optimization, Feature selection, Engineering design, Benchmarking, reproducibility.

Optimization has become a central computational requirement in engineering, computer science, operations research, artificial intelligence, energy planning, and biomedical analysis. Modern problems are rarely smooth, convex, or low-dimensional; they are frequently nonlinear, constrained, multimodal, noisy, expensive to evaluate, and subject to multiple conflicting objectives. Under these conditions, exact optimization and classical gradient-based methods may either become impractical or require assumptions that do not hold in real-world systems. Metaheuristic algorithms have therefore emerged as flexible search frameworks capable of obtaining high-quality approximate solutions with limited information about the structure of the objective function [1–4].

The Whale Optimization Algorithm (WOA), introduced by Mirjalili and Lewis [5], is one of the most recognizable swarm-intelligence algorithms developed during the last decade. Its search logic abstracts the bubble-net hunting behavior of humpback whales into three major operators: encircling the best solution, moving around it through a logarithmic spiral, and exploring the search space by referring to randomly selected agents. These mechanisms produce a simple optimizer with few control parameters, concise mathematical equations, and competitive performance on several numerical and engineering-design benchmarks.

The popularity of WOA is partly explained by its ease of implementation. A basic WOA code can be written using only population initialization, fitness evaluation, coefficient-vector calculation, boundary handling, and position updates. This accessibility encouraged rapid adoption in domains such as maximum power point tracking, economic load dispatch, feature selection, image segmentation, neural-network training, wireless sensor network clustering, task scheduling, and structural design. However, the same simplicity has also created a proliferation of variants whose technical contributions are not always clearly separated from routine operator replacement. This rapid diffusion has also been documented in WOA-specific review studies [6–8]. A stronger review of WOA must therefore move beyond the question of whether a new variant performs better on a limited benchmark set. It should ask which weakness the variant addresses, which part of the search process is modified, what computational cost is added, whether the improvement is statistically significant, and whether the mechanism remains useful outside the benchmark functions used by the original authors. Without this level of analysis, the literature can become an accumulation of algorithm names rather than a coherent body of optimization knowledge. This paper expands and strengthens the survey by organizing WOA research around search mechanisms, empirical evidence, and practical relevance. It distinguishes between fundamental design principles, modification strategies, application-driven adaptations, and methodological requirements for fair benchmarking. The analysis is especially focused on the relationship between exploration and exploitation, because almost every successful WOA variant can be interpreted as an attempt to modify this relationship either directly or indirectly.

The contributions of this expanded paper are fivefold: 1) it provides a complete and corrected presentation of the canonical WOA mathematical model, 2) it offers a mechanism-based taxonomy of WOA adaptations, 3) it critically compares the purposes and limitations of major improvement families, 4) it maps WOA variants to application domains and explains why particular variants are suitable for particular classes of problems, and 5) it proposes a reproducible benchmarking framework and identifies research directions that remain insufficiently explored. The rest of the paper is organized as follows. Section 2 presents the review scope and guiding research questions. Section 3 introduces the foundations of metaheuristic optimization. Section 4 explains the original WOA model. Section 5 develops a detailed taxonomy of WOA variants. Section 6 discusses application domains. Section 7 focuses on benchmarking and reproducibility. Section 8 provides a critical discussion of limitations and evidence. Section 9 proposes future research directions. Sections 10-13 provide design guidelines, quality assessment, domain-specific recommendations, and a research roadmap, while Section 14 concludes the paper.

This paper is designed as a structured narrative survey rather than a formal bibliometric or Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA)-based systematic review. The objective is to synthesize the technical evolution of WOA, clarify the function of major enhancements, and provide guidance for researchers who wish to select, compare, or develop WOA variants. Accordingly, the emphasis is placed on mechanism, methodological quality, and practical interpretation rather than publication counts alone.

The reviewed body of work can be divided into four evidence layers. The first layer contains foundational optimization and metaheuristic literature, including the original WOA paper and related population-based algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithms (GA), simulated annealing, and gravitational search algorithms. The second layer contains surveys and systematic reviews that summarize the development of WOA and its variants. The third layer contains algorithmic papers that introduce modifications, hybridizations, or theoretical analyses. The fourth layer contains application studies in engineering, power systems, Machine Learning (ML), image processing, communications, and other domains. The foundational layer includes classical metaheuristics that remain common baselines for WOA evaluation [9–11].

Inclusion was based on conceptual relevance and methodological clarity. A study was considered relevant when it introduced a WOA variant, analyzed the behavior of WOA, compared WOA with established optimizers, used WOA for a well-defined application, or provided a significant survey or taxonomy. Studies were treated cautiously when they reported superiority using only one dataset, omitted parameter settings, failed to conduct repeated independent runs, or did not provide statistical testing. Such studies may still be useful for identifying application trends, but they should not be used as strong evidence of general algorithmic superiority. The analytical method used in this survey is component-based. Instead of classifying variants only by their names, each modification is examined according to the part of the algorithm that it affects: initialization, coefficient scheduling, randomization, local search, representation, archive management, constraint handling, or computational parallelization. This method makes it possible to compare apparently different variants that actually target the same search deficiency. For example, chaotic initialization, opposition-based learning, and Latin hypercube sampling are different techniques, but all attempt to improve initial diversity and coverage. The survey also separates algorithmic contribution from application contribution. A paper that applies WOA to photovoltaic control may be important for renewable energy, even if it does not introduce a new optimizer. Conversely, a paper that introduces an advanced hybrid WOA may not demonstrate practical value unless it is evaluated under realistic objective functions, constraints, and computational budgets. Strong future research should ideally contribute to both dimensions: it should modify the optimizer for a clear reason and validate the modification in a credible problem context.

RQ2: What weaknesses Premature convergence, diversity Explains why modification is motivate WOA variants? loss, sensitivity to parameter necessary schedules, weak local search, and representation limits RQ3: How should WOA By the search component modified Produces a deeper and less variants be classified? rather than by the name of the repetitive taxonomy hybrid algorithm RQ4: Where is WOA most Domains with black-box, nonlinear, Links algorithm choice to problem useful? multimodal, or constrained structure objectives where derivative information is unavailable RQ5: How should WOA Repeated runs, diverse benchmarks, Improves methodological be evaluated? fair baselines, statistical tests, credibility ablation analysis, and reproducible code

Metaheuristics are general-purpose optimization strategies designed to explore large search spaces without relying on strict assumptions such as convexity, differentiability, or linearity. They are especially valuable when objective functions are discontinuous, nonseparable, noisy, multimodal, or computationally expensive. Unlike exact methods, metaheuristics do not guarantee global optimality in finite time for arbitrary problems; instead, they aim to provide robust approximations under realistic computational limits.

