WASPAS-based multi-expert decision algorithm for physical education using circular pythagorean fuzzy aggregation with prioritized weights

Abstract

Decision-making is a crucial process that involves selecting a course of action from multiple alternatives. It requires evaluating different options, weighing their potential outcomes, and making a choice based on available information and objectives. In physical education, decision-making plays a vital role in instructing teachers and students to interact with each other, facilitating understanding and retention of previously learned material. In this article discusses decision-making in physical education for teaching and learning, aiming to create engaging and inspiring opportunities for students to learn and improve their thinking skills. To achieve this, we utilize a circular Pythagorean fuzzy set (CPyFS), which captures membership and non-membership grades while exhibiting circular behavior. This approach is particularly useful for handling uncertainty in assessment processes, allowing decision-makers ($\mathcal{D}$) to discuss their opinions within a large domain. The CPyFS technique builds upon fuzzy set (FS) theory, with Pythagorean FS (PyFS) being a special case. Our goal is to define weighted averaging and geometric operators within the CPyFS framework, based on triangular norms. We demonstrate the effectiveness of these aggregation operators (AOs) through examples and establish key mathematical properties, such as idempotency and boundedness. Furthermore, we propose a new model using the weighted aggregated sum product assessment (WASPAS) method, exploring its application in multi-criteria group decision-making (MCGDM). The effectiveness and reliability of this technique are validated through sensitivity analysis and comparative study, showcasing its potential in handling uncertainties in physical education.

Introduction

In our daily life problems, we know that people are facing distinct types of decision-making problems. Decision-making is the process of selecting a course of action from multiple alternatives. It involves evaluating different options, weighing their potential outcomes, and making a choice based on the available information and objectives. It can be applied in both personal and professional settings. Considering this issue, researchers are focused on learning how to make better decisions. The multi-criteria decision-making (MCDM) approach is of supreme interest to researchers who deal with decision-making analysis. MCDM is a process that evaluates and prioritizes different options based on multiple conflicting criteria. It is often used in complex decision scenarios where there is not just one single goal but multiple objectives that need to be balanced. Wei et al.¹ suggested the MCDM VIKOR method, Liu et al.² introduced MCDM methods based on intuitionistic fuzzy sets, Ullah et al.² elaborate bipolar-valued hesitant fuzzy sets in MCDM, and Mahmood et al.³ developed power aggregation operators in MCDM.

MCGDM is a sub-field of decision-making, which has some applications in various fields for research like in fuzzy set theory. It is an extension of the MCDM process, where a group of collaboratively evaluates multiple alternatives based on various criteria. This approach is commonly used in settings where decisions have to be made collectively, such as in organizations, public policy, or community planning, where different stakeholders with varying preferences and perspectives are involved. The assessment of physical education for teaching and learning is an MCGDM problem involving several alternative and incommensurable criteria. It is also more applicable in such situations where a group of criteria is utilized, and then the MCDM approach converts to the MCGDM approach. Wu⁴ adopted the MCGDM approach based on the fuzzy VIKOR method, Wang⁵ instead MCGDM model based on the improved ARAS method, Banik et al.⁶ explored agricultural-based MCGDM problems, and Abbas et al.⁷ suggested partitioned Hamy mean aggregation for MCGDM in the MAIRCA approach.

Literature study

The first time Zadeh⁸ defined FS, having a membership grade must be equal to or less than 1. FS cannot handle complicated fuzzy information. To resolve the restriction, Atanassov⁹ developed intuitionistic FS (IFS) having non-membership grades into the FS. Later on, Yager developed the concept of Pythagorean FS (PyFS)¹⁰ which is the generalized form of existing fuzzy frameworks.

In this context, we studied some other genuine extensions in the IFS framework, Atanassov¹¹ modified the concept of IFS to circular IFS (CIFS). In CIFS, membership and non-membership grades have the radius of each element in the universal set. The sum of both grades must equal to or less than 1, while the radius also is in closed interval 0 to 1. However, CIFS also has some limitations due to its small domain. Olgun et al.¹² introduced the concept of circular PyFS (CPyFS) by increasing the domain of CIFS. In CPyFS, the sum of the squares of both grades must be equal to or less than 1, while the radius also is in closed interval 0 to 1. The above scenarios of FS theory are briefly discussed in Figure 1.

Fig. 1

Shows extensions of FS theory.

Ali et al.¹³ developed circular q-rung orthopair FS with Dombi AOs and application to symmetry analysis in artificial intelligence. Moreover, disc spherical FS having circular behavior in fuzzy information, which is the generalized form CIFS and CPyFS. Ali¹⁴ introduced WASPAS-based decision aid approach with Dombi power AOs under disc spherical fuzzy framework.

Physical education is a crucial component that affects both teaching effectiveness and student learning outcomes. It involves a range of choices made by teachers and students before, during, and after physical education lessons. There are many frameworks that a teacher could use to support students with different interests, abilities, and learning styles. By adopting a variety of strategies and tools, you can create a more dynamic, inclusive, and engaging physical education environment that fosters lifelong physical activity habits in students. These factors have their properties and are complicated and uncertain information like in FS theory. For this, the best evaluation process in fuzzy phenomena and fuzzy concepts is an MCDM and MCGDM approach because it can easily summarize the opinions of the evaluation factors. Many researchers have worked in this field. We may refer to; Bailey¹⁵ adopted physical education and sports in schools. Dyson¹⁶ elaborated on the quality of physical education. Wang¹⁷ modified the fuzzy comprehensive evaluation of physical education based on high-dimensional data mining. Yu¹⁸ explored computer information technology in college physical education using fuzzy evaluation theory. Lv¹⁹ designed the fuzzy rule set for college physical education evaluation with life education programs.

Motivation

In this research article, we extend one of the most used MCGDM approaches based on the WASPAS technique with the CPyF version. Zavadskas et al.²⁰ introduced the concept of the WASPAS technique which has two parts that are, the weighted sum model (WSM) and weighted product model (WPM). The WASPAS technique has been one of the popular decision-making methods and an advanced version of the MCGDM approach, which is used to evaluate and rearrange different individuals based on multiple criteria. As we know, the WASPAS technique can be easily replaced with WSM and WPM, since it provides us a more significant and effective solution, and yields a more composite evaluation than its components. Also, this method provides a systematic approach to decision-making, considering both qualitative and quantitative factors and allowing to assess and prioritize alternatives based on their preferences and criteria importance. The environments of FS have been evaluating the WASPAS technique since they can take the uncertainty of the CPyF linguistic terms in the assessment situation. In this manuscript, we illustrate its application through a physical education approach. The detailed summary of WASPAS technique frameworks is discussed in Table 1.

Table 1 In the past some publications based on the WASPAS model.

In modern-day scenarios, criteria are not independent but it has some connection and influences each other. For instance, decision-making in physical education might lead to increased benefits and quality. Therefore, researchers are considering interdependence between criteria to enhance the accuracy, realism, and effectiveness of the decision-making process in complex situations. However, in Table 1, we observed that the existing work based on the WASPAS method does not handle such problems where the information is in the form of CPyFS. To resolve the restriction, we defined the WASPAS method in the framework of CPyF information based on linguistic terms. Next, we discussed in detail the evaluation process and algorithm of the WASPAS method for the MCGDM approach. Therefore, we observed that the information presented in Table 1, will become the special case of the proposed WASPAS technique. Also, the developed WASPAS technique and AOs are very effective and superior because of their future based on the above advantages.

CPyFS is an extension of PyFS, which itself is an extension of IFS. The primary motivation behind using CPyFS is to address the limitations of traditional fuzzy set theory in handling complex, uncertain, and vague information. The limitation of existing FS theory we may refer to, in CIFS, membership and non-membership grades have the radius of each element in the universal set. The sum of both grades must equal to or less than 1, while the radius also is in closed interval 0 to 1. On the other hand, in CPyFS, the sum of the squares of both grades must be equal to or less than 1, while the radius also is in closed interval 0 to 1. It means that CPyFS increasing the domain of CIFS. In real-world applications is briefly discussed in the following:

• Decision-making in Economic Growth and Digital Economy: CPyFS can be applied to evaluate the impact of digital infrastructure investments on economic growth, considering multiple criteria and uncertain data.

• Risk Assessment in Finance: CPyFS can be used to model periodic patterns in financial data, such as stock prices or credit risk, and provide more accurate risk assessments.

• Environmental Sustainability Evaluation: CPyFS can be applied to assess the sustainability of environmental policies, considering multiple criteria and uncertain data.

By using CPyFS, the study can provide a more comprehensive and accurate evaluation of economic growth and digital economy, considering the complexities and uncertainties involved. The connection to real-world applications highlights the practical significance of the study and its potential impact.

Research gap

CPyFS has played a significant role in FS theory because it has circular information for each element on the universal set. Various kinds of operators and methods were proposed based on it by the researcher. During the revision and research analysis, we feel that the existing approaches can be extendable. Additionally, we observed that researchers have faced some queries like.

• 1. How do we define the WAO and WGO in the framework of CPyF information?

