An Improved Parameter Control Based on a Fuzzy System for Gravitational Search Algorithm

Yu Xianrui; Yu Xiaobing; Li Chenliang; Chen Hong

doi:10.2991/ijcis.d.200615.001

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Volume 13, Issue 1, 2020, Pages 893 - 903

An Improved Parameter Control Based on a Fuzzy System for Gravitational Search Algorithm

Authors

Yu Xianrui², Yu Xiaobing¹^{, 2}^{, *}, Li Chenliang², Chen Hong²

¹Key Laboratory of Meteorological Disaster (KLME), Ministry of Education and Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters (CIC-FEMD), Nanjing University of Information Science and Technology, Nanjing 210044, China

²School of Management Science and Engineering Nanjing University of Information Science and Technology, Nanjing 210044, China

^*Corresponding author. Email: yuxb111@163.com

Corresponding Author

Yu Xiaobing

Received 7 October 2019, Accepted 12 May 2020, Available Online 25 June 2020.

DOI: 10.2991/ijcis.d.200615.001 How to use a DOI?
Keywords: Gravitational search algorithm; Fuzzy system; Fuzzy rules; Optimization
Abstract: Recently, a kind of heuristic optimization algorithm named gravitational search algorithm (GSA) has been rapidly developed. In GSA, there are two main parameters that control the search process, namely, the number of applied agents (Kbest) and the gravity constant (G). To balance exploration and exploitation, a fuzzy system containing twelve fuzzy rules is proposed to intelligently control the parameter setting of the GSA. The proposed method can enhance the convergence ability and yield better optimization results. The performance of fuzzy GSA (FGSA) is examined by fifteen benchmark functions. Extensive experimental results are tested and compared with those of the original GSA, CGSA, CLPSO, NFGSA, PSGSA and EKRGSA.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

As optimization problems have become increasingly complicated, the traditional methods cannot effectively solve them in a high-dimensional search space. Hence, heuristic search algorithms have come into being. Heuristic optimization methods are developed based on natural or physical processes. Algorithms inspired by the behavior of natural phenomena have received widespread attention over the past few decades [1–4].

The superior performance of gravitational search algorithm (GSA) has attracted increasing attention. In recent years, many modified versions of GSA have been proposed. The most famous are the real GSA [5], introduced for real-valued variables; the binary GSA [6], which has variables with a value of either 0 or 1; the discrete GSA [7], possessing variables with discrete values; and the mixed GSA [8], containing both continuous and binary variables.

To control exploitation and exploration efficiently, some new GSA operators have been designed, including disruption [9–11], chaotic [12,13], mutation [14], crossover [15], and so on.

In the original GSA, the parameters mainly consist of the following aspects: mass of agent (M), number of agents (N), gravitational constant (G), distance between agents in the search space (R), power between distances (P) and number of applied agents (Kbest). The number of agents is generally set at the beginning of the algorithm and fixed during operation [16]. M has three attributes [5]: active gravitational mass, which determines the intensity of the gravitational field produced by an agent; passive gravitational mass, which determines the strength of an agent's interaction with the gravitational field and inertial mass, which represents the intensity of resistance to motion change when a force acts on an agent. The value of N is set according to the requirements of the specific algorithm used. The distance between agents is calculated as a function of the Euclidian distance, and the power between distances is set to one in the GSA and all its variants.

The values of the masses in the original GSA, including the passive mass, active mass and inertia mass, are set to the same value. Javidi suggested that mass calculation should be improved by defining an opportune function named sigma scaling and the Boltzmann functions [17]. These functions try to balance exploration and exploitation to prevent the algorithm from falling into local optimum. Khajooei and Rashedi proposed a function related to the concept of antigravity, which utilizes both positive and negative masses. The algorithm aims to enhance the ability to explore the search space [18].

In most GSA versions, both G and Kbest follow the same rules and decrease as the number of iterations increases [19]. In the original GSA, Kbest is obtained by the decreasing function in Eq. (1) [5]:

Kbest=0.01×N×final_per+1−tT×100−final_per,(1)

where N is the number of agents, final_per is a control parameter allowing only one agent to apply a force to others, which denotes the percentage of particles that exert gravitational force on all particles in the end, t is the number of iterations and T is the maximum number of iterations. S. He et al. proposed EKRGSA and modified the function of Kbest, which decreases exponentially with the number of iterations from N to 1 in Eq. (2) [20]:

Kbest=N×final_per100tT,(2)

where N is the number of agents, final_per is the percentage of particles that exert gravitational force on all particles in the end, t is the number of iterations and T is the maximum number of iterations.

G is controlled by an exponential function in Eq. (3) [5]:

Gt=G0×e−αtT,(3)

where G0 denotes the initial value, α denotes the gravitational coefficient and T is the maximum number of iterations.

