On the use of fuzzy rule interpolation techniques for monotonic multi-input fuzzy rule base models

Constructing a monotonicity relating function is important, as many engineering problems revolve around a monotonicity relationship between input(s) and output(s). In this paper, we investigate the use of fuzzy rule interpolation techniques for monotonicity relating fuzzy inference system (FIS). A mathematical derivation on the conditions of an FIS to be monotone is provided. From the derivation, two conditions are necessary. The derivation suggests that the mapped consequence fuzzy set of an FIS to be of a monotonicity order. We further evaluate the use of fuzzy rule interpolation techniques in predicting a consequent associated with an observation according to the monotonicity order. There are several findings in this article. We point out the importance of an ordering criterion in rule selection for a multi-input FIS before the interpolation process; and hence, the practice of choosing the nearest rules may not be true in this case. To fulfill the monotonicity order, we argue with an example that conventional fuzzy rule interpolation techniques that predict each consequence separately is not suitable in this case. We further suggest another class of interpolation techniques that predicts the consequence of a set of observations simultaneously, instead of separately. This can be accomplished with the use of a search algorithm, such as the brute force, genetic algorithm or etc.

A new dempster-shafer theory-based method with fuzzy targets for fuzzy sets ranking

In this paper, a new Fuzzy Set (FS) ranking method (for type-1 and interval type-2 FSs), which is based on the Dempster-Shafer Theory (DST) of evidence with fuzzy targets, is investigated. Fuzzy targets are adopted to reflect human viewpoints on fuzzy ranking. Two important measures in DST, i.e., the belief and plausibility measures, are used to rank FSs. The proposed approach is evaluated with several benchmark examples. The use of the belief and plausibility measures in fuzzy ranking are discussed and compared. We further analyze the capability of the proposed approach in fulfilling six reasonable fuzzy ordering properties as discussed in [9]-[11].

Vector valued similarity measures for Atanassov's intuitionistic fuzzy sets

We present a new approach for defining similarity measures for Atanassov's intuitionistic fuzzy sets (AIFS), in which a similarity measure has two components indicating the similarity and hesitancy aspects. We justify that there are at least two facets of uncertainty of an AIFS, one of which is related to fuzziness while other is related to lack of knowledge or non-specificity. We propose a set of axioms and build families of similarity measures that avoid counterintuitive examples that are used to justify one similarity measure over another. We also investigate a relation to entropies of AIFS, and outline possible application of our method in decision making and image segmentation. © 2014 Elsevier Inc. All rights reserved.

Segmentation of gray scale image based on intuitionistic fuzzy sets constructed from several membership functions

Segmentation is the process of extraction of objects from an image. This paper proposes a new algorithm to construct intuitionistic fuzzy set (IFS) from multiple fuzzy sets as an application to image segmentation. Hesitation degree in IFS is formulated as the degree of ignorance (due to the lack of knowledge) to determine whether the chosen membership function is best for image segmentation. By minimizing entropy of IFS generated from various fuzzy sets, an image is thresholded. Experimental results are provided to show the effectiveness of the proposed method.

Pobreza multidimensional na região nordeste: uma aplicação da Teoria dos Conjuntos Fuzzy (em 2010)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Probabilidades imprecisas: intervalar, fuzzy e fuzzy intuicionista

The idea of considering imprecision in probabilities is old, beginning with the Booles George work, who in 1854 wanted to reconcile the classical logic, which allows the modeling of complete ignorance, with probabilities. In 1921, John Maynard Keynes in his book made explicit use of intervals to represent the imprecision in probabilities. But only from the work ofWalley in 1991 that were established principles that should be respected by a probability theory that deals with inaccuracies. With the emergence of the theory of fuzzy sets by Lotfi Zadeh in 1965, there is another way of dealing with uncertainty and imprecision of concepts. Quickly, they began to propose several ways to consider the ideas of Zadeh in probabilities, to deal with inaccuracies, either in the events associated with the probabilities or in the values of probabilities. In particular, James Buckley, from 2003 begins to develop a probability theory in which the fuzzy values of the probabilities are fuzzy numbers. This fuzzy probability, follows analogous principles to Walley imprecise probabilities. On the other hand, the uses of real numbers between 0 and 1 as truth degrees, as originally proposed by Zadeh, has the drawback to use very precise values for dealing with uncertainties (as one can distinguish a fairly element satisfies a property with a 0.423 level of something that meets with grade 0.424?). This motivated the development of several extensions of fuzzy set theory which includes some kind of inaccuracy. This work consider the Krassimir Atanassov extension proposed in 1983, which add an extra degree of uncertainty to model the moment of hesitation to assign the membership degree, and therefore a value indicate the degree to which the object belongs to the set while the other, the degree to which it not belongs to the set. In the Zadeh fuzzy set theory, this non membership degree is, by default, the complement of the membership degree. Thus, in this approach the non-membership degree is somehow independent of the membership degree, and this difference between the non-membership degree and the complement of the membership degree reveals the hesitation at the moment to assign a membership degree. This new extension today is called of Atanassov s intuitionistic fuzzy sets theory. It is worth noting that the term intuitionistic here has no relation to the term intuitionistic as known in the context of intuitionistic logic. In this work, will be developed two proposals for interval probability: the restricted interval probability and the unrestricted interval probability, are also introduced two notions of fuzzy probability: the constrained fuzzy probability and the unconstrained fuzzy probability and will eventually be introduced two notions of intuitionistic fuzzy probability: the restricted intuitionistic fuzzy probability and the unrestricted intuitionistic fuzzy probability

