F-DeNETS: A Hybrid Methodology for Complex Multi-Criteria Decision-Making Under Uncertainty

Abstract

In the modern business environment, where uncertainty and complexity make decision-making difficult, the need for robust, transparent and adaptable support tools is highlighted. The proposed method, named Flexible Decision Navigator for Evaluating Trends and Strategies (F-DeNETS), offers a complementary perspective to classic Artificial Intelligence (AI), Big Data and Multi-Criteria Decision-Making (MCDM) tools. Despite their broad use, these methods frequently suffer from critical sensitivities in the weighting of criteria and the handling of uncertainty, leading to compromised reliability and limited practical utility in environments with limited data availability. To bridge this gap, F-DeNETS integrates intuition and uncertainty into a transparent and statistically grounded process. It introduces a balanced approach that combines statistical evidence with human judgment, extending the boundaries of classic AI, Big Data and MCDM methods. Classic MCDM methods, although useful, are sometimes limited by subjectivity, staticity and dependence on large volumes of data. To fill this gap, F-DeNETS, a hybrid framework combining Fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL), Non-Asymptotic Fuzzy Estimators (NAFEs) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), transforms expert judgments into statistically sound fuzzy quantifications, incorporates dynamic adaptation to new data, reduces bias and enhances reliability. A numerical application from the shipping industry demonstrates that F-DeNETS offers a flexible and interpretable methodology for optimal decisions in environments of high uncertainty.

1. Introduction

In the rapidly evolving business environment, the need for effective decision-making has intensified dramatically, requiring managers to rely on timely, accurate and applicable data [1]. To mitigate risks and make an organization more “intelligent” and responsive, a change in the way it perceives problems is required. This is where Big Data analytics comes into play, which, according to the definition of [2], transforms high volume, velocity and variety (3Vs) information into value. This capability allows organizations to make informed decisions that lead to competitive advantage, improved performance and increased profits [3,4], acting as a critical tool for automating processes and gaining deep market knowledge [5].

However, the extensive use of data is not without risks. The use of potentially sensitive information opens up the ethical dilemma of its strategic exploitation [6], creating a critical need for a balance between innovation and privacy protection. In this context, Artificial Intelligence (AI) emerges as a decisive factor, with the ability to make decisions in real time based on algorithms that learn and adapt automatically [7]. For example, in Human Resource Management (HRM), decisions that were previously made by humans are increasingly being made by algorithms [8]. While this can improve the employee experience and enable the prediction of important phenomena [9], the process is not without risks. The growing adoption of AI has focused attention on the ethical values and principles that guide its development [10], with concerns and ethics that developers may neglect in favor of technical priorities [11].

To address these ethical dilemmas and improve the process, qualitative analysis can play a crucial role by incorporating human perception, judgment, and framing of problems, as proposed by the Throughput model [12]. In particular, it allows for the understanding of subtle and contextual aspects, such as the emotional dimension and ethical implications, helping to create richer algorithmic pathways that take into account not only the voted data but also the human experience [13,14]. However, qualitative analysis has disadvantages, such as subjectivity that can introduce biases into algorithms [15], difficulty in scaling, and inability to generalize findings [16]. A great part of research has focused on alternatives ranking based on emotional texts, which are extracted through sentiment analysis techniques. However, due to the vague and uncertain nature of the information contained in textual reviews, it is very difficult to evaluate this qualitative, emotional information with confidence. For this reason, the processing of these textual emotions based on fuzzy set theory is an area that fascinates researchers, opening new horizons and proposing entirely new dimensions to scientific research [17].

To address the complexity of today’s decision-making problems, characterized by uncertainty, conflicting criteria, and the need for transparency, Multi-Criteria Decision-Making (MCDM) methods offer a structured and systematic framework [18,19]. The main MCDM methods mentioned include the following:

• Analytic Hierarchy Process (AHP) [20]: incorporates qualitative and quantitative factors but is sensitive to inconsistencies.
• Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [21]: easy to use but assumes linear relationships.
• Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE) [22]: flexible with good visualization but sensitive to parameter selection.
• ELimation Et Choix Traduisant la Realité (ELECTRE) [23]: covers ambiguity but can be computationally demanding.
• Fuzzy-based MCDM [23,24]: realistic in uncertain environments but with potential subjectivity.
• Hybrid Methods [25,26]: combine techniques for more robust models but can be complex.

MCDM methods bridge gaps in traditional methods, offering a way to systematically address multiple criteria and uncertainty. However, they also have limitations, such as subjectivity in weighting [27], computational complexity [28], their static nature in the face of dynamic environments [29], and the challenges of integrating them with Big Data and Artificial Intelligence (AI) technologies for fully automated decision-making.

Despite their limitations, MCDMs remain an indispensable tool for making transparent and optimal decisions, with applications in a wide range of disciplines [30,31,32]. From Business Administration and Strategic Planning to Human Resource Management (HRM), Sustainable Development, Healthcare, Public Policy and Finance, these methods help to decompose complex dilemmas and evaluate alternative solutions.

In this paper, taking under consideration the above-mentioned statements, the Flexible Decision Navigator for Evaluating Trends and Strategies (F-DeNETS) is proposed as a powerful complementary framework to traditional AI, Big Data and MCDM methods, addressing their specific and critical sensitivities. F-DeNETS is a new integrated methodological framework that combines the Fuzzy DEMATEL and Fuzzy TOPSIS methods with Non-Asymptotic Fuzzy Estimators (NAFEs). The use of NAFEs is its key feature, which transform the subjective judgments of experts into statistically valid fuzzy quantitative estimates. This process essentially “enumerates” human intuition, drastically reducing bias and arbitrariness, especially when data is incomplete, or unrepresentative, or comes from small samples, where conventional Machine Learning models fail. Furthermore, the framework does not avoid uncertainty but embraces it, incorporating fuzzy logic and confidence intervals, ensuring that this critical information is preserved and transferred throughout the decision-making chain, leading to more realistic and robust results. Despite their statistical basis and their advantages in dealing with uncertainty using small samples, there is a clear gap in the literature regarding their use. NAFEs have not, to date, been integrated or applied in the context of Multi-Criteria Decision-Making (MCDM). Existing methodologies that use fuzzy numbers for weights continue to rely mainly on subjective definitions or asymptotic approaches. The adaptation and application of NAFEs to generate statistically substantiated criterion weights are, therefore, important innovations that this work attempts to develop.

Beyond its technical robustness, F-DeNETS stands out for its ability to balance between human and computational elements. The difficulty (or challenge) that uncertainty introduces into decision-making problems is what creates the need for this categorization of problems or methods. Uncertainty usually arises from the fact that experts are not sufficiently familiar with the different alternatives or cannot clearly evaluate the relative position of each in the light of different criteria. Because the information in decision-making is often complex and uncertain, decision-makers find it difficult to define their goals and criteria clearly and precisely [33]. F-DeNETS keeps human judgment and contextual reasoning at the center, using user-friendly linguistic scales, while ensuring complete transparency and interpretability. Each step of the methodology is mathematically substantiated and transparent, allowing for sensitivity analysis to understand how different parameters affect the final result. This transparency, combined with its event-oriented architecture, makes it highly adaptable to dynamic environments, capable of dynamically updating the criteria and their weights based on new information. Therefore, it is not a replacement for AI but a substantial compromise that enhances its capabilities, especially in high-stakes areas such as human resource management, crisis management, strategic decisions, etc., where ethical dilemmas and contextual understanding are traditional weaknesses of automated systems.

The proposed F-DeNETS methodology addresses and significantly mitigates three main problems that plague existing MCDM methods, especially under conditions of uncertainty:

First, it replaces the traditional and often arbitrary weight assignment (e.g., via Entropy or Fuzzy AHP) with NAFEs. NAFEs offer statistical justification of criterion weights through confidence intervals, taking into account both the variance of expert ratings and the sample size. This eliminates the tendency of Entropy to overemphasize criteria with high dispersion (primacy attribute). NAFEs also reduce the subjectivity and biases introduced by paired comparisons in methods such as AHP, improving the objectivity and reliability of the process. More precisely, to overcome the intrinsic limitations of existing weighting methods, such as the existing bias of entropy-based techniques [34] and the subjectivity inherent in pairwise comparisons of Fuzzy AHP, this paper introduces NAFEs as a core component of the F-DeNETS framework. NAFEs provide a statistically robust foundation for criterion weighting by transforming expert judgments into fuzzy numbers grounded in confidence intervals, thereby significantly enhancing the reliability and objectivity of decisions under uncertainty and small sample sizes.

Second, F-DeNETS addresses the dual limitation of staticity and reliance on small or subjective samples. Through its dynamic, event-driven architecture, the framework allows for re-evaluation and adjustment of criteria based on new information, transforming it from a static model into an adaptive decision-making tool. The term “event-driven architecture” means that the flow of the methodology is not simply linear but proceeds based on the appearing “events” or the results of each previous stage. For example, the criteria structure resulting from Fuzzy DEMATEL (event) determines which criteria will be weighted with NAFEs. This architecture makes the model flexible and adaptable, which allows it to respond dynamically to the evolving nature of complex problems. At the same time, the use of NAFEs allows for efficient estimation even with small samples of experts as the estimators are based on statistical confidence intervals rather than large volumes of data. This mitigates the critical disadvantage of reliance on subjective estimates and limits the risk of bias.

Third, the integrated hybrid framework (F-DEMATEL/NAFEs/F-TOPSIS) enhances the transparency and interpretability of the results, a weak point of many MCDM methods. The causal analysis with DEMATEL reveals the core factors affecting the system, while the built-in sensitivity analysis through the parameters of NAFEs allows decision-makers to evaluate the robustness and stability of the decision under different uncertainty scenarios. This dual approach not only improves the quality of the decision but also offers a transparent and easily interpretable result, reducing the gap between the complexity of the model and its practical utilization by managers. F-DeNETS constitutes an approach that attempts to redefine the relationship between human and computational intelligence, proposing a framework that does not ignore uncertainty but utilizes it as valuable information in the decision-making process. By using non-asymptotic estimators, subjective judgments are statistically substantiated and move away from the arbitrariness that characterizes many traditional methods, while its dynamic, event-driven architecture allows the system to adapt in real time, absorbing new data and revising criteria. In summary of this part, the methodology emphasizes transparency and interpretability, offering clear mapping of causal relationships and the ability to perform sensitivity analysis that enhances user confidence. Thus, instead of operating in competition with Artificial Intelligence, it enhances it with tools that allow for more realistic, robust and socially conscious decisions, especially in areas where data is limited and the human dimension is irreplaceable.

The main scientific contributions of this work can be summarized as follows:

• First, F-DeNETS is proposed as a new hybrid framework that combines the advantages of Fuzzy DEMATEL, Non-Asymptotic Fuzzy Estimators (NAFE) and Fuzzy TOPSIS in a dynamic, event-driven architecture.
• Second, the use of NAFEs is introduced for the statistically robust calculation of uncertain criterion weights from small samples of expert responses, providing an objective and statistically controlled way of calculating uncertainty in contrast to its subjective definition in classical fuzzy techniques.
• Third, the proposed model ensures robustness through the inherent capability of sensitivity analysis on NAFE parameters, offering greater flexibility in decision-making.
• Fourth, the proposed methodology presents wide application in multidisciplinary fields, such as engineering, health sciences, economics, environmental management, shipping and generally any context that requires decision-making based on numerous and often competing criteria, under conditions of limited and unclear information.

This article is structured in seven main sections. Section 1 introduces the subject and briefly presents the proposed F-DeNETS methodology. Section 2 discusses the literature review, with an emphasis on hybrid approaches to Multi-Criteria Decision-Making and the identification of research gaps. Section 3 presents the proposed methodology in detail, including the fundamentals of fuzzy set theory, NAFEs and the steps of implementing the F-DeNETS framework. Section 4 concerns the numerical application of the method to a real crew selection problem, while Section 5 announces and discusses the main findings. Section 6 summarizes the conclusions, highlights the limitations and suggests directions for future research. Finally, Section 7 analyzes the theoretical and managerial implications of the work.

2. Literature Review

2.1. Fuzzy DEMATEL and Fuzzy TOPSIS Integration

Multi-criteria analysis is a widely used approach, with applications across numerous scientific fields where decisions must be made based on multiple criteria. A significant body of research has employed combinations of Fuzzy TOPSIS and Fuzzy DEMATEL—either individually, in conjunction with other Multi-Criteria Decision Analysis (MCDA) and Multi-Criteria Decision-Making (MCDM) methods or integrated with techniques from fuzzy set theory. Given the extensive literature in this area, the author aims to present, critically evaluate, and comparatively analyze a selection of representative studies that highlight key developments in the international research landscape.

2.1.1. Fuzzy DEMATEL and Fuzzy TOPSIS Approaches

Baykasoglu [35] use the combination of Fuzzy DEMATEL and Fuzzy TOPSIS in Truck Selection for a Land Transport Company. Fuzzy DEMATEL calculates weights for hierarchical criteria (e.g., the economy and spare parts cost), while Fuzzy Hierarchical TOPSIS extends classic TOPSIS for hierarchical criteria structures, using the weights from DEMATEL. Their combination incorporates criteria hierarchies and interactions between them, offering a more realistic evaluation. Bali et al. [36] again use a combination of MCDM methods this time in third-party logistics to select two cargo companies for a firm that produces heat systems and wants to sell its specific products via the internet. They use Fuzzy DEMATEL to calculate the weights of the criteria, taking into account the causal relationships between them. The result is the clarification of the criteria that are “cause” and those that are “causal”. Then, they apply Fuzzy TOPSIS to rank the alternatives based on the distance from the ideal and anti-ideal point. In other words, Fuzzy DEMATEL determines the weights of the criteria, which feed Fuzzy TOPSIS for evaluating alternatives. Özdemir [37] focuses on the analysis of occupational accidents occurring in ports. The aim is to identify the most important factors leading to accidents and to evaluate prevention priorities. The analysis is based on data from ports, with the aim of enhancing safety and reducing risks. In a methodological context, Fuzzy DEMATEL is used to identify and quantify the causal relationships between factors that cause accidents (e.g., human errors, technical failures, and environmental factors). Fuzzy TOPSIS is applied to rank risk factors in terms of their criticality. In short, DEMATEL identifies relationships and weights of criteria, and TOPSIS ranks factors based on the weights. Forozandeh [38] applies MCDM methods to identify and prioritize supply chain management (SCM) challenges in R&D projects of Iranian organizations (electronics, mechanical engineering, etc.). In the same context, the author uses Fuzzy DEMATEL to categorize challenges and analyze causal relationships/influences between them. Then, through Fuzzy TOPSIS, it prioritizes the challenges based on weights from DEMATEL and distance from ideal solutions. The logic of the combination of the methods is the following: DEMATEL identifies critical causal challenges, which then feed TOPSIS for ranking. Beyond the aforementioned works, numerous other studies have investigated the integration of Fuzzy TOPSIS and Fuzzy DEMATEL [39,40,41,42,43,44,45,46,47,48].

2.1.2. Fuzzy DEMATEL, Fuzzy TOPSIS and Fuzzy Information Representation Framework Approaches

A distinct application of this combination of methods concerns the way uncertainty and ambiguity is represented in the two MCDM methods. Typical examples are the study of Zhang and Su [49], Naz et al. [50] and Zeng et al. [51]. Zhang and Su [49] try to select the most suitable participants for a knowledge task (Knowledge-Intensive Crowdsourcing) with an example from crowdsourcing platforms such as “Threadless, InnoCentive”. In a methodological framework, a combination of Fuzzy DEMATEL and Fuzzy TOPSIS is used, using the 2-tuple linguistic method for the representation of fuzzy data. DEMATEL is used to uncover and quantify the causal relationships between the key characteristics (criteria) of the participants (interests, competence, reputation and availability), while TOPSIS is applied to evaluate and rank the pre-selected participants, based on the distances from the positive ideal and the negative ideal solution. A similar methodology was also followed by Naz et al. [50]. Zeng et al. [51] use the Human Factors Analysis and Classification System (HFACS) method to determine human factors risk indicators of flight dispatchers. More specifically, the application concerns Sichuan Airlines for the assessment of human factors risks of flight dispatchers, where critical factors such as working conditions, psychological and physical conditions and work processes are highlighted. The authors apply the extended hesitant fuzzy theory to represent uncertainty and ambiguity in the evaluation, combining the improved fuzzy TOPSIS for ranking risk factors and DEMATEL-ISM (Interpretive Structural Modeling) points for causal analysis and hierarchical factor structure.

