Introduction

Crisp set theory is essential in decision-making sciences; it offers a precise and definite context for classifying and assessing selections. This model is best for the decision-making process that involves binary or described options, as it removes uncertainty by ensuring that elements are either part of a set. This clarity enables decision-makers to assess options with assurance, thus enhancing problem-solving capabilities, particularly in situations where accuracy and clearly defined categories are essential.

Fuzzy set theory

Crisp sets prove to be helpful in situations that are clearly defined and predictable. However, their shortcomings are highlighted in the presence of ambiguity or uncertainty. Zadeh1 established the fuzzy set (FS) theory to solve this issue, explaining that DoM’s range lies between the \(0\) and \(1\) interval\(.\) This model permits information scientists and researchers to handle hesitation and vague data more efficiently. The invention of FS theory marked a significant advancement in decision-making sciences. Many researchers used this model to solve real-world challenges. For example, Ardil1 established a ranking analysis methodology based on FS to select optimal combat aircraft. Furthermore, Sahu et al.2 assessed the durability of web-based applications through the FS approach. FS theory faced challenges addressing complex situations when DoNM was involved in the data. To solve this issue, Atanassov3 established the idea of intuitionistic fuzzy sets (IFS), which defined both DoM and DoNM within the closed interval \(\left[ {0,1} \right],\) adhering to the condition \(0 \le {\Omega }\left( {\text{x}} \right) + \partial \left( {\text{x}} \right) \le 1.\) The principles of IFS have inspired numerous data scientists and mathematicians to employ this framework to address intricate real-world challenges. PyFS was proposed by Yager4 as an advancement of IFS, wherein the squared sum of membership and non-membership values is confined to a range from 0 to 1, thereby offering a more robust analytical framework. The applications of this framework have been extensively examined in the literature.

The importance of AHP and aggregation operators in multi-criteria decision-making

An MCDM framework is designed to evaluate and select from various options frequently involving competing criteria. It is crucial in the field of decision sciences, offering systematic methodologies to address complex problems that require the simultaneous consideration of multiple factors. This framework prepares decision-makers to compare various alternatives, rank criteria based on their importance, and make informed choices. The AHP is a key tool within MCDM, providing a structured method for evaluating and prioritizing the diverse factors that affect intricate decisions. By breaking down the decision-making process into a hierarchy of smaller, more manageable sub-problems, AHP allows decision-makers to assign weights to each criterion according to their relative importance through pairwise comparisons. Additionally, AHP includes a consistency check, which helps reduce bias and improves the accuracy of the decision-making process.

AHP enhances transparency, supports logical decision-making, and showcases adaptability across multiple domains by integrating qualitative and quantitative data, thus functioning as a powerful tool for tackling complex MCDM challenges. Numerous mathematicians have applied AHP in various fuzzy environments. For instance, Acar et al.5 Introduced AHP within interval-valued PyFS frameworks, while Kahraman6 focused on AHP for selecting wind turbines in a PyFS setting. Dhumras and Bajaj7 used the AHP framework for green supply chain management in the energy sector. Otay et al.8 presented an interval-valued circular PyFS-based AHP method. Furthermore, Sun et al.9 formulated an AHP model utilizing PyFS data for MCDM challenges. AOs are among the most efficient tools in Multi-criteria Decision Making (MCDM) because they consolidate individual criterion assessments into a singular score value (SV). These operators allow decision-makers to evaluate the overall performance of alternatives by merging multiple, frequently conflicting criteria into a unified decision outcome. Mathematicians have developed a variety of AOs across different fuzzy contexts. For instance, Dhumras et al.10 proposed an application for estimating risk factors for cardiovascular disease based on the MADM technique. Wasim et al.11 proposed PyFS-based AOs for evaluating the technology sector, while Rahman12 introduced Einstein AOs within the PyFS framework. Chunsong et al.13 investigated Sklar power AOs for PyFS-based data, and Rahman et al.14 developed PyFS and logarithmic AOs tailored for the PyFS context. Ali et al.15 contributed PyFS-based AOs for studies related to material selection. Dhumras et al.16 proposed a decision-making technique for industry 5.0 catalyst in the consumer electronics market evaluations, and Garg and Dhumras17 proposed a decision-making model for industry 4.0 supply chain management system in production systems.

The importance of TNM and TCNM in fuzzy set theory

Triangular norms are essential components in the realm of FS theory. This significance arises from the challenges many FS models face in accurately processing information or adhering to the fundamental definition of FS, allowing outcomes to extend beyond the unit interval \([0, 1]\). To bridge this gap, the introduction of innovative mathematical frameworks became imperative. Manger et al.18 pioneered the concepts of TNM and TCNM, leading researchers to identify distinct TNMs and TCNMs that cater to specific FS models. For example, Khan et al.19 investigated the operational rules of Archimedean TNM and TCNM, while Yahya et al.20 formulated Frank TNM and TCNM operations. Chatterjee and Seikh21 proposed a decision-making technique for solid waste management, and Chatterjee and Seikh22 proposed the MEREC-VIKOR approach for electric vehicle adoption. Dhumras et al.23 established the TOPSIS/VIKOR approach in an electronic marketing strategic plan. Huang19 also introduced Hamacher TNM and TCNM rules for data aggregation. Seikh and Chatterjee24 proposed Dombi AOs based on IFS to select the E-learning websites, Seikh and Mandal25 proposed Dombi AOs based on interval-valued Fermatean fuzzy to select the bio-medical waste management. Pal et al.26 used Dombi AOs to choose renewable energy, and Seikh and Mandal27 proposed Frank AOs for MCDM application. Many more AOs exist, but Sugeno integrals are vital for aggregating data in fuzzy decision-making when criteria are not precisely defined. Weber operators offer a flexible way to balance severity and compassion in aggregation by altering between AND and OR behavior. Both are vital in decision systems requiring significant judgment, as control systems, pattern recognition, and multi-criteria assessment. Many researcher used Sugeno-Weber AOs such as, Hussain et al.28 proposed Sugeno-Weber decision model for sustainable digital security assessment, Wang et al.29 established Sugeno-Weber decision model for solar panel selection and Ullah et al.30 proposed IFS information aggregation under Sugeno-Weber TNM.

