hierarchical pattern classification abstract the aim of this paper is to present a divisive hierarchical clustering method a chain of fuzzy partitions ordered by the refinement relation and the corresponding binary fuzzy hierarchy are obtained by this method since no a priori knowledge concerning the number of real clusters is needed the method represents a solution for the cluster validity problem an algorithm based on this method is given an adaptive distance is used to make the algorithm efficient also for unequal population clusters

fuzzy sets modeling and optimization for disaster control systems planning abstract various forms of disasters natural and man made abound although to some extent some of these have been exacerbated by modern civilization various technologies have been invoked to control or mitigate their adverse effects modeling and optimization based techniques have also been employed we outline some efforts based on fuzzy sets theory which are utilized in conjunction with classical methods to enhance the reliability and utility of these efforts in particular disasters caused by floods and storm water pollution are used as vehicles for the exemplification of our models

fuzzy probability measures abstract in 4 hohle has defined fuzzy measures on g fuzzy sets 2 where g stands for a regular boolean algebra consequently since the unit interval is not complemented fuzzy sets in the sense of zadeh 8 do not fit in this framework in a straightforward manner it is the purpose of this paper to continue the work started in 5 which deals with 0 1 fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5 we give necessary and sufficient conditions for such a measure to be a classical integral as in 9 in the case the space is generated a counterexample in the general case is also presented finally it is shown that a fuzzy probability measure is always an integral if the space is generated if we replace the operations and by the t norm to and its dual s0 see 6

fuzzy equivalence relation and its multigranulation spaces multigranulation rough sets focus on describing uncertain sets in different granularity spaces different approximation accuracy of an uncertain set can be obtained through changing knowledge granularities according to user s need however it is a bottleneck for establishing a multigranulation space to better describe an uncertain set in a proper granulation space and the typical way of a multigranulation space is a hierarchical quotient space determined by a fuzzy equivalence relation in this paper a new concept granularity of binary relation gb is put forward firstly and knowledge granulation gk is proved to be a special case of gb the changing rules of granularity series for describing a hierarchical quotient space structure are proposed and approximation accuracy series of an uncertain set is presented for automatically searching optimal knowledge space according to user s requirement a utility function with both knowledge space granularity and approximation accuracy is proposed then classification isomorphism and granularity isomorphism between two hierarchical quotient space structures are defined and discussed through the hierarchical quotient space structure of a fuzzy equivalence relation the essence of classification ability of a fuzzy equivalence relation is also revealed finally a new algorithm for establishing isomorphic fuzzy equivalence relations based on a hierarchical quotient space structure is presented and this algorithm can further explain the nature of those isomorphic fuzzy equivalence relations from a new perspective

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

bayesian estimation in forest surveys when samples or prior information are fuzzy abstract fuzzy phenomena are frequently encountered in forestry due to the complexity of the systems under investigation in this paper existing inference methods are presented to accommodate situations in which random events take place in fuzzy circumstances fuzzy concepts are incorporated into these commonly accepted approaches to account for uncertainties used by factors other than randomness which traditional statistical approaches deal primarily with based on two extensions of likelihood functions for fuzzy samples bayesian estimates are generalized for use when sample information and prior distribution of parameters are fuzzy the estimators are developed for some typical membership functions for fuzzy samples and fuzzy priors applications are illustrated using examples for forestry

effects of membership function parameters on the performance of a fuzzy signal detector this paper describes a signal detection algorithm based on fuzzy logic the detector combines evidence provided by two waveform features and explicitly considers uncertainty in the detection decision the detector classifies waveforms including a signal not including a signal or being uncertain in which case no conclusion regarding presence or absence of a signal is drawn piecewise linear membership functions are used and a method to describe the membership functions in terms of two parameters is developed the performance of the detector is compared to a bayesian maximum likelihood detector using brainstem auditory evoked potential signals in simulated noise and the effects of the steepness slope and overlap of the membership functions on detector performance are evaluated by varying the membership function steepness and overlap the fuzzy detector can almost completely eliminate classification errors at the cost of a large number of uncertain classifications or it can be made to perform similarly to the bayesian detector

