A soft computing TOPSIS framework with fractional orthopair fuzzy sine aggregation for robust model evaluation in data science

Abstract

Model selection in data science involves evaluating multiple alternatives across conflicting criteria under uncertainty, where existing fuzzy multi criteria group decision making (MCGDM) approaches often fail to capture asymmetric uncertainty and nonlinear interactions in expert judgments. To address this limitation, this study proposes a novel MCGDM framework based on fractional orthopair fuzzy sets (FOFS). The FOFS structure enables flexible and precise modeling of uncertainty by allowing independent fractional control of membership degree (MD) and non-membership degree (NMD). Furthermore, sine trigonometric aggregation operators are introduced to capture nonlinear relationships and fluctuations in expert evaluations. An integrated FOFS–TOPSIS method is then developed to rank candidate models based on their distances from positive ideal solution (PIS) and negative ideal solution (NIS). The applicability of the framework is demonstrated through a numerical study involving fifteen predictive models, fifteen evaluation criteria, and three experts. The results indicate that Alternative achieved the highest overall ranking, followed by , while mid and lower ranked models revealed trade-offs in accuracy, computational efficiency, and robustness. Comparative and sensitivity analyses confirm the framework’s robustness, stability, and superior ranking performance.

Introduction

The rapid advancement of artificial intelligence (AI) and deep learning models (DLMs) has transformed a wide range of application domains, including healthcare, finance, cybersecurity, and intelligent transportation systems^1,2. With the increasing deployment of these models in critical tasks such as medical diagnosis, fraud detection, and autonomous decision-making (DM), selecting the most appropriate model has become a crucial and non-trivial problem^3,4. Recent studies indicate that even minor differences in model performance, computational cost, and robustness can significantly influence system reliability and decision quality. In practice, this selection process involves evaluating multiple and often conflicting criteria, such as accuracy, efficiency, interpretability, scalability, and resilience to noise. Moreover, DM is typically performed by multiple experts including data scientists, engineers, and domain specialists whose assessments may be subjective, imprecise, and inconsistent. This complexity naturally frames the deep learning (DL) model selection problem as a multi criteria group decision making (MCGDM) task. Although classical approaches such as AHP⁵, TOPSIS⁶ and VIKOR⁷ provide structured mechanisms for ranking alternatives, they are not well suited to handle the uncertainty and vagueness inherent in expert driven evaluations⁸. While fuzzy extensions have improved the representation of linguistic information, many existing approaches still lack the flexibility required to capture complex and asymmetric uncertainty patterns^9–11. These limitations motivate the need for more advanced and expressive DM frameworks capable of effectively addressing uncertainty in modern data driven environments.

Despite the widespread use of classical MCGDM and fuzzy based approaches in model evaluation, these traditional frameworks still struggle to effectively capture the ambiguity, hesitation, and conflicting judgments that characterize real-world data science model selection^12–14. In practical environments, experts often provide assessments influenced by model complexity, dataset characteristics, domain requirements, and deployment constraints resulting in heterogeneous and sometimes inconsistent evaluations^15–17. To better represent this uncertainty, researchers have developed several advanced extensions of the classical fuzzy set (FS)¹⁸, including intuitionistic fuzzy sets (IFSs)¹⁹, Pythagorean fuzzy sets (PyFSs)²⁰, and rung orthopair fuzzy sets (ROFSs)²¹ and ,quasi rung orthopair fuzzy set (QOFS)²². These extended models enhance representational flexibility by allowing a wider separation between MD and NMD^23,24. However, they still enforce a rigid structural relationship between the two components, limiting their ability to capture asymmetric uncertainty a common feature in data science model assessments, where positive aspects (accuracy, robustness, scalability) and negative aspects (training time, overfitting risk, computational burden) often carry different degrees of ambiguity^25–27. Consequently, existing orthopair fuzzy models may fail to fully reflect the diverse uncertainty patterns that shape expert preferences in complex model selection scenarios^28,29. Aggregation operators play a fundamental role in MCGDM by combining individual expert evaluations into a unified assessment^30,31. Over the years, numerous operators such as the Hamy mean³², weighted geometric operators³³, power aggregation operators³⁴, and various generalized averaging schemes have been developed to enhance the flexibility and sensitivity of information fusion^35–37. These operators help capture interactions among criteria and model different types of relationships, including compensation, reinforcement, and attenuation among decision attributes^38–40. Despite their wide applicability, most existing aggregation mechanisms rely on linear or monotonic structures and therefore struggle to represent nonlinear or oscillatory variations in expert evaluations patterns that frequently arise in dynamic and uncertain decision environments^41,42. This limitation becomes more pronounced when dealing with orthopair-based fuzzy information, where MD and NMD may exhibit complex, asymmetric, or periodic behavior. Consequently, there remains a need for aggregation operators capable of handling such nonlinear patterns while maintaining compatibility with fractional orthopair fuzzy representations⁴³. In the absence of suitable aggregation mechanisms tailored to FOFSs, decision-makers may be unable to fully benefit from the enhanced expressive capacity offered by fractional orthopair modeling particularly when evaluating complex, high dimensional alternatives. The difficulty of model selection becomes even more pronounced in group MD scenarios, where experts from diverse backgrounds such as DL, machine learning engineering, and system operations often express conflicting preferences. These heterogeneous viewpoints naturally lead to inconsistent, hesitant, or partially conflicting evaluations that traditional crisp or classical fuzzy aggregation approaches cannot effectively capture^44–46. Accordingly, advanced aggregation operators with strong uncertainty-handling capabilities are necessary to fuse these diverse expert judgments into a coherent, interpretable, and balanced decision outcome^47–50.

This gap restricts the ability of existing models to exploit the full discriminatory power of FOFS-based aggregation in high dimensional MCGDM settings. To effectively address these challenges, this study proposes a novel sine empowered FOFS–based TOPSIS framework for advanced MCGDM in data science model selection. The proposed approach integrates the flexible dual parameter structure of FOFSs with newly developed sine trigonometric aggregation operators STFOFWA, STFOFOWA, and STFOFOWG which capture interactions and asymmetric uncertainty more accurately than traditional fuzzy aggregators. By incorporating FOFS theory with the ranking strength of TOPSIS, the proposed method enables more precise quantification of proximity between each data science model and its corresponding PIS and NIS. In this study, a panel of three domain experts evaluates fifteen predictive model alternatives including classical DLMs models, ensemble learners, and deep neural architectures across critical criteria such as accuracy, precision, recall, robustness, training time, scalability, interpretability, overfitting risk, explainability, and maintenance difficulty. The entropy method is adapted to the FOFS environment to compute objective criteria weights, while the proposed sine-based FOFS operators aggregate expert judgments into comprehensive decision matrices. This integration establishes a mathematically rigorous, uncertainty-resilient, and expert-driven evaluation mechanism capable of handling ambiguous, conflicting, and linguistically expressed assessments. The significance of this research lies in its ability to bridge the gap between advanced fuzzy uncertainty modeling and practical model selection requirements in real-world AI environments. As data-driven systems continue to expand in complexity and application scope, the need for reliable, transparent, and uncertainty aware model evaluation frameworks becomes increasingly urgent.

Research gap

Despite significant progress in MCGDM, existing approaches still face critical limitations when applied to data science model selection under uncertainty. Conventional aggregation operators, such as Hamy mean³², weighted geometric operators³³, power aggregation operators³⁴, provide effective computational tools for evaluating alternatives; however, they are primarily designed for precise or mildly uncertain environments and lack the capability to adequately model the complex uncertainty, vagueness, and hesitation inherent in expert evaluations^51,52. From a mathematical perspective, classical fuzzy frameworks impose restrictive structural conditions. For instance, IFS are governed by the constraint is ¹⁹, which limits the expressive capacity of uncertainty representation. PFS extend this to ²⁰, while -ROFS generalize further as, ²¹, which enables fractional control over MD and NMD, offering a significantly more flexible and generalized framework for uncertainty modeling. Despite this theoretical advancement, existing studies have not fully exploited the potential of FOFS⁵³ in practical DM environments. In particular, there is a clear absence of methodologies that integrate FOFS with nonlinear transformation mechanisms, such as sine trigonometric operators, to capture higher order interactions among evaluation criteria. This limitation is critical, as real-world data science problems often involve nonlinear relationships and complex dependencies that cannot be effectively modeled using linear aggregation schemes. Furthermore, most existing MCGDM approaches focus either on fuzzy modeling or aggregation design independently, without integrating them into a unified and robust DM framework. There is a notable lack of studies that simultaneously incorporate FOFS-based representation, sine driven aggregation operators, and TOPSIS-based ranking mechanisms for high dimensional model selection problems. This gap becomes particularly significant in modern data science applications, where multiple experts evaluate numerous machine learning and deep learning (DL) models across diverse and often conflicting criteria. Such environments require decision frameworks capable of handling heterogeneous information, asymmetric uncertainty, and nonlinear interactions, which are beyond the scope of traditional fuzzy approaches. Therefore, there is a strong need for a generalized and flexible MCGDM framework that can: (i) Mathematically capture asymmetric and nonlinear uncertainty using fractional orthopair structures; (ii) Incorporate advanced sine-based aggregation operators for modeling complex interactions; (iii) Enhance ranking accuracy through FOFS-based similarity and distance measures; and (iv) Maintain robustness and interpretability in high dimensional decision environments. To address these limitations, the proposed FOFS–TOPSIS framework introduces a novel integration of fractional orthopair modeling with sine trigonometric aggregation, providing a more powerful and mathematically consistent approach for advanced data science model selection.

Motivation

The swift expansion of AI and DLM across critical sectors has underscored the need for reliable, transparent, and uncertainty-aware model selection frameworks. As organizations increasingly rely on predictive models for automation, risk forecasting, medical diagnosis, financial analytics, and operational DM, the challenge of selecting the most appropriate data science model has grown significantly. This challenge is amplified by the diversity of available algorithms, the multidimensional nature of evaluation criteria, and inherent uncertainty in expert judgment. The motivation for this study stems from several interconnected factors:

1. Multi-criteria nature of data science model evaluation: Model selection involves balancing numerous and often conflicting attributes such as predictive accuracy, robustness, interpretability, scalability, computational efficiency, latency, overfitting risk, and stability. Traditional crisp methods are unable to provide a systematic and uncertainty resilient structure for evaluating such complex criteria simultaneously. A mathematically consistent MCGDM framework is required to ensure balanced and rational DM.

2. Heterogeneous and domain-specific expert opinions: Model evaluation typically requires consultation from data scientists, DL engineers, and system analysts each contributing unique perspectives, priorities, and technical preferences. Consolidating these diverse viewpoints without bias is essential for generating reliable and consensus driven rankings of data science models.

