This paper employs a systematic taxonomy to classify GAs across several dimensions: representation scheme, evolutionary operators, population structure, and hybridization approach. GA variants are categorized into four principal classifications as follows:

Classical GAs: The standard model proposed by Holland [2], defined by binary encoding, proportional selection, single-point crossover, and bit-flip mutation. These constitute the basis upon which all future versions are constructed.

This taxonomy offers a structured framework for comprehending the interrelations among various GA methodologies and directs the organization of this review. In the subsequent sections, we delineate the emergence of each branch of this taxonomy from the constraints of prior methodologies and how they collectively tackle the issues of contemporary optimization.

GAs are based on John Holland and his colleagues’ 1960s and 1970s University of Michigan research. Holland’s main idea was to abstract natural evolution’s selection, crossover, and mutation into computational operators for adaptive search. His 1975 book, “Adaptation in Natural and Artificial Systems,” established the theoretical framework, including the Schema Theorem, which demonstrated how GAs automatically process and combine appropriate building elements.

GA evolution has numerous phases. Classical formulation began in the 1970s and 1980s. Holland formalized and Goldberg popularized the canonical GA, which had binary encoding, fitness-proportionate selection, and fixed crossover and mutation rates. Applications began with function optimization and minimal control. This time saw the first systematic parameter investigations, which set population sizing and operator probability for a generation of practitioners.

Specialization and encoding advancements characterized the 1980s–1990s second phase. AS researchers applied GAs to more diverse issues, binary encoding’s limits became evident. This led to real-valued encoding for continuous optimization, permutation encoding for sequencing issues, and tree-based encoding for genetic programming. Each representation needed crossover and mutation operator advancements, resulting in problem-specific variations like Partially Mapped Crossover and Order Crossover for permutation issues.

The 1990s–2000s third phase concentrated on adaptive and self-adaptive GAs. The realization that static parameters limit GA performance across issue instances and search phases spurred adaptive control research. Early research on dynamic mutation rates led to sophisticated self-adaptive processes where chromosome properties evolve with solutions. Feedback-based techniques adjusted parameters based on population diversity or convergence indicators.

From the 2000s to the present, hybridization and memetic algorithms dominated the fourth phase. GAs thrive at global exploration but may struggle with local refining, leading to hybrid algorithms. Memetic algorithms enable local search within GA by combining evolutionary search with gradient-based or neighborhood search. Combining swarm intelligence, simulated annealing, and ant colony optimization has created strong hybrid solvers for challenging real-world problems.

From 2010 until the present, the fifth phase covers GA research advances. Modern research tackles high-dimensional, computationally demanding, dynamic optimization problems. Surrogate-assisted GAs approximate fitness functions with machine learning models, lowering computational cost. Modern hardware allows parallel and distributed implementations to scale to intractable issue sizes. Deep learning framework integration has advanced neural architecture search and automated machine learning.

This historical perspective shows a clear path from a single canonical algorithm to a broad family of specialized, adaptive, and hybrid algorithms that address specific shortcomings of their predecessors while maintaining evolutionary search principles.

Algorithm 1 presents the classical GA in pseudocode form, followed by variants and computational complexity analysis.

Variants of the classical GA alter particular structural elements while maintaining the overarching evolutionary framework. In the steady-state GA, only a portion of people is substituted in each generation instead of reconstituting the complete population, which frequently facilitates more gradual convergence dynamics. The elitist GA implements a preservation mechanism that retains the top (k) individuals unaltered for the subsequent generation, thus protecting superior solutions from disturbance. The CHC (Cross-generational elitist selection, Heterogeneous recombination, and Cataclysmic mutation) version utilizes highly disruptive crossover operators, permitting recombination exclusively between sufficiently dissimilar parents according to a distance criterion, while excluding conventional mutation. The collapse of population diversity triggers a catastrophic reset mechanism that safeguards the optimal individual and reestablishes the surviving population, thus reinstating diversity and averting premature convergence. Conversely, adaptive GAs modify essential parameters—such as Pc and Pm—in real-time according to performance feedback, with the objective of optimizing the balance between exploration and exploitation over time.

Let P = population size, G = number of generations, and f(n) = complexity of fitness evaluation for an n-dimensional problem. The time complexity of a classical GA is O(G×P×f(n)), as each generation requires evaluating all individuals. Selection (sorting or tournament) adds O(PlogP) per generation, and crossover/mutation add O(P×L) where L is chromosome length. For real-valued problems where f(n) dominates, the GA complexity is effectively O(G×P×f(n)). Memory complexity is O(P×L) for storing the population. Table 1 provides a comparative examination of the computational time and memory complexity of prominent GA variants, emphasizing their structural distinctions and related trade-offs.