2.1.3. Fuzzy DEMATEL, Fuzzy TOPSIS and Fuzzy ANP Approaches

An addition to the dyad of these MCDM methods is Fuzzy ANP. Buyukozkan and Cifci [52] propose a hybrid MCDM combining three techniques: fuzzy DEMATEL for analyzing causal relationships between criteria, fuzzy ANP for evaluating and weighting criteria considering dependencies and interactions and fuzzy TOPSIS for ranking alternative suppliers based on the distance from the ideal and anti-ideal points. The use of fuzzy set theory is performed to deal with uncertainty in human judgment. The application concerns a real case of green supplier evaluation at “Ford Otosan”, where criteria such as green accounting, green organizational activities, organizational performance and green supplier capabilities are specified. In the same context, for supplier selection in manufacturing companies, Alam-Tabriz et al. [53] develop a hybrid MCDM model with three stages: Initially, they use Fuzzy DEMATEL to identify the relationships between the criteria, then Fuzzy ANP to calculate the weights of the criteria with interdependence and feedback and finally fuzzy TOPSIS for the ranking of alternative suppliers. Vinodh et al. [54], for the selection of the best concept design for instrument panels in the automotive industry, combine Fuzzy DEMATEL, Fuzzy ANP and Fuzzy TOPSIS for the selection of the agile concept design. DEMATEL is used to analyze and capture the causal relationships between the agility criteria. ANP determines the weights of the criteria taking into account the interdependence between them, while TOPSIS is used for the final ranking of the alternative designs. Ocampo et al. [55] apply MCDM methods to sustainable food production in large Philippine enterprises. The study “maps” the best options (e.g., “life cycle analysis” for new products) into practices with advanced implementation manuals. In a methodological framework, they construct an integrated hybrid fuzzy multi-criteria determination to address the complexity and interdependencies in the sustainable food production strategy. The process consists of three steps. Initially, Fuzzy DEMATEL is used to identify how decision elements interact (e.g., options like “optimal batch size” or “facility location”). Then, experts express the influence between elements using fuzzy linguistic terms (e.g., “very high”), and based on this process, the main causal factors are determined (e.g., “quality culture” influences “defect avoidance”). Fuzzy ANP incorporates the interactions from DEMATEL into a hierarchical network model and calculates the weights for strategic options through paired comparisons under uncertainty. In the last phase, Fuzzy TOPSIS maps the optimal options to established sustainable production practices (SMPs). Further contributions using this combination of MCDM methods include [55,56,57,58].

2.1.4. Fuzzy DEMATEL, Fuzzy TOPSIS and Fuzzy AHP Approaches

An alternative combination is that of Fuzzy DEMATEL, Fuzzy AHP and Fuzzy TOPSIS. Below are listed some of the studies that use this combination. Sathyan et al. [59] apply this triplet of methods for prioritizing responsiveness factors in the Indian automobile industry (e.g., demand management and management commitment). Fuzzy DEMATEL maps cause-and-effect relationships between responsiveness enablers in the supply chain. Fuzzy AHP calculates weights of evaluation criteria (e.g., delivery time reduction and customer satisfaction), while Fuzzy TOPSIS prioritizes factors based on the distance from the ideal solution. In summary, DEMATEL identifies the causal factors that affect the system, while TOPSIS uses the AHP weights for the final ranking. Al-Baldawi et al. [60] conduct their study in an Iraq medium-sized company producing healthy water, juices and soft drinks, aiming to determine which lean dimension is most critical for its leanness, to identify the dimension driving the improvement process and to evaluate the level of implementation of the five lean dimensions. Fuzzy AHP calculates the importance weights of the five “Lean dimensions” (management, process, supplier, customer and employee). Fuzzy DEMATEL analyzes the causal relationships between the dimensions and determines influence weights. In the final stage, Fuzzy TOPSIS evaluates the level of leanness of the company based on the distance from the ideal/anti-ideal scenario. In other words, the weights from FAHP and FDEMATEL are combined to calculate combined weights, which are fed into FTOPSIS for the final evaluation.

2.1.5. Fuzzy DEMATEL, Fuzzy TOPSIS and Other MCDM Techniques Approaches

Less frequently, along with the combination of Fuzzy DEMATEL and Fuzzy TOPSIS, the Fuzzy ELECTRE method [61] has been used for the evaluation of the effectiveness of knowledge transfer and the Fuzzy PROMETHEE method in the food supply chain [62].

This literature review has revealed the main cons of existing MCDM models (hybrid and non-hybrid), including (1) high sensitivity to subjective expert judgments and fuzzy data processing, (2) static structures that fail to adapt to new information and (3) reliance on large data sets or complex pair-wise comparisons. The proposed F-DeNETS framework is designed to bridge these gaps. Specifically, it directly addresses the first limitation by replacing subjective weighting with statistically validated NAFEs, which transform expert judgments into fuzzy numbers derived from confidence intervals. It resolves the second through a dynamic, event-driven architecture that allows for real-time updating of criteria. Finally, it overcomes the third limitation by working effectively even with small teams of experts, constituting a flexible and adaptable methodology. A detailed analysis follows in the next section.

2.2. Identified Limitations and Research Gaps

The combinations of MCDM and MCDA methods are particularly efficient as they can provide important directions for decision-makers (DMs). A thorough analysis of the literature on the use of these methods across various scientific fields and industrial sectors reveals a number of methodological limitations. The following section presents the key limitations for which the approach proposed in this paper will attempt to offer improvements.

2.2.1. Fuzzy DEMATEL and Fuzzy TOPSIS

A significant and frequently noted challenge in these studies is that the outcomes are often highly sensitive to how fuzzy data is processed. In DEMATEL, the reliance on subjective expert judgments can introduce bias. Arbitrary estimation of the “influence” between criteria can sensitize the results to subjective errors. Incorrect estimates of effects between criteria (in DEMATEL) can distort the weights and the final ranking. The use of fuzzy linguistic terms (e.g., “very high influence”) can lead to inconsistencies. Also, the choice of membership functions (e.g., triangular vs. trapezoidal) significantly affects the results.

In [35], a lack of group decision-making is noted as the model relies on a single decision-maker, which limits its reliability. Furthermore, in the same study, the weights for the proposed approach are derived from a modified fuzzy DEMATEL method, yet the final values are crisp.

Finally, most of the papers in this category recommend a combination with other MCDM methods (such as Fuzzy AHP or Fuzzy VIKOR) for cross-validating the results.

2.2.2. Fuzzy DEMATEL and Fuzzy TOPSIS and Fuzzy Information Representation Frameworks

A review of this methodological category reveals a set of recurrent and persistent limitations. First among these is the difficulty of accurately modeling complex cause–effect relationships in fuzzy environments, particularly when intricate dependencies exist between criteria [49]. Another key issue is the inherent subjectivity and reliance on expert judgments, which can compromise reliability when dealing with fuzzy or hesitant fuzzy data. Further complicating matters is the challenge of interpreting aggregated results, especially in contexts involving human factors. Additionally, the sensitivity of outcomes to both the selection of fuzzy logic parameters and the choice of model-specific thresholds remains a notable concern [51]).

Overall, the integration of fuzzy DEMATEL and fuzzy TOPSIS offers a structured means to address Multi-Criteria Decision-Making under uncertainty, facilitating the identification of causal relationships and the ranking of alternatives. Nevertheless, the method’s effectiveness is tempered by complications arising from fuzzy data complexity, reliance on subjective inputs and pronounced parameter sensitivity limitations that demand careful implementation and further scholarly attention.

2.2.3. Fuzzy DEMATEL, Fuzzy TOPSIS and ANP

Within this methodological category, the issue of subjectivity arises as a central concern. The reliance on expert knowledge, while valuable, introduces a risk of bias, and it appears that subjectivity in expert judgments remains difficult to fully eliminate. Furthermore, accurately modeling complex causal relationships in ambiguous environments continues to pose a significant challenge, particularly when interdependencies among criteria are intricate. The generalizability of results is also noted as a limitation, especially when studies rely on data from a single organizational context [54].

In light of these recurring challenges, studies within this domain consistently emphasize the need to develop robust decision-support systems tailored for broader industrial application. Such systems could mitigate existing limitations and enhance the practicality and reliability of these methods in real-world settings.

2.2.4. Fuzzy DEMATEL, Fuzzy TOPSIS and AHP

A common issue in these methods is their subjectivity since the results depend on the opinions of a small group of experts. Another problem is that the models are static and do not consider changes over time. Many studies suggest improving these methods by combining them with other fuzzy techniques or with neural networks to handle uncertainty better and increase accuracy.

However, a key limitation, pointed out by [60], is that results may not be reliable when very few experts are used, as in their study, which included only five experts from one company. Because of these challenges, most papers recommend developing hybrid models and using big data to calculate weights in a more objective way.

2.2.5. Final Remarks and Additional Information on Criteria Weights

In most of the models mentioned above, the Fuzzy DEMATEL method is employed to calculate criterion weights by analyzing causal relationships and their interactions. Specifically, Fuzzy DEMATEL is first applied to determine the degree to which each criterion influences the others, resulting in a causal influence table composed of fuzzy numbers. Following normalization and aggregation of these data, the method produces a total relationship table that captures both direct and indirect effects among the criteria. Subsequent defuzzification and normalization steps yield the final weight values, reflecting the relative importance of each criterion within the decision-making process.

These weights are then used as inputs in the Fuzzy TOPSIS method or, in some cases, first integrated into Fuzzy ANP or AHP methods for further refinement to rank the available alternatives. In summary, within the hybrid approach, Fuzzy DEMATEL serves as the mechanism for deriving and emphasizing criterion weights based on causal relationships under conditions of uncertainty. While the method aims to provide a realistic and comprehensive evaluation of the relative importance of each criterion, it is nonetheless subject to the limitations and challenges discussed in the preceding paragraphs.

Another method for calculating the weights in Fuzzy TOPSIS is the entropy weight model which is used for fuzzy sets on problems with completely unknown weight information. Entropy weighting and TOPSIS have been widely used in the research literature [63,64,65,66,67,68,69,70,71,72,73]. According to Chen [74], the use of the entropy method to determine criteria weights in TOPSIS tends to overemphasize the criterion with the greatest data differentiation (primacy attribute) while weakening the criteria with low differentiation, which reduces the overall completeness of the evaluation and can lead to biased or unrealistic results. As a consequence, the method can significantly alter the ranking of alternatives.

Biswas and Sarkar [34] point out a significant limitation in existing TOPSIS methods. These methods require prior weight information to solve decision-making problems. Typically, these methods operate under the assumption that either the weights of the DMs are known, the weights of the criteria are known or both. The authors observe that some studies have attempted to address situations where DMs weights are unknown (but criteria weights are available), while others have dealt with unknown criteria weights (provided the DMs weights are known). A key shortcoming that they identify is the lack of an approach that can effectively handle cases where weight information for both DMs and criteria is simultaneously unavailable.

The limitations discussed above such as the reliance on subjective judgments, sensitivity to fuzzy data processing and the constraints of static modeling show a clear need for a more robust, adaptive and statistically grounded methodology. The proposed F-DeNETS framework addresses these research gaps by introducing a dynamic architecture powered by NAFEs. This architecture is designed to mitigate subjectivity, operate reliably even with small expert samples and explicitly incorporate uncertainty into the weighting process through statistical confidence intervals.

3. The Proposed Methodology

3.1. F-DeNETS Development

3.1.1. Foundations of Fuzzy Sets Theory

An introduction to the key elements of fuzzy sets theory [75] will help establish a basis for the subsequent discussion in the area of fuzzy MCDM.

Definition 1.

Let  $\mathrm{X}$ be a set of numbers or objects and  $\mathrm{x}\in \mathrm{X}$; then, a fuzzy set  $\tilde{\mathrm{K}}$ is a subset of  $\mathrm{X}$, or  $\tilde{\mathrm{K}}\subseteq \mathrm{X}$ is a set of ordered pairs,  $\mathrm{x}$ and  ${\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right):\tilde{\mathrm{K}}=\left\{\mathrm{x},{\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)|\mathrm{x}\in \mathrm{X}\right\}$, where  ${\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)$ is the membership function that defines the membership grade of element  $\mathrm{x}$ in the fuzzy set  $\tilde{\mathrm{K}}$.

Definition 2.

The domain or support of  $\tilde{\mathrm{K}}$ is the crisp (classical) set for all  $\mathrm{x}\in \mathrm{X}$, for which

$${\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)\ge 0\mathrm{or} \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\left(\tilde{\mathrm{K}}\right)=\left\{\mathrm{x},\mathrm{ }{\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)\ge 0|\mathrm{x}\in \mathrm{X}\right\}$$

Definition 3.

The a-cut (or a-level) of the fuzzy set  $\tilde{\mathrm{K}}$ is the crisp set  ${}^{\mathrm{a}}\tilde{\mathrm{K}}=\left\{\mathrm{x}\in \mathrm{X}:{\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)\ge \mathrm{a}\right\}$ where  $\mathrm{a}\in \left[\mathrm{0,1}\right]$, that is, the a-cut  ${}^{\mathrm{a}}\tilde{\mathrm{K}}$ is a subset in which its elements belong to the fuzzy set  $\tilde{\mathrm{K}}$ with a membership grade equal or greater to  $\mathrm{a}$.

$${}^{\mathrm{a}}\tilde{\mathrm{K}}=\left\{xϵ\mathrm{X}|{\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right)\ge \mathrm{a}\right\}$$

Definition 4.

Let  $\mathrm{X},\mathrm{Y}$ be two crisp sets and  $\mathrm{f}:\mathrm{X}\to \mathrm{Y}$. Let  $\tilde{\mathrm{K}}:\mathrm{X}\to \mathrm{I}=\left[\mathrm{0,1}\right]$ be a fuzzy subset of  $\mathrm{X}$; then, the fuzzy subset of  $\mathrm{Y}$ is defined by  $\tilde{\mathrm{L}}=\mathrm{f}\left(\tilde{\mathrm{K}}\right)$ as

$$\begin{gathered} \mathrm{f}\left(\tilde{\mathrm{K}}\right):\mathrm{Y}\to \mathrm{I}=\left[\mathrm{0,1}\right] \\ \mathrm{With} \mathrm{f}\left(\tilde{\mathrm{K}}\right)\left(\mathrm{y}\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\tilde{\mathrm{K}}\left(\mathrm{x}\right):\mathrm{f}\left(\mathrm{x}\right)=\mathrm{y}\right\} \\ {\mu }_{\tilde{\mathrm{L}}}\left(\mathrm{y}\right)=\underset{\mathrm{f}\left(\mathrm{x}\right)=\mathrm{y}}{\bigvee }{\mu }_{\tilde{\mathrm{K}}}\left(\mathrm{x}\right) \end{gathered}$$

If $\tilde{\mathrm{K}}$ is a finite set, then, in the above definition, it holds that $\mathrm{s}\mathrm{u}\mathrm{p}=\mathrm{m}\mathrm{a}\mathrm{x}$.

Definition 5.

