Fault Tree Importance Measures

Fault Tree Importance Measures (systems-engineering-safety/arp4761a/fault-tree-importance-measures)

Use when you must rank the basic events of a fault tree by their contribution to the top event probability. This leaf computes the four standard importance measures, Birnbaum, Fussell-Vesely, risk achievement worth (RAW) and risk reduction worth (RRW), for every basic event from the minimal cut sets and the basic-event probabilities, then sorts the events by each measure and flags the dominant contributors. It pairs with systems-engineering-safety/arp4761a/fta-fmea, which produces the minimal cut sets and event probabilities this leaf ranks; the outputs (rank lists, dominance flags) gate risk-reduction prioritization inside the ARP4761A quantitative safety assessment flow. All logic is deterministic, offline, pure stdlib.

Domain quick reference

• Inputs: cut_sets, a list of sets of basic-event names (one set per minimal cut set), and probs, a dict mapping every basic-event name to a probability in (0, 1).
• Top event probability Q: probability of the union of the cut sets under event independence, Q = sum over non-empty subsets of the cut sets of (-1)^(k+1) times the product of the event probabilities in the union of that subset (exact inclusion-exclusion over 2^n - 1 subsets, n = number of cut sets).
• Birnbaum measure of event e: B_e = Q(q_e = 1) - Q(q_e = 0), the change in top probability when the event probability moves from 0 to 1, evaluated by re-computing the union with the event forced true and then forced false.
• Fussell-Vesely measure: FV_e = (Q - Q(q_e = 0)) / Q, the fraction of the top event probability that involves the event, always in [0, 1].
• Risk achievement worth: RAW_e = Q(q_e = 1) / Q, the factor by which the top probability grows when the event is forced true, always >= 1.
• Risk reduction worth: RRW_e = Q / Q(q_e = 0), the factor by which the top probability shrinks when the event is forced false, always >= 1; unbounded (infinity) when forcing the event false removes every failure path.
• Closed-form identities of the re-evaluation method: FV_e = 1 - 1/RRW_e and RAW_e = B_e / Q + 1 / RRW_e for every contributing event; Q(q = 1) >= Q >= Q(q = 0).
• ARP4761A frames the quantitative safety assessment context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Take the analysis inputs from fta-fmea: the minimal cut sets (list of event-name sets) and the basic-event probabilities (dict), and confirm every event in every cut set has a probability in (0, 1).
2. Compute the top event probability with top_event_probability and sanity-check it against the fta-fmea cut-set result before ranking.
3. Get the full per-event picture with importance_measures, which returns {event: {birnbaum, fussell_vesely, raw, rrw}} for every event that appears in the cut sets.
4. Rank the events with rank_events(cut_sets, probs, measure), one call per measure; the result is sorted descending with alphabetical tie-break. Default measure is fussell_vesely.
5. Flag the dominant contributors with dominant_contributors(cut_sets, probs, threshold = 0.1): events whose Fussell-Vesely measure strictly exceeds the threshold, sorted descending.
6. Read the rank lists together: Fussell-Vesely and RAW/RRW order risk-reduction effort, Birnbaum shows raw sensitivity of the top event to each event probability.
7. Confirm determinism and input rejection with the contract test: python3 scripts/test_fault_tree_importance_measures.py.

Worked example

Anchor tree: cut_sets = [{"A", "B"}, {"C"}], probs = {"A": 0.01, "B": 0.02, "C": 0.03}.

• Q = 0.0002 + 0.03 - 0.000006 = 0.030194 (union of the cut sets AB and C by inclusion-exclusion).
• Event A: birnbaum 0.0194 (Q(q=1) = 0.0494 via B or C, Q(q=0) = 0.03), fussell_vesely 0.006425, raw 1.6361, rrw 1.00647.
• Event B: birnbaum 0.0097 (Q(q=1) = 0.0397 via A or C), fussell_vesely 0.006425, raw 1.3148, rrw 1.00647.
• Event C: birnbaum 0.9998 (Q(q=1) = 1, Q(q=0) = 0.0002), fussell_vesely 0.9934, raw 33.1192, rrw 150.97.
• rank_events by fussell_vesely returns C first (0.9934), then A and B (tied at 0.006425, alphabetical order for determinism); rank by raw gives the same event order (C, A, B).
• dominant_contributors at the default threshold 0.1 returns ["C"]: event C dominates the top event probability, and fixing it gives the largest risk reduction (rrw 150.97).