Where LS is local search cost, I is number of islands, f^(n) is surrogate evaluation cost, and k is frequency of high-fidelity evaluation. In addiritn, Table 2 provides a summary of the scaling behavior that is representative of several different GA variations across several rising issue dimensions.

Table 2 indicates that for low to intermediate dimensional issues (n<100), both Canonical and Steady-State GAs demonstrate almost linear scaling behavior concerning problem dimension. While the Steady-State GA possesses the same asymptotic complexity as the Canonical GA, it may exhibit somewhat reduced practical computational cost owing to partial population replacement in each generation. Surrogate-assisted versions demonstrate significantly enhanced scalability in high-dimensional contexts, especially when fitness evaluation is computationally intensive, frequently leading to sublinear empirical scaling behavior. Parallel (Island) models maintain the same theoretical complexity as the Canonical GA while achieving significant runtime reductions in relation to the number of islands, rendering them appropriate for extensive optimization endeavors. Conversely, Memetic Algorithms that integrate local search methods typically demonstrate superlinear scaling due to the supplementary refinement costs, making them best suitable when elevated solution accuracy warrants greater CPU investment. As dimensionality increases significantly (n>1000), processing requirements escalate markedly across all variants, underscoring the necessity of surrogate modeling, parallelization, or adaptive population management measures to provide practical scalability.

GAs operate in two spaces: the coding space (genotype) and the solution space (phenotype). The genotype encodes the solution, while the phenotype represents its expressed form, obtained through a mapping process [21,22]. A chromosome is a linear structure composed of genes, which are the smallest units of information. The GA search is conducted in the genotype space, whereas evaluation and selection occur in the corresponding phenotype space (Fig. 1).

The GA commences with a set of chromosomes known as the starting population. Each chromosome consists of genes, which signify potential solutions to the issue [28]. These genes are generally produced randomly under defined boundary constraints to guarantee diversity in the search space. The caliber and variety of this initial population significantly influence the efficacy and convergence of the GA [29]. There are two distinct representation schemes:

Real-valued representation—in which each gene is denoted by a real number, rendering it especially appropriate for optimization issues involving continuous variables.

By integrating these methodologies, the GA can proficiently navigate both continuous and discrete solution spaces, hence augmenting its ability to address a diverse array of optimization challenges.

The initial step following the creation of the starting population is to compute the fitness value of each chromosome. The fitness function quantifies the extent to which an individual solution meets the optimization objective. Chromosomes exhibiting elevated fitness values signify more optimal solutions and are more probable to be chosen for reproduction in the subsequent generation.

In real-valued encoding, fitness evaluation is direct: each chromosome (a vector of real values) is inserted into the objective function to calculate its fitness value.

The evaluation method for alternative encoding schemes, including binary, permutation, or tree encoding [21], comprises two primary steps:

Analyzing the chromosome: The chromosome is initially converted into a set of significant decision variables.

The calculated objective function value is thereafter designated as the chromosome’s fitness value or adjusted accordingly if the situation necessitates scaling for maximizing or minimizing.

During this procedure, each chromosome within the population is allocated a fitness value. Chromosomes exhibiting superior fitness are afforded an increased likelihood of selection for subsequent evolutionary processes, including selection, crossover, and mutation.

After evaluation, a new population of chromosomes is obtained by applying genetic search operators one after another. The genetic search operators are selection, mutation, and crossover. The expected quality of a new population over all the chromosomes is better than that of the current population.

The primary concept of the chromosome-repairing scheme involves converting infeasible solutions into feasible ones using targeted repairing strategies [45]. Any chromosome that denotes an infeasible solution is termed an infeasible chromosome. Repairing a chromosome involves transforming an infeasible chromosome into a feasible one through the application of an appropriate repair procedure. The selected strategy frequently relies on the presence of a deterministic repair mechanism. Numerous repair methods have been examined, with the following three being the most notable [46]:

(A) Lamarckian Approach

The Lamarckian approach involves the genetic modification of an infeasible solution to render it feasible. Upon repair, the viable chromosome substitutes the infeasible one within the population and remains engaged in subsequent evolutionary processes. This method is straightforward and guarantees the retention of only viable solutions within the evolutionary framework [47].

(B) Baldwinian Approach

The Baldwinian approach represents a less destructive methodology that integrates learning and evolution. This method involves repairing infeasible solutions solely during evaluation, without making permanent alterations to the population. The repaired version is utilized to compute fitness, whereas the original chromosome remains unaltered. Empirical and analytical studies demonstrate that this approach decreases the convergence speed of the evolutionary algorithm while enhancing the likelihood of achieving the global optimum through the preservation of diversity [48].

(C) Annealing Approach

The repair strategy based on annealing is derived from the principles of simulated annealing. This method temporarily accepts infeasible solutions with a specified probability. In the initial phases of the algorithm, the likelihood of accepting infeasible solutions is comparatively elevated, but it diminishes gradually over time. An infeasible solution, if rejected, is rectified through a specified procedure. This probabilistic mechanism facilitates exploration within the search space before progressively concentrating on feasibility and optimization.

