Magnetotactic bacteria optimization algorithm (MBOA) is a new optimization algorithm inspired by the characteristics of magnetotactic bacteria, which is a kind of polyphyletic group of prokaryotes with the characteristics of magnetotaxis that make them orient and swim along geomagnetic field lines. The original Magnetotactic Bacteria Optimization Algorithm (MBOA) and several new variants of MBOA mimics the interaction energy between magnetosomes chains to obtain moments for solving problems. In this paper, Magnetotactic Bacteria Optimization Algorithm with the Best Individual-guided Differential Interaction Energy (MBOA-BIDE) is proposed. We improved interaction energy calculation by using the best individual-guided differential interaction energy formation. We focus on analyzing the performance of different parameters settings. The experiment results show that the proposed algorithm is sensitive to parameters settings on some functions.
The research of algorithms have been conducted many years, the field of algorithm is very mature now. Evolutionary algorithm (EA) is a very popular research field. The common evolutionary algorithms are genetic algorithm (GA), Differential Evolution (DE) [1], Particle Swarm Optimization (PSO) [2] and Bacterial Foraging Optimization algorithm (BFOA) [3] and so on.
Magnetotactic Bacteria Optimization Algorithm (MBOA) [4] [5] which is introduced by Mo is one of the modern heuristic algorithms and inspired by the magnetotactic bacteria. In nature, magnetotactic bacteria (MTBs) is a special kind of bacteria which have many micro magnetic particles named magnetosome in their bodies. These magnetic particles can generate moments to guide the bacteria to swim along geomagnetic field lines of the earth [6] [7]. In recent years, several improved MBOA, such as BMMBOA [8], MBOA-BR [9], MBOA-BT [10], PSMBA [11], MBMMA [12], have been proposed to modify the performance of MBOA. In the Moments of the Best Individual-based Magnetotactic Bacteria Optimization Algorithm (BMMBOA), similar to DE/best/1, the problem solutions are generated by moments mechanisms based on interaction energy among solutions [8]. In the Magnetotactic Bacteria Optimization Algorithm Based On Best-R and Scheme (MBOA-BR), similar to DE/best/r and scheme, it regulates the moments based on the information exchange between best individual's moments and some randomly one [9]. In the Magnetotactic Bacteria Optimization Algorithm based on Best-Target (MBOA-BT), similar to DE/best/target scheme, some cells will receive MTS information from the interaction between the local best one and the target one to balance the local search and global search [10]. In the Power Spectrum-Based Magnetotactic Bacteria Algorithm (PSMBA), it is based on the models of power spectra of the magnetic field noise produced by Brownian rotation of nonmotile bacteria in zero magnetic field [11]. In the Magnetotactic Bacteria Moment Migration Algorithm (MBMMA), the moments of relative good solutions can migrate each other to enhance the diversity of the MBMMA [12].
In this paper, we proposed a Magnetotactic Bacteria Optimization Algorithm with the Best Individual-guided Differential Interaction Energy (MBOA-BIDE) in order to overcome the shortcomings of complicated interaction energy calculation of the original MBOA and several new variants of MBOA and focus on the study of the effect of different parameters settings.
2. Magnetotactic Bacteria Optimization Algorithm with the Best Individual-Guided Differential Interaction Energy (MBOA-BIDE)
In the following, we briefly describe the basic operators and the main steps of MBOA-BIDE. MBOA-BIDE mainly has three steps and three main operators including moment generation, moment regulation, moment replacement.
2.1. Interaction Distance
First, in the algorithm, each solution is looked as a cell containing a magnetosome chain. At first we define stands for the best cell of the population in the current generation. The distance of the cell and the
best cell, , is calculated as follows: .
After that, we can get a distance matrix. is the size of cell population, stands for dimension of every cell.
2.2. Moments Generation
Based on the distances among cells, the interaction energy is defined as
where the settings of and will be discussed in the next section. stands for randomly selected variables from.,.
After obtaining interaction energy, the moments are generated as follows:
where the settings of will be discussed in the next section.