• 2. How do we define linguistic terms for the evaluation of alternatives to select the best opinion of the ?

• 3. How do we prioritize the weighted vector for the importance of criteria?

• 4. How do we define the new model of the WASPAS technique for the aggregation?

• 5. How do we collect the data of alternative corresponding criteria into a singleton set by utilizing AOs?

• 6. How do we get a suitable selection between the collections of any finite number of alternatives?

In the above situation, the proposed work of this manuscript can be handled easily different types of problems in the fuzzy research field and fill the gap. Anyhow, we observed that the WAO WGO and WASPAS techniques can be extendable in the framework of the CPyF information, which is very authentic and flexible in FS theory. The new approach is significant to the uncertainty data in an identification process that requires a large domain for to discuss their opinion for each element on the universal set, which must belong in closed intervals from 0 to 1. The previous approaches cannot give us such information for the evaluation of alternative concerning criteria and prioritization of weights for the importance of criteria in the framework of CPyF information. Therefore, the proposed work fills this gap.

Advantages

The advantage of the WASPAS technique is that it can also solve those problems which have two types of approaches: benefit criteria and cost criteria. Alternative and criteria are not independent but it has some connection and influences each other. For instance, decision-making in physical education might lead to increased benefits and quality. Therefore, researchers are considering the relation between criteria to enhance the accuracy and effectiveness of the decision-making process in complex situations. However, the initiated work presented in Table 1 cannot evaluate the problems whose information is in the form of CPyFS. To resolve the restriction, we defined the WASPAS technique in the framework of CPyF information based on linguistic terms and prioritized weights for the importance of criteria which are the main advantages of the new work. The other advantage of the proposed work is in informing of circular information which can also address the radius of membership and non-membership grades for the evaluation of alternatives against criteria. Additionally, we observed that the information presented in Table 1, will become the special case of the proposed WASPAS technique. Also, the proposed work is effective and best because of its future scope which is briefly discussed in this context.

Main objectives of the proposed work

In this article, we consider the initiated AOs and WASPAS techniques in the CPyF environment based on linguistic terms. The major objectives are listed below:

• 1. Consideration five different types of alternatives in physical education to choose one of the best. For evaluation, we utilized the MCGDM approach.

• 2. The selected five different types of criteria for the evaluation of the corresponding alternative. The weights of the criteria must be equal to the order of criteria with a constraint sum of the weights equal to 1.

• 3. Define a solution for the prioritization of weights for the importance of criteria and utilize it to address the evaluation of the MCGDM approach.

• 4. Resolution of a numerical example based on the MCGDM approach under the CPyF context to show the reliability of the derived theory.

• 5. Check the influence result by assigning different parametric values on various options of alternatives. Interestingly, the ranking of the alternatives is obtained in the same way from each parametric value.

• 6. The initiated work is compared with earlier established works to show the reliability and effectiveness of the new work.

Contribution of the proposed work

In this manuscript, all main contribution is briefly discussed in the following:

• 1. Recall related definitions and their basic laws, score value, accuracy value, and AOs. Then initiate WAO and WGO in the environment of CPyFS.

• 2. To adopt the linguistic terms in the framework of the CPyF context for the evaluation of alternatives and the importance of concern criteria.

• 3. Prioritization of weights for the importance of criteria and utilization it for the evaluation process of MCGDM problem in the framework of CPyF information.

• 4. Based on the initiated AOs, we defined the new model of the WASPAS technique and then utilized it to aggregate the decision matrix (DM) into a singleton set.

• 5. For the satisfaction of our proposed work, we study a numerical example under consideration of physical education based on the MCGDM approach.

• 6. Limitations of sensitivity analysis to check the influence result by assigning different parametric values on various options of alternatives.

• 7. Finally, a comparative study in which we contrast the proposed approaches with earlier established work.

The geometric representation of the article is discussed in Figure 2.

Fig. 2

An overview of the proposed theory.

Structure of the Article

This article is computed based on CPyF information with some AOs and WASPAS techniques. The remaining part of this article is organized as follows: Sect. 1 presents an overview of this research work. In Sect. 2, we recall-related definitions, operation laws, score value, accuracy value, and AOs. Section 3 has new AOs with some significant mathematical properties in the environment of CPyF information that were established. Section 4, discussed the extension of the WASPAS technique for solving the MCGDM problem under the presence of linguistic sets in the CPyF context. In Sect. 5, we applied our proposed work to a numerical example, to show the reliability and effectiveness of our proposed approaches. Section 6 has a sensitivity analysis of the proposed operators and Sect. 7 highlights the comparison of the new operators with the earlier established work. Section 8 provides the conclusion, limitations, and future research scopes of the article are summarized.

Background

In this section, we recall some definitions of FS theory, basic operations, score function, and accuracy function needed in this article.

Circular pythagorean fuzzy set and their basic laws

In this subsection, we briefly discussed the idea of CPyFS and its basic operational laws, score value, and accuracy value of CPyFS. It is an essential tool for coping with uncertain and unpredictable information in daily life problems.

Definition 1

¹²: On a fixed universal set $\:\mathcal{U}$. The CPyFS $\:A$ on $\:\mathcal{U}$ is signified as: $\:A=\left\{\left({\mathcal{M}}_{A}\left(\mathcal{X}\right),{\mathcal{N}}_{A}\left(\mathcal{X}\right);{\mathcal{R}}_{A}\left(\mathcal{X}\right)\right)|\mathcal{X}\in\:\mathcal{U}\right\}$, where the membership grade is represented $\:{\mathcal{M}}_{A}\left(\mathcal{X}\right)$, non-membership grade is shown by $\:{\mathcal{N}}_{A}\left(\mathcal{X}\right)$ with a constraint for each element $\:\mathcal{X}\in\:\mathcal{U}$ in $\:A$ required that $\:Sum\left({\left({\mathcal{M}}_{A}\left(\mathcal{X}\right)\right)}^{2},{\left({\mathcal{N}}_{A}\left(\mathcal{X}\right)\right)}^{2}\right)\in\:\left[\text{0,1}\right]$, while the radius of the circle around the membership and non-membership grades on the plane is represented by $\:{\mathcal{R}}_{A}\left(\mathcal{X}\right)$. Further, $\:{\mathcal{H}}_{A}\left(\mathcal{X}\right)=\sqrt{1-Sum\left({\left({\mathcal{M}}_{A}\left(\mathcal{X}\right)\right)}^{2},{\left({\mathcal{N}}_{A}\left(\mathcal{X}\right)\right)}^{2}\right)}$ is called the refusal grade of $\:\mathcal{X}\in\:\mathcal{U}$ in $\:A$. For convenience, the mathematical shape of CPyF value (CPyFV) is denoted by $\:{A}_{\mathcal{j}}=({\mathcal{M}}_{{A}_{\mathcal{j}}},{\mathcal{N}}_{{A}_{\mathcal{j}}},{\mathcal{R}}_{{A}_{\mathcal{j}}})$, $\:\left(\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}\right)$.

Definition 2

¹²: Let $\:\mathcal{M}\left({\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)$, $\:\mathcal{N}\left({\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)$, and $\:\mathcal{R}\left({\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)$ belong to $\:\left[\text{0,1}\right]$ with a constraint for each element $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\in\:\mathcal{U}$ in $\:A$ required that $\:Sum\left({\left(\mathcal{M}\left({\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)\right)}^{2},{\left(\mathcal{N}\left({\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)\right)}^{2}\right)\in\:\left[\text{0,1}\right]$. Then, the CPyFV is denoted by $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}$=$\:\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)$=$\:\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\left(\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}\right)$.

Definition 3

¹²: Let $\:{A}_{1}=\left({\mathcal{M}}_{{A}_{1}},{\mathcal{N}}_{{A}_{1}},{\mathcal{R}}_{{A}_{1}}\right)$ and $\:{A}_{2}=\left({\mathcal{M}}_{{A}_{2}},{\mathcal{N}}_{{A}_{2}},{\mathcal{R}}_{{A}_{2}}\right)$ be the two CPyFVs on the universal set $\:\mathcal{U}$ with different radius and $\:\lambda\:>0$. Then, we described some basic algebraic operations between them, such as:

$$\:{A}_{1} \oplus {A}_{2}=\left(\begin{array}{c}\sqrt{{\left({\mathcal{M}}_{{A}_{1}}\right)}^{2}+{\left({\mathcal{M}}_{{A}_{2}}\right)}^{2}-{\left({\mathcal{M}}_{{A}_{1}}\right)}^{2}{\left({\mathcal{M}}_{{A}_{2}}\right)}^{2}},\\\:{\mathcal{N}}_{{A}_{1}}{\mathcal{N}}_{{A}_{2}};\\\:\sqrt{{\left({\mathcal{R}}_{{A}_{1}}\right)}^{2}+{\left({\mathcal{R}}_{{A}_{2}}\right)}^{2}-{\left({\mathcal{R}}_{{A}_{1}}\right)}^{2}{\left({\mathcal{R}}_{{A}_{2}}\right)}^{2}}\end{array}\right)$$

