Enhancing Dynamic Parameter Adaptation in the Bird Swarm Algorithm Using General Type-2 Fuzzy Analysis and Mathematical Functions

Abstract

The pursuit of continuous improvement across diverse processes presents a pressing challenge. Precision in manufacturing, efficient delivery route planning, and accurate diagnostics are imperative, prompting the exploration of innovative solutions. Nature-inspired algorithms offer a pathway for enhancing these processes. In this study, we address this challenge by dynamically adapting parameters in the Bird Swarm Algorithm using General Type-2 Fuzzy Systems, encompassing a range of rules and membership functions. Two complex case studies validate the effectiveness of our approach. The first evaluates Congress of Evolutionary Competition 2017 functions, while the second tackles the intricacies of Congress of Evolutionary Competition 2019 functions. Our methodology achieves an 97% improvement for Congress of Evolutionary Competition 2017 functions and a significant 70% enhancement for Congress of Evolutionary Competition 2019 functions. Notably, our results are benchmarked against the original method. Crucially, rigorous statistical analysis underscores the significant advancements facilitated by our proposed method. The comparison demonstrates clear and statistically significant improvements over the original approach. This study proves the marked impact of integrating General Type-2 Fuzzy Systems into the Bird Swarm Algorithm, presenting a promising avenue for addressing intricate optimization challenges in diverse domains.

1. Introduction

The realm of continuous process improvement has witnessed the integration of bio-inspired algorithms as a response to escalating demands for enhanced quality and precision. These algorithms serve as potent tools for seeking out optimal solutions across diverse problem domains, including the optimization of 5G networks [1], food processes [2], wireless sensor networks [3], detecting email spam [4], and lung tumor detection [5], among others.

The diversity of available metaheuristics underscores their potential, although it is crucial to acknowledge that an algorithm’s excellence in one context might not translate universally [6]. This inherent variability fuels a captivating realm of study and analysis. Among the array of contemporary metaheuristics, we encounter intriguing names such as the Orca Predation Algorithm [7], Alpine skiing optimization [8], the Horse Optimization Algorithm [9], the Starling Murmuration Optimizer [10], and the Barnacle Mating Optimizer [11]. Familiar methods such as Particle Swarm Optimization [12] and the Genetic Algorithm [13] also feature in this landscape. The underlying premise of this multifaceted exploration emphasizes the need for continual methodological enhancement, particularly when optimal solutions elude the grasp of existing algorithms. Various avenues are explored, spanning from hybridization methods [14,15,16] to the integration of fuzzy logic for dynamic parameter adaptation [17,18,19]. Fuzzy logic is an area of soft computing that involves the use of approximate rather than exact reasoning. This type of logic tries to build models of human reasoning that manifest its approximate, qualitative character. The aim is to provide the basis for approximate reasoning in handling imprecise propositions based on fuzzy set theory [20,21]. The versatility of fuzzy logic has allowed for its integration into various domains, from predictive modeling [22] and medicine [23] to construction engineering and management [24] and opinion mining [25].

In this research, we focus on the Bird Swarm Algorithm (BSA), which is inspired by how birds search for food, performing a dynamic parameter adaptation. Previous works [26,27] honed BSA performance through the application of Type-1 and Interval Type-2 fuzzy systems to address a variety of issues; tests were carried out using both classic and CEC2017 suite benchmark functions. The results demonstrated significant improvements compared to the original approach and other methods previously documented in the literature. Furthermore, our proposal was successfully applied to the optimization of various intelligent computing techniques, such as neural networks and fuzzy logic, which play a fundamental role in medical diagnosis. These advances contributed significantly to the improvement of the results obtained in this context. In our studies, we delve into the application of General Type-2 Fuzzy Systems (GT2 FS) to study this novel approach, properly named the General Bird Swarm Algorithm (GBSA). This application reveals remarkable efficacy, particularly in scenarios characterized by high levels of uncertainty.

Our innovation resides in the meticulous modification of the social acceleration (C) and cognitive acceleration (S) parameters, whereby we adjust them to exert a substantial influence on the algorithmic outcomes. This modification lays the foundation for a rigorous examination of GBSA’s potential through two distinct case studies. The first set of studies employs 30 functions from the Congress of Evolutionary Competition 2017 (CEC2017), comparing the results against the original method. To discern optimal performance, we also leverage diverse Mamdani-type fuzzy systems. The second case study turns to the evaluation of ten functions from the Congress of Evolutionary Competition 2019 (CEC2019), engaging different membership functions from those used in the preceding study. Notably, both case studies integrate four rule sets, each representing variations of increasing, decreasing, or combined rules.

The main goal of this research is to make a significant contribution to the realm of computer science by introducing and assessing the General Bird Swarm Algorithm (GBSA) as an enhanced iteration of the BSA algorithm, incorporating the innovative framework of General Type-2 fuzzy systems. The results derived from this research not only provide valuable insights into the tangible effects of these systems on optimizing intricate challenges but also serve as an impetus for future explorations in crafting intelligent adaptive algorithms.

The article is organized into the following sections. Section 2 covers the fundamental concepts, while Section 3 details the problem statement and the proposed method. Section 4 describes the results of various experiments and offers the statistical analysis and discussion. Finally, Section 5 outlines the conclusions obtained and future work to consider.

2. Basic Concepts

In this section, the basic concepts of the Bird Swarm Algorithm and General Type-2 Fuzzy Systems are presented. The BSA is an optimization algorithm based on the collective intelligence of a swarm of particles that is inspired by the behavior of birds in search of food. On the other hand, General Type-2 Fuzzy Systems are an extension of traditional fuzzy systems that allow for the capturing and managing of uncertainty in a more sophisticated way. The theoretical foundations and main characteristics of these concepts are explored, laying the foundations for understanding the improvement proposal presented in this work.

2.1. Bird Swarm Algorithm

Meng proposed the BSA in 2016 [28] to solve optimization problems [29,30,31,32,33]. It aims to imitate different bird behaviors, such as foraging, flight, and vigilance; these are based on their interactions and social behaviors.

The behaviors to be imitated by the algorithm are the following:

1. Foraging: The birds within the swarm may change their behavior. It is modelled with a stochastic decision when they carry out foraging or vigilance.

When each bird searches for food, it uses its own experience and the existing experience within the swarm. Such behavior can be analyzed as follows:

$$x_{i,j}^{t+1}=x_{i,j}^t+\left(p_{i,j}-x_{i,j}^t\right)Crand\left(0,1\right)+\left(g_j-x_{i,j}^t\right)Srand\left(0,1\right),$$

(1)

where p_i,j represents the best previous position of the i_th bird, and g_j corresponds to the best previously shared position in the swarm. C and S are the values of the coefficients of cognitive and social acceleration, respectively. $j\in \left[1,\dots ,D\right],rand\left(0,1\right)$ is used as independent numbers uniformly distributed in (0,1).

While birds are engaged in foraging behavior, they can recall and update the best experiences individually and within the swarm regarding food patches. Social information is shared instantly among the entire swarm.

When engaging in foraging behavior, birds have the ability to individually recall and update their best experiences, as well as instantly share social information among the entire swarm regarding food patches.

2. Vigilance: With this behavior, the birds will try to move to the center of the swarm to compete with others, but they will not move directly to the center of the swarm. This can be analyzed as follows:

$$x_{i,j}^{t+1}=x_{i,j}^t+A1\left(mea{n}_j-x_{i,j}^t\right)\times rand\left(0,1\right)+A2\left(p_{k,j}-x_{i,j}^t\right)\times rand\left(-1,1\right)$$

(2)

$$A1=a1\times \exp \left(-\frac{pFi{t}_i}{sumFit+\epsilon }\times N\right)$$

(3)

$$A2=a2\times exp\left(\left(\frac{pFi{t}_i-pFi{t}_k}{\left|pFi{t}_k-pFi{t}_i\right|+\epsilon }\right)\frac{N\times pFi{t}_k}{sumFit+\epsilon }\right)$$

(4)

where A1 and A2 are the effects caused by the interference when the birds move to the center of the swarm. pFit_i corresponds to the best value in the ith position; a1 and a2 are positive constants in [0, 2]. k is a positive integer between 0 and N, and sumFit represents the sum of the best fitness values in the swarm. ε corresponds to avoiding an error of zero-division.

Each bird in the swarm endeavors to move towards the center, aiming to maintain vigilance. This movement can influence the inference induced by the competition among the swarm. Birds with a larger food supply tend to position themselves closer to the center of the swarm, while those with a smaller supply may occupy relatively peripheral positions.

3. Flight: Periodically, the birds fly to other places; when this happens, they can be producers or scroungers. The birds with low reserves scrounge, and the birds with the largest food reserves are the producers. Birds with an intermediate reserve can switch between being scroungers and being producers. Mathematically, the representation of this behavior is:

$$x_{i,j}^{t+1}=x_{i,j}^t+\mathrm{rand}n\left(0,1\right)\times x_{i,j}^t,$$

(5)

$$x_{i,j}^{t+1}=x_{i,j}^t+\left(x_{k,j}^t-x_{i,j}^t\right)\times FL\times \mathrm{rand}\left(0,1\right),$$

(6)

where FQ is a positive integer, meaning the birds may move to another place every FQ interval. FL (FL ∈ [0, 2]) corresponds to the scrounger following the producer to look for food. randn (0, 1) is a Gaussian-distributed random number with a mean of 0 and a standard deviation of 1.