Suppose, we can obtain a moment vector matrix.
Then the total moments of a cell is regulated as follows:
where stands for the moment of a randomly selected MTS from.,.
2.3. Moments Regulation
After moments generation, the moments regulation is realized as follows:
If > 0.5, the moments in the cell are regulated as follows:
Otherwise, they are regulated as follows:
where stands for the th dimension of current best cell in the current generation.
2.4. Moments Replacement
After the moments regulation, we set a replacement probability 0.5, some cells with worse fitness are replaced as follows:
if > 0.5,
where is a random number between 1 and. stands for the moment of a randomly selected MTS from
3. Parameters Settings
To evaluate the performance of MBOA-BIDE, the experiments are carried out on 10 benchmark functions. In this section, the benchmark functions are presented firstly. Secondly, the simulation results obtained from different parameter settings are analyzed and discussed.
In all experiments, during each run, a maximum fitness evaluation of 200000 generations is used. To reduce statistical errors, each test is repeated 30 times independently and the mean results are used in the comparisons.
3.1. Benchmark functions
The ten basic benchmark problems summarised in Table 1, can be classified into two groups. The first five functions - are unimodal functions. The unimodal functions here are used to test if MBOA-BIDE can maintain the fast-converging feature compared with the other methods. The next five functions - are multimodal functions with many local optima. These functions can be used to test the global search ability of the algorithm in avoiding premature convergence.
3.2. The effect of population size N
We set MBOA-BIDE with different population size (= 10, 40, 50, 100, 150 and 200). The results of different population size are presented in Table 2. From Table 2, we can see that population size with 40 is providing the best results in eight of the ten selected functions.
In this study, MBOA-BIDE with different population size are ranked based on their mean performances. They are ranked according to their performances using a standard competition ranking scheme. In competition ranking, algorithms receive the same rank if their performances are same. The next performing algorithm is
Benchmark functions
Function
Range
D
Formulation
: Sphere
[−100, 100]
30
: Schwefel 2.22
[−10, 10]
30
: Schwefel 1.2
[−100, 100]
30
: Quartic Noise
[−1.28, 1.28]
30
: Rosenbrock
[−30, 30]
30
: Rastrigin
[−5.12, 5,12]
30
Generalized Schwefel
[−500, 500]
30
: Foxholes
[−65.536, 65.536]
2
: Sixhump
[−5, 5]
2
: Goldstein price
[−2, 2]
2
assigned a rank with a gap (gap is determined based on the number of equally performing algorithms). Table 3 provides the ranks of the different population size and the average rank for all the functions based on mean performances. Based on the average ranking, the order of performance obtained is = 40 followed by = 50, = 10, = 100, = 150 and = 200 respectively.
Figure 1 presents the histograms that indicate the number of times each population size have achieved the ranks in the range of 1 to 6. It can be seen that = 40 achieves the top rank as compared to the other different population size.
3.3. The effect of magnetic field <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/56599x78.png" xlink:type="simple"/></inline-formula>
To study the effects of in MBOA-BIDE, we use, and for testing the performance of MBOA- BIDE. Firstly we suppose is constant (= 1, 3, 5, 7, 10), and study the effect of on test functions. Secondly, we also study the effect of varying with generation increases as follows:
・ is linearly increases from 1 to10 (= 1 - 10 LINER).
・ is exponentially increases from 1 to 10 (= 1 - 10 EXP )
・ is linearly increases from 1 to 100 (= 1 - 100 LINER)
・ is exponentially increases from 1 to 100 (= 1 - 100 EXP).
The results are shown in Table 4. From Table 4, for, we can see that when is constant, = 10, the method can achieve better performance, when = 1 - 100 LINER can achieve better performance on. For, when = 3 and = 1 - 100 EXP, the method can achieve better performance. For, we can see that when = 1 and = 1 - 10 LINER can achieve better performance.
Figure 2 Presents the line chart and histograms that indicate the mean, best and median values each have achieved for, respectively. From Figure 2, we can see when = 1 - 100 EXP, MBOA-BIDE achieve the
Histogram of individual mean ranks
Statistical results obtained by MBOA-BIDE with different population size N
Func.
