Pythagorean Fuzzy Aggregation Operators Research Articles

Multi-Criteria Decision-Making with Novel Pythagorean Fuzzy Aggregation Operators

Unpredictability and fuzziness coexist in decision-making analysis due to the complexity of the decision-making environment. “Pythagorean fuzzy numbers” (PFNs) outperform “intuitionistic fuzzy numbers” (IFNs) when dealing with unclear data. The “Pythagorean fuzzy set” (PFS) is a useful tool because it removes the restriction that the sum of membership degrees be less than or equal to one by substituting the square sum for the sum of membership degrees. This study proposes two aggregating operators (AOs). The recommended operators outperform the already specified PFN operators. The proposed operator is utilised in the multicriteria decision-making process to identify the best candidate for instruction (MCDM).

Enhancing Interval-Valued Pythagorean Fuzzy Decision-Making through Dombi-Based Aggregation Operators

The success of any endeavor or process is heavily contingent on the ability to reconcile and satisfy balance requirements, which are often characterized by symmetry considerations. In practical applications, the primary goal of decision-making processes is to efficiently manage the symmetry or asymmetry that exists within different sources of information. In order to address this challenge, the primary aim of this study is to introduce novel Dombi operation concepts that are formulated within the framework of interval-valued Pythagorean fuzzy aggregation operators. In this study, an updated score function is presented to resolve the deficiency of the current score function in an interval-valued Pythagorean fuzzy environment. The concept of Dombi operations is used to introduce some interval-valued Pythagorean fuzzy aggregation operators, including the interval-valued Pythagorean fuzzy Dombi weighted arithmetic (IVPFDWA) operator, the interval-valued Pythagorean fuzzy Dombi ordered weighted arithmetic (IVPFDOWA) operator, the interval-valued Pythagorean fuzzy Dombi weighted geometric (IVPFDWG) operator, and the interval-valued Pythagorean fuzzy Dombi ordered weighted geometric (IVPFDOWG) operator. Moreover, the study investigates many important properties of these operators that provide new semantic meaning to the evaluation. In addition, the suggested score function and newly derived interval-valued Pythagorean fuzzy Dombi aggregation (IVPFDA) operators are successfully employed to select a subject expert in a certain institution. The proposed approach is demonstrated to be successful through empirical validation. Lastly, a comparative study is conducted to demonstrate the validity and applicability of the suggested approaches in comparison with current techniques. This research contributes to the ongoing efforts to advance the field of evaluation and decision-making by providing novel and effective tools and techniques.

A Novel SIR Approach to Closeness Coefficient-Based MAGDM Problems Using Pythagorean Fuzzy Aczel–Alsina Aggregation Operators for Investment Policy

In this study, a novel Pythagorean fuzzy aggregation operator is presented by combining the concepts of Aczel–Alsina ( A A ) T-norm and T-conorm operations for multiple attribute group decision-making (MAGDM) challenge for the superiority and inferiority ranking (SIR) approach. This approach has many advantages in solving real-life problems. In this study, the superiority and inferiority ranking method is illustrated and showed the effectiveness for decision makers by using multicriteria. The Aczel–Alsina aggregation operators on interval-valued IFSs, hesitant fuzzy sets (HFSs), Pythagorean fuzzy sets (PFSs), and T-spherical fuzzy sets (TSFSs) for multiple attribute decision-making (MADM) issues have been proposed in the literature. In addition, we propose a Pythagorean fuzzy Aczel–Alsina weighted average closeness coefficient ( PF − A A − WA − C C ) aggregation operator on the basis of the closeness coefficient for MAGDM challenges. To highlight the relevancy and authenticity of this approach and measure its validity, we conducted a comparative analysis with the method already in vogue.

Generalized Cubic Pythagorean Fuzzy Aggregation Operators and their Application to Multi-attribute Decision-Making Problems

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can hold much more information and can be used to describe both an interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) at the same time to handle data uncertainties. Based on it, the present study is classified into three phases. The first phase is to modify the existing operational laws and aggregation operators (AOs) in the article presented by Abbas et al. (Journal of Intelligent & Fuzzy Systems, vol. 37, no. 1, pp. 1529–1544, (2019)). The main objective of improved operational laws is to eliminate the flows and ambiguities in existing AOs. Secondly, based on these laws, various AOs to aggregate the information are acquired along with their requisite properties and relations. Lastly, an approach for interpreting the multi-attribute decision-making (MCDM) problem based on the stated operators is given and illustrated with an example. Some of the existing models are used to perform a comprehensive comparative analysis to demonstrate their impacts.

