© 2007 Pearson Education AQL LTPD Acceptance Sampling Plans Supplement I

© 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

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Nu

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© 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

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of

de

fec

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Nu

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© 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

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y o

f P

rob

abil

ity

of

acce

pta

nce

acce

pta

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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

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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