Evolutionary salp swarm algorithm with multi-search strategies and advanced memory mechanism for solving global optimization and complex engineering problems

Abstract

Real-world optimization problems, such as global optimization, cleaner production system, and complex design challenges are inherently complex, involving many variables and constraints. These factors make it challenging for optimizers to determine optimal solutions efficiently. Salp Swarm Algorithm (SSA) adapts easily to complex optimization problems due to its simplicity, multi-search strategy, and few control parameters. However, its search strategy lacks precision in guiding the population toward optimal regions of the solution space, which limits its effectiveness in optimizing cleaner production systems and complex design problems. This study proposes an evolutionary SSA (ESSA) to address complex optimization problems. ESSA proposes distinct innovative search strategies, including two evolutionary search strategies that enhance diversity and adaptive search, as well as an enhanced SSA search strategy that, while less exploratory, ensures steady convergence. ESSA introduces an advanced memory mechanism that stores the best and inferior solutions identified during optimization, enhancing diversity and preventing premature convergence. Moreover, it incorporates a stochastic universal selection method to regulate the archive by selecting individuals according to their fitness values. The performance of ESSA was evaluated using benchmark functions CEC 2017 and CEC 2020, compared to seven leading algorithms. Results show that ESSA outperforms SSA and others in solution quality and convergence speed. Statistical analyses confirm that ESSA ranks first and achieves the best optimization effectiveness, with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively, surpassing other optimizers. Additionally, ESSA’s practical applicability is demonstrated through its success in optimizing a cleaner production system and solving complex design problems, highlighting its effectiveness in tackling challenging optimization tasks.

Introduction

Real-world optimization problems are inherently complex, involving challenges such as non-convex search spaces that hinder the discovery of global optima, nonlinearly constrained variables that limit feasible solutions, and high-dimensional spaces that increase computational complexity and the risk of overfitting^1–3. These characteristics typically render such problems NP-hard, reflecting their substantial computational complexity and the difficulty in guaranteeing high-quality solutions for algorithms^4–6. To overcome these challenges, metaheuristic algorithms offer effective strategies for optimizing single and multiple objectives in various fields^7–11. These algorithms employ a low-cost, population-based approach to efficiently explore the solution space and minimize the risk of becoming trapped in suboptimal regions^12,13. Metaheuristic algorithms are derived from diverse foundational principles and are typically categorized into distinct approaches, including evolutionary-based^13,15, physics-based^14,15, human behavior-based^16,17, and swarm intelligence-based^18–20. This diversity of inspirations leads to the development of various mathematical models, each incorporating distinct mechanisms that balance global exploration of the search space with local refinement in promising regions¹¹. Evolutionary-based optimization algorithms mimic natural selection and genetic evolution to improve optimization outcomes. Prominent examples include the genetic algorithm (GA)²¹, genetic programming (GP)²², differential evolution (DE)^23,24, evolution strategies (ES)^25–27, and biogeography-based optimization (BBO)²⁸.

GA encodes candidate solutions as genetic structures that evolve through selection, crossover, and mutation for optimization. It is particularly effective for discrete optimization, combinatorial problems, and global search tasks²⁹. However, GA may suffer from premature convergence, reduced population diversity, and a tendency to become trapped in suboptimal solutions rather than reaching the global optimum³⁰. These issues often arise due to factors such as early dominance by high-fitness individuals, which limits the exploration of diverse solutions; insufficient variability introduced by mutation and crossover, and a gradual loss of diversity across generations, thereby reducing adaptability. Additionally, poor selection pressure can accelerate convergence and limit exploration, while overexploiting local optima can hinder global search abilities³¹. To resolve these issues, careful tuning of mutation rates, selection mechanisms, and diversity-preserving strategies is essential for robust optimization performance³². GP extends GA by evolving entire programs instead of fixed-length representations, typically using tree-based structures to encode solutions³³. However, its computational complexity, due to the vast search space, can lead to increased processing time and resource consumption^22,34. DE is a highly effective optimization algorithm designed for continuous domains, demonstrating strong performance in complex, nonlinear, and multimodal problem spaces^35,36. Nevertheless, DE’s effectiveness is sensitive to parameter settings—particularly the scaling factor and crossover rate—making precise parameter tuning critical for optimal performance across diverse optimization scenarios^37,38.

DE has undergone numerous enhancements, leading to several successful variants designed to improve performance across diverse optimization tasks. Adaptive differential evolution with an optional external archive (JADE)³⁸ introduces a novel mutation strategy that utilizes an archive and adjustable parameters to enhance the search process in promising areas. However, its performance depends on proper parameter tuning and efficient archive management to maximize adaptability³⁷. Success-history-based adaptive differential evolution (SHADE)³⁹ refines the parameter adaptation technique for DE, which utilizes a historical memory of successful control parameter settings to guide the selection of future control parameter values. Linear population size reduction-SHADE (LSHADE)³⁷ incorporates a shrinking population mechanism to optimize resource usage while maintaining solution quality. The quantum avian navigation optimization algorithm (QANA)⁴⁰ integrates quantum computing principles with avian-inspired navigation strategies to expand search capabilities. The effectiveness of QANA lies in its ability to dynamically switch between quantum search strategies, helping to prevent premature convergence. ES algorithms utilize self-adaptive mutation and selection, excelling in continuous optimization and reinforcement learning⁴¹. Although flexible, they need careful tuning of mutation step sizes to prevent slow convergence or inefficiency⁴². BBO is inspired by biogeography, focusing on the migration of species and their distribution in habitats⁴³. Candidate solutions are viewed as islands of suitability, and information migration enables stronger solutions to influence weaker ones. BBO performance is influenced by migration rates and mutation strategies, requiring fine-tuning to ensure diversity and prevent stagnation⁴⁴.

Physics-based optimization algorithms utilize search strategies that draw inspiration from fundamental physical laws and chemical reactions, enabling innovative problem-solving techniques. These algorithms often leverage principles such as energy minimization, thermodynamic processes, or dynamic system behaviors to enhance optimization efficiency. Chemical reaction optimization (CRO)⁴⁵ mimics the interactions of molecules in a chemical reaction to reach a low-energy stable state. CRO effectively navigates complex search spaces; however, its performance depends on the precise tuning of reaction parameters to maintain diversity and avoid premature convergence. The gravitational search algorithm (GSA)¹⁵ uses Newton’s law of gravitation, modeling candidate solutions as interacting masses, which is effective for high-dimensional problems but prone to premature convergence in complex and large-scale issues⁴⁶. The water cycle algorithm (WCA)⁴⁷ simulates the hydrological cycle, directing solutions toward optimal areas while needing precise parameter tuning to balance exploration and exploitation during the optimization process. The snow ablation optimizer (SAO)⁴⁷ simulates the natural phenomenon of snow ablation, a process in which snow gradually melts and sublimates due to environmental factors. Human behavior-based optimization algorithms simulate a wide range of human behaviors—such as thinking, decision-making, emotional responses, social interactions, learning, and adaptation—to propose novel movement-based search strategies. The imperialist competitive algorithm (ICA)⁴⁸ models geopolitical competition, where dominant solutions expand influence over weaker ones. ICA has a complicated structure and can struggle with premature convergence, particularly when imperialist influence is too dominant. The social-based algorithm (SBA)¹⁷ leverages human collective decision-making to refine search efficiency. SBA relies on numerous control parameters; its effectiveness is highly dependent on precise tuning to achieve an optimal balance between exploration and exploitation. The queuing search algorithm (QSA)⁴⁹ organizes search agents based on priority-driven movements inspired by real-world queuing behavior. However, QSA requires careful priority assignment to ensure an optimal balance between search breadth and convergence speed.

Swarm intelligence algorithms are inspired by the coordinated behavior of natural systems, such as flocks of birds, schools of fish, and ant colonies, where individuals act without centralized control. Particle swarm optimization (PSO)⁵⁰, which mimics the behavior of birds and fish, adjusts positions based on both individual and collective best solutions to achieve rapid convergence in continuous optimization. The salp swarm optimization algorithm (SSA)¹⁸ models the leader behavior of salps, ensuring a dynamic balance between exploration and exploitation in complex optimization landscapes. The starling murmuration optimizer (SMO)¹⁹ simulates the mesmerizing collective behavior of starlings during murmuration by incorporating three coordinated search strategies—separating, diving, and whirling—to enhance adaptability and diversity in optimization tasks. This combination of search strategies enables SMO to balance exploration and exploitation, making it adaptable to complex, multi-modal optimization challenges. However, tuning the algorithm’s parameters carefully is essential to achieve optimal performance and maintain stable behavior during the search process. The artificial lemming algorithm (ALA)⁵¹ simulates lemming behaviors, including migration, digging, foraging, and avoiding predators, to enhance optimization efficiency. Moreover, ALA employs an energy-decreasing mechanism that regulates search intensity, helping to prevent stagnation and promote convergence toward global optima.

Among the optimization algorithms discussed, SSA has garnered considerable attention due to its simple structure, which incorporates multiple search strategies and requires only a minimal number of control parameters^52,53. During the optimization process, SSA employs randomized switching among search strategies to efficiently search the problem space. The number of function evaluations in each iteration is equal to the population size, which reduces computational costs and improves efficiency, particularly in high-dimensional optimization tasks^54,55. As a result, SSA has gained popularity for solving both binary and continuous optimization problems, particularly in scenarios that require low parameter tuning, simple code matching with the problem, and low computational complexity. Despite its strengths, SSA exhibits notable limitations when applied to complex or large-scale optimization problems. Its search strategies can sometimes cause the algorithm to become trapped in local optima, leading to premature convergence and reducing the overall quality of the solution^56,57. Furthermore, SSA exhibits slower convergence rates for specific functions, which hinders performance in time-sensitive optimization scenarios. The challenge of maintaining an optimal balance between exploration and exploitation also affects its ability to identify globally optimal solutions consistently. These drawbacks restrict SSA’s ability to solve highly complex optimization tasks effectively.

Some recently developed metaheuristic algorithms still face challenges in achieving optimal performance in global optimization, particularly when applied to high-dimensional or multi-objective problems where the search space becomes increasingly complex¹⁰. These challenges often result in premature convergence to local optima and sluggish progress toward the global solution, primarily due to insufficient exploration mechanisms and limited adaptability. Cleaner production systems, such as wind farm layout optimization, must strike a balance between environmental and economic constraints. However, many algorithms lose efficiency in complex real-world scenarios due to limited solution diversity. In engineering and industrial design, there is a growing demand for adaptable algorithms that minimize computational complexity while maintaining high efficiency. Yet, some optimization methods still struggle to deliver scalable and robust solutions when faced with tightly constrained design problems. Furthermore, metaheuristic algorithms demonstrate strong performance in solving specific problems but lack universal applicability. The No Free Lunch (NFL) theorem⁵⁸ asserts that no single algorithm consistently outperforms all others across all optimization scenarios, emphasizing the necessity for careful algorithm selection and problem-specific modifications to achieve optimal results. The NFL theorem further highlights that all optimization algorithms, including SSA, have inherent trade-offs, meaning their performance is problem-dependent. SSA’s effectiveness varies based on the characteristics of the optimization task, often requiring refinements and adaptations to address its limitations.

Addressing these challenges requires advanced search strategies that enhance exploration, adaptability, and scalability across diverse applications. Therefore, the challenges discussed above motivate this study to propose an evolutionary salp swarm algorithm (ESSA) for solving global optimization, wind frame layout, and complex engineering design problems. The proposed ESSA incorporates evolutionary search strategies and an enhanced SSA search strategy. It also features two best and inferior archives, utilizing a stochastic universal selection method to manage the archive size by selecting individuals according to their fitness values. The ESSA’s performance was evaluated using test suites from the IEEE Congress on Evolutionary Computation 2017 (IEEE CEC 2017)⁵⁹. Its results were compared against several state-of-the-art optimization algorithms, including the salp swarm algorithm (SSA)¹⁸, quantum-based avian navigation optimizer algorithm (QANA)⁴⁰, sinh cosh optimizer (SCHO)⁶⁰, starling murmuration optimizer (SMO)¹⁹, parrot optimizer (PO)⁶¹, human evolutionary optimization algorithm (HEOA)⁶², and adaptive differential evolution (ADE)³⁵. The results indicate that ESSA significantly outperforms SSA and other optimization algorithms in terms of both local and global search efficiency, as well as in avoiding local optima and achieving a faster convergence speed. Statistical analyses confirm that ESSA ranks first in all test functions and achieves the best optimization effectiveness, with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively, in performance compared to competing algorithms. Furthermore, ESSA’s applicability is evident in optimizing the wind farm layout problem and addressing engineering design challenges, as evaluated using benchmarks from IEEE CEC 2020⁶³. The contributions of this study are summarized in three key points:

1. Proposing ESSA to elevate solution quality through more efficient exploration and exploitation of the search space. ESSA synergistically integrates two advanced evolutionary search strategies alongside an enhanced SSA search strategy, ensuring a balanced and adaptive approach to both global and local optimization tasks. Moreover, it features an advanced memory mechanism that maintains two distinct solution archives, leveraging the collective experience of the population to foster diversity and convergence. The memory uses the stochastic universal selection (SUS) method to manage archive size efficiently.

2. Conducting extensive experimental evaluations of ESSA demonstrated its effectiveness in solving complex global optimization problems, outperforming seven state-of-the-art contender optimizers, including SSA, QANA, SCHO, SMO, PO, HEOA, and ADE across benchmark functions with dimensions of 30, 50, and 100. The analysis confirmed its strong local and global search strategies, robust avoidance of local optima, rapid convergence rate, top ranking in statistical analyses, and overall higher optimization effectiveness compared to other contender optimizers.

3. Evaluating the applicability of the proposed ESSA through experiments in solving wind farm layout and engineering design problems demonstrated its higher efficiency over contender optimizers.

This paper is structured as follows. The Related Work section reviews existing literature and recent SSA variants. The Salp Swarm Algorithm (SSA) section outlines the mathematical foundations of SSA. The Evolutionary Salp Swarm Algorithm (ESSA) section introduces the proposed improvements. The Experimental Evaluations in Solving Global Optimization Problems section presents benchmark results comparing ESSA with other state-of-the-art algorithms. The Experimental Evaluations in Optimizing Wind Farm Layout and Engineering Designs section applies the algorithm to practical design problems. The discussion section interprets the findings, and the Conclusion section summarizes the study and suggests future research directions.

Relate works

Although SSA performs well in specific optimization scenarios, its inability to maintain a proper balance between exploration and exploitation often results in premature convergence to suboptimal solutions. To address these shortcomings, researchers have developed improved, hybrid, and adapted SSA variants. These modifications focus on refining search strategies, tuning control parameters, and integrating adaptive mechanisms to enhance SSA’s efficiency, accuracy, and robustness in solving complex optimization problems across diverse domains.

The improved SSA variants incorporate innovative search strategies and operators to strengthen the algorithm’s ability to explore promising regions and achieve high-quality solutions. Furthermore, some variants have introduced diverse population distribution techniques, effectively dispersing search agents across the problem space using various approaches. Zhang et al.⁵² introduced the enhanced SSA (ESSA), which incorporates strategies such as orthogonal learning, quadratic interpolation, and generalized oppositional learning to improve global exploration and local exploitation. Orthogonal learning helps the weakest salp escape local optima, quadratic interpolation enhances global accuracy through localized refinement, and generalized oppositional learning optimizes population quality during initialization and transitional phases. Neggaz et al.⁶⁴ enhanced the SSA using the sine cosine algorithm (SCA)⁶⁵ and a disruption operator to tackle feature selection challenges. This adjustment refines follower positions in SSA using a sinusoidal function, thereby enhancing exploration and minimizing the risk of stagnation. Additionally, the disruption operator is incorporated across solutions to boost population diversity, ensuring a well-balanced equilibrium between exploration and exploitation.

El-Ashmawi et al.⁶⁶ introduced a modified salp local refinement heuristic (MSLRH) algorithm to address task assignments and fundamental combinatorial optimization challenges in engineering and scientific fields. The MSLRH algorithm integrates with the local refinement heuristic approach to enhance assignments using various operators. Ateya et al.⁶⁷ offered a chaotic SSA to address the software-defined networking problem. Ren et al.⁶⁸ introduced an adaptive weight and Levy flight for SSA (WLSSA). WLSSA uses adaptive weights to broaden the exploratory range of the traditional SSA in the early stages and accelerate convergence in the later stages. The Lévy flight mechanism, through random walks, improves the balance and effectiveness of global search and local exploitation. Kansal et al.⁶⁹ proposed an emended SSA (ESSA) to solve the multiobjective electric power load dispatch problem. ESSA refines SSA by integrating solitary and colonial reproduction phases to improve convergence and search efficiency. It converts multiobjective optimization into a scalar objective using fuzzy set theory, resolves conflicting objectives via cardinal priority ranking, and applies variable elimination with exterior penalties to ensure compliance with operational constraints. Lin et al.⁵⁵ recommended the double mutational SSA (DMSSA), integrating the cuckoo mutation strategy (CMS) and adaptive DE mutation strategy (ADMS) into the original SSA framework. CMS employs judgment, shuffling, and mutation to enhance information exchange among search agents, ensuring diverse exploration while minimizing randomness. Meanwhile, ADMS utilizes selection, mutation, and adaptation, in which some individuals refine their movement by considering the global optimum and adjusting variation factors adaptively.

Gholami et al.⁷⁰ developed the mutated SSA (MSSA) for allocating active and reactive power sources in radial distribution systems. MSSA employs a mutation mechanism to avoid local minima and enhance population diversity. Chen et al.⁷¹ presented a quantum evolutionary salp swarm algorithm (QSSA) to optimize engineering designs in mechanical systems. QSSA employs an elite opposition-based learning approach to enhance diversity and a wavelet mutation mechanism to prevent individuals from stagnating. Zhao et al.⁷² modeled a chaotic SSA using opposition-based global optimization learning. This algorithm utilizes opposition-based learning to achieve faster convergence. The chaotic local search also enhances the algorithm’s performance in obtaining the global optimal solution. Zhang et al.⁷³ introduced an ensemble/composite mutation strategy (CMS) and a restart strategy (RS) for SSA (CMSRSSSA). CMSRSSSA enhances SSA’s ability to strike a balance between exploration and exploitation, thereby overcoming the limitations of single-search methods in continuous optimization problems. The CMS in CMSRSSSA enhances both exploration and exploitation, while RS prevents salps from becoming trapped in local optima, resulting in a more effective search process. Ahmed et al.⁷⁴ introduced an improved SSA incorporating a Gaussian random walk to enhance performance in solving multidimensional constrained global optimization problems. Additionally, a re-dispersion strategy is implemented to mitigate stagnation at local optima, significantly enhancing exploration capabilities.

Çelik et al.⁷⁵ proposed a modified SSA (mSSA) to tackle global optimization challenges. The mSSA incorporates chaotic parameter adjustment to better balance exploration and exploitation, a mutualistic interaction between two leader salps to improve information exchange and search efficiency, and a randomized follower strategy to increase diversity and independently explore unvisited regions. Selvi et al.⁷⁶ introduced a modified SSA combined with a chaotic coupled map lattices method for the secure encryption and compression of medical images during data transmission. This algorithm improves both security and efficiency in processing medical image data, ensuring robust protection while optimizing performance. Tubishat et al.⁵⁶ developed the dynamic salp swarm algorithm (DSSA) to address feature selection challenges by reducing dimensionality and selecting the optimal subset of variables. DSSA enhances the update mechanism for salp positions using Singer’s chaotic map, promoting a diverse search space and preventing premature convergence. Additionally, it integrates a local search algorithm to refine exploitation, improving overall optimization efficiency and solution accuracy. The other SSA variants are an efficient salp swarm algorithm⁵⁷, modified salp swarm optimization^77,78, Laplacian SSA⁷⁹, chaotic SSA^80,81, improved SSA based on the spiral flight search strategy⁸², fractional SSA⁸³, and quantum SSA⁸⁴.

Some SSA variants are hybridized with optimization algorithms from evolutionary-based, physics-based, human behavior-based, and swarm intelligence-based categories to enhance performance and overcome limitations. These hybrid approaches combine multiple algorithmic strengths, leading to more efficient and robust solutions. Ibrahim et al.⁸⁵ hybridized SSA with PSO (SSAPSO) for feature selection problems. Zhang et al.⁸⁶ presented chaotic and differential evolution SSA (CDESSA) to solve real-world problems. The CDESSA incorporates a differential evolution algorithm and a chaotic initialization method to enhance accuracy and convergence speed during the search process. The chaotic initialization method generates a superior initial population, enabling the algorithm to locate the global optimum more effectively. Concurrently, the differential evolution algorithm enhances each agent’s search capability, striking a balance between global exploration and local intensification. Bairathi et al.⁸⁷ proposed an improved SSA that integrates random opposition-based learning, multiple leadership strategies, and simulated annealing. The results indicate that the improved SSA demonstrates superior effectiveness in handling complex multi-modal problems, offering enhanced optimization performance and robustness.

