Real-world problems rarely involve optimizing a single objective. Zeleny’s statement (1982): “Multiple objectives are all around us” [1] reflects the ubiquity and importance of considering multiple criteria, goals, or objectives in our daily lives. Decision-making often must account for conflicting objectives, necessitating trade-offs. For instance, in automotive design, reducing fuel consumption and emissions often conflicts with cost and performance. Similarly, in supply chain optimization, minimizing costs can conflict with maximizing customer satisfaction. These complexities have led to the emergence of MOO as a crucial area of research. It wasn’t until the early 1970s that formal approaches to considering multiple objectives in decision-making processes were widely studied. A review of the historic development of this field is provided by [2]. Since the 1970s, specific conferences, newsletters, journals, and academic societies have been established for this new research area.

The foundation of MOO is the concept of Pareto-optimality, introduced by Vilfredo Pareto in the early 20th century. A solution is Pareto-optimal if improving one objective necessitates the degradation of at least one other. However, formal methods for MOO gained prominence only in the 1970s with the development of structured approaches to multi-criteria decision-making (MCDM) or multi-criteria decision analysis (MCDA) [1,2]. These methods have since evolved to address diverse applications in engineering, economics, and beyond.

In general, two primary research directions can be identified: The first focuses on determining viable solutions to a MOO problem. This inquiry led to the development of Pareto-optimal solutions (also known as Pareto-efficient or simply efficient) and methods for identifying such solutions across various problem types. Given that there is typically no single, definitive Pareto-optimal solution, a second question emerges: How can we assist a decision-maker in choosing an alternative from those that are mathematically sound (i.e., the Pareto-optimal solutions)? This question spawned numerous approaches that incorporate additional information, particularly regarding a decision maker’s preferences, into the solution process. While this second question involves considering the availability and suitability of relevant information, aspects of rational decision-making, psychological factors, and the user-friendliness of methods, the first question deals with optimization in potentially complex problems. Methods addressing the first question are referred to as type 1 approaches, while those addressing the second are called type 2 approaches. It is also possible to address both questions simultaneously in a single approach that utilizes additional (preference-based) information during the optimization process. This information can be provided beforehand or progressively throughout the optimization process (interactive methods). These combined approaches are classified as type 3 approaches. This typology not only guides algorithm design but also frames how decision-making and optimization are intertwined in practical applications.

Complex optimization problems have found promising solutions in Evolutionary Algorithms (EAs), particularly when conventional optimization and operations research methods fall short due to assumptions about problem properties like linearity, convexity, or differentiability. This makes EAs attractive for tackling intricate MOO challenges as well. However, the widespread exploration of EAs for multi-objective problems didn’t gain traction until the late 1980s. The field’s growth culminated in the organization of specialized conferences, beginning with the in-augural International Conference on Evolutionary Multi-Criterion Optimization in 2001 [3]. This event gave rise to the commonly used acronym for the field, EMO.

Multi-Objective Optimization Problems (MOOPs) present unique challenges compared to single-objective problems. These challenges can be broadly categorized as follows: •

Solution Representation: ⮚

Identifying the Pareto front, a set of Pareto-optimal solutions, is computationally intensive, particularly for high-dimensional problems. Recent advancements, such as the use of deep learning techniques to approximate the Pareto front, have shown promise in alleviating some computational burdens.

The distinction between “multi-objective” and “many-objective” optimization is primarily a matter of the number of objectives involved, which in turn significantly impacts the complexity of the problem and the effectiveness of traditional Multi-Objective Evolutionary Algorithms (MOEAs).

Multi-objective Optimization (MOOPs) involve optimizing two or three conflicting objectives simultaneously. In this context, the concept of Pareto dominance is well-defined and relatively straightforward to apply. Such problems can be characterized as follows:

Number of Objectives: Typically, 2 or 3 objectives.

Many-objective Optimization: Many-objective optimization problems (MaOPs) refer to problems with a large number of objectives, typically four or more, and often extending to tens or even hundreds of objectives. The increase in the number of objectives introduces significant challenges that differentiate MaOPs from traditional MOOPs. MaOPs can be characterized as follows: •

Number of Objectives: Generally, 4 or more objectives, often extending to a much larger number.

Table 1 summarizes the key differences between multi-objective and many-objective optimization.