Population-based metaheuristics maintain a set of candidate solutions, often called agents, particles, individuals, or search points. This population enables simultaneous exploration of multiple regions of the search space and reduces dependence on a single initial solution. Information may be exchanged through global-best memory, neighborhood structures, selection pressure, social learning, or archive-based mechanisms. WOA belongs to this population-based category because it updates a group of whales according to the current best candidate and stochastic search rules.

A central principle in metaheuristic design is the balance between exploration and exploitation. Exploration refers to wide-ranging search intended to discover promising regions and avoid premature commitment to local optima. Exploitation refers to intensifying search around high-quality solutions in order to refine objective values and accelerate convergence. A search process that overemphasizes exploration may waste evaluations, whereas one that overemphasizes exploitation may converge too early. The WOA model explicitly encodes this balance through the coefficient vector A and the probability-based selection between encircling and spiral motion.

The no-free-lunch perspective is also relevant. No optimizer can outperform all others on all possible problems when averaged over the space of all objective functions. Therefore, claims that one WOA variant is universally superior should be interpreted carefully. A more scientifically useful claim is conditional: a variant may be more suitable for high-dimensional functions, discrete feature selection, constrained engineering design, expensive simulations, or multi-objective trade-off problems. This conditional interpretation is essential for meaningful algorithm selection.

Constraint handling is another fundamental issue. Many engineering and scheduling problems include inequality constraints, equality constraints, boundary constraints, or discrete feasibility rules. WOA, like many continuous metaheuristics, requires additional mechanisms to deal with infeasible solutions. Common strategies include penalty functions, feasibility rules, repair operators, decoders, constraint-preserving representations, and multi-objective treatment of constraint violation. The choice of strategy can strongly affect results and should be reported clearly.

Computational complexity is usually dominated by objective-function evaluation rather than position updates. For a population of N whales, dimensionality D, and maximum iteration count T, the update cost of canonical WOA is approximately O(TND), while the total cost is O(TN(Cf + D)), where Cf denotes the cost of one fitness evaluation. In expensive engineering simulations, Cf may be orders of magnitude larger than the update equations. This explains why Surrogate-Assisted WOA (SA–WOA) and Parallel WOA (PWOA) have become increasingly relevant.

The design of metaheuristic experiments must also recognize stochastic variability. Because random initialization and random coefficients influence the trajectory of the search, a single run cannot represent algorithm performance. Reliable comparison requires repeated independent runs, appropriate summary statistics, and nonparametric significance tests when normality cannot be assumed. This requirement is not a minor methodological detail; it is central to the credibility of WOA research.

schedules.

Typical Strength Typical Weakness Strong population May require several diversity and flexible operators and encoding parameter tuning Simple social learning and Risk of premature effective global search convergence around a dominant leader

Useful stochastic Metaphor may not transitions and acceptance always translate into rules robust search

Intuitive learning Can become structures and adaptive problem-dependent behavior or hard to justify theoretically Potentially improved Additional balance between global complexity and risk and local search of unfair comparison

clustering-based initialization.

Weakness or Risk Simplicity may limit flexibility in complex landscapes

A fixed linear schedule may not match the difficulty of all problems

Excessive attraction to the best solution can reduce diversity quickly Discrete, binary, and constrained problems require additional representation mechanisms Objective evaluation may dominate cost in real-world simulations

WOA variants can be interpreted as attempts to correct one or more limitations of the canonical algorithm. The most frequent motivations are to maintain population diversity, accelerate convergence, improve local exploitation, escape local optima, adapt parameters automatically, handle discrete representations, manage multiple objectives, or reduce computational cost. A mechanism-based taxonomy is more useful than a name-based taxonomy because it reveals recurring design patterns across apparently different variants. Recent hybrid and multi-strategy developments show that this trend is continuing [12], [13].

The following subsections organize WOA improvements according to the search component that is modified. This organization helps researchers select an appropriate variant for a problem. For example, feature selection typically requires binary representation and sparsity-aware fitness design; expensive engineering optimization benefits from surrogate assistance; multi-objective design requires Pareto archiving; and high-dimensional continuous optimization may benefit from decomposition or multi-population cooperation.

WOA has been applied across a wide range of domains because many real-world problems can be formulated as black-box optimization tasks. However, the success of WOA in a domain depends on the suitability of its representation, constraint-handling method, and evaluation protocol. This section discusses major application areas and highlights the specific reasons WOA is attractive in each area.

A critical application review should avoid treating all successful case studies as equivalent. Some applications demonstrate only that WOA can be used, whereas others show that WOA is particularly well suited to the problem structure. Strong evidence requires comparison against domain-specific methods, sensitivity analysis, and validation on realistic data. The following subsections therefore emphasize both opportunities and methodological cautions.

Benchmarking is essential for evaluating any WOA variant, but benchmark design is often the weakest part of metaheuristic research. A fair comparison should use diverse test functions, realistic dimensions, consistent stopping criteria, equal numbers of fitness evaluations, repeated independent runs, and appropriate statistical tests. Reporting only the best result is insufficient because it hides variability and may reward lucky runs. Benchmark functions should include unimodal functions for exploitation assessment, multimodal functions for local-optima avoidance, hybrid and composition functions for landscape complexity, rotated or shifted functions for nonseparability, and constrained engineering problems for practical relevance. A variant that performs well only on simple unimodal functions may be effective at local exploitation but weak at global exploration. Conversely, a variant that performs well on multimodal functions but converges slowly may be unsuitable for expensive applications. Modern benchmark construction also emphasizes shifted, rotated, and structurally challenging functions [24].

Performance indicators should include best, mean, median, standard deviation, convergence curves, success rate, and runtime. For multi-objective variants, hypervolume, Inverted Generational Distance (IGD), spacing, spread, and archive diversity may be needed. For constrained problems, feasible rate and total constraint violation should be reported. For feature selection, accuracy should be accompanied by feature-count reduction and stability.

Statistical analysis should match the stochastic nature of WOA. The Wilcoxon signed-rank test, Friedman test, Holm post-hoc procedure, and effect-size measures are commonly used in metaheuristic comparison. Statistical significance alone is not enough; practical significance should also be discussed. A tiny improvement that requires twice the computational cost may not be meaningful in real applications.

