The main contributions of the paper include knowledge consolidation, noting the strengths and drawbacks, proposing improvements by creative approaches, and showing its usefulness in optimization domains. The contributions are as follows: (a)

Analysis of SO’s Methodology: In this research, the theoretical approach and basic search mechanisms of the SO will be considered, with its approach for maintaining a trade-off between exploration and exploitation. The effect of various control parameters on the performance of SO will be discussed, and approaches for controlling them will also be evaluated.

Various comprehensive surveys have been published on Nature-inspired and Metaheuristic optimization algorithms, providing broad taxonomies, classifications, and general performance discussions. However, most of these studies adopt a macro-level perspective, summarizing multiple algorithm families without offering an in-depth methodological and critical examination of any single algorithm, and the same has been highlighted in Table 1. In contrast, the present review adopts a micro-level analytical perspective, focusing exclusively on the SO algorithm. The distinctive features of this review are:

A detailed theoretical dissection of SO’s exploration–exploitation strategy, search operators, and control parameter influence, which is generally not addressed in broad surveys.

A survey study on the SO often considers research questions about the algorithm’s performance, modifications, applications, and comparative analysis. Typical key research questions consist of:

The principled methodological and theoretical research questions are as follows:

What are the fundamental principles of the SO, and how has a mathematical optimization framework been established based on the mating behaviour and foraging strategies of snakes in their natural environment?

The research questions pertaining to the different variants of SO are as follows:

What significant enhancements and hybridizations of the SO have been documented in the literature?

The research questions concerning the applications and performance assessment of SO are as follows:

What are the primary application domains where the SO and its variants have been successfully employed?

The research questions pertaining to the challenges and future study paths of SO are as follows:

What are the outstanding problems and ongoing scientific inquiries related to the SO?

These research questions can help build the basis for a rather comprehensive paper on the subject of the SO in terms of a literature survey.

The remaining sections of the paper are organized around seven key pillars, structured as follows: Research Methodology and survey taxonomy are discussed in Section 2. A comprehensive overview of the SO Algorithm and its mathematical formulation is presented in Section 3. Literature review of SO-based research is discussed in Section 4. A detailed overview of SO’s applications across diverse sectors is performed in Section 5. Theoretical comparison with other MAs is presented in Section 6. Performance evaluation of SO and other tested MAs has been performed in Section 7. The conclusion and future research are elaborated in Section 8.

The foremost objective of the preliminary study phase is to attain a comprehensive understanding of the scope of the literature search. This necessitates determining potential sources for further investigation and performing a preliminary search using pertinent keywords. Initially, the search was conducted based on several key terminologies associated with the algorithm, such as Snake Optimizer, Snake Optimizer Algorithm, Snake Optimizer Nature-Inspired Algorithm, (though not limited to), as illustrated in Fig. 3. Those papers that included these keywords in their title were chosen. To compile the various published articles related to SO, several well-known publishers, including IEEE, Elsevier, Springer, MDPI, Nature Portfolio, Taylor & Francis, Wiley, Frontiers, and many others, have been included. To facilitate this, Google Scholar, a commonly used web search engine making full-text research articles available across multiple disciplines, has been taken into account.

This phase basically is included to guarantee the relevance of the selected articles with regard to the research theme. The screening process focuses on identifying the most suitable publications that align with the research objectives while eliminating those that do not meet the specified criteria. This screening, without any doubt, helped eliminate irrelevant research papers. To achieve our objective, we establish specific criteria for the article screening phase, (i) inclusion and (ii) exclusion criteria. Inclusion criteria include: Research articles published only in reputable journals, Research articles written in the English language, and a full article available for download. On the other hand, Exclusion criteria comprise: Research articles published as book chapters, Research papers available in languages besides English, and the full version of the article not available for download.

The important characteristics of the selected 129 different SO-related journal and conference papers are meticulously recorded and classified based on several parameters, namely titles, authors, publication year, publishers, citations, application area, and type, throughout the data extraction phase.

Among these 129 papers, the numbers of journal and conference papers are 104 and 25, respectively. The graphical representation of this information is presented in Fig. 4. Year-wise publications of these 129 considered papers for the survey are pictorially presented in Fig. 5. The publishers of these papers are depicted in Fig. 6, and this figure clearly demonstrates that IEEE is at the top of the list. Elsevier and Springer, two major contributors to the body of literature on this subject, are closely followed. Wiley, MDPI, and Nature get the fourth rank. Fig. 7 shows year-wise citations of the original source paper of SO. The citations of the SO are increasing, which proves the popularity of the SO in the MA-based research domain. MAs that were published in 2022 and achieved excellent citations are presented in Fig. 8. The figure clearly shows that SO gets fourth rank by achieving 1206 citations. The surveys or review papers on the first three ranked MAs of 2022 have already been published in the literature. Therefore, we opt for SO for the survey.

The SO algorithm is highlighted as Algorithm 1, and the phases of the SO algorithm are detailed as follows, with its diagrammatic representation depicted in Fig. 9.

Phase 1: Population Initializing Phase

Like all other metaheuristics, SO [11] initiates the optimization process via the creation of randomly and uniformly distributed. In essence, this is done to guarantee diversity in the initial solution, giving the search mechanism a solid base. Eq. (1) can be used to obtain the initial population. Furthermore, the population is equally split into two groups, assuming a 50% distribution of male and female snakes. Eqs. (2) and (3) are used to divide the population. (1)Xi=Xmin+r×(Xmax−Xmin)where, Xi is the position of the ith individual, r is a random number within the range [0, 1] and Xmax,Xmin represents the lower and upper bounds of the problem, respectively. (2)Nm≈N2 (3)Nf=N−Nmwhere, N represents the total number of individuals, Nm denotes the number of male individuals and Nf represents the number of female individual numbers.

Phase 2: Evaluation Phase

The two groups created in phase 1 are evaluated to determine the best individual in each group, thus identifying the best male (f_best,m), female (f_best,f), and the optimal food position (f_food). Further, the temperature, Temp is calculated using Eq. (4), where t and T in the equation basically refer to the current iteration and the total number of iterations, respectively (4)Temp=exp⁡(−tT)

This phase also deals with defining the quantity of food, typically denoted as Q, and the same is obtained using Eq. (5). (5)Q=c1*exp(t−TT)where, c1 is constant with the value assigned as 0.5.

Phase 3: Exploration Phase (when food is unavailable)

The exploration phase begins whenever the food is unavailable prompting the snakes to explore the search space. In order to determine when to initiate phase 3, a threshold value of 0.25 is set. If the quantity of food, Q falls below threshold, the snakes initiate the exploration phase by choosing any random position and update their position accordingly. This process is carried forward using Eqs. (6) and (8).

In regard to male snakes, ith male position is determined using Eq. (6) as given below: (6)Xi,m(t+1)=Xrand,m(t)±c2×Am×((Xmax−Xmin)×rand+Xmin)where, Xi,m refers to ith male position, Xrand,m refers to position of random male, rand is a random value within the range [0, 1], c2 is constant with the value assigned as 0.05 and Am basically refers to male ability to find food and is calculated using Eq. (7). (7)Am=exp⁡(−frand,mfi,m)where, frand,m refers to fitness of Xrand,m and fi,m refers to fitness of ith individual in male group.

Similarly, in regard to male snakes, ith female position is determined using Eq. (8) as given below: (8)Xi,f(t+1)=Xrand,f(t)±c2×Af×((Xmax−Xmin)×rand+Xmin)where, Xi,f refers to ith female position, Xrand,f refers to position of random female, rand is a random value within the range [0, 1], c2 is constant with the value assigned as 0.05 and Af basically refers to female ability to find food and is calculated using Eq. (9). (9)Af=exp⁡(−frand,ffi,f)where, frand,f refers to fitness of Xrand,f and fi,f refers to fitness of ith individual in female group.

Phase 4: Exploitation Phase (when food is available)

This phase is executed when the food is available which means that Q > Threshold (0.25). Since, the food alone do not determine the mating process rather snakes do rely on temperature as well, so another threshold value in regard to temperature is set to 0.6 and checked accordingly.

If Q (Food Quantity) > Threshold (0.25)

If the temperature > Threshold (0.6) (hot) then

The snakes will navigate towards only food and exploitation is carried out using Eq. (10). (10)Xi,j(t+1)=Xfood(t)±c3×Temp×rand×(Xfood−Xi,j(t)where, Xi,j is the position of the individual male or female, Xfood refers to the position of the best individuals, and c3 is constant and is set to 2.

However, If the rand < Threshold (0.6) then the snake will either enter the Fighting Phase else Mating Phase.

Phase 5: Fighting Phase

Male snakes compete for mating rights, often engaging in physical battles to attract females, as represented by Eqs. (11) and (12). (11)Xi,m(t+1)=Xi,m(t)+c3×FM×rand×(Q×Xbest,f−Xi,m(t))where, Xi,m refers to ith male position, Xbest,f refers to the position of the best individual in female group, and FM is the fighting ability of male agent.

Also, (12)Xi,f(t+1)=Xi,f(t)+c3×FF×rand×(Q×Xbest,m−Xi,f(t))where, Xi,f refers to ith female position, Xbest,m refers to the position of the best individual in male group, and FF is the fighting ability of female agent.

FF and FM are further evaluated using Eqs. (13) and (14). (13)FM=exp⁡(−fbest,ffi) (14)FF=exp⁡(−fbest,mfi)where, fbest,f refers to fitness of the top performing female agent in the group, fbest,m refers to fitness of the top performing male agent in the group.

