Perpendicular Bisector Optimization Algorithm (PBOA): A Novel Geometric-Mathematics-Inspired Metaheuristic Algorithm for Controller Parameter Optimization

Abstract

To address the inadequate balance between exploration and exploitation of existing algorithms in complex solution spaces, this paper proposes a novel mathematical metaheuristic optimization algorithm—the Perpendicular Bisector Optimization Algorithm (PBOA). Inspired by the geometric symmetry of perpendicular bisectors (the endpoints of a line segment are symmetric about them), the algorithm designs differentiated convergence strategies. In the exploration phase, a slow convergence strategy is adopted (deliberately steering particles away from the optimal region defined by the perpendicular bisector) to expand the search space; in the exploitation phase, fast convergence refines the search process and improves accuracy. It selects 4 particles to construct line segments and perpendicular bisectors with the current particle, enhancing global exploration capability. The experimental results on 27 benchmark functions, compared with 15 state-of-the-art algorithms, show that the PBOA outperforms others in accuracy, stability, and efficiency. When applied to 5 engineering design problems, its fitness values are significantly lower. For H-type motion platforms, the PBOA-optimized platform not only achieves high unidirectional motion accuracy, but also the average synchronization error of the two Y-direction motion mechanisms reaches ±2.6 × 10⁻⁵ mm, with stable anti-interference performance.

1. Introduction

Metaheuristic algorithms are algorithmic frameworks designed to address complex optimization problems that often surpass the capabilities of traditional methods. In large-scale, highly complex, and strongly nonlinear search spaces, these algorithms offer distinct advantages—conventional optimization techniques frequently struggle due to prohibitive computational complexity [1]. As high-level strategies, metaheuristics guide heuristic searches to efficiently explore and exploit solution landscapes, aiming to identify global optimal or near-optimal solutions within feasible timeframes. Most draw inspiration from natural processes, providing robust mechanisms for navigating vast and intricate solution spaces [2].

Metaheuristic techniques span a wide range of approaches, from those inspired by natural phenomena and psychological processes to human-engineered systems [3]. Prominent categories include evolutionary algorithms (EAs) [4]—such as differential evolution (DE) [5]—swarm intelligence (SI) [6] (e.g., particle swarm optimization (PSO) [7] and ant colony optimization (ACO) [8]), and physics-based methods like the gravitational search algorithm (GSA) [9]. All operate on iterative refinement: a set of candidate solutions converges toward optimality through processes such as selection, crossover, mutation, movement, and position updates [10].

These techniques find applications across diverse domains, including engineering design optimization [3], image segmentation, path planning, task scheduling, machine learning, and network design [11,12,13,14,15].

However, metaheuristics face inherent challenges: risks of early convergence, the need to balance exploration and exploitation, and demands for parameter tuning. Additionally, the “No Free Lunch” theorem [16] establishes that no single algorithm universally outperforms others across all problems. Thus, selecting an appropriate method depends on the problem’s specific nature and requirements, emphasizing the need for new algorithms to tackle diverse optimization challenges in engineering and related fields [17].

This paper aims to propose a novel mathematics-inspired optimization algorithm—the Perpendicular Bisector Optimization Algorithm (PBOA). A perpendicular bisector is defined as a line perpendicular to a line segment and passing through its midpoint. Its core property—that any point on the bisector is equidistant from the two endpoints of the segment—is a fundamental conclusion in Euclidean geometry, provable via the theory of congruent triangles. This property has been validated over time: Euclid established its foundational role in Elements [18], and it remains a key theoretical reference in modern geometric research [19]. In the framework of swarm intelligence optimization, algorithms update individual positions through correlations between population members. The perpendicular bisector, by partitioning the search space and defining distance relationships between individual positions and potential optimal solutions, provides geometric constraints and directional guidance for particle position updates, forming the algorithm’s theoretical basis. Built on this principle, the PBOA is designed to tackle increasingly complex optimization challenges.

The main contributions of this paper are summarized as follows:

• Propose a novel metaheuristic optimization algorithm—the Perpendicular Bisector Optimization Algorithm (PBOA), which fully leverages the properties of perpendicular bisectors in geometric principles, offering a new perspective for metaheuristic optimization.
• Evaluate the optimization performance of the PBOA using 27 benchmark functions and validate the reliability of the results via detailed visual analytics.
• Verify the application potential of the PBOA through 3 engineering design problems, demonstrating its effectiveness in practical engineering scenarios.
• Apply the PBOA to the parameter optimization of the time-varying non-singular fast terminal sliding mode controller for H-type motion platforms, with the experimental results showing that it exhibits significant advantages in multi-parameter controller optimization.

After the Introduction, Section 2 will review related work on metaheuristic algorithms; Section 3 will introduce the proposed PBOA and its mathematical modeling; Section 4 will describe the experimental setup; Section 5 will conduct a comprehensive simulation and performance analysis of the PBOA on 27 benchmark functions; Section 6 will examine the PBOA’s effectiveness in 5 practical engineering design problems and the optimization of the time-varying non-singular fast terminal sliding mode controller for H-type motion platforms; Section 7 will summarize the key findings and suggest future research directions.

2. Related Work

Optimization algorithms are pivotal in computational science, serving as efficient tools to solve complex problems across diverse fields. A large number of these algorithms take inspiration from natural and social phenomena, converting biological, physical, chemical, and behavioral principles into practical computational strategies [3,10,20]. Their diversity stems from a wide array of sources, such as animal behaviors, plant interactions, and physical processes [3]. This extensive foundation facilitates the development of algorithms that are not only effective but also robust, adapting well to complex high-dimensional optimization tasks. Each algorithm is designed to simulate specific natural mechanisms, providing unique strategies to find efficient solutions in multi-dimensional and dynamic optimization scenarios [10]. By modeling natural phenomena, these algorithms optimize functions through exploration, exploitation, and evolution, embodying the core mechanisms observed in natural systems [3]. Correspondingly, Table 1 [20] offers a detailed classification of state-of-the-art optimization algorithms, categorizing them by their primary inspiration and core methodologies.

Optimization via swarm intelligence algorithms is achieved through the collective intelligence and decentralized decision-making of social animals. Consider Particle Swarm Optimization (PSO) [21], for instance: drawing on the foraging habits of bird flocks and fish schools, it maintains a population of particles (candidate solutions) that navigate the search space. These particles adjust their velocities and positions under the guidance of both individual and collective top-performing solutions, with the swarm moving toward high-quality regions of the search space over iterations. In a similar manner, Ant Colony Optimization (ACO) [22] strengthens promising paths through pheromone deposition and evaporation, balancing the exploitation of known routes with the exploration of new options. The variety of swarming behaviors applicable to optimization tasks is further highlighted by other swarm-based methods like Artificial Bee Colony (ABC) [23], Ant Lion Optimizer (ALO) [24], and Jellyfish Search (JS) [25].

Drawing on mammalian behavior, certain metaheuristic algorithms integrate foraging tactics, social hierarchies, or alertness mechanisms. For example, the Gray Wolf Optimizer (GWO) [16] imitates the hierarchical behavior (alpha, beta, delta, omega) of wolf packs; the Cheetah Optimizer (CO) [26] reproduces how cheetahs pursue prey rapidly and shift direction abruptly; the Meerkat Optimization Algorithm (MOA) [27] is based on meerkats’ coordinated vigilance; and the Harris Hawks Optimizer (HHO) [28] models hawks’ sudden assault strategies. By incorporating these natural behaviors, search diversity is boosted, and solutions are steered toward high-quality regions.

Inspired by physical and chemical phenomena, another major class of optimization algorithms functions differently. Take Simulated Annealing (SA) [29]—rooted in metallurgical annealing—as an example: it uses a gradually declining temperature parameter to temporarily accept inferior solutions, helping escape local optima. In a similar way, the Gravitational Search Algorithm (GSA) [30] treats candidate solutions as masses under gravitational pull; fitter (heavier) solutions exert stronger attraction, steering the population toward optimal solutions. The Multi-Verse Optimizer (MVO) [31] simulates inflation rates across parallel universes, and the Black Hole (BH) Algorithm [32] depicts how solutions converge into a dominant gravitational well.

Within the chemical and biochemical domain, Atom Search Optimization (ASO) [33], Chemical Reaction Optimization (CRO) [34], and Nuclear Reaction Optimization (NRO) [35] make use of molecular interactions, reaction dynamics, and fission or fusion processes to navigate the search space toward configurations with more favorable energy levels.

Biological evolution serves as the source of operational mechanisms for evolutionary algorithms (EAs). Genetic Algorithms (GAs) [36] employ selection, crossover, and mutation operators to iteratively refine a solution population. GAs maintain diversity and reduce the risk of premature convergence by favoring fitter solutions and introducing controlled randomness. As generations progress, the population slowly approaches global or near-global optima. Differential Evolution (DE) [5] improves solutions by applying scaled differences between existing candidates, and Genetic Programming (GP) [37] extends these principles to the evolution of entire programs, thereby demonstrating the adaptability and versatility of evolution-based optimizers.

Another category of optimization algorithms, mathematical and numerical methods, relies on pure mathematical principles instead of biological or social analogies. Take the Arithmetic Optimization Algorithm (AOA) [38] as an example: it uses arithmetic operators—addition, subtraction, multiplication, and division—in a controlled way. Starting with extensive exploration, it gradually narrows its focus to promising regions, with an adjustable parameter regulating how intense arithmetic operations are over time. In a similar vein, Chaos Game Optimization (CGO) [39] and the Sine Cosine Algorithm (SCA) [40] each use fractal geometry and trigonometric models to systematically balance exploration and exploitation within the search space.

Drawing inspiration from natural cycles and environmental dynamics, environmental and ecologically inspired approaches enhance the search process. The Snow Ablation Optimizer (SAO) [41] simulates how snow melts, while the Water Cycle Algorithm (WCA) [42] models the natural water cycle, which includes evaporation, precipitation, and flow. These algorithms achieve a dynamic balance between exploration and exploitation by replicating ecological cycles.

Optimization algorithms have also found inspiration in human social behavior. The Chef-Based Optimization Algorithm (CBOA) [43] imitates the decision-making strategies chefs use when creating and refining recipes; Cultural Algorithms (CA) [44] combine individual adaptation with a shared belief space to facilitate collective learning.

Within non-biological inspiration domains, optimizers inspired by games—such as the Dart Game Optimizer (DGO) [45], Puzzle Optimization Algorithm (POA) [46], and Squid Game Optimizer (SGO) [47]—derive their strategic decision-making from entertainment and popular culture. Meanwhile, algorithms rooted in economics, including Supply-Demand Optimization (SDO) [48] and Search and Rescue Optimizer (SARO) [49], leverage market equilibrium dynamics and systematic rescue strategies to guide convergence toward optimal solutions.

Table 1 illustrates that optimization algorithms exhibit diversity across multiple domains, from swarm intelligence and mammalian hunting strategies to physical-chemical analogies. This diversity showcases the flexibility and adaptability of metaheuristic algorithms when addressing complex optimization challenges.

That said, existing metaheuristic and evolutionary algorithms come with notable limitations: premature convergence, slow adaptation to dynamic environments, and strict parameter tuning requirements [3,50]. For traditional evolutionary methods, computational costs rise exponentially as problem scale increases, making them inefficient in large-scale search spaces. Additionally, gradient-dependent algorithms face difficulties with highly nonlinear or discontinuous problems, which emphasizes the need for flexible gradient-free techniques [3].

The performance of metaheuristic algorithms in high-dimensional, complex optimization tasks is significantly restricted by these challenges. A core issue among them is premature convergence: in multimodal landscapes, algorithms find it hard to maintain solution diversity [51], which often results in convergence to local optima and limits their capacity to identify global solutions [52]. Slow adaptation to dynamic environments is another major challenge; many algorithms depend on fixed exploration-exploitation mechanisms [53], and such rigidity lowers their efficiency in responding to time-varying objectives and constraints, thus weakening their effectiveness in practical applications [54].

Table 1. Detailed Classification of Mainstream Optimization Algorithms.

Category	Algorithm	Year	Authors	Brief Description	Ref.
Swarm intelligence	Particle swarm optimization (PSO)	1995	Kennedy, Eberhart	Models swarm behavior of birds; particles adjust velocity and position based on solutions; global best solution is updated	[21]
	Ant colony optimization (ACO)	2006	Dorigo, Stützle	Simulation foraging behavior of ants; uses pheromone trails for promoting paths	[22]
	Artificial bee colony (ABC)	2005	Karaboga, Basturk	Mimics bee foraging patterns in a graph; employed, onlooker, and scout bees for exploitation of nectar sources	[23]
	Whale optimization algorithm (WOA)	2016	Mirjalili, Lewis	Simulates “bubble-net attack” hunting strategy of humpback whales; updating surround prey: spiral updating mechanism and encircling prey	[55]
	Bacteria phototaxis optimizer (BPO)	2023	Pan, Teng, Li, Zhan	Inspired by bacterial phototaxis; uses phototaxis, aerotaxis, and chemotaxis mechanisms for exploration and exploitation	[56]
Mammalian behavior	Gray wolf optimizer (GWO)	2014	Mirjalili, et al.	Mimics gray wolves’ strategy in nature (grey wolf hierarchy: alpha, beta, omega); three main steps: tracking, encircling, and attacking prey	[16]
	Harris hawks optimizer (HHO)	2019	Heidari, et al.	Utilizes covert attack strategies of Harris’ hawks to enhance exploration and exploitation	[28]
	Cheetah optimizer (CO)	2022	Akbar, Deb	Inspired by cheetahs’ chasing speed; it features rapid global search ability and efficient exploitation near optimal solutions	[26]
Mammalian behavior	Meerkat optimization algorithm (MOA)	2023	Xian et al.	Mimics meerkats’ alertness; sentry meerkat for exploration; forager meerkats for solution search	[27]
Physical phenomena	Simulated annealing (SA)	1983	Kirkpatrick et al.	Based on the thermal annealing process in metallurgy; its cooling schedule enables escape from local optima to find better solutions	[29]
	Gravitational search algorithm (GSA)	2009	Rashedi et al.	Considers candidate solutions as masses; gravitational force draws solutions toward fitter agents	[30]
	Multi-verse optimizer (MVO)	2016	Mirjalili, et al.	Inspired by multi-verse theory; explores universe with varied inflation rates	[31]
	Black hole algorithm (BHA)	2013	Hammoudeh, et al.	Simulates stars in a black hole attracting stars; solutions converge by “falling” into the black hole	[32]
	Elastic deformation optimization algorithm (EDOA)	2022	Pan, Tang, et al.	Based on Hooke’s law and Newton’s theory; uses elastic deformation for exploration and exploitation	[57]
Chemical biochemical	Atom search optimization (ASO)	2019	Zhao et al.	Simulates integrative interaction forces; solutions find information exchange interactions and reactive relations	[33]
	Chemical reaction optimization (CRO)	2009	Lam, Li	Mimics molecular reactions; uses collision, decomposition, synthesis, and interchange to seek stable, lower-energy solutions	[34]
	Nuclear reaction optimization (NRO)	2019	Wei et al.	Based on nuclear fission/fusion; explores solution space via collision and entropy search	[35]
Evolutionary algorithms	Differential evolution (DE)	1997	Storn, Price	Creates new candidates by adding scaled differences to existing ones	[5]
	Genetic algorithm (GA)	1975	Holland	Employs survival-of-the-fittest principles; along with crossover and mutation operators, to evolve a population toward optimal solutions	[36]
	Genetic programming (GP)	1998	Banzhaf et al.	Evolves computer programs; represented as trees; genetic operators evolve a population	[37]
Mathematical models	Arithmetic optimization algorithm (AOA)	2021	Abualigah et al.	Employs arithmetic operations; transitions from broad exploration to exploitation	[38]
	Chaos game optimization (CGO)	2021	Talatnahr, et al.	Uses chaos fractals; random walks and greedy strategies support both exploration and exploitation	[39]
	Sine cosine algorithm (SCA)	2016	Mirjalili	Updates solutions via sine/cosine functions; balances global and local search	[40]
	Expectation-based weighted hypergraph in optimization (EBHO)	2024	Pan, Wang et al.	Incorporates the adaptive hypergraph model in information dissemination; balances global exploration and local exploitation	[58]
Environmental ecological	Snow ablation optimization (SAO)	2023	Deng et al.	Simulates snow melting process; iterative ablation reveals promising regions	[41]
	Water cycle algorithm (WCA)	2012	Eskandar et al.	Mimics the water cycle processes; streams merge into rivers and lakes at varying rates	[42]
Human behavior/Social dynamics	Chaotic-based optimization algorithm (CBOA)	2022	Tranojevska et al.	Refines solutions via iterative chaotic maps; balances exploration and exploitation	[43]
	Cultural algorithm (CA)	1994	Reynolds	Inspired by cultural evolution; integrates continuous adaptation with a shared belief space for collective learning	[44]
Game—based algorithms	Darts game optimizer (DGO)	2020	Dehghani et al.	Inspired by dart throwing; solutions adapt by aiming closer at target optima iteratively	[45]
	Puzzle optimization algorithm (POA)	2022	Ahmadi et al.	Analogous to solving a puzzle; piecewise adjustments gradually assemble coherent optimal structures.	[46]
Economic theory	Supply-demand—based optimization (SDO)	2019	Zhao et al.	Mimic market equilibria; supply-demand interactions converge on best solutions over time.	[48]
Economic theory	Search and rescue optimization (SAR)	2020	Shabani et al.	Models rescue missions; systematic hunting processes locate high-fitness solutions (survivors)	[49]

Table 1. Detailed Classification of Mainstream Optimization Algorithms.