(1)

$$\:{A}_{1} \otimes {A}_{2}=\left(\begin{array}{c}{\mathcal{M}}_{{A}_{1}}{\mathcal{M}}_{{A}_{2}},\\\:\sqrt{{\left({\mathcal{N}}_{{A}_{1}}\right)}^{2}+{\left({\mathcal{N}}_{{A}_{2}}\right)}^{2}-{\left({\mathcal{N}}_{{A}_{1}}\right)}^{2}{\left({\mathcal{N}}_{{A}_{2}}\right)}^{2}};\\\:{\mathcal{R}}_{{A}_{1}}{\mathcal{R}}_{{A}_{2}}\end{array}\right)$$

(2)

$$\:\lambda\:{A}_{1}=\left(\begin{array}{c}\sqrt{1-{\left(1-{\left({\mathcal{M}}_{{A}_{1}}\right)}^{2}\right)}^{\:\lambda\:}},\\\:{\left({\mathcal{N}}_{{A}_{1}}\right)}^{\:\lambda\:};\sqrt{1-{\left(1-{\left({\mathcal{R}}_{{A}_{1}}\right)}^{2}\right)}^{\:\lambda\:}}\:\end{array}\right)$$

(3)

$$\:{\left({A}_{1}\right)}^{\:\lambda\:}=\left(\begin{array}{c}{\left({\mathcal{M}}_{{A}_{1}}\right)}^{\:\lambda\:},\\\:\sqrt{{1-\left(1-{\left({\mathcal{N}}_{{A}_{1}}\right)}^{2}\right)}^{\:\lambda\:}};{\left({\mathcal{R}}_{{A}_{1}}\right)}^{\:\lambda\:}\end{array}\right)$$

(4)

$$\:{\left({A}_{1}\right)}^{c}=\left\{{\mathcal{N}}_{{A}_{1}},{\mathcal{M}}_{{A}_{1}},{\mathcal{R}}_{{A}_{1}}\right\}$$

(5)

Definition 4

³²: The score value $\:\left(\mathcal{S}\right)$ and accuracy value $\:( \mathop {\text{A}}\limits_{ \cdot }$) of the sorting CPyFV $\:A=\left({\mathcal{M}}_{A},{\mathcal{N}}_{A},{\mathcal{R}}_{A}\right)$ is defined by

$$\:\mathcal{S}\left(A\right)=\left({\left({\mathcal{M}}_{A}\right)}^{2}-{\left({\mathcal{N}}_{A}\right)}^{2}\right)\times\:{\mathcal{R}}_{A}\in\:\left[-\text{1,1}\right]$$

(6)

$$\:( \mathop {\text{A}}\limits_{ \cdot } )\left(A\right)=\left({\left({\mathcal{M}}_{A}\right)}^{2}+{\left({\mathcal{N}}_{A}\right)}^{2}\right)\times\:{\mathcal{R}}_{A}\in\:\left[\text{0,1}\right]$$

(7)

Note That: The order relation between the two CPyFVs is given below:

• $\:{A}_{1}<{A}_{2}$ if and only if $\:\mathcal{S}\left({A}_{1}\right)<\mathcal{S}\left({A}_{2}\right)$ or $\:\mathcal{S}\left({A}_{1}\right)=\mathcal{S}\left({A}_{2}\right)$

• $\:{A}_{1}<{A}_{2}$ if and only if $\mathop {\text{A}}\limits_{ \cdot } (A_{1} ) < \mathop {\text{A}}\limits_{ \cdot } (A_{2} )$ or $\mathop {\text{A}}\limits_{ \cdot } (A_{1} ) = \mathop {\text{A}}\limits_{ \cdot } (A_{2} )$

Some proposed new fuzzy aggregation models

This section aims to define WAO, WGO, and valuable properties in the framework of CPyFS. The idea of CPyFS is a more effective, reliable, and flexible way of modeling complex and uncertain information when assign the data for decision-making and problem-solving in this field. The proposed technique has also identified the radius between membership grade and non-membership grade for each element of the universal set. Further, we demonstrate the proposed work with the help of some reliable examples, which demonstrate the initiated AOs.

Circular pythagorean fuzzy weighted averaging operator

In this subsection, we proposed a CPyF WAO (CPyFWAO) and utilized it with an example. Further, some valuable properties are proposed in the framework of CPyFS.

Definition 5

The CPyFWAO for CPyFVs $\:{A}_{\mathscr{j}}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ is of the form:

$$\:CPyFWAO\left({A}_{1},{A}_{2},{A}_{3},\dots\:,{A}_{\mathcal{j}}\right)=\sum\:_{\mathcal{j}=1}^{\mathcal{n}}{\omega\:}_{\mathcal{j}}{A}_{\mathcal{j}}$$

(8)

Where the weighted vector $\:\omega\:=\left({\:\omega\:}_{1},{\:\omega\:}_{2},\dots\:,{\:\omega\:}_{\mathcal{j}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ satisfied the constraint $\:{\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\ge\:0$ and $\:\sum\:{\:\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=1$.

Theorem 1

The aggregated value of a collection of CPyFVs $\:{\text{A}}_{\mathcal{j}}$ by using the CPyFWAO is also a CPyFV and is given below:

$$\:CPyFWAO\left({A}_{1},{A}_{2},{A}_{3},\dots\:,{A}_{\mathcal{j}}\right)=\left(\sqrt{1-\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{{\mathcal{M}}^{2}}_{\mathcal{j}}\right)}^{{\omega\:}_{\mathcal{j}}},}\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{{\mathcal{N}}_{\mathcal{j}}}^{{\omega\:}_{\mathcal{j}}},\sqrt{1-\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{{\mathcal{R}}^{2}}_{\mathcal{j}}\right)}^{{\omega\:}_{\mathcal{j}}}}\right)$$

(9)

Proof

See Appendix A.

Example 1

Let $\:{\mathcal{X}}_{1}=\left(\begin{array}{c}\text{0.90,0.30,0.60}\end{array}\right)$, $\:{\mathcal{X}}_{2}=\left(\begin{array}{c}\text{0.95,0.10,0.53}\end{array}\right)$, and $\:{\mathcal{X}}_{3}=\left(\begin{array}{c}\text{0.92,0.20,0.56}\end{array}\right)$ be the three CPyFVs on the universal set $\:\mathcal{U}=\left\{{\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right\}$ and the corresponding weights of criteria$\:\:\:\omega\:=\left(\text{0.25,0.40,0.35}\right)$, $\:\sum\:_{\mathcal{\:}\mathcal{j}=1}^{3}{\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=1$. Then, evaluate $\:CPyFWAO\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right)$.

Solution:

$$\:CPyFWAO\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right)=\left(\begin{array}{c}\sqrt{1-\left({\left(1-{0.90}^{2}\right)}^{0.25}\right)\left({\left(1-{0.95}^{2}\right)}^{0.40}\right)\left({\left(1-{0.92}^{2}\right)}^{0.35}\right)},\\\:{\left(0.30\right)}^{0.25}{\left(0.10\right)}^{0.40}{\left(0.20\right)}^{0.35},\\\:\sqrt{1-\left({\left(1-{0.60}^{2}\right)}^{0.25}\right)\left({\left(1-{0.53}^{2}\right)}^{0.40}\right)\left({\left(1-{0.56}^{2}\right)}^{0.35}\right)}\end{array}\right)=\left(\text{0.9300,0.1677,0.5575}\right)$$

Therefore, we can see that after utilization of our proposed operator defined in Eq. $\:\left(9\right)$, we get again a CPyFV. Hence, this example shows the reliability of our proposed approach.

Properties of circular pythagorean fuzzy weighted averaging operator

This subsection demonstrates the main three properties of the CPyFWAO are defined as follows:

• Idempotency: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the collection of the CPyFV. Then, the idempotency of the CPyFWAO is:

  $$\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{A}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)=\mathcal{X}$$

• Boundedness: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the collection of the CPyFV. If $\:{\mathcal{X}}^{-}=\left(\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{M}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\:\:\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{N}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{R}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right)\right)$ and $\:{\mathcal{X}}^{+}=\left(\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{M}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{N}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{R}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right)\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$. Then, the boundedness of the CPyFWAO is

  $$\:{\mathcal{X}}^{-}\le\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{A}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)\le\:{\mathcal{X}}^{+}$$

• Monotonicity: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ and $\:{{\mathcal{X}}^{*}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({{\mathcal{M}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{{\mathcal{N}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{{\mathcal{R}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left({\mathcal{M}}^{*},{\mathcal{N}}^{*},{\mathcal{R}}^{*}\right)$ Such that for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the two collections of the CPyFV. If $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\le\:{{\mathcal{X}}^{*}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}$ for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}\mathcal{\:}$ such that $\:\sqrt{{{\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}<\sqrt{{{{\mathcal{M}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}},\:\sqrt{{{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}>\sqrt{{{{\mathcal{N}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}$, and $\:\sqrt{{{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}<\sqrt{{{{\mathcal{R}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}$. Then, the monotonicity of the CPyFWAO is

  $$\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{A}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)\le\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{A}\text{O}\left({{\mathcal{X}}_{1}}^{*},{{\mathcal{X}}_{2}}^{*},\dots\:,{{\mathcal{X}}_{\mathcal{n}}}^{*}\right)$$

Circular pythagorean fuzzy weighted geometric operator

In this subsection, we proposed a CPyF WGO (CPyFWGO) and utilized it with an example. Further, some valuable properties are proposed in the framework of CPyFS.