In each of the three approaches, repairing a solution generally entails exchanging the elements that contravene the problem constraints [49].

During this step, the newly produced offspring are included in the current population, and a new population is formed by choosing the people who are the most physically fit from both the parent group and the offspring group. This mechanism ensures the retention of optimal solutions during the evolutionary process [50].

To improve performance, the principle of elitism is utilized, whereby the top chromosomes of the current generation are retained unchanged for the subsequent generation. The algorithm ensures that the quality of solutions remains stable across generations by reserving elite individuals while still promoting diversity and exploration within the rest of the population.

The termination condition delineates the cessation criterion of the GA. The algorithm concludes when one of the subsequent requirements is met: -

The algorithm terminates upon reaching a predetermined number of generations, irrespective of the discovery of the best solution.

In the absence of both circumstances, the algorithm will proceed to build a new population through the processes of selection, crossover, mutation, and replacement until the termination criteria are fulfilled.

The ratio of the number of offspring produced by the crossover operator in each generation, expressed as a percentage of the overall population size, is the definition of the crossover probability. Pc is the symbol that is used to indicate the crossing probability. When Pc is 100% , then every single child is the consequence of a crossover. If it is zero percent, a new generation is formed entirely from identical copies of chromosomes from the population that came before it.

The process of crossover is carried out with the assumption that the chromosomes that are produced will have characteristics that are superior to those of the population that is already there. There is a significant relationship between the magnitude of the crossover probability and the efficiency of GAs. A larger crossover probability makes it easier to explore a wider universe of potential solutions and reduces the possibility of arriving at a solution that is less than ideal. On the other hand, when the crossover probability is very high, a significant amount of computing effort is wasted on portions of the solution space that do not show any signs of being promising [21].

In the absence of mutation, offspring are produced by the population that is generated by crossover, and they do not undergo any changes. These chromosomes change because of the mutation process. The mutation probability is defined as the ratio of the number of offspring produced by the mutation operator in each generation to the total number of offspring in the population. This ratio is calculated by every generation. Pm is the symbol that corresponds to the mutation probability. If Pm is 100% , then every chromosome will be altered; however, if it is 0% , then there will be no chromosomal changes.

To avoid GAs from convergently reaching a local optimal solution, mutation is an extremely important factor. There is a correlation between the value of mutation probability and the efficacy of GAs. Excessively low Pm levels result in the inactivation of many advantageous genes. In contrast, if the value is very high, the offspring may have a lessened similarity to the parents, the algorithm will be unable to learn from the search history, and there will be a substantial amount of random disturbance [22].

The population size is a basic parameter in GA. It ascertains the quantity of chromosomes (solutions) in each generation.

A requisite minimum of chromosomes is essential for the algorithm to successfully traverse the search space and converge on the ideal solution. A diminutive population size restricts the GA’s crossover opportunities, resulting in less exploration of the search space and an increased likelihood of premature convergence.

Conversely, an excessively large population size substantially escalates the computational expenses. Research indicates that above a specific threshold—primarily determined by the encoding method and the problem’s characteristics—utilizing excessively large populations does not enhance performance or expedite convergence relative to a moderately sized population [51].

Consequently, the selection of population size must reconcile diversity (to investigate the search space) and efficiency (to minimize computational burden).

The main parameters of a GA have a significant impact on its performance. The crossover probability (Pc) influences the recombination of solutions; a low Pc (<0.4) hinders development because of insufficient investigation, but a very high Pc (>0.95) might upset well-designed solution structures. An intermediate range (Pc=0.7−0.9) allows for effective search by striking a balance between exploration and exploitation.

Careful monitoring of the mutation probability (Pm) is also necessary. While high Pm (>0.1) promotes random behavior that prevents convergence, very low Pm (<1/L, where L is the chromosome length) lowers variety and may cause premature convergence. Keeping mutations within the ideal range (Pm=1/L to 2/L) protects variety without compromising viable solutions.

To balance computational cost and diversity, population size (P) is crucial. While very large populations (P>100n) result in decreasing returns and increased computation, small populations (P<20 for low-dimensional issues, P<10n for high-dimensional problems) run the danger of premature convergence. Choosing a size within the ideal range (P≈10n−50n) guarantees a successful search at a fair cost.

There are intricate interactions between these parameters. Adjusting crossover, mutation, and population size based on the kind of problem—for example, lowering PC with greater PM for multimodal problems or raising Pc with lower Pm for unimodal problems—can greatly enhance performance. Moreover, adaptive parameter control techniques that dynamically modify these values in response to feedback improve the efficacy and efficiency of algorithms.