In summary, while both multi-objective and many-objective optimization deal with multiple conflicting objectives, the sheer number of objectives in many-objective problems introduces a “curse of dimensionality” that fundamentally changes the behavior of Pareto dominance and necessitates different algorithmic approaches and decision-making strategies.

MOEAs have proven invaluable in diverse domains, including: •

Engineering: ⮚

Structural Design Optimization: MOEAs are employed to optimize designs under multiple constraints. For instance, Reference [14] demonstrated the use of the NSGA-II algorithm for optimizing truss structures, achieving significant improvements in weight reduction while maintaining strength standards. MOEAs are frequently used to optimize designs under multiple constraints like a study by Kalyanmoy Deb and his colleagues that explored various applications of MOEAs in structural optimization, showcasing their effectiveness in minimizing material usage while satisfying performance criteria [15].

These specific case studies illustrate the continued relevance and effectiveness of MOEAs in solving complex optimization challenges across diverse fields, underscoring their effectiveness in solving complex optimization problems involving multiple conflicting objectives. They are also reflecting their adaptability to contemporary issues and advancements in technology.

The evolution of MOEAs has been shaped by several landmark contributions, ranging from the conceptual foundation of evolutionary algorithms to the development of sophisticated hybrid techniques. These milestones represent significant progress in algorithmic strategies, problem formulation, and application scope.

Table 2 provides a chronological overview of key milestones in MOEA development. The conceptual groundwork for intelligent behavior in machines can be traced back to Alan Turing’s early ideas in the 1950s, which laid the foundation for machine learning and computational intelligence. Although not directly proposing evolutionary algorithms, Turing’s vision inspired subsequent developments in artificial intelligence. The foundation of genetic algorithms (GAs) was laid in the early 1960s by John Holland, who sought methods to automatically generate adaptive finite-state automata. His initial publications on this concept predate his widely cited book Adaptation in Natural and Artificial Systems (1975), which formalized and popularized GA methodology. A more comprehensive historical account of GA development can be found in Goldberg’s book Genetic Algorithms in Search, Optimization, and Machine Learning (1989), which outlines the foundational ideas and their early evolution [25]. The first formal evolutionary algorithms, however, were introduced later by Ingo Rechenberg and John Holland in the 1970s [26–28]. Starting from the foundational theories, the field has witnessed the emergence of influential algorithms like NSGA, SPEA, NSGA-II, and MOEA/D. These developments reflect a continuous effort to improve convergence behavior, maintain diversity, and tackle increasingly complex multi-objective problems.

Recent years have seen a shift toward integrating MOEAs with advanced techniques, such as local search, decomposition strategies, and machine learning models. This trend demonstrates the field’s dynamic nature and its responsiveness to the challenges posed by real-world, high-dimensional, and dynamic optimization scenarios.

The historical development of MOEAs is characterized by foundational theories, key algorithmic breakthroughs, and hybrid advancements. Table 2 presents a condensed timeline of these significant milestones, highlighting their contributions to the evolution of the field.

This review aims to: ⮚

Provide a comprehensive historical perspective on the evolution of MOEAs.

By synthesizing these aspects, this article seeks to serve as a resource for researchers and practitioners, advancing both theoretical understanding and practical application of MOEAs.

MOOPs are characterized by the need to optimize two or more conflicting objectives simultaneously. This complexity arises in various fields, including engineering, economics, and environmental management, where decision-makers must balance trade-offs among competing goals. The formulation of an MOOP can be expressed mathematically as follows: (1)Maximizef(x)={f1(x),f2(x),…,fq(x)},x∈X⊆Rnwhere f(x) represents a vector of q objective functions to be maximized, X is the solution or search space, and f(X) is the objective space. A solution x1 is said to dominate x2 (denoted x1≺x2) if fi(x1)≥fi(x2) for all i=1,…,q and fi(x1)⟩fi(x2) for at least one i.

The Pareto-optimal set is defined as [28]: (2)P∗={x∈X|∃/y∈Xsuch that y≺x}

This set represents solutions where no objective can be improved without degrading another, and the corresponding Pareto front is the image of P∗ in the objective space. Identifying the Pareto front is central to solving MOOPs, as it provides a comprehensive view of the trade-offs involved in the decision-making process [31].

