
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)11989-5
10.1016/j.heliyon.2024.e35958
e35958
Research Article
An improved grey wolf optimization algorithm based on scale-free network topology
Zhang Jun a
Dai Yongqiang dyq@gsau.edu.cn
a⁎
Shi Qiuhong b
a College of Information Science and Technology, Gansu Agricultural University, Lanzhou Gansu, 730070, China
b Information & Network Center, Gansu Agricultural University, Lanzhou Gansu, 730070, China
⁎ Corresponding author. dyq@gsau.edu.cn
11 8 2024
30 8 2024
11 8 2024
10 16 e359582 3 2024
2 8 2024
6 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The grey wolf optimizer is a novel intelligent optimization algorithm that has become popular due to its low number of parameters, fast convergence speed, and simplicity. However, the classical algorithm, with its update strategy allowing wolves to learn only from the alpha wolves, often leads to premature convergence and lower convergence accuracy. Therefore, in this paper, an improved grey wolf optimization algorithm based on scale-free network topology (SFGWO) is proposed to address these issues. The improved algorithm first employs a strategy for formulating a population based on a scale-free network topology, where interaction between wolves is limited to topological neighbors, which helps enhance the exploration capabilities of the algorithm. Second, a neighbor learning strategy is introduced to capture individual diversity, facilitating the solution space exploration. Finally, an adaptive individual regeneration strategy is adopted to balance the exploration and exploitation processes and reduce the risk of falling into local optima. The proposed algorithm is evaluated through simulation experiments using 23 classical and the CEC2019 benchmark functions. The experimental results demonstrate that the SFGWO algorithm excels in terms of solution accuracy and exploration capabilities. The applicability and effectiveness of the SFGWO algorithm are further validated through testing on three practical engineering problems.

Keywords

Grey wolf optimizer
Scale-free network topology
Neighborhood learning
Adaptive individual regeneration
==== Body
pmc1 Introduction

Real-world optimization problems, particularly complex multimodal optimization problems found in the fields of science and engineering, present significant challenges. These problems are typically classified as Nondeterministic Polynomial (NP)-hard, making them difficult to solve using traditional deterministic methods. Metaheuristic algorithms have emerged as a key tool for addressing these complex optimization problems due to their robustness and flexibility. These algorithms are broadly classified into various classes, including evolutionary algorithms, swarm intelligence (SI) algorithms, and others [1]. Among these, swarm intelligence algorithms, inspired by the collective behaviors of animals in nature, have shown great promise in addressing large-scale complex optimization problems [[2], [3], [4]].

In recent years, several SI algorithms have been introduced, such as Particle Swarm Optimization (PSO) [5], Artificial Bee Colony optimization (ABC) [6], Ant Colony Optimization (ACO) [7], cuckoo search (CS) [8], Sand Cat Swarm Optimization (SCSO) [9,10], Artificial Electric Field Algorithm (AEFA) [11,12], Fox Optimizer (FOX) [13,14], the Aquila optimizer [15], and the grey wolf optimizer (GWO) [16]. The GWO, specifically proposed by Mirjalili et al. in 2014 [16], draws inspiration from the hierarchical leadership and hunting strategies observed in grey wolves (GWs). To optimize the search process, the algorithm aims to simulate the collective actions of GW packs, including searching, encircling, and attacking prey. During hunting, individual wolves are led by leading wolves to update their positions to catch prey. The GWO stands out for its fast convergence, simple structure, ease of programming, and straightforward implementation, and has found successful applications in diverse areas such as path planning [17], engineering design [18], medical image analysis [19], and the stabilization of wide-area power systems [20].

However, since all the candidate wolves are guided by the three best wolves in each iteration, the GWO is prone to premature convergence and reaching only local optimality when solving complex multidimensional problems [21,22]. To address the above issues, efforts have been made to improve the algorithm, mainly from the aspects of parameter adjustment and search update mechanisms, while hybrid approaches integrating parts of other optimization algorithms have also been introduced. More specifically, some representative studies on GWO improvement based on parameter adjustment include the following: Mittal et al. [23] proposed a modified GWO algorithm based on nonlinear controllable parameters to balance the exploration and exploitation, which resulted in improved algorithm performance. However, this approach suffered from premature convergence. Wen [24] adjusted the parameter using a nonlinear strategy and initialized the population through an opposition-based learning strategy. However, this implementation cannot converge when dealing with high-dimensional problems. Hou et al. [25] proposed an improved GWO algorithm that achieved a balance between global exploration and local search by enhancing the chaotic tent mapping for population initialization and incorporating a nonlinear convergence factor refined through a Gaussian distribution curve. Nevertheless, parameter adjustment strategies suffer from the drawback that they require setting different parameter values during the iteration process. However, different functions have varying requirements for exploration and exploitation at different stages, leading the algorithm to perform well only on specific functions.

Studies presenting improved versions of the GWO based on modified update mechanisms have also been put forward. Chen et al. [26] introduced a weight factor to adjust the position update equation of the algorithm dynamically, as well as a probabilistic disturbance strategy for re-initializing the GW's position to enhance the population diversity. Heidari et al. [27] proposed an efficient modification to the GWO algorithm, in which the combination of Lévy flights and a greedy selection strategy during the hunting phase addressed the issue of stagnation at local optima during the iterative process. Saremi et al. [28] introduced an evolutionary population dynamics operator to reposition poor search agents and increase the diversity of the population, thus enhancing the exploitation dimension. However, this approach relies on random positions and cannot converge when dealing with high-dimensional problems.

Other GWO approaches that have proven to be effective include those proposing hybrid solutions mixing GWO characteristics with those of other optimization algorithms. For instance, Singh et al. [29] incorporated position update equations from the sine-cosine algorithm [30] to enhance the movement direction and speed of the alpha GWs. This approach aims to improve global convergence, exploration, and exploitation performance through the expedition of the search process, but suffers from premature convergence. Gaidhane et al. [31] combined the information-sharing strategy of the ABC with the primal hunting behavior of wolves, enhancing both exploration and exploitation capabilities. They also utilized chaotic mappings and opposition-based learning to initialize the population. Zhu et al. [32] combined the Differential Evolution (DE) algorithm with the GWO, utilizing DE's search ability to address stagnation and accelerate convergence. Hybridization with other optimization algorithms is a promising approach, but it is important to note that integrating different algorithms may lead to conflicts or incompatibilities between them.

Although the above-mentioned variants of GWO have achieved reasonably improved results over the initial version of the algorithm, updating individual wolf positions still relies on the three best wolves, and the experience of other individuals is not utilized during the search process. This limitation makes it difficult to explore and exploit other regions extensively. To fully leverage the individual experiences within the population and better balance the exploration and exploitation capabilities during the search process, network topology and neighborhood information have become an effective strategy. Recent studies have found that using topological network structures for inter-individual information sharing can effectively enhance the optimization performance of swarm intelligence algorithms [21,33,34]. Lu et al. [13] proposed a cellular GWO algorithm based on a cellular topological structure. This GWO version achieved information dissemination between neighbors by restricting each wolf from exchanging information with its neighbors, thereby maintaining diversity and enhancing the algorithm's exploration capabilities. Bilal H et al. [35] proposed a GWO variation that employed an island model to partition the population into independent small groups (islands). GWO enhancement strategies were then applied to each island, utilizing a random ring topology and best-worst migration strategies to exchange information, enhance population diversity, and improve the potential for finding global optima.

As network topologies are being introduced in optimization algorithms, the use of scale-free networks (SFNs) emerges as an effective method to improve metaheuristic algorithms. SFNs can capture more accurately the properties of real-world networks such as financial [36], social [37], and transportation [38] networks. In contemporary studies, SFNs are being widely adopted to improve SI algorithms. For example, Wu et al. [39] proposed a variant of the PSO algorithm based on a scale-free base network. Ji et al. [40] used an SFN to guide the search process of bees in an ABC algorithm. This enabled the bees near bad food sources to acquire more information from those near good food sources and limited the direct exchange of information between bees near good food sources, which resulted in a better balance of exploration and exploitation. Li et al. [41] proposed a PSO variant in which a corresponding neighborhood was created for each particle, and elite particles were selected from the neighborhood to participate in the evolution process, so that the guiding role of the elite particles in the group search process could be fully utilized. Banerjee et al. [42] proposed an adult-pup teaching-learning-based interactive GWO. In this version, the population was divided into two subgroups: adult and pup wolves, each employing distinct initialization methods and exploring the entire search space independently. The sharing of information between these two groups resulted in an improved balance between exploration and exploitation and enhanced overall algorithm performance. Therefore, in population-based metaheuristic algorithms, the sharing and correlation of information through topological networks can enhance the algorithm's optimization performance.

Based on the above considerations, this paper proposes an improved GWO algorithm based on an SFN, termed the scale-free GWO (SFGWO). This algorithm features a modified GWO topology structure, which promotes information exchange among neighborhood individuals, enhanced algorithm diversity, and improved capacity to discover the global optimal solution. The main contributions of this study are outlined below.(1) A novel topology structure is introduced based on an SFN, significantly enriching the algorithm's diversity and promoting effective information sharing among individuals.

(2) A neighbor learning strategy (NLS) is incorporated, which allows individuals to adapt and learn from their local neighbors, further enhancing the exploration of the solution space.

(3) An adaptive individual regeneration strategy (AIRS) is adopted to help maintain population diversity and reduce the risk of getting trapped in local optima.

(4) The proposed SFGWO algorithm exhibits superior performance in terms of solution accuracy and exploration ability, as demonstrated through experimentation on benchmark functions.

The remaining sections of this paper are organized as follows. Section 2 provides a detailed introduction to the GWO, SFNs, neighborhood learning and adaptive individual regeneration. In Section 3, the motivation, principles, and implementation method of the proposed algorithm are elaborated. Section 4 contains a series of diverse experiments with comparative analyses to evaluate the effectiveness of the proposed algorithm, while in Section 5, the performance of the proposed SFGWO in solving practical engineering application problems is discussed through three engineering design problems. In the concluding section, we summarise the key findings presented in this paper and provide a forward-looking perspective on future research directions.

2 Related work

2.1 Classical grey wolf optimizer

The GWO is a recent SI-based optimization algorithm derived from the simulation of the social hierarchy and hunting mechanisms of grey wolves in nature. Like other metaheuristics, in the GWO, the optimization process is initialized by generating a random set of candidate solutions. In each iteration, the best, second-best, and third-best individuals in the group are called the alpha (α), beta (β), and delta (δ) wolves, respectively, while the individuals comprising the rest of the pack are referred to as the omega (ω) wolves. The social hierarchy mechanism of GWs plays an important role in the group hunting process, and ω wolves are assumed to accept the leadership of the α, β, and δ wolves. The social hierarchy of the GW group is shown in Fig. 1.Fig. 1 Social hierarchies of grey wolves.

Fig. 1

The group hunting behavior of wolves is divided into two stages: chasing and encircling the prey and hunting.1) CHASING AND ENCIRCLING PREY

The mathematical model of the GW group gradually approaching and encircling prey is shown in Equation (1):(1) x(t+1)=xpreyt−A×|C×xpreyt−xt|

where t is the current iteration number, xpreyt is the position of the prey at the tth iteration, while x(t+1) and xt are the positions of the wolf at the (t + 1)-th and t-th iterations, respectively. A×|C×xpreyt−xt| is the enclosing step, where A and C are defined as follows:(2) A=2×a×r1−a

(3) C=2×r2

where r1 and r2 are random numbers between [0, 1]. A is in the range [−a, a] and regulates exploration and exploitation; that is, when |A| < 1, wolves attack their prey, while when |A| > 1, wolves look for prey. C is in the range [0, 2]. a is a distance control parameter that decreases linearly from 2 to 0 as the number of iterations increases:(4) a=2−2ttmax

in Equation (4), tmax is the maximum allowed number of iterations.2) HUNTING

During the hunting process, it is assumed that the three best wolves have a better knowledge of the prey's position and the other GWs follow the three leaders. Therefore, the position update equation for each wolf is as follows:(5) {x1=xαt−A1×|C1×xαt−xt|x2=xβt−A2×|C2×xβt−xt|x3=xδt−A3×|C3×xδt−xt|

(6) x(t+1)=x1+x2+x33

where xαt, xβt and xδt are the positions of the α, β, and δ wolves at the tth iteration, and x1, x2, and x3 represent the direction and step size of wolf ω towards the α, β, and δ wolves, respectively. Equation (6) is the position update equation for the GWs, calculated based on the average position of the α, β, and δ wolves. After reaching the maximum number of iterations, xα is assumed to be the prey position. The GWO process is shown in Fig. 2.Fig. 2 Flow chart of GWO.

Fig. 2

2.2 Scale-free network and the BA algorithm

While studying the topological structure of the World Wide Web, Barabási and Albert observed that the degree distribution of nodes in the network follows a power-law distribution. They referred to this configuration as an "SFN". The power-law distribution is characterized by most nodes having low degrees, while a few nodes having extremely high degrees. In SFNs, a node's degree is defined as the number of edges that are connected to a node. A small number of nodes will have higher degrees and are called "hub nodes", while nodes with lower degrees, which usually include most nodes in the network, are called "non-hub nodes". This distribution of nodes mirrors the heterogeneous nature of real-world complex systems. Observing how new nodes attach to original nodes in real-world networks, Barabási and Albert proposed a network growth model to build SFNs, referred to as the BA model [43]. The creation process of the BA model can be described as follows.1) Initial Stage: Create a fully connected network with an initial size of m0.

2) Growth Stage: At each time step, a new node is added with m (m ≤ m0) edges. The probability P(ki) that a new node will be connected to an existing node i depends on the degree ki of node i, which is given by the following equation:

(7) P(ki)=ki∑jkj

where kj is the degree of node j.3) Repetition Stage: Repeat step 2 continuously, adding new nodes and establishing connections until the network reaches the desired size or node count.

Barabási and Albert have shown that such a growing graph evolves to a stable scale-free network, with a vertex degree probability distribution that follows a power law, i.e., p(k) ∼ k−γ, where γ = 3 [44].

Fig. 3 shows the power-law distribution of the BA model when γ = 3. As can be seen from the figure, the power-law distribution allows for the existence of some nodes with a large number of links (i.e., edges), while most nodes have only a few links.Fig. 3 Scale‐free network topology.

Fig. 3

In this work, an SFN is used as the neighborhood structure of the GWO individual wolves. The BA model is used to create a population based on the SFN topology. During the network creation process, each node in the SFN corresponds to a GW individual of GWO, thus realizing an SFN-based connection between GW individuals.

