Swallow Search Algorithm (SWSO): A Swarm Intelligence Optimization Approach Inspired by Swallow Bird Behavior

Abstract

Swarm Intelligence (SI) algorithms were applied widely in solving complex optimization problems because they are simple, flexible, and efficient. The current paper proposes a new SI algorithm, which is based on the bird-like actions of swallows, which have highly synchronized behaviors of foraging and migration. The optimization algorithm (SWSO) makes use of these behaviors to boost the ability of exploration and exploitation in the optimization process. Unlike other birds, swallows are known to be so precise when performing fast directional alterations and making intricate aerial acrobatics during foraging. Moreover, the flight patterns of swallows are very efficient; they have extensive capabilities to transition between flapping and gliding with ease to save energy over long distances during migration. This allows instantaneous changes of wing shape variations to optimize performance in any number of flying conditions. The model used by the SWSO algorithm combines these biologically inspired flight dynamics into a new computational model that is aimed at enhancing search performance in rugged terrain. The design of the algorithm simulates the swallow’s social behavior and energy-saving behavior, converting it into exploration, exploitation, control mechanisms, and convergence control. In order to verify its effectiveness, (SWSO) is applied to many benchmark problems, such as unimodal, multimodal, fixed-dimension functions, and a benchmark CEC2019, which consists of some of the most widely used benchmark functions. Comparative tests are conducted against more than 30 metaheuristic algorithms that are regarded as state-of-the-art, developed so far, including PSO, MFO, WOA, GWO, and GA, among others. The measures of performance included best fitness, rate of convergence, robustness, and statistical significance. Moreover, the use of (SWSO) in solving real-life engineering design problems is used to prove (SWSO)’s practicality and generality. The results confirm that the proposed algorithm offers a competitive and reliable solution methodology, making it a valuable addition to the field of swarm-based optimization.

1. Introduction

Optimization is a mathematical and computational technique to find the best possible solution or outcome from a set of available alternatives while satisfying certain constraints [1]. This mathematical formula can be written as [2]:

$${minmize}_{x\in R^n} f_i\left(x\right)\hspace{1em}or\hspace{1em}{maximize}_{x\in R^n} f_i\left(x\right),\left(i=1,2,3,4\dots ..M\right)$$

(1)

Subject to

$$h_j\left(x\right)=0\hspace{1em}\left(j=1,2,3,4\dots \dots J\right)$$

(2)

$$g_k\left(x\right)\le 0 or g_k\left(x\right)\ge 0,\hspace{1em}\left(k=1,2,3,4\dots ..K\right)$$

(3)

where $f_i\left(x\right), h_j\left(x\right), g_k\left(x\right)$ are functions of the design vector

$$x_i={\left(x_1,x_2,x_3\dots .,x_n\right)}^T$$

(4)

Here, the components $x_i$ of Equation (4) are called design or decision variables, and they can be real-valued continuous, integer-valued discrete, or a mix of these two types. The function ${ f}_i\left(x\right)$ is called the cost function or objective function; it is the indicator of how good or bad a solution is. $R^n$ represents the space formed by the decision variables, which is called the search space, while the space formed by the objective function values is called the solution space. The equalities for $h_j\left(x\right)$ and inequalities for $g_k\left(x\right)$ are called constraints [2].

Optimization has been applied to many problems and fields, starting from simple problems and ending with solving complex ones in which the environment is constantly changing and a quick decision must be made. These fields include engineering, economics, logistics, and operations research.

A simple way to classify optimization algorithms is to look at the nature of the algorithm; this leads to dividing it into two categories: deterministic algorithms and stochastic algorithms. Deterministic algorithms follow a strict, well-defined, and repeatable procedure without incorporating any randomness or stochastic elements. These algorithms work well for problems that are linear, continuous, differentiable, and convex. However, they often struggle with real-world optimization problems, which are usually non-linear, non--convex, high-dimensional, and NP-hard, with discrete or combinatorial search spaces [3]. To solve these problems, stochastic (or probabilistic) approaches have been developed. There are two types of stochastic algorithms: heuristic and metaheuristic. The term heuristic originates from the Greek verb heurísko, which means “find out” or “to discover” [4]. Behaving like a baby discovering new things by trial and error, heuristic optimization algorithms are used when solutions to a complex optimization problem can be found in a reasonable time, but there is no guarantee that optimal solutions are reached.

Further development of heuristic algorithms led to metaheuristic algorithms. The term metaheuristic was coined by Glover in 1989 [5] by adding the Greek prefix “meta,” which means ‘beyond’ or ‘higher level,’ to “heuristic” to refer to a set of methodologies conceptually ranked above heuristics in the sense that they guide the design of heuristics. In addition, all metaheuristic algorithms use a certain tradeoff between randomization and local search. A metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a lower-level procedure or heuristic (partial search algorithm) that may provide a sufficiently good solution for an optimization problem. Metaheuristics are strategies applied to find solutions to problems with significantly large search spaces, making them very difficult to explore using traditional methods [6].

1.1. Metaheuristic Algorithms Classifications

Metaheuristic algorithms can be classified based on their design inspiration and operational structure into four main types:

• Based on search strategy: Metaheuristic can be classified into single-solution, which focuses on iteratively improving a single candidate solution. For example, Simulated Annealing (SA) [7], Tabu Search (TS) [5], or population-based [8] methods work with a group of solutions simultaneously, promoting diversity and exploration. For example, Genetic Algorithms (GAs) [9], Particle Swarm Optimization (PSO) [10], Ant Colony Optimization (ACO) [11], and Differential Evolution (DE) [2].
• Based on the nature of inspiration: Metaheuristics can be classified into three types: bio-inspired algorithms, physics-based algorithms [12], and social and cultural algorithms. The first type mimics the way living organisms solve their problems in their environments, such as how birds flock, ants find food, or how evolution drives survival through natural selection. By imitating these behaviors, bio-inspired algorithms can solve complex optimization problems where traditional algorithms might struggle. There are several examples of this type of optimization, such as evolutionary-based [13] GA, swarm-based PSO [14], ACO, artificial bee colony (ABC) [15], and behavioral-based Cuckoo Search [16] and Firefly Algorithm [17]. The second type is inspired by physical laws and phenomena. For example, Simulated Annealing (thermodynamics) [18] and the gravitational search algorithm (Newtonian gravity) [19,20]. The third one is based on social behaviors or human-inspired processes, such as Harmony Search [21] (musical improvisation) and Teaching–Learning-Based Optimization (classroom dynamics) [22].
• Based on trajectory control: Metaheuristics can be classified as deterministic, stochastic, or a mixture or hybrid of both. Deterministic algorithms follow a rigorous procedure with repeatable design variables and functions. For example, hill-climbing. On the other hand, stochastic algorithms always have some randomness to explore the search space and avoid local optima (most metaheuristics fall into this category) [3].
• Based on single or multiple solutions: Metaheuristics can be classified as trajectory-based or population-based [22]. Trajectory-based metaheuristics focus on finding a single solution and use iterative improvement to refine that solution. For example, simulated annealing [23]. Population-based metaheuristics focus on finding multiple solutions and use a population of solutions to explore the search space. For example, the genetic algorithms (GAs), Ant Colony Optimization (ACO), and particle swarm optimization (PSO) [24].

1.2. Optimization Problems Classification

Understanding different optimization problem types helps in choosing the appropriate mathematical models and solution approaches. Each type has its challenges and requires a special technique for a solution. In general, optimization problems can be divided into six main categories based on their objectives, structural properties, constraints, solution methods, computational complexity, and application domains. As a matter of fact, the optimization problems usually involve a combination of these six types.

The classification of these problem types appears in Figure 1, according to research from a literature review and additional optimization studies.

Based on the problem objective, optimization can be either a single objective [25], which occurs when M = 1 in Equation (1), in which case the goal is either to minimize or to maximize a single function, or multiple objectives [26,27], which occurs when M > 1, in which case it needs to optimize multiple conflicting objectives at the same time. For a single objective, there is a single global best for the entire swarm and a personal best for each particle. It focuses on a single objective function, and the particles update their velocity and position to minimize or maximize this objective. For example, finding the shortest route between two cities. While in the multiobjective, instead of a single solution, it aims to find a Pareto front [28], which is a set of optimal solutions where no objective can be improved without worsening another. It focuses on multiple conflicting objectives and uses an external archive to store the non-dominated solutions that form the Pareto front [28]. For example, optimizing a vehicle design to minimize fuel consumption while maximizing performance and safety. Also, there is another objective classification based on the linearity of the involved functions. If all objective and constraint functions—namely $f_i\left(x\right), h_j\left(x\right)$, and $g_k\left(x\right)$—are linear, the optimization problem is called linear, and when at least one of these functions is non-linear, it is considered a non-linear optimization problem [29].

Convex and non-convex are the two types of optimization problems that are based on problem structure. If a line is connected between two points on a convex function, the function must be under the line. In a convex optimization problem, any local minimum is also a global minimum. This is the major advantage because the algorithm can find the best solution without getting stuck in suboptimal points. Non-convex optimization problems have complex landscapes with multiple local optima, making their computations hard [30].

When working with optimization problems, one of the first distinctions to make is whether or not the problem includes constraints. If it does, the problem is considered constrained; otherwise, it is unconstrained. Among constrained problems, the restrictions placed on the variables can be either hard—meaning they must be strictly followed—or soft, meaning there is some flexibility, and violations may be tolerated within certain limits [31]. The most common approach for solving constrained optimization problems is to use a penalty function. The constrained problem is transformed to an unconstrained one by penalizing the constraints and building a single objective function. The resulting single objective function is then minimized using an unconstrained optimization algorithm [2].

Optimization problems are classified into four types based on the solution methods which depend on several factors such as the problem complexity, the solution accuracy, and the computational cost. The first type is the exact method, where the solution is guaranteed optimal, but the computations are expensive. The second type is the heuristic method, which provides good solutions within a reasonable time, but the optimality is not guaranteed. The third type is the metaheuristic method, which provides a high-level problem algorithm capable of finding a near-optimal solution to different complex problems, but not necessarily the exact solution. Finally, there is the hybrid method, which combines multiple strategies to exploit the strengths of different algorithms.

Starting with the famous question: is solving a problem as easy as checking a solution? Most experts think not, but no one has been able to prove it either way, which is why it remains one of the most important open problems in mathematics and computer science today [32]. This question highlights the importance of classifying optimization problems according to their complexity. Based on this complexity, the computational complexity of optimization problems varies significantly from easy to hard problems. The easy ones fall under what is known as the class (P), which can be solved efficiently with polynomial-time algorithms; in other words, the time it takes grows reasonably as the input gets larger, like checking if a number is even or sorting a list alphabetically [33]. Then, there are the (NP) problems, which are a bit harder. In this type of problem, we might not know how to find a solution quickly, but if someone gives us the answer, we can verify it fast, like checking whether a finished Sudoku puzzle is correct. A subset of NP, called NP-complete, contains the hardest problems in NP—if we can figure out how to solve just one of them efficiently, it would mean we can solve all NP problems efficiently, which is why they are so important [34]. NP-hard problems are even tougher in some ways; they are at least as hard as NP-complete ones but do not necessarily have solutions that are easy to check, like the infamous Halting Problem [35]. Because finding exact solutions to NP-hard problems is computationally infeasible for large instances, researchers often rely on heuristic and metaheuristic algorithms, such as genetic algorithms, PSO, ACO, DE, and other bio-inspired approaches to find approximate solutions.

Finally, based on the application area, optimization is classified as function optimization, combinatorial optimization, optimization in an unknown search space, engineering design, and decision-making optimization. Function optimization is the process of optimizing a continuous mathematical function, for instance, minimizing cost or maximizing efficiency. On the contrary, combinatorial optimization is concerned with discrete variables and looks for an optimal element of a finite set. These optimization problems arise in numerous applications [36,37,38]. However, this set is too large to be enumerated; it is implicitly given by its combinatorial structure. The goal is to develop efficient algorithms by understanding and exploiting this structure. For example, it is seen in problems like the Traveling Salesman Problem (TSP) or the Knapsack Problem [18]. Optimization in an unknown search space occurs when the space is always changing or when it is too complicated to explore it directly. The metaheuristic approach is used in this type of problem, such as genetic algorithms, particle swarm optimization, ant colony optimization, differential evolution, and bio-inspired algorithms. Engineering design optimization focuses on identifying optimal design parameters for systems or components in fields such as structural design or control systems. Lastly, decision-making optimization addresses problems involving the selection of the best course of action under uncertainty, commonly in multiobjective contexts where trade-offs between conflicting objectives must be balanced [26].

1.3. Challenges

The difficulty is that most of the swarm intelligence algorithms, like PSO and ACO, have high computational complexity, which may become a hindrance in its efficient solution for real time context critical applications. This issue is realized in the scalability where it is quite difficult to accommodate large numbers of such agents. As such, the successful implementation of swarm intelligence applications depends on two essential factors: environmental disorder management and multiagent intercommunication optimization.

1.4. Contribution

The primary contributions of this paper are as follows:

• The new Swallow Search optimization algorithm (SWSO) serves as a novel metaheuristic that finds inspiration from swallow migration patterns.
• The proposed algorithm has been evaluated using a range of benchmark functions, including unimodal, multimodal, and fixed-dimension functions, as well as the functions of the CEC2019 benchmark.
• A comparison of the (SWSO) algorithm’s performance has been made with different algorithms, such as MFO, PSO, GSA, BA, FPA, SMS, FA, GA, DA, WOA, BOA, COA, FDO, FOX, and AFO.
• The SWSO algorithm has been applied to two constrained engineering design problems, and its performance has been compared to the more recent outcomes found in the literature.

To simplify reading the paper’s abbreviations and symbols,

Table 1. Abbreviations used in this paper according to their appearance.

Abbreviation	Term
SI	Swarm Intelligence
SWSO	Swallow Search Optimization
P	Polynomial Time
NP	Nondeterministic Polynomial Time
NP-Complete	Nondeterministic Polynomial-time Complete
NP-Hard	Nondeterministic Polynomial-time Hard
TSP	Traveling Salesman Problem
MOPSO	Multiobjective Particle Swarm Optimization
MOCH-PSO	Multiobjective Constraint-Handling-PSO
AOOA	Apiary Organizational-Based Optimization Algorithm
DP	Differential Privacy
AO	Aquila Optimizer
PSAO	An enhanced Aquila Optimizer with Particle Swarm
WSNs	Wireless Sensor Networks
D3QN	Dueling Double Deep Q-Network
RP	Routing Protocol
WOAD3QN-RP	Whale Optimization Algorithm uses D3QN
SSO	Swallow Swarm Optimization
Kac	lysine acetylation
SIPSC-Kac	Swarm Intelligence and Protein Spatial Characteristics-Kac
B2C e-commerce	Business-to-Consumer electronic commerce
PSO	Particle Swarm Optimization
IPSO	Improved Particle Swarm Optimization
GA-PSO-SQP	Genetic Algorithm- Particle Swarm Optimization- Sequential Quadratic Programming
CPSO	Co-Evolutionary Particle Swarm Optimization
MPPT	Maximum Power Point Tracking
PEM	Proton Exchange Membrane
DSO	Donkey and Smuggler Optimization
AFOX	Adaptive Fox Optimization
FOX	Fox Optimization
IFOX	Improved FOX
NSCSO	Non-Dominated Sorting Chicken Swarm Optimization
GWO	Grey Wolf Optimizer
VGWO	Variant Grey Wolf Optimizer
GA	Genetic Algorithm
IAGA	Improved Adaptive Genetic Algorithm
DA	Dragon-Fly Algorithm
BDA	Binary Dragonfly Algorithm
WOA	Whale Optimization Algorithm
IWOA	Improved Whale Optimization Algorithm
BOA	Butterfly Optimization Algorithm
COA	Chimp Optimization Algorithm
FDO	Fitness Dependent Optimizer
MFO	Moth–Flame Optimization
GSA	Gravitational Search Algorithm
BA	Bat Algorithm
CEBA	Chaos-Enhanced Bat Algorithm
FPA	Flower Pollination Algorithm
SMS	State of Mater Search
FA	Firefly Algorithm
DE	Dolphin Echolocation
FEP	Fast Evolutionary Programing
ACO	Ant Colony Optimization
IACO	Improved Ant Colony Optimization
INFO	Weighted Mean of Vectors Algorithm
MINFO	Modified Weighted Mean of Vectors Algorithm
SCSO	Sand Cat Search Optimizer
AVOA	African Vultures Optimization Algorithm
SCA	Sine Cosine Algorithm
MSCA	Modified Sine Cosine Algorithm
HHO	Harris Hawks Optimization
RIME	A RIME-Ice physical phenomenon-based optimization.
ZOA	Zebra Optimization Algorithm
NRO	Nuclear Reaction Optimization
BBOA	Brown-Bear Optimization Algorithm
TSA	Tree-Seed Algorithm
FOX-TSA	Hybrid FOX- Tree-Seed Algorithm
RSO	Rat Swarm Optimization
MRSO	Modified Rat Swarm Optimization
SMR	Slime Mould Reproduction
MSSSA	Multistrategy-Sparrow Search Algorithm
ASO	Atom Search Optimization
SO	Snake Optimizer
MVO	Multiverse Optimizer
ABC	Artificial Bee Colony
MSHO	Modified Sea Horse Optimizer
AZOA	American Zebra Optimization Algorithm
FDA	Flow Direction Algorithm
CBO	Colliding Bodies Optimization
CS	Cuckoo Search
SSO-C	Social Spider Optimization
ACSA	An Improved Cuckoo Search Algorithm
WEOA	Water Evaporation Optimization Algorithm.
WCMFO	Water Cycle and Moth-Flame Optimization
EJS	Enhanced Jellyfish Search
CSS	Charged System Search
MDE	Multiple Differential Evolution

and

Table 2. List of symbols used in this paper.