3. Uncertainty, hesitation, and vagueness in expert assessments: Experts frequently express evaluations using linguistic terms such as highly accurate, moderately interpretable, or computationally heavy. These qualitative assessments introduce significant ambiguity, hesitation, and imprecision conditions that crisp methods and even many traditional fuzzy approaches fail to capture adequately. A more flexible uncertainty model is required to represent these diverse linguistic judgments with high fidelity.

4. High dimensionality of modern model selection environments: With numerous predictive models and a wide spectrum of evaluation criteria, the decision process must remain computationally scalable, stable, and adaptable to large datasets. The framework must preserve analytical clarity and ranking consistency even as the number of models and criteria increases.

Existing methods rarely incorporate fractional orthopair fuzzy structures or nonlinear sine-based operators both essential for modeling the complex uncertainty patterns inherent in data science model evaluation. Given these challenges, this study is motivated to develop a FOFS–TOPSIS framework supported by new sine trigonometric aggregation operators and FOFS-based TOPSIS approach. The proposed approach introduces a mathematically rigorous, uncertainty-aware, and expert driven mechanism for ranking data science models under high dimensional and uncertainty rich environments. In this study, three domain experts evaluate predictive model alternatives across critical criteria, ensuring a realistic and comprehensive analysis. By integrating the flexible uncertainty modeling capability of FOFSs with the discriminative ranking power of TOPSIS enhanced through nonlinear sine-based aggregation the proposed framework delivers more accurate, transparent, and resilient decision outcomes. This contributes to improved model selection reliability, optimized DM, and enhanced operational performance across modern AI-driven systems.

Contributions and research objectives

In response to the identified research gaps, particularly the limitations of existing fuzzy MCGDM approaches in handling asymmetric uncertainty and nonlinear interactions, this study develops a novel and uncertainty resilient DM framework for data science model selection. Unlike traditional fuzzy TOPSIS methods based on IFSs, PFSs, or -ROFSs, the proposed approach leverages the enhanced flexibility of FOFS, which enable independent and fractional control of MD and NMD. This feature allows more accurate representation of asymmetric and complex uncertainty patterns commonly observed in expert evaluations. Furthermore, existing aggregation operators are predominantly linear or monotonic in nature, limiting their ability to capture nonlinear or oscillatory variations in decision data. To address this limitation, this study introduces sine trigonometric aggregation operators that provide a more expressive mechanism for modeling complex interactions among criteria and expert judgments. The integration of FOFS with the TOPSIS method is not merely a combination of existing techniques, but a structured enhancement that improves the discrimination power, ranking precision, and robustness of decision outcomes under high uncertainty and multi expert environments. The key contributions of this study are summarized as follows.

1. To develop novel sine trigonometric fractional orthopair fuzzy aggregation operators, such as, STFOFWA, STFOFOWA, and STFOFOWG, capable of capturing nonlinear interactions and asymmetric uncertainty in expert evaluations.

2. To propose an integrated FOFS–TOPSIS framework that extends conventional TOPSIS by incorporating fractional uncertainty modeling and sine-based aggregation, resulting in improved discrimination power and ranking precision.

3. To demonstrate the applicability of the proposed framework through a comprehensive MCGDM problem involving three experts, fifteen predictive model alternatives, and fifteen evaluation criteria, including both benefit and cost attributes.

4. To perform a detailed sensitivity analysis with respect to fractional parameters, validating the robustness, stability, and adaptability of the proposed model under varying uncertainty conditions.

5. To conduct comparative analysis with existing fuzzy TOPSIS approaches, highlighting the superior performance of the proposed method in terms of ranking consistency, interpretability, and decision reliability.

The proposed framework is designed for real-world integration, providing actionable insights for data science practitioners and organizations. By connecting methodological innovation to practical DM, it supports robust, interpretable, and reliable model selection in complex, uncertainty rich environments.

Manuscript structure

The remainder of the manuscript is organized as follows. Section  “Introduction” presents the introduction. Section  “Preliminaries” outlines the necessary preliminaries. Section  “Proposed work” describes the proposed methodology and its associated developments. Section  “An advanced soft computing based multi criteria group decision making framework for data science model selection under uncertainty” formulates the proposed MCGDM problem. Section  “Sensitivity analysis” provides the sensitivity analysis, while section “ Comparative analysis” presents the comparative analysis. Finally, section “Conclusion” offers the conclusion, limitations, and future research directions.

Review of TOPSIS extensions under uncertainty

Over the years, numerous extensions of the TOPSIS method have been developed to address uncertainty in DM environments^54,55. The earliest extension, fuzzy TOPSIS, incorporates fuzzy numbers to represent imprecise information, but it lacks the ability to model hesitation explicitly. To overcome this limitation, intuitionistic fuzzy TOPSIS (IF-TOPSIS)⁵⁶ introduces MD and NMD; however, it is constrained by the condition , which limits its flexibility. Subsequently, Pythagorean fuzzy TOPSIS (PF-TOPSIS)⁵⁷ and rung orthopair fuzzy TOPSIS (ROF-TOPSIS)⁵⁸ were proposed to enhance uncertainty modeling by relaxing these constraints to , respectively. While these approaches improve expressive power, they still rely on fixed or integer-based structures, which restrict fine grained control over uncertainty representation. In addition, neuromorphic TOPSIS introduces truth, indeterminacy, and falsity components, offering a broader representation of uncertainty. However, it often increases computational complexity and may introduce redundancy in practical applications. Hybrid and improved TOPSIS variants have also been developed by combining different aggregation techniques, yet many of these approaches continue to rely on linear or monotonic aggregation operators, limiting their ability to capture nonlinear interactions among criteria. In contrast, the proposed FOFS–TOPSIS framework introduces a fractional orthopair fuzzy structure, allowing continuous control over uncertainty through the parameter. Furthermore, the integration of sine trigonometric aggregation operators enables nonlinear transformation and interaction modeling, which significantly enhances the flexibility, robustness, and discrimination capability of the DM process. Table 1 present the comparison of the proposed work TOPSIS approach with the existing TOPSIS approaches.

Table 1.

Comparison of existing TOPSIS with proposed FOFS–TOPSIS.

Method	Uncertainty model	Constraint	Flexibility	Nonlinear modeling	MD–NMD interaction	Aggregation type	Limitations
Fuzzy TOPSIS	FSs		Low	No	No	Linear	No hesitation modeling
IF-TOPSIS	IFS		Moderate	No	Limited	Linear	Restricted structure
PF-TOPSIS	PFSs		Moderate	No	Limited	Linear	Fixed power
ROF TOPSIS	ROFS		High	Limited	Limited	Linear	Integer-based constraint
Neuromorphic TOPSIS	NEOTFS	Independent	High	Limited	No	Linear	High complexity
Hybrid TOPSIS	Mixed models	Varies	Moderate	Limited	Limited	Mostly linear	Lack of generalization
Proposed FOFS–TOPSIS	FOFSs		Very High	Yes (Sine-based)	Strong (Coupled)	Nonlinear

Preliminaries

This section presents the fundamental concepts and mathematical foundations required for the development of the proposed framework. It introduces the key definitions, properties, and operations related to FOFSs and aggregation mechanisms. These preliminaries provide the necessary theoretical basis for modeling uncertainty, handling expert evaluations, and formulating the proposed DM approach in a consistent and rigorous manner.

Definition 5

⁵³Let be a universal set. A FOFS on can be defined as,

Where , represent the MF and NMF respectively with the condition , where p and q are positive integers such that . For the simplicity we take and the hesitancy of FOF set is given by . This model serves as the core structure of the proposed framework, offering enhanced flexibility through fractional parameters to better capture uncertainty and hesitation in expert evaluations.

Definition 6

⁵³Let and , be any FOFNs and , then

1.

2.

3.

4.

These operational laws define how FOFNs are combined and scaled, forming the basis for aggregation operators used in the proposed DM framework.

Definition 7

⁵³Consider and , be any FOFNs and , and be a positive real number, then the following properties are satisfied,

These properties ensure mathematical consistency, such as commutativity and scalability, which are essential for reliable aggregation DM.

Definition 8

⁵³Let be a FOFN then the score function of can be defined as,

4

The score function provides a ranking mechanism for FOFNs by transforming fuzzy information into a comparable scalar value, which is crucial for DM and alternative evaluation.

Proposed work

In this section, we present the sine trigonometric operational laws (STOLs) for FOFNs. These laws are designed to capture nonlinear relationships among MD and NMD, thereby enhancing the flexibility and expressiveness of the proposed DM framework. Let , the and satisfying the condition:

• (i) and ,
• (ii), for all .

These conditions ensure that the fuzzy information remains valid and bounded, forming the basis for defining new operational laws. Since the sine function is positive and bounded within the first quadrant, it can be effectively utilized to transform fuzzy values while preserving their range. In particular, the following mappings hold:

for all . These transformations ensure that both MD and NMD remain within admissible bounds after applying the sine function, enabling nonlinear modeling of uncertainty. Based on these properties, we introduce a new operator called the Standard Trigonometric Fractional Orthopair Fuzzy Set (STFOFS), defined as follows.

Definition 9

For the QOFS , the STFOFS of can be defined as,

Which is also FOFNs. This operator introduces a sine-based nonlinear transformation that enhances the modeling capability of FOFNs by capturing complex interactions between MD and NMD. It plays a central role in the proposed aggregation process, leading to more flexible and realistic DM outcomes.

Remark 1

It has noticed that and hence lies in the first quadrant only. Furthermore, the term , define the MF which belong to . On way STFOFS the range of sin function also belong to and 1. In the same way , define the NMF it lies between

Definition 10

For a , the number,

is called sine trigonometric fractional fuzzy number (STFFNS). This definition introduces a sine-based transformation of FOFNs, which enhances their ability to model nonlinear uncertainty and provides a foundation for the proposed aggregation operators.

Theorem 1

Consider , then the STFFNS that defined in the Definition 10 is FOFN.

Proof

The desired outcome can be substantiated through evidence showing that,

• (i) and
• (ii) satisfy for QOFN .

, we have and , it implies that , Hence the part is valid. Also condition and , , it implies that Thus, Part 2 is verified, completing the proof. Ensures that the transformed pair also satisfies the FOFN condition. Hence, the result holds. This theorem guarantees that the proposed sine transformation preserves the fundamental structure of FOFNs, ensuring mathematical validity of the model.

Theorem 2

For any two FOFNS and , we have

(i)

(ii)

Proof

This can be easily proof by using Definition 10. This theorem shows that the sine transformation is compatible with aggregation operations, preserving commutativity and ensuring consistency in the DM process.

Theorem 3

For any three FOFNs , and , we have

Proof

This can be easily proof by using Definition 10. This theorem ensures that the proposed transformation preserves associativity, which is essential for aggregating multiple expert evaluations in a consistent manner.

Theorem 4

Let , and be any three FOFNs and and be arbitrary real numbers then the following operational are satisfied,

1.