N = 10
N = 40
N = 50
N = 100
N = 150
N = 200
Mean
0
0
0
0
2.5201e−254
4.5896e−181
Dev
0
0
0
0
0
0
Rank
1
1
1
1
5
6
Mean
0
0
0
4.3137e−209
3.3968e−129
5.5820e−91
Dev
0
0
0
0
6.3405e−129
1.2236e−90
Rank
1
1
1
4
5
6
Mean
0
0
0
3.0876e−312
5.5975e−182
1.0773e−124
Dev
0
0
0
0
0
3.8100e−124
Rank
1
1
1
4
5
6
Mean
1.4263e−05
2.2258e−05
1.1082e−05
2.7951e−05
2.2837e−05
2.7755e−05
Dev
1.0928e−05
1.7502e−05
1.1717e−05
2.8866e−05
1.5794e−05
3. 4894e−05
Rank
2
3
1
6
4
5
Mean
21.4943
21.3003
22.1453
23.6719
24.2513
24.6209
Dev
0.7394
0.4264
0.3933
0.2229
0.2143
0.1610
Rank
2
1
3
4
5
6
Mean
0
0
0
0
0
0
Dev
0
0
0
0
0
0
Rank
1
1
1
1
1
1
Mean
−1.1199e+04
−1.1808e+04
−1.1535e+04
−1.0843e+04
−1.0664e+04
−1.0807e+04
Dev
1.0496e+03
1.0526e+03
1.5453e+03
1.9108e+03
1.7327e+03
1.5578e+03
Rank
3
1
2
4
6
5
Mean
7.8914
2.5827
2.4521
1.1949
0.9980
0.9980
Dev
4.0483
3.3070
3.6346
1.0783
4.1884e−12
5.8272e−12
Rank
6
5
4
3
1
1
Mean
−1.03162842
−1.03162845
−1.03162845
−1.03162845
−1.03162845
−1.03162845
Dev
3.2639e−08
6.5144e−10
9.0922e−10
3.3173e−10
4.8852e−10
7.3936e−10
Rank
6
1
1
1
1
1
Mean
3.0000
3.00000
3.0000
3.00000
3.00000
3.00000
Dev
3.7073e−07
5.7118e−08
3.7689e−08
5.3505e−08
2.3598e−08
3.1246e−08
Rank
1
1
1
1
1
1
Histogram of statistical results of MBOA-BIDE with different B value
Rank table for the mean values
Fun
N = 10
N = 40
N = 50
N = 100
N = 150
N = 200
1
1
1
1
5
6
1
1
1
4
5
6
1
1
1
4
5
6
2
3
1
6
4
5
2
1
3
4
5
6
1
1
1
1
1
1
3
1
2
4
6
5
6
5
4
3
1
1
6
1
1
1
1
1
1
1
1
1
1
1
Avg. rank
2.4
1.6
1.6
2.9
3.4
3.8
Statistical results obtained by MBOA-BIDE with different B values
Fun
Mean
Median
Dev
Best
Worst
B = 1
26.9030
26.9095
0.1675
26.5374
27.2250
B = 3
21.6974
21.8007
0.4795
20.6532
22.7105
B = 5
21.2422
21.2866
0.4115
19.8061
21.8931
B = 7
21.1473
21.2003
0.4291
19.7336
22.0915
B = 10
20.9895
21.0589
0.3426
19.8434
21.5301
B = 1 - 10 LINER
21.2495
21.3203
0.4827
20.4088
22.3778
B = 1 - 10 EXP
20.3502
20.3270
0.9573
18.1462
21.9417
B = 1 - 100 LINER
19.3093
19.3314
0.1863
18.7989
19.6055
B = 1 - 100 EXP
21.0213
21.0466
0.3493
20.1838
21.4598
B = 1
−1.0623e+04
−1.0600e+04
247.5844
−1.1137e+04
−1.0214e+04
B = 3
−1.0880e+04
−1.1211e+04
1.0098e+03
−1.1755e+04
−8.3927e+03
B = 5
−8.4505e+03
−8.3105e+03
1.2271e+03
−1.1428e+04
−6.8399e+03
B = 7
−7.2628e+03
−7.1560e+03
548.8644
−8.4595e+03
−6.3355e+03
B = 10
−7.4822e+03
−7.5949e+03
647.8553
−8.8086e+03
−6.0637e+03
