OC curve for the single sampling
plan N = 3000, n=89, c= 2
1
Probabilities of Acceptance for the
Single Sampling Plan n = 89, c = 2
Assumed Process Quality
Sample size, np0
n
Probability of
acceptance, Pa
Percent of
Lots Accepted
100Pa
P0
100P0
0.01
1.0
89
0.9
0.938
93.8
0.02
2.0
89
1.8
0.731
73.1
0.03
3.0
89
2.7
0.494
49.4
0.04
4.0
89
3.6
0.302
30.2
0.05
5.0
89
4.5
0.174
17.4
0.06
6.0
89
5.3
0.106
10.6
0.07
7.0
89
6.2
0.055
5.5
2
Double Sampling Plan
Inspect a sample
of 150 from lot
of 2400
If 1 or less
Nonconforming
units accept lots and
stop
If 5 or less
Nonconforming units
On both samples,
Accept the lot
If 2 or 3 nonconforming
units, inspect a second
sample of 200
If 4 or more
Nonconforming units
the lot is not accepted
and stop
If 6 or more
Nonconforming units
On both samples
The lot is not accepted
Graphical description of the double sampling plan: N=2400,n1=150, c1=1,
r1=4, n2=200, c2=5, and r2=6
3
OCC for Double Sampling Plan
4
OCC for a Multiple Sampling Plan
5
Type-A and Type-B OC curves:
6
Effect of n and c on OC curves:
7
Other Aspects of OC Curve Behavior
8
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.
9
Average Outgoing Quality
The Average Outgoing Quality (AOQ) is the average of
rejected lots (100% inspection) and accepted lots ( a
sample of items inspected)
N -n
AOQ = Pac * p(
) where
N
Pac = Probability of accepting a lot
p = Fraction defective
n = sample size
N = Lot size
Note that as the lot size N becomes large relative to the
sample size n, AOQ ≈ Pacp
10
Average Quality of Inspected Lots
Typically the term (N-n)/N is very close to 1; therefore,
the equation most often used is:
AOQ = Pac * p where
Pac = Probability of accepting a lot
p = Fraction defective
11
Example
If N = 10,000, n = 89, and c = 2, and that the
incoming lots are of quality p = 0.01. What is AOQ?
AOQ = Pacp(N-n)/N
=?
12
Average Outgoing Quality (AOQ) for
the Single Sampling Plan n = 89, c = 2
Assumed Process Quality
Sample size, np0
n
Probability of
acceptance, Pa
AOQ
100Pacp
93.8
P0
100P0
0.01
1.0
89
0.9
0.938
0.02
2.0
89
1.8
?
0.03
3.0
89
2.7
0.494
0.04
4.0
89
3.6
?
0.05
5.0
89
4.5
?
0.06
6.0
89
5.3
0.106
0.636
0.07
7.0
89
6.2
0.055
0.385
1.482
13
AOQ and Acceptance Sampling
15 lots
2% nonconforming
Producer
N=3000
n=89
c=2
11 lots
2% nonconforming
Consumer
4 lots
2% nonconforming
4 lots
0% nonconforming
Figure 9-15 How acceptance Sampling works
14
AOQ and Acceptance Sampling
Total Number
11 lots-
Number
Nonconforming
11(3000)=33,000
33,000(0.02)=660
4(3000)(0.98)=11,7
60
0%
Nonconforming
0
44,760
660
2%
Nonconforming
4 lots-
Percent Nonconforming (AOQ) =
660/44,760 X 100 =1.47%
Figure 9-15 cont’d.
Average Outgoing Quality curve for the sampling plan
N = 3000, n = 89, and c = 2
Average Outgoing Quality Level
 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 Average Outgoing Quality
Level AOQL.
17
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
pa of accepting a lot with defect level
p, we have:
ATI = n + (1 - pa) (N - n)
where N is the lot size.
18
Example
If a lot size N = 10,000, and sample size n = 89, number of
acceptance c = 2, find ATI at p = 0.01
19
20
Average Sample Number (ASN)
For a single sampling (n,c) we know each
and every lot has a sample of size n taken
and inspected or tested. For double, multiple
and sequential plans, the amount of sampling
varies depending on the number of defects
observed.
21
Average Sample Number (ASN)
For any given double, multiple or
sequential plan, a long term ASN 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 plan scheme.
ASN = n1 + n2 (1 – P1) for a double
sampling plan.
22
Sampling Plan Design
Suppose α is known and the AQL is also
known then :
 Sampling plan with stipulated producer’s risk
 Sampling plan with stipulated consumer’s risk
 Sampling plan with stipulated producer’s and
consumer’s risk
can be designed.
23
Sampling Plan Design
Stipulated Producer’s Risk
 α = 0.05
 Pa=0.95
AQL = 1.2%
P0.95= 0.012
Assume values for C, find np0.95 for this c
value, calculate n
24
Sampling Plan Design
Stipulated Consumer’s Risk
 β = 0.10
 Pa=0.10
LQ = 6.0%
P0.10= 0.060
Assume values for C, find np0.95 for this c
value, calculate n
25
Sampling Plan Design
Stipulated Producer’s and Consumer’s risk
 α = 0.10
 AQL=0.9
β = 0.10
LQ= 7.8
Find the ratio of P0.10/P0.95. From table 9-4
C is between 1 and 2. Find n for c =1 and n for
c =2 .
26
Sampling Plan Design
Have 4 plans.
Select plan based on:
 Lowest sampling size
 Greatest sampling size
 Plan exactly meets consumer’s stipulation
and is as close as possible to producer’s
stipulation
 Plan exactly meets producer’s stipulation and
is as close as possible to consumer’s
stipulation
27