An advanced fermatean fuzzy DoC MCDM architecture for comprehensive quantitative assessment of physical fitness competency across academic institutions

Abstract

The fitness of the students in the college is a critical parameter of their health, well-being, and academic output. Nonetheless, the assessment of physical fitness determinants is uncertain and subjective due to differing opinions among experts and the interconnection of the standards with one another. In addressing this complexity, the current research presents a new Fermatean Fuzzy (FF) Deck-of-Cards-Method (DoCM) Multi-Criteria Decision-Making (MCDM) model for the quantitative evaluation of physical fitness in academic settings. The model is based on the fact that FF sets have high expressiveness, enabling them to represent expert hesitation and dual uncertainty more effectively than conventional fuzzy models. The DoC approach is used to obtain the systematic and preference-based weights of the evaluation criteria, thereby achieving rational and flexible prioritization. A hypothetical case study is created to demonstrate the relevance of the framework, encompassing criteria such as endurance, strength, flexibility, body composition, and mental well-being. The findings underscore the model’s ability to produce consistent and explainable rankings of fitness determinants using incomplete or unclear information. When compared to current literature where most models use FS/IFS/PFS as their default uncertainty modeling framework with cognitively demanding weighting functions, the proposed FF-DoC framework bridges the gap of integrating high-expressiveness uncertainty modeling with transparent and human-eliciting weight elicitation to assess institutional fitness competency.

Introduction

Physical fitness is a vital component of student health, which not only impacts their physical well-being but also their cognitive abilities, emotional stability, and academic performance¹. The importance of physical fitness in institutions of higher learning has been given considerable attention, as most students lead sedentary lifestyles, spend a significant amount of time on screens, and do not engage in regular physical exercise². The measurement of students’ fitness level should be conducted in reference to several interrelated determinants, including cardiovascular endurance, muscular strength, flexibility, body composition, and mental wellness³. All determinants demonstrate diverse situations related to student health, and their weight may vary in different circumstances, institutions, and cultural contexts⁴. Thus, it follows that an all-inclusive system of evaluation is required to ensure that health promotion endeavors are evidence-based, data-driven, and contextually relevant⁵. This is also a complex context of DM, as it involves uncertain and subjective judgments, as well as conflicting ones, among fitness experts, health teachers, and policymakers⁶. Conventional methods of evaluation usually do not reflect ambiguity and hesitation inherent in human evaluations. (MCDM) Techniques⁷ offer a methodological and organized approach to addressing this kind of complexity, incorporating both quantitative and expert DM processes⁸. Nevertheless, classical crisp or even traditional fuzzy models do not necessarily capture the underlying uncertainty and uncertainties which are attributed to human perceptions⁹. To address these limitations effectively, this research paper proposes a new FF-DoCM paradigm as a quantitative measure of the determinants of physical fitness in institutions. FF sets¹⁰ are a more advanced development of intuitionistic and PFS which permit more extensive representation of degrees of membership and non-membership to characterize hesitation and imprecision better. The suggested method, combined with the (DoC) method¹¹, which allows for intuitive weighting of criteria based on ranks, creates a more viable and adaptable decision support system. DoC method increases transparency and consistency in extracting expert preferences whereas Fermatean framework¹² makes sure that the model is capable of handling uncertainty. The significance of the proposed work is that it would allow the integration of the subjective expert reasoning with quantitative assessment in a manner which will enable preserving both analytical rigor and human interpretability¹³. Although the superior fuzzy MCDM models like spherical fuzzy, plithogenic fuzzy and probabilistic-fuzzy models have been quickly developed, some of the methodological shortcomings have not been addressed in human-oriented evaluation issues like the evaluation of physical fitness. Numerous current methods utilize capabilities of normalization sensitive aggregation that may have a strong influence on ranking stability and preference interpretation especially when the decisions of experts are reluctant, inconsistent, or weakly ordered. Besides, probabilistic and plithogenic frameworks, although possessing high expressive power, can increase cognitive and computational complexity, and thus preference elicitation becomes less transparent as a way of expressing the decision-makers. Conversely, recent literature does not have a systematic combination of high-order fuzzy uncertainty modeling and cognitively explainable preference elicitation processes. Particularly, the Deck-of-Cards approach, allowing to articulate the preferences in an ordinal and gap-based manner, has not been systematically integrated with Fermatean fuzzy sets to assess soft, health-based considerations in the state of deep uncertainty. This is the gap that drives the current research that seeks to fill the expressive uncertainty modeling with the cognitively consistent weighting in a single MCDM architecture. This is the reason why Fermatean Fuzzy Sets are the best as compared to Pythagorean Fuzzy Sets. Despite the fact that the Pythagorean Fuzzy Sets (PFS) are much better than the intuitionistic fuzzy sets by virtue of the fact that it opens the admissible space of membership and non-membership degrees, it nonetheless presents restrictions in the case of a high degree of expert hesitation. When it comes to subjective and psychological standards like overall mental wellness, professionals tend to show both some agreeableness and disagreement as well as high levels of indecision. Fermatean Fuzzy Sets (FFS) are further extensions of PFS, in that the cubic sum of the degree of membership and non-membership can be in the unit interval. This long latitude allows professionals to demonstrate greater doubtfulness and indecisiveness without compelling them to adopt feigned accuracy. As a result, FFS are a more realistic and more human-focused reflection of expert decision-making, especially of the soft criteria whose perceptions are unclear, contradictory, or subjective, e.g., mental wellness in physical fitness testing.The model illustrates the ranking, weighting, and analysis of various fitness determinants, minimizing cognitive bias and maximizing reliability through a hypothetical case study. The findings reveal that the suggested FF-DoC based framework gives more stable and informative rankings than traditional fuzzy MCDM methods. Such a contribution to methodology can assist educational policymakers, health administrators¹⁴, and physical instructors in devising specific fitness improvement plans, resource allocation, and overall student health. Finally, the research paper provides a basis to use superior fuzzy-driven DM models to the context of health and wellness assessment in academic activities so that sophisticated human decisions will be modeled in a more precise and balanced way¹⁵.

