Fuzzy Set

Several fuzzifications of formal concept analysis have been proposed to deal with uncertain information. In this paper, we focus on concept lattices under a multiadjoint paradigm, which enriches the language providing greater flexibility to the user in that she can choose from a number of different connectives. Multi-adjoint concept lattices are shown to embed different fuzzy extensions of concept lattices found in the literature, the main results of the paper being the representation theorem of this paradigm and the embedding of other well-known approaches.

During the last decade, "fuzzy techniques" have been increasingly applied to the research area of construction management discipline. To date, however, no paper has attempted to summarize and present a critique of the existing "fuzzy" literature. This paper, therefore, aims to comprehensively review the fuzzy literature that has been published in eight selected top quality journals

This paper deals with the application of a soft computing technique to a medical project. The application consists in connecting a patient to an ophthalmologist situated in a hospital or to a specialized web center through a telecommunication network. The patient may be at home or at any other place agreed upon for that teleconsultation. The aim of the study concerns the evaluation of a patient's visual acuity, using web-based medical advice. Indeed an ophthalmologist actually requires specific equipment, available only in medical centers and used by specialized staff. This situation leads to limit the number of patients being examined by the physician and so to extend the waiting time necessary before you can get an appointment (you have to wait between 8 and 10 months at the moment in France). So, public health services have an interest in making these tests available to the patients, in schools, firms, at home and so on, the tests being implemented remotely by competent personnel thanks to modern means of telecommunication. In order to achieve this, a certain number of tests are made, during which optotypes (or letters of different sizes) are presented to the patient. We propose an algorithmic treatment of the binary answers, based on the use of fuzzy sets, in order to eliminate the different mistakes (voluntary or not) that can be made. This algorithm also allows to assess the quality of the examination by measuring the inaccuracy of the visual acuity. This work must be considered as a pioneering telemedicine project, and particularly as a way of developing expert systems able to generate medical advice without the physician's actual presence.

Career guidance for students, particularly in rural areas is a challenging issue in India. In the present era
of digitalization, there is a need of an automated system that can analyze a student for his/her capabilities,
suggest a career and provide related information. Keeping in mind the requirement, the present paper is an
effort in this direction. In this paper, a fuzzy based conceptual framework has been suggested. It has two
parts; in the first part a students will be analyzed for his/her capabilities and in the second part the
available courses, job aspects related to their capabilities will be suggested. To analyze a student, marks
in various subject in 10+2 standards and vocational interest in different fields have been considered and
fuzzy sets have been formed. On example basis, fuzzy inference rules have been framed for analyzing the
abilities in engineering, medical and hospitality fields only. In second part, concept of composition of
relations has been used to suggest the related courses and jobs.

An original methodology is proposed for the representa- tion, using MRI acquisitions, of cereberal structures in the presence of non-infiltrating pathology. This methodology includes the automatic detection and segmentation of brain tumors, the description of their anatomic and functional environment, the description of internal cerebral structures and their spatial relations to the tumor.

New entropy measures such as higher order fuzzy entropy and hybrid entropy of a set have been introduced. Various properties along with their proofs have been included. Applicability of these new measures to various problems has been highlighted.

In this paper, the Fuzzy Vector Median is proposed, defined as an extension of Vector Median. It is based on a novel distance definition of multidimensional fuzzy numbers (fuzzy vectors), which satisfy the property of angle decomposition. The proposed distance of two fuzzy vectors depends on the classical distance of the fuzzy set centers and on the fuzziness that every fuzzy set holds. As a result the Fuzzy Vector Median of a set of fuzzy vectors is affected by the presence of fuzziness.

This paper considers a single product inventory control in a Distribution Supply Chain (DSC). The DSC operates in the presence of uncertainty in customer demands. The demands are described by imprecise linguistic expressions that are modelled by discrete fuzzy sets. Inventories at each facility within the DSC are replenished by applying periodic review policies with optimal order up-to-quantities. Fuzzy customer demands imply fuzziness in inventory positions at the end of review intervals and in incurred relevant costs per unit time interval. The determination of the minimum of defuzzified total cost of the DSC is a complex problem which is solved by applying decomposition; the original problem is decomposed into a number of simpler independent optimisation subproblems, where each retailer and the warehouse determine their optimum periodic reviews and order up-to-quantities. An iterative coordination mechanism is proposed for changing the review periods and order up-to-quantities for each retailer and the warehouse in such a way that all parties within the DSC are satisfied with respect to total incurred costs per unit time interval. Coordination is performed by introducing fuzzy constraints on review periods and fuzzy tolerances on retailers and warehouse costs in local optimisation subproblems. Ó 2007 Published by Elsevier B.V.