Social network multiple-criteria decision-making approach for evaluating unmanned ground delivery vehicles under the Pythagorean fuzzy environment

With the rapid development of instant delivery, the shrinking labor population and prevailing contact-free economy, companies have launched unmanned ground delivery vehicles (UGDVs) to replace human distribution with machines. To meet the requirements for selecting UGDVs and achieve better applications in community delivery, a multi-criteria decision-making (MCDM) framework, combining the self-confidence aggregation approach and social trust network, is proposed in this study. Based on the internal characteristics of UGDVs, a multi-criteria comprehensive evaluation system for UGDVs is constructed. Then, a trust propagation and aggregation mechanism to yield expert weights based on a social trust network is suggested. Further, a self-confidence Pythagorean fuzzy aggregation operator is proposed to enhance the credibility of the decision results and compensate for the defects of existing methods. Finally, a practical case is considered to demonstrate the complete process of the MCDM model and to conduct a comparative analysis and sensitivity analysis of the model.

Multi-attribute decision making method using advanced Pythagorean fuzzy weighted geometric operator and their applications for real estate company selection

In this paper, a novel multi-attribute decision-making method using Advanced Pythagorean fuzzy weighted geometric operator in a Pythagorean fuzzy environment is developed. Pythagorean fuzzy aggregation operators have drawbacks that they give indeterminate results in some special cases when membership value or non-membership value gets 0 value or 1 value and the weight vector is of type (1,0)T or (0,1)T. The Advanced Pythagorean fuzzy geometric operator, the proposed operator can overcome the drawbacks. In some situations, for example, where the sum of squares of membership degree and non-membership degree gets unit value of a Pythagorean fuzzy number, multi-attribute decision making (MADM) methods using some existing aggregation operators give unreasonable ranking orders (ROs) of alternatives or can't discriminate the ROs of alternatives. But the present MADM method can get over the drawbacks of the existing MADM methods. The present MADM method is devoted to eliminate the drawbacks of the existing MADM methods and to select the best real estate company for investment.

Sine trigonometric operational laws and its based Pythagorean fuzzy aggregation operators for group decision-making process

The paper aims are to impersonate some robust sine-trigonometric operations laws to determine the group decision-making process under the Pythagorean fuzzy set (PFS) situation. The PFS has a notable feature to trade with the dubious information with a broader membership representation space than the intuitionistic fuzzy set. Based on it, the present paper is classified into three phases. The first phase is to introduce new operational laws for PFS. The main idea behind these proposed operations is to incorporate the qualities of the sine function, namely periodicity and symmetric about the origin towards the decisions of the objects. Secondly, based on these laws, numerous operators to aggregate the information are acquired along with their requisite properties and relations. Finally, an algorithm to interpret the multiattribute group decision making problem is outlined based on the stated operators and manifest it with an illustrative example. A detailed comparative interpretation is achieved with some of the existing methods to reveal their influences.

Some novel interval-valued Pythagorean fuzzy aggregation operator based on Hamacher triangular norms and their application in MADM issues

The motivation of this paper is to build up a new technique to manage multi-attribute decision-making (MADM) issues with interval-valued Pythagorean fuzzy (IVPF) set according to the concepts of Hamacher t-norm and t-conorm, and give its application in MADM problems. To begin with, we integrate the Hamacher operations on IVPF sets. Then, in view of these operations, we formulate a few IVPF aggregation operators, for example, IVPF Hamacher weighted average operator, IVPF Hamacher order weighted average operator, IVPF Hamacher hybrid average operator, IVPF Hamacher weighted geometric operator, IVPF Hamacher order weighted geometric operator, and IVPF Hamacher hybrid geometric operator. We investigate the distinguished features of these suggested operators. Moreover, we carry out the proposed aggregation operators to produce a method for solving MADM issues under IVPF data. We present an example of the emerging software system selection to elaborate on its practicality and effectiveness. We investigate the impact of the parameter for different values on decision-making outcomes. The main advantage of utilizing the suggested operator lies in the fact that this operator provides an increasingly complete perspective on the issue to the decision-makers. The technique suggested in this study gives progressively broad, improve the accuracy and exact outcomes when contrasted with the existing related methods. Therefore, this technique performs an important function in real-life MADM issues.

Complex pythagorean fuzzy aggregation operators based on confidence levels and their applications.

The most important influence of this assessment is to analyze some new operational laws based on confidential levels (CLs) for complex Pythagorean fuzzy (CPF) settings. Moreover, to demonstrate the closeness between finite numbers of alternatives, the conception of confidence CPF weighted averaging (CCPFWA), confidence CPF ordered weighted averaging (CCPFOWA), confidence CPF weighted geometric (CCPFWG), and confidence CPF ordered weighted geometric (CCPFOWG) operators are invented. Several significant features of the invented works are also diagnosed. Moreover, to investigate the beneficial optimal from a large number of alternatives, a multi-attribute decision-making (MADM) analysis is analyzed based on CPF data. A lot of examples are demonstrated based on invented works to evaluate the supremacy and ability of the initiated works. For massive convenience, the sensitivity analysis and merits of the identified works are also explored with the help of comparative analysis and they're graphical shown.

