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.     

Choquet integrals of weighted intuitionistic fuzzy information

The Choquet integral is a very useful way of measuring the expected utility of an uncertain event [G. Choquet, Theory of capacities, Annales de l’institut Fourier 5 (1953) 131-295]. In this paper, we use the Choquet integral to propose some intuitionistic fuzzy aggregation operators. 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 is worth pointing out that most of the existing intuitionistic fuzzy aggregation operators are special cases of our operators. Moreover, we propose the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator to aggregate interval-valued intuitionistic fuzzy information, and apply them to a practical decision-making problem involving the prioritization of information technology improvement projects.     

Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making

For the real decision making problems, most criteria have inter-dependent or interactive characteristics so that it is not suitable for us to aggregate them by traditional aggregation operators based on additive measures. Thus, to approximate the human subjective decision making process, it would be more suitable to apply fuzzy measures, where it is not necessary to assume additivity and independence among decision making criteria. In this paper, an intuitionistic fuzzy Choquet integral is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered. First, we introduced two operational laws on intuitionistic fuzzy values. Then, based on these operational laws, intuitionistic fuzzy Choquet integral operator is proposed. Moreover, some of its properties are investigated. It is shown that the intuitionistic fuzzy Choquet integral operator can be represented by some special t-norms and t-conorms, and it is also a generalization of the intuitionistic fuzzy OWA operator and intuitionistic fuzzy weighted averaging operator. Further, the procedure and algorithm of multi-criteria decision making based on intuitionistic fuzzy Choquet integral operator is given under uncertain environment. Finally, a practical example is provided to illustrate the developed approaches.     

An intuitionistic view of the Dempster–Shafer belief structure

We discuss the Dempster–Shafer belief theory and describe its role in representing imprecise probabilistic information. In particular, we note its use of intervals for representing imprecise probabilities. We note in fuzzy set theory that there are two related approaches used for representing imprecise membership grades: interval-valued fuzzy sets and intuitionistic fuzzy sets. We indicate the first of these, interval-valued fuzzy sets, is in the same spirit as Dempster–Shafer representation, both use intervals. Using a relationship analogous to the type of relationship that exists between interval-valued fuzzy sets and intuitionistic fuzzy sets, we obtain from the interval-valued view of the Dempster–Shafer model an intuitionistic view of the Dempster–Shafer model. Central to this view is the use of an intuitionistic statement, pair of values, (Bel(A) Dis(A)), to convey information about the value of a variable lying in the set A. We suggest methods for combining intuitionistic statements and making inferences from these type propositions.     

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.     

Generalized Atanassov’s intuitionistic fuzzy power geometric operators and their application to multiple attribute group decision making

In this paper, we extend the power geometric (PG) operator and the power ordered weighted geometric (POWG) operator [Z.S. Xu, R.R. Yager, Power-geometric operators and their use in group decision making, IEEE Transactions on Fuzzy Systems 18 (2010) 94–105] to Atanassov’s intuitionistic fuzzy environments, i.e., we develop a series of generalized Atanassov’s intuitionistic fuzzy power geometric operators to aggregate input arguments that are Atanassov’s intuitionistic fuzzy numbers (IFNs). Then, we study some desired properties of these aggregation operators and investigate the relationships among these operators. Furthermore, we apply these aggregation operators to develop some methods for multiple attribute group decision making with Atanassov’s intuitionistic fuzzy information. Finally, two practical examples are provided to illustrate the proposed methods.     

Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making

In this paper, we define the Choquet integral operator for Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet integral average (PFCIA) operator and Pythagorean fuzzy Choquet integral geometric (PFCIG) 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 is worth pointing out that most of the existing Pythagorean fuzzy aggregation operators are special cases of our operators. Meanwhile, some basic properties are discussed in detail. Later, we propose two approaches to multiple attribute group decision making with attributes involving dependent and independent by the PFCIA operator and multi‐attributive border approximation area comparison (MABAC) in Pythagorean fuzzy environment. Finally, two illustrative examples have also been taken in the present study to verify the developed approaches and to demonstrate their practicality and effectiveness.     

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.     

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.     

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.     

Fuzzy induced generalized aggregation operators and its application in multi-person decision making

We present a wide range of fuzzy induced generalized aggregation operators such as the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. They are aggregation operators that use the main characteristics of the fuzzy OWA (FOWA) operator, the induced OWA (IOWA) operator and the generalized (or quasi-arithmetic) OWA operator. Therefore, they use uncertain information represented in the form of fuzzy numbers, generalized (or quasi-arithmetic) means and order inducing variables. The main advantage of these operators is that they include a wide range of mean operators such as the FOWA, the IOWA, the induced Quasi-OWA, the fuzzy IOWA, the fuzzy generalized mean and the fuzzy weighted quasi-arithmetic average (Quasi-FWA). We further generalize this approach by using Choquet integrals, obtaining the fuzzy induced quasi-arithmetic Choquet integral aggregation (Quasi-FICIA) operator. We also develop an application of the new approach in a strategic multi-person decision making problem.     

Some Hesitant Fuzzy Einstein Aggregation Operators and Their Application to Multiple Attribute Group Decision Making

Hesitant fuzzy sets, as a new generalized type of fuzzy set, has attracted scholars’ attention due to their powerfulness in expressing uncertainty and vagueness. In this paper, motivated by the idea of Einstein operation, we develop a family of hesitant fuzzy Einstein aggregation operators, such as the hesitant fuzzy Einstein Choquet ordered averaging operator, hesitant fuzzy Einstein Choquet ordered geometric operator, hesitant fuzzy Einstein prioritized weighted average operator, hesitant fuzzy Einstein prioritized weighted geometric operator, hesitant fuzzy Einstein power weighted average operator, and hesitant fuzzy Einstein power weighted geometric operator. And we also study some desirable properties and generalized forms of these operators. Then, we apply these operators to deal with multiple attribute group decision making under hesitant fuzzy environments. Finally, a numerical example is provided to illustrate the practicality and validity of the proposed method.     