In the context of EAs, the multi-objective nature of an optimization problem often appears irrelevant during certain steps. Operations such as random initialization of solutions, mutation, crossover, and recombination generally operate solely within the solution space, without directly considering the objective functions. However, the selection process—a critical step in EAs—explicitly involves the objective function(s) via the fitness function. Frequently, the fitness function mirrors the objective function(s), making selection dependent on relative fitness values.

Selection mechanisms play a crucial role in guiding the search process of MOEAs. They determine which individuals from the population are chosen for reproduction based on their fitness relative to multiple objectives. Effective selection strategies not only enhance the convergence towards the Pareto front but also ensure a diverse set of solutions, which is essential for exploring the trade-offs among conflicting objectives. As the field of MOEAs evolves, various innovative selection techniques have emerged, each aiming to improve the efficiency and effectiveness of the optimization process [26]. Common selection strategies include: 1)

Tournament Selection: Tournament Selection is a popular method in evolutionary algorithms where a subset of individuals is randomly chosen from the population, and the best individual among them is selected for reproduction. This method introduces a competitive aspect to selection, allowing for a balance between exploration and exploitation. The size of the tournament can be adjusted to control selection pressure; larger tournaments favor stronger individuals, while smaller tournaments maintain diversity by giving weaker individuals a chance to be selected [32].

Table 3 provides a clear overview of different selection strategies used in MOEAs, highlighting their respective strengths and weaknesses. This can help understanding the trade-offs involved in choosing a selection strategy for each specific optimization problem.

EAs adapt well to MOOPs due to their population-based nature, which enables simultaneous exploration of multiple solutions. However, traditional EA selection mechanisms are unsuitable for multi-objective settings, as solutions can be incomparable. These strategies rely on the ability to clearly determine whether one solution is superior to another. In MOO, however, the comparison is complicated by the possibility of incomparability. For two solutions, and the relationship (dominates) or might not hold; instead, and could be incomparable.

To address these challenges, early researchers (e.g., Fonseca and Fleming, 1995) explored scalar-valued fitness functions that allowed standard selection techniques to be applied. Two primary approaches emerged: as discussed in the following.

Both decomposition-based methods and aggregation methods are strategies used in multi-objective optimization to simplify the problem by transforming multiple objectives into a single, or a set of single, objective problems. However, they differ fundamentally in their approach and how they handle the objectives.

Aggregation Methods (Scalarization): Aggregation methods, also known as scalarization techniques, combine multiple objectives into a single objective function. This is typically achieved by assigning weights to each objective and summing them up, or by using other mathematical formulations. The goal is to transform the multi-objective problem into a single-objective problem that can then be solved using traditional optimization techniques. Basic concept of how they work are as follows:

Simple Additive Weighting (SAW): This is the most common form, where objectives are multiplied by predefined weights and then summed. For example, for objectives f1(x),f2(x),…,fq(x), the aggregated objective would be F(x)=w1f1(x)+w2f2(x)+…+wqfq(x), where wi are the weights.

Reference-point Approaches: These methods define a “reference point” (e.g., an ideal or utopia point) and minimize the distance to this point.

Utility Functions: A utility function is constructed to represent the decision-maker’s preferences over different objective values.

Such approaches can be characterized as follows:

Simplicity: They simplify the optimization process by reducing it to a single-objective problem.

Dependence on Decision Maker Input: They often require the decision-maker to specify preferences (e.g., weights or reference points) before the optimization begins.

Concentration Bias: A significant drawback is their tendency to bias solutions towards specific regions of the Pareto front, especially convex ones, and they may fail to find solutions in concave regions of the Pareto front. This means they might not be able to generate a diverse set of Pareto-optimal solutions, potentially limiting the exploration of the entire Pareto front.

Decomposition-based Methods: Decomposition-based methods, exemplified by MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition), approach the multi-objective problem by decomposing it into a set of scalar optimization subproblems. These subproblems are then optimized simultaneously and collaboratively. Decomposition-based methods work as follows:

Subproblem Creation: The multi-objective problem is transformed into a number of single-objective subproblems, often using a set of weight vectors or reference points. Each subproblem is designed to optimize a specific aspect of the overall multi-objective problem.

The methods can be characterized as follows:

Scalability: They are particularly effective in handling many-objective problems (problems with a large number of objectives) because they transform the problem into a set of simpler subproblems, reducing the computational complexity associated with high-dimensional objective spaces.

