fuzzy differential equations and the extension principle we study the cauchy problem for differential equations considering its parameters and or initial conditions given by fuzzy sets these fuzzy differential equations are approached in two different ways a by using a family of differential inclusions and b the zadeh extension principle for the solution of the model we conclude that the solutions of the cauchy problem obtained by both are the same we also provide some illustrative examples

generalized fuzzy differentiability with lu parametric representation in the present paper we use a new generalization of the hukuhara difference and derivative for fuzzy valued functions and we study several properties of the new concepts in the setting of the lu parametric representation of fuzzy numbers assessed both from theoretical and computational points of view

numerically solution of fuzzy differential equations by adomian method in this paper a method for numerical approximation of fuzzy first order initial value problem is presented buckly and feruing fuzzy sets syst 110 2000 43 have solved fuzzy first order initial value problem they solved problem in crisp case and then extend the solution to fuzzy case in this paper we use adomian method to find the approximate solution in crisp case and extend it to fuzzy case using this method we solve problems that the classical methods could not be applied for them in crisp case

representation and application of fuzzy numbers abstract in this paper we present the theoretical background of fuzzy numbers connected with the possibility and dempster shafer theories we describe some types of representation of fuzzy numbers and we study the notions of the distance and orders between fuzzy numbers based on these representations the second part is devoted to the application of fuzzy numbers in data analysis artificial intelligence and decision making

fuzzy differential systems and malliavin calculus abstract we consider fuzzy differential systems or differential equations with fuzzy inputs and fuzzy initial conditions using a fuzzy sample path approach we derive partial differential equations for the resulting membership functions we also demonstrate sufficient continuity conditions under which the solutions exist and are fuzzy convex the membership functions are shown to be identical to those for a time varying deterministic differential equation with a fuzzy initial condition under sufficient stability conditions we also consider the membership functions in steady state

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

a comparison of vertex method with jhe method the jhe method has been recently proposed as an approximation to the vertex method for solving the fuzzy weighted average problem in this study the usefulness of the jhe method is demonstrated via illustrative examples to be rather restricted and in practice cannot be regarded as an efficient alternative to the vertex method the appropriateness of the jhe method only appears at the situation where the number of criteria exceeds 11 and the exactness of the solution is not a primary concern

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

extended fuzzy linear models and least squares estimates least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads 1 the idea is to introduce best fit difference models which exploit both the hukuhara difference and the l2 metric distance the fuzzy models use lr fuzzy numbers fitted models are compared by using the coefficient of determination in a similar way to its use in classical statistical least squares fitting the non lr fuzzy case is also considered

least squares estimates in fuzzy regression analysis abstract regression is a very powerful methodology for forecasting which is considered as an essential component of successful or applications in this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis based on the extension principle the membership function of the sum of squared errors is constructed the fuzzy sum of squared errors is a function of the regression coefficients to be determined which can be minimized via a nonlinear program formulated under the structure of the chen klein method for ranking fuzzy numbers to illustrate how the proposed method is applied three cases one crisp input fuzzy output one fuzzy input fuzzy output and one non triangular fuzzy observations are exemplified the results show that the least squares method of this paper is able to determine the regression coefficients with better explanatory power most important it works for all types of fuzzy observations not restricted to the triangular one

crank nicolson and legendre spectral collocation methods for a partial integro differential equation with a singular kernel abstract in this paper we present an efficient numerical method for the solution of a partial integro differential equation with a singular kernel in the time direction a crank nicolson finite difference scheme is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term also for space discretization we apply legendre spectral collocation method we discuss the stability and convergence of proposed method and show that the method is unconditionally stable and convergent with order o 3 2 n s where n and s are time step size number of collocation points and regularity of exact solution respectively we compare the numerical results of proposed method with the results of other schemes in the literature in terms of accuracy computational order and cpu time to show the efficiency and applicability of it

metric spaces of fuzzy sets 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