A fuzzy number  $\tilde{\mathrm{O}}$ is a convex fuzzy set  $\tilde{\mathrm{O}}$ on the line of the real numbers $\mathrm{R}$

$$\mathrm{O}\equiv {\mu }_{\tilde{\mathrm{O}}}\left(\mathrm{x}\right):\mathrm{x}\in \mathrm{R}\to \mathrm{O}\left(\mathrm{x}\right)\in \left[\mathrm{0,1}\right]$$

so that

$\tilde{\mathrm{O}}$ is a normal fuzzy set (that is $\exists \mathrm{x}\in \mathrm{R}:{\mu }_{\tilde{\mathrm{O}}}\left({\mathrm{x}}_0\right)=1$);

$\tilde{\mathrm{O}}$ is a convex fuzzy set;

$\tilde{\mathrm{O}}$ is a compact fuzzy set;

$\tilde{\mathrm{O}}$ is a semi-continuous fuzzy set.

3.1.2. Non-Asymptotic Fuzzy Estimators

Let us consider a continuous and monotonic (increasing) function to construct the NAFEs for the mean value of a fuzzy variable. The family of the linear non-asymptotic fuzzy estimators is chosen, and

$$\mathrm{f}\left(\alpha \right)=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\epsilon }{2}}\right)\alpha +{\displaystyle \frac{\epsilon }{2}} \mathrm{f}\mathrm{o}\mathrm{r} \alpha \in \left(\mathrm{0,1}\right] \mathrm{a}\mathrm{n}\mathrm{d} \epsilon \in \left(\mathrm{0,1}\right).$$

Proposition 1

([76]). Let  ${\mathrm{P}}_1,{\mathrm{P}}_2,\dots {\mathrm{P}}_{\mathrm{n}}$ be a data sample randomly created, ${\mathrm{p}}_1,{\mathrm{p}}_2,\dots {\mathrm{p}}_{\mathrm{n}}$ be the sample values, and  $\epsilon \in \left(\mathrm{0,1}\right)$. When the data sample  $\mathrm{v}$ is small and normally distributed with the population standard deviation  $\left(\sigma \right)$ to be unknown, then

$${\check{\mu }}_{\epsilon }\left(p\right)=\left\{\begin{array}{c}{\displaystyle \frac{2}{1-\mathsf{\epsilon }}}{T}_{\mathrm{v}-1}\left({\displaystyle \frac{\overline{\mathrm{p}}-\mathrm{p}}{\frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}}\right)-{\displaystyle \frac{\epsilon }{1-\mathsf{\epsilon }}}, if p\in \left[\overline{\mathrm{p}}-{\mathrm{t}}_{1-{\displaystyle \frac{\epsilon }{2}}}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathrm{v}}}},\overline{\mathrm{p}}\right]\\ \\ {\displaystyle \frac{2}{1-\mathsf{\epsilon }}}{T}_{\mathrm{v}-1}\left({\displaystyle \frac{\mathrm{p}-\overline{\mathrm{p}}}{\frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}}\right)-{\displaystyle \frac{\mathsf{\epsilon }}{1-\mathsf{\epsilon }}}, if p\in \left[\overline{\mathrm{p}},\overline{\mathrm{p}}+{\mathrm{t}}_{1-{\displaystyle \frac{\epsilon }{2}}}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}\right]\\ \\ 0 else \end{array}\right.$$

is the membership function of $p$ in $\mathsf{\epsilon }$. Τhe support of the fuzzy number $\mathsf{\epsilon }$ is the $\left(1-\epsilon \right)100\%$ confidence interval for $\mu$. The alpha-cuts of this fuzzy number are the following intervals:

$${\alpha }_{{\ddot{\mu }}_{\epsilon }=}=\left[\overline{\mathrm{p}}-{\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathsf{\nu }}}},\overline{\mathrm{p}}+{\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathsf{\nu }}}}\right],\alpha \in \left(\mathrm{0,1}\right]$$

where ${\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}={\mathrm{T}}^{-1}\left(1-\mathrm{f}\left(\alpha \right)\right),\mathrm{f}\left(\alpha \right)=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\epsilon }}{2}}\right)\alpha +{\displaystyle \frac{\epsilon }{2}}$, Tau ($\mathrm{T}$) is for the t-student distribution’s cumulative distribution function with $\mathrm{v}-1$ freedom degrees and $s$ is the standard deviation of the sample.

3.1.3. F-DeNETS Process

F-DEMATEL Steps [77]

Step 1. Investigate and identify the set of criteria for evaluating alternative options.

Step 2. Set the decision goal and form a committee. To reach a decision, we need to define the goals, collect the necessary information, identify an appropriate number of alternatives, evaluate the alternatives, select one of them and finally check whether we have achieved the goal.

Step 3. List the evaluation criteria and the fuzzy linguistic scale. The evaluation criteria are causal relationships. The evaluation scale used is the fuzzy linguistic scale as presented by Li [78], as shown in

Table 1. The linguistic terms and linguistic values.

Linguistic Terms	Linguistic Values
Very-high causality (VH)	(0.75, 1.0, 1.0)
High causality (H)	(0.5, 0.75, 1.0)
Low causality (L)	(0.25, 0.5, 0.75)
Very-low causality (VL)	(0, 0.25, 0.5)
No causality (No)	(0, 0, 0.25)

.

Step 4. Obtain the ratings from the experts. To determine the causal relationship between two criteria $\mathrm{C}=\left\{{\mathrm{C}}_{\mathrm{i}}|\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}\right\},$ the committee was asked to make pair-wise comparisons using the linguistic variables. Each expert corresponds to a table. The elements of the table are triangular fuzzy numbers. Therefore, for each of the x experts who make up the committee, we have the fuzzy table (initial direct relationship table) ${\tilde{\mathrm{Z}}}^{\mathrm{k}}(\mathrm{n}\times \mathrm{n})$:

$${\tilde{\mathrm{Z}}}^{\mathrm{k}}=\left[\begin{array}{cccc}0& {\tilde{\mathrm{z}}}_{12}^{〈\mathrm{k}〉}& \dots & {\tilde{\mathrm{z}}}_{1\mathrm{n}}^{〈\mathrm{k}〉}⋮\\ {\tilde{\mathrm{z}}}_{21}^{〈\mathrm{k}〉}& 0& \dots & {\tilde{\mathrm{z}}}_{2\mathrm{n}}^{〈\mathrm{k}〉}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\mathrm{z}}}_{n1}^{〈\mathrm{k}〉}& {\tilde{\mathrm{z}}}_{\mathrm{n}2}^{〈\mathrm{k}〉}& \dots & 0\end{array}\right]$$

where ${\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}=\left({\mathrm{l}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉},{\mathrm{m}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉},{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}\right)$ and $\mathrm{k}=\mathrm{1,2},\dots ,\mathrm{p}$.

Step 5. Next, we find the normalized direct-relation fuzzy matrix ${\tilde{\mathrm{X}}}^{\mathrm{k}}$

$${\tilde{\mathrm{X}}}^{\mathrm{k}}=\left[\begin{array}{cccc}{\tilde{\mathrm{X}}}_{11}^{〈\mathrm{k}〉}& {\tilde{\mathrm{X}}}_{12}^{〈\mathrm{k}〉}& \dots & {\tilde{\mathrm{X}}}_{1\mathrm{n}}^{〈\mathrm{k}〉}⋮\\ {\tilde{\mathrm{X}}}_{21}^{〈\mathrm{k}〉}& {\tilde{\mathrm{X}}}_{22}^{〈\mathrm{k}〉}& \dots & {\tilde{\mathrm{X}}}_{2\mathrm{n}}^{〈\mathrm{k}〉}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\mathrm{X}}}_{n1}^{〈\mathrm{k}〉}& {\tilde{\mathrm{X}}}_{\mathrm{n}2}^{〈\mathrm{k}〉}& \dots & {\tilde{\mathrm{X}}}_{\mathrm{n}\mathrm{m}}^{〈\mathrm{k}〉}\end{array}\right]$$

where ${\tilde{\mathrm{x}}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}={\displaystyle \frac{{\tilde{\mathrm{z}}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}}{{\mathrm{r}}^{〈\mathrm{k}〉}}}=\left({\displaystyle \frac{{\mathrm{l}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}}{{\mathrm{r}}^{〈\mathrm{k}〉}}},{\displaystyle \frac{{\mathrm{m}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}}{{\mathrm{r}}^{〈\mathrm{k}〉}}},{\displaystyle \frac{{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}}{{\mathrm{r}}^{〈\mathrm{k}〉}}}\right)$ with $\mathrm{k}=1,2,\dots ,\mathrm{p}$ and ${\mathrm{r}}^{〈\mathrm{k}〉}=\underset{1\le \mathrm{i}\le \mathrm{n}}{\mathit{max}\{\mathit{max}}\sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}\underset{1\le \mathrm{j}\le \mathrm{n}}{\max }\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}\}$. Then, $\tilde{\mathrm{X}}={\displaystyle \frac{\left({\tilde{\mathrm{X}}}^{〈1〉}+{\tilde{\mathrm{X}}}^{〈2〉}+\dots +{\tilde{\mathrm{X}}}^{〈\mathrm{p}〉}\right)}{\mathrm{p}}}$ leads to the normalized direct-relation fuzzy matrix $\tilde{\mathrm{X}}(\mathrm{n}\times \mathrm{n})$:

$$\tilde{\mathrm{X}}=\left[\begin{array}{cccc}{\tilde{\mathrm{x}}}_{11}& \tilde{\mathrm{x}}& \dots & {\tilde{\mathrm{x}}}_{1\mathrm{n}}⋮\\ {\tilde{\mathrm{x}}}_{21}& {\tilde{\mathrm{x}}}_{22}& \dots & {\tilde{\mathrm{x}}}_{2\mathrm{n}}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\mathrm{x}}}_{n1}& {\tilde{\mathrm{x}}}_{\mathrm{n}2}& \dots & {\tilde{\mathrm{x}}}_{\mathrm{n}\mathrm{m}}\end{array}\right]$$

where ${\tilde{\mathrm{x}}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}={\displaystyle \frac{\sum _{\mathrm{k}=1}^{\mathrm{p}}{\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{〈\mathrm{k}〉}}{\mathrm{p}}}$.

Step 6. Develop and evaluate the structural model. The total relation matrix $\tilde{\mathrm{T}}(\mathrm{n}\times \mathrm{n})$

$$\tilde{\mathrm{T}}=\left[\begin{array}{cccc}{\tilde{\mathrm{t}}}_{11}& \tilde{\mathrm{t}}& \dots & {\tilde{\mathrm{t}}}_{1\mathrm{n}}⋮\\ {\tilde{\mathrm{t}}}_{21}& {\tilde{\mathrm{t}}}_{22}& \dots & {\tilde{\mathrm{t}}}_{2\mathrm{n}}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{\mathrm{t}}}_{n1}& {\tilde{\mathrm{t}}}_{\mathrm{n}2}& \dots & {\tilde{\mathrm{t}}}_{\mathrm{n}\mathrm{m}}\end{array}\right]$$

with ${\tilde{\mathrm{t}}}_{\mathrm{i}\mathrm{j}}=\left({\mathrm{l}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }},{\mathrm{m}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }},{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }}\right)$ is found by calculating the following:

$$\begin{gathered} \mathrm{M}\mathrm{A}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{X}\left[{\mathrm{l}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }}\right]={\mathrm{X}}_{\mathrm{l}}\times {\left(\mathrm{I}-{\mathrm{X}}_{\mathrm{l}}\right)}^{-1}, \\ \mathrm{M}\mathrm{A}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{X}\left[{\mathrm{m}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }}\right]={\mathrm{X}}_{\mathrm{m}}\times {\left(\mathrm{I}-{\mathrm{X}}_{\mathrm{m}}\right)}^{-1}, \\ \mathrm{M}\mathrm{A}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{X}\left[{\mathrm{u}}_{\mathrm{i}\mathrm{j}}^{\mathrm{\prime }\mathrm{\prime }}\right]={\mathrm{X}}_{\mathrm{u}}\times {\left(\mathrm{I}-{\mathrm{X}}_{\mathrm{u}}\right)}^{-1} \end{gathered}$$

Now, the values ${\tilde{\mathrm{D}}}_{\mathrm{i}}$, ${\tilde{\mathrm{R}}}_{\mathrm{i}}$, ${\tilde{\mathrm{D}}}_{\mathrm{i}}+{\tilde{\mathrm{R}}}_{\mathrm{i}}$, ${\tilde{\mathrm{D}}}_{\mathrm{i}}-{\tilde{\mathrm{R}}}_{\mathrm{i}}$ can be calculated. To obtain the causal relationships for the defuzzification process, the Converting the Fuzzy data into Crisp Scores (CFCS) from Opricovic and Tzeng [79] was employed:

For a triangular positive fuzzy number ${\tilde{\mathrm{Z}}}_{\mathrm{i}}=\left({\mathrm{l}}_{\mathrm{i}},{\mathrm{m}}_{\mathrm{i}},{\mathrm{u}}_{\mathrm{i}}\right)$ if $\mathrm{L}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\mathrm{L}}_{\mathrm{i}}\right)$ and $\mathrm{U}=\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathrm{U}}_{\mathrm{i}}\right)$ and $\Delta =U-\mathrm{L},$ then the crisp value of ${\tilde{\mathrm{Z}}}_{\mathrm{i}}$ is as follows:

$${\tilde{\mathrm{Z}}}_{\mathrm{i}}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{p}}=\mathrm{L}+\Delta ⨯{\displaystyle \frac{\left(m-L\right){\left(\Delta +u-m\right)}^2\left(U-L\right)+{(\Delta -l+m)}^2{(u-L)}^2}{\left(\Delta -m+u\right)\left(u-L\right){(\Delta -l+m)}^2+{\left(\Delta +u-m\right)}^2\left(\Delta -l+m\right)(U-l)}}$$

Since the more critical criteria were isolated, the criteria weights can be derived. Based on ${\tilde{\mathrm{D}}}_{\mathrm{i}}+{\tilde{\mathrm{R}}}_{\mathrm{i}}$ values, criteria with highest prominence are retained for weight calculation in Step 7.

Remark 1:

At this stage, the results of fuzzy DEMATEL led to the distinction of the criteria into two groups: the causal criteria, which exert a strong influence on the system, and the effect criteria, which are influenced to a greater extent by the rest. In order to facilitate further multi-criteria analysis and in order to limit the dimension of the problem, a merger of selected effect criteria with the corresponding causal criteria that presented a high degree of interdependence was carried out. In this way, certain synthetic criteria emerged, which more fully capture the dynamics of the relationships, while some other criteria remained unchanged. This choice is justified by the fact that maintaining a large number of interrelated criteria can lead to redundancy phenomena and reduce the discriminatory power of the multi-criteria methodology. On the contrary, merging into synthetic criteria ensures on the one hand a more realistic representation of the interactions between the factors; on the other hand, it improves the interpretability of the results and reduces the complexity of the next phase of analysis (TOPSIS). In this way, the essence of the distinction between causal and effect criteria is preserved, while, at the same time, the reliability and practical utility of decision-making are enhanced. The merging process to create the final list of criteria was based on a quantitative and algorithmic approach, which was applied to the Fuzzy DEMATEL results. More specifically, all criteria that had a negative value in the index (D-R) were considered “effect” criteria as this indicates that they receive more influence than they exert in the criteria system. Then, each of these “effect” criteria was merged with the “cause” criterion that exerted the greatest direct influence on it as quantified by the highest value in the corresponding table of total fuzzy relations T̃. For example (see Section 4), C4 (Communication & Team Spirit, with D-R = −0.2679) was merged with C3 (Leadership & Team Management), which was the cause that influenced it the most. This algorithm ensured that problem dimension reduction was a structured, data-driven process, preserving the most critical causal relationships identified by DEMATEL.

NAFEs Step

Step 7. In this step, the consolidated criteria from DEMATEL analysis are subjected to expert evaluation for weight determination using NAFEs. We consider a sample of experts (academics and/or professionals) who assign values to the weight of each criterion. Using NAFEs and taking into account the different weights of the experts’ opinions, we calculate the weights of the criteria. Suppose that a decision-making committee consists of Z individuals, then the weight of the criteria and the evaluation of the alternatives with respect to each criterion are calculated by the weighted Non-Asymptotic Fuzzy Estimators. To take into account the difference in the weights of expert opinions, we reduce the significance level by expanding the range of the fuzzy estimator at the extremes.

