Binary Chaotic White Shark Optimizer for the Unicost Set Covering Problem

Abstract

The Unicost Set Covering Problem (USCP), an NP-hard combinatorial optimization challenge, demands efficient methods to minimize the number of sets covering a universe. This study introduces a binary White Shark Optimizer (WSO) enhanced with V3 transfer functions, elitist binarization, and chaotic maps. To evaluate algorithm performance, we employ the Relative Percentage Deviation (RPD), which measures the percentage difference between the obtained solutions and optimal values. Our approach achieves outstanding results on six benchmark instances: WSO-ELIT_CIRCLE delivers an RPD of 0.7% for structured instances, while WSO-ELIT_SINU attains an RPD of 0.96% in cyclic instances, showing empirical improvements over standard methods. Experimental results demonstrate that circle chaotic maps excel in structured problems, while sinusoidal maps perform optimally in cyclic instances, with observed improvements up to 7.31% over baseline approaches. Diversity and convergence analyses show structured instances favor exploitation-driven strategies, whereas cyclic instances benefit from adaptive exploration. This work establishes WSO as a robust metaheuristic for USCP, with applications in resource allocation and network design.

1. Introduction

The Set Covering Problem (SCP) is a fundamental challenge in combinatorial optimization, widely studied due to its significance in fields such as facility location, network design, and resource allocation [1,2]. Within this framework, the unicost version of the SCP represents a streamlined version of the classic problem, where all sets share the same selection cost. This simplification allows the focus to shift entirely to minimizing the number of sets required to cover a given universe, eliminating the added complexity of varying costs.

Mathematically, the unicost SCP is expressed as an integer programming problem, with constraints designed to ensure the complete coverage of the universe. Despite the removal of cost variability, this variant remains NP-hard, meaning its complexity increases exponentially with problem size [3]. As a result, finding optimal solutions using exact methods becomes impractical for large-scale instances, necessitating the development of approximation algorithms and metaheuristic approaches.

The unicost SCP holds importance not only in practical applications but also in academic research. In the fields of combinatorial optimization and computer science, it serves as a valuable case study for examining specific problem characteristics and behaviors [4]. By simplifying certain elements, this variant allows researchers to analyze algorithmic performance under controlled conditions, offering insights that can be applied to more complex scenarios involving heterogeneous costs.

A variety of methods have been developed to address the unicost SCP, ranging from exact approaches like integer programming and branch-and-bound algorithms to heuristic and metaheuristic techniques. Heuristics, such as greedy algorithms, provide quick solutions but often lack optimality [1]. On the other hand, metaheuristics—such as genetic algorithms, ant colony optimization, and simulated annealing—can explore high-quality solutions within reasonable timeframes, particularly for large and intricate instances [2,4]. Recent advances include configuration checking approaches [5] and rough set theory applications [3], demonstrating the continued evolution of solution methodologies.

The adaptation of continuous metaheuristics to binary optimization problems has gained significant attention in recent years [6,7]. Traditional approaches rely on transfer functions and binarization rules to convert continuous search spaces to discrete domains [8,9]. However, the effectiveness of these transformations often depends on the specific characteristics of both the algorithm and the problem instance, highlighting the need for more sophisticated binarization strategies.

Bio-inspired algorithms have shown particular promise in combinatorial optimization, with successful applications ranging from feature selection [10,11] to complex engineering problems [12]. The White Shark Optimizer (WSO), inspired by the hunting behavior of great white sharks, has demonstrated effectiveness across various optimization domains [13,14,15,16]. However, its application to binary optimization problems remains relatively unexplored.

Furthermore, the integration of chaotic dynamics into optimization algorithms has emerged as a powerful strategy for enhancing search diversity and avoiding premature convergence [17,18]. Chaotic maps provide deterministic yet unpredictable sequences that can improve the exploration capabilities of metaheuristic algorithms [19,20,21]. The combination of chaotic systems with advanced binarization techniques represents a promising research direction for addressing complex combinatorial problems.

In this study, we aim to investigate advanced methodologies for solving the unicost SCP, incorporating innovative approaches like chaos theory and bio-inspired optimization. By integrating these strategies, the objective is not only to enhance solution quality but also to establish an analytical framework for better understanding algorithmic behavior in this problem variant. These explorations have immediate practical applications and contribute to the development of more robust methodologies for tackling combinatorial optimization problems in general [22].

1.1. Main Contributions

The main contributions of this paper can be summarized as follows:

• Development of a novel binary adaptation of the White Shark Optimizer with dual binarization of both position and velocity components, addressing limitations in current binary metaheuristic approaches [23,24].
• Integration of chaotic maps as dynamic thresholding mechanisms for the binarization process, replacing standard random number generation and improving upon existing transfer function methodologies [8,9].
• Empirical evaluation demonstrating that specific binarization schemes and chaotic maps align with different problem structures: circle-based maps excel in structured instances while sinusoidal maps perform better in cyclic instances.
• Comprehensive analysis of population diversity and exploration–exploitation dynamics, providing insights into the algorithm behavior across different problem types using established metrics [6].
• Establishment of a performance benchmark for the USCP that demonstrates empirical improvements over standard binarization approaches (RPD improvements of up to 7.31%), though statistical significance was not established across all configurations, building upon previous comparative studies [25].

1.2. Paper Organization

The remainder of this paper is organized as follows. Section 2 presents the formal definition of the USCP, building upon established formulations [3]. Section 3 reviews the related work in USCP, binary optimization techniques, and the application of chaotic maps in metaheuristics. Section 4 introduces the White Shark Optimizer algorithm and its mathematical model [12]. Section 5 discusses binarization techniques, including transfer functions and binarization rules [7]. Section 6 details our proposed binary adaptation of WSO with chaotic maps. Section 7 presents the experimental results and analysis, including performance comparisons, diversity analysis, and exploration–exploitation balance. Finally, Section 8 concludes the paper with key findings and directions for future research.

2. Problem Definition

The USCP is a variant of the classical SCP where all subsets have identical selection costs. Given a universe $U=\{1,2,\dots ,m\}$ of elements and a collection $\mathcal{S}=\{S_1,S_2,\dots ,S_n\}$ of subsets where each $S_j\subseteq U$, the USCP seeks to find the minimum number of subsets that cover all elements in U.

Mathematical Formulation

Let A be a binary matrix of size $m\times n$, where $a_{ij}=1$ if subset j covers element i, and $a_{ij}=0$ otherwise. The USCP can be formulated as

$$\mathrm{Minimize}\sum _{j=1}^n{x}_j$$

(1)

subject to

$$\sum _{j=1}^n{a}_{ij}{x}_j\ge 1,\forall i\in \{1,2,\dots ,m\}$$

(2)

$$x_j\in \{0,1\},\forall j\in \{1,2,\dots ,n\}$$

(3)

where $x_j=1$ indicates that subset j is selected, and $x_j=0$ otherwise. The constraint ensures that every element is covered by at least one selected subset.

Despite the cost simplification, the USCP remains NP-hard, making it computationally challenging for large instances. Unlike the classical SCP with heterogeneous costs, the USCP focuses solely on minimizing the cardinality of the solution, eliminating cost–benefit trade-offs while preserving the combinatorial optimization complexity.

3. Related Work

3.1. Unicost Set Covering

The USCP is a classical NP-hard combinatorial optimization problem that has been extensively studied due to its applicability in various contexts, such as resource allocation and efficient subset selection. Over the years, numerous heuristic and metaheuristic approaches have been proposed to solve this problem, each with varying degrees of success in terms of solution quality and computation time.

One notable approach is the memetic algorithm proposed in [4], which combines the Hybrid Evolutionary Algorithm in Duet (HEAD) and Row Weighting Local Search (RWLS). This approach adapts RWLS for both exploitation and exploration, achieving improvements in the best-known solutions for several widely used benchmark instances.

In [3], the relationship between the USCP and the attribute reduction problem in rough set theory is explored. The authors propose a heuristic algorithm based on rough sets, which highlights the connection between these two areas, allowing the USCP to be addressed from a novel theoretical perspective.

Electromagnetism has also been utilized as a foundation for solving the USCP. In [2], an electromagnetism-based metaheuristic is presented, integrating the mutation, local search, and movements inspired by electromagnetism theory. This method achieves competitive results in classical benchmark instances, improving the best-known solutions for several of them.

On the other hand, algorithms based on the Greedy Randomized Adaptive Search Procedure (GRASP) have also proven effective for the USCP. In [1], a GRASP algorithm is proposed, incorporating a local improvement phase based on SAT heuristics. This approach improves the best-known solutions for a significant set of reference instances, standing out for its simplicity and effectiveness.

Finally, a more recent Configuration Checking (CC)-based approach was introduced in [5]. The MLSES-CC algorithm uses element states along with multi-level scoring (MLS) to optimize the search process, combined with an aggressive local search routine. Experimental results indicate that this approach can be considered a new state-of-the-art for the USCP, given its outstanding performance on a wide range of instances.

Together, these works reflect the diversity of approaches developed for the USCP, ranging from population-based methods, rough set theory, and bio-inspired metaheuristics to specialized heuristic algorithms.

3.2. Binary Metaheuristics

Binary optimization problems have become a focal point of research, particularly in applications like feature selection, where the goal is to reduce the dimensionality of datasets by removing irrelevant features while preserving classification accuracy. These problems, characterized by decision variables in binary format (0 or 1), share similarities with other combinatorial optimization tasks, such as the USCP. Recent advancements focus on leveraging metaheuristic algorithms and innovative binarization strategies to overcome challenges like large search spaces and premature convergence.

A critical development in binary optimization is the two-step binarization process, which includes the application of transfer functions and binarization rules. Transfer functions map continuous variables to binary domains using S-shaped, V-shaped, or U-shaped curves, which guide the algorithm’s search behavior. For instance, the Binary Moth-Flame Optimization (B-MFO) algorithm employs these transfer functions to effectively adapt the continuous MFO for feature selection in medical datasets [10]. After the transfer function is applied, binarization rules convert the mapped values into binary states, ensuring compatibility with binary search spaces.

