Fuzzy Sets and Fuzzy Logic

The American Journal of Psychology, 1993

Properties of the Min-Max Composition 79 PREFACE TO THE FOURTH EDITION xxv around fuzzy control, a concept that was very applicable, easy to understand, and, therefore, attractive to many industrial practitioners and the broad public. Since the start of computational intelligence theoretical as well as applicationoriented developments have become much more diversified and clear lead-times between theoretical development and application can no longer be recognized. I have used the opportunity of a fourth edition of this textbook, for which I am very grateful to Kluwer Academic Publishers, to adapt the book to the new developments, without exceeding the scope of a basic textbook, as follows: All chapters have been updated . The scope of part I has only been extended with respect to t-norms, other operators and uncertainty modeling because I am convinced that chapters 2 to 7 are still sufficient as a mathematical basis to understand all new developments in this area and also for part II of the book, where the major changes and extensions of this edition can be found : In chapter lathe modeling of uncertainty in expert systems was extended because this component has gained importance in practice. In chapter II primarily a section for defuzzification has been added for the same reason. Chapter 12 has been added because the application of fuzzy technology in information processing is already important and will certainly increase in importance in the future. Chapter 13 has been extended by explaining new methodological developments in dynamic fuzzy data analysis, which will also be of growing importance in the future. Eventually applications in chapter 15 have been completely restructured by deleting some, adding others and classifying all of them differently. This was necessary because the focus of applications here changed, for reasons explained in this chapter, strongly from "engineering intelligence" to "business intelligence". Of course, the index and the references have also been updated and extended. This time I would like to thank again Kluwer Academic Publishers for giving me the chance of a fourth edition and Dr. Angstenberger for her excellent research cooperation and for letting me use one application from her book. In particular, I would like to thank Ms. Katja Palczynski for her outstanding help to get the manuscripts ready for the publisher. I hope that this new edition of my textbook will help to keep respective courses in universities and elsewhere up-to-date and challenging and motivating for students as well as professors. It may also be useful for practitioners that want to update their knowledge of fuzzy technology and look for new applications in their area. Aachen, April 2001 H.-i. Zimmermann FUZZY SETS 1.1 Crispness, Vagueness, Fuzziness, Uncertainty Most of our traditional tools for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. By crisp we mean dichotomous, that is, yes-or-no-type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false-and nothing in between. In set theory, an element can either belong to a set or not; and in optimization, a solution is either feasible or not. Precision assumes that the parameters of a model represent exactly either our perception of the phenomenon modeled or the features of the real system that has been modeled. Generally, precision also implies that the model is unequivocal, that is, that it contains no ambiguities. Certainty eventually indicates that we assume the structures and parameters of the model to be definitely known, and that there are no doubts about their values or their occurrence. If the model under consideration is a formal model [Zimmermann 1980, p. 127], that is, if it does not pretend to model reality adequately, then the model assumptions are in a sense arbitrary, that is, the model builder can freely decide which model characteristics he chooses. If, however, the model or theory asserts factuality [Popper 1959; Zimmermann 1980], that is, if conclusions drawn from these models have a bearing on reality and are

Fuzzy Sets and Systems, 2003

This is an unusual book. Why and in which respect? It is amazing how Prof. Klir is able to explain a number of nontrivial facts using simple and succinct means. In 19 chapters of this small book he has succeeded in describing the essential theory of fuzzy set and fuzzy logic theory including their applications as well as explaining the main philosophical problems araising when dealing with uncertainty and vagueness. Thus, the book is very informative-one can ÿnd everything relevant there, mostly only outlined. However, the core of the problem is completed by enough references to be able to ÿnd missing information.

International Journal of Information Engineering and Electronic Business, 2013

In this art icle, we would like to revisit and comment on the definit ion of co mplementation of fu zzy sets and also on some of the theories and formulas associated with this. Furthermore, the existing probability-possibility consistency principles are also revisited and related results are v iewed fro m the standpoint of the Randomness-Fuzziness consistency principles. It is found that the existing definition of complementation as well as the probability-possibility consistency principles is not well defined. Consequently the results obtained from these would be inappropriate fro m our standpoints. Hence we would like to suggest some new defin itions for so me of the terms often used in the theory of fuzzy sets whenever possible.

Mathematical Social Sciences, 1991

2009

The uncertainty inherent in data, values of parameters, boundary conditions or variables used as inputs to mathematical models may be quantified by use of stochastic variables. As an example, let us consider the mortality of bacteria, which may be considered as a parameter useful to characterise the quality of a water sample. If the mortality has large values, then bacteria are eliminated and the water quality has a good chance of remaining acceptable. Mortality of bacteria is influenced by several factors, such as temperature, solar light, salinity and some biological characteristics. Usually, all these parameters are taken into consideration by means of the characteristic time t 90 , that is the time necessary to eliminate 90% of bacteria. Because of the various uncertainties, t 90 may be considered to be a random variable having a probability density distribution. As shown in .1, a log-normal probability density function may be used to fit the available data and represent uncertainties in the values of t 90 .

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

Advanced Fuzzy Logic Approaches in Engineering Science, 2019

Human health risk assessment is an important and a popular aid in the decision-making process. The basic objective of risk assessment is to assess the severity and likelihood of impairment to human health from exposure to a substance or activity that under plausible circumstances can cause harm to human health. One of the most important aspects of risk assessment is to accumulate knowledge on the features of each and every available data, information and model parameters involved in risk assessment. It is observed that most frequently model parameters, data, and information are tainted with aleatory and epistemic uncertainty. In such situations, fuzzy set theory or probability theory or Dempster-Shafer theory (DSS) can be explored to represent uncertainty. If all the three types of uncertainty coexist how far computation of the risk is concern, two ways to deal with the situation either transform all the uncertainties to one type of format or need for joint propagation of uncertaint...

2011

On two important counts, the Zadehian theory of fuzzy sets urgently needs to be restructured. First, it can be established that for a normal fuzzy number N = [α, β, γ] with membership function Ψ 1 (x), if α ≤ x ≤ β, Ψ 2 (x), if β ≤ x ≤ γ, and 0, otherwise, Ψ 1 (x) is in fact the distribution function of a random variable defined in the interval [α, β], while Ψ 2 (x) is the complementary distribution function of another random variable defined in the interval [β, γ]. In other words, every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. This is how a normal fuzzy number should be constructed, and this is how partial presence of an element in a fuzzy set has to be defined. Hence the measure theoretic matters with reference to fuzziness have to be studied accordingly. Secondly, the field theoretic matters related to fuzzy sets are required to be revised all over again because in the current definition of the complement of a fuzzy set, fuzzy membership function and fuzzy membership value had been taken to be the same, which led to the conclusion that the fuzzy sets do not follow the set theoretic axioms of exclusion and contradiction. For the complement of a normal fuzzy set, fuzzy membership function and fuzzy membership value are two different things, and the complement of a normal fuzzy set has to be defined accordingly. We shall further show how fuzzy randomness should be explained with reference to two laws of randomness defined for every fuzzy observation so as to make fuzzy statistical conclusions. Finally, we shall explain how randomness can be viewed as a special case of fuzziness defined in our perspective with reference to normal fuzzy numbers of the type [α, β, β]. Indeed every probability distribution function is a Dubois-Prade left reference function, and probability can be viewed in that way too.

Encyclopedia of Database Systems, 2009