Table 4 summarizes the key differences between aggregation methods (scalarization) and de-composition-based methods.

In essence, while both methods aim to simplify multi-objective problems, aggregation methods directly combine objectives into one, often requiring prior knowledge of preferences and potentially limiting the exploration of the Pareto front. Decomposition-based methods, on the other hand, break the problem into smaller, interconnected subproblems, allowing for more effective exploration of the Pareto front, especially in high-dimensional and complex scenarios.

Pareto-based methods, such as Non-dominated Sorting Genetic Algorithm (NSGA)-II [36,44], are known for their efficiency and widespread application. NSGA, despite its conceptual strengths, suffered from high computational cost due to nested loops in its sorting procedure and lacked an explicit mechanism to maintain diversity. NSGA-II was proposed to address both of these issues: it introduced a fast non-dominated sorting algorithm and the crowding distance concept for diversity preservation, significantly improving scalability and efficiency in practice.

Pareto-based ranking [45,46] is a fundamental concept in MOEAs that prioritizes solutions based on Pareto dominance. Instead of relying solely on fitness values, solutions are ranked according to their ability to dominate others in terms of multiple objectives. This approach promotes a diverse set of solutions that represent different trade-offs among objectives. In multi-objective optimization problems in engineering design, Pareto-based ranking is widely used to identify optimal de-signs that balance performance metrics. For instance, in structural optimization, engineers can use Pareto-based ranking to select designs that minimize weight while maximizing strength.

Decomposition-based methods, such as MOEA/D [47], decompose the multi-objective problem into a series of scalar optimization subproblems, which are then optimized in a collaborative manner. Traditional Pareto-based algorithms like NSGA-II and SPEA rely on dominance comparisons, which become computationally expensive and less effective in many-objective scenarios. MOEA/D was introduced to address this by decomposing a multi-objective problem into a set of scalar subproblems and optimizing them simultaneously. This allowed for better scalability and a structured search, particularly useful in high-dimensional problems. This approach allows for more efficient exploration of the Pareto front and better coverage of the solution space. MOEA/D has been successfully applied in complex engineering problems, such as multi-objective scheduling in manufacturing, where different objectives (e.g., minimizing completion time and maximizing resource utilization) need to be optimized simultaneously [48,49].

Most existing MOEAs used population-based approaches, which could be memory- or computation-intensive. Pareto Archived Evolution Strategy (PAES) provided a minimalist alternative using a single-solution evolution strategy and a Pareto archive to retain diversity. It targeted simplicity and efficiency while still achieving good convergence and coverage in smaller or resource-constrained environments.

Indicator-based methods utilize performance metrics to guide the selection process in MOEAs. Recent works have proposed using indicators such as hypervolume and generational distance to evaluate the quality of the approximated Pareto front. These metrics serve as alternative criteria for selection, enabling algorithms to focus on improving the overall quality of solutions rather than merely exploring the search space [55]. Indicators like the hypervolume indicator have been particularly effective in guiding the optimization process toward better-converged solutions, as demonstrated in recent applications in resource allocation [54].

Recent advancements have integrated performance metrics directly into the selection process of MOEAs. Techniques such as hypervolume maximization and Pareto spread minimization have been employed to enhance the effectiveness of the search process. These methods allow for a more nuanced evaluation of solutions, leading to improved convergence and diversity [56,57]. Practical applications in engineering design often leverage these techniques to achieve optimal trade-offs among competing objectives, showcasing their effectiveness in real-world scenarios [58].

Preference-based evolutionary optimization has emerged as a significant advancement in MOEA research. These methods incorporate user-defined preferences during the optimization process, allowing decision-makers to influence the search towards more desirable solutions. Recent studies have highlighted the potential of artificial preference relations, such as reference directions or preferred areas of solutions, to enhance exploration efficiency and user satisfaction [59]. Examples include applications in resource allocation problems where user preferences significantly impact solution selection, demonstrating the practical utility of these methods [54].

