Boston Consulting Group Matrix-Based Equilibrium Optimizer for Numerical Optimization and Dynamic Economic Dispatch

Abstract

Numerous optimization problems exist in the design and operation of power systems, critical for efficient energy use, cost minimization, and system stability. With increasing energy demand and diversifying energy structures, these problems grow increasingly complex. Metaheuristic algorithms have been highlighted for their flexibility and effectiveness in addressing such complex problems. To further explore the theoretical support of metaheuristic algorithms for optimization problems in power systems, this paper proposes a novel algorithm, the Boston Consulting Group Matrix-based Equilibrium Optimizer (BCGEO), which integrates the Equilibrium Optimizer (EO) with the classic economic decision-making model, the Boston Consulting Group Matrix. This matrix is utilized to construct a model for evaluating the potential of individuals, aiding in the rational allocation of computational resources, thereby achieving a better balance between exploration and exploitation. In comparative experiments across various dimensions on CEC2017, the BCGEO demonstrated superior search performance over its peers. Furthermore, in dynamic economic dispatch, the BCGEO has shown strong optimization capabilities and potential in power system optimization problems. Additionally, the experimental results in the spacecraft trajectory optimization problem suggest its potential for broader application across various fields.

1. Introduction

Optimization problems are pervasive across various fields, including economic dispatch in power systems [1], photovoltaic parameter estimation [2], wind farm layout in the new energy field [3], and feature selection in machine learning [4]. With escalating data volume, variables, and constraints, these problems grow increasingly complex, often eluding solutions from traditional algorithms. Such challenges necessitate innovative approaches, leading to the rise in metaheuristic algorithms (MHAs), which have attracted significant attention in recent decades [5,6,7]. Distinguished from traditional algorithms, MHAs offer the advantage of finding acceptable solutions even when the optimal solution remains unattainable within specified constraints. Their excellent search capability stems from the diverse sources of inspiration that inform their construction [8].

To this day, the sources of inspiration for MHAs continue to expand and can be broadly categorized into four main groups. The first category comprises evolutionary algorithms, originally inspired by Darwinian evolution and Mendelian genetics. A prime example is the genetic algorithm (GA), which employs chromosomes as search agents to iteratively explore and exploit the solution space in pursuit of optimality [9,10,11]. This foundational concept has also spurred the development of other algorithms, such as evolutionary strategies and differential evolution (DE) [12,13,14]. The second category consists of swarm intelligence algorithms, which emulate the social behaviors observed in animal populations. For instance, particle swarm optimization (PSO) simulates the collective foraging behavior of birds, where each bird’s flight is guided by both its personal best position and the best position of the swarm [15,16,17]. Similarly, ant colony optimization (ACO) simulates the pheromone-influenced foraging paths of ants [18,19,20,21]. The third category encompasses human-based algorithms, which are designed to simulate various aspects of human social behavior and cognitive processes. An example is the brain storm optimization (BSO) algorithm, which models the idea-generation process akin to the brainstorming approach proposed by Osborn [22,23], clustering ideas as search agents. The final category is physics-based algorithms, which draw inspiration from the principles of physics. A quintessential example is the gravitational search algorithm (GSA), applying gravitational laws and mass interactions to update search agents’ positions [24,25,26]. These diverse algorithms play a significant role across various domains. They contribute to constructing convolutional neural networks [27], enhancing medical image recognition [28,29,30,31], and even forecasting the spread of diseases like COVID-19, showcasing the versatility and adaptability of MHAs in addressing complex real-world optimizations [32].

In 2020, the Equilibrium Optimizer (EO) was introduced as a novel physics-based algorithm [33]. Drawing inspiration from a well-established mass balance model [34], this optimizer is founded on the principle of mass conservation within a control volume, as described in [35,36]. Since its inception, the EO has demonstrated remarkable performance across various domains. It has been effectively applied to diode parameter  estimation [37,38], solar photovoltaic parameter estimation [39,40,41], and multimodal image registration [42,43,44], among other applications. Nonetheless, adhering to the theory of No-Free-Lunch, there is a recognized scope for further enhancement of the EO algorithm.

The performance of MHAs, including the EO, is intrinsically linked to two fundamental search behaviors: exploration and exploitation [45]. Exploration is the process of seeking new solutions across the search space, aiming to find potentially globally optimal solutions [46,47]. It emphasizes diversity and breadth, introducing randomness and a variety of search strategies to escape local optima and expand the search horizon. Exploitation, on the other hand, involves the refinement and optimization of identified high-quality solutions, accelerating the convergence towards the global optimum [48,49]. This behavior prioritizes precision and focus, selectively enhancing existing solutions to focus on the global optimum. Neither behavior alone suffices for optimal MHA performance; instead, a delicate balance between the two is sought. Overemphasis on exploration may result in slow convergence, while an excessive focus on exploitation risks becoming trapped in local optima [50]. However, achieving this balance is challenging, as the solution space is often not fully known, making it difficult to discern whether a particular action constitutes exploration or exploitation [51]. This quest for balance remains an open and active area of research.

Within the MHA community, there is a widely acknowledged perspective that exploration and exploitation are intrinsically linked to population diversity [52]. Diversity reflects the variation among individuals, and a lack of diversity may hinder an algorithm from jumping out of local optima. Conversely, a diverse population usually indicates a stronger global search capability [53]. In algorithms such as the GA and DE, a diverse population also facilitates more efficient information exchange, as offspring similarity to their parents can limit the exploration of new solution spaces. While diversity is advantageous for MHAs, its precise relationship with the balance between exploration and exploitation remains unclear. This is because diversity can be achieved through pure exploration, and higher diversity does not necessarily imply better fitness in the population. Therefore, when designing MHAs, diversity should not be employed as the sole metric of performance [54,55]. Nevertheless, it is often regarded as a critical metric that must be balanced with other factors. Evidence from successful classical algorithms supports this view. For instance, the greedy selection mechanism commonly utilized in the GA and DE ensures that the most promising individuals are carried over to subsequent generations; offspring displace parents only if they demonstrate superior fitness. This mechanism not only accelerates convergence but also maintains population diversity, as each individual is an iteration from the initial generation. Additionally, in PSO, the personal best positions and the global best position serve as attractors, guiding the population towards potentially advantageous solutions [56]. This approach is also seen as a means of balancing convergence and diversity.

Building upon the established understanding of the interplay between exploration, exploitation, and population diversity, this paper introduces a novel approach to individual selection inspired by the Boston Consulting Group (BCG) Matrix. This approach, termed BCGEO, is designed to enhance the performance of the Equilibrium Optimizer (EO) by incorporating strategic principles from the BCG Matrix. The BCG Matrix, a classic decision model in economic strategy, was developed by Bruce D. Henderson in 1970 for the Boston Consulting Group [57]. It has since become a cornerstone for product analysis, aiding organizations in the rational distribution of their strategic resources. The matrix categorizes products into four quadrants—stars, question marks, dogs, and cash cows—based on relative market share and market growth rate. This classification enables companies to apply tailored strategies, prioritizing investment in more lucrative products [58,59]. The BCGEO employs a similar conceptual framework, using fitness and the magnitude of fitness improvement as key criteria for evaluating the potential of individuals. This two-dimensional evaluation allows for the identification of individuals with higher potential, ensuring that computational resources are directed towards those with the most promising prospects for improvement. Conversely, those considered to have lower potential are discarded, thereby optimizing the performance of the search process. By integrating the strategic insights of the BCG Matrix, the BCGEO aims to achieve a better balance between exploration and exploitation while maintaining diversity. In the field of metaheuristic algorithms, this application of a classic economic decision-making model is expected to contribute to the discussions on algorithm design and optimization strategies.

The motivation of this paper is to propose a novel mechanism for evaluating the potential of individuals to improve the performance of the EO and to provide more theoretical support for optimization problems existing in power systems. Similarly to most MHAs, the EO relies on fitness to evaluate the quality of individuals. However, the intermediate state of an individual does not accurately reflect its final state, which leads to the fact that relying on fitness alone may stifle individuals that have great potential but whose fitness is not good enough in the early stages. The BCGEO algorithm evaluates the potential of an individual based on its fitness and the magnitude of fitness improvement. Individuals that do not meet the criteria in both aspects will be discarded and computational resources will be shifted to the region near the individuals with higher potential through a local search. We expect such an evaluation mechanism to bring performance improvement to the EO, and provide more theoretical support for the research of MHAs. It is also our hope that these theories will assist in solving optimization problems in areas like the power systems.

The contributions of this paper can be summarized as follows: (1) This study incorporates the BCG Matrix decision model and innovatively proposes a two-dimensional particle potential evaluation mechanism based on the fitness and its improvement magnitude. (2) The BCGEO algorithm adopting this evaluation mechanism is benchmarked against the original EO as well as six other mainstream algorithms in numerical optimization comparison experiments, thus verifying the effectiveness of the evaluation mechanism and the competitiveness. (3) Through two real-world optimization problems, this paper verifies the potential of the BCGEO algorithm in the field of power systems, as well as the possibility of its application in other fields. (4) This paper also discusses the impact of this evaluation mechanism on population diversity, which provides a theoretical basis for further optimization and application of the algorithm.

The remaining sections of this paper are organized as follows: Section 2 describes the optimization process of the original EO algorithm in detail. Section 3 details the BCG Matrix decision model and describes how the model can be integrated into the BCGEO.  Section 4 shows experimental validation on numerical optimization and real-world optimization problems. Section 5 discusses the BCGEO and EO algorithms in terms of population diversity. Section 6 provides a comprehensive summary of the work in this study and an outlook on future research directions.

2. Equilibrium Optimizer

The EO derives its inspiration from a dynamic mass balance model that represents an equilibrium state within a control volume [60,61,62]. This state of equilibrium is defined by a condition where the rate of change in mass for non-reactive constituents over a unit time is balanced by the sum of the mass entering the system, the mass generated within it, and the mass exiting the system [63,64]. In the EO, particles that act as search agents are characterized by their concentrations, with the search objective being to reach this equilibrium state. The EO follows a process similar to PSO, with its main steps including population initialization, candidate update, and population update.

2.1. Initialization

In the original EO algorithm, the population initialization is executed through a random approach. Let x symbolize the particle utilized for search, with $x^u$ and $x^l$ denoting the upper and lower bounds of the search space, respectively. The population initialization equation of the EO can be represented as follows:

$$\begin{array}{c}\hfill x_{i,d}^0=x_d^l+r_{i,d}^0·(x_d^u-x_d^l)i=1,...,Nd=1,...,D\end{array}$$

(1)

where $x^0$ signifies the particles within the initial population, i represents the index of each particle, and N indicates the size of the population (i.e., the total number of particles). The dimension of the search space is denoted by D, and $r^0$ is a randomly assigned value ranging from 0 to 1.

2.2. Candidates Update

The candidate solution set in the EO, known as the equilibrium pool, fulfills a role akin to that of $pbest$ and $gbest$ in PSO. At the beginning of the search process, due to the lack of information about equilibrium states, “promising” particles are required to act as candidates for the equilibrium state, guiding the motion of the population. These “promising” particles constitute the equilibrium pool, which includes four best-so-far particles found by the population and an average particle. The average particle is obtained by taking the arithmetic mean of these four best particles. The calculation can be conducted   as follows:

$$\begin{array}{c}\hfill x_a^p={\displaystyle \frac{x_1^p+x_2^p+x_3^p+x_4^p}{4}}\end{array}$$

(2)

here, $x_a^p$ represents the average particle, and $x_1^p$ to $x_4^p$ represent the four best particles. The equilibrium pool can be represented as follows:

$$\begin{array}{cc}\hfill X^p& =\left\{x_1^p,x_2^p,x_3^p,x_4^p,x_a^p\right\}\hfill \end{array}$$

(3)

The particles in the equilibrium pool are relevant to the update of each individual in the population. Among them, the four best-so-far particles provide exploration capabilities, enabling the population to explore a wider solution space. The average particle, on the other hand, contributes to exploitation capabilities, assisting the population in progressing towards better solutions. This combination of exploration and exploitation abilities helps improve the search efficiency of the algorithm and the quality of the optimization results.

2.3. Population Update

The population update equation for the EO is as follows:

$$\begin{array}{c}{x}_{i,d}(k+1)=x_{i,d}^p\left(k\right)+(x_{i,d}\left(k\right)-x_{i,d}^p\left(k\right))·Q_{i,d}\left(k\right)+{\displaystyle \frac{E_{i,d}\left(k\right)}{{\theta }_{i,d}\left(k\right)·V}}·(1-Q_{i,d}\left(k\right))\\ i=1,2,3,...,Nd=1,2,3,...,Dk=1,2,3,...,K\end{array}$$

(4)

where $x^p$ signifies a particle randomly selected from the equilibrium pool. The index k is utilized to indicate the current generation, while K represents the maximum number of iterations. The control volume, denoted as V, is set to 1. $\theta$ is a random number that ranges between 0 and 1. Q is defined as follows:

$$\begin{array}{c}\hfill Q_{i,d}=\alpha \times g(r^Q-0.5)\times [e^{-\theta t}-1]\end{array}$$

(5)

$$\begin{array}{cc}\hfill t& ={(1-{\displaystyle \frac{k}{K}})}^{\left(\beta {\displaystyle \frac{k}{K}}\right)}\hfill \end{array}$$

(6)

In Equation (5), $r^Q$ represents a random value between 0 and 1. $g\left(\right)$ is a sign function that takes a positive value if the variable is greater than 0, and a negative value otherwise. The parameters $\alpha$ and $\beta$ are used to adjust the tendency of exploration and exploitation. A higher value of $\alpha$ increases the intensity of exploration, while a higher value of $\beta$ increases the intensity of exploitation. The recommended values in the original EO are $\alpha =2$ and $\beta =1$. The third term in Equation (4) is referred to as the generation term, which is determined by the generation rate E, which can enhance the exploitation phase to provide the exact solutions. It can be calculated using the following equations:

$$\begin{array}{cc}\hfill E& =E^{0}Q\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill E_{i,d}^0& =G_{i,d}·(x_{i,d}^p-{\theta }_{i,d}·x_{i,d})i=1,2,....Nd=1,2,...,D\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \overrightarrow{G}& =\left\{\begin{array}{ccc}0.5\overrightarrow{r^G}& & r^P\ge P\\ \overrightarrow{0}& & r^P<P\end{array}\right.\hfill \end{array}$$

(9)

Equation (9) defines the generation rate control parameter G, which determines the likelihood of the generation term contributing to the update process. The value of G is controlled by the generation probability P and two random numbers $r^G$ and $r^P$, both ranging from 0 to 1. When $r^P$ is greater than P, the value of G is the product of $r^G$ and 0.5, at which point E is activated, and the current particle uses the generation term to update its state. When $r^P$ is less than P, G is 0, deactivating E, and the current particle update does not include the generation term. The number 0, along with G, and $r^G$ are represented as vectors, as they are arrays of length D, where D corresponds to the dimensions of each particle. When P is set to 1, the generation term is completely excluded from the update process. In contrast, when P is set to 0, the generation term is always involved in the update process. In the original EO, the generation probability parameter P is set to 0.5. From the equations above, it can be observed that the second term in Equation (4) tends to induce larger displacements of the particles, promoting an explorative effect. On the other hand, the third term is responsible for smaller displacements, facilitating a more exploitative behavior. This characteristic of the algorithm allows for a balanced exploration and exploitation in the search process.

3. Boston Consulting Group Matrix-Based Equilibrium Optimizer

3.1. Boston Consulting Group Matrix

The BCG Matrix, also known as the growth/share matrix, is an analytical tool for product portfolio management that is utilized to facilitate strategic decision-making and adjustments within a company [65]. It assesses the value and potential of a product by measuring its relative market share (RMS) and market growth rate (MGR) [66,67]. The relative market share is the proportional relationship between the sales of a product and that of its main competitors in a specific market. A larger RMS usually implies greater competitiveness, which can be viewed as the degree of success of a company’s product, as well as an internal measure of the company’s strength in the market. The market growth rate refers to the change in the market size of a product, which can be used to evaluate whether the size of the market opportunity is growing or declining. Thus, the MGR is also an external measure of market attractiveness. Typically, from an investment perspective, companies with declining relative market share are disfavored, while companies with growing market share indicate a competitive advantage. The BCG Matrix categorizes products into four quadrants using RMS and MGR as the horizontal and vertical axes, respectively: stars, question marks, dogs, and cash cows, as illustrated in Figure 1.

The descriptions and strategies for the four product categories are as follows:

• Cash Cows: Products with high relative market share but low market growth rates. Renowned for their high profitability and cash-generating capabilities, Cash Cows typically generate excess cash that surpasses the amount needed to maintain business operations. Companies often strive to bolster their portfolios with a strong lineup of “Cash Cows”. The strategic approach to these products is to minimize further investment and to squeeze as much cash as possible, as significant investment in a low-growth sector may not generate commensurate returns.
• Stars: Products with both high relative market share and market growth rate. They may necessitate significant cash infusion to secure and enhance their market standing. However, for companies, this investment is worthwhile because the focus of stars is to protect their market share and gain a larger share of the market growth than competitors. Under the precondition of maintaining market leadership, “Stars” have the potential to mature into “Cash Cows” when market growth rates decline. Conversely, if they fail to preserve their competitive position, they risk degenerating into “Dogs”.
• Question Marks: Products with high market growth rates but low relative market shares. The high market growth rates suggest a need for substantial investment. However, due to their limited market share, these products do not generate substantial cash flows and may even incur a net cash outflow. If “Question Marks” can successfully increase their market share, they may evolve into “Stars”, and subsequently, “Cash Cows”; failure to do so may see them transform into “Dogs”.
• Dogs: Products with low relative market share and market growth. They are generally less profitable and consume more cash than they yield. Dogs can detract from a company’s return on investment (ROI), a critical metric used by investors to gauge the effectiveness of a company’s management. To bolster overall performance, companies need to minimize their investments in “Dogs” or even divest from these less profitable businesses.

3.2. BCGEO

In 1999, Cooper et al. concluded from their research on portfolio management and performance practices that, in order to achieve higher performance, companies of reference place more emphasis on strategic methods rather than financial approaches [68,69]. Portfolio management refers to the strategic process of optimizing overall performance through methods such as resource allocation and risk assessment. It can be considered an optimization problem with the objective of maximizing overall performance. The BCG Matrix was introduced as a solution to this problem as early as the 1970s. The increasing usage of the BCG Matrix over the decades has further demonstrated its significance as a valuable tool for addressing portfolio management issues [70].

The integration of the BCG Matrix’s principles into the EO is particularly inspired by its two-dimensional evaluation of product potential. In the BCGEO, the potential of particles within the search space is evaluated based on their fitness and the magnitude of fitness improvement. This approach is analogous to the BCG Matrix’s classification of products into “Stars”, “Question Marks”, “Cash Cows”, and “Dogs”, each with distinct strategic implications. The magnitude of improvement in fitness refers to the difference in fitness between a particle in the current generation and in the previous generation. In the context of maximization optimization, it can be represented as follows:

$$\begin{array}{cc}\hfill \Delta f\left(x_i\right)& =f\left(x_i^{cur}\right)-f\left(x_i^{pre}\right)\hfill \end{array}$$

(10)

where $f\left(x_i^{cur}\right)$ is the fitness of the particle in the current generation, and $f\left(x_i^{pre}\right)$ is the fitness of the particle in the previous generation, thus $\Delta f$ denotes the change in fitness, which is also called the magnitude of the improvement.

In the BCGEO, particles with high fitness and significant improvement are identified as having high potential, akin to “Stars” with high market growth and relative market share. Conversely, those with low fitness and minimal improvement are considered less viable, similarly to “Dogs” with low market growth and relative market share, indicating a need to reduce investment or even divest. Computational resources initially assigned to particles with low fitness and limited improvement potential will be reallocated to those exhibiting high potential. This process reflects strategic decisions in portfolio management, aiming for a strategic shift of the population towards more promising search areas. The consistency with BCG Matrix decision-making ensures that the BCGEO serves not only as an optimization algorithm but also as a strategic tool for resource allocation and risk management in complex search spaces. This process can be represented by the following equations:

$$\begin{array}{cc}\hfill X^A& =[x_1^A,x_2^A,x_3^A,...,x_m^A,...,x_M^A],1\le m\le M,1\le M\le N\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill X^B& =[x_1^B,x_2^B,x_3^B,...,x_m^B,...,x_M^B],1\le m\le M,1\le M\le N\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill X^C& =X^A\cap X^B\hfill \end{array}$$

(13)

In Equations (11) and (12), $X^A$ represents the set of M worst particles sorted by fitness. $X^B$ represents the set of M worst particles sorted by the magnitude of fitness improvement. $X^C$ represents the intersection of these two sets, where particles within $X^C$ will be discarded. As the BCG Matrix divides products into four categories, for simplicity, M is set as one-fourth of the population size. The equation for discarding particles is as follows:

$$\begin{array}{cc}\hfill x_m^C& =x^p+S\times (x^u-x^l)\times (r^C-0.5),1\le m\le M\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill S(k+1)=S\left(k\right)\ast \gamma ,1\le k\le K\end{array}$$

(15)

In Equation (14) and (15), $x_m^C$ is a particle in $X^C$, and $x^u$ and $x^l$ represent the upper and lower bounds of the search space, respectively. $r^C$ is a random number between 0 and 1. The search radius is denoted by S, and following previous research [12,71], it is initially set to 0.01. As the iterations progress, its value decreases, and the extent of its change depends on the shrinking parameter $\gamma$, which is set to 0.985. These equations illustrate that a discarded particle, once updated, loses its original information. The updated particle’s characteristics are entirely determined by the search behavior of particles in the equilibrium pool. This process differs from conventional local searches in that it does not perform function evaluations on updated particles, nor does it use comparisons to decide whether new particles are retained. The aim of this approach is to facilitate particles’ escape from the current region, thus preventing the wastage of computational resources that would arise from further exploitation of that area.

