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Etienne Kerre - One of the best experts on this subject based on the ideXlab platform.
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the impact of Fuzzy Set Theory on contemporary mathematics survey
Applied and Computational Mathematics, 2011Co-Authors: Etienne KerreAbstract:In this paper we first outline the shortcomings of classical binary logic and Cantor's Set Theory in order to handle imprecise and uncertain information. Next we briefly introduce the basic notions of Zadeh's Fuzzy Set Theory among them: definition of a Fuzzy Set, operations on Fuzzy Sets, the concept of a linguistic variable, the concept of a Fuzzy number and a Fuzzy relation. The major part consists of a sketch of the evolution of the mathematics of fuzziness, mostly illustrated with examples from my research group during the past 35 years. In this evolution I see three overlapping stages. In the first stage taking place during the seventies only straightforward fuzzifications of classical domains such as general topology, Theory of groups, relational calculus, ... have been introduced and investigated w.r.t. the main deviations from their binary originals. The second stage is characterized by an explosion of the possible fuzzifications of the classical structures which has lead to a deep study of the alternatives as well as to the enrichment of the structures due to the non-equivalence of the different fuzzifications. Finally some of the current topics of research in the mathematics of fuzziness are highlighted. Nowadays Fuzzy research concerns standardization, axiomatization, extensions to lattice-valued Fuzzy Sets, critical comparison of the different so-called soft computing models that have been launched during the past three decennia for the representation and processing of incomplete information.
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on the position of intuitionistic Fuzzy Set Theory in the framework of theories modelling imprecision
Information Sciences, 2007Co-Authors: Glad Deschrijver, Etienne KerreAbstract:Intuitionistic Fuzzy Sets [K.T. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR's Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in Bulgarian)] are an extension of Fuzzy Set Theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic Fuzzy Sets, it is useful to determine the position of intuitionistic Fuzzy Set Theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic Fuzzy Sets and other models of imprecision.
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uninorms in l Fuzzy Set Theory
Fuzzy Sets and Systems, 2004Co-Authors: Glad Deschrijver, Etienne KerreAbstract:Uninorms are an important generalization of triangular norms and conorms, having a neutral element lying anywhere in the unit interval. In this paper we introduce the notion of uninorm in interval-valued Fuzzy Set Theory, or equivalently in intuitionistic Fuzzy Set Theory in the sense of Atanassov, and investigate its properties.
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implication in intuitionistic Fuzzy and interval valued Fuzzy Set Theory construction classification application
International Journal of Approximate Reasoning, 2004Co-Authors: Chris Cornelis, Glad Deschrijver, Etienne KerreAbstract:Abstract With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two Settings: first, within intuitionistic Fuzzy Set Theory and secondly, within interval-valued Fuzzy Set Theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two models’ formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and Fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context.
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on the relationship between some extensions of Fuzzy Set Theory
Fuzzy Sets and Systems, 2003Co-Authors: Glad Deschrijver, Etienne KerreAbstract:Since Zadeh introduced Fuzzy Sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of Fuzzy Set Theory, others try to handle imprecision and uncertainty in a different (better?) way. Kerre (Computational Intelligence in Theory and Practice, Physica-Verlag, Heidelberg, 2001, pp. 55-72) has given a summary of the links that exist between Fuzzy Sets and other mathematical models such as flou Sets (Gentilhomme), two-fold Fuzzy Sets (Dubois and Prade) and L-Fuzzy Sets (Goguen). In this paper, we establish the relationships between intuitionistic Fuzzy Sets (Atanassov, VII ITKR's Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)), L-Fuzzy Sets (J. Math. Anal. Appl. 18 (1967) 145), interval-valued Fuzzy Sets (Sambuc, Ph.D. Thesis, University of Marseille, France, 1975), interval-valued intuitionistic Fuzzy Sets (Intuitionistic Fuzzy Set, Physica-Verlag, Heidelberg, New York, 1999).
Glad Deschrijver - One of the best experts on this subject based on the ideXlab platform.
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uninorms which are neither conjunctive nor disjunctive in interval valued Fuzzy Set Theory
Information Sciences, 2013Co-Authors: Glad DeschrijverAbstract:Abstract Uninorms are a generalisation of t -norms and t -conorms for which the neutral element is an element of [0, 1] which is not necessarily equal to 0 (as for t -norms) or 1 (as for t -conorms). Uninorms on the unit interval are either conjunctive or disjunctive, i.e. they aggregate the pair (0, 1) to either 0 or 1. In real-life applications, this kind of aggregation may be counter-intuitive. Interval-valued Fuzzy Set Theory and Atanassov’s intuitionistic Fuzzy Set Theory are extensions of Fuzzy Set Theory which allows to model uncertainty about the membership degrees. In these theories there exist uninorms which are neither conjunctive nor disjunctive. In this paper we study such uninorms more deeply and we investigate the structure of these uninorms. We also give several examples of uninorms which are neither conjunctive nor disjunctive.