2.3 Neighborhood learning

According to the definition of SFNs in Section 2.1, each node is connected to several other nodes, referred to as “neighbors”. A neighborhood learning strategy aims to facilitate the information exchange between these neighboring nodes and the collaboration among nodes within the neighborhood domain, thereby enhancing the algorithm's performance. As discussed in the introduction, recent research has shown that an appropriate neighborhood topology and improvements in how information is exchanged among individuals can effectively enhance the GWO's population diversity, thus leading to a better balance between exploration and exploitation. This neighborhood topology-based connectivity approach can effectively leverage individual information within the neighborhood to guide the group's search direction within the domain. In this manner, the sharing and correlating of specific information through topological networks can enhance the optimization performance of population-based metaheuristic algorithms [21,35]. For instance, Kishor et al. [45] proposed an enhanced variant of the GWO algorithm, with a basic crossover operator between randomly selected individuals. This approach aimed to promote information sharing among pack members and maintain a balance between exploration and exploitation.

2.4 Adaptive individual regeneration

The AIRS is based on the concept of the dynamic adjustment of the way in which new individuals are generated in the population. This dynamic adjustment of the individuals' generation or updating enhances the algorithm's exploration ability and ability to jump out of local optima [46]. Yan et al. [47] proposed an improved GWO, which employs an elimination-relocation strategy to eliminate and regenerate several of the worst individuals to enhance the algorithm's performance and prevent it from getting trapped in local optima. Zhang et al. [48] employed an improved GWO algorithm to enhance the performance of an electronic nose. They utilized an adaptive restart strategy to randomly re-initialize a portion of the search agents based on their fitness values during the iteration process, thereby enhancing the algorithm's search capability and stability. Zhang et al. [49] presented a modified version of the GWO that eliminated and re-initialized poorly performing wolves, effectively expanding the random search and accelerating convergence.

The individual regeneration strategy offers several advantages due to its timely intervention in stagnant individuals, which results in an increase in their activity and the dynamic updating of the population's individual states. Firstly, it significantly enhances the algorithm's global search capabilities. When individuals become stagnant, the strategy reactivates them, allowing them to explore the solution space further and potentially discover the global optimal solution. Secondly, this strategy increases the algorithm's exploration potential, since the re-initialization operation results in greater diversity in individual positions, reducing the likelihood of falling into the same local optimal solution. Finally, the individual regeneration strategy improves the algorithm's robustness, allowing it to adapt better to complex optimization problems, especially those with multiple local optimal solutions. These advantages collectively make the individual regeneration strategy an effective means of enhancing the GWO algorithms.

3 Proposed algorithm

This section provides a detailed description of the proposed SFN topology GWO algorithm (SFGWO) algorithm. The algorithm comprises three key strategies. The first is the population formulation strategy, which utilizes an SFN topology to build the topological structure of the GW population. The second strategy is the neighbor learning strategy, involving the derivation of update equations that enable individuals to learn from their neighbors within the SFN. The third and final strategy is the AIRS, where each individual is assigned a variable to keep track of the number of times its position has stopped updating. If this count exceeds a specified threshold, the individual will undergo a re-initialization operation to enhance the algorithm's ability to jump out of the local optimum. The flowchart of the SFGWO is shown in Fig. 4, with the novel strategies introduced marked in red boxes.Fig. 4 Flow chart of SFGWO.

Fig. 4

3.1 Scale-free network topology population construction strategy

Increasing the information exchange among neighboring GWs can enhance population diversity and improve the algorithm's exploration capabilities [13]. Therefore, we employ the BA model to formulate a population based on an SFN topology. The purpose of using an SFN topology is to create the population, establish a corresponding neighborhood for each GW individual, and then leverage domain information exchange to select those individuals that will participate in the evolutionary process, thus resulting in enhanced algorithm's performance.

The SFN topology creation process is as follows.1) First, the fitness values of GW individuals are calculated in each iteration, and the GW population is sorted based on fitness values in ascending order. Let the top m0 grey wolves represent the number of hub nodes.

2) All hub nodes are connected to form a fully connected network.

3) The remaining individuals are sequentially connected to the existing network nodes based on their fitness values, with each individual linking to two existing wolves with a probability based on P(ki). This ensures that wolves with more connections (higher degrees) have a higher chance of attracting new links.

4) Each wolf's neighborhood is defined by its direct connections in the SFN, which encourages localized interaction and learning.

The effect of the formulation of the SFN topology is depicted in Fig. 5. Nodes labeled as 0, 1, 2, 3, and 4 exhibit higher degrees, while other nodes have lower degrees. Not each node is directly linked to the hub nodes; instead, they form relatively sparse connections with other non-hub nodes. This characteristic prevents the rapid transmission of information from the leading wolves (the wolf with lower fitness) to the entire population. Therefore, it maintains the diversity of the population to a certain extent, and leads to a better balance between exploration and exploitation in the optimization process.Fig. 5 Scale - free network topology population.

Fig. 5

After the SFN topology space has been created, each GW individual selects some neighbors from its neighborhood to participate in the search process. The specific implementation is described in detail in Section 3.2.

3.2 Neighborhood learning strategy

In the SFN analogy, each wolf corresponds to a node. The number of neighbors a node has is determined by the number of nodes that are connected to it. According to step 3 of Section 3.1, each node has at least two nodes connected to it, but can also have more connections.

The information exchange between different types of nodes plays a crucial role in improving the performance of the SFGWO algorithm. Hub nodes ensure that high-quality solutions will be reached and guide the evolution direction. In contrast, non-hub nodes help maintain population diversity, and thus the solution space will be explored. The collaboration between different types of nodes can improve the balance between exploration and exploitation effectively. To implement this collaboration mechanism, the following position update equations are defined:(8) xinew={xi+r3×(Lxinei−xi)f(xinei)<f(xi)andp<0.4xi+2×r4×(xinei−xi)f(xinei)<f(xi)andp≥0.4xgbestf(xinei)≥f(xi)

(9) xi(t+1)={xinewf(xinew)<f(xi)xiotherwise

where f(⋅) is the fitness value function. Smaller values of the fitness function represent a smaller distance between the individual and the objective function, i.e., the individual is closer to the optimal solution and is thus a better candidate. xi and xnei i are the positions of the i-th GW and a randomly selected neighbor; r3, r4, and p are random numbers between [0, 1]. The value 0.4 is a threshold for switching between the two update equations, which was determined after multiple rounds of experiments and systematic parameter fine-tuning. The value of L is randomly chosen from {-1, 1} and causes random alterations in the search direction of the GW. xgbest is calculated using Equation (6), and is the current potentially global optimal position. These equations govern how nodes exchange information and update their positions in the SFN topology, facilitating optimization and balancing exploration and exploitation, as explained below.

In Equation (8), if f(xnei i) < f(xi), this indicates that the neighbor's position is a potentially better solution and warrants further exploration. The current individual then moves towards the neighbor's position using the corresponding update equation, facilitating local exploration. Therefore, each wolf's search will benefit from the information of its neighbors, contributing to the exploration of a broader solution space. Here, two update equations are presented to approach the neighbor's position differently. These equations are randomly switched based on the value of p, ensuring comprehensive exploitation of the neighborhood.

Else, if f(xnei i) ≥ f(xi), this implies that further exploration of the neighbor's position is unnecessary. Instead, the current individual redirects its movement towards the potential global optimum position xgbest. This movement strategy ensures that individuals search for high-quality solutions, leading to faster convergence. Finally, as shown in Equation (9), a greedy selection approach for xi and the updated solution xinew is applied based on their fitness values, retaining the better candidate solutions. This accelerates the search process and focuses on exploring the best directions obtained so far.

3.3 Adaptive individual regeneration strategy

When solving complex optimization problems, the GW population tends to gradually converge to specific regions of the solution space, leading to a loss of population diversity. To address this issue, an adaptive individual regeneration method is employed in this study to regenerate stationary GW individuals. The strategy works through the random assignment of new positions to the GWs. These new positions are selected from a region far away from the GW's current position, which ensures that the GWs will explore new areas and have a chance to find a better solution.

In this strategy, a variable Si is assigned to each GW to maintain a record of the number of iterations for which the GW has remained stationary, indicating a state of stagnation. Additionally, a variable Smax represents the maximum number of stationary iterations allowed. After each GW position update, if the new fitness value f(xnew i) is greater than or equal to the previous fitness value f(xi), the GW individual has not found a better solution, indicating a state of stagnation, and the stagnation count Si is incremented by 1. If several consecutive such stagnations occur and the stagnation count reaches or exceeds the maximum allowed Smax, the position of that GW individual is re-initialized to facilitate its "regeneration". Subsequently, the stagnation count Si is reset to zero. The value of Smax is adaptively adjusted within the range [SL, SU] based on the algorithm's stagnation status. Here, SL and SU represent the lower and upper bounds of Smax. Algorithm stagnation is determined using the following equations:(10) R=abs×(1−fbest/prefbest)

(11) IsStagnation={TrueR<rxFalseelse

where fbest represents the current best value attained by the algorithm, prefbest represents the best value obtained during the previous iteration, and abs is the absolute value function. R is a numerical value representing the degree of stagnation of the algorithm. In this paper, the threshold of the optimal value change rate rx is set to 10−4 to ensure that the algorithm will not prematurely regenerate GW positions; this value was determined experimentally as yielding the best results. When R is less than 10−4, it indicates that the algorithm has stagnated. In such cases, the value of Smax is reduced to facilitate the regeneration of more individuals, allowing the algorithm to break out of local optima. On the other hand, when the algorithm has not stagnated, the value of Smax is increased by 2 to reduce the impact of regeneration on the algorithm's exploitation capabilities, indicating that the individuals in the current population are exploring effectively and do not require adjustment. This approach enables the adaptive adjustment of Smax, balancing the exploration and exploitation processes of the algorithm and improving the algorithm's performance. The pseudo-code for the AIRS is as follows.Adaptive individual regeneration strategy	
Input: population size N., Smax, SL, SU, rx	
R = abs(1-fbest/perfbest)	
ifR < rxthen	
 Smax = max(SL, Smax− 1)	
Else	
 Smax = min(SU, Smax + 2)	
end if	
for i = 1: N do	
 Updating individual positions using equations (8) and (9).	
 iff(xnew) ≥ f(xi) then	
 Si = Si + 1	
 else	
 Si = 0	
 end if	
 ifSi ≥ Smaxthen	
 xi = r5 × (ub − lb) + lb	
 Si = 0	
 end if	
end for	

In this above, the stagnation number Si of each GW is initialized to 0, while the maximum stagnation number variable Smax is initialized to a large value, such as 10. r5 is a random number between [0, 1]. ub (upper bound) and lb (lower bound) represent the upper and lower limits of the solution space, respectively.

3.4 Framework of the SFGWO algorithm

Using the improvement strategies described above, this paper proposes an improved GWO algorithm based on an SFN topology, as depicted in Fig. 4. Building upon the traditional GWO, in each iteration, the algorithm first creates an SFN topology population based on fitness values. It then randomly selects a neighbor from the current individual's neighbors and updates the individual's position using the neighbor learning strategy. Finally, the algorithm dynamically regenerates individuals based on their stagnation counts to avoid local optima. The specific pseudo code is provided below.SFGWO algorithm pseudo code	
Algorithm: SFGWO	
Input: Population size N, Maximum Iteration Count MaxIterate,	
Output: Optimal solution xα	
Initialize:N, MaxIterate, Smax, SL, SU, rx, m	
Evaluate the fitness f(xi) at xi, select α, β and δ wolves	
while t < MaxIterate	
 formulating a SFN topology population	
 for i = 1: N do	
 Calculate xgbest using Eq. (6)	
 Randomly select xnei i from the neighbors of the i-th GW.	
 Adaptively regenerate stagnant GWs	
 select the best three wolves to update the α, β and δ wolves	
end while	

4 Experimental analysis and comparison

This section is devoted to the comprehensive assessment of SFGWO's performance. As the number of hub nodes m0 is a key SFGWO parameter need, a test on unimodal and multimodal functions is first conducted to obtain an appropriate value for m0. To accurately assess the impact of each improvement strategy employed in the SFGWO algorithm, an ablation study was also conducted. Following this, the SFGWO's performance was evaluated through 23 benchmark functions and compared with the performance of 8 other algorithms. Finally, the CEC2019 test functions and the corresponding results are discussed and analyzed.

Experiments include the following aspects: in Section 4.1, the benchmark functions are introduced, and the parameter settings of the comparison algorithms are listed. In Section 4.2, tests conducted with different numbers of hub nodes (m0) are presented, and the effect of m0 on the optimization performance is analyzed. An ablation study is presented in Section 4.3, and the contribution of each of the three improvement strategies is discussed. A detailed comparative analysis of the results of other algorithms on 23 classical benchmark functions is presented in Section 4.4, while further tests are conducted using the CEC2019 benchmark functions in Section 4.5.

4.1 Benchmark functions and parameter settings

In this section, the 23 classical and CEC2019 benchmark functions used to evaluate SFGWO's performance are presented. Table 1 displays the information for the 23 classical benchmark functions used in this study. Among these functions, f1–f7 are unimodal functions, f8–f13 are multimodal functions, and f14–f23 are fixed-dimension multimodal functions. The definitions of the CEC2019 benchmark functions are provided in Table 2.Table 1 Classical benchmark functions.