Definition 6

The CPyFWGO for CPyFVs $\:{A}_{\mathcal{j}}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ is of the form:

$$\:CPyFWGO\left({A}_{1},{A}_{2},{A}_{3},\dots\:,{A}_{\mathcal{j}}\right)=\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{{A}_{\mathcal{j}}}^{{\omega\:}_{\mathcal{j}}}$$

(10)

Where the weighted vector $\:\omega\:=\left({\:\omega\:}_{1},{\:\omega\:}_{2},\dots\:,{\:\omega\:}_{\mathcal{j}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ satisfied the constraint $\:{\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\ge\:0$ and $\:\sum\:{\:\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=1$.

Theorem 2

The aggregated value of a collection of CPyFVs $\:{\text{A}}_{\mathcal{j}}$ by using the CPyFWGO is also a CPyFV and is given below:

$$CPyFWGO\left({A}_{1},{A}_{2},{A}_{3},\dots\:,{A}_{\mathcal{j}}\right)=\left(\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{{\mathcal{M}}_{\mathcal{j}}}^{{\omega\:}_{\mathcal{j}}},\sqrt{1-\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{{\mathcal{N}}^{2}}_{\mathcal{j}}\right)}^{{\omega\:}_{\mathcal{j}}}},\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{{\mathcal{R}}_{\mathcal{j}}}^{{\omega\:}_{\mathcal{j}}}\right)$$

(11)

Proof

See Appendix B.

Example 2

Let $\:{\mathcal{X}}_{1}=\left(\begin{array}{c}\text{0.90,0.30,0.60}\end{array}\right)$, $\:{\mathcal{X}}_{2}=\left(\begin{array}{c}\text{0.95,0.10,0.53}\end{array}\right)$, and $\:{\mathcal{X}}_{3}=\left(\begin{array}{c}\text{0.92,0.20,0.56}\end{array}\right)$ be the three CPyFVs on the universal set $\:\mathcal{U}=\left\{{\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right\}$ and the corresponding weights of criteria$\:\:\:\omega\:=\left(\text{0.25,0.40,0.35}\right)$, $\:\sum\:_{\mathcal{\:}\mathcal{j}=1}^{3}{\omega\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=1$. Then, evaluate $\:CPyFWGO\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right)$.

Solution:

$$\:CPyFWGO\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{3}\right)=\left(\begin{array}{c}{\left(0.90\right)}^{0.25}{\left(0.95\right)}^{0.40}{\left(0.92\right)}^{0.35},\\\:\sqrt{1-\left({\left(1-{0.30}^{2}\right)}^{0.25}\right)\left({\left(1-{0.10}^{2}\right)}^{0.40}\right)\left({\left(1-{0.20}^{2}\right)}^{0.35}\right)},\\\:{\left(0.60\right)}^{0.25}{\left(0.53\right)}^{0.40}{\left(0.56\right)}^{0.35}\end{array}\right)=\left(\text{0.9268,0.2025,0.5552}\right)$$

Therefore, we can see that after utilization of our proposed operator defined in Eq. $\:\left(11\right)$, we get again a CPyFV. Hence, this example shows the reliability of our proposed approach.

Properties of circular pythagorean fuzzy weighted geometric operator

This subsection demonstrates the main three properties of the CPyFWGO are defined as follows:

• Idempotency: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the collection of the CPyFV. Then, the idempotency of the CPyFWGO is

  $$\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{G}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)=\mathcal{X}$$

• Boundedness: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the collection of the CPyFV. If $\:{\mathcal{X}}^{-}=\left(\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{M}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\:\:\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{N}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{R}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right)\right)$ and $\:{\mathcal{X}}^{+}=\left(\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{M}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{min}}\left({\mathcal{N}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right),\underset{\mathcal{\:j\:}}{\text{max}}\left({\mathcal{R}}_{{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}}\right)\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$. Then, the boundedness of the CPyFWGO is

  $$\:{\mathcal{X}}^{-}\le\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{G}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)\le\:{\mathcal{X}}^{+}$$

• Monotonicity: Let $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left(\mathcal{M},\mathcal{N},\mathcal{R}\right)$ such that for all $\:\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ and $\:{{\mathcal{X}}^{*}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({{\mathcal{M}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{{\mathcal{N}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}},{{\mathcal{R}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)=\left({\mathcal{M}}^{*},{\mathcal{N}}^{*},{\mathcal{R}}^{*}\right)$ such that for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}$ be the two collections of the CPyFV. If $\:{\mathcal{X}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}\le\:{{\mathcal{X}}^{*}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}$ for all $\:\mathcal{\:}\mathcal{j}=\text{1,2},\dots\:,\mathcal{n}\mathcal{\:}$ such that $\:\sqrt{{{\mathcal{M}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}<\sqrt{{{{\mathcal{M}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}},\:\sqrt{{{\mathcal{N}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}>\sqrt{{{{\mathcal{N}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}$, and $\:\sqrt{{{\mathcal{R}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}<\sqrt{{{{\mathcal{R}}^{*}}_{\mathcal{X}\mathcal{\:}\mathcal{j}\mathcal{\:}}}^{2}}$. Then, the monotonicity of the CPyFWGO is

  $$\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{G}\text{O}\left({\mathcal{X}}_{1},{\mathcal{X}}_{2},\dots\:,{\mathcal{X}}_{\mathcal{n}}\right)\le\:\text{C}\text{P}\text{y}\text{F}\text{W}\text{G}\text{O}\left({{\mathcal{X}}_{1}}^{*},{{\mathcal{X}}_{2}}^{*},\dots\:,{{\mathcal{X}}_{\mathcal{n}}}^{*}\right)$$

Extension of WASPAS technique with circular pythagorean fuzzy set based on linguistic terms

In this section, we focus WASPAS technique. The WASAS method integrates the two well-known approaches: weighted sum represented by $\:{Q}_{\mathcal{i}}^{S}$ and weighted product represented by $\:{Q}_{\mathcal{i}}^{P}$ in the environment of CPyF data, called the WASPAS technique. The extension of WASPAS technique is that it can easily handle the uncertainty data in an assessment process that requires a large domain for to discuss their opinion by assigning membership and non-membership grades whose squared sum is at most equal to 1, while the radius belongs to closed interval 0 to 1for each element on the universal set. In other words, the sum of its grades can exceed 1 in CIFS, but the sum of the power of 2 of its grades cannot. Additionally, we include the proposed linguistic terms used for the evaluation of alternatives concerning criteria are showcased in Table 2 in this context. Further, the linguistic terms provided for the importance weights of the criteria are given in Table 3 respectively:

Table 2 Linguistic terms used for evaluating the alternatives concerning criteria in the form of cpyfvs.

Table 3 Linguistic terms used for the importance weight of the criteria in the form of cpyfvs.

Evaluation process of MCGDM approach based on the WASPAS technique

In this subsection, we determined the theory of the MCGDM technique to evaluate complex and unpredictable problems under a system of CPyF information. To show the supremacy and superiority of our derived models, we utilized the theory of the MCGDM problem in the presence of CPyF information by applying the above-defined operators: CPyFWAO and CPyFWGO. The designed DM whose elements are in the form of $\:\mathcal{m}$ alternatives $\:{\mathcal{A}}_{\mathcal{i}}(\mathcal{i}=\text{1,2},\dots\:,\mathcal{m})$ and $\:\mathcal{C}$ represents the set, which has a collection of $\:\mathcal{n}$ criteria $\:{\mathcal{C}}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}(\mathcal{\:}\mathcal{j}\mathcal{\:}=\text{1,2},\dots\:,\mathcal{n})$. The set of corresponding prioritized weighted vector is $\:\stackrel{\sim}{{\omega\:}^{\mathcal{S}}}=\left(\stackrel{\sim}{{\omega\:}_{1}^{\mathcal{S}}},\stackrel{\sim}{{\omega\:}_{2}^{\mathcal{S}}},\dots\:,\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ of all criteria satisfied the constraint $\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\ge\:0$ and $\:\sum\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=1$. Suppose that $\:{\varvec{M}}_{\mathcal{m}\times\:\mathcal{n}}={\left({\mathcal{X}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)}_{\mathcal{m}\times\:\mathcal{n}}$ is the DM, for instance, $\:{\mathcal{X}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}=\left({\mathcal{M}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}},\:\:{\mathcal{R}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)$, where $\:{\mathcal{M}}_{\mathcal{i}\mathcal{j}}$ membership grade that the alternative $\:{\mathcal{A}}_{\mathcal{i}}$ satisfied the criteria $\:{\:\mathcal{C}\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}$, $\:{\mathcal{N}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}$ the non-membership grade that the alternative $\:{\mathcal{A}}_{\mathcal{i}}$ not satisfied the criteria $\:{\:\mathcal{C}\:}_{\mathcal{\:}\mathcal{j}\mathcal{\:}}$, and $\:{\mathcal{R}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}$ represent the radius grade of $\:{\mathcal{M}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}$and $\:{\mathcal{N}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}$ on the plane. Also this, $\:{\mathcal{M}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{N}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}},{\mathcal{R}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}\in\:\left[\text{0,1}\right]$ and $\:{\left({\mathcal{M}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)}^{2}+{\left({\mathcal{N}}_{\mathcal{i}\mathcal{\:}\mathcal{j}\mathcal{\:}}\right)}^{2}\in\:\left[\text{0,1}\right]$, where $\:\mathcal{i}=\text{1,2},3,\dots\:,\mathcal{\:}\mathcal{m}$ and $\:\mathcal{\:}\mathcal{j}\mathcal{\:}=\text{1,2},3,\dots\:,\:\mathcal{n}$. The decision values cover the CPyFV information of alternatives regarding each criterion. Usually, a DM has two types of criteria: cost and benefit. The decision values of the for each alternative regarding each criterion are stored in the DM and then integrated by the proposed AOs. The proposed WASPAS technique has the following steps:

