Ripple Evolution Optimizer: A Novel Nature-Inspired Metaheuristic

Abstract

This paper presents a novel Ripple Evolution Optimizer (REO) that incorporates adaptive and diversified movement—a population-based metaheuristic that turns a coastal-dynamics metaphor into principled search operators. REO augments a JADE-style current-to-p-best/1 core with jDE self-adaptation and three complementary motions: (i) a rank-aware that pulls candidates toward the best, (ii) a time-increasing that aligns agents with an elite mean, and (iii) a scale-aware sinusoidal that lead solutions with a decaying envelope; rare Lévy-flight kicks enable long escapes. A reflection/clamp rule preserves step direction while enforcing bound feasibility. On the CEC2022 single-objective suite (12 functions spanning unimodal, rotated multimodal, hybrid, and composition categories), REO attains 10 wins and 2 ties, never ranking below first among 34 state-of-the-art compared optimizers, with rapid early descent and stable late refinement. Population-size studies reveal predictable robustness gains for larger N. On constrained engineering designs, REO achieves outperforming results on Welded Beam, Spring Design, Three-Bar Truss, Cantilever Stepped Beam, and 10-Bar Planar Truss. Altogether, REO couples adaptive guidance with diversified perturbations in a compact, transparent optimizer that is competitive on rugged benchmarks and transfers effectively to real engineering problems.

1. Introduction

Optimization plays a central role across the sciences, engineering, and management because many real-world decisions involve choosing the best configuration from an immense combinatorial or continuous search space. Problems such as structural design, energy management, parameter tuning in machine learning, and operations scheduling often lead to nonlinear, multimodal, or NP-hard formulations for which deterministic exact algorithms are computationally prohibitive [1]. To circumvent these limitations, researchers have developed a broad class of metaheuristic algorithms that combine stochastic exploration with problem-specific heuristics to efficiently approximate near-optimal solutions. Metaheuristics encompass a diverse family of strategies, including evolutionary algorithms, swarm intelligence, physics-based heuristics, and other population-based approaches, all designed to escape local minima and balance exploration and exploitation in the search space [2,3].

In the past decade, the landscape of metaheuristics has expanded rapidly, spurred by inspiration from natural processes and mathematical analogies. Nature-inspired optimizers model biological or physical phenomena such as the cooperative hunting behavior of animal groups [4], the social dynamics of jackals [5], or the foraging patterns of insects and birds [6,7]. These algorithms exploit analogies with ecosystems and physical laws to design simple update rules and information-sharing mechanisms. For example, the spotted hyena optimizer simulates group hunting to update candidate solutions and avoid premature convergence [4], whereas the golden jackal optimizer mimics the dispersal and prey-detection strategies of golden jackals to enhance diversification [5]. Inspired by natural resource allocation, economic and physical principles have also given rise to supply–demand-based optimizers [8], light spectrum optimizers [9], and tornado optimizers with Coriolis force [10], highlighting the versatility of metaheuristic design.

Another hallmark of contemporary metaheuristic research is the integration of metaheuristics with problem-specific models and hybrid frameworks to tackle complex engineering tasks. For instance, researchers have combined physics-inspired search with neural network architectures to build fault-diagnosis systems for transformers [11] and gas turbines [12]. Swarm-inspired algorithms have been deployed for proton exchange membrane fuel cell flow-field optimization [13], multilevel image segmentation [1], software-defined networking [14], and malware detection in Android systems [15]. These hybrid strategies illustrate how metaheuristics can be customized to exploit domain knowledge while preserving generic search capabilities.

As the number of metaheuristics grows, so does the need for comparative studies and performance analyses. Recent research has proposed numerically efficient variants and multi-objective frameworks aimed at improving convergence speed, robustness, and scalability. Examples include modified backtracking search algorithms informed by species evolution and simulated annealing [16], nomad–migration-inspired algorithms for global optimization [17], and quantum-inspired swarm optimizers [18]. Multi-objective approaches such as the geometric mean optimizer extend the search to Pareto frontiers [19], and advances in regularization and graph filtering have improved metaheuristic-based machine-learning models [20]. Collectively, these developments illustrate the vibrancy of research in metaheuristics and the need for continued innovation.

To address this gap, this paper introduces the ripple evolution optimizer, a novel nature-inspired metaheuristic designed to handle complex and engineering design problems. The new algorithm derives its inspiration from the propagation of ripples on water surfaces, exploiting the nonlinear interactions between neighboring waves to guide candidate solutions. Following a description of the algorithm, we present extensive experimental comparisons on benchmark and engineering design problems to demonstrate the competitiveness of the ripple evolution optimizer. We also discuss the main contributions of the present work.

2. Related Work

Metaheuristics inspired by animal behavior remain a fertile source of algorithmic innovation. The hippopotamus optimization algorithmproposed by Amiri et al. [21] models the social hierarchy and foraging strategies of hippopotamuses to strike a balance between exploration and exploitation, whereas the spotted hyena optimizer by Dhiman and Kumar [4] captures cooperative hunting to search the solution space efficiently. Similar inspirations include the grasshopper optimization algorithm applied to wireless communications networks [22], the Golden Jackal Optimization algorithm for engineering applications [5], and the Humboldt squid optimization algorithm which imitates the collective intelligence of squids [23]. Researchers have also drawn from avian behavior to develop pigeon-inspired and hummingbird-based schemes for structural health monitoring and communication systems [24,25]. These works highlight the trend of designing population-based search rules grounded in zoological observations.

Another line of research develops metaheuristics based on the dynamics of physical and economic systems. Wang et al. [16] introduced a modified backtracking search algorithm that integrates species-evolution principles and simulated annealing to improve convergence in constrained engineering problems. Zhao et al. [8] proposed a supply–demand-based optimization where search agents negotiate and trade resources to achieve equilibrium in the objective landscape. Abdel-Basset et al. [9] formulated a light spectrum optimizer that leverages properties of light wavelengths, while Braik et al. [10] incorporated the Coriolis force into a tornado-inspired algorithm to enhance diversification. Other novel metaphors include the garden balsam algorithm [6], the nomad–migration-inspired algorithm [17], the hunting search heuristic [2], the light spectrum optimizer [9], and the migration search algorithm [26]. Each of these approaches introduces distinct mechanisms for information sharing and reinforcement, illustrating the creative use of analogies from disparate domains.

Hybrid and problem-specific strategies continue to expand the scope of metaheuristics. Tao et al. [11] combined a probabilistic neural network with a bio-inspired optimizer for transformer fault diagnosis, and Hou et al. [12] developed a dynamic recurrent fuzzy neural network trained via a chaotic quantum pigeon-inspired optimizer for gas turbine modelling. In power systems, Kumar et al. [27] devised a nutcracker optimizer for congestion control, while Rathee et al. [28] used a quantum-inspired genetic algorithm to deploy sink nodes in wireless sensor networks. Metaheuristics have also been applied to structural design and fuel cell engineering: Ghanbari et al. [13] optimized bio-inspired flow fields for proton exchange membrane fuel cells, and Nemati et al. [29] employed a banking-system-inspired algorithm for truss size and layout design. Within the realm of image processing, Haddadi et al. [30] restored medical images using an enhanced regularized inverse filtering approach guided by a bio-inspired search, while Bhandari et al. [1] designed a multilevel image thresholding method using nature-inspired algorithms. These examples show how metaheuristics are being integrated into domain-specific models to solve real-world tasks beyond conventional benchmarks.

Many studies focus on tailoring metaheuristics to address multiple objectives and improve numerical efficiency. Pandya et al. [19] proposed a multi-objective geometric mean optimizer that eschews metaphors and uses mathematical transformations to locate Pareto-optimal solutions. Dalla Vedova et al. [10] applied bio-inspired algorithms to prognostics of electromechanical actuators, while Li and Sun [6] introduced a numerical optimization algorithm based on garden balsam biology. Anaraki and Farzin [23] reported the Humboldt squid optimizer, Maroosi and Muniyandi [31] developed a membrane-inspired multiverse optimizer for web service composition, and Panigrahi et al. [15] used nature-inspired optimization with support vector machines to identify malicious Android data. Recent work also explored quantum-inspired swarm algorithms [18], orca predation models [32], and improved graph neural networks with nonconvex norms inspired by unified optimization frameworks [20]. The diversity of applications—from scheduling and communication networks to health monitoring and machine learning—demonstrates the versatility of metaheuristics when adapted with suitable objective functions and constraints.

Several comparative studies provide insights into the relative strengths of different algorithms and offer guidelines for metaheuristic design. Attaran et al. [33] introduced an evolutionary algorithm inspired by hunter spiders and compared its performance against established heuristics. Bhandari et al. [1] evaluated nature-inspired optimizers for color image segmentation, while Sivakumar and Kanagasabapathy [34] benchmarked multiple bio-inspired algorithms for cantilever beam parameter optimization. Chiang et al. [14] combined artificial bee colony algorithms with support vector machines for software-defined networking, and Chou and Truong [7] proposed a jellyfish-inspired optimizer and showed its competitiveness on benchmark functions. In structural engineering, Zhou et al. [35] examined bamboo-inspired multi-cell structures optimized by particle swarm algorithms, while Janizadeh et al. [36] used three novel optimizers to tune LightGBM models for wildfire susceptibility prediction. These studies reveal that no single algorithm universally outperforms others; rather, the effectiveness of a metaheuristic depends on the problem domain, search landscape, and parameter settings.

Overall, the literature demonstrates a vigorous pursuit of novel metaphors and hybrid architectures to enhance the capability of metaheuristics. Despite the proliferation of algorithms, challenges remain in balancing exploration and exploitation, avoiding premature convergence, and ensuring robustness across diverse problems. The proposed ripple evolution optimizer contributes to this ongoing evolution by introducing new interaction dynamics based on ripple propagation, aiming to deliver competitive performance on complex and engineering optimization tasks.

3. Ripple Evolution Optimizer: A Novel Nature-Inspired Metaheuristic

3.1. Inspiration from Nature (Ocean Ripples, Tides, and Undertow)

The three core inspirations in REO are: concentric ripples traveling across the sea surface, a tide that gradually pulls the water mass in a consistent direction, and an undertow that draws material back toward the sea after waves break on the shore. We use these notions to encode stochastic (ripples), time-aware (tides), and rank-aware (undertow) forces on each search agent.

3.1.1. Ripple Superposition

We model ripples as sinusoidal, decaying perturbations that encourage agents to probe along multiple directions, analogous to overlapping wavefronts. This intuition is later stated analytically in Equation (7). The visual metaphor is shown in Figure 1, where circular wavefronts fade in thickness and opacity with radius to depict amplitude decay Equation (6) and directional probing.

3.1.2. Tide and Undertow

The tide makes exploitation gradually stronger (Equation (5)), while the undertow rewards high-quality agents with a stronger pull toward the global best (Equation (4)). Together, they bias movement without removing diversity. Figure 2 renders the ocean/shore interface with a slow rightward tide arrow (growing with time $t/T$) and a leftward near-shore undertow arrow (stronger for better-ranked agents).

3.1.3. Elite Crest

The top-performing agents form a crest whose mean (Equation (10)) provides a stable, multi-point guide, mitigating premature convergence to a single attractor. Figure 3 marks the elite agents in green, the best with a star, and overlays the crest mean as a highlighted marker.

For interference between multiple ripple sources, Figure 4 adds a second source and a global swell (sinusoid), reflecting the superposition implied by Equation (7) and the multiple guides in Equation (11).

3.2. Mathematical Model

We formalize REO on a bounded domain $\mathcal{D}=[\mathit{l},\mathit{u}]\subset {\mathbb{R}}^d$ with population size N and iteration budget T. Let $\mathsf{\Delta }=\mathit{u}-\mathit{l}$ and ${\mathit{x}}_i^{\left(t\right)}\in \mathcal{D}$ be agent i at iteration t.

3.3. Initialization and Fitness

We state the randomized initialization in Equation (1), where ⊙ is the Hadamard product:

$${\mathit{x}}_i^{\left(0\right)}=\mathit{l}+{\mathrm{rand}}_i\left(d\right)\odot \mathsf{\Delta },i=1,\dots ,N.$$

(1)

Agent quality is given by the objective evaluation in Equation (2):

$$f_i^{\left(t\right)}=f\left({\mathit{x}}_i^{\left(t\right)}\right).$$

(2)

3.4. Rank- and Time-Aware Pulls

After sorting agents by ascending fitness, let $r_i^{\left(t\right)}\in \{0,\dots ,N-1\}$ denote the rank (0 for the best). We define the normalized rank share in Equation (3) and the undertow strength in Equation (4):

$${\mathrm{rankshare}}_i^{\left(t\right)}={\displaystyle \frac{r_i^{\left(t\right)}}{max(1,N-1)}}.$$

(3)

$${\eta }_i^{\left(t\right)}={\eta }_0\left(1-{\mathrm{rankshare}}_i^{\left(t\right)}\right),{\eta }_0\in [0,1].$$

(4)

The tide is a time-aware exploitation factor that grows linearly with t, as shown in Equation (5):

$${\tau }^{\left(t\right)}={\tau }_0{\displaystyle \frac{t}{T}},{\tau }_0>0.$$

(5)

3.5. Ripple (Swell) and Amplitude Decay

To encode decaying sinusoidal perturbations, we define the amplitude in Equation (6) and the swell vector in Equation (7); here $\sigma \in (0,1)$ scales the swell relative to the search span and $\omega >0$ is the angular frequency:

$$A^{\left(t\right)}=A_0{\delta }^t,0<\delta <1,$$

(6)

$${\mathit{s}}^{\left(t\right)}=\left(A^{\left(t\right)}\sigma \right)sin\left(\omega {\displaystyle \frac{t}{T}}+{\mathit{\varphi }}^{\left(t\right)}\right)\odot \mathsf{\Delta }.$$

(7)

3.6. Self-Adaptive Parameters (jDE)

We adopt the jDE mechanism [37] to update the mutation scale $F_i$ and crossover rate ${\mathrm{Cr}}_i$. The F update in Equation (8) and the $Cr$ update in Equation (9) use probabilities ${\tau }_F,{\tau }_{Cr}\in (0,1)$:

$$F_i^{(t+1)}=\left\{\begin{array}{cc}\mathrm{U}(F_{min},F_{max}),\hfill & \mathrm{if}{u}_F<{\tau }_F,\hfill \\ F_i^{\left(t\right)},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(8)

$${\mathrm{Cr}}_i^{(t+1)}=\left\{\begin{array}{cc}\mathrm{U}(0,1),\hfill & \mathrm{if}{u}_{Cr}<{\tau }_{Cr},\hfill \\ {\mathrm{Cr}}_i^{\left(t\right)},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(9)