基于Choquet 积分的直觉不确定语言信息集结算子及其应用

基于直觉不确定语言信息,针对属性间不严格相互独立且具有较大关联度的群决策问题,提出了两种基于直觉不确定语言信息的Choquet积分算子.首先,分析了因属性关联使得以往直觉不确定语言信息集结算子失效的现象,对此引入模糊测度,提出了直觉不确定语言的Choquet加权算术平均算子(IULCWA)和直觉不确定语言的Choquet加权几何平均算子(IULCGM);然后,证明了算子的相关性质,研究了属性间相关的、属性值为直觉不确定语言数的多属性群决策方法;最后,通过实例分析说明了以往直觉不确定语言信息集结算子的局限性以及新算子的有效性.     

Intuitionistic fuzzy geometric aggregation operators based on einstein operations

Intuitionistic fuzzy information aggregation plays an important part in Atanassov's intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and further develop some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. We also establish some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and give some numerical examples to illustrate the developed aggregation operators. In addition, we compare the proposed operators with the existing intuitionistic fuzzy geometric operators and get the corresponding relations. Finally, we apply the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. © 2011 Wiley Periodicals, Inc.     

Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making

In this paper, we investigate the multiple attribute group decision making (MAGDM) problems with 2-tuple linguistic information. Firstly, motivated by the ideas of Choquet integral and Shapley index, we propose three 2-tuple linguistic aggregation operators called Shapley 2-tuple linguistic Choquet averaging operator, Shapley 2-tuple linguistic Choquet geometric operator and generalized Shapley 2-tuple linguistic Choquet averaging operator. Then we discuss some properties of these operators, such as idempotency, monotonicity, boundary and commutativity. Secondly, if the information about the weights of decision makers (DMs) and attributes is incompletely known, we build two models to determine the optimal fuzzy measures on DM set and attribute set, respectively. Furthermore, we develop a new method for multiple attribute group decision making under 2-tuple linguistic environment based on the proposed operators. Finally, we apply the developed MAGDM method to select the most desirable emergency alternative and the validity of the developed method is verified by comparing the evaluation results with those obtained from the existing 2-tuple correlated aggregation operators.     

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.     

直觉模糊集的Choquet积分相关测度及其决策应用

Qu给出的直觉模糊集的Choquet积分相关系数的计算公式与相关系数的性质相矛盾.为此,通过一个实例说明Qu定义的直觉模糊Choquet积分相关系数定义存在的问题,并结合相关系数的性质证明,分析问题出现的原因;然后,针对存在的问题,以直觉模糊集的Choquet积分相关指标为基础,给出新的直觉模糊集的Choquet积分信息能量的概念,定义新的直觉模糊集的Choquet积分相关系数,并讨论相关系数的性质;最后,利用新定义的直觉模糊集的Choquet积分相关测度,推导出方案与正理想方案之间的Choquet积分相关系数计算公式,据此提出一种直觉模糊多属性决策方法,并通过实例分析以及方法对比,说明所提出方法的可行性和有效性.     

A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS

An extension of TOPSIS, a multi-criteria interval-valued intuitionistic fuzzy decision making technique, to a group decision environment is investigated, where inter-dependent or interactive characteristics among criteria and preference of decision makers are taken into account. To get a broad view of the techniques used, first, some operational laws on interval-valued intuitionistic fuzzy values are introduced. Based on these operational laws, a generalized interval-valued intuitionistic fuzzy geometric aggregation operator is proposed which is used to aggregate decision makers’ opinions in group decision making process. In addition, some of its properties are discussed. Then Choquet integral-based Hamming distance between interval-valued intuitionistic fuzzy values is defined. Combining the interval-valued intuitionistic fuzzy geometric aggregation operator with Choquet integral-based Hamming distance, an extension of TOPSIS method is developed to deal with a multi-criteria interval-valued intuitionistic fuzzy group decision making problems. Finally, an illustrative example is used to illustrate the developed procedures.     

An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect

This paper presents a new interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST). This interpretation makes it possible to represent all mathematical operations on intuitionistic fuzzy values as the operations on belief intervals. Such approach allows us to use directly the Dempster’s rule of combination to aggregate local criteria presented by intuitionistic fuzzy values in the decision making problem. The usefulness of the developed method is illustrated with the known example of multiple criteria decision making problem. The proposed approach and a new method for interval comparison based on DST, allow us to solve multiple criteria decision making problem without intermediate defuzzification when not only criteria, but their weights are intuitionistic fuzzy values.     

Belief and Plausibility Functions on Intuitionistic Fuzzy Sets

Belief and plausibility functions based on Dempster–Shafer theory have been used to measure uncertainty. They are also widely studied and applied in diverse areas. Numerous studies in the literature have presented various generalizations of belief and plausibility functions to fuzzy sets. However, there are still less generalizations of belief and plausibility functions to intuitionistic fuzzy sets. Because intuitionistic fuzzy sets can present the degrees of both membership and nonmembership with a degree of hesitancy, the knowledge and semantic representation becomes more general and applicable than fuzzy sets. In this paper, we propose a generalization of belief and plausibility functions to intuitionistic fuzzy sets based on fuzzy integral. Some numerical examples show the effectiveness of the proposed generalization. Furthermore, this generalization of belief and plausibility functions to intuitionistic fuzzy sets is able to catch more information about the change of intuitionistic fuzzy focal elements.