Two defects of biogeographybased optimization (BBO) are found out by analyzing the characteristics of its dominant migration operator. One is that, due to global topology and directcopying migration strategy, information in several goodquality habitats tends to be copied to the whole habitats rapidly, which would lead to premature convergence. The other is that the generated solutions by migration process are distributed only in some specific regions so that many other areas where competitive solutions may exist cannot be investigated. To remedy the former, a new migration operator precisely developed by modifying topology and copy mode is introduced to BBO. Additionally, diversity mechanism is proposed. To remedy the latter defect, quantitative orthogonal learning process accomplished based on space quantizing and orthogonal design is proposed. It aims to investigate the feasible region thoroughly so that more competitive solutions can be obtained. The effectiveness of the proposed approaches is verified on a set of benchmark functions with diverse characteristics. The experimental results reveal that the proposed method has merits regarding solution quality, convergence performance, and so on, compared with basic BBO, five BBO variant algorithms, seven orthogonal learningbased algorithms, and other nonOLbased evolutionary algorithms. The effects of each improved component are also analyzed.
Optimization problems have widely existed in any areas of life (economic, engineering, medicine, business, urban planning, etc.) and they play a very vital role in both academic research field and industrial production. They tend to be more complicated with the unceasing progress of science and technology. To handle such challenging optimization problems, much effort has been devoted by researchers and different techniques have been proposed during the last several decades. The proposed methods can be categorized into two main groups: derivativebased algorithms and artificial intelligence methods. Derivativebased algorithms require objective functions to be smooth and differentiable. Due to this property, they are restricted to be applied to many complex optimization problems. Oppositely, artificial intelligence methods do not require certain properties to be satisfied. In artificial intelligence methods, evolutionary algorithms (EAs) are popular and have been successfully applied to realworld problems. EAs mimic various social behaviors existing in nature to solve the optimization problems. Some popular EAs include genetic algorithm (GA) [
As an effective EA, biogeographybased optimization (BBO) [
In this paper, both migration process modification and hybridization strategy are focused on. Two defects of BBO are found out by analyzing the characteristics of the migration operator, which is the main operator of BBO. One is that, due to global topology and directcopying migration strategy, information in several goodquality habitats tends to be copied to the whole habitats rapidly, which would lead to premature convergence. The other is that the solutions generated by migration operator are distributed only in some specific regions so that other areas where competitive solutions may exist cannot be investigated. To remedy the former, a new migration operator precisely developed by taking topology modification and copymode improvement as two cutin points is proposed. Diversity mechanism is also proposed for remedying it. To remedy the latter defect, quantitative orthogonal learning (QOL) process accomplished based on space quantizing and orthogonal design is adopted. It aims to investigate the solution space thoroughly so that more competitive solutions can be obtained.
The remainder of this paper is organized as follows. Section
BBO is inspired by the equilibrium theory of island biogeography. Each individual in population is called a “habitat.” The goodness of a habitat (i.e., candidate solution) is measured by the habitat suitability index (HSI) [
There are two main operators in BBO: (
In allusion to the defects of the migration operator in basic BBO, a new BBO variant based on multitopology migration and QOL, called MTQLBBO, is proposed. The key points of our methods are described in detail in this section.
Assuming that the number of decision variables in
Migration operator is the main operator of BBO algorithm, because it determines how the new population is generated from the previous population. So, it influences BBO’s search trajectory from the initial population. To enhance the performance of BBO, it is very necessary to analyze the characteristics of the migration operator.
During the migration process, we first use
A schematic diagram of multitopology.
Global topology
Ring topology
Multitopology includes two kinds of topology: global topology and ring topology, as shown in Figure
The motivation of introducing the parameter
As for copymode modification, an indirectcopying migration operator is introduced. This operator is expressed as
The core idea of constructing this operator is based on two considerations. One is that, instead of directly copying information from
Divide the whole habitats into Group A and Group B
Implement migration operator in Group A
Select emigrated habitat
Randomly select a habitat from all habitats both in A and B group (
Implement migration operator in Group B
Randomly arrange the habitats in Group B in a ring topology
Select emigrated habitat
Randomly select a habitat from the whole habitat (
Mix the whole habitats together
There are three points that should be noted. First, the two groups are required to have roughly the same average HSI level. The grouping method in this paper is as follows:
Sort all habitats by the HSI from high to low and then number all of them from 1 to
Select
Put habitats whose numbers are those selected integers into Group B and the rest of the habitats are put into Group A.
Secondly, habitat
To verify the effectiveness and efficiency of the multitopology migration process, an enhanced BBO with the proposed migration operator (EBBO) is constructed to compare with basic BBO. Note that EBBO differs from BBO only in that it uses multitopology migration operator to replace the basic migration operator. Both EBBO and basic BBO are conducted on a 30dimensional Alpine function
Benchmark functions in experimental tests.

