Interval-valued Pythagorean Research Articles

Decision Support Methodology Based on Covering-Based Interval-Valued Pythagorean Fuzzy Rough Set Model and Its Application to Hospital Open-Source EHRs System Selection

The target of current work is to propose a new approach to deal with multiattribute decision-making (MADM) problems with interval-valued Pythagorean fuzzy set (IVPFS) based on the concepts of covering-based rough set (CRS) and TOPSIS and give its application in MADM problems. To begin with, we integrate the fuzzy rough set (FRS), IVPFS and CRS and define the covering-based interval-valued Pythagorean fuzzy rough set (CIVPFRS). Firstly, the relative notions of the CIVPFRS model are introduced. In addition, the distance measure of interval-valued Pythagorean fuzzy numbers (IVPFNs) is defined; based on the proposed distance, the rough and precision degrees of CIVPFRS are discussed. Thirdly, on the basis of the theoretical analysis for CIVPFRS models, an interval-valued Pythagorean fuzzy TOPSIS method is designed to deal with the MADM problems with interval-valued Pythagorean fuzzy information (IVPFI). Last of all, the validity and merits of the proposed approach are illustrated by an example, and the sensitivity analysis of the parameters and the comparison with the existing related methods are carried out.β

Interval-valued Pythagorean Fuzzy EDAS method: An Application to Car Selection Problem

Interval-valued Pythagorean Fuzzy EDAS method: An Application to Car Selection Problem

Correlation Coefficients of Interval-Valued Pythagorean Hesitant Fuzzy Sets and Their Applications

An interval-valued Pythagorean hesitant fuzzy set (IVPHFS) not only can be regarded as the union of some interval-valued Pythagorean fuzzy sets but also represent the Pythagorean hesitant fuzzy elements in the form of interval values. So IVPHFSs are extensions of Pythagorean hesitant fuzzy sets (PHFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs), which are powerful tools to represent more complicated, uncertain and vague information. This paper focuses on the four kinds of correlation coefficients for PHFSs, and extends them to the correlation coefficients and the weighted correlation coefficients for IVPHFSs. In the processing, we develop the least common multiple expansion (LCME) method to solve the problem that the cardinalities of Pythagorean hesitant fuzzy elements (PHFEs) (or interval-valued Pythagorean hesitant fuzzy elements (IVPHFEs)) are different. In addition, we propose score functions and accuracy functions of Pythagorean fuzzy elements (PFEs) (or interval-valued Pythagorean fuzzy elements (IVPFEs)) to rank all the PFEs (or IVPFEs) in a PHFE (or an IVPHFE). Especially, score functions and accuracy functions of IVPFEs are both presented as interval numbers. Then use the comparison method of interval numbers to compare two revised IVPHFEs in order to keep the original fuzzy information as far as possible. What’s more, we define the local correlations and local informational energies which can depict the similarity between two IVPHFEs more meticulously and completely. At last the numerical examples to show the feasibility and applicability of the proposed methods in multiple criteria decision making (MCDM) and clustering analysis.

Interval-Valued Pythagorean Normal Fuzzy Information Aggregation Operators for Multi-Attribute Decision Making

The interval-valued Pythagorean fuzzy (IVPF) sets, describing the membership and non-membership degrees from interval values, can address uncertain information, while the normal fuzzy number (NFN) can depict normal distribution information in anthropogenic activity and natural environment. By combining the advantages of both operations, in this study, we proposed the interval-valued Pythagorean normal fuzzy (IVPNF) sets by introducing the NFN into IVPF environment. Firstly, we defined the conception, the operational laws, score function, accuracy function of IVPNF sets. Secondly, we presented four information aggregation operators to aggregate IVPNF information, including the IVPNF weighted averaging (IVPNFWA) operator, IVPNF weighted geometric (IVPNFWG) operator, the generalized IVPNFWA operator, and the generalized IVPNFWG operator. In addition, we analyzed some desirable properties of monotonicity, commutativity, and idempotency for the proposed four operators. Finally, a numerical example on multi-attribute decision-making problem is given to verify the practicality of the proposed operators, and the comparative and sensitive analysis are used to show the effectiveness and flexibility of our proposed approach.

