Hybrid arithmetic

Kaufmann's (1986) formulation of hybrid numbers, which simultaneously express fuzzy and probabilistic uncertainty, allows addition and subtraction, but offers no obvious way to do multiplication, division or other operations. We describe another, more comprehensive formulation for hybrid numbers that allows the full suite of arithmetic operations, permitting them to be incorporated into complex mathematical calculations. There are two complementary approaches to computing with these hybrid numbers. The first is extremely efficient and yields theoretically optimal results in many circumstances. The second more general approach is based on Monte Carlo simulation using intervals or fuzzy numbers rather than scalar numbers.