The NFGSA [21] proposed a neural network combined with a fuzzy system to adjust the parameter α and redefined the definition of the parameter Kbest. The PSGSA [21] redefined the calculation of the parameter α combined with the plane output surface based on NFGSA.

In the binary GSA, G is considered as a linear decreasing function in Eq. (4). Rashedi et al. examined the effects of different values of G and Kbest [6]. The results of their experiments have shown that the optimal value of G is different for different functions and that G is a parameter determined by the problem. Furthermore, the value of Kbest was experimentally proven to change with time.

Gt=G0×1−tT,(4)

where G0 denotes the initial value, T denotes the maximum number of iterations and t is the iteration counter.

ED=Rave−RminRmax−Rmin(5)

CM=fave(t)−fave(t−1)fave(t)(6)

Fatemeh and Esmat use a fuzzy rule for controlling GSA setting [22]. In their opinion, the distance between agents (R) can be rewritten into a new parameter ED, calculated according to Eq. (5), which detects the ability of algorithms in terms of diversity. In Eq. (5), Rave, Rmax and Rmin are the average, maximum and minimum Euclidian distances between agents, respectively. Then, a new parameter CM (see Eq. (6)) is introduced to measure the progress of the algorithm, where fave(t) is the average value of the agent finesses at iteration t. The combination of ED and CM can be expressed by a fuzzy logic controller to prevent the algorithm from falling into local optimum and converging too early. During the operation of the algorithm, G is increased by these fuzzy rules when the optimization progress of the algorithm is stagnant. At the later stage of the algorithm, G is decreased so that the exploitation capabilities of the algorithm can be improved. This is consistent with the rules for strengthening the exploration ability in the early stage and increasing the exploitation capability in the later stage.

In another study [23], the authors deemed that the value of Kbest should be increased to escape trapping local optima when the best fitness is not improved after multiple iterations. For the parameter G, the authors suggested that a simple exponential decreasing function for all iterations is unreasonable when solving complicated problems. Therefore, eight fuzzy rules are proposed to update G and Kbest.

According to the two parameter control methods based on fuzzy theory mentioned above, the former uses two functions to control two input parameters, which represent the population diversity and population progress respectively. However, although these fuzzy rules of the former are used to control the parameter G, they cannot be used to control the parameter Kbest, which is also significant in determining the motion of agents. Conversely, the latter uses fuzzy rules to control G and Kbest, but they cannot use appropriate functions such as Eqs. (5–6) to control the input parameters. The goal of a fuzzy system is to produce output parameters through the values of the corresponding input parameters, which should be calculated in reasonable ways. These descriptive languages without specific control functions for calculating the input parameters are obviously insufficient. Based on this observation, a fuzzy system including new control functions is introduced to control G and Kbest. The aim is to balance exploration and exploitation, which can improve the performance of GSA.

It is important to control exploitation and exploration in GSA. A new fuzzy system is applied to intelligently control the parameters of GSA, which includes twelve fuzzy rules. They are adopted to increase the convergence rate and prevent the algorithm from falling into local optimum.
New control functions are designed to calculate the value of the input variables so that the agent diversity and optimization progress of GSA can be well monitored.
Compared with the original GSA, CGSA, CLPSO, NFGSA, PSGSA and EKRGSA, the proposed method has achieved better results for fifteen well-known standard functions. Additional experiments have indicated that these fuzzy rules are effective.

This paper consists of five sections. Introduction is firstly suggested as Section 1. After that, GSA is reviewed in Section 2. In Section 3 fuzzy GSA is described in detail. Experimental results are presented in Section 4. At last, conclusions are included in Section 5.

2. GRAVITATIONAL SEARCH ALGORITHM

In GSA, the objective function of an optimization problem is based on many variables. Each variable has an upper and a lower bound, as shown in Eq. (7), represented by xld and xud, respectively. A search field is set up using these boundaries.

xld≤xd≤xud,d=1,2,…,m(7)

Let Xi(i=1,2,…,k) be the agents, and the position of the ith agent can be presented in Eq. (8):

Xi=xi1,…‥,xim,i=1,2,…,k,(8)

where each value in Xi represents the position of the ith agent in the search space. At the iteration t, the total forces applied to agent i from a set of heavier agents should be defined as Eq. (9):

Fidt=∑j∈Kbest,j≠irandjFijd =∑j∈Kbest,j≠irandjG(t)MajtMpitRijt+ε(xjdt−xidt),(9)

where Majt is the active mass related to agent j, Mpit is the passive mass related to agent i, G(t) is the gravitational constant at iteration t, ε is a small value, Rijt is the Euclidian distance between the two agents i and j and randj is a random variable in the interval [0, 1]. Kbest is a function of time, which is initialized to k0 at the beginning and decreases over time.

