Interval-valued Pythagorean Fuzzy Set Research Articles

Linguistic Interval-Valued Pythagorean Fuzzy Sets and Their Application to Multiple Attribute Group Decision-making Process

The paper’s aims are to present a novel concept of linguistic interval-valued Pythagorean fuzzy set (LIVPFS) or called a linguistic interval-valued intuitionistic type-2 fuzzy set, which is a robust and trustworthy tool, and to accomplish the imprecise information while solving the decision-making problems. The presented LIVPFS is a generalization of the linguistic Pythagorean fuzzy set, by characterizing the membership and non-membership degrees as the interval-valued linguistic terms to represent the uncertain information. To explore the study, we firstly define some basic operational rules, score and accuracy functions, and the ordering relations of LIVPFS with a brief study of the desirable properties. Based on the stated operational laws, we proposed several weighted averages and geometric aggregating operators to aggregate the linguistic interval-valued Pythagorean fuzzy information. The fundamental inequalities between the proposed operators and their properties are discussed in detail. Finally, a multiple attribute group decision-making (MAGDM) algorithm is promoted to solve the group decision-making problems with uncertain information using linguistic features and the proposed operators. The fundamental inequalities between the proposed operators and their properties are discussed in detail. Also, the illustration of the stated algorithm is given through several numerical examples and compared their performance with the results of the existing algorithms. Based on the stated MAGDM algorithm and the suitable operators, the decision-makers’ can be selected their best alternatives with their own attitude character towards optimism or pessimism choice. The presented LIVPFS is an extension of the several existing sets and is more generalized to utilize the uncertain and imprecise information with a wider range of information. Based on the presented aggregation operators, a decision-maker can select the desired one as per their choices to access the finest alternatives.

A Novel Two-Stage Multi-Criteria Decision-Making Method Based on Interval-Valued Pythagorean Fuzzy Aggregation Operators with Self-Confidence Levels

Due to insufficient information in multi-criteria decision-making (MCDM) problems, the decision values given by experts are often fuzzy and uncertain. As an extension of Pythagorean fuzzy set (PFS), interval-valued Pythagorean fuzzy (IPF) set is a more effective and powerful tool to handle fuzzy information in decision problems. But, there are two key issues that needed to be solved: weights of experts in the IPF environment and IPF aggregation operators. For these issues, a two-stage MCDM method is constructed in the IPF environment. In the first stage, a novel method for determining the weights of experts is proposed by introducing IPF set (IPFS) into social networks. To do that, the concepts of trust function (TF) and trust score (TS) in the IPF environment are defined to obtain the objective weights of experts. Meanwhile, the subjective weights of experts are obtained from the number of experts. Afterward, the objective weight and subjective weight of each expert are combined to derive the weight of each expert. In the second stage, a novel weighted sum model (WSM) with novel IPF aggregation operators is constructed to rank alternatives. Considering the psychological behavior of experts, that is, self-confidence level, four IPF aggregation operators with self-confidence levels are defined, namely, the self-confidence interval-valued Pythagorean fuzzy weighted averaging (SC-IPFWA) and ordered weighted averaging (SC-IPFOWA) operator, the self-confidence interval-valued Pythagorean fuzzy weighted geometric (SC-IPFWG) and ordered weighted geometric (SC-IPFOWG) operator. Finally, a numerical case is used to verify the effectiveness of the proposed two-stage MCDM method.

Approaches to Some Induced Einstein Geometric Aggregation Operators Based on Interval-Valued Pythagorean Fuzzy Numbers and Their Application

Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper we introduce the notion of induced interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (I-IVPFEOWG) operator and induced interval-valued Pythagorean fuzzy Einstein hybrid geometric (I-IVPFEHG) operator along with their some basic properties such as, idempotency, boundedness, commutatively, monotonicity. The main advantage of the induced aggregation operators is that, these operators are more suitable to aggregate the individual preference relations into a collective preference relation. Therefore these methods play a vigorous role in daily life problems. Furthermore, a method for multi-attribute group decision-making (MAGDM) problems based on these operators was developed, and the operational procedures were explained in detail.