Phase 6: Mating Phase

However, female snakes eventually make the decision of choosing one male snake among the many from the competition. Eqs. (15) and (16) depicts the mating procedure. (15)Xi,m(t+1)=Xi,m(t)+c3×Mm×rand×(Q×Xi,f(t)−Xi,m(t)) (16)Xi,f(t+1)=Xi,f(t)+c3×Mf×rand×(Q×Xi,m(t)−Xi,f(t))where, Xi,f refers to ith female position, Xi,m refers to the position of the corresponding male agent, and Mm and Mf denotes the mating potential of the male and female snakes, respectively and are determined with the help of Eqs. (17) and (18). (17)Mm=exp⁡(−fi,ffi,m) (18)Mf=exp⁡(−fi,mfi,f)

Phase 7: Termination Phase

Upon hatching, the worst male and female snakes are identified and replaced using Eqs. (19) and (20). The algorithm terminates once the predefined number of iterations is reached. (19)Xworst,m=Xmin+rand×(Xmax−Xmin) (20)Xworst,f=Xmin+rand×(Xmax−Xmin)

The computational complexity of any algorithm typically defines the amount of time and memory an algorithm requires as the input size grows. The key complexity factors in snake optimizer algorithms include Population size (N), Problem Dimension (D), Number of Iteration (T), and Fitness function cost (f(D)). In each iteration, SO typically performs fitness evaluation for all snakes represented by O (N × f(D)), position updates such as movement, mating, fighting behavior denoted as O (N × D), which is the vector operations over D dimensions, sorting and ranking snakes denoted as O (N log N). In such a scenario, the total time complexity over T iteration is represented using Eq. (21). (21)TimeComplexicity=(O(T×(N×f(D)+N×D+Nlog⁡N)))

In most of the optimization problems wherein fitness evaluation dominates, and sorting is negligible or avoided, time complexity is denoted using Eq. (22). (22)TimeComplexity=(O(T×N×D))or(O(T×N×f(D)))

On the other hand, the space complexity of the SO is basically determined by the data structure used to store the population and the related information, such as Population position (X), Fitness values, the required O(N) memory, Temporary position matrix, and Global best storage. Hence, the overall space complexity of SO is dominated by population storage and is depicted in Eq. (23). (23)SpaceComplexity=O(N×D)

Table 3 presents a comparative analysis of the theoretical time complexity and structural properties of the Snake Optimizer (SO) and several well-established metaheuristic algorithms.

Parameter sensitivity basically refers to how the algorithm’s performance is affected by changes in its control parameters. The performance of the SO is strongly influenced by several core parameters that administrate the balance between exploration and exploitation, convergence speed, and solution diversity. Some of the control parameters are Temperature (Temp), Population size (N), Maximum Iterations (T), Energy parameter (E), Food quantity (Q), Male-Female ratio, and Threshold parameter (θ). (a)

Temperature (Temp): This is one of the crucial parameters that typically controls behavioral mode switching and determines whether a snake fights or mates. The high temperature leads to aggressive exploration, whereas the low temperature indicates stable exploitation. If the temperature drops drastically, the algorithm tends to miss out the global optimum and give priority to exploitation. Therefore, the gradual cooling schedule for this parameter is important as a slower drop guarantees well exploration in the early stages.

The Snake Optimizer algorithm, despite being implemented in various optimization tasks, generates effective results, but it suffers from a few drawbacks when used in complex or multi-modal search spaces. One major drawback is its tendency toward premature convergence, particularly when dealing with complex, multimodal, or high-dimensional search spaces. This behaviour may cause the algorithm to become trapped in local optima, thereby limiting solution quality. Additionally, the convergence speed of the basic SO algorithm may be slow in later iterations due to insufficient population diversity, which reduces its exploration capability. Additionally, the SO algorithm is quite sensitive to parameter settings, thus it needs careful fine parameter tuning to perform effectively across different problem domains. Furthermore, algorithms are not robust in noisy and dynamic situations, and different fitness landscapes may challenge the algorithm’s ability to adapt effectively and sustain a premium solution over time. Table 4 highlights the key strengths and limitations of the SO algorithm.

Thereby to address the limitations of the original SO algorithm, several research initiatives have focused on developing numerous variants of SO, namely, improved, hybrid, binary, and multi-objective SO, which aim to overcome the weaknesses of the standard SO. These enhanced versions introduce mechanisms such as adaptive parameter adjustment, diversity preservation strategies, and hybridization with other optimization techniques. Such modifications improve the balance between exploration and exploitation, enabling the algorithm to escape local optima and converge more efficiently. Table 5 is tabulated to clearly highlight the differences and characteristics of improved SO algorithms and standard SO.

An improvised variant of the SO is an enhanced version of the original algorithm, developed to polish the algorithm, guaranteeing it is more well-aligned with the specific features of the problem at hand. Metaheuristic improvements are typically intended to cater to the inadequacies of the original algorithm, in so doing, strengthening its capacity to recognize solutions that are optimal or close to optimal. These changes could include operator tweaks, parameter tuning, or modifications to the search protocols. Since real-world optimization problems vary in complexity, and a more polished algorithm can produce faster, more accurate, and more reliable solutions in a variety of applications, creating improvised versions becomes imperative. The major improvement strategies that are utilized in the literature of SO are depicted in Fig. 10. However, it is noticed that orthogonal learning, fitness distance balance-based selection mechanism, Weibull distribution, local escaping operator, and multi-population strategies are not utilized for the enhancement of SO.

