On some new fuzzy entropy measure of Pythagorean fuzzy sets for decision-making based on an extended TOPSIS approach

Fuzzy entropy measures are valuable tools in decision-making when dealing with uncertain or imprecise information.There exist many entropy measures for Pythagorean Fuzzy Sets(PFS)in the literature that fail to deal with the problem of providing reasonable or consistent results to the decision-makers.To deal with the shortcomings of the existing measures,this paper proposes a robust fuzzy entropy measure for PFS to facilitate decision-making under uncertainty.The usefulness of the measure is illustrated through an illustration of decision-making in a supplier selection problem and compared with existing fuzzy entropy measures.The Technique for Order Performance by Similarity to Ideal Solution(TOPSIS)approach is also explored to solve the decision-making problem.The results demonstrate that the proposed measure can effectively capture the degree of uncertainty in the decision-making process,leading to more accurate decision outcomes by providing a reliable and robust ranking of alternatives.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

Green supplier selection is an important debate in green supply chain management(GSCM),attracting global attention from scholars,especially companies and policymakers.Companies frequently search for new ideas and strategies to assist them in realizing sustainable development.Because of the speculative character of human opinions,supplier selection frequently includes unreliable data,and the interval-valued Pythagorean fuzzy soft set(IVPFSS)provides an exceptional capacity to cope with excessive fuzziness,inconsistency,and inexactness through the decision-making procedure.The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers(IVPFSNs)and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average(IVPFSIWA)and the interval-valued Pythagorean fuzzy soft interaction weighted geometric(IVPFSIWG)operators,and analyze their properties.These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals,providing a superior solution to other methods.Moreover,specialist judgments were calculated by the MCGDM technique,supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects.Lastly,a statistical clarification of the planned method for green supplier selection is presented.

Fuzzy Multi-Criteria Decision Support System for the Best Anti-Aging Treatment Selection Process through Normal Wiggly Hesitant Fuzzy Sets

This socialized environment among educated and developed people causes themto focusmore on their appearance and health,which turns them towards medical-related treatments,leading us to discuss anti-aging treatment methods for each age group,particularly for urban people who are interested in this.Some anti-aging therapies are used to address the alterations brought on by aging in human life without the need for surgery or negative effects.Five anti-aging therapies such as microdermabrasion or dermabrasion,laser resurfacing anti-aging skin treatments,chemical peels,dermal fillers for aged skin,and botox injections are considered in this study.Based on the criteria of safety risk,investment cost,customer happiness,and side effects,the optimal alternative is picked.As a result,a NormalWiggly Hesitant Pythagorean Fuzzy Set(NWHPFS)is constructed and used in Multi-Criteria Decision-Making(MCDM)using traditional wavy mathematical approaches.The entropy approach is utilized to determine weight values,and the Normal Wiggly Hesitant Pythagorean-VlseKriterijumska Optimizacija I Kompromisno Resenje(NWHPF-VIKOR)method is utilized to rank alternatives using MCDM methodologies.Sensitivity analysis and comparative analysis were performed to ensure the robustness and reliability of the proposed method.The smart final choice will undoubtedly assist Decision Makers(DM)in making the right judgments,and the MCDM approach will undoubtedly assist individuals in understanding the medicine.

Multi-attribute decision making with Pythagorean fuzzy sets via conversions to intuitionistic fuzzy sets and ORESTE method

The aim of this paper is to study the conversions between Pythagorean fuzzy sets and Atanassov’s intuitionistic fuzzy sets.Besides,an ORESTE method based on multi-attribute decision making with Pythagorean fuzzy sets is developed by utilising the developed conversions.In this paper,according to the geometric representations of Pythagorean fuzzy sets and Atanassov’s intuitionistic fuzzy sets,two types of conversions between the two fuzzy sets are constructed,which are further used to derive information measures include entropy and cross-entropy measures of Pythagorean fuzzy sets.Then,by combining with the ORESTE method,a direct decision procedure for multi-attribute decision making with Pythagorean fuzzy information is developed.Finally,a numerical example of the evaluation of regional energy efficiency is shown to illustrate the feasibility and validity of the developed decision procedure.

