on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

clustering of fuzzy data and simultaneous feature selection a model selection approach abstract fuzzy data occurs frequently in the fields of decision making social sciences and control theory we consider the problem of clustering fuzzy data along with automatic component number detection and feature selection a model selection criterion called minimum message length is used to address the problem of component number selection the bayesian framework can be adopted here by applying an explicit prior distribution over the parameter values we discuss both uninformative and informative priors for the latter a gradient descent algorithm for automatic optimization of the prior hyper parameters is presented the problem of simultaneous feature selection involves ordering the discriminative features according to their relative importance and at the same time eliminating non discriminative features the feature selection problem is also formulated as a parameter estimation problem by extending the concept of feature saliency then the estimation can be computed simultaneously with the clustering steps by combining the clustering the cluster number detection and the feature selection into one estimation problem we modified the fuzzy expectation maximization em algorithm to perform all of the estimation evaluation criteria are proposed and empirical study results are reported to showcase the efficacy of our proposals

maximum likelihood estimation from fuzzy data using the em algorithm a method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample this method is based on maximizing the observed data likelihood defined as the probability of the fuzzy data it is shown that the em algorithm may be used for that purpose which makes it possible to solve a wide range of statistical problems involving fuzzy data this approach called the fuzzy em fem method is illustrated using three classical problems normal mean and variance estimation from a fuzzy sample multiple linear regression with crisp inputs and fuzzy outputs and univariate finite normal mixture estimation from fuzzy data

clustering and classification of fuzzy data using the fuzzy em algorithm in this article we address the problem of clustering imprecise data using a finite mixture of gaussians we propose to estimate the parameters of the model using the fuzzy em algorithm this extension of the em algorithm allows us to handle imprecise data represented by fuzzy numbers first we briefly recall the principle of the fuzzy em algorithm then we provide closed forms for the parameter estimates in the case of gaussian fuzzy data we also describe a monte carlo procedure for estimating the parameter updates in the general case experiments carried out on synthetic and real data demonstrate the interest of our approach for taking into account attribute and label uncertainty

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

the s differentiability of a fuzzy valued mapping several approaches to define the differential of fuzzy valued mappings can be found in the literature in this paper we introduce a new concept of differential of a fuzzy valued mapping which is based on the notion of differential of the support function associated with such a mapping properties of the new concept as well as the relation between it and previous approaches are also analyzed in detail

on the vectoral ekeland s variational principle and minimal points in product spaces abstract phelps 13 noticed that the scalar ekeland s variational principle evp is equivalent to the existence of a minimal point of the epigraph of the corresponding function with respect to an appropriate order attouch and riahi 1 showed that evp is equivalent to the existence of maximal points with respect to cones satisfying some additional conditions taking these into account gopfert and tammer 6 7 established a maximal point theorem in a product space the aim of this paper is to obtain several minimal point theorems in product spaces and the corresponding variants of the vectorial evp

multi period mixed production planning with uncertain demands fuzzy and interval fuzzy sets approach this paper shows a general model of a mixed production planning problem with fuzzy demands the main focus is the development of a model for production planning using fuzzy sets in order to use classical mathematical programming techniques to reach an optimal solution over a multiple criteria context the classical fuzzy linear programming model namely the soft constraints model is used to involve flexibility in the problem moreover an interval fuzzy set approach is used to involve uncertainty in the problem

a brezis browder principle on partially ordered spaces and related ordering theorems abstract through a simple extension of brezis browder principle to partially ordered spaces a very general strong minimal point existence theorem on quasi ordered spaces is proved this theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems and three general ekeland variational principles in optimization problems where the objective space is quasi ordered then they are applied to prove strong minimal point existence results generalizations of bishop phelps lemma in linear spaces and ekeland variational principles in set valued optimization problems through a set solution criterion

probabilistic foundations for measurement modelling with fuzzy random variables we propose fuzzy random variables as a tool for modelling measurements when both aleatory and fuzzy uncertainty have to be taken into account uncertainty propagation follows the ordinary scheme for the random part and uses a t normed extension principle for the fuzzy part we concentrate on the probabilistic theoretical underpinnings of the model particularly limit theorems and discuss their implications to the model

the fuzzy hyperbolic inequality index associated with fuzzy random variables the aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy valued random variables in general populations for this purpose the fuzzy hyperbolic inequality index associated with general fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index extending those for classical inequality indices is given two examples illustrating the computation of the fuzzy inequality index are also considered some comments and suggestions are finally included

bayesian abstract fuzzy economies random quasi variational inequalities with random fuzzy mappings and random fixed point theorems abstract the main purpose of this paper is to introduce an abstract fuzzy economy model with differential asymmetric information and a measure space of agents it generalizes a previous model proposed by the author in 2009 the applications concern the fuzzy equilibrium existence and the existence of the solutions for two types of random quasi variational inequalities furthermore the paper presents random fixed point theorems with random fuzzy mappings extensions of the ones with random data our results either generalize or improve corresponding ones present in literature they consider the bochner integrability setting a measure space of indices and use random fuzzy mappings which describe the uncertainties