a fuzzy adaptive network approach to parameter estimation in cases where independent variables come from an exponential distribution in a regression analysis it is assumed that the observations come from a single class in a data cluster and the simple functional relationship between the dependent and independent variables can be expressed using the general model y f x email protected however a data cluster may consist of a combination of observations that have different distributions that are derived from different clusters when faced with issues of estimating a regression model for fuzzy inputs that have been derived from different distributions this regression model has been termed the switching regression model and it is expressed with y l f l x email protected l email protected i 1 pl i here l i indicates the class number of each independent variable and p is indicative of the number of independent variables j r jang anfis adaptive network based fuzzy inference system ieee transaction on systems man and cybernetics 23 3 1993 665 685 m michel fuzzy clustering and switching regression models using ambiguity and distance rejects fuzzy sets and systems 122 2001 363 399 e q richard a new approach to estimating switching regressions journal of the american statistical association 67 338 1972 306 310 in this study adaptive networks have been used to construct a model that has been formed by gathering obtained models there are methods that suggest the class numbers of independent variables heuristically alternatively in defining the optimal class number of independent variables the use of suggested validity criterion for fuzzy clustering has been aimed in the case that independent variables have an exponential distribution an algorithm has been suggested for defining the unknown parameter of the switching regression model and for obtaining the estimated values after obtaining an optimal membership function which is suitable for exponential distribution

conditional fuzzy entropy of fuzzy dynamical systems abstract entropy of fuzzy dynamical systems has been investigated as an isomorphism invariant in dumitrescu 1995 13 in this paper we will introduce a new invariant called the conditional fuzzy entropy of fuzzy dynamical systems with respect to a finite fuzzy partition and an invariant sub algebra which is an extension of fuzzy entropy on fuzzy dynamical systems this new invariant possesses some basic properties such as power rule fuzzy version of abramov rokhlin entropy additive formula and generating sequence

three way fuzzy partitions defined by shadowed sets abstract pedrycz introduces the notion of a shadowed set as a three way approximation of a fuzzy set given a pair of thresholds a shadowed set approximates a fuzzy set by three regions an core of the fuzzy set the difference between a support and the core and the complement of the support a fuzzy partition is a family of nonempty fuzzy sets satisfying certain conditions when applying ideas of shadowed sets to a family of nonempty fuzzy sets the result is a three way approximation of the family that is a family of shadowed sets this enables us to study a new type of three way fuzzy partitions we introduce and examine three pairs of properties for defining a three way fuzzy partition disjointness property overlap property and coverage property of cores and supports we propose a general definition of a three way fuzzy partition by using two sets of properties by considering all possible non equivalent subsets of properties we obtain 21 classes of three way fuzzy partition we study three classes of three way fuzzy partition and establish their connections to partitions fuzzy partitions and interval set clusters

on the perimeter and area of fuzzy sets abstract the note of area comes naturally as a measure of a fuzzy set recently a rosenfeld defined the permeter diameter height and width of a fuzzy set athough these do not generalize the corresponding notions for crisp sets they lack inner conformity in that they do not satisfy simple interrelations we are used to in the case of crisp sets in the present note we modify definitions and show that our perimeter diameter and height fit more comfortably with the notion of area we also define a projection operator that should be used along with the multiplicative conjunction in the process of fuzzy reasoning

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

an efficient method combining active learning kriging and monte carlo simulation for profust failure probability abstract for more and more complicated engineering structures it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption by combining active learning kriging with monte carlo simulation ak mcs an efficient method is proposed to estimate the profust failure probability firstly the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption this integral is further reorganized as a weighted sum of a series of classical failure probabilities by gaussian quadrature and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds secondly mcs is used according to the probability input distribution to generate the sample pool in which the active learning kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds an improved learning function is proposed by minimizing the u learning function minima corresponding to all limit state functions so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the kriging model once the updating process of the kriging model converges all limit state functions can be identified by the kriging model and the profust failure probability can be estimated by using the kriging model without any extra model evaluation several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability

issues on adjointness in multiple valued logics we contribute to the theory of implications and conjunctions related by adjointness in multiple valued logics we suggest their use in zadeh s compositional rule of inference to interpret generalized modus ponens inference schemata we provide new complete characterizations of implications that distinguish left arguments implications that satisfy the exchange principle divisible conjunctions commutative conjunctions associative conjunctions and triangular norms we also introduce and characterize pseudo strict and pseudo continuous implications and conjunctions and we explore the close relationship between these two notions

specificity shift in solving fuzzy relational equations abstract studied is a system of n fuzzy relational equations with a max t composition x k r y k where the input output fuzzy sets x k and y k are available while the fuzzy relation r needs to be determined the solution to these equations is derived through a new paradigm of specificity shift the main objective is to modify a level of specificity of the fuzzy sets relational constraint so that the modified constraints allow for the use of some standard theoretical results of the theory of fuzzy relational equations that otherwise would have been found totally unjustifiable in more detail the specificity of the available input fuzzy sets becomes gradually increased while an opposite tendency is observed for the output fuzzy sets the optimization of the specificity levels is discussed in detail numerical studies are also included finally the use of the specificity shift is discussed in a conjunction with the exploitation of standard gradient based techniques