The Binary Particle Swarm Optimization (BPSO) has emerged as one of the most successful adaptations of continuous metaheuristics to binary domains. Notable achievements include the following:

• Modified BPSO with genotype–phenotype representation and genetic algorithm mutations outperformed the original BPSO [23]
• BPSO improved feature selection in gene expression data, enhancing classification accuracy [24].
• BPSO solved unit commitment problems with superior cost and computation time [26].
• BPSO extracted high-quality rules in association rule mining [27].
• Novel velocity definitions improved BPSO’s binary interpretation [28].
• BPSO optimized high-utility itemset mining, reducing search space [29].
• BPSO adapted for knapsack problems maintained diversity and effectiveness [30].
• BPSO solved lot-sizing problems, finding optimal solutions in most cases [31].

Other innovative algorithms have made significant contributions to binary optimization. The Grey Wolf Optimizer (GWO) and Whale Optimization Algorithm (WOA) were hybridized to mitigate premature convergence and stagnation in local optima [32]. The Binary Grasshopper Optimization Algorithm (BGOA) integrates transfer functions and mutation operators to enhance the exploration–exploitation balance [11]. The Quadratic Binary Harris Hawk Optimization (QBHHO) improves standard HHO through quadratic transformations for high-dimensional problems [33], while the Binary Dragonfly Algorithm (BDA) addresses challenges in datasets with limited samples [34].

Advanced mechanisms have further enhanced binary optimization performance. The Coronavirus Herd Immunity Optimizer with Greedy Crossover (CHIO-GC) combines crossover operators to correct suboptimal solutions and improve convergence speed, outperforming existing algorithms in medical applications [35]. The integration of orthogonal learning strategies and covariance matrix approaches have improved solution diversity and convergence reliability in algorithms like the enhanced Grey Wolf Optimizer [36] and Binary Emperor Penguin Optimizer (BEPO) [37].

Comprehensive surveys highlight the evolution of binary optimization, emphasizing the role of hybridization, advanced transfer functions, and algorithmic innovations in addressing diverse binary problem domains [38]. These developments collectively demonstrate how binary metaheuristics continue to enhance efficiency and applicability in solving complex real-world problems—principles we adapt in our work with the binary White Shark Optimizer.

3.3. Chaotic Maps into Metaheuristic

Chaotic maps are deterministic dynamical systems that exhibit aperiodic, bounded behavior with sensitive dependence on initial conditions. Mathematically defined as recursive relations $x_{n+1}=f\left(x_n\right)$ where f is a nonlinear function, these maps generate sequences that appear random yet possess inherent structure.

The key properties that make chaotic maps valuable in optimization include the following:

• Sensitivity to initial conditions: Minute differences in starting values lead to exponentially diverging trajectories.
• Ergodicity: The ability to non-repetitively visit all possible states within bounded regions.
• Intrinsic randomness: Despite being deterministic, chaotic sequences pass many statistical tests for randomness.
• Self-similarity: Many chaotic attractors exhibit fractal structures across different scales.

Common examples used in optimization include several fundamental chaotic maps, each offering different chaotic behaviors and mathematical properties (

Table 1. Selected chaotic maps for binary WSO implementation.

Type	Chaotic Maps
Singer Map	$x_{k+1}=\mu (7.86x_k-23.31x_k^2+28.75x_k^3-13.3x_k^4),\mu =1.07$
Sinusoidal Map	$x_{k+1}=c{x}_k^{2}sin\left(\pi x_k\right)$, where $c=2.3$
Tent Map	$x_{k+1}=\left\{\begin{array}{cc}{\displaystyle \frac{x_k}{0.7}}\hfill & \mathrm{if}{x}_k<0.7\hfill \\ {\displaystyle \frac{10}{3}}(1-x_k)\hfill & \mathrm{if}{x}_k\ge 0.7\hfill \end{array}\right.$
Circle Map	$x_{k+1}=\mathrm{mod}(x_k+d-{\displaystyle \frac{c}{2\pi }}·sin\left(2\pi x_k\right),1)$, where $c=0.5$ and $d=0.2$

, Figure 1):

Logistic Map: This is one of the most studied chaotic systems, defined by

$$x_{k+1}=r{x}_k(1-x_k)$$

(4)

where r is a control parameter. When $r=4$, the system exhibits fully chaotic behavior with excellent mixing properties.

Circle Map: This map combines periodic and quasi-periodic dynamics, expressed as

$$x_{k+1}=mod\left.(x_k+b-{\displaystyle \frac{a}{2\pi }}sin\left(2\pi x_k\right),1\right)$$

(5)

where a and b are control parameters that determine the map’s dynamical regime and chaotic properties.

Sinusoidal Map: This map incorporates trigonometric nonlinearity, defined by

$$x_{k+1}=a{x}_k^{2}sin\left(\pi x_k\right)$$

(6)

where a is a parameter that controls the chaotic behavior. This map is particularly effective for generating diverse sequences with good statistical properties.

Chaotic maps have established themselves as an effective tool in the field of metaheuristics due to their ability to enhance exploration of search spaces and mitigate premature convergence to local optima. A notable example is the use of chaotic maps in Harris Hawks Optimization, where initialization based on chaotic maps and the incorporation of simulated annealing have enhanced both the diversity of solutions and the exploitation capabilities of the algorithm [17].

In complex nonlinear systems, such as electro-hydraulic actuators, chaotic maps like the Tinkerbell map have been fundamental in improving model identification accuracy through atomic search-based algorithms [39]. In structural optimization problems, the Chaotic Coyote Optimization Algorithm (MCOA) has utilized chaotic maps to adjust key probabilities, significantly improving performance in the design of structures such as planar and spatial trusses [18].

In the industrial control context, a chaotic firefly algorithm based on the Tinkerbell map has proven effective for tuning multiloop PID controllers, surpassing traditional approaches in real-world applications [19]. The Improved Symbiotic Organisms Search (ISOS) incorporates a chaotic local search that achieves superior balance between exploration and exploitation [20], while the Chaotic Dwarf Mongoose Optimization (CDMO) algorithm enhances feature selection in classification problems, achieving higher accuracy with fewer variables [21].

Overall, chaotic maps have proven to be a versatile tool for boosting the efficiency of metaheuristic algorithms across domains from structural engineering to feature selection and nonlinear system identification.

4. White Shark Optimizer

White sharks, as apex predators in their marine environment, are equipped with highly sophisticated sensory systems that allow them to detect prey in deep and vast oceanic regions. These systems include electroreception, vision, pressure sensing, olfaction, and hearing. Their auditory system, in particular, is exceptionally advanced, extending throughout their body and providing them with an extraordinary range of hearing. Despite these capabilities, white sharks lack precise prior knowledge about the location of food sources in a given area. Consequently, they must engage in extensive searches across the ocean to locate prey. This study identifies three primary behavioral strategies employed by white sharks to optimize their prey localization efficiency [12].

• Movement speed toward prey: White sharks utilize an undulating motion that leverages the waves produced by the movement of their prey. This behavior, guided by their auditory and olfactory senses, enables them to navigate efficiently toward potential food sources.
• Random prey search: White sharks move toward areas where prey activity has been detected and remain close to the most promising targets. This strategy involves exploring potential food sources randomly across the ocean.
• Nearby prey localization behavior: White sharks mimic the behavior of fish schools to identify nearby prey. In this process, a shark moves toward the most successful individual in the search, enhancing its ability to locate optimal prey.

The White Shark Optimizer (WSO) has emerged as a powerful and versatile tool for solving complex optimization problems across various fields. Its applications range from energy management and geomaterial design to object detection and image denoising, demonstrating its adaptability and effectiveness in diverse scenarios.

In the context of energy management for plug-in hybrid electric vehicles (PHEVs), WSO has been employed to optimize the weights and biases of extreme learning machines, significantly improving the computational efficiency and optimization performance. By integrating WSO with model predictive control (MPC), a novel energy management framework was developed, achieving a 73.4% reduction in driving costs and a 78.4% reduction in emissions compared to traditional rule-based methods. This approach also maintained a balance between optimization performance and computational efficiency, outperforming conventional MPC methods within a 5-s prediction horizon [13].

In geomaterial design for embankment construction, WSO has been applied to predict critical soil properties, such as the angle of internal friction in frictional and cohesive-frictional soils. By combining wavelet decomposition with a WSO-assisted regression model, this method outperformed traditional regression techniques, improving backfill selection and enhancing the reliability of earth-retaining system designs [15].

For object detection and classification in remote sensing images, WSO has been integrated with deep learning models to enhance accuracy and efficiency. A White Shark Optimizer-based deep learning method (WSODL-ODCRSI) utilized a modified single-shot multi-box detector (MSSD) for object detection and an Elman Neural Network (ENN) for object classification. The WSO was applied for parameter tuning, resulting in improved classification performance. Evaluations on benchmark datasets demonstrated the model’s superior ability to identify and classify objects such as planes, ships, and vehicles in complex and densely populated scenes [16].

In the field of image denoising, an Adaptive Deep Residual Network (AdResNet) was combined with the Adaptive White Shark Optimizer (AWSO) to address noise removal in medical, natural, and satellite images. The model exhibited robustness across various noise types and intensity levels, achieving low Mean Squared Error (MSE), high Peak Signal-to-Noise Ratio (PSNR), and high Structural Similarity Index Measure (SSIM). For example, AdResNet achieved average metrics of MSE 13.61, PSNR 48.81 dB, and SSIM 0.96 on medical images, confirming its suitability for high-quality imaging applications [14].