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arithmetic operators in interval valued Fuzzy Set Theory
Information Sciences, 2007Co-Authors: Glad DeschrijverAbstract:We introduce the addition, subtraction, multiplication and division on L^I, where L^I is the underlying lattice of both interval-valued Fuzzy Set Theory [R. Sambuc, Fonctions @F-floues. Application a l'aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Universite de Marseille, France, 1975] and intuitionistic Fuzzy Set Theory [K.T. Atanassov, Intuitionistic Fuzzy Sets, 1983, VII ITKR's Session, Sofia (deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the Lukasiewicz t-norm and the product t-norm on the unit interval to L^I and some related operators can be written in a similar way as their counterparts on ([0,1],=<).
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on the position of intuitionistic Fuzzy Set Theory in the framework of theories modelling imprecision
Information Sciences, 2007Co-Authors: Glad Deschrijver, Etienne KerreAbstract:Intuitionistic Fuzzy Sets [K.T. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR's Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in Bulgarian)] are an extension of Fuzzy Set Theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic Fuzzy Sets, it is useful to determine the position of intuitionistic Fuzzy Set Theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic Fuzzy Sets and other models of imprecision.
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uninorms in l Fuzzy Set Theory
Fuzzy Sets and Systems, 2004Co-Authors: Glad Deschrijver, Etienne KerreAbstract:Uninorms are an important generalization of triangular norms and conorms, having a neutral element lying anywhere in the unit interval. In this paper we introduce the notion of uninorm in interval-valued Fuzzy Set Theory, or equivalently in intuitionistic Fuzzy Set Theory in the sense of Atanassov, and investigate its properties.
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implication in intuitionistic Fuzzy and interval valued Fuzzy Set Theory construction classification application
International Journal of Approximate Reasoning, 2004Co-Authors: Chris Cornelis, Glad Deschrijver, Etienne KerreAbstract:Abstract With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two Settings: first, within intuitionistic Fuzzy Set Theory and secondly, within interval-valued Fuzzy Set Theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two models’ formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and Fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context.
Chris Aldrich - One of the best experts on this subject based on the ideXlab platform.
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new methodology for hazardous waste classification using Fuzzy Set Theory part ii intelligent decision support system
Journal of Hazardous Materials, 2008Co-Authors: Ndeke Musee, Chris Aldrich, Leon LorenzenAbstract:In part 1 of this paper, factors that influence hazards and eco/toxicity in composite hazardous wastes were described. In part 2, a computer-aided decision support tool based on Fuzzy Set Theory is proposed to support the classification of composite wastes. Given the chemical properties, the nature of microorganisms that may be present, the behaviour of chemicals in humans and ecosystems, and the quantities of wastes, the computer-aided tool automatically classifies the waste as benign, partially hazardous, hazardous or highly hazardous. The functionality of the computer-aided decision tool is demonstrated through nine worked examples and the results are discussed in detail.
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New methodology for hazardous waste classification using Fuzzy Set Theory Part I. Knowledge acquisition.
Journal of Hazardous Materials, 2008Co-Authors: Ndeke Musee, Leon Lorenzen, Chris AldrichAbstract:In the literature on hazardous waste classification, the criteria used are mostly based on physical properties, such as quantity (weight), form (solids, liquid, aqueous or gaseous), the type of processes generating them, or a Set of predefined lists. Such classification criteria are inherently inadequate to account for the influence of toxic and hazard characteristics of the constituent chemicals in the wastes, as well as their exposure potency in multimedia environments, terrestrial mammals and other biota. Second, none of these algorithms in the literature has explicitly presented waste classification by examining the contribution of individual constituent components of the composite wastes. In this two-part paper, we propose a new automated algorithm for waste classification that takes into account physicochemical and toxicity effects of the constituent chemicals to humans and ecosystems, in addition, to the exposure potency and waste quantity. In part I, available data on the physicochemical and toxicity properties of individual chemicals in humans and ecosystems, their exposure potency in environmental systems and the effect of waste quantity are described, because they fundamentally contribute to the final waste ranking. Knowledge acquisition in this study was accomplished through the extensive review of published and specialized literature to establish facts necessary for the development of Fuzzy rule-bases. Owing to the uncertainty and imprecision of various forms of data (both quantitative and qualitative) essential for waste classification, and the complexity resulting from knowledge incompleteness, the use of Fuzzy Set Theory for the aggregation and computation of waste classification ranking index is proposed. A computer-aided intelligent decision tool is described in part II of this paper and the functionality of the Fuzzy waste classification algorithm is illustrated through nine worked examples.