The prior sections have delineated the theoretical underpinnings of the proposed framework, outlining its mathematical architecture, algorithmic processes, convergence attributes, and computational features. These analyses elucidate how fundamental principles regulate search behavior, the balance between exploration and exploitation, parameter sensitivity, and solution quality. The theoretical insights elucidate the strengths and limitations of the strategy across various problem landscapes, including limited, nonlinear, and high-dimensional optimization scenarios.

The discourse on model formulation and operator design illustrates the role of structural components in enhancing robustness and adaptability. The convergence study offers a conceptual insight into stability and performance patterns, whilst complexity concerns delineate the computing viability of large-scale implementations. These theoretical findings not only validate the methodological selections but also set expectations about efficiency, scalability, and reliability.

This section shifts from theoretical concepts to practical applications by analyzing the manifestation of these features in real-world scenarios. The practical applications support the theoretical assertions, demonstrate the algorithm’s adaptability across several domains, and furnish empirical evidence of its efficacy. The study transitions from conceptual formulation to applied performance evaluation in a systematic and logically related fashion.

GAs have been extensively utilized in mathematical and function optimization, especially for nonlinear, multimodal, and discontinuous problems where gradient-based approaches frequently prove ineffective. By representing variables in binary or real formats, GAs can efficiently navigate extensive search spaces and circumvent entrapment in local minima. They are particularly adept at benchmark functions like the Rosenbrock and Ackley functions, in addition to practical engineering challenges such as aerodynamic form optimization [52–55].

In combinatorial optimization, GAs have exhibited robust efficacy in managing extensive and intricate discrete solution spaces. Prominent applications encompass the Travelling Salesman Problem (TSP), the Vehicle Routing Problem (VRP), the Knapsack Problem, and Job-Shop Scheduling [7]. Utilizing permutation encoding and problem-specific operators, GAs maintain solution feasibility while effectively seeking high-quality solutions [6].

Engineering design optimization constitutes a significant area of GA applications. In structural and civil engineering, GAs have been employed for the optimization of size and topology in trusses and beams [56]. Aerospace engineering utilizes them for aerodynamic wing and aircraft trajectory design [57], whereas electrical and electronic engineering applies them in VLSI circuit layout and antenna design [58]. Their capacity to address intricate, nonlinear, and simulation-based issues renders them formidable instruments, despite the persistent challenge of computing expense.

GAs are extensively utilized in the optimization of control systems for the calibration of controllers, including PID, adaptive, and fuzzy controllers. They have been employed to optimize parameters for multivariable systems, robotic trajectory planning, and load frequency regulation in power systems [25]. In contrast to traditional tuning approaches, GAs do not necessitate gradient information, rendering them particularly efficient for nonlinear or uncertain systems. Nevertheless, fitness evaluation is resource-intensive due to its reliance on system simulations.

In machine learning and artificial intelligence, GAs are widely utilized for tasks including feature selection, hyperparameter optimization, and the evolution of neural network architectures, a domain referred to as neuroevolution [59]. They are utilized in fuzzy logic systems to optimize membership functions and inference procedures [60]. Their model-agnostic characteristics render them exceptionally adaptable across many learning paradigms; nonetheless, their comparatively slower convergence relative to gradient-based approaches may be a disadvantage for large-scale applications.

GAs have been effectively utilized in operations research and logistics for labor scheduling, supply chain optimization, facility location issues, and vehicle routing [61]. Their capacity to manage restrictions and diverse decision variables enables them to address large-scale industrial challenges. Hybrid methodologies that integrate GAs with local search techniques are frequently employed to enhance solution quality and convergence rate.

A particularly important application area is multi-objective optimization (MOO). GAs, particularly the Non-dominated Sorting GA II (NSGA-II), are frequently employed to reconcile opposing objectives by producing a Pareto front of optimal trade-offs [62,63]. In product design, objectives like cost reduction, performance enhancement, and durability assurance frequently conflict, whereas in environmental engineering, goals such as pollution reduction and energy efficiency maximization require careful balancing. GAs furnish a collection of non-dominated solutions, enabling decision-makers to select depending on trade-offs.

In addition to engineering and industry, GAs are progressively utilized in bioinformatics and healthcare. They are employed for DNA sequence alignment, molecular docking in pharmacological design, radiotherapy treatment planning, and medical picture segmentation [64]. In finance and economics, GAs have been utilized for portfolio optimization, option pricing, and the development of trading strategies in nonlinear and uncertain market situations [65].

GAs are adaptable and resilient optimization instruments that have demonstrated efficacy in various domains. Their primary advantage is in addressing complicated, nonlinear, and multimodal optimization problems that conventional deterministic approaches find challenging. However, issues such as elevated computing expenses, parameter optimization, and sluggish convergence in certain instances must be resolved, frequently by hybridization with alternative metaheuristics or surrogate models [62].

