A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making

Precision measurement for effective pollution mitigation by evaluating air quality monitoring systems in linguistic Pythagorean fuzzy dombi environment

Air quality is a major concern for human health, with pollutants linked to respiratory problems and chronic illnesses. Air quality monitoring systems are essential for measuring and tracking pollutants in indoor and outdoor environments. In the various disciplines of fuzzy environments, the aggregation operators are indispensable components of the decision-making process and possess a significant capacity to manage unpredictable and ambiguous data. This study utilizes the linguistic Pythagorean fuzzy set to address the aforementioned environmental scenarios, which improve comprehension of air quality through the application of AOs. This work introduces two new aggregation operators: the linguistic Pythagorean fuzzy Dombi ordered weighted averaging (LPFDOWA) and the language Pythagorean fuzzy Dombi ordered weighted geometric (LPFDOWG), and examines their structural properties. Furthermore, we develop a novel scoring function for multiple attribute decision-making (MADM) issues within the context of linguistic Pythagorean fuzzy knowledge. We provide a systematic mathematical procedure to address MADM issues within the context of the linguistic Pythagorean fuzzy Dombi framework. Furthermore, we effectively employ these approaches to address the MADM issue of selecting an efficient Air quality monitoring systems for air pollution monitoring. Additionally, we present a thorough comparative analysis to demonstrate the effectiveness of the proposed methodology relative to conventional techniques.

Fermatean fuzzy Linguistic term set based on linguistic scale function with Dombi aggregation operator and their application to multi criteria group decision -making problem

The selection of an industrial location is a challenging multiple-criteria decision-making (MCDM) problem that depends on taking a variety of locations as well as incompatible and inconsistent criteria. This paper proposed a comprehensive framework for the strategic selection of industrial locations, considering both quantitative and qualitative aspects. Decision-makers (DMs) have to deal with ambiguous information throughout this process due to a complex decision environment or their insufficient knowledge. We present a new Fermatean Fuzzy (FF) Linguistic term set based on the Dombi aggregation operators (AOs). By combining the FF set with Linguistic variables, the FF Linguistic (FFL) set is an effective approach for thoroughly representing uncertain evaluation information. We establish a basic operational principles and certain aggregation operator under FFL information, such as the FF Linguistic Dombi weighted averaging (FFLDWA) operator FF Linguistic Dombi weighted geometric (FFLDWG) operator and some fundamental properties of these operators with appropriated elaboration. Based on these operators, a multi-criteria group decision-making technique is developed. Finally, we used a numerical example to compare the flexibility of the suggested technique with other existing methods. Thus, by knowing priorities industries, the best site can be selected.

Selection of third-party logistics in supply chain finance under probabilistic complex hesitant fuzzy sets and distance measures

Financial institutions attach great importance to the multiple attribute decision-making (MADM) problems of selecting an effective third-party logistics (3PL) service provider in supply chain finance. A suitable 3PL service provider can not only assist financial institutions in carrying out supply chain finance activity but can also replace financial institutions in supervising the operation of a target financing supply chain, therefore minimizing financial institutions' operating risks. However, there is no sufficient study on the selection of a 3PL service provider. Hence, the objective of this paper is to establish an innovative concept of the "probabilistic complex hesitant fuzzy set" (PCHFS), which is a hybrid structure of the complex hesitant fuzzy set and the probabilistic fuzzy set to manage complex information in real-decision theory. This paper examines the periodicity of probabilistic fuzzy information and extends the range from [ 0 , 1 ] to the unit disc in the complex plane to provide more ability to describe the full meaning of the information in a mathematical model. Based on the internal structure of the set and to find the degree of discrimination between the pairs of PCHFSs, the generalized distance measures and modified generalized distance measures are defined. Several properties and their relationships between them are derived in detail. Also, several cases of the proposed measures are exposed, which reduces them to the existing studies. Furthermore, based on these proposed measures, a multiple-attribute decision-making (MADM) approach is established in an uncertain environment, and several numerical examples are given to examine the feasibility and validity of the explored measures. Finally, the modified and parameterized distance measures based on PCHFSs are verified by comparing them with some existing measures.

Development of Aczel-Alsina t-norm based linear Diophantine fuzzy aggregation operators and their applications in multi-criteria decision-making with unknown weight information

Aczel-Alsina t-norm and t-conorm are intrinsically flexible and endow Aczel-Alsina aggregation operators with greater versatility and robustness in the aggregation process than operators rooted in other t-norms and t-conorm families. Moreover, the linear Diophantine fuzzy set (LD-FS) is one of the resilient extensions of the fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PyFSs), and q-rung orthopair fuzzy sets (q-ROFSs), which has acquired prominence in decision analysis due to its exceptional efficacy in resolving ambiguous data. Keeping in view the advantages of both LD-FSs and Aczel-Alsina aggregation operators, this article aims to establish Aczel-Alsina operation rules for LD-FSs, such as Aczel-Alsina sum, Aczel-Alsina product, Aczel-Alsina scalar multiplication, and Aczel-Alsina exponentiation. Based on these operation rules, we expose the linear Diophantine fuzzy Aczel-Alsina weighted average (LDFAAWA) operator, and linear Diophantine fuzzy Aczel-Alsina weighted geometric (LDFAAWG) operator and scrutinize their distinctive characteristics and results. Additionally, based on these aggregation operators (AOs), a multi-criteria decision-making (MCDM) approach is designed and tested with a practical case study related to forecasting weather under an LD-FS setting. The developed model undergoes a comparative analysis with several prevailing approaches to demonstrate the superiority and accuracy of the proposed model. Besides, the influence of the parameter Λ on the ranking order is successfully highlighted.