Importance of Pythagorean fuzzy set

The key advantage of the PyFS model lies in its capability to capture higher degrees of ambiguity and hesitancy compared to traditional IFS. In PyFS, the squared sum of Dom and DoNM is acceptable to be less than or equal to 1, as \(0 \le \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right) \le 1\). It offers more meaningful skills and flexibility in demonstrating vague and imprecise information, and is also essential for complex decision-making situations. Regarding practical significance, MCDM based on PyFS is helpful when human decisions are fundamentally ambiguous or when experts face difficulty providing precise assessments. This contains applications in supplier selection, risk assessment, healthcare decision-making, and environmental management.

Motivation and proposed work

Numerous mathematicians have investigated various AHP models for FS and PyFS theories, as documented in the current literature. Many researchers have proposed AOs that incorporate prioritization and power AOs. In this article, we present innovative AOs based on AHP. The key contributions of this study are as follows:

  1. i.

    Development of an AHP-based algorithm for evaluating weight vectors of attributes.

  2. ii.

    Introduction of new AOs, particularly the PyFSWWA and PyFSWWG operators.

  3. iii.

    Examine the essential axioms of the proposed methodology, including monotonicity, boundedness, and idempotency.

  4. iv.

    Conducting a thorough comparative analysis with existing methodologies to assess their applicability and effectiveness.

Organization of the proposed work

The article is structured as follows: Section “Preliminaries” provides essential definitions and operational principles. Section “AHP for PyFS-based information” examines the Analytic Hierarchy Process (AHP) in the context of PyFS-based information. Section “Proposed AOs based on PyFS” introduces the new Aggregation Operators (AOs) and their characteristics. Section “AHP-based MCDM algorithm for PyFS” details the Multi-criteria Decision-Making (MCDM) algorithm developed from the proposed methodology. Section “Case study” presents a case study that emphasizes the sustainable utilization of urban land. Section “Numerical example” addresses a practical numerical problem based on the suggested method. Section “Comparison” performs a comparative analysis to evaluate the effectiveness of the proposed approach. Finally, Section “Conclusion” summarizes the conclusions, and Fig. 1 illustrates the overall framework of the proposed theory.

Fig. 1
figure 1

Show the framework of the proposed model.

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Preliminaries

This section outlines the essential definitions required for comprehending our proposed theory. The introduction of FS theory, which was later expanded to include PyFS, marked a significant milestone in decision sciences. It enhanced mathematicians’ precision and effectiveness in tackling intricate problems that traditional CS theories could not adequately address. Here, we will define the concepts of PyFS, score function, and accuracy function, along with the fundamental operational laws governing PyFS with Sugeno-Weber TNM and TCNM.

Definition 1

Consider \({\Phi }\) to be the non-empty set. A PyFS \({\zeta }\) is stated as4:

$${\upzeta } = \left\{ {\left( {x,{{ \Omega }}\left( x \right),\partial \left( x \right)} \right)|x \in {\Phi }} \right\}$$

where \({{ \Omega }}\left( {\text{x}} \right) \in \left[ {0,1} \right]\) is the DoM, and \(\partial \left( {\text{x}} \right) \in \left[ {0,1} \right]\) is the DoNM and satisfies that:

$$0 \le \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right) \le 1$$

Here, the hesitancy degree \(\hbar \left( x \right)\) is defined as:

$$\hbar \left( x \right) = \sqrt {1 - \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right)}$$

The score function translates fuzzy data into scalar values. The score is a standard measurement used to compare one fuzzy element with another when ranking or making decisions in the human sciences.

Definition 2

Consider \({\text{C}}^{{\text{o}}} \left( x \right) = \left( {{\Omega }\left( x \right),\partial \left( x \right)} \right)\) be the PyFVs. The score value \({\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}} } \right)\) is given as follows31:

$${\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}} } \right) = \left( {\left( {\Omega \left( {\text{x}} \right)} \right)^{2} - \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right),{\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}} } \right) \in \left[ {0,1} \right]$$

where, \({\Omega }\left( {\text{x}} \right)\) be the DoM of the aggregated value and \(\partial \left( {\text{x}} \right)\) be the DoNM of aggregated value. To take the square difference of DoM and DoNM, calculate the value of the score function.

Definition 3

Consider \({\text{C}}^{{\text{o}}} \left( x \right) = \left( {{\Omega }\left( x \right),\partial \left( x \right)} \right)\) be the PyFVs. The accuracy function \({\Lambda }\left( {{\text{C}}^{{\text{o}}} } \right)\) is given as follows31:

$${\Lambda }\left( {{\text{C}}^{{\text{o}}} } \right) = \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right),{{ \Lambda }}\left( {{\text{C}}^{{\text{o}}} } \right) \in \left[ {0,1} \right]$$

Remark

Let \({\text{C}}^{{\text{o}}}_{1} \left( x \right)\) and \({\text{C}}^{{\text{o}}}_{2} \left( x \right)\) Represent two PyFVs. The comparison of PyFVs can be made using the following principle:

  1. i.

    If \({\text{ a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{1} } \right) > {\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{2} } \right)\), then \({\text{C}}^{{\text{o}}}_{1}\) is considered greater than \({\text{C}}^{{\text{o}}}_{2}\).

  2. ii.

    If \({\text{ a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{1} } \right) < {\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{2} } \right)\), then \({\text{C}}^{{\text{o}}}_{1}\) is considered smaller than \({\text{C}}^{{\text{o}}}_{2}\).

