Biogeography based optimization (BBO) is a new competitive populationbased algorithm inspired by biogeography. It simulates the migration of species in nature to share information. A new hybrid BBO (HBBO) is presented in the paper for constrained optimization. By combining differential evolution (DE) mutation operator with simulated binary crosser (SBX) of genetic algorithms (GAs) reasonably, a new mutation operator is proposed to generate promising solution instead of the random mutation in basic BBO. In addition, DE mutation is still integrated to update one half of population to further lead the evolution towards the global optimum and the chaotic search is introduced to improve the diversity of population. HBBO is tested on twelve benchmark functions and four engineering optimization problems. Experimental results demonstrate that HBBO is effective and efficient for constrained optimization and in contrast with other stateoftheart evolutionary algorithms (EAs), the performance of HBBO is better, or at least comparable in terms of the quality of the final solutions and computational cost. Furthermore, the influence of the maximum mutation rate is also investigated.
With the development of science and engineering, related optimization problems become more and more complex. Optimization methods are being confronted with great challenges brought by some undesirable but unavoidable characteristics of optimization problems such as being highdimensional, nondifferentiable, nonconvex, noncontinuous, and so on. Efficient optimization methods are urgently required by the complicated optimization problems in the real world. Therefore, various evolutionary algorithms have been applied to solve difficult optimization problems in recent decades, which include GAs [
Biogeography based optimization (BBO) is a new populationbased algorithm. It simulates the blossom and extinction of species in different habitats based on the mathematical model of biogeography. Decision variables of better solutions tend to be shared in the migration operation and decision variables of each solution are probabilistically replaced to improve the diversity of population in mutation operation. Due to good search ability, BBO has been applied to PID parameter tuning [
In comparison with other EAs, owing to directcopyingbased migration and random mutation, exploration ability of BBO is not so efficient despite outstanding exploitation. In other words, BBO can be easily trapped into local optimum and suffer from premature convergence owing to lack of corresponding exploration to balance its exploitation.
In order to overcome the weakness of BBO, lots of improved BBO variants have been proposed. Ma [
In order to balance the exploration and exploitation of BBO, a new hybrid BBO called as HBBO is proposed in the paper. The unique points of HBBO are shown as the following. On one hand, a new hybrid mutation operator combining DE mutation and SBX is presented in HBBO while operators of EAs are often hybridized with migration operator in most of BBO variants. On the other hand, HBBO provides a new method to extend BBO to optimize constrained problems well due to only a few BBO variants available for constrained optimization in previous literatures. In addition, DE is applied to evolve one half of population to improve convergence speed further and chaotic search is introduced to enhance the diversity of population. Experiments have been conducted on twelve benchmark functions and four engineering optimization problems, and HBBO is compared with many other stateoftheart algorithms from the quality of solutions obtained and computational cost. Furthermore, the influence of maximum mutation rate on HBBO is studied.
The rest of the paper is organized as follows. Constrained optimization, basic BBO, mutation strategies of DE, and SBX are briefly introduced in Section
Constrained optimizations are always inevitable in scientific study and engineering design. A general constrained optimization problem can be written as the following
Biogeography is the study of the distribution of species on earth surface over time. BBO is proposed based on the mathematical model of biogeography by Simon in 2008 [
In BBO, each individual evolves by immigration and mutation operator. The SIVs of individuals are probabilistically shared in migration operator as shown in Algorithm
Target individual
For
Select
If
For
Select
If
Replace
End if
End for
End if
End for
The following mathematical model is applied to calculate immigration rate and emigration rate owing to its outstanding performance in [
In mutation operator, it is probabilistically decided whether or not to replace each SIV in a solution by a randomly generated SIV in the light of mutation rate. The detail of mutation operator is shown in Algorithm
Target individual
For
Select
If
Replace it with a randomly generated SIV
End if
End for
More details about basic BBO can be found in [
DE algorithm is a populationbased stochastic search method proposed by Storn and Price in 1997 [
Genetic algorithms simulate the evolutional process in nature to solve optimization problems. In GA, some good individuals are chosen based on Deb’s feasiblebased rule. Different individuals can share information in crosser operator. SBX is one of the most popular crosser operators which can explore the neighborhood region of parent individual as follows:
In mutation operator of basic BBO, SIVs are replaced probabilistically by new SIVs randomly generated. Although the mutation of BBO can improve the diversity of population, the random operation brings blindness to search. To modify the defect, a new hybrid mutation operator is proposed, in which DE mutation operator and SBX are mixed to generate promising SIV as shown in Algorithm
Target individual
For
If
Get two candidate SIVs of offspring;
(1) Get a temp SIV by DE
(2) If
Get another temp SIV by (
Else
Get another temp SIV by (
End if
Else
the
(population
End if
End for
Two temp offspring individuals are gotten for
In order to speed up convergence, DE is further hybridized with BBO. The first half of parent population also evolves by two DE mutation strategies (
For convenience and easy use, selfadaption mechanism for mutation scaling factor of DE proposed in [
Based on unique ergodicity, inherent stochastic property, and irregular chaos, chaotic search can reach each situation in given space so that it can contribute to the escape from the local optimum and is often integrated into EAs to enhance global search ability. Hence, the chaotic search is brought in for the first half of population. In the paper, logistic maps are used to generate chaotic sequences as follows:
In the initial phase, large chaotic search radium is helpful for escape from the local optimum; small chaotic search radium can improve the accuracy of search at the later stage of evolution. The search radium
In order to maintain solutions feasible, any new decision variable generated in evolution process should be repaired if it violates boundary. Suppose that
The whole procedure of HBBO is described in Algorithm
Generate the initial population
Evaluate the fitness and constraint violations of each individual in
For each generation do
Sort the individuals in
For each one in
Get two new individuals by two DE mutation strategies (
Evaluate the fitness value and constraint violations of these two new individuals;
Among these two new individuals and corresponding parent individual, the best one is stored into population
End for
Update the vector
For each one in the first half of
Generate a new individual by Algorithm
End for
For each one in the first half of
Get one offspring by Algorithm
End for
Go on chaotic search for the first half of POP and the new individuals generated are stored into population
Make a contrast between the corresponding ones in
as the corresponding one in
The population
Update
End for
In order to validate the performance of the proposed HBBO on numerical optimization, twelve benchmark test functions are adopted. The selected benchmark problems propose a good challenge and measure for constrained optimization techniques. Main characteristics of the selected benchmark functions are shown in detail in Table
Main characteristics of the twelve selected benchmark functions.
Benchmark function 

