Decision support system based on AHP and PROMETHEE under rough pythagorean fuzzy set information for selection of basketball team

Abstract

The selection of a team in team-based sporting activities and professional situations is a crucial decision that requires an optimal choice from multiple conditions. The solutions currently available do not simplify the issues of the best team selection under different alternatives. In the modern age, the theory of multi-attribute decision-making (MADM) is a well-known approach for assessing decision-making problems. The main objective of this article is to introducing new aggregation operator (AO) called rough Pythagorean Fuzzy Dombi weighted averaging (RPyFDWA) operator, analytic hierarchy process (AHP) for measuring the weights of alternatives and also investigated the organizing preference rankings for enrichment evaluation (PROMETHEE) for ranking of alternatives by imposing a lower and upper approximations, under rough Pythagorean fuzzy set (RPyFS) framework. The RPyFS framework has a superior structure for handling uncertain data compared to existing frameworks. We also establish significant mathematical characteristics of the operator, including idempotency, monotonicity, and boundedness, to determine its flexibility. We have provided a case study based on a basketball team. The proposed theory is applied to the solution of a non-theoretical example related to assessing team selection issues. We can use our proposed AOs to select the best team from the list of four teams based on various attributes like physical fitness, performance criteria, experience, age, and injury prevention. We notice that the team It is the best team among all other considered teams using the proposed RPyFDWA operator and PROMETHEE method. We provided a comparison between other current approaches and established AOs for analyzing the authenticity and validity of the proposed approach. Also, provide a sensitivity analysis of the proposed study to observe the change in input variables that affects the model’s or decision’s dependability. A solid conclusion is provided at the end.

Introduction

The selection of basketball teams is a crucial endeavor that has a direct impact on team integration, performance, and ultimate success. Traditionally, we select players based on their past performances, experience, and individual judgment by coaches or those in charge of the selection process. Although such traditional methods make a specific contribution, they do not enable the development of a systematic estimation framework, and they can lead to a suboptimal or biased decision-making process. To address these shortcomings, this paper proposes an MADM process for selecting basketball teams. It consists of diffusion-driven player behavior synthesis and a practical MADM approach in a Pythagorean fuzzy environment, utilizing the PROMETHEE approach, Dombi operators, and AHP.

By creating and simulating realistic game scenarios, the diffusion-based model enables the generation of different patterns of player behavior, facilitating a thorough assessment of a player’s physical, experiential, psychological, and leadership qualities. This fusion approach enables careful analysis by combining quantitative performance data and expert judgment, thereby overcoming the uncertainty and ambiguity inherent in human decision-making. Zadeh¹ introduced the fuzzy set (FS) and intuitionistic fuzzy set (IFS), proposed by Atanassov², which include a non-membership degree (NMD). The rough FS was initially proposed by Dubois and Prade³, who established extensions to the idea of FS involving the introduction of the upper approximation space and lower approximation space on [0, 1]. Perez-Toledano et al.⁴ proposed selecting basketball teams based on player performance using a Performance Index Rating and a multi-criteria evolutionary algorithm. Strategies of Talent Selection and Correlation with Success in European Basketball National Team Programs, suggested by Kalén et al.⁵. Evaluation of air purifiers to enhance the Air Quality Index based on the Circular instead of fuzzy Heronian means is proposed by Guo⁶. Rukhsar et al.⁷ introduced intelligent decision analysis for green supplier selection, incorporating multiple attributes with circular intuitionistic fuzzy information aggregation and Frank triangular norms. The information about the IF dombi aggregation with upper and lower approximation is suggested by Khan et al.⁸.

Aggregation operator

In AOs, steps are developed to combine multiple requirements and expert observations into a single result. It is for this reason that scholars have introduced several AOs through the use of various types of fuzzy models, e.g., some Dombi AOs (DAOs) formulated using complex q-rung ortho-pair fuzzy sets and applied to MADM, as described by Ali and Mahmood⁹. Hamacher AOs on PyFS and its application in the MADM problem, and Asif et al.¹⁰, as well as the novel method of solving the MADM problems with the help of Fermatean fuzzy Hamacher AOs by Hadi et al.¹¹. Imagine having fuzzy interactional partitioned Heronian mean AOs: an example in MADM process analyzed by Lin et al.¹² and Aczel-Alsina AOs and their application in intuitionistic fuzzy MADM by Senapati et al.¹³ and use of refined spherical fuzzy DAOs in decision aids system suggested by Khan et al.¹⁴ and t-spherical fuzzy power aggregation operator (TSFPAOs) and its application in MADM proposed by Garg et al.¹⁵. Ashraf et al.¹⁶ proposed the Einstein MADM approach to examine the effects of technological innovation. Riaz et al.¹⁷ developed linear Diophantine fuzzy AOs and MADM. The concept of t-spherical fuzzy information and its similarity measure are mentioned by Hussain et al.¹⁸. Decision-based algorithm for circular Pythagorean fuzzy framework and advanced petroleum exploration methods created by Nazir et al.¹⁹.

Multi-attribute decision making

MADM is one of the most potent tools in decision-making sciences for data aggregation. The MADM is regarded as the methodology in FS theory, in which the ranking of alternative options is based on the assessment of different attributes within an atmosphere of uncertainty and imprecision. Different scholars have developed numerous MADM methods with varied structural frameworks. Hybrid models for decision-making based on rough Pythagorean fuzzy bipolar soft information proposed by Akram and Ali²⁰. Pythagorean Fuzzy Overlap Functions and Corresponding Fuzzy Rough Sets for Multi-Attribute Decision Making Yan et al.²¹, .

The MADM methodology is developed by using an interval-valued Pythagorean fuzzy set and differential evolutionary algorithm suggested by Chander and Das²². Linguistic Pythagorean fuzzy sets and their applications in MADM process proposed by Garg²³ and Novel Aczel–Alsina operators for PyFS with Application in MADM developed by Hussain et al.²⁴. The MADM policy-maker framework, which implements a non-numerical approach to analysis, as discussed in the interview using interval-valued triangular fuzzy numbers, as represented by Mohammadian et al.²⁵. Hesitant fuzzy prioritized operators and their application to MADM were introduced by Wei²⁶. The theory of MADM under the spherical trapezoidal data for the assessment of electric vehicles provided by Irvanizam et al.²⁷. The application of MADM in the evaluation of purification strategy discussed by Ali et al.²⁸ and the investigation of security gate system using the MADM given by Ali and Popa²⁹. The theory of Maclurin symmetric mean for the assessment of the MADM problem offered by Azeem et al.³⁰.

Theory of t-norm and t-conorm

The triangular norm (TN) and triangular conorm (TCN) are essential in fuzzy set theory, particularly in uncertain and imprecise applications of MADM.

The initial concept was proposed by Menger³¹, and several superior versions have been proposed over time. For example, Dombi operations have a substantial importance in the fuzzy set theory because they are very flexible when it comes to modeling the interaction between the values of any membership. Their parameters are adjustable, which enables them to shift easily among different TN and TCN, making them suitable for decision-making in various situations. Their generalizability property of other AOs increases their therapy vagueness in complex issues, fuzzy MADM, in which the compromise degree between criteria has to be fine-tuned. Many research scholars proposed multiple operational laws under Dombi TN and TCN for different fuzzy frameworks, such as the theory of p, q quasirung orthopair fuzzy sets for solving group decision-making problems under Dombi operations provided by Ali and Mehmood³². The theory of dombi operation and PROMETHEE II method under interval-valued Fermatean fuzzy set discussed by Seikh and Mandal³³ and Mandal and Seikh³⁴ proposed the MABAC approach for assessment of uncertain and fuzzy information using the MADM method. The solution of the MADM problem using the q-rung orthopair fuzzy set concept offered by Seikh and Mandal³⁵ and Mandal and Seikh³⁶ developed the MADM method using the Dombi operational laws and confidence level-based AOs for the investigation of fuzzy data.

Other variants of TN and TCN operations, such as Einstein, Aczel-Alsina, and Hamacher, have also received wide applications in decision analysis. Some pictures of fuzzy AO based on Frank TN and TCN were introduced by Seikh and Mandal³⁷ and Wang et al.³⁸, respectively, due to their control properties of parameters. Moreover, Khan et al.³⁹ utilized Dombi AOs in data consolidation, incorporating upper and lower approximations to achieve improved granularity in the decision-making process. In conclusion, TN, or TCN, represents the mathematical pillars of TN, primarily in the Pythagorean fuzzy system, which enhances the accuracy, flexibility, and reliability of mathematical models of decision-making in UAM (uncertainty-aware environments).

Importance of Dombi t-norm and t-conorm

Selection of Dombi operations is motivated by the fact that they are flexible in terms of parameterization, whereby the decision-makers have a chance to control the magnitude of sacrifice between optimism and pessimism during the process of aggregation. However, it is considered to be a broader range of possible applications than fixed operators like the Aczel Alsina, Frank, and Sklar Schweiger norms because the parameter of the Dombi t-norm and t-conorm may be adjusted to suit varying risk and uncertainty attitudes. This flexibility introduces a more generalized model, so that the model captures a broader range of decision-making preferences, and its applicability is especially suitable in complex and uncertain environments.