Table 1Benchmark Expressions	Dimensions	Range	fmin	
f1(x)=∑i=1nxi2	30	[−100,100]	0	
f2(x)=∑i=1n|xi2|+∏i=1n|xi|	30	[−10,10]	0	
f3(x)=∑i=1n(∑j−1ixj)2	30	[−100,100]	0	
f4(x)=maxi{|xi|,1≤i≤n}	30	[−100,100]	0	
f5(x)=∑i=1n−1[100(xi+1−xi2)2+(xi−1)2]	30	[−30,30]	0	
f6(x)=∑i=1n([xi+0.5])2	30	[−100,100]	0	
f7(x)=∑i=1nixi4+random[0,1)	30	[1.28,1.28]	0	
f8(x)=∑i=1n−xisin(|xi|)	30	[−500,500]	−418.9829
 × dim	
f9(x)=∑i=1n[xi2−10cos(2πxi)+10]	30	[−5.12,5.12]	0	
f10(x)=−20exp(−0.21n∑i=1nxi2)−exp(1n∑i=1ncos(2πxi))+20+e	30	[−32,32]	0	
f11(x)=14000∑i=1nxi2−∏i=1ncos(xii)+1	30	[−600,600]	0	
f12(x)=πn{10sin(πy1)+∑i=1n−1(yi−1)2[1+10sin2(πyi+1)]+(yn−1)2}+∑i=1nu(xi,10,100,4)yi=1+xi+14u(xi,a,k,m)={k(xi−a)mxi>a0−a<xi<ak(−xi−a)mxi<−a	30	[−50,50]	0	
f13(x)=0.1{sin2(3πx1)+∑i=1n(xi−1)2[1+sin2(3πxi+1)]+(xn−1)2[1+sin2(2πxn)]}+∑i=1nu(xi,5,100,4)	30	[−50,50]	0	
f14(x)=(1500+∑j=1251j+∑i=12(xi−ai)6)−1	2	[−65,65]	0.998	
f15(x)=∑i=111[ai−x1(bi2+bix2)bi2+bix3+x4]2	4	[−5,5]	0.00030	
f16(x)=4x12−2.1x14+13x16+x1x2−4x22+4x24	2	[−5,5]	−1.0316	
f17(x)=(x2−5.14π2x12+5πx1−6)2+10(1−18π)cosx1+10	2	[−5,5]	0.398	
f18(x)=[1+(x1+x2+1)2(19−14x1+3x12−14x2+6x1x2+3x22)][30+(2x1−3x2)2(18−32x1+12x12+48x2−36x1x2+27x22)].	2	[−2,2]	3	
f19(x)=−∑i=14ciexp(−∑j=13aij(xj−pij)2)		[1,3]	−3.86	
f20(x)=−∑i=14ciexp(−∑j=16aij(xj−pij)2)	6	[0,1]	−3.32	
f21(x)=−∑i=15[(X−ai)(X−ai)T+ci]−1	4	[0,10]	−10.1532	
f22(x)=−∑i=17[(X−ai)(X−ai)T+ci]−1	4	[0,10]	−10.4028	
f23(x)=−∑i=110[(X−ai)(X−ai)T+ci]−1	4	[0,10]	−10.5363	

Table 2 CEC2019 benchmark functions.

Table 2	Functions	fmin	Dimensions	Range	
C1	Storn's Chebyshev Polynomial Fitting Problem	1	9	[−8192, 8192]	
C2	Inverse Hilbert Matrix Problem	1	16	[−16384, 16384]	
C3	Lennard-Jones Minimun Energy Cluster	1	18	[−4, 4]	
C4	Rastrigin's Function	1	10	[−100, 100]	
C5	Griewangk's Function	1	10	[−100, 100]	
C6	Weierstrass Function	1	10	[−100, 100]	
C7	Modified Schwefel's Function	1	10	[−100, 100]	
C8	Expanded Schaffer's F6 Function	1	10	[−100, 100]	
C9	Happy Cat Function	1	10	[−100, 100]	
C10	Ackley Function	1	10	[−100, 100]	

To ensure the fairness of the comparisons, all simulation experiments in this study were conducted on a single host machine equipped with an Intel® Xeon® Gold 5118 CPU at 2.30 GHz and with 16 Gb of RAM. The experiments were developed and performed using Python 3.7. To validate the effectiveness and superiority of the SFGWO algorithm, we compared it with the classical GWO algorithm [8], AGWO [36], SOGWO [37], AGWOCS [38], MFO [39], SCA [17], AOA [40], and TSA [41]. To ensure a fair comparison, the population size was set to 50, the numbers of dimensions were set to 30 and 50, and the number of iterations was set to 1000 for all algorithms. The termination criterion was the reaching of the maximum number of iterations.

During each iteration, the SFGWO fitness function was invoked only once, with the maximum fitness evaluation number (maxFES) being equal to the maximum number of iterations. The evaluation metrics used were the best fitness value (Min), the mean fitness value (Mean), and the standard deviation (Std.) of the fitness values obtained over 30 independent runs of each algorithm. Other parameters were configured according to their original settings, detailed in Table 3.Table 3 Algorithm parameter setting.

Table 3Algorithm	Parameter setting	
MFO	b = 1; a = [−2, −1]	
SCA	a = [2, 0] (decreasing linearly along with the number of iterations)	
AOA	alpha = 5, miu = 0.5, moa_min = 0.2, moa_max = 0.9	
TSA	Pmin = 1; Pmax = 4	
AGWO, AGWOCS, SOGWO	Same as the original papers.	
GWO	a = [2, 0]	
SFGWO	a = [2, 0], m0 = 5, Smax = 9, SL = 4, SU = 15, rx = 10−4	

4.2 SFGWO performance for different values of m0

In SFNs, hub nodes with relatively high degrees play an important role in the network structure and the exchange of information in the neighborhood. In this section, we will analyze the effect of the number of hub nodes m0 on the optimization performance when formulating SFNs. To study m0's impact on the optimization results, its value was varied from 2 to 8 while keeping the other parameters consistent with the values given in Section 4.1 of this Section.

The SFGWO's test results on the unimodal and multimodal functions for 30 and 50 dimensions are presented in Table 4, Table 5. The comparison results indicate that when m0 = 5, the SFGWO achieves the optimal value most frequently, demonstrating significant advantages. The effect is especially apparent in functions f1−f13, with a particularly pronounced performance in function f3. Table 6 contains SFGWO's test results under different values m0 of fixed-dimensional multimodal functions. It is evident that when m0 = 5, the SFGWO attains the same optimal values in f14, f16, f17, f18, f19, f21, f22, and f23, but with a smaller standard deviation. Through the experiments, it can be concluded that the scale-free topology parameter exhibits excellent robustness on fixed-dimensional multimodal functions, with the overall performance being optimal when m0 is set to 5.Table 4 SFGWO 30-dimensional unimodal function test results under different m0 values.

Table 4F		m0 = 2	m0 = 3	m0 = 4	m0 = 5	m0 = 6	m0 = 7	m0 = 8	
f1	Mean	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f2	Mean	9.68E−194	3.65E−194	9.37E−194	2.54E−193	8.41E−195	2.60E−192	1.90E−191	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f3	Mean	1.03E−05	7.86E−12	3.14E−176	1.87E−290	7.16E−23	5.20E−291	1.46E−287	
Std.	5.66E−05	4.30E−11	0.00E+00	0.00E+00	3.92E−22	0.00E+00	0.00E+00	
f4	Mean	1.25E−174	1.74E−176	3.14E−176	2.29E−176	3.63E−176	3.38E−175	7.85E−174	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f5	Mean	1.07E−02	1.68E−02	2.47E−02	3.00E−02	1.09E−02	1.57E−02	1.03E−02	
Std.	1.40E−02	2.76E−02	5.53E−02	1.10E−01	1.89E−02	1.76E−02	1.21E−02	
f6	Mean	2.28E−06	5.38E−10	3.69E−07	6.06E−07	1.66E−07	1.62E−05	1.22E−07	
Std.	1.25E−05	1.40E−09	1.96E−06	2.94E−06	6.27E−07	6.85E−05	6.68E−07	
f7	Mean	8.55E−04	6.75E−04	6.10E−04	7.18E−04	9.25E−04	8.44E−04	7.91E−04	
Std.	6.46E−04	4.99E−04	4.00E−04	4.28E−04	6.12E−04	5.19E−04	6.05E−04	
f8	Mean	−1.26E+04	−1.26E+04	−1.26E+04	−1.26E+04	−1.26E+04	−1.26E+04	−1.26E+04	
Std.	2.17E−02	5.57E−02	2.77E−02	1.87E−02	6.04E−02	3.15E−02	2.52E−02	
f9	Mean	−1.26E+04	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	2.17E−02	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f10	Mean	4.35E−15	4.59E−15	4.35E−15	4.12E−15	4.35E−15	4.35E−15	4.12E−15	
Std.	1.08E−15	1.35E−15	1.08E−15	6.49E−16	1.43E−15	1.43E−15	6.49E−16	
f11	Mean	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f12	Mean	4.84E−10	3.08E−07	1.82E−07	1.36E−06	4.28E−13	1.22E-07	6.11E-08	
Std.	2.65E-09	1.69E-06	7.24E-07	5.14E-06	1.73E-12	6.66E-07	2.31E-07	
f13	Mean	6.52E-05	5.30E-05	6.01E-05	3.53E-05	9.29E-05	1.11E-04	7.49E-05	
Std.	1.32E-04	9.66E-05	9.34E-05	7.14E-05	1.68E-04	3.34E-04	1.67E-04	

Table 5 SFGWO 50-dimensional unimodal function test results under different values of m0.

Table 5F		m0 = 2	m0 = 3	m0 = 4	m0 = 5	m0 = 6	m0 = 7	m0 = 8	
f1	Mean	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f2	Mean	4.15E-189	4.73E-189	6.31E-189	4.24E-187	7.74E-189	8.27E-188	5.38E-188	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f3	Mean	2.91E−17	1.75E−09	7.74E−13	2.22E−236	2.03E−10	1.36E−09	2.17E−271	
Std.	1.60E−16	9.61E−09	3.94E−12	0.00E+00	1.11E−09	7.43E−09	0.00E+00	
f4	Mean	1.75E−170	1.29E−167	1.53E−169	1.76E−169	1.30E−170	2.52E−169	2.78E−167	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f5	Mean	5.00E−02	2.45E−02	2.84E−02	2.47E−02	2.57E−02	1.80E−02	1.83E−02	
Std.	9.97E−02	4.41E−02	6.08E−02	4.42E−02	5.12E−02	2.23E−02	2.79E−02	
f6	Mean	1.48E−04	4.75E−04	4.96E−04	1.88E−03	2.77E−05	5.22E−04	3.80E−04	
Std.	2.53E−04	1.06E−03	1.65E−03	5.50E−03	9.89E−05	1.42E−03	1.36E−03	
f7	Mean	1.06E−03	6.27E−04	6.82E−04	9.79E−04	8.33E−04	8.40E−04	8.26E−04	
Std.	7.24E−04	4.60E−04	4.08E−04	8.69E−04	6.15E−04	5.68E−04	5.35E−04	
f8	Mean	−2.09E+04	−2.09E+04	−2.09E+04	−2.09E+04	−2.09E+04	−2.09E+04	−2.09E+04	
Std.	4.02E−02	5.50E−02	4.97E−02	4.10E−02	8.93E−02	7.62E−02	5.18E−02	
f9	Mean	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f10	Mean	4.71E−15	4.59E−15	4.47E−15	4.94E−15	5.42E−15	5.18E−15	4.83E−15	
Std.	1.45E−15	1.35E−15	1.23E−15	1.60E−15	1.77E−15	1.70E−15	1.53E−15	
f11	Mean	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
f12	Mean	1.36E−06	6.52E−06	1.75E−06	1.15E−06	3.58E−06	2.93E−06	2.55E−08	
Std.	3.68E−06	2.82E−05	5.82E−06	2.17E−06	1.50E−05	1.33E−05	3.78E−08	
f13	Mean	9.81E−04	6.05E−04	1.74E−03	4.63E−04	8.64E−04	1.09E-03	5.14E-04	
Std.	2.77E-03	1.66E-03	5.81E-03	1.73E-03	2.15E-03	3.40E-03	1.96E-04	

Table 6 SFGWO multimodal function test results under different values of m0.

Table 6F		m0 = 2	m0 = 3	m0 = 4	m0 = 5	m0 = 6	m0 = 7	m0 = 8	
f14	Mean	9.98E-01	9.98E-01	9.98E-01	9.98E-01	9.98E-01	9.98E-01	9.98E-01	
Std.	5.95E-11	5.20E-11	4.20E-11	3.28E-11	3.64E-11	3.60E-11	3.55E-11	
f15	Mean	5.97E-04	5.86E-04	5.86E-04	5.57E-04	5.30E-04	5.63E-04	6.48E-04	
Std.	2.02E-04	2.05E-04	1.85E-04	1.69E-04	1.65E-04	2.61E-04	1.20E-04	
f16	Mean	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	
Std.	2.23E-07	1.52E-07	2.44E-07	4.23E-07	3.35E-08	2.80E-08	2.17E-07	
f17	Mean	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	3.98E-01	
Std.	1.32E-07	3.24E-04	1.45E-04	1.33E-04	6.60E-07	8.23E-05	4.12E-04	
f18	Mean	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	
Std.	5.74E−11	1.02E−12	5.34E−11	1.64E−12	8.18E−11	3.41E−11	1.57E−12	
f19	Mean	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	
Std.	5.94E−09	1.68E−10	2.48E−10	1.45E−11	5.69E−11	2.56E−10	7.22E−11	
f20	Mean	−3.27E+00	−3.26E+00	−3.27E+00	−3.28E+00	−3.26E+00	−3.28E+00	−3.28E+00	
Std.	5.92E−02	6.03E−02	6.03E−02	5.83E−02	6.03E−02	5.83E−02	5.83E−02	
f21	Mean	−1.02E+01	−1.02E+01	−1.02E+01	−1.02E+01	−1.02E+01	−1.02E+01	−1.02E+01	
Std.	1.83E−05	5.36E−05	5.00E−05	1.01E−04	3.39E−05	5.73E−05	1.90E−05	
f22	Mean	−1.04E+01	−1.04E+01	−1.04E+01	−1.04E+01	−1.04E+01	−1.04E+01	−1.04E+01	
Std.	7.28E−05	5.80E−05	3.86E−05	3.70E−05	9.04E−05	5.10E−05	9.48E−05	
f23	Mean	−1.05E+01	−1.05E+01	−1.05E+01	−1.05E+01	−1.05E+01	−1.05E+01	−1.05E+01	
Std.	4.25E−05	1.92E−05	6.53E−05	3.93E−05	4.87E−05	4.72E−05	7.83E−05	

4.3 Ablation study

The results of an ablation study are presented in this section to assess the impact of each improvement strategy employed in the SFGWO algorithm accurately. The SFGWO incorporates three improvement strategies; the SFN topology population creation strategy establishes a network topology structure that serves as the foundation for the neighborhood learning strategy, and these two strategies are intertwined within the SFGWO framework. Since the Scale-Free Network (SFN) serves as the foundation for the Neighborhood Learning Strategy (NLS), the scale-free networks created through SFN form the domain of each grey wolf in the NLS. Therefore, SFN and NLS work together to improve the algorithm and are analyzed as a whole during ablation experiments. Furthermore, the AIRS was also introduced. Therefore, in this section, the ablation experiments results are presented for three versions of the SFGWO: SFGWO-A (where the AIRS has been removed and SFN and NLS are adopted) and SFGWO-B (where only the AIRS is added, without SFN and NLS), and SFGWO (which includes all three strategies). The experiments were conducted on 23 classic benchmark test functions using the parameter settings outlined in Section 4.2. Table 7, Table 8 present the experimental results.Table 7 Ablation study results (f1–f13).