Ablation analysis is particularly important for hybrid WOA. When a variant includes chaotic initialization, adaptive control, Lévy flights, and local search, the paper should test each component separately. Otherwise, it is impossible to know which component produced the improvement. Ablation also prevents unnecessary complexity by identifying operators that contribute little or only help on narrow benchmark classes.

Reproducibility requires detailed reporting. Authors should disclose population size, maximum iterations, stopping criteria, dimensionality, bounds, penalty functions, random seeds, software environment, hardware, dataset splits, and all parameter values. When source code cannot be released, pseudocode should be precise enough to reimplement the method. Ambiguous descriptions such as “parameters were set experimentally” weaken the credibility of the study.

Runs At least 30 independent runs for Captures variability and stochastic comparison reduces dependence on a lucky run Stopping criterion Equal fitness-evaluation budget across Prevents hybrids from algorithms gaining unfair advantage through extra evaluations Metrics Report best, mean, median, standard Shows both performance deviation, and convergence curves level and reliability Statistics Use nonparametric tests and post-hoc Supports evidence-based analysis when comparing many claims of superiority algorithms Parameters Disclose all settings and tuning Allows replication and fair strategy interpretation Datasets Use training/validation/test separation Reduces overfitting and in ML tasks inflated accuracy Constraints Report feasibility rate and violation Distinguishes good feasible values solutions from infeasible numerical artifacts Ablation Test each added component of a Identifies whether hybrid variant complexity is justified

The WOA literature demonstrates that the algorithm is flexible and widely applicable, but it also reveals several recurring weaknesses in how metaheuristic contributions are framed. Many papers introduce a new WOA variant by combining several operators from other algorithms, then evaluate it on a limited benchmark set. Although such studies may produce improved numerical results, they do not always clarify the mechanism responsible for the improvement or the conditions under which the variant should be preferred.

A major issue is the overuse of broad claims. Statements such as “the proposed WOA is superior to all existing algorithms” are rarely defensible. A more accurate statement would specify the benchmark suite, dimensionality, objective type, constraint structure, and computational budget under which the variant performed well. Algorithmic superiority is context-dependent, and the survey should encourage precise claims rather than universal statements.

Another limitation is inadequate comparison with strong baselines. Some WOA papers compare against outdated algorithms or weak parameter settings. A strong evaluation should compare against both classical baselines and recent competitive algorithms, including non-WOA methods. Otherwise, the study may show only that a variant is better than a weak reference, not that it represents a meaningful advance.

The metaphor-driven nature of metaheuristics also requires caution. Biological inspiration can motivate algorithm design, but performance must be justified through mathematical behavior, empirical evidence, and problem suitability. A new metaphor is not itself a contribution unless it leads to a distinct and effective search mechanism. For WOA, the most valuable improvements are those that can be explained in terms of diversity, convergence, perturbation, representation, or computational efficiency.

Despite these criticisms, WOA remains valuable. Its compact structure makes it an excellent base algorithm for experimentation, teaching, and rapid prototyping. It is also suitable for hybridization because its main operators are easy to replace or augment. The challenge for future research is not simply to create more variants, but to create better-justified variants supported by transparent evidence.

In practice, WOA should be selected when the problem is black-box, nonlinear, derivative-free, and moderately expensive, and when a population-based search is appropriate. It may be less suitable when exact mathematical structure is available, when constraints require specialized solvers, when real-time guarantees are strict, or when the problem is extremely high-dimensional without decomposition. In such cases, WOA may still be useful as part of a hybrid system, but not necessarily as the primary solver.

Only best values are reported Best values hide instability and Report mean, median, deviation, random luck convergence, and all-run statistics Unclear parameter tuning Results may depend on hidden State tuning protocol and use equal manual adjustment Too few benchmark Generalization cannot be functions assessed No ablation analysis Contribution of each component is unknown No runtime or evaluation Improvement may come from budget extra computation Weak constraint reporting Infeasible solutions may appear strong Overgeneralized claims Misleads readers about applicability effort across algorithms Use diverse landscapes and real-world cases Evaluate the base algorithm plus each added operator Normalize comparisons by fitness evaluations and report runtime Report feasibility rate and violation magnitude State the domain and conditions where improvement is observed

A stronger WOA variant should be designed from an explicit diagnosis of the search problem. The developer should first identify whether the main difficulty is poor initial coverage, premature convergence, weak exploitation, infeasible solutions, high dimensionality, expensive evaluation, discrete representation, or multiple conflicting objectives. Only after this diagnosis should an operator be added. This principle prevents arbitrary hybridization and encourages mechanism-driven algorithm design.

If the weakness is poor initial coverage, the most appropriate interventions are initialization and diversity strategies. Chaotic maps, opposition-based learning, quasi-random sequences, clustering-based seeding, and Latin hypercube sampling can improve early coverage. However, these methods should be evaluated with the same number of objective evaluations as the baseline. If opposition-based learning doubles the number of initial evaluations, the comparison should either count those evaluations or provide the baseline with the same budget.

If the weakness is premature convergence, the solution is not necessarily to add more randomness. A better approach is to monitor diversity or improvement rate and apply exploration only when needed. Restarting stagnant whales, perturbing the best solution, maintaining subpopulations, or using entropy-based feedback can restore diversity without destroying useful information. This is more principled than applying a large mutation probability throughout the search.

If the weakness is weak local refinement, local search or exploitation-enhancing operators may be appropriate. Gaussian perturbation, pattern search, Nelder-Mead refinement, differential mutation around elite solutions, or gradient-informed steps can improve final accuracy. These methods are especially useful in continuous engineering design, where a good approximate solution must be refined to satisfy tight constraints. However, local search should be applied selectively because it can increase computational cost.

If the problem is constrained, the variant should include a constraint-handling mechanism from the beginning. Static penalties are easy to implement but may be sensitive to penalty coefficients. Adaptive penalties, feasibility rules, repair methods, decoders, and constraint-domination principles are often more reliable. Strong constrained WOA studies should report not only the objective value but also feasibility rate, maximum violation, average violation, and robustness across penalty settings.

If the problem is discrete or binary, the representation is the core contribution. A continuous update followed by arbitrary rounding may damage the search because small position differences may not correspond to meaningful solution changes. Transfer functions, probability vectors, permutation operators, swap sequences, or decoder-based representations should be selected according to the problem. For example, routing problems need permutation-preserving moves, while feature selection requires binary masks and sparsity-aware fitness.

