Multi attribute group decision-making based on quasirung orthopair fuzzy Frank aggregation operators for optimal vehicle selection

Abstract

This study proposes novel operational laws that extend the Frank t-norm and t-conorm to develop a new class of aggregation operators (AOs), namely the quasirung orthopair fuzzy Frank weighted average, weighted geometric, ordered weighted average, and ordered weighted geometric operators. These operators are specifically designed to manage uncertain and imprecise information within multi-attribute group decision-making (MGADM) environments. The proposed operators exhibit desirable mathematical properties such as flexibility, robustness, and compatibility, making them highly suitable for complex fuzzy decision contexts. Flexibility is notably enhanced through the independent tuning of the parameters , , and , allowing for more refined control over membership (MD), non-membership (NMD), and interaction behaviors. An entropy-based approach is employed to objectively determine unknown attribute weights, minimizing subjective bias. A real-world case study on the selection of an optimal investment location demonstrates the practical applicability of the proposed method. The results show an improvement in decision-making accuracy by approximately 7.5% compared to traditional approaches. Sensitivity analysis confirms the stability and reliability of the proposed operators under varying conditions. Comparative results further highlight the method’s superiority in terms of accuracy, interpretability, and adaptability to input variations. The paper concludes by outlining special cases and acknowledging certain limitations, offering directions for future research.

Introduction

Decision-making is a fundamental cognitive and analytical process that involves selecting the most suitable option from a set of alternatives based on certain criteria or preferences. It plays a crucial role across a wide array of fields, including business management¹, engineering design², healthcare planning³, environmental sustainability⁴, and governmental policy-making⁵. In real-world scenarios, decision-making often involves dealing with complex, imprecise, and uncertain information, making it necessary to adopt sophisticated mathematical tools to support rational judgments^6,7. Fuzzy Set (FS) theory, introduced by Zadeh⁸ in 1965, has emerged as a powerful tool in this context, enabling the modeling of uncertainty, vagueness, and ambiguity inherent in human reasoning. The primary advantage of fuzzy sets in decision-making lies in their ability to represent partial truths, offering a flexible framework where membership grades between and can capture the imprecision of real-world data, thereby enhancing the robustness and realism of decision-making models. However, despite their significant contributions, classical FSs exhibit certain limitations, particularly their inability to simultaneously and independently model the MD, NMD, and hesitation or indeterminacy. This shortcoming necessitated the development of several extensions to fuzzy sets to better address more complex and layered uncertainties. Notable among these extensions are Intuitionistic Fuzzy Sets (IFSs), proposed by Atanassov⁹, which introduced an explicit NMD alongside MD, thereby accounting for hesitation. Pythagorean Fuzzy Sets (PFSs)¹⁰ further relaxed the relationship between MD and NMD, allowing their squares’ sum to remain bounded within one, thus offering greater flexibility. Subsequently, rung orthopair fuzzy sets ( ROFSs)¹¹ generalized PFSs by permitting higher sums of powered MD and NMD, thus providing even more expressive capacity. In the framework of ROFSs, the parameter serves as a controlling factor that simultaneously influences both the MD and NMD. While this unified control mechanism offers a certain degree of flexibility, it can also negatively affect the accuracy and sensitivity of the decision-making process, particularly in complex scenarios where more nuanced modeling is required. Specifically, when decision-makers need the ability to adjust the MD and NMD independently using distinct parameters for each aspect ROFSs become inadequate, as they lack the structural flexibility to accommodate such independent tuning. Recognizing this significant limitation, Seikh and Mandal ¹² introduced the concept of quasirung orthopair fuzzy sets ( QOFSs). This novel extension allows for the separate management of the MD degree through the parameter and the NMD through the parameter q, thereby offering a more refined and adaptable framework for handling complex and asymmetric uncertainties in decision-making problems. By decoupling the control over MD and NMD, QOFSs provide a more sophisticated mathematical structure that better aligns with the real-world needs of multi-criteria decision-making environments, where uncertainty, hesitation, and conflict among attributes must be independently and accurately represented. We provided clear numerical example to better explain why QOFSs are more flexible and advantageous compared to ROFSs. In ROFSs, the parameter q q uniformly controls the influence of both MD and NMD, which can limit flexibility in decision-making scenarios where different levels of control are needed. For example, if a decision-maker evaluates a project with a MD (trust) of and a NMD (risk) of under , the condition is satisfied, allowing its use under ROFSs. However, if the decision-maker desires stricter management for trust and more relaxed handling for risk, ROFSs are insufficient because they do not allow separate parameters. To overcome this, QOFSs were introduced, permitting distinct parameters and for MD and NMD, respectively. In the same example, using for trust and for risk in QOFSs leads to , offering greater flexibility and aligning more accurately with practical decision-making needs. Thus, QOFSs enhance the modeling capability by enabling independent control over MD and NMD evaluations, better reflecting the real-world complexity of decision-making processes.

AOs offer several key advantages in decision-making processes. They systematically combine multiple criteria into a single representative value, enhancing the rationality and consistency of evaluations. Their flexibility allows adaptation to different decision contexts, while advanced fuzzy-based operators effectively handle uncertainty and imprecision. Aggregation operators also support customizable weighting, maintain essential mathematical properties like monotonicity and boundedness, and are scalable to complex problems with many alternatives and criteria. Furthermore, they promote transparency, facilitate group decision-making, and ensure reliable and reproducible outcomes. Considering the advantages of AOs, various types have been proposed by researchers, such as Senapati et al.¹³ introduced intuitionistic fuzzy AOs based on Aczel-Alsina operations, demonstrating an enhanced ability to manage ambiguity compared to classical approaches. Akram et al.¹⁴ extended aggregation methodologies to the Pythagorean fuzzy environment through the application of Dombi operations, thereby improving the modeling of uncertainty and vagueness. In the context of spherical fuzzy sets, Garg et al.¹⁵ proposed power-based AOs, while Shit and Ghorai¹⁶ developed Dombi-based operators under the Fermatean fuzzy framework, significantly advancing the ability to capture hesitation and conflicting information. Further contributions by Garg¹⁷ introduced a comprehensive suite of AOs designed to integrate collected information while preserving critical relational properties essential for robust decision analysis. Beyond these developments, Rahim et al.¹⁸ proposed Bonferroni mean-based AOs within a fuzzy cubic Pythagorean environment, enhancing the aggregation process under high degrees of uncertainty, while Amin et al.¹⁹ introduced generalized cubic Pythagorean fuzzy AOs to effectively address complex MCDM problems. More comprehensive discussions on the advancements and applications of AOs can be found in^{20, 21, 22, 23, 24, 25, 26–27}.

The Frank norm and Frank conorm are important mathematical operations used in fuzzy logic to model the concepts of intersection and union between fuzzy sets with a high degree of flexibility. These operations are parameterized by a real number, allowing them to smoothly adjust between different logical behaviors depending on the choice of the parameter. Building upon these foundations, AOs based on the Frank norm and conorm have been developed to combine multiple fuzzy values into a single representative value in decision-making processes. These Frank-based AOs inherit the adaptability of the underlying norm and conorm, enabling a fine-tuned balance between conjunctive and disjunctive aggregation behaviors according to the decision-makers’ preferences. Compared to traditional aggregation methods, Frank AOs offer several significant advantages: they provide greater flexibility through parameter adjustment, allowing the model to represent optimistic, pessimistic, or balanced decision attitudes; they preserve essential properties like associativity, commutativity, and monotonicity; and they are particularly effective in managing uncertainty, interdependencies among criteria, and diverse decision-making scenarios (see Table 1). As a result, Frank AOs are highly suitable for solving complex MCDM problems under fuzzy environments, offering improved accuracy and realism over classical aggregation approaches. In recent years, various Frank-based AOs have been proposed by researchers to enhance decision-making processes across different fuzzy set environments. For instance, Frank AOs have been developed for spherical fuzzy numbers²⁸, rung orthopair fuzzy sets²⁹, probabilistic hesitant fuzzy sets³⁰, and interval-valued picture fuzzy sets³¹. Moreover, Frank AOs have been applied in multi-criteria group decision-making under Pythagorean cubic fuzzy sets³² and picture fuzzy sets³³. Extensions have also been introduced for complex rung orthopair fuzzy sets³⁴, complex spherical fuzzy sets³⁵, and complex intuitionistic fuzzy sets³⁶. Recent studies have utilized Frank AOs for soil fertility assessment³⁷, renewable energy classification³⁸, , and electric vehicle evaluation³⁹. These works demonstrate the flexibility and effectiveness of Frank norm and conorm operations in handling uncertain, imprecise, and hesitant information within multi-criteria decision-making frameworks. By utilizing advanced AOs, such as Einstein and soft-max-based approaches for clustering, DM, and real-time applications^{40, 41, 42–43}.

Table 1. Comparison of important AOs.

Aspect	Frank44	Dombi45	Yager46	Bonferroni47	Einstein48	Aczel–Alsina49	Hamacher50
Type	Parametric t-norm and t-conorm	Parametric t-norm and t-conorm	Parametric t-norm and t-conorm	Mean-type operator	Algebraic-based operator	Parametric t-norm and t-conorm	Algebraic-based operator
Parameter control	Strong (adjusts between different behaviors)	Strong (controls flexibility in conjunction/disjunction)	Strong (manages degree of aggregation)	Weak (fixed structure, limited tunability)	Weak (fixed structure)	Strong (tunable for different aggregation behaviors)	Moderate (partial tunability)
Handling of uncertainty	Excellent	Very good	Very good	Moderate	Moderate	Very good	Good
Flexibility	Very high (smoothly adjustable between AND/OR)	High	High	Low to moderate	Low	High	Moderate
Associativity, commutativity, monotonicity	Satisfied	Satisfied	Satisfied	Partially satisfied (depending on settings)	Satisfied	Satisfied	Satisfied
Behavior control (pessimistic/optimistic)	Easily adjustable via parameter	Adjustable via parameter	Adjustable via parameter	Difficult	Fixed behavior	Adjustable via parameter	Limited adjustability
Suitability for complex MCDM problems	Excellent	Very good	Very good	Moderate	Moderate	Very good	Good
Computational complexity	Moderate	Moderate	Moderate	Low	Low	Moderate	Low
Main limitation	Requires proper tuning of parameter	Sensitive to parameter values	Sensitive to parameter values	Limited adaptability	Less flexible	Sensitive to parameter settings	Moderate flexibility only

Gaps and justification for Frank operators in QOFSs

From the above review, several research gaps and shortcomings are evident. Existing Frank-based aggregation models in ROFS provide only a single degree of flexibility through the parameter and cannot independently fine-tune the behavior of MD and NMD, thereby limiting their adaptability to real-world decision scenarios characterized by complex and asymmetrical uncertainty. Furthermore, no unified framework has yet been developed to integrate Frank algebraic structures with QOFSs, and the benefits of introducing dual parameters ( and ) for separately governing MD and NMD within a Frank operational framework remain unexplored. Additionally, the literature lacks numerical studies comparing the sensitivity, adaptability, and convergence behavior of dual-parameter versus single-parameter AOs. The theoretical justification for integrating Frank operators with QOFSs is compelling, as Frank norms and conorms provide smooth and continuous control over the interaction of degrees, enabling dynamic adjustment during information aggregation. Their parameterized nature offers fine-grained control, which is essential for decision-making models that must accommodate highly volatile or asymmetric data patterns. Moreover, combining Frank operations with the dual-parameter ( , ) structure enhances the capacity to model optimistic and pessimistic aspects of uncertainty independently, an achievement that traditional models fail to realize.

Major contributions and novelty of the proposed framework

In light of the identified research gaps, this study presents several key contributions. First, it introduces novel operational laws based on Frank norms and conorms, specifically tailored for the QOFS framework. Second, four new AOs are developed: the quasirung orthopair fuzzy frank ( QOFF) weighted averaging ( QOFFWA), QOFF weighted geometric ( QOFFWG), QOFF ordered weighted averaging ( QOFFOWA), and QOFF ordered weighted geometric ( QOFFOWG) operators. Third, the dual-parameter ( and ) design enhances adaptability by allowing independent adjustment of MD and NMD, offering significantly greater flexibility compared to traditional single-parameter ROF Frank AOs. Fourth, extensive numerical validation and comparative analyses demonstrate the superior performance of the proposed operators in terms of adaptability, convergence, and alignment with realistic decision outcomes. Fifth, the practical effectiveness of the proposed framework is illustrated through its application to a real-world investment company selection problem. Sixth, to ensure objectivity in decision-making, an entropy-based weighting method is employed to derive unknown attribute weights, thereby minimizing subjective bias. Unlike existing ROF Frank AOs, where a single parameter controls both MD and NMD behaviors, the proposed QOFF framework enables independent control through separate parameters and . Numerical experiments confirm that adjusting only cannot replicate the flexibility and convergence speed achieved when and are tuned independently. Therefore, the dual-parameter approach introduced herein marks a significant theoretical and practical advancement, particularly by allowing more accurate modeling of asymmetrical uncertainties, enhancing the sensitivity and realism of multi-criteria decision outcomes, and providing superior convergence behavior when MD and NMD distributions differ markedly challenges that traditional ROFF models are ill-equipped to address.

The remainder of this paper is organized as follows: “Materials and methods” section introduces the fundamental concepts and operational laws related to QOFS. “Proposed p,q-quasi rung orthopair fuzzy Frank AOs” section presents the construction of the proposed QOFF AOs along with their properties. “Proposed MCGDM approach” section outlines the proposed algorithm based on the new AOs for solving MADM problems. “Application” section provides a detailed numerical example and comparative analysis to validate the effectiveness of the proposed method. Finally, “Conclusion” section concludes the paper by summarizing the key findings and outlining future research directions.

Materials and methods

Some basic concepts related to -ROFSs and -QOFSs are discussed. The notations employed throughout this paper are clarified in Table 2.Table 2

Explanation of mathematical symbols.

Symbol	Explanation
	Universal set
	Any fuzzy set
	Membership function
	Non-membership function
	Any element of universal set
	Belong to
	Parameter of ROFS
	Less than
	Greater than
	Any positive real number
	Score value
	Accuracy value
	Weight vector
	Set of alternatives
	Set of criteria
	Set of experts

-Rung orthopair fuzzy set

Definition 1

¹¹ Let denote a universal set. A -ROF set concerning an element is defined as follows:

1

where , denote the MD and NMD of an element such that for . The degree of hesitancy between them is calculated as:

2

We call the pair is a -ROFN.

Definition 2

¹¹ Let , and be any three -ROFNs. Then, the operational laws between them are defined as follows:

1. ,

2. if and only if and ,

3. if and ,

4. ,

5. ,

6. ,

7. .

where represent the complement of -ROFN and is any positive real number.

Definition 3

¹¹ Consider a -ROFN. The score function of is described as follows:

3

where .

Definition 4

¹¹ Consider a -ROFN. The accuracy function of is described as follows:

4

where .

Definition 5

¹¹ Let and are two -ROFN, then.

1. If then ,

2. If then ,

3. If and,

4. If then ,

5. If then ,

6. If then .

Definition 6

¹¹ Suppose represents a family of -ROFNs and be the weight vector with constraints and . The -ROFWA operator defined as follows:

5

where .

quasirung orthopair fuzy sets

Definition 7

¹² Consider a nonempty set . A -QOFS concerning an element is described as follows:

6

Definition 8

¹² Let , and are three -QOFNs, and then,

1. ,

2. ,

3. ,

4. ,

5. if and only if and ,

6. if and only if and .

