Measure of Roughness for Rough Approximation of Fuzzy Sets and Its Topological Interpretation

2014 ◽  
Author(s):  
Alexander Šostak
Keyword(s):  
Fuzzy Sets ◽  
Topological Interpretation ◽  
Rough Approximation

scholarly journals Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Haidong Zhang ◽  
Lan Shu ◽  
Shilong Liao
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Mathematical Tool ◽  
Intuitionistic Fuzzy ◽  
Rough Approximation ◽  
Soft Rough Set

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present concepts of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, and investigate some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Furthermore, classical representations of intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to intuitionistic fuzzy soft rough sets based on decision making and a numerical example is provided to illustrate the developed approach.

On the measure of M-rough approximation of L-fuzzy sets

2017 ◽  
Vol 22 (12) ◽  
pp. 3843-3855 ◽  
Author(s):  
Sang-Eon Han ◽  
Alexander Šostak
Keyword(s):  
Fuzzy Sets ◽  
Rough Approximation

scholarly journals Modified Uncertainty Measure of Rough Fuzzy Sets from the Perspective of Fuzzy Distance

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Jie Yang ◽  
Taihua Xu ◽  
Fan Zhao
Keyword(s):  
Fuzzy Sets ◽  
Rough Sets ◽  
Boundary Region ◽  
Uncertainty Measure ◽  
Approximation Spaces ◽  
Rough Sets Theory ◽  
Fuzzy Distance ◽  
Rough Approximation ◽  
Sets Theory

As an extension of Pawlak’s rough sets, rough fuzzy sets are proposed to deal with fuzzy target concept. As we know, the uncertainty of Pawlak’s rough sets is rooted in the objects contained in the boundary region, while the uncertainty of rough fuzzy sets comes from three regions (positive region, boundary region, and negative region). In addition, in the view of traditional uncertainty measures, the two rough approximation spaces with the same uncertainty are not necessarily equivalent, and they cannot be distinguished. In this paper, firstly, a fuzziness-based uncertainty measure is proposed. Meanwhile, the essence of the uncertainty for rough fuzzy sets and its three regions in a hierarchical granular structure is revealed. Then, from the perspective of fuzzy distance, we introduce a modified uncertainty measure based on the fuzziness-based uncertainty measure and present that our method not only is strictly monotonic with finer approximation spaces, but also can distinguish the two rough approximation spaces with the same uncertainty. Finally, a case study is introduced to demonstrate that the modified uncertainty measure is more suitable for evaluating the significance of attributes. These works are useful for further study on rough sets theory and promote the development of uncertain artificial intelligence.

Topological interpretation of fuzzy sets and intervals

2003 ◽  
Vol 135 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Rudolf F. Albrecht
Keyword(s):  
Fuzzy Sets ◽  
Topological Interpretation

Hesitant fuzzy β covering rough sets and applications in multi-attribute decision making

2021 ◽  
pp. 1-16
Author(s):  
Jia-Jia Zhou ◽  
Xiang-Yang Li
Keyword(s):  
Decision Making ◽  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Rough Sets ◽  
Fuzzy Information ◽  
Hesitant Fuzzy Sets ◽  
Application Model ◽  
Rough Approximation ◽  
Multi Attribute Decision Making ◽  
The Universe

 In present paper, we put forward four types of hesitant fuzzy β covering rough sets (HFβCRSs) by uniting covering based rough sets (CBRSs) and hesitant fuzzy sets (HFSs). We firstly originate hesitant fuzzy β covering of the universe, which can induce two types of neighborhood to produce four types of HFβCRSs. We then make further efforts to probe into the properties of each type of HFβCRSs. Particularly, the relationships of each type of rough approximation operators w.r.t. two different hesitant fuzzy β coverings are groped. Moreover, the relationships between our proposed models and some other existing related models are established. Finally, we give an application model, an algorithm, and an illustrative example to elaborate the applications of HFβCRSs in multi-attribute decision making (MADM) problems. By making comparative analysis, the HFβCRSs models proposed by us are more general than the existing models of Ma and Yang and are more applicable than the existing models of Ma and Yang when handling hesitant fuzzy information.

scholarly journals Distinguishing Vagueness from Ambiguity in Rough Set Approximations

2018 ◽  
Vol 26 (Suppl. 2) ◽  
pp. 89-125 ◽  
Author(s):  
Salvatore Greco ◽  
Benedetto Matarazzo ◽  
Roman Słowiński
Keyword(s):  
Fuzzy Sets ◽  
Rough Set ◽  
Rough Sets ◽  
The Other ◽  
New Approach ◽  
Variable Precision ◽  
Rough Approximation ◽  
Imperfect Knowledge ◽  
Disjoint Sets ◽  
The Universe

In this paper we present a new approach to rough set approximations that permits to distinguish between two kinds of “imperfect” knowledge in a joint framework: on one hand, vagueness, due to imprecise knowledge and uncertainty typical of fuzzy sets, and on the other hand, ambiguity, due to granularity of knowledge originating from the coarseness typical of rough sets. The basic idea of our approach is that each concept is represented by an orthopair, that is, a pair of disjoint sets in the universe of knowledge. The first set in the pair contains all the objects that are considered as surely belonging to the concept, while the second set contains all the objects that surely do not belong to the concept. In this context, following some previous research conducted by us on the algebra of rough sets, we propose to define as rough approximation of the orthopair representing the considered concept another orthopair composed of lower approximations of the two sets in the first orthopair. We shall apply this idea to the classical rough set approach based on indiscernibility, as well as to the dominance-based rough set approach. We discuss also a variable precision rough approximation, and a fuzzy rough approximation of the orthopairs. Some didactic examples illustrate the proposed methodology.

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