Design of Migration Operators for Biogeography-Based ...embeddedlab.csuohio.edu/BBO/BBO_Papers/GuoMigration.pdfDesign of Migration Operators for Biogeography-Based Optimization and

Design of Migration Operators for Biogeography-Based Optimization andMarkov Analysis

Weian Guoa,b,c, Ming Chend, Lei Wanga∗, Shuzhi Sam Geb,c, Qidi Wua

a.Department of Electronics and Information, Tongji University, Shanghai, China, 201804b.Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 119617

c.Social Robotics Laboratory, Interactive and Digital Media Institute, National University of Singapore, Singapore, 119077d.Sino-German College of Applied Sciences, Tongji University, Shanghai, China, 200092

Abstract

Biogeography-Based Optimization (BBO) inspired from the science of biogeography is a population-based evolu-tionary algorithm and has been developed both in theory and practice. As a crucial operator, migration operator inBBO plays a role of sharing features in solutions, which significantly effects BBO’s performance. Motivated by thecrossover operators in Genetic Algorithm, three kinds of migration operators with non-uniform and uniform versionsare proposed in this paper, which could enhance BBO’s performance. In addition, we present a Markov analysis as amathematical proof to illustrate that the proposed migration operators are superior to the original migration operatorin searching. A set of fourteen benchmarks is employed in numerical simulations to test proposed migration operatorsand the results validate their feasibility and effectiveness. In particular, the proposed migration operators with unifor-m versions are more adaptive in optimization and could achieve good performances in simulations. A 3-b one-maxproblem is used to test the Markov modeling and the results are in agreement with our analysis.

Keywords: Biogeography-Based Optimization; Migration Operator; Genetic Algorithm; Markov Analysis

1. Introduction

Evolutionary Algorithms (EAs) play an important role in optimization and are well implemented in various areas[10, 2, 30, 25, 3]. Other than several classic algorithms such as Genetic Algorithm (GA) [13, 22], Particle SwarmOptimization (PSO) [15], Ant Colony Optimization (ACO) [7] and etc., many novel evolutionary algorithms are pro-posed recently. Biogeography-Based Optimization (BBO) inspired from the science of biogeography was proposedby Dan Simon and has drawn worldwide attentions [31, 11, 19, 17]. It mimics the species’ migration and mutation indispersed areas during a period of evolution. As a novel Evolutionary Algorithm, it exhibits a strong ability in solvingnumerical optimization [19, 11] and practical problems, including aircraft engine sensor selection [31], shipboardpower system [14], image processing [37, 4], mechanical design [27] and etc..

In BBO, the features sharing in solutions is analogous to the species’ migration in biogeography. The mutationof features in BBO is analogous to the mutation in nature species. The migration model decides the probabilities ofmigration and has been investigated in [19] and [11]. In paper [19], Ma proposed six migration models which areinspired by the science of biogeography. The performance of the six migration models were compared by numericalsimulations. In paper [11], Guo investigated the migration model by stochastic approaches and revealed the relation-ship between migration rates and algorithm’s performance. The simulation results in [19, 11] show that BBO exhibitsa great potential in optimization. Besides the numerical simulations, the theoretical researches are conducted in recentyears. In [32], Simon presented a Markov modeling for the standard BBO. In [29], Simon employed Markov modelsto conduct analytical comparisons of BBO and Genetic Algorithms. In [20], Ma and Simon investigated the migration

∗Corresponding Author. Phone:+86 13917262915. Email:[email protected]

Preprint submitted to Information Sciences September 11, 2013

model of Biogeography-Based Optimization by Markov theory.

In previous work, the migration model which decides the probability of migration are investigated in [19, 11, 20].In this paper, we focus on the research about migration operators. To distinguish migration model and migration op-erator, we give an explanation that migration model is to decide whether the migration occurs or not, while migrationoperator decides how to migrate. The previous design of BBO migration operators are mostly derived from the oper-ators in other famous algorithm. Paper [21] proposed blended migration operator which is similar with the blendedcrossover introduced by Eshelman and Schaffer [8]. In [18], the idea of the multi-parent migration operator came fromthe multi-operators in Genetic Algorithm (GA) and Differential Evolution (DE) [33]. Orthogonal migration operatorwas introduced in [9] by employing an orthogonal rule. Based on the researches, we are inspired to propose novel andeffective migration operators by studying the recombination operators of other algorithms.

In the research with regard to Genetic Algorithms, crossover operator was investigated in a comprehensive way.The heuristic crossover was proposed in [35], which generates the offsprings with a bias towards the better parent.Flat crossover produced offsprings in a random way [26]. Arithmetical crossover was proposed and developed in [22].Extended line crossover and extended intermediate crossover which are the variations of the blend crossover wereproposed in [24]. A fuzzy recombination operator was presented in [34]. Motivated by the ideas of the crossoveroperator, we propose three novel migration operators with non-uniform and uniform versions. The main contributionsin this paper are to propose three types of migration operators with non-uniform and uniform versions which areadaptive to optimization problems. In addition, we present Markov modeling to conduct analytical comparisons forproposed migration operators and original migration operator.

The remainder of this paper is organized as follows. In Section 2, we give an overview of Biogeography-Based Op-timization and summarize the previous work of migration operators. The problem that the original migration operatorcould result a poor diversity in population is explained in this section. Motivated by the ideas from other EvolutionaryAlgorithms (EAs), Section 3 propose three novel kinds of migration operators and their uniform versions. In Section4, we present the Markov analysis for BBO with the proposed migration operator and the analytical comparisonsare given in this section. The numerical benchmarks are employed to test the proposed migration operators and theresults are compared with previous work in Section 5. To show the effectiveness of uniform migration operators, weemploy two improved evolutionary algorithms to conduct comparisons in this section. A 3-b one-max problem is usedto test the Markov modeling and the simulation results are in agreement with our analysis. We end this paper withconclusions and present the future work in Section 6.