Selecting the foremost big data tool to optimize YouTube data in dynamic Fermatean fuzzy knowledge

Big data pertains to extensive and intricate compilations of information that necessitate the implementation of proficient and cost-effective evaluation and analysis tools to derive insights and support decision making. The Fermatean fuzzy set theory possesses remarkable capability in capturing imprecision due to its capacity to accommodate complex and ambiguous problem descriptions. This paper presents the study of the concepts of dynamic ordered weighted aggregation operators in the context of Fermatean fuzzy environment. In numerous practical decision making scenarios, the term "dynamic" frequently denotes the capability of obtaining decision-relevant data at various time intervals. In this study, we introduce two novel aggregation operators: Fermatean fuzzy dynamic ordered weighted averaging and geometric operators. We investigate the attributes of these operators in detail, offering a comprehensive description of their salient features. We present a step-by-step mathematical algorithm for decision making scenarios in the context of proposed methodologies. In addition, we highlight the significance of these approaches by presenting the solution to the decision making problem and determining the most effective big data analytics platform for YouTube data analysis. Finally, we perform a thorough comparative analysis to assess the effectiveness of the suggested approaches in comparison to a variety of existing techniques.

A multiple attribute group decision making model based on 2-tuple linguistic pythagorean fuzzy dombi aggregation operators for optimal selection of potential global suppliers

Multiple-Attribute Group Decision-Making (MAGDM) is a significant area of research in decision-making, and its principles and methodologies are widely implemented. A Pythagorean Fuzzy Set (PFS) is an extension of an Intuitionistic Fuzzy Set (IFS) that is highly valuable for representing uncertain information in real-world scenarios. The 2-Tuple Linguistic Pythagorean Fuzzy Number (2TLPFN) is a specific type of Pythagorean Fuzzy Number (PFN) that can be used to represent uncertainty in real-world decision making through the use of 2-Tuple Linguistic Terms (2TLTs). This paper focuses on the examination of Multiple Attribute Group Decision Making (MAGDM) using 2TLPFNs. Dombi's t-norm and t-conorm operations were commonly referred to as Dombi operations, which might have been greater degree of applicability if offered in a new form of flexibility within the general parameter. In this research, we implement Dombi operations to construct some 2-Tuple Linguistic Pythagorean Fuzzy (2TLPF) Dombi Aggregation operators. These operators include the 2TLPF Dombi Weighted Averaging (2TLPFDWA) operator, 2TLPF Dombi Ordered Weighted Averaging (2TLPFDOWA) operator, 2TLPF Dombi Weighted Geometric (2TLPFDWG) operator, and 2TLPF Dombi Ordered Weighted Geometric (2TLPFDOWA) operator. An analysis is conducted to examine the unique characteristics of these suggested operators. Subsequently, we leveraged the proposed operators to develop a model aimed at tackling the MAGDM problems in the 2TLPF environment. Eventually, a suitable instance has been demonstrated to validate the formation of the model as well as exhibit its implementation and resilience.

Optimizing decision-making with aggregation operators for generalized intuitionistic fuzzy sets and their applications in the tech industry

Intuitionistic fuzzy sets (IFSs) represent a significant advancement in classical fuzzy set (FS) theory. This study advances IFS theory to generalized intuitionistic fuzzy sets (GIFSBs) and introduces novel operators GIFWAA, GIFWGA, GIFOWAA, and GIFOWGA, tailored for GIFSBs. The primary aim is to enhance decision-making capabilities by introducing aggregation operators within the GIFSB framework that align with preferences for optimal outcomes. The article introduces new operators for GIFSBs characterized by attributes like idempotency, boundedness, monotonicity and commutativity, resulting in aggregated values aligned with GIFNs. A comprehensive analysis of the relationships among these operations is conducted, offering a thorough understanding of their applicability. These operators are practically demonstrated in a multiple-criteria decision-making process for evaluating startup success in the Tech Industry.

Enhanced decision technique for optimized crude oil pretreatment under disc spherical fuzzy Aczel Alsina aggregation information

Crude oil, the backbone of modern industry, holds unparalleled significance as a global energy cornerstone. Unlocking its potential hinges on effective pretreatment techniques, ensuring purity, and maximizing efficiency. This study extends the established Spherical Fuzzy Set paradigm to explore the domain of Disc Spherical Fuzzy Sets (D-SFSs) in critical decision-making for crude oil preparation. Investigating D-SFSs within the Aczel Alsina norm, the research employs comparison rules, conversion rules, and distance metrics. Primary operations of the Aczel Alsina norm in D-SFSs are examined, laying the groundwork for introducing unique aggregation operations within this framework. The paper's primary aim is to propose a hybrid method, termed MEREC-SWARA-MARCOS-D-SFSs Multiple Attribute Group Decision Making, which integrates the aforementioned aggregation procedures. A case study on crude oil pretreatment validates the effectiveness of the proposed method. Furthermore, a comprehensive comparison with CoCoSo underscores the reliability of the method. This study represents a significant stride in enhancing decision-making by providing a robust framework to tackle complex situations, particularly in the critical domain of crude oil pretreatment.

