Pythagorean Fuzzy Numbers Research Articles

Distance Measure of Pythagorean Fuzzy Sets

The main feature of Pythagorean fuzzy sets is that it is characterized by four parameters, namely membership degree, nonmembership degree, strength of commitment about membership, and direction of commitment. In this paper, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the four parameters of Pythagorean fuzzy sets. Finally, a numerical example is provided to illustrate the validity and applicability of the presented distance measures.

Pythagorean Fuzzy Clustering Analysis: A Hierarchical Clustering Algorithm with the Ratio Index-Based Ranking Methods

The Pythagorean fuzzy set introduced by R. R. Yager in 2014 is a useful tool to model imprecise and ambiguous information appearing in decision and clustering problems. In this study, we present a general type of distance measure for Pythagorean fuzzy numbers (PFNs) and propose a novel ratio index-based ranking method of PFNs. The novel ranking method of PFNs has more powerful ability to discriminate the magnitude of PFNs than the existing ranking methods for PFNs, which is further extended to compare the magnitude of interval-valued Pythagorean fuzzy numbers (IVPFNs). The IVPFN is a new extension of PFN, which is parallel to interval-valued intuitionistic fuzzy number. We introduce a general type of distance measure for IVPFNs. Afterwards, we study a kind of clustering problems in Pythagorean fuzzy environments in which the evaluation values are expressed by PFNs and/or IVPFNs and develop a novel Pythagorean fuzzy agglomerative hierarchical clustering approach. In the proposed clustering method, we define the concept of the dissimilarity degree between two clusters for each criterion and introduce the clustering procedure in the criteria level. To take all the criteria into account, we also introduce the overall clustering procedure, which is based on the overall dissimilarity degrees for a fixed aggregation operator such as the commonly used weighted arithmetic average operator or the ordered weighted averaging operator. In the overall clustering process, (1) we present a deviation degree-based method to derive the weights of criteria and further obtain the overall clustering results if the weights of criteria are completely unknown; (2) we employ the ratio index-based ranking method of IVPFNs to obtain the overall clustering results if the weights of criteria are given in advance and are expressed by IVPFNs. The salient feature of the proposed clustering method is that it not only can address the clustering problems in which the weights of criteria are not given precisely in advance but also can manage simultaneously the PFNs and IVPFNs data.

Multiple Criteria Decision Making with Probabilities in Interval-Valued Pythagorean Fuzzy Setting

Interval-valued Pythagorean fuzzy set (IVPFS), as an extension of interval-valued intuitionistic fuzzy set (IVIFS) and Pythagorean fuzzy set to deal with uncertainty, has attracted much attention since its introduction, in both theory and application aspects. The present work aims at investigating new distance measures in the IVPFSs and then employing them into multiple criteria decision-making application. To begin with, generalized interval-valued Pythagorean fuzzy weighted distance measure and generalized interval-valued Pythagorean fuzzy ordered weighted distance measure are firstly introduced in the IVPFSs. Afterward, we propose generalized probabilistic interval-valued Pythagorean fuzzy weighted averaging distance (P-GIVPFWAD) operator, generalized probabilistic interval-valued Pythagorean fuzzy order weighted averaging distance (P-GIVPFOWAD) operator and immediate generalized probabilistic interval-valued Pythagorean fuzzy ordered weighted averaging distance (IP-GIVPFOWAD) operator which are new distance measures and are able to integrate the ordered weighted averaging operator, probabilistic weight and individual distance of two interval-valued Pythagorean fuzzy numbers (IVPFNs) in the same formulation. These distance measures are very suitable to deal with the situation where the input data are represented in IVPFNs. Then we present a kind of multiple criteria decision-making method with interval-valued Pythagorean fuzzy information based on the developed distance measures. Finally, a numerical example is provided to explain the feasibility of the proposed method and the validity of the developed method is also analyzed according to the validity criterion of multiple criteria decision making.

