Fuzzy Set and Possibility Theory in Optimization: L. Zadeh’s Contributions

2013 ◽  
pp. 383-387
Author(s):  
Weldon A. Lodwick ◽  
K. David Jamison
Keyword(s):  
Fuzzy Set ◽  
Possibility Theory

Fundamental Concepts

2019 ◽  
pp. 27-77
Keyword(s):  
Decision Making ◽  
Probability Theory ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Distributions ◽  
Possibility Theory ◽  
Multi Objective ◽  
Basic Ideas ◽  
Fuzzy Stochastic Programming

In this chapter, the authors discuss some basic concepts of probability theory and possibility theory that are useful when reading the subsequent chapters of this book. The multi-objective fuzzy stochastic programming models developed in this book are based on the concepts of advanced topics in fuzzy set theory and fuzzy random variables (FRVs). Therefore, for better understanding of these advanced areas, the authors at first presented some basic ideas of probability theory and probability density functions of different continuous probability distributions. Afterwards, the necessity of the introduction of the concept of fuzzy set theory, some important terms related to fuzzy set theory are discussed. Different defuzzification methodologies of fuzzy numbers (FNs) that are useful in solving the mathematical models in imprecisely defined decision-making environments are explored. The concept of using FRVs in decision-making contexts is defined. Finally, the development of different forms of fuzzy goal programming (FGP) techniques for solving multi-objective decision-making (MODM) problems is underlined.

UNCERTAIN AND FUZZY OBJECT BASES: A DATA MODEL AND ALGEBRAIC OPERATIONS

2011 ◽  
Vol 19 (02) ◽  
pp. 275-305 ◽  
Author(s):  
TRU H. CAO ◽  
HOA NGUYEN
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Object Oriented ◽  
Possibility Theory ◽  
Object Properties ◽  
Common Framework ◽  
Formal Basis ◽  
Object Oriented Systems ◽  
Is Value

Fuzzy set theory and probability theory are complementary for soft computing, in particular object-oriented systems with imprecise and uncertain object properties. However, current fuzzy object-oriented data models are mainly based on fuzzy set theory or possibility theory, and lack of a rigorous algebra for querying and managing uncertain and fuzzy object bases. In this paper, we develop an object base model that incorporates both fuzzy set values and probability degrees to handle imprecision and uncertainty. A probabilistic interpretation of relations on fuzzy sets is introduced as a formal basis to coherently unify the two types of measures into a common framework. The model accommodates both class attributes, representing declarative object properties, and class methods, representing procedural object properties. Two levels of property uncertainty are taken into account, one of which is value uncertainty of a definite property and the other is applicability uncertainty of the property itself. The syntax and semantics of the selection and other main data operations on the proposed object base model are formally defined as a full-fledged algebra.

scholarly journals Fuzzy Epidemiology: Measures for Observational Epidemiological Studies

2020 ◽  
Author(s):  
Ana Cláudia Oliveira Melo ◽  
Laisa Ribeiro de Sá ◽  
Rodrigo Pinheiro de Toledo Vianna ◽  
Ronei Marcos de Moraes
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Observational Studies ◽  
Risk Measures ◽  
Possibility Theory ◽  
Epidemiological Studies ◽  
Case Control ◽  
Case Scenario ◽  
Fuzzy Measures ◽  
Study Population

Abstract Background Epidemiological studies bring forth classic epidemiological measures calculation that are based on resulting quantities of dichotomic categorization of individuals, such as in events: diseased, non-diseased, exposed or unexposed. Dichotomic categorizations discard inherent uncertainties and subjectivities from the illness process the exposure which generate information losses on the measures. The fuzzy set theory categorizes each individual, allowing the soft transit amongst the events and considering the uncertainties and subjectivities. For the calculation of these measures, the fuzzy possibility theory is useful. Although there is already the proposition of making use of this methodology to the calculation of association and risk measures, there are no additional studies, in the literature, that characterize or apply the measures in epidemiological studies. Neither there are proposed calculations of other epidemiological measures or studies explaining the contribution of the resulting epidemiological measure of this methodology. This paper aims to increase the epidemiological measures sets to observational studies, using fuzzy set and possibility theories in the calculation of the denominated fuzzy epidemiological measures, featuring them in an original way.Methodology The proposed fuzzy measures were based in classic epidemiological measures. An observational study was simulated on a case-control type and fuzzy theories on the categorization of the individuals and for the calculation of fuzzy measures were applied. The simulations and calculations were performed by the software R.Results It was graphically observed the incorporation of uncertainties and subjectivities in the study population categorization. Comparing the classic to the fuzzy measures, it was observed that the contribution of the embedded uncertainties and subjectivities on the fuzzy measure presented a more complete final information about the illness process and exposure of the individuals. The graphic behavior of the proposed measures and of the already existent ones were characterized.Conclusion The fuzzy set epidemiological measures changes the paradigm of measures restricted to one numerical value. The information of the new fuzzy measures is seen as more trustworthy and helpful to decision making health managers, regarding which policies must be considered in accordance to the susceptible of harm and exposure in every population of each case scenario.

FLEXIBILITY AND FUZZY CASE-BASED EVALUATION IN QUERYING: AN ILLUSTRATION IN AN EXPERIMENTAL SETTING

2003 ◽  
Vol 11 (01) ◽  
pp. 43-66 ◽  
Author(s):  
MARTINE DE CALMÈS ◽  
DIDIER DUBOIS ◽  
EYKE HULLERMEIER ◽  
HENRI PRADE ◽  
FLORENCE SEDES
Keyword(s):  
Information System ◽  
Fuzzy Set ◽  
Possibility Theory ◽  
Experimental Setting ◽  
Unified Framework ◽  
Case Based

Queries to a database can be made more powerful by allowing flexibility in the specification of what has to be retrieved, and by referring to cases either for expressing the request, or for computing the answer. In this paper, we present an implemented information system (applied to a database describing houses to let), based on an approach developed in the fuzzy set and possibility theory setting. This provides a unified framework for expressing users' preferences about what they are looking for, for weighting the importance of requirements, for referring to examples that they like and/or counter-examples that they dislike, and for making case-based predictions. Thus information querying goes beyond the retrieving of items from a database, and involves associated tools which help the user to figure out the actual contents of the database.

Timed Possibilistic Logic

1991 ◽  
Vol 15 (3-4) ◽  
pp. 211-234
Author(s):  
Didier Dubois ◽  
Jérôme Lang ◽  
Henri Prade
Keyword(s):  
Temporal Logic ◽  
Fuzzy Set ◽  
Lattice Structure ◽  
Possibility Theory ◽  
Temporal Scale ◽  
Logical Framework ◽  
Classical Formula ◽  
Possibilistic Logic ◽  
Fuzzy Subsets

This paper is an attempt to cast both uncertainty and time in a logical framework. It generalizes possibilistic logic, previously developed by the authors, where each classical formula is associated with a weight which obeys the laws of possibility theory. In the possibilistic temporal logic we present here, each formula is associated with a time set (a fuzzy set in the more general case) which represents the set of instants where the formula is certainly true (more or less certainly true in the general case). When a particular instant is fixed we recover possibilistic logic. Timed possibilistic logic generalizes possibilistic logic also in the sense that we substitute the lattice structure of the set of the (fuzzy) subsets of the temporal scale to the lattice structure underlying the certainty weights in possibilistic logic. Thus many results from possibilistic logic can be straightforwardly generalized to timed possibilistic logic. Illustrative examples are given.

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