Type of objective function 

LI  NI  NE 


G01  13  Quadratic  0.0003%  9  0  0  6 
G02  20  Nonlinear  99.9970%  1  1  0  1 
G03  10  Nonlinear  0.0000%  0  0  1  1 
G04  5  Quadratic  26.9668%  0  6  0  2 
G05  4  Nonlinear  0.0000%  2  0  3  3 
G06  2  Nonlinear  0.0064%  0  2  0  2 
G07  10  Quadratic  0.0002%  3  5  0  6 
G08  2  Nonlinear  0.8575%  0  2  0  2 
G09  7  Nonlinear  0.5235%  0  4  0  2 
G10  8  Linear  0.0007%  3  3  0  3 
G11  2  Quadratic  0.0000%  0  0  1  1 
G12  3  Quadratic  4.774%  0  9^{3}  0  0 
For each test function, we performed 30 independent runs in matlab 7.0. The parameters of HBBO for experiments are set as follows:
Through various tests, an appropriate set of population size NP for all the selected functions is found with which HBBO can present desirable performance. In the set found, population size NP for each benchmark function is given as the following: 200 for G02, 150 for G07, and 100 for the rest of benchmark functions. In each run, the maximum generations are given as the following: 200 for G01 and G06, 150 for G02 and G04, 600 for G03, 1500 for G05, 334 for G07, 40 for G08, 300 for G09 and G11, 550 for G10, and 50 for G12. In G03 and G05, the toleration value for equation constraint equals 0.001 as recommended in [
Table
Statistic results for twelve benchmark functions obtained by HBBO.
Function  Optimal 




