The bat algorithm (BA) is a heuristic algorithm that globally optimizes by simulating the bat echolocation behavior. In order to improve the search performance and further improve the convergence speed and optimization precision of the bat algorithm, an improved algorithm based on chaotic map is introduced, and the improved bat algorithm of Levy flight search strategy and contraction factor is proposed. The optimal chaotic map operator is selected based on the simulation experiments results. Then, a multipopulation parallel bat algorithm based on the island model is proposed. Finally, the typical test functions are used to carry out the simulation experiments. The simulation results show that the proposed improved algorithm can effectively improve the convergence speed and optimization accuracy.
Optimization is the selection of the best elements for a particular set of criteria from a range of effective choices, which shows many different advantages and disadvantages in terms of computational efficiency and global optimization probability, but has a wide range of applications in industry and scientific research [
The bat algorithm (BA) is a heuristic search algorithm proposed by Professor Yang in 2010 based on swarm intelligence [
An elite crossover binary bat algorithm proposed in [
In this paper, a multipopulation parallel bat algorithm based on island model is proposed and applied to three strategies including chaos [
The bat algorithm (BA) is a new swarm intelligence optimization algorithm, which simulates the foraging behavior of bats. Its principle is to use the bat’s advanced echolocation capability [
BA’s development takes advantage of existing algorithms and other interesting features, inspired by the wonderful behavior of miniature bat echolocation. Based on these assumptions, this algorithm generates a set of solutions in a random manner and then uses the loop search to find the optimal solution. During this period, the local search is adopted. That is to say that around the optimal solution, the local solution is generated by random flight and produces a global optimal solution. For bats, if their foraging space belongs to the
In the initial setting process, the frequency of the bat’s emitted sound waves is uniformly distributed in
By analyzing the loudness
The pseudocode of the bat algorithm is described as follows [
Input: Bat population
Output: The best population
init_bat ();
eval = evaluate the new population;
while termination_condition_not_meet do
for
if rand (0, 1) >
end if {local search}
if
eval = eval + 1;
if
end if {simulated annealing}
end for
end while
The chaotic motion is a highly unstable motion in a deterministic system that is limited to a finite phase space. Chaos is a form of aperiodic motion, which is unique and extensive in nonlinear systems [
There are 10 typical kinds of chaotic mapping [
Simulation test functions.
Function  Expression  Range  Minimum value 



[−100, 100]  0 


[−600, 600]  0 


[−32, 32]  0 


[0, 
0 


[−5.12, 5.12]  


[−65.536, 65.536]  0 


[−5 10]  0 


[−50, 50]  0 


[−5, 5]  0.00030 


[0, 1]  −3.32 


[0, 10]  −10.536 
In this section, the first six functions are adopted to carry out the simulation experiments. The simulation results of BA based on the different chaotic mappings is shown in Figure
Convergence curves under ten chaotic mappings. (a)
Simulation results on function optimization problems under different chaotic mappings.
Function  Chaotic mapping  Avg.  Best  Std. 


Chebyshev map  2.46 
6.07 
7.37 
Circle map  4.08 
9.77 
1.12  
Gauss/mouse map  1.77 
6.33 
5.25  
Iterative map  1.92 
1.22 
5.76  
Logistic map  1.74 
1.10 
5.28  
Piecewise map  1.91 
5.70 
7.10  
Sine map  1.60 
9.53 
4.76  
Singer map  3.38 
1.14 
1.01  
Sinusoidal map  8.43 
1.07 
2.53  
Tent map 

1.21 
7.34  



Chebyshev map  1.22 
1.67 
3.01 
Circle map  2.06 
1.27 
5.76  
Gauss/mouse map  6.44 
1.28 
1.47  
Iterative map  3.04 
1.28 
6.60  
Logistic map  7.70 
8.32 
1.66  
Piecewise map 