Confidence level based complex polytopic fuzzy Einstein aggregation operators and their application to decision-making process

A complex Polytopic fuzzy set (CPoFS) extends a Polytopic fuzzy set (PoFS) by handling vagueness with degrees that range from real numbers to complex numbers within the unit disc. This extension allows for a more nuanced representation of uncertainty. In this research, we develop Complex Polytopic Fuzzy Sets (CPoFS) and establish basic operational laws of CPoFS. Leveraging these laws, we introduce new operators under a confidence level, including the confidence complex Polytopic fuzzy Einstein weighted geometric aggregation (CCPoFEWGA) operator, the confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (CCPoFEOWGA) operator, the confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (CCPoFEHGA) operator, the induced confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (I-CCPoFEOWGA) operator and the induced confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (I-CCPoFEHGA) operator, enhancing decision-making precision in uncertain environments. We also investigate key properties of these operators, including monotonicity, boundedness, and idempotency. With these operators, we create an algorithm designed to solve multiattribute decision-making problems in a Polytopic fuzzy environment. To demonstrate the effectiveness of our proposed method, we apply it to a numerical example and compare its flexibility with existing methods. This comparison will underscore the advantages and enhancements of our approach, showing its efficiency in managing complex decision-making scenarios. Through this, we aim to demonstrate how our method provides superior performance and adaptability across different situations.

Supplier selection in green supply chain management using correlation-based TOPSIS in a q-rung orthopair fuzzy soft environment

Fuzzy hybrid models are efficient mathematical tools for managing unclear and vague data in real-world scenarios. This research explores the q-rung orthopair fuzzy soft set (q-ROFSS), which presents incomplete and ambiguous details in decision-making problems. The main intention of this study is to describe and evaluate the characteristics of the correlation coefficient (CC) and weighted correlation coefficient (WCC) for q-ROFSS. Also, the technique for order preference should be enhanced by similarity to the ideal solution (TOPSIS) with extended measures in q-ROFSS settings. Furthermore, we integrated mathematical formulations of correlation obstructions to confirm the consistency of the planned technique. It helps handle difficulties involving multi-attribute group decision-making (MAGDM). Moreover, a numerical illustration is presented to clarify how the advocated decision-making methodology can be implemented in evaluating suppliers in green supply chain management (GSCM). As a result, each alternative is assessed using multiple criteria, such as quality and reliability, capacity and scalability, compliance and certifications, and sustainability practices. The technique proposed in this study retains the selected research's specific structure more effectively than current techniques. A comparative analysis further substantiates the feasibility and effectiveness of the proposed approach over other decision-making techniques.

Aczel Alsina t-norm and t-conorm-based aggregation operators under linguistic interval-valued intuitionistic fuzzy setting with application

This article uses the Aczel-Alsina t-norm and t-conorm to make several new linguistic interval-valued intuitionistic fuzzy aggregation operators. First, we devised some rules for how linguistic interval-valued intuitionistic fuzzy numbers should work. Then, using these rules as a guide, we created a set of operators, such as linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted averaging (LIVIFAAWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina weighted geometric (LIVIFAAWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted averaging (LIVIFAAOWA) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina ordered weighted geometric (LIVIFAAOWG) operator, linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted averaging (LIVIFAAHWA) operator and linguistic interval-valued intuitionistic fuzzy Aczel-Alsina hybrid weighted geometric (LIVIFAAHWG) operators are created. Several desirable qualities of the newly created operators are thoroughly studied. Moreover, a multi-criteria group decision-making (MCGDM) method is proposed based on the developed operators. The proposed operators are then applied to real-world decision-making situations to demonstrate their applicability and validity to the reader. Finally, the suggested model is contrasted with the currently employed method of operation.

A novel perspective on the selection of an effective approach to reduce road traffic accidents under Fermatean fuzzy settings

Road traffic accidents (RTAs) pose a significant hazard to the security of the general public, especially in developing nations. A daily average of more than three thousand fatalities is recorded worldwide, rating it as the second most prevalent cause of death among people aged 5-29. Precise and reliable decisionmaking techniques are essential for identifying the most effective approach to mitigate road traffic incidents. This research endeavors to investigate this specific concern. The Fermatean fuzzy set (FFS) is a strong and efficient method for addressing ambiguity, particularly when the concept of Pythagorean fuzzy set fails to provide a solution. This research presents two innovative aggregation operators: the Fermatean fuzzy ordered weighted averaging (FFOWA) operator and the Fermatean fuzzy dynamic ordered weighted geometric (FFOWG) operator. The salient characteristics of these operators are discussed and important exceptional scenarios are thoroughly delineated. Furthermore, by implementing the suggested operators, we develop a systematic approach to handle multiple attribute decisionmaking (MADM) scenarios that involve Fermatean fuzzy (FF) data. In order to show the viability of the developed method, we provide a numerical illustration encompassing the determination of the most effective approach to alleviate road traffic accidents. Lastly, we conduct a comparative evaluation of the proposed approach in relation to a number of established methodologies.

A comprehensive study for selecting optimal treatment modalities for blood cancer in a Fermatean fuzzy dynamic environment

Cancer is characterized by uncontrolled cell proliferation, leading to cellular damage or death. Acute lymphoblastic leukemia (ALL), a kind of blood cancer, that affects lymphoid cells and is a challenging malignancy to treat. The Fermatean fuzzy set (FFS) theory is highly effective at capturing imprecision due to its capacity to incorporate extensive problem descriptions that are unclear and periodic. Within the framework of this study, two innovative aggregation operators: The Fermatean fuzzy Dynamic Weighted Averaging (FFDWA) operator and the Fermatean fuzzy Dynamic Weighted Geometric (FFDWG) operator are presented. The important attributes of these operators, providing a comprehensive elucidation of their significant special cases has been discussed in details. Moreover, these operators are utilized in the development of a systematic approach for addressing scenarios involving multiple attribute decision-making (MADM) problems with Fermatean fuzzy (FF) data. A numerical example concerning on finding the optimal treatment approach for ALL using the proposed operators, is provided. At the end, the validity and merits of the new method to illustrate by comparing it with the existing methods.