Definition 9

¹² Let be a -QOFN. The score value of can be established using the subsequent function.

7

where .

Definition 10

¹² Let be a -QOFN. The accuracy of can be determined by the following function.

8

where

Definition 11

¹²

Let ,and are two -QOFNs, then.

1. If then ,

2. If then ,

3. If and

4. If then ,

5. If then ,

6. If then .

Definition 12

¹² Let , and are three -QOFN, and , and are any positive integers then the following properties are held.

1. ,

2. ,

3. ,

4. ,

5. ,

6. .

Definition 13

⁵¹ The Frank t-norm and t-conorm are mathematical operations defined for real numbers and , where and fall in the interval . The formulas for these operations are as follows:

1. Frank t-norm (Fr):

2. Frank t-conorm ( ):

Here and is any positive value except 1 i.e., and .

By applying limit theory, the following is derived⁵²:

As approaches to 1, the Frank t-conorm tends to and Frank t-norm tend to .

As approaches infinity, the Frank t-conorm tends to and Frank t-norm tend to

Proposed -quasi rung orthopair fuzzy Frank AOs

We have presented some basic laws for working with the Frank t-norm and the t-conorm in this section. Based on these laws, we have constructed AOs, which we call -QOFFWA, -QOFFWG, -QOFFOWA and -QOFFOWG AOs.

Definition 14

Consider three -QROFNs and . These -QROFNs are characterized by their MD and NMD functions and For the sake of our discussion, let be any real number except 1, and . The ensuing operational laws have been established within the framework of the Frank norm and conorm:

3. ,

4. .

Theorem 1

Let , , be any three positive real numbers Then for three -QOFNs and , then FTN and FTCN are defined as follows:

6. .

Proof

Using Definition 14, we get.

Similarly,

Now,

Therefore, .

-quasirung orthopair fuzzy Frank arithmetic AOs

We present a variety of AOs in this section that use the rules from Sect. 3. These operators can effectively combine and simplify information, which facilitates data analysis and DM.

Definition 15

Let be a set of -QOFNs with their corresponding weight vector such that and . Then the operator.

is defined as

9

Theorem 2

The aggregated value obtained by operator of -QOFNs is still a -QOFNs, and

10

Proof

This theorem is established by using mathematical induction.

Step 1 For , we have where

Consequently, when equals 2, the assertion holds true.

Step 2 Assume the validity of the result for , i.e.,

11

Step 3 In order to demonstrate Eq. (13) is true for the case where i.e.,

12

where

Thus, Eq. (12) hold for . By mathematical induction we conclude that the result is true for all values of

Theorem 3

(Idempotency) If the -QOFNs are identical, i.e., for all , where , then .

Proof

As , for all , then we obtain.

Therefore, the result can be derived from the information provided.

Theorem 4

(Boundedness) Let be a family of -QOFNs. If.

and , then.

.

Proof

Let and . Herefore, we have ,

Therefore, .

Theorem 5

(Monotonicity) Let and be two sets of -QOFNs, where of and for . If and for all , then .

Proof

Given that and for all then we have,

Similarly, we can prove.

Thus,

Let and . Then by Definition 16, we have

If then we have, i.e., . then, we get then, by condition and for all we have and.

.

Therefore,

13

Thus, we have .

-quasirung orthopair Frank geometric aggregation operator

Definition 16

Let be a family of -QOFNs with weight vector such that and . Then is defined as follows.

14

Theorem 6

The aggregated value of -QROFNs utilizingp,q-QOFFWG operator is still ap,q-QOFN, and

15

Proof

Straight forward.

Theorem 7

(Idempotency) If the -QROFNs are identical, i.e., be a for all , where , then .

Theorem 8

(Boundedness) Let be a collection of -QOFNs. If

and , then

Theorem 9

(Monotonicity) Let and be two sets of -QOFNs, where of and for If and for all , then

quasirung orthopair order weighted averaging/geometric AOs

Definition 17

Let be a family of QOFNs with corresponding weight vector with the condition that and .

Then and operators are the functions from defined as follows

16

17

respectively, where ( ) is the permuation of satisfying , for all

Theorem 10

The aggregated value by using and operators are still -QOFNs, and

18

19

Proof

The proof is same as Theorem 2.

Theorem 11

(Idempotency) If the -QOFNs are identical, i.e., be a for all , where , then and

Theorem 12

(Boundedness) Let be a family of -QOFNs. If and ,

then p,q-QOFFOWA and p,q-QOFFOWG .

Theorem 13

(Monotonicity) Let and be two sets of -QOFNs, where of and . If and for all , thenp,q-QOFFOWA

p,q-QOFFOWA andp,q-QOFFOWG p,q-QOFFOWG .

Theorem 14

(Commutativity) Let and be two sets of -QOFNs, then and , where is any permuatation of

Proposed MCGDM approach

To address MAGDM problems under uncertainty, the proposed quasirung orthopair fuzzy frank weighted averaging and quasirung orthopair fuzzy Frank weighted geometric AOs are employed. These operators effectively integrate unknown attribute weights and evaluation values represented as QOFNs. Let a group of experts evaluate the alternatives with respect to the criteria . Initially, attribute weights are determined using the entropy method, which quantifies the degree of uncertainty and variation among criteria, resulting in a normalized weight vector , where and . Subsequently, a decision matrix ​ is constructed, where each element represents the preference value of alternative ​ with respect to criterion ​ in terms of QOFNs, satisfying the condition to maintain consistency within the QOF framework. The proposed AOs are then applied to synthesize the individual preference values and attribute weights into a comprehensive score for each alternative, facilitating a robust and flexible ranking process. This approach provides greater adaptability to decision-makers by accommodating varying risk attitudes through the adjustment of parameters p, q, and , thereby offering more insightful and customized decision support compared to existing methods. The proposed MAGDM approach proceeds through the following steps:

Step 1 Construct the individual decision matrices based on the experts’ assessments of the alternatives with respect to each criterion.

Step 2 Normalization is essential when both cost and benefit criteria are involved in a MAGDM problem because these criteria have different evaluation directions and measurement scales. Benefit criteria are those where higher values are preferred (e.g., profit, efficiency), while for cost criteria, lower values are more desirable (e.g., expenses, risks). Without normalization, directly aggregating or comparing raw values could lead to misleading results, as it would not account for these opposing objectives. This adjustment allows consistent and fair aggregation of different types of criteria, preserving the decision model’s integrity and leading to a more accurate and interpretable ranking of alternatives.

20

Step 3 The selection of parameters p and q in a quasirung orthopair fuzzy environment is crucial for controlling the aggregation behavior of MD and NMD independently, thereby enhancing the model’s adaptability to different types of uncertainty and decision-makers’ risk attitudes. Generally, smaller values of p make the model more sensitive to high MDs, favoring optimistic assessments, while larger values produce more conservative aggregation. Similarly, lower values of q heighten sensitivity to NMDs, emphasizing risk or pessimism, whereas higher values moderate this influence. When uncertainty is highly asymmetric, different values of p and q should be selected to capture this imbalance accurately; otherwise, setting p = q is appropriate for balanced scenarios. In practice, decision-makers can initially choose default values such as and adjust them through empirical tuning or sensitivity analysis to achieve desired aggregation behaviors. The parameters can also be fine-tuned to reflect historical decision patterns or specific risk preferences, ensuring that the final decision process remains flexible, realistic, and well-aligned with the characteristics of the data.

Step 4 To find the aggregated decision matrix in a quasirung orthopair fuzzy environment, the assessments provided by multiple experts for each alternative under each criterion must first be collected, typically represented as QOFNs. Once individual decision matrices are constructed for each expert, AOs such as the quasirung orthopair fuzzy frank weighted averaging or the quasirung orthopair fuzzy frank weighted geometric operators are applied to combine these assessments into a single, unified decision matrix. This process involves computing the aggregated MD and NMD by applying the proposed operational laws under the Frank t-norm and t-conorm, weighted appropriately if experts have different levels of importance.

Step 5 Calculate attributes weights. Here is the calculation formula¹²:

21

Step 6 Compute the weighted aggregated decision matrix by combining the criteria weights with the aggregated decision matrix.

Step 7 Determine the performance values for each alternative and subsequently compute the score value of each alternative based on these performance values.

Step 8 In cases where multiple alternatives yield identical score values, Eq. (9) is employed to calculate the accuracy function, enabling a more precise comparison among these alternatives. If, after this evaluation, the alternatives still exhibit equal score and accuracy values, any of them may be considered the most suitable choice based on decision-maker preference or secondary criteria. The overall structure and workflow of the proposed method are illustrated in Fig. 1, providing a clear representation of the decision-making process.

Fig. 1 [Images not available. See PDF.]

Proposed method.

Application

We are dealing with a case where several experts give their opinion on several companies based on several attributes. To obtain a complete assessment, we apply the suggested mathematical method to merge the experts’ points of view while considering their relative expertise.

Numerical example

To demonstrate the applicability and effectiveness of the proposed method, we consider a practical scenario involving the selection of an optimal vehicle for fleet acquisition. Five vehicle models—Toyota Camry ( ), Honda Accord ( ), Ford Fusion ( ), Nissan Altima ( ), and Hyundai Sonata ( ) are evaluated by a panel of three experts, denoted as , , and . The assessment is carried out based on four critical criteria: (Fuel Efficiency, measured in miles per gallon or liters per 100 km), ​ (Safety Features, including crash ratings and advanced safety technologies), (Maintenance Cost, reflecting the expected annual maintenance expenses), and ​ (Performance and Comfort, encompassing engine power, ride quality, and interior amenities). In order to appropriately account for the varying expertise levels of the evaluators, expert weights are assigned as . Each expert provides assessments in the form of quasirung orthopair fuzzy numbers with parameters and , resulting in three individual evaluation matrices , , and , which are detailed in Tables 3, 4, 5. The primary aim of this case study is to determine the most suitable vehicle for fleet procurement by systematically analyzing and aggregating expert opinions using the MAGDM approach based on the proposed AOs. This structured methodology enables comprehensive consideration of crucial factors such as operational efficiency, cost-effectiveness, reliability, driving performance, and user comfort. By rigorously applying the MAGDM process, fleet managers and decision-makers can enhance the quality of their selection process, ensure optimal resource allocation, and maximize long-term benefits through strategic investment in superior vehicle options. This case study thus highlights the practical value and robustness of the proposed framework in real-world multi-criteria decision-making environments.Table 3

Decision matrix provided expert .

Alternatives

Table 4

Decision matrix provided expert .

Alternatives

Table 5

Decision matrix provided expert .

Alternatives

In this study, normalization of the data is not required because all four attributes under consideration belong to similar categories (benefit criteria). As a result, the evaluation and comparison of alternatives across different criteria become more straightforward and intuitive without the need for additional data transformation. This uniformity ensures that no attribute disproportionately influences the decision-making process due to differences in scale. Nevertheless, it is important to highlight that if the attributes had varied significantly in nature such as combining cost-related factors with performance metrics or qualitative assessments normalization would be crucial to balance the influence of each attribute and to maintain fairness in the aggregation process. Since, the weights of the attribute are not known, Eq. (21) can be used to compute the weight assign to each attribute. This equation is a commonly used method in MAGDM processes to determine the relative importance of each attribute. the weight vector for the attributes is given by: . It can be observed that the total of is equal to one.

Equation (16) is used to calculate the aggregated decision matrix. The resultant aggregated decision matrix is presented in Table 6.Table 6

The aggregated values of experts by using QOFFWA operator ( , and ).

To combine the rating values of alternatives, we once again utilize the QOFFWA operator, which has proven to be effective in previous calculations. Specifically, we set the parameters and to 2 and 3, respectively, to achieve the desired level of aggregation. The resulting aggregated values are as follows: , , , , These aggregated values can now be used to rank the alternatives and make a more informed decision. In the end, the score values of the alternatives can be computed by employing Eq. (11) as illustrated below:

The ranking order of the alternatives shows us that is ranked the highest, followed by , , , and , respectively i.e., . Hence, it can be concluded that is the most suitable and optimal alternative among all the options. Considering the determined ranking order based on the score values, stands out as the superior choice, possessing qualities and characteristics that are superior to the rest of the alternative. Therefore, is the best option available for the intended purpose or situation.

To further validate the effectiveness of the proposed approach, an additional aggregation operator, namely the QOFFWG operator, is employed. By following all the steps outlined in the proposed MCGDM framework and using the same set of criteria weights, the alternatives are ranked accordingly. The results of this evaluation are summarized in Table 7.Table 7

Aggregated values of experts.

Alternatives

To amalgamate the rating values of the alternatives, we will more utilize -QOFFWG operator again. For this scenario, we establish the parameter values of and as 2 and 3, respectively. The resulting aggregated values are as follows: , , , , These aggregated values can now be used to rank the alternatives and make a more informed decision. Ultimately, by applying Eq. (11) in the following way, the score values as:, = 0.6199, = 0.6374, = 0.4777, = 0.6447, = 0.5261.

The ranking results indicate that ​ is identified as the most preferred alternative, followed sequentially by , , , and ( ). Consequently, it can be concluded that ​ is the optimal choice for the given objective or context. Decision-makers retain the flexibility to adjust the parameters p and q within the proposed AOs, tailoring them to the specific characteristics of the decision problem. It is important to examine how variations in the parameters p and q influence the overall decision-making process. Accordingly, we computed the score values of the alternatives under different settings of p and q. The resulting ranking orders of the alternatives under various parameter configurations are graphically illustrated in Fig. 2.

Fig. 2 [Images not available. See PDF.]

Ranking of alternatives.

Validity assessment

To evaluate the reliability and applicability of the proposed decision-making approach in dynamic environments, a validity assessment is conducted based on the test criteria suggested by Wang and Triantaphyllou⁵³. The following conditions are considered to validate the method:

1. Criterion 1: If the evaluation scores of a non-optimal alternative are replaced with inferior values while keeping the relative weights of the criteria unchanged, the ranking of the optimal alternative should remain unaffected.

2. Criterion 2: The decision-making method must exhibit the property of transitivity.

3. Criterion 3: When the decision problem is broken down into smaller subproblems and solved independently using the same MADM method, the combined rankings should be consistent with those obtained from the original, undecomposed problem.

Validation against criterion 1

The initial ranking order obtained from the proposed approach is ​. To verify compliance with Criterion 1, the non-optimal alternative ​ is replaced with a worsened version, denoted as ​. The updated ratings of ​ under the three criteria are provided as: , , , and . Applying the proposed method to this updated data, the resulting score values are calculated as: , , , , . Thus, the new ranking order becomes , confirming that ​, the original best alternative, remains unchanged. This consistency demonstrates that the proposed method satisfies Criterion 1.

Validation against criteria 2 and 3

To assess the method’s validity concerning Criteria 2 and 3, several subproblems are constructed, each consisting of subsets of the alternatives: , , and . Applying the proposed decision-making process to each subset yields the following ranking orders: For : ​, For : , for : , and for : By aggregating these individual results, the overall ranking is found to be ​, which matches the ranking order obtained from the original (undecomposed) problem. This consistency confirms that the proposed approach maintains transitivity and that the method remains valid under both Criterion 2 and Criterion 3. Thus, the results of the validity assessment strongly support the robustness, consistency, and reliability of the proposed MADM framework.

Sensitivity analysis

Sensitivity analysis is a crucial step to examine the robustness and reliability of the proposed decision-making approach. In this study, we conduct a comprehensive sensitivity analysis by investigating the effects of varying the parameters p, q, and T on the final ranking of alternatives. By systematically adjusting these parameters, we assess how sensitive the decision outcomes are to changes in their values, thereby providing valuable insights into the stability and adaptability of the proposed method.

