acceptance sampling plans

© 2007 Pearson Education

AQL LTPD

Acceptance Sampling Plans

Supplement ISupplement 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.

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.


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.


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


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)

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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.


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


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Constructing an OC CurveExample I.1


ProbabilityProportion of c or lessdefective defects


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


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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



ProbabilityProportion of c or lessDefective defects


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


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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





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


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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





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


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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





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


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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)


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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



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


Pro

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ceConstructing an OC Curve

Example I.1


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.


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 (.


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)


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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


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)


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= 0.126



Drawing the OC CurveApplication I.1


Finding (probability of rejecting AQL quality:

p = .03

np = 5.79 Pa = 0.965

= 1 – .965 = 0.035


Cumulative Poisson Probabilities


Finding (probability of accepting LTPD quality:

p = .08

np = 15.44

Pa = 0.10

= Pa = 0.10


Cumulative Poisson Probabilities


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)


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Producer’s Consumer’sRisk Risk

n (p = AQL) (p = LTPD)


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)


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60 0.122 0.126



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)


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60 0.122 0.12680 0.191 0.048



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)


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60 0.122 0.12680 0.191 0.048

100 0.264 0.017



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)


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60 0.122 0.12680 0.191 0.048

100 0.264 0.017120 0.332 0.006



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)


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n = 60, c = 1

n = 80, c = 1

n = 100, c = 1

n = 120, c = 1

Operating Characteristic Curves (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)


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c (p = AQL) (p = LTPD)

Understanding Changes in the OC Curve (with n = 60)


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)


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1 0.122 0.126



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)


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1 0.122 0.1262 0.023 0.303



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)


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1 0.122 0.1262 0.023 0.3033 0.003 0.515



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)


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1 0.122 0.1262 0.023 0.3033 0.003 0.5154 0.000 0.726



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)


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n = 60, c = 1n = 60, c = 2

n = 60, c = 3

n = 60, c = 4

Operating Characteristic Curves (with n = 60)


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 ))()((


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




(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(.




(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




(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



1.6 –

1.2 –

0.8 –

0.4 –

0 –| | | | | | | |1 2 3 4 5 6 7 8

Defectives in lot (percent)

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(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


AOQL1.6 –

1.2 –

0.8 –

0.4 –

0 –| | | | | | | |1 2 3 4 5 6 7 8

Defectives in lot (percent)

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AOQL = Average Outgoing Quality

Limit


AOQ CalculationsApplication I.2

Management has selected the following parameters:


AOQ CalculationsApplication I.2


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)

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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


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

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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

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acceptance sampling plans

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