Analytic hierarchy process

The analytical hierarchy process (AHP) is a well-structured method of regulating complicated MCDM problems. The organized decision-making technique known as AHP was created by Saaty⁴⁰. It is applied to analyze and rank options in multi-criteria situations. The mathematical measure places an emphasis that validates reliable and rational judgments. Effectiveness and diversity have led this method to expand its scope, encompassing business and management models. To be more specific, the problem of goods transportation in connection with AHP under fuzzy data was solved by Kanj et al.⁴¹, Khosravi and Izbirak⁴² developed the form of an index system to calculate the sustainability of the Iranian provinces by AHP, and a model of rough set and Optimal testing resource allocation for software system: an approach combining interval-valued IFS and AHP given by Mittal et al.⁴³. The presence of persistent IFS AHP and CODAS during the choice of automation degree was explored by Alkan et al.⁴⁴.

Research question

To enhance the positive effects and scientific method of selecting the team, this suggested method examines the capabilities of the players’ recombinant behavior and utilizes new decision-making algorithms to address existing disadvantages. The research aims to enhance decision accuracy and decrease subjectivity. The presented key question leads the study.

• 1. How can a player’s behavior synthesis based on diffusion improve the evaluation of the qualities of players?

• 2. What is the advantage of using RPyFS in combination with Dombi-AHP and PROMETHEE in the selection of a team?

• 3. Is the identified method capable of providing a more effective and objective framework to discover the optimal team environment?

Motivation and contribution

In the recent past, many MADM approaches have been proposed for precise and accurate solutions of MADM problems. In most approaches, researchers have assigned weights to attributes based on the suggestions of decision experts. There is a need to define new methods for evaluating the weight of alternatives. Also, we noticed that in past literature, no approach has been described for RPyFS-based information for investigating MADM problems. The issue of selecting the team is a complex choice that requires a trade-off between physical fitness, performance criteria, experience, age, and injury prevention. Current methods of selection are based on opinion, which is inherently subjective, leading to conflicts and biases. The current model of decision-making is hampered by the management of uncertainty and ambiguity when measuring subjective actions.

It is time to define some new methodologies for the precise investigation of MADM models. The main objective of this article is to suggest a more advanced MCDM method by utilizing RPyFS data. We use the Dombi operation because it is flexible in modeling interaction between the membership values, and therefore, it is a significant application in RPyFS. The following are the contributions:

• 1. An AHP approach is suggested to work on the weight vector and correlate all elements using it in pairs.

• 2. Defined new Dombi operations for RPyFS.

• 3. Provide new RPyFDWA and PROMETHEE methods for ranking a set of alternatives.

• 4. We provided the MADM algorithms based on the RPyFDWA and PROMETHEE methods.

• 5. Provided a case study on the optimal team selection issue based on some criteria.

• 6. We offered a numerical solution of the best team selection under the consideration of different attributes using the proposed RPyFDWA and PROMETHEE methods.

• 7. Compare our proposed MADM approaches to current ones.

Organization of the proposed theory

The structure of the manuscript is as follows: some key definitions are discussed in Sect. 2, which is the preliminary section. In Sect. 3, the Dombi TNM and TCN are established, and in Sect. 4, an AHP model is based on RPyFS. The recently established RPyFDWA explains and mentions its basic properties in Sect. 4. Discuss the MADM algorithm of the AHP model and the RPyFDWA operator in Sect. 5. The new proposed PROMETHEE and AHP MADM algorithm and model are explained in Sect. 6. The case study and numerical examples are described in Sect. 7 to evaluate the best candidates based on the RPyFDWA and PROMETHEE approach. Section 8 presents a comparative analysis of the developed method and available methods, and the sensitivity testing is given in Sect. 9. Finally, in Sect. 10, we offered a conclusion.

Preliminaries

This section comprises fundamental definitions, main functions, and values of scores necessary to understand the proposed piece of work.

Definition 1

² Let $X$ be a non-empty set, and PyFS Ψ on the set$X$ is written as in Eq. 1:

$${\Psi}=\left\{\left(\varphi\varphi,{\mu }_{{\Psi }}\left(\varphi \right), {\varpi }_{{\Psi }}\left(\varphi \right)|\varphi \in X\right)\right\}$$

(1)

Where $\left({\mu }_{{\Psi }}\left(\varphi \right), {\varpi }_{{\Psi }}\left(\varphi \right)\right): X\to \left[0, 1\right]$ and the MD be ${\mu }_{{\Psi }}\left(\varphi \right)\in \left[0, 1\right]$ and NMD ${\varpi }_{{\Psi }}\left(\varphi \right)\in \left[0, 1\right]$ and $\varphi \in X$ to the set of ${\Psi }$ with $0\le {\mu }_{{\Psi }}^{2}\left(\varphi \right)+{\varpi }_{{\Psi }}^{2}\left(\varphi \right)\le 1.$ So, ${\text{H}}_{\Psi } \left( \phi \right) = 1 - \left( {\mu _{\Psi }^{2} \left( \phi \right) + \varpi _{\Psi }^{2} \left( \phi \right)} \right)$ It is the hesitancy degree.

Operational laws for PyFS are vital, as they offer a more comprehensive context for handling ambiguity by extending existing FS operations to address MD and NMD. Atanassov (1986) proposed several vital operations, including scalar multiplication, intersection, sum, power operation, union, and product, for PyFVs.

Theorem 1

² Suppose ${{\Psi }}_{1}=\left({\mu }_{{{\Psi }}_{1}}, {\varpi }_{{{\Psi }}_{1}} \right)$ and ${{\Psi }}_{2}=\left({\mu }_{{{\Psi }}_{2}}, {\varpi }_{{{\Psi }}_{2}}\right)$ There are two PyFVs, so some essential operations are listed based on PyFVs, as below.

• 1.
  $${{\Psi }}_{1}\cup {{\Psi }}_{2}=\left(max\left({\mu }_{{{\Psi }}_{1}},{\mu }_{{{\Psi }}_{2}}\right), min\left({\varpi }_{{{\Psi }}_{1}} ,{\varpi }_{{{\Psi }}_{2}}\right)\right)$$
• 2.
  $${{\Psi }}_{1}\cap {{\Psi }}_{2}=\left(min\left({\mu }_{{{\Psi }}_{1}},{\mu }_{{{\Psi }}_{2}}\right), max\left({\varpi }_{{{\Psi }}_{1}} ,{\varpi }_{{{\Psi }}_{2}}\right)\right)$$
• 3.
  $$\Psi _{1} \oplus \Psi _{2} = \left( {\mu _{{\Psi _{1} }} + \mu _{{\Psi _{2} }} - \mu _{{\Psi _{1} }} \mu _{{\Psi _{2} }} , \varpi _{{\Psi _{1} }} ,\varpi _{{\Psi _{2} }} } \right)$$
• 4.
  $$\Psi _{1} \otimes \Psi _{2} = \left( {\mu _{{\Psi _{1} }} \mu _{{\Psi _{2} }} ,\varpi _{{\Psi _{1} }} + \varpi _{{\Psi _{2} }} - \varpi _{{\Psi _{1} }} \varpi _{{\Psi _{2} }} } \right)$$
• 5.
  $$P{{\Psi }}_{1}=\left(1-{\left(1-{\mu }_{{{\Psi }}_{1}}\right)}^{P}, {{\varpi }_{{{\Psi }}_{1}}}^{P}\right)\forall P\ge 1$$
• 6.
  $${{{\Psi }}_{1}}^{P}=\left({{\mu }_{{{\Psi }}_{1}}}^{P}\left(1-{\left(1-{\varpi }_{{{\Psi }}_{1}} \right)}^{P}\right)\right)$$

Definition 2

⁴⁵ Consider a universal set $X$ and crisp function $T\in PyFS(X\times X)$ so,

• 1. $\forall \varphi \in X,$the relation $T$ is reflexive, if ${\mu }_{T}\left(\varphi , \varphi \right)=1$ and ${\varpi }_{T}\left(\varphi , \varphi \right)=0.$.

• 2. $\forall \left(\varphi , a\right)\in X\times X,$ is a symmetric relation if ${\mu }_{T}\left(\varphi , a\right)={\mu }_{T}\left(a, \varphi \right)$ and${\varpi }_{T}\left(\varphi , a\right)={\varpi }_{T}\left(a, \varphi \right)$.

• 3. $\forall \left(\varphi , k\right)\in X\times X$, is the transitive relation if ${\mu }_{T}\left(\varphi , k\right)\ge {\bigvee }_{a\in X}\left\{{\mu }_{T}\left(\varphi , a\right)\vee {\mu }_{T}\left(a, k\right)\right\}$ and ${\varpi }_{T}\left(\varphi , k\right)\ge {\bigwedge }_{a\in X}\left\{{\varpi }_{T}\left(\varphi , a\right)\wedge {\varpi }_{T}\left(a, k\right)\right\}.$.