If the problem is multi-objective, the design should include archive management, leader selection, and diversity preservation. A single global best solution is not suitable because the goal is a set of trade-off solutions. The archive should avoid becoming too large while preserving representative nondominated solutions. Grid density, crowding distance, reference vectors, or decomposition methods can guide leader selection and maintain diversity along the Pareto front.

If the objective evaluation is expensive, the variant should focus on evaluation efficiency rather than adding complex search equations. Surrogate modeling, parallel evaluation, adaptive sampling, early stopping, and transfer learning may provide larger benefits than another perturbation operator. The optimizer should also avoid unnecessary reevaluation of identical or infeasible solutions. Caching fitness values is a simple but often neglected technique.

Finally, every new WOA variant should include a complexity statement. Authors should state how many additional evaluations, memory structures, archive operations, model-training steps, or local-search calls are introduced. Without this information, readers cannot judge whether improved accuracy is worth the extra cost. Algorithmic quality is not only measured by solution value; it also includes efficiency, stability, simplicity, and reproducibility.

Low initial diversity Use chaotic, opposition- Initial diversity metrics and based, quasi-random, or equal evaluation-budget stratified initialization comparison Premature convergence Use adaptive exploration, Convergence curves, restarts, entropy feedback, or diversity curves, and multi-swarm search stagnation analysis Weak exploitation Use local search, Gaussian Ablation showing improved perturbation, elite refinement, final precision without or gradient hints excessive runtime High dimensionality Use decomposition, variable Scaling tests across increasing grouping, cooperative dimensions coevolution, or subspace search Discrete decisions Use transfer functions, Feasibility rate and decoders, or problem-specific comparison against discrete moves baselines Constrained feasibility Use repair, feasibility rules, Objective values plus adaptive penalties, or violation and feasibility constraint domination statistics Multiple objectives Use Pareto archive, diversity Hypervolume, IGD, spread, preservation, and leader and archive-quality plots selection Expensive evaluations Use surrogate models, Wall-clock time, evaluation caching, or parallel evaluation count, and surrogate-error analysis

The strength of a WOA paper depends not only on the proposed algorithm but also on the quality of evidence used to support it. Evidence quality can be examined through four validity dimensions: internal validity, external validity, construct validity, and conclusion validity. Applying these dimensions to WOA research helps identify why some empirical claims are persuasive while others remain weak.

Internal validity concerns whether the observed improvement is actually caused by the proposed method. In WOA research, internal validity is threatened when several operators are added simultaneously without ablation analysis, when parameter tuning is inconsistent across algorithms, or when additional evaluations are not counted. To improve internal validity, each component should be tested separately and all algorithms should operate under equal evaluation budgets.

External validity concerns whether results generalize beyond the tested problems. A WOA variant evaluated only on a small set of low-dimensional benchmark functions may not generalize to high-dimensional, constrained, noisy, or discrete problems. External validity improves when studies include several benchmark categories, multiple dimensions, real-world case studies, and comparisons across different problem structures. Construct validity concerns whether the metrics actually measure the intended performance. For example, classification accuracy alone may not be an adequate metric for feature selection because it ignores feature subset size and stability. Similarly, best objective value alone is not enough for constrained optimization because it may ignore feasibility. Construct validity improves when metrics are aligned with problem goals and domain requirements.

Conclusion validity concerns whether the statistical conclusions are justified. A small number of runs, missing variance measures, or inappropriate statistical tests can lead to unreliable claims. Since WOA is stochastic, repeated independent runs are mandatory. Statistical testing should be accompanied by effect sizes, because a statistically significant difference may still be too small to matter practically.

The quality of implementation is another major issue. Two implementations of the same WOA variant may differ in boundary control, random-number generation, fitness caching, initialization, or constraint treatment. These differences can change performance. Therefore, open-source code and detailed pseudocode are not optional extras; they are essential for cumulative scientific progress.

A useful quality assessment score for WOA papers could include items such as clear problem formulation, complete equations, reproducible parameter settings, fair baselines, sufficient runs, statistical testing, ablation study, runtime reporting, code availability, and domain-specific validation. Such a scoring system would help reviewers evaluate whether a proposed variant provides genuine evidence rather than only numerical tables.

Construct validity Metrics may not reflect the true goal of the problem Conclusion validity Claims may rely on too few runs or weak statistical evidence

Implementation Different coding details may change validity algorithm behavior Recommended Mitigation Use ablation studies, equal budgets, and transparent tuning

Test across benchmark categories, dimensions, and real applications Use domain-specific metrics and report feasibility, cost, and stability Use repeated runs, nonparametric tests, confidence intervals, and effect sizes Release code or provide precise pseudocode and boundary-handling rules

The next generation of WOA research should be less focused on naming new variants and more focused on building reliable optimization frameworks. A short-term priority is reproducibility: common benchmark suites, reference implementations, and reporting standards would immediately improve the value of empirical comparisons. Such resources would also make it easier for researchers to identify genuinely useful modifications.

A medium-term priority is scalability. Many practical problems involve hundreds or thousands of variables, multiple constraints, and expensive evaluations. WOA variants should therefore incorporate decomposition, cooperative coevolution, adaptive sampling, and parallel evaluation. These mechanisms should be tested under increasing dimensionality and realistic computational budgets.

A long-term priority is theory-guided design. Metaheuristic research often advances empirically, but stronger theoretical analysis can help explain when and why an algorithm works. For WOA, theoretical work should examine diversity loss, attraction dynamics, boundary effects, convergence probabilities, and the impact of adaptive schedules. Even partial theory can improve design by identifying failure modes.

Another long-term direction is trustworthy optimization for high-stakes systems. When WOA is used in medical, power, security, or autonomous systems, performance must be reliable under uncertainty. This requires robust optimization, uncertainty quantification, explainability, and safety-aware constraints. WOA should not only find good solutions but also provide evidence that those solutions remain acceptable under perturbation.

The field should also develop stronger connections with automated algorithm design. Instead of manually combining operators, researchers can use algorithm configuration, reinforcement learning, or hyper-heuristics to select operators based on landscape features. However, automated design must be evaluated carefully to avoid overfitting to benchmark suites. Generalization across problem classes should become a central evaluation criterion.

Short term Ablation-based evaluation of Clearer understanding of which existing hybrid variants. operators actually help.