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

analysis of air traffic incidents using event trees with fuzzy probabilities this paper presents a methodology for analysing air traffic incidents based on an estimate of the probability of an air traffic incident developing into an air traffic accident air traffic incidents are officially defined as events which are dangerous but without a catastrophic impact a new approach is presented as previous research in this area has focused more on defining accident occurrence probabilities access to detailed information on air traffic incidents is limited and for this reason the methodology developed was based on fuzzy set theory analysis was carried out using event trees leading from a real incident to a hypothetical accident with the probabilities of occurrence of the various scenarios being defined by fuzzy sets the results of this analysis enable the calculation of the fuzzy probability of the incident being transformed into an accident performing this analysis new measuring techniques for the comparison of fuzzy sets were developed and partially verified the case study presented in the paper analyses a serious runway incursion incident this serious incident is analysed for influencing factors such as pilot and flight controller skill levels airport traffic volume weather conditions airport procedures and airport geometry applying the methods presented in this paper enables an assessment of the effectiveness of preventive recommendations of accident incident investigation commissions additionally probability estimations can be developed for different incident situations allowing the identification of security system weak points thus enabling appropriate proactive measures to be taken for their elimination fuzzy probability of a serious air traffic incident developing into an accident new measures for comparing fuzzy sets and their properties events importance and sensitivity analysis of the incident analysis of the state commission for aircraft accident investigation recommendation

on polygonal fuzzy sets and numbers in this paper we propose a new mathematical formalization of the concept of polygonal fuzzy numbers and an extension of this notion to fuzzy sets on r n we study the mathematical structure of these families of fuzzy sets and show that each family is a complete and separable metric space when endowed with the generalized hausdorff metric moreover we show that for n 1 the families of polygonal fuzzy numbers are isomorphic to some convex and closed convex cone of a finite dimensional space we obtain generalizations and extensions of some previous results on polygonal fuzzy numbers and simplified proofs of some well known results about approximation of fuzzy n dimensional quantities finally some developments about the approximation of families of fuzzy sets are introduced

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

simulation of discrete linear time invariant fuzzy dynamic systems in this paper a new approach to simulation of discrete linear time invariant dynamic systems with parameters and initial conditions described by fuzzy numbers is proposed this approach solves the problem of fuzzy simulation by determining at each iteration a nested family of intervals corresponding to different levels of certainty a cuts that enclose the possible system states using optimisation the originality of this approach consists in considering uncertain parameters as time invariant during the simulation process the optimisation problems associated with the approach proposed must be solved globally in order to guarantee that the minimum volume box enclosing the region of system states for each level of certainty has been derived then a global optimisation algorithm based on the theory of moments and linear matrix inequality relaxations suitable for the type of problems polynomial programs that appear in the fuzzy simulation algorithm is used finally two real engineering application examples allow assessing the performance of the approach proposed

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

linear regression with random fuzzy variables extended classical estimates best linear estimates least squares estimates this paper deals with problems which arise if for well justified statistical models like linear regression only fuzzy data are available three approaches are discussed the first one is an application of zadeh s extension principle to optimal classical estimators here the result is that they in general do not keep their optimality properties the second one is the attempt to develop a certain kind of linear theory for fuzzy random variables w r t extended addition and scalar multiplication the main problem in this connection is that fuzzy sets with these extended operations do not constitute a linear space therefore in the third approach the least squares approximation principle for fuzzy data is investigated which leads to most acceptable results

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

revisiting fuzzy differential equations abstract we revisit the original formulation of fuzzy differential equations fdes and provide examples to show that the theory of fdes is richer than the corresponding theory of ordinary differential equations odes and therefore need not be rejected the alternative formulation in terms of differential inclusions does not always have advantage over odes as claimed we also discuss a stability result via lyapunov functions in this setup

an algorithm for linear least squares problems with equality and nonnegativity constraints we present a new algorithm for solving a linear least squares problem with linear constraints these are equality constraint equations and nonnegativity constraints on selected variables this problem while appearing to be quite special is the core problem arising in the solution of the general linearly constrained linear least squares problem the reduction process of the general problem to the core problem can be done in many ways we discuss three such techniques

computing expensive multivariate functions of fuzzy numbers using sparse grids fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models with zadeh s extension principle one can obtain a fuzzy extension of any objective function computing expensive multivariate functions of fuzzy numbers however often poses a difficult problem due to non applicability of common fuzzy arithmetic algorithms severe overestimation or very high computational complexity this paper proposes a new approach based on sparse grids consisting of two parts first we compute a surrogate function using sparse grid interpolation second we perform the fuzzy valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic the new approach gives accurate results and requires only few function evaluations

numerical solutions of fuzzy differential equations abstract numerical algorithms for solving fuzzy ordinary differential equations fode are considered a scheme based on the classical euler method is discussed in detail and this is followed by a complete error analysis the algorithm is illustrated by solving several linear and nonlinear fuzzy cauchy problems