Table 7F		GWO	SFGWO-A	SFGWO-B	SFGWO	
f1	Mean	4.94E−77	0.00E+00	1.28E−76	0.00E+00	
Min	1.86E−79	0.00E+00	4.26E−80	0.00E+00	
Std.	1.32E−76	0.00E+00	2.92E−76	0.00E+00	
f2	Mean	6.59E−45	1.35E−194	8.21E−45	2.91E-191	
Min	9.75E-46	1.99E-201	3.21E-46	1.26E-200	
Std.	9.61E-45	0.00E+00	8.70E-45	0.00E+00	
f3	Mean	4.86E−17	8.88E−288	3.93E−08	1.47E−266	
Min	3.94E−23	0.00E+00	6.81E−22	0.00E+00	
Std.	1.98E−16	0.00E+00	1.07E−07	0.00E+00	
f4	Mean	1.59E−16	3.56E−174	1.25E−12	7.19E−176	
Min	2.21E−18	1.41E−180	3.00E−18	5.86E−183	
Std.	2.70E−16	0.00E+00	5.02E−12	0.00E+00	
f5	Mean	2.65E+01	2.39E+01	2.57E+01	1.20E−02	
Min	2.52E+01	2.35E+01	2.52E+01	2.50E−06	
Std.	7.82E−01	1.90E−01	1.94E−01	2.58E−02	
f6	Mean	4.20E−01	1.15E−14	7.72E−06	1.65E−09	
Min	4.96E−06	6.25E−18	4.47E−06	2.48E−18	
Std.	2.77E−01	2.91E−14	2.01E−06	8.87E−09	
f7	Mean	6.52E−04	1.24E−04	1.60E−03	8.72E−04	
Min	1.43E−04	1.48E−05	3.26E−04	2.59E−05	
Std.	3.72E−04	7.54E−05	8.61E−04	7.06E−04	
f8	Mean	−6.31E+03	−7.95E+03	−6.67E+03	−1.26E+04	
Min	−7.79E+03	−8.93E+03	−8.45E+03	−1.26E+04	
Std.	8.32E+02	4.61E+02	1.45E+03	2.50E−02	
f9	Mean	2.14E+00	0.00E+00	1.70E+01	0.00E+00	
Min	0.00E+00	0.00E+00	5.68E−14	0.00E+00	
Std.	4.16E+00	0.00E+00	3.93E+01	0.00E+00	
f10	Mean	1.42E−14	4.12E−15	1.95E−14	4.59E−15	
Min	7.55E−15	4.00E−15	1.11E−14	4.00E−15	
Std.	1.54E−15	6.49E−16	4.01E−15	1.35E−15	
f11	Mean	3.10E−03	0.00E+00	3.83E−03	0.00E+00	
Min	0.00E+00	0.00E+00	0.00E+00	0.00E+00	
Std.	6.70E−03	0.00E+00	8.30E−03	0.00E+00	
f12	Mean	2.88E−02	3.46E−03	1.07E−06	1.94E-09	
Min	6.60E-03	1.18E-19	4.79E-07	1.70E-19	
Std.	1.52E-02	1.89E-02	3.69E-07	9.03E-09	
f13	Mean	2.96E-01	5.67E-02	3.80E-04	6.41E-05	
Min	1.22E-05	1.35E-18	5.53E-06	6.40E-19	
Std.	2.01E-01	7.17E-02	2.01E-03	1.47E-04	
Mean rank	3.30	1.65	3.46	1.57	
Final rank	3	2	4	1	

Table 8 Ablation study results (f14–f23).

Table 8F		GWO	SFGWO-A	SFGWO-B	SFGWO	
f14	Mean	4.07E+00	1.13E+00	9.98E−01	9.98E−01	
Min	9.98E−01	9.98E−01	9.98E−01	9.98E−01	
Std.	4.02E+00	5.03E−01	1.50E−11	5.49E−11	
f15	Mean	3.72E−03	1.07E−03	1.07E−03	5.52E−04	
Min	3.07E−04	3.07E−04	3.07E−04	3.09E−04	
Std.	7.57E−03	3.66E−03	3.65E−03	1.51E−04	
f16	Mean	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	
Min	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	
Std.	1.87E−09	2.26E−16	2.60E−09	1.51E−06	
f17	Mean	3.98E−01	3.98E−01	3.98E−01	3.98E−01	
Min	3.98E−01	3.98E−01	3.98E−01	3.98E−01	
Std.	6.09E−07	0.00E+00	1.72E−06	9.90E−04	
f18	Mean	3.00E+00	3.00E+00	3.00E+00	3.00E+00	
Min	3.00E+00	3.00E+00	3.00E+00	3.00E+00	
Std.	4.08E−06	1.33E−15	2.07E−07	3.56E−14	
f19	Mean	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	
Min	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	
Std.	1.23E−03	5.47E−16	3.88E−06	1.58E−11	
f20	Mean	−3.25E+00	−3.28E+00	−3.25E+00	−3.28E+00	
Min	−3.32E+00	−3.32E+00	−3.32E+00	−3.32E+00	
Std.	7.60E−02	5.83E−02	5.99E−02	5.70E-02	
f21	Mean	−9.23E+00	−8.62E+00	−8.97E+00	−1.02E+01	
Min	−1.02E+01	−1.02E+01	−1.02E+01	−1.02E+01	
Std.	2.14E+00	2.38E+00	2.18E+00	7.93E−05	
f22	Mean	−1.02E+01	−9.87E+00	−1.00E+01	−1.04E+01	
Min	−1.04E+01	−1.04E+01	−1.04E+01	−1.04E+01	
Std.	9.63E−01	1.62E+00	1.34E+00	7.24E−05	
f23	Mean	−1.02E+01	−1.02E+01	−1.05E+01	−1.05E+01	
Min	−1.05E+01	−1.05E+01	−1.05E+01	−1.05E+01	
Std.	1.37E+00	1.37E+00	8.63E−05	4.17E−05	
	Mean rank	2.90	2.85	2.5	1.75	
	Final rank	4	3	2	1	

Table 7 shows that in the unimodal functions (f1−f7), SFGWO-A outperformed the other versions of the algorithm on f2, f3, f6, and f7 in terms of average values, indicating the excellent exploitation capability of SFGWO-A. However, SFGWO-B demonstrates lower performance than GWO on functions (f1−f4) because while AIRS enhances the exploration capability, it weakens the exploitation capability of SFGWO-B. In the multimodal functions (f8−f23), the SFGWO performed better than other algorithms on f8, f12, and f13 in terms of average values, demonstrating that the AIRS can enhance the algorithm's exploration capability and helps it escape local optima when the algorithm stagnates. At the bottom of Table 7, the mean and final ranks of the compared algorithms are shown, where SFGWO ranks first, SFGWO-A ranks second, and SFGWO-B ranks fourth. Therefore, it can be concluded that using all three strategies in SFGWO results in optimal performance, while independently using AIRS may negatively affect the algorithm. Table 8 presents the ablation experiment results for fixed-dimension multimodal functions, where it can be seen that the SFGWO-A performs best on five functions, SFGWO-B performs best on seven functions, and the SFGWO leads or performs best on all ten functions. This indicates that for fixed-dimension multimodal functions, the exploration capability of SFGWO-B surpasses that of SFGWO-A.

In summary, the comprehensive performance of the algorithm is optimal when all three strategies are employed simultaneously.

4.4 Comparison with other algorithms on 23 classical benchmark functions

1) Improve strategy and algorithm performance analysis

To demonstrate the effectiveness of the proposed improvement strategies, in this section, the results of tests of the proposed improvements are presented and compared across the 23 benchmark functions. Table 9, Table 10 present the results of the nine algorithms in 30 and 50 dimensions, respectively, while Table 11 displays the results for solving fixed−dimensional multi−modal function problems.Table 9 Results of the compared algorithms on functions (f1–f13); the number of dimensions was fixed to 30.

Table 9F		GWO	SOGWO	AGWO	AGWOCS	MFO	SCA	AOA	TSA	SFGWO	
f1	Mean	4.94E−77	2.05E−70	4.20E−104	1.15E−102	1.67E+03	5.43E−03	0.00E+00	4.43E−58	0.00E+00	
Min	1.86E−79	7.77E−73	4.37E−109	1.58E−108	4.57E−09	5.48E−08	0.00E+00	4.43E−58	0.00E+00	
Std.	1.32E−76	4.20E−70	1.19E−103	6.14E−102	3.79E+03	1.84E−02	0.00E+00	7.20E−74	0.00E+00	
f2	Mean	6.59E−45	3.72E−41	1.80E−61	1.26E−61	3.60E+01	4.21E−06	0.00E+00	1.43E−36	2.91E−191	
Min	9.75E−46	2.21E−42	3.90E−64	8.74E−63	2.22E−06	3.28E−08	0.00E+00	1.43E−36	1.26E−200	
Std.	9.61E−45	3.24E−41	6.47E−61	2.10E−61	1.73E+01	1.08E−05	0.00E+00	5.10E−52	0.00E+00	
f3	Mean	4.86E−17	1.90E−18	5.92E−30	1.06E−29	1.30E+04	3.24E+03	8.68E−221	2.09E−10	1.47E−266	
Min	3.94E−23	1.19E−24	3.35E−37	3.13E−36	1.25E+02	9.60E+00	0.00E+00	2.09E−10	0.00E+00	
Std.	1.98E−16	9.44E−18	2.48E−29	4.15E−29	1.02E+04	3.37E+03	0.00E+00	5.26E−26	0.00E+00	
f4	Mean	1.59E−16	1.55E−17	1.91E−27	7.75E−28	5.58E+01	1.14E+01	1.33E−03	6.30E−05	7.19E−176	
Min	2.21E−18	5.92E−19	4.47E−30	1.82E−30	2.49E+01	9.15E−01	2.07E−264	6.30E−05	5.86E−183	
Std.	2.70E−16	3.11E−17	7.81E−27	1.91E−27	1.21E+01	7.57E+00	7.28E−03	2.76E−20	0.00E+00	
f5	Mean	2.65E+01	2.65E+01	2.67E+01	2.65E+01	2.68E+06	4.47E+01	2.69E+01	2.85E+01	1.20E−02	
Min	2.52E+01	2.50E+01	2.60E+01	2.54E+01	1.02E+00	2.78E+01	2.58E+01	2.85E+01	2.50E−06	
Std.	7.82E−01	6.24E−01	5.42E−01	5.14E−01	1.46E+07	3.08E+01	6.51E−01	7.23E−15	2.58E−02	
f6	Mean	4.20E−01	3.82E−01	1.00E+00	1.15E+00	3.33E+03	4.13E+00	1.45E+00	4.47E+00	1.65E−09	
Min	4.96E−06	7.48E−06	3.61E−01	5.57E−01	2.08E−08	3.22E+00	9.89E−01	4.47E+00	2.48E−18	
Std.	2.77E−01	3.05E−01	3.37E−01	3.42E−01	5.48E+03	3.82E−01	2.09E−01	0.00E+00	8.87E−09	
f7	Mean	6.52E−04	5.10E−04	3.52E−04	4.14E−04	3.03E+00	1.69E−02	8.98E−07	8.73E−05	8.72E−04	
Min	1.43E−04	2.64E−04	1.43E−05	8.74E−05	2.86E−02	1.55E−03	5.70E−08	8.73E−05	2.59E−05	
Std.	3.72E−04	2.23E−04	2.27E−04	4.33E−04	6.46E+00	1.89E−02	8.00E−07	2.76E−20	7.06E−04	
f8	Mean	−6.31E+03	−6.69E+03	−3.93E+03	−3.87E+03	−8.43E+03	−4.07E+03	−7.48E+03	−3.64E+03	−1.26E+04	
Min	−7.79E+03	−7.75E+03	−4.46E+03	−4.54E+03	−9.88E+03	−4.88E+03	−7.98E+03	−3.64E+03	−1.26E+04	
Std.	8.32E+02	5.35E+02	2.96E+02	3.50E+02	6.97E+02	3.73E+02	3.13E+02	9.25E−13	2.50E−02	
f9	Mean	2.14E+00	5.50E−01	1.89E−15	0.00E+00	1.66E+02	1.03E+01	0.00E+00	0.00E+00	0.00E+00	
Min	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.05E+02	4.35E−06	0.00E+00	0.00E+00	0.00E+00	
Std.	4.16E+00	2.31E+00	1.04E−14	0.00E+00	4.02E+01	1.61E+01	0.00E+00	0.00E+00	0.00E+00	
f10	Mean	1.42E−14	1.30E−14	7.19E−15	7.08E−15	1.34E+01	1.26E+01	4.44E−16	7.55E−15	4.59E−15	
Min	7.55E−15	7.99E−15	4.00E−15	4.00E−15	4.58E−05	3.20E−04	4.44E−16	7.55E−15	4.00E−15	
Std.	1.54E−15	2.74E−15	1.08E−15	1.54E−15	8.84E+00	9.48E+00	0.00E+00	0.00E+00	1.35E−15	
f11	Mean	3.10E−03	1.64E−03	0.00E+00	0.00E+00	3.61E+01	2.01E−01	1.31E−02	0.00E+00	0.00E+00	
Min	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.56E−08	1.84E−07	0.00E+00	0.00E+00	0.00E+00	
Std.	6.70E−03	5.16E−03	0.00E+00	0.00E+00	6.09E+01	2.86E−01	2.03E−02	0.00E+00	0.00E+00	
f12	Mean	2.88E−02	2.15E−02	6.87E−02	8.06E−02	8.53E+06	1.53E+01	1.00E−01	1.08E+00	1.94E−09	
Min	6.60E−03	5.33E−03	3.52E−02	1.77E−02	4.78E−10	3.52E−01	7.58E−02	1.08E+00	1.70E−19	
Std.	1.52E−02	9.79E−03	2.92E−02	9.51E−02	4.67E+07	7.87E+01	2.07E−02	2.26E−16	9.03E−09	
f13	Mean	2.96E−01	2.96E−01	8.15E−01	8.75E−01	1.37E+07	4.11E+00	2.65E+00	2.74E+00	6.41E−05	
Min	1.22E−05	7.73E−02	5.58E−01	5.35E−01	2.09E−07	1.91E+00	2.40E+00	2.74E+00	6.40E−19	
Std.	2.01E−01	1.56E−01	1.61E−01	1.74E−01	7.49E+07	5.04E+00	1.36E−01	9.03E−16	1.47E−04	

Table 10 Results of the comparative algorithms on functions (f1–f13); the number of dimensions was fixed to 50.

