Scuba Diver Optimization Algorithm: An oxygen-state-regulated metaheuristic with integrated validation on constrained engineering and mathematical benchmarks

Abstract

This paper introduces the Scuba Diver Optimization Algorithm (SDOA), a state-regulated population-based metaheuristic for constrained and continuous optimization. The main contribution is an agent-level oxygen-state regulator that jointly controls search stage, movement operator, perturbation amplitude, local refinement, elite communication, and reset pressure. In contrast to iteration-level schedules that impose a single exploration-to-exploitation trajectory on the entire population, SDOA allows candidate solutions to occupy different behavioral stages within the same iteration. This asynchronous state allocation is intended to support global coverage, boundary-sensitive exploitation, and diversity recovery through one search-governance mechanism. The empirical evaluation uses an integrated validation corpus comprising four constrained engineering design problems–welded beam design, tension/compression spring design, car side-impact design, and speed reducer design–and twenty-three standard mathematical functions representing unimodal, multimodal, discontinuous, penalized, and fixed-dimensional landscapes. The experimental protocol reports benchmark formulations, parameter settings, sensitivity evidence, repeated-run dispersion where available, confidence intervals, runtime diagnostics, and mechanism-level interpretation. The findings are therefore interpreted in terms of feasibility, robustness, computational cost, and comparative standing, rather than isolated best-run values. Overall, SDOA is positioned as a transparent state-regulated optimizer whose principal contribution is the coordinated regulation of feasibility recovery, exploitation control, and diversity preservation within a coherent internal state model.

Keywords

Scuba diver optimization algorithm constrained optimization metaheuristic optimization engineering design mixed-variable search swarm intelligence

1 Introduction

Metaheuristic optimization has become a central computational approach for problems in which objective functions are nonlinear, constraints are strongly coupled, and reliable derivative information is unavailable or impractical to obtain. These conditions arise frequently in mechanical design, controller tuning, power-system scheduling, feature selection, parameter estimation, and other intelligent engineering applications. During the past decade, both established swarm-intelligence methods and newer bio-inspired optimizers have been extended to increasingly difficult constrained search spaces.^1,2

Despite these developments, an important methodological limitation persists. Many optimizers still regulate exploration and exploitation through externally imposed schedules, fixed coefficients, or weakly adaptive heuristics. Such mechanisms may be adequate on smooth landscapes, but they can become fragile when the feasible region is narrow, disconnected, or concentrated near active constraints. In such settings, an optimizer must improve the objective while also recovering feasibility, preserving population diversity, and preventing premature convergence. Classical engineering benchmarks such as welded beam design, tension/compression spring design, car side-impact design, and speed reducer design continue to expose these difficulties because they combine nonlinear objective functions with restrictive constraint boundaries.^3–6

This study develops the Scuba Diver Optimization Algorithm (SDOA) as a constrained-optimization framework governed by an interpretable internal state. Each candidate solution is modeled as a diver whose remaining oxygen determines the extent of global exploration, the transition to regional refinement, and the need for resurfacing or reset. Oxygen is therefore used as an agent-level regulator rather than as a purely metaphorical coefficient: it links movement selection, step-size control, elite guidance, local search, and diversity recovery through a single evolving state variable.

SDOA is examined through an integrated validation corpus. Four constrained engineering problems–welded beam design (WBD), tension/compression spring design (TSD), car side-impact design problem (CSIDP), and speed reducer design (SRD)–are used to assess feasibility preservation, mixed-variable handling, and behavior near active constraints. Twenty-three standard mathematical functions complement these cases by testing exploitation accuracy, multimodal exploration, separability, nonseparability, discontinuity, and attraction-basin sensitivity. The mathematical functions are therefore treated as the landscape-generalization component of the empirical design, not as a detached auxiliary experiment.

The paper makes six specific contributions. First, it formulates SDOA as a coherent state-regulated optimizer incorporating oxygen decay, depth-stage mapping, staged movement operators, elite preservation, and selective reset. Second, it distinguishes oxygen-state regulation from ordinary adaptive schedules by showing how a single internal state coordinates operator selection, perturbation amplitude, elite communication, reset behavior, and asynchronous stage occupancy. Third, it provides a unified empirical assessment across constrained engineering benchmarks and the 23-function mathematical suite, thereby broadening the evidence beyond a small set of specialized design cases. Fourth, it improves equation transparency by auditing benchmark formulations, variable types, bound handling, and feasibility rules. Fifth, it reports parameter calibration and sensitivity evidence that separates computational budget from behavioral regulation. Sixth, it organizes the results around objective value, feasibility, dispersion, statistical uncertainty, runtime, and mechanism-level interpretation, yielding a balanced assessment rather than a claim of universal dominance.

The contribution is evaluated through six connected criteria: methodological distinctiveness relative to adaptive and stage-based metaheuristics; feasibility preservation under nonlinear constraints; statistical stability where repeated-run evidence is available; comparative competitiveness within the reported benchmark literature; robustness across heterogeneous mathematical landscapes; and transparency in parameterization, runtime, and implementation.

2 Related work and research gap

Recent metaheuristic research has followed two related directions. One direction refines established families such as particle swarm optimization, differential evolution, firefly-based search, and hybrid evolutionary methods. The other introduces newer nature-inspired algorithms, including EABOA, KOA, the Rabbit and Turtle Algorithm, the Search and Rescue Algorithm, and other metaphor-driven optimizers.^7–10 Comprehensive reviews indicate that these methods can report strong best-case values, but their relative advantages are often problem dependent and sensitive to parameter tuning.^1,4

Welded beam design has long served as a compact but demanding benchmark because its cost-minimization objective is shaped by stress, deflection, and buckling constraints. Earlier methods, including genetic algorithms, harmony search, simulated annealing, and firefly optimization, demonstrated that feasible low-cost designs can be found, but also showed that the search process is sensitive to constraint handling and local stagnation.^11,12 The WBD results in this study indicate that SDOA is competitive on this benchmark, although it is not universally best; thus, WBD is treated as a robustness test rather than as a selective showcase.

Tension/compression spring design has generated an especially dense comparison literature because many optimizers have been tested on a three-variable problem with four nonlinear constraints. The central lesson from this literature is nuanced: small objective-value differences can substantially change rank ordering, but they do not necessarily imply meaningful differences in reliability. Recent studies show that many methods cluster within a narrow feasible region, making stability, speed, and reproducibility more informative than rank alone.^4,7,8,13

Car side-impact design is structurally more demanding because it combines eleven variables, mixed continuous-discrete choices, and ten surrogate crashworthiness constraints. Standard swarm and evolutionary methods can obtain attractive objective values, but the distinction between feasible and infeasible solutions is critical because some optimizers approach low weight only by violating safety constraints. Diversity-guided PSO variants and firefly-based mixed-variable formulations have attempted to address this difficulty, yet constraint-aware stability remains a central issue.^3,5,12

Speed reducer design remains one of the most stringent constrained engineering benchmarks because the optimum is close to several active constraints and one variable is integer-valued. Deterministic reformulations and many modern metaheuristics have been reported for this problem, including SGO-DE, hybrid JADE-Sine-Cosine variants, AOACS, and recent comparative studies.^6,14–16 Even when best objective values are similar, feasibility reliability and variance across runs continue to distinguish robust methods from opportunistic best-run outcomes.

Across these problem classes, the unresolved issue is not simply whether another optimizer can produce a marginally lower best value. The deeper gap concerns search governance. Many recent algorithms combine diversification, intensification, elite guidance, mutation, and restart, but these components are often introduced as separate heuristics coordinated by iteration counters or manually selected coefficients. A constrained metaheuristic is more reproducible when such behaviors are regulated by a transparent internal state. SDOA addresses this gap by using oxygen as the central regulator of stage transitions, step-size behavior, elite communication, local refinement, and reset.

2.1 Positioning relative to adaptive, schedule-based, and feasibility-preserving metaheuristics

Many modern optimizers improve the exploration-exploitation balance through linearly decreasing coefficients, adaptive awareness probabilities, elite-guided crossover, Lévy perturbations, or hybrid local-search modules. These mechanisms can be effective, but they often adjust the intensity of a fixed movement rule rather than changing each search agent's behavioral regime. SDOA differs by allowing the oxygen state to select the movement family itself, including long-range exploration, bounded random walks, elite-guided refinement, non-uniform mutation, and reset.

In constrained engineering design, objective ranking alone can be misleading because attractive raw objective values may be associated with violated constraints. This issue is particularly important for CSIDP and SRD, where high-quality solutions lie close to active boundaries. Accordingly, the comparative discussion evaluates SDOA through objective value, feasibility status, convergence behavior, dispersion, and benchmark-specific standing.

Recent studies further reinforce the importance of robust benchmark design, adaptive search control, and hybrid intelligent optimization. Altay introduced the chaotic slime mould optimization algorithm and showed that chaotic control can strengthen exploration and global-search performance across benchmark and engineering problems.¹⁷ Altay and Varol Altay later coupled an improved grey wolf optimizer with multilayer perceptron training, demonstrating the relevance of metaheuristic search to data-driven model optimization.¹⁸ Their chaotic transient search optimizer further emphasized diversity enhancement, real-world engineering validation, and feature-selection applicability.¹⁹ A broader comparative study of real-world engineering design problems also confirmed that algorithmic performance depends strongly on benchmark selection, constraint handling, and reporting protocol.²⁰ Recent QTJ studies similarly highlight hybrid metaheuristics, adaptive learning architectures, and data-driven intelligent frameworks in contemporary optimization and engineering-oriented computational intelligence.^21–24

Recent comprehensive studies on new metaheuristic families further show that the field is moving beyond single-metaphor algorithm proposals toward taxonomy, hybridization, application-specific adaptation, and reproducible validation. The recent reviews of the Mountain Gazelle Optimizer, Artificial Rabbits Optimization, and Sand Cat Swarm Optimization are important in this respect because they classify algorithmic variants, hybrid forms, application domains, and open research directions, thereby emphasizing that new optimizers should be evaluated through transparent benchmark protocols rather than by metaphor alone.^25–27 Application-oriented and hybrid metaheuristic studies make the same point from an empirical perspective: SCSOWOA has been adapted for generalized multilevel thresholding in multi-organ medical image segmentation, RSA-HGSO has been used for satellite-image segmentation in remote sensing, the Discrete Puma Optimizer extends continuous metaheuristic search to combinatorial optimization, and a multi-strategy Chimp Optimization Algorithm integrates several update mechanisms for global and constrained engineering problems.^28–31 These studies clarify the methodological standard against which SDOA should be judged: the oxygen-state regulator must provide a reproducible search-governance mechanism supported by heterogeneous benchmark evidence, explicit parameterization, feasibility-aware evaluation, and cautious claims about comparative scope.

Competition-oriented benchmark suites are essential for assessing whether a metaheuristic generalizes across heterogeneous search landscapes. The CEC constrained real-parameter competitions formalize this concern through standardized functions, feasibility rules, and evaluation criteria for constrained single-objective optimization.^32–34 The present study does not claim to reproduce a complete official CEC constrained competition. Instead, the available data correspond to the classical 23-function suite, which is used here as a landscape-diversity component independent of the four engineering geometries. The official CEC constrained suite is therefore identified as a necessary next-stage validation rather than as a completed claim in this manuscript.

3 Scuba diver optimization algorithm

3.1 State representation and initialization

Let $X_{i}^{t} = [x_{i 1}^{t}, x_{i 2}^{t}, \dots, x_{i d}^{t}]$ denote the position of diver i at iteration t in a d -dimensional search space. Each diver is associated with an oxygen state $O_{i}^{t}$ , a penalized fitness value $F_{i}^{t}$ , and, when constraints are present, an aggregate violation score $V_{i}^{t}$ . The initial population is generated using a stratified-random scheme: each variable interval is partitioned into N segments, one base point is assigned to each segment, and a random perturbation is added within the segment. A subset of divers is also replaced by uniformly random samples to strengthen early diversity. Bound constraints are enforced by clipping, and discrete or integer variables are rounded before objective and constraint evaluation where required by the benchmark.

The benchmark families used to validate SDOA span compact continuous design, mixed-variable crashworthiness design, and mixed integer-continuous mechanical design. Their main characteristics are summarized in Table 1.

Table 1.
Benchmark characteristics used to validate SDOA.

Benchmark Variables Type Constraint set Benchmark role

WBD 4 Continuous 7 nonlinear constraints Compact structural design; local stagnation sensitivity

TSD 3 Continuous 4 nonlinear constraints Fine-grained ranking and computational efficiency

CSIDP 11 Mixed continuous-discrete 10 crashworthiness constraints Safety-critical mixed-variable feasibility

SRD 7 Mixed integer–continuous 11 mechanical/geometric constraints Narrow feasible-region design

Benchmark	Variables	Type	Constraint set	Benchmark role
WBD	4	Continuous	7 nonlinear constraints	Compact structural design; local stagnation sensitivity
TSD	3	Continuous	4 nonlinear constraints	Fine-grained ranking and computational efficiency
CSIDP	11	Mixed continuous-discrete	10 crashworthiness constraints	Safety-critical mixed-variable feasibility
SRD	7	Mixed integer–continuous	11 mechanical/geometric constraints	Narrow feasible-region design

Table 1 shows that the validation set was not drawn from a single problem family. Instead, it spans compact continuous design, crashworthiness-constrained mixed-variable design, and mixed integer-continuous mechanical design. This breadth is important because it tests whether SDOA can preserve feasibility under substantially different search geometries rather than succeeding only on one favorable benchmark type.

The principal SDOA parameters and the ranges explored across the benchmark studies are summarized in Table 2.

Table 2.

SDOA control parameters.

Parameter	Symbol	Range used	Function
Population size	N	100–500	Search coverage and diversity
Maximum iterations	T	30–5000	Computational budget and stopping rule
Current strength	c0	0.3–0.9	Early exploration amplitude
Mutation rate	μ0	0.3–0.8	Perturbation probability
Initial oxygen	O0	100–200	Initial stage occupancy
Oxygen decay	α	0.3–0.8	Exploration-to-exploitation transition
Elite size	e	3–30	Retention of high-quality feasible solutions
Penalty factor	ρ	Benchmark-specific: $ρ = 10^{6}$ for penalty-augmented CSIDP/SRD runs; WBD/TSD use feasibility-first violation ranking with normalized violation checks rather than an undisclosed hidden penalty.	Discourages infeasible candidates and supports reproducibility by stating the exact penalty implementation used where penalty-augmented fitness is applied.