fuzzy control from the logical point of view abstract this paper is a discussion of the logical aspect in fuzzy control which is an application of the theory of approximate reasoning first we analyze approximate reasoning and show that it is a concise logical theory based on many valued logic integrated with some parts of linguistics then we demonstrate that fuzzy control based on equality relations can transparently be formulated as a result of the logical theory finally we demonstrate that fuzzy control can be seen as a repeated derivation in a continuously modified logical theory and propose a hypothesis on the role of linguistics in the controllability problem

an efficient algorithm for globally minimizing sum of quadratic ratios problem with nonconvex quadratic constraints in this paper a new branch and bound algorithm based on the rectangular partition and the lagrangian relaxation for solving sum of quadratic ratios problem with nonconvex quadratic constraints qsrp is proposed firstly the problem qsrp is converted into an equivalent sum of linear ratios problem with quadratic constraints lsrp then the bounding procedure is investigated by minimizing the restricted lagrangian function of lsrp with a given estimate of the lagrange multipliers minimizing a linear relaxation of the restricted lagrangian overcomes the difficulty that the restricted lagrangian is often nonconvex and facilitates the use of lagrangian duality within a global optimization framework in the algorithm the interval newton method is used to facilitate convergence in the neighborhood of the global solution the proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems finally numerical experiments are reported to show the efficiency of our algorithm

uncertainty assessment of hydrological models with fuzzy extension principle evaluation of a new arithmetic operator the most important drawback of standard fuzzy arithmetic is unrealistic accumulation of input uncertainties which results in divergence of fuzzy outputs some of the currently available methods of simulating fuzzy systems provide the results which tend to naive large values after several time steps of the system simulation in this paper a new fuzzy arithmetic operator based on fuzzy extension principle has been proposed for simulation and assessment of uncertainty in hydrological systems implementing the concept of fuzzy approximate reasoning in the proposed approach in this study represents acceptable behavior in uncertainty propagation from the parameters and structures of the models to the outputs to show the efficiency of the proposed fuzzy arithmetic operator in the context of hydrologic modeling two nonlinear monthly water balance models have been examined and their outputs have been compared with the results obtained by standard fuzzy arithmetic and the vertex method one small humid basin in france and a middle size basin in a semiarid region in iran have been the case studies of this research in this paper the lower and upper bounds and the most frequent values of the model parameters inferred from the sampling simulation procedure have been used to define triangular fuzzy membership functions three statistical indicators have been used to evaluate efficiency of the methods based on the bracket observations and coverage of the uncertainty bounds the estimated values of these indicators have shown that both vertex and the proposed methods outperform standard fuzzy arithmetic also the proposed method has provided better or roughly equal efficiencies compared with vertex method over both basins

fuzzy finite element approach for the analysis of imprecisely defined systems many engineering systems are too complex to be defined in precise mathematical terms they often contain information and features that are vague imprecise qualitative linguistic or incomplete the traditional deterministic and probabilistic techniques are not adequate to analyze such systems this paper aims at developing a fuzzy finite element approach for the analysis of imprecisely defined systems the development of the methodology starts from the basic concepts of fuzzy numbers of fuzzy arithmetic and implements suitably defined fuzzy calculus concepts such as differentiation and integration for the derivation manipulation and solution of the finite element equations simple stress analysis problems involving vaguely defined geometry material properties external loads and boundary conditions are solved to establish and to illustrate the new procedure the approach developed is applicable to systems that are described in linguistic terms as well as those that are described by incomplete information if complete data are known the method handles the information similar to that of a probabilistic approach the present approach represents a unique methodology that enables us to handle certain types of imprecisely known data more realistically compared with the existing procedures

decompositions and range for additive fuzzy measures the aim of the present paper is to prove the existence of hahn and lebesgue decompositions for additive fuzzy measures in the sense of 3 it is shown that the radon nikodym theorem can be carried over from classical measures to additive fuzzy measures with their corresponding integrals these results are applied to obtain a generalization of liapounoff s convexity theorem 12 whose utility in the theory of fuzzy games is argued in the last section

