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Abstract-Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. In this paper, we introduce a new notion of picture fuzzy sets (PFS), which are directly extensions of fuzzy sets and of intuitonistic fuzzy sets (Atanassov) . Then some operations on picture fuzzy sets are defined and some properties of these operations are considered. Here the basic preliminaries of PFS theory are presented.

Probabilistic uncertainty and imprecision in structural parameters and in environmental conditions and loads are challenging phenomena in engineering analyses. They require appropriate mathematical modeling and quantication to obtain realistic results when predicting the behavior and reliability of engineering structures and systems. But the modeling and quantication is complicated by the characteristics of the available information, which involves, for example, sparse data, poor measurements and subjective information. This raises the question whether the available information is sucient for probabilistic modeling or rather suggests a set-theoretical approach. The framework of imprecise probabilities provides a mathematical basis to deal with these problems which involve both probabilistic and non-probabilistic information. A common feature of the various concepts of imprecise probabilities is the consideration of an entire set of probabilistic models in one analysis. The theoretical dierences between the concepts mainly concern the mathematical description of the set of probabilistic models and the connection to the probabilistic models involved.This paper provides an overview on developments which involve imprecise probabilities for the solution of engineering problems. Evidence theory, probability bounds analysis with p-boxes, and fuzzy probabilities are discussed with emphasis on their key features and on their relationships to one another.

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When we have only interval ranges Ix i, ~i] of sample values xl,..., xm what is the interval IV, V-~ of possible values for the variance V of these values? We prove that the problem of computing the upper bound V is NP-hard. We provide a feasible (quadratic time) algorithm for computing the lower bound V on the variance of interval data. We also provide a feasible algorithm that computes V under reasonable easily verifiable conditions.

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In data processing, we often encounter the following problem: Suppose that we have processed the measurement resultsx 1 ,