An Entropy of Interval Type-2 Fuzzy Sets

2013 ◽  
Vol 321-324 ◽  
pp. 1999-2003 ◽  
Author(s):  
Gao Zheng ◽  
Shi Wei Yin
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Fuzzy Entropy ◽  
Calculation Formula ◽  
Membership Functions ◽  
Axiomatic Definition ◽  
Fuzzy Degree ◽  
Interval Type

The entropy shows the fuzzy degree of a fuzzy set (FS) and can be used in various areas. Aiming at the characteristics of the fuzzy entropy and type-2 fuzzy sets (IT2 FSs), we introduce a new entropy of IT2 FSs in this paper. At first, we select an axiomatic definition for it. Then, considering a fact that the operations of IT2 FSs depend on the upper membership functions (UMFs) and lower membership functions (LMFs), we propose a calculation formula and verify it accords with the four axioms of the selected definition. Finally, we use an example to illuminate its reasonable performance.

Interval Type-2 Fuzzy Sets Constructed From Several Membership Functions: Application to the Fuzzy Thresholding Algorithm

2013 ◽  
Vol 21 (2) ◽  
pp. 230-244 ◽  
Author(s):  
Miguel Pagola ◽  
Carlos Lopez-Molina ◽  
Javier Fernandez ◽  
Edurne Barrenechea ◽  
Humberto Bustince
Keyword(s):  
Fuzzy Sets ◽  
Membership Functions ◽  
Thresholding Algorithm ◽  
Interval Type ◽  
Fuzzy Thresholding

A Comparative Investigation of Type-2 Fuzzy Sets, Nonstationary Fuzzy Sets and Cloud Models

2016 ◽  
Vol 24 (02) ◽  
pp. 213-227 ◽  
Author(s):  
Han-Chen Huang ◽  
Xiaojun Yang
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Cloud Model ◽  
Comparative Investigation ◽  
Mathematical Methods ◽  
Moderate Amount ◽  
Cloud Models ◽  
Interval Type

Since Zadeh introduced fuzzy sets, a lot of extensions of this concept have been proposed, such as type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models, to represent higher levels of uncertainty. This paper provides a comparative investigation of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Type-2 fuzzy sets study the fuzziness of the membership function (MF) using primary MF and secondary MF based on analytic mathematical methods; nonstationary fuzzy sets study the randomness of the MF using primary MF and variation function based on type-1 fuzzy sets theory; cloud models study the randomness of the distribution of samples in the universe and generate random membership grades (MGs) using two random variables based on probability and statistic mathematical methods. They all concentrate on dealing with the uncertainty of the MF or the MG which type-1 fuzzy sets do not consider, and thus have many similarities. Moreover, we find out that, the same qualitative concept “moderate amount” can be represented by an interval type-2 fuzzy set, a nonstationary fuzzy set or a normal cloud model, respectively. Then, we propose a unified mathematical expression for the interval type-2 fuzzy set, nonstationary fuzzy set and normal cloud model. On the other hand, we also find out that, the theory fundament and underlying motivations of these models are quite different. Therefore, We summarize detailed comparisons of distinctive properties of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Further, we study their diverse characteristics of distributions of MGs across vertical slices. The comparative investigation shows that these models are complementary to describe the uncertainty from different points of view. Thus, this paper provides a fundamental contribution and makes a basic reference for knowledge representation and other applications with uncertainty.

Assessment of national innovation capabilities of OECD countries using trapezoidal interval type-2 fuzzy ELECTRE III method

2020 ◽  
Author(s):  
Geetha Selvaraj ◽  
Jeonghwan Jeon
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Research Work ◽  
Oecd Countries ◽  
Content Type ◽  
Electre Iii ◽  
Standard Values ◽  
Interval Type ◽  
Credibility Degree

PurposeFor a nation to become a superpower, it's scientific and technological advancement is essential. Each country is exploring how to improve themselves in terms of science and technology. The authors analyzed the innovation capabilities of 35 OECD countries that have not recently joined Lithuania.Design/methodology/approachIn recent years, a lot of research work has been done on trapezoidal interval type-2 fuzzy sets (TIT-2 FS), and many research works have been published. The trapezoidal interval type-2 fuzzy set helps effectively to represent the uncertainty comparatively than the type-1 fuzzy set. Taking advantage of this effectiveness, the authors extend the best multi-criteria decision making method (MCDM) for trapezoidal interval type-2 fuzzy sets. Here, ELimination and Choice Expressing REality III (ELECTRE III) method in the trapezoidal interval type-2 fuzzy set environment is proposed.FindingsThis analysis helps to the OECD countries to develop their level of innovation in the criteria. The authors are making this evaluation for the year 2018 based on the 31 criteria. Application of the proposed method expressed by evaluation of the national innovation capability problem. Based on the obtained results, the top five countries are United States, Switzerland, Canada, Germany and Japan.Originality/valueThe authors collected required data from different available data sources like OECD, IMD, USPTO, ITU and surveyed data reported by KISTEP. After collecting all the data from different sources, the authors calculated the standard values as KISTEP. After converting the standard values into trapezoidal interval type-2 fuzzy values, the authors construct a decision matrix based on these values. Then, the authors determined the possibility mean values and preference. Then, they calculated the concordance and discordance credibility degree values. Finally, they ranked OECD countries by the net credibility degree. The results are computed by using the MATLAB software.