Definition 3

⁴⁶ Consider a universal set $X$, let $\alpha$ denote a RPyFS relation on $X$, so $X$ ∈ RPyFS ($X\times X$). A RPyFS approximation interval is then known to be for the ordered pair$\left(X, \alpha \right)$. The upper and lower approximation of any decision purpose ${\text{H}} \underline{\in } \text{R}\text{P}\text{y}\text{F}\text{S}(X$) regarding the rough Pythagorean fuzzy approximated interval $\left(X, \alpha \right)$ are signified by $\overline{\alpha }\left({\text{H}}\right)$ and $\underline{\alpha } \left({\text{H}}\right)$ correspondingly, and are stated as below:

$$\underline{\alpha } \left({\text{H}}\right)=\left\{\left(\varphi ,{\mu }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right),{\varpi }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)\right)\varphi \in X\right\}$$

$$\overline{\alpha }\left({\text{H}}\right)=\left\{\left(\varphi ,{\mu }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right),{\varpi }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)\right)\varphi \in X\right\}$$

where,

$${{\mu }^{2}}_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)={\bigwedge }_{a\in X}\left\{{\mu }_{\alpha }\left(\varphi , a\right)\wedge {\mu }_{{\text{H}}}\left(a\right)\right\},{\varpi }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)={\bigvee }_{a\in X}\left\{{\varpi }_{\alpha }\left(\varphi , a\right)\vee {\varpi }_{{\text{H}}}\left(a\right)\right\}$$

$${{\mu }^{2}}_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)={\bigvee }_{a\in X}\left\{{\mu }_{\alpha }\left(\varphi , a\right)\vee {\mu }_{{\text{H}}}\left(a\right)\right\},{\varpi }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)={\bigwedge }_{a\in X}\left\{{\varpi }_{\alpha }\left(\varphi , a\right)\wedge {\varpi }_{{\text{H}}}\left(a\right)\right\}$$

Where $0\le {\left({\mu }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)\right)}^{2}+{\left({\varpi }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)\right)}^{2}\le 1$ and $0\le {\left({\mu }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)\right)}^{2}+{\left({\varpi }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)\right)}^{2}\le 1.$ So $\underline{\alpha } \left({\text{H}}\right),\overline{\alpha }\left({\text{H}}\right) :\text{R}\text{P}\text{y}\text{F}\text{S}\left(X\right)\to \text{R}\text{P}\text{y}\text{F}\text{S}\left(X\right)$are be upper and lower estimate operators. The pair $\alpha \left({\text{H}}\right)= \left( \underline{\alpha } \left({\text{H}}\right),\overline{\alpha }\left({\text{H}}\right) \right)=\left\{\left(\varphi ,{\mu }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right),{\varpi }_{ \underline{\alpha } \left({\text{H}}\right)}\left(\varphi \right)\right),\left(\varphi ,{\mu }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right),{\varpi }_{\overline{\alpha }\left({\text{H}}\right)}\left(\varphi \right)\right)\right\}$ be the RPyFS. For our support $\left( \underline{\alpha } \left({\text{H}}\right),\overline{\alpha }\left({\text{H}}\right) \right)$ can be recorded as $\left\{\left( \underline{\mu }, \underline{\varpi }\right),\left(\overline{\mu },\overline{\varpi }\right)\right\}$ which is called RPyFS values (RPyFVs).

Definition 4

⁴⁷ Suppose ${\alpha }_{{\text{i}}}=\left\{\left({ \underline{\mu }}_{{\text{i}}},{ \underline{\varpi }}_{{\text{i}}}\right),\left({\overline{\mu }}_{{\text{i}}},{\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the RPyFVs, so the score function (SF) $\alpha$ is defined as in Eq. 2:

$$\varpi \left(\alpha \right)=\left(\frac{2+\left({ \underline{\mu }}_{{\text{i}}}^{2}- {{ \underline{\varpi }}^{2}}_{{\text{i}}}\right)+ \left({\overline{\mu }}_{{\text{i}}}^{2}- {\overline{\varpi }}_{{\text{i}}}^{2}\right)}{4}\right), \varpi \left(\alpha \right)\in \left[-1, 1\right]$$

(2)

Definition 5

⁴⁷ Suppose ${\alpha }_{{\text{i}}}=\left\{\left({ \underline{\mu }}_{{\text{i}}},{ \underline{\varpi }}_{{\text{i}}}\right),\left({\overline{\mu }}_{{\text{i}}},{ \underline{\varpi }}_{{\text{i}}}\right)\right\}$ be the RPyFVs, so the accuracy function of $\alpha$ is designated as in Eq. 3:

$${\text{A}}\left(\alpha \right)=\left(\frac{2+\left({ \underline{\mu }}_{{\text{i}}}^{2}+ {{ \underline{\varpi }}^{2}}_{{\text{i}}}\right)+ \left({\overline{\mu }}_{{\text{i}}}^{2}+ {\overline{\varpi }}_{{\text{i}}}^{2}\right)}{4}\right), {\text{A}}\left(\alpha \right)\in \left[0, 1\right]$$

(3)

Dombi operations for RPyFS

The Dombi TN and TCN discovered by Dombi⁴⁸ are referred to as Dombi operations. Dombi laws of RPyFS are vital as they present a flexible approach to model uncertainty and provide an MD and NMD greater freedom, which is explicated below:

Definition 6

The Dombi TN and TCN of two numbers, ${\upsigma }$ and ${\Phi },$ based on the range of $\left[0, 1\right]$ for RPyFS, can be listed below in Eqs. 4 and 5 with ${\uplambda }\ge 1$:

$${R}_{n}\left({\upsigma }, {\Phi }\right)=\frac{1}{1+{\left\{{\left(\frac{1-{\upsigma }}{{\upsigma }}\right)}^{\lambda }+{\left(\frac{1-{\Phi }}{{\Phi }}\right)}^{\lambda }\right\}}^{1/\lambda }}$$

(4)

$${R}_{\stackrel{\sim}{n}}\left({\upsigma }, {\Phi }\right)=1-\frac{1}{1+{\left\{{\left(\frac{{\upsigma }}{1-{\upsigma }}\right)}^{\lambda }+{\left(\frac{{\Phi }}{1-{\Phi }}\right)}^{\lambda }\right\}}^{1/\lambda }}$$

(5)

Under the Dombi TN and TCN laws, we have introduced new operations for the RPyFVs, including scalar multiplication, sum, power rule, and product.

These are expressed in the following Definition 7:

Definition 7

Assume $\alpha \left({{\text{H}}}_{1}\right)=\left\{\left({ \underline{\mu }}_{1},{ \underline{\varpi }}_{1}\right),\left({\overline{\mu }}_{1},{\overline{\varpi }}_{1}\right)\right\}$ and $\alpha \left({{\text{H}}}_{2}\right)=\left\{\left({ \underline{\mu }}_{2},{ \underline{\varpi }}_{2}\right),\left({\overline{\mu }}_{2},{\overline{\varpi }}_{2}\right)\right\}$ There are two RPyFVs with$\lambda \ge 1$. So, a few crucial operations are defined based on the Dombi TN and TCN below.

• 1.
  $$\alpha \left({{\text{H}}}_{1}\right) \oplus \alpha \left({{\text{H}}}_{2}\right)=\left\{\begin{array}{c}\left(\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{ \underline{\mu }}_{1}^{2}}{1-{ \underline{\mu }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{ \underline{\mu }}_{2}^{2}}{1-{ \underline{\mu }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}},\sqrt{\frac{1}{1+{\left\{{\left(\frac{{ \underline{\varpi }}_{1}^{2}}{1-{ \underline{\varpi }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{ \underline{\varpi }}_{2}^{2}}{1-{ \underline{\varpi }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\overline{\mu }}_{1}^{2}}{1-{\overline{\mu }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{\overline{\mu }}_{2}^{2}}{1-{\overline{\mu }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}},\sqrt{\frac{1}{1+{\left\{{\left(\frac{{\overline{\varpi }}_{1}^{2}}{1-{\overline{\varpi }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{\overline{\varpi }}_{2}^{2}}{1-{\overline{\varpi }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right)\end{array}\right\}$$
• 2.
  $$\alpha \left({{\text{H}}}_{1}\right) \otimes \alpha \left({{\text{H}}}_{2}\right)=\left\{\begin{array}{c}\left(\sqrt{\frac{1}{1+{\left\{{\left(\frac{{ \underline{\mu }}_{1}^{2}}{1-{ \underline{\mu }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{ \underline{\mu }}_{2}^{2}}{1-{ \underline{\mu }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}},\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{ \underline{\varpi }}_{1}^{2}}{1-{ \underline{\varpi }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{ \underline{\varpi }}_{2}^{2}}{1-{ \underline{\varpi }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right),\\ \left(\sqrt{\frac{1}{1+{\left\{{\left(\frac{{\overline{\mu }}_{1}^{2}}{1-{\overline{\mu }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{\overline{\mu }}_{2}^{2}}{1-{\overline{\mu }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}},\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\overline{\varpi }}_{1}^{2}}{1-{\overline{\varpi }}_{1}^{2}}\right)}^{\lambda }+{\left(\frac{{\overline{\varpi }}_{2}^{2}}{1-{\overline{\varpi }}_{2}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right)\end{array}\right\}$$
• 3.
  $$\varpi \alpha \left({{\text{H}}}_{1}\right)=\left\{\begin{array}{c}\left(\sqrt{1-\frac{1}{1+{\left\{\varpi {\left(\frac{{ \underline{\mu }}_{1}^{2}}{1-{ \underline{\mu }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{\frac{1}{1+{\left\{\varpi {\left(\frac{{ \underline{\varpi }}_{1}^{2}}{1-{ \underline{\varpi }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right),\\ \left(\sqrt{1-\frac{1}{1+{\left\{\varpi {\left(\frac{{\overline{\mu }}_{1}^{2}}{1-{\overline{\mu }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{\frac{1}{1+{\left\{\varpi {\left(\frac{{\overline{\varpi }}_{1}^{2}}{1-{\overline{\varpi }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right)\end{array}\right\}$$
• 4.
  $${\left(\alpha \left({{\text{H}}}_{1}\right)\right)}^{\varpi }=\left\{\begin{array}{c}\left(\sqrt{\frac{1}{1+{\left\{\varpi {\left(\frac{{ \underline{\mu }}_{1}^{2}}{1-{ \underline{\mu }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{1-\frac{1}{1+{\left\{\varpi {\left(\frac{{ \underline{\varpi }}_{1}^{2}}{1-{ \underline{\varpi }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right),\\ \left(\sqrt{\frac{1}{1+{\left\{\varpi {\left(\frac{{\overline{\mu }}_{1}^{2}}{1-{\overline{\mu }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{1-\frac{1}{1+{\left\{{\varpi \left(\frac{{\overline{\varpi }}_{1}^{2}}{1-{\overline{\varpi }}_{1}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right)\end{array}\right\}$$