Using proposition 1, let ${\mathrm{W}}_1,\mathrm{W},\dots {\mathrm{W}}_{\mathrm{n}}$ be a data sample representing the expert opinions for the criteria weights, ${\mathrm{w}}_1,{\mathrm{w}}_2,\dots {\mathrm{w}}_{\mathrm{n}}$ be the criteria weights values, and $\mathsf{\epsilon }\in \left(\mathrm{0,1}\right)$. When the data sample $\mathrm{v}$ is small and normally distributed with the population standard deviation $\left(\mathsf{\sigma }\right)$ to be unknown, then

$${\check{\mu }}_{\epsilon }\left(w\right)=\left\{\begin{array}{c}{\displaystyle \frac{2}{1-\mathsf{\epsilon }}}{T}_{\mathrm{v}-1}\left({\displaystyle \frac{\overline{\mathrm{w}}-\mathrm{w}}{\frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}}\right)-{\displaystyle \frac{\epsilon }{1-\mathsf{\epsilon }}}, if w\in \left[\overline{\mathrm{w}}-{\mathrm{t}}_{1-{\displaystyle \frac{\epsilon }{2}}}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathrm{v}}}},\overline{\mathrm{w}}\right]\\ \\ {\displaystyle \frac{2}{1-\mathsf{\epsilon }}}{T}_{\mathrm{v}-1}\left({\displaystyle \frac{\mathrm{w}-\overline{\mathrm{w}}}{\frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}}\right)-{\displaystyle \frac{\mathsf{\epsilon }}{1-\mathsf{\epsilon }}}, if w\in \left[\overline{\mathrm{w}},\overline{\mathrm{w}}+{\mathrm{t}}_{1-{\displaystyle \frac{\epsilon }{2}}}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathrm{v}}}}\right]\\ \\ 0 else \end{array}\right.$$

is the membership function of $w$ in $\mathsf{\epsilon }$.Τhe support of the fuzzy number $\mathsf{\epsilon }$ is the $\left(1-\epsilon \right)100\%$ confidence interval for $\mu$. The alpha-cuts of this fuzzy number are the following intervals:

$${\alpha }_{{\ddot{\mu }}_{\epsilon }=}=\left[\overline{\mathrm{w}}-{\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathsf{\nu }}}},\overline{\mathrm{w}}+{\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}{\displaystyle \frac{\mathrm{s}}{\sqrt{\mathsf{\nu }}}}\right],\alpha \in \left(\mathrm{0,1}\right]$$

where ${\mathrm{t}}_{\mathrm{f}\left(\alpha \right)}={\mathrm{T}}^{-1}\left(1-\mathrm{f}\left(\alpha \right)\right),\mathrm{f}\left(\alpha \right)=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\epsilon }}{2}}\right)\alpha +{\displaystyle \frac{\epsilon }{2}}$, Tau (Τ) is for the t-student distribution’s cumulative distribution function with $\mathrm{v}-1$ degrees of freedom and $s$ is the sample’s standard deviation.

Fuzzy TOPSIS Steps [80]

Step 8. Decision matrix.

Let us consider the set $A_i=\left(A_1,A_2,\dots ,A_m\right)$ for alternatives. Each alternative $A_i$ is scored for several criteria $C_i,i=\mathrm{1,2},\dots ,n$. This process leads to the formulation of the decision matrix $X=\left[x_{ij}\right]$ where the rows are for alternatives and the columns for the criteria is constructed.

$$D=\begin{array}{c}{A}_1\\ A_2\\ A_3\\ .\\ .\\ .\\ A_m\\ \end{array}\left[\begin{array}{c}{\tilde{x}}_{11}{\tilde{x}}_{12}{\tilde{x}}_{13}\dots {\tilde{x}}_{1n}\\ {\tilde{x}}_{21 }{\tilde{x}}_{22}{\tilde{x}}_{23}\dots {\tilde{x}}_{2n}\\ {\tilde{x}}_{31}{\tilde{x}}_{32}{\tilde{x}}_{33}\dots {\tilde{x}}_{3n}\\ .\\ .\\ .\\ {\tilde{x}}_{m1}{\tilde{x}}_{m2}{\tilde{x}}_{m3}\dots {\tilde{x}}_{mn}\\ \end{array}\right]$$

$${\overline{x}}_{ij},\mathrm{ }i=1,\mathrm{ }2,\dots ,m\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }j=1,\mathrm{ }2,\dots ,n$$

and for the weights, the matrix containing the NAFEs for the weights of each criterion is as follows:

$$\tilde{W}=\left[{\check{\mu }}_{\epsilon }\left(w_1\right),{\check{\mu }}_{\epsilon }\left(w_2\right),\dots ,{\check{\mu }}_{\epsilon }\left(w_n\right)\right]$$

The linguistic variables employed to evaluate the alternatives are represented by Tri-angular Fuzzy Numbers, as shown in

Table 2. The linguistic terms for rating alternatives.

Linguistic Terms	Linguistic Values
Very-High (VH)	(9, 10, 10)
High (H)	(7, 9 10)
Medium High (MH)	(5, 7, 9)
Fair (F)	(3, 5, 7)
Medium Low (ML)	(1, 3, 6)
Low (L)	(0, 1, 3)
Very Low (VL)	(0, 0, 1)

.

Step 9. Calculation of normalized scores.

The normalization of triangular fuzzy numbers is performed as follows:

$${\overline{r}}_{ij}=\left({\displaystyle \frac{l_{ij}}{m_j^{\mathrm{*}}}},{\displaystyle \frac{m_{ij}}{m_j^{\mathrm{*}}}},{\displaystyle \frac{u_{ij}}{m_j^{\mathrm{*}}}}\right)$$

where $j\in BenefitCriteria$ and $m_j^{\mathrm{*}}=\underset{\mathrm{i}}{\max } u_{ij}$,

$${\overline{r}}_{ij}=\left({\displaystyle \frac{{l_j}^{-}}{u_{ij}}},{\displaystyle \frac{{m_j}^{-}}{m_{ij}}},{\displaystyle \frac{{u_j}^{-}}{l_{ij}}}\right)$$

where $j\in CostCriteria$ and ${l_j}^{-}=\underset{\mathrm{i}}{\min } l_{ij}$, ${m_j}^{-}=\underset{\mathrm{i}}{\min } m_{ij}$,

$${u_j}^{-}=\underset{\mathrm{i}}{\min } u_{ij}$$

The normalization process aims to ensure that the ranges of the normalized triangular fuzzy numbers remain within the interval [0, 1].

Step 10. Calculation of the weighted normalized scores.

Taking into account the different importance of each criterion, the weighted normalized fuzzy decision matrix can be constructed as follows:

$$\tilde{V}={\left[{\tilde{v}}_{ij}\right]}_{m\times n}$$

for each element $x_{ij}$, the weighted normalized value is computed as ${\tilde{v}}_{ij}={\tilde{w}}_j{\tilde{r}}_{ij},$ where $w_j{\overline{w}}_j$ is the weight of criterion $C_j$.

Step 11. Calculation of FPIS and FNIS.

The weighted normalized fuzzy decision matrix indicates that the elements ${\overline{v}}_j^i$ are normalized positive Triangular Fuzzy Numbers and its ranges belong to the closed interval $\left[\mathrm{0,1}\right]$. Then, the Fuzzy Positive Ideal Solution (FPIS) and the Fuzzy Negative Ideal Solution (FNIS) can be calculated in terms of weighted normalized scores.

$${\overline{A}}^{+}=\left({{\overline{v}}_1}^{+},{{\overline{v}}_2}^{+},{{\overline{v}}_3}^{+}\dots ,{{\overline{v}}_n}^{+}\right)$$

$${\overline{A}}^{-}=\left({{\overline{v}}_1}^{-},{{\overline{v}}_2}^{-},{{\overline{v}}_3}^{-}\dots ,{{\overline{v}}_n}^{-}\right)$$

where

$$\begin{gathered} {{\overline{v}}_{\mathrm{j}}}^{+}=\mathrm{ }\left(\underset{\mathrm{i}}{\max }\left\{v_{ij1}\right\},\underset{\mathrm{i}}{\max }\left\{v_{ij2}\right\},\underset{\mathrm{i}}{\max }\left\{v_{ij3}\right\},\underset{\mathrm{i}}{\max }\left\{v_{ij4}\right\}\right), \\ {{\overline{v}}_j}^{-}=\mathrm{ }\left(\underset{\mathrm{i}}{\min }\left\{v_{ij1}\right\},\underset{\mathrm{i}}{\min }\left\{v_{ij2}\right\},\underset{\mathrm{i}}{\min }\left\{v_{ij3}\right\},\underset{\mathrm{i}}{\min }\left\{v_{ij4}\right\}\right) \end{gathered}$$

Step 12. Calculation of the distances ${{\overline{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}}}_{\mathrm{i}}}^{+}$ and ${{\overline{\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}}}_{\mathrm{i}}}^{-}$ of the Alternatives ${\mathrm{A}}_{\mathrm{i}}$ from Fuzzy PIS and Fuzzy NIS.

The Euclidean distance of each alternative $A_i$ from FPIS and FNIS is computed as

$$\begin{gathered} {{\overline{Dist}}_i}^{+}=\sum _{j=1}^{n}dist\left({\overline{v}}_{ij},{{\overline{v}}_j}^{+}\right),\mathrm{ }i=\mathrm{1,2},\dots ,mandj=\mathrm{1,2},\dots ,n \\ {{\overline{Dist}}_i}^{-}=\sum _{j=1}^{n}dist\left({\overline{v}}_{ij},{{\overline{v}}_j}^{-}\right),\mathrm{ }i=\mathrm{1,2},\dots ,mandj=\mathrm{1,2},\dots ,n \end{gathered}$$

where ${\overline{Dist}}_i$ represents the Euclidean distance between fuzzy numbers.

Step 13. Calculation of the similarity (degree of closeness) to the ideal solution

A closeness coefficient is defined to establish the ranking of all alternatives after the values of ${{\overline{Dist}}_i}^{+}$ and ${{\overline{Dist}}_i}^{-}$ have been determined for each alternative. The closeness coefficient for each alternative $A_i$ is then computed using these values.

$${\overline{CC}}_i={\displaystyle \frac{{{\overline{Dist}}_i}^{-}}{{{\overline{Dist}}_i}^{+}+{{\overline{Dist}}_i}^{-}}},\mathrm{ }i=1,\mathrm{ }2,\dots ,m$$

The proposed F-DeNETS framework is an innovative fuzzy MCDM methodology that integrates F-DEMATEL, NAFEs and F-TOPSIS into an integrated analytical approach. By combining causal criteria analysis, non-asymptotic fuzzy weight estimation and multi-criteria alternative ranking, the framework provides a comprehensive and systematic approach to decision-making under uncertainty and incomplete information. The framework architecture ensures the transfer of causal information from the DEMATEL phase to weight determination via NAFEs, while the final TOPSIS phase utilizes the estimated fuzzy weights to evaluate the alternatives. Furthermore, the built-in sensitivity analysis enhances the reliability and robustness of the model, allowing the stability of the results to be checked at different significance levels and providing decision-makers with a global picture of the uncertainty characterizing the decision problem.

3.2. Sensitivity Analysis Approaches

The application of NAFEs provides a powerful tool for sensitivity analysis in statistical and decision-making problems. Since NAFEs are constructed through families of triangular fuzzy numbers based on confidence intervals, the form of the membership function and the range of support can be controlled by the choice of the function $f\left(a\right)$ and the parameter $\epsilon$. This allows us to study how small changes in critical parameters affect the final conclusions. The proposed processes for sensitivity analysis are the following:

• Construction of multiple NAFE families. Different forms of $f\left(a\right)$ are chosen (e.g., linear, quadratic, cubic, root and Buckley type). Each choice produces an alternative version of the fuzzy estimator for the same statistical parameter. This allows the sensitivity of the system to the form of the estimator to be investigated. We then estimate an unknown parameter $\theta$ from a sample $K_1,{ K}_2,\dots ,K_n$. The family of NAFEs is defined by the α-cuts as follows:

  $$a\widetilde{{\theta }_{\epsilon }}=\left[{\theta }_1\left(2f\left(a\right)\right),{\theta }_2\left(2f\left(a\right)\right)\right], a\in (\mathrm{0,1})$$

  where $f\left(a\right)$ is a monotonic increasing and continuous function with $f:\left(\mathrm{0,1}\right)\to \left({\displaystyle \frac{\epsilon .}{2}},{\displaystyle \frac{1}{2}}\right),$ and $c$ is the confidence interval. The sensitivity analysis is based on the selection of the form of $f\left(a\right)$ for different shapes of fuzzy estimators (linear, cubic, square root, etc.).

2.

Change in the confidence level ε. By changing the value of ε. (e.g., from 90% to 95%), the support of the fuzzy number changes. This way it can be assessed how stable the results remain when more “tight” or more “loose” estimates are made. There is a change in the parameter ε so as to change the support of the fuzzy number.

3.

False Alarm Rate Index. The False Alarm Rate Index [76] can be used as a metric to compare which NAFE format gives more consistent results. In practice, the format with the lowest false alarm rate is also the most robust to uncertainties and is, therefore, appropriate for high-sensitivity applications. For two parameters ${\theta }_1,{\theta }_2$ with fuzzy estimators ${\tilde{\theta }}_{1,\epsilon },{\tilde{\theta }}_{2,\epsilon }$ the fuzzy comparison is ${\displaystyle \frac{AR}{AT}}$ where $AT$ is the total area under ${\tilde{\theta }}_{2,\epsilon }$ and $AR$ is the part of it that exceeds ${\tilde{\theta }}_{1,\epsilon }$. The sensitivity is checked via the False Alarm Index, $FA= {\displaystyle \frac{N_r}{N_p}},$ where $N_r$ is the number of cases that the fuzzy comparison rejects the hypothesis $H_0$ (${\theta }_1<{\theta }_2),$ while real ordering relation is ${\theta }_1<{\theta }_2$, and $N_p$ the number of cases that indeed ${\theta }_1<{\theta }_2$

3.3. Dynamic Integration of New Information

To make the proposed framework more dynamic and adaptive, the criteria derived through DEMATEL do not remain static. On the contrary, at regular intervals, the set of criteria $\mathrm{C}=\left\{{\mathrm{C}}_{\mathrm{i}}|\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}\right\}$ can be revised in order to add new $\mathrm{C}=\left\{{\mathrm{C}}_{\mathrm{i}}|\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}\right\}$ or to remove existing criteria, to re-scale causal relationships, weights and alternatives and then to perform sensitivity analysis. In this way, the results are always based on the most recent knowledge while remaining transparent and user-friendly. Furthermore, it is proposed to use an event-driven criteria set, where a “core” of basic criteria $C_{core}\subset C$ (always active) is maintained, while a “peripheral” of criteria $C_{peripheral}\subset C$ can be activated $C_{active}=C_{core}\cup C_{peripheral}$ or deactivated $C_{active}=C_{core}$ depending on the appearance of new conditions or risks. In the TOPSIS stage, the incorporation of these changes is simple. The decision table is refreshed, the weights are updated and the closeness of the alternatives is recalculated.

3.4. F-DeNETS Contribution to Deficiency Mitigation and Methodology Improvement

Problem 1—Sensitivity to Fuzzy Data: There is a difficulty in accurately modeling complex cause–effect relationships due to subjectivity resulting from the dependence on expert opinions. Consequently, there is a sensitivity of the result to fuzzy logic parameter choices.

