A Review Study of Fuzzy Cognitive Maps in Engineering: Applications, Insights, and Future Directions

Abstract

Fuzzy Cognitive Maps (FCMs) have emerged as powerful tools for addressing diverse engineering challenges, leveraging their cognitive nature and ability to encapsulate causal relationships. This paper provides a comprehensive review of FCM applications across 15 engineering sub-domains, categorizing 80 studies by their learning family, task type, and case-specific application. We analyze the methodological advancements and practical implementations of FCMs, showcasing their strengths in areas such as decision-making, classification, time-series, diagnosis, and optimization. Qualitative criteria are systematically applied to classify FCM-based methodologies, highlighting trends, practical implications of varying complexity, and human intervention across task types and learning families. However, this study also identifies key limitations, including scalability challenges, reliance on expert knowledge, and sensitivity to data distribution shifts in real-world settings. To address these issues, we outline key areas and directions for future research focusing on adaptive learning mechanisms, hybrid methodologies, and scalable computational frameworks to enhance FCM performance in dynamic and evolving contexts. The findings of this review offer a structured roadmap for advancing FCM methodologies and broadening their application scope in both contemporary and emerging engineering domains.

1. Introduction

Artificial intelligence (AI) has become a cornerstone in addressing complex challenges across the engineering and manufacturing domains [1]. From optimizing production lines to enhancing system reliability, AI techniques such as machine learning (ML) and deep learning (DL) are reshaping traditional methods, offering innovative solutions to dynamic and chaotic environments. The capacity of AI to analyze large-scale data, learn from intricate patterns, and make autonomous decisions has led to significant improvements in efficiency, sustainability, and cost-effectiveness in engineering processes [2]. Among soft computing methodologies, Fuzzy Cognitive Maps (FCMs) [3] have emerged as a versatile approach for tackling intricate, dynamic problems. Combining elements of fuzzy logic, neural networks, and expert systems, FCMs offer a unique blend of knowledge interpretability, dynamic adaptability, and learning capability. These attributes position them as both simulation tools and adaptive systems capable of responding to evolving challenges.

In the broader context of artificial intelligence (AI), transparency and explainability are becoming as crucial as performance [4]. While cutting-edge techniques like machine learning (ML) and deep learning (DL) often dominate, their opaque decision-making processes present limitations in areas requiring interpretability. FCMs, by contrast, encapsulate causal relationships within graph-based structures, enabling stakeholders to visualize and understand the interplay between system components. Their ability to incorporate human knowledge, adapt through learning mechanisms, and evolve dynamically further enhances their suitability for applications demanding both interpretability, adaptability, and reliable performance.

From a computational perspective, FCMs can be considered as single-layer recurrent neural networks equipped with squashing activation functions, typically sigmoid ones. Their iterative updating process continues until a convergence criterion is met, resulting in a fixed-point attractor. While undesirable behaviors such as limit cycles or chaotic attractors can occur, proper convergence criteria and design considerations can mitigate these challenges [5,6]. This dynamic nature, coupled with enhanced interpretability, makes FCMs a powerful tool for real-world applications.

FCMs have been widely applied in engineering domains, addressing diverse challenges such as time-series prediction [7,8,9,10], classification [10,11,12,13], decision making [14,15,16], and diagnostics [17,18,19,20]. Their capacity to model non-linear relationships and adapt to changes has made them valuable in fields as varied as energy systems, transportation, medicine, environmental science, educational technologies, finance, business, and industrial automation. However, as FCM applications grow, there is a need for systematic analysis to assess their capabilities, limitations, and potential for future advancements.

Over the years, numerous review studies within the FCM research domain have successfully explored various aspects and perspectives of these versatile methodologies. A representation of knowledge frameworks, their application in decision-making, and their integration with FCMs has been provided in [21]. An outline of FCM models applied to time-series forecasting is presented in [22], while an overview of FCM extensions is discussed in [23]. Other studies have reported on extensions, learning methods, and diverse applications of FCMs [24,25], with a specific focus on learning algorithms found in [26,27]. An extended list of FCM variants and extensions is present in the literature, including but not limited to Fuzzy Grey Cognitive Maps (FGCM) [28], Intuitionistic Fuzzy Cognitive Maps (iFCM) [29], Fuzzy Cognitive Networks (FCNs) [5,30], Rough Cognitive Maps (RCM) [31], Neutrosophic Cognitive Maps (NCMs) [32], Dynamic Random Fuzzy Cognitive Maps (DRFCMs) [33], High-order Fuzzy Cognitive Maps (HFCMs) [9,34], Sylov’s FCMs [35,36], and others.

The strengths and limitations of FCMs for future studies are assessed in [37], while advances and challenges in FCM-based models for pattern classification are summarized in [38]. A review of FCM methods and software is provided in [39], and an overview of their use in systems risk analysis is presented in [40]. The role of FCMs in Explainable Artificial Intelligence (XAI) is analyzed in [41], and a systematic review of their applications in medicine over the last two decades is found in [42]. Additionally, experiences and lessons learned in using FCMs in participatory research and decision-making across health-related contexts are shared in [43].

This review paper provides a comprehensive overview of the applications of FCMs in engineering, systematically analyzing 80 published works across 15 identified sub-domains. Our contributions include:

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A comprehensive literature review categorizing FCM applications in engineering over the past two decades.

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A multi-perspective analysis of the reviewed works, organized by learning families, task types, and sub-domain-specific applications.

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Critical insights and qualitative criteria for evaluating complexity and human intervention, complemented by radar plots to visualize these dimensions.

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Future directions for advancing FCM methodologies and expanding their impact in engineering applications.

By presenting a structured analysis and identifying opportunities for innovation, this paper aims to serve as a foundation for further research in FCMs, fostering their continued evolution as a reliable and interpretable tool for tackling complex engineering challenges. The rest of the paper is structured as follows: Section 2 introduces the foundational aspects of FCMs, including learning families, their applications across various domains, and key limitations. Section 3 outlines the methodology for the literature review, providing an analysis of the reviewed works and distributions of publishers. Section 4 categorizes the identified subdomains in engineering, offering visual breakdowns based on learning families and task types. Section 5 delves into the subdomain-specific applications of FCMs in engineering, showcasing practical implementations. Section 6 presents the discussion, including a critical analysis of applications, qualitative criteria for complexity and human intervention, and potential future directions. Finally, Section 7 concludes the paper, summarizing the findings and insights.

2. Fuzzy Cognitive Maps

2.1. Foundational Aspects of Fuzzy Cognitive Maps

Fuzzy Cognitive Maps (FCMs) are referred to as human knowledge-based inference networks, cyclic directed weighted graphs, cognition influence graphs, and recurrent neural networks. These definitions-descriptions complete the graphical, computational, and memory state representation of FCMs. They have been used in many scientific fields for various tasks such as modeling, control, pattern recognition applications, decision making, and forecasting, highlighting their capability, flexibility, and effectiveness. FCMs were introduced by Kosko [3], based on Axelrod’s work on cognitive maps [44], in order to model complex behavioral systems using causal relations. One of the unique aspects of FCMs is their ability to integrate expert domain knowledge into the model, facilitating a hybrid intelligence approach. This feature enhances their transparency, as the nodes (concepts) and the weighted edges connecting them explicitly represent relationships and causal influences pertinent to the problem being addressed.

The concept of Cognitive Maps (CMs) was first introduced by Axelrod, as mentioned, who applied this concept to decision-making processes in political science by modeling cause-and-effect relationships between concepts. In a CM, concepts are represented as nodes, and causal relationships between any two concepts are shown as directed edges. These causal links can have positive or negative weights, where a positive link implies that an increase in one concept activates another, and a negative link implies the opposite effect. The basic CM formulation consists of a binary system, where concept states $A_i\in \{0,1\}$ and the weights $w_{ij}\in \{-1,0,1\}$ represent either positive, null, or negative relations.

FCMs extend CMs by incorporating fuzzy logic, allowing for more complex and uncertain relationships between concepts. In FCMs, the states of the concepts $A_i\in [0,1]$ are continuous, and the causal relationships between concepts are described by fuzzy values $w_{ij}\in [-1,1]$. The weight matrix W, which is an essential part of FCM, defines the influence each concept has on others. The values in this matrix quantify the intensity of these influences, with positive values ( $w_{ij}>0$) denoting a positive relationship and negative values ( $w_{ij}<0$) indicating an inverse relationship. A weight of zero ( $w_{ij}=0$) implies no direct influence between the two concepts.

These network implementations inherit selected properties from both fuzzy logic and neural networks. FCMs naturally handle different system elements such as the variables, concepts, states, and goals of the system they describe, and express their dynamical relationships, degrees of interaction, and causal relations between them through signed weighted arcs. In standard FCMs, a set of linguistic rules is defined capturing the qualitative relationships and the general behavior of the system under examination. Additionally, these networks support feedback connections for each individual concept value, introducing a memory state representation feature through the diagonal values of weight matrix W. However, the iterative update of concept activations continues until the system reaches one of the following states: (a) fixed-point attractor (the network state becomes fixed after a number of iterations); (b) limit cycle (oscillations between state vectors); or (c) chaotic attractor (chaotic behavior). Stability in the system is generally desirable, as convergence to an equilibrium indicates a consistent pattern in the influence of concepts. However, in cases where the system oscillates between states or behaves chaotically, the results may become unreliable. The convergence analysis and conditions have been studied in the literature by various works [5,45,46,47,48].

Graphically, an FCM can be represented as a directed weighted graph as presented in Figure 1a. Each node $C_i$ is a concept (e.g., variables, states, or goals), and the directed edges between nodes are the weighted causal influences quantified by weights $w_{ij}$. More specifically, Figure 1a depicts a small FCM with five nodes and their corresponding causal relationships. The weighted arcs demonstrate varying strengths and polarities (positive or negative), enabling the representation of system dynamics. Figure 1b shows the weight matrix W corresponding to the FCM, where each entry $w_{ij}$ quantifies the influence of node $C_j$ on node $C_i$. At any point in time k, the state of the system is represented by a state vector $\mathbf{A}\left(k\right)=[A_1\left(k\right),A_2\left(k\right),\dots ,A_n\left(k\right)]$, where n is the number of concepts. The update of the state of each concept is governed by a transition rule that accumulates the influence of all other connected concepts, such as Kosko’s rule, which is one of the most commonly applied. The rule is expressed as:

$$A_i(k+1)=f\left(\sum _{j=1,j\ne i}^n{w}_{ij}·A_j\left(k\right)\right)$$

(1)

where $A_i\left(k\right)$ is the activation level of concept $C_i$ at time k, $w_{ij}$ represents the causal influence of concept $C_j$ on concept $C_i$, and $f(·)$ is a squashing function. The latter is a nonlinear continuous function, called activation function, mapping the weighted sum of the node states. The sigmoid and hyperbolic tangent functions are mainly used as continuous transfer function candidates, but discrete ones can be used also.

In standard FCMs, the knowledge base is composed of fuzzy rule databases that capture the qualitative relationships between concepts. These rules try to express the system’s operational conditions and behavior. The linguistic rules encode conditional relationships between concepts in the form of “if-then” statements. The rules can be defined by human experts, generated in a data-driven manner, or created as a combination of both approaches. For example, in Figure 1c, if concepts $C_1$, $C_2$, and $C_3$ have specific activation levels, their combined influence on $C_5$ can be represented by the corresponding fuzzy weights $w_{52},w_{53},w_{54},w_{55}$. This representation may be derived from an expert-defined rule. In contrast, if the weights were learned in a data-driven manner, the example in Figure 1b would incorporate concepts $C_2$, $C_3$, $C_4$, and $C_5$ in the “if” part of the rules shown in Figure 1c.