Then, the law of motion is used to calculate the acceleration of the agents, as shown in Eq. (10):

aidt=Fid(t)MIi(t),(10)

where aidt is the acceleration of the ith agent in the dth dimension at iteration t. MIi(t) is the inertia mass related to the ith agent at iteration t. Afterward, the next velocity and its next position are calculated using Eqs. (11–12):

vidt+1=randi×vidt+aidt(11)

xidt+1=xidt+vidt+1,(12)

where randi is a uniform random variable in the interval [0, 1], vidt is the velocity of the ith agent in the dth dimension at iteration t and xidt is the position of the ith agent in the dth dimension at iteration t.

The gravitational constant G is a function of time t (see Eq. (13)), which takes an initial value G0. It is reduced with the iterative time t to control the search accuracy.

G(t)=G(G0,t)(13)

The masses of the agents are calculated by the fitness evaluation. Supposing that the gravitational mass is numerically equal to the inertia mass, the mass Mit is computed by Eqs. (14–17):

Mi=MIi,i=1,2,…,m

mit=fitit−fmaxtfmin(t)−fmaxt(14)

Mit=mit∑j=1kmjt(15)

fmin(t)=minfitjtj∈{1,…,m}(16)

fmaxt=maxfitjtj∈{1,…,m},(17)

where MIi denotes the inertia mass of the ith agent, fitit is the fitness value of the ith agent at iteration t and fmin(t) and fmaxt represent the minimum and maximum fitness values at iteration t, respectively (for a minimization problem). The better the function value is, the larger the value of the mass will be. A heavier mass represents a more efficient agent, which has a greater attractive force and runs more slowly.

To control important parameters, the original GSA uses a mathematical model, which includes linearity, an exponential character or both. As the iteration proceeds, the exploration becomes weaker and the exploitation becomes stronger. However, when faced with complex engineering problems, such as big data analysis, the problem becomes nonlinear and complex. It is not enough to control the algorithm process through a mathematical model [23]. An effective solution is to establish a balance between exploration and exploitation through parameter control.

3. FUZZY GSA

Currently, many scholars combine heuristics methods with fuzzy logic to improve algorithms [22,24–26]. They integrate algorithm processes with the linguistic description to create a fuzzy system to intelligently control parameters. In GSA, the larger the agent is, the slower the convergence rate is and the better the diversity is; the smaller the agent is, the easier it is for the agent to fall into the local optimum.

In the original GSA, there are two important parameters, i.e., Kbest and G, which are used to control the search process. Among them, G can be dominated by the gravitational coefficient α.

3.1. Kbest and G

Kbest denotes the number of agents that effectively apply force on other agents. The larger Kbest is, the more attractive it is between agents. By changing Kbest, the performance of GSA can be guaranteed by controlling the balance between exploration and exploitation. In the original GSA, all agents initially apply a force to each other. As the iteration evolves, Kbest is gradually reduced by a linear function. Finally, only one maximum fitness value exists, and only the heaviest agent applies a force to the other agents, which is shown in Eq. (9). The main idea of the Kbest formula is that when the iteration time reaches the later stage, only agents with a large weight can exert force on other agents. When Kbest increases, there is more attraction, more movement and a lower convergence rate. On the other hand, when Kbest decreases, the attraction and movement decrease, which may result in premature convergence and easily falling into the local optimum.

The parameter G is directly related to the acceleration of the agents. Its size has a great influence on the value of the acceleration, and the acceleration affects the position of agents after the gravitational force is applied. In the original GSA, G is gradually reduced by a monotonic exponential function. G is determined by the gravitational coefficient α. When α is larger, then G is smaller, the acceleration of the agent is smaller, the positional change is less, the exploration ability is weaker, and it is easy to fall into the local optimum. On the other hand, when α is smaller, then G is larger, the acceleration of the agent is greater, the positional change is more obvious, the exploration capability is stronger, and the convergence rate is slower.

3.2. Agent Diversity and Optimization Progress

Agent diversity and optimization progress are the most effective tools for monitoring the performance of the algorithm [22]. The parameter SD is introduced to measure the agent diversity, which is calculated by Eq. (18):

SD=favet−fmin(t)fmaxt−fmin(t),(18)

where favet represents the average fitness value obtained from the tth iteration, fmin(t) and fmaxt represent the minimum and maximum fitness values in the tth iteration, respectively. When SD is small, the fitness values are similar, indicating the following situations:

The positions of the agents in the search space are not far apart. In this case, the objective function has fallen into the local optimum at similar locations in the search space.
The positions of the agents in the search space are far apart. In this case, the objective function has at least 2 local optima at different locations in the search space.