Representing the operation mentioned above, we will be able to demonstrate the following regulations described in the second theorem. In the case of two PFVs.

Theorem 2

Assume $\alpha \left({{\text{H}}}_{1}\right)=\left\{\left({ \underline{\mu }}_{1},{ \underline{\varpi }}_{1}\right),\left({\overline{\mu }}_{1},{\overline{\varpi }}_{1}\right)\right\}$ and $\alpha \left({{\text{H}}}_{2}\right)=\left\{\left({ \underline{\mu }}_{2},{ \underline{\varpi }}_{2}\right),\left({\overline{\mu }}_{2},{\overline{\varpi }}_{2}\right)\right\}$ Are there two RPyFVs with ${\lambda }_{1},{\lambda }_{2}\ge 1$. So, the offered laws are valid.

• a.
  $$\alpha \left({{\text{H}}}_{1}\right) \oplus \alpha \left({{\text{H}}}_{2}\right)= \alpha \left({{\text{H}}}_{2}\right) \oplus \alpha \left({{\text{H}}}_{1}\right)$$
• b.
  $$\alpha \left({{\text{H}}}_{1}\right) \otimes \alpha \left({{\text{H}}}_{2}\right)= \alpha \left({{\text{H}}}_{2}\right) \otimes \alpha \left({{\text{H}}}_{1}\right)$$
• c.
  $${\lambda }_{1}\left(\alpha \left({{\text{H}}}_{1}\right) \oplus \alpha \left({{\text{H}}}_{2}\right)\right)={\lambda }_{1}\alpha \left({{\text{H}}}_{1}\right) \oplus {\lambda }_{1}\alpha \left({{\text{H}}}_{2}\right)$$

• d. ${\left(\alpha \left({{\text{H}}}_{1}\right) \otimes \alpha \left({{\text{H}}}_{2}\right)\right)}^{{\lambda }_{1}}={\left(\alpha \left({{\text{H}}}_{1}\right)\right)}^{{\lambda }_{1}}$ \otimes ${\left(\alpha \left({{\text{H}}}_{2}\right)\right)}^{{\lambda }_{1}}$.

Proof

The proof is visible in the Appendix.

AHP for RPyFS-based information

The AHP involves pair-wise comparisons of the alternatives according to the targets that have specified weights. It begins with the development of the primary goal. Alternatives, as well as their weights, are thoroughly evaluated on the third level, which provides an opportunity to learn more about the information designated as RPyFVs in greater detail. The given step is a complete description of the AHP process:

• Step 1: The first step is to build an AHP model, termed a decision matrix (DM), and deconstruct the MCDM problem using RPyFVs.

• Step 2: In the given step, we do a pair comparison within DM.

• Step 3: Evaluate DM consistency ratio (CR). Continue until the consistency index (CI) is less than 0.10; otherwise, restructure the DM accordingly.

• Step 4: Obtain WV of the alternatives. Figure 1 contains the entire procedure of the AHP constructed on RPyFVs.

As shown in Fig. 1, the primary application of linguistic information in creating an AHP structure is illustrated in a decision matrix that enables pair-wise evaluation comparisons of several options. Then the CR is assessed. If CI is at least 0.10, the process continues; otherwise, there is a modification of the DM or the hierarchical structure. The linguistic scale index (LSI) that was calculated through pair-wise comparisons is shown in Table 1 and AHP is presented in Fig. 1.

Table 1 Linguistic scale.

Fig. 1

The diagram illustrates the AHP methodology that was developed to determine the weight of criteria.

Rough pythagorean fuzzy Dombi weighted averaging operators

The RPyFDWA operator is being developed to enhance decision-making in uncertain and complex environments. It combines Dombi operations with AHP-weighted averaging, which is more flexible in controlling the impacts of each component. It provides an advantage when it comes to multi-attribute decision-making in a fuzzy environment, where the combination of expertise views and values of the criteria quantity is required.

Definition 8

Suppose $\gamma \left({\xi }_{{\text{i}}}\right)=\left\{\left({ \underline{\mu }}_{{\text{i}}}, { \underline{\varpi }}_{{\text{i}}}\right), \left({\overline{\mu }}_{{\text{i}}}, {\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the set of RPyFVs. Suppose $\mathcal{T}={\left({\mathcal{T}}_{1}, {\mathcal{T}}_{2}, \dots , {\mathcal{T}}_{l}\right)}^{t}$ is the weight vector (WV). Where $t\ge 1,$ and also the aggregation of all WV is one, as $\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}=1$ and $0\le {\mathcal{T}}_{{\text{i}}}\le 1.$ So, the RPyFDWA operator is a function ${\gamma \left(\xi \right)}^{l}\to \gamma \left(\xi \right)$ given below:

$$\text{R}\text{P}\text{y}\text{F}\text{P}\text{D}\text{W}\text{A}\left(\gamma \left({\xi }_{1}\right), \gamma \left({\xi }_{2}\right), \dots , \gamma \left({\xi }_{l}\right)\right)={\oplus }_{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}\gamma \left({\xi }_{{\text{i}}}\right)$$

Since the verification of the formula, making use of the mathematical induction may prove the strict and orderly way of proving the formula correct in all other cases, the attributes of the operator RPyFDWA must be determined. Induction supports the reliability and applicability of the proposed AOs by establishing the base case and then demonstrating the premise that if a fact holds on one occasion, it also holds on the other.

Theorem 3

Suppose $\gamma \left({\xi }_{{\text{i}}}\right)=\left\{\left({ \underline{\mu }}_{{\text{i}}}, { \underline{\varpi }}_{{\text{i}}}\right), \left({\overline{\mu }}_{{\text{i}}}, {\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the set of RPyFVs. The WV be $\mathcal{T}={\left({\mathcal{T}}_{1}, {\mathcal{T}}_{2}, \dots , {\mathcal{T}}_{l}\right)}^{t}$. Where $t\ge 1,$ and also the aggregation of all WV is one, as $\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}=1$ and $0\le {\mathcal{T}}_{{\text{i}}}\le 1.$ So the RPyFDWA operators are given below in Eq. 6.

$$\begin{aligned} RPyFDWA\left( {\gamma \left( {\xi _{1} } \right), \gamma \left( {\xi _{2} } \right), \ldots ,\gamma \left( {\xi _{l} } \right)} \right) & = \oplus _{{{\text{i}} = 1}}^{l} {\mathcal{T}}_{{\text{i}}} \gamma \left( {\xi _{{\text{i}}} } \right) \\ & \begin{array}{*{20}c} {\left( {\sqrt {1 - \frac{1}{{1 + \left\{ {\sum {_{{{\text{i}} = 1}}^{l} } {\mathcal{T}}_{{\text{i}}} \left( {\frac{{\underline{\mu } _{{\text{i}}}^{2} }}{{1 - \underline{\mu } _{{\text{i}}}^{2} }}} \right)^{\lambda } } \right\}^{{\frac{1}{\lambda }}} }}} , \sqrt {\frac{1}{{1 + \left\{ {\sum {_{{{\text{i}} = 1}}^{l} } {\mathcal{T}}_{{\text{i}}} \left( {\frac{{\underline{\varpi } _{{\text{i}}}^{2} }}{{1 - \underline{\varpi } _{{\text{i}}}^{2} }}} \right)^{{\lambda }} } \right\}^{{\frac{1}{\lambda }}} }}} } \right), } \\ { \left( {\sqrt {1 - \frac{1}{{1 + \left\{ {\sum { _{{{\text{i}} = 1}}^{l} } {\mathcal{T}}_{{\text{i}}} \left( {\frac{{\overline{{\mu }} _{{\text{i}}}^{2} }}{{1 - \overline{{\mu }} _{{\text{i}}}^{2} }}} \right)^{{\lambda }} } \right\}^{{\frac{1}{{\lambda }}}} }}} , \sqrt {\frac{1}{{1 + \left\{ {\sum { _{{{\text{i}} = 1}}^{l} } {\mathcal{T}}_{{\text{i}}} \left( {\frac{{1 - \overline{{\varpi }} _{{\text{i}}}^{2} }}{{\overline{{\varpi }} _{{\text{i}}}^{2} }}} \right)^{{\lambda }} } \right\}^{{\frac{1}{{\lambda }}}} }}} } \right)} \\ \end{array} \\ \end{aligned}$$

(6)

Proof

The proof is visible in the Appendix.