2. Brief of Biogeography-based Optimization and Problems Description

2.1. Brief of Biogeography-Based Optimization

Biogeography-based Optimization is inspired from the science of Biogeography. The habitats in biogeographyare analogous to the solutions in optimization problems. The suitability of one island for living is evaluated by HighSuitability Index (HSI). The variables that characterize the habitability are called Suitability Index Variables (SIVs).In optimization problems, a good solution is analogous to an island with a high HSI, while a poor solution representsan island with a low HSI. An island with high HSI has more species than an island with low HSI. This means theimmigration to a good island is difficult for the aliens. Therefore, the emigration is dominant in good islands. Incontrast, since few species live in poor islands, immigration is dominant in poor islands. To sum up, good solutionstend to share their features with poor solutions while poor solutions tend to accept features from good solutions. Weuse the notation ∆t to denote a time period which is so small that during ∆t at most one migration occurs. In addition,three assumptions are described as follows,

1. there were K species at time t and no any immigration or emigration occurred during the period from t to t + ∆t;2. there were K + 1 species at time t and at most only one species emigrate;3. there were K − 1 species at time t and at most one species immigrated.

2

Considering that the probability PK denotes the habitat contains exactly K species, PK updates in each generationfrom t to t + ∆t as (1),

PK(t + ∆t) = PK(t)(1 − λK∆t − µK∆t) + PK−1λK−1∆t + PK+1µK+1∆t (1)

where λK and µK are the immigration rate and emigration rate in the habitat that has K species respectively.

By defining

PK =

−(λK + µK)PK + µK+1PS +1, K = 0−(λK + µK)PK + λK−1PK−1 + µK+1PK+1, 1 ≤ K ≤ Kmax − 1−(λK + µK)PK + λK−1PK−1 K = Kmax

(2)

we obtainP = AP (3)

whereP = [P0, ..., PN]T (4)

A =

−(λ0 + µ0) µ1 0 · · · 0

λ0 −(λ1 + µ1) µ2. . .

......

. . .. . .

. . ....

.... . . λn−2 −(λn−1 + µn−1) µn

0 · · · 0 λn−1 −(λn + µn)

(5)

Figure 1: Linear Migration Model in Biogeography-Based Optimization

Fig. 1 illustrates a linear model of species distribution in a single habitat [31]. The researches on migration mod-els have been investigated in [20]. In the linear migration model, I is the maximum possible immigration rate and Erepresents the maximum possible emigration rate. The immigration rate and the emigration rate are functions of thenumber of species in the habitat. For one island, the immigration rate increases and the emigration rate decreases asthe number of species increases. The pseudo-codes of migration are shown in Algorithm 1.

Like most other Evolutionary Algorithms, Biogeography-Based Optimization has the mutation operator whichcould help algorithms break away local optimum and explore the searching space. In BBO [31], the mutation rate iscalculated by (6).

m(K) = mmax

(Pmax − PK

Pmax

)(6)

where mmax is defined by users in advance and Pmax and PK can be obtained in (1). The pseudo-codes of mutation areshown in Algorithm 2.

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Algorithm 1 Pseudo-codes of Biogeography-Based Optimization MigrationSelect Hi according to immigration rate λi

if Hi is selected thenfor j = 1 to n do

Select H j according to emigration rate µi

if H j is selected thenReplace SIV in Hi with SIV in H j

end ifend for

end if

Algorithm 2 Pseudo-codes of Biogeography-Based Optimization Mutationfor i = 1 to n do

Use λi and µi to compute the probability Pi and mutation rate mi

Select Hi(SIV) with probability mi

if Hi(SIV) is selected thenReplace Hi(SIV) with a random value

end ifend for

2.2. Problems Descriptions

In the original Biogeography-Based Optimization [31], the migration operator is given as (7).

Hi(S IV)← H j(S IV) (7)

In (7), it is obvious that Hi(SIV) will be replaced by H j(SIV), which could make the diversity of population worse andresult in a local optimal. To overcome the problem, paper [21] proposed a blended migration operator (BMO) shownin (8), which is inspired from [24].

Hi(S IV)← αHi(S IV) + (1 − α)H j(S IV) (8)

where α ∈ [0, 1] is a random or deterministic value or a function. The blended migration operator which is widelyused in other population-based optimization algorithms is easily implemented for continuous optimization problems.Information sharing between solutions is competitive but not preponderant, which could be better in exploring thesearching space. Noted that when α = 0, (8) is the same as (7). In blended migration operator, the good solutionscould degraded due to migration while the poor solutions could obtain new features from good ones. Hence comparedwith (7), the information sharing between solutions in blended migration operator is bidirectional. The simulationresults in [21] show the performance of blended migration operator is better than the original migration operator.However, according to the description and simulation in [21], the assignment of α depends on users’ experience andthe values of α are influential to algorithms’ performance. Hence how to design the values of α is problem whichshould be further discussed. In addition, to pursue a better performance of BBO, novel migration operators are pro-posed in Section 3.

3. Design of Migration Operator

To explore the migration operator in Biogeography-Based Optimization, we propose three migration operatorsand their uniform versions, which are inspired from the operators in other classic evolutionary algorithms.

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3.1. Uniform blended migration operator

In [21], the blended migration operator employed a non-uniform blended parameter α, which the value of α is setas a const. In the simulation of [21], Ma and Simon set the values of α as 0, 0.5 and 0.8 respectively and the resultsshow that the value of α is influential on BBO’s performance. Blended migration operator with α = 0.5 is better thanthat with α = 0.8 which shows that when a new solution feature is contributed equally from itself and the selectedemigrant, the performance is the best. In this subsection, we extend the work of [21] to propose a uniform blendedmigration operator (UBMO), which set the blended parameter termed α ∈ [0, 1] as a random value. The uniformblended migration operator is demonstrated as (9).

Hi(S IV)← αHi(S IV) + (1 − α)H j(S IV) (9)

In uniform blended migration operator, the value of α is uniformly distributed in interval [0, 1] and needn’t to beset by users in advance. Thereby, the uniform blended migration operator is easy to implement. Meanwhile, accordingto the numerical simulation in Section 5, uniform blended migration operator is competitive to non-uniform blendedmigration operator. The pseudo-codes of uniform blended migration operator is given in Algorithm 3.