Once the FCM structure is established, the next step involves training the model to optimize the weights $w_{ij}$ and improve its predictive capabilities. The process of learning these fuzzy weights $w_{ij}$ often requires a training phase to improve the FCM’s accuracy and stability. Depending on the application, this training can be performed using historical data, simulation scenarios, or hybrid approaches that combine expert-defined rules with data-driven learning algorithms. Various learning methods, such as Hebbian-based, Gradient-based, Metaheuristic, or Hybrid approaches, can be employed to adjust the weights iteratively, ensuring that the model reflects the underlying system dynamics and operational conditions effectively. Overall, FCMs provide a powerful and flexible framework for modeling complex systems by combining human knowledge with learning algorithms, enabling both interpretability and adaptability to real-world applications.

2.2. Learning Families in Fuzzy Cognitive Maps

Fuzzy Cognitive Maps can be categorized into distinct learning families based on the underlying training algorithms employed. These learning approaches are designed to adjust the weights of the FCM model, enhancing its ability to represent dynamic systems or improve its performance in predictive tasks. Below is a brief description of the major learning families used in FCMs.

Table 1. Advantages and disadvantages of learning families in FCMs.

Learning Family	Advantages	Disadvantages
Rule-Based	Leverages domain expertise for high interpretability.	Lacks adaptability to dynamic environments.
	Suitable for systems with clear and deterministic rules.	Relies heavily on expert knowledge, making it resource-intensive.
	Handles uncertainty effectively in decision-support systems.	Limited scalability for complex systems.
Hebbian-Based	Inspired by biological processes, enabling unsupervised learning.	Prone to instability due to over-reinforcement of weights.
	Simple to implement with minimal computational requirements.	May fail to capture complex relationships beyond linear correlations.
	Effective for reinforcing strong correlations.	Prone to weight saturation, reducing the model’s capacity to adapt to new data.
Metaheuristics	Effective for solving nonlinear, high-dimensional optimization problems.	Computationally expensive, especially for large systems.
	Capable of escaping local optima through global search strategies.	Results may vary due to stochastic nature.
	Broad applicability across diverse domains.	Requires careful tuning of hyperparameters.
Gradient-Based or Gradient-Like	Suitable for optimizing weights based on a clear loss function.	Computationally intensive for large datasets or deep networks.
	Widely applicable in predictive and classification tasks.	Sensitive to hyperparameters like learning rate and momentum.
	Supported by well-established mathematical foundations.	May get stuck in local minima.
Gradient-Free	Faster convergence in scenarios where gradients are difficult to compute.	Less precise compared to gradient-based methods.
	Effective for high-dimensional and complex landscapes.	May require a large number of iterations to achieve optimal performance.
	Avoids issues with gradient vanishing or exploding.	Limited theoretical guarantees for convergence.
Hybrid	Combines strengths of multiple methods, enhancing robustness.	Complexity increases due to integration of multiple approaches.
	Flexible and adaptable to diverse problem domains.	Challenging to design and tune hybrid frameworks effectively.
	Can balance trade-offs between accuracy, interpretability, and computational efficiency.	May require significant computational resources.

provides a consolidated summary of the potential advantages and disadvantages of the learning families used in FCMs.

2.2.1. Rule-Based Learning

Rule-based learning involves systems that rely on predefined expert rules to make decisions or infer outcomes. Unlike other learning methods, these systems do not always optimize based on data, but instead follow a set of established rules to govern behavior. This category is particularly relevant for Rule-Based Fuzzy Cognitive Maps (RBFCMs), where the weights are determined through expert knowledge rather than data-driven or other learning mechanism.

2.2.2. Hebbian-Based Learning

Rooted in Hebbian theory, this family of algorithms adjusts the strength of connections between concepts based on the correlation of their activities. Hebbian-based learning operates on the principle that “neurons that fire together, wire together”, meaning that the connections between highly correlated concepts are reinforced. This approach is biologically inspired and is often used in unsupervised learning scenarios.

2.2.3. Metaheuristics

Metaheuristic approaches encompass both population-based and single-solution-based optimization techniques. These methods typically employ heuristic or stochastic processes to explore the solution space and identify optimal or near-optimal weight configurations for the FCM. Metaheuristics are particularly useful in handling complex, nonlinear problems where the solution landscape is vast and may contain multiple local optima.

2.2.4. Gradient-Based or Gradient-like Learning

These methods utilize gradient information to optimize the parameters of the FCM by iteratively adjusting the weights in the direction that minimizes a loss function. This family includes well-known algorithms such as backpropagation and stochastic gradient descent (SGD), as well as variants like the Adam optimizer. Although the implementations of gradient-based, gradient-like, and least-squares methods can be distinguished in a more dedicated identification, in this taxonomy, they are mapped under the same cluster, recognizing a common origin with Widrow’s ADALINE. A more specialized analysis could further separate these approaches, but their foundational principle remains aligned.

2.2.5. Gradient-Free Learning

This category includes algorithms that do not rely on gradient information to optimize the model. Gradient-free methods are typically faster and may work well in scenarios where computing gradients is difficult or impractical. These approaches often explore the solution space through random or heuristic techniques, making them suitable for high-dimensional problems with complex landscapes.

2.2.6. Hybrid Learning

Hybrid learning methods combine elements from two or more of the above categories to leverage their respective strengths. By doing so, these approaches can address various aspects of a problem more effectively. For example, hybrid learning may involve integrating gradient-based updates with evolutionary strategies or combining rule-based reasoning with metaheuristic optimization.

2.3. Projection of Fuzzy Cognitive Maps in Various Domains

Fuzzy Cognitive Maps (FCMs) have established their presence across diverse domains by offering a flexible and interpretable framework for modeling complex systems, supporting decision-making, and enabling predictive analysis. Their applications span computational simulation, pattern recognition, time-series prediction, environmental science, medicine, bioinformatics, business, finance, educational technology, and more, addressing domain-specific challenges with notable success.

In computational simulation and modeling, FCMs have been applied to optimize dynamic systems, reduce model complexity, and enhance interpretability. In [49], a reduction in the need for expert intervention was employed using Evolution Strategies (ES) to optimize the parameters of FCMs. Concept reduction techniques, such as Big Bang–Big Crunch (BB–BC) optimization, have been utilized to simplify large-scale FCMs while maintaining decision-making accuracy, particularly in integrated waste management systems [50]. Additionally, [51] introduced a hybrid model that combines the Extreme Learning Machine (ELM) with a curiosity-driven mechanism to adapt FCMs in dynamic environments. This method reduces prediction errors and improves modeling accuracy, showcasing the ability of FCMs to dynamically adjust to changing system conditions. State-space models were integrated into FCMs in [52], focusing on dynamic behavior modeling in complex systems. An application of FCMs in decision-making was demonstrated in [31], introducing Rough Cognitive Networks (RCNs) integrating rough set theory with FCMs. Other works aimed at developing Granular Cognitive Maps (GCMs) in advanced system modeling [53], and optimization techniques have also been introduced in FCM learning [54], highlighting the scalability of FCMs in simulation tasks.

Indicative works in pattern recognition include Fuzzy Cognitive Networks with Functional Weights (FCNs-FW) for both classification and time-series prediction [10], image classification in Caltech-101, Caltech-256, and 15-Scenes using Interpretable Intuitionistic Fuzzy Cognitive Maps (I²FCM) [55], and an interpretable image-based classification framework that automatically determines FCM structure from data called xFCM [13], validated on Caltech-101, COVID CT scans, and iRoads datasets. Hybrid deep learning structures have been combined with FCNs-FW for handwritten digit recognition, stock prediction, and index tracking portfolio management [56]. Other works include Fuzzy Probabilistic FCMs [57] for classifying time-series data from the UCR repository, achieving faster execution and enhanced interpretability, and a speech emotion recognition model using a dedicated FCM (e-FCM) integrated with the Pleasure-Arousal-Dominance (PAD) model, improving speech-based emotion detection on databases like TYUT2.0 and EMO-DB [58].

In time-series analysis, FCMs have been successfully integrated with advanced techniques to improve forecasting accuracy. For instance, High-Order FCMs (HFCMs) combined with empirical wavelet transforms have demonstrated enhanced performance for non-stationary and noisy datasets [59]. Hybrid models incorporating Long Short-Term Memory (LSTM) networks and attention mechanisms have enabled effective multivariate time-series prediction, capturing both short-term fluctuations and long-term trends [60]. Additionally, Sparse Autoencoder (SAE)-based FCMs have been applied to optimize feature representation for complex time-series data, ensuring improved scalability and prediction accuracy [61]. An Elastic Net and HFCM framework has been produced for EEG-based human action prediction [62], demonstrating significant improvements in capturing complex temporal dynamics from EEG signals.

In healthcare and medicine, FCMs have proven valuable for disease diagnosis, treatment planning, and mental health assessment. For example, FCM-based systems have been used to grade brain tumors using Nonlinear Hebbian Learning (NHL), providing interpretable tools for clinical decision-making [63]. In cognitive neuroscience, FCMs have been applied to predict depression severity based on EEG data, offering robust solutions for mental health diagnostics [64]. Moreover, hybrid approaches combining FCMs with Possibilistic Fuzzy C-Means (PFCM) clustering have been effective in improving the classification of celiac disease [65]. In [66] a hybrid FCM-Agent-Based Modeling (ABM) framework has been proposed to predict the spatial spread of infectious diseases, including COVID-19.

Fuzzy Cognitive Maps (FCMs) have emerged as a valuable tool in bioinformatics, particularly for Gene Regulatory Network (GRN) reconstruction and disease modeling. To optimize large-scale GRNs, advanced learning approaches such as the Dynamic Multiagent Genetic Algorithm (dMAGA) [67] and Evolutionary Algorithm-based FCMs [68] have been employed, enhancing scalability and accuracy. Mutual Information-based Multi-Agent FCMs (MIMA-FCM) further improved computational efficiency by reducing the search space for critical GRN elements [69]. Additionally, FCMs have been applied to predict HIV-1 drug resistance using optimization techniques like Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO), improving prediction accuracy and interpretability [70].

Environmental science has seen extensive use of FCMs for scenario development, sustainability assessment, and environmental restoration. FCMs have been applied to analyze deforestation dynamics in the Brazilian Amazon, enabling better understanding of ecosystem feedback loops for policy development [71]. In air quality forecasting, robust causal FCM models have integrated Particle Swarm Optimization (PSO) to predict ozone levels, outperforming traditional prediction techniques in urban environments [72]. Similarly, FCMs combined with nature-based solutions have contributed to water quality management by engaging local communities and stakeholders in decision-making processes [73].

In the business and financial domains, FCMs have facilitated enterprise resource planning, supply chain optimization, and financial forecasting. FCMs combined with optimization algorithms such as the Artificial Bee Colony (ABC) method have improved the efficiency of enterprise resource planning systems, enhancing the accuracy and generalization of decision-making processes [74]. Similarly, FCMs integrated with interval-valued intuitionistic fuzzy sets have been applied for stock index forecasting, providing robust solutions for financial market predictions [75]. In supply chain resilience, FCMs have modeled the domino effects of disruptions, helping industries design recovery strategies for maintaining stability [76].

In education, FCMs have been applied to support personalized learning and adaptive instruction. By integrating FCMs into e-learning systems, models have been developed to tailor instructional content based on learners’ cognitive abilities and knowledge states, improving engagement and knowledge retention [77]. Game-based learning systems have also benefited from FCMs by dynamically adjusting educational objectives to learner performance, as demonstrated in adaptive automobile driving simulators [78].

2.4. Limitations of Fuzzy Cognitive Maps

Fuzzy Cognitive Maps (FCMs) have demonstrated significant utility in modeling complex systems and supporting decision-making across various domains. However, several inherent limitations affect their broader applicability and performance. A critical challenge is their dependence on expert knowledge, where the construction of the weight matrix relies heavily on domain expertise, introducing subjectivity and limiting generalizability. Additionally, FCMs often exhibit scalability issues as the complexity grows with an increasing number of nodes and interconnections, leading to reduced interpretability and significant computational burdens in large-scale systems. Large-scale problems often require extensive fuzzy rule databases, which further increase memory requirements and computational effort. Their static weight representation further restricts the ability to capture time-varying relationships in dynamic environments, while their limited expressiveness for nonlinear interactions reduces their accuracy in complex systems.

