On the calculation of extended max and min operations between convex fuzzy sets of the real line

doi:10.1016/j.fss.2009.06.005

Fuzzy Sets and Systems

Volume 160, Issue 21, 1 November 2009, Pages 3103-3114

https://doi.org/10.1016/j.fss.2009.06.005 Get rights and content

Abstract

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.

References (27)

R. Ambrosio et al.
Maximum and minimum between fuzzy symbols in non-interactive and weakly non-interactive situations
Fuzzy Sets and Systems
(1984)
C.H. Chiu et al.
A simple computation of MIN and MAX operations for fuzzy numbers
Fuzzy Sets and Systems
(2002)
S. Coupland et al.
New geometric inference techniques for type-2 fuzzy sets
International Journal of Approximate Reasoning
(2008)
D. Dubois et al.
Operations in a fuzzy-valued logic
Information and Control
(1979)
Z. Gera et al.
Exact calculations of extended logical operations on fuzzy truth values
Fuzzy Sets and Systems
(2008)
M. Mizumoto et al.
Some properties of fuzzy sets of type-2
Information and Control
(1976)
J.T. Starczewski
Extended triangular norms
Information Sciences
(2009)
C.L. Walker et al.
The algebra of fuzzy truth values
Fuzzy Sets and Systems
(2005)
L.A. Zadeh
Fuzzy sets
Information and Control
(1965)
L.A. Zadeh
The concept of a linguistic variable and its application to approximate reasoning—I
Information Sciences
(1975)

L.A. Zadeh

The concept of a linguistic variable and its application to approximate reasoning—II

Information Sciences

(1975)

R. Boukezzoula et al.

Min and max operators for fuzzy intervals and their potential use in aggregation operators

IEEE Transactions on Fuzzy Systems

(2007)

C.H. Chiu, The study of fuzziness for fuzzy sets, Ph.D. Thesis,...

Cited by (27)

Approximation by Kantorovich-type max-min operators and its applications
2022, Applied Mathematics and Computation
In this study, we construct Kantorovich variant of max-min kind operators, which are nonlinear. By using these new operators, we obtain some uniform approximation results in $N$ -dimension ( $N \geq 1$ ). Then, we estimate the error with the help of Hölder continuous functions and modulus of continuity. Furthermore, we give some illustrative applications to verify our theory and also investigate some shape-preserving properties of Kantorovich-type max-min Bernstein operator. Lastly, we examine the image processing implementation of our results via Kantorovich-type max-min Shepard operator.
Regular summability methods in the approximation by max-min operators
2022, Fuzzy Sets and Systems
Citation Excerpt :
The main motivation for this study is to develop our recent results regarding max-min operators in [22]. Although max-min operations are often used in fuzzy logic theory (see, for instance, [15,24,34,36]), there are only a few works including them in approximation theory (see [11,22]). We know that max-min operators verify the pseudo-linearity condition which is a weaker concept than the usual linearity (see, for details, [11]).
In this paper, by using nonnegative regular summability methods we improve and generalize the approximation properties of max-min operators which have been investigated systematically in our recent paper published in 2020 in this journal. We also discuss the rate of convergence in the approximation. Applications and concluding remarks at the end of the paper explain why we need such summability methods.
Approximation by max-min operators: A general theory and its applications
2020, Fuzzy Sets and Systems
Citation Excerpt :
In the present paper, inspired mainly from the paper [12], we study a general form of pseudo linear operators constructed with the classical max-min operations. We should note that max-min operations are frequently used in the theory of fuzzy logic (see, for instance, [13,28,30,31]. Now we use them in the approximation theory.
In this study, we obtain a general approximation theorem for max-min operators including many significant applications. We also study the error estimation in this approximation by using Hölder continuous functions. The main motivation for this work is the paper by Bede et al. (2008) [12]. As a special case of our results, we explain how to approximate nonnegative continuous functions of one and two variables by means of the max-min Shepard operators. We also study the approximation by the max-min Bernstein operators. Furthermore, to verify the theory we display graphical illustrations.
On the extension of nullnorms and uninorms to fuzzy truth values
2018, Fuzzy Sets and Systems
Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions
2014, Applied Soft Computing Journal
Citation Excerpt :
T2FS enables handling additional levels of uncertainty by introducing the fuzzy membership function, which characterizes the membership value of an element as a T1FS [29,47]. Despite various efforts on making the cost of using T2FSs affordable, e.g., see [47,41,48,50,12,14,11], to compensate the computational complexities of T2FSs some variations are proposed; notably interval T2FS (IT2FS) [30] and shadowed fuzzy set (SFS) [43,45]. In IT2FS, the membership grade of an element is an interval that enables modeling the first degree of uncertainty.
In this paper, we deal with the problem of classification of interval type-2 fuzzy sets through evaluating their distinguishability. To this end, we exploit a general matching algorithm to compute their similarity measure. The algorithm is based on the aggregation of two core similarity measures applied independently on the upper and lower membership functions of the given pair of interval type-2 fuzzy sets that are to be compared. Based on the proposed matching procedure, we develop an experimental methodology for evaluating the distinguishability of collections of interval type-2 fuzzy sets. Experimental results on evaluating the proposed methodology are carried out in the context of classification by considering interval type-2 fuzzy sets as patterns of suitable classification problem instances. We show that considering only the upper and lower membership functions of interval type-2 fuzzy sets is sufficient to (i) accurately discriminate between them and (ii) judge and quantify their distinguishability.
Multivariate modeling and type-2 fuzzy sets
2011, Fuzzy Sets and Systems
This paper explores the link between type-2 fuzzy sets and multivariate modeling. Elements of a space $X$ are treated as observations fuzzily associated with values in a multivariate feature space. A category or class is likewise treated as a fuzzy allocation of feature values (possibly dependent on values in $X$ ). We observe that a type-2 fuzzy set on $X$ generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations. In practice many type-2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task. We then show that an arbitrary type-2 fuzzy set can be so constructed, by taking as a feature space a set of membership functions on $X$ . This construction presents a new perspective on the Representation Theorem of Mendel and John. The multivariate modeling underpinning the type-2 fuzzy sets can also constrain realizable forms of membership functions. Because averaging operators such as centroid and subsethood on type-2 fuzzy sets involve a search for optima over membership functions, constraining this search can make computation easier and tighten the results. We demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool, alongside standard operations such as intersection and subsethood, for concept fusion and computing with words.

View all citing articles on Scopus

View full text

On the calculation of extended max and min operations between convex fuzzy sets of the real line

Abstract

Fuzzy Sets and Systems

Fuzzy Sets and Systems

International Journal of Approximate Reasoning

Information and Control

Fuzzy Sets and Systems

Information and Control

Information Sciences

Fuzzy Sets and Systems

Information and Control

Information Sciences

Information Sciences

Min and max operators for fuzzy intervals and their potential use in aggregation operators

IEEE Transactions on Fuzzy Systems