Algorithm 3 Pseudo-codes of Uniform Blended Biogeography-Based Optimization MigrationSelect Hi according to immigration rate λi

if Hi is selected thenfor j = 1 to n do

Select H j according to emigration rate µi

if Hi is selected thenReplace SIV in Hi with SIV in H j

end ifend for

end if

3.2. Heuristic Migration Operator

In Genetic Algorithm, the heuristic crossover was proposed in [35] and developed by [23]. In this subsection, weimport the idea to Biogeography-Based Optimization to propose the Heuristic Migration Operator (HMO). Assumingthat H j(SIV) is selected to immigrate from Hi(SIV), the heuristic migration operator is given as (10),

Hi(S IV)← Hi(S IV) + γ(H j(S IV) − Hi(S IV)

), F(H j) ≥ F(Hi) (10)

where γ ∈ [0, 1] and F(·) is a fitness function. In Heuristic Migration Operator, the individual fitness is taken accountof the migration operator which is different from the idea of blended migration operator. The better individuals willinfluence the worse individuals, while the worse individuals cannot affect the better ones. The pseudo-codes of heuris-tic migration operator is given in Algorithm 4.

Algorithm 4 Pseudo-codes of Heuristic Migration OperatorSelect Hi according to immigration rate λi

if Hi is selected thenfor j = 1 to i do

Select H j according to emigration rate µ j

if H j is selected thenSIV in H j migrate to Hi based on Eq. (10)

end ifend for

end if

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In (10), the value of γ is set as a const. Like the uniform blended migration operator, we design a uniform heuristicmigration operator (UHMO) shown in (11).

Hi(S IV)← Hi(S IV) + γ(H j(S IV) − Hi(S IV)

), F(H j) ≥ F(Hi) (11)

γ ∈ [0, 1] is a random value and F(·) is a fitness function. The pseudo-codes of uniform heuristic migration operator issimilar with Algorithm 4. The difference between blended migration operator (BMO) and heuristic migration operator(HMO) is that the fitness of solutions is considered in HMO, while not in BMO.

3.3. Extended Migration Operator

In [12, 28] , BLX-ρ crossover operator was proposed for Genetic Algorithm, which is an extension of flat crossover[26]. By defining x1 and x2 are two parents, x1,i and x2,i present the ith gene and jth gene in x1 and x2 respectively.The offsprings could be located outside the following interval

[min(x1,i, x2,i)] − I · ρ,max(x1,i, x2,i) + I · ρ] (12)

where I = max(x1,i, x2,i) − min(x1,i, x2,i) and ρ are defined in advance. Without loss of generality, assuming x1,i > x2,iin (12), then (12) can be described as

o = (x2,i − (x1,i − x2,i) · ρ) + ψ[(x1,i + I · ρ) − (x2,i − I · ρ)]= (ψ + 2ψρ − ρ)x1,i + (1 − ψ − 2ψρ + ρ)x2,i (13)

where o is the one of the offsprings. By defining λ = ψ + 2ψρ − ρ, (13) can be rewritten as (14),

o = λx1,i + (1 − λ)x2,i (14)

By comparing (14) and (8), BLX-ρ crossover is similar with the blended crossover, but λ does not belong to [0, 1].Based on the idea of BLX-ρ, [24] proposed Extended Intermediate Crossover as an extension of BLX-ρ, which isdemonstrated in (15).

o = x1,i + ρi(x1,i − x2,i) (15)

where ρi ∈ [−0.25, 1.25]. Based on the idea of extended intermediate crossover, we proposed a novel migrationoperator termed Extended Migration Operator(EMO) in this subsection. By defining the Hi being selected to migrateand H j being emigrant, EMO is described as (16),

Hi(S IV) = Hi(S IV) + ζ(H j(S IV) − Hi(S IV)), F(H j) ≥ F(Hi) (16)

where ζ ∈ [−0.25, 1.25] is a const and and F(·) is a fitness function. The uniform version of extended migrationoperator (UEMO) is given as (17).

Hi(S IV) = Hi(S IV) + ζ(H j(S IV) − Hi(S IV)), F(H j) ≥ F(Hi) (17)

where ζ ∈ [−0.25, 1.25] is a random value and F(·) is a fitness function.

To compare the various migration operators visually, we give a diagram shown as Fig. 2 to illustrate uniformversions of blended migration operator, heuristic migration operator and extended migration operator. In Fig. 2,parents1 and parents2 are two parents and the bold line presents the domain of offsprings. For uniform heuristicmigration operator and uniform extended migration operator, the fitness of parent2 is better than that of parent1.It is obvious that, for uniform blended migration operator and uniform heuristic migration operator, the domain ofoffsprings is the interval of two parents, while the domain of offsprings exceed the interval of parents for uniformextended migration operator.

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Figure 2: Diagram of Different Migration Operators

4. Markov Analysis for BBO with Proposed Migration Operators

A Markov Chain model is helpful to analyze the probability pi j that the state si transfer to s j. All of the transitionprobabilities could be denoted in a transition matrix P = [pi j]. To analysis the migration operator in a mathematicalway, we derive the Markov model of BBOs with proposed migration operators. In [32], a Markov modeling for theoriginal BBO was presented. In this paper, we discuss the Markov modeling of BBOs with proposed migration oper-ators in a binary search space. Two assumptions are made in the modeling. One is all the new solutions are producedbefore old solutions are replaced. Another one is any solution is allowed to migrate to itself although it occurs withonly a small probability.

We present the Markov modeling of BBO with uniform blended migration operator first. Suppose that for aproblem with m dimensions in a binary search space, there are n possible solutions which are represented by xi,where i = 1, . . . , n. For BBO, the population contains N candidate solutions which are represented by y j, wherej = 1, . . . ,N. We use the notation υi to denote the number of xi individuals in the population. Hence

∑ni=1 υi = N and

the the population can be depicted as

Population = {y1, ..., yN}

= {x1, x1, . . . , x1︸︷︷︸υ1 copies

, x2, x2, . . . , x2︸︷︷︸υ2 copies

, . . . , xn, xn, . . . , xn︸︷︷︸υn copies

} (18)

where yi has been ordered to group identical individuals. Since we take into account the binary space, the operationof proposed migration operator could produce a value which does not belong to the set {0,1}. Hence we round theproduced value to make sure the validity of offsprings. We use the notation Ji(s) to denote the set of population indicesj as in (19).

Ji(s) ={j : RA

(αxi(s) + (1 − α)x j(s)

)= xi(s)

}(19)

where α is a random value and the notation “RA(X)” rounds the absolute value of X to the nearest integer. By orderingyk in the same order as xi, we obtain

yk =

x1, for k = 1, ..., υ1

x2, for k = υ1, ..., υ1 + υ2...

...

xn, for k =∑n−1

i=1 {υi + 1, . . . ,N}

(20)

For the sake of convenience, we define σ(k) as (21).

