; 3) there were 1S + species at time t , and one species emigrated.

If time tΔ is small enough so that the probability of more

than one immigration or emigration can be ignored then

taking the limit of equation (1) as 0tΔ → gives the following

equation

0 0 1 1

1 1 1 1 max

1 1 max

0( ) 1 1S S S S S S S S

n n n n

P P SP P P P S S

P P S S

λ μλ μ λ μ

μ λ− − + +

− −

− + =��= − + + + ≤ ≤ −�� − + =�

� (2)

From the graph of Figure1 of [1], the equation for emigration

rate kμ and immigration rate kλ for k number of species is

derived as per following way

1

k

k

Ekn

kIn

μ

λ

=

� �= −� �

(3)

When , 1.k kE I λ μ= + =

Where E and I are the maximum emigration rate and

maximum immigration rate respectively. ‘n’ is the total

number of species in the habitat.

III. IMPROVED OPPOSITION-BASED BIOGEOGRAPHY OPTIMIZATION

A. Opposition-Based Learning OBL Tizhoosh first proposed opposition-based learning

(opposition-based learning, OBL) concept [6]. The main principal of OBL is to utilize opposite numbers to approach the solution. The inventors of OBL claim that a number’s opposite is probably closer than a random number to a solution. Thus, by comparing a number to its opposite, a smaller search space is needed to converge to the right solution[7]. Despite initial opposition-based learning is proposed differential evolution algorithm to speed up the convergence rate, but in essence, opposition-based learning have universal applicability. Rahnamayan also proposed the concept of learning based on quasi-confrontation, and it has been applied to many evolutionary algorithms such as differential evolution [15], particle swarm optimization [12]-[13].

De�nition 1: Let 1 2( , , )DP x x x= � be a point in

D -dimensional space, where ,x a bi i i� ∈ � � and

1,2, ,i D= � .The opposite point 1 2( , , )DP x x x=� � � �� is

completely de�ned by its components i i i ix a b x= + −� . Now, by employing the opposite point de�nition, the

opposition-based optimization can be de�ned as follows.

De�nition 2: Let 1 2( , , )DP x x x= � be a point in

D -dimensional space(i.e. a candidate solution).Assume

( )f � is a �tness function which is used to measure the

candidate’s �tness. According to the de�nition of the opposite

point, 1 2( , , )DP x x x=� � � �� is the opposite of

1 2( , , )DP x x x= � .Now, if ( ) ( )f P f P≥�

, then point P can

be replaced with P�

, otherwise, we continue with P . Hence,

the point and its opposite point are evaluated simultaneously

in order to continue with the �tter one.

a) Opposite Population Initialization

Step 1. generate initial population ( )P popsize randomly,

where popsize is population size.

Step 2. generate opposite population

, , , 1,2, , ; 1,2, ,i j j j i jOP a b P i popsize j D= + − = =� � (4)

Where ,i jP is the jth vector of the ith individual in the

population, ,i jOP is the opposite individual of ,i jP .

�K immigration rate �K emigration rate

Fig. 1. Linear migration rates plotted against the sorted population

643

Step 3. calculate the fitness value of population, ,i jP and

,i jOP ,select Individual of popsize which has better fitness.

b) Implementation of Opposite Population Jumping

In order to control the step size of opposition, the

calculation of opposition is based on a dynamic interval

boundaries ( ) ( )( ),j ja t b t� � � , the dynamic opposition is

computed as:

( ) ( ), ,( ) ( )

( ) ( ) , ( ) ( )

1,2, , ; 1,2, ,

i j j j i j

j ij j ij

OP a t b t P

a t Min x t b t Max x t

i popsize j D

= + −

= =

= =� �

(5)

Where ( ) ( ),j ja t b t are the minimum and maximum values

of the jth dimension in current search space respectively, and

1,2, ,t = � indicates the generations.

By staying with variable’s interval static boundaries, the

opposite point would jump outside of the already shrunken

search space and the converged space would be expanded.

Hence, calculating opposite point by using variables’ current

interval in the population ( ) ( )( ),j ja t b t� � � , which is as the

search does progress, increasingly smaller than the

corresponding initial range ( ),j ja b� � � .

B. Improved OBBO When the global optimum deviation from domain of the

geometric center of function. The candidate solution is easy

to reverse away from the global optimal solution, resulting in

a lot of invalid search, it will obviously reduce the utilization

of opposite population and performance of the algorithm, so

by changing the symmetry point between the candidate

solutions and its corresponding opposite solution, can

overcome serious shortcomings that implicit in OBBO.

This paper presents an opposition-based learning strategy

using the current optimum to further improve the

performance of the algorithm. By this way, the geometric

center of the search space from2

a b+ transfer to bestx .In

addition, combined with the method of paper [17] by using

multiple-point to generate opposite points. We modify the

original OBL used in OBBO by conducting opposition on

multiple points. For each individual iP , we randomly select a

different individual 1iP in the population, where 1i i≠ , and

, 1 1,2, ,i i popsize= � , ps. And then we recombine iP and 1iP

to get a new point. The opposition is conducted on the

recombined point.

, , 1,2 ( 1 2 )besti j i j i jOP x m P m P′ = − ∗ + ∗ (6)

Where 1 2( , , , )best best best bestDx x x x= � is the optimal solution of

the current population. 1, 2m m are two random numbers

within [0,1], and 1 2 1m m+ = .

In order to achieve the purpose that individuals from

different environment, rather than all together in the same

environment. The search space is evenly divided into several

areas, in each region to take part in random individuals as

initial population. This allows the algorithm to search the

entire solution space to find more of the best individual. This

approach not only well to avoid premature convergence, and

can quickly search the global optimal solution.