3.7. Elite Crest and Mutant Construction

Let the top-k agents ($k=\lceil pN\rceil$) constitute the crest. We use the mean of the top-m ($m=\lceil \rho N\rceil$) for stability, defined in Equation (10):

$${\overline{\mathit{c}}}^{\left(t\right)}={\displaystyle \frac{1}{m}}\sum _{j=1}^m{\mathit{x}}_{\left(j\right)}^{\left(t\right)},\left(·\right)\mathrm{sorts}\mathrm{by}\mathrm{fitness}\mathrm{asc}.$$

(10)

Combine Equations (4)–(7) with a JADE-like current-to-p-best/1 [38]. The mutant vector is defined by Equation (11), where $p\in \{1,\dots ,k\}$ indexes a random member of the crest, $r1\ne r2\ne i$ are random indices, and ${\mathit{x}}_{★}^{\left(t\right)}$ is the global best:

$$\begin{array}{cc}\hfill {\mathit{v}}_i^{\left(t\right)}& ={\mathit{x}}_i^{\left(t\right)}+F_i^{\left(t\right)}\left({\mathit{x}}_p^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+F_i^{\left(t\right)}\left({\mathit{x}}_{r1}^{\left(t\right)}-{\mathit{x}}_{r2}^{\left(t\right)}\right)\hfill \\ \hfill & +{\eta }_i^{\left(t\right)}\left({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+{\tau }^{\left(t\right)}\left({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+{\mathit{s}}^{\left(t\right)}.\hfill \end{array}$$

(11)

3.8. Crossover, Lévy Drift, and Selection

Binomial crossover forms the trial vector per dimension j as in Equation (12) (with a forced dimension $j_{\mathrm{rand}}$):

$$u_{i,j}^{\left(t\right)}=\left\{\begin{array}{cc}{v}_{i,j}^{\left(t\right)},\hfill & \mathrm{if}\mathrm{rand}\left(\right)<{\mathrm{Cr}}_i^{\left(t\right)}\mathrm{or}j=j_{\mathrm{rand}},\hfill \\ x_{i,j}^{\left(t\right)},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(12)

With probability in Equation (13), we apply a Lévy-flight kick [39,40] as in Equation (14), using Mantegna’s recipe Equations (15) and (16) with stability $\alpha \in (1,2)$ and scale $\kappa >0$:

$$p_{\mathrm{drift}}^{\left(t\right)}=p_0\left(1-{\displaystyle \frac{t}{T}}\right),p_0\in (0,1).$$

(13)

$${\mathit{u}}_i^{\left(t\right)}\leftarrow {\mathit{u}}_i^{\left(t\right)}+\kappa {\mathcal{\ell }}^{\left(t\right)}\odot \mathsf{\Delta }.$$

(14)

$${\ell }_j^{\left(t\right)}={\displaystyle \frac{u_j}{|v_j{|}^{1/\alpha }}},u_j\sim \mathcal{N}(0,{\sigma }_u^2),v_j\sim \mathcal{N}(0,1),$$

(15)

$${\sigma }_u={\left[{\displaystyle \frac{\Gamma (1+\alpha )sin(\pi \alpha /2)}{\Gamma \left(\frac{1+\alpha }{2}\right)\alpha 2^{(\alpha -1)/2}}}\right]}^{1/\alpha }.$$

(16)

We use reflection for boundary handling [41], dimension-wise as in Equation (17) (applied twice for long jumps):

$${\tilde{x}}_{i,j}^{\left(t\right)}=\left\{\begin{array}{cc}2l_j-u_{i,j}^{\left(t\right)},\hfill & u_{i,j}^{\left(t\right)}<l_j,\hfill \\ 2u_j-u_{i,j}^{\left(t\right)},\hfill & u_{i,j}^{\left(t\right)}>u_j,\hfill \\ u_{i,j}^{\left(t\right)},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(17)

Finally, greedy selection in Equation (18) ensures non-worsening replacement:

$${\mathit{x}}_i^{(t+1)}=\left\{\begin{array}{cc}{\tilde{\mathit{u}}}_i^{\left(t\right)},\hfill & f({\tilde{\mathit{u}}}_i^{\left(t\right)})<f({\mathit{x}}_i^{\left(t\right)}),\hfill \\ {\mathit{x}}_i^{\left(t\right)},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(18)

3.9. High-Level Pseudocode

Algorithm 1 shows the high-level pseudocode.

Algorithm 1 REO: Ripple Evolution with adaptive and diversified movement

Require:  N (pop), T (Max_iter), $\mathit{l},\mathit{u}$ (lb, ub), d (dim), f (fobj) Ensure:  Best_score, Best_pos, curve 1: Initialize $\mathsf{\Delta }\leftarrow \mathit{u}-\mathit{l}$ and agents ${\mathit{x}}_i^{\left(0\right)}$ by Equation (1); evaluate $f_i^{\left(0\right)}$ by Equation (2) 2: Set jDE params and initialize $F_i,{\mathrm{Cr}}_i$ [37] (updates via Equations (8) and (9)) 3: Initialize $A^{\left(0\right)}=A_0$ and $\delta$ for amplitude decay (6) 4: for $t=0$$T-1$ do 5:     Sort population by $f_i^{\left(t\right)}$; compute ${\mathrm{rankshare}}_i^{\left(t\right)}$ by Equation (3) 6:     Compute undertow ${\eta }_i^{\left(t\right)}$ using Equation (4) and tide ${\tau }^{\left(t\right)}$ by Equation (5) 7:     Update amplitude $A^{\left(t\right)}$ by Equation (6) and swell ${\mathit{s}}^{\left(t\right)}$ by Equation (7) 8:     Form crest set and elite mean ${\overline{\mathit{c}}}^{\left(t\right)}$ via Equation (10) 9:     Update $F_i,{\mathrm{Cr}}_i$ using jDE Equations (8) and (9) 10:    for $i=1$ N do 11:         Construct mutant ${\mathit{v}}_i^{\left(t\right)}$ by Equation (11) 12:         Perform binomial crossover to obtain ${\mathit{u}}_i^{\left(t\right)}$ by Equation (12) 13:         With probability (13), apply Lévy drift using Equations (14)–(16) 14:         Reflect to bounds by Equation (17); evaluate and select via Equation (18) 15:     end for 16:     Record best fitness in curve[t] 17: end for 18: return global best, best fitness

3.10. Movement Strategy

REO update decomposes into five forces: (i) current-to-p-best/1 (first two terms of Equation (11)), (ii) undertow to the global best (Equation (4)), (iii) tide toward the crest mean (Equations (5) and (10)), and (iv) sinusoidal swell (Equation (7)). Their vector superposition is the mutant step in Equation (11). Figure 5 illustrates these components as arrows from the current agent.

3.11. Exploration and Exploitation Behavior

REO balances exploration and exploitation via (Equation (6)) amplitude decay, the drift probability in Equation (13), and growing tide Equation (5). Early iterations have larger $A^{\left(t\right)}$, higher $p_{\mathrm{drift}}^{\left(t\right)}$, and weaker tide, yielding long-range, multi-directional search; late iterations invert these proportions. We depict the progression in two separate figures: Figure 6 (early) and Figure 7 (late).

3.12. Ripple Evolution Optimizer (REO) Novelty

The metaphor of ripples serves only as an intuitive analogy; the novelty of REO lies in three explicit operators and their interaction schedules, which collectively generate a two-anchor vector field atop a Differential Evolution (DE) backbone. As formalized in Equation (11), REO’s mutation decomposes the step for agent i at iteration t as follows:

$$\begin{array}{cc}\hfill {\mathit{v}}_i^{\left(t\right)}& =F_i^{\left(t\right)}\left({\mathit{x}}_p^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+F_i^{\left(t\right)}\left({\mathit{x}}_{r1}^{\left(t\right)}-{\mathit{x}}_{r2}^{\left(t\right)}\right)\hfill \\ \hfill & +{\eta }_i^{\left(t\right)}\left({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+{\tau }^{\left(t\right)}\left({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+{\mathit{s}}^{\left(t\right)}.\hfill \end{array}$$

(19)

3.12.1. Rank-Aware Undertow

The first unique operator is the rank-aware undertow ${\eta }_i^{\left(t\right)}({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)})$, which scales the exploitation pressure according to each agent’s fitness rank. As defined in Equations (3) and (4), the undertow coefficient ${\eta }_i^{\left(t\right)}={\eta }_0(1-{\mathrm{rankshare}}_i^{\left(t\right)})$ ensures that high-quality agents are pulled more strongly toward the global best, while low-quality ones retain more freedom to explore. Unlike the scalar selection pressure in Genetic Algorithms (GA) or the uniform global attraction in Particle Swarm Optimization (PSO), REO’s undertow operates as a vector field that modulates both direction and magnitude per agent and per coordinate.

3.12.2. Time-Growing Tide Toward Elite Mean

The second mechanism is the tide, a time-dependent attraction toward the mean of the top-m individuals (the “crest’’). As shown in Equations (5)–(10), the term ${\tau }^{\left(t\right)}({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)})$ grows linearly with time t. This creates a second attractor that represents collective elite knowledge, contrasting with DE’s reliance on a single sampled p-best or PSO’s dual global/local anchors. By incorporating two simultaneous attractors—the global best and the elite mean—REO mitigates premature convergence and stabilizes the search trajectory.

3.12.3. Zero-Mean Range-Scaled Swell

The third operator is the swell term ${\mathit{s}}^{\left(t\right)}$, a zero-mean sinusoidal perturbation whose amplitude decays geometrically as $A^{\left(t\right)}=A_0{\delta }^t$. Defined in Equations (6) and (7), it introduces structured, range-scaled oscillations with random phase, providing controlled exploration early in the run while vanishing as convergence progresses. Unlike the constant stochastic variation in GA, PSO, or DE, REO’s swell adds a deterministic, frequency-controlled component that improves early coverage of the search space without destabilizing exploitation.

3.12.4. Complementary Schedules and Two-Anchor Dynamics

Each operator follows a complementary schedule: the undertow increases with solution quality, the tide increases with iteration count, the swell amplitude decreases with time, and Lévy kicks are applied only early in the search. These opposing schedules define a consistent exploration–exploitation trajectory: broad exploration in early iterations followed by controlled exploitation near convergence. The combined “two-anchor’’ behavior—bias toward both ${\mathit{x}}_{★}$ and $\overline{\mathit{c}}$—is absent in GA, PSO, and classical DE.

3.12.5. Analytical Expectation of the Search Bias

Taking expectations over random terms in Equation (19), and using $\mathbb{E}[{\mathit{x}}_{r1}-{\mathit{x}}_{r2}]=\mathit{0}$ and $\mathbb{E}\left[{\mathit{s}}^{\left(t\right)}\right]=\mathit{0}$, the mean step simplifies to:

$$\mathbb{E}\left[{\mathit{v}}_i^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\mid {\mathcal{P}}^{\left(t\right)}\right]={\eta }_i^{\left(t\right)}({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)})+{\tau }^{\left(t\right)}({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}).$$

(20)

This demonstrates that exploration arises from the variance of the random terms (differential mutation, swell, and Lévy drift), while exploitation is driven by two biased forces toward distinct anchors. Hence, REO’s exploration/exploitation balance is structurally different from GA, DE, and PSO.

3.13. Standard-Terms Presentation and Comparative Analysis with DE and PSO

This subsection presents REO strictly in standard optimization terms (populations, candidate solutions, and move operators), and compares its update rule side by side with well-established algorithms. We avoid metaphorical phrasing and focus on functional operators, schedules, and resulting biases/variances.

3.13.1. Notation and Baseline Updates

Let ${\mathit{x}}_i^{\left(t\right)}\in [\mathit{l},\mathit{u}]\subset {\mathbb{R}}^d$ denote the position of individual i at iteration t; ${\mathit{x}}_{★}^{\left(t\right)}$ is the best individual, and ${\overline{\mathit{c}}}^{\left(t\right)}$ is the mean of the top-m elites. The JADE-style DE mutant and PSO updates are:

JADE (Current-to-p-best/1) [38]

$${\mathit{v}}_i^{\left(t\right)}={\mathit{x}}_i^{\left(t\right)}+F_i^{\left(t\right)}\left({\mathit{x}}_p^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+F_i^{\left(t\right)}\left({\mathit{x}}_{r1}^{\left(t\right)}-{\mathit{x}}_{r2}^{\left(t\right)}\right).$$

(21)

• A binomial crossover (12) and greedy selection (18) produce the next population.

PSO (Global Best Variant)

$$\begin{array}{cc}\hfill {\mathit{v}}_i^{(t+1)}& =\omega {\mathit{v}}_i^{\left(t\right)}+c_1{\mathit{r}}_1^{\left(t\right)}\odot \left({\mathbf{p}}_i^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+c_2{\mathit{r}}_2^{\left(t\right)}\odot \left({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathit{x}}_i^{(t+1)}& ={\mathit{x}}_i^{\left(t\right)}+{\mathit{v}}_i^{(t+1)}.\hfill \end{array}$$

(23)

• Here ${\mathbf{p}}_i^{\left(t\right)}$ is the personal best, $\omega$ is inertia, $c_1,c_2$ are cognitive/social coefficients, and ${\mathit{r}}_1,{\mathit{r}}_2$ are i.i.d. uniform vectors.

3.13.2. REO Update in Standard Terms

REO is inspired by the JADE core with two bias terms and one structured, scheduled perturbation:

$$\begin{array}{cc}\hfill {\mathit{v}}_i^{\left(t\right)}& ={\mathit{x}}_i^{\left(t\right)}+F_i^{\left(t\right)}\left({\mathit{x}}_p^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)+F_i^{\left(t\right)}\left({\mathit{x}}_{r1}^{\left(t\right)}-{\mathit{x}}_{r2}^{\left(t\right)}\right)\hfill \\ \hfill & +\underset{\mathrm{rank}-\mathrm{aware}\mathrm{undertow};\mathrm{Equations}\left(3\right)\mathrm{and}\left(4\right)}{\underbrace{{\eta }_i^{\left(t\right)}\left({\mathit{x}}_{★}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)}}+\underset{\mathrm{time}-\mathrm{growing}\mathrm{tide};\mathrm{Equations}\left(5\right)\mathrm{and}\left(10\right)}{\underbrace{{\tau }^{\left(t\right)}\left({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)}\right)}}\hfill \\ \hfill & +\underset{\mathrm{range}-\mathrm{scaled},\mathrm{zero}-\mathrm{mean}\mathrm{swell};\mathrm{Equations}\left(6\right)\mathrm{and}(7)}{\underbrace{{\mathit{s}}^{\left(t\right)}}}.\hfill \end{array}$$

(24)

REO then applies the same binomial crossover (12) and greedy selection (18) as DE, with reflection/clamp for bounds (17). Parameters $F_i^{\left(t\right)}$ and ${\mathrm{Cr}}_i^{\left(t\right)}$ self-adapt via jDE (8) and (9). Optional Lévy kicks are scheduled early (Equations (13)–(16)).

