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Random fuzzy variable is mapping from a possibility space to a collection of random variables. This paper first presents a new definition of the expected value operator of a random fuzzy variable, and proves the linearity of the operator. Then, a random fuzzy simulation approach, which combines fuzzy simulation and random simulation, is designed to estimate the expected value of a random fuzzy variable. Based on the new expected value operator, three types of random fuzzy expected value models are presented to model decision systems where fuzziness and randomness appear simultaneously. In addition, random fuzzy simulation, neural networks and genetic algorithm are integrated to produce a hybrid intelligent algorithm for solving those random fuzzy expected valued models. Finally, three numerical examples are provided to illustrate the feasibility and the effectiveness of the proposed algorithm.

This paper considers a renewal process in which the interarrival times and rewards are characterized as fuzzy variables. A fuzzy elementary renewal theorem shows that the expected number of renewals per unit time is just the expected reciprocal of the interarrival time. Furthermore, the expected reward per unit time is provided by a fuzzy renewal reward theorem. Finally, a numerical example is presented for illustrating the theorems introduced in the paper.

A fuzzy variable is a function from a possibility space to the set of real numbers, while a bifuzzy variable is a function from a possibility space to the set of fuzzy variables. In this paper, a concept of chance distribution is originally presented for bifuzzy variable, and the linearity of expected value operator of bifuzzy variable is proved. Furthermore, bifuzzy simulations are designed and illustrated by some numerical experiments.

Possibility measures and credibility measures are widely used in fuzzy set theory. Compared with possibility measures, the advantage of credibility measures is the self-duality property. This paper gives a relation between possibility measures and credibility measures, and proves a sufficient and necessary condition for credibility measures. Finally, the credibility extension theorem is shown.

This paper presents the independence of fuzzy variables as well as its applications in fuzzy random optimization. First, the independence of fuzzy variables is defined based on the concept of marginal possibility distribution function, and a discussion about the relationship between the independent fuzzy variables and the noninteractive (unrelated) fuzzy variables is included. Second, we discuss some properties of the independent fuzzy variables, and establish the necessary and sufficient conditions for the independent fuzzy variables. Third, we propose the independence of fuzzy events, and deal with its fundamental properties. Finally, we apply the properties of the independent fuzzy variables to a class of fuzzy random programming problems to study their convexity.

A random fuzzy variable is a function from a credibility space to the set of random variables. Chance distribution is a type of mathematical description of random fuzzy variables. This paper presents a sufficient and necessary condition for chance distributions of random fuzzy variables.

The concept of fuzzy entropy is used to provide a quantitative measure of the uncertainty associated with every fuzzy variable. This paper proposes the maximum entropy principle for fuzzy variables, that is, out of all the membership functions satisfying given constraints, choose the one that has maximum entropy. The problem is what is the specific formulation of the maximum entropy membership function. The purpose of this paper is to solve this problem by Euler–Lagrange equation.

Fuzzy variables are functions from credibility spaces to the set of real numbers. The set of fuzzy variables is a linear space with the classic operations of addition and multiplication by numbers. Its subspace formed by fuzzy variables with finite p^th absolute moments is showed to be a complete para-normed space. The concept of para-normed space is novel, and is an extension of normed space. It is seen that most properties of normed spaces hold in para-normed spaces. Also some useful inequalities in para-normed spaces are obtained.

This note presents the characterization of a fuzzy variable whose possibility distribution is a convex fuzzy set.

This paper proposes fuzzy-based multi-objective design optimization approach for the optimal analysis of buried pipe based on the expected value of a fuzzy output variable when the membership function is computed. The design of pipe structures is usually associated with uncertainties. Therefore, the principle of fuzzy set and a multi-objective optimization algorithm is applied to account for the variabilities associated with the uncertain parameters to ensure an acceptable performance against the impact of uncertainties. Different methods such as deterministic and non-deterministic methods have been proposed to model the effect of uncertainties and analyse the performance of engineering structure in the literature. Herein, a fuzzy-based uncertainty modelling approach that employs the optimal performance of a hybrid GA-GAM for the analysis of a buried pipeline is proposed. The purpose of the strategy is to optimise the design variable while considering the adverse effect of the uncertain fuzzy variables and variability of the structural performance. The uncertain fuzzy variable is used in the analysis to take into account the subjective nature of the corrosion process, while the entropy of a fuzzy variable used as a global measure of variable dispersion is employed to measure the variability and sensitivity of the structural response. The outcome of the fuzzy-based multi-objective design optimization provides a set of optimal solution for the analysis of fuzzy structure. Finally, the applicability and characteristic of this method are demonstrated using a numerical example, and the outcome denotes acceptable analytical tools for design engineers and can be applied to analyse other engineering structures.

