Setting the Reject Threshold from Escape Cost vs. Scrap Cost
A pharma blister-pack line has these cost parameters: cost of an escape
(a defective blister reaching a pharmacy) C_escape = $2,000 (recall
exposure, regulatory reporting cost, amortized brand risk); cost of a
false reject C_scrap = $3 (material plus rework). The line produces
40,000 blisters/day and the plant has a hard scrap-budget ceiling of
1.5% of daily output.
- Compute the break-even probability
p*usingreject if p(s)·C_escape > (1-p(s))·C_scrap. - Your anomaly detector's score distribution on golden (normal) parts
has a long right tail: roughly 2.5% of genuinely normal parts score
above the level that corresponds to
p(s) = p*. What does this mean for the reject policy, and how do you resolve the conflict with the scrap-budget ceiling? - Propose a concrete three-band policy (pass / review / reject) that respects both the cost-derived threshold and the scrap budget, and explain what happens to the parts that would have been auto-rejected under the pure cost rule but can't be under the scrap budget.
1. Break-even probability:
Setting the two sides equal: p·2000 = (1-p)·3 → 2000p = 3 - 3p →
2003p = 3 → p* = 3/2003 ≈ 0.0015 (about 0.15%). Because escape cost
so heavily outweighs scrap cost, even a very small estimated chance of
a true defect (well under 1%) already justifies rejecting on pure
expected-cost grounds.
2. The conflict:
If 2.5% of genuinely normal parts already score above the p*
threshold, a pure cost-rule auto-reject policy would reject roughly
2.5% of all output as a floor — before counting any parts that reject
because they are genuinely defective on top of that. Against a 1.5%
scrap-budget ceiling, this policy is not deployable as stated: the
threshold that expected-cost math says is "worth it" per-decision
still produces an aggregate false-reject rate the plant's margin
cannot absorb. This is exactly the tension the assumptions table
flags: escape cost dominates false-reject cost per decision, but false
rejects are not free in aggregate, and a scrap budget is a real,
separate constraint the per-decision cost inequality doesn't know
about.
3. A three-band resolution:
Set t2 (auto-reject) higher than the raw p*-implied threshold —
high enough that the auto-reject band's false-reject rate alone stays
comfortably under the scrap budget (e.g., calibrated so auto-reject
only fires for scores in, say, the top 0.5% of the normal
distribution). Set t1 (route to review) at or near the p*-implied
level, so that parts whose expected cost favors rejection but whose
score isn't extreme enough to auto-reject land in the operator-review
band instead of being auto-scrapped. The parts that would have been
auto-rejected under the pure cost rule but fall between t1 and t2
are not silently passed — they go to a human, whose real-time judgment
substitutes for the certainty the score alone doesn't provide, and
whose capacity (from the assumptions table) is exactly what should be
checked to confirm this band's expected volume is reviewable. This is
the same move case-study-fraud-detection's four-way policy makes:
the middle band exists precisely because the extremes of a policy
derived from one constraint (cost) can violate a different, real
constraint (capacity, or here, scrap budget), and a human-reviewed
middle band is cheaper than either getting the aggregate rate wrong or
hand-waving the conflict away.
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