3.13.3. REO Mechanism in Comparison to PSO or JADE

REO’s operator in comparison to PSO or JADE:

(i) No velocity state: REO does not maintain a ${\mathit{v}}_i$ state or inertia $\omega$ (Equations (22) and (23)); it is a direct position update with DE-style crossover and greedy selection. (ii) Anchors: PSO uses ${\mathbf{p}}_i$ (personal best) and ${\mathit{x}}_{★}$; REO uses ${\mathit{x}}_{★}$ and $\overline{\mathit{c}}$ (elite mean), a multi-point anchor not present in PSO. (iii) Scheduling: REO’s undertow depends on rank (quality) and its tide grows with time; PSO’s attraction coefficients $c_1,c_2$ are static (or annealed but not rank-conditioned). (iv) Structured perturbation: REO’s ${\mathit{s}}^{\left(t\right)}$ is range-scaled, zero-mean, sinusoidal with decaying amplitude, unlike PSO’s stochastic terms, which enter as multiplicative random scalars of point-to-point differences and velocity memory.

REO In comparison with JADE (or classic DE: (i) Two anchors vs one: JADE’s step uses a single sampled p-best (Equation (21)). REO adds rank-aware pull to ${\mathit{x}}_{★}$ and a time-growing pull to $\overline{\mathit{c}}$, producing a two-anchor bias not expressible as $F\left({\mathit{x}}_p-{\mathit{x}}_i\right)$ with any scalar F. (ii) Rank/time coupling: ${\eta }_i^{\left(t\right)}$ depends on rank; ${\tau }^{\left(t\right)}$ depends on time; these cannot be absorbed into a constant or jDE-updated $F_i^{\left(t\right)}$ without changing the anchor point(s) and introducing quality conditioning. (iii) Swell: ${\mathit{s}}^{\left(t\right)}$ adds a zero-mean, span-aware, scheduled oscillation that is not part of DE’s differential term and cannot be re-parameterized as $F({\mathit{x}}_{r1}-{\mathit{x}}_{r2})$.

3.13.4. REO Operator Mapping with Standard Terms

Table 1. Operator-level comparison in standard terms.

Aspect	JADE (DE)	PSO	REO
State	Positions only	Positions + velocities	Positions only
Primary move	$F({\mathit{x}}_p-{\mathit{x}}_i)+F({\mathit{x}}_{r1}-{\mathit{x}}_{r2})$	$\omega \mathit{v}+c_1{\mathit{r}}_1\odot ({\mathbf{p}}_i-{\mathit{x}}_i)+c_2{\mathit{r}}_2\odot ({\mathit{x}}_{★}-{\mathit{x}}_i)$	JADE core + rank-aware ${\eta }_i({\mathit{x}}_{★}-{\mathit{x}}_i)$ + time-growing $\tau (\overline{\mathit{c}}-{\mathit{x}}_i)$ + ${\mathit{s}}^{\left(t\right)}$
Anchors	Single sampled p-best	Personal best + global best	Global best + elite mean (two anchors)
Scheduling	jDE for $F,\mathrm{Cr}$	Optional annealing of $\omega$	jDE for $F,\mathrm{Cr}$; ${\eta }_i$ rank-conditioned; $\tau$ increases with t; $\parallel {\mathit{s}}^{\left(t\right)}\parallel$ decays
Exploration source	Differential term	Random coefficients + inertia	Differential term + structured ${\mathit{s}}^{\left(t\right)}$ + early Lévy
Exploitation source	p-best term	Cognitive/social pulls	Rank-aware best pull + time-growing elite-mean pull
Selection	Greedy, binomial crossover	None (direct state update)	Greedy, binomial crossover

summarizes the Operator-level comparison in standard terms of metaheuristic field.

3.14. Rationale for Component Design and Expected Advantages

While REO follows a standard population-based optimizer structure—initialization, variation, evaluation, and selection—each of its components is deliberately constructed to address known weaknesses of existing metaheuristics. This subsection explains the individual purpose of each operator and why their combination is expected to outperform other optimizers on specific classes of problems.

3.14.1. Initialization and Diversity Preservation

The population is initialized uniformly over the search domain to ensure unbiased coverage and maximal entropy at iteration 0. Because REO introduces no velocity state or historical memory, the diversity of the initial sampling directly affects global exploration. The subsequent swell term (Equation (7)) maintains controlled diversity through range-scaled oscillations even after several generations, preventing early clustering that often limits PSO or GA in high dimensions.

3.14.2. Differential and Rank-Conditioned Exploitation

The DE core (Equation (11)) provides efficient local search through difference vectors between randomly chosen agents. To enhance convergence stability, REO augments this with a rank-aware undertow ${\eta }_i^{\left(t\right)}({\mathit{x}}_{★}-{\mathit{x}}_i)$ that adaptively scales exploitation intensity: high-ranking individuals are drawn strongly toward the global best, while lower-ranking ones explore more widely. This selective pressure accelerates convergence on unimodal landscapes and maintains exploration on multimodal ones.

3.14.3. Time-Dependent Collective Guidance

The tide component ${\tau }^{\left(t\right)}({\overline{\mathit{c}}}^{\left(t\right)}-{\mathit{x}}_i^{\left(t\right)})$ increases over time, guiding all agents toward the elite mean once the population begins to converge. This collective attractor mitigates oscillations around the best solution—a common issue in PSO—and allows cooperative exploitation of multiple promising regions. On composite functions or constrained engineering designs where multiple local minima exist, this term stabilizes the search trajectory and ensures steady improvement.

3.14.4. Structured Exploration via the Swell

The swell term ${\mathit{s}}^{\left(t\right)}$ injects sinusoidal, range-scaled perturbations with a geometrically decaying amplitude. Its early large amplitude promotes long-range exploration akin to Lévy flights but in a deterministic and bounded manner; its late small amplitude supports fine-grained local refinement. Empirically, this yields faster initial descent than DE and smoother final convergence than PSO.

3.14.5. Complementary Scheduling of Operators

The three dynamic coefficients—${\eta }_i^{\left(t\right)}$ (rank-based), ${\tau }^{\left(t\right)}$ (time-based), and $A^{\left(t\right)}$ (amplitude decay)—are complementary:

• early iterations: large $A^{\left(t\right)}$, small ${\tau }^{\left(t\right)}$⇒ exploration dominant;
• mid-phase: balanced undertow and tide ⇒ exploration–exploitation equilibrium;
• late iterations: small $A^{\left(t\right)}$, large ${\tau }^{\left(t\right)}$⇒ refined exploitation.

This predictable scheduling produces the rapid early progress and stable late convergence observed in CEC2022 experiments.

3.14.6. Theoretical Analysis of the Outperforming Performance

REO’s integration of a two-anchor bias (best and elite mean), rank-dependent adaptation, and structured perturbation provides an adaptive exploration–exploitation trajectory absent in GA, DE, and PSO. GA’s stochastic mutation can lose gradient direction; PSO’s global–local duality can oscillate; DE’s fixed differential scale can stagnate near optima. REO’s coordinated design offers:

• faster descent on smooth unimodal functions due to rank-aware undertow;
• improved robustness on rotated multimodal and hybrid functions through oscillatory exploration;
• superior consistency on constrained engineering problems via the tide’s collective stabilization.

Hence, although REO conforms to the generic population-based template, its operator coupling and scheduling yield demonstrable advantages for landscapes where both global coverage and precise final refinement are required.

3.15. Computational Cost

Operation Counts per Iteration

Let N be the population size, d the dimensionality, and T the iteration budget. Denote by $C_f$ the time to evaluate the objective $f\left(·\right)$ once on a single agent. One iteration of REO (Section 1) performs: (i) a population sort (to obtain the best and the elite set), (ii) N objective evaluations, and (iii) $\mathsf{\Theta }\left(Nd\right)$ vector operations for variation, recombination, and bound handling. Using $\mathsf{\Theta }\left(·\right)$-notation,

$$C_{\mathrm{sort}}=\mathsf{\Theta }(NlogN),C_{\mathrm{eval}}=\mathsf{\Theta }\left(N{C}_f\right),C_{\mathrm{arith}}=\mathsf{\Theta }\left(Nd\right).$$

(25)

Hence the per-iteration and total costs are

$$C_{\mathrm{iter}}=\mathsf{\Theta }\left(NlogN+N{C}_f+Nd\right),$$

(26)

$$C_{\mathrm{total}}=\mathsf{\Theta }\left(T\left(NlogN+N{C}_f+Nd\right)\right)(\mathrm{plus}N{C}_f\mathrm{for}\mathrm{initialization}).$$

(27)

Space usage is linear in the population:

$$S=\mathsf{\Theta }\left(Nd\right).$$

(28)

Cheap Objectives (Closed-Form Mathematical Relations)

When f is a deterministic, closed-form mapping with low arithmetic cost, $C_f=\mathsf{\Theta }\left(d\right)$ or $C_f=\mathsf{\Theta }\left(1\right)$, the iteration time is dominated by $NlogN+Nd$. In this regime, two engineering choices keep overhead small: (i) partial selection instead of full sorting. REO only needs the global best and the top-m agents to compute the elite mean; replacing a full sort by an m-element heap or nth_element yields

$$C_{\mathrm{select}}=\mathsf{\Theta }(Nlogm)(m\ll N),$$

(29)

which tightens (26) to $\mathsf{\Theta }(Nlogm+Nd)$ for cheap f; and (ii) vectorization. The variation and crossover steps are BLAS-1/2 style and run efficiently in SIMD/GPU, making $C_{\mathrm{arith}}$ bandwidth-bound in practice. Empirically, for common analytic benchmarks, the wall-clock is then driven by memory traffic rather than transcendental calls (e.g., sin in the perturbation), and REO scales near-linearly with N.

Expensive Objectives (Simulators, Black-Box Models, or API Calls)

When $C_f$ dominates, (26) reduces to $C_{\mathrm{iter}}=\mathsf{\Theta }\left(N{C}_f\right)$, and the wall-clock is controlled by how evaluations are dispatched. Let P be the number of parallel workers (threads/processes/remote slots). Under synchronous batches,

$$C_{\mathrm{iter},\mathrm{wall}}\approx \mathsf{\Theta }\left(\frac{N}{P}{C}_f\right)+\mathsf{\Theta }(Nlogm+Nd),$$

(30)

where m is the elite set size used by REO. For remote services (APIs), decompose the evaluation time per agent as

$$C_f^{\mathrm{api}}=L_{\mathrm{net}}+C_{\mathrm{srv}}+L_{\mathrm{parse}},$$

(31)

with network latency $L_{\mathrm{net}}$, service processing time $C_{\mathrm{srv}}$, and client-side serialization/parsing $L_{\mathrm{parse}}$. If the platform imposes a rate limit $R_{max}$ (requests/sec) and a maximum concurrency $P_{max}$, the effective parallelism is $P=min(P_{\mathrm{local}},P_{max})$, and the sustainable throughput is bounded by $R_{max}$. In this case, REO behaves like a standard population-based optimizer whose wall-clock is dominated by external I/O; the algorithmic overhead (selection and vector ops) is negligible.

Asynchronous and Batched Execution

To reduce idle time when $C_f$ varies across agents (e.g., heterogeneous API latencies), an asynchronous variant evaluates agents as workers become free and updates the elite set on arrival; in practice, this preserves REO’s behavior while improving utilization. If the external platform supports batched queries, grouping the N candidates per iteration into a few large requests amortizes $L_{\mathrm{net}}$ and $L_{\mathrm{parse}}$, effectively shrinking $C_f^{\mathrm{api}}$.

4. Experimental Setup

All experiments were conducted in MATLAB R2024 using a common protocol across REO and all baselines: a population of $\mathbf{N}=\mathbf{50}$ agents, a budget of $\mathbf{T}=\mathbf{1000}$ iterations per run, and $\mathbf{30}$ independent runs per problem with distinct seeds (rng(seed,’twister’)). We evaluate on the CEC2022 single-objective suite (12 functions spanning unimodal, rotated multimodal, hybrid, and composition) and five constrained engineering designs (Welded Beam, Spring Design, Three-Bar Truss, Cantilever Stepped Beam, Ten-Bar Planar Truss), treating all as minimization with canonical bounds/constraints; initialization is uniform in $[\mathit{l},\mathit{u}]$, bound violations are handled by reflection/clamp (long jumps reflected twice), feasibility has priority in replacement (feasible ≻ infeasible; among infeasible, lower total violation), and required discrete variables are projected to the nearest admissible value before evaluation. REO employs a JADE current-to-p-best/1 core with jDE self-adaptation $({\tau }_F={\tau }_{\mathrm{Cr}}=0.1,F_{min}=0.1,F_{max}=0.9)$, elite sampling $p=0.1$ and elite-mean fraction $\rho =0.2$, a rank-aware undertow of strength ${\eta }_0=0.6$, a time-growing tide ${\tau }^{\left(t\right)}={\tau }_0(t/T)$ with ${\tau }_0=0.6$, a range-scaled swell $A^{\left(t\right)}=A_0{\delta }^t$ with $A_0=0.2$, $\delta =0.995$, $\omega =\pi$, $\sigma =0.05$, and early Lévy kicks $p_{\mathrm{drift}}^{\left(t\right)}=p_0(1-t/T)$ with $p_0=0.2$, $\alpha =1.5$, $\kappa =0.01$; these settings are fixed across problems unless sensitivity analyses are explicitly reported. Baselines (e.g., DE/JADE, PSO, CMA–ES, GA, ABC, CS, and recent nature-inspired methods) follow canonical parameterizations from their primary sources but strictly match the shared budget/population/runs, stopping at T or when the suite’s tolerance to the known optimum is achieved. For each algorithm and problem, we report mean, standard deviation (when known), error to the optimum, and a per-function rank (lower-is-better) with tie-breaking by better mean, then better best, then lower std; global summaries include average rank, Top-k counts, and head-to-head tallies. Statistical significance is assessed via two-sided Wilcoxon signed-rank tests over paired per-function results ($n=12$ for CEC2022). Convergence is visualized with mean best-so-far curves and shaded variability bands, complemented by boxplots of final scores, ECDFs across functions, and a Dolan–Moré performance profile.

