acceptance sampling plans
DESCRIPTION
a. . AQL LTPD. Acceptance Sampling Plans. Supplement I. Acceptance Sampling. Acceptance sampling is a statistical process for determining whether to accept or reject a lot of products by testing a random sample of parts taken from the lot. - PowerPoint PPT PresentationTRANSCRIPT
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AQL LTPD
Acceptance Sampling Plans
Supplement ISupplement I
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Acceptance Sampling
Acceptance sampling is a statistical process for determining whether to accept or reject a lot of products by testing a random sample of parts taken from the lot.
An acceptance sampling plan is specified by n and c, where,n = the sample size, andc = the critical number of defectives in the
sample up to which the lot will be accepted.
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OC Curve
Let Pd = Probability of defectives in the lot
Pa = Probability of accepting the lot P(x< c),
where x = number of defectives in the sample
OC Curve is a graph with values of Pd on the x-axis and the corresponding values of Pa in the y-axis.
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Computing Pa for a given sampling plan and Pd value
Compute nPd
Use Poisson Probability Table and lookup the value of Pa for the value of c
Example: Given a sampling plan of n = 60 and c = 2, if Pd = 1%, nPd = 60(.01) = .6
np 0 1 2
.40 .670 .938 .992
.45 .638 .925 .989
.50 .607 .910 .986
.55 .577 .894 .982
.60 .549 .878 .977
.65 .522 .861 .972
Pa = .977
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OC Curve
1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
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Constructing OC Curve
The Noise King Muffler Shop, a high-volume installer of replacement exhaust muffler systems, just received a shipment of 1,000 mufflers. The sampling plan for inspecting these mufflers calls for a sample size n=60 and an acceptance number c=1. Construct the OC curve for this sampling plan.
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ProbabilityProportion of c or less defective defects
(p) np (Pa) Comments
n = 60c = 1
1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
Constructing an OC CurveExample I.1
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ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
n = 60c = 1
1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
np 0 1 2
.05 .951 .999 1.000
.10 .905 .995 1.000
.15 .861 .990 .999
.20 .819 .982 .999
.25 .779 .974 .998
.30 .741 .963 .996
.35 .705 .951 .994
.40 .670 .938 .992
.45 .638 .925 .989
.50 .607 .910 .986
.55 .577 .894 .982
.60 .549 .878 .977
.65 .522 .861 .972
Constructing an OC CurveExample I.1
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ProbabilityProportion of c or lessDefective defects
(p) np (Pa) Comments
0.01 0.6
n = 60c = 11.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
np 0 1 2
.05 .951 .999 1.000
.10 .905 .995 1.000
.15 .861 .990 .999
.20 .819 .982 .999
.25 .779 .974 .998
.30 .741 .963 .996
.35 .705 .951 .994
.40 .670 .938 .992
.45 .638 .925 .989
.50 .607 .910 .986
.55 .577 .894 .982
.60 .549 .878 .977
.65 .522 .861 .972
Constructing an OC CurveExample I.1
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ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
0.01 0.6 0.878
n = 60c = 1
1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
np 0 1 2
.05 .951 .999 1.000
.10 .905 .995 1.000
.15 .861 .990 .999
.20 .819 .982 .999
.25 .779 .974 .998
.30 .741 .963 .996
.35 .705 .951 .994
.40 .670 .938 .992
.45 .638 .925 .989
.50 .607 .910 .986
.55 .577 .894 .982
.60 .549 .878 .977
.65 .522 .861 .972
Constructing an OC CurveExample I.1
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ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
0.01 0.6 0.878
n = 60c = 11.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
np 0 1 2
.05 .951 .999 1.000
.10 .905 .995 1.000
.15 .861 .990 .999
.20 .819 .982 .999
.25 .779 .974 .998
.30 .741 .963 .996
.35 .705 .951 .994
.40 .670 .938 .992
.45 .638 .925 .989
.50 .607 .910 .986
.55 .577 .894 .982
.60 .549 .878 .977
.65 .522 .861 .972
Constructing an OC CurveExample I.1
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ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
0.01 0.6 0.878
n = 60c = 11.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
Constructing an OC CurveExample I.1
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 –
0.663
| | | | | | | | | |1 2 3 4 5 6 7 8 9 10
0.308
0.199
0.048
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
0.01 0.6 0.8780.02 1.2 0.6630.03 1.8 0.4630.04 2.4 0.3080.05 3.0 0.1990.06 3.6 0.1260.07 4.2 0.0780.08 4.8 0.0480.09 5.4 0.0290.10 6.0 0.017
n = 60c = 1
Constructing an OC CurveExample I.1
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
0.878
0.663
0.463
0.308
0.1990.126 0.078
0.048 0.0290.017
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ceConstructing an OC Curve
Example I.1
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AQL and LTPD
Acceptable Quality Level (AQL)The poorest level of quality that is acceptable to
the customer. It is specified as a percentage of defectives in the lot.