Wang et al. [36] in their work suggested an enhanced version of SO, named ISO (Improved Snake Optimizer), designed to enhance the convergence ability and stability for solving complex continuous optimization problems. The proposed ISO integrates a chaotic mapping strategy for generating a uniform and randomly distributed initial population, a forced switching mechanism to balance SO’s updating phases and improve convergence speed, a Whale Optimization Algorithm (WOA) variant to enhance exploration, and an optimal domain perturbation strategy to refine solutions near the best results. ISO outperforms classical SO and eight other classical or recent algorithms, showcasing higher convergence efficiency (94/160) and stability (77/160) on CEC-2021 test functions, also achieving 100% convergence efficiency and 96.25% stability in applicable functions. Three real-world continuous optimization problems were used to further confirm its usefulness, and discretization may be used in combinatorial optimization in the future. Zhang et al. [37] propose a multi-strategy Integrated Snake Optimizer (ISO) for solving task scheduling problems in cloud-fog computing environments, where efficient computational resource allocation is crucial to minimizing transmission delay and total costs. To enhance the standard Snake Optimizer, several optimization strategies are integrated. Circle map initialization enhances population diversity as well as traversal, guaranteeing a well-searched space distribution. Synergistic Optimization Strategy further supports the global search by boosting population diversity, while the Elite Reverse Learning Mechanism (ERLM) accelerates convergence, thus improving solution quality by pulling out the best-performing individuals. Furthermore, Evolutionary Population Dynamics presents mutation to additionally expand the search and avoid premature convergence. The proposed ISO is compared with Genetic Algorithm (GA), Wildebeest Herd Optimization (WHO), Grey Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), Harris Hawks Optimization (HHO), Arithmetic Optimization Algorithm (AOA), and African Vultures Optimization Algorithm (AVOA) on multiple datasets and the experimental result clearly indicate that ISO demonstrates better convergence, lower costs, improved response speed, and optimized transfer paths, making it a more efficient approach for cloud-fog computing resource allocation. Li et al. [38] introduced the Multi-Task Snake Optimization (MTSO) algorithm, an improved variant of SO basically designed for Multi-Task Optimization (MTO), in order to address challenges such as low optimization precision and high computational costs. The proposed algorithm typically operates in two phases: first, independently solving each optimization task, and second, enhancing performance through knowledge transfer between tasks. The transfer mechanism is solely guided by the probability of knowledge exchange and elite selection process that allows the algorithm to either integrate superior knowledge from other tasks or else improve solutions through the means of self-perturbation. Experimental results demonstrate that the proposed MTSO outperforms existing MTO algorithms, achieving superior accuracy on benchmark functions and real-world engineering problems, such as planar kinematic arm control, robot gripper optimization, and car side-impact design, proving its effectiveness in handling multiple optimization tasks simultaneously. Song et al. [39] proposed a hybrid algorithm termed the Modified Snake Optimizer (MSO) specifically designed for color image thresholding segmentation, with a primary application in agriculture for early identification, diagnosis, and treatment of rice disease. MSO integrates numerous techniques: a dynamic adaptive parameter tuning method to speed up convergence by fine-tuning the algorithm’s parameters; an improved global position update formula incorporates guidance from the optimal individual and random disturbances to balance global and local searches; Lévy flight is introduced in the flight mode to help the optimizer escape local optima; a balancing strategy in the mating mode enables effective sharing of information among sub-populations; and a hill-climbing jump operation is applied to the optimal individual to overcome local stagnation. These integrations jointly boost the algorithm’s performance, resulting in more accurate segmentation outcomes compared to traditional methods and other optimization techniques. Kong et al. [40] proposed an Improved Snake Optimizer (ISO)-based MPPT control strategy, typically designed for renewable energy applications, precisely for optimizing power converters in photovoltaic systems. The proposed methods hybridize the traditional Snake Optimizer with several enhancement techniques, namely a Lévy flight strategy to enhance global search capabilities and quickly bypass local optima, a dynamic exploitation probability to prevent premature convergence, and chaos theory to increase the diversity of snake positions for a more robust search. Each considered in the proposed method contributes significantly to refining the tracking performance, thus accelerating convergence, reducing oscillations, and further minimizing power loss under real-world conditions. Experimental results show that the ISO achieves a maximum tracking efficiency of 99.99% and a response time of only 0.08 s thereby significantly outperforming the conventional SO. An enhanced Snake Optimization algorithm’s variant, called SO-OBL, that integrates Opposition-Based Learning (OBL) to improve global optimization and for multilevel image segmentation was proposed by Houssein et al. [41]. Specially designed for medical imaging applications, specifically for accurate liver disease segmentation from CT scans. In this proposed hybridized approach, the traditional SO algorithm is boosted by OBL to improve the search process and also to avoid local optima, while an optimized multilevel thresholding technique leveraging Otsu’s function is employed to effectively demarcate liver boundaries despite uneven presence and unclear edges. Experiments on CEC’2022 test functions and segmentation tests demonstrate that the proposed SO-OBL outperforms eleven state-of-the-art metaheuristics, achieving superior metrics, thereby enhancing segmentation performance in the field of medical imaging. Mai et al. [42] in their work proposed an enhanced Snake algorithm, termed as ISASO, for accurate parameter identification of photovoltaic (PV) models, basically to optimize the operation and control of PV systems. The proposed ISASO hybridizes the traditional Snake Optimization algorithm with Subtraction Average-Based Optimization (SABO) to improve global search capability by updating agent positions by means of a consistent arithmetic mean, thus avoiding premature convergence towards local optima. In addition, a tent chaotic map is employed for population initialization for diversity improvement, while a dynamic learning factor and adaptive inertia weight strategy are blended together to accelerate convergence and exploit and explore trade-offs. Tested on CEC2005 benchmark functions and many PV models, the proposed ISASO exhibits better parameter identification accuracy with the lowest values of Root Mean Square Error (RMSE) and superior performance compared to other metaheuristics. Yang et al. [43] in their work tried to optimize the design of Hybrid Wind-Wave Energy Converters (HWWECs) in order to maximize power output and operating stability. By integrating a V27-225 kW wind turbine with a wave energy converter, the system enhances energy efficiency and reliability. The work suggests an Enhanced Snake Optimizer (ESO), based on chaotic initialization, asynchronous learning factors, and Lévy flight, to optimize the array configuration of several HWWECs. Simulation outcomes using various configurations (three, six, and twelve buoys) prove ESO superior to other algorithms, producing the greatest absorption of power, with a gain of 144.337 kW compared to the baseline Snake Optimizer. Wang and Wang [44] proposed a modified variant of SO known as Multi-Strategy Enhanced Snake Optimizer (EMSO), designed to improve Quantitative Structure-Activity Relationship (QSAR) modelling by optimizing hyperparameters and selecting relevant molecular descriptors. It introduces new food emergence and temperature change mechanisms to balance exploration and exploitation, along with a death and regeneration strategy to escape local optima. Moreover, new food searching, fighting, mating, and eating formulas improve search accuracy and efficiency. Compared on benchmark functions, engineering design problems, and feature selection tasks, the model performs better than algorithms currently available. Used in QSAR modelling for predicting environmental toxicity, it performs improved hyperparameter tuning and selects a lower number of more informative features, constituting a valuable tool for improving QSAR models. Nafeh et al. [45] introduce an optimization approach for the sizing of a photovoltaic (PV) battery grid-connected system for electric vehicle fast charging stations in Cairo, Egypt. The major aims are minimizing the total system cost and maintaining energy balance. A new energy management strategy is formulated based on two price models to maximize power flow and find optimal component sizes. Furthermore, a modified Snake Optimization (MSO) algorithm is introduced and simulated in MATLAB to solve the optimization problem. A comparative study with four other meta-heuristic algorithms validates the better performance of MSO. A techno-economic analysis also validates the feasibility of the proposed system throughout its lifetime with efficient energy management and optimal cost-effectiveness. Zheng et al. [46] proposed a modified version of the traditional SO, named the Snake Optimizer based on Sobol Sequential Nonlinear Factors and Different Learning Strategies (SNDSO), aimed at solving issues such as vulnerability to local optima, bad convergence in high-dimensional problems, and adversity in discretized and multi-constraint situations. The proposed algorithm delivers three main enhancements: Sobol sequences to optimize initial population distribution, nonlinear factors employing the inverse tangent function to induce a balance between exploration and exploitation, and learning strategies to promote population diversity and avoid premature convergence. The efficacy of SNDSO is established through extensive testing, including CEC2015 and CEC2017 benchmark functions for optimization in high-dimensional problems, twelve feature selection problems for discretized situations, and five real-world multi-constraint optimization problems. Experimental outcomes affirm SNDSO’s excellence in outperforming the shortcomings of SO, positioning it as a robust algorithm for sophisticated optimization problems. An Improved Snake Optimizer-based Deployment Method (DMISO) for sea disaster monitoring, suggested by Jin et al. [47], uses an AHP-based hierarchical deployment technique to coordinate sensor positioning with sea disaster spatial distribution. Subsequently, an Improved Snake Optimizer (ISO), reinforced with Opposition-Based Learning (OBL), is utilized to deploy nodes efficiently while maintaining the trade-off between network coverage and cumulative energy cost. Moreover, a Path Allocation Strategy (PAS) is used to redistribute nodes so that energy consumption is evenly distributed throughout the network. Simulation results show that DMISO decreases energy consumption by 35.46% over the Underwater Fruit Fly Optimization Algorithm (UFOA) while enhancing coverage by 4.77% over UFOA and 17.4% over random deployment. In addition, DMISO balances energy consumption efficiently and prolongs the entire network lifetime, hence making it a stable solution for underwater monitoring purposes. Bao et al. [48] in their work proposed an Improved Binary Snake Optimizer (IBSO) that improves the existing Snake Optimizer for binary optimization, particularly for the purpose of feature selection. The proposed algorithm incorporated Hamming distance and a Gaussian-based Mutation Transfer Function (MTF), thus enhancing search diversity, local exploration, and computational efficiency. Tested on approximately 27 benchmark datasets, IBSO outperforms other binary swarm intelligence algorithms, achieving better accuracy and lower computational cost. Its effectiveness is further validated against different SO variants, proving its superiority in high-dimensional data selection and machine learning tasks. Sarkhi and Koyuncu [49] introduced an algorithm called Energy Serpent Optimizer (ESO), an enhanced version of the Snake Optimizer, typically intended to improve Deep Reinforcement Learning (DRL) models for game AI, especially in dynamic environments such as Pac-Man. Through the optimization of hyperparameters, the introduced ESO improves AI adaptability, response time, and efficiency, thereby making DRL models more efficient in real-time gaming. It improves the algorithm thereby refining the whole learning process, making it converge faster and make better decisions. This combination of metaheuristic optimization with DRL not only improves gaming AI but also has potential in autonomous systems and robotics. Zheng et al. [50] present an improved variant of the Snake Optimization Algorithm (ISO) that integrates multiple strategies like Sobol initialization, Lévy flight, adaptive step size, and Brownian motion that improves the Snake Optimization algorithm by enhancing exploration, convergence speed, and avoiding local optima. Tests on benchmark functions show better stability, faster convergence, and stronger global optimization than SO. ISO also achieves superior results in engineering applications such as UAV and robot path planning, wireless sensor deployment, and pressure vessel design, demonstrating strong robustness and optimization capability. Liu et al. [51] in their research presented an enhanced Snake Optimization Algorithm (SOEA) to overcome the deficiencies of the original Snake Optimization Algorithm. The developed SOEA improves performance through the use of an elite opposition-based learning strategy, which optimizes population placement to achieve optimal global search efficacy and iteration speed. Moreover, the adaptive threshold technique is utilized, which enhances local search power and speeds up convergence. Experimental comparisons emphasize that the introduced SOEA considerably outperforms SOA, decreasing iterations to 34% and capturing a lesser Mean Squared Error. The usefulness of SOEA is further ensured through its usage in path planning for multiple UAVs, proving the algorithm’s capability in handling high-dimensional and nonlinear optimization problems. Liao and Wang [52] improve Time-Difference-of-Arrival (TDOA) localization with an Improved Snake Optimization algorithm to solve problems in accuracy and convergence speed. The enhanced SO combines a chaotic system for multi-purpose initialization, a better exploration strategy for accelerated early-stage convergence, an adaptive evolutionary threshold to deal with variations in noise, and a snake oviposition strategy based on genetic principles for precise search performance. Simulations validate that this method enhances localization accuracy and robustness and is worth using in wireless sensor networks, surveillance systems, radar, and underwater acoustic localization. Yang et al. [53] proposed a Snake Optimization (MSSO)-based routing protocol for Wireless Sensor Networks (WSN), employing Fuzzy C-Means (FCM) to competently select the cluster head and the Minimum Spanning Tree (MST) to route amongst clusters. The proposed MSSO enhances traditional SO with the help of dynamic updating of parameters, adaptive alpha mutation, and bi-directional search with an increase in convergence speed and efficiency. The new protocol reduces energy usage (26.64%), increases network lifespan (25.84%), stabilizes the clusters (52.43%), and increases throughput (40.99%) as opposed to previous methods, hence becoming very efficient for energy-saving WSN routing. An improved variant of SO for cloud task scheduling was proposed by Damera et al. [54]. In the suggested approach, SO is enhanced using sine chaos mapping, spiral search method, and dynamic adaptive weights, enhancing global search ability and avoiding local optima. The findings from the experiment show that the algorithm performs better than the existing methods by effectively optimizing Quality-of-Service (QoS) parameters such as make span and energy efficiency, with performance gains of 6%, 4.6%, and 3.27% compared to current methods, making it very efficient for dynamic cloud environments. Zhao and Zhou [55] in their paper offer a multi-step short-term wind power forecasting model combining Complementary Ensemble Empirical Mode Decomposition (CEEMD), an Improved Snake Optimization Algorithm (ISCASO), and Kernel Extreme Learning Machine (KELM) to improve forecasting performance. CEEMD separates complicated wind power data into smoother parts, eliminating instability. ISCASO tunes the KELM parameters, enhancing predictive accuracy. By integrating individual component forecasts, the final model attains better accuracy in the forecast of short-term wind power trends, outperforming present models. Zhang et al. [56] introduced an Improved Snake Optimizer (ISO) to optimize task offloading and resource scheduling in Ultra-Dense Edge Computing (UDEC) networks. The Power Allocation (PA) issue of UDEC network is addressed using a quasi-convex method that basically reduces energy consumption, while the Joint Request Offloading and Resource Scheduling (JRORS) issue is represented as a mixed-integer nonlinear programming problem to reduce request delay while enhancing welfare. The ISO incorporates an oscillation term to optimize weight parameter updates and enhance search efficiency. Simulation results indicate that the presented approach surpasses DE, PSO, ALO, BOA, and GWO in terms of minimum energy consumption and maximum welfare and is, therefore, an economical alternative for edge computing task management. Kong and Liu [57] in their research introduced an Improved Snake Optimization (ISO) algorithm to improve the melting point of zinc smelting slag, which is an important factor influencing the smelting efficiency and quality of products. A predictive model based on CatBoost is created, whose parameters are optimized with Tree-structured Parzen Estimator (TPE) to provide higher accuracy. The suggested ISO algorithm is then used to calculate optimal production parameters, thereby achieving an average melting point reduction of 65°C. Experimental verification on actual smelter data in Shaanxi, China, validates the superiority of this method over conventional methods, with the TPE-optimized CatBoost model achieving an R² of 93.89%. Ghamari et al. [58] proposed a technique that combined Reinforcement Learning (RL) and Snake Optimization (SO) to achieve optimized gain initialization for an adaptive PI controller for DC/DC Boost converters. The proposed method improved robustness, real-time adaptability, and frequency matching utilizing the Deterministic Policy Gradient (DPG) technique while avoiding RL’s exploration limitation. The SO algorithm enhances initial parameter choice, resulting in accelerated convergence and dependable performance. Experimental verification on a hardware platform verifies higher control efficiency. Mai et al. [59] introduce a dual-layer MPPT control technology using a dynamic adaptive Snake Optimization Algorithm (ISO), and variable-step perturb & observe (IP&O) to effectively monitor maximum power in photovoltaic systems under partial shading and dynamic weather. The hybrid method utilizes the global search of ISO to prevent local optima and IP&O accuracy for high-speed convergence, with 99.91% average tracking efficiency (simulation) and 99.68% (experimental) at speeds of 0.07 and 0.66 s, respectively, being faster, more stable, and more adaptive to shading compared to traditional methods (P&O, IC). Verified on an HIL+RCP platform, the technique reduces power oscillations and performs exceptionally well in real-world partial shading conditions, providing a robust solution for complex PV system conditions. Zhou et al. [60] developed an adaptive switching control scheme with anti-disturbance attitude for quadrotor Unmanned Aerial Vehicles (UAVs) based on an Improved Snake Optimizer (ISO) to improve performance with measurement noise. The control scheme, or the Adaptive Switching Disturbance Rejection Controller (AWDRC), integrates linear active disturbance rejection control with an adaptive switching extended state observer to effectively reconstruct noisy signals. The ISO algorithm uses quadratic interpolation and holistic learning methods to improve the quadrotor’s attitude controller parameters. Experimental verification using state-of-the-art meta-heuristic algorithms confirms the efficacy of the proposed ISO-based AWDRC in quadrotor attitude tracking control and presents its robustness using Monte Carlo experiments. Wang et al. [61] recommended a short-term photovoltaic power forecasting technique using a combination of K-means clustering, an enhanced Snake Optimization (ISO) algorithm, and a CNN-Bidirectional Long Short-Term Memory (BiLSTM) network. The K-means algorithm is employed for weather classification; model parameter optimization is improved using ISO. The proposed method provides high regression coefficients of 0.99216 for sunny, 0.95772 for cloudy, and 0.93163 for rainy days, thus providing better prediction precision and flexibility across different weather conditions, thus supporting renewable energy management. Zhi et al. [62] proposed an Improved Snake Optimization (I-SO) algorithm to optimize the parameters of Active Disturbance Rejection Control (ADRC), which uses chaotic elite opposition learning and sine-cosine search for diverse initial solutions and enhanced local optimization, respectively. Further, the experimental result highlights that the proposed I-SO is efficient in finding ideal parameters, achieving optical path scanning stability of 99.2%. Singh and Kaushik [63] suggested a technique that combines anisotropic filtering, segmentation using Mask R-CNN, and a better adaptive snake optimizer (ASO) with SqueezeNet for leaf classification and a deep Q-network for disease detection. ASO increases adaptability, which enhances accuracy (0.924 for classification, 0.919 for detection). Agrawal and Mahapatra [64] proposed a novel Meta-Heuristic Snake Optimization Algorithm, an improved variant of SO, designed with the purpose of enhancing the exploration and exploitation, optimizing PSS, TCSC, and their coordinated control. The coordinated model achieves superior stability, with oscillations settling in under 2 s and a 99.30% damping ratio. To optimize the SVM’s parameters, an Improved Snake Optimizer (ISO) is proposed by Lu et al. [65] and his team members to address the limitations such as weak population initialization, slow early convergence, and local optima trapping, of the original Snake Optimization Algorithm. The proposed ISO combines Mirror Opposition-Based Learning (MOBL) to enhance population diversity, a Novel Evolutionary Population Dynamics (NEPD) model for precise searching, and Differential Evolution Strategy (DES) to prevent getting stuck in local optima. Further, the experiments performed on numerous benchmark functions and CEC2022 datasets validate ISO’s better optimization accuracy and speed. Zhang et al. [66] and his team members announced an Improved Snake Optimization (ISO) algorithm for beamforming design in sparse conformal arrays, to boost array sparsity and as well optimize radiation patterns. The proposed ISO integrated Sobol sequences for better population initialization, Cauchy mutation to escape local optima, and a nonlinear time-varying factor inspired by the Whale Algorithm, along with a modified flag control function to improve global exploration. The simulations performed on a sparse cylindrical conformal array highlights that 50% array sparsity, lower sidelobe levels, and faster convergence-10% faster than SO and 46% faster than GA. Kaliraj et al. [67] offered a metaheuristic method based on reinforcement learning called Snake Swarm Optimization (SSO) for actual resource allocation and offloading. The technique integrates operator data with IoT infrastructure and employs neural caching to maximize execution efficiency. Optimal resource allocation through a cost mapping table and incentive factor-based optimal resource allocation ensures efficient distribution among edge users. Performance is measured based on criteria such as delivery ratio, energy expenditure, throughput, and delay, with results proving to be better than conventional optimization methods like Gray Wolf Optimization (GWO), Ant Colony Optimization, and Genetic Algorithms. Mohammed et al. [68] proposed an Improved Snake Optimizer Algorithm (ISO) for fast and accurate Maximum Power Point Tracking (MPPT). The proposed strategy differentiates between Uniform Shading Conditions (USCs) and PSCs, avoiding unnecessary searches and guaranteeing rapid tracking. Additionally, a method for detecting load variations with high convergence speed is introduced. Proposed technique was experimentally confirmed using a buck–boost converter with a sampling time of 0.05 s, achieving 99.86% efficiency and an average tracking time of less than 0.75 s under varying weather conditions. Comparative analysis against recent metaheuristic confirms its superior performance. Zheng et al. [69] proposed a Compact Snake Optimization (cSO) algorithm, that employed a compact strategy into the standard Snake Optimization algorithm, making it more efficient especially in situations where limited computing and memory resources were required. Performance evaluation performed across 28 test functions from CEC2013 confirms that cSO outperforms existing intelligent computing algorithms. Furthermore, integrating cSO with WKNN and Received Signal Strength Indicator (RSSI) positioning significantly reduces localization errors, signifying its effectiveness in refining indoor positioning accuracy. Braik et al. [70] anticipated three improved adaptive variants of the Snake Optimizer namely Exponential SO (ESO), Power SO (PSO), and delayed S-shaped SO (SSO), to address the optimizer’s tendency to get trapped in local optima and to enhance the search performance. These adaptive models efficiently maintain the balance between exploration and exploitation, guaranteeing more efficient feature selection. Their binary versions were developed for Feature Selection (FS) tasks using the K-Nearest Neighbor (KNN) classifier and tested on 24 datasets. Experimental results demonstrated that the Binary Power Snake Optimizer (BPSO) outperformed competing FS methods across multiple evaluation metrics, including classification accuracy, sensitivity, specificity, and fitness scores, achieving over 90% accuracy in several datasets. Khurma et al. [71] in his proposed work integrated a logarithm operator with SO, to enhance exploitation through a cosine-based separation metric. Furthermore, three selection mechanisms namely, Tournament Logarithmic Snake Optimizer (TLSO), Proportional Logarithmic Snake Optimizer (PLSO), and Linear Order Logarithmic Snake Optimizer (LLSO) are introduced to improve the exploration phase. The experiment performed over 22 medical datasets demonstrated that TLSO achieved the highest accuracy in 86% of cases and the best feature reduction in 82%, proving its reliability and efficiency in optimizing feature selection for medical diagnosis. An enhanced version of the Snake Optimizer which incorporates chaotic maps into its parameter choice instead of random strings resulted in creating Chaotic Snake Optimizer (CSO) variants based on four different mappings with chaos, presented by Yıldızdan [72]. The performance of this proposed variant was evaluated across classical as well as CEC2019 test functions, which highlighted noteworthy improvements in SO’s efficiency. The proposed CSO attained higher mean values in most of the cases and based on the analysis it ranked second overall among competing algorithms, proving to be not the best but a promising optimization method. An improved version of the Snake Optimizer, known as Mixed Strategy Snake Optimizer (MSSO) was introduced by Qi et al. [73] that employed a dual inverse learning strategy using Tent chaotic mapping and mirror imaging learning. In addition, a spiral foraging strategy is applied to enhance population diversity and for premature convergence avoidance. Further, a sinusoidal exponential food index is also introduced to replace the original algorithm’s limited benchmark, thereby boosting global search efficiency. The proposed MSSO is tested on eight benchmark functions and compared with several metaheuristic, including Whale, Gray Wolf, Grasshopper, Harris Hawk, and Pelican algorithms. The experimental result highlighted that the proposed MSSO outperforms these algorithms in terms of convergence speed and search efficiency. Belabbes et al. [74] proposed the Snake Optimizer metaheuristic for parameter estimation of three different PV cells: monocrystalline silicon, amorphous silicon, and RTC France with one-diode and two-diode models. Though the default SO algorithm is effective, a better version shows better accuracy, decreasing root mean square error (RMSE) up to 16% and computing time significantly. The enhanced SO is highly accurate at smaller population sizes and a lower number of iterations and hence is one of the most effective techniques for PV parameter extraction. Yao et al. [75] introduced the Enhanced Snake Optimizer (ESO), which integrates an opposition-based learning strategy and dynamic update mechanisms, including parameter adjustments, sine–cosine composite perturbation factors, and Tent-chaos & Cauchy mutation. The usefulness of ESO has been verified with 23 classic benchmark functions, the CEC 2019 function set, and four real world engineering design problems. These results from the experiments, along with statistical calculations, show that ESO has better results than 13 other current algorithms, including SO, proving ESO’s optimization skill is unmatched. Wang et al. [76] proposed a Multi-Strategy Snake Optimizer (MSO) incorporating opposition-based learning, adaptive weight factor, and Lévy flight strategy is proposed. Testing on 23 benchmark functions confirms MSO’s superior accuracy and convergence speed. Among 11 optimization algorithms evaluated for HESS capacity configuration, MSO demonstrated the best performance, reducing LCC by 4.5% while ensuring reliable power supply even during wind turbine and PV unit failures. Li et al. in [77] developed a Snake Optimizer which incorporates a dual mutation mechanism with the intention of automating the scheduling of workflows within cloud environments. The algorithm attempts to minimize the makespan while observing the contractual budget limitations defined by the user. Typically, SO is designed for continuous optimization problems, but this research converts it into a discrete version which is more appropriate for workflow scheduling by implementing a task execution order aware fitness function that reduces waiting times for dependent tasks. As the study also aims to improve SO’s performance, it was found necessary to change the parameter control strategies in different stages of evolution for the problem at hand. A dual mutation mechanism is proposed where “snakes” that do not improve undergo bit mutations to ensure diversity and greater exploration of potential solutions. Based on real-world scientific workflows, the algorithm was validated and showed greater efficacy in finding feasible solutions in less time compared to other available approaches. Yan and Razmjooy [78] presented a new approach for lung cancer detection using CNNs with lung CT images, where preprocessing is done prior to analysis. A variant of the Snake Optimizer, termed iSO, is also introduced to improve CNN performance. The model is validated on the IQ-OTH/NCCD Lung Cancer Dataset and compares the results with other existing methods to demonstrate higher accuracy and effectiveness. Li et al. [79] presented a variable step multiscale single threshold SloEn (VSM-StSloEn) that enhances complexity analysis at several time scales while streamlining the calculations. In addition, a snake optimization-based VSM-StSloEn (SO-VSM-StSloEn) is proposed for better threshold selection, ensuring greater robustness and accuracy. The SO-VSM-StSloEn has been experimentally validated on simulated and real data, proving its insensitivity to signal length and threshold variations, while outperforming existing methods based on entropy in classifying different signal types. Li and Ye [80] created a new description of Snake Optimizer based multi-level image segmentation which uses SO with an enhanced version of Otsu’s thresholding technique, named SO-Otsu. SO-Otsu enhances SO by extracting important regions from TPD images and improving the diagnostic accuracy and efficiency. The authors compare their results with five other optimization algorithms: the Fruit Fly Optimization Algorithm, Sparrow Search Algorithm, Grey Wolf Optimizer, Whale Optimization Algorithm, and Harris Hawks Optimization. The authors justify their choice of optimization algorithms in the study based on experimental results which conclude that SO-Otsu, unlike the other methods, has superior speed, detail extraction, and fidelity, efficiency in TPD image segmentation. Shi et al. [81] in their research solves the issues of multi-AGV scheduling in automated electric meter verification workshops by introducing a better Snake Optimization Algorithm (ISOA). The model seeks to optimize completion time and charging expenses with collision-avoidance constraints incorporated. The algorithm applies AGV-order-address three-level mapping to task assignment and sorting, uses an improved A algorithm for shortest pathfinding and conflict avoidance, and uses large neighborhood search to reduce charging costs. Simulations with real data indicate a 16.4% improvement in completion time and a 60.3% reduction in charging occurrences compared to First-In First-Out (FIFO) scheduling. The approach outperforms alternative state-of-the-art algorithms, providing evidence of its usefulness in optimizing multi-AGV scheduling. Khurma et al. [82] suggested an enhanced Snake Optimizer by alleviating its tendency towards highly fit solutions, which may limit search space diversity. Inspired by Evolutionary Algorithms (EA) and the Darwinian “survival of the fittest” concept, the paper introduces novel selection operators to replace the standard global best operator in SO. The suggested variations are SO-Roulette Wheel, SO-Tournament, SO-Linear Rank, and SO-Exponential Rank, all intended to enhance solution diversity and performance. These variants were tested under experimental comparison to determine their efficacy, and the pivotal role played by selection mechanisms in enhancing the efficiency of SO. Further, parameter analysis was performed to determine their effect on optimization performance. Dai et al. [83] gives an improved thermal error prediction for motorized spindles via a Kernel Extreme Learning Machine (KELM) optimized using the Snake Optimization algorithm. Simulation tools ANSYS software are used to simulate the spindle’s thermal property and measure the temperature field distribution. The data of axial temperature and thermal displacement are acquired with a designed specific experimental platform. The FCM algorithm and grey relation analysis are utilized to improve feature selection, lowering the number of measurement points from ten to four. The SO-KELM model is put forward and compared to KELM and PSO-KELM models, and it proves to be more accurate and stable in thermal error forecasting. Liu et al. [84] presented an Adaptive Chaotic Gaussian Variant Snake Optimization Algorithm (ACGSOA) algorithm is proposed here. The technique includes a chaotic operator to improve the position parameter for optimization as well as accelerate convergence speed and an adaptive Gaussian variant operator for avoiding the algorithm being stuck at local optima. Comparison simulations to SO, WOA, ABC, and FOA exhibit that ACGSOA converges quickly and enhances coverage rates by 7.20%, 7.32%, 7.96%, and 11.03%, respectively, solidifying its superiority in the optimization of SEMWSN deployment.