σ(k) = min(r) (21)

7

where∑r

i=1 υi ≥ k. Hence yk could be depicted as

yk = xσ(k), k = 1, . . . ,N (22)

According to the definition of migration in [31], the probabilities that the immigration occurs and does not occurare given in (23) and (24).

Pr(immigration) = λ (23)

Pr(no immigration) = (1 − λ) (24)

By defining yk(s)t+1 as the sth feature of yk at generation t + 1 and xk(s) as the sth feature in the kth parent at generationt, if the migration does not occur,

yk(s)t+1 = xk(s) (25)

In (25), it shows that there is no change from generation t to generation t + 1. The probability that yk(s)t+1 = xi(s) isshown in (26).

Pr(yk(s)t+1 = xi(s)|no immigration

)= 1 (26)

However, if the sth feature of xk is selected for immigration during generation t, then the probability that yk(s)t+1 isequal to xk(s) is proportional to the combined emigration rates of all possible solutions whose indices are in the set of(19). The probability can be written as in (27).

Pr(yk(s)t+1 = xi(s)|immigration

)=

∑j∈Ji(s) υ jµi∑n

j=1 υ jµ j(27)

By combining (24), (23), (26) and (27), along with the probability of immigration to xk(s) is equal to λk, to obtain thetotal probability

Pr(yk(s)t+1 = xi(s)) = Pr(no immigration) × Pr(yk(s)t+1 = xi(s)|no immigration

)+Pr(immigration) × Pr

(yk(s)t+1 = xi(s)|immigration

)(28)

=(1 − λm(k)

)10

(xm(k)(s) − xi(s)

)+ λm(k)


j=1 υ jµ j

where 10 is the indicator function on the set {0}.

By defining Θ = [θ1, θ1, . . . , θn]T as the population and Pki (Θ) as the probability that immigration results in yk = xi,we obtain (29).

Pki(Θ) = Pr(yk,t+1 = xi)

=

m∏s=1

(1 − λσ(k))

10(xσ(k)(s) − xi(s)

)+ λσ(k)


j=1 υ jµ j

(29)

where k ∈ [1,N] and i ∈ [1, n]. To compute Pki(Θ) for all k and i, we obtain P(Θ). We define Θ as the population weobtained after one generation, which we start with a population vector Θ. We use the notation PrUBMO(Θ|Θ) as thetransition probability matrix for UBMO and deduce it in (30).

PrUBMO(Θ|Θ) =∑

J∈YUBMO

N∏k=1

n∏i=1

[Pki (Θ)]Jki (30)

where

YUBMO =

J ∈ R : Jki ∈ {0, 1},n∑

i=1

Jki = 1 for all k,N∑

k=1

Jki = θi for all i

(31)

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Similarly, we can obtain the transition probability matrix for UHMO and UEMO respectively. Since in (11),F(H j) ≥ F(Hi), we obtain Pki(Θ) for UHMO as follows,


=

m∏s=1

(1 − λσ(k))

10(xσ(k)(s) − xi(s)

)+ λσ(k)

∑q∈Qi(s) υqµq∑n

q=1 υqµq

(32)

where Qi(s) ={q : RA

(xi(s) + γ

(xq(s) − xi(s)

))= xi(s), q ≥ i

}, k ∈ [1,N] and i ∈ [1, n]. The transition probability

matrix for UHMO is given in (33).

PrUHMO(Θ|Θ) =∑

Q∈YUHMO

i∏k=1

n∏i=1

[Pki (Θ)]Qki (33)

where

YUHMO =

Q ∈ R : Qki ∈ {0, 1},n∑

i=1

Qki = 1 for all k,i∑

k=1

Qki = θi for all i

(34)

For UEMO,


=

m∏s=1

[(1 − λσ(k)

)10

(xσ(k)(s) − xi(s)

)+ λσ(k)

∑e∈Ei(s) υeµe∑n

e=1 υeµe

](35)

where Ei(s) ={e : RA

(xi(s) + ζ (xe(s) − xi(s))

)= xi(s), e ≥ i

}, k ∈ [1,N] and i ∈ [1, n]. The transition probability

matrix for UEMO is given in (36).

PrUEMO(Θ|Θ) =∑

E∈YUEMO

i∏k=1

n∏i=1

[Pki (Θ)]Eki (36)

where

YUEMO =

E ∈ R : Eki ∈ {0, 1},n∑

i=1

Eki = 1 for all k,i∑

k=1

Eki = θi for all i

(37)

To compare with the original BBO, we recall the definition of J in [32] as in (38).

Ji(s) ={j : x j(s) = xi(s)

}(38)

We use C(·) to denote the cardinality of a set. It is obvious that C (Ji(s)) ≥ C (Ji(s)), which enlarge the value of Pki(Θ).A large value of Pki(Θ) is helpful to boost the information sharing in a population, which makes BBO with uniformblended migration operator has a stronger ability in searching. By comparing C (Qi(s)) and C (Ji(s)), although thesearch spaces of Qi(s) is not larger than that of Ji(s), C (Qi(s)) is not necessarily smaller than C (Ji(s)), whichmeans the value of (32) is not necessarily smaller than that of (29). This means although BBO with uniform heuristicmigration operator searches in a small space but has a more effective searching ability. The analysis of BBO withuniform extended migration operator is similar. We also can conclude that the effectiveness of UEMO is better thanthat of original migration operator.

5. Simulation and Analysis

5.1. Simulations for BenchmarksIn this section, 14 functions are employed as numerical benchmarks to test the performance of the proposed mi-

gration operators and the results are compared with previous migration operators. The functions are given in Table 1

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Table 1: Benchmarks functions.The Granularity of Each Domain Was 0.1. Except for the Quartic Function, Which has a Granularity of 0.01Function Name Dimension Domain

f01 Ackleys Function 20 [-30,30]f02 Fletcher-Powell 20 [-π,π]f03 Generalized Griewanks function 20 [-600,600]f04 Generalized Penalized function 1 20 [-50,50]f05 Generalized Penalized function 2 20 [-50,50]f06 Quartic function 20 [-1.28,1.28]f07 Generalised Rastrigins function 20 [-5.12,5.12]f08 Generalized Rosenbrocks function 20 [-2.048,2.048]f09 Schwefel Problem 1.2 20 [-65.535,65.535]f10 Schwefel Problem 2.21 20 [-100,100]f11 Schwefel Problem 2.22 20 [-10,10]f12 Schwefel Problem 2.26 20 [-512,512]f13 Sphere Model 20 [-5.12,5.12]f14 Step Function 20 [-200,200]

Table 2: Properties of Each Benchmarks Functions.Function Name Multimodal? Separable? Regular?Ackleys Function Yes No YesFletcher-Powell Yes No NoGeneralized Griewanks function Yes No YesGeneralized Penalized function 1 Yes No YesGeneralized Penalized function 2 Yes No YesQuartic function No Yes YesGeneralised Rastrigins function Yes Yes YesGeneralized Rosenbrocks function No No YesSchwefel Problem 1.2 No No YesSchwefel Problem 2.21 No No NoSchwefel Problem 2.22 Yes No NoSchwefel Problem 2.26 Yes Yes NoSphere Model No Yes YesStep Function No Yes No

and the properties of the 14 benchmarks are given in Table 2. More details can be found in [36].