Lot Tolerance Percent Defective (LTPD)The quality level at which the lot is considered
bad. It is specified as a percentage of defectives in the lot.
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Risks
Producer’s riskThe probability of rejecting a good lot (i.e. Pd =
AQL) based on the acceptance sampling plan. This is also known as Type I error ().
Consumer’s riskThe probability of accepting a bad lot (i.e. Pd =
LTPD) based on the acceptance sampling plan. This also known as Type II error (.
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 –
0.663
| | | | | | | | | |1 2 3 4 5 6 7 8 9 10
0.308
0.199
0.048
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
ProbabilityProportion of c or lessdefective defects
(p) np (Pa) Comments
0.01 (AQL) 0.6 0.878 = 1.000 – 0.878 = 0.1220.02 1.2 0.6630.03 1.8 0.4630.04 2.4 0.3080.05 3.0 0.1990.06 (LTPD) 3.6 0.126 = 0.1260.07 4.2 0.0780.08 4.8 0.0480.09 5.4 0.0290.10 6.0 0.017
n = 60c = 1
Consumer’s and Producer’s risks - Example I.1
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
0.878
0.663
0.463
0.308
0.1990.126 0.078
0.048 0.0290.017
= 0.122
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
= 0.126
Constructing an OC CurveExample I.1
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Drawing the OC CurveApplication I.1
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Finding (probability of rejecting AQL quality:
p = .03
np = 5.79 Pa = 0.965
= 1 – .965 = 0.035
Drawing the OC CurveApplication I.1
Cumulative Poisson Probabilities
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Finding (probability of accepting LTPD quality:
p = .08
np = 15.44
Pa = 0.10
= Pa = 0.10
Drawing the OC CurveApplication I.1
Cumulative Poisson Probabilities
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Drawing the OC CurveApplication I.1
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Drawing the OC CurveApplication I.1
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
Producer’s Consumer’sRisk Risk
n (p = AQL) (p = LTPD)
60 0.122 0.12680 0.191 0.048
100 0.264 0.017120 0.332 0.006
Understanding Changes in the OC Curve (with c = 1)
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
n = 60, c = 1
n = 80, c = 1
n = 100, c = 1
n = 120, c = 1
Operating Characteristic Curves (with c = 1)
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
Producer’s Consumer’sRisk Risk
c (p = AQL) (p = LTPD)
1 0.122 0.1262 0.023 0.3033 0.003 0.5154 0.000 0.726
Understanding Changes in the OC Curve (with n = 60)
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1.0 –
0.9 –
0.8 –
0.7 –
0.6 –
0.5 –
0.4 –
0.3 –
0.2 –
0.1 –
0.0 – | | | | | | | | | |1 2 3 4 5 6 7 8 9 10
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
n = 60, c = 1n = 60, c = 2
n = 60, c = 3
n = 60, c = 4
Operating Characteristic Curves (with n = 60)
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Average Outgoing Quality
AOQ =
where,Pd = probability of defectives in the lot
Pa = probability of accepting the lot
N = Lot sizen = sample size
N
nNPP ad ))()((
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Average Outgoing QualityExample I.2
Noise King example with rectified inspection for its single-sampling plan with
n = 110, c = 3, N = 1000
Proportion ProbabilityDefective of Acceptance
(p) np (Pa)
0.01 1.10 0.9740.02 2.20 0.8190.03 3.30 0.581 = (0.603 + 0.558)/20.04 4.40 0.3590.05 5.50 0.202 = (0.213 + 0.191)/20.06 6.60 0.1050.07 7.70 0.052 = (0.055 + 0.048)/20.08 8.80 0.024
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Average Outgoing QualityExample I.2
Proportion ProbabilityDefective of Acceptance
(p) np (Pa) AOQ
0.01 1.10 0.9740.02 2.20 0.8190.03 3.30 0.5810.04 4.40 0.3590.05 5.50 0.2020.06 6.60 0.1050.07 7.70 0.0520.08 8.80 0.024
For p = 0.01, Pa = 0.974
AOQ =
= 0.0087
1000
)1101000)(974.0)(01(.