B = 1 - 10 LINER
−1.1722e+04
−1.2147e+04
1.3761e+03
−1.2320e+04
−6.9487e+03
B = 1 - 10 EXP
−1.1799e+04
−1.1768e+04
134.5198
−1.2219e+04
−1.1609e+04
B = 1 - 100 LINER
−9.1931e+03
−9.1373e+03
1.3684e+03
−1.2563e+04
−6.5952e+03
B = 1 ? 100 EXP
−1.2099e+04
−1.2530e+04
1.3411e+03
−1.2553e+04
−7.2080e+03
B = 1
3.3981
0.9980
4.1130
0.9980
12.6705
B = 3
5.6198
4.5408
3.8717
0.9980
12.6705
B = 5
7.1337
8.2029
4.7696
0.9980
12.6705
B = 7
5.8563
3.9683
4.4765
0.9980
12.6705
B = 10
6.3849
6.1614
3.6604
0.9980
12.6705
B = 1 - 10 LINER
1.8220
0.9980
2.4891
0.9980
12.6705
B = 1 - 10 EXP
2.8240
0.9980
3.8456
0.9980
12.6705
B = 1 - 100 LINER
2.7863
0.9980
3.8309
0.9980
12.6705
B = 1 - 100 EXP
2.2053
0.9980
3.0987
0.9980
12.6705
performance on.
3.4. The Effect of c<sub>1</sub> and c<sub>2</sub>
Firstly we suppose that and is constant, and study the effect of and on three test functions. We set + = 1, and with different values (0.1, 0.3, 0.5, 0.7, 0.9). Secondly, we also study the effect of
and varying with generation increases. The parameter settings are as follows:
・ = 0 - 1 L: is linearly increases from 0 to1, = 1 -;
From Table 5 and Table 6, we can see that for, = 0.1 - 1 E obtain the best performance. For and, = 0 - 2 L obtain the best performance. and based on the average ranking, the order of performance obtained is = 0 - 1 L followed by = 0 - 2 L, = 0 - 1 L, = 0 - 2 L, = 0.3, = 0.1, = 0.1 - 1 E, = 0.9, = 0.7, = 0.1-1E and = 0.5, respectively.
4. Conclusion
In this paper, by analyzing the performance of different parameters settings of MBOA-BIDE, we can see that when = 40 - 50, is exponentially increases from 1 to 100, is linearly increases from 0 to 1. = 1 -, MBOA-BIDE obtain the better performance. The experiment results show that the proposed algorithm is
Statistical results obtained by MBOA-BIDE with different C<sub>1</sub> and C<sub>2</sub> values
Fun
Mean
Median
Dev
Best
Worst
Rank
c1 = 0.1
21. 3901
21.4491
0.2710
20.6426
21.8700
9
c1 = 0.3
21.0181
21.0926
0.3837
20.1813
21.6728
6
c1 = 0.5
21.0303
21.0385
0.3369
20.2972
21.7053
7
c1 = 0.7
20.9564
20.9955
0.3636
20.0590
21.5185
5
c1 = 0.9
21.3102
21.3443
0.2975
20.4737
21.8988
8
c1 = 0-1 L
21.0213
21.0466
0.3493
20.1838
21.4598
3
c2 = 0-1 L
21.0500
21.0776
0.4023
20.2052
21.7740
4
c1 = 0.1-1E
20.8318
20.8450
0.3135
20.2436
21.6018
2
c2 = 0.1-1E
20.7453
20.7975
0.3485
19.8905
21.3564
1
c1 = 0-2L
21.8727