Problem statement

Assessment of physical fitness among college students is a complex and uncertain activity which is affected ferby physiological, psychological and environmental aspects. The conventional evaluation methods and conventional statistical procedures are usually poor to carry out such reviews since they fail to well depict the vagueness and subjectivity involved in human judgments. Health instructors and experts often offer conflicting or reluctant responses in deciding on the relative significance of the determinants of fitness that include strength, endurance, flexibility and mental well-being. The fuzzy MCDM models that are conventional have enhanced decision-making by capturing the element of uncertainty but still lacked the capability to capture more profound degrees of hesitation and inaccuracy. In addition, most of the existing models rely on arbitrary or inconsistent weighting schemes, which do not represent the thought processes of experts. Lack of a structured and intuitive weighting method causes biased prioritization, and decreases the accuracy of the final examinations. Thus, it is strongly required that a strong and human-based decision-making system can manage a high degree of uncertainty, expert reluctance, and preference disparity and remain interpretable and reasonable. To satisfy this requirement, this paper has proposed a FF-DoC MCDM model, that combines the sophisticated uncertainty-processing power of FF sets and the intuitive and preference-oriented nature of the DoC approach. The suggested model will yield higher accuracy, consistency, and interpretability of physical fitness determinants in the academic setting and thus contribute to evidence-based health DM and policy-making in educational institutions.

Objectives and contribution

In this section we will discuss the Objectives.

• To come up with a hybrid FF-DoC MCDM that can model profound uncertainty and expert hesitation under physical fitness assessment.

• To investigate the implication of the proposed framework in the ranking of discrimination and stability as opposed to classical fuzzy, intuitionistic fuzzy and Pythagorean fuzzy DoC models.

Here we will discuss the contribution of this study.

• Theoretical proposal of an Advanced FF-DoC hybrid MCDM framework to assess fitness competency.

• Calculates weights using DoC based-preference intensity which is easier to interpret and less cognitive load.

• Models uncertainty and expert hesitation (Fermatean fuzzy sets), thus allowing more representation compared to FS/IFS/PFS.

• Checks the strengths through sensitivity analysis, enhancing the reliability of decision making.

The current developments in the field of the fuzzy (MCDM) have enhanced the accuracy and reliability of the outcomes of decision making in the real world by a significant level. Modern research is placing more emphasis in improving uncertainty representation, robustness and ranking stability through extensions of classical fuzzy models. As an example, complex healthcare and environmental decision problems have been solved using fractional fuzzy-based MCDM methods, with enhanced stability and interpretability with superior fuzzy distance metrics and mixed weighting-ranking methods. Specifically, recent studies in Scientific Reports have demonstrated that advanced fractional fuzzy decision-support systems can be used to handle highly uncertain and heterogeneous information in areas such as medical waste management and sustainable energy planning, leading to more credible and practically relevant assessments. The models enrich the conventional approaches to methodology with richer fuzzy information structures, better weighting mechanisms and robustness-driven validation plans. More than that, more sophisticated hybrid fuzzy models such as quantum-based and spherical fuzzy extensions have been proposed to deal with more ambiguous and interdependent criteria in a deep way. Such models have been shown to be superior with respect to capturing the complex patterns of expert judgments, and also sensitivity to rankings is less very sensitive as opposed to traditional fuzzy MCDM techniques. Together, these recent advancements underscore a distinct methodological change to more expressive, stable, and application-oriented fuzzy MCDM models, which has a great basis in enhancing the conventional assessment approaches in the real-life decision-making situations.

