Constrained mathematics analysis of uncertain variables

Analyses (e.g., safety analyses and reliability analyses) that are based on variable or uncertain inputs need to reflect the ranges of potential output variability or uncertainty. A challenging problem in mathematically processing the operands is that constraints inherent in the problem definition can require computations that are difficult to implement. Examples of possible constraints are that the sum of the probabilities of partitioned possible outcomes must be one, and repeated appearances of the same variable must all have the identical value. The latter, called the “repeated variable problem” will be addressed in this paper in order to show how range-based probabilistic evaluation of Boolean logic expressions, such as those describing the outcomes of fault trees and event trees, can be facilitated. The results are applicable to fuzzy mathematics, interval analysis, Monte Carlo analysis and other range-based techniques. The problem is important, because unconstrained computations result in wider ranges of outputs than those properly obtained with constrained mathematics. We illustrate techniques that can be used to transform complex constrained problems into trivial problems in most tree logic expressions, and into tractable problems in most other cases. The approach is based on the Boolean logic characteristics of “unateness” and “minimal compactness”, and differential calculus characteristics related to regions of monotonicity. Example problems are used to demonstrate the techniques and the advantages of constrained mathematics