Complex Fermatean fuzzy extended TOPSIS method and its applications in decision making

The fuzzy set has its own limitations due to the membership function only. The fuzzy set does not describe the negative aspects of an object. The Fermatean fuzzy set covers the negative aspects of an object. The complex Fermatean fuzzy set is the most effective tool for handling ambiguous and uncertain information. The aim of this research work is to develop new techniques for complex decision-making based on complex Fermatean fuzzy numbers. First, we construct different aggregation operators for complex Fermatean fuzzy numbers, using Einstein t-norms. We define a series of aggregation operators named complex Fermatean fuzzy Einstein weighted average aggregation (CFFEWAA), complex Fermatean fuzzy Einstein ordered weighted average aggregation (CFFEOWAA), and complex Fermatean fuzzy Einstein hybrid average aggregation (CFFEHAA). The fundamental properties of the proposed aggregation operators are discussed here. The proposed aggregation operators are applied to the decision-making technique with the help of the score functions. We also construct different algorithms based on different aggregation operators. The extended TOPSIS method is described for the decision-making problem. We apply the proposed extended TOPSIS method to MAGDM problem "selection of an English language instructor". We also compare the proposed models with the existing models.

Analyzing of optimal classifier selection for EEG signals of depression patients based on intelligent fuzzy decision support systems

Electroencephalograms (EEG) is used to assess patients' clinical records of depression (EEG). The disorder of human thinking is a very complex problem caused by heavy-duty in daily life. We need some future and optimal classifier selection by using different techniques for depression data extraction using EEG. Intelligent decision support is a decision-making process that is automated based on some input information. The primary goal of this proposed work is to create an artificial intelligence-based fuzzy decision support system (AI-FDSS). Based on the given criteria, the AI-FDSS is considered for classifier selection for EEG under depression information. The proposed intelligent decision technique examines classifier alternatives such as Gaussian mixture models (GMM), k-nearest neighbor algorithm (k-NN), Decision tree (DT), Nave Bayes classification (NBC), and Probabilistic neural network (PNN). For analyzing optimal classifiers selection for EEG in depression patients, the proposed technique is criterion-based. First, we develop a general algorithm for intelligent decision systems based on non-linear Diophantine fuzzy numbers to examine the classifier selection technique using various criteria. We use classifier methods to obtain data from depression patients in normal and abnormal situations based on the given criteria. The proposed technique is criterion-based for analyzing optimal classifier selection for EEG in patients suffering from depression. The proposed model for analyzing classifier selection in EEG is compared to existing models.

Multi-attribute decision-making based on novel Fermatean fuzzy similarity measure and entropy measure

To deal with situations involving uncertainty, Fermatean fuzzy sets are more effective than Pythagorean fuzzy sets, intuitionistic fuzzy sets, and fuzzy sets. Applications for fuzzy similarity measures can be found in a wide range of fields, including clustering analysis, classification issues, medical diagnosis, etc. The computation of the weights of the criteria in a multi-criteria decision-making problem heavily relies on fuzzy entropy measurements. In this paper, we employ t-conorms to suggest various Fermatean fuzzy similarity measures. We have also discussed all of their interesting characteristics. Using the suggested similarity measurements, we have created some new entropy measures for Fermatean fuzzy sets. By using numerical comparison and linguistic hedging, we have established the superiority of the suggested similarity metrics and entropy measures over the existing measures in the Fermatean fuzzy environment. The usefulness of the proposed Fermatean fuzzy similarity measurements is shown by pattern analysis. Last but not least, a novel multi-attribute decision-making approach is described that tackles a significant flaw in the order preference by similarity to the ideal solution, a conventional approach to decision-making, in a Fermatean fuzzy environment.

Nonlinear distance measures under the framework of Pythagorean fuzzy sets with applications in problems of pattern recognition, medical diagnosis, and COVID-19 medicine selection

Background

The concept of Pythagorean fuzzy sets (PFSs) is an utmost valuable mathematical framework, which handles the ambiguity generally arising in decision-making problems. Three parameters, namely membership degree, non-membership degree, and indeterminate (hesitancy) degree, characterize a PFS, where the sum of the square of each of the parameters equals one. PFSs have the unique ability to handle indeterminate or inconsistent information at ease, and which demonstrates its wider scope of applicability over intuitionistic fuzzy sets.

Results

In the present article, we opt to define two nonlinear distances, namely generalized chordal distance and non-Archimedean chordal distance for PFSs. Most of the established measures possess linearity, and we cannot incorporate them to approximate the nonlinear nature of information as it might lead to counter-intuitive results. Moreover, the concept of non-Archimedean normed space theory plays a significant role in numerous research domains. The proficiency of our proposed measures to overcome the impediments of the existing measures is demonstrated utilizing twelve different sets of fuzzy numbers, supported by a diligent comparative analysis. Numerical examples of pattern recognition and medical diagnosis have been considered where we depict the validity and applicability of our newly constructed distances. In addition, we also demonstrate a problem of suitable medicine selection for COVID-19 so that the transmission rate of the prevailing viral pandemic could be minimized and more lives could be saved.

