Computational Intelligence and Neuroscience
Volume 2018 (2018), Article ID 9827372, 13 pages
https://doi.org/10.1155/2018/9827372

Research Article

Modification of Fish Swarm Algorithm Based on Lévy Flight and Firefly Behavior

Zhenrui Peng, Kangli Dong, Hong Yin, and Yu Bai

School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou, China

Correspondence should be addressed to Zhenrui Peng; pzrui@163.com

Received 11 June 2018; Revised 18 July 2018; Accepted 9 August 2018; Published 13 September 2018

Academic Editor: Raşit Köker

Copyright © 2018 Zhenrui Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Artificial fish swarm algorithm easily converges to local optimum, especially in solving the global optimization problem of multidimensional and multiextreme value functions. To overcome this drawback, a novel fish swarm algorithm (LFFSA) based on Lévy flight and firefly behavior is proposed. LFFSA incorporates the moving strategy of firefly algorithm into two behavior patterns of fish swarm, i.e., chasing behavior and preying behavior. Furthermore, Lévy flight is introduced into the searching strategy. To limit the search band, nonlinear view and step size based on dynamic parameter are considered. Finally, the proposed algorithm LFFSA is validated with several benchmark problems. Numerical results demonstrate that LFFSA has a better performance in convergence speed and optimization accuracy than the other test algorithms.

1. Introduction

Optimization plays an important role in modern industry as well as in scientific world. Due to the computational costs of the existing numerical methods, researchers have to rely on metaheuristic algorithms to solve complex optimization problems.

Artificial fish swarm algorithm (AFSA) was proposed as a metaheuristic algorithm based on fish swarm behaviors—preying behavior, swarming behavior, and chasing behavior [1]. With its simple principle and such good features as robustness and tolerance of parameter setting, AFSA has become an increasingly important tool in swarm intelligence optimization. Some algorithms derived from AFSA have been presented: Wang et al. eliminated the step restriction and added new leaping behavior to improve the stability of the algorithm [2]. Another modified form of AFSA used particle swarm optimization to reformulate AFSA, integrated AFSA with communication behavior, and created formulas for major AFSA parameters [3]. Luan et al. adopted normal distribution function, Cauchy distribution function, multiparent crossover operator, mutation operator, and modified minimal generation gap model to overcome the drawback of slow convergence speed in later iterations [4]. Another improved AFSA with crossover and culture algorithm was proposed to enhance its optimization efficiency [5]. Hu et al. integrated the merits of the self-adaptation strategy, mutation strategy, and hybrid strategy into the social behaviors of AFSA [6].

Over the last decades, AFSA and its improved algorithms have been successfully applied to various engineering optimization problems. Kumar et al. adopted AFSA to optimize renewable energy sources in a microgrid [7]. By categorizing the social behaviors of fish swarm into foraging, reproductive, and random behaviors, a novel artificial fish swarm algorithm was advocated for solving large-scale reliability-redundancy application problem [8]. A derivative of AFSA was used to solve the multiobjective disassembly line balancing problem with fuzzy disassembly times [9]. A binary fish swarm algorithm was presented to solve profit-based unit commitment problem in generation companies [10].

Firefly algorithm (FA) is another metaheuristic algorithm based on the idealized behavior of the flashing characteristics of fireflies [11]. FA can adaptively adjust the radius of the induction and parallel search the optimum in multiple peaks. FA has natural advantages in solving multimodal optimization problem. Some drawbacks of searching strategy and parameters in FA were targeted and improved by researchers. A derivative of FA with directed movement of fireflies was proposed by Farahani et al. [12]. Chaotic sequence was introduced into the basic FA to adjust parameters γ and α by Coelho et al. [13]. Eagle Strategy, which combines Lévy flight search with FA, was introduced by Yang and Deb [14]. Elitist FA was presented [15], which tried to enhance the best solution position by generating m uniform random vectors and moving in the direction of best solution.

Firefly algorithm and its modified forms were also successfully applied to numerous practical problems. Jagatheesan et al. used firefly algorithm to design a controller for an automatic generation control of multiarea power thermal systems [16]. An FA-inspired band selection and optimized extreme learning machine were proposed for hyperspectral image classification [17]. A self-adaptive firefly algorithm was developed for placement of FACTS devices [18]. Teshome et al. modified FA to counteract some inherent problems that may hinder the performance of the maximum power point tracking [19]. Alb et al. used FA to solve a shielding/shunting electromagnetic problem [20]. Mishra et al. proposed a method for optimal placement of interline power flow controller by using FA [21]. Other applications of improved FA include image compression [22], financial forecasting [23], image segmentation [24], structural optimization [25], classification problem [26], unconstrained optimization [27], economic dispatch problems [28], clustering [29], image retrieval [30], and mechanical optimal design [31]. Some researchers presented comprehensive reviews of existing FA and its modified forms to encourage new researchers to employ FA for solving their own problems [32, 33].

