Intuitionistic fuzzy PRI-AND and PRI-OR aggregation operators

In this paper, we investigate the multi-attribute decision making (MADM) problem under Atanassov’s intuitionistic fuzzy environment in which the attributes are in different priority levels. We develop the intuitionistic fuzzy prioritized “and” operator and intuitionistic fuzzy prioritized “or” operator, which are motivated by the idea of Yager’s prioritized “and” operator and prioritized “or” operator. These intuitionistic fuzzy prioritized aggregation operators can be applied to aggregate intuitionistic fuzzy information when the attributes are in different priority levels. A practical example is used to illustrate the applicability and effectiveness of the proposed intuitionistic fuzzy prioritized “or” operator.     

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.     

Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making

The uncertainty measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) is important for information discrimination under intuitionistic fuzzy environment. Although many entropy measures and knowledge measures haven been proposed to depict uncertainty of AIFSs, how to measure the uncertainty of AIFSs is still an open topic. The relation between uncertainty and other measures like entropy measures, fuzziness and intuitionism is not clear. This paper introduces uncertainty measures by using new defined divergence-based cross entropy measure of AIFSs. Axiomatic properties of the developed uncertainty measure are analysis, together with the monotony property of uncertainty degree with respect to fuzziness and intuitionism. To adjust the contribution of fuzzy entropy and intuitionistic entropy on the total uncertainty, the proposed cross entropy and uncertainty measures are parameterized. Numerical examples indicate the effectiveness and agility of the biparametric uncertainty measure in quantifying uncertainty degree. Then we apply the cross entropy and uncertainty measures into an optimal model to determine attribute weights in multi-attribute group decision making (MAGDM) problems. A new method for intuitionistic fuzzy MAGDM problems is proposed to show the efficiency of proposed measures in applications. It is demonstrated by application examples that the proposed measures can get reasonable results coinciding with other existing methods.     

Approaches to multiple attribute group decision making based on intuitionistic fuzzy power aggregation operators

Intuitionistic fuzzy numbers (IFNs) are very suitable to be used for depicting uncertain or fuzzy information. Motivated by the idea of power aggregation [R.R. Yager, The power average operator, IEEE Transactions on Systems, Man, and Cybernetics-Part A 31 (2001) 724–731], in this paper, we develop a series of operators for aggregating IFNs, establish various properties of these power aggregation operators, and then apply them to develop some approaches to multiple attribute group decision making with Atanassov’s intuitionistic fuzzy information. Moreover, we extend these aggregation operators and decision making approaches to interval-valued Atanassov’s intuitionistic fuzzy environments.     

On averaging operators for Atanassov’s intuitionistic fuzzy sets

Atanassov’s intuitionistic fuzzy set (AIFS) is a generalization of a fuzzy set. There are various averaging operators defined for AIFSs. These operators are not consistent with the limiting case of ordinary fuzzy sets, which is undesirable. We show how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. We provide two generalizations of the existing methods for other averaging operators. We relate operations on AIFS with operations on interval-valued fuzzy sets. Finally, we propose a new construction method based on the ?ukasiewicz triangular norm, which is consistent with operations on ordinary fuzzy sets, and therefore is a true generalization of such operations.     

Prioritized Information Fusion Method for Triangular Intuitionistic Fuzzy Set and its Application to Teaching Quality Evaluation

In this article, we examine the issue of triangular intuitionistic fuzzy information fusion. We first propose some new triangular intuitionistic fuzzy aggregation operators based on the prioritized average operator, such as the triangular intuitionistic fuzzy prioritized weighted average and the triangular intuitionistic fuzzy prioritized weighted geometric operators. We study some desired properties of the proposed operators, such as idempotency, noncompensatory, and boundary. We then develop an approach to deal with group decision‐making problems under triangular intuitionistic fuzzy environments. Finally, a practical example about teaching quality evalution is provided to illustrate the group decision‐making process.     

Prioritized intuitionistic fuzzy aggregation operators

In some multi-attribute decision making problems, distorted conclusions will be generated due to the lack of considering various relationships among the attributes of decision making. In this paper, we investigate the prioritization relationship of attributes in multi-attribute decision making with intuitionistic fuzzy information (i.e., partial or all decision information, like attribute values and weights, etc., is represented by intuitionistic fuzzy values (IFVs)). Firstly, we develop a new method for comparing two IFVs, based on which the basic intuitionistic fuzzy operations satisfy monotonicities. In addition, we devise a method to derive the weights with intuitionistic fuzzy forms, which can indicate the importance degrees of the corresponding attributes. Then we develop a prioritized intuitionistic fuzzy aggregation operator, which is motivated by the idea of the prioritized aggregation operators [R.R. Yager, Prioritized aggregation operators, International Journal of Approximate Reasoning 48 (2008) 263–274]. Furthermore, we propose an intuitionistic fuzzy basic unit monotonic (IF-BUM) function to transform the derived intuitionistic fuzzy weights into the normalized weights belonging to the unit interval. Finally, we develop a prioritized intuitionistic fuzzy ordered weighted averaging operator on the basis of the IF-BUM function and the transformed weights.     