  3. iii.

    If \({\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{1} } \right) = {\text{a}}^{{\text{o}}} \left( {{\text{C}}^{{\text{o}}}_{2} } \right)\) Then check the accuracy function:

    1. a.

      If \({\Lambda }\left( {{\text{C}}^{{\text{o}}}_{1} } \right) > {\Lambda }\left( {{\text{C}}^{{\text{o}}}_{2} } \right)\), then \({\text{C}}^{{\text{o}}}_{1}\) is considered greater than \({\text{C}}^{{\text{o}}}_{2} .\)

    2. b.

      If \({\Lambda }\left( {{\text{C}}^{{\text{o}}}_{1} } \right) < {\Lambda }\left( {{\text{C}}^{{\text{o}}}_{2} } \right)\), then \({\text{C}}^{{\text{o}}}_{1}\) is considered smaller than \({\text{C}}^{{\text{o}}}_{2}\).

    3. c.

      If \({{ \Lambda }}\left( {{\text{C}}^{{\text{o}}}_{1} } \right) = {\Lambda }\left( {{\text{C}}^{{\text{o}}}_{2} } \right)\), for two PyFVs, then \({\text{C}}^{{\text{o}}}_{1}\) is similar to \({\text{C}}^{{\text{o}}}_{2}\).

Definition 4

The Sugeno-Weber TN and TCN are defined as follows32:

$$\gamma^{\phi } = \left\{ \begin{gathered} \gamma_{d} \left( {b,a} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \;if \vee = - 1 \hfill \\ \max \left( {0,\frac{b + a - 1 + \phi ba}{{1 + \phi }}} \right),\quad if - 1 < \phi < + \infty \hfill \\ \gamma_{d} \left( {b,a} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \phi = + \infty \hfill \\ \end{gathered} \right.$$
$$Z^{\phi } = \left\{ \begin{gathered} Z_{d} \left( {b,a} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if \vee = - 1 \hfill \\ \min \left( {1,b + a - \left( {\frac{\phi }{1 + \phi }} \right)ba} \right),\quad if - 1 < \phi < + \infty \hfill \\ Z_{d} \left( {b,a} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\phi = + \infty \hfill \\ \end{gathered} \right.$$

where \({\upgamma }_{d} \left( {b,a} \right)\) and \(Z_{d} \left( {b,a} \right)\) Known as drastic TN and TCN.

AHP for PyFS-based information

The AHP employs pairwise comparisons among alternatives to ascertain the weights of chosen criteria. The process begins with the establishment of the primary goal. At the third level, alternatives are systematically compared, and their respective weights are assessed, facilitating an in-depth analysis of the data depicted as PyFVs. Figure 2 provides a geometric representation of the objective of AHP based on criteria and alternatives for a better understanding.

Fig. 2
figure 2

Show the objective of the AHP technique under the criteria and alternatives.

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The following is a detailed outline of the AHP process:

Step 1: Construct an AHP model represented as a decision matrix and articulate the MCDA problem utilizing PyFVs.

Step 2: Perform pairwise comparisons within the decision matrix.

Step 3: Evaluate the CR of the decision matrix. If the CI is below \(0.10\), continue; if not, revise the decision matrix accordingly.

Step 4: Ultimately, compute the weight vectors for the alternatives. Figure 3 shows the complete AHP model, outlining the steps from crucially the goal and criteria to performing pairwise comparisons and calculating priority weights. It achieves this by ranking alternatives to support effective decision-making.

Fig. 3
figure 3

Show the AHP model to calculate the weight vector for all criteria.

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The diagram illustrates the initial application of linguistic measurements in developing a hierarchical framework represented as a decision matrix, which facilitates the pairwise comparison of various options. Subsequently, the consistency ratio (CR) is assessed. If the CR is below \(0.10\), the process proceeds; if not, modifications are implemented to the decision matrix or the hierarchical framework. Table 1 presents the linguistic scale index (LSI) employed for the pairwise comparisons.

Table 1 Linguistic Scale with PyFVs.
Full size table

This study changes linguistic terms into PyFVs using a designed scale display in Table 1. In this, each term was linked with a specific MoD \(\Omega_{{\text{r}}}\) and MoNM \(\partial_{{\text{r}}}\) degree, fulfilling the Pythagorean form \(0 \le \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right) \le 1\). The scale consists of \(\left( {1 - 7} \right)\) Linguistic variables range from "Absolutely more essential" to "Less essential," with PyFVs pairs and LSI values to express varying degrees of importance and inverse importance. This mapping permits for significant illustration of expert decisions, taking both the strength and ambiguity of preferences. The PyFVs were determined based on existing literature and advanced through expert discussions, where professionals were asked to rank criteria using these linguistic terms. A detailed mapping process and validation ensured that the linguistic terms accurately reproduced the experts’ intended meaning, allowing the integration of qualitative insights into the decision-making model with clarity and consistency.

Proposed AOs based on PyFS

This subsection introduces newly proposed AOs based on PyFVs and the AHP. The weight vectors are calculated using AHP, and the resulting AOs are referred to as PyFSWWA and PyFSWWG operators. Additionally, we examine some fundamental axioms of AOs, including idempotency, monotonicity, and boundedness.