At the same time, numerous studies have shown that Lévy flight is similar to flight characteristics of given animals and insects [34–36], which has been widely used in swarm intelligence algorithms. Subsequently, Lévy flight has been applied to optimization, and preliminary results showed its potential capabilities. Jensi and Jiji proposed an enhanced particle swarm optimization with Lévy flight [37]. Tang et al. proposed a new framework of shuffled frog-leaping algorithm based on the exploration and exploitation mechanism by using Lévy flight [38]. Yahya and Saka proposed a multiobjective artificial bee colony algorithm with Lévy flight and applied it to construction site layout planning [39].

Hybridization is recognized to be an important aspect of high performing algorithms in recent years [40]. Owing to some drawbacks of traditional AFSA and FA, they are not suitable for solving highly nonlinear and multimodal problems. By integrating the merits of AFSA with Lévy flight and FA, this paper proposes a novel hybrid algorithm, named LFFSA (fish swarm algorithm based on Lévy flight and firefly behavior), for global optimization. The highlights of the new algorithm are as follows:

(i)Attraction degree is involved in the definition of artificial fish(ii)Lévy flight is used to adjust the search route of artificial preying fishes(iii)By analyzing the relationship between swarming behavior and chasing behavior, unnecessary behavior (swarming behavior) is excluded instead of improving AFSA through adding new behaviors [3–6](iv)Time complexity of the improved algorithm is also further analyzed to demonstrate the effectiveness of the improvement

The remainder of this paper is organized as follows: Section 2 describes the basic AFSA, FA, and Lévy flight respectively; Section 3 proposes and explains LFFSA algorithm in details; in Section 4, the superiority of proposed algorithm LFFSA is validated by several benchmark problems; Section 5 outlines the conclusion.

2. Background

2.1. Artificial Fish Swarm Algorithm

AFSA is a swarm intelligence algorithm, which can be employed to solve the optimization problem by imitating swarming, chasing, and preying behaviors of artificial fishes [1]. As shown in Figure 1, let be the current position of one artificial fish, be the viewpoint of artificial fish at one moment, be the visual scope of each individual, and be fishes within the of , be the biggest step of artificial fish, and δ be the congestion factor of fish swarm. The food concentration is proportional to the fitness function . The behavior patterns of fish swarms can be described as follows.

Figure 1: Vision concept of the artificial fish.

Swarming behavior: if , where is the central point inside the of the point , swarming behavior is to be executed. Take as . The fish at will take a step toward the point .

Chasing behavior: if the point (denoted by ) having the best objective function value inside the satisfies , and if the of is not crowded, chasing behavior is to be executed. Take as . The fish at will take a step toward the point .

Preying behavior: preying behavior is tried in the following situations:

(1), , and the is not crowded(2)The is crowded

Here, a point inside the of is randomly selected. If , the preying behavior is to be executed. Take as . The fish at will take a step toward the point . Otherwise, it will move a step randomly within its .

The best solution obtained in each iteration is marked as “board.” After the specified iterations, search process is terminated and the result on the “board” is regarded as the final solution.

For artificial preying fishes, the position-updating can be formulated as

where is the next position of artificial fish; is the current position of artificial fish; is the position which has a better objective function value; rand is a random number in [−1, 1]; and is the distance between two position vectors.

For artificial swarming fishes, the position-updating can be formulated as

For artificial chasing fishes, the position-updating can be formulated as

The flowchart of AFSA is shown in Figure 2.

Figure 2: Flowchart of AFSA.

2.2. Firefly Algorithm

Firefly algorithm (FA) [11] is another swarm intelligence algorithm. It achieves swarming phenomenon by using the fluorescent signal between two firefly individuals.

The attraction between fireflies depends on their light intensities and attraction degree. The light intensity is in proportion to the objective function value of firefly’s position. Attraction degree is in proportion to the light intensity. The brighter the light intensity is, the higher attraction degree will be. Besides, the farther the distance is, the lower the light intensity and attraction degree will be.

In the simplest form, the light intensity varies with the distance r monotonically and exponentially as

where is the original light intensity and γ is the light absorption coefficient. As firefly attraction degree is proportional to the light intensity seen by adjacent fireflies, the attraction degree of a firefly can be defined as

where is the attraction degree at .

The distance between any two fireflies i and j at and , respectively, is the Cartesian distance, which is calculated as

where n is the dimensionality of the given problem.

2.3. Lévy Flight

Lévy flight is one kind of random searching strategy [35]. Flying step satisfies a heavy-tailed Lévy distribution, which can be represented by a clear power-law equation as

where is random Lévy step. For searching problems inside a wide range of unknown space, the variance of Lévy movement increases faster than the dimensional Brownian movement.

To some extent, the foraging behavior of nature animals is a kind of random movement behavior. Because next movement usually depends on the current position and the probability of moving to next position, the effectiveness of each random movement becomes greatly important. Recent studies show that Lévy flight is one of the best searching strategies in random movement model [35, 41–43].