Kundu et al.⁵⁴ hybridized the SSA with a teaching–learning-based optimization to address reliability redundancy allocation challenges under nonlinear resource constraints. Fan et al.⁸⁸ introduced the genetic SSA (GSSA) with population partitioning to solve numerical optimization problems. It initializes the process with a chaotic dyad-based learning method to create an initial population, reduces dimensionality using population partitioning, and carries out crossover and mutation in the reduced dimension. Finally, GSSA applies three mutation operators to the optimal solution to prevent stagnation in local optima. Zhang et al.⁵³ combined SSA with the Harris eagle foraging strategy. Additionally, it employs a multi-point leadership crossover strategy to improve the effective exploitation of potential areas. Other hybrid SSA variants include the teaching–learning guided SSA⁸⁹, an improved hybrid SSA and Aquila optimizer⁹⁰, and SSA based on a Firefly algorithm⁹¹.

Salp swarm algorithm (SSA)

Inspired by the collective movement of deep-sea salp chains, Mirjalili et al.¹⁸ developed the Salp Swarm Algorithm (SSA) to mimic their cooperative navigation for solving optimization problems. SSA replicates salps’ interactions, utilizing their collective movement principles to tackle complex optimization problems across various fields. SSA begins the first optimization process by distributing N salps in a D-dimensional problem space, as presented in Eq. (1).

1

In SSA, each salp has a position in a multidimensional space. The notation indicates the initial position of the i-th salp in the j-th dimension at the first time step. The position is initialized randomly using a number generated from a uniform distribution between 0 and 1. The variables  and denote the upper and lower limits for the j-th dimension of the i-th salp, where i = 1, 2, …, N and j = 1, 2, …, D. After initialization, the fitness values of the salps’ positions are computed. Based on their indices, SSA then divides the population into two distinct groups. Individuals with indices less than or equal to half the total number of salps () are designated as leaders. The remaining individuals () are considered followers and update their positions based on the movements of the leaders.

For salps with indices from 1 to , the updated position in the j-th dimension is determined using Eqs. (2) and (3). In this formulation, F_j (t) represents the location of the food source in the j-th dimension, derived from the best position found by the swarm so far. Parameter c₁(t) is the most significant in the SSA, as it helps balance exploration and exploitation, defined using Eq. (4), where t and MaxIt represent the current iteration and the maximum number of iterations, respectively. Two control parameters, c₂(t), and c₃(t), are each drawn randomly from the interval 0, and 1.

2

3

4

For , the next position of the followers is computed using Eq. (5), where and denotes the current and previous salp positions at time t, respectively.

5

Equation (6) shows the method used to bring the salps back into the search area when they venture outside the search space.

6

The SSA pseudocode is presented in Algorithm 1. Initially, the salps are positioned within a D-dimensional problem space using Eq. (1). Subsequently, their fitness values are evaluated, and the location of the top-performing salp is designated as the food source. Then, the salps collaborate to optimize their positions for predetermined iterations. The position of the salps for the indices where  is determined using Eqs. (2) and (3). In contrast, the positions of the salps for indices where are computed using Eq. (5). At the end of each iteration, the upper and lower bounds of the positions are checked. Any violated positions are corrected using Eq. (6). Throughout the iterations, the quality of the positions is evaluated, and the best position found during this process is returned as the final solution.

Algorithm 1.

The pseudocode of SSA.

Evolutionary salp swarm algorithm (ESSA)

SSA employs three search strategies to explore and exploit the problem space effectively. A conditional statement evaluates the value of c₃ to determine how to randomly reinitialize part of the population toward the food position when c₃ < 0.5 and away from the food position when c₃ ≥ 0.5. Using coefficients c₁ and c₂, along with defined bounds, facilitates a more diverse exploration of the optimization landscape. For the latter half of the population, movement is achieved by averaging the positions of each salp with that of its predecessor. It is expected that this mechanism helps maintain diversity in the vicinity of the best solution. However, reports from SSA variants indicate that the SSA struggles with an inefficient search mechanism. This results in several weaknesses, such as insufficient exploration and exploitation, as well as a lack of diversity in solving various complex problems^69–71,81. Therefore, this section presents a new SSA variant, the evolutionary salp swarm algorithm (ESSA), designed to enhance solution quality by addressing these weaknesses.

The proposed ESSA introduces innovative search strategies, including an evolutionary search approach composed of two distinct evolutionary search strategies and an enhanced SSA search strategy, which features a single search strategy. Additionally, it employs two specialized archives—best and inferior—enhanced with a stochastic universal selection method to effectively regulate the archive by selecting individuals according to their fitness values.

Let the salp population  be the matrix containing the positions of all salps vectors, where is the position of the i-th salp. The corresponding fitness values for the matrix are represented by the vector , where is the fitness value of . The finest solution found in each iteration is stored in the vector and .

Evolutionary SSA search strategies

The proposed ESSA utilizes two evolutionary search strategies to enhance the SSA’s performance in identifying promising areas. For the portion of the salp population with indices lower than half of the total population , ESSA directs movement using Eq. (7)³⁹ when coefficient c₃ < 0.5 and Eq. (8) when coefficient c₃ ≥ 0.5. In Eq. (7), vector is the current position, is a value computed using a Cauchy distribution, is a randomly chosen position from , is a randomly chosen position from the archive based on Definition 1 and is the i-th best solution from the matrix .

Let matrix is the sorted version of the matrix , ordered by their smallest fitness values . The best  vector solutions from are considered as the matrix .

Equation (8) is a rand/1/bin search strategy²³, where , and are distinct randomly selected vectors from and F is a constant value which is specified in Table 1.

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Table 1.

Initial parameters of the proposed ESSA and the other contender optimizers.

Algorithm	Parameters
SSA18	c2 and c3 are uniformly distributed random numbers in the open interval (0, 1)
QANA40	The number of flocks (k) = 10. K′ = 9, and K′′ = 50
SCHO60	u = 0.388, m = 0.45, n = 0.5, p = 10, q = 9, α = 4.6, β = 1.55, ct = 3.6, BSi = 0, and r1 to r5 are random numbers in the open interval (0, 1)
SMO19	The number of flocks is 10, λ = 20, µ = 0.5, and θ, and ϕ belong to the interval (0, 1.8]
PO61	H is a random number in the open interval (0, 1), v is a random number from a standard normal distribution, which has a mean of 0 and a variance of 1, and β = 1.5
HEOA62	A = 0.6, LN = 0.4, EN = 0.4, FN = 0.1, R is a random number in the open interval (0, 1), v is a random number from a standard normal distribution, which has a mean of 0, and a variance of 1, and β = 1.5
ADE35	NPopmin = 4, NPopinit = 28, βmin = 0.2, βmax = 0.8, pCR = 0.2, µ = 4, and β2 is the random numbers from a continuous uniform distribution
ESSA	The maximum size of the archives (k) is 700, c is a random value in the open interval (0, 1), and F = [1.0, 1.0, 0.8]

An enhanced SSA search strategy

The proposed ESSA applies this search strategy on the last half of the salp population, precisely when the i-th salp satisfies . The enhanced SSA search strategy is calculated using Eq. (9), where is a random position from the best archive solution, as defined in Definition 1, and denotes the random positions of salps indexed as .

9

Advanced memory mechanism

The proposed ESSA introduces an advanced memory mechanism for information storage, as defined in Definition 1.

Definition 1 (ESSA archives)

The proposed ESSA utilizes two archives, referred to as the best archive () and inferior archive (). Let matrix as the best archive with a finite size that store the near-optimal solutions found by the entire population at each iteration, as represented in Eq. (10), where  represents the finest solution originate by the N salps at iteration t, and the value of k is specified in Table 1. The is updated to retain the  best solutions. If the size of the archive exceeds , stochastic universal selection method, as presented in Algorithm 2, is applied to maintain the archive size such that .

10

Let matrix as the inferior archive stores the positions of salps that cannot be improved during each iteration. It is defined mathematically using Eq. (11) for a minimization problem. If the archive’s size surpasses , The stochastic universal selection method, illustrated in Algorithm 2, ensures that the archive size remains within the limit.

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Stochastic universal selection (SUS) method

The SUS method is a selection technique used in genetic algorithms that chooses individuals based on their fitness values, ensuring proportional representation in the selection process^92,93. This method is beneficial as it decreases randomness during selection and encourages greater diversity among the individuals chosen. The SUS method pseudocode is provided in Algorithm 2, assuming that lower fitness values are preferable. As indicated in the algorithm, the first step involves calculating the total fitness  of the salps using Eq. (12), where  represents the fitness value of individual i and N stands for the total number of individuals.

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Next, the selection probabilities are calculated using Eq. (13), which normalizes the inverse fitness values.

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Following this, the cumulative probabilities for each individual are determined using Eq. (14).

14

Then, the distance between pointers is calculated using Eq. (15), with k indicates the number of individuals to be selected. A random starting point r is then generated within the range [0, d), as calculated in Eq. (16) and the positions of the pointers are defined as {r, r + d, r + 2 × d, …, r + (n − 1) × d}.

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16

For each pointer, compute the using Eq. (17). Then, compare this value with the , which are computed using Eq. (14) to determine if they are greater than or equal to the . Finally, identify the individual corresponding to the current pointer and remove that individual from the archive.

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Algorithm 2.

The pseudocode of the stochastic universal selection method.

Greedy selection

The ESSA employs a greedy selection strategy, as indicated in Eq. (18), by evaluating the fitness value of a solution with that of its previous position . This indicates that if the solution’s fitness is improved over that of the previous solution, the solution is accepted; otherwise, the last solution is retained.

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Parameter is adaptively updated according to the Reference³⁹. Algorithm 3 depicts the pseudocode for ESSA.

The flowchart of the proposed ESSA is presented in Fig. 1.

Fig. 1.

The flowchart of the proposed ESSA.

Algorithm 3.

The pseudocode of ESSA.

Computational complexity of ESSA

As demonstrated in Algorithms 2 and 3, the computational complexity of the proposed ESSA is structured into three primary phases: initialization, movement, and advanced memory mechanism.

In the initialization phase (Line 1), the salp population is distributed within the D-dimensional problem space using Eq. (1), with a time complexity of O (N × D). Their fitness values are then computed (Line 2), also with a complexity of O (N × D). As a result, the total complexity of the initialization phase is O (N × D) + O (N × D) = O (N × D). The movement phase (Lines 4–26) updates the positions of salps based on specific equations and conditions over a predefined number of iterations, MaxIt. The best salp in the population is assigned as the food source (Line 5), with a time complexity of O(N). Next, the parameter c₁ is updated (Line 6) using Eq. (4), with a complexity of O (1). The algorithm runs through N salps (Lines 7–20). If (Line 8), a condition is checked with complexity O(N). For half of the population, a random value c₃ is generated within the open interval [0, 1] (Line 10), with a complexity of O (N × D). The condition c₃ < 0.5 (Line 11) is checked with a complexity of O (N/2). If true, the salp’s position is updated using Eq. (7) (Line 12), with a complexity of O (N/2 × D). Otherwise, if c₃ ≥ 0.5 (Line 13), the condition is checked with a complexity of O (N/2), and the salp’s position is updated using Eq. (8) (Line 14), with a complexity of O (N/2 × D). For the remaining half of the population (N/2 < i ≤ N, Line 17), the condition is checked with a complexity of O (N/2), and the salp’s position is updated using Eq. (9) (Line 18), with a time complexity of O (N/2 × D). Equation (6) is then applied to check and adjust the positions of salps that have violated the valid range (Line 21) with a time complexity of O (N × D). Afterward, the new fitness values are calculated (Line 22), with a time complexity of O (N × D). The greedy selection method, as described in Eq. (18) (Line 23), also has a time complexity of O(N × D). Thus, the worst-case computational complexity of the movement phase is O (MaxIt × (N × D)).

The advanced memory mechanisms phase is performed using Algorithm 2 (Line 24), with a time complexity of O (MaxIt × k × D), where k is the archive size. Combining all three phases, the total computational complexity of ESSA is O (N × D) + O (MaxIt × (N × D + k × D)). Thus, the overall computational complexity of ESSA is O(MaxIt × (N × D + k × D)), which simplifies to O(MaxIt × N × D) when the archive size k is small relative to the population size N.

Experimental evaluations in solving global optimization problems

This section outlines the experimental setup, evaluation procedures, and supporting analyses and comparisons used to demonstrate the performance and advantages of the proposed algorithm. All evaluations were performed under identical software and hardware conditions to ensure consistency and enable fair comparisons with competing optimization algorithms. All algorithms were implemented 20 times on a laptop with a 13th Gen Intel® Core™ i9-13900H processor running at 2.60 GHz and a 64-bit operating system to ensure a fair comparison. The MATLAB R2024a programming language was utilized for execution. In this comparison, the IEEE CEC 2017 benchmark functions⁵⁹, consisting of 29 test functions, were considered at different dimensional sizes of 30, 50, and 100. The test suite includes unimodal, multimodal, hybrid, and composite functions, designed to evaluate algorithm performance in terms of local and global search capabilities, avoidance of local optima, and convergence speed. The proposed algorithm was compared with contender optimizers, including the salp swarm algorithm (SSA)¹⁸, quantum-based avian navigation optimizer algorithm (QANA)⁴⁰, sinh cosh optimizer (SCHO)⁶⁰, starling murmuration optimizer (SMO)¹⁹, parrot optimizer (PO)⁶¹, human evolutionary optimization algorithm (HEOA)⁶², and adaptive differential evolution (ADE)³⁵. Table 1 outlines the initial parameter settings for ESSA and the competing optimizers. The primary parameter values for each algorithm align with the recommendations provided in their respective original publications.

The proposed ESSA is validated through a comprehensive sensitivity analysis aimed at identifying the optimal values for its key parameters. The results of this analysis are presented in Tables 2, 3, 4, and 5, where each table summarizes performance metrics, including the average value (Avg), standard deviation (Std), and the best result (Min), across multiple runs. The best-performing outcomes are highlighted in bold, and a comparative summary of Wins (W), Ties (T), and Losses (L) for each algorithm is included in the final row of each table. Following this, ESSA is benchmarked against competing algorithms across varying problem dimensions—30, 50, and 100—to evaluate its robustness and adaptability. These comparison results are reported in Tables 6, 7, 8, 9, 10, and 11. To determine the statistical significance of performance differences, the Friedman test is applied, with results shown in Tables 12, 13, and 14. In addition, Table 15 presents the mean absolute error (MAE) values as another indicator of solution quality. Finally, Table 16 presents the overall effectiveness of optimization, providing a comprehensive view of ESSA’s capabilities across global optimization.

Table 2.

Impact of memory size on ESSA performance with fixed 3000 iterations, N = 100, and D = 30.

Functions	Statistical metrics	k = 300	k = 500	k = 700	Functions	Statistical metrics	k = 300	k = 500	k = 700
F1	Mean	1.0000E + 02	1.0000E + 02	1.0000E + 02	F17	Mean	1.7580E + 03	1.7595E + 03	1.7500E + 03
	Std	0.0000E + 00	0.0000E + 00	0.0000E + 00		Std	1.8187E + 01	2.3937E + 01	1.3260E + 01
	Min	1.0000E + 02	1.0000E + 02	1.0000E + 02		Min	1.7389E + 03	1.7420E + 03	1.7210E + 03
F3	Mean	3.8331E + 03	3.0000E + 02	3.0000E + 02	F18	Mean	1.8587E + 03	1.8756E + 03	1.8670E + 03
	Std	1.5801E + 04	0.0000E + 00	0.0000E + 00		Std	3.2374E + 01	4.7391E + 01	3.9410E + 01
	Min	3.0000E + 02	3.0000E + 02	3.0000E + 02		Min	1.8251E + 03	1.8268E + 03	1.8280E + 03
F4	Mean	4.0239E + 02	4.0601E + 02	4.1460E + 02	F19	Mean	1.9445E + 03	1.9545E + 03	1.9460E + 03
	Std	2.0038E + 00	1.4687E + 01	2.5140E + 01		Std	2.2919E + 01	3.3285E + 01	2.9670E + 01
	Min	4.0000E + 02	4.0000E + 02	4.0000E + 02		Min	1.9152E + 03	1.9106E + 03	1.9120E + 03
F5	Mean	5.1769E + 02	5.1520E + 02	5.1570E + 02	F20	Mean	2.1153E + 03	2.0899E + 03	2.0960E + 03
	Std	3.1142E + 00	3.4997E + 00	3.1330E + 00		Std	4.8032E + 01	4.7726E + 01	5.1930E + 01
	Min	5.1200E + 02	5.0968E + 02	5.0820E + 02		Min	2.0459E + 03	2.0378E + 03	2.0310E + 03
F6	Mean	6.0000E + 02	6.0000E + 02	6.0000E + 02	F21	Mean	2.3167E + 03	2.3174E + 03	2.3170E + 03
	Std	9.6000E-03	1.0000E-03	2.4000E-03		Std	2.7613E + 00	4.9445E + 00	2.9390E + 00
	Min	6.0000E + 02	6.0000E + 02	6.0000E + 02		Min	2.3125E + 03	2.3109E + 03	2.3100E + 03
F7	Mean	7.4755E + 02	7.4595E + 02	7.4540E + 02	F22	Mean	2.3001E + 03	2.3000E + 03	2.3000E + 03
	Std	2.7979E + 00	3.1491E + 00	3.1890E + 00		Std	5.4860E-01	0.0000E + 00	0.0000E + 00
	Min	7.4279E + 02	7.4001E + 02	7.4040E + 02		Min	2.3000E + 03	2.3000E + 03	2.3000E + 03
F8	Mean	8.1915E + 02	8.1767E + 02	8.1700E + 02	F23	Mean	2.6611E + 03	2.6634E + 03	2.6650E + 03
	Std	3.6852E + 00	3.1481E + 00	2.6530E + 00		Std	6.4259E + 00	6.2819E + 00	7.2590E + 00
	Min	8.1487E + 02	8.1285E + 02	8.1250E + 02		Min	2.6522E + 03	2.6515E + 03	2.6560E + 03
F9	Mean	9.0036E + 02	9.0024E + 02	9.0030E + 02	F24	Mean	2.8359E + 03	2.8363E + 03	2.8370E + 03
	Std	8.2370E-01	2.9970E-01	3.2300E-01		Std	4.8945E + 00	4.2428E + 00	5.0220E + 00
	Min	9.0000E + 02	9.0000E + 02	9.0000E + 02		Min	2.8297E + 03	2.8272E + 03	2.8300E + 03
F10	Mean	3.0436E + 03	3.0333E + 03	3.0240E + 03	F25	Mean	2.8871E + 03	2.8871E + 03	2.8870E + 03
	Std	1.7184E + 02	2.1386E + 02	1.8030E + 02		Std	4.3390E-01	4.5160E-01	3.2180E-01
	Min	2.6880E + 03	2.6004E + 03	2.6970E + 03		Min	2.8868E + 03	2.8868E + 03	2.8870E + 03
F11	Mean	1.1757E + 03	1.1756E + 03	1.1770E + 03	F26	Mean	3.6497E + 03	3.7567E + 03	3.7020E + 03
	Std	3.4689E + 01	2.9249E + 01	2.4190E + 01		Std	2.9640E + 02	1.0266E + 02	2.1870E + 02
	Min	1.1308E + 03	1.1259E + 03	1.1370E + 03		Min	2.8000E + 03	3.6201E + 03	2.8000E + 03
F12	Mean	6.5093E + 03	6.8127E + 03	6.8190E + 03	F27	Mean	3.2128E + 03	3.2116E + 03	3.2110E + 03
	Std	2.5356E + 03	2.8786E + 03	2.0960E + 03		Std	6.1841E + 00	8.2735E + 00	6.2750E + 00
	Min	3.0641E + 03	2.6487E + 03	4.0620E + 03		Min	3.2034E + 03	3.1959E + 03	3.1990E + 03
F13	Mean	1.3392E + 03	1.3322E + 03	1.3380E + 03	F28	Mean	3.1230E + 03	3.1440E + 03	3.1170E + 03
	Std	1.7000E + 01	1.5505E + 01	2.3960E + 01		Std	4.7766E + 01	5.5374E + 01	3.7690E + 01
	Min	1.3128E + 03	1.3114E + 03	1.3140E + 03		Min	3.1000E + 03	3.1000E + 03	3.1000E + 03
F14	Mean	1.4940E + 03	1.4999E + 03	1.4950E + 03	F29	Mean	3.3915E + 03	3.3839E + 03	3.3890E + 03
	Std	2.5934E + 01	3.1152E + 01	1.9530E + 01		Std	3.6887E + 01	3.6318E + 01	5.3430E + 01
	Min	1.4458E + 03	1.4449E + 03	1.4410E + 03		Min	3.3404E + 03	3.3273E + 03	3.3360E + 03
F15	Mean	1.5662E + 03	1.5660E + 03	1.5820E + 03	F30	Mean	5.1520E + 03	5.0933E + 03	5.1190E + 03
	Std	3.7936E + 01	5.8740E + 01	4.3520E + 01		Std	9.4580E + 01	9.1178E + 01	1.2450E + 02
	Min	1.5065E + 03	1.5182E + 03	1.5200E + 03		Min	5.0003E + 03	4.9523E + 03	4.9600E + 03
F16	Mean	2.1062E + 03	1.9825E + 03	2.0510E + 03	Count	W/T/L	9/2/18	9/4/16	7/4/18
	Std	1.4912E + 02	1.4897E + 02	1.6400E + 02	Friedman test		1.982	1.962	2.056
	Min	1.8600E + 03	1.7861E + 03	1.7710E + 03	Optimization effectiveness		34.48%	37.93%	31.03%

Table 3.