Furthermore, the equilibrium pool in the BCGEO introduces a distinctive approach compared to the original EO algorithm. Here, the particle exhibiting the most significant fitness improvement is selected for inclusion in the equilibrium pool. This strategy is driven by two rationales. First, as previously established, a significant increase in fitness is a strong indicator of a particle’s potential, hence an in-depth search around its location is worthwhile. Second, directly allocating computational resources to the most elite particles risks falling into a local optimum. To mitigate this, the infusion of new particles into the equilibrium pool is intended to enhance population diversity and encourage exploration. The equilibrium pool in the BCGEO is mathematically represented as follows:

$$\begin{array}{c}\hfill x_a^p={\displaystyle \frac{x_1^p+x_2^p+x_3^p+x_4^p}{4}}\end{array}$$

(16)

$$\begin{array}{cc}\hfill X^p& =\left\{x_1^p,x_2^p,x_3^p,x_4^p,x_a^p,x_b^p\right\}\hfill \end{array}$$

(17)

here, $x_b^p$ represents the particle with the highest magnitude of fitness improvement in the current iteration. Algorithm 1 shows the pseudo code of the BCGEO, where the positions of some key equations in the search process are indicated.

Figure 2 illustrates the descriptive process of the BCGEO. In the case of a minimization evaluation function, the orange dots represent particles in the k-th iteration and the red dots represent particles in the $(k+1)$-th iteration. The green hexagram signifies a randomly selected particle from the equilibrium pool. The length of the blue arrows indicates the magnitude of fitness improvement. Particle $x_2$ is discarded due to insufficient fitness and the magnitude of fitness improvement. The newly generated particle $x_2^{\prime new}$ emerges within the search range of $x^p$, escaping from the previous region. Figure 3 depicts the general flowchart of the BCGEO.

Algorithm 1: BCGEO

4. Experimental Analysis

To evaluate the performance of the BCGEO algorithm, a comprehensive set of 29 benchmark functions from the CEC2017 competition was utilized. These functions encompass a range of complexities, including unimodal functions F1 to F2, simple multimodal functions F3 to F9, hybrid functions F10 to F19, and composition functions F20 to F29 [72]. In particular, hybrid functions combine different types of optimization problems or functions, typically by merging unimodal and multimodal components, to create more challenging problem scenarios. This structure is designed to reflect the complexity of real-world problems, where different subcomponents of the variables may exhibit distinct properties. Composition functions combine multiple basic functions, each with its own shifted optimum position and coverage range, to form a more complex and multimodal landscape. The composition function is defined as a weighted sum of these basic functions, with weights determined by the distance from the new shifted optimum positions.

In addition, to evaluate the effectiveness of BCGEO in addressing real-world challenges, two real-world optimization problems from the CEC2011 competition were included in the test, which are a dynamic economic dispatch problem in power systems and a spacecraft trajectory optimization problem [73].

Each experiment was independently run 51 times to ensure the accuracy and reliability of the results. Performance comparisons against other algorithms were statistically validated using the Wilcoxon signed-rank test at a significance level of 0.05 [74]. Superior performance by the BCGEO was denoted by “+”, underperformance by “−”, and no significant difference by “≈”. The cumulative counts of these indicators were denoted as w, t, and l, representing the number of “wins”, “tosses”, and “losses”, respectively. Furthermore, the Friedman test was utilized to rank the performance of the algorithms. The experiments were performed on a system equipped with an Intel(R) Core(TM) i7-12700H CPU at 2.30 GHz, running the Windows 11 operating system, with MATLAB R2020a serving as the computational platform.

4.1. Effectiveness Testing

The improvements in the BCGEO are divided into two main aspects: discarding unpromising particles and increasing the number of particles in the equilibrium pool. To ensure that these modifications enhance the algorithmic performance without adverse effects, we conducted an experimental comparison with two alternative versions: “OnlyPoolChange”, discarding no particles, and “NoPoolChange”, making no alterations to the equilibrium pool. Moreover, to confirm the benefits of assessing particle potential based on two dimensions rather than one, we introduced additional experimental configurations: “FitnessDiscard”, focusing solely on fitness as the criterion for discarding particles, and “FitnessImproveDiscard”, considering only the magnitude of fitness improvement as the criterion for discarding particles. The experiments were conducted on problems from the CEC2017 benchmark on 30 dimensions. The population size was set to 100, and the termination criterion was defined as the total number of function evaluations reaching $D\times {10}^4$. Here, D refers to the dimension of the evaluation function. The results of the experiment are presented in

Table 1. Experimental results of effectiveness testing.

	BCGEO		Fitness Discard		Fitness Improve Discard		Only Pool Change		No Pool Change		EO
	Mean	std	Mean	std	Mean	std	Mean	std	Mean	std	Mean	std
F1	$4.028\times {10}^3$	$4.732\times {10}^3$	$6.079\times {10}^3$	$6.560\times {10}^3$	$5.453\times {10}^3$	$5.075\times {10}^3$	$4.153\times {10}^3$	$4.970\times {10}^3$	$4.019\times {10}^3$	$4.806\times {10}^3$	$3.535\times {10}^3$	$3.996\times {10}^3$
F2	$8.612\times {10}^3$	$3.653\times {10}^3$	$8.711\times {10}^3$	$3.478\times {10}^3$	$9.309\times {10}^3$	$3.923\times {10}^3$	$9.299\times {10}^3$	$4.030\times {10}^3$	$1.185\times {10}^4$	$4.223\times {10}^3$	$5.660\times {10}^1$	$8.386\times {10}^1$
F3	$8.774\times {10}^1$	$6.386\times {10}^0$	$9.334\times {10}^1$	$1.335\times {10}^1$	$9.106\times {10}^1$	$1.060\times {10}^1$	$9.137\times {10}^1$	$1.025\times {10}^1$	$9.251\times {10}^1$	$1.354\times {10}^1$	$7.715\times {10}^1$	$2.398\times {10}^1$
F4	$2.056\times {10}^1$	$5.529\times {10}^0$	$2.024\times {10}^1$	$5.710\times {10}^0$	$1.977\times {10}^1$	$5.425\times {10}^0$	$2.047\times {10}^1$	$6.327\times {10}^0$	$2.102\times {10}^1$	$5.864\times {10}^0$	$6.108\times {10}^1$	$1.766\times {10}^1$
F5	$4.383\times {10}^{-5}$	$4.633\times {10}^{-5}$	$3.439\times {10}^{-5}$	$4.623\times {10}^{-5}$	$5.383\times {10}^{-5}$	$4.559\times {10}^{-5}$	$4.700\times {10}^{-5}$	$6.393\times {10}^{-5}$	$4.526\times {10}^{-5}$	$4.688\times {10}^{-5}$	$5.462\times {10}^{-3}$	$1.602\times {10}^{-2}$
F6	$5.631\times {10}^1$	$7.148\times {10}^0$	$5.623\times {10}^1$	$7.852\times {10}^0$	$5.525\times {10}^1$	$7.619\times {10}^0$	$5.692\times {10}^1$	$9.084\times {10}^0$	$5.560\times {10}^1$	$7.998\times {10}^0$	$8.607\times {10}^1$	$1.773\times {10}^1$
F7	$2.317\times {10}^1$	$6.187\times {10}^0$	$2.205\times {10}^1$	$5.713\times {10}^0$	$2.196\times {10}^1$	$4.661\times {10}^0$	$2.250\times {10}^1$	$5.767\times {10}^0$	$2.099\times {10}^1$	$6.367\times {10}^0$	$6.216\times {10}^1$	$1.442\times {10}^1$
F8	$3.511\times {10}^{-3}$	$1.755\times {10}^{-2}$	$1.229\times {10}^{-2}$	$3.590\times {10}^{-2}$	$8.908\times {10}^{-3}$	$6.362\times {10}^{-2}$	$3.511\times {10}^{-3}$	$1.755\times {10}^{-2}$	$8.908\times {10}^{-3}$	$6.362\times {10}^{-2}$	$5.585\times {10}^0$	$5.919\times {10}^0$
F9	$2.012\times {10}^3$	$4.675\times {10}^2$	$2.074\times {10}^3$	$6.018\times {10}^2$	$2.119\times {10}^3$	$4.143\times {10}^2$	$2.083\times {10}^3$	$4.563\times {10}^2$	$1.964\times {10}^3$	$4.721\times {10}^2$	$3.432\times {10}^3$	$7.442\times {10}^2$
F10	$3.682\times {10}^1$	$2.903\times {10}^1$	$4.535\times {10}^1$	$2.834\times {10}^1$	$5.393\times {10}^1$	$2.203\times {10}^1$	$4.373\times {10}^1$	$2.854\times {10}^1$	$4.962\times {10}^1$	$2.600\times {10}^1$	$4.305\times {10}^1$	$2.958\times {10}^1$
F11	$8.469\times {10}^4$	$4.558\times {10}^4$	$7.528\times {10}^4$	$3.589\times {10}^4$	$8.230\times {10}^4$	$5.120\times {10}^4$	$9.030\times {10}^4$	$4.717\times {10}^4$	$9.215\times {10}^4$	$5.222\times {10}^4$	$8.837\times {10}^4$	$1.270\times {10}^5$
F12	$1.541\times {10}^4$	$1.773\times {10}^4$	$1.989\times {10}^4$	$2.088\times {10}^4$	$2.034\times {10}^4$	$1.976\times {10}^4$	$2.476\times {10}^4$	$1.797\times {10}^4$	$1.959\times {10}^4$	$1.830\times {10}^4$	$2.230\times {10}^4$	$2.061\times {10}^4$
F13	$2.128\times {10}^4$	$1.261\times {10}^4$	$2.295\times {10}^4$	$1.548\times {10}^4$	$2.337\times {10}^4$	$1.527\times {10}^4$	$2.352\times {10}^4$	$1.679\times {10}^4$	$3.458\times {10}^4$	$2.462\times {10}^4$	$7.908\times {10}^3$	$8.610\times {10}^3$
F14	$3.377\times {10}^3$	$5.865\times {10}^3$	$4.619\times {10}^3$	$6.687\times {10}^3$	$4.596\times {10}^3$	$6.502\times {10}^3$	$3.638\times {10}^3$	$4.513\times {10}^3$	$5.440\times {10}^3$	$8.126\times {10}^3$	$6.723\times {10}^3$	$1.015\times {10}^4$
F15	$3.449\times {10}^1$	$3.990\times {10}^1$	$4.813\times {10}^1$	$7.069\times {10}^1$	$4.058\times {10}^1$	$5.003\times {10}^1$	$2.589\times {10}^1$	$1.446\times {10}^1$	$3.614\times {10}^1$	$4.739\times {10}^1$	$6.280\times {10}^2$	$2.601\times {10}^2$
F16	$3.289\times {10}^1$	$1.435\times {10}^1$	$3.109\times {10}^1$	$1.379\times {10}^1$	$3.132\times {10}^1$	$1.090\times {10}^1$	$2.884\times {10}^1$	$5.712\times {10}^0$	$3.183\times {10}^1$	$1.873\times {10}^1$	$2.250\times {10}^2$	$1.478\times {10}^2$
F17	$3.921\times {10}^5$	$1.743\times {10}^5$	$4.092\times {10}^5$	$2.100\times {10}^5$	$4.001\times {10}^5$	$1.802\times {10}^5$	$4.422\times {10}^5$	$1.696\times {10}^5$	$5.126\times {10}^5$	$1.586\times {10}^5$	$1.252\times {10}^5$	$1.048\times {10}^5$
F18	$5.017\times {10}^3$	$6.193\times {10}^3$	$8.250\times {10}^3$	$1.213\times {10}^4$	$5.759\times {10}^3$	$9.558\times {10}^3$	$8.477\times {10}^3$	$1.559\times {10}^4$	$3.830\times {10}^3$	$4.894\times {10}^3$	$6.135\times {10}^3$	$9.934\times {10}^3$
F19	$7.777\times {10}^1$	$5.412\times {10}^1$	$7.832\times {10}^1$	$8.133\times {10}^1$	$6.481\times {10}^1$	$5.729\times {10}^1$	$6.034\times {10}^1$	$5.356\times {10}^1$	$7.663\times {10}^1$	$5.655\times {10}^1$	$2.391\times {10}^2$	$1.543\times {10}^2$
F20	$2.199\times {10}^2$	$5.473\times {10}^0$	$2.202\times {10}^2$	$5.297\times {10}^0$	$2.197\times {10}^2$	$4.339\times {10}^0$	$2.189\times {10}^2$	$5.109\times {10}^0$	$2.196\times {10}^2$	$5.471\times {10}^0$	$2.572\times {10}^2$	$1.750\times {10}^1$
F21	$1.000\times {10}^2$	$2.040\times {10}^{-13}$	$1.000\times {10}^2$	$2.470\times {10}^{-13}$	$1.000\times {10}^2$	$2.046\times {10}^{-13}$	$1.000\times {10}^2$	$2.306\times {10}^{-13}$	$1.000\times {10}^2$	$2.309\times {10}^{-13}$	$1.209\times {10}^3$	$1.693\times {10}^3$
F22	$3.623\times {10}^2$	$9.457\times {10}^0$	$3.612\times {10}^2$	$8.319\times {10}^0$	$3.631\times {10}^2$	$7.859\times {10}^0$	$3.611\times {10}^2$	$8.358\times {10}^0$	$3.577\times {10}^2$	$7.783\times {10}^0$	$4.095\times {10}^2$	$2.177\times {10}^1$
F23	$4.359\times {10}^2$	$7.736\times {10}^0$	$4.348\times {10}^2$	$6.629\times {10}^0$	$4.356\times {10}^2$	$6.002\times {10}^0$	$4.340\times {10}^2$	$5.145\times {10}^0$	$4.315\times {10}^2$	$6.087\times {10}^0$	$4.811\times {10}^2$	$1.929\times {10}^1$
F24	$3.860\times {10}^2$	$1.504\times {10}^0$	$3.858\times {10}^2$	$1.570\times {10}^0$	$3.861\times {10}^2$	$1.431\times {10}^0$	$3.865\times {10}^2$	$9.210\times {10}^{-1}$	$3.857\times {10}^2$	$1.627\times {10}^0$	$3.888\times {10}^2$	$1.203\times {10}^1$
F25	$9.561\times {10}^2$	$1.424\times {10}^2$	$9.454\times {10}^2$	$2.030\times {10}^2$	$9.712\times {10}^2$	$1.444\times {10}^2$	$9.505\times {10}^2$	$1.364\times {10}^2$	$9.254\times {10}^2$	$1.838\times {10}^2$	$1.580\times {10}^3$	$2.792\times {10}^2$
F26	$5.053\times {10}^2$	$8.418\times {10}^0$	$5.084\times {10}^2$	$8.792\times {10}^0$	$5.069\times {10}^2$	$8.208\times {10}^0$	$5.068\times {10}^2$	$7.716\times {10}^0$	$5.083\times {10}^2$	$9.469\times {10}^0$	$5.130\times {10}^2$	$8.095\times {10}^0$
F27	$3.813\times {10}^2$	$4.230\times {10}^1$	$3.830\times {10}^2$	$3.892\times {10}^1$	$3.802\times {10}^2$	$3.978\times {10}^1$	$3.794\times {10}^2$	$3.863\times {10}^1$	$3.658\times {10}^2$	$4.659\times {10}^1$	$3.412\times {10}^2$	$4.823\times {10}^1$
F28	$4.821\times {10}^2$	$3.279\times {10}^1$	$4.914\times {10}^2$	$4.187\times {10}^1$	$4.879\times {10}^2$	$4.166\times {10}^1$	$4.773\times {10}^2$	$3.630\times {10}^1$	$4.967\times {10}^2$	$4.672\times {10}^1$	$6.372\times {10}^2$	$1.416\times {10}^2$
F29	$4.814\times {10}^3$	$2.707\times {10}^3$	$3.807\times {10}^3$	$2.183\times {10}^3$	$5.556\times {10}^3$	$3.536\times {10}^3$	$4.721\times {10}^3$	$2.818\times {10}^3$	$4.529\times {10}^3$	$2.597\times {10}^3$	$5.843\times {10}^3$	$3.528\times {10}^3$
w/t/l	19/4/6		18/5/6		18/5/6		18/5/6		17/4/8		-
Friedman Rankings	2.8793		3.2241		3.1552		3.5172		3.569		4.6552

.

In the effectiveness testing, the BCGEO was compared alongside FitnessDiscard, FitnessImproveDiscard, OnlyPoolChange, and NoPoolChange against the EO algorithm. Table 1 presents the results, showing that the BCGEO achieved 19 wins, while FitnessDiscard, FitnessImproveDiscard, and OnlyPoolChange each secured 18 wins, and NoPoolChange obtained 17 wins. These outcomes indicate that each modification in the BCGEO positively contributes to the algorithm’s performance. According to the Friedman rankings, the BCGEO’s performance ranks at the top. Consequently, the conclusion is reached that the two-dimensional evaluation of particle potential, factoring in both the fitness and the magnitude of fitness improvement, proves to be an effective approach.

To validate the effectiveness of the proposed enhancement approach in practical problems, we conducted an additional experiment on multi-threshold image segmentation. The experiment utilized three images, namely starfish, parrot, and boats, from commonly used test sets in the field of image processing [75,76]. The objective function chosen for the experiment was Kapur’s entropy, with a population size of 20 and a termination criterion of 100 iterations. The segmented images were evaluated using the Peak Signal to Noise Ratio (PSNR), Quality Index based on Local Variance (QILV), and Haar wavelet-based Perceptual Similarity Index (HPSI) to assess the image quality [77].

Figure 4 displays the original images and their segmented counterparts, obtained with threshold counts set to 6, 7, and 8. Table 2 presents the mean and best values of the objective function from 51 experiments, along with the average PSNR, QILV, and HPSI scores. Superior values within the table are indicated in bold. It can be observed that in optimizing the Kapur entropy, the BCGEO outperforms the EO in terms of both mean and best objective function values, indicating that the BCGEO is capable of segmenting images into regions with distinct features more effectively. Furthermore, based on the assessment results of the PSNR, QILV, and HPSI indicators, in most cases, the images segmented by the BCGEO algorithm have a higher degree of similarity to the original images. These experimental results demonstrate that the improvement scheme in the BCGEO is also effective for practical problems.

4.2. Benchmark Function Test

To evaluate the competitiveness of the BCGEO, comparative experiments were conducted against CBSO [78], RGBSO [79], GGSA [80], HGSA [81], GLPSO [82], and WFS [83]. The evaluation utilized the CEC2017 benchmark functions, tested over dimensions of 30, 50, and 100. Except for WFS, the population size for the algorithms was set to 100. Due to its particular nature, WFS operated with a population size of 10,000. The termination criterion was set as reaching $D\ast {10}^4$ function evaluations, where D represents the dimensions. Table 3 displays the parameter settings for the algorithms in this experiment.

Table 4, Table 5 and Table 6 show the experimental results of the algorithm on 30, 50, and 100 dimensions, respectively, across the 29 functions from the CEC2017 benchmark. In comparison with the EO, the BCGEO achieved 19, 20, and 20 wins on benchmark functions with these dimensions, respectively. Most of these wins came from simple multimodal functions, hybrid functions, as well as composition functions. The results clearly indicate that the improvement scheme in the BCGEO is effective across both low and high dimensions. In the comparison with CBSO, RGBSO, GGSA, HGSA, GLPSO, and WFS, the BCGEO achieved 23, 22, 23, 19, 24, and 27 wins on the 30 dimensions, 26, 25, 22, 19, 25, and 28 wins on the 50 dimensions, and 24, 21, 21, 18, 25, and 28 wins on the 100 dimensions, respectively. Additionally, BCGEO’s top ranking in the Friedman test further demonstrates its exceptional competitiveness.

Table 2. Experimental results of image segmentation.

	Number of Thresholds	EO		BCGEO		EO	BCGEO	EO	BCGEO	EO	BCGEO	EO	BCGEO
		Mean	std	Mean	std	Best Value		PSNR		QILV		HPSI
Starfish	6	31.4265	0.129302	31.4615	0.14732	31.61982	31.6242	19.58764	19.8526	0.920396	0.92691	0.594064	0.60912
	7	35.30936	0.154267	35.3607	0.15658	35.50873	35.5135	21.28211	21.3289	0.940816	0.94409	0.681785	0.68428
	8	38.90176	0.165766	39.003	0.14645	39.17261	39.1829	22.60519	22.6649	0.95238	0.95425	0.742732	0.74772
Parrot	6	31.6176	0.10648	31.6587	0.07892	31.7085	31.7087	20.56261	20.6221	0.913635	0.92045	0.65416	0.651928
	7	35.37867	0.127182	35.4445	0.04752	35.47665	35.4823	21.86745	21.9952	0.934729	0.93834	0.694495	0.69719
	8	38.78185	0.198123	38.8512	0.17848	39.06617	39.0674	22.95959	23.12	0.950392	0.95444	0.726741	0.73209
Boats	6	30.74964	0.121762	30.7969	0.10031	30.86662	30.8686	20.72047	20.8099	0.918029	0.92267	0.627963	0.63262
	7	34.39853	0.066497	34.431	0.04927	34.48967	34.5095	21.7466	21.74105	0.927998	0.92854	0.65152	0.649653
	8	37.77482	0.111386	37.8144	0.09347	37.95857	38.0678	23.15214	23.2084	0.948718	0.94974	0.71893	0.72054

Table 2. Experimental results of image segmentation.