Evaluating Recognitive Balanced Scorecard-Based Quality Improvement Strategies of Energy Investments With the Integrated Hesitant 2-Tuple Interval-Valued Pythagorean Fuzzy Decision-Making Approach to QFD

This study aims to identify quality improvement strategies for energy investments. For this purpose, a model is proposed which includes 4 different stages. In the first stage, the MCDM problem is identified for evaluating the service development for energy industry. In this framework, quality function deployment (QFD) approach is taken into consideration which includes both customer expectations and technical requirements at the same time to improve the quality in the organization. The second stage is related to the calculation of the correlation coefficients of decision matrices for the criteria by considering hesitant 2-tuple interval-valued Pythagorean fuzzy sets. The third stage includes the weighting of the customer expectations with hesitant 2-tuple interval-valued Pythagorean fuzzy (HIVPF) DEMATEL. In the final stage, TRIZ-based quality improvement strategies of energy investments are ranked by using 2-tuple HIVPF TOPSIS. Thus, the motivation of this study is to figure out the weights of the criteria for quality improvement strategies in energy investments. Also, the most important contribution of this study to the literature is related to the originality in the methodology by proposing a new MCDM model while using hesitant linguistic term sets, linguistic 2-tuple information, interval-valued Pythagorean fuzzy sets properly. The findings indicate that empathy is the most significant criterion for the customer expectations in the energy investments. In addition to this issue, it is also identified that customization is the best factor among the technical requirements of energy investments. Moreover, information and communication facilities and organizational background are found as the best competencies of new service development in energy investments. Furthermore, that prior action and periodic action are the most prominent strategies for quality improvement. While considering these results, it can be said that the pricing policy of the energy companies should be fair to increase customer satisfaction. Additionally, offering flexible payment opportunities on energy bills can have a positive influence on the customer satisfaction in this process. Also, preliminary planning of the project should be done in detail in energy investments. Owing to this issue, customers' preferences can be identified before the product is placed on the market. In addition, it can be possible to identify the risks that may arise in energy investments with the help of the periodical audits.

A Novel Multiple Attribute Group Decision-Making Approach Based on Interval-Valued Pythagorean Fuzzy Linguistic Sets

This article investigates multi-attribute group decision-making (MAGDM) problems based on interval-valued Pythagorean fuzzy linguistic sets (IVPFLSs). The IVPFLSs are regarded as an efficient tool to describe decision makers’ (DMs’) evaluation information from both quantitative and qualitative aspects. However, existing IVPFLSs based MAGDM methods are still insufficient and inadequate to deal with complicated practical situations. This article aims to propose a novel MAGDM method and the main contributions of the present work are three-fold. First, we propose new operations of interval-valued Pythagorean fuzzy linguistic numbers (IVPFLNs) based on linguistic scale function. Second, we propose new aggregation operators (AOs) of IVPFLNs based on power average operator and Muirhead mean. The proposed AOs take the interrelationship among any numbers of attributes into account and eliminate the bad influence of DMs’ unreasonable evaluation values on the final decision results. Third, based on the new operations and AOs of IVPFLNs, we introduce a novel approach to MAGDM and present its main steps. Finally, we discuss the effectiveness of the proposed approach and investigates their advantages through numerical examples.

A new decision approach for the sustainable transport investment selection based on the generalized entropy and knowledge measure under an interval-valued Pythagorean fuzzy environment

Finding the most suitable transport project is one of the most important tasks in transport planning. This task gets more complicated as the sustainable criteria get involved in the process. In this paper, a new multi-criteria group decision-making method with unknown expert and attribute weights is proposed to address the sustainable transport investment selection problem. To make the method more powerful in dealing with uncertain elements, an Interval-Valued Pythagorean Fuzzy (IVPF) set is used as an attractive and useful tool to handle uncertainty. First, a generalized entropy measure under an IVPF environment is introduced, which enables the method to determine the fuzziness of the attribute values, which are expressed by Interval-Valued Pythagorean Fuzzy Numbers (IVPFNs). To determine the fuzziness of IVPFNs with identical membership and non-membership degrees, a generalized knowledge measure of the IVPFNs is also introduced. Based on this measure and considering the deviation between attribute assessments, a new optimization model is presented to obtain unknown attribute weights. In addition, based on the extension of the VIKOR method, a new algorithm is presented to determine the unknown expert weights. Finally, a real case study is considered to show the efficiency of the proposed methods.