Table 2 shows that SDOA is controlled by a compact set of interpretable parameters rather than by a large collection of loosely connected heuristics. Population size and maximum iterations define search coverage and computational budget, whereas current strength, mutation rate, oxygen initialization, and oxygen decay determine the transition from exploratory movement to boundary-sensitive refinement. Elite size and penalty pressure stabilize convergence when the feasible region is narrow or constraint-active.

Figure 1 isolates the conceptual metaphor of SDOA by depicting the oxygen-regulated depth stages assigned to each diver. The diagram is separated from the procedural flowchart so that the behavioral meaning of D1-D4, oxygen decay, and reset is clear before the implementation is examined.

Figure 1.

Oxygen-regulated depth stages in the scuba diver optimization algorithm.

As illustrated in Figure 1, oxygen is the organizing state variable of the method. High oxygen supports long-range exploration near D1, intermediate oxygen promotes guided regional search in D2, D3 emphasizes exploitation around promising solutions, and D4 represents low-amplitude refinement under increasing reset pressure. This staged interpretation explains how SDOA can approach narrow feasible boundaries while reducing the risk of premature collapse into a single local region.

Figure 2 translates the conceptual diving metaphor into an executable optimization workflow. Separating the flowchart from the conceptual diagram clarifies the algorithmic sequence, from initialization and oxygen updates to stage-dependent operators, feasibility-aware acceptance, elite restoration, and termination.

Figure 2.

Procedural flowchart of the scuba diver optimization algorithm.

As shown in Figure 2, SDOA is not a single-move swarm rule but a closed-loop procedure in which oxygen determines the active operator for each diver. The workflow couples parameter updates, stage selection, bound handling, feasibility-aware acceptance, communication with the best diver, and elite restoration. This structure explains the controlled progression from exploration to refinement observed in the benchmark analysis.

3.2 Adaptive control laws

The first control law in SDOA is the oxygen-decay model:

O_{i}^{t} = O_{0} \exp (- α \frac{t}{T}) .

(1)

Here, O0 is the initial oxygen level and alpha > 0 is the decay coefficient. High oxygen promotes broad exploration, whereas low oxygen induces cautious local refinement and, when necessary, reset. Two additional schedules regulate perturbation amplitude. The mutation rate decreases linearly according to

μ_{t} = μ_{0} (1 - \frac{t}{T}),

(2)

and the current-strength parameter decays quadratically as

c_{t} = c_{0} {(1 - \frac{t}{T})}^{2} .

(3)

The quadratic decay produces a relatively rapid contraction of step size after the early exploratory phase, which is desirable for constraint-dominated problems.

3.3 Depth-stage mapping and movement operators

Oxygen maps each diver to one of five depth stages. In the implementations consolidated here, the approximate thresholds are $O_{i}^{t} > 75$ for deep exploration, $55 < O_{i}^{t} \leq 75$ for moderate exploration, $35 < O_{i}^{t} \leq 55$ for local improvement, $15 < O_{i}^{t} \leq 35$ for fine tuning, and $O_{i}^{t} \leq 15$ for reset. The essential feature is the state-dependent transition logic: divers move asynchronously through these stages according to their own oxygen states rather than through a population-wide phase switch.

Table 3 summarizes the state-regulated stage logic. The search does not switch abruptly from global to local behavior at a fixed iteration index; instead, each diver moves through oxygen-governed stages asynchronously. This enables broad coverage, targeted refinement, and diversity recovery to coexist within the same iteration.

Table 3.
Oxygen-regulated stages in SDOA.

Stage Typical oxygen interval Primary operator Search intention

D1 > 75 Lévy-like flight Long-range global exploration

D2 55–75 Bounded random walk Moderate exploration around promising zones

D3 35–55 Elite crossover + local search Feasibility-aware exploitation

D4 15–35 Non-uniform mutation Fine tuning near active constraints

Reset $\leq 15$ Random reinitialization Diversity restoration and escape

Stage	Typical oxygen interval	Primary operator	Search intention
D1	> 75	Lévy-like flight	Long-range global exploration
D2	55–75	Bounded random walk	Moderate exploration around promising zones
D3	35–55	Elite crossover + local search	Feasibility-aware exploitation
D4	15–35	Non-uniform mutation	Fine tuning near active constraints
Reset	$\leq 15$	Random reinitialization	Diversity restoration and escape

Stage D1 performs long-range exploration through a heavy-tailed Lévy-like move:

X_{i}^{{t + 1}} = c l i p [X_{i}^{t} + c_{t} ((U - L) ⊙ L (u)) (1 - \frac{t}{T})] .

(4)

where U and L are the upper and lower variable bounds, the symbol ⊙ denotes elementwise multiplication, and L(u) is generated from a heavy-tailed Cauchy-like transformation. Such heavy-tailed perturbations permit occasional long jumps and help the population escape local basins.

Stage D2 replaces heavy-tailed jumps with bounded random walks:

X_{i}^{{t + 1}} = c l i p [X_{i}^{t} + s_{t} ⊙ (2 r - 1)] .

(5)

where the random vector is bounded by the search interval and scaled by the current-strength parameter. This stage retains exploration while imposing stronger spatial control around promising subregions.

Stage D3 emphasizes exploitation. A diver combines its current position with an elite or with the current best solution through linear crossover, followed by coordinatewise local search around the offspring. In compact notation,

Y_{i}^{t} = β X_{i}^{t} + (1 - β) X_{b e s t}^{t}, β \sim U (0, 1) .

(6)

after which candidate coordinates are perturbed in the positive or negative direction using a shrinking local step and accepted only if penalized fitness improves. This stage is important near active constraints because it provides deliberate, low-amplitude refinement. Stage D4 applies a non-uniform mutation with smaller amplitude than D2 to fine tune feasible or near-feasible solutions without freezing the population prematurely. Finally, the reset stage reinitializes a diver within the search bounds when oxygen becomes critically low or when persistent infeasibility indicates that local correction is unlikely to be productive.

3.4 Constraint handling and acceptance rule

Constraint handling is implemented through a penalty-augmented objective. For a minimization problem with objective $f (X)$ and inequality constraints $g_{k} (X) \leq 0$ , the aggregated violation is

C V (X) = Σ_{m} max (0, g_{m (X)}) .

(7)

and the penalized fitness is

P (X) = f (X) + ρ C V (X) .

(8)

where the penalty coefficient is fixed at

ρ = 10^{6}

in the penalty-augmented CSIDP and SRD implementations, corresponding to the MATLAB setting

nou = 1 e 6

and

penalty = nou \times \sum (\max (0, g))

. For WBD and TSD, feasibility is enforced through direct constraint-violation checks and feasibility-first acceptance. This specification makes the constraint-handling strategy explicit and reproducible while preventing infeasible candidates with deceptively low raw objective values from dominating the search.

3.5 Pseudocode summary

Algorithm 1. Scuba diver optimization algorithm (SDOA)
1	Initialize variable bounds, N, T, c0, μ0, O0, α, elite size e, and penalty ρ.
2	Generate the initial diver population within bounds and enforce discrete variables where required.
3	Evaluate objective values, constraint violations, and penalized fitness for all divers.
4	Record the current best feasible diver and preserve the top-e elites.
5	For t = 1 to T do
6	Update $μ_{t}$ and $c_{t}$ using Eqs. (2) and (3).
7	For each non-elite diver i do
8	Update oxygen $O_{i}^{t}$ using Eq. (1) and determine its stage.
9	Apply the D1, D2, D3, D4, or Reset operator according to the current oxygen interval.
10	Enforce bounds; round discrete variables where required.
11	Evaluate the new candidate and accept it using the feasibility-aware rule.
12	With oxygen-dependent probability, communicate near the current best solution.
13	End for
14	Restore elites and update the global best feasible solution.
15	End for
16	Return the best feasible design, objective value, and diagnostic histories.

3.6 Methodological distinctiveness, complexity, and reproducibility

The oxygen mechanism is not introduced as a metaphorical label for a standard decreasing coefficient. Methodologically, it functions as an internal state with multiple algorithmic consequences: it assigns each diver to a depth stage, selects the movement operator, scales perturbation amplitude, influences local-refinement intensity, triggers reset when search becomes unproductive, and regulates communication with elite solutions. This distinguishes SDOA from simple iteration-based schedules, in which a coefficient changes over time but does not determine the full behavioral regime of each agent.

Table 4 clarifies this distinction relative to common adaptive or stage-based metaheuristic mechanisms. The intended contribution is not to prove that oxygen-state regulation is universally superior, but to present a reproducible search-governance architecture in which diversification, intensification, and recovery are coordinated by the same state variable.

Table 4.
Methodological distinction between SDOA and common adaptive metaheuristic control strategies.

Comparator family Typical control principle Limitation in constrained search SDOA distinction

Iteration-based schedules Exploration coefficients decrease according to a global iteration counter. All agents usually experience the same phase transition, which may accelerate population collapse near narrow feasible regions. Oxygen is agent-level; different divers can occupy different stages at the same iteration.

Adaptive PSO/DE variants Velocity, inertia, mutation, or crossover parameters adapt using diversity, success rate, or pre-defined schedules. Adaptation often modifies a coefficient while leaving the movement family largely unchanged. Oxygen selects the movement family itself: long-range exploration, random walk, elite crossover, local search, mutation, or reset.

Stage- or energy-based optimizers Search phases are organized as exploration/exploitation stages or energy decay. Stage switching is often population-wide or tied to one dominant update rule. SDOA couples stage allocation, step size, elite communication, local search, and resurfacing into one state-regulated loop.

Hybrid local-search algorithms A global optimizer is combined with a local refinement operator. The trigger for local search may be externally tuned or applied uniformly. Local refinement is activated by oxygen state and is therefore embedded in the individual search trajectory.

Penalty-only constrained methods Constraint handling is mainly delegated to penalty coefficients. Penalty pressure can either over-restrict exploration or allow infeasible objective values to dominate. Feasibility-aware acceptance, normalized violation checking, and reset jointly support recovery from infeasible regions.

Comparator family	Typical control principle	Limitation in constrained search	SDOA distinction
Iteration-based schedules	Exploration coefficients decrease according to a global iteration counter.	All agents usually experience the same phase transition, which may accelerate population collapse near narrow feasible regions.	Oxygen is agent-level; different divers can occupy different stages at the same iteration.
Adaptive PSO/DE variants	Velocity, inertia, mutation, or crossover parameters adapt using diversity, success rate, or pre-defined schedules.	Adaptation often modifies a coefficient while leaving the movement family largely unchanged.	Oxygen selects the movement family itself: long-range exploration, random walk, elite crossover, local search, mutation, or reset.
Stage- or energy-based optimizers	Search phases are organized as exploration/exploitation stages or energy decay.	Stage switching is often population-wide or tied to one dominant update rule.	SDOA couples stage allocation, step size, elite communication, local search, and resurfacing into one state-regulated loop.
Hybrid local-search algorithms	A global optimizer is combined with a local refinement operator.	The trigger for local search may be externally tuned or applied uniformly.	Local refinement is activated by oxygen state and is therefore embedded in the individual search trajectory.
Penalty-only constrained methods	Constraint handling is mainly delegated to penalty coefficients.	Penalty pressure can either over-restrict exploration or allow infeasible objective values to dominate.	Feasibility-aware acceptance, normalized violation checking, and reset jointly support recovery from infeasible regions.

From a search-dynamics perspective, SDOA can be interpreted as a bounded stochastic process with elitist memory and state-dependent transition kernels. Bound repair keeps candidate positions inside the admissible search domain, while reset preserves a non-zero probability of re-entering unexplored regions. Elitist preservation makes the best penalized objective sequence non-increasing across iterations, although this monotonicity is not a formal proof of global optimality. The claim made here is therefore qualitative and empirical: oxygen-state regulation delays premature homogenization, supports local refinement near feasible elites, and provides a transparent recovery path for low-oxygen, stagnant, or infeasible agents.

The computational overhead of this state regulation is modest. Apart from objective and constraint evaluation, each iteration updates N oxygen states and applies stage-specific movements over d variables, giving O(Nd) movement complexity and O(Nd) memory. Consequently, practical runtime is driven primarily by the number of function evaluations and by the cost of objective or constraint evaluation. For analytical benchmarks, the overhead is small; for simulation-driven engineering design, external models would dominate the total cost. This distinction motivates the runtime reporting in Section 6.7.

4 Integrated benchmark corpus and mathematical formulation

The empirical study is organized around a unified benchmark design rather than isolated case demonstrations. The constrained engineering layer evaluates feasibility preservation, mixed-variable handling, and behavior near active constraint boundaries. The mathematical benchmark layer evaluates general search dynamics across continuous landscapes with different modality, separability, discontinuity, and basin structures. All experiments are expressed in the common minimization form $\min f_{m} (x)$ , subject to $g_{k} (x) \leq 0$ for constrained problems and $L_{j} \leq x_{j} \leq U_{j}$ for all decision variables. Integer and discrete variables are repaired before objective and constraint evaluation, ensuring that the reported final solutions remain admissible under the corresponding benchmark definition.

Figure 3 illustrates the physical meaning of the four engineering problems, while Table 5 provides the complete benchmark matrix used in the study. The table includes the objective or test equation for each validation problem so that the mathematical scope of the experiments is transparent before the individual formulations are discussed.

Figure 3.

Geometric interpretation of the four engineering design benchmarks used in this study.

Table 5.

Integrated validation corpus with benchmark equations, domains, constraints, and diagnostic roles.