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

A novel outranking method for multiple criteria decision making with interval-valued Pythagorean fuzzy linguistic information

This paper aim to propose a novel outranking method based on interval-valued Pythagorean fuzzy linguistic set (IVPFLS) to estimate alternatives for decision makers. With the rapid increase of the complexity, the challenges of a group of decision makers, a high degree of uncertainty and conflicting criteria are associated with multiple criteria decision making (MCDM) problems and few studies for these. In this regard, the outranking-VIKOR (O-VIKOR) method with the construction of compromise solution considers “group utility” and “individual regret” in pairwise alternatives. To fully show the compromise outranking relation, the O-VIKOR method allows decision makers reach consensus by mutual concessions when the dominant position from alternatives is extant. Subsequently, we define the credibility formula that constitutes a complementary judgment matrix for devoting to ranking the alternatives more precision. When it is necessary to process the evaluation information of multiple decision makers, the interval-value Pythagorean fuzzy linguistic entropic induced ordered weighted averaging (IVPFLEIOWA) operator is proposed with entropic order-inducing variable which includes a new interval-value Pythagorean fuzzy linguistic entropy for measuring uncertainty to capture the interrelationship among the primary preference of decision makers. Finally, a case study on site selection problem for a manufacturer is presented to illustrate the efficiency of our proposed method and make comparative analysis with other methods.

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.

Interval-valued pythagorean fuzzy rough approximation operators and its application

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model.

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.

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.

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.

Interval-valued pythagorean fuzzy power average-based MULTIMOORA method for multi-criteria decision-making

ABSTRACT In the multi-criteria decision-making (MCDM), multi-objective optimisation by ratio analysis (MOORA) plus the full multiplicative form (MULTIMOORA) are a useful method, which has the flexibility and robustness properties. The aim of this paper is to present a more robust method of MULTIMOORA to solve the MCDM evaluations with interval-valued Pythagorean fuzzy sets (IVPFSs). More specifically, considering the mutual support relationship among the input arguments, we firstly extend the power average (PA) and power geometric (PG) operators to accommodate the interval-valued Pythagorean fuzzy environment. For this, we propose four aggregation operators, including the interval-valued Pythagorean fuzzy power average (IVPFPA), the weighted interval-valued Pythagorean fuzzy power average (WIVPFPA), the interval-valued Pythagorean fuzzy power geometric (IVPFPG) and the weighted interval-valued Pythagorean fuzzy power geometric (WIVPFPG). With the aid of WIVPFPA and WIVPFPG, we further put forward a new robustness of the ranking method, i.e., the interval-valued Pythagorean fuzzy power average-based MULTIMOORA (IVPFPA-MULTIMOORA). Due to the availability and vast diversity among open-source electronic health record (EHR) systems, choosing an appropriate system is a challenge for many health-care institutions under the interval-valued Pythagorean fuzzy environment. Based on these above-mentioned research findings, we successfully utilise the IVPFPA-MULTIMOORA method to solve the selection problem of hospital open-source EHRs systems for MedLab in Ghana. Meanwhile, the effectiveness of the proposed method is verified by some comparative analyses.

D-WASPAS: Addressing Social Cognition in Uncertain Decision-Making with an Application to a Sustainable Project Portfolio Problem