1.33 
6.04  
Sine map  3.75 
4.52 
1.12  
Singer map  8.91 
1.16 
1.33  
Sinusoidal map  4.62 
2.06 
9.24  
Tent map  9.79 
8.08 
2.58  



Chebyshev map  2.11 
1.16 
5.77 
Circle map  2.13 
1.16 
9.04  
Gauss/mouse map  2.01 
1.42 
7.21  
Iterative map  2.24 
1.16 
7.99  
Logistic map  2.23 
1.91 
9.84  
Piecewise map 

1.59 
6.80  
Sine map  2.41 
1.16 
6.13  
Singer map  1.95 
1.64 
7.85  
Sinusoidal map  2.22 
1.76 
9.29  
Tent map  1.90 
1.31 
7.28  



Chebyshev map  −4.3381  −3.4831  0.4727 
Circle map  −4.2815  −2.8800  0.9298  
Gauss/mouse map  −4.4270  −3.3986  0.7517  
Iterative map  −4.3976  −3.2347  0.7557  
Logistic map  −4.6527  −3.5362  0.9810  
Piecewise map 

−3.3934  0.1789  
Sine map  −4.1871  −3.3551  0.5061  
Singer map  −3.9540  −3.2770  0.5729  
Sinusoidal map  −4.1621  −2.6385  0.7561  
Tent map  −4.0751  −2.6231  0.9906  



Chebyshev map  12.3377  6.9650  4.9388 
Circle map  14.6261  6.9650  5.0547  
Gauss/mouse map  13.4322  5.9700  5.1170  
Iterative map  11.6412  4.9750  4.6038  
Logistic map  12.0392  3.9801  4.3472  
Piecewise map 

6.9649  1.6074  
Sine map  13.1337  4.9750  5.5895  
Singer map  14.0292  7.9600  3.9410  
Sinusoidal map  13.0342  6.9649  6.4242  
Tent map  13.5317  6.9650  5.2686  



Chebyshev map  2.64 
3.69 
2.98 
Circle map  5.17 
1.08 
8.15  
Gauss/mouse map  1.84 
4.49 
1.45  
Iterative map  1.92 
1.64 
1.69  
Logistic map  3.22 
2.76 
3.84  
Piecewise map  1.49 
6.80 
1.90  
Sine map  1.75 
2.25 
2.58  
Singer map  2.17 
2.46 
2.55  
Sinusoidal map 

2.12 
1.72  
Tent map  2.02 
7.22 
2.21 
It can be seen from the simulation results on six function optimization problems that the convergence speed and optimization ability of the piecewise chaotic mapping are the best, the optimal value can be found, and the fluctuation is relatively small. Although the optimal value is not obtained in function
Expression of piecewise map.
Chaotic mapping  Expression  Range 