Impact of memory size on ESSA performance for D = 30, 50 and 100.

Functions	Statistical metrics	D = 30		D = 50		D = 100
		k = 500	k = 700	k = 500	k = 700	k = 500	k = 700
F1	Mean	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02
	Std	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	4.6000E-03	4.6000E-03
	Min	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02	1.0000E + 02
F3	Mean	3.0000E + 02	3.0000E + 02	1.8495E + 04	3.0000E + 02	5.3128E + 02	5.3130E + 02
	Std	0.0000E + 00	0.0000E + 00	5.6061E + 04	0.0000E + 00	1.6652E + 02	1.6650E + 02
	Min	3.0000E + 02	3.0000E + 02	3.0000E + 02	3.0000E + 02	3.5527E + 02	3.5530E + 02
F4	Mean	4.0601E + 02	4.1460E + 02	4.2879E + 02	4.1460E + 02	5.5762E + 02	5.5760E + 02
	Std	1.4687E + 01	2.5140E + 01	4.8873E + 01	2.5140E + 01	3.7791E + 01	3.7790E + 01
	Min	4.0000E + 02	4.0000E + 02	4.0000E + 02	4.0000E + 02	4.8092E + 02	4.8090E + 02
F5	Mean	5.1520E + 02	5.1570E + 02	5.4297E + 02	5.1570E + 02	6.3878E + 02	6.3880E + 02
	Std	3.4997E + 00	3.1330E + 00	6.2120E + 00	3.1330E + 00	1.3889E + 01	1.3890E + 01
	Min	5.0968E + 02	5.0820E + 02	5.3238E + 02	5.0820E + 02	6.1132E + 02	6.1130E + 02
F6	Mean	6.0000E + 02	6.0000E + 02	6.0002E + 02	6.0000E + 02	6.0054E + 02	6.0050E + 02
	Std	1.0000E-03	2.4000E-03	1.6700E-02	2.4000E-03	3.5640E-01	3.5640E-01
	Min	6.0000E + 02	6.0000E + 02	6.0000E + 02	6.0000E + 02	6.0017E + 02	6.0020E + 02
F7	Mean	7.4595E + 02	7.4540E + 02	7.8954E + 02	7.4540E + 02	9.4202E + 02	9.4200E + 02
	Std	3.1491E + 00	3.1890E + 00	4.4754E + 00	3.1890E + 00	1.6118E + 01	1.6120E + 01
	Min	7.4001E + 02	7.4040E + 02	7.7830E + 02	7.4040E + 02	9.0473E + 02	9.0470E + 02
F8	Mean	8.1767E + 02	8.1700E + 02	8.4588E + 02	8.1700E + 02	9.3952E + 02	9.3950E + 02
	Std	3.1481E + 00	2.6530E + 00	6.5004E + 00	2.6530E + 00	1.4714E + 01	1.4710E + 01
	Min	8.1285E + 02	8.1250E + 02	8.3457E + 02	8.1250E + 02	9.1455E + 02	9.1450E + 02
F9	Mean	9.0024E + 02	9.0030E + 02	9.0494E + 02	9.0030E + 02	9.7127E + 02	9.7130E + 02
	Std	2.9970E-01	3.2300E-01	4.6086E + 00	3.2300E-01	5.5308E + 01	5.5310E + 01
	Min	9.0000E + 02	9.0000E + 02	9.0072E + 02	9.0000E + 02	9.2361E + 02	9.2360E + 02
F10	Mean	3.0333E + 03	3.0240E + 03	4.6858E + 03	3.0240E + 03	1.0995E + 04	1.1000E + 04
	Std	2.1386E + 02	1.8030E + 02	3.1333E + 02	1.8030E + 02	5.4887E + 02	5.4890E + 02
	Min	2.6004E + 03	2.6970E + 03	3.7046E + 03	2.6970E + 03	9.6826E + 03	9.6830E + 03
F11	Mean	1.1756E + 03	1.18E + 03	1.2486E + 03	1.1770E + 03	1.9586E + 03	1.9590E + 03
	Std	2.9249E + 01	2.42E + 01	2.1165E + 01	2.4190E + 01	2.1240E + 02	2.1240E + 02
	Min	1.1259E + 03	1.14E + 03	1.2078E + 03	1.1370E + 03	1.6217E + 03	1.6220E + 03
F12	Mean	6.8127E + 03	6.82E + 03	2.8658E + 04	6.8190E + 03	8.1311E + 04	8.1310E + 04
	Std	2.8786E + 03	2.10E + 03	1.7262E + 04	2.0960E + 03	3.4546E + 04	3.4550E + 04
	Min	2.6487E + 03	4.06E + 03	6.5511E + 03	4.0620E + 03	1.7256E + 04	1.7260E + 04
F13	Mean	1.3322E + 03	1.34E + 03	1.6199E + 03	1.3380E + 03	3.1566E + 03	3.1570E + 03
	Std	1.5505E + 01	2.40E + 01	2.8869E + 02	2.3960E + 01	1.3929E + 03	1.3930E + 03
	Min	1.3114E + 03	1.31E + 03	1.3219E + 03	1.3140E + 03	1.5341E + 03	1.5340E + 03
F14	Mean	1.4999E + 03	1.50E + 03	1.6373E + 03	1.4950E + 03	2.9165E + 03	2.9160E + 03
	Std	3.1152E + 01	1.95E + 01	5.5668E + 01	1.9530E + 01	1.1259E + 03	1.1260E + 03
	Min	1.4449E + 03	1.44E + 03	1.5375E + 03	1.4410E + 03	1.7496E + 03	1.7500E + 03
F15	Mean	1.5660E + 03	1.58E + 03	1.7234E + 03	1.5820E + 03	2.0042E + 03	2.0040E + 03
	Std	5.8740E + 01	4.35E + 01	1.0253E + 02	4.3520E + 01	5.5985E + 02	5.5990E + 02
	Min	1.5182E + 03	1.52E + 03	1.6009E + 03	1.5200E + 03	1.6867E + 03	1.6870E + 03
F16	Mean	1.9825E + 03	2.05E + 03	2.2332E + 03	2.0510E + 03	3.8831E + 03	3.8830E + 03
	Std	1.4897E + 02	1.64E + 02	1.7343E + 02	1.6400E + 02	3.3870E + 02	3.3870E + 02
	Min	1.7861E + 03	1.77E + 03	1.9594E + 03	1.7710E + 03	3.1757E + 03	3.1760E + 03
F17	Mean	1.7595E + 03	1.75E + 03	2.3401E + 03	1.7500E + 03	3.6765E + 03	3.6760E + 03
	Std	2.3937E + 01	1.33E + 01	1.4180E + 02	1.3260E + 01	2.7239E + 02	2.7240E + 02
	Min	1.7420E + 03	1.72E + 03	2.0232E + 03	1.7210E + 03	3.1672E + 03	3.1670E + 03
F18	Mean	1.8756E + 03	1.87E + 03	2.5244E + 03	1.8670E + 03	2.3862E + 04	2.3860E + 04
	Std	4.7391E + 01	3.94E + 01	1.0266E + 03	3.9410E + 01	1.3039E + 04	1.3040E + 04
	Min	1.8268E + 03	1.83E + 03	1.9447E + 03	1.8280E + 03	5.9020E + 03	5.9020E + 03
F19	Mean	1.9545E + 03	1.95E + 03	2.0061E + 03	1.9460E + 03	2.1048E + 03	2.1050E + 03
	Std	3.3285E + 01	2.97E + 01	3.8216E + 01	2.9670E + 01	2.9835E + 01	2.9830E + 01
	Min	1.9106E + 03	1.91E + 03	1.9417E + 03	1.9120E + 03	2.0507E + 03	2.0510E + 03
F20	Mean	2.0899E + 03	2.10E + 03	2.3790E + 03	2.0960E + 03	3.8795E + 03	3.8800E + 03
	Std	4.7726E + 01	5.19E + 01	1.4351E + 02	5.1930E + 01	2.1390E + 02	2.1390E + 02
	Min	2.0378E + 03	2.03E + 03	2.1106E + 03	2.0310E + 03	3.4993E + 03	3.4990E + 03
F21	Mean	2.3174E + 03	2.32E + 03	2.3414E + 03	2.3170E + 03	2.4649E + 03	2.4650E + 03
	Std	4.9445E + 00	2.94E + 00	7.5743E + 00	2.9390E + 00	2.2921E + 01	2.2920E + 01
	Min	2.3109E + 03	2.31E + 03	2.3287E + 03	2.3100E + 03	2.4210E + 03	2.4210E + 03
F22	Mean	2.3000E + 03	2.30E + 03	4.6003E + 03	2.3000E + 03	1.3651E + 04	1.3650E + 04
	Std	0.0000E + 00	0.00E + 00	2.0919E + 03	0.0000E + 00	4.7278E + 02	4.7280E + 02
	Min	2.3000E + 03	2.30E + 03	2.3000E + 03	2.3000E + 03	1.2728E + 04	1.2730E + 04
F23	Mean	2.6634E + 03	2.67E + 03	2.7672E + 03	2.6650E + 03	2.9354E + 03	2.9350E + 03
	Std	6.2819E + 00	7.26E + 00	9.0300E + 00	7.2590E + 00	2.4888E + 01	2.4890E + 01
	Min	2.6515E + 03	2.66E + 03	2.7542E + 03	2.6560E + 03	2.9079E + 03	2.9080E + 03
F24	Mean	2.8363E + 03	2.84E + 03	2.9390E + 03	2.8370E + 03	3.4312E + 03	3.4310E + 03
	Std	4.2428E + 00	5.02E + 00	1.0970E + 01	5.0220E + 00	2.3501E + 01	2.3500E + 01
	Min	2.8272E + 03	2.83E + 03	2.9228E + 03	2.8300E + 03	3.3960E + 03	3.3960E + 03
F25	Mean	2.8871E + 03	2.89E + 03	3.0488E + 03	2.8870E + 03	3.2180E + 03	3.2180E + 03
	Std	4.5160E-01	3.22E-01	3.1399E + 01	3.2180E-01	5.1632E + 01	5.1630E + 01
	Min	2.8868E + 03	2.89E + 03	2.9607E + 03	2.8870E + 03	3.0772E + 03	3.0770E + 03
F26	Mean	3.7567E + 03	3.70E + 03	4.2357E + 03	3.7020E + 03	7.5390E + 03	7.5390E + 03
	Std	1.0266E + 02	2.19E + 02	1.2811E + 02	2.1870E + 02	3.8250E + 02	3.8250E + 02
	Min	3.6201E + 03	2.80E + 03	4.0454E + 03	2.8000E + 03	7.0282E + 03	7.0280E + 03
F27	Mean	3.2116E + 03	3.21E + 03	3.2547E + 03	3.2110E + 03	3.3952E + 03	3.3950E + 03
	Std	8.2735E + 00	6.28E + 00	2.6049E + 01	6.2750E + 00	4.2393E + 01	4.2390E + 01
	Min	3.1959E + 03	3.20E + 03	3.2272E + 03	3.1990E + 03	3.3033E + 03	3.3030E + 03
F28	Mean	3.1440E + 03	3.12E + 03	3.2958E + 03	3.1170E + 03	3.3424E + 03	3.3420E + 03
	Std	5.5374E + 01	3.77E + 01	1.8288E + 01	3.7690E + 01	2.7737E + 01	2.7740E + 01
	Min	3.1000E + 03	3.10E + 03	3.2588E + 03	3.1000E + 03	3.2935E + 03	3.2940E + 03
F29	Mean	3.3839E + 03	3.39E + 03	3.4063E + 03	3.3890E + 03	5.1104E + 03	5.1100E + 03
	Std	3.6318E + 01	5.34E + 01	1.4665E + 02	5.3430E + 01	4.0857E + 02	4.0860E + 02
	Min	3.3273E + 03	3.34E + 03	3.2739E + 03	3.3360E + 03	4.3055E + 03	4.3050E + 03
F30	Mean	5.0933E + 03	5.12E + 03	6.2780E + 05	5.1190E + 03	7.2149E + 03	7.2150E + 03
	Std	9.1178E + 01	1.25E + 02	4.2537E + 04	1.2450E + 02	1.8973E + 03	1.8970E + 03
	Min	4.9523E + 03	4.96E + 03	5.8241E + 05	4.9600E + 03	5.3472E + 03	5.3470E + 03
Friedman value		1.452	1.548	1.506	1.494	1.517	1.483

Table 4.

Impact of population size on ESSA performance with fixed 3000 iterations and D = 30.

Functions	Statistical metrics	N = 50	N = 100	N = 200	Functions	Statistical metrics	N = 50	N = 100	N = 200
F1	Mean	1.0000E + 02	1.0000E + 02	1.0000E + 02	F17	Mean	1.7979E + 03	1.7500E + 03	1.7422E + 03
	Std	0.0000E + 00	0.0000E + 00	0.0000E + 00		Std	5.5808E + 01	1.3260E + 01	7.0937E + 00
	Min	1.0000E + 02	1.0000E + 02	1.0000E + 02		Min	1.7466E + 03	1.7210E + 03	1.7315E + 03
F3	Mean	7.9070E + 03	3.0000E + 02	3.0000E + 02	F18	Mean	2.3367E + 03	1.8670E + 03	1.8674E + 03
	Std	2.3733E + 04	0.0000E + 00	0.0000E + 00		Std	6.4051E + 02	3.9410E + 01	3.6921E + 01
	Min	3.0000E + 02	3.0000E + 02	3.0000E + 02		Min	1.8440E + 03	1.8280E + 03	1.8282E + 03
F4	Mean	4.3580E + 02	4.1460E + 02	4.0279E + 02	F19	Mean	1.9643E + 03	1.9460E + 03	1.9324E + 03
	Std	2.9296E + 01	2.5140E + 01	1.8744E + 00		Std	3.6922E + 01	2.9670E + 01	1.3731E + 01
	Min	4.0006E + 02	4.0000E + 02	4.0000E + 02		Min	1.9182E + 03	1.9120E + 03	1.9141E + 03
F5	Mean	5.2230E + 02	5.1570E + 02	5.1337E + 02	F20	Mean	2.1429E + 03	2.0960E + 03	2.0852E + 03
	Std	4.6732E + 00	3.1330E + 00	2.8616E + 00		Std	5.6431E + 01	5.1930E + 01	4.2327E + 01
	Min	5.1428E + 02	5.0820E + 02	5.0828E + 02		Min	2.0426E + 03	2.0310E + 03	2.0441E + 03
F6	Mean	6.0003E + 02	6.0000E + 02	6.0000E + 02	F21	Mean	2.3251E + 03	2.3170E + 03	2.3120E + 03
	Std	5.9400E-02	2.4000E-03	2.0000E-04		Std	4.3535E + 00	2.9390E + 00	3.0925E + 00
	Min	6.0000E + 02	6.0000E + 02	6.0000E + 02		Min	2.3183E + 03	2.3100E + 03	2.3041E + 03
F7	Mean	7.5344E + 02	7.4540E + 02	7.4141E + 02	F22	Mean	2.3004E + 03	2.3000E + 03	2.3000E + 03
	Std	4.2465E + 00	3.1890E + 00	1.9517E + 00		Std	9.0540E-01	0.0000E + 00	0.0000E + 00
	Min	7.4285E + 02	7.4040E + 02	7.3735E + 02		Min	2.3000E + 03	2.3000E + 03	2.3000E + 03
F8	Mean	8.2430E + 02	8.1700E + 02	8.1537E + 02	F23	Mean	2.6773E + 03	2.6650E + 03	2.6578E + 03
	Std	5.3141E + 00	2.6530E + 00	3.1817E + 00		Std	7.9531E + 00	7.2590E + 00	6.3351E + 00
	Min	8.1334E + 02	8.1250E + 02	8.1008E + 02		Min	2.6607E + 03	2.6560E + 03	2.6445E + 03
F9	Mean	9.0262E + 02	9.0030E + 02	9.0001E + 02	F24	Mean	2.8513E + 03	2.8370E + 03	2.8300E + 03
	Std	2.7669E + 00	3.2300E-01	3.2800E-02		Std	1.2378E + 01	5.0220E + 00	3.6201E + 00
	Min	9.0027E + 02	9.0000E + 02	9.0000E + 02		Min	2.8359E + 03	2.8300E + 03	2.8234E + 03
F10	Mean	3.0317E + 03	3.0240E + 03	2.9801E + 03	F25	Mean	2.8873E + 03	2.8870E + 03	2.8869E + 03
	Std	3.5775E + 02	1.8030E + 02	1.9422E + 02		Std	5.3150E-01	3.2180E-01	8.2100E-02
	Min	2.3067E + 03	2.6970E + 03	2.4229E + 03		Min	2.8868E + 03	2.8870E + 03	2.8868E + 03
F11	Mean	1.1777E + 03	1.1770E + 03	1.1596E + 03	F26	Mean	3.8456E + 03	3.7020E + 03	3.7036E + 03
	Std	3.2584E + 01	2.4190E + 01	3.2894E + 01		Std	2.1868E + 02	2.1870E + 02	5.2294E + 01
	Min	1.1279E + 03	1.1370E + 03	1.1070E + 03		Min	3.0349E + 03	2.8000E + 03	3.5948E + 03
F12	Mean	1.1442E + 04	6.8190E + 03	2.2688E + 03	F27	Mean	3.2171E + 03	3.2110E + 03	3.2118E + 03
	Std	6.4211E + 03	2.0960E + 03	4.0070E + 02		Std	7.1206E + 00	6.2750E + 00	5.5354E + 00
	Min	3.5486E + 03	4.0620E + 03	1.6728E + 03		Min	3.2050E + 03	3.1990E + 03	3.2027E + 03
F13	Mean	1.8369E + 03	1.3380E + 03	1.3243E + 03	F28	Mean	3.1412E + 03	3.1170E + 03	3.1232E + 03
	Std	1.1325E + 03	2.3960E + 01	8.5204E + 00		Std	5.7479E + 01	3.7690E + 01	4.8636E + 01
	Min	1.3434E + 03	1.3140E + 03	1.3050E + 03		Min	3.1000E + 03	3.1000E + 03	3.1000E + 03
F14	Mean	1.5208E + 03	1.4950E + 03	1.4869E + 03	F29	Mean	3.4347E + 03	3.3890E + 03	3.3764E + 03
	Std	4.1934E + 01	1.9530E + 01	1.3684E + 01		Std	8.5322E + 01	5.3430E + 01	4.4070E + 01
	Min	1.4559E + 03	1.4410E + 03	1.4578E + 03		Min	3.3469E + 03	3.3360E + 03	3.3328E + 03
F15	Mean	1.6617E + 03	1.5820E + 03	1.5404E + 03	F30	Mean	5.2718E + 03	5.1190E + 03	5.1020E + 03
	Std	1.3356E + 02	4.3520E + 01	3.9841E + 01		Std	2.0491E + 02	1.2450E + 02	1.1626E + 02
	Min	1.5375E + 03	1.5200E + 03	1.5063E + 03		Min	5.0152E + 03	4.9600E + 03	4.9447E + 03
F16	Mean	2.1181E + 03	2.0510E + 03	1.8886E + 03	Count	W/T/L	0/1/28	3/4/22	20/4/5
	Std	1.2854E + 02	1.6400E + 02	1.8519E + 02	Friedman value		2.706	1.917	1.377
	Min	1.8831E + 03	1.7710E + 03	1.6070E + 03	Optimization effectiveness		1.72%	17.24%	75.86%

Table 5.

Impact of iterations on ESSA performance with a fixed population size N = 100 and D = 30.