A hybridized variant basically means combining the advantages of various algorithms and intending to improve the performance. A hybridized SO is capable of yielding better results as it improves exploration, exploitation, and prevents premature convergence, thereby leading towards better solutions and improving the efficiency. By integrating with other complementary algorithms and strategies, a hybrid variant is capable of adapting complex problem landscapes more effectively, leading to more optimal and reliable outcomes. Guan et al. [85] introduced a hybrid model, SO-VMD-LSTM (Snake Optimization-Variational Mode Decomposition-Long Short-Term Memory), to perform ultra short-term forecasting of Electric Vehicle (EV) charging demand to ensure an efficient power grid management. VMD breaks down the past data into several Intrinsic Mode Functions (IMFs) for decreasing complexity, and SO automatically optimizes VMD parameters with improved accuracy and efficiency. LSTM networks subsequently forecast loads from such IMFs, and their results are added for the ultimate prediction. This strategy strengthens feature extraction, eliminates noise, and notably increases prediction accuracy, decreasing RMSE and MAE by 30.1% and 32.9% over VMD-LSTM, and 59.3% and 62.6% over the baseline LSTM technique. Puri et al. [86] introduced a hybrid Reptile Search Algorithm (RSA) and Snake Optimizer for EEG channel selection called RSO to detect Alzheimer’s Disease (AD) and Mild Cognitive Impairment (MCI). Due to the introduction of redundancy and complexity by the multi-channel EEG data, channel selection is optimized by RSO to improve the classification accuracy. Whereas SO performs fine-tuning of feature selection, the RSA search space is optimally searched. EMD, LCOWFB, VMD, and DWT decomposition methods are employed to analyze EEG sub-bands. RSO performed better than standalone metaheuristic and existing AD detection approaches when applied to two publicly available AD EEG datasets. It demonstrated high accuracy with reduced channels-optimal binary classification with 4 out of 16 channels (EMD) and 89.99% accuracy for three-class classification with 7 out of 19 channels (LCOWFB). Bölükbaş et al. [87] presented a hybrid method known as Scatter Search Snake Optimization (SSSO) intended to identify the most relevant feature, minimizing redundancy, however, without compromising on accuracy. While Snake Optimizer enhances the balance between exploration and exploitation, Scatter Search (SS) is good for global search but can become stuck in local optima. SSSO guarantees a better balance between local and global searches by combining SS for structured solution creation with SO for improved search capacity. Tested on UCI Machine Learning and epileptic disease classification datasets, the strategy outperforms popular optimization strategies in feature selection, exhibiting a robust metaheuristic approach with improved accuracy with fewer features. The paper by Fan et al. [88] presents a novel Snake Optimization–Tangential Functional Link Artificial Neural Network (ASO-TFLANN) model to improve the linear range of the Linear Variable Differential Transformer (LVDT), which is a core device for vibration noise measurement and active vibration isolation. The ASO component is an enhanced form of the Snake Optimization Algorithm that combines Latin hypercube sampling and the Levy flight technique to improve global search ability and maintain population diversity, thereby avoiding overfitting and local optimality problems often found with gradient descent. A voltage–displacement test bench is constructed to measure input and output values under different excitation conditions, which are utilized by the TFLANN to calculate the weight vectors for precise LVDT output mapping. Simulation and online testing prove that this hybrid model lowers error significantly, increases the LVDT’s linear operating range, and improves overall measurement precision, thus creating a robust approach for enhanced measurement quality. Duraibi [89] introduced a hybrid framework that integrates a Snake Optimization Algorithm with deep learning for image-based malware classification (SODCNN-IMC). Its primary goal is to accurately detect and classify malware images such as binary representations or screenshots, thereby enhancing cybersecurity defences against evolving threats. In this setup, ShuffleNet is used to extract feature vectors from malware images, and the Snake Optimization Algorithm is used to optimize the hyperparameters of ShuffleNet to ensure better feature extraction. Moreover, an attention-based bi-directional long short-term memory (LSTM) model is used to identify and classify the malware based on the optimized features. Evaluated on the Malimg malware dataset, the combined approach profoundly enhances classification accuracy, with the highest accuracy reported at 98.42%, thus proving that it is capable of improving the malware detection effectiveness of current models. Alawad et al. [90] proposed a Hybrid Snake Optimizer Algorithm (HSOA) aimed at enhancing population diversity, search effectiveness and avoiding premature convergence. It basically comprises of two primary techniques namely, Oppositional-Mutual Learning for better initialization and Dynamic Polynomial Mutation to progress searchability during optimization. The proposed HSOA is applied to the Economic Load Dispatch (ELD) problem with Valve Point Effects (VPE), a non-convex, complex optimization problem in power systems. Further, it is compared to 47 optimization algorithms on five real-world ELD instances, and HSOA ranks at the top in many scenarios, proving its efficiency. Further evaluations using the IEEE-CEC 2014 benchmark functions confirm its competitiveness among metaheuristic. HSOA is a hybrid variant of SO, integrating opposition-based learning and mutation techniques to enhance its optimization capabilities. Ersali et al. [91] in their research, proposed a novel hybrid SO algorithm, referred to as the Opposition-Based Snake Optimizer with Pattern Search (OSOPS). The OSOPS improves the Snake Optimizer by incorporating Opposition-Based Learning (OBL) and Pattern Search (PS), thus improving exploration and exploitation abilities. In addition, a crossover frequency constraint is proposed to mitigate high-frequency noise and ensure reliable performance under disturbances. The effectiveness of OSOPS is demonstrated using statistical box plot analysis and convergence performance comparisons with the baseline SO algorithm. The experimental outcome emphasized that the proposed OSOPS-based system exhibits quicker rise times (14.21% improvement over SO, 32.10% improvement over PP), quicker settling times (15.38% improvement over SO, 84.95% improvement over PP), and greater bandwidths (18.74% improvement over PP, 17.03% improvement over SO). Alkahtani et al. [92] introduced a Hybrid Snake Optimizer-based Route Selection Approach for Unmanned Aerial Vehicles Communication (HSO-RSAUAVC) to increase UAV deployment effectiveness. The designed HSO-RSAUAVC method blends the Snake Optimizer with Bernoulli Chaotic Mapping and Lévy Flight (LF) to optimize route selection in UAV communication. Additionally, a fitness function involving residual energy, distance, and UAV degree is used to dynamically adjust UAV routes, reduce communication interference, and optimize energy consumption. Simulation findings revealed that the proposed method achieves significant improvements in UAV communication performance and reliability compared with the current approaches. Jithendra et al. [93] introduced a hybrid Adaptive Neuro-Fuzzy Inference System (ANFIS) framework enhanced with Snake Optimizer and Induced Ordered Weighted Average (IOWA) for hourly atmospheric pressure prediction. The proposed IOWA-ANFIS-SO model effectively handles dimensionality issues and optimizes ANFIS performance by incorporating meteorological factors such as air temperature, sea surface temperature, wind speed, and wind direction. Based on the observations from weather buoy network stations of Ireland, the model was trained (70%) and validated (30%) to make it reliable. The suggested IOWA-ANFIS-SO model performed much better than conventional IOWA-ANFIS and ANFIS-SO models with RMSE (0.4698), MAE (0.3593), MAPE (0.0003), and R² (0.9903) at the best alpha values. Amor et al. [94] designed a hybrid Grey Theory-Snake Optimizer (GT-SO) for improving the machinability (cutting force) and surface roughness of nTiO₂-GFRPC. Incorporating grey theory to reduce more than one response to a single objective function and the snake optimizer to perform the optimization, the work produces the optimum. Validation validates enhanced output performance, and comparison with other metaheuristics identifies GT-SO’s superiority in cutting, milling, and shaping composite materials, which renders it a useful tool in manufacturing and materials engineering. Fan et al. [95] and his team members, in their work, proposed a hybrid Advanced Snake Optimizer-Linear Quadratic Regulator (ASO-LQR) for the purpose of optimization of vehicle active suspension systems through the enhancement of LQR weight coefficient tuning. The proposed ASO is typically the enhancement of Snake Optimizer with Genetic Algorithm crossover-mutation, Lévy Flight perturbation, and adaptive oscillating weights to ensure improved convergence and precision. Further, the simulation results for different road conditions indicate that the proposed ASO-LQR reduces vibration damping performance over SO, GA, and APSO, rendering it very efficient in automotive suspension control.

