Acceptance Sampling - Help

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This web based tool is implemented by Quasydoc© with algorithms and R-code provided by dr. Stijn Luca. The tool aims at providing a user-friendly way to develop single and double stage sampling plan for attributes and variables inspection. We implemented the use and visualization of four possible sampling plans.

Sampling plans for attributes inspection

In these sampling plans items are sampled from the lot. Each item is inspected and classified as conforming or nonconforming.

1. Single stage sampling plan for attributes inspection
   A random sample of size n is taken from the lot. When the number of nonconforming units d in the sample does not exceed the acceptance number c, the lot is accepted; When d>c, the lot is rejected.
2. Double stage sampling plan for attributes inspection
   A random sample of size n is taken from the lot. If the number of nonconforming items d does not exceed the acceptance number c of stage 1, the lot is accepted; If d exceeds the acceptance number c of stage 2, the lot is rejected; If d exceeds the acceptance number of stage 1, but does not exceed the acceptance number of stage 2, a second sample of size n is taken from the lot. In this case the total number of nonconforming units that is found in stage 1 and stage 2 is considered. When this total amount does not exceed the acceptance number from stage 2, the lot is accepted; Otherwise, it is rejected.

Sampling plans for variables inspection (with known variance)

In these sampling plans items are sampled from the lot. A measurement is performed for each item of the sample and a sample mean is considered. A lower (resp. upper) specification limit specifies a minimum (resp. maximum) target value for the individual measurements. We assume that enough data from the process of the supplier is available such that a good estimate of the variance can be obtained. Otherwise a sampling plan for unknown variance should be used, which is not implemented in this tool.

3. Single stage sampling plan for variables inspection
   When the standardized difference between the sample mean and the lower (or upper) specification limit is more than the acceptance constant k, the lot is accepted. Otherwise, it is rejected. Equivalently, one can look at an estimate of the percentage p of nonconforming items that do not meet the specification limit and accept the lot only when p<M.
4. Double stage sampling plan for variables inspection (with known variance)
   A random sample of size n is taken from the lot. When the standardized difference between the sample mean and the lower (or upper) specification limit is more than the acceptance constant k from stage 2, the lot is accepted; When it is lower than the acceptance constant k from stage 1, the lot is rejected; When it is between the two criteria, a second sample of size n is taken. In this case the mean of all measurements from sample 1 and 2 is considered. When the standardized difference between this overall mean and the lower (or upper) specification limit is lower than the acceptance constant from the first stage, the lot is rejected. Otherwise, it is accepted.

The two panels

The interface for all sampling plans consists of two main panels. The left plan gives an overview of the parameters of the plan. The right panel visualizes the plan by plots of the OC-curve and the average sample number.

Left panel:

In the left panel the AQL (acceptable quality level) and RQL (rejectable quality level) can be specified. The AQL is the maximum fraction nonconforming that is allowed to accept the lot. The RQL is the minimum fraction nonconforming that is required to reject the lot. When the defect ratio p equals AQL, the lot should be accepted with a high probability 1 – α. The probability α to reject a lot with p=AQL is termed the supplier’s risk. The probability to accept a lot with p=RQL is termed the customer’s risk.
The left panel works in two directions: (i) a corresponding sampling plan is calculated given some risks, or (ii) corresponding risks are calculated given some sampling plan.

Right panel:

The OC-curves shows the acceptance probability as a function of the defect ratio. As more nonconforming items are present in the lot, the probability that the lot is accepted using the sampling plan decreases. As well the AQL as the RQL is indicated by a vertical line. The acceptance probability at AQL should be approximately 1-suppliers risk; The acceptance probability at RQL should be approximately the customer’s risk. These identities will almost never be exact as sample sizes are discrete numbers.

The curve of the average sample number shows the average number of required items that have to be inspected as a function of the fraction nonconforming. For the single stage sampling plans, the size is constant. For a double stage sampling plan, the lots will usually be accepted in the first stage when the quality is very good and rejected in the first stage when the quality is very bad. This gives an average sample number that is smaller than the sample size used in single sampling plans for lots with either a very good or very bad quality. When lots are of intermediate quality, a second sample will be required and the total number of items that need to be inspected will increase.

Summarized terminology

Common

AQL: Acceptable ratio of nonconforming items in a lot
RQL: Unacceptable ratio of nonconforming items in a lot
Supplier: Risk to reject an acceptable lot with a fraction nonconforming below AQL
Customer: Risk to accept a rejectable lot with a fraction nonconforming above RQL
n: Sample size

Single attribute

c: Acceptable number of nonconforming items

Double attribute

c (stage 1): Acceptable number of nonconforming items for the first sample
c (stage 2): Acceptable number of nonconforming items for both samples together
Curtailed: The inspection is ended when the observed nonconforming items in the combined sample exceeds the second acceptance number.

Single variables sampling scheme

k: Required deviation between sample mean and specification limit (the acceptance constant)
M: Maximum percentage of measurements in the sample that are beyond the specification limit

Double variables sampling scheme

k (stage 1): Required deviation between sample mean and specification limit for the first sample and both samples together
k (stage 2): Acceptable deviation between sample mean and specification limit for the second sample