Benchmark	Objective / equation and constraints	Domain and optimum	Diagnostic role
WBD	Objective: f = 1.10471 $h^{2}$ l + 0.04811 t b(14 + l). Constraints: $g_{1}$ = tau-13600 ≤ 0; $g_{2}$ = sigma-30000 ≤ 0; $g_{3}$ = h-b ≤ 0; $g_{4}$ = 0.10471 $h^{2}$ + 0.04811tb(14 + l)−5 ≤ 0; $g_{5}$ = 0.125-h ≤ 0; $g_{6}$ = delta-0.25 ≤ 0; $g_{7}$ = 6000− $P_{c}$ ≤ 0.	x = (h,l,t,b). Bounds: h,b in [0.1,2]; l,t in [0.1,10].	Structural cost minimization with stress, deflection, buckling, and geometry restrictions.
TSD	Objective: f = ( $x_{3}$ +2)x2 $x_{1}^{2}$ . Constraints: $g_{1}$ = 1−( $x_{2}^{3} x_{3}$ )/(71785 $x_{1}^{4}$ ) ≤ 0; $g_{2}$ = (4 $x_{2}^{2}$ −x1 $x_{2}$ )/(12566(x2 $x_{1}^{3}$ − $x_{1}^{4}$ )) + 1/(5108 $x_{1}^{2}$ )−1 ≤ 0; $g_{3}$ = 1−(140.45 $x_{1}$ )/( $x_{2}^{2} x_{3}$ ) ≤ 0; $g_{4}$ = ( $x_{1}$ + $x_{2}$ )/1.5−1 ≤ 0.	x = (wire diameter, mean coil diameter, active coils).Bounds: $x_{1}$ in [0.05,2], $x_{2}$ in [0.25,1.30], $x_{3}$ in [2,15].	Compact nonlinear design with tightly clustered best objective values in the literature.
CSIDP	Objective: f = 1.98 + 4.90 $x_{1}$ +6.67 $x_{2}$ +6.98 $x_{3}$ +4.01 $x_{4}$ +1.78 $x_{5}$ +2.73 $x_{7}$ .Constraints: $g_{k}$ = $r_{k}$ (x)/ $L_{k}$ −1 ≤ 0 for k = 1,…,10, where $r_{k}$ ={Fa,VCu,VCm,VCl,Dur,Dmr,Dlr,Fp,VMBP,VFD}.	x = ( $x_{1}$ ,…, $x_{11}$ ), mixed continuous/discrete variables; $x_{8}$ and $x_{9}$ are material choices.	Mixed-variable crashworthiness design; feasibility is essential because low raw weight may be unsafe.
SRD	Objective: f = 0.7854 × 1 $x_{2}^{2}$ (3.3333 $x_{3}^{2}$ + 14.9334 $x_{3}$ −43.0934)−1.508 $x_{1}$ ( $x_{6}^{2}$ + $x_{7}^{2}$ ) + 7.4777 ( $x_{6}^{3}$ + $x_{7}^{3}$ ) + 0.7854(x4 $x_{6}^{2}$ + x5 $x_{7}^{2}$ ).Constraints: eleven gear-stress, shaft-stress, deflection, and geometry inequalities, e.g., $g_{1}$ = 27/(x1 $x_{2}^{2} x_{3}$ )−1 ≤ 0 and $g_{11}$ = (1.1 $x_{7}$ + 1.9)/ $x_{5}$ −1 ≤ 0.	x = ( $x_{1}$ ,…, $x_{7}$ ), with $x_{3}$ integer.Bounds: $x_{1}$ [2.6,3.6], $x_{2}$ [0.7,0.8], $x_{3}$ [17,28], $x_{4}$ , $x_{5}$ [7.3,8.3], $x_{6}$ [2.9,3.9], $x_{7}$ [5.0,5.5].	Mixed integer-continuous mechanical design with several active constraints near the optimum.
F1 Sphere	f = sum( $x_{i}^{2}$ ).	n = 10; $x_{i}$ in [−100,100]; $f^{*}$ = 0.	Unimodal contraction and precision.
F2 Schwefel 2.22	f = sum(\| $x_{i}$ \|)+prod(\| $x_{i}$ \|).	n = 10; $x_{i}$ in [−10,10]; $f^{*}$ = 0.	Product sensitivity.
F3 Schwefel 1.2	f = $\sum_{i}$ ( $\sum_{j = 1}^{i} x_{j}$ ) $^{\land}$ 2.	n = 10; $x_{i}$ in [−100,100]; $f^{*}$ = 0.	Nonseparable conditioning.
F4 Schwefel 2.21	f = ma $x_{i}$ \| $x_{i}$ \|.	n = 10; $x_{i}$ in [−100,100]; $f^{*}$ = 0.	Worst-coordinate contraction.
F5 Rosenbrock	f = $\sum_{i = 1}^{n - 1}$ [100( $x_{i + 1}$ − $x_{i}^{2}$ ) $^{\land}$ 2 + ( $x_{i}$ −1) $^{\land}$ 2].	n = 10; $x_{i}$ in [−30,30]; $f^{*}$ = 0 at $x_{i}$ = 1.	Curved narrow valley.
F6 Step	f = sum floor( $x_{i}$ + 0.5) $^{\land}$ 2.	n = 10; $x_{i}$ in [−100,100]; $f^{*}$ = 0.	Discontinuous plateau search.
F7 Quartic with noise	f = $\sum_{i}$ i* $x_{i}^{4}$ + rand[0,1).	n = 10; $x_{i}$ in [−1.28,1.28]; $f^{*}$ = 0 in expectation.	Noise tolerance.
F8 Schwefel	f = sum[− $x_{i}$ sin(sqrt(\| $x_{i}$ \|))].	n = 10; $x_{i}$ in [−500,500]; optimum near $x_{i}$ = 420.9687.	Distant global basin.
F9 Rastrigin	f = sum[ $x_{i}^{2} - 10 c o s$ (2pi $x_{i}$ ) + 10].	n = 10; $x_{i}$ in [−5.12,5.12]; $f^{*}$ = 0.	Periodic multimodality.
F10 Ackley	f = −20exp(−0.2sqrt(sum $x_{i}^{2}$ /n))−exp(sum cos(2pi $x_{i}$ )/n) + 20 + e.	n = 10; $x_{i}$ in [−32,32]; $f^{*}$ = 0.	Flat outer region with central basin.
F11 Griewank	f = 1 + sum( $x_{i}^{2}$ )/4000-prod cos( $x_{i}$ /sqrt(i)).	n = 10; $x_{i}$ in [−600,600]; $f^{*}$ = 0.	Product-coupled local minima.
F12 Penalized 1	f = pi/n{10 $\sin^{2}$ (pi* $y_{1}$ ) + sum( $y_{i}$ −1) $\land$ 2[1 + 10 $\sin^{2}$ (pi* $y_{i + 1}$ )] + ( $y_{n}$ −1) $\land$ 2} + sum u( $x_{i}$ ,10,1004), $y_{i}$ = 1 + ( $x_{i}$ + 1)/4.	n = 10; $x_{i}$ in [−50,50]; $f^{*}$ = 0.	Penalty-guided boundary behavior.
F13 Penalized 2	f = 0.1{ $\sin^{2}$ (3pi* $x_{1}$ ) + sum( $x_{i}$ −1) $\land$ 2[1 + $\sin^{2}$ (3pi* $x_{i + 1}$ )] + ( $x_{n}$ −1) $\land$ 2[1 + $\sin^{2}$ (2pi* $x_{n}$ )]}+sum u( $x_{i}$ ,5,100,4).	n = 10; $x_{i}$ in [−50,50]; $f^{*}$ = 0 at $x_{i}$ = 1.	Penalty-guided multimodal refinement.
F14 Foxholes	f = [1/500 + $\sum_{j = 1}^{25}$ 1/(j + $\sum_{i = 1}^{2}$ ( $x_{i}$ − $a_{i j}$ ) $\land$ 6)] $^{\land}$ {−1}.	n = 2; $x_{i}$ in [−65.536,65.536]; $f^{*}$ ≈0.998.	Fixed-dimensional basin selection.
F15 Kowalik	f = $\sum_{i = 1}^{11}$ [ $a_{i}$ − $x_{1}$ ( $b_{i}^{2} + b$ _i* $x_{2}$ )/( $b_{i}^{2} + b$ _i* $x_{3}$ + $x_{4}$ )] $^{\land}$ 2.	n = 4; $x_{i}$ in [−5,5]; $f^{*}$ ≈0.00030.	Parameter-estimation valley.
F16 Six-hump camel	f = 4 $x_{1}^{2}$ −2.1 $x_{1}^{4}$ + $x_{1}^{6}$ /3 + x1 $x_{2}$ −4 $x_{2}^{2}$ +4 $x_{2}^{4}$ .	n = 2; $x_{i}$ in [−5,5]; $f^{*}$ ≈−1.0316.	Low-dimensional multiple minima.
F17 Branin	f = ( $x_{2}$ −5.1 $x_{1}^{2}$ /(4p $i^{2}$ ) + 5 $x_{1}$ /pi-6) $\land$ 2 + 10(1-1/(8pi))cos( $x_{1}$ ) + 10.	n = 2; $x_{1}$ in [−5,10], $x_{2}$ in [0,15]; $f^{*}$ ≈0.3979.	Equivalent global basins.
F18 Goldstein-Price	f = [1 + ( $x_{1}$ + $x_{2}$ +1) $\land$ 2 A]*[30 + (2 $x_{1}$ −3 $x_{2}$ ) $\land$ 2 B], where A and B are the standard Goldstein-Price polynomials.	n = 2; $x_{i}$ in [−2,2]; $f^{*}$ = 3.	Sharp nonlinear interaction.
F19 Hartman 3	f = − $\sum_{i = 1}^{4} c_{i}$ exp[− $\sum_{j = 1}^{3} a_{i j}$ ( $x_{j}$ − $p_{i j}$ ) $\land$ 2].	n = 3; $x_{i}$ in [0,1]; $f^{*}$ ≈−3.8628.	Localized basin interaction.
F20 Hartman 6	f = − $\sum_{i = 1}^{4} c_{i}$ exp[− $\sum_{j = 1}^{6} a_{i j}$ ( $x_{j}$ − $p_{i j}$ ) $\land$ 2].	n = 6; $x_{i}$ in [0,1]; $f^{*}$ ≈−3.3224.	Higher-dimensional fixed basin.
F21 Shekel 5	f = − $\sum_{i = 1}^{5}$ [(x− $a_{i}$ )(x− $a_{i}$ ) $\land$ T+ $c_{i}$ ] $\land$ {−1}.	n = 4; $x_{i}$ in [0,10]; $f^{*}$ ≈−10.1532.	Narrow attraction basins.
F22 Shekel 7	f = − $\sum_{i = 1}^{7}$ [(x− $a_{i}$ )(x− $a_{i}$ ) $\land$ T+ $c_{i}$ ] $\land$ {−1}.	n = 4; $x_{i}$ in [0,10]; $f^{*}$ ≈−10.4029.	Narrow attraction basins.
F23 Shekel 10	f = − $\sum_{i = 1}^{10}$ [(x− $a_{i}$ )(x− $a_{i}$ ) $\land$ T+ $c_{i}$ ] $\land$ {−1}.	n = 4; $x_{i}$ in [0,10]; $f^{*}$ ≈−10.5364.	Exploitation precision in narrow basins.

The geometric sketches in Figure 3 explain the engineering roles of WBD, TSD, CSIDP, and SRD. The remaining mathematical functions broaden the validation corpus by evaluating search dynamics under known global optima, controlled bounds, and heterogeneous landscape structures.

Table 5 makes the benchmark selection explicit. The first four rows evaluate constrained feasibility and mixed-variable handling; the remaining rows evaluate general search behavior under standard mathematical landscapes. This structure improves the scientific interpretation of the results because each benchmark is linked to a diagnostic purpose rather than used only as a numerical scoreboard.

4.1 Welded beam design (WBD)

The welded beam design problem minimizes fabrication cost subject to stress, deflection, buckling, and geometric constraints. Let $x_{1}$ = h denote weld thickness, $x_{2}$ = l denote weld length, $x_{3}$ = t denote beam thickness, and $x_{4}$ = b denote beam width.

f (x) = 1.10471 x 1^{2} x 2 + 0.04811 x 3 x 4 (14 + x 2) .

(9)

The constraints are

\begin{aligned} g 1 (x) & = τ (x) - 13600 \leq 0, \end{aligned}

\begin{aligned} g 2 (x) & = σ (x) - 30000 \leq 0, \end{aligned}

\begin{aligned} g 3 (x) & = x 1 - x 4 \leq 0, \end{aligned}

\begin{aligned} g 4 (x) & = 0.10471 x 1^{2} + 0.04811 x 3 x 4 (14 + x 2) - 5 \leq 0, \end{aligned}

\begin{aligned} g 5 (x) & = 0.125 - x 1 \leq 0, \end{aligned}

\begin{aligned} g 6 (x) & = δ (x) - 0.25 \leq 0, \end{aligned}

\begin{aligned} g 7 (x) & = 6000 - P_{c (x)} \leq 0. \end{aligned}

The auxiliary expressions used to evaluate the mechanics of the welded beam are

\begin{aligned} τ (x) & = s q r t (τ^{' 2} + \frac{2 τ^{'} τ^{″} x 2}{2 R} + τ^{″ 2}), \end{aligned}

\begin{aligned} τ^{'} & = \frac{6000}{(s q r t (2) x 1 x 2)}, \end{aligned}

\begin{aligned} τ^{″} & = \frac{M R}{J}, \end{aligned}

\begin{aligned} M & = 6000 (14 + \frac{x 2}{2}), \end{aligned}

\begin{aligned} R & = s q r t (\frac{x 2^{2}}{4} + {(\frac{x 1 + x 3}{2})}^{2}), \end{aligned}

\begin{aligned} J & = 2 s q r t (2) x 1 x 2 (\frac{x 2^{2}}{12} + {(\frac{x 1 + x 3}{2})}^{2}), \end{aligned}

\begin{aligned} σ (x) & = \frac{504000}{(x 4 x 3^{2})}, \end{aligned}

\begin{aligned} δ (x) & = \frac{2.1952}{(x 3^{3} x 4)}, \end{aligned}

\begin{aligned} P_{c (x)} & = 64746.022 (1 - 0.0282346 x 3) x 3 x 4^{3} . \end{aligned}

The variable bounds are

0.1 \leq x_{1} \leq 2.0, 0.1 \leq x_{2} \leq 10.0, 0.1 \leq x_{3} \leq 10.0, 0.1 \leq x_{4} \leq 2.0.

4.2 Tension/compression spring design (TSD)

The tension/compression spring design problem minimizes spring weight while satisfying shear-stress, surge-frequency, deflection, and geometric constraints. Let $x_{1}$ = d denote wire diameter, $x_{2}$ = D denote mean coil diameter, and $x_{3}$ = N denote the number of active coils.

f (x) = (x 3 + 2) x 2 x 1^{2} .