	Number of Thresholds	EO		BCGEO		EO	BCGEO	EO	BCGEO	EO	BCGEO	EO	BCGEO
		Mean	std	Mean	std	Best Value		PSNR		QILV		HPSI
Starfish	6	31.4265	0.129302	31.4615	0.14732	31.61982	31.6242	19.58764	19.8526	0.920396	0.92691	0.594064	0.60912
	7	35.30936	0.154267	35.3607	0.15658	35.50873	35.5135	21.28211	21.3289	0.940816	0.94409	0.681785	0.68428
	8	38.90176	0.165766	39.003	0.14645	39.17261	39.1829	22.60519	22.6649	0.95238	0.95425	0.742732	0.74772
Parrot	6	31.6176	0.10648	31.6587	0.07892	31.7085	31.7087	20.56261	20.6221	0.913635	0.92045	0.65416	0.651928
	7	35.37867	0.127182	35.4445	0.04752	35.47665	35.4823	21.86745	21.9952	0.934729	0.93834	0.694495	0.69719
	8	38.78185	0.198123	38.8512	0.17848	39.06617	39.0674	22.95959	23.12	0.950392	0.95444	0.726741	0.73209
Boats	6	30.74964	0.121762	30.7969	0.10031	30.86662	30.8686	20.72047	20.8099	0.918029	0.92267	0.627963	0.63262
	7	34.39853	0.066497	34.431	0.04927	34.48967	34.5095	21.7466	21.74105	0.927998	0.92854	0.65152	0.649653
	8	37.77482	0.111386	37.8144	0.09347	37.95857	38.0678	23.15214	23.2084	0.948718	0.94974	0.71893	0.72054

Table 3. Parameter settings on CEC2017 benchmark functions.

Algorithm	Parameters
CBSO	$P_{rep}=0.2, P_{clus/1}=0.8, P_{cen/1}=0.4, P_{cen/2}=0.5$
RGBSO	$K=5, P_{clus/1}=0.8, P_{cen/1}=0.4, P_{cen/2}=0.5$
GGSAS	$G_0=100, \alpha =20, w_1\left(t\right)=2-2t^3/T^3, w_2\left(t\right)=2t^3/T^3$
HGSA	$G_0=100, L=100, w_1\left(t\right)=1-t^6/T^6, w_2\left(t\right)=t^6/T^6$
GLPSO	$\omega =0.7298, pm=0.01, c=1.49618, sg=7$
WFS	$v>0$

Table 3. Parameter settings on CEC2017 benchmark functions.

Algorithm	Parameters
CBSO	$P_{rep}=0.2, P_{clus/1}=0.8, P_{cen/1}=0.4, P_{cen/2}=0.5$
RGBSO	$K=5, P_{clus/1}=0.8, P_{cen/1}=0.4, P_{cen/2}=0.5$
GGSAS	$G_0=100, \alpha =20, w_1\left(t\right)=2-2t^3/T^3, w_2\left(t\right)=2t^3/T^3$
HGSA	$G_0=100, L=100, w_1\left(t\right)=1-t^6/T^6, w_2\left(t\right)=t^6/T^6$
GLPSO	$\omega =0.7298, pm=0.01, c=1.49618, sg=7$
WFS	$v>0$

Table 4. Experimental results on CEC2017 benchmark functions with 30 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$4.028\times {10}^3$	$4.732\times {10}^3$		$3.535\times {10}^3$	$3.996\times {10}^3$	≈	$3.455\times {10}^3$	$2.883\times {10}^3$	≈	$2.539\times {10}^3$	$2.795\times {10}^3$	≈
F2	$8.612\times {10}^3$	$3.653\times {10}^3$		$5.660\times {10}^1$	$8.386\times {10}^1$	−	$2.801\times {10}^0$	$3.076\times {10}^0$	−	$\mathbf{1}.\mathbf{894}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{1}.\mathbf{350}\times {\mathbf{10}}^{\mathbf{1}}$	−
F3	$8.774\times {10}^1$	$6.386\times {10}^0$		$\mathbf{7}.\mathbf{715}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{398}\times {\mathbf{10}}^{\mathbf{1}}$	−	$9.278\times {10}^1$	$1.907\times {10}^1$	≈	$8.010\times {10}^1$	$3.150\times {10}^1$	−
F4	$\mathbf{2}.\mathbf{056}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{5}.\mathbf{529}\times {\mathbf{10}}^{\mathbf{0}}$		$6.108\times {10}^1$	$1.766\times {10}^1$	+	$1.916\times {10}^2$	$3.655\times {10}^1$	+	$2.184\times {10}^2$	$4.395\times {10}^1$	+
F5	$\mathbf{4}.\mathbf{383}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{4}.\mathbf{633}\times {\mathbf{10}}^{-\mathbf{5}}$		$5.462\times {10}^{-3}$	$1.602\times {10}^{-2}$	+	$4.901\times {10}^1$	$7.888\times {10}^0$	+	$5.755\times {10}^1$	$1.011c{10}^1$	+
F6	$5.631\times {10}^1$	$7.148\times {10}^0$		$8.607\times {10}^1$	$1.773\times {10}^1$	+	$4.274\times {10}^2$	$9.785\times {10}^1$	+	$7.225\times {10}^2$	$1.530\times {10}^2$	+
F7	$2.317\times {10}^1$	$6.187\times {10}^0$		$6.216\times {10}^1$	$1.442\times {10}^1$	+	$1.441\times {10}^2$	$2.804\times {10}^1$	+	$1.591\times {10}^2$	$2.882\times {10}^1$	+
F8	$3.511\times {10}^{-3}$	$1.755\times {10}^{-2}$		$5.585\times {10}^0$	$5.919\times {10}^0$	+	$3.181\times {10}^3$	$7.398\times {10}^2$	+	$3.916\times {10}^3$	$1.192\times {10}^3$	+
F9	$\mathbf{2}.\mathbf{012}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{675}\times {\mathbf{10}}^{\mathbf{2}}$		$3.432\times {10}^3$	$7.442\times {10}^2$	+	$4.254\times {10}^3$	$5.554\times {10}^2$	+	$4.451\times {10}^3$	$5.690\times {10}^2$	+
F10	$\mathbf{3}.\mathbf{682}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{903}\times {\mathbf{10}}^{\mathbf{1}}$		$4.305\times {10}^1$	$2.958\times {10}^1$	≈	$1.327\times {10}^2$	$4.708\times {10}^1$	+	$1.527\times {10}^2$	$5.218\times {10}^1$	+
F11	$\mathbf{8}.\mathbf{469}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{4}.\mathbf{558}\times {\mathbf{10}}^{\mathbf{4}}$		$8.837\times {10}^4$	$1.270\times {10}^5$	−	$1.930\times {10}^6$	$1.182\times {10}^6$	+	$8.852\times {10}^5$	$6.113\times {10}^5$	+
F12	$1.541\times {10}^4$	$1.773\times {10}^4$		$2.230\times {10}^4$	$2.061\times {10}^4$	+	$5.136\times {10}^4$	$3.499\times {10}^4$	+	$5.641\times {10}^4$	$2.480\times {10}^4$	+
F13	$2.128\times {10}^4$	$1.261\times {10}^4$		$7.908\times {10}^3$	$8.610\times {10}^3$	−	$\mathbf{2}.\mathbf{097}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{322}\times {\mathbf{10}}^{\mathbf{3}}$	−	$3.556\times {10}^3$	$3.019\times {10}^3$	−
F14	$3.377\times {10}^3$	$5.865\times {10}^3$		$6.723\times {10}^3$	$1.015\times {10}^4$	+	$2.654\times {10}^4$	$1.527\times {10}^4$	+	$3.676\times {10}^4$	$3.419\times {10}^4$	+
F15	$\mathbf{3}.\mathbf{449}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{3}.\mathbf{990}\times {\mathbf{10}}^{\mathbf{1}}$		$6.280\times {10}^2$	$2.601\times {10}^2$	+	$1.189\times {10}^3$	$2.768\times {10}^2$	+	$1.557\times {10}^3$	$4.261\times {10}^2$	+
F16	$\mathbf{3}.\mathbf{289}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{435}\times {\mathbf{10}}^{\mathbf{1}}$		$2.250\times {10}^2$	$1.478\times {10}^2$	+	$4.767\times {10}^2$	$1.883\times {10}^2$	+	$8.679\times {10}^2$	$3.156\times {10}^2$	+
F17	$3.921\times {10}^5$	$1.743\times {10}^5$		$1.252\times {10}^5$	$1.048\times {10}^5$	−	$8.636\times {10}^4$	$4.595\times {10}^4$	−	$1.108\times {10}^5$	$7.374\times {10}^4$	−
F18	$5.017\times {10}^3$	$6.193\times {10}^3$		$6.135\times {10}^3$	$9.934\times {10}^3$	≈	$8.726\times {10}^4$	$5.287\times {10}^4$	+	$5.951\times {10}^4$	$2.562\times {10}^4$	+
F19	$\mathbf{7}.\mathbf{777}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{5}.\mathbf{412}\times {\mathbf{10}}^{\mathbf{1}}$		$2.391\times {10}^2$	$1.543\times {10}^2$	+	$5.045\times {10}^2$	$1.292\times {10}^2$	+	$8.956\times {10}^2$	$2.159\times {10}^2$	+
F20	$\mathbf{2}.\mathbf{199}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{5}.\mathbf{473}\times {\mathbf{10}}^{\mathbf{0}}$		$2.572\times {10}^2$	$1.750\times {10}^1$	+	$3.771\times {10}^2$	$4.322\times {10}^1$	+	$4.176\times {10}^2$	$3.928\times {10}^1$	+
F21	$\mathbf{1}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{040}\times {\mathbf{10}}^{-\mathbf{13}}$		$1.209\times {10}^3$	$1.693\times {10}^3$	+	$3.123\times {10}^3$	$2.118\times {10}^3$	+	$4.539\times {10}^3$	$1.663\times {10}^3$	+
F22	$\mathbf{3}.\mathbf{623}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{457}\times {\mathbf{10}}^{\mathbf{0}}$		$4.095\times {10}^2$	$2.177\times {10}^1$	+	$7.008\times {10}^2$	$1.260\times {10}^2$	+	$1.005\times {10}^3$	$1.210\times {10}^2$	+
F23	$\mathbf{4}.\mathbf{359}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{7}.\mathbf{736}\times {\mathbf{10}}^{\mathbf{0}}$		$4.811\times {10}^2$	$1.929\times {10}^1$	+	$7.231\times {10}^2$	$1.255\times {10}^2$	+	$1.155\times {10}^3$	$1.021\times {10}^2$	+
F24	$\mathbf{3}.\mathbf{860}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{504}\times {\mathbf{10}}^{\mathbf{0}}$		$3.888\times {10}^2$	$1.203\times {10}^1$	+	$3.896\times {10}^2$	$8.816\times {10}^0$	+	$3.904\times {10}^2$	$1.295\times {10}^1$	≈
F25	$9.561\times {10}^2$	$1.424\times {10}^2$		$1.580\times {10}^3$	$2.792\times {10}^2$	+	$3.696\times {10}^3$	$2.039\times {10}^3$	+	$6.068\times {10}^3$	$1.041\times {10}^3$	+
F26	$\mathbf{5}.\mathbf{053}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{8}.\mathbf{418}\times {\mathbf{10}}^{\mathbf{0}}$		$5.130\times {10}^2$	$8.095\times {10}^0$	+	$6.713\times {10}^2$	$1.593\times {10}^2$	+	$1.233\times {10}^3$	$2.523\times {10}^2$	+
F27	$3.813\times {10}^2$	$4.230\times {10}^1$		$3.412\times {10}^2$	$4.823\times {10}^1$	−	$3.830\times {10}^2$	$4.589\times {10}^1$	≈	$3.434\times {10}^2$	$5.768\times {10}^1$	−
F28	$\mathbf{4}.\mathbf{821}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{279}\times {\mathbf{10}}^{\mathbf{1}}$		$6.372\times {10}^2$	$1.416\times {10}^2$	+	$1.324\times {10}^3$	$2.881\times {10}^2$	+	$1.675\times {10}^3$	$3.453\times {10}^2$	+
F29	$\mathbf{4}.\mathbf{814}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{707}\times {\mathbf{10}}^{\mathbf{3}}$		$5.843\times {10}^3$	$3.528\times {10}^3$	≈	$3.925\times {10}^5$	$2.109\times {10}^5$	+	$2.167\times {10}^5$	$1.611\times {10}^5$	+
w/t/l			-			19/4/6			23/3/3			22/2/5
Friedman Rankings			2.2241			2.8966			5.4828			6.069
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$\mathbf{1}.\mathbf{845}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{9}.\mathbf{625}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$2.840\times {10}^3$	$2.568\times {10}^3$	≈	$9.855\times {10}^4$	$4.741\times {10}^5$	≈	$7.061\times {10}^8$	$3.322\times {10}^8$	+
F2	$5.816\times {10}^4$	$6.555\times {10}^3$	+	$4.483\times {10}^4$	$3.599\times {10}^3$	+	$2.191\times {10}^4$	$5.145\times {10}^3$	+	$1.500\times {10}^4$	$4.385\times {10}^3$	+
F3	$1.305\times {10}^2$	$1.950\times {10}^1$	+	$1.191\times {10}^2$	$2.141\times {10}^0$	+	$2.914\times {10}^2$	$9.243\times {10}^1$	+	$2.454\times {10}^2$	$5.224\times {10}^1$	+
F4	$1.104\times {10}^2$	$1.156\times {10}^1$	+	$1.512\times {10}^2$	$1.364\times {10}^1$	+	$1.761\times {10}^2$	$1.916\times {10}^1$	+	$1.447\times {10}^2$	$2.978\times {10}^1$	+
F5	$7.976\times {10}^0$	$3.914\times {10}^0$	+	$8.907\times {10}^0$	$5.858\times {10}^0$	+	$5.087\times {10}^0$	$2.062\times {10}^0$	+	$2.550\times {10}^1$	$5.934\times {10}^0$	+
F6	$\mathbf{3}.\mathbf{708}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{971}\times {\mathbf{10}}^{\mathbf{0}}$	−	$4.026\times {10}^1$	$2.350\times {10}^0$	−	$1.620\times {10}^2$	$5.406\times {10}^1$	+	$2.344\times {10}^2$	$3.747\times {10}^1$	+
F7	$8.525\times {10}^1$	$1.105\times {10}^1$	+	$1.042\times {10}^2$	$8.549\times {10}^0$	+	$1.535\times {10}^2$	$3.818\times {10}^1$	+	$1.350\times {10}^2$	$3.097\times {10}^1$	+
F8	$1.115\times {10}^{-13}$	$1.592\times {10}^{-14}$	+	$\mathbf{5}.\mathbf{466}\times {\mathbf{10}}^{-\mathbf{14}}$	$\mathbf{5}.\mathbf{736}\times {\mathbf{10}}^{-\mathbf{14}}$	−	$1.409\times {10}^1$	$9.247\times {10}^0$	+	$1.769\times {10}^3$	$1.126\times {10}^3$	+
F9	$3.310\times {10}^3$	$3.822\times {10}^2$	+	$3.183\times {10}^3$	$4.904\times {10}^2$	+	$6.542\times {10}^3$	$3.351\times {10}^2$	+	$4.599\times {10}^3$	$6.497\times {10}^2$	+
F10	$1.452\times {10}^2$	$3.227\times {10}^1$	+	$9.613\times {10}^1$	$2.965\times {10}^1$	+	$1.322\times {10}^2$	$6.007\times {10}^1$	+	$3.211\times {10}^2$	$6.943\times {10}^1$	+
F11	$9.557\times {10}^5$	$2.920\times {10}^6$	+	$1.352\times {10}^5$	$7.064\times {10}^4$	+	$7.840\times {10}^6$	$1.328\times {10}^7$	+	$9.802\times {10}^7$	$7.885\times {10}^7$	+
F12	$1.773\times {10}^4$	$4.815\times {10}^3$	+	$\mathbf{1}.\mathbf{246}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{5}.\mathbf{210}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$5.504\times {10}^4$	$2.297\times {10}^5$	≈	$7.474\times {10}^5$	$8.142\times {10}^5$	+
F13	$2.307\times {10}^5$	$9.825\times {10}^4$	+	$7.247\times {10}^3$	$5.010\times {10}^3$	−	$3.526\times {10}^4$	$8.101\times {10}^4$	≈	$7.673\times {10}^3$	$7.822\times {10}^3$	−
F14	$2.897\times {10}^3$	$1.518\times {10}^3$	≈	$\mathbf{7}.\mathbf{404}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{5}.\mathbf{749}\times {\mathbf{10}}^{\mathbf{2}}$	−	$8.491\times {10}^3$	$8.313\times {10}^3$	+	$1.553\times {10}^5$	$1.581\times {10}^5$	+
F15	$1.183\times {10}^3$	$2.290\times {10}^2$	+	$1.153\times {10}^3$	$1.836\times {10}^2$	+	$1.359\times {10}^3$	$2.055\times {10}^2$	+	$9.717\times {10}^2$	$2.496\times {10}^2$	+
F16	$1.020\times {10}^3$	$2.020\times {10}^2$	+	$1.044\times {10}^3$	$1.898\times {10}^2$	+	$2.777\times {10}^2$	$1.646\times {10}^2$	+	$3.222\times {10}^2$	$1.068\times {10}^2$	+
F17	$1.577\times {10}^5$	$7.468\times {10}^4$	−	$\mathbf{6}.\mathbf{117}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{930}\times {\mathbf{10}}^{\mathbf{4}}$	−	$6.947\times {10}^5$	$7.499\times {10}^5$	≈	$2.156\times {10}^5$	$1.481\times {10}^5$	−
F18	$4.249\times {10}^3$	$1.451\times {10}^3$	≈	$\mathbf{2}.\mathbf{867}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{1}.\mathbf{167}\times {\mathbf{10}}^{\mathbf{3}}$	−	$9.548\times {10}^3$	$1.393\times {10}^4$	≈	$1.104\times {10}^6$	$1.130\times {10}^6$	+
F19	$8.958\times {10}^2$	$1.703\times {10}^2$	+	$9.078\times {10}^2$	$1.905\times {10}^2$	+	$2.793\times {10}^2$	$1.377\times {10}^2$	+	$3.918\times {10}^2$	$1.084\times {10}^2$	+
F20	$3.157\times {10}^2$	$1.821\times {10}^1$	+	$3.209\times {10}^2$	$3.538\times {10}^1$	+	$3.743\times {10}^2$	$2.345\times {10}^1$	+	$3.332\times {10}^2$	$2.850\times {10}^1$	+
F21	$1.000\times {10}^2$	$1.475\times {10}^{-10}$	+	$1.911\times {10}^2$	$6.439\times {10}^2$	+	$1.021\times {10}^2$	$2.319\times {10}^0$	+	$3.027\times {10}^2$	$8.436\times {10}^1$	+
F22	$5.600\times {10}^2$	$3.596\times {10}^1$	+	$4.731\times {10}^2$	$1.275\times {10}^2$	+	$5.930\times {10}^2$	$2.099\times {10}^1$	+	$5.310\times {10}^2$	$3.988\times {10}^1$	+
F23	$5.082\times {10}^2$	$3.327\times {10}^1$	+	$5.182\times {10}^2$	$3.901\times {10}^1$	+	$6.565\times {10}^2$	$2.180\times {10}^1$	+	$5.791\times {10}^2$	$3.661\times {10}^1$	+
F24	$4.271\times {10}^2$	$1.216\times {10}^1$	+	$3.917\times {10}^2$	$8.595\times {10}^0$	+	$4.330\times {10}^2$	$2.126\times {10}^1$	+	$5.235\times {10}^2$	$3.617\times {10}^1$	+
F25	$3.644\times {10}^2$	$5.810\times {10}^2$	−	$\mathbf{2}.\mathbf{529}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{991}\times {\mathbf{10}}^{\mathbf{1}}$	−	$2.945\times {10}^3$	$9.361\times {10}^2$	+	$2.537\times {10}^3$	$7.018\times {10}^2$	+
F26	$6.768\times {10}^2$	$4.510\times {10}^1$	+	$5.552\times {10}^2$	$2.274\times {10}^1$	+	$6.666\times {10}^2$	$2.153\times {10}^1$	+	$6.188\times {10}^2$	$3.091\times {10}^1$	+
F27	$4.294\times {10}^2$	$2.264\times {10}^1$	+	$\mathbf{3}.\mathbf{097}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{678}\times {\mathbf{10}}^{\mathbf{1}}$	−	$5.465\times {10}^2$	$7.722\times {10}^1$	+	$6.325\times {10}^2$	$7.868\times {10}^1$	+
F28	$1.406\times {10}^3$	$2.283\times {10}^2$	+	$1.197\times {10}^3$	$2.116\times {10}^2$	+	$8.682\times {10}^2$	$1.783\times {10}^2$	+	$1.005\times {10}^3$	$1.440\times {10}^2$	+
F29	$4.022\times {10}^4$	$1.594\times {10}^4$	+	$7.428\times {10}^3$	$1.707\times {10}^3$	+	$9.170\times {10}^4$	$1.539\times {10}^5$	+	$6.895\times {10}^6$	$5.630\times {10}^6$	+
w/t/l			23/3/3			19/2/8			24/5/0			27/0/2
Friedman Rankings			4.3103			3.4655			5.6552			5.8966

