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New Mathematical Derivations Applicable to Safety and Reliability Analysis

Description: Boolean logic expressions are often derived in safety and reliability analysis. Since the values of the operands are rarely exact, accounting for uncertainty with the tightest justifiable bounds is important. Accurate determination of result bounds is difficult when the inputs have constraints. One example of a constraint is that an uncertain variable that appears multiple times in a Boolean expression must always have the same value, although the value cannot be exactly specified. A solution for this repeated variable problem is demonstrated for two Boolean classes. The classes, termed functions with unate variables (including, but not limited to unate functions), and exclusive-or functions, frequently appear in Boolean equations for uncertain outcomes portrayed by logic trees (event trees and fault trees).
Date: April 19, 1999
Creator: Cooper, J.A. & Ferson, S.
Partner: UNT Libraries Government Documents Department

Hybrid processing of stochastic and subjective uncertainty data

Description: Uncertainty analyses typically recognize separate stochastic and subjective sources of uncertainty, but do not systematically combine the two, although a large amount of data used in analyses is partly stochastic and partly subjective. We have developed methodology for mathematically combining stochastic and subjective data uncertainty, based on new ``hybrid number`` approaches. The methodology can be utilized in conjunction with various traditional techniques, such as PRA (probabilistic risk assessment) and risk analysis decision support. Hybrid numbers have been previously examined as a potential method to represent combinations of stochastic and subjective information, but mathematical processing has been impeded by the requirements inherent in the structure of the numbers, e.g., there was no known way to multiply hybrids. In this paper, we will demonstrate methods for calculating with hybrid numbers that avoid the difficulties. By formulating a hybrid number as a probability distribution that is only fuzzy known, or alternatively as a random distribution of fuzzy numbers, methods are demonstrated for the full suite of arithmetic operations, permitting complex mathematical calculations. It will be shown how information about relative subjectivity (the ratio of subjective to stochastic knowledge about a particular datum) can be incorporated. Techniques are also developed for conveying uncertainty information visually, so that the stochastic and subjective constituents of the uncertainty, as well as the ratio of knowledge about the two, are readily apparent. The techniques demonstrated have the capability to process uncertainty information for independent, uncorrelated data, and for some types of dependent and correlated data. Example applications are suggested, illustrative problems are worked, and graphical results are given.
Date: November 1, 1995
Creator: Cooper, J.A.; Ferson, S. & Ginzburg, L.
Partner: UNT Libraries Government Documents Department

Constrained Mathematics for Calculating Logical Safety and Reliability Probabilities with Uncertain Inputs

Description: Calculating safety and reliability probabilities with functions of uncertain variables can yield incorrect or misleading results if some precautions are not taken. One important consideration is the application of constrained mathematics for calculating probabilities for functions that contain repeated variables. This paper includes a description of the problem and develops a methodology for obtaining an accurate solution.
Date: January 21, 1999
Creator: Cooper, D.K.; Cooper, J.A. & Ferson, S.
Partner: UNT Libraries Government Documents Department