Definition 5

Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots ,\Psi } \right)\) and \(\tau_{r}\) be a WV with a condition \(\mathop \sum \limits_{r}^{\Psi } \tau_{r} = 1\) and the parameter \(\vee \ge 1\):

$$PyFSWWA\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{3} } \right) = \oplus_{r = 1}^{\Psi } \tau_{r} \Gamma_{r}$$

Theorem 1

Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots ,\Psi } \right)\) and \(\tau_{r}\) be a WV. Its sum is one, and \(\vee \ge 1\) . The aggregation outcomes are also PyFVs and are as follows:

$$PyFSWWA\left( {\Gamma_{1} ,\Gamma_{2} , \ldots ,\Gamma_{\Psi } } \right) = \left( {\begin{array}{*{20}c} {\sqrt {\frac{1 + \vee }{ \vee }\left( {1 - \mathop \prod \limits_{r = 1}^{\Psi } \left( {1 - \Omega_{r}^{2} \frac{ \vee }{1 + \vee }} \right)} \right)^{{\tau_{r} }} } ,} \\ {\frac{1}{ \vee }\left( {\left( {1 + \vee } \right)\mathop \prod \limits_{r = 1}^{\Psi } \left( {\frac{{ \vee \partial_{r} + 1}}{1 + \vee }} \right)^{{\tau_{r} }} - 1} \right)} \\ \end{array} } \right)$$

Proof

The proof of Theorem 1 is given in the appendix.

Theorem 2

(Idempotency) Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots , \Psi } \right)\) . PyF \(\Gamma_{1} = \Gamma_{2} = \cdots = \Gamma_{i} = \Gamma .\) So

$$PyFSWWA\left( { \Gamma_{1} , \Gamma_{2} , \ldots ,\Gamma_{i} } \right) = \Gamma$$

Proof

The proof of Theorem 2 is given in the appendix.

Theorem 3

(Boundedness) Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots , \Psi } \right)\). PyF \(\Gamma^{ - } = \left( {\min \left[ {\Omega_{r} } \right], \max \left[ {\partial_{r} } \right]} \right)\) and \(\Gamma^{ + } = \left( {\max \left[ {\Omega_{r} } \right], \min \left[ {\partial_{r} } \right]} \right)\) then we obtain,

$$\Gamma^{ - } \le PyFSWWA\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{i} } \right) \le \Gamma^{ + }$$

Proof

The proof of Theorem 3 is given in the appendix.

Theorem 4

(Monotonicity) Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs and \(\Gamma_{r}^{\prime } = \left( {\Omega^{\prime},\partial^{\prime}} \right)\), where \(\left( {r = 1,2,3, \ldots ,\Psi } \right)\) and hold condition:

$$PyFSWWA\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{i} } \right) \le PyFSWWA\left( { \Gamma_{1}^{\prime } , \Gamma_{2}^{\prime } , \ldots , \Gamma_{i}^{\prime } } \right)$$

Proof

The proof of Theorem 4 is given in the appendix.

Definition 6

Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots ,\Psi } \right)\) and \(\tau_{r}\) be a WV with a condition \(\mathop \sum \limits_{r}^{\Psi } \tau_{r} = 1\) and the parameter \(\vee \ge 1\):

$$PyFSWWG\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{\Psi } } \right) = \otimes_{r = 1}^{\Psi } \tau_{r} \Gamma_{r}$$

Theorem 5

Let \(\Gamma_{r} = \left( {\Omega_{r} ,\partial_{r} } \right)\) be a collection of PyFVs where \(\left( {r = 1,2,3, \ldots ,\Psi } \right)\) and \(\tau_{r}\) be a WV. Its sum is one, and \(\vee \ge 1\). The aggregation outcomes are also PyFVs and are as follows:

$$PyFSWWG\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{\Psi } } \right) = \left( {\left( {\begin{array}{*{20}c} {\frac{1}{ \vee }\left( {\left( {1 + \vee } \right)\mathop \prod \limits_{r = 1}^{\Psi } \left( {\frac{{ \vee \Omega_{r} + 1}}{1 + \vee }} \right)^{{\tau_{r} }} - 1} \right)^{{\tau_{r} }} ,} \\ {\sqrt {\frac{ \vee }{1 + \vee }\left( {1 - \mathop \prod \limits_{r = 1}^{\Psi } \left( {1 - \partial_{r}^{2} \left( {\frac{ \vee }{1 + \vee }} \right)} \right)} \right)^{{\tau_{r} }} } } \\ \end{array} } \right)} \right)$$

Proof

The proof of Theorem 5 is the same as Theorem 1, and it satisfies the necessary characteristics of AOs.

AHP-based MCDM algorithm for PyFS

This section presents a Multi-criteria Decision-Making (MCDM) algorithm based on the Analytic Hierarchy Process (AHP) within the PyFS framework. We establish a decision-making matrix and assess the consistency index (CI) before calculating the weight vectors (WV). Subsequently, we employ the proposed PyFSWWA and PyFSWWG operators to evaluate and rank the alternatives. The detailed steps for this procedure are provided below.

Weight vector calculation

Step 1: Initially, we create a decision matrix (DM) by employing linguistic scale values and comparing the alternatives pairwise:

$$\acute{\text{M}} = \left( L \right)_{{{\varvec{a}} \times {\varvec{b}}}}$$

where \(i \times j\) represents the relative importance of standards \(a\) to \(b\).

Step 2: Calculate the geometric mean \({\text{G}}MM\) of each row in the decision matrix \(\left( {D\dot{M}} \right)\):

$${\text{G}}M_{i} = \mathop \prod \limits_{i,j = 1}^{\Psi } L_{i \times j}$$

Step 3: Compute the sum \(\left( {P_{i}{\prime} } \right)\) of all \(\left( {{\text{G}}MM} \right)\) values:

$$P_{i}^{\prime } = \mathop \sum \limits_{i = 1}^{\Psi } \left( {{\text{G}}MM_{i} } \right)$$

Step 4: Compute the inverse \(\left( {IV_{i} } \right)\) of \(\left( {{\text{G}}MM} \right)\):

$$IV_{i} = \left( {P_{i}^{\prime } } \right)^{ - 1} = \frac{1}{{\mathop \sum \nolimits_{i = 1}^{\Psi } \left( {{\text{G}}MM_{i} } \right)}}$$

Then arrange \(\left( {IV_{1} ,IV_{2} , \ldots ,IV_{n} } \right)\) In ascending order.

