  Fault tree analysis is a vital method of assessing safety risks. It helps to
identify potential causes of accidents, assess their likelihood and severity,
and suggest preventive measures. Quantitative analysis of fault trees is often
done via the dependability metrics that compute the system's failure behaviour
over time. However, the lack of precise data is a major obstacle to
quantitative analysis, and so to reliability analysis. Fuzzy logic is a popular
framework for dealing with ambiguous values and has applications in many
domains. A number of fuzzy approaches have been proposed to fault tree
analysis, but -- to the best of our knowledge -- none of them provide rigorous
definitions or algorithms for computing fuzzy unreliability values. In this
paper, we define a rigorous framework for fuzzy unreliability values. In
addition, we provide a bottom-up algorithm to efficiently calculate fuzzy
reliability for a system. The algorithm incorporates the concept of
$\alpha$-cuts method. That is, performing binary algebraic operations on
intervals on horizontally discretised $\alpha$-cut representations of fuzzy
numbers. The method preserves the nonlinearity of fuzzy unreliability. Finally,
we illustrate the results obtained from two case studies.