Function name  Formula  Dimension  Search domain 


Elliptic 




Exponential 




Quartic 




Rosenbrock 




Schwefel 2 




Schwefel 3 




Schwefel 4 




Sphere 




Step 




SumPower 




SumSquare 




Zakharov 




Ackley 




Alpine 




Griewank 




Levy 




Penalty 1 




Penalty 2 




Rastrigin 




Schwefel 




Weierstrass 




Bohachevsky_2 




Expansion of 






NCRRastrigin 




For a fair comparison, both BBO and EBBO start their iterative process from the same initial population. The values of
Variables
Initial population by Latin Hypercube Sampling
BBO at generation = 50
BBO at generation = 100
BBO at generation = 150
BBO at generation = 200
EBBO at generation = 50
EBBO at generation = 100
EBBO at generation = 150
EBBO at generation = 200
Assume solutions are threedimensional, and denote
Spatial location of
It can be seen from Figure
The process is described as follows. Suppose
To skip out on local optimal solutions, diversity mechanism is introduced. When the average HSI of the current population does not change over some certain generations in the evolutionary process, which is expressed as
By incorporating the multitopology migration operator, QOL strategy, and diversity mechanism into BBO, the proposed MTQLBBO is developed and its main procedures are described as follows:
Set the parameters of MTQLBBO, mainly including
Initialize the population based on Latin Hypercube Sampling.
Sort the population from best to worst and obtain each habitat’s species number according to its ranking in the population.
Calculate
Implement QOL operator.
Similar to BBO, perform elite reservation strategy.
If required, apply the diversity mechanism.
Check the stopping criteria. If met, stop and output the result; otherwise, go back to step (
To verify the performance of MTQLBBO, a test suit, consisting of 24 scalable benchmark functions [
Parameter settings of the MTQLBBO algorithm used in this paper.
Parameters  Notation  Value 

Maximum immigration rate 

1 
Maximum emigration rate 

1 
Population size 

100 
Orthogonal array 


Mixed degree of ring topology 

0.3 
Maximum evaluations for each function.
FEs_Max/  Functions 

150,000 

200,000 

250,000 

300,000 

400,000 

For fair comparison between different algorithms, to eliminate the contingency, all experiment results are obtained based on 40 independent runs. The mean and the standard deviation of the bestofrun errors over 40 independent runs are provided to measure the performance. In addition, to compare the statistically significant differences between the computational results of two algorithms, MannWhitney
To fully evaluate the performance of the proposed algorithm, several group experiments are carried out. The experiment structure is described as Figure
The experiment structure diagram.
The experiment can be divided into two parts: (
It is known from Section
To validate the effectiveness of MTQLBBO, the optimization problems are solved 40 times independently for each benchmark function using BBO and MTQLBBO. The number of dimensions
Comparison of BBO and MTQLBBO for all benchmark functions (

BBO  MTQLBBO  

Mean  Standard deviation  Mean  Standard deviation  

1.98 
1.43 



5.12 
2.21 



2.95 
1.12 





1.43 
6.87 



6.87 
4.52 

8.40 
6.72 



3.11 
5.23 



1.34 
8.32 



1.23 
2.21 



7.42 
2.64 



8.98 
1.24 



4.14 
1.03 



8.82 
2.98 



1.01 
2.76 



3.68 
1.98 



2.89 
2.36 



2.52 
3.70 



2.57 
1.21 



1.21 
2.58 



6.82 
5.23 



9.51 
1.23 



8.63 
1.02 



1.36 
1.89 



5.74 
4.23 





22/0/2 
Besides solution quality, convergence property is also an important evaluation indicator for judging the performance of an algorithm. To evaluate the convergence performance, convergence curves of some representative benchmark functions are plotted and shown in Figure
Convergence graphs of MTQLBBO and other five BBO variants for six representative test functions.
To explore the effect of the problem dimension on the performance of MTQLBBO, a scalability study based on
Mean and standard deviation of optimal results of BBO and MTQLBBO on different dimensions (

BBO  MTQLBBO  

Mean (std. dev.)  Mean (std. dev.)  