Fuzzy Risk Assessment Method for Airborne Network Security Based on AHP-TOPSIS

With the exponential increase in information security risks,ensuring the safety of aircraft heavily relies on the accurate performance of risk assessment.However,experts possess a limited understanding of fundamental security elements,such as assets,threats,and vulnerabilities,due to the confidentiality of airborne networks,resulting in cognitive uncertainty.Therefore,the Pythagorean fuzzy Analytic Hierarchy Process(AHP)Technique for Order Preference by Similarity to an Ideal Solution(TOPSIS)is proposed to address the expert cognitive uncertainty during information security risk assessment for airborne networks.First,Pythagorean fuzzy AHP is employed to construct an index system and quantify the pairwise comparison matrix for determining the index weights,which is used to solve the expert cognitive uncertainty in the process of evaluating the index system weight of airborne networks.Second,Pythagorean fuzzy the TOPSIS to an Ideal Solution is utilized to assess the risk prioritization of airborne networks using the Pythagorean fuzzy weighted distance measure,which is used to address the cognitive uncertainty in the evaluation process of various indicators in airborne network threat scenarios.Finally,a comparative analysis was conducted.The proposed method demonstrated the highest Kendall coordination coefficient of 0.952.This finding indicates superior consistency and confirms the efficacy of the method in addressing expert cognition during information security risk assessment for airborne networks.

Improved approach to quality function deployment based on Pythagorean fuzzy sets and application to assembly robot design evaluation

Quality function deployment(QFD)is an effective method that helps companies analyze customer requirements(CRs).These CRs are then turned into product or service characteristics,which are translated to other attributes.With the QFD method,companies could design or improve the quality of products or services close to CRs.To increase the effectiveness of QFD,we propose an improved method based on Pythagorean fuzzy sets(PFSs).We apply an extended method to obtain the group consensus evaluation matrix.We then use a combined weight determining method to integrate former weights to objective weights derived from the evaluation matrix.To determine the exact score of each PFS in the evaluation matrix,we develop an improved score function.Lastly,we apply the proposed method to a case study on assembly robot design evaluation.

Pythagorean probabilistic hesitant triangular fuzzy aggregation operators with applications in multiple attribute decision making

As a generalization of fuzzy set,hesitant probabilistic fuzzy set and pythagorean triangular fuzzy set have their own unique advantages in describing decision information.As modern socioeconomic decision-making problems are becoming more and more complex,it also becomes more and more difficult to appropriately depict decision makers’cognitive information in decision-making process.In order to describe the decision information more comprehensively,we define a pythagorean probabilistic hesitant triangular fuzzy set(PPHTFS)by combining the pythagorean triangular fuzzy set and the probabilistic hesitant fuzzy set.Firstly,the basic operation and scoring function of the pythagorean probabilistic hesitant triangular fuzzy element(PPHTFE)are proposed,and the comparison rule of two PPHTFEs is given.Then,some pythagorean probabilistic hesitant triangular fuzzy aggregation operators are developed,and their properties are also studied.Finally,a multi-attribute decision-making(MADM)model is constructed based on the proposed operators under the pythagorean probabilistic hesitant triangular fuzzy information,and an illustration example is given to demonstrate the practicability and validity of the proposed decision-making method.

Novel Scheme for Robust Confusion Component Selection Based on Pythagorean Fuzzy Set

The substitution box,often known as an S-box,is a nonlinear component that is a part of several block ciphers.Its purpose is to protect cryptographic algorithms from a variety of cryptanalytic assaults.A Multi-Criteria Decision Making(MCDM)problem has a complex selection procedure because of having many options and criteria to choose from.Because of this,statistical methods are necessary to assess the performance score of each S-box and decide which option is the best one available based on this score.Using the Pythagorean Fuzzy-based Technique for Order of Preference by Similarity to Ideal Solution(TOPSIS)method,the major objective of this investigation is to select the optimal S-box to be implemented from a pool of twelve key choices.With the help of the Pythagorean fuzzy set(PFS),the purpose of this article is to evaluate whether this nonlinear component is suitable for use in a variety of encryption applications.In this article,we have considered various characteristics of S-boxes,including nonlinearity,algebraic degree,strict avalanche criterion(SAC),absolute indicator,bit independent criterion(BIC),sum of square indicator,algebraic immunity,transparency order,robustness to differential cryptanalysis,composite algebraic immunity,signal to noise ratio-differential power attack(SNR-DPA),and confusion coefficient variance on some standard S-boxes that are Advanced Encryption Following this,the findings of the investigation are changed into Pythagorean fuzzy numbers in the shape of a matrix.This matrix is then subjected to an analysis using the TOPSIS method,which is dependent on the Pythagorean fuzzy set,to rank the most suitable S-box for use in encryption applications.