This paper focuses on hardware implementations of fuzzy inference systems which provide low silicon cost, high operational speed and adaptability to different application domains. The architecture and basic building blocks of two fuzzy logic controllers are described and their functionality is illustrated with experimental results showing the capability of these systems to be applied as function approximators.

This paper proposes a model for the reoccupation of ants in a region of attraction, using evolutive partial differential diffusion-advection equations, in which the population dispersion and velocity in directions x and y are fuzzy parameters. The domain under study contains an attractive region, representing areas with high concentrations of palatable host plants with better nutritious qualities. The algorithm developed here uses information about the foraging behavior of a leaf-cutting ant colony of the Amazon region in northern Brazil. In the first model, we determined the numerical solution of the partial differential deterministic equation with constant dispersion and velocity, without incorporating uncertainties that may occur in this type of biological phenomenon. In the second model, we calculated the numerical solution of the partial differential equation, determining the dispersion at each iteration and for each triangular finite element. The dispersion was determined from a fuzzy rule-based system that depends on the number of individuals of the population and on the characteristics of the terrain. In this model, we also considered the velocity along the x and y directions as a fuzzy parameter that depends on the ant movement characteristics on the terrain in the y and x directions, respectively. The two models produced solutions consistent with ant behavior in response to the proximity of an attractive region, as stated in the papers by Shepherd (1982) [2] and Vasconcelos (1999) [3], but the fuzzy model incorporates uncertainties that do occur in the biological phenomenon under study.

In this paper, we present a view of kernels from a fuzzy set theoretical perspective. Indeed, it turns out that kernels which are positive-definite functions have to fulfill a consistency property given by the so-called T-transitivity of a fuzzy T-equivalence relation with respect to the triangular norm T. As a result, we introduce a triangular norm T cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are T cos-equivalences. Finally, some applied relevant aspects are discussed as a starting point for further research.

As a generalization of fuzzy sets, the concept of interval-valued fuzzy sets was introduced by Gorzalczany [Fuzzy Sets and Systems 21 (1987) 1]. In this paper, we shall extend the concept of “fuzzy inclusion”, introduced by Šostak [Supp. Rend. Circ. Mat. Palermo (Ser. II) 11 (1985) 89], to the interval-valued fuzzy setting and study its fundamental properties for some extent.

The evolutionary based computational approach has been matured into an extensive collection of concepts and techniques with the hype in the complexities of the problems which could not be easily dealt by the classical approach. The effectiveness and efficiency of these modern approaches are making them popular in the field of research and development. In this paper, we have focused on the involvement of a multi valued logical evolutionary based computational approach in the development of an effective system. This paper describes the area where fuzzy logic can be implemented effectively in system development life cycle. This paper explicitly mentions the application of Fuzzy Inference System (FIS) in one of the fact finding techniques of requirement analysis viz. rating scale question method to find the percentage level of the satisfaction of the user of an existing system or a new system.

... Springer-Verlag Berlin Heidelberg 2006 Fuzzy Genetic System for Modelling Investment Portfolio Rahib H. Abiyev and Mustafa Menekay ... (5) Here )(y i B μ is membership degree of the constraint. qi is maximum violation (tol-erance) designated for the constraint. ...

Quantifying or, more generally, estimating the severity of the possible consequences of occupational accidents is a decisive step in any occupational risk assessment process. Because of the lack of historic information (accident data collection and recording are incipient and insufficient, particularly in construction) and the lack of practical tools in the construction industry, the estimation/quantification of occupational accident severity is a notably arbitrary process rather than a systematic and rigorous assessment.