4.81 
+ 


3.87 
+ 



−  4.97 

3.92 
+ 


6.70 
+ 


9.11 
+ 


6.49 
+ 



=  0.00 

7.17 
+ 


5.48 
+ 


1.91 
+ 


2.42 
+ 


1.15 
+ 


2.02 
+ 


1.18 
+ 


8.27 
+ 


7.05 
+ 


4.21 
+ 



=  0.00 

8.87 
+ 


3.58 
+ 


1.14 
+ 


6.10 
+ 


2.29 
+ 




21/2/1  





4.04 
+ 


8.32 
+ 


5.15 
+ 


2.18 
+ 


2.09 
+ 


6.15 
+ 



−  8.34 

3.24 
+ 



−  2.86 

1.21 
+ 


1.23 
+ 


1.84 
+ 


9.99 
+ 


3.21 
+ 


6.12 
+ 


5.32 
+ 


2.83 
+ 


3.54 
+ 


1.43 
+ 


8.12 
+ 


1.53 
+ 


1.23 
+ 


2.69 
+ 


2.50 
+ 




22/0/2  





6.12 
+ 


2.97 
+ 



−  8.81 

3.17 
+ 


5.49 
+ 


9.91 
+ 


2.96 
+ 


8.71 
+ 


9.97 
+ 


7.79 
+ 


5.19 
+ 


2.56 
+ 


5.43 
+ 


8.92 
+ 


8.32 
+ 


9.14 
+ 


7.02 
+ 


6.98 
+ 


7.51 
+ 


3.31 
+ 


3.19 
+ 


2.87 
+ 


4.87 
+ 


1.23 
+ 




23/0/1  





9.25 
+ 


6.95 
+ 


2.10 
+ 


4.91 
+ 


7.09 
+ 


4.24 
+ 


1.49 
+ 


1.32 
+ 



−  7.86 

5.64 
+ 


8.13 
+ 


2.14 
+ 


1.73 
+ 


2.21 
+ 


1.21 
+ 


4.30 
+ 


4.42 
+ 


3.45 
+ 


4.93 
+ 


5.63 
+ 


3.57 
+ 


2.45 
+ 


5.47 
+ 


3.21 
+ 




23/0/1 
From the results, it can be seen that MTQLBBO performs better than BBO in all different dimensions. For
From the above comparison, we can conclude that MTQLBBO has better performance than BBO for solving both low dimensional and high dimensional optimization problems on the selected instances. In addition, it is noticed that, for MTQLBBO, although it is able to provide better results on some functions when the dimension increases from 10 to 160, its performance deteriorates on more functions with the growth of problem dimension. The main reason is that the search space increases dramatically, thus raising the difficulty of solving the problem.
The performance of MTQLBBO is compared with other five BBO variant algorithms at
Comparison with other BBO variant algorithms (

