1. Quality Control Methods16
Quality Control Methods
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16.6
Acceptance Sampling
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3. Acceptance Sampling
Items coming from a production process are often sent in groups to another company or commercial establishment.
A group might consist of all units from a particular production run or shift, in a shipping container of some sort, sent in response to a particular order, and so on.
The group of items is usually called a lot, the sender is referred to as a producer, and the recipient of the lot is the consumer.

4. Acceptance Sampling
Our focus will be on situations in which each item is either defective or non defective, with p denoting the proportion of defective units in the lot.
The consumer would naturally want to accept the lot only if the value of p is suitably small.
Acceptance sampling is that part of applied statistics dealing with methods for deciding whether the consumer should accept or reject a lot.

5. Acceptance Sampling
Until quite recently, control chart procedures and acceptance sampling techniques were regarded by practitioners as equally important parts of quality control methodology. This is no longer the case.
The reason is that the use of control charts and other recently developed strategies offers the opportunity to design quality into a product, whereas acceptance sampling deals with what has already been produced and thus does not provide for any direct control over process quality.

6. Acceptance Sampling
This led the late American quality control expert W. E. Deming, a major force in persuading the Japanese to make substantial use of quality control methodology, to argue strongly against the use of acceptance sampling in many situations.
In a similar vein, the recent book by Ryan devotes several chapters to control charts and mentions acceptance sampling only in passing.
As a reflection of this deemphasis, we content ourselves here with a brief introduction to basic concepts.

7. Single-Sampling Plans

8. Single-Sampling Plans
The most straightforward type of acceptance sampling plan involves selecting a single random sample of size n and then rejecting the lot if the number of defectives in the sample exceeds a specified critical value c.
Let the rv X denote the number of defective items in the lot and A denote the event that the lot is accepted.
Then P(A) = P(X  c) is a function of p; the larger the value of p, the smaller will be the probability of accepting the lot.

9. Single-Sampling Plans
If the sample size n is large relative to N, P(A) is calculated using the hyper-geometric distribution (the number of defectives in the lot is Np):

10. Single-Sampling Plans
When n is small relative to N (the rule of thumb suggested previously was n  .05N, but some authors employ the less conservative rule n  .10N), the binomial distribution can be used:
P(X  c) =

11. Single-Sampling Plans
Finally, if P(A) is large only when p is small (this depends on the value of c), the Poisson approximation to the binomial distribution is justified:
P(X  c) 
The behavior of a sampling plan can be nicely summarized by graphing P(A) as a function of p. Such a graph is called the operating characteristic (OC) curve for the plan.

12. Example 16.11
Consider the sampling plan with c = 2 and n = 50. If the lot size N exceeds 1000, the binomial distribution can be used.
This gives
P(A) = 5 P(X  2)
= (1 – p) p (1 – p) p2(1 – p)48

13. Example 16.11
cont’d
The accompanying table shows P(A) for selected values of p, and the corresponding operating characteristic (OC) curve is shown in Figure 16.12
OC curve for sampling plan with c = 2, n = 50
Figure 16.12

14. Single-Sampling Plans
The OC curve for the plan of Example has P(A) near 1 for p very close to 0. However, in many applications a defective rate of 8% [for which P(A) = .226] or even just 5% [P(A) = .541] would be considered excessive, in which case the acceptance probabilities are too high.
Increasing the critical value c while holding n fixed gives a plan for which P(A) increases at each p (except 0 and 1), so the new OC curve lies above the old one.

15. Single-Sampling Plans
This is desirable for p near 0 but not for larger values of p. Holding c constant while increasing n gives a lower OC curve, which is fine for larger p but not for p close to 0.
We want an OC curve that is higher for very small p and lower for larger p.
This requires increasing n and adjusting c.

16. Designing a Single-Sampling Plan

17. Designing a Single-Sample Plan
An effective sampling plan is one with the following characteristics:
1. It has a specified high probability of accepting lots that the producer considers to be of good quality.
2. It has a specified low probability of accepting lots that the consumer considers to be of poor quality.