Symbols	Definition
$\alpha$	Probability of following the leader
$\theta$	Energy threshold for leader switching
${\omega }^{\left(t\right)}$	Adaptive inertia weight
${\delta }^{\left(t\right)}$	Exploration decay factor
${\epsilon }_i$	Local random Gaussian noise

summarize all used abbreviations and symbols used, respectively.

The rest of the paper is organized as follows: Section 2 presents the literature review, which includes an overview of previous studies. These studies are classified into three main groups: swarm intelligence and optimization algorithms, metaheuristic and hybrid algorithms, and finally, bio-inspired algorithms. In Section 3, the inspiration behind the proposed (SWSO) algorithm is discussed, and the biological behavior of the swallow bird is described. Section 4 describes the methodology and algorithm used in this paper, which is inspired by the remarkable social behaviors, such as foraging, migration, social interaction, flying in a V-shape flock, and power saving. Section 5 comprises an assessment of the performance of the SWSO technique, including a comparison with other approaches and an examination of its efficacy in specific scenarios. The proposed SWSO algorithm is applied in Section 6 on two of the standard engineering problems as a test case. Finally, we provide a conclusion of the presented work and some recommendations for future work in Section 7. Additional figures and the complete calculations of the relative ranks of the proposed algorithm over its competitors are provided in Appendix A.

2. Literature Review

In order to give a complete overview of recent developments in this area, the references are categorized into three major sections with regard to their content and topicality, the first category is swarm intelligence and optimization algorithms, the second category is metaheuristic and hybrid algorithms, and lastly, bio-inspired and hybrid optimization models. There is one paragraph that gives an overview of the main points made in each reference.

2.1. Swarm Intelligence and Optimization Algorithms

A novel nature-inspired metaheuristic optimization technique called “Apiary Organizational-Based Optimization Algorithm (AOOA)” is introduced [39]. The algorithm simulates the systematic behavior of honeybees within the apiary and focuses on their roles as queens, drones, and workers. The authors highlight AOOA’s ability to explore and exploit solution spaces effectively, showing an advancement in the optimization field. The problem of incorporating differential privacy (DP) into swarm intelligence (SI) to improve optimization was examined in Ref. [40]. The approach showed enhanced algorithmic efficiency by preventing local optima stagnation, thus relevant to precautionary domains such as health care. To overcome the limitations of the original Aquila Optimizer (AO), such as insufficient local exploitation ability, low optimization precision, and slow convergence rate, the authors in Ref. [41] improved the original Aquila Optimizer by enhancing the position updating and combining the golden sine operator with self-learning and social learning mechanisms from particle swarm algorithms. A non-linear equilibrium factor was introduced to balance global search and local exploitation capabilities, thereby improving convergence speed and accuracy. The PSAO was tested on different benchmark functions and engineering design problems to show the strength of the improved optimizer compared with the old one. In Ref. [42], the authors presented a new mutation-driven swarm intelligence algorithm with translation and rotation invariance for global optimization. The studied algorithm provided better results than the conventional methods in solving independent as well as coupled multimodal problems with stability and efficiency as was evident in the engineering optimization problems. Ref. [43] enhanced machine learning classifiers used to detect code smell by applying optimizers based on swarm intelligence. The approach raised prediction accuracy and computer performance and was useful for software engineering. Both heuristic-based and swarm intelligence techniques were used in Ref. [44] to optimize work–life balance issues. It was shown that the algorithms produce a positive outcome in the field of the task scheduling and the resources management, to the problems of personal and organizational efficiency. Ref. [14] presented a two-stage greedy auction algorithm for target assignment in large-scale UAV swarms. This method helped in well-coordination of tasks at the operational level especially surveillance and reconnaissance duties. The WOAD3QN-RP protocol, which integrates swarm intelligence and deep reinforcement learning for smart routing in wireless sensor networks, was proposed in Ref. [45]. This innovation increased energy efficiency, data rates, and network lifetime, which geared the solution for IoT applications. The paper [46] aimed to review the method of swarm intelligence in enhancing of sliding wear in nanocomposites. The latter demonstrated good performance in terms of balancing between the required computational precision and complexity, and pointed toward the further applications of the algorithms in material design and tribology. In Ref. [47], the authors introduce Swallow Swarm Optimization (SSO), a biological inspired swarm intelligence algorithm in which explorer, aimless, and leader particles are modeled having different behaviors and variable search radii. It is a role-based architecture that allows the swarm to come out of local minima and converge due to the decentralized decision-making system as a reflection of simple social interactions in nature. Effective and intuitive as it is, the design of SSO remains rather static and not as much mathematically versatile as more recent swarm-related solutions.

2.2. Metaheuristic and Hybrid Algorithms

In Ref. [48], complex optimization problems have been solved using the hybrid dolphin and sparrow optimization (DSO) algorithm. The integration of adaptive and stochastic schemes makes DSO exhibit good global search characteristics, fast convergence, and stability in solving constrained problems such as airfoil design. In Ref. [49], the authors propose a swarm intelligence and protein spatial characteristics algorithm (SIPSC-Kac) that integrates swarm intelligence to predict the sites of lysine acetylation by analyzing protein spatial features. The success of swarm intelligence in the problem of determining biomolecular protein modification sites was demonstrated through this bioinformatics application, thereby highlighting the potential of the concept in computational biology. Regarding a multimodal transportation system, the authors have proposed an optimizer for logistics distribution in Ref. [50], exploiting swarm intelligence. In Ref. [51], an electrical automation control system is optimized using an AI-based optimization method, drastically reducing response time and improving stability amid power and frequency disturbances, with a failure rate of 0.02 (turbine) due to optimization, indicating its efficiency and resilience in various engineering contexts.

2.3. Bio-Inspired and Hybrid Optimization Models

A literature review of features of algorithms and control techniques in swarm robotics for autonomous aerial robots was prepared by Ref. [52]. The single sensor implicitly underlined issues of cooperative robotics, power consumption, and real-time reactivity specific to environments. Thus, the elements are combined to enhance the characteristics of the approach in terms of improving the monitoring of livestock and controlling the pattern of grazing, which has (connection) a positive impact on the efficiency of cattle farming and its contribution to the type of the environment that the livestock is placed in. Field tests support the model to demonstrate the effectiveness of automating time-honored grazing patterns. The model incorporated SIA for routing and coverage to demonstrate the capacity and future that precision farming has. A novel approach to pathfinding inspired by the cooperative behavior between donkeys and smugglers is presented in Ref. [53]. The Donkey and Smuggler Optimization Algorithm (DSO) has dual modes: the Smuggler mode, which explores all possible paths to find the shortest one, and the Donkeys mode, which uses various donkey behaviors to enhance the search process. The Smuggler mode is useful at the beginning of the optimization process while the Donkeys mode is activated when the algorithm has already identified a promising solution and needs to improve it. The results show the DSO potential for solving complex optimization problems and unfamiliar search spaces. The [54] paper introduces a novel adaptive nature-inspired optimization algorithm called “Adaptive Fox Optimization (AFOX) Algorithm”, which is inspired by the hunting behavior of foxes. AFOX shows good performance in comparison to traditional one, particularly in speeding up convergence to the global solution and avoiding local optima optimization problems. The AFOX study was tested to solve real-world optimization problems as well as various benchmark functions, showing a significant advancement in the field of optimization, making it a promising tool for a wide range of real-world applications. Chicken swarm’s nature was the base for an innovative NSCSO optimization model that is developed in Ref. [55]. It utilizes non-dominated sorting for the integration of diverse objectives, and for solving multiobjective functions, the algorithm yields a better performance than other related work. Through NSCSO, the authors demonstrate that bio-inspired algorithms can effectively solve high-dimensional optimization problems. Its uses extend to the planning of various systems, including engineering design and resource allocation, to be a useful tool for decision making.

3. Swallow Search Optimization Algorithm (SWSO)

Flight of birds may represent one of the most fascinating wonders of nature. This flight has been a dream of mankind since ancient times, as people used to imitate how birds fly in the air by observing them and analyzing their interactions during flight, as well as studying the various factors that affect the mechanics of flight. Engineers have used this dream as inspiration to create novel technologies specifically within optimization algorithm development. In this section, the inspiration behind the proposed (SWSO) algorithm is discussed, and the biological behavior is described.

3.1. Inspiration

Swallows are medium-sized birds [56], they are 17 to 19 cm long, including 2 to 7 cm of elongated outer tail feathers. Their wingspan is 32 to 34.5 cm, and they weigh 16 to 22 g [57], with pointed narrow wings, short bills, and small weak feet [58]. The tail extends well beyond the wingtips, and the long outer feathers give the tail a deep fork, assisting in maneuvering and acrobatics during foraging [59]. Barn swallows are aerial-insectivores that breed and roost in a colony.

The barn swallow, as shown in Figure 2, is a bird of open country that normally nests in man-made structures, spreading with human expansion. It builds a cup nest from mud pellets in barns or similar structures. They fly with fluid wingbeats in bursts of straight flight, rarely gliding, and can execute quick, tight turns and dives [60]. These birds feed on the wing, snagging insects from just above the ground or water to heights of 100 feet or more. When aquatic insects hatch, Barn Swallows may join other swallow species in mixed foraging flocks [59].

Migratory swallows are a prominent model system in evolutionary, ecological, and behavioral research. They have evolved less pointed, more rounded wings compared to their resident counterparts. This is unusual because, traditionally, long-distance migratory birds are expected to have pointed wings to reduce drag and improve flight efficiency. However, in swallows, wing roundedness increases with migratory distance. This deviation from the expected pattern may be due to the need for migratory swallows to forage in low food availability and inclement weather, requiring more maneuverability at low speeds, or because their roosting habits in dense, low vegetation necessitate better take-off performance, favoring rounded wings [58].

3.2. Biological Behavior of Swallows

Swallows exhibit remarkable social behaviors, including the following:

• Foraging: Swallows forage in groups, dynamically adjusting their flight patterns to locate and capture prey efficiently.
• Migration: Swallows undertake long migratory journeys, navigating complex environments and adapting to changing conditions.
• Social Interaction: Swallows communicate and coordinate within flocks, sharing information about food sources and dangers.
• Fly in a V-shape flock: This method allows swallows to cover the greatest possible distance by taking advantage of the total air flow formed by the entire flock, thus reducing the effort expended during flight. Figure 3 shows the V-shaped formation during flight. This means that when each bird flaps its wings in the air, it generates energy that helps it fly and also helps the bird following it, which enables all the birds in the flock to cooperate in the flight process.
• Power saving: Studies have shown that the V-shape formation in flight enables the entire flock to cover a distance of 71% more than if the bird flew alone. Not only that, but researchers have found that the birds alternate their places in the flock in a strange way. When one of the birds feels tired, it returns back so that it can take advantage of the air current resulting from the flock as a whole, thus saving power by reducing the effort expended to the maximum degree, then after resting a while, it returns to its place again, making way for another bird to rest.

4. Methodology and Mathematical Model

Understanding optimization problem types is important for selecting an effective algorithm because it determines the most suitable search strategy design. This problem type classification is based on multiple dimensions, including solution methods, objective types, problem structures, constraints, and complexity levels.

This paper presents a Swallow Search Optimization (SWSO) algorithm that functions as a non-deterministic bio-inspired metaheuristic method that tackles NP-hard single-objective optimization problems with non-convex and constrained structures and partial or total non-separability, especially during obstacle avoidance and formation constraint situations. Unlike Ref. [47], the proposed method gives new ways of controlling agent coordination as well as finding optimal solutions to constrained and complex optimization problems. Not only does this model the social action of swallows but it also adds energy-driven rotations of leadership, the PSO-style velocity update and even V-formation flight dynamics to guide swarm coordination. The algorithm starts by initializing both position and velocity parameters in continuous bounded domains that focus on continuous search areas. The adaptive cognitive-social parameters (c₁, c₂) together with energy-based leader selection work effectively in dynamic environments that require the continuous evolution of leadership and behavior. The dual feature of storing multiple solutions along with best position memory enables SWSO to optimize multimodal functions while maintaining diversity through stochastic exploration decay. SWSO implements a repulsion-based penalty mechanism to resolve constrained problems that contain spatial constraints such as obstacles. The mechanism guides agents through feasible regions to prevent them from entering prohibited zones while sustaining diversity among the population. The V-formation control system, together with leader energy management, enables efficient agent coordination that sets this SWSO variant apart from others when dealing with structured optimization landscapes found in real-world applications.

Mathematical Model and Algorithm

In nature, excellent examples of engineering solutions are found. These engineering solutions have been perfected over millions of years of evolution. By studying and understanding lessons from nature, new or better designs of materials and structures can be made. For biomimetic applications, it is important to have a clear understanding of biology [61,62], which is why this paper examines and mathematically models the swallow’s biological behavior to gain insights that can inspire the SWSO optimization algorithm.

The Swallow Search Optimization Algorithm (SWSO) is a nature-inspired metaheuristic based on the social and energetic behavior of swallow birds. It simulates their foraging, migration, and social communication behaviors to balance global exploration and local exploitation in solving complex optimization problems. The algorithm models a population of N agents (swallows), each represented by a position vector $x\in R^d$, where $d$ is the problem dimension. The goal is to minimize a real-valued objective function $f:R^d\to R$.

4.1. Swallow Foraging Behavior

The foraging behavior of swallows is modeled to enhance exploration. Swallows search for food in groups, adjusting their positions based on successful foraging events.

• Initialization

Each swallow’s position is initialized uniformly within the search space:

$$x_{i,j}^{\left(0\right)}\sim u\left({lb}_j,{ub}_j\right), \mathrm{f}\mathrm{o}\mathrm{r} i=1,2,\dots \dots ,N \mathrm{a}\mathrm{n}\mathrm{d} j=1,2,\dots \dots ,d$$

(5)

where ${lb}_j$ and ${ub}_j$ are the lower and upper bounds for dimension $j$.

Each agent is also assigned a normalized energy value ${E_i}^{\left(0\right)}\in \left[0,1\right]$, representing its readiness to lead the swarm.

• Exploration (with probability 1 − α)

$$x_i^{\left(t+1\right)}=x_i^{\left(t\right)}+{\delta }^{\left(t\right)}.\left({2r}_i-1\right)\circ \left(ub-lb\right)$$

(6)

where,

• ${\delta }^{\left(t\right)}={explore\text{\_}decay}^{\left({\displaystyle \frac{t}{T}}\right)}$ is a decaying exploration factor, encouraging convergence over time.
• $r_i \sim u{\left(0,1\right)}^d$ is a random vector.

This simulates distributed search, preventing premature convergence.

4.2. Swallow Migration Behavior

Migrating birds have been monitored to fly across the globe with a minimum amount of food. For our design, migration behavior is modeled to enhance exploitation. Swallows move towards optimal regions while avoiding suboptimal areas.

• Fitness Evaluation

At each iteration $t$, the fitness of each agent $i$ is computed as follows:

$$f_i^{\left(t\right)}=f\left(x_i^{\left(t\right)}\right)$$

(7)

The globally best solution found so far is tracked:

$$x_{best}=arg min\hspace{1em}{}_{x_i^{\left(t\right)}}f_i^{\left(t\right)}, f_{best}=arg min\hspace{1em}{}_{i}f_i^{\left(t\right)}$$

(8)

• Position Update Formula

Each non-leader swallow updates its position based on either exploitation (following the leader) or exploration (random search), governed by a probability $\alpha \in \left[0,1\right]$.

$$x_i^{\left(t+1\right)}=x_i^{\left(t\right)}+r_i\circ \left(x_L^{\left(t\right)}-x_i^{\left(t\right)}\right)+{\epsilon }_{i }$$

(9)

where,

• ${x_L}^{\left(t\right)}$ is the position of the leader.
• ${\epsilon }_i\sim N\left(0,{\sigma }^2\right)$ is a small Gaussian perturbation (for local search).

This strategy exploits learned successful paths while maintaining variability for refinement.

• Velocity Update Formula

All metaheuristics share a common concept of achieving a balance between exploration and exploitation [63]. Exploration is the ability of members to generate a variety of solutions in the search area, which results in a global search feature of the metaheuristic algorithms. Exploitation, on the other hand, indicates the ability of members to make the best use of past iterations’ solutions information to narrow the search, leading to a local search behavior of the metaheuristic. Different metaheuristic problems emphasize either exploratory approaches or exploitation methods in their problem-solving strategies. It is important to strike a balance between these two aspects to optimize the performance of a metaheuristic algorithm. Several strategies for balancing exist, such as static control, which uses fixed parameters [64], dynamic adjustment with parameters change over time [65], greedy selection that used for pure exploitation [66], randomized search for pure exploration [67], and feedback-based or self-adaptive methods that respond to agent performance or environmental cues [68]. The ε-greedy strategy, together with Bayesian methods [69], including Upper Confidence Bound (UCB) and Thompson Sampling, forms the advanced category of balancing strategies [70,71,72].

The proposed Swallow Search Optimization (SWSO) algorithm uses hybrid dynamic strategies which derive their principles from Particle Swarm Optimization (PSO). The algorithm regulates exploration–exploitation transitions using time-variable cognitive and social coefficients together with velocity-based update rules. This design fosters both rapid convergence and sustained diversity within the population.