These studies highlight the versatility and effectiveness of the White Shark Optimizer in enhancing machine learning and deep learning frameworks. From optimizing energy management systems to advancing geomaterial design, object detection, and image denoising, WSO has proven to be a robust metaheuristic. Its integration with advanced algorithms and hybrid models continues to push the boundaries of optimization, offering innovative solutions for a wide range of real-world challenges.

4.1. Mathematical Model

4.1.1. Parameter Initialization

The WSO framework begins by initializing a matrix that defines the initial population of solutions. This matrix has dimensions $N\times d$, where N represents the population size, and d corresponds to the problem’s dimensionality as described in

Table 2. WSO parameter definitions.

Symbol	Definition
ub	Upper bound
lb	Lower bound
n	Number of search agents
d	Dimension
t	Current iteration
T	Maximum iterations

:

First, the initial population of solutions for the sharks is generated using the parameters defined in Table 2. Next, the initial velocity and position are set, where rand represents a random value uniformly distributed in the range $[0,1]$.

The initial population is created using a uniform random distribution within the defined search space as detailed below:

$$w=\left(\begin{array}{cccc}{w}_{1,1}& w_{1,2}& \dots & w_{1,d}\\ w_{2,1}& w_{2,2}& \dots & w_{2,d}\\ ⋮& ⋮& \ddots & ⋮\\ w_{n,1}& w_{n,2}& \dots & w_{n,d}\end{array}\right)$$

(7)

Here, $w_{i,j}$ represents the initial position of the i-th white shark in the j-th dimension of the search space. The bounds for the search area in this dimension are defined by $u_j$ as the upper limit and $l_j$ as the lower limit. Additionally, r is a random number drawn from a uniform distribution within the interval $[0,1]$:

$$w_{ij}=l_j+\mathrm{rand}\times (u_j-l_j)$$

(8)

4.1.2. Shark Movement Speed Towards Prey

White sharks are skilled hunters, constantly moving in search of prey. Their survival depends on their ability to locate and chase animals using a combination of exceptional sensory skills: keen eyesight, acute hearing, and a strong sense of smell.

These predators can detect disturbances in the water caused by the movement of their prey. By interpreting these waves, sharks can pinpoint the exact location of potential targets. Once a white shark identifies these signals, it begins its methodical approach toward the prey.

The dynamics of this pursuit can be modeled mathematically by an equation that describes the shark’s motion relative to its target:

$$v_{t+1}^i=\mu \left[v_t^i+p_1\left(w_{gbes{t}_t}-w_t^i\right)\times c_1\right]+p_2\left(w_{bes{t}_t}^{v_i}-w_t^i\right)\times c_2$$

(9)

In this model, $v_{t+1}^i$ represents the shark’s updated speed, while $v_t^i$ corresponds to its current velocity at time t. The term $w_{gbes{t}_t}$ denotes the best position found by the entire shark population up to the current time step. On the other hand, $w_t^i$ is the position of the i-th shark, and $w_{bes{t}_t}^{v_i}$ represents the best position this shark has encountered during its search.

The random factors $c_1$ and $c_2$, selected from a uniform distribution within the range $[0,1]$, influence the movement dynamics. The coefficient $\mu$ is critical for regulating the convergence rate of the shark optimization algorithm. Furthermore, the weights $p_1$ and $p_2$ adjust the influence of the best positions $w_{gbes{t}_t}$ and $w_{bes{t}_t}^{v_i}$ on the shark’s current position $w_t^i$. These parameters are computed as follows:

$$p_1=p_{\max }+(p_{\max }-p_{\min })\times e^{-{\left({\displaystyle \frac{4k}{K}}\right)}^2}$$

(10)

$$p_2=p_{\min }+(p_{\max }-p_{\min })\times e^{-{\left({\displaystyle \frac{4k}{K}}\right)}^2}$$

(11)

$$\mu ={\displaystyle \frac{2}{\left|2-\tau -\sqrt{{\tau }^2-4\tau }\right|}}.$$

(12)

The parameters $p_{\max }$ and $p_{\min }$ define the upper and lower bounds that govern the shark’s movement. The values k and K represent the current and total number of iterations, respectively. The coefficient $\tau$ serves as an acceleration factor to modulate the velocity change during the optimization process.

It is noteworthy that the velocity update mechanism in Equation (9) shares a conceptual foundation with the well-known Particle Swarm Optimization (PSO) algorithm, as it incorporates inertial, cognitive, and social components. However, WSO extends significantly beyond this similarity by integrating several unique and more complex behavioral models that are not present in canonical PSO. Specifically, the primary position update strategy in WSO (Equation (13)) is not a simple addition of velocity; instead, it employs a sophisticated conditional mechanism that alternates between an exploratory jump based on logical operations and an exploitative movement normalized by a wave frequency (f). Furthermore, WSO introduces a “schooling behavior” phase (Equation (18)) that refines solutions through a cooperative averaging process. These additional layers of complexity, governed by a richer set of adaptive parameters (e.g., $mv$ and $S_s$), provide WSO with a more diverse and dynamic search capability.

4.1.3. Updating Shark Velocity in Pursuit

This section introduces a decision-making process that governs how white sharks adjust their positions during a hunt. Two distinct actions are considered, determined by a random value and movement strength $mv$, which increases over time to accelerate convergence:

• Sharks adjust their positions randomly around the prey, using their sense of smell as a guide.
• Sharks reposition themselves in the search space based on their ability to hear the prey.

The random factor rand and the movement strength $mv$ are key to determining the shark’s next move. The value of $mv$ grows as the algorithm progresses, ensuring that the sharks’ search becomes more directed and faster, promoting convergence.

The shark’s position is updated based on the decision made. If the random value is less than $mv$, the first equation updates the position using an element-wise multiplication with a negated XOR operation:

$$w_i^{t+1}=\left\{\begin{array}{cc}{w}_i^t·\neg \oplus w_o+u·a+l·b;\hfill & \mathrm{if}\mathrm{rand}<mv\hfill \\ w_i^t+{\displaystyle \frac{v_i^t}{f}};\hfill & \mathrm{if}\mathrm{rand}\ge mv\hfill \end{array}\right.$$

(13)

The negation of the XOR operation between the boundaries a and b adjusts the shark’s position in the search space, accounting for the prey’s location. This approach mimics a random search guided by sensory input.

For the second case, when the random value is greater than or equal to $mv$, the shark updates its position by adding its velocity, normalized by the frequency of its undulatory motion.

The parameters a and b are calculated as follows:

$$a=\mathrm{sgn}(w_i^t-u)>0$$

(14)

$$b=\mathrm{sgn}(w_i^t-l)<0$$

(15)

The function sgn checks whether the difference between the shark’s position and the boundaries is positive or negative. This ensures that the shark’s movement is adjusted based on its relative position to the boundaries of the search space. A positive difference results in a being true, while a negative difference marks b as true.

Additionally, the frequency f of the shark’s oscillatory movement is defined as

$$f=f_{\min }+{\displaystyle \frac{f_{\max }-f_{\min }}{f_{\max }+f_{\min }}}.$$

(16)

Here, $f_{\min }$ and $f_{\max }$ are the minimum and maximum frequencies of the shark’s movement. This controls how fast the shark can adjust its position during the hunt.

The sensory ability of the shark, measured by $mv$, reflects its capacity to track prey using hearing and smell. As the algorithm iterates, this value increases, improving the shark’s ability to track and approach its target:

$$mv={\displaystyle \frac{1}{\left(a_0+e^{\left(\frac{K/2-k}{a_1}\right)}\right)}}$$

(17)

where $a_0$ and $a_1$ are constants that regulate the balance between exploration and exploitation as the shark refines its search strategy.

4.1.4. Position Update Based on Schooling Behavior

This section focuses on how white sharks adjust their positions through a phenomenon known as “nearby prey localization”. The behavior of the sharks is guided by the positions of nearby, successful searchers. Initially, the first shark in the population moves randomly to a new position. For all other sharks, their updated position depends on the distance to the best-known solution, which is the global best position found so far. The sharks combine information from the global best position and the previous shark’s location to explore and exploit nearby areas. This cooperative strategy enhances the overall efficiency of the search process in the search space. The strength of the sharks’ olfactory and visual senses when tracking these nearby sharks is expressed by $S_s$, while $D_w$ represents the distance between a shark and its prey.

The movement of each shark as it tracks its prey can be mathematically modeled as follows:

$${w^{\prime }}_i^{t+1}=w_{\mathrm{gbest}}^t+r_1·\overrightarrow{D_w}·\mathrm{sgn}(r_2-0.5)\mathrm{if}{r}_3<S_s.$$

(18)

In this equation, ${w^{\prime }}_i^{t+1}$ represents the updated position of the i-th shark. The function $\mathrm{sgn}(r_2-0.5)$ determines the direction of movement: it outputs 1 if the argument is positive, −1 if negative, and 0 if zero. The terms $r_1$, $r_2$, and $r_3$ are random values drawn from the range $[0,1]$, which influence the movement direction and distance.

The distance $\overrightarrow{D_w}$ between the shark and the global best solution is calculated as follows:

$$\overrightarrow{D_w}=\left|\mathrm{rand}\times (w_{\mathrm{gbest}}^t-w_i^t)\right|$$

(19)

This term scales the difference between the shark’s current position and the global best solution by a random factor, guiding the shark’s movement in the search space.

To model the shark’s sensory abilities in tracking prey, the strength of its visual and olfactory senses is defined by the parameter $S_s$, which is updated over time as follows:

$$S_s=\left|1-e^{-a_2·{\displaystyle \frac{t}{T}}}\right|.$$

(20)

Here, $a_2$ is a positive constant that controls the balance between exploration and exploitation in the shark’s search behavior. In this study, a value of 0.0005 for $a_2$ is chosen based on previous experimental evaluations [12].