In all simulations, the linear migration model [31, 11] is employed in all versions of BBO. We use a populationsize of 50 and the maximum generation is 1000. For each benchmark the dimension is 20. Since we do not takeaccount of the mutation, the mutation rate is set as 0. For each benchmark, we run 50 times to obtain an averageperformance. The values in “Mean” row is the average performance in the 50 runs and “Best” is the best performancein the 50 runs.

First,we compare blended migration operators with different values of α. The uniform blended migration operatoris also considered in the comparisons. The results are given in Table 3. The values of α is set as 0.2, 0.5 and 0.8respectively. To compare the results convenience, for each benchmark, we assign 3 points for the best operator, 2points for the second best operator, 1 point for the third best operator, and 0 point for the worst one. Then we add thepoints for all benchmarks and obtain Table 3, where the blue column presents the scores for “Mean” value and the redcolumn presents the “Best” value.

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Table 3: Simulation results obtained by Blended BBO (α=0.2), B-BBO (α=0.5), B-BBO (α=0.8) and uniform blended BBO with α on 14 functionsover 50 independent runs. The table shows the best and mean values.Function α = 0.2 α = 0.5 α = 0.8 α

f01 Mean 3.4598E-01 2.4274E-02 1.5552E-04 3.3997E-03Best 1.1422E-01 4.2145E-03 7.7316E-06 9.6889E-04Stdev 1.5718E-01 9.5561E-03 1.3567E-04 8.5456E-04

f02 Mean 3.3846E+04 2.1637E+04 2.1247E+04 2.4833E+04Best 4.8129E+03 6.2465E+02 6.2293E+02 1.9552E+03Stdev 2.1089E+04 9.9231E+03 1.1329E+04 7.1349E+03

f03 Mean 1.0043E+00 1.0001E+00 1.0000E+00 1.0000E+00Best 1.0008E+00 1.0000E+00 1.0000E+00 1.0000E+00Stdev 1.3039E-03 8.4086E-05 1.0516E-07 9.9519E-07




f07 Mean 5.7463E+00 7.8484E-01 5.2961E-01 1.2747E-01Best 1.8911E+00 5.7803E-03 2.9815E-08 2.8427E-05Stdev 1.9487E+00 4.3032E-01 3.9384E-01 2.9723E-01


f09 Mean 2.5127E+02 1.6956E+02 6.8846E+01 7.7015E+01Best 4.6787E+01 9.9155E+00 2.3956E-01 2.5050E+00Stdev 1.5245E+02 9.8369E+01 4.1461E+01 6.3021E+01



f12 Mean 3.2775E+00 2.5628E+00 2.7340E+00 3.5882E+00Best 1.4765E+00 1.0690E+00 7.8769E-01 1.6609E+00Stdev 8.0605E-01 1.0231E+00 6.9505E-01 7.3428E-01


f14 Mean 1.7000E+00 1.9800E+00 5.4000E-01 6.6000E-01Best 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00Stdev 1.4832E+00 1.0000E+00 4.5826E-01 6.4031E-01

11

Figure 3: Performances of Blended Migration Operators (α = 0.2, 0.5, 0.8) and Uniform Blended Migration Operator

According to the results in Fig. 3, it is obvious that BMO with α = 0.8 achieves the best performance both for“Mean” value and “Best” value. BMO with α = 0.8 and BMO with α = 0.5 are both better than the BMO withα = 0.2, which has the same conclusion with [21]. However, in [21], BMO with α = 0.5 performs better than BMOwith alpha = 0.8, while our results show that BMO with α = 0.8 is outperforms BMO with α = 0.5. We do not usethe same benchmarks as [21] and the test condition is not the same as well. Hence we draw the conclusion that underdifferent conditions and for different benchmarks , it is hard to decide how to choose the value of α. In Fig. 3, UBMOperforms the second best, which shows that UBMO is a competitive strategy in optimization. Considering that incommon, the test environment is uncertain and there exist differences between benchmarks, UBMO, which needn’t todesign the value of α in advance, is a wisdom choice when no more information obtained.

Second, we compare heuristic migration operators with different values of γ. The uniform heuristic migrationoperator is taken into account as well. The results are shown in Table 4. The values of γ are set as 0.2, 0.5 and 0.8respectively. We assign points for each operator as in Fig. 3 and obtain Fig. 4.

Figure 4: Performances of Heuristic Migration Operators (γ = 0.2, 0.5, 0.8) and Uniform Heuristic Migration Operator

In Table 4, HMO with γ = 0.8 performs the best both in “Mean” values and “Best” values and the performance ofHMO with γ = 0.2 performs the worst. The performance of HMO with γ = 0.5 is superior to HMO with γ = 0.2 andinferior to HMO with γ = 0.8. In addition, it is also obvious that UHMO performs the second best in the comparisons,which means that the uniform strategy is feasible and effective. Since in uniform strategy, the value of γ needn’t beset in advance, it is easier to implement than others.

