scholarly journals Arithmetics of Vectors of Fuzzy Sets

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1614
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decomposition Theorem ◽  
Extension Principle ◽  
Membership Functions ◽  
Arithmetic Operations ◽  
Mild Conditions ◽  
Different Types ◽  
Scalar Products ◽  
Α Level

The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.

Generating fuzzy sets from the families of sets

2021 ◽  
pp. 1-22
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decision Makers ◽  
Membership Functions ◽  
Engineering Problems ◽  
Original Family ◽  
Mechanical Procedure ◽  
Subjective Bias ◽  
Α Level

The main purpose of this paper is to establish a mechanical procedure to determine the membership functions using the data collected from the economic and engineering problems. Determining the membership functions from the collected data may depend on the subjective viewpoint of decision makers. The mechanical procedure proposed in this paper can get rid of the subjective bias of decision makers. The concept of solid families is also proposed by regarding the sets in a family to be continuously varied. The desired fuzzy sets will be generated in the sense that its α-level sets will be identical to the sets of the original family. In order to achieve this purpose, any arbitrary families will be rearranged as the nested families by applying some suitable functions to the original families that are formulated from the collected data.

scholarly journals Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 11 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decomposition Theorem ◽  
Dual Decomposition ◽  
Α Level

The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.

scholarly journals Inner Product of Fuzzy Vectors

2021 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Potential Application ◽  
Level Sets ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decomposition Theorem ◽  
Inner Product ◽  
Extension Principle ◽  
Membership Functions ◽  
Linear Optimization Problems ◽  
Fuzzy Interval

Abstract The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using the extension principle and the form of decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the a-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear optimization problems.

scholarly journals METHOD FOR OBTAINING THE PARAMETERS OF MEMBERSHIP FUNCTIONS OF FUZZY SETS BASED ON REAL DATA FOR AUTOMATED INFORMATION PROCESSING SYSTEMS

2019 ◽  
Vol 46 (3) ◽  
pp. 79-86
Author(s):  
E. E. Bisyanov ◽  
A. A. Gutnik
Keyword(s):  
Information Processing ◽  
Fuzzy Sets ◽  
Real Data ◽  
Linguistic Variables ◽  
Membership Functions ◽  
Automated Systems ◽  
Accessory Function ◽  
Different Types ◽  
Automated Processing ◽  
The Membership Function

Objectives Development of a method for selecting the type of accessory function and obtaining its parameters to allow subjective personal influences in automated information processing to be excluded.Method. Existing methods for constructing membership functions were analysed. The research was based on the methods of fuzzy logic and data analysis.Results. A method for obtaining the parameters of membership functions of fuzzy sets using real data is suggested. It is proposed to use the data obtained from the object under study to determine the kernel of the fuzzy number, as well as derive theoretical information about the object for the carrier. Triangular, trapezoidal, bell-shaped and Gaussian membership functions are considered. The appearance of the membership function can be defined using the criterion of the relations of the kernel to the carrier of a fuzzy set. The results of calculations for obtaining the membership functions based on data on the power consumption of electric motors of different types are given.Conclusion. The developed method can be used both in decision support systems as well as in automated systems for controlling technological processes. If necessary, the values of the criterion proposed in the article can be revised to take the values included in the set of measured real data into account or to simplify the procedure of automated processing. Further research will use the described method to obtain the terms of linguistic variables. 

scholarly journals On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Power Series ◽  
Uniform Convergence ◽  
Membership Function ◽  
Level Sets ◽  
Radius Of Convergence ◽  
Extension Principle ◽  
Membership Functions ◽  
Function Series ◽  
Convergence Of Sequences ◽  
The Relationship

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

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