2.

3.

4.

5.

Proof

The proofs of parts and 3 are presented here; the remaining cases follow by analogous reasoning, Consider , , then from the definition 10, we have,

and

by using the definition of addition laws of FOFNs, we get,

. Hence the part 1 of the is proved.

Now for the , we have,

and

, using the operational laws of FOFNs we get,

, Hence part 3 proved. This theorem confirms that the sine-transformed FOFNs preserve scaling and exponentiation properties, which are crucial for weighted aggregation and parameter tuning in the proposed framework.

Theorem 5

For any two FOFNs and such that . Similarly, , which implies that, .

Hence

Implies that

Hence

This theorem establishes a monotonicity property of the proposed sine transformation, ensuring that the ordering of FOFNs is preserved after transformation. This is essential for maintaining consistent ranking behavior in DM.

Theorem 6

For any FOFNs and we have

Proof

For the and , we take , and

, As we know , , and , Hence we have , which implies that , and thus we gets , such that

, Similarly

, such that,

Hence proved the required results. This theorem shows that the additive aggregation () dominates the multiplicative aggregation (), which is useful for controlling aggregation behavior and ensuring balanced decision outcomes.

Theorem 7

For any positive and FOFNs , , iff , and , iff .

Proof

The proof follows similarly to Theorem 6 by utilizing the properties of FOFNs and monotonic behavior of the sine-based transformation under scalar operations. This theorem describes the effect of scalar parameters on the transformed FOFNs, which is important for weight adjustment and tuning in aggregation operators.

Role and mechanism of sine trigonometric aggregation operators

The sine trigonometric aggregation operators play a central role in the proposed framework by providing a flexible and nonlinear mechanism for combining expert evaluations expressed in the FOFS environment. In real-world DM scenarios, expert judgments are often inconsistent, nonlinear, and subject to fluctuations due to uncertainty and hesitation. Traditional linear aggregation operators are not sufficient to capture such complex patterns. To address this limitation, the proposed operators incorporate sine functions to transform and aggregate input values. Specifically, individual FOFS evaluations provided by multiple experts are first processed through sine-based transformations, which adjust the MD and NMD in a nonlinear manner. These transformed values are then aggregated using weighted schemes to produce a final collective assessment. The use of sine functions introduces several important properties. First, it enables the modeling of nonlinear relationships among criteria and expert opinions. Second, it captures oscillatory and fluctuating behavior in human judgments, which is often observed in uncertain environments. Third, it reduces the dominance of extreme values, leading to more balanced and stable aggregation outcomes. As a result, the sine trigonometric aggregation operators enhance the expressive power of the FOFS framework and improve the reliability, robustness, and interpretability of the final DM results compared to conventional linear aggregation methods.

Development of aggregation operators within the STFOFNS framework

Consider be the collection of FOFNs and be the wights vector such that and . Subsequently, by employing STFOFNS, we formulated several operators as follows.

Definition 10

Consider be the collection of FOFNs. As sine trigonometric fractional orthopair fuzzy weighted averaging (STFOFWA) operators is a mapping STFOFWA: and defined as,

5

Theorem 8

The results produced by the STFOFWA operator yield aggregated values that are again FOFNs, defined as follows,

6

Proof

To prove Theorem 8, we employ mathematical induction with respect to .

Step 1. For , we have.

STFOFWA, from the Definition 10, we have

and

Hence for , Eq. (6) is satisfied.

Step 2. We assume that Eq. (6) is true for , such that:

Step 3. Based on the induction assumption for , we demonstrate that the proposition remains valid for , such that STFOFWA

Using the method of mathematical induction, we conclude that Eq. (6) is valid for every n.

Property 1

If for all i then STFOFWA.

Property 2

For a collection of n FOFNs , , defined and then .

Property 3

For a FOFNs and , if , , then we have

Definition 11

Consider a set of n FOFNs , ,. The sine trigonometric fractional orthopair fuzzy ordered weighted averaging (STFOFOWA) operator is defined as a mapping given by:

7

In Eq. (7), the permutation map, denote as having the condition for all value .

Theorem 9

The results produced by the STFOFOWA operator yield aggregated values that are again FOFNs, defined as follows,

STFOFOWOA

8

Proof

The proof follows similarly to Theorem 8.

Property 4

If for all i then STFOFOWA

Property 5

For a collection of n FOFNs , , defined and then .

Property 6

For a FOFNs and , if , , then we have

Definition 12

Let a set of n FOFNs , . The sine trigonometric fractional orthopair fuzzy ordered weighted geometric (STFOFOWG) operator is defined as a mapping given by:

9

Theorem 10

The results produced by the STFOFOWG operator yield aggregated values that are again FOFNs, defined as follows,

10

Proof

The proof follows similarly to Theorem 8.

Property 7

If for all i then STFOFOWG.

Property 8

For a collection of n FOFNs , , defined and then .

Property 9

For a FOFNs and , if , , then we have

Mathematical distinction from existing trigonometric operators

Existing trigonometric fuzzy operators are generally defined by applying simple transformations such as, , directly to MD values, without considering the structural dependency between MD and NMD. Such formulations do not incorporate orthopair constraints and typically treat MD and NMD independently. In contrast, the proposed sine trigonometric operators are formulated through a multi stage nonlinear transformation within the FOFS framework. In particular, the MD is transformed as, , while the NMD is defined through a nested fractional trigonometric structure: . This formulation ensures that: (i) Both transformed components remain within the unit interval, (ii) The fractional orthopair constraint is preserved, and (iii) A nonlinear dependency between MD and NMD is introduced. Therefore, unlike conventional trigonometric operators, the proposed approach provides a structured, constraint preserving, and interaction aware transformation, which significantly enhances the modeling capability of fuzzy information in complex DM environments.

Classical TOPSIS and its limitations under uncertainty

The TOPSIS is one of the most widely used MCDM methods due to its simplicity and effectiveness. The fundamental principle of TOPSIS is based on the concept that the optimal alternative should have the minimum distance from the PIS and the maximum distance from the NIS. Typically, Euclidean distance measures are used to compute the separation of each alternative from these ideal points, and a closeness coefficient is calculated to determine the final ranking. Despite its effectiveness in deterministic environments, classical TOPSIS suffers from several limitations when applied to real-world DM problems involving uncertainty and imprecision. First, it requires crisp numerical inputs, which makes it difficult to incorporate subjective judgments and linguistic evaluations provided by experts. Second, it does not account for hesitation or ambiguity in DM, as each evaluation must be expressed as a single precise value. Third, classical TOPSIS assumes a symmetric and linear structure of information, which is inadequate for modeling complex relationships between criteria, especially when positive and negative attributes exhibit different uncertainty behaviors. To overcome these limitations, various fuzzy extensions of TOPSIS have been proposed. However, existing approaches based on IFS, PFS, and ROFSs still impose structural constraints that limit their ability to capture nonlinear and asymmetric uncertainty patterns. In this context, the proposed FOFS–TOPSIS framework introduces a fractional orthopair fuzzy structure combined with sine-based aggregation operators, enabling more flexible, expressive, and robust modeling of uncertain decision environments.

Proposed algorithm

Let a decision-making problem (DMP) involving m alternatives ,…, ​ and n criteria , ,…,, assessed by decision-makers. Let and denote the normalized weight information corresponding to the decision-maker and the criteria, respectively. Here, and . Each alternative ​ is evaluated by every decision-maker under each criterion using the parameters of FOFSs. The evaluation information is expressed as , where , and . In summary, the aggregated preference information for alternative can be represented as,

11

To identify the optimal alternative among the available options, the steps of the proposed algorithm are described as follows:

Step 1. For each decision-maker, construct the decision matrix . These matrices compile and organize the assessment data, thereby summarizing the preferences and evaluations of all decision-makers over the given alternatives,

Step 2. For each pair , the individual assessments ​ are combined into a unified evaluation using either the STFOFOWA or STFOFOWG operator. The aggregated evaluations are subsequently organized into the decision matrix , representing the integrated judgments of all decision-makers across the set of alternatives.

Step 3. In MCGDM problem, both benefit and cost criteria are used to evaluate and compare alternatives. Benefit criteria reflect the favorable characteristics of an alternative, while cost criteria quantify its unfavorable or costly aspects. These two perspectives together ensure a balanced assessment of each option’s viability. When normalization is required, each ​ is transformed into according to the rule:

12

Step 4. To determine the importance of each criterion, the classical Shannon entropy method (Shannon, 1948) is extended and adapted to the FOF environment. The enhanced entropy-based procedure for computing the criteria weights is outlined as follows:

13

Step 5. Compute the entropy for each criterion ​ using Eq. (14). Entropy quantifies the degree of uncertainty or randomness associated with a given attribute. By evaluating the entropy of each attribute, one can assess the extent of diversity or uniformity among the available alternatives. This analysis helps identify the criteria that exert the most significant influence on the DM process and determine which alternatives are most likely to yield the desired outcomes.

14

Here, ​ is a normalization constant ensuring that . If the criteria weights​ are not predetermined, they can be derived using Eq. (14):

15

This ensures that criteria with lower entropy receive greater weights in the DM process.

Step 6. For each alternative, the proposed operator is employed to aggregate the normalized evaluations , resulting in the comprehensive aggregated value .

Step 7. The positive and negative ideal solutions are determined using the score and accuracy functions, which act as reference points for assessing the performance of all alternatives. Let . For each criterion the ideal values are obtained as,

16

17

If all score values for a criterion are identical, the accuracy function is applied to provide an additional level of discrimination.

Step 8. The separation of each alternative from the PIS and the NIS is determined using the proposed weighted distance measure . These separations quantify how closely each alternative aligns with the optimal and least desirable outcomes, and are formally defined as follows:

,

Based on these separation measures, the closeness index for each alternative is calculated as:

18

Step 9. To evaluate the relative performance of all alternatives, the separation measure is computed for each option. This value reflects how suitable an alternative is with respect to the decision objective. A smaller denotes a better performing and more desirable alternative. Consequently, the alternative ​ corresponding to the minimum is identified as the optimal solution, as it yields the most favorable result according to the defined evaluation criteria.

Step 10. Compute the score function for each alternative , as defined by Eq. (19):

19

In Eq. (15), with p and q permitted to take any relationship where p and q are positive integers such that . This generality allows the formulation to accommodate diverse modeling conditions.

Step 11. The alternatives ​ are ranked according to their corresponding preference scores, and the alternative with the highest ranking are selected as the most favorable.

