Fussell-Vesely importance
Fussell-Vesely importance (FV) is the fraction of top-event probability attributable to minimal cut sets containing a given basic event. In plain English: if you could magically eliminate this one event, how much of the system's current risk would disappear with it? It's the standard "where is risk actually accumulating right now?" measure for prioritising inspection, proof-test scheduling, and mitigation effort.
Formula
I_FV(i) = P(union of cut sets containing event i) / P(top)
The numerator is the probability that at least one minimal cut set containing event i is "completed" (all its events failed); the denominator is the overall top-event probability. The ratio is bounded between 0 and 1.
Worked example
Suppose a tree has three minimal cut sets after MOCUS:
MCS1 = {a, b} P = 0.01·0.02 = 2.0e-4
MCS2 = {a, c} P = 0.01·0.005 = 5.0e-5
MCS3 = {d} P = 0.003 = 3.0e-3
P(top) ≈ Σ ≈ 3.25e-3
FV importance of a: a appears in MCS1 and MCS2, contributing 2.0e-4 + 5.0e-5 = 2.5e-4. Divide by P(top): I_FV(a) ≈ 0.077 — about 7.7% of total risk.
FV importance of d: d only appears in MCS3, contributing 3.0e-3. I_FV(d) ≈ 0.92 — about 92% of total risk.
Even though d's individual probability isn't dramatic, it dominates because its cut set is order-1 — a single-point-of-failure that drives the whole system. FV makes that visible at a glance.
FV vs Birnbaum
Both rank basic events by influence, but they answer different questions:
- Birnbaum — structural sensitivity, ignoring how likely the event is. Best for "if I could improve any one component, which one helps most?"
- Fussell-Vesely — actual contribution given current event probabilities. Best for "where is the most risk to attack today?"
Most operating procedures (in nuclear PRA, IEC 61511 SIS verification, ARP 4761 SSA) prioritise FV ranking because they're allocating real maintenance and inspection budget against present risk, not hypothetical re-design.