SD  FFEs 

G01  −15  −15  −14.799953  −15  −13 

50,100 
G02  −0.803619  −0.8036179  −0.7805965  −0.7852652  −0.7330360 

75,200 
G03  −1  −1.0050100  −1.0050100  −1.0050100  −1.0050100 

150,100 
G04  −30665.539  −30665.53867  −30665.5387  −30665.5387  −30665.5387 

37,600 
G05  5126.4981  5126.4842  5126.4842  5126.4842  5126.4842 

375,100 
G06  −6961.81388  −6961.81388  −6961.81388  −6961.81388  −6961.81388 

50,100 
G07  24.3062091  24.3062091  24.3062091  24.3062091  24.3062091 

125,400 
G08  −0.095825  −0.095825  −0.095825  −0.095825  −0.095825 

10,100 
G09  680.630057  680.630057  680.630057  680.630057  680.630057 

75,100 
G10  7049.248021  7049.248021  7049.248021  7049.248021  7049.248021 

137,600 
G11  0.75  0.74990  0.74990  0.74990  0.74990 

75,100 
G12  −1  −1  −1  −1  −1  0  12,600 
In this part, the proposed approach HBBO is compared with other six stateoftheart optimization technologies.
The following are the six stateofthe art optimization technologies: conventional BBO with DE mutation technology (CBODM) [
Statistical features of results for twelve benchmark functions obtained by HBBO and other six stateoftheart algorithms.
Function  Metrics  HBBO  CBODM  PSODE  CRGA  SAPF  SMES  CDE 

G01  Best 







Mean  −14.799953 


−14.9850  −14.552 



Worst  −13 


−14.9467  −13.097 





G02  Best 

−0.803557  −0.8036145  −0.802959  −0.803202  −0.803601  −0.794669 
Mean  −0.7805965 

−0.756678  −0.764494  −0.755798  −0.785238  −0.785480  
Worst  −0.7330360 

−0.6367995  −0.722109  −0.745712  −0.751322  −0.779837  


G03  Best  −1.0050100 

−1.0050100  −0.9997 


NA 
Mean  −1.0050100 

−1.0050100  −0.9972  −0.964 

NA  
Worst  −1.0050100 

−1.0050100  −0.9931  −0.887 

NA  


G04  Best 



−30665.520  −30665.401 


Mean 



−30664.398  −30665.922 

−30665.536  
Worst 



−30660.313  −30656.471 

−30665.509  


G05  Best  5126.4842 


5126.500  5126.907  5126.599  NA 
Mean  5126.4842 


5507.041  5214.232  5174.492  NA  
Worst  5126.4842 


6112.075  5564.642  5304.167  NA  


G06  Best 



−6956.251  −6961.046 


Mean 



−6740.288  −6953.061  −6961.284  −6960.603  
Worst 



−6077.123  −6943.304  −6952.482  −6901.285  


G07  Best 

24.326 

24.882  24.838  24.327  NA 
Mean 

24.345  24.306210  25.746  27.328  24.475  NA  
Worst 

24.378  24.3062172  27.381  33.095  24.843  NA  


G08  Best 


−0.095826 



NA 
Mean 


−0.095826  −0.095819  −0.095635 

NA  
Worst 


−0.095826  −0.095808  −0.092697 

NA  


G09  Best 



680.726  680.773  680.632  680.771 
Mean 



681.347  681.246  680.643  681.503  
Worst 



682.965  682.081  680.719  685.144  


G10  Best 

7059.802 

7114.743  7069.981  7051.903  NA 
Mean 

7075.832 

8785.149  7238.964  7253.047  NA  
Worst 

7098.254  7049.2482  10826.09  7489.406  7638.366  NA  


G11  Best  0.74990 

0.749999 

0.749 

NA 
Mean  0.74990 

0.749999  0.752  0.751 

NA  
Worst  0.74990 

0.749999  0.757  0.757 

NA  


G12  Best 







Mean 




−0.99994 



Worst 




−0.999548 


With respect to CBODM, a variant of BBO, similar results are obtained by HBBO for five functions (G04, G06, G08, G09, and G12); in two functions (G07, G10), HBBO has better performance in the respect of considered metrics (
In contrast with other five stateoftheart methods, the performance of HBBO is obviously inferior for function G01; HBBO can get better or similar solutions for the selected test functions except for G01, G03, and G11. In G03 and G11, the results obtained by HBBO are only lightly inferior to those of SMES. Furthermore, the computational cost is very competitive with respect to other methods for all selected test functions except G05.
In this part, four wellknown engineering optimization problems are utilized to validate the performance of HBBO on solving realworld optimization problems. The four engineering optimization problems contain welded beam design problem, tension/compression spring design problem, speed reducer design problem, and threebar truss design problem, which are listed in Appendix
Statistic results for four engineering optimization problems solved by HBBO.
Engineering optimization problem 




SD  FFEs 

Welded beam design  1.724852309  1.724852309  1.724852309  1.724852309 

25,050 
Tension/compression spring design  0.012665233  0.012665393  0.012665234  0.012666698 

