Theory of Fuzzy Sets: An Overview
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paper_title: ON ENTROPY OF FUZZY SETS

paper_content:

In this paper we deal with the entropy of fuzzy sets. We first review several defined entropies of fuzzy sets and then propose a new one. Some comparisons are made with some existing entropies to show the effectiveness of the proposed one.
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paper_title: Fuzzy sets as a basis for a theory of possibility

paper_content:

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μF, then a proposition of the form “X is F”, where X is a variable taking values in U, induces a possibility distribution tx which equates the possibility of X taking the value u to μF(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle. ::: ::: A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form “X is F is α-possible”, where α is a number in the interval [0,1], is formulated and illustrated by examples.
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paper_title: Fuzzy sets as a basis for a theory of possibility

paper_content:

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μF, then a proposition of the form “X is F”, where X is a variable taking values in U, induces a possibility distribution tx which equates the possibility of X taking the value u to μF(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle. ::: ::: A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form “X is F is α-possible”, where α is a number in the interval [0,1], is formulated and illustrated by examples.
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paper_title: A process monitoring module based on fuzzy logic and pattern recognition

paper_content:

This article presents a plastic injection moulding monitoring module based on knowledge built on-line using feedback from production data. A fuzzy classifier was especially developed for this application. It is based on unsupervised and supervised classification methods. The role of the first one is to identify the functioning modes of the process whereas the role of the second one is to associate the state of the process to one of the identified functioning modes at the moment where a workpiece is injected. Furthermore this diagnosis module integrates an on-line learning method which allows to enrich and upgrade the initial knowledge during production. The results obtained show that the monitoring system is a solution for quality and productivity control having serious economical advantages. For example maintenance tasks can be anticipated and the size of the training set can be considerably reduced. The computing times show that the monitoring system can be used for the purpose of industrial applications without any decrease of production rate.
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Title: Theory of Fuzzy Sets: An Overview

Section 1: Introduction
Description 1: Provide an introduction to the concept of fuzzy set theory, its origins, and its foundational differences from classical set theory.

Section 2: Baruah's Definition of Complementation of Fuzzy Sets
Description 2: Discuss Baruah's new definition of fuzzy set complementation using a membership function and a reference function.

Section 3: Some Results of Fuzzy Sets
Description 3: Present some existing definitions and theories of fuzzy sets along with the new definitions proposed by Baruah.

Section 4: Cardinality of Fuzzy Sets
Description 4: Explain the concept of cardinality in the context of fuzzy sets and the generalizations applied.

Section 5: Entropy of Fuzzy Numbers
Description 5: Discuss the importance of fuzzy entropy and the different measures proposed by various researchers.

Section 6: Cartesian Product of Two Fuzzy Sets
Description 6: Provide insights into the Cartesian product as applied to fuzzy sets.

Section 7: Fuzzy Matrices
Description 7: Describe fuzzy matrices, their properties, and how they differ from classical matrices.

Section 8: Probability-Possibility Consistency Principle
Description 8: Review the principles of converting between probability and possibility theories and their mathematical formulations.

Section 9: Randomness-Fuzziness Consistency Principles
Description 9: Introduce the framework for reasoning with probability and possibility consistency, as well as the new principles proposed by Baruah.

Section 10: New Definition of Entropy of Fuzzy Numbers
Description 10: Provide the new definition of fuzzy entropy based on the Randomness-Fuzziness Consistency principle and discuss its properties.

Section 11: Fuzzy Ranking
Description 11: Discuss the methods and challenges in ranking fuzzy numbers, and why current methods may be inadequate.

Section 12: Conclusions
Description 12: Summarize the revisited results on fuzzy set complementation, existing principles, and the introduction of new concepts and principles.