Triangular Interval Type-2 Interpolative Fuzzy Reasoning

2012 ◽  
Vol 198-199 ◽  
pp. 261-266
Author(s):  
Yang Chen ◽  
Tao Wang
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Inference ◽  
Fuzzy Reasoning ◽  
Membership Functions ◽  
Inference Models ◽  
Definition Of ◽  
Interval Type ◽  
Triangular Type

This paper first gives the definition of interval type-2 fuzzy sets,then investigates interval type-2 interpolative fuzzy reasoning under Triangular type membership functions. Two interpolative fuzzy reasoning algorithms responding to interval type-2 fuzzy inference models in the line of type-1 interpolative fuzzy reasoning algorithms are proposed.

scholarly journals Properties of interval type-2 fuzzy sets in decision support systems

2021 ◽  
pp. 75-81
Author(s):  
Alexander Zakovorotniy ◽  
Artem Kharchenko
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Inference ◽  
Systems Design ◽  
Membership Functions ◽  
Fuzzy Inference Systems ◽  
Footprint Of Uncertainty ◽  
Expert Assessments ◽  
Interval Type ◽  
Inference Systems

Definitions and methods of designing interval type-2 fuzzy sets in fuzzy inference systems for control problems of complex technical objects in conditions of uncertainty are considered. The main types of uncertainties, that arise when designing fuzzy inference systems and depend on the number of expert assessments, are described. Methods for assessing intra-uncertainty and inter-uncertainty are proposed, taking into account the different number of expert assessments at the stage of determining the types and number of membership functions. Factors influencing the parameters and properties of interval type-2 fuzzy during experimental studies are determined. Such factors include the number of experiments performed, external factors, technical parameters of the control object, and the reliability of the components of the computer system decision support system. The properties of the lower and upper membership functions of interval type-2 fuzzy sets are investigated on the example of the Gaussian membership function, which is one of the most used in the problems of fuzzy inference systems design. The main features and differences in the methods of determining the lower and upper membership functions of interval type-2 fuzzy sets for different types of uncertainties are taken into account. Methods for determining the footprint of uncertainty, as well as the dependence of its size on the number of expert assessments, are considered. The footprint of uncertainty is characterized by the lower and upper membership functions, and its size directly affects the accuracy of the obtained solutions. Methods for determining interval type-2 fuzzy sets using regulation factors of membership function parameters for intra-uncertainty and weighting factors of membership functions for inter-uncertainties have been developed. The regulation factor of the function parameters can be used to describe the lower and upper membership functions while determining the size of the footprint of uncertainty. Complex interval type-2 sets are determined to take into account inter-uncertainties in the problems of fuzzy inference systems design.

Design of single and double acceptance sampling plans based on interval type-2 fuzzy sets

2021 ◽  
pp. 1-13
Author(s):  
Gürkan Işık ◽  
İhsan Kaya
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Measurement Errors ◽  
Uncertainty Modeling ◽  
Characteristic Functions ◽  
Acceptance Sampling ◽  
Sampling Plans ◽  
Acceptance Sampling Plans ◽  
Interval Type

Defectiveness of items is generally considered as a certain value in acceptance sampling plans (ASPs). It is clear that, it may not be certainly known in some real-case problems. Uncertainties of the inspection process such as measurement errors, inspectors’ hesitancies or vagueness of the process etc. should be taken into account to obtain more reliable results. The fuzzy set theory (FST) is one of the best methods to overcome these problems. There are some studies in the literature formulating the ASPs with the help of FST. Deciding the right membership functions of the fuzzy sets (FSs) has a vital importance on the quality of the uncertainty modeling. Additionally, the fuzzy set extensions have been offered to model more complicated uncertainties to achieve better modeling. As one of these extensions, type-2 fuzzy sets (T2FSs) gives an ability to model uncertainty in situations where it is not possible to determine exact membership function parameters. In this study, single and double ASPs based on interval T2FSs (IT2FSs) have been designed for binomial and Poisson distributions. Thus, it becomes possible to make more flexible, sensitive and descriptive sensitivity analyzes. The main characteristic functions of ASPs have been derived and the suggested formulations have been illustrated on a comparative application from manufacturing process. Results allowing for more comprehensive analysis as against to the traditional and T1FSs based plans have been obtained.

Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions

2014 ◽  
Vol 17 ◽  
pp. 79-89 ◽  
Author(s):  
Lorenzo Livi ◽  
Hooman Tahayori ◽  
Alireza Sadeghian ◽  
Antonello Rizzi
Keyword(s):  
Fuzzy Sets ◽  
Membership Functions ◽  
Interval Type
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