Piecewise map 

(0, 1) 
It can be seen from the simulation results on six function optimization problems that the convergence speed and optimization ability of the piecewise chaotic mapping are the best, the optimal value can be found, and the fluctuation is relatively small. Although the optimal value is not obtained in functions
The study found that Levy’s flight behavior in nature is based on the ideal way for food seekers to find food in an unfocused and unpredictable environment, which includes shortrange exploratory bounce and occasional long walks. Viswanathan et al. [
From a mathematical point of view, Levy flight behavior reflects a class of nonGaussian stochastic processes, whose steady increments obey the stable distribution of Levy and whose flight path simulation is shown in Figure
Levy flight tracks.
Combined with the bat’s echolocation feature, it helps to significantly and effectively improve the performance of the bat algorithm. Therefore, the improved algorithm replaces equation (
The Levy flight is used to replace the local searching for the optimal position of an individual bat, which generates a larger matching and optimization iterations in the global search process so as to make the situation of falling into local optimum improve and also make the convergence accuracy of the algorithm improve.
For the bat algorithm, since the individual is flying toward the optimal solution during optimization, there is usually a problem of early convergence of the bats, so it is difficult to obtain better optimization results. In order to avoid the premature convergence problem of bat algorithm and make the individual converge to the global optimal solution quickly, a contraction factor is proposed to realize the shrink factor bat algorithm (SBA), which not only maintains the diversity of the population but also improves the convergence efficiency. The proposed SBA can be realized by the following equations:
The algorithm based on the parallel model has the following two characteristics. The first is to break down a group into multiple groups by realizing a divideandconquer approach. The second is to control and manage the information exchange among the subgroups. From the perspective of parallel algorithms, this structural difference produces three parallel population models: masterslave parallel model, island model, and adjacency model [
In this paper, an island multipopulation parallel bat algorithm (IBA) is proposed by adopting the parallel optimization scheme and introducing a centralized information migration strategy. The entire population is divided into many subgroups. Each subgroup performs a global search only on the island, and the suitability of each individual in the island is calculated and evaluated to produce the best individual in the island. The entire evolution of the island is realized by using a separate subprocess to reduce the degree of coupling. Each subprocess uses a centralized migration strategy to periodically send the best individual in the island to the main process to form the main process, and the main process selects the global best individual to from the entire population so as to broadcast to the subprocess, which will force the subpopulation to perform the global most excellent evolution. The flow chart of the algorithm is shown in Figure
Bat algorithm flow chart based on island model.
The multipopulation idea based on the island model is integrated into the other three improved bat algorithms (SBA, LBA, and CBA) to form the island multipopulation chaotic bat algorithm (CBAS), the island multipopulation Levy flight bat algorithm (LBAS), and the island multipopulation shrink factor bat algorithm (SBAS). The seven algorithms (IBA, SBA, LBA, CBA, CBAS, LBAS, and SBAS) were adopted to carry out the simulation experiments on twelve typical test functions (
Parameter settings of the algorithm.
Name of parameter  Parameter values 

Population size 

Maximum number of iterations 

Loudness 

Rate 

Maximum frequency 

Minimum frequency 

The performance comparison results are listed in Table
Performance comparison results under seven algorithms.
Function  Optimization method  Optimal solution  Average  Standard deviation 


IBA  3.4364 
3.3842 
6.7955 
SBA  3.0580 
3.0762 
3.0405  
LBA  4.4310 
1.0789 
5.6368  
CBA  2.9439 
1.3670 
3.1453  
SBAS  4.2826 
9.0180 
2.7054  
LBAS  9.1300 

2.9546  
CBAS  3.3146 
2.0426 
1.4284  



IBA  6.6230 
5.9714 
1.7914 
SBA  7.3645 
8.5002 
1.2582  
LBA  2.1114 
9.9413 
2.4116  
CBA  2.2553 
1.9696 
1.3041  
SBAS  3.3806 

1.8490  
LBAS  6.3657 
1.0919 
2.4972  
CBAS  1.8559 
6.6582 
1.9975  



IBA  2.3169  2.9481  0.4250 
SBA  0.0009 

0.8860  
LBA  1.1552  2.1514  0.6486  
CBA  1.1551  2.3774  0.5599  
SBAS  2.0133  2.3912  0.3715  
LBAS  2.3169  2.8544  0.4626  
CBAS  1.6462  2.7592  0.7613  



IBA  5.9699  15.5719  13.7849 
SBA  3.9899  9.9497  5.6459  
LBA  7.9598  13.4321  3.3074  
CBA  3.9798  12.8350  6.8421  
SBAS 

7.2633  2.3566  
LBAS  5.9699  8.5568  1.7909  
CBAS  4.9748  19.5561  16.6536  



IBA  3.8782 
4.4456 
9.5610 
SBA  5.8307 
1.1123 
2.9158  
LBA  8.6362 
1.1877 
2.3393  
CBA  1.0664 

6.0237  
SBAS  1.3461 
5.0505 
2.2872  
LBAS  2.8347 
1.7624 
5.2873  
CBAS  3.2492 
2.5676 
1.5720  



IBA  −0.0916  −0.8837  0.3961 
SBA  −1.0833  −1.0833  0.0000  
LBA  −1.0833  −1.0833  0.0000  
CBA  −1.0833  −1.0833  0.0000  
SBAS  −0.6009  −1.0315  0.2153  
LBAS  1.0416 