Since the uniform migration operators are easily implemented, in Table 5, we compare the the uniform types ofthe proposed migration operators. The column of “Original” presents the performance of original migration operatorshown in (7); The column of “UBMO” presents the performance of uniform blended migration operator shown in

12

Table 4: Simulation results obtained by Heuristic BBO (γ=0.2), H-BBO (γ=0.5), H-BBO (γ=0.8) and Uniform Heuristic BBO with γ on 14functions over 50 independent runs. The table shows the best and mean values.Function γ = 0.2 γ = 0.5 γ = 0.8 γ







f07 Mean 6.8618E+00 3.8339E-01 2.3251E-01 1.5291E-01Best 8.5678E-01 7.0577E-04 2.2924E-08 2.0774E-05Stdev 2.0783E+00 8.4235E-01 7.9203E-01 4.8687E-01


f09 Mean 2.5721E+00 7.8170E+01 4.1260E+01 4.0952E+01Best 5.1643E+00 8.9445E-01 4.7349E-03 8.9673E-02Stdev 1.6388E+02 1.3635E+02 2.6178E+01 6.5889E+01


f11 Mean 1.8294E-01 2.9957E+00 2.6394E-06 4.8982E-04Best 6.9623E-02 7.9900E-04 3.4364E-07 1.0880E-04Stdev 4.3925E-02 8.3813E-04 3.7323E-06 2.9141E-04

f12 Mean 2.2868E+00 1.9675E+00 1.9286E+00 2.9597E+00Best 1.1595E+00 7.1704E+00 6.8400E-01 1.1603E+00Stdev 7.4799E-01 5.0603E-01 1.2245E+00 1.0527E+00


f14 Mean 9.0000E-01 9.6000E-01 5.6000E-01 4.2000E-01Best 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00Stdev 7.8102E-01 4.8990E-01 5.0000E-01 4.5826E-01

13

(8);The column of “UHMO” presents the performance of uniform heuristic migration operator shown in (10); Thecolumn of “UEMO” presents the performance of uniform extended migration operator shown in (15). To give ageneral view of the ability of BBO in optimization, we also compared BBO and its variants with several improvedwell-known evolutionary algorithms, including stud Genetic Algorithm (SGA) which employed the best individuals inthe population for crossover [16] and coefficient Particle Swarm Optimization (CPSO) [5, 6] which took account intothe affect of a constriction factor and be given in (39). For the SGA, we use single point crossover with a crossoverprobability of 1, and a mutation probability of 0.01. For CPSO, c1 and c2 are both set as 2.1.

vi, j = χ[vi, j + c1 ∗ r1 ∗ (yi, j − xi, j) + c2 ∗ r2 ∗ (y j − xi, j)] (39)

where yi, j is the best position xi achieved and y j is the best position in the whole swarm up to time j. The value of χis set as (40).

χ =2∣∣∣∣2 − φ − √(φ2 − 4φ

)∣∣∣∣ (40)

where φ = c1 + c2, φ > 4.

To exhibit the performances of each algorithm in Table 5, we assign the scores for them. For each benchmark, thealgorithm which obtains the best performance gets 5 points, the one obtain the second performance gets 4 points,...,theone performs the worst gets 0 points. The scores are given in Fig. 5.

Figure 5: Performances of Original Migration Operator and other uniform migration operators

First, we compare BBO with different migration operators. For “Mean”, original migration operator only winsfor f09 and for “Best”, original migration operator wins for f07, f11 and f13. For f14, all kinds of migration operatorarchive the best performance in the column of “Best”; For “Mean”, “UBMO” only wins for f7 and f10. For “Best”,“UBMO” does not perform the best for any benchmark. For f03, “UBMO”, “UHMO” and “UEMO” all archive thebest performance; For “UHMO”, it wins for f10 in the column of “Best” and does not performs the best for any bench-mark in the column of “Mean”. For “UEMO”, it wins for f01, f02, f04, f05, f06, f08, f11, f12, f13, f14 totally 10benchmarks, in the column of “Mean”, and it also wins f01, f02, f03, f04, f05, f06, f08, f09, f12, totally 9 benchmarks,in the column of “Best”. The test results show that the improvements of migration operators are feasible and effectiveto enhance BBO’s performance. In the comparisons of BBO with different migration operators, “UEMO” performsthe best, which means to extend the domain of offsprings can help algorithms explore the searching space and pursuea good performance. In addition, by comparing the “UBMO” with “UHMO”, it is in general that the “UHMO” is su-perior to “UBMO”. Considering that in “UHMO” only good solutions can migrate to poor solutions, “UHMO” couldimprove poor solutions by good solutions. Since in BBO the migration occurs in the whole population, which meansin any generation, any individual could be affected by any other individuals, the influence of local optimal solutionsis weaken. Second, according to Fig. 5, the performance of original BBO is similar with that of SGA. However, theBBOs with improved migration operators outperform StudGA and CPSO which indicates that the proposed uniformmigration operators are helpful to enhance BBO’ performance.

14

Table 5: Simulation results obtained by Original BBO, UBMO, UHMO, UEMO, SGA and CPSO on 14 functions over 50 independent runs. Thetable shows the best and mean values.Function Original UBMO UHMO UEMO SGA CPSO

f01 Mean 6.5863E-01 3.3997E-03 1.4071E-03 1.2432E-04 6.1651E-01 1.3482E+01Best 3.2574E-01 9.6889E-04 4.7880E-04 3.1381E-05 2.3109E-01 1.2505E+01Stdev 2.1169E-01 8.5456E-04 2.6238E-04 3.3592E-05 2.3379E-01 5.7573E-01

f02 Mean 1.3006E+04 2.4833E+04 2.0884E+04 1.2979E+04 7.0140E+03 2.8559E+05Best 2.0425E+03 1.9552E+03 3.1931E+03 1.7371E+03 1.7965E+03 1.9151E+05Stdev 6.2198E+03 7.1349E+03 1.5780E+04 5.6596E+03 4.0564E+03 6.8814E+04

f03 Mean 1.0168E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0072E+00 3.9419E+01Best 1.0047E+00 1.0000E+00 1.0000E+00 1.0000E+00 1.0032E+00 3.4637E+01Stdev 1.6502E-02 9.9519E-07 2.8271E-07 5.1976E-09 2.9372E-03 3.4013E+00

f04 Mean 1.9788E-02 2.2383E-03 2.0216E-06 3.2648E-09 1.9754E-03 1.5692E+05Best 1.0834E-03 3.2002E-08 5.5094E-09 1.2263E-10 4.0067E-04 2.3524E+04Stdev 2.2141E-02 1.0582E-06 8.3461E-08 2.7752E-09 2.7441E-03 8.7884E+04