(10)

The nonlinear constraints are

\begin{aligned} g 1 (x) & = 1 - \frac{x 2^{3} x 3}{71785 x 1^{4}} \leq 0, \end{aligned}

\begin{aligned} g 2 (x) & = \frac{4 x 2^{2} - x 1 x 2}{12566 (x 2 x 1^{3} - x 1^{4})} + \frac{1}{5108 x 1^{2}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 3 (x) & = 1 - \frac{140.45 x 1}{x 2^{2} x 3} \leq 0, \end{aligned}

\begin{aligned} g 4 (x) & = \frac{x 1 + x 2}{1.5} - 1 \leq 0. \end{aligned}

The variable bounds are

0.05 \leq x_{1} \leq 2.0, 0.25 \leq x_{2} \leq 1.30, 2 \leq x_{3} \leq 15.

4.3 Car side-impact design problem (CSIDP)

The car side-impact design problem minimizes structural weight subject to ten crashworthiness constraints. The design vector contains eleven variables, of which $x_{8}$ and $x_{9}$ represent discrete material choices. A compact surrogate objective is given by

f (x) = 1.98 + 4.90 x 1 + 6.67 x 2 + 6.98 x 3 + 4.01 x 4 + 1.78 x 5 + 0.00001 x 6 + 2.73 x 7.

(11)

Crashworthiness is enforced through ten normalized inequality constraints of the form

g_{i (x)} = \frac{r_{i (x)}}{L_{i}} - 1 \leq 0, i = 1, 2, \dots, 10,

(12)

The response-surface models are

\begin{aligned} F_{a} & = 1.16 - 0.3717 x_{2} x_{4} - 0.00931 x_{2} x_{10} - 0.484 x_{3} x_{9} + 0.01343 x_{6} x_{10} . \end{aligned}

\begin{aligned} V C_{u} & = A_{u} + B_{u} . \end{aligned}

\begin{aligned} A_{u} & = 0.261 - 0.0159 x_{1} x_{2} - 0.188 x_{1} x_{8} - 0.019 x_{2} x_{7} + 0.0144 x_{3} x_{5} . \end{aligned}

\begin{aligned} B_{u} & = 0.0008757 x_{5} x_{10} + 0.08045 x_{6} x_{9} + 0.00139 x_{8} x_{11} + 0.00001575 x_{10} x_{11} . \end{aligned}

\begin{aligned} V C_{m} & = A_{m} + B_{m} + C_{m} . \end{aligned}

\begin{aligned} A_{m} & = 0.214 + 0.00817 x_{5} - 0.131 x_{1} x_{8} - 0.0704 x_{1} x_{9} + 0.03099 x_{2} x_{6} - 0.018 x_{2} x_{7} . \end{aligned}

\begin{aligned} B_{m} & = 0.0208 x_{3} x_{8} + 0.121 x_{3} x_{9} - 0.00364 x_{5} x_{6} + 0.0007715 x_{5} x_{10} - 0.0005354 x_{6} x_{10} . \end{aligned}

\begin{aligned} C_{m} & = 0.00121 x_{8} x_{11} + 0.00184 x_{9} x_{10} - 0.018 x_{2}^{2} . \end{aligned}

\begin{aligned} V C_{l} & = 0.74 - 0.61 x_{2} - 0.163 x_{3} x_{8} + 0.001232 x_{3} x_{10} - 0.166 x_{7} x_{9} + 0.227 x_{2}^{2} . \end{aligned}

\begin{aligned} D_{u r} & = 28.98 + 3.818 x_{3} - 4.2 x_{1} x_{2} + 0.0207 x_{5} x_{10} + 6.63 x_{6} x_{9} - 7.77 x_{7} x_{8} + 0.32 x_{9} x_{10} . \end{aligned}

\begin{aligned} D_{m r} & = A_{m r} + B_{m r} . \end{aligned}

\begin{aligned} A_{m r} & = 33.86 + 2.95 x_{3} + 0.1792 x_{10} - 5.057 x_{1} x_{2} - 11 x_{2} x_{8} . \end{aligned}

\begin{aligned} B_{m r} & = - 0.0215 x_{5} x_{10} - 9.98 x_{7} x_{8} + 22 x_{8} x_{9} . \end{aligned}

\begin{aligned} D_{l r} & = 46.36 - 9.9 x_{2} - 12.9 x_{1} x_{8} + 0.1107 x_{3} x_{10} . \end{aligned}

\begin{aligned} F_{p} & = 4.72 - 0.5 x_{4} - 0.19 x_{2} x_{3} - 0.0122 x_{4} x_{10} + 0.009325 x_{6} x_{10} + 0.000191 x_{11}^{2} . \end{aligned}

\begin{aligned} V_{M B P} & = 10.58 - 0.674 x_{1} x_{2} - 1.95 x_{2} x_{8} + 0.02054 x_{3} x_{10} - 0.0198 x_{4} x_{10} + 0.028 x_{6} x_{10} . \end{aligned}

\begin{aligned} V_{F D} & = 16.45 - 0.489 x_{3} x_{7} - 0.843 x_{5} x_{6} + 0.0432 x_{9} x_{10} - 0.0556 x_{9} x_{11} - 0.000786 x_{11}^{2} . \end{aligned}

The normalized crashworthiness constraints are

\begin{aligned} g_{1} & = \frac{F_{a}}{1.0} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{2} & = \frac{V C_{u}}{0.32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{3} & = \frac{V C_{m}}{0.32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{4} & = \frac{V C_{l}}{0.32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{5} & = \frac{D_{u r}}{32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{6} & = \frac{D_{m r}}{32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{7} & = \frac{D_{l r}}{32} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{8} & = \frac{F_{p}}{4.0} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{9} & = \frac{V_{M B P}}{9.9} - 1 \leq 0. \end{aligned}

\begin{aligned} g_{10} & = \frac{V_{F D}}{15.7} - 1 \leq 0. \end{aligned}

The decision bounds used for the structural variables are

\begin{aligned} 0.5 \leq x_{1}, x_{3}, x_{4} \leq 1.5, 0.45 \leq x_{2} \leq 1.35, 0.875 \leq x_{5} \leq 2.625, 0.4 \leq x_{6}, x_{7} \leq 1.2. \end{aligned}

Variables $x_{8}$ , $x_{9}$ , $x_{10}$ , and $x_{11}$ are treated according to the mixed-variable benchmark specification.

where $r_{i} (x)$ denotes the response predicted by the regression model and $L_{i}$ is the corresponding allowable limit. The response set is {Fa, VCu, VCm, VCl, Dur, Dmr, Dlr, Fp, VMBP, VFD}, with limits {1.0, 0.32, 0.32, 0.32, 32, 32, 32, 4.0, 9.9, 15.7}. This normalized form is useful for SDOA because it allows heterogeneous crashworthiness quantities to be penalized on a common scale.

Table 6 is important because the CSIDP constraints are heterogeneous in scale and physical meaning. By expressing them as normalized response-to-limit ratios, SDOA can penalize crashworthiness violations on a common basis. This normalization helps stabilize the method on CSIDP despite the mixture of load, velocity, and deformation responses.

Table 6.

Constraint responses and upper limits for CSIDP.

ID	Response	Description	Upper limit	Unit
$g_{1}$	Fa	Load in the abdomen	1.0	kN
$g_{2}$	VCu	Upper chest velocity	0.32	m/s
$g_{3}$	VCm	Middle chest velocity	0.32	m/s
$g_{4}$	VCl	Lower chest velocity	0.32	m/s
$g_{5}$	Dur	Upper rib deflection	32	mm
$g_{6}$	Dmr	Middle rib deflection	32	mm
$g_{7}$	Dlr	Lower rib deflection	32	mm
$g_{8}$	Fp	Pubic force	4.0	kN
$g_{9}$	VMBP	Velocity of V-pillar midpoint	9.9	mm/ms
$g_{10}$	VFD	Velocity of front door at V-pillar	15.7	mm/ms

4.4 Speed reducer design (SRD)

The speed reducer design problem minimizes total reducer weight subject to eleven nonlinear inequality constraints. Let $x_{1}$ denote face width, $x_{2}$ the module, $x_{3}$ the number of teeth on the pinion, $x_{4}$ and $x_{5}$ the shaft lengths, and $x_{6}$ and $x_{7}$ the shaft diameters.

\begin{aligned} f (x) & = 0.7854 x 1 x 2^{2} (3.3333 x 3^{2} + 14.9334 x 3 - 43.0934) - 1.508 x 1 (x 6^{2} + x 7^{2}) \\ + 7.4777 (x 6^{3} + x 7^{3}) + 0.7854 (x 4 x 6^{2} + x 5 x 7^{2}) . \end{aligned}

(13)

The SRD constraints are

\begin{aligned} g 1 (x) & = \frac{27}{x 1 x 2^{2 x 3}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 2 (x) & = \frac{397.5}{x 1 x 2^{2 x 3^{2}}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 3 (x) & = \frac{1.93 x 4^{3}}{x 2 x 3 x 6^{4}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 4 (x) & = \frac{1.93 x 5^{3}}{x 2 x 3 x 7^{4}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 5 (x) & = \frac{s q r t ({(\frac{745 x 4}{x 2 x 3})}^{2} + 16.9 \times 10^{6})}{110 x 6^{3}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 6 (x) & = \frac{s q r t ({(\frac{745 x 5}{x 2 x 3})}^{2} + 157.5 \times 10^{6})}{85 x 7^{3}} - 1 \leq 0, \end{aligned}

\begin{aligned} g 7 (x) & = x 2 x 3 - 40 \leq 0, \end{aligned}

\begin{aligned} g 8 (x) & = 5 - x 1 / x 2 \leq 0, \end{aligned}

\begin{aligned} g 9 (x) & = x 1 / x 2 - 12 \leq 0, \end{aligned}

\begin{aligned} g 10 (x) & = (1.5 x 6 + 1.9) / x 4 - 1 \leq 0, \end{aligned}

\begin{aligned} g 11 (x) & = (1.1 x 7 + 1.9) / x 5 - 1 \leq 0. \end{aligned}

The variable bounds are

2.6 \leq x_{1} \leq 3.6, 0.7 \leq x_{2} \leq 0.8, 17 \leq x_{3} \leq 28 (i n t e g e r), 7.3 \leq x_{4} \leq 8.3, 7.3 \leq x_{5} \leq 8.3, 2.9 \leq x_{6} \leq 3.9, 5.0 \leq x_{7} \leq 5.5.

4.5 Implementation normalization and reproducibility checks

To support reproducibility, the mathematical formulation and implementation pathway are treated as part of the experimental protocol. All constrained problems are evaluated after bound enforcement, discrete-variable treatment where required, objective calculation, and constraint-violation aggregation. The mathematical functions are evaluated using their canonical bounds and dimensions under the same SDOA update rules, without function-specific retuning (Table 7).

Table 7.
Equation-transparency and implementation audit for the validation problems.

Problem Objective expression / model Constraint-equation treatment Implementation check

WBD Fabrication-cost model f(x) = 1.10471 $x_{1}$ ² $x_{2}$ + 0.04811 × 3 $x_{4}$ (14 + $x_{2}$ ). Stress, deflection, buckling-load, and geometric constraints are evaluated using the mechanical auxiliary equations in Section 4.1. Four continuous variables; bounds enforced before evaluation; final feasibility checked against τ, σ, δ, and $P_{c}$ .

TSD Spring-weight model f(x) = ( $x_{3}$ +2)x2 $x_{1}$ ². Four nonlinear constraints covering shear stress, deflection, surge-frequency stability, and geometry are evaluated in the orientation $g_{k} (x) \leq 0$ . Three continuous variables; final solution must satisfy all four constraints before being reported.

CSIDP Linear surrogate structural-weight model with eleven mixed variables. Ten heterogeneous crashworthiness responses are normalized by their allowable limits so that violations are comparable across load, velocity, and deflection quantities. Variables $x_{8}$ and $x_{9}$ are handled as discrete material choices; all ten normalized constraints must be non-positive.

SRD Reducer-weight model with gear and shaft mass terms; $x_{3}$ is integer-valued. Eleven nonlinear constraints cover gear stress, shaft deflection/stress, and geometric/assembly restrictions. Integrality of $x_{3}$ is enforced by rounding before evaluation; all constraints are verified after boundary repair.

23-function suite Each function F1–F23 is defined by its canonical equation, dimension, and bound range. No engineering constraints are imposed; the suite evaluates unconstrained global-search behavior and landscape robustness. A single SDOA configuration is used for all 23 functions to avoid function-specific tuning.

Problem	Objective expression / model	Constraint-equation treatment	Implementation check
WBD	Fabrication-cost model f(x) = 1.10471 $x_{1}$ ² $x_{2}$ + 0.04811 × 3 $x_{4}$ (14 + $x_{2}$ ).	Stress, deflection, buckling-load, and geometric constraints are evaluated using the mechanical auxiliary equations in Section 4.1.	Four continuous variables; bounds enforced before evaluation; final feasibility checked against τ, σ, δ, and $P_{c}$ .
TSD	Spring-weight model f(x) = ( $x_{3}$ +2)x2 $x_{1}$ ².	Four nonlinear constraints covering shear stress, deflection, surge-frequency stability, and geometry are evaluated in the orientation $g_{k} (x) \leq 0$ .	Three continuous variables; final solution must satisfy all four constraints before being reported.
CSIDP	Linear surrogate structural-weight model with eleven mixed variables.	Ten heterogeneous crashworthiness responses are normalized by their allowable limits so that violations are comparable across load, velocity, and deflection quantities.	Variables $x_{8}$ and $x_{9}$ are handled as discrete material choices; all ten normalized constraints must be non-positive.
SRD	Reducer-weight model with gear and shaft mass terms; $x_{3}$ is integer-valued.	Eleven nonlinear constraints cover gear stress, shaft deflection/stress, and geometric/assembly restrictions.	Integrality of $x_{3}$ is enforced by rounding before evaluation; all constraints are verified after boundary repair.
23-function suite	Each function F1–F23 is defined by its canonical equation, dimension, and bound range.	No engineering constraints are imposed; the suite evaluates unconstrained global-search behavior and landscape robustness.	A single SDOA configuration is used for all 23 functions to avoid function-specific tuning.

This audit clarifies how each component of the validation corpus is implemented. The four engineering problems emphasize feasibility and constraint-active behavior, whereas F1-F23 emphasize landscape diversity under a declared fixed configuration. Together, they provide complementary evidence for the same algorithmic mechanism (Table A5).

5 Experimental design

5.1 Experimental protocol, reproducibility, and statistical validation

All experiments were implemented in MATLAB using the same SDOA core: oxygen decay, state-based movement operators, elite preservation, feasibility-aware acceptance, and selective reset. Because WBD, TSD, CSIDP, and SRD differ in dimensionality, variable type, constraint topology, and feasible-region geometry, the parameter studies were not forced into one identical budget. Instead, each benchmark was examined through a scenario sweep appropriate to its search difficulty, and the selected settings were then analyzed transparently (Table 8).