Research question and motivation

In this section we will discuss the research question:

• In this section we will discuss the Research Question and Motivation of this study Do Fermatean fuzzy sets integration with the Deck-of-Cards approach improve ranking discrimination in uncertain expert opinion

• How much are the obtained rankings stable to small changes in criteria weights and dominance scores?

Here are some motivations of this study

The selection of FFS over PFS is motivated by its enhanced ability to model complex expert cognition under high uncertainty. While PFSs restrict the squared sum of membership and non-membership degrees to , FFSs extend this flexibility by allowing the cubic sum constraint, which enlarges the feasible decision space and enables experts to express stronger hesitation and nonlinear perception differences. In human-centric evaluation problems such as physical fitness assessment, expert judgments are often imprecise, hesitant, and psychologically biased; therefore, FFSs provide a richer representation structure that captures deeper uncertainty compared to PFSs. This additional expressive power improves robustness of aggregation and makes the FF-DoC framework more suitable for subjective–quantitative integration in academic health analytics.

Literature review

The assessment of physical fitness among college students has been a major point of studies in the field of sports sciences, health education, as well as, behavioral studies. Many researchers have come up with a framework and models used to establish the important factors that affect student fitness such as aerobic endurance, muscular strength, and flexibility, body composition, and psychological well-being. Conventional methods used to interpret the data on physical fitness tended to be based on statistical or descriptive analysis. Although such techniques could be very helpful, they lacked to deal with subjective judgments, imprecise measurements and contrasting expert judgment that is often the case in health and behavioral evaluations. As a result, scholars started incorporating (MCDM) techniques to assess the importance of various indicators of fitness in a systematic way and rank their significance to each other. In the past ten years, MCDM approaches like AHP¹⁶, TOPSIS¹⁷, VIKOR¹⁸, WASPAS¹⁹ and EDAS²⁰ have been utilized in health and sports related appraisals. As an example, fuzzy AHP/fuzzy TOPSIS were used to evaluate athletic performance, exercise programs or health risk factors. Such models improved decision consistency in the sense that expert preferences were integrated in case of uncertainty. They usually, though, used fuzzy sets, that is, where there is minimal flexibility in representing the hesitation or dual uncertainty. The fact that traditional fuzzy sets²¹ cannot be used to describe greater levels of vagueness and overlapping perceptions makes them limited in the context of realistic human judgments- especially on areas such as student health where judgments are affected by behavioral, emotional, and physiological variability. To overcome this drawback, researchers proposed superior extensions of Fuzzy sets (FS)²¹, Intuitionistic Fuzzy Sets (IFS)²², Pythagorean Fuzzy Sets (PFS)²³, and q-Rung Orthopair Fuzzy Sets (q-ROFS)²⁴. Every evolution was expected to offer a more general space to represent the MD and NMD so that the DM can express uncertainty more precisely. Even these models, however, were rather rigid in higher-dimensional ambiguity and expert hesitation. The (FFS)²⁵ became an important development in this development. It generalizes the representational strength of q-ROFS in that it allows the cubic addition of the membership and non-membership degrees to lie inside the unit interval hence providing a better tradeoff between flexibility and interpretability. There is the option of the FFS model to support complex patterns of uncertainty and is therefore quite practical when dealing with psychological and physiological assessments like determining the physical fitness of students: specialists tend to give contradictory or overlapping verdicts. Despite all the developments in methodologies, several limitations persist in existing studies. The majority of previous studies have either applied the single fuzzy MCDM methods or applied arbitrary weighting schemes that do not consider expert consistency and cognitive behavior. The subjective allocation of weights within fuzzy systems tends to bring prejudice and irregularity in the results of decisions. Also, little research has included ranking-based cognitive weighting schemes like the DoCM that enable the DM to give his or her preferences in intuitive terms as opposed to numerical terms. A research gap is still evident in the absence of the ability to combine sophisticated fuzzy logic with structured preference elicitation. These gaps are filled in the current study through the proposal of an integrated FF-DoCM a MCDM framework in which the determinants of physical fitness among college students are to be assessed. The FF theory, combined with the DoC methodology, ensures that the subjective uncertainty, as well as the expertise intuition, can be adequately represented in the process of weighting and ranking. This hybrid framework not only increases the transparency and reliability of the decision but also reduces the cognitive load when evaluating expertise. The proposed model has been shown to provide consistent, interpretable, and environment-related information on the ranking of fitness determinants through a hypothetical case study. This paper, therefore, plays a crucial role in the current development of fuzzy MCDM procedures, as it fills the gap between human-based decision-making and state-of-the-art uncertainty modeling, providing a strong analytical foundation for future studies in educational health assessment. Recent developments in fuzzy MCDM studies focus on hybrid models of uncertainty and architectures of evaluation based on robustness. An example is the better decision reliability of the fractional fuzzy and the hybrid CRITIC-TOPSIS²⁶ frameworks employed in complicated management contexts through the use of sophisticated uncertainty structures and sensitivity verification. Equally, integrated fuzzy FUCOM-FMARCOS models reflect the increasing tendency to integrate weighting consistency with compromise ranking models to support the policy-based decision analysis²⁷. Furthermore, the more recent extensions like complex Pythagorean hesitant fuzzy aggregation stress more cognitive representation of the expert judgment in educational assessment issues²⁸. Although this has occurred, the current literature has been more concerned with hybrid aggregation/optimization approaches as opposed to incorporating expressive FF modeling with DoC-based weight elicitation. Consequently, the offered FF-DoCM architecture adds to the existing paradigms of hybrid fuzzy MCDM by expanding them to the more human-based approach to uncertainty representation along with the structured learning of preferences²⁹.