Approaches to Pythagorean Fuzzy Stochastic Multi-criteria Decision Making Based on Prospect Theory and Regret Theory with New Distance Measure and Score Function

In this paper, we initiate a new axiomatic definition of Pythagorean fuzzy distance measure, which is expressed by Pythagorean fuzzy number that will reduce the information loss and remain more original information. Then, the objective weights of various criteria are determined via grey system theory. Combining objective weights with subjective weights, we present the combined weights, which can reflect both the subjective considerations of the decision maker and the objective information. Meanwhile, a novel score function is proposed. Later, we present two algorithms to solve stochastic multicriteria decision making problem, which takes prospect preference and regret aversion of decision makers into consideration in the decision process. Finally, the effectiveness and feasibility of approach is demonstrated by a numerical example.

Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process

Pythagorean fuzzy set, an extension of the intuitionistic fuzzy set which relax the condition of sum of their membership function to square sum of its membership functions is less than one. Under these environment and by incorporating the idea of the confidence levels of each Pythagorean fuzzy number, the present study investigated a new averaging and geometric operators namely confidence Pythagorean fuzzy weighted and ordered weighted operators along with their some desired properties. Based on its, a multi criteria decision-making method has been proposed and illustrated with an example for showing the validity and effectiveness of it. A computed results are compared with the aid of existing results.

Some Induced Averaging Aggregation Operators Based on Pythagorean Fuzzy Numbers

In this paper we present two new types aggregation operators such as, induced Pythagorean fuzzy ordered weighted averaging aggregation operator and induced Pythagorean fuzzy hybrid averaging aggregation operator. We also discuss of important properties of these proposed operators and construct some examples to develop these operators.

Airline Passenger Profiling Based on Fuzzy Deep Machine Learning

Passenger profiling plays a vital part of commercial aviation security, but classical methods become very inefficient in handling the rapidly increasing amounts of electronic records. This paper proposes a deep learning approach to passenger profiling. The center of our approach is a Pythagorean fuzzy deep Boltzmann machine (PFDBM), whose parameters are expressed by Pythagorean fuzzy numbers such that each neuron can learn how a feature affects the production of the correct output from both the positive and negative sides. We propose a hybrid algorithm combining a gradient-based method and an evolutionary algorithm for training the PFDBM. Based on the novel learning model, we develop a deep neural network (DNN) for classifying normal passengers and potential attackers, and further develop an integrated DNN for identifying group attackers whose individual features are insufficient to reveal the abnormality. Experiments on data sets from Air China show that our approach provides much higher learning ability and classification accuracy than existing profilers. It is expected that the fuzzy deep learning approach can be adapted for a variety of complex pattern analysis tasks.

Symmetric Pythagorean Fuzzy Weighted Geometric/Averaging Operators and Their Application in Multicriteria Decision-Making Problems

Pythagorean fuzzy sets PFSs, originally proposed by Yager, are a new tool to deal with vagueness with the square sum of the membership degree and the nonmembership degree equal to or less than 1, which have much stronger ability than Atanassov's intuitionistic fuzzy sets to model such uncertainty. In this paper, we modify the existing score function and accuracy function for Pythagorean fuzzy number to make it conform to PFSs. Associated with the given operational laws, we define some novel Pythagorean fuzzy weighted geometric/averaging operators for Pythagorean fuzzy information, which can neutrally treat the membership degree and the nonmembership degree, and investigate the relationships among these operators and those existing ones. At length, a practical example is provided to illustrate the developed operators and to make a comparative analysis.

A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making

The Pythagorean fuzzy set, as a new extension of intuitionistic fuzzy set, has recently been developed to manage the complex uncertainty in practical group decision problems. The purpose of this article is to develop a new decision method based on similarity measure to address multiple criteria group decision making problems within Pythagorean fuzzy environment based on Pythagorean fuzzy numbers (PFNs). The contribution of this article is fivefold: (1) An accuracy function of PFNs is defined and a new ranking method for PFNs is proposed; (2) new Pythagorean fuzzy aggregating operators are developed; (3) a novel similarity measure for PFNs is presented, and some desirable properties are discussed; (4) a simple and effective Pythagorean fuzzy group decision method is introduced; and (5) The proposed method is applied to address the selection problem of photovoltaic cells.