Verification

• Confirm top_event_probability([{"A", "B"}, {"C"}], {"A": 0.01, "B": 0.02, "C": 0.03}) returns 0.030194 and that a single cut set {"A"} returns the event probability itself.
• Confirm the anchor measures: C birnbaum 0.9998, A 0.0194, B 0.0097; C fussell_vesely 0.9934, A and B 0.006425; raw 33.1192 / 1.6361 / 1.3148; rrw 150.97 / 1.00647.
• Confirm the closed-form identities FV = 1 - 1/RRW and RAW = B/Q + 1/RRW hold on the anchor tree, and that Q(q=1) >= Q >= Q(q=0) for every event.
• Confirm a lone single-event cut set {"A"} with p = 0.5 gives birnbaum 1.0, fussell_vesely 1.0, raw 2.0 = 1/Q, and rrw infinity (the risk reduction is unbounded because forcing the event false eliminates the only failure path).
• Confirm ValueError rejection of non-physical inputs: empty cut_sets, an empty cut set entry, an unknown event name, a probability outside (0, 1), an event that appears in no cut set, and an unknown measure name.
• Run the contract test offline: python3 scripts/test_fault_tree_importance_measures.py (35 tests, deterministic, exit 0).

Related leaves

• systems-engineering-safety/arp4761a/fta-fmea: produces the minimal cut sets and the event probabilities that feed this leaf, with cut-set probability sanity and FMEA severity mapping on the same analysis set.
• systems-engineering-safety/arp4761a/markov-analysis: state-probability dynamics for the failure conditions this leaf ranks at the fault-tree level.
• systems-engineering-safety/arp4761a/reliability-block-diagram: series/parallel network reliability as the alternative top-event model.
• systems-engineering-safety/arp4761a/failure-rate-estimation: derives the basic-event probabilities from demonstration data before the importance ranking.

Pitfalls

• Ranking on Birnbaum alone: event A has the higher Birnbaum measure (0.0194 vs B 0.0097) but the same Fussell-Vesely value as B (0.006425); Birnbaum is a raw sensitivity, not a fractional contribution, so risk-reduction prioritization should use FV, RAW or RRW as the primary order.
• Treating an unbounded RRW as an error or capping it: when the event sits in the only single-event cut set, Q(q=0) = 0 and the risk reduction worth is mathematically unbounded; the module reports positive infinity and the report should say so rather than inventing a finite value.
• Threshold semantics: dominance is strict, FV > threshold; an event whose Fussell-Vesely value equals the threshold is not flagged (use threshold 0.99 with FV 1.0 and the list comes back empty).
• Feeding redundant or non-minimal cut sets: duplicated or superset cut sets distort the per-event measures and inflate the inclusion-exclusion term count (2^n - 1 subsets), so take the minimal cut sets from fta-fmea and keep n small in quick studies.
• Reading significance into tie order: equal measure values are ordered alphabetically for determinism only, A before B in the example, so a tie is not a ranking statement.
• Passing degenerate event probabilities as nominal inputs: probs must lie strictly inside (0, 1) and cover every event in every cut set; forced true and forced false are internal re-evaluations, not inputs.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_fault_tree_importance_measures.py

The test covers the anchor tree (Q = 0.030194, each measure inside the spec magnitude bounds), single and four cut-set union probabilities against closed forms, Birnbaum truth values (C 0.9998, p = 0.5 lone event 1.0), Fussell-Vesely <= 1, RAW and RRW >= 1, rank ordering with alphabetical tie-break, strict dominance at the threshold, closed-form identities linking all four measures, Q(q=1) >= Q >= Q(q=0), the unbounded RRW lone-cause case, determinism, exact dict keys, and ValueError rejection of empty cut_sets, empty cut set entries, unknown events, events in no cut set, probabilities outside (0, 1) and unknown measure names. All 35 tests pass offline in under 20 seconds.

Compliance

• Standards referenced, not reproduced: ARP4761A is proprietary (SAE, circa USD 180), name + paraphrase only; the importance-measure relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false. Inputs (cut sets, probabilities) come from the sibling fta-fmea leaf; this leaf adds the ranking layer only.