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.     

Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making

Considering that there may exist some interactions between membership function and non-membership function of different intuitionistic fuzzy sets, we present some new operational laws from the probability point of view and give a geometric interpretation of the new operations. Based on which, a new class of generalized intuitionistic fuzzy aggregation operators are developed, including the generalized intuitionistic fuzzy weighted geometric interaction averaging (GIFWGIA) operator, the generalized intuitionistic fuzzy ordered weighted geometric interaction averaging (GIFOWGIA) operator and the generalized intuitionistic fuzzy hybrid geometric interaction averaging (GIFHGIA) operator. The properties of these new generalized aggregation operators are investigated. Moreover, approaches to multiple attributes decision making are given based on the generalized aggregation operators under intuitionistic fuzzy environment, and an example is illustrated to show the validity and feasibility of new approach. Finally, we give a systematic comparison between the work of this paper and that of other papers.     

Intuitionistic Fuzzy Hybrid Weighted Aggregation Operators

Intuitionistic fuzzy sets (IFSs) have attracted more and more scholars’ attention due to their powerfulness in expressing vagueness and uncertainty. In the course of decision making with IFSs, aggregation operators play a very important role since they can be used to synthesize multidimensional evaluation values represented as intuitionistic fuzzy values into collective values. This paper proposes a family of intuitionistic fuzzy hybrid weighted aggregation operators, such as the intuitionistic fuzzy hybrid weighted averaging operator, the intuitionistic fuzzy hybrid weighted geometric operator, the generalized intuitionistic fuzzy hybrid weighted averaging operator, and the generalized intuitionistic fuzzy hybrid weighted geometric operator. All these newly developed operators not only can weight both the arguments and their ordered positions simultaneously but also have some desirable properties, such as idempotency, boundedness, and monotonicity. To show the applications of our proposed intuitionistic fuzzy hybrid weighted aggregation operators, a simple schema for decision making with intuitionistic fuzzy information is developed. An example concerning the human resource management is given to illustrate the validity and applicability of the proposed method and also the hybrid weighted aggregation operators.     

Some geometric aggregation operators based on intuitionistic fuzzy sets

The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.     

Intuitionistic Fuzzy Aggregation Operators

An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators.     

The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making

In this paper, we present the induced generalized intuitionistic fuzzy ordered weighted averaging (I-GIFOWA) operator. It is a new aggregation operator that generalized the IFOWA operator, including all the characteristics of both the generalized IFOWA and the induced IFOWA operators. It provides a very general formulation that includes as special cases a wide range of aggregation operators for intuitionistic fuzzy information, including all the particular cases of the I-IFOWA operator, GIFOWA operator and the induced intuitionistic fuzzy ordered geometric (I-IFOWG) operator. We also present the induced generalized interval-valued intuitionistic fuzzy ordered weighted averaging (I-GIIFOWA) operator to accommodate the environment in which the given arguments are interval-valued intuitionistic fuzzy sets. Further, we develop procedures to apply them to solve group multiple attribute decision making problems with intuitionistic fuzzy or interval-valued intuitionistic fuzzy information. Finally, we present their application to show the effectiveness of the developed methods.     

Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making

With respect to multi-attribute group decision making (MAGDM) problems in which both the attribute weights and the decision makers (DMs) weights take the form of real numbers, attribute values provided by the DMs take the form of intuitionistic fuzzy numbers, a new group decision making method is developed. Some operational laws, score function and accuracy function of intuitionistic fuzzy numbers are introduced at first. Then a new aggregation operator called induced generalized intuitionistic fuzzy ordered weighted averaging (IG-IFOWA) operator is proposed, which extend the induced generalized ordered weighted averaging (IGOWA) operator introduced by Merigo and Gil-Lafuente [Merigo, J. M., & Gil-Lafuente, A. M. (2009). The induced generalized OWA operator. Information Sciences, 179, 729-741] to accommodate the environment in which the given arguments are intuitionistic fuzzy sets that are characterized by a membership function and a non-membership function. Some desirable properties of the IG-IFOWA operator are studied, such as commutativity, idempotency, monotonicity and boundary. And then, an approach based on the IG-IFOWA and IFWA (intuitionistic fuzzy weighted averaging) operators is developed to solve MAGDM problems with intuitionistic fuzzy information. Finally, a numerical example is used to illustrate the developed approach.     

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.     

Generalized point operators for aggregating intuitionistic fuzzy information

We first develop a series of intuitionistic fuzzy point operators, and then based on the idea of generalized aggregation (Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3:93–107 and Zhao H, Xu ZS, Ni MF, Liu SS. Generalized aggregation operators for intuitionistic fuzzy sets. Int J Intell Syst 2010;25:1–30), we develop various generalized intuitionistic fuzzy point aggregation operators, such as the generalized intuitionistic fuzzy point weighted averaging (GIFPWA) operators, generalized intuitionistic fuzzy point ordered weighted averaging (GIFPOWA) operators, and generalized intuitionistic fuzzy point hybrid averaging (GIFPHA) operators, which can control the certainty degrees of the aggregated arguments with some parameters. Furthermore, we study the properties and special cases of our operators. © 2010 Wiley Periodicals, Inc.     