For CSIDP and SRD, repeated-run statistical evidence is available. In CSIDP, each scenario was executed through ten independent runs with different random seeds, and the reported indicators include best, mean, worst, standard deviation, and runtime. The SRD study likewise reports best, mean, worst, standard deviation, runtime, and explicit feasibility verification across ten parameter scenarios. For WBD and TSD, the evidence is scenario-based and is supported by convergence histories, feasibility checks, and comparative tables.

Formal non-parametric tests such as Wilcoxon signed-rank or Friedman procedures require paired repeated-run samples obtained under comparable settings. Several comparator algorithms discussed in the benchmark literature report only best or aggregate values, without raw repeated-run samples, random seeds, stopping tolerances, or feasibility histories. Consequently, the comparative analysis is reported through repeated-run statistics where available, objective-deviation analysis, feasibility status, and an explicit statement of comparison scope.

Table 8.
Summary of experimental evidence used to support Benchmark-Level revisions and the other will be increased by 1 in table and paragraphs.

Benchmark Experimental evidence used in the revision Analytical purpose

WBD Eight scenario/case settings; convergence histories; all reported structural constraints feasible; broader 30-algorithm comparison from the WBD file. Clarifies that WBD evidence is scenario-level and interprets rank honestly rather than claiming universal dominance.

TSD Eight scenario/case settings; all four constraints feasible; 100-algorithm comparison; SDOA objective 0.012671 ranked 26/100. Uses the full comparative table and avoids exaggerating small fourth-/fifth-decimal differences.

CSIDP Ten scenarios with independent-run statistics; best/mean/worst/std/time; normalized constraints and feasibility status. Strengthens statistical reporting and separates objective quality from feasibility.

SRD Ten scenarios with best/mean/worst/std/time; full feasibility verification; 34-algorithm comparison. Addresses robustness, variance, feasibility reliability, and broad comparison.

Benchmark	Experimental evidence used in the revision	Analytical purpose
WBD	Eight scenario/case settings; convergence histories; all reported structural constraints feasible; broader 30-algorithm comparison from the WBD file.	Clarifies that WBD evidence is scenario-level and interprets rank honestly rather than claiming universal dominance.
TSD	Eight scenario/case settings; all four constraints feasible; 100-algorithm comparison; SDOA objective 0.012671 ranked 26/100.	Uses the full comparative table and avoids exaggerating small fourth-/fifth-decimal differences.
CSIDP	Ten scenarios with independent-run statistics; best/mean/worst/std/time; normalized constraints and feasibility status.	Strengthens statistical reporting and separates objective quality from feasibility.
SRD	Ten scenarios with best/mean/worst/std/time; full feasibility verification; 34-algorithm comparison.	Addresses robustness, variance, feasibility reliability, and broad comparison.

The empirical study is therefore organized as a single benchmark corpus rather than as unrelated case studies. The corpus combines constrained engineering formulations with standard mathematical functions so that SDOA can be examined from complementary perspectives: feasibility preservation, mixed-variable handling, exploitation accuracy, multimodal exploration, and robustness across attraction-basin structures. All problems are written in the common form $\min f (x)$ , with constraint functions $g_{k} (x) \leq 0$ where applicable.

Parameter studies were not imposed uniformly across benchmarks because the problems differ substantially in scale and difficulty. Instead, each benchmark was tested under scenario sweeps tailored to its search budget and feasibility characteristics.

The experimental evidence consolidated here was produced with a MATLAB implementation on 16-GB workstations. Although the case studies are benchmark-specific, the algorithmic backbone remains constant: oxygen decay, state-based movement operators, elite preservation, feasibility-aware acceptance, and selective reset.

In all reported results, only feasible designs are retained as final solutions. The discussion therefore emphasizes objective quality together with run-to-run dispersion, active-constraint behavior, runtime, and the distinction between the numerically best scenario and the statistically most stable scenario.

5.2 Integrated validation protocol across engineering and mathematical landscapes

The validation protocol is designed as a unified sequence. WBD, TSD, CSIDP, and SRD evaluate SDOA under explicit engineering constraints, while F1-F23 evaluate the same search mechanism under known mathematical landscapes. The two groups are interpreted jointly: the engineering benchmarks test feasibility and practical design relevance, whereas the mathematical functions test whether the oxygen-regulated search policy transfers across smooth, discontinuous, multimodal, penalized, and fixed-dimensional structures.

The mathematical-function component used a fixed 200-iteration budget and one declared parameter setting across all functions. This choice was intended to evaluate the transferability of the search mechanism rather than to optimize each function through case-specific retuning.

The official CEC constrained benchmark suites remain the appropriate next standard for a full competition-style comparison under uniform source code, feasibility tolerance, function-evaluation budget, and paired repeated runs.

5.3 Parameter calibration and sensitivity design

The parameter-calibration protocol separates issues that are often conflated in metaheuristic papers: parameter disclosure, internal sensitivity evidence, fair comparison boundaries, and safeguards against tuning bias (Table 9).

Table 9.
Parameter-calibration protocol and safeguards against tuning bias.

Protocol element	Revision added to the manuscript	Purpose
Parameter disclosure	All SDOA settings are reported for each engineering benchmark and for the 23-function suite.	Prevents hidden or undocumented tuning.
Internal sensitivity analysis	Scenario sweeps are summarized using best, mean, worst, standard deviation, runtime, coefficient of variation, and confidence intervals where repeated runs are available.	Shows whether performance is stable or tied to one favorable configuration.
Fair comparison boundary	Literature-based comparisons are explicitly separated from internally controlled SDOA runs.	Avoids overstating historical comparisons as identical-budget reruns.
Recommended fair tuning protocol	For future direct comparisons, all algorithms should receive the same function-evaluation budget, the same number of independent runs, the same feasibility tolerance, and a pre-declared tuning grid.	Makes future Wilcoxon/Friedman testing statistically valid.
Hold-out validation principle	Parameter settings should be selected on calibration cases and then tested on unseen functions or engineering problems.	Reduces overfitting to a single benchmark.

The present study reports two types of evidence with different inferential strength: internally generated SDOA scenario statistics and literature-based comparator values. These evidence types are interpreted separately to avoid presenting historical comparisons as fully controlled experiments.

5.4 Statistical analysis and computational-efficiency protocol

The statistical protocol distinguishes valid within-study analysis from invalid historical over-comparison. For CSIDP and SRD, the available repeated-run summaries permit confidence intervals, coefficients of variation, best-to-mean gaps, and approximate comparisons between selected SDOA scenarios. For literature algorithms, formal Wilcoxon signed-rank, Friedman, or post-hoc Holm tests are not applied because most comparator studies do not provide the raw paired samples, random seeds, feasibility histories, or stopping conditions required for those tests. Applying such tests to isolated best values would be statistically misleading. The manuscript therefore reports only the analyses justified by the available data and specifies the protocol required for a future fully paired comparison (Table 10, Table A1).

Table 10.
Statistical and computational-efficiency protocol.

Evidence type	Metric reported	Interpretation
Scenario-level repeated runs	Best, mean, worst, standard deviation, runtime, 95% confidence interval, and coefficient of variation.	Assesses robustness and dispersion within SDOA.
Best versus stable settings	Comparison between the numerically best scenario and the scenario with the narrowest dispersion.	Separates peak performance from reproducibility.
Runtime analysis	Wall-clock runtime ranges and mean runtime by benchmark/scenario group.	Clarifies computational cost of the reported SDOA runs.
Literature comparisons	Objective-deviation ratio and feasibility status.	Used as benchmark-positioning evidence only, not as direct runtime-normalized competition.
Future full statistical comparison	Wilcoxon signed-rank, Friedman ranking, and post-hoc Holm adjustment over identical raw repeated runs.	Provides a reproducible route for fully controlled multi-algorithm validation.

6 Results and discussion

The results are presented as an integrated empirical narrative. The engineering cases establish feasibility, mixed-variable handling, and performance near active constraints; the mathematical functions then expose the same SDOA mechanism to standard landscape structures with known optima. Subsequent subsections synthesize comparative standing, parameter sensitivity, computational cost, statistical reliability, and mechanism-level behavior. This organization supports scientific interpretation rather than emphasizing isolated best-run values (Table A2).

The scenario sweeps reveal a consistent pattern across the four engineering problems. SDOA benefits from relatively strong early movement, moderate oxygen decay, and small-to-moderate elite sets when the feasible region is narrow. Longer iteration budgets generally reduce dispersion, but the improvement is most meaningful when additional evaluations are combined with selective reset rather than merely larger populations. This effect is most visible in SRD and CSIDP, where several constraints are active near high-quality solutions.

Table 11 provides a concise cross-benchmark summary of the best reported SDOA settings and outcomes. A common pattern emerges: when the feasible region is narrow, moderate oxygen decay and restrained elite preservation are preferable to overly aggressive exploitation. The table also indicates that the strongest empirical standing of SDOA appears on SRD and CSIDP, while WBD and TSD demonstrate competitiveness and feasibility stability.

Table 11.
Cross-benchmark best settings and outcomes.

Benchmark	Best settings	Best objective	Feasibility evidence
WBD	N = 100, T = 300, c0 = 0.3, μ0 = 0.5, O0 = 150, α = 0.5, e = 3	2.22221	All 8 cases feasible
TSD	N = 200, T = 400, c0 = 0.8, μ0 = 0.5, O0 = 100, α = 0.8, e = 20	0.012671	All 8 cases feasible
CSIDP	N = 500, T = 5000, c0 = 0.9, μ0 = 0.5, O0 = 200, α = 0.3, e = 25	22.842983	All 10 scenarios feasible
SRD	N = 500, T = 2000, c0 = 0.3, μ0 = 0.3, O0 = 200, α = 0.4, e = 20	2992.856695	All 10 scenarios feasible

Figure 4 synthesizes the benchmark-specific performance profiles into one visual comparison. Rather than relying only on isolated best values, the figure shows how the objective changes across cases and scenarios, making it easier to judge whether the algorithm improves consistently as the search budget and parameter settings vary.

Figure 4.

Objective-value trajectories across the benchmark-specific case and scenario studies.

The trends in Figure 4 support three observations developed in the subsequent subsections. First, WBD exhibits a clear reduction in cost as the computational budget increases. Second, TSD remains concentrated within a narrow performance band, indicating stability under multiple settings. Third, CSIDP and SRD show that SDOA repeatedly locates high-quality feasible regions across scenarios rather than depending on a single exceptional run.

6.1 Welded beam design results

For welded beam design, performance improved as the iteration budget increased and the search became less aggressive. The best reported result, 2.22221, was obtained using 100 divers, 300 iterations, current strength 0.3, mutation rate 0.5, initial oxygen 150, oxygen decay 0.5, and elite size 3.

The detailed case-by-case results are reported in Table 12. The progression from Case 1 to Case 8 demonstrates a clear reduction in cost without loss of feasibility. The best configuration converged to $h = 0.205431, l = 7.110996, t = 9.033377, a n d b = 0.206075$ , with constraint values $τ = 13598.95 \leq 13600, σ = 29971.22 \leq 30000, δ = 0.014451 \leq 0.25, a n d P_{c} = 6028.87 \geq 6000$ .

Table 12.
Detailed SDOA case study for welded beam design.

Case N T C0 Mu0 O0 Alpha e Best cost Time (s)

1 100 30 0.5 0.3 100 0.8 10 2.39952 0.1

2 100 30 0.5 0.5 100 0.5 10 2.978323 0.12

3 100 30 0.3 0.5 100 0.5 10 2.90747 0.04

4 100 30 0.5 0.3 150 0.3 10 2.30545 0.04

5 100 30 0.3 0.5 150 0.5 3 2.265897 0.06

6 100 100 0.3 0.5 150 0.5 3 2.231723 0.1

7 100 200 0.3 0.5 150 0.5 3 2.22926 0.32

8 100 300 0.3 0.5 150 0.5 3 2.22221 0.42

Case	N	T	C0	Mu0	O0	Alpha	e	Best cost	Time (s)
1	100	30	0.5	0.3	100	0.8	10	2.39952	0.1
2	100	30	0.5	0.5	100	0.5	10	2.978323	0.12
3	100	30	0.3	0.5	100	0.5	10	2.90747	0.04
4	100	30	0.5	0.3	150	0.3	10	2.30545	0.04
5	100	30	0.3	0.5	150	0.5	3	2.265897	0.06
6	100	100	0.3	0.5	150	0.5	3	2.231723	0.1
7	100	200	0.3	0.5	150	0.5	3	2.22926	0.32
8	100	300	0.3	0.5	150	0.5	3	2.22221	0.42

Table 12 shows a systematic reduction in welded-beam cost as the computational budget increases. This improvement was achieved without loss of feasibility, which is more informative than a single final best value. The table therefore supports the view that oxygen-regulated staging can guide the search from coarse exploration toward fine structural-design refinement.

6.2 Tension/compression spring results

For tension/compression spring design, the strongest configuration used 200 divers, 400 iterations, current strength 0.8, mutation rate 0.5, oxygen level 100, oxygen decay 0.8, and elite size 20. This setting produced a best feasible objective of 0.012671.

The scenario spread on the spring benchmark is extremely small, consistent with the broader literature: many state-of-the-art optimizers cluster tightly in the fourth and fifth decimal places. The practical significance of SDOA on this benchmark therefore lies in reproducibility and feasibility reliability rather than in a large numerical separation from all competitors.

Table 13 reveals a narrow spread of best weights across the spring-design cases. This indicates that SDOA does not depend on a fragile parameter setting to obtain a competitive answer; instead, it repeatedly converges to the same small feasible region under several configurations. For a benchmark where many algorithms differ only in the fourth or fifth decimal place, this stability is important.

Table 13.
Detailed SDOA case study for tension/compression spring design.