Medium term Large-scale, constrained, and Better applicability to realistic DWOA frameworks. engineering systems.

Medium term Surrogate-assisted and Reduced computational cost PWOA for expensive and improved scalability.

objectives.

Long term Theory-guided analysis of More principled design of convergence and diversity adaptive mechanisms.

dynamics.

Long term Trustworthy WOA for high- Optimization methods that are stakes domains. robust, explainable, and safe.

This expanded survey has presented a stronger and more critical review of the WOA. It has clarified the canonical model, corrected the mathematical presentation, explained the algorithm’s strengths and weaknesses, and organized WOA variants according to the search components they modify. The survey shows that WOA remains attractive because of its simplicity, intuitive search behavior, and adaptability across many problem domains.

The analysis also shows that most successful WOA variants respond to specific limitations: premature convergence, loss of diversity, weak local search, poor discrete representation, insufficient constraint handling, expensive evaluation, or conflicting objectives. Adaptive schedules, chaotic maps, mutation strategies, hybrid operators, binary transformations, Pareto archives, surrogate models, and parallel subpopulations should therefore be understood as targeted design responses rather than arbitrary additions. For application domains, WOA has demonstrated value in engineering design, power systems, ML, biomedical analysis, image processing, IoT, cloud/edge computing, cybersecurity, logistics, and transportation. Nevertheless, practical usefulness depends on formulation quality, feasibility handling, validation design, and computational cost. A WOA solution is strong only when it is both algorithmically justified and empirically validated under realistic conditions.

The paper recommends that future WOA research emphasize mechanism-driven design, rigorous benchmarking, statistical testing, ablation analysis, open-source implementation, and careful domain validation. Rather than multiplying variants without clear justification, the field should move toward theoretically informed, reproducible, scalable, and explainable WOA frameworks. Such a direction will strengthen the scientific contribution of WOA and increase its reliability in real-world optimization.

All aspects of the research and manuscript preparation were carried out by the author. The author has read and approved the final version of the manuscript.

Not applicable.

All data supporting the reported findings in this research paper are provided within the manuscript.

The author declares that they do not have any conflict of interest.

The author confirms consent for the publication of this work.

This article does not contain any studies with human participants performed by the author.

Appendix A. Practical checklist for designing a WOA study.

A practical WOA study should begin by defining the optimization problem with sufficient precision. The objective function, decision variables, constraints, bounds, and evaluation procedure must be stated clearly. If the problem is discrete or mixed-variable, the encoding and decoding scheme should be explained before any algorithmic result is presented. If the problem is multi-objective, the trade-off objectives and decision-maker preferences should be defined.

The next step is selecting an appropriate WOA variant. The selection should follow the problem structure. Continuous unconstrained problems may use canonical or adaptive WOA; high-dimensional problems may require decomposition; binary feature selection requires transfer functions; constrained engineering problems require repair or feasibility rules; and expensive simulations may require surrogate assistance. The variant should not be selected merely because it is recent or popular.

Experimental design should then establish fair comparison. Competing algorithms should receive equal evaluation budgets, and parameter settings should be obtained from original papers or tuned using a transparent protocol. Random seeds, independent runs, and hardware environment should be recorded. For machine-learning applications, the data split must prevent leakage between training, validation, and testing. Finally, the results should be interpreted cautiously. Statistical significance should be combined with effect size and computational cost. If a variant improves accuracy by a very small margin while doubling runtime, the practical value may be limited. Strong conclusions should be restricted to the tested problem classes, and limitations should be acknowledged explicitly.

Table A1. Submission checklist for rigorous WOA research.

Checklist Item Question to Answer Before Submission Problem definition Are objective functions, variables, constraints, and bounds fully specified?

Variant justification Does each modification address a clearly identified weakness of WOA?

Baseline fairness Are all algorithms compared under equal fitness-evaluation budgets?

Parameter reporting Are population size, iterations, coefficients, and tuning method disclosed?

Statistical validity Are enough independent runs and suitable statistical tests used?

Ablation Is the contribution of each added operator tested separately?

Reproducibility Are code, seeds, datasets, and experimental settings available or described?

Practical relevance Are runtime, feasibility, scalability, and domain-specific metrics reported?

Appendix B. Reporting template for WOA experiments.

The following template is included to make the survey more directly useful for authors preparing WOA manuscripts. A WOA experiment should first state the optimization formulation in a compact mathematical form. This includes the objective function, decision vector, lower and upper bounds, equality and inequality constraints, dimensionality, and whether the problem is continuous, binary, integer, permutation-based, or mixed-variable. When the objective function is derived from simulation, the simulation settings should be documented so that the evaluation is reproducible.

The algorithm configuration should then be reported in a separate subsection. This subsection should include population size, maximum iteration count, total fitness-evaluation budget, initialization method, boundary-handling method, coefficient schedule, probability settings, mutation or local-search probabilities, archive size for multi-objective problems, and stopping conditions. If a parameter is tuned, the tuning range and selection criterion should be stated. If parameters are adopted from previous literature, the reference should be cited and any changes should be justified. The comparison protocol should be written before results are discussed. It should identify baseline algorithms, explain why they were selected, and state whether their parameters were taken from original papers, tuned by the authors, or set using a common protocol. All algorithms should be run under comparable computational budgets. When a hybrid WOA includes additional local search or opposition-based evaluations, these evaluations should be counted unless a clear reason is provided.

The result tables should include more than the best solution. Recommended columns include best, worst, mean, median, standard deviation, success rate, average runtime, and rank. For constrained problems, additional columns should show feasible rate and average constraint violation. For machine-learning problems, results should include accuracy or error plus model complexity, feature count, and validation protocol. For multi-objective problems, hypervolume and IGD should be accompanied by representative Pareto-front plots. The discussion section should interpret results rather than only repeating numerical rankings. Authors should explain why the proposed WOA variant performs well or poorly on specific functions or applications. For example, strong performance on multimodal functions may indicate effective exploration, while weak performance on rotated functions may indicate sensitivity to variable interaction. This interpretation is what transforms a numerical experiment into a scientific contribution.

Finally, limitations should be stated explicitly. A WOA variant may be strong on medium-dimensional continuous problems but untested on discrete or dynamic problems. It may produce accurate solutions but require high runtime. It may work well on benchmark functions but need additional validation on real data. Acknowledging such boundaries strengthens the manuscript because it shows that the authors understand the scope of their evidence.

Table B1. Recommended reporting template for WOA experimental manuscripts.