Both conditions indicate that the objective function is trapped in the local optimum regardless of whether the locations of the agents are similar or different, which implies that the diversity of the agents is poor, and vice versa.

In addition to SD, the parameter ZM is considered for measuring the optimization progress. ZM is calculated as Eq. (19) in minimization problems:

ZM=fbestt−1−fbest(t)fbest_avet−fbestt,(19)

where t is the number of iterations, fbest(t) is the optimal fitness value in the tth iteration, and fbest_avet is the average optimal fitness value of the previous t iterations. When ZM is small, the current iterative optimal value is not very different from that of the previous iteration, which indicates that the optimization progress is not good, and vice versa.

The smaller SD is, the worse the diversity of the agents is. The larger SD is, the better the diversity of the agents is. The smaller ZM is, the worse the optimization process is. The larger ZM is, the better the algorithm optimization process is.

3.3. Membership Functions

A fuzzy inference system is composed of three input variables and two output variables. SD, ZM and t are the input variables. t is the current iteration. α and Kbest are the output variables. Fuzzy theory uses linguistic descriptions instead of mathematical language. In this paper, five membership functions are proposed to describe the input and output variables. The role of the membership functions is to label t, α, and Kbest as low, medium and high. SD and ZM are converted to low and high labels. The membership functions are depicted in Figure 1. In most cases, the membership functions are devised by experts. The range of SD is the interval [0, 0.6], the range of ZM is the interval [0, 1], the range of t is the interval [0, 1000], the range of α is the interval [28, 30] and the range of Kbest is the interval [2, 50].

3.4. Fuzzy Rules

Twelve fuzzy rules are extracted from the previous subsections with three input and two output variables, as shown in Table 1. These rules are proposed to intelligently control the search process of GSA, prevent the algorithm from falling into a local optimum and avoid premature convergence. Each of the fuzzy rules is set up to enhance the performance of the algorithm.

Rule		SD	T	ZM		α	Kbest
1		high	low	high		high	medium
2		low	low	high		medium	high
3		low	low	low		low	medium
4		high	low	low		high	medium
5		high	medium	high		high	medium
6	If	low	medium	high	Then	medium	high
7		low	medium	low		low	high
8		high	medium	low		medium	medium
9		high	high	high		high	low
10		low	high	high		medium	medium
11		low	high	low		low	medium
12		high	high	low		high	low

Table 1

Fuzzy rules for controlling the parameter of α and Kbest.

The first rule in Table 1 indicates that in the early stage of the iterations, the optimization progress and the diversity are good. At this time, when α increases, G decreases; then, the gravitation between agents decreases, and the motion becomes slower. When Kbest decreases, the number of agents that exert a force is reduced, and the movement decreases. Hence, α takes a large value and Kbest takes a median value, which can enhance the local search ability and convergence ability of the algorithm.

The second rule indicates that in the early stage of the iterations, the optimization progress is good and the diversity is bad. This may reveal that the algorithm is prematurely trapped; thus, α and Kbest should increase. Therefore, α gives a medium value, and Kbest gives a high value. By the rule, the acceleration increases, and the movement is faster. As a result, the convergence rate is lower.

The third rule is that at early iterations, the optimization progress and the diversity are poor. This shows that the algorithm is trapped in local optimum and suffers from premature convergence. At this moment, by decreasing α and increasing Kbest, the acceleration and velocity are increased. Then, agents can escape from the local optimum.

The fourth rule is that at early iterations, the optimization progress is poor and the diversity is good. This scenario shows that the algorithm converges slowly. It is necessary to increase α and decrease Kbest; hence, α takes a high value and Kbest takes a medium value to increase the convergence speed.

The seventh rule indicates that at medium iterations, the optimization progress and the diversity are poor. Similar to the third rule, the algorithm falls into local optimum, and premature convergence occurs. α gives a low value, and Kbest is taken as a high value in the medium term. Then, the acceleration increases, the position changes more quickly and escape from the local optimum occurs as soon as possible.

Regarding the ninth rule, when the algorithm is at a later iteration, the optimization progress and the diversity are good. In this case, the exploitation capability should be enhanced. Therefore, α gives a high value, and Kbest takes a low value, which speeds up the convergence.

The difference between the twelfth rule and the ninth rule is that the optimization progress is low. Because the diversity is good, it is necessary to strengthen the convergence ability, which requires a high value of α and a low value of Kbest.

In this paper, inputs are combined logically using the AND operator to obtain a crisp decision front, and the center-of-gravity method is utilized for defuzzification.