f05 Mean 1.0205E-01 1.3052E-04 3.5851E-05 1.1186E-07 1.8484E-02 3.3204E+06Best 3.4786E-02 3.2683E-07 7.3382E-07 2.4154E-09 8.3090E-03 6.5043E+05Stdev 3.7674E-02 1.1245E-04 1.7326E-05 2.2910E-07 8.4322E-03 1.4606E+06

f06 Mean 5.9660E-07 3.9358E-13 3.6917E-14 1.2961E-15 0.0000E+00 7.0386E-01Best 3.0000E-08 3.3010E-16 6.4715E-16 1.3731E-18 0.0000E+00 3.7367E-01Stdev 2.0402E-07 1.0185E-13 9.1092E-14 8.9366E-16 0.0000E+00 2.4216E-01

f07 Mean 8.0872E+02 1.2747E-01 1.5291E-01 2.0134E-01 0.0000E+00 1.2371E+02Best 0.0000E+00 2.8427E-05 2.0774E-05 6.1739E-08 0.0000E+00 1.1589E+02Stdev 1.1316E+01 2.9723E-01 4.8687E-01 1.9836E-05 0.0000E+00 4.4074E+00

f08 Mean 3.5625E+01 1.7770E+01 1.7709E+01 1.7058E+01 5.1711E+01 2.7285E+02Best 9.3181E+00 1.6840E+01 1.6732E+01 1.2593E+01 1.3131E+01 2.2548E+02Stdev 2.3533E+01 2.8400E-01 2.8909E-01 8.5342E-01 2.8442E+01 3.3520E+01

f09 Mean 4.4812E+00 7.7015E+01 4.0952E+01 4.6887E+00 5.3409E+03 5.2695E+03Best 1.6951E+00 2.5050E+00 8.9673E-02 2.6094E-04 5.0889E+03 4.7274E+03Stdev 1.9637E+00 6.3021E+01 6.5889E+01 3.4516E+01 1.4601E+01 3.2661E+02

f10 Mean 5.6366E+02 1.6653E+02 1.8878E+02 2.6910E+02 2.6967E+03 2.5948E+03Best 1.3354E+02 3.7711E+01 3.2155E+01 3.8476E+01 1.7533E+03 2.0921E+03Stdev 2.8717E+02 4.4016E+01 1.0778E+02 1.2098E+02 4.3785E+02 3.3944E+02

f11 Mean 1.3400E-01 2.3695E-03 4.8982E-04 2.7357E-05 2.2280E+01 2.2624E+01Best 0.0000E+00 5.5519E-04 1.0880E-04 4.6377E-06 1.8380E+01 2.0074E+01Stdev 7.7460E-02 1.2843E-03 2.9141E-04 2.2767E-05 2.0701E+00 1.7187E+00

f12 Mean 6.8460E+00 3.5882E+00 2.9597E+00 1.3942E+00 2.7104E+01 2.9565E+01Best 3.1000E+00 1.6609E+00 1.1603E+00 4.2796E-01 2.4134E+01 2.5372E+01Stdev 1.7990E+00 7.3428E-01 1.0527E+00 3.9279E-01 2.0232E+00 2.0201E+00

f13 Mean 2.4189E-03 4.1370E-07 7.0628E-08 8.6881E-10 1.2144E+01 1.1744E+01Best 0.0000E+00 2.0931E-08 3.5954E-09 5.9876E-11 9.6863E+00 9.5129E+00Stdev 3.0236E-03 3.8354E-07 6.0630E-08 3.6172E-10 1.2317E+00 2.2127E+00

f14 Mean 1.5200E+02 6.6000E-01 4.2000E-01 2.2000E-01 0.0000E+00 4.0665E+03Best 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 3.0670E+03Stdev 1.9621E+00 6.4031E-01 4.5826E-01 8.0000E-01 0.0000E+00 7.5859E+02

15

5.2. Simulations for BBO Markov Modeling

To confirm the Markov model, we employ the 3-b one-max problem with a search space cardinality of eight and apopulation size of four. As a popular benchmark, one-max problem has a fitness function which is proportional to thenumber of ones in the population [1]. The fitness of values of the 3-b one-max problem are given in (41).

f (000) = 1, f (001) = 2, f (010) = 2, f (011) = 3,f (100) = 2, f (101) = 3, f (110) = 3, f (111) = 4. (41)

The solution is denoted by a vector {a1, a2, a3, . . . , a8}, which presents the numbers of individuals that equal to{000, 001, 010, . . . , 111} respectively. In the simulation, we conduct 50 Monte Carlo runs and for run it has 20000generations with rate of 0.1, 0.01 and 0.001 respectively. The performance of BBO with original migration operatoris presented in Table 6.

Table 6: The simulation results for the 3-b one-max problem based on BBO with original migration operator. Table shows the most probablepopulations, and the combined probabilities of convergence to population that contain no optimal solutions. “∗” means the bit is an arbitrary valueof the set {0,1}Mutation Rates Population Vector Probability(Original)

0.1 0 0 0 0 0 0 1 3 0.04090 0 0 0 0 1 1 2 0.03560 0 0 0 0 1 0 3 0.0351∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.2678

0.01 0 0 0 0 0 0 0 4 0.51360 0 0 0 0 0 1 3 0.10720 0 0 0 0 0 4 0 0.0744∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1152

0.01 0 0 0 0 0 0 0 4 0.84210 0 0 0 0 0 4 0 0.05420 0 0 0 0 4 0 0 0.0462∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1025

In Table 6, it is seen that a mutation rate of 0.1 is too large to result in a good performance. The probability that thepopulation does not have any optimal individuals is more than 25% in most cases. However, BBO with a reasonablesmaller mutation rate could achieve a better performance. With the mutation rate 0.01 and 0.001, the probabilitythat the algorithm achieves a global optimal increase to 51% and 84% respectively. The performances of BBO withmutation rate 0.01 and 0.001 are shown in Fig. 6(a) and Fig. 6(b) respectively.