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

on weighted possibilistic mean and variance of fuzzy numbers dubois and prade defined an interval valued expectation of fuzzy numbers viewing them as consonant random sets carlsson and fuller defined an interval valued mean value of fuzzy numbers viewing them as possibility distributions in this paper we shall introduce the notation of weighted interval valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval valued probabilistic mean we shall also introduce the notations of crisp weighted possibilistic mean value variance and covariance of fuzzy numbers which are consistent with the extension principle furthermore we show that the weighted variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

metrics and orders in space of fuzzy numbers abstract two classes of metrics p q and p where 1 p 0 q 1 in space of fuzzy numbers are introduced then a method of ranking fuzzy numbers based on these metrics is proposed and investigated not only fuzzy numbers but also the left sided and right sided fuzzy numbers are considered applications in making decisions and statistics are suggested

on possibilistic mean value and variance of fuzzy numbers dubois and prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions in this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval valued possibilistic mean and investigate its relationship to the interval valued probabilistic mean we also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions which are consistent with the extension principle we also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

robust ergodic properties in partially hyperbolic dynamics we study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting earlier work of bonatti and viana 2000 about existence and finitude of physical measures is extended to the case of local diffeomorphisms moreover we prove that such systems constitute a c 2 open set in which statistical stability is a dense property in contrast all mostly contracting systems are shown to be stable under small random perturbations

fuzzy regression methods a comparative assessment abstract the fundamental differences between fuzzy regression and ordinary regression are identified in this paper fuzzy regression can be used to fit fuzzy data and crisp data into a regression model whereas ordinary regression can only fit crisp data through a comprehensive literature review three approaches of fuzzy regression are summarized the first approach of fuzzy regression is based on minimizing fuzziness as an optimal criterion the second approach uses least squares of errors as a fitting criterion and two methods are summarized in this paper the third approach can be described as an interval regression analysis for each fuzzy regression method numerical examples and graphical presentations are used to evaluate their characteristic and differences with ordinary least squares regression based on the comparative assessment the fundamental differences between ordinary least squares regression and conventional fuzzy regression are concluded that is ordinary least squares regression modeling data with randomness type of uncertainty and conventional fuzzy regression modeling data with fuzziness type of uncertainty in order to integrate both randomness and fuzziness types of uncertainty into one regression model the concept of hybrid fuzzy least squares regression analysis is proposed in this paper and the details of its method are derived in the accompanying paper

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

approximating a cuts with the vertex method if f rn r is continuous and monotonic in eachvariable and if i is a fuzzy number on the ith coordinate then the membership on r induced by and by the membership onfrn given by x min 1 x1 n xn can be evaluated by determining the membership at the endpoints of the level cuts of each i here more general conditions are given for both the function and the manner in which the fuzzy numbers i are combined so that this simple method for computing induced membership may be used in particular a geometric condition is given so that the cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

approximate fuzzy arithmetic operations using monotonic interpolations abstract we suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number or interval and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers in order to reduce the distance between the true result of the operation and its approximation the monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic several computational experiments are given to show the good performance of the proposed procedure

an energy minimization framework for monotonic cubic spline interpolation this paper describes the use of cubic splines for interpolating monotonic data sets interpolating cubic splines are popular for fitting data because they use low order polynomials and have c2 continuity a property that permits them to satisfy a desirable smoothness constraint unfortunately that same constraint often violates another desirable property monotonicity it is possible for a set of monotonically increasing or decreasing data points to yield a curve that is not monotonic i e the spline may oscillate in such cases it is necessary to sacrifice some smoothness in order to preserve monotonicity the goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint we first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper the conditions are simplified and consolidated to yield a fast method for determining monotonicity this result is applied within an energy minimization framework to yield linear and nonlinear optimization based methods we consider various energy measures for the optimization objective functions comparisons among the different techniques are given and superior monotonic c2 cubic spline interpolation results are presented extensions to shape preserving splines and data smoothing are described

least squares fuzzy regression with fuzzy random variables let x y be two convex fuzzy random variables on rn using a suitable metric we prove that the conditional expectation e y x is the best approximation of y by measurable functions of x this generalizes the analogous and well known property for real random variables a further topic is the approximation of y by a linear function of x in special cases and by use of hukuhara s difference between fuzzy sets we obtain formulas which are analogous to the classical structure contrary to the classical fact however the conditional expectation of gaussian fuzzy random variables in general does not coincide with the linear regression function