Functions	Statistical metrics	MaxIt = 1500	MaxIt = 3000	MaxIt = 4000	Functions	Statistical metrics	MaxIt = 1500	MaxIt = 3000	MaxIt = 4000
F1	Mean	1.0000E + 02	1.0000E + 02	1.0000E + 02	F17	Mean	1.7907E + 03	1.7500E + 03	1.7489E + 03
	Std	0.0000E + 00	0.0000E + 00	0.0000E + 00		Std	3.7563E + 01	1.3260E + 01	2.7990E + 01
	Min	1.0000E + 02	1.0000E + 02	1.0000E + 02		Min	1.7443E + 03	1.7210E + 03	1.7260E + 03
F3	Mean	3.0003E + 02	3.0000E + 02	3.0000E + 02	F18	Mean	1.8965E + 03	1.8670E + 03	1.8669E + 03
	Std	6.0300E-02	0.0000E + 00	0.0000E + 00		Std	7.8058E + 01	3.9410E + 01	4.0282E + 01
	Min	3.0000E + 02	3.0000E + 02	3.0000E + 02		Min	1.8304E + 03	1.8280E + 03	1.8254E + 03
F4	Mean	4.3884E + 02	4.1460E + 02	4.1418E + 02	F19	Mean	1.9594E + 03	1.9460E + 03	1.9501E + 03
	Std	3.0185E + 01	2.5140E + 01	2.3565E + 01		Std	2.6120E + 01	2.9670E + 01	2.9451E + 01
	Min	4.0002E + 02	4.0000E + 02	4.0000E + 02		Min	1.9177E + 03	1.9120E + 03	1.9104E + 03
F5	Mean	5.2052E + 02	5.1570E + 02	5.1510E + 02	F20	Mean	2.1600E + 03	2.0960E + 03	2.0891E + 03
	Std	3.8360E + 00	3.1330E + 00	3.8462E + 00		Std	4.3728E + 01	5.1930E + 01	5.1232E + 01
	Min	5.1449E + 02	5.0820E + 02	5.0437E + 02		Min	2.0644E + 03	2.0310E + 03	2.0357E + 03
F6	Mean	6.0000E + 02	6.0000E + 02	6.0000E + 02	F21	Mean	2.3228E + 03	2.3170E + 03	2.3168E + 03
	Std	1.0000E-03	2.4000E-03	4.7000E-03		Std	5.3964E + 00	2.9390E + 00	6.1408E + 00
	Min	6.0000E + 02	6.0000E + 02	6.0000E + 02		Min	2.3096E + 03	2.3100E + 03	2.3111E + 03
F7	Mean	7.5473E + 02	7.4540E + 02	7.4377E + 02	F22	Mean	2.3001E + 03	2.3000E + 03	2.3002E + 03
	Std	4.3721E + 00	3.1890E + 00	1.8735E + 00		Std	5.4860E-01	0.0000E + 00	7.6250E-01
	Min	7.4655E + 02	7.4040E + 02	7.4039E + 02		Min	2.3000E + 03	2.3000E + 03	2.3000E + 03
F8	Mean	8.2502E + 02	8.1700E + 02	8.1649E + 02	F23	Mean	2.6733E + 03	2.6650E + 03	2.6661E + 03
	Std	3.8065E + 00	2.6530E + 00	3.8254E + 00		Std	4.7709E + 00	7.2590E + 00	6.5676E + 00
	Min	8.1576E + 02	8.1250E + 02	8.0783E + 02		Min	2.6623E + 03	2.6560E + 03	2.6566E + 03
F9	Mean	9.0043E + 02	9.0030E + 02	9.0021E + 02	F24	Mean	2.8399E + 03	2.8370E + 03	2.8368E + 03
	Std	8.0480E-01	3.2300E-01	2.2060E-01		Std	4.1752E + 00	5.0220E + 00	4.0731E + 00
	Min	9.0000E + 02	9.0000E + 02	9.0000E + 02		Min	2.8330E + 03	2.8300E + 03	2.8274E + 03
F10	Mean	3.6293E + 03	3.0240E + 03	2.8378E + 03	F25	Mean	2.8870E + 03	2.8870E + 03	2.8871E + 03
	Std	2.9106E + 02	1.8030E + 02	2.2198E + 02		Std	1.2480E-01	3.2180E-01	3.0570E-01
	Min	2.8930E + 03	2.6970E + 03	2.4613E + 03		Min	2.8868E + 03	2.8870E + 03	2.8868E + 03
F11	Mean	1.1660E + 03	1.1770E + 03	1.1764E + 03	F26	Mean	3.8072E + 03	3.7020E + 03	3.6404E + 03
	Std	2.1322E + 01	2.4190E + 01	2.0506E + 01		Std	9.9954E + 01	2.1870E + 02	2.9294E + 02
	Min	1.1358E + 03	1.1370E + 03	1.1388E + 03		Min	3.5771E + 03	2.8000E + 03	2.8000E + 03
F12	Mean	1.0358E + 04	6.8190E + 03	6.1972E + 03	F27	Mean	3.2117E + 03	3.2110E + 03	3.2128E + 03
	Std	3.7190E + 03	2.0960E + 03	3.0785E + 03		Std	6.5937E + 00	6.2750E + 00	7.1323E + 00
	Min	5.5689E + 03	4.0620E + 03	2.1780E + 03		Min	3.1956E + 03	3.1990E + 03	3.1979E + 03
F13	Mean	1.3631E + 03	1.3380E + 03	1.3420E + 03	F28	Mean	3.1487E + 03	3.1170E + 03	3.1416E + 03
	Std	2.7653E + 01	2.3960E + 01	2.0283E + 01		Std	5.1520E + 01	3.7690E + 01	5.9692E + 01
	Min	1.3283E + 03	1.3140E + 03	1.3159E + 03		Min	3.1000E + 03	3.1000E + 03	3.1000E + 03
F14	Mean	1.4922E + 03	1.4950E + 03	1.4991E + 03	F29	Mean	3.3870E + 03	3.3890E + 03	3.3852E + 03
	Std	2.7645E + 01	1.9530E + 01	2.5795E + 01		Std	3.5944E + 01	5.3430E + 01	4.9317E + 01
	Min	1.4280E + 03	1.4410E + 03	1.4389E + 03		Min	3.3150E + 03	3.3360E + 03	3.3532E + 03
F15	Mean	1.5725E + 03	1.5820E + 03	1.5803E + 03	F30	Mean	5.1497E + 03	5.1190E + 03	5.1049E + 03
	Std	4.2617E + 01	4.3520E + 01	4.7115E + 01		Std	1.2214E + 02	1.2450E + 02	1.2532E + 02
	Min	1.5133E + 03	1.5200E + 03	1.5190E + 03		Min	4.9897E + 03	4.9600E + 03	4.9741E + 03
F16	Mean	2.1951E + 03	2.0510E + 03	2.0285E + 03	Count	W/T/L	3/3/23	7/4/18	15/3/11
	Std	1.3013E + 02	1.6400E + 02	1.0684E + 02	Friedman test		2.470	1.803	1.728
	Min	1.9737E + 03	1.7710E + 03	1.8694E + 03	Optimization effectiveness		15.52%	31.03%	56.90%

Table 6.

Impact analysis of each improvement strategy via ablation experiments (D = 30).

Functions	Statistical metrics	SSA	ESSA-V1	ESSA-V2	ESSA-V3	ESSA-V4	ESSA
F1	Mean	3.0140E + 03	2.3583E + 03	1.0000E + 02	1.4958E + 10	1.2749E + 10	1.0000E + 02
	Std	3.8420E + 03	1.5916E + 03	0.0000E + 00	3.7590E + 09	2.9102E + 09	0.0000E + 00
	Min	1.0000E + 02	3.8319E + 02	1.0000E + 02	7.7783E + 09	6.2146E + 09	1.0000E + 02
F3	Mean	3.0000E + 02	8.3893E + 03	3.0000E + 02	5.9584E + 04	5.2961E + 04	3.0000E + 02
	Std	0.0000E + 00	1.8455E + 04	1.0000E-04	1.6281E + 04	1.0233E + 04	0.0000E + 00
	Min	3.0000E + 02	3.0000E + 02	3.0000E + 02	3.1986E + 04	4.0848E + 04	3.0000E + 02
F4	Mean	4.9080E + 02	4.8749E + 02	4.5318E + 02	2.4799E + 03	2.0151E + 03	4.1460E + 02
	Std	2.0750E + 01	1.9645E + 01	1.7562E + 01	6.3475E + 02	5.2337E + 02	2.5140E + 01
	Min	4.2010E + 02	4.3919E + 02	4.0002E + 02	1.3193E + 03	1.4254E + 03	4.0000E + 02
F5	Mean	6.3210E + 02	5.2571E + 02	5.3304E + 02	8.0991E + 02	7.8479E + 02	5.1570E + 02
	Std	3.1480E + 01	3.6028E + 00	5.4476E + 00	2.9117E + 01	1.5876E + 01	3.1330E + 00
	Min	5.9150E + 02	5.1969E + 02	5.2079E + 02	7.5751E + 02	7.5881E + 02	5.0820E + 02
F6	Mean	6.3380E + 02	6.0000E + 02	6.0000E + 02	6.5285E + 02	6.4308E + 02	6.0000E + 02
	Std	1.3990E + 01	0.0000E + 00	8.0000E-04	4.5597E + 00	6.8004E + 00	2.4000E-03
	Min	6.1320E + 02	6.0000E + 02	6.0000E + 02	6.4300E + 02	6.3181E + 02	6.0000E + 02
F7	Mean	8.6840E + 02	7.6568E + 02	7.6596E + 02	1.2546E + 03	1.2105E + 03	7.4540E + 02
	Std	4.6380E + 01	6.9724E + 00	3.5772E + 00	6.1656E + 01	8.2063E + 01	3.1890E + 00
	Min	7.8930E + 02	7.4726E + 02	7.5746E + 02	1.1753E + 03	1.0644E + 03	7.4040E + 02
F8	Mean	9.1780E + 02	8.2603E + 02	8.3406E + 02	1.0776E + 03	1.0692E + 03	8.1700E + 02
	Std	3.5380E + 01	4.3909E + 00	5.3479E + 00	1.6530E + 01	2.1503E + 01	2.6530E + 00
	Min	8.5170E + 02	8.1668E + 02	8.2565E + 02	1.0329E + 03	1.0124E + 03	8.1250E + 02
F9	Mean	4.2140E + 03	9.3128E + 02	9.0162E + 02	6.8180E + 03	6.0287E + 03	9.0030E + 02
	Std	1.6880E + 03	2.1881E + 01	4.6399E + 00	1.4229E + 03	1.6407E + 03	3.2300E-01
	Min	1.4210E + 03	9.0841E + 02	9.0000E + 02	3.3609E + 03	3.8294E + 03	9.0000E + 02
F10	Mean	4.6050E + 03	3.1623E + 03	2.9216E + 03	8.0367E + 03	7.9439E + 03	3.0240E + 03
	Std	6.1430E + 02	2.7428E + 02	2.2140E + 02	3.3605E + 02	3.1116E + 02	1.8030E + 02
	Min	3.4260E + 03	2.5504E + 03	2.5458E + 03	7.3125E + 03	7.3130E + 03	2.6970E + 03
F11	Mean	1.2800E + 03	1.1932E + 03	1.1788E + 03	2.3084E + 03	2.2788E + 03	1.1770E + 03
	Std	6.3480E + 01	3.8715E + 01	3.0724E + 01	3.6597E + 02	4.0836E + 02	2.4190E + 01
	Min	1.1750E + 03	1.1408E + 03	1.1179E + 03	1.7851E + 03	1.6037E + 03	1.1370E + 03
F12	Mean	3.1620E + 06	7.8727E + 05	9.8142E + 03	1.0213E + 09	8.4076E + 08	6.8190E + 03
	Std	3.2150E + 06	3.9997E + 05	6.5424E + 03	3.2861E + 08	3.6971E + 08	2.0960E + 03
	Min	1.8170E + 05	1.6849E + 05	2.6505E + 03	4.5803E + 08	3.0158E + 08	4.0620E + 03
F13	Mean	1.0950E + 05	1.0477E + 04	1.3872E + 03	1.2642E + 08	9.8543E + 07	1.3380E + 03
	Std	5.8350E + 04	4.2560E + 03	4.5628E + 01	6.3875E + 07	6.4119E + 07	2.3960E + 01
	Min	3.7090E + 04	3.5396E + 03	1.3454E + 03	4.3906E + 07	1.8188E + 07	1.3140E + 03
F14	Mean	6.5370E + 03	3.0135E + 03	1.4558E + 03	1.6365E + 05	1.1486E + 05	1.4950E + 03
	Std	4.2090E + 03	2.2258E + 03	1.9236E + 01	1.1812E + 05	7.5057E + 04	1.9530E + 01
	Min	1.8390E + 03	1.5342E + 03	1.4329E + 03	2.1748E + 04	1.9263E + 04	1.4410E + 03
F15	Mean	5.8610E + 04	2.7268E + 03	1.5613E + 03	5.3832E + 06	4.4630E + 06	1.5820E + 03
	Std	6.0300E + 04	1.2068E + 03	3.1546E + 01	5.4773E + 06	6.0010E + 06	4.3520E + 01
	Min	6.1890E + 03	1.5742E + 03	1.5184E + 03	1.6824E + 05	6.2938E + 05	1.5200E + 03
F16	Mean	2.4970E + 03	2.1594E + 03	2.1151E + 03	3.8978E + 03	3.6934E + 03	2.0510E + 03
	Std	2.5850E + 02	1.5363E + 02	1.1995E + 02	1.8486E + 02	2.4422E + 02	1.6400E + 02
	Min	2.1580E + 03	1.9173E + 03	1.9346E + 03	3.6041E + 03	3.2034E + 03	1.7710E + 03
F17	Mean	2.0180E + 03	1.7862E + 03	1.7730E + 03	2.4829E + 03	2.4611E + 03	1.7500E + 03
	Std	1.4060E + 02	4.2955E + 01	3.0914E + 01	1.6248E + 02	1.9847E + 02	1.3260E + 01
	Min	1.8470E + 03	1.7462E + 03	1.7349E + 03	2.2395E + 03	2.0246E + 03	1.7210E + 03
F18	Mean	1.6120E + 05	1.0122E + 05	1.9069E + 03	2.0074E + 06	2.0369E + 06	1.8670E + 03
	Std	1.1180E + 05	5.5760E + 04	1.0543E + 02	1.3980E + 06	1.5041E + 06	3.9410E + 01
	Min	3.7030E + 04	2.2724E + 04	1.8461E + 03	3.9417E + 05	4.2854E + 05	1.8280E + 03
F19	Mean	4.5610E + 05	5.7457E + 03	1.9519E + 03	8.5459E + 06	5.6602E + 06	1.9460E + 03
	Std	2.4100E + 05	2.3857E + 03	2.4417E + 01	7.2308E + 06	4.7909E + 06	2.9670E + 01
	Min	3.1800E + 04	1.9591E + 03	1.9152E + 03	1.3533E + 06	7.9423E + 05	1.9120E + 03
F20	Mean	2.4490E + 03	2.1485E + 03	2.1579E + 03	2.5803E + 03	2.5642E + 03	2.0960E + 03
	Std	1.6820E + 02	4.7882E + 01	5.9791E + 01	1.3823E + 02	1.3551E + 02	5.1930E + 01
	Min	2.1450E + 03	2.0451E + 03	2.0421E + 03	2.3156E + 03	2.3194E + 03	2.0310E + 03
F21	Mean	2.4170E + 03	2.3252E + 03	2.3354E + 03	2.5738E + 03	2.5720E + 03	2.3170E + 03
	Std	3.1370E + 01	3.3112E + 00	5.4031E + 00	1.7147E + 01	2.1523E + 01	2.9390E + 00
	Min	2.3700E + 03	2.3173E + 03	2.3272E + 03	2.5363E + 03	2.5251E + 03	2.3100E + 03
F22	Mean	3.9390E + 03	2.3002E + 03	2.3001E + 03	3.8441E + 03	3.8149E + 03	2.3000E + 03
	Std	2.0800E + 03	8.2880E-01	5.4860E-01	2.5536E + 02	3.6709E + 02	0.0000E + 00
	Min	2.3000E + 03	2.3000E + 03	2.3000E + 03	3.3351E + 03	3.1712E + 03	2.3000E + 03
F23	Mean	2.7560E + 03	2.6745E + 03	2.6912E + 03	3.0050E + 03	2.9739E + 03	2.6650E + 03
	Std	3.6770E + 01	5.4827E + 00	1.1464E + 01	3.3082E + 01	3.5395E + 01	7.2590E + 00
	Min	2.7030E + 03	2.6600E + 03	2.6632E + 03	2.9441E + 03	2.8940E + 03	2.6560E + 03
F24	Mean	2.9010E + 03	2.8449E + 03	2.8644E + 03	3.1700E + 03	3.1300E + 03	2.8370E + 03
	Std	2.3310E + 01	5.0580E + 00	1.5154E + 01	2.8637E + 01	1.8602E + 01	5.0220E + 00
	Min	2.8570E + 03	2.8321E + 03	2.8375E + 03	3.0927E + 03	3.0908E + 03	2.8300E + 03
F25	Mean	2.9000E + 03	2.8895E + 03	2.8871E + 03	3.6132E + 03	3.5177E + 03	2.8870E + 03
	Std	1.8640E + 01	1.0419E + 01	2.3320E-01	1.6333E + 02	1.9067E + 02	3.2180E-01
	Min	2.8840E + 03	2.8834E + 03	2.8868E + 03	3.3411E + 03	3.2797E + 03	2.8870E + 03
F26	Mean	4.2670E + 03	3.5606E + 03	4.0639E + 03	7.0202E + 03	6.4570E + 03	3.7020E + 03
	Std	8.7400E + 02	5.1973E + 02	1.7384E + 02	9.1709E + 02	1.0120E + 03	2.1870E + 02
	Min	2.8000E + 03	2.8000E + 03	3.7430E + 03	5.4958E + 03	4.5495E + 03	2.8000E + 03
F27	Mean	3.2300E + 03	3.2156E + 03	3.2134E + 03	3.4732E + 03	3.4326E + 03	3.2110E + 03
	Std	1.7330E + 01	5.7941E + 00	7.3342E + 00	5.1706E + 01	5.2520E + 01	6.2750E + 00
	Min	3.2020E + 03	3.2024E + 03	3.2015E + 03	3.4142E + 03	3.3448E + 03	3.1990E + 03
F28	Mean	3.2170E + 03	3.1931E + 03	3.1574E + 03	4.3239E + 03	4.1053E + 03	3.1170E + 03
	Std	2.0230E + 01	1.0809E + 01	6.3851E + 01	2.6382E + 02	2.3646E + 02	3.7690E + 01
	Min	3.1870E + 03	3.1704E + 03	3.1000E + 03	3.9544E + 03	3.6842E + 03	3.1000E + 03
F29	Mean	3.9360E + 03	3.3973E + 03	3.4569E + 03	4.9043E + 03	4.7421E + 03	3.3890E + 03
	Std	1.3620E + 02	3.8747E + 01	4.1761E + 01	2.5945E + 02	1.6757E + 02	5.3430E + 01
	Min	3.6470E + 03	3.3606E + 03	3.3839E + 03	4.2736E + 03	4.4396E + 03	3.3360E + 03
F30	Mean	1.6110E + 06	8.5559E + 03	5.2064E + 03	1.7096E + 07	1.8642E + 07	5.1190E + 03
	Std	1.4530E + 06	2.0712E + 03	1.6574E + 02	7.2446E + 06	9.7133E + 06	1.2450E + 02
	Min	1.6010E + 05	6.1982E + 03	4.9760E + 03	5.5247E + 06	9.3022E + 06	4.9600E + 03
Count	W/T/L	0/1/28	1/1/27	3/1/25	0/0/29	0/0/29	22/3/4
Friedman test value		3.8655	2.5534	2.2690	5.6121	5.2966	1.4034

Table 7.

