Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders

In this work we study a new class of similarity measures between interval-valued fuzzy sets. The novelty of our approach lays, firstly, on the fact that we develop all the notions with respect to total orders of intervals; and secondly, on that we consider the width of intervals so that the uncertainty of the output is strongly related to the uncertainty of the input. For constructing the new interval-valued similarity, interval valued aggregation functions and interval-valued restricted equivalence functions which take into account the width of the intervals are needed, so we firstly study these functions, both in line with the two above stated features. Finally, we provide an illustrative example which makes use of an interval-valued similarity measure in stereo image matching and we show that the results obtained with the proposed interval-valued similarity measures improve numerically (according to the most widely used measures in the literature) the results obtained with interval valued similarity measures which do not consider the width of the intervals

Interval Type-2 Fuzzy Sets are Generalization of Interval-Valued Fuzzy Sets: Toward a Wider View on Their Relationship

In this paper, we will present a wider view on the relationship between interval-valued fuzzy sets and interval type-2 fuzzy sets, where we will show that interval-valued fuzzy sets are a particular case of the interval type-2 fuzzy sets. For this reason, both concepts should be treated in a different way. In addition, the view presented in this paper will allow a more general perspective of interval type-2 fuzzy sets, which will allow representing concepts that could not be presented by interval-valued fuzzy sets

Grey sets and greyness

This paper discusses the application of grey numbers for uncertainty representation. It highlights the difference between grey sets and interval-valued fuzzy sets, and investigates the degree of greyness for grey sets. It facilitates the representation of uncertainty not only for elements of a set, but also the set itself as a whole. Our results show that a grey set could be specified for interval-valued fuzzy sets or rough sets under special conditions. With the notion of grey sets and their associated degrees of greyness, various set operations between grey sets are discussed

R- Fuzzy sets in factor space

Interval-valued fuzzy sets, both-branch fuzzy sets and R-Fuzzy sets are three extended forms of ordinary fuzzy sets. Compared to traditional fuzzy sets, interval-valued fuzzy sets have a stronger ability to express uncertainty because they provide more choices for the descriptions of attributes. Using both-branch fuzzy sets has solved some problems in engineering decision and engineering control. For R-Fuzzy sets, a new fuzzy system is constructed, with both-branch fuzzy sets as a special case. The resulting theorems improve the flexibility of uncertainty system models and expand the range of fuzzy system applications as well

Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets

AbstractWe begin the paper studying the axioms that the indicators of the grade of inclusion of a fuzzy set in another fuzzy set must satisfy. Next, we present an expression of such indicator, first for fuzzy sets and then for interval-valued fuzzy sets, analyzing in both cases their main properties. Then, we suggest an expression for the similarity measure between interval-valued fuzzy sets. Besides, we study two methods for inference in approximate reasoning based on interval-valued fuzzy sets, the inclusion grade indicator and the similarity measure. Afterwards, we expose some of the most important properties of the methods of inference presented and we compare these methods to Gorzalczany's. Lastly, we use the indicator of the grade of inclusion for interval-valued fuzzy sets as an element that selects from the different methods of inference studied, the one that will be executed in each case

Fast Computation of Centroids for Constant-Width Interval-Valued Fuzzy Sets

Interval-valued fuzzy sets provide a more adequate description of uncertainty than traditional fuzzy sets; it is therefore important to use interval-valued fuzzy sets in applications. One of the main applications of fuzzy sets is fuzzy control, and one of the most computationally intensive part of fuzzy control is defuzzification. Since a transition to interval-valued fuzzy sets usually increases the amount of computations, it is vitally important to design faster algorithms for the corresponding defuzzification. In this paper, we provide such an algorithm for a practically important case of constant-width interval-valued fuzzy set

Triangular norms which are join-morphisms in 3-dimensional fuzzy set theory

The n-dimensional fuzzy sets have been introduced as a generalization of interval-valued fuzzy sets, Atanassov's intuitionistic and interval-valued intuitionistic fuzzy sets. In this paper we investigate t-norms on 3-dimensional sets which are join-morphisms. Under some additional conditions we show that they can be represented using a representation which generalizes a similar representation for t-norms in interval-valued fuzzy set theory

On the incorporation of interval-valued fuzzy sets into the Bousi-Prolog system: declarative semantics, implementation and applications

In this paper we analyse the benefits of incorporating interval-valued fuzzy sets into the Bousi-Prolog system. A syntax, declarative semantics and im- plementation for this extension is presented and formalised. We show, by using potential applications, that fuzzy logic programming frameworks enhanced with them can correctly work together with lexical resources and ontologies in order to improve their capabilities for knowledge representation and reasoning

Interval Type-2 Fuzzy Sets are generalization of Interval-Valued Fuzzy Sets: Towards a Wider view on their relationship

Abstract-In this paper, we will present a wider view on the relationship between interval-valued fuzzy sets and interval type-2 fuzzy sets where we will show that interval-valued fuzzy sets are a particular case of the interval type-2 fuzzy sets. For this reason, both concepts should be treated in a different way. In addition, the view presented in this paper will allow a more general perspective of interval type-2 fuzzy sets which will allow representing concepts which could not be presented by intervalvalued fuzzy sets. Index Terms-Interval-valued fuzzy sets; Interval type-2 fuzzy sets; Type-2 fuzzy sets