inverse possibility analysis method for possibility based design optimization doi 10 2514 1 16546 structural analysis and design optimization have recently been extended to the stochastic approach to consider variousuncertainties however inareaswhereitisnotpossibletoproduceaccurate statistical information forinput data the probabilistic method is not appropriate for stochastic structural analysis and design optimization because improper modeling of uncertainty could cause a greater degree of statistical uncertainty than those of physical uncertainty for systems with insufficient information for input data possibility based or fuzzy set methods have recently been introduced in structural analysis and design optimization using possibility methods the extended fuzzy operations are much simpler than of random variables possibility based design optimization will provide more conservative designs than those from probability methods andwill provide a system level of failure possibility automatically this paper proposes a new formulation of possibility based design optimization using the performance measure approach for the inverse possibility analysis the maximal possibility search method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the multilevel cut method two mathematical examples including a nonmonotonic response and a physical example of vehicle side impact are used to demonstrate the proposed maximal possibility search method and possibility based design optimization

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

fuzzy sets and membership functions abstract in this paper we try to provide an analytical basis for the theory of fuzzy sets the main idea is to introduce the the notions of consensus sets and subjective variables which strongly resemble respectively the events and random variables in the probability theory the resemblance is a natural consequence from the fact that one of the most important roles of the fuzzy set theory is to present a method for the analysis of the uncertainty originated in human subjectivity while the probability theory for that of randomness

a practicable contraction approach for the sum of the generalized polynomial ratios problem abstract in this paper a practicable contraction approach is proposed for solving the sum of the generalized polynomial ratios problem p with generalized polynomial constraints due to the intrinsic difficulty of problem p less work has been devoted to solving this problem the proposed approach is based on reducing the original nonconvex problem p as a standard geometric programming gp problem by utilizing simple transformation and contraction strategies the resulting optimization problem can be solved effectively by utilizing the solutions of a series of gp problems the tractability and effectiveness of the proposed successive contraction approach are demonstrated by several numerical examples and the performance comparison of the proposed approach and other methods published is also presented in terms of solution quality

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

entropy of fuzzy dynamical systems abstract entropy of a finite fuzzy partition had been defined dumitrescu 1983 1993 using an additive fuzzy measure butnariu 1983 in this paper we define the independence of fuzzy partitions with respect to a fuzzy measure some properties concerning the entropy of independent fuzzy partitions are proved this concept of independence is also relevant for the study of the entropy of a fuzzy process dumitrescu 1993 we prove some new results concerning the entropy of a fuzzy process the fuzzy measure preserving transformations dumitrescu 1993 may be related with the fuzzy ergodic theory the main problem in ergodic theory is to build the isomorphism invariants kolmogorov s 1958 transformation entropy is such an invariant in this paper we define the entropy of a fuzzy dynamical system we prove that this entropy is an isomorphism invariant

global optimization for sum of linear ratios problem with coefficients the global optimization of the sum of linear ratios problem has attracted the interest of researchers and practitioners for a number of years since these types of optimization problems are nonconvex various specialized algorithms have been proposed for globally solving these problems however these algorithms are only for the case that sum of linear ratios problem without coefficients and may be difficult to implement in this paper a branch and bound algorithm is proposed for globally solving the sum of linear ratios problem with coefficients by utilizing an equivalent problem and linearization technique the initial nonconvex programming problem is reduced to a sequence of linear programming problems the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are given to show the feasibility and effectiveness of the proposed algorithm

the structural properties and weak convergence of fuzzy orthogonal measures the aim of this paper is to study the fuzzy measures proposed by butnariu 1 3 first the concept of fuzzy orthogonal measure on a regular fuzzy measurable space is presented then the structural characteristics of fuzzy orthogonal measure which shows the relation between the classical finite measure and the fuzzy orthogonal measure is demonstrated finally the weak convergence of fuzzy orthogonal measures is investigated

fuzzy boundary element method for the analysis of imprecisely defined systems many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches the uncertainties present may be associated with parameters that are vague imprecise or linguistic for example the geometry material properties external actions loads and boundary conditions may be imprecise in a practical system a fuzzy boundary element method for the analysis of imprecisely defined systems is developed starting from the basic concepts of fuzzy sets and fuzzy arithmetic the various steps of the boundary element method are redefined using fuzzy concepts the resulting fuzzy equations are solved using a fuzzified version of gaussian elimination procedure coupled with truncation the truncation method limits the growth of interval ranges of response parameters to obtain realistic and accurate solutions the procedure is illustrated by considering the analysis of a potential flow problem the methodology is applicable to uncertain systems that are described in linguistic terms as well as those that are described by incomplete information the approach represents a unique methodology that enables the analysis and design of many engineering systems more realistically

contribution to application of energy measure of fuzzy sets abstract the works of de luca termini continued by for example knopfmacher loo and gottwald are the most important on the topic of determination of measures of fuzzy sets the matter is to evaluate how fuzzy a fuzzy set is there are two general concepts of measures of fuzzy set i e entropy and energy measures we show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations applications of the use of energy measure discussed in detail include decision making fuzzy process control and prediction in fuzzy systems

