A new interval-valued knowledge measure for interval-valued intuitionistic fuzzy sets and application in decision making

In this paper, firstly we discuss some entropy measures for the interval-valued intuitionistic fuzzy sets (IvIFSs). Then we extend the knowledge measure for the intuitionistic fuzzy sets (IFSs) to propose a new interval-valued knowledge measure for the IvIFSs. Based on the proposed knowledge measure we construct a new interval-valued information entropy measure for IvIFSs, which is an extended notion of the entropy measures for IFSs. The proposed knowledge measure is defined as an interval of amounts of knowledge measured on an IvIFS, related to the uncertain information in terms of interval membership degree and interval non-membership degree. In comparison with other existing measures, it seems to be simpler and more intuitively appealing. Several illustrative examples are performed to demonstrate the effectiveness and practicality of the proposed method in handling with the increasing complexity of the decision making problems.     

直觉区间值模糊集的熵、距离测度和相似测度

熵、距离测度和相似测度是模糊集理论中的三个基本概念.本文中,我们给出了直觉区间值模糊集的熵、距离测度和相似测度的公理化定义并讨论了它们之间的基本关系.有别于以往的交、并运算,我们提出了直觉区间值模糊集的加法、乘法运算,并在此基础上讨论了直觉区间值模糊集的熵、距离测度和相似测度的性质.     

区间犹豫模糊熵和区间犹豫模糊相似度

基于犹豫模糊熵的概念,提出了区间犹豫模糊熵和相似度的概念,同时研究了它们之间的相互关系。给出了区间犹豫模糊熵的公理化定义,在此基础上构造了两种形式的熵测度公式,并且证明了它们满足区间犹豫模糊熵的四条公理化准则;依据区间犹豫模糊熵引入了区间犹豫模糊加权熵的概念;提出了区间犹豫模糊相似度的概念,并且研究了区间犹豫模糊环境下的熵和相似度之间的关系。     

Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship

In this paper, we introduce an axiomatic definition of an interval-valued fuzzy sets’ inclusion measure which is different from Bustince’s [H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets, Applications to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23 (2000) 137-209]. The relationship among the normalized distance, the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets is investigated in detail. Furthermore, six theorems are proposed showing how the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets can be deduced by the interval-valued fuzzy sets’ normalized distance based on their axiomatic definitions. Some formulas have also been put forward to calculate the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets.     

Interval-valued intuitionistic fuzzy cosine similarity measures for multiple attribute decision-making

In this work, considering all information carried by the membership degrees, nonmembership degree, and hesitancy degree in interval-valued intuitionistic fuzzy sets (IVIFSs) as 3-D vector representations, we propose a cosine similarity measure and a weighted cosine similarity measure for IVIFSs based on the extension of the cosine similarity measure (angular coefficient) between intuitionistic fuzzy sets in 2-D vector space. Then, the weighted cosine similarity measure for IVIFSs is applied to multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. Through the similarity measure between the ideal alternative and each alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, two illustrative examples are given to demonstrate the applications and efficiency of the proposed decision-making method.     

一种区间直觉模糊熵公式及其应用

改进了区间直觉模糊集的标准Hamming距离公式,使改进后的公式不仅考虑了区间直觉模糊集的隶属度和非隶属度,还考虑了犹豫度。基于该距离公式定义了一个区间直觉模糊集的熵公式。将提出的距离和熵公式应用于解决属性权重未知的区间直觉模糊多属性决策问题。     

A method based on distance measure for interval-valued intuitionistic fuzzy group decision making

In this paper we introduce some relations and operations of interval-valued intuitionistic fuzzy numbers and define some types of matrices, including interval-valued intuitionistic fuzzy matrix, interval-valued intuitionistic fuzzy similarity matrix and interval-valued intuitionistic fuzzy equivalence matrix. We study their properties, develop a method based on distance measure for group decision making with interval-valued intuitionistic fuzzy matrices and, finally, provide an illustrative example.     