Decision-making is an interdisciplinary area that has roots in mathematics, economics, and social science. Multiple-criteria group decision-making (MCGDM) is one of the most applicable areas of decision-making. Social cognition is involved in group decision-making. Therefore, it is necessary to address how decision makers (DMs) process and apply judgments and information during the process. In recent years, many approaches have been applied to MCGDM. As an important aspect of this process, uncertainty has led to the application of fuzzy sets. However, utilizing various decision-making approaches can result in different results and confusion among DMs. Moreover, using classic fuzzy sets and expressing degrees of belonging by crisp values has proven to be inadequate for uncertain decision-making environments. This paper presents a novel MCGDM approach, double-weighted aggregated sum product assessment (D-WASPAS), under interval-valued Pythagorean fuzzy (IVPF) uncertainty. The proposed approach applies knowledge measures to address the objective weights of criteria. Then, subjective and objective weights of criteria are aggregated to create a more appropriate weight. This approach considers three decision-making methods. In the first, an IVPF-ARAS (additive ratio assessment) method is extended to rank the alternatives. In the second, an IVPF-EDAS (evaluation based on distance from average solution) method is developed to rank the alternatives. In the third, a novel IVPF-COADAP (complex adequate appraisal) method is utilized for a third ranking. To aggregate the results, two steps are carried out using the WASPAS method. First, the results of the ranking approaches are aggregated. This process starts with computing the objective weights of the ranking approaches and aggregating the outcome with the subjective weights of the approaches. Then, the WASPAS method is applied to aggregate the obtained rankings and obtain a set of rankings for each DM. The second aggregation is utilized to aggregate the results for the DMs and reach a final set of rankings. Similarly, the subjective and objective weights of the DMs are applied in the WASPAS to aggregate the results. It should be noted that since the WASPAS method is utilized twice to aggregate the results, this approach is called D-WASPAS. A case study of the application of the proposed method shows that it is applicable to many multiple-criteria analysis and decision-making processes. Moreover, the results are more reliable because various decision-making methods are taken into consideration, and it is a last-aggregation process. Double-weighted aggregated sum product assessment offers a novel decision-making framework that is applicable in real-world decision-making situations. The proposed method is based on interval-valued Pythagorean fuzzy sets (IVPFSs), which would be especially applicable to uncertain situations. Also, it would enhance calculations of the process by offering more flexibility in dealing with uncertainty. Consequently, introducing this new decision-making framework and applying extended fuzzy sets would make the proposed method more widely applicable. The last-aggregation nature of this method avoids loss of cognitive information and assigning weights to the DMs, and the different ranking methods address the social cognition that leads to the judgments expressed and the final decisions.

Evaluating large, high-technology project portfolios using a novel interval-valued Pythagorean fuzzy set framework: An automated crane project case study

The contemporary organization relies increasingly on developing large, high technology projects in order to gain local and global competitive advantage. Uncertainty and the complexity of project evaluation requires improved and tailored decision making support systems. A new framework for high technology project portfolio evaluation is introduced. Novel development of an interval-valued Pythagorean fuzzy set (IVPFS) approach is shown to accommodate degrees of membership, non-membership and hesitancy in the evaluation process. Developed methods of linear assignment, IVPFS ranking, IVPFS knowledge index, and IVPFS comparison provide a new framework for group evaluation based on a weighting for each decision expert. The framework is developed as a last aggregation which avoids information loss and introduces a new aggregation process. A novel multi-objective model is then introduced to address project portfolio selection while optimizing the value of the portfolio in terms of resilience (the risk of disruption and delays) and skill utilization (assignment of human resources). The applicability of this framework is demonstrated through a case study in high technology portfolio evaluation. The case study shows that the presented framework can be applied as the core to a high technology evaluation decision support system.

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.

Distance measures for cubic Pythagorean fuzzy sets and its applications to multicriteria decision making

The main objective of this paper is to develop a sophisticated mathematical expression which can carry much more information than the general intuitionistic fuzzy set (IFS), interval-valued intuitionistic fuzzy set (IVIFS) and cubic intuitionistic fuzzy set (CIFS). CIFS is one of the powerful tools to handle uncertainty in complex situation. It is the simultaneous consideration of both the IVIFS and IFS. As in many real life situation, interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) are more capable than IVIFS and IFS to represent the vagueness or ill-defined information; therefore, it motivates us to enhance the capability of CIFS in complex decision-making problems. This paper presents a novel notion of cubic Pythagorean fuzzy set (CPFS) incorporating IVPFS and PFS simultaneously, to encounter uncertainty in a more specific manner. Furthermore, a family of distance measures for CPFSs is defined and applications of the proposed distance measures are shown in medical decision-making problem.

