Lot-by-Lot Acceptance Sampling for Attributes

Introduction to Statistical Quality Control, 4th E
Chapter 14
Lot-by-Lot Acceptance Sampling
for Attributes

14-1. The Acceptance-Sampling
Problem
• Acceptance sampling is concerned with
inspection and decision making regarding
products.
• Three aspects of sampling are important:
1. Involves random sampling of an entire “lot”
2. Accept and Reject Lots (does not achieve
quality improvement) “Lot sentencing”
3. Audit tool

Problem
• Three approaches to lot sentencing:
1. Accept with no inspection
2. 100% inspection
3. Acceptance sampling

Problem
Why Acceptance Sampling and Not 100%
Inspection?
• Testing is destructive
• Cost of 100% inspection is high
• 100% inspection is not feasible (require
too much time)
• If vendor has excellent quality history

Problem
14-1.1 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

Problem
14-1.2 Types of Sampling Plans
• There are variables sampling plans and attribute
sampling plans (this chapter concentrates on
attributes)
1. Single sampling plan
2. Double-sampling plan
3. Multiple-sampling plan
4. Sequential-sampling

Problem
14-1.3 Lot Formation
• Considerations before inspection:
– Lots should be homogeneous
– Larger lots more preferable than smaller lots
– Lots should be conformable to the materials-
handling systems used in both the vendor and
consumer facilities.

Problem
14-1.4 Random Sampling
• The units selected for inspection should be
chosen at random.
• 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.

14-2. Single-Sampling Plans For
Attributes
14-2.1 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 at least 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.

Attributes
14-2.2 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.

Attributes
14-2.3 Designing a Single-Sampling Plan with a
Specified OC Curve
• Let the probability of acceptance be 1 - α for lots
with fraction defective p1.
• Let the probability of acceptance be β for lots with
fraction defective p2.
• Assume binomial sampling (with type-B OC curves)
is appropriate.

Attributes
14-2.3 Designing a Single-Sampling Plan with a
Specified OC Curve
• The sample size n and acceptance number c are the
solution to
∑ −
−
=
∑ −
−
=−
=
−
=
−
c
0d
dn
2
d
2
c
0d
dn
1
d
1
)p1(p
)!dn(!d
!n
)p1(p
)!dn(!d
!n
1
β
α

Attributes
Example
• Consider constructing a sampling plan for which
– p1 = 0.01
α = 0.05
– p2 = 0.06
β = 0.10
– N = 1000
• Using computer software or a graphical approach (using an
appropriate binomial nomograph) it can be shown that the
necessary values of n and c are 85 and 2, respectively.

14-4. Military Standard 105E
(ANSI/ASQC Z1.4 ISO 2859)
14-4.1 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

• Three types of sampling are provided for:
1. Single
2. Double
3. Multiple
• Provisions for each type of sampling plan
include
1. Normal inspection
2. Tightened inspection
3. Reduced inspection

• 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 provide are used to determine the
appropriate sampling scheme.

• Switching Rules
– Normal to tightened
– Tightened to normal
– Normal to reduced
– Reduced to normal
– Discontinuance of inspection

14-4.2 Procedure
1. Choose the AQL
2. Choose the inspection level
3. Determine the lot size
4. Find the appropriate sample size code letter from Table
14-4
5. Determine the appropriate type of sampling plan to use
(single, double, multiple)
6. Enter the appropriate table to find the type of plan to be
used.
7. Determine the corresponding normal and reduced
inspection plans to be used when required.

Example
• Suppose a product is submitted in lots of size
N = 2000. 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 14-4 indicates the appropriate
sample size code letter is K.
• From Table 14-5 for single-sampling plans
under normal inspection, the normal inspection
plan is n = 125, c = 2.

14-4.3 Discussion
• There are several points about the standard that
should be emphasized:
1. MIL STD 105E is AQL-oriented
2. The sample sizes selected for use in MIL STD 105E
are limited
3. The sample sizes are related to the lot sizes.
4. Switching rules from normal to tightened and from
tightened to normal are subject to some criticism.
5. A common abuse of the standard is failure to use the
switching rules at all.

14-4.3 Discussion
• ANSI/ASQC Z1.4 or ISO 2859 is the civilian standard
counterpart of MIL STD 105E.
• Differences include:
1. Terminology “nonconformity”, “nonconformance”, and
“percent nonconforming” is used.
2. Switching rules were changed slightly to provide an option for
reduced inspection without the use of limit numbers
3. Several tables that show measures of scheme performance were
introduced
4. A section was added describing proper use of individual
sampling plans when extracted from the system.
5. A figure illustrating switching rules was added.

Lot-by-Lot Acceptance Sampling for Attributes

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