Wang et al. [96] proposed an energy-aware multi-hop routing protocol, ESO-GJO, for Wireless Sensor Networks (WSNs) to improve network lifetime and save energy. The protocol combines an Enhanced Snake Optimizer (ESO) with Golden Jackal Optimization (GJO). ESO enhances Snake Optimizer using a Brownian motion function within the exploitation step and employs variables such as CH energy, degree of nodes, and distance from BS to find the best CHs. GJO is subsequently applied to create multi-hop routing structures between CHs and BS. Simulations verify that ESO-GJO significantly surpasses currently used protocols such as LSA, LEACH-IACA, and LEACH-ANT through its energy efficiency and network duration. Agrawal et al. [97] introduced the Advanced Heffron-Phillips Model (AHPM), implemented through a higher-order Synchronous Generator Model 1.1 with ten K-Constants, which improves power system stability. The proposed model combines the Snake Optimization Algorithm and Linear Quadratic Regulator (LQR) for optimal control. AHPM achieves a 99.98% damping ratio and a settling time between 1.5 and 2.0 s, providing fast and robust stability. Ablin and Prabin [98] suggested a Gated Graph Attention-based Crossover Snake (GGA-CS) algorithm that incorporated Graph Neural Networks (GNN) to incorporate spectral-spatial connections and a gated attention mechanism to amplify spectral bands. A crossover Snake Optimization Algorithm tunes parameters and enhances classification precision. As SOA is paired with GNN and attention mechanisms, it is a hybridized variant. The approach is validated on Indian Pines, University of Pavia, and Salinas datasets and is hence beneficial for remote sensing, environmental monitoring, precision agriculture, and land cover classification. Al-Qazzaz et al. [99] introduced a Snake-optimized Xception model using the Xception deep CNN model for effective feature extraction, along with a Snake Optimization Algorithm adapting parameters dynamically to conduct enhanced search space exploration to defend against highly realistic forgeries applied in fraud, misinformation, and identity crimes. This hybridized SOA version enhances the accuracy of DeepFake detection up to 98.53% mean accuracy, precision, recall, and F1-score rates over the classic Xception, as well as compared to other mechanisms. Approved on several DeepFake datasets, the model substantially minimizes misdetection and false positives, which is extremely beneficial for digital media forensics, media verification, legal evidence authentication, and prevention of fake news and identity fraud. Chen et al. [100] investigate pavement performance model development through the use of ensemble machine learning methods for forecasting pavement distress. Employing the Research Institute of the Highway Ministry of Transport track data, a comparison of predictive models such as Random Forest (RF), Extreme Gradient Boosting (XGB), Snake Optimizer-Random Forest (SO-RF), and Snake Optimizer-Extreme Gradient Boosting (SO-XGB) was conducted. Reliable evaluation was ensured by using repeated K-Fold cross-validation. Among the models, SO-XGB showed higher accuracy, especially in rutting depth prediction, with better performance on various performance indices like R², MAE, RMSE, MAPE, and PI. The results affirm that SO-XGB is a good and efficient tool for automated forecasting of pavement deterioration and present a strong solution for pavement performance evaluation. Aljebreen et al. [101] suggested an automated attack detection improvement by DDoS attack detection through a Snake Optimizer with Ensemble Learning (DDAD-SOEL) technique. The Snake Optimizer is used for feature selection (FS) to reduce the size of the data and to improve model effectiveness. A blend of deep learning algorithms like Long Short-Term Memory (LSTM), Bidirectional LSTM (BiLSTM), and Deep Belief Network (DBN) is applied for classification, while model parameters are optimized by Adadelta optimizer. Simulation on benchmark datasets confirms that DDAD-SOEL is superior to existing models in the detection of malicious activities with greater accuracy and effectiveness. Wang et al. [102] proposed a new Ultra-Wide-Band (UWB) location system combined with the Snake Optimizer Long Short-Term Memory (SO-LSTM) architecture. It is a multi-base station and multi-tag UWB configuration, wherein Time Division Multiple Access (TDMA) and Two-Way Ranging (TWR) are utilized to increase real-time distance estimation. The hyperparameters of LSTM are also optimized by the Snake Optimizer, enhancing positioning estimation accuracy. Experimental results verify that the SO-LSTM model performs much better than traditional methods such as the Least Squares (LS) method and the Kalman Filter (KF), lowering root mean square error (RMSE) by as much as 63.39% and maximum positioning error (MPE) by 60.77%, illustrating its higher accuracy and reliability.