The framework incorporates velocity-based updates, which apply momentum-based searches instead of random moving through the solution space. The algorithm uses historical positional data to enhance agent steering behavior as an adaptive system within the solution space. This facilitates a balanced search behavior, enabling global exploration in early iterations and focused exploitation in later stages, thereby improving solution quality and search efficiency.

$${Velocity}_i^{\left(t+1\right)}=\omega .{Velocity}_i^{\left(t\right)}+{c_1.r}_1.{{(P}_{best,i}^{\left(t\right)}-x}_i^{\left(t\right)})+{c_2.r}_2.\left(x_L^{\left(t\right)}-x_i^{\left(t\right)}\right)$$

(10)

• The term $\omega .{Velocity}_i^{\left(t\right)}$ retains a portion of the previous velocity, which controls the momentum, allows for smoother motion in the search space, and helps the agent to keep exploring in a similar direction. The typical range for the parameter w falls between 0 and 1, while its common use value amounts to 0.7.
• The term ${c_1.r}_1. {{(P}_{best,i}^{\left(t\right)}-x}_i^{\left(t\right)})$ is the cognitive component which reflects the personal experience of the agent and pulls the agent toward its own historically best position. $c_1$ is the cognitive learning coefficient, and $r_1$ is a random value in [0, 1], which adds stochasticity. In this case, ${(P}_{best,i}^{\left(t\right)})$ is the personal best position found by particle iii on its search history and it augments the particle in exploiting parts of the search space that gave good fitness values previously.
• The term ${c_2.r}_2. \left(x_L^{\left(t\right)}-x_i^{\left(t\right)}\right)$ is the social component that encourages social learning by attracting the agent to the current leader’s position. $c_2$ is the social learning coefficient, and $r_2$ is another random value in [0, 1].

The resulting velocity is used to update the agent’s position to make the agent move in the direction of its new velocity.

$$x_i^{\left(t+1\right)}=x_i^{\left(t\right)}+{Velocity}_i^{\left(t\right)}$$

(11)

The algorithm produces better dynamic behavior through time and creates a shift from independent search toward swarm convergence, by decreasing $c_1$, which reduces exploration and increases $c_2$ for better exploitation of the search space.

4.3. Social Interaction and Information Sharing

Swallows communicate to share information about food sources and predators, enhancing the search efficiency. The Figure 4 representation highlights key behavior of the SWSO and demonstrates how the velocity vector, energy decay mechanism, leader switching behavior, and repulsion-based constraint handling uniquely influence the position update of agents in SWSO—distinguishing it from earlier role-based methods with static agent types.

• Leader Selection

When a bird changes its position from follower to leader status or vice versa, the entire flock behavior pattern transforms. The leader must actively direct flock movements because other birds will adapt their positions accordingly.

In the SWSO algorithm, a leader is selected based on combining both fitness and energy, encouraging high-performance and well-energized individuals:

$$leader index={arg min}_i\left(f_i^{\left(t\right)}+\left(1-{E_i}^{\left(t\right)}\right)\right)$$

(12)

Without Leader Switching, the problem is likely separable (with minimal interdependencies).

• Energy Dynamics

Each agent’s energy is updated to simulate fatigue and partial recovery:

$$E_i^{\left(t+1\right)}=min\left(max\left(E_i^{\left(t\right)}-0.01+0.05.u_i,0\right),1\right),\hspace{1em}{u}_i\sim u\left(0,1\right)$$

(13)

If the leader’s energy $E_L^{\left(t\right)}$ falls below a threshold $\theta$, a new leader is chosen:

$$New leader={arg min}_i\left(f_i^{\left(t\right)}+\left(1-{E_i}^{\left(t\right)}\right)\right)$$

(14)

The biological inspiration drawn from swallows—such as foraging in groups, energy-efficient V-formation flight, leader rotation, and rapid maneuvering—has been rigorously formalized in the proposed Swallow Search Optimization (SWSO) algorithm through a set of mathematical models. Swarm is modeled to have foraging behavior by distributed random exploration (Equation (6)) so that it can diversify its search in initial iterations. Directional migration (i.e., movement towards the best-performing individual) together with local refinement based on Gaussian noise (Equation (9)) is used to model migration and exploitation. Momentum, self-learning, and social influence are all added in velocity and position updates (Equations (10) and (11)) just as it has been observed with swallows changing direction based on memory and group dynamics. The biological process of alternating flight roles of the V-formation to minimize energy consumption and maximize coordination is simulated by leader switching and energy dynamics (Equations (12–14)). The constraints handling through repulsion-based constraints mimics obstacle avoidance, and hence, the model adapts to a constrained setting dynamically.

The pseudo-code of the SWSO algorithm is presented:

Algorithm 1. Pseudo code of the SWSO algorithm

1- Initialize Swallow_pos randomly within [lb, ub] 2- Set Velocity to zero matrix 3- Set Best_score = ∞ and Best_pos = zero vector, 4- Initialize P_best = Swallow_pos and P_best_fitness = ∞ 5- Initialize Energy randomly in [0, 1] for all agents 6- Generate V_formation using generate_v_formation() 7- FOR t = 1 to Max_iter DO 8- FOR each swallow i = 1 to N DO 9- Clamp Swallow_pos[i] within [lb, ub] 10- Evaluate Swallow_fitness[i] = fobj(Swallow_pos[i]) 11- IF Swallow_fitness[i] < Best_score THEN 12- Best_score ← Swallow_fitness[i] 13- Best_pos ← Swallow_pos[i] 14- END IF 15- IF Swallow_fitness[i] < P_best_fitness[i] THEN 16- P_best[i] ← Swallow_pos[i] 17- P_best_fitness[i] ← Swallow_fitness[i] 18- END IF 19- END FOR 20- Select Leader ← agent with min(fitness + (1 − Energy)) 21- Update adaptive parameters c1 and c2 over time 22- FOR each swallow i ≠ Leader DO 23- Generate random values r1, r2 ∈ [0, 1] 24- Update Velocity[i] using: w × Velocity[i] + c1 × r1 × (P_best[i] − Swallow_pos[i]) + c2 × r2 × (Leader − Swallow_pos[i]) 25- Update Swallow_pos[i] += Velocity[i] 26- IF rand ≥ alpha THEN 27- Apply decaying random exploration 28- END IF 29- FOR each obstacle DO 30- IF distance to obstacle < 1 THEN 31- Apply repulsion force to avoid collision 32- END IF 33- END FOR, END FOR 34- Update Energy[i] ← Energy[i] − 0.01 + 0.05 × rand 35- Clamp Energy[i] within [0, 1] 36- IF Energy [Leader] < 0.5 THEN 37- Find nearest neighbor and swap with Leader 38- END IF 39- Apply maintain_v_formation() to Swallow_pos 40- Store Best_score in cg_curve[t] 41- IF t == 1 or mod(t,10) == 0 THEN 42- Print: Iteration t | Best Score = Best_score 43- END IF, END FOR 44- RETURN Best_score, Best_pos, cg_curve

5. Validation and Comparison

5.1. Benchmark Functions

A set of benchmark functions, which are groups of functions that can be used to test the performance of any optimization problem [73], serve to evaluate the efficiency of the proposed SWSO algorithm. These benchmark functions cover all categories of unimodal, multimodal, fixed-dimension multimodal, and composite functions.

• Unimodal functions: The single global optimum of unimodal functions, as shown in Figure 5, provides an excellent platform for testing both convergence efficiency and exploitation capabilities of algorithms.
• Multimodal functions: Multimodal functions serve as crucial evaluation tools to test Swallow Search Optimization Algorithm’s (SWSO) ability to prevent premature convergence while escaping local minima during evaluations. The search space complexity is measured through these functions which contain multiple local optima to test SWSO’s exploration capabilities.
• Fixed-dimension multimodal functions: Multimodal functions with fixed dimensions differ from those that provide scalable dimensional capabilities. The test functions with fixed dimensions present predetermined dimensions that limit their ability to evaluate problems of different sizes. Although using the multimodal functions provides adjustable design variable counts, their landscape structures still differ from those provided by the multimodal with fixed dimensions [74].
• The composite functions: The complexity of composite test functions increases because they use transformations including random shifts of the global optimum along with search space rotations and boundary-based placement of optima. Composite functions provide excellent benchmarking capabilities for SWSO under dynamic and deceptive conditions because they include features which enhance robustness and adaptability across various optimization scenarios [30,73]. Table 3, Table 4 and Table 5 describe the unimodal, multimodal, and fixed-dimension multimodal, respectively. These tables give a detailed overview of the equations, problem dimensions $Dim(n)$, upper and lower bands in the boundaries, and minimum goal values of each function ($F_{min})$ [75].

Table 3. Unimodal benchmark functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Sphere	$F_1\left(X\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	30	[−100, 100]	0
Schwefel 2.22	$F_2\left(X\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30	[−10, 10]	0
Schwefel 1.2	$F_3\left(X\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^i}{x_j}^2$	30	[−100, 100]	0
Schwefel 2.21	$F_4\left(X\right)=\underset{1\le i\le N}{\mathit{max}}\left|x_i\right|$	30	[−100, 100]	0
Rosenbrock	$F_5\left(X\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(1-x_i\right)}^2\right]$	30	[−30, 30]	0
Step	$F_6\left(X\right)={\displaystyle \sum _{i=1}^n}{\left(⌊x_i+0.5⌋\right)}^2$	30	[−100, 100]	0
Quartic	$F_7\left(X\right)={\displaystyle \sum _{i=1}^n}i{x}_i^4+random[0,1)$	30	[−1.28, 1.28]	0

Table 3. Unimodal benchmark functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Sphere	$F_1\left(X\right)={\displaystyle \sum _{i=1}^n}{x}_i^2$	30	[−100, 100]	0
Schwefel 2.22	$F_2\left(X\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30	[−10, 10]	0
Schwefel 1.2	$F_3\left(X\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^i}{x_j}^2$	30	[−100, 100]	0
Schwefel 2.21	$F_4\left(X\right)=\underset{1\le i\le N}{\mathit{max}}\left|x_i\right|$	30	[−100, 100]	0
Rosenbrock	$F_5\left(X\right)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(1-x_i\right)}^2\right]$	30	[−30, 30]	0
Step	$F_6\left(X\right)={\displaystyle \sum _{i=1}^n}{\left(⌊x_i+0.5⌋\right)}^2$	30	[−100, 100]	0
Quartic	$F_7\left(X\right)={\displaystyle \sum _{i=1}^n}i{x}_i^4+random[0,1)$	30	[−1.28, 1.28]	0

Table 4. Multimodal basic functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Schwefel 2.26	$F_8\left(X\right)={\displaystyle \sum _{i=1}^n}-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	−418.9829 $n$
Shifted Rastrigin	$F_9\left(X\right)={\displaystyle \sum _{i=1}^n}\left[x_i^2-10cos\left(2\pi x_i\right)+10n\right]$	30	[−5.12, 5.12]	0
Ackley	$\begin{array}{cc}\hfill F_{10}\left(X\right)=-20\mathit{exp}& \left(-{\displaystyle \frac{1}{5}}\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{x}_i^2}\right)\hfill \\ & -\mathit{exp}\left({\displaystyle \frac{1}{n}}\sum _{i=1}^{n}cos\left(2\pi x_i\right)\right)+20+e\hfill \end{array}$	30	[−32, 32]	0
Griewangk	$F_{11}\left(X\right)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\displaystyle \prod _{i=1}^n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0
Penalized function one	$\begin{array}{cc}\hfill F_{12}\left(X\right)={\displaystyle \frac{\pi }{n}}{\displaystyle \{}10{sin}^2& \left(\pi y_1\right)+{\displaystyle \sum _{i=1}^{n-1}}{\left(y_i-1\right)}^2\left[1+10{sin}^2\left(\pi y_{i+1}\right)\right]\hfill \\ & +{\left(y_n-1\right)}^2{\displaystyle \}}+{\displaystyle \sum _{i=1}^n}{u}_i\hfill \end{array}$ $\begin{array}{cc}\mathrm{Where}:& y_i=1+{\displaystyle \frac{x_i+1}{4}}, u_i\left(x_i,a,k,m\right)=\left(x_i,10,100,4\right)\hfill \\ & u_i=\left\{\begin{array}{ccc}k{\left(x_i-a\right)}^m\hfill & ,& x_i>a\\ 0\hfill & ,& -a\le x_i\le a\\ k{(-x_i-a)}^m\hfill & ,& x_i<-a\end{array}\right.\end{array}$	30	[−50, 50]	0
Penalized function two	$\begin{array}{cc}\hfill F_{13}\left(X\right)={\displaystyle \frac{1}{10}}{\displaystyle \{}{sin}^2& \left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^n}{\left(x_i-1\right)}^2\left[1+{sin}^2\left(3\pi x_{i+1}\right)\right]\hfill \\ & +{\left(x_n-1\right)}^2\left[1+{sin}^2\left(2\pi x_n\right)\right]{\displaystyle \}}\hfill \\ & +{\displaystyle \sum _{i=1}^n}{u}_i\left(x_i,5,100,4\right)\hfill \end{array}$	30	[−50, 50]	0

Table 4. Multimodal basic functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Schwefel 2.26	$F_8\left(X\right)={\displaystyle \sum _{i=1}^n}-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	−418.9829 $n$
Shifted Rastrigin	$F_9\left(X\right)={\displaystyle \sum _{i=1}^n}\left[x_i^2-10cos\left(2\pi x_i\right)+10n\right]$	30	[−5.12, 5.12]	0
Ackley	$\begin{array}{cc}\hfill F_{10}\left(X\right)=-20\mathit{exp}& \left(-{\displaystyle \frac{1}{5}}\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{x}_i^2}\right)\hfill \\ & -\mathit{exp}\left({\displaystyle \frac{1}{n}}\sum _{i=1}^{n}cos\left(2\pi x_i\right)\right)+20+e\hfill \end{array}$	30	[−32, 32]	0
Griewangk	$F_{11}\left(X\right)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\displaystyle \prod _{i=1}^n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0
Penalized function one	$\begin{array}{cc}\hfill F_{12}\left(X\right)={\displaystyle \frac{\pi }{n}}{\displaystyle \{}10{sin}^2& \left(\pi y_1\right)+{\displaystyle \sum _{i=1}^{n-1}}{\left(y_i-1\right)}^2\left[1+10{sin}^2\left(\pi y_{i+1}\right)\right]\hfill \\ & +{\left(y_n-1\right)}^2{\displaystyle \}}+{\displaystyle \sum _{i=1}^n}{u}_i\hfill \end{array}$ $\begin{array}{cc}\mathrm{Where}:& y_i=1+{\displaystyle \frac{x_i+1}{4}}, u_i\left(x_i,a,k,m\right)=\left(x_i,10,100,4\right)\hfill \\ & u_i=\left\{\begin{array}{ccc}k{\left(x_i-a\right)}^m\hfill & ,& x_i>a\\ 0\hfill & ,& -a\le x_i\le a\\ k{(-x_i-a)}^m\hfill & ,& x_i<-a\end{array}\right.\end{array}$	30	[−50, 50]	0
Penalized function two	$\begin{array}{cc}\hfill F_{13}\left(X\right)={\displaystyle \frac{1}{10}}{\displaystyle \{}{sin}^2& \left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^n}{\left(x_i-1\right)}^2\left[1+{sin}^2\left(3\pi x_{i+1}\right)\right]\hfill \\ & +{\left(x_n-1\right)}^2\left[1+{sin}^2\left(2\pi x_n\right)\right]{\displaystyle \}}\hfill \\ & +{\displaystyle \sum _{i=1}^n}{u}_i\left(x_i,5,100,4\right)\hfill \end{array}$	30	[−50, 50]	0

Table 5. Fixed-dimension benchmark functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
De Jong $5^{th}$ Function	$F_{14}\left(X\right)={\left({\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}}{\displaystyle \frac{1}{j+{\sum }_{i=1}^2{\left(x_i-a_{ij}\right)}^6}}\right)}^{-1}$	2	[−65, 65]	1
Kowalik’s Function	$F_{15}\left(X\right)={\displaystyle \sum _{i=1}^{11}}{\left[a_i-{\displaystyle \frac{x_1\left(b_i^2+b_i{x}_2\right)}{b_i^2+b_i{x}_3+x_4}}\right]}^2$	4	[−5, 5]	0.00030
Six-Hump Camel	$F_{16}\left(X\right)={4x}_1^2-2.1x_1^4+{\displaystyle \frac{x_1^6}{3}}+x_1{x}_2-{4x}_2^2+{4x}_2^4$	2	[−5, 5]	−1.0316
Branin Function	$F_{17}\left(X\right)={\left(x_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{x}_1^2+{\displaystyle \frac{5}{\pi }}{x}_1-6\right)}^2+10\left(1-{\displaystyle \frac{1}{8\pi }}\right)cos{x}_1+10$	2	[−5, 5]	0.397887
Goldstein-Price Function	$\begin{array}{cc}\hfill F_{18}\left(X\right)=[1+(x_1& +x_2+1{)}^2(19-14x_1+3x_1^2-14x_2\hfill \\ & +6x_1{x}_2+3x_2^2)][30\hfill \\ & +{\left(2x_1-3x_2\right)}^2(18-32x_1+12x_1^2+48x_2\hfill \\ & -36x_1{x}_2+27x_2^2)]\hfill \end{array}$	2	[−2, 2]	3
Hartmann 3-Dimensional Function	$F_{19}\left(X\right)=-{\displaystyle \sum _{i=1}^4}{\alpha }_{i}exp\left(-{\displaystyle \sum _{j=1}^3}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	3	[0, 1]	−3.86278
Hartmann 6-Dimensional Function	$F_{20}\left(X\right)=-{\displaystyle \sum _{i=1}^4}{\alpha }_{i}exp\left(-{\displaystyle \sum _{j=1}^6}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	6	[0, 1]	−3.32237
Shekel Function	$F_{21}\left(X\right)=-{\displaystyle \sum _{i=1}^5}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.1532
Shekel Function	$F_{22}\left(X\right)=-{\displaystyle \sum _{i=1}^7}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.4028
Shekel Function	$F_{23}\left(X\right)=-{\displaystyle \sum _{i=1}^{10}}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.5364

Table 5. Fixed-dimension benchmark functions.