Case N T C0 Mu0 O0 Alpha e Best weight Time (s)

1 200 400 0.3 0.5 100 0.8 30 0.012722 2.15

2 100 400 0.3 0.5 100 0.8 30 0.012674 1.03

3 100 400 0.3 0.5 100 0.5 30 0.012689 0.8

4 100 400 0.3 0.5 150 0.5 30 0.012724 0.77

5 100 400 0.5 0.5 100 0.5 30 0.012762 0.84

6 100 400 0.5 0.5 100 0.5 10 0.01272 0.63

7 200 400 0.8 0.5 100 0.8 20 0.012671 1.32

8 200 400 0.8 0.8 100 0.8 20 0.012696 1.15

Case	N	T	C0	Mu0	O0	Alpha	e	Best weight	Time (s)
1	200	400	0.3	0.5	100	0.8	30	0.012722	2.15
2	100	400	0.3	0.5	100	0.8	30	0.012674	1.03
3	100	400	0.3	0.5	100	0.5	30	0.012689	0.8
4	100	400	0.3	0.5	150	0.5	30	0.012724	0.77
5	100	400	0.5	0.5	100	0.5	30	0.012762	0.84
6	100	400	0.5	0.5	100	0.5	10	0.01272	0.63
7	200	400	0.8	0.5	100	0.8	20	0.012671	1.32
8	200	400	0.8	0.8	100	0.8	20	0.012696	1.15

6.3 Car side-impact design results

CSIDP is a stringent test of feasibility management because low structural weight is meaningful only when all crashworthiness constraints are satisfied. The best reported configuration used 500 divers, 5000 iterations, current strength 0.9, mutation rate 0.5, oxygen level 200, oxygen decay 0.3, and elite size 25, producing a best feasible objective value of 22.842983.

Two complementary observations emerge from the CSIDP scenario study. Scenario 7 delivered the best objective value, whereas Scenario 9 produced the lowest standard deviation. The numerically best setting and the statistically most concentrated setting are therefore not identical, a distinction that strengthens the empirical interpretation.

Table 14 documents the CSIDP scenario settings and shows that the reported performance differences are tied to interpretable changes in search intensity, oxygen regulation, and elite retention. The table separates the question of which configuration produced the best objective from the broader question of which configurations maintained statistically controlled behavior over repeated runs.

Table 14.
Scenario settings for car side-impact design.

Scenario N T C0 Mu0 O0 Alpha e

1 500 2000 0.8 0.5 150 0.6 25

2 500 2000 0.5 0.5 150 0.3 25

3 500 2000 0.5 0.5 150 0.4 15

4 500 2000 0.4 0.4 150 0.4 15

5 500 2000 0.4 0.4 150 0.4 30

6 500 2000 0.5 0.5 150 0.3 30

7 500 5000 0.9 0.5 200 0.3 25

8 500 5000 0.5 0.5 200 0.3 25

9 500 5000 0.5 0.5 200 0.5 25

10 500 5000 0.4 0.4 200 0.4 30

Scenario	N	T	C0	Mu0	O0	Alpha	e
1	500	2000	0.8	0.5	150	0.6	25
2	500	2000	0.5	0.5	150	0.3	25
3	500	2000	0.5	0.5	150	0.4	15
4	500	2000	0.4	0.4	150	0.4	15
5	500	2000	0.4	0.4	150	0.4	30
6	500	2000	0.5	0.5	150	0.3	30
7	500	5000	0.9	0.5	200	0.3	25
8	500	5000	0.5	0.5	200	0.3	25
9	500	5000	0.5	0.5	200	0.5	25
10	500	5000	0.4	0.4	200	0.4	30

Table 15 shows that CSIDP should be evaluated through both objective quality and dispersion. Scenario 7 delivered the best value, while Scenario 9 produced an exceptionally small standard deviation and therefore a tighter concentration of repeated-run outcomes. This separation of peak performance from reproducibility is important for constrained design.

Table 15.

Statistical performance of SDOA on car side-impact design.

Scenario	Best	Mean	Worst	Std. dev.	Time (s)
1	22.843728	23.051693	23.282567	0.218856	141.08
2	22.843048	22.930754	23.289461	0.18366	152.61
3	22.84307	22.969158	23.267367	0.2003	166.5
4	22.843077	23.04059	23.275235	0.207717	172.56
5	22.843549	22.882887	23.224619	0.120079	146.41
6	22.843033	23.038161	23.49324	0.257713	147.99
7	22.842983	22.965292	23.26162	0.196327	405.35
8	22.843029	22.885579	23.262802	0.132544	411.69
9	22.843384	22.844077	22.844944	0.000566	406.55
10	22.843163	23.047397	23.375911	0.219812	398.71

The best design vector from Scenario 7 is reported in Appendix A. Across the feasibility tables, all ten crashworthiness constraints remained satisfied, and several active constraints were approached closely without violation.

Figure 5 focuses on dispersion rather than absolute objective value. This figure is important because a constrained optimizer should not be judged only by its single best run; it should also be assessed by how tightly repeated runs cluster around that performance.

Figure 5.

Standard-deviation profiles across the CSIDP and SRD scenario studies.

Figure 5 reinforces the same point. For CSIDP, Scenario 7 produced the best objective value, whereas Scenario 9 yielded the smallest standard deviation. For SRD, Scenario 9 achieved the best objective value, while Scenario 4 provided the narrowest dispersion. The figure therefore separates absolute solution quality from repeated-run stability.

6.4 Speed reducer design results

SRD provides the strongest support in this study for SDOA on narrow feasible-region design. The best configuration used 500 divers, 2000 iterations, current strength 0.3, mutation rate 0.3, oxygen level 200, oxygen decay 0.4, and elite size 20, yielding a best feasible objective value of 2992.856695.

As in CSIDP, the numerically best SRD scenario and the most stable scenario were not the same. Scenario 9 produced the best objective value, whereas Scenario 4 yielded the lowest standard deviation. The results therefore support a nuanced interpretation in which SDOA combines strong solution quality with controlled statistical spread.

Table 15 shows that the SRD study explored a structured range of search budgets and oxygen-state configurations rather than a single arbitrary parameter set. The table helps explain why the SRD results are especially informative: the optimum is close to several active constraints, so differences in current strength, oxygen decay, and elite size visibly affect the balance between accuracy and stability.

Table 16.
Scenario settings for speed reducer design.

Scenario N T C0 Mu0 O0 Alpha e

1 500 500 0.5 0.5 200 0.5 20

2 500 500 0.8 0.5 200 0.5 20

3 500 1000 0.8 0.6 200 0.3 30

4 500 1000 0.5 0.5 200 0.3 30

5 500 1000 0.5 0.5 200 0.5 25

6 500 2000 0.5 0.5 200 0.5 25

7 500 2000 0.5 0.5 200 0.4 25

8 500 2000 0.5 0.5 200 0.4 20

9 500 2000 0.3 0.3 200 0.4 20

10 500 2000 0.3 0.3 150 0.4 20

Scenario	N	T	C0	Mu0	O0	Alpha	e
1	500	500	0.5	0.5	200	0.5	20
2	500	500	0.8	0.5	200	0.5	20
3	500	1000	0.8	0.6	200	0.3	30
4	500	1000	0.5	0.5	200	0.3	30
5	500	1000	0.5	0.5	200	0.5	25
6	500	2000	0.5	0.5	200	0.5	25
7	500	2000	0.5	0.5	200	0.4	25
8	500	2000	0.5	0.5	200	0.4	20
9	500	2000	0.3	0.3	200	0.4	20
10	500	2000	0.3	0.3	150	0.4	20

Table 17 confirms that SDOA remained statistically well behaved on SRD across all ten scenarios. Scenario 9 achieved the best objective value, while Scenario 4 showed the smallest standard deviation. This difference between best and most stable settings is consistent with careful benchmark analysis and should not be collapsed into a single ranking claim.

Table 17.

Statistical performance of SDOA on speed reducer design.

Scenario	Best	Mean	Worst	Std. dev.	Time (s)
1	2992.9134	2993.1066	2993.6292	0.1969	33.77
2	2992.9729	2993.0905	2993.2088	0.0683	33.65
3	2992.9145	2993.0035	2993.1438	0.0683	73.98
4	2992.9169	2992.9808	2993.1066	0.0551	75.16
5	2992.8919	2993.0195	2993.1569	0.0936	66.86
6	2992.8891	2992.9954	2993.1049	0.0596	139.83
7	2992.8833	2992.9929	2993.1495	0.0822	139.48
8	2992.8626	2992.9953	2993.1279	0.0918	138.44
9	2992.8567	2993.0097	2993.1189	0.0762	139.09
10	2992.9108	2992.9962	2993.1200	0.0829	147.32

6.5 Comparative interpretation

The comparative set is organized by benchmark rather than by a single universal list of algorithms, because the available literature differs across WBD, TSD, CSIDP, and SRD. Table 18 states the baseline-selection logic and the limitation of each comparison layer. This framing makes clear that the manuscript claims benchmark positioning, not a fully controlled rerun of every modern optimizer under identical code and hardware.

Table 18.
Baseline-selection rationale and comparison limits.

Benchmark layer Baselines used Reason for inclusion Limitation

WBD Classical and modern methods from the welded-beam comparison set Long-standing structural benchmark with many reported feasible designs Literature values may differ in stopping budget and constraint tolerance

TSD Dense spring-design comparison set including 100 reported algorithms Provides a large ranking context where small fourth-/fifth-decimal differences are common Not all algorithms provide raw repeated-run samples

CSIDP PSO, DE, GA, FA, and SDOA with feasibility status Captures mixed-variable crashworthiness comparison where infeasible solutions must be separated Comparator set is narrower than WBD/TSD but includes feasibility labels

SRD Speed-reducer comparison set from the associated study Tests narrow feasible-region mechanical design with integer variable Runtime and seeds for all historical comparators are not uniformly available

23-function suite Internal SDOA evaluation under fixed parameter setting Tests general search dynamics without function-specific tuning Not a full official CEC constrained competition and not a multi-algorithm rerun

Benchmark layer	Baselines used	Reason for inclusion	Limitation
WBD	Classical and modern methods from the welded-beam comparison set	Long-standing structural benchmark with many reported feasible designs	Literature values may differ in stopping budget and constraint tolerance
TSD	Dense spring-design comparison set including 100 reported algorithms	Provides a large ranking context where small fourth-/fifth-decimal differences are common	Not all algorithms provide raw repeated-run samples
CSIDP	PSO, DE, GA, FA, and SDOA with feasibility status	Captures mixed-variable crashworthiness comparison where infeasible solutions must be separated	Comparator set is narrower than WBD/TSD but includes feasibility labels
SRD	Speed-reducer comparison set from the associated study	Tests narrow feasible-region mechanical design with integer variable	Runtime and seeds for all historical comparators are not uniformly available
23-function suite	Internal SDOA evaluation under fixed parameter setting	Tests general search dynamics without function-specific tuning	Not a full official CEC constrained competition and not a multi-algorithm rerun

The comparative study uses the problem-specific information available in the associated benchmark studies. WBD is compared against the algorithms listed in the welded-beam study, TSD against the spring-design comparison set, CSIDP against the established PSO/DE/GA/FA comparison with explicit feasibility status, and SRD against the speed-reducer comparison set. These are literature-based reference comparisons rather than perfectly normalized head-to-head reruns on identical hardware; therefore, the interpretation emphasizes objective deviation, feasibility reliability, and practical standing within each benchmark.

Taken together, the results suggest that the main value of SDOA is methodological rather than a claim of universal first-place ranking. The algorithm delivers high-quality feasible solutions with stable behavior across structurally different constrained benchmarks. This distinction matters because a metaheuristic may report an impressive best value while failing to reproduce it or while approaching it through weak feasibility.

A dedicated comparative study is necessary because strong single-run values alone do not establish methodological competitiveness. Consolidating these results in one place makes it possible to evaluate SDOA against the best reported methods on each benchmark while preserving a fair distinction among objective value, feasibility, and statistical reliability.

For minimization problems, the normalized objective-deviation ratio of algorithm j is defined as

O D_{j} = \frac{f_{j}^{b e s t}}{f_{r e f}},

(27)

where

f_{j}^{best}

is the best feasible objective obtained by algorithm j and

f_{ref}^{best}

is the best feasible objective in the comparison set for the same benchmark. A value of 1.000000 indicates parity with the best reported result, whereas larger values indicate deterioration relative to the leading method.

Table 19 sharpens the interpretation of the preceding results. For welded beam design, SDOA should be characterized as competitive rather than dominant, since harmony-search-based variants still report lower objective values in the broader literature. At the same time, SDOA remains aligned with several contemporary constrained optimizers, placing it within the modern competitive range.

Table 19.

Consolidated comparative standing of SDOA across the four benchmark families.

Benchmark	Comparison basis	SDOA best	Standing	Key finding
WBD	30 algorithms reported in the welded-beam study	2.22221	Competitive	Competitive against a wide range of methods; remains close to several modern metaheuristics although harmony-search-based variants report lower objective values.
TSD	Dense spring-design comparison set reported in the TSD study	0.01267	Competitive within a tightly clustered objective band	The obtained value lies very close to a large cluster of high-performing spring-design optimizers.
CSIDP	PSO, DE, GA, FA, and SDOA comparison with feasibility status	22.842983	Near-best feasible group	SDOA attains a near-best feasible objective and compares favorably with the strongest feasible methods.
SRD	Algorithms reported in the speed-reducer comparison set	2992.856695	Best reported in the comparison set	SDOA yields the strongest objective among the reported SRD comparators.

The spring benchmark leads to a different interpretation. In the comparison set of one hundred algorithms, SDOA, with an objective value of 0.012671, lies within the leading performance cluster. The best-reported method improves on SDOA by only about 0.55%, which is small relative to the scale and saturation of this benchmark. This supports the conclusion that SDOA is competitive on compact constrained mechanical design even when it is not the numerical best.

The comparative outcome is most favorable for CSIDP and SRD. In CSIDP, SDOA reaches a near-best feasible objective while preserving constraint satisfaction, whereas some literature methods approach similar raw objective levels only with feasibility concerns. In SRD, SDOA obtains the best objective value within the reported 34-algorithm comparison set. Because SRD is a mature benchmark with small performance margins, this result should be interpreted as meaningful within the available comparison scope, not as evidence of universal superiority over all possible implementations.

Overall, the comparative study supports a balanced assessment. SDOA is not uniformly ranked first on every benchmark, but it is consistently positioned near the leading group and shows its clearest advantage when feasibility reliability and narrow feasible regions are central to the optimization task.

6.5.1 Detailed comparative tables

To make the benchmark-by-benchmark comparison explicit, Tables 20–23 report SDOA against ten other algorithms for each engineering problem. For welded beam design, tension/compression spring design, and speed reducer design, the comparator values are reproduced from the comparative study of Verma and Parouha.³⁵ For the car side-impact design problem, the comparator values are reproduced from the mixed-variable benchmark compilation of Bayzidi et al..³⁶ The SDOA row in each table comes from the present study. Because historical studies may differ in stopping budgets, constraint tolerances, and benchmark variants, the tabulated literature values should be interpreted as reported reference points rather than perfectly normalized head-to-head experiments.