Furthermore, the iterative nature of FCMs introduces convergence and stability issues, where models may stabilize at fixed points, exhibit limit cycles, or even diverge toward chaotic behavior, complicating their reliability in predictive tasks. FCMs are also sensitive to initial conditions, where minor variations in starting states can produce significantly different outcomes. Addressing these limitations through advancements in learning algorithms, hybrid methodologies, and integration with machine learning techniques remains a promising research direction to extend the applicability and effectiveness of FCMs across dynamic, large-scale, and uncertain systems.

3. Literature Review of FCM Applications in Engineering

This study is based on a comprehensive analysis of the literature in the field of Engineering focusing on the application of Fuzzy Cognitive Maps (FCMs). To construct the pool of papers, a systematic search was conducted using scientifically relevant databases, including IEEE, Elsevier, Springer, MDPI, Web of Science, Scopus, and Semantic Scholar. The primary keyword/topic used for the search was “Fuzzy Cognitive Maps”.

The initial search yielded a pool of top papers comprising approximately 500 publications. To ensure the relevance and quality of the material, the following filtering process was applied:

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Only journal articles and conference papers published after the year 2000 were considered.

•

Papers were manually reviewed by examining their abstracts and, where necessary, the full content to assess their relevance to the scope of this research, which focuses specifically on FCM applications in Engineering.

•

Works outside the engineering domain or unrelated to the research focus were excluded.

This selection process resulted in a refined pool of 80 papers, which form the core review material for this study. The selected publications represent diverse applications of FCMs across various engineering subdomains, enabling a thorough investigation of their contributions, methodologies, and limitations.

3.1. Publisher Distribution

The pool of selected papers spans several prominent publishers, reflecting the diverse sources contributing to the research on FCMs.

Table 2. Publisher distribution of selected articles and proceedings.

Publisher	IEEE	Elsevier	Springer	MDPI	Other	Wiley	IOS Press	World Scientific	Taylor & Francis	Preprint
Journal Articles	7	24	6	7	4	3	1	1	1	1
Proceedings	20	1	1	0	3	0	0	0	0	0
Total	27	25	7	7	7	3	1	1	1	1

presents the distribution of articles and proceedings across the major publishers.

The distribution shows that IEEE contributes the largest proportion of conference proceedings (20 out of 25, approximately 80%), reflecting its prominence in engineering conferences. Elsevier accounts for the highest number of journal articles (24 out of 55, approximately 43.6%), highlighting its significant role in publishing FCM research. Other publishers such as Springer, MDPI, and Wiley also contribute notable shares, indicating the multidisciplinary interest in FCM applications. Smaller contributions from publishers such as IOS Press, World Scientific, and Taylor & Francis illustrate the broader engagement with FCMs across diverse contexts. To complement this data, Figure 2 provides a visual breakdown of publisher contributions, distinguishing between journal articles, conference proceedings, and the combined totals. The leading roles of IEEE and Elsevier are highlighted, while the distribution’s diversity is also illustrated. This combined analysis underscores the significant presence of FCM research in both journals and conferences, with IEEE and Elsevier playing leading roles in disseminating advancements in this field. The visual representation further emphasizes the relative contributions of various publishers, enabling a clearer understanding of the publication landscape.

3.2. Temporal Distribution of Publications

The selected pool of papers also demonstrates the temporal evolution of research on FCMs in the field of engineering. Figure 3 illustrates the distribution of publications by year, highlighting notable trends in research activity. The analysis reveals that research on FCMs has seen significant growth in recent years, particularly in 2023, which accounts for the largest share with 12 publications. Notable activity is also observed in 2019 and 2022, each contributing nine papers. This upward trend reflects the increasing relevance of FCMs in addressing engineering challenges. Conversely, fewer contributions were observed in earlier years, indicating that the field has gained substantial momentum in the past decade. The consistent research output since 2018 underscores the growing interest and applicability of FCMs in various engineering domains.

4. Applications of FCMs in Engineering

4.1. Sub-Domains Identification

To better understand the diverse applications of Fuzzy Cognitive Maps (FCMs) in Engineering, the domain was divided into 15 distinct sub-domains. The sub-domains were designated as follows: (1) Control Systems; (2) Decision Support Systems; (3) Energy Systems; (4) Robotics; (5) Fault Detection and Diagnosis; (6) Maritime Systems; (7) Reliability and Safety Systems; (8) Energy Economics; (9) Production Management; (10) Software; (11) Industry 4.0; (12) Networking and Communications; (13) Prognostics and Health Management (PHM)/Predictive Maintenance; (14) Transportation Systems; and (15) Remote Sensing Systems. This classification enables a structured analysis of the methods, tasks, and trends in FCM research, ensuring comprehensive coverage and providing insights into underexplored areas. The sub-domains were selected based on the reviewed literature, with a focus on areas where FCMs have been applied to solve engineering problems. These include traditional fields like Control Systems and Decision Support Systems, as well as emerging areas like Industry 4.0, Transportation Systems, Fault Detection and Diagnosis, and Energy Systems.

Figure 4 presents a treemap visualization of the 15 sub-domains, highlighting their relative prevalence in percentages based on the number of studies reviewed. Larger blocks indicate higher research activity, with Control Systems and Decision Support Systems being the most explored areas. Conversely, smaller blocks represent yet under-explored areas, such as Remote Sensing Systems. Control Systems and Decision Support Systems account for a significant portion of FCM research, reflecting their critical role in engineering decision-making and automation. On the other hand, sub-domains like Prognostics and Health Management(PHM)/Predictive Maintenance show limited studies, presenting opportunities for future research. This categorization forms the basis for the subsequent detailed review of tasks and methodologies within each sub-domain. By understanding the distribution of FCM applications, we can identify gaps and guide future research directions.

4.2. Clustering of Engineering Works Under Learning Families

The reviewed FCM-based engineering works were clustered into distinct learning families to better understand the approaches adopted in solving engineering problems. This clustering reflects the dominant methods used to train and implement FCMs and highlights trends in learning approaches over time. Note that the learning family candidates are as already presented in Section 2.2; thus, the six distinct learning cases are: (1) Rule-based; (2) Hebbian-based; (3) Metaheuristics; (4) Gradient-based or Gradient-like; (5) Gradient-free; and (6) Hybrid.

Figure 5 shows the temporal evolution of publications across these learning families, along with the total number of studies per year. Notably, there has been an increase in works utilizing Metaheuristics, Hybrid methods, and Rule-based approaches in recent years, reflecting their flexibility and robustness in engineering applications. Additionally, more Gradient-based or Gradient-like approaches have been proposed over the last few years, as a general observation.

As a further analysis of the distribution of learning families, Figure 6 provides a breakdown of the learning families in percentages, revealing their relative prevalence. Rule-based methods account for the largest share (30%), demonstrating their enduring importance in engineering systems, mainly due to practically oriented implementations incorporating operational rules. Metaheuristics constitute 25% of the approaches, highlighting their adaptability to complex optimization problems. Gradient-based methods follow at 16.2%, while Hybrid techniques (18.8%) reflect the integration of multiple paradigms to enhance performance. Hebbian-based (8.8%) and Gradient-free methods (1.2%) remain less prevalent, but offer specialized advantages in specific tasks.

The breakdown of Metaheuristics reveals that 90% of the works rely on Population-based methods, with Single-Solution-based approaches accounting for only 5% (see Figure 6). The dominance of Population-based methods highlights their suitability for exploring large solution spaces in engineering problems. In the hybrid learning breakdown, 40% of the methods integrate Rule-based approaches with Population-based Metaheuristics, while 26.7% combine Population-based Metaheuristics with Gradient-based optimization. Smaller contributions include causal inference, parallel GPU computing, and Hebbian methods, illustrating the diverse combinations explored to improve learning efficiency and accuracy.

From an insight perspective, the clustering analysis indicates a growing trend toward Hybrid approaches and Metaheuristics in recent years, driven by their flexibility in handling complex, non-linear engineering systems. Rule-based methods, while traditional, remain widely adopted due to their simplicity and interpretability. These insights offer a foundation for identifying dominant learning trends and guiding future research toward underexplored combinations, such as Gradient-free methods integrated with emerging techniques.

4.3. Task Types of Application Classes

This study identifies eight distinct task types, each corresponding to a specific function within FCM-based applications and highlighting key focus areas. Decision Making involves selecting optimal actions or decisions based on available information, often applied in strategic or operational settings. Control tasks focus on regulating, adjusting, and identifying systems or processes to achieve desired outcomes, such as managing dynamic systems. Modeling refers to constructing representations of systems, behaviors, or phenomena to enhance understanding and simulation. In addition, Optimization tasks aim to minimize or maximize specific objectives, such as cost or efficiency. Time-Series Prediction plays a critical role in forecasting future states using historical data, which is especially relevant in predictive analytics. Diagnosis focuses on identifying faults, errors, or abnormal conditions within systems to enable early intervention. Classification tasks deal with categorizing data into predefined groups, which is widely applied in pattern recognition. Finally, Anomaly Detection focuses on identifying deviations from normal patterns, which is crucial for detecting rare events, malfunctions or faults. The distribution of these task types across the applications is presented in Figure 7, where Decision Making and Control tasks are notably prevalent. It is important to note that the total count in Figure 7 is 90, as some of the 80 works have been associated with more than one task type.

4.4. Detailed Distribution of Learning Families and Task Types

To provide a more granular analysis of how learning families align with task types in engineering applications, we present the following visualization and analysis. This complements the earlier discussions on task types and learning families, offering insights into the relationships between these elements.

4.4.1. Visual Analysis of Distribution

Figure 8 presents a bubble chart showcasing the percentage share of studies for each combination of task types and learning families. The size of the bubbles represents the relative contribution of each pair. Notably, Metaheuristics account for 100% of Anomaly Detection and Classification tasks, reflecting their strong adaptability to these domains. Modeling has been tackled with candidates from all learning families, showcasing its broad applicability across methodologies. Optimization tasks have primarily been addressed using Metaheuristics and Hybrid methods, with Metaheuristics holding the largest share. In Decision Making tasks, Rule-based learning algorithms emerge as the most utilized family, highlighting their structured approach in decision support frameworks. Additionally, Hebbian-based algorithms have not yet been applied to Time-Series Prediction or Diagnosis tasks, indicating potential areas for future exploration. Figure 9 and Figure 10 provide complementary perspectives illustrating the distribution of learning families for each task type and the distribution of task types within each learning family, respectively. Metaheuristics dominates tasks like Anomaly Detection, Classification, and Optimization, and is also the only learning family that has been adopted in all task type problems. Rule-based approaches, while more straightforward, are used primarily for three task type problems in engineering, and have been applied mainly in Decision Making.

4.4.2. Tabular Summary of Distribution

To complement the visualizations,

Table 3. Percentage distribution of task types across learning families.

Task Type	Rule-Based	Hebbian-Based	Metaheuristics	Gradient-Based	Gradient-Free	Hybrid
Decision Making	41.67%	8.33%	8.33%	8.33%	8.33%	25.0%
Control	16.67%	16.67%	16.67%	33.33%	0.0%	16.67%
Modeling	16.67%	16.67%	33.33%	0.0%	16.67%	16.67%
Optimization	0.0%	0.0%	55.56%	0.0%	0.0%	44.44%
Classification	0.0%	0.0%	100.0%	0.0%	0.0%	0.0%
Anomaly Detection	0.0%	0.0%	100.0%	0.0%	0.0%	0.0%
Time-Series Prediction	0.0%	0.0%	50.0%	50.0%	0.0%	0.0%
Diagnosis	0.0%	0.0%	33.33%	33.33%	0.0%	33.33%

presents a numerical summary of the distribution of task types across learning families, expressed in percentages. This tabular representation ensures clarity and provides precise values for readers who prefer detailed numerical data. This subsection integrate tabular perspectives, enhancing the understanding of how FCM learning families are employed to address diverse engineering tasks. The findings align with trends observed in earlier sections, further solidifying the role of FCMs in engineering applications. Note that the percentages in Table 3 represent the relative distribution of each task type across different learning families. Each row sums to 100%, reflecting how a specific task type is distributed among the learning families. However, percentages should not be summed vertically, as they do not represent total contributions of each learning family across all task types.