Function Name	Function	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
De Jong $5^{th}$ Function	$F_{14}\left(X\right)={\left({\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}}{\displaystyle \frac{1}{j+{\sum }_{i=1}^2{\left(x_i-a_{ij}\right)}^6}}\right)}^{-1}$	2	[−65, 65]	1
Kowalik’s Function	$F_{15}\left(X\right)={\displaystyle \sum _{i=1}^{11}}{\left[a_i-{\displaystyle \frac{x_1\left(b_i^2+b_i{x}_2\right)}{b_i^2+b_i{x}_3+x_4}}\right]}^2$	4	[−5, 5]	0.00030
Six-Hump Camel	$F_{16}\left(X\right)={4x}_1^2-2.1x_1^4+{\displaystyle \frac{x_1^6}{3}}+x_1{x}_2-{4x}_2^2+{4x}_2^4$	2	[−5, 5]	−1.0316
Branin Function	$F_{17}\left(X\right)={\left(x_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{x}_1^2+{\displaystyle \frac{5}{\pi }}{x}_1-6\right)}^2+10\left(1-{\displaystyle \frac{1}{8\pi }}\right)cos{x}_1+10$	2	[−5, 5]	0.397887
Goldstein-Price Function	$\begin{array}{cc}\hfill F_{18}\left(X\right)=[1+(x_1& +x_2+1{)}^2(19-14x_1+3x_1^2-14x_2\hfill \\ & +6x_1{x}_2+3x_2^2)][30\hfill \\ & +{\left(2x_1-3x_2\right)}^2(18-32x_1+12x_1^2+48x_2\hfill \\ & -36x_1{x}_2+27x_2^2)]\hfill \end{array}$	2	[−2, 2]	3
Hartmann 3-Dimensional Function	$F_{19}\left(X\right)=-{\displaystyle \sum _{i=1}^4}{\alpha }_{i}exp\left(-{\displaystyle \sum _{j=1}^3}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	3	[0, 1]	−3.86278
Hartmann 6-Dimensional Function	$F_{20}\left(X\right)=-{\displaystyle \sum _{i=1}^4}{\alpha }_{i}exp\left(-{\displaystyle \sum _{j=1}^6}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	6	[0, 1]	−3.32237
Shekel Function	$F_{21}\left(X\right)=-{\displaystyle \sum _{i=1}^5}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.1532
Shekel Function	$F_{22}\left(X\right)=-{\displaystyle \sum _{i=1}^7}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.4028
Shekel Function	$F_{23}\left(X\right)=-{\displaystyle \sum _{i=1}^{10}}{\left[\left(x-a_i\right){\left(x-a_i\right)}^T+C_i\right]}^{-1}$	4	[0, 10]	−10.5364

In addition to functional classification based on landscape characteristics, these benchmark functions are arranged into four primary categories based on variable interaction and separability patterns, which range from simple to complex combinations. Separable functions are the first and simplest type, since their variables operate independently; the sphere function serves as a representative example of a separable function [76]. The second is a more complex one involving partially non-separable functions, where a subset of variables is interdependent while the remaining ones remain independent, introducing moderate difficulty. The third type is the m-non-separable, which emerges frequently in benchmark suites proposed by the CEC [76]. The fourth type displays the most challenging optimization characteristics since its fully non-separable functions cause all variables to entangle with one another throughout the search space [75]. Functions like the rotated versions of Rastrigin and Ackley belong to this class and are designed to strictly test the global search capability of algorithms [76], as summarized in

Table 6. Cec2019 benchmark functions.

No.	Functions	$\mathit{D}\mathit{i}\mathit{m}(\mathit{n})$	Range	${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$
Cec01	Storn’s Chebyshev Polynomial Fitting Problem	9	[−8192, 8192]	1
Cec02	Inverse Hilbert Matrix Problem	16	[−16,384, 16,384]	1
Cec03	Lennard–Jones Minimum Energy Cluster	18	[−4, 4]	1
Cec04	Rastrigin’s Function	10	[−100, 100]	1
Cec05	Griewank’s Function	10	[−100, 100]	1
Cec06	Weierstrass Function	10	[−100, 100]	1
Cec07	Modified Schwefel’s Function	10	[−100, 100]	1
Cec08	Expanded Schaffer’s F6 Function	10	[−100, 100]	1
Cec09	Happy Cat Function	10	[−100, 100]	1
Cec10	Ackley Function	10	[−100, 100]	1

.

5.2. Comparison with Other Algorithms

Simulation tests are implemented using MATLAB R2020a on Lenovo laptop with the following specifications: system model: 20VD, 11th Generation Intel Core i5-1135G7 processor with a base speed of 2.4 GHz (8 logical cores), 16 GB RAM, Windows 11 Pro operating system, and manufactured by Lenovo, headquartered in Beijing, China. In the SWSO algorithm, the following probability is set to α = 0.8, the exploration decay rate is 0.9, the energy threshold for leader switching is 0.3, and the adaptive inertia is sampled randomly in the range [0.6, 1.0]. A Gaussian perturbation scales the random exploration term ${\epsilon }_i\sim N\left(0,{0.052}^2\right)$, and convergence is tracked at each iteration (see Appendix A, Figure A1). Particularly, the iteration was set to 1000 in both the conventional benchmark functions and the complex real-world engineering problems, and the number of independent runs was 30. To make all the comparative evaluations consistent, the mean values of the results were reported. The SWSO algorithm was tested as well using a number of benchmark functions of unimodal, multimodal, fixed-dimension multimodal, and CEC 2019 composite functionality.

For SWSO, the following parameter values were used throughout all benchmark tests (based on extensive tuning and empirical studies): Population size (30), Initial energy (E₀ = 1.0), Energy decay factor (λ = 0.005), Cognitive coefficient (c₁) linearly decreasing from 2.5 to 0.5, Social coefficient (c₂) linearly increasing from 0.5 to 2.5, Gaussian disturbance (σ = 0.1), and Repulsion threshold (ε = 10⁻⁶).

For all comparative algorithms, it used parameter values as originally recommended in their respective publications or widely accepted benchmarks, including the PSO (c₁ = c₂ = 2.0, w = 0.729), GA (Crossover rate = 0.8, mutation rate = 0.01), MFO (b = 1, t = iteration index), WOA (linearly decreases from 2 to 0), GWO (α, β, δ coefficients as per default settings), and DE (F = 0.5, CR = 0.9).

All algorithms used a population size of 50 and a maximum of 1000 iterations to ensure fair comparison.

A total of 29 functions were operated to test them. The behavior of the algorithm was compared with over 30 standard optimization strategies, grouped into six distinct categories of algorithms (group-A to group-F). These are PSO, MFO, GSA, FA, GA, WOA, INFO, HHO, RIME, TSA, and FOX. Some of the evaluation measures involved average fitness values, standard deviation of fitness values, convergence rate, and durability of multiple runs. Analysis was conducted by Wilcoxon rank-sum tests to ensure significance is confirmed. Comparison of the results obtained is consistent and indicates that SWSO performs competitively or above in all types of functions, especially in exploration (multimodal) and convergence accuracy (unimodal or fixed dimension problems). In the CEC2019 benchmarks, it passed its tests, assuring its quality in feasible situations of simulation necessities.

Based on the mentioned different benchmark function types, the results of the SWSO are compared with other SI algorithms, and the comparison results are illustrated in four sections:

5.2.1. The Unimodal Benchmark Test Functions (F1–F7): Exploitation Capability

Based on unimodal benchmark functions, which mainly measure the ability of an algorithm to find the best global solution, the SWSO algorithm compared with two different groups of SI algorithms, group-A which are the following: MFO [77], PSO [78], GSA [79], BA [80], FPA [81], SMS [82], FA [83], and IFOX [84], and group-B which are: WOA [85], FEP [86], IACO [87], VGWO, GWO [88], INFO [89], SCA [90], and RIME [12]. The comparison with group-A and group-B is presented in

Table 7. (a): Comparison result of Average (Ave) and Standard deviation (SD) for the proposed model using classical unimodal benchmark functions, with results of group-A. (b): Comparison result of Average (Ave) and Standard deviation (SD) for the proposed model using classical unimodal benchmark functions, with results of group-B.

(a)
	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	IFOX
F	Ave
F1	1.2466 × 10−84	1.1700 × 10−4	1.36 × 10−4	2.53 × 10−16	2.0792 × 10+4	2.0364 × 10+2	1.2000 × 10+2	7.4807 × 10+3	−2.0 × 10+2
F2	1.5366 × 10−48	6.3900 × 10−4	4.21 × 10−2	5.57 × 10−2	8.9786 × 10+1	1.1169 × 10+1	2.0531 × 10−2	3.9325 × 10+1	3.6 × 10−2
F3	3.5466 × 10−86	6.9673 × 10+2	7.01 × 10+1	8.96 × 10+2	6.2481 × 10+4	2.3757 × 10+2	3.7820 × 10+4	1.7357 × 10+4	3.5 × 10−3
F4	9.2386 × 10−53	7.0686 × 10+1	1.09 × 10+0	7.35 × 10+0	4.9743 × 10+1	1.2573 × 10+1	6.9170 × 10+1	3.3954 × 10+1	−9.6 × 10+2
F5	4.214 × 10+0	1.3915 × 10+2	9.67 × 10+1	6.75 × 10+1	1.9951 × 10+6	1.0975 × 10+4	6.3822 × 10+6	3.7950 × 10+6	7.4 × 10+0
F6	1.039 × 10+0	1.1300 × 10−4	1.02 × 10−4	2.5 × 10−16	1.7053 × 10+4	1.7538 × 10+2	4.1439 × 10+4	7.8287 × 10+3	2.2 × 10−2
F7	3.027 × 10−4	9.1155 × 10−2	1.23 × 10−1	8.94 × 10−2	6.0451 × 10+0	1.3594 × 10−1	4.9520 × 10−2	1.9063 × 10+0	3.7 × 10−3
F	SD
F1	6.326 × 10−84	1.5000 × 10−4	2.02 × 10−4	9.67 × 10−17	5.8924 × 10+3	7.8398 × 10+1	0.0000 × 10+0	8.9485 × 10+2	2.2 × 10−1
F2	6.4714 × 10−48	8.7700 × 10−4	4.54 × 10−2	1.94 × 10−1	4.1958 × 10+1	2.9196 × 10+0	4.7180 × 10−3	2.4659 × 10+0	3.3 × 10−2
F3	2.4891 × 10−85	1.8853 × 10+2	2.21 × 10+1	3.18 × 10+2	2.9769 × 10+4	1.3665 × 10+2	0.0000 × 10+0	1.7401 × 10+3	4.8 × 10−2
F4	6.5246 × 10−52	5.2751 × 10+0	3.17 × 10−1	1.74 × 10+0	1.0144 × 10+1	2.2900 × 10+0	3.8767 × 10+0	1.8697 × 10+0	1.1 × 10+2
F5	1.360 × 10+0	1.2026 × 10+2	6.01 × 10+1	6.22 × 10+1	1.2524 × 10+6	1.2057 × 10+4	7.2997 × 10+5	7.5903 × 10+5	1.6 × 10+2
F6	2.990 × 10−1	9.8700 × 10−5	8.28 × 10−5	1.74 × 10−16	4.9176 × 10+3	6.3453 × 10+1	3.2952 × 10+3	9.7521 × 10+2	3.4 × 10−1
F7	1.993 × 10−4	4.6420 × 10−2	4.50 × 10−2	4.34 × 10−2	3.0453 × 10+0	6.1212 × 10−2	2.4015 × 10−2	4.6006 × 10−1	5.3 × 10−2
(b)
	SWSO	WOA	FEP	IACO	VGWO	INFO	SCA	GWO	RIME
F	Ave
F1	1.2466 × 10−84	1.41 × 10−30	5.70 × 10−4	−2.0 × 10+2	−2.0 × 10+2	5.453 × 10−53	2.125 × 10+2	7.667 × 10−22	6.491 × 10+0
F2	1.5366 × 10−48	1.06 × 10−21	8.10 × 10−3	3.3 × 10−3	5.4 × 10−2	3.880 × 10−26	4.671 × 10−1	1.344 × 10−13	4.601 × 10+0
F3	3.5466 × 10−86	5.39 × 10−7	1.60 × 10−2	1.4 × 10−2	8.0 × 10−3	6.698 × 10−50	2.290 × 10+4	2.249 × 10−3	3.317 × 10+3
F4	9.2386 × 10−53	7.26 × 10−2	3.00 × 10−1	−1.6 × 10+2	−3.3 × 10+2	6.986 × 10−27	5.866 × 10+1	5.559 × 10−5	1.952 × 10+1
F5	4.214 × 10+0	2.79 × 10+1	5.06 × 10+0	3.2 × 10+1	1.3 × 10+1	2.542 × 10+1	2.570 × 10+6	2.877 × 10+1	2.489 × 10+3
F6	1.039 × 10+0	3.12 × 10+0	0	7.4 × 10−2	1.5 × 10−1	6.157 × 10−6	3.62 × 10+2	1.764 × 10+0	6.476 × 10+0
F7	3.027 × 10−4	1.43 × 10−3	1.42 × 10−1	1.5 × 10−2	8.1 × 10−3	4.792 × 10−3	1.944 × 10+0	5.162 × 10−3	9.443 × 10−2
F	SD
F1	6.326 × 10−84	4.91 × 10−30	1.30 × 10−4	5.8 × 10−1	4.1 × 10−1	1.48 × 10−53	5.474 × 10+1	1.84 × 10−22	1.285 × 10+0
F2	6.4714 × 10−48	2.39 × 10−21	7.70 × 10−4	3.8 × 10−2	3.7 × 10−2	7.744 × 10−27	1.78 × 10−1	3.639 × 10−14	9.219 × 10−1
F3	2.4891 × 10−85	2.93 × 10−6	1.40 × 10−2	1.8 × 10−1	1.1 × 10−1	1.956 × 10−50	6.724 × 10+3	6.62 × 10−4	6.295 × 10+2
F4	6.5246 × 10−52	3.97 × 10−1	5.00 × 10−1	7.8 × 10+1	2.2 × 10+2	1.587 × 10−27	1.170 × 10+1	1.372 × 10−5	4.246 × 10+0
F5	1.360 × 10+0	7.64 × 10−1	5.87 × 10+0	5.5 × 10+2	2.8 × 10+2	5.659 × 10−1	5.99 × 10+5	9.021 × 10−1	5.231 × 10+2
F6	2.990 × 10−1	5.32 × 10−1	0	9.2 × 10−1	2.7 × 10+0	1.370 × 10−6	8.40 × 10+1	4.020 × 10−1	1.130 × 10+0
F7	1.993 × 10−4	1.15 × 10−3	3.52 × 10−1	2.5 × 10−1	1.4 × 10−1	1.133 × 10−3	4.345 × 10−1	1.355 × 10−3	1.997 × 10−2

, parts a and b, respectively, where the best global minimum solution for each function is shown in bold in the tables.

5.2.2. The Multimodal Benchmark Test Functions (F8–F13): Exploration Capability

To evaluate the SWSO algorithm on multimodal benchmark functions (F8–F13), which are designed to challenge an algorithm’s exploration capabilities by featuring numerous local optima, the SWSO algorithm was compared with two different groups of SI algorithms, group-C which are MFO, PSO, GSA, BA, FPA, SMS, FA, and GA [77] and group-D, which are presented in Ref. [89], MINFO, INFO, SCSO, AVOA, SCA, HHO, GWO, RIME, and ZOA. The comparison between group-C and group-D is presented in

Table 8. (a): Comparison result of Average (Ave) and Standard deviation (SD) for the proposed model using classical multimodal benchmark functions, with results of group-C. (b): Comparison result of Average (Ave) and Standard deviation (SD) for the proposed model using classical multimodal benchmark functions, with results of group-D.