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

some properties of k step exclusion processes we introduce k step exclusion processes as generalizations of the simple exclusion process we state their main equilibrium properties when the underlying stochastic matrix corresponds to a random walk or is positive recurrent and reversible finally we prove laws of large numbers for tagged and second class particles

fuzzy measurability and integrability developpement d une nouvelle theorie relative au probleme de la mesure des objets flous continuite des fonctions sigma additives sur une classe sigma additive d ensembles flous extension de la notion d integrale floue methode de calcul des integrales floues problemes ouverts

evaluation of design alternatives in mechanical engineering abstract the paper gives a description of a structure dedicated to evaluation of design alternatives in mechanical engineering a list of particular features of the system such as its ability to cope with fuzziness and a uniform way of treating various criteria involved in design processes is given and discussed in length special attention is focused on empirical engineering oriented aspects of membership functions estimation with a particular emphasis on expressing achieved precision the paper re analyzes some existing methods of multiobjective decision making some original contributions of the study to this area are focussed on developing new indices significantly facilitating interpretation of results of ranking such as e g specificity of different alternatives as well as strength of dominance observed among them an illustration of the performance of the system is completed with the aid of an evaluation of different types of joints found in hydraulic cylinders

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

characterization of fuzzy measures constructed by means of triangular norms abstract following the ideas presented by the author e p klement j math anal appl 85 1982 543 565 finite t fuzzy measures are introduced t being a measurable triangular norm we show that a t fuzzy measure is always a fuzzy measure as considered earlier e p klement j math anal appl 25 1980 330 339 then we study the relation to the integral with respect to some classical measure finally for some special triangular norms t we give precise characterizations of the corresponding classes of t fuzzy measures

monotonic optimization problems and solution approaches problems of maximizing or minimizing monotonic functions of n variables under monotonic constraints are discussed a general framework for monotonic optimization is presented in which a key role is given to a property analogous to the separation property of convex sets the approach is applicable to a wide class of optimization problems including optimization problems dealing with functions representable as differences of increasing functions d i functions

on fuzzy neighbourhood spaces abstract we study the behaviour of the fuzzy closure operator of a fuzzy neighbourhood space fns for short with regard to the two operations of sum and conjunction on fuzzy subsets we supply sufficient conditions on a cech fuzzy closure operator to be the fuzzy closure operator of a fns finally we study quotient fuzzy neighbourhood systems with the help of the notion of dual fuzzy neighbourhood spaces of 11

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

the approximation of piecewise linear membership functions and łukasiewicz operators in this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations the gradient based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives with this construction we can even approximate the lukasiewicz operator family which plays an important role in fuzzy logic first of all from the theoretical point of view although in practice in optimization and learning it is rarely used because the lack of good analytical properties e g a continuous gradient the proposed approximation enlarges the applicability of fuzzy methods to the operators and membership functions where the differentiability is desirable

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

naive possibilistic classifiers for imprecise or uncertain numerical data in real world problems input data may be pervaded with uncertainty in this paper we investigate the behavior of naive possibilistic classifiers as a counterpart to naive bayesian ones for dealing with classification tasks in the presence of uncertainty for this purpose we extend possibilistic classifiers which have been recently adapted to numerical data in order to cope with uncertainty in data representation here the possibility distributions that are used are supposed to encode the family of gaussian probabilistic distributions that are compatible with the considered dataset we consider two types of uncertainty i the uncertainty associated with the class in the training set which is modeled by a possibility distribution over class labels and ii the imprecision pervading attribute values in the testing set represented under the form of intervals for continuous data moreover the approach takes into account the uncertainty about the estimation of the gaussian distribution parameters due to the limited amount of data available we first adapt the possibilistic classification model previously proposed for the certain case in order to accommodate the uncertainty about class labels then we propose an algorithm based on the extension principle to deal with imprecise attribute values the experiments reported show the interest of possibilistic classifiers for handling uncertainty in data in particular the probability to possibility transform based classifier shows a robust behavior when dealing with imperfect data

solving the sum of ratios problems by a harmony search algorithm the sum of ratios problems have numerous applications in economy and engineering the sum of ratios problems are considered to be difficult as these functions are highly nonconvex and multimodal in this study we propose a harmony search algorithm for solving a sum of ratios problem numerical examples are also presented to demonstrate the effectiveness and robustness of the proposed method in all cases the solutions obtained using this method are superior to those obtained from other methods

