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© 2007 Pearson Education
AQL LTPD
Acceptance Sampling Plans
Supplement ISupplement I
© 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
er
of
de
fec
tiv
es
© 2007 Pearson Education
Operating Characteristic Curve
Ideal OC curveIdeal OC curve
Pro
bab
ilit
y o
f P
rob
abil
ity
of
acce
pta
nce
acce
pta
nce
Proportion defectiveProportion defective
© 2007 Pearson Education
Operating Characteristic Curve
Ideal OC curveIdeal OC curve
Typical OC curveTypical OC curve
Pro
bab
ilit
y o
f P
rob
abil
ity
of
acce
pta
nce
acce
pta
nce
Proportion defectiveProportion defective
© 2007 Pearson Education
Operating Characteristic Curve
1.01.0
Ideal OC curveIdeal OC curve
Typical OC curveTypical OC curve
AQL LTPDAQL LTPD
Pro
bab
ilit
y o
f P
rob
abil
ity
of
acce
pta
nce
acce
pta
nce
Proportion defectiveProportion defective
© 2007 Pearson Education
Constructing an OC CurveExample I.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
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ce
© 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
(AQL) (LTPD)
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
(AQL) (LTPD)
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 (AQL) 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
(AQL) (LTPD)
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 (AQL) 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
(AQL) (LTPD)
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 (AQL) 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
(AQL) (LTPD)
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 (AQL) 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
(AQL) (LTPD)
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 (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
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
(AQL) (LTPD)
Proportion defective (hundredths)
Pro
bab
ilit
y o
f ac
cep
tan
ceConstructing an OC Curve
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
Understanding Changes in the OC Curve (with c = 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
(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)
© 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 0.126
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
Producer’s Consumer’sRisk Risk
n (p = AQL) (p = LTPD)
60 0.122 0.12680 0.191 0.048
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
Producer’s Consumer’sRisk Risk
n (p = AQL) (p = LTPD)
60 0.122 0.12680 0.191 0.048
100 0.264 0.017
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
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)
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
Producer’s Consumer’sRisk Risk
c (p = AQL) (p = LTPD)
1 0.122 0.126
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
Producer’s Consumer’sRisk Risk
c (p = AQL) (p = LTPD)
1 0.122 0.1262 0.023 0.303
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
Producer’s Consumer’sRisk Risk
c (p = AQL) (p = LTPD)
1 0.122 0.1262 0.023 0.3033 0.003 0.515
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
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
Acceptance Sampling Plan Data
AQL Based LTPD Based
0 0.0509 5 2.2996 381 0.3552 36 3.8875 652 0.8112 81 5.3217 893 1.3675 137 6.6697 1114 1.9680 197 7.9894 1335 2.6256 263 9.2647 1546 3.2838 328 10.5139 1757 3.9794 398 11.7726 1968 4.6936 469 12.9903 2179 5.4237 542 14.2042 237
10 6.1635 616 15.4036 257
Acceptance Expected Sample Expected SampleNumber Defectives Size Defectives Size
© 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)
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
For p = 0.01: 0.01(0.974)(1000 – 110)/1000 = 0.0087For p = 0.02: 0.02(0.819)(1000 – 110)/1000 = 0.0146For p = 0.03: 0.03(0.581)(1000 – 110)/1000 = 0.0155For p = 0.04: 0.04(0.359)(1000 – 110)/1000 = 0.0128For p = 0.05: 0.05(0.202)(1000 – 110)/1000 = 0.0090For p = 0.06: 0.06(0.105)(1000 – 110)/1000 = 0.0056For p = 0.07: 0.07(0.052)(1000 – 110)/1000 = 0.0032For p = 0.08: 0.08(0.024)(1000 – 110)/1000 = 0.0017
Noise King uses rectified inspection for its single-sampling plan with
n = 110, c = 3, N = 1000
Estimate the probabilities of
acceptance for portion defective values from
0.01 to 0.08 in steps of 0.01
© 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)
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
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)
Average Outgoing QualityExample I.2
© 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
© 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