Comparison of local and global abilities analysis on D = 30.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	Mean	3.014E + 03	1.000E + 02	5.547E + 09	1.235E + 02	5.215E + 06	3.542E + 09	4.471E + 03	1.000E + 02
	Std	3.842E + 03	0.000E + 00	3.619E + 09	5.067E + 01	2.901E + 06	1.261E + 09	4.787E + 03	0.000E + 00
	Min	1.000E + 02	1.000E + 02	1.802E + 09	1.000E + 02	7.008E + 05	1.876E + 09	1.145E + 02	1.000E + 02
F3	Mean	3.000E + 02	3.713E + 03	4.175E + 04	3.006E + 02	1.215E + 04	6.363E + 04	8.285E + 04	3.000E + 02
	Std	0.000E + 00	1.283E + 03	1.200E + 04	5.452E-01	4.748E + 03	6.764E + 03	1.756E + 04	0.000E + 00
	Min	3.000E + 02	1.545E + 03	1.390E + 04	3.000E + 02	2.585E + 03	4.876E + 04	5.332E + 04	3.000E + 02
F4	Mean	4.908E + 02	4.553E + 02	9.182E + 02	4.511E + 02	5.247E + 02	7.330E + 02	4.891E + 02	4.146E + 02
	Std	2.075E + 01	3.432E + 01	3.466E + 02	3.153E + 01	2.574E + 01	7.673E + 01	1.433E + 00	2.514E + 01
	Min	4.201E + 02	4.000E + 02	5.564E + 02	4.040E + 02	4.898E + 02	6.403E + 02	4.878E + 02	4.000E + 02
F5	Mean	6.321E + 02	5.673E + 02	6.903E + 02	5.641E + 02	7.077E + 02	8.308E + 02	6.357E + 02	5.157E + 02
	Std	3.148E + 01	2.289E + 01	2.821E + 01	1.828E + 01	2.917E + 01	4.575E + 01	1.258E + 01	3.133E + 00
	Min	5.915E + 02	5.338E + 02	6.228E + 02	5.348E + 02	6.614E + 02	7.652E + 02	6.085E + 02	5.082E + 02
F6	Mean	6.338E + 02	6.007E + 02	6.395E + 02	6.005E + 02	6.565E + 02	6.713E + 02	6.000E + 02	6.000E + 02
	Std	1.399E + 01	8.662E-01	8.571E + 00	4.770E-01	7.469E + 00	1.264E + 01	0.000E + 00	2.400E-03
	Min	6.132E + 02	6.001E + 02	6.211E + 02	6.000E + 02	6.446E + 02	6.399E + 02	6.000E + 02	6.000E + 02
F7	Mean	8.684E + 02	7.998E + 02	1.030E + 03	8.217E + 02	1.072E + 03	1.290E + 03	8.910E + 02	7.454E + 02
	Std	4.638E + 01	2.375E + 01	6.107E + 01	2.578E + 01	7.395E + 01	4.622E + 01	1.384E + 01	3.189E + 00
	Min	7.893E + 02	7.616E + 02	9.333E + 02	7.757E + 02	9.804E + 02	1.203E + 03	8.663E + 02	7.404E + 02
F8	Mean	9.178E + 02	8.617E + 02	9.493E + 02	8.748E + 02	9.618E + 02	1.078E + 03	9.382E + 02	8.170E + 02
	Std	3.538E + 01	1.458E + 01	3.617E + 01	1.786E + 01	2.820E + 01	3.320E + 01	1.289E + 01	2.653E + 00
	Min	8.517E + 02	8.368E + 02	8.962E + 02	8.488E + 02	9.066E + 02	1.017E + 03	9.151E + 02	8.125E + 02
F9	Mean	4.214E + 03	9.256E + 02	7.266E + 03	1.044E + 03	4.821E + 03	1.062E + 04	9.023E + 02	9.003E + 02
	Std	1.688E + 03	1.817E + 01	2.868E + 03	1.382E + 02	8.202E + 02	1.182E + 03	4.452E + 00	3.230E-01
	Min	1.421E + 03	9.022E + 02	3.870E + 03	9.137E + 02	3.475E + 03	8.063E + 03	9.000E + 02	9.000E + 02
F10	Mean	4.605E + 03	4.125E + 03	5.202E + 03	4.320E + 03	5.663E + 03	7.532E + 03	6.477E + 03	3.024E + 03
	Std	6.143E + 02	1.050E + 03	5.873E + 02	5.643E + 02	8.675E + 02	6.704E + 02	2.801E + 02	1.803E + 02
	Min	3.426E + 03	2.757E + 03	4.153E + 03	3.103E + 03	3.845E + 03	6.504E + 03	5.790E + 03	2.697E + 03
Count	W/T/L	0/1/8	0/1/8	0/0/9	0/0/9	0/0/9	0/0/9	0/1/8	6/3/0

Table 8.

Comparison of local and global abilities analysis on D = 50.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	Mean	4.915E + 03	1.121E + 02	2.249E + 10	5.781E + 03	3.126E + 07	1.388E + 10	1.757E + 04	1.000E + 02
	Std	7.020E + 03	1.485E + 01	7.802E + 09	7.419E + 03	1.461E + 07	3.962E + 09	2.221E + 04	0.000E + 00
	Min	1.000E + 02	1.000E + 02	6.994E + 09	1.103E + 02	6.540E + 06	7.684E + 09	8.688E + 02	1.000E + 02
F3	Mean	3.000E + 02	3.286E + 04	8.380E + 04	1.850E + 03	4.127E + 04	1.342E + 05	2.101E + 05	3.000E + 02
	Std	0.000E + 00	6.080E + 03	1.315E + 04	2.022E + 03	6.968E + 03	1.308E + 04	2.933E + 04	0.000E + 00
	Min	3.000E + 02	2.368E + 04	6.481E + 04	3.854E + 02	2.856E + 04	1.130E + 05	1.490E + 05	3.000E + 02
F4	Mean	5.520E + 02	4.747E + 02	3.393E + 03	5.313E + 02	6.461E + 02	2.208E + 03	5.656E + 02	4.146E + 02
	Std	5.262E + 01	4.985E + 01	1.384E + 03	5.805E + 01	5.412E + 01	5.018E + 02	3.972E + 01	2.514E + 01
	Min	4.285E + 02	4.000E + 02	1.014E + 03	4.286E + 02	4.874E + 02	1.432E + 03	5.151E + 02	4.000E + 02
F5	Mean	7.691E + 02	6.481E + 02	8.696E + 02	6.456E + 02	8.604E + 02	1.064E + 03	8.193E + 02	5.157E + 02
	Std	3.951E + 01	3.706E + 01	3.355E + 01	4.056E + 01	4.518E + 01	5.600E + 01	2.377E + 01	3.133E + 00
	Min	6.771E + 02	5.876E + 02	8.133E + 02	5.876E + 02	7.862E + 02	9.697E + 02	7.719E + 02	5.082E + 02
F6	Mean	6.479E + 02	6.049E + 02	6.543E + 02	6.052E + 02	6.661E + 02	6.830E + 02	6.000E + 02	6.000E + 02
	Std	1.072E + 01	2.230E + 00	9.991E + 00	2.476E + 00	4.976E + 00	7.673E + 00	0.000E + 00	2.400E-03
	Min	6.271E + 02	6.013E + 02	6.365E + 02	6.018E + 02	6.560E + 02	6.709E + 02	6.000E + 02	6.000E + 02
F7	Mean	1.027E + 03	9.050E + 02	1.401E + 03	9.964E + 02	1.522E + 03	1.841E + 03	1.112E + 03	7.454E + 02
	Std	6.179E + 01	4.173E + 01	7.966E + 01	5.839E + 01	1.208E + 02	9.834E + 01	2.246E + 01	3.189E + 00
	Min	9.187E + 02	8.376E + 02	1.276E + 03	9.025E + 02	1.298E + 03	1.667E + 03	1.047E + 03	7.404E + 02
F8	Mean	1.072E + 03	9.549E + 02	1.163E + 03	9.355E + 02	1.191E + 03	1.417E + 03	1.126E + 03	8.170E + 02
	Std	6.036E + 01	5.458E + 01	4.141E + 01	3.235E + 01	4.244E + 01	5.934E + 01	1.902E + 01	2.653E + 00
	Min	1.006E + 03	9.005E + 02	1.103E + 03	8.806E + 02	1.115E + 03	1.275E + 03	1.087E + 03	8.125E + 02
F9	Mean	1.100E + 04	1.361E + 03	2.632E + 04	2.643E + 03	1.387E + 04	3.452E + 04	1.019E + 03	9.003E + 02
	Std	2.517E + 03	3.545E + 02	5.186E + 03	9.890E + 02	2.399E + 03	2.820E + 03	1.118E + 02	3.230E-01
	Min	6.752E + 03	1.054E + 03	1.716E + 04	1.330E + 03	1.008E + 04	2.913E + 04	9.055E + 02	9.000E + 02
F10	Mean	7.675E + 03	6.787E + 03	8.939E + 03	7.088E + 03	9.322E + 03	1.294E + 04	1.215E + 04	3.024E + 03
	Std	7.557E + 02	1.496E + 03	9.050E + 02	9.689E + 02	1.191E + 03	8.359E + 02	3.857E + 02	1.803E + 02
	Min	5.939E + 03	4.466E + 03	7.484E + 03	4.961E + 03	7.435E + 03	1.173E + 04	1.095E + 04	2.697E + 03
Count	W/T/L	0/1/8	0/0/9	0/0/9	0/0/9	0/0/9	0/0/9	0/1/8	7/2/0

Table 9.

Comparison of local and global abilities analysis on D = 100.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	Mean	1.602E + 04	4.178E + 02	8.607E + 10	2.124E + 04	1.331E + 08	5.816E + 10	2.597E + 04	1.000E + 02
	Std	1.833E + 04	1.139E + 03	1.704E + 10	1.886E + 04	6.497E + 07	7.778E + 09	1.959E + 04	4.600E-03
	Min	1.009E + 02	1.000E + 02	4.932E + 10	2.407E + 03	5.304E + 07	4.208E + 10	1.990E + 02	1.000E + 02
F3	Mean	5.109E + 03	1.591E + 05	2.215E + 05	3.629E + 04	1.497E + 05	3.036E + 05	5.949E + 05	5.313E + 02
	Std	1.627E + 03	1.258E + 04	2.334E + 04	8.370E + 03	1.589E + 04	8.008E + 03	4.689E + 04	1.665E + 02
	Min	2.547E + 03	1.322E + 05	1.732E + 05	2.110E + 04	1.257E + 05	2.876E + 05	4.908E + 05	3.553E + 02
F4	Mean	6.593E + 02	5.921E + 02	1.122E + 04	7.171E + 02	9.827E + 02	7.295E + 03	6.302E + 02	5.576E + 02
	Std	3.592E + 01	3.747E + 01	3.977E + 03	4.970E + 01	2.042E + 02	1.234E + 03	1.316E + 01	3.779E + 01
	Min	6.088E + 02	4.838E + 02	6.915E + 03	6.340E + 02	8.187E + 02	5.397E + 03	6.004E + 02	4.809E + 02
F5	Mean	1.212E + 03	9.843E + 02	1.475E + 03	9.849E + 02	1.342E + 03	1.880E + 03	1.405E + 03	6.388E + 02
	Std	1.101E + 02	7.672E + 01	8.621E + 01	7.860E + 01	4.892E + 01	8.394E + 01	3.124E + 01	1.389E + 01
	Min	1.040E + 03	8.582E + 02	1.344E + 03	8.403E + 02	1.208E + 03	1.682E + 03	1.310E + 03	6.113E + 02
F6	Mean	6.567E + 02	6.212E + 02	6.665E + 02	6.235E + 02	6.697E + 02	7.000E + 02	6.000E + 02	6.005E + 02
	Std	4.287E + 00	4.099E + 00	5.751E + 00	6.615E + 00	3.455E + 00	5.121E + 00	7.500E-03	3.564E-01
	Min	6.493E + 02	6.122E + 02	6.564E + 02	6.159E + 02	6.635E + 02	6.913E + 02	6.000E + 02	6.002E + 02
F7	Mean	1.648E + 03	1.495E + 03	2.734E + 03	1.957E + 03	3.044E + 03	3.590E + 03	1.816E + 03	9.420E + 02
	Std	1.667E + 02	1.796E + 02	1.051E + 02	1.914E + 02	2.272E + 02	1.204E + 02	4.169E + 01	1.612E + 01
	Min	1.345E + 03	1.287E + 03	2.541E + 03	1.701E + 03	2.576E + 03	3.312E + 03	1.749E + 03	9.047E + 02
F8	Mean	1.530E + 03	1.199E + 03	1.864E + 03	1.273E + 03	1.809E + 03	2.427E + 03	1.699E + 03	9.395E + 02
	Std	1.231E + 02	5.660E + 01	7.536E + 01	9.254E + 01	9.511E + 01	9.497E + 01	2.437E + 01	1.471E + 01
	Min	1.311E + 03	1.098E + 03	1.743E + 03	1.087E + 03	1.651E + 03	2.203E + 03	1.652E + 03	9.145E + 02
F9	Mean	2.423E + 04	8.256E + 03	7.371E + 04	1.302E + 04	2.951E + 04	7.007E + 04	3.168E + 03	9.713E + 02
	Std	3.368E + 03	2.476E + 03	1.099E + 04	3.333E + 03	3.251E + 03	6.147E + 03	9.492E + 02	5.531E + 01
	Min	1.840E + 04	5.681E + 03	5.344E + 04	7.897E + 03	2.540E + 04	5.786E + 04	1.767E + 03	9.236E + 02
F10	Mean	1.538E + 04	1.849E + 04	2.158E + 04	1.519E + 04	1.940E + 04	2.711E + 04	2.908E + 04	1.100E + 04
	Std	1.545E + 03	6.532E + 03	1.553E + 03	1.103E + 03	1.798E + 03	1.487E + 03	5.269E + 02	5.489E + 02
	Min	1.233E + 04	1.221E + 04	1.905E + 04	1.298E + 04	1.606E + 04	2.371E + 04	2.816E + 04	9.683E + 03
Count	W/T/L	0/0/9	0/0/9	0/0/9	0/0/9	0/0/9	0/0/9	1/0/8	8/0/1

Table 10.

Comparison of local and global abilities analysis of the proposed algorithm and contender algorithms for D = 30.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F11	Mean	1.280E + 03	1.181E + 03	2.080E + 03	1.188E + 03	1.382E + 03	2.297E + 03	1.204E + 03	1.177E + 03
	Std	6.348E + 01	3.270E + 01	9.142E + 02	3.518E + 01	1.016E + 02	4.165E + 02	1.988E + 01	2.419E + 01
	Min	1.175E + 03	1.132E + 03	1.266E + 03	1.126E + 03	1.257E + 03	1.670E + 03	1.162E + 03	1.137E + 03
F12	Mean	3.162E + 06	1.745E + 04	2.063E + 08	2.468E + 04	4.271E + 07	1.903E + 08	4.038E + 06	6.819E + 03
	Std	3.215E + 06	1.490E + 04	3.066E + 08	1.584E + 04	3.149E + 07	1.193E + 08	3.227E + 06	2.096E + 03
	Min	1.817E + 05	2.633E + 03	6.700E + 06	7.274E + 03	6.206E + 06	7.839E + 07	1.380E + 05	4.062E + 03
F13	Mean	1.095E + 05	1.610E + 03	6.863E + 07	7.192E + 03	1.471E + 05	3.437E + 07	9.917E + 04	1.338E + 03
	Std	5.835E + 04	5.746E + 02	1.700E + 08	7.904E + 03	1.042E + 05	2.766E + 07	1.422E + 05	2.396E + 01
	Min	3.709E + 04	1.336E + 03	4.525E + 05	1.482E + 03	2.655E + 04	6.405E + 06	9.305E + 03	1.314E + 03
F14	Mean	6.537E + 03	1.457E + 03	3.634E + 05	1.484E + 03	7.829E + 04	1.053E + 06	1.009E + 05	1.495E + 03
	Std	4.209E + 03	2.873E + 01	6.039E + 05	2.819E + 01	4.874E + 04	5.652E + 05	7.323E + 04	1.953E + 01
	Min	1.839E + 03	1.430E + 03	1.361E + 04	1.445E + 03	2.994E + 04	1.371E + 05	2.394E + 04	1.441E + 03
F15	Mean	5.861E + 04	2.230E + 03	3.158E + 06	1.618E + 03	6.715E + 04	6.749E + 05	2.801E + 04	1.582E + 03
	Std	6.030E + 04	1.439E + 03	1.304E + 07	6.864E + 01	6.617E + 04	5.221E + 05	1.983E + 04	4.352E + 01
	Min	6.189E + 03	1.516E + 03	1.534E + 04	1.536E + 03	1.362E + 04	8.589E + 04	1.760E + 03	1.520E + 03
F16	Mean	2.497E + 03	2.079E + 03	2.899E + 03	2.307E + 03	3.143E + 03	3.507E + 03	2.309E + 03	2.051E + 03
	Std	2.585E + 02	1.762E + 02	2.820E + 02	2.368E + 02	3.925E + 02	3.242E + 02	1.400E + 02	1.640E + 02
	Min	2.158E + 03	1.745E + 03	2.467E + 03	1.855E + 03	2.470E + 03	2.923E + 03	2.088E + 03	1.771E + 03
F17	Mean	2.018E + 03	1.863E + 03	2.163E + 03	1.921E + 03	2.397E + 03	2.396E + 03	1.925E + 03	1.750E + 03
	Std	1.406E + 02	1.353E + 02	2.254E + 02	1.157E + 02	2.202E + 02	2.404E + 02	6.430E + 01	1.326E + 01
	Min	1.847E + 03	1.743E + 03	1.851E + 03	1.763E + 03	2.077E + 03	2.027E + 03	1.837E + 03	1.721E + 03
F18	Mean	1.612E + 05	4.287E + 04	1.261E + 06	2.853E + 03	1.055E + 06	3.722E + 06	8.062E + 05	1.867E + 03
	Std	1.118E + 05	2.419E + 04	2.007E + 06	1.296E + 03	7.919E + 05	2.898E + 06	5.231E + 05	3.941E + 01
	Min	3.703E + 04	1.342E + 04	1.210E + 05	1.853E + 03	1.269E + 05	7.410E + 05	1.597E + 05	1.828E + 03
F19	Mean	4.561E + 05	1.948E + 03	1.759E + 06	1.943E + 03	1.417E + 06	3.853E + 06	3.258E + 04	1.946E + 03
	Std	2.410E + 05	2.072E + 01	6.897E + 06	2.181E + 01	9.276E + 05	3.318E + 06	2.045E + 04	2.967E + 01
	Min	3.180E + 04	1.918E + 03	3.040E + 04	1.916E + 03	2.114E + 04	2.878E + 05	2.472E + 03	1.912E + 03
F20	Mean	2.449E + 03	2.187E + 03	2.671E + 03	2.211E + 03	2.522E + 03	2.792E + 03	2.234E + 03	2.096E + 03
	Std	1.682E + 02	1.173E + 02	1.984E + 02	1.320E + 02	1.617E + 02	1.952E + 02	9.257E + 01	5.193E + 01
	Min	2.145E + 03	2.027E + 03	2.216E + 03	2.044E + 03	2.290E + 03	2.476E + 03	2.124E + 03	2.031E + 03
F21	Mean	2.417E + 03	2.356E + 03	2.476E + 03	2.362E + 03	2.489E + 03	2.583E + 03	2.435E + 03	2.317E + 03
	Std	3.137E + 01	1.519E + 01	3.981E + 01	1.847E + 01	4.533E + 01	3.986E + 01	1.238E + 01	2.939E + 00
	Min	2.370E + 03	2.330E + 03	2.410E + 03	2.330E + 03	2.431E + 03	2.510E + 03	2.412E + 03	2.310E + 03
F22	Mean	3.939E + 03	2.300E + 03	5.675E + 03	2.830E + 03	2.966E + 03	6.952E + 03	4.571E + 03	2.300E + 03
	Std	2.080E + 03	9.193E-01	1.808E + 03	1.308E + 03	1.440E + 03	2.493E + 03	5.544E + 02	0.000E + 00
	Min	2.300E + 03	2.300E + 03	2.776E + 03	2.300E + 03	2.312E + 03	2.709E + 03	3.820E + 03	2.300E + 03
F23	Mean	2.756E + 03	2.718E + 03	2.920E + 03	2.717E + 03	2.976E + 03	3.107E + 03	2.789E + 03	2.665E + 03
	Std	3.677E + 01	2.461E + 01	6.034E + 01	1.690E + 01	1.026E + 02	1.022E + 02	1.236E + 01	7.259E + 00
	Min	2.703E + 03	2.676E + 03	2.836E + 03	2.691E + 03	2.748E + 03	2.966E + 03	2.768E + 03	2.656E + 03
F24	Mean	2.901E + 03	2.866E + 03	3.126E + 03	2.888E + 03	3.095E + 03	3.307E + 03	2.990E + 03	2.837E + 03
	Std	2.331E + 01	1.493E + 01	4.667E + 01	1.771E + 01	7.563E + 01	9.326E + 01	1.388E + 01	5.022E + 00
	Min	2.857E + 03	2.840E + 03	3.033E + 03	2.859E + 03	2.994E + 03	3.176E + 03	2.963E + 03	2.830E + 03
F25	Mean	2.900E + 03	2.890E + 03	3.107E + 03	2.897E + 03	2.934E + 03	3.048E + 03	2.887E + 03	2.887E + 03
	Std	1.864E + 01	1.266E + 01	1.179E + 02	2.127E + 01	2.779E + 01	4.367E + 01	3.227E-01	3.218E-01
	Min	2.884E + 03	2.884E + 03	2.951E + 03	2.884E + 03	2.892E + 03	2.962E + 03	2.887E + 03	2.887E + 03
F26	Mean	4.267E + 03	3.948E + 03	6.503E + 03	4.278E + 03	6.315E + 03	6.050E + 03	4.950E + 03	3.702E + 03
	Std	8.740E + 02	6.603E + 02	8.578E + 02	5.711E + 02	1.644E + 03	1.740E + 03	1.109E + 02	2.187E + 02
	Min	2.800E + 03	2.900E + 03	4.158E + 03	2.818E + 03	2.850E + 03	3.929E + 03	4.692E + 03	2.800E + 03
F27	Mean	3.230E + 03	3.219E + 03	3.385E + 03	3.222E + 03	3.330E + 03	3.398E + 03	3.209E + 03	3.211E + 03
	Std	1.733E + 01	9.877E + 00	7.093E + 01	1.185E + 01	7.788E + 01	9.968E + 01	4.446E + 00	6.275E + 00
	Min	3.202E + 03	3.207E + 03	3.291E + 03	3.204E + 03	3.229E + 03	3.280E + 03	3.202E + 03	3.199E + 03
F28	Mean	3.217E + 03	3.168E + 03	3.648E + 03	3.179E + 03	3.293E + 03	3.565E + 03	3.222E + 03	3.117E + 03
	Std	2.023E + 01	6.623E + 01	3.350E + 02	5.883E + 01	2.841E + 01	9.031E + 01	1.350E + 01	3.769E + 01
	Min	3.187E + 03	3.100E + 03	3.346E + 03	3.100E + 03	3.241E + 03	3.454E + 03	3.209E + 03	3.100E + 03
F29	Mean	3.936E + 03	3.601E + 03	4.107E + 03	3.570E + 03	4.583E + 03	4.939E + 03	3.724E + 03	3.389E + 03
	Std	1.362E + 02	1.657E + 02	2.622E + 02	1.607E + 02	3.436E + 02	3.542E + 02	1.296E + 02	5.343E + 01
	Min	3.647E + 03	3.384E + 03	3.740E + 03	3.373E + 03	4.092E + 03	4.419E + 03	3.465E + 03	3.336E + 03
F30	Mean	1.611E + 06	6.150E + 03	6.259E + 06	6.819E + 03	1.747E + 07	2.335E + 07	2.481E + 04	5.119E + 03
	Std	1.453E + 06	1.697E + 03	5.108E + 06	1.892E + 03	1.699E + 07	1.548E + 07	1.090E + 04	1.245E + 02
	Min	1.601E + 05	5.053E + 03	4.116E + 05	5.162E + 03	2.046E + 06	5.776E + 06	1.006E + 04	4.960E + 03
Count	W/T/L	0/0/20	1/0/19	0/0/20	1/0/19	0/0/20	0/0/20	1/0/19	17/0/3

Table 11.

