In many real-life applications (e.g., in aircraft maintenance), we need to estimate the probability of failure of a complex system (such as an aircraft as a whole or one of its subsystems). Complex systems are usually built with redundancy allowing them to withstand the failure of a small number of components. In this paper, we assume that we know the structure of the system, and, as a result, for each possible set of failed components, we can tell whether this set will lead to a system failure. In some cases, for each component A, we know the probability P(A) of its failure; in other cases, however, we only know the lower and upper bounds for this probability. Sometimes, we only have expert estimates for these probabilities, estimates that can be described as fuzzy numbers.
Usually, it is assumed that failures of different components are independent events, but sometimes, we know that they are dependent -- in which case we usually do not have any specific information about their correlation. Our objective is to use all this information to estimate the probability of failure of the entire the complex system. In this paper, we describe methods for such estimation.