A New Kind of Generalized Pythagorean Fuzzy Soft Set and Its Application in Decision-Making

The aim of this paper is to introduce the concept of a generalized Pythagorean fuzzy soft set(GPFSS),which is a combination of the generalized fuzzy soft sets and Pythagorean fuzzy sets.Several of important operations of GPFSS including complement,restricted union,and extended intersection are discussed.The basic properties of GPFSS are presented.Further,an algorithm of GPFSSs is given to solve the fuzzy soft decision-making.Finally,a comparative analysis between the GPFSS approach and some existing approaches is provided to show their reliability over them.

Einstein Weighted Geometric Operator for Pythagorean Fuzzy Hypersoft with Its Application in Material Selection

Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications.Pythagorean fuzzy hypersoft set(PFHSS)is the most influential and capable leeway of the hypersoft set(HSS)and Pythagorean fuzzy soft set(PFSS).It is also a general form of the intuitionistic fuzzy hypersoft set(IFHSS),which provides a better and more perfect assessment of the decision-making(DM)process.The fundamental objective of this work is to enrich the precision of decision-making.A novel mixed aggregation operator called Pythagorean fuzzy hypersoft Einstein weighted geometric(PFHSEWG)based on Einstein’s operational laws has been developed.Some necessary properties,such as idempotency,boundedness,and homogeneity,have been presented for the anticipated PFHSEWG operator.Multi-criteria decision-making(MCDM)plays an active role in dealing with the complications of manufacturing design for material selection.However,conventional methods of MCDM usually produce inconsistent results.Based on the proposed PFHSEWG operator,a robust MCDM procedure for material selection in manufacturing design is planned to address these inconveniences.The expected MCDM method for material selection(MS)of cryogenic storing vessels has been established in the real world.Significantly,the planned model for handling inaccurate data based on PFHSS is more operative and consistent.

Aggregation Operators for Interval-Valued Pythagorean Fuzzy Hypersoft Set with Their Application to Solve MCDM Problem

Experts use Pythagorean fuzzy hypersoft sets(PFHSS)in their investigations to resolve the indeterminate and imprecise information in the decision-making process.Aggregation operators(AOs)perform a leading role in perceptivity among two circulations of prospect and pull out concerns from that perception.In this paper,we extend the concept of PFHSS to interval-valued PFHSS(IVPFHSS),which is the generalized form of intervalvalued intuitionistic fuzzy soft set.The IVPFHSS competently deals with uncertain and ambagious information compared to the existing interval-valued Pythagorean fuzzy soft set.It is the most potent method for amplifying fuzzy data in the decision-making(DM)practice.Some operational laws for IVPFHSS have been proposed.Based on offered operational laws,two inventive AOs have been established:interval-valued Pythagorean fuzzy hypersoft weighted average(IVPFHSWA)and interval-valued Pythagorean fuzzy hypersoft weighted geometric(IVPFHSWG)operators with their essential properties.Multi-criteria group decision-making(MCGDM)shows an active part in contracts with the difficulties in industrial enterprise for material selection.But,the prevalent MCGDM approaches consistently carry irreconcilable consequences.Based on the anticipated AOs,a robust MCGDMtechnique is deliberate formaterial selection in industrial enterprises to accommodate this shortcoming.A real-world application of the projectedMCGDMmethod for material selection(MS)of cryogenic storing vessels is presented.The impacts show that the intended model is more effective and reliable in handling imprecise data based on IVPFHSS.

Novel Decision Making Methodology under Pythagorean Probabilistic Hesitant Fuzzy Einstein Aggregation Information