(a)
	SWSO		MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F	Ave
F8	−2.1935 × 10+3		−8.505 × 10+3	−3.575 × 10+3	−2.355 × 10+3	6.555 × 10+4	−8.095 × 10+3	−3.945 × 10+3	−3.665 × 10+3	−6.335 × 10+3
F9	0		8.465 × 10+1	1.245 × 10+2	3.105 × 10+1	9.625 × 10+1	9.275 × 10+1	1.535 × 10+2	2.155 × 10+2	2.375 × 10+2
F10	8.8825 × 10−16		1.265 × 10+0	9.175 × 10+0	3.745 × 10+0	1.595 × 10+1	6.845 × 10+0	1.915 × 10+1	1.465 × 10+1	1.785 × 10+1
F11	0		1.915 × 10−2	1.245 × 10+1	4.875 × 10−1	2.205 × 10+2	2.725 × 10+0	4.215 × 10+2	6.975 × 10+1	1.805 × 10+2
F12	5.0135 × 10−2		8.945 × 10−1	1.395 × 10+1	4.635 × 10−1	2.895 × 10+7	4.115 × 10+0	8.745 × 10+6	3.685 × 10+5	3.415 × 10+7
F13	1.715 × 10−1		1.165 × 10−1	1.185 × 10+4	7.625 × 10+0	1.095 × 10+8	6.245 × 10+1	1.005 × 10+8	5.565 × 10+6	1.085 × 10+8
F	SD
F8	2.3185 × 10+2		7.265 × 10+2	4.315 × 10+2	3.825 × 10+2	0	1.555 × 10+2	4.045 × 10+2	2.145 × 10+2	3.335 × 10+2
F9	0		1.625 × 10+1	1.435 × 10+1	1.375 × 10+1	1.965 × 10+1	1.425 × 10+1	1.865 × 10+1	1.725 × 10+1	1.905 × 10+1
F10	0		7.305 × 10−1	1.575 × 10+0	1.715 × 10−1	7.755 × 10−1	1.255 × 10+0	2.395 × 10−1	4.685 × 10−1	5.315 × 10−1
F11	0		2.175 × 10−2	4.175 × 10+0	4.985 × 10−2	5.475 × 10+1	7.285 × 10−1	2.535 × 10+1	1.215 × 10+1	3.245 × 10+1
F12	1.2485 × 10−2		8.815 × 10−1	5.855 × 10+0	1.385 × 10−1	2.185 × 10+6	1.045 × 10+0	1.415 × 10+6	1.725 × 10+5	1.895 × 10+6
F13	3.035 × 10−2		1.935 × 10−1	3.075 × 10+4	1.235 × 10+0	1.055 × 10+8	9.485 × 10+1	0	1.695 × 10+6	3.855 × 10+6
(b)
	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
F	Ave
F8	−2.1935 × 10+3	−1.2575 × 10+4	−6.9225 × 10+3	−4.8135 × 10+3	−1.525 × 10+4	−3.3845 × 10+3	−1.2565 × 10+4	−4.5665 × 10+3	−8.7595 × 10+3	−5.425 × 10+3
F9	0	0	0	0	0	1.2665 × 10+2	0	1.4675 × 10+1	9.3545 × 10+1	0
F10	8.8825 × 10−16	8.8825 × 10−16	8.8825 × 10−16	8.8825 × 10−16	8.8825 × 10−16	2.345 × 10+ 01	8.8825 × 10−16	4.3005 × 10−12	3.925 × 10+0	8.8825 × 10−16
F11	0	0	0	0	0	5.6205 × 10+0	0	1.9015 × 10−2	1.4605 × 10+0	0
F12	5.0135 × 10−2	2.4215 × 10−5	1.375 × 10−1	2.3375 × 10−1	9.175 × 10−7	1.785 × 10+7	4.9735 × 10−5	1.465 × 10−1	1.3565 × 10+1	4.2465 × 10−1
F13	1.715 × 10−1	4.175 × 10−1	4.655 × 10−1	2.895 × 10+0	1.145 × 10−7	4.245 × 10+6	2.085 × 10−4	1.195 × 10+0	8.905 × 10−1	2.885 × 10+0
F	SD
F8	2.3185 × 10+2	5.2525 × 10−4	9.2505 × 10+2	9.7165 × 10+2	7.295 × 10+2	2.6695 × 10+2	2.8885 × 10+0	8.1675 × 10+2	5.5385 × 10+2	6.7835 × 10+2
F9	0	0	0	0	0	3.8275 × 10+1	0	3.6605 × 10+0	1.4315 × 10+1	0
F10	0	0	0	0	0	7.895 × 10+0	0	8.9845 × 10−13	5.365 × 10−1	0
F11	0	0	0	0	0	9.2435 × 10−1	0	5.5835 × 10−3	3.2275 × 10−2	0
F12	1.2485 × 10−2	5.4135 × 10−6	2.3185 × 10−2	5.9155 × 10−2	1.9025 × 10−7	2.5155 × 10+6	1.6985 × 10−5	4.0165 × 10−2	3.4205 × 10+0	9.1455 × 10−2
F13	3.035 × 10−2	1.165 × 10−1	1.075 × 10−1	2.755 × 10−1	3.945 × 10−8	1.3405 × 10+6	5.535 × 10−5	2.465 × 10−1	1.645 × 10−1	3.475 × 10−1

, parts a and b, respectively.

5.2.3. The Fixed-Dimension Benchmark Test Functions (F14–F19): Balanced Optimization

To analyze the SWSO algorithm on the F14–F19 benchmark functions, which concentrate on how well it balances exploration and exploitation, we used group-C, as it had been used in comparable research and made fair comparisons possible.

Table 9. Comparison result of Average (Ave) and Standard deviation (SD) for the proposed model using classical fixed-dimension benchmark functions, with results of group-C.

	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F	Ave
F14	9.980 × 10−1	8.25 × 10−31	1.38 × 10+2	5.43 × 10−19	1.30 × 10+2	1.01 × 10+1	1.06 × 10+2	1.76 × 10+2	9.21 × 10+1
F15	4.661 × 10−4	6.67 × 10+1	1.67 × 10+2	2.04 × 10+1	5.44 × 10+2	1.14 × 10+1	1.56 × 10+2	3.54 × 10+2	9.67 × 10+1
F16	−1.03+0	1.19 × 10+2	3.95 × 10+2	2.45 × 10+2	6.97 × 10+2	2.35 × 10+2	4.07 × 10+2	3.08 × 10+2	3.69 × 10+2
F17	4.286 × 10−1	3.45 × 10+2	4.86 × 10+2	3.15 × 10+2	7.45 × 10+2	3.55 × 10+2	5.19 × 10+2	5.49 × 10+2	4.51 × 10+2
F18	3.0908+0	1.04 × 10+1	2.57 × 10+2	7.00 × 10+1	5.44 × 10+2	5.48 × 10+1	1.54 × 10+2	1.75 × 10+2	9.59 × 10+1
F19	−3.8617+0	7.07 × 10+2	7.90 × 10+2	8.82 × 10+2	8.96 × 10+2	5.73 × 10+2	6.12 × 10+2	8.30 × 10+2	5.24 × 10+2
F	SD
F14	1.235 × 10−4	1.08 × 10−30	1.16 × 10+2	1.35 × 10−19	1.19 × 10+2	3.16 × 10+1	2.69 × 10+1	8.69 × 10+1	2.79 × 10+1
F15	7.774 × 10−5	5.32 × 10+1	1.64 × 10+2	6.31 × 10+1	1.49 × 10+2	3.38 × 10+0	6.82 × 10+1	1.03 × 10+2	9.70 × 10+0
F16	8.761 × 10−4	2.83 × 10+1	1.22 × 10+2	4.91 × 10+1	1.91 × 10+2	3.96 × 10+1	6.54 × 10+1	3.74 × 10+1	4.28 × 10+1
F17	2.721 × 10−2	4.31 × 10+1	6.73 × 10+1	1.01 × 10+2	1.43 × 10+2	2.06 × 10+1	4.27 × 10+1	1.63 × 10+2	3.15 × 10+1
F18	9.449 × 10−2	3.75 × 10+0	2.00 × 10+2	4.83 × 10+1	1.99 × 10+2	4.21 × 10+1	9.69 × 10+1	8.32 × 10+1	5.38 × 10+1
F19	3.57 × 10−4	1.95 × 10+2	1.89 × 10+2	4.52 × 10+1	8.63 × 10+1	1.49 × 10+2	1.55 × 10+2	1.57 × 10+2	2.29 × 10+1

presents the results obtained for group-C.

5.2.4. CEC2019 Benchmark Test Functions (CEC01–CEC10): Surrogate Optimization in the Real World [25]

In order to provide a comprehensive assessment of the SWSO algorithm, experimental data is compared with three Swarms Intelligence (SI) opponent groups on the CEC2019 benchmarkable functions. In

Table 10. (a): Comparing the results of the proposed algorithm with group-C algorithms on CEC2019 test functions. (b) Comparing the results of the proposed algorithm with group-E algorithms on CEC2019 test functions. (c) Comparing the results of the proposed algorithm with group-F algorithms on CEC2019 test functions.

(a)
	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
F	Ave
Cec01	9.55 × 10+8	3.84 × 10+3	5.50 × 10+4	5.37 × 10+4	5.46 × 10+4	5.45 × 10+10	8.73 × 10+3	3.47 × 10+8	1.45 × 10+10	5.53 × 10+4
Cec02	1.84 × 10+1	1.83 × 10+1	1.83 × 10+1	1.87 × 10+1	1.83 × 10+1	1.92 × 10+1	1.84 × 10+1	1.87 × 10+1	2.03 × 10+1	1.92 × 10+1
Cec03	1.370 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1	1.37 × 10+1
Cec04	4.84 × 10+1	1.18 × 10+2	2.47 × 10+2	2.63 × 10+3	3.39 × 10+2	3.89 × 10+2	9.17 × 10+2	1.14 × 10+2	6.33 × 10+1	3.59 × 10+2
Cec05	2.09 × 10+0	2.29 × 10+0	2.35 × 10+0	3.28 × 10+0	3.24 × 10+0	3.84 × 10+0	4.65 × 10+0	2.85 × 10+0	2.49 × 10+0	4.24 × 10+0
Cec06	1.07 × 10+1	1.02 × 10+1	1.16 × 10+1	1.16 × 10+1	1.08 × 10+1	1.30 × 10+1	1.26 × 10+1	1.32 × 10+1	1.11 × 10+1	1.09 × 10+1
Cec07	2.82 × 10+2	3.21 × 10+2	5.97 × 10+2	9.64 × 10+2	9.37 × 10+2	1.21 × 10+3	7.51 × 10+2	8.76 × 10+2	4.10 × 10+2	3.02 × 10+2
Cec08	2.88 × 10+0	5.65 × 10+0	6.33 × 10+0	6.57 × 10+0	7.16 × 10+0	6.73 × 10+0	7.27 × 10+0	7.25 × 10+0	5.82 × 10+0	5.60 × 10+0
Cec09	3.47 × 10+0	3.99 × 10+0	4.54 × 10+0	3.58 × 10+2	5.58 × 10+0	4.03 × 10+2	6.62 × 10+0	7.98 × 10+0	3.81 × 10+0	3.88 × 10+2
Cec10	1.97 × 10+1	2.12 × 10+1	2.14 × 10+1	2.14 × 10+1	2.12 × 10+1	2.16 × 10+1	2.16 × 10+1	2.16 × 10+1	2.12 × 10+1	2.13 × 10+1
F	SD
Cec01	8.48 × 10+8	1.19 × 10+3	1.33 × 10+4	3.52 × 10+3	3.49 × 10+3	1.38 × 10+10	8.79 × 10+3	1.11 × 10+8	3.49 × 10+9	4.27 × 10+3
Cec02	5.75 × 10−2	3.65 × 10−15	3.46 × 10−15	1.27 × 10−1	1.31 × 10−4	1.75 × 10−1	9.78 × 10−2	1.39 × 10−1	4.37 × 10−1	3.00 × 10−1
Cec03	1.11 × 10−9	1.82 × 10−15	2.80 × 10−15	2.82 × 10−6	1.50 × 10−10	8.37 × 10−5	8.70 × 10−6	9.36 × 10−6	3.80 × 10−9	2.40 × 10−5
Cec04	1.06 × 10+1	3.27 × 10+1	5.28 × 10+1	7.55 × 10+2	6.74 × 10+1	6.88 × 10+2	1.98 × 10+2	1.72 × 10+1	1.15 × 10+1	1.79 × 10+2
Cec05	5.83 × 10−2	6.23 × 10−1	7.29 × 10−1	2.83 × 10−1	3.76 × 10−1	1.86 × 10−1	5.93 × 10−1	2.43 × 10−1	1.18 × 10−1	5.63 × 10−1
Cec06	6.00 × 10−1	1.86 × 10+0	2.26 × 10+0	1.60 × 10+0	2.20 × 10+0	7.68 × 10−1	1.25 × 10+0	6.62 × 10−1	1.37 × 10+0	8.72 × 10−1
Cec07	1.72 × 10+2	1.19 × 10+2	2.39 × 10+2	2.88 × 10+1	2.58 × 10+2	2.52 × 10+2	1.52 × 10+2	2.38 × 10+2	1.67 × 10+2	1.22 × 10+2
Cec08	7.45 × 10−1	8.95 × 10−1	7.54 × 10−1	6.25 × 10−1	6.94 × 10−1	3.44 × 10−1	5.76 × 10−1	9.34 × 10−1	6.48 × 10−1	3.40 × 10−1
Cec09	5.39 × 10−2	1.38 × 10−1	2.78 × 10−1	9.89 × 10+1	6.42 × 10−1	1.01 × 10+2	6.14 × 10−1	1.06 × 10+0	1.00 × 10−1	1.20 × 10+2
Cec10	5.10 × 10+0	3.58 × 10+0	1.46 × 10−1	1.15 × 10−1	5.66 × 10−1	9.52 × 10−1	1.60 × 10−1	2.96 × 10+0	5.73 × 10−1	1.44 × 10+0
(b)
	SWSO	IWOA	NRO	BDA	CEBA	IPSO	IAGA	BBOA	FOX	IFOX
F	Ave
Cec01	9.55 × 10+8	1.1 × 10+8	4.5 × 10+6	6.6 × 10+7	9.8 × 10+8	1.1 × 10+7	5.5 × 10+7	7.2 × 10+6	6.2 × 10+2	7.0 × 10+5
Cec02	1.84 × 10+1	5.0 × 10+3	1.3 × 10+2	8.2 × 10+2	4.7 × 10+3	5.9 × 10+2	1.4 × 10+3	8.9 × 10+1	5.3 × 10+0	1.5 × 10+1
Cec03	1.370 × 10+1	5.8 × 10+0	6.1 × 10+0	8.0 × 10+0	1.2 × 10+1	5.4 × 10+0	5.8 × 10+0	4.8 × 10+0	1.2 × 10+1	7.5 × 10+0
Cec04	4.84 × 10+1	3.7 × 10+3	2.9 × 10+2	3.4 × 10+3	1.8 × 10+4	2.7 × 10+2	3.7 × 10+2	9.9 × 10+3	2.4 × 10+4	2.3 × 10+2
Cec05	2.09 × 10+0	2.6 × 10+0	1.3 × 10+0	2.5 × 10+0	6.4 × 10+0	1.4 × 10+0	1.7 × 10+0	3.4 × 10+0	5.2 × 10+0	1.7 × 10+0
Cec06	1.07 × 10+1	1.1 × 10+1	1.1 × 10+1	1.2 × 10+1	1.4 × 10+1	1.1 × 10+1	8.1 × 10+0	1.1 × 10+1	1.2 × 10+1	1.1 × 10+1
Cec07	2.82 × 10+2	4.4 × 10+2	2.8 × 10+1	3.2 × 10+2	1.3 × 10+3	9.9 × 10+1	5.3 × 10+1	6.0 × 10+2	1.3 × 10+3	1.6 × 10+2
Cec08	2.88 × 10+0	1.1 × 10+0	1.0 × 10+0	1.1 × 10+0	1.8 × 10+0	1.0 × 10+0	1.0 × 10+0	1.3 × 10+0	1.6 × 10+0	1.0 × 10+0
Cec09	3.47 × 10+0	5.6 × 10+1	9.3 × 10+0	1.1 × 10+2	8.8 × 10+2	9.3 × 10+0	1.4 × 10+1	2.2 × 10+2	8.7 × 10+2	6.3 × 10+0
Cec10	1.97 × 10+1	2.1 × 10+1	2.1 × 10+1	2.2 × 10+1	2.2 × 10+1	2.2 × 10+1	2.1 × 10+1	2.1 × 10+1	2.2 × 10+1	2.2 × 10+1
F	SD
Cec01	8.48 × 10+8	1.5 × 10+8	3.5 × 10+7	2.5 × 10+8	4.2 × 10+7	5.8 × 10+7	5.9 × 10+7	5.9 × 10+7	1.4 × 10+4	1.3 × 10+7
Cec02	5.75 × 10−2	4.2 × 10+2	3.9 × 10+2	2.0 × 10+3	2.0 × 10+2	4.5 × 10+2	4.3 × 10+2	4.2 × 10+2	5.6 × 10+0	1.6 × 10+2
Cec03	1.11 × 10−9	1.5 × 10+0	2.2 × 10+0	1.6 × 10+0	9.8 × 10−2	2.2 × 10+0	1.5 × 10+0	1.1 × 10+0	5.0 × 10−2	9.2 × 10−1
Cec04	1.06 × 10+1	2.1 × 10+3	1.5 × 10+3	6.0 × 10+3	2.0 × 10+3	1.3 × 10+3	1.4 × 10+3	1.4 × 10+3	3.4 × 10+2	1.2 × 10+3
Cec05	5.83 × 10−2	3.8 × 10−1	4.8 × 10−1	1.1 × 10+0	2.8 × 10−1	3.8 × 10−1	4.1 × 10−1	2.9 × 10−1	1.4 × 10−1	3.7 × 10−1
Cec06	6.00 × 10−1	8.4 × 10−1	8.0 × 10−1	3.9 × 10−1	7.3 × 10−2	7.6 × 10−1	1.2 × 10+0	7.0 × 10−1	5.8 × 10−1	6.2 × 10−1
Cec07	1.72 × 10+2	1.6 × 10+2	1.2 × 10+2	4.1 × 10+2	5.3 × 10+1	1.2 × 10+2	1.1 × 10+2	1.1 × 10+2	3.2 × 10+1	9.5 × 10+1
Cec08	7.45 × 10−1	1.0 × 10−1	5.4 × 10−2	2.6 × 10−1	6.1 × 10−2	5.1 × 10−2	6.0 × 10−2	4.9 × 10−2	4.9 × 10−2	4.9 × 10−2
Cec09	5.39 × 10−2	8.9 × 10+1	4.7 × 10+1	2.2 × 10+2	1.7 × 10+1	5.0 × 10+1	5.4 × 10+1	5.6 × 10+1	4.9 × 10+0	3.7 × 10+1
Cec10	5.10 × 10+0	1.7 × 10−1	6.2 × 10−1	8.6 × 10−2	1.0 × 10−2	1.1 × 10−1	1.3 × 10−1	3.3 × 10−1	7.4 × 10−2	8.0 × 10−2
(c)
	SWSO	Hybrid FOX−TSA		FOX	TSA		PSO	GWO	MRSO	RSO
F	Ave
Cec01	9.55 × 10+8	2.33 × 10−6		6.51 × 10+3	1.60 × 10+3		1.33 × 10+3	3.05 × 10+3	1.588 × 10+5	6.263 × 10+4
Cec02	1.84 × 10+1	1.71 × 10+1		1.83 × 10+1	1.74 × 10+1		1.73 × 10+1	1.73 × 10+1	1.83 × 10+1	1.84 × 10+1
Cec03	1.37 × 10+1	1.27 × 10+1		1.370 × 10+1	1.27 × 10+1		1.27 × 10+1	1.27 × 10+1	1.37 × 10+1	1.37 × 10+1
Cec04	4.84 × 10+1	1.37 × 10+3		1.80 × 10+1	4.76 × 10+1		6.37 × 10+1	4.13 × 10+1	9.20 × 10+3	8.86 × 10+3
Cec05	2.09 × 10+0	1.92 × 10+0		6.30 × 10+0	1.55 × 10+0		1.26 × 10+0	1.35 × 10+0	4.57 × 10+0	4.631 × 10+0
Cec06	1.07 × 10+1	8.39 × 10+0		5.68 × 10+0	1.03 × 10+1		6.51 × 10+0	1.06 × 10+1	1.09 × 10+1	1.16 × 10+1
Cec07	2.82 × 10+2	3.12 × 10+2		4.56 × 10+2	2.96 × 10+2		1.65 × 10+2	3.85 × 10+2	6.11 × 10+2	7.89 × 10+2
Cec08	2.88 × 10+0	5.28 × 10+0		5.68 × 10+0	5.71 × 10+0		5.19 × 10+0	4.60 × 10+0	6.31 × 10+0	6.32 × 10+0
Cec09	3.47 × 10+0	1.35 × 10+0		3.80 × 10+0	2.43 × 10+0		2.79 × 10+0	3.99 × 10+0	4.96 × 10+2	5.86 × 10+2
Cec10	1.97 × 10+1	2.01 × 10+1		2.10 × 10+1	2.04 × 10+1		2.08 × 10+1	2.03 × 10+1	2.13 × 10+1	2.14 × 10+1
F	SD
Cec01	8.48 × 10+8	2.65 × 10−3		2.73 × 10+4	2.68 × 10+2		8.21 × 10+2	1.93 × 10+3	3.199 × 10+5	1.392 × 10+4
Cec02	5.75 × 10−2	7.52 × 10−2		4.60 × 10−4	1.35 × 10−2		6.63 × 10−15	1.13 × 10−4	7.231 × 10−3	1.981 × 10−1
Cec03	1.11 × 10−9	1.02 × 10−9		8.42 × 10−4	1.13 × 10−3		6.78 × 10−4	1.04 × 10−3	1.337 × 10−6	1.828 × 10−4
Cec04	1.06 × 10+0	8.77 × 10+2		6.98 × 10+0	8.78 × 10+0		3.38 × 10+1	1.53 × 10+1	3.203 × 10+3	2.152 × 10+3
Cec05	5.83 × 10−2	1.15 × 10−1		7.49 × 10−1	2.39 × 10−1		9.23 × 10−2	2.00 × 10−1	4.133 × 10−1	4.290 × 10−1
Cec06	6.00 × 10−1	6.39 × 10−1		1.59 × 10+0	4.19 × 10−1		1.44 × 10+0	4.90 × 10−1	1.011 × 10+0	8.597 × 10−1
Cec07	1.72 × 10+2	1.45 × 10+2		1.97 × 10+2	1.62 × 10+2		1.05 × 10+2	3.33 × 10+2	2.291 × 10+2	2.154 × 10+2
Cec08	7.45 × 10−1	3.46 × 10−1		3.62 × 10−1	6.05 × 10−1		4.87 × 10−1	1.00 × 10+0	4.138 × 10−1	4.334 × 10−1
Cec09	5.39 × 10−2	9.61 × 10+0		6.09 × 10−3	2.97 × 10−2		3.61 × 10−1	9.07 × 10−1	1.493 × 10+2	1.362 × 10+2
Cec10	5.10 × 10+0	8.16 × 10−2		8.72 × 10−3	8.11 × 10−2		4.76 × 10+0	4.89 × 10−1	1.490 × 10−1	1.116 × 10−1