• Step 1 The will take the linguistic terms for the evaluation of alternatives against the criteria provided in Table 2. Assume that there are $\:kth$ , which he/she investigates his/her, the $\:\mathcal{m}$ alternatives against $\:\mathcal{n}$ criteria according to the following $\:kth$ DM.

  $$\:{{M}^{k}}_{\mathcal{m}\times\:\mathcal{n}}=\begin{array}{c}{\mathcal{A}}_{1}\\\:{\mathcal{A}}_{2}\\\: \vdots \\\:{\mathcal{A}}_{\mathcal{m}}\end{array}\:\left[\begin{array}{cccc}{\mathcal{C}}_{1}&\:{\mathcal{C}}_{2}&\:\dots\:&\:{\mathcal{C}}_{\mathcal{n}}\\\:{{\mathcal{X}}^{k}}_{11}&\:{{\mathcal{X}}^{k}}_{12}&\:\dots\:&\:{{\mathcal{X}}^{k}}_{1\mathcal{n}}\\\:{{\mathcal{X}}^{k}}_{21}&\:{{\mathcal{X}}^{k}}_{22}&\:\dots\:&\:{{\mathcal{X}}^{k}}_{2\mathcal{n}}\\\: \vdots &\: \vdots &\: \ddots \:&\: \vdots \\\:{{\mathcal{X}}^{k}}_{\mathcal{m}1}&\:{{\mathcal{X}}^{k}}_{\mathcal{m}2}&\:\dots\:&\:{{\mathcal{X}}^{k}}_{\mathcal{m}\mathcal{n}}\end{array}\right]$$

  (12)

  Where $\:{{\mathcal{X}}^{k}}_{\mathcal{i}\mathcal{j}}=\left({{\mathcal{M}}^{k}}_{\mathcal{i}\mathcal{j}},{{\mathcal{N}}^{k}}_{\mathcal{i}\mathcal{j}},{{\mathcal{R}}^{k}}_{\mathcal{i}\mathcal{j}}\right)$ such that $\:\mathcal{i}\left(\mathcal{i}=\text{1,2},3,\dots\:,\mathcal{m}\right)$ and $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ represent the CPyFVs of each alternative $\:\mathcal{i}$ against each criterion $\:\mathcal{j}$. There are many ways to demonstrate the DM: (1) assess the alternative regard criteria. Remember that if the criteria are beneficial, then can take the upper linguistic term, while in the case of cost criteria, the lower linguistic term is preferred. (2) can directly simplify the preferences based to determine the membership, non-membership, and radius grades of $\:{{\mathcal{X}}^{k}}_{\mathcal{i}\mathcal{j}}$.

• Step 2 Suppose that the DM has benefit and cost criteria. For this contrast, it is necessary to normalize the DM. For normalization, utilize the following equations.

  $$\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{\mathcal{i}\mathcal{j}}=\left\{\begin{array}{c}\left({\mathcal{M}}_{\mathcal{i}\mathcal{j}},{\mathcal{N}}_{\mathcal{i}\mathcal{j}},{\mathcal{R}}_{\mathcal{i}\mathcal{j}}\right);\:\:\:\:\:\:\:\:\:for\:benefit\:criteria\:\\\:\:\begin{array}{c}\left({\mathcal{N}}_{\mathcal{i}\mathcal{j}},{\mathcal{M}}_{\mathcal{i}\mathcal{j}},{\mathcal{R}}_{\mathcal{i}\mathcal{j}}\right)\end{array};\:\:\:\:\:\:\:\:\:for\:cost\:criteria\:\:\end{array}\right.$$

  (13)

  The complement operation is performed based on Eq. ($\:5$). Then, the DM is presented in Eq. $\:\left(12\right)$ converted into the normalized matrix presented in Eq. (14).

  $$\:{{\stackrel{\sim}{M}}^{k}}_{\mathcal{m}\times\:\mathcal{n}}=\begin{array}{c}{\mathcal{A}}_{1}\\\:{\mathcal{A}}_{2}\\\: \vdots \\\:{\mathcal{A}}_{\mathcal{m}}\end{array}\:\left[\begin{array}{cccc}{\mathcal{C}}_{1}&\:{\mathcal{C}}_{2}&\:\dots\:&\:{\mathcal{C}}_{\mathcal{n}}\\\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{11}&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{12}&\:\dots\:&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{1\mathcal{n}}\\\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{21}&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{22}&\:\dots\:&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{1\mathcal{n}}\\\: \vdots &\: \vdots &\:\ddots\:&\: \vdots \\\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{\mathcal{m}1}&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{\mathcal{m}2}&\:\dots\:&\:{{\stackrel{\sim}{\mathcal{X}}}^{k}}_{\mathcal{m}\mathcal{n}}\end{array}\right]$$

  (14)

  Keep in mind that if the taken only benefit criteria, then there will be no need for normalization.

• Step 3 Another issue is to identify the prioritized weighted vector for the importance of all criteria. Also, note that all the weights of criteria need not be considered equal. To identify the criteria weights, all opinions of for the importance of each criterion must be aggregated. For this, take the linguistic terms provided in Table 3 for the importance of criteria according to the following DM.

  $$\:\omega\:=\begin{array}{c}{\mathcal{D}}_{1}\\\:{\mathcal{D}}_{2}\\\: \vdots \\\:{\mathcal{D}}_{k}\end{array}\:\left[\begin{array}{cccc}{\mathcal{C}}_{1}&\:{\mathcal{C}}_{2}&\:\dots\:&\:{\mathcal{C}}_{\mathcal{n}}\\\:{\mathcal{X}}_{11}&\:{\mathcal{X}}_{12}&\:\dots\:&\:{\mathcal{X}}_{1\mathcal{n}}\\\:{\mathcal{X}}_{21}&\:{\mathcal{X}}_{22}&\:\dots\:&\:{\mathcal{X}}_{2\mathcal{n}}\\\: \vdots &\: \vdots &\:\ddots\:&\: \vdots \\\:{\mathcal{X}}_{k1}&\:{\mathcal{X}}_{k2}&\:\dots\:&\:{\mathcal{X}}_{k\mathcal{n}}\end{array}\right]$$

  (15)

  Next, we aggregate the above DM based on Eq. $\:\left(9\right)$ according to the weights of $\:kth$ having different experience levels. Then defuzzify the aggregated weights of all criteria by using Eq. (16) and normalized it by using Eq. (17).

  $$\:{\omega\:}_{\mathcal{j}}^{\mathcal{S}}=\left(1-{\left({\mathcal{M}}_{\mathcal{j}}\right)}^{2}-{\left({\mathcal{N}}_{\mathcal{j}}\right)}^{2}\right)\times\:{\mathcal{R}}_{\mathcal{j}}$$

  (16)
  $$\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=\frac{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}{\sum\:_{\mathcal{j}=1}^{\mathcal{n}}{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}$$

  (17)

  After that, the normalized aggregated prioritized weighted vector $\:\stackrel{\sim}{{\omega\:}^{\mathcal{S}}}=\left(\stackrel{\sim}{{\omega\:}_{1}^{\mathcal{S}}},\stackrel{\sim}{{\omega\:}_{2}^{\mathcal{S}}},\dots\:,\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ of all criteria satisfied the constraint $\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\ge\:0$ and $\:\sum\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=1$.