Manuscript Component Minimum Information to Include Problem formulation Objective, variables, dimensionality, bounds, constraints, representation, and evaluation procedure.

Algorithm settings Population size, iterations, evaluation budget, coefficient schedules, initialization, boundary handling, and stopping criteria.

Variant components Exact description of each added mechanism and how it modifies canonical WOA.

Baselines Algorithm names, parameter settings, references, tuning method, and equal-budget policy.

Statistical protocol Number of runs, random seeds, summary statistics, tests, post-hoc analysis, and effect sizes.

Application validation Domain-specific metrics, feasibility, runtime, scalability, and sensitivity analysis.

Reproducibility package Code, datasets, seed files, benchmark definitions, and raw result tables when possible.

Appendix C. Operator-level interpretation of WOA variants.

Many WOA variants can be understood as operator-level changes to a small number of algorithmic functions. The initialization operator determines the first population. The leader operator determines which solution guides the movement. The movement operator determines how candidate positions change. The perturbation operator determines how diversity is restored. The selection operator determines which solutions survive. The constraint operator determines whether a candidate is feasible or repairable. The archive operator stores nondominated or elite solutions. Describing a variant in terms of these operators makes the method easier to compare and reproduce.

For example, a chaotic adaptive WOA may be represented as a canonical WOA in which the initialization operator uses a chaotic map and the control operator uses a nonlinear schedule. A hybrid WOA-DE may be represented as a canonical WOA in which some movement steps are replaced by differential mutation and crossover. A binary WOA may be represented as a canonical WOA with an added transfer operator that maps continuous position values into binary decisions. This operator-level description is clearer than simply naming the hybrid.

Operator-level interpretation also supports modular experimentation. Researchers can replace one operator at a time while keeping the rest of the algorithm fixed. This produces cleaner ablation studies and helps identify which operator is responsible for improvement. It also makes it possible to build reusable WOA libraries in which initialization, movement, perturbation, selection, and constraint handling are interchangeable modules.

A final advantage is conceptual transparency. When a paper says that a new WOA variant improves exploration, the reader should be able to identify which operator creates that improvement. If exploration improves because of a Lévy perturbation, this should be visible in the algorithm description. If exploitation improves because of local search, the frequency and cost of local search should be reported. Transparent operator descriptions reduce ambiguity and make WOA research easier to evaluate.

Table C1. Operator-level view for describing and comparing WOA variants.

Operator Role in WOA Framework Initialization Generates the first population within the search bounds.

Leader selection Determines the solution or archive member that guides movement.

Movement Updates whale positions using encircling, spiral, or exploration rules.

Perturbation Restores diversity or escapes local optima.

Representation Maps candidate positions into valid decisions.

Constraint Controls feasibility and violation handling treatment.

Archive/memory Stores useful solutions for later selection.

allows readers to identify evidence gaps more quickly.

they should not be evaluated using the same criteria.

Examples of Possible Modifications Chaotic maps, Latin hypercube sampling, opposition-based learning, orthogonal design.

Elite selection, random elite choice, Pareto archive leader, crowding-based leader.

Nonlinear coefficients, PSO velocity, DE mutation, GWO hierarchy, SCA movement.

Gaussian mutation, Cauchy mutation, Lévy flight, random walk, restart.

Binary transfer functions, permutation decoders, integer repair, mixed-variable encoding.

Static/adaptive penalty, repair, feasibility rules, constraint domination.

External Pareto archive, elite memory, historical best population, surrogate training set.

Appendix D. Evaluation matrix for future WOA surveys.

Future surveys of WOA can be strengthened by using a structured evaluation matrix instead of summarizing papers only by year, author, and application. The matrix should record the modified component of the algorithm, the motivation for modification, the benchmark type, the dimensionality, the comparison baselines, the statistical tests, the application domain, and whether source code or datasets are available. Such a matrix

A survey matrix should also distinguish between methodological novelty and application novelty. A paper may be methodologically novel if it introduces a new adaptive schedule, a new leader-selection mechanism, or a new discrete representation. Another paper may be application-novel if it applies an existing WOA variant to a new engineering problem with a realistic formulation. Both kinds of contribution can be valuable, but

The evidence level of each paper can be categorized qualitatively. A low-evidence paper may provide a new variant but test it on few functions without statistical analysis. A medium-evidence paper may include repeated runs and several baselines but lack ablation or code. A high-evidence paper should include diverse benchmarks, fair evaluation budgets, statistical testing, ablation, runtime analysis, and reproducible implementation. This classification would make future surveys more critical and less descriptive.

Another useful survey dimension is problem suitability. WOA is not equally suitable for every optimization task. Future reviews should record whether the problem is continuous, binary, integer, permutation-based, constrained, dynamic, noisy, expensive, multi-objective, or many-objective. This would help readers see which WOA mechanisms are most often used for each problem type and where research remains underdeveloped. Finally, future surveys should examine negative or neutral results when available. Metaheuristic literature often emphasizes successful improvements, but unsuccessful or marginal results are scientifically informative because they reveal limits. A balanced survey should not only report where WOA performs well but also identify problem types where it struggles, conditions where simpler algorithms are competitive, and cases where added complexity does not justify the improvement.

Table D1. Evaluation matrix for future high-quality WOA surveys.

Evaluation Dimension Recommended Coding Categories for Future Surveys Modified WOA component Initialization, control schedule, movement, perturbation, representation, archive, constraint handling, parallelization.

Problem type Continuous, binary, integer, permutation, constrained, noisy, dynamic, expensive, multi-objective, many-objective.

Evidence level Low: limited tests; medium: repeated tests and baselines; high: statistics, ablation, runtime, and reproducibility.

Benchmark coverage Unimodal, multimodal, rotated, shifted, composition, engineering design, real-world application.

Validation method Repeated independent runs, cross-validation, sensitivity analysis, feasibility analysis, Pareto-front indicators.

Reproducibility Complete pseudocode, parameter settings, seeds, source code, datasets, and raw results.

Practical relevance Runtime, scalability, interpretability, domain constraints, and deployment feasibility.

Appendix E. Glossary of key WOA terms.

This glossary clarifies recurring terms used throughout the survey. It is included to improve readability for researchers who are new to WOA while preserving technical precision for advanced readers. Consistent terminology is important because many WOA papers use similar words such as exploration, diversification, perturbation, mutation, and randomization without clearly separating their meanings.