, part a, you can see the results for group-C algorithms [89], which have previously been used for previous categories. Table 10, part b, shows the results from group-E algorithms, which consist of IWOA, NRO, BDA, CEBA, IPSO, IAGA, BBOA, FOX, and IFOX [84]. Group-F algorithms, including Hybrid FOX-TSA, FOX, TSA, PSO, GWO, MRSO, and RSO [91,92], are compared to the results in Table 10c.

5.3. Result Analysis and Discussion

In this section, we analyze the SWSO’s performance by running it on different benchmark functions. These benchmark functions are designed to measure separate capabilities of the algorithm. To check its performance and consistency, the SWSO algorithm is compared with some popular SI algorithms, and the results tables are illustrated in Appendix A.

Table A1. (a): A comparison between the average ranks of group-A optimization algorithms on unimodal benchmark functions. (b): A comparison between the standard deviation ranks of group-A optimization algorithms on unimodal benchmark functions. (c): Comparison of group-A optimization algorithms on unimodal benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	WOA	FEP	ACO	VGWO	INFO	SCA	GWO	RIME
F1	1	3	5	7	7	2	8	4	6
F2	1	3	6	5	7	2	8	4	9
F3	1	3	7	6	5	2	9	4	8
F4	1	4	5	8	9	2	7	3	6
F5	1	5	2	7	3	4	9	6	8
F6	5	7	1	3	4	2	9	6	8
F7	1	2	8	6	5	3	9	4	7
Average Rank	1.57	3.85	4.85	6.00	5.71	2.42	8.42	4.42	6.28
(b)
Test Function	SWSO	WOA	FEP	ACO	VGWO	INFO	SCA	GWO	RIME
F1	1	3	5	7	6	2	9	4	8
F2	1	3	5	7	6	2	8	4	9
F3	1	3	5	7	6	2	9	4	8
F4	1	4	5	8	9	2	7	3	6
F5	4	2	5	7	6	1	9	3	8
F6	3	4	1	5	7	2	8	3	6
F7	1	3	8	7	6	2	9	4	5
Average Rank	1.71	3.14	4.85	6.85	6.57	1.85	8.42	3.57	7.14
(c)
Test Function			1st	2nd	3rd	4th	5th	SWSO Rank	Subtotal
Average Ranks									11
F1			SWSO	INFO	WOA	GWO	FEP	1
F2			SWSO	INFO	WOA	GWO	IACO	1
F3			SWSO	INFO	WOA	GWO	VGWO	1
F4			SWSO	INFO	GWO	WOA	FEP	1
F5			SWSO	FEP	VGWO	INFO	WOA	1
F6			FEP	INFO	IACO	VGWO	SWSO	5
F7			SWSO	WOA	INFO	GWO	VGWO	1
Standard Deviation Ranks									12
F1			SWSO	INFO	WOA	GWO	FEP	1
F2			SWSO	INFO	WOA	GWO	FEP	1
F3			SWSO	INFO	WOA	GWO	FEP	1
F4			SWSO	INFO	GWO	WOA	FEP	1
F5			INFO	WOA	GWO	SWSO	FEP	4
F6			FEP	INFO	SWSO	WOA	IACO	3
F7			SWSO	INFO	WOA	GWO	RIME	1
Average SWSO rank for average performance (F1–F7): 11/7 = 1.57 Average SWSO rank for standard deviation (F1–F7): 12/7 = 1.71

a and

Table A2. (a): A comparison between the average ranks of group-B optimization algorithms on unimodal benchmark functions. (b): A comparison between the standard deviation ranks of group-B optimization algorithms on unimodal benchmark functions. (c): Comparison of group-B optimization algorithms on unimodal benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	IFOX
F1	1	3	4	2	9	7	6	8	5
F2	1	2	5	9	7	8	3	7	4
F3	1	5	3	6	9	4	8	7	2
F4	1	8	2	3	8	5	6	4	9
F5	1	5	4	3	7	6	9	8	5
F6	5	3	2	1	9	6	8	7	4
F7	1	5	6	4	9	7	3	8	2
Average Rank	1.57	4.42	3.71	3.57	8.14	5.85	6.28	7.28	4.14
(b)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	IFOX
F1	1	3	4	2	9	7	6	8	5
F2	1	2	5	6	9	8	3	7	4
F3	2	6	4	7	9	5	1	8	3
F4	1	7	2	3	8	5	6	4	9
F5	1	4	2	3	9	6	7	8	5
F6	4	3	2	1	9	6	8	7	5
F7	1	5	4	3	9	7	2	8	6
Average Rank	1.57	4.28	3.28	3.57	8.85	6.28	4.71	7.14	5.28
(c)
Test Function		1st	2nd	3rd	4th	5th		SWSO Rank	Subtotal
Average Ranks									11
F1		SWSO	GSA	MFO	PSO	SMS		1
F2		SWSO	MFO	SMS	IFOX	PSO		1
F3		SWSO	IFOX	PSO	FPA	MFO		1
F4		SWSO	PSO	GSA	FPA	FA		1
F5		SWSO	IFOX	GSA	PSO	MFO		1
F6		GSA	PSO	MFO	IFOX	SWSO		5
F7		SWSO	IFOX	SMS	GSA	MFO		1
Standard Deviation Ranks									11
F1		SWSO	GSA	MFO	PSO	IFOX		1
F2		SWSO	MFO	SMS	IFOX	PSO		1
F3		SMS	SWSO	IFOX	PSO	FPA		2
F4		SWSO	PSO	GSA	FA	FPA		1
F5		SWSO	PSO	GSA	MFO	IFOX		1
F6		GSA	PSO	MFO	SWSO	IFOX		4
F7		SWSO	SMS	GSA	PSO	MFO		1
Average SWSO rank for average performance (F1–F7): 11/7 = 1.57 Average SWSO rank for standard deviation (F1–F7): 11/7 = 1.57

a show the average performance ranks of group-A and B optimizers, respectively. The algorithms are ranked according to how well they reduced the objective function for every test, so a lower rank means a better algorithm.

In addition, Table A1b and Table A2b present the standard deviation ranks for the same groups, highlighting how consistently each algorithm performs across multiple runs. Here, a lower rank indicates higher stability and less variation in results. The SWSO algorithm not only maintained its leading position in terms of average performance but also demonstrated excellent consistency, frequently ranking first or very close to it in the standard deviation tables. This dual strength—accuracy and reliability—further reinforces SWSO’s robustness across the unimodal benchmark functions.

Meanwhile, Table A1c and Table A2c serve as a summarized view of both aspects—highlighting which algorithms appeared most frequently in the top five positions for both performance and stability. It condenses the ranking information into a more comparative format, reaffirming SWSO’s superiority by tallying how often it ranked first. Ultimately, across all three tables, SWSO maintains a leading position, offering both accuracy and reliability, which are critical traits in optimization problem solving.

5.3.1. Unimodal Functions (F1–F7)

First, we compared SWSO with group-A, which includes WOA, FEP, ACO, VGWO, INFO, SCA, GWO, and RIME, as seen in Table A1. Here, SWSO dominated, consistently taking the top spot for average performance on six out of seven unimodal benchmark functions (F1, F2, F3, F4, F5, and F7). The only exception was on F6, where it fell in fifth place and FEP and INFO took the highest marks. Overall, SWSO had the highest score of 1.57 on average, outdoing other algorithms in this group. For consistency, measured by standard deviation, SWSO was equally impressive, often ranking first or very close to it across most functions, ending with an excellent average rank of 1.71. INFO was its closest competitor in both average and standard deviation in this set.

Then, we brought in group-B, consisting of MFO, PSO, GSA, BA, FPA, SMS, FA, and IFOX, with their results in Table A2. The behavior was much the same. SWSO scored the highest on average for six unimodal functions (F1, F2, F3, F4, F5, and F7), appearing at the top once again. Just like with group-A, F6 was the exception, ranking fifth, with GSA taking the lead. For this group, SWSO’s average ranking was still 1.57, which was much better than the other methods. When checking consistency, SWSO led on five of the functions by securing the first position for standard deviation. Otherwise, the algorithm consistently did well and reached a strong average rank of 1.57. It seems that F6 simply behaves differently than the rest of the unimodal functions, giving other algorithms a chance to shine, but for almost all other unimodal problems, SWSO proves to be exceptionally robust and accurate across both comparison groups.

5.3.2. Multimodal Functions (F8–F13)

To evaluate the SWSO algorithm on multimodal benchmark functions (F8–F13), the SWSO algorithm was compared against two distinct sets of optimization algorithms. The first, group-C, includes MINFO, INFO, SCSO, AVOA, SCA, HHO, GWO, RIME, and ZOA, with their ranks detailed in

Table A3. (a): A comparison between the average ranks of group-C optimization algorithms on multimodal benchmark functions. (b): A comparison between the standard deviation ranks of group-C optimization algorithms on multimodal benchmark functions. (c): Comparison of group-C optimization algorithms on multimodal benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
F8	1	9	6	4	10	2	8	3	7	5
F9	1	1	1	1	1	4	1	2	3	1
F10	1	1	1	1	1	4	1	2	3	1
F11	1	1	1	1	1	4	1	2	3	1
F12	4	2	5	7	1	10	3	6	9	8
F13	3	4	5	9	1	10	2	7	6	8
Average Rank	1.83	3.00	3.16	3.83	2.50	5.66	2.66	3.66	5.16	4.00
(b)
Test Function	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
F8	2	1	9	10	7	3	4	8	5	6
F9	1	1	1	1	1	4	1	2	3	1
F10	1	1	1	1	1	4	1	2	3	1
F11	1	1	1	1	1	4	1	2	3	1
F12	4	2	5	7	1	10	3	6	9	8
F13	3	5	4	8	1	10	2	7	6	9
Average Rank	2.00	1.83	3.50	4.66	2.00	5.83	2.00	4.50	4.83	4.33
(c)
Test Function		1st	2nd	3rd	4th	5th		SWSO Rank		Subtotal
Average Ranks										11
F8		SWSO	SCA	GWO	SCSO	ZOA		1
F9		SWSO	GWO	RIME	SCA	-		1
F10		SWSO	GWO	RIME	SCA	-		1
F11		SWSO	GWO	RIME	SCA	-		1
F12		AVOA	MINFO	HHO	SWSO	INFO		4
F13		AVOA	HHO	SWSO	MINFO	INFO		3
Standard Deviation Ranks										12
F8		MINFO	SWSO	SCA	HHO	RIME		2
F9		SWSO	GWO	RIME	SCA	-		1
F10		SWSO	GWO	RIME	SCA	-		1
F11		SWSO	GWO	RIME	SCA	-		1
F12		AVOA	MINFO	HHO	SWSO	INFO		4
F13		AVOA	HHO	SWSO	INFO	MINFO		3
Average SWSO rank for average performance (F8–F13): 11/6 = 1.83 Average SWSO rank for standard deviation (F8–F13): 12/6 = 2.00

. Here, SWSO demonstrated a strong showing, frequently ranking first for average performance on functions like F8 through F11. While it placed a bit lower on F12 and F13, its overall average rank in group-C was an impressive 1.83. For the standard deviation, which represents the consistency of the algorithm, SWSO also performed admirably, often securing the top spot and maintaining an average rank of 2.00, placing it among the most reliable in this group.