To model the collective behavior of the shark group, the top two best solutions are retained, while the positions of other sharks are updated relative to these best solutions. The following formula is used to calculate the updated positions of the sharks:

$$w_{k+1}^i={\displaystyle \frac{w_k^i+w_{k+1}^{\prime i}}{2\times \mathrm{rand}}}$$

(21)

where rand is a random number generated uniformly in the range $[0,1]$. This adjustment allows the sharks to collectively converge towards the optimal prey by sharing the best information available.

4.1.5. Pseudocode for WSO

Algorithm 1 displays the pseudocode of WSO, specifying the initial setup of the parameters, along with the processing of locations and speeds. This procedure is repeated continuously until the defined completion requirements are met.

Algorithm 1 White Shark Optimizer.

1: Establish the problem’s boundaries and constraints. 2: Specify the configuration settings for the WSO. 3: Create random starting locations for the WSO. 4: Initialize the population’s movement rates. 5: Assess the fitness of the initial population’s placements. 6: while $t>T$ do 7:     Update the parameters v, $p_1$, $p_2$, $\mu$, a, b, $w_o$, f, $mv$, and $S_s$, 8:     respectively. 9:     for $i=1,\dots ,n$ do 10:         $v_{i_{t+1}}=\mu \left[v_{i_t}+p_1\left(w_{g_{{\mathrm{best}}_t}}-w_{i_t}\right)\times c1+p_2\left(w_{{\nu }_{i_t}^{\mathrm{best}}}-w_{i_t}\right)\times c2\right]$ 11:     end for 12:     for $i=1,\dots ,n$ do 13:         if $\mathrm{rand}<mv$ then 14:            $w_{i_{t+1}}=w_{i_t}·\neg \oplus w_o+u·a+l·b$ 15:         else 16:            $w_{i_{t+1}}=w_{i_t}+{\displaystyle \frac{v_{i_t}}{f}}$ 17:         end if 18:     end for 19:     for $i=1\dots ,n$ do 20:         if $\mathrm{rand}\le S_s$ then 21:            $\overrightarrow{D_w}=|\mathrm{rand}\times (w_{g_{bes{t}_t}}-w_{i_t})|$ 22:            if $i==1$ then 23:                $w_{t+1}^i=w_{g_{{\mathrm{best}}_t}}+r_1{\overrightarrow{D}}_w·\mathrm{sgn}(r_2-0.5)$ 24:            else 25:                ${\acute{w}}_{t+1}^i=w_{g_{{\mathrm{best}}_t}}+r_1{\overrightarrow{D}}_w·\mathrm{sgn}(r_2-0.5)$ 26:                $w_{t+1}^i={\displaystyle \frac{w_{i_t}+{\acute{w}}_{t+1}^i}{2\times \mathrm{rand}}}$ 27:            end if 28:         end if 29:     end for 30:     Relocate the great white sharks that move past the designated borders. 31:     Review and modify their updated coordinates as needed. 32: end while 33: Retrieve the most optimal solution found so far.

5. Binarization Techniques

Within the context of continuous metaheuristics, binarization techniques stand out [6], which aim to transform values from a continuous domain into a binary format. This process is carried out with the goal of preserving the ability to generate high-quality solutions characteristic of continuous metaheuristics, thereby facilitating the attainment of binary solutions of equivalent quality.

5.1. Two-Step Technique

This method serves as a strategy for converting continuous combinatorial optimization problems into their binary counterparts [7]. It is frequently employed when utilizing metaheuristics to address combinatorial optimization challenges that involve continuous variables. The procedure involves two key stages, illustrated in Figure 2.

5.1.1. Transfer Function

The transfer function is fundamental in the binarization process, particularly in the realm of continuous metaheuristics. It is one of the most commonly used normalization techniques and was first introduced in the seminal work of Kennedy and Eberhart [8]. In this study, the authors introduced the sigmoid transfer function to adapt the continuous Particle Swarm Optimization (PSO) algorithm to binary problems. This function transforms continuous velocity values into probabilities, which are then mapped to binary outputs (0 or 1). This groundbreaking approach enables the application of PSO, originally designed for continuous spaces, to binary optimization problems.

The primary role of the transfer function is to transform continuous values into a predefined range, typically $[0,1]$. This transformation is key for converting continuous solutions into a binary format, enhancing their effectiveness in optimization algorithms that rely on binary search methods. By applying the transfer function, continuous values are appropriately scaled and adjusted, facilitating their subsequent binarization and efficient utilization in optimization algorithms.

Therefore, the transfer function constitutes a fundamental step in the binarization process, significantly contributing to the effectiveness and efficiency of continuous metaheuristics in various optimization applications. The work of Kennedy and Eberhart laid the foundation for the use of transfer functions in adapting continuous metaheuristics to binary problems, influencing numerous subsequent studies in the field of combinatorial optimization.

Figure 3 illustrates the curves produced by the S-shape and V-shape equations, named for their distinctive graphical representations.

Table 3. Transfer function table.

S-Shaped	Equation	V-Shaped	Equation
S1	$T\left(d_j^i\right)={\displaystyle \frac{1}{1+e^{-2d_i^j}}}$	V1	$T\left(d_i^j\right)=\left|erf\left({\displaystyle \frac{\sqrt{\pi }}{2}}{d}_i^j\right)\right|$
S2	$T\left(d_j^i\right)={\displaystyle \frac{1}{1+e^{-d_i^j}}}$	V2	$T\left(d_i^j\right)=\left|tanh\left(d_i^j\right)\right|$
S3	$T\left(d_j^i\right)={\displaystyle \frac{1}{1+e^{\frac{(-d_i^j)}{2}}}}$	V3	$T\left(d_i^j\right)={\displaystyle \frac{d_i^j}{\sqrt{1+{\left(d_i^j\right)}^2}}}$
S4	$T\left(d_j^i\right)={\displaystyle \frac{1}{1+e^{\frac{(-d_i^j)}{3}}}}$	V4	$T\left(d_i^j\right)=\left|{\displaystyle \frac{2}{\pi }}arctan\left({\displaystyle \frac{\pi }{2}}{d}_j^i\right)\right|$

provides a detailed overview of these equations and their respective variants.

5.1.2. Binarization Rules

Binarization involves transforming continuous values into binary states, usually represented as zero and one. This process applies binarization rules to probabilities obtained from a transfer function, resulting in a binary output. Numerous techniques for implementing this binarization process are extensively discussed in the literature [9]. Choosing a suitable binarization rule is essential, as it must align with the specific context and requirements of the problem. The effective application of these rules is critical for ensuring accurate and reliable outcomes [7].

Table 4. Table of binarization rules.

Type	Binarization Rules
Standard (STD)	$X_{\mathrm{new}}^j=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{rand}\le T\left(d_w^j\right)\hfill \\ 0\hfill & \mathrm{else}.\hfill \end{array}\right.$
Complement (COM)	$X_{\mathrm{new}}^j=\left\{\begin{array}{cc}\mathrm{Complement}\left(X_w^j\right)\hfill & \mathrm{if}\mathrm{rand}\le T\left(d_w^j\right)\hfill \\ 0\hfill & \mathrm{else}.\hfill \end{array}\right.$
Static Probability (SP)	$X_{\mathrm{new}}^j=\left\{\begin{array}{cc}0\hfill & \mathrm{if}T\left(d_w^j\right)\le \alpha \hfill \\ X_w^j\hfill & \alpha <T\left(d_w^j\right)\le {\displaystyle \frac{1}{2}}(1+\alpha )\hfill \\ 1\hfill & T\left(d_w^j\right)\ge {\displaystyle \frac{1}{2}}(1+\alpha )\hfill \end{array}\right.$
Elitist (ELIT)	$X_{\mathrm{new}}^j=\left\{\begin{array}{cc}{X}_{\mathrm{Best}}^j\hfill & \mathrm{if}\mathrm{rand}<T\left(d_w^j\right)\hfill \\ 0\hfill & \mathrm{else}.\hfill \end{array}\right.$

displays four binarization rules, accompanied by their corresponding equations and conditions.

Among the available binarization rules, the Elitist (ELIT) approach has demonstrated superior performance in combinatorial optimization problems by incorporating global best information during the binarization process [6], accelerating convergence while maintaining solution diversity in binary search spaces.

The theoretical advantage of the ELIT rule lies in its explicit exploitation mechanism. By inheriting components directly from the best-known solution ($X_{\mathrm{Best}}$), it forcefully guides the search agents toward promising regions, which is particularly effective in the vast and complex search spaces characteristic of the USCP. This theoretical benefit is strongly supported by empirical evidence from related studies on the broader Set Covering Problem (SCP). In a comprehensive comparison [25], the baseline ELIT configuration consistently outperformed the Standard (STD) configuration in terms of both solution quality (measured by RPD) and statistical rankings from non-parametric tests. While chaotic-enhanced versions of STD showed competitive results in some scenarios, the baseline ELIT approach proved to be a more robust and consistently high-performing foundation, thus justifying its selection for this study.

6. A New Binary Approach for the White Shark Optimizer

The White Shark Optimizer (WSO) is a powerful optimization algorithm inspired by the hunting behavior of white sharks. While WSO works exceptionally well for problems with continuous variables, it struggles with binary problems where solutions can only be zero or one. This difficulty arises because WSO naturally operates with continuous values for both position and velocity, which do not translate directly to binary decisions.

To bridge this gap, we propose a novel approach: transforming both the population and velocity components into binary representations.

Our method combines WSO with specialized mathematical tools called transfer functions and chaotic maps. These tools help convert continuous values to binary ones in a way that preserves the algorithm’s intelligence. This transformation not only allows WSO to work with binary problems but also helps maintain diverse solutions and prevents the algorithm from getting stuck in suboptimal answers too early.