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.     

An alternative to fuzzy methods in decision-making problems

In this work we present a construction method for Atanassov’s intuitionistic fuzzy preference relations starting from fuzzy preference relations and taking into account the ignorance of the expert in the construction of the latter. Moreover, we propose two generalizations of the weighted voting strategy to work with Atanassov’s intuitionistic fuzzy preference relations. An advantage of these algorithms is that they start from fuzzy preference relations and their results can be compared with those of any other decision-making algorithm based on fuzzy sets theory. We verify that our proposal is able to provide a unique solution in some cases in which the voting strategy is not able to order the alternatives.     

毕达哥拉斯模糊优先集成算子及其决策应用

作为直觉模糊集的推广形式,毕达哥拉斯模糊数能更好地刻画现实中的不确定性,此外在某些问题上,方案的属性之间往往具有优先关系,针对此类信息的集成问题,将毕达哥拉斯模糊数与优先集成算子相结合,提出了毕达哥拉斯模糊优先集成算子,包括毕达哥拉斯模糊优先加权平均算子和毕达哥拉斯模糊优先加权几何算子,并讨论了这些算子的性质。在此基础上,提出了毕达哥拉斯模糊优先集成算子的多属性决策方法,最后将其应用于国内四家航空公司服务质量评价中,说明了该算子的有效性和可行性。     

基于ε-修正的直觉模糊信息集成方法及其在决策中的应用

已有的一些直觉模糊集成算子在处理一些特殊直觉模糊数时会出现反直觉现象。首先介绍了两个直觉模糊集成算子和直觉模糊数的比较方法。接着,举例说明了这些集成算子在某些情况下出现的反直觉现象。然后提出了基于ε-修正的直觉模糊集成算子,并讨论了ε取值对此算子结果的影响。之后建立了一种基于ε-修正的直觉模糊集成算子的决策方法。最后通过一个实例比较了原集成算子和本文提出的修正集成算子的集成结果,验证基于ε-修正的直觉模糊集成算子可以修正这些反直觉现象,这也拓宽了原集成算子的使用范围。     

模糊数直觉模糊几何集成算子及其在决策中的应用

模糊数直觉模糊集是直觉模糊集的拓展.针对模糊数直觉模糊信息的集成问题,定义了模糊数直觉模糊数的一些运算法则,基于这些法则给出了一些新的几何集成算子,即模糊数直觉模糊加权几何(FIFWG)算子、模糊数直觉模糊有序加权几何(FIFOWG)算子和模糊数直觉模糊混合几何(FIFHG)算子.在此基础上,提出一种属性权重确知且属性值以模糊数直觉模糊数形式给出的多属性群决策方法.最后通过实例分析结果证明了该方法的有效性.     

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.     

On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.     

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same 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.     

区间直觉模糊信息的集成方法及其在决策中的应用

对区间直觉模糊信息的集成方法进行了研究.定义了区间直觉模糊数的一些运算法则,并基于这些运算法则,给出区间直觉模糊数的加权算术和加权几何集成算子.定义了区间直觉模糊数的得分函数和精确函数,进而给出了区间直觉模糊数的一种简单的排序方法.最后提供了一种基于区间直觉模糊信息的决策途径,并进行了实例分析.     

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.     

An intuitionistic fuzzy group decision making method using entropy and association coefficient

Group decision making is a process in which experts rank and choose the most desirable alternatives based on some accepted criteria. The aim of this paper was to introduce a method to solve group decision making problems with Atanassov’s intuitionistic fuzzy sets. First, the weight of each criterion is calculated using intuitionistic fuzzy entropy. Then, the total criteria weight vector is calculated by aggregating the calculated weights. Using the obtained weight vector, the alternatives are ranked based on the association coefficient of the performance of alternatives related to each criterion and the positive ideal intuitionistic fuzzy set value and the negative ideal intuitionistic fuzzy set value. Finally, to show the application of the proposed method, it is implemented in software vendor selection.     

Intuitionistic preference relations and their application in group decision making

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.