a hypothesis testing based discussion on the sensitivity of means of fuzzy data with respect to data shape abstract in assessing fuzzy numbers to model imprecise data associated with random experiments trapezoidal fuzzy numbers are often considered such an assessment is mainly due to easing both interpretation and computation this becomes especially noticeable when those assessing fuzzy numbers to data have a weak knowledge low background and little or no expertise in using fuzzy sets as it happens when questionnaires whose responses involve a free fuzzy rating are conducted since the required training to explain the meaning and use of trapezoidal fuzzy numbers is definitely lower than that associated with other shapes nevertheless a question that constantly arises in connection with this trapezoidal assessment is whether it can importantly affect the conclusions of the study involving such data this paper aims to answer the last question from a statistical perspective more concretely the analysis of the influence of the choice of trapezoidal fuzzy numbers to model data is to be based on the conclusions from statistical hypothesis testing about the mean values of the involved fuzzy datasets for this purpose the p values of tests have been compared for trapezoidal assessment vs other frequently used ones like some lu assessments the analysis is first performed by developing simulation based pairwise comparisons and it is later illustrated and corroborated to some extent with a real life example the analysis indicates that the shape of the fuzzy assessment scarcely affects statistical conclusions

a characterization of a general class of ranking functions on triangular fuzzy numbers this paper aims to the discussion of properties for ranking fuzzy numbers an approach of axiomatic type is followed providing the definition of two groups of requirements which characterize a ranking function through its behaviour on triangular fuzzy numbers particularly the problem of making the evaluations on triangular fuzzy numbers sensitive to their spread has been deeply analyzed

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

accurate c 2 rational interpolants in tension a family of rational interpolants is defined for a set of real data points x i f i 1 leqq i leqq n with x 1 x 2 cdots x n each interpolant of the family is a c 2 spline with a cubic numerator and a quadratic denominator which is identified uniquely by the value of a positive tension parameter c 0 the solution constructed is nonparametric a value of c 0 close to zero will produce piecewise linear behaviour while a large value of c 0 will give approximately the cubic spline solution the method used arises from a study of other rational functional forms for effecting monotonic or convex interpolation a convergence analysis establishes an error bound in terms of c 0 and shows that o h 4 accuracy is obtained when exact boundary derivatives are prescribed several examples are supplied to support the practical value of the method

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

multidimensional least squares fitting with a fuzzy model the aim of this paper is to deal with the question of fuzzy multivariable linear regression by least squares similar to the traditional least squares method we develop a fuzzy analogue by a distance defined on a fuzzy number space and propose a fuzzy multivariable linear regression model it is shown that the model has a unique solution and the solution can be given in an analytic expression in this paper we also give an index to evaluate the goodness of fit between the observed value and the estimated value

comparison results for fuzzy differential equations in this paper we present several comparison results for the solutions of fuzzy differential equations by using the hukuhara derivative these results constitute the extension to the fuzzy context of some comparison results for the solutions of linear differential equations besides some side results about the hukuhara difference and the partial orderings are given we illustrate the applicability of the new results by showing several examples

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

on the issue of defuzzification and selection based on a fuzzy set abstract we are concerned with the problem of selecting a crisp element based on information provided by a fuzzy set a problem which manifests itself in the defuzzification step in fuzzy logic controllers we provide a unifying approach to this selection process among other characteristics this unification puts the defuzzification methods of mean of maxima and center of gravity in the same framework we show that this selection can be viewed as a three step operation transformation of the decision fuzzy set normalization to probability distribution selection based on the probability distribution a number of different procedures for selection are discussed

on the reliability of the influence measure in the transformation method of fuzzy arithmetic the transformation method has been proposed as a practical tool for the simulation and the analysis of systems with uncertain parameters using fuzzy arithmetic a major advantage of this method among others is the fact that in the analysis part of the method the degree of influence for each uncertain parameter can be computed with a rather simple formula however a thorough examination of how accurate this measure of the degree of influence really is has not yet been performed in the paper the influence measure provided by the transformation method is compared to a rather classical analytical approach based on differential calculus furthermore an additional measure is introduced which can serve as an indicator for the effectiveness and the reliability of the influence measure calculated by the transformation method

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers