Supplemental Material for Chapter 14
S14-1. Consideration of Inspection Error
A basic assumption in the construction of acceptance sampling plans is that the
inspection process is error-free. However, this is often not the case with inspection
activities; indeed, many inspection processes are known to be error-prone. While these
errors are usually unintentional, they have the effect of distorting (sometimes severely)
the performance measures of the inspection or acceptance sampling plan that has been
adopted. In complex inspection activities, error rates of 20-30 % are not terribly unusual.
In this section we illustrate some of the effects of inspection error on single sampling
plans for attributes.
Two types of errors are possible in attributes inspection. An item that is good can be
classified as defective (this is a type I error), or an item that is defective can be classified
as good (this is a type II error). Define
E1  event that a good item is classified as a defective
E2  event that a defective item is classified as good
A  event that an item is defective
B  event that an item is classified as a defective
Then
P( B)  P( A) P( E2 )  P( A) P( E1 )
By defining the quantities
p  P( A) the true fraction efective
pe  P( B) the apparent fraction defective
e1  P ( E1 ) the probability that a type I error is made
e2  P( E2 ) the probability that a type II error is made
the expression for apparent fraction defective can be written as
pe  p(1  e2 )  (1  p)e1
The OC curve of a sampling plan plots the probability of lot acceptance versus the lot
fraction defective. When inspection error is present, the probability of lot acceptance is
c
n
Pa ( e )     ped (1  pe )n d
d 0  d 
Thus the OC curve of a sampling plan when inspection error is present can be easily
obtained.
Now consider how inspection error impacts AOQ and ATI. For error-free inspection, the
AOQ for single sampling inspection for attributes is
AOQ 
( N  n) pPa
N
1
If defective items are replaced, and if the inspection of the replacements also involves
inspection error, then the AOQ becomes
AOQ 
npe2  p( N  n)(1  pe ) Pa ( e )  p( N  n)(1  Pa ( e ) )e2
N (1  pe )
If the defectives discovered are not replaced, then
AOQ 
npe2  p ( N  n) Pa ( e )  p ( N  n)(1  Pa ( e ) )e2
N  npe  (1  P( e ) )( N  n) pe
The ATI assuming error-free inspection and 100% screening of rejected lots is
ATI  n  (1  Pa )( N  n)
The corresponding ATI when the discovered defectives are replaced and the replacement
process is subject to inspection error is
ATI 
n  (1  Pa ( e ) )( N  n)
1  pe
If the discovered defectives are not replaced, then
ATI  n  (1  Pa ( e ) )( N  n)
The design of a sampling plan is usually accomplished by choosing a set of parameters
for the plan such that the OC curve passes through two points, generally at the AQL and
LTPD levels of quality. When there is inspection error, the probabilities of acceptance no
longer have the values that would have been obtained with perfect inspection. However,
it is possible to modify the design procedure by forcing the actual OC curve to fit the
desired points.
To illustrate this process, suppose that we want the OC curve to match the points (AQL,
1-  ) and (LTPD,  ). Then in designing the sampling plan we need to use
AQLe  AQL(1  e2 )  (1  AQL)e1
LTPDe  LTPD(1  e2 )  (1  LTPD)e1
If the observed OC curve fits these points, then the actual OC curve will fit the desired
points.
To illustrate, suppose that AQL  0.01,1    0.95, LTPD  0.06,   0.1. The single
sampling plan without any considerations of inspection error has n  89, c  2 . Now
suppose that there is inspection error and e1  0.01, e2  0.15 . Then we can calculate
AQLe  AQL(1  e2 )  (1  AQL)e1
 0.01(1  0.15)  (1  0.01)0.01
 0.0184
and
2
LTPDe  LTPD(1  e2 )  (1  LTPD )e1
 0.06(1  0.15)  (1  0.06)0.01
 0.0604
The single sampling plan that has an OC curve passing through the points (0.0184, 0.95),
(0.0604, 0.10) has the parameters n  150, c  5 . Notice that the effect of inspection error
is to increase the sample size required to obtain the desired protection at the specified
points on the OC curve.
Inspection error has an effect on AOQ. Incorrect classification of a good item reduces
the AOQ because more screening inspection occurs. Incorrect classification of a
defective item results in higher AOQ values because it reduces the likelihood that lots
will be screened. The general impact of a type I inspection error is to increase the ATI,
while a type II inspection error decreases the ATI.
S14-2. A Lot Sensitive Compliance (LTPD) Sampling Plan
In this section we briefly describe a sampling plan that is particularly useful in
compliance testing, or sampling for product safety related purposes. The procedure is
also very useful in any situation where minimum sample sizes are required. Developed by
Schilling (1978), the plans provide for rejection of the lot if any defectives are found in
the sample. They are based on a well-defined relationship between the sampling plan and
the size of the lots submitted for inspection. The sampling procedure gives the
proportion of the lot that must be sampled to guarantee that the fraction defective in the
lot is less than a prescribed limit with probability 0.90. That is, for a specified LTPD the
probability of lot acceptance is 0.10. Sample sizes are based on the hypergeometric
distribution. Schilling gives a table to assist in implementing the procedure.
In general, an acceptance-sampling plan with a zero acceptance number has a very
undesirable shape at low fractions nonconforming. Since these lot sensitive compliance
plans have c = 0, the manufacturer’s process must operate at an average level of quality
that is less than about 5% of the LTPD in order to have a reasonably small probability of
a good lot being rejected. If the process average is closer than that to the LTPD a plan
with c = 0 is not a good choice and a Dodge-Romig LTPD plan would likely be
preferable. However, these plans have much larger sample sizes then Schilling’s plans.
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