In the proposed model, fuzzy DEMATEL is used to find the “cause” criteria, and then the weights are calculated using the NAFEs method that takes into account all statistical information. The use of Non-Asymptotic Fuzzy Estimators (NAFEs) to calculate weights in Fuzzy TOPSIS (when expert opinions are expressed with linguistic variables or scales) offers significant advantages over classic fuzzy means or crisp approaches:

• Better Handling of Uncertainty and Ambiguity: The 1–5 rating scales are converted into fuzzy numbers (e.g., triangular), but NAFEs incorporate statistical reliability through the chosen ε, avoiding arbitrariness.
• Flexibility in Defining the Shape: The optimal shape of the membership function (linear, square root, etc.) for the weights can be chosen based on the False Alarm Rate. This minimizes incorrect decisions when the weights are similar.
• Improved Comparison of Similar Weights: In Fuzzy TOPSIS, a critical issue is the discrimination of alternatives with similar weights. NAFEs allow for more accurate comparison through the False Alarm Rate, which selects the scheme that minimizes errors at nearly equal weights.
• Stringency Control via the ε Parameter: The parameter ε ∈ (0,1) controls its range. The “stringency” of the estimator can be adjusted. A small ε (e.g., 0.05) leads to a wider estimator and, therefore, a more conservative weight estimate. A larger ε (e.g., 0.2) leads to a narrower estimator and, therefore, a riskier estimate.
• Suitable Integration in TOPSIS: NAFEs are proven fuzzy numbers (convex, normal and compact support), so they are acceptable inputs for fuzzy number operations in Fuzzy TOPSIS (fuzzy sums, multiplications and distances). The membership function is calculated analytically (Proposition 1), without numerical errors.

In summary, NAFEs transform expert opinions into statistically sound fuzzy estimators with controlled uncertainty, optimized scheme for comparison and explicit interpretation through confidence intervals. This makes Fuzzy TOPSIS less sensitive to data noise and more reliable in critical decisions where criteria have similar weights.

Problem 2—Sample Suitability: There is a lack of group decisions, the expert sample is usually small or the data may come from a single organization. The two aforementioned facts limit the generalizability of the model since it is based on estimates of a number of experts and sometimes from a specific pool.

The proposed model is structured on a group of decision-makers. By employing NAFEs, experts provide crisp opinions as they are typically more comfortable with crisp values than with fuzzy scores. Nevertheless, the final output remains a fuzzy number. As a result, both the weights and the alternative ratings in TOPSIS are fuzzy, preserving the fuzzy nature of the methodology.

Additionally, the Non-Asymptotic Fuzzy Estimators used to derive the weights can accommodate various types of fuzzy numbers. This is particularly advantageous when the sample size of experts is small as the NAFE method remains capable of producing reliable estimates even under such constraints.

Problem 3—Extension of methods to improve accuracy: The vast majority of previous papers clearly give the direction for new combinations of MCDM methods and fuzzy information representation frameworks. The goal is to increase the accuracy of the results by properly incorporating fuzzy information, cross-validating results and reducing the static nature of existing models since they do not take into account dynamic changes over time.

To enhance the accuracy of results and effectively incorporate ambiguity, the proposed model employs NAFEs as a robust framework for representing fuzzy information. Unlike approaches that rely on predefined fuzzy values, the NAFE method accounts for variation and dispersion in the crisp evaluations provided by experts, generating fuzzy estimates with statistical validity. By utilizing confidence intervals, the method also quantifies the degree of confidence associated with each criterion, thereby reducing the reliance on unsupported estimates.

A key advantage of NAFEs is their ability to preserve ambiguity throughout the analysis, deferring defuzzification until the final stages. This avoids premature simplification and maintains the integrity of the estimates. The framework naturally accommodates linguistic judgments, often preferred by human evaluators, without requiring arbitrary numerical conversions.

Furthermore, NAFEs address the static limitations of conventional models by enabling dynamic adaptation. The model can adjust its “shape” and “degree of rigor” in response to the most recent data, rather than remaining fixed on historical averages. This flexibility is achieved through the use of real confidence intervals derived from current observations, ensuring estimates reflect contemporary trends.

The method also supports the selection of an optimal membership function shape, such as linear or square root, based on the False Alarm Rate. An adjustable parameter, $\epsilon$∈ (0, 1), controls the range and stringency of the estimator: a smaller $\epsilon$ (e.g., 0.05) results in a wider, more conservative confidence interval, while a larger $\epsilon$ (e.g., 0.2) yields a narrower, riskier estimate.

Finally, to ensure validity and robustness, the model undergoes sensitivity analysis using different types of fuzzy numbers, confirming its reliability under varying assumptions.

Problem 4—Calculation of the weight according to the dispersion of its values (entropy method): The entropy method calculates the weight of each criterion according to the dispersion of its values among the alternatives [74]. In simple words, if a criterion has large differences in its values from alternative to alternative, it receives a very high weight. On the contrary, if a criterion has close values for all alternatives, the method gives it a very low weight, so its effect on the final result is almost zero.

In the current paper, the entropy method is not used to calculate the weights of the criteria. The experts’ evaluations are collected, and these are converted into NAFEs in order to capture the uncertainty and variation in their judgments. In this way, the weights of the criteria are derived not only from the dispersion of the data but from a flexible fuzzy representation of the subjective information that takes into account the dispersion of the crisp evaluations, makes use of confidence intervals and can be dynamically adjusted through the choice of the membership function shape and the parameter $\epsilon$.

Problem 5—A common limitation in current decision-making methods is their reliance on prior knowledge of weights. In most existing approaches, solutions are feasible only when the weights of the DMs, the criteria or both are known in advance. This dependence on predefined weights restricts the applicability of these methods in real-world scenarios where such information may be incomplete or unavailable.

The application of NAFEs to expert opinions is performed in a single integrated step, eliminating the need for a separate calculation of expert weights. The resulting fuzzy estimate captures both the importance of each criterion and the relative influence of each expert as their collective judgments are synthesized directly into the fuzzy structure. This synthesis is facilitated by the flexibility in the estimator’s shape and the adjustable strictness of the parameter $\epsilon$.

By addressing these aspects concurrently, the method avoids additional weighting stages, streamlines the process and ensures a consistent and robust representation of decision-making information.

4. A Numerical Application

A numerical application of the proposed method is presented to clearly illustrate its procedural steps. In addition to implementing the method, a sensitivity analysis is conducted. This analysis serves to evaluate the stability and reliability of the results, identify which criteria or parameters exert the most influence on the decisions and enhance user confidence in the methodology by demonstrating that the conclusions remain valid and transparent across different scenarios. The application addresses a critical issue in international shipping. This concerns the selection of suitable seafarers. It should be noted that this paper does not provide an in-depth analysis of the problem itself as the primary aim at this stage is to demonstrate the functionality and applicability of the proposed method. For better readability and replicability of the proposed method, the mathematical notation of F-DeNETS is included in the Appendix A.

4.1. Application Description

The application of F-DeNETS concerns the selection of the appropriate seafarer. For this reason, 12 basic seafarer selection criteria were collected from the latest literature. More specifically, the studies of Koutra et al. [81] and Arslan et al. [82] were used to extract the criteria. After a merging process, 12 criteria were extracted. The criteria are the following:

C1.

Technical Expertise & Knowledge;

C2.

Education & Certifications;

C3.

Leadership & Team Management;

C4.

Communication & Team Spirit;

C5.

Problem-Solving & Decision-Making;

C6.

Adaptability & Flexibility;

C7.

Professional Conduct & Ethics;

C8.

Emotional Resilience & Motivation;

C9.

Safety & Environmental Awareness;

C10.

Organizational & Cognitive Abilities;

C11.

Cultural Fluency;

C12.

Crisis Management & Self-Management.

Based on these criteria and applying the F-DeNETS steps, five candidate seafarers are evaluated and ranked for in-ship recruitment. Candidate seafarers (SFs) are denoted as ${SF}_1, { SF}_2,\dots ,{SF}_5$.

4.2. Formation of a Final List of Criteria

In this stage, we use Fuzzy DEMATEL to find the causal relationships between the criteria.

Step 1. Τhe list of criteria extracted from the literature is used.

Step 2. Goal setting and committee formation.

The decision-making process begins with the clear definition of the goals and the formation of the appropriate committee. The goal is to find the list of the most appropriate criteria for the selection of a sailor. The committee consists of both academics and professionals in order to ensure reliability.

Step 3. At this stage, the evaluation criteria (causal relationships) are defined, along with the fuzzy linguistic scale to be used. This scale is adopted from the proposal of Li [78] and is summarized in Table 1.

Step 4. In the next stage, the experts’ estimates are collected. In order to capture the causal relationship between two criteria $\mathrm{C}=\left\{{\mathrm{C}}_{\mathrm{i}}|\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{n}\right\}$, the committee is asked to perform pairwise comparisons using linguistic variables. Each expert produces a table (based on Table 1), whose elements are represented by triangular fuzzy numbers. Therefore, for each of the x experts in the committee, the fuzzy table, “initial direct relationship table” is constructed. To avoid unnecessary space consumption, the initial direct relationship matrix and its normalization are presented as an example (

Table 3. (a) Initial direct-relation matrix for expert 1. (b) Normalized direct-relation matrix for expert 1.

(a)
	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
C1	(0, 0, 0)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)
C2	(0.5, 0.75, 1)	(0, 0, 0)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)
C3	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0, 0, 0)	(0.75, 1, 1)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.75, 1, 1)
C4	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0.25, 0.5, 0.75)
C5	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.75, 1, 1)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)
C6	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0, 0.25, 0.5)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.5, 0.75, 1)
C7	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.5, 0.75, 1)	(0.75, 1, 1)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)
C8	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.75, 1, 1)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.75, 1, 1)
C9	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)
C10	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0, 0, 0)	(0.25, 0.5, 0.75)	(0.5, 0.75, 1)
C11	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0, 0.25, 0.5)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0, 0.25, 0.5)	(0, 0, 0)	(0.25, 0.5, 0.75)
C12	(0.25, 0.5, 0.75)	(0, 0.25, 0.5)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0.75, 1, 1)	(0.75, 1, 1)	(0.5, 0.75, 1)	(0.75, 1, 1)	(0.5, 0.75, 1)	(0.5, 0.75, 1)	(0.25, 0.5, 0.75)	(0, 0, 0)
(b)
	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
C1	(0, 0, 0)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)
C2	(0.0513, 0.0769, 0.1026)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)
C3	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0, 0, 0)	(0.0769, 0.1026, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0769, 0.1026, 0.1026)
C4	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)
C5	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)
C6	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0, 0.0256, 0.0513)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)
C7	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0513, 0.0769, 0.1026)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)
C8	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)
C9	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)
C10	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)	(0.0513, 0.0769, 0.1026)
C11	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0, 0.0256, 0.0513)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0, 0.0256, 0.0513)	(0, 0, 0)	(0.0256, 0.0513, 0.0769)
C12	(0.0256, 0.0513, 0.0769)	(0, 0.0256, 0.0513)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0.0769, 0.1026, 0.1026)	(0.0769, 0.1026, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0769, 0.1026, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0513, 0.0769, 0.1026)	(0.0256, 0.0513, 0.0769)	(0, 0, 0)

a,b). The same procedure occurs for the remaining experts.

Step 5. Then, the aggregated normalized direct-relation fuzzy matrix is calculated. Normalization is necessary to ensure that all values are on the same scale and that the experts’ fuzzy estimates are comparable to each other (

Table 4. Aggregated normalized direct-relation matrix.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
C1	(0, 0, 0)	(0.0559, 0.0807, 0.0994)	(0.0187, 0.0435, 0.0684)	(0.0187, 0.0435, 0.0684)	(0.0746, 0.0994, 0.0994)	(0.0217, 0.0466, 0.0714)	(0.0124, 0.0373, 0.0621)	(0.0156, 0.0404, 0.0653)	(0.0559, 0.0807, 0.0994)	(0.0590, 0.0839, 0.0994)	(0.0092, 0.0341, 0.0589)	(0.0249, 0.0497, 0.0746)
C2	(0.0559, 0.0807, 0.0994)	(0, 0, 0)	(0.0310, 0.0559, 0.0807)	(0.0093, 0.0342, 0.0590)	(0.0249, 0.0497, 0.0746)	(0.0062, 0.0310, 0.0559)	(0.0590, 0.0839, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0310, 0.0559, 0.0807)	(0.0310, 0.0559, 0.0807)	(0.0249, 0.0497, 0.0746)	(0.0093, 0.0342, 0.0590)
C3	(0.0092, 0.0341, 0.0589)	(0.0123, 0.0372, 0.0621)	(0, 0, 0)	(0.0746, 0.0994, 0.0994)	(0.0559, 0.0807, 0.0994)	(0.0528, 0.0777, 0.0994)	(0.0528, 0.0777, 0.0994)	(0.0590, 0.0839, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0559, 0.0807, 0.0994)	(0.0528, 0.0777, 0.0994)	(0.0746, 0.0994, 0.0994)
C4	(0, 0.0249, 0.0497)	(0.0061, 0.0309, 0.0558)	(0.0342, 0.0590, 0.0839)	(0, 0, 0)	(0.0187, 0.0435, 0.0684)	(0.0310, 0.0559, 0.0807)	(0.0249, 0.0497, 0.0746)	(0.0559, 0.0807, 0.0994)	(0, 0.0249, 0.0497)	(0.0249, 0.0497, 0.0746)	(0.0746, 0.0994, 0.0994)	(0.0310, 0.0559, 0.0807)
C5	(0.0341, 0.0589, 0.0838)	(0.0062, 0.0310, 0.0559)	(0.0559, 0.0807, 0.0994)	(0.0249, 0.0497, 0.0746)	(0, 0, 0)	(0.0559, 0.0807, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0280, 0.0528, 0.0777)	(0.0559, 0.0807, 0.0994)	(0.0746, 0.0994, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0590, 0.0839, 0.0994)
C6	(0.0062, 0.0310, 0.0559)	(0.0062, 0.0310, 0.0559)	(0.0279, 0.0528, 0.0776)	(0.0310, 0.0559, 0.0807)	(0.0310, 0.0559, 0.0807)	(0, 0, 0)	(0.0061, 0.0309, 0.0558)	(0.0590, 0.0839, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0310, 0.0559, 0.0807)	(0.0528, 0.0777, 0.0994)	(0.0559, 0.0807, 0.0994)
C7	(0.0249, 0.0497, 0.0746)	(0.0342, 0.0590, 0.0839)	(0.0559, 0.0807, 0.0994)	(0.0497, 0.0746, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0156, 0.0404, 0.0653)	(0, 0, 0)	(0.0497, 0.0746, 0.0994)	(0.0746, 0.0994, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0249, 0.0497, 0.0746)	(0.0341, 0.0589, 0.0838)
C8	(0.0062, 0.0310, 0.0559)	(0.0093, 0.0342, 0.0590)	(0.0310, 0.0559, 0.0807)	(0.0559, 0.0807, 0.0994)	(0.0497, 0.0746, 0.0994)	(0.0746, 0.0994, 0.0994)	(0.0249, 0.0497, 0.0746)	(0, 0, 0)	(0.0125, 0.0374, 0.0622)	(0.0497, 0.0746, 0.0994)	(0.0310, 0.0559, 0.0807)	(0.0746, 0.0994, 0.0994)
C9	(0.0310, 0.0559, 0.0807)	(0.0249, 0.0497, 0.0746)	(0.0187, 0.0435, 0.0684)	(0, 0.0249, 0.0497)	(0.0341, 0.0589, 0.0838)	(0.0249, 0.0497, 0.0746)	(0.0746, 0.0994, 0.0994)	(0.0156, 0.0404, 0.0653)	(0, 0, 0)	(0.0249, 0.0497, 0.0746)	(0, 0.0249, 0.0497)	(0.0280, 0.0528, 0.0777)
C10	(0.0559, 0.0807, 0.0994)	(0.0341, 0.0589, 0.0838)	(0.0528, 0.0777, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0746, 0.0994, 0.0994)	(0.0497, 0.0746, 0.0994)	(0.0249, 0.0497, 0.0746)	(0.0558, 0.0807, 0.0994)	(0.0310, 0.0559, 0.0807)	(0, 0, 0)	(0.0249, 0.0497, 0.0746)	(0.0528, 0.0777, 0.0994)
C11	(0, 0.0249, 0.0497)	(0.0062, 0.0310, 0.0559)	(0.0342, 0.0590, 0.0839)	(0.0746, 0.0994, 0.0994)	(0, 0.0249, 0.0497)	(0.0310, 0.0559, 0.0807)	(0.0249, 0.0497, 0.0746)	(0.0280, 0.0528, 0.0777)	(0, 0.0249, 0.0497)	(0.0062, 0.0310, 0.0559)	(0, 0, 0)	(0.0249, 0.0497, 0.0746)
C12	(0.0310, 0.0559, 0.0807)	(0.0092, 0.0341, 0.0589)	(0.0559, 0.0807, 0.0994)	(0.0342, 0.0590, 0.0839)	(0.0746, 0.0994, 0.0994)	(0.0746, 0.0994, 0.0994)	(0.0497, 0.0746, 0.0994)	(0.0746, 0.0994, 0.0994)	(0.0559, 0.0807, 0.0994)	(0.0559, 0.0807, 0.0994)	(0.0249, 0.0497, 0.0746)	(0, 0, 0)

).