Hybrid methods that combine MOEAs with other optimization techniques, such as swarm intelligence and local search algorithms, are becoming more prevalent. These combinations can enhance the search capabilities of MOEAs, enabling them to tackle more complex and high-dimensional optimization problems effectively [54,60–62]. A recent study proposed a Hybrid Selection based MOEA (HS-MOEA) that combines dominance, decomposition, and indicator-based strategies to enhance the balance between diversity and convergence in multi-objective optimization problems. This approach demonstrated superior performance on various test suites, including DTLZ and WFG [54]. A notable study by Singh & Chaturvedi [60] integrated Particle Swarm Optimization with MOEAs to enhance convergence speed while maintaining solution diversity.

Combining evolutionary methods with machine learning or metaheuristics to enhance performance, improve convergence, and solution quality. For example, combining MOEAs with reinforcement learning has demonstrated promising results in dynamic optimization environments. While MOEA/D offered a powerful decomposition strategy, it lacked fine-tuning capabilities in local regions. MOEA/D-H hybridized MOEA/D with local search methods to combine global exploration with local exploitation. This hybrid approach improved solution quality in complex real-world problems with rugged or discontinuous landscapes. As optimization problems became more dynamic and data-driven, classical MOEAs showed limitations in adaptability and computational efficiency. The integration of machine learning techniques (e.g., neural networks, surrogate models, reinforcement learning) into MOEAs aimed to guide the search process more intelligently, improve convergence in real-time applications, and handle expensive objective evaluations through prediction and adaptation mechanisms.

MOOPS, which involve more than three objectives, introduce significant challenges due to the exponential growth in the number of Pareto-optimal solutions. This phenomenon complicates the search for optimal solutions and necessitates the development of specialized techniques. The term “many-objective optimization” has been introduced to characterize these complex problems, emphasizing the need for innovative approaches to maintain solution diversity and convergence [70,71]. The implications of many-objective optimization extend beyond mere solution count; they affect computational resources and the effectiveness of algorithms in converging to a diverse set of optimal solutions.

As the complexity of real-world problems increases, many applications now involve more than three or four conflicting objectives. This shift has given rise to the field of MaO, typically referring to optimization problems with four or more objectives. The conventional MOEA frameworks, especially Pareto-based methods like NSGA-II and SPEA2, often face scalability issues in such contexts due to challenges like loss of selection pressure, insufficient diversity, and difficulties in Pareto ranking when most solutions tend to become non-dominated.

This has led to the development of EMaO algorithms, which extend MOEAs with specialized techniques for high-dimensional objective spaces. Examples include:

NSGA-III, which introduces reference points to maintain diversity in many-objective problems.

Additionally, machine learning techniques and large language models (LLMs) have been recently incorporated into EMaO frameworks to guide the search process intelligently, predict performance, or reduce computational complexity. The growing importance of EMaO is reflected in the increasing number of benchmark suites (e.g., MaF test problems), dedicated workshops, and focused research on real-world, large-scale optimization tasks in fields such as energy systems, supply chains, and bioinformatics.

The application of MOEAs in real-time decision-making scenarios is on the rise. Researchers are exploring how MOEAs can be utilized in systems requiring immediate responses, such as autonomous systems and smart cities. This trend emphasizes the need for algorithms that can deliver quick and reliable solutions in dynamic and uncertain environments [72–74].

Large-Scale Decision Variable Analysis [75,76]: Research on an algorithm denoted as LMEA-DVQA highlights the importance of decision variable analysis in improving the performance of multi-objective evolutionary algorithms. This study compares several algorithms and emphasizes the effectiveness of the proposed method in various test scenarios [20].

Fairness, transparency, and ethical implications in algorithmic decisions are becoming key concerns. With the growing importance of ethical considerations in decision-making, future research may also address the fairness and equity of solutions generated by MOEAs. Future MOEA research will likely incorporate ethical criteria directly into fitness functions or develop mechanisms to ensure equity and sustainability in generated solutions. Ensuring that optimization processes consider social and ethical implications will be vital in areas such as resource allocation and environmental sustainability [73,77–79].

The utilization of parallel computing resources, including GPUs and distributed architectures, has significantly improved the scalability of MOEAs. Recent advancements emphasize the importance of leveraging parallelization to enhance computational efficiency, particularly in high-dimensional optimization problems [58]. This trend is expected to continue as computational re-sources become more accessible and powerful.

While Evolutionary Multi-Objective Optimization (EMO) techniques are adept at generating a diverse set of trade-off solutions (Pareto fronts), the ultimate selection of a single solution or a subset of solutions often involves a human decision-maker. This process bridges the gap between optimization and real-world application. As such, the integration of decision-making mechanisms into EMO algorithms has become a key area of interest.