Table 10F		GWO	SOGWO	AGWO	AGWOCS	MFO	SCA	AOA	TSA	SFGWO	
f1	Mean	1.74E−56	2.15E−51	1.07E−72	4.20E−71	8.67E+03	2.64E+01	3.40E−77	3.91E−36	0.00E+00	
Min	3.71E−59	1.48E−53	4.97E−75	1.94E−75	2.13E−02	2.01E−02	0.00E+00	3.91E−36	0.00E+00	
Std.	3.71E−56	3.96E−51	4.09E−72	2.18E−70	9.00E+03	4.52E+01	1.86E−76	0.00E+00	0.00E+00	
f2	Mean	1.84E−33	1.24E−30	1.15E−44	2.49E−44	8.03E+01	5.31E−03	0.00E+00	2.45E−25	2.89E−189	
Min	3.13E−34	2.58E−31	1.50E−46	3.19E−46	4.67E−02	3.97E−05	0.00E+00	2.45E−25	2.58E−197	
Std.	1.27E−33	7.97E−31	1.35E−44	3.47E−44	3.84E+01	7.99E−03	0.00E+00	4.67E−41	0.00E+00	
f3	Mean	6.54E−06	1.68E−07	3.75E−14	4.64E−15	4.47E+04	2.84E+04	1.13E−02	1.72E−05	1.04E−260	
Min	8.85E−10	6.01E−13	3.31E−20	6.22E−21	7.79E+03	7.84E+03	1.60E−252	1.72E−05	7.77E−295	
Std.	1.72E−05	8.49E−07	1.49E−13	1.92E−14	2.60E+04	9.82E+03	1.14E−02	3.45E−21	0.00E+00	
f4	Mean	6.20E−10	2.24E−11	6.16E−15	2.00E−17	7.99E+01	5.18E+01	4.14E−02	1.45E−04	4.47E−170	
Min	6.34E−11	3.15E−12	4.75E−19	2.20E−20	7.06E+01	3.16E+01	1.27E−02	1.45E−04	1.89E−179	
Std.	5.85E−10	1.61E−11	3.12E−14	3.70E−17	4.88E+00	9.77E+00	5.61E−03	2.76E−20	0.00E+00	
f5	Mean	4.66E+01	4.68E+01	4.66E+01	4.68E+01	8.00E+06	4.70E+05	4.79E+01	4.89E+01	1.35E−02	
Min	4.52E+01	4.52E+01	4.58E+01	4.57E+01	1.43E+02	3.22E+02	4.72E+01	4.89E+01	6.91E−05	
Std.	8.48E−01	8.77E−01	6.05E−01	7.07E−01	2.44E+07	7.39E+05	4.35E−01	7.23E−15	2.06E−02	
f6	Mean	1.41E+00	1.75E+00	3.81E+00	3.65E+00	6.99E+03	2.64E+01	5.06E+00	1.07E+01	2.40E−04	
Min	5.03E−01	7.56E−01	3.03E+00	2.44E+00	2.79E−02	9.80E+00	4.16E+00	1.07E+01	1.65E−08	
Std.	4.58E−01	5.23E−01	4.40E−01	4.17E−01	1.15E+04	2.22E+01	2.91E−01	3.61E−15	5.59E−04	
f7	Mean	1.13E−03	7.69E−04	6.85E−04	6.20E−04	2.24E+01	3.01E−01	4.87E−07	2.63E−05	9.22E−04	
Min	2.50E−04	2.43E−04	1.07E−04	1.98E−04	1.70E−01	6.15E−02	2.44E−08	2.63E−05	6.63E−05	
Std.	5.06E−04	5.08E−04	4.51E−04	4.04E−04	2.81E+01	2.52E−01	3.90E−07	6.89E−21	6.20E−04	
f8	Mean	−9.83E+03	−9.64E+03	−5.09E+03	−4.77E+03	−1.29E+04	−5.25E+03	−9.47E+03	−6.86E+03	−2.09E+04	
Min	−1.20E+04	−1.22E+04	−5.95E+03	−5.67E+03	−1.61E+04	−6.08E+03	−1.08E+04	−6.86E+03	−2.09E+04	
Std.	1.71E+03	1.83E+03	3.92E+02	4.22E+02	1.41E+03	3.90E+02	6.05E+02	1.85E−12	7.53E−02	
f9	Mean	3.14E+00	3.02E−01	0.00E+00	0.00E+00	3.00E+02	4.98E+01	0.00E+00	0.00E+00	0.00E+00	
Min	5.68E−14	0.00E+00	0.00E+00	0.00E+00	1.90E+02	2.49E−01	0.00E+00	0.00E+00	0.00E+00	
Std.	5.60E+00	1.19E+00	0.00E+00	0.00E+00	6.55E+01	4.14E+01	0.00E+00	0.00E+00	0.00E+00	
f10	Mean	2.80E−14	2.61E−14	8.85E−15	8.62E−15	1.97E+01	1.64E+01	4.44E−16	4.00E−15	4.83E−15	
Min	2.18E−14	1.87E−14	7.55E−15	7.55E−15	1.60E+01	6.81E−02	4.44E−16	4.00E−15	4.44E−16	
Std.	2.75E−15	4.51E−15	2.38E−15	2.31E−15	7.92E−01	7.62E+00	0.00E+00	0.00E+00	1.53E−15	
f11	Mean	0.00E+00	8.73E−04	0.00E+00	0.00E+00	5.75E+01	1.08E+00	1.17E−01	0.00E+00	0.00E+00	
Min	0.00E+00	0.00E+00	0.00E+00	0.00E+00	1.53E−02	1.03E−01	1.00E−02	0.00E+00	0.00E+00	
Std.	0.00E+00	4.78E−03	0.00E+00	0.00E+00	6.89E+01	5.37E−01	7.17E−02	0.00E+00	0.00E+00	
f12	Mean	6.28E−02	5.77E−02	1.74E−01	1.61E−01	2.56E+07	5.69E+05	4.21E−01	9.14E−01	6.25E−07	
Min	2.02E−02	2.89E−02	1.04E−01	9.49E−02	1.51E+00	3.77E+00	3.46E−01	9.14E−01	8.21E−11	
Std.	2.69E−02	3.13E−02	6.43E−02	5.27E−02	7.81E+07	1.25E+06	3.67E−02	1.13E−16	1.66E−06	
f13	Mean	1.32E+00	1.36E+00	2.58E+00	2.64E+00	8.20E+07	8.99E+05	4.65E+00	4.72E+00	2.00E−04	
Min	7.56E−01	9.36E−01	2.29E+00	2.18E+00	1.97E+00	9.56E+00	4.36E+00	4.72E+00	1.61E−09	
Std.	3.78E−01	2.49E−01	2.26E−01	2.61E−01	1.67E+08	1.74E+06	1.55E−01	0.00E+00	3.23E−04	

Table 11 Results of the algorithms’ application on functions (f14–f23).

Table 11F		GWO	SOGWO	AGWO	AGWOCS	MFO	SCA	AOA	TSA	SFGWO	
f14	Mean	4.07E+00	2.05E+00	1.79E+00	3.29E+00	1.49E+00	1.39E+00	5.35E+00	1.08E+01	9.98E−01	
Min	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	9.98E−01	1.08E+01	9.98E−01	
Std.	4.02E+00	1.91E+00	1.90E+00	3.52E+00	1.15E+00	8.07E−01	3.48E+00	5.42E−15	5.49E−11	
f15	Mean	3.72E−03	1.68E−03	1.71E−03	3.18E−04	1.04E−03	9.20E−04	8.78E−03	9.49E−04	5.52E−04	
Min	3.07E−04	3.07E−04	3.08E−04	3.08E−04	5.16E−04	3.56E−04	3.57E−04	9.49E−04	3.09E−04	
Std.	7.57E−03	5.08E−03	5.08E−03	2.56E−05	3.85E−04	3.96E−04	1.21E−02	1.10E−19	1.51E−04	
f16	Mean	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	
Min	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	−1.03E+00	
Std.	1.87E−09	4.16E−09	1.34E−06	1.03E−06	4.52E−16	1.26E−05	2.08E−08	0.00E+00	1.51E−06	
f17	Mean	3.98E−01	3.98E−01	3.98E−01	3.98E−01	4.75E−01	3.99E−01	4.05E−01	4.00E−01	3.98E−01	
Min	3.98E−01	3.98E−01	3.98E−01	3.98E−01	3.98E−01	3.98E−01	3.98E−01	4.00E−01	3.98E−01	
Std.	6.09E−07	4.19E−07	1.62E−04	1.71E−04	4.21E−01	1.00E−03	7.41E−03	0.00E+00	9.90E−04	
f18	Mean	3.00E+00	5.70E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	4.80E+00	3.00E+00	3.00E+00	
Min	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	3.00E+00	
Std.	4.08E−06	1.48E+01	4.13E−06	1.25E−06	1.65E−15	2.10E−05	6.85E+00	9.03E−16	3.56E−14	
f19	Mean	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.82E+00	−3.86E+00	
Min	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.86E+00	−3.82E+00	−3.86E+00	
Std.	1.23E−03	2.36E−03	3.57E−03	1.41E−03	9.03E−16	2.76E−03	2.44E−03	0.00E+00	1.58E−11	
f20	Mean	−3.25E+00	−3.25E+00	−3.19E+00	−3.30E+00	−3.22E+00	−3.01E+00	−3.21E+00	−3.19E+00	−3.28E+00	
Min	−3.32E+00	−3.32E+00	−3.32E+00	−3.31E+00	−3.32E+00	−3.21E+00	−3.28E+00	−3.19E+00	−3.32E+00	
Std.	7.60E−02	8.14E−02	1.12E−01	5.87E−03	4.51E−02	1.97E−01	4.78E−02	4.52E−16	5.70E−02	
f21	Mean	−9.23E+00	−9.82E+00	−7.42E+00	−7.75E+00	−6.72E+00	−3.53E+00	−5.03E+00	−5.29E+00	−1.02E+01	
Min	−1.02E+01	−1.02E+01	−9.77E+00	−9.72E+00	−1.02E+01	−7.70E+00	−8.94E+00	−5.29E+00	−1.02E+01	
Std.	2.14E+00	1.29E+00	1.97E+00	1.75E+00	3.38E+00	2.08E+00	1.74E+00	2.71E−15	7.93E−05	
f22	Mean	−1.02E+01	−1.04E+01	−8.28E+00	−8.66E+00	−8.41E+00	−4.32E+00	−5.44E+00	−4.97E+00	−1.04E+01	
Min	−1.04E+01	−1.04E+01	−1.03E+01	−1.03E+01	−1.04E+01	−9.43E+00	−8.71E+00	−4.97E+00	−1.04E+01	
Std.	9.63E−01	1.70E−04	1.60E+00	1.30E+00	3.16E+00	2.00E+00	1.83E+00	9.03E−16	7.24E−05	
f23	Mean	−1.02E+01	−1.04E+01	−8.71E+00	−8.66E+00	−8.43E+00	−4.47E+00	−5.78E+00	−5.01E+00	−1.05E+01	
Min	−1.05E+01	−1.05E+01	−1.02E+01	−1.01E+01	−1.05E+01	−7.27E+00	−9.83E+00	−5.01E+00	−1.05E+01	
Std.	1.37E+00	9.87E−01	1.24E+00	9.50E−01	3.31E+00	1.70E+00	1.73E+00	9.03E−16	4.17E−05	

Based on the 30-dimensional comparison results in Table 9, both the AOA and SFGWO algorithms achieved the optimal solution for the unimodal function f1, indicating their excellent performance. For the unimodal functions f2 and f7, AOA exhibited superior performance, with its optimal values outperforming those of the SFGWO, which in turn performed better than other algorithms on other unimodal test functions, whether viewed from the perspective of mean or standard deviation. This suggests that the SFGWO algorithm performs stably and efficiently on unimodal functions. Since unimodal benchmark functions have only one optimal value, they serve as benchmarks for the exploitation ability. The main reason SFGWO has good performance on unimodal functions is that the information exchange between GWs and their neighbors (hub nodes or non-hub nodes) enhances the local exploitation ability of the neighborhood domain. Despite the SFGWO's superior performance on function f6 compared to other algorithms, a significant gap exists between its mean and minimum values, indicating a lack of stability when handling function f6. This inconsistency in performance could stem from the reliance of the AIRS on random values, leading to a variability in algorithm outcomes.

In the optimization experiments on multimodal functions (f8−f13) and fixed-dimension multimodal functions (f14−f23), the SFGWO algorithm outperformed the other algorithms in most cases. On multimodal functions f9 and f11, both the SFGWO and AGWOCS algorithms achieved the optimal solution, which indicates that these two algorithms possess strong exploration capabilities, enabling them to find the optimal solution in the complex search space of multimodal functions. On function f8, the SFGWO algorithm was the only one to achieve the optimal solution, which suggests that it has an advantage in exploration and can find the optimal solution in multimodal functions where other algorithms fail. On f12 and f13, the SFGWO algorithm performed relatively well, surpassing the other eight algorithms. However, the significant gap between the mean and minimum values suggests that the algorithm may be prone to premature convergence to local optima during the optimization process. The AOA algorithm showed excellent performance on multimodal function f10, which implies that it has an advantage over specific multimodal functions. As shown in Table 11, the SFGWO algorithm achieved the global optimum on functions f14, f16, f17, f18, f19, f21, f22, and f23. The AGWOCS algorithm demonstrated relatively better performance on f15, surpassing the other algorithms. However, it is worth noting that many other algorithms also achieved good optimization results on the fixed-dimension multi-modal functions, as these functions are relatively simple. Multimodal functions and fixed-dimension multimodal functions are generally more complex because they contain multiple peaks, one of which is the global optimum, while the others are local optima. Therefore, multimodal benchmarks are suitable for measuring the exploration abilities of optimization algorithms.

The experimental results of SFGWO on the multimodal functions and fixed-dimension multimodal functions demonstrate its effectiveness in exploring the entire search space and finding different types of optimal solutions. The main reason for the good performance of SFGWO is that at each iteration, the population based on SFN is rebuilt according to the optimization results of the previous iteration. This updates the neighbors of each GW, which is beneficial to the full utilization of the information exchange between wolves and neighbors and to expand the search range. When the algorithm is trapped in a local optimum, the AIRS can reactivate the stagnant wolves dynamically and change their positions, thereby escaping the optimum.