Table 4. Experimental results on CEC2017 benchmark functions with 30 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$4.028\times {10}^3$	$4.732\times {10}^3$		$3.535\times {10}^3$	$3.996\times {10}^3$	≈	$3.455\times {10}^3$	$2.883\times {10}^3$	≈	$2.539\times {10}^3$	$2.795\times {10}^3$	≈
F2	$8.612\times {10}^3$	$3.653\times {10}^3$		$5.660\times {10}^1$	$8.386\times {10}^1$	−	$2.801\times {10}^0$	$3.076\times {10}^0$	−	$\mathbf{1}.\mathbf{894}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{1}.\mathbf{350}\times {\mathbf{10}}^{\mathbf{1}}$	−
F3	$8.774\times {10}^1$	$6.386\times {10}^0$		$\mathbf{7}.\mathbf{715}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{398}\times {\mathbf{10}}^{\mathbf{1}}$	−	$9.278\times {10}^1$	$1.907\times {10}^1$	≈	$8.010\times {10}^1$	$3.150\times {10}^1$	−
F4	$\mathbf{2}.\mathbf{056}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{5}.\mathbf{529}\times {\mathbf{10}}^{\mathbf{0}}$		$6.108\times {10}^1$	$1.766\times {10}^1$	+	$1.916\times {10}^2$	$3.655\times {10}^1$	+	$2.184\times {10}^2$	$4.395\times {10}^1$	+
F5	$\mathbf{4}.\mathbf{383}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{4}.\mathbf{633}\times {\mathbf{10}}^{-\mathbf{5}}$		$5.462\times {10}^{-3}$	$1.602\times {10}^{-2}$	+	$4.901\times {10}^1$	$7.888\times {10}^0$	+	$5.755\times {10}^1$	$1.011c{10}^1$	+
F6	$5.631\times {10}^1$	$7.148\times {10}^0$		$8.607\times {10}^1$	$1.773\times {10}^1$	+	$4.274\times {10}^2$	$9.785\times {10}^1$	+	$7.225\times {10}^2$	$1.530\times {10}^2$	+
F7	$2.317\times {10}^1$	$6.187\times {10}^0$		$6.216\times {10}^1$	$1.442\times {10}^1$	+	$1.441\times {10}^2$	$2.804\times {10}^1$	+	$1.591\times {10}^2$	$2.882\times {10}^1$	+
F8	$3.511\times {10}^{-3}$	$1.755\times {10}^{-2}$		$5.585\times {10}^0$	$5.919\times {10}^0$	+	$3.181\times {10}^3$	$7.398\times {10}^2$	+	$3.916\times {10}^3$	$1.192\times {10}^3$	+
F9	$\mathbf{2}.\mathbf{012}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{675}\times {\mathbf{10}}^{\mathbf{2}}$		$3.432\times {10}^3$	$7.442\times {10}^2$	+	$4.254\times {10}^3$	$5.554\times {10}^2$	+	$4.451\times {10}^3$	$5.690\times {10}^2$	+
F10	$\mathbf{3}.\mathbf{682}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{903}\times {\mathbf{10}}^{\mathbf{1}}$		$4.305\times {10}^1$	$2.958\times {10}^1$	≈	$1.327\times {10}^2$	$4.708\times {10}^1$	+	$1.527\times {10}^2$	$5.218\times {10}^1$	+
F11	$\mathbf{8}.\mathbf{469}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{4}.\mathbf{558}\times {\mathbf{10}}^{\mathbf{4}}$		$8.837\times {10}^4$	$1.270\times {10}^5$	−	$1.930\times {10}^6$	$1.182\times {10}^6$	+	$8.852\times {10}^5$	$6.113\times {10}^5$	+
F12	$1.541\times {10}^4$	$1.773\times {10}^4$		$2.230\times {10}^4$	$2.061\times {10}^4$	+	$5.136\times {10}^4$	$3.499\times {10}^4$	+	$5.641\times {10}^4$	$2.480\times {10}^4$	+
F13	$2.128\times {10}^4$	$1.261\times {10}^4$		$7.908\times {10}^3$	$8.610\times {10}^3$	−	$\mathbf{2}.\mathbf{097}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{322}\times {\mathbf{10}}^{\mathbf{3}}$	−	$3.556\times {10}^3$	$3.019\times {10}^3$	−
F14	$3.377\times {10}^3$	$5.865\times {10}^3$		$6.723\times {10}^3$	$1.015\times {10}^4$	+	$2.654\times {10}^4$	$1.527\times {10}^4$	+	$3.676\times {10}^4$	$3.419\times {10}^4$	+
F15	$\mathbf{3}.\mathbf{449}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{3}.\mathbf{990}\times {\mathbf{10}}^{\mathbf{1}}$		$6.280\times {10}^2$	$2.601\times {10}^2$	+	$1.189\times {10}^3$	$2.768\times {10}^2$	+	$1.557\times {10}^3$	$4.261\times {10}^2$	+
F16	$\mathbf{3}.\mathbf{289}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{435}\times {\mathbf{10}}^{\mathbf{1}}$		$2.250\times {10}^2$	$1.478\times {10}^2$	+	$4.767\times {10}^2$	$1.883\times {10}^2$	+	$8.679\times {10}^2$	$3.156\times {10}^2$	+
F17	$3.921\times {10}^5$	$1.743\times {10}^5$		$1.252\times {10}^5$	$1.048\times {10}^5$	−	$8.636\times {10}^4$	$4.595\times {10}^4$	−	$1.108\times {10}^5$	$7.374\times {10}^4$	−
F18	$5.017\times {10}^3$	$6.193\times {10}^3$		$6.135\times {10}^3$	$9.934\times {10}^3$	≈	$8.726\times {10}^4$	$5.287\times {10}^4$	+	$5.951\times {10}^4$	$2.562\times {10}^4$	+
F19	$\mathbf{7}.\mathbf{777}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{5}.\mathbf{412}\times {\mathbf{10}}^{\mathbf{1}}$		$2.391\times {10}^2$	$1.543\times {10}^2$	+	$5.045\times {10}^2$	$1.292\times {10}^2$	+	$8.956\times {10}^2$	$2.159\times {10}^2$	+
F20	$\mathbf{2}.\mathbf{199}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{5}.\mathbf{473}\times {\mathbf{10}}^{\mathbf{0}}$		$2.572\times {10}^2$	$1.750\times {10}^1$	+	$3.771\times {10}^2$	$4.322\times {10}^1$	+	$4.176\times {10}^2$	$3.928\times {10}^1$	+
F21	$\mathbf{1}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{040}\times {\mathbf{10}}^{-\mathbf{13}}$		$1.209\times {10}^3$	$1.693\times {10}^3$	+	$3.123\times {10}^3$	$2.118\times {10}^3$	+	$4.539\times {10}^3$	$1.663\times {10}^3$	+
F22	$\mathbf{3}.\mathbf{623}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{457}\times {\mathbf{10}}^{\mathbf{0}}$		$4.095\times {10}^2$	$2.177\times {10}^1$	+	$7.008\times {10}^2$	$1.260\times {10}^2$	+	$1.005\times {10}^3$	$1.210\times {10}^2$	+
F23	$\mathbf{4}.\mathbf{359}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{7}.\mathbf{736}\times {\mathbf{10}}^{\mathbf{0}}$		$4.811\times {10}^2$	$1.929\times {10}^1$	+	$7.231\times {10}^2$	$1.255\times {10}^2$	+	$1.155\times {10}^3$	$1.021\times {10}^2$	+
F24	$\mathbf{3}.\mathbf{860}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{504}\times {\mathbf{10}}^{\mathbf{0}}$		$3.888\times {10}^2$	$1.203\times {10}^1$	+	$3.896\times {10}^2$	$8.816\times {10}^0$	+	$3.904\times {10}^2$	$1.295\times {10}^1$	≈
F25	$9.561\times {10}^2$	$1.424\times {10}^2$		$1.580\times {10}^3$	$2.792\times {10}^2$	+	$3.696\times {10}^3$	$2.039\times {10}^3$	+	$6.068\times {10}^3$	$1.041\times {10}^3$	+
F26	$\mathbf{5}.\mathbf{053}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{8}.\mathbf{418}\times {\mathbf{10}}^{\mathbf{0}}$		$5.130\times {10}^2$	$8.095\times {10}^0$	+	$6.713\times {10}^2$	$1.593\times {10}^2$	+	$1.233\times {10}^3$	$2.523\times {10}^2$	+
F27	$3.813\times {10}^2$	$4.230\times {10}^1$		$3.412\times {10}^2$	$4.823\times {10}^1$	−	$3.830\times {10}^2$	$4.589\times {10}^1$	≈	$3.434\times {10}^2$	$5.768\times {10}^1$	−
F28	$\mathbf{4}.\mathbf{821}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{279}\times {\mathbf{10}}^{\mathbf{1}}$		$6.372\times {10}^2$	$1.416\times {10}^2$	+	$1.324\times {10}^3$	$2.881\times {10}^2$	+	$1.675\times {10}^3$	$3.453\times {10}^2$	+
F29	$\mathbf{4}.\mathbf{814}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{707}\times {\mathbf{10}}^{\mathbf{3}}$		$5.843\times {10}^3$	$3.528\times {10}^3$	≈	$3.925\times {10}^5$	$2.109\times {10}^5$	+	$2.167\times {10}^5$	$1.611\times {10}^5$	+
w/t/l			-			19/4/6			23/3/3			22/2/5
Friedman Rankings			2.2241			2.8966			5.4828			6.069
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$\mathbf{1}.\mathbf{845}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{9}.\mathbf{625}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$2.840\times {10}^3$	$2.568\times {10}^3$	≈	$9.855\times {10}^4$	$4.741\times {10}^5$	≈	$7.061\times {10}^8$	$3.322\times {10}^8$	+
F2	$5.816\times {10}^4$	$6.555\times {10}^3$	+	$4.483\times {10}^4$	$3.599\times {10}^3$	+	$2.191\times {10}^4$	$5.145\times {10}^3$	+	$1.500\times {10}^4$	$4.385\times {10}^3$	+
F3	$1.305\times {10}^2$	$1.950\times {10}^1$	+	$1.191\times {10}^2$	$2.141\times {10}^0$	+	$2.914\times {10}^2$	$9.243\times {10}^1$	+	$2.454\times {10}^2$	$5.224\times {10}^1$	+
F4	$1.104\times {10}^2$	$1.156\times {10}^1$	+	$1.512\times {10}^2$	$1.364\times {10}^1$	+	$1.761\times {10}^2$	$1.916\times {10}^1$	+	$1.447\times {10}^2$	$2.978\times {10}^1$	+
F5	$7.976\times {10}^0$	$3.914\times {10}^0$	+	$8.907\times {10}^0$	$5.858\times {10}^0$	+	$5.087\times {10}^0$	$2.062\times {10}^0$	+	$2.550\times {10}^1$	$5.934\times {10}^0$	+
F6	$\mathbf{3}.\mathbf{708}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{971}\times {\mathbf{10}}^{\mathbf{0}}$	−	$4.026\times {10}^1$	$2.350\times {10}^0$	−	$1.620\times {10}^2$	$5.406\times {10}^1$	+	$2.344\times {10}^2$	$3.747\times {10}^1$	+
F7	$8.525\times {10}^1$	$1.105\times {10}^1$	+	$1.042\times {10}^2$	$8.549\times {10}^0$	+	$1.535\times {10}^2$	$3.818\times {10}^1$	+	$1.350\times {10}^2$	$3.097\times {10}^1$	+
F8	$1.115\times {10}^{-13}$	$1.592\times {10}^{-14}$	+	$\mathbf{5}.\mathbf{466}\times {\mathbf{10}}^{-\mathbf{14}}$	$\mathbf{5}.\mathbf{736}\times {\mathbf{10}}^{-\mathbf{14}}$	−	$1.409\times {10}^1$	$9.247\times {10}^0$	+	$1.769\times {10}^3$	$1.126\times {10}^3$	+
F9	$3.310\times {10}^3$	$3.822\times {10}^2$	+	$3.183\times {10}^3$	$4.904\times {10}^2$	+	$6.542\times {10}^3$	$3.351\times {10}^2$	+	$4.599\times {10}^3$	$6.497\times {10}^2$	+
F10	$1.452\times {10}^2$	$3.227\times {10}^1$	+	$9.613\times {10}^1$	$2.965\times {10}^1$	+	$1.322\times {10}^2$	$6.007\times {10}^1$	+	$3.211\times {10}^2$	$6.943\times {10}^1$	+
F11	$9.557\times {10}^5$	$2.920\times {10}^6$	+	$1.352\times {10}^5$	$7.064\times {10}^4$	+	$7.840\times {10}^6$	$1.328\times {10}^7$	+	$9.802\times {10}^7$	$7.885\times {10}^7$	+
F12	$1.773\times {10}^4$	$4.815\times {10}^3$	+	$\mathbf{1}.\mathbf{246}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{5}.\mathbf{210}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$5.504\times {10}^4$	$2.297\times {10}^5$	≈	$7.474\times {10}^5$	$8.142\times {10}^5$	+
F13	$2.307\times {10}^5$	$9.825\times {10}^4$	+	$7.247\times {10}^3$	$5.010\times {10}^3$	−	$3.526\times {10}^4$	$8.101\times {10}^4$	≈	$7.673\times {10}^3$	$7.822\times {10}^3$	−
F14	$2.897\times {10}^3$	$1.518\times {10}^3$	≈	$\mathbf{7}.\mathbf{404}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{5}.\mathbf{749}\times {\mathbf{10}}^{\mathbf{2}}$	−	$8.491\times {10}^3$	$8.313\times {10}^3$	+	$1.553\times {10}^5$	$1.581\times {10}^5$	+
F15	$1.183\times {10}^3$	$2.290\times {10}^2$	+	$1.153\times {10}^3$	$1.836\times {10}^2$	+	$1.359\times {10}^3$	$2.055\times {10}^2$	+	$9.717\times {10}^2$	$2.496\times {10}^2$	+
F16	$1.020\times {10}^3$	$2.020\times {10}^2$	+	$1.044\times {10}^3$	$1.898\times {10}^2$	+	$2.777\times {10}^2$	$1.646\times {10}^2$	+	$3.222\times {10}^2$	$1.068\times {10}^2$	+
F17	$1.577\times {10}^5$	$7.468\times {10}^4$	−	$\mathbf{6}.\mathbf{117}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{930}\times {\mathbf{10}}^{\mathbf{4}}$	−	$6.947\times {10}^5$	$7.499\times {10}^5$	≈	$2.156\times {10}^5$	$1.481\times {10}^5$	−
F18	$4.249\times {10}^3$	$1.451\times {10}^3$	≈	$\mathbf{2}.\mathbf{867}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{1}.\mathbf{167}\times {\mathbf{10}}^{\mathbf{3}}$	−	$9.548\times {10}^3$	$1.393\times {10}^4$	≈	$1.104\times {10}^6$	$1.130\times {10}^6$	+
F19	$8.958\times {10}^2$	$1.703\times {10}^2$	+	$9.078\times {10}^2$	$1.905\times {10}^2$	+	$2.793\times {10}^2$	$1.377\times {10}^2$	+	$3.918\times {10}^2$	$1.084\times {10}^2$	+
F20	$3.157\times {10}^2$	$1.821\times {10}^1$	+	$3.209\times {10}^2$	$3.538\times {10}^1$	+	$3.743\times {10}^2$	$2.345\times {10}^1$	+	$3.332\times {10}^2$	$2.850\times {10}^1$	+
F21	$1.000\times {10}^2$	$1.475\times {10}^{-10}$	+	$1.911\times {10}^2$	$6.439\times {10}^2$	+	$1.021\times {10}^2$	$2.319\times {10}^0$	+	$3.027\times {10}^2$	$8.436\times {10}^1$	+
F22	$5.600\times {10}^2$	$3.596\times {10}^1$	+	$4.731\times {10}^2$	$1.275\times {10}^2$	+	$5.930\times {10}^2$	$2.099\times {10}^1$	+	$5.310\times {10}^2$	$3.988\times {10}^1$	+
F23	$5.082\times {10}^2$	$3.327\times {10}^1$	+	$5.182\times {10}^2$	$3.901\times {10}^1$	+	$6.565\times {10}^2$	$2.180\times {10}^1$	+	$5.791\times {10}^2$	$3.661\times {10}^1$	+
F24	$4.271\times {10}^2$	$1.216\times {10}^1$	+	$3.917\times {10}^2$	$8.595\times {10}^0$	+	$4.330\times {10}^2$	$2.126\times {10}^1$	+	$5.235\times {10}^2$	$3.617\times {10}^1$	+
F25	$3.644\times {10}^2$	$5.810\times {10}^2$	−	$\mathbf{2}.\mathbf{529}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{991}\times {\mathbf{10}}^{\mathbf{1}}$	−	$2.945\times {10}^3$	$9.361\times {10}^2$	+	$2.537\times {10}^3$	$7.018\times {10}^2$	+
F26	$6.768\times {10}^2$	$4.510\times {10}^1$	+	$5.552\times {10}^2$	$2.274\times {10}^1$	+	$6.666\times {10}^2$	$2.153\times {10}^1$	+	$6.188\times {10}^2$	$3.091\times {10}^1$	+
F27	$4.294\times {10}^2$	$2.264\times {10}^1$	+	$\mathbf{3}.\mathbf{097}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{678}\times {\mathbf{10}}^{\mathbf{1}}$	−	$5.465\times {10}^2$	$7.722\times {10}^1$	+	$6.325\times {10}^2$	$7.868\times {10}^1$	+
F28	$1.406\times {10}^3$	$2.283\times {10}^2$	+	$1.197\times {10}^3$	$2.116\times {10}^2$	+	$8.682\times {10}^2$	$1.783\times {10}^2$	+	$1.005\times {10}^3$	$1.440\times {10}^2$	+
F29	$4.022\times {10}^4$	$1.594\times {10}^4$	+	$7.428\times {10}^3$	$1.707\times {10}^3$	+	$9.170\times {10}^4$	$1.539\times {10}^5$	+	$6.895\times {10}^6$	$5.630\times {10}^6$	+
w/t/l			23/3/3			19/2/8			24/5/0			27/0/2
Friedman Rankings			4.3103			3.4655			5.6552			5.8966