The performances of BBOs with uniform blended migration operator, uniform heuristic migration operator anduniform extended migration operator are presented in Table 7, Table 8 and Table 9. From the three tables, we cansee that the probability to obtain an optimal increases with the decrease of the mutation rate from 0.1 to 0.001, whichmeans the mutation rates are influential to algorithms’ performance and a reasonable mutation is helpful to enhancealgorithms. In addition, it is seen that with a reasonable mutation rate 0.01, for BBO with UBMO, UHMO and UEMO,the probabilities that the best solution is considered as the optimal is around 0.3, which is smaller than that in BBOwith original migration operator. This is an agreement with the Markov modeling in Section 4. BBOs with UBMO,UHMO and UEMO have a stronger ability in searching. The information sharing has been enhanced, which means theprobability for states transition is enlarged. This could help BBO be more effective and achieve a good performance.Fig. 7, Fig. 8 and Fig. 9 show the performances of BBOs with UBMO, UHMO and UEMO respectively. Accordingthe simulation results in Table 7, Table 8 and Table 9, it is obvious that mutation rates of 0.01 and 0.001 are morereasonable, we employ the rates in Fig. 7, Fig. 8 and Fig. 9.

16

(a) Original BBO with mutation of 0.01 (b) Original BBO with mutation of 0.001

Figure 6: Typical performance of BBO with original migration operator for a 3-b one-max optimization problem. The three most probablepopulations are shown, along with the cumulative probability of all populations that have no optimal individuals.

Table 7: The simulation results for the 3-b one-max problem based on BBO with uniform blended migration operator. Table shows the mostprobable populations, and the combined probabilities of convergence to population that contain no optimal solutions. “∗” means the bit is anarbitrary value of the set {0,1}Mutation Rates Population Vector Probability(UBMO)

0.1 0 0 0 0 0 1 1 2 0.02020 0 0 0 0 0 2 2 0.01740 0 0 1 0 0 1 2 0.0161∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.3813

0.01 0 0 0 0 0 0 0 4 0.30880 0 0 0 0 0 1 3 0.11750 0 0 0 0 1 0 3 0.0883∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1318

0.01 0 0 0 0 0 0 0 4 0.85250 0 0 0 0 0 1 3 0.03520 0 0 0 0 1 0 3 0.0341∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.0371

Table 8: The simulation results for the 3-b one-max problem based on BBO with uniform heuristic migration operator. Table shows the mostprobable populations, and the combined probabilities of convergence to population that contain no optimal solutions. “∗” means the bit is anarbitrary value of the set {0,1}Mutation Rates Population Vector Probability(UHMO)

0.1 0 0 0 0 0 1 1 2 0.01950 0 0 0 0 0 2 2 0.01660 0 0 0 1 1 1 1 0.0162∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.3936

0.01 0 0 0 0 0 0 0 4 0.27010 0 0 0 0 0 1 3 0.11100 0 0 0 0 1 0 3 0.0786∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1923

0.01 0 0 0 0 0 0 0 4 0.81180 0 0 0 0 0 4 0 0.05220 0 0 0 0 0 1 3 0.0369∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.0896

17

(a) UBMO with mutation of 0.01 (b) UBMO with mutation of 0.001

Figure 7: Typical performance of BBO with uniform blended migration operator for a 3-b one-max optimization problem. The three most probablepopulations are shown, along with the cumulative probability of all populations that have no optimal individuals.

(a) UHMO with mutation of 0.01 (b) UHMO with mutation of 0.001

Figure 8: Typical performance of BBO with uniform heuristic migration operator for a 3-b one-max optimization problem. The three most probablepopulations are shown, along with the cumulative probability of all populations that have no optimal individuals.

18

Table 9: The simulation results for the 3-b one-max problem based on BBO with uniform extended migration operator. Table shows the mostprobable populations, and the combined probabilities of convergence to population that contain no optimal solutions. “∗” means the bit is anarbitrary value of the set {0,1}Mutation Rates Population Vector Probability(UEMO)

0.1 0 0 0 0 0 0 2 2 0.01670 0 0 0 0 1 1 2 0.01660 0 0 0 1 1 1 1 0.0154∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.3962

0.01 0 0 0 0 0 0 0 4 0.29000 0 0 0 0 0 1 3 0.11050 0 0 0 0 1 0 3 0.0772∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1840

0.01 0 0 0 0 0 0 0 4 0.79040 0 0 0 0 0 4 0 0.10030 0 0 0 0 0 1 3 0.0320∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0.1104

(a) UEMO with mutation of 0.01 (b) UEMO with mutation of 0.001

Figure 9: Typical performance of BBO with uniform extended migration operator for a 3-b one-max optimization problem. The three most probablepopulations are shown, along with the cumulative probability of all populations that have no optimal individuals.

19

6. Conclusions

In this paper, we extended the blended migration operator and proposed its uniform type. Two novel migrationoperator and their uniform types are proposed as well, which are motivated by the variation of crossover in GeneticAlgorithm. By analyzing the mathematical expressions of migration operator, we summarized the characterizes ofeach migration operator. In numerical simulation, we performed BBO’s competitive ability in optimization problems.Blended migration operator, heuristic migration operator and extended migration operator are compared and the re-sults show that the parameters in the migration operators are sensitive to the optimization functions, which means fordifferent optimization problems, various parameters could result different performance. By extending the migrationoperators, we propose their uniform versions which needn’t to set the operator parameters. The simulation results alsoshow that the performances of uniform migration operator are feasible and effective. The general performances aresuperior to that of the non-uniform migration operators. The improvement of migration operators are also superior toother improved Evolutionary Algorithms (EAs) which exhibits that the proposed migration operators are competitivein solving optimization problems. To present the advances of proposed migration operator, we conduct Markov mod-eling as a mathematical proof to illustrate them. In proposed migration operator, the probabilities for state transitionare larger which enhance the searching effectiveness in optimization.

As a novel evolutionary algorithms, Biogeography-Based Optimization has a huge potential to explore and exploit.In future work, the migration operator could be investigated by importing fuzzy strategy which was successfullyhybridized with crossover in Genetic Algorithm to overcome the difficulty of premature convergence. Since differentmigration models could effect BBO’s performance, it remains to consider influence of different migration models onthe migration operators. In this paper, 14 functions are employed as benchmarks to test the performance of migrationoperators. However, the practical problems with high-dimension or big data should be considered in future. Inaddition, it is challenging but necessary to built the theoretical analysis of convergency of BBO, which is benefit tofurther develop BBO.

Acknowledgement

This work was sponsored by the National Natural Science Foundation of China under Grant No.70871091,No.61075064, No.61034004 and No.61005090, Program for New Century Excellent Talents in University of Min-istry of Education of China, Ph.D. Programs Foundation of Ministry of Education of China (20100072110038). Theauthor Weian GUO appreciates China Scholarship Council for sponsoring his living expenses in Singapore.

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