Fuzzy decision support modeling for internet finance soft power evaluation based on sine trigonometric Pythagorean fuzzy information

The Pythagorean fuzzy set (PFS) is one of the most important concepts to accommodate more uncertainties than the intuitionistic fuzzy sets, fuzzy sets and hence its applications are more extensive. The well-known sine trigonometric function ensures the periodicity and symmetry of the origin in nature and thus satisfies the expectations of decision-makers over the parameters of the multi-time process. Keeping the features of sine function and the importance of the PFS, introduce the novel sine trigonometric operational laws (STOLs) under Pythagorean Fuzzy Settings. In addition, novel sine-trigonometric Pythagorean fuzzy aggregation operators are established based on these STOLs. The core of the study is the decision-making algorithm for addressing multi-attribute decision-making problems based on the proposed aggregation operators with unknown weight information of the given criteria. Finally, an illustrative example on internet finance soft power evaluation is provided to verify the effectiveness. Sensitivity and comparative analyses are also implemented to assess the stability and validity of our method.

Induced generalized pythagorean fuzzy aggregation operators and their application based on t-norm and t-conorm

In this paper, we present some new operators, namely induced Pythagorean fuzzy Einstein ordered weighted geometric operator, induced Pythagorean fuzzy Einstein hybrid geometric operator, induced generalized Pythagorean fuzzy Einstein ordered weighted geometric operator, induced generalized Pythagorean fuzzy Einstein hybrid geometric operator. Moreover, these operators are applied to decision-making, in which the specialists of the problems deliver their favorites to display the practicality, and value of the new presented work. We also study the applicability in a decision-making concerning strategic choice of the best information system and give an example to show the effectiveness of the developed methods.

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.

Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making

<p style='text-indent:20px;'>Although intuitionistic fuzzy preference relations have become powerful techniques to express the decision makers' preference information over alternatives or criteria in group decision making, some limitations of them are pointed out in this paper, then they are overcame by developed the group decision making with Pythagorean fuzzy preference relations (PFPRs). Specially, we provide a partial order on the set of all the PFPRs, based on which, a deviation measure is defined. Then, we check and reach the acceptably multiplicative consistency and consensus of PFPRs associated with the partial order and mathematical programming. Concretely, acceptably multiplicative consistent PRPRs are defined by the deviation between a given PFPR and a multiplicative consistent PFPR constructed by a normal Pythagorean fuzzy priority vector. Then acceptable consensus of a collection of PFPRs is defined by the deviation of each PFPR and the aggregated result from symmetrical Pythagorean fuzzy aggregation operators. Based on which, a method which can simultaneously modify the unacceptable consistency and consensus of PFPRs in a stepwise way is provided. Particularly, we also prove that the collective PFPR obtained by aggregating several individual acceptably consistent PFPRs with various symmetric aggregation operators is still acceptably consistent. Then, a procedure is provided to solve group decision making with PRPRs and a numerical example is given to illustrate the effectiveness of our method.

Group Generalized Pythagorean Fuzzy Aggregation Operators and Their Application in Decision Making

Fuzzy information is generally represented by fuzzy set (FS). Pythagorean fuzzy set (PFS), as a new extension of FS, can represent fuzzy information more effectively. The accuracy of fuzzy information represented by PFS directly affects the result of information processing. Generalized parameter (GP), expressed by Pythagorean fuzzy number, can judge the accuracy of fuzzy information represented by PFS. However, for complex fuzzy application problems, a single GP cannot comprehensively judge the fuzzy information, and the GP may also be wrong. Therefore, the concept of group generalized parameters (GGPs) is proposed. The GGPs can comprehensively judge the accuracy of fuzzy information, and can avoid the influence caused by the error of a certain GP. In order to effectively aggregate the fuzzy information and the GGPs, some new group generalized Pythagorean fuzzy aggregation operators are proposed, including group generalized Pythagorean fuzzy weighted average (GGPFWA) operator and group generalized Pythagorean fuzzy weighted geometric (GGPFWG) operator. The definitions, theorems and properties of GGPFWA operator and GGPFWG operator are given and proved. In this paper, GGPFWA operator and GGPFWG operator are applied to solve decision making problems, and their effectiveness and feasibility have been verified by three cases of pattern recognition, medical diagnosis and multiple criteria decision making.