3.5. Pseudocode of FGSA

Pseudocode of FGSA

Input: N: the number of agents;

T: maximum number of iterations;

Rpower: power of the Euclidian distance between two agents

F_index: the index of the test function

Min_flag: type of problem (min_flag = 1, minimization or min_flag = 0, maximization)

(1) Rnorm = 2 (norm in the Euclidian distance);

(2) Generate the initial position of the ith agent by randomly forming from search space;

(3) for iteration = 1: T (the maximum iteration)

(4) Checks the search boundaries for agents;

(5) Evaluate the objective function values of agents;

(6) If min_flag = 1

(7) Select the minimum value in the calculation result;

(8) else

(9) Select the maximum value in the calculation result;

(10) end If

(11) Calculate the values of masses of each agent (M);

(12) Update gravitational coefficient (α) and the number of effective agents (Kbest) by twelve fuzzy rules;

(13) Calculate gravitational constant (G);

(14) Calculate acceleration and velocity;

(15) Update agents' position;

(16) end for

Output: Fbest: the best result of the objective function values;

Lbest: the best solution (the location of Fbest in search space);

BestChart: the best function values over iterations.

4. EXPERIMENTS AND DISCUSSION

The performance of the proposed algorithm can be tested by fifteen standard benchmark functions [27], including some unimodal and some multimodal criteria, which are presented in Table 2. The proposed algorithm is tested for minimization and compared with the original GSA, CGSA [13] and CLPSO [28]. The CGSA (chaotic GSA) embeds chaotic maps into the gravitational constant (G) and proposes an adaptive normalization method to make the transition from the exploration stage to the exploitation stage smoother, which are all used to balance exploration and exploitation. The CLPSO (comprehensive learning particle swarm optimizer) introduces a novel learning strategy and combines the particles' historical best information to update the particle velocity in PSO. In all algorithms, the number of agents is 50 (N = 50). The dimensions of the test function are set to 30, the maximum number of iterations is 1000 and G0 is set to 100. From f1 to f13, the minimum value is 0, except for f8, which has a minimum value of −418.9823 × n. The optimum locations for the functions of Table 2 can be found in [0]ⁿ, except for f5, f12, and f13 in [1]ⁿ and f8. in [420.96]ⁿ. For f14 and f15, the minimum values are 1 and 0.0003, respectively. The optimum locations of f14 and f15 are (−32, 32) and (0.1928, 0.1908, 0.1231, 0.1358), respectively.

Test Functions	Bounds
f1X=∑i=1nxi2	[−100, 100]ⁿ
f2X=∑i=1n\|xi\|+∏i=1n\|xi\|	[−10, 10]ⁿ
f3X=∑i=1n∑j=1ixj2	[−100, 100]ⁿ
f4X=maxi\|xi\|,1≤i≤n	[−100, 100]ⁿ
f5X=∑i=1n−1100xi+1−xi22+xi−12	[−30, 30]ⁿ
f6X=∑i=1nxi+0.52	[−100, 100]ⁿ
f7X=∑i=1nixi 4+random0,1	[−1.28, 1.28]ⁿ
f8X=∑i=1n−xisin\|xi\|	[−500, 500]ⁿ
f9X=∑i=1nxi 2−10cos2πxi+10	[−5.12, 5.12]ⁿ
f10X=−20exp−0.2∑i=1nxi 2n−exp1n∑i=1ncos2πxi+20+e	[−32, 32]ⁿ
f11X=14000∑i=1nxi2−∏i=1ncosxii+1	[−600, 600]ⁿ
f12X=πn10sinπy1+∑i=1nyi−121+10sin2πyi+1+yn−12yi=1+xi+14 +∑i=1nuxi,10,100,4	[−50, 50]ⁿ
f13X=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn +∑i=1nuxi,5,100,4	[−50, 50]ⁿ
f14X=1500+∑j=1251j+∑i=12xi−aij6−1	[−65.53, 65.53]²
f15X=∑i=111ai−x1bi2+bix2bi2+bix3+x42	[−5, 5]⁴

Table 2

Test functions.

The results are averaged over 25 runs. In all tables, the mean represents the average of the 25 best fitness values, the median represents the median of the 25 best fitness values, and the variance represents the variance of the 25 best fitness values, among which the best results have been highlighted in bold.

The results are shown in Table 3 and Figure 2. Table 3 illustrates that the FGSA provides a better mean, median and variance than those of the original GSA in f1, f2, f3, f4, f5, f8, f10, f12, f13 and f14. The results of the mean, median and variance in f6 indicate that the FGSA is the same as the original GSA. At the same time, the results of the original GSA are worse than those of the FGSA in f7, f9, f11 and f15.

