Normal fuzzy probability for generalized triangular fuzzy sets

Chul Kang(강철),Yong Sik Yun(윤용식) 한국지능시스템학회 2012 한국지능시스템학회논문지 Vol.22 No.2

확률공간 (Ω,￡,Ρ) 위에 정의된 퍼지집합을 퍼지이벤트라 한다. Zadeh는 확률 Ρ를 이용하여 퍼지이벤트 A에 대한 확률을 정의하였다. 우리는 일반화된 삼각퍼지집합을 정의하고 거기에 확장된 대수적 작용소를 적용하였다. 일반화된 삼각퍼지집합은 대칭적이지만 함숫값으로 1을 갖이 않을 수 있다. 두 개의 일반화된 삼각퍼지집합 A와 B에 대하여 A(+)B와 A(-)B는 일반화된 사다리꼴퍼지집합이 되었지만, A(·)B와 A(/)B는 일반화된 삼각퍼지집합도 되지 않았고 일반화된 사다리꼴퍼지집합도 되지 않았다. 그리고 정규분포를 이용하여 R 위에서 정규퍼지확률을 정의하였다. 그리고 일반화된 삼각퍼지집합에 대한 정규퍼지확률을 계산하였다. A fuzzy set A defined on a probability space (Ω,￡. P) is called a fuzzy event. Zadeh defines the probability of the fuzzy event A using the probability P. We generalized triangular fuzzy set and apply the extended algebraic operations to these fuzzy sets. A generalized triangular fuzzy set is symmetric and may not have value 1. For two generalized triangular fuzzy sets A and B, A(+)B and A(-)B become generalized trapezoidal fuzzy sets, but A(·)B and A(/)B need not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set. We define the normal fuzzy probability on R using the normal distribution. And we calculate the normal fuzzy probability for generalized triangular fuzzy sets.

Some remarks on fuzzy Baire sets

G. Thangaraj,N. Raji 원광대학교 기초자연과학연구소 2023 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.26 No.2

In this paper, the means by which fuzzy Baire sets are obtained from fuzzy simply$^{\ast}$ open sets in fuzzy hyperconnected spaces are discussed. It is obtained that fuzzy Baire sets in fuzzy fraction dense spaces are not fuzzy dense sets. Also the conditions under which fuzzy Baire sets are generated from fuzzy nowhere dense sets, fuzzy dense and fuzzy open sets in fuzzy fraction dense and fuzzy DG$_\delta$-spaces are obtained. It is established that existence of a fuzzy co-$\sigma$-boundary set in fuzzy weakly Baire spaces ensures the existence of a pair of disjoint fuzzy Baire sets and fuzzy open sets in fuzzy hyperconnected and fuzzy nodef spaces are fuzzy Baire sets.

다각형 구간 타입-2 퍼지집합을 이용한 퍼지 공학시스템의 신뢰도 분석

조상엽 한국지식정보기술학회 2022 한국지식정보기술학회 논문지 Vol.17 No.5

공학시스템을 운영하는 경우 사용자의 실수, 입력 오류, 입력 데이터의 불확실성 등에 의해 신뢰도가 떨어지는 출력을 생산하게 된다. 이러한 문제를 제어하기 위한 방법은 시스템의 구성요소의 신뢰도를 반영하여 전체 공학시스템이 신뢰도를 출력에 반영하는 것이다. 본 논문에서는 공학시스템의 신뢰도를 평가하기 위하여 퍼지집합 이론을 기반으로 신뢰도를 계산하는 방법을 사용한다. 퍼지집합의 종류에는 소속값을 실수로 표현하는 전통적인 퍼지집합, 과 소속값을 구간으로 표현하는 직관퍼지집합, 구간으로 표현되는 소속값의 상한은 1로 고정하고 하한을 λ로 표현하는 수준 (λ,1) 구간값 퍼지집합, 불확정성을 표현할 수 있는 뉴트로소픽 집합, 다양한 퍼지집합을 표현하기 위한 다각형 타입-1 퍼지집합, 전통적인 퍼지지합의 소속값을 소속함수로 표현하는 타입-2퍼지집합 등이 제안되었다. 본 연구에서 우리는 다각형 구간 타입-2 퍼지집합을 이용하여 공학시스템의 신뢰도를 평가하는 방법을 제안한다. 구간 타입-2 퍼지집합은 이차 소속값이 모두 1과 같은 값을 가진다. 다각형 구간 타입-2 퍼지집합은 하한 소속함수와 상한 소속함수가 다각형으로 표현된다. 그러므로 기존의 퍼지집합들 보다 더 다양한 모양의 퍼지집합을 표현하는 것이 가능하게 된다. In the case of operating the engineering systems, the low reliable output of the systems may produced due to users error, input error, uncertainty of input data, etc. The way to overcome these problem is that the reliability of the entire engineering systems is reflected in the output by reflecting the reliability of the components of the systems. In this paper, in order to analyze the reliability of the engineering systems, we use the fuzzy set theory for calculating the reliability. The fuzzy sets have various types of sets such as the conventional fuzzy sets that represents the degree of membership as a real number, the intuitive fuzzy sets that expresses the degree of membership as an interval, the level (λ,1) interval valued fuzzy set that describe the upper bound of interval is fixed at 1 and the lower bound represent λ, the neutrosophic sets that can express the indeterminacy, the polygonal type-1 fuzzy sets that can describe various shape of conventional fuzzy sets, the type-2 fuzzy sets that represent the membership degree of conventional fuzzy sets as the membership function. In this study, we propose a method to analysis the reliability of engineering systems using polygonal interval type-2 fuzzy sets. In the interval type-2 fuzzy sets, secondary grades are all equal to 1. In the polygon interval type-2 fuzzy sets, the lower membership function and the upper membership function are expressed as polygons. Therefore, it becomes possible to represent fuzzy sets of more various shapes than the conventional fuzzy sets.