Step 5: Compute the product \(\left( {{\text{p}}_{i} } \right)\) of \(\left( {P_{i}^{\prime } } \right)\) and arrange \(\left( {IV_{i} } \right)\):

$${\text{p}}_{i} = \mathop \prod \limits_{i = 1}^{r} \left( {P_{i}{\prime} ,IV_{i} } \right)$$

Step 6: Calculate the average \(\left( {\ddot{\text{A}}_{i} } \right)\) of \(\left( {{\text{p}}_{i} } \right)\):

$$\ddot{\text{A}}_{i} = \frac{{\mathop \sum \nolimits_{i = 1}^{\Psi } \left( {{\text{p}}_{i} } \right)}}{{\varvec{r}}}$$

where \(i = 1,2,3, \ldots ,\Psi\).

Step 7: Normalize each priority weight by dividing \(\left( {\ddot{\text{A}}_{i} } \right)\) by the sum of all \(\mathop \sum \limits_{i = 1}^{\Psi } \left( {\ddot{\text{A}}_{i} } \right)\) Values to obtain the weight vectors (WVs):

$$\omega = \frac{{\ddot{\text{A}}_{i} }}{{\mathop \sum \nolimits_{i = 1}^{\Psi } \left( {\ddot{\text{A}}_{i} } \right)}}$$

Step 8: Verify the consistency index (CI) of the calculated \(\omega\) using the following formula:

$$CI = \frac{{{\uppi }_{\max } - 1}}{\Psi - 1}$$

where

$${\uppi }_{\max } = \mathop \sum \limits_{i = 1}^{\Psi } \left( {\ddot{\text{A}}_{i} } \right)$$

And \(\Psi\) be the number of alternatives.

Step 9: Apply the developed PyFSWWA and PyFSWWG operators as follows:

$$PyFSWWA\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{\Psi } } \right) = \left( {\begin{array}{*{20}c} {\sqrt {\frac{1 + \vee }{ \vee }\left( {1 - \mathop \prod \limits_{r = 1}^{\Psi } \left( {1 - \Omega_{r}^{2} \frac{ \vee }{1 + \vee }} \right)} \right)^{{\tau_{r} }} } ,} \\ {\frac{1}{ \vee }\left( {\left( {1 + \vee } \right)\mathop \prod \limits_{r = 1}^{\Psi } \left( {\frac{{ \vee \partial_{r} + 1}}{1 + \vee }} \right)^{{\tau_{r} }} - 1} \right)} \\ \end{array} } \right)$$

and

$$PyFSWWG\left( { \Gamma_{1} , \Gamma_{2} , \ldots , \Gamma_{\Psi } } \right) = \left( {\left( {\begin{array}{*{20}c} {\frac{1}{ \vee }\left( {\left( {1 + \vee } \right)\mathop \prod \limits_{r = 1}^{\Psi } \left( {\frac{{ \vee \Omega_{r} + 1}}{1 + \vee }} \right)^{{\upsilon \tau_{r} }} - 1} \right)^{{\tau_{r} }} ,} \\ {\sqrt {\frac{ \vee }{1 + \vee }\left( {1 - \mathop \prod \limits_{r = 1}^{\Psi } \left( {1 - \partial_{r}^{2} \left( {\frac{ \vee }{1 + \vee }} \right)} \right)} \right)^{{\tau_{r} }} } } \\ \end{array} } \right)} \right)$$

Step 10: Calculate the aggregated results into a single value using the SF formula defined in Definition 2:

$${\text{a}}^{{\text{o}}} F\left( x \right) = \left( {\left( {\Omega \left( x \right)} \right)^{2} - \left( {\partial \left( x \right)} \right)^{2} } \right)$$

If the score value of all alternatives is equal, then we use the accuracy function to find the ranking. Accuracy function defined in definition 3:

$${\Lambda }\left( {{\text{C}}_{{\text{i}}}^{{\text{o}}} } \right) = \left( {\left( {{\Omega }\left( {\text{x}} \right)} \right)^{2} + \left( {\partial \left( {\text{x}} \right)} \right)^{2} } \right)$$

Step 11: Arrange all the aggregated results in descending order.

Case study

Athletes identify that participating in sports offers numerous benefits. It improves physical health and nurtures a sense of camaraderie among team members, which explains the widespread appreciation for sports in various cultures. Additionally, it is essential to highlight that engaging in sports can positively impact personal and professional dimensions of life. Participation in sports develops critical attributes such as leadership, confidence, and discipline, essential for securing employment, progressing in careers, and achieving professional growth. Employers often seek individuals who can react promptly and make sound decisions, qualities commonly found in athletes. Participation in sports significantly improves an individual’s ability to communicate effectively with teammates and colleagues, a skill that proves invaluable in a professional environment. Athletes frequently demonstrate creativity in overcoming challenges and appreciate the importance of teamwork qualities that are highly transferable and attractive to prospective employers.

Furthermore, engaging in sports fosters strong organizational skills and the capacity to manage multiple tasks simultaneously, traits in high demand across various industries. Figure 4 shows kids involved in diverse sports like running, tennis, yoga, cycling, weight lifting, and more. It highlights the importance of physical activity for health, fun, and skill growth.

Fig. 4
figure 4

Presents children actively participating in many sports and physical activities. Address: https://www.euroschoolindia.com/blogs/benefits-of-sports-for-students/.