MOBBO  PBBOG  DBBO  RCBBOG  POLBBO  MTQLBBO  

Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  

2.30 
+  6.99 
+  7.37 
+  3.72 
+  8.69 
+ 


5.35 
+  4.18 
+  4.31 
+  7.95 
+ 

= 


7.83 
+  5.53 
+  4.02 
+  5.62 
+  8.74 
+ 


5.57 
+  6.99 
+  4.90 
+  2.64 
+  1.46 
+ 


1.45 
−  6.16 
− 

−  6.82 
+  8.80 
−  6.87 

5.80 
+  5.95 
+  5.23 
+  2.03 
+  6.13 
+ 


1.39 
−  5.19 
+  9.38 
+  1.54 
+ 

−  4.23 

7.44 
+  4.78 
+  4.61 
+  1.12 
+  5.59 
+ 



= 

= 

= 

= 

= 


8.30 
+  3.93 
+  4.51 
+  1.55 
+ 

−  6.78 

1.60 
+  2.32 
+  4.20 
+  2.52 
+  5.24 
+ 


5.14 
+  2.45 
+  2.98 
+  6.54 
+  9.25 
+ 


3.33 
+  3.95 
+  6.64 
+  2.12 
+  4.00 
+ 


8.23 
+  8.32 
+  9.08 
+  1.83 
+ 

−  5.04 

4.05 
+  5.62 
+  4.73 
−  4.22 
+ 

−  2.36 

1.43 
+  4.30 
+  3.10 
+  3.01 
+  7.43 
+ 


5.96 
+  1.92 
+  2.25 
+  5.36 
+  1.57 
+ 


5.96 
+  2.29 
+  8.26 
+  3.31 
+  2.46 
+ 



=  5.29 
+ 

=  3.61 
+ 

= 


8.58 
+  7.75 
+  6.18 
+  6.12 
+  7.79 
+ 


1.38 
+  2.46 
+  2.76 
+  1.86 
+ 

= 


5.86 
+  8.12 
+  2.45 
+  1.45 
+ 

= 



−  4.98 
+  1.56 
+  2.41 
+  2.58 
+  1.89 

2.13 
−  4.12 
−  2.45 
+  3.14 
+ 

−  5.98 



18/2/4  21/1/2  20/2/2  23/1/0  13/5/6 
It can be seen from Table
Convergence graphs of MTQLBBO and other five BBO variants for six representative test functions.
Due to the advantages of orthogonal learning (OL) strategy, it has been employed in many EAs to improve algorithms’ performance. In this subsection, the results achieved by MTQLBBO are compared with those achieved by other OLbased evolutionary algorithms, which include OXDE [
Comparison with other OLbased algorithms.

OXDE 
ODE/2  OPSO 
OLPSOG 
OLPSOL 
OXBBO  POLBBO 
MTQLBBO 

Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  

NA 
1.45 
5.50 
1.16 
1.64 
2.01 
8.74 


4.78 

4.96 
2.15 
1.26 
1.89 
4.40 
1.43 

NA 

NA  NA  NA  3.21 
8.80 
6.87 

NA  1.43 
1.26 
9.85 
7.67 
1.12 
6.13 


NA  2.72 
NA  NA  NA  2.61 

4.23 

5.21 
2.06 
6.45 
4.12 
1.11 
3.96 
5.59 


NA 

NA  NA  NA 




2.66 
4.67 
6.23 
7.98 
4.14 
6.79 
4.00 


1.82 

2.29 
4.83 



2.36 

1.03 
6.73 
1.56 
1.59 
1.57 
7.06 
1.57 


2.25 
4.37 
1.46 
4.39 
1.35 
7.06 
2.46 


8.99 

6.97 
1.07 






NA  2.93 
3.84 
3.82 
NA  7.79 
8.94 



6/0/2  7/1/4  10/0/0  10/0/0  7/1/2  7/2/3  8/2/3 
MTQLBBO outperforms OXDE, ODE/2, OPSO, OLPSOG, OLPSOL, OXBBO, and POLBBO on 6, 7, 10, 10, 7, 7, and 8 functions, respectively. The algorithm is surpassed by OXDE, ODE/2, OLPSOL, OXBBO, and POLBBO on 2, 4, 2, 3, and 3 functions, respectively. For OPSO and OLPSOG, the algorithm wins both of them on all functions. Overall, MTQLBBO provides relatively competitive solutions compared to other OLbased algorithms.
The proposed algorithm MTQLBBO is further compared with some other nonOLbased EAs, which include CLPSO [
Comparison with other nonOLbased EAs.

