Constrained Mathematics and the Repeated Variable Problem. J. A. Cooper, Sandia Nat’l Labs, Albuquerque, NM; Scott Ferson, Applied Biomathematics, Setauket, NY; and Douglas K. Cooper, Analogy, Inc., Beaverton, OR
Safety analysis uncertainties are typically represented by probability density functions, bounded intervals, possibilistic variables, or fuzzy numbers. In each of these cases, methodology is well known. However, there are numerous examples where less than optimum results and even incorrect results are derived under misapplication of the conventional approaches to expressions that contain repeated variables. For example, if the probability for the union of two independent variables is solved by any of the above processing techniques with no constraints, it is possible to obtain values that are less than zero or greater than one. There are other ramifications of this phenomenon, such as apparent violation of the distributive and identity laws. We have developed correct solution algorithms and systematic approaches that dramatically reduce the computational burden by pre-screening to indicate the appropriate algorithm that can be applied. For example, the interval-based probability of a Boolean function derived for a unate expression can provably be accomplished by simply not inverting the subtrahend bounds limits of repeated variables from the sense appearing in the unate expression. This contrasts with the conventional solution, where the bounds are inverted. We will demonstrate our techniques through example problems, and will describe practical applications, where lack of a constrained solution can cause a misleading result.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company.
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