Step 6. In this phase, the inverse matrix (

Table 5. Inverse matrix.

Criterion	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
C1	(1.0171, 1.0902, 1.5413)	(0.0652, 0.1547, 0.6020)	(0.0414, 0.1561, 0.7046)	(0.0360, 0.1493, 0.6708)	(0.0969, 0.2138, 0.7239)	(0.0451, 0.1631, 0.7007)	(0.0339, 0.1440, 0.6635)	(0.0391, 0.1598, 0.7124)	(0.0747, 0.1818, 0.6842)	(0.0814, 0.1965, 0.7289)	(0.0260, 0.1339, 0.6448)	(0.0494, 0.1703, 0.7163)
C2	(0.0660, 0.1553, 0.6142)	(1.0112, 1.0735, 1.4963)	(0.0488, 0.1564, 0.6953)	(0.0274, 0.1343, 0.6460)	(0.0457, 0.1580, 0.6829)	(0.0252, 0.1372, 0.6673)	(0.0735, 0.1752, 0.6765)	(0.0439, 0.1572, 0.7005)	(0.0487, 0.1506, 0.6499)	(0.0500, 0.1596, 0.6931)	(0.0380, 0.1390, 0.6402)	(0.0304, 0.1450, 0.6830)
C3	(0.0268, 0.1390, 0.6662)	(0.0252, 0.1310, 0.6360)	(1.0338, 1.1428, 1.7272)	(0.1037, 0.2315, 0.7824)	(0.0880, 0.2242, 0.8075)	(0.0880, 0.2225, 0.8110)	(0.0774, 0.2029, 0.7751)	(0.0953, 0.2323, 0.8311)	(0.0508, 0.1763, 0.7402)	(0.0865, 0.2198, 0.8135)	(0.0798, 0.2021, 0.7608)	(0.1090, 0.2462, 0.8256)
C4	(0.0086, 0.1000, 0.5566)	(0.0128, 0.0986, 0.5350)	(0.0520, 0.1586, 0.6845)	(1.0223, 1.1043, 1.5787)	(0.0369, 0.1487, 0.6627)	(0.0522, 0.1619, 0.6765)	(0.0390, 0.1416, 0.6411)	(0.0757, 0.1866, 0.7092)	(0.0136, 0.1164, 0.6073)	(0.0421, 0.1513, 0.6731)	(0.0887, 0.1865, 0.6511)	(0.0528, 0.1654, 0.6885)
C5	(0.0500, 0.1539, 0.6596)	(0.0197, 0.1190, 0.6045)	(0.0813, 0.2039, 0.7824)	(0.0503, 0.1723, 0.7268)	(1.0334, 1.1390, 1.6844)	(0.0851, 0.2111, 0.7768)	(0.0489, 0.1675, 0.7220)	(0.0601, 0.1898, 0.7767)	(0.0778, 0.1939, 0.7310)	(0.1005, 0.2244, 0.7801)	(0.0470, 0.1629, 0.7067)	(0.0896, 0.2187, 0.7907)
C6	(0.0170, 0.1113, 0.5872)	(0.0140, 0.1030, 0.5583)	(0.0482, 0.1589, 0.7081)	(0.0520, 0.1607, 0.6801)	(0.0528, 0.1679, 0.7026)	(1.0255, 1.1163, 1.6311)	(0.0243, 0.1313, 0.6529)	(0.0808, 0.1956, 0.7383)	(0.0398, 0.1451, 0.6561)	(0.0516, 0.1643, 0.7078)	(0.0688, 0.1712, 0.6771)	(0.0786, 0.1942, 0.7339)
C7	(0.0387, 0.1392, 0.6429)	(0.0447, 0.1390, 0.6205)	(0.0776, 0.1955, 0.7722)	(0.0714, 0.1877, 0.7389)	(0.0517, 0.1758, 0.7431)	(0.0426, 0.1658, 0.7368)	(1.0247, 1.1155, 1.6441)	(0.0752, 0.1993, 0.7846)	(0.0918, 0.2027, 0.7206)	(0.0502, 0.1716, 0.7486)	(0.0450, 0.1566, 0.6977)	(0.0617, 0.1871, 0.7662)
C8	(0.0208, 0.1237, 0.6190)	(0.0193, 0.1163, 0.5908)	(0.0576, 0.1783, 0.7487)	(0.0794, 0.1970, 0.7312)	(0.0768, 0.2016, 0.7553)	(0.1020, 0.2242, 0.7586)	(0.0455, 0.1615, 0.7035)	(1.0330, 1.1366, 1.6864)	(0.0344, 0.1497, 0.6806)	(0.0753, 0.1974, 0.7610)	(0.0545, 0.1673, 0.6961)	(0.1025, 0.2279, 0.7719)
C9	(0.0422, 0.1302, 0.5781)	(0.0341, 0.1173, 0.5457)	(0.0369, 0.1422, 0.6609)	(0.0164, 0.1211, 0.6144)	(0.0530, 0.1620, 0.6672)	(0.0419, 0.1502, 0.6596)	(0.0871, 0.1854, 0.6536)	(0.0350, 0.1458, 0.6684)	(1.0192, 1.0955, 1.5537)	(0.0432, 0.1502, 0.6640)	(0.0134, 0.1130, 0.5966)	(0.0468, 0.1577, 0.6749)
C10	(0.0714, 0.1778, 0.6995)	(0.0467, 0.1479, 0.6534)	(0.0804, 0.2069, 0.8136)	(0.0525, 0.1783, 0.7568)	(0.1055, 0.2361, 0.8060)	(0.0816, 0.2118, 0.8075)	(0.0499, 0.1723, 0.7514)	(0.0866, 0.2193, 0.8266)	(0.0571, 0.1779, 0.7442)	(1.0339, 1.1407, 1.7214)	(0.0487, 0.1682, 0.7358)	(0.0862, 0.2196, 0.8220)
C11	(0.0057, 0.0903, 0.5175)	(0.0110, 0.0905, 0.4984)	(0.0472, 0.1459, 0.6380)	(0.0875, 0.1826, 0.6252)	(0.0144, 0.1188, 0.6010)	(0.0460, 0.1477, 0.6301)	(0.0357, 0.1306, 0.5978)	(0.0457, 0.1494, 0.6440)	(0.0099, 0.1052, 0.5646)	(0.0200, 0.1218, 0.6111)	(1.0160, 1.0852, 1.5183)	(0.0411, 0.1457, 0.6362)
C12	(0.0489, 0.1617, 0.6885)	(0.0242, 0.1315, 0.6372)	(0.0868, 0.2195, 0.8204)	(0.0652, 0.1967, 0.7713)	(0.1081, 0.2453, 0.8120)	(0.1089, 0.2445, 0.8141)	(0.0757, 0.2030, 0.7785)	(0.1087, 0.2471, 0.8337)	(0.0824, 0.2081, 0.7657)	(0.0894, 0.2244, 0.8176)	(0.0523, 0.1777, 0.7417)	(1.0414, 1.1594, 1.7385)

) is calculated and, from it, the total relationship matrix (

Table 6. Total-relation matrix T.

Criterion	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
C1	(0.0171, 0.0902, 0.5413)	(0.0652, 0.1547, 0.6020)	(0.0414, 0.1561, 0.7046)	(0.0360, 0.1493, 0.6708)	(0.0969, 0.2138, 0.7239)	(0.0451, 0.1631, 0.7007)	(0.0339, 0.1440, 0.6635)	(0.0391, 0.1598, 0.7124)	(0.0747, 0.1818, 0.6842)	(0.0814, 0.1965, 0.7289)	(0.0260, 0.1339, 0.6448)	(0.0494, 0.1703, 0.7163)
C2	(0.0660, 0.1553, 0.6142)	(0.0112, 0.0735, 0.4963)	(0.0488, 0.1564, 0.6953)	(0.0274, 0.1343, 0.6460)	(0.0457, 0.1580, 0.6829)	(0.0252, 0.1372, 0.6673)	(0.0735, 0.1752, 0.6765)	(0.0439, 0.1572, 0.7005)	(0.0487, 0.1506, 0.6499)	(0.0500, 0.1596, 0.6931)	(0.0380, 0.1390, 0.6402)	(0.0304, 0.1450, 0.6830)
C3	(0.0268, 0.1390, 0.6662)	(0.0252, 0.1310, 0.6360)	(0.0338, 0.1428, 0.7272)	(0.1037, 0.2315, 0.7824)	(0.0880, 0.2242, 0.8075)	(0.0880, 0.2225, 0.8110)	(0.0774, 0.2029, 0.7751)	(0.0953, 0.2323, 0.8311)	(0.0508, 0.1763, 0.7402)	(0.0865, 0.2198, 0.8135)	(0.0798, 0.2021, 0.7608)	(0.1090, 0.2462, 0.8256)
C4	(0.0086, 0.1000, 0.5566)	(0.0128, 0.0986, 0.5350)	(0.0520, 0.1586, 0.6845)	(0.0223, 0.1043, 0.5787)	(0.0369, 0.1487, 0.6627)	(0.0522, 0.1619, 0.6765)	(0.0390, 0.1416, 0.6411)	(0.0757, 0.1866, 0.7092)	(0.0136, 0.1164, 0.6073)	(0.0421, 0.1513, 0.6731)	(0.0887, 0.1865, 0.6511)	(0.0528, 0.1654, 0.6885)
C5	(0.0500, 0.1539, 0.6596)	(0.0197, 0.1190, 0.6045)	(0.0813, 0.2039, 0.7824)	(0.0503, 0.1723, 0.7268)	(0.0334, 0.1390, 0.6844)	(0.0851, 0.2111, 0.7768)	(0.0489, 0.1675, 0.7220)	(0.0601, 0.1898, 0.7767)	(0.0778, 0.1939, 0.7310)	(0.1005, 0.2244, 0.7801)	(0.0470, 0.1629, 0.7067)	(0.0896, 0.2187, 0.7907)
C6	(0.0170, 0.1113, 0.5872)	(0.0140, 0.1030, 0.5583)	(0.0482, 0.1589, 0.7081)	(0.0520, 0.1607, 0.6801)	(0.0528, 0.1679, 0.7026)	(0.0255, 0.1163, 0.6311)	(0.0243, 0.1313, 0.6529)	(0.0808, 0.1956, 0.7383)	(0.0398, 0.1451, 0.6561)	(0.0516, 0.1643, 0.7078)	(0.0688, 0.1712, 0.6771)	(0.0786, 0.1942, 0.7339)
C7	(0.0387, 0.1392, 0.6429)	(0.0447, 0.1390, 0.6205)	(0.0776, 0.1955, 0.7722)	(0.0714, 0.1877, 0.7389)	(0.0517, 0.1758, 0.7431)	(0.0426, 0.1658, 0.7368)	(0.0247, 0.1155, 0.6441)	(0.0752, 0.1993, 0.7846)	(0.0918, 0.2027, 0.7206)	(0.0502, 0.1716, 0.7486)	(0.0450, 0.1566, 0.6977)	(0.0617, 0.1871, 0.7662)
C8	(0.0208, 0.1237, 0.6190)	(0.0193, 0.1163, 0.5908)	(0.0576, 0.1783, 0.7487)	(0.0794, 0.1970, 0.7312)	(0.0768, 0.2016, 0.7553)	(0.1020, 0.2242, 0.7586)	(0.0455, 0.1615, 0.7035)	(0.0330, 0.1366, 0.6864)	(0.0344, 0.1497, 0.6806)	(0.0753, 0.1974, 0.7610)	(0.0545, 0.1673, 0.6961)	(0.1025, 0.2279, 0.7719)
C9	(0.0422, 0.1302, 0.5781)	(0.0341, 0.1173, 0.5457)	(0.0369, 0.1422, 0.6609)	(0.0164, 0.1211, 0.6144)	(0.0530, 0.1620, 0.6672)	(0.0419, 0.1502, 0.6596)	(0.0871, 0.1854, 0.6536)	(0.0350, 0.1458, 0.6684)	(0.0192, 0.0955, 0.5537)	(0.0432, 0.1502, 0.6640)	(0.0134, 0.1130, 0.5966)	(0.0468, 0.1577, 0.6749)
C10	(0.0714, 0.1778, 0.6995)	(0.0467, 0.1479, 0.6534)	(0.0804, 0.2069, 0.8136)	(0.0525, 0.1783, 0.7568)	(0.1055, 0.2361, 0.8060)	(0.0816, 0.2118, 0.8075)	(0.0499, 0.1723, 0.7514)	(0.0866, 0.2193, 0.8266)	(0.0571, 0.1779, 0.7442)	(0.0339, 0.1407, 0.7214)	(0.0487, 0.1682, 0.7358)	(0.0862, 0.2196, 0.8220)
C11	(0.0057, 0.0903, 0.5175)	(0.0110, 0.0905, 0.4984)	(0.0472, 0.1459, 0.6380)	(0.0875, 0.1826, 0.6252)	(0.0144, 0.1188, 0.6010)	(0.0460, 0.1477, 0.6301)	(0.0357, 0.1306, 0.5978)	(0.0457, 0.1494, 0.6440)	(0.0099, 0.1052, 0.5646)	(0.0200, 0.1218, 0.6111)	(0.0160, 0.0852, 0.5183)	(0.0411, 0.1457, 0.6362)
C12	(0.0489, 0.1617, 0.6885)	(0.0242, 0.1315, 0.6372)	(0.0868, 0.2195, 0.8204)	(0.0652, 0.1967, 0.7713)	(0.1081, 0.2453, 0.8120)	(0.1089, 0.2445, 0.8141)	(0.0757, 0.2030, 0.7785)	(0.1087, 0.2471, 0.8337)	(0.0824, 0.2081, 0.7657)	(0.0894, 0.2244, 0.8176)	(0.0523, 0.1777, 0.7417)	(0.0414, 0.1594, 0.7385)

), which combines the direct and indirect effects between the criteria. This table allows the development of the structural model and the capture of the overall interdependence of the criteria in

Table 7. Causal diagram calculations.

