- Email: [email protected]
A fuzzy hierarchical operator in the grey wolf optimizer algorithm
Accepted Manuscript Title: A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm Authors: Luis Rodr´ıguez, Oscar Castillo, Jos´e Soria, Patricia Melin, Fevrier Valdez, Claudia I. Gonzalez, Gabriela E. Martinez, Jesus Soto PII: DOI: Reference:
S1568-4946(17)30182-5 http://dx.doi.org/doi:10.1016/j.asoc.2017.03.048 ASOC 4143
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
10-8-2016 14-2-2017 5-3-2017
Please cite this article as: Luis Rodr´ıguez, Oscar Castillo, Jos´e Soria, Patricia Melin, Fevrier Valdez, Claudia I.Gonzalez, Gabriela E.Martinez, Jesus Soto, A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm, Applied Soft Computing Journalhttp://dx.doi.org/10.1016/j.asoc.2017.03.048 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm
Luis Rodríguez, Oscar Castillo*, José Soria, Patricia Melin, Fevrier Valdez, Claudia I. Gonzalez, Gabriela E. Martinez, Jesus Soto Tijuana Institute of Technology, Calzada Tecnologico s/n, CP 22379, B. C., Tijuana, Mexico,
*Email corresponding autor: [email protected] fig Graphical abstract
Highlights
The main goal is to study the performance of the Grey Wolf Optimizer algorithm when a new hierarchical operator is introduced. This new operator is basically a hierarchical transformation that is inspired in the hierarchical social pyramid of the grey wolf. This proposed operator is applied to the simulation of the hunting process in the algorithm and has 5 variants are presented. Notably the variants having the greatest impact in the GWO performance are based on the use fuzzy logic.
Abstract—The main goal of this paper is to study the performance of the Grey Wolf Optimizer (GWO) algorithm when a new hierarchical operator is introduced in the algorithm. This new operator is basically a hierarchical transformation that is inspired in the hierarchical social pyramid of the grey wolf. This proposed operator is applied to the simulation of the hunting process in the algorithm and has 5 variants that are explained in more detail in this paper (centroid, weighted, based on the fitness and two variants using fuzzy logic). Notably the variants having the greatest impact in the GWO performance are based on the use of fuzzy logic. We also present the motivation and results of experiments, as well as the benchmark functions that were used for the tests that are presented. In addition we are presenting a comparison among all methods for 30, 64 and 128 dimensions and we conclude that the performance of the Hierarchical GWO algorithm is better when using a fuzzy variant of the hierarchical operator. Keywords—dynamic adaptation; fuzzy logic; performance; GWO; benchmark functions; new operator; hierarchical, pyramid;
1. –Introduction
Today it is more common that researchers in the area of computer science work with bio-inspired meta-heuristics to solve optimization problems. There are a lot papers and algorithms that researchers have developed based on natural behavior that have been recently proposed. The main challenge in developing a new algorithm is to represent and simulate as much as possible each of the most important behaviors and characteristics of real phenomena. This representation can be implemented by the operators in the algorithms, for example in genetic algorithms we use operators, like crossover, mutation and selection. In this paper we are presenting a modification to the grey wolf optimizer algorithm based on the implementation of a new hierarchical operator in the algorithm, inspired by the hierarchical pyramid of the grey wolf, which is explained in more detail in Section 4. We are presenting four new approaches for the implementation of this hierarchy. Of these four new methods, two of them are represented by mathematical models and the other two are based on fuzzy logic, and this paper explains in detail each of these proposed operators. In this paper we are presenting tables and hypothesis testing results to demonstrate the performance of the algorithm with the 4 methods implemented for the new operator. The rest of the paper is organized into the following sections. Section 2 shows a literature review, in Section 3 we present the original grey wolf optimizer algorithm, Section 4 describes in detail the proposed hierarchical operator, in Section 5 the results and discussions are presented, and finally in Section 6 the conclusions are presented. 2. - Literature review Researchers have turned to meta-heuristics [13] especially because they can have superior abilities than conventional optimization techniques in complex and non-linear problems and
these techniques have become very popular lately because of the following features: simplicity, flexibility, derivative free approaches, and because they have shown more ability to avoid local optima. For example some of the optimization techniques based on meta-heuristics that have been studied lately are: the Genetic Algorithm (GA) [2], Particle Swarm Optimization (PSO) [11], Ant Colony Optimization (ACO) [10] and Artificial Bee Colony (ABC) [3]. In addition to the huge amount of theoretical work, these optimization techniques have been used in various areas of application. Meta-heuristic techniques can be classified as follows:
Evolutionary (based on concepts of evolution in nature) [8]: Genetic Algorithms (GAs), Evolutionary Programming (EP) [24] and Genetic Programming (GP).
Based on Physics (imitate the physical rules) [6]: Big-Bang Big Crunch (BBBC), Gravitational Search Algorithm (GSA) [18] and Artificial Chemical Reactions Optimization Algorithm (ACROA).
Swarm Intelligence (Social Behavior of swarms, herds, flocks or schools of creatures in nature) [23]: Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO) and the Bat-inspired Algorithm (BA).
The NFL Theorem (No Free Lunch) [21] has logically proven that there is no meta-heuristic appropriately suited for solving all optimization problems. For example, a particular meta-heuristic can produce very promising results for a set of problems, but the same algorithm can show poor performance in a set of different problems. Therefore we can find a diversity of techniques and new meta-heuristics are born for solving optimization problems.
The formal definition of fuzzy sets was proposed by Zadeh in 1965 [25] and it is as follows: Let 𝑿 be a space of points (objects), with a generic element of 𝑿 denoted by 𝒙. Thus, 𝑿 = {𝒙}. A fuzzy set 𝑨 in 𝑿 is characterized by a membership (characteristic) 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒇𝑨 (𝒙) which associates with each point in 𝑿 a real number in the interval [0, 1], with the value of 𝒇𝑨 (𝒙) at 𝒙 representing the "degree of membership" of 𝒙 in 𝑨. Thus, the nearer the value of 𝒇𝑨 (𝒙) to unity, the higher the degree of membership of 𝒙 in 𝑨. When 𝑨 is a set in the ordinary sense of the term, its membership function can take on only two values 0 and 1, with 𝒇𝑨 (𝒙) = 1 or 0 according as 𝒙 does or does not belong to 𝑨 respectively. Thus, in this case 𝒇𝑨 (𝒙) reduces to the familiar characteristic function of 𝒂 in set 𝑨. (When there is a need to differentiate between such sets and fuzzy sets, the sets with two-valued characteristic functions will be referred to as ordinary sets or simply sets.) 3. - Grey Wolf Optimizer The Grey Wolf Optimizer algorithm (GWO) [15] is a meta-heuristic that was originated in 2014 created by Seyedali Mirjalili, and inspired basically because in the literature there was not a Swarm Intelligence (SI)[5] technique based on the hierarchy of leadership of the Grey Wolf. The hierarchy of leadership and the hunting mechanism of the grey wolf are illustrated in the following pyramid in Fig. 1. In addition we can mention that the social hierarchy of the grey wolf in the group of hunters has a very interesting social behavior. According to C. Muro [17] the main phases of the grey wolf hunting are:
Tracking, chasing, and approaching the prey.
Pursuing, encircling, and harassing the prey until it stops moving.
Attack towards the prey
In order to mathematically model the social hierarchy of wolves when designing the GWO, we consider the fittest solutions as the alpha (𝛼) wolves. Consequently, the second and third best solutions are called beta (𝛽) and delta (𝛿)wolves, respectively. The rest of the candidate solutions are assumed to be omega (𝜔) wolves. In the GWO algorithm the hunting (optimization) is guided by 𝛼, 𝛽, and 𝛿. The 𝜔 wolves follow these three wolves. In the next example we find in more detail how the three best wolves and their positions are obtained. Suppose that we have a minimization problem (f1 of Table 4) with 3 dimensions, that we have 5 wolves and the minimum of the objective function is zero, then in Table 1 we show the corresponding results.
So based on the fitness values of Table 1 we have the following locations for:
alpha = (-3.9697, -3.0645, 1.9932)
beta = (6.1873, -0.0954,4.0133)
delta = (5.5931, 0.1255, -7.4112)
As mentioned above, grey wolves encircle prey during the hunting process. In order to mathematically model the encircling behavior the following equations are proposed: D = ‖ C ∙ Xp (t)- X(t)‖
(1)
X(t+1) =Xp (t)– AD
(2)
Where t indicates the current iteration, A and C are coefficients, 𝐗 𝐩 is the position vector of the prey, and 𝐗 indicates the position vector of a grey wolf. The A and C coefficients are calculated as follows: A = 2a ∙ r1 –a
(3)
C = 2 ∙ r2
(4)
Where the components of 𝒂 are linearly decreased from 2 to 0 over the course of the iterations and 𝑟1, 𝑟2 are random numbers in [0, 1]. In nature, grey wolves have the ability to recognize the location of prey and encircle them. The hunt is usually guided by the alpha wolf. In order to mathematically simulate the hunting behavior of the grey wolves, we assume that the alpha (best candidate solution), beta and delta have better knowledge about the potential location of prey. Therefore, we have to save the first three best solutions obtained so far, and force the other search agents (including the omegas) to update their positions according to the position of the best search agents. The following Equations have been proposed in this regard. Dα =‖C1 ∙ Xα -X‖,Dβ =‖C2 ∙ Xβ -X‖,
X1 =Xα - A1 ∙(Dα ),
X2 =Xβ - A2 ∙(Dβ ),
X(t+1) =
Dδ =‖C3 ∙ Xδ -X‖
(5)
X3 =Xδ - A3 ∙(Dδ ) (6)
X1 + X2 + X3 3
(7)
We can note that Equations 5 and 6 are the same as Equations 1 and 2 respectively, but in this case (Equations 5 and 6) we now have the positions of the alpha, beta and delta wolves to update the positions of the omega wolves and X1, X2 and X3 in Equation 7 and this represents the results obtained for updating the omega wolves based on alpha (X1), beta (X2) and delta(X3) respectively. Fig. 2 illustrates in a graphical way the process of updating the positions based on the best three wolves, where the blue, pink and green circles, represent the alpha, beta and delta wolves respectively, and the red circle represents an omega wolf that should move where the leaders of the pack (center), then it is assumed that the next estimated position should be the position represented by the yellow circle. As mentioned above the grey wolves finish the hunt by attacking the prey when it stops moving. In order to mathematically model the process of approaching the prey, we sequentially decrease the value of “a”. Note that the fluctuation range of A is also decreased by “a”. When random values of A are in [-1, 1], the next position of a search agent can be in any position between its current position and the position of the prey. Fig. 3(a) shows that |A| < 1 forces the wolves to attack towards the prey.
a) If |A| < 1 then attacking prey (exploitation) b) If |A| > 1 then searching for prey (exploration) Grey wolves mostly search according to the position of the alpha, beta, and delta wolves. They diverge from each other to search for prey and converge to attack the prey. In order to mathematically model divergence, we utilize A with random values greater than 1 or less than -1
to force the search agent to diverge from the prey. This emphasizes exploration and allows the GWO algorithm to search globally. Fig. 3(b) also shows that |A| > 1 forces the grey wolves to diverge from the prey to hopefully find a fitter prey. Fig. 4 shows the corresponding pseudocode of the algorithm. 4. - Hierarchical Operator The main inspiration of the GWO algorithm is based mainly on the social behavior of the grey wolf, its hierarchical pyramid and the corresponding hunting mechanism. The hierarchy pyramid was presented in Figure 1. Where we mentioned that the dominant wolf is α followed by β, and δ, which are the following leaders in the pyramid respectively, and the ω wolves are search agents in the algorithm. Below we present the five variants in the hierarchical operator that we mentioned in the introduction of this paper, the first is called centroid and this operator was proposed by the author of the original paper. The second and third variants are the weighted average and the weighted based on fitness respectively, which are proposed in this paper and are two different ways to assign a weight to the leaders of the pack based on the hierarchical pyramid. Finally, we explain in detail the last two variants that use fuzzy logic for dynamic adaptation of the weights that the leading wolves should have throughout the iterations.
In addition we can mention that a brief justification of adding these operators in the algorithm is because in the weighted average and weighted based on the fitness we are only adding a product for each best result (alpha, beta and delta), so that the computational complexity is not affected
and the execution time is minimum, is almost imperceptible, and in the case of using fuzzy logic is well known that the execution time is more because more operations are performed, but the complexity does not increase significantly because fuzzy logic only requires first order operations, such as maximum and minimum, addition, subtraction and products. Centroid This is the first operator that was implemented in the original paper and as presented in Equation 7 is simply the average of the estimated positions of the three best search agents, in this case alpha, beta and delta. 𝐗(𝑡 + 1) =
𝑿1 + 𝑿2 + 𝑿3
(7)
3
Weighted average The main idea for the weighted average is to assign a numerical value to each of the three wolf leaders in order to represent the hierarchy of the pyramid, and for this example, α has a weighting of 50%, β with 30% for the weighting and finally δ with a weight of 20%. This weighted average is reflected in Equation 8. 𝐗(𝑡 + 1) =
𝟓 𝑿1 + 3 𝑿2 +2 𝑿3 10
(8)
Where we can notice that in this case the numbers are fixed, but could be a normalized weight between zero and one and we can also have a weighting with bigger numbers, for example between 0 and 100. In this case this is a static weighting and can treated in different ways, values and ranges depending on the characteristics of the problem to be solved. For example in this
work we have that the alpha wolf has a weighting of 5, beta and delta 3 and 2 respectively, then the sum of the weights equals 10. Weighted based on the fitness When we say based on the fitness, immediately we think on the fitness that the best individual has for the problem to be solved, so the following equations are proposed for a weighting based on the fitness of the α , β, and δ wolves. 𝑾𝛼 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 ) 𝑆𝛼
𝑾𝛽 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 )
𝐗(𝑡 + 1) =
𝑆𝛽
𝑾𝛿 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 ) 𝑆𝛿
𝑿1 ∗ 𝑾𝛼 + 𝑿2 ∗𝑾𝛽 + 𝑿3 ∗ 𝑾𝛿 (𝑾𝛼 + 𝑾𝛽 +𝑾𝛿 )
(9)
(10)
Where: 𝑾𝛼 , 𝑾𝛽 ,𝑾𝛿 , represent the weight of alpha, beta and delta wolves respectively 𝑺𝛼 , 𝑺𝛽 , 𝑺𝛿 , express the fitness of the alpha, beta and delta individual respectively Equations 9 and 10 are the equations responsible for mathematically representing the weights assigned to each of the three leader wolves based on their fitness to solve the problem, thus having the alpha wolf with more weight in the hunting process, then having the second leading beta and ending with the lowest weight for delta, which in theory is the one that has less knowledge about the location of potential prey (optimization). Fuzzy weights
In addition it is possible to implement a dynamic pyramid in the algorithm for adjusting the weights for the three best wolves. We can define the equation that represents the mathematical model of the dynamic pyramid as follows:
𝐗(𝑡 + 1) =
𝑿1 ∗ 𝑭𝑾𝛼 + 𝑿2 ∗ 𝑭𝑾𝛽 + 𝑿3 ∗ 𝑭𝑾𝛿 𝑭𝑾𝛼 + 𝑭𝑾𝛽 + 𝑭𝑾𝛿
(11)
Where: 𝑭𝑾𝛼 , 𝑭𝑾𝛽 , 𝑭𝑾𝛿 represent the dynamic weights of the alpha, beta and delta wolves respectively. It is important to remember that when we say dynamic weights we refer to the idea that these weights are not fixed parameters and that are changing in each iteration of the algorithm as mentioned above. Figure 5 shows the general structure of the fuzzy system, designed exclusively for the fuzzy hierarchical pyramid with one input, which are the iterations of the algorithm, and three outputs, which correspond to the fuzzy weights of alpha, beta and delta respectively. The fuzzy system is of Mamdani type and has a defuzzification by the centroid method [4][7][14][19][20].
