A ranking function based on principal‐value Pythagorean fuzzy set in multicriteria decision making

Multicriteria decision making (MCDM) has been attracting attention in recent years. There are two essential directions in the research territory, one direction is the research of representation of evaluation information and another is the construction of ranking function. In this paper, we consider some nonstandard fuzzy sets, intuitionistic, and interval‐valued fuzzy sets. Especially, the Pythagorean membership grade and Pythagorean fuzzy set receive much attention. Then, to reflect the importance of principal value, we shall propose the principal‐value Pythagorean fuzzy number (p‐PFN) and principal‐value Pythagorean fuzzy set. Furthermore, a novel ranking function is constructed to select the ideal alternative(s) based on p‐PFNs in MCDM. Finally, an investment strategy decision‐making problem is proposed to reveal the availability and practicability of the function under the new environment.     

Interval‐Valued Intuitionistic Fuzzy Multiattribute Group Decision Making Based on Cross Entropy Measure and Choquet Integral

In this paper, a new operator called the arithmetic interval‐valued intuitionistic fuzzy Choquet aggregation (AIVIFCA) operator is defined. Since interactions between elements might exist in all their combinations, the generalized Shapley AIVIFCA (GSAIVIFCA) operator is introduced. Further, to simplify the complexity of solving a fuzzy measure, the 2‐additive generalized Shapley AIVIFCA (2AGSAIVIFCA) operator is presented. Moreover, a decision procedure to interval‐valued intuitionistic fuzzy multiattribute group decision making is developed. When the weight vectors on attribute set and expert set are not exactly known, the models for obtaining the optimal fuzzy measures are established by using the defined cross entropy measure and the Shapley function. Finally, a numerical example is provided to illustrate the developed procedure.     

Interval‐valued Pythagorean fuzzy GRA method for multiple‐attribute decision making with incomplete weight information

In this paper, the concept of multiple‐attribute group decision‐making (MAGDM) problems with interval‐valued Pythagorean fuzzy information is developed, in which the attribute values are interval‐valued Pythagorean fuzzy numbers and the information about the attribute weight is incomplete. Since the concept of interval‐valued Pythagorean fuzzy sets is the generalization of interval‐valued intuitionistic fuzzy set. Thus, due the this motivation in this paper, the concept of interval‐valued Pythagorean fuzzy Choquet integral average (IVPFCIA) operator is introduced by generalizing the concept of interval‐valued intuitionistic fuzzy Choquet integral average operator. To illustrate the developed operator, a numerical example is also investigated. Extended the concept of traditional GRA method, a new extension of GRA method based on interval‐valued Pythagorean fuzzy information is introduced. First, utilize IVPFCIA operator to aggregate all the interval‐valued Pythagorean fuzzy decision matrices. Then, an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method is established, to get the weight vector of the attributes. Based on the traditional GRA method, calculation steps for solving interval‐valued Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of grey relation between every alternative and positive‐ideal solution and negative‐ideal solution is calculated. To determine the ranking order of all alternatives, a relative relational degree is defined by calculating the degree of grey relation to both the positive‐ideal solution and negative ideal solution simultaneously. Finally, to illustrate the developed approach a numerical example is to demonstrate its practicality and effectiveness.     

The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making

In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) is developed by integrating the cross-entropy of intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulae are discussed and their properties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment. For both methods, an optimisation model is established in order to determine the weight vector for MCDM problems with incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality and effectiveness of the proposed approaches.     

Generalized aggregation operators for intuitionistic fuzzy sets

The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval‐valued intuitionistic fuzzy weighted averaging operator, generalized interval‐valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval‐valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval‐valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.     

