The application of constrained mathematics in probabilistic uncertainty analysis

Safety and reliability analyses often depend on Boolean logic combinations of input variables that have uncertainty (imperfect knowledge) or variability (probabilistically described outcomes). 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. An 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. It has been recognized that using interval-based computations such as interval arithmetic and fuzzy or possibilistic mathematics in an unconstrained mode (applied by sequentially parsing equation solutions), and even Monte Carlo analysis can significantly misrepresent extreme values. This phenomenon, its ramifications, and a solution for the problem are discussed.