Soft rough fuzzy sets and soft fuzzy rough sets

Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed.     

Sine-square embedded fuzzy sets versus type-2 fuzzy sets

The uncertainty is an inherent part of real-world applications. Type-2 fuzzy sets minimize the effects of uncertainties that cannot be modeled using type-1 fuzzy sets. However, the computational complexity of the type-2 fuzzy sets is very high and it is more difficult than type-1 fuzzy sets to use and understand. This paper proposes sine-square embedded fuzzy sets and gives a comparison with type-2 and nonstationary fuzzy sets. The sine-square embedded fuzzy sets consist of type-1 fuzzy sets and the sine function. The footprint of uncertainty in the type-2 fuzzy sets is provided with amplitude and frequency of sine-square function in the proposed algorithm. The proposed sine-square embedded fuzzy sets are much simpler than the type-2 fuzzy sets and the nonstationary fuzzy sets. Two control applications that are chosen as position control of a dc motor and simulation of human lifting motion using five-segment human model are carried out to demonstrate the effectiveness of the proposed approach.     

Superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems

We analyse P-superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems and we derive a simple estimation for the fuzzy sets. This paper presents a method to calculate an optimal value for Snell's optimal stopping problem.     

Hesitant fuzzy sets

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.     

Ordinal fuzzy sets

Fuzzy set theory has been used as a framework for interpreting imprecise linguistic expressions. In general, a linguistic term is described by the compatibility ordering induced in some universe of discourse (UoD). A membership function in fuzzy set theory serves to reflect this ordering by assignment of values in [0, 1] for objects in UoD. When we compute the meaning of a linguistic expression such as "young and tall" using fuzzy membership functions, two implicit assumptions are made. First, we assume the membership values have quantitative meaning so that they can be quantitatively manipulated, for example, by adding or subtracting (the extensive scale assumption). Second, we assume that the scales of the membership values used in describing the different linguistic terms are comparable and the same (the common scale assumption). In many cases, these assumptions cannot be justified. Some proposals have been made to address the first issue by using ordinal scale in defining fuzzy membership functions. However, the second issue has not been properly investigated. In this paper, we propose a framework that does not depend on both of these assumptions. Such framework will facilitate our understanding and investigation of qualitative reasoning without the extensive scale and common scale assumptions.     

Balanced fuzzy sets

The paper presents a new approach to fuzzy sets and uncertain information based on an observation of asymmetry of classical fuzzy operators. Parallel is drawn between symmetry and negativity of uncertain information. The hypothesis is raised that classical theory of fuzzy sets concentrates the whole negative information in the value 0 of membership function, what makes fuzzy operators asymmetrical. This hypothesis could be seen as a contribution to a broad range discussion on unification of aggregating operators and uncertain information processing rather than an opposition to other approaches. The new approach “spreads” negative information from the point 0 into the interval [−1, 0] making scale and operators symmetrical. The balanced counterparts of classical operators are introduced. Relations between classical and balanced operators are discussed and then developed to the hierarchies of balanced operators of higher ranks. The relation between balanced norms, on one hand, and uninorms and nullnorms, on the other, are quite close: balanced norms are related to equivalence classes of some equivalence relation build on linear dependency in the spaces of uninorms and nullnorms. It is worth to stress that this similarity is raised by two entirely different approaches to generalization of fuzzy operators. This observation validates the generalized hierarchy of fuzzy operators to which both approaches converge. The discussion in this paper is aimed at presenting the idea and does not aspire to detailed exploration of all related aspects of uncertainty and information processing.     

Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology

Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another extension concerns bipolar fuzzy sets, where both positive information and negative information are handled, along with their imprecision. We detail these extensions from the point of view of the underlying lattice structure. In the case of bipolarity, its two-components nature raises the question of defining a proper partial ordering. In this paper, we consider Pareto (component-wise) and lexicographic orderings.     

犹豫模糊软集

犹豫模糊集是对模糊集的一种推广,它是一类关于域中每个元素所含隶属度的集合,常应用于群决策中,但由于其本身在参数工具上的缺乏使得难于处理不确定数据。为了提高决策的精确性,将软集与犹豫模糊集结合起来,提出犹豫模糊软集的概念,并给出犹豫模糊软集的基本运算法则和性质。     

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。     

多值直觉模糊软集

将多值直觉模糊集和软集相结合,提出了多值直觉模糊软集的概念。给出了多值直觉模糊软集的补、交、并“、且”“、或”运算的概念,并基于这些概念研究了若干性质。     

双极值模糊软集

提出了双极值模糊软集的概念,给出双极值模糊软集的补、并、交、"且"及"或"运算,并讨论了它们的性质。     

Distances between fuzzy sets

Given two fuzzy subsets μ and ν of a metric space S (e.g., the Euclidean plane), we define the ‘shortest distance’ between μ and ν as a density function on the non-negative reals; our definition is applicable both when μ and ν are discrete-valued and when they are ‘smooth’ (i.e., differentiable), and it generalizes the definition of shortest distance for crisp sets in a natural way. We also define the mean distance between μ and ν, and show how it relates to the shortest distance. the relationship to earlier definitions of distance between fuzzy sets [1,3] is also discussed.     

Categories of fuzzy sets

In this paper we look at two categories, the category of fuzzy subsets and a quotient category / of fuzzy sets. The category / is an extension of the category of sets, and the standard constructions in fuzzy set theory arise naturally within this category. This paper was presented at a workshop in the Czech Republic, hosted by Professors Vilém Novák and Irina Perfilieva. I thank them for their warm hospitality and the opportunity to participate in an inspiring workshop.     

Generalised fuzzy soft sets

In this paper, we define generalised fuzzy soft sets and study some of their properties. Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown.     

软模糊粗糙集

软集理论是1999年Molodtsov为了克服传统数学在处理不确定性问题时所遇到的困难而提出的一种新的数学工具。将软集理论与Z.Pawlak粗糙集结合起来,提出了软模糊粗糙集和软模糊粗糙群及它们的同态的概念,讨论了它们相关的性质。     

Shadowed sets: representing and processing fuzzy sets

This study introduces a new concept of shadowed sets that can be regarded as a certain operational framework simplifying processing carried out with the aid of fuzzy sets and enhancing interpretation of results obtained therein. Some conceptual links between this idea and some others known in the literature are established. In particular, it is demonstrated how fuzzy sets can induce shadowed sets. Subsequently, shadowed sets reveal interesting conceptual and algorithmic relationships existing between rough sets and fuzzy sets. Detailed computational aspects of shadowed sets are discussed. Several illustrative examples are provided.     

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.     

一种Vague集转化为Fuzzy集的方法

已存在的Vague集向Fuzzy集转化的多种模型都存在着一定缺陷。论文从另一种角度探讨了Vague集到Fuzzy集的转化,它把模糊隶属度看成是参数k的函数,这样在计算模糊隶属度时,只需给定满足条件的参数k即可;另一方面,在进行模糊信息处理时,会遇到一些不确定因素,如果所得到的模糊隶属度与实际结果有些偏差,可根据参数k的数学性质有目的地调整k,而不必去重新建立方法模型,因此,此方法比已存在方法灵活,适用;文章也对已有的转化方法与所提模型进行了比较,说明了所提模型的正确性及有效性。