Impact of the parameter p on the ranking order of alternatives in decision-making

Understanding how the parameter p influences the ranking of alternatives is crucial when applying the proposed AOs in MADM. It is important to note that setting p and q to low values, such as 1 or 2, is not feasible in this context because it can lead to violations of the fundamental constraint of the QOFNs, specifically that the sum of the notational values must not exceed 1. For instance, considering preference values of 0.75 and 0.70, we observe that (1 or ( ), both of which are invalid under the QOFN framework. Consequently, to ensure consistency and compliance with the operational laws, the parameter q is fixed at 3. With q fixed, we proceed to investigate the influence of varying p on the final ranking of the alternatives. By systematically adjusting the parameter p while holding q constant, we can observe how the ranking sequence evolves. This analysis provides valuable insights for decision-makers, enabling them to better understand the sensitivity of the decision-making process to changes in the aggregation parameters. Specifically, it assists in identifying the most appropriate value of p that aligns with the particular characteristics and risk preferences of the decision environment. Table 8 summarizes the ranking orders of the alternatives for different values of p ranging from 2 to 10, with q consistently set at 3. Furthermore, the impact of p on the ranking orders is graphically depicted in Fig. 3, offering a visual representation of how alternative preferences shift with variations in p. This detailed investigation helps decision-makers recognize the critical role of parameter tuning in enhancing the reliability and accuracy of MAGDM outcomes. By analyzing the score values and observing the changes in rankings corresponding to different p values, decision-makers can strategically select an optimal p that achieves a desired level of aggregation sensitivity, thereby improving the decision quality and ensuring more robust and informed selections.Table 8

Score values and ranking order for different values of for fixed and .

Parameter	Score values					Ranking order
	0.6031	0.6260	0.4689	0.6308	0.4869

Fig. 3 [Images not available. See PDF.]

Impact of parameter on ranking order of alternatives.

Role of parameter in the decision-making process

The subject of this inquiry is the QOFFWG operator. In this case, we varied between 3 and 10, while holding constant at 2. Table 9, display a summary of the resultant values of QOFFWG operator.

Table 9

Ranking order of different values of for fixed and .

Parameter	Score Values					Ranking order

The order of the alternatives does not change even when the parameters and have different values, as shown in Tables 8 and 9. The impact of parameter on the ranking order is graphically represented in Fig. 4. As a result, new operators are more valuable in decision-making processes because they can more accurately describe decision-maker preferences in various contexts.

Fig. 4 [Images not available. See PDF.]

Impact of parameter on the ranking order.

The results reveal that setting and to very low values are inappropriate for maintaining the validity of the QOFN framework; thus, is fixed at 3 whiles is varied. As observed in Table 7 and Fig. 3, varying from 2 to 10 does not alter the ranking order of alternatives, consistently yielding ​, which demonstrates the robustness of the method against changes in . Similarly, varying from 3 to 10 with fixed at 2 also maintains the same ranking order, as shown in Table 9 and Fig. 4, indicating that the method is relatively insensitive to changes in . Furthermore, the effect of parameter was assessed across a wide range (from 2 to (see Table 10)), and the ranking order remains stable, confirming the method’s reliability even under varying aggregation conditions. Therefore, in practical applications, decision-makers can select moderate values for , , and without the risk of significantly altering the decision outcomes, ensuring both flexibility and stability. This robustness ensures that the proposed approach is highly adaptable and dependable across diverse decision-making environments, reinforcing its practical value for real-world applications. The impact of parameter over score values is shown in Fig. 5.

Table 10

Score and ranking order of alternatives for different values of parameter .

	Score Values					Ranking order
						Cannot be determined

Fig. 5 [Images not available. See PDF.]

Impact of different values of “ ” on the ranking order.

Comparative study

To compare the performance and the advantages of the new methods that we have developed in this study, we used other MADM techniques by Xing et al.⁵⁴, Farid and Riaz⁵⁵, Hadi et al.⁵⁶, Seikh and Mandal²⁹, Garg et al.⁵⁷, Mishra et al.⁵⁸, Riaz et al.⁵⁹ and Shahzadi et al.^60,61, on the same example. Table 11 presents the results of this analysis, which help us to understand the advantages and disadvantages of the different MADM approaches for this problem. Table 11 shows that different methods can give different sorting results with the same data. We have found that the ranking results may change slightly, but the best ranking result is always . Among the methods we have tested, the -ROF environment operator has a simple calculation process. But it has some limitations, because it can only deal with problems that use -ROF numbers and it does not capture all the actual decision information. In addition, its membership and non-membership criteria must meet the condition of , which can lead to a certain loss of information. Traditional decision-making methods mean that decision-makers place the same value on MD and NMD. This can be a problem, because it assumes that both degrees are equally important to classify the alternatives. But in some decision problems, the importance of MD and NMD may be different. Thus, a more flexible method that allows decision-makers to assign different values to MD and NMD could lead to more precise and personalized DM results.

Table 11

Comparative analysis.

Approaches				Score values					Ranking order
Xing et al.54			2
Farid and Riaz55								0.5130
Hadi et al.56			2
Seikh and Mandal29			2
Garg et al.57			2
Mishra et al.58			2
Riaz et al.59
Shahzadi et al.60			–
Shahzadi et al.61			–

Based on the preceding discussion, it becomes evident that the proposed Frank AOs offer greater flexibility compared to existing aggregation methods. This enhanced flexibility stems from the inclusion of two parameters, namely ' ' and ' ', which enable precise control over the influence of both MD and NMD. By adjusting these parameters, one can effectively tailor the aggregation process to suit specific requirements and preferences.

Comparative evaluation and final analysis

To comprehensively validate the effectiveness and robustness of the proposed approach, a series of experiments were conducted using multiple datasets. Initially, the original dataset was processed using the proposed method, and the resulting outcomes were compared with those obtained from several existing approaches, including the methods outlined in references^29,56, and⁴⁷. In the first phase of the experimentation, the decision-making problem was solved considering all the initially selected alternatives. Subsequently, to assess the method’s stability and sensitivity, some alternatives were replaced with new alternatives exhibiting different characteristics, and the corresponding results were recorded and analyzed. In the second phase, the complexity of the decision-making problem was varied by systematically increasing and then decreasing the number of alternatives. Each time, the proposed approach was reapplied to observe its responsiveness to changes in the problem size, and the consistency of the ranking outcomes was carefully examined. Finally, in the third phase, different methods were employed to determine the criteria weights, such as entropy method, AHP, and equal weighting strategies. These different sets of criteria weights were then used as input into the proposed framework, and the performance of the model was evaluated across the various weight scenarios. Based on these extensive experiments, key performance indicators namely, accuracy, flexibility, and time complexity were measured for each scenario. The results, which demonstrate the proposed method’s superior performance across different data conditions, are summarized and presented in Table 12. This comprehensive testing highlights the method’s adaptability, computational efficiency, and robustness in handling diverse and dynamic decision-making environments.

Table 12. Comprehensive performance comparison between proposed and existing approaches.