In this paper, exploration refers to search behavior that increases the chance of discovering new promising regions, while exploitation refers to behavior that refines solutions near already promising regions. Diversity describes the spread of candidate solutions in the search space. Stagnation means that the population no longer produces meaningful fitness improvement. A leader is the solution or archive member that guides the position update. These concepts are useful for comparing WOA variants because they describe search behavior rather than algorithm names.

Table E1. Glossary of important terms used in WOA research.

Term Meaning in WOA Research Exploration Global search behavior intended to discover new regions and reduce local-optimum trapping.

Exploitation Local refinement around promising solutions, usually guided by the current best whale or archive leader.

Diversity The degree to which population members occupy different regions of the search space.

Stagnation A state in which repeated iterations produce little or no improvement in objective value.

Leader The best solution or selected archive member used to guide the movement of other whales.

Transfer function A mapping that converts continuous WOA positions into binary or discrete decisions.

Archive A memory structure used to store elite or nondominated solutions, especially in multi-objective WOA.

Ablation study An experiment that removes or isolates components to measure their individual contribution.

Using this terminology consistently improves the clarity of WOA studies. For instance, a method that changes the initial population should be described as an initialization or diversity mechanism, not simply as an improved optimizer. Likewise, a method that adds Gaussian disturbance near elite solutions should be interpreted primarily as a local refinement mechanism, whereas a method based on heavy-tailed jumps should be interpreted as an escape or diversification mechanism.

Clear terminology also supports fair comparison among variants. Two algorithms may have different names but perform similar functions if both modify the same operator. Conversely, two algorithms may share the same label, such as adaptive WOA, while using very different feedback rules. Describing the exact search role of each component reduces ambiguity and helps readers understand whether a contribution is genuinely new. For authors, the glossary can be used as a checklist while writing the methodology section. Each proposed component should be connected to one or more of the listed terms, and the experiment should include evidence that the intended effect occurred. For example, if a component is claimed to improve diversity, the paper should report a diversity indicator or at least a convergence analysis that supports the claim.

Appendix F. Comprehensive catalogue of WOA modifications and hybrid variants.

This appendix integrates the extensive variant catalogue from the original manuscript with the mechanism-oriented taxonomy developed in the expanded survey. The aim is not to treat every named variant as equally validated, but to preserve the breadth of reported WOA modifications while clarifying that each one should be assessed by its search role, computational cost, and empirical evidence.

Table F1. Combined catalogue of WOA modifications and hybrid variants retained from the original manuscript and aligned with the expanded taxonomy.

Number Modification / Hybrid Type Description 1 Adaptive WOA (AWOA) Dynamically adjusts control parameters to maintain balance between exploration and exploitation throughout iterations.

2 Improved WOA (IWOA) Refines the encircling and spiral equations to improve local search efficiency.

3 Chaotic WOA (CWOA) Employs chaotic maps for parameter updates to improve population diversity.

4 Inertia-Based WOA Introduces inertia-like terms to prevent stagnation and accelerate (ILWOA/MWOA) convergence.

Table F1. Continued.

Number Modification / Hybrid Type Description 5 WOA–PSO Hybrid Combines WOA’s exploration and PSO’s exploitation for balanced performance.

6 WOA–DE Hybrid Integrates DE’s mutation–crossover operators to enhance convergence speed.

7 WOA–GA Hybrid Embeds genetic crossover and selection operators for improved diversity.

8 Binary WOA (BWOA) Adapts WOA for discrete problems using binary transfer functions.

9 Multi-Objective WOA Extends WOA for Pareto-based multi-objective optimization.

(MOWOA) 10 Quantum WOA (QWOA) Incorporates qubit representation and quantum superposition principles.

11 Levy Flight WOA (LWOA) Uses Lévy flights for improved global exploration.

12 Opposition-Based WOA Applies opposition-based learning for faster and more accurate (OBWOA) convergence.

13 Hybrid WOA–GWO Combines WOA with GWO to balance exploration and exploitation.

14 Hybrid WOA–SCA Integrates SCA dynamics to strengthen global search.

15 Chaos-Driven Adaptive WOA Combines chaotic initialization and adaptive parameter control. (CAWOA) 16 Random Walk WOA Incorporates stochastic random walks for enhanced exploration. (RWWOA) 17 Elite Opposition-Based WOA Integrates elite opposition learning for faster convergence.

(EOBWOA) 18 Fuzzy WOA (FWOA) Uses fuzzy logic to adjust parameters adaptively based on search performance.

19 MBWOA Introduces Gaussian or Cauchy mutations to diversify the population.

20 Ensemble WOA (EWOA) Combines multiple control strategies or sub-algorithms concurrently.

21 PWOA Distributes computation across multiple nodes for faster convergence.

22 DL–WOA Uses WOA as an optimizer for neural network training and hyperparameter tuning.

23 Hybrid WOA–SA Combines simulated annealing’s probabilistic acceptance with WOA’s search.

24 Hybrid WOA–ACO Incorporates pheromone-guided path construction for routing and scheduling.

25 DWOA Employs time-varying control coefficients for adaptive exploration intensity.

26 Orthogonal WOA (OWOA) Uses orthogonal experimental design for efficient initialization. 27 Enhanced WOA (EWOA) General category for improved exploitation or parameter re-scaling variants.

28 Self-Adaptive WOA (SAWOA) Self-regulates control parameters using feedback from fitness evolution.

29 Improved Convergence WOA Implements nonlinear convergence control for faster (ICWOA) optimization.

30 Gaussian WOA (GWOA) Adds Gaussian perturbation to intensify local refinement.

Table F1. Continued.

Number Modification / Hybrid Type Description 31 Cauchy WOA (CWOA2) Employs Cauchy distribution mutation to improve search diversity.

32 Exponential Decay WOA Introduces exponential decay control on parameters for (EDWOA) smoother convergence.

33 Differential WOA (DWOA2) Integrates differential update mechanisms for enhanced diversity.

34 Entropy-Controlled WOA Utilizes information entropy to balance search adaptively.

(EnWOA) 35 Gradient-Guided WOA Uses gradient direction information for faster (GGWOA) convergence.