Birds engage in different activities within the BSA framework. Producers are actively involved in foraging for food, while scroungers randomly follow a producer in their search for food. Algorithm 1 presents the pseudocode of the BSA, outlining the step-by-step procedure for its execution.

Algorithm 1: Bird Swarm Algorithm, BSA Pseudocode [28]

1:   Input N: the number of individuals (birds) contained by the population

2:        M: the maximum number of iterations

3:        FQ: the frequency of birds’ flight behaviors

4:        P: the probability of foraging for food

5:        C, S, a1, a2, FL: five constant parameters

6:   t=0; Initialize the population and define the related parameters

7:  Evaluate the N individuals’ fitness value, and find the best solution

8:  While (t < M)

9:     If (t % FQ ≠ 0)

10:       For i = 1 : N

11:        If rand (0,1) < P

12:          Birds forage for food (Equation (1))

13:        Else

14:          Birds keep vigilance (Equation (2))

15:        End if

16:       End for

17:      Else

18:       Divide the swarm into two parts: producers and scroungers.

19:         For i = 1 : N

20:         If i is a producer

21:             Producing (Equation (5))

22:          Else

23:          Scrounging (Equation (6))

24:       End if   End For

25:     End If  Evaluate new solutions

26:    if the new solutions are better than their previous ones, update then

27:     Find the best solutions

28:   t=t+1; End while

29:   Output: the individual with the best objective function value in the

2.2. General Type-2 Fuzzy System

General Type-2 Fuzzy Logic (GT2FL) is an extension of Type-1 Fuzzy Logic (T1FL) that allows for the modeling of uncertainties that cannot be modeled by Type-1 Fuzzy Sets [34,35,36].

This type of fuzzy system permits uncertainty to be modeled in an efficient form. This is expressed mathematically as:

$$\tilde{A}=\{\left(x,u\right), {\mu }_{\tilde{A}}\left(x,u\right)|{\forall }_x \in X, {\forall }_u\in \left[0,1\right]\}$$

(7)

where X is the universe of the primary variable of $\tilde{A},x.$ The representation of the 3D membership function is denoted by ${\mu }_{\tilde{A}}\left(x,u\right),$ where $x\in X$ and $u\in J_x\subseteq \left[0,1\right]$ and $0\le {\mu }_{\tilde{A}}\left(x,u\right)\le 1$. In a mathematical form, this is represented as follows:

$$\tilde{A}={{\displaystyle \int }}_{x \in X}^0{{\displaystyle \int }}_{u\in J^u{}_{x\subseteq \left[0,1\right]}}^0\frac{{\mu }_{\tilde{A}}\left(x,u\right)}{\left(x,u\right)}$$

(8)

where ∫∫ denotes the union for $x$ and $u$.

For readers seeking a deeper understanding, references [34,35,36] provide valuable insights into the theoretical foundations of GT2FL. By focusing on Equations (7) and (8) onwards, we aim to offer a clear and concise introduction to GT2FL.

3. Problem Statement and Proposed Method

Previously, we worked with Type-1 and Interval Type-2 Fuzzy Systems for the dynamic parameter adaptation of the BSA to determine whether, with this modification, there is an improvement in the results provided [26]. This new proposal applies General Type-2 Fuzzy Systems (GT2 FS) in a similar study to analyze their performance. The parameters to which this modification is made are the social (S) and cognitive (C) acceleration coefficients. Figure 1 shows the flowchart indicating the section to be modified, corresponding to the inspiration of the birds’ foraging.

The first fuzzy system designed presents GaussGauss membership functions (MFs). The inputs correspond to the iterations and the diversity, while the outputs are the parameters C and S. The iterations input and the diversity input have the MFs “low”, “medium”, and “high” as linguistic variables. For outputs C and S, there are five MFs, to which “low”, “medium low”, “medium”, “medium high” and “high” correspond as the linguistic variables.

Regarding the corresponding ranges, the input variable “Iteration” spans from 0 to 1, reflecting the normalized percentage of iterations. In the case of the “Diversity” variable, its range also extends between 0 and 1, considering the normalization of individual dispersion. As for the value range of the “C” and “S” outputs, it encompasses values from 2 to 5. This choice is grounded in comprehensive tests conducted on the method, consistently yielding positive results within this specific interval.

The design of the fuzzy system with GaussGauss MFs is shown in Figure 2.

Equation (9) presents the parametrization of a Gaussian primary MF with an uncertain mean and a Gaussian secondary MF “gaussmgausstype2”. The parameter σ represents the standard deviation of the primary MF. Additionally, $m_1$ and $m_2$ refer to the left and right means, respectively, of the Gaussian MF with uncertain mean [37,38].

$$\begin{gathered} \tilde{\mu }\left(x,u\right)=gaussmgausstype2\left(x,u,\left[\sigma ,m_1,m_2,\rho \right]\right) \\ \tilde{\mu }\left(x,u\right)=\mathit{exp}\left[-\frac{1}{2}{\left(\frac{u-p_x}{{\sigma }_u}\right)}^2\right] \\ {\mu }_1(x)=\exp \left[-\frac{1}{2}{\left(\frac{x-m_1}{\sigma }\right)}^2\right] \\ {\mu }_2(x)=\exp \left[-\frac{1}{2}{\left(\frac{x-m_2}{\sigma }\right)}^2\right] \\ \overline{\mu }\left(x\right)=\{\begin{array}{cc}{\mu }_1\left(x\right)& x<m_1\\ 1& m_1\le x\le m_2\\ {\mu }_2\left(x\right)& x>m_2\end{array} \\ \underset{\_}{\mu }\left(x\right)=\{\begin{array}{cc}{\mu }_2\left(x\right)& x\le \frac{m_1+m_2}{2}\\ {\mu }_1\left(x\right)& x>\frac{m_1+m_2}{2}\end{array} \\ {p}_x=\mathit{exp}\left[-\frac{1}{2}{\left(\frac{x-m}{\sigma }\right)}^2\right]\mathrm{where}m=\frac{m_1+m_2}{2} \\ \delta =\overline{\mu }\left(x\right)-\underset{\_}{\mu }\left(x\right) \\ {\sigma }_u=\left(1+\rho \right)\frac{\delta }{2\sqrt{3}}+\epsilon \end{gathered}$$

(9)

The parameter ρ denotes the fraction of uncertainty associated with the support of the secondary MF. The second fuzzy system analyzed has the same structure, designed with TrianGauss membership functions. The design of this fuzzy system is presented in Figure 3.

The parameterization of a triangular primary General Type-2 MF with a Gaussian secondary MF, referred to as “trigausstype2,” is presented in Equation (10). The parameters $a_1$, $b_1,$ and $c_1$ correspond to the upper MF, while $a_2$, $b_2,$ and $c_2$ represent the lower MFs [37,39].

$$\begin{gathered} \mu \left(x,u\right)=trigausstype2\left(x,u,\left[a_1,b_1,c_1,a_2,b_2,c_2,\rho \right]\right) \\ \mu \left(x,u\right)=exp\left[-\frac{1}{2}{\left(\frac{u-p_x}{{\sigma }_u}\right)}^2\right] \mathrm{where} \\ {\mu }_1\left(x\right)=\max \left(\min \left(\frac{x-a_1}{b_1-a_1},\frac{c_1-x}{c_1-b_1}\right),0\right)\mathrm{and} \\ {\mu }_2\left(x\right)=\max \left(\min \left(\frac{x-a_2}{b_2-a_2},\frac{c_2-x}{c_2-b_2}\right),0\right) \\ \overline{\mu }\left(x\right)=\{\begin{array}{cc}max\left({\mu }_1\left(x\right),{\mu }_2\left(x\right)\right)& \forall x\notin \left(b1,b2\right)\\ 1& \forall x\in \left(b1,b2\right)\end{array} \\ \underset{\_}{\mu }\left(x\right)=min\left({\mu }_1\left(x\right),{\mu }_2\left(x\right)\right) \\ {p}_x=\max \left(\min \left(\frac{x-a_x}{b_x-a_x},\frac{c_x-x}{c_x-b_x}\right),0\right), \\ \mathrm{where}{a}_x=\frac{a_1+a_2}{2}, b_x=\frac{b_1+b_2}{2}, c_x=\frac{c_1+c_2}{2} \\ \delta =\overline{\mu }\left(x\right)-\underset{\_}{\mu }\left(x\right) \\ {\sigma }_u=\frac{1+\rho }{2\sqrt{3}}\delta +\epsilon \end{gathered}$$

(10)

The parameter ρ represents the fraction of uncertainty in the support of the secondary MF, while δ denotes the support of the triangular or trapezoidal MFs. For the third analysis, the GbellGbell MFs are used. The design of this fuzzy system is presented in Figure 4.