5. Statistical Comparison Results over CEC2022

The CEC2022 test suite comprises 12 benchmark functions ($\mathrm{F}1$–$F12$) designed to evaluate single-objective optimization algorithms. Each function provides a challenging landscape with different modalities and complexities, requiring algorithms to explore and exploit the search space effectively. In dynamic multimodal optimization research, the presence of multiple global or local optima is well recognized, and benchmark test suites combine multimodal functions with different change modes to generate diverse environments for algorithm evaluation. The populations used in CEC competitions often include both “simple” environments with several global and local peaks and “composition” functions whose landscapes include a large number of local peaks that can mislead evolutionary search. The results provided here are static single-objective data from the CEC2022 suite; however, the insights from dynamic multimodal problems illustrate the importance of robustness and adaptability across heterogeneous functions.

5.1. Evaluation Metrics

For each test function, four statistics were recorded for every optimizer: the mean objective value over multiple runs, the standard deviation (Std.), an error measure (difference between the observed mean value and the known optimum), and a rank. Lower mean and error values are better, and lower standard deviations indicate more consistent performance. Ranks are assigned per function; the best mean receives rank 1, and higher numbers denote worse performance. Average ranks across all functions allow holistic comparison among optimizers.

5.2. Performance of the REO Optimizer Compared with Other Optimizers

The REO algorithm consistently achieved the best performance across all 12 CEC2022 functions. As shown in the following tables, REO obtained the lowest mean objective value, zero or very small error, and negligible standard deviation on every function. Consequently, it was ranked first on all functions, giving it an average rank of 1 (

Table 2. Average performance metrics across all CEC2022 functions.

	Avg. Mean	Avg. Std	Avg. Error	Avg. Rank
REO	1630.522	7.755	41.106	1.000
RUN	1762.260	178.927	131.167	6.417
ALO	1780.948	197.615	161.287	6.917
MVO	1992.062	408.728	197.207	7.750
DO	1906.588	323.255	182.939	8.917
COA	1879.176	295.843	206.338	8.917
GWO	2182.404	599.070	325.335	9.167
RTH	1671.519	88.186	37.758	9.417
AVOA	1840.468	257.135	196.813	10.417
AO	2478.426	895.092	639.601	11.000
SHIO	2118.822	535.489	510.489	12.750
SCSO	1965.670	382.337	363.100	14.000
SHO	2086.689	503.356	330.634	14.083
ZOA	1857.656	274.322	264.262	14.500
GJO	2327.343	744.010	264.127	14.500
HHO	1867.917	284.584	197.485	16.750
WOA	3237.238	1653.905	942.996	18.833
HGSO	138,432.792	136,849.459	89,396.787	19.417
HLOA	1891.831	308.498	311.604	19.500
TTHHO	2085.397	502.064	352.049	19.500
GBO	4996.690	3413.356	3444.596	19.667
CPO	1900.950	317.616	319.717	19.917
DOA	5,613,083.270	5,611,499.936	25,094,275.378	20.167
FOX	1973.348	390.015	262.732	21.500
SMA	3431.706	1848.372	1280.892	21.667
TSO	2306.774	723.441	698.201	21.667
Chimp	73,031.639	71,448.306	45,862.493	21.750
AOA	2682.215	1098.881	768.000	25.083
ROA	84,639.090	83,055.757	170,522.389	25.750
RSA	4,481,662.783	4,480,079.450	2,028,245.370	29.083
FLO	1,894,426.298	1,892,842.965	2,303,568.232	29.583
SPBO	35,531,797.147	35,530,213.813	23,511,193.308	30.500
SSOA	13,891,516.288	13,889,932.954	18,466,078.923	32.000
OHO	67,798,230.993	67,796,647.660	41,782,384.436	32.750

). The second-best algorithms overall were RUN, ALO, MVO, DO and COA. Nonetheless, their average ranks were much higher (6.42–8.92), and their mean objective values and variances were noticeably worse than those of REO.

On each benchmark function ${\mathrm{F}}_i$, the mean objective value of REO was either equal to or lower than that of every competitor. For example, on the shifted and fully rotated Zakharov function $F1$, the mean of REO was $300.0$ with zero standard deviation, whereas the nearest competitors (RTH, DO, and several others) obtained the same mean value but had non-zero standard deviations. For $\mathrm{F}2$, REO achieved a mean of $402.58$, while the second-best algorithm (ALO) produced $403.98$; REO also had an almost zero error measure compared with larger errors for other methods. Similarly, on $\mathrm{F}3$, the mean of REO was $600.0$ (with zero error), whereas the best competitor (GWO) attained $600.30$ with an error of approximately $0.30$. These small yet consistent margins illustrate the superiority of REO.

A more pronounced gap appears on functions with higher dimensions or more complex landscapes. On $\mathrm{F}6$ REO yielded a mean of $1809.77$, whereas RTH achieved $1826.65$ and COA produced $1916.73$. For $\mathrm{F}7$, the mean of REO was $2004.24$, while the next best algorithm (COA) delivered $2020.41$, and ALO produced $2053.60$. Similar patterns are observed on $\mathrm{F}8$–$\mathrm{F}12$, where REO maintained the lowest means and standard deviations; for instance, on $\mathrm{F}12$ the mean of REO was $2860.20$, whereas MVO (second) achieved $2862.36$ and RUN produced $2876.17$. Across all functions, the error measures of REO were minimal (often below 40), while many competing algorithms recorded errors in the hundreds or thousands (

Table A1. REO Results in comparison with state-of-the-art optimizers over CEC2022 (Group 1).

Function	Measure	REO	SMA	GBO	RTH	CPO	COA	SCSO	DOA	ZOA	SPBO	TSO	AO	TTHHO
F1	mean	300.000	16,900.597	7868.615	300.000	981.508	317.024	1801.966	1895.276	697.862	26,176.837	4750.826	872.700	312.415
	Std.	0.000	16,600.597	7568.615	0.000	681.508	17.024	1501.966	1595.276	397.862	25,876.837	4450.826	572.700	12.415
	error measure	0.000	11,874.182	4956.494	0.000	1451.845	26.010	2273.235	2553.890	935.614	6927.866	5508.664	463.019	19.171
F1	rank	1	33	25	1	15	12	16	17	13	34	24	14	11
F2	mean	402.581	452.966	444.742	411.699	423.869	406.033	433.995	491.903	431.409	1094.518	424.986	418.084	470.581
	Std.	3.940	52.966	44.742	11.699	23.869	6.033	33.995	91.903	31.409	694.518	24.986	18.084	70.581
	error measure	2.581	31.299	25.924	20.353	32.345	2.949	26.176	106.030	30.078	274.228	32.515	26.194	94.370
F2	rank	1	23	22	7	12	3	19	26	18	30	13	10	24
F3	mean	600.000	620.655	620.413	611.848	637.061	603.206	615.503	623.220	616.166	668.404	631.587	612.471	628.735
	Std.	0.000	20.655	20.413	11.848	37.061	3.206	15.503	23.220	16.166	68.404	31.587	12.471	28.735
	error measure	0.000	13.196	9.874	10.923	11.770	4.874	12.032	11.162	7.298	9.788	12.698	6.659	12.167
F3	rank	1	17	16	10	26	4	14	18	15	34	23	12	21
F4	mean	810.083	838.870	836.529	822.457	832.587	828.879	827.825	824.514	812.466	902.295	844.571	823.668	825.772
	Std.	1.432	38.870	36.529	22.457	32.587	28.879	27.825	24.514	12.466	102.295	44.571	23.668	25.772
	error measure	10.083	11.420	9.979	9.812	1.764	5.820	6.430	9.526	3.611	12.137	16.366	8.681	7.384
F4	rank	1	25	23	10	19	17	16	12	2	34	29	11	13
F5	mean	900.000	1484.652	1050.145	1049.876	1551.412	1019.747	1010.111	1173.483	1014.690	3451.315	1827.040	997.873	1388.892
	Std.	0.000	584.652	150.145	149.876	651.412	119.747	110.111	273.483	114.690	2551.315	927.040	97.873	488.892
	error measure	0.000	472.104	152.080	90.371	199.784	215.990	117.853	195.992	66.519	667.842	874.950	52.252	143.077
F5	rank	1	28	15	14	30	13	11	17	12	34	33	9	23
F6	mean	1809.765	5677.840	33,919.673	1826.654	3120.834	4423.698	3854.209	67,336,761.737	3597.939	426,332,821.762	4031.066	11,087.131	6258.135
	Std.	13.852	3877.840	32,119.673	26.654	1320.834	2623.698	2054.209	67,334,961.737	1797.939	426,331,021.762	2231.066	9287.131	4458.135
	error measure	9.765	2616.177	35816.917	22.403	1543.796	1990.347	1606.781	301127840.444	1798.520	282,125,839.580	1352.132	6922.344	3597.598
F6	rank	1	19	25	2	3	15	10	31	6	33	12	24	21
F7	mean	2004.235	2052.650	2069.420	2037.401	2115.933	2020.408	2046.890	2063.633	2039.307	2161.409	2073.196	2039.008	2071.799
	Std.	8.827	52.650	69.420	37.401	115.933	20.408	46.890	63.633	39.307	161.409	73.196	39.008	71.799
	error measure	4.235	26.499	18.999	15.013	52.264	4.312	24.979	41.437	14.054	40.985	22.249	13.079	23.743
F7	rank	1	17	22	8	29	2	15	19	13	34	24	11	23
F8	mean	2219.091	2228.475	2240.069	2233.442	2278.199	2223.370	2227.916	2228.705	2231.548	2348.422	2237.610	2226.484	2234.139
	Std.	5.552	28.475	40.069	33.442	78.199	23.370	27.916	28.705	31.548	148.422	37.610	26.484	34.139
	error measure	19.091	6.876	27.160	36.711	65.027	7.685	4.280	13.281	27.091	139.749	9.133	3.117	12.334
F8	rank	1	14	23	20	28	2	12	15	16	31	22	9	21
F9	mean	2529.284	2594.821	2571.185	2529.284	2552.204	2536.631	2572.129	2560.766	2599.920	2766.442	2567.088	2567.694	2609.746
	Std.	0.000	294.821	271.185	229.284	252.204	236.631	272.129	260.766	299.920	466.442	267.088	267.694	309.746
	error measure	229.284	47.874	27.707	0.000	46.950	32.855	33.585	47.179	46.939	63.028	61.840	33.092	55.262
F9	rank	1	24	20	1	11	8	21	15	25	32	16	17	27
F10	mean	2531.034	2581.995	2541.277	2550.504	2634.426	2573.403	2555.738	2579.061	2559.907	2573.886	2603.991	2536.191	2572.175
	Std.	59.074	181.995	141.277	150.504	234.426	173.403	155.738	179.061	159.907	173.886	203.991	136.191	172.175
	error measure	153.034	67.972	63.625	62.531	215.038	61.563	62.708	72.968	60.981	34.190	258.399	55.028	94.052
F10	rank	1	22	5	9	28	19	12	21	14	20	26	3	18
F11	mean	2600.000	2874.215	2926.663	2815.979	2785.229	2732.789	2775.718	2885.080	2757.808	3716.126	2796.474	2693.787	2755.431
	Std.	0.000	274.215	326.663	215.979	185.229	132.789	175.718	285.080	157.808	1116.126	196.474	93.787	155.431
	error measure	0.000	191.420	212.963	171.395	181.294	121.744	184.306	313.356	153.406	303.444	195.562	89.838	135.942
F11	rank	1	23	26	20	15	8	13	25	12	33	18	4	11
F12	mean	2860.196	2872.729	2871.544	2869.085	2898.136	2864.921	2866.041	2911.859	2932.846	2884.343	2892.856	2866.018	2896.950
	Std.	0.382	172.729	171.544	169.085	198.136	164.921	166.041	211.859	232.846	184.343	192.856	166.018	196.950
	error measure	65.196	11.690	13.426	13.582	34.722	1.912	4.841	99.275	27.037	6.855	33.902	1.916	29.487
F12	rank	1	15	14	11	23	4	9	26	28	17	20	8	21

,

Table A2. REO Results in comparison with state-of-the-art optimizers over CEC2022 (Group 2).

