BIZOMAT MAKEOS · Quality
Acceptance Sampling Plan (n and c for Producer and Consumer Risk)
What sample size and acceptance number protect both me and my customer at these quality levels, and does my current plan do it?
Design of a single attributes sampling plan: the smallest sample size and acceptance number whose operating characteristic accepts lots at the acceptable quality level with at least 1 − α probability and accepts lots at the rejectable quality level with at most β probability (binomial, as the sampling OC tool), its average outgoing quality limit, and the same checks for a plan you choose.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Acceptable quality level (AQL) | % defective | Defect level that should be accepted nearly always. |
| Rejectable quality level (LTPD) | % defective | Defect level that should be rejected nearly always (lot tolerance percent defective). |
| Producer’s risk α | % | Largest chance of rejecting a good lot at the AQL. |
| Consumer’s risk β | % | Largest chance of accepting a bad lot at the RQL. |
| Your sample size (0 = use the designed plan) | Sample size of a plan you want to check (0 checks the designed plan). | |
| Your acceptance number | Acceptance number of that plan (accept if the sample has this many defectives or fewer). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Acceptable quality level (AQL)
- What it is
- The percentage of defective items in a lot that you regard as acceptable quality; lots at this level should almost always pass.
- Units and format
- Enter a number in % defective: % defective. The form accepts 0.01 % defective to 20 % defective. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 % defective as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Rejectable quality level (LTPD)
- What it is
- The percentage defective at which a lot is regarded as bad and should almost always be rejected.
- Units and format
- Enter a number in % defective: % defective. The form accepts 0.1 % defective to 50 % defective. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 % defective as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Must be above the AQL; the closer the two levels, the larger the sample needed.
What this means: Producer’s risk α
- What it is
- The highest chance you accept of rejecting a lot that is actually at the AQL: the producer’s risk.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.1% to 50%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Consumer’s risk β
- What it is
- The highest chance you accept of passing a lot that is actually at the RQL: the consumer’s risk.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.1% to 50%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Your sample size (0 = use the designed plan)
- What it is
- The sample size of an existing or proposed plan you want to check against both risks; zero checks the plan the tool designs instead.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 5000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Your acceptance number
- What it is
- The largest number of defectives in the sample at which your plan still accepts the lot.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 200. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Only used when your sample size is above zero, and it must be less than that sample size.
Method and formulas
- Probability of acceptance P_a(p) = Σ_{k≤c} C(n, k) p^k (1 − p)^(n−k) (as the sampling OC tool).
- For each c from 0 upward, find the smallest n with P_a(RQL) ≤ β; the first such plan with P_a(AQL) ≥ 1 − α is the design.
- Average outgoing quality limit = largest p × P_a(p) over p.
- Example: sample 132 and accept on 3 or fewer defectives: P_a is 95.575% at the AQL and 9.9228% at the RQL.
Limitations
- Binomial model for large lots; small lots need the hypergeometric model.
- Single sampling only; double and sequential plans need fewer samples on average.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design an incoming inspection plan
- Check whether a supplier’s plan protects you
- Trade inspection cost against risk
Who it is for
- Quality engineers
- Supplier quality engineers
- Inspectors
What you get
- Designed sample size and acceptance number
- Acceptance probabilities and AOQL
- Check of your plan
Interpreting the result
- The closer the AQL and RQL, the larger the sample must be to tell them apart.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 128–133. Topic reference; no source tables or recommended settings are reproduced.
- ISO 2859-1 Sampling procedures for inspection by attributes
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Accepting or rejecting a lot
In Accept or Reject by the Numbers
What it covers: Decide about lots by sampling: designing a plan for producer's and consumer's risk, operating characteristics, lot acceptance from process capability, and what a sample adds to prior data.
Method explained here · BIZOMAT original book
Charting and sampling defect counts
What it covers: Charting daily defect counts with c and u charts, why clustering between parts causes false alarms and how to widen the limits, how inspection misses defects, and planning sampling of outgoing parts.
Method explained here · BIZOMAT original book
Testing tubes and bars with encircling coils
In Eddy-Current Inspection by the Numbers
What it covers: Encircling-coil testing of tubes: fill factor, frequency for the wall, differential and absolute coils, line speed, reference tubes and end zones, and auditing a supplier with a sampling plan.
Method explained here · BIZOMAT original book
Handling, storing and proving magnets
In Permanent Magnets in Machines by the Numbers
What it covers: Work safely with strong magnets: pinching, shattering and implants; storage and shipping; incoming inspection by moment with a sampling plan; and baselines that prove magnetic parts in service.
Method explained here · BIZOMAT original book
Procedures, people, sampling and cost
In Ultrasonic Testing by the Numbers
What it covers: What a written ultrasonic procedure must fix, how inspectors are qualified, what a 10% weld sample really protects against, random selection, trustworthy records and what testing costs.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
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