Algorithm 1: Improved OBBO (IOBBO)

Step 1: Initialize Population

Step 2: Remove/Replace Duplicates

Step 3: Fitness Evaluation

Step 4: Initialize Opposite Population

Step 5: Start the BBO Generation Loop

Step 6: Store Elite Individuals

Step 7: BBO Migrations

Step 8: Remove/Replace Duplicates

Step 9: Fitness Evaluation

Step 10: Opposite Population Jumping

Step 11: Restore Elite Individuals

Step 12: Remove/Replace Duplicates

Step 13: Sort Population

Step 14: Go to Step5

IV. EMPIRICAL ANALYSIS Five benchmark functions are implemented to compare the

performance of BBO, OBBO, IOBBO and IODE. For each

algorithm, the same parameter settings were used for test as

follows. The population size ( )popsize is set to 100. The

644

dimension is 30. For OBBO and IOBBO, jumping rate

constant, rJ is set to 0.3, and a mutation probability of 0.005

per independent variable per generation. We used a function

evaluation limit of 100. The results achieved by each

algorithm are averaged over 30 trails.

The performance of the different EAs as follow.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7x 10

4

number of function evaluations

f(x)

Sphere

IODE

BBOOBBO

IOBBO

Fig. 6. BBO, OBBO, IODE and IOBBO results for the Sphere benchmark function.

0 10 20 30 40 50 60 70 80 90 100-14000

-12000

-10000

-8000

-6000

-4000

-2000

number of function evaluations

f(x)

Schwefel

IODE

BBOOBBO

IOBBO

Fig. 5. BBO, OBBO, IODE and IOBBO results for the Schwefel benchmark function

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

350

400

450

number of function evaluations

f(x)

Rastrigin

IODE

BBOOBBO

IOBBO

Fig. 4. BBO, OBBO, IODE and IOBBO results for the Rastrigin benchmark function.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

number of function evaluations

f(x)

Griewank

IODE

BBOOBBO

IOBBO

Fig. 3. BBO, OBBO, IODE and IOBBO results for the Griewank benchmark function

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

number of function evaluations

f(x)

Ackley

IODE

BBOOBBO

IOBBO

Fig. 2. BBO, OBBO, IODE and IOBBO results for the Ackley benchmark function

645

,Singapore, pp. 2229-2236, 2010. utionaryComputationIEEE Congress on Evol- tional diferential evolution,”

S. Rahnamayan, H.R. Tizhoosh, and M. M. Salama,“Quasi-opposi-

Z. Cai, W. Gong and C. Ling, “Research on a novel

Table , Table shows the comparison of several algorithms, in which Mean is the best function value, Std dev is the standard deviation.

Comparison of results from Table and Table can be

seen, the performance of IOBBO on the five test functions have shown good optimization ability, whether it is the solution, are superior to standard BBO. Compared with OBBO, through the introduction of the current search point and another random point opposite strategy to improve the reorganization, IOBBO global optimization capability has been further improved, for these test functions, its solution quality is clearly superior to OBBO. Compared with IODE, IOBBO in the current generation is served as a symmetry

point between an estimate and the corresponding opposite estimate, resulting in a high rate of opposite population usage. IOBBO is significantly better than IODE in optimization speed and the optimal value of function.

V. CONCLUSION This paper presented a strategy to improve opposition

BBO, which IOBBO. Opposition-based learning strategy used the current optimal solution. In the process of taking the opposite populations, the current optimal solution is instead with the range of the upper and lower limits. In order to make initial population more evenly distributed, the proposed method dividing the range of values into several areas. In order to verify IOBBO, we selected five standard functions for testing experiments. Simulation results showed that in most test problems, the performance of IOBBO is better than OBBO. But only for five standard functions to verify and can not fully explain IOBBO is certainly better than the other algorithms. In future, we will try to conduct more experiments to analyze the performance of IOBBO, and propose more effective strategies to further improve the algorithm.

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TABLE . THE RESULTS ACHIEVED BY BBO,OBBO AND IOBBO

function IODE IOBBO

Mean Std dev Mean Std dev

Ackley 5.1896 1.0255 0.5179 0.0697

Griewank 2.4519 13.7213 0.5371 2.1098

Rastrigin 203.6842 1.3557 19.6368 0.6046

Schwefel -5.3586e+0

03

286.5661 -1.2540e+0

04

2.1682

Sphere 281.2128 2.1386 1.0471 0.1875

TABLE . THE RESULTS ACHIEVED BY BBO,OBBO AND IOBBO

function IOBBO OBBO BBO

Mean Std

dev

Mean Std

dev

Mean Std

dev

Ackley 0.5179 0.06

97

1.8177 0.22

38

11.871

5

3.743

0

Griewank 0.5371 2.10

98

0.9115 3.37

85

27.961

1

59.30

87

Rastrigin 19.636

8

0.60

46

35.858

3

0.96

76

81.465

5

1.152

8

Schwefel -1.254

0e+004

2.16

82

-1.213

6e+004

169.

4886

-1.080

4e+004

297.1

284

Sphere 1.0471 0.18

75

5.8502 0.44

49

3.8707

e+003

11.54

29

TABLE I BENCHMARK FUNCTION

function Domain argmin minf(x)

Ackley (�32 32)n 0n 0

Griewank (600,600)n 0n 0

Rastrigin (5.12,5.12)n 0n 0

Schwefel (�500, 500)n 420.9687n (-418.9829n)n

Sphere (�100, 100)n 0n 0

WHERE N IS THE PROBLEM DIMENSION

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http://academic.csuohio.edu/simond/bbo/markov/

http://academic.csuohio.edu/simond/bbo/simplied/

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