Research gap

Although fuzzy-MCDM models have increasingly been applied in educational and health-based decision-making, there is still a definite methodological gap on the quantitative evaluation of the physical fitness competency in academic institutions. In particular, the available literature is mostly based on classical fuzzy models (e.g., FS, IFS, PFS) that might not be sufficient to represent strong hesitation and contradictory expert opinions that are common in assessing physical fitness factors, especially soft/behavioral and psychological ones. In addition, most reported models use either cognitively challenging (e.g., pairwise comparisons) or poorly explained weight-derivation procedures, thereby reducing transparency and reproducibility. Besides, there is a lack of integration between high-expressiveness uncertainty modeling and human-centered preference elicitation mechanisms in the literature, which is needed in institutional decision-making contexts that involve multiple stakeholders. To overcome these shortcomings, this paper proposes an Advanced FF-DoC MCDM architecture that bridges the above gap by combining the expressiveness of Fermatean fuzzy sets with the understandability and simplicity of the DoC weighting scheme. The main novelty of the research is in: (i) the proposal of a hybrid FF-DoC weighting-evaluation pipeline to measure the physical fitness competency under uncertainty; (ii) the presentation of explainable and structured weight elicitation mechanism based on the intensity of card-gap preference; and (iii) the promotion of methodological transparency through stepwise implementation of the algorithm and validation of its robustness by sensitivity analysis. Therefore, the suggested framework enhances its methodological (a new hybrid uncertainty-weighting integration) and practical (a dependable decision-support tool in academic institutions) aspects.

Layout

The study is structured as follows. Section "Introduction" represents the Introduction of the study. Section "Preliminaries" represents the Existing work of fuzzy MCDM. Section "Fermatean fuzzy deck of cards (FF-DoC) approach:" illustrates our proposed work. In section "On checking the consistency, compute the weights of the items using the following formula with Beginning at 1." a hypothetical case study is discussed. Section “Conclusion” represents the Conclusion of this study.

Preliminaries

In this Section we will discuss the Basic Definitions of Intutionistic Fuzzy set (IFS), Pythagorean fuzzy set (PFS), Fermatean Fuzzy set and its Score Function.

Definition 1

Suppose be a non-empty seta (IFS) is defined on it as follows²²:

1

Where represent the MD is and NMD is and to the collection with So, be the hesitancy degree. For our ease written in the form is known as intuitionistic fuzzy values (IFVs).

Definition 2

Suppose be a non-empty set and Pythagorean fuzzy (PF) set is defined on it as follows²³:

2

Where represent the MD is and NMD is and to the collection with So, be the hesitancy degree. For our ease written in the form is known as (PFVs).

Definition 3

Suppose be a non-empty set and fuzzy set is defined on it as follows²⁵:

3

Where represent the MD is and NMD is and to the collection with So, be the hesitancy degree. For our ease written in the form is known as Fermatean Fuzzy values (FFVs).