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

simulation of random fuzzy variables an empirical approach to statistical probabilistic studies with fuzzy experimental data in this paper we simulate different types of random fuzzy variables to get some conclusions concerning fuzzy valued random variables this simulation was carried out to illustrate certain limit results formalizing the convergence of the arithmetic mean of sampled fuzzy data to the population mean or an expected value of the random fuzzy variable like the well known strong law of large numbers and the law of iterated logarithm since the theoretical results have recently been proved this simulation analysis represents a complementary study in which we can determine sample sizes providing us with suitable approximations finally future directions regarding statistical inference with fuzzy data and other applications are commented

interactive multiobjective fuzzy random programming through the level set based probability model this paper considers a multiobjective linear programming problem involving fuzzy random variable coefficients a new fuzzy random programming model is proposed by extending the ideas of level set based optimality and a stochastic programming model the original problem involving fuzzy random variables is transformed into a deterministic equivalent problem through the proposed model an interactive algorithm is provided to obtain a satisficing solution for a decision maker from among a set of newly defined pareto optimal solutions it is shown that an optimal solution of the problem to be solved iteratively in the interactive algorithm is analytically obtained by a combination of the bisection method and the simplex method

the kolmogorov sinai entropy in the setting of fuzzy sets for image texture analysis and classification the kolmogorov sinai k s entropy is used to quantify the average amount of uncertainty of a dynamical system through a sequence of observations sequence probabilities therefore play a central role for the computation of the entropy rate to determine if the dynamical system under study is deterministically non chaotic deterministically chaotic or random this paper extends the notion of the k s entropy to measure the entropy rate of imprecise systems using sequence membership grades in which the underlying deterministic paradigm is replaced with the degree of fuzziness while constructing sequential probabilities for the calculation of the k s entropy is difficult in practice the estimate of the k s entropy in the setting of fuzzy sets in an image is feasible and can be useful for modeling uncertainty of pixel distributions in images the fuzzy k s entropy is illustrated as an effective feature for image analysis and texture classification highlightsdevelopment of a new method for feature extraction using chaos and fuzzy sets modeling kolmogorov sinai k s entropy with fuzzy logic is feasible fuzzy k s entropy is an effective feature for analysis and classification of textural images

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

building confidence interval based fuzzy random regression models in real world regression analysis statistical data may be linguistically imprecise or vague given the co existence of stochastic and fuzzy uncertainty real data cannot be characterized by using only the formalism of random variables in order to address regression problems in the presence of such hybrid uncertain data fuzzy random variables are introduced in this study to serve as an integral component of regression models a new class of fuzzy regression models that is based on fuzzy random data is built and is called the confidence interval based fuzzy random regression model ci frrm first a general fuzzy regression model for fuzzy random data is introduced then using expectations and variances of fuzzy random variables sigma confidence intervals are constructed for fuzzy random input output data the ci frrm is established based on the sigma confidence intervals the proposed regression model gives rise to a nonlinear programming problem that consists of fuzzy numbers or interval numbers since sign changes in the fuzzy coefficients modify the entire programming structure of the solution process the inherent dynamic nonlinearity of this optimization makes it difficult to exploit the techniques of linear programming or classical nonlinear programming therefore we resort to some heuristics finally an illustrative example is provided

pricing a contingent claim with random interval or fuzzy random payoff in one period setting this article proposes a method for pricing a contingent claim with random interval and fuzzy random payoff on introduction of the acceptability concept based on classical no arbitrage argument a price interval and a fuzzy price are obtained in random interval market and fuzzy random market respectively new definitions on replicative strategies sub replicative and sup replicative ones in two market setting are given some interesting results similar to those in the classical random market are presented

