Matematik Bölümü / Department of Mathematicshttp://hdl.handle.net/20.500.11851/2612020-08-12T18:28:14Z2020-08-12T18:28:14ZSome Results On Hausdorff Metric For Intuitionistic Fuzzy SetsAkın, ÖmerBayeğ, Selamihttp://hdl.handle.net/20.500.11851/35662020-07-02T08:54:49Z2018-01-01T00:00:00ZSome Results On Hausdorff Metric For Intuitionistic Fuzzy Sets
Akın, Ömer; Bayeğ, Selami
Fuzzy set theory was frstly introduced by L. A. Zadeh in 1965 [1]. In fuzzy sets,
every element in the set is accompanied with a function μ(x): X → [0, 1], called
membership function. The membership function may have uncertainty in some
applications because of the subjectivity of the expert or the missing information in the
model. Hence some extensions of fuzzy set theory were proposed [2-4]. One of these
extensions is Atanassov’s intuitionistic fuzzy set (IFS) theory [2].
In 1986, Atanassov [2] introduced the concept of intuitionistic fuzzy sets and
carried out rigorous researches to develop the theory. In this set concept, he
introduced a new degree ν(x) : X → [0, 1], called non-membership function, such that
the sum μ+ν is less than or equal to 1. Hence the difference 1 −(μ+ν) is regarded as
degree of hesitation. Since intuitionistic fuzzy set theory contains membership
function, non-membership function and the degree of hesitation, it can be regarded
as a tool which is more flexible and closer to human reasoning in handling
uncertainty due to imprecise knowledge or data.
The Hausdorff metric distance between the alpha cuts of fuzzy numbers is one
of the most used metric on fuzzy set theory. It measures how far fuzzy numbers are
[5]. In this work we will give some results based on the maximum metric involving
Hausdorff metric to measure the distance between intuitionistic fuzzy numbers. We
will show that intuitionistic fuzzy numbers are complete under the maximum metric
based on Hausdorff metric. As a result we will prove that that space of continuous
intuitionistic fuzzy number valued functions are complete under this metric.
International Conference On Mathematics And Mathematics Educatİon ICME-2018, (27-28 June 2018: Ordu,Turkey)
2018-01-01T00:00:00ZAn Indicator Operator Algorithm for Solving a Second
Order Intuitionistic Fuzzy Initial Value ProblemAkın, ÖmerBayeğ, Selamihttp://hdl.handle.net/20.500.11851/35652020-07-02T09:36:42Z2016-01-01T00:00:00ZAn Indicator Operator Algorithm for Solving a Second
Order Intuitionistic Fuzzy Initial Value Problem
Akın, Ömer; Bayeğ, Selami
L. Zadeh [1] was the first who introduced the concept of fuzzy settheory as an
extension of the classical notion of the set theory. He remindedpeople that things are
not always black or white; there may besome grey colours in life. Hence he simply
assigned so called the membershipfunction to each element in a classical set and
started the fuzzyset theory. However in some cases the membership concept in a
fuzzyset is itself uncertain. This uncertainty may be because of the subjectivityof
expert knowledge, complexity of data or imprecision of themodels. Since the fuzzy
set theory considers only the degree of membership,it does not involve the degree of
uncertainty for the membership.To handle such situations, the generalized concepts
of fuzzy set theory are used [1]-[5]. One of these generalizations is intuitionistic fuzzy
settheory which was given by Atanassov [2]. Atanassov introduced intuitionisticfuzzy
set concept by extending the definition of fuzzy set afteranalyzing the shortcomings
of it. He defined the intuitionistic fuzzy setconcept by introducing the the nonmembership function into the fuzzyset such that sum of both is less than one. And in
his further researches,he showed the exclusive properties of intuitionistic fuzzy sets
[6]-[13].In the past decades, the intuitionistic fuzzy set theory has penetratedinto
different research areas, such as decision making [14]-[17], clusteringanalysis [18],
medical diagnosis [19]-[20], pattern recognition [21]-[23] In recent years the topic of
fuzzy differential equations has beenrapidly grown to model the real life situations
where the observed datais insufficient [24]-[27]. Especially to describe the relation
between velocityand acceleration, second order differential equations has
greaterimportance in science. Therefore many approaches [27]-[29] were givento
solve second order fuzzy differential equations. However there areonly few works [30]-[32] to observe the intuitionistic fuzzy differentialequations.In this work, we have
examined the solution of the following second orderintuitionistic fuzzy initial value
problems given in Eq. (1)-(2) usingintuitionistic Zadeh’s Extension Principle [13].
𝑦
′′(𝑥) + 𝑎1
𝑖 𝑦
′
(𝑥) + 𝑎2
𝑖 𝑦(𝑥) = ∑𝑏 ̅𝑗
𝑖𝑔𝑗(𝑥)
𝑟
𝑗=1
𝑦(0) = 𝛾 ̅0
𝑖
; 𝑦′(0) = 𝛾 ̅1
𝑖
Here 𝛾 ̅0
𝑖
; 𝛾 ̅1
𝑖 and 𝑏̅
𝑗
𝑖
(j=1, 2, 3,…, r ) are intuitionistic fuzzy numbers. 𝑔𝑗(𝑥) (j=1, 2,
3,…,r ) are continuous forcing functions on the interval [0, ∞).
International Conference On Mathematics And Mathematics Educatİon (ICMME-2016) (12-14 May 2016: Elazığ, Turkey)
2016-01-01T00:00:00ZSome Results on the Fundamental Concepts of Fuzzy Set Theory in Intuitionistic Fuzzy Environment by Using α and β cutsAkın, ÖmerBayeğ, Selamihttp://hdl.handle.net/20.500.11851/35642020-07-02T08:24:21Z2019-01-01T00:00:00ZSome Results on the Fundamental Concepts of Fuzzy Set Theory in Intuitionistic Fuzzy Environment by Using α and β cuts
Akın, Ömer; Bayeğ, Selami
In this paper we have frstly examined the properties of α and β cuts of intuitionistic fuzzy numbers in Rn with the help of well-known Stacking and Characterization theorems in fuzzy set theory. Then, we have studied the generalized Hukuhara diﬀerence in intuitionistic fuzzy environment by using the properties of α and β cuts and support function. Finally, we have extended the strongly generalized diﬀerentiability concept from fuzzy set theory to intuitionistic fuzzy environment and proved the related theorems with this concept.
2019-01-01T00:00:00ZApproximation by max-min operators: A general theory and its applicationsGökçer, Yeliz TürkanDuman, Oktayhttp://hdl.handle.net/20.500.11851/35632020-06-26T13:05:24Z2020-09-01T00:00:00ZApproximation by max-min operators: A general theory and its applications
Gökçer, Yeliz Türkan; Duman, Oktay
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.
2020-09-01T00:00:00Z