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Additionally, sports provide essential lessons about success and failure. Regardless of one’s level of proficiency in a sport, facing challenges and unforeseen circumstances is a shared experience. These situations enhance one’s adaptability in unfamiliar or high-pressure work environments. Incorporating sports into one’s routine supports physical and mental health and develops skills that can offer a competitive edge in one’s career. In this context, we will examine several traditional sports and their contributions to human development as follows:

  1. (1)

    Physical health

Physical health is defined by the body’s capacity to operate effectively without difficulties. Participating in sports is crucial for preserving and improving human health. Participation in physical activities and sports functions as a means of exercise, enhancing stamina. Regular movement of the body and its muscles fosters better coordination between body parts and the brain, improving overall physical performance efficiency. Cardiovascular Health: When students engage in physical activities such as swimming or running throughout their educational experience, these sports facilitate increased blood circulation, which helps reduce the likelihood of cardiovascular problems. Muscle Strength: A lack of participation in sports can weaken muscles, making it difficult to perform daily tasks in the future. Figure 5. show the health benefits of running, including improved heart, lung, joint, and mental health.

Fig. 5
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Show the impact of Exercise on Physical and Mental Well-Being. (https://medium.com/@rajamujtaba900/advantage-of-sports-activities-f987767461c3).

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In contrast, sports involvement enhances muscle strength. For example, repetitive arm movements in swimming lead to muscle fiber hypertrophy, where small tears are repaired and rebuilt, resulting in firmer, larger, and more defined muscles. Increasing muscle mass enables individuals to perform tasks more effectively and lift heavier weights, ultimately enhancing their overall physical abilities.

  1. (2)

    Mental health and discipline

Sports play an essential role in enhancing mental health. Participation in physical activities causes the brain to coach glands to release hormones that positively impact the body. These are improved overall mental wellness, reducing stress and promoting mood, attention, and self-discipline improvements. The human body makes endorphins, which act as natural mood boosters by modifying stress and unease. These are especially beneficial for students. It helps them to handle academic challenges and personal issues. Furthermore, sports involvement increases serotonin and dopamine levels, both neurotransmitters connected to happiness and gladness. It helps in reducing symptoms of depression and encourages a more positive perspective.

  1. (3)

    Social skills and teamwork

Effective communication is vital for the success of any student. A lack of skill to generate ideas clearly can result in difficulties across various life fields. But sports involvement provides students with essential opportunities to build relationships, impact others, and navigate social connections. Teamwork plays a vital role in a student’s accomplishments. Participation in team activities can improve the cultivation of leadership skills. Through collaboration in a team, students learn to achieve group dynamics and struggle towards common goals. The foundation, communication, and leadership are key elements significantly affecting teamwork.

  1. (4)

    Character development

Character development can improve positive traits and moral values within an individual. This process is vital for students because it supports their ethical development and shapes their behaviors, attitudes, and habits. The main criteria of character development are moral values, self-discipline, accountability, and self-confidence. Ethical principles are critical for every student. They cover the ability to differentiate between right and wrong, and the appearance of qualities such as honesty, fairness, respect, and empathy. Self-discipline is vital for every student. The most talented students balance their time between academic responsibilities, sports, and other activities while remaining focused on their goals. Accountability should develop a sense of accountability for their academic duties and daily responsibilities in students. Participation in sports can help to take ownership of one’s tasks. Self-confidence is the trust in one’s abilities and skills, accompanied by a positive outlook. This declaration is vital for success and is raised through teamwork, leadership, influencing others, and meeting deadlines.

Next, we will explore various sports directly associated with the factors above and contribute to career satisfaction, such as football, swimming, taekwondo, and golf.

  1. (1)

    Football

Football requires significant effort and commitment from its players, with continuous training and passion vital for success. The path to becoming a professional in this sport involves both physical and psychological challenges. The second division is essential in nurturing footballers as they chase their goals. Effective football is based on values such as devotion and flexibility. In football, participants face daily difficulties that require solid obligation and determination. Figure 6 shows the spirit of ground football, where skills and sportsmanship are cultivated from an early age.

Fig. 6
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Show the Football game: promoting physical well-being, coordination, and Social Abilities in Developing Minds. (https://www.chinadaily.com.cn/a/201804/25/WS5adfdbdca3105cdcf651a5b4.html).

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Footballers engage in intense training, tackling, physical exhaustion, boosting their stamina, and honing their speed to enhance their performance on the field. Mentally, they must maintain focus, motivation, and discipline, overcoming periods of fatigue to achieve peak performance.

  1. (2)

    Swimming

Swimming has numerous benefits, particularly in terms of enhancing our physical health. However, our focus today will be on its capacity to boost workplace performance. A key factor in productivity is how we begin our day. Engaging in a pleasurable activity before heading to the office creates a positive environment and encourages the release of endorphins. Swimming, or simply being in water, is an excellent way to achieve this. A short 30-min swim, especially outdoors, can significantly elevate your mood. Figure 7 shows the key criteria established by swimmers during competition. It reveals the intense physical and technical demands of competitive swimming.

Fig. 7
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Show Stamina, Speed, and Skill at the Swimming Competition. (https://www.olympics.com/en/news/the-history-of-olympic-swimming).

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To improve happiness, swimming improves awareness and cognitive abilities, allowing students to approach analytical tasks more effectively from the start of the workday. Also, swimming helps to decrease tension and stress and develop overall well-being. This positive mindset improves your outlook and inspires teamwork and effective networking, which are crucial in today’s workplace. Additionally, the discipline of regular swimming practice is essential. Including swim training on your resume indicates to potential employers your students’ commitment and capability to complete tasks. It is necessary for managing large projects that require patience, sustained focus, and goal-oriented strategies, traits often related to successful swimmers.

  1. (3)

    Tennis

Tennis shows a lifestyle that many consider a lifelong journey. This estimation is covered by studies that indicate that tennis is among the top activities to participate. The sport uniquely combines physical activity, mental challenge, and social interaction, making it an appealing choice for individuals across all age groups. Its attractiveness as a lifelong sport stems from its flexibility to accommodate varying skill levels and physical capabilities. Unlike many other sports that may become more difficult as one ages, tennis is adjusted to fit personal preferences, certifying it remains manageable and enjoyable for everyone involved. Figure 8 shows the thrill of playing tennis.