Leon Lorenzen - One of the best experts on this subject based on the ideXlab platform.
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new methodology for hazardous waste classification using Fuzzy Set Theory part ii intelligent decision support system
Journal of Hazardous Materials, 2008Co-Authors: Ndeke Musee, Chris Aldrich, Leon LorenzenAbstract:In part 1 of this paper, factors that influence hazards and eco/toxicity in composite hazardous wastes were described. In part 2, a computer-aided decision support tool based on Fuzzy Set Theory is proposed to support the classification of composite wastes. Given the chemical properties, the nature of microorganisms that may be present, the behaviour of chemicals in humans and ecosystems, and the quantities of wastes, the computer-aided tool automatically classifies the waste as benign, partially hazardous, hazardous or highly hazardous. The functionality of the computer-aided decision tool is demonstrated through nine worked examples and the results are discussed in detail.
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New methodology for hazardous waste classification using Fuzzy Set Theory Part I. Knowledge acquisition.
Journal of Hazardous Materials, 2008Co-Authors: Ndeke Musee, Leon Lorenzen, Chris AldrichAbstract:In the literature on hazardous waste classification, the criteria used are mostly based on physical properties, such as quantity (weight), form (solids, liquid, aqueous or gaseous), the type of processes generating them, or a Set of predefined lists. Such classification criteria are inherently inadequate to account for the influence of toxic and hazard characteristics of the constituent chemicals in the wastes, as well as their exposure potency in multimedia environments, terrestrial mammals and other biota. Second, none of these algorithms in the literature has explicitly presented waste classification by examining the contribution of individual constituent components of the composite wastes. In this two-part paper, we propose a new automated algorithm for waste classification that takes into account physicochemical and toxicity effects of the constituent chemicals to humans and ecosystems, in addition, to the exposure potency and waste quantity. In part I, available data on the physicochemical and toxicity properties of individual chemicals in humans and ecosystems, their exposure potency in environmental systems and the effect of waste quantity are described, because they fundamentally contribute to the final waste ranking. Knowledge acquisition in this study was accomplished through the extensive review of published and specialized literature to establish facts necessary for the development of Fuzzy rule-bases. Owing to the uncertainty and imprecision of various forms of data (both quantitative and qualitative) essential for waste classification, and the complexity resulting from knowledge incompleteness, the use of Fuzzy Set Theory for the aggregation and computation of waste classification ranking index is proposed. A computer-aided intelligent decision tool is described in part II of this paper and the functionality of the Fuzzy waste classification algorithm is illustrated through nine worked examples.
J Maiti - One of the best experts on this subject based on the ideXlab platform.
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modeling uncertainty in risk assessment an integrated approach with Fuzzy Set Theory and monte carlo simulation
Accident Analysis & Prevention, 2013Co-Authors: N S Arunraj, Saptarshi Mandal, J MaitiAbstract:Modeling uncertainty during risk assessment is a vital component for effective decision making. Unfortunately, most of the risk assessment studies suffer from uncertainty analysis. The development of tools and techniques for capturing uncertainty in risk assessment is ongoing and there has been a substantial growth in this respect in health risk assessment. In this study, the cross-disciplinary approaches for uncertainty analyses are identified and a modified approach suitable for industrial safety risk assessment is proposed using Fuzzy Set Theory and Monte Carlo simulation. The proposed method is applied to a benzene extraction unit (BEU) of a chemical plant. The case study results show that the proposed method provides better measure of uncertainty than the existing methods as unlike traditional risk analysis method this approach takes into account both variability and uncertainty of information into risk calculation, and instead of a single risk value this approach provides interval value of risk values for a given percentile of risk. The implications of these results in terms of risk control and regulatory compliances are also discussed.
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modeling uncertainty in risk assessment an integrated approach with Fuzzy Set Theory and monte carlo simulation
Accident Analysis & Prevention, 2013Co-Authors: N S Arunraj, Saptarshi Mandal, J MaitiAbstract:Modeling uncertainty during risk assessment is a vital component for effective decision making. Unfortunately, most of the risk assessment studies suffer from uncertainty analysis. The development of tools and techniques for capturing uncertainty in risk assessment is ongoing and there has been a substantial growth in this respect in health risk assessment. In this study, the cross-disciplinary approaches for uncertainty analyses are identified and a modified approach suitable for industrial safety risk assessment is proposed using Fuzzy Set Theory and Monte Carlo simulation. The proposed method is applied to a benzene extraction unit (BEU) of a chemical plant. The case study results show that the proposed method provides better measure of uncertainty than the existing methods as unlike traditional risk analysis method this approach takes into account both variability and uncertainty of information into risk calculation, and instead of a single risk value this approach provides interval value of risk values for a given percentile of risk. The implications of these results in terms of risk control and regulatory compliances are also discussed. Language: en