Comparison of local and global abilities analysis of the proposed algorithm and contender algorithms for D = 50.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F11	Mean	1.385E + 03	1.261E + 03	5.041E + 03	1.276E + 03	1.631E + 03	3.561E + 03	1.298E + 03	1.177E + 03
	Std	6.448E + 01	4.017E + 01	2.307E + 03	4.877E + 01	1.354E + 02	7.501E + 02	3.077E + 01	2.419E + 01
	Min	1.257E + 03	1.204E + 03	1.591E + 03	1.204E + 03	1.427E + 03	2.577E + 03	1.262E + 03	1.137E + 03
F12	Mean	1.908E + 07	2.885E + 05	5.210E + 09	7.528E + 05	1.610E + 08	2.171E + 09	2.590E + 07	6.819E + 03
	Std	8.038E + 06	3.377E + 05	3.966E + 09	8.892E + 05	8.852E + 07	7.377E + 08	2.121E + 07	2.096E + 03
	Min	3.017E + 06	6.369E + 04	1.939E + 08	1.711E + 05	4.119E + 07	8.309E + 08	7.215E + 06	4.062E + 03
F13	Mean	1.214E + 05	8.443E + 03	1.172E + 09	7.963E + 03	5.466E + 05	2.125E + 08	5.930E + 04	1.338E + 03
	Std	8.369E + 04	8.570E + 03	2.100E + 09	6.182E + 03	5.457E + 05	1.057E + 08	7.273E + 04	2.396E + 01
	Min	4.053E + 04	1.580E + 03	1.285E + 07	1.913E + 03	7.228E + 04	8.775E + 07	4.553E + 03	1.314E + 03
F14	Mean	5.252E + 04	7.462E + 03	4.776E + 05	1.752E + 03	5.386E + 05	2.217E + 06	1.112E + 06	1.495E + 03
	Std	5.190E + 04	8.906E + 03	4.115E + 05	3.964E + 02	3.202E + 05	1.477E + 06	5.865E + 05	1.953E + 01
	Min	5.347E + 03	1.641E + 03	1.158E + 05	1.524E + 03	8.614E + 04	2.545E + 05	1.269E + 05	1.441E + 03
F15	Mean	5.503E + 04	8.417E + 03	7.434E + 07	6.700E + 03	8.386E + 04	1.980E + 07	1.599E + 04	1.582E + 03
	Std	4.706E + 04	6.878E + 03	2.182E + 08	5.003E + 03	6.071E + 04	1.544E + 07	1.387E + 04	4.352E + 01
	Min	1.526E + 04	1.705E + 03	2.209E + 05	1.681E + 03	2.444E + 04	3.274E + 06	2.663E + 03	1.520E + 03
F16	Mean	3.082E + 03	2.792E + 03	3.729E + 03	3.179E + 03	4.568E + 03	4.780E + 03	3.436E + 03	2.051E + 03
	Std	3.876E + 02	4.123E + 02	3.306E + 02	4.746E + 02	5.766E + 02	8.292E + 02	2.769E + 02	1.640E + 02
	Min	2.332E + 03	2.162E + 03	2.964E + 03	2.376E + 03	3.594E + 03	3.215E + 03	2.785E + 03	1.771E + 03
F17	Mean	3.019E + 03	2.637E + 03	3.365E + 03	2.814E + 03	3.659E + 03	4.087E + 03	2.824E + 03	1.750E + 03
	Std	3.100E + 02	2.513E + 02	3.184E + 02	2.759E + 02	3.770E + 02	3.576E + 02	1.508E + 02	1.326E + 01
	Min	2.444E + 03	2.264E + 03	2.772E + 03	2.295E + 03	2.904E + 03	3.269E + 03	2.609E + 03	1.721E + 03
F18	Mean	3.158E + 05	4.638E + 05	7.377E + 06	1.877E + 04	3.215E + 06	8.977E + 06	4.135E + 06	1.867E + 03
	Std	2.449E + 05	4.153E + 05	8.944E + 06	1.039E + 04	2.455E + 06	1.116E + 07	2.986E + 06	3.941E + 01
	Min	1.091E + 05	1.333E + 05	3.882E + 05	7.525E + 03	7.619E + 05	2.055E + 06	1.286E + 06	1.828E + 03
F19	Mean	9.178E + 05	2.038E + 03	1.814E + 07	6.988E + 03	3.374E + 06	6.865E + 06	2.113E + 04	1.946E + 03
	Std	3.139E + 05	4.076E + 01	5.560E + 07	6.671E + 03	1.749E + 06	4.475E + 06	1.131E + 04	2.967E + 01
	Min	2.557E + 05	1.957E + 03	3.478E + 05	2.028E + 03	4.792E + 05	2.087E + 06	3.096E + 03	1.912E + 03
F20	Mean	2.968E + 03	2.728E + 03	3.337E + 03	2.867E + 03	3.364E + 03	3.905E + 03	2.914E + 03	2.096E + 03
	Std	3.259E + 02	2.234E + 02	3.282E + 02	3.101E + 02	2.817E + 02	2.258E + 02	1.943E + 02	5.193E + 01
	Min	2.499E + 03	2.216E + 03	2.657E + 03	2.206E + 03	2.866E + 03	3.408E + 03	2.621E + 03	2.031E + 03
F21	Mean	2.525E + 03	2.414E + 03	2.683E + 03	2.432E + 03	2.741E + 03	2.896E + 03	2.638E + 03	2.317E + 03
	Std	3.281E + 01	3.214E + 01	3.532E + 01	2.708E + 01	6.890E + 01	7.140E + 01	1.998E + 01	2.939E + 00
	Min	2.485E + 03	2.367E + 03	2.627E + 03	2.386E + 03	2.641E + 03	2.734E + 03	2.576E + 03	2.310E + 03
F22	Mean	8.690E + 03	3.496E + 03	1.144E + 04	8.977E + 03	1.107E + 04	1.424E + 04	1.360E + 04	2.300E + 03
	Std	2.500E + 03	2.489E + 03	1.044E + 03	1.000E + 03	1.197E + 03	1.106E + 03	7.952E + 02	0.000E + 00
	Min	2.300E + 03	2.300E + 03	9.774E + 03	7.032E + 03	9.387E + 03	1.240E + 04	1.081E + 04	2.300E + 03
F23	Mean	2.956E + 03	2.863E + 03	3.351E + 03	2.911E + 03	3.457E + 03	3.708E + 03	3.061E + 03	2.665E + 03
	Std	5.530E + 01	3.519E + 01	1.076E + 02	3.157E + 01	1.311E + 02	1.524E + 02	1.974E + 01	7.259E + 00
	Min	2.848E + 03	2.823E + 03	3.237E + 03	2.866E + 03	3.245E + 03	3.458E + 03	3.028E + 03	2.656E + 03
F24	Mean	3.089E + 03	3.003E + 03	3.582E + 03	3.055E + 03	3.584E + 03	3.899E + 03	3.274E + 03	2.837E + 03
	Std	4.662E + 01	2.189E + 01	1.029E + 02	3.991E + 01	1.008E + 02	1.351E + 02	1.474E + 01	5.022E + 00
	Min	2.999E + 03	2.973E + 03	3.431E + 03	3.004E + 03	3.383E + 03	3.713E + 03	3.248E + 03	2.830E + 03
F25	Mean	3.040E + 03	3.041E + 03	4.466E + 03	3.047E + 03	3.136E + 03	3.982E + 03	2.997E + 03	2.887E + 03
	Std	3.345E + 01	5.024E + 01	4.479E + 02	3.355E + 01	3.320E + 01	2.184E + 02	1.894E + 01	3.218E-01
	Min	2.969E + 03	2.929E + 03	3.626E + 03	2.961E + 03	3.090E + 03	3.677E + 03	2.981E + 03	2.887E + 03
F26	Mean	3.715E + 03	5.241E + 03	9.874E + 03	5.600E + 03	9.216E + 03	1.039E + 04	6.975E + 03	3.702E + 03
	Std	1.450E + 03	6.662E + 02	8.911E + 02	4.849E + 02	3.139E + 03	2.068E + 03	1.542E + 02	2.187E + 02
	Min	2.900E + 03	2.900E + 03	8.152E + 03	5.017E + 03	4.233E + 03	7.367E + 03	6.645E + 03	2.800E + 03
F27	Mean	3.412E + 03	3.360E + 03	4.000E + 03	3.404E + 03	3.884E + 03	4.305E + 03	3.257E + 03	3.211E + 03
	Std	8.592E + 01	7.701E + 01	2.012E + 02	1.110E + 02	2.570E + 02	2.706E + 02	1.314E + 01	6.275E + 00
	Min	3.282E + 03	3.275E + 03	3.638E + 03	3.288E + 03	3.569E + 03	3.951E + 03	3.232E + 03	3.199E + 03
F28	Mean	3.301E + 03	3.310E + 03	5.293E + 03	3.303E + 03	3.380E + 03	5.184E + 03	3.306E + 03	3.117E + 03
	Std	3.025E + 01	3.857E + 01	6.730E + 02	1.324E + 01	4.205E + 01	3.307E + 02	8.278E + 01	3.769E + 01
	Min	3.262E + 03	3.259E + 03	4.300E + 03	3.279E + 03	3.319E + 03	4.469E + 03	3.259E + 03	3.100E + 03
F29	Mean	4.802E + 03	3.993E + 03	5.399E + 03	4.076E + 03	6.641E + 03	6.569E + 03	4.254E + 03	3.389E + 03
	Std	4.236E + 02	3.447E + 02	5.429E + 02	3.166E + 02	7.246E + 02	4.958E + 02	2.191E + 02	5.343E + 01
	Min	3.925E + 03	3.526E + 03	4.354E + 03	3.498E + 03	4.950E + 03	5.792E + 03	3.701E + 03	3.336E + 03
F30	Mean	2.772E + 07	6.566E + 05	2.631E + 08	7.823E + 05	1.059E + 08	1.460E + 08	1.449E + 06	5.119E + 03
	Std	5.682E + 06	7.157E + 04	5.125E + 08	1.451E + 05	3.843E + 07	5.986E + 07	3.739E + 05	1.245E + 02
	Min	2.086E + 07	5.825E + 05	5.337E + 07	6.087E + 05	4.905E + 07	6.616E + 07	9.138E + 05	4.960E + 03
Count	W/T/L	0/0/20	0/0/20	0/0/20	0/0/20	0/0/20	0/0/20	0/0/20	20/0/0

Table 12.

Comparison of local and global abilities analysis of the proposed algorithm and contender algorithms for D = 100.

Functions	Statistical metrics	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F11	Mean	2.488E + 03	2.039E + 03	5.860E + 04	2.148E + 03	7.674E + 03	8.410E + 04	6.359E + 03	1.959E + 03
	Std	2.530E + 02	1.832E + 02	1.544E + 04	1.362E + 02	9.205E + 02	1.134E + 04	1.544E + 03	2.124E + 02
	Min	2.057E + 03	1.666E + 03	3.259E + 04	1.835E + 03	5.635E + 03	6.353E + 04	4.462E + 03	1.622E + 03
F12	Mean	4.957E + 07	2.891E + 06	2.378E + 10	5.875E + 06	6.344E + 08	1.070E + 10	1.788E + 08	8.131E + 04
	Std	2.031E + 07	1.620E + 06	9.763E + 09	3.047E + 06	2.561E + 08	3.970E + 09	4.362E + 07	3.455E + 04
	Min	1.637E + 07	9.671E + 05	9.491E + 09	1.742E + 06	1.928E + 08	5.188E + 09	9.038E + 07	1.726E + 04
F13	Mean	6.243E + 04	8.133E + 03	5.492E + 09	6.761E + 03	5.847E + 05	1.087E + 09	7.720E + 03	3.157E + 03
	Std	1.533E + 04	5.015E + 03	3.476E + 09	3.992E + 03	4.634E + 05	6.097E + 08	7.219E + 03	1.393E + 03
	Min	3.284E + 04	3.094E + 03	1.763E + 09	2.488E + 03	8.899E + 04	4.338E + 08	1.834E + 03	1.534E + 03
F14	Mean	2.615E + 05	1.918E + 05	5.198E + 06	6.934E + 04	2.260E + 06	1.076E + 07	1.619E + 07	2.916E + 03
	Std	1.537E + 05	1.068E + 05	3.016E + 06	3.523E + 04	8.823E + 05	2.574E + 06	4.991E + 06	1.126E + 03
	Min	6.219E + 04	4.981E + 04	1.365E + 06	1.534E + 04	6.520E + 05	6.955E + 06	7.639E + 06	1.750E + 03
F15	Mean	6.483E + 04	4.306E + 03	1.584E + 09	4.920E + 03	7.339E + 04	2.414E + 08	1.001E + 04	2.004E + 03
	Std	2.773E + 04	3.860E + 03	2.207E + 09	5.954E + 03	2.453E + 04	1.598E + 08	6.903E + 03	5.599E + 02
	Min	2.660E + 04	1.887E + 03	2.877E + 07	1.962E + 03	3.641E + 04	3.771E + 07	3.310E + 03	1.687E + 03
F16	Mean	5.782E + 03	4.917E + 03	8.009E + 03	5.086E + 03	8.110E + 03	1.111E + 04	9.086E + 03	3.883E + 03
	Std	9.069E + 02	5.593E + 02	9.890E + 02	6.807E + 02	8.280E + 02	9.665E + 02	3.822E + 02	3.387E + 02
	Min	4.349E + 03	4.030E + 03	6.546E + 03	4.002E + 03	6.755E + 03	9.392E + 03	8.389E + 03	3.176E + 03
F17	Mean	4.976E + 03	4.536E + 03	1.041E + 04	4.761E + 03	6.596E + 03	9.449E + 03	5.490E + 03	3.676E + 03
	Std	5.966E + 02	5.526E + 02	5.454E + 03	2.531E + 02	7.721E + 02	2.650E + 03	2.484E + 02	2.724E + 02
	Min	3.684E + 03	3.486E + 03	5.113E + 03	4.324E + 03	5.021E + 03	6.670E + 03	5.047E + 03	3.167E + 03
F18	Mean	3.959E + 05	5.259E + 05	8.230E + 06	2.232E + 05	2.527E + 06	9.518E + 06	2.354E + 07	2.386E + 04
	Std	1.577E + 05	2.635E + 05	4.494E + 06	1.353E + 05	1.097E + 06	3.498E + 06	5.694E + 06	1.304E + 04
	Min	1.229E + 05	2.027E + 05	2.203E + 06	6.611E + 04	1.157E + 06	4.788E + 06	1.417E + 07	5.902E + 03
F19	Mean	2.153E + 06	5.917E + 03	7.253E + 08	7.057E + 03	9.274E + 06	2.125E + 08	9.123E + 03	2.105E + 03
	Std	5.576E + 05	4.705E + 03	1.511E + 09	5.861E + 03	5.354E + 06	1.280E + 08	1.046E + 04	2.983E + 01
	Min	1.252E + 06	2.182E + 03	1.396E + 07	2.197E + 03	1.726E + 06	7.498E + 07	2.528E + 03	2.051E + 03
F20	Mean	4.857E + 03	4.7550E + 03	5.454E + 03	4.831E + 03	5.761E + 03	6.759E + 03	5.539E + 03	3.880E + 03
	Std	6.025E + 02	5.5355E + 02	4.630E + 02	3.159E + 02	6.114E + 02	4.597E + 02	2.867E + 02	2.139E + 02
	Min	3.833E + 03	3.8002E + 03	4.576E + 03	4.348E + 03	4.849E + 03	5.948E + 03	4.913E + 03	3.499E + 03
F21	Mean	2.999E + 03	2.6726E + 03	3.589E + 03	2.814E + 03	3.681E + 03	3.899E + 03	3.231E + 03	2.465E + 03
	Std	1.341E + 02	6.0096E + 01	8.958E + 01	8.323E + 01	1.492E + 02	1.033E + 02	2.578E + 01	2.292E + 01
	Min	2.816E + 03	2.6006E + 03	3.458E + 03	2.675E + 03	3.425E + 03	3.716E + 03	3.170E + 03	2.421E + 03
F22	Mean	1.649E + 04	1.7990E + 04	2.590E + 04	1.813E + 04	2.148E + 04	2.948E + 04	3.096E + 04	1.365E + 04
	Std	3.528E + 03	1.2997E + 04	1.068E + 03	1.425E + 03	1.639E + 03	1.487E + 03	4.955E + 02	4.728E + 02
	Min	2.304E + 03	2.3000E + 03	2.415E + 04	1.543E + 04	1.868E + 04	2.713E + 04	2.985E + 04	1.273E + 04
F23	Mean	3.442E + 03	3.2197E + 03	4.415E + 03	3.359E + 03	4.415E + 03	4.790E + 03	3.550E + 03	2.935E + 03
	Std	6.939E + 01	6.3163E + 01	1.614E + 02	1.124E + 02	2.001E + 02	2.914E + 02	3.077E + 01	2.489E + 01
	Min	3.331E + 03	3.0893E + 03	4.011E + 03	3.205E + 03	4.093E + 03	4.376E + 03	3.491E + 03	2.908E + 03
F24	Mean	3.908E + 03	3.6872E + 03	5.715E + 03	3.989E + 03	5.632E + 03	5.607E + 03	4.103E + 03	3.431E + 03
	Std	1.257E + 02	5.6659E + 01	2.183E + 02	1.438E + 02	4.043E + 02	1.908E + 02	3.719E + 01	2.350E + 01
	Min	3.745E + 03	3.5872E + 03	5.239E + 03	3.751E + 03	4.946E + 03	5.188E + 03	4.028E + 03	3.396E + 03
F25	Mean	3.284E + 03	3.2764E + 03	8.150E + 03	3.406E + 03	3.603E + 03	6.815E + 03	3.367E + 03	3.218E + 03
	Std	6.578E + 01	7.8953E + 01	1.197E + 03	7.904E + 01	6.492E + 01	4.508E + 02	4.881E + 01	5.163E + 01
	Min	3.118E + 03	3.0602E + 03	6.822E + 03	3.272E + 03	3.488E + 03	5.954E + 03	3.302E + 03	3.077E + 03
F26	Mean	1.154E + 04	1.0216E + 04	2.541E + 04	1.367E + 04	2.400E + 04	2.752E + 04	1.472E + 04	7.539E + 03
	Std	3.970E + 03	6.9590E + 02	1.632E + 03	2.376E + 03	6.308E + 03	5.426E + 03	2.047E + 02	3.825E + 02
	Min	2.900E + 03	8.8207E + 03	2.246E + 04	1.082E + 04	8.941E + 03	1.558E + 04	1.428E + 04	7.028E + 03
F27	Mean	3.604E + 03	3.5977E + 03	4.934E + 03	3.687E + 03	4.131E + 03	5.084E + 03	3.447E + 03	3.395E + 03
	Std	9.091E + 01	9.1556E + 01	3.961E + 02	9.083E + 01	3.296E + 02	3.958E + 02	2.955E + 01	4.239E + 01
	Min	3.406E + 03	3.4243E + 03	4.350E + 03	3.561E + 03	3.701E + 03	4.571E + 03	3.385E + 03	3.303E + 03
F28	Mean	3.394E + 03	3.3349E + 03	1.131E + 04	3.495E + 03	3.630E + 03	9.920E + 03	3.617E + 03	3.342E + 03
	Std	3.858E + 01	4.6966E + 01	2.253E + 03	4.552E + 01	5.450E + 01	1.103E + 03	3.939E + 02	2.774E + 01
	Min	3.332E + 03	3.2427E + 03	8.054E + 03	3.411E + 03	3.505E + 03	7.753E + 03	3.360E + 03	3.294E + 03
F29	Mean	8.425E + 03	6.6218E + 03	1.087E + 04	6.766E + 03	1.224E + 04	1.287E + 04	7.320E + 03	5.110E + 03
	Std	7.798E + 02	4.5161E + 02	2.883E + 03	4.653E + 02	1.453E + 03	9.657E + 02	2.598E + 02	4.086E + 02
	Min	7.247E + 03	5.7397E + 03	8.113E + 03	6.025E + 03	9.689E + 03	1.140E + 04	6.718E + 03	4.305E + 03
F30	Mean	1.262E + 07	6.2174E + 03	3.153E + 09	1.155E + 04	1.746E + 08	9.640E + 08	1.809E + 04	7.215E + 03
	Std	5.417E + 06	3.5816E + 02	2.212E + 09	3.849E + 03	1.063E + 08	3.918E + 08	4.053E + 03	1.897E + 03
	Min	4.469E + 06	5.7832E + 03	1.125E + 08	6.581E + 03	5.373E + 07	2.164E + 08	1.313E + 04	5.347E + 03
Count	W/T/L	0/0/20	2/0/18	0/0/20	0/0/20	0/0/20	0/0/20	0/0/20	18/0/2

Table 13.