Some induced interval-valued Pythagorean trapezoidal fuzzy averaging aggregation operators based on Einstein operations and their application in group decision-making

The aim of this paper is to investigate the information aggregation methods under interval-valued Pythagorean trapezoidal fuzzy environment. Some Einstein operational laws on Pythagorean trapezoidal fuzzy numbers are defined based on Einstein sum and Einstein product. We define interval-valued Pythagorean trapezoidal fuzzy aggregation operators, induced interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging (I-IVTFEOWA) operator and induced interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-IVPTFEHA) operator. We discuss some basic properties of the proposed operator, including idempotency, commutativity and monotonicity. We construct an algorithm for multiple-attribute group decision-making problem, and apply the proposed aggregation operator to deal with multiple-attribute group decision-making. Finally, we construct a numerical example for multiple-attribute group decision-making.

Soft Computing-based New Interval-valued Pythagorean Triangular Fuzzy Multi-criteria Group Assessment Method without Aggregation: Application to a Transport Projects Appraisal

In this paper, an interval-valued Pythagorean triangular fuzzy number (IVPTFN) as a useful tool to handle decision-making problems with vague quantities is defined. Then, their operational laws are developed. By introducing a novel method of making a decision on the concept of possibility theory, a multi-attribute group decision-making (MAGDM) problem is considered, in which the attribute values are expressed with the IVPTFN and the information on the decision makers’ (DM) weights is completely unknown. Two novel forms of a multi-attributive border approximation area comparison (MABAC) technique are proposed to solve the problem. One of them is applied to compute the weights of the decision makers, and the other is used to rank the preference order of alternatives, that is based on the possibility expected value and standard deviation and has no aggregation of information. Finally, to illustrate the practicality and effectiveness of proposed method in real-world problems, the proposed method is applied in a real case study of an Iranian transport complex to sustainability assessment of its transport projects.

Models for Multiple Attribute Decision Making with Interval-Valued Pythagorean Fuzzy Muirhead Mean Operators and Their Application to Green Suppliers Selection

Models for Multiple Attribute Decision Making with Interval-Valued Pythagorean Fuzzy Muirhead Mean Operators and Their Application to Green Suppliers Selection

Green Supplier selection for plastic industry using integrated model based on Pythagorean fuzzy AHP and fuzzy TOPSIS

Purpose –Supply Chain Management (SCM) has emerged in recent years and is now increasing in popularity. It aims to reduce production costs, maximize earnings, improve customer relations, improve inventory management and increase customer satisfaction. Reducing total cost of procurement is one of the most important parameters in aligning the objectives of partners in a supply network. Design/methodology/approach – This paper proposed Pythagorean Fuzzy Analytic Hierarchy Process (PFAHP) and Fuzzy Technique for Order Preference by Similarity to Ideal Solution (FTOPSIS) integrated model for green supplier selection for plastic industry. In the first phase of the study, the parameters to be used in selecting suppliers are determined. Then, by making pairwise comparisons, the weights of these parameters are determined with PFAHP method. Finally, the supplier that is most suitable for 3 suppliers is determined using the FTOPSIS method. Findings – Among the criteria evaluated by 5 experts, the most important criteria, the parameters of the Inspection methods and management and organizations are determined as the most important parameters respectively. The supplier no. 3 has been identified as the most suitable supplier. Discussion –The case study is performed under a fuzzy environment to reduce uncertainty and vagueness, and linguistic variables parameterized by interval-valued Pythagorean and triangular fuzzy numbers are used. Through the case study, 8 main and 45 sub-criteria supplier selection evaluation criteria used to assess 3 suppliers by FTOPSIS. However, our study has some disadvantages and possible further work is recommended. Other possible different fuzzy sets can be used in the projected method.