0.8996  
CBAS  0.2604  −0.8475  0.5539  



IBA  3.8153 
1.3534 
1.2480 
SBA  1.8240 

5.6482  
LBA  1.3140 
2.0437 
2.1852  
CBA  2.1564 
1.9632 
2.5663  
SBAS  1.2837 
2.4524 
2.0517  
LBAS  1.4992 
1.3792 
1.4911  
CBAS  1.3498 
1.3242 
1.9408  



IBA  0.0003  0.0090  0.0159 
SBA  0.0003  0.0016  0.0026  
LBA  0.0003  0.0033  0.0073  
CBA  0.0003  0.0006  0.0003  
SBAS  0.0003 

5.6425  
LBAS  0.0005  0.0028  0.0029  
CBAS  0.0003  0.0007  0.0005  



IBA  −3.3220  −3.0019  0.9602 
SBA  −3.3220  −2.8942  0.6939  
LBA  −3.3220  −3.0492  0.7143  
CBA  −3.2031  −3.2507  0.0582  
SBAS  −3.2031 

0.0545  
LBAS  −3.2031 

0.0545  
CBAS  −3.2031  −3.2982  0.0476  



IBA  −5.1285  −4.2011  1.8551 
SBA  −5.1285 

3.6134  
LBA  −5.1285  −4.8311  0.8920  
CBA  −5.1285  −5.1285  4.5201  
SBAS  −5.1285 

1.5732  
LBAS  −5.1285  −2.8510  1.9806  
CBAS  −5.1285 

5.7559 
Convergence curves for typical function optimization problems under seven algorithms. (a)
It can be seen from the simulation results that the overall searching ability of IBA, SBAS, LBAS, and CBAS is better than that of the original algorithms (BA, SBA, LBA, and CBA), which have less volatility and relatively stable performance. From the perspective of a single function, most algorithms corresponding to each function are different, which proves that the seven different algorithms have different optimization capabilities for the same problem. SBAS obtains the optimal value for test functions
It can be seen from the simulation results that the overall searching ability of IBA, SBAS, LBAS, and CBAS is better than that of the original algorithms (BA, SBA, LBA, and CBA), which have less volatility and relatively stable performance. From the perspective of a single function, most algorithms corresponding to each function are different, which proves that the seven different algorithms have different optimization capabilities for the same problem. SBAS obtains the optimal value for test functions
Based on the concept of chaotic mapping, the optimal chaotic mapping is selected to produce the chaotic bat algorithm. The LBA algorithm is proposed by adopting the Levy flight searching strategy. The shrink factor is proposed to realize SBA algorithm, and the multipopulation parallel bat algorithm (IBA) based on island is proposed. The performance is compared by using seven test functions under seven optimization algorithms (IBA, SBA, LBA, CBA, SBAS, LBAS, and CBAS). The comparison results show that the multipopulation algorithms (IBA, SBAS, LBAS, and CBAS) are better than the original algorithms and have relatively stable performance in both the optimization ability and the convergence speed. The experiment proves that the improved algorithm has better optimization precision and convergence speed, which can make up for the deficiency of the original algorithm.
There are no data available for this paper.
The authors declare no conflicts of interest.
ShaSha Guo participated in the draft writing and critical revision of this paper; JieSheng Wang participated in the concept, design, and interpretation of results and commented on the manuscript; and XiaoXu Ma participated in the data collection, analysis, and algorithm simulation.
This work was supported by the Basic Scientific Research Project of Institution of Higher Learning of Liaoning Province (grant no. 2017FWDF10) and the project by Liaoning Provincial Natural Science Foundation of China (grant no. 20180550700).