Framework Of Fuzzy Set Theory Research Articles

Characterization of Nested Lattice Codes Through Quotient Groups Relative to Normal Fuzzy Subgroups for Channel Quantization

Interference is in general, an undesirable obstacle to communication in wireless communication networks. Therefore, in order to realize interference alignment, this work intends to characterize classical codes through the framework of fuzzy set theory for quantizing real-valued channel coefficients. Thenceforward, from the classical framework nested real lattice codes are constructed by using construction A of nested real ideal lattices and, therefore, through normal fuzzy subgroups this work shows that the constructed quotient groups of the nested real ideal lattices relative to normal fuzzy subgroups of these nested real ideal lattices are nested vector spaces and, consequently, such nested quotient groups have the structure of a linear code, are isomorphic to the nested real lattice codes and defined by linear fuzzy codes. Through such an isomorphism and the construction of the nested quotient groups relative to normal fuzzy subgroups, it is possible to conclude that the codewords of these nested linear fuzzy codes are in a one-to-one correspondence to membership vectors. Through the constructed nested quotient groups, that is, nested linear fuzzy codes, this work supplies how to quantize real-valued channel coefficients onto these nested quotient groups in order to realize interference alignment and, consequently, it is possible to encode an arbitrary real-valued channel coefficient by a codeword from a linear fuzzy code.

A VIKOR-based approach to evaluate river contamination risks caused by wastewater treatment plant discharges

Wastewater treatment plants (WWTPs) rarely eliminate emerging contaminants from effluents they discharged into waterways, and therefore, represent significant contaminations sources with deleterious environmental risks. This paper presents a VIKOR-based model to assess the contamination risk posed by a cluster of WWTPs. A risk index was defined via building a membership function embodying the performance degrees of WWTPs and risks levels within the framework of fuzzy set theory. The proposed approach was tested using a case study of WWTPs cluster along the Pearl River. Sensitivity analyses were carried out to investigate the robustness of the model. The results confirmed the ability of the proposed approach to reveal the risk level of a given treatment point. Further, the comparison with a TOPSIS scheme as well as sensitivity analysis results substantiate the consistency, accuracy, and reliability of the proposed approach. It is therefore bounds to improve the decentralized management of WWTPs-induced river contamination.

Multicriteria Approach for Supplier Selection: Evidence from a Case Study in the Fashion Industry

Worldwide manufactures are redesigning supply chains, often outsourcing with the aim of maintaining a competitive advantage and gaining market share. When selecting and purchasing fabrics, firms should actively cooperate with suppliers, to meet customers’ needs. In this view, the supplier selection process plays a key role in keeping a competitive edge in global markets. Therefore, this study proposes a multicriteria decision-making model (MCDM) to ease supplier evaluation and selection. Supply chain operation reference metrics (SCOR metrics) and fuzzy technique for order of preference by similarity to ideal solution (FTOPSIS) are combined to build a model for supplier selection. The former allows us to conduct a very thorough fact-based analysis of all features in the supply chain, while the combination of fuzzy theory and SCOR model allows us to deal with uncertainty. The main novelty of this method is that it incorporates a consolidated supply chain management criteria within the framework of fuzzy set theory and multicriteria decision-making model (MCDM) facilitating their application into practice. The proposed approach is tested by considering the case of a manufacturing firm in the fashion industry willing to designate the most appropriate supplier within a set of three potential ones.