Justification for selecting TOPSIS

Among the various MCDM methods, the TOPSIS is selected in this study due to its strong theoretical foundation and suitability for handling multi-criteria problems involving conflicting attributes. The core principle of TOPSIS is based on evaluating alternatives with respect to their distances from the PIS and NIS, which provides a clear and intuitive ranking mechanism. Compared to other MCDM approaches, such as VIKOR, PROMETHEE, ELECTRE, and COPRAS, TOPSIS offers several advantages. VIKOR emphasizes compromise solutions, which may not fully capture the relative performance of all alternatives. PROMETHEE and ELECTRE are based on outranking relations, requiring additional preference thresholds and often leading to increased computational complexity. COPRAS relies on proportional assessment, which may not effectively represent nonlinear relationships among criteria. In contrast, TOPSIS provides a distance based and compensatory framework, which is highly suitable for integration with advanced fuzzy models. In particular, the proposed FOFS–TOPSIS framework benefits from the ability of TOPSIS to incorporate customized distance measures and aggregation operators, allowing seamless integration of fractional orthopair fuzzy information and sine based nonlinear transformations. This compatibility enables more accurate modeling of uncertainty and improves the discrimination power of the ranking process. Therefore, TOPSIS is selected as the underlying ranking mechanism due to its conceptual simplicity, computational efficiency, and strong adaptability to the proposed nonlinear fuzzy framework, making it particularly appropriate for complex DM scenarios.

Stability and transparency of FOFS–TOPSIS

The distance-based mechanism of TOPSIS offers inherent advantages in stability and transparency when combined with the proposed FOFS framework. By computing the closeness of each alternative to PIS and NIS, TOPSIS provides consistent rankings even under high uncertainty. When evaluated with FOFS representations and sine-based aggregation operators, TOPSIS preserves the ordering of alternatives across multiple fractional parameter settings, demonstrating robust performance. Furthermore, the method’s clear mapping from fuzzy evaluations to numerical distances ensures that the DM process remains interpretable, allowing decision-makers to understand the contribution of each criterion and expert evaluation. Comparative observations with other MCDM techniques confirm that TOPSIS maintains superior stability and transparency under FOFS conditions, making it a reliable choice for uncertainty resilient model selection.

Justification for using sine based operators

The selection of sine trigonometric functions in the proposed aggregation framework is motivated by their distinctive mathematical properties and suitability for modeling uncertainty. Unlike linear or monotonic nonlinear functions, the sine function provides a bounded, smooth, and nonlinear transformation within the interval , ensuring that the transformed MD and NMD values remain valid without additional normalization constraints. Furthermore, sine functions exhibit controlled curvature and gradual variation, which enables effective modeling of uncertainty, hesitation, and nonlinear interactions among criteria. In contrast, alternative nonlinear functions such as exponential transformations may introduce excessive sensitivity or distortion, particularly near boundary values. The sine-based transformation maintains stability and proportionality, preserving the relative differences among alternatives while enhancing discrimination capability. Additionally, the inherent structure of sine functions allows better representation of subtle interaction effects between MD and NMD, which is essential in fractional orthopair fuzzy environments. These characteristics make sine-based operators particularly suitable for integration with FOFS, leading to improved robustness, flexibility, and interpretability in MCDM processes.

Linguistic terms

In real-world MCGDM problems, experts often face uncertainty, ambiguity, and incomplete information, particularly when evaluating complex systems such as data science models. Assigning precise numerical values to expert judgments in such scenarios is impractical. Therefore, qualitative linguistic terms such as low, average, high, or excellent are commonly used to express expert opinions intuitively. Within the proposed STFOFS-based MCGDM framework, these linguistic terms are systematically mapped to FOFNs, providing a quantitative representation of subjective evaluations. Each linguistic term is defined by a MD range, representing the degree to which a particular evaluation belongs to that category, and a NMD, representing the degree of hesitation or uncertainty. This approach allows the framework to capture expert hesitancy, vagueness, and dual uncertainty while preserving the nuanced meaning of qualitative assessments. Table 2 summarizes the mapping of linguistic terms to FOFNs used in this study. The term excellent corresponds to a MD range of , reflecting a very high-performance level, while its fractional structure models any hesitation or uncertainty in the expert’s judgment. Similarly, terms such as average and low represent neutral and below average performance, respectively, with corresponding FOFN ranges. This transformation enables consistent, structured, and interpretable aggregation of expert evaluations using the STFOFWA and STFOFOWA operators, forming the foundation for constructing decision matrices and generating reliable rankings of alternatives. By explicitly linking linguistic descriptors to FOFNs, the framework ensures that qualitative expert knowledge is quantitatively represented, bridging the gap between human cognition and computational reasoning and enhancing the transparency, realism, and accuracy of the DM process.

Table 2.

Linguistic term and its key feature for the decision-making process.

Performance category	Linguistic term	Notation	Corresponding FOFNs	Remarks
Low performance zone	Extremely Low			Minimal capability, nearly unacceptable
	Very Low			Very poor quality or weak contribution
	Low	L		Clearly below desired level
	Slightly Low			Slight shortfall, marginal performance
	Below Average			Slightly under normal expectations
Neutral zone	Average	A		Neutral or baseline; acceptable performance
	Moderate	M		Balanced state; consistent but unremarkable
	Fair	F		Slightly above average; modest quality
High performance zone	Slightly High			Mildly better than average
	Above Average			Reliable and generally good performance
	Good	G		Solid and positive results
	High	H		Strong, dependable, and effective
	Very High			Superior quality or consistency
	Excellent	E		Exceptional; very high satisfaction
	Extremely High			Ideal or near perfect outcome

The Fig. 1 illustrates the overall architecture of the proposed FOFS-MCGDM framework developed for the evaluation and selection of deep learning models. The framework integrates inputs from multiple experts, each providing FOFS-based assessments for a set of benefit and cost criteria related to deep learning performance. Expert judgments are first aggregated using sine-based fractional orthopair fuzzy operators, while objective criterion weights are computed through the entropy method. These weighted evaluations form a comprehensive decision matrix that captures the uncertainty, variability, and interaction among criteria. The aggregated information is then mapped across all alternatives, and the FOFS-TOPSIS procedure identifies the model that exhibits the highest closeness to the ideal solution. As shown, the process systematically transforms expert input and criterion importance into a robust and mathematically grounded optimal model choice, ensuring reliable and explainable DM in complex AI model evaluation scenarios.

Fig. 1.

Systematic over view of the proposed MCGDM problem.

Justification for choosing fractional orthopair fuzzy set

FOFS are particularly suitable for the present study due to their ability to handle complex and asymmetric uncertainty in data science model selection. Traditional fuzzy frameworks, such as IFS, q- ROFS and -QROFS, are limited in representing fractional uncertainty and nuanced expert hesitation. For instance, q-ROFS and -QROFS extend this range but remain largely symmetric and are insufficient to capture the full spectrum of uncertainty present in high dimensional, multi expert evaluations. In contrast, FOFS allows independent fractional control of MD and NMD, enabling the precise modeling of asymmetric, vague, and hesitant expert judgments. This flexibility is critical when integrating opinions from multiple experts with diverse perspectives and conflicting evaluations. Moreover, FOFS supports robust aggregation and ranking, providing more reliable and discriminative decision outcomes in practical applications. By accommodating fractional uncertainty and enhancing expressiveness, FOFS directly contributes to improved model selection workflows, ensuring that the final rankings reflect both the magnitude and the confidence of expert assessments under uncertainty. In this study, the FOFS framework is integrated with sine trigonometric aggregation operators and the TOPSIS method to systematically evaluate fifteen predictive models across fifteen criteria. Expert evaluations are weighted and aggregated to determine robust rankings, demonstrating the practical applicability of FOFS in real-world data science DM workflows, where uncertainty, multiple criteria, and expert disagreement are inherent.

Justification of case study selection

The selected case study represents a challenging MCGDM problem, where multiple alternatives must be evaluated under uncertainty and conflicting expert opinions. This scenario was chosen intentionally because it reflects real-world decision environments in which our proposed FOFS-based MCGDM framework excels. Specifically, the case study allows for the demonstration of the framework’s ability to handle expert hesitation, fuzzy information, and non-linear interactions among criteria. By applying the methodology to this problem, we not only illustrate the practical relevance of the approach but also highlight its robustness, stability, and superior performance compared to traditional DM techniques.

An advanced soft computing based multi criteria group decision making framework for data science model selection under uncertainty

In the rapidly evolving landscape of AI and data driven DM, selecting an appropriate predictive model has become a crucial yet complex task. The growing number of machine learning (ML) and deep learning (DL) algorithms, each characterized by distinct strengths, limitations, and computational requirements, poses a major challenge for practitioners and researchers. Hence, a systematic and intelligent DM framework is essential to evaluate and rank potential models effectively across multiple, often conflicting, criteria. This study applies the proposed STFOFS based MCGDM framework to identify the most suitable predictive model for data science applications. The framework incorporates both quantitative and qualitative evaluation parameters, capturing uncertainty and vagueness in expert judgments through fractional orthopair fuzzy information. A panel of three experts (data scientist), (ML engineer), and (domain analyst), collaboratively assess predictive model alternatives, including Linear Regression, Logistic Regression, Decision Tree, Random Forest, Gradient Boosting, LightGBM, CatBoost, Support Vector Machine, K-Nearest Neighbors, Naïve Bayes, Deep Neural Network, Convolutional Neural Network, Recurrent Neural Network, Autoencoder, and Bayesian Network. The evaluation process is based on criteria: Accuracy, Precision, Recall, Score, Training Time, Computational Complexity, Scalability, Overfitting Risk, Interpretability, Robustness, Hyperparameter Sensitivity, Reproducibility, Maintenance Difficulty, Explainability, and Real Time Performance.

Expert contribution and integration within the proposed MCGDM framework

To ensure the reliability and comprehensiveness of the evaluation process, this study incorporates the expertise of three domain professionals with diverse yet complementary specializations. Each expert contributes unique insights aligned with their professional background, ensuring a balance between algorithmic precision, computational feasibility, and real-world applicability. Qualitative and quantitative judgments are aggregated using the STFOFWA operator, synthesizing multiple expert opinions into a coherent decision outcome. This integration enhances the robustness, transparency, and interpretability of the proposed framework, leading to a rational and well-founded selection of the optimal data science model.

Methodology for the expert’s weights

In group DM, experts contribute different perspectives and experience. We need experts’ weights , to reflect each expert’s relative influence when aggregating judgments. The AHP derives these weights from pairwise comparisons using Saaty’s scale, and checks consistency of the judgments⁵⁹. In the proposed MCGDM problem we three experts, , and . The Saaty is scale from 1 to 9: equal importance; moderate; strong; very strong; 9 = extreme (2,4,6,8 are intermediates).

Step 1. Build the pairwise comparison matrix. Let compare and :

Step 2. Derive the weight vector . Using the AHP principal eigenvector method such as,

Step 3. Normalization decision matrix

Step 4. Consistency check. Find the then,

Using the above steps, we get the expert weights that is . The weights of the experts and key features are presented in Table 3.

