A fuzzy linear programming model with fuzzy parameters and. A more intuitive interpretation is given of the notion of fuzzy random variables. This function is called a random variable or stochastic variable or more precisely a random function stochastic function. We then have a function defined on the sample space.
Estimation of a simple linear regression model for fuzzy. Fuzzy probabilistic structural analysis considering fuzzy random. Fuzzy approaches for multiobjective fuzzy random linear. Estimation of hazard rate and mean residual life ordering. His method is a direct application of classical methods in probability theory to fuzzy random variables. 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.
Mathematic description about the second kind fuzzy random variable namely the random variable with crisp eventfuzzy probability was studied. Received january 1977 revised june 1977 the purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of. Denote cut of the fuzzy variables as, where and are the lower and upper bounds of the cut of. Since their first introduction, fuzzy random variables have been given a num. Therefore the random variables with fuzzy parameters can be important.
Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered. Pdf overview on the development of fuzzy random variables. Parametric testing statistical hypotheses for fuzzy random variables gholamreza hesamian 0 mehdi shams 0 0 m. Pdf this paper presents a backward analysis on the interpretation, modelling and impact of the concept of fuzzy random variable.
A dynamic average valueatrisk portfolio model with fuzzy. Reliability analysis in presence of random variables and. A possibilistic interpretation of the expectation of a fuzzy random. In this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable. Fuzzy random variables or fuzzy variables generalize random variables and random vectors. 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. It is shown that a finite set of observables always possesses a joint distribution. This extension of a non fuzzy random variable is the appropriate way to model imprecise or fuzzy results of a probabilistic experiment. Fuzzy random variables x fuzzy random functions xs, t fuzzy stochastic analysis uncertain input data mapping uncertain result data. On the variability of the concept of variance for fuzzy random variables abstract. Possibilitynecessitybased probabilistic expectation. We study simple categorical situations and discuss relationships between such maps and conditional probabilities. A fuzzy random variable is a variable whose distribution parameters such as mean or standard deviation are fuzzy numbers.
Random variables and probability distributions random variables suppose that to each point of a sample space we assign a number. Probabilistic aspects of these random elements have been deeply discussed in the literature. Puri indiana university, indiana 47405 and ralescu cincinnati, blooming ton, department of mathematics, dan a. Examples and generalizations 3 the classical case we start with a probability space. A practical method for calculating structural reliability. This paper presents a new definition of fuzzy random variable, and gives a novel defi. Shapiro smeal college of business, penn state university, university park, pa 16802, usa phone.
Parametric testing statistical hypotheses for fuzzy random. The concept of fuzzy random variable was introduced by kwakernaak 5,6 and puri and ralescu8. 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. A perceptionbased portfolio model under uncertainty is presented. N2 the results obtained in part i of the paper are specialized to the case of discrete fuzzy random variables. 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. A fuzzy random variable is just a random variable that takes on values in a space of fuzzy sets. Kwakernaak 143 introduced the notion of a fuzzy random variable as a. Comparing to the reliability membership function for measuring the safety in case of mixed uncertainties, the proposed indices are more intuitive and easier to. Puri and others published fuzzy random variables find, read and cite all the research you need on researchgate. 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. Probabilistic inventory model with fuzzy cost components. 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. 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.
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. Harmonizing two approaches to fuzzy random variables. Implementing fuzzy random variables some preliminary observations. On the variability of the concept of variance for fuzzy.
The definition and properties of the discrete fuzzy random variable are discussed. Northholland publishing company fuzzy variables steven nahmias department of industrial engineering systems management engineering and operations research, university of pittsburgh, pittsburgh, pa 15261, u. 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. Rfvs are used in koissi and shapiro 2006 in a fuzzy regression context. The notion of fuzzy random variable has been introduced to model random mechanisms generating impreciselyvalued data. 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. Introducing average valueatrisks under conditions, this paper formulates average valueatrisks in dynamic stochastic environment. 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. Notes of fuzzy random variables center for machine. 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. Overview on the development of fuzzy random variables. Reliability assessment with fuzzy random variables using.