Equation (11) presents the parametrization of a general bell-shaped primary MF and a general bell shape secondary MF “gbellmgbelltype2”:

$$\begin{gathered} \mu \left(x,u\right)=gbellmgbelltype2\left(x,u,\left[a,b,c_1,c_2,b_u\right]\right) \\ {p}_x=\frac{1}{1+{\left[{\left(\frac{x-c}{a}\right)}^2\right]}^b},\mathrm{where}c=\frac{c_1+c_2}{2} \\ {\mu }_1\left(x\right)=\frac{1}{1+{\left[{\left(\frac{x-c_1}{a}\right)}^2\right]}^b} \\ {\mu }_2\left(x\right)=\frac{1}{1+{\left[{\left(\frac{x-c_2}{a}\right)}^2\right]}^b} \\ \overline{\mu }\left(x\right)=\{\begin{array}{cc}{\mu }_1\left(x\right)& x<c_1\\ 1& c_1\le x\le c_2\\ {\mu }_2\left(x\right)& x>c_2\end{array} \\ \underset{\_}{\mu }\left(x\right)=\{\begin{array}{cc}{\mu }_2\left(x\right)& x\le c\\ {\mu }_1\left(x\right)& x>c\end{array} \\ \delta =\overline{\mu }\left(x\right)-\underset{\_}{\mu }\left(x\right) \\ {a}_u=\frac{\delta }{2\sqrt{6}}+\epsilon \\ \mu \left(x,u\right)=\frac{1}{1+{\left[{\left(\frac{u-p_x}{a_u}\right)}^2\right]}^{b_u}} \end{gathered}$$

(11)

$b_u$ is a parameter of the secondary MF that modifies the support and the core.

The differently designed fuzzy systems were tested with four rules sets. These sets have nine rules, either ascending, descending, or a combination of both, depending on the case to be analyzed.

Table 1. Fuzzy rules used in the first GT2 FS.

Rule	Antecedent		Consequent
	Iteration	Diversity	C	S
1	Low	Low	High	Low
2	Low	Medium	Medium High	Medium
3	Low	High	Medium High	Medium Low
4	Medium	Low	Medium High	Medium Low
5	Medium	Medium	Medium	Medium
6	Medium	High	Medium Low	Medium High
7	High	Low	Medium	High
8	High	Medium	Medium Low	Medium High
9	High	High	Low	High

presents the first set of rules, where C is decreasing and S is increasing.

The second set of rules is presented in

Table 2. Fuzzy rules used in the second GT2 FS.

Rule	Antecedent		Consequent
	Iteration	Diversity	C	S
1	Low	Low	Low	High
2	Low	Medium	Medium	Medium High
3	Low	High	Medium Low	Medium High
4	Medium	Low	Medium Low	Medium High
5	Medium	Medium	Medium	Medium
6	Medium	High	Medium High	Medium Low
7	High	Low	Medium	High
8	High	Medium	Medium High	Medium Low
9	High	High	High	Low

, where C increases and S decreases.

Table 3. Fuzzy rules used in the third GT2 FS.

Rule	Antecedent		Consequent
	Iteration	Diversity	C	S
1	Low	Low	Low	High
2	Low	Medium	Low	Medium
3	Low	High	Medium	Medium Low
4	Medium	Low	Medium Low	Medium Low
5	Medium	Medium	Medium	Medium
6	Medium	High	Medium	Medium High
7	High	Low	Medium	High
8	High	Medium	Medium High	Medium High
9	High	High	High	High

shows the third combination of rules; C works with medium-low iterations, while S uses medium-high iterations.

The last set of rules is illustrated in

Table 4. Fuzzy rules used in the four GT2 FS.

Rule	Antecedent		Consequent
	Iteration	Diversity	C	S
1	Low	Low	High	High
2	Low	Medium	High	Medium
3	Low	High	Medium	Medium High
4	Medium	Low	High	Medium High
5	Medium	Medium	Medium	Medium
6	Medium	High	Medium	Medium Low
7	High	Low	Medium Low	Medium
8	High	Medium	Medium	Medium Low
9	High	High	Low	Low

, where C and S work with high and medium-high iterations.

Regarding the calculation of the percentage of iterations, the current iteration is taken into account over the total of these, which are calculated as follows:

$$Iter=\frac{Citer}{Titer}$$

(12)

Citer corresponds to the current iteration, while Titer refers to the total iterations of the algorithm.

On the other hand, diversity is defined as the dispersion of individuals within the population [40,41], as represented by the following expression:

$$Diversity\left(S\left(t\right)\right)=\frac{1}{n_s}{\displaystyle \sum }_{i=1}^{n_s}\sqrt{{\displaystyle \sum }_{j=1}^{n_x}{\left(X_{ij}\left(t\right)-{\overline{X}}_j\left(t\right)\right)}^2}$$

(13)

In the context provided, the notation used is as follows: S represents the population, $n_s$ represents the number of individuals within the population, and $n_x$ signifies the number of dimensions of each individual. The variable $X_{ij}$ denotes the position of individual i, while $\overline{X}$ represents the position of the best individual in the population.

4. Results and Discussion

In this section, the results obtained from the experiments carried out for both case studies are described.

4.1. First Study Case

In the first case study, 30 functions of CEC2017 are used; for this purpose, both the original and the proposed methods were experimented with.

Table 5. Parameters used by the algorithms to perform the experiments.

	Iteration	Population	Dim	FQ	a1	a2	C	S
BSA	1000	40	30	3	1	1	1.5	1.5
GBSA	1000	40	30	3	1	1	Dynamic	Dynamic

presents and defines the parameters used by both methods, and it can be seen that the difference between these is the values taken by the parameters Cand S. The experiments of the proposed method were implemented in Matlab^® 2021b and were executed on a computer with an Intel I7 processor, with 16 GB of RAM and the Windows 11 operating system. It is worth mentioning that special programs are used for the generation and design of General Type-2 fuzzy systems, which were developed by the research group [34,37,39].

Table 6. CEC2017 functions description.

Function	No.	Name Function	Fi
Unimodal Functions	1	Shifted and Rotated Bent Cigar	100
	2	Shifted and Rotated Sum of Different Power	200
	3	Shifted and Rotated Zakharov	300
Simple Multimodal Functions	4	Shifted and Rotated Rosenbrock	400
	5	Shifted and Rotated Rastrigin’s	500
	6	Shifted and Rotated Expanded Scaffer’s F6	600
	7	Shifted and Rotated Lunacek Bi-Rastrigin	700
	8	Shifted and Rotated Non-Continuous Rastrigin’s	800
	9	Shifted and Rotated Levy	900
	10	Shifted and Rotated Schwefel’s	1000
Hybrid Functions	11	Hybrid Function 1 (N = 3)	1100
	12	Hybrid Function 2 (N = 3)	1200
	13	Hybrid Function 3 (N = 3)	1300
	14	Hybrid Function 4 (N = 4)	1400
	15	Hybrid Function 5 (N = 4)	1500
	16	Hybrid Function 6 (N = 4)	1600
	17	Hybrid Function 6 (N = 5)	1700
	18	Hybrid Function 6 (N = 5)	1800
	19	Hybrid Function 6 (N = 5)	1900
	20	Hybrid Function 6 (N = 6)	2000
Composition Functions	21	Composition Function 1 (N = 3)	2100
	22	Composition Function 2 (N = 3)	2200
	23	Composition Function 3 (N = 4)	2300
	24	Composition Function 4 (N = 4)	2400
	25	Composition Function 5 (N = 5)	2500
	26	Composition Function 6 (N = 5)	2600
	27	Composition Function 7 (N = 6)	2700
	28	Composition Function 8 (N = 6)	2800
	29	Composition Function 9 (N = 3)	2900
	30	Composition Function 10 (N = 3)	3000

defines the mathematical functions of the CEC2017; in the first column, the classification is described. Column 3 provides each function name, and Column 4 shows the global minimum value that each function reaches.

Table 7. Results obtained for the GaussGauss MFs designs, tested with CEC2017 functions.

No.		Original	E×p3
1	Average	3.180 × 1010	1.781 × 109
	STD	8.650 × 109	8.623 × 108
2	Average	1.027 × 1046	5.981 × 1030
	STD	7.631 × 1046	2.745 × 1031
3	Average	7.630 × 104	4.382 × 104
	STD	1.159 × 104	8.945 × 103
4	Average	6.745 × 103	7.782 × 102
	STD	3.096 × 103	1.336 × 102
5	Average	8.488 × 102	7.365 × 102
	STD	4.287 × 101	4.007 × 101
6	Average	6.754 × 102	6.472 × 102
	STD	8.922 × 10+	1.155 × 101
7	Average	1.356 × 103	1.083 × 103
	STD	8.106 × 101	6.379 × 101
8	Average	1.088 × 103	9.958 × 102
	STD	3.624 × 101	2.760 × 101
9	Average	7.636 × 103	4.369 × 103
	STD	1.432 × 103	1.482 × 103
10	Average	7.434 × 103	7.146 × 103
	STD	6.253 × 102	8.492 × 102
11	Average	5.847 × 103	1.628 × 103
	STD	2.296 × 103	1.689 × 102
12	Average	2.988 × 109	9.269 × 107
	STD	2.085 × 109	7.411 × 107
13	Average	5.683 × 108	2.794 × 106
	STD	1.437 × 109	1.658 × 107
14	Average	2.576 × 105	4.109 × 104
	STD	5.443 × 105	1.714 × 105
15	Average	1.592 × 107	4.673 × 104
	STD	4.983 × 107	3.576 × 104
16	Average	4.126 × 103	3.251 × 103
	STD	6.388 × 102	4.145 × 102
17	Average	2.918 × 103	2.410 × 103
	STD	3.704 × 102	2.557 × 102
18	Average	2.237 × 106	7.948 × 105
	STD	4.123 × 106	4.848 × 106
19	Average	3.017 × 107	2.534 × 106
	STD	6.582 × 107	1.313 × 107
20	Average	2.831 × 103	2.544 × 103
	STD	2.266 × 102	1.794 × 102
21	Average	2.649 × 103	2.504 × 103
	STD	5.301 × 101	4.408 × 101
22	Average	8.312 × 103	4.059 × 103
	STD	1.123 × 103	2.387 × 103
23	Average	3.352 × 103	3.027 × 103
	STD	1.546 × 102	1.096 × 102
24	Average	3.537 × 103	3.256 × 103
	STD	1.540 × 102	1.492 × 102
25	Average	4.205 × 103	3.094 × 103
	STD	4.789 × 102	7.383 × 101
26	Average	9.949 × 103	6.481 × 103
	STD	9.100 × 102	1.370 × 103
27	Average	3.768 × 103	3.486 × 103
	STD	2.755 × 102	1.971 × 102
28	Average	5.354 × 103	3.459 × 103
	STD	6.885 × 102	8.772 × 101
29	Average	6.121 × 103	4.749 × 103
	STD	9.945 × 102	5.622 × 102
30	Average	7.522 × 107	6.593 × 106
	STD	1.239 × 108	9.192 × 106

shows the results of the experimentation carried out with the GT2 FS, where the GaussGauss membership functions were used. The average and the standard deviation are presented. Column 3 shows the results obtained with the original method, while Column 4 presents the results of the experiments conducted with GBSA using rule set 3, because better results were obtained with this particular variant of GBSA.