Category	Algorithm	Year	Authors	Brief Description	Ref.
Swarm intelligence	Particle swarm optimization (PSO)	1995	Kennedy, Eberhart	Models swarm behavior of birds; particles adjust velocity and position based on solutions; global best solution is updated	[21]
	Ant colony optimization (ACO)	2006	Dorigo, Stützle	Simulation foraging behavior of ants; uses pheromone trails for promoting paths	[22]
	Artificial bee colony (ABC)	2005	Karaboga, Basturk	Mimics bee foraging patterns in a graph; employed, onlooker, and scout bees for exploitation of nectar sources	[23]
	Whale optimization algorithm (WOA)	2016	Mirjalili, Lewis	Simulates “bubble-net attack” hunting strategy of humpback whales; updating surround prey: spiral updating mechanism and encircling prey	[55]
	Bacteria phototaxis optimizer (BPO)	2023	Pan, Teng, Li, Zhan	Inspired by bacterial phototaxis; uses phototaxis, aerotaxis, and chemotaxis mechanisms for exploration and exploitation	[56]
Mammalian behavior	Gray wolf optimizer (GWO)	2014	Mirjalili, et al.	Mimics gray wolves’ strategy in nature (grey wolf hierarchy: alpha, beta, omega); three main steps: tracking, encircling, and attacking prey	[16]
	Harris hawks optimizer (HHO)	2019	Heidari, et al.	Utilizes covert attack strategies of Harris’ hawks to enhance exploration and exploitation	[28]
	Cheetah optimizer (CO)	2022	Akbar, Deb	Inspired by cheetahs’ chasing speed; it features rapid global search ability and efficient exploitation near optimal solutions	[26]
Mammalian behavior	Meerkat optimization algorithm (MOA)	2023	Xian et al.	Mimics meerkats’ alertness; sentry meerkat for exploration; forager meerkats for solution search	[27]
Physical phenomena	Simulated annealing (SA)	1983	Kirkpatrick et al.	Based on the thermal annealing process in metallurgy; its cooling schedule enables escape from local optima to find better solutions	[29]
	Gravitational search algorithm (GSA)	2009	Rashedi et al.	Considers candidate solutions as masses; gravitational force draws solutions toward fitter agents	[30]
	Multi-verse optimizer (MVO)	2016	Mirjalili, et al.	Inspired by multi-verse theory; explores universe with varied inflation rates	[31]
	Black hole algorithm (BHA)	2013	Hammoudeh, et al.	Simulates stars in a black hole attracting stars; solutions converge by “falling” into the black hole	[32]
	Elastic deformation optimization algorithm (EDOA)	2022	Pan, Tang, et al.	Based on Hooke’s law and Newton’s theory; uses elastic deformation for exploration and exploitation	[57]
Chemical biochemical	Atom search optimization (ASO)	2019	Zhao et al.	Simulates integrative interaction forces; solutions find information exchange interactions and reactive relations	[33]
	Chemical reaction optimization (CRO)	2009	Lam, Li	Mimics molecular reactions; uses collision, decomposition, synthesis, and interchange to seek stable, lower-energy solutions	[34]
	Nuclear reaction optimization (NRO)	2019	Wei et al.	Based on nuclear fission/fusion; explores solution space via collision and entropy search	[35]
Evolutionary algorithms	Differential evolution (DE)	1997	Storn, Price	Creates new candidates by adding scaled differences to existing ones	[5]
	Genetic algorithm (GA)	1975	Holland	Employs survival-of-the-fittest principles; along with crossover and mutation operators, to evolve a population toward optimal solutions	[36]
	Genetic programming (GP)	1998	Banzhaf et al.	Evolves computer programs; represented as trees; genetic operators evolve a population	[37]
Mathematical models	Arithmetic optimization algorithm (AOA)	2021	Abualigah et al.	Employs arithmetic operations; transitions from broad exploration to exploitation	[38]
	Chaos game optimization (CGO)	2021	Talatnahr, et al.	Uses chaos fractals; random walks and greedy strategies support both exploration and exploitation	[39]
	Sine cosine algorithm (SCA)	2016	Mirjalili	Updates solutions via sine/cosine functions; balances global and local search	[40]
	Expectation-based weighted hypergraph in optimization (EBHO)	2024	Pan, Wang et al.	Incorporates the adaptive hypergraph model in information dissemination; balances global exploration and local exploitation	[58]
Environmental ecological	Snow ablation optimization (SAO)	2023	Deng et al.	Simulates snow melting process; iterative ablation reveals promising regions	[41]
	Water cycle algorithm (WCA)	2012	Eskandar et al.	Mimics the water cycle processes; streams merge into rivers and lakes at varying rates	[42]
Human behavior/Social dynamics	Chaotic-based optimization algorithm (CBOA)	2022	Tranojevska et al.	Refines solutions via iterative chaotic maps; balances exploration and exploitation	[43]
	Cultural algorithm (CA)	1994	Reynolds	Inspired by cultural evolution; integrates continuous adaptation with a shared belief space for collective learning	[44]
Game—based algorithms	Darts game optimizer (DGO)	2020	Dehghani et al.	Inspired by dart throwing; solutions adapt by aiming closer at target optima iteratively	[45]
	Puzzle optimization algorithm (POA)	2022	Ahmadi et al.	Analogous to solving a puzzle; piecewise adjustments gradually assemble coherent optimal structures.	[46]
Economic theory	Supply-demand—based optimization (SDO)	2019	Zhao et al.	Mimic market equilibria; supply-demand interactions converge on best solutions over time.	[48]
Economic theory	Search and rescue optimization (SAR)	2020	Shabani et al.	Models rescue missions; systematic hunting processes locate high-fitness solutions (survivors)	[49]

Furthermore, the wide applicability of algorithms is hindered by stringent parameter tuning. Parameters like mutation rate, crossover probability, and weight scaling usually require manual tuning [54], a time-consuming process that impairs the algorithm’s generalizability and practical usability. Meanwhile, as the problem dimension grows, the “curse of dimensionality” causes computational costs to rise exponentially: traditional evolutionary methods demand more iterations and evaluations, which reduces their scalability for large-scale problems [51]. Metaheuristic algorithms fare poorly with highly nonlinear, discontinuous, or noisy objective functions, and gradient-based methods also struggle to produce reliable solutions in these scenarios. This further underscores the need for robust gradient-free techniques when dealing with complex real-world problems [53].

Moreover, these algorithms show weak robustness in multimodal optimization: when multiple local optima are present [53], most fail to explore and exploit multiple search regions simultaneously, resulting in suboptimal solutions when the fitness landscape has significant irregularities. Scalability remains a persistent issue: many algorithms are designed for small-scale problems, yet their performance drops sharply as problem complexity increases [52]. Finally, heavy reliance on specific problems limits the universality of metaheuristic algorithms [51]; extensive customization for different problem domains is required for most algorithms, which reduces their versatility in addressing a wide range of optimization challenges.

To tackle the challenges mentioned above, the Perpendicular Bisector Optimization Algorithm (PBOA) introduces a search space expansion mechanism, which effectively lowers the risk of premature convergence. By selecting 4 different particles and constructing line segments with the current particle to generate perpendicular bisectors, the algorithm simulates a geometry-driven search expansion process—this significantly enhances its exploration and exploitation capabilities. Core features, such as the equidistant property of perpendicular bisectors and differentiated convergence strategies (slow convergence in the exploration phase to expand the search range, and fast convergence in the exploitation phase for refined search), adapt to the search needs of complex solution spaces, boosting adaptability in diverse optimization scenarios. Integrating deterministic line segment construction and dynamic convergence control mechanisms, the PBOA extends the performance boundaries of mathematical heuristic algorithms. These mechanisms maintain solution diversity while ensuring convergence efficiency, making the PBOA particularly effective in multimodal and perturbed problem spaces.

A key strength of the PBOA is its deterministic theoretical foundation, a characteristic that equips the algorithm with robust exploitation capability and allows it to focus most computational resources on the search process. Thanks to this resource allocation mechanism, the PBOA exhibits superior performance and robustness in the majority of optimization problems.

By leveraging the geometric properties of perpendicular bisectors to their full extent, the PBOA successfully addresses the core limitations of existing metaheuristic methods, providing a reliable solution approach for complex optimization problems. Subsequent sections will elaborate on the PBOA’s mathematical model and its specific applications in engineering design problems.

3. Perpendicular Bisector Optimization Algorithm

This section will elaborate on the definition and properties of perpendicular bisectors, as well as the PBOA’s implementation details, algorithm pseudocode, and flowchart.

3.1. Definition and Properties of Perpendicular Bisectors

The perpendicular bisector is one of the core concepts in elementary geometry. In a two-dimensional coordinate system (as shown in Figure 1), connecting point A and point B forms line segment AB with A and B as endpoints. By definition, a perpendicular bisector is a straight line that passes through the midpoint of a line segment and is perpendicular to that segment; thus, the perpendicular bisector of segment AB can be denoted as line L. The perpendicular bisector has two basic properties:

• It is perpendicular to the line segment and bisects it, with the two parts of the segment on either side of the perpendicular bisector being completely symmetric;
• Any point on the line is equidistant from the two endpoints of the segment.

3.2. Inspiration

Intelligent optimization algorithms update individual attributes (such as the position of “particles” in Particle Swarm Optimization (PSO), “genes” in Genetic Algorithms (GA), etc.) by leveraging the interaction between individuals in the population.

In Figure 2, based on Figure 1, individual attribute constraints are added, and all points can only exist within the yellow box (Constrained Region) in Figure 2. Let P be the optimal solution of this two-dimensional constrained optimization problem within the limited region. At this point, the perpendicular bisector L divides the constrained region into two parts; since the distance from point P to point A is less than that to point B, P must be located in the green and pink regions on the left side of the perpendicular bisector L. Take any point B′ in the grid region; in the case of Figure 2, the distance from point B′ to P is less than that from point A to P. By constructing the perpendicular bisector L′ of segment AB′, P must lie within the pink region in Figure 2.

In three-dimensional space, it can be decomposed into three two-dimensional subspaces, namely the Y-Z plane, X-Y plane, and X-Z plane. For points P, A, and B in three-dimensional space, they can be projected onto these three two-dimensional subspaces, respectively, to obtain projection points: P′, A′, B′ on the Y-Z plane; P″, A″, B″ on the X-Y plane; and P‴, A‴, B‴ on the X-Z plane. Taking the Y-Z plane as an example, we construct line segment A′B′ and its perpendicular bisector. As shown in Figure 3, since point P′ is closer to point A′, it can be known that P′ lies in the region containing A′ as divided by the perpendicular bisector.

By performing the same operation on the X-Y plane and X-Z plane, it can be determined that P″ exists in the specific region divided by the perpendicular bisector of the corresponding plane, and P‴ also exists in the corresponding region divided by the perpendicular bisector of the X-Z plane. Finally, the position region of point P in three-dimensional space is the intersection of the corresponding regions in the three two-dimensional subspaces:

$${\mathit{\tau }}_2={{\mathit{\tau }}_2}^{\prime }\cap {{\mathit{\tau }}_2}^{″}\cap {{\mathit{\tau }}_2}^{‴},$$

(1)

Based on the above analysis, complex multi-dimensional solutions can be decomposed into multiple two-dimensional solutions through projection. The properties of perpendicular bisectors in two-dimensional space can also be extended to solve multi-dimensional problems.

3.3. Mathematical Correlation Analysis and Convergence Proof

3.3.1. Correlation Between Geometric Characteristics and Optimization Performance

The core of the PBOA lies in mapping the fitness difference between individuals to the geometric distance relationship in the solution space and using the properties of the perpendicular bisector to narrow the search range. To establish the mathematical correlation between geometric characteristics and optimization performance, we first define the following variables:

Let the solution space be a bounded convex set $\Omega \in {ℝ}^n$, and let the objective function $f:\Omega \to ℝ$ be continuous and differentiable. For any two points $A,B\in \Omega$, their fitness values satisfy $f(A)<f(B)$ (indicating that A is closer to the optimal solution P* than B). According to the geometric properties described in Section 3.2, the perpendicular bisector L of segment AB divides the space into two half-spaces H_A and H_B, where H_A is the region containing A. The key mathematical correlation is: In an ideal scenario, due to the strict consistency between the geometric division by the perpendicular bisector and the direction of the fitness gradient, the probability that $P^{\ast }\in H_A$ holds is 1; in actual engineering, affected by disturbance factors, this probability is not less than 0.5 and approaches 1 as the iteration accuracy improves.

This correlation can be quantified by the fitness gradient. Let $\nabla f(X)$ be the gradient of the objective function at point X, which reflects the direction of the fastest increase in fitness. For the optimal solution P*, there exists a neighborhood $U(P^{\ast })$ such that for any $X\in U(P^{\ast })$, $\nabla f(X)\cdot (X-P^{\ast })>0$ (that is, in the minimization problem, the gradient points away from the optimal solution). When constructing the perpendicular bisector L, the angle θ between the normal vector of L and the gradient direction $\nabla f(X)$ satisfies:

$$\cos \theta ={\displaystyle \frac{(B-A)\cdot \nabla f(A)}{\parallel B-A\parallel \cdot \parallel \nabla f(A)\parallel }}$$

(2)

When $f(A)<f(B)$, $(B-A)\cdot \nabla f(A)>0$ holds, so $\theta <{90}^{\circ }$, which means that the normal direction of the perpendicular bisector is consistent with the gradient direction in the local region. This ensures that the H_A region divided by the perpendicular bisector contains the search direction of the optimal solution, thereby improving the optimization efficiency.

3.3.2. Convergence Proof Between Geometric Characteristics and Optimization Performance

To prove the convergence of the PBOA, it is necessary to show that as the number of iterations increases, the algorithm can approach the global optimal solution with a probability of 1.

Assumption 1.

The objective function f has a lower bound in the solution space Ω, that is, there exists a constant m such that $f(X)\ge m$ for all $X\in \Omega$

Assumption 2.

The solution space Ω is a compact set (closed and bounded), so any sequence in Ω has a convergent subsequence.

Assumption 3.

The objective function has monotonicity with respect to the distance to the optimal solution in the local neighborhood of the optimal solution.

Definition 1.

Let ${\left\{X_k\right\}}_{k=1}^{\infty }$be the sequence of solutions generated by the PBOA, where X_k is the optimal solution found in the k-th iteration. The algorithm is said to converge if $\underset{k\to \infty }{\lim }f(X_k)=f(P^{\ast })$ holds with a probability of 1 (where P* is the global optimal solution).

Proof. 

• Feasibility of region shrinkage: In each iteration, the PBOA constructs a new perpendicular bisector based on the current optimal individual X_k and a randomly selected individual Y_k, and updates the search region to ${\Omega }_{k+1}={\Omega }_k\cap H_{X_k}$, where $H_{X_k}$ is the half-space containing X_k divided by the perpendicular bisector of X_kY_k. Since $f(X_k)\le f(Y_k)$, combined with the monotonicity of the objective function and the geometric properties of the perpendicular bisector, $P^{\ast }\in H_{X_k}$ must hold in an ideal scenario, and a high probability of holding is ensured in practice. Therefore, $P^{\ast }\in {\Omega }_{k+1}\subseteq {\Omega }_k$; that is, the search region is always a nested sequence containing the optimal solution.
• Convergence of the region sequence: Due to the compactness of Ω, according to the finite intersection property of compact sets, the intersection of the nested sequence $\left\{{\Omega }_k\right\}$ is a non-empty compact set ${\Omega }^{\ast }={\displaystyle \underset{k=1}{\overset{\infty }{\cap }}{\Omega }_k}$, and $P^{\ast }\in {\Omega }^{\ast }$. If the diameter of ${\Omega }_k$ (the maximum distance between any two points in the set) satisfies $\underset{k\to \infty }{\lim }\mathrm{diam}({\Omega }_k)=0$, then ${\Omega }^{\ast }=\left\{P^{\ast }\right\}$.
• Probability of approaching the optimal solution: In each iteration, based on the monotonicity of the objective function, the probability that the new search region ${\Omega }_{k+1}$ contains the neighborhood of P* is high. According to the Borel-Cantelli lemma, the probability that the algorithm eventually enters and stays in any neighborhood of P* is 1, that is, ${\lim }_{k\to \infty }{X}_k=P^{\ast }$ holds with a probability of 1.
• Convergence of fitness values: Since f is continuous $\underset{k\to \infty }{\lim }f(X_k)=f(P^{\ast })$, holds, and the convergence proof is completed. □

3.3.3. Limitations and Supplementary Strategies

The above convergence proof is based on the unimodal assumption and monotonicity assumption of the objective function. For multimodal problems, although the properties of the perpendicular bisector cannot directly ensure convergence to the global optimal solution, the introduction of mutation operators (such as Gaussian mutation) can help the algorithm jump out of local optima. Let the mutation probability be $p_m\in (0,1)$, and the mutation step size decreases with the number of iterations. Then, the probability that the algorithm explores the neighborhood of the global optimal solution within a limited number of iterations is still 1, thus extending the convergence to multimodal problems. At the same time, in practical applications, to address the impact of disturbance factors, by increasing the number of samples and optimizing the individual selection strategy, the probability that $P^{\ast }\in H_A$ can be made to continuously approach 1 as the iteration accuracy improves, ensuring the optimization performance of the algorithm.

3.4. Convergence Proof of PBOA

The PBOA starts from a set of random solutions, as shown in Equation (3).

$$X=\left[\begin{array}{cccccc}{x}_{1,1}& \dots & x_{1,j}& \dots & x_{1,Dim-1}& x_{1,Dim}\\ x_{2,1}& \dots & x_{2,j}& \dots & x_{2,Dim-1}& x_{2,Dim}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \dots & \dots & x_{i,j}& \dots & \dots & \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ x_{N-1,1}& \dots & x_{N-1,j}& \dots & x_{N-1,Dim-1}& x_{N-1,Dim}\\ x_{N,1}& \dots & x_{N,j}& \dots & x_{N,Dim-1}& x_{N,Dim}\end{array}\right],$$

(3)

where x_i represents the corresponding position information of individuals in the ith subgroup. Dim denotes the dimension of the problem, and N denotes the number of individuals in the subpopulation. Each x_i,j in matrix X can be calculated using Equation (4).

$$x_{i,j}=r\times \left(U{B}^j-L{B}^j\right)+L{B}^j,i=1,2,\dots ,N,j=1,2,\dots ,Dim,$$

(4)

where r is a random number in the interval [0, 1], and UB_j and LB_j are the upper and lower bounds of the jth dimension of the given problem, respectively.

3.5. Types of Objects for Constructing Perpendicular Bisectors

Based on the principle of perpendicular bisectors, two particles are required to form a line segment for generating a perpendicular bisector. The PBOA provides 4 types of particles for each particle to be updated to construct line segments:

• Type 1: Particle with the best global fitness (α);
• Type 2: Particle with the second-best global fitness (β);
• Type 3: Particle with the third-best global fitness (γ);
• Type 4: Randomly selected particle.

In each iteration, the probabilities of selecting these 4 types of particles are P₁, P₂, P₃, and P₄, respectively. Compared with single-type endpoints, perpendicular bisectors constructed with 4 types of endpoints can form a broader search area (as shown in Figure 4), enhancing the algorithm’s exploration capability in multi-modal spaces.

3.6. Exploitation Phase

This subsection presents the equation of the perpendicular bisector and designs the exploitation phase for particle update in the PBOA based on this equation.

3.6.1. Equation of the Perpendicular Bisector in Two-Dimensional Space

In a two-dimensional space, let the coordinates of point A be (a₁, a₂) and the coordinates of point B be (b₁, b₂); then the equation of the perpendicular bisector L of segment AB can be expressed as:

$$\left\{\begin{array}{c}{\lambda }_1{l}_1+{\lambda }_2{l}_2+{\lambda }_3=0\\ {\lambda }_1=2\times (b_1-a_1)\\ {\lambda }_2=2\times (b_2-a_2)\\ {\lambda }_3=a_1^2-b_1^2+a_2^2-b_2^2\end{array}\right.,$$

(5)

The region close to point A can be expressed as Equation (6), which provides geometric constraints for particle position update.

$$l_2<-{\displaystyle \frac{{\lambda }_1{l}_1+{\lambda }_3}{{\lambda }_2}}$$

(6)

3.6.2. Particle Update Method in the Exploitation Phase

Based on the properties of perpendicular bisectors, the particle position update of the PBOA is divided into 3 steps:

• Step 1: Update of the first-dimensional variable using Equation (7),

$$x_i^1(t+1)=(x_m^1(t)+x_i^1(t))/2,$$

(7)

where x_m is the particle with which x_i constructs a line segment.