Function	Measure	HHO	SSOA	RUN	GWO	MVO	AOA	GJO	HLOA	WOA	RSA	SHO	FLO	DO
F1	mean	301.806	11,293.075	300.000	2029.523	300.010	8697.436	2066.980	300.014	16,022.132	8777.105	2539.039	8917.237	300.003
	Std.	1.806	10,993.075	0.000	1729.523	0.010	8397.436	1766.980	0.014	15,722.132	8477.105	2239.039	8617.237	0.003
	error measure	1.071	2433.432	0.000	1697.369	0.005	6884.058	1447.968	0.025	8428.573	2347.469	1764.691	1273.796	0.005
F1	rank	10	30	6	18	8	27	19	9	32	28	21	29	7
F2	mean	426.318	1438.447	409.757	420.801	406.559	826.861	438.023	406.872	427.450	1102.226	435.404	1634.352	413.445
	Std.	26.318	1038.447	9.757	20.801	6.559	426.861	38.023	6.872	27.450	702.226	35.404	1234.352	13.445
	error measure	33.210	400.643	21.266	20.970	2.671	323.795	41.151	3.126	30.346	653.659	57.937	423.507	23.044
F2	rank	14	32	6	11	4	29	21	5	15	31	20	33	8
F3	mean	630.755	655.976	612.156	600.303	601.182	637.163	606.660	648.059	634.659	645.812	609.226	645.670	604.583
	Std.	30.755	55.976	12.156	0.303	1.182	37.163	6.660	48.059	34.659	45.812	9.226	45.670	4.583
	error measure	8.743	7.434	5.689	0.516	1.545	7.722	4.496	11.397	13.674	3.335	5.510	9.529	5.516
F3	rank	22	32	11	2	3	27	8	30	25	29	9	28	5
F4	mean	826.076	855.488	821.889	814.461	817.482	832.485	821.232	844.108	841.133	848.923	820.322	849.216	825.810
	Std.	26.076	55.488	21.889	14.461	17.482	32.485	21.232	44.108	41.133	48.923	20.322	49.216	25.810
	error measure	8.522	11.615	6.226	6.750	7.823	8.321	7.655	18.068	16.793	6.373	5.834	11.961	11.096
F4	rank	15	33	8	3	5	18	7	28	26	31	6	32	14
F5	mean	1319.910	1604.662	978.973	907.954	900.057	1308.511	962.528	1402.142	1423.981	1441.385	1072.166	1456.632	983.810
	Std.	419.910	704.662	78.973	7.954	0.057	408.511	62.528	502.142	523.981	541.385	172.166	556.632	83.810
	error measure	145.528	193.921	36.631	12.782	0.154	157.513	70.099	232.996	286.952	158.522	113.335	191.607	118.720
F5	rank	21	32	7	3	2	20	5	24	25	26	16	27	8
F6	mean	3806.335	166,665,619.625	3182.351	6395.797	5971.458	4037.837	7945.659	3861.126	4434.161	53,751,268.631	4464.440	22,703,425.419	4761.776
	Std.	2006.335	166,663,819.625	1382.351	4595.797	4171.458	2237.837	6145.659	2061.126	2634.161	53,749,468.631	2664.440	22,701,625.419	2961.776
	error measure	1747.588	221,589,176.904	1335.925	1871.969	2008.888	1192.688	1293.822	2850.706	2188.785	24,335,005.208	1647.107	27,640,219.168	1754.379
F6	rank	9	32	4	22	20	13	23	11	16	30	17	29	18
F7	mean	2052.247	2131.198	2037.971	2024.498	2035.516	2093.584	2041.290	2105.708	2064.570	2124.884	2026.420	2103.249	2027.173
	Std.	52.247	131.198	37.971	24.498	35.516	93.584	41.290	105.708	64.570	124.884	26.420	103.249	27.173
	error measure	24.262	26.348	9.137	9.185	39.677	26.763	16.680	29.136	23.912	22.848	10.415	20.378	6.669
F7	rank	16	32	10	3	7	26	14	28	20	31	4	27	5
F8	mean	2232.476	2355.481	2223.899	2224.083	2228.138	2248.708	2226.523	2296.505	2233.059	2253.877	2223.942	2244.260	2223.770
	Std.	32.476	155.481	23.899	24.083	28.138	48.708	26.523	96.505	33.059	53.877	23.942	44.260	23.770
	error measure	13.226	96.688	1.278	5.163	28.368	37.895	3.170	62.612	6.456	21.895	1.830	24.891	5.450
F8	rank	18	32	4	7	13	26	10	29	19	27	5	25	3
F9	mean	2568.046	2787.297	2529.285	2560.093	2536.647	2690.040	2569.191	2531.502	2557.904	2719.079	2587.331	2758.261	2529.284
	Std.	268.046	487.297	229.285	260.093	236.647	390.040	269.191	231.502	257.904	419.079	287.331	458.261	229.284
	error measure	30.923	49.753	0.000	39.037	32.851	38.995	23.473	6.142	35.698	27.554	22.182	44.256	0.000
F9	rank	18	33	5	14	9	29	19	7	12	30	23	31	4
F10	mean	2597.987	2708.862	2552.328	2535.133	2569.878	2609.437	2561.466	2679.896	2536.786	2643.823	2547.393	2658.647	2549.937
	Std.	197.987	308.862	152.328	135.133	169.878	209.437	161.466	279.896	136.786	243.823	147.393	258.647	149.937
	error measure	133.827	107.957	58.762	62.167	87.109	115.888	61.797	342.143	63.867	128.505	58.188	127.829	62.602
F10	rank	25	32	11	2	17	27	15	31	4	29	7	30	8
F11	mean	2754.203	3675.586	2635.064	2810.817	2675.451	3217.882	2818.181	2724.192	2780.018	3175.503	2823.400	3345.298	2791.527
	Std.	154.203	1075.586	35.064	210.817	75.451	617.882	218.181	124.192	180.018	575.503	223.400	745.298	191.527
	error measure	177.997	364.810	97.513	173.872	155.101	372.814	188.813	139.182	183.967	459.242	253.988	367.387	201.031
F11	rank	9	32	2	19	3	29	21	7	14	28	22	31	17
F12	mean	2898.851	3069.755	2863.450	2865.382	2862.361	2986.633	2870.383	2901.851	2891.004	2952.150	2891.185	3077.339	2867.937
	Std.	198.851	369.755	163.450	165.382	162.361	286.633	170.383	201.851	191.004	252.150	191.185	377.339	167.937
	error measure	44.926	77.570	1.580	4.235	2.294	49.548	10.404	43.709	36.930	109.831	26.590	104.473	6.752
F12	rank	24	32	3	6	2	30	12	25	18	29	19	33	10

and

Table A3. REO Results in comparison with state-of-the-art optimizers over CEC2022 (Group 3).

Function	Measure	FOX	ROA	ALO	AVOA	Chimp	SHIO	OHO	HGSO
F1	mean	300.000	8477.647	300.000	300.000	2274.443	3701.791	14,724.679	4246.992
	Std.	0.000	8177.647	0.000	0.000	1974.443	3401.791	14,424.679	3946.992
	error measure	0.000	1280.074	0.000	0.000	1154.214	3884.769	6187.418	1236.568
F1	rank	5	26	3	4	20	22	31	23
F2	mean	415.301	753.296	403.980	431.224	567.695	431.257	2451.052	482.848
	Std.	15.301	353.296	3.980	31.224	167.695	31.257	2051.052	82.848
	error measure	24.232	324.950	4.089	33.680	117.304	30.103	996.906	15.775
F2	rank	9	28	2	16	27	17	34	25
F3	mean	649.560	633.858	606.490	614.177	627.504	605.393	660.693	626.433
	Std.	49.560	33.858	6.490	14.177	27.504	5.393	60.693	26.433
	error measure	9.920	12.257	6.739	9.534	7.169	4.151	4.013	4.909
F3	rank	31	24	7	13	20	6	33	19
F4	mean	838.068	844.107	821.908	833.631	836.525	816.676	845.389	832.948
	Std.	38.068	44.107	21.908	33.631	36.525	16.676	45.389	32.948
	error measure	10.857	8.865	10.448	12.201	8.195	7.972	2.933	4.052
F4	rank	24	27	9	21	22	4	30	20
F5	mean	1486.474	1370.511	974.625	1215.933	1269.705	948.649	1581.290	1000.787
	Std.	586.474	470.511	74.625	315.933	369.705	48.649	681.290	100.787
	error measure	60.739	219.527	136.945	219.161	182.519	94.230	83.253	29.627
F5	rank	29	22	6	18	19	4	31	10
F6	mean	4294.815	988,112.679	3333.468	3718.850	855,099.288	3759.744	813,540,668.079	1,638,867.010
	Std.	2494.815	986,312.679	1533.468	1918.850	853,299.288	1959.744	813,538,868.079	1,637,067.010
	error measure	2067.655	2,044,013.325	1541.820	1853.750	548,275.257	1754.143	501,380,245.039	1,071,348.661
F6	rank	14	27	5	7	26	8	34	28
F7	mean	2152.445	2074.398	2037.702	2031.001	2059.501	2039.192	2123.132	2067.928
	Std.	152.445	74.398	37.702	31.001	59.501	39.192	123.132	67.928
	error measure	53.509	34.532	15.691	8.508	10.688	19.596	9.035	8.958
F7	rank	33	25	9	6	18	12	30	21
F8	mean	2405.798	2243.123	2227.573	2223.952	2304.731	2226.446	2457.029	2232.235
	Std.	205.798	43.123	27.573	23.952	104.731	26.446	257.029	32.235
	error measure	116.963	17.001	6.147	3.212	61.602	5.090	139.094	2.024
F8	rank	33	24	11	6	30	8	34	17
F9	mean	2549.245	2660.820	2529.928	2529.284	2558.473	2585.655	2824.990	2608.553
	Std.	249.245	360.820	229.928	229.284	258.473	285.655	524.990	308.553
	error measure	36.709	57.006	1.842	0.000	20.854	40.159	107.423	30.530
F9	rank	10	28	6	3	13	22	34	26
F10	mean	2875.742	2596.302	2559.364	2568.017	2592.985	2551.934	2965.790	2544.649
	Std.	475.742	196.302	159.364	168.017	192.985	151.934	565.790	144.649
	error measure	556.849	86.167	60.672	69.926	349.904	58.524	266.220	61.169
F10	rank	33	24	13	16	23	10	34	6
F11	mean	2717.808	2978.927	2710.784	2754.540	3317.333	2879.024	4231.245	2786.157
	Std.	117.808	378.927	110.784	154.540	717.333	279.024	1631.245	186.157
	error measure	128.091	168.036	148.480	150.486	149.370	203.864	435.998	13.195
F11	rank	6	27	5	10	30	24	34	16
F12	mean	2994.919	2923.412	2865.555	2865.009	2871.482	2880.102	3238.549	2896.966
	Std.	294.919	223.412	165.555	165.009	171.482	180.102	538.549	196.966
	error measure	87.264	46.929	2.565	1.297	12.838	23.264	135.905	5.975
F12	rank	31	27	7	5	13	16	34	22

).

The stability of an optimizer can be inferred from the standard deviation across runs. REO had an exceptionally low average standard deviation (about $7.75$) compared with other algorithms; the second-best method (RTH) had an average standard deviation of approximately $88.19$, and most algorithms exhibited much larger variances (hundreds to millions). The low variance of REO demonstrates that it consistently converged to high-quality solutions across repeated runs.

The next strongest optimizers were RUN, ALO, and MVO. RUN achieved an average rank of $6.42$ and an average mean value of $1762.26$. ALO had a slightly higher average mean ($1780.95$) and rank ($6.92$). While these algorithms occasionally approached REO on simpler functions (for instance, RUN equaled REO on $\mathrm{F}1$ and produced only slightly higher means on $\mathrm{F}4$ and $\mathrm{F}12$), none matched REO’s uniform dominance across the entire suite. Algorithms such as SMA, GBO, SPBO, SSOA, and OHO performed poorly, with extremely large error measures and standard deviations, leading to average ranks greater than 29.

As it can be seen in

Table 3. Optimizers results vs. REO across 12 functions and number of Top-3 finishes (rank ≤ 3). W/D/L counts functions where an algorithm had a better/same/worse rank than REO.

Algorithm	W/D/L vs. REO			Top-3 Finishes
	W	D	L	(Rank $\le 3$)
RUN	0	0	12	2
ALO	0	0	12	2
MVO	0	0	12	4
DO	0	0	12	1
COA	0	0	12	3
GWO	0	0	12	5
RTH	0	2	10	3
AVOA	0	0	12	1
AO	0	0	12	1
SHIO	0	0	12	0
SCSO	0	0	12	0
SHO	0	0	12	0
ZOA	0	0	12	1
GJO	0	0	12	0
HHO	0	0	12	0
WOA	0	0	12	0
HGSO	0	0	12	0
HLOA	0	0	12	0
TTHHO	0	0	12	0
GBO	0	0	12	0
CPO	0	0	12	1
DOA	0	0	12	0
FOX	0	0	12	0
SMA	0	0	12	0
TSO	0	0	12	0
Chimp	0	0	12	0
AOA	0	0	12	0
ROA	0	0	12	0
RSA	0	0	12	0
FLO	0	0	12	0
SPBO	0	0	12	0
SSOA	0	0	12	0
OHO	0	0	12	0

, REO does not lose to any competitor on any function; only RTH ties REO (two functions: $\mathrm{F}1$ and $\mathrm{F}9$). The most frequent non-REO top 3 finishes are observed for GWO (5), followed by MVO (4), and COA/RTH (3 each).

The rank-based summaries in

Table 4. Global summary over the 12 CEC2022 functions using the reported ranks (lower is better). Columns show: Avg. rank, median rank, SD of ranks across functions, the number of wins (#rank $=1$), Top-3/Top-5 counts, Bottom-3 (#rank $\ge 32$), and the average-rank gap to REO (${\mathsf{\Delta }}_{\mathrm{REO}}$).

Algorithm	Avg.	Med.	SD	Wins	Top-3	Top-5	Bottom-3	${\mathbf{\Delta }}_{\mathbf{REO}}$
REO	1.00	1.0	0.00	12	12	12	0	0.00
RUN	6.42	6.0	2.93	0	2	5	0	5.42
ALO	6.92	6.5	3.04	0	2	4	0	5.92
MVO	7.75	6.0	5.76	0	4	6	0	6.75
COA	8.92	8.0	5.85	0	3	5	0	7.92
DO	8.92	8.0	4.75	0	1	4	0	7.92
GWO	9.17	6.5	7.06	0	5	5	0	8.17
RTH	9.42	9.5	6.14	2	3	3	0	8.42
AVOA	10.42	8.5	5.88	0	1	3	0	9.42
AO	11.00	10.5	5.37	0	1	2	0	10.00
SHIO	12.75	11.0	6.94	0	0	2	0	11.75
SCSO	14.00	13.5	3.44	0	0	0	0	13.00
CPO	15.17	15.0	8.36	0	0	0	0	14.17
TSO	21.67	22.5	5.99	0	0	0	1	20.67
SMA	21.67	22.5	5.34	0	0	0	1	20.67
Chimp	21.75	21.0	5.51	0	0	0	0	20.75
ROA	25.75	26.5	1.83	0	0	0	0	24.75
AOA	25.08	27.0	5.04	0	0	0	0	24.08
RSA	29.08	29.0	1.55	0	0	0	0	28.08
FLO	29.58	29.5	2.43	0	0	0	3	28.58
SPBO	30.50	33.0	5.55	0	0	0	8	29.50
SSOA	32.00	32.0	0.71	0	0	0	11	31.00
OHO	32.75	34.0	1.64	0	0	0	8	31.75
FOX	21.50	26.5	11.17	0	0	1	3	20.50
DOA	20.17	18.5	5.44	0	0	0	0	19.17
ZOA	18.50	15.0	6.79	0	0	1	0	17.50
TTHHO	19.50	21.0	4.39	0	0	0	0	18.50
HHO	18.92	17.0	7.02	0	0	0	0	17.92
GJO	17.42	16.0	6.01	0	0	0	0	16.42
HLOA	19.92	18.0	10.14	0	0	0	0	18.92
WOA	23.92	23.5	7.16	0	0	0	1	22.92
SHO	24.50	24.0	6.99	0	0	0	0	23.50
GBO	16.75	16.0	4.83	0	0	1	0	15.75
HGSO	17.75	16.5	7.08	0	0	1	0	16.75

and

Table 5. Best algorithm(s) per function according to the global rank rows in Table A1, Table A2 and Table A3. Ties shown comma-separated.

Function	Winner(s)
F1	REO, RTH
F2	REO
F3	REO
F4	REO
F5	REO
F6	REO
F7	REO
F8	REO
F9	REO, RTH
F10	REO
F11	REO
F12	REO

show a clear overall winner: REO attains an average rank of $1.00$ with 12/12 wins, indicating consistent dominance across all CEC2022 functions considered. The only ties at the top occur on F1 and F9, where RTH matches REO’s best rank.