Fig. 8
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Show the Thrill of Playing Tennis. (https://longislandtennismagazine.com/article/underhand-serve-effective-alternative/).

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In addition to its physical benefits, tennis offers many mental health advantages. The game’s strategic components require quick thinking, the ability to predict an opponent’s actions, and swift decision-making under pressure. This mental engagement can boost cognitive functions, enhance problem-solving skills, and improve overall mood and well-being. Furthermore, the concentration and focus demanded during play can act as a form of mindfulness, helping to alleviate stress and foster a sense of tranquility.

  1. (4)

    Golf

Golf is a sport that highlights precision, concentration, and mental focus. It adopts creativity and imaginative thinking, primarily when imagining the path and distance of one’s shots. Hand-eye management is crucial in golf. It is the ability to track the ball’s landing. Golf provides an ideal environment for developing these skills, away from the distractions of excited crowds and the sounds of cheering. For many, engaging in a round of golf presents a significant opportunity to relieve stress. Hitting the ball during tense moments allows individuals to channel their anxiety into a productive activity.

Furthermore, the physical activity involved in golf activates the release of endorphins. It can uplift mood, alleviate pain, and help reduce feelings of depression. Golf is a stimulating pursuit, and the enhanced circulation during play supports improved blood flow to the brain. Studies suggest that golfing can influence cognitive functions. The repetitive nature of swings aids in developing muscle memory, while navigating the course sharpens one’s sense of distance and depth perception.

Numerical example

Investigating the elements prompting career satisfaction in sports participation is complex yet essential. MCDM is a practical method to certify a complete and precise assessment. MCDM allows for assessing alternatives based on various attributes to identify the most advantageous choice. In this framework, we begin by gathering data with the assistance of decision-making specialists, utilizing PyFVs to develop a decision matrix. \({\left({\varvec{L}}\right)}_{{\varvec{a}}\times {\varvec{b}}}.\) The alternatives under consideration include four traditional sports:

\({\upbeta}_{1}\) = football

\({\upbeta}_{2}\) = swimming

\({\upbeta}_{3}\) = tennis

\({\upbeta}_{4}\) = golf

We seek to rank various sports according to their significance, taking into account the following weights assigned to each criterion \({\text{y}}_{1} = 0.1\) for physical health, \({\text{y}}_{2} = 0.4\) for mental health and discipline, \({\text{y}}_{3} = 0.4\) for social skills and teamwork, and \({\text{y}}_{4} = 0.1\) for character development. The weight vectors are determined using the Analytic Hierarchy Process (AHP) within the PyFS framework, guaranteeing that the sum of the weight vectors equals one. Additionally, the Consistency Index (CI) is calculated to assess the reliability of the derived weight vectors. To evaluate the alternatives, we employ the PyFSWWA and PyFSWWG operators. The decision-making process is detailed in the subsequent steps.

Step 1: Collect data from sports experts and create a PyFV-based decision matrix \({\left(L\right)}_{a\times b}\). Table 2 shows the decision matrix in the form of PyFVs taken from an expert.

Table 2 PyFVs-based pairwise comparison decision matrix.
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Step 2: Using proposed AOs such as PyFSWWA and PyFSWWG on Table 2. The aggregated results are shown in Table 3.

Table 3 Aggregated outcomes by using PyFSWWA and PyFSWWG.
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Step 3: Compute the aggregated results’ SVs using Definition 2. All score values are presented in Table 4.

Table 4 Score Values of all alternatives.
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Step 4: In Table 5, we present the rankings of score values.

Table 5 Ranking of alternatives.
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Table 5 outlines the optimal alternatives arranged in sequence. It was noted that the proposed operator, such as the PyFSWWA operator, identifies an alternative. \({\upbeta}_{3}\) as the superior choice, with the ranking order being \({\upbeta}_{3} > {\upbeta}_{2} > {\upbeta}_{4} > {\upbeta}_{1}\). Conversely, when the PyFSWWG operator is employed, \({\upbeta}_{3}\) It is recognized as the top option, with the ranking sequence being \({\upbeta}_{3} > {\upbeta}_{2} > {\upbeta}_{4} > {\upbeta}_{1}\). The geometrical representation of Table 4 is shown in Fig. 9.

Fig. 9
figure 9

Shows the graphical representation of weighted averaging and weighted geometric means.

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Figure 9 presents a graphical depiction of the aggregated outcomes. It is evident from Fig. 9 that our proposed AHP-based PyFSWWA and PyFSWWG operators lead to the conclusion that \({\upbeta}_{2}\) is the best alternative using both AOs.

Comparison

A comprehensive comparison was undertaken to assess the applicability and versatility of various extended AOs. It was observed that several FS-based AOs, including the MCDM approach rooted in FS theory as proposed by Shahzadi et al.33 and the MCDM solution based on FS theory by Al Shami34, are insufficient for processing information based on PyFS due to their failure to incorporate the DoNM structure. In contrast, various AOs, such as the Pythagorean fuzzy Hamacher weighted averaging (PyFHWA) and Pythagorean fuzzy Hamacher weighted geometric (PyFHWG) developed by Shahzadi et al.33 and Pythagorean fuzzy Yager weighted averaging (PyFYWG) and Pythagorean fuzzy Yager weighted geometric (PyFYWG) developed by Wu and Wei35, exhibit compatibility with PyFS-based data. For a more comprehensive understanding, the ranking derived from the PyFSWWA and PyFSWWG operators and the existing model are presented in Table 6.

