Sampling Oc

Sampling Procedures Inspection by Attributes SAFM Consulting Services

Learning Objectives Ø Explain the purpose of acceptance sampling Ø Contrast acceptance sampling and process control Ø Compare and contrast single and multiple sampling plans Ø Determine the average outgoing quality of inspected lots 2

Sampling Plans Ø Acceptance sampling: Form of

inspection applied to lots or batches of items before or after a process, to judge conformance with predetermined standards

Ø Sampling plans: Plans that

specify lot size, sample size, number of samples, and acceptance/rejection criteria Ø Single-sampling Ø Double-sampling Ø Multiple-sampling 3

Acceptance Sampling Ø Acceptance Sampling most useful when ØA large number of items must be processed in a short time ØThe cost consequences of passing defects are low ØDestructive testing is required ØFatigue or boredom leads to inspection errors

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Introduction Ø BS 6001-1 1999 (ISO 2859-1:1999) Sampling  procedures for inspection by attributes. Part 1:  Sampling schemes indexed by acceptance quality  limit (AQL) for lot-by-lot inspection. Ø ISO 2859-2:1985 Sampling procedures for  inspection by attributes. Sampling plans indexed  by limiting quality (LQ) for isolated lot inspection. Ø ISO 2859-0:1995 - Part 0: Introduction to

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ISO 2859 Part 1 Ø Acceptance sampling system for inspection by attributes. Ø It is indexed in terms of acceptance quality level (AQL) Ø Aim - to induce a supplier to maintain a process average at least as good as the specified acceptance AQL, whilst at the same time providing an upper limit for the risk to the consumer of accepting the occasional poor lot. Ø Applicable to end products, raw materials, 6 operations, maintenance operations,

Usage of the Part 1 scheme. Ø Intended to be used for a continuing series of lots, which will allow the application of switching  rules. The rules provide:  - protection to the consumer should a deterioration in quality be detected.  - an incentive to the supplier to reduce  inspection costs should consistently good  quality be achieved. Ø Can also be used for inspection of lots in  isolation

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Terminology Ø Inspection by attributes: inspection where an  item is classified as conforming or nonconforming with respect to a specified requirement or set of requirements. Ø Nonconformity: Non fulfillment of a specified  requirement. Usually classified according to the  degree of seriousness. More serious  nonconformities will usually be assigned a very  small AQL, whilst less serious 8

Terminology (cont) Ø Normal inspection: Use of a sampling plan with  acceptance criteria devised to secure the producer a high probability of acceptance when the process average of the lot is better than the AQL. Used when there is no reason to suspect the process average differs from an acceptable level. Ø Tightened inspection: Use of a sampling plan  with an acceptance criteria that is tighter than that for the corresponding plan for normal inspection. Invoked when the inspection results of consecutive lots indicate that the process 9 average might be poorer than the AQL.

Terminology (cont) Ø Acceptable Quality Limit (AQL): Quality level  that is the worst tolerable process average when a continuing series of lots is submitted for acceptance sampling. AQL does not mean ’desirable level’. ISO 2859 is designed to encourage suppliers to have process averages consistently better than the AQL, otherwise there is a risk of switching to tighter inspection. Ø The designation of an AQL does not imply that the supplier has the right knowingly to supply any  nonconforming items. 10

Terminology (cont) Ø AQL values shall not exceed 10% nonconforming. Ø When the quality level is expressed as number of  nonconformities per 100 items, AQL values up to  1000 nonconformities per 100 items may be used.  Sampling: Ø Sample selection must be drawn from the lot by simple random sampling. Ø When double or random sampling is to be used, each subsequent sample shall be selected 11

Inspection Levels Ø 4 special inspection levels - S1, S2, S3, S4. Ø 3 general inspection levels - I, II, III. Ø Special inspection levels used when sample size  must be kept small and larger sampling risks can  be tolerated. Ø Level II will be used unless another inspection level is specified. Ø Level I is used when less discrimination is required, Level III when greater discrimination is  required.

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steps in the use of the standard be summarized as follows: Ø Decide on the AQL. Ø Decide on the inspection level. Ø Determine the lot size. Ø Enter the table to find sample size code letter. Ø Decide on type of sampling to be used. Ø Enter proper table to find the plan to be used. Ø Begin with normal inspection, follow the switching rules and the rule for stopping the inspection (if needed). 

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Sample size code letters Lot Size

Special Inspection Levels

General Insp. Levels

S1

S2

S3

S4

I

II

III

2-8

A

A

A

A

A

A

B

9-15 16-25 26-50 51-90 91-150 151-280 281-500 501-1200

A A A B B B B C

A A B B B C C C

A B B C C D D E

A B C C D E E F

A B C C D E F G

B C D E F G H J

C D E F G H J K

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Sample size code letters (cont.) Lot Size

Special Inspection Levels

General Insp. Levels

S1

S2

S3

S4

I

II

III

C

D

E

G

H

K

L

3201-10000 C 10001-35000 C 35001-150000 D 150000D 500000 >=500000 D

D D E E E

F F G G H

G H J J K

J K L M N

L M N P Q

M N P Q R

1201-3200

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Single Sampling plans selected segment of normal inspection table