• Step 4 In the case of MCGDM approaches, first we convert the $\:kth$ DM $\:{{\stackrel{\sim}{M}}^{k}}_{\mathcal{m}\times\:\mathcal{n}}$ to $\:{\stackrel{\sim}{M}}_{\mathcal{m}\times\:\mathcal{n}}$. This step can be done by using Eq. (9) according to the weights of $\:kth$ having different experience levels and we will get the following aggregated DM.

  $$\:{\stackrel{\sim}{M}}_{\mathcal{m}\times\:\mathcal{n}}=\begin{array}{c}{\mathcal{A}}_{1}\\\:{\mathcal{A}}_{2}\\\: \vdots \\\:{\mathcal{A}}_{\mathcal{m}}\end{array}\:\left[\begin{array}{cccc}{\mathcal{C}}_{1}&\:{\mathcal{C}}_{2}&\:\dots\:&\:{\mathcal{C}}_{\mathcal{n}}\\\:{\stackrel{\sim}{\mathcal{X}}}_{11}&\:{\stackrel{\sim}{\mathcal{X}}}_{12}&\:\dots\:&\:{\stackrel{\sim}{\mathcal{X}}}_{1\mathcal{n}}\\\:{\stackrel{\sim}{\mathcal{X}}}_{21}&\:{\stackrel{\sim}{\mathcal{X}}}_{22}&\:\dots\:&\:{\stackrel{\sim}{\mathcal{X}}}_{1\mathcal{n}}\\\: \vdots &\: \vdots &\:\ddots\:&\: \vdots \\\:{\stackrel{\sim}{\mathcal{X}}}_{\mathcal{m}1}&\:{\stackrel{\sim}{\mathcal{X}}}_{\mathcal{m}2}&\:\dots\:&\:{\stackrel{\sim}{\mathcal{X}}}_{\mathcal{m}\mathcal{n}}\end{array}\right]$$

  (18)

  According to the WASPAS technique suggestion, the result of the weighted sum, which is CPyFWA denoted by $\:{Q}_{\mathcal{i}}^{S}$ as presented in Eq. (19).

  $$\:{Q}_{\mathcal{i}}^{S}=CPyFWA=\sum\:_{\mathcal{j}=1}^{\mathcal{n}}\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}$$

  (19)

  Since Eq. (19) has two parts. The first part multiplication operator and the second part addition operator are performed. Therefore, both parts are done based on Eq. (9) as presented in Eq. (20).

  $$\:{Q}_{\mathcal{i}}^{S}=CPyFWA=\sum\:_{\mathcal{j}=1}^{\mathcal{n}}\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=\left(\sqrt{1-\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{\mathcal{M}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{2}\right)}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}}},\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\mathcal{N}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}},\sqrt{1-\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{\mathcal{R}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{2}\right)}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}}}\right)$$

  (20)

  The next suggestion of the WASPAS method is a weighted product, which is CPyFWG denoted by $\:{Q}_{\mathcal{i}}^{P}$ as presented in Eq. (21).

  $$\:{Q}_{\mathcal{i}}^{P}=CPyWG=\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}}$$

  (21)

  We can see that Eq. (21) has also two parts. The first part exponential operator and the second part multiplication operator are performed. Therefore, both parts are done based on Eq. ($\:11$) as presented in Eq. (22).

  $$\:{Q}_{\mathcal{i}}^{P}=CPyFWG=\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}}=\left(\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\mathcal{M}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}},\sqrt{1-{\prod\:}_{\mathcal{j}=1}^{\mathcal{n}}{\left(1-{\mathcal{N}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{2}\right)}^{\stackrel{\sim}{{\omega\:}^{\mathcal{S}}}}},\prod\:_{\mathcal{j}=1}^{\mathcal{n}}{\mathcal{R}}_{\stackrel{\sim}{{\mathcal{X}}_{\mathcal{i}\mathcal{j}}}}^{\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}}\right)$$

  (22)

  Now, we aggregate the DM presented in Eq. (18) based on Eqs. (20) and (22) according to the prioritized weighted vector of all criteria, then we will get $\:{Q}_{\mathcal{i}}^{S}$ and $\:{Q}_{\mathcal{i}}^{P}$ respectively. After that, the convex formula of the WASPA technique denoted by $\:{Q}_{\mathcal{i}}$ and presented in Eq. (23).

  $$\:{Q}_{\mathcal{i}}=WASPAS=\gamma\:{Q}_{\mathcal{i}}^{S}+\left(1-\gamma\:\right){Q}_{\mathcal{i}}^{P}$$

  (23)

  Since, Eq. (23) has two terms, both terms have scalar multiplication operator. For this, both terms are done based on Eq. ($\:3$) and then utilized the addition operator defined in Eq. ($\:1$).

• Step 5 Calculated the score values of $\:{Q}_{\mathcal{i}}^{S}$, $\:{Q}_{\mathcal{i}}^{P}$, and $\:{Q}_{\mathcal{i}}$ based on Eq. (6) of each alternative. If the score values of any two alternatives are the same, then we have to compute Eq. (7). Finally, the rank of alternatives and identify a suitable alternative from the set of alternatives.

  The flow chart of the evaluation of MCGDM approaches based on the WASPAS technique is shown in Figure 3 for easy understanding.

Fig. 3

Procedure of the WASPAS technique for decision-making problem.

Algorithm

The aim of this subsection is to solve the MCGDM problems in the environment of CPyFS by using the proposed WASPAS technique; the following steps should be followed:

• Step 1 Combining two categories of the CPyF information and investigate the DM in the following form.

  $$\:{M}_{\mathcal{m}\times\:\mathcal{n}}=\begin{array}{c}{\mathcal{A}}_{1}\\\:{\mathcal{A}}_{2}\\\: \vdots \\\:{\mathcal{A}}_{\mathcal{m}}\end{array}\:\left[\begin{array}{cccc}{\mathcal{C}}_{1}&\:{\mathcal{C}}_{2}&\:\dots\:&\:{\mathcal{C}}_{\mathcal{n}}\\\:\left({\mathcal{M}}_{11},{\mathcal{N}}_{11},{\mathcal{R}}_{11}\right)&\:\left({\mathcal{M}}_{12},{\mathcal{N}}_{12},{\mathcal{R}}_{12}\right)&\:\dots\:&\:\left({\mathcal{M}}_{1\mathcal{n}},{\mathcal{N}}_{1\mathcal{n}},{\mathcal{R}}_{1\mathcal{n}}\right)\\\:\left({\mathcal{M}}_{21},{\mathcal{N}}_{21},{\mathcal{R}}_{21}\right)&\:\left({\mathcal{M}}_{22},\mathcal{N},{\mathcal{R}}_{22}\right)&\:\dots\:&\:\left({\mathcal{M}}_{2\mathcal{n}},{\mathcal{N}}_{2\mathcal{n}},{\mathcal{R}}_{2\mathcal{n}}\right)\\\: \vdots &\: \vdots &\:\ddots\:&\: \vdots \\\:\left({\mathcal{M}}_{\mathcal{m}1},{\mathcal{N}}_{\mathcal{m}1},{\mathcal{R}}_{\mathcal{m}1}\right)&\:\left({\mathcal{M}}_{\mathcal{m}2},{\mathcal{N}}_{\mathcal{m}2},{\mathcal{R}}_{\mathcal{m}2}\right)&\:\dots\:&\:\left({\mathcal{M}}_{\mathcal{m}\mathcal{n}},{\mathcal{N}}_{\mathcal{m}\mathcal{n}},{\mathcal{R}}_{\mathcal{m}\mathcal{n}}\right)\end{array}\right]$$

• Step 2 The DM should be normalized if CPyF information has two types of criteria: benefit criteria and cost criteria according to the following expression.

  $$\:{\stackrel{\sim}{M}}_{\mathcal{m}\times\:\mathcal{n}}=\left\{\begin{array}{c}\left({\mathcal{M}}_{\mathcal{i}\mathcal{j}},{\mathcal{N}}_{\mathcal{i}\mathcal{j}},{\mathcal{R}}_{\mathcal{i}\mathcal{j}}\right);\:\:\:\:\:\:\:\:\:for\:benefit\:criteria\:\\\:\:\begin{array}{c}\left({\mathcal{N}}_{\mathcal{i}\mathcal{j}},{\mathcal{M}}_{\mathcal{i}\mathcal{j}},{\mathcal{R}}_{\mathcal{i}\mathcal{j}}\right)\end{array};\:\:\:\:\:\:\:\:\:for\:cost\:criteria\:\:\end{array}\right.$$

  It is not necessary every time to assess the data when it is determined in the DM in the form of benefit criteria.

• Step 3 Find out the prioritized weighted vector for the importance of criteria. For this, we take the linguistic terms provided in Table 3. The prioritized weighted vector $\:\stackrel{\sim}{{\omega\:}^{\mathcal{S}}}=\left(\stackrel{\sim}{{\omega\:}_{1}^{\mathcal{S}}},\stackrel{\sim}{{\omega\:}_{2}^{\mathcal{S}}},\dots\:,\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ of all criteria satisfied the constraint $\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\ge\:0$ and $\:\sum\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=1$.

• Step 4 Integrate the normalized DM which is in the form of CPyF information by using the proposed operators presented in Eqs. (20), (22), and (23).