Table 5. Experimental results on CEC2017 benchmark functions with 50 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$1.722\times {10}^3$	$1.606\times {10}^3$		$2.832\times {10}^3$	$2.711\times {10}^3$	≈	$6.145\times {10}^3$	$5.608\times {10}^3$	+	$2.379\times {10}^3$	$3.402\times {10}^3$	≈
F2	$5.215\times {10}^4$	$1.239\times {10}^4$		$2.844\times {10}^3$	$2.216\times {10}^3$	−	$\mathbf{5}.\mathbf{507}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{370}\times {\mathbf{10}}^{\mathbf{1}}$	−	$1.708\times {10}^3$	$3.176\times {10}^3$	−
F3	$9.947\times {10}^1$	$3.579\times {10}^0$		$\mathbf{8}.\mathbf{635}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{4}.\mathbf{305}\times {\mathbf{10}}^{\mathbf{1}}$	−	$1.737\times {10}^2$	$5.016\times {10}^1$	+	$1.448\times {10}^2$	$5.685\times {10}^1$	+
F4	$\mathbf{5}.\mathbf{332}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{020}\times {\mathbf{10}}^{\mathbf{1}}$		$1.508\times {10}^2$	$2.866\times {10}^1$	+	$3.258\times {10}^2$	$5.261\times {10}^1$	+	$3.648\times {10}^2$	$7.374\times {10}^1$	+
F5	$\mathbf{2}.\mathbf{543}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{2}.\mathbf{338}\times {\mathbf{10}}^{-\mathbf{3}}$		$1.323\times {10}^{-1}$	$2.717\times {10}^{-1}$	+	$5.772\times {10}^1$	$6.903\times {10}^0$	+	$6.180\times {10}^1$	$5.644\times {10}^0$	+
F6	$1.196\times {10}^2$	$1.384\times {10}^1$		$1.932\times {10}^2$	$3.798\times {10}^1$	+	$9.181\times {10}^2$	$1.494\times {10}^2$	+	$1.391\times {10}^3$	$2.702\times {10}^2$	+
F7	$\mathbf{5}.\mathbf{673}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{158}\times {\mathbf{10}}^{\mathbf{1}}$		$1.549\times {10}^2$	$2.480\times {10}^1$	+	$3.328\times {10}^2$	$4.648\times {10}^1$	+	$3.614\times {10}^2$	$5.549\times {10}^1$	+
F8	$\mathbf{1}.\mathbf{083}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{7}.\mathbf{195}\times {\mathbf{10}}^{-\mathbf{1}}$		$1.856\times {10}^2$	$3.628\times {10}^2$	+	$9.884\times {10}^3$	$1.646\times {10}^3$	+	$1.163\times {10}^4$	$2.228\times {10}^3$	+
F9	$\mathbf{4}.\mathbf{661}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{7}.\mathbf{639}\times {\mathbf{10}}^{\mathbf{2}}$		$6.178\times {10}^3$	$9.814\times {10}^2$	+	$7.365\times {10}^3$	$8.475\times {10}^2$	+	$7.652\times {10}^3$	$7.801\times {10}^2$	+
F10	$\mathbf{3}.\mathbf{841}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{4}.\mathbf{907}\times {\mathbf{10}}^{\mathbf{0}}$		$1.287\times {10}^2$	$6.057\times {10}^1$	+	$2.034\times {10}^2$	$4.760\times {10}^1$	+	$2.021\times {10}^2$	$5.242\times {10}^1$	+
F11	$2.012\times {10}^6$	$1.321\times {10}^6$		$8.909\times {10}^5$	$6.633\times {10}^5$	−	$1.502\times {10}^7$	$9.505\times {10}^6$	+	$3.772\times {10}^6$	$1.665\times {10}^6$	+
F12	$9.020\times {10}^3$	$9.437\times {10}^3$		$7.071\times {10}^3$	$6.386\times {10}^3$	≈	$6.368\times {10}^4$	$3.754\times {10}^4$	+	$7.850\times {10}^4$	$5.173\times {10}^4$	+
F13	$5.227\times {10}^4$	$4.677\times {10}^4$		$5.164\times {10}^4$	$4.412\times {10}^4$	≈	$2.681\times {10}^4$	$1.715\times {10}^4$	−	$2.592\times {10}^4$	$2.025\times {10}^4$	−
F14	$1.125\times {10}^4$	$5.999\times {10}^3$		$1.234\times {10}^4$	$7.610\times {10}^3$	≈	$2.760\times {10}^4$	$1.684\times {10}^4$	+	$3.196\times {10}^4$	$1.698\times {10}^4$	+
F15	$\mathbf{3}.\mathbf{781}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{485}\times {\mathbf{10}}^{\mathbf{2}}$		$1.253\times {10}^3$	$3.849\times {10}^2$	+	$2.182\times {10}^3$	$4.718\times {10}^2$	+	$2.455\times {10}^3$	$5.077\times {10}^2$	+
F16	$\mathbf{2}.\mathbf{825}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{456}\times {\mathbf{10}}^{\mathbf{2}}$		$1.061\times {10}^3$	$3.141\times {10}^2$	+	$1.607\times {10}^3$	$3.539\times {10}^2$	+	$2.077\times {10}^3$	$3.478\times {10}^2$	+
F17	$1.179\times {10}^6$	$3.223\times {10}^5$		$2.973\times {10}^5$	$1.923\times {10}^5$	−	$1.720\times {10}^5$	$8.712\times {10}^4$	−	$\mathbf{1}.\mathbf{233}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{7}.\mathbf{027}\times {\mathbf{10}}^{\mathbf{4}}$	−
F18	$1.790\times {10}^4$	$1.023\times {10}^4$		$1.749\times {10}^4$	$1.376\times {10}^4$	≈	$3.859\times {10}^5$	$2.366\times {10}^5$	+	$1.709\times {10}^5$	$6.359\times {10}^4$	+
F19	$\mathbf{2}.\mathbf{460}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{082}\times {\mathbf{10}}^{\mathbf{2}}$		$8.543\times {10}^2$	$3.131\times {10}^2$	+	$1.287\times {10}^3$	$2.898\times {10}^2$	+	$1.778\times {10}^3$	$3.733\times {10}^2$	+
F20	$\mathbf{2}.\mathbf{533}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{036}\times {\mathbf{10}}^{\mathbf{1}}$		$3.280\times {10}^2$	$2.814\times {10}^1$	+	$6.313\times {10}^2$	$8.281\times {10}^1$	+	$6.859\times {10}^2$	$8.144\times {10}^1$	+
F21	$\mathbf{2}.\mathbf{830}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{315}\times {\mathbf{10}}^{\mathbf{3}}$		$6.650\times {10}^3$	$9.778\times {10}^2$	+	$8.048\times {10}^3$	$9.646\times {10}^2$	+	$8.162\times {10}^3$	$8.804\times {10}^2$	+
F22	$\mathbf{4}.\mathbf{595}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{528}\times {\mathbf{10}}^{\mathbf{1}}$		$5.471\times {10}^2$	$3.438\times {10}^1$	+	$1.110\times {10}^3$	$2.050\times {10}^2$	+	$1.693\times {10}^3$	$2.045\times {10}^2$	+
F23	$\mathbf{5}.\mathbf{290}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{541}\times {\mathbf{10}}^{\mathbf{0}}$		$6.123\times {10}^2$	$2.705\times {10}^1$	+	$1.073\times {10}^3$	$2.399\times {10}^2$	+	$1.821\times {10}^3$	$1.982\times {10}^2$	+
F24	$\mathbf{5}.\mathbf{239}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{154}\times {\mathbf{10}}^{\mathbf{1}}$		$5.488\times {10}^2$	$3.619\times {10}^1$	+	$5.686\times {10}^2$	$3.089\times {10}^1$	+	$5.458\times {10}^2$	$4.204\times {10}^1$	+
F25	$1.330\times {10}^3$	$2.403\times {10}^2$		$2.382\times {10}^3$	$4.244\times {10}^2$	+	$9.242\times {10}^3$	$2.068\times {10}^3$	+	$1.124\times {10}^4$	$1.242\times {10}^3$	+
F26	$\mathbf{5}.\mathbf{198}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{168}\times {\mathbf{10}}^{\mathbf{1}}$		$6.328\times {10}^2$	$5.521\times {10}^1$	+	$1.323\times {10}^3$	$3.804\times {10}^2$	+	$2.873\times {10}^3$	$5.555\times {10}^2$	+
F27	$\mathbf{4}.\mathbf{588}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{146}\times {\mathbf{10}}^{-\mathbf{3}}$		$4.936\times {10}^2$	$2.380\times {10}^1$	+	$5.088\times {10}^2$	$2.250\times {10}^1$	+	$5.011\times {10}^2$	$2.155\times {10}^1$	+
F28	$\mathbf{5}.\mathbf{589}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{016}\times {\mathbf{10}}^{\mathbf{1}}$		$9.485\times {10}^2$	$3.237\times {10}^2$	+	$2.553\times {10}^3$	$4.503\times {10}^2$	+	$2.701\times {10}^3$	$4.779\times {10}^2$	+
F29	$\mathbf{8}.\mathbf{389}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{1}.\mathbf{330}\times {\mathbf{10}}^{\mathbf{5}}$		$1.035\times {10}^6$	$2.039\times {10}^5$	+	$1.147\times {10}^7$	$3.525\times {10}^6$	+	$8.910\times {10}^6$	$1.168\times {10}^6$	+
w/t/l			−			20/5/4			26/0/3			25/1/3
Friedman Rankings			2.069			2.6897			5.4138			5.8966
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$8.151\times {10}^2$	$1.190\times {10}^3$	−	$\mathbf{7}.\mathbf{945}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{3}}$	−	$6.138\times {10}^6$	$4.045\times {10}^7$	≈	$1.672\times {10}^9$	$5.857\times {10}^8$	+
F2	$1.357\times {10}^5$	$1.019\times {10}^4$	+	$1.180\times {10}^5$	$9.514\times {10}^3$	+	$7.827\times {10}^4$	$1.101\times {10}^4$	+	$3.396\times {10}^4$	$5.752\times {10}^3$	−
F3	$1.852\times {10}^2$	$5.426\times {10}^1$	+	$1.958\times {10}^2$	$3.970\times {10}^1$	+	$8.643\times {10}^2$	$2.489\times {10}^2$	+	$4.725\times {10}^2$	$1.192\times {10}^2$	+
F4	$2.268\times {10}^2$	$2.059\times {10}^1$	+	$2.676\times {10}^2$	$2.065\times {10}^1$	+	$3.566\times {10}^2$	$4.186\times {10}^1$	+	$2.754\times {10}^2$	$4.782\times {10}^1$	+
F5	$2.455\times {10}^1$	$4.689\times {10}^0$	+	$2.362\times {10}^1$	$4.394\times {10}^0$	+	$1.472\times {10}^1$	$2.961\times {10}^0$	+	$3.170\times {10}^1$	$8.117\times {10}^0$	+
F6	$\mathbf{6}.\mathbf{565}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{808}\times {\mathbf{10}}^{\mathbf{0}}$	−	$7.097\times {10}^1$	$4.037\times {10}^0$	−	$3.677\times {10}^2$	$8.170\times {10}^1$	+	$4.310\times {10}^2$	$6.070\times {10}^1$	+
F7	$2.322\times {10}^2$	$2.094\times {10}^1$	+	$2.918\times {10}^2$	$1.582\times {10}^1$	+	$3.620\times {10}^2$	$3.855\times {10}^1$	+	$2.727\times {10}^2$	$4.181\times {10}^1$	+
F8	$6.505\times {10}^2$	$4.508\times {10}^2$	+	$1.133\times {10}^1$	$8.013\times {10}^1$	−	$1.488\times {10}^3$	$1.104\times {10}^3$	+	$7.835\times {10}^3$	$4.195\times {10}^3$	+
F9	$5.856\times {10}^3$	$5.261\times {10}^2$	+	$5.747\times {10}^3$	$5.356\times {10}^2$	+	$1.231\times {10}^4$	$4.412\times {10}^2$	+	$8.582\times {10}^3$	$1.036\times {10}^3$	+
F10	$4.027\times {10}^2$	$8.683\times {10}^1$	+	$1.262\times {10}^2$	$1.352\times {10}^1$	+	$7.270\times {10}^2$	$6.007\times {10}^2$	+	$6.736\times {10}^2$	$1.194\times {10}^2$	+
F11	$1.426\times {10}^6$	$3.676\times {10}^5$	−	$\mathbf{8}.\mathbf{335}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{3}.\mathbf{748}\times {\mathbf{10}}^{\mathbf{5}}$	−	$1.251\times {10}^8$	$3.145\times {10}^8$	+	$3.315\times {10}^8$	$1.775\times {10}^8$	+
F12	$1.333\times {10}^4$	$2.085\times {10}^3$	+	$\mathbf{5}.\mathbf{693}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{358}\times {\mathbf{10}}^{\mathbf{2}}$	−	$2.880\times {10}^6$	$1.276\times {10}^7$	≈	$3.384\times {10}^6$	$3.226\times {10}^6$	+
F13	$7.864\times {10}^4$	$3.492\times {10}^4$	+	$\mathbf{2}.\mathbf{290}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{282}\times {\mathbf{10}}^{\mathbf{4}}$	−	$2.809\times {10}^5$	$4.382\times {10}^5$	+	$1.115\times {10}^5$	$9.378\times {10}^4$	+
F14	$7.238\times {10}^3$	$2.067\times {10}^3$	−	$7.845\times {10}^3$	$1.791\times {10}^3$	−	$\mathbf{6}.\mathbf{184}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{6}.\mathbf{436}\times {\mathbf{10}}^{\mathbf{3}}$	−	$9.225\times {10}^5$	$1.021\times {10}^6$	+
F15	$1.759\times {10}^3$	$3.008\times {10}^2$	+	$1.828\times {10}^3$	$3.048\times {10}^2$	+	$2.785\times {10}^3$	$4.031\times {10}^2$	+	$1.793\times {10}^3$	$4.811\times {10}^2$	+
F16	$1.733\times {10}^3$	$3.044\times {10}^2$	+	$1.670\times {10}^3$	$3.095\times {10}^2$	+	$1.505\times {10}^3$	$2.726\times {10}^2$	+	$1.285\times {10}^3$	$2.466\times {10}^2$	+
F17	$8.779\times {10}^5$	$6.773\times {10}^5$	−	$1.815\times {10}^5$	$6.779\times {10}^4$	−	$3.868\times {10}^6$	$4.309\times {10}^6$	+	$1.601\times {10}^6$	$9.382\times {10}^5$	+
F18	$1.469\times {10}^4$	$2.796\times {10}^3$	≈	$\mathbf{1}.\mathbf{423}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{336}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$6.479\times {10}^4$	$3.813\times {10}^5$	−	$1.954\times {10}^6$	$1.403\times {10}^6$	+
F19	$1.219\times {10}^3$	$2.864\times {10}^2$	+	$1.315\times {10}^3$	$3.107\times {10}^2$	+	$1.313\times {10}^3$	$3.192\times {10}^2$	+	$9.669\times {10}^2$	$2.346\times {10}^2$	+
F20	$4.495\times {10}^2$	$3.068\times {10}^1$	+	$4.561\times {10}^2$	$2.659\times {10}^1$	+	$5.756\times {10}^2$	$2.577\times {10}^1$	+	$4.730\times {10}^2$	$4.939\times {10}^1$	+
F21	$7.729\times {10}^3$	$4.755\times {10}^2$	+	$7.902\times {10}^3$	$5.189\times {10}^2$	+	$1.122\times {10}^4$	$3.842\times {10}^3$	+	$8.180\times {10}^3$	$2.080\times {10}^3$	+
F22	$8.633\times {10}^2$	$9.392\times {10}^1$	+	$1.071\times {10}^3$	$1.964\times {10}^2$	+	$9.861\times {10}^2$	$4.881\times {10}^1$	+	$8.064\times {10}^2$	$6.412\times {10}^1$	+
F23	$8.271\times {10}^2$	$4.575\times {10}^1$	+	$8.866\times {10}^2$	$4.895\times {10}^1$	+	$1.055\times {10}^3$	$4.607\times {10}^1$	+	$8.493\times {10}^2$	$7.058\times {10}^1$	+
F24	$6.558\times {10}^2$	$2.651\times {10}^1$	+	$5.813\times {10}^2$	$1.493\times {10}^1$	+	$9.207\times {10}^2$	$1.121\times {10}^2$	+	$9.038\times {10}^2$	$9.022\times {10}^1$	+
F25	$3.168\times {10}^2$	$1.197\times {10}^2$	−	$\mathbf{3}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{7}.\mathbf{596}\times {\mathbf{10}}^{-\mathbf{13}}$	−	$5.958\times {10}^3$	$6.938\times {10}^2$	+	$4.742\times {10}^3$	$5.017\times {10}^2$	+
F26	$1.341\times {10}^3$	$1.647\times {10}^2$	+	$1.401\times {10}^3$	$2.788\times {10}^2$	+	$1.419\times {10}^3$	$1.126\times {10}^2$	+	$1.069\times {10}^3$	$1.120\times {10}^2$	+
F27	$6.278\times {10}^2$	$8.239\times {10}^1$	+	$5.023\times {10}^2$	$2.110\times {10}^1$	+	$1.207\times {10}^3$	$2.199\times {10}^2$	+	$1.260\times {10}^3$	$3.099\times {10}^2$	+
F28	$2.230\times {10}^3$	$2.890\times {10}^2$	+	$1.706\times {10}^3$	$2.706\times {10}^2$	+	$2.037\times {10}^3$	$4.919\times {10}^2$	+	$1.981\times {10}^3$	$3.938\times {10}^2$	+
F29	$3.002\times {10}^7$	$4.668\times {10}^6$	+	$1.340\times {10}^6$	$9.608\times {10}^4$	+	$1.236\times {10}^7$	$6.858\times {10}^6$	+	$1.995\times {10}^8$	$3.754\times {10}^7$	+
w/t/l			22/1/6			19/1/9			25/2/2			28/0/1
Friedman Rankings			5.8966			3.5862			6.3103			5.8621