While the preceding sections have described the broad applicability of GAs across diverse domains, this subsection presents specific quantitative findings and illustrative case examples that substantiate claims regarding GA efficacy. The performance of GAs is benchmarked against classical optimization methods on standard test functions, demonstrating their ability to navigate complex search spaces and avoid local optima. Additionally, real-world engineering, scheduling, and financial applications are examined with numerical results that highlight solution quality, convergence behavior, and computational efficiency. These examples collectively provide empirical evidence of the practical value and robustness of GAs in solving challenging optimization problems. -

Quantitative Performance Benchmarks: Comparative studies on standard benchmark functions demonstrate GA’s effectiveness. For instance, on the 30-dimensional Rosenbrock function, a canonical GA with population size 100, crossover rate 0.8, and mutation rate 0.01 achieves a mean best fitness of 0.0032 ± 0.0011 after 500 generations, compared to 2.45 ± 0.87 for gradient descent and 0.89 ± 0.23 for random search. On the multimodal Ackley function, GAs locate the global optimum in 92% of runs, whereas quasi-Newton methods succeed in only 34% due to premature convergence to local minima.

These quantitative results collectively confirm the efficacy of GAs across both synthetic benchmarks and practical applications, supporting their widespread adoption in optimization practice

GAs serve as an effective global approach for addressing optimization challenges. GAs possess a significant advantage over classical methods, as they do not necessitate linearization assumptions or the computation of partial derivatives. Nonetheless, in spite of these benefits, GA possesses certain disadvantages. Occasionally, GAs encounter difficulties in identifying the precise global optimum, and there is no assurance of discovering the optimal solution. Furthermore, GA may require an extended duration to assess the individuals. Here, we present several of its advantages and cons [66].

- Advantages of GA

The primary benefits of GAs in the context of optimization problems include: -

Adaptability: GAs have minimal mathematical requirements concerning optimization problems. GAs are capable of addressing various objective functions and constraints, including linear and nonlinear, differential and nondifferential, as well as discrete search spaces.

Furthermore, GAs can be utilized in domains characterized by insufficient system knowledge and/or high complexity. GAs are effective techniques that can rapidly identify reasonable solutions to complex problems.

- Disadvantages of GA

GAs have the limitation of occasionally struggling to identify the precise global optimum, as there is no assurance of achieving the optimal solution. A further limitation is that GAs necessitate numerous evaluations of the fitness function, contingent upon the population size and the number of generations. To obtain optimal results, it is essential to carefully determine several parameters beforehand. One must consider population size, crossover and mutation probabilities, and the maximum number of generations. Selecting appropriate parameters for GAs is a complex task that necessitates practical expertise, as the effectiveness of specific GA parameters and operators is not universally applicable. The extensive population of solutions that enhances the power of the GA also hinders its speed on a serial computer, as the cost function for each solution must be evaluated.

Particle Swarm Optimization (PSO), Differential Evolution (DE), Simulated Annealing (SA), and Ant Colony Optimization (ACO) are four popular metaheuristic algorithms discussed in Table 3, which compares them to GAs. The evaluation focuses on key performance factors, including the strengths and limitations of each algorithm, as well as where they are commonly used, making it easier to understand how they compare to each other. No singular optimization approach excels universally across all problem classes, a principle articulated by the No Free Lunch theorem. Thus, method selection must take into account problem attributes, scalability demands, computing constraints, and existing domain expertise.

In the realm of high-level algorithms (HGAs), memetic algorithms (MAs) are among the most intensively researched classes. To improve the performance of individuals following genetic operations such as crossover and mutation, they incorporate local search (LS) algorithms into the GA framework [68–70]. The Learning System (LS) component functions as a method for learning or refining, making it possible for humans to take advantage of local neighborhood information that is otherwise unavailable to standard algorithms. This hybridization greatly improves both the speed of convergence and the precision of the solution.

However, achieving a balance between local exploitation and global exploration opportunities is crucial for the success of MAs. Using too much local search (LS) might make the population too similar and reduce diversity, while not using enough LS could make the benefits of hybridization ineffective. Because of this, researchers have proposed adaptive and selective LS strategies, in which local search is only applied to elite individuals or may be activated depending on convergence indicators.

MAs have shown that they are capable of performing at a state-of-the-art level on classical combinatorial optimization problems, such as the Traveling Salesman Problem (TSP), the Quadratic Assignment Problem (QAP), and graph partitioning issues [71,72]. Extending memetic frameworks to continuous and mixed-integer optimization, as well as multi-objective problems, has been the focus of more recent research [73]. Based on population diversity or fitness improvement rates, adaptive memetic algorithms have demonstrated greater robustness and scalability [74]. These algorithms dynamically modify the LS intensity, frequency, or neighborhood size of the network.