There are three main paradigms for incorporating decision-making in EMO:

A priori approaches, where decision-makers specify preferences (e.g., weights, aspiration levels) before the optimization begins.

Recent advancements include preference-based EMO algorithms (e.g., reference-point-based NSGA-III) and interactive EMO systems, which employ machine learning or surrogate models to predict and incorporate user preferences on the fly. These approaches enhance the practical usability of EMO methods, especially in domains such as environmental policy planning, engineering design, healthcare, and financial portfolio selection.

Incorporating decision-making into EMO frameworks also facilitates the resolution of many-objective problems, where visualizing or interpreting large Pareto sets becomes increasingly difficult. Decision-making support systems, visualization tools, and dimensionality reduction techniques now play a pivotal role in making EMO results actionable.

Bibliometric analysis provides valuable insights into the historical trajectory and scholarly attention surrounding the development of MOEAs. While previous sections have discussed the algorithmic innovations in MOEAs from a technical standpoint, this section complements that perspective by revealing how such developments are reflected in the research landscape through publication trends, journal focus, and geographic distribution.

The annual volume of publications on MOEAs across major academic databases (ScienceDirect, IEEE Xplore, Web of Science) demonstrates a significant growth trend over the past two decades. This increase correlates with key algorithmic breakthroughs such as the introduction of NSGA-II, MOEA/D, and recent hybrid and deep-learning-based MOEAs, showing a tangible link between algorithmic milestones and scholarly output.

Moreover, the distribution of MOEA-related research across journals reveals the interdisciplinary nature of these algorithms. Journals focusing on evolutionary computation, soft computing, applied mathematics, operations research, and domain-specific applications (e.g., energy, healthcare, logistics) have all contributed to shaping the algorithmic evolution of MOEAs. This trend underlines how methodological advancements have been driven both by theoretical interest and practical demand.

Highly cited authors and institutions identified through citation analysis also mirror the timeline of algorithmic development. For instance, the rise in citations of foundational works by Deb (e.g., NSGA-II) and Zhang (e.g., MOEA/D) corresponds to the widespread adoption of these algorithms in subsequent research. Similarly, countries with significant contributions—such as India, China, Germany, and the USA—have played central roles in both theoretical and application-oriented advancements.

Thus, rather than merely presenting descriptive bibliometric statistics, this section contextualizes the scholarly evolution of MOEAs in parallel with their technical advancements. The bibliometric patterns observed serve as a macro-level indicator of how the field has matured, diversified, and responded to new computational and societal challenges.

In this section, the visibility of MOEAs in the ScienceDirect website is explored. By the search of (“Multi-objective Evolutionary Algorithm”) keywords in this science base, the following results were obtained. The number of journal papers in each year is shown in Fig. 2:

The number of all journal papers in this field has been 6955 papers that quotas of different journals are as follows and illustrated in Fig. 3: Applied Soft Computing (699), Expert Systems with Applications (473), Swarm and Evolutionary Computation (449), Information Sciences (383), Computers & Industrial Engineering (238), Engineering Applications of Artificial Intelligence (202), Energy (195), Knowledge-Based Systems (170), Computers & Operations Research (124), Neurocomputing (123), Applied Energy (120), Journal of Cleaner Production (112), Energy Conversion and Management (107), Europe-an Journal of Operational Research (104), International Journal of Electrical Power & Energy Systems (90), Journal of Hydrology (86), Renewable Energy (65), Applied Thermal Engineering (64), Procedia Computer Science (61), Renewable and Sustainable Energy Reviews (60), IFAC-Papers On-Line (56), Environmental Modelling & Software (55), Future Generation Computer Systems (54), Ocean Engineering (52), Reliability Engineering & System Safety (49).

An increasing number of publications per year are observed that can allow us to say the field of MOEAs has now reached a stage of maturity after the earliest papers published at 2001, and there are also many basic issues yet to be resolved and there is an active and vibrant worldwide community of researchers working on these issues.

We repeated the search this time with the keywords of (“Multi-objective” and “Evolutionary Algorithm”) keywords in this ScienceDirect website. The number of all journal papers in this field is 20,814 papers and the number of Journal papers in each year is as Fig. 4. In this case, the quotas of different journals are as follows and illustrated in Fig. 5.

