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 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 The Collocation Method For Fourth Order Fuzzy Boundary Value Problems

2019 ◽  
Vol 21 (85) ◽  
pp. 1-10
Author(s):  
Rasha H.Ibraheem
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Fourth Order ◽  
Level Sets ◽  
Boundary Value ◽  
Method Of Solution ◽  
Α Level ◽  
Fuzzy Boundary Value Problems ◽  
Fuzzy Boundary

In this paper, the collocation method is considered to solve the nonhomogeneous  for fourth order fuzzy boundary value problems, In which the fuzziness appeared together in the boundary conditions and in the nonhomogeneous term of the differential equation. The method of solution depends on transforming the fuzzy problem to equivalent crisp problems using the concept of α-level sets.

scholarly journals Convexity and level sets for interval-valued fuzzy sets

2022 ◽  
Author(s):  
Pedro Huidobro ◽  
Pedro Alonso ◽  
Vladimír Janiš ◽  
Susana Montes
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Level Sets ◽  
Fuzzy Optimization ◽  
Interval Valued

AbstractConvexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems.

scholarly journals The Decomposition for the α-level Fuzzy Soft Sets

2020 ◽  
pp. 62-73
Author(s):  
Li Fu ◽  
Luo Tai Qie ◽  
Qiao Yun Liu
Keyword(s):  
Skin Disease ◽  
Decomposition Theorem ◽  
Disease Diagnosis ◽  
Soft Set ◽  
Simple Application ◽  
Soft Sets ◽  
Fuzzy Soft Sets ◽  
Α Level

In this paper, the properties of α-level fuzzy soft sets and α-level fuzzy soft lattices are discussed. Firstly, some soft operations between α-level fuzzy soft sets (lattices) are dened, such as the soft union and intersection operations, and illustrates them by the examples. Secondly, the relation between the α-level fuzzy soft lattices and the α-level fuzzy soft sets are studied, we mainly verify the properties not valid in the case discussed, but tenable in the classical soft set theory.Thirdly, the decomposition theorem for the α-level fuzzy soft sets is proved. Lastly, a simple application in skin disease diagnosis is illustrated.

Multi-Level Control of Fuzzy-Constraint Propagation via Evaluations with Linguistic Truth Values in Generalized-Mean-Based Inference

2016 ◽  
Vol 20 (2) ◽  
pp. 355-377 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Level Set ◽  
Constraint Propagation ◽  
Fuzzy Rules ◽  
Truth Values ◽  
Fuzzy Constraints ◽  
Control Scheme ◽  
Fuzzy Constraint ◽  
Generalized Mean ◽  
Α Level

A method is proposed for fuzzy inference which can propagate convex fuzzy-constraints from given facts to consequences in various forms by applying a number of fuzzy rules, particularly when asymmetric fuzzy sets are used for given facts and/or fuzzy rules. The conventionalmethod, α-GEMS (α-level-set and generalized-mean-based inference in synergy with composition), cannot be performed with asymmetric fuzzy sets; it can be conducted only with symmetric fuzzy sets. In order to cope with asymmetric fuzzy sets as well as symmetric ones, a control scheme is proposed for the fuzzy-constraint propagation, which is α-cut based and can be performed independently at each level of α. It suppresses an excessive specificity decrease in consequences, particularly stemming from the asymmetricity. Thereby, the fuzzy constraints of given facts are reflected to those of consequences, to a feasible extent. The theoretical aspects of the control scheme are also presented, wherein the specificity of the support sets of consequences is evaluated via linguistic truth values (LTVs). The proposed method is named α-GEMST (α-level-set and generalized-meanbased inference in synergy with composition via LTV control) in order to differentiate it from α-GEMS. Simulation results show that α-GEMST can be properly performed, particularly with asymmetric fuzzy sets. α-GEMST is expected to be applied to the modeling of given systems with various fuzzy input-output relations.

Multi-Level Control of Fuzzy-Constraint Propagation in Inference Based on α-Cuts and Generalized Mean

2013 ◽  
Vol 17 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Level Set ◽  
Constraint Propagation ◽  
Fuzzy Rules ◽  
Nonlinear Mapping ◽  
Fuzzy Input ◽  
Fuzzy Constraint ◽  
Multi Level ◽  
Generalized Mean ◽  
Α Level

An inference method is proposed, which can perform nonlinear mapping between convex fuzzy sets and present a scheme of various fuzzy-constraint propagation from given facts to deduced consequences. The basis of nonlinear mapping is provided by α-GEMII (α-level-set and generalized-mean-based inference) whereas the control of fuzzy-constraint propagation is based on the compositional rule of inference (CRI). The fuzzy-constraint propagation is controlled at the multi-level of α in its α-cut-based operations. The proposed method is named α-GEMS (α-level-set and generalized-mean-based inference in synergy with composition). Although α-GEMII can perform the nonlinear mapping according to a number of fuzzy rules in parallel, it has limitations in the control of fuzzy-constraint propagation and therefore has difficulty in constructing models of various given systems. In contrast, CRI-based inference can rather easily control fuzzy-constraint propagation with high understandability especially when a single fuzzy rule is used. It is difficult, however, to perform nonlinear mapping between convex fuzzy sets by using a number of fuzzy rules in parallel. α-GEMS can solve both of these problems. Simulation results show that α-GEMS is performed well in the nonlinear mapping and fuzzy-constraint propagation. α-GEMS is expected to be applied to modeling of given systems with various fuzzy input-output relations.

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