Table 3.

Description of experts involved in the proposed MCGDM framework.

Expert	Role	Area of expertise	Experience	Assigned weighted
	Data Scientist	Model accuracy, interpretability, and optimization
	Machine Learning Engineer	Training efficiency, computational complexity, and scalability	8
	Domain Analyst	Real-time deployment, integration, and maintainability	7

Remark 1

While classical AHP is applied for expert weighting in this study, the proposed FOFS–TOPSIS framework is fully compatible with fuzzy or fractional expert weights. If expert evaluations themselves are uncertain, their weights can be modeled using FOFS extensions, ensuring methodological consistency and maintaining the uncertainty resilient nature of the DM framework. This design provides flexibility for practitioners to adopt crisp or fuzzy weighting schemes without altering the core aggregation and ranking process.

Comprehensive description of the alternatives

The 15 alternatives represent a diverse range of classical and modern algorithms frequently applied in predictive analytics and machine learning tasks. Each alternative was analyzed across 15 evaluation criteria, covering both technical performance metrics and operational aspects. This structured comparison ensures a fair and comprehensive evaluation within the fuzzy multi expert DM environment. Summarized descriptions of the alternatives are presented in Table 4.

Table 4.

Summary of data science model alternatives considered in the STFOFS–MCGDM framework.

Model name	Model type	Learning nature	Interpretability	Computational efficiency	Key characteristics
Linear Regression	Statistical	Supervised	High	Very High	Simple, interpretable baseline for regression tasks
Logistic Regression	Statistical	Supervised	High	High	Effective for binary classification; interpretable coefficients
Decision Tree	Non-linear	Supervised	High	Medium	Rule-based model with strong interpretability
Random Forest	Ensemble	Supervised	Medium	Medium	Robust against overfitting; strong generalization ability
Gradient Boosting	Ensemble	Supervised	Medium	Medium	High accuracy and flexibility with parameter tuning
LightGBM	Ensemble	Supervised	Medium	Very High	Efficient and scalable for large datasets
CatBoost	Ensemble	Supervised	Medium	High	Handles categorical variables effectively; minimal preprocessing
Support Vector Machine	Non-linear	Supervised	Medium	Medium	Performs well with high dimensional data
K-Nearest Neighbors	Instance-based	Supervised	Medium	Low	Simple algorithm but computationally expensive for large datasets
Naïve Bayes	Probabilistic	Supervised	High	High	Efficient with small datasets; assumes feature independence
Deep Neural Network	Neural	Supervised	Low	Low	Powerful feature extraction and complex pattern recognition
Convolutional Neural Network	Neural	Supervised	Low	Medium	Ideal for structured data such as images and signals
Recurrent Neural Network	Neural	Supervised	Low	Medium	Effective for sequential and time- series data
Autoencoder	Neural	Unsupervised	Low	Medium	Used for dimensionality reduction and anomaly detection
Bayesian Network	Probabilistic	Supervised	High	Medium	Models probabilistic dependencies; suitable for causal inference

Scalability and computational efficiency

Although the current study focuses on fifteen models, fifteen criteria, and three experts, the proposed STFOFS TOPSIS framework is designed to scale efficiently to larger and more complex DM problems. Operations within the FOFS environment and the sine trigonometric aggregation operators are computationally efficient and can be vectorized for high dimensional datasets. The TOPSIS ranking procedure remains effective even with a substantial increase in the number of alternatives, criteria, or experts. Preliminary experiments on larger synthetic datasets demonstrate that the framework maintains stable and consistent rankings while preserving computational feasibility, confirming its applicability to real-world, large scale data science model selection scenarios.

Criteria for evaluation

In the context of modern AI applications, selecting an optimal predictive model requires a balanced consideration of multiple technical, operational, and interpretability related factors. With the rapid development of diverse algorithms ranging from classical statistical models to advanced DL frameworks, decision makers must evaluate alternatives across numerous conflicting performance dimensions. Therefore, a comprehensive and systematic evaluation framework is necessary to ensure fairness, transparency, and reliability in model selection. In this study, critical evaluation criteria were identified through expert consultation and literature analysis to assess the effectiveness and adaptability of data science models. These criteria are divided into two categories: benefit criteria (BC), where higher values indicate better performance, and cost criteria (CC), where lower values are preferred. The selected criteria cover both quantitative aspects (such as accuracy, training time, and robustness) and qualitative dimensions (such as interpretability and explainability), ensuring that both technical efficiency and practical usability are captured within the DM process. Although some criteria are inherently qualitative, such as interpretability () and explainability.

(), they are converted into FOFNs to effectively handle expert uncertainty and hesitation in linguistic assessments. This fuzzification allows qualitative judgments to be consistently aggregated, compared, and analyzed alongside quantitative criteria within the proposed MCGDM framework. By adopting this approach, the framework preserves the intrinsic ambiguity of human evaluations while enabling systematic mathematical treatment, ultimately facilitating more robust and reliable model selection.

Criteria identification and expert selection

The evaluation criteria considered in this study were identified through a comprehensive literature review of recent studies on data science model evaluation and MCDM frameworks. Relevant criteria frequently used in prior research were initially collected and subsequently refined to eliminate redundancy and ensure relevance to the problem context. To enhance the practical validity of the selected criteria, a panel of domain experts was consulted. The experts were selected based on their academic background, research experience, and professional involvement in data science, ML, and decision support systems. Each expert possesses substantial experience in evaluating predictive models and contributed to validating, refining, and confirming the final set of criteria. This combined approach ensures that the criteria are both theoretically justified and practically meaningful, thereby strengthening the reliability of the proposed DM framework (Table 5).

Table 5.

Enhanced evaluation criteria for data science model selection using the proposed MCGDM framework.

Code	Criterion name	Evaluation nature	Description	Expected impact	Type
	Accuracy	Quantitative	Measures overall correctness of predictions on test data	High accuracy leads to reliable model selection
	Precision	Quantitative	Ratio of true positives among all predicted positives	Prevents overestimation of positive outcomes
	Recall	Quantitative	Ability to capture true positive cases in datasets	Ensures model sensitivity in critical tasks
	Score	Quantitative	Balances precision and recall for robust evaluation	Provides a unified view of model efficiency
	Training Time	Computational	Measures computational time needed for training	Impacts scalability and cost-efficiency
	Computational Complexity	Computational	Reflects resource intensity during execution	Affects deployment feasibility in large systems
	Scalability	Technical	Assesses capability to handle large datasets	Critical for big data and cloud-based systems
	Overfitting Risk	Statistical	Indicates the model’s ability to generalize to unseen data	Reduces model bias and improves reliability
	Interpretability	Qualitative	Measures how easily model decisions can be understood	Builds trust and regulatory compliance
	Robustness	Quantitative	Evaluates stability under noise and data perturbations	Enhances reliability in real-world environments
	Hyperparameter Sensitivity	Technical	Examines dependence on fine-tuned parameters	High sensitivity reduces reproducibility
	Reproducibility	Experimental	Ensures consistency of results across different runs	Essential for scientific transparency
	Maintenance Difficulty	Operational	Evaluates update and integration effort	Affects lifecycle management of AI systems
	Explainability	Qualitative	Determines how well results can be explained to users	Enables ethical and interpretable AI decisions
	Real-Time Performance	Technical	Measures efficiency and response speed in live systems	Essential for real-time analytics and automation

The assessments provided by the three experts are summarized in Tables 6 and 7, and 8.

Table 6.

The decision matrix provided by expert ​ in terms of linguistic variables.

Table 7.

The decision matrix provided by expert ​ in terms of linguistic variables.

Table 8.

The decision matrix provided by expert ​ in terms of linguistic variables.

The Fig. 2 presents the overall workflow of the proposed FOFS-based MCGDM framework designed for the evaluation and selection of deep learning models. The process begins with defining the DM problem and collecting assessments from multiple experts, each providing an FOFS-based decision matrix. These individual evaluations are integrated using a sine-based aggregation operator to construct the aggregated decision matrix. Criterion weights are then computed objectively through the entropy method, followed by normalization and weighted aggregation to obtain the refined decision matrix. Using the FOFS-TOPSIS procedure, the framework determines the PIS and NIS, computes distances, and evaluates the closeness index for each model. Finally, the model with the highest closeness value is selected as the optimal deep learning alternative. This systematic structure ensures transparency, robustness, and accurate handling of uncertainty throughout the model selection process.

Fig. 2.

Flowchart of the proposed MCGDM problem.

Selection of the parameters and

In the proposed STFOFS-based MCGDM framework, the parameters p and q act as fundamental control exponents that regulate the relative influence and nonlinearity of the MF and NMF. These parameters play a decisive role in defining the degree of flexibility, sensitivity, and interpretability of the model, directly influencing how expert evaluations are aggregated and how subtle differences among alternatives are captured. The general condition ensures logical consistency within the fractional orthopair fuzzy structure, preserving the mathematical integrity of the model while providing adaptability to various uncertainty levels encountered in real-world decision scenarios.

1. When p and q assume smaller values (e.g., 1 or 2), the aggregation process becomes flatter, reducing sensitivity to small differences in expert opinions and potentially concealing important but fine-grained variations in model performance. In contrast, larger exponents amplify these differences, thereby improving discrimination between alternatives but at the cost of magnifying minor noise or measurement uncertainty. For the current study, which involves evaluating data science models under multiple conflicting criteria, the parameters are set to and to achieve a balanced and stable response. This moderate configuration provides sufficient sensitivity to highlight meaningful variations among models such as differences in accuracy, scalability, and interpretability without introducing instability or bias in the ranking process.

2. Moreover, maintaining equality between p and q ensures that the aggregation mechanism remains neutral, giving equal importance to both MF and NMF.

3. If , the model becomes optimistic, favoring alternatives with higher MF; conversely, if , it becomes conservative, overemphasizing uncertainty.

4. By keeping , the proposed STFOFS–MCGDM model achieves symmetry, fairness, and unbiased evaluation, which are essential for robust decision analysis in complex data-driven environments. Additionally, adopting and , aligns the proposed framework with the standard practices observed in related fuzzy modeling studies, including ROFS²¹, p,QOFS²².

These prior works have consistently demonstrated that this parameter configuration offers the optimal balance between sensitivity, interpretability, and computational stability. Beyond maintaining methodological consistency, it also facilitates comparative validation with existing fuzzy models and provides a solid foundation for future sensitivity experiments, where varying p and q independently can help analyze the influence of optimism and conservatism in the decision process. Hence, the selection of and under the condition establishes a neutral, literature consistent, and analytically stable baseline, enabling precise and interpretable rankings of data science models within the proposed STFOFS–MCGDM framework. The aggregated decision matrix is summarized in Tables 9, 10 and 11. The values od and ,

Table 9.

Aggregated decision matrix for the criteria , , , and .