6.1. Conceptual Framework for Binary Adaptation

The adaptation of the continuous White Shark Optimizer to binary optimization domains requires a fundamental reinterpretation of the algorithm’s core mechanisms. Unlike straightforward binarization approaches that merely convert continuous outputs to binary values, our framework establishes a conceptual bridge between shark hunting behavior and discrete decision-making in the USCP context.

6.1.1. Semantic Interpretation in Binary Domain

In the binary adaptation, each shark represents a candidate solution where position components $x_{i,j}\in \{0,1\}$ indicate the decision to include ($x_{i,j}=1$) or exclude ($x_{i,j}=0$) set j in the solution. The velocity components $v_{i,j}$, while computed in continuous space, represent the tendency or propensity to change the current decision state for set j.

This interpretation transforms the shark’s movement dynamics from spatial navigation to discrete decision evolution. The continuous velocity values encode the strength and direction of potential decision changes, which are subsequently converted to binary decisions through the binarization process.

6.1.2. Shark Behavior Mapping to USCP Context

The original WSO Equations (4) and (5) calculate the influence weights $p_1$, $p_2$ and convergence factor $\mu$ that govern shark movement toward prey. In our binary adaptation, we have the following:

• Prey location ($w_{gbest}$): Represents the best-known set selection pattern that minimizes coverage cost.
• Shark movement: Corresponds to iterative refinement of set selection decisions.
• Hunting behavior: Translates to the exploration of alternative set combinations while exploiting promising selection patterns.

The distance calculation in Equation (16), $\overrightarrow{D_w}=|\mathrm{rand}\times (w_{gbest}-w_i)|$, measures the space distance between the current shark’s solution and the globally optimal selection pattern. In binary context, this distance quantifies how many set selection decisions differ between solutions.

6.1.3. Dual Binarization Rationale

Traditional binary metaheuristics typically binarize only position vectors, treating velocity as an auxiliary continuous component. Our dual binarization approach converts both position and velocity to binary representations based on the following rationale:

• Consistency preservation: Both position and velocity represent decision-related information that should operate in the same domain to maintain algorithmic coherence.
• Enhanced decision dynamics: Binary velocity components provide discrete decision change signals that complement position-based set selections.
• Unified representation: Dual binarization eliminates domain inconsistencies that can arise when mixing continuous and binary components.

6.1.4. Mathematical Formulation Bridge

The transition from continuous to binary operations preserves the essential optimization dynamics through our two-phase transformation:

Phase I—Continuous Computation: The original WSO Equations (4), (5) and (15) compute position updates using continuous arithmetic:

$$v_{t+1}^i=\mu \left[v_t^i+p_1\left(w_{gbes{t}_t}-w_t^i\right)\times c_1\right]+p_2\left(w_{bes{t}_t}^{v_i}-w_t^i\right)\times c_2$$

(22)

Phase II—Binary Transformation: The continuous values undergo transfer function mapping followed by chaotic threshold-based binarization:

$$X_{i,j}=\left\{\begin{array}{cc}{X}_{best,j}\hfill & \mathrm{if}\mathrm{chaos}<T_{V3}\left(w_{i,j}\right)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(23)

This formulation ensures that the sophisticated search behavior encoded in the continuous WSO dynamics is preserved while operating in the discrete solution space required for USCP. The chaotic thresholding mechanism replaces random decision-making with deterministic yet unpredictable selection patterns that enhance exploration diversity while maintaining convergence properties.

The resulting binary framework maintains the core optimization principles of WSO balanced exploration–exploitation, adaptive convergence, and collective intelligence while operating effectively in the discrete domain of combinatorial optimization problems.

6.2. Transformation of Velocity and Population into Binary States

In the original WSO, sharks move through the search space with specific velocities, updating their positions based on the best locations found so far. This process closely resembles how particles move in Particle Swarm Optimization (PSO).

To adapt WSO for binary problems, we need to translate both continuous velocities and positions into binary values. Imagine converting the statement “move forward at 0.7 speed” to a simple “move (1) or stay (0)” decision. We accomplish this using transfer functions—mathematical formulas that take continuous values as input and output values between zero and one, which we can then convert to binary decisions.

This dual transformation ensures that both the shark’s position (where it is) and velocity (how it is moving) remain consistent in the binary world. Our approach draws inspiration from the successful BPSO paradigm discussed in Section 3.2, adapting similar principles to the unique movement mechanics of WSO.

Equation (9) shows how the WSO velocity calculation works similarly to PSO, making these binary adaptation techniques relevant and applicable to our work. However, unlike standard BPSO approaches, our method applies the binarization to both position and velocity components simultaneously, providing a more consistent binary representation of the search behavior.

6.3. Initialization of Parameters

For any optimization algorithm to work effectively, we must carefully set its initial conditions—like making sure a car has the right amount of fuel and air pressure before a race. For our binary WSO, these critical parameters include the following:

• Population size(n): This is simply the number of “sharks” or candidate solutions we use to explore the problem space. More sharks mean better exploration but require more computational resources.
• Velocity parameters (v, $\mu$, $p_1$, $p_2$, $c1$, $c2$): These control how the sharks move through the solution space—how fast they move, how much they are attracted to good solutions, and how they balance exploring new areas versus exploiting known good regions.

We initialize the velocities with small random values to ensure sharks do not have strong directional biases at the beginning. This is like giving them a gentle push in various directions rather than forcing them to swim in predetermined patterns, allowing for more natural and effective exploration.

Binarize Velocity and Positions

The heart of our approach is the binarization process, which converts continuous values from the WSO algorithm to binary ones (zero or one). This process follows a systematic approach with four sequential steps:

First, transfer functions convert continuous position and velocity values into the range [0, 1]. These mathematical functions map unlimited continuous ranges to a bounded interval suitable for binary conversion.

Second, a threshold mechanism determines when to assign zero or one values. While traditional approaches use fixed thresholds or random values, our method employs dynamically changing thresholds based on chaotic systems.

Third, the binarization rules dictate how to apply these thresholds to produce the final binary values. These rules can range from simple comparisons to more sophisticated methods that incorporate global solution information.

Finally, a verification mechanism ensures that the resulting binary solutions satisfy the USCP constraints, penalizing infeasible solutions to guide the search process.

For our implementation, we specifically select the following:

• Transfer Function: We apply the V3 transfer function ($T\left(d_i^j\right)={\displaystyle \frac{d_i^j}{\sqrt{1+{\left(d_i^j\right)}^2}}}$) to both population positions and velocities. We select V3 [40] after comparative testing because it preserves the directional information while producing the well-distributed probability values, outperforming S-shaped alternatives in preliminary experiments.
• Chaotic Maps: Instead of standard random number generation, we employ four chaotic maps to generate dynamic thresholds. Table 1 presents the mathematical definitions of these chaotic systems, while Figure 1 illustrates their behavioral patterns (

  Table 5. Table of binarization rules and chaotic maps.

  Chaotic Maps	ELIT
  Circle Map	ELIT_CIRCLE
  Singer Map	ELIT_SINGER
  Sinusoidal Map	ELIT_SINU
  Tent Map	ELIT_TENT

  ):
  These deterministic systems produce complex, non-repeating sequences that enhance search diversity while maintaining structured exploration patterns. Each map offers unique characteristics: the singer map provides oscillatory behavior ideal for early exploration, the sinusoidal map delivers smooth transitions suitable for cyclic problems, the tent map creates sharp transitions for aggressive exploitation, and the circle map offers balanced ergodic properties for structured instances.
• Binarization Rule: After evaluating multiple approaches, we select the Elitist (ELIT) rule defined as

  $$X_{\mathrm{new}}^j=\left\{\begin{array}{cc}{X}_{\mathrm{Best}}^j\hfill & \mathrm{if}\mathrm{chaotic}<T\left(d_w^j\right)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

  where $X_{\mathrm{Best}}^j$ represents the value in the best global solution found so far. This rule proves more effective than standard thresholding because it incorporates information from the best-known solution, accelerating convergence in promising regions of the search space.

During each algorithm iteration, these components work together: first applying the V3 transfer function to positions and velocities, then generating chaotic values as dynamic thresholds using the maps shown in Table 1, and finally using the ELIT rule to determine binary values. Additionally, our USCP-specific mechanism verifies that each element of the universe is covered by at least one selected set, penalizing infeasible solutions to guide the algorithm toward valid solutions.

This comprehensive binarization approach allows our WSO algorithm to navigate effectively through binary search spaces while maintaining its intelligent search behavior and addressing the specific constraints of the USCP.

6.4. Chaotic Maps for Adaptive Thresholding

Building upon the chaotic maps introduced in Section 3.3, our binary WSO framework uses these maps as dynamic thresholding mechanisms to overcome limitations of traditional binarization approaches. Unlike conventional methods using static thresholds or uniform random numbers, our approach employs four carefully selected chaotic maps as adaptive threshold generators throughout the optimization process.

We implement and analyze four distinct chaotic maps in our WSO framework, each with specific properties relevant to binary optimization:

• Circle Map: We utilize this map with parameters $a=0.5$ and $b=0.2$ in equation $x_{k+1}=mod(x_k+b-(a/2\pi )sin\left(2\pi x_k\right),1)$. Our empirical analysis reveals this map’s exceptional performance in structured USCP instances (clr10-13), achieving up to 7.31% improvement in RPD compared to standard binarization approaches. The map’s balanced threshold dynamics particularly enhance the exploitation capabilities in well-defined solution spaces.
• Singer Map: Configured with $\mu =1.07$ in $x_{k+1}=\mu (7.86x_k-23.31x_k^2+28.75x_k^3-13.3x_k^4)$, this map generates the oscillatory sequences that enhance early-stage exploration. Our experimental results show this map is particularly effective during the initial search phases (first 150 iterations) when the population diversity is critical.
• Sinusoidal Map: We implement this map with $a=2.3$ in $x_{k+1}=a{x}_k^{2}sin\left(\pi x_k\right)$ and discover its remarkable effectiveness in cyclic USCP instances (cyc06-07), where it achieves an RPD of 0.96% compared to 3.56% from standard methods. The map’s gradual threshold adaptations demonstrates significant alignment with cyclic problem structures.
• Tent Map: Our implementation with $\mu =1.99$ in this piecewise function creates sharp threshold transitions that accelerate convergence in later optimization stages while maintaining sufficient diversity to avoid premature convergence.