Notably, monotonicity is central to the truth of the presented RPyFDWA operator, as it ensures that the aggregate output cannot diminish with an increase in input. This is rational, given the sequence of the data. This is a significant feature of consistency, as it implies that the aggregation process will be more consistent in terms of the relative input size.

Theorem 4

(Idempotency) Suppose $\gamma \left({\xi }_{{\text{i}}}\right)=\left\{\left({ \underline{\mu }}_{{\text{i}}}, { \underline{\varpi }}_{{\text{i}}}\right), \left({\overline{\mu }}_{{\text{i}}}, {\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the set of RPyFVs. Suppose $\mathcal{T}={\left({\mathcal{T}}_{1}, {\mathcal{T}}_{2}, \dots , {\mathcal{T}}_{l}\right)}^{t}$ is the WV. Where $t\ge 1,$ and also the aggregation of all WV is one, as $\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}=1$ and $0\le {\mathcal{T}}_{{\text{i}}}\le 1,$ following that, the RPyFDWA operator is fulfilled to have specific essential characteristics. Consider $\gamma \left({\xi }_{{\text{i}}}\right)=W\left(\xi \right)$ through $W\left(\xi \right)=\left\{\left( \underline{\mu }, \underline{\varpi }\right), \left(\overline{\mu }, \overline{\varpi }\right)\right\}$ is provided in Eq. 7.

$$RPyFDWA\left(\gamma \left({\xi }_{1}\right), \gamma \left({\xi }_{2}\right), \dots , \gamma \left({\xi }_{l}\right)\right)= W\left(\xi \right)$$

(7)

Proof

Proof of this theorem provided in the Appendix.

The RPyFDWA operator shall also have a limit that is checked to ensure the output of the assigned aggregation is limited in a way that it does not surpass the range of another when the output is collated, including the lower and upper levels of input. This characteristic makes it moderate and forecastable, as little can emerge as an outlier or surprise during the aggregation process.

Theorem 5

(Boundedness) Suppose $\gamma \left({\xi }_{{\text{i}}}\right)=\left\{\left({ \underline{\mu }}_{{\text{i}}}, { \underline{\varpi }}_{{\text{i}}}\right), \left({\overline{\mu }}_{{\text{i}}}, {\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the set of Py-PVs. Suppose $\mathcal{T}={\left({\mathcal{T}}_{1}, {\mathcal{T}}_{2}, \dots , {\mathcal{T}}_{l}\right)}^{t}$ is the WV. Where $t\ge 1,$ and also the aggregation of all WV is one, as $\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}=1$ and $0\le {\mathcal{T}}_{{\text{i}}}\le 1.$ After that, the RPyFDWA operator satisfies specific essential characteristics. Consider ${\gamma \left({\xi }_{{\text{i}}}\right)}^{-}=\left\{\left(min{ \underline{\mu }}_{{\text{i}}},max { \underline{\varpi }}_{{\text{i}}}\right), \left(min{\overline{\mu }}_{{\text{i}}}, max{\overline{\varpi }}_{{\text{i}}}\right)\right\}$ and ${\gamma \left({\xi }_{{\text{i}}}\right)}^{+}=\left\{\left(max{ \underline{\mu }}_{{\text{i}}},min { \underline{\varpi }}_{{\text{i}}}\right), \left(max{\overline{\mu }}_{{\text{i}}}, min{\overline{\varpi }}_{{\text{i}}}\right)\right\}$ is provided in Eq. 8.

$${\gamma \left({\xi }_{{\text{i}}}\right)}^{-}\le \text{R}\text{P}\text{y}\text{F}\text{D}\text{W}\text{A} \le \left(\gamma \left({\xi }_{1}\right), \gamma \left({\xi }_{2}\right), \dots , \gamma \left({\xi }_{l}\right)\right){\gamma \left({\xi }_{{\text{i}}}\right)}^{+}$$

(8)

Proof

The proof is visible in the Appendix.

The monotonicity process also plays a critical role in the effectiveness of the proposed RPyFDWA operator, as it guarantees that the value resulting from an aggregate outcome will not be smaller than that resulting from an increase in the input values, provided the order in the data is maintained. This is vital as consistency implies that there is no breach of the relative size of inputs during the process of aggregation.

Theorem 6

(Monotonicity) Suppose $\gamma \left({\xi }_{{\text{i}}}\right)=\left\{\left({ \underline{\mu }}_{{\text{i}}}, { \underline{\varpi }}_{{\text{i}}}\right), \left({\overline{\mu }}_{{\text{i}}}, {\overline{\varpi }}_{{\text{i}}}\right)\right\}$ be the set of RPyFVs. Suppose $\mathcal{T}={\left({\mathcal{T}}_{1}, {\mathcal{T}}_{2}, \dots , {\mathcal{T}}_{l}\right)}^{t}$ is the WV. Where $t\ge 1,$ and also the aggregation of all WV is one, as $\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}=1$ and $0\le {\mathcal{T}}_{{\text{i}}}\le 1.$ Subsequently, to possess the necessary qualities, the RPyFDWA operator must be fulfilled. Consider other collections of RPyFVs $\overbrace {{\gamma \left( {\xi _{{\text{i}}} } \right)}}^{{}} = \left\{ {\left( {\overbrace {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{{\text{i}}} }}^{{}},\overbrace {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varpi } _{{\text{i}}} }}^{{}}} \right),\left( {\overbrace {{\bar{\mu }_{{\text{i}}} }}^{{}},\overbrace {{\bar{\varpi }_{{\text{i}}} }}^{{}}} \right)} \right\}$. As $\left( {\overbrace {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{{\text{i}}} }}^{{}},\overbrace {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varpi } _{{\text{i}}} }}^{{}}} \right) \ge \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{{\text{i}}} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varpi } _{{\text{i}}} } \right)$ and $\left( {\overbrace {{\bar{\mu }_{{\text{i}}} }}^{{}},\overbrace {{\bar{\varpi }_{{\text{i}}} }}^{{}}} \right) \ge \left( {\bar{\mu }_{{\text{i}}} ,\bar{\varpi }_{{\text{i}}} } \right)$ is provided in Eq. 9.

$${\text{RPyFDWA}}\left( {\overbrace {{\gamma \left( {\xi _{1} } \right)}}^{{}},\overbrace {{\gamma \left( {\xi _{2} } \right)}}^{{}}, \ldots ,\overbrace {{\gamma \left( {\xi _{l} } \right)}}^{{}}} \right) \ge R{\text{PyFDWA}}\left( {\gamma \left( {\xi _{1} } \right),\gamma \left( {\xi _{2} } \right), \ldots ,\gamma \left( {\xi _{l} } \right)} \right)$$

(9)

Proof

The proof is visible in the Appendix.

The preference for the order of inputs is taken into consideration by decision-makers under the RPyFDWA operator, as opposed to dealing with each input on an equal basis. The most significant tool, in terms of assessing complicated information, in DM is the RPyFDWA operator.

AHP-based and PROMETHEE MADM algorithm for RPyFVs

This is a demonstration of an algorithm of MADM within the AHP RPyFVs structure. We create a decision matrix, through which we can obtain CI, after which it is possible to determine the WVs. Then we inspected and ordered the alternatives based on our modeled RPyFDWA operator. The process in this procedure is the following:

MADM algorithm and evaluation of weight vector:

• Step 1: We construct a value-based decision matrix using the opinion-based linguistic scale and make comparisons on a pair-wise basis of the available options using Eq. 10.

  $$N={\left(S\right)}_{c\times \mathcal{d}}$$

  (10)

  Where $ii\times j$ shows the relative significance of standard $\text{a} \text{t}\text{o} \text{b}$.

• Step 2: In the decision matrix, the geometric mean (GM) is calculated for each row using Eq. 11.

  $${GM}_{i}={\prod }_{i=1}^{\mathcal{Z}}{S}_{i\times j}$$

  (11)

• Step 3: Add up $\left({S}_{i}\right)$ All the GM values are using Eq. 12.

  $${S}_{i}=\sum _{i=1}^{\mathcal{Z}}\left({GM}_{i}\right)$$

  (12)

• Step 4: In this step, we compute the inverse $\left({I}_{i}\right)$ The GM is calculated using Eq. 13.

  $${I}_{i}={\left({S}_{i}\right)}^{-1}=\frac{1}{\sum _{i=1}^{\mathcal{Z}}\left({GM}_{i}\right)}$$

  (13)

  Then sort in $\left({I}_{1}, {I}_{2}, \dots {I}_{n}\right)$ Descending order.