From the results obtained and analyzing the average in each set of experiments, we can see that, for Experiment 1, five improvements were achieved in Functions 1, 3, 10, 18, and 22; for Experiment 2, the method only achieved two improvements, in Functions 8, and 25. For Experiment 3, fourteen improvements were achieved in Functions 2, 4, 6, 7, 9, 11, 12, 19, 20, 21, 23, 26, 28, and 29, and, for Experiment 4, eight improvements were obtained in Functions 5, 13, 14, 15, 16, 17, 27, and 30. As Experiment 3 is the set of rules that obtains the most significant number of improvements in the results of the mathematical functions, these are taken as a basis for experimenting with other membership functions. It is worth mentioning that each improvement in the results is highlighted in bold. It is important to highlight that the choice of the mean as a reference point for comparing the original method and the proposed method is made with the intent of identifying potential significant changes in the obtained outcomes. This approach enables us to evaluate the differences between the approaches more precisely and robustly. The specifics and outcomes of this comparison are addressed and analyzed in the following section through a detailed statistical analysis.

Table 8. Results obtained for the different FS designs; experiment carried out with CEC2017 functions.

No.		GaussGauss	TrianGauss	GbellGbell
1	Average	1.781 × 109	1.778 × 109	1.657 × 109
	STD	8.623 × 108	8.214 × 108	6.997 × 108
2	Average	5.981 × 1030	2.267 × 1031	1.487 × 1041
	STD	2.745 × 1031	1.147 × 1032	1.487 × 1042
3	Average	4.382 × 104	4.202 × 104	4.440 × 104
	STD	8.945 × 103	9.658 × 103	9.395 × 103
4	Average	7.782 × 102	7.943 × 102	7.759 × 102
	STD	1.336 × 102	1.329 × 102	1.894 × 102
5	Average	7.365 × 102	7.356 × 102	7.242 × 102
	STD	4.007 × 10+1	3.820 × 101	3.934 × 101
6	Average	6.472 × 10+2	6.501 × 102	6.477 × 102
	STD	1.155 × 10+1	1.108 × 101	1.028 × 101
7	Average	1.083 × 10+3	1.089 × 103	1.073 × 103
	STD	6.379 × 10+1	6.114 × 101	5.731 × 101
8	Average	9.958 × 10+2	9.897 × 102	9.966 × 102
	STD	2.760 × 10+1	2.908 × 101	2.844 × 101
9	Average	4.369 × 10+3	4.581 × 103	4.464 × 103
	STD	1.482 × 103	1.330 × 103	1.526 × 103
10	Average	7.146 × 103	6.933 × 103	6.936 × 103
	STD	8.492 × 102	7.594 × 102	8.918 × 102
11	Average	1.628 × 103	1.644 × 103	1.635 × 103
	STD	1.689 × 102	1.794 × 102	1.657 × 102
12	Average	9.269 × 107	1.236 × 108	1.166 × 108
	STD	7.411 × 107	1.045 × 108	9.353 × 107
13	Average	2.794 × 106	9.930 × 105	2.785 × 107
	STD	1.658 × 107	1.831 × 106	2.718 × 108
14	Average	4.109 × 104	4.986 × 104	3.721 × 104
	STD	1.714 × 105	1.402 × 105	1.131 × 105
15	Average	4.673 × 104	4.731 × 104	1.767 × 107
	STD	3.576 × 104	4.330 × 104	1.763 × 108
16	Average	3.251 × 103	3.313 × 103	3.225 × 103
	STD	4.145 × 102	4.546 × 102	3.943 × 102
17	Average	2.410 × 103	2.440 × 103	2.422 × 103
	STD	2.557 × 102	2.868 × 102	2.333 × 102
18	Average	7.948 × 105	5.130 × 105	4.112 × 105
	STD	4.848 × 106	1.314 × 106	5.422 × 105
19	Average	2.534 × 106	4.943 × 105	5.061 × 105
	STD	1.313 × 107	7.708 × 105	7.777 × 105
20	Average	2.544 × 103	2.536 × 103	2.561 × 103
	STD	1.794 × 102	1.697 × 102	1.899 × 102
21	Average	2.504 × 103	2.521 × 103	2.511 × 103
	STD	4.408 × 101	4.128 × 101	3.535 × 101
22	Average	4.059 × 103	3.759 × 103	4.207 × 103
	STD	2.387 × 103	1.970 × 103	2.333 × 103
23	Average	3.027 × 103	3.023 × 103	3.040 × 103
	STD	1.096 × 102	1.226 × 102	1.192 × 102
24	Average	3.256 × 103	3.269 × 103	3.236 × 103
	STD	1.492 × 102	1.664 × 102	1.607 × 102
25	Average	3.094 × 103	3.092 × 103	3.087 × 103
	STD	7.383 × 101	6.931 × 101	6.474 × 101
26	Average	6.481 × 103	6.693 × 103	6.632 × 103
	STD	1.370 × 103	1.661 × 103	1.478 × 103
27	Average	3.486 × 103	3.474 × 103	3.470 × 103
	STD	1.971 × 102	2.024 × 102	1.773 × 102
28	Average	3.459 × 103	3.465 × 103	3.470 × 103
	STD	8.772 × 101	8.233 × 101	9.007 × 101
29	Average	4.749 × 103	4.743 × 103	4.744 × 103
	STD	5.622 × 102	5.443 × 102	5.336 × 102
30	Average	6.593 × 106	6.566 × 106	6.482 × 106
	STD	9.192 × 106	6.232 × 106	8.134 × 106

compares the results obtained with different membership functions using rule set number 3, corresponding to parameter C working with medium-low iterations, while S works with medium-high iterations.

The average of the experimental set is considered to analyze the result obtained, where the best result is highlighted in bold. Column 3 shows the results from the GaussGauss membership functions, for which ten improvements were obtained. Column 4 shows the results obtained using the TrianGauss membership functions, achieving nine improvements, while Column 5 presents the results obtained using GbellGbell membership function, achieving eleven improvements.

A comparative analysis is conducted between the proposed method and other bio-inspired algorithms found in the literature. Similar scenarios were considered for the comparison, with each algorithm operating under the conditions of 1000 iterations and 30 dimensions. Specifically, the Fibonacci search-based Moth Flame Optimizer [42] and the Bacterial Foraging Optimization with Chaotic Chemotaxis Step Length, Gaussian Mutation, and Chaotic Local Search [43] were selected as the comparison algorithms. Remarkably, our proposed method exhibits a greater number of performance improvements in the presented results. The comparative results for these methods are presented in

Table 9. Comparison between bio-inspired methods.