On Fuzzy Implication Classes - Towards Extensions of Fuzzy Rule-Based Systems

Atualmente, há diferentes definições de implicações fuzzy aceitas na literatura. Do ponto de vista teórico, esta falta de consenso demonstra que há discordâncias sobre o real significado de "implicação lógica" nos contextos Booleano e fuzzy. Do ponto de vista prático, isso gera dúvidas a respeito de quais "operadores de implicação" os engenheiros de software devem considerar para implementar um Sistema Baseado em Regras Fuzzy (SBRF). Uma escolha ruim destes operadores pode implicar em SBRF's com menor acurácia e menos apropriados aos seus domínios de aplicação. Uma forma de contornar esta situação e conhecer melhor os conectivos lógicos fuzzy. Para isso se faz necessário saber quais propriedades tais conectivos podem satisfazer. Portanto, a m de corroborar com o significado de implicação fuzzy e corroborar com a implementação de SBRF's mais apropriados, várias leis Booleanas têm sido generalizadas e estudadas como equações ou inequações nas lógicas fuzzy. Tais generalizações são chamadas de leis Boolean-like e elas não são comumente válidas em qualquer semântica fuzzy. Neste cenário, esta dissertação apresenta uma investigação sobre as condições suficientes e necessárias nas quais três leis Booleanlike like — y ≤ I(x, y), I(x, I(y, x)) = 1 e I(x, I(y, z)) = I(I(x, y), I(x, z)) — se mantém válidas no contexto fuzzy, considerando seis classes de implicações fuzzy e implicações geradas por automorfismos. Além disso, ainda no intuito de implementar SBRF's mais apropriados, propomos uma extensão para os mesmos

On fuzzy ideals and fuzzy filters of fuzzy lattices

In the literature there are several proposals of fuzzi cation of lattices and ideals concepts. Chon in (Korean J. Math 17 (2009), No. 4, 361-374), using the notion of fuzzy order relation de ned by Zadeh, introduced a new notion of fuzzy lattice and studied the level sets of fuzzy lattices, but did not de ne a notion of fuzzy ideals for this type of fuzzy lattice. In this thesis, using the fuzzy lattices de ned by Chon, we de ne fuzzy homomorphism between fuzzy lattices, the operations of product, collapsed sum, lifting, opposite, interval and intuitionistic on bounded fuzzy lattices. They are conceived as extensions of their analogous operations on the classical theory by using this de nition of fuzzy lattices and introduce new results from these operators. In addition, we de ne ideals and lters of fuzzy lattices and concepts in the same way as in their characterization in terms of level and support sets. One of the results found here is the connection among ideals, supports and level sets. The reader will also nd the de nition of some kinds of ideals and lters as well as some results with respect to the intersection among their families. Moreover, we introduce a new notion of fuzzy ideals and fuzzy lters for fuzzy lattices de ned by Chon. We de ne types of fuzzy ideals and fuzzy lters that generalize usual types of ideals and lters of lattices, such as principal ideals, proper ideals, prime ideals and maximal ideals. The main idea is verifying that analogous properties in the classical theory on lattices are maintained in this new theory of fuzzy ideals. We also de ne, a fuzzy homomorphism h from fuzzy lattices L and M and prove some results involving fuzzy homomorphism and fuzzy ideals as if h is a fuzzy monomorphism and the fuzzy image of a fuzzy set ~h(I) is a fuzzy ideal, then I is a fuzzy ideal. Similarly, we prove for proper, prime and maximal fuzzy ideals. Finally, we prove that h is a fuzzy homomorphism from fuzzy lattices L into M if the inverse image of all principal fuzzy ideals of M is a fuzzy ideal of L. Lastly, we introduce the notion of -ideals and - lters of fuzzy lattices and characterize it by using its support and its level set. Moreover, we prove some similar properties in the classical theory of - ideals and - lters, such as, the class of -ideals and - lters are closed under intersection. We also de ne fuzzy -ideals of fuzzy lattices, some properties analogous to the classical theory are also proved and characterize a fuzzy -ideal on operation of product between bounded fuzzy lattices L and M and prove some results.