• Step 5: Compute the product $\left({p}_{i}\right)$ of $\left({S}_{i}\right)$ and establish $\left({I}_{i}\right)$. Choose an appropriate product. $\left({p}_{i}\right)$of$\left({S}_{i}\right)$ and $\left({I}_{i}\right)$ compute using Eq. 14.

  $${p}_{i}=\prod _{i=1}^{t}\left({S}_{i},{I}_{i}\right)$$

  (14)

• Step 6: Calculate the mean $\left({A}_{i}\right)$ of $\left({p}_{i}\right)$ using Eq. 15.

  $${A}_{i}=\frac{\sum _{i=1}^{\mathcal{Z}}\left({p}_{i}\right)}{t}$$

  (15)

  Where $I=1, 2, 3, \dots , n$.

• Step 7: Now, to calculate the weight vector (WVs), normalize each priority weight by dividing $\left({A}_{i}\right)$ by the total value of all $\sum _{i=1}^{\mathcal{Z}}\left({A}_{i}\right)$ values using Eq. 16.

  $$w=\frac{{A}_{i}}{\sum _{i=1}^{\mathcal{Z}}\left({A}_{i}\right)}$$

  (16)

• Step 8: Check CI of calculated $w$ using the formula below. The value of CI of the computed $w$ is checked using the assistance of the following formula given in Eqs. 17 and 18.

  $$\text{C}\text{I}=\frac{{H}_{max}-1}{\mathcal{Z}-1}$$

  (17)
  $${H}_{max}=\sum _{i=1}^{\mathcal{Z}}\left({A}_{i}\right)$$

  (18)

  and $\mathcal{Z}\mathcal{ }$is the alternative number.

• Step 9: Use the formulated RPyFDWA operators as follows.

  $$=\left\{\begin{array}{c}\left(\sqrt{1-\frac{1}{1+{\left\{\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}{\left(\frac{{ \underline{\mu }}_{{\text{i}}}^{2}}{1-{ \underline{\mu }}_{{\text{i}}}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{\frac{1}{1+{\left\{\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}{\left(\frac{{ \underline{\varpi }}_{{\text{i}}}^{2}}{1-{ \underline{\varpi }}_{{\text{i}}}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right), \\ \left(\sqrt{1-\frac{1}{1+{\left\{\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}{\left(\frac{{\overline{\mu }}_{{\text{i}}}^{2}}{1-{\overline{\mu }}_{{\text{i}}}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}, \sqrt{\frac{1}{1+{\left\{\sum _{{\text{i}}=1}^{l}{\mathcal{T}}_{{\text{i}}}{\left(\frac{1-{\overline{\varpi }}_{{\text{i}}}^{2}}{{\overline{\varpi }}_{{\text{i}}}^{2}}\right)}^{\lambda }\right\}}^{\frac{1}{\lambda }}}}\right)\end{array}\right\}$$

• Step 10: The SF formula will be used to calculate the cumulative sum as a single value.

  $$\varpi \left(\alpha \right)=\left(\frac{2+\left({ \underline{\mu }}_{{\text{i}}}^{2}- {{ \underline{\varpi }}^{2}}_{{\text{i}}}\right)+ \left({\overline{\mu }}_{{\text{i}}}^{2}- {\overline{\varpi }}_{{\text{i}}}^{2}\right)}{4}\right)$$

• Step 11: Arrange all the cumulative findings in descending order and select the best alternative with the highest score value.

The graphical view of MADM algorithm is offered in Fig. 2.

Fig. 2

The diagram illustrates the MADM algorithm based on the proposed approach.

Proposed PROMETHEE method for RPyFVs

This section will outline the defined methodology, which involves applying the PROMETHEE approach and RPyFVs. The procedure of the PROMETHEE procedure is as follows:

• Step 1. Build the RPyFVs decision matrix based on the linguistic table.

• Step 2. A decision matrix should be normalized, where the data are of two kinds: beneficial and non-beneficial. To this, we apply the following formulas:

  Beneficial type criteria are defined in Eq. 19.

  $$L_{{i{\text{i}}}} = \frac{{\left[ {a_{{i{\text{i}}}} - {\text{min}}\left( {a_{{i{\text{i}}}} } \right)} \right]}}{{\left[ {{\text{max}}\left( {a_{{i{\text{i}}}} } \right) - {\text{min}}\left( {a_{{i{\text{i}}}} } \right)} \right]}}; \left( {i = 1, 2, \ldots ,n;{\text{i}} = 1, 2, \ldots ,n} \right)$$

  (19)

  Non-beneficial type criteria are defined in Eq. 20.

  $$L_{{i{\text{i}}}} = \frac{{\left[ {{\text{min}}\left( {a_{{i{\text{i}}}} } \right) - a_{{i{\text{i}}}} } \right]}}{{\left[ {{\text{max}}\left( {a_{{i{\text{i}}}} } \right) - {\text{min}}\left( {a_{{i{\text{i}}}} } \right)} \right]}}; \left( {i = 1, 2, \ldots ,n;{\text{i}} = 1, 2, \ldots ,n} \right)$$

  (20)

• Step 3. Calculate the evaluation difference of ${i}^{th}$ alternatives concerning other choices.

• Step 4. With the following equation, the preference function$p\left(a, \beta \right)$ given in Eq. 21.

  $$p\left(a, \beta \right)=0; \text{i}\text{f} { L}_{a{\text{i}}}<{ L}_{\beta {\text{i}}}; \left(D{M}_{a}-D{M}_{\beta }\right)\le 0$$
  $$p\left(a, \beta \right)=\left({ L}_{a{\text{i}}}-{L}_{\beta {\text{i}}}\right); \text{I}\text{f} { L}_{a{\text{i}}}>{ L}_{\beta {\text{i}}}; \left(D{M}_{a}-D{M}_{\beta }\right)>0$$

  (21)

• Step 5. To apply the following equation, we find the weight preference function$\upphi\left(a, \beta \right)$ given in Eq. 22.

  $$\upphi\left(a, \beta \right)=\frac{\sum _{{\text{i}}=1}^{n} \overline{\overline{R}} p\left(a, \beta \right)}{\sum _{{\text{i}}=1}^{n} \overline{\overline{R}} }$$

  (22)

• Step 6. To compute the entering and leaving outranking flow, we find the entering flow (sign negative) ${f}^{-}$ for ${a}^{th}$ alternative, explain below: where $\beta \ne a$ and $D$ show the total alternatives. To estimate the entering and exiting outranking flow, we calculate the entering flow (sign negative) ${f}^{-}$ for ${a}^{th}$ alternative, which consists of the following: where $\beta \ne a$ and $D$ indicate the total options provided in Eq. 23.

  $${f}^{-}=\frac{1}{D-1}\sum _{i=1}^{D}\upphi\left(\beta , a\right)$$

  (23)

  As entering flow (sign positive) ${f}^{+}$ for ${a}^{th}$ alternative, explain below in Eq. 24.

  $${f}^{+}=\frac{1}{D-1}\sum _{i=1}^{D}\upphi\left(a, \beta \right)$$

  (24)

• Step 7. To determine the net outranking of each alternative, we apply the formula shown below in Eq. 25.

  $$f={f}^{+}-{f}^{-}$$

  (25)

• Step 8. We finally organized all the consequences to come up with the best alternative.

How proposed theories enhance the decision-making process

This sub-section describes the reason why the boundedness property of the RPyFDWA operator guarantees stability and consistency in the aggregation process, so that the results do not get outside of the logical boundaries. Such a property bolsters the power of the results of a decision, especially when they are uncertain. Moreover, we have also explained how the preference functions within PROMETHEE assist decision-making by enabling pairwise comparisons between alternatives and allowing one to rank the preferences in the form of clear-cut preference structures. The subsection now incorporates referenced equations used to depict the boundedness proof of RPyFDWA, as well as the mathematical derivation of PROMETHEE preference functions, hence making their overall outcome substitution in multi-criteria decision-making much easier.

Case study

The RPyFS on a given case study on a basketball team presents an appropriate framework to describe the uncertainty and vagueness that come with human-centered attributes, such as physical fitness, game performance, experience, age, and injury prevention. These qualities can sometimes not always be measured accurately, but they are some of the most important factors in determining the success of a given team. The inclusion of RPyFS within the MADM framework gives the latter decision-making method a firmer foundation because it enables one to take into account not only the margin of hesitation but also the uncertainty of expert assessment. This relationship explains why RPyFS is relatively applicable in the basketball scenario, allowing for a better representation of the subjective variables that affect player choice and performance in sports.

Basketball is a real-time game that requires quality decision-making, teamwork, and constant adjustments. Artificial intelligence has altered the behavior tests and adaptation plans of players in recent years. Diffusion models (DMs) have become popular for synthesizing player movements, predicting game outcomes, and enhancing training simulations using various artificial intelligence techniques. The patterns suggest that they are highly adjustable, realistic, and scalable demonstrations of player behavior, making them highly suitable for sports analytics, coach development, and automated game simulation. We employed the MADM method to select the best players within the RPyFS model in our methodology. To select this model, we considered four key criteria: physical fitness, game performance, experience, age, and injury prevention.

In artificial intelligence-based game development, analytics, and simulation, player behavior synthesis is an emerging technique in diffusion modeling. DMs are typically used to create high-quality images and texts, although they can also be employed in simulating and predicting player actions in various games. Some of the best criteria of this approach can be discussed in the following section. The graphical view of team unity shows in Fig. 3.