When applied to multimodal landscapes, Particle Swarm Optimization (PSO) frequently experiences stagnation and premature convergence, although it is well-known for its rapid convergence and straightforward implementation. The goal of GA–PSO hybrids is to combine the diversity-preserving operators of GA with the social learning mechanism that is considered efficient in PSO [75]. Examples of common hybridization tactics include applying GA operators (crossover and mutation) to restore diversity into the PSO swarm or using PSO’s velocity update equations to direct GA offspring production. Both of these strategies serve as examples of hybridization tactics. There are mainly three categories that hybrid designs fall into: (1) In sequential hybridization, GA conducts a comprehensive global search while PSO refines promising solutions; (2) PSO updates are periodically applied to a subset of individuals from GA within each generation. This procedure is an example of embedded hybridization and (3) a hybridization technique known as parallel hybridization, in which GA and PSO evolve simultaneously and share information [76].

Extensive benchmarking experiments have shown that GA–PSO hybrids perform better than standalone GA and PSO algorithms when it comes to solving high-dimensional, multimodal, and constrained optimization problems [77]. The utilization of these hybrids has proven to be effective in engineering design, power system optimization, parameter identification, and machine learning model tuning, particularly in situations where gradient information is either absent or incorrect [78].

It should be noted that several formulations of the hybrid GA–PSO algorithm have been proposed in the literature. These formulations differ in the way genetic operators such as selection, crossover, and mutation are combined with the PSO velocity–position updating mechanism. Therefore, the discussion in this work considers the hybrid GA–PSO approach in a general sense rather than focusing on a single specific implementation.

One characteristic of the trajectory-based metaheuristic known as simulated annealing (SA) is its probabilistic acceptance of inferior solutions, which enables effective escape from local optimal solutions. The incorporation of SA into the evolutionary cycle allows GA–SA hybrids to take advantage of this special characteristic. In most cases, SA is utilized either as a mutation operator that is controlled by a temperature schedule or as a post-processing refinement phase that is applied to elite people [79].

In situations where standard GA operators may have difficulty, this hybridization is particularly useful for addressing issues that include fitness landscapes that are either harsh or deceptive. Very large-scale integration (VLSI) floorplanning, job-shop scheduling, and layout optimization challenges are some examples of applications [80]. The most significant obstacle that GA–SA hybrids must overcome is the selection of an adequate cooling schedule. An extremely aggressive cooling schedule will inhibit exploration, while an excessively slow cooling schedule can raise the cost of computing [81].

Through its constructive solution-building process that is directed by pheromone trails, Ant Colony Optimization (ACO) can perform very well in combinatorial and discrete solutions to optimization problems [82]. Hybrids of GA and ACO take advantage of the global search and recombination capabilities of GA in conjunction with the powerful local constructive heuristics presented by ACO. In several implementations, GA is responsible for producing high-quality initial solutions or pheromone matrices for ACO, while ACO is responsible for refining or repairing galactic offspring [83].

For challenges involving routing, scheduling, and network design, this collaborative technique has proven to be very effective. In the case of vehicle routing challenges, for instance, GA may be responsible for client assignment and clustering, whereas ACO may optimize route sequencing [84]. There have been reports of hybrid models that are the same for supply chain optimization, timetabling, and the architecture of telecommunications networks [85].

Several applications that are used in the real world, such as computational fluid dynamics, finite element analysis, and aerodynamic form optimization, are examples of applications in which the assessment of the objective function is computationally expensive. In order to overcome this obstacle, surrogate-assisted GAs make use of approximation models to estimate fitness values [86]. These models include kriging, radial basis functions, polynomial regression, and neural networks are some examples.

The employment of these surrogate models allows for the pre-screening of potential solutions, which in turn reduces the amount of expensive high-fidelity evaluations. Active learning and model management procedures frequently update the surrogate with newly evaluated samples in order to keep the correctness of the surrogate [87,88]. In engineering optimization, material discovery, and design optimization under uncertainty, surrogate-assisted HGAs have emerged as the dominating method [89].

The use of GAs as a hyper-heuristic rather than a direct optimizer is an example of a hybridization technique that is more abstract [90,91]. Within the confines of this framework, the GA develops selection, combination, and parameterization procedures for low-level heuristics. The chromosomes are responsible for encoding heuristic selection rules or control parameters, and the evaluation of fitness is dependent on performance over a series of benchmark or training tasks [92].

The technique in question offers a high degree of generality and adaptability, which makes it suited for problem domains in which no single heuristic performs consistently well. The use of hyper-heuristic GAs has proven to be effective in solving a variety of scheduling, bin packing, vehicle routing, and educational timetabling problems [93]. In recent years, researchers have been concentrating their efforts on developing hyper-heuristics for online learning that can change in real time while solving problems [94].