Below we provide more details of the input and outputs in the fuzzy system for the implementation of the fuzzy hierarchical pyramid.
Figure 6 shows the input to the fuzzy system for the hierarchical pyramid, the input is the number of total iterations in the algorithm, and we can notice that it has three triangular
membership functions labeled as low, medium and high respectively, and has a normalized range between 0 and 1. In contrast, Figure 7 shows an example of the three outputs of the fuzzy system, which is the weight that each of the three wolves will have to guide them during the hunting process, and the output contains three triangular membership functions classified into low, medium, and high respectively. The range of each output is between 0 and 100 in order to represent the percentage values, so low is near 0%, medium is near 50% of the weight and high is the most weight during the hunt that is close to 100%, and this is the justification for using the range between 0 and 100. Figure 7 shows the output for the alpha wolf, and it is important to say that the outputs for the beta and delta wolves, the structure and characteristics of the output are represented in the same way as shown in Figure 7.
When we think of the rules for the fuzzy system that will represent the hierarchical pyramid, the main goal is to exploit the ability of fuzzy logic for adjusting the dynamic parameters, so we have two options for the behavior of the pyramid that we are presenting below. The first option is to assume that at the beginning of the hunt, the three best wolves have the same weight to guide the pack to the prey, in the second step only the alpha and beta guide the pack because they are those who have better knowledge about the position of the prey and finally at the end of the hunt, the decisions and responsibilities in the pack are guided only by the alpha wolf. Thus the corresponding pyramid can be as illustrated in Figure 8. Figure 8 shows that in the initial iterations the weight for the three wolves in the hunt and the guide are equal, when the algorithm has achieved half of the iterations, the weight in the guide is
to alpha and delta and when the algorithm is in the final iterations and is starting the phase of exploitation, the alpha wolf has the best knowledge of the prey therefore and this is the wolf who completely dominates the hunt. The pyramid is called in “increase” because alpha increases the weight when passing iterations.
The second option is the opposite to the first philosophy of hunting, where the main goal is that the alpha wolf starts with the full weight of the hunting process respecting its hierarchy and making decisions to start the hunt and lead the pack, in the second step or in the middle of the hunt the beta wolf helps in the hunt and in the encircling process of the prey, and finally in the end of the hunt the three best wolfs work equally to attack and to end the hunt.
Figure 9 shows that at the beginning of the iterations the alpha wolf is the only one who guides the hunting because this wolf has the best knowledge of the possible position of the prey, in the middle of the iterations beta helps to lead the pack, and in the end of the iterations when the algorithm is focused on exploiting, the best option is to use the average of the best three wolves that are closest to the objective. The pyramid is presented in a decrease fashion because alpha loses weight in the hunting process along the iterations. Figure 10 shows the comparison of the pyramids and its direct relationship with the iterations in the algorithm.
Tables 2 and 3 show the rules used for each of the fuzzy pyramids that were presented above.
5. - Results and Discussion For this work specifically we evaluate the performance of each methods with 23 benchmark functions [9][16][22] with which the original algorithm was also tested. These benchmark functions are classified as unimodal, multimodal and fixed-dimension multimodal and are functions that have been already tested in other meta-heuristics, especially to test new algorithms and verify that they have a good performance. The functions are described below in Tables 4, 5 and 6 respectively, where “Range” represents the boundary of the function´s search space and fmin is the optimal value. We also present plots of the 2-D versions of the benchmark functions in Figures 11, 12 and 13 with the goal of analyzing the form and space of search of the functions that we test with the GWO algorithm. The main justification to test the algorithm with these benchmark functions is because there exists a great variety of meta-heuristics that use these benchmark functions to validate their performance and mainly because is important test the proposed methods in this work with the same conditions that the author describes in the original paper. Although the original author only tested functions with 30 dimensions (unimodal and multimodal functions), in this work we present a comparison with 30 and 64 dimensions and a hypothesis test with 128 dimensions because it is possible to increase the number of dimensions and for the fixed-dimension multimodal functions it is possible to test, develop and analyze with the number of dimensions that are shown in Table 6.
Table 4 and Figure 11 show the equations with their characteristics and the 2-D versions of the unimodal benchmark functions respectively, and these benchmark functions are evaluated in 30, 64 and 128 dimensions.
In Table 5 and Figure 12 we can find the equations with their characteristics and the 2-D versions of the multimodal benchmark functions respectively, and these benchmark functions are evaluated in 30, 64 and 128 dimensions.
Table 6 and Figure 13 show the equations with their characteristics and the 2-D versions of the fixed-dimension multimodal benchmark functions respectively and these benchmark functions are evaluated only in the dimensions that Table 6 indicates.
It is important to say that in the all experiments we use 30 wolves as the total population and a maximum of 500 iterations in the algorithm that corresponds to 15,000 function evaluations. In Table 7 we present a brief comparison among the first three variants proposed in this paper, and this table shows the averages and standard deviations of the first 13 benchmark functions with 30 dimensions.
The main goal of Table 7 is to show the results obtained in the 30 independent executions that were performed for the first 13 benchmark functions described above with the aim of testing each variant described in the previous section and show the performance that the algorithm
achieves when using the proposed modification of the hierarchical pyramid. We can note that there is a significant difference between the methods although for this simulation the idea is to show the results of the algorithm when we implement the new variants and analyze the performance of the GWO algorithm.
In Table 8 we can find a comparison among the fuzzy methods analyzed in this paper and the original method, and this table shows the averages and standard deviations of the first 13 benchmark functions, which are unimodal and multimodal respectively with 30 dimensions.
It is important to mention that Table 8 shows the results of the variant that includes fuzzy logic for dynamic adaptation of the weights in the hierarchical pyramid as mentioned before, in increase and decrease fashion respectively, and that we can identify this in the table with the labels “Fuzzy D” for a decrease fashion and “Fuzzy I” for an increasing fashion. Table 9 presents a comparison of the methods: centroid, fitness, weighted for unimodal and multimodal benchmark functions, but in this case the benchmark functions are with 64 dimensions because we want to analyze the performance of the algorithm when the problems are more complex and analyze the performance of the proposed variants presented in this paper.
Tables 9 and 10 also show the averages and standard deviations obtained by each method with the goal of showing the results and the accuracy of the different variants. In Table 10 we can find the results when the weights are implemented with fuzzy logic for the case of 64 Dimensions.
In this paper we are also presenting the performance of the grey wolf optimizer algorithm for more complex problems, and in the following tables we present the first 13 benchmark functions with a 128 dimensions. Below in the following tables we are also presenting a hypothesis test comparing each new method with the original method. In this case each table shows the averages and standard deviations obtained in 100independent runs of each method and finally the value of z obtained in hypothesis testing [12]. The goal is to statistically demonstrate if the new methods obtained better results than the original method. We considered the Z-test because for realizing this test we need a minimum 30 samples and we have 100 samples. So we performed this test with a 95% of confidence and 5% of significance (α = 0.05) and according to the tables of critical values of the Z-test with α = 0.05, and the range of rejecting the null hypothesis is in Z0 = -1.645. The parameters, the claim and data for the hypothesis testing are as follows:
𝜇1 = 𝑁𝑒𝑤 𝑀𝑒𝑡ℎ𝑜𝑑
𝜇2 = 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑀𝑒𝑡ℎ𝑜𝑑
The mean of the new method is less than mean of the centroid method (claim).
𝐻0 : 𝜇1 ≥ 𝜇2
𝐻𝑎 : 𝜇1 < 𝜇2 (𝐶𝑙𝑎𝑖𝑚)
𝛼 = 0.05
sample size (n) = 100
𝑍0 = −1.645
In addition we need the value of Z for realizing the comparison with Z0 and conclude if there exists enough evidence to say that the new method is better than the original one.
The value of Z is obtained by the following equations.
𝜎𝑥̅1− 𝑥̅2 ≈ √
𝑍 ≈
(𝑥̅1 − 𝑥̅ 2 ) 𝜎𝑥̅ − 𝑥̅ 1 2
𝑠12 𝑛1
+
𝑠22 𝑛2
(12)
(13)
Where: 𝑠1 , 𝑠2 : are the standard deviations of the two methods respectively. 𝑛1 , 𝑛2 : are the sizes of the sample of the methods, respectively. 𝑥̅1 , 𝑥̅2 : are the averages obtained for each method respectively.
Figure 14 shows the rejection zone of the null hypothesis (H0), so if the value of Z is less than Z0 or -1.645 we can reject the null hypothesis and we can state that the mean of the new method is lower than mean of the centroid method with a 95% of confidence and this value of Z is shown in the following tables to conclude if the variants in the weight of the leaders are better and have a better performance than the original method.
In Table 11 we can find the results of the hypothesis test between the original method (Centroid) and the first proposed method in this paper that is the weighted method. Then according to the hypothesis testing explained above the proposed method has better performance in 7 of 13 benchmark functions considered in this paper because according of the hypothesis test (Z-test) the Z values are lower than Z0 and this means that while the values of Z are lower than -1.645
there exist enough evidence to conclude that the proposed method is better than the original one. So if we analyze Table 11 we can find that for the F1, F2, F3, F5, F6, F9 and F10 functions the values of Z are lower that de Z0 (-1.645). In addition is important to say that while the Z value is farther to the value of Z0 (-1.645) we have more evidence to conclude that the new method has better performance. For example in Function 1 the value of Z is -12.6917, so it is in the rejection zone (according to hypothesis test) and we can say that for this specific benchmark function the new method is better than the original one and this is the comparison that we performed with each analyzed function.
Table 12 shows an analysis of a hypothesis test between the centroid method and the second method analyzed in this paper, which was the fitness method and we can conclude that the proposed method is better than the original method in 4 of the first 13 analyzed benchmark functions.
In Tables 13 and 14 we can find the results of hypothesis testing between the original method (Centroid) and the two fuzzy methods described in this paper, that is the fuzzy pyramid in decrease and the fuzzy pyramid in increase respectively, and we can note that in both methods the performance of the algorithm is better in 12 of the 13 analyzed benchmark functions. In other words, both methods show better performance in all the unimodal benchmark functions and for
the multimodal benchmark functions the original method is better than the fuzzy method only in function number 12.
Finally in the following tables we are presenting a comparison between the original method (centroid) and the new methods described above. In addition we also present the next group the benchmark functions analyzed in this paper, which are the fixed-dimension multimodal benchmark functions and their characteristics were previously described. It is important to say that the tables show the average and the standard deviations of 100 independent runs for each function. Table 15 shows a comparison among the original method (centroid) and the first two methods presented in this paper, which are the fitness method and weighted method, for the fixeddimension multimodal benchmark functions that we presented at the beginning of this section.
In Table 16 we can find the averages and the standard deviations of the original method (centroid) and the fuzzy methods that we presented in this paper. We also present a comparison between these methods for the fixed-dimension multimodal benchmark functions.
It is important to mention that we consider that there is not enough difference between these methods because some results are very similar and this is the reason that we do not present a hypothesis test for these benchmark functions because there is not enough evidence to perform it. But we present a brief comparison based on the average and the standard deviation that we have obtained in each method.
Processing Time for each operator In order to clarify the impact that each operator has in the algorithm with respect to the execution time, we are presenting Tables 17, 18 and 19, where we can find the averages and standard deviations of processing times for each analyzed operator. These averages and standard deviations are the results to 100 independent executions for each operator in the first 13 benchmark functions that we are presenting in this paper with 30, 64 and 128 dimensions respectively.
In Table 17 we can note that the better method regarding computing time for the case of 30 dimensions is the centroid method and the results are very similar to the Weighted method, and the worst results are with the Fitness method and the Fuzzy logic methods produce similar results and time is approximately twice when compared with the Centroid method. In addition it is important to mention that these experiments were executed in an Acer Aspire M3970 computer with Intel Core i5-2300 CPU @ 2.80 GHz processor and 6144 MB of RAM memory.
Table 18 shows the results with 64 dimensions and the results have very similar performance with 30 dimensions, so the best methods in regards to computing time are the Weighted and Centroid methods respectively and the worst is the Fitness method, so the Fuzzy Logic method did maintained in the middle of the methods with respect to the required time.
Table 19 shows the performance in time for 128 dimensions for each analyzed method in this paper. In addition we show in the Figure 15 a plot, where we can find the average times that each method requires to evaluate all benchmark functions (first 13 functions) that we evaluated to analyze the time required for each method.
We can conclude based on Table 19 and Figure 15 that for complex problems, the performance of the Fuzzy Logic operators is improving because the difference in the percentage of the time between the Centroid method an Fuzzy Logic methods gets increasingly lower when the problem becomes more complex. It is expected that with a number of dimensions high enough the fuzzy logic method would require less time than the other methods.
Finally, we can conclude that although the execution time is more with the Fuzzy logic methods, this difference in time is very small and does not increase in an excessive way, so this option of implementing the fuzzy operators is viable for improving the results of the algorithm. 6. –Conclusion In this work we have analyzed the grey wolf optimizer algorithm that is considered a relatively new meta-heuristic because it was proposed in 2014, so the algorithm has an area of opportunity for adding new characteristics, like the author of the method mentioned in the original paper. In this paper we presented a new operator that affects directly one of the original inspirations of this algorithm and this characteristic is the hierarchical pyramid of the wolf. If we remember the two main characteristics in this algorithm are the social behavior and the hierarchical pyramid. The main contribution of this paper consists of the three proposed methods to implement the hierarchical pyramid that directly affect the new position of the omega wolves based on the best three wolves (alpha, beta and delta). We focus on the hierarchical pyramid and we are presenting three different options to implement the pyramid, which are the weighted average, based on the
fitness, and based on fuzzy logic, that were described in this paper with a mathematical model and a fuzzy system. According to the presented results we can conclude that in the methods based on mathematical models, each method has better performance depending of the problem to solve, but we can prove with a hypothesis test that when we use fuzzy logic in the algorithm, we can note a better performance than with the mathematical models in a majority of the analyzed benchmark functions. In this work we justify the problems that were used for testing the new operators, although it would be important as future work to test these new operators with other types of problems or applications, such as in Encryption [1], simulators and plants and testing the performance of new proposed methods in other applications.