Three-way decision with conflict analysis approach in the framework of fuzzy set theory

Three-way decision with conflict analysis approach in the framework of fuzzy set theory

Quantification of qualitative assessments using computing with words: in framework of fuzzy set theory

When qualitative data are collected, which is a widespread situation encountered in several disciplines ranging from social sciences to decision making to engineering as a result of dealing with ill-defined concepts or with complex tasks to assess, there is a certain need to crunch them and infer from them. Hence, quantification, namely computing with words (CW), emerges as a research area. Even though social science disciplines deal with these types of data numerically using various Likert scales, fuzzy set theory first proposed by Zadeh dealing words or sentences with mathematically oriented manner in order to create machines that mimic the reasoning of human being brings new perspectives. Since then, its applicability has expanded into various disciplines for different purposes to transform subjectivity into objectivity. In this manuscript, the effort of transformation from subjectivity into objectivity will be reviewed based on the available proposed methods in the literature. Two illustrative examples will be employed. While the first illustrative example, which consists of balanced and unbalanced linguistic term sets, is used for each reviewed method in the literature to show its computations step by step, the second of which, balanced linguistic term set, is an example used in the literature that is employed. Therefore, a comprehensive review of the proposed methods with illustrative examples is presented.

Comments on some theories of fuzzy computation

In classical computability theory, there are several (equivalent) definitions of computable function, decidable subset and semi-decidable subset. This paper is devoted to the discussion of some proposals for extending these definitions to the framework of fuzzy set theory. The paper mainly focuses on the notions of fuzzy Turing machine and fuzzy computability by limit processes. The basic idea of this paper is that the presence of real numbers in the interval [0,1] forces us to refer to endless approximation processes (as in recursive analysis) and not to processes terminating after a finite number of steps and giving the exact output (as in recursive arithmetic). In accordance with such a point of view, an extension of the famous Church thesis is proposed.

Fuzzy—probabilistic seismic hazard assessment, case study: Tehran region, Iran

This study presents a new model for probabilistic seismic hazard assessment based on fuzzy sets theory. To accomplish seismic hazard analysis in the framework of fuzzy sets theory, all of the variables are first converted into Gaussian fuzzy sets using α-cut method. Then, fuzzified variables are used in seismic hazard analysis. Finally, the outputs are defuzzified using center of area method. The method is applied to Tehran region, Iran, and the fuzzy hazard curve is obtained for the site. The curve displays a fuzzy-probabilistic estimate of peak ground acceleration (PGA) over bedrock for the various return periods. PGA values for the region are estimated to be 0.18–0.20 g and 0.42–0.48 g for 50- and 475-year return periods, respectively. The results are given as fuzzy intervals which accommodate the vagueness inherent in the data.

Seismic liquefaction potential assessed by fuzzy comprehensive evaluation method

Liquefaction of loose, saturated granular soils during earthquakes poses a major hazard in many regions of the world. The determination of liquefaction potential of soils induced by earthquake is a major concern and an essential criterion in the design process of the civil engineering structures. A large number of factors that affect the occurrence of liquefaction during earthquake exist a form of uncertainty of non-statistical nature. Fuzzy systems are used to handle uncertainty from the data that cannot be handled by classical methods. It uses the fuzzy set to represent a suitable mathematical tool for modeling of imprecision and vagueness. The pattern classification of fuzzy classifiers provides a means to extract fuzzy rules for information mining that leads to comprehensible method for knowledge extraction from various information sources. Therefore, it is necessary to handle the soil liquefaction problem in a rational framework of fuzzy set theory. This study investigates the feasibility of using fuzzy comprehensive evaluation model for predicting soil liquefaction during earthquake. In the fuzzy comprehensive evaluation model of soil liquefaction, the following factors, such as earthquake intensity, standard penetration number, mean diameter and groundwater table, are selected as the evaluating indices. The results show that the method is a useful tool to assess the potential of soil liquefaction.

CAPM with fuzzy returns and hypothesis testing

Over the last four decades, several estimation issues of the beta have been discussed extensively in many articles. An emerging consensus is that the betas are time-dependent and their estimates are impacted by the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold. After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test is carried out to investigate whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks suggest that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change.