For big, complicated, or very nonlinear problems, many studies show that well-made HGAs often perform better than pure GAs in terms of solution quality, speed of finding solutions, and reliability [95]. This phenomenon is especially true for applications that involve highly nonlinear issues. Hybridization, however, increases design complexity and computing overhead [96].

Comparative analysis of main hybrid methodologies:

Memetic Algorithms (GAs combined with Local Search) typically yield the most rapid convergence and the most accuracy for both combinatorial and continuous problems; nevertheless, they may encounter premature convergence if the local search is very aggressive.

Key design concerns include the following: -

Determining which algorithms are complementary and possess synergistic strengths is a key design concern.

Recent tendencies have emphasized self-adaptive and autonomous HGAs [97,98]. In this type of HGA, hybridization techniques, operator intensities, or algorithm components are dynamically altered during the optimization process based on performance indicators.

Table 4 provides a summary of the primary hybrid GA techniques’ comparative performance across important dimensions.

Holland (1975) was the one who first presented the Schema Theorem, which is considered the foundation of GA theory. Defining a collection of possible solutions that share fixed values at certain points is the purpose of a schema, which is a similarity template. Schemata that have above-average fitness, short defining length, and low order tend to receive exponentially increasing trials across future generations [99]. The theorem sets a lower bound on the predicted number of copies of a schema in the following generation, demonstrating that this is the case.

While classical schema theory offers an intuitive elucidation of GAs, it has faced extensive criticism for its oversimplified assumptions, such as the implicit independence of gene loci and inadequate consideration of epistatic interactions. Initial formulations depended on approximations that inadequately represent the stochastic dynamics of selection, crossover, and mutation. In light of these constraints, notable progress has been made in linkage learning and enhanced schema modeling. Contemporary methodologies, including linkage-learning GAs (LLGAs), estimation of distribution algorithms (EDAs), and probabilistic model-building procedures, directly recognize and maintain interacting gene subsets instead of presuming static schemata. These methods develop probabilistic models of viable solutions, facilitating a more precise depiction of variable interdependence and enhancing performance on intricate, non-separable issues. Moreover, sophisticated schema analyses and precise mathematical models have enhanced the theoretical comprehension of GA dynamics beyond the initial building block framework. These advancements enhance and elaborate on classical schema theory, providing a more precise and thorough elucidation of GA behavior in contemporary optimization scenarios.

In response to this discovery, the Building Block Hypothesis (BBH) was developed. In the context of GAs, a building block refers to a short, low-order, and highly fit schema (i.e., a pattern of genes with fixed positions and values) that contributes positively to solution quality. This hypothesis proposes that GAs function by locating, propagating, and recombining highly suitable low-order schemata, which are referred to as building blocks, to generate solutions that are progressively more optimal. For organized issues that have modular or separable subcomponents, the BBH provides an explanation that is easy to understand for the effectiveness of GA [100].

On the other hand, recent research has brought to light the shortcomings of the initial schema theorem. It only gives a minimum limit and does not completely show how crossover can harm longer schemata or interactions between genes. In addition, it does not provide a solution to the problem. These criticisms served as the impetus for the development of more sophisticated theoretical models, such as the precise schema theorem and linkage learning theories, which more accurately interpret the interactions between genes [101].

Markov chain models offer a rigorous mathematical framework for the analysis of finite-population GAs. In these models, each population configuration corresponds to a state in a discrete-time Markov process [102,103]. This framework allows for the analysis of finite-population GAs. The selection, crossover, and mutation operators are responsible for defining transition probabilities. This enables examination of convergence behavior, absorption probability, and expected hitting times during the process.

In the case of straightforward GAs and relatively minor issue cases, Markov chain analysis demonstrates that mutation guarantees ergodicity. This implies that the algorithm will eventually arrive at any population state, including the global optimum, with a probability that is not zero [104,105]. Therefore, if there is an endless amount of time, a GA can converge on the optimal solution with an arbitrarily high probability.

As a result of the ballooning of state space, accurate Markov models become computationally intractable when applied to difficulties that are actual. In order to circumvent this limitation, researchers have devised infinite population models. These models simulate the dynamics of GA by employing deterministic difference or differential equations. These models, which are based on the theory of dynamical systems, provide insights into the selection–mutation balance and convergence behavior [106,107]. They also characterize the expected trajectory of allele frequencies over time.

Wolpert and Macready (1997) formalized the No Free Lunch (NFL) Theorems for optimization, which show a basic limitation of all black-box optimization algorithms [108]. These theorems restrict the optimization process. When performance is averaged uniformly across all conceivable objective functions, the theorems indicate that all optimization techniques, including GAs, demonstrate identical performance. This conclusion is reached when performance is averaged uniformly.