Table 10.

Aggregated decision matrix for the criteria , , , and .

Table 11.

Aggregated decision matrix for the criteria , , , and .

STFOFOWG

We apply the above geometric aggregation operators to construct the aggregated decision matrix, using the parameter values of and and experts weights , , .

For the membership degree:

For the non-membership degree:

Therefore, we have . Similarly, the remaining components can be determined using the same procedure.

To ensure consistent interpretation of performance across all attributes, the aggregated FOFS evaluations were normalized using Eq. (17). This normalization distinguishes between benefit criteria, where higher values indicate better performance, and cost criteria, where lower values are more desirable. In the proposed DLMs evaluation framework, criteria , , , and , are identified as cost type attributes, while all other criteria function as benefit attributes. Following the prescribed normalization rule, benefit criteria retain their original MD and NMD, whereas cost criteria undergo MD-NMD reversal. This ensures that the normalized decision matrix expresses all criteria in a directionally consistent manner, forming a unified basis for weighted aggregation and TOPSIS-based ranking. The normalized decision matrix is presented in Tables 12, 13 and 14.

Table 12.

Normalized decision matrix for the criteria , , , and .

Table 13.

Normalized decision matrix for the criteria , , , and .

Table 14.

Normalized decision matrix for for the criteria , , , and .

The above formula is employed for normalization. Since is a cost criterion, it is normalized accordingly, yielding the normalized value .

In MCGDM problems, determining the relative importance of evaluation criteria is a crucial step for achieving objective and balanced decision outcomes. Since expert judgments and data distributions often vary in uncertainty and informational richness, the entropy method provides a robust, data driven mechanism to compute criterion weights without relying solely on subjective inputs. It evaluates how much information each criterion contributes to differentiating among alternatives by quantifying key indicators weight . The of each criterion is obtained by normalizing the diversity values across all criteria. This ensures that the total importance is proportionally distributed based on their informational contribution, satisfying the condition, .These weights form the foundation of the objective weighting process in the proposed STFOFS–MCGDM framework. They ensure that criteria with higher discriminative capacity, such as computational complexity or overfitting risk, are assigned higher importance, while those with more uniform evaluations receive lower weights. This approach enhances transparency, minimizes bias, and aligns the decision model with the inherent data variability, leading to more reliable and scientifically grounded criterion prioritization. The Eq. (15), can be used to find the weights of criteria, the results is presented in Table 15.

Table 15.

Criteria and its corresponding weights.

Criteria	Weights	Criteria	Weights	Criteria	Weights

The criteria weights are computed using the aforementioned formula. The step by step procedure is presented as follows.

For the alternative 1:

In the way we can find the following parts,

In the proposed FOFS-based MCGDM framework, the objective criterion weights and the aggregated decision matrix are jointly employed to generate the weighted aggregated decision matrix, as shown in Tables 16, 17 and 18. This step is accomplished through the sine-powered STFOFOWA aggregation operator, which seamlessly incorporates the relative significance of each evaluation criterion into the collective expert assessments. By embedding the entropy derived weights within the FOFS aggregation structure, the operator ensures that criteria with greater discriminative capability exert stronger influence on the final evaluations, while still preserving the underlying uncertainty and fractional orthopair structure. This weighted transformation provides a more balanced, realistic, and interpretable foundation for subsequent ranking and distance-based analysis in the deep learning model selection process.

Table 16.

Weighted aggregated for the criteria , , , and .

Table 17.

Weighted aggregated for the criteria , , , and .

Table 18.

Weighted aggregated for the criteria , , , and .

STFOFOWG

The above geometric aggregation operators are employed to compute the weighted aggregated decision matrix, using and , and the criteria weights provided in Table 15.

For the membership degree:

For the non-membership degree:

Therefore, we have

In the proposed FOFS-based MCGDM framework, the PIS and NIS are constructed from the weighted aggregated decision matrix to represent the best and worst achievable performance levels for each criterion. Using the Eqs. (16) and  (17), to find the PIS, the PIS captures the maximum desirable evaluations, while the NIS reflects the minimum values across all alternatives. The PIS and NIS are presented in Table 19.

Table 19.

Positive and negative ideal solutions.

Criteria

The aforementioned formula is applied to determine the PIS.

Similarly, the remaining parts of the PIS are determined following the same procedure.

The aforementioned formula is applied to determine the NIS.

Similarly, the remaining parts of the NIS are determined following the same procedure.

In the proposed FOFS-based MCGDM framework, after identifying the PIS and NIS, the separation distances of each alternative from these ideal references are computed using the weighted FOFS distance measures defined in Eqs. (18) and  (19). These distance calculations quantify how far each deep learning model deviates from the optimal PIS and how close it is to the least desirable NIS performance patterns. The resulting distances, which form the basis for closeness coefficient computation, are summarized in Table 20.

Table 20.

Distance measure between Positive and Negative Ideal solutions.

Alternatives				Ranking

The following formula is employed to calculate the distances from the PIS and NIS:

. The remaining parts can be show in the same way.

. The remaining parts can be show in the same way.

Now the closeness index:

. The remaining parts can be show in the same way.

Table 20 presents the distance measures between the PIS and NIS for the evaluated alternatives. quantifies how far a model’s performance deviates from the ideal benchmark across all criteria, whereas measures proximity to the worst-case performance spectrum. These two distances were fused into the closeness coefficient representing the overall suitability of each model for the target task. The results clearly indicate that achieves the highest ranking, with the smallest distance to the PIS and the largest distance from the NIS , yielding a dominant closeness coefficient of . This confirms that exhibits the most favorable balance across key criteria such as accuracy, generalization ability, computational efficiency, robustness, and scalability, making it the optimal model under the proposed framework. follows as the second-best option , demonstrating strong proximity to the ideal solution despite slightly higher divergence compared to . Models , , and occupy ranks , with competitive performance reflected by their relatively low , and high values. Mid-range alternatives such as , ,, , , and display moderate closeness coefficients, indicating reasonable but not optimal performance. Lower-ranked models, and , with values of and , show substantial deficiencies, likely due to weaker training stability, overfitting resistance, inference speed, or reliability, rendering them less suitable relative to other alternatives.

Practical implications

The proposed FOFS–based MCGDM framework offers substantial utility for practitioners and industry applications. By systematically capturing expert judgments under uncertainty, the framework allows data scientists, ML engineers, and domain analysts to evaluate predictive models across multiple dimensions, including technical performance (accuracy, robustness, computational efficiency) and operational feasibility (scalability, interpretability, deployment readiness). In real-world settings, such as AI driven analytics, financial forecasting, and healthcare predictive systems, this approach supports informed and robust DM, reducing risks associated with model overfitting, operational inefficiency, and computational bottlenecks. Consequently, the framework not only identifies theoretically optimal models but also ensures that selected models are actionable, reliable, and suitable for practical implementation.

The Fig. 3 presents the final ranking of the deep learning models based on the closeness coefficients computed using the proposed FOFS-based MCGDM framework. The results clearly show that is the best-performing model, followed by and , while alternatives such as and rank lower due to weaker overall performance. This visualization provides a quick and intuitive comparison of all models, confirming the effectiveness of the proposed approach in distinguishing high-quality deep learning solutions.

Fig. 3.

Final ranking of deep learning model alternatives using the proposed work.

Sensitivity analysis

In the proposed FOFS-based MCGDM framework for evaluating deep learning models, the parameters and play a central role in regulating the behavior, stability, and robustness of the DM process. Within the STFOF-based aggregation structure, these parameters determine how MD and NMD information is fused, thereby controlling the degree to which the framework amplifies or suppresses variability in expert assessments. This flexibility is particularly important in the domain of deep learning model evaluation, where uncertainty, imprecision, and conflicting expert judgments frequently arise due to differences in model architectures, dataset characteristics, training procedures, and performance metrics. By adjusting and , the decision-maker can tune the sensitivity level of the aggregation operator, ensuring that the FOFS-based model appropriately reflects real world variation in expert evaluations. Higher values of and typically yield a more conservative decision behavior, reducing the influence of extreme assessments and making the ranking process more stable under noisy or highly uncertain conditions an important feature when experts disagree on model performance or when the metrics exhibit stochastic variability. Conversely, lower values of and enhance discriminative power, enabling the framework to highlight subtle performance differences between deep learning models, which is desirable when evaluating architectures with comparable accuracy but different computational costs, robustness levels, or resource requirements.

In sensitivity analysis, the parameters and therefore act as stability regulators. They govern how sensitively the closeness coefficient responds to changes in the weighted decision matrix, expert preferences, or criterion weights. This is particularly relevant for model evaluation criteria such as training stability, inference speed, robustness to noise, generalization capability, and scalability attributes that inherently introduce uncertainty into the assessment process. By adjusting and , the decision framework ensures that small perturbations in these measurements do not disproportionately distort the final ranking. Furthermore, the tunable nature of and also allows the FOFS–MCGDM framework to incorporate different risk attitudes. For instance, risk-averse decision-makers (institutions prioritizing reliability over speed) can adopt higher parameter values to smooth out variations, whereas risk seeking or performance driven environments may prefer lower values to maximize differentiation and model competitiveness. The effects of varying the parameter on the ranking outcomes are summarized in Table 20, while the influence of adjusting the parameter is presented in Table 21. Throughout the analysis, each parameter pair is expressed as, where the first component corresponds to the MD scaling and the second to the NMD scaling in the STFOF aggregation. This systematic sensitivity assessment demonstrates the robustness of the proposed FOFS-based MCGDM methodology and confirms that the ranking of deep learning models remains highly stable across a broad range of parameter settings.

Table 21.

The influence of the parameter while fixed .

The Fig. 4 illustrates the sensitivity behavior of the proposed FOFS-based MCGDM model under varying values of the parameter while keeping , fixed. Each contour band represents the closeness coefficient range of the evaluated deep learning models across, . This visualization highlights how changes in the parameter which controls the non-linear amplification within the sine-based FOFS aggregation affect the stability and ranking consistency of the alternatives. Models such as, , and remain consistently dominant across all values, demonstrating strong robustness, whereas alternatives like and ​ show high sensitivity with notable fluctuations. The contour representation provides a clear and intuitive understanding of parameter influence, confirming that the proposed method maintains stable decision behavior even under wide parametric variation.

Fig. 4.

Sensitivity of the proposed FOFS-based MCGDM model to variations in parameter with fixed .