Among the compared optimizers other than REO, the strongest overall performers by average rank are RUN ($\overline{R}=6.42$), ALO ($6.92$), MVO ($7.75$), COA ($8.92$), DO ($8.92$), GWO ($9.17$), and RTH ($9.42$). Notably, GWO and MVO accumulate the largest number of Top-3 finishes among non-REO algorithms (five and four, respectively). In contrast, OHO, SSOA, and SPBO reside at the bottom of the ranking distribution, with high Bottom-3 counts ( 8, 11, and 8, respectively ) and average ranks above 30.

Stability varies substantially across optimizers as reflected by the SD of ranks. Some methods are consistently good or bad: e.g., RUN (SD $\approx 2.93$) is tightly clustered in the single-digit ranks, while SSOA (SD $\approx 0.71$) is consistently near the bottom. Others are more volatile: FOX shows the largest variability (SD $\approx 11.17$), switching between mid-pack and near-bottom ranks depending on the function, whereas HLOA (SD $\approx 10.14$) also exhibits wide fluctuations.

5.3. REO vs. Traditional Optimizers on CEC2022

Table A4. REO Results vs. Traditional Optimizers over CEC2022).

Function	Measure	REO	PSO	CMAES	GA	DE	CS	ABC
F1	mean	300.000	625.402	27,818.803	35,989.483	1927.604	17,652.882	5828.933
	Std.	0.000	1029.013	13,531.172	8055.384	564.140	13,285.973	2065.394
	error measure	0.000	325.402	27,518.803	35,689.483	1627.604	17,352.882	5528.933
F1	rank	1	2	6	7	3	5	4
F2	mean	402.581	460.904	651.355	640.462	402.615	411.132	405.085
	Std.	3.940	34.327	93.846	204.299	1.731	10.361	0.069
	error measure	2.581	60.904	251.355	240.462	2.615	11.132	5.085
F2	rank	1	5	7	6	2	4	3
F3	mean	600.000	601.413	614.347	655.913	600.000	607.309	600.000
	Std.	0.000	1.353	21.044	14.131	0.000	7.738	0.000
	error measure	0.000	1.413	14.347	55.913	0.000	7.309	0.000
F3	rank	1	4	6	7	1	5	3
F4	mean	810.083	818.356	821.340	878.501	817.500	855.322	823.101
	Std.	1.432	8.993	7.804	11.314	2.612	25.330	7.211
	error measure	10.083	18.356	21.340	78.501	17.500	55.322	23.101
F4	rank	1	3	4	7	2	6	5
F5	mean	900.000	923.360	900.000	1570.241	900.159	1210.375	1088.901
	Std.	0.000	73.173	0.000	372.009	0.121	471.622	124.480
	error measure	0.000	23.360	0.000	670.241	0.159	310.375	188.901
F5	rank	1	4	1	7	3	6	5
F6	mean	1809.765	5967.520	10,971,599.786	47,007,215.646	2295.720	245,240.607	1922.310
	Std.	13.852	1781.664	26,588,179.508	79,306,727.240	275.759	612,571.849	101.382
	error measure	9.765	4167.520	10,969,799.786	47,005,415.646	495.720	243,440.607	122.310
F6	rank	1	4	6	7	3	5	2
F7	mean	2004.235	2022.144	2093.553	2124.336	2005.567	2035.599	2007.375
	Std.	8.827	1.149	65.357	50.292	0.626	18.355	0.455
	error measure	4.235	22.144	93.553	124.336	5.567	35.599	7.375
F7	rank	1	4	6	7	2	5	3
F8	mean	2219.091	2222.901	2253.049	2252.068	2212.126	2230.522	2212.797
	Std.	5.552	7.842	15.460	27.558	3.367	7.595	7.194
	error measure	19.091	22.901	53.049	52.068	12.126	30.522	12.797
F8	rank	3	4	7	6	1	5	2
F9	mean	2529.284	2556.692	2569.673	2697.656	2529.284	2531.657	2556.563
	Std.	0.000	51.959	52.862	42.942	0.000	6.327	70.619
	error measure	229.284	256.692	269.673	397.656	229.284	231.657	256.563
F9	rank	1	5	6	7	2	3	4
F10	mean	2531.034	2542.240	2790.693	2742.117	2500.748	2545.071	2570.119
	Std.	59.074	57.933	463.781	374.060	0.246	69.988	0.046
	error measure	153.034	142.240	390.693	342.117	100.748	145.071	170.119
F10	rank	2	3	7	6	1	4	5
F11	mean	2600.000	2964.193	2963.725	3277.093	2648.544	2788.517	2601.718
	Std.	0.000	248.061	134.297	346.909	54.081	219.091	2.413
	error measure	0.000	364.193	363.725	677.093	48.544	188.517	1.718
F11	rank	1	6	5	7	3	4	2
F12	mean	2860.196	2871.808	2875.160	3037.545	2864.518	2864.013	2865.237
	Std.	0.382	10.508	4.307	41.264	0.411	0.700	1.373
	error measure	65.196	171.808	175.160	337.545	164.518	164.013	165.237
F12	rank	1	5	6	7	3	2	4

compares REO against Traditional Optimizers, including PSO, CMAES, GA, DE, CS, and ABC, on the CEC2022 suite. REO attains the top rank on 10 of 12 functions (with ties on F3 with DE and on F5 with CMAES), yielding the best average rank of $1.25$ (versus DE $2.17$, ABC $3.50$, PSO $4.08$, CS $4.50$, CMAES $5.58$, and GA $6.75$). In terms of solution quality and robustness, REO achieves the smallest error measure on nearly all problems—zero error on F1, F3, F5, and F11—and exhibits very low variability (Std. = 0 on F1, F3, F5, F9, F11 and $\approx 0.38$ on F12), while substantially outperforming classical methods on challenging instances such as F6. The only clear exceptions are F8 and F10, where DE leads (with ABC second on F8), yet REO remains competitive (third and second, respectively). Overall, the evidence indicates that REO provides consistently superior accuracy and stability across diverse landscapes.

6. Visual Results over CECC2022

This section provides a discussion of the visual results obtained by running the optimizer over the CEC2022 benchmark functions. The following subsections will cover different aspects of the optimization process, including convergence behavior, performance distribution, and the impact of population size on optimization.

6.1. Search History, Trajectory, Fitness and Convergence Curve Comparison Comparison of Different Population Sizes

Figure 8 and Figure 9 presents Search history, trajectory, fitness and convergence curve comparison and plots comparing the final objective values across population sizes for each benchmark function. The plot confirms that larger population sizes generally lead to better performance, especially for complex functions. This result further emphasizes the importance of selecting an appropriate population size to balance between computational cost and optimization quality.

The optimizer’s convergence performance is illustrated across CEC2022 12 benchmark functions. Figure 10 shows the convergence of the best objective value with respect to the number of iterations for each of the 12 functions. These plots illustrate the optimizer’s ability to find solutions that improve over time. Generally, it can be observed that the optimizer exhibits rapid convergence initially, especially for simpler functions such as F1, F2, and F5, where the best objective rapidly approaches the optimal value within a few hundred iterations.

For more complex functions like F3, F7, and F12, the convergence slows significantly, and the optimizer requires more iterations to improve its objective value. For these functions, the algorithm’s exploration phase seems to dominate, as it spends more time searching for better solutions. Notably, functions such as F10 and F11 show a plateau after initial improvement, suggesting that the optimizer may become stuck in local optima before continuing to improve after several iterations. The performance of the optimizer across these functions is generally consistent, but the complexity of the objective landscape plays a significant role in the rate of convergence.

6.2. Final Objective Values

Figure 11 shows the Empirical Cumulative Distribution Function (ECDF) of the final objective values for the different population sizes (pop = 20, 50, 100). This plot provides insight into how the optimizer performs across different population sizes. The results show that larger populations (pop = 100) generally result in a higher fraction of benchmark functions achieving better final objective values. The ECDF indicates that the larger population tends to explore the search space more thoroughly, leading to improved optimization outcomes. Conversely, the smaller population sizes (pop = 20) often result in suboptimal final objective values, as they are less capable of escaping local optima.

Interestingly, while larger populations have a clear advantage in terms of final objective values, the overall trend is relatively consistent across population sizes, with a noticeable shift towards better performance with increased population size. This result emphasizes the trade-off between computational expense and solution quality, suggesting that while a larger population improves performance, it requires more resources.

6.3. Final Objective Scatter Plot

In Figure 12, a scatter plot of the final objective values is presented across all benchmark functions. This plot shows the best objective values achieved by the optimizer for different population sizes. From the scatter plot, it is clear that the optimizer’s performance varies significantly across functions, with certain functions showing excellent optimization results regardless of population size, while others exhibit a more erratic performance.

For instance, functions such as F1 and F2 show a very tight clustering of objective values for all population sizes, indicating that the optimizer consistently finds high-quality solutions. However, functions like F3 and F12 show a wider spread of results, particularly for smaller population sizes, which suggests that the optimizer struggles to find optimal solutions on these more complex functions. The scatter plot also confirms that larger population sizes (pop = 100) generally lead to better optimization results, but the magnitude of improvement is not uniform across all functions.

6.4. Impact of Population Size on Optimization

Figure 13 and Figure 14 present boxplots for the final objective values across different population sizes. These boxplots offer a summary of the distribution of objective values achieved by the optimizer for each function. From the plots, it is evident that the optimizer’s performance is significantly affected by population size. Larger population sizes lead to a higher probability of achieving lower objective values, as reflected by the lower median values and narrower interquartile ranges in the boxplots for pop = 50 and pop = 100 compared to pop = 20.

However, the effect of population size varies depending on the complexity of the function. For simpler functions (F1, F2), the performance differences between population sizes are minimal, suggesting that these functions do not require large populations to achieve near-optimal solutions. In contrast, more complex functions (F3, F6, F10) show a clear advantage for larger population sizes, as the larger populations allow for better exploration and more consistent convergence.

6.5. Performance Profile

Figure 15 shows the performance profile, which tracks the fraction of functions solved within a given factor of the best objective value across all population sizes. The plot illustrates the relative performance of the optimizer for each population size. The population size of 100 consistently outperforms the other sizes, showing that it solves a higher fraction of functions within a given performance threshold. The population size of 50 shows moderate performance, while the size of 20 is the least effective.

6.6. Sensitivity Analysis

Population-based optimization algorithms often have several control parameters that regulate the balance between exploration and exploitation. As noted in the literature, proper parameter tuning is critical for competitive performance yet is notoriously difficult. To follow the reviewer’s suggestion for a thorough assessment of the original Ripple Evolution (REO) algorithm, we conducted a sensitivity analysis across all 12 benchmark functions. Two key parameters were investigated:

• Amplitude decay rate $\rho$: controls how quickly the ripple amplitude diminishes over iterations.
• Pull strength $\gamma$: scales the attraction of each individual toward the current global best.

For each parameter, three values were tested ($\rho \in \{0.80,0.95,0.99\}$ and $\gamma \in \{0.30,0.50,0.80\}$). The REO algorithm was run with population size 50, 50 iterations, dimension 10 and bounds $[-100,100]$. Each configuration was repeated three times, and the final objective values were averaged.

6.7. Sensitivity to the Amplitude Decay Rate $\rho$

Table 6. Effect of the amplitude decay rate $\rho$ on the REO algorithm. Each entry reports the mean final objective value over three runs. Smaller values indicate better performance.

Function	$\mathit{\rho }=0.80$	$\mathit{\rho }=0.95$	$\mathit{\rho }=0.99$
Sphere (F1)	0.0890	9.6981	46.7919
Schwefel (F2)	3648.03	3655.35	3656.01
Rosenbrock (F3)	1654.72	0.3182	16,250.39
Rastrigin (F4)	48.01	62.20	109.09
Griewank (F5)	0.4143	0.5039	0.6817
Ackley (F6)	19.32	19.96	19.98
Alpine (F7)	10.67	14.26	15.86
Schaffer (F8)	2.70	2.73	2.64
Zakharov (F9)	914.20	974.44	964.53
Levy (F10)	141.76	269.82	570.13
Dixon–Price (F11)	629.94	1838.14	2338.68
Styblinski–Tang (F12)	1.09	0.59	0.44

lists the averaged best objective values for each function and value of $\rho$. Figure 16 visualizes the same data. Overall, a faster decay ($\rho =0.80$) generally leads to better performance on most functions, particularly Sphere, Rastrigin, Griewank, and Alpine. On the Schwefel, Rosenbrock, and Zakharov functions, the differences are more nuanced, but very slow decay ($\rho =0.99$) tends to degrade performance by injecting too much stochasticity.

6.8. Sensitivity to the Pull Strength $\gamma$

Table 7. Effect of the pull strength $\gamma$ on the REO algorithm. Smaller values indicate better performance.

Function	$\mathit{\gamma }=0.30$	$\mathit{\gamma }=0.50$	$\mathit{\gamma }=0.80$
Sphere (F1)	5.30	5.68	6.23
Schwefel (F2)	3663.03	3658.88	3684.40
Rosenbrock (F3)	1626.17	0.3217	12,256.93
Rastrigin (F4)	67.48	59.22	63.12
Griewank (F5)	0.5501	0.2693	0.2768
Ackley (F6)	19.45	19.38	15.98
Alpine (F7)	20.11	14.06	10.50
Schaffer (F8)	3.17	3.03	2.72
Zakharov (F9)	1138.79	598.88	262.22
Levy (F10)	357.16	144.19	227.18
Dixon–Price (F11)	645.50	186.41	625.33
Styblinski–Tang (F12)	0.29	137.30	536.70

and Figure 17 summarize the effect of varying the pull strength. A moderate pull ($\gamma =0.50$) yields the best or near-best performance on several functions (Rastrigin, Griewank, Ackley, Alpine, and Levy). Too low a pull ($\gamma =0.30$) leads to under-exploitation and higher errors on many problems, whereas an excessively strong pull ($\gamma =0.80$) causes premature convergence and poor performance on Sphere, Schwefel, Rosenbrock, and Dixon–Price.

The sensitivity analysis confirms that the original RE algorithm is highly sensitive to its control parameters across diverse problem landscapes. On most functions, a relatively fast amplitude decay ($\rho =0.80$) is beneficial because it reduces oscillations quickly and concentrates search effort; however, on functions with complex topologies like Levy and Styblinski–Tang, a slightly slower decay retains diversity and prevents premature convergence. Overall, very slow decay ($\rho =0.99$) tends to inject too much random perturbation, causing stagnation or divergence.

