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.     

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.     

Induced generalized intuitionistic fuzzy operators

We study the induced generalized aggregation operators under intuitionistic fuzzy environments. Choquet integral and Dempster–Shafer theory of evidence are applied to aggregate inuitionistic fuzzy information and some new types of aggregation operators are developed, including the induced generalized intuitionistic fuzzy Choquet integral operators and induced generalized intuitionistic fuzzy Dempster–Shafer operators. Then we investigate their various properties and some of their special cases. Additionally, we apply the developed operators to financial decision making under intuitionistic fuzzy environments. Some extensions in interval-valued intuitionistic fuzzy situations are also pointed out.     

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.     

Induced choquet ordered averaging operator and its application to group decision making

Yager (Fuzzy Sets Syst 2003;137:59–69) extended the idea of order‐induced aggregation to the Choquet aggregation and defined a more general type of Choquet integral operator called the induced Choquet ordered averaging (I‐COA) operator, which take as their argument pairs, in which one component called order‐inducing variable is used to induce an ordering over the second components called argument variable and then aggregated. The aim of this paper is to develop the I‐COA operator. Some of its properties are investigated. We show its relationship to the induced‐ordered weighted averaging operator. Finally, we provide some I‐COA operators to aggregate fuzzy preference relations in group decision‐making problems. © 2009 Wiley Periodicals, Inc.     

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.     

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.     

A novel induced aggregation method for intuitionistic fuzzy set and its application in multiple attribute group decision making

In this paper, by unifying the induced ordered weighted averaging (IOWA) and the weighted average, a novel induced aggregation method for intuitionistic fuzzy set is investigated. More specifically, a new intuitionistic fuzzy (IF) induced aggregation operator called weighted intuitionistic fuzzy IOWA weighted average (WIFIOWAWA) operator is introduced. A significant advantage of the WIFIOWAWA operator is that it can eliminate the drawback of the existing operators by the dual roles of its order‐inducing variables. In addition, some of its desired properties and families are explored. Furthermore, using the proposed operator, a procedure is developed to solve multiple attribute group decision making problems in the case of IF situation. Finally, an illustrative example is provided to demonstrate the effectiveness and practicality of the developed method.     

基于Zhenyuan积分的直觉模糊多属性决策方法

针对属性之间具有相互关联关系的直觉模糊多属性决策问题,提出一种基于Zhenyuan积分的决策方法.首先提出直觉模糊Zhenyuan积分平均(IFZA)算子;然后探讨IFZA算子的优良性质以及与现有直觉模糊集成算子的关系,研究表明,IFZA算子可以改进现有直觉模糊奇异积分算子的缺陷,能够全面度量属性之间的相互关联关系;最后提出一种基于IFZA算子的属性间具有相互关联关系的直觉模糊多属性决策方法,并通过实例验证所提出方法的有效性和可行性.     

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.     

基于前景ITFNCI 算子的群体MULTIMOORA决策方法

利用进一步完善的直觉梯形模糊数运算法则,提出基于模糊测度和Choquet积分的直觉梯形模糊数的信息集成算子,并证明了该集成算子的相关性质.根据前景理论定义直觉梯形模糊前景效应和前景价值函数,构造前景ITFNCI算子.针对多方参与且决策者信息关联的多属性群决策问题,利用前景ITFNCI算子集结群体直觉梯形前景价值函数,进而采用MULTIMOORA理论进行方案比选,基于优势理论获得方案的综合排序以确定最优方案.案例分析表明了所提出方法的可行性和有效性.     

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.     

Network-based decision making via generalized fuzzy integral operators

Recent advances in network-based decision making methods have given rise to computationally efficient solution methodologies for intelligent systems. One type of hierarchical network implementation, the fuzzy integral operator approach, is investigated. In this approach, we generalized the Choquet fuzzy integral as an excellent component for decision analysis and making. This involves extending the standard operators in information aggregation with generalized operators, resulting in increased flexibility. The characteristics of the Choquet fuzzy integrals and their generalizations are addressed and network-based decision making frameworks are then proposed. The trainable hierarchical networks are able to perceive and interpret complex decisions by using those processing elements called neurons. We also present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain. ©1999 John Wiley & Sons, Inc.     

基于累积前景理论和Choquet积分的直觉梯形模糊多属性决策

针对属性值为直觉梯形模糊数且属性存在关联性的风险决策问题, 提出一种基于累积前景理论和Choquet积分的直觉梯形模糊多属性决策方法。根据直觉梯形模糊数距离公式定义了直觉梯形模糊信息的前景价值函数, 通过价值函数和决策权重函数计算方案单属性前景值, 并运用Choquet积分融合属性间存在关联性的前景价值信息获得方案综合前景值, 根据综合前景值的大小实现方案的排序和优选。风险投资实例分析说明了该方法的可行性。     

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.     

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.     

A New Intuitionistic Fuzzy Linguistic Hybrid Aggregation Operator and Its Application for Linguistic Group Decision Making

The linguistic aggregation operator is an important decision‐making model that is proving effective for dealing with the input data that takes the form of uncertain information. In this paper, considering the principal component of the intuitionistic fuzzy linguistic variables, we develop a new intuitionistic fuzzy linguistic hybrid aggregation (NIFLHA) operator to solve group decision‐making problems under the situation with intuitionistic fuzzy linguistic information. Then, we study some of its main properties by utilizing some operational laws of intuitionistic fuzzy linguistic variables and the different families of the NIFLHA operator. Moreover, the multiperson NIFLHA (MP‐NIFLHA) operator is introduced to evaluate the opinions of experts. Finally, an illustrative example about a multiperson decision‐making problem is developed to reveal the applicability and the availability of the raised operator.     

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.     

Choquet fuzzy integral-based hierarchical networks for decisionanalysis

A Choquet fuzzy integral-based approach to hierarchical network implementation is investigated. In this approach, we generalized the fuzzy integral as an excellent component for decision analysis. The generalization involves replacing the max (or min) operator in information aggregation with a fuzzy integral-based neuron, resulting in increased flexibility. The characteristics of the Choquet fuzzy integral are studied and a network-based decision-analysis framework is proposed. The trainable hierarchical network can be implemented utilizing the fuzzy integral-based neurons and connectives. The training algorithms are derived and several examples given to illustrate the behaviors of the networks. Also, we present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain     

Intuitionistic fuzzy geometric Heronian mean aggregation operators

In this paper, the multi-criteria decision making problem with the assumption that the criteria are correlative is studied under intuitionistic fuzzy environment. Some new aggregation operators for intuitionistic fuzzy information are proposed, including the intuitionistic fuzzy geometric Heronian mean (IFGHM) operator and the intuitionistic fuzzy geometric weighed Heronian mean (IFGWHM) operator. We investigate the properties of the proposed operators, such as idempotency, monotonicity, permutation and boundary. Moreover, an approach is proposed for multi-criteria decision making based on IFGWHM operator. An example about talent introduction is given to illustrate the proposed method.