On fuzzy maximal, minimal and mean open sets

A. Swaminathan,S. Sivaraja 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.1

We have observed that there exist certain fuzzy topological spaces with no fuzzy minimal open sets. This observation motivates us to investigate fuzzy topological spaces with neither fuzzy minimal open sets nor fuzzy maximal open sets. We have observed if such fuzzy topological spaces exist and if it is connected are not fuzzy cut-point spaces. We also study and characterize certain properties of fuzzy mean open sets in fuzzy T_1-connected fuzzy topological spaces.

Optimization of Fuzzy Set-Fuzzy Systems based on IG by Means of GAs with Successive Tuning Method

Keon-Jun Park,Sung-Kwun Oh,Hyun-Ki Kim 대한전기학회 2008 Journal of Electrical Engineering & Technology Vol.3 No.1

We introduce an optimization of fuzzy set-fuzzy systems based on IG (Information Granules). The proposed fuzzy model implements system structure and parameter identification by means of IG and GAs. The concept of information granulation was coped with to enhance the abilities of structural optimization of the fuzzy model. Granulation of information realized with C-Means clustering helps determine the initial parameters of the fuzzy model such as the initial apexes of the membership functions in the premise part and the initial values of polynomial functions in the consequence part of the fuzzy rules. The initial parameters are adjusted effectively with the help of the GAs and the standard least square method. To optimally identify the structure and the parameters of the fuzzy model we exploit GAs with successive tuning method to simultaneously search the structure and the parameters within one individual. We also consider the variant generation-based evolution to adjust the rate of identification of the structure and the parameters in successive tuning method. The proposed model is evaluated with the performance of the conventional fuzzy model.

On fuzzy sets having the fuzzy Baire property in fuzzy topological spaces

G. Thangaraj,N. Raji 원광대학교 기초자연과학연구소 2021 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.21 No.2

In this paper, fuzzy sets having the property of fuzzy Baire in the fuzzy topological spaces are introduced by means of fuzzy first category sets. The conditions for the existence of fuzzy Baireness and fuzzy resolvability of fuzzy topological spaces are established by means of fuzzy sets having the property of fuzzy Baire. It is established that there are no fuzzy sets having the property of fuzzy Baire in fuzzy hyperconnected spaces and fuzzy sets having the property of fuzzy Baire in fuzzy Baire spaces are fuzzy second category sets.

타입-2 퍼지집합을 이용한 퍼지 공학시스템의 신뢰도 분석

전병찬,조상엽 한국지식정보기술학회 2022 한국지식정보기술학회 논문지 Vol.17 No.5

공학시스템의 설계에서 시스템의 신뢰도를 보장하는 것은 중요한 주제 중에 하나이다. 공학시스템의 신뢰도는 하위시스템의 순차 또는 병렬 구성요소를 반영하여 각각 방법으로 신뢰도를 계산한 후, 전체 공학시스템의 신뢰도를 평가하게 되면 시스템의 신뢰도를 계산하는 것이 가능하게 된다. 본 연구에서는 타입-2 퍼지집합을 이용하여 공학시스템의 신뢰도를 분석하는 방법을 제안한다. 타입-2 퍼지집합은 퍼지집합을 삼차원적으로 표현하므로 기존의 퍼지집합-퍼지집합, 직관 퍼지집합, 모호집합, 수준 (λ,1) 구간값 퍼지집합, 뉴트로소픽 집합, 팔각형 직관 퍼지집합, 다각형 타입-1 퍼지집합 등-보다 더 다양한 형태의 퍼지집합을 표현하는 것이 가능하다. 또한 소속값을 단위 구간 [0, 1] 사이의 임의의 값을 가질 수 있는 이차 소속값으로 표현하므로 신뢰도를 하나의 실수로 표현하는 타입-1 퍼지집합보다 소속값을 더 유연하게 표현하는 것이 가능하다. 그리고 신뢰도공학에서 제시하는 가능성척도 기반의 두 가지 가정을 수용하므로 신뢰도를 평가하는데 타입-2 퍼지집합을 사용하는 이론적 근거를 가지고 있다. 타입-2 퍼지집합을 이용하여 신뢰도를 계산하는 방법은 퍼지 공학시스템, 퍼지 의사결정시스템, 퍼지 신뢰도공학, 퍼지 추론시스템 등에 적용하는 것이 가능하다. The guarantee for the reliability of systems in the design of engineering systems is one of the important issues. The reliability of the engineering systems can be analyzed by evaluating the reliability of the entire engineering systems after calculating the reliability in each method by reflecting the sequential or parallel components of the subsystems. In this study, we propose the method for analyzing the reliability of the engineering systems using type-2 fuzzy sets. Since type-2 fuzzy sets represent the fuzzy set as the three dimensional, it is possible to represent more various shape of fuzzy sets than the conventional fuzzy sets-fuzzy sets, intuitionistic fuzzy sets, vague sets, level (λ,1) interval-valued fuzzy sets, neutrosophic sets, octagonal intuitionistic fuzzy sets, polygonal type-1 fuzzy sets, etc-. In addition, since the degree of membership is represented as the secondary grade which may take any value from unit interval [0, 1], it becomes possible to express the degree of membership more flexible than the type-1 fuzzy sets that represent the reliability as one real number. And then it accepts the two assumptions based on the possibility measure of fuzzy reliability engineering, it has theoretical bases for using type-2 fuzzy sets to evaluate reliability. The proposed method of obtaining the reliability based type-2 fuzzy sets can be applied to fuzzy engineering systems, fuzzy decision making systems, fuzzy reliability engineering, fuzzy inference systems, etc.