References [1] F. Aladwan, M. Alshraideh and M. Rasol. “A Genetic Algorithm Approach for Breaking of Simplified Data Encryption Standard”, International Journal of Security and Its Applications 9(9), 2015, pp. 295-304. [2] M. Alshraideh. “A complete Automation of Unit Testing for JavaScript Programs” Journal of Computer Science 4(12), 2008, pp. 1012-1019. [3] B. Basturk and D. Karaboga, “An artificial bee colony (ABC) algorithm for numeric function optimization”, IEEE, swarm intelligence symposium, 2006, pp. 12-4. [4] J. Barraza, P. Melin and F. Valdez “Fuzzy FWA with dynamic adaptation of parameters”, IEEE CEC 2016, “accepted for publication” [5] G. Beni and J. Wang, “Swarm intelligence in cellular robotic systems”, Robots and biological systems: towards a new bionics?, Springer, 1993, pp. 703–12. [6] U. Can and B. Alatas, “Physics based metaheuristic algorithms for global optimization”, American Journal of Information Science and Computer Engineering, vol. 1, 2015, pp. 94-106. [7] O. Castillo, H. Neyoy, J. Soria, P. Melin and F. Valdez. A new approach for dynamic fuzzy logic parameter tuning in Ant Colony Optimization and its application in fuzzy control of a mobile robot, Applied Soft Computing 2015; 28; 150–159. [8] H. Dias de Mello Jr, L. Martí and M. Rebuzzi Vellasco, “Evolutionary algorithms and elliptical copulas applied to continuous optimization problems,” Information Sciences, vol. 369, pp. 419-440, 2016. [9]J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms”, Int J Comput Math, vol. 77, 2001, pp. 481–506. [10]M. Dorigo, M. Birattari and T. Stutzle, “Ant colony optimization”, IEEE, Comput Intell Magaz, 2006, pp. 2839. [11] J. Kennedy and R. Eberhart, “Particle swarm optimization, in Neural Networks”, IEEE international conference, 1995, pp. 1942-1948. [12]R. Larson and B. Farber, (2003), Elementary Statistics Picturing the World, Pearson Education Inc. (428-433). [13]Melián B., Moreno J., Metaheurísticas: una visión global, Revista Iberoamericana de Inteligencia Artificial 2003; 19; 7-28. [14] P. Melin, F. Olivas, O. Castillo, F. Valdez, J. Soria and M. Valdez, Optimal design of fuzzy classification systems using PSO with dynamic parameter adaptation through fuzzy logic, Expert Systems with Applications 2013; 40; 3196–3206. [15] S. Mirjalili, M. Mirjalili and A. Lewis, “Grey Wolf Optimizer”, Advances in Engineering Software, vol. 69, 2014, pp. 46-61. [16] M. Molga and C. Smutnicki, “Test functions for optimization needs”, unpublished. [17] C. Muro, R. Escobedo, L. Spector and R. Coppinger, “Wolf-pack (Canis lupus) hunting strategies emerge from simple rules”, Computational simulations, Behav Process, vol. 88, 2011, pp. 192–7.
[18]E. Reshedi, H. Nezamabadi-Pour and S. Saryazdi, “GSA: a gravitational search algorithm”, Inf Sci 2009;179:2232-48. [19] L. Rodriguez, O. Castillo and J. Soria, “Grey Wolf Optimizer (GWO) with dynamic adaptation of parameters using fuzzy logic”, IEEE CEC 2016, “accepted for publication”. [20] Salah, Bareqa et al. “Skin Cancer Recognition by Using a Neuro-Fuzzy System.” Cancer Informatics 10 (2011): 1–11. PMC.Web. 6 Dec. 2016. [21]DH. Wolpert, and WG. Macready, “No free lunch theorems for optimization”, Evolut Comput, IEEE Trans, 1997, pp. 67–82. [22] X-S. Yang, “Test problems in optimization”, arXiv, preprint arXiv:1008.0549, 2010 [23] X. Yang and M. Karamanoglu, “Swarm intelligence and bio-inspired computation: an overview”, Swarm Intelligence and Bio-Inspired Computation, 2013, pp. 3-23. [24]X. Yao, Y. Liu and G. Lin, “Evolutionary programming made faster” Evolut Comput, IEEE Trans, vol. 3, 1999, pp. 82–102. . [25]L. Zadeh, “Fuzzy sets,” Information and control, No. 8, pp. 338-353, 1965.
Fig.1Hierarchy pyramid
Fig.2 Updating wolf positions in the algorithm
Fig.3Attacking prey versus searching for prey
Initialize the grey wolf population Xi(i = 1, 2, …, n) Initialize a, A and C Calculate the fitness of each search agent Xα = the best search agent Xβ = the second best agent Xδ = the third best search agent while (t < Max number of iterations) for each search agent Update the position of the current search agent by Equation (7) end for Update a, A and C Calculate the fitness of all search agents Update Xα , Xβ and Xδ t = t+ 1 endwhile return Xα Fig.4 Pseudocode of the algorithm
Fig. 5 General structure of the fuzzy system
Membership Grades
1
Low
Medium
High
0.5
0
0
0.1
0.2
0.3
0.4
0.5 0.6 Iterations
0.7
0.8
0.9
1
Membership Grades
Fig. 6 Input of the FIS for the fuzzy pyramid
1 Low
Medium
High
0.5
0
0
10
20
30
40
50 Alpha
Fig. 7 Output of the FIS for the fuzzy pyramid
60
70
80
90
100
Fig. 8 Pyramid in increase
Fig. 9 Pyramid in decrement
Fig. 10 Comparison of the two pyramids
Fig. 11 2-D versions of the unimodal benchmark functions
Fig. 12 2-D versions of the multimodal benchmark functions
Fig. 13 2-D versions of the fixed-dimension multimodal benchmark functions
Fig. 14 Reject zone of z-test with α = 0.05
4
Fitness
Time in Seconds
3.5
3
2.5
2
Fuzzy I 1.5
Fuzzy D Centroid 1
Weighted Hierarchical Operators
Fig. 15 Comparison in processing time for each operator with 128 dimensions
Table1Example of the leaders obtained in the GWO algorithm
d1 -5.5390 -3.9667 -8.3744 6.1873 5.5931
d2 -9.5147 -3.0645 -3.2446 -0.9254 0.1255
d3 -6.9951 1.9932 6.9530 4.0133 -7.4112
fitness 170.1424 29.0986 129.0029 55.2451 86.2248
Table 2 Rules in increase of the fuzzy pyramid
Rules in increase 1.- if(iterations is low) then (alpha is medium) (beta is medium) (delta is medium) 2.- if(iterations is medium) then (alpha is medium) (beta is medium) (delta is low) 3.- if(iterations is high) then (alpha is high) (beta is medium) (delta is low)
Table 3 Rules in decrease of the fuzzy pyramid
Rules in decrease 1.- if(iterations is low) then (alpha is high) (beta is medium) (delta is low) 2.- if(iterations is medium) then (alpha is medium) (beta is medium) (delta is low) 3.- if(iterations is high) then (alpha is medium) (beta is medium) (delta is medium)
Table 4 Unimodal Benchmark functions
Function 𝑛
𝑓1 (𝑥) = ∑ 𝑥𝑖 2
Range [-100, 100]
fmin 0
[-10, 10]
0
[-100, 100]
0
[-100, 100]
0
[-30, 30]
0
[-100, 100]
0
[-1.28, 1.28]
0
𝑖=1 𝑛
𝑛
𝑓2 = ∑|𝑥1 | + ∏|𝑥1 | 𝑖=1
𝑖=1 𝑛
2
𝑖
𝑓3 = ∑ (∑ 𝑥𝑗 ) 𝑖=1
𝑗−1
𝑓4 = 𝑚𝑎𝑥𝑖 {|𝑥1 |, 1 ≤ 𝑖 ≤ 𝑛 } 𝑛−1 2 2
2
𝑓5 (𝑥) = ∑[100(𝑥𝑖+1 − 𝑥𝑖 ) + (𝑥1 − 1) ] 𝑖=1 𝑛
𝑓6 (𝑥) = ∑([𝑥-1 + 0.5])2 𝑖=1
𝑛
𝑓7 (𝑥) = ∑ 𝑖𝑥𝑖 4 + 𝑟𝑎𝑛𝑑𝑜𝑚[0,1] 𝑖=1
Table 5 Multimodal benchmark functions
Function 𝑛
𝑓8 (𝑥) = ∑ −𝑥𝑖 sin (√|𝑥𝑖 |)
Range [-500, 500]
fmin -2,094.91
[-5.12, 5.12]
0
[-32, 32]
0
[-600, 600]
0
[-50, 50]
0
[-50, 50]
0
𝑖=1 𝑛
𝑓9 (𝑥) = ∑[𝑥𝑖 2 − 10 cos(2𝜋𝑥𝑖 ) + 10] 𝑖=1 𝑛
𝑛
1 1 𝑓10 (𝑥) = 20 𝑒𝑥𝑝 (−0.2√ ∑ 𝑥𝑖 2 ) − 𝑒𝑥𝑝 ( ∑ cos(2𝜋𝑥𝑖 )) + 20 + 𝑒 𝑛 𝑛 𝑖=1
𝑓11 (𝑥) =
𝑓12 (𝑥) =
𝑛
𝑛
𝑖=1
𝑖=1
𝑖=1
1 𝑥𝑖 ∑ 𝑥1 2 − ∏ cos ( ) + 1 4000 √𝑖
𝜋 {10 sin(𝜋𝑦1 ) 𝑛 𝑛−1
+ ∑(𝑦𝑖 − 1)2 [1 + 10 sin2 (𝜋𝑦𝑦+1 )] + (𝑦𝑛 − 1)2 } 𝑖=1 𝑛
+ ∑ 𝑢(𝑥𝑖 , 10, 100, 4) 𝑖=1
𝑥𝑖 + 1 4 𝑘(𝑥𝑖 − 𝑎)𝑚 , 𝑥𝑖 > 𝑎 − 𝑎 < 𝑥𝑖 < 𝑎 𝑢(𝑥1 , 𝑎, 𝑘, 𝑚) = { 0 , 𝑘(−𝑥𝑖 − 𝑎)𝑚 , 𝑥𝑖 < −𝑎 𝑦1 = 1 +
𝑓13 (𝑥) = 0.1 {sin2(3𝜋𝑥𝑖 ) 𝑛
+ ∑(𝑥1 − 1)2 [ 1 + sin2 (3𝜋𝑥𝑖 + 1)] 𝑖=1
+ (𝑥𝑛 − 1)2 [1 + sin2 (2𝜋𝑥𝑛 )]} 𝑛
+ ∑ 𝑢(𝑥𝑖 , 5, 100, 4) 𝑖=1
Table 6 Fixed-dimension multimodal benchmark functions
Function
Dim 2
Range [-65, 65]
fmin 1
𝑥1 (𝑏12 + 𝑏𝑖 𝑥2 ) 𝑓15 (𝑥) = ∑ [𝑎𝑖 − 2 ] 𝑏1 + 𝑏𝑖 𝑥3 + 𝑥4
4
[-5, 5]
0.00030
1 𝑓16 (𝑥) = 4𝑥12 − 2.1𝑥14 + 𝑥16 + 𝑥1 𝑥2 − 4𝑥22 + 4𝑥24 3
2
[-5, 5]
-1.0316
2
[-5, 5]
0.398
2
[-2, 2]
3
3
[1, 3]
-3.86
6
[0, 1]
-3.32
4
[0, 10]
-10.1532
4
[0, 10]
-10.4028
4
[0, 10]
-10.5363
−1
25
1 1 𝑓14 (𝑥) = ( + ∑ 6) 2 500 ∑ 𝑗 + (𝑥 − 𝑎 ) 𝑖𝑗 𝑖=2 𝑖 𝑗=1 2
11
𝑖=1
𝑓17 (𝑥) = (𝑥2 −
2 5.1 2 5 1 𝑥 + 𝑥 − 6 ) + 10 (1 − ) cos 𝑥1 4𝜋 2 1 𝜋 1 8𝜋 + 10
𝑓18 (𝑥) = [1 + (𝑥1 + 𝑥2 + 1)2 (19 − 14𝑥1 + 3𝑥12 − 14𝑥2 + 6𝑥1 𝑥2 + 3𝑥22 )] × [30 + (2𝑥1 − 3𝑥2 )2 × (18 − 32𝑥1 + 12𝑥12 + 48𝑥2 − 36𝑥1 𝑥2 + 27𝑥22 )] 3
4
2
𝑓19 (𝑥) = − ∑ 𝑐1 𝑒𝑥𝑝 (− ∑ 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ) 𝑖=1
𝑗=1
4
6
2
𝑓20 (𝑥) = − ∑ 𝑐1 𝑒𝑥𝑝 (− ∑ 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ) 𝑖=1
𝑗=1
5
𝑓21 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1 𝑖=1 7
𝑓22 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖
)𝑇
+ 𝑐𝑖
]−1
𝑖=1 10
𝑓23 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1 𝑖=1
Table 7 Comparison among the methods: centroid, fitness and weighted with unimodal and multimodal benchmark functions
30 Dimensions Function Centroid F1 6.59E-28 F2 7.18E-17 F3 3.29E-06 F4 5.61E-07 F5 26.8126 F6 0.8166 F7 0.0022 F8 -6123.10 F9 0.3105 F10 1.06E-13 F11 0.0045 F12 0.0534 F13 0.6545
STD 6.34E-05 0.0290 79.1496 1.3151 69.9050 1.26E-04 0.1003 4087.44 47.3561 0.0778 0.0067 0.0207 0.0045
Fitness 9.28E-11 3.55E-07 2806.72 4.9906 125.9725 14.74 0.0091 -19230.07 11.3785 9.03E-07 0.0046 0.4757 9.7919
STD 8.26E-11 1.15E-07 2320 3.8208 0.5694 1.1695 0.0035 3756.75 7.9794 2.94E-07 0.0063 0.0984 0.4707
Weighted 7.12E-31 8.07E-19 3.30E-06 1.66E-07 26.8580 0.7563 0.0019 -6006.70 2.1193 4.83E-14 0.0042 0.04897 0.6135
STD 2.40E-30 7.1511E-19 1.26E-05 4.00E-07 0.7177 0.3735 0.0010 790.5799 4.81229 7.76E-15 0.0082 0.0272 0.2601
Table 8 Comparison among the methods: centroid, fuzzy in decrease and fuzzy in increase with unimodal and multimodal benchmark functions
30 Dimensions Function Centroid F1 6.59E-28 F2 7.18E-17 F3 3.29E-06 F4 5.61E-07 F5 26.8126 F6 0.8166 F7 0.0022 F8 -6123.10 F9 0.3105 F10 1.06E-13 F11 0.0045 F12 0.0534 F13 0.6545
STD 6.34E-05 0.0290 79.1496 1.3151 69.9050 1.26E-04 0.1003 4087.44 47.3561 0.0778 0.0067 0.0207 0.0045
Fuzzy D. 4.69E-30 1.50E-18 0.5337 7.45E-08 51.4764 5.7266 0.0017 -5160.88 0.4618 2.75E-14 0.0016 0.2800 3.4073
STD 1.43E-29 3.18E-18 3.96 1.90E-07 16.0976 3.2391 0.0010 1008.48 2.8904 2.22E-14 0.0055 0.1682 1.7817
Fuzzy I. 9.47E-32 1.73E-19 0.5337 4.81E-08 51.4594 5.7212 0.0017 -5011.22 0.9691 2.38E-14 7.29E-04 0.2795 3.3789
STD 4.74E-31 3.50E-19 3.9604 1.05E-07 16.1231 3.2469 9.21E-04 869.24 2.9832 1.60E-14 0.0030 0.1684 1.8247
Table 9 Comparison of the methods: centroid, fitness and weighted for 64 dimensions
64 Dimensions Function Centroid F1 7.78E-17 F2 1.18E-10 F3 9.29 F4 0.0080 F5 61.6661 F6 4.3925 F7 0.0047 F8 -11059.92 F9 5.7832 F10 9.83E-10 F11 0.0028 F12 0.1714 F13 3.3109