An approach for exploring the dynamical behaviour of inhomogeneous structural inclusions under consideration of epistemic uncertainty

PurposeThe purpose of this paper is to analyze the wave scattering behaviour of an inhomogeneous and eccentric inclusion in a homogeneous matrix material. Another purpose is to evaluate the influence of epistemic uncertainty on the wave scattering behaviour, particularly on the lack of knowledge about this eccentricity. This task calls for a multidisciplinary model.Design/methodology/approachThe inclusion is modelled as a multi‐layered obstacle, with all layers being eccentric with respect to each other. The material behaviour of the embedding matrix is linear elastic and isotropic. In a multidisciplinary approach, the interaction of the inhomogeneous inclusion and the embedding matrix with respect to an incoming shear wave of arbitrary shape is solved analytically. The purely analytical solution process takes place in the frequency‐domain. Due to the lack of knowledge about the eccentric configuration of the matrix inclusion and its influence on the total wave field inside the matrix material, the mechanical model is coupled with fuzzy set theory for modelling this non‐stochastic uncertainty.FindingsAn analytical model for describing the wave scattering behaviour of an elastic matrix inclusion with eccentric set‐up is found and intimately connected with the framework of fuzzy set theory. Hence it is shown that the treatment of epistemic uncertainty with the derived analytical model is possible and fruitful. Additionally, it is shown that eccentric configurations lead to highly increased amplitudes with respect to the reference case of a concentric or even homogenous set‐up of the inclusion.Originality/valueThe value of this contribution is in the analytical model, which allows one to predict the wave scattering behaviour of eccentric configurations of multi‐layered fibres including the surrounding interphase, and its coupling with fuzzy set theory to cope with the epistemic uncertainty inherent in the geometric set‐up of the matrix inclusion.

Towards a Fuzzy Expert System on Toxicological Data Quality Assessment.

Quality assessment (QA) requires high levels of domain-specific experience and knowledge. QA tasks for toxicological data are usually performed by human experts manually, although a number of quality evaluation schemes have been proposed in the literature. For instance, the most widely utilised Klimisch scheme1 defines four data quality categories in order to tag data instances with respect to their qualities; ToxRTool2 is an extension of the Klimisch approach aiming to increase the transparency and harmonisation of the approach. Note that the processes of QA in many other areas have been automatised by employing expert systems. Briefly, an expert system is a computer program that uses a knowledge base built upon human expertise, and an inference engine that mimics the reasoning processes of human experts to infer new statements from incoming data. In particular, expert systems have been extended to deal with the uncertainty of information by representing uncertain information (such as linguistic terms) as fuzzy sets under the framework of fuzzy set theory and performing inferences upon fuzzy sets according to fuzzy arithmetic. This paper presents an experimental fuzzy expert system for toxicological data QA which is developed on the basis of the Klimisch approach and the ToxRTool in an effort to illustrate the power of expert systems to toxicologists, and to examine if fuzzy expert systems are a viable solution for QA of toxicological data. Such direction still faces great difficulties due to the well-known common challenge of toxicological data QA that "five toxicologists may have six opinions". In the meantime, this challenge may offer an opportunity for expert systems because the construction and refinement of the knowledge base could be a converging process of different opinions which is of significant importance for regulatory policy making under the regulation of REACH, though a consensus may never be reached. Also, in order to facilitate the implementation of Weight of Evidence approaches and in silico modelling proposed by REACH, there is a higher appeal of numerical quality values than nominal (categorical) ones, where the proposed fuzzy expert system could help. Most importantly, the deriving processes of quality values generated in this way are fully transparent, and thus comprehensible, for final users, which is another vital point for policy making specified in REACH. Case studies have been conducted and this report not only shows the promise of the approach, but also demonstrates the difficulties of the approach and thus indicates areas for future development.