Definition 4

The score function for FFs is defined as:

4

Fermatean fuzzy deck of cards (FF-DoC) approach

In this paper, the DoCM of FFS is applied to tackle a MCDM problem in Quantitative Evaluation of Physical Fitness in Academic Environments. In order to resolve these MCDM issues³⁰ a powerful DM method is introduced The objective is to rank the possible alternative Quantitative Evaluation of Physical Fitness in Academic Environments based on various factors. Considering a set of alternative and : a set of criteria, where optional weights for each criterion, and alternatives in this analysis are FF values and since we are considering a FF framework so . The DoCM offers a cognitively compatible method having a visually interpretable way of ranking options, which is less complex than the old pairwise approaches to DM. It is a good means to connote the vagueness and indecisiveness of the human lingual and policy judgment, and when combined with FFS, it can offer a clear and adaptable method of approaching the intricate and multicultural language planning DM process. Moreover, it can fit the nonlinear associations between linguistic factors, which increases the strength of strategic prioritization. Therefore, FF-DoCM model provides a balance, open, and evidence-based decision-making in sustainable management of the language landscape. This is a hybrid approach that we will apply to solve our example problem. This approach is more effective in resolving the ambiguity and uncertainty of the problem. These steps will be adhered to in the DoCM.

First of all we will discuss h w we will calculate weights in DOC method.

1. From the most desirable to the least preferred, the decision-makers rank the choices (or criteria). Where represents the alternatives, and the criteria represent the choices. Additionally, we will rank these solutions according to our preferences.

2. They create a comparison table and specify how many blank cards (gaps) exist between the two pieces. The matrix contains every comparison, including those involving interval and unknown values. These figures may vary depending on the situation.

3. To be able to create a consistent table, the test consistency of the table that results from the application of the rule and the table values must meet the consistency requirement. Here The values on blank cards are e and and . The following card in the table should have the sum of the values that we will compare with one another. The following formula will be used to determine the table’s consistency.

   5

For all

Where inconsistencies exist, amend by engaging the relevant decision-makers. And modify the value till it satisfies the requirement. There is one thing that should be considered.

• 4.On checking the consistency, compute the weights of the items using the following formula with Beginning at 1.

6

Normalize the obtained weights using the following formula.

7

• 5.Assemble a decision table that includes FF numbers and tolerable hesitant values on the FF values with the help of the given formula.

8

where the hesitation value is denoted by and it is found from the FF number where presents the degree of membership and represents the degree of non-membership.

9

such that

• 6.Considering each criterion , construct matrix according to the following steps and create a pairwise comparison table of each criterion. The values are supposed to satisfy the following requirement.

10

• 7.Calculate the local scores. Determine the local criteria and calculate the local score by adding up all of the available numbers of pairwise comparison tables for each criterion and adding up each row of criteria with the other rows.

11

for to for Unweighted: overall and for Weighted:   

• 8.Priorities of rank alternatives on the basis of decreasing values. Referring to the scores obtained.
• 9.This represents visual illustration of ranks.

The DM algorithm based on FF-DoCM is given in Figure 1.

Fig. 1.

Illustration of FF-DoCM decision process.

Case study

The case study that will be used in this research is deliberately hypothetical. The main goal of the research is to prove the mathematical framework, consistency, and uncertainty-processing capacity of the suggested Fermatean Fuzzy Deck-of-Cards (FF–DoC) MCDM framework instead of performing an analysis on a particular institutional data set. Physical fitness data in the real world can be quite non-homogeneous, specific to an institution and limited by privacy and ethical factors that can be misleading to test a newly developed decision-making methodology. Using a regulated hypothetical dataset that is representative of actual determinants of fitness and expert pattern of judgment, it is possible to systematically test the proposed model to prove its robustness, interpretability, and stability of ranking. This practice is firmly entrenched in fuzzy MCDM literature especially in validating methods methodologically and comparing the results.. Here is a detailed description of the Criteria and alternatives shown in Tables 1 and 2.

Table 1.

Description of alternative.

Alternative code	Alternative description
A₁	Group alpha: Students participating regularly in structured physical training programs including cardio, stretching, and strength routines.
A₂	Group beta: Students involved mainly in recreational sports such as basketball, badminton, or football but without a fixed training schedule.
A₃	Group gamma: Students engaged in minimal physical activities, focusing more on academic and sedentary tasks.
A₄	Group delta: Students who balance daily exercise with academic workload, including yoga, cycling, and jogging.
A₅	Group epsilon: Students following personalized fitness and diet plans under professional supervision.

Table 2.

Description of criteria.

