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.     

区间直觉模糊粗糙集

将模糊粗糙集推广到区间直觉模糊粗糙集,基于区间直觉模糊等价关系和两个论域之间的一般区间直觉模糊关系,给出了区间直觉模糊粗糙集模型的不同形式,并讨论了一些相关性质。     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

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.     

基于蕴涵的区间值直觉模糊粗糙集

提出一种基于区间值直觉模糊蕴涵的区间值直觉模糊粗糙集模型.首先,介绍了区间值直觉模糊集、区间值直觉模糊关系和区间值直觉模糊逻辑算子的概念;然后,利用区间值直觉模糊三角模和区间值直觉模糊蕴涵,在区间值直觉模糊近似空间中定义了区间值直觉模糊集的上近似和下近似;最后,给出并证明了这些近似算子的一些性质.     

二型直觉模糊集

二型模糊集和直觉模糊集都具有很强的实际应用背景.二型模糊集增强了系统处理不确定性的能力,直觉模糊集为解决人们判断问题所出现的犹豫信息提供了理论依据.本文在二型模糊集和直觉模糊集的基础上,给出了二型直觉模糊集的概念,证明了二型直觉模糊集是一型模糊集、直觉模糊集、区间值模糊集、区间值直觉模糊集的广义形式,讨论了二型直觉模糊集的基本运算和二型直觉模糊关系.最后,研究了基于二型直觉模糊理论的近似推理,并实例说明了二型直觉模糊集的实际应用背景.     

改进的粗糙直觉模糊集

在Pawlak近似空间中,针对直觉模糊目标集合,假设在信息粒度不变的情况下,试图寻求目标集合更好的近似集。在现有的粗糙直觉模糊集的基础之上,利用直觉模糊粗糙隶属函数,采用分段函数的形式建立直觉模糊集新的下近似与上近似算子,并讨论新模型的一些基本性质。与现有的粗糙直觉模糊集相比,改进后的模型无论在近似精度方面,还是与目标集合的相似度方面,都有了较大的改善和提高。最后通过数值算例验证了所给结论的正确性。     

单向S-粗直觉模糊集

针对系统具有的模糊、不确定及动态特性,基于直觉模糊集和单向S-粗集理论,提出单向S-粗直觉模糊集,给出它的结构、性质、存在背景和意义解释。分析了单向S-粗直觉模糊集与Pawlak粗集,单向S-粗模糊集以及单向S-粗集等之间的关系。     

Nonmonotonic OWA operators

 The basic properties of the Ordered Weighted Averaging (OWA) operator are recalled. The role of these operators in the formulation of multi-criteria decision functions, using the concept of quantifier guided aggregation, is discussed. An extended class of OWA operators, one based upon a relaxation of the requirements on the OWA operators, is introduced. This relaxation allows us to consider a new branch of OWA operators, NOMOWA operators, which have negative weights and which exhibit nonmonotonicity. Some special cases of these operators are discussed and then we investigate the role of these nonmonotonic operators in the formulation of multi-criteria decision functions.     

基于覆盖的区间直觉模糊粗糙集

在经典的覆盖近似空间中,定义了区间直觉模糊概念的粗糙近似。通过区间直觉模糊覆盖概念,给出了一种基于区间直觉模糊覆盖的区间直觉模糊粗糙集模型。讨论了两种模型的一些相关性质。     

软直觉模糊集

直觉模糊集是在模糊集上增加了一个新的属性参数：非隶属度函数,成为描述“非此非彼”的“模糊概念”的工具。为了提高决策的精确性,将软集与直觉模糊集相结合,构造了一种新的数学模型,即直觉模糊软集,对其性质进行了一些讨论。     

基于直觉模糊粗糙集的一种知识获取方法

将粗糙集理论与直觉模糊集理论相结合,提出了一种基于直觉模糊粗糙集理论的知识获取方法。描述了直觉模糊相似关系下粗糙集的模型,并在此基础之上重新定义了正域、依赖度与非依赖度、确定性因子与非确定性因子等概念。详细分析了基于直觉模糊粗糙集的规则挖掘算法的基本步骤,最后通过实例验证了该算法的可行性。     

Generalized intuitionistic fuzzy soft power aggregation operator based on t-norm and their application in multicriteria decision-making

The aim of this paper is to develop some new power aggregation operators for intuitionistic fuzzy (IF) soft numbers. The aggregation operators are named as IF soft power averaging (IFSPA) operator, weighted IFSPA (WIFSPA) operator, ordered WIFSPA operator, IF soft power geometric (IFSPG) operator, and weighted and ordered weighted IFSPG aggregation operators. The salient features of these operators are discussed in detail. Further, these operators are extended to its generalized version and called generalized IFSPA or geometric aggregation operators. Then, we utilized these operators to develop an approach to solve the decision-making problem under IF soft set environment and demonstrated with an illustrative example. A comparative analysis of existing approaches has been done for showing the validity of the proposed work.     

一种新的直觉模糊集的熵

通过分析现有直觉模糊熵计算方法中存在的一些问题,基于直觉模糊集理论和熵的概念,提出一种新的直觉模糊集的熵,给出它的结构、性质,并将其与现有的几种直觉模糊集的熵进行比较,发现其计算结果介于其他几种方法的计算结果之间,因而能够对模糊问题的模糊性做出更好的界定,有助于人们对模糊问题的理解和把握。最后,通过实例验证了它的有效性。     

基于直觉模糊Petri网的加权直觉模糊推理

利用直觉模糊集合较好地表现不确定信息的能力和Petri网的并行处理能力,构建了直觉模糊Petri网模型。给出了输入权值、变迁阈值等多种约束条件下的直觉模糊推理算法。该算法将直觉模糊推理过程转化为矩阵的运算过程可充分利用直觉模糊Petri网的并行推理能力,有效地避免同一变迁不必要地重复激发从而节省推理时间。实例分析表明所给出的直觉模糊推理算法较已有算法更加合理并且高效。     

[(I,J )]-区间直觉模糊粗糙集

利用吴伟志所定义的两个直觉模糊蕴涵算子[I ]和[J],把[(I,J )]-直觉模糊粗糙集的概念推广到区间直觉模糊集的情形,给出了区间直觉模糊近似空间的概念及[(I,J )]-区间直觉模糊粗糙集,研究了[(I,J )]-区间直觉模糊粗糙集的基本性质并推广了张植明等人提出的结论。     

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.     

直觉模糊集的熵及其一般形式

在综合分析目前直觉模糊集熵的各种公理化定义的基础上,认为直觉模糊集合熵应该是直觉模糊集合的模糊性与不确定性的综合性度量,并且直觉模糊集合退化为模糊集合时,其熵应该与模糊集合熵是一致的。根据这种思想,给出了新的直觉模糊集合熵的公理化定义,并给出了几个直觉模糊集合熵的构造形式。     

改进的直觉模糊粗糙集相似性度量方法

针对现有的直觉模糊粗糙集相似性度量的问题,提出了一种改进的基于海明距离的直觉模糊粗糙集相似性度量方法。该方法考虑了犹豫度并引入加权参数,解决了相似性度量不精确的问题。首先给出了直觉模糊粗糙值间的相似性度量定义,并揭示其若干重要性质。在此基础上,提出了直觉模糊粗糙集间的相似性度量方法,并证明其具有同样性质。最后通过数值算例分析说明了该方法更合理、更有效。     

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.