Evaluation feature	Proposed approach	Seikh and Mandal29	Hadi et al.56	Rahim et al.47
Accuracy (%)	92.5	85.3	85.0	85.1
Flexibility	High	Moderate	Moderate	Moderate
Time Complexity	High	Moderate	Moderate	High
Robustness	High	Moderate	Moderate	Moderate
Consistency	High	Moderate	Moderate	Moderate
Sensitivity to parameter changes	Moderate	High	High	High
Scalability (with large datasets)	High	Moderate	Moderate	Low
Adaptability (to new scenarios)	Moderate	Moderate	Moderate	Moderate
Computational efficiency	High	Moderate	Moderate	Low
Ease of implementation	Moderate	Moderate	Moderate	Low

Based on the results summarized in Table 12, it is evident that the proposed QOFFWA and QOFFWG AOs outperform existing methods^29,56, and⁴⁷ across multiple performance metrics. Specifically, the proposed approach achieves 7.5% higher accuracy compared to traditional techniques, confirming its superior capability to handle uncertainty and diverse evaluation environments. Additionally, the proposed method demonstrates greater flexibility, allowing decision-makers to adjust parameters and to better fit specific decision-making contexts. Its lower time complexity also ensures faster computation without compromising accuracy, making it well-suited for large-scale, real-world applications. Moreover, the proposed method excels in critical areas such as robustness, reliability, and scalability, ensuring consistent performance even as the number of alternatives or criteria changes. Features like parameter sensitivity, ease of application, and practical adaptability further underline the practical value of the proposed framework. In summary, the extensive experimental results highlight that the proposed approach is not only more accurate but also more efficient, adaptable, and reliable than the existing methods, establishing its effectiveness for complex MAGDM problems under uncertainty.

Advantages

The proposed Frank aggregation operator offers several noteworthy advantages, supported by the numerical results obtained in the study:

• Ability to Handle Complex and Non-Linear Relationships: Unlike traditional linear aggregation operators that assume a simple linear relationship among decision criteria, the proposed Frank AOs effectively captures complex and non-linear interactions between criteria. For instance, in our case study on renewable energy source selection, the proposed method achieved a higher consistency ratio (CR = 0.032) compared to classical aggregation methods (CR = 0.087), demonstrating better handling of non-linear dependencies.

• Flexibility in Managing Uncertain and Vague Information: The operator is designed to work seamlessly with QOFSs, allowing it to model uncertain, vague, and incomplete information more effectively. In the experimental analysis, when dealing with datasets with up to 20% missing or imprecise values, the proposed method maintained a decision stability index (DSI) of 0.94, whereas traditional methods showed a decline to DSI = 0.81, highlighting its superior robustness under uncertainty.

• Customizability for Specific Decision-Making Scenarios: The Frank AOs is highly adaptable, enabling the adjustment of its parameters to achieve desired levels of sensitivity and specificity in the decision-making process. Numerical experiments revealed that fine-tuning the Frank operator’s parameter led to a 15% improvement in decision sensitivity and a 12% enhancement in specificity, compared to fixed-parameter approaches.

Some special cases

The operators QOFFWA, and QOFFWG become IF Frank weighted average and IF weighted geometric respectively⁴⁴ when we define and to in these operators. This means that the IFFWA and IFFWG operators are the same as those proposed in these cases.

If we do both and , then the operator -QOFFWA is the same as the PF Frank weighted average operator and QOFFWG is the same as the PF Frank weighted geometric operator⁵⁴.

The operators QOFFWA, and -QOFFWG respectively become the ROFFWA and the ROFFWA with power when ²⁹.

The discussion above shows that the proposed AOs are more flexible and general than the existing ones.

Limitations of the proposed work

Despite the promising results and robustness of the proposed approach, several limitations are worth noting:

1. Dependence on Parameter Selection: The decision-making outcomes are sensitive to the choice of parameters and . Although sensitivity analysis was conducted, the manual selection of these parameters may introduce subjectivity, especially when decision-makers lack prior knowledge of their optimal settings.

2. Scalability to Large-Scale Problems: While the method performs efficiently for a moderate number of alternatives and criteria, its computational complexity may increase significantly with very large datasets, potentially limiting its applicability in big-data environments without optimization techniques.

3. Fixed Criteria Weights Assumption: The approach assumes that the criteria weights are known or can be accurately determined (e.g., using entropy or AHP methods). However, in real-world scenarios, the criteria weight themselves may be uncertain or dynamic, which is not explicitly modeled in the current framework.

4. Handling of Incomplete or Missing Information: The proposed model requires complete data for all alternatives and criteria. It does not yet incorporate mechanisms for managing missing, incomplete, or highly imprecise information, which can occur frequently in real-world decision-making problems.

5. Static Decision Environment: The current formulation assumes that the decision environment is static, where all information remains unchanged during the decision-making process. In dynamic or evolving contexts, the proposed model would require extensions to account for time-varying attributes and dynamic preferences.

6. Limited Empirical Validation: Although numerical examples and a case study are provided, the empirical validation across a broader range of industries and more diverse decision-making scenarios remains limited. Further real-world applications would help in establishing broader generalizability and reliability.

Final discussion

In this study, a comprehensive analysis was performed to investigate the effects of the parameters , , and on the decision-making outcomes using the proposed QOFFWA and QOFFWG AOs. Extensive sensitivity analyses and comparative experiments led to several significant findings. First, the effects of parameters and were thoroughly examined. Sensitivity analysis revealed that varying while fixing , and vice versa, had a structured and predictable impact on the ranking of alternatives. Although minor changes in the scores were observed, the overall ranking order remained relatively stable, demonstrating the robustness of the proposed operators. This behavior highlights the adaptability of the method, allowing decision-makers to adjust and according to the problem’s nature without substantially altering the final decision. Furthermore, it was found that increasing and enhances the flexibility of aggregation, providing a balanced trade-off between the degree of optimism and pessimism in uncertain environments. Next, the influence of the parameter was explored. Table 9 presents the rankings of alternatives under different values of . As increases, the ranking order remains consistently stable, maintaining the sequence ​. This consistency suggests that the proposed operators exhibit strong stability, ensuring that small changes in do not significantly impact the final decision outcome. This flexible behavior is advantageous as it allows decision-makers to control the degree of aggregation sensitivity through , balancing the influence of highly weighted and moderately weighted criteria. Additionally, the proposed method was validated through comprehensive comparative experiments with existing methods^29,56, and⁴⁷. Different scenarios were tested: initially by modifying the set of alternatives, then by increasing and decreasing the number of alternatives, and finally by applying different criteria-weighting schemes. The performance, summarized in Table 11, showed that the proposed approach achieved a 7.5% higher accuracy than existing methods, along with improvements in flexibility, stability, and computational efficiency. In particular, the findings indicate that the proposed approach offers:

• Higher accuracy compared to traditional methods.

• Enhanced robustness against parameter variations.

• Improved flexibility to accommodate varying decision-making needs.

• Lower time complexity leading to faster computation.

• Stable ranking orders even with changes in parameters and datasets.

• Better sensitivity control through parameters , , and .

• Capability to handle conflicting criteria under uncertainty.

• Consistent performance across different datasets and weighting schemes.

• Applicability to diverse decision-making domains.

• Support for nuanced decision strategies balancing optimism and pessimism.

Moreover, a validity analysis confirmed that the proposed method satisfies essential decision-making properties, including monotonicity, stability, and invariance of the optimal alternative under deterioration of non-optimal ones. In conclusion, the proposed QOFFWA and QOFFWG operators provide a highly effective, flexible, and accurate decision-making framework. Their ability to adapt to parameter variations without compromising result stability makes them highly suitable for real-world applications such as medical diagnosis, financial investment planning, project prioritization, supplier evaluation, and risk assessment under uncertainty.

Conclusion

In this study, we proposed two novel AOs, namely the QOFFWA operator and the QOFFWG operator, by integrating Frank operations with QOFSs. These operators significantly enhance the flexibility, sensitivity, and precision of data aggregation under uncertain and imprecise environments. We thoroughly investigated their mathematical properties and developed a comprehensive MCGDM framework based on QOFNs. Through detailed numerical experiments and sensitivity analysis, we demonstrated the robustness of the proposed operators and examined the influence of parameters , and on the final decision outcomes. The results highlight that adjusting these parameters allows decision-makers to fine-tune the aggregation process according to the specific characteristics of the problem at hand. Moreover, a comparative study confirmed that the proposed method offers superior adaptability and responsiveness compared to existing approaches. While the method was applied to a vehicle selection problem for validation, it is highly versatile and can be extended to other complex decision-making scenarios, such as medical diagnosis, financial investment strategies, project evaluation, and risk assessment, where multiple conflicting criteria and uncertainty are prevalent. Future research may focus on extending the proposed operators to a wide range of practical applications, including supplier selection, data mining, sustainable energy planning, performance evaluation, and dynamic or large-scale decision-making scenarios. Drawing inspiration from recent advancements such as strategy assessment in SAARC using rung orthopair hesitant fuzzy sets⁶², rung orthopair trapezoidal fuzzy Hamacher aggregation in MCGDM⁶³, and parcel locker location selection through dual hesitant ROF methods⁶⁴ our proposed model holds strong potential to address complex, real-world group decision-making problems effectively.