Novel neutrality operation–based Pythagorean fuzzy geometric aggregation operators for multiple attribute group decision analysis

Pythagorean fuzzy sets (PFSs) accommodate more uncertainties than L_x the intuitionistic fuzzy sets and hence its applications are more extensive. Under the PFS, the objective of this paper is to develop some new operational laws and their corresponding weighted geometric aggregation operators. For it, we define some new neutral multiplication and power operational laws by including the feature of the probability sum and the interaction coefficient into the analysis to get a neutral or a fair treatment to the membership and nonmembership functions of PFSs. Associated with these operational laws, we define some novel Pythagorean fuzzy weighted, ordered weighted, and hybrid neutral geometric operators for Pythagorean fuzzy information, which can neutrally treat the membership and nonmembership degrees. The desirable relations and the characteristics of the proposed operators are studied in details. Furthermore, a multiple attribute group decision-making approach based on the proposed operators under the Pythagorean fuzzy environment is developed. Finally, an illustrative example is provided to show the practicality and the feasibility of the developed approach.     

Fundamental Properties of Interval‐Valued Pythagorean Fuzzy Aggregation Operators

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems with interval‐valued Pythagorean fuzzy sets (IVPFSs). First, the concept, operational laws, score function, and accuracy function of IVPFSs are defined. Then, based on the operational laws, two interval‐valued Pythagorean fuzzy aggregation operators are developed for aggregating the interval‐valued Pythagorean fuzzy information, such as interval‐valued Pythagorean fuzzy weighted average (IVPFWA) operator and interval‐valued Pythagorean fuzzy weighted geometric (IVPFWG) operator. A series of inequalities of aggregation operators are studied. Later, we develop some interval‐valued Pythagorean fuzzy point operators. Moreover, combining the interval‐valued Pythagorean fuzzy point operators with IVPFWA operator, we present some interval‐valued Pythagorean fuzzy point weighted averaging (IVPFPWA) operators, which can adjust the degree of the aggregated arguments with some parameters. Then, we propose an interval‐valued Pythagorean fuzzy ELECTRE method to solve uncertainty MAGDM problem. Finally, an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Hesitant interval-valued Pythagorean fuzzy VIKOR method

In this article, we investigate multiple attribute decision-making problems with hesitant interval-valued Pythagorean fuzzy information. First, the concepts of hesitant interval-valued Pythagorean fuzzy set are defined, and the operation laws, the score function, and accuracy function have been developed. Then several distance measures for hesitant interval-valued Pythagorean fuzzy values have been presented including the Hamming distance, Euclidean distance, and generalized distance, and so on. Based on the operational laws, a series of aggregation operators have been developed including the hesitant interval-valued Pythagorean fuzzy weighted averaging (HIVPFWA) operator, the hesitant interval-valued Pythagorean fuzzy geometric weighted averaging (HIVPFGWA) operator, the hesitant interval-valued Pythagorean fuzzy ordered weighed averaging (HIVPFOWA) operator, and hesitant interval-valued Pythagorean fuzzy ordered weighed geometric averaging (HIVPFOWGA) operator. By using the generalized mean operator, we also develop the generalized hesitant interval-valued Pythagorean fuzzy weighed averaging (GHIVPFWA) operator, the generalized hesitant interval-valued Pythagorean fuzzy weighed geometric averaging (GHIVPFWGA) operator, the generalized hesitant interval-valued Pythagorean fuzzy ordered weighted averaging (GHIVPFOWA) operator, and generalized hesitant interval-valued Pythagorean fuzzy ordered weighted geometric averaging (GHIVPFOWGA) operator operator. We further develop several hybrid aggregation operators including the hesitant interval-valued Pythagorean fuzzy hybrid averaging (HIVPFHA) operator and the generalized hesitant interval-valued Pythagorean fuzzy hybrid averaging (GHIVPFHA) operator. Based on the distance measures and the aggregation operators, we propose a hesitant interval-valued Pythagorean fuzzy VIKOR method to solve multiple attribute decision problems with multiple periods. Finally, an illustrative example for evaluating the metro project risk is given to demonstrate the feasibility and effectiveness of the proposed method.     

Generalized power aggregation operators and their applications in group decision making

In this paper, we investigate a generalized power average (GPA) operator and its weighted form, which are on the basis of the power average (PA) operator and the generalized mean, and develop a generalized power ordered weighted average (GPOWA) operator based on the power ordered weighted average (POWA) operator. Then, we extend these operators to uncertain environments and present an uncertain generalized power average (UGPA) operator and its weighted form, and an uncertain generalized power ordered weighted average (UGPOWA) operator to aggregate the input arguments taking the form of interval of numerical values. We also extend the GPA operator and the GPOWA operator to intuitionistic fuzzy environment, and obtain the generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Moreover, some properties of these operators are studied. We also present new approaches on the basis of the proposed operators in an example of strategic decision making.     

