1. Lot-by-Lot Acceptance Sampling for Attributes
Chapter 15
Lot-by-Lot Acceptance Sampling for Attributes

2. The Acceptance-Sampling Problem
Acceptance sampling is concerned with inspection and decision making regarding products.

3. Three aspects of sampling
The purpose of acceptance sampling is to sentence lots, not to estimate the lot quality
Although, some plans do this
Acceptance sampling is not quality control
Reject or accept lots only
Even if lots are of the same quality, sampling will accept some lots and reject others

4. Three aspects of sampling
Quality cannot be inspected into the product
Acceptance sampling is an audit tool that insures that the output of a process conforms to requirements

5. The Acceptance-Sampling Problem
Three approaches to lot sentencing:
Accept with no inspection
100% inspection
Acceptance sampling

6. The Acceptance-Sampling Problem
Why Acceptance Sampling and Not 100% Inspection?
Testing can be destructive
Cost of 100% inspection is high
100% inspection is not feasible
Requires too much time
Can be inaccurate
If vendor has excellent quality history

7. The Acceptance-Sampling Problem
Advantages and Disadvantages of Sampling
Advantages
Less expensive
Reduced damage
Reduces the amount of inspection error
Disadvantages
Risk of accepting “bad” lots, rejecting “good” lots
Less information generated
Requires planning and documentation

8. The Acceptance-Sampling Problem
Types of Sampling Plans
There are variables sampling plans and attribute sampling plans (this chapter is about attributes)
Single sampling plan
Double-sampling plan
Multiple-sampling plan
Sequential-sampling

9. The Acceptance-Sampling Problem
Lot Formation
Considerations before inspection:
Lots should be homogeneous
Produced by the same machine, same operators, common raw materials, approximately the same time

10. Considerations before inspection:
Lot Formation
Considerations before inspection:
Larger lots more preferable than smaller lots
More economical
Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities.

11. The Acceptance-Sampling Problem
Random Sampling
The units selected for inspection should be chosen at random.
If random samples are not used, bias can be introduced.
If any judgment methods are used to select the sample, the statistical basis of the acceptance-sampling procedure is lost.

12. Guidelines for Using acceptance Sampling
It is a statement of the sample size to be used and the associated acceptance or rejection criteria.
Sampling scheme is defined as the set of procedures consists of acceptance sampling plans in which lot sizes, sample sizes, and acceptance criteria along with the 100% inspection and sampling are related.

13. Acceptance sampling procedure depends upon the objectives and the history of the organization.
Application of methodology is not static. It keep on moving from one level to another. e.g, we might begin with attribute sampling, and as our experience with the supplier increase then move to much less inspection and variable sampling

14. Single-Sampling Plans For Attributes
Definition of a Single-Sampling Plan
A single sampling plan is defined by sample size, n, and the acceptance number c. Say there are N total items in a lot. Choose n of the items at random. If more than c of the items are unacceptable, reject the lot.
N = lot size
n = sample size
c = acceptance number
d = observed number of defectives
The acceptance or rejection of the lot is based on the results from a single sample - thus a single-sampling plan.

15. Example
N = 10000, n = 89, c = 2
From a lot of 10,000, take a sample of size 89
Observe the number of defectives, d
If d < 2, accept
Otherwise, reject

16. Single-Sampling Plans For Attributes
The OC Curve
The operating-characteristic (OC) curve measures the performance of an acceptance-sampling plan.
The OC curve plots the probability of accepting the lot versus the lot fraction defective.
The OC curve shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected.

17. Example
If p = .01, Pa = .9397
If p = .02, Pa = means that 73.66% of lots will be expected to be accepted and 26.34% will be rejected

18. Effect of n and c on OC curves
Fig is the ideal OC curve
Pa = 1.0 until a level of quality that is considered ‘bad’ is reached
But it can never be attained in practice.

19. Effect of n and c on OC curves
OC curve for different values of n
By increasing the sample size, we get closer to the ideal OC curve

20. Effect of n and c on OC curves
OC curve for different values of c
As c is decreased, the OC curve shifts to the left
When c = 0, it is very hard on the vendor

21. Type A or Type B OC curves
In the type B OC curve, it is assumed that the samples come from the large lot or from a stream of lots selected at random.
Binomial distribution is used as p is constant
In the Type A OC curve, isolated lot of finite size is used with size N
The exact probability distribution is ‘hypergeometric’ as p is not constant.
Type A OC curve will lie below Type B

22. Rectifying inspection
Acceptance sampling require corrective action when lots are rejected
100% screening of rejected lots
Defective items are removed, returned to the supplier, or replaced.
Such sampling programs are known as “rectifying inspection programs”
Affects the outgoing quality
Fraction defective =po, average fraction defective = p1