Transient stability analysis of electrical power systems using a neural network based on fuzzy ARTMAP

This work presents a methodology to analyze transient stability for electric energy systems using artificial neural networks based on fuzzy ARTMAP architecture. This architecture seeks exploring similarity with computational concepts on fuzzy set theory and ART (Adaptive Resonance Theory) neural network. The ART architectures show plasticity and stability characteristics, which are essential qualities to provide the training and to execute the analysis. Therefore, it is used a very fast training, when compared to the conventional backpropagation algorithm formulation. Consequently, the analysis becomes more competitive, compared to the principal methods found in the specialized literature. Results considering a system composed of 45 buses, 72 transmission lines and 10 synchronous machines are presented. © 2003 IEEE.

Fuzzy environmental quality index

This paper presents the construction of a fuzzy environmental quality index for decision support in municipal environmental management. Five groups of indicators were selected in order to obtain an equation that best represented reality in terms of environmental quality. The calculation was carried out using fuzzy mathematical concepts, with the aid of the package Fuzzy Logical Toolbox 2.1 for Matlab ® 6.1, which provides functions and some applications of the theory of fuzzy sets. The work seeks to create a method of inference concerning the nature of urban areas that are unsustainable with respect to the environment, an issue that is often relegated to the background during public policy discussions. The development of this index, together with its implementation and dissemination, could improve public awareness of environmental issues, and promote mobilization towards the use of best practices in local development. © 2010 IEEE.

Traffic flow breakdown prediction using feature reduction through Rough-Neuro fuzzy Networks

The prediction of the traffic behavior could help to make decision about the routing process, as well as enables gains on effectiveness and productivity on the physical distribution. This need motivated the search for technological improvements in the Routing performance in metropolitan areas. The purpose of this paper is to present computational evidences that Artificial Neural Network ANN could be use to predict the traffic behavior in a metropolitan area such So Paulo (around 16 million inhabitants). The proposed methodology involves the application of Rough-Fuzzy Sets to define inference morphology for insertion of the behavior of Dynamic Routing into a structured rule basis, without human expert aid. The dynamics of the traffic parameters are described through membership functions. Rough Sets Theory identifies the attributes that are important, and suggest Fuzzy relations to be inserted on a Rough Neuro Fuzzy Network (RNFN) type Multilayer Perceptron (MLP) and type Radial Basis Function (RBF), in order to get an optimal surface response. To measure the performance of the proposed RNFN, the responses of the unreduced rule basis are compared with the reduced rule one. The results show that by making use of the Feature Reduction through RNFN, it is possible to reduce the need for human expert in the construction of the Fuzzy inference mechanism in such flow process like traffic breakdown. © 2011 IEEE.

Application of an iterative method and an evolutionary algorithm in fuzzy optimization

This work develops two approaches based on the fuzzy set theory to solve a class of fuzzy mathematical optimization problems with uncertainties in the objective function and in the set of constraints. The first approach is an adaptation of an iterative method that obtains cut levels and later maximizes the membership function of fuzzy decision making using the bound search method. The second one is a metaheuristic approach that adapts a standard genetic algorithm to use fuzzy numbers. Both approaches use a decision criterion called satisfaction level that reaches the best solution in the uncertain environment. Selected examples from the literature are presented to compare and to validate the efficiency of the methods addressed, emphasizing the fuzzy optimization problem in some import-export companies in the south of Spain. © 2012 Brazilian Operations Research Society.

Selecting green suppliers based on GSCM practices: Using fuzzy TOPSIS applied to a Brazilian electronics company

Due to an increased awareness and significant environmental pressures from various stakeholders, companies have begun to realize the significance of incorporating green practices into their daily activities. This paper proposes a framework using Fuzzy TOPSIS to select green suppliers for a Brazilian electronics company; our framework is built on the criteria of green supply chain management (GSCM) practices. An empirical analysis is made, and the data are collected from a set of 12 available suppliers. We use a fuzzy TOPSIS approach to rank the suppliers, and the results of the proposed framework are compared with the ranks obtained by both the geometric mean and the graded mean methods of fuzzy TOPSIS methodology. Then a Spearman rank correlation coefficient is used to find the statistical difference between the ranks obtained by the three methods. Finally, a sensitivity analysis has been performed to examine the influence of the preferences given by the decision makers for the chosen GSCM practices on the selection of green suppliers. Results indicate that the four dominant criteria are Commitment of senior management to GSCM; Product designs that reduce, reuse, recycle, or reclaim materials, components, or energy; Compliance with legal environmental requirements and auditing programs; and Product designs that avoid or reduce toxic or hazardous material use. © 2013 Elsevier B.V. All rights reserved.

Epidemiological Models of Directly Transmitted Diseases: An Approach via Fuzzy Sets Theory

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)