Interval valued Pythagorean fuzzy aggregation operators based Malcolm Baldrige National Quality Award assessment

Malcolm Baldrige National Quality Award (MBNQA) is a quality assessment and rewarding system that aims to increase the awareness of quality management. Although the award is launched in the USA in 1989 and only given to the U.S based companies, it is recognized internationally. There are 7 types of categories in the award system (Leadership, Strategic planning, Customer focus, Measurement, analysis, and knowledge management, Workforce focus, Process management, and Results) where the evaluation is made over 1000 points and each category has its own weight. Since almost all the publications in the literature are based on crisp measurements and evaluations of the system performances, we proposed a multi attribute decision making (MADM) method using interval valued Pythagorean fuzzy weighted averaging (IVPFWA) and interval valued Pythagorean fuzzy weighted geometric (IVPFWG) aggregation operators for MBNQA assessment to represent the decision makers’ subjective evaluations better. A comparison of the results with Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method and an illustrative example are presented in the study.

Some new generalized interval-valued Pythagorean fuzzy aggregation operators using einstein t-norm and t-conorm

Some new generalized interval-valued Pythagorean fuzzy aggregation operators using einstein t-norm and t-conorm

Neutrality operations-based Pythagorean fuzzy aggregation operators and its applications to multiple attribute group decision-making process

Pythagorean fuzzy sets accommodate more uncertainties than the intuitionistic fuzzy sets and hence it is one of the most important concepts to describe the fuzzy information in the process of decision making. Under this environment, the main objective of the work is to develop some new operational laws and their corresponding weighted aggregation operators. For it, we define some new neutral addition and scalar multiplication operational laws by incorporating the features of a neutral character towards the membership degrees of the set and the probability sum. Some properties of the proposed laws are investigated. Then, associated with these operational laws, we define some novel Pythagorean fuzzy weighted, ordered weighted and hybrid neutral averaging aggregation operators for Pythagorean fuzzy information, which can neutrally treat the membership and non-membership degrees. The various relations and the characteristics of the proposed operators are discussed. Further, in order to ease with the possible application, we present an algorithm to solve the multiple attribute group decision-making problems under the Pythagorean fuzzy environment. Finally, a practical example is provided to illustrate the approach and show its superiority, advantages by comparing their performance with some several existing approaches.

Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making

Continuous interval-valued Pythagorean fuzzy aggregation operators for multiple attribute group decision making

The Pythagorean fuzzy Einstein Choquet integral operators and their application in group decision making

Since Pythagorean fuzzy set is a powerful tool as it relaxing the condition that the sum of membership degrees is less than or equal to one with the square sum is less than or equal to one. Also Choquet integral is a very useful way of measuring the expected utility of an uncertain event. Therefore, in this paper we use the Choquet integral to develop Pythagorean fuzzy aggregation operators, namely Pythagorean fuzzy Einstein Choquet integral averaging operator and Pythagorean fuzzy Einstein Choquet integral geometric operator. The operators not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements or their ordered positions. It must be noted that several existing operators are the special cases of the developed operators. Further the properties such as boundedness, monotonicity and idempotency of the proposed operators have been studied in detail. Furthermore based on the developed operators a multi-criteria group decision-making method has been presented. Finally an illustrative example is presented to illustrate the validity and effectiveness of the proposed method.

Multiattribute Group Decision-Making Approach With Linguistic Pythagorean Fuzzy Information

The purpose of this study is to construct the multi-attribute group decision making (MAGDM) approach with linguistic Pythagorean fuzzy information (LPFI) based on generalized linguistic Pythagorean fuzzy aggregation operators (GLPFA). To begin with, we define the generalized indeterminacy degree-preference distance of linguistic Pythagorean fuzzy numbers (LPFNs), on the basis of it, we build a new approach for ranking the alternatives after analysing the existed comparison rule. In addition, we introduce the new version of t-norms (TNs) and t-conorms (TCs) named linguistic Pythagorean t-norms (LPTNs) and linguistic Pythagorean t-conorms (LPTCs), which can be used to handle the LPFI; some special cases for LPTNs and LPTCs are obtained and they can deal with Pythagorean fuzzy information (PFI). Thirdly, we introduce the generalized linguistic Pythagorean fuzzy average aggregation operator (GLPFAA) based on LPTN and LPTC along with their properties are also investigated, whilst, some special cases of GLPAA are obtained when LPTN and LPTC take some special TNs and TCs. Finally, a MAGDM approach based on some LPTNs and LPTCs is constructed to deal with some MAGDM problems with unknown attributes'weights and experts' weights, before building the MAGDM approach, we define new cross-entropy to fix the experts's weights and use the maximizing deviation to calculate the attributes' weights based on the proposed indeterminacy degree-preference distance. Consequently, an illustrative example is provided in order to show the effectiveness and advantages of the proposed method and some comparisons are also carried out.