We propose two processes to obtain L-fuzzy concepts based on finite L-fuzzy contexts and the theory of Cousot and Cousot [Pacific J. Math. 82 (1979) 43]. The first algorithm calculates the L-fuzzy concepts derived from an L-fuzzy set and the second one constructs the whole L-fuzzy concept lattice. We also represent graphically the L-fuzzy concept lattice.

Fermented sausages are dried in natural or artificial driers. Air relative humidity (RH) and temperature (T) are regulated in artificial driers. Nowadays the set points of RH and T are manually readjusted in a staggered form by an expert on the basis of product evaluation and on his own experience. The main drawbacks of this control system are the discontinuity of the product evaluation and the cost of the experts' training. This work proposes an approach based on the theory of fuzzy sets to readjust the set points of RH and T with the help of on-line instrumental measurements related with the expert evaluation. The input parameters evaluated for the fuzzy control system were: an estimated water activity at the product surface (from heat balance) and the working time of the fan drier. The output parameters were the set points of RH (low and high). The fuzzy control system was applied to the drying process of fermented sausages in an industrial drier at constant temperature. Results show that this system can improve the manual control system and it is helpful for quality assurance during the drying process. It reduces the risk of crust development, one of the most important defects in fermented sausages.

A noncooperative model of a normal form game is developed using tools from fuzzy set theory. The paper extends the decision theory framework of to a game-theoretic setting. We identify conditions for the existence of equilibrium in this setting and provide a duopoly example. The paper also provides a review of the existing literature on noncooperative fuzzy games and their applications.

This paper considers the automatic design of fuzzy rule-based classification systems from labeled data. The classification accuracy and interpretability of generated rules are of major importance in fuzzy classification systems. We propose a weighting function for compatibility grade of patterns that improves the performance of fuzzy classification system without degrading the interpretability of fuzzy rules. Our approach divides the covering subspace of each fuzzy rule into two subdivisions based on a threshold. Any pattern with compatibility grade above this threshold should be classified truly so the weighting function enhances their association degree. For patterns below threshold, their compatibility grades remain unchanged. The splitting threshold for each rule (i.e. the compatibility grade of a specific pattern) is found using distribution of patterns in the covering subspace of that rule. We also show that how the proposed approach is applicable when fuzzy rules have certainty grades. Experiments on some well-known data sets are used to evaluate the performance of our approach.

This study examines estimates of extinction rates for the current purported biotic crisis and from the fossil record. Studies that compare current and geological extinctions sometimes use metrics that confound different sources of error and reflect different features of extinction processes. The per taxon extinction rate is a standard measure in paleontology that avoids some of the pitfalls of alternative approaches. Extinction rates reported in the conservation literature are rarely accompanied by measures of uncertainty, despite many elements of the calculations being subject to considerable error. We quantify some of the most important sources of uncertainty and carry them through the arithmetic of extinction rate calculations using fuzzy numbers. The results emphasize that estimates of current and future rates rely heavily on assumptions about the tempo of extinction and on extrapolations among taxa. Available data are unlikely to be useful in measuring magnitudes or trends in current extinction rates.

The purpose of this paper is to present a view of a telemonitoring system based on fuzzy logic for distress situations detection of elderly people living alone in the home environment. This system includes three different sensors: physiological sensors (cardiac frequency, activity or agitation, posture and fall detection sensor), microphones and infrared sensors. Taking into account the difficulty of the statistical modeling of abnormal situation and considering that the Fuzzy Logic has been actually used with success in the development of classifier and analysis of control systems, its use in the decision fusion module of our home elderly people monitoring system is presented and evaluated in this paper.

The theory of fuzzy sets and the development of qualitative reasoning have had similar motivations: coping with complexity in reasoning about the properties of physical systems. An approach is described that utilizes fuzzy sets to develop a fuzzy qualitative simulation algorithm that allows a semiquantitative extension to qualitative simulation, providing three significant advantages over existing techniques. Firstly, it allows a more detailed description of physical variables, through an arbitrary, but finite, discretisation of the quantity space. The adoption of fuzzy sets also allows common-sense knowledge to be represented in defining values through the use of graded membership, enabling the subjective element in system modelling to be incorporated and reasoned with in a formal way. Secondly, the fuzzy quantity space allows more detailed description of functional relationships in that both strength and sign information can be represented by fuzzy relations holding against two or multivariables. Thirdly, the quantity space allows ordering information on rates of change to be used to compute temporal durations of the state and the possible transitions. Thus, an ordering of the evolution of the states and the associated temporal durations are obtained. This knowledge is used to develop an effective temporal filter that significantly reduces the number of spurious behaviors. Experimental results with the algorithm are presented and comparison with other recently proposed methods is made.