Function		FGSA	GSA	CGSA	CLPSO
f1	Mean	6.74E−24	2.069E−17	1.620E−05	5.502E−02
	Median	6.32E−24	1.978E−17	1.430E−5	4.516E−01
	Variance	4.2E−48	2.847E−35	2.088E−11	7.247E−03
f2	Mean	3.55E−12	2.272E−08	2.014E−02	4.612E−02
	Median	3.56E−12	2.244E−08	1.996E−02	4.382E−02
	Variance	2.66E−25	1.687E−17	7.659E−06	8.210E−05
f3	Mean	1.05E+02	2.546E+02	1.534E−03	8.585E+01
	Median	9.007E+01	2.286E+02	1.298E−02	8.726E+01
	Variance	2.271E+03	6.670E+03	6.764E−07	1.639E+02
f4	Mean	1.26E−12	3.188E−09	4.267E−03	2.509E+01
	Median	1.19E−12	3.213E−09	4.429E−03	2.519E+01
	Variance	4.49E−26	3.655E−19	4.558E−07	3.369E+00
f5	Mean	2.66E+01	2.857E+01	2.508E+01	4.021E+02
	Median	2.65E+01	2.610E+01	2.499E+01	3.997E+02
	Variance	1.028E−01	1.576E+02	9.226E−02	7.377E+03
f6	Mean	0	0	0	4.721E−02
	Median	0	0	0	4.497E−02
	Variance	0	0	0	1.380E−04
f7	Mean	4.57E−02	1.798E-02	4.919E−02	3.123E−02
	Median	4.412E−02	1.808E-02	4.762E−02	3.106E−02
	Variance	1.82E−03	3.836E−05	7.182E−05	5.638E−05
f8	Mean	−2.928E+03	−2.840E+03	−2.789E+03	−1.254E+04
	Median	−2.877E+03	−2.865E+03	−2.722E+03	−1.255E+04
	Variance	2.130E+05	3.766E+05	9.139E+04	1.130E+03
f9	Mean	2.69E+01	1.496E+01	2.080E+01	1.001E+01
	Median	2.587E+03	1.492E+01	2.139E+01	1.046E+01
	Variance	2.799E+03	1.068E+01	1.054E+01	2.805E+00
f10	Mean	2.08E−12	3.732E−9	3.038E−03	1.581E−01
	Median	2.15E−12	3.671E−09	3.059E−03	1.485E−01
	Variance	1.39E−25	1.986E−19	3.014E−07	1.049E−03
f11	Mean	2.06E+01	3.777	0.022	0.152
	Median	2.017E+01	3.433	0.013	0.145
	Variance	2.29E+01	2.071	0.001	0.002
f12	Mean	4.32E−26	1.431E−19	1.126E−07	1.818E+01
	Median	4.32E−26	1.484E−19	1.103E−07	1.828E+01
	Variance	1.39E−52	1.804E−39	1.141E−15	9.706E−01
f13	Mean	6.53E−25	1.900E−18	1.940E−06	5.245E−02
	Median	5.8E−25	1.747E−18	1.794E−06	4.956E−02
	Variance	7.05E−50	2.006E−37	6.331E−13	5.622E−04
f14	Mean	3.506	3.656	2.530	0.998
	Median	2.209	2.982	2.317	0.998
	Variance	6.916	6.926	1.020	0
f15	Mean	2.11E−03	2.007E−03	1.889E−03	5.807E−04
	Median	2.102E−03	1.991E−03	1.219E−03	5.857E−04
	Variance	2.44E−07	4.99263E−07	2.59053E−06	5.45262E−09

GSA, gravitational search algorithm; FGSA, fuzzy GSA.

Table 3

The best results of 15 benchmark function with GSA, CGSA and CLPSO.

For CGSA and CLPSO, the results of FGSA are better in most functions. In particular, the mean, median and variance of f1, f2, f4, f10, f12, and f13 are significantly better than those of the CGSA and CLPSO. In f3, f9, f11, f14 and f15, FGSA is less effective than CGSA and CLPSO; FGSA is worse than CGSA and better than CLPSO in f5. In the result of f6, FGSA is the same as CGSA and better than CLPSO. For f7 and f8, FGSA is better than CGSA and worse than CLPSO. Further analysis of the results by a statistical method is shown in later paragraphs.

The progress of GSA, CGSA and CLPSO for f1, f2, f4, f10, f12 and f13 are plotted in Figure 2. It can be observed that the convergence rate of FGSA is faster than that of the other algorithms and that the mean of the iterative full period is the smallest. This fully demonstrates that FGSA tends to look for the global optimum faster than the other algorithms. Additionally, FGSA shows better convergence ability than that of the others. Therefore, FGSA has excellent performance and a high convergence rate.

The non-parametric statistical method Wilcoxon test was used to make a compete comparison. Table 4 lists the results of the Wilcoxon test on every test function between FGSA and the other algorithms. Rows “+,” “=” and “−” denote the number of test functions of the other algorithms that FGSA performs better than, almost the same as, and worse than, respectively. Row “Score” indicates the difference between the number of “1”s and “−1”s. It is used to make an overall comparison between two algorithms.