• Step 5 Calculate the score value of all initiated AOs and WASPAS techniques by using Eq. (6) of each alternative. Also, identify the ranking of each alternative and select the best alternative.

The flow chart of the algorithm based on the proposed WASPAS technique in the framework of CPyF information is showcased in Figure 4 for easy understanding.

Fig. 4

Evaluation process of MCGDM problem by using the proposed model.

Case study with numerical example

In this case study, we examine the challenges of the physical education problem by utilizing the proposed WASPAS technique in the framework of CPyF information. For this, we can use the evaluation process of the MCGDM approach.

Proposed problem statement

In This subsection aims to understand how to evaluate decision-making in physical education for teaching and learning with the help of numerical examples based on fuzzy information. Physical education is a crucial component that affects both teaching effectiveness and student learning outcomes. It involves a range of choices made by teachers and students before, during, and after physical education lessons. By adopting a variety of strategies and tools, you can create a more dynamic, inclusive, and engaging physical education environment that fosters lifelong physical activity habits in students.

In this article, we can promote decision-making in physical education for teaching and learning. For this, we can apply the proposed work based on MCGDM techniques and utilize the evaluation process to evaluate the numerical example whose data is in the form of CPyF information. The following numerical example shows the reliability of our proposed new approaches and illustrates the best alternative for physical education. Here considering these five different types of alternatives to decision-making in physical education for teaching and learning. That is game-based learning, experiential learning, problem-based learning, inquiry-based learning, and cooperative learning. This alternative has several key criteria that differentiate each alternative. Let us assume that each alternative has these five different types of criteria that is, student engagement, skill development, accessibility, opportunities for reflection, and transferability.

This manuscript addresses the challenges of physical education by introducing the proposed work. The new work improves the reliability and accuracy of aggregating the hypothetical data of energy sources. Due to unlike the existing approach, the proposed work plays a significant role in achieving accuracy by reducing the complexities of energy sources. It means that the novel idea of the initiated WASPAS technique will give the chance to improve the accuracy, and it can deal with the inherited complexities in integrating heterogeneous data. Based on these factors, we want to accumulate the performance of physical education with the help of the CPyF linguistic set.

Proposed problem solution

In this subsection the initiated work proposed in this manuscript is applied to deal with the problem stated above; we are required to follow the following steps:

• 1. Enlisted the DM of alternatives and criteria. The prioritized weighted vector of criteria must be determined according to their criteria. Assume that, the mentioned types of physical education are alternatives and the challenges affecting their performances as the criteria.

• 2. If the DM accrues both beneficial and cost types criteria, then required to normalize the DM based on Eq. $\:\left(13\right)$ and Eq. $\:\left(14\right)$.

• 3. Utilize the elaborated AOs defined in Eq. $\:\left(\text{21,23,24}\right)$ to aggregate the information provided in the DM.

• 4 Calculate the score of each alternative regarding criteria based on Eq. (6) and arrange them in increasing order.

• 5. Rank the other options based on the computed score of the alternative. If the score of the two alternatives is the same, then we have to compute Eq. (7).

We can follow the above steps to find out the best alternative practically. The proposed extension of the WASPAS technique in the framework of CPyF information is presented in Eq. $\:\left(\text{21,23,24}\right)$, which can be integrated into the problem of physical education by acting on the challenges mentioned above. They can play a vital role in real-life scenarios with the help of FS information and then aggregating that data for better estimation.

Example 3

Suppose a firm wants to assess the performance of five various decision-making in physical education for teaching and learning that is, game-based learning, experiential learning, problem-based learning, inquiry-based learning, and cooperative learning based on the five criteria that is, student engagement, skill development, accessibility, opportunities for reflection, and transferability to select the best alternative in physical education. Note that the data assumed in this study is hypothetical. Let$\:\:\left\{{\mathcal{A}}_{1},\:{\mathcal{A}}_{2},\:{\mathcal{A}}_{3},\:{\mathcal{A}}_{4},{\mathcal{A}}_{5}\right\}$ and $\:\beta\:=\left\{{\mathcal{C}}_{\:1},{\mathcal{C}}_{\:2\:},{\mathcal{C}}_{\:3\:},{\mathcal{C}}_{\:4\:},{\mathcal{C}}_{\:5}\right\}$ be the alternative and the corresponding criteria, respectively. The uses the mentioned criterion to evaluate the alternatives under consideration prioritized weighted vector $\:\stackrel{\sim}{{\omega\:}^{\mathcal{S}}}=\left(\stackrel{\sim}{{\omega\:}_{1}^{\mathcal{S}}},\stackrel{\sim}{{\omega\:}_{2}^{\mathcal{S}}},\dots\:,\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\right)$ such that $\:\mathcal{j}\left(\mathcal{j}=\text{1,2},3,\dots\:,\mathcal{n}\right)$ of all criteria satisfied the constraint $\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}\ge\:0$ and $\:\sum\:\stackrel{\sim}{{\omega\:}_{\mathcal{j}}^{\mathcal{S}}}=1$. The evaluation information on the five alternatives $\:\left\{{\mathcal{A}}_{1},\:{\mathcal{A}}_{2},\:{\mathcal{A}}_{3},\:{\mathcal{A}}_{4},{\mathcal{A}}_{5}\right\}$ under the five criteria$\:\:\beta\:=\left\{{\mathcal{C}}_{\:1},{\mathcal{C}}_{\:2\:},{\mathcal{C}}_{\:3\:},{\mathcal{C}}_{\:4\:},{\mathcal{C}}_{\:5}\right\}$ such that values are assigned to each alternative in the form of CPyFVs showcased in the DM.

• Step 1 The decision values provided by the for each alternative assessed in the light of five criteria are stored in Table 4. For this constraint, the will use the linguistic terms provided in Table 2.

  Table 4 Linguistic terms provided for each energy source by the .

• Step 2 The have classified the criteria into two groups: The criteria first, second, and fourth are benefit while the third and fifth criteria are cost. For this constraint, we will utilize Eq. (13) for the normalization. The normalized decision values are showcased in Table 5 respectively.

  Table 5 Normalized decision values are provided for each energy source by the .

• Step 3 To illustrate the prioritized weighted vector for the importance of criteria. The will use the linguistic terms provided in Table 3 according to the DM showcased in Table 6.

  Table 6 Importance weights of the criteria for evaluation.

  The weights of three for all criteria having different experience levels are 0.38, 0.30, and 0.32 respectively showcased in Table 7.

  Table 7 Importance weights of the criteria for evaluation and experience levels of the .

  First, we aggregate importance weights of the criteria concerning experience levels of the presented in Table 7 by using the CPyFWAO defined in Eq. (9). Then, based on Eqs. (16) and (17), defuzzified and normalized the aggregated criteria weights, which are presented in Table 8.

  Table 8 Defuzzified and then normalized criteria weights of the .

• Step 4 Aggregate the normalized decision values provided for each energy source presented in Table 5 based on Eq. $\:\left(9\right)$ under consideration of different experience levels, 0.38, 0.30, and 0.32 of the and the calculated results are shown in Table 9.

  Table 9 Aggregated normalized DM for renewable energy source.

  Based on Table 9 applied the proposed AOs presented in Eqs. (20) and (22) according to the normalized aggregated prioritized weight for each criterion presented in Table 8, the accumulated results of $\:{Q}_{\mathcal{i}}^{S}$ and $\:{Q}_{\mathcal{i}}^{P}$ are shown in Table 10.

  Table 10 Accumulated result of weighted sum and weighted product.

  Therefore, based on Table 10 and Eq. ($\:3$), $\:\gamma\:{Q}_{\mathcal{i}}^{S}$ and $\:\left(1-\gamma\:\right){Q}_{\mathcal{i}}^{P}$ whenever $\:\gamma\:=0.5$ are obtained as in Table 11. Based on Eq. ($\:1$), the fourth column of Table 11, $\:{Q}_{\mathcal{i}}$ is obtained by using the convex formula of the WASPAS technique.

  Table 11 Convex formula of WASPAS method.

• Step 5 Based on Eq. (6), evaluate the score values of $\:{Q}_{\mathcal{i}}^{S}$, $\:{Q}_{\mathcal{i}}^{P}$, and $\:{Q}_{\mathcal{i}}$ presented in Tables 10 and 11 of each alternative and then the ranking of each alternative is showcased in Table 12.

  Table 12 Representation of score values and ranking.

  In Table 12, we can see that the overall ranking of alternatives, in the case of CPyFWA is $\:{\mathcal{A}}_{4}>{\mathcal{A}}_{2}>{\mathcal{A}}_{3}\:>{\mathcal{A}}_{1}\:>{\mathcal{A}}_{5}$, in the case of CPyFWG is $\:{\mathcal{A}}_{1}>{\mathcal{A}}_{4}>{\mathcal{A}}_{2}\:>{\mathcal{A}}_{3}\:>{\mathcal{A}}_{5}$, and in the case of WASPAS is $\:{\mathcal{A}}_{2}>{\mathcal{A}}_{4}>{\mathcal{A}}_{3}\:>{\mathcal{A}}_{1}\:>{\mathcal{A}}_{5}$. Thus, we observed that $\:{\mathcal{A}}_{4}$ is the best alternative in physical education which represents inquiry-based learning with the help of initiated CPyFWA and $\:{\mathcal{A}}_{1}$ is the best alternative in physical education which represents game-based learning with the help of CPyFWG while $\:{\mathcal{A}}_{2}$ is the best optimal alternative in physical education which represents experiential learning in the case of derived theory of WASPAS model, but it all depends on the consideration. From Figure 5, the readers easily understand the behavior and ranking of alternatives.