In this article, we introduce a general concept of fuzzy operators. These operators are then used to generalize the possibilistic conditioning formulation proposed by Nguyen [1]. This generalization depends on the relation which exists between this conditioning and the probabilistic t-norm. By using other t-norms, other conditionings are obtained, and their properties are studied. One application of normalized possibilistic conditioning is the measure of the dependency between two fuzzy statistical variables. The measure constructed can be considered as the possibilistic counterpart of mutual information commonly used in statistics.

When fuzzy information is taken into consideration in design, it is difficult to analyze the reliability of machine parts because we usually must deal with random information and fuzzy information simultaneously. Therefore, in order to make it easy to analyze fuzzy reliability, this paper proposes the transformation between discrete fuzzy random variable and discrete random variable based on a fuzzy reliability analysis when one of the stress and strength is a discrete fuzzy variable and the other is a discrete random variable. The transformation idea put forwards in this paper can be extended to continuous case, and can also be used in the fuzzy reliability analysis of repairable system.

This study introduces the concept of statistically pre-Cauchy sequences of fuzzy variables in five directions of credibility theory: almost surely, in measure, in mean, in distribution, and uniformly almost surely. However, the main focus is kept on statistically pre-Cauchy sequences in measure, in mean, and in distribution. Furthermore, a correlation between statistically pre-Cauchy sequences and statistically convergent sequences is established using fuzzy variables. Additionally, the exploration of statistically pre-Cauchy sequences of fuzzy variables is initiated through the application of Orlicz functions. The objective is to enhance the understanding of the statistical behavior and convergence properties of fuzzy variables in various credibility scenarios.

Possibility, necessity and credibility measures are used in the literature in order to deal with imprecision. Recently, Yang and Iwamura [L. Yang and K. Iwamura, Applied Mathematical Science2(46) (2008) 2271–2288] introduced a new measure as convex linear combination of possibility and necessity measures and they determined some of its axioms. In this paper, we introduce characteristics (parameters) of a fuzzy variable based on that measure, namely, expected value, variance, semi-variance, skewness, kurtosis and semi-kurtosis. We determine some properties of these characteristics and we compute them for trapezoidal and triangular fuzzy variables. We display their application for the determination of optimal portfolios when assets returns are described by triangular or trapezoidal fuzzy variables.

This paper explores the study of fuzzy transportation problem (FTP) using multi-choice goal-programming approach. Generally, the decision variable in transportation problem (TP) is considered as real variable, but here the decision variable in each node is chosen from a set of multi-choice fuzzy numbers. Here, we formulate a mathematical model of FTP considering fuzzy goal to the objective function. Thereafter, the solution procedure of the proposed model is developed through multi-choice goal programming approach. The proposed approach is not only improved the applicability of goal programming in real world situations but also provided useful insight about the solution of a new class of TP. A real-life numerical experiment is incorporated to analyze the feasibility and usefulness of this paper. The conclusions about our proposed work including future studies are discussed last.

The quantity notions in mathematics and metrology and their relation and interaction are considered. The quantity in mathematics belong to the modelling field and is an ideal object while the quantity in metrology has an experimental character and so is an uncertain object. Every metrological model object including measurement aim, measurand, metrological characteristic of a measuring instrument, are expressed by mathematical quantities. When the model object is evaluated by using measurement, so the experimental quantity is obtained, that is metrological quantity. Because of the principal uncertain character of metrological quantities, measuring data and relating metrological quantities have to be processed using firstly non-classic mathematical means but of special type, taking into account the above-mentioned character. There are approximate linear equations theory, interval arithmetic, fuzzy set, and so on. There is not wide use of these means. The reasons are traditions, and absence of data structure analysis, and special place of stochastic tools. The latter is conditioned by some peculiarities of probabilistic-stochastic models. Main, and wide spread, mistakes or faults in use of these models are discussed. Indirect measurement is considered as the field of most complicated interaction of experimental and mathematical quantities.