Fuzzy random variables

Puri, Madan L.; Ralescu, Dan A.

doi:10.1016/0022-247x(86)90093-4

Cited by 1,591 publications

(425 citation statements)

References 12 publications

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“…Puri and Ralescu [29,30], Klement et al [21]). In the latter class, the d ∞ metric is employed to characterize the measurability condition, since it seems to be the most adequate one to extend the Hausdorff metric and provide this class with a metric structure.…”

Section: Introductionmentioning

confidence: 99%

A generalized strong law of large numbers

Colubi¹,

López-Díiaz²,

Domíinguez-Menchero³

et al. 1999

Probab Theory Relat Fields

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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.

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Section: Introductionmentioning

confidence: 99%

A generalized strong law of large numbers

Colubi¹,

López-Díiaz²,

Domíinguez-Menchero³

et al. 1999

Probab Theory Relat Fields

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“…In this case, it may be more appropriate to interpret the experts' ambiguous understanding of the realized values of random parameters as fuzzy numbers. From such a point of view, a fuzzy random variable was first introduced by Kwakernaak [20], and its mathematical basis was constructed by Puri and Ralescu [21]. An overview of the developments of fuzzy random variables was found in the recent article of Gil et al [27].…”

Section: Random Fuzzy Variablesmentioning

confidence: 99%

“…Further extensions to two-level linear programming problems with random variables, called stochastic two-level linear programming problems [15,16], two-level integer programming problems [17], and twolevel linear programming problems involving fuzzy random variables, called fuzzy random two-level programming problems [18,19], have also been considered. It should be noted here that fuzzy random variables [20][21][22] are considered to be random variables whose realized values are not real values but fuzzy numbers or fuzzy sets. Arecent survey paper of Sakawa and Nishizaki [23] is devoted to reviewing and classifying the numerous major papers in the area of so-called cooperative multilevel programming.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, from a viewpoint of ambiguity and randomness different from fuzzy random variables [20][21][22], by considering the experts' ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and mathematical programming problems with random fuzzy variables were formulated together with the development of a simulation-based approximate solution method [24].…”

Section: Introductionmentioning

confidence: 99%

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Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility

Sakawa¹,

Matsui²

2013

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This paper considers two-level integer programming problems involving random fuzzy variables with cooperative behavior of the decision makers. Considering the probabilities that the decision makers' objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile criteria to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level integer programming problems are reduced to deterministic ones. Through the introduction of genetic algorithms with double strings for nonlinear integer programming problems, 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 demonstrates the feasibility and efficiency of the proposed method.

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“…Since Puri and Ralescu [2] introduced the concept of fuzzy random variables as a natural generalization of random sets, several authors have studied laws of large numbers for fuzzy random variables. Among others, several variants of strong law of large numbers (SLLN) for independent fuzzy random variables were built on the basis of SLLN for independent random sets.…”

Section: Introductionmentioning

confidence: 99%

Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables

Kim¹

2013

International Journal of Fuzzy Logic and Intelligent Systems

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In this paper, we present some results on weak laws of large numbers for weighted sums of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable real Banach space. First, we give weak laws of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables. Then, we consider the case that the weighted averages of expectations of fuzzy random variables converge. Finally, weak laws of large numbers for weighted sums of strongly tight or identically distributed fuzzy random variables are obtained as corollaries.

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Fuzzy random variables

Cited by 1,591 publications

References 12 publications

A generalized strong law of large numbers

A generalized strong law of large numbers

Interactive Fuzzy Programming for Random Fuzzy Two-Level Integer Programming Problems through Fractile Criteria with Possibility

Weak laws of large numbers for weighted sums of Banach space valued fuzzy random variables

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