Ismail [103] proposed a new Snake-Optimized Framework (CKD-SO) that improves CKD prediction accuracy by combining Snake Optimizer with five ML algorithms for feature selection and classification. Through effective determination of the most useful medical data, CKD-SO significantly enhances diagnostic accuracy to an impressive 99.7% accuracy rate. This framework supports early intervention strategies, minimizing CKD-related mortality and enhancing healthcare outcomes. Samiayya et al. [104] presented a Hybrid Snake Whale Optimization (HSWO) Algorithm to optimize CH selection to improve energy efficiency and network life. The three-phase approach includes initialization, where the network parameters are set; cluster head selection, where CHs are optimally selected using HSWO with respect to energy, distance, and delay; and route maintenance, which maintains stable transmission of data. Performance assessment verifies that HSWO has a dramatic network lifetime enhancement (5600 rounds) and energy efficiency (0.98) over current practices. Masood et al. [105] proposed a new ELM-SO model, which combines Extreme Learning Machine (ELM) with Snake Optimization Algorithm to improve the accuracy of predictions. The model makes use of air quality, weather parameters and is further compared with numerous ML models, such as SVR, RF, ELM, GBR, XGBoost, and deep learning-based LSTM. The experimental results show that the proposed ELM-SO performs better than other models, with a squared correlation coefficient (R²) of 0.928 and a Root Mean Square Error (RMSE) of 30.325 µg/m³, indicating its effectiveness in PM2.5 forecasting. Jiang et al. [106] came up with the idea of applying a hybrid deep belief network (HDBN) combined with an improved snake optimization algorithm (ISOA) to enhance accuracy in forecasting. The proposed HDBN applies deep learning and probabilistic graphical models in identifying complex patterns in data and ISOA, inspired by snake movement, optimizing forecasting performance. Proven using real-world data of a commercial building, the HDBN-ISOA methodology performs better than traditional techniques and proves to be feasible for the optimization of energy consumption and reduced operational expenses of air conditioning systems. Kassem [107] proposed Snake Optimization with Deep Learning Enabled Disease Detection Model for Colorectal Cancer (SODL-DDCC) to detect colorectal cancer (CC) in histopathological images. The approach starts with bilateral filtering (BF) to remove noise, followed by Inception v3 to extract features, in which the hyperparameters are optimized using the Snake Optimization Algorithm. A Graph Convolution Network (GCN) is applied for classification. The validity of the model is tested through a benchmark dataset, exhibiting better performance than other methods. Al-Shourbaji et al. [108] in their study introduce a Feature Selection (FS) approach called Reptile Search Algorithm–Snake Optimizer (RSA-SO) to enhance Machine Learning (ML) model performance by selecting the Optimal Feature Subset (OFS). RSA-SO integrates the Reptile Search Algorithm (RSA) and Snake Optimizer in a parallel mechanism, minimizing the risk of being stuck in local optima while balancing exploration and exploitation. The effectiveness of RSA-SO is validated through experiments on ten UCI datasets and two real-world engineering problems. Comparison against seven popular Meta-Heuristic (MH) algorithms confirms RSA-SO’s competitive performance in optimizing FS tasks. Fu et al. [109] in their research put forward a gas outburst prediction model using Multiple Strategy Fusion and Improved Snake Optimization (MFISO) and Temporal Convolutional Network (TCN) to improve the accuracy of prediction in underground mines. The Snake Optimization Algorithm is upgraded using sine chaos mapping, spiral search strategy, and dynamic adaptive weight to improve global search ability and escape from local optima. Moreover, the Tangent-based ReLU (ThLU) enhances TCN’s generalization capacity. The MFISO algorithm fine-tunes TCN’s hyperparameters, improving prediction accuracy. Comparative experiments compared to GRU, LSTM, SO-TCN, WOA-TCN, and PSO-TCN demonstrate that MFISO-TCN has lower MAE (3.11%), MAPE (0.47%), and RMSE (3.31%), affirming its better performance in gas outburst prediction. Rawa [110] proposed a hybrid meta-heuristic technique combining Snake Optimizer and Sine Cosine Algorithm (SCA) (SO-SCA). This hybridization enhances exploration and exploitation, improving solution accuracy and efficiency. Additionally, the study integrates energy storage systems (ESS) and fault current limiters (FCLs) into the planning process, ensuring network reliability under different cascading failure conditions. The suggested SO-SCA model is also utilized for load forecasting, which performs better than other methods. Simulation studies were conducted using the Garver and IEEE 24-bus systems to confirm the efficacy of SO-SCA in optimizing TEP and accurately forecasting load growth, which proves its efficiency in improving power grid resilience. Major MAs that are hybridized with SO are depicted in Fig. 11. There are numerous MAs exist in the literature. However, only six well-known MAs are utilized to develop hybrid SO variants. Therefore, researchers must utilize other popular MAs to develop hybrid SO variants in different optimization fields.