22.0773
0.6619
19.7414
22.8133
11
c2 = 0-2L
21.7704
21.9754
0.5780
20.3235
22.8893
10
c1 = 0.1
−1.2352e+04
−1.2532e+04
985.4425
−1.2549e+04
−7.1345e+03
4
c1 = 0.3
−1.2406e+04
−1.2533e+04
690.3886
−1.2547e+04
−8.7509e+03
3
c1 = 0.5
−1.2295e+04
−1.2536e+04
912.3674
−1.2548e+04
−8.8768e+03
6
c1 = 0.7
−1.2170e+04
−1.2535e+04
1.1179e+03
−1.2556e+04
−8.6381e+03
7
c1 = 0.9
−1.2105e+04
−1.2529e+04
1.3193e+03
−1.2548e+04
−7.3420e+03
8
c1 = 0 - 1 L
−1.2099e+04
−1.2530e+04
1.3411e+03
−1.2553e+04
−7.2080e+03
9
c2 = 0 - 1 L
−1.2417e+04
−1.2535e+04
646.0001
−1.2552e+04
−8.9972e+03
2
c1 = 0.1 - 1 E
−1.2091e+04
−1.2534e+04
1.3939e+03
−1.2548e+04
−6.6607e+03
10
c2 = 0.1 - 1 E
−1.1775e+04
−1.2533e+04
1.7649e+03
−1.2547e+04
−6.7527e+03
11
c1 = 0 - 2 L
−1.2430e+04
−1.2436e+04
34.6550
−1.2488e+04
−1.2349e+04
1
c2 = 0 - 2 L
−1.2297e+04
−1.2434e+04
763.0817
−1.2498e+04
−8.2595e+03
5
c1 = 0.1
2.2362
0.9980
3.3244
0.9980
12.6705
4
c1 = 0.3
2.7849
0.9980
3.7353
0.9980
12.6705
8
c1 = 0.5
4.2983
0.9980
4.6656
0.9980
12.6705
11
c1 = 0.7
2.9363
0.9980
3.7453
0.9980
12.6705
9
c1 = 0.9
2.4267
0.9980
3.7530
0.9980
12.6705
6
c1 = 0 - 1 L
2.2053
0.9980
3.0987
0.9980
12.6705
3
c2 = 0 - 1 L
2.3454
0.9980
3.5817
0.9980
12.6705
5
c1 = 0.1 - 1 E
2.7436
0.9980
3.6848
0.9980
12.6705
7
c2 = 0.1 - 1 E
3.1675
0.9980
4.0601
0.9980
12.6705
10
c1 = 0 - 2 L
1.0102
0.9980
0.0563
0.9980
1.2898
1
c2 = 0 - 2 L
1.5118
0.9980
2.0119
0.9980
12.6705
2
Rank table for the mean values
c1, c2
Individual ranking of benchmark functions
Avg. RANK (R)
c1 = 0.1
9
4
4
5.67
c1 = 0.3
6
3
8
5.67
c1 = 0.5
7
6
11
8
c1 = 0.7
5
7
9
7
c1 = 0.9
8
8
6
6.67
c1 = 0 - 1 L
3
9
3
5
c2 = 0 - 1 L
4
2
5
3.67
c1 = 0.1 - 1 E
2
10
7
6.33
c2 = 0.1 - 1 E
1
11
10
7.33
c1 = 0 - 2 L
11
1
1
4.33
c2 = 0 - 2 L
10
5
2
5.67
sensitive to parameters settings on some functions.
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China under Grant No.61075113, the Excellent Youth Foundation of Heilongjiang Province of China under Grant No. JC201212, the Fundamental Research Funds for the Central Universities No. HEUCFX041306.
Cite this paper
Hongwei Mo,Lili Liu,Jiao Zhao, (2015) Performance Research on Magnetotactic Bacteria Optimization Algorithm with the Best Individual-Guided Differential Interaction Energy. Journal of Computer and Communications,03,127-136. doi: 10.4236/jcc.2015.35016
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