Let and be two nonnegative fuzzy random variables with probability density functions and, respectively. This function is called a random variableor stochastic variable or more precisely a random function stochastic function. By considering a suitable generalization of the hausdorff metric, we derive the lebesguedominated convergence type theorem. The new definition of expectation generalizes the integral of a setvalued function. Integral calculus for setvalued functions fuzzy random variables m a d a n l. 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. 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. It has an internal possibility distribution and an external possibility distribution called membership functions. Fuzzy random variables possess several interpretations. 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. Pdf a method to simulate fuzzy random variables maria.
Possibilitynecessitybased probabilistic expectation models. Statistics with fuzzy random variables article pdf available in metron lxv3. This leads to fuzzy random variables in the bugajskigudder sense. Example fuzzy sets, fuzzy values and fuzzy variables. This function is called a random variableor stochastic variable or more precisely a. Further results on the perfectness of probability distributions of fuzzy random variables are presented. A multiple linear regression model for lr fuzzy random. If the probability distribution functions of and are and, then the probability density function of the fuzzy random variable is. Puri redefines the definition of fuzzy random variables and expectations 19. A fuzzy variable defines the language that will be used to discuss a fuzzy concept such as temperature, pressure, age, or height. 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.
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. Received january 1977 revised june 1977 the purpose of this study is to explore a possible axiomatic framework from which a rigorous. Pdf constructive definitions of fuzzy random variables. Buckley 3, 4 exhibit an approach to fuzzy probability and. Statistics with fuzzy random variables semantic scholar. Fuzzy random variable has been defined in several ways in literature. Fuzzy random variables hereafter frvs for short in puri and ralescus sense 31 were introduced as an extension of random sets to model this kind of situation, and they can be employed in many contexts see, for instance, examples in 3,15,11. P and a measurable function f of into the real line r.
A multiple linear regression model for lr fuzzy random variables maria brigida ferraro. Denote ps the category of probability spaces and random maps. Fuzzy random variable, fuzzy random function, fuzzy stochastic finite element method. It is a simple matter to conceptualize fuzzy regression. This paper presents a new definition of fuzzy random variable, and gives a novel definition of scalar expected value operator for fuzzy random variables. Overviews of fuzzy random variables have been provided by gil et al. Interval monte carlo is compared with existing search algorithms used in the reliability assessment of fuzzy random structural systems for both efficiency and accuracy. Some properties concerning the measurability of fuzzy random variable are also discussed.
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. Arnold 1, 2 presented an approach to test fuzzily formulated hypotheses, in which he considered fuzzy constraints on the type i and ii errors. 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. Liu 22 defined the fuzzy random failure function of the structure with fuzzy random variables as the basic variables. Constructive definitions of fuzzy random variables. Abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable. Apr 24, 2009 on the variability of the concept of variance for fuzzy random variables abstract.
A minimax distribution free procedure for mixed inventory models invoving variable lead time with fuzzy lost sales are analyzed by ouyang and chang2002. 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. Effects fuzzy events, observables fuzzy random variables and their distributions are defined. Ro0, 1, that is, uu rsuch that ua is a nonempty compact interval, where. A unified approach to fuzzy random variables are considered by kratschmer2001 kao and hsu2002 discussed a singleperiod inventory model with fuzzy demand. Random variable with fuzzy probability springerlink. In the proposed model, randomness and fuzziness are evaluated respectively by probabilistic expectation and the mean values with evaluation weights and \\lambda \mean functions. Historically, they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about. Historically, they were proposed either as a tool for handling linguistic label information in statistics or to represent uncertainty about classical random variables. 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. 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. In addition, the concept of independent and identically distributed fuzzy random. 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.
This extension of a nonfuzzy random variable is the appropriate way to model imprecise or fuzzy results of a probabilistic experiment. The outcomes of kwakernaaks fuzzy random variables are fuzzy real subsets and the extreme points. Probabilistic inventory model with fuzzy cost components and. Fuzzy logic is then a logic of partial degrees of truth. Examples and generalizations martin papco abstract. 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.
542 1051 1136 219 991 281 406 478 31 346 66 1342 96 776 1515 766 1376 1601 827 116 1347 812 1040 238 1266 721 218 1495