The Properties of Continuous Pythagorean Fuzzy Information

In practical decision-making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support membership degree and the against nonmembership degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets PFS, as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers PFNs, which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail.

Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods

Pythagorean fuzzy set initially developed by Yager (2014) is a new tool to model imprecise and ambiguous information in multicriteria decision making problems. In this paper, we propose a novel closeness index for Pythagorean fuzzy number (PFN) and also introduce a closeness index-based ranking method for PFNs. Next, we extend the Pythagorean fuzzy set to present the concept of interval-valued Pythagorean fuzzy set (IVPFS) which is parallel to interval-valued intuitionistic fuzzy set. The elements in IVPFS are called interval-valued Pythagorean fuzzy numbers (IVPFNs). We further introduce the basic operations of IVPFNs and investigate their desirable properties. Meanwhile, we also explore the ranking method and the distance measure for IVPFNs. Afterwards, we develop a closeness index-based Pythagorean fuzzy QUALIFLEX method to address hierarchical multicriteria decision making problems within Pythagorean fuzzy environment based on PFNs and IVPFNs. This hierarchical decision problem includes the main-criteria layer and the sub-criteria layer in which the relationships among main-criteria are interdependent, the relationships among sub-criteria are independent and the weights of sub-criteria take the form of IVPFNs. Therefore, in the developed method we first define the concept of concordance/discordance index based on the closeness index-based ranking methods and compute the sub-weighted concordance/discordance indices by employing the weighted averaging aggregation operator based on the closeness indices of IVPFNs. In order to take main-criteria interactions into account, we further employ Choquet integral to calculate the main-weighted concordance/discordance indices. By investigating all possible permutations of alternatives with the level of concordance and discordance of the complete preference order, we finally obtain the optimal rankings of alternatives. The proposed method is implemented in a risk evaluation problem in order to demonstrate its applicability and superiority. The salient features of the proposed method, compared to the state-of-the-art QUALIFLEX-based methods, are: (1) it can take the interactive phenomena among criteria into account; (2) it can manage simultaneously the PFN and IVPFN decision data; (3) it can deal effectively with the hierarchal structure of criteria. The proposed method provides us with a useful way for hierarchical multicriteria decision making problems within Pythagorean fuzzy contexts. In addition, we also extend the proposed method to manage heterogeneous information which includes five different types of information such as real numbers, interval numbers, fuzzy numbers, PFNs and hesitant fuzzy elements.

A Note on Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

In this note, we point out an error to the proof of Theorem 3.4 in Zhang and Xu Int J Intell Syst 2014;2912:1061-1078 by a counterexample. We find that the inequality i.e., |πβ12-πβ22|i¾?|πβ12-πβ32| with respect to the degrees of indeterminacy of any three Pythagorean fuzzy numbers in the proof of Theorem 3.4 in Zhang and Xu's paper is not valid. A new proof is provided in this note.

The Maximizing Deviation Method Based on Interval-Valued Pythagorean Fuzzy Weighted Aggregating Operator for Multiple Criteria Group Decision Analysis

As a new extension of Pythagorean fuzzy set (also called Atanassov’s intuitionistic fuzzy set of second type), interval-valued Pythagorean fuzzy set which is parallel to Atanassov’s interval-valued intuitionistic fuzzy set has recently been developed to model imprecise and ambiguous information in practical group decision making problems. The aim of this paper is to put forward a novel decision making method for handling multiple criteria group decision making problems within interval-valued Pythagorean fuzzy environment based on interval-valued Pythagorean fuzzy numbers (IVPFNs). There are three key issues being addressed in this approach. The first is to introduce an interval-valued Pythagorean fuzzy weighted arithmetic averaging (IVPF-WAA) operator to aggregate the decision data in order to get the overall preference values of alternatives. Some desirable properties of the IVPF-WAA operator are also investigated. Based on the idea of the maximizing deviation method, the second is to establish an optimization model for determining the weights of criteria for each expert. The third is to construct a minimizing consistency optimal model to derive the weights of criteria for the group. Finally, an illustrating example is given to verify the proposed approach.