A Tour of Type-1 and Interval Type-2 Fuzzy Sets Theory

2020 ◽  
pp. 9-63
Author(s):  
Rosana Sueli da Motta Jafelice ◽  
Ana Maria Amarillo Bertone
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Sets Theory ◽  
Interval Type ◽  
Sets Theory

An interval type-2 hesitant fuzzy best-worst method

2021 ◽  
pp. 1-28
Author(s):  
Ashraf Norouzi ◽  
Hossein Razavi hajiagha
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Group Decision Making ◽  
Group Decision ◽  
Fuzzy Ahp ◽  
Uncertain Data ◽  
Interval Type ◽  
Best Worst Method

Multi criteria decision-making problems are usually encounter implicit, vague and uncertain data. Interval type-2 fuzzy sets (IT2FS) are widely used to develop various MCDM techniques especially for cases with uncertain linguistic approximation. However, there are few researches that extend IT2FS-based MCDM techniques into qualitative and group decision-making environment. The present study aims to adopt a combination of hesitant and interval type-2 fuzzy sets to develop an extension of Best-Worst method (BWM). The proposed approach provides a flexible and convenient way to depict the experts’ hesitant opinions especially in group decision-making context through a straightforward procedure. The proposed approach is called IT2HF-BWM. Some numerical case studies from literature have been used to provide illustrations about the feasibility and effectiveness of our proposed approach. Besides, a comparative analysis with an interval type-2 fuzzy AHP is carried out to evaluate the results of our proposed approach. In each case, the consistency ratio was calculated to determine the reliability of results. The findings imply that the proposed approach not only provides acceptable results but also outperforms the traditional BWM and its type-1 fuzzy extension.

An interval type-2 fuzzy ANP approach to evaluate enterprise technological innovation ability

Kybernetes ◽  
2016 ◽  
Vol 45 (9) ◽  
pp. 1486-1500 ◽  
Author(s):  
Tong Wu ◽  
Xinwang Liu
Keyword(s):  
Fuzzy Sets ◽  
Evaluation Model ◽  
Influence Factors ◽  
Content Type ◽  
Fuzzy Anp ◽  
Enterprise Technology ◽  
Innovation Ability ◽  
Interval Type

Purpose The purpose of this paper is to overcome the drawbacks of analytic hierarchy process in solving complex decision-making problems, especially for the evaluation of enterprise technology innovation ability (ETIA). Because interval type-2 fuzzy sets (IT2 FSs) can handle uncertainty linguistic variables in a more flexible and precise way than type-1 fuzzy sets with their second fuzzy membership functions, a fuzzy ANP method with IT2 FSs is proposed to evaluate the ETIA. Design/methodology/approach The criteria of evaluation on ETIA are identified and an evaluation model for ETIA is constructed on the basis of the application analysis of ETIA and theoretical design of ANP. In addition, two different ranking methods of IT2 FSs are applied in processing the relationships between influence factors of ETIA. Findings By using the proposed interval type-2 fuzzy ANP (IT2 FANP) method, the efficiencies of the whole evaluation of ETIA can be measured and the important factors in the ETIA can also be determined. Compared with the type-1 FANP through the ranking results, the proposed IT2 FANP is more reasonable and robust for the evaluation of ETIA. Practical implications The proposed IT2 FANP method is applied on the evaluation of ETIA. With respect to the application, the proposed method can be used to evaluate many more complex problems that contain feedback and circular relationships. Originality/value The proposed IT2 FANP approach can solve the complexities and uncertainties at the same time. Considering the subjective initiative of decision-makers and the feedback between influence factors, the proposed method is more efficient than the existing type-1 approaches in the literature.

Multi-criteria decision-making method based on type-2 fuzzy sets

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 431-450 ◽  
Author(s):  
Jing Wang ◽  
Qing-Hui Chen ◽  
Hong-Yu Zhang ◽  
Xiao-Hong Chen ◽  
Jian-Qiang Wang
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Computational Models ◽  
Fuzzy Numbers ◽  
Multi Criteria Decision Making ◽  
Interval Numbers ◽  
Interval Type ◽  
Uncertainty Information

Type-2 fuzzy sets (T2FSs) are the extension of type-1 fuzzy sets (T1FSs), which can convey more uncertainty information in solving multi-criteria decision-making (MCDM) problems. Motivated by the extension from interval numbers to triangular fuzzy numbers, three-trapezoidal-fuzzy-number-bounded type-2 fuzzy numbers (TT2FNs) are defined on the basis of interval type-2 trapezoidal fuzzy numbers (IT2TFNs), and they can convey more uncertainty information than T1FSs and IT2FSs. Moreover, the drawbacks of the existing computational models of generalized fuzzy numbers are analyzed, and a new computational model of fuzzy numbers is proposed, which is further extended to TT2FNs. Besides, a MCDM method is proposed to deal with the evaluation information given in the form of TT2FNs. Finally, an illustrative example and comparison analysis are provided to demonstrate the feasibility and validity of the proposed method.