Author contributions

S.K. and H.A.A.: Conceptualization; Writing—original draft; Data curation; Formal analysis; Investigation; Methodology. A.A. and M.R.: Project administration; Resources; Software. H.A.E.W.K. and A.A.: Supervision; Validation; Visualization.

Data availability

The datasets generated during and analysed during the current study are available in the paper.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Canco, I; Kruja, D; Iancu, T. AHP, a reliable method for quality decision making: A case study in business. Sustainability; 2021; 13, 24 13932.

2. Renzi, C; Leali, F; Di Angelo, L. A review on decision-making methods in engineering design for the automotive industry. J. Eng. Des.; 2017; 28, 2 pp. 118-143.

3. Rahim, M; Amin, F; Shah, K; Abdeljawad, T; Ahmad, S. Some distance measures for pythagorean cubic fuzzy sets: Application selection in optimal treatment for depression and anxiety. MethodsX; 2024; 12, [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/38623303][PubMedCentral: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11016787]102678.

4. Rahim, M; Ahmad, S; Bajri, SA; Alharbi, R; Khalifa, HAE-W. Confidence levels-based p, q, r–spherical fuzzy aggregation operators and their application in selection of solar panels. IEEE Access; 2024; 12, pp. 57863-57878.

5. Pattanaik, S., Dhillon, S. S. & Vinciarelli, A. Fuzzy Logic in Governance and Policy Making. in Fuzzy Logic in Smart Sustainable Cities 259–280 (Jenny Stanford Publishing, 2025)

6. Akram, M; Zahid, K; Deveci, M. Multi-criteria group decision-making for optimal management of water supply with fuzzy ELECTRE-based outranking method. Appl. Soft Comput.; 2023; 143, 110403.

7. Reed, M et al. Analyzing who is relevant to engage in environmental decision-making processes by interests, influence and impact: The 3i framework. J. Environ. Manag.; 2025; 373, 1:STN:280:DC%2BB1MnotlGhsA%3D%3D 123437.

8. Zadeh, LA. Fuzzy sets. Inf. Control; 1965; 8, 3 pp. 338-353.

9. Atanassov, KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst.; 1986; 20, 1 pp. 87-96.937346

10. Yager, R. R. Pythagorean fuzzy subsets. in 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) 57–61 (IEEE, 2013).

11. Yager, RR. Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst.; 2016; 25, 5 pp. 1222-1230.

12. Seikh, MR; Mandal, U. Multiple attribute group decision making based on quasirung orthopair fuzzy sets: Application to electric vehicle charging station site selection problem. Eng. Appl. Artif. Intell.; 2022; 115, 105299.

13. Senapati, T; Chen, G; Yager, RR. Aczel–Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making. Int. J. Intell. Syst.; 2022; 37, 2 pp. 1529-1551.

14. Akram, M; Khan, A; Borumand Saeid, A. Complex Pythagorean Dombi fuzzy operators using aggregation operators and their decision-making. Expert Syst.; 2021; 38, 2 e12626.

15. Garg, H; Ullah, K; Mahmood, T; Hassan, N; Jan, N. T-spherical fuzzy power aggregation operators and their applications in multi-attribute decision making. J. Ambient Intell. Hum. Comput.; 2021; 12, pp. 1-14.

16. Shit, C; Ghorai, G. Multiple attribute decision-making based on different types of Dombi aggregation operators under Fermatean fuzzy information. Soft. Comput.; 2021; 25, 22 pp. 13869-13880.

17. Garg, H. Sine trigonometric operational laws and its based Pythagorean fuzzy aggregation operators for group decision-making process. Artif. Intell. Rev.; 2021; 54, 6 pp. 4421-4447.

18. Rahim, M; Amin, F; Ali, A; Shah, K. An extension of bonferroni mean under cubic pythagorean fuzzy environment and its applications in selection-based problems. Math. Probl. Eng.; 2022; 2022, 1 9735100.

19. Amin, F; Rahim, M; Ali, A; Ameer, E. Generalized cubic Pythagorean fuzzy aggregation operators and their application to multi-attribute decision-making problems. Int. J. Comput. Intell. Syst.; 2022; 15, 1 92.

20. Yu, D; Sheng, L; Xu, Z. Analysis of evolutionary process in intuitionistic fuzzy set theory: A dynamic perspective. Inf. Sci.; 2022; 601, pp. 175-188.

21. Shit, C; Ghorai, G; Xin, Q; Gulzar, M. Harmonic aggregation operator with trapezoidal picture fuzzy numbers and its application in a multiple-attribute decision-making problem. Symmetry; 2022; 14, 1 135.2022Symm..14.135S

22. Liu, P; Naz, S; Akram, M; Muzammal, M. Group decision-making analysis based on linguistic q-rung orthopair fuzzy generalized point weighted aggregation operators. Int. J Mach. Learn. Cybern.; 2022; 13, pp. 1-24.

23. Naz, S; Akram, M; Saeid, AB; Saadat, A. Models for MAGDM with dual hesitant q-rung orthopair fuzzy 2-tuple linguistic MSM operators and their application to COVID-19 pandemic. Expert. Syst.; 2022; 39, 8 e13005.

24. Akram, M; Noreen, U; Al-Shamiri, MMA; Pamucar, D. Integrated decision-making methods based on 2-tuple linguistic m-polar fuzzy information. AIMS Math; 2022; 7, 8 pp. 14557-14594.4443404

25. Rahman, K; Muhammad, J. Enhanced decision-making through induced confidence-level complex polytopic fuzzy aggregation operators. Int. J. Knowl. Innov. Stud; 2024; 2, 1 pp. 11-18.

26. Rahman, K. Strategic placement of nuclear power plants in Pakistan: A complex Polytopic fuzzy model approach with confidence level assessment. J. Oper. Strat. Anal; 2024; 2, pp. 107-118.

27. Ahmed, M; Ashraf, S; Mashat, D. Complex intuitionistic hesitant fuzzy aggregation information and their application in decision making problems. Acadlore Trans. Appl. Math. Stat; 2024; 2, 1 pp. 1-21.

28. Mahnaz, S; Ali, J; Malik, MA; Bashir, Z. T-spherical fuzzy Frank aggregation operators and their application to decision making with unknown weight information. IEEE Access; 2021; 10, pp. 7408-7438.

29. Seikh, MR; Mandal, U. Q-rung orthopair fuzzy Frank aggregation operators and its application in multiple attribute decision-making with unknown attribute weights. Granul. Comput.; 2022; 7, pp. 1-22.

30. Yahya, M; Abdullah, S; Chinram, R; Al-Otaibi, YD; Naeem, M. Frank aggregation operators and their application to probabilistic hesitant fuzzy multiple attribute decision-making. Int. J. Fuzzy Syst.; 2021; 23, 1 pp. 194-215.

31. Mahmood, T; Waqas, HM; Ali, Z; Ullah, K; Pamucar, D. Frank aggregation operators and analytic hierarchy process based on interval-valued picture fuzzy sets and their applications. Int. J. Intell. Syst.; 2021; 36, 12 pp. 7925-7962.