On Accuracy Function and Distance Measures of Interval-valued Pythagorean Fuzzy Sets with Application in Decision Making

The notion of interval-valued Pythagorean fuzzy sets permits four important parameters, i.e., membership degree, non-membership degree, and a pair of values strength of commitment and direction of commitment, to a given set to have an interval value in dealing with imprecise information. In the present communication, a new accuracy function is being provided to overcome the shortcomings of the existing score and available accuracy functions for interval-valued Pythagorean fuzzy sets. The validity of the proposed accuracy function has been discussed in detail through the illustrative examples. Further, a new interval-valued Pythagorean fuzzy $p$-distance measure for interval-valued Pythagorean fuzzy numbers has been proposed and used in context with the existing weighted averaging operators. Finally, in view of the proposed accuracy function, distance measure and weighted averaging operators, a numerical example of multi-criteria decision-making problem has been solved to validate the proposed methodology.

An interval-valued Pythagorean prioritized operator-based game theoretical framework with its applications in multicriteria group decision making

Multicriteria decision-making process explicitly evaluates multiple conflicting criteria in decision making. The conventional decision-making approaches assumed that each agent is independent, but the reality is that each agent aims to maximize personal benefit which causes a negative influence on other agents' behaviors in a real-world competitive environment. In our study, we proposed an interval-valued Pythagorean prioritized operator-based game theoretical framework to mitigate the cross-influence problem. The proposed framework considers both prioritized levels among various criteria and decision makers within five stages. Notably, the interval-valued Pythagorean fuzzy sets are supposed to express the uncertainty of experts, and the game theories are applied to optimize the combination of strategies in interactive situations. Additionally, we also provided illustrative examples to address the application of our proposed framework. In summary, we provided a human-inspired framework to represent the behavior of group decision making in the interactive environment, which is potential to simulate the process of realistic humans thinking.

Geometric aggregation operators with interval-valued Pythagorean trapezoidal fuzzy numbers based on Einstein operations and their application in group decision making

The aim of this paper is to investigate information aggregation methods under interval-valued Pythagorean trapezoidal fuzzy environment. Some Einstein operational laws on interval-valued Pythagorean trapezoidal fuzzy numbers are defined based on Einstein sum and Einstein product. Based on Einstein operations, we define interval-valued Pythagorean trapezoidal fuzzy aggregation operators, such as interval-valued Pythagorean trapezoidal fuzzy Einstein weighted geometric operator, interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted geometric operator and interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid geometric operator. Furthermore, we apply the proposed aggregation operators to deal with multiple attribute group decision making problem. Finally we construct a numerical example for multiple attribute group decision making problem and compare the result with existing methods.

Models for Multiple Attribute Decision Making with Interval-Valued Pythagorean Fuzzy Muirhead Mean Operators and Their Application to Green Suppliers Selection

In this paper, we extend MM operator and dual MM (DMM) operator to process the interval-valued Pythagorean fuzzy numbers (IVPFNs) and then to solve the MADM problems. Firstly, we develop some interval-valued Pythagorean fuzzy Muirhead mean operators by extending MM and DMM operators to IVPFNs. Then, we prove some properties and discuss some special cases with respect to the parameter vector. Moreover, we present some new methods to deal with MADM problems with the IVPFNs based on the proposed MM and DMM operators. Finally, we verify the validity and reliability of our methods by using an application example for green supplier selections, and analyse the advantages of our methods by comparing it with other existing methods.

Green Supplier Selection Based on DEA Model in Interval-Valued Pythagorean Fuzzy Environment

The green supplier selection has gradually become a hot topic around the world. To achieve the purpose of optimal decision-making, it is of great importance to process the imprecise and ambiguous information that may exist in the evaluation and selection of green suppliers. In the present article, first, converting the imprecise and ambiguous information into interval-valued Pythagorean fuzzy numbers (IVPFNs) through the Likert scale. Then, using the interval-valued Pythagorean fuzzy weighted average (IVPFWA) operator and multiplicative theory to introduce the IVPFNs into the non-radial SE-BCC model of Data Envelopment Analysis (DEA), we developed the DEA model in the interval-valued Pythagorean fuzzy environment. Later, ranking all decision-making units (DMUs) to be evaluated and selecting the best of them. Moreover, The proposed model can not only deal with the case where the quantitative indicators are interval-valued numbers, but also solve the problem of qualitative analysis in the evaluation process. Finally, a numerical example of green suppliers selection is provided to demonstrate the feasibility and usefulness of the model.