Friedman ranks for each algorithm based on their performance for D = 30.

Function	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	6.058	1.737	7.583	3.032	5.447	6.530	4.322	1.293
F3	3.839	4.530	5.261	2.475	4.165	6.632	7.991	1.107
F4	6.208	2.649	7.583	2.092	5.532	6.473	4.128	1.334
F5	5.797	2.898	5.549	1.894	6.413	7.701	4.515	1.233
F6	6.147	2.992	5.404	3.690	6.653	7.734	1.839	1.540
F7	5.744	1.880	5.485	2.939	6.558	7.670	4.485	1.240
F8	5.926	1.971	4.984	2.853	6.073	7.563	5.376	1.255
F9	6.158	2.322	6.496	3.554	5.323	7.885	3.256	1.006
F10	5.717	5.146	3.910	2.010	4.526	7.047	6.152	1.491
F11	6.149	2.107	6.352	2.914	5.240	7.489	4.717	1.031
F12	5.973	2.014	7.462	3.023	5.556	6.460	4.490	1.022
F13	6.086	2.482	7.516	2.541	4.559	6.450	5.310	1.057
F14	4.934	2.472	6.640	1.329	4.744	7.883	5.755	2.242
F15	6.556	3.041	7.400	2.031	4.956	6.182	4.780	1.055
F16	6.284	1.491	5.223	2.600	6.539	7.477	4.247	2.138
F17	6.168	2.000	5.309	2.830	7.574	6.574	4.218	1.327
F18	5.648	3.119	5.288	1.976	6.224	7.522	5.150	1.075
F19	6.782	3.038	5.597	1.622	6.054	7.191	4.266	1.450
F20	6.228	2.275	6.464	2.571	5.356	7.549	4.166	1.391
F21	5.859	1.976	5.594	2.756	6.391	7.561	4.569	1.295
F22	6.021	1.904	6.637	3.012	4.007	7.747	5.567	1.104
F23	5.200	2.799	5.604	1.972	6.742	7.922	4.506	1.254
F24	4.947	1.993	6.766	2.793	5.711	7.984	4.562	1.244
F25	6.258	2.709	7.504	3.822	5.569	6.403	2.710	1.024
F26	5.139	1.815	7.322	3.230	7.026	5.585	4.595	1.288
F27	5.783	2.624	6.856	3.758	5.705	7.568	2.365	1.340
F28	5.720	2.013	7.549	3.040	5.495	6.497	4.656	1.029
F29	6.110	3.012	5.308	1.932	6.707	7.621	4.190	1.120
F30	6.351	2.554	5.375	2.440	6.947	7.176	4.082	1.075

Table 14.

Friedman ranks for each algorithm based on their performance for D = 50.

Function	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	5.801	2.011	7.613	3.211	5.438	6.565	4.354	1.006
F3	3.453	5.138	4.831	2.456	4.295	6.464	7.992	1.370
F4	5.789	1.975	7.612	3.021	5.503	6.561	4.493	1.047
F5	5.981	2.805	6.129	2.029	5.574	7.532	4.750	1.200
F6	6.373	2.880	5.396	3.955	6.520	7.665	1.688	1.524
F7	5.728	1.924	5.544	3.025	6.574	7.636	4.476	1.092
F8	5.884	2.918	5.429	1.910	6.300	7.648	4.716	1.194
F9	6.240	2.366	6.495	3.699	5.332	7.842	3.021	1.005
F10	5.894	4.339	4.128	2.077	4.714	7.068	6.393	1.387
F11	6.141	2.149	7.446	2.940	5.237	6.309	4.694	1.084
F12	5.885	2.001	7.649	3.012	5.453	6.542	4.447	1.010
F13	6.168	2.803	7.645	2.217	5.407	6.499	4.249	1.012
F14	5.404	3.026	4.524	1.972	5.723	7.378	6.901	1.071
F15	6.477	2.919	7.550	2.087	5.163	6.491	4.277	1.036
F16	5.470	2.051	4.818	3.230	6.894	7.088	5.205	1.243
F17	6.315	1.938	5.299	2.763	6.515	7.525	4.195	1.449
F18	5.242	3.452	6.133	1.979	4.853	7.055	6.215	1.070
F19	6.603	2.452	7.517	2.549	5.387	6.329	4.101	1.063
F20	6.043	2.709	5.245	2.564	5.977	7.543	4.589	1.330
F21	5.803	1.919	5.481	2.847	6.558	7.617	4.523	1.250
F22	5.467	1.544	5.426	3.364	4.534	7.058	6.871	1.737
F23	5.093	1.876	5.677	2.931	6.790	7.932	4.495	1.206
F24	4.796	1.964	6.072	2.857	6.582	7.985	4.544	1.200
F25	5.047	2.080	7.572	3.607	5.537	6.510	2.169	3.477
F26	3.691	2.485	6.819	3.549	5.729	7.430	4.665	1.633
F27	5.774	2.612	6.697	3.734	5.677	7.760	2.607	1.139
F28	4.959	3.127	7.540	2.933	5.410	6.529	4.492	1.012
F29	6.129	1.981	5.464	2.932	7.683	6.583	4.099	1.129
F30	6.359	1.939	7.572	3.017	5.895	6.023	4.096	1.099

Table 15.

Friedman ranks for each algorithm based on their performance for D = 100.

Function	SSA (2017)	QANA (2021)	SCHO (2022)	SMO (2022)	PO (2024)	HEOA (2024)	ADE (2024)	ESSA
F1	5.749	2.007	7.628	4.229	5.452	6.589	3.343	1.003
F3	3.492	5.878	4.655	2.462	4.042	6.423	7.996	1.052
F4	5.777	2.008	7.618	4.323	5.509	6.574	3.188	1.004
F5	6.028	1.968	6.370	2.955	4.443	7.561	5.580	1.097
F6	6.367	2.946	5.394	3.980	6.425	7.776	1.560	1.551
F7	5.369	2.004	5.560	4.217	6.582	7.622	3.641	1.004
F8	5.768	1.925	6.423	2.954	5.502	7.814	4.484	1.130
F9	6.696	3.178	7.479	3.415	5.266	6.401	2.563	1.004
F10	5.489	5.641	4.877	1.781	3.581	5.926	7.342	1.364
F11	4.924	2.510	6.115	2.740	4.702	7.362	6.607	1.041
F12	5.902	2.004	7.650	3.011	5.377	6.572	4.479	1.005
F13	6.254	3.000	7.724	2.173	5.385	6.595	3.864	1.004
F14	5.637	3.053	5.475	1.993	4.561	6.506	7.727	1.049
F15	6.305	2.002	7.711	3.005	5.362	6.579	4.032	1.005
F16	5.777	1.926	4.683	2.755	5.393	7.517	6.614	1.336
F17	5.443	1.873	7.763	2.806	5.426	6.704	4.657	1.328
F18	5.435	3.380	5.715	1.996	4.494	6.236	7.694	1.050
F19	6.355	2.002	7.656	3.009	5.360	6.568	4.043	1.007
F20	5.552	4.210	4.180	1.951	5.140	7.171	6.465	1.331
F21	5.660	1.902	5.555	2.968	6.683	7.615	4.480	1.137
F22	5.004	4.357	5.507	2.290	3.786	6.168	7.510	1.378
F23	5.100	1.923	6.070	3.017	6.556	7.799	4.449	1.086
F24	4.690	1.925	7.778	3.448	6.841	5.788	4.447	1.083
F25	5.365	2.011	7.561	3.578	5.439	6.512	4.531	1.004
F26	4.955	1.946	6.766	3.480	5.749	7.559	4.485	1.060
F27	5.547	2.619	6.809	4.236	5.582	7.525	2.668	1.014
F28	5.256	1.269	7.547	3.494	4.926	6.518	5.245	1.745
F29	6.165	1.934	5.590	2.965	6.889	7.291	4.050	1.117
F30	6.174	1.168	7.704	3.005	5.512	6.577	4.021	1.840

Table 16.

The overall MAE values and ranks in the IEEE CEC 2017 test suites.

Optimizers	MAE D = 30	Rank D = 30	MAE D = 50	Rank D = 50	MAE D = 100	Rank D = 100
SSA18	1.504E + 04	5	8.103E + 05	5	7.457E + 05	4
QANA40	7.237E + 02	3	2.690E + 04	3	4.632E + 04	2
SCHO60	6.032E + 07	7	2.419E + 08	7	2.024E + 09	8
SMO19	4.765E + 02	2	2.666E + 04	2	6.354E + 04	3
PO61	3.058E + 05	6	3.275E + 06	6	1.012E + 07	6
HEOA62	6.561E + 07	8	2.892E + 08	8	1.602E + 09	7
ADE35	1.330E + 04	4	3.240E + 05	4	3.759E + 06	5
ESSA	3.291E + 02	1	1.986E + 04	1	1.924E + 03	1

Sensitivity analyses

A comprehensive sensitivity analysis was conducted to examine how three key parameters—memory size (k), population size (N), and the maximum number of iterations (MaxIt)—affect the performance of the proposed ESSA in solving optimization problems.

The first experiment focused specifically on the impact of memory size, with findings summarized in Table 2. In this evaluation, memory sizes of 300, 500, and 700 were tested to assess their influence on ESSA’s optimization effectiveness. All trials were conducted under consistent settings: a maximum of 3000 iterations, a population size of N = 100, and a problem dimension of 30. The results indicate that a memory size of k = 500 yielded the best overall performance for a problem dimension of D = 30. This setup yielded a Friedman test value of 1.962 and achieved an optimization effectiveness of 37.93%, surpassing the results obtained with both smaller and larger memory configurations. These findings suggest that k = 500 offers an optimal balance between memory utility and search efficiency, resulting in superior solution quality compared to other settings. To validate the generalizability of this configuration, ESSA with k = 500 was further evaluated on higher-dimensional problems. Table 3 presents the results for dimensions D = 30, 50, and 100 using the same population size.

According to Table 3, for dimension 30, the Friedman value increases from 1.42 at k = 500 to 1.548 at k = 700. In dimension 50, it shows a slight decrease from 1.506 to 1.494 as k increases. Similarly, for dimension 100, the value drops from 1.517 at k = 500 to 1.484 at k = 700. These consistently strong outcomes, even in more complex scenarios, underscore the effectiveness and scalability of using a memory size of k = 700 across varied optimization landscapes.

Table 4 presents the impact of varying population sizes (N = 50, 100, and 200) on the performance of ESSA, evaluated under fixed conditions of 3000 iterations and a problem dimension of D = 30. The results reveal that N = 200 delivers the best performance, achieving a Friedman test value of 1.377 and an optimization effectiveness of 75.86%. Following this, N = 100 ranks second, with a Friedman test value of 1.917 and an effectiveness of 17.24%. These findings suggest that increasing the population size improves the algorithm’s optimization capability, likely due to greater diversity and exploration potential in the search space. However, while N = 200 yields superior results, it also entails a higher computational cost. To strike a balance between performance and efficiency, the population size was set to N = 100 for the remaining evaluations. This decision reflects a practical trade-off, maintaining satisfactory performance while reducing computational overhead.

Table 5 reports the impact of varying the maximum number of iterations (MaxIt = 1500, 3000, and 4000) on the performance of ESSA, evaluated with a fixed population size of N = 100 and a problem dimension D = 30. The results demonstrate that MaxIt = 4000 delivers the best performance, with a Friedman test value of 1.728 and an optimization effectiveness of 56.90%. Following that, MaxIt = 3000 yields the second-best results, achieving a Friedman value of 1.803 and an effectiveness of 31.03%. These findings suggest that increasing the number of iterations enhances ESSA’s performance by allowing more exploration and refinement of candidate solutions. However, longer iterations also result in increased computational time and resource consumption. Therefore, it is essential to maintain a practical balance between computational cost and solution quality. Based on this trade-off, the maximum number of iterations (MaxIt) for all subsequent experiments was calculated using the formula (D × 10⁴)/N, where D is the problem dimension and N is fixed at 100.

Impact analysis of each improvement strategy via ablation experiments

In this subsection, ablation experiments are conducted to evaluate the impact of each proposed search strategy on the performance of ESSA. Four ESSA variants, labeled ESSA-V₁ through ESSA-V₄, are introduced for analysis. In these variants, the proposed search strategies are selectively applied to isolate and assess their contributions to overall performance. The variants are described as follows:

• ESSA-V₁: In this version, the proposed evolutionary search strategies in ESSA are removed, and only Eqs. (2) and (3) from the SSA are retained.

• ESSA-V₂: This variant retains the evolutionary search components defined in Eqs. (7) and (8), while replacing the proposed enhanced SSA search strategy with the original formulation given in Eq. (5) of the canonical SSA.

• ESSA-V₃: This variant replaces Eq. (2) of the canonical SSA with Eq. (7) from the proposed ESSA, while all other components remain unchanged.

• ESSA-V₄: This version substitutes Eq. (3) of the canonical SSA with Eq. (8) from the proposed ESSA, preserving the rest of the algorithm.

The results of the canonical SSA, the proposed ESSA, and its four variants are compared and analyzed in Table 6. The proposed ESSA achieves the highest number of wins, with 22 wins, three ties, and only four losses among all tested algorithms. To statistically validate these performance differences, Friedman test statistics were computed for each algorithm, yielding the following values: SSA = 3.8655, ESSA-V₁ = 2.5534, ESSA-V₂ = 2.2690, ESSA-V₃ = 5.6121, ESSA-V₄ = 5.2966, and ESSA = 1.4034. A lower Friedman ranking score indicates stronger overall performance across the benchmark set. Accordingly, ESSA ranks highest among all variants and baselines. These findings provide compelling evidence that integrating the proposed search strategies significantly enhances the performance of ESSA.

Figure 2 presents a comparison of convergence curves, highlighting the impact of each improvement strategy on the performance of ESSA. In this comparison, the canonical SSA, the proposed ESSA, and its variants (ESSA-V₁ to ESSA-V₄) are evaluated based on their convergence behavior. The results demonstrate that the proposed ESSA, which incorporates all suggested search strategies, achieves faster and more stable convergence compared to both the baseline SSA and the partial variants. This confirms the effectiveness of the integrated enhancements in accelerating convergence and improving solution quality.

Fig. 2.

Comparison of convergence curves illustrating the impact of each improvement strategy.

Evaluation of local and global search capabilities

Functions F₁ and F₃ are categorized as unimodal, meaning they possess a single global optimum with no other local minima present in the search space. This characteristic makes them ideal for measuring exploitation, local search capabilities, and performance. Evaluating these tests provides essential insights into how effectively the algorithm can achieve optimal solutions and refine them in the vicinity of the global optimum. The results presented in Tables 7, 8, and 9 for a dimensions of 30, 50, and 100 indicate that ESSA outperforms competing optimizers in effectively exploiting the unimodal tests. Conversely, functions F₄ to F₁₀ are suitable for assessing the exploration abilities of the proposed algorithm. These multimodal functions contain multiple global optima and several local optima, making them particularly useful for testing the exploration ability within the problem search. The results summarized in the tables demonstrate that ESSA exhibits strong performance in handling multimodal functions when compared to its competitor algorithms. Specifically, in Table 7 (D = 30), ESSA secures six wins and three ties across nine benchmark functions, achieving consistent dominance. In Table 8 (D = 50), the algorithm improves further, with seven wins and two ties. In Table 9 (D = 100), it achieves its highest success rate, with eight wins and only one loss (in function F₆). These outcomes collectively highlight ESSA’s robustness and adaptability in navigating complex search landscapes, positioning it as one of the top-performing algorithms in the benchmark suite.

Evaluation of local optima avoidance

The skill of optimization algorithms to overcome local optima and their overall robustness is evaluated using hybrid and composite test functions by varying the problem’s dimensionality. Hybrid test functions F₁₁-F₂₀ combine different mathematical types, including linear, nonlinear, convex, and non-convex functions, resulting in complex domains with varied optimal sub-regions. These functions are specifically designed to measure the robustness of optimization algorithms. Likewise, composite test functions F₁₂-F₃₀ are created by merging more straightforward functions or transformations. They frequently include characteristics such as shifted optima and varying scales, which present significant challenges for optimization methods. The outcomes summarized in Tables 10, 11, and 12 further underscore the strong performance of ESSA across various dimensions. For a dimensionality of D = 30, ESSA secured 17 wins and three ties out of 20 benchmark test functions. At D = 50, it demonstrated flawless performance, achieving 20 wins out of 20 test functions. Even at the highest dimensionality tested (D = 100), ESSA remained competitive with 18 wins and only two losses. These results consistently position ESSA among the top-performing optimization algorithms in the benchmark suite.

In dimension 30, as shown in Table 10, QANA reached a low average result of 1.457E + 03 for F₁₃, SMO attained a low average result of 1.943E + 03 for test function F₁₈, and ADE achieved an average result of 3.209E + 03 for F₂₆. The proposed algorithm consistently attained the lowest mean value across all runs for the other test functions in this dimension. In dimension 50, as testified in Table 11, ESSA outperforms all competing algorithms, attaining success across all evaluated test functions. In Table 12, which focuses on a problem dimension of D = 100, ESSA shows strong performance by securing the best results in 18 out of 20 test functions. The exceptions are functions F₂₈ and F₃₀, where the QANA algorithm achieves superior outcomes compared to ESSA. These results underscore ESSA’s overall dominance.

Convergence speed analysis

Figure 3 presents the convergence curves for both unimodal and multimodal test functions, namely F₃, F₅, and F₁₀. Figure 4 displays the convergence behavior for hybrid and composite test functions F₂₀, F₂₃, and F₃₀. As shown in these figures, ESSA exhibits two distinct behaviors during the convergence process: rapid exploration in the early stages and steady refinement in the later stages, both of which contribute to its overall effectiveness.

Fig. 3.

Comparison of convergence speed analysis in unimodal and multimodal test functions.

Fig. 4.