Next, we expanded our comparison to group-D, which includes MFO, PSO, GSA, BA, FPA, SMS, FA, and GA, with their performance laid out in

Table A4. (a): A comparison between the average ranks of group-D optimization algorithms on multimodal benchmark functions. (b): A comparison between the standard deviation ranks of group-D optimization algorithms on multimodal benchmark functions. (c): Comparison of group-D optimization algorithms on multimodal benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F8	1	8	3	2	9	7	5	4	6
F9	1	3	6	2	5	4	7	8	9
F10	1	2	5	3	7	4	9	6	8
F11	1	2	5	3	8	4	9	6	7
F12	1	3	5	2	8	4	7	6	9
F13	2	1	5	3	9	4	7	6	8
Average Rank	1.16	3.16	4.83	2.5	7.66	4.5	7.33	6.0	7.83
(b)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F8	4	9	8	6	1	2	7	3	5
F9	1	5	4	2	9	3	7	6	8
F10	1	6	9	2	7	8	3	4	5
F11	1	2	5	3	9	4	7	6	8
F12	1	3	5	2	9	4	7	6	8
F13	2	3	6	4	9	5	1	7	8
Average Rank	1.66	4.66	6.16	3.16	7.33	4.33	5.33	5.33	7.00
(c)
Test Function		1st	2nd	3rd	4th		5th	SWSO Rank	Subtotal
Average Ranks									7
F8		SWSO	GSA	PSO	FA		SMS	1
F9		SWSO	GSA	MFO	FPA		BA	1
F10		SWSO	MFO	GSA	FPA		PSO	1
F11		SWSO	MFO	GSA	FPA		PSO	1
F12		SWSO	GSA	MFO	FPA		PSO	1
F13		MFO	SWSO	GSA	FPA		PSO	2
Standard Deviation Ranks									10
F8		BA	FPA	FA	SWSO		GA	4
F9		SWSO	GSA	FPA	PSO		MFO	1
F10		SWSO	GSA	SMS	FA		GA	1
F11		SWSO	MFO	GSA	FPA		PSO	1
F12		SWSO	GSA	MFO	FPA		PSO	1
F13		SMS	SWSO	MFO	GSA		FPA	2
Average SWSO rank for average performance (F8–F13): 7/6 = 1.16 Average SWSO rank for standard deviation (F8–F13): 10/6 = 1.66

. Against this group, SWSO’s average performance was even more remarkable. It took the first rank on F8, F9, F10, F11, and F12, only slipping to second on F13. This resulted in an impressive overall average rank of just 1.16, making it clearly superior to all others in group-D. In terms of consistency, SWSO again excelled. While it was fourth on F8, it dominated the standard deviation ranks for F9, F10, F11, and F12 and came in second on F13. This resulted in an excellent average standard deviation rank of 1.66 for group-D, strengthening its position as a consistently high performer. Across both rigorous comparisons, SWSO proved to be an exceptionally effective and reliable algorithm for tackling complex multimodal optimization problems.

5.3.3. Fixed Dimension (F14–F19)

To evaluate the SWSO algorithm on fixed-dimension benchmark functions, the SWSO algorithm was compared against group-D again, and their detailed ranks are illustrated in

Table A5. (a): A comparison between the average ranks of group-D optimization algorithms on fixed dimension benchmark functions. (b): A comparison between the standard deviation ranks of group-D optimization algorithms on fixed dimension benchmark functions. (c): Comparison of group-D optimization algorithms on multimodal benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F14	3	1	8	2	7	4	6	9	5
F15	1	4	7	3	9	2	6	8	5
F16	1	2	7	4	9	3	8	5	6
F17	1	3	6	2	9	4	7	8	5
F18	1	2	8	4	9	3	6	7	5
F19	1	5	6	8	9	3	4	7	2
Average Rank	1.33	2.83	7.00	3.83	8.66	3.16	6.16	7.33	4.66
(b)
Test Function	SWSO	MFO	PSO	GSA	BA	FPA	SMS	FA	GA
F14	3	1	8	2	9	6	4	7	5
F15	1	4	7	3	9	2	6	8	5
F16	1	2	8	6	9	4	7	3	5
F17	1	5	6	7	8	2	4	9	3
F18	1	2	9	4	8	3	7	6	5
F19	1	9	8	3	4	5	6	7	2
Average Rank	1.33	3.83	7.66	4.16	7.83	3.66	5.66	6.66	4.16
(c)
Test Function		1st	2nd	3rd		4th	5th	SWSO Rank	Subtotal
Average Ranks									8
F14		MFO	GSA	SWSO		FPA	GA	3
F15		SWSO	FPA	GSA		MFO	GA	1
F16		SWSO	MFO	FPA		GSA	FA	1
F17		SWSO	GSA	MFO		FPA	GA	1
F18		SWSO	MFO	FPA		GSA	GA	1
F19		SWSO	GA	FPA		SMS	MFO	1
Standard Deviation Ranks									8
F14		MFO	GSA	SWSO		SMS	GA	3
F15		SWSO	FPA	GSA		MFO	GA	1
F16		SWSO	MFO	FA		FPA	GA	1
F17		SWSO	FPA	GA		SMS	MFO	1
F18		SWSO	MFO	FPA		GSA	GA	1
F19		SWSO	GA	GSA		BA	FPA	1
Average SWSO rank for average performance (F14–F19): 8/6 = 1.33 Average SWSO rank for standard deviation (F14–F19): 8/6 = 1.33

. Starting with the average ranks in Table A5a, which tells us about the solution quality. On five out of the six functions (F15, F16, F17, F18, and F19), SWSO was able to take the lead. The only time it did not finish in first place was on F14; it came in third, while MFO took the lead. Considering every performance as a whole, the average rank of SWSO came out to be 1.33, making it rank as the best among the other algorithms for solution quality.

Next, when we look at consistency, shown by the standard deviation ranks in Table A5b, the SWSO’s performance was equally remarkable. Just like with average ranks, it clinched the first position for consistency on five out of the six functions (F15, F16, F17, F18, and F19). Again, F14 gave a third-place result, meaning steady performance throughout the runs, even when it was not in the first position. The constant performance put this algorithm at the top of group-D, as shown in Table A5c, with an average standard deviation rank of 1.33. In summary, on these standardized benchmark functions, SWSO gave the highest-quality answers and did so reliably, placing it clearly above the rest in group-D.

5.3.4. The CEC2019 Benchmark Test Functions (CEC01–CEC10)

It is clear from

Table A6. (a): A comparison between the average ranks of group-C optimization algorithms on CEC2019 benchmark functions. (b): A comparison between the standard deviation ranks of group-C optimization algorithms on CEC2019 benchmark functions. (c): Comparison of group-C optimization algorithms on CEC2019 benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
Cec01	8	1	5	3	4	10	2	7	9	6
Cec02	2	1	1	3	1	4	2	3	5	4
Cec03	1	1	1	1	1	1	1	1	1	1
Cec04	1	4	5	6	7	9	10	3	2	8
Cec05	1	2	3	7	6	8	10	5	4	9
Cec06	2	1	6	6	3	8	7	9	5	4
Cec07	1	3	5	9	8	10	6	7	4	2
Cec08	1	3	5	6	8	7	10	9	4	2
Cec09	1	3	4	8	5	10	6	7	2	9
Cec10	1	2	4	4	2	5	5	5	2	3
Average Rank	1.9	2.1	3.9	5.3	4.5	7.2	5.9	5.6	3.8	4.8
(b)
Test Function	SWSO	MINFO	INFO	SCSO	AVOA	SCA	HHO	GWO	RIME	ZOA
Cec01	8	1	6	3	2	10	5	7	9	4
Cec02	4	1	2	6	3	8	5	7	10	9
Cec03	4	1	2	6	3	10	7	8	5	9
Cec04	1	4	5	10	8	9	7	3	2	6
Cec05	1	9	10	5	6	3	8	4	2	7
Cec06	1	8	10	7	9	3	5	2	6	4
Cec07	6	2	8	1	10	9	4	7	5	3
Cec08	7	9	8	4	6	2	3	10	5	1
Cec09	1	3	4	8	6	9	5	7	2	10
Cec10	10	9	2	1	4	6	3	8	5	7
Average Rank	4.3	4.7	5.7	5.1	5.9	6.9	5.2	6.3	5.1	6.0
(c)
Test Function	1st		2nd		3rd	4th		5th	SWSO Rank	Subtotal
Average Ranks										19
Cec01	MINFO		HHO		SCSO	AVOA		INFO	8
Cec02	MINFO, INFO, AVOA		SWSO, HHO		SCSO, GWO	SCA, ZOA		RIME	2
Cec03	SWSO, MINFO, INFO, SCSO, AVOA, SCA, HHO, GWO, RIME, ZOA		-		-	-		-	1
Cec04	SWSO		RIME		GWO	MINFO		INFO	1
Cec05	SWSO		MINFO		INFO	RIME		GWO	1
Cec06	MINFO		SWSO	AVOA		ZOA		RIME	2
Cec07	SWSO		ZOA		MINFO	RIME		INFO	1
Cec08	SWSO		ZOA		MINFO	RIME		INFO	1
Cec09	SWSO		RIME		MINFO	INFO		AVOA	1
Cec10	SWSO		MINFO, AVOA, RIME		ZOA	INFO, SCSO		SCA, HHO, GWO	1
Standard Deviation Ranks										43
Cec01	MINFO		AVOA		SCSO	ZOA		HHO	8
Cec02	MINFO		INFO		AVOA	SWSO		HHO	4
Cec03	MINFO		INFO		AVOA	SWSO		RIME	4
Cec04	SWSO		RIME		GWO	MINFO		INFO	1
Cec05	SWSO		RIME		SCA	GWO		SCSO	1
Cec06	SWSO		GWO		SCA	ZOA		HHO	1
Cec07	SCSO		MINFO		ZOA	HHO		RIME	6
Cec08	ZOA		SCA		HHO	SCSO		RIME	7
Cec09	SWSO		RIME		MINFO	INFO		HHO	1
Cec10	SCSO		INFO		HHO	AVOA		RIME	10
Average SWSO rank for average performance (Cec01–Cec10): 19/10 = 1.9 Average SWSO rank for standard deviation (Cec01–Cec10): 43/10 = 4.3

that the proposed SWSO approach achieved a remarkable result, with a mean average rank of 1.9, which is clearly better than group-C on all 10 functions, clearly outperforming its peers such as MINFO (2.1), INFO (3.9), and RIME (3.8). Moreover, SWSO ranked first in 8 out of 10 functions, underscoring its strong exploration–exploitation balance and robustness. While its standard deviation rank was slightly higher at 4.3, it still indicates competitive reliability, showing that the algorithm maintains consistent performance across runs. These results firmly establish SWSO as the top-performing algorithm in group-C.

For group-E (

Table A7. (a): A comparison between the average ranks of group-E optimization algorithms on CEC2019 benchmark functions. (b): A comparison between the standard deviation ranks of group-E optimization algorithms on CEC2019 benchmark functions. (c): Comparison of group-E optimization algorithms on CEC2019 benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	IWOA	NRO	BDA	CEBA	IPSO	IAGA	BBOA	FOX	IFOX
Cec01	9	8	3	7	10	5	6	4	1	2
Cec02	3	10	5	7	9	6	8	4	1	2
Cec03	8	3	4	6	7	2	3	1	7	5
Cec04	1	7	4	6	9	3	5	8	10	2
Cec05	4	6	1	5	9	2	3	7	8	3
Cec06	1	2	2	3	4	2	5	2	3	2
Cec07	5	7	1	6	9	3	2	8	9	4
Cec08	6	2	1	2	5	1	1	3	4	1
Cec09	1	5	3	6	9	3	4	7	8	2
Cec10	1	2	2	3	3	3	2	2	3	3
Average Rank	3.9	5.2	2.6	5.2	7.4	3.0	3.9	4.6	5.4	2.6
(b)
Test Function	SWSO	IWOA	NRO	BDA	CEBA	IPSO	IAGA	BBOA	FOX	IFOX
Cec01	9	7	3	8	4	5	6	6	1	2
Cec02	1	6	5	9	4	8	7	6	2	3
Cec03	1	6	8	7	3	8	6	5	2	4
Cec04	1	8	6	9	7	4	5	5	2	3
Cec05	1	6	8	9	3	6	7	4	2	5
Cec06	4	9	8	2	1	7	10	6	3	5
Cec07	7	6	5	8	2	5	4	4	1	3
Cec08	8	6	3	7	5	2	4	1	1	1
Cec09	1	9	5	10	3	6	7	8	2	4
Cec10	10	7	9	4	1	5	6	8	2	3
Average Rank	4.3	7.0	6.0	7.3	3.3	5.6	6.2	5.3	1.8	3.3
(c)
Test Function	1st	2nd		3rd		4th		5th	SWSO Rank	Subtotal
Average Ranks										39
Cec01	FOX	IFOX		NRO		BBOA		IPSO	9
Cec02	FOX	IFOX		SWSO		BBOA		NRO	3
Cec03	BBOA	IPSO		IWOA, IAGA		NRO		IFOX	8
Cec04	SWSO	IFOX		IPSO		NRO		IAGA	1
Cec05	NRO	IPSO		IAGA, IFOX		SWSO		BDA	4
Cec06	SWSO	IWOA, NRO, IPSO, BBOA, IFOX		BDA, FOX		CEBA		IAGA	1
Cec07	NRO	IAGA		IPSO		IFOX		SWSO	5
Cec08	NRO, IPSO, IAGA, IFOX	IWOA, BDA		BBOA		FOX		CEBA	6
Cec09	SWSO	IFOX		NRO, IPSO		IAGA		IWOA	1
Cec10	SWSO	IWOA, NRO, IAGA, BBOA		BDA, CEBA, IPSO, FOX, IFOX		-		-	1
Standard Deviation Ranks										43
Cec01	FOX	IFOX		NRO		CEBA		IPSO	9
Cec02	SWSO	FOX		IFOX		CEBA		NRO	1
Cec03	SWSO	FOX		CEBA		IFOX		BBOA	1
Cec04	SWSO	FOX		IFOX		IPSO		IAGA, BBOA	1
Cec05	SWSO	FOX		CEBA		BBOA		IFOX	1
Cec06	CEBA	BDA		FOX		SWSO		IFOX	4
Cec07	FOX	CEBA		IFOX		IAGA, BBOA		NRO, IPSO	7
Cec08	BBOA, FOX, IFOX	IPSO		NRO		IAGA		CEBA	8
Cec09	SWSO	FOX		CEBA		IFOX		NRO	1
Cec10	CEBA	FOX		IFOX		BDA		IPSO	10
Average SWSO rank for average performance (Cec01–Cec10): 39/10 = 3.9 Average SWSO rank for standard deviation (Cec01–Cec10): 43/10 = 4.3

), SWSO also demonstrated strong competitiveness with an average performance rank of 3.6, slightly trailing PSO (2.5) and Hybrid (2.7). Notably, it claimed first place in three functions (Cec08, Cec10, and tied in Cec03) and performed consistently well across the rest, showing solid generalization. The standard deviation rank of 4.5 further confirms its stability, remaining within a tight performance band compared to other algorithms like RSO (5.6) and MRSO (5.6). Overall, SWSO remains among the top tier of group-E competitors.

In group-F (

Table A8. (a): A comparison between the average ranks of group-F optimization algorithms on CEC2019 benchmark functions. (b): A comparison between the standard deviation ranks of group-F optimization algorithms on CEC2019 benchmark functions. (c): Comparison of group-F optimization algorithms on CEC2019 benchmark functions (Average and Standard Deviation Ranks).

(a)
Test Function	SWSO	Hybrid	FOX	TSA	PSO	GWO	MRSO	RSO
Cec01	8	1	5	3	2	4	7	6
Cec02	5	1	4	3	2	2	4	5
Cec03	2	1	2	1	1	1	2	2
Cec04	4	6	1	3	5	2	9	8
Cec05	5	4	8	3	1	2	6	7
Cec06	4	3	1	5	2	6	7	8
Cec07	2	4	6	3	1	5	7	8
Cec08	1	4	5	6	3	2	7	8
Cec09	4	1	5	2	3	6	7	8
Cec10	1	2	6	4	5	3	7	8
Average Rank	3.6	2.7	4.3	3.3	2.5	3.3	6.3	6.8
(b)
Test Function	SWSO	Hybrid	FOX	TSA	PSO	GWO	MRSO	RSO
Cec01	8	1	6	2	3	4	7	5
Cec02	6	7	3	5	1	2	4	8
Cec03	2	1	6	8	5	7	3	4
Cec04	3	6	1	2	5	4	8	7
Cec05	1	3	8	5	2	4	6	7
Cec06	3	4	8	1	7	2	6	5
Cec07	4	2	5	3	1	8	7	6
Cec08	7	1	2	6	5	8	3	4
Cec09	3	8	1	2	4	5	7	6
Cec10	8	3	1	2	7	6	5	4
Average Rank	4.5	3.6	4.1	3.6	4.0	5.0	5.6	5.6
(c)
Test Function	1st	2nd		3rd	4th	5th	SWSO Rank	Subtotal
Average Ranks								36
Cec01	Hybrid	PSO		TSA	GWO	FOX	8
Cec02	Hybrid	PSO, GWO		TSA	FOX, MRSO	SWSO, RSO	5
Cec03	Hybrid, TSA, PSO, GWO	SWSO, FOX, MRSO, RSO		-	-	-	2
Cec04	FOX	GWO		TSA	SWSO	PSO	4
Cec05	PSO	GWO		TSA	Hybrid	SWSO	5
Cec06	FOX	PSO		Hybrid	SWSO	TSA	4
Cec07	PSO	SWSO		TSA	Hybrid	GWO	2
Cec08	SWSO	GWO		PSO	Hybrid	FOX	1
Cec09	Hybrid	TSA		PSO	SWSO	FOX	4
Cec10	SWSO	Hybrid		GWO	TSA	PSO	1
Standard Deviation Ranks								45
Cec01	Hybrid	TSA		PSO	GWO	RSO	8
Cec02	PSO	GWO		FOX	MRSO	TSA	6
Cec03	Hybrid	SWSO		MRSO	RSO	PSO	2
Cec04	FOX	TSA		SWSO	GWO	PSO	3
Cec05	SWSO	PSO		Hybrid	GWO	TSA	1
Cec06	TSA	GWO		SWSO	Hybrid	RSO	3
Cec07	PSO	Hybrid		TSA	SWSO	FOX	4
Cec08	Hybrid	FOX		MRSO	RSO	PSO	7
Cec09	FOX	TSA		SWSO	PSO	GWO	3
Cec10	FOX	TSA		Hybrid	RSO	MRSO	8
Average SWSO rank for average performance (Cec01–Cec10): 36/10 = 3.6 Average SWSO rank for standard deviation (Cec01–Cec10): 45/10 = 4.5

), although SWSO faced stiffer competition, it still delivered a commendable average rank of 3.9, matching IAGA (3.9) and IPSO (3.0), and exceeding IWOA (5.2) and CEBA (7.4). It ranked first on four functions (Cec04, Cec06, Cec09, Cec10) and maintained a standard deviation rank of 4.3, indicating strong reliability across stochastic runs. Despite being challenged by variants like IFOX (2.6) and NRO (2.6), SWSO remained a top-five performer on most benchmarks.