Algorithms for Interval-Valued Pythagorean Fuzzy Sets in Emergency Decision Making Based on Multiparametric Similarity Measures and WDBA

Emergency decision making is critically important for countries or communities to enhance the effectiveness and validity of the emergency response, which can greatly lower environmental damage, casualties, and economic loss. In the case of emergency decision evaluation, the essential problems that arise serious inexactness, fuzziness, and ambiguity. Interval-valued Pythagorean fuzzy set (IVPFS), portrayed by membership and non-membership with the interval form, is an effective and flexible way to seize indeterminacy. In this paper, primarily, a novel score function for an interval-valued Pythagorean fuzzy number is initiated for managing some comparative issues. Then, a new distance measure for IVPFSs with multiple parameters is studied for solving the counter-intuitive situations. The interesting properties among the developed similarity measures, distance measures, and entropy have also been derived. Then, the objective weights of diverse attributes are ascertained by a novel entropy approach. Also, we explore the combination weight, which can reveal both objective preference and subjective preference. In addition, two interval-valued Pythagorean fuzzy decision making methods based on weighted distance-based approximation and multiparametric similarity measure are presented. Later, the validity of the algorithms is illustrated by a mine emergency decision making issue with the influence of diverse parameters on the ordering. Finally, a comparison with some existing decision making methods has been executed by the counter-intuitive phenomena and discrimination problems for verifying their effectiveness.

An Interval-Valued Pythagorean Fuzzy Compromise Approach with Correlation-Based Closeness Indices for Multiple-Criteria Decision Analysis of Bridge Construction Methods

The purpose of this paper is to develop a novel compromise approach using correlation-based closeness indices for addressing multiple-criteria decision analysis (MCDA) problems of bridge construction methods under complex uncertainty based on interval-valued Pythagorean fuzzy (IVPF) sets. The assessment of bridge construction methods requires the consideration of multiple alternatives and conflicting tangible and intangible criteria in intricate and varied circumstances. The concept of IVPF sets is capable of handling imprecise and ambiguous information and managing complex uncertainty in real-world applications. Inspired by useful ideas concerning information energies, correlations, and correlation coefficients, this paper constructs new concepts of correlation-based closeness indices for IVPF characteristics and investigates their desirable properties. These indices can be utilized to achieve anchored judgments in decision-making processes and to reflect a certain balance between connections with positive and negative ideal points of reference. Moreover, these indices can fully consider the amount of information associated with higher degrees of uncertainty and effectively fuse imprecise and ambiguous evaluative ratings to construct a meaningful comparison approach. By using the correlation-based closeness index, this paper establishes effective algorithmic procedures of the proposed IVPF compromise approach for conducting multiple-criteria evaluation tasks within IVPF environments. The proposed methodology is implemented in a practical problem of selecting a suitable bridge construction method to demonstrate its feasibility and applicability. The practicality and effectiveness of the proposed methodology are verified through a comparative analysis with well-known compromise methods and other relevant nonstandard fuzzy models.

Interval-valued Pythagorean Fuzzy Graph

Interval-valued Pythagorean Fuzzy Graph

Induced interval-valued Pythagorean trapezoidal fuzzy aggregation operators based on Einstein operations and their application in group decision making.

The aim of this paper is to investigate the information aggregation methods under induced interval-valued Pythagorean trapezoidal fuzzy environment. Some Einstein operational laws on Pythagorean trapezoidal fuzzy numbers are defined based on Einstein sum and Einstein product. In this paper, we introduce the idea of induced interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted geometric (I-IVPTFEOWG) operator and induced interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid geometric (I-IVPTFEHG) operator. We discuss some basic properties of the proposed operator, including idempotency, commutativity and monotonicity. We construct an algorithm for multiple attribute group decision making problem, and apply the proposed aggregation operator to deal with multiple attribute group decision making. Finally we construct a numerical example for multiple attribute group decision making and compare the result with existing methods.