36 Momentum-Driven WOA (M- Applies momentum to reduce oscillation in position WOA) updates.

37 K-Means Assisted WOA (KM– Employs clustering to maintain diversity around promising WOA) regions.

38 Hybrid WOA–BAT Combines echolocation-inspired BAT algorithm with WOA’s mechanisms.

39 Hybrid WOA–Firefly (WOA–FA) Uses light-intensity-based attraction to improve exploration.

40 Hybrid WOA–ABC Merges foraging-based search from ABC to refine local search.

41 Hybrid WOA–HS Incorporates harmony memory and pitch adjustment mechanisms.

42 Hybrid WOA–TLBO Leverages teaching–learning phases to strengthen WOA’s learning.

43 Hybrid WOA–CSA Combines crow memory strategy with WOA’s encircling behavior.

44 Hybrid WOA–MFO Integrates Moth Flame Optimization’s flight path search pattern.

45 Hybrid WOA–SMA Employs slime mould oscillatory behavior to enhance adaptability.

46 Hybrid WOA–ALO Uses trapping and random walk of ALO for exploratory improvement.

47 Hybrid WOA–SOA Combines seagull migration and attack phases with WOA’s model.

48 Hybrid WOA–CS Embeds Lévy-flight-based cuckoo dynamics for random search.

49 Hybrid WOA–MPA Fuses marine predator–prey dynamics with WOA’s exploitation.

50 Hybrid WOA–EPO Incorporates penguin grouping and thermal dynamics principles.

51 Hybrid WOA–GJO Combines golden jackal chasing strategy with WOA’s bubble-net phase.

52 Multi-Population WOA Divides population into multiple sub-swarms for diversity (MPWOA) maintenance.

53 Multi-Swarm Cooperative WOA Enables information sharing among multiple WOA (MSWOA) subgroups.

54 Ensemble Learning–Driven Uses ensemble decision mechanisms among WOA WOA (EL–WOA) variants.

55 SA–WOA Employs surrogate modeling to handle computationally expensive evaluations.

Table F1. Continued.

Number Modification / Hybrid Type Description 56 Transfer Learning–Based WOA Transfers prior optimization knowledge to new problem (TL–WOA) instances.

57 Reinforcement Learning–WOA Utilizes RL agents to dynamically adjust parameters and (RL–WOA) strategies.

58 Dynamic Population WOA Adjusts population size based on diversity and (DPWOA) convergence indicators.

59 Energy-Balanced WOA Applies bio-inspired energy consumption model to (EBWOA) guide search.

60 Thermal Exchange WOA Adapts heat transfer theory for dynamic search (TEWOA) adaptation.

61 Modified Spiral Search WOA Introduces variable-radius spiral equations for refined (SSWOA) exploitation.

62 Neighborhood Search WOA Incorporates local neighbor interactions for improved (NSWOA) convergence.

63 Orthogonal Chaotic WOA Combines orthogonal learning and chaos for (OCWOA) initialization.

64 Improved Discrete WOA Employs customized mapping for integer and (IDWOA) permutation optimization.

65 Hybrid WOA–EO Integrates equilibrium optimizer’s exploration rules.

66 Hybrid WOA–SFO Uses sailfish-inspired attacking–migrating strategies.

67 Improved Multi-Objective WOA Incorporates fuzzy dominance and crowding distance (IMOWOA) ranking.

68 Hybrid WOA–Jellyfish Search Combines time-controlled motion from JS for adaptive (WOA–JS) exploration.

The catalogue shows that WOA research has expanded from simple parameter adaptation and chaotic randomization toward hybrid, multi-population, surrogate-assisted, and learning-guided search. However, future studies should avoid presenting a new label as sufficient evidence of novelty; the mechanism, ablation evidence, and suitability for a specific problem class must be made explicit.

Appendix G. Extended application catalogue of WOA.

This appendix preserves the broad application inventory from the original manuscript and links it to the stronger discussion of evaluation criteria in the expanded version. The table is useful as a practical index for researchers selecting a WOA variant for a new domain.

Table G1. Combined application catalogue of WOA across scientific, engineering, and intelligent-system domains.

Number Application Domain Specific Application 1 Structural optimization Truss structure design, welded beam design, pressure vessel optimization, and tension/compression spring design 2 Energy systems Photovoltaic MPPT, renewable energy integration, economic load dispatch, and microgrid operation 3 Power systems Optimal power flow, generator scheduling, capacitor placement, and load forecasting 4 Image processing Image segmentation, contrast enhancement, thresholding, and image reconstruction 5 Medical diagnosis Liver segmentation, brain tumor detection, toxicity classification, and COVID-19 diagnosis 6 Data clustering and feature High-dimensional feature selection, clustering, and selection dimensionality reduction Table G1. Continued.

Number Application Domain Specific Application 7 Control systems PID parameter tuning, adaptive control, and load frequency control 8 WSNs Node deployment, energy-efficient clustering, and lifetime maximization 9 ML and DL Hyperparameter tuning for SVM, CNN, and hybrid WOA–DL frameworks 10 Manufacturing and scheduling Job-shop scheduling, process planning, and production sequencing 11 Communication networks Resource allocation, channel estimation, and QoS optimization in 5G 12 Supply chain and logistics Inventory optimization, routing, and transport cost minimization 13 Robotics and path planning Path optimization and swarm-based multi-robot coordination 14 Financial systems Portfolio optimization, risk management, and stock trend prediction 15 Water resources engineering Reservoir operation and water distribution optimization 16 Transportation systems Traffic control, route optimization, and intelligent scheduling 17 Cybersecurity and cryptography Image encryption, key generation, and network intrusion detection 18 IoT Energy-efficient clustering, resource optimization, and device deployment 19 Cloud and edge computing Task allocation, VM scheduling, and resource provisioning 20 Bioinformatics Gene selection, protein folding, and DNA sequence alignment 21 Renewable energy forecasting Wind and solar power prediction using hybrid WOA–ML models 22 Structural health monitoring Damage detection and modal parameter estimation 23 Environmental systems Pollution control, waste management, and ecosystem modeling 24 Network security IDS optimization, DoS prevention, and secure communication 25 Signal processing Filter design, spectral estimation, and biomedical signal analysis 26 Software engineering Test case prioritization and software effort estimation

27 Hydrology Flood forecasting and watershed modeling

28 Aerospace engineering UAV path planning and aerodynamic shape optimization 29 Smart grids Load management, power prediction, and demand response 30 Combinatorial optimization TSP, knapsack, and assignment problems

The application catalogue confirms that WOA is most defensible when the target problem is nonlinear, derivative-free, multimodal, constrained, discrete, or expensive to evaluate. For deployment-oriented research, the algorithm should be assessed not only by best objective value but also by feasibility, runtime, robustness, and reproducibility.