This research proposes multicriteria decision-making(MCDM)-based real-time Mesenchymal stem cells(MSC)transfusion framework.The testing phase of the methodology denotes the ability to stick to plastic surfaces,the upregulation and downregulation of certain surface protein markers,and lastly,the ability to differentiate into various cell types.First,two scenarios of an enhanced dataset based on a medical perspective were created in the development phase to produce varying levels of emergency.Second,for real-timemonitoring ofCOVID-19 patients with different emergency levels(i.e.,mild,moderate,severe,and critical),an automated triage algorithmbased on a formal medical guideline is proposed,taking into account the improvement and deterioration procedures fromone level to the next.For this strategy,Einstein aggregation information under the Pythagorean probabilistic hesitant fuzzy environment(PyPHFE)is developed.Einstein operations on PyPHFE such as Einstein sum,product,scalar multiplication,and their properties are investigated.Then,several Pythagorean probabilistic hesitant fuzzy Einstein aggregation operators,namely the Pythagorean probabilistic hesitant fuzzy weighted average(PyPHFWA)operator,Pythagorean probabilistic hesitant fuzzy Einstein weighted geometric(PyPHFEWG)operator,Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted average(PyPHFEOWA)operator,Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted geometric(PyPHFEOWG)operator,Pythagorean probabilistic hesitant fuzzy Einstein hybrid average(PyPHFEHA)operator and Pythagorean probabilistic hesitant fuzzy Einstein hybrid geometric(PyPHFEHG)operator are investigated.All the above-mentioned operators are helpful in design the algorithm to tackle uncertainty in decision making problems.In last,a numerical case study of decision making is presented to demonstrate the applicability and validity of the proposed technique.Besides,the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique.

Bibliometric analysis on Pythagorean fuzzy sets during 2013-2020

Purpose-The purpose of this paper is to make a comprehensive analysis of 354 publications about Pythagorean fuzzy sets(PFSs)from 2013 to 2020 in order to comprehensively understand their historical progress and current situation,as well as future development trend.Design/methodology/approach-First,this paper describes the fundamental information of these publications on PFSs,including their data information,annual trend and prediction and basic features.Second,the most productive and influential authors,countries/regions,institutions and the most cited documents are presented in the form of evaluation indicators.Third,with the help of VOSviewer software,the visualization analysis is conducted to show the development status of PFSs publications at the level of authors,countries/regions,institutions and keywords.Finally,the burst detection of keywords,timezone review and timeline review are exported from CiteSpace software to analyze the hotspots and development trend on PFSs.Findings-The annual PFSs publications present a quickly increasing trend.The most productive author is Wei Guiwu(China).Wei Guiwu and Wei Cun have the strongest cooperative relationship.Research limitations/implications-The implication of this study is to provide a comprehensive perspective for the scholars who take a fancy to PFSs,and it is valuable for scholars to grasp the hotspots in this field in time.Originality/value-It is the first paper that uses the bibliometric analysis to comprehensively analyze the publications on PFSs.It can help the scholars in the field of PFSs to quickly understand the development status and trend of PFSs.

A hybrid heterogeneous Pythagorean fuzzy group decision modelling for crowdfunding development process pathways of fintech‑based clean energy investment projects

This study aims to evaluate the crowdfunding alternatives regarding new service development process pathways of clean energy investment projects.In this framework,a new model has been generated by considering the consensus-based group decisionmaking with incomplete preferences,Pythagorean fuzzy decision-making trial and evaluation laboratory(DEMATEL)and technique for order preference by similarity to ideal solution(TOPSIS).Moreover,a comparative evaluation has been performed with Vise Kriterijumska Optimizacija I.Kompromisno Resenje methodology and sensitivity analysis has been made by considering 4 different cases.The main contribution is to identify appropriate crowdfunding-based funding alternatives for the improvement of the clean energy investments with a novel MCDM model.By considering the iteration technique and consensus-based analysis,the missing parts in the evaluations can be completed and opposite opinion problems can be reduced.Furthermore,with the help of hybrid MCDM model by combining DEMATEL and TOPSIS,more objective results can be reached.It is concluded that the analysis results are coherent and reliable.The findings indicate that the full launch is the most significant criterion for equity and debt-based crowdfunding alternatives.On the other side,the analysis has the highest weight for reward and donation-based alternatives whereas design is the most essential item regarding the royalty-based alternative.Additionally,it is also defined that equity-based crowdfunding alternative is the most significant for the service development process of clean energy investment projects.In this way,it will be possible to provide a continuous resource for clean energy investment projects.On the other hand,by providing financing with equity,there will be no fixed financing cost for clean energy investors.If these investors make a profit,they distribute dividends with the decision of their authorized bodies.