Table 5. Experimental results on CEC2017 benchmark functions with 50 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$1.722\times {10}^3$	$1.606\times {10}^3$		$2.832\times {10}^3$	$2.711\times {10}^3$	≈	$6.145\times {10}^3$	$5.608\times {10}^3$	+	$2.379\times {10}^3$	$3.402\times {10}^3$	≈
F2	$5.215\times {10}^4$	$1.239\times {10}^4$		$2.844\times {10}^3$	$2.216\times {10}^3$	−	$\mathbf{5}.\mathbf{507}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{370}\times {\mathbf{10}}^{\mathbf{1}}$	−	$1.708\times {10}^3$	$3.176\times {10}^3$	−
F3	$9.947\times {10}^1$	$3.579\times {10}^0$		$\mathbf{8}.\mathbf{635}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{4}.\mathbf{305}\times {\mathbf{10}}^{\mathbf{1}}$	−	$1.737\times {10}^2$	$5.016\times {10}^1$	+	$1.448\times {10}^2$	$5.685\times {10}^1$	+
F4	$\mathbf{5}.\mathbf{332}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{020}\times {\mathbf{10}}^{\mathbf{1}}$		$1.508\times {10}^2$	$2.866\times {10}^1$	+	$3.258\times {10}^2$	$5.261\times {10}^1$	+	$3.648\times {10}^2$	$7.374\times {10}^1$	+
F5	$\mathbf{2}.\mathbf{543}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{2}.\mathbf{338}\times {\mathbf{10}}^{-\mathbf{3}}$		$1.323\times {10}^{-1}$	$2.717\times {10}^{-1}$	+	$5.772\times {10}^1$	$6.903\times {10}^0$	+	$6.180\times {10}^1$	$5.644\times {10}^0$	+
F6	$1.196\times {10}^2$	$1.384\times {10}^1$		$1.932\times {10}^2$	$3.798\times {10}^1$	+	$9.181\times {10}^2$	$1.494\times {10}^2$	+	$1.391\times {10}^3$	$2.702\times {10}^2$	+
F7	$\mathbf{5}.\mathbf{673}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{158}\times {\mathbf{10}}^{\mathbf{1}}$		$1.549\times {10}^2$	$2.480\times {10}^1$	+	$3.328\times {10}^2$	$4.648\times {10}^1$	+	$3.614\times {10}^2$	$5.549\times {10}^1$	+
F8	$\mathbf{1}.\mathbf{083}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{7}.\mathbf{195}\times {\mathbf{10}}^{-\mathbf{1}}$		$1.856\times {10}^2$	$3.628\times {10}^2$	+	$9.884\times {10}^3$	$1.646\times {10}^3$	+	$1.163\times {10}^4$	$2.228\times {10}^3$	+
F9	$\mathbf{4}.\mathbf{661}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{7}.\mathbf{639}\times {\mathbf{10}}^{\mathbf{2}}$		$6.178\times {10}^3$	$9.814\times {10}^2$	+	$7.365\times {10}^3$	$8.475\times {10}^2$	+	$7.652\times {10}^3$	$7.801\times {10}^2$	+
F10	$\mathbf{3}.\mathbf{841}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{4}.\mathbf{907}\times {\mathbf{10}}^{\mathbf{0}}$		$1.287\times {10}^2$	$6.057\times {10}^1$	+	$2.034\times {10}^2$	$4.760\times {10}^1$	+	$2.021\times {10}^2$	$5.242\times {10}^1$	+
F11	$2.012\times {10}^6$	$1.321\times {10}^6$		$8.909\times {10}^5$	$6.633\times {10}^5$	−	$1.502\times {10}^7$	$9.505\times {10}^6$	+	$3.772\times {10}^6$	$1.665\times {10}^6$	+
F12	$9.020\times {10}^3$	$9.437\times {10}^3$		$7.071\times {10}^3$	$6.386\times {10}^3$	≈	$6.368\times {10}^4$	$3.754\times {10}^4$	+	$7.850\times {10}^4$	$5.173\times {10}^4$	+
F13	$5.227\times {10}^4$	$4.677\times {10}^4$		$5.164\times {10}^4$	$4.412\times {10}^4$	≈	$2.681\times {10}^4$	$1.715\times {10}^4$	−	$2.592\times {10}^4$	$2.025\times {10}^4$	−
F14	$1.125\times {10}^4$	$5.999\times {10}^3$		$1.234\times {10}^4$	$7.610\times {10}^3$	≈	$2.760\times {10}^4$	$1.684\times {10}^4$	+	$3.196\times {10}^4$	$1.698\times {10}^4$	+
F15	$\mathbf{3}.\mathbf{781}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{485}\times {\mathbf{10}}^{\mathbf{2}}$		$1.253\times {10}^3$	$3.849\times {10}^2$	+	$2.182\times {10}^3$	$4.718\times {10}^2$	+	$2.455\times {10}^3$	$5.077\times {10}^2$	+
F16	$\mathbf{2}.\mathbf{825}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{456}\times {\mathbf{10}}^{\mathbf{2}}$		$1.061\times {10}^3$	$3.141\times {10}^2$	+	$1.607\times {10}^3$	$3.539\times {10}^2$	+	$2.077\times {10}^3$	$3.478\times {10}^2$	+
F17	$1.179\times {10}^6$	$3.223\times {10}^5$		$2.973\times {10}^5$	$1.923\times {10}^5$	−	$1.720\times {10}^5$	$8.712\times {10}^4$	−	$\mathbf{1}.\mathbf{233}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{7}.\mathbf{027}\times {\mathbf{10}}^{\mathbf{4}}$	−
F18	$1.790\times {10}^4$	$1.023\times {10}^4$		$1.749\times {10}^4$	$1.376\times {10}^4$	≈	$3.859\times {10}^5$	$2.366\times {10}^5$	+	$1.709\times {10}^5$	$6.359\times {10}^4$	+
F19	$\mathbf{2}.\mathbf{460}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{082}\times {\mathbf{10}}^{\mathbf{2}}$		$8.543\times {10}^2$	$3.131\times {10}^2$	+	$1.287\times {10}^3$	$2.898\times {10}^2$	+	$1.778\times {10}^3$	$3.733\times {10}^2$	+
F20	$\mathbf{2}.\mathbf{533}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{036}\times {\mathbf{10}}^{\mathbf{1}}$		$3.280\times {10}^2$	$2.814\times {10}^1$	+	$6.313\times {10}^2$	$8.281\times {10}^1$	+	$6.859\times {10}^2$	$8.144\times {10}^1$	+
F21	$\mathbf{2}.\mathbf{830}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{315}\times {\mathbf{10}}^{\mathbf{3}}$		$6.650\times {10}^3$	$9.778\times {10}^2$	+	$8.048\times {10}^3$	$9.646\times {10}^2$	+	$8.162\times {10}^3$	$8.804\times {10}^2$	+
F22	$\mathbf{4}.\mathbf{595}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{528}\times {\mathbf{10}}^{\mathbf{1}}$		$5.471\times {10}^2$	$3.438\times {10}^1$	+	$1.110\times {10}^3$	$2.050\times {10}^2$	+	$1.693\times {10}^3$	$2.045\times {10}^2$	+
F23	$\mathbf{5}.\mathbf{290}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{541}\times {\mathbf{10}}^{\mathbf{0}}$		$6.123\times {10}^2$	$2.705\times {10}^1$	+	$1.073\times {10}^3$	$2.399\times {10}^2$	+	$1.821\times {10}^3$	$1.982\times {10}^2$	+
F24	$\mathbf{5}.\mathbf{239}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{154}\times {\mathbf{10}}^{\mathbf{1}}$		$5.488\times {10}^2$	$3.619\times {10}^1$	+	$5.686\times {10}^2$	$3.089\times {10}^1$	+	$5.458\times {10}^2$	$4.204\times {10}^1$	+
F25	$1.330\times {10}^3$	$2.403\times {10}^2$		$2.382\times {10}^3$	$4.244\times {10}^2$	+	$9.242\times {10}^3$	$2.068\times {10}^3$	+	$1.124\times {10}^4$	$1.242\times {10}^3$	+
F26	$\mathbf{5}.\mathbf{198}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{168}\times {\mathbf{10}}^{\mathbf{1}}$		$6.328\times {10}^2$	$5.521\times {10}^1$	+	$1.323\times {10}^3$	$3.804\times {10}^2$	+	$2.873\times {10}^3$	$5.555\times {10}^2$	+
F27	$\mathbf{4}.\mathbf{588}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{146}\times {\mathbf{10}}^{-\mathbf{3}}$		$4.936\times {10}^2$	$2.380\times {10}^1$	+	$5.088\times {10}^2$	$2.250\times {10}^1$	+	$5.011\times {10}^2$	$2.155\times {10}^1$	+
F28	$\mathbf{5}.\mathbf{589}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{016}\times {\mathbf{10}}^{\mathbf{1}}$		$9.485\times {10}^2$	$3.237\times {10}^2$	+	$2.553\times {10}^3$	$4.503\times {10}^2$	+	$2.701\times {10}^3$	$4.779\times {10}^2$	+
F29	$\mathbf{8}.\mathbf{389}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{1}.\mathbf{330}\times {\mathbf{10}}^{\mathbf{5}}$		$1.035\times {10}^6$	$2.039\times {10}^5$	+	$1.147\times {10}^7$	$3.525\times {10}^6$	+	$8.910\times {10}^6$	$1.168\times {10}^6$	+
w/t/l			−			20/5/4			26/0/3			25/1/3
Friedman Rankings			2.069			2.6897			5.4138			5.8966
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$8.151\times {10}^2$	$1.190\times {10}^3$	−	$\mathbf{7}.\mathbf{945}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{3}}$	−	$6.138\times {10}^6$	$4.045\times {10}^7$	≈	$1.672\times {10}^9$	$5.857\times {10}^8$	+
F2	$1.357\times {10}^5$	$1.019\times {10}^4$	+	$1.180\times {10}^5$	$9.514\times {10}^3$	+	$7.827\times {10}^4$	$1.101\times {10}^4$	+	$3.396\times {10}^4$	$5.752\times {10}^3$	−
F3	$1.852\times {10}^2$	$5.426\times {10}^1$	+	$1.958\times {10}^2$	$3.970\times {10}^1$	+	$8.643\times {10}^2$	$2.489\times {10}^2$	+	$4.725\times {10}^2$	$1.192\times {10}^2$	+
F4	$2.268\times {10}^2$	$2.059\times {10}^1$	+	$2.676\times {10}^2$	$2.065\times {10}^1$	+	$3.566\times {10}^2$	$4.186\times {10}^1$	+	$2.754\times {10}^2$	$4.782\times {10}^1$	+
F5	$2.455\times {10}^1$	$4.689\times {10}^0$	+	$2.362\times {10}^1$	$4.394\times {10}^0$	+	$1.472\times {10}^1$	$2.961\times {10}^0$	+	$3.170\times {10}^1$	$8.117\times {10}^0$	+
F6	$\mathbf{6}.\mathbf{565}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{2}.\mathbf{808}\times {\mathbf{10}}^{\mathbf{0}}$	−	$7.097\times {10}^1$	$4.037\times {10}^0$	−	$3.677\times {10}^2$	$8.170\times {10}^1$	+	$4.310\times {10}^2$	$6.070\times {10}^1$	+
F7	$2.322\times {10}^2$	$2.094\times {10}^1$	+	$2.918\times {10}^2$	$1.582\times {10}^1$	+	$3.620\times {10}^2$	$3.855\times {10}^1$	+	$2.727\times {10}^2$	$4.181\times {10}^1$	+
F8	$6.505\times {10}^2$	$4.508\times {10}^2$	+	$1.133\times {10}^1$	$8.013\times {10}^1$	−	$1.488\times {10}^3$	$1.104\times {10}^3$	+	$7.835\times {10}^3$	$4.195\times {10}^3$	+
F9	$5.856\times {10}^3$	$5.261\times {10}^2$	+	$5.747\times {10}^3$	$5.356\times {10}^2$	+	$1.231\times {10}^4$	$4.412\times {10}^2$	+	$8.582\times {10}^3$	$1.036\times {10}^3$	+
F10	$4.027\times {10}^2$	$8.683\times {10}^1$	+	$1.262\times {10}^2$	$1.352\times {10}^1$	+	$7.270\times {10}^2$	$6.007\times {10}^2$	+	$6.736\times {10}^2$	$1.194\times {10}^2$	+
F11	$1.426\times {10}^6$	$3.676\times {10}^5$	−	$\mathbf{8}.\mathbf{335}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{3}.\mathbf{748}\times {\mathbf{10}}^{\mathbf{5}}$	−	$1.251\times {10}^8$	$3.145\times {10}^8$	+	$3.315\times {10}^8$	$1.775\times {10}^8$	+
F12	$1.333\times {10}^4$	$2.085\times {10}^3$	+	$\mathbf{5}.\mathbf{693}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{358}\times {\mathbf{10}}^{\mathbf{2}}$	−	$2.880\times {10}^6$	$1.276\times {10}^7$	≈	$3.384\times {10}^6$	$3.226\times {10}^6$	+
F13	$7.864\times {10}^4$	$3.492\times {10}^4$	+	$\mathbf{2}.\mathbf{290}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{282}\times {\mathbf{10}}^{\mathbf{4}}$	−	$2.809\times {10}^5$	$4.382\times {10}^5$	+	$1.115\times {10}^5$	$9.378\times {10}^4$	+
F14	$7.238\times {10}^3$	$2.067\times {10}^3$	−	$7.845\times {10}^3$	$1.791\times {10}^3$	−	$\mathbf{6}.\mathbf{184}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{6}.\mathbf{436}\times {\mathbf{10}}^{\mathbf{3}}$	−	$9.225\times {10}^5$	$1.021\times {10}^6$	+
F15	$1.759\times {10}^3$	$3.008\times {10}^2$	+	$1.828\times {10}^3$	$3.048\times {10}^2$	+	$2.785\times {10}^3$	$4.031\times {10}^2$	+	$1.793\times {10}^3$	$4.811\times {10}^2$	+
F16	$1.733\times {10}^3$	$3.044\times {10}^2$	+	$1.670\times {10}^3$	$3.095\times {10}^2$	+	$1.505\times {10}^3$	$2.726\times {10}^2$	+	$1.285\times {10}^3$	$2.466\times {10}^2$	+
F17	$8.779\times {10}^5$	$6.773\times {10}^5$	−	$1.815\times {10}^5$	$6.779\times {10}^4$	−	$3.868\times {10}^6$	$4.309\times {10}^6$	+	$1.601\times {10}^6$	$9.382\times {10}^5$	+
F18	$1.469\times {10}^4$	$2.796\times {10}^3$	≈	$\mathbf{1}.\mathbf{423}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{336}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$6.479\times {10}^4$	$3.813\times {10}^5$	−	$1.954\times {10}^6$	$1.403\times {10}^6$	+
F19	$1.219\times {10}^3$	$2.864\times {10}^2$	+	$1.315\times {10}^3$	$3.107\times {10}^2$	+	$1.313\times {10}^3$	$3.192\times {10}^2$	+	$9.669\times {10}^2$	$2.346\times {10}^2$	+
F20	$4.495\times {10}^2$	$3.068\times {10}^1$	+	$4.561\times {10}^2$	$2.659\times {10}^1$	+	$5.756\times {10}^2$	$2.577\times {10}^1$	+	$4.730\times {10}^2$	$4.939\times {10}^1$	+
F21	$7.729\times {10}^3$	$4.755\times {10}^2$	+	$7.902\times {10}^3$	$5.189\times {10}^2$	+	$1.122\times {10}^4$	$3.842\times {10}^3$	+	$8.180\times {10}^3$	$2.080\times {10}^3$	+
F22	$8.633\times {10}^2$	$9.392\times {10}^1$	+	$1.071\times {10}^3$	$1.964\times {10}^2$	+	$9.861\times {10}^2$	$4.881\times {10}^1$	+	$8.064\times {10}^2$	$6.412\times {10}^1$	+
F23	$8.271\times {10}^2$	$4.575\times {10}^1$	+	$8.866\times {10}^2$	$4.895\times {10}^1$	+	$1.055\times {10}^3$	$4.607\times {10}^1$	+	$8.493\times {10}^2$	$7.058\times {10}^1$	+
F24	$6.558\times {10}^2$	$2.651\times {10}^1$	+	$5.813\times {10}^2$	$1.493\times {10}^1$	+	$9.207\times {10}^2$	$1.121\times {10}^2$	+	$9.038\times {10}^2$	$9.022\times {10}^1$	+
F25	$3.168\times {10}^2$	$1.197\times {10}^2$	−	$\mathbf{3}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{7}.\mathbf{596}\times {\mathbf{10}}^{-\mathbf{13}}$	−	$5.958\times {10}^3$	$6.938\times {10}^2$	+	$4.742\times {10}^3$	$5.017\times {10}^2$	+
F26	$1.341\times {10}^3$	$1.647\times {10}^2$	+	$1.401\times {10}^3$	$2.788\times {10}^2$	+	$1.419\times {10}^3$	$1.126\times {10}^2$	+	$1.069\times {10}^3$	$1.120\times {10}^2$	+
F27	$6.278\times {10}^2$	$8.239\times {10}^1$	+	$5.023\times {10}^2$	$2.110\times {10}^1$	+	$1.207\times {10}^3$	$2.199\times {10}^2$	+	$1.260\times {10}^3$	$3.099\times {10}^2$	+
F28	$2.230\times {10}^3$	$2.890\times {10}^2$	+	$1.706\times {10}^3$	$2.706\times {10}^2$	+	$2.037\times {10}^3$	$4.919\times {10}^2$	+	$1.981\times {10}^3$	$3.938\times {10}^2$	+
F29	$3.002\times {10}^7$	$4.668\times {10}^6$	+	$1.340\times {10}^6$	$9.608\times {10}^4$	+	$1.236\times {10}^7$	$6.858\times {10}^6$	+	$1.995\times {10}^8$	$3.754\times {10}^7$	+
w/t/l			22/1/6			19/1/9			25/2/2			28/0/1
Friedman Rankings			5.8966			3.5862			6.3103			5.8621

Table 6. Experimental results on CEC2017 benchmark functions with 100 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$4.225\times {10}^3$	$3.153\times {10}^3$		$6.123\times {10}^3$	$8.098\times {10}^3$	≈	$3.526\times {10}^6$	$1.046\times {10}^6$	+	$5.465\times {10}^3$	$6.283\times {10}^3$	≈
F2	$2.616\times {10}^5$	$2.722\times {10}^4$		$5.614\times {10}^4$	$1.004\times {10}^4$	−	$\mathbf{8}.\mathbf{793}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{177}\times {\mathbf{10}}^{\mathbf{3}}$	−	$5.128\times {10}^4$	$3.194\times {10}^4$	−
F3	$2.454\times {10}^2$	$2.599\times {10}^1$		$\mathbf{2}.\mathbf{250}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{684}\times {\mathbf{10}}^{\mathbf{1}}$	−	$2.876\times {10}^2$	$5.196\times {10}^1$	+	$2.407\times {10}^2$	$5.821\times {10}^1$	≈
F4	$\mathbf{2}.\mathbf{137}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{081}\times {\mathbf{10}}^{\mathbf{1}}$		$4.507\times {10}^2$	$6.283\times {10}^1$	+	$8.253\times {10}^2$	$7.597\times {10}^1$	+	$8.663\times {10}^2$	$8.069\times {10}^1$	+
F5	$3.057\times {10}^{-1}$	$1.396\times {10}^{-1}$		$4.842\times {10}^0$	$3.568\times {10}^0$	+	$6.557\times {10}^1$	$4.422\times {10}^0$	+	$6.500\times {10}^1$	$4.578\times {10}^0$	+
F6	$3.522\times {10}^2$	$3.928\times {10}^1$		$6.247\times {10}^2$	$1.065\times {10}^2$	+	$2.423\times {10}^3$	$3.097\times {10}^2$	+	$3.645\times {10}^3$	$5.085\times {10}^2$	+
F7	$\mathbf{2}.\mathbf{101}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{583}\times {\mathbf{10}}^{\mathbf{1}}$		$4.190\times {10}^2$	$6.295\times {10}^1$	+	$9.342\times {10}^2$	$9.183\times {10}^1$	+	$9.889\times {10}^2$	$1.006\times {10}^2$	+
F8	$\mathbf{3}.\mathbf{608}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{379}\times {\mathbf{10}}^{\mathbf{1}}$		$1.009\times {10}^4$	$3.534\times {10}^3$	+	$2.676\times {10}^4$	$2.753\times {10}^3$	+	$2.394\times {10}^4$	$2.259\times {10}^3$	+
F9	$1.248\times {10}^4$	$1.296\times {10}^3$		$1.413\times {10}^4$	$1.441\times {10}^3$	+	$1.576\times {10}^4$	$1.304\times {10}^3$	+	$1.556\times {10}^4$	$1.140\times {10}^3$	+
F10	$2.914\times {10}^3$	$8.104\times {10}^2$		$\mathbf{7}.\mathbf{473}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{280}\times {\mathbf{10}}^{\mathbf{2}}$	−	$1.216\times {10}^3$	$1.718\times {10}^2$	−	$1.297\times {10}^3$	$2.819\times {10}^2$	−
F11	$6.109\times {10}^6$	$2.857\times {10}^6$		$1.987\times {10}^6$	$9.078\times {10}^5$	−	$1.112\times {10}^8$	$2.731\times {10}^7$	+	$1.687\times {10}^7$	$6.254\times {10}^6$	+
F12	$5.925\times {10}^3$	$6.931\times {10}^3$		$6.740\times {10}^3$	$4.174\times {10}^3$	+	$3.861\times {10}^4$	$1.208\times {10}^4$	+	$4.117\times {10}^4$	$1.389\times {10}^4$	+
F13	$4.926\times {10}^5$	$8.436\times {10}^4$		$3.187\times {10}^5$	$1.136\times {10}^5$	−	$2.857\times {10}^5$	$1.121\times {10}^5$	−	$\mathbf{9}.\mathbf{207}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{070}\times {\mathbf{10}}^{\mathbf{4}}$	−
F14	$2.377\times {10}^3$	$3.822\times {10}^3$		$3.534\times {10}^3$	$3.067\times {10}^3$	+	$3.233\times {10}^4$	$1.642\times {10}^4$	+	$3.764\times {10}^4$	$1.784\times {10}^4$	+
F15	$\mathbf{1}.\mathbf{793}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{185}\times {\mathbf{10}}^{\mathbf{2}}$		$3.311\times {10}^3$	$7.184\times {10}^2$	+	$5.268\times {10}^3$	$8.399\times {10}^2$	+	$5.188\times {10}^3$	$8.145\times {10}^2$	+
F16	$\mathbf{1}.\mathbf{128}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{025}\times {\mathbf{10}}^{\mathbf{2}}$		$3.169\times {10}^3$	$6.124\times {10}^2$	+	$3.839\times {10}^3$	$5.209\times {10}^2$	+	$3.828\times {10}^3$	$5.856\times {10}^2$	+
F17	$1.888\times {10}^6$	$2.654\times {10}^5$		$6.365\times {10}^5$	$2.866\times {10}^5$	−	$4.603\times {10}^5$	$1.420\times {10}^5$	−	$\mathbf{2}.\mathbf{092}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{6}.\mathbf{282}\times {\mathbf{10}}^{\mathbf{4}}$	−
F18	$2.641\times {10}^3$	$3.372\times {10}^3$		$2.461\times {10}^3$	$2.185\times {10}^3$	+	$2.676\times {10}^6$	$1.438\times {10}^6$	+	$7.477\times {10}^5$	$2.264\times {10}^5$	+
F19	$\mathbf{1}.\mathbf{210}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{854}\times {\mathbf{10}}^{\mathbf{2}}$		$2.735\times {10}^3$	$6.335\times {10}^2$	+	$3.619\times {10}^3$	$4.043\times {10}^2$	+	$3.743\times {10}^3$	$6.343\times {10}^2$	+
F20	$\mathbf{3}.\mathbf{951}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{902}\times {\mathbf{10}}^{\mathbf{1}}$		$5.660\times {10}^2$	$6.790\times {10}^1$	+	$1.590\times {10}^3$	$2.503\times {10}^2$	+	$1.971\times {10}^3$	$2.052\times {10}^2$	+
F21	$1.305\times {10}^4$	$9.778\times {10}^2$		$1.555\times {10}^4$	$1.563\times {10}^3$	+	$1.712\times {10}^4$	$1.577\times {10}^3$	+	$1.724\times {10}^4$	$1.229\times {10}^3$	+
F22	$\mathbf{6}.\mathbf{770}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{144}\times {\mathbf{10}}^{\mathbf{1}}$		$8.651\times {10}^2$	$5.472\times {10}^1$	+	$2.262\times {10}^3$	$6.163\times {10}^2$	+	$3.205\times {10}^3$	$3.317\times {10}^2$	+
F23	$\mathbf{9}.\mathbf{631}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{1}}$		$1.187\times {10}^3$	$6.989\times {10}^1$	+	$2.456\times {10}^3$	$7.656\times {10}^2$	+	$4.002\times {10}^3$	$4.750\times {10}^2$	+
F24	$8.172\times {10}^2$	$4.801\times {10}^1$		$7.833\times {10}^2$	$6.953\times {10}^1$	−	$8.023\times {10}^2$	$6.034\times {10}^1$	≈	$\mathbf{7}.\mathbf{458}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{228}\times {\mathbf{10}}^{\mathbf{1}}$	−
F25	$3.713\times {10}^3$	$5.352\times {10}^2$		$7.041\times {10}^3$	$1.373\times {10}^3$	+	$2.257\times {10}^4$	$5.565\times {10}^3$	+	$2.836\times {10}^4$	$2.398\times {10}^3$	+
F26	$\mathbf{6}.\mathbf{732}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{342}\times {\mathbf{10}}^{\mathbf{1}}$		$7.451\times {10}^2$	$4.961\times {10}^1$	+	$2.381\times {10}^3$	$8.598\times {10}^2$	+	$4.840\times {10}^3$	$1.516\times {10}^3$	+
F27	$5.827\times {10}^2$	$3.470\times {10}^1$		$\mathbf{5}.\mathbf{668}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{454}\times {\mathbf{10}}^{\mathbf{1}}$	−	$6.242\times {10}^2$	$4.049\times {10}^1$	+	$5.705\times {10}^2$	$3.150\times {10}^1$	−
F28	$\mathbf{1}.\mathbf{931}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{598}\times {\mathbf{10}}^{\mathbf{2}}$		$3.189\times {10}^3$	$5.669\times {10}^2$	+	$6.382\times {10}^3$	$7.283\times {10}^2$	+	$5.736\times {10}^3$	$6.244\times {10}^2$	+
F29	$\mathbf{9}.\mathbf{005}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{3}}$		$1.754\times {10}^4$	$1.304\times {10}^4$	+	$1.233\times {10}^7$	$3.963\times {10}^6$	+	$3.239\times {10}^6$	$9.437\times {10}^5$	+
w/t/l			−			20/1/8			24/1/4			21/2/6
Friedman Rankings			2.5517			2.9655			6.0345			5.6897
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$3.908\times {10}^3$	$2.648\times {10}^3$	≈	$\mathbf{3}.\mathbf{707}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{211}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$9.755\times {10}^4$	$1.848\times {10}^5$	+	$3.904\times {10}^9$	$1.440\times {10}^9$	+
F2	$2.911\times {10}^5$	$1.258\times {10}^4$	+	$2.744\times {10}^5$	$1.552\times {10}^4$	+	$1.492\times {10}^5$	$4.428\times {10}^4$	−	$1.118\times {10}^5$	$1.208\times {10}^4$	−
F3	$5.764\times {10}^2$	$1.431\times {10}^2$	+	$2.736\times {10}^2$	$3.846\times {10}^1$	+	$3.081\times {10}^2$	$5.051\times {10}^1$	+	$8.701\times {10}^2$	$1.063\times {10}^2$	+
F4	$6.212\times {10}^2$	$2.684\times {10}^1$	+	$7.287\times {10}^2$	$3.137\times {10}^1$	+	$3.946\times {10}^2$	$6.658\times {10}^1$	+	$5.664\times {10}^2$	$8.519\times {10}^1$	+
F5	$3.717\times {10}^1$	$3.515\times {10}^0$	+	$3.113\times {10}^1$	$2.947\times {10}^0$	+	$\mathbf{1}.\mathbf{438}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3}.\mathbf{827}\times {\mathbf{10}}^{-\mathbf{2}}$	−	$2.766\times {10}^1$	$7.590\times {10}^0$	+
F6	$1.685\times {10}^2$	$1.155\times {10}^1$	−	$\mathbf{1}.\mathbf{518}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{8}.\mathbf{209}\times {\mathbf{10}}^{\mathbf{0}}$	−	$7.417\times {10}^2$	$8.550\times {10}^1$	+	$1.191\times {10}^3$	$1.388\times {10}^2$	+
F7	$6.725\times {10}^2$	$4.438\times {10}^1$	+	$7.954\times {10}^2$	$3.085\times {10}^1$	+	$3.766\times {10}^2$	$5.894\times {10}^1$	+	$5.921\times {10}^2$	$9.029\times {10}^1$	+
F8	$7.016\times {10}^3$	$9.191\times {10}^2$	+	$2.081\times {10}^3$	$7.172\times {10}^2$	+	$8.940\times {10}^3$	$3.517\times {10}^3$	+	$1.137\times {10}^4$	$4.462\times {10}^3$	+
F9	$1.238\times {10}^4$	$9.580\times {10}^2$	≈	$1.225\times {10}^4$	$8.714\times {10}^2$	≈	$\mathbf{1}.\mathbf{119}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{167}\times {\mathbf{10}}^{\mathbf{3}}$	−	$1.926\times {10}^4$	$1.661\times {10}^3$	+
F10	$1.630\times {10}^4$	$2.509\times {10}^3$	+	$4.394\times {10}^3$	$1.405\times {10}^3$	+	$3.231\times {10}^4$	$1.353\times {10}^4$	+	$4.568\times {10}^3$	$7.797\times {10}^2$	+
F11	$3.893\times {10}^6$	$6.060\times {10}^6$	−	$\mathbf{1}.\mathbf{329}\times {\mathbf{10}}^{\mathbf{6}}$	$\mathbf{4}.\mathbf{346}\times {\mathbf{10}}^{\mathbf{5}}$	−	$3.595\times {10}^7$	$1.746\times {10}^7$	+	$1.296\times {10}^9$	$2.863\times {10}^8$	+
F12	$1.364\times {10}^4$	$2.461\times {10}^3$	+	$\mathbf{3}.\mathbf{031}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{063}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$2.420\times {10}^4$	$1.928\times {10}^4$	+	$7.821\times {10}^6$	$4.666\times {10}^6$	+
F13	$4.431\times {10}^5$	$1.567\times {10}^5$	−	$2.037\times {10}^5$	$3.621\times {10}^4$	−	$3.769\times {10}^6$	$3.642\times {10}^6$	+	$2.035\times {10}^6$	$8.195\times {10}^5$	+
F14	$2.292\times {10}^3$	$6.814\times {10}^2$	+	$\mathbf{8}.\mathbf{601}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{547}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$5.793\times {10}^3$	$4.997\times {10}^3$	+	$1.915\times {10}^6$	$1.674\times {10}^6$	+
F15	$4.809\times {10}^3$	$4.435\times {10}^2$	+	$4.904\times {10}^3$	$5.010\times {10}^2$	+	$4.085\times {10}^3$	$7.461\times {10}^2$	+	$4.491\times {10}^3$	$5.926\times {10}^2$	+
F16	$3.093\times {10}^3$	$4.370\times {10}^2$	+	$3.203\times {10}^3$	$3.943\times {10}^2$	+	$3.391\times {10}^3$	$4.840\times {10}^2$	+	$2.907\times {10}^3$	$4.913\times {10}^2$	+
F17	$3.684\times {10}^5$	$7.676\times {10}^4$	−	$2.813\times {10}^5$	$5.077\times {10}^4$	−	$4.552\times {10}^6$	$4.224\times {10}^6$	+	$2.467\times {10}^6$	$1.219\times {10}^6$	+
F18	$1.813\times {10}^3$	$1.230\times {10}^3$	≈	$\mathbf{1}.\mathbf{226}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{9}.\mathbf{443}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$5.520\times {10}^3$	$5.474\times {10}^3$	+	$6.476\times {10}^6$	$3.870\times {10}^6$	+
F19	$3.811\times {10}^3$	$4.202\times {10}^2$	+	$3.891\times {10}^3$	$3.827\times {10}^2$	+	$2.877\times {10}^3$	$5.376\times {10}^2$	+	$2.899\times {10}^3$	$4.790\times {10}^2$	+
F20	$8.780\times {10}^2$	$5.272\times {10}^1$	+	$9.193\times {10}^2$	$4.243\times {10}^1$	+	$6.539\times {10}^2$	$5.677\times {10}^1$	+	$8.649\times {10}^2$	$9.716\times {10}^1$	+
F21	$1.678\times {10}^4$	$7.995\times {10}^2$	+	$1.694\times {10}^4$	$8.212\times {10}^2$	+	$\mathbf{1}.\mathbf{252}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{109}\times {\mathbf{10}}^{\mathbf{3}}$	−	$2.067\times {10}^4$	$1.341\times {10}^3$	+
F22	$1.854\times {10}^3$	$2.519\times {10}^2$	+	$3.091\times {10}^3$	$2.906\times {10}^2$	+	$8.129\times {10}^2$	$4.525\times {10}^1$	+	$1.500\times {10}^3$	$1.437\times {10}^2$	+
F23	$1.580\times {10}^3$	$1.199\times {10}^2$	+	$1.264\times {10}^3$	$7.487\times {10}^1$	+	$1.375\times {10}^3$	$6.757\times {10}^1$	+	$2.000\times {10}^3$	$1.707\times {10}^2$	+
F24	$1.293\times {10}^3$	$7.217\times {10}^1$	+	$8.390\times {10}^2$	$5.914\times {10}^1$	+	$8.454\times {10}^2$	$5.749\times {10}^1$	+	$1.672\times {10}^3$	$1.093\times {10}^2$	+
F25	$1.131\times {10}^3$	$2.205\times {10}^3$	−	$\mathbf{5}.\mathbf{524}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{803}\times {\mathbf{10}}^{\mathbf{3}}$	−	$8.555\times {10}^3$	$5.981\times {10}^2$	+	$1.096\times {10}^4$	$1.097\times {10}^3$	+
F26	$1.562\times {10}^3$	$1.559\times {10}^2$	+	$1.453\times {10}^3$	$2.107\times {10}^2$	+	$8.547\times {10}^2$	$5.908\times {10}^1$	+	$1.421\times {10}^3$	$1.312\times {10}^2$	+
F27	$9.826\times {10}^2$	$1.003\times {10}^2$	+	$6.247\times {10}^2$	$2.500\times {10}^1$	+	$6.658\times {10}^2$	$4.056\times {10}^1$	+	$2.151\times {10}^3$	$3.566\times {10}^2$	+
F28	$4.809\times {10}^3$	$3.773\times {10}^2$	+	$4.462\times {10}^3$	$3.878\times {10}^2$	+	$3.396\times {10}^3$	$5.700\times {10}^2$	+	$5.322\times {10}^3$	$6.490\times {10}^2$	+
F29	$1.487\times {10}^5$	$1.095\times {10}^5$	+	$9.267\times {10}^3$	$2.152\times {10}^3$	≈	$1.318\times {10}^5$	$9.161\times {10}^4$	+	$2.988\times {10}^8$	$7.818\times {10}^7$	+
w/t/l			21/3/5			18/6/5			25/0/4			28/0/1
Friedman Rankings			4.5862			3.7586			4.3448			6.069