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

some properties of strong solutions to stochastic fuzzy differential equations we consider stochastic fuzzy differential equations driven by m dimensional brownian motion such equations can be useful in modeling of hybrid dynamic systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously under a boundedness condition which is weaker than linear growth condition and the lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations solutions which are fuzzy stochastic processes and their uniqueness are considered to be in a strong sense an estimation of error of the picard approximate solution is established we give a boundedness type result for the solution defined on finite time interval also the stabilities of solution on initial condition and coefficients of the equation are shown the existence and uniqueness of a solution defined on infinite time interval is proven finally some applications of fuzzy stochastic differential equations are considered all the results presented in this paper apply to set valued stochastic differential equations

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

existence of equilibria via ekeland s principle abstract in the literature when dealing with equilibrium problems and the existence of their solutions the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity together with some weak continuity assumptions of the function in this paper we focus on conditions that do not involve any convexity concept neither for the domain nor for the function involved starting from the well known ekeland s theorem for minimization problems we find a suitable set of conditions on the function f that lead to an ekeland s variational principle for equilibrium problems via the existence of e solutions we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems

ekeland type variational principle with applications to quasi variational inclusion problems abstract in this paper we study the ekeland type variational principle a caristi kirk type fixed point theorem and a maximal element theorem in the setting of uniform spaces by using these results we establish some existence results for solutions of quasi variational inclusion problems quasi optimization problems and equilibrium problems defined on separated and sequentially complete uniformly spaces

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

on some nonlinear problems induced by an abstract maximal element principle abstract in this paper by applying an abstract maximal element principle on quasi ordered sets established by lin and du we obtain a generalized brezis browder principle system vectorial versions of ekeland s variational principle and maximal element principle and a vectorial version of takahashi s nonconvex minimization theorem moreover we investigate the equivalence between scalar versions and vectorial versions of these results

on possibilistic representations of fuzzy intervals it is acknowledged that a fuzzy interval has two equivalent representations given in terms of the so called left and right sides of the membership function lr representation or in terms of the lower and upper branches defining the endpoints of the cuts lu representation in this paper we suggest an additional representation of fuzzy intervals called acf representation using an average cumulative function instead of the membership function based on possibility theory we illustrate how to build the new representation and we state its basic properties the main result is that the average cumulative ac function can be uniquely defined for any fuzzy interval and it is possible to move from one representation to the others through appropriate transformations an interesting link can be established between acf representation and quantile functions with a possible statistical interpretation useful in real application we also recommend a parametric form of the ac function

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

an ekeland s variational principle for set valued mappings with applications in this paper we obtain a general ekeland s variational principle for set valued mappings in complete metric space which is different from those in g y chen x x huang ekeland s e variational principle for set valued mapping mathematical methods of operations research 48 1998 181 186 g y chen x x huang s h hou general ekeland s variational principle for set valued mappings journal of optimization theory and applications 106 2000 151 164 s j li w y zhang on ekeland s variational principle for set valued mappings acta mathematicae application sinica english series 23 2007 141 148 by the result we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain respectively moreover we give some equivalent results to the variational principle

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

fuzzy rule based models with randomized development mechanisms abstract fuzzy rule based models have attracted attention because of their modular architectures well developed design methodologies and practices as well as interpretability aspects methods exploiting factors of randomness offer significant efficiency and implementation simplicity that are essential in numerous application areas in this study we propose an original development of fuzzy rule based models established with the aid of concepts of randomization algorithms several design strategies involving different random prototypes generation and basis functions approximation are studied we investigate performance aspects of randomized rule base and look at the performance versus the key components of the models such as the number of rules and the use of the randomized algorithms in the development furthermore a comparative study is offered to quantify the efficiency of randomized algorithms experimental studies are reported for a series of publicly available data sets to illustrate the effectiveness of the proposed method and discuss its main features

strong law of large numbers for t normed arithmetics we prove a strong law of large numbers for sequences of pairwise i i d fuzzy random variables addition is given by the extension principle associated with a general continuous triangular norm our result includes the slln for levelwise sums obtained when the chosen triangular norm is the minimum

central extensions of sphere groups closed forms for the central extensions of the infinite dimensional lie algebras of smooth maps from two and three spheres to compact lie groups are given

stackelberg solutions for fuzzy random two level linear programming through probability maximization with possibility this paper considers stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments taking into account vagueness of judgments of decision makers fuzzy goals are introduced into the formulated fuzzy random two level linear programming problems on the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal together with the introduction of probability maximization criterion in stochastic programming we propose new two level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values through the proposed models it is shown that the original two level linear programming problems with fuzzy random variables can be transformed into deterministic two level linear fractional programming problems for the transformed problems extended concepts of stackelberg solutions are defined and computational methods are also presented a numerical example is provided to illustrate the proposed methods