The pull strength $\gamma$ regulates exploitation. Moderate values ($\gamma =0.50$) offer a good balance between exploring new regions and exploiting the current best solutions. Very small pulls ($\gamma =0.30$) fail to guide the population effectively on many functions, while excessive pulls ($\gamma =0.80$) can drive the swarm toward local optima prematurely, producing large errors on several test problems. These observations mirror general findings in the metaheuristic community: algorithms that rely solely on exploitation risk premature convergence, whereas those that overemphasize exploration may converge too slowly. Similar trade-offs are seen in genetic algorithms with Lévy flights, where increased exploration improves escape from local minima but introduces parameter sensitivity and may hinder convergence.

6.9. Wilcoxon Signed-Rank Summary for REO

As it can be seen in

Table A5. Wilcoxon Signed Results part 1.

Function	RUN	ALO	MVO	DO	COA	GWO	AVOA	AO	SHIO	SCSO	SHO
F1	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F2	0.501591	0.411465	0.851925	0.433048	0.167184	0.000681	0.147416	0.000593	0.002821	0.005734	0.000681
	T+: 87, T-: 123	T+: 83, T-: 127	T+: 100, T-: 110	T+: 126, T-: 84	T+: 142, T-: 68	T+: 196, T-: 14	T+: 146, T-: 64	T+: 197, T-: 13	T+: 185, T-: 25	T+: 179, T-: 31	T+: 196, T-: 14
	Z: 0.6720, SRN: 123.0000	Z: 0.8213, SRN: 127.0000	Z: 0.1867, SRN: 110.0000	Z: −0.7840, SRN: 84.0000	Z: −1.3813, SRN: 68.0000	Z: −3.3973, SRN: 14.0000	Z: −1.4487, SRN: 59.0000	Z: −3.4346, SRN: 13.0000	Z: −2.9866, SRN: 25.0000	Z: −2.7626, SRN: 31.0000	Z: −3.3973, SRN: 14.0000
F3	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F4	8.86$\times {10}^{-5}$	0.000219	0.000593	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000293	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.00012	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 204, T-: 6	T+: 197, T-: 13	T+: 210, T-: 0	T+: 210, T-: 0	T+: 202, T-: 8	T+: 210, T-: 0	T+: 210, T-: 0	T+: 208, T-: 2	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.6959, SRN: 6.0000	Z: −3.4346, SRN: 13.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.6213, SRN: 8.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8453, SRN: 2.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F5	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F6	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F7	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000338	0.000103	0.178956	0.00078	0.00014	8.86$\times {10}^{-5}$	0.000103	0.00014	0.000219
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 201, T-: 9	T+: 209, T-: 1	T+: 141, T-: 69	T+: 195, T-: 15	T+: 207, T-: 3	T+: 210, T-: 0	T+: 209, T-: 1	T+: 207, T-: 3	T+: 204, T-: 6
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.5839, SRN: 9.0000	Z: −3.8826, SRN: 1.0000	Z: −1.3440, SRN: 69.0000	Z: −3.3599, SRN: 15.0000	Z: −3.8079, SRN: 3.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.8079, SRN: 3.0000	Z: −3.6959, SRN: 6.0000
F8	0.00078	0.000103	0.001507	0.000254	0.00014	0.000254	0.000254	0.000681	0.000103	8.86$\times {10}^{-5}$	0.00039
	T+: 195, T-: 15	T+: 209, T-: 1	T+: 190, T-: 20	T+: 203, T-: 7	T+: 207, T-: 3	T+: 203, T-: 7	T+: 203, T-: 7	T+: 196, T-: 14	T+: 209, T-: 1	T+: 210, T-: 0	T+: 200, T-: 10
	Z: −3.3599, SRN: 15.0000	Z: −3.8826, SRN: 1.0000	Z: −3.1733, SRN: 20.0000	Z: −3.6586, SRN: 7.0000	Z: −3.8079, SRN: 3.0000	Z: −3.6586, SRN: 7.0000	Z: −3.6586, SRN: 7.0000	Z: −3.3973, SRN: 14.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000	Z: −3.5466, SRN: 10.0000
F9	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F10	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000103	0.00012	0.000189	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000103	0.000103	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 209, T-: 1	T+: 208, T-: 2	T+: 205, T-: 5	T+: 210, T-: 0	T+: 210, T-: 0	T+: 209, T-: 1	T+: 209, T-: 1	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.8453, SRN: 2.0000	Z: −3.7333, SRN: 5.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000
F11	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F12	0.000681	8.86$\times {10}^{-5}$	0.011129	0.00012	8.86$\times {10}^{-5}$	0.000103	0.000517	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 196, T-: 14	T+: 210, T-: 0	T+: 173, T-: 37	T+: 208, T-: 2	T+: 210, T-: 0	T+: 209, T-: 1	T+: 198, T-: 12	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
Total	+:11, -:0, =:1	+:11, -:0, =:1	+:11, -:0, =:1	+:11, -:0, =:1	+:10, -:0, =:2	+:12, -:0, =:0	+:11, -:0, =:1	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0

,

Table A6. Wilcoxon Signed Results part 2.

Function	ZOA	GJO	HHO	WOA	HGSO	HLOA	TTHHO	GBO	CPO	DOA	FOX
F1	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F2	0.001019	8.86$\times {10}^{-5}$	0.002204	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.001713	0.005111	8.86$\times {10}^{-5}$	0.016881	8.86$\times {10}^{-5}$	0.036561
	T+: 193, T-: 17	T+: 210, T-: 0	T+: 187, T-: 23	T+: 210, T-: 0	T+: 210, T-: 0	T+: 189, T-: 21	T+: 180, T-: 30	T+: 210, T-: 0	T+: 169, T-: 41	T+: 210, T-: 0	T+: 161, T-: 49
	Z: −3.2853, SRN: 17.0000	Z: −3.9199, SRN: 0.0000	Z: −3.0613, SRN: 23.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.1359, SRN: 21.0000	Z: −2.8000, SRN: 30.0000	Z: −3.9199, SRN: 0.0000	Z: −2.3893, SRN: 41.0000	Z: −3.9199, SRN: 0.0000	Z: −2.0906, SRN: 49.0000
F3	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F4	0.000681	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000103	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000103	8.86$\times {10}^{-5}$
	T+: 196, T-: 14	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 209, T-: 1	T+: 210, T-: 0	T+: 210, T-: 0	T+: 209, T-: 1	T+: 210, T-: 0
	Z: −3.3973, SRN: 14.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000
F5	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F6	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F7	8.86$\times {10}^{-5}$	0.000103	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 209, T-: 1	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F8	0.000219	0.000189	8.86$\times {10}^{-5}$	0.00014	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	0.000103	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 204, T-: 6	T+: 205, T-: 5	T+: 210, T-: 0	T+: 207, T-: 3	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 209, T-: 1	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.6959, SRN: 6.0000	Z: −3.7333, SRN: 5.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8079, SRN: 3.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.8826, SRN: 1.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F9	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F10	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F11	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F12	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
Total	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0

and

Table A7. Wilcoxon Signed Results part 3.

Function	SMA	TSO	Chimp	AOA	ROA	RSA	FLO	SPBO	SSOA	OHO
F1	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F2	0.004045	0.027621	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 182, T-: 28	T+: 164, T-: 46	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −2.8746, SRN: 28.0000	Z: −2.2026, SRN: 46.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F3	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F4	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F5	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F6	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F7	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F8	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F9	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F10	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F11	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000	Z: −3.9199, SRN: 0.0000
F12	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$	8.86$\times {10}^{-5}$
	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0	T+: 210, T-: 0
Total	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0	+:12, -:0, =:0

for the (12 test functions × 32 peer optimizers), REO shows overwhelmingly superior performance: out of 12 functions, REO is significant on almost all of them with no significant losses, and only 7 ties—all on F2 against RUN, ALO, MVO, DO, COA, and AVOA, plus one on F7 against COA—while every other case favors REO, typically with very small p-values ($p=8.86\times {10}^{-5}$) and large negative test statistics ($Z\approx -3.92$; e.g., on F1, F3–F6, F9–F11 we observe $T^{-}=0$ versus every competitor). Aggregating per competitor, REO achieves 12–0–0 (wins–losses–ties) against all compared optimizers, 11–0–1 against RUN, ALO, MVO, DO, and AVOA, and 10–0–2 against COA, confirming that REO’s advantage is consistent, statistically robust, and practically large across the entire benchmark suite.

7. Applications of REO in Solving Engineering Design Problems

7.1. Welded Beam Engineering Design Problem

The welded beam design problem seeks the minimum fabrication cost of a welded beam subject to constraints on shear stress, bending stress, buckling load, and end deflection. Typical decision variables are weld size ($x_1$), weld length ($x_2$), beam thickness ($x_3$), and beam width ($x_4$) (see Figure 18).

REO attains the top rank (Rank 1) on Welded Beam with best objective $=1.72485$ and mean $=1.72485$ (std $=6.84\times {10}^{-7}$). Compared with the next best method, POA, the first optimizer improves the best objective by 0.30% (from 1.73003 to 1.72485) and reduces the mean by 0.36% (from 1.73117 to 1.72485). Relative to the median across the remaining optimizers, the first optimizer yields a 42.46% lower mean objective, indicating stable performance. These figures (see

Table 8. Optimization results for Welded Beam.

Algorithm	Best	Mean	Std	Sem	Ranking
REO	1.72485	1.72485	6.84344$8\times {10}^7$	4.83904$8\times {10}^7$	1
POA	1.73003	1.73117	0.00161066	0.00113891	2
ChOA	1.73599	1.74053	0.00641954	0.0045393	3
MFO	1.7303	1.74362	0.0188384	0.0133208	4
TTHHO	2.02435	2.18608	0.228711	0.161723	5
ZOA	1.89849	2.31309	0.586327	0.414596	6
SCA	2.07043	2.31321	0.343332	0.242772	7
TSO	1.8501	2.42664	0.815356	0.576544	8
RSA	2.95651	2.96356	0.00997739	0.00705508	9
SHO	2.47667	3.03139	0.7845	0.554725	10
SMA	3.07915	3.0883	0.0129357	0.0091469	11
FLO	2.90831	3.22846	0.45276	0.320149	12
MPA	3.32489	3.33212	0.0102287	0.00723282	13
BOA	2.83457	3.49045	0.927553	0.655879	14
WOA	3.8404	4.81798	1.38252	0.977588	15
ROA	2.41104	5.40773	4.23796	2.99669	16
SSOA	112,592	188,975	108,022	76,382.9	17

) consistently highlight the superior performance of REO over the competing optimizers on this problem.

7.2. Spring Engineering Design Problem

The tension/compression spring design problem (See Figure 19) minimizes the spring weight while satisfying constraints on shear stress, surge frequency, and limits on geometry. The decision variables are wire diameter ($x_1$), mean coil diameter ($x_2$), and the number of active Lcoils ($x_3$).

REO attains the top rank (Rank 1) on Spring Design with best Objective $=\mathrm{180,806}$ and mean $=180,806$ (std $=0.0263$). Compared with the next best method, POA, the first optimizer improves the best objective by $-\mathbf{0.00}\%$ (from 180,806 to 180,806) and reduces the mean by 0.00% (from 180,806 to 180,806). Relative to the median across the remaining optimizers, the first optimizer yields a 19.26% lower mean objective, indicating stable performance. These figures (see

Table 9. Optimization results for Spring Design.

Algorithm	Best	Mean	Std	Sem	Ranking
REO	180,806	180,806	0.0263029	0.018599	1
POA	180,806	180,806	0.959752	0.678647	2
ChOA	180,945	181,057	157.649	111.474	3
WOA	181,491	182,103	864.784	611.495	4
MFO	180,865	182,564	2402.46	1698.8	5
ZOA	183,473	183,547	104.832	74.1274	6
TSO	180,806	198,341	24,798.2	17,535	7
SCA	197,590	198,388	1129.08	798.383	8
TTHHO	206,916	211,573	6586.4	4657.29	9
MPA	236,266	236,304	54.0556	38.2231	10
SHO	268,748	276,016	10,279.4	7268.63	11
ROA	221,451	333,833	158,931	112,381	12
SMA	284,594	428,570	203,612	143,976	13
RSA	269,537	431,878	229,585	162,341	14
BOA	428,572	439,556	15,533.2	10,983.6	15
FLO	483,334	505,207	30,933.2	21,873.1	16
SSOA	489,755	583,727	132,897	93,972.6	17

) consistently highlight the superior performance of RE over the competing optimizers on this problem.

7.3. Three-Bar Truss Engineering Design Problem

The three-bar truss problem (See Figure 20) minimizes structural weight with stress and displacement constraints. A symmetric V-shaped truss supports a vertical load; design variables typically parameterize the cross-sectional areas.

REO attains the top rank (Rank 1) on Three-Bar Truss with best Objective $=263.896$ and mean $=263.896$ (std $=1.32\times {10}^{-7}$). Compared with the next best method, POA, the first optimizer improves the best objective by $-\mathbf{0}.\mathbf{00}\%$ (from 263.896 to 263.896) and reduces the mean by 0.00% (from 263.896 to 263.896). Relative to the median across the remaining optimizers, the first optimizer yields a 0.31% lower mean objective, indicating stable performance. These figures (see

Table 10. Optimization results for Three-Bar Truss.

Algorithm	Best	Mean	Std	Sem	Ranking
RE	263.896	263.896	1.32176$\times {10}^{-7}$	9.34624$\times {10}^{-8}$	1
POA	263.896	263.896	8.93859$\times {10}^{-7}$	6.32054$\times {10}^{-7}$	2
MPA	263.898	263.901	0.00315227	0.00222899	3
ZOA	263.899	263.907	0.0118833	0.00840275	4
ChOA	263.951	263.972	0.0292866	0.0207087	5
MFO	263.905	263.988	0.117595	0.083152	6
SCA	263.933	264.087	0.21749	0.153788	7
TTHHO	263.978	264.286	0.436233	0.308464	8
BOA	264.282	264.609	0.461766	0.326518	9
FLO	264.624	264.828	0.287783	0.203493	10
SHO	265.164	265.35	0.263387	0.186243	11
SSOA	265.062	265.478	0.588168	0.415898	12
ROA	265.129	265.923	1.12373	0.794598	13
WOA	264.383	266.118	2.45444	1.73555	14
RSA	267.161	267.293	0.186617	0.131958	15
SMA	266.368	268.523	3.04693	2.15451	16
TSO	264.158	273.5	13.2124	9.34259	17

) consistently highlight the superior performance of RE over the competing optimizers on this problem.