43,800 
Speed reducer design  2996.348165  2996.348165  2996.348165  2996.348165 

25,100 
Threebar truss design  263.89584338  263.89584338  263.89584338  263.89584338 

7,550 
In order to demonstrate the superiority of HBBO, it is compared with other stateoftheart algorithms on the four engineering problems. Welded beam and tension/compression spring design problems are also attempted by PSODE [
Statistic results for welded beam design obtained by HBBO and other six stateoftheart methods.
Method 



FFEs 

HBBO 




PSODE  1.7248531  1.7248579  1.7248811  33,000 
CDE  1.733461  1.768158  1.824105  240,000 
CPSO  1.728024  1.748831  1.782143  200,000 
( 

1.777692  NA  30,000 
UPSO  1.92199  2.83721  NA  100,000 
ABC 

1.741913  NA  30,000 
Statistic results of HBBO and other six stateoftheart methods for tension/compression spring design.
Method 



FFEs 

HBBO 

0.012665393  0.012666698  43,800 
PSODE 



42,100 
CDE  0.0126702  0.012703  0.012790  240,000 
CPSO  0.0126747  0.01273  0.012924  200,000 
( 
0.012689  0.013165  NA  30,000 
UPSO  0.01312  0.02294  NA  100,000 
ABC 

0.012709  NA 

Statistic results of HBBO and other six stateoftheart methods for speed reducer design.
Method 



FFEs 

HBBO 




PSODE 


2996.348166  70,100 
( 


NA  30,000 
ABC  2997.058  2997.058  NA  30,000 
Statistic results of HBBO and other six stateoftheart methods for threebar truss design.
Method 



FFEs 

HBBO 




PSODE 



17,600 
Ray and Liew  263.89584654  263.90335672  263.96975638  17,610 
From Tables
In this part, HBBO is compared with the original BBO and selfadapting DE (SADE) to demonstrate the searching efficiency of HBBO further. In addition, the influence of maximum mutation rate on searching efficiency of HBBO is investigated.
The detail of the original BBO can be gotten from [
Figure
Objective function value curves of four test functions solved by HBBO, SADE, and BBO.
G02
G03
G07
G09
The maximum value of mutation rate
The other parameters are in accordance with description in Section
Statistical features of the results obtained by HBBO with different
Function  Metrics  HBBO  







G01 







−14.666665  −14.633610  −14.600000 

−14.632939  

−12.999949  −12.008289  −13 

−12  
SD 








G02 

−0.8036071  −0.8036171 

−0.8036179  −0.8036173 

−0.7780832  −0.7787019 

−0.7805965  −0.7775819  

−0.71807066 

−0.7248475  −0.7330360  −0.7225584  
SD 








G03 

−1.0044781 





−0.7515293 






−0.2971747 





SD 








G07 







24.3062100 






24.3062351 





SD 








G10 







7054.178611  7049.248022 


7049.248027  

7194.608835  7049.248076 


7049.248200  
SD  26.52 




From Table
The paper proposes a new hybrid biogeography based optimization (HBBO) for constrained optimization. For the presented algorithm HBBO, a new mutation operator was proposed to generate promising solutions by merging DE mutation with SBX; a half of the population also evolved by two mutation strategies of DE. Chaotic search was introduced for escape from stagnation and Deb’s feasibilitybased rule was applied to handle constraints. Furthermore, selfadaption mechanism for the mutation scaling factor of DE was utilized to avoid bothering of choosing an appropriate parameter.
Simulation experiments were performed on twelve benchmark test functions and four wellknown engineering optimization problems. HBBO can obtain better or comparable results in contrast with other stateoftheart optimization technologies. At the same time, the low computation cost is the obvious advantage of our HBBO. In short, HBBO is an effective and efficient method for constrained optimization. In addition, the influence of maximum mutation rate was investigated and the results demonstrate HBBO with maximum mutation rate of middle value has better comprehensive performance.
Maximum immigration rate
A welded beam is designed for the minimum cost subject to constraints on shear stress (
In this problem, the objective is to minimize the weight of a tension/compression spring subject to constraints on minimum deflection, shear stress, surge frequency, and limits on outside diameter and on design variables. The design variables are the mean coil diameter
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20120036130001, the Fundamental Research Funds for the Central Universities of China under Grant no. 2014MS93, and the Independent Research Funds of State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources of China under Grant no. 201414.