18. Designing a Single-Sample Plan
A plan of this sort can be developed by proceeding as follows.
Let’s designate two different values of p, one for which P(A) is a specified value close to 1 and the other for which P(A) is a specified value near 0.
These two values of p—say, p1 and p2— are often called the acceptable quality level (AQL) and the lot tolerance percent defective (LTPD).

19. Designing a Single-Sample Plan
That is, we require a plan for which
1. P(A) = 1 –  when p = p1 = AQL ( small)
2. P(A) =  when p = p2 = LTPD ( small)
This is analogous to seeking a hypothesis testing procedure with specified type I error probability  and specified type II error probability .

20. Designing a Single-Sample Plan
For example, we might have
AQL =  = (P(A) = .95)
LTPD =  = (P(A) = .10)
Because X is discrete, we must typically be content with values of n and c that approximately satisfy these conditions.

21. Designing a Single-Sample Plan
Table 16.6 gives information from which n and c can be determined in the case  = .05,  = .10.
Factors for Determining n and c for a Single-Sample Plan with  = .05,  = .10.
Table 16.6

22. Example 16.12 Let’s determine a plan for which AQL = p1 = .01 and
LTPD = p2 = The ratio of p2 to p1 is
This value lies between the ratio 4.89 given in Table 16.6, for which c = 4, and 4.06, for which c = 4.
Once one of these values of c is chosen, n can be determined either by dividing the np1 value in Table 16.6 by p1 or via np2/p2.

23. Example 16.12
cont’d
Thus four different plans (two values of c, and for each two values of n) give approximately the specified value of  and . Consider, for example, using c = 3 and
n =
Then
 = 1 – P (X  3 when p = p1) = .050
(the Poisson approximation with  = 1.37 also gives .050) and
 = P(X  3 when p = p2) = .131

24. Example 16.12 The plan with c = 4 and n determined from np2 = 7.99 has
cont’d
The plan with c = 4 and n determined from np2 = 7.99 has
n = 178,  = .034, and  = .094.
The larger sample size results in a plan with both  and  smaller than the corresponding specified values.

25. Designing a Single-Sample Plan
The book by Douglas Montgomery cited in the chapter bibliography contains a chart from which c and n can be determined for any specified 𝛼 and β. It may happen that the number of defective items in the sample reaches c + 1 before all items have been examined. For example, in the case c = 3 and n = 137, it may be that the 125th item examined is the fourth defective item, so that the remaining 12 items need not be examined. However, it is generally recommended that all items be examined even when this does occur, in order to provide a lot-by-lot quality history and estimates of p over time.

26. Double-Sampling Plans

27. Double-Sampling Plans
In a double-sampling plan, the number of defective items x1 in an initial sample of size n1 is determined.
There are then three possible courses of action:
Immediately accept the lot, immediately reject the lot, or take a second sample of n2 items and reject or accept the lot depending on the total number of x1 + x2 defective items in the two samples.

28. Double-Sampling Plans
Besides the two sample sizes, a specific plan is characterized by three further numbers—c1, r1, and c2—as follows:
1. Reject the lot if x1  r1.
2. Accept the lot if x1  c1.
3. If c1 < x1 < r1 , take a second sample; then accept the lot if x1 + x2  c2 and reject it otherwise.

29. Example 16.13 Consider the double-sampling plan with n1 = 80, n2 = 80,
c1 = 2, r1 = 5, and c2 = 6.
Thus the lot will be accepted if (1) x1 = 0, 1, or 2; (2)x1 = 3 and x2 = 0, 1, 2 or 3; or (3)x1 = 4 and x2 = 0, 1 or 2.

