Historically, they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about. A fuzzy linear programming model with fuzzy parameters and. Random variables and probability distributions random variables suppose that to each point of a sample space we assign a number. Fuzzy random variables possess several interpretations. We then have a function defined on the sample space. In this paper we define the concepts of fuzzy random variable. Pdf a method to simulate fuzzy random variables maria. This function is called a random variableor stochastic variable or more precisely a. Reliability assessment with fuzzy random variables using. Since their first introduction, fuzzy random variables have been given a num. A multiple linear regression model for lr fuzzy random variables maria brigida ferraro.
One important facet of this relationship is the fuzzy random variable frv, which is a measurable function 2from a probability space to the set of fuzzy variables. Estimation of a simple linear regression model for fuzzy. Mathematic description about the second kind fuzzy random variable namely the random variable with crisp eventfuzzy probability was studied. Random variable with fuzzy probability springerlink. Estimation of hazard rate and mean residual life ordering. Fuzzy random variable, fuzzy random function, fuzzy stochastic finite element method. This extension of a nonfuzzy random variable is the appropriate way to model imprecise or fuzzy results of a probabilistic experiment. Department of mathematics, university of cincinnati, ohio 45221 the concept of an integral of a setvalued function was first introduced by k u d o 1 2 in connection with the theory of experiments.
A possibilistic interpretation of the expectation of a fuzzy random. A dynamic average valueatrisk portfolio model with fuzzy. Fuzzy random variables x fuzzy random functions xs, t fuzzy stochastic analysis uncertain input data mapping uncertain result data. Puri redefines the definition of fuzzy random variables and expectations 19. In this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable. Arnold 1, 2 presented an approach to test fuzzily formulated hypotheses, in which he considered fuzzy constraints on the type i and ii errors. We prove a measurability result which implies that the measurable events concerning the values of a fuzzy random variable, in two related mathematical approaches wherein the codomains of the variables are different spaces, are the same provided both approaches apply. Integral calculus for setvalued functions fuzzy random variables m a d a n l. Probabilistic inventory model with fuzzy cost components. A perceptionbased portfolio model under uncertainty is presented. Fuzzy probabilistic structural analysis considering fuzzy random.
Parametric testing statistical hypotheses for fuzzy random variables gholamreza hesamian 0 mehdi shams 0 0 m. Abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable. The definition and properties of the discrete fuzzy random variable are discussed. Fuzzy random variables reconcile both the concept of 30 imprecision modelled by means of fuzzy sets and the concept of stochastic 31 variability 10, 16, 17,23,32, and have been used in various. Examples and generalizations 3 the classical case we start with a probability space. Random linear programming problems through a probability maximization model hitoshi yano and kota matsui abstractin this paper, two kinds of fuzzy approaches are proposed for not only multiobjective stochastic linear programming problems, but also multiobjective fuzzy random linear programming problems through a probability maximization model. Examples and generalizations martin papco abstract. This paper presents a new definition of fuzzy random variable, and gives a novel defi. It has an internal possibility distribution and an external possibility distribution called membership functions.
Fuzzy approaches for multiobjective fuzzy random linear. Overviews of fuzzy random variables have been provided by gil et al. Parametric testing statistical hypotheses for fuzzy random. Reliability analysis in presence of random variables and. Rfvs are used in koissi and shapiro 2006 in a fuzzy regression context. Pdf constructive definitions of fuzzy random variables. Interval monte carlo is compared with existing search algorithms used in the reliability assessment of fuzzy random structural systems for both efficiency and accuracy. Based on a natural extension of the concept of variance, he extended the kolmogorovs inequality to independent fuzzy random variables and obtained some limit theorems.
This function is called a random variable or stochastic variable or more precisely a random function stochastic function. We then study some of the properties of these fuzzy connectives. Liu 22 defined the fuzzy random failure function of the structure with fuzzy random variables as the basic variables, and analyzed the fuzziness of the structural reliability state and the. A more intuitive interpretation is given of the notion of fuzzy random variables. On the variability of the concept of variance for fuzzy. This leads to fuzzy random variables in the bugajskigudder sense. A multiple linear regression model for lr fuzzy random. Uncertain variables are modeled as fuzzy random variables and interval monte carlo simulation along with interval finite element method is used to evaluate failure probability.
Denote cut of the fuzzy variables as, where and are the lower and upper bounds of the cut of. Fuzzy random variable has been defined in several ways in literature. Comparing to the reliability membership function for measuring the safety in case of mixed uncertainties, the proposed indices are more intuitive and easier to. Pdf overview on the development of fuzzy random variables. Example fuzzy sets, fuzzy values and fuzzy variables. Based on the interval probability and using the fuzzy resolution theorem, the feasible condition about a probability fuzzy number set was given go a step further the definition and characters of random variable with fuzzy probability rvfp and the.
Liu 22 defined the fuzzy random failure function of the structure with fuzzy random variables as the basic variables. Statistics with fuzzy random variables semantic scholar. Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered. Shams department of mathematical sciences, university of kashan, isfahan, iran in this paper, a method is proposed for testing statistical hypotheses about the fuzzy parameter of the underlying parametric population. Buckley 3, 4 exhibit an approach to fuzzy probability and. Fuzzy logic is then a logic of partial degrees of truth. A fuzzy random variable is just a random variable that takes on values in a space of fuzzy sets.