Group Decision Making Under Interval‐Valued Multiplicative Intuitionistic Fuzzy Environment Based on Archimedean t‐Conorm and t‐Norm

The main focus of this paper is to investigate group decision‐making (GDM) method under interval‐valued multiplicative intuitionistic fuzzy environment based on Archimedean t‐conorm and t‐norm. First of all, some operations laws are proposed for interval‐valued multiplicative intuitionistic fuzzy elements, which is an extension of multiplicative intuitionistic fuzzy operations developed earlier by other scholars. The effectiveness of these proposed operations is illustrated with some numerical examples. Then, a series of aggregation operators are proposed and the desirable properties are also studied. This paper reveals that some existing multiplicative intuitionistic fuzzy and interval‐valued multiplicative intuitionistic fuzzy aggregation operators are the special cases of the operators proposed in this paper. Finally, a GDM method based on proposed operators under interval‐valued multiplicative intuitionistic fuzzy environment is proposed, and a real case about annual evaluation for personnel of Zhejiang University of Finance and Economics is presented to illustrate the effectiveness of the proposed method.     

Correlation for Dual Hesitant Fuzzy Sets and Dual Interval‐Valued Hesitant Fuzzy Sets

Ever since fuzzy set has been introduced, several extensions have been established, such as interval‐valued fuzzy sets, Atanassov's intuitionistic fuzzy sets, interval‐valued Atanassov's intuitionistic fuzzy sets, fuzzy multisets, hesitant fuzzy sets, interval‐valued hesitant fuzzy sets, and dual hesitant fuzzy sets. In this contribution, we propose dual interval‐valued hesitant fuzzy sets, which encompass fuzzy sets and its aforementioned extensions as special cases. Because of the importance of correlation measure in data analysis, we propose an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and then extend the approach to the dual interval‐valued hesitant fuzzy set theory. We also put forward some formulas to create new correlation coefficients for fuzzy sets and its extensions in a general way. In addition, we give a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.     

A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems

The objective of this paper is divided into two folds. Firstly, a new generalized improved score function has been presented in the interval-valued intuitionistic fuzzy sets (IVIFSs) environment by incorporating the idea of weighted average of the degree of hesitation between their membership functions. Secondly, an IVIFSs based method for solving the multi-criteria decision making (MCDM) problem has been presented with completely unknown attribute weights. A ranking of the different attributes is based on the proposed generalized improved score functions and the sensitivity analysis on the ranking of the system has been done based on the decision-making parameters. An illustrative examples have been studied to show that the proposed function is more reasonable in the decision-making process than other existing functions.     

Study on the ranking problems in multiple attribute decision making based on interval‐valued intuitionistic fuzzy numbers

This paper puts forward a new ranking method for multiple attribute decision‐making problems based on interval‐valued intuitionistic fuzzy set (IIFS) theory. First, the composed ordered weighted arithmetic averaging operator and composed ordered weighted geometric averaging operator are extended to the IIFSs in which they are, respectively, named interval‐valued intuitionistic fuzzy composed ordered weighted arithmetic averaging (IIFCOWA) operator and interval‐valued intuitionistic composed ordered weighted geometric averaging (IIFCOWG) operator. Afterwards, to compare interval‐valued intuitionistic fuzzy numbers, we define the concepts of the maximum, the minimum, and ranking function. Some properties associated with the concepts are investigated. Using the IIFCOWA or IIFCOWG operator, we establish the detailed steps of ranking alternatives (or attributes) in multiple attribute decision making. Finally, an illustrative example is provided to show that the proposed ranking method is feasible in multiple attribute decision making.     

基于变精度粗糙直觉模糊集的决策规则获取

以直觉模糊目标信息系统为研究对象,以粗糙集和直觉模糊集为工具,以知识发现为目的,给出了从直觉模糊决策表中获取决策规则的一种有效方法。即通过对Pawlak粗糙隶属函数的定义进行推广,给出粗糙直觉模糊隶属函数,利用新的粗糙隶属函数,建立了变精度粗糙直觉模糊集模型。在此模型基础上定义了变精度粗糙直觉模糊集的近似质量和近似约简,由近似约简导出概率决策规则集,从而给出了直觉模糊决策表的概率决策规则获取方法。最后,以实例说明了这一方法的有效性。关键词：     

Switching between type-2 fuzzy sets and intuitionistic fuzzy sets: an application in medical diagnosis