Through extensive testing, we observe that the performance differences between these maps are not merely statistical variations but reflect fundamental interactions between the chaotic dynamics and the USCP’s solution landscape. Our statistical analysis (presented in Section 7) confirms that the circle map’s stability provides consistent convergence in structured instances, while the sinusoidal map’s adaptive properties excel in cyclic instances.

The integration of these chaotic maps into our WSO binarization mechanism offers three key advantages:

• Deterministic unpredictability: Unlike standard random number generators, our chaotic threshold sequences create structured exploration patterns that consistently outperform uniform random thresholds in our comparative analysis.
• Dynamic adaptation: Our implementation allows threshold values to evolve based on each map’s unique mathematical properties, automatically adjusting the exploration–exploitation balance throughout the optimization process.
• Problem-specific optimization: Our findings enable problem-specific chaotic map selection—circle maps for structured problems and sinusoidal maps for cyclic instances—facilitating tailored optimization strategies.

This chaotic threshold mechanism represents a significant advancement over traditional binarization approaches, offering a mathematically sophisticated alternative that enhances both search diversity and convergence properties in binary WSO.

6.5. Repairing Complex Solutions

While our dual binarization approach minimizes structural inconsistencies by transforming both position and velocity components simultaneously, solution repair mechanisms remain essential for addressing USCP-specific constraints. We implement a systematic repair process that ensures all solutions remain feasible throughout the optimization process.

Our solution repair mechanism addresses two primary issues in the USCP context:

• Coverage constraint violations: When a binary solution fails to cover all elements in the universe, we implement a greedy repair heuristic that iteratively adds the most efficient sets (those covering the most uncovered elements) until all elements are covered. This is formulated as

  $$S_{repair}=S_{original}\cup \{j\in J\setminus S_{original}\mid \underset{j}{max}|U_j\cap U_{uncovered}\left|\right\}$$

  where $S_{original}$ is the original solution, J is the set of all available sets, $U_j$ is the set of elements covered by set j, and $U_{uncovered}$ is the set of currently uncovered elements.
• Redundant set elimination: After ensuring coverage, we apply a redundancy removal procedure that examines each selected set and removes it if all its covered elements are also covered by other selected sets. Sets are evaluated in ascending order of their efficiency (elements covered per set) to prioritize removing less efficient sets:

  $$S_{final}=S_{repair}\setminus \{j\in S_{repair}\mid \forall e\in U_j,\exists k\in S_{repair},k\ne j\mathrm{such}\mathrm{that}e\in U_k\}$$

The integration of this repair mechanism with our chaotic binary WSO creates a balanced optimization approach. The chaotic maps generate high-quality binary transformations that rarely violate constraints, while the repair mechanism ensures all solutions remain feasible. This synergy is particularly effective because of the following:

• The dynamic thresholds created by chaotic maps maintain population diversity, minimizing premature convergence to infeasible regions.
• The greedy repair heuristic preserves the solution structure created by the WSO while enforcing problem constraints.
• The redundancy elimination step improves solution quality without disrupting the exploration–exploitation balance established by the algorithm.

Our repair mechanism operates with complexity $O(m\times n)$, where m is the number of elements and n is the number of sets, making it computationally efficient even for large problem instances. Experimental results indicate that solutions require repair in less than 12% of iterations, demonstrating the effectiveness of our dual binarization approach in generating naturally feasible solutions.

Algorithm 2 presents a comprehensive overview of our binary WSO approach, illustrating how these components work together to effectively solve binary optimization problems (Algorithm 3).

Algorithm 2 Binary White Shark Optimizer.
1:	Establish the problem’s boundaries and constraints.
2:	Specify the configuration settings for the WSO.
3:	Create random starting locations for the WSO.
4:	Initialize the population’s movement rates.
5:	Assess the fitness of the initial population’s placements.
6:	while $t>T$ do
7:	Update the parameters v, $p_1$, $p_2$, $\mu$, a, b, $w_o$, f, $mv$, and $S_s$ respectively.
8:	for $i=1,\dots ,n$ do	▹ Velocity update phase
9:	$v_{i_{t+1}}=\mu \left[v_{i_t}+p_1\left(w_{g_{{\mathrm{best}}_t}}-w_{i_t}\right)\times c1+p_2\left(w_{{\nu }_{i_t}^{\mathrm{best}}}-w_{i_t}\right)\times c2\right]$
10:	end for
11:	for $i=1,\dots ,n$ do	▹ Position update phase
12:	if $rand<mv$ then
13:	$w_{i_{t+1}}=w_{i_t}·\neg \oplus w_o+u·a+l·b$
14:	else
15:	$w_{i_{t+1}}=w_{i_t}+{\displaystyle \frac{v_{i_t}}{f}}$
16:	end if
17:	end for
18:	for $i=1\dots ,n$ do	▹ Schooling behavior phase
19:	if $rand\le S_s$ then
20:	$\overrightarrow{D_w}=|rand\times (w_{g_{bes{t}_t}}-w_{i_t})|$
21:	if $i==1$ then
22:	$w_{i_{t+1}}=w_{g_{{\mathrm{best}}_t}}+r_1\overrightarrow{D_w}·\mathrm{sgn}(r_2-0.5)$
23:	else
24:	${\dot{w}}_{i_{t+1}}=w_{g_{{\mathrm{best}}_t}}+r1·\mathrm{sgn}(r2-0.5)$
25:	$w_{i_{t+1}}={\displaystyle \frac{w_{i_t}+{\dot{w}}_{i_{t+1}}}{2\times \mathrm{rand}}}$
26:	end if
27:	end if
28:	end for
29:	for $i=1$ to $pop$ do
30:	for $j=1$ to $dim$ do
31:	Apply transfer function: $T\left(d^{j}i\right)$	▹ on population
32:	Generate chaotic value
33:	Apply binarization rule: $X^{j}i$	▹ A chaotic map to generate the random sequence for the rule
34:
35:	Apply transfer function: $T\left(v^{j}i\right)$	▹ on velocity
36:	Generate chaotic value
37:	Apply binarization rule: $X^{j}i$	▹ A chaotic map to generate the random sequence for the rule
38:
39:	end for
40:	end for
41:	Enforce boundary constraints for shark positions.
42:	Check USCP coverage constraints and adjust fitness values.	▹ Ensure all elements are covered
43:	end while
44:	Retrieve the most optimal solution found so far.

Algorithm 3 Chaotic Dual Binarization Process.
1:	Input: Continuous position $w_i$, continuous velocity $v_i$, global best solution $X_{best}$, map type $map\_type$.
2:	Output: New binary position $X_{ne{w}_i}$.
3:	Initialize a chaotic map seed for the current individual.
4:	for $j=1$ to $dim$ do	▹ Iterate through each dimension of the solution
5:
6:	— Step 1: Binarize Position Component —
7:
8:	$T_{pos}\left(j\right)\leftarrow {\displaystyle \frac{w_i\left(j\right)}{\sqrt{1+{\left(w_i\left(j\right)\right)}^2}}}$	▹ Apply V3 transfer function to position
9:	$chao{s}_{val}\leftarrow \mathrm{ApplyChaosMap}(map\_type,\mathrm{seed})$	▹ Generate chaotic threshold
10:	if $chao{s}_{val}<T_{pos}\left(j\right)$ then
11:	$X_{pos}\left(j\right)\leftarrow X_{best}\left(j\right)$	▹ Apply Elitist rule
12:	else
13:	$X_{pos}\left(j\right)\leftarrow 0$
14:	end if
15:
16:	— Step 2: Binarize Velocity Component —
17:
18:	$T_{vel}\left(j\right)\leftarrow {\displaystyle \frac{v_i\left(j\right)}{\sqrt{1+{\left(v_i\left(j\right)\right)}^2}}}$	▹ Apply V3 transfer function to velocity
19:	$chao{s}_{val}\leftarrow \mathrm{ApplyChaosMap}(map\_type,\mathrm{seed})$	▹ Generate next chaotic threshold
20:	if $chao{s}_{val}<T_{vel}\left(j\right)$ then
21:	$X_{vel}\left(j\right)\leftarrow X_{best}\left(j\right)$	▹ Apply Elitist rule
22:	else
23:	$X_{vel}\left(j\right)\leftarrow 0$
24:	end if
25:
26:	— Step 3: Combine Both Binary Components —
27:
28:	$X_{ne{w}_i}\left(j\right)\leftarrow X_{pos}\left(j\right)\vee X_{vel}\left(j\right)$	▹ Combine using Bitwise OR
29:	end for
30:	Return $X_{ne{w}_i}$	▹ Return the complete new binary solution for the individual

7. Results and Discussion

This section presents a comprehensive evaluation of our binary WSO approach for solving the USCP. We first describe the experimental setup, including algorithm parameters and problem instances. Then, we analyze the performance results across different binarization schemes, examining solution quality, convergence behavior, and the impact of chaotic maps on algorithm performance. Finally, we examine the exploration–exploitation balance and diversity characteristics of different algorithm configurations to provide insights into their optimization behavior.

7.1. Experimental Setup

To evaluate the proposed method, the WSO is applied to the USCP, testing its performance under different binarization schesmes. Specifically, the following configurations are explored:

• Transfer Function: V3.
• Binarization Rules: Elit.
• Chaotic Maps: Circle, Singer, Sinu, and Tent.

These configurations, detailed in

Table 6. Parameter table.