fixed point theorems for multi valued contractive mappings and multi valued caristi type mappings abstract in this paper the famous banach contraction principle and caristi s fixed point theorem are generalized to the case of multi valued mappings our results are extensions of the well known nadler s fixed point theorem s b nadler jr multi valued contraction mappings pacific j math 30 1969 475 487 as well as of some caristi type theorems for multi valued operators see n mizoguchi w takahashi fixed point theorems for multivalued mappings on complete metric spaces j math anal appl 141 1989 177 188 j p aubin optima and equilibria an introduction to nonlinear analysis grad texts in math springer verlag berlin 1998 p 17 s s zhang q luo set valued caristi fixed point theorem and ekeland s variational principle appl math mech 10 2 1989 111 113 in chinese english translation appl math mech english ed 10 2 1989 119 121 etc

existence theorems for variational inclusion problems and the set valued vector ekeland variational principle in a complete metric space abstract in this paper we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in ekeland s sense and the existence results for some variants of set valued vector ekeland variational principles in a complete metric space our results contain ekeland s variational principle as a special case and our approaches are different to those for any existence theorems for such problems

interactive multiobjective fuzzy random linear programming maximization of possibility and probability this paper considers multiobjective linear programming problems with fuzzy random variables coefficients a new decision making model is proposed to maximize both possibility and probability which is based on possibilistic programming and stochastic programming an interactive algorithm is constructed to obtain a satisficing solution satisfying at least weak pareto optimality

limit theorems for fuzzy random variables this paper deals with limit theorems for fuzzy valued measurable mappings which provide as a whole a foundation of statistical analysis with fuzzy data a strong law of large numbers a central limit theorem and a gliwenko cantelli theorem are proved the results are formulated simultaneously with respect to the lp metrics on the fuzzy sample spaces investigated by diamond and kloeden in particular these versions of the limit theorems are related to identical compatible concepts of convergence and measurability in the fuzzy sample spaces the proofs of the theorems are based heavily on isomorphic isometric embeddings of the fuzzy sample spaces endowed with lp metrics into respective lp spaces which are banach spaces of type 2 these embeddings provide the application of convergence results in banach spaces

strong laws of large numbers for t norm based addition of fuzzy set valued random variables this paper proposes a convergence theorem of fuzzy set valued random variables under addition which is given by the general triangular normed extension principles applying the result with the known results of the various types for strong laws of large numbers of fuzzy set valued random variables we obtain a few interesting results of the strong laws of large numbers for triangular norm based addition of fuzzy set valued random variables the finding of teran strong law of large numbers for t normed arithmetics fuzzy sets and systems 159 2008 343 360 is a special case of our results

central limit theorems for generalized set valued random variables abstract we give central limit theorems for generalized set valued random variables whose level sets are compact both in r d or in a banach space under milder conditions than those obtained recently by the latter two authors

the l a mean squared dispersion associated with a fuzzy random variable in this paper we introduce a parameterized real valued measure of the mean dispersion of a fuzzy random variable with respect to an arbitrary fuzzy number this measure extends the second moment of a classical random variable and is based on a parameterized distance between fuzzy numbers properties of the measure presented are analyzed and the extension of the variance of a classical random variable which particularizes the mean squared dispersion is also examined some examples illustrating the computation and possible applications of the measure are included finally a brief discussion about the interest of using a parameterized distance and about some future directions of this study is developed

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

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

solving influence diagrams with fuzzy chance and value nodes in this paper we discuss how influence diagrams are affected by the presence of fuzziness in chance and value nodes in this way by modelling fuzzy valued random variables and utilities in terms of fuzzy random variables we analyze the statistical rules corresponding to the affected value preserving transformations namely chance node removal and decision node removal some supporting results on fuzzy random variables introduced in this paper will provide us with the required mathematical tool to formalize the new statistical rules finally an example is included to illustrate the study in the paper

fixed point theorems for multivalued mappings on complete metric spaces on obtient une version multivaluee du theoreme de point fixe de caristi qui est equivalent au principe e variationnel du a ekeland on utilise ce resultat pour obtenir un theoreme qui deduit le theoreme de nadler et un theoreme de point fixe commun pour une application a 1 valeur et une application multivaluee