Table 6. Experimental results on CEC2017 benchmark functions with 100 dimensions.

	BCGEO			EO			CBSO			RGBSO
	Mean	std		Mean	std		Mean	std		Mean	std
F1	$4.225\times {10}^3$	$3.153\times {10}^3$		$6.123\times {10}^3$	$8.098\times {10}^3$	≈	$3.526\times {10}^6$	$1.046\times {10}^6$	+	$5.465\times {10}^3$	$6.283\times {10}^3$	≈
F2	$2.616\times {10}^5$	$2.722\times {10}^4$		$5.614\times {10}^4$	$1.004\times {10}^4$	−	$\mathbf{8}.\mathbf{793}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{177}\times {\mathbf{10}}^{\mathbf{3}}$	−	$5.128\times {10}^4$	$3.194\times {10}^4$	−
F3	$2.454\times {10}^2$	$2.599\times {10}^1$		$\mathbf{2}.\mathbf{250}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{4}.\mathbf{684}\times {\mathbf{10}}^{\mathbf{1}}$	−	$2.876\times {10}^2$	$5.196\times {10}^1$	+	$2.407\times {10}^2$	$5.821\times {10}^1$	≈
F4	$\mathbf{2}.\mathbf{137}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{081}\times {\mathbf{10}}^{\mathbf{1}}$		$4.507\times {10}^2$	$6.283\times {10}^1$	+	$8.253\times {10}^2$	$7.597\times {10}^1$	+	$8.663\times {10}^2$	$8.069\times {10}^1$	+
F5	$3.057\times {10}^{-1}$	$1.396\times {10}^{-1}$		$4.842\times {10}^0$	$3.568\times {10}^0$	+	$6.557\times {10}^1$	$4.422\times {10}^0$	+	$6.500\times {10}^1$	$4.578\times {10}^0$	+
F6	$3.522\times {10}^2$	$3.928\times {10}^1$		$6.247\times {10}^2$	$1.065\times {10}^2$	+	$2.423\times {10}^3$	$3.097\times {10}^2$	+	$3.645\times {10}^3$	$5.085\times {10}^2$	+
F7	$\mathbf{2}.\mathbf{101}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{583}\times {\mathbf{10}}^{\mathbf{1}}$		$4.190\times {10}^2$	$6.295\times {10}^1$	+	$9.342\times {10}^2$	$9.183\times {10}^1$	+	$9.889\times {10}^2$	$1.006\times {10}^2$	+
F8	$\mathbf{3}.\mathbf{608}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{379}\times {\mathbf{10}}^{\mathbf{1}}$		$1.009\times {10}^4$	$3.534\times {10}^3$	+	$2.676\times {10}^4$	$2.753\times {10}^3$	+	$2.394\times {10}^4$	$2.259\times {10}^3$	+
F9	$1.248\times {10}^4$	$1.296\times {10}^3$		$1.413\times {10}^4$	$1.441\times {10}^3$	+	$1.576\times {10}^4$	$1.304\times {10}^3$	+	$1.556\times {10}^4$	$1.140\times {10}^3$	+
F10	$2.914\times {10}^3$	$8.104\times {10}^2$		$\mathbf{7}.\mathbf{473}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{280}\times {\mathbf{10}}^{\mathbf{2}}$	−	$1.216\times {10}^3$	$1.718\times {10}^2$	−	$1.297\times {10}^3$	$2.819\times {10}^2$	−
F11	$6.109\times {10}^6$	$2.857\times {10}^6$		$1.987\times {10}^6$	$9.078\times {10}^5$	−	$1.112\times {10}^8$	$2.731\times {10}^7$	+	$1.687\times {10}^7$	$6.254\times {10}^6$	+
F12	$5.925\times {10}^3$	$6.931\times {10}^3$		$6.740\times {10}^3$	$4.174\times {10}^3$	+	$3.861\times {10}^4$	$1.208\times {10}^4$	+	$4.117\times {10}^4$	$1.389\times {10}^4$	+
F13	$4.926\times {10}^5$	$8.436\times {10}^4$		$3.187\times {10}^5$	$1.136\times {10}^5$	−	$2.857\times {10}^5$	$1.121\times {10}^5$	−	$\mathbf{9}.\mathbf{207}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{070}\times {\mathbf{10}}^{\mathbf{4}}$	−
F14	$2.377\times {10}^3$	$3.822\times {10}^3$		$3.534\times {10}^3$	$3.067\times {10}^3$	+	$3.233\times {10}^4$	$1.642\times {10}^4$	+	$3.764\times {10}^4$	$1.784\times {10}^4$	+
F15	$\mathbf{1}.\mathbf{793}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{185}\times {\mathbf{10}}^{\mathbf{2}}$		$3.311\times {10}^3$	$7.184\times {10}^2$	+	$5.268\times {10}^3$	$8.399\times {10}^2$	+	$5.188\times {10}^3$	$8.145\times {10}^2$	+
F16	$\mathbf{1}.\mathbf{128}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{025}\times {\mathbf{10}}^{\mathbf{2}}$		$3.169\times {10}^3$	$6.124\times {10}^2$	+	$3.839\times {10}^3$	$5.209\times {10}^2$	+	$3.828\times {10}^3$	$5.856\times {10}^2$	+
F17	$1.888\times {10}^6$	$2.654\times {10}^5$		$6.365\times {10}^5$	$2.866\times {10}^5$	−	$4.603\times {10}^5$	$1.420\times {10}^5$	−	$\mathbf{2}.\mathbf{092}\times {\mathbf{10}}^{\mathbf{5}}$	$\mathbf{6}.\mathbf{282}\times {\mathbf{10}}^{\mathbf{4}}$	−
F18	$2.641\times {10}^3$	$3.372\times {10}^3$		$2.461\times {10}^3$	$2.185\times {10}^3$	+	$2.676\times {10}^6$	$1.438\times {10}^6$	+	$7.477\times {10}^5$	$2.264\times {10}^5$	+
F19	$\mathbf{1}.\mathbf{210}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{854}\times {\mathbf{10}}^{\mathbf{2}}$		$2.735\times {10}^3$	$6.335\times {10}^2$	+	$3.619\times {10}^3$	$4.043\times {10}^2$	+	$3.743\times {10}^3$	$6.343\times {10}^2$	+
F20	$\mathbf{3}.\mathbf{951}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{902}\times {\mathbf{10}}^{\mathbf{1}}$		$5.660\times {10}^2$	$6.790\times {10}^1$	+	$1.590\times {10}^3$	$2.503\times {10}^2$	+	$1.971\times {10}^3$	$2.052\times {10}^2$	+
F21	$1.305\times {10}^4$	$9.778\times {10}^2$		$1.555\times {10}^4$	$1.563\times {10}^3$	+	$1.712\times {10}^4$	$1.577\times {10}^3$	+	$1.724\times {10}^4$	$1.229\times {10}^3$	+
F22	$\mathbf{6}.\mathbf{770}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{144}\times {\mathbf{10}}^{\mathbf{1}}$		$8.651\times {10}^2$	$5.472\times {10}^1$	+	$2.262\times {10}^3$	$6.163\times {10}^2$	+	$3.205\times {10}^3$	$3.317\times {10}^2$	+
F23	$\mathbf{9}.\mathbf{631}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{1}}$		$1.187\times {10}^3$	$6.989\times {10}^1$	+	$2.456\times {10}^3$	$7.656\times {10}^2$	+	$4.002\times {10}^3$	$4.750\times {10}^2$	+
F24	$8.172\times {10}^2$	$4.801\times {10}^1$		$7.833\times {10}^2$	$6.953\times {10}^1$	−	$8.023\times {10}^2$	$6.034\times {10}^1$	≈	$\mathbf{7}.\mathbf{458}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{228}\times {\mathbf{10}}^{\mathbf{1}}$	−
F25	$3.713\times {10}^3$	$5.352\times {10}^2$		$7.041\times {10}^3$	$1.373\times {10}^3$	+	$2.257\times {10}^4$	$5.565\times {10}^3$	+	$2.836\times {10}^4$	$2.398\times {10}^3$	+
F26	$\mathbf{6}.\mathbf{732}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{342}\times {\mathbf{10}}^{\mathbf{1}}$		$7.451\times {10}^2$	$4.961\times {10}^1$	+	$2.381\times {10}^3$	$8.598\times {10}^2$	+	$4.840\times {10}^3$	$1.516\times {10}^3$	+
F27	$5.827\times {10}^2$	$3.470\times {10}^1$		$\mathbf{5}.\mathbf{668}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{454}\times {\mathbf{10}}^{\mathbf{1}}$	−	$6.242\times {10}^2$	$4.049\times {10}^1$	+	$5.705\times {10}^2$	$3.150\times {10}^1$	−
F28	$\mathbf{1}.\mathbf{931}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{598}\times {\mathbf{10}}^{\mathbf{2}}$		$3.189\times {10}^3$	$5.669\times {10}^2$	+	$6.382\times {10}^3$	$7.283\times {10}^2$	+	$5.736\times {10}^3$	$6.244\times {10}^2$	+
F29	$\mathbf{9}.\mathbf{005}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{115}\times {\mathbf{10}}^{\mathbf{3}}$		$1.754\times {10}^4$	$1.304\times {10}^4$	+	$1.233\times {10}^7$	$3.963\times {10}^6$	+	$3.239\times {10}^6$	$9.437\times {10}^5$	+
w/t/l			−			20/1/8			24/1/4			21/2/6
Friedman Rankings			2.5517			2.9655			6.0345			5.6897
	GGSA			HGSA			GLPSO			WFS
	mean	std		mean	std		mean	std		mean	std
F1	$3.908\times {10}^3$	$2.648\times {10}^3$	≈	$\mathbf{3}.\mathbf{707}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{211}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$9.755\times {10}^4$	$1.848\times {10}^5$	+	$3.904\times {10}^9$	$1.440\times {10}^9$	+
F2	$2.911\times {10}^5$	$1.258\times {10}^4$	+	$2.744\times {10}^5$	$1.552\times {10}^4$	+	$1.492\times {10}^5$	$4.428\times {10}^4$	−	$1.118\times {10}^5$	$1.208\times {10}^4$	−
F3	$5.764\times {10}^2$	$1.431\times {10}^2$	+	$2.736\times {10}^2$	$3.846\times {10}^1$	+	$3.081\times {10}^2$	$5.051\times {10}^1$	+	$8.701\times {10}^2$	$1.063\times {10}^2$	+
F4	$6.212\times {10}^2$	$2.684\times {10}^1$	+	$7.287\times {10}^2$	$3.137\times {10}^1$	+	$3.946\times {10}^2$	$6.658\times {10}^1$	+	$5.664\times {10}^2$	$8.519\times {10}^1$	+
F5	$3.717\times {10}^1$	$3.515\times {10}^0$	+	$3.113\times {10}^1$	$2.947\times {10}^0$	+	$\mathbf{1}.\mathbf{438}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3}.\mathbf{827}\times {\mathbf{10}}^{-\mathbf{2}}$	−	$2.766\times {10}^1$	$7.590\times {10}^0$	+
F6	$1.685\times {10}^2$	$1.155\times {10}^1$	−	$\mathbf{1}.\mathbf{518}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{8}.\mathbf{209}\times {\mathbf{10}}^{\mathbf{0}}$	−	$7.417\times {10}^2$	$8.550\times {10}^1$	+	$1.191\times {10}^3$	$1.388\times {10}^2$	+
F7	$6.725\times {10}^2$	$4.438\times {10}^1$	+	$7.954\times {10}^2$	$3.085\times {10}^1$	+	$3.766\times {10}^2$	$5.894\times {10}^1$	+	$5.921\times {10}^2$	$9.029\times {10}^1$	+
F8	$7.016\times {10}^3$	$9.191\times {10}^2$	+	$2.081\times {10}^3$	$7.172\times {10}^2$	+	$8.940\times {10}^3$	$3.517\times {10}^3$	+	$1.137\times {10}^4$	$4.462\times {10}^3$	+
F9	$1.238\times {10}^4$	$9.580\times {10}^2$	≈	$1.225\times {10}^4$	$8.714\times {10}^2$	≈	$\mathbf{1}.\mathbf{119}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{167}\times {\mathbf{10}}^{\mathbf{3}}$	−	$1.926\times {10}^4$	$1.661\times {10}^3$	+
F10	$1.630\times {10}^4$	$2.509\times {10}^3$	+	$4.394\times {10}^3$	$1.405\times {10}^3$	+	$3.231\times {10}^4$	$1.353\times {10}^4$	+	$4.568\times {10}^3$	$7.797\times {10}^2$	+
F11	$3.893\times {10}^6$	$6.060\times {10}^6$	−	$\mathbf{1}.\mathbf{329}\times {\mathbf{10}}^{\mathbf{6}}$	$\mathbf{4}.\mathbf{346}\times {\mathbf{10}}^{\mathbf{5}}$	−	$3.595\times {10}^7$	$1.746\times {10}^7$	+	$1.296\times {10}^9$	$2.863\times {10}^8$	+
F12	$1.364\times {10}^4$	$2.461\times {10}^3$	+	$\mathbf{3}.\mathbf{031}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{063}\times {\mathbf{10}}^{\mathbf{3}}$	≈	$2.420\times {10}^4$	$1.928\times {10}^4$	+	$7.821\times {10}^6$	$4.666\times {10}^6$	+
F13	$4.431\times {10}^5$	$1.567\times {10}^5$	−	$2.037\times {10}^5$	$3.621\times {10}^4$	−	$3.769\times {10}^6$	$3.642\times {10}^6$	+	$2.035\times {10}^6$	$8.195\times {10}^5$	+
F14	$2.292\times {10}^3$	$6.814\times {10}^2$	+	$\mathbf{8}.\mathbf{601}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{547}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$5.793\times {10}^3$	$4.997\times {10}^3$	+	$1.915\times {10}^6$	$1.674\times {10}^6$	+
F15	$4.809\times {10}^3$	$4.435\times {10}^2$	+	$4.904\times {10}^3$	$5.010\times {10}^2$	+	$4.085\times {10}^3$	$7.461\times {10}^2$	+	$4.491\times {10}^3$	$5.926\times {10}^2$	+
F16	$3.093\times {10}^3$	$4.370\times {10}^2$	+	$3.203\times {10}^3$	$3.943\times {10}^2$	+	$3.391\times {10}^3$	$4.840\times {10}^2$	+	$2.907\times {10}^3$	$4.913\times {10}^2$	+
F17	$3.684\times {10}^5$	$7.676\times {10}^4$	−	$2.813\times {10}^5$	$5.077\times {10}^4$	−	$4.552\times {10}^6$	$4.224\times {10}^6$	+	$2.467\times {10}^6$	$1.219\times {10}^6$	+
F18	$1.813\times {10}^3$	$1.230\times {10}^3$	≈	$\mathbf{1}.\mathbf{226}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{9}.\mathbf{443}\times {\mathbf{10}}^{\mathbf{2}}$	≈	$5.520\times {10}^3$	$5.474\times {10}^3$	+	$6.476\times {10}^6$	$3.870\times {10}^6$	+
F19	$3.811\times {10}^3$	$4.202\times {10}^2$	+	$3.891\times {10}^3$	$3.827\times {10}^2$	+	$2.877\times {10}^3$	$5.376\times {10}^2$	+	$2.899\times {10}^3$	$4.790\times {10}^2$	+
F20	$8.780\times {10}^2$	$5.272\times {10}^1$	+	$9.193\times {10}^2$	$4.243\times {10}^1$	+	$6.539\times {10}^2$	$5.677\times {10}^1$	+	$8.649\times {10}^2$	$9.716\times {10}^1$	+
F21	$1.678\times {10}^4$	$7.995\times {10}^2$	+	$1.694\times {10}^4$	$8.212\times {10}^2$	+	$\mathbf{1}.\mathbf{252}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{109}\times {\mathbf{10}}^{\mathbf{3}}$	−	$2.067\times {10}^4$	$1.341\times {10}^3$	+
F22	$1.854\times {10}^3$	$2.519\times {10}^2$	+	$3.091\times {10}^3$	$2.906\times {10}^2$	+	$8.129\times {10}^2$	$4.525\times {10}^1$	+	$1.500\times {10}^3$	$1.437\times {10}^2$	+
F23	$1.580\times {10}^3$	$1.199\times {10}^2$	+	$1.264\times {10}^3$	$7.487\times {10}^1$	+	$1.375\times {10}^3$	$6.757\times {10}^1$	+	$2.000\times {10}^3$	$1.707\times {10}^2$	+
F24	$1.293\times {10}^3$	$7.217\times {10}^1$	+	$8.390\times {10}^2$	$5.914\times {10}^1$	+	$8.454\times {10}^2$	$5.749\times {10}^1$	+	$1.672\times {10}^3$	$1.093\times {10}^2$	+
F25	$1.131\times {10}^3$	$2.205\times {10}^3$	−	$\mathbf{5}.\mathbf{524}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{803}\times {\mathbf{10}}^{\mathbf{3}}$	−	$8.555\times {10}^3$	$5.981\times {10}^2$	+	$1.096\times {10}^4$	$1.097\times {10}^3$	+
F26	$1.562\times {10}^3$	$1.559\times {10}^2$	+	$1.453\times {10}^3$	$2.107\times {10}^2$	+	$8.547\times {10}^2$	$5.908\times {10}^1$	+	$1.421\times {10}^3$	$1.312\times {10}^2$	+
F27	$9.826\times {10}^2$	$1.003\times {10}^2$	+	$6.247\times {10}^2$	$2.500\times {10}^1$	+	$6.658\times {10}^2$	$4.056\times {10}^1$	+	$2.151\times {10}^3$	$3.566\times {10}^2$	+
F28	$4.809\times {10}^3$	$3.773\times {10}^2$	+	$4.462\times {10}^3$	$3.878\times {10}^2$	+	$3.396\times {10}^3$	$5.700\times {10}^2$	+	$5.322\times {10}^3$	$6.490\times {10}^2$	+
F29	$1.487\times {10}^5$	$1.095\times {10}^5$	+	$9.267\times {10}^3$	$2.152\times {10}^3$	≈	$1.318\times {10}^5$	$9.161\times {10}^4$	+	$2.988\times {10}^8$	$7.818\times {10}^7$	+
w/t/l			21/3/5			18/6/5			25/0/4			28/0/1
Friedman Rankings			4.5862			3.7586			4.3448			6.069