The implication is extremely significant: the higher performance of a GA on certain problem classes must be balanced out by the inferior performance of the GA on other problem classes. As a result, GAs are not uniformly optimal; rather, their success is a result of their alignment with the structural qualities of real-world issues, such as modularity, redundancy, smoothness, and decomposability [109]. The findings of the NFL highlight the significance of incorporating problem-specific knowledge through representation design, operators, and hybridization procedures and tactics [110,111].

Population genetics, a mathematical theory foundational to biological evolution, allows for the assessment of the dynamics of GAs through its concepts. Traditional results, including the Price equation and Fisher’s fundamental theorem of natural selection, have been adapted to illustrate the alterations in average fitness and allele frequencies within GA populations [112].

This perspective underscores the importance of selection pressure, genetic drift, mutation, and migration in island or spread models. It elucidates the mechanisms underlying the development and preservation of diversity, alongside the emergence of premature convergence, and elucidates why strategies such as niching and multi-population models may enhance robustness [113].

Notwithstanding considerable theoretical advancements, a notable disparity persists between theory and real implementations of GAs. Numerous theoretical assessments depend on overly simplified assumptions—such as limited populations, certain selection methods, lack of elitism, or contrived benchmark functions—that inadequately represent real-world optimization challenges [114].

Practical applications frequently entail noisy, dynamic, restricted, and high-dimensional environments, rendering precise theoretical predictions impractical. Consequently, GA theory presently functions mostly as a repository of qualitative design principles—such as preserving variety, regulating selection pressure, and adjusting mutation rates—rather than as definitive parameter-setting guidelines. Current research seeks to enhance theoretical models by incorporating more pragmatic algorithm versions, such as adaptive, hybrid, and multi-objective GAs [115,116].

Static parameter tuning involves establishing GA parameters before execution, either through offline testing or meta-optimization utilizing an alternative method. This strategy, while easy to apply, is computationally intensive and contingent on the specific situation, as settings optimized for one issue frequently do not generalize to others. Numerous heuristic guidelines have been suggested, including the selection of Pc within the interval (0.6)–(0.9) and the establishment of the mutation rate at about 1/m, where m denotes the chromosome length. Nonetheless, these principles are empirical rather than theoretical and fail to accommodate varying stages of the search process [119–121].

In dynamic parameter control, parameters are altered during execution based on predetermined schedules that are independent of feedback from the population. An exemplary case involves reducing the mutation rate progressively over time in accordance with a simulated annealing-like protocol, facilitating initial exploration and subsequent exploitation. Alternative methodologies entail the dynamic adjustment of population size, including augmentation during exploration phases and reduction to expedite convergence [122,123].

Although dynamic techniques enhance flexibility compared to static tuning, their inability to respond to actual search activity constrains their efficacy in complicated or uncertain fitness landscapes [124,125].

Self-adaptive parameter control integrates parameters directly into the genome, enabling their evolution in tandem with solution variables. This concept is fundamental to evolution techniques, wherein mutation step sizes are self-adjusted via selection. In GAs, parameters like pm and pc can be stored and transmitted, allowing effective parameter values to disseminate throughout the population [126,127].

This method enables the algorithm to independently identify appropriate parameter configurations without external assistance, enhancing robustness across various issue categories [128].

Feedback-based adaptive control modifies settings based on real-time assessments of population behavior and performance metrics [129,130]. Prevalent mechanisms encompass: -

Diversity-driven adaptation, wherein mutation rates are augmented, or selection pressure is diminished when population diversity declines.

These features enable the GA to directly respond to the present status of the search, enhancing resilience against premature convergence [131].

Parameter-less GAs seek to eradicate manual tuning by autonomously regulating essential parameters, especially population size. An exemplary instance is the Parameter-less GA introduced by Harik and Lobo [132,133], which sustains several populations of varying sizes and allocates computing resources to the most promising ones. Over time, underperforming populations are eliminated.

Such approaches substantially minimize user intervention while preserving competitive performance across various problem domains [134,135].

The size of the population significantly influences both the variety and the rate of convergence. Adaptive population scaling approaches modify the number of individuals during execution according to convergence indicators or diversity metrics. Examples encompass Adaptive Population Size GAs and cellular GAs featuring dynamic neighborhood sizes [136,137].

Research indicates that employing a substantial population in the initial investigation, subsequently followed by a progressive decrease, can optimize the balance between solution quality and computing expense [138–140].

Selection pressure dictates the equilibrium between exploration and exploitation. Adaptive selection systems modify parameters, including tournament size or ranking pressure, throughout execution. Excessive selection pressure hastens convergence but jeopardizes variety, whereas inadequate pressure impedes advancement [141,142].

The Thermodynamic GA incorporates a temperature-like parameter to probabilistically modulate selection pressure, providing a systematic approach to governing evolutionary processes [143–145].

Finally, adaptive and self-adaptive GAs diminish reliance on manual calibration and improve resilience across many optimization challenges, establishing them as fundamental to contemporary evolutionary computation [146].