• Step 2: Update variables in other dimensions (j ≥ 2) using Equation (8),

$$\left\{\begin{array}{c}{x}_i^j(t+1)=\eta r_1{\displaystyle \frac{-{\sigma }_1{x}_i^{j-1}(t)-{\sigma }_3}{{\sigma }_2}},{\sigma }_1\ne 0,{\sigma }_2\ne 0\\ \begin{array}{c}{x}_i^{j-1}=L{B}^{j-1}+(U{B}^{j-1}-L{B}^{j-1})\ast r_2\\ x_i^j=x_m^j-k_1{r}_1(x_i^j-x_m^j)/2\end{array},{\sigma }_1=0\\ \begin{array}{c}{x}_i^{j-1}=x_m^{j-1}-k_1{r}_1(x_i^{j-1}-x_m^{j-1})/2\\ x_i^j=L{B}^j+(U{B}^j-L{B}^j)\ast r_2\end{array},{\sigma }_2=0\end{array}\right.$$

(8)

where

$$\left\{\begin{array}{l}{\sigma }_1=2(x_m^{j-1}-x_i^{j-1})\hfill \\ {\sigma }_2=2(x_m^j-x_i^j)\hfill \\ {\sigma }_3={(x_m^{j-1})}^2-{(x_i^{j-1})}^2+{(x_m^j)}^2-{(x_i^j)}^2\hfill \end{array}\right.$$

(9)

where r₁ is a random number in the range of [−1, 1], r₂ is a random number in the range of [0, 1], and k₁ is the motion range adjustment factor, whose calculation formula is as follows:

$$\eta =\left\{\begin{array}{l}(1-{\displaystyle \frac{t-1^{-300}}{Max\_iter}})e^{x_i^j-x_m^j},k_1\ne 0\hfill \\ 1^{-300},k_1\ne 0\hfill \end{array}\right.$$

(10)

where t is the current iteration number, and Max_iter is the maximum number of iterations. k₁ is regulated by a dual regulation mechanism: its value decreases as the number of iterations increases and decreases as the distance between the two particles shrinks. The core logic of this design is: in the early stage of PBOA iteration, the algorithm needs to expand the search range; at this time, k₁ takes a larger value, which can support particles to move in a wider range. In the later stage of iteration, the algorithm needs to reduce the movement range of particles to enhance the accuracy and efficiency of local search.

• Step 3: Re-update of the first-dimensional variable: After the particle updates the last dimension variable, a line segment is constructed between the last-dimensional variable and the first-dimensional variable, and then the first-dimensional variable is updated via Equation (8).

3.7. Exploration Phase

The exploration phase is used to prevent particles from rapidly approaching the selected object according to the convergence strategy of the perpendicular bisector, which would result in insufficient search in most of the space. In this phase, the update steps of particles are the same as the 3 steps in the exploitation phase, but the update method of variables in other dimensions (j ≥ 2) is different: in the exploration phase, variables must be updated based on Equation (11) to make particles approach the midpoint of the line segment, thereby slowing down the aggregation speed and preserving search diversity.

$$\left\{\begin{array}{c}{x}_i^j(t+1)={\displaystyle \frac{-{\sigma }_1{x}_i^{j-1}(t)-{\sigma }_3}{{\sigma }_2}},{\sigma }_1\ne 0,{\sigma }_2\ne 0\\ \begin{array}{c}{x}_i^{j-1}=L{B}^{j-1}+(U{B}^{j-1}-L{B}^{j-1})\ast r_2\\ x_i^j=x_m^j-(x_i^j-x_m^j)/2\end{array},{\sigma }_1=0\\ \begin{array}{c}{x}_i^{j-1}=x_m^{j-1}-(x_i^{j-1}-x_m^{j-1})/2\\ x_i^j=L{B}^j+(U{B}^j-L{B}^j)\ast r_2\end{array},{\sigma }_2=0\end{array}\right.$$

(11)

Consistent with the exploitation phase, after updating all variables of x_i, it is necessary to re-update x_i¹, but this time Equation (11) is used instead of Equation (8).

3.8. Selection Method for Exploration Phase and Exploitation Phase

The probability P in Equation (12) is used to regulate mode selection and convergence mode. In the early stage of iteration (t is small), the P value is small, the exploration phase has a higher probability of being selected, and global exploration is prioritized. In the later stage of iteration (t is large), the P value is large, and the exploitation phase has a higher probability of being selected to enhance local search.

$$P=\lambda t/Max\_iter$$

(12)

In the early stages of algorithm iteration, even as the iteration count t increases, the parameter λ effectively restrains the excessive growth of probability P. This ensures that the algorithm conducts sufficient global exploration and avoids premature transition to the exploitation phase. In the later stages of iteration, with the cumulative increase in t and the synergistic effect of λ, the weight of the exploitation phase gradually increases. Meanwhile, the regulatory mechanism of λ ensures that this transition process is smooth and orderly rather than abrupt. Consequently, λ not only effectively prevents the over-prolongation of the exploration phase (which would delay convergence efficiency) but also avoids the premature initiation of the exploitation phase (which may cause the algorithm to fall into suboptimal local solutions), ultimately achieving a dynamic balance between exploration and exploitation throughout the entire iterative cycle.

3.9. Pseudocode and Algorithm Flow Chart of PBOA

The pseudocode of the PBOA is shown in Algorithm 1, and the algorithm flow chart is shown in Figure 5.

Algorithm 1: Perpendicular Bisector Optimization Algorithm (PBOA)
1:	Initialize parameters: population size N, particle dimension d, maximum number of iterations Max_iter, probability parameters P, P1, P2, P3, P4 and adjustment factor λ.
2:	Randomly initialize the population and calculate the fitness values of all particles.
3:	Sort by fitness to determine the globally optimal particle (first in fitness), the second optimal particle (second in fitness), and the third optimal particle (third in fitness).
4:	Set the current number of iterations as t = 1.
5:	while t ≤ Max_iter do
6:	Set the particle index as i = 1.
7:	While i ≤ N do
8:	Generate two random numbers r3 and r4 in the range of [0, 1].
9:	Calculate the probability P according to Equation (12).
10:	Update the first-dimensional variable xi1 of the ith particle according to Equation (7).
11:	Select another endpoint particle of the line segment according to r3.
12:	if r3 < P1 then
13:	Select the global optimal particle (α) and the current particle to form a line segment.
14:	else if r3 < P2
15:	Select the second optimal particle (β) and the current particle to form a line segment.
16:	else if r3 < P3
17:	Select the third optimal particle (γ) and the current particle to form a line segment.
18:	else
19:	Randomly select a particle from the population to form a line segment with the current particle.
20:	end if
21:	Select the update mode according to r4;
22:	if r4 < P then
23:	Update the j ≥ 1 dimensional variables of the particle according to Equation (8).
24:	Re-update xi1 using xiDim.
25:	else
26:	Update the j ≥ 1 dimensional variables of the particle according to Equation (11).
27:	Re-update xi1 using xiDim.
28:	end if
29:	i = i + 1.
30:	Calculate the fitness value of the ith particle after update.
31:	If the new fitness is better, update the records of the globally optimal, second optimal, and third optimal particles.
32:	end while
33:	t = t + 1.
34:	end while
35:	Output the position of the global optimal particle and its corresponding best fitness.

3.10. Computational Complexity of PBOA

The computational complexity of an optimization algorithm can be characterized by a function that relates the running time of the algorithm to the input size of the problem. In complexity analysis, the O-notation (big O notation) is a widely recognized representation method. Based on this, the time complexity of the PBOA can be described as follows:

$$\begin{array}{l}O(\mathrm{PBOA})=O(\mathrm{Initialize} \mathrm{the} \mathrm{population})+O(\mathrm{Update} \mathrm{particle} \mathrm{positions})\\ +O(\mathrm{Calculate} \mathrm{the} \mathrm{fitness} \mathrm{of} \mathrm{updated} \mathrm{particles})\\ +O(\mathrm{Update} \mathrm{the} \mathrm{particles} \mathrm{ranked} \mathrm{first}, \sec \mathrm{ond}, \mathrm{and} \mathrm{third} \mathrm{in} \mathrm{fitness})\end{array}$$

(13)

In addition to the factors involved in Equation (13), the time complexity of the PBOA is also affected by the following parameters: maximum number of iterations (Max_iter), population size (N), problem dimension (Dim), and computational cost of the objective function (f). Among them, the time complexity of the particle update process can be expressed as:

$$\begin{array}{l}O(\mathrm{Update} \mathrm{particle} \mathrm{positions})=O(\mathrm{Determine} \mathrm{the} \mathrm{endpoints})\\ +O(\mathrm{Determine} \mathrm{the} \mathrm{update} \mathrm{method})\\ +O(\mathrm{Update} \mathrm{the} \mathrm{particle} \mathrm{positions})\end{array}$$

(14)

Therefore, considering the above influencing factors, the overall time complexity of the Perpendicular Bisector Optimization Algorithm (PBOA) can be derived as:

$$\begin{array}{l}O(\mathrm{PBOA})=O(N\times d)+O(Max\_iter\times N)+O(Max\_iter\times N)\\ +O(Max\_iter\times N\times (d+1))+O(Max\_iter\times N\times d\times f)\\ +O(Max\_iter\times N)\end{array}$$

(15)

In O-notation, the time complexity of an algorithm is dominated by the term with the fastest growth rate (the remaining lower-order terms can be ignored). Based on this, the time complexity of the Perpendicular Bisector Optimization Algorithm (PBOA) can be simplified as:

$$O(PBOA)=O(Max\_iter\times N\times d\times f)$$

(16)

4. Experimental Setup

This section provides an overview of the experimental setup, including the selection of benchmark test functions, the choice of comparison algorithms, the setting of evaluation metrics, and parameter configuration.

4.1. Benchmark Test Functions

In this paper, 27 optimization functions from reference [59] are selected as the test set (

Table 2. Test function information table.

Test Function	D	[LB, UB]	fmin
$F_1(x)={\displaystyle \sum _{i=1}^D}{x}_i^2$	30	[−100, 100]D	0
$F_2(x)={\displaystyle \sum _{i=1}^D}\left|x_i\right|+{\displaystyle \prod _{i=1}^D}\left|x_i\right|$	30	[−10, 10]D	0
$F_3(x)={\displaystyle \sum _{i=1}^D}{\left({\displaystyle \sum _{j=1}^i}{x}_j\right)}^2$	30	[−100, 100]D	0
$F_4(x)={\max }_i\left\{\right|x_i|,1\le i\le D\}$	30	[−100, 100]D	0
$F_5(x)={\displaystyle \sum _{i=1}^{D-1}}[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2]$	30	[−30, 30]D	0
$F_6(x)={\displaystyle \sum _{i=1}^D}{[x_i+0.5]}^2$	30	[−100, 100]D	0
$F_7(x)={\displaystyle \sum _{i=1}^D}i{x}_i^4+random[0,1)$	30	[−1.28, 1.28]D	0
$F_8(x)={\displaystyle \sum _{i=1}^D}-x_i\sin (\sqrt{\left|x_i\right|})$	30	[−500, 500]D	−12,569.5
$F_9(x)={\displaystyle \sum _{i=1}^D}[x_i^2-10\cos (2\pi x_i)+10]$	30	[−5.12, 5.12]D	0
$F_{10}(x)=-20\exp (-0.2\sqrt{{\displaystyle \frac{1}{D}}{\displaystyle \sum _{i=1}^D}{x}_i^2})-\exp ({\displaystyle \frac{1}{D}}{\displaystyle \sum _{i=1}^D}\cos 2\pi x_i)+20+e$	30	[−32, 32]D	0
$F_{11}(x)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^D}{x}_i^2-{\displaystyle \prod _{i=1}^D}\cos ({\displaystyle \frac{x_i}{\sqrt{i}}})+1$	30	[−600, 600]D	0
$\begin{array}{c}{F}_{12}(x)={\displaystyle \frac{\pi }{D}}\{10{\sin }^2(\pi y_i)+{\displaystyle \sum _{i=1}^{D-1}}{(y_i-1)}^2[1+10{\sin }^2(\pi y_{i+1})]+\\ {(y_D-1)}^2\}+{\displaystyle \sum _{i=1}^D}u(x_i,10,100,4),y_i=1+{\displaystyle \frac{1}{4}}(x_i+1),\\ u(x_i,a,k,m)=\left\{\begin{array}{cc}k{(x_i-a)}^m,& x_i>a\\ 0,& -a\le x_i\le a\\ k{(-x_i-a)}^m,& x_i<-a\end{array}\right.\end{array}$	30	[−50, 50]D	0
$\begin{array}{c}{F}_{13}(x)=0.1\{{\sin }^2(3\pi x_1)+{\displaystyle \sum _{i=1}^{D-1}}{(x_i-1)}^2[1+{\sin }^2(3\pi x_{i+1})]+(x_D-\\ 1{)}^2[1+{\sin }^2(2\pi x_D)]\}+{\displaystyle \sum _{i=1}^D}u(x_i,5,100,4)\end{array}$	30	[−50, 50]D	0
$F_{14}(x)={\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}}{\displaystyle \frac{1}{[j+{\displaystyle \sum _{i=1}^2}{(x_j-a_{ij})}^6]}}$	2	[−65.536, 65.536]D	1
$F_{15}(x)={\displaystyle \sum _{i=1}^{11}}{[a_i-{\displaystyle \frac{x_1(b_i^2+b_i{x}_2)}{b_i^2+b_i{x}_3+x_4}}]}^2$	4	[−5, 5]D	0.0003075
$F_{16}(x)=4x_1^2-2.1x_1^4+{\displaystyle \frac{1}{3}}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[−5, 5]D	−1.0316285
$F_{17}(x)={(x_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{x}_1^2+{\displaystyle \frac{5}{\pi }}{x}_1-6)}^2+10(1-{\displaystyle \frac{1}{8\pi }})\cos x_1+10$	2	[−5, 10] × [0, 15]	0.398
$\begin{array}{l}{F}_{18}(x)=[1+{(x_1+x_2+1)}^2(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+\\ 3x_2^2)][30+{(2x_1-3x_2)}^2(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2)]\end{array}$	2	[−2, 2]D	3
$F_{19}(x)=-{\displaystyle \sum _{i=1}^4}{c}_i\exp [-{\displaystyle \sum _{j=1}^4}{a}_{ij}{(x_j-p_{ij})}^2]$	4	[0, 1]D	−3.86
$F_{20}(x)=-{\displaystyle \sum _{i=1}^4}{c}_i\exp [-{\displaystyle \sum _{j=1}^6}{a}_{ij}{(x_j-p_{ij})}^2]$	6	[0, 1]D	−3.32
$F_{21}(x)=-{\displaystyle \sum _{i=1}^5}{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0, 10]D	−10
$F_{22}(x)=-{\displaystyle \sum _{i=1}^7}{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0, 10]D	−10
$F_{23}(x)=-{\displaystyle \sum _{i=1}^{10}}{[(x-a_i){(x-a_i)}^T+c_i]}^{-1}$	4	[0, 10]D	−10
$F_{24}(x)=F_1(y),y=x-2$	30	[−100, 100]D	0
$F_{25}(x)=F_3(y),y=x-2$	30	[−100, 100]D	0
$F_{26}(x)=F_9(y),y=x-2$	30	[−5.12, 5.12]D	0
$F_{27}(x)=F_{11}(y),y=x-2$	30	[−600, 600]D	0

) to evaluate the efficiency and stability of the PBOA. Among them, the first 23 functions (F1~F23) are derived from the CEC2005 test suite, aiming to analyze the global search ability and local search ability of the algorithm; the remaining 4 functions (F24~F27) are translation variants of F1, F3, F9, and F11, which are used to verify the adaptability of the algorithm when perturbations are introduced into the independent variables.

The 23 benchmark functions in the CEC2005 test suite mentioned above can be divided into three categories:

• Unimodal functions (F1~F7): Containing only a single global optimal value, they are mainly used to test the convergence performance of the algorithm.
• Multimodal functions (F8~F13): Including multiple local optimal solutions with complex structures, they focus on evaluating the algorithm’s ability to balance exploration and exploitation to avoid premature convergence.
• Fixed-dimensional multimodal functions (F14~F23): With low dimensions but a large number of local optimal solutions, they are mainly used to test the robustness of the algorithm in low-dimensional spaces.

4.2. Comparison Algorithms

To evaluate the performance of the PBOA, 15 optimization algorithms summarized in

Table 3. List of 13 compared optimization algorithms.

Year	Algorithm	Proposed by
2025	Divine Religion Optimization Algorithm (DRA)	Ali Toufanzadeh Mozhdehi et al. [60]
2025	Plant Rhizome Growth Optimization Algorithm (PRGO)	Jin Zhang et al. [61]
2023	Sea-horse optimizer (SHO)	Zhao et al. [62]
2023	Geometric mean optimizer (GMO)	Rezaei et al. [63]
2022	White shark optimizer (WSO)	Braik et al. [64]
2021	Artificial gorilla troops optimizer (GTO)	Abdollahzadeh et al. [65]
2019	Henry gas solubility optimization (HGSO)	Hashim et al. [66]
2016	Whale optimization algorithm (WOA)	Mirjalili et al. [55]
2016	Sine cosine algorithm (SCA)	Mirjalili et al. [40]
2015	Moth-flame optimization (MFO)	Mirjalili et al. [67]
2014	Gray wolf optimizer (GWO)	Mirjalili et al. [16]
1997	Differential evolution (DE)	Storn and Price [5]
1995	Particle swarm optimization (PSO)	Kennedy and Eberhart [21]
1965	Nelder-Mead simplex algorithm (NEL)	Nelder and Mead [68]
2000s	Voronoi diagram-based optimization algorithm (VOR)	Various researchers [69,70]

are selected as comparison objects. These algorithms are all representative methods published in recent years, including two classical meta-heuristic algorithms that update based on geometric principles, with high citation counts.

4.3. Evaluation Metrics

To evaluate the performance of the algorithms, the Average Optimal Fitness (AOF) is adopted as the core evaluation metric. AOF is defined as the average of the optimal fitness values obtained from each independent run of the algorithm in multiple independent tests.

In terms of the meaning of the metric, a smaller AOF value indicates a lower average optimal fitness of the algorithm, meaning that the algorithm is more likely to find better solutions in multiple tests. In the algorithm performance ranking, the higher the ranking (closer to 1), the smaller the corresponding AOF value; the lower the ranking, the larger the AOF value.

4.4. Parameter Settings

The specific parameter configurations of each algorithm are detailed in

Table 4. Parameter settings for compared algorithms.

Algorithm	Parameter
Perpendicular Bisector Optimization Algorithm (PBOA)	λ = 0.4
	P1 = P2 = P3 = 0.28, P4 = 0.16
Divine Religion Optimization Algorithm (DRA)	Faith factor = 0.6
	Ritual coefficient = 0.4
	Propagation probability = 0.3
	Reform probability = 0.1
	Control parameter a = 2 (linearly decreases to 0)
Plant Rhizome Growth Optimization Algorithm (PRGO)	Root spread rate = 0.3
	Growth intensity = 0.5
	Geotropism coefficient = 2
	Decay factor = 0.9
Sea-horse optimizer (SHO)	U = 0.05
	V = 0.05
	L = 0.05
Geometric mean optimizer (GMO)	Geometric rate = 0.01
	Update frequency = 5
White shark optimizer (WSO)	Hunt depth = 0.2
	Speed factor = 0.8
Artificial gorilla troops optimizer (GTO)	Tension rate = 0.8
	Aggression level = 0.2
Henry gas solubility optimization (HGSO)	l1 = 5 × 10−3
	l2 = 100
	l3 = 1 × 10−2
	alpha = 1
	beta = 1
	M1 = 0.1
	M2 = 0.2
Moth-flame optimization (MFO)	a linearly decreases from −1 to −2
Whale optimization algorithm (WOA)	Convergence constant a = [2, 0]
	b = 1
Sine cosine algorithm (SCA)	r1 = random (0, 1)
	r2 = random (0, 1)
	r3 = random (0, 1)
	r4 = random (0, 1)
Gray wolf optimizer (GWO)
Convergence constant a = [2, 0]	Mutation factor F = 0.5
	Crossover rate CR = 0.9
Particle swarm optimization (PSO)	Inertia weight w = [0.9, 0.4]
	Cognitive coefficient c1 = 2
	Social coefficient c2 = 2
Voronoi diagram-based optimization algorithm (VOR)	Local search probability = 0.7
	Global exploration probability = 0.3
	Refinement factor = 0.95 (linearly decreases to 0.5)
	Jitter factor = 1 × 10−6
Nelder-Mead simplex algorithm (NEL)	Reflection coefficient (alpha) = 1
	Expansion coefficient (gamma) = 2
	Contraction coefficient (rho) = 0.5
	Shrink coefficient (sigma) = 0.5

. To ensure the fairness and impartiality of the comparative experiments, the parameter values listed in Table 4 are determined based on the following criteria: (1) For most algorithms, the parameters are set to the default values recommended in their original studies, which have been validated to achieve optimal performance in general optimization scenarios. (2) For a few algorithms where the original studies suggest a range of parameter values, we conducted preliminary experiments to select the values that yield stable and representative performance within that range, avoiding extreme values that might artificially degrade their optimization capabilities. (3) All parameters are kept consistent across different benchmark functions and engineering design problems to eliminate the influence of parameter variability on the comparison results. This is to ensure the transparency of the experimental process and the reproducibility of the results.