4.5. Flow Across Domains, Learning Families, and Task Types

The Sankey diagram presented in Figure 11 illustrates the relationships between engineering subdomains, learning families, and task types in FCM-based research. The following observations are drawn directly from the dataset and accurately reflect the distribution of contributions:

•

Insights from Subdomains Perspective: The subdomains Control Systems, Energy Systems, and Fault Detection and Diagnosis emerge as prominent subdomains, with diverse connections to multiple learning families, reflecting their broad applicability and the varied methodologies used to address their challenges. Additionally, subdomains such as Industry 4.0, Maritime Systems, and Reliability and Safety Systems are primarily tackled using Rule-based approaches, while Software, Energy Economics, and Robotics are predominantly addressed using Hybrid Methods.

–

The Control Systems subdomain is associated with all learning families, showcasing its versatility in addressing a wide range of engineering challenges.

–

Energy Systems demonstrates connections mainly to Metaheuristics, Gradient-based or Gradient-like, and Hybrid methods, highlighting the utility of these approaches in predictive analysis for energy applications.

–

Fault Detection and Diagnosis spans primarily across Hybrid and Gradient-based or Gradient-like methods, emphasizing the importance of these families in detection and predictive diagnostics.

–

Additionally, Decision Support Systems span primarily between Rule-based and Metaheuristics methods, indicating a preference for structured reasoning and heuristic optimization in these applications.

–

Industry 4.0, Maritime Systems, and Reliability and Safety Systems are mainly tackled using Rule-based approaches, while Software, Energy Economics, and Robotics are primarily addressed using Hybrid family approaches.

•

Learning Families as Intermediaries: Learning families act as bridges, linking subdomains to task types. The key trends observed are:

–

Gradient-based or Gradient-like methods are frequently employed for Control, leveraging their capacity for iterative refinement and dynamic adjustment. They also serve as a dominant candidate for Time-Series Prediction and Diagnosis tasks.

–

Metaheuristics exhibits diversified applications across all tasks, showcasing the method’s adaptability and ease of use for addressing various engineering problems.

–

Hybrid methods demonstrate versatility by addressing Control, Modeling, and Time-Series Prediction, integrating strengths from multiple paradigms to solve complex problems.

–

Rule-based methods, while more focused, are primarily linked to Decision Making, Control, and Modeling, reflecting their role in structured decision-support frameworks.

•

Prevalent and Emerging Task Types: The Sankey diagram highlights the prevalence of certain task types across learning families:

–

Decision Making and Control are the most frequently addressed tasks, reflecting their centrality in engineering applications.

–

Modeling also appears prominently, underscoring the role of FCMs in representing and simulating complex systems.

–

Emerging tasks, such as Time-Series Prediction and Anomaly Detection, have fewer connections, suggesting growing but still limited application areas.

•

Interdisciplinary and Multi-tasking Applications: Several subdomains and learning families contribute to multiple task types, reflecting the interconnected and multidisciplinary nature of FCM applications.

The Sankey diagram provides a clear visualization of these relationships from an oversight perspective, offering valuable insights into the methodological diversity and distribution of FCM-based research. By bridging domains, learning families, and task types, it highlights current trends and potential areas for further exploration.

5. Subdomain-Specific Applications of Fuzzy Cognitive Maps in Engineering

This section delves into the diverse applications of Fuzzy Cognitive Maps (FCMs) across specific engineering subdomains. By systematically categorizing and analyzing their implementation, we aim to uncover patterns, highlight strengths, and identify domain-specific challenges. Note that, in total, 95 works will be presented, as some of the 80 identified studies are associated with more than one task type or subdomain. This exploration not only showcases the versatility of FCMs but also provides insights into their practical impact and future potential in engineering.

5.1. Control Systems

Fuzzy Cognitive Maps (FCMs) have been extensively applied in control systems to tackle dynamic, nonlinear, and multi-variable challenges. This section reviews a set of contributions that demonstrate advancements in frameworks, algorithms, and applications.

Table 4. Applications of Fuzzy Cognitive Maps in Control Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Control	Adaptive Control in a Simulated Hydro-Electric Plant	Gradient-based or Gradient-like	Fuzzy Cognitive Networks (FCNs) using fuzzy rule database	[79]	2007
	Chemical Process Control	Hebbian-based	Active Hebbian Learning Algorithm (AHL)	[80]	2004
	Chemical Process Control	Rule-based	Intuitionistic Fuzzy Cognitive Map (iFCM)	[29]	2012
	Chemical Process Control Problem	Hebbian-based	Fuzzy Cognitive Map (FCM) with Nonlinear Hebbian Learning (NHL)	[81]	2003
	Chemical Process Control Problem	Metaheuristics (Population-based)	Fuzzy Cognitive Map (FCM) with Particle Swarm Optimization (PSO)	[82]	2003
	Chemical Process Control Problem	Hebbian-based	Data-Driven Nonlinear Hebbian Learning (DD-NHL)	[83]	2008
	Control Process System	Metaheuristics (Population-based)	Divide and Conquer Real-Coded Genetic Algorithm (RCGA)	[84]	2010
	Controlling Complex Dynamic Systems	Metaheuristics (Population-Based)	Evolutionary Fuzzy Cognitive Maps (EFCMs)	[85]	2003
	DC Motor Speed Control and System Identification	Gradient-based or Gradient-like	Fuzzy Cognitive Network with Functional Weights (FCNs-FW)	[86]	2018
	HVAC Systems	Hybrid (Rule-based and Population-based)	Fuzzy Cognitive Maps (FCMs)	[87]	2018
	Industrial Process Control	Rule-based	Dynamic Fuzzy Grey Cognitive Maps (DFGCM) and Dynamic Fuzzy General Grey Cognitive Maps (DFGGCM)	[88]	2021
	Industrial Tank-Valves Control, Heat Exchanger	Metaheuristics (Population-based)	Memetic Particle Swarm Optimization (MPSO)	[89]	2009
	Supervisory Control of Fermentation Process	Rule-based	Dynamic Fuzzy Cognitive Map (D-FCM)	[90]	2013
	Supervisory Control System for a Liquid Mixing Tank	Hybrid (Rule-based and Population-based Metaheuristic)	Interactive Evolutionary Optimization of Fuzzy Cognitive Maps (IEO-FCM)	[91]	2017
	System Identification and Indirect Inverse Control	Gradient-based or Gradient-like	Fuzzy Cognitive Network with Functional Weights (FCNs-FW)	[92]	2018
Modeling	Chemical Process, Two Tank System and Heat Exchanger Control	Hebbian-based	Hebbian Learning Algorithms for FCMs	[93]	2011
	Dynamic System Simulation and Analysis	Hebbian-based	Improved Nonlinear Hebbian Rule for FCMs (INHL)	[94]	2004
	Eco-Industrial Park	Metaheuristics (Population-based)	Memetic Particle Swarm Optimization (MPSO)	[89]	2009
	Supervisory Control System for a Liquid Mixing Tank	Hybrid (Rule-based and Population-based Metaheuristic)	Interactive Evolutionary Optimization of Fuzzy Cognitive Maps (IEO-FCM)	[91]	2017
	Multiphase Liquid-Gas Plant Modeling	Metaheuristics (Population-Based)	Fuzzy Cognitive Maps combined with Gray Wolf Optimization (FCM-GWO)	[95]	2023
	Slurry Rheology	Gradient-based or Gradient-like	Maximum Entropy-based Learning Method for FCMs (LEFCM)	[96]	2019
	Slurry Rheology	Metaheuristics	Divide and Conquer Real-Coded Genetic Algorithm (RCGA)	[84]	2010

provides a summary of these contributions, detailing the specific applications, associated learning families, algorithms used, references, and publication years. This overview enables a structured understanding of the wide-ranging applications of FCMs in control systems, emphasizing the key developments and their practical implementations.

A general framework for Fuzzy Cognitive Networks (FCNs) was developed to support continuous feedback and adaptive decision-making, showcasing its effectiveness in managing hydroelectric plants by reducing computational burdens and enhancing control accuracy [79]. The Active Hebbian Learning (AHL) algorithm addressed issues of convergence and dependence on expert-defined weights, introducing sequences of activation concepts and mathematical formulations for weight adjustments, successfully applied to process control systems [80]. An intuitionistic FCM (iFCM) model was developed to incorporate intuitionistic fuzzy sets, enhancing the handling of uncertainty in decision support systems [29]. The model was demonstrated to be effective in process control and medical decision support. Nonlinear Hebbian Learning (NHL) was proposed to dynamically update causal relationships in FCMs, improving robustness and stability in process control applications [81]. Additionally, a learning method based on Particle Swarm Optimization (PSO) optimized FCM weights for desired steady states in industrial processes, demonstrating robustness where traditional methods struggled [82]. To refine NHL, the Data-Driven Nonlinear Hebbian Learning (DD-NHL) algorithm was developed, utilizing historical data to improve learning quality and achieving lower error rates and enhanced generalization capabilities [83]. A divide-and-conquer approach was proposed to accelerate the learning of large-scale FCMs using Real-Coded Genetic Algorithms (RCGA) [84]. This method split data into subsets, learned submodels in parallel, and merged them into a final model, achieving significant scalability and efficiency gains. Another framework, Evolutionary Fuzzy Cognitive Maps (EFCMs), incorporated evolutionary strategies to design and tune FCMs for multi-stimulus situations, synergies, and conditional effects, improving robustness and adaptability in dynamic systems [85].

A switching control scheme for DC motors was proposed, using multiple Fuzzy Cognitive Network models equipped with functional weights (FCN-FW) [86]. The approach provided robust and flexible control mechanisms by adapting online and switching models based on performance indices. Compared to conventional PID controllers, this method achieved reduced tracking errors and smoother control signals. A comprehensive review of control techniques for HVAC systems emphasized FCMs as advanced methods capable of addressing nonlinear and multi-input/multi-output (MIMO) structures [87]. That work highlighted the adaptability and efficiency of FCMs in real-world control applications. Additionally, dynamic extensions to Fuzzy Grey Cognitive Maps (FGCMs) were introduced to handle uncertainty and dynamic environments, particularly in industrial control systems, through environment-adaptive weight updates [88]. A memetic PSO algorithm was also introduced, combining global PSO with deterministic and stochastic local search methods to improve efficiency and accuracy in FCM learning tasks. This approach outperformed traditional PSO, Differential Evolution (DE), and Genetic Algorithm (GA) methods across multiple real-life applications [89]. A supervisory system based on Dynamic Fuzzy Cognitive Maps (D-FCMs) was developed for chemical process control, addressing both normal and critical operation modes [90]. This system allowed dynamic generation of set-points for PID controllers, improving process performance in modular and critical industrial environments. In another study, Interactive Evolutionary Computing was integrated with FCMs to iteratively optimize FCM structures based on qualitative expert evaluations [91]. This approach, called IEO-FCM, demonstrated its potential in supervisory control and navigation tasks, even under incomplete or uncertain expert knowledge. Fuzzy Cognitive Networks with functional weights (FCN-FW) were applied for system identification and indirect inverse control in a coupled two-tank system, achieving superior convergence and control accuracy compared to traditional methods [92].