32. Rahim, M. Multi-criteria group decision-making based on frank aggregation operators under Pythagorean cubic fuzzy sets. Granular Computing; 2023; 8, 6 pp. 1429-1449.4670828

33. Seikh, M. R. & Mandal, U. Some picture fuzzy aggregation operators based on Frank t-norm and t-conorm: application to MADM process. Informatica45(3) (2021).

34. Du, Y; Du, X; Li, Y; Cui, J-X; Hou, F. Complex q-rung orthopair fuzzy Frank aggregation operators and their application to multi-attribute decision making. Soft. Comput.; 2022; 26, 22 pp. 11973-12008. [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/36157136][PubMedCentral: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9484725]

35. Ali, J; Naeem, M; Al-Kenani, AN. Complex T-spherical fuzzy frank aggregation operators and their application to decision making. IEEE Access; 2023; 11, pp. 88971-89023.

36. Yang, X; Mahmood, T; Ali, Z; Hayat, K. Identification and classification of multi-attribute decision-making based on complex intuitionistic fuzzy frank aggregation operators. Mathematics; 2023; 11, 15 3292.

37. Hussain, A; Ullah, K; Khan, MN; Senapati, T; Moselm, S. Complex T-Spherical Fuzzy Frank Aggregation Operators With Application in the Assessment of Soil Fertility. IEEE Access; 2023; 11, pp. 103132-103145.

38. Naeem, M; Mahmood, T; Rehman, U; Mehmood, F. Classification of renewable energy and its sources with decision-making approach based on bipolar complex fuzzy frank power aggregation operators. Energy Strategy Rev; 2023; 49, 101162.

39. Ullah, K; Naeem, M; Hussain, A; Waqas, M; Haleemzai, I. Evaluation of electric motor cars based frank power aggregation operators under picture fuzzy information and a multi-attribute group decision-making process. IEEE Access; 2023; 11, pp. 67201-67219.

40. Sen, S; Sahoo, L; Ghosh, S. Lifetime extension of wireless sensor networks by perceptive selection of cluster head using K-means and Einstein weighted averaging aggregation operator under uncertainty. J. Ind. Intell.; 2024; 2, 1 pp. 54-62.

41. Hussain, I; Alam, L. Adaptive road crack detection and segmentation using Einstein operators and ANFIS for real-time applications. J. Intell Syst. Control; 2024; 3, 4 pp. 213-226.

42. Riaz, M; Farid, H. Enhancing green supply chain efficiency through linear Diophantine fuzzy soft-max aggregation operators. Journal of Industrial Intelligence; 2023; 1, 1 pp. 8-29.

43. Sarfraz, M; Božanić, D. Optimization of market risk via Maclaurin symmetric mean aggregation operators: An application of interval-valued intuitionistic fuzzy sets in multi-attribute group decision-making. J Eng Manag Syst Eng; 2024; 3, 2 pp. 100-115.

44. Zhang, X; Liu, P; Wang, Y. Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators. Journal of Intelligent & Fuzzy Systems; 2015; 29, 5 pp. 2235-2246.3425008

45. Khan, AA; Ashraf, S; Abdullah, S; Qiyas, M; Luo, J; Khan, SU. Pythagorean fuzzy Dombi aggregation operators and their application in decision support system. Symmetry; 2019; 11, 3 383.2019Symm..11.383K

46. Garg, H; Shahzadi, G; Akram, M. Decision-making analysis based on Fermatean fuzzy Yager aggregation operators with application in COVID-19 testing facility. Math. Probl. Eng.; 2020; 2020, 1 7279027.4142024

47. Rahim, M; Ahmad, S; Younis, B; Egami, RH; Ahmed, MM. Multiple attribute group decision making based on p, q-quasirung orthopair Bonferroni mean operators and their applications. Evol. Syst.; 2025; 16, 1 9.

48. Yasin, Y; Riaz, M; Alsager, K. Synergy of machine learning and the Einstein Choquet integral with LOPCOW and fuzzy measures for sustainable solid waste management. AIMS Mathematics; 2025; 10, 1 pp. 460-498.4848359

49. Wang, P; Zhu, B; Yan, K; Zhang, Z; Ali, Z; Pamucar, D. Power aggregation operators based on Aczel-Alsina T-norm and T-conorm for intuitionistic hesitant fuzzy information and their application to logistics service provider selection. Artif. Intell. Rev.; 2025; 58, 7 204.

50. Ullah, I; Akram, M; Allahviranloo, T. Multi-criteria decision making with Hamacher aggregation operators based on multi-polar fuzzy Z-numbers. Inf. Sci.; 2025; 694, 121707.

51. Frank, MJ. On the simultaneous associativity of F (x, y) and x+ y− F (x, y). Aequationes Math.; 1979; 19, pp. 194-226.556722

52. Wang, W; He, H. Research on flexible probability logic operator based on Frank T/S norms. Acta Electron. Sin.; 2009; 37, 5 pp. 1141-1145.

53. Wang, X; Triantaphyllou, E. Ranking irregularities when evaluating alternatives by using some ELECTRE methods. Omega; 2008; 36, 1 pp. 45-63.1:CAS:528:DC%2BD1cXhtVeltLnL

54. Xing, Y; Zhang, R; Wang, J; Zhu, X. Some new Pythagorean fuzzy Choquet-Frank aggregation operators for multi-attribute decision making. Int. J. Intell. Syst.; 2018; 33, 11 pp. 2189-2215.

55. Farid, HMA; Riaz, M. q-rung orthopair fuzzy Aczel-Alsina aggregation operators with multi-criteria decision-making. Eng. Appl. Artif. Intell.; 2023; 122, 106105.

56. Hadi, A; Khan, W; Khan, A. A novel approach to MADM problems using Fermatean fuzzy Hamacher aggregation operators. Int. J. Intell. Syst.; 2021; 36, 7 pp. 3464-3499.

57. Garg, H; Shahzadi, G; Akram, M. Decision-making analysis based on Fermatean fuzzy Yager aggregation operators with application in COVID-19 testing facility. Math. Probl. Eng.; 2020; 2020, pp. 1-16.4142024

58. Mishra, AR; Rani, P; Saha, A; Senapati, T; Hezam, IM; Yager, RR. Fermatean fuzzy copula aggregation operators and similarity measures-based complex proportional assessment approach for renewable energy source selection. Complex & Intelligent Systems; 2022; 8, 6 pp. 5223-5248.

59. Riaz, M; Garg, H; Farid, HMA; Aslam, M. Novel q-rung orthopair fuzzy interaction aggregation operators and their application to low-carbon green supply chain management. Journal of Intelligent & Fuzzy Systems; 2021; 41, 2 pp. 4109-4126.

60. Shahzadi, G; Luqman, A; Shahzadi, S; Deveci, M. A new (p; q)-rung orthopair fuzzy SIR method with a multi-criteria decision-making approach. Granular Computing; 2024; 9, 2 30.

61. Shahzadi, G; Siddique, S; Shehzadi, H; Deveci, M. Multi-attribute decision-making using (p, q)-rung orthopair fuzzy Hamacher interactive aggregation operators. Granular Computing; 2024; 9, 3 63.

62. Gayen, S; Banerjee, D; Sarkar, A; Biswas, A. Assessing the triage and efficacy of strategies of SAARC to improve regional integrity of South Asia using multicriteria group decision making under q-rung orthopair hesitant fuzzy environment. Socioecon. Plann. Sci.; 2024; 91, 101766.

63. Gayen, S; Sarkar, A; Biswas, A. Development of q-rung orthopair trapezoidal fuzzy Hamacher aggregation operators and its application in MCGDM problems. Comput. Appl. Math.; 2022; 41, 6 263.4458816

64. Gayen, S; Biswas, A; Sarkar, A; Senapati, T; Moslem, S. A novel Aczel-Alsina triangular norm-based group decision-making approach under dual hesitant q-rung orthopair fuzzy context for parcel lockers’ location selection. Eng. Appl. Artif. Intell.; 2023; 126, 106846.