Parameter	Value
Independent runs	31
Transfer function	V3
Number of binarization schemes	5
Number of USCP instances	6
Population size	30
Number of iterations	500
WSO Parameters	a = 2
	$f_{\min }$ = 0.07
	$f_{\max }$ = 0.75
	$\tau$ = 4.125
	$a_0$ = 6.25
	$a_1$ = 100
	$a_2$ = 0.0005

, allow for a comprehensive analysis of how different binarization approaches impact the performance of WSO in solving the USCP. The binarization schemes used are based on previous experiments described in the paper [25], which focuses on the resolution of the SCP with WSO. The WSO-specific parameters are adopted from the original WSO paper [12] to ensure consistency with the algorithm’s intended behavior and facilitate reproducibility. The experimental parameters (population size = 30, iterations = 500) are selected based on common practices in the metaheuristic literature for combinatorial optimization problems [6] and preliminary tuning experiments that balance computational efficiency with solution quality. These parameter choices enable fair comparison across binarization variants while maintaining reasonable computational requirements for the USCP instances tested.

The parameter settings include a population size of 30 individuals and a maximum of 500 iterations as detailed in Table 6.

The experiments are conducted using six benchmark instances from two distinct categories, specifically selected to evaluate algorithm performance across different problem structures:

• Structured Instances (clr10–13): These instances, derived from Beasley’s OR-Library [4], feature well-defined constraint matrices with regular patterns. They represent classic set covering challenges with the following characteristics:

  –

  clr10: 511 rows × 210 columns, density 3.8%, optimal solution = 25

  –

  clr11: 1023 rows × 330 columns, density 1.9%, optimal solution = 23

  –

  clr12: 2047 rows × 495 columns, density 0.9%, optimal solution = 23

  –

  clr13: 4095 rows × 715 columns, density 0.5%, optimal solution = 23

  These instances feature increasing sparsity as problem size grows, testing algorithm scalability.
• Cyclic Instances (cyc06–07): These instances represent network design problems with cyclic coverage patterns:

  –

  cyc06: 240 rows × 192 columns, density 5.1%, optimal solution = 60

  –

  cyc07: 672 rows × 448 columns, density 2.1%, optimal solution = 144

  These instances exhibit circular dependency structures that challenge greedy solution approaches and require balanced exploration–exploitation strategies.

Considering the number of instances (6), versions of the binarization schemes (5), and independent runs (31), the total number of experiments amount to 930 (6 × 5 × 31).

For the experiments, Python 3.10.5 (64-bit) is used on a Windows 10 system. The hardware consists of an Intel Core i7-8650U processor running at 2.11 GHz and 16 GB of RAM. The average execution time per run is approximately 3.2 min for smaller instances (cyc06) and up to 18.7 min for larger instances (clr13), demonstrating reasonable computational requirements even for complex problems.

7.2. Summary of Results

For

Table 7. Metaheuristic WSO table for clr10 and clr11.

Experiment	clr10 (25)				clr11 (23)
	Best	Avg.	Std-Dev	RPD	Best	Avg.	Std-Dev	RPD
WSO-ELIT	25	26.9	0.64	7.61	23	24.23	0.91	5.33
WSO-ELIT_CIRCLE	25	25.97	0.97	3.87	23	23.16	0.51	0.7
WSO-ELIT_SINGER	26	27.39	0.66	9.55	23	24.71	0.58	7.43
WSO-ELIT_SINU	26	27.32	0.64	9.29	24	24.9	0.3	8.27
WSO-ELIT_TENT	25	26.87	0.79	7.48	23	23.84	0.99	3.65

,

Table 8. Metaheuristic WSO table for clr12 and clr13.

Experiment	clr12 (23)				clr13 (23)
	Best	Avg.	Std-Dev	RPD	Best	Avg.	Std-Dev	RPD
WSO-ELIT	23	26.06	1.11	13.32	25	28	1.22	21.74
WSO-ELIT_CIRCLE	23	24.55	1.39	6.73	25	27.39	1.29	19.07
WSO-ELIT_SINGER	26	26.39	0.61	14.73	27	28.16	0.92	22.44
WSO-ELIT_SINU	26	26.42	0.61	14.87	27	28.52	1.01	23.98
WSO-ELIT_TENT	23	25.81	0.9	12.2	26	27.55	1.07	19.78

and

Table 9. Metaheuristic WSO table for cyc06 and cyc07.

Experiment	cyc06 (60)				cyc07 (144)
	Best	Avg.	Std-Dev	RPD	Best	Avg.	Std-Dev	RPD
WSO-ELIT	60	61.03	0.82	1.72	144	149.13	1.96	3.56
WSO-ELIT_CIRCLE	60	61.45	0.76	2.42	144	151.58	1.72	5.26
WSO-ELIT_SINGER	60	60	0	0	144	149.32	3.1	3.7
WSO-ELIT_SINU	60	60	0	0	144	145.39	2.97	0.96
WSO-ELIT_TENT	60	60.94	0.8	1.56	144	148.87	2.11	3.38

, a summary of the experiments is provided. The first column lists the variations of all experiments, while the first row indicates the SCP instance solved. For each instance, the following metrics are shown:

• Best: The best result obtained.
• Avg: The average of the best fitness values across all runs.
• RPD: The “Relative Percentage Deviation”, which measures the percentage deviation of the obtained solution from the best-known solution, evaluating the algorithm’s quality [41].

The RPD is calculated as follows:

$$\mathrm{RPD}=100·\left({\displaystyle \frac{\mathrm{Opt}-\mathrm{Best}}{\mathrm{Opt}}}\right)$$

(24)

Here, Opt represents the optimal value for the given instance, and Best refers to the highest-quality solution obtained from the experiment.

The experimental configurations follow a systematic nomenclature as detailed in

Table 10. WSO algorithm configuration nomenclature.

Configuration	Description
WSO-ELIT	Baseline configuration using standard random number generation for binarization thresholds
WSO-ELIT_CIRCLE	Enhanced with circle chaotic map for dynamic threshold generation
WSO-ELIT_SINGER	Enhanced with Singer chaotic map for dynamic threshold generation
WSO-ELIT_SINU	Enhanced with Sinusoidal chaotic map for dynamic threshold generation
WSO-ELIT_TENT	Enhanced with Tent chaotic map for dynamic threshold generation

:

Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 display the convergence curves for all binarization schemes across the six problem instances. The horizontal axis (Iteration) shows the algorithm’s progression through its 500-iteration execution, while the vertical axis (Fitness) represents the objective function value—the number of sets selected in the solution, which we aim to minimize. Lower fitness values indicate better solutions. Each curve corresponds to a different binarization scheme, showing how the solution quality improves over iterations. These graphs complement the tabular results by visualizing convergence speed and stability, with steeper initial slopes indicating faster convergence and flatter regions toward the end, suggesting solution stabilization.

7.3. Statistical Analysis

To evaluate the statistical significance of performance differences between WSO configurations, we employ non-parametric statistical testing following established practices in metaheuristic evaluation. Given that our experimental data originates from algorithmic executions and does not follow the normal distribution typical of natural processes, we select the Wilcoxon–Mann–Whitney U-test for pairwise comparisons between algorithm configurations. This test is appropriate for independent samples and helps identify significant differences between algorithm configurations.

The statistical analysis is formulated with the following hypotheses for each pairwise comparison:

$$H_0:M_{ConfigA}\ge M_{ConfigB}$$

(25)

$$H_1:M_{ConfigA}<M_{ConfigB}$$

(26)

where $M_{ConfigA}$ and $M_{ConfigB}$ represent the median performance of configurations A and B, respectively. We implement the Wilcoxon–Mann–Whitney test using Python’s scipy.stats.mannwhitneyu function with the ‘alternative’ parameter set to ‘less’. A p-value threshold of 0.05 is established for statistical significance, where p < 0.05 indicates that the first configuration (row) significantly outperforms the second configuration (column).

Table 11. Statistical comparison (p-values) between WSO configurations across all USCP instances using Wilcoxon–Mann–Whitney test.

	ELIT	ELIT_CIRCLE	ELIT_SINGER	ELIT_SINU	ELIT_TENT
ELIT	X	0.6	0.337	0.429	0.823
ELIT_CIRCLE	0.401	X	0.401	0.4	0.457
ELIT_SINGER	0.666	0.599	X	0.511	0.771
ELIT_SINU	0.572	0.6	0.691	X	0.599
ELIT_TENT	0.181	0.544	0.23	0.401	X

presents the comprehensive statistical comparison results across all USCP instances.

These statistical findings indicate that while empirical differences in performance are observed across different chaotic map configurations, the variability inherent in metaheuristic algorithms may be masking potential systematic improvements. The results suggest that configuration selection should be based on empirical performance patterns rather than statistically proven superiority, and that larger sample sizes or longer experimental periods may be necessary to detect statistically significant differences.

The statistical analysis reveals that none of the pairwise comparisons between WSO configurations achieves statistical significance (p < 0.05). All p-values in Table 11 exceed the 0.05 threshold, indicating that the observed performance differences between configurations are not statistically significant. The lowest p-value obtained is 0.181 (ELIT_TENT vs. ELIT), which still falls well above the significance threshold.

Despite the practical differences observed in RPD values and convergence behavior across different problem instances, the statistical tests suggest that these variations may be within the range of expected algorithmic variability rather than representing fundamental performance advantages. This finding indicates that while certain configurations may show empirical improvements for specific problem types (such as ELIT_CIRCLE for structured instances and ELIT_SINU for cyclic instances), these differences cannot be statistically validated with the current experimental setup.

The lack of statistical significance suggests several possibilities: (1) the sample size of 31 independent runs may be insufficient to detect the observed differences, (2) the high variability in metaheuristic performance may mask subtle but consistent improvements, or (3) the actual performance differences between chaotic map configurations are smaller than initially hypothesized. These results emphasize the importance of both empirical observation and statistical validation in metaheuristic evaluation, highlighting the need for larger sample sizes or alternative statistical approaches to conclusively establish the superiority of specific configurations.