5. Experimental Results and Analysis

5.1. Parameter Sensitivity Analysis

The core parameters involved in this paper include: the number of optimal points among the types of particles to be selected as endpoints of line segments, the key factor λ for selecting particle movement modes, and the probability parameters P₁, P₂, P₃, and P₄ that control the probability of particles following different targets. To evaluate the impact of these parameters on the algorithm performance, a parameter sensitivity analysis is conducted.

The experiment adopts a unified setup: the population size N = 50, and the maximum number of iterations Max_iter = 1000. For each parameter combination, the PBOA runs independently 50 times on each test function. The results of the sensitivity analysis are shown in Figure 6, Figure 7 and Figure 8. Based on these results, this paper determines the optimal setting method for each parameter, which is as follows:

5.1.1. Selection of Test Functions

To further evaluate the impact of parameters on the algorithm performance, this study selects 8 representative benchmark functions (F1, F3, F5, F9, F10, F12, F14, F18) for analysis. The specific parameters of each test function are detailed in Table 1, covering different characteristics (such as unimodal/multimodal, high-dimensional/low-dimensional, etc.), which can support a comprehensive evaluation of parameter impacts.

5.1.2. Sensitivity Analysis of the Number of Optimal Points

To quantify the impact of the number of optimal points among particle types selected as endpoints of line segments (hereinafter referred to as “endpoint candidate optimal points”) on the optimization performance of the PBOA, this experiment systematically adjusts this number from 1 to 10, resulting in 10 distinct parameter combinations. The remaining parameters of the PBOA are fixed as follows: each endpoint candidate optimal point has an equal probability of being selected; the control parameter λ is set to 0.4. For each combination of the number of endpoint candidate optimal points and test functions, 50 independent runs are conducted to ensure statistical robustness. The comprehensive ranking of algorithm performance is adopted as the core evaluation metric, where a smaller ranking indicates superior overall performance under that parameter configuration.

Based on the optimization results of the PBOA on benchmark functions with different numbers of endpoint candidate optimal points, Figure 6 presents the final rankings of each function under the 10 quantity configurations. The key findings are summarized as follows:

Unimodal functions (F1, F3, F5): For F1, the ranking is optimal with 3 endpoint candidate optimal points, followed by 2 and 4, while the performance is the poorest with 10. This indicates that a moderate number of endpoint candidate optimal points (3–4) facilitates efficient local exploitation for this unimodal problem. In F3, the performance is optimal with 1 endpoint candidate optimal point, followed by 2 and 3, with the worst performance observed with 7, suggesting that fewer endpoint candidate optimal points can improve the convergence efficiency of this function. For F5, the ranking is optimal with 7 endpoint candidate optimal points, with 2 and 8 also performing well, and the lowest ranking with 10. Overall, these results demonstrate that for unimodal function optimization, the number of endpoint candidate optimal points in the range (3–7) yields superior comprehensive rankings, reflecting strong adaptability to local exploitation scenarios.

Multimodal functions (F9, F10, F12): In F9, the algorithm achieves optimal rankings when the number of endpoint candidate optimal points is 3, 4, 5, 6, 7, or 9, indicating high robustness to variations in this number for this multimodal function; however, performance is the poorest with 8. For F10, the best rankings are observed with 7 and 9 endpoint candidate optimal points, followed by 8, with the lowest ranking with 10, suggesting that a moderate-to-large number of endpoint candidate optimal points (7–9) effectively balances exploration across multiple peaks. In F12, the optimal ranking is attained with 4 endpoint candidate optimal points, followed by 7 and 3/6, with the worst performance with 10. Overall, for multimodal function optimization, the number of endpoint candidate optimal points in the range (3–7) results in balanced rankings across most functions, effectively synergizing global exploration and local exploitation.

Fixed-dimensional multimodal functions (F14, F18): For F14, optimal rankings are achieved with 4, 6, or 7 endpoint candidate optimal points, followed by 3/9, with the poorest performance with 10. In F18, the algorithm exhibits exceptional stability: the number of endpoint candidate optimal points ranging from 3 to 9 all achieve optimal rankings, while 1 and 2 perform significantly worse. These results indicate that fixed-dimensional multimodal functions benefit from a moderate-to-large number of endpoint candidate optimal points (3–9), with low sensitivity to specific values within this range.

In summary, integrating the ranking results across unimodal, multimodal, and fixed-dimensional multimodal functions, when the number of endpoint candidate optimal points is in the range (3–7), the PBOA achieves superior comprehensive rankings on most test functions with strong performance stability. This range demonstrates robust performance across diverse optimization scenarios, thus recommending (3–7) as the optimal range for the number of endpoint candidate optimal points in the PBOA. Considering the algorithm’s outstanding performance with 3 optimal points in unimodal functions and its good adaptability in multimodal and fixed-dimensional multimodal functions, this paper ultimately selects 3 as the number of endpoint candidate optimal points.

5.1.3. Sensitivity Analysis of Parameter λ

To quantify the impact of parameter λ on the search ability of the PBOA, the experiment sets λ to increase with a fixed step size of 0.02 in the interval [0, 1], forming 51 groups of parameter experiments. Other parameters of the PBOA are set as follows: P₁, P₂, P₃ are all 0.1, and P₄ is 0.7. For each combination of parameters and test functions, 51 independent optimization experiments are conducted, and the comprehensive ranking of algorithm performance is recorded as the core basis for parameter evaluation.

Based on the performance of the PBOA on benchmark functions under different λ values, Figure 7 illustrates the final rankings of the PBOA for optimizing each function with 51 λ values (a smaller ranking indicates better comprehensive performance of the algorithm under that parameter combination). The experimental results show that:

Unimodal functions (F1, F3, F5): In F1, when λ ∈ [0.02, 1], the algorithm’s ranking is overall in a better range; in F3, the ranking is stable and high when λ ∈ [0.02, 1], with the optimal ranking when λ ∈ [0.10, 1]. This indicates that in unimodal function optimization, λ can maintain good performance of the PBOA within a large range, especially when λ is in the above range, the comprehensive ranking is better, reflecting that the algorithm has strong adaptability to λ in local exploitation scenarios. In F5, the final ranking of the algorithm is the smallest (optimal) when λ ∈ [0.02, 0.2], indicating that a low λ value is more conducive to improving the convergence accuracy of unimodal functions. Based on the performance of unimodal functions, it is recommended that λ be preferentially selected from [0.02, 0.5].

Multimodal functions (F9, F10, F12): In F9, the algorithm rankings corresponding to 51 λ values show disordered fluctuations, indicating that this function has low sensitivity to λ; in F10, the ranking is significantly better and more stable when λ ∈ [0.18, 1]; in F12, lower λ values in [0.02, 0.3] correspond to better rankings. Overall, in multimodal function optimization, when λ ∈ [0.18, 0.5], the algorithm’s ranking performance is more balanced across most functions, which can well balance the synergy between global exploration and local exploitation.

Fixed-dimensional multimodal functions (F14, F18): Under all λ values, the final rankings of the algorithm show random distribution characteristics, and the overall ranking is low. This indicates that when the PBOA handles fixed-dimensional multimodal functions, its performance has low dependence on λ and has certain limitations.

In summary, combining the ranking results of unimodal and multimodal functions, when λ ∈ [0.18, 0.5], the PBOA achieves better comprehensive rankings on most test functions with strong performance stability. Therefore, this paper recommends this interval as the optimal value range for parameter λ.

5.1.4. Sensitivity Analysis of Parameter P₁, P₂, P₃, and P₄

In the PBOA, parameters P₁, P₂, P₃, and P₄ are key factors affecting the construction of perpendicular bisectors between particles and other objects. Among them, P₁, P₂, and P₃ represent the probabilities of particles associating with the globally optimal (rank 1), second optimal (rank 2), and third optimal (rank 3) particles, respectively, while P₄ is the probability of particles associating with randomly selected remaining particles. This section explores the impact of these parameters on PBOA performance through experiments, with the specific design as follows: P₁ = P₂ = P₃ is set to increase from 0 to 0.33 with a step size of 0.01 (34 parameter combinations in total), and P₄ is calculated as P₄ = 1 − P₁ − P₂ − P₃; to ensure experimental validity, other parameters are fixed as λ = 0.4; for each combination of parameters and test functions, 50 independent optimization experiments are conducted, with the comprehensive ranking as the core performance evaluation index (a smaller ranking indicates better comprehensive performance).

Figure 8 shows the ranking trends of the 34 parameter combinations. Based on the above results, the analysis is as follows:

Unimodal functions (F1, F3, F5): In F1 and F3, when the values of P₁, P₂, and P₃ are greater than 0.08, the algorithm rankings are overall high and stable, indicating that the algorithm performs excellently in unimodal function optimization; in F5, higher values of P₁, P₂, and P₃ correspond to better rankings, suggesting that increasing these parameters helps improve the search accuracy of the algorithm on this function. Overall, larger values of P₁, P₂, and P₃ are more conducive to the algorithm’s performance in unimodal function optimization.

Multimodal functions (F9, F10, F12): F9 achieves better rankings when P₁, P₂, P₃ ∈ [0.06, 0.33]; in F10, rankings improve significantly when P₁, P₂, and P₃ are greater than 0.07; in F12, rankings show an optimizing trend as parameter values increase. Overall, in multimodal function optimization, larger values of P₁, P₂, and P₃ lead to better rankings, with the combination P₁ = P₂ = P₃ = 0.28 achieving the optimal ranking and the strongest performance stability.

Fixed-dimensional multimodal functions (F14, F18): Rankings of all parameter combinations show disordered distribution, with overall low rankings, indicating that adjusting P₁, P₂, P₃, and P₄ cannot effectively improve the algorithm’s performance on this type of function.

In summary, considering the ranking performance across all function types, when P₁ = P₂ = P₃ = 0.28 and P₄ = 0.16, the PBOA achieves the optimal comprehensive ranking on most test functions with strong performance stability, which is beneficial for enhancing the overall robustness of the algorithm. Therefore, this parameter combination is recommended as the optimal setting for P₁, P₂, P₃, and P₄.

5.2. Qualitative Analysis

This section systematically presents the execution process of the PBOA on selected test functions. Figure 9 shows the qualitative analysis results of the PBOA on 8 test functions. The analysis uses four recognized indicators to intuitively evaluate the algorithm performance: search history, average fitness of the population, first-dimensional trajectory, and convergence curve. The specific meanings of each indicator are as follows:

• Search history: Used to display the spatial distribution of the population during the search process;
• First-dimensional trajectory: Reflects the position change law of solutions along the first dimension during algorithm iteration;
• Average fitness: Characterizes the evolution trend of the overall fitness of the population as iterations progress;
• Convergence curve: Describes the dynamic process of the algorithm approaching the optimal solution.

For detailed observation, the number of iterations of the PBOA is set to 1000, and the population size is set to 50.

From the search history, it can be seen that the PBOA exhibits strong search ability in both global and local search spaces and can quickly converge to the optimal solution region. The first-dimensional trajectory clearly shows that in the initial iteration stage, the position change range of the algorithm is wide to quickly locate potential optimal regions; in subsequent iterations, the frequency and amplitude of position changes gradually decrease, reflecting the strategic transition from global exploration to local exploitation. The average fitness curve shows a continuous downward trend, indicating that the population gradually converges to the optimal solution with iterations. These characteristics collectively indicate that the proposed PBOA has high efficiency in solution space search and can effectively balance the synergy between exploitation and exploration.

Figure 10 shows the iterative variation law of PBOA population diversity on 8 test functions. The results indicate that the population does not exhibit homogenization during iteration; especially on F14 and F18, the population diversity shows an upward trend with the increase in the number of iterations, demonstrating that the PBOA has strong global search capability and can effectively escape local optimal traps.

In the PBOA, particles are reset to random positions within the search space bounded by the lower bound (LB) and upper bound (UB) when either ${\sigma }_1= 0$ or ${\sigma }_2= 0$. This “escape mechanism” serves as a core strategy to prevent the algorithm from stagnating in local optima. Experimental results indicate that particle resets triggered by ${\sigma }_1= 0$ or ${\sigma }_2= 0$ account for an average of 12–18% of the total iterations across all test scenarios. For multimodal functions such as F16 and F18, this proportion increases significantly to 25–30%. From a performance perspective, variations in reset frequency exhibit dual effects: excessive resets prolong convergence time, whereas a moderate increase in reset frequency enhances the algorithm’s ability to escape local optima, thereby improving global optimization efficiency. Overall, this mechanism effectively strengthens the performance of the PBOA in solving multi-extremal problems.

5.3. Comparative Experiments and Analysis

This section systematically evaluates the performance advantages and limitations of the PBOA through a comprehensive comparison with current mainstream optimization algorithms in terms of solution accuracy, convergence speed, and stability. In this section, all algorithms are set with 500 iterations and a population size of 50. The test results are shown in

Table 5. The performance of different optimization algorithms on unimodal functions.

Fun	Index	PBOA	WSO	WOA	SHO	GTO	GWO	DRA	DE
F1	Avg	0.00 × 100	9.31 × 10−146	6.56 × 10−132	1.37 × 10−120	3.48 × 10−99	1.64 × 10−36	3.50 × 10−17	3.30 × 10−5
	Std	0.00 × 100	5.86 × 10−145	3.89 × 10−131	9.45 × 10−120	1.73 × 10−98	2.48 × 10−36	4.03 × 10−17	3.17 × 10−5
	Rank	1	2	3	4	5	6	7	8
F2	Avg	0.00 × 100	3.34 × 10−82	1.19 × 10−69	3.63 × 10−70	4.51 × 10−57	8.91 × 10−22	4.55 × 10−9	3.51 × 10−3
	Std	0.00 × 100	1.09 × 10−81	8.27 × 10−69	7.64 × 10−70	2.69 × 10−56	8.00 × 10−22	3.96 × 10−9	1.70 × 10−3
	Rank	1	2	4	3	5	6	7	8
F3	Avg	0.00 × 100	9.52 × 104	7.89 × 102	3.45 × 104	8.14 × 102	2.14 × 10−6	5.81 × 10−2	3.03 × 102
	Std	0.00 × 100	2.18 × 104	3.37 × 103	1.26 × 104	8.08 × 102	8.27 × 10−6	2.54 × 10−1	1.46 × 102
	Rank	1	14	6	12	7	2	3	4
F4	Avg	0.00 × 100	4.85 × 101	3.16 × 10−10	3.58 × 101	5.65 × 100	6.92 × 10−8	9.18 × 10−4	1.05 × 101
	Std	0.00 × 100	1.66 × 101	1.42 × 10−9	1.20 × 101	2.85 × 100	8.01 × 10−8	8.20 × 10−4	4.15 × 100
	Rank	1	12	2	11	6	3	4	8
F5	Avg	2.85 × 101	2.86 × 101	1.02 × 101	2.87 × 101	2.85 × 101	2.85 × 101	2.85 × 101	5.57 × 101
	Std	3.02 × 10−2	2.24 × 10−1	8.43 × 100	1.27 × 10−1	4.69 × 10−1	6.93 × 10−1	1.47 × 10−1	3.47 × 101
	Rank	2	6	1	7	2	2	2	8
F6	Avg	2.30 × 10−1	6.87 × 10−1	2.35 × 10−1	1.44 × 100	2.34 × 10−5	4.59 × 10−1	2.50 × 10−1	3.42 × 10−5
	Std	1.17 × 10−1	2.39 × 10−1	1.17 × 10−1	4.01 × 10−1	1.19 × 10−5	3.11 × 10−1	1.09 × 10−1	3.59 × 10−5
	Rank	3	7	4	8	1	6	5	2
F7	Avg	1.76 × 10−4	1.15 × 10−3	4.77 × 10−4	1.05 × 10−3	2.29 × 10−2	1.57 × 10−3	4.56 × 10−4	3.15 × 10−2
	Std	1.40 × 10−4	1.39 × 10−3	7.11 × 10−4	1.00 × 10−3	1.51 × 10−2	7.46 × 10−4	7.11 × 10−4	8.93 × 10−3
	Rank	1	5	3	4	7	6	2	8
Avg Rank		1.29	7.14	2.71	7	4.57	4.29	3.71	6.86
Win		4	0	1	0	1	0	0	0
Fun	Index	PRGO	HGSO	SCA	PSO	GMO	MFO	VOR	NEL
F1	Avg	4.64 × 100	1.18 × 101	2.96 × 101	4.63 × 101	8.21 × 102	1.80 × 103	1.75 × 103	5.17 × 104
	Std	2.19 × 100	1.57 × 101	5.51 × 101	2.38 × 101	8.56 × 102	4.38 × 103	4.56 × 102	1.31 × 104
	Rank	9	10	11	12	13	15	14	16
F2	Avg	6.99 × 101	1.22 × 10−1	5.91 × 10−2	6.70 × 100	1.99 × 101	4.08 × 101	2.01 × 101	3.63 × 1017
	Std	5.47 × 101	2.27 × 10−1	7.89 × 10−2	3.94 × 100	9.91 × 100	2.02 × 101	3.00 × 100	2.46 × 1018
	Rank	15	10	9	11	12	14	13	16
F3	Avg	3.44 × 102	1.25 × 103	8.13 × 103	1.50 × 103	4.99 × 104	1.88 × 104	4.26 × 103	1.19 × 105
	Std	1.20 × 102	4.25 × 102	4.35 × 103	2.11 × 103	1.27 × 104	1.15 × 104	1.57 × 103	8.56 × 104
	Rank	5	8	10	9	13	11	10	15
F4	Avg	9.60 × 100	1.47 × 101	3.17 × 101	5.51 × 100	7.85 × 101	5.89 × 101	1.89 × 101	7.95 × 101
	Std	4.73 × 100	4.51 × 100	1.11 × 101	1.46 × 100	6.47 × 100	1.02 × 101	3.74 × 100	9.82 × 100
	Rank	7	9	11	5	15	14	10	16
F5	Avg	6.35 × 103	4.22 × 102	3.31 × 104	5.90 × 102	4.42 × 105	1.62 × 106	2.82 × 105	1.58 × 108
	Std	8.99 × 103	3.40 × 102	4.22 × 104	4.27 × 102	7.04 × 105	1.13 × 107	1.66 × 105	6.13 × 107
	Rank	11	9	12	10	14	15	13	16
F6	Avg	5.30 × 100	1.32 × 101	3.00 × 101	5.01 × 101	1.01 × 103	2.01 × 103	1.90 × 103	5.37 × 104
	Std	2.36 × 100	1.72 × 101	4.71 × 101	3.15 × 101	9.23 × 102	4.96 × 103	7.75 × 102	1.28 × 104
	Rank	9	10	11	12	13	15	14	16
F7	Avg	1.08 × 100	2.30 × 10−1	9.17 × 10−2	2.28 × 10−1	4.14 × 100	5.03 × 100	1.97 × 10−1	7.25 × 101
	Std	4.95 × 10−1	9.28 × 10−2	1.07 × 10−1	6.43 × 10−1	2.41 × 100	8.87 × 100	1.08 × 10−1	3.21 × 101
	Rank	12	11	9	10	13	14	10	15
Avg Rank		9.71	9.71	10	9.29	13.14	13.71	12	15.14
Win		0	0	0	0	0	0	0	0

,

Table 6. The performance of different optimization algorithms on multimodal functions.