From the comparison results of the 50-dimensional data shown in Table 10, we can conclude that the SFGWO algorithm was still able to obtain the optimal solution in function f1. At the same time, AOA could not achieve the optimal solution in 50 dimensions as it did in 30 dimensions. In functions f3−f5, f8, f9, and f11−f13, SFGWO's performance was excellent and surpassed that of the other eight algorithms in terms of both accuracy and stability. Out of the total 13 functions, SFGWO outperformed the other algorithms in terms of the mean value in nine of them. In summary, the SFGWO algorithm exhibited strong performance and outstanding results even in high-dimensional spaces.2) Nonparametric test analysis

Relying solely on the mean and standard deviation to compare algorithms is insufficient. To comprehensively assess the performance of the SFGWO algorithm, in this section, the mean value from 30 independent runs of each algorithm is utilized, and the Wilcoxon rank-sum and Friedman tests are employed as non-parametric statistical methods. Table 12 presents the results of the Wilcoxon rank-sum tests between the SFGWO and the other compared algorithms, while Table 13 contains the rank mean values obtained from the Friedman test for algorithm ranking.Table 12 Wilcoxon rank sum test results.

Table 12SFGWO vs.	+	–	≈	
GWO	21	2	0	
SOGWO	21	2	0	
AGWO	21	1	1	
AGWOCS	20	1	2	
MFO	21	2	0	
SCA	23	0	0	
AOA	21	0	2	
TSA	21	0	2	

Table 13 Friedman test results.

Table 13Algorithm	Rank	Final Rank	
GWO	4.43	5	
SOGWO	4.23	4	
AGWO	4.13	3	
AGWOCS	3.65	2	
MFO	6.82	9	
SCA	6.78	8	
AOA	4.91	6	
TSA	5.86	7	
SFGWO	2.08	1	

The Wilcoxon rank-sum test was employed to evaluate whether significant differences existed between the two algorithms in terms of performance. This test is based on comparing the rank distribution of algorithm performance and is suitable for situations where the data do not follow a normal distribution. The significance level p = 0.05 was used to evaluate the algorithm performance. If p < 0.05, the null hypothesis is rejected, suggesting a significant difference between the two algorithms; conversely, p ≥ 0.05 indicates that the two tested algorithms do not exhibit significant differences. In the results of the Wilcoxon rank-sum test, "+", "−", and "≈" denote whether the accuracy and stability of the SFGWO algorithm were better, worse, or equal to the corresponding algorithm's test results.

The Friedman test was used to rank the performance of multiple algorithms across repeated runs and assess their overall performance. Lower ranks indicate better algorithm performance, while higher ranks indicate poorer performance. We can determine whether the rank differences between algorithms are statistically significant through the Friedman test.

According to Table 12, SFGWO demonstrated a high overall solving capability across the 23 benchmark functions, outperforming the GWO in 21 of them. Compared to AGWO, the SFGWO exhibited significantly better performance on 21 of the 23 benchmark functions, with only a slight decrease in performance on one function and comparable performance on another one. Compared to AOA, SFGWO performed better in 21 functions and approached AOA's performance level in the remaining 2. Therefore, there was a significant difference between SFGWO and the other compared algorithms in most cases.

According to the rank mean results obtained from the Friedman test shown in Table 13, the SFGWO was ranked at 2.08, taking the first position among the nine algorithms. The AGWOCS was ranked at 3.65, coming in second place, while the AGWO was ranked at 4.13, coming in third position. This indicates that the proposed SFGWO algorithm has good optimization accuracy and stability.3) Exploration and exploitation analysis

In optimization algorithms, exploration and exploitation are two crucial phases. The algorithm's optimization efficiency is largely determined by the balance between exploration and exploitation capabilities. Excessive exploration can reduce search efficiency and slow down convergence speed, while excessive exploitation may lead to premature entrapment in local optima, preventing the discovery of better solutions. In this section, Eq. (12), (13), (14), (15) are employed to determine the proportions of the exploration and exploitation phases [50]. This analysis was then used to characterize the algorithm's exploration and exploitation processes during operation and compare them with the classical GWO.(12) Divj=1n∑i=1nmedian(xj)−xij

(13) Div=1dim∑j=1dimDivj

(14) Exploration%=DivDivmax×100

(15) Exploration%=|Div−Divmax|Divmax×100

The population diversity values were determined using Eqs. (12), (13)), where Divj represents the diversity of the j-th dimension in the population; median(xj) represents the median value of the j-th dimension among all n population members; xj i represents the value of the j-th dimension of the i-th individual in the population. Div is the overall diversity of the population in the current iteration, while dim is the number of dimensions of the problem. Equations (14), (15)) determine the percentages of exploration and exploitation during the iteration, where Divmax represents the maximum value of diversity during the entire iteration process; Exploration% represents the percentage of exploration in the current iteration, while Exploitation% represents the percentage of exploitation in the current iteration.

In Fig. 6, we employed unimodal functions f3, f5, and f6, and multimodal functions f10, f12, and f13 to separately analyze the exploitation and exploration capabilities of the optimization algorithms during the iteration process. The first column in Fig. 6 shows the convergence curves of GWO and SFGWO, while the second and third columns present the exploration and exploitation curves of the GWO and the SFGWO, respectively.Fig. 6 Convergence Exploration and exploitation comparison of SFGWO and GWO.

Fig. 6

For functions f3 and f6, the average exploitation of the SFGWO was greater than that of the GWO, indicating that the SFGWO had comparatively enhanced exploitation capabilities. Examining the convergence curve of function f5, we observe that after around 300 iterations, the GWO algorithm stagnates and becomes trapped in a local optimum. The balance analysis graph shows that the SFGWO's Exploration% increases around this point and finally escapes the local optimum. In contrast, the Exploration% of the GWO remains unchanged, leading to optimization termination. This demonstrates that the SFGWO can dynamically balance exploration and exploitation to achieve optimal optimization performance. The graphs for functions f10 and f13 show that the SFGWO has enhanced exploration capabilities while maintaining a balanced exploitation approach.

The above analysis indicates that SFGWO demonstrates the ability to dynamically balance exploration and exploitation during the optimization process, thereby improving overall optimization performance across different types of functions.4) Convergence Speed Analysis

To evaluate the algorithms’ convergence performance and solution speed, Fig. 7 displays the corresponding curves for the 23 benchmark functions, with 30 dimensions for f1 to f13.Fig. 7 Convergence curves of the compared algorithms.

Fig. 7

The convergence performance of the algorithms on different benchmark functions can be analyzed from these results.

For the unimodal functions (f1−f7), it is evident that the SFGWO and AOA both achieve the optimal values for f1, but the SFGWO demonstrated a faster convergence speed. For functions f3, f4, f5, and f6, the SFGWO not only converged faster but also achieved higher accuracy. However, for f2 and f7, the convergence accuracy of SFGWO was not as good as that of AOA. Overall, for single-peaked functions (f1−f7), except for f2 and f7, SFGWO outperformed the other eight algorithms in terms of both convergence speed and accuracy.

For the multimodal functions and fixed-dimension multimodal functions (f8−f23), the SFGWO exhibited significantly faster convergence speeds and higher accuracy on functions f8, f9, f11, f12, and f13 compared to the other algorithms. For the curves of f12 and f13 specifically, although the algorithm experienced multiple stagnations during the solution process, it eventually managed to escape, further demonstrating its excellent ability to escape local optima. In functions f14, f16−f19, and f21−f23, the SFGWO quickly approached the optimal solution and captured high-precision final solutions effectively. Analyzing the convergence curves for multi-modal functions shows that SFGWO has outstanding search performance.

In summary, SFGWO outperformed other algorithms in terms of both convergence speed and final convergence accuracy across various benchmark functions.5) Box plot analysis

To evaluate the optimization capability and stability of the improved algorithm, boxplots for the 23 classical benchmark functions are presented in this section, as shown in Fig. 8. The boxplots illustrate the distribution characteristics of the data by displaying the data distribution as quartiles. The bottom and top of the boxplot represent the minimum and maximum values of the data, respectively, while the bottom and top of the box represent the lower (Q1) and upper (Q3) quartiles, respectively. The height of each box reflects the dispersion of the dataset; a higher box indicates that the results of the dataset were more dispersed, meaning that the results of the algorithm were less stable, while a lower box means that most data points were concentrated in a similar numerical range. The results obtained by the algorithm were relatively stable.Fig. 8 Boxplot on benchmark functions F1 to F23.

Fig. 8

For the unimodal functions (f1−f7), we observe that on functions f1, f5, and f6, the boxplot of the SFGWO algorithm is lower than that of the other eight algorithms, demonstrating its superior performance in terms of optimization capability. However, for functions f5 and f6, the SFGWO algorithm exhibited some outliers, indicating that while it performed well in terms of optimization, there was still room for improvement in terms of data stability. On function f4, the SFGWO algorithm exhibited a small and low box, which means that the performance of SFGWO was more stable. In contrast, although the AOA algorithm achieved the lowest function value, it exhibited some outliers, indicating that its data stability was not as good as that of the SFGWO.

For the multimodal functions (f8−f13), it is observed that on functions f8, f9, and f11, the boxplots of the SFGWO algorithm are very narrow and lower than those of other algorithms, indicating that the SFGWO can not only maintain stability but also handle multimodal functions with great optimization difficulty effectively. For functions f12 and f13, although the boxplots of the SFGWO are lower than those of other algorithms, some outliers also appeared. This indicates that the AIRS improved the algorithm's exploration potential, but due to its random nature, it may occasionally lead to instability.

For the fixed-dimensional multimodal functions (f14−f23), it is observed that on functions f14, f17, f19, f21, f22, and f23, the data of the SFGWO algorithm was more concentrated and had fewer outliers. This indicates that SFGWO had good stability and global search capability.

Overall, the SFGWO showed excellent performance and stability on most functions, but there were some outliers. Therefore, the SFGWO needs further improvement, especially in its ability to handle outliers and ensure data stability, so that it can deal with various problems and datasets more stably and reliably. Such improvements will help improve the applicability and practicality of the SFGWO algorithm.6) Time Consumption Test of SFGWO for Classical Benchmarks

In this section, a detailed analysis of the computational time complexity of the SFGWO was conducted. The runtime for the GWO and the SFGWO in solving the 23 benchmark functions (f1−f23) in 30-dimensional space was recorded. Each function was run 30 times, and the average runtime for each function was calculated. As observed in Table 14, the SFGWO algorithm's average running time slightly exceeded that of GWO on functions (f1−f13). However, on functions (f15−f23), the SFGWO's average running time is significantly longer than GWO's. This difference stems from the presence of multiple local optima in functions (f15−f23). The adaptive individual regeneration strategy requires more individuals to undergo regeneration operations to escape local optima, resulting in increased computational times. Although the improvements introduced in the SFGWO incurred additional computations, this increased computational burden is aimed at enhancing the algorithm's performance, particularly in achieving better precision and global search capabilities. Therefore, despite the increased runtime of SFGWO, significant improvements have been made in solution accuracy, making the additional execution time acceptable.Table 14 Average runtime (Seconds).

Table 14F	GWO Average runtime	SFGWO Average runtime	
f1	23.01	23.08	
f2	23.32	23.57	
f3	29.10	29.25	
f4	23.04	23.09	
f5	23.51	26.39	
f6	23.02	23.30	
f7	23.21	27.04	
f8	23.18	27.22	
f9	22.93	23.44	
f10	23.83	26.89	
f11	23.74	24.53	
f12	25.10	25.59	
f13	25.63	27.22	
f14	19.39	25.18	
f15	4.15	9.11	
f16	1.78	6.76	
f17	1.82	6.76	
f18	2.00	7.03	
f19	4.46	9.62	
f20	6.84	11.77	
f21	10.85	16.88	
f22	13.54	19.43	
f23	17.45	23.58	
average of all functions	17.17	20.29	

4.5 Results on CEC2019 test functions

This section presents further tests on the algorithm's performance using the CEC2019 benchmark functions. Table 15 presents the test results of SFGWO and the other eight algorithms on the CEC2019 benchmark test functions C1 to C10, along with the mean rank values and final rankings from the Friedman test at the bottom. The results of the Friedman test indicate that the proposed SFGWO outperformed the other algorithms in handling these challenging test functions.Table 15 Results of the compared methods on the CEC2019 test functions.

Table 15F		GWO	SOGWO	AGWO	AGWOCS	MFO	SCA	AOA	TSA	SFGWO	
C1	Mean	6.10E+03	5.86E+03	4.21E+04	2.06E+03	1.20E+07	1.64E+06	1.00E+00	6.45E+02	1.00E+00	
Min	1.00E+00	1.00E+00	1.00E+00	1.00E+00	1.69E+05	1.00E+00	1.00E+00	6.45E+02	1.00E+00	
Std.	2.13E+04	2.30E+04	1.77E+05	5.19E+03	1.55E+07	2.56E+06	0.00E+00	2.31E−13	2.04E−07	
C2	Mean	1.35E+02	2.87E+02	1.36E+02	1.52E+02	7.38E+02	5.22E+02	4.04E+02	7.28E+02	4.94E+00	
Min	3.74E+01	1.69E+01	8.72E+00	2.94E+01	1.19E+02	1.46E+02	1.18E+01	9.34E+00	4.45E+00	
Std.	8.05E+01	1.76E+02	1.28E+02	9.72E+01	5.24E+02	1.80E+02	2.25E+02	4.09E+02	8.57E−01	
C3	Mean	3.61E+00	2.21E+00	3.13E+00	4.14E+00	7.13E+00	8.06E+00	7.05E+00	5.90E+00	5.10E+00	
Min	1.02E+00	1.41E+00	1.00E+00	1.92E+00	1.41E+00	4.86E+00	4.40E+00	5.90E+00	1.41E+00	
Std.	2.55E+00	1.06E+00	2.33E+00	1.83E+00	2.45E+00	1.49E+00	1.41E+00	9.03E−16	1.36E+00	
C4	Mean	4.81E+01	1.56E+01	1.41E+02	2.24E+02	9.05E+01	9.14E+02	3.15E+03	5.25E+03	4.30E+01	
Min	1.20E+01	5.98E+00	7.42E+00	1.73E+02	8.96E+00	3.79E+02	1.93E+03	5.25E+03	1.49E+01	
Std.	1.83E+01	8.57E+00	1.16E+02	3.89E+01	1.99E+02	3.48E+02	8.28E+02	1.85E−12	1.77E+01	
C5	Mean	1.42E+00	1.66E+00	1.60E+00	1.82E+00	1.20E+00	2.11E+00	2.36E+00	3.04E+00	1.23E+00	
Min	1.12E+00	1.12E+00	1.05E+00	1.56E+00	1.03E+00	1.84E+00	1.36E+00	3.04E+00	1.01E+00	
Std.	2.01E−01	6.80E−01	2.53E−01	9.26E−02	1.72E−01	8.83E−02	6.08E−01	1.36E−15	2.98E−01	
C6	Mean	1.05E+01	2.06E+00	1.01E+01	1.03E+01	5.61E+00	1.03E+01	6.97E+00	1.07E+01	1.04E+01	
Min	9.06E+00	1.15E+00	7.85E+00	8.59E+00	1.25E+00	9.05E+00	5.36E+00	1.07E+01	8.57E+00	
Std.	5.80E−01	7.69E−01	8.16E−01	7.53E−01	1.89E+00	5.94E−01	6.60E−01	5.42E−15	6.64E−01	
C7	Mean	1.66E+03	6.82E+02	1.10E+03	1.16E+03	9.59E+02	1.44E+03	1.41E+02	1.21E+03	1.08E+00	
Min	1.05E+03	1.41E+02	5.41E+02	6.14E+02	3.49E+02	6.48E+02	7.88E+01	3.30E+02	1.55E+02	
Std.	2.48E+02	3.44E+02	2.95E+02	2.87E+02	3.08E+02	2.64E+02	1.75E+02	3.57E+02	1.25E+02	
C8	Mean	1.00E+00	3.55E+00	1.00E+00	1.01E+00	1.01E+00	1.04E+00	1.04E+00	1.13E+00	1.00E+00	
Min	1.00E+00	2.26E+00	1.00E+00	1.00E+00	1.00E+00	1.02E+00	1.00E+00	1.13E+00	1.00E+00	
Std.	2.15E−03	5.12E−01	6.02E−03	4.36E−03	3.17E−02	1.27E−02	3.33E−02	4.52E−16	7.00E−07	
C9	Mean	4.56E+00	1.16E+00	4.79E+00	6.10E+00	5.25E+00	1.68E+01	1.83E+00	1.96E+01	1.49E+00	
Min	1.42E+00	1.09E+00	1.13E+00	4.16E+00	1.11E+00	8.88E+00	1.40E+00	1.96E+01	1.21E+00	
Std.	1.94E+00	4.48E−02	7.30E+00	1.16E+00	2.16E+01	6.64E+00	5.71E−01	3.61E−15	1.43E−01	
C10	Mean	2.14E+01	2.14E+01	2.00E+01	2.03E+01	2.11E+01	2.14E+01	2.10E+01	2.16E+01	2.09E+01	
Min	2.13E+01	2.12E+01	1.00E+00	9.01E+00	2.10E+01	2.12E+01	2.08E+01	2.16E+01	4.03E+00	
Std.	6.79E−02	8.25E−02	5.12E+00	3.31E+00	1.24E−01	8.89E−02	6.15E−02	0.00E+00	3.18E+00	
Mean Rank		4.7	3.8	3.8	4.9	5.25	7.3	4.9	7.7	2.65	
Final Rank		3	2	2	4	5	6	4	7	1	

5 Applications of SFGWO in practical engineering problems

To further corroborate the efficacy of the SFGWO algorithm and assess its optimization performance in practical applications, it was applied to three practical engineering design problems: a speed reducer design, a cantilever beam design, and a welded beam design. The algorithm's performance on these engineering problems was evaluated to ascertain its effectiveness and applicability in practical scenarios. The experimental parameters outlined in Section IV were also adopted for these applications.

5.1 Speed reducer design

The speed reducer design problem aims to minimize the weight of a speed reducer under certain constraints. The design structure is illustrated in Fig. 9 [51]. This problem involves seven design variables and 11 constraints. The design variables are denoted as follows: face width – x1, module of teeth – x2, number of teeth on pinion –x3, length of first shaft between bearings – x4, length of second shaft between bearings – x5, diameter of first shaft – x6, and diameter of second shaft – x7. The objective function aims to minimize the total weight of the speed reducer while meeting the specified constraints.Fig. 9 Schematic of speed reducer design.

Fig. 9

The mathematical model is expressed as follows:ConsiderX=[x1,x2,x3,x4,x5,x6,x7].

Minimizef(x)=0.7854x1x22(3.3333x32+14.9334x3−43.0934)−1.508x1(x62+x72)+7.4770(x63+x72)+0.7854(x4x62+x5x72).

Subjectto

g1(x)=27x1x12x3−1≤0,g2(x)=397.5x1x22x32−1≤0,

g3(x)=1.93x43x2x64x3−1≤0,g4(x)=1.93x53x2x74x3−1≤0,g5(x)=(745x4/(x2x3))2+16.9×106110x63−1≤0,

g6(x)=(745x5/(x2x3))2+157.5×10685.0x73−1≤0,

g7(x)=x2x340−1≤0,g8(x)=5x2x1−1≤0,g9(x)=x112x2−1≤0,

(16) g10(x)=1.5x6+1.9x4−1≤0,g11(x)=1.1x7+1.9x5−1≤0.

where:2.6≤x1≤3.6,0.7≤x2≤0.8,17≤x3≤28

7.3≤x4≤8.3,7.8≤x5≤8.3,2.9≤x6≤3.9,5≤x7≤5.5

The optimal results obtained using the SFGWO algorithm, along with those of AOA [52], SCA [30], MFO [53], TSA [54], MGWO [55], RW-GWO [56], GWO-XOBL [57], WF-GWO [58], MDM-GWO [51], and the classical GWO algorithm, are presented in Table 16, from which it is evident that the SFGWO achieved an optimal solution of 2894.4920, significantly outperforming the second-ranked AOA (2997.9157), demonstrating a substantial competitive advantage. The SFGWO's solution confirms its effectiveness in reducing engineering costs associated with the speed reducer design.Table 16 Results of the compared algorithms on the speed reducer design problem.

Table 16Algorithm	Optimal values for variables	Optimal weight	
x1	x2	x3	x4	x5	x6	x7	
AOA [52]	3.50384	0.7	17	7.3	7.72933	3.35649	5.2867	2997.9157	
SCA [30]	3.508755	0.7	17	7.3	7.8	3.46102	5.289213	3030.563	
MFO [53]	3.49745	0.7	17	7.82775	7.71245	3.35178	5.28635	2998.9408	
TSA [54]	3.519098	0.7	17	7.3	7.8	3.3680264	5.3151837	3013.5435	
MGWO [55]	3.499820	0.7	17.01306	7.4166313	7.838926	3.3584894	5.2873136	3002.9519	
RW-GWO [56]	3.500645	0.7	17	7.309897	7.808962	3.350328	5.286125	2997.0683	
GWO-XOBL [57]	3.502246	0.7	17	7.3	7.8	3.353244	5.288644	2999.2508	
WF-GWO [58]	3.502501	0.7	17	7.437258	7.806525	3.350118	5.286136	2999.3043	
MDM-GWO [51]	3.501335	0.7	17	7.3	7.8	3.357801	5.285996	2999.1348	
GWO	3.526045	0.7	17	7.392843	7.816034	3.3748634	5.3132109	3017.9426	
SFGWO	3.50003026	0.7	17. 0000496	7.30077377	7.72119559	2.9	5.286584	2894.4920	

5.2 Cantilever beam design problem

This design problem aims to minimize the weight of a cantilever beam with five hollow blocks. This problem comprises five design variables, aiming to find the set of values that minimizes the overall mass of the beam, as depicted in Fig. 10 [59].Fig. 10 Schematic view of cantilever beam design problem.

Fig. 10

The mathematical model is expressed as follows:ConsiderX=[x1,x2,x3,x4,x5].

Minimizef(x)=0.0624(x1+x2+x3+x4+x5).

(17) Subjectto:g1(x)=64x13+27x23+19x33+7x43+1x53−1≤0.

where 0.01≤x1,x2,x3,x4,x5≤100.

Table 17 lists the optimal solutions obtained by the SFGWO algorithm, along with those of AOA [52], MFO [53], AO [15], WOA [60], MVO [59], SSA [61], AGWO [62], BGWO [63], and the classical GWO algorithm. The data in the table shows that the SFGWO can achieve a better solution for the optimal mass compared to other algorithms. Furthermore, the AO algorithm exhibited strong competitiveness with a solution of 1.3390, which was very close to that of the SFGWO. It is important to note that the solution of AOA was above 1.5, i.e. significantly higher than the solutions of the other nine algorithms. This indicates that the AOA is not suitable for solving the cantilever beam design problem.Table 17 Results of the compared algorithms for solving the cantilever beam design problem.

Table 17Algorithm	Optimal values for variables	Optimal weight	
x1	x2	x3	x4	x5	
AOA [52]	6.581606	6.257284	6.074678	3.756637	2.699341	1.583059668	
MFO [53]	5.9849	5.3167	4.4973062	3.5136954	2.1616550	1.33999	
AO [15]	5.8881	5.5451	4.3798	3.5973	2.1026	1.3390	
WOA [60]	6.438105	4.416771	4.545978	4.05337	3.067875	1.405379027	
MVO [59]	5.909317	5.50287	4.427878	3.512903	2.142224	1.341300009	
SSA [61]	5.996229	5.338329	4.50466	3.480124	2.155083	1.340004072	
AGWO [62]	6.015647	5.310255	4.500563	3.494626	2.15301	1.339983936	
BGWO [63]	6.01303	5.3112	4.4953	3.5079	2.1461	1.33996	
GWO	5.99473	5.342051	4.5137	3.477375	2.146893	1.34002433	
SFGWO	5.94311616	4.8502788	4.48257505	3.44010772	2.19484345	1.301495734	

5.3 Welded beam design problem

This design problem aims to minimize the manufacturing cost of a welded beam. This problem involves four design variables and seven constraints. The four design variables are denoted as follows: weld thickness h – x1, length l – x2, height t – x3) and thickness b – x4. Fig. 11 [63] illustrates the basic construction diagram of the welded beam model.Fig. 11 Welded Beam design problem.

Fig. 11

The mathematical model is expressed as follows:ConsiderX=[x1,x2,x3,x4]=[h,l,t,b].

Minimizef(x)=1.10471x12x2+0.04811x3x4(14.0+x2).

Subjectto:

g1(x)=τ(x)−τmax≤0,g2(x)=σ(x)−σmax≤0,g3(x)=δ(x)−δmax≤0,g4(x)=x1−x4≤0,g5(x)=P−Pc(x)≤0,g6(x)=0.125−x1≤0,

g7(x)=1.10471x12+0.04811x3x4(14.0+x2)−5.0≤0.

where:τ(x)=(τ′)+2τ′τ″x22R+(τ″)2,τ′=P2x1x2,τ″=MRJ,M=P(L+x22),

R=x224+(x1+x32)2,J=2{2x1x2[x224+(x1+x32)2]},

σ(x)=6PLx4x32,δ(x)=6PL3Ex32x4,Pc(x)=4.013Ex32x4636L2(1−x32LE4G),

P=6000lb,L=14in.,δmax=0.25in.,E=30×106psi,

G=12×106psi,τmax=13600psi,σmax=30000pis.

(18) 0.1≤x1,x4≤2and0.1≤x2,x3≤10.

Table 18 presents the optimal solutions obtained by the SFGWO algorithm, along with those of AOA [52], MFO [53], SCA [30], WOA [60], MVO [59], SSA [61], AGWO [62], BGWO [63], and the classical GWO algorithm. From this table, it is evident that the results of the different algorithms are very close; however, the SFGWO achieved the optimal cost value of 1.6903262, which was the best among all the compared algorithms. The BGWO obtained an optimal cost of 1.6972, which was very close to that of SFGWO.Table 18 Results of the algorithm comparison for solving the welded beam design problem.

Table 18Algorithm	Optimal values for variables	Optimal cost	
x1	x2	x3	x4	
AOA [52]	0.194475	2.57092	10.000	0.201827	1.7164	
MFO [53]	0.203567	3.443025	9.230278	0.212359	1.732541	
SCA [30]	0.204695	3.536291	9.004290	0.210025	1.759173	
WOA [60]	0.205396	3.484293	9.037426	0.206276	1.730499	
MVO [59]	0.205463	3.473193	9.044502	0.205695	1.72645	
SSA [61]	0.2057	3.4714	9.0366	0.2057	1.72491	
AGWO [62]	0.20555	3.4744	9.0378	0.20572	1.7253	
BGWO [63]	0.2055	3.2607	9.0361	0.2059	1.6972	
GWO	0.205678	3.471403	9.036964	0.205729	1.724995	
SFGWO	0.20353578	3.28124602	9.07873122	0.20404719	1.6903262	

Based on the experimental results of the practical optimization problem mentioned above, the SFGWO met the design requirements effectively, finding the lowest-cost solutions, demonstrating its advantages in solving real-world problems and particularly in addressing complex constrained optimization problems.

6 Conclusion

To address the issues of premature convergence and lower accuracy encountered by the classical GWO algorithm when dealing with complex high-dimensional problems, in this paper, an improved GWO algorithm is proposed based on an SFN topology (SFGWO), which combines three strategies aimed at improving its performance in solving optimization problems. First, a SFN topology is used to build the topological structure of the GW population in each iteration and improve the balance between exploration and exploitation. Second, a neighborhood learning strategy, which also aims to offer an improved exploration and exploitation balance compared to the GWO. Third is an AIRS, which helps prevent stagnation and allows escaping from local optima.

To validate the effectiveness of SFGWO, tests were conducted using 23 classical benchmark functions, ten functions from the CEC2019 benchmark, and three engineering problems. The test results demonstrate that the SFGWO algorithm excels in terms of solution accuracy, convergence speed, and the ability to address complex optimization problems, including practical engineering application problems. However, the SFGWO also has certain shortcomings. Although its optimization capability has improved, the optimization time has increased, indicating a trade-off where the additional computational load is incurred to enhance its optimization ability, and the AIRS may have negative effects when the algorithm is applied to certain functions.

In this work, we focus on exploring more effective strategy improvement methods to shorten optimization time and enhance the algorithm efficiency. Simultaneously, we will delve into other complex topological structures to further strengthen SFGWO's robustness. By applying the SFGWO algorithm to a broader range of real-world optimization problems, such as (1) water resource allocation and reservoir scheduling optimization, (2) operational scheduling optimization in power systems, and (3) feed formulation optimization, we will further validate its effectiveness and robustness and promote the wider application of this algorithm in various fields.

