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University of Hail
College of Engineering
ME 418 – Quality in Manufacturing
ISE 320 - Quality Control and Industrial Statistics
CHAPTER 06
ACCEPTANCE SAMPLING PLANS
Professor Mohamed Aichouni
http://faculty.uoh.edu.sa/m.aichouni/me418-quality
http://faculty.uoh.edu.sa/m.aichouni/ise230-quality/
Acceptance Sampling
• Acceptance sampling is a method used to accept or reject
product based on a random sample of the product.
• The purpose of acceptance sampling is to sentence lots (accept
or reject) rather than to estimate the quality of a lot.
• Acceptance sampling plans do not improve quality. The nature
of sampling is such that acceptance sampling will accept some
lots and reject others even though they are of the same quality.
• The most effective use of acceptance sampling is as an auditing
tool to help ensure that the output of a process meets
requirements.
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Acceptance Sampling
Take a Sample
Size ‘n’,
Accept if ‘c’ or less.
Producer
Consumer
Risk is a ‘bad’ lot
will be accepted.
Risk is a ‘good’ lot will be
rejected and sent back.
Lot Received
from Supplier
Acceptance
Sampling Flow
Chart
Random Sample
of Material
Selected
Items Inspected
and Analyzed
Results
Compared with
Acceptance
Criteria
2
Define and
Analyze
Sampling Plan
Accept the Lot
Reject the Lot
Send Lot to
Inventory or
Production
Decide
Disposition,
Return to
Supplier
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Acceptance Sampling
N : sample size
sampling
Inspect the
sample
‫عدد القبول‬c :
Acceptance number
b
Accept the lot
5
Reject the lot
N
Terminology
• As mentioned acceptance sampling can reject “good”
lots and accept “bad” lots. More formally:
• Producers risk refers to the probability of rejecting a
good lot.
• AQL (Acceptable Quality Level) - the numerical
definition of a good lot; associated with Producer`s risk.
• The ANSI/ASQC standard describes AQL as “the
maximum percentage or proportion of nonconforming
items or number of nonconformities in a batch that can
be considered satisfactory as a process average”
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Terminology
• Consumers Risk refers to the probability of accepting a
bad lot where:
• LTPD (Lot Tolerance Percent Defective) - the numerical
definition of a bad or poor lot .
• described by the ANSI/ASQC standard as “the
percentage or proportion of nonconforming items or
noncomformities in a batch for which the customer
wishes
i h the
th probability
b bilit off acceptance
t
to
t be
b a specified
ifi d low
l
value.
• Limiting Quality Level - Numerical definition of a ‘poor’
lot, associated with the consumer’s risk.
Types of Sampling Plans
Classification of acceptance sampling plans
Acceptance sampling plans
By attributes
By variables
Item sampling
Si l
Single
Sampling Plans by
Attributes:
• Single sampling plan by
attributes
•
Double sampling plan by
attributes
Double, multiple
Sequential
8
4
Bulk sampling
• Sequential sampling plan
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Single Sampling
Operating Characteristic (OC) curve
OC Curve
1.2
Producers Risk
1
P r o b a b i l i tty o f A c c e p ta n c e
The Operating
Ch
Characteristic
t i ti Curve
C
is
i
typically used to
represent the four
parameters (Producers
Risk, Consumers Risk,
AQL and LTPD) of the
sampling plan
plan.
P on the x axis represents
the percent defective in
the lot.
0.8
0.6
Consumers Risk
0.4
0.2
0
AQL
5
P
LTPD
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Ideal Operating Characteristics Curve
(100% Inspection)
P(Accept Whole
Shipment)
P off Acceptance
100%
0%
Always
Accept
0
1
2
Always Reject
3 4
5
6
7 8
Proportion of non Conforming
Actual OC Curves
• Are determined by sample size [n] and
acceptance
p
number [[c].
]
– Accept the lot if ‘c’ or fewer
nonconforming are obtained, reject if
more.
• OK to assume Binomial distribution (if lot
size is 10x sample size).
• Calculate P accept for range of incoming
p levels.
l
l
6
9 10
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Actual OC Curves
P(Accept Whole Shipment)
P < 100 %
Risk to accept bad lot
and/or reject a good lot
100%
Accept the
lot
0%
0
1
Reject the lot
2 3 4
Cut-Off
5
6
7
8
9
10
Proportion of non Conforming
13
Actual OC Curves and the Producer`s Risk and
Consumer`s Risk
P of acceptance
100
95
 = 0.05
AQL
75
50
25
10
= 0.10 0
LTPD for0
1
2
Good
lot
14
7
3
4
5
6
AQL
7
8
LTPD
Indifference zone
Proportion of non
conforming
Bad
lot
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Sample problem
• Given a lot size of N=2000, a
sample size n=50, and an
acceptance number c=2.
• Calculate the OC curve for this
plan.
()
px (1-p)(n-x)
1.2
x = 0,1,..n
1
Probabilityy of accepting
p g is
obtaining c=2 or less nonconforming items in samples
of size n=50.