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

fuzzy control with fuzzy inputs this paper is concerned with the use of fuzzy inputs in fuzzy logic controllers a precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed the proposed concepts are applied to control a first order process with a pi controller this simple process is chosen to clearly illustrate the behavior of the closed loop system using fuzzy inputs for fuzzy reference and fuzzy measurement finally a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed loop fuzzy control

solving sum of quadratic ratios fractional programs via monotonic function in this paper a branch reduce bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints due to its intrinsic difficulty less work has been devoted to globally solving this problem the proposed algorithm is based on reformulating the problem as a monotonic optimization problem and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm

the possibilistic moments of fuzzy numbers and their applications this paper introduces the nearest weighted interval and point approximations of a fuzzy number and then suggests weighted possibilistic moments about these points of fuzzy numbers the possibilistic moments play an important role in fuzzy sets and systems specifically in physics mathematics and statistics we provide the definition of the moments of fuzzy numbers as well as the definition of moments in probability theory some of their applications are mentioned

fuzzy particle filtering for uncertain systems in this paper we propose a novel fuzzy particle filtering method for online estimation of nonlinear dynamic systems with fuzzy uncertainties this approach uses a sequential fuzzy simulation to approximate the possibilities of the state intervals in the state space and estimates the state by fuzzy expected value operator to solve the degeneracy problem of the fuzzy particle filter one corresponding resampling technique is introduced in addition we compare the fuzzy particle filter with ordinary particle filter in both aspects of the theoretical basis and algorithm design and demonstrate that the proposed filter outperforms standard particle filters especially when the number of the particles is small the numerical simulations of two continuous state nonlinear systems and a jump markov system are employed to show the effectiveness and robustness of the proposed fuzzy particle filter

the fuzzy delphi method via fuzzy statistics and membership function fitting and an application to the human resources a new fuzzy delphi method is proposed the method employs the fuzzy statistics and technique of the conjugate gradient search to fit membership functions membership functions besides triangles may be derived for the fuzzy forecasts a procedure termed as stabilization rather than convergence of the iterative interval valued surveys is utilized in which the a level set of the fuzzy forecasts is used the testing of stability of the process is conducted on the fuzzy forecasts between two consecutive iterations instead of within each iteration an application of this method to the problem of managerial talent assessment for a company located in taiwan and the obtained results are provided

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

inferring a possibility distribution from empirical data several transformations from probabilities to possibilities have been proposed in particular dubois and prade s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution in this paper this method is generalized to the case where the probabilities are unknown the only information being a data sample represented by a histogram it is proposed to characterize the probabilities of the different classes by simultaneous confidence intervals with a given confidence level email protected from this imprecise specification a procedure for constructing a possibility distribution is described insuring that the resulting possibility distribution will dominate the true probability distribution in at least 100 email protected of the cases finally a simple efficient algorithm is given which makes the computations tractable even if the number of classes is high

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

fuzzy rule based models for infiltration modeling water movement in the unsaturated zone requires a great number of parameters such as the initial water content the saturated moisture content and the saturated hydraulic conductivity which can relatively easily be assessed other functions such as the suction head and the relative hydraulic conductivity which are highly nonlinear functions of the actual water content are more difficult to find furthermore infiltration experiments carried out at a few locations can only define these parameters at selected locations and a reasonable interpolation of these parameters is extremely difficult in this paper two fuzzy logic based models are presented for the infiltration process basic definitions of fuzzy sets are given and basic operations are explained principles of the rule based infiltration models are presented the rules for the first model are based on the moisture content of the adjacent layers and the depth of the wetting front using these rules the flows can be calculated as exact quantities the rules are derived from a training set which was obtained from several test runs of a green and ampt ga infiltration model separate rules are derived for the flow dynamics in the upper layer runoff and infiltration and for water flow between the layers examples of the application of the model are given the model performs with similar accuracy as the ga model using only 40 of its parameters and running 2 orders of magnitude faster a training set obtained by numerical solution of the richards equation was used to create a second fuzzy rule based model here the rules are based on the moisture content of the adjacent layers results obtained by the use of the models are compared with measured values showing a very accurate reproduction of the values

