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LASP is a sampling scheme and a set of rules
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A lot acceptance sampling plan (LASP) is a sampling scheme
and a set of rules for making decisions. The decision, based on counting
the number of defectives in a sample, can be to accept the lot, reject
the lot, or even, for multiple or sequential sampling schemes, to take
another sample and then repeat the decision process.
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Types of acceptance plans to choose from
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LASPs fall into the following categories:
- Single sampling plans: One
sample of items is selected at random from a lot and the
disposition of the lot is determined from the resulting
information. These plans are usually denoted as (\(n,c\))
plans for a sample size \(n\),
where the lot is rejected if there are more than \(c\)
defectives. These are the most
common (and easiest) plans to use although not the most efficient
in terms of average number of samples needed.
- Double sampling plans: After the first sample is tested,
there are three possibilities:
- Accept the lot
- Reject the lot
- No decision
If the outcome is (3), and a second sample is taken, the procedure
is to combine the results of both samples and make a final
decision based on that information.
- Multiple sampling plans:
This is an extension of the double sampling plans where more than
two samples are needed to reach a conclusion. The advantage of
multiple sampling is smaller sample sizes.
- Sequential sampling plans:
This is the ultimate extension of multiple sampling where
items are selected from a lot one at a time and after inspection
of each item a decision is made to accept or reject the lot or
select another unit.
- Skip lot sampling plans:
Skip lot sampling means that only a fraction of the submitted lots
are inspected.
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Definitions of basic Acceptance Sampling terms
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Deriving a plan, within one of the categories listed above, is discussed
in the pages that follow. All derivations depend on the properties you
want the plan to have. These are described using the following terms:
- Acceptable Quality
Level (AQL): The AQL is a percent defective that is the
base line requirement for the quality of the producer's product.
The producer would like to design a sampling plan such that there
is a high probability of accepting a lot that has a defect
level less than or equal to the AQL.
- Lot Tolerance
Percent Defective (LTPD): The LTPD is a designated high
defect level that would be unacceptable to the consumer. The
consumer would like the sampling plan to have a low probability
of accepting a lot with a defect level as high as the
LTPD.
- Type I Error
(Producer's Risk): This is the probability, for a given
(\(n,c\))
sampling plan, of rejecting a lot that has a defect
level equal to the AQL. The producer suffers when this occurs,
because a lot with acceptable quality was rejected. The symbol \(\alpha\)
is commonly used for the Type I error and typical values for \(\alpha\)
range from 0.2 to 0.01.
- Type II Error
(Consumer's Risk): This is the probability, for a given
(\(n,c\))
sampling plan, of accepting a lot with a defect level
equal to the LTPD. The consumer suffers when this occurs, because
a lot with unacceptable quality was accepted. The symbol \(\beta\)
is commonly used for the Type II error and typical values range
from 0.2 to 0.01.
- Operating
Characteristic (OC) Curve: This curve plots the
probability of accepting the lot (Y-axis) versus the lot fraction
or percent defectives (X-axis). The OC curve is the primary
tool for displaying and investigating the properties
of a LASP.
- Average Outgoing
Quality (AOQ): A common procedure, when sampling and
testing is non-destructive, is to 100 % inspect rejected lots and
replace all defectives with good units. In this case, all rejected
lots are made perfect and the only defects left are those in lots
that were accepted. AOQs
refer to the long term defect
level for this combined LASP and 100 % inspection of rejected lots
process. If all lots come in with a defect level of exactly \(p\),
and the OC curve for the chosen (\(n,c\))
LASP indicates a probability \(p_a\)
of accepting such a lot, over the long run the AOQ
can easily be shown to be:
$$ \mbox{AOQ} = \frac{p_a p (N - n)}{N} \, ,$$
where \(N\)
is the lot size.
- Average
Outgoing Quality Level (AOQL): A plot of the AOQ
(Y-axis) versus the incoming lot \(p\)
(X-axis) will start at 0 for \(p=0\),
and return to 0 for \(p = 1\)
(where every lot is 100 % inspected and rectified). In between, it will rise
to a maximum. This maximum, which is the worst possible long term
AOQ,
is called the AOQL.
- Average Total
Inspection (ATI): When rejected lots are 100 % inspected,
it is easy to calculate the ATI
if lots come consistently
with a defect level of \(p\).
For a LASP (\(n,c\))
with a probability \(p_a\)
of accepting a lot with defect level \(p\),
we have
$$ \mbox{ATI} = n + (1-p_a)(N-n) \, , $$
where \(N\)
is the lot size.
- Average Sample Number
(ASN): For a single sampling LASP (\(n,c\))
we know each and every lot has a sample of size \(n\)
can be calculated assuming all lots come in with a defect level of \(p\).
A plot of the ASN,
versus the incoming defect level \(p\),
describes the sampling efficiency of a given LASP scheme.
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