Criterion	Criterion name	Description
C₁	Cardiovascular endurance	Measures the ability of the heart and lungs to supply oxygen efficiently during prolonged exercise (e.g., running, cycling).
C₂	Muscular strength	Reflects the maximum amount of force a muscle or group of muscles can produce during a single effort.
C₃	Flexibility	Evaluates the range of motion around joints, crucial for preventing injuries and maintaining posture.
C₄	Body composition	Represents the proportion of body fat to lean muscle mass; lower fat and higher muscle ratio indicate better fitness.
C₅	Speed and agility	Measures quickness and coordination during movement or athletic tasks.
C₆	Balance and stability	Assesses control over body movement, essential for both athletic and daily physical activities.
C₇	Mental wellness	Examines psychological factors such as stress management, motivation, and emotional balance related to physical fitness.
C₈	Lifestyle habits	Includes diet, sleep patterns, and daily activity routines that directly affect physical health and performance.

Numerical evaluation

The criteria (or alternatives) are ranked by the decision-makers from most desirable to least wanted. Let and be alternatives. The alternatives are ranked according to the previously stated criteria in Table 2.

A comparison table is produced by defining the quantity of blank cards, or gaps, that exist between two parts. The matrix contains all pairwise comparisons, and as Table 3 shows, it may contain interval or unknown numbers.

Table 3.

Alternative ranking that is performance based.

		4
		3
		1
		2
		5

Verify that the table complies with the rule shown in Table 4 and Table 5.

Table 5.

Consistency table.

Consistency verification

Table 4.

Blank cards based comparison table.

		7	9	13	14
	0	0	4	8	9
	0	0	0	3	4
	0	0	0	0	1
	0	0	0	0	0

From 1 to i-1, k = 1.

Consequently, normalize, as shown in Table 6.

Table 6.

Normalization of the weights value.

Final weights value

Each evaluation’s q-rung fuzzy number is shown in Table 7.

Table 7.

Decision matrix.

	0.2	0.7	0.5	0.1			0.4	0.1
	0.35	0.5	0.46	0.4			0.3	0.14
	0.6	0.2	0.21	0.3			0.5	0.2
	0.7	0.1	0.36	0.2			0.7	0.1
	0.36	0.12	0.1	0.4			0.4	0.3
	0.4	0.1	0.7	0.5	0.6	0.9	0.43	0.8
	0.3	0.14	0.3	0.4	0.8	0.4	0.6	0.7
	0.5	0.2	0.8	0.2	0.9	0.8	0.8	0.5
	0.7	0.1	0.7	0.1	0.7	0.1	0.7	0.1
	0.4	0.3	0.4	0.3	0.4	0.3	0.45	0.3

The Fermatean fuzzy decision matrix represents the information about the membership degree , non-membership degree is represented in Table 8.and the resulting score value is calculated with the help of the equation and shown in Table 8.

Table 8.

Calculated scores.

	−0.34	0.12		0.06	0.06	0.22	−0.51	−0.43
	−0.08	0.03		0.02	0.02	−0.04	0.45	−0.13
	0.21	−0.02		0.12	0.12	0.50	0.22	0.39
	0.34	0.04		0.34	0.34	0.34	0.34	0.34
	0.04	−0.06		0.04	0.04	0.04	0.04	0.06

Step 6. Construct matrix in Table 9, for each criterion

Table 9.

Pairwise comparison matrix.

Step 7. Calculate Local Scores as at Table 10 below.

Table 10.

Find the local scores.

Sum of rows of each criteria

for to for Unweighted: over all and for Weighted:

Step 8. Rank the alternatives in Table 11 in descending order of .

Table 11.

Rank the alternatives.

Ranking of alternatives

Step 9. The rank is presented visually in Figure 2.

Fig. 2.

Ranking of alternatives.

Though the proposed Fermatean Fuzzy Deck-of-Cards (FFS-DoC) framework exhibits high measure-methodological performance in a hypothetical study, there are a number of issues which can be encountered when using it to measure real-life situations with physical education instructors. To begin with, the expert opinions can be different because of the disparity in the experience, teaching philosophy and understanding the criteria of fitness, which can make these judgments more subjective and inconsistent. Second, the teachers might experience the problem of assigning the Fermatean fuzzy membership and non-membership degrees directly, which will require the application of properly designed linguistic scales and guidance procedures. Besides, the Deck-of-Cards technique is cognitively intuitive yet might need a close facilitation to achieve a stable spacing of preferences especially in cases where the criteria are numerous. Practical information can be incomplete or inaccurate too because of the short time of observation or institutional limitations. To effectively solve these problems with the help of professional training, consensus-making methods, and uniform elicitation procedures, the real-world implementation of the proposed framework will not be possible.

