Fictional case: Meridian Components' quality alert
Decide whether to recall a flagged batch before a distributor breaks it up and sends it to customers.
Use defect base rates, test accuracy, and recall costs to choose a response to a quality signal at a manufacturing division.
The noon pickup
At 9 a.m., Meridian Components gets a quality alert on a batch of printed faceplates already at a distributor. The distributor will hold the cartons until noon. Meridian can use its afternoon pickup to recall the batch, or it can let the distributor send it on to customers. Once customers open the cartons and mix the faceplates into inventory, a replacement becomes much more expensive.
The production supervisor favors a recall: “Our screen catches three out of four bad batches.” The distribution manager favors release: “Most alerts are false alarms.” Both statements could be true. You lead the division and must decide what to do with this batch.
Meridian, the figures, and the commercial arrangements are fictional. The issue is a shade of printing outside the customer's agreed color range. It has no safety or functional consequences. In this case both responses are permitted, and your objective is to minimize expected additional cost.
Recall Lesson 040: If someone offers a second test, what decision would you compare it with to judge what the test is worth?
What the screen measures
A batch is one production run. For this case its print color is consistent throughout: either the batch meets the color requirement or the whole batch needs replacement. The screen measures a retained sample from each batch. Today's alert means the reading crossed a warning threshold. It does not establish that the batch failed the customer's color requirement.
Quality staff checked comparable batches against the customer's reference color, including batches that raised no alert. Use the following rates from that work as planning assumptions for this line.
| What quality staff measured | Rate |
|---|---|
| Share of all batches outside the color requirement before seeing a screen result | 4% |
| Share of defective batches that trigger an alert | 75% |
| Share of acceptable batches that trigger an alert | 12.5% |
The first number is the base rate. The second tells you how often the screen detects an actual defect. The third is its false alarm rate: even an acceptable batch sometimes triggers the screen. The division needs a fourth number: among batches that trigger an alert, how many are defective? Bayes' theorem connects these two directions of conditional probability. You can do the calculation by counting batches. Penn State: Bayes’ Theorem.
What each response costs
All amounts below are additional dollars for this one batch, measured from 9 a.m. The original production cost has already been incurred. Both choices are affordable. The division has included freight, replacements, handling and customer credits wherever they apply.
| Action by noon | If batch is defective | If batch is acceptable |
|---|---|---|
| Recall now and supply an acceptable replacement | $6,000 | $6,000 |
| Release to customers | $40,000 | $0 |
An immediate recall completely avoids the later $40,000 field replacement. These costs are alternatives; they never both occur. No further test can finish before noon today.
Your decision
Allow 10–12 minutes. Use paper and a calculator. Work in batches throughout; the 4% is not a fraction of the faceplates inside today's cartons.
- Imagine 1,000 comparable batches. Start with 40 defective and 960 acceptable. How many of each trigger an alert? Given today's alert, calculate the probability this batch is defective.
- Draw a decision tree with recall and release as the two choices. Under each, label the two possible batch conditions and their costs. Calculate expected cost for each choice after the alert.
- Find the defect probability at which the choices cost the same on average. Would you make the same decision using only the 4% base rate? Write a two-sentence instruction to the distributor explaining today's choice.
- For future alerted batches, a lab offers a definitive color check before noon for $5,000 per batch. It would identify the true condition without error or delay. Calculate the maximum fee worth paying under the same assumptions.
Try the decision before reading the debrief.
Debrief
Of 40 defective batches, 0.75 × 40 = 30 trigger an alert. Of 960 acceptable batches, 0.125 × 960 = 120 trigger one. That gives 150 alerts altogether. The chance of a defect given an alert is 30 ÷ 150 = 20%. The supervisor's 75% answers a different question: it uses the 40 defective batches as its denominator.
| Batch condition | Alert | No alert | Total |
|---|---|---|---|
| Defective | 30 | 10 | 40 |
| Acceptable | 120 | 840 | 960 |
| Total | 150 | 850 | 1,000 |
The distribution manager is right that most alerts are false: 120 of the 150 flagged batches are acceptable. But an alert still raises the chance of a defect from 4% to 20%. Whether that warrants a recall depends on the cost of being wrong in each direction.
On the recall branch, both outcomes cost $6,000, so the expected cost is $6,000. On the release branch, weight each cost by its probability: (0.20 × $40,000) + (0.80 × $0) = $8,000. Recall the batch. This reduces expected cost by $2,000. There is no $8,000 invoice on the release branch: the actual cost would be either $0 or $40,000.
To find the decision threshold, ask what chance of a $40,000 loss would justify spending $6,000 to prevent it. Divide $6,000 by $40,000: the choices tie at 15%. Before the screen, release would have cost 0.04 × $40,000 = $1,600 on average, well below the recall cost. Today's 20% clears the threshold. The defect can still be unlikely in ordinary language and likely enough to justify action.
A suitable instruction is: “Keep the cartons together for our afternoon pickup; we will send replacement faceplates. Today's alert puts the estimated chance of a color defect at one in five, which makes a $6,000 replacement now preferable to risking a $40,000 replacement after distribution.”
For a future alerted batch, a perfect check would let you recall only when a defect is confirmed. Before the lab fee, expected cost would be (0.20 × $6,000) + (0.80 × $0) = $1,200. Recall answer: compare that with the best action already available after the alert, a $6,000 recall. The most the check is worth is $6,000 − $1,200 = $4,800. At $5,000, testing would bring expected cost to $6,200. Decline the offer at that price.
You can check the $4,800 another way. On the 20% of alerts that are real, the lab would confirm the recall you already planned. On the other 80%, it would save an unnecessary $6,000 recall. That saving averages 0.80 × $6,000 = $4,800 across alerted batches.
What to verify after the pickup
Have the quality lead compare the recalled faceplates with the customer's reference color and record the result. An acceptable batch would be disappointing, but it was the more likely outcome even when recall was the better decision. To assess the policy, track outcomes across comparable batches, including a sample that never triggered an alert. If the ink, production line, or screen settings change, check the rates again.
Keep your two-branch tree and the $4,800 limit beside it. The next lesson brings these calculations together with spreadsheet scenarios and the cost of passing up an alternative.
Sources
Comments
Loading comments…