Table 6 Represents the comparison between other existing approaches and the proposed approach.
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The information displayed in the Table 6 demonstrates that many FS-based aggregation operators such as, Wang and Liu36, Sheikh and Mandal37, Al Shami34, and Kesharvarz Ghorabee et al.32 are insufficient for aggregating PyFS-based data, underscoring the importance of PyFS. As a result, we compared our results with those of Hamacher TNM, TCNM, and Yager TNM, TCNM-based AOs. A graphical representation of the comparative analysis with the current aggregation operators is presented in Fig. 10 to enhance comprehension.

Fig. 10
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Show the comparative study between the proposed model and the existing model.

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Figure 10 provides a comparative analysis with other AOs. The data demonstrates the identification of the proposed PyFSWWA and PyFSWWG operators with \({\upbeta}_{3}\) recognized as the optimal choice. The established ranking is \({\upbeta}_{3} > {\upbeta}_{2} > {\upbeta}_{4} > {\upbeta}_{1}\). In contrast, when applying Hamacher and Yager TNM and TCNM-based AOs, \({\upbeta}_{3}\) It is also identified as the leading alternative. Additionally, it is essential to note that no results are generated when Wang and Liu36, Sheikh and Mandal37, Al Shami34, and Kesharvarz Ghorabee et al.32 based AOs are utilized with our PyFS-based data. These existing models cannot handle the PyFS because the sum of DoM and DoNM exceeds \(\left[0, 1\right]\). As compared to these models, the range of PyFS is wide.

Practical implementations of the proposed model

In this, we explain practical implementations of a Pythagorean fuzzy framework under the MCDM model and propose AOs.

Academic and career counseling

In academic and career counseling scenarios, the PyFS, based on the MCDM model, can be a powerful decision-support tool to guide students and ambitious athletes toward sports that support their continuing happiness and personal interests. Career counselors can take subjective data from students under numerous criteria, such as income expectations, level of passion, physical and mental demand tolerance, and career durability. These data stated over linguistic terms are renewed into PyFVs to handle doubt and ambiguity in human decision-making. Our proposed model then uses these values, using our established AOs such as PyFSWWA and PyFSWWG operator to rank sports according to suitability for each individual and psychologically aligned recommendations.

Parental decision support for children’s sports involvement

Parents often struggle to select the right sport for their children, balancing their children’s comfort with long-term deliberations, such as safety, school compatibility, and career chances. Our proposed model suggests a planned and objective way to guide that decision. Parents and children jointly assess numerous sports using linguistic terms to rate criteria as enjoyment, physical strain, cost, and educational flexibility. These data are renewed into PyFVs, balancing doubt and emotional distinctions in such conclusions. Our proposed technique processes these data to rank sports. In this way, it helps families make well-informed, balanced selections that support long-term happiness and growth for the children.

Athlete retirement planning and transition counseling

Retired athletes face hesitation when transitioning to new careers, such as sports or a shift to new fields. Our proposed model can be applied to estimate and mention secondary career options such as coaching, media involvement, and switching to less demanding sports under an athlete’s interests, physical capabilities, and life goals. Using the PyFS framework to capture data on emotional readiness, community impact preference, and desire for public assignment. To use our AOs and then rank post-retirement paths by their potential to bring happiness and satisfaction. Our proposed guidance can facilitate retiring athletes’ psychological and professional transition, reducing post-career stress and identity loss.

Conclusion

In the contemporary technological landscape, selecting the optimal alternative presents a significant challenge. Within the ___domain of decision-making sciences, Multi-criteria Decision Making (MCDM) serves as a valuable tool for synthesizing complex information and identifying the best choice based on various attributes. Numerous mathematical frameworks have been developed under the FS theory for MCDM models utilizing the DoM concept. However, these frameworks have not successfully aggregated information where the DoNM is involved. In this article, we introduce new Aggregation Operators (AOs) derived from Sugeno-Weber Triangular Norm (TNM) and Triangular Conorm (TCNM) operations. The Analytic Hierarchy Process (AHP) is recognized as a precise and widely used methodology for determining the Weight Values (WVs) of attributes through pairwise comparisons within the decision matrix. Drawing inspiration from these fundamental characteristics, we have developed AHP-based PyFSWWA and PyFSWWG operators. Subsequently, we formulated a comprehensive algorithm for AHP to compute the Weight Values and presented an MCDM algorithm utilizing the PyFSWWA and PyFSWWG operators within the PyFS framework. Through this methodology, we have provided solutions to real-world numerical examples that rank factors influencing career satisfaction in sports participation. Finally, to assess the validity and effectiveness of our proposed approach, we conducted a thorough comparison with existing methodologies.

Limitation

PyFS offers improved capability to model doubt and hesitancy compared to existing fuzzy and IFS; they are not without limitations. One major challenge is unreliability in the increased estimation complexity, especially in large-scale decision-making complications where multiple criteria and alternatives are involved. Additionally, when the square sum of DoM and DoNM exceeds \(\left[0, 1\right]\) The proposed model fails to aggregate information.

Future direction

To overcome these challenges, in Future research, we can focus on expanding our theoretical framework to encompass various fuzzy structures and TNM and TCNM operations, including the concept of q-rung orthopair Fuzzy Sets (q-ROFS) as defined by Khan et al.38 using Aczel-Alsina TNM and TCNM operations, and Khan et al.39 developed the power AOs using q-ROFS theory. Jaleel40 proposed an agricultural robotics system, and Garg and Rehman41 developed an application for analyzing risk evaluation of hepatitis with sine trigonometric laws-based decision making. The novel idea of spherical SFS was developed by Mahmood et al.42, and the Pythagorean FS-based Aczel-Alsina AOs were introduced by Hussain et al.43.

Singh et al.44 established TOPSIS with q-norm picture fuzzy discriminant measure to select a green supplier problem, and Aggarwal et al.45 invented decision making to choose a banking site based on similarity measures. And interval valued t-spherical by Nazeer et al.46.