Conclusions

Although the issues concerning the COVID-19 pandemic are very much challenging, yet it is the current need of the hour to save the human race. Furthermore, the justifiable structure of our proposed distances and also their feasible nature suggest that their applications are not only limited to some specific research domains, but decision-makers from other spheres as well shall hugely benefit from them and possibly come up with some further extensions of the ideas.

MULTIMOORA Method-Based Schweizer–Sklar Operations for CO2 Geological Storage Site Selection Under Pythagorean Fuzzy Environment

The site selection of CO₂ geological storage facilities is essential for the development of safe and efficient carbon capture, utilization, and storage (CCUS) projects. Normally, CO₂ geological storage site selection can be regarded as a complex multi-criteria decision-making (MCDM) problem. The aim of this paper is to present an integrated decision-making method for solving the site selection problem for CO₂ geological storage. To achieve this goal, this method is based on multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA) method and prioritized aggregation operators in Pythagorean fuzzy environment. The academic contributions of this study include: first, some Pythagorean fuzzy Schweizer–Sklar prioritized aggregation (PFSSPA) operators are proposed, which take into account the priority levels of criteria and the risk preferences of decision makers. The excellent properties of these operators are given. Then this study extends the classical MULTIMOORA method based on the developed aggregation operators (named PFSSPA-MULTIMOORA), and the calculation process of this method is described in detail. Subsequently, on the basis of the constructed criteria system, the PFSSPA-MULTIMOORA method is applied to rank the alternatives. Finally, we successfully utilized the PFSSPA-MULTIMOORA method to solve the site selection problem of CO₂ geological storage in China. A comparative analysis of existing methods verifies the effectiveness and robustness of the proposed method. This work can provide advanced decision support for researchers and practitioners.

New cosine similarity and distance measures for Fermatean fuzzy sets and TOPSIS approach

The most straightforward approaches to checking the degrees of similarity and differentiation between two sets are to use distance and cosine similarity metrics. The cosine of the angle between two n-dimensional vectors in n-dimensional space is called cosine similarity. Even though the two sides are dissimilar in size, cosine similarity may readily find commonalities since it deals with the angle in between. Cosine similarity is widely used because it is simple, ideal for usage with sparse data, and deals with the angle between two vectors rather than their magnitude. The distance function is an elegant and canonical quantitative tool to measure the similarity or difference between two sets. This work presents new metrics of distance and cosine similarity amongst Fermatean fuzzy sets. Initially, the definitions of the new measures based on Fermatean fuzzy sets were presented, and their properties were explored. Considering that the cosine measure does not satisfy the axiom of similarity measure, then we propose a method to construct other similarity measures between Fermatean fuzzy sets based on the proposed cosine similarity and Euclidean distance measures and it satisfies the axiom of the similarity measure. Furthermore, we obtain a cosine distance measure between Fermatean fuzzy sets by using the relationship between the similarity and distance measures, then we extend the technique for order of preference by similarity to the ideal solution method to the proposed cosine distance measure, which can deal with the related decision-making problems not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods.

A multi-attribute decision-making method for the location selection of emergency rescue centers based on improved cumulative prospect theory under the background of ecological sustainable development

The location decision of the emergency rescue center, as a necessary link in the early stage to ensure the smooth development of emergency management, has become increasingly important. This paper analyzes the research theories and methods of location problem at home and abroad. By considering the decision-making psychological behavior of decision-makers, based on the idea of cumulative prospect theory, combined with the Pythagorean fuzzy multi-attribute evaluation method, this paper proposes a Pythagorean fuzzy multi-attribute decision-making evaluation method based on improved cumulative prospect theory. This method is to sort the results of each decision-making scheme by calculating the comprehensive cumulative prospect value. Based on the utility curve improved cumulative prospect theory, the research first depicts the psychological behavior characteristics of various decision-making groups under different risk preferences, and then designs a distance measurement method based on the geometric center of Pythagorean fuzzy right triangle. The main core of the distance measurement method is to convert Pythagorean fuzzy numbers into Pythagorean fuzzy right triangles. In the aspect of attribute weight assignment, a subjective and objective weighting method based on the combination of value function and deviation method of improved cumulative prospect theory is proposed. Finally, the Pythagorean fuzzy multi-attribute decision-making method based on the improved cumulative prospect theory is realized through the selection of reference objects, the calculation of value function value, weight function value and cumulative prospect value. This study takes the site selection of emergency rescue center in the construction period of Ya’an Linzhi Section of Sichuan Tibet Railway in China as an example, and proposes 10 emergency rescue centers as alternatives. Considering the different preferences of decision-makers, the improved decision-making method proposed in this paper is used to obtain the most optimal site selection scheme under different decision-making preferences.

Some Enhanced Distance Measuring Approaches Based on Pythagorean Fuzzy Information with Applications in Decision Making

The construct of Pythagorean fuzzy distance measure (PFDM) is a competent measuring tool to curb incomplete information often encountered in decision making. PFDM possesses a wider scope of applications than distance measure under intuitionistic fuzzy information. Some Pythagorean fuzzy distance measure approaches (PFDMAs) have been developed and applied in decision making, albeit with some setbacks in terms of accuracy and precision. In this paper, some novel PFDMAs are developed with better accuracy and reliability rates compared to the already developed PFDMAs. In an effort to validate the novel PFDMAs, some of their properties are discussed in terms of theorems with proofs. In addition, some applications of the novel PFDMAs in problems of disease diagnosis and pattern recognition are discussed. Furthermore, we present comparative studies of the novel PFDMAs in conjunction to the existing PFDMAs to buttress the merit of the novel approaches in terms of consistency and precision. To end with, some new Pythagorean fuzzy similarity measuring approaches (PFDSAs) based on the novel PFDMAs are presented and applied to solve the problems of disease diagnosis and pattern recognition as well.