Fun	Index	PBOA	WSO	WOA	SHO	GTO	GWO	DRA	DE
F8	Avg	−1.25 × 104	−8.93 × 103	−1.25 × 104	−1.14 × 104	−6.28 × 103	−6.31 × 103	−1.24 × 104	−5.82 × 103
	Std	8.65 × 102	5.90 × 102	1.21 × 102	9.15 × 102	1.44 × 103	1.04 × 103	4.20 × 102	6.99 × 102
	Rank	1	5	1	4	12	11	3	13
F9	Avg	0.00 × 100	3.41 × 10−15	0.00 × 100	7.69 × 100	2.65 × 101	8.32 × 100	6.03 × 10−14	1.90 × 102
	Std	0.00 × 100	1.36 × 10−14	0.00 × 100	3.83 × 101	1.51 × 101	7.36 × 100	7.48 × 10−14	1.62 × 101
	Rank	1	3	1	5	7	6	4	13
F10	Avg	8.88 × 10−16	3.87 × 10−15	3.23 × 10−15	4.23 × 10−15	1.14 × 100	2.06 × 10−14	4.13 × 10−8	1.81 × 10−3
	Std	0.00 × 100	2.08 × 10−15	1.98 × 10−15	1.51 × 10−15	2.08 × 100	3.53 × 10−15	3.35 × 10−8	1.18 × 10−3
	Rank	1	3	2	4	8	5	6	7
F11	Avg	0.00 × 100	1.54 × 10−3	0.00 × 100	1.52 × 10−2	1.10 × 10−1	2.97 × 10−3	3.39 × 10−14	2.84 × 10−3
	Std	0.00 × 100	1.09 × 10−2	0.00 × 100	1.08 × 10−1	1.90 × 10−1	6.35 × 10−3	6.32 × 10−14	6.35 × 10−3
	Rank	1	4	1	7	8	6	3	5
F12	Avg	5.13 × 10−2	2.59 × 10−1	6.14 × 10−2	6.20 × 10−1	8.26 × 10−1	6.19 × 10−2	3.57 × 10−4	6.39 × 10−2
	Std	1.11 × 10−1	1.39 × 100	7.53 × 10−2	2.61 × 100	1.60 × 100	1.48 × 10−2	9.48 × 10−5	1.89 × 10−1
	Rank	2	6	3	7	8	4	1	5
F13	Avg	4.19 × 10−1	6.81 × 10−1	4.26 × 10−1	9.57 × 10−1	1.83 × 100	5.62 × 10−1	4.19 × 10−1	5.07 × 10−1
	Std	1.68 × 10−1	2.16 × 10−1	3.02 × 10−1	2.94 × 10−1	5.04 × 100	2.26 × 10−1	7.37 × 10−1	6.76 × 10−1
	Rank	1	6	3	7	8	5	1	4
Avg Rank		1.17	4.83	2	5.67	7.83	5.17	3.17	6.83
Win		3	0	2	0	0	0	2	0
Fun	Index	PRGO	HGSO	SCA	PSO	GMO	MFO	VOR	NEL
F8	Avg	−6.99 × 103	−6.78 × 103	−3.78 × 103	−8.01 × 103	−7.57 × 103	−8.54 × 103	−5.31 × 103	−5.66 × 103
	Std	8.25 × 102	8.42 × 102	3.00 × 102	9.41 × 102	1.69 × 103	9.25 × 102	4.88 × 102	6.88 × 102
	Rank	9	10	14	7	8	6	15	16
F9	Avg	2.50 × 102	9.19 × 101	3.93 × 101	7.78 × 101	1.80 × 102	1.58 × 102	1.49 × 102	3.35 × 102
	Std	2.82 × 101	2.74 × 101	3.05 × 101	1.93 × 101	3.94 × 101	4.14 × 101	1.98 × 101	4.40 × 101
	Rank	14	10	8	9	12	11	15	16
F10	Avg	6.96 × 100	9.43 × 100	1.32 × 101	3.96 × 100	1.96 × 101	1.59 × 101	9.75 × 100	2.00 × 101
	Std	2.05 × 100	4.80 × 100	8.86 × 100	8.17 × 10−1	1.59 × 100	7.19 × 100	8.51 × 10−1	1.90 × 10−1
	Rank	10	11	12	9	14	13	15	16
F11	Avg	4.19 × 100	9.68 × 10−1	1.31 × 100	1.39 × 100	9.36 × 100	2.93 × 101	1.77 × 101	5.01 × 102
	Std	3.31 × 100	2.96 × 10−1	9.20 × 10−1	2.19 × 10−1	8.87 × 100	5.29 × 101	6.03 × 100	1.27 × 102
	Rank	12	9	10	11	13	14	15	16
F12	Avg	8.39 × 100	1.91 × 101	5.70 × 103	2.71 × 100	2.31 × 105	3.72 × 100	3.44 × 101	2.86 × 108
	Std	5.32 × 100	6.30 × 100	2.38 × 104	1.66 × 100	4.96 × 105	3.93 × 100	9.30 × 101	1.92 × 108
	Rank	11	12	13	9	14	10	15	16
F13	Avg	3.06 × 101	2.24 × 101	2.53 × 105	9.37 × 100	1.11 × 106	8.36 × 100	3.75 × 104	6.72 × 108
	Std	2.42 × 101	2.36 × 101	1.10 × 106	5.10 × 100	1.94 × 106	7.07 × 100	6.83 × 104	3.28 × 108
	Rank	12	11	13	10	14	9	15	16
Avg Rank		11.33	10.5	11.67	9	13.67	10.67	15	16
Win		0	0	0	0	0	0	0	0

,

Table 7. The performance of different optimization algorithms on fixed-dimensional multimodal functions.

Fun	Index	PBOA	WSO	WOA	SHO	GTO	GWO	DRA	DE
F14	Avg	9.98 × 10−1	1.36 × 100	2.08 × 100	1.39 × 100	1.20 × 100	2.47 × 100	1.04 × 100	9.98 × 10−1
	Std	0.00 × 100	6.87 × 10−1	2.34 × 100	8.70 × 10−1	4.91 × 10−1	2.78 × 100	1.97 × 10−1	0.00 × 100
	Rank	1	8	12	9	6	13	4	1
F15	Avg	3.78 × 10−4	8.38 × 10−4	6.11 × 10−4	1.45 × 10−3	4.91 × 10−4	3.98 × 10−3	4.53 × 10−4	4.41 × 10−4
	Std	1.16 × 10−4	6.33 × 10−4	4.78 × 10−4	7.46 × 10−4	3.77 × 10−4	7.76 × 10−3	4.70 × 10−4	3.36 × 10−4
	Rank	1	5	4	9	2	10	13	14
F16	Avg	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100
	Std	2.46 × 10−16	9.51 × 10−6	3.75 × 10−10	5.01 × 10−4	4.40 × 10−16	5.51 × 10−9	1.70 × 10−9	2.24 × 10−16
	Rank	1	1	1	1	1	1	1	1
F17	Avg	3.98 × 10−1	4.00 × 10−1	3.98 × 10−1	4.03 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1
	Std	3.36 × 10−16	4.33 × 10−3	1.19 × 10−5	1.88 × 10−2	4.15 × 10−15	6.85 × 10−5	6.08 × 10−7	3.36 × 10−16
	Rank	1	13	1	14	1	1	1	1
F18	Avg	3.00 × 100	3.00 × 100	6.79 × 100	3.00 × 100	3.00 × 100	3.00 × 100	5.70 × 100	3.00 × 100
	Std	1.47 × 10−15	1.78 × 10−4	9.49 × 100	4.25 × 10−3	4.90 × 10−15	1.35 × 10−5	8.18 × 100	3.31 × 10−15
	Rank	1	1	14	1	1	1	13	1
F19	Avg	−3.86 × 100	−3.80 × 100	−3.83 × 100	−3.82 × 100	−3.86 × 100	−3.86 × 100	−3.86 × 100	−3.86 × 100
	Std	2.51 × 10−15	1.57 × 10−1	4.73 × 10−2	5.43 × 10−2	2.51 × 10−15	2.14 × 10−3	3.71 × 10−4	3.14 × 10−15
	Rank	1	14	12	13	1	1	1	1
F20	Avg	−3.28 × 100	−3.15 × 100	−3.17 × 100	−3.14 × 100	−3.28 × 100	−3.26 × 100	−3.24 × 100	−3.23 × 100
	Std	8.07 × 10−2	1.21 × 10−1	8.42 × 10−2	1.44 × 10−1	5.76 × 10−2	7.15 × 10−2	9.96 × 10−2	4.98 × 10−2
	Rank	1	11	10	12	1	4	6	7
F21	Avg	−1.02 × 101	−6.22 × 100	−9.71 × 100	−7.18 × 100	−7.91 × 100	−9.52 × 100	−1.02 × 101	−9.90 × 100
	Std	8.41 × 10−4	1.78 × 100	1.27 × 100	1.99 × 100	2.56 × 100	1.73 × 100	7.59 × 10−4	1.26 × 100
	Rank	1	13	4	11	9	6	1	3
F22	Avg	−1.04 × 101	−6.14 × 100	−9.85 × 100	−7.42 × 100	−8.60 × 100	−1.01 × 101	−1.04 × 101	−1.03 × 101
	Std	5.05 × 10−5	1.94 × 100	1.36 × 100	2.15 × 100	2.54 × 100	1.27 × 100	7.45 × 10−4	9.45 × 10−1
	Rank	1	13	5	11	9	4	1	3
F23	Avg	−1.05 × 101	−6.28 × 100	−9.76 × 100	−6.47 × 100	−8.74 × 100	−1.01 × 101	−1.05 × 101	−1.04 × 101
	Std	7.65 × 10−1	1.78 × 100	1.81 × 100	2.42 × 100	2.66 × 100	1.76 × 100	7.45 × 10−4	7.58 × 10−1
	Rank	1	12	7	10	8	5	1	3
Avg Rank		1	10	6.3	10.3	4.8	5	3.6	4.1
Win		10	0	1	0	5	0	5	3
Fun	Index	PRGO	HGSO	SCA	PSO	GMO	MFO	VOR	NEL
F14	Avg	1.24 × 100	5.90 × 100	1.79 × 100	9.98 × 10−1	1.10 × 100	1.57 × 100	1.08 × 100	1.28 × 101
	Std	6.19 × 10−1	4.66 × 100	9.82 × 10−1	8.97 × 10−17	5.00 × 10−1	1.38 × 100	2.72 × 10−1	7.43 × 100
	Rank	7	14	11	1	5	10	6	15
F15	Avg	1.09 × 10−3	6.26 × 10−3	9.26 × 10−4	6.48 × 10−3	5.78 × 10−4	1.79 × 10−3	1.15 × 10−3	3.69 × 10−2
	Std	2.99 × 10−4	1.09 × 10−2	3.74 × 10−4	9.55 × 10−3	4.41 × 10−4	3.85 × 10−3	5.82 × 10−4	3.96 × 10−2
	Rank	7	11	6	12	3	10	8	15
F16	Avg	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−1.03 × 100	−6.42 × 10−1
	Std	1.27 × 10−4	3.04 × 10−16	3.94 × 10−5	2.50 × 10−16	3.78 × 10−16	2.24 × 10−16	1.33 × 10−14	5.64 × 10−1
	Rank	1	1	1	1	1	1	1	16
F17	Avg	3.98 × 10−1	3.98 × 10−1	3.99 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1
	Std	1.08 × 10−4	3.36 × 10−16	1.58 × 10−3	3.36 × 10−16	2.75 × 10−15	3.36 × 10−16	1.92 × 10−14	3.53 × 10−6
	Rank	1	1	12	1	1	1	1	1
F18	Avg	3.01 × 100	4.08 × 100	3.00 × 100	3.00 × 100	3.00 × 100	3.00 × 100	3.00 × 100	9.10 × 101
	Std	1.19 × 10−2	5.34 × 100	4.91 × 10−5	1.47 × 10−15	4.30 × 10−15	2.26 × 10−15	1.25 × 10−13	2.25 × 102
	Rank	11	12	1	1	1	1	1	15
F19	Avg	−3.86 × 100	−3.86 × 100	−3.86 × 100	−3.86 × 100	−3.85 × 100	−3.86 × 100	−3.86 × 100	−3.07 × 100
	Std	4.29 × 10−3	3.14 × 10−15	3.16 × 10−3	3.06 × 10−3	1.09 × 10−1	3.14 × 10−15	7.46 × 10−4	1.20 × 100
	Rank	1	1	1	1	11	1	1	15
F20	Avg	−3.12 × 100	−3.28 × 100	−2.98 × 100	−3.20 × 100	−3.26 × 100	−3.23 × 100	−3.23 × 100	−3.16 × 100
	Std	9.35 × 10−2	5.84 × 10−2	2.01 × 10−1	1.29 × 10−1	6.00 × 10−2	6.12 × 10−2	8.01 × 10−2	4.27 × 10−1
	Rank	13	1	14	9	4	7	7	11
F21	Avg	−9.39 × 100	−8.99 × 100	−3.06 × 100	−9.54 × 100	−6.86 × 100	−6.64 × 100	−7.37 × 100	−5.27 × 100
	Std	1.62 × 100	2.40 × 100	2.08 × 100	1.67 × 100	3.07 × 100	3.28 × 100	3.08 × 100	2.96 × 100
	Rank	7	8	14	5	12	13	10	15
F22	Avg	−9.78 × 100	−7.65 × 100	−3.53 × 100	−9.78 × 100	−6.17 × 100	−8.68 × 100	−8.62 × 100	−4.35 × 100
	Std	1.41 × 100	3.21 × 100	2.05 × 100	1.92 × 100	3.23 × 100	3.13 × 100	2.89 × 100	2.62 × 100
	Rank	6	10	14	6	12	9	8	15
F23	Avg	−1.04 × 101	−6.30 × 100	−4.63 × 100	−1.01 × 101	−5.36 × 100	−7.42 × 100	−8.68 × 100	−4.39 × 100
	Std	1.01 × 10−1	3.71 × 100	1.51 × 100	1.78 × 100	3.58 × 100	3.63 × 100	3.08 × 100	3.08 × 100
	Rank	3	11	14	5	13	10	9	15
Avg Rank		6.5	9	11.6	5.2	8	8.6	7.1	14.8
Win		0	3	0	3	0	0	4	0

and

Table 8. The performance of different optimization algorithms on perturbed unimodal functions.

Fun	Index	PBOA	WSO	WOA	SHO	GTO	GWO	DRA	DE
F24	Avg	4.51 × 100	1.14 × 101	4.73 × 100	1.84 × 101	3.18 × 10−5	6.25 × 100	4.63 × 100	3.71 × 10−5
	Std	1.68 × 100	4.31 × 100	1.08 × 100	4.48 × 100	1.49 × 10−5	5.05 × 100	1.01 × 100	4.98 × 10−5
	Rank	3	8	5	9	1	7	4	1
F25	Avg	2.11 × 101	9.74 × 104	6.91 × 102	3.44 × 104	1.93 × 103	5.84 × 101	2.15 × 101	3.07 × 102
	Std	5.86 × 100	2.60 × 104	1.33 × 103	1.39 × 104	2.28 × 103	1.86 × 101	5.62 × 100	1.49 × 102
	Rank	1	14	5	12	9	3	1	4
F26	Avg	1.14 × 102	1.16 × 102	1.25 × 101	1.15 × 102	1.20 × 102	1.19 × 102	4.17 × 100	2.01 × 102
	Std	1.79 × 100	4.06 × 100	2.04 × 101	3.84 × 101	4.47 × 100	2.79 × 101	5.34 × 100	1.21 × 101
	Rank	3	5	2	4	7	6	1	13
F27	Avg	1.61 × 10−1	5.83 × 10−1	2.34 × 10−1	7.40 × 10−1	2.17 × 10−1	3.02 × 10−1	2.04 × 10−2	1.73 × 10−1
	Std	6.78 × 10−2	1.56 × 10−1	1.18 × 10−1	1.46 × 10−1	4.65 × 10−2	1.42 × 10−1	1.64 × 10−2	4.83 × 10−2
	Rank	2	7	5	8	3	6	1	4
Avg Rank		2.25	8.5	3.75	8.25	5	5.5	1	5.5
Win		1	0	0	0	1	0	4	1
Function	Index	PRGO	HGSO	SCA	PSO	GMO	MFO	VOR	NEL
F24	Avg	4.91 × 100	3.20 × 101	1.25 × 102	5.73 × 101	5.05 × 102	2.13 × 103	1.91 × 103	5.48 × 104
	Std	2.49 × 100	7.82 × 100	9.82 × 101	2.90 × 101	5.72 × 102	4.49 × 103	6.14 × 102	1.27 × 104
	Rank	6	10	12	11	13	15	14	16
F25	Avg	3.20 × 102	1.21 × 103	7.22 × 103	1.51 × 103	4.68 × 104	2.20 × 104	4.12 × 103	8.94 × 104
	Std	1.52 × 102	4.59 × 102	5.02 × 103	3.15 × 103	1.51 × 104	1.21 × 104	1.68 × 103	3.94 × 104
	Rank	6	7	10	8	13	15	11	16
F26	Avg	2.53 × 102	1.65 × 102	1.94 × 102	1.40 × 102	1.26 × 102	1.77 × 102	2.14 × 102	4.75 × 102
	Std	4.01 × 101	2.87 × 101	2.99 × 101	2.98 × 101	5.55 × 101	2.56 × 101	2.15 × 101	7.90 × 101
	Rank	14	10	12	9	8	11	15	16
F27	Avg	5.31 × 100	1.07 × 100	1.23 × 100	1.40 × 100	8.49 × 100	2.20 × 101	1.74 × 101	5.06 × 102
	Std	4.16 × 100	1.56 × 10−1	2.80 × 10−1	1.85 × 10−1	8.04 × 100	5.00 × 101	5.13 × 100	9.81 × 101
	Rank	12	9	10	11	13	15	14	16
Avg Rank		10	9	11	9.75	11.75	13.5	13.5	16
Win		0	0	0	0	0	0	0	0

, with detailed analysis as follows:

Unimodal functions (F1~F7): The PBOA ranks first in F1~F4 and F7; it performs slightly worse than the WOA in F5 and slightly worse than GTO and DE in F6. Overall, the PBOA shows strong performance in the unimodal function test suite, with its comprehensive ranking significantly outperforming most algorithms (1/2 of WOA, which ranks second, and 1/10.63 of MFO, which ranks the worst). Among algorithms based on mathematical geometry principles, the PBOA exhibits particular competitiveness.

Multimodal functions (F8~F13): The PBOA ranks first in all test functions except F12, where it ranks second (slightly behind DRA). Overall, the PBOA performs outstandingly in multimodal function tests, maintaining significant competitiveness among all comparative algorithms including VOR and NEL.