Result discussion

The use of the proposed FF-DoC framework generated an understandable and interpretable ranking of the five student groups assessed on eight physical-fitness criteria. The summative dominance scores indicated that was an outstanding performance with the highest score and Rank 1 due to the fact that students with systematic routines and balanced fitness levels always do better than others. scored and was ranked 2 which indicates performance of high quality in most areas that are important like mental wellness, endurance and stability. , and were positioned as the third, fourth and fifth respectively as they were not strong or consistent in various criteria. These findings prove the usefulness of the framework to make a significant difference between options even when expert judgment is uncertain or has contradicting views. The advantage of the proposed solution lies in the ability to deal with ambivalent and wavering judgments using FF modeling, whereas the DoC algorithm provides the derivation of the weights in an understandable and cognitively consistent way. In comparison to classical fuzzy, IF or PF models, which in many cases have difficulty with tie situations or poor discrimination, the FFS-DoC structure is much more expressive and has better stability in the final rankings. The strength of the method was also justified by the consistency of the ranking order in cases of small changes of scores. In sum, the results indicate that the offered FFS-DoC framework provides a more reliable, realistic and humanistic manner of DM and allows educational establishments to define the predominant fitness behavior and establish specific interventions to enhance the health of students. Group Delta is the best ranked company because it has well balanced and good performance on the criteria that were evaluated. According to the description of the criteria, Group Delta showed steady high levels of membership of valuable qualities like physical strength, psychological well-being, and general adaptability, and rather low non-membership and ambivalence rates. Particularly, the group exhibited positive psychological stability and stress-management capacity, both of which are essential in the overall fitness testing. Group Delta had well-distributed performance profile unlike other groups that exhibited well performance in specific criteria and poor performance in the rest of the performance criteria. This balance yielded greatly to its dominance score in the suggested Fermatean fuzzy decision-making model and thus scored the highest in general ranking.

Rank robustness and sensitivity analysis

The sensitivity analysis was used to test the strength of the proposed FF -DoC framework by applying small perturbations (5 percent) to the obtained weights of the criteria. The rankings on top-ranked alternatives remained the same, which means that rank stability is high in case of moderate preference variation. Such action proves that the suggested framework is not too sensitive to the slight changes in expert judgments. The FFDoC approach had greater resilience to rank reversal than the classical fuzzy and intuitionistic fuzzy DoC-based models and showed that it was robust in uncertain decision situations.

Qualitative comparative analysis

In this section we will discuss the qualitative comparative analysis of this study. The qualitative comparative analysis of this study with existing frameworks is shown in Table 12.

Table 12.

Qualitative comparative analysis.

Criterion	Fuzzy-DoC	IFS–DoC	PFS–DoC	FFS–DoC (Proposed)
Models’ explicit hesitation/indeterminacy	×
Expressiveness of membership space (capacity to encode uncertainty)	×
Deep uncertainty handling (high hesitation + conflicting judgments)	×
Robustness with incomplete/patchy expert inputs	×
Weight elicitation transparency (DoC cognitive fit)
Consistency control in weights (via DoC gap checks)
Stability in sensitivity analysis (rank stability under small shocks)	×
Interpretability for non-technical stakeholders
Scalability to many criteria/alternatives
Suitability for health/education (soft factors + psychometrics)
Avoids arbitrary parameter tuning
Discrimination power among close alternatives	×

Numerical comparison of ranking score achieved by various models are given below in Table 13.

Table 13.

Comparison of ranking score.

Method	Best alternative	Dominance/Utility score
Fuzzy–DoC		0.681
IFS–DoC		0.712
PFS–DoC		0.734
FFS–DoC (Proposed)

The quantitative analysis shown in Table 12 is the numerical evidence of the performance improvement that the proposed FF-DoC framework has. Although DoC-based models produce a similar ranking, the Fermatean fuzzy extension gives more dominance to the best ranked alternatives, meaning that more discrimination is revealed amongst student groups that are competitive. Specifically, the offered approach provides the greatest utility of the optimal alternative, showing a higher capacity to reproduce expert hesitation and profound uncertainty. These numerical findings support the qualitative assertions and prove that the given framework can improve traditional fuzzy, IF and PF analogs, which can be measured.

Sensitivity analysis

In this section we will discuss the sensitivity analysis of this study with respect to vary weights The results are shown in Table 14.

Table 14.

Sensitivity analysis.