3. Fish Swarm Algorithm Based on Lévy Flight and Firefly Behavior

AFSA has several disadvantages in solving nonlinear and multimodal problems. Firstly, AFSA uses swarming behavior and chasing behavior to execute parallel search in a simple and fast way. However, after determining the direction, each artificial fish moving with random step will be unable to approach the target point effectively. Secondly, artificial fish will execute preying behavior when it does not meet the conditions of swarming behavior and chasing behavior. This kind of searching strategy is inefficient and can easily miss the optimum point. Thirdly, too many moving patterns can increase the algorithm complexity, which may cause slow convergence speed.

The above-listed drawbacks of AFSA are improved in the proposed LFFSA. FA has the unique moving strategy using attraction between fireflies, which can be used to fix the random moving after determining the direction in AFSA. And, the preying behavior can be improved using Lévy flight to specify the behavior of artificial fishes. In LFFSA, attraction degree is involved in the definition of artificial fishes, which allows each individual move according to attraction degree; Lévy flight is also considered in the definition of the artificial preying fish to avoid falling into the local optimum; chasing behavior is excluded to decrease the algorithm complexity.

The flowchart of LFFSA is shown in Figure 3. Pseudo code of LFFSA is described in Algorithm 1. To show the difference between AFSA and LFFSA vividly, mechanisms of both algorithms are provided in Figure 4.

Figure 3: Flowchart of LFFSA.

(1)
Initialization of ;

(2)
Evaluate fitness;

(3)
while the stop criterion is not met do

(4)
    while do

(5)
        if meets the condition of chasing behavior

(6)
            Move with attraction degree and get a new based on Equation (13);

(7)
        else

(8)
            Prey with Lévy flight and get a new based on Equations (11) and (14);

(9)
        end if

(10)
    end while

(11)
    Update the “board”;

(12)
end while

(13)
Obtain the optimum solution

Algorithm 1: LFFSA.

Figure 4: Mechanisms of AFSA and LFFSA.

The main improvements of LFFSA are summarized as follows:

(a)Improvement 1: FA-based moving strategy. Attraction degree is involved in the definition of artificial fishes, which can be formulated as

where is the Cartesian distance between artificial fishes i and j given by Equation (6); γ is the light intensity coefficient, which can be set as a constant; and , the largest attraction degree, is attraction degree of an artificial fish at .

The position-updating with preying behavior established by attraction degree can be formulated as

where is the current position of artificial fish i; is the attraction degree; α is the step factor which is a constant between 0 and 1; is a number chosen randomly in [−1, 1]; and is to avoid falling into the local optimum.

(b)Improvement 2: inertia weight. A linear inertia weight is added into Equation (9) as

where is the weight size inherited from the last position of an artificial fish; denotes the biggest weight; is the minimum weight; represents the current iteration; and is the ultimate iteration.

Similarly, the position-updating with chasing behavior can be formulated as

where is the position with the highest food concentration in the view of artificial fish and other parameters are defined in Equations (10) and (11).

(c)Improvement 3: Lévy flight-based search strategy. Lévy flight is involved in the definition of artificial fishes are executing preying behavior. The movement can be formulated as

where is the current position of artificial fish i; is the point to point multiplication; denotes a random vector generated by Lévy flight; represents the best fish on the “board”; ; and μ and have the standard normal distribution , , respectively, where

where is the standard Gamma function.(d)Improvement 4: nonlinear visual and step. and change nonlinearly and dynamically in LFFSA. The updating equations are as follows:

where , the minimum visual, takes 0.001; , the minimum step, takes 0.0002; ρ is a nonlinear weight; and s represents an integer, . Here, . Other parameters are the same as those in Equation (11). The relationship between s and ρ is as shown in Figure 5.

Figure 5: Value of ρ.

Besides, define the biggest distance between two artificial fishes as

where, and represent the upper bound and the lower bound of searching range respectively; D denotes the D-dimension searching space. The initial is equal to , and initial is equal to . Then and change dynamically according to Equations (16)–(18).

(e)Improvement 5: decrease of time complexity. Swarming behavior in AFSA is excluded to decrease the algorithm complexity.

4. Numerical Simulation

4.1. Comparison of Convergence Accuracy

LFFSA is validated by numerical simulations. Fish swarm algorithm with firefly behavior (FFSA), differential evolution (DE) algorithm, self-adaptive differential evolution (jDE) algorithm, and the two basic algorithms (AFSA and FA) are compared. All the algorithms are coded in Matlab 2014b. The operating system is windows 7. Simulation hardware is a PC with 2.50 GHz Inter Core i5 and 2.00 GB Memory.

Parameters shown in Table 1 are determined by trial and error.

Table 1: Parameter settings.