7.4. Cantilever Stepped Beam Engineering Design Problem

The cantilever stepped beam problem (See Figure 21) minimizes the material weight (or cost) of a beam divided into a number of segments, under stress and deflection constraints. The design variables ($x_1$–$x_{10}$) are the cross-sectional dimensions of each segment.

REO attains the top rank (Rank 1) on Cantilever Stepped Beam with best Objective $=\mathrm{62,772.8}$ and mean $=\mathrm{62,795.2}$ (std $=31.8$). Compared with the next best method, POA, the first optimizer improves the best objective by 1.97% (from 64,036 to 62,772.8) and reduces the mean by 2.05% (from 64,110.5 to 62,795.2). Relative to the median across the remaining optimizers, the first optimizer yields a 22.62% lower mean objective, indicating stable performance. These figures (see

Table 11. Optimization results for Cantilever Stepped Beam.

Algorithm	Best	Mean	Std	Sem	Ranking
RE	62,772.8	62,795.2	31.7591	22.457	1
POA	64,036	64,110.5	105.417	74.541	2
ChOA	64,047.7	64,722.5	954.266	674.768	3
ZOA	64,971.2	65,964.9	1405.33	993.715	4
MPA	70,538.5	71,321.5	1107.4	783.047	5
MFO	63,969.8	71,935.2	11,264.7	7965.37	6
TTHHO	71,948.2	72,510	794.495	561.793	7
ROA	76,904.5	78,652.3	2471.8	1747.83	8
FLO	78,149.8	80,258.7	2982.47	2108.93	9
SHO	70,568.4	82,051	16,238.9	11,482.6	10
SMA	82,673.8	86,060	4788.74	3386.15	11
SCA	86,389.5	86,715.6	461.139	326.074	12
WOA	82,236.8	87,037.6	6789.28	4800.75	13
TSO	87,611.3	92,027.4	6245.26	4416.07	14
SSOA	88,391.3	103,404	21,230.5	15,012.2	15
RSA	107,990	121,050	18,469.2	13,059.7	16
BOA	107,209	385,248	393,207	278,040	17

) consistently highlight the superior performance of RE over the competing optimizers on this problem.

7.5. Ten Bar Planar Truss Engineering Design Problem

The classical 10-bar planar truss problem (See Figure 22) minimizes the truss weight under stress and displacement constraints. The design variables are the cross-sectional areas of the ten members.

REO attains the top rank (Rank 1) on Ten-Bar Planar Truss with best Objective $=597.999$ and mean $=598.153$ (std $=0.217$). Compared with the next best method, REA, the first optimizer improves the best objective by 1.91% (from 609.626 to 597.999) and reduces the mean by 2.65% (from 614.41 to 598.153). Relative to the median across the remaining optimizers, the first optimizer yields a 29.03% lower mean objective, indicating stable performance. These figures (see

Table 12. Optimization results for Ten-Bar Planar Truss.

Algorithm	Best	Mean	Std	Sem	Ranking
RE	597.999	598.153	0.217428	0.153745	1
REA	609.626	614.41	6.76587	4.78419	2
ChOA	629.66	631.075	2.00103	1.41495	3
SCA	618.151	638.098	28.2097	19.9472	4
POA	639.131	690.144	72.1435	51.0132	5
TTHHO	726.011	772.247	65.3875	46.2359	6
BOA	756.704	796.892	56.8347	40.1882	7
ZOA	802.904	822.195	27.2823	19.2915	8
MFO	598.581	829.543	326.63	230.962	9
RSA	816.156	842.873	37.783	26.7166	10
FLO	908.464	910.248	2.52336	1.78428	11
ROA	941.918	968.643	37.796	26.7258	12
SHO	910.257	1043.38	188.268	133.126	13
WOA	606.65	1059.41	640.303	452.763	14
MPA	1652.48	1699.17	66.0339	46.693	15
TSO	1754.83	2092.94	478.167	338.115	16
SSOA	2947.66	3022.2	105.415	74.5397	17
SMA	3896.64	4206.47	438.152	309.82	18

) consistently highlight the superior performance of RE over the competing optimizers on this problem.

7.6. Pressure-Vessel Design Problem

A pressure vessel (See Figure 23) stores fluid under pressure; it consists of a cylindrical shell capped by two hemispherical heads. The optimization task is to choose the shell thickness $T_s$, head thickness $T_h$, inner radius R, and cylindrical section length L so that the cost of material, forming, and welding is minimized. Rolled steel plate is only available in increments of 0.0625 in, so $T_s$ must be a multiple of 0.0625 in and $T_h$ must be selected from a discrete set, while R and L are continuous. NEORL’s problem description summarizes the variables and bounds.

The cost function (in U.S. dollars) includes material, forming, and welding costs and is defined as

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

where $\overrightarrow{x}=(x_1,x_2,x_3,x_4)$ represents $(T_s,T_h,R,L)$ in inches. The constraints ensure minimum thickness, sufficient volume, and a maximum length:

$$\begin{array}{cc}\hfill -x_1+0.0193x_3& \le 0,\hfill \\ \hfill -x_2+0.00954x_3& \le 0,\hfill \\ \hfill -\pi x_3^2{x}_4-\frac{4}{3}\pi x_3^3+\mathrm{1,296,000}& \le 0,\hfill \\ \hfill x_4-240& \le 0,\hfill \end{array}$$

with bounds $0.0625\le x_1\le 6.1875$ (multiples of 0.0625), $x_2\in \{0.0625,0.125,\dots ,0.625\}$, $10\le x_3\le 200$ and $10\le x_4\le 200$.

As can be seen in

Table 13. Pressure-Vessel Design Problem.

Algorithm	Mean	Std	Sem	Best_ Objective	x1	x2	x3	x4	Ranking
RE	6106.462	88.44507	62.54011	6043.922	0.834645	0.421283	43.13177	164.7269	1
TTHHO	6389.283	80.42098	56.86622	6332.417	0.937164	0.489922	48.51611	110.572	2
MFO	6468.769	308.4362	218.0973	6250.672	0.950479	0.469907	49.24617	104.4488	3
ChOA	6615.45	173.0552	122.3685	6493.082	0.893977	0.488128	46.7041	133.4076	4
MPA	6728.932	902.8383	638.4031	6090.529	0.871793	0.433362	45.13376	142.7826	5
ZOA	6850.71	259.4634	183.4684	6667.241	1.08779	0.539701	56.33734	54.87912	6
BOA	7431.184	318.6951	225.3515	7205.833	1.123468	0.565372	55.30213	64.37156	7
SHO	7457.611	221.8144	156.8465	7300.764	1.092471	0.644523	56.00177	56.87267	8
SCA	8563.674	527.8143	373.2211	8190.453	0.945005	0.669178	41.81536	195.794	9
FLO	9325.825	253.9611	179.5776	9146.247	1.199675	0.850448	52.88779	77.33427	10
ROA	10,007.93	392.8583	277.7927	9730.138	1.090717	1.070255	55.8731	59.24966	11
WOA	10,691.29	4116.735	2910.971	7780.319	1.221813	0.491293	50.61004	93.5787	12
TSO	12,007.79	7287.183	5152.817	6854.969	0.959659	0.597251	49.20803	104.7554	13
SMA	12,727.88	8309.569	5875.752	6852.123	0.900551	0.518306	45.92008	147.3161	14
RSA	17,493.81	6137.341	4339.756	13,154.06	1.387056	1.355401	61.74348	27.09978	15
SSOA	31,657.96	5171.079	3656.505	28,001.45	1.666317	3.576956	54.34977	95.46167	16

, of the compared 16 metaheuristic algorithms. The RE optimizer achieved the lowest mean objective value (about 6106.46) and the best single solution (around 6043.92). Its modest standard deviation implies reliable convergence. The second-best (TTHHO) had a mean cost of approximately 6389.28, about 5% higher than RE. Algorithms such as MFO and ChOA had larger variability and higher costs. The poorest performers (SCA, FLO, ROA, WOA, TSO, SMA, RSA, SSOA) exhibited mean costs 40–420% higher than the best and large standard deviations. RE’s best design uses a thin shell ($x_1\approx 0.83$ in), a moderate head thickness ($x_2\approx 0.42$ in), an inner radius $x_3\approx 43.13$ in, and a cylinder length $x_4\approx 164.73$ in, satisfying all constraints and minimizing cost.

7.7. Stepped Transmission-Shaft Design

A stepped shaft transmits power through gears and pulleys mounted on different diameters (See Figure 24). The benchmark problem considers a three-segment shaft with fixed lengths $L_1,L_2,L_3$ and design variables $(d_1,d_2,d_3)=(x_1,x_2,x_3)$ for the diameters. The goal is to minimize the shaft weight while satisfying fatigue strength, minimum step size, and maximum-deflection constraints. Rodríguez-Cabal et al. formulated the model using the Modified Goodman fatigue criterion.

The objective function is the total weight of the three cylindrical segments, given by

$$\underset{\overrightarrow{x}}{min}f\left(\overrightarrow{x}\right)=\gamma {\displaystyle \frac{\pi }{4}}\left(L_1{d}_1^2+L_2{d}_2^2+L_3{d}_3^2\right),$$

where $\gamma =0.2834\mathrm{lb}/{\mathrm{in}}^3$ and $L_1,L_2,L_3$ are fixed segment lengths. Six constraints enforce safety and manufacturability. First, each diameter must exceed a minimum safe diameter $k\left(d_i\right)$ derived from the Modified Goodman criterion, expressed as $g_i=d_i-k\left(d_i\right)\ge 0$ for $i=1,2,3$ Second, bearing manufacturers require the middle diameter to exceed each adjacent diameter by at least 0.0787 in, giving $g_4=d_2-d_1\ge 0.0787$ and $g_5=d_2-d_3\ge 0.0787$. Third, the deflection of the shaft must not exceed 0.005 in: $|y_{max}|\le 0.005$. Bounds on the diameters are $1\le d_i\le 4$ in.

As can be seen in

Table 14. Stepped Transmission-Shaft Design.

Algorithm	Mean	Std	Sem	Best Objective	x1	x2	Ranking
RE	0.002086	0.000259	0.000183	0.001903	0.087247	0.25	1
FLO	0.002268	0	0	0.002268	0.067356	0.5	2
MFO	0.002268	4.33$\times {10}^{-12}$	3.06$\times {10}^{-12}$	0.002268	0.067356	0.5	3
TSO	0.002268	4.68$\times {10}^{-10}$	3.31$\times {10}^{-10}$	0.002268	0.067356	0.5	4
TTHHO	0.002268	2.4$\times {10}^{-9}$	1.7$\times {10}^{-9}$	0.002268	0.067356	0.5	5
WOA	0.002268	1.39$\times {10}^{-8}$	9.83$\times {10}^{-9}$	0.002268	0.067356	0.5	6
SHO	0.002268	9.39$\times {10}^{-8}$	6.64$\times {10}^{-8}$	0.002268	0.067356	0.5	7
MPA	0.002269	2.06$\times {10}^{-7}$	1.46$\times {10}^{-7}$	0.002268	0.067356	0.5	8
ZOA	0.002269	8.95$\times {10}^{-7}$	6.33$\times {10}^{-7}$	0.002269	0.067359	0.5	9
BOA	0.002272	3.22$\times {10}^{-6}$	2.28$\times {10}^{-6}$	0.002269	0.067371	0.5	10
ChOA	0.002272	1.19$\times {10}^{-6}$	8.4$\times {10}^{-7}$	0.002272	0.067402	0.5	11
SCA	0.002292	2.62$\times {10}^{-5}$	1.85$\times {10}^{-5}$	0.002274	0.067437	0.5	12
SSOA	0.002308	4.16$\times {10}^{-5}$	2.94$\times {10}^{-5}$	0.002279	0.06751	0.5	13
SMA	0.002317	1.24$\times {10}^{-6}$	8.79$\times {10}^{-7}$	0.002316	0.067888	0.502561	14
ROA	0.002763	0.000684	0.000484	0.00228	0.067521	0.5	15
RSA	0.005168	0.000204	0.000144	0.005024	0.081003	0.765617	16

, RE obtains the lowest mean objective value (about 0.002086) and the best single solution (≈0.001903), indicating the lightest weight shaft. Algorithms such as FLO, MFO, TSO, TTHHO, WOA, and SHO converge to a common mean objective of about 0.002268 with negligible variance; their designs use smaller diameters ($d_1\approx 0.06736$ in, $d_2=0.5$ in). RE’s design uses a slightly larger $d_1\approx 0.08725$ in but a smaller $d_2=0.25$ in, reducing weight by roughly 8%. Lower-ranked algorithms show gradually larger mean objectives and variability, suggesting difficulties in handling the nonlinear constraints. The worst performers (SMA, ROA, RSA) yield substantially heavier shafts and higher standard deviations. Overall, RE offers superior performance on both benchmark problems, delivering the best solutions with low variability.

8. Conclusions

We introduced REO (Ripple Evolution), a nature inspired, population based optimizer that combines a JADE-style current to p-best/1 core with jDE self adaptation and three complementary motions—rank aware undertow toward the incumbent best, a time varying tide toward the elite mean, and a scale aware swell with occasional Lévy kicks—supplemented by reflection/clamp boundary handling; we provided a compact, equation labeled model, pseudocode tied to those equations, visual diagrams, and a complexity analysis showing evaluation dominated cost. Empirically, REO achieved first or tied first across the CEC2022 functions considered, with notable strength on rotated multimodal, hybrid, and composition landscapes; convergence band and performance profile plots showed rapid early descent and stable late refinement, and population studies indicated that larger populations (e.g., 100) improve robustness at predictable computational expense. On constrained engineering problems (Welded Beam, Spring Design, Three Bar Truss, Cantilever Stepped Beam, Ten Bar Planar Truss), REO attained the best objectives among compared methods, demonstrating transfer from benchmarks to real designs. Collectively, these results underscore REO’s contributions in (i) design—a principled composition that couples adaptive guidance with diversified perturbations atop a strong DE backbone; (ii) transparency—fully specified equations, pseudocode, and bounds; and (iii) effectiveness—consistent top tier performance with practical guidance on population sizing and schedules. For practitioners, we recommend moderate to large populations when budgets allow, an increasing tide weight with concurrent decay of swell amplitude and Lévy probability, and jDE self-adaptation to reduce manual tuning; when evaluations are costly, consider smaller populations with restarts or early stopping on stagnation. Limitations include evaluation budget sensitivity typical of wrapper methods; promising extensions include discrete/mixed encodings, surrogate and early acceptance schemes, multi-objective variants, restart/niching for multimodality, and scalable vectorized/GPU implementations, alongside formal convergence analyses under stochastic schedules.