Methodology (Proposed model)	weight ω	Ranking of alternatives

Implications of study

The results acquired prove that the suggested FF-DoCM framework offers a systematic process of converting subjective assessments by the expert to objective quantitative rankings in the face of uncertainty. FF information enables more modeling of the hesitation and cognitive variability than the conventional assessment methods, which also improves the soundness of the institutional fitness assessment. The sensitivity analysis also shows that the ranking is not dependent on changes in parameters, which implies the credibility of the suggested decision architecture.

In the view of the policy, the FF-DoCM model could help the educational administrators to develop the strategies of the data-based health assessments by defining the key dimensions of performance that should be countered by the institution. The framework can be used by the decision-makers to distribute the training resources, change physical fitness policy, and concentrate on improving programs in accordance with the aggregated expertise instead of the individual judgments. In addition to this, the model also encourages open assessment standards, which may decrease bias and enhance the trust of the stakeholders of academic health monitoring systems. These findings point to the possibility of FF-DoCM as a decision-support instrument that could be used in strategic planning and evidence-based policy development in the educational settings.

Conclusion

The use of the suggested FF-DoCM has valuable applications to the policy-making in schools. The framework, through a systematic combination of professional judgment and uncertainty and hesitation, provides the evaluation of student fitness and wellness profiles to be transparent and fair. This justifies the policy decisions that extend beyond the quantitative measurements of performance and include the psychological and behavioral aspects. Moreover, FF-DoCM may help administrators and policymakers prioritize groups of people and distribute resources more efficiently and create specific intervention programs concerning physical education and mental health. The characteristic of the model of managing subjective and ambiguous information increases the strength of the policy decisions to be more adaptive to the real educational environment in which human judgment is the major role. This paper aimed at creating a dependable and human-focused DM model that would be used in assessing the determinants of physical fitness among college students, a field where the judgment of experts is always unpredictable, inaccurate and biased. The proposed framework designed by combining FFS and the (DoC) technique was able to overcome the drawbacks of the traditional fuzzy and orthopair-based MCDM models. The Fermatean construction was such that it enabled specialists to capture more deeply rooted degrees of uncertainty and ambiguity, whereas DoC approach guaranteed the uniformity and cognitively sensible criteria weighting. The results of the hypothetical case study that involved eight criteria and five groups of students confirmed that the FFS-DoC method provides more consistent, realistic, and interpretable rankings, which provide a better understanding of which fitness characteristics have the most significant impact on the overall physical well-being of students. The benefits of this framework are that it has a better ability to manage uncertainty, enlistment of preferences is transparent, and it has a high degree of flexibility in various evaluation environments. It minimizes bias in professional determinations, validates logical conclusions in decision making and enhances the analytical basis of development of student-centered strategies of improving fitness. Even though the proposed FF-DoCM framework is highly interpretable and robust on medium scale academic environments, there are other future enhancements that can be made to enhance the scalability of the framework to large and diverse educational institutions. One option is that hierarchical or modular FF-DoCM structure can be constructed by clustering fitness criteria into higher-level dimensions (e.g., physical, psychological, and lifestyle factors) so that local weights can be assigned to clusters and aggregated on a global scale. This helps to greatly decrease the cognitive and computational load as the criteria to consider becomes larger. Second, the model can be generalized to the group decision-making setting through the integration of Fermatean fuzzy aggregation operators to integrate judgments made by various expert panels in different departments, campuses, or cultural settings. Third, to scale better and to be empirically grounded, hybrid weighting schemes that combine Deck-of-Cards-based subjective preferences with data-driven objective information (wearable fitness data or institutional health statistics, etc.) will be beneficial. Moreover, efficient deployment of the framework into the large-scale institutional environment would be possible with the help of the algorithmic optimization of the DoC comparison process and software-based decision-support implementations. Lastly, the FF-DoCM model can be utilized in providing education health evaluation to the whole nation or the entire world through the introduction of normalization strategies that enhance cross-institutional benchmarking. In the future, the model may be applied to real-life evaluations based on empirical fitness measures, which can be implemented into digital health monitors, or integrated with optimization and machine-learning algorithms to make predictive analysis. Possible future research is also the implementation of them in other human-related areas like evaluation of mental health, sports talent recognition and analysis of lifestyle behavior. On the whole, the suggested solution will add a sophisticated and applicable decision-support system to academic institutions that are in need of evidence-based and uncertainty-resistant solutions to fitness evaluation.

Author contributions

Liyi Xie conceived the study idea. Yali Huon and XianLiang Wang contributed to the Data Analysis to data investigation and writing.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request

Declarations

Competing interests

The authors declare no competing interests.

Ethics

This study did not involve human participants, data, or tissue.