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

an embedding theorem for convex fuzzy sets in this paper we embed the space of upper semicontinuous convex fuzzy sets on a banach space into a space of continuous functions on a compact space the following structures are preserved by the embedding convex cone metric sup semilattice the indicator function of the unit ball is mapped to the constant function 1 two applications are presented strong laws of large numbers for fuzzy random variables and korovkin type approximation theorems

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

interactive fuzzy random two level linear programming based on level sets and fractile criterion optimization in this paper assuming cooperative behavior of the decision makers two level linear programming problems under fuzzy random environments are considered by introducing a level sets of fuzzy random variables and defining an a stochastic two level linear programming problem for guaranteeing the degree of realization of the problem through a new decision making model proposed based on the fractile criterion optimization model considering fuzzy goals the a stochastic two level programming problem can be equivalently transformed into a deterministic problem interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented an illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method

fuzzy random variables expectation and martingales abstract banach space valued fuzzy random variables on a complete finite measure space f are studied their fuzzy valued integral is defined and its properties are examined some connections between fuzzy integral and conditional expectation are given the definition and some properties of fuzzy martingales are investigated

fuzzy periodic review system with fuzzy random variable demand in this paper a periodic review inventory system has been analyzed in a mixed imprecise and uncertain environment where fuzziness and randomness appear simultaneously a model has been developed with customer demand assumed to be a fuzzy random variable the lead time has been assumed to be a constant the lead time demand and the lead time plus one period s demand have also been assumed to be fuzzy random variables a methodology has been developed to determine the optimal inventory level and the optimal period of review such that the total expected annual cost in the fuzzy sense is minimized a numerical example has been presented to illustrate the model

interactive random fuzzy two level programming through possibility based probability model this paper considers interactive decision making methods for random fuzzy two level linear programming problems assuming that the decision makers concern about the probabilities that their own objective function values are smaller than or equal to certain target values fuzzy goals of the decision makers for the probabilities are introduced then the possibility based probability model to maximize the degrees of possibility with respect to the attained probability is considered interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented an illustrative numerical example demonstrates the feasibility and efficiency of the proposed method

the drop theorem the petal theorem and ekeland s variational principle on montre que le principe variationnel d ekeland ou plus precisement une forme legerement modifiee est une consequence d un resultat geometrique connu sous le terme de theoreme de la goutte

dynamical systems over the space of upper semicontinuous fuzzy sets given a locally compact separable metric space e a perfect transformation defined on e induces a zadeh extension which is a transformation from the space of all upper semicontinuous fuzzy sets defined on e to itself here the latter space is equipped with the hit or miss topology in this setting each upper semicontinuous fuzzy set is identified with its hypograph a closed subset in the product space of e and 0 1 this approach does not require the relevant fuzzy sets to hold a compact support and it also overcomes the drawback of the traditional level set method further it is proved that the zadeh extension with this setting is continuous in the hit or miss topology and dynamical properties of zadeh extension regarding iteration have been explored

area preserving diffeomorphisms of riemann surfaces a number of riemann surfaces are constructed by factoring the plane under the action of discontinuous groups expressions are found for the algebras and curvatures of the groups of area preserving diffeomorphisms of the riemann surfaces the difficulties involved in finding finite algebras which yield the diffeomorphism algebras in the infinite dimension limit are discussed

a generalized distance and enhanced ekeland s variational principle for vector functions we propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation this result is shown to contain probably a large part of existing versions of ekeland s variational principle evp we introduce the notion of a weak function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced evp for various settings under relaxed lower semicontinuity assumptions these principles conclude the existence not only of p strict minimizers of p perturbations of the considered vector function but also p sharp and p strong minimizers our results are proved to be stronger than the classical evp and many generalizations in the literature even in the usual one dimensional case by numerous corollaries and examples we include equivalent formulations of our enhanced evp as well

two sample hypothesis tests of means of a fuzzy random variable abstract in this paper we will consider some two sample hypothesis tests for means concerning a fuzzy random variable in two populations for this purpose we will make use of a generalized metric for fuzzy numbers and we will develop an exact study for the case of normal fuzzy random variables and an asymptotic study for the case of simple general fuzzy random variables