entropy of a fuzzy process abstract using an additive fuzzy measure 1 in 5 a general concept of entropy of a fuzzy partition has been considered this concept is used to define the entropy of a fuzzy process in this paper some properties of this entropy are summarized these results could be used to build a fuzzy ergodic theory in a further paper the entropy of a measure preserving transformation of a fuzzy measure space will be studied

r fuzzy sets abstract the theory of fuzzy sets was founded by zadeh as an approach to cope with the pressing need to deal with phenomena which cannot be modelled properly by conventional mathematics because they contain factors which are fuzzy in nature however in this frame of the theory of fuzzy sets we cannot discern two conflicting fuzzy conceptions properly and therefore the excluded middle law is violated in this theory one does not differ fuzzy sets from their membership functions that is a fuzzy set in the sense of zadeh is equivalent to its membership function this may be the main reason why the excluded middle law was violated in this paper we try to provide a mathematical frame of fuzzy sets theory such that it can not only eliminate the suspicion for the objective reality of membership functions of fuzzy sets but also avoid violating the excluded middle law in our frame fuzzy sets and their membership functions are not equivalent conceptions

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

fuzzy sets induced by consensus and set operations abstract the idea of consensus is introduced to construct a fuzzy set system the two different approaches to the fuzzy set theory the falling shadow and the fuzzy set system based on consensus are shown to be closely related and the latter gives another analysis of those three types of set operations that were investigated by the former and proves them to be subject to the combinations of concerning representative consensus sets a notion of homomorphism is defined and applied to two fuzzy set systems to clarify similarity between them this notion shows itself very useful in the study of fuzzy set systems

lattice valued f transforms and similarity relations abstract the relationships between lattice valued f transforms and similarity relations are investigated several similarity relations are naturally associated with f transforms three types of such similarity relations related to lattice valued f transforms are investigated namely similarity relationships that are related to spaces with fuzzy partitions as ground structures of the f transform relationships between extensional hulls with respect to similarity relations and f transforms and relationships between similarities of pairs of functions on the one hand and similarities of the f transforms of these functions on the other some examples of possible applications are also presented

fuzzy dynamic programming fuzzy adaptive neuro control and the general medical diagnosis problem abstract the fuzzy medical diagnosis decision models of esogbue and elder employed fuzzy sets theory to directly and more correctly model all the information nets useful in reaching a scientific understanding of a patient s health status this knowledge was utilized in constructing a sensible medical hypothesis decision problem in this paper we show how fuzzy dynamic programming and neural networks can be used to extend the process to other phases by linking the four stages of medical hypothesis physician s observation preliminary diagnosis and final diagnosis in an adroit manner

after the fuzzy wave reached europe abstract we briefly describe the situation of the fuzzy community in europe before the fuzzy wave we show that there was a lot of expertise in almost all subareas of fuzzy logic then we describe some aspects of the fuzzy wave itself in particular fuzzy control and first generation fuzzy systems that played a role during the time the fuzzy wave reached europe we also make some remarks about european companies and their activities during this time then we discuss the situation after the fuzzy wave some aspects of second generation fuzzy systems the combination of fuzzy logic with other theories and techniques and some new application areas finally we will discuss the chances of fuzzy systems in europe in the future

digital signal restoration using fuzzy sets a new signal restoration method with considerable flexibility in incorporating a priori information is developed the method defines a fuzzy set for each piece of information to restrict the set of acceptable solutions using fuzzy sets makes it possible to model partially defined information as well as exact knowledge the intersection of all the fuzzy sets is the feasibility set the original signal is a member of this set with a high membership value and any high membership valued element of this set is a nonrejectable solution such solutions can be computed by using optimization techniques ideally the feasibility set contains only the original signal the chance of recovering the original signal decreases as the feasibility set gets larger thus the size of the feasibility set gives a quality measure for the solution the method generated successful results in many restorations for which the conventional techniques have failed and may be applied in image coding and tomography

a stochastic model for growing sandpiles and its continuum limit we study a stochastic model for the sandpile growth the height of the sandpile above every site of a d dimensional lattice is denoted by an integer the height changes randomly and after a suitable scaling of the time and space we show that the rescaled height converges to a deterministic function the limit is governed by a nonlinear evolution equation