	Di			Ri			Di + Ri			Di-Ri			Crisp Di + Ri	Crisp Di-Ri
C1	0.6062	1.9135	8.0933	0.4133	1.5727	7.3705	1.0195	3.4862	15.4638	−6.7643	0.3407	7.6800	5.4677	0.3473
C2	0.5089	1.7414	7.8452	0.3281	1.4222	6.9779	0.8370	3.1636	14.8232	−6.4691	0.3192	7.5171	5.1180	0.3571
C3	0.8645	2.3705	9.1764	0.6919	2.0650	8.7559	1.5564	4.4355	17.9323	−7.8915	0.3054	8.4845	6.5469	0.2755
C4	0.4967	1.7199	7.6644	0.6641	2.0157	8.3226	1.1608	3.7355	15.9870	−7.8259	−0.2958	7.0003	5.7383	−0.2679
C5	0.7437	2.1563	8.7416	0.7632	2.1912	8.6485	1.5068	4.3475	17.3900	−7.9048	−0.0349	7.9784	6.4046	0.0126
C6	0.5534	1.8199	8.0334	0.7440	2.1564	8.6702	1.2974	3.9763	16.7035	−8.1168	−0.3365	7.2893	6.0280	−0.2872
C7	0.6751	2.0357	8.6161	0.6156	1.9307	8.2598	1.2907	3.9664	16.8760	−7.5847	0.1050	8.0005	6.0462	0.1396
C8	0.7013	2.0813	8.5032	0.7790	2.2187	8.9119	1.4802	4.3001	17.4152	−8.2107	−0.1374	7.7242	6.3732	−0.1243
C9	0.4692	1.6707	7.5371	0.6003	1.9033	8.0982	1.0695	3.5741	15.6353	−7.6290	−0.2326	6.9368	5.5614	−0.2142
C10	0.8004	2.2569	9.1381	0.7241	2.1221	8.7202	1.5246	4.3790	17.8583	−7.9198	0.1348	8.4139	6.4949	0.1667
C11	0.3801	1.5137	7.0821	0.5781	1.8635	8.0668	0.9582	3.3772	15.1488	−7.6866	−0.3498	6.5040	5.3344	−0.3562
C12	0.8919	2.4189	9.2193	0.7895	2.2371	8.8476	1.6814	4.6560	18.0669	−7.9556	0.1819	8.4298	6.7274	0.1900

(the table refers to the original 12 criteria). At this stage, defuzzification has been applied using the CFCS method of Opricovic and Tzeng [79].

At this point, a merger (Remark 1) of the effect (passive criteria) with the causal criteria was carried out. Consequently, some synthetic criteria were created, while some others remained constant. Applying the procedure described in Remark 1, the following final set of criteria for rating emerges.

New List of Seven Criteria:

C’1. Technical Expertise & Knowledge (unchanged)

• Contains: Only its original meaning.

C’2. Education & Certifications (unchanged)

• Contains: Only its original meaning.

C’3. Leadership & Team Management + Communication & Team Spirit = Leadership & Communication Excellence

• Contains: (C3 + C4).
• Leadership absorbs team communication.

C’4. Problem-Solving & Decision-Making (from C5—renamed)

• Contains: Only its original meaning of C5.

C’5. Adaptability & Flexibility (from C6—renamed)

• Contains: Only its original meaning of C6.

C’6. Emotional Resilience & Motivation + Professional Conduct & Ethics + Safety & Environmental Awareness = Professional Integrity & Safety Consciousness

• Contains: (C8 + C7 + C9).
• Combination of emotional maturity, ethics and safety consciousness.

C’7. Organizational & Cognitive Abilities + Crisis Management & Self-Management + Cultural Fluency = Organizational Excellence & Crisis Leadership

• Includes: (C10 + C12 + C11).
• Organizational abilities that include crisis management and cultural adaptation.

4.3. Criteria Weights Calculation

At this stage, the consolidated criterion resulting from the DEMATEL analysis is evaluated by a group of experts in order to determine their weights through the use of NAFEs.

Step 7. Use of NAFEs to calculate the weights of the criteria.

The study involves a sample of experts (academics and shipping professionals) who determine the weight of each criterion, without assigning them different weights in advance in order to avoid the additional subjectivity that this process would introduce. Instead, the experts’ assessments are transformed into NAFEs, which incorporate both the uncertainty and the variance of their judgments. If a decision committee includes Z members, then the weight of the criteria and the assessment of the alternatives with respect to each criterion are obtained through the NAFEs. This methodology allows the evaluation criteria to be weighted through the structure of the fuzzy sets, without requiring a priori weighting of the individual opinions of each expert. In order to incorporate the differences in the weight of the expertise, the statistical significance level is reduced by expanding the range of the fuzzy estimator at the boundary points. This approach proves to be particularly effective in situations where the opinions of the experts present significant deviations, allowing a more adaptive and dynamic evaluation process. Finally, the NAFEs are shown in

Table 8. NAFEs a−cuts for criteria weights.

Criterion	$\mathit{a}$	$\mathbf{f}\left(\mathit{\alpha }\right)$	${\mathit{t}}_{\mathit{f}\left(\mathit{a}\right)}$	$\mathit{a}-\mathit{C}\mathit{u}\mathit{t}$ (Left Tail)	$\mathit{\alpha }-\mathit{C}\mathit{u}\mathit{t}$ (Right Tail)
C’1. Technical Expertise & Knowledge	0.00	0.02500	−2.364624	0.837467	0.902533
	0.25	0.14375	−1.151069	0.854163	0.885837
	0.50	0.26250	−0.668896	0.860797	0.879203
	0.75	0.38125	−0.314236	0.865677	0.874323
	1.00	0.50000	0.000000	0.870000	0.870000
C’2. Education & Certifications	0.00	0.02500	−2.364624	0.713934	0.758566
	0.25	0.14375	−1.151069	0.725387	0.747113
	0.50	0.26250	−0.668896	0.729937	0.742563
	0.75	0.38125	−0.314236	0.733284	0.739216
	1.00	0.50000	0.000000	0.736250	0.736250
C’3. Leadership & Communication Excellence	0.00	0.02500	−2.364624	0.804522	0.845478
	0.25	0.14375	−1.151069	0.815031	0.834969
	0.50	0.26250	−0.668896	0.819207	0.830793
	0.75	0.38125	−0.314236	0.822279	0.827721
	1.00	0.50000	0.000000	0.825000	0.825000
C’4. Problem-Solving & Decision-Making	0.00	0.02500	−2.364624	0.863637	0.893863
	0.25	0.14375	−1.151069	0.871393	0.886107
	0.50	0.26250	−0.668896	0.874475	0.883025
	0.75	0.38125	−0.314236	0.876742	0.880758
	1.00	0.50000	0.000000	0.878750	0.878750
C’5. Adaptability & Flexibility	0.00	0.02500	−2.364624	0.763280	0.796720
	0.25	0.14375	−1.151069	0.771861	0.788139
	0.50	0.26250	−0.668896	0.775270	0.784730
	0.75	0.38125	−0.314236	0.777778	0.782222
	1.00	0.50000	0.000000	0.780000	0.780000
C’6. Professional Integrity &Safety Consciousness	0.00	0.02500	−2.364624	0.892163	0.920337
	0.25	0.14375	−1.151069	0.899393	0.913107
	0.50	0.26250	−0.668896	0.902265	0.910235
	0.75	0.38125	−0.314236	0.904378	0.908122
	1.00	0.50000	0.000000	0.906250	0.906250
C’7. Organizational Excellence & Crisis Leadership	0.00	0.02500	−2.364624	0.822163	0.850337
	0.25	0.14375	−1.151069	0.829393	0.843107
	0.50	0.26250	−0.668896	0.832265	0.840235
	0.75	0.38125	−0.314236	0.834378	0.838122
	1	0.5	6.87851 × 10−17	0.836250	0.836250

.

4.4. Seafarers’ Final Ranking

In this phase, Fuzzy TOPSIS is used to rank the alternatives (SFs).

Step 8. Decision matrix. This step involves organizing all alternatives and evaluation criteria into a structured matrix (

Table 9. Integrated matrix.

	C1			C2			C3			C4			C5			C6			C7
SF1	4.00	5.25	6.38	3.63	5.00	6.50	5.50	6.88	8.00	3.25	4.88	6.50	3.25	4.88	6.38	3.50	5.38	7.13	5.25	7.00	8.38
SF2	3.00	3.88	5.00	4.13	5.75	7.13	7.63	8.75	9.13	2.63	4.00	5.63	7.00	8.50	9.25	3.00	3.88	5.00	2.50	3.75	5.38
SF3	3.13	5.00	6.88	2.75	4.75	6.75	3.63	4.75	5.88	0.88	1.75	3.25	5.38	7.00	8.13	4.25	5.00	5.75	3.25	4.75	6.50
SF4	7.25	8.88	9.63	4.63	6.25	7.63	0.88	2.75	4.75	5.25	7.13	8.38	5.38	7.13	8.50	4.75	6.50	8.00	0.75	1.25	2.50
SF5	3.00	4.38	5.75	4.38	6.00	7.50	1.63	3.13	4.88	4.63	6.38	7.75	3.25	4.63	6.13	1.25	2.25	3.88	3.25	5.25	7.00

). The importance of this step lies in creating a systematic and organized representation of the decision problem, allowing for the oversight of all elements that will influence the final choice.

Step 9. Calculation of normalized scores. In this step, the initial ratings are converted to a common measurement scale, usually from 0 to 1 (

Table 10. Normalized matrix.

	C1			C2			C3			C4			C5			C6			C7
SF1	0.34	0.48	0.60	0.34	0.48	0.65	0.49	0.62	0.74	0.34	0.51	0.69	0.27	0.41	0.55	0.39	0.61	0.82	0.52	0.70	0.85
SF2	0.26	0.35	0.47	0.39	0.56	0.71	0.67	0.79	0.85	0.27	0.42	0.60	0.58	0.72	0.80	0.33	0.44	0.58	0.25	0.37	0.55
SF3	0.27	0.45	0.64	0.26	0.46	0.67	0.32	0.43	0.54	0.09	0.18	0.35	0.44	0.59	0.70	0.47	0.57	0.66	0.32	0.47	0.66
SF4	0.63	0.80	0.90	0.43	0.60	0.76	0.08	0.25	0.44	0.54	0.75	0.89	0.44	0.60	0.73	0.53	0.74	0.92	0.07	0.12	0.25
SF5	0.26	0.40	0.54	0.41	0.58	0.75	0.14	0.28	0.45	0.48	0.67	0.83	0.27	0.39	0.53	0.14	0.25	0.45	0.32	0.52	0.71

). The criticality of this process is seen in the fact that it allows the comparison of criteria that have different measurement units or scales, creating a homogeneous basis for further analysis.

Step 10. Calculation of the weighted normalized scores. The normalized scores are multiplied by the corresponding importance weights of each criterion (

Table 11. Weighted normalized matrix.

	W1			W2			W3			W4			W5			W6			W7
	0.83	0.87	0.91	0.71	0.74	0.76	0.81	0.83	0.85	0.87	0.88	0.9	0.76	0.78	0.8	0.89	0.91	0.92	0.82	0.84	0.85
		C1			C2			C3			C4			C5			C6			C7
SF1	0.42	0.55	0.66	0.48	0.66	0.85	0.60	0.75	0.88	0.39	0.58	0.78	0.35	0.53	0.69	0.44	0.67	0.89	0.63	0.84	1.00
SF2	0.31	0.40	0.52	0.54	0.75	0.93	0.84	0.96	1.00	0.31	0.48	0.67	0.76	0.92	1.00	0.38	0.48	0.63	0.30	0.45	0.64
SF3	0.32	0.52	0.71	0.36	0.62	0.89	0.40	0.52	0.64	0.10	0.21	0.39	0.58	0.76	0.88	0.53	0.63	0.72	0.39	0.57	0.78
SF4	0.75	0.92	1.00	0.61	0.82	1.00	0.10	0.30	0.52	0.63	0.85	1.00	0.58	0.77	0.92	0.59	0.81	1.00	0.09	0.15	0.30
SF5	0.31	0.45	0.60	0.57	0.79	0.98	0.18	0.34	0.53	0.55	0.76	0.93	0.35	0.50	0.66	0.16	0.28	0.48	0.39	0.63	0.84

). This step is crucial as it reflects the true importance of each criterion in the decision-making process, ensuring that the most important criteria have the greatest impact on the final result.

Step 11. Calculation of FPIS and FNIS. Ideal solutions are identified that represent the best and worst possible performance across all criteria. The importance of this step lies in creating benchmarks that will be used to compare and evaluate real alternatives. In this study, the fuzzy ratings have been normalized to the [0,1] scale, which allows the definition of FPIS and FNIS with simple and distinct extremes: for “benefit” criteria, FPIS is set to 1 and FNIS to 0, while for “cost” criteria, FPIS is 0 and FNIS is 1. This choice simplifies the calculations, maintains compatibility with the normalization process and ensures that the calculations of distances from the ideal points are consistent and easily interpreted, without affecting the relative ranking of the alternatives.

Step 12. The distances of each alternative from the positive and negative ideal points are calculated (Table 10). This step is critical as it quantifies how close or far each alternative is from the ideal or worst-case solution, providing the basis for the final ranking.

Step 13. Calculation of the similarity (degree of closeness) to the ideal solution: The final step involves calculating an index that expresses the degree to which each alternative approaches the ideal solution (

Table 12. (a) Distance to Fuzzy Positive Ideal Solution. (b) Distance to Fuzzy Negative Ideal Solution.

(a)
	C1	C2	C3	C4	C5	C6	C7
SF1	0.536	0.526	0.398	0.508	0.602	0.431	0.341
SF2	0.646	0.469	0.241	0.585	0.316	0.559	0.624
SF3	0.565	0.563	0.576	0.800	0.435	0.439	0.534
SF4	0.248	0.423	0.759	0.308	0.425	0.315	0.853
SF5	0.612	0.443	0.719	0.371	0.614	0.731	0.508
(b)
	C1	C2	C3	C4	C5	C6	C7
SF1	0.484	0.505	0.625	0.534	0.425	0.631	0.702
SF2	0.370	0.566	0.773	0.451	0.703	0.460	0.408
SF3	0.481	0.492	0.441	0.233	0.587	0.572	0.504
SF4	0.787	0.613	0.295	0.742	0.604	0.746	0.169
SF5	0.414	0.594	0.319	0.673	0.409	0.307	0.542

a,b). The importance of this stage is crucial as it produces the final ranking of the alternatives (

Table 13. Alternatives’ coefficient of closeness and ranking.

Seafarer	CC	Rank
SF1	0.5390	2
SF2	0.5202	3
SF3	0.4583	4
SF4	0.5428	1
SF5	0.4491	5

), allowing decision-makers to select the optimal solution based on objective criteria.

4.5. Sensitivity Analysis

Table 14. Sensitivity analysis for NAFEs families $f\left(a\right)$ and confidence level ε.