When dealing with vagueness, there are situations when there is insufficient information available, making it impossible to satisfactorily evaluate membership. The intuitionistic fuzzy set theory is more suitable than fuzzy sets to deal with such problem. In 1996, Atanassov proposed the mapping from intuitionistic fuzzy sets to fuzzy sets. Furthermore, intuitionistic fuzzy sets are isomorphic to interval valued fuzzy sets, and interval valued fuzzy sets are regarded as the special cases of type-2 fuzzy sets in recently studies. However, their discussions are not only hardly comprehending but also lacking the reliable applications. In this study, the advantage of type-2 fuzzy sets is employed, and the switching relation between type-2 fuzzy sets and intuitionistic fuzzy sets is defined axiomatically. The switching results are applied to show the usefulness of the proposed method in pattern recognition and medical diagnosis reasoning.     

A Novel Method for Multiattribute Decision Making with Interval‐Valued Pythagorean Fuzzy Linguistic Information

Pythagorean fuzzy set (PFS), proposed by Yager (2013), is a generalization of the notion of Atanassov's intuitionistic fuzzy set, which has received more and more attention. In this paper, first, we define the weighted Minkowski distance with interval‐valued PFSs. Second, inspired by the idea of the Pythagorean fuzzy linguistic variables, we define a new fuzzy variable called interval‐valued Pythagorean fuzzy linguistic variable set (IVPFLVS), and the operational laws, score function, accuracy function, comparison rules, and distance measures of the IVPFLVS are defined. Third, some aggregation operators are presented for aggregating the interval‐valued Pythagorean fuzzy linguistic information such as the interval‐valued Pythagorean fuzzy linguistic weighted averaging (IVPFLWA), interval‐valued Pythagorean fuzzy linguistic ordered weighted averaging (IVPFLOWA) , interval‐valued Pythagorean fuzzy linguistic hybrid averaging, and generalized interval‐valued Pythagorean fuzzy linguistic ordered weighted average operators. Fourth, some desirable properties of the IVPFLWA and IVPFLOWA operators, such as monotonicity, commutativity, and idempotency, are discussed. Finally, based on the IVPFLWA or interval‐valued Pythagorean fuzzy linguistic geometric weighted operator, a practical example is provided to illustrate the application of the proposed approach and demonstrate its practicality and effectiveness.     

Weight computation of criteria in a decision-making problem by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set

The aim of this study is to propose an objective method for determining weights of criteria (also called attributes) based on a new measure of intuitionistic fuzzy information, called knowledge measure, in a real-world multi-criteria decision-making problem under intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. To address this issue, we first analyze the existing entropy measures and show that their use in objective weight determination process may lead us to produce unreliable weights of criteria by citing appropriate examples. Then we analyze important properties of knowledge measure of intuitionistic fuzzy set (IFS) and also define knowledge measure for interval-valued intuitionistic fuzzy set. Then a new method to determine the weights of criteria is developed on the basis of knowledge measure where information about criteria weights is completely unknown and partly known. A real-life example is presented to illustrate the proposed weight determination method and a comparative analysis is carried out to indicate the practicality and effectiveness of knowledge-based weight-generation method under both intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. Finally, we formulate the axioms for knowledge measure associated with IFSs and we also propose families (classes) of knowledge measures.     

Algorithms for complex interval‐valued q‐rung orthopair fuzzy sets in decision making based on aggregation operators,AHP, and TOPSIS

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.     

A Novel Approach for Linguistic Group Decision Making Based on Generalized Interval‐Valued Intuitionistic Fuzzy Linguistic Induced Hybrid Operator and TOPSIS

In the multiple attribute linguistic group decision making analysis with interval‐valued intuitionistic fuzzy linguistic information, seeking highly efficient aggregation method and order relation play a crucial role. In this paper, we redefine an interval‐valued intuitionistic fuzzy linguistic variable that considers principal component and propose generalized interval‐valued intuitionistic fuzzy linguistic induced hybrid aggregation (GIVIFLIHA) operator with entropic order‐inducing variable and interval‐valued intuitionistic fuzzy linguistic technique for order preference by similarity to an ideal solution (TOPSIS) order relation based on interval‐valued intuitionistic fuzzy linguistic distance measure. Then, some primary properties of the GIVIFLIHA operator are discussed, and a linguistic group decision‐making approach based on GIVIFLIHA operator and interval‐valued intuitionistic fuzzy linguistic TOPSIS order relation is proposed. Finally, a numerical example concerning the investment strategy is given to illustrate the validity and applicability of the proposed method, and then the method is compared with the existing method to further illustrate its flexibility.     