Fuzzy rough data reduction in large datasets with building application software (FRDRS)

Fuzzy rough data reduction software (FRDRS) is designed to facilitate make reduction of high dimensional data. It is a toolkit for analysing tabular data within the framework of fuzzy set theory. It is designed to support the overall data mining and knowledge discovery process: from initial browsing and pre-processing of the data, via computation of minimal attributes sets and generation of if-then rules or descriptive patterns, to validation and analysis of the induced rules or patterns. This computer programme is used for reduction of information system tables by using Java programming language and fuzzy-rough sets approach. Many researchers achieved a lot of very strong papers in the areas of applications and generalisations of rough set theory with fuzzy set theory or topological space theory. But they do not make good experimental results that support their results because the difficulties that they faced due to the large amount of data need to study and to analyse. Therefore, our software is interesting for those researchers to achievement their experiments. This programme is used under operating systems ‘windows_98, windows_2000, windows me, windows xp’ not used under DOS.

Fuzzy set theory

AbstractSince its inception in 1965, the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of this theory can be found, for example, in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Mathematical developments have advanced to a very high standard and are still forthcoming to day. In this review, the basic mathematical framework of fuzzy set theory will be described, as well as the most important applications of this theory to other theories and techniques. Since 1992 fuzzy set theory, the theory of neural nets and the area of evolutionary programming have become known under the name of ‘computational intelligence’ or ‘soft computing’. The relationship between these areas has naturally become particularly close. In this review, however, we will focus primarily on fuzzy set theory. Applications of fuzzy set theory to real problems are abound. Some references will be given. To describe even a part of them would certainly exceed the scope of this review. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

Computing additive [formula omitted]-diversity from presence and absence scores: A critique and alternative parameters

Whittaker first proposed to measure the variation in species composition among plots or β -diversity as the ratio between regional diversity ( γ -diversity) and average local diversity ( α -diversity). More recently, an alternative way of partitioning diversity for which β -diversity is obtained as the difference between γ -diversity and average α -diversity has become very popular for linking the structure of species assemblages to ecosystem functioning in a spatially explicit manner. Unfortunately, additive β -diversity computed from species presences and absences suffers from the major drawback of being dependent on regional species richness. For instance, if the separation between β -diversity and γ -diversity is incomplete, so that variation in species composition is affected by species richness, then differences in β -diversity values among different sets of plots could reflect differences in the species count rather than any fundamental difference in species composition among the plots. Based on the above observation, in this paper I will first propose a basic requirement for β -diversity measures that adequately captures our intuitive notion of independence of species richness. Next, I will show that additive β -diversity computed from species presence and absence scores can be interpreted within the framework of fuzzy set theory. Finally, based on this unusual “fuzzy” interpretation of additive β -diversity, I will introduce two families of parametric β -diversity measures whose members have varying sensitivities to the presence of rare and frequent species.

English

Since the descriptions and judgements on weapon systems are usually linguistic and fuzzy, it is more realistic to evaluate weapon systems in the framework of fuzzy sets theory. In this paper, a new method to evaluate the best main battle tank is proposed. It can be seen that the proposed method is more efficient due to the fact that, by canonical representation of arithmetic operation on fuzzy numbers, simple arithmetic operations on crisp numbers are used, instead of complicated fuzzy numbers operations. In addition, the final scores of each alternative can be represented as crisp numbers. As a result, the order of alternatives can be determined without the procedure of ranking fuzzy numbers. Finally, a numerical example to evaluate the best main battle tanks is used to illustrate the efficiency of the proposed method.

Conversion of probabilistic information into fuzzy sets for engineering decision analysis

To carry out seismic hazard analysis in the framework of fuzzy set theory, it may become necessary to convert probabilistic information regarding some of the variables into triangular or trapezoidal fuzzy sets. In this paper, three approaches for converting probabilistic information, represented by a probability distribution, into an equivalent triangular or trapezoidal fuzzy set are discussed. In all the three approaches, the probability distribution is first converted into a probabilistic fuzzy set, which is then converted into the equivalent triangular or trapezoidal fuzzy set. The first approach is based on the method of least-square curve fitting, the second approach is based on the conservation of uncertainty (represented by the entropy) associated with the probabilistic fuzzy set in a mean square sense, and the third approach is based on the minimisation of Hausdorff distance (HD) between the probabilistic and the equivalent fuzzy sets. The effectiveness of these approaches in preserving the entropy as well as in preserving the elements of the fuzzy set and their corresponding grades of membership are also discussed with the help of a numerical example of obtaining equivalent fuzzy set for peak ground acceleration. It is found that the approach based on minimisation of Hausdorff distance provides a simple and efficient way for converting the probabilistic information into an equivalent fuzzy set.