A multi-objective version of the Snake Optimizer improves the original algorithm by solving multiple conflicting objectives at the same time rather than concentrating on a single objective. In contrast to the original SO, which finds a single optimum, these versions provide a Pareto-optimal solution set, efficiently trading-off among conflicting objectives. Gao and Liu [111] in their work introduced a novel method named the Integrated Multi-Objective Snake Optimizer (IMOSO) to improve convergence precision and population diversity in a multi-objective optimization scenario. The proposed algorithm enhances the Snake Optimization Algorithm, basically using two primary strategies: adaptive mating among subpopulations, mimicking snakes’ multi-partner mating behavior, and an external archive local disturbance mechanism, which improves non-inferior solutions with low diversity. Comparison against seven leading-edge algorithms on WFG test functions shows that IMOSO performs better in convergence and diversity and thus is very efficient on challenging multi-objective optimization problems. Li et al. [112] introduce an Artificial Potential Field-Enhanced Improved Multiobjective Snake Optimization Algorithm (APF-IMOSO) to overcome the limitations of premature convergence and solution diversity deficiency in the conventional path-planning techniques of mobile robots. The algorithm introduces four major improvements to SO and incorporates multiple fitness functions to optimize path length, safety, energy usage, and time efficiency. Experimental outcomes in both static and dynamic settings exhibit significant enhancements, such as an 8.02% decrease in path length, 7.61% improvement in safety, 50.71% improvement in energy efficiency, and 12.74% time saving over the original SO. Robotic testing in real-world scenarios verifies the algorithm’s performance, recording an average path length error of only 1.19%, echoing its robust potential for dynamic and complicated robotic navigation operations.

Abu Khurma et al. [113] proposed two wrapper-based feature selection methods employing the Snake Optimizer, namely Binary Snake Optimizer (BSO) and an enhanced variant, BSO-CV, that combines crossover operators (one-point, two-point, and uniform crossover) governed by a switch probability for enhancing search effectiveness. The Experimental evaluation was performed across the COVID-19 dataset and 23 disease benchmark datasets, which clearly highlights that the proposed BSO-CV is superior in comparison to that of regular BSO in terms of accuracy and execution time, lowering the dimension of the COVID-19 dataset by 89% vs. BSO’s 79%. In addition, proposed BSO-CV improves the exploration-exploitation balance significantly, surpassing state-of-the-art wrapper-based FS techniques like HLBDA, LBMFO-V3, and CHIO-GC, with a performance greater than 90% on the majority of datasets. Braik et al. [70] proposed three improved adaptive variants of the Snake Optimizer, namely Exponential SO (ESO), Power SO (PSO), and delayed S-shaped SO (SSO), to address the optimizer’s tendency to get trapped in local optima and improve search performance. These adaptive models provide a better trade-off between exploration and exploitation and guarantee more effective feature selection. Their binary variants were formulated for FS applications in the context of the KNN classifier and validated on 24 datasets. The experimental outcomes confirmed that BPSO was superior to alternative FS techniques under several evaluation metrics, including classification accuracy, sensitivity, specificity, and fitness scores, and attained over 90% accuracy in various datasets.

Finally, the proportional distribution of the different variants is graphically presented in Fig. 12. It can be seen that the Improved variants of SO are a major development in the literature. Whereas, hybridized variants of SO are the second most developed variants in the literature. There has been little effort put to develop Mult-objective and Binary variants of SO in the literature.

While numerous variants of the Snake Optimizer (SO) have been proposed in recent years, a closer examination reveals recurring design patterns and common improvement strategies. Rather than viewing these developments as isolated contributions, this section synthesizes the literature to identify dominant modification trends, comparative advantages, and underlying research directions. A systematic analysis of existing SO variants indicates that most improvements fall into four principal categories: (i) exploration enhancement, (ii) exploitation refinement, (iii) adaptive parameter control, and (iv) hybridization mechanisms. The same is highlighted in Table 6.

Exploration Enhancement Strategies: A significant number of studies focus on strengthening SO’s global search capability to prevent premature convergence. Common techniques include: Chaotic map-based initialization, Lévy flight mechanisms, Random walk strategies, Opposition-based learning (OBL), and Mutation-based perturbation operators. These modifications primarily aim to increase population diversity during early iterations. The recurring pattern suggests that researchers perceive exploration insufficiency as a primary limitation of the original SO formulation. Empirical comparisons across studies consistently report improved global optima discovery when diversity-enhancing mechanisms are incorporated.

There are a number of gaps in the literature on the SO, particularly with regard to its theoretical foundation, resilience, and effectiveness for specific types of issues. The original SO has issues with convergence and scalability, particularly in high-dimensional or extremely complicated optimization situations.

Robustness and parameter sensitivity: The classical SO is unable to maintain the key balance between exploring search space and exploiting the best solutions. This problem could lead to early convergence, which would prevent the SO from finding the global optimal solution and instead trap it in a local optimum. While current adaptive versions of the SO are intended to tackle this challenge, comprehensive comparison studies of the best methods for various problem types are lacking in the literature.

In accordance with the particular problem, SO necessitates careful adjustment of a number of control parameters. Self-adaptive methods that can dynamically alter these parameters over the course of search are still the subject of extensive research. They would become less reliant on manual adjustment and more resilient to a greater variety of optimization problems as a result.

Theoretical framework and convergence analysis: The literature does not define the theoretical convergence features of the SO in as much detail. A systematic mathematical study that guarantees convergence to the global optimum for a particular class of problems is still a significant shortcoming, despite multiple studies showing good empirical convergence rates. When used for complex, high-dimensional, non-linear optimization problems, the conventional SO has stability issues. Thus, more research is needed in this area.

To increase intra-cluster similarity and decrease inter-cluster similarity, pixels are separated into clusters during image clustering. Each pixel in crisp partitional clustering is allocated to a single cluster, indicating a binary connection between pixels and clusters. Assume that the input image I represents with L gray levels and N pixels or zi. The objective function J is [128] if the number of clusters is nc. (24)J=argmin∑i=1N∑j=1ncuij‖zi−mj‖2where ‖⋅‖ symbolize the inner product norm in d dimensional space to obtain the distance between the ith pixel zi and the jth cluster center mj. If the pixel zi is a part of cluster Cj, its membership value is uij=1. In every other instance, uij=0.

The vulnerability of traditional crisp clustering techniques like K-Means (KM) to local optima trapping brought on by random center initialization, which results in empty clusters, is a major shortcoming. Therefore, the convergence of KM to a global optimal is a significant obstacle. The Stirling numbers of the second kind provide the number of ways to split N pixels into nc non-empty clusters [128]: (25)S(N,nc)=1nc!∑i=0nc(−1)nc−i(nci)iN

The number of possible partitions can be determined using ncNnc!. Therefore, an exhaustive examination of all potential clustering methods to ascertain the optimal global solution is computationally impractical. Furthermore, it has been established that this non-convex minimization problem is NP-hard, regardless of nc=2. As a result, researchers frequently employ metaheuristic algorithms (MAs) to effectively tackle this challenge [11].

Metaheuristic algorithms (MAs) constitute essential approaches for solving complex optimization problems. These methods are effective and operate independently of gradient information. MAs are distinguished by their autonomous functionality, which facilitates adaptability and broad applicability across various real-world optimization scenarios [128].

In Algorithm 2, the objective function defined in Eq. (21) is initially considered, followed by the generation of a randomly initialized population of candidate solutions. Each solution corresponds to a specific set of cluster centers. Subsequently, each pixel is assigned to its nearest cluster center. The fitness of each solution is then evaluated based on the objective function. Since this clustering task is formulated as a minimization problem, lower fitness values correspond to better solutions. The solutions are subsequently refined through the application of MA’s operators to achieve improved outcome. The pixels are subsequently assigned to clusters according to the minimal distance. Additionally, the efficacy of the solutions is assessed, and the optimal solution is retained. This method is executed repeatedly until the stopping requirements are not met. The optimal global solution, serving as the superior cluster center, is attained, and segmentation is performed accordingly.

In the proposed framework, the SO algorithm serves as an optimization-driven mechanism for enhancing clustering-based color image segmentation. The primary role of SO is to determine optimal cluster centroids by minimizing intra-cluster variance and maximizing inter-cluster separability. By efficiently balancing exploration and exploitation during the search process, SO mitigates the limitations of conventional clustering approaches, such as sensitivity to initialization and premature convergence to local optima. Unlike traditional K-means or fuzzy clustering methods, which often depend heavily on initial centroid selection, SO introduces a global search capability that enables more stable and robust convergence toward optimal clustering configurations. This results in improved region homogeneity, clearer object boundaries, reduced pixel misclassification, and enhanced preservation of structural details. Consequently, the segmented image better represents meaningful regions within the original image. Image clustering is not an isolated task but a fundamental preprocessing stage in many real-world computer vision and image analysis systems. The quality of clustering directly affects the performance and reliability of subsequent computational processes. The practical significance of SO-based image clustering can be understood through the following application domains:

Medical Image Analysis: Accurate clustering facilitates the separation of abnormal tissues (e.g., tumors, lesions) from healthy regions. This enables quantitative measurements such as region size, shape, and intensity distribution, which are critical for diagnosis, treatment planning, and disease monitoring.