Confidence Levels Complex q-Rung Orthopair Fuzzy Aggregation Operators and Its Application in Decision Making Problem

The theory investigated in this analysis is substantially more suitable for evaluating the dilemmas in real life to manage complicated, risk-illustrating, and asymmetric information. The complex Pythagorean fuzzy set is expanded upon by the complex q-rung orthopair fuzzy set (Cq-ROFS). They stand out by having a qth power of the real part of the complex-valued membership degree and a qth power of the real part and imaginary part of the complex-valued non-membership degree that is equal to or less than 1. We define the comparison method for two complex q-rung orthopair fuzzy numbers as well as the score and accuracy functions (Cq-ROFNs). Some averaging and geometric aggregation operators are examined using the Cq-ROFSs operational rules. Additionally, their main characteristics have been fully illustrated. Based on the suggested operators, we give a novel approach to solve the multi-attribute group decision-making issues that arise in environmental contexts. Making the best choice when there are asymmetric types of information offered by different specialists is the major goal of this work. Finally, we used real data to choose an ideal extinguisher from a variety of options in order to show the effectiveness of our decision-making technique. The effectiveness of the experimental outcomes compared to earlier research efforts is then shown by comparing them to other methods.

Another view on tolerance based multigranulation q-rung orthopair fuzzy rough sets with applications

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.

Performance analysis of Pythagorean fuzzy entropy and distance measures in selecting software reliability growth models using TOPSIS framework

Purpose

The purpose of this paper is to create a numerical technique to tackle the challenge of selecting software reliability growth models (SRGMs).

Design/methodology/approach

A real-time case study with five SRGMs tested against a set of four selection indexes were utilised to show the functionality of TOPSIS approach. As a result of the current research, rating of the different SRGMs is generated based on their comparative closeness.

Findings

An innovative approach has been developed to generate the current SRGMs selection under TOPSIS environment by blending the entropy technique and the distance-based approach.

Originality/value

In any multi-criteria decision-making process, ambiguity is a crucial issue. To deal with the uncertain environment of decision-making, various devices and methodologies have been explained. Pythagorean fuzzy sets (PFSs) are perhaps the most contemporary device for dealing with ambiguity. This article addresses novel tangent distance-entropy measures under PFSs. Additionally, numerical illustration is utilized to ascertain the strength and authenticity of the suggested measures.

A novel approach on spherical fuzzy rough set based-EDA𝒮 method for group decision support system

In daily life, the decision making problem is a complicated work related to uncertainties and vagueness. To overcome this vagueness and uncertainties, many fuzzy sets and theories have been presented by different scholars and researchers. EDA𝒮 (Evaluation based on distance from average solution) method plays a major role in decision-making problems. Especially, when multi-attribute group decision-making (MAGDM) problems have more conflicting attribute. In this paper, a new approach known as Spherical fuzzy rough-EDA𝒮 (SFR-EDA𝒮) method is used to handle these uncertainties in the MAGDM problem. The aggregation operators have the ability to combine different sources of information, which plays an essential role in decision making (DM) problem. Keeping in view the increasing complexity of the DM problem, it will be useful to combine the aggregation operators with the fuzzy sets in solving DM problem. Therefore, an aggregation operator known as SFR-EDA𝒮 method is utilized. For this propounded some new averaging and geometric aggregation is investigated. Moreover, the essential and desirable properties with some particular cases are deliberated and discussed detail. To evaluate the emergency program, a MAGDM approach is used based on the new introduced operators. Later on, the viability and applicability the proposed method is certified by a detailed analysis with the other existing approaches.

Multi-attribute group decision-making problem based on some induced Einstein aggregation operators under complex fuzzy environment

The aim of the paper is to introduce some complex Einstein aggregation operators for aggregating the different complex Pythagorean fuzzy sets (CPFSs) by considering the dependency between the pairs of its membership degrees. In the existing studies of fuzzy and its extensions, the uncertainties present in the data are handled with the help of degrees of membership that are the subset of real numbers, which may also loss some valuable data and hence consequently affect the decision results. A modification to these, complex Pythagorean fuzzy set handles the uncertainties with the degree whose ranges are extended from real subset to the complex subset with unit disc and hence handle the two dimensional information in a single set. Thus motivated by this and this paper we present some novel Einstein aggregation operators, namely complex Pythagorean fuzzy Einstein weighted averaging (CPFEWA) operator, complex Pythagorean fuzzy Einstein ordered weighted averaging (CPFEOWA) operator, complex Pythagorean fuzzy Einstein hybrid averaging (CPFEHA) operator, induced complex Pythagorean fuzzy Einstein ordered weighted averaging (I-CPFEOWA) operator, and induced complex Pythagorean fuzzy Einstein hybrid averaging (I-CPFEHA) operator. Also develop some of their desirable properties. Furthermore, based on these operators a multi-attribute group decision making problems developed. An illustrative example related to the selection of the best alternative is considered to show the effectiveness, of the novel developed methods.