Fig. 3

Show the unity in the game. Address. e096c533268623b8e41ca6b5e195c430.jpg (736 × 552).

Figure 2 shows the unity, determination, and shared commitment that drive them toward success.

Physical fitness

One of the basic requirements in selecting basketball teams is physical fitness because the game requires individuals to be very fit in terms of stamina, strength, agility, and speed. More athletic players can perform well in bursts over long periods, make a quick recovery between games, and meet the physical demands of defending and attacking. Fitness testing, including sprint times, vertical jump height, agility drills, and endurance tests, is used, offering an objective measurement by which to rank candidates. Not only does a high fitness level contribute to a player’s personal efficiency, but it also lifts the team’s pace, defensive coverage, and endurance in stressful games.

Being physically fit is a vital attribute for a basketball player. It encompasses a player’s power, speed, stamina, and mobility on the field. During a basketball match, a player must be able to run up and down the court, jump to take rebounds, defend the ball, and play for an extended period. A physically fit player is less prone to fatigue and, as such, is less easily affected, able to respond swiftly, and able to maintain a high standard of play throughout. During the selection process, players will be chosen in the best physical shape, making them more effective and reliable during intense matches. The graphical view of physical fitness is shown in Fig. 4.

Fig. 4

Shows the physical fitness in basketball. Address. jfmk-06-00067-g003.png (3283 × 2021).

Figure 3 classifies basketball performance into key physical components for both male and female players. This figure highlights continuous efforts, explosive efforts, jumping ability, curvilinear movement, and deceleration capacity. These factors together define an athlete’s overall performance and game-specific physical demands.

Performance criteria

Game performance can be defined as the way a basketball player plays in real games. It consists of key skills such as shooting accuracy, passing, dribbling, defense, and overall contribution to the team’s success. A player who performs well in the game is always able to score points, assist his teammates, defend effectively, and make the right choices throughout the game. This criterion is quite essential, as it indicates a player’s actual performance on the court. It is also used to help coaches understand how reliable a player can be in decisive moments of the match. An illustration of high game performance can be seen in a player who can score themselves under pressure, make plays, and defend themselves against an aggressor. This is one of the key elements when choosing ideal talents in a competitive basketball team.

Game performance reflects the ability of a player to transfer the skills to concrete achievements in a contest. These are shooting precision, defending, passing, and quick tactical decision-making under pressure. Quantitative data, such as points per game, assists, rebounds, and steals, provide statistical data, with coach and peer reviews providing a qualitative angle. Good game performance is a consistent indicator of reliability and influence on the success of the team, which is critical in decision-making. Gamers who perform well in competition situations tend to be flexible, time-sensitive, and skilled in situations where there are high stakes.

Experience and age

Experience and age inform us of how old a player is, the duration they have played in a high-level game, and their maturity level during gameplay. These experienced players had played numerous games, and therefore, when it comes to making good decisions and acting in stressful situations, it is no mystery to them. The question of age is also essential; younger players can be more active and quicker than older ones, and often possess better decision-making and leadership skills. A team that can perform well typically requires a combination of both. During selection, they prefer players who have valuable experience and a good mix of ages, as these players can bring confidence and knowledge to the team.

A combination of experience and age contributes to the development of a player and his preparedness and prospects in the team settings. Elderly players tend to have these traits, such as composure, an understanding of the game, and leadership strategy, that can influence the less experienced players in the team. Age may refer to the physical peak, where a younger player has potential for long-term development, whereas an older player provides stability and guidance. It is decisive to balance such factors: when overly dependent on a specific age group, the team cannot be as adaptive. A balance between young enthusiasm and mature wisdom creates the most effective balance of sides to address both short-term needs and long-term development.

Injury prevention

Injury prevention is the state of well-being and physical solidity of a player. Due to the nature of certain players, some of whom are highly gifted, they may not always be available when a team requires their contribution, often as a result of injuries. Injured players who have shorter recuperation times tend to be more reliable. This characteristic examines a player’s past injury record, their recovery, and the likelihood of remaining fit throughout the entire season or tournament. The players selected should be those who maintain good health, hence making the team strong and steady in all matches.

A combination of experience and age contributes to the development of a player and his preparedness and prospects in the team settings. Elderly players tend to have these traits, such as composure, an understanding of the game, and leadership strategy, that can influence the less experienced players in the team. Age may refer to maturity and also the physical peak, where a younger player has potential for long-term development, whereas an older player provides stability and guidance. It is decisive to balance such factors: when overly dependent on a specific age group, the team cannot be as adaptive. A balance between young enthusiasm and mature wisdom creates the most effective balance of sides to address both short-term needs and long-term development.

Numerical example

To establish the validity of our developed operator, we utilized it with a professional basketball team, aiming to select the best team for the upcoming season. MADM can be considered a practical method for achieving a comprehensive and valid evaluation. The MADM enables the assessment of the different options based on various criteria to determine the most advantageous alternative. This numerical example will begin by gathering data facilitated by decision-making experts and the use of RPyFVs to build a decision matrix.

In this regard, the alternatives under consideration include four teams ${{\Phi }}_{i}=\left({{{\Phi }}_{1}, {\Phi }}_{2},{{\Phi }}_{3},\dots ,{{\Phi }}_{\text{n}}\right)$ and consider the four criteria listed below:

• 1. Physical fitness $\left({E}_{1}\right)$.

• 2. Performance criteria$\left({E}_{2}\right)$.

• 3. Experience & age $\left({E}_{3}\right)$.

• 4. Injury prevention $\left({E}_{4}\right)$.

We assign numeric values to alternatives and attributes in Table 2.

We applied hypothetical information in this work rather than real-world player records, which can prove the efficiency of the given PROMETHEE approach and the operator. The set of data that was constructed would present the reality-like conditions of physical fitness performance criteria, experience, age, and injury prevention, physical fitness, performance measures, and other related features, thus enabling us to demonstrate how the developed decision-making framework can be applied. The presence of hypothetical data guarantees the clarity of presenting the methodology and is aimed at specifying the theoretical contributions of the work.

Table 2 Decision matrix.

To use Table 2 to find the weight vector by using the AHP technique.

We apply the constructed theory of the workings of RPyFDWA Table 2 above, and the results are indicated in Table 3.

Table 3 The result of the aggregation finding was achieved through the proposed RPyFDWA operator. Aggregated values by using the RPyFDWA operator.

Table 4 Displays scores of aggregated information.

To gain a clearer picture, Table 4 is presented in descending order in Table 5, and a graphical representation is provided in Figure 5.

Table 5 Display the ranking of the score value.

Fig. 5

Shows the graphical representation of Table 5.

To notice the Fig. 5; Table 5, the optimal recommendation of the final ranking setting implied that: ${\varPhi }_{1}$ can be ranked first concerning the value of the score, followed by ${\varPhi }_{4}$,${\varPhi }_{3}$ and ${\varPhi }_{2}$. It is ascertained that the customary AHP and RPyFDWA are followed in the aggregation of the results, which verifies the greater working performance and reduced unfairness in the performance of the hybrid simulation setup, as graphically illustrated in Fig. 4.

PROMETHEE method

We apply the PROMETHEE method to rank and select the best team based on multiple criteria. By comparing alternative pair-wise, we compute preference flows. The team with the highest net flow value is considered the best choice.

• Step 1. Take a decision matrix based on the RPyFVs provided in Table 2.

• Step 2. Since the selected attributes are of the same type, we do not need to normalize the decision matrix.

• Step 3. Calculate the difference matrix for each alternative, corresponding to the alternatives. Table 6 presents a difference matrix, which represents the comparison of alternatives based on deviations in criteria.

  Table 6 Difference matrix.

• Step 4. We calculate the preference matrix. Table 7 represents the Preference matrix, which can be used to illustrate the relative preferences among alternatives according to specific criteria.

  Table 7 Preference matrix.

• Step 5. Now, we find the weight preference function. Table 8 displays a summary of the weight preference, which is the combination of individual weight preferences to arrive at the overall ranking of the alternatives.

  Table 8 Aggregated weighted preference function.

• Step 6. Now, we calculate the leaving and entering outranking by using the sum of the weight preference function. Table 9 also displays the values of net outranking according to the leaving flow and entering flow, which aids in the final ranking of the decisions.

  Table 9 Leaving and entering outranking.

• Step 7. Now, we compute the net outranking outcome. The outranking result, expressed as the net outranking of the PROMEHEE method, is presented in Table 10.

  Table 10 Net outranking outcome.

• Step 8. Finally, we consider all the consequences to determine the best alternative. The ranking is provided in Table 11. The ranking of alternatives by using the PROMATHEE approach is displayed in Fig. 6.

  Table 11 Ranking of alternatives.

  Fig. 6

  

  Display the outranking outcome by the PROMETHEE model.

The optimal recommendation of the final ranking setting implied that: ${\varPhi }_{1}$ were ranked first concerning the value of the score, followed by ${\varPhi }_{4}$,${\varPhi }_{3}$ and ${\varPhi }_{2}$. It is ascertained that the customary AHP and PROMETHEE methods are followed in aggregating the results, which verifies the greater working performance and reduced unfairness in the performance of the hybrid simulation setup graphically presented in Fig. 7.