A study evaluated Hebbian-like algorithms, including Differential Hebbian, Nonlinear Hebbian, Data-Driven Nonlinear Hebbian, and Active Hebbian Learning, highlighting their generalization capabilities and convergence efficiency [93]. Another study proposed an improved Nonlinear Hebbian Learning rule to enhance learning efficiency and stability, demonstrating its effectiveness in modeling and simulating complex systems [94]. Additionally, a hybrid approach integrating FCMs with Gray Wolf Optimization (GWO) successfully modeled multiphase liquid-gas plants, demonstrating high reliability in industrial processes [95]. Another approach transformed FCM learning into a convex optimization problem, ensuring rapid and robust learning from noisy data [96]. This method demonstrated superior noise suppression, better weight distribution, and improved generalization compared to traditional methods like PSO and NHL.

5.2. Decision Support Systems

Fuzzy Cognitive Maps (FCMs) have been widely applied in decision support systems, addressing challenges such as multi-criteria decision-making, risk assessment, and modeling complex systems. This subsection reviews several advancements in this area, highlighting diverse contributions.

Table 5. Applications of Fuzzy Cognitive Maps in Decision Support Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Classification	Pattern Classification	Metaheuristics (Population-based using PSO, DE, RCGA, VMO)	FCM Expert: Software Tool for Scenario Analysis and Pattern Classification Based on Fuzzy Cognitive Maps	[11]	2018
Decision Making	Cognitive Modeling and Decision-Making Systems	Rule-based	Fuzzy Cognitive Maps (FCMs) with Synergies and Conditional Effects	[97]	2001
	Construction Labor Productivity Improvement Strategies Prioritization	Rule-based	Fuzzy Multi-Criteria Decision Making (Fuzzy MCDM) and Fuzzy Cognitive Maps (FCMs)	[98]	2021
	Multi-Criteria Decision Making for Robot Selection	Rule-based	Fuzzy Cognitive Maps-Based Multi-Criteria Decision Making Method (MCDM)	[99]	2020
	Multi-Stimulus Reasoning and System Dynamics Modeling	Hybrid (Rule-based and Hebbian-based)	Fuzzy Cognitive Maps (FCMs) with Rule-based and Modified Hebbian Learning	[100]	2001
	Non-Monotonic and Uncertain Cause-Effect Systems	Rule-based	Rule-Based Fuzzy Cognitive Maps (RBFCMs) with new reasoning mechanisms	[101]	2017
	Optimization of Decision Support Systems	Metaheuristics (Population-Based)	Evolutionary Fuzzy Cognitive Maps (E-FCM) with Graph Theory Metrics	[102]	2019
	Scenario Analysis	Metaheuristics (Population-based using PSO, DE, RCGA, VMO)	FCM Expert: Software Tool for Scenario Analysis and Pattern Classification Based on Fuzzy Cognitive Maps	[11]	2018
	Risk Assessment in Manufacturing	Metaheuristics (Population-based)	Z-number Multi-Stage Fuzzy Cognitive Map with fuzzy learning algorithm and PSO	[103]	2021
	Security Risk Assessment for E-health Systems	Rule-based	Fuzzy Cognitive Maps (FCMs) for Security Risk Assessment	[104]	2014
	Security Risk Assessment for Video Surveillance Systems	Rule-based	Fuzzy Cognitive Maps for Risk Assessment	[105]	2016
	Supply Chain Risk Management	Rule-based	Fuzzy Cognitive Maps (FCMs)	[106]	2023
Modeling	Cognitive Modeling and Decision-Making Systems	Rule-based	Fuzzy Cognitive Maps (FCMs) with Synergies and Conditional Effects	[97]	2001
	Modeling of Complex System Phenomena	Rule-based	Aggregation Functions in Computing With Words (CWW)	[107]	2019
Optimization	Automatic FCM Construction	Metaheuristics (Mixed with Population-based and Single-Solution-based)	Fuzzy Cognitive Map (FCM) with Genetic Algorithm (GA) and Simulated Annealing (SA)	[108]	2007
	Partitioning of Complex Fuzzy Cognitive Maps	Metaheuristics (Population-Based)	Immune Algorithm for Fuzzy Cognitive Map Partitioning	[109]	2009
	Privacy-preserving Distributed Learning in Healthcare and Finance	Metaheuristics (Population-based using PSO)	Concurrent vertical and horizontal federated learning with Fuzzy Cognitive Maps (FCMs)	[110]	2024
	System Modeling	Metaheuristics (Single-Solution-based)	Fuzzy Cognitive Maps (FCMs) learned by Tabu Search (TS)	[111]	2007

reflects the versatility of FCMs in addressing different decision-making scenarios, ranging from risk assessment and cognitive modeling to optimization tasks in distributed environments. It also demonstrates the integration of FCMs with various learning families and algorithms. FCM Expert, a software tool capable of designing, learning, and simulating FCM-based systems, was introduced in [11]. This tool supports scenario analysis and pattern classification, integrating various machine learning algorithms to optimize system performance and topology. Extensions to FCMs have been proposed in [97], incorporating synergies and conditional effects into the inference engine, enhancing their representation of real-world decision-making processes. A decision support model integrating fuzzy AHP, fuzzy TOPSIS, and FCMs has been proposed in [98] to quantify the impact of construction labor productivity improvement strategies. The study identified 16 key strategies through literature and expert surveys and validated the model using structural and behavioral tests. Additionally, ref. [99] presented a new FCM-based method for multi-criteria decision-making (MCDM), introducing a dynamic approach for ranking alternatives by considering different scenarios regarding causal relationships among criteria. This method demonstrated its utility through a simulated robot selection problem.

Enhancements to FCM reasoning in multi-stimulus environments were proposed in [100], incorporating rule-based and Hebbian-inspired dynamics to improve robustness in real-world decision-making scenarios. Moreover, new reasoning and impact accumulation mechanisms for rule-based FCMs (RBFCMs) were introduced in [101], improving their effectiveness in handling non-monotonic and uncertain cause-effect relationships, particularly in cyber defense applications. To simplify and optimize complex systems, ref. [102] integrated evolutionary algorithms with graph theory metrics to create a bank of FCMs. This approach improved accuracy and reduced complexity in decision support systems across multiple domains. Risk assessment has been a focal area for FCM applications. For instance, ref. [103] proposed a Z-number Multi-Stage FCM approach for improving risk assessment in discrete production processes, leveraging a Particle Swarm Optimization-based learning algorithm. Similarly, ref. [104] introduced a lightweight risk assessment method suitable for small-to-medium systems, including e-health applications. This work was extended in [105], where FCMs were used for continuous feedback on vulnerabilities in automated video surveillance systems.

During the COVID-19 crisis, ref. [106] used FCMs to model causal relationships between web traffic KPIs and pandemic metrics, offering actionable insights for managing supply chain risks. This study demonstrated the effectiveness of web analytics and FCMs in informing strategic decisions during global crises. In another contribution, ref. [107] introduced an approach using Computing with Words and threshold aggregation functions to model complex, discontinuous system behaviors. Additionally, ref. [108] developed a hybrid learning method combining Genetic Algorithms (GA) and Simulated Annealing (SA) for automatic FCM construction, effectively balancing the strengths of both algorithms for improved learning speed and quality. To address complexity in large-scale systems, an immune algorithm for partitioning FCMs was proposed in [109], enhancing their manageability in decision support contexts. Meanwhile, a federated learning framework integrating FCMs was introduced in [110], addressing privacy concerns and non-IID data challenges in distributed environments, demonstrating its applicability in healthcare and finance. Finally, a Tabu Search-based method for FCM learning was proposed in [111], iteratively improving the map by exploring the solution space while avoiding cycles, and enhancing the quality of the resulting FCM for decision support applications.

5.3. Energy Economics

Fuzzy Cognitive Maps (FCMs) have been effectively applied to address challenges in energy economics, particularly in modeling and time-series prediction tasks. A hybrid approach combining gradient-based local search and evolutionary algorithms, including Real-Coded Genetic Algorithms (RCGA) and Differential Evolution (DE), was proposed to enhance the automatic construction of FCMs [112]. This method demonstrated sufficient performance over traditional techniques, offering better accuracy and time efficiency across both synthetic and real-world energy models. In the realm of energy forecasting, three FCM-based prediction models—Modified Genetic Model (MGM), RCGA, and Single Objective Genetic Algorithm (SOGA)—were developed and compared for household electricity consumption prediction [113]. Among these, the SOGA model achieved the lowest Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) in short-term forecasting. The study illustrated the capability of FCMs in managing nonlinearities and multivariate time-series data, outperforming conventional Artificial Neural Network (ANN) and neuro-fuzzy models.

Additionally, a time series forecasting model integrating Empirical Wavelet Transformation (EWT) with High-Order Fuzzy Cognitive Maps (HFCM) was introduced [114]. This approach showcased robustness against nonstationary time series and outliers, delivering enhanced performance compared to baseline models like ARIMA, Support Vector Regression (SVR), and Weighted High-Order Fuzzy Cognitive Maps (WHFCM). The model further validated the utility of FCMs in complex energy forecasting scenarios. These contributions underscore the potential of FCM-based frameworks in advancing energy economics by improving prediction accuracy and computational efficiency while addressing the inherent challenges in energy data.

Table 6. Applications of Fuzzy Cognitive Maps in Energy Economics, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Modeling	Electricity Markets	Hybrid (Population-based and Gradient-based or Gradient-like)	Hybrid Gradient-Based Evolutionary Algorithms	[112]	2012
Time Series Prediction	Electricity Consumption Prediction	Hybrid (Population-based and Gradient-based or Gradient-like)	Fuzzy Cognitive Maps (FCMs) using three different learning algorithms: Multi-step Gradient Method (MGM), Real Coded Genetic Algorithm (RCGA), and Structure Optimization Genetic Algorithm (SOGA)	[113]	2015
	Electricity Load Prediction	Gradient-based or Gradient-like	Robust Empirical Wavelet Fuzzy Cognitive Map (REW-FCM)	[114]	2020

summarizes the corresponding works in the under-examination sub-domain.

5.4. Energy Systems

Fuzzy Cognitive Maps (FCMs) have been extensively utilized in the domain of energy systems, addressing challenges in anomaly detection, control, decision-making, optimization, and time-series prediction. These contributions highlight the versatility and effectiveness of FCMs in managing complex, dynamic, and data-intensive energy applications.

Table 7. Applications of Fuzzy Cognitive Maps in Energy Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Anomaly Detection	Anomaly Detection in Oil and Gas Plants	Metaheuristics (Population-based)	Gray Wolf Optimization (GWO) Algorithm combined with Fuzzy Cognitive Maps	[115]	2022
Control	Maximum Power Point Tracking for Photovoltaic Arrays	Gradient-based or Gradient-like	Fuzzy Cognitive Networks (FCNs) using fuzzy rule database	[116]	2006
	Maximum Power Point Tracking for Photovoltaic Arrays	Gradient-based or Gradient-like	Fuzzy Cognitive Networks (FCNs) using fuzzy rule database	[117]	2007
Decision Making	Wind Energy Deployment Pathways	Rule-based	Fuzzy Cognitive Maps (FCM)-based scenario planning method	[118]	2022
Optimization	Energy Management in Autonomous Polygeneration Microgrids	Metaheuristics (Population-Based)	Hybrid Fuzzy Cognitive Map (FCM) and Petri Nets (PN) using PSO	[119]	2012
Time Series Prediction	Energy Use Forecasting	Metaheuristics (Population-based)	Nested Structure of Fuzzy Cognitive Maps (FCM) and Artificial Neural Networks (ANN) using RCGA and SOGA	[120]	2022
	Gas Consumption Prediction	Metaheuristics (Population-based)	Ensemble Method Combining Fuzzy Cognitive Maps and Neural Networks using RCGA and SOGA	[121]	2019
	Photovoltaic Power Forecasting	Gradient-based or Gradient-like	Time Series Expansion and High-order Fuzzy Cognitive Maps (TSE-HFCM)	[122]	2023
	Prediction of Key Parameters in Coal Gasification Process	Hybrid (Hebbian-based and Population-Based)	Time Delay Mining Fuzzy Time Cognitive Maps (TM-FTCM)	[123]	2021
	Solar Energy Forecasting	Metaheuristics (Population-Based)	High-Order Fuzzy Cognitive Maps (HFCM) with Fuzzy Time Series (FTS)	[124]	2020
	Wind Power Forecasting	Hybrid (Population-based and Gradient-based or Gradient-like)	IVMDHFCM (Improved Variational Mode Decomposition and High-Order Fuzzy Cognitive Maps)	[125]	2022

summarizes this stream of works. A methodology integrating FCM and Gray Wolf Optimization (GWO) was developed for anomaly detection in industrial plants [115]. This approach achieved high accuracy in detecting anomalies, especially in complex scenarios involving multiple interacting variables. Simulations demonstrated the model’s ability to prevent catastrophic failures through early detection. In the context of control, a Maximum Power Point Tracking (MPPT) method was developed by combining fuzzy logic with Fuzzy Cognitive Networks (FCNs) [116,117]. This method improved the speed and accuracy of reaching the maximum power point in photovoltaic systems under varying environmental conditions, leading to significant energy gains compared to traditional techniques.