				GBSA
No.		Es-MFO	CCGBFO	GaussGauss	TrianGauss	GbellGbell
1	Average	6.21 × 1010	6.244 × 1010	1.781 × 109	1.778 × 109	1.657 × 109
	STD	1.01 × 1010	5.453 × 109	8.623 × 108	8.214 × 108	6.997 × 108
3	Average	1.25 × 105	8.408 × 104	4.382 × 104	4.202 × 104	4.440 × 104
	STD	3.07 × 104	6.060 × 103	8.945 × 103	9.658 × 103	9.395 × 103
4	Average	1.66 × 104	1.834 × 104	7.782 × 102	7.943 × 102	7.759 × 102
	STD	3.93 × 104	2.685 × 103	1.336 × 102	1.329 × 102	1.894 × 102
5	Average	8.80 × 102	9.36 × 102	7.365 × 102	7.356 × 102	7.242 × 102
	STD	1.94 × 102	2.973 × 101	4.007 × 101	3.820 × 101	3.934 × 101
6	Average	6.52 × 102	6.870 × 102	6.472 × 102	6.501 × 102	6.477 × 102
	STD	2.82 × 101	7.613 × 100	1.155 × 101	1.108 × 101	1.028 × 101
7	Average	1.46 × 103	1.417 × 103	1.083 × 103	1.089 × 103	1.073 × 103
	STD	2.40 × 101	2.584 × 101	6.379 × 101	6.114 × 101	5.731 × 101
8	Average	1.08 × 103	1.151 × 103	9.958 × 102	9.897 × 102	9.966 × 102
	STD	1.50 × 102	1.630 × 101	2.760 × 101	2.908 × 101	2.844 × 101
9	Average	1.25 × 104	9.377 × 103	4.369 × 103	4.581 × 103	4.464 × 103
	STD	1.13 × 104	1.462 × 103	1.482 × 103	1.330 × 103	1.526 × 103
10	Average	6.83 × 103	7.695 × 103	7.146 × 103	6.933 × 103	6.936 × 103
	STD	7.88 × 102	5.802 × 102	8.492 × 102	7.594 × 102	8.918 × 102
11	Average	5.31 × 103	9.693 × 103	1.628 × 103	1.644 × 103	1.635 × 103
	STD	1.95 × 103	2.120 × 103	1.689 × 102	1.794 × 102	1.657 × 102
12	Average	2.31 × 109	1.555 × 1010	9.269 × 107	1.236 × 108	1.166 × 108
	STD	7.40 × 109	3.501 × 109	7.411 × 107	1.045 × 108	9.353 × 107
13	Average	1.04 × 108	1.504 × 1010	2.794 × 106	9.930 × 105	2.785 × 107
	STD	2.24 × 108	3.501 × 109	1.658 × 107	1.831 × 106	2.718 × 108
14	Average	1.39 × 106	7.806 × 106	4.109 × 104	4.986 × 104	3.721 × 104
	STD	1.99 × 106	6.612 × 106	1.714 × 105	1.402 × 105	1.131 × 105
15	Average	8.93 × 108	1.35 × 109	4.673 × 104	4.731 × 104	1.767 × 107
	STD	2.25 × 109	5.767 × 18	3.576 × 104	4.330 × 104	1.763 × 108
16	Average	3.23 × 103	6.699 × 103	3.251 × 103	3.313 × 103	3.225 × 103
	STD	3.80 × 102	1.296 × 103	4.145 × 102	4.546 × 102	3.943 × 12
17	Average	2.46 × 103	6.446 × 103	2.410 × 103	2.440 × 103	2.422 × 103
	STD	2.35 × 102	4.536 × 103	2.557 × 102	2.868 × 102	2.333 × 102
18	Average	1.02 × 107	1.24 × 108	7.948 × 105	5.130 × 105	4.112 × 105
	STD	1.54 × 107	9.270 × 107	4.848 × 106	1.314 × 106	5.422 × 105
19	Average	1.33 × 109	1.189 × 109	2.534 × 106	4.943 × 105	5.061 × 105
	STD	2.70 × 109	5.865 × 108	1.313 × 107	7.708 × 105	7.777 × 105
20	Average	2.73 × 103	2.949 × 103	2.544 × 103	2.536 × 103	2.561 × 103
	STD	2.52 × 102	1.818 × 102	1.794 × 102	1.697 × 102	1.899 × 102
21	Average	2.52 × 103	2.783 × 103	2.504 × 103	2.521 × 103	2.511 × 103
	STD	3.36 × 101	4.853 × 101	4.408 × 101	4.128 × 101	3.535 × 101
22	Average	6.73 × 103	9.460 × 103	4.059 × 103	3.759 × 103	4.207 × 103
	STD	2.30 × 103	4.739 × 102	2.387 × 103	1.970 × 103	2.333 × 103
23	Average	2.87 × 103	3.639 × 103	3.027 × 103	3.023 × 103	3.040 × 103
	STD	3.65 × 101	1.570 × 102	1.096 × 102	1.226 × 102	1.192 × 102
24	Average	3.05 × 103	3.869 × 103	3.256 × 103	3.269 × 103	3.236 × 103
	STD	4.10 × 101	1.454 × 102	1.492 × 102	1.664 × 102	1.607 × 102
25	Average	6.39 × 103	5.859 × 103	3.094 × 103	3.092 × 103	3.087 × 103
	STD	2.92 × 103	5.048 × 102	7.383 × 101	6.931 × 101	6.474 × 101
26	Average	7.74 × 103	1.233 × 104	6.481 × 103	6.693 × 103	6.632 × 103
	STD	3.44 × 103	7.258 × 102	1.370 × 103	1.661 × 103	1.478 × 103
27	Average	3.29 × 103	3.404 × 103	3.486 × 103	3.474 × 103	3.470 × 103
	STD	2.52 × 101	1.983 × 102	1.971 × 102	2.024 × 102	1.773 × 102
28	Average	7.09 × 103	3.327 × 13	3.459 × 103	3.465 × 103	3.470 × 103
	STD	3.03 × 103	2.867 × 101	8.772 × 101	8.233 × 101	9.007 × 101
29	Average	4.36 × 103	1.090 × 104	4.749 × 103	4.743 × 103	4.744 × 103
	STD	3.06 × 102	4.053 × 103	5.622 × 102	5.443 × 102	5.336 × 102
30	Average	3.97 × 106	2.160 × 109	6.593 × 106	6.566 × 106	6.482 × 106
	STD	3.29 × 106	1.264 × 109	9.192 × 106	6.232 × 106	8.134 × 106

.

The most favorable results are highlighted, and it can be observed that, with the different versions of the proposed method, there are 23 improved outcomes out of the 30 functions of the CEC2017.

It is worth mentioning that, for this comparison, Function 2 is not taken into account, since many authors do not consider it due to the level of stability it presents when experimenting with it.

4.2. Second Study Case

For the second case study, the CEC2019 functions are experimented with, which present greater complexity; as in the previous analysis, both method parameter settings are performed, as presented in Table 5.

The CEC2019 functions are listed in

Table 10. CEC2019 functions description.

ID	Formulation	Dimensions	Range
CEC01	Storn’s Chebyshev Polynomial Fitting Problem	9	[−8192, 8192]
CEC02	lnverse Hilbert Matrix Problem	16	[−16,384, 16,384]
CEC03	Lennard–Jones Minimum Energy Cluster	18	[−4, 4]
CEC04	Rastrigin’s Function	10	[−100, 100]
CEC05	Griewank’s Function	10	[−100, 100]
CEC06	Weierstrass Function	10	[−100, 100]
CEC07	Modified Schwefel’s Function	10	[−100, 100]
CEC08	Expanded Schaffer’s F6 Function	10	[−100, 100]
CEC09	Happy Cat Function	10	[−100, 100]
CEC10	Ackley’s Function	10	[−100, 100]

. The first column shows the function ID, the second column is the function name; the third column represents the dimensions, and the fourth column presents the limits reached by the function. It is worth mentioning that, for all these functions, their global minimum is 1.

Different experiments are carried out to determine the set of rules with which the best results are obtained, and these are listed in

Table 11. Results obtained for the GaussGauss MFs designs for experiments with with CEC2019 functions.

No.		Original	Exp1
1	Average	9.278 × 104	4.098 × 104
	STD	7.631 × 104	2.720 × 103
2	Average	1.763 × 101	1.734 × 101
	STD	2.328 × 10−1	6.916 × 10−4
3	Average	1.270 × 101	1.270 × 101
	STD	1.563 × 104	4.797 × 10−6
4	Average	5.832 × 103	3.152 × 102
	STD	3.345 × 103	3.322 × 102
5	Average	3.299 × 10+0	1.450 × 100
	STD	9.886 × 10−1	2.223 × 10−1
6	Average	1.042 × 101	1.041 × 101
	STD	8.138 × 10−1	7.585 × 10−1
7	Average	4.233 × 102	3.741 × 102
	STD	2.222 × 102	2.193 × 102
8	Average	5.249 × 10+0	5.403 × 100
	STD	6.627 × 10−1	1.056 × 100
9	Average	6.068 × 102	3.177 × 100
	STD	4.416 × 102	4.774 × 10−1
10	Average	2.038 × 101	2.023 × 101
	STD	2.912 × 10−1	1.160 × 100

. For this case, the GaussGauss membership functions are used. Column 3 shows the original method results, while Column 4 shows the GBSA results using rule set 1.

A comparison is made between the results of the original and proposed methods to determine which of the experiments obtains a better result. With Experiment 1, presented in Column 4, four improved results were achieved, with improvements in Functions 1, 2, 9, and 10; with Experiment 2, three improvements were achieved, in Functions 6, 7, and 8. Meanwhile, in Experiment 3, only two improvements were obtained in Functions 4 and 5. Based on this analysis, it can be determined that the most significant number of improvements is obtained with rule set 1, which will be used to experiment with the different membership functions.

The experimentation with different membership functions is presented in

Table 12. Results obtained for the different FS designs in experiments with CEC2019 functions.

No.		GaussGauss	TrianGauss	GbellGbell
1	Average	4.098 × 10+04	4.701 × 10+04	4.841 × 10+04
	STD	2.720 × 10+03	5.991 × 10+03	7.620 × 10+03
2	Average	1.734 × 10+01	1.736 × 10+01	1.739 × 10+01
	STD	6.916 × 10−04	1.799 × 10−02	3.914 × 10−02
3	Average	1.270 × 10+01	1.270 × 10+01	1.270 × 10+01
	STD	4.797 × 10−06	1.246 × 10−05	1.140 × 10−05
4	Average	3.152 × 102	1.453 × 102	1.547 × 102
	STD	3.322 × 102	8.810 × 101	1.195 × 102
5	Average	1.450 × 10	1.322 × 10+	1.346 × 10
	STD	2.223 × 10−1	1.581 × 10−1	1.711 × 10−1
6	Average	1.041 × 101	1.062 × 101	1.044 × 101
	STD	7.585 × 10−1	5.790 × 10−1	7.951 × 10−1
7	Average	3.741 × 102	6.176 × 102	6.307 × 102
	STD	2.193 × 102	2.121 × 102	2.011 × 102
8	Average	5.403 × 10	6.245 × 10	6.422 × 10
	STD	1.056 × 10	6.207 × 10−1	4.004 × 10−1
9	Average	3.177 × 10	3.144 × 10	3.181 × 1
	STD	4.774 × 10−1	3.908 × 10−1	3.730 × 10−1
10	Average	2.023 × 101	2.035 × 101	2.035 × 10+1
	STD	1.160 × 10	2.245 × 10−1	3.226 × 10−1

. In Column 1, the function number is presented, in Column 2, the description of the results obtained is given, in Column 3, the results of using the GaussGauss membership functions are given, in Column 4, the results with the TrianGauss membership functions are presented, and, in Column 5, the results using the GbellGbell membership functions are given.