一种区间直觉模糊熵的构造方法

研究了区间直觉模糊熵。证明了三个直觉模糊熵公式的等价性。对直觉模糊熵公式进行推广,引入一个新的区间直觉模糊熵公式,该熵公式满足区间糊熵的公理化直觉模定义的4个条件。     

Weight computation of criteria in a decision-making problem by knowledge measure with intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set

The aim of this study is to propose an objective method for determining weights of criteria (also called attributes) based on a new measure of intuitionistic fuzzy information, called knowledge measure, in a real-world multi-criteria decision-making problem under intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. To address this issue, we first analyze the existing entropy measures and show that their use in objective weight determination process may lead us to produce unreliable weights of criteria by citing appropriate examples. Then we analyze important properties of knowledge measure of intuitionistic fuzzy set (IFS) and also define knowledge measure for interval-valued intuitionistic fuzzy set. Then a new method to determine the weights of criteria is developed on the basis of knowledge measure where information about criteria weights is completely unknown and partly known. A real-life example is presented to illustrate the proposed weight determination method and a comparative analysis is carried out to indicate the practicality and effectiveness of knowledge-based weight-generation method under both intuitionistic fuzzy and interval-valued intuitionistic fuzzy environment. Finally, we formulate the axioms for knowledge measure associated with IFSs and we also propose families (classes) of knowledge measures.     

区间值模糊软集的信息测度及其聚类算法

针对区间值模糊软集信息测度难以精确定义的问题,提出了区间值模糊软集的距离测度、相似度、熵、包含度、子集度的公理化定义,给出了区间值模糊软集的信息测度公式,并讨论了它们的转换关系。然后提出了一个基于相似度的聚类算法,该算法结合区间值模糊软集的特性,着重对给出评价对象的具有相似知识水平的专家进行聚类,同时讨论了算法的计算复杂度。最后通过实例说明该算法能有效地处理专家聚类问题。     

覆盖粗糙直觉模糊集模型的研究

粗糙集和直觉模糊集的融合是一个研究热点。在粗糙集、直觉模糊集和覆盖理论基础上,给出了模糊覆盖粗糙隶属度和非隶属度的定义。考虑到元素自身与最小描述元素的隶属度和非隶属度之间的关系,构建了两种新的模型——覆盖粗糙直觉模糊集和覆盖粗糙区间值直觉模糊集,证明了这两种模型的一些重要性质,与此同时定义了一种新的直觉模糊集的相似性度量公式,并用实例说明其应用。     

Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure

This article proposes a new axiomatic definition of entropy of interval-valued fuzzy sets (IVFSs) and discusses its relation with similarity measure. First, we propose an axiomatic definition of entropy for IVFS based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by De Luca, Termini and Liu. Next, some formulae are derived to calculate this kind of entropy. Furthermore we investigate the relationship between entropy and similarity measure of IVFSs and prove that similarity measure can be transformed by entropy. Finally, a numerical example is given to show that the proposed entropy measures are more reasonable and reliable for representing the degree of fuzziness of an IVFS.     

区间直觉模糊连续熵及其多属性决策方法

提出了区间直觉模糊连续熵,并且研究了一种新的处理区间直觉模糊多属性决策问题的方法。基于连续有序加权平均（COWA）算子,给出了区间直觉模糊连续熵的概念,并且证明了区间直觉模糊连续熵满足区间直觉模糊熵的公理化定义的四个条件。在此基础上,针对属性权重信息完全未知的决策问题,通过衡量每一属性所含的信息量来确定属性权重。依据备选方案与理想方案间的加权相关系数,给出了一种新的区间直觉模糊多属性决策方法。实验结果验证了新的决策方法的可行性和有效性。     