For decision-making, plausible wind energy deployment scenarios in Iran were developed using FCMs under various macroeconomic and political uncertainties [118]. The study emphasized the effects of economic stagnation, government support, sanctions, and political instability on the sector’s future growth. In optimization, a hybrid energy management system combining FCMs and Petri Nets was introduced for autonomous polygeneration microgrids [119]. The methodology optimized both system design and operation, enabling efficient part-load operation of devices, which resulted in significant reductions in component sizing and overall system cost. Significant advancements have been made in the application of FCMs for time-series prediction. A nested structure integrating FCM with Artificial Neural Networks (ANN) and Long Short-Term Memory (LSTM) networks was developed for energy use forecasting [120]. This model achieved lower forecasting errors and higher correlation coefficients compared to standard techniques. Another study combined FCMs with Neural Networks for natural gas consumption prediction, resulting in improved accuracy through an ensemble learning approach [121]. A hybrid framework integrating fuzzy information granulation, improved variational mode decomposition, and high-order FCMs was proposed for short-term photovoltaic (PV) power forecasting [122]. This method demonstrated enhanced prediction accuracy by effectively handling noise in time-series data. Similarly, a novel time-delay mining framework was integrated into FCMs for predicting key operational parameters in coal gasification, capturing process dynamics more effectively and enabling real-time industrial applications [123].

For solar energy forecasting, a hybrid approach combining high-order FCMs with Fuzzy Time Series (FTS) was introduced. This method integrated genetic algorithms to optimize weight matrices, resulting in higher prediction accuracy for solar radiation [124]. Lastly, a hybrid method combining Improved Variational Mode Decomposition (IVMD) with HFCMs was proposed for wind power forecasting, showcasing superior forecasting performance compared to state-of-the-art methods [125]. These studies collectively underscore the significant potential of FCMs in energy systems, from improving control strategies to enhancing forecasting accuracy, thereby contributing to the advancement of sustainable energy solutions.

5.5. Fault Detection and Diagnosis

The field of fault detection and diagnosis has benefited significantly from the application of Fuzzy Cognitive Maps (FCMs), as summarized in

Table 8. Applications of Fuzzy Cognitive Maps in Fault Detection and Diagnosis, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Diagnosis	Fault Detection and Diagnosis in Industrial Robotics	Hybrid (Causal Inference with Metaheuristics)	Information Flow-Based Fuzzy Cognitive Maps (IF-FCM) with Social Learning Particle Swarm Optimization (SL-PSO)	[126]	2024
	Fault Prediction in Industrial Bearings	Metaheuristics (Population-Based)	Learning Fuzzy Cognitive Maps (LFCMs) using Modified Asexual Reproduction Optimization (MARO)	[18]	2023
	Incipient Inter-Turn Short-Circuit Fault Detection in Induction Generators applied in Wind Turbines	Gradient-based or Gradient-like	Multiple Cognitive Classifier System (MCCS) incorporating Fuzzy Cognitive Networks with Functional Weights (FCN-FW)	[19]	2021
	Industrial Anomaly Detection and Root Cause Analysis	Hybrid (Causal Inference with Metaheuristics)	Information Flow-Based Fuzzy Cognitive Maps (IF-FCM)	[127]	2023
	Motor Bearing Fault Detection and Diagnosis	Gradient-based or Gradient-like	Fuzzy Cognitive Network with Functional Weights (FCNs-FW)	[17]	2018

. These methodologies showcase the ability of FCMs to enhance automation, improve interpretability, and adapt to real-time industrial environments. A FCM model was developed using Information Flow analysis and Social Learning Particle Swarm Optimization (SL-PSO) to detect and diagnose faults in industrial robots [126]. This approach filters spurious causalities, enhances system optimization, and reduces reliance on continuous expert oversight, proving valuable for automating causal analysis in complex settings. In another advancement, an explainable fault prediction system leveraging Learning Fuzzy Cognitive Maps (LFCMs) was introduced [18]. This system not only predicts faults but also provides interpretability for decision-making processes, which is critical in industrial applications where understanding the reasoning behind predictions is essential. The model’s effectiveness was validated using real-world industrial data, demonstrating early fault detection capabilities.

For the detection of incipient short-circuit faults in induction generators used in wind turbines, a Multiple Cognitive Classifier System (MCCS) was proposed [19]. This system integrates Fuzzy Cognitive Networks with functional weights (FCN-FW) into various multi-classifier topologies, such as parallel, serial, and hierarchical configurations. Achieving high accuracy and a perfect distinction of normal conditions across six faulty states, this approach proves suitable for real-time industrial applications, with reduced complexity and computational requirements. Further, a construction framework for FCMs utilizing Liang–Kleeman Information Flow (L–K IF) analysis was introduced to improve interpretability and accuracy by eliminating spurious correlations [127]. This framework demonstrated better performance compared to a few FCM models, making it a robust tool for industrial anomaly detection and root cause analysis.

Lastly, advancements in motor fault detection and diagnosis were achieved through the integration of functional weights into Fuzzy Cognitive Networks (FCN-FW). This approach enhances classification accuracy for motor fault conditions by leveraging statistical features from current and vibration signals. Comparative analyses with other machine learning models highlighted the efficiency and scalability of this method, particularly in combination with statistical features [17].

5.6. Industry 4.0

The transformative potential of Fuzzy Cognitive Maps (FCMs) has been demonstrated in the context of Industry 4.0, particularly for readiness assessment and maturity level prediction in manufacturing organizations.

Table 9. Applications of Fuzzy Cognitive Maps in Industry 4.0, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Decision Making	Industry 4.0 Implementation and Readiness Assessment	Rule-based	Conventional Fuzzy Cognitive Maps (FCMs)	[128]	2023
	Industry 4.0 Maturity Level Assessment	Rule-based	Integrated Fuzzy DEMATEL and FCMs	[129]	2023
Modeling	Industry 4.0 Implementation and Readiness Assessment	Rule-based	Conventional Fuzzy Cognitive Maps (FCMs)	[128]	2023

provides a summary of these applications, highlighting their contributions to decision-making and modeling in Industry 4.0 environments.

An Industry 4.0 readiness model was proposed to evaluate the preparedness of manufacturing organizations for Industry 4.0 adoption [128]. The model employs FCMs to identify key concepts influencing readiness, perform static and dynamic analyses, and offer actionable recommendations for managers. By incorporating both technological and non-technological dimensions, the model provides a comprehensive framework validated through exploratory and confirmatory factor analysis. This approach effectively guides organizations in identifying readiness gaps and prioritizing improvement areas.

A hybrid methodology integrating Fuzzy DEMATEL and FCMs was developed to assess Industry 4.0 maturity levels under varying management scenarios [129]. This approach identifies inter-concept relationships and simulates maturity transitions, emphasizing the critical role of technology, operations, and strategy in achieving successful Industry 4.0 transformations. The integration of these methods enables more accurate and actionable insights for organizations navigating Industry 4.0 transitions.

5.7. Maritime Systems

Fuzzy Cognitive Maps (FCMs) have been effectively utilized in maritime systems to assess navigational safety, predict maintenance requirements, and analyze accident causes.

Table 10. Applications of Fuzzy Cognitive Maps in Maritime Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Decision Making	Effectiveness Assessment of Ship Navigation Safety Countermeasures	Hebbian-based	Fuzzy Cognitive Maps (FCM) with Non-linear Hebbian Learning	[130]	2019
	Maintenance Prediction for Rubber Fenders	Rule-based	Fuzzy Rule-based Decision-making System for Rubber Fender Lifetime Evaluation	[131]	2015
	Maritime Accident Analysis	Rule-based	Marine Accident Learning with Fuzzy Cognitive Maps (MALFCMs)	[132]	2022
	Maritime Accident Analysis and Prevention	Rule-based	Integration of HFACS and Cognitive Mapping (CM) Technique for Human Error Analysis	[133]	2014
	Maritime Collision Accident Analysis	Rule-based	Fuzzy Cognitive Maps (FCM)	[134]	2018

provides a comprehensive summary of these applications, showcasing the versatility of FCMs in addressing challenges in the maritime domain. One study developed an FCM model integrated with fuzzy logic and evidential reasoning to evaluate the effectiveness of navigational safety countermeasures [130]. That approach demonstrated high efficiency and stability in simulating safety dynamics, providing valuable insights for enhancing ship navigational systems. In the field of maintenance prediction, a fuzzy rule-based decision-making system was proposed to assess the lifetime of rubber fenders [131]. By integrating expert knowledge with physical property analysis, the system effectively predicts maintenance schedules and diagnoses faults, accounting for physical deterioration under varying environmental conditions.

The Maritime Accident Learning with Fuzzy Cognitive Maps (MALFCM) method was developed to incorporate historical accident data and expert opinions in weighting human-related contributing factors in maritime accidents [132]. This method improves the objectivity and reliability of accident analyses, identifying and ranking key human factors and enabling targeted safety interventions. Another study combined the Human Factors Analysis and Classification System (HFACS) with cognitive mapping techniques to prioritize human error factors in marine accidents [133]. This hybrid model was applied to a real-world maritime case study, demonstrating its effectiveness in generating actionable preventive measures for enhancing safety. An FCM model was also developed to investigate contributing factors to maritime collision accidents, with a particular focus on human factors [134]. This study highlighted the dominant role of human errors and their interactions, offering a novel framework for maritime safety assessments.

5.8. Networking and Communications

Fuzzy Cognitive Maps (FCMs) have demonstrated potential in addressing challenges within the networking and communications domain, particularly for network management and optimization tasks.

Table 11. Applications of Fuzzy Cognitive Maps in Networking and Communications, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Decision Making	Cognitive Software-Defined Networking for Network Management	Hebbian-based	Enhanced Hebbian-based Fuzzy Cognitive Maps (FCMs)	[135]	2019
Optimization	Cross-Layer Optimization for LPWAN Management	Hybrid (Rule-based and Population-based Metaheuristic)	Fuzzy Cognitive Maps with Adaptive Glowworm Swarm Optimization (AGSO-FCM)	[136]	2023

summarizes the two applications of FCMs in this sub-domain. A cognitive software-defined networking (SDN) architecture was proposed using FCMs enhanced with Hebbian-based learning [135]. This system demonstrated efficient decision-making and self-adaptation to dynamic network conditions without requiring human intervention. By integrating FCMs into SDN, the architecture achieved high accuracy in decision-making and minimal latency overhead compared to legacy SDN systems. The approach also exhibited robust convergence properties, ensuring efficient network management in emulated scenarios via Mininet. In another study, a cross-layer optimization framework was developed by combining FCMs with Adaptive Glowworm Swarm Optimization (AGSO) [136]. This hybrid approach was applied to the management of energy-harvesting low-power wide area networks (EH-LPWANs). The framework effectively balanced multiple conflicting objectives, including power consumption, network throughput, and communication quality. By automating the optimization process, the model demonstrated superior performance in simulated scenarios based on LoRaWAN parameters, outperforming traditional methods in terms of convergence speed and network lifetime.