P a c c e p ta n c e
b(x) = n
x
0.8
0.6
0.4
0.2
0
Vary p from 0 to 0.15
(what if p = ….)
0
0.03
0.06
0.09
0.12
(p)
OC for possible sampling plans
Operating Curve
Probabiility of acceptance: P(.)
100%
90%
80%
c=0
70%
c=1
60%
c=2
50%
c=3
40%
30%
20%
10%
0%
0.00% 0.02% 0.04% 0.06% 0.08% 0.10% 0.12% 0.14%
Expected proportion of defectives
16
8
0.15
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Vary n and c
1.2
P accept
1
n=50 c=2
0.8
n=80 c=2
0.6
n=50 c=1
0.4
n=100 c=2
0.2
0
0
0.05
0.1
0.15
p
Double Sampling
• In an effort to reduce the amount of inspection double
(or multiple) sampling is used. Whether or not the
sampling effort will be reduced depends on the
defective proportions of incoming lots. Typically, four
parameters are specified:
n1 =
c1 =
n2 =
c2 =
9
number of units in the first sample
acceptance number for the first sample
number of units in the second sample
acceptance number for both samples
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Procedure
• A double sampling plan proceeds as follows:
A random sample off size n1 is drawn from
f
the lot.
If the number of defective units (say d1 )  c1 the lot
is accepted.
If d1 c2 the lot is rejected.
If neither of these conditions are satisfied a second
p of size n2 is drawn from the lot.
sample
If the number of defectives in the combined samples
(d1 + d2) > c2 the lot is rejected. If not the lot is
accepted.
Double Sampling Plan
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Double Sampling Plan
• Application of double sampling requires that a first sample of
size n1 is taken at random from the (large) lot. The number of
defectives is then counted and compared to the first sample's
acceptance number a1 and rejection number r1. Denote the
number of defectives in sample 1 by d1 and in sample 2 by d2,
then:
– If d1<= a1, the lot is accepted.
If d1 >= r1, the lot is rejected.
If a1 < d1 < r1, a second sample is taken.
• If a second sample of size n2 is taken, the number of defectives,
d2 is counted.
d2,
counted The total number of defectives is D2 = d1 + d2.
d2
Now this is compared to the acceptance number a2 and the
rejection number r2 of sample 2. In double sampling, r2 = a2 + 1
to ensure a decision on the sample.
– If D2 <= a2, the lot is accepted.
If D2 >= r2, the lot is rejected.
–
Designing Acceptance Plans
• This should be performed on agreement between the
producer and the consumer.
• Each party work to reduce the risk, by varying n and c
to obtain different OC curves.
• Single and multiple sampling plans can be used.
• Refer to standard published Standards (MIL
(MIL-STDSTD
105D, Dodge Romig Tables).
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Acceptance Sampling Plans on Minitab
• Minitab perform all the
necessary calculations to
d i acceptance
design
t
sampling plans.
• http://blog.minitab.com/bl
og/applying-statistics-inquality-projects/attributeacceptance-samplingfor-an-acceptancenumber-of-0
Examples will be worked out next
sessions
•
International Standards on Sampling Plans
ISO 2859
- 0:1995
ISO 2859
- 1:1999
ISO 2859
ISO 2859
- 1:1999/Cor 1:2001
- 2:1985
ISO 2859
- 3:1991
ISO 2859
- 4:2002
ISO 3951:1989
ISO 8422:1991
ISO 8422:1991/Cor 1:1993
ISO 8423:1991
ISO 8423:1991/Cor 1:1993
ISO/TR 8550:1994
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Sampling procedures for inspection by
attributes
Introduction to the ISO 2859 attribute sampling system
Sampling procedures for inspection by attri
Sampling schemes indexed by acceptance quality limit
(AQL) for lot
- by - lot inspection
butes
Sampling procedures for inspection by attributes
Sampling plans indexed by limiting quality (LQ) for isolated
nspection
-- Part 0:
-- Part 1:
-- Part 2:
lot i
Sampling procedures for inspection by attributes
Skip - lot sampling procedures
Sampling procedures for inspection by attributes
Procedures for assessment of declared quality levels
-- Part 3:
-- Part 4:
Sampling procedures and charts for inspection by variables
for percent nonconforming
Sequential sampling plans for inspection by attributes
Sequential sampling plans for inspection by variables for
percent nonconforming (known stan
dard deviation)
Guide for the selection of an acceptance sampling system,
scheme or plan for inspection of discrete items in lots
ISO 10725:2000
Acceptance sampling plans and procedures for the
inspection of bulk materials
ISO 11648
- 1:2003
Statistical aspects of sampling from bulk materials
1: General principles
-- Part
ISO 11648
- 2:2001
Statistical aspects of sampling from bulk materials
2: Sampling of particulate materials
-- Part
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Conclusion
"Quality control truly
begins
g
and ends
with education",
K. Ishikawa (1990).
Lecture Finished
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Yes
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(See you next lecture)
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