30. Example 16.13
cont’d
Assuming that the lot size is large enough for the binomial approximation to apply, the probability P(A) of accepting the lot is
P(A) = P(X1 = 0, 1, or 2) + P(x1 = 3, x2 = 0 ,1, 2, or 3)
+ P(X1 = 4, X2 = 0, 1 or 2)

31. Example 16.13
cont’d
Again the graph of P(A) versus p is the plan’s OC curve. The OC curve for this plan appears in Figure
OC curve for the double-sampling plan of Example 16.13
Figure 16.13

32. Double-Sampling Plans
One standard method for designing a double-sampling plan involves proceeding as suggested earlier for single-sample plans.
Specify values p1 and p2 along with corresponding acceptance probabilities 1 –  and . Then find a plan that satisfies these conditions.
The book by Montgomery provides tables similar to Table 16.6 for this purpose in the cases n2 = n1 and n2 = 2n1 with 1 –  = .95,  = .10. Much more extensive tabulations of plans are available in other sources.

33. Double-Sampling Plans
Analogous to standard practice with single-sample plans, it is recommended that all items in the first sample be examined even when the (r1 + 1)st defective is discovered prior to inspection of the n1th item.
However, it is customary to terminate inspection of the second sample if the number of defectives is sufficient to justify rejection before all items have been examined.
This is referred to as curtailment in the second sample.

34. Double-Sampling Plans
Under curtailment, it can be shown that the expected number of items inspected in a double-sampling plan is smaller than the number of items examined in a single-sampling plan when the OC curves of the two plans are close to being identical.
This is the major virtue of double-sampling plans.
For more on these matters as well as a discussion of multiple and sequential sampling plans (which involve selecting items for inspection one by one rather than in groups), a book on quality control should be consulted.

35. Rectifying Inspection and other Design Criteria

36. Rectifying Inspection and Other Design Criteria
In some situations, sampling inspection is carried out using rectification.
For single-sample plans, this means that each defective item in the sample is replaced with a satisfactory one, and if the number of defectives in the sample exceeds the acceptance cutoff c, all items in the lot are examined and good items are substituted for any defectives
Let N denote the lot size. One important characteristic of a sampling plan with rectifying inspection is average outgoing quality, denoted by AOQ.

37. Rectifying Inspection and Other Design Criteria
This is the long-run proportion of defective items among those sent on after the sampling plan is employed. Now defectives will occur only among the N – n items not inspected in a lot judged acceptable on the basis of a sample.
Suppose, for example, that P(A) = P (X  c) = .985 when p = .01.Then, in the long run, 98.5% of the items not in the sample will not be inspected, of which we expect 1% to be defective. This implies that the expected number of defectives in a randomly selected batch is
(N – n)  P(A)  p = (N – n).

38. Rectifying Inspection and Other Design Criteria
Dividing this by the number of items in a lot gives average outgoing quality:
Because AOQ = 0 when either p = 0 or p = 1 [ P(A) = 0 in the latter case], it follows that there is a value of p between 0 and 1 for which AOQ is a maximum. The maximum value of AOQ is called the average outgoing quality limit, AOQL.

39. Rectifying Inspection and Other Design Criteria
For example, for the plan with n = 137 and c = 3 discussed previously, AOQL = , the value of AOQ at p  .02.
Proper choices of n and c will yield a sampling plan for which AOQL is a specified small number.
Such a plan is not, however, unique, so another condition can be imposed. Frequently this second condition will involve the average (i.e., expected) total number inspected, denoted by ATI.

40. Rectifying Inspection and Other Design Criteria
The number of items inspected in a randomly chosen lot is a random variable that takes on the value n with probability P(A) and N with probability 1 – P(A).
Thus the expected number of items inspected in a randomly selected lot is
ATI = n  P(A) + N  (1 – P(A))
It is common practice to select a sampling plan that has a specified AOQL and, in addition, minimum ATI at a particular quality level p.

41. Standard Sampling Plans

42. Standard Sampling Plans
It may seem as though the determination of a sampling plan that simultaneously satisfies several criteria would be quite difficult.
Fortunately, others have already laid the groundwork in the form of extensive tabulations of such plans.
MIL STD 105D, developed by the military after World War II, is the most widely used set of plans.

43. Standard Sampling Plans
A civilian version, ANSI/ASQC Z1.4, is quite similar to the military version.
A third set of plans that is quite popular was developed at Bell Laboratories prior to World War II by two applied statisticians named Dodge and Romig.
The book by Montgomery (see the chapter bibliography) contains a readable introduction to the use of these plans.