Comparison of convergence speed analysis in hybrid and composite test functions.

As shown in these figures, ESSA exhibits two main behaviors throughout its optimization process. In the first behavior, the ESSA algorithm swiftly converges to promising regions with steep slopes, as observed for the multimodal and hybrid test functions F₁₀ and F₂₀ across dimensions 30, 50, and 100. This rapid convergence highlights the algorithm’s capability to approximate these areas and identify optimal solutions accurately. In the second behavior, the proposed ESSA begins by broadly exploring the problem space during the first half of the iterations to identify promising regions. After this phase, it reduces its speed and emphasizes the local refinement of the discovered areas more precisely, resulting in a gradual descent that levels off into a flat region. This convergence behavior is evident in test functions F₃, F₅, F₂₃, and F₃₀ across dimensions D = 30 and D = 50. However, at D = 100, ESSA loses its leading position in function F₃₀, where the QANA algorithm outperforms it.

Friedman statistical test and mean absolute error analyses

Friedman’s test is a non-parametric statistical test used to detect differences among multiple groups. It is calculated using Eq. (19), where Q represents Friedman’s test statistic, N denotes the count of test functions, k indicates the count of algorithms being compared, and ​ refers to the sum of ranks for the j-th optimizer.

19

Given the stochastic behavior of algorithms, a statistical test is applied to rank the algorithms based on their effectiveness and superiority in obtaining the best solution. The optimization values are presented in Tables 13, 14, and 15 for dimensions 30, 50, and 100. At a dimensionality of 30, as shown in Table 13, the proposed ESSA achieves the best Friedman rank across all test functions, except for F₁₄ and F₁₆, where SMO and QANA obtain the top ranks, respectively. As shown in Table 14 for dimension 50, In the case of dimension 50, ESSA maintains superior rankings on all test functions except F₂₂ and F₂₅, where SMO and QANA again outperform it. For the higher dimension of 100, as presented in Table 15, ESSA continues to show competitive performance, achieving the best Friedman ranks on most test functions. However, QANA surpasses ESSA in F₂₈ and F₃₀. These results consistently affirm ESSA’s strong and stable performance across various dimensional complexities, with only a few instances where specific algorithms outperform it on isolated functions.

The computed Q statistic is estimated against a chi-squared distribution with k − 1 degrees of freedom to determine significance. When the Q statistic exceeds the critical value from this distribution, it indicates a statistically significant difference among the groups. The findings reveal significant differences among the algorithms. The proposed ESSA consistently secures first place in dimensions 30, 50, and 100. This demonstrates that ESSA effectively addresses complex problems and provides better solutions than the other tested optimizers.

Figures 5, 6, and 7 illustrate the Friedman ranking of each test function along with the overall scores for dimensions 30, 50, and 100. The proposed ESSA consistently achieved the lowest average ranks. For dimension 30, it attained an average rank of 1.24, as shown in Fig. 5. For dimension 50, the same rank of 1.24 is reported in Fig. 6. In dimension 100, ESSA achieved an even lower average rank of 1.13, as illustrated in Fig. 7. These results demonstrate its superior performance relative to the competing algorithms.

Fig. 5.

Friedman rankings for test functions and their overall scores in dimension 30.

Fig. 6.

Friedman ranks for each test function and their overall rankings in dimension 50.

Fig. 7.

Friedman ranks for each test function and their overall rankings in dimension 100.

The mean absolute error (MAE) measures the accuracy of algorithms by quantifying the difference between optimal values and those obtained at the end of the process. MAE is determined using Eq. (20), where n represents the total count of test functions,  is the ideal value of the i-th test function, and  is the value obtained by the algorithm for the same i-th test function.

20

Table 16 presents the MAE values and corresponding ranks for the IEEE CEC 2017 test suite. The results demonstrate that the proposed ESSA achieves the lowest MAE across all evaluated dimensions: 3.291E + 02 for the 30-dimensional problem, 1.986E + 04 for 50 dimensions, and 1.924E + 03 for 100 dimensions—securing the top rank in each case. Notably, SMO and QANA consistently hold the second and third positions across all three dimensions. These rankings highlight the reliability and superiority of ESSA.

Evaluation of optimization effectiveness

The optimization effectiveness (OE) of the proposed ESSA, along with other comparative optimization algorithms, is rigorously assessed using Eq. (21) and presented in Table 17 for problem dimensions 30, 50, and 100 (expressed in percentage terms). Wins signify the number of test functions in which the algorithm demonstrates superior performance compared to others, Ties indicate instances where multiple algorithms achieve identical results, and Losses refer to cases where the algorithm does not perform as well as competing methods. To ensure a fair and balanced assessment, rather than treating ties as a full win—which could artificially inflate the algorithm’s effectiveness—or as a full loss—which could unjustly diminish its performance—ties are assigned a fractional value by being divided by 2. This approach ensures a more accurate reflection of the algorithm’s comparative efficiency and prevents bias in performance rankings. The results indicate that ESSA achieves high effectiveness, with values of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively.

20

Table 17.

The effectiveness values and ranks in the IEEE CEC 2017 test suits.

Algorithms	Optimization Effectiveness
	D = 30	D = 50	D = 100
SSA18	1.72%	1.72%	0%
QANA40	5.17%	0%	0%
SCHO60	0%	0%	0%
SMO19	3.45%	0%	0%
PO61	0%	0%	0%
HEOA62	0%	0%	0%
ADE35	5.17%	1.72	3.45%
ESSA	84.48%	96.55%	89.66%

Experimental evaluations in optimizing wind farm layout and engineering designs

This section evaluates the performance of the proposed ESSA in addressing the wind farm layout optimization problem and various complex engineering design challenges. To assess its effectiveness, robustness, and real-world applicability, ESSA’s results are benchmarked against those of several state-of-the-art competitor algorithms. The constrained optimization challenges considered include wind farm layout optimization, speed reducer weight minimization, pressure vessel design, welded beam design, and tension/compression spring design (Case 1). Additional test cases involve solving the process flow sheeting problem, configuring a robot gripper, and design of the rolling element bearing. These engineering problems incorporate a mix of equality and inequality constraints, reflecting the complexity and realism of practical design scenarios. The aim is to minimize the objective function , subject to a set of constraints that must be satisfied. Equation (21) defines the real-world constrained optimization problems considered in this study, where the decision variables are represented as . Each inequality constraint must satisfy , while each equality constraint must satisfy .

21

Subject to the constraints:

An equality constraint can be converted into an inequality constraints by utilizing Eq. (22), where ε serves as a tolerance value for the transformation, generally set close to zero.

22

Table 18 outlines the specifics of the problems. The parameter D indicates the problem’s dimensionality, g represents the inequality constraint functions, and denotes the ideal solution to the problem.

Table 18.

Explanation of parameters in the wind farm layout and complex design problems.

Problems	Dimension	g
Wind farm layout	30	91	− 6.2607000000E + 03
Weight minimization of a speed reducer	7	11	2.9944244658E + 03
Pressure vessel design	4	4	5.8853327736E + 03
Welded beam design	4	5	1.6702177263E + 00
Tension/compression spring design (case 1)	3	3	1.2665232788E-02
Process flow sheeting	3	3	1.0765430833E + 00
Robot gripper	7	7	2.5287918415E + 00
Rolling element bearing	10	9	1.4614135715E + 04

Optimizing the layout of wind farms is a crucial aspect of cleaner production systems, aimed at enhancing efficiency and sustainability in energy generation. Figure 8³⁶ depicts the wind farm layout problem. The results are summarized in Table 19. In this challenge, ADE achieves the lowest mean value of −5.9102238400E + 03.

Fig. 8.

Wind farm layout problem³⁶.

Table 19.

Optimization results for complex engineering design problems.

Algorithm	Wind farm layout problem			Weight minimization of a speed reducer problem
	Average	Std	Min	Average	Std	Min
SSA	−5.8363194820E + 03	9.2806629000E + 01	−6.0047360360E + 03	3.0217525427E + 03	1.4525201345E + 01	3.0036696287E + 03
QANA	−5.7489623650E + 03	2.7615228000E + 02	−6.1481893640E + 03	2.9944710661E + 03	5.4653829700E-13	2.9944710661E + 03
SCHO	−5.6236718710E + 03	8.7927946000E + 01	−5.7714623630E + 03	3.0082967481E + 03	3.6881627748E + 00	3.0006884350E + 03
SMO	−5.3834120290E + 03	1.0363605400E + 02	−5.6034050260E + 03	2.9944710661E + 03	4.0104946660E-13	2.9944710661E + 03
PO	−5.6118135990E + 03	1.2098759500E + 02	−5.8335336810E + 03	3.0102755199E + 03	5.1358175674E + 00	3.0025281620E + 03
HEOA	−5.5417963300E + 03	1.9116375800E + 02	−5.7140323840E + 03	4.4942062665E + 03	9.2820007235E + 02	3.2192793338E + 03
ADE	−5.9102238400E + 03	3.9891825000E + 01	−5.9916116580E + 03	2.9944710661E + 03	5.4653829700E-13	2.9944710661E + 03
ESSA	−5.7350615070E + 03	3.0350710600E + 02	−6.1559321810E + 03	2.9944710661E + 03	4.7934579624E-13	2.9944710661E + 03

Algorithm	Pressure vessel design problem			Welded beam design problem
	Average	Std	Min	Average	Std	Min
SSA	6.5749411717E + 03	7.7183734368E + 02	6.0783592583E + 03	1.8825885700E + 00	1.3711655371E-01	1.7680813724E + 00
QANA	5.9301175392E + 03	1.1007766410E + 02	5.8853328157E + 03	1.7248523086E + 00	2.8164982938E-12	1.7248523086E + 00
SCHO	6.6786836790E + 03	4.5710597574E + 02	5.9998435861E + 03	1.8184053146E + 00	9.5207611478E-02	1.7318535739E + 00
SMO	5.8853327814E + 03	2.4118711220E-05	5.8853327736E + 03	1.7248523086E + 00	2.5734760544E-12	1.7248523086E + 00
PO	6.4948416446E + 03	5.4691631572E + 02	5.8948967274E + 03	1.7425607350E + 00	1.9698759785E-02	1.7288666310E + 00
HEOA	3.4454869190E + 04	4.5154436990E + 04	6.4990284487E + 03	3.7156816800E + 00	8.8518559497E-01	1.9335398605E + 00
ADE	6.0881808175E + 03	2.2171063574E + 02	5.8884556321E + 03	1.8952224608E + 00	1.5563779926E-01	1.7271872932E + 00
ESSA	5.8853327736E + 03	8.4974495766E-08	5.8853327736E + 03	1.7248523086E + 00	1.2299084006E-12	1.7248523086E + 00

Algorithm	Tension/compression spring design (case 1) problem			Process flow sheeting problem
	Average	Std	Min	Average	Std	Min
SSA	1.3121788366E-02	7.0365040061E-04	1.2704745379E-02	1.1806173063E + 00	8.9572672357E-02	1.0765431146E + 00
QANA	1.2681273233E-02	2.0790021669E-05	1.2665571841E-02	1.1652662956E + 00	8.9492006478E-02	1.0765430978E + 00
SCHO	1.3388603784E-02	6.8615331463E-04	1.2717548385E-02	1.3753140877E + 03	2.0328776643E + 03	1.0772974047E + 00
SMO	1.2665526427E-02	4.6664152278E-07	1.2665232931E-02	1.0765430833E + 00	2.3405556457E-16	1.0765430833E + 00
PO	1.2774910781E-02	1.3104892031E-04	1.2666706070E-02	1.1809317499E + 00	8.9167929793E-02	1.0765818371E + 00
HEOA	1.9785752805E-02	2.7101868986E-03	1.4953184936E-02	1.2415894591E + 00	2.6596465736E-02	1.1658945906E + 00
ADE	1.2789773713E-02	1.1597340097E-04	1.2697719359E-02	1.1367160578E + 00	7.9487687598E-02	1.0765430833E + 00
ESSA	1.2665239862E-02	1.7440159907E-08	1.2665232788E-02	1.0765430833E + 00	2.3405556457E-16	1.0765430833E + 00

Algorithm	Robot gripper problem			Rolling element bearing problem
	Average	Std	Min	Average	Std	Min
SSA	3.4273908274E + 00	4.9055980097E-01	2.7392393558E + 00	1.7122377196E + 04	2.0880877694E + 02	1.6985856868E + 04
QANA	2.5608629661E + 00	2.8975452942E-02	2.5442554486E + 00	1.6958202287E + 04	3.8347663699E-12	1.6958202287E + 04
SCHO	4.2152549904E + 00	1.3631385255E + 00	3.4865478479E + 00	1.7082602533E + 04	3.4823178061E + 01	1.7038730796E + 04
SMO	2.5512220090E + 00	2.2787279207E-02	2.5437871153E + 00	1.6958202287E + 04	3.8347663699E-12	1.6958202287E + 04
PO	3.8301955611E + 00	7.8757472482E-01	2.6817255252E + 00	1.7049035154E + 04	2.5298025043E + 01	1.6978141939E + 04
HEOA	8.1042470228E + 00	1.4588094255E + 00	5.3741808447E + 00	2.0685397194E + 04	4.9508812936E + 03	1.7058766922E + 04
ADE	3.4873975174E + 00	2.8961544622E-01	2.9741542155E + 00	1.6958202287E + 04	3.8347663699E-12	1.6958202287E + 04
ESSA	2.5437855670E + 00	2.0690042074E-10	2.5437855668E + 00	1.6958202287E + 04	3.8347663699E-12	1.6958202287E + 04

Meanwhile, the proposed ESSA reaches the lowest minimum value of −6.1559321810E + 03. For the speed reducer weight minimization challenge, SMO, ADE, and ESSA all yield an excellent mean value of 2.9944244658E + 03. In the case of the pressure vessel design problem, only the ESSA successfully finds the finest value of 5.8853327736E + 03. Similarly, the ESSA again demonstrates its effectiveness for the tension/compression spring and welded beam design by attaining the finest value of 1.7248523086E + 00. In the process flow sheeting problem, the SMO and ESSA algorithms achieved a value of 1.0765430833E + 00. For the robot gripper problem, the proposed ESSA obtained a value of 2.5437855670E + 00. Regarding the rolling element bearing problem, the SMO, ADE, and ESSA algorithms reached a best value of 1.6958202287E + 04.

Discussion

This section offers a thorough analysis of the findings, explaining how the proposed evolutionary search strategies, an enhanced SSA search strategy, and an advanced memory mechanism enhance ESSA’s optimization efficiency and superiority.

The performance and applicability of ESSA were evaluated using two distinct sets of benchmark functions. The section Experimental Evaluations in Solving Global Optimization Problems examines ESSA’s capability to identify optimal solutions across a variety of search landscapes. The section Experimental Evaluations in Optimizing Wind Farm Layout and Engineering Designs highlights ESSA’s real-world applicability by demonstrating its effectiveness in addressing practical challenges, such as clean production systems and complex engineering design tasks. Meanwhile, in these evaluations, ESSA’s performance was systematically assessed against seven leading optimization algorithms—SSA, QANA, SCHO, SMO, PO, HEOA, and ADE—providing a comprehensive benchmark of its strengths and advancements over competing optimizers. The findings of the first evaluation set are summarized as follows.

• Table 6 and the convergence curves in Fig. 2 show the impact analysis of each improvement strategy of the proposed ESSA through ablation experiments, where the effectiveness of individual components is systematically evaluated. The results highlight that the proposed ESSA, incorporating all suggested enhancements, consistently outperforms its variants and the original SSA in terms of convergence speed and solution quality.

• ESSA’s optimization performance, as detailed in Tables 7, 8, and 9, underscores its robust local and global search capabilities across problems. In dimension 30, ESSA outperformed all competitors, securing six wins, three ties, and zero losses. Its dominance further increased in dimension 50, achieving seven wins and two ties without a single loss. Its success is largely credited to the evolutionary SSA search strategies and enhanced SSA search strategy, which together boost its exploration and exploitation abilities, enabling efficient convergence on high-quality global optima.

• Tables 10, 11, and 12 highlight ESSA’s exceptional capability to escape local optima and deliver consistently strong performance across test functions F₁₁ to F₃₀. In the 30-dimensional setting, ESSA achieved 17 wins, with no ties and only three losses, underscoring its effectiveness in navigating complex search landscapes. At dimension 50, it demonstrated complete dominance by winning all 20 test cases, while at dimension 100, it maintained top-tier performance with 18 wins and only two losses. This outstanding performance is largely attributed to its advanced memory mechanisms, which dynamically capture and retain both high-quality and suboptimal solutions throughout the optimization process.

• Figs. 3 and 4 illustrate two key convergence behaviors in ESSA, showcasing its strong optimization capability. The first behavior exhibits rapid convergence, where ESSA efficiently navigates toward steep-sloped optimal regions, guided by its proposed evolutionary SSA search strategies and advanced memory mechanism, which track promising solutions. The second behavior follows an exploration-refinement approach, where ESSA initially expands its search space to identify optimal regions across diverse landscapes. As the search progresses, ESSA gradually shifts focus, refining solutions through its enhanced SSA search strategy, effectively preventing premature convergence while ensuring higher solution accuracy.

• Tables 13, 14, and 15 highlight ESSA’s exceptional performance, securing the top Friedman rank across 27 test functions in dimensions 30, 50, and 100. This consistent ranking underscores ESSA’s optimization superiority, demonstrating its ability to efficiently explore search spaces, escape local optima, and accelerate convergence toward high-quality solutions.

• Table 17 highlights the optimization effectiveness of the proposed ESSA, which achieved success rates of 84.48% at dimension 30, 96.55% at dimension 50, and 89.66% at dimension 100. These consistently high success rates emphasize ESSA’s competitive advantage over alternative optimization algorithms, particularly in handling complex, high-dimensional problem spaces.

The second evaluation set focuses on optimizing cleaner production systems and complex design challenges, assessed using CEC 2020-constrained engineering problems. These problems evaluate the applicability of the proposed ESSA, ensuring it effectively balances feasibility and performance in complex engineering scenarios. Table 19 presents the optimization results across various engineering design problems, demonstrating ESSA’s strong performance. ESSA achieves the minimum value for the wind farm layout problem, surpassing other methods. For the speed reducer problem, ESSA, SMO, and ADE collectively attain the minimum and best average values. In the pressure vessel design problem, ESSA records the best average value, showcasing its efficiency. Similarly, ESSA achieves the best average value in the welded beam design problem. For the tension/compression spring design (case 1) problem, ESSA maintains the best average value, further reinforcing its optimization capability. In the process flow sheeting problem, ESSA and SMO both secure the best average value, demonstrating their effectiveness. The robot gripper problem sees ESSA achieving the best average value. In contrast, in the rolling element bearing problem, ESSA, SMO, and ADE achieve the best average value, indicating strong optimization performance across various complex scenarios.

Conclusion and future work

This study introduces ESSA, a new algorithm designed to address complex real-world optimization challenges. ESSA integrates evolutionary search strategies with an improved SSA framework and incorporates dual archive structures—best and inferior—managed through a stochastic universal selection mechanism that prioritizes individuals based on fitness. Experimental results demonstrate that ESSA significantly outperforms the original SSA and other benchmark algorithms in both local and global search efficiency, effectively avoiding local optima and achieving faster convergence. Statistical analyses confirm that ESSA consistently ranks first across all test functions, attaining top optimization effectiveness scores of 84.48%, 96.55%, and 89.66% for dimensions 30, 50, and 100, respectively, surpassing the scores of competing optimizers. Moreover, ESSA demonstrates strong practical applicability, yielding effective results in real-world scenarios such as wind farm layout optimization and complex engineering design problems.

ESSA demonstrates remarkable optimization capabilities; however, its computational complexity presents challenges, as the integration of multi-search strategies and an advanced memory mechanism can significantly increase processing demands, particularly in highly complex scenarios. Additionally, parameter sensitivity remains a critical factor, requiring extensive experimentation to refine configurations across diverse problem domains and ensure optimal efficiency. To address ESSA’s shortcomings and enhance its optimization performance, reducing computational complexity can be achieved by utilizing parallel computing techniques or distributed processing, thereby enabling more efficient handling of complex optimization tasks. To address the issue of parameter sensitivity, future work could focus on developing self-adaptive parameter tuning mechanisms that will allow ESSA to dynamically adjust key settings based on problem-specific characteristics. This would reduce reliance on manual tuning and enhance the algorithm’s robustness across diverse optimization scenarios, including sustainability-oriented engineering, the integration of renewable energy technologies, or the optimization of intelligent manufacturing systems.

Author contributions

H. Z. wrote the conceptualization, methodology, software, validation, investigation, and resources and contributed to the original draft preparation, writing, review, and editing.

Data availability

The source code is available at https://hodazamani.github.io.

Declarations

Competing interests

The author declares no competing interests.