Presented is a completeness theorem for fuzzy equational logic with truth values in a complete residuated lattice: Given a fuzzy set Σ of identities and an identity p≈q, the degree to which p≈q syntactically follows (is provable) from Σ equals the degree to which p≈q semantically follows from Σ. Pavelka style generalization of well-known Birkhoff's theorem is therefore established.

In this paper, we establish the local and global existence and uniqueness results for fuzzy functional differential equations. For the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle. Also, we apply these results to fuzzy differential equations with distributed delays and fuzzy population models.

Fuzzy set ordination was used to examine relationships between tree species and site factors on Mt. Ascutney, Vermont, USA. Prism plots were established at four elevations (455, 610, 760 and 915 m) along contour lines that encircled the mountain. In addition to elevation, which was the most important factor affecting tree species composition, slope, aspect and exposure to solar radiation all affected composition; however, these latter three factors were important only at 455 m. Topographic position was important at higher elevations as well. The responses of tree species to any of these factors were highly individualistic. Plots of the upper bounds of relative basal area of each species against the apparent elevation of sites where that species was found were useful in predicting how important an individual species could become over the course of forest succession. For example, northern hardwoods, which comprise the most common forest type of this region, are not very common on this mountain. This approach shows, however, that they are currently at levels well below their potential maximum; this is probably due to past disturbance.

A new comprehensive model of information retrieval (IR) based on Zadeh's calculus of linguistic statements is proposed. Its characteristic and novel feature is the capability to take into account both the imprecision and uncertainty pervading the textual information representation. It extends earlier IR models based on broadly meant fuzzy logic. Moreover, some techniques for indexing documents and queries in the framework of this model are proposed. The results of the computational experiments on standard document collections are reported.

In this article we firstly summarize some notions on L−fuzzy sets, where L denotes a complete lattice. We then study a special case of L−fuzzy sets, namely the "intuitionistic fuzzy sets". The importance of these sets comes from the fact that the negation is being defined independently from the fuzzy membership function. The latter implies both flexibility and effectiveness in fuzzy inference applications. We additionally show several practical applications on intuitionistic fuzzy sets, in the context of computational intelligence.

Many color vision systems require a first step of classifying pixels in a given image into a discrete set of color classes. In this paper we describe a human perceptionbased approach to pixel color segmentation. Fuzzy sets are defined on the H, S and V components of the HSV color space and provide a fuzzy logic model that aims to follow the human intuition of color classification. Experiments suggest that the classification performed by the proposed algorithm introduces an improvement over some other basic color classification techniques, especially in outdoor natural scenes, which are considered more challenging to color segmentation methods. The knowledge-driven model allows simple modification of the classification based on the needs of a specific application, and the efficiency of the algorithm in terms of computational complexity makes the proposed method suitable for applications where efficiency is a primary issue.

8. HANDBOOK OF MONETARY ECONOMICS (in 2 volumes) Edited by Benjamin Friedman and Frank Hahn 9. HANDBOOK OF DEVELOPMENT ECONOMICS (in 4 volumes) Volumes 1 and 2 edited by Hollis B. Chenery and TN Srinivasan Volumes 3A and 3B edited by Jere ...

Bow-tie analysis is a fairly new concept in risk assessment that can describe the relationships among different risk control parameters, such as causes, hazards and consequences to mitigate the likelihood of occurrence of unwanted events in an industrial system. It also facilitates the performance of quantitative risk analysis for an unwanted event providing a detailed investigation starting from basic causes to final consequences. The credibility of quantitative evaluation of the bow-tie is still a major concern since uncertainty, due to limited or missing data, often restricts the performance of analysis. The utilization of expert knowledge often provides an alternative for such a situation. However, it comes at the cost of possible uncertainties related to incompleteness (partial ignorance), imprecision (subjectivity), and lack of consensus (if multiple expert judgments are used). Further, if the bow-tie analysis is not flexible enough to incorporate new knowledge or evidence, it may undermine the purpose of risk assessment.

