SOP: Acceptance Sampling of Lots by Attributes — Double and Multiple Sampling

GIFSQ — Global Institute for Food Safety & Quality

Principle

Double and multiple sampling plans cut inspection on lots that are clearly good or clearly poor, by deciding in stages instead of examining the whole sample at once. In double sampling, a first sample n₁ is judged against Ac₁/Re₁: the lot is accepted if defectives are at or below Ac₁, rejected if at or above Re₁, and only if the count falls between the two is a second sample drawn — the decision then rests on the cumulative count against cumulative acceptance and rejection numbers. Multiple sampling extends this to up to seven stages, comparing cumulative defective counts against each stage’s Ac and Re until a decision is reached.

Procedure

Double sampling:

From the code letter and the AQL, find the first-sample size n₁ with Ac₁/Re₁ and the cumulative Ac/Re for the combined samples in the double-sampling table

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Inspect a random first sample of n₁ units

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Accept the lot if defectives are at or below Ac₁; reject if at or above Re₁

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If the count falls between Ac₁ and Re₁, draw a second random sample of n₂ units

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Add the defective counts of both samples; accept if the cumulative count is at or below the cumulative Ac, reject if at or above the cumulative Re

Multiple sampling:

From the code letter and the AQL, find the per-stage sample size and the cumulative Ac/Re for up to seven stages in the multiple-sampling table

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Draw and inspect one stage’s random sample; compare the cumulative defective count with that stage’s Ac and Re

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Accept at or below Ac, reject at or above Re; otherwise draw the next stage’s sample and continue until a decision is reached

Worked double-sampling example (18,000 mangoes, code M, AQL 1.0%): first sample 200 fruits, Ac = 3, Re = 7; cumulative sample size 400, cumulative Ac = 8, cumulative Re = 9. Three or fewer spoiled in the first 200 means accept; seven or more means reject. If 4, 5 or 6 are defective, inspect a second 200: 8 or fewer spoiled from both samples combined means accept, 9 or more means reject.

Worked multiple-sampling example (same lot): 80 fruits per stage. Cumulative sample sizes and acceptance/rejection numbers by stage:

• First — 80: Ac 0, Re 4
• Second — 160: Ac 1, Re 6
• Third — 240: Ac 3, Re 8
• Fourth — 320: Ac 5, Re 10
• Fifth — 400: Re 11 (the acceptance number is illegible in the scan and is not carried)
• Sixth — 480: Ac 10, Re 12
• Seventh — 560: Ac 13, Re 14

So: inspect 80 fruits first — 0 spoiled means accept, 4 or more means reject; 1 to 3 spoiled means draw a second 80 and continue. From both samples (160 fruits), 1 or fewer spoiled means accept and 6 or more means reject; 2 to 5 spoiled means continue to further stages until a satisfactory decision is reached.

Note: the scanned double and multiple sampling tables are garbled and are not reproduced here; use the printed MIL-STD 105D double- and multiple-sampling tables for the code letter and AQL found above, with the worked examples carrying the book’s printed numbers.

Precautions

• Every stage’s sample must be random — every piece in the lot needs an equal likelihood of selection at each draw.
• Keep the samples of successive stages separate in counting; the decision always rests on the cumulative count, not on any single stage alone.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.

MIL-STD 105D attribute sampling (ISO/DIS 2859) — cited in the book.