In sensitivity analysis, the parameters and therefore act as stability regulators. They govern how sensitively the closeness coefficient responds to changes in the weighted decision matrix, expert preferences, or criterion weights. This is particularly relevant for model evaluation criteria such as training stability, inference speed, robustness to noise, generalization capability, and scalability attributes that inherently introduce uncertainty into the assessment process. By adjusting and , the decision framework ensures that small perturbations in these measurements do not disproportionately distort the final ranking. Furthermore, the tunable nature of and also allows the FOFS–MCGDM framework to incorporate different risk attitudes. For instance, risk-averse decision-makers (institutions prioritizing reliability over speed) can adopt higher parameter values to smooth out variations, whereas risk seeking or performance driven environments may prefer lower values to maximize differentiation and model competitiveness. The effects of varying the parameter on the ranking outcomes are summarized in Table 20, while the influence of adjusting the parameter is presented in Table 22. Throughout the analysis, each parameter pair is expressed as, where the first component corresponds to the MD scaling and the second to the NMD scaling in the STFOF aggregation. This systematic sensitivity assessment demonstrates the robustness of the proposed FOFS-based MCGDM methodology and confirms that the ranking of deep learning models remains highly stable across a broad range of parameter settings.

Table 22.

The influence of the parameter while fixed .

The second sensitivity analysis Fig. 5 illustrates how variations in the parameter , while keeping , influence the closeness coefficients of all deep learning model alternatives. Since the parameter governs the penalization strength of NMD information within the FOFS-based aggregation process, its variation provides insight into how sensitive the decision outcomes are to increasing levels of uncertainty, hesitation, or conflicting evaluations. As increases from to , the contour patterns show how stable or fluctuating each alternative’s performance remains across different uncertainty-handling intensities. This visualization helps validate the robustness of the proposed MCGDM model by verifying that top ranked alternatives remain consistent under parameter perturbations, ensuring reliability in deep learning model evaluation and selection.

Fig. 5.

Sensitivity of the proposed FOFS-based MCGDM model to variations in parameter with fixed .

Sensitivity analysis of expert weights

In MCGDM problems, the relative weights of experts can influence the aggregation of evaluations. To ensure the robustness of the proposed fractional orthopair fuzzy MCGDM framework, a sensitivity analysis was conducted by varying the weights of each expert within a realistic range around their baseline values. The baseline weights were: : , : and : . For the analysis, the weights of and were varied slightly within of their baseline values such as, : , : , while remained the dominant expert to reflect practical DM scenarios. The total weights were normalized to for each variation. The sensitivity analysis of the expert’s weights is presented in Table 23.

Table 23.

Sensitivity analysis of rankings under realistic expert weight variation.

Scenario				Ranked alternative	Notes
Baseline					Baseline ranking
					Minor variation in and
					Minor variation in and
					Balanced small variations

Interpretation

The sensitivity analysis shows that the top ranked alternative () remains unchanged across all realistic variations of expert weights. Extreme weight shifts, where or would surpass , are practically unlikely and therefore not considered. These results confirm that the proposed FOFSA–TOPSIS framework provides stable and robust rankings, reinforcing its reliability for real-world DM scenarios. This analysis demonstrates that minor fluctuations in expert influence do not affect the final outcome, highlighting the practical robustness of the proposed approach in handling subjective expert judgments.

Comparative analysis

In the proposed FOFS TOPSIS–based MCGDM framework, enhanced with sine-based aggregation operators to capture nonlinear interactions among evaluation attributes, the determination of criterion importance is a crucial foundation for reliable deep learning model assessment. To objectively quantify the influence of each criterion, the entropy diversity weight method is employed and adapted into the FOF and sine aggregation environment. Table 24 summarizes the three core parameters entropy, diversity, and weights that collectively shape the weighting mechanism used in the subsequent sine-driven aggregation phases of the model. The entropy column measures the degree of uncertainty or uniformity across the performance evaluations submitted by multiple experts for each criterion. In the context of deep learning model assessment, a high entropy value indicates that all models perform similarly on that criterion (comparable training time or accuracy stability), making it less informative for distinguishing model quality. Conversely, lower entropy values represent criteria where models show highly variable behaviors (robustness under noise, generalization across datasets), which provide richer discriminatory information. The diversity column , translates entropy into a direct measure of a criterion’s discriminative capacity. Higher diversity values identify criteria that highlight significant differences among competing deep learning models, making them especially important for DM. For instance, high-diversity criteria such as, , , and , may correspond to computational complexity, training stability, or failure-risk metrics factors that deeply influence practical deployment decisions. The weights column, obtained by normalizing diversity values, establishes the relative importance of each criterion in the overall MCGDM process. These weights directly govern the behavior of the proposed sine-based aggregation operators (STFOFWA and STFOFOWA), which use nonlinear sine transformations to preserve subtle variations in MDs and NMDs. By incorporating the computed weights into these operators, the framework ensures that criteria with higher discriminative strength exert stronger influence during the aggregation of expert evaluations and the construction of weighted decision matrices.

Table 24.

The comparative analysis of the proposed work with existing approaches.

Authors	Aggregation operator	Best model	Stability (%)	Accuracy (%)	Time complexity	CBI	Robustness index	Sensitivity to criteria uncertainty	Fuzzy information handling strength	Overall class	Rank
Propose					Very High Efficiency			Low	Very Strong	Excellent
Ullah et al.60	Hamacher AO				Medium			Medium	Strong	Very Good
Rahim et al.61	Hammy mean AO				Medium			Medium	Strong	Very Good
Özlü62	Dombi AO				High			High	Moderate	Good
Basarik63	Choquet Integral AO				High			High	Strong	Good
Rahim et al.64	Power Bonferroni AO				Medium			Medium	Moderate	Good
Saqlain et al.65	Einstein AO				Medium			Medium	Strong	Very Good
Khan and Ali66	Copula AO				Medium			Medium	Strong	Very Good
Khan et al.67	Hybrid AO				Medium			Medium	Strong	Very Good
Farid et al.68	Fairly AO				Medium			Medium	Strong	Good
Palanikumar et al.69	trigonometric AO				High			High	Moderate	Very Good
Deva70	Aczel–Alsina AO				High			High	Moderate	Good

Computation of stability and accuracy measures

To ensure transparency and reproducibility of the comparative analysis presented in Table 24, the definitions of stability and accuracy are explicitly provided.

Stability (%): Stability reflects the robustness of the ranking results under variations in input parameters, such as criteria weights and expert judgments. It is computed as the proportion of scenarios in which the ranking order remains unchanged relative to the baseline case. Mathematically, it is defined as,

Where represents the number of scenarios in which the ranking remains consistent, and denotes the total number of sensitivity scenarios considered. A higher stability value indicates that the method produces consistent and reliable rankings under varying decision conditions.

Accuracy : Accuracy measures the degree to which the obtained ranking aligns with a reference or benchmark solution. In this study, it is computed based on the closeness of the obtained ranking scores to the ideal solution using normalized closeness coefficients. It is defined as,

Where represents the computed closeness coefficient of alternative , ​ denotes the ideal or benchmark value, and is the total number of alternatives. Higher accuracy values indicate that the method yields result closer to the optimal decision solution. These measures provide complementary insights: stability evaluates the robustness of rankings, while accuracy assesses the precision of the decision outcomes, thereby offering a comprehensive evaluation of the proposed framework in comparison with existing approaches.

The computational performance of the proposed FOFS–Sine Aggregation MCGDM framework was rigorously analyzed across multiple software environments, as summarized in Table 25. Unlike traditional MCGDM studies that provide only runtime comparisons, we introduce a comprehensive multi-dimensional evaluation including memory efficiency, numerical stability, scalability, parallelization capability, and uncertainty-handling strength all of which are crucial for deep learning model assessment. MATLAB demonstrated the highest numerical stability and reproducibility, making it ideal for implementing the STFOFS operators and TOPSIS pipeline. Python, particularly with NumPy/SciPy, outperformed MATLAB in execution time and scalability, while PyTorch achieved unparalleled parallelization and automatic differentiation support, making it highly suitable for FOFS-based uncertainty modeling in neural networks. The GPU-accelerated MATLAB variant achieved the fastest execution for large FOFS matrices, proving beneficial for batch evaluation of multiple deep learning models. Overall, the combined analysis shows that both MATLAB and Python complement each other, providing a robust computational backbone for executing the proposed MCGDM framework efficiently and accurately.

Table 25.

Comprehensive computational performance analysis across software environments for the proposed FOFS–Sine aggregation MCGDM framework.

Platform environment	Execution time (sec)	Memory efficiency	Numerical stability	Scalability index	Parallelization support	Uncertainty-handling strength	Overall suitability for proposed MCGDM
MATLAB (R2025b)		High	Very High (floating- point precision + stable solvers)	Medium High	Parfor, GPU	Excellent	Ideal for prototyping & reproducibility
Python (NumPy + SciPy)		Very High	High (depends on libraries)	Very High	Multiprocessing, Dask, Ray	Very High	Best for large-scale deployments
Python (PyTorch)		Medium	Extremely High	Excellent	CUDA, cuDNN	High	Perfect for deep learning seamless integration
MATLAB GPU-Accelerated		Medium	Very High	High	CUDA direct	Excellent	Fastest environment for STFOFS batch operations
Google Colab (Python)		Medium	Medium–High	High	shared GPU	High	Good for experiments, not ideal for production
Jupyter + Anaconda (Python)		High	High	Very High	Very High	Very High	Best for MCGDM research workflows

Discussion

The results obtained from the proposed FOFS–TOPSIS framework provide several important insights into data science model selection under uncertainty. The ranking outcomes demonstrate that the integration of FOFSs with sine trigonometric aggregation operators enables a more refined evaluation of alternatives by effectively capturing uncertainty, hesitation, and nonlinear relationships in expert judgments. The top ranked model () consistently exhibits superior performance across multiple criteria, including accuracy, robustness, and computational efficiency. Its strong closeness coefficient indicates a well-balanced trade off among conflicting evaluation factors. Similarly, and other high-ranking alternatives show competitive performance, confirming the reliability of the proposed framework in identifying suitable models. In contrast, lower ranked alternatives such as and reveal significant performance limitations, particularly in terms of stability, efficiency, and predictive reliability. The comparative analysis further highlights that the proposed FOFS–TOPSIS approach provides improved discrimination among alternatives compared to existing fuzzy MCGDM methods. This improvement is mainly attributed to the flexibility of FOFS in modeling asymmetric uncertainty and the capability of sine trigonometric aggregation operators to capture nonlinear variations in expert evaluations. As a result, the proposed framework generates more consistent and interpretable rankings. The sensitivity analysis confirms the robustness of the proposed method, as the top ranked alternatives remain stable across different parameter settings. This stability indicates that the framework is reliable even when there is uncertainty in parameter selection, which is a common challenge in real-world DM problems. From a practical perspective, the proposed framework offers valuable support for data science practitioners and decision-makers. It enables systematic evaluation of predictive models by incorporating multiple criteria and expert opinions, ensuring that selected models are not only accurate but also efficient, scalable, and suitable for deployment. This makes the framework highly applicable in domains such as healthcare, finance, and intelligent systems, where DM under uncertainty is critical.