直觉模糊知识粒的分解与合成研究

针对近似空间笛卡尔积粗糙集模型及其可分解性问题,采用直觉模糊三角模算子构成新的直觉模糊积近似空间,研究了基于直觉模糊知识粒下积粗糙集模型的分解及合成问题.首先,运用直觉模糊三角模运算构造出新的直觉模糊关系,验证了其符合等价关系的条件,并给出新的等价关系的算法原理;其次,构建了直觉模糊积粗糙集模型,对其模型结构及数学特性...     

Pythagorean Dombi fuzzy aggregation operators with application in multicriteria decision-making

A Pythagorean fuzzy set, an extension of intuitionistic fuzzy sets, is very helpful in representing vague information that occurs in real world scenarios. The Dombi operators with operational parameters, have excellent flexibility. Due to the flexible nature of these Dombi operational parameters, this research paper introduces some new aggregation operators under Pythagorean fuzzy environment, including Pythagorean Dombi fuzzy weighted arithmetic averaging (PDFWAA) operator, Pythagorean Dombi fuzzy weighted geometric averaging (PDFWGA) operator, Pythagorean Dombi fuzzy ordered weighted arithmetic averaging operator and Pythagorean Dombi fuzzy ordered weighted geometric averaging operator. Further, this paper presents several advantageous characteristics, including idempotency, monotonicity, boundedness, reducibility and commutativity of preceding operators. By utilizing PDFWAA and PDFWGA operators, this article describes a multicriteria decision-making (MCDM) technique for solving MCDM problems. Finally, a numerical example related to selection of a leading textile industry is presented to illustrate the applicability of our proposed technique.     

On intuitionistic fuzzy decision-making using soft likelihood functions

Inspired by Yager, in this paper, we present the concept of likelihood for intuitionistic fuzzy sets (IFSs), and propose an approach for flexible computation of likelihood functions of IFSs for multicriteria decision-making (MCDM). We employ ordered weighted average (OWA) aggregation method to soften the strong likelihood constraint condition. The OWA measure can be considered as the attitudinal character, which determines OWA weights, including optimistic or pessimistic likelihood values. Then the reliability-based soft likelihood function is developed by considering the reliability of intuitionistic fuzzy information. Some examples are conducted by using the proposed (reliability-based) soft likelihood functions in intuitionistic fuzzy environment for MCDM problem, and the results are analyzed in detail.     

An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis

This paper presents an interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions for managing multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. First, certain likelihood-based preference functions are proposed using the likelihoods of interval-valued intuitionistic fuzzy preference relationships. Next, selected practical indices of concordance/discordance are established to evaluate all possible permutations of the alternatives. The optimal priority order of the alternatives is determined by comparing all comprehensive concordance/discordance values based on score functions. Furthermore, this paper considers various preference types and develops another interval-valued intuitionistic fuzzy permutation method using programming models to address multiple criteria decision-making problems with incomplete preference information. The feasibility and applicability of the proposed methods are illustrated in the problem of selecting a suitable bridge construction method. Moreover, certain comparative analyses are conducted to verify the advantages of the proposed methods compared with those of other decision-making methods. Finally, the practical effectiveness of the proposed methods is validated with a risk assessment problem in new product development.     

Extension of the ELECTRE method based on interval-valued fuzzy sets

Decision-making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy ELECTRE method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For the purpose of proving the validity of the proposed model, we present a numerical example and build a practical maintenance strategy selection problem.