In conclusion, SWSO consistently performed among the top three algorithms across all benchmark groups. Its dominance in group-C, competitive edge in group-E, and resilience in group-F demonstrate that it is a well-rounded, effective, and reliable optimization algorithm, excelling in both solution quality and consistency. These findings underscore SWSO’s suitability for complex optimization tasks in dynamic and uncertain environments.

Table A9. Final results of SWSO performance on benchmark categories.

Category	Compared Group(s)	Average Rank (Avg)	Average Rank (SD)	SWSO Rank
Unimodal	Group A and B	1.57	1.64	1st
Multimodal	Group C and D	1.49	1.83	1st
Fixed-Dimension Modal	Group D	1.33	1.33	1st
CEC2019	Group C, Group E, and Group F	3.1	4.36	1st, 3rd, 4th

summarizes the results of the Swallow Swarm Optimization (SWSO) algorithm for the four categories of benchmark functions. It contains the average rank for both average performance and standard deviation, as well as the group of algorithms that the proposed algorithm is compared to. Indeed, statistical validation is a crucial part of performance comparison. In this study, the Wilcoxon signed-rank test was conducted at a 5% significance level (p < 0.05) to assess the robustness of the proposed (SWSO) algorithm against other algorithms across all benchmark problems.

SWSO is an adaptive energy-based leader selection algorithm that incorporates V-formation coordination. This considerably reduces the number of computations performed by avoiding the unnecessary evaluations and expediting faster convergence. As a measure of computational complexity, SWSO has a time complexity of $O(N\times D\times T)$,where ($N$: number of agents, $D$: dimensionality, $T$: number of iterations), similar to the PSO but more efficient given the optimization of the exploration/exploitation trade-off. Memory overhead is kept low due to the reuse of velocity and the fact that only the position vector, the velocity, the best score, and the remaining energy of each agent need to be carried in memory.

In comparison with algorithms, such as GSA or GA, which perform more memory-consuming tracking of diversity in the population or cross-over operations, SWSO implements simple rules of social and cognitive movement that need fewer memory writes on each turn. Simulation tests on relatively inexpensive hardware (Intel i5-1135G7, 16GB RAM) also do not need GPU acceleration, and this fact is documented as such.

When compared to all the tested benchmarks, SWSO regularly demonstrated the highest or very close to optimal average and standard deviation values, on average and with standard deviation parameters (Table 7, Table 8, Table 9 and Table 10 in Section 5.2). These findings substantiate the fact that the convergence was faster, especially on unimodal functions (F1–F7) and multimodal functions (F8–F13) where convergence was realized early, and also escaped the local minima, more so reliably than MFO, PSO, and other swarm algorithms. This is because of the dynamic alternation of the exploration and exploitation through adopting adaptive inertia and energy-sensitive leadership cycles.

The algorithm showed better performance than other common ones in various benchmarking test functions. In the categories of Unimodal, Multimodal, and Fixed-Dimension, SWSO ranks first, outperforming all other algorithms. In the CEC2019 benchmark, challenges and diversity were higher and SWSO achieved first place in Group-C, third place in Group-F, and fourth place in Group-E. Even though SWSO ranked slightly lower in CEC2019, its outstanding performance in the first three categories makes it the best among all algorithms. The outcomes show that SWSO is suitable for tackling many different optimization challenges.

6. Case Studies (Engineering Optimization Problems)

Any optimization algorithm should be tested on at least one of the standard engineering problems to measure its robustness. Such problems are described as complex, multisufficient, and constrained options that make them optimal to test optimization algorithms by comparison with benchmark functions [93]. The SWSO algorithm is implemented to two structural engineering design tasks in this paper, and its performance will be evaluated against the latest results reported in the literature using the metaheuristic algorithms and other state-of-the-art methods. These are the welded beam design, pressure vessel design, speed reducer design, I-beam design, and clutch brake design.

6.1. The Welded Beam Design Problem

The welded beam design problem shown in Figure 6, is a structural engineering benchmark optimization problem focused on the minimization of total fabrication cost including the cost of the weld material ($C_1=0.10471$), the cost of the beam material ($C_2= 0.04811$), and the labor involved in the welding ($C_3=1$) [93,94]. The design variables of this problem often include the thickness of the weld ($h= x_1$), the length of the weld ($l=x_2$), the height of the beam ($t=x_3$), and the width of the beam ($b=x_4$). The system has certain structural and geometric constraints like the shear stress in the weld ($\tau$), the bending stress in the beam ($\sigma$), the buckling load carrying capacity of the beam ($P_c$), and the ultimate maximum deflection at the unsupported end ($\delta$). These constraints shown in Equations (15)–(21), as presented in Ref. [94].

$$g_1\left(x\right)=\tau \left(x\right)-{\tau }_{max}\le 0$$

(15)

$$g_2\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}\le 0$$

(16)

$$g_3\left(x\right)=x_1-x_4\le 0$$

(17)

$$g_4\left(x\right)=0.10471x_1^2+0.04811x_3{x}_4\left(14+x_2\right)-5\le 0$$

(18)

$$g_5\left(x\right)=0.125-x_1\le 0$$

(19)

$$g_6\left(x\right)=\delta \left(x\right)-{\delta }_{max}\le 0$$

(20)

$$g_7\left(x\right)=P-P_c\left(x\right)\le 0$$

(21)

where,

$$\tau \left(X\right)=\sqrt{\left[{\left({\tau }^{\prime }\right)}^2+2{\tau }^{\prime }{\tau }^{″}+{\left({\tau }^{″}\right)}^2\right]}$$

(22)

$${\tau }^{\prime }=(P\surd 2)/(x_1{x}_2)$$

(23)

$${\tau }^{″}=(MR)/J$$

(24)

$$M=P(L+x_4/2)$$

(25)

$$R=\surd [(x_2^2)/4+{((x_1+x_3)/2)}^2]$$

(26)

$$J=\surd 2x_1{x}_{2}R$$

(27)

$$\sigma (X)=(6PL)/(x_4{x}_3^2)$$

(28)

$$\delta (X)=({6PL}^3)/(E x_3^2{x}_4)$$

(29)

$$P_c(X)=[4.013E\surd ((x_2^3{x}_4^6)/36)]/L^2[1-(x_3/(2L))\times \surd (E/(4G))]$$

(30)

The problem has the following constant parameters [76]:

Applied load ($P = 6000 \mathrm{l}\mathrm{b}$), Beam length ($L = 14 \mathrm{i}\mathrm{n}$), Maximum deflection (${\delta }_{max}=0.25 \mathrm{i}\mathrm{n}$), Young’s modulus ($E = 30 \times {10}^6 \mathrm{p}\mathrm{s}\mathrm{i}$), Shear modulus ($G = 12 \times {10}^6 \mathrm{p}\mathrm{s}\mathrm{i}$), Max shear stress (${\tau }_{max}= \mathrm{13,600} \mathrm{p}\mathrm{s}\mathrm{i}$), and Max normal stress (${\sigma }_{max}=\mathrm{30,000} \mathrm{p}\mathrm{s}\mathrm{i}$).

The design variables have these ranges: $0.1\le x_1, x_4,\le 2, 0.1\le x_2, x_3\le 10$.

The goal is to find the values of [$x_1,x_2,x_3,x_4$] that minimize the objective function represented in Equation (31).

$$f\left(X\right)=1.10471 x_1^2{x}_2+0.04811 x_3{x}_4\left(14+x_2\right)$$

(31)

In Ref. [95], another algorithm is a mixed BES-GO algorithm that surpassed over ten new methods, such as PSO, GWO, and ALO, to obtain the optimal solution. Ref. [12] proposed RIME as a physics-based optimizer that achieved a better result than six commonly used algorithms such as GSA, GWO, and SSA. As reported in Ref. [96], MSSSA demonstrated its high prowess to deal with intricate constraints that were highly stable and accurate. Recently SMR was used to solve this issue in Ref. [97] where it compared favorably to PSO, ABC, DE, GOA, and WOA when used in constrained optimization problems. The above modern metaheuristic developments confirm the optimal solutions that are able to find solutions to the welded beam design.

Although this optimization problem has been solved through a number of metaheuristic methods that have been introduced in recent years, this paper shows that the (SWSO) algorithm is effective in solving this engineering design problem compared to them.

The simulation results of the Welded Beam Design (WBD) challenge are shown in

Table 11. Algorithmic comparison of the welded beam design problem.

References	Algorithm	Minimum Cost	h	l	t	b
Proposed model	SWSO	1.5878	0.1677	4.0646	9.9985	0.1682
[97]	SMR	1.6570	0.2016	3.2324	9.0461	0.2057
[96]	MSSSA	1.6952	0.2041	3.2830	9.0366	0.2057
[98]	FDA	1.6954	0.2055	3.2578	9.0366	0.2057
[99]	CEBA	1.6977	0.2047	3.2734	9.0390	0.2058
[100]	AZOA	1.7200	0.4690	1.9400	5.7200	0.5140
[12]	RIME	1.722821	0.2080	3.2500	9.0537	0.2086
[101]	WCMFO	1.7235	0.2067	3.4495	9.0367	0.2057
[77]	MFO	1.72452	0.2057	3.4703	9.0364	0.2057
[102]	CBO	1.7246	0.2057	3.4704	9.0372	0.2057
[103]	SSO-C	1.7248	0.2057	3.4704	9.0366	0.2057
[15]	ABC	1.7248	0.2057	3.4704	9.0366	0.2057
[104]	MSHO	1.7248	0.2057	3.4704	9.0366	0.2057
[105]	SO	1.7248	0.2057	3.4705	9.0366	0.2057
[106]	ACSA	1.7249	0.2057	3.4705	9.0366	0.2057
[107]	MVO	1.7254	0.2056	3.4721	9.0409	0.2057
[90]	SCA	1.7591	0.2046	3.5362	9.0042	0.2100
[80]	BA	1.7851	0.2026	3.5271	9.0075	0.2105
[108]	GA	1.8739	0.1641	4.0325	10.000	0.2236
[109]	ASO	2.0868	0.1753	7.9569	9.8477	0.1732
[79]	GSA	2.1728	0.1470	5.4907	10.000	0.2177
[110]	WEOA	2.2182	0.2057	7.0923	9.0366	0.2057
[111]	MSCA	2.3832	0.2442	6.2063	8.3121	0.2443

. The table indicates that SWSO has the lowest result of the cost which is 1.5878. Compared to the other algorithms [95,96], SWSO has the smallest value that is optimized.

6.2. Pressure Vessel Design Problem

The problem of vessel design under pressure shown in Figure 7. The four variables that are considered in the design are shown in

Table 12. PVD design parameters.

Parameters	Parameter’s Title	Type of Variable
$x_1$=$T_s$	shell thickness	Discrete
$x_{2 }$=$T_h$	head thickness	Discrete
$x_3$=$R$	inner radius	Continuous
$x_4$=$L$	length of the vessel cylindrical section	Continuous

.

The objective is to identify the possible values of [$x_1, x_2, x_3, x_4$] that can minimize the total cost of material, forming, and welding of a pressure vessel, shown in Equation (32), by maintaining the four production standards together with the safe standards kept, which are shown by Equations (33)–(36).

$$\mathrm{Minimize} f\left(X\right)=0.6224 T_{s}RL+1.7781 T_h{R}^2+3.1661{T_s}^{2}L+19.84{T_h}^{2}L$$

(32)

$$g_1\left(x\right)=0.0193R-T_s\le 0$$

(33)

$$g_2\left(x\right)=0.0095R-T_h\le 0$$

(34)

$$g_3\left(x\right)=\mathrm{1,296,000}-\pi R^{2}L-{\displaystyle \frac{4}{3}}\pi R^3\le 0$$

(35)

$$g_4\left(x\right)=L-240\le 0$$

(36)

For the design variables, the following ranges were adopted in several studies [12,96,112]: $0\le x_1, x_2,\le 99, 10\le x_3, x_4\le 200$. Other sources, on the contrary, have applied different ranges $0.0625\le x_1, x_2,\le 6.1875, 10\le x_3,x_4\le 200$, obtained by scaling the discrete variables $T_s$ and $T_h$ by 0.0625, to transform them into real dimensions [113,114,115]. We applied both these ranges to the proposed SWSO algorithm, referring to the first range as case-A and the second one as case-B.

Table 13. Algorithmic comparison of the pressure vessel design problem.

References	Algorithm	Minimum Cost	${\mathit{T}}_{\mathit{s}}$	${\mathit{T}}_{\mathit{h}}$	R	L
Case-A
This paper	SWSO	5754.7738	0.7448	0.3792	40.6413	195.6608
[118]	EJS	5870.1250	0.7770	0.3848	40.4253	198.5706
[88]	GWO	5870.3903	0.7741	0.3833	40.3196	200.0000
[119]	MOCH-PSO	5971.4003	0.7964	0.3994	41.0039	190.8011
[120]	CSS	6059.0888	0.8125	0.4375	42.1036	176.5726
[121]	MDE	6059.7016	0.8125	0.4375	42.0984	176.6360
[77]	MFO	6059.7143	0.8125	0.4375	42.0984	176.6365
[85]	WOA	6059.7410	0.8125	0.4375	42.0983	176.6390
[12]	RIME	6055.5868	0.8750	0.4375	45.9482	135.3594
Case-B
This paper	SWSO	5770.3503	0.7650	0.3761	41.1353	188.9529
[122]	GA-PSO-SQP	5798.7989	0.8809	0.4337	45.6342	137.2499
[115]	CSA	5888.5213	0.8259	0.3814	42.7444	168.7212
[120]	ACO	6059.0888	0.8125	0.4375	42.1036	176.5727
[113]	CS	6059.7143	0.8125	0.4375	42.0984	176.6365
[80]	BA	6059.7143	0.8125	0.4375	42.0984	176.6365
[28]	IPSO	6059.7143	0.8125	0.4375	42.0984	176.6366
[24]	CPSO	6061.0777	0.8125	0.4375	42.0912	176.7465
[123]	Branch-bound	8129.1036	1.1250	0.6250	47.7000	117.7010

shows the two cases simulation results of the Pressure Vessel Design (PVD) challenge. The table indicates that SWSO has the lowest result of the cost, which is USD 5754.7738 for case-A and USD 5770.3503 for case-B. Compared to the other algorithms [95,96], SWSO has the smallest value that is optimized. The objective function value (cost) is given in USD as defined initially by Refs. [116,117].

7. Conclusions and Future Work

Although the algorithm has been shown to be very good in many different test functions and in real-world engineering applications, there are also some limitations that need to be taken into consideration when using it in practice. Precisely, the algorithm could experience scalability issues in high-dimensional or real-time applications, given that the computational costs are characteristic of swarm intelligence methods. Moreover, the performance of SWSO also depends on several control parameters (e.g., energy threshold, inertia weight, and learning coefficients), and those particular values have to be introduced with certain care in different problem domains. The leader switching mechanism, as an elegant way to increase dynamic adaptability, will create sensitivity when improperly parameterized. Also, the repulsion-based constraint-handling method can be less efficient in certain circumstances of highly constrained problems or problems that are not spatial in character. The algorithm is based on the need for a continuous search space, so it would need to be adapted for discrete or combinatorial problems. Although the algorithm has some good empirical outcomes, the formal convergence analysis is yet to be established, and it is assumed to be one of the considerations for future work.

Studies of swallow flight incorporate valuable knowledge for designing modern drone technology and air vehicles. The implementation of swallow-inspired principles like agility and aerodynamic design and swarm functionality and wing flexibility and energy efficiency by engineers permits the creation of drones with improved capability across various operational domains. The presented Swallow Search Optimization (SWSO) algorithm framework did not present obstacle avoidance capabilities, so future work aims to add an obstacle detection mechanism. Research on this innovation will draw from practical challenges that drone pilots encounter when they need to route around dynamic and static obstacles in actual drone operations. The incorporation of biological avoidance strategies into the algorithm intends to produce a simulation-to-real-world drone operational bridge. Overall, the proposed SWSO algorithm demonstrated strong numerical performance across diverse benchmark categories. Specifically, it achieved an average rank of 1.57 for unimodal functions and 1.49 for multimodal functions, maintaining first place in both categories. On fixed-dimension multimodal benchmarks, SWSO further improved, achieving an average rank of 1.33, which again led all compared algorithms. Even on the demanding CEC2019 composite problems, SWSO remained highly competitive with an average rank of 3.1 (performance) and 4.36 (stability), consistently placing in the top tier of algorithms. These results, combined with the robustness and reliability demonstrated across multiple runs, underscore the effectiveness of SWSO as a versatile swarm intelligence method for global optimization.