Linear $\mathit{f}\left(\mathit{a}\right)$, Quadratic $\mathit{f}\left(\mathit{a}\right)$ and Exponential $\mathit{h}\left(\mathit{a}\right)$ support for $\mathit{a}=0$
	$\mathit{\epsilon }=0.05$		$\mathit{\epsilon }=0.1$		$\mathit{\epsilon }=0.15$		$\mathit{\epsilon }=0.2$
Seafarer No.	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking
SF1	0.5390	2	0.5387	2	0.5385	2	0.5384	2
SF2	0.5202	3	0.5199	3	0.5198	3	0.5197	3
SF3	0.4583	4	0.4578	4	0.4575	4	0.4573	4
SF4	0.5428	1	0.5427	1	0.5426	1	0.5425	1
SF5	0.4491	5	0.4486	5	0.4483	5	0.4481	5
Linear $\mathit{f}\left(\mathit{a}\right)$ support for $\mathit{a}=0.25$
		$\mathit{\epsilon }=0.05$		$\mathit{\epsilon }=0.1$		$\mathit{\epsilon }=0.15$		$\mathit{\epsilon }=0.2$
Seafarer No.	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking
SF1	0.5382	2	0.5381	2	0.5381	2	0.5380	2
SF2	0.5195	3	0.5194	3	0.5194	3	0.5193	3
SF3	0.4571	4	0.4570	4	0.4569	4	0.4568	4
SF4	0.5424	1	0.5423	1	0.5423	1	0.5423	1
SF5	0.4479	5	0.4478	5	0.4477	5	0.4476	5
Quadratic $\mathit{f}\left(\mathit{a}\right)$support for $\mathit{a}=0.25$
	$\mathit{\epsilon }=0.05$		$\mathit{\epsilon }=0.1$		$\mathit{\epsilon }=0.15$		$\mathit{\epsilon }=0.2$
Seafarer No.	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking
SF1	0.5386	2	0.5385	2	0.5383	2	0.5382	2
SF2	0.5199	3	0.5198	3	0.5196	3	0.5196	3
SF3	0.4578	4	0.4575	4	0.4573	4	0.4572	4
SF4	0.5426	1	0.5425	1	0.5425	1	0.5424	1
SF5	0.4485	5	0.4483	5	0.4481	5	0.4480	5
Exponential $\mathit{f}\left(\mathit{a}\right)$support for $\mathit{a}=0.25$
		$\mathit{\epsilon }=0.05$		$\mathit{\epsilon }=0.1$		$\mathit{\epsilon }=0.15$		$\mathit{\epsilon }=0.2$
Seafarer No.	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking	Closeness coefficient ${\overline{\mathbf{C}\mathbf{C}}}_{\mathbf{i}}$	SFs Ranking
SF1	0.5385	2	0.5382	2	0.5382	2	0.5381	2
SF2	0.5198	3	0.5196	3	0.5195	3	0.5194	3
SF3	0.4575	4	0.4572	4	0.4571	4	0.4570	4
SF4	0.5425	1	0.5424	1	0.5424	1	0.5423	1
SF5	0.4483	5	0.4480	5	0.4478	5	0.4477	5

presents a sensitivity analysis for the performance evaluation of five seafarers (SF1 to SF5) based on the closeness coefficient ${\overline{CC}}_i$ and their ranking, examining the effect of different values of the confidence interval ($\epsilon =0.5, 0.1, 0.15, 0.2)$ and different types of $\mathrm{f}\left(\alpha \right)$ functions (linear, quadratic and exponential) for both $\alpha =0 \mathrm{a}\mathrm{n}\mathrm{d} 0.25$. The main and most important finding is the absolute stability and robustness of the ranking: regardless of the size of $\epsilon$ or the type of $\mathrm{f}\left(\alpha \right)$, the ranking order remains unchanged, with ${SF}_4$ always being first, followed by ${SF}_1$, ${SF}_2$, ${SF}_3$ and lastly ${SF}_5$. The absolute values of ${\overline{CC}}_i$ for all seafarers show a minimal, almost negligible, decrease as $\epsilon$ increases, indicating a very small effect of uncertainty in the measurements, without affecting their relative performance. Similarly, changing the function $\mathrm{f}\left(\alpha \right)$ or the parameter $\alpha$ causes minimal fluctuations in the absolute values, without ever changing the final ranking. This result suggests that the conclusions of the evaluation are highly reliable and robust as the superiority of ${SF}_4$ and the overall hierarchy of seafarers remain constant under a wide range of parametric changes and modeling assumptions.

The dual robustness investigation in F-DeNETS highlights its reliability as a methodological framework. At a statistical level, the validation showed extremely low variability (CV = 0.072%, below the acceptable threshold of 0.20%), perfect ranking stability (RSI = 1.00), non-significant differences according to ANOVA (F = 0.024, p> 0.05) and perfect Spearman correlation (ρ = 1.00) in all pairs of parameters ε. These results confirm the internal consistency of the system and the absence of sensitivity to small changes. At the same time, the theoretical analysis of the functions $\mathrm{f}\left(\alpha \right)$ for NAFEs documented that the support (α = 0) is invariant in all families (linear, quadratic and exponential), with differences appearing only at intermediate levels α. This finding constitutes a “robustness theorem” that confirms that the choice of membership grade α = 0 constitutes a methodologically safe and rational basis as it ensures convergence of all approaches and prevents artificial differentiations in the final decision-making (Table 14). Combined, the two lines of evidence establish the reliability, stability and practical suitability of F-DeNETS in real decision-making environments.

At this stage, this paper’s goal is limited to presenting the methodology, without a quantitative assessment of reliability through the False Alarm Rate indicator. The data is hypothetical, making any threshold arbitrary. The robustness of the method has already been documented. Therefore, the focus remains on understanding and applying F-DeNETS, with documented methodological safety.

5. Results and Discussion

The most important finding from the application of F-DeNETS is the great robustness of the ranking for different uncertainty scenarios, as demonstrated in the Sensitivity Analysis section (Table 14). The ranking of seafarers (SF4 > SF1 > SF2 > SF3 > SF5) remained absolutely stable at all confidence levels (ε) and for every form of the NAFE function f(α) tested. This stability is a strong validation of the central methodological innovation, namely, the use of NAFEs to calculate weights. In contrast to methods such as Entropy, which are sensitive to the dispersion of the data, or Fuzzy AHP, which can be disturbed by subjective pair-wise comparisons, the statistically substantiated confidence intervals of NAFEs’ produced weights that were both semantically meaningful and robust. This result empirically confirms that F-DeNETS effectively mitigates the “ranking instability” often observed in traditional MCDM methods, offering decision-makers a reliable and valid result, even under conditions of parametric uncertainty.

The development of the integrated F-DeNETS framework represents a significant advance in the field of MCDM under uncertainty as it combines the analytical power of Fuzzy DEMATEL and Fuzzy TOPSIS with the statistical reliability of NAFEs. The DEMATEL–TOPSIS combination has emerged in the literature as a reliable approach for analyzing complex causal systems and ranking alternative solutions [35,36]. DEMATEL allows for the clarification of the interactions between criteria and the identification of the driving factors (“causes”) that affect the system as a whole, while TOPSIS utilizes this information for the final ranking of alternatives, ensuring the consistency and objectivity of the evaluation [37,38].

The main innovation of F-DeNETS lies in the integration of NAFEs for the calculation of criteria weights. This approach essentially addresses limitations observed in traditional methods:

• Instead of Entropy, NAFEs avoid the overemphasis on Criteria with high variance, which can degrade the value of homogeneous but important criteria [74]. The methodology provides a more balanced and comprehensive assessment, ensuring that all critical criteria are taken into account.
• Instead of Fuzzy AHP/ANP, F-DeNETS reduces the time-consuming and subjective process of paired comparisons, which introduces a significant risk of bias and limits practical application in complex systems. However, in recent fuzzy AHP variants, pairwise comparisons can be automatically generated from objective input data or aggregated expert evaluations, and consistency-driven algorithms can minimize subjectivity [52,53,54]. But, NAFEs quantify expert uncertainty statistically. The main advantage of NAFEs is that, instead of considering experts’ answers as absolute data, they measure and statistically quantify their uncertainty. This leads to two main benefits: the minimization of random subjective deviations and the contribution to a more reliable aggregate assessment of multiple experts, taking into account not only their average opinion but also the statistical dispersion around it.
• Instead of simple fuzzy means, NAFEs incorporate statistical reliability through confidence intervals, based on the true variations of expert ratings. Furthermore, the parameter ε reduces the arbitrariness in converting crisp data into fuzzy numbers, providing an objective and transparent framework.

Summing up, the high stability of the rankings under sensitivity analysis (Table 14) demonstrates a key advantage of the F-DeNETS framework: its resilience to the ranking volatility often caused by the bias of entropy weighting or the inherent subjectivity of Fuzzy AHP. By leveraging NAFEs, the proposed model provides a more balanced and statistically reliable weighting mechanism, which is particularly crucial in high-stakes domains, where expert data is often limited and decisions have significant consequences (

Table 15. Comparison of weighting methods for Fuzzy MCDM.

Feature	Entropy Method	Fuzzy AHP	Proposed NAFEs
Basis of weight	Data dispersion	Expert pair-wise comparisons	Expert judgment + statistical intervals
Statistical foundation	Information theory	Priority theory	Inferential statistics (confidence Intervals)
Handles small samples	Poor (requires data)	Moderate (but subjective)	Yes (designed for it)
Subjectivity control	Low (data-driven)	High (prone to bias)	Managed and quantified
Primacy bias (variance sensitivity)	High (over-emphasizes high variance)	Moderate	Low (balanced via statistics)
Handles dynamic environments	No/Limited	No/Limited	Yes

).

F-DeNETS also addresses important limitations that have been identified in the literature:

• Reliance on subjective estimates and small samples: NAFEs were designed to work effectively even with limited expert samples, reducing the sensitivity to subjective errors and preserving the fuzzy nature of the data [49,60].
• Model stability: The introduction of dynamic, event-driven sets of criteria (e.g., C_core and C_peripheral allows the system to adapt to new information or changes in the environment, transforming it from a static tool into a flexible decision-support system.
• Interpretability and sensitivity: The built-in sensitivity analysis through different forms of NAFEs (linear, quadratic and exponential) and the parameter ε allows the adjustment of the “stringency” of the weights and enhances the transparency of the results, offering a more complete picture of the reliability and robustness of the ranking [51].

Overall, F-DeNETS combines the proven effectiveness of traditional hybrid MCDM methods with modern statistical tools, offering a tool that is simultaneously reliable, flexible and transparent. Its ability to adapt to dynamic environments, reduce subjectivity and provide statistically valid estimates of criterion weights makes it particularly suitable for addressing complex, uncertain decision-making problems. This approach not only confirms the value of hybrid MCDM methods but also paves the way for more advanced applications in multi-criteria and dynamic systems.

6. Theoretical and Managerial Implications

F-DeNETS makes a significant contribution to the theory of Multi-Criteria Decision-Making by introducing NAFEs as a new class of statistically substantiated fuzzy numbers based on confidence intervals. This approach goes beyond traditional fuzzy means methods and subjective membership functions, offering a more objective and transparent way of transforming expert assessments into fuzzy estimators. The single three-stage hybrid framework (F-DEMATEL/NAFEs/F-TOPSIS) integrates causal analysis, statistically substantiated weight estimation and alternative ranking, addressing well-known problems of entropy and AHP methods. Furthermore, the integration of dynamic, event-driven sets of criteria offers a theoretically innovative tool that allows the adaptation of MCDM systems to changing environments, indicating new directions for research applications in dynamic, multi-criteria frameworks. The framework also provides the possibility of sensitive analysis and uncertainty control through parameters such as ε and ε, enhancing the theoretical robustness and reliability of the results. In practice, F-DeNETS offers managers and decision-makers significant advantages in terms of reliability, transparency and efficiency. Replacing arbitrary membership functions with confidence intervals reduces subjectivity in the calculation of weights and increases confidence in the results, while the built-in sensitivity analysis allows for the assessment of the stability of decisions in different scenarios. The framework works effectively with small groups of experts, making it suitable for limited data and practical scenarios where collecting extensive evaluations is difficult. The flexibility of F-DeNETS allows applications in many areas, such as supply chain management, HR, finance, healthcare and many more, while the ability to introduce new criteria or re-evaluate existing ones makes the tool adaptable to rapidly changing environments. Furthermore, causal factor analysis through DEMATEL helps to identify the real roots of problems and not just the symptoms, leading to more strategic and sustainable solutions while reducing complexity and cost compared to traditional hybrid methods such as AHP/ANP.

7. Conclusions, Limitations and Future Research

The application of the F-DeNETS framework leads to clear and multiple conclusions regarding the effectiveness and reliability of hybrid MCDM methods under uncertainty. First, the integration of Fuzzy DEMATEL and Fuzzy TOPSIS confirms that simultaneous causal analysis and ranking of alternatives achieves greater objectivity, consistency and transparency in complex systems. The results clearly show that F-DeNETS clarifies the interactions between criteria, highlighting the guiding “causes” that affect the overall performance while at the same time ensuring that the final ranking of alternatives is stable and scientifically substantiated. A key conclusion is that the use of Non-Asymptotic Fuzzy Estimators (NAFEs) significantly reduces subjectivity and uncertainty compared to traditional methods such as Entropy or Fuzzy AHP/ANP, providing statistically valid confidence intervals for the weights of the criteria and enhancing the reliability of the results.

Furthermore, F-DeNETS proves to be highly flexible and adaptable to dynamic and changing environments, allowing the incorporation of new criteria or the adjustment of existing ones based on new information. A crucial conclusion is that this capacity makes the tool capable of supporting decision-making even with limited expert samples, reducing the reliance on subjective estimates and enhancing reliability and transparency. Furthermore, the built-in sensitivity analysis through parameters such as ε and ε provides the possibility to estimate the robustness of decisions, confirming that F-DeNETS not only evaluates options but also allows for strategic risk management. Overall, the findings indicate that F-DeNETS effectively combines theoretical and practical innovation, providing a fully integrated MCDM tool that offers reliable, transparent and flexible solutions to complex multi-criteria and uncertain problems.

Despite its advantages, F-DeNETS presents some limitations. First, the accuracy of the results depends on the quality of the expert assessments, even if the method can work with small samples. Furthermore, the complexity of the calculations and the incorporation of parameters such as f(α) and ε may require advanced knowledge or support software for effective implementation. Another limitation is the absence of an analysis of alternative normalization techniques (e.g., Euclidean, sum or linear normalization). This research focused on the sensitivity analysis of the results with respect to the critical parameters of the core F-DeNETS methodology. More specifically, it focused on the shape f(α) and the confidence level (ε) of the NAFEs in the criteria weights, as presented in detail in Section 3.2 and Table 14. This choice was based on the fact that the normalization method used is standardized and widely accepted, while the primary innovation of the model lies in the statistical foundation of the weights. A limitation could also be considered the use of CFCS in Step 6 of DEMATEL, which can be interpreted as “premature elimination of the fuzziness” and seems to contradict the philosophy of preserving uncertainty. However, this choice is not made by chance and is based on the clear separation of roles between the phases of F-DeNETS. Fuzziness elimination in DEMATEL is applied exclusively for the unification of criteria. This operation completes the “fuzzy” phase of causality and does not affect the weights. The real innovation and preservation of uncertainty appears next where NAFEs introduce a new, statistically substantiated fuzziness on the weights, based on confidence intervals, which is never eliminated during Fuzzy TOPSIS. This fuzziness is combined with the evaluations via α-cuts and is maintained until the final calculation of the distances. Therefore, F-DeNETS avoids premature elimination of fuzziness at critical points (weights and alternatives ratings), maintaining its statistical reliability and robustness while using elimination only as a tool to optimize the problem structure. Another possible limitation is the absence of a direct comparison with other established Multi-Criteria Decision-Making (MCDM) approaches. However, in the context of this article, which focuses on introducing methodological innovation, a theoretical and qualitative comparison has already been incorporated through Table 15, which clearly demonstrates the inherent advantages of NAFE over classical methods (such as avoiding the subjectivity of Fuzzy AHP and the primacy bias of Entropy). Furthermore, the absolute robustness of the results presented in the sensitivity analysis (Table 14) is in itself strong indirect evidence of the reliability of the method. The present study focuses on demonstrating the methodological innovation of F-DeNETS. To this end, the application of the framework is carried out in a controlled, hypothetical scenario. This approach allows for a clear demonstration of the steps and advantages of the methodology, without however implying that the framework is limited to such data. On the contrary, its structure makes it equally applicable to real problems. Finally, while the framework is general and adaptable, its application in some areas may require specialization or adjustments to the criteria and data collection procedures.

Future research can consider further improving and extending F-DeNETS. In particular, the integration of Machine Learning or simulation techniques could enhance the predictive ability and adaptability in dynamic systems. The development of automated software tools could reduce the difficulty of implementation and facilitate extensive use in operational environments. Furthermore, investigating the effectiveness of F-DeNETS in different fields (such as finance, medical sciences, engineering, etc.) will provide further practical evidence and highlight the flexibility and applicability of the framework to complex, multi-criteria problems. A comparative analysis with alternative normalization methods may constitute also a particular direction for future research. Another direction could be an extensive comparative study with other network models (ANP, WINGS, etc.). Finally, it is of utmost importance the implementation of F-DeNETS in real-world problems, tested via real data and compared real-world settings with existing methods.