23. Rectifying inspection
Rejected lots
Fraction defective 0
Incoming lots
Inspection
activity
Outgoing lots
Fraction defective p0
Fraction defective p1<p0
Fraction defective p0
Accepted lots

24. Average outgoing quality
AOQ is the quality in the lot resulting from applying rectifying inspection
In a lot of size N, there will be
n items in the sample that, after inspection, contain no defectives (all of the defectives were replaced)
N-n items that, if the lot is rejected, also contain no defectives (the balance of the lot was inspected 100%)
N-n items that, if the lot is accepted, contain p(N-n) defectives

25. Average outgoing quality
AOQ = [Pa p (N-n)]/N
Example
N = 10000, n = 89, c = 2, p = .01
Previously determined that Pa = .9397
AOQ [(.9397)(.01)( )]/10000
AOQ = .0093
Since (N-n)/N , AOQ ~ Pap

26. AOQ curve for rectifying inspection
for n = 89, c = 2
When incoming quality is very good, average fraction defective of outgoing lots is low
When incoming quality is very poor, average fraction defective of outgoing lots is low

28. Average outgoing quality limit
AOQL = .0155
No matter how bad the incoming lots are, the outgoing quality level will never be worse than 1.55% fraction defective

29. Military Standard 105E (ANSI/ASQC Z1.4 ISO 2859)
Description of the Standard
Developed during World War II
MIL STD 105E is the most widely used acceptance-sampling system for attributes
Gone through four revisions since 1950.
MIL STD 105E is a collection of sampling schemes making it an acceptance-sampling system

30. Military Standard 105E (ANSI/ASQC Z1.4 ISO 2859)
Description of the Standard
Three types of sampling are provided for:
Single
Double
Multiple
Provisions for each type of sampling plan include
Normal inspection
Tightened inspection
Reduced inspection

31. Military Standard 105E Description of the Standard
The acceptable quality level (AQL) is a primary focal point of the standard
The AQL is generally specified in the contract or by the authority responsible for sampling.
Different AQLs may be designated for different types of defects.
Defects include critical defects, major defects, and minor defects.
Tables for the standard provided are used to determine the appropriate sampling scheme.

32. Military Standard 105E Description of the Standard Switching Rules
Normal to tightened
Tightened to normal
Normal to reduced
Reduced to normal
Discontinuance of inspection

33. Military Standard 105E Procedure Choose the AQL
Choose the inspection level
Determine the lot size
Find the appropriate sample size code letter from Table 15-4
Determine the appropriate type of sampling plan to use (single, double, multiple)
Enter the appropriate table to find the type of plan to be used.
Determine the corresponding normal and reduced inspection plans to be used when required.

34. Military Standard 105E Example
Suppose a product is submitted in lots of size N = The AQL is 0.65%. Say we wanted to generate normal single-sampling plans.
For lots of size 2000, (and general inspection level II) Table 15-4 indicates that the appropriate sample size code letter is K.
From Table 15-5 for single-sampling plans under normal inspection, the normal inspection plan is n = 125, c = 2.

36. Military Standard 105E Discussion
There are several points about the standard that should be emphasized:
MIL STD 105E is AQL-oriented
The sample sizes selected for use in MIL STD 105E are limited
The sample sizes are related to the lot sizes.
Switching rules from normal to tightened and from tightened to normal are subject to some criticism.
A common abuse of the standard is failure to use the switching rules at all.

37. Switching rules Start “And” conditions accepted 2 out of 5
O Production steady
O 10 consecutive lots
accepted
O Approved by
responsible authority
2 out of 5
consecutive lots
rejected
Tightened
Normal
Reduced
“Or” conditions
O Lot rejected
O Irregular production
O Lot meets neither accept
nor reject criteria
O Other conditions
warrant return to
normal inspection
5 consecutive
lots accepted
10 consecutive
lots remain on
tightened inspection
Discontinue
inspection

38. Dodge-Romig Plans For rectifying inspection
See Table 15-8 for an example for
AOQL = 3%
Indexed by lot size (N) and process average (p)

39. Example
N = 5000, p = .01
Want a single sampling plan (w/rectifying inspection) with AOQL = 3%
Read n = 65, c = 3 from the table
These plans minimize ATI
Pa = at p = .01 (determined as previously)
ATI = 65 + ( )(5000 – 65) = 86.22

40. Example, cont. Also, note that LTPD = 10.3%
This is the point on the OC curve for which Pa = .10
That is, this plan provides that 90% of incoming lots that are as bad as 10.3% defective will be rejected

41. LTPD plans Can also develop a plan for a specified LTPD
Table 14-9 is for LTPD = 1%

42. Example
N = 5000, p = .25%
We want a single sampling plan (w/rectifying inspection) with LTPD of 1%
Find n = 770, c = 4