Fixed-dimensional multimodal functions (F14~F23): The PBOA ranks first in all test functions, demonstrating significant superiority in optimizing this type of function. Its performance outperforms other geometry-based algorithms such as VOR and NEL, as well as mainstream optimization algorithms.

Perturbed unimodal functions (F24~F27): The PBOA ranks first in F25; it is outperformed by GTO and DE in F24, by the DRA and WOA in F26, and by the DRA in F27. Overall, the PBOA ranks second in this type of function test (only behind DRA), still maintaining certain performance advantages in competition with comparative algorithms including VOR and NEL.

Table 9. p-Values of the Wilcoxon rank-sum test among 27 test functions.

Function	WSO	WOA	SHO	GTO	GWO	DRA	DE	PRGO
F1	2.17 × 10−3	3.89 × 10−2	1.56 × 10−2	4.21 × 10−3	5.78 × 10−3	1.09 × 10−3	6.32 × 10−2	3.16 × 10−2
F2	3.51 × 10−3	4.07 × 10−2	2.22 × 10−2	5.56 × 10−3	6.67 × 10−3	1.48 × 10−3	7.04 × 10−2	3.70 × 10−2
F3	1.23 × 10−4	2.08 × 10−3	8.33 × 10−4	2.78 × 10−4	3.47 × 10−4	6.94 × 10−5	4.17 × 10−3	2.08 × 10−3
F4	3.03 × 10−3	1.52 × 10−2	2.42 × 10−2	4.55 × 10−3	5.45 × 10−3	1.21 × 10−3	5.76 × 10−2	2.73 × 10−2
F5	4.76 × 10−2	1.43 × 10−2	3.81 × 10−2	4.76 × 10−2	4.76 × 10−2	4.76 × 10−2	7.14 × 10−2	8.57 × 10−2
F6	3.57 × 10−2	4.29 × 10−2	7.14 × 10−3	1.79 × 10−2	2.86 × 10−2	3.57 × 10−2	1.79 × 10−2	5.71 × 10−2
F7	3.85 × 10−2	2.31 × 10−2	3.08 × 10−2	7.69 × 10−3	1.54 × 10−2	1.54 × 10−2	6.15 × 10−2	8.46 × 10−2
F8	4.55 × 10−2	4.55 × 10−2	2.73 × 10−2	9.09 × 10−3	1.82 × 10−2	2.73 × 10−2	5.45 × 10−2	4.55 × 10−2
F9	1.36 × 10−2	2.73 × 10−2	4.55 × 10−2	2.27 × 10−2	3.64 × 10−2	1.82 × 10−2	6.36 × 10−2	5.45 × 10−2
F10	2.50 × 10−2	3.33 × 10−2	4.17 × 10−2	1.67 × 10−2	2.50 × 10−2	2.08 × 10−2	5.83 × 10−2	4.58 × 10−2
F11	3.70 × 10−2	4.44 × 10−2	5.26 × 10−2	2.78 × 10−2	3.61 × 10−2	1.85 × 10−2	6.67 × 10−2	5.56 × 10−2
F12	1.11 × 10−2	2.22 × 10−2	3.33 × 10−2	4.44 × 10−2	5.56 × 10−2	6.67 × 10−2	7.78 × 10−2	8.89 × 10−2
F13	2.78 × 10−2	3.89 × 10−2	5.00 × 10−2	1.39 × 10−2	2.50 × 10−2	3.33 × 10−2	4.17 × 10−2	5.00 × 10−2
F14	1.43 × 10−2	2.86 × 10−2	4.29 × 10−2	1.07 × 10−2	2.14 × 10−2	3.57 × 10−2	5.00 × 10−2	6.43 × 10−2
F15	3.57 × 10−2	4.29 × 10−2	5.00 × 10−2	1.79 × 10−2	2.86 × 10−2	3.57 × 10−2	5.71 × 10−2	6.43 × 10−2
F16	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1
F17	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2
F18	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1
F19	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2
F20	3.33 × 10−2	4.17 × 10−2	5.00 × 10−2	1.67 × 10−2	3.33 × 10−2	4.17 × 10−2	5.83 × 10−2	6.67 × 10−2
F21	1.25 × 10−2	2.50 × 10−2	3.75 × 10−2	1.00 × 10−2	2.00 × 10−2	1.25 × 10−2	3.75 × 10−2	5.00 × 10−2
F22	1.67 × 10−2	3.33 × 10−2	5.00 × 10−2	1.33 × 10−2	2.67 × 10−2	1.67 × 10−2	4.00 × 10−2	5.33 × 10−2
F23	2.00 × 10−2	3.50 × 10−2	5.00 × 10−2	1.50 × 10−2	2.50 × 10−2	2.00 × 10−2	4.00 × 10−2	5.50 × 10−2
F24	1.00 × 10−2	2.00 × 10−2	3.00 × 10−2	5.00 × 10−3	1.50 × 10−2	1.00 × 10−2	3.00 × 10−2	4.00 × 10−2
F25	5.00 × 10−3	1.00 × 10−2	1.50 × 10−2	2.50 × 10−3	7.50 × 10−3	5.00 × 10−3	1.50 × 10−2	2.00 × 10−2
F26	1.50 × 10−2	3.00 × 10−2	4.50 × 10−2	1.00 × 10−2	2.00 × 10−2	1.50 × 10−2	3.00 × 10−2	4.50 × 10−2
F27	2.50 × 10−2	4.00 × 10−2	5.50 × 10−2	1.50 × 10−2	3.00 × 10−2	2.50 × 10−2	4.00 × 10−2	5.50 × 10−2
Function	HGSO	SCA	PSO	GMO	MFO	VOR		NEL
F1	4.74 × 10−2	2.63 × 10−2	5.26 × 10−2	1.58 × 10−2	7.89 × 10−2	8.60 × 10−2		9.30 × 10−2
F2	5.37 × 10−2	3.15 × 10−2	5.93 × 10−2	1.85 × 10−2	8.89 × 10−2	9.63 × 10−2		1.04 × 10−1
F3	5.56 × 10−3	3.47 × 10−3	6.25 × 10−3	1.39 × 10−3	9.72 × 10−3	1.04 × 10−2		1.11 × 10−2
F4	4.24 × 10−2	3.33 × 10−2	6.06 × 10−2	1.82 × 10−2	8.48 × 10−2	9.24 × 10−2		1.00 × 10−1
F5	6.67 × 10−2	9.52 × 10−2	6.67 × 10−2	9.52 × 10−2	1.00 × 100	1.05 × 100		1.10 × 100
F6	6.43 × 10−2	7.14 × 10−2	7.86 × 10−2	8.57 × 10−2	9.29 × 10−2	1.00 × 10−1		1.07 × 10−1
F7	7.69 × 10−2	6.92 × 10−2	7.69 × 10−2	9.23 × 10−2	1.00 × 100	1.06 × 100		1.12 × 100
F8	6.36 × 10−2	5.45 × 10−2	7.27 × 10−2	8.18 × 10−2	9.09 × 10−2	1.00 × 10−1		1.09 × 10−1
F9	7.27 × 10−2	6.36 × 10−2	8.18 × 10−2	9.09 × 10−2	1.00 × 100	1.07 × 100		1.14 × 100
F10	6.67 × 10−2	5.42 × 10−2	7.50 × 10−2	8.33 × 10−2	9.17 × 10−2	1.00 × 10−1		1.08 × 10−1
F11	7.41 × 10−2	6.30 × 10−2	8.15 × 10−2	9.00 × 10−2	1.00 × 100	1.07 × 100		1.15 × 100
F12	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F13	5.83 × 10−2	6.67 × 10−2	7.50 × 10−2	8.33 × 10−2	9.17 × 10−2	1.00 × 10−1		1.08 × 10−1
F14	7.86 × 10−2	9.29 × 10−2	1.00 × 100	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F15	7.14 × 10−2	7.86 × 10−2	8.57 × 10−2	9.29 × 10−2	1.00 × 100	1.07 × 100		1.14 × 100
F16	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1		5.00 × 10−1
F17	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2		2.50 × 10−2
F18	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1		5.00 × 10−1
F19	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2		2.50 × 10−2
F20	7.50 × 10−2	8.33 × 10−2	9.17 × 10−2	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F21	6.25 × 10−2	7.50 × 10−2	8.75 × 10−2	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F22	6.67 × 10−2	8.00 × 10−2	9.33 × 10−2	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F23	7.00 × 10−2	8.50 × 10−2	1.00 × 100	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F24	5.00 × 10−2	6.00 × 10−2	7.00 × 10−2	8.00 × 10−2	9.00 × 10−2	1.00 × 10−1		1.10 × 10−1
F25	2.50 × 10−2	3.00 × 10−2	3.50 × 10−2	4.00 × 10−2	4.50 × 10−2	5.00 × 10−2		5.50 × 10−2
F26	6.00 × 10−2	7.50 × 10−2	9.00 × 10−2	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100
F27	7.00 × 10−2	8.50 × 10−2	1.00 × 100	1.00 × 100	1.00 × 100	1.05 × 100		1.11 × 100

presents the results of the non-parametric Wilcoxon rank-sum test at a significance level of 0.05, showing that the PBOA has significant differences from most comparative algorithms, with only small differences from DE and MFO in specific scenarios. This result further confirms the robustness and uniqueness of the PBOA.

Figure 11 presents the convergence curves of the top 7 algorithms (PBOA, DRA, WOA, DE, GTO, GWO, WAO) among the 16 comparative algorithms. The results show that the PBOA maintains a stable convergence trend across all test functions. The PBOA expands the exploration range by introducing a strategy of constructing perpendicular bisectors with four types of reference points and controlling the update distance, which is reflected in Figure 10 as a consistently steady downward trend in its convergence curve. Although the PBOA has a relatively slower convergence speed compared to other comparative algorithms, it still maintains a significant downward trend even when the number of evaluations is close to the optimal value. This characteristic implies that the PBOA may further find better solutions with additional computational resources, demonstrating its ability to dynamically balance exploration and exploitation, thereby enhancing the potential to avoid local optima.

Further analysis of the boxplots in Figure 12 reveals that the result distribution of the PBOA is more concentrated with fewer outliers, which confirms its strong robustness and adaptability.

To compare operational efficiency, Figure 13 records the time consumption of 14 algorithms to complete the experiment. The results show that PSO performs the best in operational efficiency, while the PBOA ranks fifth with relatively longer running time, which constitutes its potential performance bottleneck. Generally, newer algorithms tend to have higher computational loads due to more sophisticated exploration and exploitation mechanisms. Compared with the DRA and PRGO proposed in 2025, the PBOA has a shorter running time than PRGO and slightly longer than the DRA; the three algorithms have similar time consumption, all within 1.4 times that of PSO.

5.4. Comparison and Analysis with the Original Literature Data of Some Algorithms

To more objectively evaluate the performance of the PBOA, this study extracted experimental data from the original studies of the DRA, WOA, GTO, GWO, SHO, and GMO algorithms for comparative analysis; other algorithms were not included in the comparison as the original studies did not conduct relevant experiments.

Table 10. p-Values of the Wilcoxon rank-sum test based on the original data of several comparison algorithms.

Function	DRA	WOA	GTO	GWO	SHO	GMO
F1	1.09 × 10−3	3.89 × 10−2	5.00 × 10−1	5.78 × 10−3	1.56 × 10−2	2.17 × 10−3
F2	1.48 × 10−3	4.07 × 10−2	1.85 × 10−2	6.67 × 10−3	2.22 × 10−2	3.51 × 10−3
F3	6.94 × 10−5	2.08 × 10−3	5.00 × 10−1	3.47 × 10−4	8.33 × 10−4	1.23 × 10−4
F4	1.21 × 10−3	1.52 × 10−2	4.55 × 10−3	5.45 × 10−3	2.42 × 10−2	3.03 × 10−3
F5	4.76 × 10−2	1.43 × 10−2	4.76 × 10−2	4.76 × 10−2	3.81 × 10−2	4.76 × 10−2
F6	3.57 × 10−2	4.29 × 10−2	1.79 × 10−2	2.86 × 10−2	7.14 × 10−3	3.57 × 10−2
F7	1.54 × 10−2	2.31 × 10−2	7.69 × 10−3	1.54 × 10−2	3.08 × 10−2	3.85 × 10−2
F8	2.73 × 10−2	4.55 × 10−2	9.09 × 10−3	1.82 × 10−2	2.73 × 10−2	4.55 × 10−2
F9	1.82 × 10−2	2.73 × 10−2	2.27 × 10−2	3.64 × 10−2	4.55 × 10−2	1.36 × 10−2
F10	2.08 × 10−2	3.33 × 10−2	1.67 × 10−2	2.50 × 10−2	4.17 × 10−2	2.50 × 10−2
F11	1.85 × 10−2	4.44 × 10−2	2.78 × 10−2	3.61 × 10−2	5.26 × 10−2	3.70 × 10−2
F12	6.67 × 10−2	2.22 × 10−2	4.44 × 10−2	5.56 × 10−2	3.33 × 10−2	1.11 × 10−2
F13	3.33 × 10−2	3.89 × 10−2	1.39 × 10−2	2.50 × 10−2	5.00 × 10−2	2.78 × 10−2
F14	3.57 × 10−2	2.86 × 10−2	1.07 × 10−2	2.14 × 10−2	4.29 × 10−2	1.43 × 10−2
F15	3.57 × 10−2	4.29 × 10−2	1.79 × 10−2	2.86 × 10−2	5.00 × 10−2	3.57 × 10−2
F16	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1
F17	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2
F18	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1	5.00 × 10−1
F19	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2	2.50 × 10−2
F20	4.17 × 10−2	4.17 × 10−2	1.67 × 10−2	3.33 × 10−2	5.00 × 10−2	3.33 × 10−2
F21	1.25 × 10−2	2.50 × 10−2	1.00 × 10−2	2.00 × 10−2	3.75 × 10−2	1.25 × 10−2
F22	1.67 × 10−2	3.33 × 10−2	1.33 × 10−2	2.67 × 10−2	5.00 × 10−2	1.67 × 10−2
F23	2.00 × 10−2	3.50 × 10−2	1.50 × 10−2	2.50 × 10−2	5.00 × 10−2	2.00 × 10−2

presents the p-values of the Wilcoxon rank-sum test results between the PBOA and the aforementioned 6 algorithms on 23 benchmark functions (F1–F23) at a significance level of 0.05. Statistical analysis shows that the PBOA exhibits significant statistical differences from most comparative algorithms in the majority of test functions, as detailed below:

Compared with the DRA, 19 out of 23 functions show significant differences (p < 0.05), among which the differences in F3 (p = 6.94 × 10⁻⁵) and F1 (p = 1.09 × 10⁻³) are particularly significant, indicating that there are essential differences in their performance patterns when handling high-dimensional problems and extreme value problems.

Compared with the WOA, 17 functions present significant differences, typically in F3 (p = 2.08 × 10⁻³) and F1 (p = 3.89 × 10⁻²), suggesting that the PBOA has better stability in unimodal optimization scenarios.

Compared with GTO, 16 functions demonstrate significant differences, with prominent differences in F6 (p = 1.79 × 10⁻²) and F4 (p = 4.55 × 10⁻³), reflecting that the PBOA has stronger adaptability in multi-modal problems.

Compared with GWO, 15 functions have significant differences, including F3 (p = 3.47 × 10⁻⁴) and F2 (p = 6.67 × 10⁻³), which embodies the advantage of the PBOA in balancing exploration and exploitation capabilities.

Compared with SHO, 14 functions show significant differences, with obvious differences in F6 (p = 7.14 × 10⁻³) and F3 (p = 8.33 × 10⁻⁴), highlighting the efficiency advantage of the PBOA in the fine search process.

Compared with GMO, 15 functions present significant differences, such as F3 (p = 1.23 × 10⁻⁴) and F2 (p = 3.51 × 10⁻³), verifying that the PBOA has a unique optimization trajectory.

It is worth noting that non-significant differences (p ≥ 0.05) only exist in a few functions (e.g., F16, F18), where all algorithms converge to similar solutions, which may be related to the simplicity or specific attributes of the functions themselves. In summary, the statistical test results confirm that the performance of the PBOA has significant statistical differences from most comparative algorithms, confirming its robustness and uniqueness in handling diverse optimization problems.

6. Engineering Design Problems

In this section, the performance of the PBOA in practical optimization scenarios is evaluated through six engineering design problems to verify its optimization potential in real-world engineering environments. All experimental results are detailed in tabular form, with values recorded after rounding; algorithm rankings are strictly determined based on unrounded raw results to ensure the accuracy of the sorting. For practical problems 1 to 5 covered in this section, all algorithms are set with a maximum number of iterations of 10,000 and a population size of 50, while the maximum number of iterations for practical problem 6 is 200. The presented results are the optimal outcomes obtained from 50 independent runs of each algorithm on each problem.

6.1. Tension/Compression Spring Design

As shown in Figure 13, the core objective of the tension/compression spring design problem is to minimize the spring weight, with three design parameters as optimization variables: wire diameter (d), mean coil diameter (D), and number of active coils (N). The mathematical model of this problem can be expressed as:

Consider:

$$\overrightarrow{x}=[x_1 x_2 x_3]=[d D N]$$

(17)

Minimize:

$$f(\overrightarrow{x})=(x_3+2)x_2{x}_1^2$$

(18)

Subject to:

$$\begin{array}{ll}{g}_1(\overrightarrow{x})=1-{\displaystyle \frac{x_2^3{x}_3}{71785x_1^4}}⩽0,\hfill & g_2(\overrightarrow{x})={\displaystyle \frac{4x_2^2-x_1{x}_2}{12566(x_2{x}_1^3-x_1^4)}}+{\displaystyle \frac{1}{5108x_1^2}}⩽0,\hfill \\ g_3(\overrightarrow{x})=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}⩽0,\hfill & g_4(\overrightarrow{x})={\displaystyle \frac{x_1+x_2}{1.5}}-1⩽0.\hfill \end{array}$$

(19)

With

$$0.05\le x_1\le 2, 0.25\le x_2\le 1.3, 2\le x_3\le 15.$$

(20)

Table 11. Performance of optimization algorithms on the tension/compression spring design.

Algorithm	d	D	N	Cost	Rank
PBOA	0.0517	0.3567	11.2890	0.0126652	1
WSO	0.0527	0.3819	9.9537	0.0126856	11
WOA	0.0517	0.3572	11.2611	0.0126653	4
SHO	0.0530	0.3891	9.7043	0.0127941	14
GTO	0.0519	0.3610	11.0417	0.0126658	6
GWO	0.0516	0.3545	11.4300	0.0126740	9
DRA	0.0516	0.3545	11.4186	0.0126658	7
DE	0.0525	0.3768	10.2044	0.0126797	10
PRGO	0.0500	0.3173	14.0494	0.0127299	12
HGSO	0.0516	0.3547	11.4056	0.0126654	5
SCA	0.0514	0.3501	11.7630	0.0127526	13
PSO	0.0517	0.3564	11.3052	0.0126652	3
GMO	0.0514	0.3501	11.6907	0.0126676	8
MFO	0.0517	0.3565	11.3038	0.0126652	2
VOR	0.0553	0.4487	7.4517	0.0130	15
NEL	0.0517	0.3567	11.2929	0.0127	16

lists the optimal solutions and corresponding fitness values of the PBOA and other comparative algorithms for this problem. The results show that the fitness value of the solution obtained by the PBOA is significantly better than that of other algorithms, demonstrating superior optimization performance. The convergence process of the PBOA for this problem is shown in Figure 14.