5.9. Prognostics and Health Management (PHM)/Predictive Maintenance

Fuzzy Cognitive Maps (FCMs) have shown significant potential in Prognostics and Health Management (PHM) and predictive maintenance tasks, particularly for diagnosing faults and predicting the Remaining Useful Life (RUL) of complex systems.

Table 12. Applications of Fuzzy Cognitive Maps in PHM/Predictive Maintenance, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Diagnosis *	Predictive Maintenance, Remaining Useful Life Prediction	Gradient-based or Gradient-like	Hybrid deep learning structures combining Fuzzy Cognitive Networks with Functional Weights (FCNs-FW) with CNNs, ESNs, and Autoencoders	[56]	2023
	Predictive Maintenance	Hybrid (Population-based and Gradient-based or Gradient-like)	Fuzzy Cognitive Maps (FCM) for Health Indicator Prognostics using PSO, ABC, GWO, and SGDM	[137]	2022

* Note: While this class of applications focuses on prognostic operations, they are categorized under the Diagnosis task type for consistency.

provides an overview of the key contributions in this subdomain. In recent advancements, hybrid deep learning structures integrating Fuzzy Cognitive Networks with Functional Weights (FCNs-FW) with Convolutional Neural Networks (CNNs), Echo State Networks (ESNs), and Autoencoders were proposed [56]. These models demonstrated excellent performance across diverse applications, including predictive maintenance and RUL prediction for turbofan engines. The use of functional weights reduced the need for expert intervention and memory requirements, making the models highly efficient and scalable. The approach was validated using the NASA C-MAPSS dataset and exhibited superior accuracy and RMSE performance compared to traditional methods. This work highlights the adaptability of FCMs in transitioning to advanced deep learning frameworks.

Additionally, FCMs were introduced as a method for predicting engine health status and RUL, enhanced with the concept-by-concept learning method [137]. This approach combined FCMs with several global optimization techniques, including Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Grey Wolf Optimization (GWO), and Stochastic Gradient Descent Method (SGDM). The hybrid learning approach demonstrated competitive performance compared to Artificial Neural Networks (ANNs). Using the C-MAPSS dataset, this method achieved good results, evaluated using metrics such as Mean Square Error (MSE).

5.10. Production Management

Fuzzy Cognitive Maps (FCMs) have been effectively employed in production management to model complex systems and optimize processes.

Table 13. Applications of Fuzzy Cognitive Maps in Production Management, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Modeling	Analysis of Process Quality Control Variables	Rule-based	Fuzzy Cognitive Maps (FCM)	[138]	2022
	Process Control	Metaheuristics (Population-based)	Extended Great Deluge Algorithm (EGDA) for FCMs	[139]	2011
Optimization	Job Shop Scheduling	Metaheuristics (Population-based)	Extended Great Deluge Algorithm (EGDA) for FCMs	[139]	2011

summarizes their applications in this subdomain, highlighting key advancements in modeling and optimization tasks. A methodology for analyzing process quality control variables using FCMs was developed, demonstrating its capability to identify critical factors influencing finished product quality [138]. The study focused on a beverage carbonation process, where variables such as filler pressure and beverage temperature were identified as key determinants of product quality. This approach emphasized the causal reasoning capabilities of FCMs and their effectiveness in modeling interrelationships between process variables. The study utilized carbonation process data from a beverage production company, showcasing the practical relevance of FCMs in industrial settings.

In another contribution, a training algorithm for FCMs was proposed using the Extended Great Deluge Algorithm (EGDA) [139]. The method was tested on two applications: an industrial process control problem and a job shop scheduling problem. EGDA-based FCM learning demonstrated improvements over previous methods, such as Particle Swarm Optimization (PSO), in terms of error minimization, feasibility of the weight matrix, and steady-state convergence. This contribution highlights the potential of FCMs in addressing optimization challenges within production management, offering a robust solution with minimal human intervention.

5.11. Reliability and Safety Systems

The integration of Fuzzy Cognitive Maps (FCMs) into reliability and safety systems has enabled advancements in modeling, decision-making, and time-series prediction.

Table 14. Applications of Fuzzy Cognitive Maps in Reliability and Safety Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Decision Making	Maritime Accident Analysis and Prevention	Rule-based	Integration of HFACS and Cognitive Mapping (CM) Technique for Human Error Analysis	[133]	2014
	Occupational Safety	Rule-based	Fuzzy Cognitive Maps (FCM)	[140]	2019
	Reliability Engineering for Transformer Systems	Rule-based	Fuzzy Grey Cognitive Maps (FGCM)	[141]	2012
Modeling	Health Management System	Rule-based	Fuzzy Cognitive Maps (FCM)	[140]	2019
Time Series Prediction	Boiler Heat-Conducting Oil Temperature Prediction	Gradient-based or Gradient-like	Multi-Modality Fuzzy Cognitive Maps (MMFCMs)	[142]	2023

presents a summary of key applications in this domain, categorized by task type, learning family, and algorithmic approach. A hybrid model was proposed combining the Human Factors Analysis and Classification System (HFACS) with Cognitive Mapping (CM) to analyze and prioritize human error factors contributing to maritime accidents [133]. Applied to a real-world case study involving a lifeboat drill on a cruise ship, this approach demonstrated its utility in generating actionable preventive measures, enhancing maritime safety through targeted interventions. The influence of various processes on the effectiveness of Occupational Safety and Health Management Systems was modeled using FCMs [140]. This work demonstrated how different safety measures impact decision-making and system efficiency, offering valuable insights for optimizing occupational safety management. Fuzzy Grey Cognitive Maps (FGCMs) were introduced as a method to model uncertainty and expert hesitancy in reliability engineering [141]. Applied to transformer systems, FGCMs identified and analyzed failure causes across six scenarios, integrating grey system theory with FCMs to enhance reliability assessments in complex systems. A multi-modality FCM approach was developed for predicting boiler heat-conducting oil temperature using an industrial boiler dataset [142]. By preprocessing data to handle outliers and dividing time series into granular subsequences, the model combined FCM outputs to achieve high accuracy. RMSE evaluations revealed superior interpretability and performance compared to LSTM, highlighting the strengths of FCMs in industrial time-series prediction tasks.

5.12. Remote Sensing Systems

Fuzzy Cognitive Maps (FCMs) have been applied in remote sensing systems for tasks such as image classification and decision-making.

Table 15. Applications of Fuzzy Cognitive Maps in Remote Sensing Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Classification	Remote Sensing Image Classification	Metaheuristics (Population-based)	Fuzzy Cognitive Maps with Bird Swarm Optimization (FCMBS)	[12]	2022
Decision Making	Lunar South Pole Landing Site Selection	Rule-based	Fuzzy Cognitive Map (FCM) Algorithm	[143]	2022

summarizes the key applications in this subdomain, including the learning families and algorithms used. A hybrid model combining Fuzzy Cognitive Maps with Bird Swarm Optimization (BSA) was used for remote sensing image classification [12]. This approach included preprocessing, feature extraction using RetinaNet, and parameter tuning with BSA. It was evaluated on the UCM21 and AID datasets, showing improved classification performance in terms of accuracy, precision, and recall compared to other methods. For lunar exploration, a multi-factor FCM model was introduced to evaluate and select landing sites at the lunar south pole [143]. The model incorporated factors such as PSR distribution, slope, rock distribution, illumination intensity, and temperature. By integrating engineering constraints and multi-source remote sensing data, it identified feasible areas that balance safety and scientific goals.

5.13. Robotics

Fuzzy Cognitive Maps (FCMs) have been extensively used in robotics for tasks ranging from navigation to fault diagnosis.

Table 16. Applications of Fuzzy Cognitive Maps in Robotics, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Control	Autonomous Vehicle Navigation and Route Planning	Hybrid (Rule-based and Population-based Metaheuristic)	Fuzzy Cognitive Maps (FCMs) adjusted by Particle Swarm Optimization (PSO)	[144]	2019
	Multi-Robot Systems in Semi-Unknown Environments	Rule-based	Fuzzy Cognitive Map (FCM)	[145]	2019
	Adaptive Level of Autonomy for Human-UAV Collaborative Surveillance	Gradient-based or Gradient-like	Fuzzy Cognitive Map (FCM)	[146]	2020
	Reactive Navigation, Path Planning	Hybrid (Rule-based and Population-based Metaheuristic)	Fuzzy Cognitive Maps (FCMs) with Rule-based learning combined with Particle Swarm Optimization (PSO) and Migration Algorithm (MA)	[147]	2021
	Mobile Robotics Navigation	Hybrid (Rule-based and Population-based Metaheuristic)	Interactive Evolutionary Optimization of Fuzzy Cognitive Maps (IEO-FCM)	[91]	2017
Decision Making	Adaptive Level of Autonomy for Human-UAV Collaborative Surveillance	Gradient-based or Gradient-like	Fuzzy Cognitive Map (FCM)	[146]	2020
	Multi-Criteria Decision Making for Robot Selection	Rule-based	Fuzzy Cognitive Maps-Based Multi-Criteria Decision Making Method (MCDM)	[99]	2020
Diagnosis	Fault Detection and Diagnosis in Industrial Robotics	Hybrid (Causal Inference with Metaheuristics)	Information Flow-Based Fuzzy Cognitive Maps (IF-FCM) with Social Learning Particle Swarm Optimization (SL-PSO)	[126]	2024
Modeling	Mobile Robotics Navigation	Hybrid (Rule-based and Population-based Metaheuristic)	Interactive Evolutionary Optimization of Fuzzy Cognitive Maps (IEO-FCM)	[91]	2017

summarizes key applications, highlighting the diversity of learning approaches and their applications in robotics. For autonomous vehicle navigation and route planning, a method was proposed to optimize FCM parameters using Particle Swarm Optimization (PSO) [144]. A new double simplex initialization method improved the efficiency of the optimization process, demonstrating its effectiveness in IoT-enabled environments for vehicle navigation tasks. Similarly, an FCM-based controller was developed for multi-robot systems performing foraging tasks in semi-unknown environments, showing better computational performance and robustness compared to traditional fuzzy logic controllers [145].

An adaptive cognitive model utilizing Situated Fuzzy Cognitive Maps (SiFCMs) was introduced to dynamically adjust Levels of Autonomy (LOA) in human-UAV teams [146]. This approach employed the Truncated Backpropagation Through Time (TBPTT) learning algorithm, achieving enhanced task efficiency and balanced operator workload in dynamic environments. A hybrid approach for reactive navigation used FCMs adapted by PSO and the Self-Organizing Migration Algorithm (SOMA), enabling effective real-time path planning in dynamic environments [147]. For decision-making, an FCM-based multi-criteria decision-making method was presented to rank alternatives in robot selection problems [99]. This approach effectively handled dependence and feedback among criteria, offering a flexible and dynamic influence function. In fault detection and diagnosis, an Information Flow-based FCM model was developed using Social Learning Particle Swarm Optimization (SL-PSO) [126]. This model identified and filtered spurious causalities in industrial robotics, enhancing optimization and reducing reliance on continuous expert oversight. Another contribution integrated Interactive Evolutionary Optimization (IEO-FCM) to optimize FCM structures iteratively, demonstrating its utility in mobile robotics navigation and supervisory control tasks [91].

5.14. Software

Fuzzy Cognitive Maps (FCMs) have been applied in the software domain to address complex challenges in decision support and large-scale modeling.