Comparative Analysis of Pythagorean MCDM Methods for the Risk Assessment of Childhood Cancer

According to the World Health Organization(WHO),cancer is the leading cause of death for children in low and middle-income countries.Around 400,000 kids get diagnosed with this illness each year,and their survival rate depends on the country in which they live.In this article,we present a Pythagorean fuzzy model that may help doctors identify the most likely type of cancer in children at an early stage by taking into account the symptoms of different types of cancer.The Pythagorean fuzzy decision-making techniques that we utilize are Pythagorean Fuzzy TOPSIS,Pythagorean Fuzzy Entropy(PF-Entropy),and Pythagorean Fuzzy PowerWeighted Geometric(PFPWG).Ourmodel is fed with nineteen symptoms and it diagnoses the risk of eight types of cancers in children.We develop an algorithm for each method and calculate its complexity.Additionally,we consider an example to make a clear understanding of our model.We also compare the final results of various tests that prove the authenticity of this study.

Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Set with Their Application to Solve Multi-Attribute Group Decision Making Problem

Interval-valued Pythagorean fuzzy soft set(IVPFSS)is a generalization of the interval-valued intuitionistic fuzzy soft set(IVIFSS)and interval-valued Pythagorean fuzzy set(IVPFS).The IVPFSS handled more uncertainty comparative to IVIFSS;it is the most significant technique for explaining fuzzy information in the decision-making process.In this work,some novel operational laws for IVPFSS have been proposed.Based on presented operational laws,two innovative aggregation operators(AOs)have been developed such as interval-valued Pythagorean fuzzy soft weighted average(IVPFSWA)and interval-valued Pythagorean fuzzy soft weighted geometric(IVPFSWG)operators with their fundamental properties.A multi-attribute group decision-making(MAGDM)approach has been established utilizing our developed operators.A numerical example has been presented to ensure the validity of the proposed MAGDM technique.Finally,comparative studies have been given between the proposed approach and some existing studies.The obtained results through comparative studies show that the proposed technique is more credible and reliable than existing approaches.

A novel approach to emergency risk assessment using FMEA with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment

Purpose-The application of the traditional failure mode and effects analysis(FMEA)technique has been widely questioned in evaluation information,risk factor weights and robustness of results.This paper develops a novel FMEA framework with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment to solve these problems.Design/methodology/approach-This paper introduces innovatively interval-value Pythagorean fuzzy weighted averaging(IVPFWA)operator,Tchebycheff metric distance and interval-value Pythagorean fuzzy weighted geometric(IVPFWG)operator into the MULTIMOORA submethods to obtain the risk ranking order for emergencies.Finally,an illustrative case is provided to demonstrate the practicality and feasibility of the novel fuzzy FMEA framework.Findings-The feasibility and validity of the proposed method are verified by comparing with the existing methods.The calculation results indicate that the proposed method is more consistent with the actual situation of project and has more reference value.Practical implications-The research results can provide supporting information for risk management decisions and offer decision-making basis for formulation of the follow-up emergency control and disposal scheme,which has certain guiding significance for the practical popularization and application of risk management strategies in the infrastructure projects.Originality/value–A novel approach using FMEA with extended MULTIMOORA method is developed under IVPF environment,which considers weights of risk factors and experts.The method proposed has significantly improved the integrity of information in expert evaluation and the robustness of results.

Knowledge diffusion trajectories in the Pythagorean fuzzy field based on main path analysis

Purpose-The purpose of this article is to conduct a main path analysis of 627 articles on the theme of Pythagorean fuzzy sets(PFSs)in the Web of Science(WoS)from 2013 to 2020,to provide a conclusive and comprehensive analysis for researchers in this field,and to provide a study on preliminary understanding of PFSs.Design/methodology/approach-The research topic of Pythagorean fuzzy fields,through keyword extraction and describing the changes in characteristic themes over the past eight years,are firstly examined.Main path analysis,including local and global main paths and key route paths,is then used to reveal the most influential relationships between papers and to explore the trajectory and structure of knowledge transmission.Findings-The application of Pythagorean fuzzy theory to the field of decision-making has been popular,and combinations of the traditional Pythagorean fuzzy decision-making method with other fuzzy sets have attracted widespread attention in recent years.In addition,over the past eight years,research interest has shifted to different types of PFSs,such as interval-valued PFSs.Research limitations/implications-This paper implicates to investigate the growth in certain trends in the literature and to explore the main paths of knowledge dissemination in the domain of PFSs in recent years.Originality/value-This paper aims to identify the topics in which researchers are currently interested,to help scholars to keep abreast of the latest research on PFSs.