Function	GSA	CGSA	CLPSO
Function	Wilcoxon	Wilcoxon	Wilcoxon
f1	+	+	+
f2	+	+	+
f3	+	−	=
f4	+	+	+
f5	+	−	=
f6	=	=	+
f7	−	=	−
f8	=	=	−
f9	−	−	−
f10	+	+	+
f11	−	−	−
f12	+	+	+
f13	+	+	+
f14	=	=	−
f15	=	−	−
+	8	6	7
=	4	4	2
−	3	5	6
Score	5	1	1

GSA, gravitational search algorithm; FGSA, fuzzy GSA.

Table 4

The Wilcoxon test of the best results for 15 benchmark function.

For example, FGSA outperforms GSA on 8 functions (f1, f2, f3, f4, f5, f10, f12, f13), performs the same as GSA on 4 functions (f6, f8, f14, f15) and performs worse on 3 functions (f7, f9, f11). Therefore, the score is 8−3=5, which indicates that FGSA is better than GSA.

4.1. Comparison with the Latest GSA Variants

To further validate the performance of the proposed algorithm, the three latest GSA variants, i.e., NFGSA [21], PSGSA [21] and EKRGSA [20], are selected for comparison.

In all algorithms, the test functions are the same as in Table 2. The number of agents is 50 (N = 50). The dimension of the test function is set to 30, the maximum number of iterations is 1000 and G0 is set to 100 in FGSA, NFGSA and PSGSA and to 1000 in the EKRGSA. The results are averaged over 25 runs. Note that the results of NFGSA, PSGSA and EKRGSA are taken from their respective reference papers. Because there are no experimental results on the function f6 for the EKRGSA, the comparison with EKRGSA on f6 is omitted.

Table 5 illustrates that the mean of FGSA is the best for f1. The results of the mean rank second compared to that of NFGSA, PSGSA and EKRGSA on f3, f8, f14 and f15. For f5, FGSA provides better result than does PSGSA but worse than that of NFGSA and EKRGSA. In general, the results of FGSA are better on most functions.

Function		FGSA	NFGSA	PSGSA	EKRGSA
f1	Mean	6.74E−24	7.51E−23	1.04E−02	3.68E−19
f2	Mean	3.55E−12	7.69E−09	3.77E−02	3.74E−09
f3	Mean	1.05E+02	9.13E +01	2.72E+02	2.46E+03
f4	Mean	1.26E−12	3.42E−08	1.67E−02	2.16E−08
f5	Mean	2.66E+01	2.50E+01	5.53E+01	2.457E+01
f6	Mean	0	0=	3.33E−02	−
f7	Mean	4.57E−02	5.06E−02	1.05E−01	6E−02
f8	Mean	−2.928E+03	−2.68E+03	−3.39E+03	−
f9	Mean	2.69E+01	2.33E+01	2.13E+01	−
f10	Mean	2.08E−12	5.16E−12	1.48E−02	3.61E−10
f11	Mean	2.06E+01	1.50E+00	1.01E+01	−
f12	Mean	4.32E−26	2.07E−02	1.17E−02	1.91E−21
f13	Mean	6.53E−25	1.10E−03	1.30E−01	3.13E−20
f14	Mean	3.506	4.32	5.93	1.02
f15	Mean	2.11E−03	2.78E−03	4.52E−03	9.05E−04

Table 5

The best results of 15 benchmark function with NFGSA, PSGSA and EKRGSA.

4.2. Discussion

To set the parameters of GSA more reasonably, a fuzzy system with twelve fuzzy rules is introduced. However, the setting of the three input parameters needs to confirm the rationality. Thus, an experiment is designed to test the performance of three different combinations of input parameters: SD+t, ZM+t, and SD+ZM+t. Three groups are tested independently. The contrast is tested for minimization, and the results are averaged over 25 runs. The experimental results are shown in Table 6.

Function	SD + t	ZM + t	SD + ZM + t
f1	2.91E−21	1.43E−23	6.74E−24
f2	6.99E−11	1.60E−11	3.55E−12
f3	1.83E+02	1.93E+02	1.05E+02
f4	2.73E−11	1.67E−12	1.26E−12
f5	2.68E+01	2.88E+01	2.66E+01
f6	0	0	0
f7	3.98E−02	1.96E−02	4.57E−02
f8	−2.84E+03	−2.73E+03	−2.93E+03
f9	2.95E+01	1.91E+01	2.69E+01
f10	3.80E−11	2.77E−12	2.08E−12
f11	1.60E+01	1.15E+01	2.06E+01
f12	1.69E−23	8.48E−26	4.32E−26
f13	2.27E−22	1.34E−24	6.53E−25
f14	5.17E+00	3.91E+00	3.82E+00
f15	3.27E−03	3.27E−03	2.11E−03

Table 6

Mean of three ways of input parameters.

