1. OC curve for the single sampling plan N = 3000, n=89, c= 2

2. Probabilities of Acceptance for the Single Sampling Plan n = 89, c = 2
Assumed Process Quality
Sample size, n
np0
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
1.8
0.731
73.1
0.03
3.0
2.7
0.494
49.4
0.04
4.0
3.6
0.302
30.2
0.05
5.0
4.5
0.174
17.4
0.06
6.0
5.3
0.106
10.6
0.07
7.0
6.2
0.055
5.5

3. Double Sampling Plan Inspect a sample of 150 from lot of 2400
If 1 or less
Nonconforming
units accept lots and
stop
If 4 or more
Nonconforming units
the lot is not accepted
and stop
If 2 or 3 nonconforming
units, inspect a second
sample of 200
If 5 or less
Nonconforming units
On both samples,
Accept the lot
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

4. OCC for Double Sampling Plan

5. OCC for a Multiple Sampling Plan

6. Type-A and Type-B OC curves:
fg1506

7. Effect of n and c on OC curves:

8. Other Aspects of OC Curve Behavior

9. 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.

10. Average Outgoing Quality
The Average Outgoing Quality (AOQ) is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected)
Note that as the lot size N becomes large relative to the sample size n, AOQ ≈ Pacp

11. Average Quality of Inspected Lots
Typically the term (N-n)/N is very close to 1; therefore, the equation most often used is:

12. Example
If N = 10,000, n = 89, and c = 2, and that the incoming lots are of quality p = What is AOQ?
AOQ = Pacp(N-n)/N
= ?

13. Average Outgoing Quality (AOQ) for the Single Sampling Plan n = 89, c = 2
Assumed Process Quality
Sample size, n
np0
Probability of acceptance, Pa
AOQ 100Pacp P0
100P0
0.01
1.0 89
0.9
0.938
93.8
0.02
2.0
1.8 ?
0.03
3.0
2.7
0.494
1.482
0.04
4.0
3.6
0.05
5.0
4.5
0.06
6.0
5.3
0.106
0.636
0.07
7.0
6.2
0.055
0.385

14. AOQ and Acceptance Sampling
11 lots
2% nonconforming
15 lots
2% nonconforming
Producer
N=3000
n=89
c=2
Consumer
4 lots
2% nonconforming
4 lots
0% nonconforming
Figure 9-15 How acceptance Sampling works

15. AOQ and Acceptance Sampling
Total Number
Number Nonconforming
11 lots-
2% Nonconforming
11(3000)=33,000
33,000(0.02)=660
4 lots-
0% Nonconforming
4(3000)(0.98)=11,760
44,760
660
Percent Nonconforming (AOQ) = 660/44,760 X 100 =1.47%
Figure 9-15 cont’d.

16. Average Outgoing Quality curve for the sampling plan N = 3000, n = 89, and c = 2

17. 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.

18. 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.

19. Example
If a lot size N = 10,000, and sample size n = 89, number of acceptance c = 2, find ATI at p = 0.01

21. 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.

22. 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.

23. 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.

24. Sampling Plan Design Stipulated Producer’s Risk
α = AQL = 1.2%
Pa= P0.95= 0.012
Assume values for C, find np0.95 for this c value, calculate n

25. Sampling Plan Design Stipulated Consumer’s Risk
β = LQ = 6.0%
Pa= P0.10= 0.060
Assume values for C, find np0.95 for this c value, calculate n

26. Sampling Plan Design Stipulated Producer’s and Consumer’s risk
α = β = 0.10
AQL= 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 .

27. 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