CLPSO  CMAES  GL25  SLPSO 
MGBGE 
iMEABC  MSODPSOG  MSODPSOL  MTQLBBO 

Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  

9.10 
2.35 
1.80 
NA  2.14 
1.50 
NA 
NA 


5.47 
5.32 
2.12 
2.06 
1.69 
2.81 

1.22 
1.43 

8.19 

2.47 
7.93 
6.10 
1.14 
1.54 
1.02 
6.87 

3.14 
1.80 
2.56 
1.35 
8.50 
7.15 
1.46 
1.35 


3.35 

4.04 
2.54 
1.31 
5.13 
NA  NA  4.23 

1.90 
5.84 
2.87 
2.78 
8.79 
1.08 
NA  NA 


2.10 
6.49 
1.86 
NA 


NA  NA 


2.68 
1.95 
1.03 
3.47 
7.69 
4.94 
3.73 
2.90 


1.42 
6.57 
1.24 
2.27 




2.36 

4.56 
2.40 
1.03 
1.57 
1.57 
2.16 
1.57 
1.57 


1.10 
1.40 
4.20 
NA  1.35 
3.36 
NA  NA 


9.19 
2.44 
2.47 

3.98 





4.87 
5.55 
3.57 
3.82 
4.87 
3.82 
3.82 
1.18 




13/0/0  10/0/3  11/0/2  7/1/2  6/1/3  8/2/3  4/1/3  4/1/3 
In this section, we have analyzed effects of modified components, that is, multitopology based migration strategy and QOL strategy, on the performance of the algorithm. The influences of three additional parameters, brought by the modified components, are also investigated in this section.
Convergence property is one of the most important characteristics for an optimization algorithm. To analyze the effects different components have on convergence property, two variants of MTQLBBO, that is, MTBBO and QOLBBO, are developed. They differ from MTQLBBO only in that MTBBO does not contain the QOL process, while QLBBO implements BBO’s original migration operator instead of the multitopology migration operator. Both of them are used to solve all benchmark functions, and their performance is compared with that of BBO and MTQLBBO. A representative convergence graph is plotted in Figure
Convergence graphs of BBO, MBBO, QOLBBO, and MTQLBBO on function
Comparing the convergence curves of BBO and MTBBO, we can see that MTBBO converges to a better solution than BBO, but its convergence speed in early stages is slower than BBO. It shows that multitopology migration strategy exactly improves exploration ability and can help converge to a better solution. However, the paid expense is reducing the convergence speed in the early iterative stages. QOLBBO converges faster and achieves a better solution than BBO by investigating the solution space more systematically. The paid expense is extra computational overhead brought by the QOL process.
Moreover, it is worth noting that, different from slower convergence speed in the early stages resulting from adding multitopology migration process to BBO, adding BBO to QOLBBO accelerates the convergence speed instead. The reason might be as follows. Good habitats guide the searching directions of the population. Information in good habitats spreads to the whole habitats very rapidly in BBO, and the whole habitats can move toward a more promising direction quickly. Multitopology reduces the speed of information spread and thus decelerates the convergence speed. However, MTBBO has good population diversity. When QOL process is implemented among habitats with better population diversity, the space that can be potentially investigated becomes much wider. Hence, the solution space can be investigated more thoroughly and a betterquality solution is more likely to be obtained at each iteration, which accelerates the convergence speed. It also proves that the mutually beneficial cooperation between the multitopology migration operator and QOL process can significantly help MTQLBBO improve the searching process toward the promising directions.
To analyze the effects the modified components have on solution quality, both MTBBO and QOLBBO are used to optimize all test functions and the results are summarized in Table
Optimal results of MTBBO, QOLBBO, and MTQLBBO on all test functions (

BBO  MTBBO  QOLBBO  MTQLBBO  

Mean  Standard deviation  Mean  Standard deviation  Mean  Standard deviation  Mean  Standard deviation  

1.98 
1.43 
1.23 
2.32 
+  2.43 
7.43 
+ 



5.12 
2.21 
5.09 
1.23 
+ 


= 



2.95 
1.12 
7.35 
9.21 
+  6.43 
5.23 
+ 





7.23 
2.91 
+  9.23 
3.12 
+  1.43 
6.87 

6.50 
8.52 


−  7.35 
9.15 
+  6.87 
4.52 

8.40 
6.72 
6.01 
6.12 
+  9.34 
4.92 
+ 



3.11 
5.23 
6.39 
7.29 
+ 


−  4.23 
7.23 

1.34 
8.32 
9.14 
4.23 
+  4.87 
3.42 
+ 



1.23 
2.21 
1.17 
8.93 
+  1.43 
5.43 
+ 



7.42 
2.64 
9.07 
2.12 
+  6.87 
7.32 
+ 



8.98 
1.24 
5.34 
4.23 
+  5.54 
4.19 
+ 



4.14 
1.03 
9.09 
1.94 
+  8.82 
9.71 
+ 



8.82 
2.98 
4.34 
5.87 
+  6.94 
9.23 
+ 



1.01 
2.76 
1.23 
5.87 
+  6.12 
9.23 
+ 



3.68 
1.98 
2.03 
1.01 
+ 


= 



2.89 
2.36 
4.95 
3.29 
+  3.13 
4.38 
+ 



2.52 
3.70 
3.56 
3.47 
+  1.29 
3.21 
+ 



2.57 
1.21 
2.37 
2.18 
+  1.87 
7.34 
+ 



1.21 
2.58 
8.23 
3.23 
+  9.74 
8.54 
+ 



6.82 
5.23 
3.82 
2.89 
+  6.64 
1.32 
+ 



9.51 
1.23 
1.61 
2.71 
+  1.01 
9.48 
+ 



8.63 
1.02 
6.15 
4.23 
+  4.14 
8.04 
+ 



1.36 
1.89 
1.03 
3.01 
+  7.12 
5.97 
+ 



5.74 
4.23 
2.12 
5.12 
+  6.47 
4.79 
+ 





23/0/1  22/2/1  

21/0/3  21/0/3 
Although multitopology migration operator could improve the performance, the effects depend on the determination of the parameter
Influence of the parameter

