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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 Presentation
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© 2007 Pearson Education
AQL LTPD
Acceptance Sampling Plans
Supplement ISupplement I
© 2007 Pearson Education
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.
© 2007 Pearson Education
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.
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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.
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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.
© 2007 Pearson Education
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 (.
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
Drawing the OC CurveApplication I.1
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
Drawing the OC CurveApplication I.1
© 2007 Pearson Education
Drawing the OC CurveApplication I.1
© 2007 Pearson Education
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)
© 2007 Pearson Education
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)
© 2007 Pearson Education
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)
© 2007 Pearson Education
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)
© 2007 Pearson Education
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 ))()((
© 2007 Pearson Education
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
© 2007 Pearson Education
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(.
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
AOQ CalculationsApplication I.2
Management has selected the following parameters:
© 2007 Pearson Education
AOQ CalculationsApplication I.2
© 2007 Pearson Education
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
© 2007 Pearson Education
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
© 2007 Pearson Education
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
de
fec
tiv
es
Nu
mb
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of
de
fec
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