Table 20.
Comparative results for the welded beam design problem (10 literature algorithms plus SDOA).

Rank Algorithm Best objective OD Source

1 HS 1.724800 1.000000 Mahdavi et al.

2 HS-SQP 1.724800 1.000000 Fesanghary et al.

3 FA 1.731200 1.003711 Yang

4 GA 1.748300 1.013624 Coello

5 WEOA 2.218200 1.286062 Ahmed

6 BB-BC 2.218600 1.286294 Almufti

7 ABC 2.218700 1.286352 Almufti

8 SDOA 2.222210 1.288387 This study

9 SA-GA 2.250000 1.304499 Hwang and He

10 BFO 2.339800 1.356563 Montes and Ocana

Rank	Algorithm	Best objective	OD	Source
1	HS	1.724800	1.000000	Mahdavi et al.
2	HS-SQP	1.724800	1.000000	Fesanghary et al.
3	FA	1.731200	1.003711	Yang
4	GA	1.748300	1.013624	Coello
5	WEOA	2.218200	1.286062	Ahmed
6	BB-BC	2.218600	1.286294	Almufti
7	ABC	2.218700	1.286352	Almufti
8	SDOA	2.222210	1.288387	This study
9	SA-GA	2.250000	1.304499	Hwang and He
10	BFO	2.339800	1.356563	Montes and Ocana

Table 21.

Comparative results for the tension/compression spring design problem (10 literature algorithms plus SDOA).

Order	Algorithm	Best objective	OD	Source
1	KOA	0.0126019	1.000000	Literature
2	CCO	0.01266345	1.004884	Literature
3	RBKA	0.01266500	1.005007	Literature
4	PDWOA	0.01266500	1.005007	Literature
5	QPSO	0.01266500	1.005007	Literature
6	EHO	0.01266500	1.005007	Literature
7	CODEQ	0.012665238	1.005026	Literature
8	BSA	0.01266900	1.005325	Literature
9	SDOA	0.01267000	1.005404	This study
10	SFOA	0.01267000	1.005404	Literature
11	PSO	0.01267000	1.005404	Literature
12	BAT	0.01267000	1.005404	Literature

Table 22.

Comparative results for the car side-impact design problem (10 literature algorithms plus SDOA).

Order	Algorithm	Best objective	Feasibility	Observation
1	DE	22.842980	Feasible	Near-best feasible objective
2	SDOA	22.842983	Feasible	Near-best feasible objective; this study
3	PSO	22.844740	Feasible	Competitive feasible solution
4	GA	22.856530	Infeasible	Competitive raw objective but violates constraints
5	FA	22.842980	Infeasible	Competitive raw objective but violates constraints

Table 23.

Comparative results for the speed reducer design problem (10 literature algorithms plus SDOA).

Rank	Algorithm	Best objective	OD	Source
1	SDOA	2992.856695	1.000000	This study
2	SGO	2994.381000	1.000509	Literature
3	SGO-DE	2994.381000	1.000509	Literature
4	DEDS	2994.471000	1.000539	Literature
5	DELC	2994.471000	1.000539	Literature
6	WCA	2994.471000	1.000539	Literature
7	LCA	2994.471000	1.000539	Literature
8	IAPSO	2994.471000	1.000539	Literature
9	MHS-PCLS	2994.471000	1.000539	Literature
10	I-ABC	2994.471000	1.000539	Literature

Table 20 shows that welded beam design remains the most difficult benchmark for SDOA in the present manuscript. Relative to the stronger classical and hybrid baselines reported in,³⁵ SDOA does not approach the leading minimum cost. This result should therefore be interpreted as evidence that SDOA is feasible and stable on WBD, but not yet competitive with the most specialized optimizers reported for this benchmark.

Table 21 leads to a more favorable interpretation. Although SDOA is not the numerical best method in the broader spring-design comparison, its best objective value differs only slightly from the elite group of algorithms. The OD value remains close to one, confirming that SDOA belongs to the leading performance cluster for the TSD benchmark.

Table 22 shows that SDOA is effectively tied with the strongest CSIDP methods. Its best objective value is numerically almost identical to the best literature values reported in,³⁶ and the difference is negligible at the practical scale of the problem. This supports the claim that SDOA is highly competitive on narrow, mixed-variable crashworthiness search spaces.

Table 23 indicates that SDOA performs strongly on the speed reducer problem, ranking behind only the best two methods in the classical comparison source.³⁵ This broader historical view should be read together with the modern SRD comparison set, where SDOA ranked first among the methods compiled in that manuscript. Taken together, the two views suggest that SDOA is highly competitive on SRD, even when the exact ordering varies with the comparison set and benchmark formulation.

Overall, the detailed comparative tables clarify the position of SDOA. The method is weakest on WBD, close to the state of the art on TSD, effectively tied with the strongest feasible CSIDP solvers, and strongly competitive on SRD. This profile is consistent with the central argument of the paper: SDOA's main strength lies in state-regulated feasibility preservation and constrained-search stability rather than universal first-place performance on every engineering benchmark.

6.5.2 Integrated comparative interpretation across the four benchmark problems

The comparative interpretation draws directly from the four benchmark-specific studies. For WBD, SDOA achieves 2.22221 and remains competitive relative to several classical and modern methods, although it does not surpass the best harmony-search-based variants. For TSD, the obtained value of 0.01267 lies within a very narrow objective band populated by many strong optimizers. For CSIDP, SDOA attains a near-best feasible objective and compares favorably with PSO and DE while preserving constraint satisfaction. For SRD, SDOA yields the best objective within the reported comparison set, highlighting the value of state-regulated search on narrow feasible-region problems.

Taken together, these results support a cautious conclusion: SDOA is not uniformly dominant on every constrained design problem, yet it is consistently competitive and particularly effective when feasibility preservation and controlled refinement are decisive. This pattern is consistent with the algorithm design, in which oxygen-regulated state transitions couple diversification, intensification, and recovery within one coherent framework.

6.6 Landscape-level robustness within the integrated validation corpus

The mathematical-function component broadens the empirical basis of the paper beyond specialized engineering geometries while remaining part of the same validation corpus. Table 24 summarizes the diagnostic categories used to interpret F1-F23 alongside the constrained engineering problems rather than as a detached experiment.

Table 24.
Diagnostic role of the mathematical-function component within the integrated validation corpus.

Category Functions Dimension Diagnostic role

Unimodal F1-F7 10 Exploitation accuracy, conditioning sensitivity, and convergence speed near a single optimum.

Multimodal F8-F13 10 Exploration ability, avoidance of premature stagnation, and recovery from local optima.

Fixed-dimensional F14-F23 2–6 Precision on low-dimensional but highly structured landscapes with localized minima.

Category	Functions	Dimension	Diagnostic role
Unimodal	F1-F7	10	Exploitation accuracy, conditioning sensitivity, and convergence speed near a single optimum.
Multimodal	F8-F13	10	Exploration ability, avoidance of premature stagnation, and recovery from local optima.
Fixed-dimensional	F14-F23	2–6	Precision on low-dimensional but highly structured landscapes with localized minima.

Table 25 reports the aggregate performance profile obtained from the mathematical-function component of the validation corpus. Table 26 extends this evidence by comparing the SDOA best values with representative metaheuristic algorithms on the same 23-function suite.

Table 25.

Aggregate SDOA outcomes on the mathematical-function component of the validation corpus.

Category	No. of functions	Success cases	Mean improvement (%)	Median convergence iteration	Mean runtime (s)
Unimodal	7	2	99.01	47	15.84
Multimodal	6	5	98.98	44	40.85
Fixed-Dim	10	7	30.24	24	67.24
All functions	23	14	69.10	31	44.71

Table 26.

Comparative results of SDOA and representative algorithms on the 23 mathematical benchmark functions.

Function	Benchmark	SDOA	PSO	DE	GWO	WOA	GA	Best method
F1	Sphere	7.00E-04	6.40E-02	6.40E-03	1.93E-13	3.33E-12	6.05E-02	GWO
F2	Schwefel 2.22	4.39E-03	4.19E-02	4.04E-02	6.67E-09	2.36E-08	6.13E-02	GWO
F3	Schwefel 1.2	1.06E-02	8.2634	2.3452	6.01E-05	4.37E-06	9.2928	WOA
F4	Schwefel 2.21	3.03E-02	8.75E-01	5.86E-01	7.70E-05	4.14E-03	2.28E-01	GWO
F5	Rosenbrock	4.23E-02	9.9471	13.1501	6.5678	7.6400	9.4487	SDOA
F6	Step	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	Tie
F7	Quartic with Noise	1.18E-03	1.99E-03	1.34E-02	3.29E-02	7.10E-04	2.42E-02	WOA
F8	Schwefel	1.75E-03	908.2633	1163.2746	2600.6965	1100.7091	1085.9503	SDOA
F9	Rastrigin	2.63E-04	14.2783	29.5953	2.17E-13	6.0862	4.0253	GWO
F10	Ackley	8.06E-03	1.71E-01	1.01E-01	8.19E-08	8.48E-07	1.37E-01	GWO
F11	Griewank	5.83E-02	5.69E-01	4.83E-01	3.62E-11	1.40E-02	3.53E-01	GWO
F12	Penalized 1	2.10E-05	6.83E-04	3.78E-04	3.40E-05	1.50E-02	2.41E-04	SDOA
F13	Penalized 2	5.11E-04	1.19E-02	4.16E-03	2.69E-05	1.14E-02	3.78E-03	GWO
F14	Foxholes	0.998004	0.998004	0.998004	11.718700	0.998004	0.998004	Tie
F15	Kowalik	3.08E-04	6.19E-04	3.21E-04	7.35E-04	5.59E-04	5.76E-04	SDOA
F16	Six-hump Camel	−1.031628	−1.031628	−1.031628	−1.031627	−1.031628	−1.031628	Tie
F17	Branin	0.397887	0.397887	0.397887	0.397915	0.397888	0.397890	Tie
F18	Goldstein-Price	3.000000	3.000000	3.000000	3.000128	3.000000	3.000007	Tie
F19	Hartman 3	−3.862782	−3.862782	−3.862782	−3.862508	−3.862779	−3.862782	Tie
F20	Hartman 6	−3.203162	−3.322368	−3.322368	−3.322140	−3.320362	−3.322356	DE
F21	Shekel 5	−10.153199	−10.153200	−10.153200	−10.150998	−10.150227	−10.152864	DE
F22	Shekel 7	−10.402941	−10.402940	−10.402941	−10.392423	−10.402445	−10.401988	SDOA/DE
F23	Shekel 10	−10.536410	−3.835427	−10.536410	−10.525285	−10.534895	−10.535199	SDOA/DE

Table 26 compares SDOA with five representative metaheuristic algorithms: particle swarm optimization (PSO), differential evolution (DE), grey wolf optimizer (GWO), whale optimization algorithm (WOA), and genetic algorithm (GA). The table provides a clearer empirical context for the mathematical benchmark suite by showing that SDOA obtains the best or tied-best result on several functions, including Rosenbrock, Schwefel, Penalized Function 1, Kowalik, Six-hump Camel, Branin, Goldstein-Price, Hartman 3, Shekel 7, and Shekel 10.

The comparison also indicates that SDOA is not uniformly superior across all mathematical landscapes. GWO reports stronger values on several unimodal and multimodal functions, including Sphere, Schwefel 2.22, Rastrigin, Ackley, and Griewank, whereas DE is stronger on Hartman 6 and Shekel 5. This pattern supports a balanced interpretation: SDOA is competitive and robust, but its main contribution remains the state-regulated coordination of exploration, exploitation, elite communication, and diversity recovery rather than universal numerical dominance on every function.

Because comparator values may originate from implementations with different stopping criteria, random seeds, and boundary-handling rules, the table is used as a contextual benchmark comparison. A fully controlled statistical comparison would require rerunning all algorithms under the same population size, iteration budget, number of independent trials, stopping rule, and feasibility or boundary-handling protocol.

The detailed SDOA-only function-level results are provided in Appendix A to preserve readability in the main text. Several values approach the known global-optimum level of the corresponding benchmark functions, including Sphere, Step, Rastrigin, Ackley, penalized functions, Goldstein-Price, Hartman, and Shekel functions. The additional comparison in Table 26 complements these internal results by positioning SDOA relative to representative metaheuristic baselines.Figure 6).

Figure 6.

Representative convergence behavior of SDOA on selected mathematical functions within the integrated validation corpus.

Taken together, the engineering-design and mathematical benchmark results support a bounded but stronger interpretation of SDOA. The engineering problems demonstrate feasibility-aware constrained performance, while the 23-function suite provides broader evidence that the oxygen-regulated state mechanism remains effective across heterogeneous mathematical landscapes. The claim is therefore deliberately limited: SDOA is a competitive and robust state-regulated optimizer, but full generality across official CEC constrained competitions should be established through future paired multi-algorithm experiments.

6.7 Parameter sensitivity and computational efficiency

Tables 26 and 27 reorganize the empirical evidence around sensitivity and computational cost. The aim is to make the effect of SDOA's own parameters visible rather than to rely on one best configuration. Population size and iteration count primarily influence computational budget, while current strength, mutation rate, oxygen level, oxygen decay, and elite size influence behavioral regulation. The results show that longer budgets generally reduce dispersion, but the strongest improvements occur when additional evaluations are accompanied by suitable oxygen decay and reset pressure rather than by larger populations alone.

Table 27.
Scenario-level sensitivity and runtime summary for SDOA.

Benchmark Sensitivity evidence Best / dispersion outcome Runtime evidence Interpretation

WBD 8 cases varying T, c0, μ0, O0, α, and e. Best cost 2.22221; case range 2.22221–2.978323. 0.04–0.42 s; mean 0.15 s. Increasing T from 30 to 300 steadily improved the best feasible cost; the best case used small elite size and moderate oxygen decay.

TSD 8 cases with fixed T = 400 and varied N, c0, μ0, O0, α, and e. Best 0.012671; range only 0.012671–0.012762; CV ≈ 0.24%. 0.63–2.15 s; mean 1.09 s. Performance is insensitive over the tested settings, suggesting a stable feasible basin rather than one fragile parameter point.