The results show that, with the GaussGauss membership functions, six improvements are obtained, which are highlighted in bold. In contrast, with the TrianGauss functions, the result is improved in three of the mathematical functions; in this case, with the membership function GbellGbell, no improvement was obtained in the membership functions studied. These results are statistically analyzed to obtain a clearer picture and determine whether there is evidence of improvement.

Table 13. Comparison between DA, CSO, and GBSA for CEC2019 experiments.

				GBSA
No.		DA	CSO	GaussGauss	TrianGauss	GbellGbell
1	Average	4.68 × 104	1.58 × 109	4.098 × 104	4.701 × 104	4.841 × 104
	STD	8.99 × 103	1.71 × 109	2.720 × 103	5.991 × 103	7.620 × 103
2	Average	1.83 × 101	1.97 × 101	1.734 × 101	1.736 × 101	1.739 × 101
	STD	4.19 × 10−2	5.81 × 10−1	6.916 × 10−4	1.799 × 10−2	3.914 × 10−2
3	Average	1.27 × 101	1.37 × 101	1.270 × 101	1.270 × 101	1.270 × 101
	STD	1.50 × 10−12	2.35 × 10−6	4.797 × 10−6	1.246 × 10−5	1.140 × 10−5
4	Average	1.03 × 102	1.79 × 102	3.152 × 102	1.453 × 102	1.547 × 102
	STD	2.00 × 101	5.54 × 101	3.322 × 102	8.810 × 101	1.195 × 102
5	Average	1.18 × 10	2.67 × 10	1.450 × 10	1.322 × 10	1.346 × 10
	STD	5.76 × 10−2	1.72 × 10−1	2.223 × 10−1	1.581 × 10−1	1.711 × 10−1
6	Average	5.65 × 10	1.12 × 101	1.041 × 10+1	1.062 × 101	1.044 × 101
	STD	4.27 × 10−8	7.08 × 10−1	7.585 × 10−1	5.790 × 10−1	7.951 × 10−1
7	Average	8.99 × 102	3.65 × 102	3.741 × 102	6.176 × 102	6.307 × 102
	STD	4.02 × 10	1.65 × 102	2.193 × 102	2.121 × 102	2.011 × 102
8	Average	6.21 × 10	5.50 × 10	5.403 × 10	6.245 × 10	6.422 × 10
	STD	1.66 × 10−3	4.85 × 10−1	1.056 × 10	6.207 × 10−1	4.004 × 10−1
9	Average	2.60 × 10	6.33 × 10	3.177 × 10	3.144 × 10	3.181 × 10
	STD	2.33 × 10−1	1.30 × 10	4.774 × 10−1	3.908 × 10−1	3.730 × 10−1
10	Average	2.01 × 101	2.14 × 101	2.023 × 101	2.035 × 101	2.035 × 101
	STD	7.09 × 10−2	6.90 × 10−2	1.160 × 10	2.245 × 10−1	3.226 × 10−1

shows s comparison of the average and standard deviation results obtained with the proposed method of the CEC2019 functions with the Dragonfly Algorithm [44] and the Cat Swarm Optimization Algorithm [45], corresponding to Columns 3 and 4, from Columns 5 and 3. Column 7 present the results with the different GBSA variants.

The optimal outcomes are highlighted; upon examination, it becomes evident that, in this instance, there are four improvements out of the ten CEC2019 functions.

4.3. Statistical Test

To compare the results obtained, statistical tests are carried out. Specifically, the Z parametric test is used, which uses the following mathematical expression:

$$Z=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{{\sigma }_{{\overline{x}}_1-{\overline{x}}_2}}$$

(14)

where ${\overline{x}}_1-{\overline{x}}_2$ is the difference between the sample means, ${\mu }_1-{\mu }_2$ represents the difference between the population means, ${\sigma }_{{\overline{x}}_1-{\overline{x}}_2}$ = square root ($\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2})$ corresponds to the population standard deviation, and $(n_1, n_2)$ are the sample sizes.

4.3.1. Statistical Test for the CEC2017 Functions

Based on the results obtained in the CEC2017 functions, and considering the different membership functions used, the null hypothesis establishes that the results obtained by the GBSA algorithm are greater than or equal to those obtained with the original method. In contrast, the alternative hypothesis expresses that the results obtained by the GBSA method are lower than the original method. The statistical parameters utilized to carry out this study are shown in

Table 14. Z-test parameters.

Parameters of Z-Test GBSA vs. BSA
Critical Value (Zc)	−1.64
Significance Level (α)	0.05
H0	µ1 ≥ µ2
Ha (Claim)	µ1 < µ2
Level of significance	95%

.

Table 15. Z-test result for the CEC2017 experiments performed with GaussGauss MFs.

Fx	Original		GBSA GT2 GaussGauss
	Average	STD	Average	STD	Z Value	Evidence
1	3.180 × 1010	8.650 × 109	1.781 × 109	8.623 × 108	−18.917	S
2	1.027 × 1046	7.631 × 1046	5.981 × 1030	2.745 × 1031	−0.737	N.S
3	7.630 × 104	1.159 × 104	4.382 × 104	8.945 × 103	−12.153	S
4	6.745 × 103	3.096 × 103	7.782 × 102	1.336 × 102	−10.547	S
5	8.488 × 102	4.287 × 101	7.365 × 102	4.007 × 101	−10.476	S
6	6.754 × 102	8.922 × 10	6.472 × 102	1.155 × 101	−10.569	S
7	1.356 × 103	8.106 × 101	1.083 × 103	6.379 × 101	−14.482	S
8	1.088 × 103	3.624 × 101	9.958 × 102	2.760 × 101	−11.034	S
9	7.636 × 103	1.432 × 103	4.369 × 103	1.482 × 103	−8.683	S
10	7.434 × 103	6.253 × 102	7.146 × 103	8.492 × 102	−1.496	N.S
11	5.847 × 103	2.296 × 103	1.628 × 103	1.689 × 102	−10.04	S
12	2.988 × 109	2.085 × 109	9.269 × 107	7.411 × 107	−7.601	S
13	5.683 × 108	1.437 × 109	2.794 × 106	1.658 × 107	−2.155	S
14	2.576 × 105	5.443 × 105	4.109 × 104	1.714 × 105	−2.078	S
15	1.592 × 107	4.983 × 107	4.673 × 104	3.576 × 104	−1.744	S
16	4.126 × 103	6.388 × 102	3.251 × 103	4.145 × 102	−6.296	S
17	2.918 × 103	3.704 × 102	2.410 × 103	2.557 × 102	−6.175	S
18	2.237 × 106	4.123 × 106	7.948 × 105	4.848 × 106	−1.241	N.S
19	3.017 × 107	6.582 × 107	2.534 × 106	1.313 × 107	−2.255	S
20	2.831 × 103	2.266 × 102	2.544 × 103	1.794 × 102	−5.43	S
21	2.649 × 103	5.301 × 101	2.504 × 103	4.408 × 101	−11.47	S
22	8.312 × 103	1.123 × 103	4.059 × 103	2.387 × 103	−8.83	S
23	3.352 × 103	1.546 × 102	3.027 × 103	1.096 × 102	−9.391	S
24	3.537 × 103	1.540 × 102	3.256 × 103	1.492 × 102	−7.165	S
25	4.205 × 103	4.789 × 102	3.094 × 103	7.383 × 101	−12.563	S
26	9.949 × 103	9.100 × 102	6.481 × 103	1.370 × 103	−11.546	S
27	3.768 × 103	2.755 × 102	3.486 × 103	1.971 × 102	−4.572	S
28	5.354 × 103	6.885 × 102	3.459 × 103	8.772 × 101	−14.961	S
29	6.121 × 103	9.945 × 102	4.749 × 103	5.622 × 102	−6.578	S
30	7.522 × 107	1.239 × 108	6.593 × 106	9.192 × 106	−3.025	S

shows the results obtained by implementing the Z-test. The information mentioned corresponds to experiments with the General Type-2 fuzzy system with GaussGauss MFs. Columns 2 and 3 show the results obtained from the original algorithm, while Columns 4 and 5 present the results of the proposed method. Column 6 contains the values of Z, with which it can be determined whether the differences in the comparison of methods are significant. Finally, the column of evidence is presented, where S is significant evidence and NS means that no significant evidence is found. In this first experiment, it is observed that 27 of the 30 functions of the CEC2017 show significant improvements with the proposed method.

Regarding the experimentation with the General Type-2 fuzzy inference system using the TrianGauss membership functions, 29 of the 30 functions of the CEC2017 show significant improvements with the GBSA. The results can be seen in

Table 16. Z-test result for the CEC2017 experiments performed with TrianGauss MFs.