By similar search of (“Multi-objective Evolutionary Algorithm”) keywords in IEEE website, the following results were obtained. From 2003 up to 2025: Conference Publications (980), Journals & Magazines, Early Access Articles (234). The time extension of publications in IEEE is as follows (Fig. 6):

2003–2010: Conference Publications (245), Journals & Magazines, Early Access Articles (9).

2011–2020: Conference Publications (481), Journals & Magazines, Early Access Articles (98).

2021–2024: Conference Publications (239), Journals & Magazines, Early Access Articles (127).

The number of all journal papers in this field has been 234 papers that quotas of different journals are as follows and illustrated in Fig. 7.

We repeated the search this time with the keywords of (“Multi-objective” and “Evolutionary Algorithm”) keywords in IEEE too. The number of all journal papers in this field has been 1195 papers and the number of Journal papers in each time extension is as Fig. 8. The quotas of different journals in this case are illustrated in Fig. 9.

As can be seen in Figs. 6 and 8, although the final time extension covers a shorter range (only 2021 to 2025), the number of journal articles in this field has grown compared to previous larger time extensions, indicating the growing attention of the scientific community to this research field.

The authors with the most publication in IEEE, are brought in the following: Yaochu Jin(62), Qingfu Zhang (44), Xingyi Zhang (39), Kay Chen Tan (39), Ye Tian (35), Gary G. Yen (33), Jun Zhang (26), Xin Yao (24), Maoguo Gong (23), Ling Wang (21), Shengxiang Yang (20), Qiuzhen Lin (19), Hisao Ishibuchi (19), Hai-Lin Liu (19), Lei Zhang (19), Dunwei Gong (18), Ran Cheng (17), Yong Wang (16), Aimin Zhou (16), Jing Liu (15), Licheng Jiao (15), Fan Cheng (14), Ke Tang (13), Zexuan Zhu (13), Carlos A. Coello Coello (13).

Also in these years, the number of papers in IEEE conferences with this topic in different countries has been as: China (350), Canada (80), Australia (67), USA (60), Japan (55), Singapore (50), Spain (49), UK (41), New Zealand (38), Mexico (28), Poland (28), Norway (27), Brazil (24).

By search of (“Multi-objective Evolutionary Algorithm”) keywords in ISI WOS, 3241 documents are found which include 2062 Article, 1220 Proceeding Paper, 26 Early Access, 23 Review Article, 7 Book Chapters, 4 Retracted Publication, 2 Editorial Material, 2 Meeting Abstract, 1 Correction, and 1 Letter. Also, the number of published papers in each year in this database is shown in Fig. 10.

In this knowledge database, the authors with the most published papers are: Coello, Carlos A Coello (62), Kim, Kwang-Yong (29), Koziel, Slawomir (23), Tan, Kay Chen (22), GAO, Liang (21), Jin, Yaochu (20), Wang, Ling (19), Bekasiewicz, Adrian (18), London, Joao Bosco Augusto (17), Delbem, Alexandre C B (17), Yao, Xin (17), While, L. (16), Zhang, Hui Jie (16), Jiao, Licheng (16), Rui, Wang (16), Cheng, Fan (15), Husain, Afzal (13), Guo, Xiwang (13), Liu, Jing (13), Konstantinidis, Andreas (13).

The search on the ISI Web of Science allows us to get the most cited papers that can provide a picture on the important contributions on the topic that are representative approaches of different categorization areas. One notable point about the most cited articles is that 4 of the 6 most cited articles were published in the Journal of Evolutionary Computation.

Regarding the search methodology, let us mention that the citation data reported in this subsection is based on a keyword-specific bibliometric search in the ISI Web of Science database using the exact term “Multi-objective Evolutionary Algorithm”. This keyword was chosen to ensure consistency and specificity in identifying publications that explicitly fall under the MOEA category. However, it is important to note that some of the most impactful publications in the broader EMO field—such as the NSGA-II paper by Deb et al. (2002)—may not contain this exact phrase in their title or abstract and thus may not appear at the top of this particular search. To address this limitation and better reflect the true academic impact, we have included a dedicated subentry for the NSGA-II paper, which, as of 2024, has accumulated over 33,000 citations, making it the most cited paper in the field of evolutionary computation.