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

one to one correspondences between ɛ partitions 1 ɛ equivalences and ɛ pseudometrics the generalization of the concept of a classical or crisp partition to that of a fuzzy partition requires in our researches the choice of a family of fuzzy subsets which preserves the conditions characterizing crisp partitions however the usual constraints for the union and intersection of subsets are very limiting and seem to be far from the fuzzy spirit so that the union of subsets in the partitions is forced to be close to the universe of discourse and the intersection of any two subsets in the partition is forced to be close to the empty set furthermore since the classical definition of partition is very related to the union and intersection the generalization of this concept has to be applicable for any extension of these operations for fuzzy sets apart from this we need to redefine the concept of fuzzy partition to establish a one to one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations the different definitions that we found in the literature did not exactly verify these requirements although some of them were more appropriate than others finally we have given a definition of fuzzy partition of any fuzzy subset of the universe which is a generalization of crisp partitions and of most of the different definitions of fuzzy partitions which preserves the classical ideas of covering and disjointness and such that it is appropriate in our researches classification problems fuzzy questionnaires fuzzy information theory once our fuzzy partitions epartitions are defined it is necessary to find the fuzzy binary relation which generates and characterizes the fuzzy partitions this fuzzy relation has to be by analogy with the crisp case a fuzzy equivalence relation two points in the same fuzzy subset of the partition will be related and two points in different subsets will not in relation with this equivalence relation it is also possible to obtain a way to measure the distance between two points by means of the degree of relation that there exists between them and therefore by means of the membership degree to a same subset in the fuzzy partition the definitions of these concepts and the relationships between them are the purpose of this paper

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

equilibrium fluctuations of nongradient reversible particle systems by taking the generalized symmetric exclusion process as an example we present a unified martingale approach to study the equilibrium fluctuations of nongradient reversible interacting particle systems the hydrodynamic limit of the generalized symmetric exclusion process has been derived by kipnis landim and 011a we show that the limit of the density fluctuation fields of this process is an infinite dimensional ornstein uhlenbeck process

compatibility of t norms with the concept of e partition in this paper we study the behaviour of a kind of partitions formed by fuzzy sets the e partitions with respect to three important operations refinement union and product of partitions in the crisp set theory the previous operations lead to new partitions every refinement of a partition is also a partition the union of partitions of disjoint sets is a partition of the union set the product of two partitions of two sets is a partition of the intersection of the partitioned sets it has been proven that e partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t norm and the union by the maximum t conorm in this paper we consider any t norm defining the intersection of fuzzy sets and we characterize those t norms for which refinements unions and products of e partitions are e partitions we pay special attention to these characterizations in the case of continuous t norms

fuzzy δ ɛ partitions in this paper the concept of a fuzzy e partitions of any fuzzy subset of the universe is introduced as a generalization of that of a classical partition the approach is based on four observations the first one is that for any two members of a classical partition their intersection is the empty set if they are not equal the second one is that the union of all members of a classical partition is equal to the partitioned set the third one is that this definition has to be applicable for any extension of the union and intersection for fuzzy subsets and the fourth one is that this definition has to allow us to establish a one to one correspondence between fuzzy partitions and some kind of fuzzy equivalence relations once this new concept is introduced by considering that the constrains for union and intersection are too limiting and that they have to be slightly modified some equivalence definitions of e partitions are proposed in particular two very important cases are studied they are the case when is equal to e because this is very useful in practical applications and the case when is equal to 1 e that is when the degree of fuzziness for the union and the intersection are equal this case is very useful in theoretical applications and it will be related to the concept of e equal fuzzy subsets apart from the fact that they are a generalization of that of a classical partition the new concept of e partition is also a generalization of that of the best known definitions of fuzzy partitions proposed in the literature

global optimization of nonlinear sum of ratios problem this article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem p on nonconvex feasible region first a problem q is derived which is equivalent to problem p in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and the constraint functions of the problem q over the feasible region the proposed branch and bound algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of linear programming problems the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm

triangular norm based measures and their markov kernel representation abstract we approach the problem whether left continuous triangular norm based valuations called t measures or t probability measures defined on triangular normbased tribes of the unit cube can be disintegrated by markov kernels we prove that each t measure based on a fundamental triangular norm these triangular norms t together with their corresponding triangular conorms s satisfy the functional equation t x y s x y x y can be uniquely represented as a sum of a disintegrable t measure and a hard core which is either identically zero or which is monotonically irreducible i e cannot be disintegrated

estimation of fuzzy memberships from histograms based on the conclusions drawn in the bijective transformation between possibility and probability a method is proposed to estimate the fuzzy membership function for pattern recognition purposes a rational function approximation to the probability density function is obtained from the histogram of a finite and sometimes very small number of samples this function is normalized such that the highest ordinate is one the parameters representing the rational function are used for classifying the pattern samples based on a max min decision rule the method is illustrated with examples