Sensitive analysis of the proposed theory

We have inserted various values of parameters $\lambda$ to discuss the hybrid effects of parameters on a ranking of alternatives. When different values of the parameter $\lambda$ were assigned in the proposed RPyFDWA operator, the ranking of the options became ${\varPhi }_{1}>{\varPhi }_{2}>{\varPhi }_{3}>{\varPhi }_{4}$. At different values of the parameter$\lambda$, the same order of the alternatives is achieved, as well as the best one is ${\varPhi }_{1}$ and this is indicated in Table 12. In this query, we have received that in the event of a change in the parameter value, the ranking does not make any difference. In this way, the professional will be capable of selecting the appropriate value of the parameter to be used in the respective calculation.

Table 12 Influence of $\lambda$ on the ranking of alternatives through the aggregation operator-based method.

Table 12 indicates that the ranking of the best alternative remains the same when the parameter value is altered. The geometrical illustration of the sensitivity analysis of the parameter by using the RPyFDWA operator is provided in Fig. 8.

Fig. 7

Sensitivity analysis of the parameter by using the RPyFDWA operator.

Based on Fig. 7, we have seen that by changing the value of the parameter (taking as $\lambda =\text{3,4},\text{5,6},\text{7,8},\text{9,10,20}$ ). We have noted a slight variation in the ranking of alternatives, but no change in the best alternative.

The RPyFDWA operator presents a pragmatic strategy to reflect uncertainty and hesitancy by distinguishing the rough MD and NMD definition that enables the model to remain stable even as the data gets imprecise or incomplete. Dombi operational laws in the design of AOs are pretty flexible, and their parametric potential allows adjusting the results of a decision to the degree of risk propensity or tolerance to uncertainty. Such adaptability means that this model does not behave excessively sensitively due to the drastic change in input parameters and hence does not alter drastically in its performance based on the scenario. As shown in Table 12 and the related spider graph Fig. 7, the ranking pattern of the alternatives is very similar under a pretty broad range of parameter values, which confirms that the suggested framework is reliable and robust.

Comparative analysis

In general, this part aims to demonstrate that the POMETHEE approach enables us to efficiently and accurately generate formulation and averaging AO. In this connection, we find a relationship between our MADM methodologies and known methods, such as the RPyFDWA and PROMETHEE approach, which is compared with the Pythagorean hesitant fuzzy rough dombi weighted averaging (PyHFRDWA) operators presented by Attaullah et al.⁴⁹ and the dual hesitant Pythagorean fuzzy Hamacher weighted averaging (DHPFHWA) operator proposed by Wei and Lu⁵⁰.

The PROMETHEE approaches are effective in addressing MADM problems and are valuable and commercially applicable. Their advantage lies in combining the specific description of the RPyFDWA and PROMETHEE with the most popular AHP technique. Such a method is particularly effective in complex situations, providing more detailed and comprehensive results. It can be beneficial in situations involving opinions and uncertainty, making it a valuable tool in the real world, which relies heavily on decision-making.

Our proposed theory, compared with the Spearman ranking method, gives a better evaluation since it also takes into consideration the magnitude of the preferences as well as erring in judgments, in addition to looking at the order of alternatives. Although the method presented by Li et al.⁵¹ for fuzzy values helps verify consistency and correlation of the ranks, the technique we have developed is more valuable in drawing detailed analytical inferences, as it captures quantitative differences as well as ranking patterns, which is more useful in complex scenarios of decision making in MADM. We also notice that during the aggregation of fuzzy information, many AOs are not applicable to handle rough Pythagorean fuzzy information-based data sets. Ashraf et al.⁵² proposed the theory of q-rung orthopair rough fuzzy Einstein weighted averaging (q-RORFEWA) operator, and Ghani et al.⁵³ developed the complex q-rung orthopair rough fuzzy Einstein weighted averaging (Cq-RORFEWA) operator for the assessment of fuzzy information. Also, the comparative analysis between the established model is presented in Table 13.

Table 13 The comparison between the established and existing operators.

Table 13 shows the ranking outcome using the newly developed approaches and other existing methods. Where we noticed that ${{\Phi }}_{1}$ is the best alternative by using the proposed RPyFDWA and PROMETHEE method. The option ${{\Phi }}_{1}$ is the finest option using the PyHFRDWA and Spearman’s ranking method, while ${{\Phi }}_{2}$ It is the best option using the DHPFHWA operator. We also noticed that many other AOs are unable to handle RPyFS-based data, such as DHFWA, IFDWA, q-RORFEWA, and Cq-RORFEWA, due to limitations in their structure. The graphical view is provided in Fig. 8.

Fig. 8

Show the comparative analysis between the established model and the existing model.

Figure 8 presents a comparative analysis of our proposed theory with other existing approaches, such as RPyFDWA, PROMETHEE, PyFDWA, DHPFHWA, and Spearman’s ranking method, to validate our developed theory.

Runtime and scalability discussion

Another aspect of the value of the proposed solution is its computational efficiency, which is essential in testing its feasibility on a larger dataset. Expressing the complexity of the run, it is noted that in the case of the PROMETHEE method, the same is mainly determined by the number of alternatives. $\left({{\Phi }}_{\text{i}}\right)$ and criteria $\left(E\right)$. The complexity of PROMETHEE lies in the evaluation of preference functions and comparison between alternatives on each of the requirements that are to be carried out. $\left({{\Phi }}_{\text{i}}\times E\right)$ in the aggregate. Such quadratic growth is acceptable only for small problem sizes, but can prove to be computationally demanding when there are a huge number of alternatives.

For the RPyFDWA, the complexity depends on the computation of MD and NMD along with rough boundary intervals for each criterion. These steps generally involve linear aggregation across attributes and hence scale as ${\mu }_{{\Psi }}$ and ${\varpi }_{{\Psi }}$ for each aggregation operation. However, when combined with pairwise comparison-based evaluation, the effective complexity can approach. ${\mu }_{{\Psi }}$ and ${\varpi }_{{\Psi }}$.

Conclusion

This paper has established an informative and robust process of decision-making in selecting the most efficient team from a large number of players by integrating the approaches of RPyFS, AHP, and PROMETHEE. We have proposed new MADM models given as follows:

• Introduced new operational laws for DTNM and DTCNM using the RPyFS-based information.

• Developed the AHP based on the RPyFS, which makes the model increasingly flexible in managing non-linear relationships, particularly those involving complex interactions among different selection criteria. The AHP technique is effective in organizing and ranking the evaluation attributes, including physical fitness, skill level, teamwork, and experience.

• Motivated by the theory of AHP, Dombi operations under RPyFS, we constructed the RPyFDWA operator for the aggregation of complicated information.

• Proposed the PROMETHEE approach for effective ranking of the basketball team, the developed approach makes the selection much more realistic and objective.

• In the proposed method, we describe MADM algorithms in detail for the RPyFDWA operator and PROMETHEE approach, and address fundamental issues of basketball evaluation by solving them through proposed techniques.

• As a case study analysis in real life, the selection of the best basketball team involves considering different attributes, such as physical fitness, performance criteria, experience, age, and injury prevention.

• To prove the effectiveness of the model, we carried out a comparative study with other existing MADM models.

• Sensitivity analysis, backed by a graphical representation, shows the flexibility of the generated model and the consistency in making decisions concerning team selection carried out by experts.

Benefits and limitations of developed theories

The main objective of the diagnosed theories is to make new modifications in decision support systems and gain more realistic and trustworthy results under the MADM approach. In this regard, we have diagnosed the PROMETHEE approach and RPyFDWA operator for precise analysis of MADM problems. The proposed methodologies are reliable tools for aggregating uncertain and ambiguous information. Both developed approaches are applicable only for RPyFVs-based details with accuracy. Our proposed techniques help investigate all real-life problems where decision-making is involved.

Besides a list of benefits, there are a few limitations in our proposed approach. The PROMETHEE method is efficient in managing the MADM, but it is susceptible to the choice of functions of preferences and levels of threshold, which may determine the final ranking. It might also have problems in representing uncertainty and vagueness in decision data, since it mainly operates on things that are given concretely in terms of fuzzy values. Also, our developed theories apply to RPyFVs-based data but cannot handle other fuzzy sets, such as rough q-rung orthopair fuzzy set, rough picture fuzzy set, rough spherical fuzzy set, and rough t-spherical fuzzy sets.

Future directions

We extend our proposed theory in the future for different fuzzy environments, such as the theory of complex q-rung orthopair fuzzy sets discussed by Mahmood et al.⁵⁷. The theory of hesitant bipolar fuzzy sets, as proposed by Wei et al.⁵⁸. A hybrid model, which consists of fuzzy AHP and fuzzy WASPAS under construction site selection, has been studied by Turskis et al.⁵⁹. We will attempt to incorporate the established AOs with another model, such as the Fuzzy TOPSIS model by Nadeen et al.⁶⁰. The idea of a spherical fuzzy set-based CRITIC approach for accurately solving MADM problems, given by Irvanizam et al.⁶¹. The theory of quasirung orthopair fuzzy set given by Seikh and Mandal⁶² and Seikh and Mandal⁶³ provided the solution of decision-making problems using the quasirung orthopair fuzzy set.