Acceptance Sampling Plans by
Variables
CH 16
Contents
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Advantages and Disadvantages of
Acceptance Sampling by Variables.
Types of Acceptance Sampling by
Variables.
Chain Sampling.
Continuous Sampling.
Advantages of Variables Sampling
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The same OC curve can be obtained with a smaller
sample size than would be required by an
attributes sampling plan.
Measurement data provide more information about
the manufacturing process.
When AQLs are very small, the sample sizes
required by attributes sampling plans are very
large.
Disadvantages of Variables Sampling
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The distribution of the quality characteristic must be
known.
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A separate sampling plan must be employed for each
quality characteristic that is being inspected.
Types of Variables Sampling Plans
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Plans that control the lot or process fraction
defective.
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Plans that control a lot or process mean.
Plans to Control Process Fraction Defective
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Since the quality characteristic is a variable, there will exist
either LSL, USL, or both, that define the acceptable values
of this parameter.
Fig. 1 illustrates the situation in which the quality
characteristic x is normally distributed and there is LSL on
this parameter.
Fig. (1)
Plans to Control Process Fraction Defective
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Procedure 1 (k-Method)
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Take a random sample of n items from the lot and
compute
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If there is a critical value of p of interest that should
not be exceeded with stated probability, we can
translate this value of p into critical distance k.
If ZLSL ≥ k, we would accept the lot because the sample
data imply that the lot mean is sufficiently far above
LSL to insure that p is satisfactory.
Plans to Control Process Fraction Defective
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Procedure 2 (M-Method)
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Compute ZLSL .
Use ZLSL to estimate the fraction defective of the lot or
process .
Determine the max. allowable fraction defective M
(using specific values of n, k).
If exceeds M, reject the lot; otherwise, accept it.
Plans to Control Process Fraction Defective
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Notes
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In the case of an USL, we compute
If
is unknown, it is estimated by s.
When there is only a single specification limit (LSL or
USL), either procedure may be used.
When there are both LSL and USL, M-method should be
used by computing ZLSL and ZUSL, finding the
corresponding fraction defective estimates ^pLSL and ^pUSL
^LSL + p
^USL ≤ M, the lot will be
Then, if p
accepted.
Designing a variables sampling plan with
a specified OC curve
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Let
interest.
be the two points on the OC curve of
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p1 and p2 are the levels of lot or process fraction
nonconforming that correspond to acceptable and rejectable
levels of quality, respectively.
Designing a variables sampling plan
with a specified OC curve
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Example 1
Designing a variables sampling plan
with a specified OC curve
Designing a variables sampling plan with
a specified OC curve
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Example 2 :Design a sampling plan using M-method
Designing a variables sampling plan with
a specified OC curve
MIL STD 414
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There are five general levels of inspection, and level IV is
designated as “normal”.
MIL STD 414
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As MIL STD 105E, sample size code letters are
used, but the same code letter does not imply the
same sample size in both standards.
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Sample sizes are a function of the lot size and the
inspection level.
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All the sampling plans in the standards assume that
the quality characteristic is normally distributed.
MIL STD 414
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Organization of MIL STD 414
MIL STD 414
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Example 3: Using MIL STD 414
Solution
From table, if we use IV level, the sample size code letter is O.
From a second table, we find n=100.
For AQL of 1%, on normal inspection, k=2.
For AQL of 1%, on tightened inspection, k=2.14
MIL STD 414
Plans to Control A Process Mean
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Example 4
Solution
Let -XA be the value of the sample average below witch the lot
will be accepted.
If lots have 0.95 probability of acceptance, then
P (X ≤ XA ) = 0.95
Plans to Control A Process Mean
P (Z
≤
) = 0.95
=1.64
If lots have 0.1 probability of acceptance, then
P (X ≤ -XA ) = 0.1
p (Z ≤
) = 0.1
= -1.28
These two equations can be solved for n and XA , giving n=9 and
-XA=0.356
Chain Sampling Plan
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It is used for small sample size plans that have acceptance number
of zero.
Lots should be produced under the same conditions, and be
expected to be of the same quality.
There should be a good record of quality performance on the part
of supplier.
The general procedure for ChSP-1:
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For each lot, select the sample of size n and observe the defectives.
If sample has zero defectives, accept the lot .
If sample has two or more defectives, reject the lot.
If sample has 1 defective, accept the lot provided that there have been
no defectives in the previous i lots.
Chain Sampling Plan
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The effect of chain sampling on OC curve:
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It is more difficult to reject lots
with very small p with ChSp-1
Plan than it is with ordinary single
Plan.
In practice, values of i vary between
3 and 5, since OC curves of such plans
approximate OC curve for S-S plan.
The points on the OC curve of
a ChSP-1 plan are given by:
Continuous Sampling Plan
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Used when production is continuous. That is, the
manufacturing operation do not result in the natural
formation of lots.
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It consists of altering sequences of sampling
inspection and screening (100% inspection).
Continuous Sampling Plan
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Procedure for CSP-1 plans
Continuous Sampling Plan
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The average number of units inspected in a 100% screening is
1qi
equal to u  pqi , where q=1-p
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The average number of units passed under the sampling inspection
1
is v 
fp
The average fraction of total manufactured units inspected in the
u  fv
long run is AFI 
u v
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The average fraction of manufactured units passed under the
sampling procedure is pa  v
u v
Continuous Sampling Plan
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OC curves for various continuous sampling plans, CSP-1.