bootstrap techniques and fuzzy random variables synergy in hypothesis testing with fuzzy data in previous studies we have stated that the well known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one dimensional euclidean space in this paper we show that the one sample method of testing about the mean of a fuzzy random variable can be extended to general ones more precisely to those whose range is not necessarily finite and whose values are fuzzy subsets of finite dimensional euclidean space this extension is immediately developed by combining some tools in the literature namely bootstrap techniques on banach spaces a metric between fuzzy sets based on the support function and an embedding of the space of fuzzy random variables into a banach space which is based on the support function

vectorial form of ekeland type variational principle with applications to vector equilibrium problems and fixed point theory abstract this paper introduces a vectorial form of equilibrium version of ekeland type variational principle some equivalent results to our variational principle are given as applications we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi metric spaces with a w distance caristi kirk fixed point theorem for multivalued maps is also established in a more general setting

belief functions induced by random fuzzy sets a general framework for representing uncertain and fuzzy evidence abstract we revisit zadeh s notion of evidence of the second kind and show that it provides the foundation for a general theory of epistemic random fuzzy sets which generalizes both the dempster shafer theory of belief functions and possibility theory in this perspective dempster shafer theory deals with belief functions generated by random sets while possibility theory deals with belief functions induced by fuzzy sets the more general theory allows us to represent and combine evidence that is both uncertain and fuzzy we demonstrate the application of this formalism to statistical inference and show that it makes it possible to reconcile the possibilistic interpretation of likelihood with bayesian inference

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

least squares fitting of compact set valued data in contexts where set valued data occur in a natural manner it is reasonable to ask whether such data can be directly processed for example to develop numerical techniques for constructing multifunction models which are a best fit to a body of set valued observations this note proposes methods of least squares fitting of such data for the case where each datum is a nonempty compact interval of the real line two models will be considered a interval valued output observed from interval valued input and b interval valued output resulting from exact real input

on the problem of characterizing derivatives for the fuzzy valued functions in this paper the concept of locally mean fuzzy henstock integrability for the fuzzy valued functions is presented several necessary and sufficient conditions of differentiability for the fuzzy valued functions are given by means of this concept

supply chain coordination based on a buyback contract under fuzzy random variable demand abstract the demand of production is random in the supply chain furthermore it is difficult to measure the demand in the real world hence the interval is more likely to be used to present the demand the interval can be viewed as a random variable this paper introduces this random variable as the demand of production into the model of supply chain we represent uncertain and fuzzy demand by a fuzzy random variable in a supply chain system based on a two level buyback contract for a newsvendor model with a single cycle for the sake of computational convenience the crisp possibilistic mean which has been widely used in supply chain is employed to defuzzify the fuzzy random demand in this paper the expected profit is defuzzified using a crisp possibilistic mean value the optimal order quantities in decentralized and centralized systems are analyzed and the conditions for supply chain coordination are obtained a realistic example is presented to illustrate the effectiveness of supply chain coordination and the impact of mixed imprecise and uncertain demand on the retailer order quantities and the expected profits for all sides

some φ coupled fixed point results via modified f control function s concept in metric spaces and its applications abstract the aim of this paper is to propose a new coupled fixed point theorem for a mixed monotone mapping in a metric space which is taken no account of the continuity of the mapping to this end we introduce the notions of coupled fixed point and modified f control function our obtained results are applied to derive some coupled fixed point theorems in partial metric spaces and quasi metric spaces respectively some applications to matrix equations and integral equations are also presented so as to illustrate our results

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

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

gaussian processes and martingales for fuzzy valued random variables with continuous parameter abstract the purpose of this paper is to discuss fuzzy valued gaussian processes and martingales with continuous parameter we mainly obtain a representation theorem for fuzzy valued gaussian processes after proving a new embedding theorem based on previous results about set valued and fuzzy valued martingales cf s li y ogura j fuzzy math 4 1996 905 s li y ogura fuzzy sets and systems 101 1999 453 s li y ogura ann probab 26 1998 1384 1402 s li y ogura in proc ifsa 97 vol 4 1997 pp 9 13 s li y ogura in c bertoluzza m a gil d a ralescu eds statistical modeling analysis and management of fuzzy data physica in press y ogura s li fuzzy sets and systems in press we prove optional sampling theorems for set valued and fuzzy valued martingales with continuous parameter

ekeland s variational principle in locally convex spaces and the density of extremal points in this paper we prove a general version of ekeland s variational principle in locally convex spaces where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function from this we obtain a number of special versions of ekeland s principle which include all the known extensions of the principle in locally convex spaces moreover we give a general criterion for judging the density of extremal points in the general ekeland s principle which extends and improves the related known results