STD 8.00E-17 4.72E-11 17.92 0.0138 0.8593 0.7908 0.0022 1920.1685 5.78776 4.55E-10 0.0081 0.0753 0.3288
Fitness 2.94E-17 9.65E-11 9.9873 0.0068 61.6087 4.3178 0.0039 -10996.72 4.6677 8.05E-10 0.0036 0.27048 3.6451
STD 3.88E-17 5.37E-11 22.13 0.0126 0.8255 0.7304 0.0018 1844.1950 4.73 4.51E-10 0.0095 0.1170 0.4317
Weighted 5.69E-19 6.26E-12 5.33 0.0064 61.4798 4.4938 0.0041 -11122.48 4.3519 9.80E-11 0.0032 0.1647 3.3866
STD 7.79E-19 3.35E-12 11.35 0.0096 0.8794 0.6553 0.0019 1466.88 7.26821 5.76E-11 0.0070 0.0525 0.3535
Table 10 Comparison of the methods: centroid, fuzzy in decrease and fuzzy in increase for 64 dimensions
64 Dimensions Function Centroid F1 7.78E-17 F2 1.18E-10 F3 9.29 F4 0.0080 F5 61.6661 F6 4.3925 F7 0.0047 F8 -11059.92 F9 5.7832 F10 9.83E-10 F11 0.0028 F12 0.1714 F13 3.3109
STD 8.00E-17 4.72E-11 17.92 0.0138 0.8593 0.7908 0.0022 1920.1685 5.78776 4.55E-10 0.0081 0.0753 0.3288
Fuzzy D. 1.85E-18 5.64E-12 4.16 0.0174 61.8753 6.8445 0.0025 -6613.67 1.8943 9.42E-11 0.0010 0.3206 4.2253
STD 4.48E-18 9.80E-12 15.08 0.1391 0.7334 1.6324 0.0017 3272.5693 7.37467 1.74E-10 0.0048 0.1253 0.5845
Fuzzy I. 6.22E-20 1.38E-12 1.60 7.13E-04 61.8082 6.8298 0.0023 -6496.35 2.0865 1.87E-11 0.0019 0.3126 4.1995
STD 1.40E-19 2.49E-12 5.20 0.0028 0.7565 1.6476 0.0015 2992.15 5.32146 3.38E-11 0.0056 0.1300 0.6368
Table 11 Hypothesis test between the centroid method and the weighted method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06 F9 11.8031 F10 9.98E-07 F11 0.0056 F12 0.3791 F13 9.5433
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
Weighted 5.09E-12 6.80E-08 2382.68 6.4470 125.7715 14.7852 0.0088 -19040.08 9.2998 1.98E-07 0.0048 0.36557 9.6057
STD 4.28E-12 1.97E-08 1960 4.9446 0.7346 1.2106 0.0036 3768.8485 7.09064 8.00E-08 0.0120 0.0581 0.4982
Value of Z -12.6917 -29.1863 -1.6888 2.0907 -2.0909 -2.018 -0.3768 -0.9446 -2.4327 -24.6154 -0.4694 -1.505 0.8625
Table 12 Hypothesis test between the centroid method and the fitness method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92
STD 1.07E-10 1.85E-07 2610.00
Fitness 9.28E-11 3.55E-07 2806.72
STD 8.26E-11 1.15E-07 2320
Value of Z -3.5658 -11.7523 -0.3643
F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
5.1865 125.9696 15.1367 0.0090 -19558.06 11.8031 9.98E-07 0.0056 0.3791 9.5433
3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
4.9906 125.9725 14.7400 0.0091 -19230.07 11.3785 9.03E-07 0.0046 0.4757 9.7919
3.8208 0.5694 1.1695 0.0035 3756.75 7.9794 2.94E-07 0.0063 0.0984 0.4707
-0.3806 0.0351 -2.3151 0.1908 -0.5991 -0.3888 -2.2048 -0.7330 8.0532 3.5272
Table 13 Hypothesis test between the centroid method and the fuzzy in decrease method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06 F9 11.8031 F10 9.98E-07 F11 0.0056 F12 0.3791 F13 9.5433
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
Fuzzy D. 7.69E-12 4.86E-08 751.2593 2.0329 81.1047 10.1128 0.0040 -9105.10 3.2140 1.47E-07 0.0018 0.3851 6.1473
STD 1.50E-11 8.78E-08 1527.22 4.7895 29.4134 3.6583 0.0039 7300.89 6.9663 2.40E-07 0.0076 0.0838 2.4682
Value of Z -12.3382 -27.4639 -7.2178 -5.3427 -15.2501 -12.9926 -9.0655 -12.5687 -8.4167 -21.4893 -2.6594 0.5540 -13.4584
Table 14 Hypothesis test between the centroid method and the fuzzy in increase method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93
Fuzzy I. 6.55E-13 1.51E-08 673.8766 1.2525 81.1221 10.0137 0.0038 -8712.21
STD 1.17E-12 2.56E-08 1422.2729 2.5651 29.4403 3.5254 0.0039 6498.40
Value of Z -13.1156 -31.9068 -7.6035 -9.1511 -15.2302 -13.6933 -9.4281 -14.2299
F9 F10 F11 F12 F13
11.8031 9.98E-07 0.0056 0.3791 9.5433
7.4572 3.15E-07 0.0121 0.0686 0.5246
3.1345 3.96E-08 0.0018 0.3736 6.0642
5.9764 6.81E-08 0.0069 0.0834 2.3403
-9.0709 -29.7384 -2.7281 -0.5093 -14.5061
Table 15 Comparison between the methods: centroid, fitness and weighted with the fixed-dimension multimodal benchmark functions
Function F14 F15 F16 F17 F18 F19 F20 F21 F22 F23
Centroid 4.2582 0.0048 -1.0316 0.3979 5.4300 -3.8615 -3.2841 -9.3441 -10.3479 -10.4811
STD 4.1457 0.0094 2.54E-08 1.73E-04 13.8872 0.0025 0.0628 1.9578 0.5313 0.5361
Fitness 3.8876 0.0059 -1.0111 0.3979 5.4300 -3.8612 -3.2584 -9.1646 -10.1558 -10.3182
STD 3.9466 0.0089 0.0874 2.51E-06 13.8872 0.0026 0.0877 2.1462 1.2227 1.2560
Weighted 5.0584 0.0058 -1.0316 0.3979 3.0000 -3.8612 -3.2463 -9.2684 -10.1895 -9.9937
STD 4.4456 0.0100 2.49E-08 2.97E-06 3.40E-05 0.0028 0.0940 2.0688 1.0408 1.8828
Table 16 Comparison between the methods: centroid, fuzzy in decrease and fuzzy in increase with the fixeddimension multimodal benchmark functions
Function F14 F15 F16 F17 F18 F19 F20 F21 F22 F23
Centroid 4.2582 0.0048 -1.0316 0.3979 5.4300 -3.8615 -3.2841 -9.3441 -10.3479 -10.4811
STD 4.1457 0.0094 2.54E-08 1.73E-04 13.8872 0.0025 0.0628 1.9578 0.5313 0.5361
Fuzzy D. 5.8640 0.0039 -1.0316 0.4026 3.8100 -3.8615 -3.2666 -7.7454 -8.9120 -9.1294
STD 4.7449 0.0075 8.22E-06 0.0284 8.1002 0.0024 0.0691 2.3013 1.5826 1.3086
Fuzzy I. 6.1356 0.0038 -1.0316 0.4026 3.8100 -3.8619 -3.2553 -8.0997 -9.0410 -9.1564
STD 4.8359 0.0075 8.22E-06 0.0284 8.1002 0.0017 0.0672 2.1677 1.4157 1.3648
Table 17 Comparison in processing time for 30 Dimensions
30 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.2461 (0.0025) 0.2807 (0.0036) 1.1004 (0.0346) 0.3246 (0.0138) 0.3741 (0.0075) 0.3765 (0.0225) 0.4610 (0.0100) 0.4061 (0.0065) 0.3919 (0.0068) 0.4055 (0.0059) 0.4424 (0.0053) 0.7578 (0.0046) 0.7513 (0.0053) 0.4860
Fitness 0.2535 (0.0107) 0.2800 (0.0026) 1.1116 (0.0254) 0.3294 (0.0034) 3.3325 (0.0547) 3.1505 (0.0233) 3.2359 (0.0287) 0.3936 (0.0047) 0.3744 (0.0052) 0.3925 (0.0029) 0.4207 (0.0072) 0.7275 (0.0033) 0.7258 (0.0074) 1.1329
Weighted 0.2595 (0.0149) 0.2831 (0.0119) 1.1091 (0.0226) 0.3247 (0.0078) 0.3675 (0.0084) 0.3558 (0.0094) 0.4661 (0.0065) 0.4076 (0.0038) 0.3909 (0.0076) 0.412 (0.0077) 0.4446 (0.0083) 0.7675 (0.0117) 0.7630 (0.0092) 0.4886
Fuzzy I. 0.6482 (0.0220) 0.6802 (0.0195) 1.5040 (0.0126) 0.7014 (0.0137) 0.7583 (0.0103) 0.7484 (0.0076) 0.8537 (0.0078) 0.7995 (0.0087) 0.7793 (0.0138) 0.7645 (0.0270) 0.7866 (0.0082) 1.1054 (0.0057) 1.1019 (0.0086) 0.8640
Fuzzy D. 0.6381 (0.0399) 0.6647 (0.0084) 1.5006 (0.0161) 0.7043 (0.0147) 0.7585 (0.0118) 0.7510 (0.016) 0.8562 (0.0100) 0.8016 (0.0258) 0.7800 (0.0110) 0.7964 (0.0077) 0.8455 (0.0186) 1.182 (0.0181) 1.1696 (0.0264) 0.8807
Fuzzy I. 0.7272 (0.0222) 0.7506 (0.0051) 2.5758 (0.0863) 0.8174 (0.0209) 0.8864 (0.0396) 0.8718 (0.0443) 1.0318 (0.0037) 0.9228 (0.0302) 0.8906 (0.0125) 0.9033 (0.0046) 0.9482 (0.0061) 1.4519 (0.0267) 1.4341 (0.0051) 1.0932
Fuzzy D. 0.7274 (0.0200) 0.7505 (0.0040) 2.489 (0.0212) 0.7878 (0.0061) 0.8405 (0.0054) 0.8302 (0.0046) 1.0102 (0.0064) 0.9179 (0.0370) 0.8904 (0.0350) 0.8860 (0.0055) 0.9316 (0.0100) 1.4041 (0.0067) 1.3946 (0.0117) 1.0662
Table 18 Comparison in processing time for 64 Dimensions
64 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.4056 (0.0206) 0.4119 (0.017) 2.2367 (0.0728) 0.4583 (0.0274) 0.5200 (0.0199) 0.5069 (0.0225) 0.6927 (0.0124) 0.5501 (0.0114) 0.5288 (0.0047) 0.5466 (0.0094) 0.5789 (0.0042) 1.068 (0.0083) 1.0445 (0.0164) 0.7346
Fitness 0.3934 (0.0121) 0.4155 (0.0093) 2.2191 (0.0577) 0.4473 (0.0025) 6.4466 (0.0208) 6.3529 (0.1440) 6.4613 (0.0263) 0.5483 (0.0043) 0.5261 (0.0039) 0.5440 (0.0045) 0.5874 (0.0085) 1.0557 (0.0040) 1.0479 (0.0083) 2.0804
Weighted 0.3721 (0.0069) 0.3979 (0.0033) 2.1265 (0.0138) 0.4345 (0.0054) 0.4841 (0.0036) 0.4735 (0.0054) 0.6525 (0.0042) 0.5325 (0.0042) 0.5066 (0.0041) 0.5285 (0.0034) 0.5672 (0.0067) 1.036 (0.0053) 1.0267 (0.0057) 0.7030
Table 19 Comparison in processing time for 128 Dimensions
128 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.6171 (0.0056) 0.6447 (0.0042) 4.2734 (0.0780) 0.6803 (0.0053) 0.7427 (0.0073) 0.7300 (0.0058) 1.0579 (0.0056) 0.8216 (0.0223) 0.7755 (0.0064) 0.7975 (0.0069) 0.8489 (0.0065) 1.6856 (0.0647) 1.7465 (0.0289) 1.1863
Fitness 0.6322 (0.0125) 0.6542 (0.0037) 4.3411 (0.0181) 0.6862 (0.0052) 12.5764 (0.0365) 12.6336 (0.0389) 12.9683 (0.0807) 0.8439 (0.0017) 0.7995 (0.0038) 0.8197 (0.0017) 0.8761 (0.0031) 1.6912 (0.0056) 1.6605 (0.0125) 3.9372
Weighted 0.6487 (0.0243) 0.6706 (0.0084) 4.4372 (0.1050) 0.7212 (0.0163) 0.7736 (0.0091) 0.7597 (0.0106) 1.1117 (0.0194) 0.8649 (0.0246) 0.8223 (0.0297) 0.8398 (0.0313) 0.8866 (0.0127) 1.7026 (0.0548) 1.6764 (0.0267) 1.2243
Fuzzy I. 1.0056 (0.0526) 1.0488 (0.0599) 4.6376 (0.0682) 1.0938 (0.0355) 1.1495 (0.0318) 1.1196 (0.0303) 1.4424 (0.0034) 1.1876 (0.0070) 1.1540 (0.0126) 1.1560 (0.0203) 1.2013 (0.0068) 2.0090 (0.0083) 2.0692 (0.0891) 1.5596
Fuzzy D. 0.9749 (0.0226) 1.0222 (0.0415) 4.6159 (0.0754) 1.0384 (0.0098) 1.0978 (0.0126) 1.0825 (0.0045) 1.4157 (0.0064) 1.1742 (0.0072) 1.1320 (0.0056) 1.1541 (0.0054) 1.2088 (0.0117) 2.0096 (0.0067) 2.0398 (0.0852) 1.5358
S1568-4946(17)30182-5 http://dx.doi.org/doi:10.1016/j.asoc.2017.03.048 ASOC 4143
To appear in:
Applied Soft Computing
Received date: Revised date: Accepted date:
10-8-2016 14-2-2017 5-3-2017
Please cite this article as: Luis Rodr´ıguez, Oscar Castillo, Jos´e Soria, Patricia Melin, Fevrier Valdez, Claudia I.Gonzalez, Gabriela E.Martinez, Jesus Soto, A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm, Applied Soft Computing Journalhttp://dx.doi.org/10.1016/j.asoc.2017.03.048 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Fuzzy Hierarchical Operator in the Grey Wolf Optimizer Algorithm
Luis Rodríguez, Oscar Castillo*, José Soria, Patricia Melin, Fevrier Valdez, Claudia I. Gonzalez, Gabriela E. Martinez, Jesus Soto Tijuana Institute of Technology, Calzada Tecnologico s/n, CP 22379, B. C., Tijuana, Mexico,
*Email corresponding autor: [email protected] fig Graphical abstract
Highlights
The main goal is to study the performance of the Grey Wolf Optimizer algorithm when a new hierarchical operator is introduced. This new operator is basically a hierarchical transformation that is inspired in the hierarchical social pyramid of the grey wolf. This proposed operator is applied to the simulation of the hunting process in the algorithm and has 5 variants are presented. Notably the variants having the greatest impact in the GWO performance are based on the use fuzzy logic.
Abstract—The main goal of this paper is to study the performance of the Grey Wolf Optimizer (GWO) algorithm when a new hierarchical operator is introduced in the algorithm. This new operator is basically a hierarchical transformation that is inspired in the hierarchical social pyramid of the grey wolf. This proposed operator is applied to the simulation of the hunting process in the algorithm and has 5 variants that are explained in more detail in this paper (centroid, weighted, based on the fitness and two variants using fuzzy logic). Notably the variants having the greatest impact in the GWO performance are based on the use of fuzzy logic. We also present the motivation and results of experiments, as well as the benchmark functions that were used for the tests that are presented. In addition we are presenting a comparison among all methods for 30, 64 and 128 dimensions and we conclude that the performance of the Hierarchical GWO algorithm is better when using a fuzzy variant of the hierarchical operator. Keywords—dynamic adaptation; fuzzy logic; performance; GWO; benchmark functions; new operator; hierarchical, pyramid;
1. –Introduction
Today it is more common that researchers in the area of computer science work with bio-inspired meta-heuristics to solve optimization problems. There are a lot papers and algorithms that researchers have developed based on natural behavior that have been recently proposed. The main challenge in developing a new algorithm is to represent and simulate as much as possible each of the most important behaviors and characteristics of real phenomena. This representation can be implemented by the operators in the algorithms, for example in genetic algorithms we use operators, like crossover, mutation and selection. In this paper we are presenting a modification to the grey wolf optimizer algorithm based on the implementation of a new hierarchical operator in the algorithm, inspired by the hierarchical pyramid of the grey wolf, which is explained in more detail in Section 4. We are presenting four new approaches for the implementation of this hierarchy. Of these four new methods, two of them are represented by mathematical models and the other two are based on fuzzy logic, and this paper explains in detail each of these proposed operators. In this paper we are presenting tables and hypothesis testing results to demonstrate the performance of the algorithm with the 4 methods implemented for the new operator. The rest of the paper is organized into the following sections. Section 2 shows a literature review, in Section 3 we present the original grey wolf optimizer algorithm, Section 4 describes in detail the proposed hierarchical operator, in Section 5 the results and discussions are presented, and finally in Section 6 the conclusions are presented. 2. - Literature review Researchers have turned to meta-heuristics [13] especially because they can have superior abilities than conventional optimization techniques in complex and non-linear problems and
these techniques have become very popular lately because of the following features: simplicity, flexibility, derivative free approaches, and because they have shown more ability to avoid local optima. For example some of the optimization techniques based on meta-heuristics that have been studied lately are: the Genetic Algorithm (GA) [2], Particle Swarm Optimization (PSO) [11], Ant Colony Optimization (ACO) [10] and Artificial Bee Colony (ABC) [3]. In addition to the huge amount of theoretical work, these optimization techniques have been used in various areas of application. Meta-heuristic techniques can be classified as follows:
Evolutionary (based on concepts of evolution in nature) [8]: Genetic Algorithms (GAs), Evolutionary Programming (EP) [24] and Genetic Programming (GP).
Based on Physics (imitate the physical rules) [6]: Big-Bang Big Crunch (BBBC), Gravitational Search Algorithm (GSA) [18] and Artificial Chemical Reactions Optimization Algorithm (ACROA).
Swarm Intelligence (Social Behavior of swarms, herds, flocks or schools of creatures in nature) [23]: Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO) and the Bat-inspired Algorithm (BA).
The NFL Theorem (No Free Lunch) [21] has logically proven that there is no meta-heuristic appropriately suited for solving all optimization problems. For example, a particular meta-heuristic can produce very promising results for a set of problems, but the same algorithm can show poor performance in a set of different problems. Therefore we can find a diversity of techniques and new meta-heuristics are born for solving optimization problems.
The formal definition of fuzzy sets was proposed by Zadeh in 1965 [25] and it is as follows: Let 𝑿 be a space of points (objects), with a generic element of 𝑿 denoted by 𝒙. Thus, 𝑿 = {𝒙}. A fuzzy set 𝑨 in 𝑿 is characterized by a membership (characteristic) 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒇𝑨 (𝒙) which associates with each point in 𝑿 a real number in the interval [0, 1], with the value of 𝒇𝑨 (𝒙) at 𝒙 representing the "degree of membership" of 𝒙 in 𝑨. Thus, the nearer the value of 𝒇𝑨 (𝒙) to unity, the higher the degree of membership of 𝒙 in 𝑨. When 𝑨 is a set in the ordinary sense of the term, its membership function can take on only two values 0 and 1, with 𝒇𝑨 (𝒙) = 1 or 0 according as 𝒙 does or does not belong to 𝑨 respectively. Thus, in this case 𝒇𝑨 (𝒙) reduces to the familiar characteristic function of 𝒂 in set 𝑨. (When there is a need to differentiate between such sets and fuzzy sets, the sets with two-valued characteristic functions will be referred to as ordinary sets or simply sets.) 3. - Grey Wolf Optimizer The Grey Wolf Optimizer algorithm (GWO) [15] is a meta-heuristic that was originated in 2014 created by Seyedali Mirjalili, and inspired basically because in the literature there was not a Swarm Intelligence (SI)[5] technique based on the hierarchy of leadership of the Grey Wolf. The hierarchy of leadership and the hunting mechanism of the grey wolf are illustrated in the following pyramid in Fig. 1. In addition we can mention that the social hierarchy of the grey wolf in the group of hunters has a very interesting social behavior. According to C. Muro [17] the main phases of the grey wolf hunting are:
Tracking, chasing, and approaching the prey.
Pursuing, encircling, and harassing the prey until it stops moving.
Attack towards the prey
In order to mathematically model the social hierarchy of wolves when designing the GWO, we consider the fittest solutions as the alpha (𝛼) wolves. Consequently, the second and third best solutions are called beta (𝛽) and delta (𝛿)wolves, respectively. The rest of the candidate solutions are assumed to be omega (𝜔) wolves. In the GWO algorithm the hunting (optimization) is guided by 𝛼, 𝛽, and 𝛿. The 𝜔 wolves follow these three wolves. In the next example we find in more detail how the three best wolves and their positions are obtained. Suppose that we have a minimization problem (f1 of Table 4) with 3 dimensions, that we have 5 wolves and the minimum of the objective function is zero, then in Table 1 we show the corresponding results.
So based on the fitness values of Table 1 we have the following locations for:
alpha = (-3.9697, -3.0645, 1.9932)
beta = (6.1873, -0.0954,4.0133)
delta = (5.5931, 0.1255, -7.4112)
As mentioned above, grey wolves encircle prey during the hunting process. In order to mathematically model the encircling behavior the following equations are proposed: D = ‖ C ∙ Xp (t)- X(t)‖
(1)
X(t+1) =Xp (t)– AD
(2)
Where t indicates the current iteration, A and C are coefficients, 𝐗 𝐩 is the position vector of the prey, and 𝐗 indicates the position vector of a grey wolf. The A and C coefficients are calculated as follows: A = 2a ∙ r1 –a
(3)
C = 2 ∙ r2
(4)
Where the components of 𝒂 are linearly decreased from 2 to 0 over the course of the iterations and 𝑟1, 𝑟2 are random numbers in [0, 1]. In nature, grey wolves have the ability to recognize the location of prey and encircle them. The hunt is usually guided by the alpha wolf. In order to mathematically simulate the hunting behavior of the grey wolves, we assume that the alpha (best candidate solution), beta and delta have better knowledge about the potential location of prey. Therefore, we have to save the first three best solutions obtained so far, and force the other search agents (including the omegas) to update their positions according to the position of the best search agents. The following Equations have been proposed in this regard. Dα =‖C1 ∙ Xα -X‖,Dβ =‖C2 ∙ Xβ -X‖,
X1 =Xα - A1 ∙(Dα ),
X2 =Xβ - A2 ∙(Dβ ),
X(t+1) =
Dδ =‖C3 ∙ Xδ -X‖
(5)
X3 =Xδ - A3 ∙(Dδ ) (6)
X1 + X2 + X3 3
(7)
We can note that Equations 5 and 6 are the same as Equations 1 and 2 respectively, but in this case (Equations 5 and 6) we now have the positions of the alpha, beta and delta wolves to update the positions of the omega wolves and X1, X2 and X3 in Equation 7 and this represents the results obtained for updating the omega wolves based on alpha (X1), beta (X2) and delta(X3) respectively. Fig. 2 illustrates in a graphical way the process of updating the positions based on the best three wolves, where the blue, pink and green circles, represent the alpha, beta and delta wolves respectively, and the red circle represents an omega wolf that should move where the leaders of the pack (center), then it is assumed that the next estimated position should be the position represented by the yellow circle. As mentioned above the grey wolves finish the hunt by attacking the prey when it stops moving. In order to mathematically model the process of approaching the prey, we sequentially decrease the value of “a”. Note that the fluctuation range of A is also decreased by “a”. When random values of A are in [-1, 1], the next position of a search agent can be in any position between its current position and the position of the prey. Fig. 3(a) shows that |A| < 1 forces the wolves to attack towards the prey.
a) If |A| < 1 then attacking prey (exploitation) b) If |A| > 1 then searching for prey (exploration) Grey wolves mostly search according to the position of the alpha, beta, and delta wolves. They diverge from each other to search for prey and converge to attack the prey. In order to mathematically model divergence, we utilize A with random values greater than 1 or less than -1
to force the search agent to diverge from the prey. This emphasizes exploration and allows the GWO algorithm to search globally. Fig. 3(b) also shows that |A| > 1 forces the grey wolves to diverge from the prey to hopefully find a fitter prey. Fig. 4 shows the corresponding pseudocode of the algorithm. 4. - Hierarchical Operator The main inspiration of the GWO algorithm is based mainly on the social behavior of the grey wolf, its hierarchical pyramid and the corresponding hunting mechanism. The hierarchy pyramid was presented in Figure 1. Where we mentioned that the dominant wolf is α followed by β, and δ, which are the following leaders in the pyramid respectively, and the ω wolves are search agents in the algorithm. Below we present the five variants in the hierarchical operator that we mentioned in the introduction of this paper, the first is called centroid and this operator was proposed by the author of the original paper. The second and third variants are the weighted average and the weighted based on fitness respectively, which are proposed in this paper and are two different ways to assign a weight to the leaders of the pack based on the hierarchical pyramid. Finally, we explain in detail the last two variants that use fuzzy logic for dynamic adaptation of the weights that the leading wolves should have throughout the iterations.
In addition we can mention that a brief justification of adding these operators in the algorithm is because in the weighted average and weighted based on the fitness we are only adding a product for each best result (alpha, beta and delta), so that the computational complexity is not affected
and the execution time is minimum, is almost imperceptible, and in the case of using fuzzy logic is well known that the execution time is more because more operations are performed, but the complexity does not increase significantly because fuzzy logic only requires first order operations, such as maximum and minimum, addition, subtraction and products. Centroid This is the first operator that was implemented in the original paper and as presented in Equation 7 is simply the average of the estimated positions of the three best search agents, in this case alpha, beta and delta. 𝐗(𝑡 + 1) =
𝑿1 + 𝑿2 + 𝑿3
(7)
3
Weighted average The main idea for the weighted average is to assign a numerical value to each of the three wolf leaders in order to represent the hierarchy of the pyramid, and for this example, α has a weighting of 50%, β with 30% for the weighting and finally δ with a weight of 20%. This weighted average is reflected in Equation 8. 𝐗(𝑡 + 1) =
𝟓 𝑿1 + 3 𝑿2 +2 𝑿3 10
(8)
Where we can notice that in this case the numbers are fixed, but could be a normalized weight between zero and one and we can also have a weighting with bigger numbers, for example between 0 and 100. In this case this is a static weighting and can treated in different ways, values and ranges depending on the characteristics of the problem to be solved. For example in this
work we have that the alpha wolf has a weighting of 5, beta and delta 3 and 2 respectively, then the sum of the weights equals 10. Weighted based on the fitness When we say based on the fitness, immediately we think on the fitness that the best individual has for the problem to be solved, so the following equations are proposed for a weighting based on the fitness of the α , β, and δ wolves. 𝑾𝛼 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 ) 𝑆𝛼
𝑾𝛽 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 )
𝐗(𝑡 + 1) =
𝑆𝛽
𝑾𝛿 =
(𝑆𝛼 +𝑆𝛽 +𝑆𝛿 ) 𝑆𝛿
𝑿1 ∗ 𝑾𝛼 + 𝑿2 ∗𝑾𝛽 + 𝑿3 ∗ 𝑾𝛿 (𝑾𝛼 + 𝑾𝛽 +𝑾𝛿 )
(9)
(10)
Where: 𝑾𝛼 , 𝑾𝛽 ,𝑾𝛿 , represent the weight of alpha, beta and delta wolves respectively 𝑺𝛼 , 𝑺𝛽 , 𝑺𝛿 , express the fitness of the alpha, beta and delta individual respectively Equations 9 and 10 are the equations responsible for mathematically representing the weights assigned to each of the three leader wolves based on their fitness to solve the problem, thus having the alpha wolf with more weight in the hunting process, then having the second leading beta and ending with the lowest weight for delta, which in theory is the one that has less knowledge about the location of potential prey (optimization). Fuzzy weights
In addition it is possible to implement a dynamic pyramid in the algorithm for adjusting the weights for the three best wolves. We can define the equation that represents the mathematical model of the dynamic pyramid as follows:
𝐗(𝑡 + 1) =
𝑿1 ∗ 𝑭𝑾𝛼 + 𝑿2 ∗ 𝑭𝑾𝛽 + 𝑿3 ∗ 𝑭𝑾𝛿 𝑭𝑾𝛼 + 𝑭𝑾𝛽 + 𝑭𝑾𝛿
(11)
Where: 𝑭𝑾𝛼 , 𝑭𝑾𝛽 , 𝑭𝑾𝛿 represent the dynamic weights of the alpha, beta and delta wolves respectively. It is important to remember that when we say dynamic weights we refer to the idea that these weights are not fixed parameters and that are changing in each iteration of the algorithm as mentioned above. Figure 5 shows the general structure of the fuzzy system, designed exclusively for the fuzzy hierarchical pyramid with one input, which are the iterations of the algorithm, and three outputs, which correspond to the fuzzy weights of alpha, beta and delta respectively. The fuzzy system is of Mamdani type and has a defuzzification by the centroid method [4][7][14][19][20].
Below we provide more details of the input and outputs in the fuzzy system for the implementation of the fuzzy hierarchical pyramid.
Figure 6 shows the input to the fuzzy system for the hierarchical pyramid, the input is the number of total iterations in the algorithm, and we can notice that it has three triangular
membership functions labeled as low, medium and high respectively, and has a normalized range between 0 and 1. In contrast, Figure 7 shows an example of the three outputs of the fuzzy system, which is the weight that each of the three wolves will have to guide them during the hunting process, and the output contains three triangular membership functions classified into low, medium, and high respectively. The range of each output is between 0 and 100 in order to represent the percentage values, so low is near 0%, medium is near 50% of the weight and high is the most weight during the hunt that is close to 100%, and this is the justification for using the range between 0 and 100. Figure 7 shows the output for the alpha wolf, and it is important to say that the outputs for the beta and delta wolves, the structure and characteristics of the output are represented in the same way as shown in Figure 7.
When we think of the rules for the fuzzy system that will represent the hierarchical pyramid, the main goal is to exploit the ability of fuzzy logic for adjusting the dynamic parameters, so we have two options for the behavior of the pyramid that we are presenting below. The first option is to assume that at the beginning of the hunt, the three best wolves have the same weight to guide the pack to the prey, in the second step only the alpha and beta guide the pack because they are those who have better knowledge about the position of the prey and finally at the end of the hunt, the decisions and responsibilities in the pack are guided only by the alpha wolf. Thus the corresponding pyramid can be as illustrated in Figure 8. Figure 8 shows that in the initial iterations the weight for the three wolves in the hunt and the guide are equal, when the algorithm has achieved half of the iterations, the weight in the guide is
to alpha and delta and when the algorithm is in the final iterations and is starting the phase of exploitation, the alpha wolf has the best knowledge of the prey therefore and this is the wolf who completely dominates the hunt. The pyramid is called in “increase” because alpha increases the weight when passing iterations.
The second option is the opposite to the first philosophy of hunting, where the main goal is that the alpha wolf starts with the full weight of the hunting process respecting its hierarchy and making decisions to start the hunt and lead the pack, in the second step or in the middle of the hunt the beta wolf helps in the hunt and in the encircling process of the prey, and finally in the end of the hunt the three best wolfs work equally to attack and to end the hunt.
Figure 9 shows that at the beginning of the iterations the alpha wolf is the only one who guides the hunting because this wolf has the best knowledge of the possible position of the prey, in the middle of the iterations beta helps to lead the pack, and in the end of the iterations when the algorithm is focused on exploiting, the best option is to use the average of the best three wolves that are closest to the objective. The pyramid is presented in a decrease fashion because alpha loses weight in the hunting process along the iterations. Figure 10 shows the comparison of the pyramids and its direct relationship with the iterations in the algorithm.
Tables 2 and 3 show the rules used for each of the fuzzy pyramids that were presented above.
5. - Results and Discussion For this work specifically we evaluate the performance of each methods with 23 benchmark functions [9][16][22] with which the original algorithm was also tested. These benchmark functions are classified as unimodal, multimodal and fixed-dimension multimodal and are functions that have been already tested in other meta-heuristics, especially to test new algorithms and verify that they have a good performance. The functions are described below in Tables 4, 5 and 6 respectively, where “Range” represents the boundary of the function´s search space and fmin is the optimal value. We also present plots of the 2-D versions of the benchmark functions in Figures 11, 12 and 13 with the goal of analyzing the form and space of search of the functions that we test with the GWO algorithm. The main justification to test the algorithm with these benchmark functions is because there exists a great variety of meta-heuristics that use these benchmark functions to validate their performance and mainly because is important test the proposed methods in this work with the same conditions that the author describes in the original paper. Although the original author only tested functions with 30 dimensions (unimodal and multimodal functions), in this work we present a comparison with 30 and 64 dimensions and a hypothesis test with 128 dimensions because it is possible to increase the number of dimensions and for the fixed-dimension multimodal functions it is possible to test, develop and analyze with the number of dimensions that are shown in Table 6.
Table 4 and Figure 11 show the equations with their characteristics and the 2-D versions of the unimodal benchmark functions respectively, and these benchmark functions are evaluated in 30, 64 and 128 dimensions.
In Table 5 and Figure 12 we can find the equations with their characteristics and the 2-D versions of the multimodal benchmark functions respectively, and these benchmark functions are evaluated in 30, 64 and 128 dimensions.
Table 6 and Figure 13 show the equations with their characteristics and the 2-D versions of the fixed-dimension multimodal benchmark functions respectively and these benchmark functions are evaluated only in the dimensions that Table 6 indicates.
It is important to say that in the all experiments we use 30 wolves as the total population and a maximum of 500 iterations in the algorithm that corresponds to 15,000 function evaluations. In Table 7 we present a brief comparison among the first three variants proposed in this paper, and this table shows the averages and standard deviations of the first 13 benchmark functions with 30 dimensions.
The main goal of Table 7 is to show the results obtained in the 30 independent executions that were performed for the first 13 benchmark functions described above with the aim of testing each variant described in the previous section and show the performance that the algorithm
achieves when using the proposed modification of the hierarchical pyramid. We can note that there is a significant difference between the methods although for this simulation the idea is to show the results of the algorithm when we implement the new variants and analyze the performance of the GWO algorithm.
In Table 8 we can find a comparison among the fuzzy methods analyzed in this paper and the original method, and this table shows the averages and standard deviations of the first 13 benchmark functions, which are unimodal and multimodal respectively with 30 dimensions.
It is important to mention that Table 8 shows the results of the variant that includes fuzzy logic for dynamic adaptation of the weights in the hierarchical pyramid as mentioned before, in increase and decrease fashion respectively, and that we can identify this in the table with the labels “Fuzzy D” for a decrease fashion and “Fuzzy I” for an increasing fashion. Table 9 presents a comparison of the methods: centroid, fitness, weighted for unimodal and multimodal benchmark functions, but in this case the benchmark functions are with 64 dimensions because we want to analyze the performance of the algorithm when the problems are more complex and analyze the performance of the proposed variants presented in this paper.
Tables 9 and 10 also show the averages and standard deviations obtained by each method with the goal of showing the results and the accuracy of the different variants. In Table 10 we can find the results when the weights are implemented with fuzzy logic for the case of 64 Dimensions.
In this paper we are also presenting the performance of the grey wolf optimizer algorithm for more complex problems, and in the following tables we present the first 13 benchmark functions with a 128 dimensions. Below in the following tables we are also presenting a hypothesis test comparing each new method with the original method. In this case each table shows the averages and standard deviations obtained in 100independent runs of each method and finally the value of z obtained in hypothesis testing [12]. The goal is to statistically demonstrate if the new methods obtained better results than the original method. We considered the Z-test because for realizing this test we need a minimum 30 samples and we have 100 samples. So we performed this test with a 95% of confidence and 5% of significance (α = 0.05) and according to the tables of critical values of the Z-test with α = 0.05, and the range of rejecting the null hypothesis is in Z0 = -1.645. The parameters, the claim and data for the hypothesis testing are as follows:
𝜇1 = 𝑁𝑒𝑤 𝑀𝑒𝑡ℎ𝑜𝑑
𝜇2 = 𝐶𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑀𝑒𝑡ℎ𝑜𝑑
The mean of the new method is less than mean of the centroid method (claim).
𝐻0 : 𝜇1 ≥ 𝜇2
𝐻𝑎 : 𝜇1 < 𝜇2 (𝐶𝑙𝑎𝑖𝑚)
𝛼 = 0.05
sample size (n) = 100
𝑍0 = −1.645
In addition we need the value of Z for realizing the comparison with Z0 and conclude if there exists enough evidence to say that the new method is better than the original one.
The value of Z is obtained by the following equations.
𝜎𝑥̅1− 𝑥̅2 ≈ √
𝑍 ≈
(𝑥̅1 − 𝑥̅ 2 ) 𝜎𝑥̅ − 𝑥̅ 1 2
𝑠12 𝑛1
+
𝑠22 𝑛2
(12)
(13)
Where: 𝑠1 , 𝑠2 : are the standard deviations of the two methods respectively. 𝑛1 , 𝑛2 : are the sizes of the sample of the methods, respectively. 𝑥̅1 , 𝑥̅2 : are the averages obtained for each method respectively.
Figure 14 shows the rejection zone of the null hypothesis (H0), so if the value of Z is less than Z0 or -1.645 we can reject the null hypothesis and we can state that the mean of the new method is lower than mean of the centroid method with a 95% of confidence and this value of Z is shown in the following tables to conclude if the variants in the weight of the leaders are better and have a better performance than the original method.
In Table 11 we can find the results of the hypothesis test between the original method (Centroid) and the first proposed method in this paper that is the weighted method. Then according to the hypothesis testing explained above the proposed method has better performance in 7 of 13 benchmark functions considered in this paper because according of the hypothesis test (Z-test) the Z values are lower than Z0 and this means that while the values of Z are lower than -1.645
there exist enough evidence to conclude that the proposed method is better than the original one. So if we analyze Table 11 we can find that for the F1, F2, F3, F5, F6, F9 and F10 functions the values of Z are lower that de Z0 (-1.645). In addition is important to say that while the Z value is farther to the value of Z0 (-1.645) we have more evidence to conclude that the new method has better performance. For example in Function 1 the value of Z is -12.6917, so it is in the rejection zone (according to hypothesis test) and we can say that for this specific benchmark function the new method is better than the original one and this is the comparison that we performed with each analyzed function.
Table 12 shows an analysis of a hypothesis test between the centroid method and the second method analyzed in this paper, which was the fitness method and we can conclude that the proposed method is better than the original method in 4 of the first 13 analyzed benchmark functions.
In Tables 13 and 14 we can find the results of hypothesis testing between the original method (Centroid) and the two fuzzy methods described in this paper, that is the fuzzy pyramid in decrease and the fuzzy pyramid in increase respectively, and we can note that in both methods the performance of the algorithm is better in 12 of the 13 analyzed benchmark functions. In other words, both methods show better performance in all the unimodal benchmark functions and for
the multimodal benchmark functions the original method is better than the fuzzy method only in function number 12.
Finally in the following tables we are presenting a comparison between the original method (centroid) and the new methods described above. In addition we also present the next group the benchmark functions analyzed in this paper, which are the fixed-dimension multimodal benchmark functions and their characteristics were previously described. It is important to say that the tables show the average and the standard deviations of 100 independent runs for each function. Table 15 shows a comparison among the original method (centroid) and the first two methods presented in this paper, which are the fitness method and weighted method, for the fixeddimension multimodal benchmark functions that we presented at the beginning of this section.
In Table 16 we can find the averages and the standard deviations of the original method (centroid) and the fuzzy methods that we presented in this paper. We also present a comparison between these methods for the fixed-dimension multimodal benchmark functions.
It is important to mention that we consider that there is not enough difference between these methods because some results are very similar and this is the reason that we do not present a hypothesis test for these benchmark functions because there is not enough evidence to perform it. But we present a brief comparison based on the average and the standard deviation that we have obtained in each method.
Processing Time for each operator In order to clarify the impact that each operator has in the algorithm with respect to the execution time, we are presenting Tables 17, 18 and 19, where we can find the averages and standard deviations of processing times for each analyzed operator. These averages and standard deviations are the results to 100 independent executions for each operator in the first 13 benchmark functions that we are presenting in this paper with 30, 64 and 128 dimensions respectively.
In Table 17 we can note that the better method regarding computing time for the case of 30 dimensions is the centroid method and the results are very similar to the Weighted method, and the worst results are with the Fitness method and the Fuzzy logic methods produce similar results and time is approximately twice when compared with the Centroid method. In addition it is important to mention that these experiments were executed in an Acer Aspire M3970 computer with Intel Core i5-2300 CPU @ 2.80 GHz processor and 6144 MB of RAM memory.
Table 18 shows the results with 64 dimensions and the results have very similar performance with 30 dimensions, so the best methods in regards to computing time are the Weighted and Centroid methods respectively and the worst is the Fitness method, so the Fuzzy Logic method did maintained in the middle of the methods with respect to the required time.
Table 19 shows the performance in time for 128 dimensions for each analyzed method in this paper. In addition we show in the Figure 15 a plot, where we can find the average times that each method requires to evaluate all benchmark functions (first 13 functions) that we evaluated to analyze the time required for each method.
We can conclude based on Table 19 and Figure 15 that for complex problems, the performance of the Fuzzy Logic operators is improving because the difference in the percentage of the time between the Centroid method an Fuzzy Logic methods gets increasingly lower when the problem becomes more complex. It is expected that with a number of dimensions high enough the fuzzy logic method would require less time than the other methods.
Finally, we can conclude that although the execution time is more with the Fuzzy logic methods, this difference in time is very small and does not increase in an excessive way, so this option of implementing the fuzzy operators is viable for improving the results of the algorithm. 6. –Conclusion In this work we have analyzed the grey wolf optimizer algorithm that is considered a relatively new meta-heuristic because it was proposed in 2014, so the algorithm has an area of opportunity for adding new characteristics, like the author of the method mentioned in the original paper. In this paper we presented a new operator that affects directly one of the original inspirations of this algorithm and this characteristic is the hierarchical pyramid of the wolf. If we remember the two main characteristics in this algorithm are the social behavior and the hierarchical pyramid. The main contribution of this paper consists of the three proposed methods to implement the hierarchical pyramid that directly affect the new position of the omega wolves based on the best three wolves (alpha, beta and delta). We focus on the hierarchical pyramid and we are presenting three different options to implement the pyramid, which are the weighted average, based on the
fitness, and based on fuzzy logic, that were described in this paper with a mathematical model and a fuzzy system. According to the presented results we can conclude that in the methods based on mathematical models, each method has better performance depending of the problem to solve, but we can prove with a hypothesis test that when we use fuzzy logic in the algorithm, we can note a better performance than with the mathematical models in a majority of the analyzed benchmark functions. In this work we justify the problems that were used for testing the new operators, although it would be important as future work to test these new operators with other types of problems or applications, such as in Encryption [1], simulators and plants and testing the performance of new proposed methods in other applications.
References [1] F. Aladwan, M. Alshraideh and M. Rasol. “A Genetic Algorithm Approach for Breaking of Simplified Data Encryption Standard”, International Journal of Security and Its Applications 9(9), 2015, pp. 295-304. [2] M. Alshraideh. “A complete Automation of Unit Testing for JavaScript Programs” Journal of Computer Science 4(12), 2008, pp. 1012-1019. [3] B. Basturk and D. Karaboga, “An artificial bee colony (ABC) algorithm for numeric function optimization”, IEEE, swarm intelligence symposium, 2006, pp. 12-4. [4] J. Barraza, P. Melin and F. Valdez “Fuzzy FWA with dynamic adaptation of parameters”, IEEE CEC 2016, “accepted for publication” [5] G. Beni and J. Wang, “Swarm intelligence in cellular robotic systems”, Robots and biological systems: towards a new bionics?, Springer, 1993, pp. 703–12. [6] U. Can and B. Alatas, “Physics based metaheuristic algorithms for global optimization”, American Journal of Information Science and Computer Engineering, vol. 1, 2015, pp. 94-106. [7] O. Castillo, H. Neyoy, J. Soria, P. Melin and F. Valdez. A new approach for dynamic fuzzy logic parameter tuning in Ant Colony Optimization and its application in fuzzy control of a mobile robot, Applied Soft Computing 2015; 28; 150–159. [8] H. Dias de Mello Jr, L. Martí and M. Rebuzzi Vellasco, “Evolutionary algorithms and elliptical copulas applied to continuous optimization problems,” Information Sciences, vol. 369, pp. 419-440, 2016. [9]J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms”, Int J Comput Math, vol. 77, 2001, pp. 481–506. [10]M. Dorigo, M. Birattari and T. Stutzle, “Ant colony optimization”, IEEE, Comput Intell Magaz, 2006, pp. 2839. [11] J. Kennedy and R. Eberhart, “Particle swarm optimization, in Neural Networks”, IEEE international conference, 1995, pp. 1942-1948. [12]R. Larson and B. Farber, (2003), Elementary Statistics Picturing the World, Pearson Education Inc. (428-433). [13]Melián B., Moreno J., Metaheurísticas: una visión global, Revista Iberoamericana de Inteligencia Artificial 2003; 19; 7-28. [14] P. Melin, F. Olivas, O. Castillo, F. Valdez, J. Soria and M. Valdez, Optimal design of fuzzy classification systems using PSO with dynamic parameter adaptation through fuzzy logic, Expert Systems with Applications 2013; 40; 3196–3206. [15] S. Mirjalili, M. Mirjalili and A. Lewis, “Grey Wolf Optimizer”, Advances in Engineering Software, vol. 69, 2014, pp. 46-61. [16] M. Molga and C. Smutnicki, “Test functions for optimization needs”, unpublished. [17] C. Muro, R. Escobedo, L. Spector and R. Coppinger, “Wolf-pack (Canis lupus) hunting strategies emerge from simple rules”, Computational simulations, Behav Process, vol. 88, 2011, pp. 192–7.
[18]E. Reshedi, H. Nezamabadi-Pour and S. Saryazdi, “GSA: a gravitational search algorithm”, Inf Sci 2009;179:2232-48. [19] L. Rodriguez, O. Castillo and J. Soria, “Grey Wolf Optimizer (GWO) with dynamic adaptation of parameters using fuzzy logic”, IEEE CEC 2016, “accepted for publication”. [20] Salah, Bareqa et al. “Skin Cancer Recognition by Using a Neuro-Fuzzy System.” Cancer Informatics 10 (2011): 1–11. PMC.Web. 6 Dec. 2016. [21]DH. Wolpert, and WG. Macready, “No free lunch theorems for optimization”, Evolut Comput, IEEE Trans, 1997, pp. 67–82. [22] X-S. Yang, “Test problems in optimization”, arXiv, preprint arXiv:1008.0549, 2010 [23] X. Yang and M. Karamanoglu, “Swarm intelligence and bio-inspired computation: an overview”, Swarm Intelligence and Bio-Inspired Computation, 2013, pp. 3-23. [24]X. Yao, Y. Liu and G. Lin, “Evolutionary programming made faster” Evolut Comput, IEEE Trans, vol. 3, 1999, pp. 82–102. . [25]L. Zadeh, “Fuzzy sets,” Information and control, No. 8, pp. 338-353, 1965.
Fig.1Hierarchy pyramid
Fig.2 Updating wolf positions in the algorithm
Fig.3Attacking prey versus searching for prey
Initialize the grey wolf population Xi(i = 1, 2, …, n) Initialize a, A and C Calculate the fitness of each search agent Xα = the best search agent Xβ = the second best agent Xδ = the third best search agent while (t < Max number of iterations) for each search agent Update the position of the current search agent by Equation (7) end for Update a, A and C Calculate the fitness of all search agents Update Xα , Xβ and Xδ t = t+ 1 endwhile return Xα Fig.4 Pseudocode of the algorithm
Fig. 5 General structure of the fuzzy system
Membership Grades
1
Low
Medium
High
0.5
0
0
0.1
0.2
0.3
0.4
0.5 0.6 Iterations
0.7
0.8
0.9
1
Membership Grades
Fig. 6 Input of the FIS for the fuzzy pyramid
1 Low
Medium
High
0.5
0
0
10
20
30
40
50 Alpha
Fig. 7 Output of the FIS for the fuzzy pyramid
60
70
80
90
100
Fig. 8 Pyramid in increase
Fig. 9 Pyramid in decrement
Fig. 10 Comparison of the two pyramids
Fig. 11 2-D versions of the unimodal benchmark functions
Fig. 12 2-D versions of the multimodal benchmark functions
Fig. 13 2-D versions of the fixed-dimension multimodal benchmark functions
Fig. 14 Reject zone of z-test with α = 0.05
4
Fitness
Time in Seconds
3.5
3
2.5
2
Fuzzy I 1.5
Fuzzy D Centroid 1
Weighted Hierarchical Operators
Fig. 15 Comparison in processing time for each operator with 128 dimensions
Table1Example of the leaders obtained in the GWO algorithm
d1 -5.5390 -3.9667 -8.3744 6.1873 5.5931
d2 -9.5147 -3.0645 -3.2446 -0.9254 0.1255
d3 -6.9951 1.9932 6.9530 4.0133 -7.4112
fitness 170.1424 29.0986 129.0029 55.2451 86.2248
Table 2 Rules in increase of the fuzzy pyramid
Rules in increase 1.- if(iterations is low) then (alpha is medium) (beta is medium) (delta is medium) 2.- if(iterations is medium) then (alpha is medium) (beta is medium) (delta is low) 3.- if(iterations is high) then (alpha is high) (beta is medium) (delta is low)
Table 3 Rules in decrease of the fuzzy pyramid
Rules in decrease 1.- if(iterations is low) then (alpha is high) (beta is medium) (delta is low) 2.- if(iterations is medium) then (alpha is medium) (beta is medium) (delta is low) 3.- if(iterations is high) then (alpha is medium) (beta is medium) (delta is medium)
Table 4 Unimodal Benchmark functions
Function 𝑛
𝑓1 (𝑥) = ∑ 𝑥𝑖 2
Range [-100, 100]
fmin 0
[-10, 10]
0
[-100, 100]
0
[-100, 100]
0
[-30, 30]
0
[-100, 100]
0
[-1.28, 1.28]
0
𝑖=1 𝑛
𝑛
𝑓2 = ∑|𝑥1 | + ∏|𝑥1 | 𝑖=1
𝑖=1 𝑛
2
𝑖
𝑓3 = ∑ (∑ 𝑥𝑗 ) 𝑖=1
𝑗−1
𝑓4 = 𝑚𝑎𝑥𝑖 {|𝑥1 |, 1 ≤ 𝑖 ≤ 𝑛 } 𝑛−1 2 2
2
𝑓5 (𝑥) = ∑[100(𝑥𝑖+1 − 𝑥𝑖 ) + (𝑥1 − 1) ] 𝑖=1 𝑛
𝑓6 (𝑥) = ∑([𝑥-1 + 0.5])2 𝑖=1
𝑛
𝑓7 (𝑥) = ∑ 𝑖𝑥𝑖 4 + 𝑟𝑎𝑛𝑑𝑜𝑚[0,1] 𝑖=1
Table 5 Multimodal benchmark functions
Function 𝑛
𝑓8 (𝑥) = ∑ −𝑥𝑖 sin (√|𝑥𝑖 |)
Range [-500, 500]
fmin -2,094.91
[-5.12, 5.12]
0
[-32, 32]
0
[-600, 600]
0
[-50, 50]
0
[-50, 50]
0
𝑖=1 𝑛
𝑓9 (𝑥) = ∑[𝑥𝑖 2 − 10 cos(2𝜋𝑥𝑖 ) + 10] 𝑖=1 𝑛
𝑛
1 1 𝑓10 (𝑥) = 20 𝑒𝑥𝑝 (−0.2√ ∑ 𝑥𝑖 2 ) − 𝑒𝑥𝑝 ( ∑ cos(2𝜋𝑥𝑖 )) + 20 + 𝑒 𝑛 𝑛 𝑖=1
𝑓11 (𝑥) =
𝑓12 (𝑥) =
𝑛
𝑛
𝑖=1
𝑖=1
𝑖=1
1 𝑥𝑖 ∑ 𝑥1 2 − ∏ cos ( ) + 1 4000 √𝑖
𝜋 {10 sin(𝜋𝑦1 ) 𝑛 𝑛−1
+ ∑(𝑦𝑖 − 1)2 [1 + 10 sin2 (𝜋𝑦𝑦+1 )] + (𝑦𝑛 − 1)2 } 𝑖=1 𝑛
+ ∑ 𝑢(𝑥𝑖 , 10, 100, 4) 𝑖=1
𝑥𝑖 + 1 4 𝑘(𝑥𝑖 − 𝑎)𝑚 , 𝑥𝑖 > 𝑎 − 𝑎 < 𝑥𝑖 < 𝑎 𝑢(𝑥1 , 𝑎, 𝑘, 𝑚) = { 0 , 𝑘(−𝑥𝑖 − 𝑎)𝑚 , 𝑥𝑖 < −𝑎 𝑦1 = 1 +
𝑓13 (𝑥) = 0.1 {sin2(3𝜋𝑥𝑖 ) 𝑛
+ ∑(𝑥1 − 1)2 [ 1 + sin2 (3𝜋𝑥𝑖 + 1)] 𝑖=1
+ (𝑥𝑛 − 1)2 [1 + sin2 (2𝜋𝑥𝑛 )]} 𝑛
+ ∑ 𝑢(𝑥𝑖 , 5, 100, 4) 𝑖=1
Table 6 Fixed-dimension multimodal benchmark functions
Function
Dim 2
Range [-65, 65]
fmin 1
𝑥1 (𝑏12 + 𝑏𝑖 𝑥2 ) 𝑓15 (𝑥) = ∑ [𝑎𝑖 − 2 ] 𝑏1 + 𝑏𝑖 𝑥3 + 𝑥4
4
[-5, 5]
0.00030
1 𝑓16 (𝑥) = 4𝑥12 − 2.1𝑥14 + 𝑥16 + 𝑥1 𝑥2 − 4𝑥22 + 4𝑥24 3
2
[-5, 5]
-1.0316
2
[-5, 5]
0.398
2
[-2, 2]
3
3
[1, 3]
-3.86
6
[0, 1]
-3.32
4
[0, 10]
-10.1532
4
[0, 10]
-10.4028
4
[0, 10]
-10.5363
−1
25
1 1 𝑓14 (𝑥) = ( + ∑ 6) 2 500 ∑ 𝑗 + (𝑥 − 𝑎 ) 𝑖𝑗 𝑖=2 𝑖 𝑗=1 2
11
𝑖=1
𝑓17 (𝑥) = (𝑥2 −
2 5.1 2 5 1 𝑥 + 𝑥 − 6 ) + 10 (1 − ) cos 𝑥1 4𝜋 2 1 𝜋 1 8𝜋 + 10
𝑓18 (𝑥) = [1 + (𝑥1 + 𝑥2 + 1)2 (19 − 14𝑥1 + 3𝑥12 − 14𝑥2 + 6𝑥1 𝑥2 + 3𝑥22 )] × [30 + (2𝑥1 − 3𝑥2 )2 × (18 − 32𝑥1 + 12𝑥12 + 48𝑥2 − 36𝑥1 𝑥2 + 27𝑥22 )] 3
4
2
𝑓19 (𝑥) = − ∑ 𝑐1 𝑒𝑥𝑝 (− ∑ 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ) 𝑖=1
𝑗=1
4
6
2
𝑓20 (𝑥) = − ∑ 𝑐1 𝑒𝑥𝑝 (− ∑ 𝑎𝑖𝑗 (𝑥𝑗 − 𝑝𝑖𝑗 ) ) 𝑖=1
𝑗=1
5
𝑓21 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1 𝑖=1 7
𝑓22 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖
)𝑇
+ 𝑐𝑖
]−1
𝑖=1 10
𝑓23 (𝑥) = − ∑[(𝑋 − 𝑎𝑖 )(𝑋 − 𝑎𝑖 )𝑇 + 𝑐𝑖 ]−1 𝑖=1
Table 7 Comparison among the methods: centroid, fitness and weighted with unimodal and multimodal benchmark functions
30 Dimensions Function Centroid F1 6.59E-28 F2 7.18E-17 F3 3.29E-06 F4 5.61E-07 F5 26.8126 F6 0.8166 F7 0.0022 F8 -6123.10 F9 0.3105 F10 1.06E-13 F11 0.0045 F12 0.0534 F13 0.6545
STD 6.34E-05 0.0290 79.1496 1.3151 69.9050 1.26E-04 0.1003 4087.44 47.3561 0.0778 0.0067 0.0207 0.0045
Fitness 9.28E-11 3.55E-07 2806.72 4.9906 125.9725 14.74 0.0091 -19230.07 11.3785 9.03E-07 0.0046 0.4757 9.7919
STD 8.26E-11 1.15E-07 2320 3.8208 0.5694 1.1695 0.0035 3756.75 7.9794 2.94E-07 0.0063 0.0984 0.4707
Weighted 7.12E-31 8.07E-19 3.30E-06 1.66E-07 26.8580 0.7563 0.0019 -6006.70 2.1193 4.83E-14 0.0042 0.04897 0.6135
STD 2.40E-30 7.1511E-19 1.26E-05 4.00E-07 0.7177 0.3735 0.0010 790.5799 4.81229 7.76E-15 0.0082 0.0272 0.2601
Table 8 Comparison among the methods: centroid, fuzzy in decrease and fuzzy in increase with unimodal and multimodal benchmark functions
30 Dimensions Function Centroid F1 6.59E-28 F2 7.18E-17 F3 3.29E-06 F4 5.61E-07 F5 26.8126 F6 0.8166 F7 0.0022 F8 -6123.10 F9 0.3105 F10 1.06E-13 F11 0.0045 F12 0.0534 F13 0.6545
STD 6.34E-05 0.0290 79.1496 1.3151 69.9050 1.26E-04 0.1003 4087.44 47.3561 0.0778 0.0067 0.0207 0.0045
Fuzzy D. 4.69E-30 1.50E-18 0.5337 7.45E-08 51.4764 5.7266 0.0017 -5160.88 0.4618 2.75E-14 0.0016 0.2800 3.4073
STD 1.43E-29 3.18E-18 3.96 1.90E-07 16.0976 3.2391 0.0010 1008.48 2.8904 2.22E-14 0.0055 0.1682 1.7817
Fuzzy I. 9.47E-32 1.73E-19 0.5337 4.81E-08 51.4594 5.7212 0.0017 -5011.22 0.9691 2.38E-14 7.29E-04 0.2795 3.3789
STD 4.74E-31 3.50E-19 3.9604 1.05E-07 16.1231 3.2469 9.21E-04 869.24 2.9832 1.60E-14 0.0030 0.1684 1.8247
Table 9 Comparison of the methods: centroid, fitness and weighted for 64 dimensions
64 Dimensions Function Centroid F1 7.78E-17 F2 1.18E-10 F3 9.29 F4 0.0080 F5 61.6661 F6 4.3925 F7 0.0047 F8 -11059.92 F9 5.7832 F10 9.83E-10 F11 0.0028 F12 0.1714 F13 3.3109
STD 8.00E-17 4.72E-11 17.92 0.0138 0.8593 0.7908 0.0022 1920.1685 5.78776 4.55E-10 0.0081 0.0753 0.3288
Fitness 2.94E-17 9.65E-11 9.9873 0.0068 61.6087 4.3178 0.0039 -10996.72 4.6677 8.05E-10 0.0036 0.27048 3.6451
STD 3.88E-17 5.37E-11 22.13 0.0126 0.8255 0.7304 0.0018 1844.1950 4.73 4.51E-10 0.0095 0.1170 0.4317
Weighted 5.69E-19 6.26E-12 5.33 0.0064 61.4798 4.4938 0.0041 -11122.48 4.3519 9.80E-11 0.0032 0.1647 3.3866
STD 7.79E-19 3.35E-12 11.35 0.0096 0.8794 0.6553 0.0019 1466.88 7.26821 5.76E-11 0.0070 0.0525 0.3535
Table 10 Comparison of the methods: centroid, fuzzy in decrease and fuzzy in increase for 64 dimensions
64 Dimensions Function Centroid F1 7.78E-17 F2 1.18E-10 F3 9.29 F4 0.0080 F5 61.6661 F6 4.3925 F7 0.0047 F8 -11059.92 F9 5.7832 F10 9.83E-10 F11 0.0028 F12 0.1714 F13 3.3109
STD 8.00E-17 4.72E-11 17.92 0.0138 0.8593 0.7908 0.0022 1920.1685 5.78776 4.55E-10 0.0081 0.0753 0.3288
Fuzzy D. 1.85E-18 5.64E-12 4.16 0.0174 61.8753 6.8445 0.0025 -6613.67 1.8943 9.42E-11 0.0010 0.3206 4.2253
STD 4.48E-18 9.80E-12 15.08 0.1391 0.7334 1.6324 0.0017 3272.5693 7.37467 1.74E-10 0.0048 0.1253 0.5845
Fuzzy I. 6.22E-20 1.38E-12 1.60 7.13E-04 61.8082 6.8298 0.0023 -6496.35 2.0865 1.87E-11 0.0019 0.3126 4.1995
STD 1.40E-19 2.49E-12 5.20 0.0028 0.7565 1.6476 0.0015 2992.15 5.32146 3.38E-11 0.0056 0.1300 0.6368
Table 11 Hypothesis test between the centroid method and the weighted method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06 F9 11.8031 F10 9.98E-07 F11 0.0056 F12 0.3791 F13 9.5433
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
Weighted 5.09E-12 6.80E-08 2382.68 6.4470 125.7715 14.7852 0.0088 -19040.08 9.2998 1.98E-07 0.0048 0.36557 9.6057
STD 4.28E-12 1.97E-08 1960 4.9446 0.7346 1.2106 0.0036 3768.8485 7.09064 8.00E-08 0.0120 0.0581 0.4982
Value of Z -12.6917 -29.1863 -1.6888 2.0907 -2.0909 -2.018 -0.3768 -0.9446 -2.4327 -24.6154 -0.4694 -1.505 0.8625
Table 12 Hypothesis test between the centroid method and the fitness method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92
STD 1.07E-10 1.85E-07 2610.00
Fitness 9.28E-11 3.55E-07 2806.72
STD 8.26E-11 1.15E-07 2320
Value of Z -3.5658 -11.7523 -0.3643
F4 F5 F6 F7 F8 F9 F10 F11 F12 F13
5.1865 125.9696 15.1367 0.0090 -19558.06 11.8031 9.98E-07 0.0056 0.3791 9.5433
3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
4.9906 125.9725 14.7400 0.0091 -19230.07 11.3785 9.03E-07 0.0046 0.4757 9.7919
3.8208 0.5694 1.1695 0.0035 3756.75 7.9794 2.94E-07 0.0063 0.0984 0.4707
-0.3806 0.0351 -2.3151 0.1908 -0.5991 -0.3888 -2.2048 -0.7330 8.0532 3.5272
Table 13 Hypothesis test between the centroid method and the fuzzy in decrease method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06 F9 11.8031 F10 9.98E-07 F11 0.0056 F12 0.3791 F13 9.5433
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93 7.4572 3.15E-07 0.0121 0.0686 0.5246
Fuzzy D. 7.69E-12 4.86E-08 751.2593 2.0329 81.1047 10.1128 0.0040 -9105.10 3.2140 1.47E-07 0.0018 0.3851 6.1473
STD 1.50E-11 8.78E-08 1527.22 4.7895 29.4134 3.6583 0.0039 7300.89 6.9663 2.40E-07 0.0076 0.0838 2.4682
Value of Z -12.3382 -27.4639 -7.2178 -5.3427 -15.2501 -12.9926 -9.0655 -12.5687 -8.4167 -21.4893 -2.6594 0.5540 -13.4584
Table 14 Hypothesis test between the centroid method and the fuzzy in increase method
128 Dimensions Function Centroid F1 1.41E-10 F2 6.11E-07 F3 2933.92 F4 5.1865 F5 125.9696 F6 15.1367 F7 0.0090 F8 -19558.06
STD 1.07E-10 1.85E-07 2610.00 3.4498 0.5983 1.2524 0.0039 3982.93
Fuzzy I. 6.55E-13 1.51E-08 673.8766 1.2525 81.1221 10.0137 0.0038 -8712.21
STD 1.17E-12 2.56E-08 1422.2729 2.5651 29.4403 3.5254 0.0039 6498.40
Value of Z -13.1156 -31.9068 -7.6035 -9.1511 -15.2302 -13.6933 -9.4281 -14.2299
F9 F10 F11 F12 F13
11.8031 9.98E-07 0.0056 0.3791 9.5433
7.4572 3.15E-07 0.0121 0.0686 0.5246
3.1345 3.96E-08 0.0018 0.3736 6.0642
5.9764 6.81E-08 0.0069 0.0834 2.3403
-9.0709 -29.7384 -2.7281 -0.5093 -14.5061
Table 15 Comparison between the methods: centroid, fitness and weighted with the fixed-dimension multimodal benchmark functions
Function F14 F15 F16 F17 F18 F19 F20 F21 F22 F23
Centroid 4.2582 0.0048 -1.0316 0.3979 5.4300 -3.8615 -3.2841 -9.3441 -10.3479 -10.4811
STD 4.1457 0.0094 2.54E-08 1.73E-04 13.8872 0.0025 0.0628 1.9578 0.5313 0.5361
Fitness 3.8876 0.0059 -1.0111 0.3979 5.4300 -3.8612 -3.2584 -9.1646 -10.1558 -10.3182
STD 3.9466 0.0089 0.0874 2.51E-06 13.8872 0.0026 0.0877 2.1462 1.2227 1.2560
Weighted 5.0584 0.0058 -1.0316 0.3979 3.0000 -3.8612 -3.2463 -9.2684 -10.1895 -9.9937
STD 4.4456 0.0100 2.49E-08 2.97E-06 3.40E-05 0.0028 0.0940 2.0688 1.0408 1.8828
Table 16 Comparison between the methods: centroid, fuzzy in decrease and fuzzy in increase with the fixeddimension multimodal benchmark functions
Function F14 F15 F16 F17 F18 F19 F20 F21 F22 F23
Centroid 4.2582 0.0048 -1.0316 0.3979 5.4300 -3.8615 -3.2841 -9.3441 -10.3479 -10.4811
STD 4.1457 0.0094 2.54E-08 1.73E-04 13.8872 0.0025 0.0628 1.9578 0.5313 0.5361
Fuzzy D. 5.8640 0.0039 -1.0316 0.4026 3.8100 -3.8615 -3.2666 -7.7454 -8.9120 -9.1294
STD 4.7449 0.0075 8.22E-06 0.0284 8.1002 0.0024 0.0691 2.3013 1.5826 1.3086
Fuzzy I. 6.1356 0.0038 -1.0316 0.4026 3.8100 -3.8619 -3.2553 -8.0997 -9.0410 -9.1564
STD 4.8359 0.0075 8.22E-06 0.0284 8.1002 0.0017 0.0672 2.1677 1.4157 1.3648
Table 17 Comparison in processing time for 30 Dimensions
30 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.2461 (0.0025) 0.2807 (0.0036) 1.1004 (0.0346) 0.3246 (0.0138) 0.3741 (0.0075) 0.3765 (0.0225) 0.4610 (0.0100) 0.4061 (0.0065) 0.3919 (0.0068) 0.4055 (0.0059) 0.4424 (0.0053) 0.7578 (0.0046) 0.7513 (0.0053) 0.4860
Fitness 0.2535 (0.0107) 0.2800 (0.0026) 1.1116 (0.0254) 0.3294 (0.0034) 3.3325 (0.0547) 3.1505 (0.0233) 3.2359 (0.0287) 0.3936 (0.0047) 0.3744 (0.0052) 0.3925 (0.0029) 0.4207 (0.0072) 0.7275 (0.0033) 0.7258 (0.0074) 1.1329
Weighted 0.2595 (0.0149) 0.2831 (0.0119) 1.1091 (0.0226) 0.3247 (0.0078) 0.3675 (0.0084) 0.3558 (0.0094) 0.4661 (0.0065) 0.4076 (0.0038) 0.3909 (0.0076) 0.412 (0.0077) 0.4446 (0.0083) 0.7675 (0.0117) 0.7630 (0.0092) 0.4886
Fuzzy I. 0.6482 (0.0220) 0.6802 (0.0195) 1.5040 (0.0126) 0.7014 (0.0137) 0.7583 (0.0103) 0.7484 (0.0076) 0.8537 (0.0078) 0.7995 (0.0087) 0.7793 (0.0138) 0.7645 (0.0270) 0.7866 (0.0082) 1.1054 (0.0057) 1.1019 (0.0086) 0.8640
Fuzzy D. 0.6381 (0.0399) 0.6647 (0.0084) 1.5006 (0.0161) 0.7043 (0.0147) 0.7585 (0.0118) 0.7510 (0.016) 0.8562 (0.0100) 0.8016 (0.0258) 0.7800 (0.0110) 0.7964 (0.0077) 0.8455 (0.0186) 1.182 (0.0181) 1.1696 (0.0264) 0.8807
Fuzzy I. 0.7272 (0.0222) 0.7506 (0.0051) 2.5758 (0.0863) 0.8174 (0.0209) 0.8864 (0.0396) 0.8718 (0.0443) 1.0318 (0.0037) 0.9228 (0.0302) 0.8906 (0.0125) 0.9033 (0.0046) 0.9482 (0.0061) 1.4519 (0.0267) 1.4341 (0.0051) 1.0932
Fuzzy D. 0.7274 (0.0200) 0.7505 (0.0040) 2.489 (0.0212) 0.7878 (0.0061) 0.8405 (0.0054) 0.8302 (0.0046) 1.0102 (0.0064) 0.9179 (0.0370) 0.8904 (0.0350) 0.8860 (0.0055) 0.9316 (0.0100) 1.4041 (0.0067) 1.3946 (0.0117) 1.0662
Table 18 Comparison in processing time for 64 Dimensions
64 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.4056 (0.0206) 0.4119 (0.017) 2.2367 (0.0728) 0.4583 (0.0274) 0.5200 (0.0199) 0.5069 (0.0225) 0.6927 (0.0124) 0.5501 (0.0114) 0.5288 (0.0047) 0.5466 (0.0094) 0.5789 (0.0042) 1.068 (0.0083) 1.0445 (0.0164) 0.7346
Fitness 0.3934 (0.0121) 0.4155 (0.0093) 2.2191 (0.0577) 0.4473 (0.0025) 6.4466 (0.0208) 6.3529 (0.1440) 6.4613 (0.0263) 0.5483 (0.0043) 0.5261 (0.0039) 0.5440 (0.0045) 0.5874 (0.0085) 1.0557 (0.0040) 1.0479 (0.0083) 2.0804
Weighted 0.3721 (0.0069) 0.3979 (0.0033) 2.1265 (0.0138) 0.4345 (0.0054) 0.4841 (0.0036) 0.4735 (0.0054) 0.6525 (0.0042) 0.5325 (0.0042) 0.5066 (0.0041) 0.5285 (0.0034) 0.5672 (0.0067) 1.036 (0.0053) 1.0267 (0.0057) 0.7030
Table 19 Comparison in processing time for 128 Dimensions
128 Dimensions Function Centroid F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Average
0.6171 (0.0056) 0.6447 (0.0042) 4.2734 (0.0780) 0.6803 (0.0053) 0.7427 (0.0073) 0.7300 (0.0058) 1.0579 (0.0056) 0.8216 (0.0223) 0.7755 (0.0064) 0.7975 (0.0069) 0.8489 (0.0065) 1.6856 (0.0647) 1.7465 (0.0289) 1.1863
Fitness 0.6322 (0.0125) 0.6542 (0.0037) 4.3411 (0.0181) 0.6862 (0.0052) 12.5764 (0.0365) 12.6336 (0.0389) 12.9683 (0.0807) 0.8439 (0.0017) 0.7995 (0.0038) 0.8197 (0.0017) 0.8761 (0.0031) 1.6912 (0.0056) 1.6605 (0.0125) 3.9372
Weighted 0.6487 (0.0243) 0.6706 (0.0084) 4.4372 (0.1050) 0.7212 (0.0163) 0.7736 (0.0091) 0.7597 (0.0106) 1.1117 (0.0194) 0.8649 (0.0246) 0.8223 (0.0297) 0.8398 (0.0313) 0.8866 (0.0127) 1.7026 (0.0548) 1.6764 (0.0267) 1.2243
Fuzzy I. 1.0056 (0.0526) 1.0488 (0.0599) 4.6376 (0.0682) 1.0938 (0.0355) 1.1495 (0.0318) 1.1196 (0.0303) 1.4424 (0.0034) 1.1876 (0.0070) 1.1540 (0.0126) 1.1560 (0.0203) 1.2013 (0.0068) 2.0090 (0.0083) 2.0692 (0.0891) 1.5596
Fuzzy D. 0.9749 (0.0226) 1.0222 (0.0415) 4.6159 (0.0754) 1.0384 (0.0098) 1.0978 (0.0126) 1.0825 (0.0045) 1.4157 (0.0064) 1.1742 (0.0072) 1.1320 (0.0056) 1.1541 (0.0054) 1.2088 (0.0117) 2.0096 (0.0067) 2.0398 (0.0852) 1.5358