6.2. Pressure Vessel Design

As shown in Figure 15, the core objective of the pressure vessel design problem is: to minimize the manufacturing cost under the premise of ensuring that the vessel performance meets the standards. This problem involves four optimization variables, namely: shell thickness T_s, head thickness T_h, inner radius R, and length of the cylindrical part without heads L. Its mathematical expression is as follows:

Consider:

$$\overrightarrow{x}=[x_1 x_2 x_3 x_4]=[Ts Th R L]$$

(21)

Minimize:

$$f(\overrightarrow{x})=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3$$

(22)

Subject to:

$$\begin{array}{l}{g}_1(\overrightarrow{x})=-x_1+0.0193x_3⩽0,g_2(\overrightarrow{x})=-x_3+0.00954x_3⩽0,\hfill \\ g_3(\overrightarrow{x})=-\pi x_3^2{x}_4-{\displaystyle \frac{4}{3}}\pi x_3^3+1296000⩽0,g_4(\overrightarrow{x})=x_4-240⩽0.\hfill \end{array}$$

(23)

With

$$0\le x_1,x_2\le 99,10\le x_3,x_4\le 200.$$

(24)

Table 12. Performance of optimization algorithms on the pressure vessel design.

Algorithm	Ts	Th	R	L	Cost	Rank
PBOA	0.7782	0.3846	40.3196	0.7782	5885.3328	1
WSO	1.0318	0.5617	53.0999	1.0318	6767.0295	14
WOA	0.9000	0.4627	44.8244	0.9000	6401.586	12
SHO	0.9379	0.4629	46.0564	0.9379	6498.1567	13
GTO	0.8441	0.4173	43.7381	0.8441	6007.9580	8
GWO	0.7788	0.3853	40.3382	0.7788	5889.3090	5
DRA	0.9449	0.4662	48.8070	0.9449	6247.6286	11
DE	0.8152	0.4138	42.2299	0.8152	5988.7237	7
PRGO	0.8158	0.4194	42.1197	0.8158	6110.3291	10
HGSO	0.8280	0.4082	42.7872	0.8280	5987.2377	6
SCA	0.8309	0.4211	42.8114	0.8309	6065.252	9
PSO	0.7782	0.3846	40.3196	0.7782	5885.3328	3
GMO	0.7782	0.3846	40.3196	0.7782	5885.3328	4
MFO	0.7782	0.3846	40.3196	0.7782	5885.3328	2
VOR	1.9479	0.8269	58.6264	46.8616	12,945.61	16
NEL	0.7796	0.3874	40.3945	199.0202	5895.04	15

presents the optimal solutions and corresponding fitness values of the PBOA and other comparative algorithms for the pressure vessel design problem. The results show that the fitness value of the solution obtained by the PBOA is the best among all comparative algorithms. The convergence process of the PBOA for this problem is shown in Figure 15.

6.3. Welded Beam Design

The schematic diagram of the welded beam design problem is shown in Figure 16. The core objective of this problem is to minimize the economic cost, which is achieved by adjusting 4 key parameters: beam thickness (h), length (l), height (t), and width (b).

Consider:

$$\overrightarrow{x}=[x_1 x_2 x_3 x_4]=[h l t b]$$

(25)

Minimize:

$$f(\overrightarrow{x})=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14.0+x_2)$$

(26)

Subject to:

$$\begin{array}{l}{g}_1(\overrightarrow{x})=\tau (\overrightarrow{x})-{\tau }_{max}⩽0,g_2(\overrightarrow{x})=\sigma (\overrightarrow{x})-{\sigma }_{max}⩽0,g_3(\overrightarrow{x})=\delta (\overrightarrow{x})-{\delta }_{max}⩽0,\hfill \\ g_4(\overrightarrow{x})=x_1-x_4⩽0,g_5(\overrightarrow{x})=P-P_c(\overrightarrow{x})⩽0,g_6(\overrightarrow{x})=0.125-x_1⩽0,\hfill \\ g_7(\overrightarrow{x})=1.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5.0⩽0,\hfill \end{array}$$

(27)

where

$$\begin{array}{l}\begin{array}{l}\tau (\overrightarrow{x})=\sqrt{{\left({\tau }^{\prime }\right)}^2+2{\tau }^{\prime }{\tau }^{″}{\displaystyle \frac{x_2}{2R}}+{\left({\tau }^{\prime }\right)}^2},{\tau }^{\prime }={\displaystyle \frac{p}{\sqrt{2x_1{x}_2}}},{\tau }^{″}={\displaystyle \frac{MR}{J}},\hfill \\ M=P\left(L+{\displaystyle \frac{x_2}{2}}\right),R=\sqrt{{\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2},\hfill \end{array}\\ J=2\left(\sqrt{2}{x}_1{x}_2\left({\displaystyle \frac{x_2^2}{12}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2\right)\right),\sigma (\overrightarrow{x})={\displaystyle \frac{6PL}{x_4{x}_3^2}},\delta (x)={\displaystyle \frac{4P{L}^3}{E{x}_3^3{x}_4}},\\ P_c(x)={\displaystyle \frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}}\left(1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}\right),P=6000\mathrm{lb},L=14\mathrm{in},\\ \begin{array}{l}{\delta }_{\max }=0.25\mathrm{in},E=30\times 1^6\mathrm{psi},G=12\times {10}^6\mathrm{psi},\hfill \\ {\tau }_{max}=13600\mathrm{psi},{\sigma }_{max}=30000\mathrm{psi}.\hfill \end{array}\end{array}$$

(28)

With

$$0.1\le x_1,x_4\le 2,0.1\le x_2,x_3\le 10.$$

(29)

Table 13. Performance of optimization algorithms on the welded beam design.

Algorithm	h	l	t	b	Cost	Rank
PBOA	0.2057	3.2349	9.0366	0.2057	1.6928	1
WSO	0.1764	4.9107	8.4736	0.1764	1.9725	14
WOA	0.2016	3.3421	8.9937	0.2016	1.7266	8
SHO	0.1538	4.9460	8.8542	0.1538	1.9559	12
GTO	0.1631	4.4533	9.0135	0.1631	1.7856	11
GWO	0.2055	3.2419	9.0370	0.2055	1.6937	5
DRA	0.2353	3.1273	8.0573	0.2353	1.9565	13
DE	0.2070	3.2535	9.0168	0.2070	1.7041	7
PRGO	0.2065	3.2392	9.0339	0.2065	1.7406	9
HGSO	0.2054	3.2123	9.1037	0.2054	1.6981	6
SCA	0.1827	3.9134	9.0064	0.1827	1.7527	10
PSO	0.2057	3.2349	9.0366	0.2057	1.6928	3
GMO	0.2057	3.2359	9.0370	0.2057	1.6929	4
MFO	0.2057	3.2349	9.0366	0.2057	1.6928	2
VOR	0.1828	3.9143	8.9757	0.2085	1.7576	15
NEL	0.2077	3.2136	8.9932	0.2077	1.7003	16

presents the optimization schemes and corresponding fitness values of various algorithms for the welded beam design problem. The results show that the fitness value of the solution obtained by the PBOA is the best among all comparative algorithms. The convergence process of the PBOA for this problem is shown in Figure 16.

6.4. Wireless Sensor Network Coverage Optimization

Wireless Sensor Networks (WSNs) play a crucial role in fields such as environmental monitoring and smart agriculture, where the rationality of node deployment directly affects monitoring efficiency. The core objective of the wireless sensor network coverage optimization problem is to minimize the fitness value. The optimization variables are the coordinates of 20 sensor nodes: the x and y coordinates of each node ($x_i$, $y_i$ for $i=1,2,\dots ,20$). The mathematical model of this problem can be expressed as:

Consider:

$$\overrightarrow{x}=[\begin{array}{ccccccc}{x}_1& y_1& x_2& y_2& \dots & x_{20}& y_{20}\end{array}]$$

(30)

Minimize:

$$f(\overrightarrow{x})=(1-S)+P$$

(31)

where

$$\begin{array}{l}S=0.6\cdot C_R+0.3\cdot (1-{\sigma }_E)+0.1\cdot C_N\hfill \\ P=10\cdot I(C_R<0.95)+10\cdot ({\sigma }_E>0.1)+10\cdot I\left(\mathsf{\exists }i,n(i)<2\right)\hfill \end{array}$$

(32)

Subject to:

$$\begin{array}{l}{C}_R\ge 0.95,\hfill \\ {\sigma }_E\le 0.1,\hfill \\ C_N=1,\hfill \\ 0\le x_i\le 100,0\le y_i\le 100,\hspace{1em}i=1,2,\dots ,20\hfill \end{array}$$

(33)

With:

• $C_R$: Coverage rate. $I_{\mathrm{cover}}(g)$ is an indicator function (1 if grid point g is covered, 0 otherwise), and G is the set of grid points.

$$C_R={\displaystyle \frac{\sum _{g\in G}{I}_{\mathrm{cover}}(g)}{\left|G\right|}}$$

(34)

• ${\sigma }_E$: Standard deviation of normalized remaining energy.

$$e(i)={\displaystyle \frac{d_{\mathrm{edge}}(i)}{\underset{j=1,\dots ,20}{\max }{d}_{\mathrm{edge}}(j)}}$$

(35)

where $d_{\mathrm{edge}}(i)$ is the minimum distance from node i to the area edge, and $\overline{e}$ is the mean of e(i)

$${\sigma }_E=\sqrt{{\displaystyle \frac{1}{20}}\sum _{i=1}^{20}{\left(e(i)-\overline{e}\right)}^2}$$

(36)

• $C_N$: Connectivity. $n(i)$ is the number of neighbors of node i (nodes within communication radius).

$$C_N={\displaystyle \frac{\sum _{i=1}^{20}I(n(i)\ge 2)}{20}}$$

(37)

Table 14. Performance of optimization algorithms on the wireless sensor network coverage optimization.

Algorithm	x1	y1	x2	y2	Cost	Rank
PBOA	10.5115	40.3429	65.2083	84.9795	10.0765	1
WSO	83.7670	11.6806	28.6629	21.8187	20.1858	14
WOA	55.5789	23.8674	47.3117	77.4691	10.2972	7
SHO	64.6173	29.8110	41.6177	42.6812	10.3409	10
GTO	32.6733	42.5154	55.6760	72.7533	10.3327	8
GWO	87.9851	54.2295	11.2848	33.6120	10.0785	2
DRA	77.6323	15.7926	18.4501	26.9842	10.2140	6
DE	38.9435	28.9058	69.7581	69.3842	10.3712	11
PRGO	26.2388	9.5811	58.4581	97.1406	10.0863	3
HGSO	57.8266	68.7201	67.4936	40.6281	20.0996	12
SCA	85.8792	24.2662	22.1519	45.3583	20.1926	15
PSO	63.4411	27.7791	8.7834	61.1598	20.1076	13
GMO	6.4521	49.2349	34.5628	86.1071	10.0941	4
MFO	30.5274	50.2275	49.3120	67.2670	10.3363	9
VOR	75.5312	69.6071	44.0973	64.9408	10.3136	5
NEL	63.8887	31.9920	54.5713	54.8354	20.1443	16

presents the optimization schemes of various algorithms for the wireless sensor network coverage optimization problem (due to space constraints, only the first 4 variables of each algorithm’s result are shown) along with their corresponding fitness values. Among them, the complete solution of the PBOA is (10.51, 40.34, 65.21, 84.98, 86.92, 50.81, 86.25, 13.19, 31.70, 24.82, 9.66, 86.05, 22.33, 60.90, 87.17, 69.22, 66.53, 55.50, 67.53, 10.33, 44.29, 48.97, 87.39, 87.41, 51.60, 85.82, 31.69, 88.48, 85.54, 32.69, 11.87, 63.17, 34.37, 9.21, 10.52, 12.92, 45.95, 78.65, 55.84, 25.89).

The results show that among all the comparative algorithms, the fitness value of the solution obtained by the PBOA is the optimal. The convergence process of the PBOA for this problem is illustrated in Figure 17.

6.5. Unmanned Aerial Vehicle (UAV) Path Planning Problem

In scenarios like environmental monitoring and logistics distribution, the rationality of Unmanned Aerial Vehicle (UAV) path planning directly affects task execution efficiency and energy consumption. The core objective of the UAV path planning problem is to minimize the fitness value, which comprehensively considers flight distance, energy consumption, and penalties for violating constraints. The optimization variables consist of values related to the visit order (first 5 variables) and coordinates of waypoints (next 20 variables, with each pair representing an $(x,y)$ coordinate). The mathematical model of this problem, with known parameters specified, is as follows:

Known Parameters:

• Start point coordinates: $S=(0,0)$;
• Target points coordinates (5 in total):

  $$T_1=(200,300),T_2=(500,100),T_3=(300,600),T_4=(700,400),T_5=(600,200);$$
• Flight height: Fixed at 100 (unit: meters, does not affect 2D path calculation);
• Maximum flight distance: $D_{\max }=1800$ (unit: meters, including redundancy);
• Maximum load: $L_{\max }=2.0$ (unit: kg);
• Load reduction per target point: $\Delta L=0.3$ (unit: kg, load decreases by this value after reaching each target point);
• Energy consumption coefficient: $k=0.01$;
• No-fly zones (each defined by center coordinates $(x,y)$ and radius r):

  $$NF{Z}_1=(350,250,50),NF{Z}_2=(550,350,50),NF{Z}_3=(250,500,50) (\mathrm{unit}:\mathrm{meters});$$
• Safe distance from no-fly zones: $s=10$ (unit: meters, the minimum allowed distance between flight path and no-fly zones).

Consider:

$$\overrightarrow{x}=[x_1 x_2 x_3 x_4 x_5 w_{1x} w_{1y} w_{2x} w_{2y}\dots w_{10x} w_{10y}]$$

(38)

$x_1,x_2,x_3,x_4,x_5$: Values used to determine the visit order of the 5 target points (converted into a permutation of {1, 2, 3, 4, 5} via sorting). $w_{ix},w_{iy}\left(i=1,2,\dots ,10\right)$: x and y coordinates of 10 waypoints (inserted between consecutive points in the path).

Minimize:

$$f(\overrightarrow{x})=D+E+P$$

(39)

where

• Total flight distance (D):

The sum of Euclidean distances between consecutive points in the full path. The full path is constructed as:

$$S\to {\mathrm{Waypoint}}_1\to T_{o_1}\to {\mathrm{Waypoint}}_2\to T_{o_2}\to \cdots \to T_{o_5}\to {\mathrm{Waypoint}}_{10}\to S$$

(40)

where $o_1,o_2,\dots ,o_5$ is the visit order of target points (a permutation of {1, 2, 3, 4, 5}). Mathematically $D={\displaystyle \sum _{i=1}^{m-1}}\sqrt{{(p_{i+1,x}-p_{i,x})}^2+{(p_{i+1,y}-p_{i,y})}^2}$, where $p_1,p_2,\dots ,p_m$ are consecutive points in the full path.

• Total energy consumption (E):

Calculated using the model $E=k{\displaystyle \sum _{i=1}^{m-1}}\left(d_i+0.5\cdot l_i\cdot d_i\right)$, where $d_i$ is the distance of the i-th segment, and $l_i$ is the load during the i-th segment. The load changes as:$l_1=L_{\max }{l}_{t+1}=l_t-\Delta L$ after visiting each target point t.

• Penalty term (P):

If any flight segment is within the range of a no-fly zone ($\mathrm{distance}<r+s$), P increases by 1000 per violation. If total distance $D>D_{\max }$, P increases by $10\cdot (D-D_{\max })$.

Subject to:

• No-fly zone avoidance: For each no-fly zone $NF{Z}_j=(c_x,c_y,r)$ and each flight segment between points $p_a$ and $p_b$, the minimum distance from $NF{Z}_j$ to the segment must be $\ge r+s$:

  $${\min }_{p\in \mathrm{segment}(p_a,p_b)}\sqrt{{(p_x-c_x)}^2+{(p_y-c_y)}^2}\ge r+s.$$

  (41)
• Maximum distance limit: $D\le D_{\max }$.
• Unique visit order: The visit order $o_1,o_2,\dots ,o_5$ must be a permutation of {1, 2, 3, 4, 5} (all target points are visited exactly once).

Table 15. Performance of optimization algorithms on the UAV path planning problem.

Algorithm	x1	x2	x3	x4	Cost	Rank
PBOA	1.0044	5.0000	1.0287	2.0938	4185.3462	1
WSO	2.3791	4.8366	2.3872	4.3635	7639.7230	14
WOA	1.1666	1.5726	1.1666	1.1666	5923.5029	9
SHO	3.8016	2.2880	3.6737	2.2916	8708.8240	15
GTO	3.8854	1.8514	3.4595	2.9495	4499.8534	7
GWO	1.3281	4.4264	2.0459	3.1476	4186.6557	6
DRA	4.8470	4.6330	4.8149	4.7319	4196.7911	7
DE	3.4017	2.2360	2.7971	2.6401	4692.7808	8
PRGO	1.0000	4.0276	1.1858	2.3793	4430.7179	6
HGSO	1.0200	4.2953	1.1637	1.5627	5017.4585	10
SCA	5.0000	1.4530	3.0511	3.0329	7078.9039	13
PSO	5.0000	1.0000	4.9255	1.4434	4185.7440	3
GMO	1.0000	1.1186	1.0000	1.0000	4185.5565	2
MFO	1.0000	4.8359	1.0000	1.0000	4185.4297	2
VOR	2.4934	3.4997	3.0254	3.2174	4598.6124	8
NEL	4.8904	3.4521	3.1326	2.9174	11,452.0116	16

presents the optimization schemes of various algorithms for the UAV path planning problem (due to space constraints, only the first 4 variables of each algorithm’s result are shown) along with their corresponding fitness values. Among them, the complete solution of the PBOA is (1.00, 5.00, 1.03, 2.09, 3.03, 0.03, 0.04, 311.73, 486.28, 300.28, 599.86, 640.89, 281.84, 562.35, 162.36, 258.95, 51.83, 400.99, 252.66, 405.41, 782.37, 586.97, 489.63, 0.00, 54.58).

The results show that among all the comparative algorithms, the fitness value of the solution obtained by the PBOA is the optimal. The convergence process of the PBOA for this problem is illustrated in Figure 18.

6.6. Design of Time-Varying Nonsingular Fast Terminal Sliding Mode Controller for H-Type Motion Platform

Figure 19 shows the schematic diagram and structural sketch of the H-type motion platform. The motion mechanism of the platform is as follows: two parallel Y-direction ball screws drive the worktable collaboratively to realize Y-direction motion; the X-direction ball screw is connected across the two Y-axis ball screws, responsible for driving the worktable to move in the X-direction; the entire motion system is powered by a servo motor. Benefiting from its stable structural design, the H-type motion platform has a wide range of application scenarios.

In terms of control accuracy, the synchronization error of the two Y-direction ball screws must be strictly controlled within a very small range, which is the key to ensuring the Y-direction displacement accuracy of the platform. As proposed in Reference [71], the actual displacement in the X-direction d_x′ and Y-direction d_y′ of the worktable can be expressed as:

$$\left\{\begin{array}{c}{d}_x^{\prime }=d_x\\ d_y^{\prime }=d_{y1}+d_x{\displaystyle \frac{d_{y2}-d{y}_1}{L}}\end{array}\right.$$

(42)

where d_y1 is the displacement of the Y₁ motion mechanism, dy₂ is the displacement of the Y₂ motion mechanism, d_x is the displacement of the X motion mechanism, θ is the angle between the X motion mechanism and the line perpendicular to the Y₁ and Y₂ motion mechanisms, and L is the distance between Y₁ and Y₂.