Advantages of the proposed work

The proposed FOFSA–TOPSIS framework offers several practical advantages over traditional model selection approaches:

• Effective handling of uncertainty and hesitation: Unlike classical methods that rely on precise numerical inputs, the proposed approach models uncertainty, vagueness, and expert hesitation using fractional orthopair fuzzy information.

• Capability to represent asymmetric uncertainty: The FOFS structure allows independent and fractional control of MD and NMD, enabling more accurate modeling of real-world evaluation scenarios.

• Integration of multiple expert opinions: The framework systematically combines assessments from multiple experts, resulting in more balanced, reliable, and consensus driven decision outcomes.

• Modeling of nonlinear relationships: The use of sine trigonometric aggregation operators captures complex and nonlinear interactions among criteria and expert judgments, which are often overlooked in traditional linear aggregation methods.

• Improved ranking accuracy and discrimination power: The integration with TOPSIS enhances the ability to distinguish between closely competing alternatives, leading to more precise and meaningful rankings.

• Direct applicability to real-world workflows: Supports practical data science processes, enabling organizations to select models in domains such as healthcare, finance, and intelligent systems where uncertainty, multiple criteria, and expert disagreement are common.

Practical implications

1. The proposed FOFSA–TOPSIS framework bridges the gap between theoretical advancements and real-world application:

2. Actionable model rankings: Provides clear guidance on which predictive models to select under uncertain or incomplete information.

3. Robust integration of expert knowledge: Systematically incorporates multiple expert opinions to improve decision reliability

4. Stability under uncertainty: Sensitivity analysis ensures top ranked models remain stable despite variations in fractional parameters.

5. Enhanced interpretability: Structured outputs facilitate communication and DM among stakeholders

6. Real world domain applicability: Framework can be directly applied in healthcare, finance, intelligent systems, and other domains with high uncertainty and MCGDM challenges.

By explicitly linking methodological innovations to workflow outcomes, the framework offers tangible benefits for practitioners, making it a valuable tool for data-driven model selection.

Computational complexity analysis of the proposed FOFS–TOPSIS framework

The computational complexity of the proposed FOFS–TOPSIS framework is analyzed by considering its major computational components, including expert weight determination, sine-based fuzzy aggregation, entropy-based criteria weighting, normalization, distance computation, and final ranking. Let m, and k denote the number of alternatives, criteria, and experts, respectively. The most computationally intensive step is the construction and aggregation of the FOFS decision matrix using sine-based operators, which requires , operations. The application of sine trigonometric transformations to MD and NMD also follows the same order, as each evaluation is processed independently. Subsequent steps, including normalization and entropy-based criteria weight computation, require operations. The determination of PIS and NIS and the associated distance calculations also incur a complexity of . The final ranking of alternatives based on closeness coefficients requires due to sorting. Therefore, the overall computational complexity of the proposed framework can be expressed as: . Compared to existing fuzzy TOPSIS variants, the proposed method introduces additional sine-based transformations; however, these operations are computationally lightweight and do not significantly increase the overall runtime. Unlike AHP, which requires pairwise comparisons, the proposed framework avoids excessive computational overhead while maintaining scalability. Furthermore, compared to methods such as VIKOR and other fuzzy TOPSIS extensions, the proposed FOFS–TOPSIS approach provides a favorable balance between computational efficiency and enhanced modeling capability, particularly in handling nonlinear and asymmetric uncertainty. This makes the framework suitable for medium to large scale decision-making problems. To further enhance computational efficiency in practical implementations, optimization strategies such as parallel processing, vectorized operations, and precomputation of sine transformations can be employed (Table 26).

Table 26.

Computational complexity comparison of the proposed FOFS–TOPSIS framework.

Method	Proposed FOFS–TOPSIS	AHP	VIKOR	Fuzzy TOPSIS
AHP71
TOPSIS72
VIKOR73
WASPAS74
COPRAS75
ELECTRE76
PROMETHEE77
MULTIMO0RA78
FOFS-TOPSIS based MCGDM

Conclusion

This study proposed a novel MCGDM framework for data science model selection under uncertainty by integrating FOFS, sine trigonometric aggregation operators, and the TOPSIS method. The proposed approach effectively addresses key challenges in model evaluation, including uncertainty, expert hesitation, and the presence of multiple conflicting criteria. The results of the numerical analysis clearly demonstrate that the proposed framework produces highly stable, consistent, and discriminative rankings of alternatives. In particular, the top ranked alternative (), consistently achieves the highest closeness coefficient compared to competing models, confirming its superior performance across multiple evaluation criteria. Furthermore, the stability of rankings under different sensitivity scenarios highlights the robustness and reliability of the proposed method in handling parameter variations and uncertain environments. A key strength of the framework lies in its enhanced capability to model complex uncertainty. The FOFS structure provides flexible and precise representation of ambiguous and hesitant information through fractional parameterization, while the sine trigonometric aggregation operators effectively capture nonlinear interactions among criteria and expert evaluations. As a result, the proposed method achieves improved discrimination power, higher ranking accuracy, and better consistency compared to existing fuzzy TOPSIS approaches. From a practical perspective, the proposed framework offers a transparent, flexible, and reliable decision support tool for real-world applications. It is particularly suitable for complex DM environments involving multiple experts, diverse evaluation criteria, and uncertain information, such as healthcare analytics, financial forecasting, intelligent systems, and automated model selection in data science workflows.

Extensibility of the proposed framework

The proposed FOFS framework, combined with sine trigonometric aggregation operators, is designed in a modular manner and is not restricted to the TOPSIS method alone. The underlying structure of FOFS allows it to be integrated with other MCDM methods, such as VIKOR, PROMETHEE, ELECTRE, and COPRAS, by adapting their evaluation and aggregation procedures to the FOFS environment. For instance, in VIKOR, the compromise ranking mechanism can be reformulated using FOFS-based distance measures; in PROMETHEE and ELECTRE, preference relations can be extended using fractional orthopair fuzzy comparisons; and in COPRAS, proportional assessment can incorporate FOFS-based weighted aggregation. This demonstrates the generality and adaptability of the proposed framework for a wide range of DM scenarios. However, TOPSIS is selected in this study due to its distance based ranking mechanism, which provides a natural and efficient structure for integrating FOFS representations and sine-based nonlinear transformations. This compatibility enables more accurate modeling of uncertainty and enhances the discrimination power of the ranking results. Therefore, while the proposed framework is extendable to other MCDM methods, TOPSIS offers the most suitable foundation for demonstrating its effectiveness in this study.

Limitations

Despite its strengths, the framework has several limitations. First, although FOFS reduces subjectivity, the final decision remains influenced by expert judgment accuracy. Second, the method assumes independence among criteria, whereas deep learning metrics such as accuracy, training time, parameter count, and overfitting risk may exhibit interdependencies. Third, while the framework performs efficiently for moderate problem sizes, extremely large-scale decision environments involving hundreds of models and criteria may require additional computational optimization. Finally, the selection of fractional parameters may require domain-specific tuning, and a generalized parameter-setting strategy has not yet been established.

Impact of the proposed work

The need for this study arises from the increasing complexity of data science model selection, where multiple competing models must be evaluated under uncertainty, conflicting criteria, and expert hesitation. Existing fuzzy MCGDM approaches often struggle to capture asymmetric uncertainty and nonlinear interactions, limiting their effectiveness in real-world DM scenarios. Despite the strengths of the proposed FOFS–TOPSIS framework, certain limitations remain. The DM process is still influenced by the accuracy and consistency of expert judgments. Additionally, the current model assumes independence among evaluation criteria, whereas, in practice, performance metrics such as accuracy, computational complexity, and overfitting risk may exhibit interdependencies. Furthermore, while the framework is computationally efficient for moderate scale problems, large scale applications may require additional optimization. The selection of fractional parameters also depends on domain expertise, and a generalized parameter selection strategy is yet to be established. Notwithstanding these limitations, the proposed framework has significant practical impact. It provides a robust, flexible, and interpretable decision support tool for selecting predictive models in complex environments. By effectively handling uncertainty, integrating multiple expert opinions, and capturing nonlinear relationships, the framework enhances decision reliability and supports real-world applications in domains such as healthcare, finance, and intelligent systems.

Future directions

Future research may address the identified limitations in several ways. First, incorporating interdependencies among evaluation criteria using approaches such as fuzzy cognitive maps or DEMATEL can provide more realistic modeling of complex decision environments. Second, data driven or learning based strategies may be developed to automatically determine criterion weights and fractional parameters, reducing reliance on expert input. Additionally, improving computational scalability through parallel, GPU based, or cloud-based implementations would enable the application of the framework to large scale decision problems. The proposed methodology can also be extended to dynamic DM environments where model performance evolves over time. Furthermore, applying the framework to diverse real-world domains such as medical diagnosis, intelligent transportation, cybersecurity, renewable energy, and smart city systems can further validate its effectiveness and versatility. Integrating the proposed approach with modern MLOps pipelines may also support continuous and automated model evaluation in practical deployment settings.

Appendix A

Definition 1

¹⁹Let be a universal set an IFS I on can be defined as,

1

In Eq. (1), , represented the MF and NMF respectively having the condition . This definition introduces the basic framework for handling uncertainty by simultaneously considering MD and NMD, forming the foundation for subsequent fuzzy set extensions.

Definition 2

²⁰Consider be a universal set a PFS on can be defined as,

2

In Eq. (2), denote the MF and NMF respectively with the condition . This extension provides greater flexibility than IFS by allowing a wider range of uncertainty representation through a relaxed constraint based on squared terms.

Definition 3

²¹Let be a universal set an ROFS on can be expressed as,

Where , represent the MF and NMF respectively having the condition and q is the positive integer. This model generalizes previous PFSs by introducing a parameter q, enabling more flexible control over uncertainty and decision-maker preferences.

Definition 4

²²Consider be a universal set p,QOFS on can be defined as,

3

In Eq. (3), , denote the MF and NMF respectively such that , where p and q are positive integers. This definition further enhances flexibility by allowing independent control of MD and NMD through two parameters, improving modeling capability in complex decision environments.

Author contributions

Shah Zeb Khan: Conceptualization, Writing original draft, Data curation, Formal analysis, Investigation, Methodology. Wael Mahmoud Mohammad Salameh: Project administration, Resources, Software. Yasir Akhtar: Conceptualization, Writing original draft, Data curation, Formal analysis, Investigation, Methodology. Muhammed I. Syam: Supervision, Validation, Visualization.

Data availability

No datasets were generated or analyzed during the current study.

Competing interests

The authors declare no competing interests.

Declaration of Generative AI

The authors acknowledge the use of ChatGPT to assist in refining grammar and language during the preparation of this manuscript. All content was thoroughly reviewed and revised by the authors, who take full responsibility for the final version of the paper.