A multi-attribute decision-making method for the location selection of emergency rescue centers based on improved cumulative prospect theory under the background of ecological sustainable development

This research proposes a Pythagorean fuzzy multi-attribute decision-making evaluation method based on the improved cumulative prospect theory. The method ranks the decision-making results by calculating the comprehensive cumulative prospect value. Firstly, the research improves the cumulative prospect theory based on the utility curve, and describes the psychological and behavioral characteristics of various decision-making groups with different risk preferences. Then, a distance measure method based on the geometric center of the Pythagorean fuzzy right triangle is designed. The main core of the distance measure method is that it converts the Pythagorean fuzzy number into a Pythagorean fuzzy right triangle. In terms of attribute weighting, this research proposes a subjective and objective weighting method based on the combination of value function and deviation method of improved cumulative prospect theory. Finally, the Pythagorean fuzzy multi-attribute decision-making method based on the improved cumulative prospect theory is realized through the selection of reference object, the calculation of value function value, weight function value and cumulative prospect value. The results analysis and the comparison with other methods verify the effectiveness and advancement of the proposed decision-making method, especially that the proposed method has good applicability for the decision-making cases where the attribute value is Pythagorean fuzzy number, the attribute weight is unknown, and the psychological behavior of decision makers cannot be reflected.

Cosine similarity measures between q-rung orthopair linguistic sets and their application to group decision making problems

The q-rung orthopair linguistic set (q-ROLS), a combined version of linguistic term sets and q-rung orthopair fuzzy set, is an efficient mathematical tool to accomplish the imprecise information while solving the decision-making problems. Under this environment, we propose additional operations and relations to deal with the decision information, and some properties are well proved. Furthermore, we propound some cosine similarity measures and weighted cosine similarity measures for q-ROLSs based on the traditional cosine similarity measures with a brief study of related properties. In the proposed similarity measures, various linguistic scale functions are utilized in order to take into account the semantics of linguistic terms. Besides this, we employ the stated q-rung orthopair linguistic similarity measures to multi-criteria group decision making problems, in which the weights of DMs are delineated by the projection of individual decisions on the ideal decision results. At last, a numerical example is used to demonstrate the practicality of the suggested technique, and its efficacy is validated by comparison with prevailing techniques. The suggested method’s sensitivity and stability analyses are also provided.

Multiple attribute decision making based on Pythagorean fuzzy Aczel-Alsina average aggregation operators

A useful expansion of the intuitionistic fuzzy set (IFS) for dealing with ambiguities in information is the Pythagorean fuzzy set (PFS), which is one of the most frequently used fuzzy sets in data science. Due to these circumstances, the Aczel-Alsina operations are used in this study to formulate several Pythagorean fuzzy (PF) Aczel-Alsina aggregation operators, which include the PF Aczel-Alsina weighted average (PFAAWA) operator, PF Aczel-Alsina order weighted average (PFAAOWA) operator, and PF Aczel-Alsina hybrid average (PFAAHA) operator. The distinguishing characteristics of these potential operators are studied in detail. The primary advantage of using an advanced operator is that it provides decision-makers with a more comprehensive understanding of the situation. If we compare the results of this study to those of prior strategies, we can see that the approach proposed in this study is more thorough, more precise, and more concrete. As a result, this technique makes a significant contribution to the solution of real-world problems. Eventually, the suggested operator is put into practise in order to overcome the issues related to multi-attribute decision-making under the PF data environment. A numerical example has been used to show that the suggested method is valid, useful, and effective.

Some t-conorm-based distance measures and knowledge measures for Pythagorean fuzzy sets with their application in decision-making

The Pythagorean fuzzy sets are more robust than fuzzy sets and intuitionistic fuzzy sets in dealing with the problems involving uncertainty. To compare two Pythagorean fuzzy sets, distance measures play a crucial role. In this paper, we have proposed some novel distance measures for Pythagorean fuzzy sets using t-conorms. We have also discussed their various desirable properties. With the help of suggested distance measures, we have introduced some new knowledge measures for Pythagorean fuzzy sets. Through numerical comparison and linguistic hedges, we have established the effectiveness of the suggested distance measures and knowledge measures, respectively, over the existing measures in the Pythagorean fuzzy setting. At last, we have demonstrated the application of the suggested measures in pattern analysis and multi-attribute decision-making.

q-Rung orthopair fuzzy soft aggregation operators based on Dombi t-norm and t-conorm with their applications in decision making

Recently, some improvement has been made in the dominant notion of fuzzy set that is Yager investigated the generalized concept of fuzzy set, Intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) and called it q-rung orthopair fuzzy (q-ROF) set (q-ROFS). The aim of this manuscript is to present the concept of q-ROF soft (q-ROFSt) set (q-ROFStS) based on the Dombi operations. Since Dombi operational parameter possess natural flexibility with the resilience of variability. Some new operational laws are defined based on hybrid study of soft sets and q-ROFS. The advantage of Dombi operational parameter is very important to express the experts’ attitude in decision making. In this paper, we present q-ROFSt Dombi average (q-ROFSt DA) aggregation operators including q-ROFSt Dombi weighted average (q-ROFSt DWA), q-ROFSt Dombi ordered weighted average (q-ROFSt DOWA) and q-ROFSt Dombi hybrid average (q-ROFSt DHA) operators. Moreover, we investigate q-ROFSt Dombi geometric (q-ROFSt DG) aggregation operators including q-ROFSt Dombi weighted geometric (q-ROFSt DWG), q-ROFSt Dombi ordered weighted geometric (q-ROFSt DOWG), and q-ROFSt Dombi hybrid geometric (q-ROFSt DHG) operators. The basic properties of these operators are presented with detail such us Idempotency, Boundedness, Monotonicity, Shift invariance, and Homogeneity. Thus from the analysis and advantages of proposed model, it is clear that the investigated q-ROFSt DWA operator is the generalized form of IF St DWA, PFSt DWA and q-ROFDWA operators. Similarly, the investigated q-ROFSt DWG operator is the generalized form of IF St DWG, PFSt DWG and q-ROFDWG operators. By applying the develop approach, this manuscript contains the technique and algorithm for multicriteria decision making (MCDM). Further a numerical example is developed to illustrate the flexibility and applicability of the developed operators.