Two approaches for constructing control charts to monitor multivariate attribute processes when data is presented in linguistic form are suggested. Two monitoring statistics T \(^{\rm 2}_{f}\) and W 2 are developed based on fuzzy and probability theories. The first is similar to the Hotteling’s T 2 statistic and is based on representative values of fuzzy sets. The W 2 statistic, being a linear combination of dependent chi-square variables, its distribution is derived by Satterthwaite’s approximation. Resulting multivariate control charts are compared based on the average run length (ARL). A numerical example is given to illustrate the application of the proposed multivariate control charts and the interpretation of out-of-control signals.

A new impulse noise reduction method for colour images, called histogram-based fuzzy colour filter (HFC), is presented in this paper. The HFC filter is particularly effective for reducing high-impulse noise in digital images while preserving edge sharpness. Colour images that are corrupted with noise are generally filtered by applying a greyscale algorithm on each colour component separately. This approach causes artefacts especially on edge or texture pixels. Vector-based filtering methods were successfully introduced to overcome this problem. In this paper, we discuss an alternative technique so that no artefacts are introduced. The main difference between the new proposed method and the classical vector-based methods is the usage of colour component differences for the detection of impulse noise and the preservation of the colour component differences. The construction of the HFC filter involves three steps: (1) the estimation of the original histogram of the colour component differences, (2) the construction of suitable fuzzy sets for representing the linguistic values of these differences and (3) the construction of fuzzy rules that determine the output. Extensive simulation results show that the proposed filter outperforms many well-known filters (including vector-based approaches).

Utility mining discovers the most profitable item in transaction database which is of great importance in the upgrade of revenue. There are various algorithms for mining the high utility itemset such as CTU-PRO, Two phase and also UMMI algorithm. These algorithms mine the high utility itemset in an efficient way, but they do not reflect the fuzzy degree of purchased quantity which is essential for making decision in various applications like sales analysis and stock control. Itemsets with profit slightly less than the threshold is also discarded. To overcome these problems, fuzzy set theory is applied to the utility mining problem and a novel algorithm namely fuzzy high utility item-mine (FHUI-Mine) is introduced to mine the fuzzy high utility itemset. The quantity information from transaction database is fuzzified in order to reflect the fuzzy degree of purchased quantity. FHUI-Mine also provides a fuzzy threshold range that may include the itemsets whose support is slightly less than the designated threshold. More over this algorithm also provides the utility information due to fuzzification of threshold. To prove the feasibility of FHUI-Mine, it was compared with the well known UMMI algorithm through experimental evaluation. The results show that FHUI-Mine delivers higher mining capability as it can not only mine all high utility itemset but also discover additional itemsets that are potentially high utility ones.

In this paper, an attempt has been made to introduce cardinality of an intuitionistic fuzzy graph. Also, the definition of bipartite, complete bipartite, strong arc, strength of the connectedness, dominating set, domination number, independent set, independent domination number, total dominating and total domination number in intuitionistic fuzzy graphs are given. Some properties of these concepts are studied.

In this paper we focus on the aggregation of IDS alerts, an important component of the alert fusion process. We exploit fuzzy measures and fuzzy sets to design simple and robust alert aggregation algorithms. Exploiting fuzzy sets, we are able to robustly state whether or not two alerts are ''close in time", dealing with noisy and delayed detections. A performance metric for the evaluation of fusion systems is also proposed. Finally, we evaluate the fusion method with alert streams from anomaly-based IDS.

Axiomatic design gives a design map to designers in order to establish a scientific basis, which improves design activities. In the classical axiomatic design (AD) approach, the relationships between functional requirements (FRs) and design parameters (DPs) are defined in crisp numbers notably ''0" or ''1". This definition restricts relevant designers due to the incomplete information or uncertainty in various circumstances. This study debates the adaptation of the current AD principles into fuzzy sets theory. Those transformations are expressed by the developed formulas to prove the independence degree of the design in quantitative manner. A numerical study illustrates the logic of the proposed ideas clearly. Consequently, the proposed methodology provides a genuine link to explain the correspondences among FRs and DPs in linguistic form.

Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.