基于信息测度的勾股模糊决策模型及其应用

相对于直觉模糊集，勾股模糊集能够更为全面和有效地表达描述复杂问题中的不确定和非一致信息，使其受到了广泛研究。对于属性评价值为勾股模糊数并且属性指标权重信息数据完全未知的多属性决策问题，以提出的勾股模糊信息测度为基础，设计了新的多属性决策模型。该模型运用对数函数设计了一种新的勾股模糊数信息熵计算方法；引入了勾股模糊相似度概念，并结合对数行数提出勾股模糊数相似度的衡量方法，随后挖掘出勾股模糊数的信息熵和相似度之间的内在联系；运用提出的勾股模糊熵和相似度计算方法，构建新的多属性决策模型，并进行应用研究。实验结果表明，提出的模型合理有效，同时拓展了模型的使用范围。     

一种新的区间直觉模糊熵及其应用

提出了区间值直觉模糊集的区间直觉模糊交叉熵,这种交叉熵充分考虑了区间值直觉模糊集的隶属度,非隶属度以及犹豫度。给出一种区间值直觉模糊集的区间直觉模糊熵的公理化体系,并且基于直觉模糊交叉熵公式给出一种区间直觉模糊熵的具体测度公式。利用区间值直觉模糊集的加权相关系数,将提出的熵公式应用于解决属性权重完全未知的区间直觉模糊多属性决策问题。     

A note on measuring fuzziness for intuitionistic and interval-valued fuzzy sets

We first review the concept of fuzziness provided by DeLuca and Termini, and the view of fuzziness as related to the lack of distinction between a set and its negation. We then suggest a measure of fuzziness for intuitionistic fuzzy sets. We show that this measure satisfies the required conditions for an intuitionistic measure of fuzziness. We then take advantage of the connection between intuitionistic and interval-valued membership grades to transfer the intuitionistic measure of fuzziness to one suitable for interval-valued fizzy sets.     

用区间直觉模糊集方法对属性权重未知的群求解其多属性决策

本文首先提出群区间直觉模糊有序加权几何(groupinterval-valuedintuitionistic fuzzy orderedweighted geometric,GIVIFOWG)算子和群区间直觉模糊有序加权平均(group interval-valued intuitionistic fuzzy ordered weighted averaging,GIVIFOWA)算子.利用GIVIFOWG算子或GIVIFOWA算子聚集群的决策矩阵以获得方案在属性上的综合区间直觉模糊决策矩阵(collectiveinterval-valuedintuitionistic fuzzy decision-matrix,CIVIFDM).然后定义了一个考虑犹豫度的区间直觉模糊熵(interval-valuedintuitionistic fuzzyentropy,IVIFE);通过熵衡量每个属性所含的信息来求解属性权重.最后,提出基于可能度的接近理想解的区间排序法(interval technique for order preference by similarity to an ideal solution,ITOPSIS)和区间得分函数法.在ITOPSIS法中,依据区间距离公式计算候选方案和理想方案的属性加权区间距离,进而采用ITOPSIS准则对各方案进行排序;在区间得分函数法中,算出CIVIFDM中各方案的得分值以及精确值,然后利用区间得分准则对各方案进行排序.实验结果验证了决策方法的有效性和可行性.     

Some information measures for interval-valued intuitionistic fuzzy sets

A new information entropy measure of interval-valued intuitionistic fuzzy set (IvIFS) is proposed by using membership interval and non-membership interval of IvIFS, which complies with the extended form of Deluca-Termini axioms for fuzzy entropy. Then the cross-entropy of IvIFSs is presented and the relationship between the proposed entropy measures and the existing information measures of IvIFSs is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed entropy and cross-entropy of IvIFSs to pattern recognition and decision-making.     

基于改进熵和新得分函数的区间直觉模糊多属性决策

针对决策信息为区间直觉模糊数且属性权重完全未知的多属性决策问题, 提出基于改进的区间直觉模糊熵和新得分函数的决策方法. 首先, 利用改进的区间直觉模糊熵确定属性权重; 然后, 利用区间直觉模糊加权算术平均算子集成信息, 得到各备选方案的综合属性值, 进而指出现有得分函数存在排序失效或排序不符合实际的不足, 同时给出一个新的得分函数, 并以此对方案进行排序; 最后, 通过实例表明了所提出方法的有效性.