Data availability

Data will be made available on request.

CRediT authorship contribution statement

Jun Zhang: Writing – review & editing, Writing – original draft, Validation, Software. Yongqiang Dai: Resources, Funding acquisition, Conceptualization. Qiuhong Shi: Writing – review & editing, Validation, Software, Investigation.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Yongqiang Dai reports financial support was provided by 10.13039/501100007932 Gansu Agricultural University . If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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References

1 Stork J. Eiben A.E. Bartz-Beielstein T. A new taxonomy of global optimization algorithms Nat. Comput. 21 2022 219 242 10.1007/s11047-020-09820-4
2 Abbaszadeh Shahri A. Pashamohammadi F. Asheghi R. Abbaszadeh Shahri H. Automated intelligent hybrid computing schemes to predict blasting induced ground vibration Eng. Comput. 38 2021 3335 3349 10.1007/s00366-021-01444-1
3 Abbaszadeh Shahri A. Khorsand Zak M. Abbaszadeh Shahri H. A modified firefly algorithm applying on multi-objective radial-based function for blasting Neural Comput. Appl. 34 2021 2455 2471 10.1007/s00521-021-06544-z
4 Mirfallah Lialestani S.P. Parcerisa D. Himi M. Abbaszadeh Shahri A. A novel modified bat algorithm to improve the spatial geothermal mapping using discrete geodata in Catalonia-Spain Model. Earth Syst. Environ. 10 2024 4415 4428 10.1007/s40808-024-01992-7
5 R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, IEEE, n.d. 10.1109/mhs.1995.494215.
6 Akay B. Karaboga D. Artificial Bee colony algorithm Swarm Intelligence Algorithms 2020 CRC Press 17 30 10.1201/9780429422614-2
7 Dorigo M. Gambardella L.M. Ant colony system: a cooperative learning approach to the traveling salesman problem IEEE Trans. Evol. Comput. 1 1997 53 66 10.1109/4235.585892
8 Yang X.-S. Deb Suash Cuckoo search via Lévy flights 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC) 2009 IEEE Coimbatore, India 210 214 10.1109/NABIC.2009.5393690
9 Adegboye O.R. Feda A.K. Ojekemi O.R. Agyekum E.B. Khan B. Kamel S. DGS-SCSO: enhancing Sand Cat swarm optimization with dynamic Pinhole imaging and Golden sine algorithm for improved numerical optimization performance Sci. Rep. 14 2024 1491 10.1038/s41598-023-50910-x 38233528
10 Seyyedabbasi A. Kiani F. Sand Cat swarm optimization: a nature-inspired algorithm to solve global optimization problems Eng. Comput. 39 2022 2627 2651 10.1007/s00366-022-01604-x
11 Adegboye O.R. Deniz Ülker E. Hybrid artificial electric field employing cuckoo search algorithm with refraction learning for engineering optimization problems Sci. Rep. 13 2023 10.1038/s41598-023-31081-1
12 Yadav A. others, AEFA: artificial electric field algorithm for global optimization Swarm Evol. Comput. 48 2019 93 108
13 Feda A.K. Adegboye M. Adegboye O.R. Agyekum E.B. Fendzi Mbasso W. Kamel S. S-shaped grey wolf optimizer-based FOX algorithm for feature selection Heliyon 10 2024 e24192 10.1016/j.heliyon.2024.e24192
14 Mohammed H. Rashid T. FOX: a FOX-inspired optimization algorithm Appl. Intell. 53 2023 1030 1050
15 Abualigah L. Yousri D. Abd Elaziz M. Ewees A.A. Al-qaness M.A.A. Gandomi A.H. Aquila Optimizer: a novel meta-heuristic optimization algorithm Comput. Ind. Eng. 157 2021 107250 10.1016/j.cie.2021.107250
16 Mirjalili S. Mirjalili S.M. Lewis A. Grey wolf optimizer Adv. Eng. Software 69 2014 46 61 10.1016/j.advengsoft.2013.12.007
17 Dong L. Yuan X. Yan B. Song Y. Xu Q. Yang X. An improved grey wolf optimization with multi-strategy Ensemble for Robot path planning Sensors 22 2022 6843 10.3390/s22186843 36146192
18 Kumar V. Kumar D. An astrophysics-inspired Grey wolf algorithm for numerical optimization and its application to engineering design problems Adv. Eng. Software 112 2017 231 254 10.1016/j.advengsoft.2017.05.008
19 K S. S.K L. Khanna A. Tanwar S. Rodrigues J.J.P.C. Roy N.R. Alzheimer detection using Group Grey Wolf Optimization based features with convolutional classifier Comput. Electr. Eng. 77 2019 230 243 10.1016/j.compeleceng.2019.06.001
20 Shakarami M.R. Faraji Davoudkhani I. Wide-area power system stabilizer design based on Grey Wolf Optimization algorithm considering the time delay Elec. Power Syst. Res. 133 2016 149 159 10.1016/j.epsr.2015.12.019
21 Lu C. Gao L. Yi J. Grey wolf optimizer with cellular topological structure Expert Syst. Appl. 107 2018 89 114 10.1016/j.eswa.2018.04.012
22 Nadimi-Shahraki M.H. Taghian S. Mirjalili S. An improved grey wolf optimizer for solving engineering problems Expert Syst. Appl. 166 2021 113917 10.1016/j.eswa.2020.113917
23 Mittal N. Singh U. Sohi B.S. Modified grey wolf optimizer for global engineering optimization Applied Computational Intelligence and Soft Computing 2016 2016 1 16 10.1155/2016/7950348
24 Long W. Grey wolf optimizer based on nonlinear adjustment control parameter Proceedings of the 2016 4th International Conference on Sensors, Mechatronics and Automation (ICSMA 2016) 2016 Atlantis Press 10.2991/icsma-16.2016.111
25 Hou Y. Gao H. Wang Z. Du C. Improved grey wolf optimization algorithm and application Sensors 22 2022 3810 10.3390/s22103810 35632219
26 Chen Chuang Ryad Chellali X.Y. Improved grey wolf optimizer algorithm using dynamic weighting and probabilistic disturbance strategy J. Comput. Appl. 37 2017 3493 3497 http://www.joca.cn/EN/10.11772/j.issn.1001-9081.2017.12.3493
27 Heidari A.A. Pahlavani P. An efficient modified grey wolf optimizer with Lévy flight for optimization tasks Appl. Soft Comput. 60 2017 115 134 10.1016/j.asoc.2017.06.044
28 Saremi S. Mirjalili S.Z. Mirjalili S.M. Evolutionary population dynamics and grey wolf optimizer Neural Comput. Appl. 26 2015 1257 1263 10.1007/s00521-014-1806-7
29 Singh N. Singh S.B. A novel hybrid GWO-SCA approach for optimization problems Engineering Science and Technology, an International Journal 20 2017 1586 1601 10.1016/j.jestch.2017.11.001
30 Mirjalili S. SCA: a Sine Cosine Algorithm for solving optimization problems Knowl. Base Syst. 96 2016 120 133 10.1016/j.knosys.2015.12.022
31 Gaidhane P.J. Nigam M.J. A hybrid grey wolf optimizer and artificial bee colony algorithm for enhancing the performance of complex systems Journal of Computational Science 27 2018 284 302 10.1016/j.jocs.2018.06.008
32 Zhu A. Xu C. Li Z. Wu J. Liu Z. Hybridizing grey wolf optimization with differential evolution for global optimization and test scheduling for 3D stacked SoC J. Syst. Eng. Electron. 26 2015 317 328 10.1109/JSEE.2015.00037
33 Zhang C. Yi Z. Scale-free fully informed particle swarm optimization algorithm Inf. Sci. 181 2011 4550 4568 10.1016/j.ins.2011.02.026
34 Yue C. Qu B. Liang J. A Multiobjective particle swarm optimizer using ring topology for solving multimodal Multiobjective problems IEEE Trans. Evol. Comput. 22 2018 805 817 10.1109/TEVC.2017.2754271
35 BilalHAbedalguni B. MalekBarhoush M. Distributed GREY WOLF OPTIMIZER FOR NUMERICAL OPTIMIZATION PROBLEMS JJCIT 1 2018 10.5455/jjcit.71-1532897697
36 Cont R. Moussa A. Santos E.B. Network structure and systemic risk in Banking systems Handbook on Systemic Risk 2013 Cambridge University Press 327 368 10.1017/cbo9781139151184.018
37 Liu D. Fodor V. Rasmussen L.K. Will scale-free Popularity Develop scale-free Geo-social networks? http://arxiv.org/abs/2003.01489 2020
38 Wen H. Jin R. Research on BA scale-free networks Congestion and control Proceedings of the 2nd International Conference on Computer Science and Application Engineering 2018 ACM 10.1145/3207677.3278033
39 Wu D. Jiang N. Du W. Tang K. Cao X. Particle swarm optimization with moving particles on scale-free networks IEEE Trans. Netw. Sci. Eng. 7 2020 497 506 10.1109/TNSE.2018.2854884
40 Ji J. Song S. Tang C. Gao S. Tang Z. Todo Y. An artificial bee colony algorithm search guided by scale-free networks Inf. Sci. 473 2019 142 165 10.1016/j.ins.2018.09.034
41 Li W. Sun B. Huang Y. Mahmoodi S. Adaptive particle swarm optimization using scale-free network topology Journal of Network Intelligence 6 2021 500 517
42 Banerjee N. Mukhopadhyay S. AP-TLB-IGWO: adult-pup teaching–learning based interactive grey wolf optimizer for numerical optimization Appl. Soft Comput. 124 2022 109000 10.1016/j.asoc.2022.109000
43 Barabási A.-L. Albert R. Emergence of scaling in random networks Science 286 1999 509 512 10.1126/science.286.5439.509 10521342
44 Bollobás B. Riordan O. Spencer J. Tusnády G. The degree sequence of a scale-free random graph process: degree Sequence of a Random Graph Random Struct. Algorithm 18 2001 279 290 10.1002/rsa.1009
45 Kishor Avadh Singh P.K. Empirical study of grey wolf optimizer Pant M. Deep K. Bansal J.C. Nagar A. Das K.N. Proceedings of Fifth International Conference on Soft Computing for Problem Solving 2016 Springer Singapore Singapore 1037 1049 10.1007/978-981-10-0448-3_87
46 Adegboye O.R. Feda A.K. Ojekemi O.S. Agyekum E.B. Hussien A.G. Kamel S. Chaotic opposition learning with mirror reflection and worst individual disturbance grey wolf optimizer for continuous global numerical optimization Sci. Rep. 14 2024 4660 10.1038/s41598-024-55040-6 38409189
47 Yan F. Xu X. Xu J. Grey wolf optimizer with a novel weighted distance for global optimization IEEE Access 8 2020 120173 120197 10.1109/ACCESS.2020.3005182
48 Zhang C. Wang W. Pan Y. Enhancing electronic Nose performance by feature selection using an improved grey wolf optimization based algorithm Sensors 20 2020 4065 10.3390/s20154065 32707788
49 Zhang X. Ming Z. An optimized grey wolf optimizer based on a mutation operator and eliminating-reconstructing mechanism and its application Frontiers Inf Technol Electronic Eng 18 2017 1705 1719 10.1631/FITEE.1601555
50 Hussain K. Salleh M.N.M. Cheng S. Shi Y. On the exploration and exploitation in popular swarm-based metaheuristic algorithms Neural Comput. Appl. 31 2019 7665 7683 10.1007/s00521-018-3592-0
51 Singh S. Bansal J.C. Mutation-driven grey wolf optimizer with modified search mechanism Expert Syst. Appl. 194 2022 116450 10.1016/j.eswa.2021.116450
52 Abualigah L. Diabat A. Mirjalili S. Abd Elaziz M. Gandomi A.H. The Arithmetic optimization algorithm Comput. Methods Appl. Mech. Eng. 376 2021 113609 10.1016/j.cma.2020.113609
53 Mirjalili S. Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm Knowl. Base Syst. 89 2015 228 249 10.1016/j.knosys.2015.07.006
54 Kaur S. Awasthi L.K. Sangal A.L. Dhiman G. Tunicate Swarm Algorithm: a new bio-inspired based metaheuristic paradigm for global optimization Eng. Appl. Artif. Intell. 90 2020 103541 10.1016/j.engappai.2020.103541
55 Gupta S. Deep K. A memory-based Grey Wolf Optimizer for global optimization tasks Appl. Soft Comput. 93 2020 106367 10.1016/j.asoc.2020.106367
56 Gupta S. Deep K. A novel random Walk grey wolf optimizer Swarm Evol. Comput. 44 2019 101 112 10.1016/j.swevo.2018.01.001
57 Singh S. Chand Bansal J. Grey wolf optimizer with crossover and opposition-based learning Nigdeli S.M. Kim J.H. Bekdaş G. Yadav A. Proceedings of 6th International Conference on Harmony Search, Soft Computing and Applications 2021 Springer Singapore Singapore 401 410
58 Rodríguez L. Castillo O. Soria J. Melin P. Valdez F. Gonzalez C.I. Martinez G.E. Soto J. A fuzzy hierarchical operator in the grey wolf optimizer algorithm Appl. Soft Comput. 57 2017 315 328 10.1016/j.asoc.2017.03.048
59 Mirjalili S. Mirjalili S.M. Hatamlou A. Multi-Verse Optimizer: a nature-inspired algorithm for global optimization Neural Comput. Appl. 27 2016 495 513 10.1007/s00521-015-1870-7
60 Camacho‐Villalón C.L. Dorigo M. Stützle T. Exposing the grey wolf, moth‐flame, whale, firefly, bat, and antlion algorithms: six misleading optimization techniques inspired by bestial metaphors Int. Trans. Oper. Res. 30 2023 2945 2971 10.1111/itor.13176
61 Mirjalili S. Gandomi A.H. Mirjalili S.Z. Saremi S. Faris H. Mirjalili S.M. Salp Swarm Algorithm: a bio-inspired optimizer for engineering design problems Adv. Eng. Software 114 2017 163 191 10.1016/j.advengsoft.2017.07.002
62 Ma C. Huang H. Fan Q. Wei J. Du Y. Gao W. Grey wolf optimizer based on Aquila exploration method Expert Syst. Appl. 205 2022 117629 10.1016/j.eswa.2022.117629
63 Fan Q. Huang H. Li Y. Han Z. Hu Y. Huang D. Beetle antenna strategy based grey wolf optimization Expert Syst. Appl. 165 2021 113882 10.1016/j.eswa.2020.113882