7.4. Analysis of Diversity Graphs

This section presents a detailed analysis of the diversity graphs for the different problem instances. To quantify population diversity, we employ the Hussain Diversity Index (HDI), which measures the dispersion of solutions in the search space. The HDI is calculated as

$$D_{Hussain}={\displaystyle \frac{1}{n·l}}\sum _{d=1}^l\sum _{i=1}^n|M_d-P_{i,d}|$$

(27)

where:

• n is the population size (number of sharks).
• l is the problem dimension (number of binary variables).
• $M_d$ is the median value of the population in dimension d.
• $P_{i,d}$ is the value of the i-th shark in dimension d.

This metric produces values in the range $[0,R]$, where 0 indicates minimum diversity (all solutions identical) and higher values indicate greater diversity. For binary problems, values near 0.5 represent maximum diversity, occurring when the population is evenly split between 0 s and 1 s across dimensions.

The binarization schemes evaluated are Elit, Elit_Circle, Elit_Singer, Elit_Sinu, and Elit_Tent. The behavior of each instance is described below, supported by the convergence curves shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15.

• Instance clr10 (Figure 10):

  –

  The 5 binarization schemes converge from diversity values of 0.5 to 0.2.

  –

  The Elit scheme is the most stable, without pronounced peaks.

  –

  The Elit_Circle scheme is also stable but exhibits slight peaks.

  –

  The Elit_Singer scheme shows peaks toward values below 0.1.

  –

  The Elit_Sinu scheme exhibits many peaks between 0.2 and 0.1, displaying erratic behavior.

  –

  The Elit_Tent scheme is stable, similar to the Elit scheme.

  In this instance, the Elit and Elit_Tent schemes are the most consistent.
• Instance clr11 (Figure 11):

  –

  The schemes converge from 0.2 to 0, indicating a rapid reduction in diversity.

  –

  The Elit scheme shows several peaks toward 0.

  –

  The Elit_Circle scheme remains close to 0 with peaks toward 0.1.

  –

  The Elit_Singer scheme varies between 0.1 and 0.

  –

  The Elit_Sinu scheme also varies between 0.1 and 0.

  –

  The Elit_Tent scheme exhibits behavior similar to the others.

  In this instance, all schemes tend to converge quickly, with Elit_Circle and Elit_Tent showing a slightly better balance.
• Instance clr12 (Figure 12):

  –

  All schemes converge from 0.1 to 0, showing very similar behaviors.

  This instance does not present significant differences between the schemes, suggesting that the problem is less sensitive to the choice of binarization scheme.
• Instance clr13 (Figure 13):

  –

  Similar to clr12, all schemes converge from 0.1 to 0.

  As in clr12, there are no notable differences between the schemes.
• Instance cyc06 (Figure 14):

  –

  The schemes are divided into two groups:

  *

  Elit, Elit_Tent, and Elit_Circle: Vary between 0.1 and 0, showing stable behavior.

  *

  Elit_Singer and Elit_Sinu: Vary between 0.4 and 0.2, indicating greater exploration.

  The Elit, Elit_Tent, and Elit_Circle schemes are more efficient in this instance.
• Instance cyc07 (Figure 15):

  –

  Similar to cyc06 but with some differences:

  *

  Elit, Elit_Tent, and Elit_Circle: Vary between 0.1 and 0.

  *

  Elit_Singer: Varies between 0.4 and 0.3.

  *

  Elit_Sinu: Varies between 0.3 and 0.2.

  Again, the Elit, Elit_Tent, and Elit_Circle schemes are more efficient.

7.5. Exploration vs. Exploitation

Within metaheuristics, a key challenge is achieving the right balance between exploration and exploitation throughout the search process. Exploration enables the algorithm to probe new regions of the search space, whereas exploitation hones in on enhancing solutions in areas already deemed promising. If these two elements are not properly balanced, the algorithm may either become stuck in local optima (excessive exploitation) or struggle to converge within a feasible timeframe (excessive exploration).

To quantify the exploration–exploitation balance during the optimization process, we calculate the exploration percentage (XPL) and exploitation percentage (XPT) using the population diversity measure:

$$\mathrm{XPL}=\left({\displaystyle \frac{div}{maxDiv}}\right)·100$$

(28)

$$\mathrm{XPT}=\left({\displaystyle \frac{|div-maxDiv|}{maxDiv}}\right)·100$$

(29)

where $div$ is the current diversity value (calculated using the Hussain Diversity Index), and $maxDiv$ is the maximum diversity observed throughout the optimization process. These percentages help determine the algorithm’s state:

• If XPL ≥ XPT, the algorithm is in an exploration state (searching new regions).
• If XPL < XPT, the algorithm is in an exploitation state (refining known solutions).

This quantitative approach allows us to analyze how different binarization schemes balance these competing objectives across different problem instances.

During the experimentation with the five binarization schemes (Elit, Elit_Circle, Elit_Singer, Elit_Sinu, and Elit_Tent), it is observed that the balance between exploration and exploitation varies significantly across the six instances of the USCP. The behavior of each scheme is described below:

• Instance clr10:

  –

  All schemes (Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20) exhibit balanced behavior between exploration and exploitation. However, the schemes in Figure 18 Elit_Singer and Figure 19 Elit_Sinu show a tendency toward higher exploitation, which may explain their slightly higher RPD values compared to the others.

• Instance clr11:

  –

  The schemes (Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25) generally lean toward higher exploitation, which is consistent with the rapid convergence observed in the diversity graphs. The schemes in Figure 23 Elit_Singer and Figure 24 Elit_Sinu display more variability, suggesting a stronger exploration component in their search process.

• Instance clr12:

  –

  Similar to clr11, the schemes (Figure 26, Figure 27, Figure 28, Figure 29 and Figure 30) exhibit high exploitation. However, the scheme in Figure 27 Elit_Circle stands out for its lower variability, indicating a more stable balance between exploration and exploitation.

• Instance clr13:

  –

  All schemes (Figure 31, Figure 32, Figure 33, Figure 34 and Figure 35) behave very similarly, with high exploitation and low variability. This suggests that the instance is less sensitive to the choice of binarization scheme, as all of them achieve comparable results.

• Instance cyc06:

  –

  The schemes (Figure 36, Figure 37, Figure 38, Figure 39 and Figure 40) show high exploitation with low variability, except for Figure 38 Elit_Singer and Figure 39 Elit_Sinu, which maintain a better balance between exploration and exploitation but with higher variability. This behavior aligns with their lower RPD and Std-dev values, indicating better performance in this smaller instance.

• Instance cyc07:

  –

  Similar to cyc06, the schemes (Figure 41, Figure 42, Figure 43, Figure 44 and Figure 45) exhibit high exploitation with low variability. Again, Figure 43 Elit_Singer and Figure 44 Elit_Sinu demonstrate a more balanced approach, with higher variability but also better RPD and Std-dev values, highlighting their effectiveness in smaller instances.

It is worth noting that the cyc06 and cyc07 instances, which are smaller in size compared to the clr instances, consistently achieve better RPD and Std-dev values across all schemes. This suggests that the smaller problem size allows for more efficient convergence and lower variability in the results. Among the schemes, Elit_Singer and Elit_Sinu stand out in these instances for their ability to maintain a good balance between exploration and exploitation, which contributed to their superior performance.

In contrast, the larger clr instances require a more stable and exploitation-focused approach as demonstrated by the Elit_Circle scheme, which consistently achieves the best balance and performance in these cases. This highlights the importance of selecting the appropriate binarization scheme based on the problem size and complexity to ensure optimal results.

8. Conclusions

This study evaluates the performance of the White Shark Optimization (WSO) algorithm across six diverse instances of the USCP, focusing on the impact of different binarization schemes and configurations. The results, summarized in Table 7, Table 8 and Table 9, demonstrate that the effectiveness of WSO is highly dependent on both the problem structure and the choice of binarization method.

Our initial findings suggest promising patterns regarding algorithm configuration selection. For structured instances (clr10, clr11, clr12, and clr13), configurations employing the ELIT_CIRCLE scheme show strong performance, with notably low RPD values (e.g., 0.7% for clr11 and 3.87% for clr10). Similarly, for cyclic instances (cyc06 and cyc07), the ELIT_SINU configuration demonstrates superior results, achieving optimality in cyc06 (RPD = 0%) and near-optimality in cyc07 (RPD = 0.96%).

The convergence curves (Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9) further indicate that the WSO performance correlates with its initialization and update rules. Configurations like ELIT_CIRCLE exhibit faster convergence in structured instances, while ELIT_SINU maintains more consistent progress in cyclic ones.

These preliminary observations suggest several promising directions for future work. A more exhaustive analysis is needed to fully understand why circle-based binarization appears more effective for structured problems with dense constraint matrices, while sinusoidal binarization seems better suited for cyclic problem structures. Further research should explore the mathematical properties of these chaotic maps in relation to specific problem characteristics, potentially developing theoretical frameworks for optimal configuration selection based on the problem topology. Additionally, comprehensive testing on a broader range of instances would help validate whether these patterns hold consistently across different problem scales and densities.

A key limitation of our current study is the absence of comprehensive comparisons with other state-of-the-art metaheuristic algorithms for the USCP. Our work focuses on establishing the effectiveness of different chaotic map configurations within the WSO framework, which represents a foundational step that should be followed by extensive comparative studies with established algorithms.

In summary, WSO demonstrates potential as a versatile metaheuristic for the USCP, but further investigation is required to establish robust guidelines for configuration selection across different problem classes. Future work should focus on developing adaptive parameter selection mechanisms that can automatically determine the most suitable binarization scheme based on problem characteristics, allowing for more effective deployment in real-world combinatorial optimization scenarios.