New correlation coefficients of Pythagorean fuzzy set and its application to extended TODIM method

With the increasing complexity of decision making (DM) problems, powerful mathematical tools are needed to represent and process fuzzy and uncertain DM information, and Pythagorean fuzzy set (PFS) is such a mathematical tool. PFS has been successfully applied in the field of fuzzy multiple criteria decision making (MCDM). Correlation coefficient is an information measure of PFS, and plays an important role in the application of PFS. At present, there is a problem that the existing correlation coefficients cannot moderately measure the correlation degree between PFSs, so this paper proposes the new correlation coefficients of PFS. The TODIM method has been proved to be effective in dealing with MCDM problems that consider the psychological behavior of decision makers. This paper extends the TODIM method with the new correlation coefficients of PFS, and the extended TODIM method is called Pythagorean fuzzy CC-TODIM method. By numerical examples, it is verified that the new correlation coefficients of PFS are more reasonable and valid. By case analysis, it is verified that the Pythagorean fuzzy CC-TODIM method can effectively solve the MCDM problems, and the Pythagorean fuzzy CC-TODIM method based on the new correlation coefficients is more accurate and reliable.

Research on service quality evaluation of sports clubs with Pythagorean fuzzy information

In this paper, considering the traditional geometric operation laws and Pythagorean fuzzy information, a varies of new distance measures of Pythagorean fuzzy set are proposed, such as generalized Pythagorean fuzzy ordered geometric distance (GPFOWGD) measures and generalized Pythagorean fuzzy hybrid weighted geometric distance (GPFHWGD) measures. Besides, some special issues including Hamming distance, Euclidean distance and Hausdorff distance of these raised geometric distance measures are investigated. To testify the valid of these new presented distance measures, a decision-making model is built and illustrated by a mathematical calculation example to evaluate the service quality of sports clubs using Pythagorean fuzzy information. The results display that the approach is uncomplicated, valid and simple to compute. The example illustrates that the method presented in the paper can be used to deal with problems of uncertainty such as dynamic multiple attribute decision making.

An Inner Dependence Analysis Dynamic Decision-Making Framework

During the last decade, with the rapid development of information technology, the immense volume of data poses a challenge to decision-makers. We use a combined dynamic decision-making approach based on the analytic hierarchy process (AHP) to select the best supplier. In this paper, we discuss the interaction between criteria that can lead to expanding our proposed dynamic framework to consider the inner dependencies among criteria. The main contributions are: (1) identifying the most important criteria of supplier selection in a steel bar manufacturer in Taiwan; (2) proposing a simple and rapid analysis of the appropriate supplier selection evaluation framework; and (3) using the AHP and transformation matrix to present the inner dependence among the criteria.

Fuzzy VIKOR approach to identify COVID-19 vulnerability region to control third wave in Assam, India

These days, the appraisal of the COVID-19 vulnerability has become a difficult errand for the whole world. The COVID-19 administration dynamic issue frequently includes numerous elective arrangements clashing standards. In this paper, we present a multi-criteria decision-making (MCDM) procedure based on the fuzzy VIKOR method to survey the COVID-19 vulnerability in the state of Assam, India. The trapezoidal fuzzy number is utilized to evaluate the rating of the loads for the set-up models. We have observed environment, social, and Medical factors after observing the spread of COVID-19. To study and to have comments, a committee of five experts has been formed from a different region of Assam to observe and comment to identify Coronavirus’s weakest factors. For a better survey, we have divided the state into four areas namely Rural Area, Urban Area, Market Area in Rural Area, and Market Area in Urban Area. The current research looked at how the fuzzy VIKOR selects provinces for urgent adaptation needs differently than a traditional MCDM technique.

Rough fermatean fuzzy ideals in semigroups

In multiple criteria decision making (MCDM) problem, the decision maker deal with a situation where the sum of membership and non-membership grade of an attributes does not belongs to [0, 1]. To avoid such a situation, we proposed a new type of fuzzy system known as fermatean fuzzy system. More precisely, we presented the notion of fermatean fuzzy ideal theory and rough fermatean fuzzy sets in semigroups. The idea of lower and upper approximation in fermatean fuzzy sets has been initiated. The study has been further extended to rough fermatean fuzzy left(resp. right, interior) ideals in semigroup. Several results related to this notion are determined.

Algorithms for multiple attribute decision making with dual hesitant Pythagorean fuzzy information and their application to supplier selection

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Hamacher aggregation operators and Choquet integral with dual Pythagorean hesitant fuzzy information. Then, motivated by the ideal of Hamacher operation and Choquet integral, we have developed some Hamacher correlated operators for aggregating dual hesitant Pythagorean fuzzy information. The prominent characteristic of these proposed operators is studied. Then, we have utilized these two operators to develop some approaches to solve the dual hesitant Pythagorean fuzzy MADM problems. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Models for multiple attribute decision making with dual hesitant pythagorean fuzzy information

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Muirhead Mean (MM) operators with dual Pythagorean hesitant fuzzy information. Then, motivated by the ideal of MM operators, we have developed some MM operators for aggregating dual hesitant Pythagorean fuzzy information. The prominent characteristic of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the dual hesitant Pythagorean fuzzy multiple attribute decision making problems. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.