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Average Outgoing QualityExample I.2
Proportion ProbabilityDefective of Acceptance
(p) np (Pa) AOQ
0.01 1.10 0.974 0.00870.02 2.20 0.8190.03 3.30 0.5810.04 4.40 0.3590.05 5.50 0.2020.06 6.60 0.1050.07 7.70 0.0520.08 8.80 0.024
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Average Outgoing QualityExample I.2
Proportion ProbabilityDefective of Acceptance
(p) np (Pa) AOQ
0.01 1.10 0.974 0.00870.02 2.20 0.819 0.01460.03 3.30 0.581 0.01550.04 4.40 0.359 0.01280.05 5.50 0.202 0.00900.06 6.60 0.105 0.00560.07 7.70 0.052 0.00320.08 8.80 0.024 0.0017
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Average Outgoing QualityExample I.2
1.6 –
1.2 –
0.8 –
0.4 –
0 –| | | | | | | |1 2 3 4 5 6 7 8
Defectives in lot (percent)
Ave
rag
e o
utg
oin
g q
ual
ity
(per
cen
t)
Proportion ProbabilityDefective of Acceptance
(p) np (Pa) AOQ
0.01 1.10 0.974 0.00870.02 2.20 0.819 0.01460.03 3.30 0.581 0.01550.04 4.40 0.359 0.01280.05 5.50 0.202 0.00900.06 6.60 0.105 0.00560.07 7.70 0.052 0.00320.08 8.80 0.024 0.0017
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AOQL1.6 –
1.2 –
0.8 –
0.4 –
0 –| | | | | | | |1 2 3 4 5 6 7 8
Defectives in lot (percent)
Ave
rag
e o
utg
oin
g q
ual
ity
(per
cen
t)
Average Outgoing QualityExample I.2
AOQL = Average Outgoing Quality
Limit
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AOQ CalculationsApplication I.2
Management has selected the following parameters:
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AOQ CalculationsApplication I.2
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Solved Problem
1.0 —
0.9 —
0.8 —
0.7 —
0.6 —
0.5 —
0.4 —
0.3 —
0.2 —
0.1 —
0 — | | | | | | | | | |
1 2 3 4 5 6 7 8 9 10
Proportion defective (hundredths)(p)
Pro
bab
ilit
y o
f ac
cep
tan
ce (
Pa)
(AQL) (LTPD)
1.000 0.996
0.9510.810
0.587
0.363
0.194
0.0920.039
0.015 = 0.092
= 0.049
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Sequential Sampling Chart
8 8 –
7 7 –
6 6 –
5 5 –
4 4 –
3 3 –
2 2 –
1 1 –
0 0 –
Reject
Continue sampling
Accept
Cumulative sample sizeCumulative sample size
| | | | | | |1010 2020 3030 4040 5050 6060 7070
Nu
mb
er
of
de
fec
tiv
es
Nu
mb
er
of
de
fec
tiv
es
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Sequential Sampling Chart
8 8 –
7 7 –
6 6 –
5 5 –
4 4 –
3 3 –
2 2 –
1 1 –
0 0 –
RejectDecision to reject
Continue sampling
Accept
Cumulative sample sizeCumulative sample size
| | | | | | |1010 2020 3030 4040 5050 6060 7070
Nu
mb
er
of
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