Fx	Original		GBSA GT2 TrainGauss
	Average	STD	Average	STD	Z Value	Evidence
1	3.180 × 1010	8.650 × 109	1.778 × 109	8.214 × 108	−18.928	S
2	1.027 × 1046	7.631 × 1046	2.267 × 1031	1.147 × 102	−0.737	N.S
3	7.630 × 104	1.159 × 104	4.202 × 104	9.658 × 103	−12.446	S
4	6.745 × 103	3.096 × 103	7.943 × 102	1.329 × 102	−10.519	S
5	8.488 × 102	4.287 × 101	7.356 × 102	3.820 × 101	−10.799	S
6	6.754 × 102	8.922 × 100	6.501 × 102	1.108 × 101	−9.738	S
7	1.356 × 103	8.106 × 101	1.089 × 103	6.114 × 101	−14.409	S
8	1.088 × 103	3.624 × 101	9.897 × 102	2.908 × 101	−11.542	S
9	7.636 × 103	1.432 × 103	4.581 × 103	1.330 × 103	−8.562	S
10	7.434 × 103	6.253 × 102	6.933 × 103	7.594 × 102	−2.787	S
11	5.847 × 103	2.296 × 103	1.644 × 103	1.794 × 102	−9.998	S
12	2.988 × 109	2.085 × 109	1.236 × 108	1.045 × 108	−7.516	S
13	5.683 × 108	1.437 × 109	9.930 × 105	1.831 × 106	−2.162	S
14	2.576 × 105	5.443 × 105	4.986 × 104	1.402 × 105	−2.024	S
15	1.592 × 107	4.983 × 107	4.731 × 104	4.330 × 104	−1.744	S
16	4.126 × 103	6.388 × 102	3.313 × 103	4.546 × 102	−5.683	S
17	2.918 × 103	3.704 × 102	2.440 × 103	2.868 × 102	−5.582	S
18	2.237 × 106	4.123 × 106	5.130 × 105	1.314 × 106	−2.182	S
19	3.017 × 107	6.582 × 107	4.943 × 105	7.708 × 105	−2.469	S
20	2.831 × 103	2.266 × 102	2.536 × 103	1.697 × 102	−5.705	S
21	2.649 × 103	5.301 × 101	2.521 × 103	4.128 × 101	−10.419	S
22	8.312 × 103	1.123 × 103	3.759 × 103	1.970 × 103	−11	S
23	3.352 × 103	1.546 × 102	3.023 × 103	1.226 × 102	−9.128	S
24	3.537 × 103	1.540 × 102	3.269 × 103	1.664 × 102	−6.46	S
25	4.205 × 103	4.789 × 102	3.092 × 103	6.931 × 101	−12.597	S
26	9.949 × 103	9.100 × 102	6.693 × 103	1.661 × 103	−9.416	S
27	3.768 × 103	2.755 × 102	3.474 × 103	2.024 × 102	−4.72	S
28	5.354 × 103	6.885 × 102	3.465 × 103	8.233 × 101	−14.923	S
29	6.121 × 103	9.945 × 102	4.743 × 103	5.443 × 102	−6.653	S
30	7.522 × 107	1.239 × 108	6.566 × 106	6.232 × 106	−3.03	S

.

For the experiments carried out with the GBSA using the GbellGbell membership functions, it is found that, in 28 of 30 CEC2017 functions, the result improved significantly, as can be observed in

Table 17. Z-test results for the CEC2017 experiments performed with GbellGbell MFs.

Fx	Original		GBSA GT2 GbellGbell
	Average	STD	Average	STD	Z Value	Evidence
1	3.180 × 1010	8.650 × 109	1.657 × 109	6.997 × 108	−19.028	S
2	1.027 × 1046	7.631 × 1046	1.487 × 1041	1.487 × 1042	−0.737	N.S
3	7.630 × 104	1.159 × 104	4.440 × 104	9.395 × 103	−11.714	S
4	6.745 × 103	3.096 × 103	7.759 × 102	1.894 × 102	−10.541	S
5	8.488 × 102	4.287 × 101	7.242 × 102	3.934 × 101	−11.73	S
6	6.754 × 102	8.922 × 10	6.477 × 102	1.028 × 101	−11.153	S
7	1.356 × 103	8.106 × 101	1.073 × 103	5.731 × 101	−15.574	S
8	1.088 × 103	3.624 × 101	9.966 × 102	2.844 × 101	−10.818	S
9	7.636 × 103	1.432 × 103	4.464 × 103	1.526 × 103	−8.301	S
10	7.434 × 103	6.253 × 102	6.936 × 103	8.918 × 102	−2.506	S
11	5.847 × 103	2.296 × 103	1.635 × 103	1.657 × 102	−10.025	S
12	2.988 × 109	2.085 × 109	1.166 × 108	9.353 × 107	−7.536	S
13	5.683 × 108	1.437 × 109	2.785 × 107	2.718 × 108	−2.024	S
14	2.576 × 105	5.443 × 105	3.721 × 104	1.131 × 105	−2.171	S
15	1.592 × 107	4.983 × 107	1.767 × 107	1.763 × 108	0.052	N.S
16	4.126 × 103	6.388 × 102	3.225 × 103	3.943 × 102	−6.576	S
17	2.918 × 103	3.704 × 102	2.422 × 103	2.333 × 102	−6.196	S
18	2.237 × 106	4.123 × 106	4.112 × 105	5.422 × 105	−2.405	S
19	3.017 × 107	6.582 × 107	5.061 × 105	7.777 × 105	−2.468	S
20	2.831 × 103	2.266 × 102	2.561 × 103	1.899 × 102	−4.992	S
21	2.649 × 103	5.301 × 101	2.511 × 103	3.535 × 101	−11.873	S
22	8.312 × 103	1.123 × 103	4.207 × 103	2.333 × 103	−8.684	S
23	3.352 × 103	1.546 × 102	3.040 × 103	1.192 × 102	−8.744	S
24	3.537 × 103	1.540 × 102	3.236 × 103	1.607 × 102	−7.396	S
25	4.205 × 103	4.789 × 102	3.087 × 103	6.474 × 101	−12.668	S
26	9.949 × 103	9.100 × 102	6.632 × 103	1.478 × 103	−10.467	S
27	3.768 × 103	2.755 × 102	3.470 × 103	1.773 × 102	−4.985	S
28	5.354 × 103	6.885 × 102	3.470 × 103	9.007 × 101	−14.868	S
29	6.121 × 103	9.945 × 102	4.744 × 103	5.336 × 102	−6.683	S
30	7.522 × 107	1.239 × 108	6.482 × 106	8.134 × 106	−3.031	S

.

It can be concluded from the previous experiments that the fuzzy system with TrianGauss membership functions showed significant improvements in a greater number of mathematical functions used.

4.3.2. Statistical Test for the CEC2019 Functions

The experimentation carried out with the CEC2019 membership functions establishes a null hypothesis that the results obtained by the proposed method are greater than or equal to those obtained with the original method. As an alternative hypothesis, we suggest that the results obtained with the proposed method are lower than those obtained with the original method; for this case, the statistical parameters used are the same as for the CEC2017 membership functions, as presented in Table 14.

The first experiment involved performing dynamic parameter fitting using GaussGauss membership functions; these are presented in

Table 18. Z-test results for the CEC2019 experiments performed with GaussGauss MFs.

Fx	Original		GBSA GT2 GaussGauss
	Average	STD	Average	STD	Z Value	Evidence
1	9.278 × 104	7.631 × 104	4.098 × 104	2.720 × 103	−6.7847	S
2	1.763 × 101	2.328 × 10−1	1.734 × 101	6.916 × 10−4	−12.8755	S
3	1.270 × 101	1.563 × 10−4	1.270 × 101	4.797 × 10−6	0	NS
4	5.832 × 103	3.345 × 103	3.152 × 102	3.322 × 102	−16.3824	S
5	3.299 × 10	9.886 × 10−1	1.450 × 10	2.223 × 10−1	−18.2516	S
6	1.042 × 101	8.138 × 10−1	1.041 × 101	7.585 × 10−1	0	NS
7	4.233 × 102	2.222 × 102	3.741 × 102	2.193 × 102	−1.5713	NS
8	5.249 × 10	6.627 × 10−1	5.403 × 10	1.056 × 10	1.1997	NS
9	6.068 × 102	4.416 × 102	3.177 × 10	4.774 × 10−1	−13.661	S
10	2.038 × 101	2.912 × 10−1	2.023 × 101	1.160 × 10	−1.6723	S

. Column 6 provides the result of the Z parametric test, where it is observed that the result improved significantly for 6 of the 10 mathematical functions studied.

Table 19. Z-test results for the CEC2019 experiments performed with TrianGauss MFs.