ON FUZZY MAXIMAL, MINIMAL AND MEAN OPEN SETS

SWAMINATHAN, A.,SIVARAJA, S. The Korean Society for Computational and Applied M 2022 Journal of applied and pure mathematics Vol.4 No.1/2

We have observed that there exist certain fuzzy topological spaces with no fuzzy minimal open sets. This observation motivates us to investigate fuzzy topological spaces with neither fuzzy minimal open sets nor fuzzy maximal open sets. We have observed if such fuzzy topological spaces exist and if it is connected are not fuzzy cut-point spaces. We also study and characterize certain properties of fuzzy mean open sets in fuzzy T₁-connected fuzzy topological spaces.

구간값 피타고라스 퍼지집합에 기반을 둔 퍼지 공학시스템 신뢰도 분석

조상엽 한국지식정보기술학회 2023 한국지식정보기술학회 논문지 Vol.18 No.6

The reliability of the engineering systems operating in the real world has an important meaning. Such reliability becomes difficult to obtain an accurate reliability due to inaccurate data, wrong manipulation, etc. To solve this problems, the fuzzy sets have been used. There are several fuzzy sets used to calculate reliability. In the fuzzy sets, it is expressed as a real number , which is the belongingness degree of the fuzzy set. ∈ . In the intuitionistic fuzzy sets, the belongingness degree of the fuzzy set is expressed as , and the non-belongingness degree is described as respectively. , ∈ , 0 ≤ + ≤ 1. In the interval-valued intuitionistic fuzzy sets, the belongingness degree of the fuzzy set is expressed as [, ] and the non-belongingness degree is represented as [, ] respectively. , , , ∈ , + ≤ 1, 0 ≤ + ≤ 1, 0 ≤ + ≤ 1. In the Pythagorean fuzzy sets, the belongingness degree in a fuzzy set is expressed as , and the non-belongingness degree is expressed as . , ∈ , 0 ≤ + ≤ 1. In this paper, we propose a method for calculating the reliability of the fuzzy engineering systems using the interval-valued Pythagorean fuzzy sets that can express flexibility in the Pythagorean fuzzy sets.

구 퍼지집합에 기반을 둔 퍼지시스템 신뢰도 분석

김동혁,조상엽 한국지식정보기술학회 2023 한국지식정보기술학회 논문지 Vol.18 No.6

Since the fuzzy set proposed by Zadeh has been applied to evaluate the reliability of fuzzy systems, various fuzzy sets have been used to analyze the reliability of fuzzy systems. Expressing the reliability of the system as an accurate value is a difficult problem due to the ambiguity of the data. It is possible to overcome these problems with fuzzy sets. Therefore, fuzzy sets provide a way to appropriately express the reliability of inaccurate data that occurs in the real world. In fuzzy sets, reliability is expressed as a real number, which is the degree of membership. In intuitionistic fuzzy sets, reliability is expressed as an interval using positive and negative membership degrees. In the Pythagorean fuzzy sets, reliability is expressed by expressing the positive and negative membership degrees as squares, respectively. Therefore, it is possible to solve the problem where the sum of the positive and negative membership degrees exceeds 1. In picture fuzzy sets, reliability is expressed based on positive membership, neutral membership, and negative membership. Since the picture fuzzy set uses a neutral degree of membership, it becomes possible to express uncertainty that is neither positive nor negative. In the spherical fuzzy sets, reliability is expressed by expressing the positive membership degree, neutral membership degree, and negative membership degree as squares, respectively. Therefore, it is possible to solve the problem in which the sum of the positive membership degree, the neutral membership degree, and the negative membership degree exceeds 1. And since the spherical fuzzy set can express all the properties of conventional fuzzy sets, it can be used in various environments. Therefore, it becomes possible to express the reliability of fuzzy systems in various environments using sphere fuzzy sets, thereby enabling more flexible reliability analysis.