A time-varying nonsingular fast terminal sliding mode controller is designed in Reference [71]:

The control rate w_x for the X-direction displacement is:

$$w_x={\displaystyle \frac{d_i+\frac{k_1{\dot{e}}_x+k_3|{\dot{e}}_x{|}^{\alpha }\tanh ({\dot{e}}_x)+\delta \tanh (s_x)+k_6{s}_x}{k_2}}{K}}$$

(43)

where

$$\begin{array}{l}{s}_x=k_1{x}_1(t)+k_2{\displaystyle {\int }_0^t{x}_1}(t)dt+k_3{\displaystyle {\int }_0^t|x_2(t){|}^{\alpha }\tanh (x_2(t))dt}\\ e_i=d_i-d_{io}\\ x_1=e_x\\ x_2={\dot{x}}_1=v_i-v_o\\ \alpha =\left\{\begin{array}{c}{\displaystyle \frac{\tanh (x_2)k_4}{\tanh (\frac{1}{x_2})k_5}},x_2\ne 0\\ \hspace{1em}\tau ,x_2=0\end{array}\right.\end{array}$$

(44)

where d_i, d_o, v_i, and v_o represent the desired displacement, output displacement, desired velocity, and output velocity in the i-direction (where i can denote X, Y₁, and Y₂), respectively; δ, k₁, k₂, k₃, k₄, k₅, and k₆ are constants greater than 0; τ takes values in the range of 1–300; K is a constant affected by the screw lead and gearbox reduction ratio; and tanh is the hyperbolic tangent function.

The control rate w_Y1 for the Y₁-direction is:

$$w_{y1}={\displaystyle \frac{d_i+\frac{k_1({\dot{e}}_{y1}+\lambda {\dot{\epsilon }}_{y1})+k_3|{\dot{e}}_{y1}+\lambda {\dot{\epsilon }}_{y1}{|}^{\alpha }\tanh ({\dot{e}}_{y1}+\lambda {\dot{\epsilon }}_{y1})+\sigma \tanh (s_{y1})+k_6{s}_{y1}}{k_2}}{K}}$$

(45)

The control rate w_Y2 for the Y₂-direction is:

$$w_{y2}={\displaystyle \frac{d_i+\frac{k_1({\dot{e}}_{y2}+\lambda {\dot{\epsilon }}_{y2})+k_3|{\dot{e}}_{y2}+\lambda {\dot{\epsilon }}_{y2}{|}^{\alpha }\tanh ({\dot{e}}_{y2}+\lambda {\dot{\epsilon }}_{y2})+\sigma \tanh (s_{y2})+k_6{s}_{y2}}{k_2}}{K}}$$

(46)

where

$$\left\{\begin{array}{c}{\epsilon }_{y1}=e_{y1}-e_{y2}\\ {\epsilon }_{y2}=e_{y2}-e_{y1}\end{array}\right.$$

(47)

For the Y₁-direction:

$$\left\{\begin{array}{c}{x}_1=e_{y1}+\beta {\epsilon }_{y1}\\ x_2={\dot{x}}_1=v_{\mathrm{y}1}-v_{y1}\end{array}\right.$$

(48)

For the Y₂-direction:

$$\left\{\begin{array}{c}{x}_1=e_{y2}+\beta {\epsilon }_{y2}\\ x_2={\dot{x}}_1=v_{y2}-v_{y2}\end{array}\right.$$

(49)

Among the three controllers for the X, Y₁, and Y₂ directions, although the symbols δ, β, k₁, k₂, k₃, k₄, k₅, and k₆ are consistent, parameter settings of different controllers need to be differentiated to achieve optimal control effects. Based on this, the problem can be formulated as the “parameter optimization problem of time-varying nonsingular fast terminal sliding mode controller for H-type motion platform”. Its core objectives are: to improve the X-direction control accuracy by optimizing the parameters δ, k₁, k₂, k₃, k₄, k₅, and k₆ of the X-direction controller; and to enhance the Y-direction control accuracy while reducing the synchronization error between the Y₁ and Y₂ motion mechanisms by optimizing the parameters δ, β, k₁, k₂, k₃, k₄, k₅, and k₆ of the two Y-direction controllers. The mathematical formulation of this problem is as follows:

Consider for X-direction controller:

$$\overrightarrow{x}=[x_1 x_2 x_3 x_4 x_5 x_6 x_7]=[\delta k_1 k_2 k_3 k_4 k_5 k_6]$$

(50)

Consider for Y₁-direction controller or Y₂-direction controller:

$$\overrightarrow{x}=[x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8]=[\delta \beta k_1 k_2 k_3 k_4 k_5 k_6]$$

(51)

Minimize for X-direction controller:

$$J_x={\displaystyle {\int }_0^{\infty }t\left|e_{\mathrm{X}}(t)\right|}dt$$

(52)

Minimize for Y₁-direction controller or Y₂-direction controller:

$$J_{\mathrm{Y}}={\displaystyle {\int }_0^{\infty }t\left|e_{{\mathrm{Y}}_1}(t)\right|}+t\left|e_{{\mathrm{Y}}_2}(t)\right|+10,000t\left|{\epsilon }_{y1}\right|dt$$

(53)

All variables range from [0.0001, 10,000]. Considering factors such as transmission efficiency and disturbances, the X-direction motion mechanism is set with K = 0.99, the Y₁-direction motion mechanism with K = 0.99, and the Y₂-direction motion mechanism with K = 0.98. Additionally, random numbers following a normal distribution (mean = 0, variance = 0.001) are added to the system output to simulate external disturbances.

Table 16. Performance of the Optimization Algorithm in X-direction Controller Design.

Algorithm	k1	k2	k3	k4	k5	k6	δ	Optimal Fitness	Rank
PBOA	0.0065	1630.9684	14.1045	0.0262	12.1451	0.0020	0.0410	128.3857	1
WSO	0.0001	0.6783	0.0058	0.0001	0.3530	0.0001	0.0001	128.6596	11
WOA	0.0001	0.5838	0.0045	0.0001	0.1856	0.0001	0.0001	128.7012	13
SHO	0.0002	1.3428	0.0109	0.0001	0.7533	0.0001	0.0001	128.6560	10
GTO	0.0002	1.1287	0.0090	0.0001	0.0448	0.0001	0.0001	128.4290	3
GWO	0.0186	77.7536	0.7334	0.0403	25.0876	0.0211	0.0048	128.6646	12
DRA	0.0001	4412.2092	39.6330	0.0001	1421.9812	0.0001	0.0001	128.4453	8
DE	0.0001	9999.9966	89.8268	0.0001	9999.6518	0.0001	0.0001	128.4453	5
PRGO	0.0001	100.1021	0.8598	0.0687	31.8915	0.0407	0.0001	128.4246	2
HGSO	0.0198	99.4545	0.8326	0.0113	31.9401	0.0010	0.0016	128.7319	14
SCA	0.0001	9529.2188	85.6982	0.0003	6897.6487	0.0001	0.0005	128.4453	9
PSO	0.0001	9994.1571	89.7754	0.0001	2527.9940	0.0001	0.0001	128.4453	6
GMO	0.0001	9999.9994	89.8237	0.0189	10,000.0000	0.0001	0.0001	128.4453	7
MFO	0.0001	10,000.0000	89.8268	0.0001	10,000.0000	0.0001	0.0001	128.4453	4

,

Table 17. Performance of the Optimization Algorithm in Y₁-direction Controller Design.

Algorithm	k1	k2	k3	k4	k5	k6	δ	β	Optimal Fitness	Rank
PBOA	0.0002	722.0741	0.0003	0.1667	3007.0794	0.0009	0.0009	0.0002	1.00 × 103	1
WSO	0.0001	0.3123	0.0061	0.0001	0.1738	0.0001	0.0001	0.0656	9.77 × 104	8
WOA	0.0153	4199.8672	0.0005	0.2617	687.2270	0.0113	0.0123	63.4790	5.43 × 104	7
SHO	0.0085	54.5407	0.3919	0.0048	11.6895	0.0003	0.0007	4.4437	4.55 × 105	11
GTO	0.0001	0.1427	0.0001	0.0001	0.0266	0.0001	0.0001	0.0098	5.13 × 104	6
GWO	0.0067	126.2736	0.0855	0.0041	21.5336	0.0001	0.0024	16.9382	3.94 × 103	4
DRA	0.0001	20.0275	0.1059	0.0020	4.0116	0.0040	0.0029	1.3155	2.30 × 104	5
DE	0.0200	100.0000	0.9102	0.0114	32.2396	0.0010	0.0018	10.1245	6.60 × 105	13
PRGO	0.0001	99.9778	0.9523	0.0001	32.1736	0.0181	0.0001	9.9913	1.98 × 105	10
HGSO	0.0200	100.0000	0.9102	0.0114	32.2396	0.0010	0.0018	10.1245	6.60 × 105	13
SCA	0.0113	46.7326	0.4265	0.0065	15.0648	0.0052	0.0114	4.7313	4.72 × 105	12
PSO	0.0001	4611.9449	0.0001	0.0001	2815.7395	0.0001	0.0001	25.7426	1.61 × 103	2
GMO	0.0001	4638.8406	0.0001	0.0001	6579.7200	0.0001	0.0001	1325.3761	1.12 × 105	9
MFO	0.0001	5.5919	0.0001	0.0001	138.0058	0.0001	0.0001	18.0072	2.78 × 103	3

and

Table 18. Performance of the Optimization Algorithm in Y₂-direction Controller Design.

Algorithm	k1	k2	k3	k4	k5	k6	δ	β	Optimal Fitness	Rank
PBOA	0.0002	722.0741	0.0003	0.1667	3007.0794	0.0009	0.0009	0.0002	1.00 × 103	1
WSO	0.0001	0.3123	0.0061	0.0001	0.1738	0.0001	0.0001	0.0656	9.77 × 104	8
WOA	0.0153	4199.8672	0.0005	0.2617	687.2270	0.0113	0.0123	63.4790	5.43 × 104	7
SHO	0.0085	54.5407	0.3919	0.0048	11.6895	0.0003	0.0007	4.4437	4.55 × 105	11
GTO	0.0001	0.1427	0.0001	0.0001	0.0266	0.0001	0.0001	0.0098	5.13 × 104	6
GWO	0.0067	126.2736	0.0855	0.0041	21.5336	0.0001	0.0024	16.9382	3.94 × 103	4
DRA	0.0001	20.0275	0.1059	0.0020	4.0116	0.0040	0.0029	1.3155	2.30 × 104	5
DE	0.0200	100.0000	0.9102	0.0114	32.2396	0.0010	0.0018	10.1245	6.60 × 105	13
PRGO	0.0001	99.9778	0.9523	0.0001	32.1736	0.0181	0.0001	9.9913	1.98 × 105	10
HGSO	0.0200	100.0000	0.9102	0.0114	32.2396	0.0010	0.0018	10.1245	6.60 × 105	13
SCA	0.0113	46.7326	0.4265	0.0065	15.0648	0.0052	0.0114	4.7313	4.72 × 105	12
PSO	0.0001	4611.9449	0.0001	0.0001	2815.7395	0.0001	0.0001	25.7426	1.61 × 103	2
GMO	0.0001	4638.8406	0.0001	0.0001	6579.7200	0.0001	0.0001	1325.3761	1.12 × 105	9
MFO	0.0001	5.5919	0.0001	0.0001	138.0058	0.0001	0.0001	18.0072	2.78 × 103	3

Since the Y1-direction controller and Y2-direction controller involve the problem of synchronization error monitoring, the controllers of Y1 and Y2 need to be optimized jointly. This experiment contains only one set of data, so the fitness values and rankings in Table 18 are the same as those in Table 17, but the optimized parameters are different.

present the optimization schemes and corresponding fitness values of the PBOA and other comparative algorithms for the H-type motion platform optimization problem. The results indicate that among all comparative algorithms, the fitness value of the solution obtained by the PBOA is optimal. Figure 20 illustrates the convergence process of the PBOA for this problem.

After parameter optimization using the PBOA, the input-output curves (actual displacements of the X-axis, Y1-axis, and Y2-axis) of the H-type motion platform, as well as the synchronization error between the two Y-direction motion mechanisms, are shown in Figure 21. Experimental data reveal that, with a standard displacement of 1 mm as the reference, the control errors between the actual displacements of the X-axis, Y1-axis, Y2-axis and the standard value are all stabilized within the range of ±2 × 10⁻⁵ mm. Meanwhile, the synchronization error of the two Y-direction mechanisms is controlled within ±2.6 × 10⁻⁵ mm without significant fluctuations.

These results fully demonstrate that in engineering scenarios with disturbances, the PBOA can effectively ensure the motion accuracy and synchronization stability of the H-type motion platform, which fully meets the requirements of high-precision motion control in engineering applications.

6.7. Summary of PBOA’s Performance in Practical Problems

Table 19. Wilcoxon rank-sum test p-values (PBOA vs. other algorithms).

Problem	WSO	WOA	SHO	GTO	GWO	DRA	DE	PRGO
P1	1.82 × 10−3	2.56 × 10−3	1.67 × 10−3	2.14 × 10−3	9.35 × 10−4	2.31 × 10−3	3.02 × 10−3	2.78 × 10−3
P2	6.42 × 10−5	8.17 × 10−5	7.33 × 10−5	1.21 × 10−4	1.56 × 10−4	1.38 × 10−4	9.59 × 10−5	1.08 × 10−4
P3	3.27 × 10−6	1.15 × 10−9	2.98 × 10−6	4.03 × 10−6	5.72 × 10−4	8.21 × 10−10	6.94 × 10−5	7.53 × 10−5
P4	2.25 × 10−3	2.89 × 10−3	2.01 × 10−3	3.16 × 10−3	1.78 × 10−3	2.67 × 10−3	3.35 × 10−3	2.94 × 10−3
P5	8.62 × 10−4	9.37 × 10−4	8.95 × 10−4	1.05 × 10−3	7.81 × 10−4	9.83 × 10−4	1.12 × 10−3	1.01 × 10−3
Problem	HGSO	SCA	PSO	GMO	MFO	VOR		NEL
P1	2.95 × 10−3	3.12 × 10−3	2.64 × 10−3	2.47 × 10−3	2.81 × 10−3	5.26 × 10−3		3.89 × 10−3
P2	1.42 × 10−4	1.67 × 10−4	1.29 × 10−4	1.35 × 10−4	1.48 × 10−4	6.83 × 10−5		4.17 × 10−6
P3	8.92 × 10−5	9.35 × 10−10	6.47 × 10−5	6.89 × 10−5	7.21 × 10−5	3.16 × 10−6		9.74 × 10−10
P4	3.18 × 10−3	3.42 × 10−3	2.86 × 10−3	2.73 × 10−3	3.05 × 10−3	2.17 × 10−3		4.08 × 10−3
P5	1.17 × 10−3	1.23 × 10−3	1.05 × 10−3	1.09 × 10−3	1.13 × 10−3	9.26 × 10−4		5.73 × 10−4

presents the results of the non-parametric Wilcoxon rank-sum test at a significance level of 0.05, analyzing the performance differences between the PBOA and other comparative algorithms across 5 problems (P1 to P5). The p-values in the table indicate the statistical significance of these differences, with a p-value < 0.05 suggesting a significant performance gap between the PBOA and the corresponding algorithm.

The results show that the PBOA exhibits significant differences from most comparative algorithms across all 5 problems. Specifically, algorithms such as WSO, WOA, SHO, GTO, GWO, DRA, PRGO, HGSO, SCA, PSO, GMO, VOR, and NEL all yield p-values far below 0.05 when compared with PBOA. For instance, in P3, the extremely small p-values of the WOA (1.15 × 10⁻⁹) and SHO (2.98 × 10⁻⁶) indicate that the performance advantage of the PBOA over these algorithms is statistically highly significant. Similarly, NEL demonstrates remarkably low p-values across multiple problems, such as 4.17 × 10⁻⁶ (P2) and 9.74 × 10⁻¹⁰ (P3), reflecting a distinct performance gap with the PBOA.

Notably, the PBOA shows relatively smaller differences from DE and MFO in specific scenarios. For example, in P1, although the p-values of DE (3.02 × 10⁻³) and MFO (2.81 × 10⁻³) are still below 0.05, they are higher than those of algorithms like the WOA (2.56 × 10⁻³) and SHO (1.67 × 10⁻³). In P4, the p-values of DE (3.35 × 10⁻³) and MFO (3.05 × 10⁻³) are also larger than those of GWO (1.78 × 10⁻³) and DRA (2.67 × 10⁻³), indicating a relatively less pronounced performance gap. These results suggest that while the PBOA still outperforms DE and MFO, the statistical significance of this advantage is relatively weaker in certain problems.

Overall, the results of the Wilcoxon rank-sum test further confirm the robustness and uniqueness of the PBOA: it maintains a significant performance advantage over most comparative algorithms across different problems, and even in cases where the differences with individual algorithms (DE and MFO) are relatively smaller, it still demonstrates stable superiority. This consistency underscores the effectiveness of the PBOA in solving the target optimization problems.

7. Summary and Outlook

This paper proposes a novel mathematical heuristic optimization algorithm—the Perpendicular Bisector Optimization Algorithm (PBOA). The design inspiration of the PBOA originates from the geometric property of the perpendicular bisector, i.e., any point on the perpendicular bisector is equidistant from the two endpoints of the line segment. Based on this property, the PBOA designs two differentiated convergence strategies: the exploration phase mainly adopts a slow convergence strategy to explore potential high-quality solutions by expanding the search range; the exploitation phase focuses on a fast convergence strategy to improve solution accuracy through refined search. Meanwhile, relying on the deterministic principle, the algorithm selects 4 different particles to form line segments with the current particle and constructs perpendicular bisectors, thereby expanding the search space and enhancing global exploration capability. To comprehensively evaluate the performance of the PBOA, experiments are conducted using 27 benchmark test functions. The results show that the PBOA exhibits outstanding performance in solution accuracy and stability, outperforming most existing optimization algorithms.

Despite its excellent overall performance, the PBOA has potential limitations of slow convergence speed and long running time: the slow convergence speed stems from the design mechanism that the algorithm actively reduces particle displacement distance in the early stage to ensure global search; the long running time is due to the increased computational load caused by refined steps in the optimization process. Future research needs to further optimize the PBOA’s spatial exploration planning and computational efficiency to balance convergence speed with search range, as well as solution quality with running time. In addition, this study applies the PBOA to five engineering design problems with varying complexity and diversity, constructing a strong verification platform to validate the algorithm’s applicability. Experiments demonstrate that the PBOA can effectively explore unknown search spaces and find better solutions, thus validating its effectiveness and robustness in practical engineering scenarios. Finally, in the parameter optimization of the time-varying nonsingular fast terminal sliding mode controller for the H-type motion platform, the controller optimized by the PBOA outperforms comparative algorithms in both control accuracy and robustness, fully demonstrating its application potential in the field of controller parameter optimization.

In summary, as a new optimization method, the PBOA not only provides reliable solutions for traditional optimization problems but also exhibits strong performance in practical engineering applications. Despite limitations in operation, its advantages in solution quality and stability make it highly valuable for multiple scenarios. Future work can further explore binary and multi-objective versions of the PBOA, focusing on optimizing its convergence speed and computational efficiency to enhance applicability and effectiveness in complex optimization problems. We believe that the proposal of the PBOA will provide new directions and insights for research and applications in related fields.