$\alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications

2009 ◽  
Vol 17 (5) ◽  
pp. 1189-1207 ◽  
Author(s):  
J.M. Mendel ◽  
Feilong Liu ◽  
Daoyuan Zhai
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Sets Theory ◽  
Sets Theory

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.

scholarly journals Uncertainty of Interval Type-2 Fuzzy Sets Based on Fuzzy Belief Entropy

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1265
Author(s):  
Sicong Liu ◽  
Rui Cai
Keyword(s):  
Fuzzy Sets ◽  
Membership Function ◽  
Specific Objective ◽  
Open Issue ◽  
Belief Entropy ◽  
Interval Type ◽  
Fuzzy Subsets ◽  
The Membership Function

Interval type-2 fuzzy sets (IT2 FS) play an important part in dealing with uncertain applications. However, how to measure the uncertainty of IT2 FS is still an open issue. The specific objective of this study is to present a new entropy named fuzzy belief entropy to solve the problem based on the relation among IT2 FS, belief structure, and Z-valuations. The interval of membership function can be transformed to interval BPA [Bel,Pl]. Then, Bel and Pl are put into the proposed entropy to calculate the uncertainty from the three aspects of fuzziness, discord, and nonspecificity, respectively, which makes the result more reasonable. Compared with other methods, fuzzy belief entropy is more reasonable because it can measure the uncertainty caused by multielement fuzzy subsets. Furthermore, when the membership function belongs to type-1 fuzzy sets, fuzzy belief entropy degenerates to Shannon entropy. Compared with other methods, several numerical examples are demonstrated that the proposed entropy is feasible and persuasive.

scholarly journals Adaptive Non-singleton Type-2 Fuzzy Logic Systems: A Way Forward for Handling Numerical Uncertainties in Real World Applications

2011 ◽  
Vol 6 (3) ◽  
pp. 503 ◽  
Author(s):  
Nazanin Sahab ◽  
Hani Hagras
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Real World ◽  
Superior Performance ◽  
Great Success ◽  
Real World Applications ◽  
Interval Type ◽  
Linguistic Labels

Real world environments are characterized by high levels of linguistic and numerical uncertainties. A Fuzzy Logic System (FLS) is recognized as an adequate methodology to handle the uncertainties and imprecision available in real world environments and applications. Since the invention of fuzzy logic, it has been applied with great success to numerous real world applications such as washing machines, food processors, battery chargers, electrical vehicles, and several other domestic and industrial appliances. The first generation of FLSs were type-1 FLSs in which type-1 fuzzy sets were employed. Later, it was found that using type-2 FLSs can enable the handling of higher levels of uncertainties. Recent works have shown that interval type-2 FLSs can outperform type-1 FLSs in the applications which encompass high uncertainty levels. However, the majority of interval type-2 FLSs handle the linguistic and input numerical uncertainties using singleton interval type-2 FLSs that mix the numerical and linguistic uncertainties to be handled only by the linguistic labels type-2 fuzzy sets. This ignores the fact that if input numerical uncertainties were present, they should affect the incoming inputs to the FLS. Even in the papers that employed non-singleton type-2 FLSs, the input signals were assumed to have a predefined shape (mostly Gaussian or triangular) which might not reflect the real uncertainty distribution which can vary with the associated measurement. In this paper, we will present a new approach which is based on an adaptive non-singleton interval type-2 FLS where the numerical uncertainties will be modeled and handled by non-singleton type-2 fuzzy inputs and the linguistic uncertainties will be handled by interval type-2 fuzzy sets to represent the antecedents’ linguistic labels. The non-singleton type-2 fuzzy inputs are dynamic and they are automatically generated from data and they do not assume a specific shape about the distribution associated with the given sensor. We will present several real world experiments using a real world robot which will show how the proposed type-2 non-singleton type-2 FLS will produce a superior performance to its singleton type-1 and type-2 counterparts when encountering high levels of uncertainties.

scholarly journals Vibration Control Design for a Plate Structure with Electrorheological ATVA Using Interval Type-2 Fuzzy System

2017 ◽  
Author(s):  
Chih-Jer Lin ◽  
Chun-Ying Lee ◽  
Ying Liu
Keyword(s):  
Electric Field ◽  
Vibration Control ◽  
Fuzzy Sets ◽  
Fuzzy Modeling ◽  
Smart Structure ◽  
Thin Plates ◽  
Fuzzy Controllers ◽  
Interval Type

This study presents a vibration control using actively tunable vibration absorbers (ATVA) to suppress vibration of a thin plate. The ATVA’s is made of a sandwich hollow structure embedded with the electrorheological fluid (ERF). ERF is considered to be one of the most important smart fluids and it is suitable to be embedded in a smart structure due to its controllable viscosity property. ERF’s apparent viscosity can be controlled in response to the electric field and the change is reversible in 10 microseconds. Therefore, the physical properties of the ERF-embedded smart structure, such as the stiffness and damping coefficients, can be changed in response to the applied electric field. A mathematical model is difficult to be obtained to describe the exact characteristics of the ERF embedded ATVA because of the nonlinearity of ERF’s viscosity. Therefore, a fuzzy modeling and experimental validations of ERF-based ATVA from stationary random vibrations of thin plates are presented in this study. Because Type-2 fuzzy sets generalize Type-1 fuzzy sets so that more modelling uncertainties can be handled, a semi-active vibration controller is proposed based on Type-2 fuzzy sets. To investigate the different performances by using different types of fuzzy controllers, the experimental measurements employing type-1 fuzzy and interval type-2 fuzzy controllers are implemented by the Compact RIO embedded system. The fuzzy modeling framework and solution methods presented in this work can be used for design, performance analysis, and optimization of ATVA from stationary random vibration of thin plates.

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