Figure 5, Figure 6 and Figure 7 present boxplots illustrating the distribution of optimal solutions for the algorithms on the CEC2017 benchmark with 30, 50, and 100 dimensions. The boxplots include simple multimodal functions F4 and F7, hybrid functions F16 and F19, and composition functions F22 and F23. It is observed that the BCGEO consistently achieves the lowest and shortest graph position, indicating superior competitiveness and stability. Conversely, the original EO does not always show an advantage, particularly at higher dimensions. Specifically, when the dimension reaches 100, the EO’s median or minimum values are sometimes higher than those of its competitors. For instance, on F7, the EO’s median is higher than GLPSO, and on F16, it is higher than both GLPSO and WFS. On F22, both the median and minimum values of the EO are higher than GLPSO. The BCGEO, however, consistently maintains the lowest position and the shortest box shape, indicating that the improvement scheme enhances robustness, particularly in handling high-dimensional problems.

Figure 8, Figure 9 and Figure 10 show the convergence processes of the BCGEO and its opponents on 30, 50, and 100 dimensions. In the figures, the horizontal axis represents the number of function evaluations, and the vertical axis represents the average optimization error of 51 experiments. It can be observed that most of the BCGEO’s adversaries converge to a standstill after spending about one-sixth of the number of evaluations. WFS is a special case, which always maintains a converging trend, but is so slow that it is still unable to find a satisfactory solution after spending the full number of evaluations. On the other hand, for most problems, during an extended period of convergence, the solutions found by the BCGEO are not as good as those found by other algorithms, and may even exhibit a tendency to stagnate early on. However, in the later stages of the search, the BCGEO often exhibits a sudden acceleration in convergence, resulting in better solutions. This indicates that the improved equilibrium pool scheme indeed brings higher population diversity to the BCGEO, aiding in the coverage of the space during high-dimensional optimization. This increases the likelihood of the algorithm escaping the suboptimal region and entering better regions. On the other hand, the steepness of the convergence curve in the final stage suggests that the dynamic resource allocation strategy enhances the efficiency of exploitation, allowing the BCGEO to quickly find the optimal solution within a new region after entering it.

4.3. Real-World Optimization Test

4.3.1. Dynamic Economic Dispatch

The economic dispatch problem, as a core optimization problem in power system operation, aims to minimize the total generation cost by rationally allocating the output of each generating unit while satisfying the constraints of system operation [84,85]. It is mainly divided into two forms: static economic dispatch (SED) and dynamic economic dispatch (DED).

SED is mainly concerned with how to allocate the output of each generator to minimize the total generation cost of the system at a certain moment while satisfying the system load demand and various operational constraints [1,86]. It is a one-time decision-making process based on the current system state, without considering changes in the time series.The model of SED usually includes two major parts, the objective function and constraints. The objective function is mainly the generation cost function, and the constraints include the power balance constraints, generator output power constraints, ramp rate limits, etc. On the basis of static economic dispatch, DED further considers the load changes and generation cost changes in the time series, aiming to optimize the power allocation of generating units in multiple time periods to minimize the system operation cost in the whole dispatch cycle. It usually considers the scheduling plan for each hour in a 24 h period, thus making it 24 times more dimensional than the SED problem [87]. The cost function for the DED problem used in this experiment can be expressed as follows:

$$\begin{array}{c}\hfill Minimize:F=\sum _{t=1}^T\sum _{i=1}^{N^g}{R}_i\left(E_{i,t}\right)i=1,2,3,\cdots ,N^{g}t=1,2,3,\cdots ,T\end{array}$$

(18)

where

$$\begin{array}{c}\hfill R_i\left(E_{i,t}\right)=a_i+b_i{E}_{i,t}+c_i{E}_{i,t}^2+\left|g_{i}sin\left(h_i(E_i^l-E_{i,t})\right)\right|\end{array}$$

(19)

Equation (18) expresses the objective function of this problem, where t represents the current time period, and T refers to the total number of time period. $N^g$ denotes the total number of generators in the genset. Accordingly, $E_{i,t}$ describes the power output of generator i in the t-th time period. In Equation (19), $R\left(E\right)$ expresses the cost function, where $a_i$, $b_i$, $c_i$, $g_i$, and $h_i$ represent the cost coefficients, respectively, and $E_i^l$ denotes the lower bound for power output of generator i.

Power balance constraints are expressed as follows:

$$\begin{array}{c}\hfill \sum _{i=1}^{N^g}{E}_{i,t}=E_t^{load}+E_t^{loss}\end{array}$$

(20)

where $E_t^{load}$ and $E_t^{loss}$ are the total load demand and total transmission loss at time t. $E_t^{loss}$ can be obtained through the loss coefficient B, shown as follows:

$$\begin{array}{c}\hfill E_t^{loss}=\sum _{i=1}^{N^g}\sum _{j=1}^{N^g}{E}_{i,t}{B}_{i,j}{E}_{j,t}j=1,2,3\cdots ,N^g\end{array}$$

(21)

Generator constraints are expressed as follows:

$$\begin{array}{c}\hfill E_i^l\le E_{i,t}\le E_i^u\end{array}$$

(22)

where $E_i^l$ and $E_i^u$ are lower and upper bounds of the power that generator i can output.

Ramp rate limits can be expressed as follows:

$$\begin{array}{c}\hfill E_{i,t}-E_{i,t-1}\le L_i^u\end{array}$$

(23)

$$\begin{array}{c}\hfill E_{i,t-1}-E_{i,t}\le L_i^l\end{array}$$

(24)

The ramp rate limit refers to the maximum rate at which a generator’s power output can change in a time period. This limit ensures that the generator does not damage equipment or affect the stability of the grid due to a too rapid change during startup, shutdown, or power regulation. It consists of two types: up ramp rate limits and down ramp rate limits, which are denoted by $L^u$ and $L^l$ in Equations (23) and (24).

In order to evaluate the fitness value of each particle in these algorithms, the following equations are used as the evaluation function in this experiment:

$$W=\sum _{t=1}^T\sum _{i=1}^{N^g}{R}_i\left(E_{i,t}\right)+\alpha (\sum _{t=1}^T\sum _{i=1}^{N^g}{E}_{i,t}-E_t^{load}{)}^2+\beta (\sum _{t=1}^T\sum _{i=1}^{N^g}{E}_{i,t}-E^m{)}^2$$

(25)

$$E^m=\left\{\begin{array}{c}{E}_{i,t-1}-L_i^l,E_{i,t}<E_{i,t-1}-L_i^l\hfill \\ E_{i,t-1}+L_i^u,E_{i,t}>E_{i,t-1}+L_i^u\hfill \\ E_{i,t},otherwise\hfill \end{array}\right.$$

(26)

The last two terms of Equation (25) are penalty terms, which are used to assign a higher cost to the individuals that do not meet the constraints, instead of rejecting them directly. Alpha and beta are the parameters used to regulate the penalty.

In this DED experiment, there are nine generating units in the genset with a dimension of 216 ($D=9\times 24$), the termination condition of the algorithm is $D\ast {10}^4$ function evaluations, and the maximum and minimum output power of each generating unit is shown in

Table 7. Parameters of the DED problem.

${\mathit{E}}^{\mathit{l}}$	[150, 135, 73, 60, 73, 57, 20, 47, 20]
$E^u$	[470, 460, 340, 300, 243, 160, 130, 120, 80]
Dimension	$9\times 24$

. The experimental results, as shown in

Table 8. Result of the DED problem.

	Mean	std
BCGEO	$\mathbf{1}.\mathbf{73}\times {\mathbf{10}}^{\mathbf{7}}$	$\mathbf{1}.\mathbf{93}\times {\mathbf{10}}^{\mathbf{4}}$
EO	$1.75\times {10}^7$	$1.71\times {10}^5$
CBSO	$1.86\times {10}^7$	$2.29\times {10}^5$
RGBSO	$1.81\times {10}^7$	$1.20\times {10}^5$
GGSA	$2.51\times {10}^7$	$7.76\times {10}^5$
HGSA	$2.05\times {10}^7$	$1.77\times {10}^5$
GLPSO	$3.58\times {10}^7$	$7.93\times {10}^5$
WFS	$2.49\times {10}^7$	$4.74\times {10}^5$

, indicate that the BCGEO can achieve smaller fitness values. Figure 11 presents the convergence and boxplot analysis for the DED problem, where the convergence plot demonstrates that the BCGEO maintains the lowest convergence curve by the end of the search, and the boxplot shows that the BCGEO is not only the lowest in position but also has the shortest shape. These results suggest that the BCGEO outperforms its competitors in finding optimal solutions and exhibits a high degree of stability. It can be inferred that the BCGEO holds promising potential for addressing the DED challenge in power systems, and further theoretical exploration could provide valuable insights for the field of power systems optimization.

4.3.2. Cassini 2: Spacecraft Trajectory Optimization

The spacecraft trajectory optimization problem is a highly specialized and complex problem for optimizing the flight path of a spacecraft from Earth to a target planet [88]. The Cassini 2 problem is a specific instance of optimizing the trajectory from Earth to Saturn, which belongs to the multiple gravity assist deep space maneuver (MGADSM) type. MGADSM is a technology that uses the gravity of planets or other celestial bodies to adjust the orbit of a spacecraft. This approach can significantly reduce the amount of propellant required by a spacecraft, thereby reducing mission cost. During a deep-space exploration mission, a spacecraft flies over one or more planets on a predetermined trajectory, utilizing the gravitational pull of those planets to change its speed and direction. Each time it flies by a planet, the spacecraft gains additional kinetic energy, known as a multi-gravity assist. Through a precisely engineered gravitational boost, spacecraft are capable of complex interplanetary travel. The MGADSM enhances the efficiency of spaceflight and extends the reach of missions to remote celestial bodies, establishing it as an essential technology in contemporary deep-space exploration endeavors.

The goal of Cassini 2 is to design a trajectory that starts from Earth, passes through Venus–Venus–Earth–Jupiter–Saturn, and ends up in an elliptical orbit around Saturn (as depicted in Figure 12). Each trajectory segment is defined by various variables, such as departure time, initial velocity, swing conditions, etc. The objective function is to minimize the total velocity change, including the initial velocity, the velocity change in each trajectory segment, and the final velocity change upon entering Saturn’s orbit. This problem comprises 22 variables, each with specific upper and lower bounds, and is characterized by numerous local optima, posing a considerable challenge for global solution optimization [89].

Table 9. Result of the STO problem.

	Mean	std
BCGEO	$\mathbf{1}.\mathbf{72}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{3}.\mathbf{24}\times {\mathbf{10}}^{\mathbf{0}}$
EO	$1.86\times {10}^1$	$3.67\times {10}^0$
CBSO	$2.71\times {10}^1$	$3.49\times {10}^0$
RGBSO	$3.06\times {10}^1$	$6.44\times {10}^0$
GGSA	$3.91\times {10}^1$	$7.65\times {10}^0$
HGSA	$3.74\times {10}^1$	$5.82\times {10}^0$
GLPSO	$2.38\times {10}^1$	$2.99\times {10}^0$
WFS	$2.65\times {10}^1$	$3.39\times {10}^0$

shows the experimental results for the STO problem, and it can be seen that the BCGEO can search for smaller values compared to its rivals. The convergence graph in Figure 13 illustrates that the EO’s convergence nearly stagnates from around the 36,000th evaluation, while the BCGEO maintains a more distinct convergence trend. Furthermore, the boxplot indicates that the BCGEO is positioned lower, which signifies superior search performance. These findings suggest that the BCGEO has promising application potential in the field of deep-space exploration.

5. Discussion

5.1. Parameter Analysis of $\gamma$

In the BCGEO, the discard of low-potential particles is accompanied by the generation of an equivalent number of particles near the equilibrium pool. This process, similarly to local search, utilizes a shrinking parameter $\gamma$, which allows the radius of the search to decrease iteratively, thereby improving the precision of local exploitation. The analysis of this parameter was conducted using the CEC2017 benchmark, with test values set at 0.9, 0.925, 0.95, 0.975, 0.985, 0.988, and 0.999. Among these, 0.988 was identified as the optimal value in prior research [90,91], and 0.985 is a fine-tuned value based on it. In

Table 10. Statistical results of BCGEO with different $\gamma$ values.

$\mathit{\gamma }$	0.9	0.925	0.95	0.975	0.985	0.988	0.999
Friedman Ranking	4.2586	3.9828	3.8793	4.0862	3.4138	4.2069	4.1724
Wilcoxon Test p-value	2.49 $\times {10}^{-2}$	2.90 $\times {10}^{-1}$	6.41 $\times {10}^{-1}$	6.35 $\times {10}^{-2}$	−	3.68 $\times {10}^{-1}$	4.73 $\times {10}^{-1}$

, the Friedman ranking results indicate that the best performance occurs at $\gamma =0.985$, but the Wilcoxon test p-values show that, except for $\gamma =0.9$, there is no significant difference between $\gamma =0.985$ and the other tested values. Therefore, the BCGEO’s performance is not very sensitive to values of $\gamma$ between 0.9 and 0.999. Based on the Friedman ranking results, we select $\gamma =0.985$.

5.2. Population Diversity Analysis

Population diversity refers to the variability among individuals in the population during the iteration of the algorithm [92]. Healthy population diversity ensures that the algorithm is sufficiently exploratory in searching the solution space, thus preventing premature convergence to local optima [93]. In the early stages of the algorithm, higher diversity helps to quickly explore different regions of the solution space, increasing the likelihood of finding a global optimum. However, as the algorithm proceeds, if the diversity is too low, the algorithm may fall into a local optimum, causing the search to stagnate. Therefore, maintaining proper population diversity is critical to the success of the algorithm. In addition, population diversity is also related to the convergence speed of the algorithm. Excessive diversity may cause the algorithm to converge at a slower rate because there is too much variability among individuals to form an effective exchange of information. Conversely, insufficient diversity may cause the algorithm to converge quickly, but it may only converge to a local optimum. Therefore, algorithm designers need to carefully design the population update strategy to achieve a balance between diversity and convergence speed.

Comparisons of population diversity can help analyze the effectiveness of the scheme proposed in the BCGEO. This analysis builds on the use of the CEC 2017 benchmark function with 30 dimensions. The equation for calculating population diversity V is given below.

$$\begin{array}{c}V={\displaystyle \frac{1}{N}}·\sqrt{{\sum }_{i=1}^N{\left(∥x_i-\overline{x}∥\right)}^2}\end{array}$$

(27)

where x denotes the individual, which is the particle in the BCGEO, and $\overline{x}$ denotes the mean value of the individuals.

$$\begin{array}{c}\overline{x}={\displaystyle \frac{1}{N}}·\sum _{i=1}^N{x}_i\end{array}$$

(28)

Figure 14 illustrates the population diversity curves for the BCGEO versus the EO on partial functions. It is evident that, at the beginning of the search, the population diversity of both the BCGEO and EO algorithms is comparable. However, as the algorithm progresses, the BCGEO’s diversity curve remains consistently higher than the EO’s, indicating that the BCGEO’s scheme for maintaining diversity is effective. This sustained high diversity helps the algorithm avoid premature sticking to local optima throughout the search process. The difference in diversity suggests that the BCGEO has a stronger ability to maintain broad coverage of the solution space, thus preserving a high level of exploration. Towards the end of the algorithm’s convergence, the BCGEO exhibits a rapid decrease in diversity, aligning with the EO’s level. This shift may signify that the improvements bolstered the algorithm’s local search ability in later stages, enabling it to focus more intensively on refining the optimal solutions identified. In summary, the BCGEO’s improvement schemes facilitate extensive exploration in the early and middle stages of the algorithm, while also enhancing the fine-tuning of optimal solutions in the final stage, showcasing its capability to balance global exploration with local exploitation.

6. Conclusions

In this paper, we introduce a novel algorithm, the BCGEO, which integrates the BCG Matrix decision model to achieve a more effective balance between exploration and exploitation while preserving population diversity. A suite of experiments confirms that the BCGEO outperforms its counterparts in numerical optimization and demonstrates significant potential in addressing real-world optimization needs, including power system optimization. This research offers fresh insights into the theoretical framework of MHAs and offers a promising approach to addressing issues in power systems and related fields.

Despite its merits, the BCGEO has certain limitations. For instance, the parameter determining the number of discarded individuals is set empirically, which presents an opportunity for future research. Future works will concentrate on enhancing the BCG mechanism, aiming to make algorithm parameters adaptively adjustable based on feedback received during the search process. Furthermore, we aim to extend the application of the BCG mechanism to a broader spectrum of MHAs and explore its potential in a wider array of optimization problems within the power systems domain.