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.     

Some induced intuitionistic fuzzy aggregation operators applied to multi-attribute group decision making

In this paper, we define various induced intuitionistic fuzzy aggregation operators, including induced intuitionistic fuzzy ordered weighted averaging (OWA) operator, induced intuitionistic fuzzy hybrid averaging (I-IFHA) operator, induced interval-valued intuitionistic fuzzy OWA operator, and induced interval-valued intuitionistic fuzzy hybrid averaging (I-IIFHA) operator. We also establish various properties of these operators. And then, an approach based on I-IFHA operator and intuitionistic fuzzy weighted averaging (WA) operator is developed to solve multi-attribute group decision-making (MAGDM) problems. In such problems, attribute weights and the decision makers' (DMs') weights are real numbers and attribute values provided by the DMs are intuitionistic fuzzy numbers (IFNs), and an approach based on I-IIFHA operator and interval-valued intuitionistic fuzzy WA operator is developed to solve MAGDM problems where the attribute values provided by the DMs are interval-valued IFNs. Furthermore, induced intuitionistic fuzzy hybrid geometric operator and induced interval-valued intuitionistic fuzzy hybrid geometric operator are proposed. Finally, a numerical example is presented to illustrate the developed approaches.     

Multicriteria decision making based on intuitionistic fuzzy prioritized arithmetic mean

Atanassov’s intuitionistic fuzzy sets (AIFSs), characterized by a membership function, a nonmembership function, and a hesitancy function, is a generalization of a fuzzy set. Various aggregation operators are defined for AIFSs to deal with multicriteria decision‐making problems in which there exists a prioritization of criteria. However, these existing intuitionistic fuzzy prioritized aggregation operators are not monotone with respect to the total order on Atanassov’s intuitionistic fuzzy values (AIFVs), which is undesirable. We propose an intuitionistic fuzzy prioritized arithmetic mean based on the ?ukasiewicz triangular norm, which is monotone with respect to the total order on AIFVs, and therefore is a true generalization of such operations. We give an example that a consumer selects a car to illustrate the validity and applicability of the proposed method aggregation operator.     

Some interval-valued q-rung orthopair weighted averaging operators and their applications to multiple-attribute decision making

Q-rung orthopair fuzzy sets (q-ROFSs), initially proposed by Yager, are a new way to reflect uncertain information. The existing intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets are special cases of the q-ROFSs. However, due to insufficiency in available information, it is difficult for decision makers to exactly express the membership and nonmembership degrees by crisp numbers, and interval membership degree and interval nonmembership degree are good choices. In this paper, we propose the concept of interval-valued q-rung orthopair fuzzy set (IVq-ROFS) based on the ideas of q-ROFSs and some operational laws of q-rung orthopair fuzzy numbers (q-ROFNs). Then, some interval-valued q-rung orthopair weighted averaging operators are presented based on the given operational laws of q-ROFNs. Further, based on these operators, we develop a novel approach to solve multiple-attribute decision making (MADM) problems under interval-valued q-rung orthopair fuzzy environment. Finally, a numerical example is provided to illustrate the application of the proposed method, and the sensitivity analysis is further carried out for the parameters.     

Induced intuitionistic fuzzy Choquet integral operator for multicriteria decision making

Yager (Fuzzy Sets, Syst 2003;137:59–69) extended the idea of order‐induced aggregation to the Choquet aggregation and defined induced Choquet ordered averaging operator. In this paper, an induced intuitionistic fuzzy Choquet (IFC) integral operator is proposed for the multiple criteria decision making. Some of its properties are investigated. Furthermore, an induced generalized IFC integral operator is introduced. It is worth mentioning that most of the existing intuitionistic fuzzy aggregation operators are special cases of this induced aggregation operator. A decision procedure based on the proposed induced aggregation operator is developed for solving the multicriteria decision‐making problem in which all the decision information is represented by intuitionistic fuzzy values. An illustrative example is given for demonstrating the applicability of the proposed decision procedure. © 2011 Wiley Periodicals, Inc.     

Intuitionistic Fuzzy Interval‐Valued Linguistic Entropic Combined Weighted Averaging Operator for Linguistic Group Decision Making

Entropy, a basic concept of measuring the amount of information and the degree of confusion, has been applied in many weighted averaging operators in the linguistic group decision making. In the paper, we construct an intuitionistic fuzzy linguistic entropy based on the intuitionistic fuzzy entropy and the intuitionistic fuzzy linguistic variable. Then, inspired by operations of concentration and dilation (De SK, Biswas R, and Roy AR, Fuzzy Sets Syst. 2000;114(3):477?484), we extend the intuitionistic fuzzy linguistic entropy to the intuitionistic fuzzy interval‐valued linguistic entropy. After that, the intuitionistic fuzzy interval‐valued linguistic entropic combined weighted averaging (IFIVLECWA) operator is proposed for multiple attribute linguistic group decision making. Finally, a numerical example about the selection of optimal alternative(s) is presented to illustrate the applicability and effectiveness of the proposed method.     

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.