  Fig. 5

  

  Geometrical representation of alternatives which are obtained by utilizing the initiated work.

  In Figure 5, we can see the score values and ranking of each alternative, which are obtained by the CPyFWA and CPyFWG operators. Also, the graph of the initiated WASPAS technique in the environment of CPyFS is included in Figure 5. Therefore, we observed that $\:{\mathcal{A}}_{4}$ shows a more suitable alternative in the case of CPyFWA and $\:{\mathcal{A}}_{1}$ is the best alternative in the case of CPyFWG while in case of WASPAS model, $\:{\mathcal{A}}_{2}$ is the best alternative. Also, we note that the green color is the optimal option in Figure 5, which represents the initiated WASPAS technique.

Sensitivity analysis

In this part of the article, we check the influence of our proposed work shown in Eq. $\:\left(\text{19,21,23,24}\right)$ by considering the data in Table 5 with the help of a prioritized weighted vector. The influence results for physical education, by assigning different parametric values $\:\gamma\:$ on various options of alternatives. Interestingly, the ranking of the alternatives is obtained in the same way from the parametric value 0.1 to 0.9 as in Table 13.

Table 13 Sensitivity score values and ranking.

In Table 13, we can see that the best alternative is $\:{\mathcal{A}}_{2}$ by taking different parametric values. Therefore, we illustrate the sensitivity analysis shows that the best alternative to physical education is $\:{\mathcal{A}}_{2}$.

Comparative analysis

In this part of the article, we will check the validation and compatibility of the proposed theory. This comparison demonstrated the feasibility and use of the suggested approaches. To achieve this goal, we applied the initiated work in the information given in Table 5. The ranking results are compared with some existing works to check the supremacy and validity of the initiated techniques. For comparison, we use the following existing techniques, such as Stanujkic et al.²⁶ introduced arithmetic and geometric operators in IFS. Xu³³ adopted weighted averaging and WGO in the environment of IFS. Zhao et al.³⁴ developed generalized AOs in the framework of IFS. Peng et al.³⁵ exposed the fundamental properties of AOs in the framework of PyFS. Khan et al.³⁶ developed Pythagorean fuzzy Dombi AOs. Fahmi et al.³⁷ explored the circular intuitionistic fuzzy Hamacher operator and WGO. Thus, utilizing the information given in Table 5, the comparison view is showcased in Table 14.

Table 14 Representation of comparative analysis of proposed approach with existing approaches.

From Table 14, we observed that the initiated work is very feasible and powerful because of its features while the existing techniques cannot evaluate the data presented in Table 5 due to limited features. The main existing techniques limitation main reason is that they are operators, generalizations, and fundamental properties in the frameworks of IFS, PyFS, and CIFS. However, the initiated works are computed based on the CPyF environment which has a large domain and also circular behavior as well and the existing technique will become the special case of the proposed work. The limitations and weaknesses of existing techniques are briefly discussed as given below:

• Stanujkic et al.²⁶introduced arithmetic and geometric operators in the environment of IFS. This means that these failed to evaluate the data presented in Table 5 because IFS has only membership and non-membership grades without radius information.

• Xu³³ and Zhao et al.³⁴ adopted WAO and WGO in the environment of IFS. However, CPyFS is the advanced version of IFS which has a large domain as compared to IFS and circular behavior as well. This means that both failed to evaluate the data presented in Table 5.

• Peng et al.³⁵ exposed the fundamental properties of AOs in the framework of PyFS. Khan et al.³⁶ developed Pythagorean fuzzy Dombi AOs. We can see that, in each article the initiated AOs framework are same that is PyFS, without circular behavior and it is the special case of CPyF information. Therefore, the existing work cannot deal with the information presented in Table 5.

• Fahmi et al.³⁷ developed circular intuitionistic fuzzy Hamacher and WGO. However, it has failed to cope with the CPyF information because of the limitation problem and the proposed theory is the modified version of the existing techniques of Fahmi et al.³⁷.

Hence, we observed that the initiated work of this manuscript generalized and feasible than the earlier established works.

Conclusion and future work

In this article discusses decision-making in physical education for teaching and learning, aiming to create engaging and inspiring opportunities for students to learn and improve their thinking skills. To achieve this, we utilize a CPyFS, which captures membership and non-membership grades while exhibiting circular behavior. This approach is particularly useful for handling uncertainty in assessment processes, allowing to discuss their opinions within a large domain. The CPyFS technique builds upon FS theory, with PyFS being a special case. Our goal is to define weighted averaging and geometric operators within the CPyFS framework, based on triangular norms. We demonstrate the effectiveness of these AOs through examples and establish key mathematical properties, such as idempotency and boundedness. Furthermore, we propose a new model using the weighted aggregated WASPAS method, exploring its application in multi-criteria group MCGDM.

Next, we defined CPyF linguistic terms information, which is very authentic and reliable in FS theory and modern-day life as well. The techniques of the WASPAS method in the framework of CPyFS have many advantages. The initiated AOs and WASPAS models are very flexible and strong enough to solve the MCGDM problem and also deal with the relationships between all types of criteria according to the parametric value in the framework of CPyF information. Utilized these defined works in a numerical example in which we illustrate the best alternatives for physical education concerning each criterion based on the CPyF MCGDM technique. To confirm the reliability of our explored work, we accumulate the influence result of physical education, by assigning different parametric values $\:\gamma\:$ on various options of alternatives. Further, confirms the superiority and compatibility of the decision-making problem based on proposed AOs with some former AOs. This research delivers several important findings:

• 1. We developed the novel idea of CPyFWAO and CPyFWGO in the environment of CPyFS with examples.

• 2. Significant mathematical properties were established to show consistency and reliability levels in the system.

• 3. We adopt the linguistic terms in the framework of the CPyF context for the evaluation of alternatives and the importance of concern criteria.

• 4. Prioritization of weights for the importance of criteria and utilization in the evaluation process of the MCGDM problem in the framework of CPyF information.

• 5. Based on the above-initiated operators, we proposed the WASPAS technique and then utilized it to aggregate the DM into a singleton set.

• 6. Presentation of a systematic evaluation process and algorithm, which are used to solve any type of numerical example based on the MCGDM approach related to real-world applications.

• 7. The proposed technique has been used in physical education assessment, demonstrating effective management of uncertain criteria related to each other.

• 8. Finally, we diagnosed the sensitivity and compatibility of the new work with earlier established work.

The study resolves fundamental uncertainty modeling issues to produce an accurate evaluation support system usable in complex decision-making processes.

Limitation of the proposed WASPAS method

CPyF information is very proficient due to its large domain as compared to the CIF framework. The extension of the WASPAS technique in the framework of CPyF information is the advanced model that can easily deal with complex issues in real-life scenarios. However, the proposed model can only deal with information in the format given in Definition 1 and has limitations in certain situations, like in the presence of circular q-rung orthopair fuzzy, complex fuzzy, and bipolar fuzzy information. Then the technique is enough, which is not possible to evaluate by using this model. Moreover, the weights of the criteria must be non-zero, and their sum of grades must be equal to 1. In any other case, the models will not provide appropriate. In the presence of CPyF information, the proposed model plays a valuable and dominant role in FS theory.

Future direction

Future research will expand this study by integrating enhanced AOs and WASPAS techniques, including Aczel-Alsina power, Einstein, Dombi, and Frank operators, to improve both flexibility and accuracy of decision-making outcomes. We also try to discuss some new techniques, like the TOPSIS method, AHP method, and MARCOS method to enhance the worth of the proposed theory. However, the concept can be extended to MCGDM methods based on Aczel Alsina hamy mean operators³⁸, dombi hamy mean operators³⁹, a novel approach towards bipolar soft sets⁴⁰, frank AOs⁴¹, Einstein AOs⁴², TOPSIS method for MCGDM⁴³, q-rung orthopair fuzzy Einstein interactive geometric aggregation operators⁴⁴, Aczel–Alsina power AOs⁴⁵, Similarity measures for T-spherical fuzzy sets and their application⁴⁶, Picture fuzzy Maclaurin symmetric mean operators⁴⁷, decision-making approach in q-rung orthopair picture fuzzy environment⁴⁸, interval-valued spherical fuzzy MEREC-VIKOR approach⁴⁹, group decision-making method in interval valued Fermatean fuzzy environment⁵⁰, Fermatean Fuzzy CRADIS Approach⁵¹, Triangular divergence-based VIKOR method⁵².