0.05  0.1  0.2  0.3  0.4  0.5  0.8  1.0 

Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  Mean (std. dev.)  

2.16 
5.97 

3.45 
4.39 
6.85 
6.71 
4.85 





9.56 
3.27 
1.97 
1.68 

7.04 
2.43 


1.66 
5.87 
1.05 
4.36 

2.94 
7.78 

1.43 
8.54 
1.40 
3.52 
8.09 

5.21 
2.66 

6.87 
6.11 
8.61 
5.32 
5.35 

6.75 
8.11 
5.25 

4.16 
8.52 
6.56 
8.62 

2.33 

7.40 
4.23 
1.91 
8.06 
4.20 
9.67 

1.55 
3.54 


4.17 
2.67 
8.07 
9.14 

4.89 
3.83 


1.67 
6.00 
7.00 
1.02 

9.00 
2.34 

6.78 
4.45 
6.20 
9.30 
7.45 

3.90 
2.94 
4.74 

5.98 
4.62 
5.02 
1.09 

6.14 
6.78 
6.70 

1.18 
2.80 
1.11 
8.98 

5.63 
7.22 
9.08 


3.68 
3.10 
8.23 

1.48 
9.07 

5.04 
3.92 
1.69 
1.70 
6.99 

6.02 
4.72 

2.36 
5.91 
2.16 
5.67 
1.34 

1.27 
1.27 
4.01 

2.89 
4.40 
7.43 
2.65 

4.85 
7.68 
7.27 

3.30 
1.38 
4.26 
1.50 

6.72 
9.11 
7.62 

6.17 
6.25 
1.69 
6.21 

8.11 
6.92 
2.12 

3.26 
4.53 
3.14 
7.89 

8.87 
7.68 
4.54 

7.62 
6.60 
5.95 
1.17 

5.80 
6.80 
4.27 

9.51 
6.47 
9.38 
7.70 

2.09 
6.55 
6.67 

3.14 
5.44 
2.26 
2.99 

7.12 
3.56 
9.61 

6.70 
1.23 
4.06 
3.89 

1.22 
7.27 

5.98 
4.64 
7.99 
3.51 
5.48 



1/24  2/24  11/24  15/24  1/24  0/24  0/24  0/24 
The value of
Most of the best solutions for unimodal functions, that is, 8 out of 12, are achieved when the parameter equals 0.2 while most of the best ones for multimodal functions, 9 out of 12, are obtained when the parameter equals 0.3. The main reason might be as follows. A larger value of parameter
As for the other two parameters, that is, the number of levels (
A new variant of BBO, referred to as MTQLBBO, is proposed to solve the global numerical optimization problems. On one hand, the proposed multitopology migration operator can avoid the homogenization of habitats and enhance the exploration ability. On the other hand, QOL operator is able to investigate the solution space thoroughly and systematically so that more competitive solutions can be obtained.
To verify the effectiveness of the proposed algorithm, benchmark functions with various characteristics are employed. Experiment tests, including basic comparison with BBO, comparison with five BBO variant algorithms, comparison with seven OLbased algorithms, comparison with six evolutionary algorithms, the effect of problem dimension, the effects of different modified components, and the influence of three additional parameters, are conducted. The experimental results demonstrate that the improvement strategies exactly enhance the performance of BBO in terms of solution quality, convergence rate, and so on. Moreover, comparisons also show that MTQLBBO performs better than some other evolutionary algorithms on the majority of the selected test functions.
For future work, this paper can be extended in several directions. This work only considers unconstrained optimization problems, and it can be extended to solve multiobjective optimization problems or constrained optimization problems. Another future work can be conducted to solve some practical engineering problems. Additionally, the influences of parameter
The authors declare that they have no conflicts of interest.
This work is supported by the National Key Research and Development Program of China (2016YFB0900100).