CSIDP 10 scenarios with repeated-run statistics. Best 22.842983; best-value range 0.000745; lowest std. dev. 0.000566. 141.08–411.69 s; mean 254.95 s. Longer runs improve best-case quality, while Scenario 9 gives the strongest reproducibility.

SRD 10 scenarios with repeated-run statistics. Best 2992.8567; best-value range 0.1162; lowest std. dev. 0.0551. 33.65–147.32 s; mean 98.76 s. The best objective and the narrowest dispersion occur in different scenarios, so quality and stability must be judged together.

23-function suite 23 functions under one fixed setting: N = 150, T = 200, c0 = 0.5, μ0 = 0.4, O0 = 100, α=0.2, e = 20. 14/23 success cases; mean improvement 69.10%. 0.53–79.57 s; mean 44.71 s. The fixed setting supports general search behavior but does not replace repeated-run significance testing.

Benchmark	Sensitivity evidence	Best / dispersion outcome	Runtime evidence	Interpretation
WBD	8 cases varying T, c0, μ0, O0, α, and e.	Best cost 2.22221; case range 2.22221–2.978323.	0.04–0.42 s; mean 0.15 s.	Increasing T from 30 to 300 steadily improved the best feasible cost; the best case used small elite size and moderate oxygen decay.
TSD	8 cases with fixed T = 400 and varied N, c0, μ0, O0, α, and e.	Best 0.012671; range only 0.012671–0.012762; CV ≈ 0.24%.	0.63–2.15 s; mean 1.09 s.	Performance is insensitive over the tested settings, suggesting a stable feasible basin rather than one fragile parameter point.
CSIDP	10 scenarios with repeated-run statistics.	Best 22.842983; best-value range 0.000745; lowest std. dev. 0.000566.	141.08–411.69 s; mean 254.95 s.	Longer runs improve best-case quality, while Scenario 9 gives the strongest reproducibility.
SRD	10 scenarios with repeated-run statistics.	Best 2992.8567; best-value range 0.1162; lowest std. dev. 0.0551.	33.65–147.32 s; mean 98.76 s.	The best objective and the narrowest dispersion occur in different scenarios, so quality and stability must be judged together.
23-function suite	23 functions under one fixed setting: N = 150, T = 200, c0 = 0.5, μ0 = 0.4, O0 = 100, α=0.2, e = 20.	14/23 success cases; mean improvement 69.10%.	0.53–79.57 s; mean 44.71 s.	The fixed setting supports general search behavior but does not replace repeated-run significance testing.

Table 28.

Empirical SDOA runtime summary across the evaluated benchmark groups.

Benchmark group	Mean runtime	Minimum runtime	Maximum runtime	Comment
WBD	0.15 s	0.04 s	0.42 s	Small continuous problem; runtime dominated by iteration budget.
TSD	1.09 s	0.63 s	2.15 s	Compact problem; objective values remain tightly clustered.
CSIDP	254.95 s	141.08 s	411.69 s	Mixed-variable crashworthiness constraints substantially increase cost.
SRD	98.76 s	33.65 s	147.32 s	Runtime increases with T; feasibility remains stable across scenarios.
23-function suite	44.71 s	0.53 s	79.57 s	Times vary by function complexity and convergence iteration.

The runtime evidence clarifies the practical cost of SDOA. The compact engineering benchmarks are inexpensive, whereas CSIDP and SRD require longer runs because they involve narrower feasible regions and more demanding constraint evaluation. Runtime is therefore treated as an evaluation criterion rather than as a secondary implementation detail.

A complete empirical runtime comparison with all competing methods is not claimed because the available historical comparator studies do not uniformly report hardware, implementation language, stopping tolerance, random seeds, or raw execution logs. The manuscript therefore reports SDOA runtime transparently and defines the conditions required for a fair runtime comparison: identical programming environment, identical function-evaluation budget, identical feasibility tolerance, and the same number of independent runs.

6.8 Statistical reliability and confidence-interval analysis

Where repeated-run statistics are available, the manuscript reports uncertainty rather than relying only on the minimum objective value. For each selected scenario, the 95% confidence interval is computed as mean ± t0.975,9 x SD/sqrt(10), because each scenario contains ten independent runs. These intervals should be interpreted as within-SDOA robustness indicators, not as definitive multi-algorithm superiority tests (Table 29, Table A3).

Table 29.
Confidence intervals and coefficient-of-variation indicators for selected repeated-run scenarios.

Benchmark	Scenario	Role	Mean	Std. dev.	95% CI of mean	CV (%)
CSIDP	7	Best objective scenario	22.965292	0.196327	22.824858–23.105726	0.8549
CSIDP	9	Most stable scenario	22.844077	0.000566	22.843672–22.844482	0.0025
SRD	9	Best objective scenario	2993.0097	0.0762	2992.955194–2993.064206	0.0025
SRD	4	Most stable scenario	2992.9808	0.0551	2992.941387–2993.020213	0.0018

(Table 30).

Table 30.

Approximate statistical comparisons and limits of significance testing.

Comparison	Test / basis	Statistic	p-value	Interpretation
CSIDP Scenario 7 vs Scenario 9	Approximate Welch comparison from summary statistics	t = 1.952, df ≈ 9.00	0.0826	Scenario 9 is clearly more stable; the mean difference is not significant at 0.05 under this approximate test.
SRD Scenario 9 vs Scenario 4	Approximate Welch comparison from summary statistics	t = 0.972, df ≈ 16.39	0.3452	The best-objective and most-stable scenarios are statistically close in mean objective value.
23-function suite	No formal test applied	One outcome per function	Not applicable	The suite broadens landscape coverage but does not provide repeated-run significance.
Literature algorithms	Wilcoxon/Friedman not applied	Raw paired samples unavailable	Not applicable	Avoids invalid statistical claims from non-paired historical best values.

This statistical treatment strengthens the interpretation in two ways. First, it shows that SDOA's best values on CSIDP and SRD are accompanied by narrow dispersion under several settings. Second, it prevents overclaiming: historical best-value tables are retained as contextual references, while formal significance testing is reserved for datasets that satisfy the assumptions of paired repeated-run analysis.

6.9 Search-dynamics, ablation interpretation, and constraint-handling behavior

The mechanism-level analysis clarifies the role of the oxygen-based design. A complete statistical ablation would require rerunning SDOA after disabling individual operators under identical random seeds across all benchmarks; those full ablation reruns are not available for every problem in the present dataset. Nevertheless, the convergence histories, depth-stage logic, feasibility records, and scenario sweeps allow a meaningful qualitative ablation interpretation, summarized in Table 31.

Table 31.
Mechanism-level ablation interpretation of the oxygen-regulated SDOA design.

SDOA component	Mechanistic role	Expected effect if removed	Empirical evidence supporting the role
Oxygen state	Coordinates stage transition, movement amplitude, local refinement, and reset pressure.	Search would collapse into a conventional schedule with weaker interpretability and less adaptive staging.	The 23-function diagnostics show early D1/D2 dominance followed by D3/D4 refinement.
D1 heavy-tailed exploration	Provides long-range jumps before contraction.	Higher risk of missing distant basins on multimodal or fixed-dimensional functions.	Multimodal functions show strong success under the fixed 23-function setting.
D3 elite-guided local refinement	Improves feasible candidates around elite regions.	Slower final convergence near active constraints.	CSIDP and SRD approach tight feasible boundaries with low dispersion.
D4 non-uniform mutation	Fine tunes near constraint-active optima.	Late-stage oscillation or premature freezing may increase.	SRD scenarios show small best-to-mean gaps and stable final objective values.
Reset mechanism	Restores diversity when oxygen is exhausted or local correction is unproductive.	Higher stagnation risk in narrow or disconnected feasible regions.	The best CSIDP/SRD outcomes occur under settings that preserve diversity while retaining elite solutions.
Feasibility-aware acceptance	Prevents infeasible low-objective candidates from dominating.	Raw objective improvements could be misleading under constraint violation.	All final engineering designs are reported only after feasibility verification.

This analysis explains why the oxygen mechanism is more than a metaphor. Oxygen acts as a state variable that couples diversification, intensification, and recovery. The most important implication is asynchronous stage occupancy: different divers can search globally, refine locally, or reset at the same iteration. This produces a population-level mixture of transition kernels, which differs from a single population-wide coefficient schedule. Constraint handling also benefits because infeasible or stagnant agents can be redirected without forcing the entire population to abandon promising feasible regions.

6.10 Synthesis of empirical evidence

The results are interpreted through six complementary lenses: best feasible objective value, feasibility verification, dispersion and confidence intervals, parameter sensitivity, runtime cost, and mechanism-level behavior. This organization prevents the discussion from depending primarily on isolated best outcomes. On WBD and TSD, SDOA should be read as feasible and competitive. On CSIDP, the distinction between Scenario 7 and Scenario 9 separates peak performance from repeated-run stability. On SRD, the method shows its strongest constrained-design behavior, while the manuscript still avoids universal superiority claims because several comparisons are literature-derived. On the 23-function suite, the fixed setting across all functions reduces the risk of problem-by-problem parameter overfitting and identifies opportunities for future adaptive threshold learning (Figure A1, Table A4).

6.11 Limitations and threats to validity

Several threats to validity should be considered when interpreting the results. First, not all comparator algorithms were rerun under identical computational budgets, software environments, random-seed protocols, stopping criteria, and feasibility tolerances. Consequently, literature-based comparison tables should be read as benchmark-positioning evidence rather than as fully controlled head-to-head statistical tests.

Second, WBD and TSD are supported mainly by scenario-level evidence, whereas CSIDP and SRD include repeated-run summaries. This difference in evidence strength is reported explicitly to avoid treating all benchmark layers as statistically equivalent. Third, the 23-function suite provides useful landscape-diversity evidence but does not replace an official CEC constrained-optimization competition. Future work should therefore evaluate SDOA using paired repeated runs, identical function-evaluation budgets, common feasibility tolerances, and formal non-parametric tests such as Wilcoxon signed-rank and Friedman procedures with appropriate post-hoc correction.

7 Conclusion

This study reformulated the Scuba Diver Optimization Algorithm as an oxygen-state-regulated metaheuristic for constrained and continuous optimization. The principal contribution is the use of oxygen as an agent-level state variable that coordinates exploration, exploitation, elite communication, local refinement, and reset. This design provides an interpretable alternative to purely iteration-driven schedules and supports asynchronous stage occupancy within the same population.

The integrated validation corpus combines four constrained engineering benchmarks with twenty-three mathematical functions. The engineering results show that SDOA is feasible and competitive on WBD and TSD, near-best among feasible methods on CSIDP, and strongest on SRD within the reported comparison set. The mathematical-function component further indicates that the same search mechanism can operate across heterogeneous landscapes without function-specific retuning.

The empirical evidence is intentionally interpreted with caution. The study reports repeated-run dispersion, confidence intervals, runtime diagnostics, and sensitivity evidence where available, but it does not claim a fully controlled multi-algorithm competition against every recent optimizer. Literature-based comparisons are therefore used for benchmark positioning, whereas formal statistical superiority claims are reserved for settings with paired repeated-run samples.

The results suggest that SDOA is most valuable in constrained problems where feasibility recovery, diversity preservation, and controlled local refinement must be coordinated. Its strongest contribution is not a claim of universal dominance, but a transparent search-governance architecture that links diversification, intensification, and reset through a single internal state.

Future research should extend SDOA to official CEC constrained benchmark suites, simulation-driven engineering models, surrogate-assisted optimization, multi-objective formulations, and discrete or categorical decision spaces. Additional work should also examine adaptive learning of oxygen thresholds, operator-level ablation under paired random seeds, and fully controlled comparisons with recent adaptive and hybrid metaheuristics.

Footnotes

Ethics approval

Not applicable. The study involves benchmark optimization problems only and does not include human participants, animals, or identifiable personal data.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Author contributions

Saman M. Almufti conceived the study, developed the algorithmic framework, implemented the computational models, and conducted the experiments. Amira Bibo Sallow contributed to benchmark formulation, validation, analysis of results, and manuscript preparation. Both authors approved the final manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of data and materials

All data analyzed in this manuscript are contained in the manuscript tables and in the associated MATLAB outputs used to prepare the benchmark studies. The benchmark-specific implementations have also been made available by the authors through MATLAB Central File Exchange in the underlying case-study materials.

Supplemental material

Supplemental material for this article is available online.

Appendix A. Reproducibility Evidence

This appendix provides compact supporting evidence for reproducibility without duplicating all raw experimental files. It reports the best parameter settings, best feasible design vectors, statistical evidence where repeated runs are available, representative feasibility checks for the four engineering problems, and function-level results for the 23-function mathematical benchmark suite.

The full convergence histories, scenario-specific plots, and feasibility tables remain in the supporting files. The appendix includes only the representative evidence needed to make the main manuscript reproducible and self-contained.

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Scuba Diver Optimization Algorithm: An oxygen-state-regulated metaheuristic with integrated validation on constrained engineering and mathematical benchmarks

Abstract

Keywords

1 Introduction

2 Related work and research gap

2.1 Positioning relative to adaptive, schedule-based, and feasibility-preserving metaheuristics

3 Scuba diver optimization algorithm

3.1 State representation and initialization

3.6 Methodological distinctiveness, complexity, and reproducibility

5.1 Experimental protocol, reproducibility, and statistical validation

5.3 Parameter calibration and sensitivity design

Table 9. Parameter-calibration protocol and safeguards against tuning bias.

Table 10. Statistical and computational-efficiency protocol.

Table 11. Cross-benchmark best settings and outcomes.

6.6 Landscape-level robustness within the integrated validation corpus

Table 29. Confidence intervals and coefficient-of-variation indicators for selected repeated-run scenarios.

Table 31. Mechanism-level ablation interpretation of the oxygen-regulated SDOA design.

6.11 Limitations and threats to validity

7 Conclusion

Footnotes

Ethics approval

Consent to participate

Consent for publication

Author contributions

Funding

Declaration of conflicting interests

Availability of data and materials

Supplemental material

Appendix A. Reproducibility Evidence

References

Table 9.
Parameter-calibration protocol and safeguards against tuning bias.

Table 10.
Statistical and computational-efficiency protocol.

Table 11.
Cross-benchmark best settings and outcomes.

Table 29.
Confidence intervals and coefficient-of-variation indicators for selected repeated-run scenarios.

Table 31.
Mechanism-level ablation interpretation of the oxygen-regulated SDOA design.