Fx	Original		GBSA GT2 TrianGauss
	Average	STD	Average	STD	Z Value	Evidence
1	9.278 × 104	7.631 × 104	4.701 × 10+4	5.991 × 103	−5.9842	S
2	1.763 × 101	2.328 × 10−1	1.736 × 10+1	1.799 × 10−2	−8.5582	S
3	1.270 × 101	1.563 × 10−4	1.270 × 10+1	1.246 × 10−5	0	NS
4	5.832 × 103	3.345 × 103	1.453 × 10+2	8.810 × 101	−16.9643	S
5	3.299 × 10	9.886 × 10−1	1.322 × 10	1.581 × 10−1	−19.7695	S
6	1.042 × 101	8.138 × 10−1	1.062 × 101	5.790 × 10−1	2.0022	NS
7	4.233 × 102	2.222 × 102	6.176 × 102	2.121 × 102	6.3525	NS
8	5.249 × 10	6.627 × 10−1	6.245 × 10	6.207 × 10−1	10.8982	NS
9	6.068 × 102	4.416 × 102	3.144 × 10	3.908 × 10−1	−13.662	S
10	2.038 × 101	2.912 × 10−1	2.035 × 101	2.245 × 10−1	0	NS

explains the results obtained from the Z-test when experimenting with GT2 FS TrianGauss membership functions. These results show a significant improvement for 5 of the 10 mathematical functions of the CEC2019.

The last experiments carried out use the GbellGbell functions in the dynamic parameter adaptation of the GBSA, where it can be observed that, again, in 5 of the 10 mathematical functions, the result is significantly better. This analysis is elucidated in

Table 20. Z-test results for the CEC2019 experiments performed with GbellGbell MFs.

Fx	Original		GBSA GT2 GbellGbell
	Average	STD	Average	STD	Z Value	Evidence
1	9.278 × 104	7.631 × 104	4.841 × 10+4	7.620 × 103	−5.79	S
2	1.763 × 101	2.328 × 10−1	1.739 × 10+1	3.914 × 10−2	−8.465	S
3	1.270 × 101	1.563 × 10−4	1.270 × 10+1	1.140 × 10−5	0	NS
4	5.832 × 103	3.345 × 103	1.547 × 10+2	1.195 × 102	−16.929	S
5	3.299 × 10	9.886 × 10−1	1.346 × 10	1.711 × 10−1	−19.429	S
6	1.042 × 101	8.138 × 10−1	1.044 × 10+1	7.951 × 10−1	0	NS
7	4.233 × 102	2.222 × 102	6.307 × 10+2	2.011 × 102	6.945	NS
8	5.249 × 10	6.627 × 10−1	6.422 × 10	4.004 × 10−1	15.11	NS
9	6.068 × 102	4.416 × 10+2	3.181 × 10	3.730 × 10−1	−13.661	S
10	2.038 × 101	2.912 × 10−1	2.035 × 101	3.226 × 10−1	0	NS

.

From these experiments, we can conclude that, for the three case studies, the results are gradually improved; it would be necessary to experiment with changing the rules or conducting an optimization in the parameterization of the membership functions to analyze whether the result obtained can be improved in more mathematic functions.

4.3.3. ANOVA Test for the Comparison of Bio-Inspired Optimization Methods

In this section, an analysis of variance (ANOVA) is performed with the purpose of comparing and statistically evaluating the performance of different bio-inspired algorithms. The use of ANOVA allows us to determine whether there are significant differences between the results obtained by these algorithms in the corresponding comparisons.

Table 21. ANOVA comparing results of bio-inpired algorithms experimented with CEC2017.

Source of Variance	SS	df	MS	F	p-Value	F Critic
Between groups	2.84 × 1020	4.00 × 10	7.11 × 1019	1.28 × 10	2.80 × 10−1	2.44 × 10
Within Groups	7.76 × 1021	1.40 × 102	5.55 × 1019
Total	8.05 × 1021	1.44 × 102

presents the comparison between the five bio-inspired methods, which experiment with the functions of the CEC2017. In general, the mean of each group and the total of them must first be calculated; then, we calculate the sum of the squares, and the degrees of freedom of the factor, error, and total are determined. In addition, the mean square errors and the F statistical value are calculated.

Therefore, if we take a significance level α = 0.05, we have to reject the null hypothesis and accept the alternative hypothesis, since the p-value of the test is lower than the significance level. Based on the ANOVA results, we can confidently state that there is a significant difference between the groups in terms of their population means.

The next comparison to be made is the one corresponding to the different bio-inspired algorithms used to experiment with the functions of the CEC2019.

Table 22. ANOVA comparing the results of bio-inspired algorithms in experiments with CEC2019.

Source of Variance	SS	df	MS	F	p-Value	F Critic
Between Groups	2.00 × 1017	4.00 × 10+	4.99 × 1016	1.00 × 10	4.18 × 10−1	2.58 × 10+
Within Groups	2.25 × 1018	4.50 × 101	4.99 × 1016
Total	2.45 × 1018	4.90 × 101

presents the ANOVA statistical analysis.

Based on the ANOVA results, we do not have enough evidence to confirm that there are significant differences between the groups in terms of their population means. This means that, in this case, there is no statistical basis for concluding that at least one group performs significantly differently from the others.

4.4. Discussion

The contemporary landscape of process optimization underscores the persistent challenge of achieving higher levels of accuracy and efficiency across diverse applications. Industries ranging from manufacturing to logistics and diagnostics demand innovative solutions to address this multifaceted problem. Nature-inspired algorithms, which draw inspiration from biological systems, have emerged as a promising avenue for pushing the boundaries of optimization methodologies. The utilization of these algorithms offers the possibility of overcoming the inherent limitations of traditional techniques, particularly in scenarios characterized by uncertainty and complexity. In this study, we investigated the enhancement of optimization processes through the dynamic parameter adaptation of the Bird Swarm Algorithm (BSA) using General Type-2 Fuzzy Systems (GT2 FS). The integration of GT2 FS introduces a novel layer of adaptability and robustness to the BSA framework, allowing it to traverse intricate solution landscapes with greater efficiency. The essence of this approach lies in its ability to finely tune the algorithm’s parameters, particularly the social and cognitive acceleration coefficients, in response to varying environmental conditions.

The significance of our findings is underscored by the comprehensive evaluation of two distinct case studies. The application of the proposed General Bird Swarm Algorithm (GBSA) to the functions of CEC2017 and CEC2019 demonstrates its consistent and noteworthy performance improvements. Notably, the GBSA approach outperforms the original BSA method across both case studies, with an impressive 97% improvement for CEC2017 functions and a substantial 70% enhancement for CEC2019 functions.

Furthermore, the successful integration of GT2 FS into the BSA framework opens avenues for future research and exploration. The choice of membership functions and rule sets has been shown to significantly impact the algorithm’s performance. Future investigations could focus on optimizing these parameters further to extract even more improvements, potentially expanding the scope of the domains in which GBSA can be effectively applied. This study advances the field of optimization through the integration of General Type-2 Fuzzy Systems into the Bird Swarm Algorithm. The demonstrated improvements in performance across different case studies highlight the potential of this approach to address challenges posed by intricate optimization problems in various domains. The insights gleaned from this research pave the way for the further exploration and refinement of algorithmic techniques in the pursuit of continuous improvement.

The robustness of the proposed algorithm, the General Bird Swarm Algorithm (GBSA), can be attributed to its integration of General Type-2 fuzzy systems (GT2 FS) within the Bird Swarm Algorithm (BSA) framework. This integration introduces a dynamic parameter adaptation mechanism that enhances the algorithm’s ability to navigate intricate solution landscapes with varying degrees of uncertainty and complexity.

The use of General Type-2 fuzzy systems allows GBSA to model uncertainties more efficiently and accurately than traditional methods. By representing uncertainties in a more flexible manner, GBSA can adapt its parameters, such as the social and cognitive acceleration coefficients, in response to changing environmental conditions. This adaptability enhances the algorithm’s ability to converge to optimal solutions, even in scenarios characterized by dynamic and uncertain constraints.

Additionally, GBSA’s robustness is evident in its ability to handle different membership functions and rule sets, as indicated by the variations of the algorithm applied in the case studies. This adaptability shows GBSA’s capacity to accommodate various scenarios and optimize its performance accordingly.

5. Conclusions

In this work, an improvement to the BSA is proposed; this is referred to as the General Bird Swarm Algorithm, in which a dynamic parameter adaptation is performed using General Type-2 fuzzy systems. These systems are designed with two inputs, corresponding to iteration and diversity, as well as two outputs, i.e., the coefficients of cognitive and social acceleration.

To test the performance of both methods, mathematical functions were used, corresponding to the competition of CEC2017 and CEC2019. Based on the experiments we carried out, the results obtained from both the original method and the proposed one are compared, with a significant improvement noted in the latter. Additionally, we performed a statistical analysis to provide a clearer conclusion. There are different challenges in which we can put GBSA to the test, such as in the medical field or in industry; it can even be tested in the control area to improve times or performances in a certain process.

The General Type-2 Fuzzy approach for dynamic parameter adaptation in the BSA is an effective solution to enhancing the performance of the obtained results. In the two case studies carried out, the results of the proposed method were compared with the original method, with significant improvements obtained in both cases. This shows that the GT2 FS approach is a promising technique for algorithm optimization.

As for the functions used in the case studies, four sets of rules ( increasing, decreasing, or a combination of these) were analyzed. In addition, three different fuzzy systems were used, varying them with GaussGaussMFs. In the presented proposal, a fuzzy inference system of the Mamdani Type is used, with two inputs and two outputs. Other issues to be analyzed in this research area are the operation and performance of membership functions.

In summary, the General Type-2 fuzzy approach for dynamic parameter adaptation in the BSA is a promising technique for algorithm optimization, and this work is expected to inspire future research in this field. In future work, experiments will be carried out that use the algorithm for the control of an autonomous mobile car, allowing us to develop a clearer picture of the algorithm’s performance when the dynamic parameter adaptation is applied to different types of fuzzy systems.