NSGA-II (33,478 Citations) [36]: This seminal paper by Deb et al. (2002) introduced the NSGA-II algorithm, which significantly improved upon earlier multi-objective genetic algorithms in terms of computational efficiency and diversity preservation. Although the paper may not explicitly use the phrase “Multi-objective Evolutionary Algorithm” in its title or keywords, it is universally recognized as one of the foundational and most impactful works in the EMO domain, and as such, it is included here outside of the strict keyword-based filtering for the sake of completeness and clarity.

HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization (1532 Citations) [45]: This paper addresses the problem of the high computational effort required to calculate Hyper-volume (as the only single set quality measure that is known to be strictly monotonic with regard to Pareto dominance), which has prevented the full exploitation of the potential of this index. It proposes a fast search algorithm that uses Monte Carlo simulation to approximate the exact values of Hypervolume. The main idea is based on the ranking of solutions induced by the Hyper-volume index, rather than the actual values of the indicator. As a result, HypE is presented in detail as an estimation algorithm for MOO that reduces the available computational resources, increases the accuracy of the estimates, and provides a trade-off for the execution times.

Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization (1137 Citations) [80]: In this paper, for the first time, the optimization of MOOP using the Gray Wolf Optimizer (GWO) algorithm is investigated. For this purpose, the Multi-Objective Gray Wolf Optimizer (MOGWO) is introduced by integrating a fixed-size external archive for saving and retrieving Pareto-optimal solutions with GWO. This archive is used to define the GWO optimization parameters such as social hierarchy and simulate the hunting behavior of wolves. It shows improved accuracy compared to some other previous MOEAs.

Borg: An Auto-Adaptive Many-Objective Evolutionary Computing Framework (536 Citations) [81]: This paper introduces a method for many-objective, multimodal optimization by combining epsilon dominance, epsilon-progress as a measure of convergence speed, randomized restarts, and auto-adaptive multi-operator recombination in a unified optimization framework called Borg MOEA. Borg MOEA represents a class of algorithms whose operators are adaptively selected based on the problem and it is not a single algorithm.

MOEA/D with Adaptive Weight Adjustment (523 Citations) [82]: This paper introduces a MOEA based on decomposition (MOEA/D) with Adaptive Weight Vec-tor Adjustment (MOEA/D-AWA) to deal with the complex Pareto front in target MOOP. The considered MOOP is decomposed into a set of scalar subproblems using uniformly distributed aggregation weight vectors. Then, by analyzing the geometric relationship between weight vectors and optimal solutions under the Chebyshev decomposition scheme, a novel weight vector initialization method and an adaptive weight vector adjustment strategy with periodic weight adjustment capability are proposed.

Strategies for finding good local guides in multi-objective particle swarm optimization (MOPSO) (499 Citations) [83]: This paper introduces the Sigma method as a new method for finding the best local guides (global best particle) for each particle in a population of a set of Pareto optimal solutions in Multi-Objective Particle Swarm Optimization (MOPSO). This selection method has a significant impact on the convergence and diversity of solutions, especially when optimizing problems with a large number of objectives.

Evaluating the ε-domination based multi-objective evolutionary algorithm for a quick computation of pareto-optimal solutions (498 Citations) [84]: This paper attempts to present a computational solution for Pareto-optimal solutions to MOOP based on the concept of epsilon dominance, such that offers a good compromise in terms of con-vergence close to the Pareto-optimal front, solution diversity, and computational time. This method allows decision makers to control the achievable accuracy of the obtained Pareto-optimal solutions.

While the bibliometric analysis in this section was conducted using the specific keyword “Multi-objective Evolutionary Algorithm,” we acknowledge that this approach may not have captured all seminal publications and contributors in the broader EMO and MCDM communities. To address this, we highlight below the contributions of several prominent researchers whose work has significantly shaped the field:

Kalyanmoy Deb: Widely regarded as a foundational figure in EMO, he introduced several key algorithms including NSGA and NSGA-II [36,44], which are central to the evolution of MOEAs. His work is extensively cited and referenced throughout this paper.

We recognize that a more comprehensive representation of the field may require extending the keyword strategy to include alternative terms like “evolutionary multi-objective optimization,” “multi-criterion decision-making,” or “interactive EMO.” This limitation is acknowledged and will be addressed in future extensions of this study.