Fusion of spatial relationships for guiding recognition, example of brain structure recognition in 3D MRI

Spatial relations play an important role in recognition of structures embedded in a complex environment and for reasoning under imprecision. Several types of relationships can be modeled in a unified way using fuzzy mathematical morphology. Their combination benefits from the powerful framework of fuzzy set theory for fusion tasks and decision making. This paper presents several methods of fusion of information about spatial relationships and illustrates them on the example of model-based recognition of brain structures in 3D magnetic resonance imaging.

A FUZZY OBJECT-BASED DATA MODEL FOR IMPERFECT SPATIAL INFORMATION INTEGRATING EXACT OBJECTS AND FIELDS

In this contribution a fuzzy object-based data model for imperfect spatial information is proposed within the formal framework of fuzzy set theory and possibility theory. The proposal integrates the two opposite views of geographic entities seen as either exact objects or continuous fields in current Geographic Information Systems. By assuming a Fuzzy Object-Oriented Data modelling approach new data types for representing geographic entities characterized by either unsharp boundaries or indeterminate boundaries and relationships are defined.Linguistic qualifiers are adopted both to express the extent by which an element of the spatial domain satisfies a gradual property, and to identify regions with indeterminate boundaries.

Neuro-fuzzy modeling in bankruptcy prediction

For the past 30 years the problem of bankruptcy prediction had been thoroughly studied. From the paper of Altman in 1968 to the recent papers in the '90s, the progress of prediction accuracy was not satisfactory. This paper investigates an alternative modeling of the system (firm), combining neural networks and fuzzy controllers, i.e. using neuro-fuzzy models. Classical modeling is based on mathematical models that describe the behavior of the firm under consideration. The main idea of fuzzy control, on the other hand, is to build a model of a human control expert who is capable of controlling the process without thinking in a mathematical model. This control expert specifies his control action in the form of linguistic rules. These control rules are translated into the framework of fuzzy set theory providing a calculus, which can stimulate the behavior of the control expert and enhance its performance. The accuracy of the model is studied using datasets from previous research papers.

Fuzzy Closure Operators

Closure operators (and closure systems) play a significant role in both pure and applied mathematics. In the framework of fuzzy set theory, several particular examples of closure operators and systems have been considered (e.g. so-called fuzzy subalgebras, fuzzy congruences, fuzzy topology etc.). Recently, fuzzy closure operators and fuzzy closure systems themselves (i.e. operators which map fuzzy sets to fuzzy sets and the corresponding systems of closed fuzzy sets) have been studied by Gerla et al., see e.g. [3, 4, 6, 7]. As a matter of fact, a fuzzy set A is usually defined as a mapping from a universe set X into the real interval [0, 1] in the above mentioned works. Therefore, the structure of truth values of the “logic behind” is fixed to [0, 1] equipped usually with minimum as the operation corresponding to logical conjunction. As it appeared recently in the investigations of fuzzy logic in narrow sense [9, 10] (i.e. fuzzy logic as a many-valued logical calculus), there are several logical calculi formalizing the intuitive idea of “fuzzy reasoning” which are complete with respect to the semantics over special structures of truth values. Among these structures, the most general one is that of a residuated lattice (it is worth noticing that residuated lattices (introduced originally in [12] as an abstraction in the study of ideal systems of rings) have been proposed as a suitable structure of truth values by Goguen in [8]). From this point of view, the need for a general notion of a “fuzzy closure” concept becomes apparent. The aim of this paper is to outline a general theory of fuzzy closure operators and fuzzy closure systems. In the next section we introduce the necessary concepts. In Section 3, fuzzy closure operators and systems are