The clustered output produced by the SO-based framework serves as a structured input for higher-level analytical processes. After clustering, the segmented regions can be utilized for: Feature extraction (texture descriptors, shape metrics, color histograms), Object detection and recognition, Image classification using machine learning or deep learning models, Region-based statistical analysis, Morphological processing and boundary refinement and Decision-support systems in medical and industrial environments. By improving clustering accuracy and robustness, SO enhances the reliability of these downstream tasks. Poor segmentation often propagates errors to later stages, reducing overall system performance. Therefore, the optimization capability of SO contributes not only to improved segmentation metrics but also to enhanced effectiveness of the complete image processing pipeline.

This section delineates the experimental outcomes pertaining to the SO model alongside eighteen additional MAs-based image clustering models. A comparative analysis of these MAs was conducted using a dataset comprising 100 standard color images from the BSD500 dataset [129]. This BSD500 dataset is a publicly available open access datasets. The ground truth images of the BSD500 dataset are publicly available with the dataset. The eighteen tested MAs are Cuckoo Search (CS) [10], Differential Evolution (DE) [128], Grey Wolf Optimizer (GWO) [32], Sine-Cosine Algorithm (SCA) [130], Aquila Optimizer (AO) [131], Arithmetic Optimization Algorithm (AOA) [132], Artemisinin Optimizer (ATRO) [133], Black-winged Kite Algorithm (BKA) [134], Crested Porcupine Optimizer (CPO) [135], Dwarf Mongoose Optimization (DMO) [136], Educational Competition Optimizer (ECO) [137], Fata Morgana Algorithm (FATA) [138], Hiking Optimization Algorithm (HOA) [139], Moss Growth Optimization (MGO) [140], Polar Light Optimizer (PLO) [141], Puffer-fish Optimisation Algorithm (POA) [142], Parrot Optimizer (PARO) [143], and Reptile Search Algorithm (RSA) [144]. These algorithms were selected to ensure a fair and comprehensive evaluation of the proposed method. The set includes both well-established benchmark algorithms and recent state-of-the-art optimization techniques. Classical algorithms were chosen due to their widespread use and well-documented performance, enabling reliable and meaningful comparisons. In addition, several recently proposed algorithms were included to reflect current developments in metaheuristic optimization and to assess the competitiveness of the proposed approach against modern methods. These algorithms represent diverse search strategies and convergence behaviors, helping to avoid bias toward any specific optimization mechanism. All algorithms were implemented using consistent experimental settings and parameter configurations to ensure fairness and reproducibility, allowing for a robust evaluation of the proposed method across different optimization scenarios.

Parameter tuning is essential because of the great sensitivity of metaheuristics to parameters; suitable parameter configurations dictate their efficiency, stability, convergence rate, and fairness in comparison. However, there is no single best way to select parameters for all problems. In metaheuristics research, it is typical to use the parameter settings from the source paper, which means following the original authors’ suggestions. Use of source parameters is permissible when: Use of source parameters is permissible when you aim to replicate or validate something. This is especially true if previous studies have used the same settings. Some algorithms have defaults that a majority agree on. Hence, we utilize parameter setting by following the source paper’s suggestions, and the parameter settings for the tested MAs are reported in Table 10. MatlabR2018b, Windows-10 OS, an x64-based PC with an Intel Core i5 CPU and 8 GB RAM were used for the experiment. In the comparison of MAs, it is essential to set population size and maximum iterations to ensure a fair and scientifically valid assessment. The fundamental premise is that MAs should be evaluated under equivalent computational constraints rather than identical parameter values. Each MA must utilize the same number of function evaluations (NFE) for a fair comparison. NFE equals the product of population size and the maximum number of iterations. To ensure a fair comparison, the population size and the maximum number of iterations were fixed at 50 and 300, respectively. All algorithms have been executed with an identical number of function evaluations (FEs) under a constant population size and maximum iterations configuration. Researchers employ identical maximum iterations or identical function evaluations (FEs) for all MAs throughout execution for better comparison among them [145]. This research met both criteria, namely the same maximum iterations and the same number of function evaluations. Each MA must be executed multiple times per image because of its stochastic nature, as varying executions may yield different outcomes. A single run conveys an isolated occurrence. Several iterations reveal the following conclusions: Does the MA converge prematurely? Is it often stuck in local optima? Is the difference between the results high? An MA may sometimes find the best solutions or become stuck in local optima. A single successful outcome may simply be a chance event. In optimization research, it is common to do 20 to 30 iterations. For better comparison, it is important to include the mean, standard deviation, best, and worst outcomes. The mean shows how well the MA works on average over several runs. The standard deviation quantifies the variability of findings across multiple iterations. The best outcome demonstrates the MA’s ultimate potential. The worst outcome illustrates the MA’s failure mode. After full execution of the MAs, the values of fitness and convergence curves are recorded to evaluate their convergence characteristics. Each MA was executed 30 times per image. Subsequently, the optimization performance of the evaluated MAs was assessed by calculating the mean fitness (mf), standard deviation (σ), best fitness (bf), and worst fitness (wf) values derived from these 30 runs. Furthermore, the segmentation efficacy of the MAs-based image clustering models was evaluated using four reference image-based quality metrics and four ground truth image-based quality metrics [35,146]. A concise description of both categories of quality metrics is provided in Table 11. In this study, the number of clusters has been computed using the Density Balance (DB) [147] algorithm for all kinds of images.

Figs. 14 and 15 present the segmentation outcomes of the assessed MAs-based clustering models. Specifically, Fig. 14 illustrates the results for images numbered 3036, 296028, 79073, 100099, and 130014 from the BSD500 dataset, while Fig. 15 depicts the segmentation results for images numbered 107045, 288024, 306052, 69022, and 347031 from the same dataset. The DB method calculated nc values to be 4, 4, 3, 3, 4, 3, 2, 3, 3, and 3 for the images with the numbers 3036, 296028, 79073, 100099, 130014, 107045, 288024, 306052, 69022, and 347031, respectively. The provided values for the sample images include bf, wf, mf, and std which can be found in Tables 12–16. The tables emphasize the most optimal results in bold. The numerical data presented in Tables 12–16 demonstrate that SO is the most effective optimizer compared to other eighteen MAs. Therefore, it can be inferred that the SO algorithm represents the most reliable and stable optimizer for determining the optimal solution. Given its status as the leading optimizer, Wilcoxon’s rank-sum test [148] was used to evaluate the performance of SO against other MAs. The Wilcoxon rank-sum test is a nonparametric test that is often used to compare metaheuristic algorithms. It can tell if performance differences are statistically significant without assuming normal distribution. Since most metaheuristics are stochastic (contain randomness), multiple runs produce different results. The Wilcoxon test allows for a rigorous comparison of these performance distributions. The test was done mostly at a 5% level of significance. A p-value lower than 0.05 usually means that the algorithms are very different, which means that one is better than the other. The test uses the information from the 30 independent runs to compare pairs. The p-values presented in Table 17 predominantly fall below the 0.05 threshold, demonstrating strong statistical support that the superior fitness values achieved by SO are significant and not attributable to random variation. Metaheuristics require the evaluation of computational time because these algorithms aim to tackle complex optimization problems. Given that metaheuristics yield approximate rather than assured optimal solutions, time efficiency serves as the principal criterion for assessing the MA’s practicality in real-world applications. A fundamental challenge in metaheuristics is the balance between exploitation and exploration. Researchers can assess the computational time to see if an MA excessively prioritizes refining without producing superior outcomes. Measuring time helps us figure out how scalable the MA is when the problem size increases. Therefore, Fig. 16 graphically represents the average execution time of the examined MAs-based clustering models in ascending order. AOA and RSA consume the smallest amount of time. On the other hand, PARO needs the highest computational effort. SOO achieved sixth position among all the MAs.

An optimization process is deemed successfully completed only when it satisfies one or more specified conditions, commonly referred to as convergence criteria. These criteria encompass various factors, including the completion of a maximum number of iterations, achieving a predetermined fitness threshold, the total number of function evaluations performed, or adherence to a predefined time constraint.

The study employs identical maximum iterations and function evaluation counts to evaluate the convergence characteristic. Convergence curves, based on the same maximum number of iterations, are widely employed to demonstrate the convergence characteristic of MAs [35,149]. The convergence curve illustrates the fitness value of the globally optimal solution at each iteration. Alternatively, it illustrates the ability of MA to identify the optimal solutions with a set number of iterations. The convergence performance of an MA is considered good if it attains the best solution with fewer iterations than other competing MAs. It is also observed that some MAs converge quickly with a moderate solution, i.e., premature convergence, and MAs get trapped in a local optimum. In this context, efficiency denotes the MA’s ability to quickly find accurate solutions, its convergence behavior, and its overall computational performance. These three aspects can be analyzed using a convergence curve. Fig. 17 illustrates the convergence trajectories of the evaluated MAs across ten sample images. The convergence behavior of the SO is satisfactory, as evidenced in Fig. 17. On the other hand, the convergence curves of CPO, RSA, SCA, and AOA are not satisfactory. Therefore, these MAs exhibit suboptimal convergence patterns, suggesting a likelihood of getting trapped in local optima.

By measuring average quality metrics, we assessed the ability of MA-based clustering algorithms to segment images. The numerical values of these quality metrics, based on the reference and ground truth, are provided in Table 18. The image clustering model that is based on SO achieved the highest PSNR, MSE, SSIM, FSIM, and BDE values. AOA-based image clustering models, on the other hand, achieved the highest PRI and BDE score. The best VoI is attained by the RSA-based image clustering approach. The best GCE is achieved by POA based clustering method. Hence, it can be concluded that a superior ability to minimize the objective function does not always lead to a superior image segmentation model in terms of all quality measures. Although the RSA algorithm has no optimization results, the RSA-based image clustering model achieves the highest values of VoI. However, it is apparent that SO consistently achieved good numerical values of all quality metrics compared to other MA-based image clustering models.