Table 17. Applications of Fuzzy Cognitive Maps in Software, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Decision Making	Decision Support for Microservices Adoption	Hybrid (Rule-based and Population-based Metaheuristic)	Multi-Layer Fuzzy Cognitive Map	[148]	2022
Modeling	Decision Support for Microservices Adoption	Hybrid (Rule-based and Population-based Metaheuristic)	Multi-Layer Fuzzy Cognitive Map	[148]	2022
	Large-Scale Simulation for Self-Adaptive Systems	Hybrid (Rule-Based with Parallel GPU Computing)	Hybrid Agent-Based Models (ABM) and Fuzzy Cognitive Maps (FCM) with CUDA Acceleration	[149]	2023

presents notable contributions in this area. For decision support, a framework integrating Multi-Layer Fuzzy Cognitive Maps (MLFCMs) was developed to assist organizations in migrating to microservices [148]. This model combines rule-based reasoning with a genetic algorithm to optimize FCM initialization levels. The framework enhances transparency and interpretability in decision-making processes and allows for what-if scenario analyses. It has been successfully applied to both synthetic and real-world scenarios, providing actionable insights for operational and business complexities.

In the modeling domain, a hybrid approach combining Agent-Based Models (ABM) and FCMs with CUDA acceleration was introduced to enable large-scale simulations for self-adaptive systems [149]. This open-source Python library supports parallel execution of hybrid ABM/FCM models, scaling up to 108,000 agents. The system demonstrated substantial improvements in speed and scalability, overcoming computational limitations in large-scale simulations. It has been validated through case studies and benchmarks, proving its applicability in areas such as social dynamics and predictive modeling for large populations.

5.15. Transportation Systems

The application of Fuzzy Cognitive Maps (FCMs) in transportation systems has shown promising results in addressing traffic congestion and optimizing traffic flow.

Table 18. Applications of Fuzzy Cognitive Maps in Transportation Systems, categorized by task type, application, learning family, and algorithmic approach.

Task Type	Application	Learning Family	Algorithm	Ref.	Date
Modeling	Traffic Congestion Detection	Rule-based	Fuzzy Cognitive Map (FCM)	[150]	2021
	Traffic Flow, Freeway On-Ramp Traffic Control	Gradient-free (standard Q-learning algorithm)	Fuzzy Cognitive Map (FCM)	[151]	2023

provides an overview of key studies in this area. Traffic congestion detection was modeled using FCMs with rule-based learning [150]. This approach incorporated three input variables—flow rate, segment length, and the number of lanes—to determine congestion severity levels. The model employed Mamdani fuzzy logic to handle uncertainties in freeway networks, aiding in traffic control and infrastructure planning. The study utilized data from Hungarian freeways to validate the approach, demonstrating its applicability in real-world scenarios. A hybrid approach combining FCMs and Q-learning was introduced to optimize freeway on-ramp traffic control [151]. This method integrated reinforcement learning with FCM structures to maintain traffic density below critical thresholds. By analyzing seven traffic-related variables, the system improved freeway capacity utilization and reduced congestion. Simulations comparing controlled and uncontrolled scenarios, based on real-world data from the Hungarian e-toll system, highlighted the method’s effectiveness in enhancing traffic management.

6. Discussion

6.1. Critical Analysis of Applications

The reviewed applications of FCMs demonstrate their adaptability and versatility across diverse domains. However, a comparative evaluation of FCMs relative to other methodologies reveals both strengths and limitations that merit further investigation. For instance, FCMs provide a higher degree of interpretability compared to black-box models like deep neural networks, particularly in domains requiring stakeholder transparency. Conversely, in scenarios demanding extensive scalability or real-time responsiveness, FCMs may underperform due to computational limitations or challenges associated with iterative convergence. Indeed, the constant demand for enhanced performance in a wide range of applications remains ever-present. Most real-world problems must address shifts in data distribution, requiring appropriate mechanisms to maintain model effectiveness. Neural networks often rely on retraining strategies in such cases; hence, online adaptive FCM approaches need to be explored. While interpretability is a key strength, it should not come at the cost of achieving robust and reliable performance, as ultimately, both aspects are critical for successful real-world deployments. In specific applications, FCMs exhibit robustness under uncertainty, a trait less evident in optimization-based or statistical models. However, FCMs often rely heavily on expert knowledge to define initial rules and relationships, a process that can introduce subjectivity and limit scalability. The integration of machine learning techniques for automated weight adjustment and rule generation offers a promising avenue to address these limitations.

6.2. Qualitative Criteria for Complexity and Human Intervention

This study applies qualitative criteria to classify and analyze the complexity and human intervention levels of FCM-based methodologies across task types and learning families. The radar plots (Figure 12) visually summarize these classifications, offering a comparative view of algorithmic intricacy and required human oversight. The rationale for quantifying each work in terms of complexity and human intervention is unfolded as follows:

Complexity reflects the scale and intricacy of the system modeled, including the number of nodes or concepts, algorithmic complexity (e.g., dynamic adaptation or multi-stage processes), and computational demands (e.g., execution time or memory requirements). Based on these factors, complexity was categorized as:

•

Low Complexity: Systems with fewer than 10 nodes/concepts, characterized by simple or isolated interactions and minimal feedback loops.

•

Moderate Complexity: Systems with 10 to 50 nodes/concepts, involving moderate interactions, some feedback loops, or aggregation of interconnected subsystems.

•

High Complexity: Systems with more than 50 nodes/concepts or highly interconnected networks, incorporating non-linear relationships, significant feedback loops, or optimization processes.

Human Intervention assesses the level of manual input required during the setup, tuning, or operational phases of an FCM-based methodology. The criteria for classification are:

•

Low Human Intervention: Predominantly automated processes with minimal need for manual adjustments or tuning.

•

Moderate Human Intervention: Some human intervention required, such as initial parameter tuning or occasional manual adjustments.

•

High Human Intervention: Significant manual input necessary, including defining fuzzy partitions, constructing fuzzy rules, or extensive parameter adjustments during execution.

These criteria were systematically applied to the reviewed works to ensure consistent and objective classification. The radar plots provide a qualitative visualization of these classifications, offering insights into the balance between algorithmic complexity and human intervention. Note that each cluster, whether categorized by task type or learning family, may consist of multiple papers. As a result, the plotted values in Figure 12 represent the mean categorical scores across papers rather than individual discrete levels. This aggregation naturally results in intermediate values when different papers within a learning family receive mixed classifications (e.g., some rated as Moderate, others as High). Therefore, the radar plots illustrate the overall tendency of each learning family rather than strict categorical boundaries. More specifically, for each task type and learning family, we assigned numerical values to the qualitative classifications (Low = 1, Moderate = 2, High = 3) and computed the mean of these values across all reviewed works. This approach provides an aggregated representation of complexity and human intervention, enabling a comparative visualization of methodological trends within each category. Consequently, the presence of intermediate values in the radar plots reflects the distribution of classifications across multiple works rather than an imposed finer-grained scale. Figure 12a,b presents the radar plots that highlight the distribution of complexity and human intervention. Figure 12a illustrates these variations across task types, revealing trends such as the higher complexity and human input associated with tasks like Anomaly Detection, Classification, and Time-series Prediction. These tasks inherently demand intricate modeling frameworks, particularly when handling dynamic, non-linear relationships or high-dimensional data, which often necessitate greater manual configuration and adaptation.

Figure 12b explores these dimensions across learning families, distinguishing between Hebbian-based, Gradient-like, and Metaheuristic approaches. Gradient-based and hybrid methods are observed to exhibit slightly higher complexity due to their reliance on advanced mathematical formulations, iterative optimization processes, or integration of multiple algorithmic paradigms. Similarly, Metaheuristics often involve sophisticated search mechanisms requiring demanding computational resources, contributing to their higher complexity. Conversely, Hebbian-based approaches tend to prioritize simplicity and interpretability, resulting in relatively lower complexity and human intervention levels. However, this focus on simplicity usually comes with a compromise in performance, particularly in demanding scenarios requiring advanced capabilities.

These observations align with the inherent characteristics of the methodologies, providing a logical basis for their distribution across the dimensions of complexity and human intervention. Therefore, the radar plots serve as a valuable tool to visually compare these attributes across task types and learning families, offering insights into the trade-offs and practical implications for FCM applications. These qualitative insights provide a comprehensive understanding of how FCM methodologies vary in terms of algorithmic complexity and required manual effort, shedding light on their strengths and practical applicability in real-world scenarios.

6.3. Future Directions for FCM Research

Building on the findings of this review, several future research directions emerge that could significantly enhance the applicability and performance of FCMs:

• Advancements in FCM Methodology: Future studies could explore integrating FCMs with state-of-the-art machine learning and deep learning paradigms, such as reinforcement learning and residual neural networks, to dynamically adapt weights in real-time systems. Hybrid models, such as FCMs combined with neural networks, also offer potential for capturing complex nonlinear patterns while maintaining interpretability. Moreover, the development of online adaptive FCM approaches is critical for addressing shifts in data distribution and ensuring model robustness over time.
• Addressing Scalability and Complexity: Developing scalable algorithms capable of handling large-scale FCMs is critical for extending their applicability to complex systems, such as urban energy grids or large-scale industrial networks. Methods leveraging distributed computing or parallel processing could address computational bottlenecks, enabling real-time applications. Additionally, scalability must align with ensuring interpretability and performance, as both are crucial for practical implementation.
• Exploration of Emerging Domains: Although FCMs have been successfully applied in traditional engineering and decision-support systems, their potential in emerging fields such as autonomous systems, precision agriculture, policy modeling, or other emerging fields remains largely untapped. Expanding the scope of applications can provide insights into the adaptability and limitations of FCMs in these domains.
• Reduction in Expert Dependency: Techniques to minimize reliance on expert knowledge, such as automated relationship discovery using data-driven approaches, could significantly improve the efficiency and objectivity of FCM development. The adoption of unsupervised or semi-supervised learning frameworks may enhance the initialization and training processes of FCMs. However, these advancements must be coupled with mechanisms to ensure that improvements in automation do not compromise interpretability or performance.
• Leveraging Advances in Computational Technologies: The integration of FCMs with edge computing, Internet of Things (IoT) devices, and quantum computing could redefine their real-time processing capabilities. These technologies offer opportunities for deploying FCMs in distributed, resource-constrained environments while maintaining computational efficiency.
• Robustness in Dynamic Environments: Further research is needed to improve FCM performance in dynamic and evolving systems where input–output relationships may shift over time. Adaptation mechanisms that ensure stability and accuracy under changing conditions will be critical for long-term deployment. Real-world problems often demand solutions that address shifts in data distribution; therefore, FCM approaches must be designed to adapt effectively while maintaining high performance.
• Balancing Interpretability and Performance: While interpretability is a key strength of FCMs, it should not come at the expense of robust and reliable performance. Some works highlight interpretability as a focus, but achieving strong results in practical applications remains equally critical. Striking a balance between these aspects is essential to ensure the relevance of FCMs in addressing real-world challenges.

By addressing these future directions, FCM research can advance towards overcoming existing challenges and expanding its impact across diverse scientific and engineering domains. This progression will further establish FCMs as a valuable tool for modeling, decision-making, classification, and time-series diagnostics and prediction in complex systems.

7. Conclusions

This review demonstrates the versatility and adaptability of Fuzzy Cognitive Maps (FCMs) in addressing diverse real-world problems, from decision-support systems to predictive modeling and optimization tasks. While FCMs offer unique advantages in interpretability and causal representation, the review highlights critical challenges, such as limited scalability, dependency on expert input, and the inability to handle shifts in data distributions effectively.

Addressing these challenges requires concerted efforts to develop adaptive learning algorithms capable of dynamically adjusting weights in response to evolving data and system requirements. Furthermore, hybrid approaches, such as integrating FCMs with neural networks or reinforcement learning, could enhance their capacity to model nonlinear and complex systems. Real-time deployment in distributed environments could also benefit from leveraging advances in computational technologies such as edge computing and parallel processing.

The future of FCMs lies in expanding their applicability to emerging domains like autonomous systems, energy sustainability, and precision agriculture while improving robustness in dynamic and data-driven environments. By addressing these critical areas, FCMs have the potential to remain relevant and impactful tools for modeling and solving complex problems across various domains. This study, beyond offering a comprehensive review of engineering applications and the challenges addressed within these domains, serves as a foundation for advancing FCM methodologies and their applications. It underscores the importance of methodological innovation and empirical validation as key drivers for future research.