P and a measurable function f of into the real line r. Further results on the perfectness of probability distributions of fuzzy random variables are presented. The class fuzzyvariable is used to create instances of a fuzzy variable, providing a name for example, temperature, the units of the variable if required for example, degrees c, the universe. Probabilistic aspects of these random elements have been deeply discussed in the literature. It is a simple matter to conceptualize fuzzy regression. By considering a suitable generalization of the hausdorff metric, we derive the lebesguedominated convergence type theorem. It is shown that a finite set of observables always possesses a joint distribution. Therefore the random variables with fuzzy parameters can be important. Kwakernaak 143 introduced the notion of a fuzzy random variable as a. Randomness and fuzziness can be merged to formulate a fuzzy random variable frv, that is, a function that assigns a fuzzy subset to each possible random. Northholland publishing company fuzzy variables steven nahmias department of industrial engineering systems management engineering and operations research, university of pittsburgh, pittsburgh, pa 15261, u. Some properties concerning the measurability of fuzzy random variable are also discussed.
Ro0, 1, that is, uu rsuch that ua is a nonempty compact interval, where. Another important facet is the random fuzzy variable rfv, which is a function from a possibility space to the set of random variables, that is, the parameters of the fuzzy variables are random. In addition, the concept of independent and identically distributed fuzzy random. Received january 1977 revised june 1977 the purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of. A unified approach to fuzzy random variables are considered by kratschmer2001 kao and hsu2002 discussed a singleperiod inventory model with fuzzy demand. The notion of fuzzy random variable has been introduced to model random mechanisms generating impreciselyvalued data which can be properly described by means of fuzzy sets. Shapiro smeal college of business, penn state university, university park, pa 16802, usa phone. Fuzzy random variables or fuzzy variables generalize random variables and random vectors. This extension of a non fuzzy random variable is the appropriate way to model imprecise or fuzzy results of a probabilistic experiment. Constructive definitions of fuzzy random variables. Harmonizing two approaches to fuzzy random variables. If the probability distribution functions of and are and, then the probability density function of the fuzzy random variable is.
There are random experiments in which the notion of a classical random variable, as a map sending each elementary event to a real number, does not capture their nature. N2 the results obtained in part i of the paper are specialized to the case of discrete fuzzy random variables. This paper presents a new definition of fuzzy random variable, and gives a novel definition of scalar expected value operator for fuzzy random variables. Possibilitynecessitybased probabilistic expectation models. In the proposed model, randomness and fuzziness are evaluated respectively by probabilistic expectation and the mean values with evaluation weights and \\lambda \mean functions. Implementing fuzzy random variables some preliminary observations. Introducing average valueatrisks under conditions, this paper formulates average valueatrisks in dynamic stochastic environment. Randomfuzzy variable rfv is a type 2 fuzzy variable, defined using the mathematical possibility theory, used to represent the entire information associated to a measurement result. A fuzzy random variable is a variable whose distribution parameters such as mean or standard deviation are fuzzy numbers. Apr 24, 2009 on the variability of the concept of variance for fuzzy random variables abstract. The expected value of a fuzzy variable is a natural generalization of the integral of a set valued function 2. We study simple categorical situations and discuss relationships between such maps and conditional probabilities. Effects fuzzy events, observables fuzzy random variables and their distributions are defined.
Random fuzzy variable rfv is a type 2 fuzzy variable, defined using the mathematical possibility theory, used to represent the entire information associated to a measurement result. His method is a direct application of classical methods in probability theory to fuzzy random variables. A practical method for calculating structural reliability. Notes of fuzzy random variables center for machine. A fuzzy variable defines the language that will be used to discuss a fuzzy concept such as temperature, pressure, age, or height. The outcomes of kwakernaaks fuzzy random variables are fuzzy real subsets and the extreme points. A minimax distribution free procedure for mixed inventory models invoving variable lead time with fuzzy lost sales are analyzed by ouyang and chang2002. Both of these models are explored in the rest of this article, but, for this conceptualization, we focus on models where the data is crisp and the relationship of the variables is fuzzy. Received january 1977 revised june 1977 the purpose of this study is to explore a possible axiomatic framework from which a rigorous.
Probabilistic inventory model with fuzzy cost components and. The notion of fuzzy random variable has been introduced to model random mechanisms generating impreciselyvalued data. Admitting some simpli fication, we cotlider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less true. The concept of fuzzy random variable was introduced by kwakernaak 5,6 and puri and ralescu8. For mixed uncertainties of random variables and fuzzy variables in engineering, three indices, that is, interval reliability index, mean reliability index, and numerical reliability index, are proposed to measure safety of structure. Puri indiana university, indiana 47405 and ralescu cincinnati, blooming ton, department of mathematics, dan a. Pdf this paper presents a backward analysis on the interpretation, modelling and impact of the concept of fuzzy random variable. Overview on the development of fuzzy random variables. We propose a new method for solving fuzzy lp flp problems in which the righthand side parameters and the decision variables are represented by fuzzy numbers. Jul 01, 2010 a unified approach to fuzzy random variables are considered by kratschmer2001 kao and hsu2002 discussed a singleperiod inventory model with fuzzy demand. This function is called a random variableor stochastic variable or more precisely a random function stochastic function. Then combining optimization or the traversal method with the traditional random reliability analysis methods, the cut of and the cut of can be obtained, where and are the lower bounds of the cut of and that of, respectively. A fuzzy lp model with fuzzy parameters and decision variables 3 ambiguous, uncertain or imprecise data in lp by formalising the inaccuracies inherent in human decisionmaking. The new definition of expectation generalizes the integral of a setvalued function.
Denote ps the category of probability spaces and random maps. However, statistical analysis of fuzzy random variables has not received so much attention, in spite that implications of this analysis range over many fields, including medicine. Possibilitynecessitybased probabilistic expectation. Let and be two nonnegative fuzzy random variables with probability density functions and, respectively.
1187 277 533 1005 478 584 95 735 84 616 734 413 393 1127 857 348 449 1203 960 913 1028 680 254 350 1362 672 843 1402 1201 881 482 1111 1339 567 1185