Algorithms
Parameters
Values

FA, FFSA, LFFSA

1.0

FA, FFSA, LFFSA
γ
1.0

AFSA, FFSA, LFFSA
δ
0.618

AFSA, FFSA, LFFSA
Trynumber
5

DE, jDE
Scaling constant
0.5

DE, jDE
Crossover constant
0.9

All 6 algorithms
Population
50

All 6 algorithms
Maximum function evaluations (FEs)

The following benchmarks are chosen carefully according to their features. Functions Sphere, Quartic, and Rosenbrock, etc., are simple unimodal problems. Functions Ackley, Rastrigin, and Schwefel, etc., are highly complex multimodal problems with many local minima. Schwefel function has a maximum value and other functions have minimum values. These benchmarks are listed in Table 2 [44, 45]. The solutions of 17 test functions obtained by different algorithms are compared. To compare the convergence speed and accuracy of the algorithms clearly and correctly, all functions are run 50 times for each algorithm, respectively. The results are averaged and plotted in Figure 6.

Table 2: Test functions.

No.
Test functions
Expression
Optimum value
Domain
D

f1
Sphere

0

30

f2
Quartic

0

30

f3
Ackley

0

30

f4
Rosenbrock

0

30

f5
Rastrigin

0

30

f6
Rastrigin

0

30

f7
Schwefel

30

f8
Griewank

0

30

f9
Quadric

0

30

f10
Schaffer

0

30

f11
Schaffer

0

30

f12
Maxmod

0

30

f13
Dixon and price

0

30

f14
Powell

0

28

f15
Zakharov

0

30

f16
Sin

0

30

f17
Sin

100

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

Figure 6: Iterative curves of test functions. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10. (k) f11. (l) f12. (m) f13. (n) f14. (o) f15. (p) f16. (q) f7.

From Figure 6, LFFSA can avoid local optimum and have better convergence accuracy compared with the other algorithms. For AFSA and FA, the solutions of most functions are unsatisfactory; the DE cannot find ideal solutions of f3, f4, f5, f6, f10, f11, f12, f14, and f15; the jDE has good accuracy while solving some of those functions, e.g. f1, f2, f3, and f8; however, solutions of f4, f10, f11, and f15 obtained by jDE are not so precise; the LFFSA can obtain the ideal accuracy for almost all functions, although it cannot achieve a high precision level like solutions of f2 obtained by jDE; the FFSA is slightly worse than LFFSA. The LFFSA outperforms jDE in 10 benchmark functions, while 2 functions are comparative, and 5 functions are worse.

LFFSA is better than AFSA because Lévy flight is able to restrict the movement step of AFSA to a very small area around the current position. Furthermore, the attraction degree guides the fish moving. Besides, LFFSA can quickly lead the fish individual to the close-by optimal point. Considering all the advantages discussed above, the optimum solution can be found successfully by using LFFSA, which outperforms the basic algorithms for all test functions, and outperforms jDE for several functions. To observe the searching capabilities of different algorithms directly, the average, median, best, and worst values obtained by different algorithms are listed in Table 3. Results indicate that LFFSA can find ideal solutions and have a better robustness.

Table 3: Comparison of optimization results.

No.
Items
AFSA
Std.
FA
Std.
FFSA
Std.
LFFSA
Std.
DE
Std.
jDE
Std.

f1
Worst
3.014
0.181
−2.798
0.120
−5.132
0.071
−6.743
0.071
−8.544
0.132
−15.021
0.121

Best
2.402
−3.257
−5.233
−8.500
−10.371
−17.242

Average
2.656
−3.048
−5.145
−7.278
−8.924
−16.326

Median
2.651
−3.102
−5.193
−6.325
−8.586
−16.706

f2
Worst
−2.322
0.171
−1.9245
0.135
−11.178
0.074
−14.207
0.134
−20.506
0.400
−31.644
0.535

Best
−4.126
−2.813
−11.313
−15.585
−21.611
−33.250

Average
−2.812
−2.278
−11.273
−14.812
−21.054
−32.687

Median
−2.562
−2.197
−11.271
−14.834
−20.970
−32.844

f3
Worst
1.386
0.001
−1.776
0.124
−3.908
0.197
−6.121
0.138
−0.027
0.438
−8.934
0.167

Best
1.307
−2.415
−4.647
−6.938
−1.281
−9.471

Average
1.587
−2.168
−4.147
−6.546
−0.049
−9.163

Median
1.586
−2.177
−4.225
−6.325
−0.035
−9.164

f4
Worst
2.143
0.087
2.2733
0.097
−2.0348
0.125
−1.076
0.376
1.520
0.003
1.454
0.025

Best
1.565
1.043
−2.416
−3.405
1.412
1.363

Average
1.946
1.476
−2.158
−1.946
1.385
1.312

Median
1.854
1.385
−2.235
−2.325
1.363
1.287

f5
Worst
2.310
0.077
1.864
0.856
−2.846
0.044
−4.385
0.054
−0.579
0.323
0.898
6.009

Best
2.096
1.243
−2.982
−5.145
−1.591
−12.831

Average
2.236
1.454
−2.946
−4.643
−0.999
−3.303

Median
2.136
1.285
−2.435
−4.325
−0.963
0.148

f6
Worst
2.445
0.038
1.716
0.133
−2.382
0.076
−3.414
0.048
1.131
0.042
0.698
5.049

Best
2.318
1.255
−2.618
−3.606
1.012
−11.404

Average
2.408
1.571
−2.486
−3.489
1.077
−5.466

Median
2.419
1.601
−2.462
−3.487
1.094
−7.632

f7
Worst
3.815
0.018
3.7846
0.031
4.186

4.156

4.099

4.087

Best
3.945
3.978
4.099
4.099
4.099
4.099

Average
3.813
3.848
4.099
4.099
4.099
4.097

Median
3.736
3.785
−4.099
−4.099
4.099
4.097

f8
Worst
0.956
0.133
−0.960
0.223
−5.301
0.051
−6.271
0.055
−9.89
0.114
−Inf
0

Best
−0.644
−0.500
−5.444
−6.455
−10.255
−Inf

Average
−0.735
−0.697
−5.357
−6.372
−10.071
−Inf

Median
0.736
0.685
−5.435
−6.325
−10.074
−Inf

f9
Worst
−11.665
1.057
−9.76
1.324
−9.347
0.843
−8.695
1.323
−6.848
1.124
−6.131
2.697

Best
−14.433
−10.574
−12.194
−11.937
−10.680
−14.831

Average
−12.786
−10.456
−10.764
−10.137
−8.178
−9.938

Median
1.136
1.085
−7.435
−6.325
−7.745
−9.663

f10
Worst
0.957
0.131
0.974
0.223
−5.375
0.049
−6.274
0.056
0.673
0.032
0.280
0.110

Best
0.644
0.497
−5.448
−6.486
0.483
−0.022

Average
0.747
0.649
−5.376
−6.348
0.526
0.230

Median
1.136
1.085
−7.435
−6.325
0.547
0.211

f11
Worst
−0.301

−0.303
0.009
−4.751
0.071
−5.647
0.085
−0.896
0.103
−1.106
0.135

Best
−0.301
−0.331
−5.011
−5.965
−1.107
−1.429

Average
−0.301
−0.313
−4.838
−5.804
−0.975
−1.364

Median
−0.301
−0.311
−4.835
−5.804
−0.896
−1.402

f12
Worst
0.779
0.037
−1.177
0.163
−2.756
0.035
−3.158
0.037
−0.136
0.052
−1.904
0.221

Best
0.658
−1.638
−2.892
−3.287
−0.339
−2.526

Average
0.711
−1.388
−2.811
−3.221
−0.252
−2.291

Median
0.716
−1.376
−2.801
−3.216
0.248
−2.374

f13
Worst
4.286
0.173
0.574
0.297
−0.602
0.007
−0.602
0.005
0.039
0.068
−0.176

Best
3.706
−0.175
−0.605
−0.605
−0.162
−0.176

Average
4.147
−0.013
−0.603
−0.603
−0.088
−0.176

Median
4.213
−0.135
−0.603
−6.603
−0.086
−0.176

f14
Worst
2.404
0.084
0.405
0.318
−3.598
0.117
−4.514
0.101
0.276
0.211
−2.192
0.401

Best
2.173
−0.576
−4.001
−4.869
−0.369
−3.409

Average
2.275
0.069
−3.773
−4.656
−0.117
−2.867

Median
2.246
0.133
−3.719
−4.651
−0.134
−2.924

f15
Worst
2.588
0.0705
1.505
0.198
−4.281
0.117
−4.975
0.241
2.241
0.077
0.635
0.372

Best
2.366
−4.679
−5.792
−7.2474
1.972
−0.370

Average
2.490
−4.561
−5.408
−6.3259
2.140
0.148

Median
2.506
−4.601
−5.456
−6.3259
2.136
0.218

f16
Worst
1.438
0.058
−1.463
0.253
−2.762
0.026
−3.239
0.032
−2.354
0.042
−3.392
0.433

Best
1.268
−2.296
−2.838
−3.334
−2.482
−4.646

Average
1.366
−1.987
−2.793
−3.282
−2.408
−3.868

Median
1.375
−2.016
−2.793
−3.284
−2.402
−3.751

f17
Worst
−22.954
0.643
−25.673
1.721
−70.748
1.908
−79.645
0.131
−47.025
1.403
−63.031
1.942

Best
−24.885
−32.075
−77.054
−80.098
−51.440
−68.738

Average
−23.772
−28.641
−74.598
−79.996
−48.982
−66.539

Median
−23.731
−28.759
−74.907
−80.001
−49.035
−66.559

4.2. Computational Complexity Analysis

Time complexity is also an important indicator in the analysis of algorithms. If an algorithm is composed of several parts, then its complexity is the sum of the complexities of these parts. The algorithm may consist of a loop executed many times, and each time is with a different complexity. Time complexity of the algorithm is used to estimate the efficiency of the algorithm. It is defined that the time complexity of the algorithm, or the running time, is [46]. Define N as the population.

In the definition of time complexity, and are at different levels. If the time complexity of one algorithm is , the time complexity of the other one is , then the former algorithm is more complex. In the other case, if the time complexity of one algorithm is while the time complexity of the other one is , their complexities are both .

The time complexity analysis of AFSA is provided in Table 4.

Table 4: Time complexity analysis of AFSA.

Procedure of AFSA
Time complexity

(1) Initialization of N artificial fishes

(2) Initialization of ‘board’

(3) Swarming behavior

(4) Chasing behavior

(5) Preying behavior

(6) Judging of terminal condition

(7) Information output of ‘board’

From Table 4, the time complexity of AFSA is

Swarming behavior has N times of calculating congestion factor, 1 time of judging, and 1 time of moving. Therefore, time complexity of swarming behavior is . Chasing behavior has N times of calculating congestion factor, N times of searching, 1 time of judging, and 1 time of moving. Therefore, time complexity of chasing behavior is .

Time complexity analysis of LFFSA is listed in Table 5.

Table 5: Time complexity analysis of LFFSA.

Procedure of LFFSA
Time complexity

(1) Initialization of N artificial fishes

(2) Initialization of ‘board’

(3) Chasing behavior

(4) Preying behavior

(5) Judging of terminal condition

(6) Information output of ‘board’

Due to the lack of swarming behavior, time complexity of LFFSA can be calculated as

We can also obtain time complexity of FA:

A conclusion can be obtained that time complexities of the three algorithms are at the same level. Their computational complexities in the worst case are only the square of the training sample size.

4.3. Experimental Complexity Analysis

Time complexity is a rough estimate of time cost. The more accurate time cost of an algorithm can only be validated by running it on computer. Since different algorithms cannot reach the same convergence accuracy, the test with fixed convergence accuracy is not available. Therefore, the test with max function evaluations is conducted. Running time of each algorithm is counted by the explorer of MATLAB. Parameter settings of algorithms are the same in Section 4.1. Average running time of different algorithms is listed in Table 6. When function evaluations are the same, running speed of LFFSA is faster than that of AFSA, while DE has the fastest running speed. Results are quite in accord with those obtained by computational complexity analysis. LFFSA and jDE are comparative in experimental complexity. Running time of FFSA is almost twice as much as that of LFFSA. The improvement of LFFSA decreases time complexity to some extent.

Table 6: Average running time of algorithms.

No.
Running time (s)

AFSA
FA
FFSA
LFFSA
DE
jDE

f1
10.23
7.47
18.5
7.53
4.64
6.08

f2
12.37
7.73
21.11
7.10
4.32
6.53

f3
10.87
7.80
20.73
8.96
4.12
8.25

f4
11.40
7.67
20.49
10.27
5.63
6.98

f5
10.53
7.66
19.47
8.03
3.28
5.96

f6
19.72
6.10
25.77
16.37
7.71
10.65

f7
10.53
7.43
18.77
8.56
3.27
6.56

f8
12.33
8.03
22.72
9.23
3.79
6.63

f9
20.83
7.20
21.36
7.71
4.01
6.84

f10
16.27
8.46
20.62
7.56
4.65
7.15

f11
9.26
8.95
13.74
6.35
3.63
6.67

f12
9.65
5.34
13.75
6.51
3.04
6.28

f13
9.12
5.26
14.36
10.41
4.13
7.15

f14
13.95
5.74
19.02
10.31
6.48
9.86

f15
9.45
5.39
13.07
7.36
4.62
7.74

f16
8.73
5.21
13.88
11.46
8.52
18.87

f17
14.53
7.28
18.41
5.96
3.32
6.44

4.4. Parameter Analysis of LFFSA

The effect of parameters on optimization is analyzed in this section. Taking Ackley function as example, Figure 7 shows the change of the objective function value in the case of varying parameters. and are proportional to the optimization result. can impact the time complexity of the algorithm, so the value should be appropriate not to affect the running speed. The best value of γ and δ is 2.5 and 1.2, respectively.

(a)

(b)

(c)

(d)

Figure 7: Test curves of parameters. (a) Test curve of β₀. (b) Test curve of γ. (c) Test curve of δ. (d) Test curve of Trynumber.

5. Conclusion

LFFSA is proposed to improve the capability of AFSA which integrates the merits of both AFSA and FA. Firstly, the searching characteristic of AFSA is studied by calculating the time complexity. Secondly, 17 benchmark test functions are used to verify LFFSA. Then time complexity of LFFSA is estimated. Numerical results demonstrate that LFFSA has a better performance in accuracy and speed of optimization to solve nonlinear optimization problems than the other test algorithms. However, the solution obtained by LFFSA can be more precise, and the way of modification could provide reference for those efficient algorithms, e.g. DE and GWO.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (61463028). The authors wish to thank Dr. Yanliang Cui for his fruitful comments and suggestions.

References

X. L. Li, Z. J. Shao, J. X. Qian et al., “An optimizing method based on autonomous animats: fish-swarm algorithm,” Systems Engineering-Theory and Practice, vol. 22, pp. 188–200, 2002, in Chinese.

C. R. Wang, C. L. Zhou, and J. W. Ma, “An improved artificial fish-swarm algorithm and its application in feed-forward neural networks,” in Proceedings of the Fourth International Conference on Machine Learning & Cybernetics, Guangzhou, China, August 2005.

H. C. Tsai and Y. H. Lin, “Modification of the fish swarm algorithm with particle swarm optimization formulation and communication behavior,” Applied Soft Computing, vol. 11, no. 8, pp. 5367–5374, 2011.

X. Y. Luan, Z. P. Li, and T. Z. Liu, “A novel attribute reduction algorithm based on rough set and improved artificial fish swarm algorithm,” Neurocomputing, vol. 174, pp. 522–529, 2016.

Y. Wu, X. Z. Gao, Z. Kai et al., “Knowledge-based artificial fish-swarm algorithm,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 188–200, 2011.

X. T. Hu, H. Q. Zhang, Z. C. Li et al., “A novel self-adaptation hybrid artificial fish-swarm algorithm,” IFAC Proceedings Volumes, vol. 46, no. 5, pp. 583–588, 2013.

K. P. Kumar, B. Saravanan, and K. S. Swarup, “Optimization of renewable energy sources in a microgrid using artificial fish swarm algorithm,” Energy Procedia, vol. 90, pp. 107–113, 2016.

Q. He, X. T. Hu, H. Ren et al., “A novel artificial fish swarm algorithm for solving large-scale reliability-redundancy application problem,” ISA Transactions, vol. 59, pp. 105–113, 2015.

Z. Q. Zhang, K. P. Wang, L. X. Zhu et al., “A Pareto improved artificial fish swarm algorithm for solving a multi-objective fuzzy disassembly line balancing problem,” Expert Systems with Applications, vol. 86, pp. 165–176, 2017.

P. K. Singhal, R. Naresh, and V. Sharma, “Binary fish swarm algorithm for profit-based unit commitment problem in competitive electricity market with ramp rate constraints,” Generation Transmission and Distribution IET, vol. 9, no. 13, pp. 1697–1707, 2015.

X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, London, 2008.

S. M. Farahani, B. Nasiri, A. A. Abshouri et al., “An improved firefly algorithm with directed movement,” in Proceedings of IEEE International Conference on Computer Science & Information Technology, Sichuan, China, June 2011.

L. D. S. Coelho, D. L. D. A. Bernert, and V. C. Mariani, “A chaotic firefly algorithm applied to reliability-redundancy optimization,” Evolutionary Computation, vol. 30, pp. 517–521, 2011.

X. S. Yang and S. Deb, “Eagle strategy using Lévy walk and firefly algorithms for stochastic optimization,” in Studies in Computational Intelligence, vol. 284, pp. 101–111, Springer, Berlin, Germany, 2010.

S. L. Tilahun and C. O. Hong, “Modified firefly algorithm,” Journal of Applied Mathematics, vol. 12, pp. 2428–2439, 2012.

K. Jagatheesan, B. Anand, S. Samanta et al., “Design of a proportional-integral-derivative controller for an automatic generation control of multi-area power thermal systems using firefly algorithm,” IEEE/CAA Journal of Automatica Sinica, pp. 1–14, 2016.

H. Su, Y. Cai, and Q. Du, “Firefly-algorithm-inspired framework with band selection and extreme learning machine for hyperspectral image classification,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 10, no. 1, pp. 309–320, 2016.

S. Ranganathan, M. S. Kalavathi, and A. R. C. Christober, “Self-adaptive firefly algorithm based multi-objectives for multi-type FACTS placement,” IET Generation Transmission and Distribution, vol. 10, no. 11, pp. 188–200, 2016.

D. F. Teshome, C. H. Le, Y. W. Lin et al., “A modified firefly algorithm for photovoltaic maximum power point tracking control under partial shading,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 2, pp. 661–671, 2017.

M. Alb, P. Alotto, C. Magele et al., “Firefly algorithm for finding optimal shapes of electromagnetic devices,” IEEE Transactions on Magnetics, vol. 52, no. 3, pp. 1–4, 2016.

A. Mishra and V. N. K. Gundavarapu, “Line utilisation factor-based optimal allocation of IPFC and sizing using firefly algorithm for congestion management,” Generation Transmission and Distribution IET, vol. 10, no. 1, pp. 115–122, 2016.

M. H. Horng, “Vector quantization using the firefly algorithm for image compression,” Expert Systems with Applications, vol. 39, no. 1, pp. 078–1091, 2012.

A. Kazem, E. Sharifi, F. K. Hussain et al., “Support vector regression with chaos-based firefly algorithm for stock market price forecasting,” Applied Soft Computing, vol. 13, no. 2, pp. 947–958, 2013.

L. F. He and S. W. Huang, “Modified firefly algorithm based multilevel thresholding for color image segmentation,” Neurocomputing, vol. 240, pp. 152–174, 2017.

A. H. Gandomi, X. S. Yang, and A. H. Alavi, “Mixed variable structural optimization using Firefly Algorithm,” Computers and Structures, vol. 89, no. 23-24, pp. 2325–2336, 2011.

M. Alweshah and S. Abdullah, “Hybridizing firefly algorithms with a probabilistic neural network for solving classification problems,” Applied Soft Computing, vol. 35, pp. 513–524, 2015.

R. M. Rizk-Allah, E. M. Zaki, A. A. El-Sawy et al., “Hybridizing ant colony optimization with firefly algorithm for unconstrained optimization problems,” Applied Mathematics and Computation, vol. 224, pp. 473–483, 2013.

X. S. Yang, S. S. S. Hosseini, and A. H. Gandomi, “Firefly algorithm for solving non-convex economic dispatch problems with valve loading effect,” Applied Soft Computing, vol. 12, no. 3, pp. 1180–1186, 2012.

J. Senthilnath, S. N. Omkar, and V. Mani, “Clustering using firefly algorithm: performance study,” Swarm and Evolutionary Computation, vol. 1, no. 3, pp. 164–171, 2011.

T. Kanimozhi and K. Latha, “An integrated approach to region based image retrieval using firefly algorithm and support vector machine,” Neurocomputing, vol. 151, pp. 1099–1111, 2015.

A. Baykasoglu and F. B. Ozsoydan, “Adaptive firefly algorithm with chaos for mechanical design optimization problems,” Applied Soft Computing, vol. 36, pp. 152–164, 2015.

I. Fister, M. Perc, S. M. Kamal et al., “A review of chaos-based firefly algorithms: perspectives and research challenges,” Applied Mathematics and Computation, vol. 252, pp. 155–165, 2015.

I. Fister, I. Fister, X. S. Yang et al., “A comprehensive review of firefly algorithms,” Swarm and Evolutionary Computation, vol. 13, pp. 34–46, 2013.

C. T. Brown, L. S. Liebovitch, and R. Glendon, “Lévy flights in Dobe Ju/’hoansi foraging patterns,” Human Ecology, vol. 35, no. 1, pp. 129–138, 2007.

I. Pavlyukevich, “Lévy flights, non-local search and simulated annealing,” Mathematics, vol. 226, no. 2, pp. 1830–1844, 2012.

I. Pavlyukevich, “Cooling down Lévy flights,” Journal of Physics A: Mathematical and Theoretical, vol. 40, no. 41, pp. 12299–12313, 2007.

R. Jensi and G. W. Jiji, “An enhanced particle swarm optimization with Lévy flight for global optimization,” Applied Soft Computing, vol. 43, pp. 248–261, 2016.

D. Tang, J. Yang, S. Dong et al., “A Lévy flight-based shuffled frog-leaping algorithm and its applications for continuous optimization problems,” Applied Soft Computing, vol. 49, pp. 641–662, 2016.

M. Yahya and M. P. Saka, “Construction site layout planning using multi-objective artificial bee colony algorithm with Lévy flights,” Automation in Construction, vol. 38, pp. 14–29, 2014.

C. Blum and A. Roli, Hybrid Metaheuristics: An Introduction, Springer, Berlin, Germany, 2008.

A. M. Reynolds and M. A. Frye, “Free-flight odor tracking in drosophila is consistent with an optimal intermittent scale-free search,” PLos One, vol. 2, no. 4, p. e354, 2007.

M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, Lévy Flights and Related Topics in Physics, Springer, Berlin, Heidelberg, Germany, 1995.

M. F. Shlesinger, “Mathematical physics: search research,” Nature, vol. 443, no. 7109, pp. 281-282, 2006.

S. T. Hsieh, T. Y. Sun, C. C. Liu et al., “Efficient population utilization strategy for particle swarm optimizer,” IEEE Transactions on Systems Man & Cybernetics Part B, vol. 39, no. 2, pp. 444–456, 2009.

D. Bratton and J. Kennedy, “Defining a standard for particle swarm optimization,” in Proceedings of IEEE Swarm Intelligence Symposium, vol. 107, pp. 120–127, Honolulu, Hawaii, April 2007.

U. Manber, Introduction to Algorithms: A Creative Approach, Addison-Wesley Longman Publishing Co. Inc., Boston, MA, USA, 1989.