A Pearson-like correlation-based TOPSIS method with interval-valued Pythagorean fuzzy uncertainty and its application to multiple criteria decision analysis of stroke rehabilitation treatments

This paper extends one of the most extensively used multiple criteria decision analysis (MCDA) methods, the technique for order preference by similarity to ideal solutions (TOPSIS), to adapt to highly complicated uncertain environments based on interval-valued Pythagorean fuzzy (IVPF) sets. In contrast to classical TOPSIS methods, this paper develops a novel concept of Pearson-like IVPF correlation coefficients instead of distance measures to not only construct a useful and effective association measure between two IVPF characteristics but also depict the outranking relationship of IVPF information. Moreover, this paper proposes the (weighted) IVPF correlation-based closeness coefficients to establish a Pearson-like correlation-based TOPSIS model to manage MCDA problems within the IVPF environment. In particular, there is a definite improvement in determining the closeness coefficient required in the TOPSIS procedure. This paper considers anchored judgments with respect to the positive- and negative-ideal IVPF solutions and provides new approach- and avoidance-oriented definitions for the IVPF correlation-based closeness coefficient, which is entirely different from the traditional definition of relative closeness in TOPSIS. Furthermore, this paper proposes a comprehensive IVPF correlation-based closeness index to balance the consequences between ultra-approach orientation and ultra-avoidance orientation and acquire the ultimate compromise solution for decision support and aid. The feasibility and practicability of the developed methodology are illustrated by a practical MCDA problem of rehabilitation treatment for hospitalized patients with acute stroke. The application results, along with experimentations and comparative analyses, demonstrate that the developed methods are rational and effective.

Simplified interval-valued Pythagorean fuzzy graphs with application

Interval-valued Pythagorean fuzzy set (IVPFS) as a generalization of Pythagorean fuzzy set (PFS) increases its elasticity drastically. However, the expressions and calculations of IVPFS are slightly complicated. To overcome this drawback, in this research study, we greatly simplify the expressions of IVPFS by introducing a new concept of simplified interval-valued Pythagorean fuzzy set (SIVPFS), constituted by two Pythagorean fuzzy numbers (PFNs) with the relationships of intersection and union simultaneously. We develop systematic aggregation operators to aggregate simplified interval-valued Pythagorean fuzzy information. Meanwhile, we propose a new generalization of fuzzy graph, called simplified interval-valued Pythagorean fuzzy graph (SIVPFG), to describe uncertain information in graph theory. We develop a series of operations on two SIVPFGs and investigate their desirable properties. Finally, we develop a SIVPFG-based multi-agent decision-making approach to solve a common kind of situation where the graphic structure of agents is obscure. A numerical example is provided to illustrate the proposed approach as well as the applicability of SIVPFS and SIVPFG in decision making.

A novel interval-valued Pythagorean fuzzy QFD method and its application to solar photovoltaic technology development

Quality function deployment (QFD) is a focused methodology for carefully listening to the voice of the customer and then effectively responding to those needs or customer requirements, (CRs) and expectations through design requirements (DRs). In classical QFD, exact numbers are used to determine the priorities of the CRs and the position of the company among the competitors. However, vagueness and impreciseness are inevitable uncertainties in those kinds of human evaluations, which are generally realized by linguistic terms. In this paper, the uncertainty in design processes is captured by a relatively new extension of ordinary fuzzy sets, Pythagorean fuzzy sets (PFS), aiming at presenting a larger domain to experts for assigning a membership degree and a non-membership degree together with their hesitancy. We perform all the evaluation processes in the house of quality (HOQ) based on interval-valued PFS (IVPFS) and present some novel definitions for measuring and prioritizing CRs, DRs, and determining the position of the company among the competitors. We give an application of the proposed IVPF-HOQ model for solar photovoltaic technology development.

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.