The average of the 25 best fitness values is chosen as the standard for testing the pros and cons. The combination of SD, ZM and t provides smaller results than those of the others on f1, f2, f3, f4, f5, f8, f10, f12, f13, f14 and f15. The results of the three methods for the input parameters are the same as for f6. The results of the combination of SD, ZM and t are worse than the others for f7 and f11. For f9, the combination of SD, ZM and t is more efficient than the combination of SD and t and less efficient than the combination of ZM and t. In general, the combination of SD, ZM and t is better than the others on most functions, which represents its excellent performance and effectiveness.

The inference mechanism behind these fuzzy rules in Table 1 is herein explained. The process of controlling parameters by the corresponding fuzzy rules for f3 is selected to describe the inference mechanism. The results are represented in Table 7, which includes the early, median and late iterations separately. For the parameters of Table 7, the inputs are t (representing the current iteration), SD (measuring the agent diversity) and ZM (weighing the optimization progress), and the outputs are Kbest, α and the corresponding rule of the tth iteration. For example, when t is 199, SD is 0.47 and ZM is 0.746, it is revealed that at early iterations, SD and ZM are described as high, which is verified by the member functions in Figure 1. The agent diversity measured by SD and the optimization progress weighed by ZM are good, which corresponds to rule 1 in Table 1. As a result, the value of Kbest is 25, which is consistent with taking a median value, and the value of α is 29.49, meeting the requirements of taking a large value. The other iteration stages all have corresponding rules, which are shown in Table 7. Through the above inference mechanism, the algorithm can escape from the local optimum and enhance the rate of convergence.

Inputs	t	199	200	201	499	500	501	799	800	801
	SD	0.470	0.580	0.294	0.394	0.228	0.433	0.458	0.430	0.452
	ZM	0.746	0.903	0.276	0.656	0.367	0.603	0.405	0.177	0.377
Outputs	Kbest	25	25	25	31	40	25	11	11	11
Outputs	α	29.49	29.47	28.36	28.76	28.31	29.28	29.50	29.49	29.50
Corresponding rule		1	1	3	6	7	5	12	12	12

Table 7

Control parameters by corresponding fuzzy rules for f3.

Due to the addition of a fuzzy system, the computational cost may be slightly worse. Nowadays, the performance of computer increasingly becomes improved. The gap of response time between FGSA and GSA is within seconds. However, we admit that our proposed algorithm has increased some computational cost. This is the limitation of the paper.

5. CONCLUSIONS

The idea of intelligently controlling the parameters of GSA is introduced, and a fuzzy system is designed for this purpose. In this article, an improved idea of controlling the important parameters of GSA is proposed, which shows that a fuzzy inference system containing 12 fuzzy rules can be used to control the parameters Kbest and G.

The new strategy gives GSA a higher convergence rate and the ability to escape the local minimum. A balance between exploration and exploitation is also realized. The performance of FGSA is examined on fifteen well-known standard functions. The experimental results are compared with those of the original GSA, CGSA, CLPSO, NFGSA, PSGSA and EKRGSA. The results indicate that FGSA achieves superior performances on most functions.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

AUTHORS' CONTRIBUTIONS

Xianrui Yu conceived the research, drafted the manuscript and performed the experiments; Xiaobing Yu edited, and revised the manuscript. Chenliang Li and Hong Chen designed the experiments and analyzed the data.

ACKNOWLEDGMENTS

This research was funded by the National Natural Science Foundation of China (No.71974100, No.71503134), Natural Science Foundation of Jiangsu Province (No. BK20191402), Major Project of Philosophy and Social Science Research in Colleges and Universities of Jiangsu province (2019SJZDA039), Key Project of National Social and Scientific Fund Program (16ZDA047), Qing Lan project (R2019Q05), Postgraduate research and innovation plan project of Jiangsu Province (SJKY19_0983) and National Undergraduate Training Program for Innovation and Entrepreneurship (201910300016Z), the Joint Open Project of KLME & CIC-FEMD, NUIST (KLME202004).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 13 - 1
Pages: 893 - 903
Publication Date: 2020/06/25
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.200615.001 How to use a DOI?
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TY  - JOUR
AU  - Yu Xianrui
AU  - Yu Xiaobing
AU  - Li Chenliang
AU  - Chen Hong
PY  - 2020
DA  - 2020/06/25
TI  - An Improved Parameter Control Based on a Fuzzy System for Gravitational Search Algorithm
JO  - International Journal of Computational Intelligence Systems
SP  - 893
EP  - 903
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200615.001
DO  - 10.2991/ijcis.d.200615.001
ID  - Xianrui2020
ER  -

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