Code Letter Sample Size AQL 0.010

0.015

0.025

0.040

0.065

0.10

0.15

Ac Re

AC Re

AC Re

AC Re

AC Re

AC Re

AC Re

G

32

↓

↓

↓

↓

↓

↓

↓

H

50

↓

↓

↓

↓

↓

↓

↓

J

80

↓

↓

↓

↓

↓

↓

01

K

125

↓

↓

↓

↓

↓

01

↑

L

200

↓

↓

↓

↓

01

↑

↓

M

315

↓

↓

↓

01

↑

↓

12

N

500

↓

↓

01

↑

↓

12

23

P

800

↓

01

↑

↓

12

23

34

Q

1250

01

↑

↓

12

23

34

56

R

2000

↑

↑

12

23

34

56

78 16

Double Sampling Plans normal Inspection Code Letter

Sample

Sample Size Cumulative Samples

AQL 1.5

2.5

4.5

6.5

10

Ac Re AC Re

AC Re

AC Re

AC Re

G

FIRST

20

20

02

03

13

25

36

G

SECOND

20

40

12

34

45

67

910

H

FIRST

32

32

03

13

25

36

59

H

SECOND

32

64

34

45

67

910

1213

J

FIRST

50

50

13

25

36

59

711

J

SECOND

50

100

45

67

910

1213

1819 17

OC Curves Operating Characteristic Curves



Ø OC curve is a graph showing what any particular sampling plan can be expected to do in terms of accepting and rejecting batches. Ø An understanding of the implications of an OC curve helps understand the risks to the manufacturer, consumer, and in deciding inspection levels and batch sizes. Ø Each possible plan has its own OC curve. Ø Horizontal scale - shows the percentage defective. Ø Vertical scale - shows the percentage of batches that may be expected to be accepted if18

OC Curve Ø Operating Characteristic (OC) Curve: Probability curve that shows the probabilities of accepting lots with various fractions defective.

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Probability of accepting lot

Typical OC Curve 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

3%

0

.05

.10

.15

.20

Lot quality (fraction defective)

.25

20

Probability of accepting lot

Decision Criteria 1.00

Ideal

Not very discriminating

“Good”

“Bad”

0

Lot quality (fraction defective) 21

Sampling Terms Ø Acceptance quality level (AQL): the

percentage of defects at which consumers are willing to accept lots as “good”

Ø Lot tolerance percent defective (LTPD): the upper limit on the

percentage of defects that a consumer is willing to accept

Ø Consumer’s risk: the probability that a lot contained defectives exceeding the LTPD will be accepted

Ø Producer’s risk: the probability that a lot containing the acceptable quality level will be rejected

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Probability of accepting lot

Consumer’s and Producer’s Risk 1 α = .10 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 “Good” Indifferent 0.1 β = .10 0 0 .05 .10 AQL

LTPD “Bad”

.15

.20

Lot quality (fraction defective)

.25

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Probability of acceptance

OC Curve e.g. (n = 10, c = 1) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

.9139 .7361 .5443 .3758 .2440 .1493 .0860

0

.10

.20

.30

.40

.50

Fraction defective in lot 24

Average Quality Ø Average outgoing quality (AOQ): Average of inspected lots (100%) and uninspected lots  N − n AOQ = Pac × p   N 

Pac = Probability of accepting lot p = Fraction defective N = Lot size n = Sample size

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AOQ ( Fraction defective out )

Example : AOQ 0 0 .0 5 0.0 9 0 .1 0.0 8 0 .1 5 0.0 7 0.0 6 0 .2 0.0 50 .2 5 0.0 4 0 .3 0.0 3 0 .3 5 0.0 2 0.0 1 0 .4

0 0 .0 4 6 0 .0 7 4 0 .0 8 2 0 .0 7 5 0 .0 6 1 0 .0 4 5 0 .0 3 0 .0 1 9

Approximate AOQL = . 082

0 0

0.0 5

0.1

0.1 5

0.2

0.2 5

0.3

0.3 5

0.4

Incoming fraction defective

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OC Curve

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OC Curve example Code Letter M AQL 1.5 Sample size - 315 Accept: 10, Reject: 11 What happens if a batch with 3% defectives is submitted ? Ø Find 3% on the X axis, and follow a vertical line up until it meets the 1.5% curve. Ø Take a horizontal line across to the Y axis, and read off the value of 65%. Ø I.e., 65% of batches would be accepted and 35% rejected. Ø Ø Ø Ø Ø

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More OC curve examples Pa 99 95 90 75 50 25 10 5 1

AQL for Sample Plan M 1.0 1.5 2.5 4.0 0.929 1.53 2.40 4.05 1.27 1.97 2.96 4.78 1.48 2.24 3.29 5.20 1.89 2.74 3.90 5.95 2.43 3.38 4.65 6.87 3.06 4.11 5.49 7.87 3.71 4.85 6.33 8.84 4.13 5.33 6.86 9.46 5.01 6.29 7.93 10.7

AQL for Sample Plan J 1.0 1.5 2.5 4.0 0.55 1.04 2.28 3.73 1.03 1.73 3.32 5.07 1.39 2.20 3.99 5.91 2.16 3.18 5.30 7.50 3.33 4.57 7.06 9.55 4.84 6.30 9.14 11.9 6.52 8.16 11.3 14.3 7.66 9.41 12.7 15.8 10.1 12.0 15.6 18.9

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More examples Ø Pinholes Ø General inspection level 1, AQL 1.5%. Ø Minimum Sample size code lettre L (200 Sample) Ø If a batch with 3% pinholes is submitted, there is a 75% chance of them being accepted.

Ø Container Ø Agreed AQL of 0.65%, general Inspection level I. Ø Sample code L (200 Sample). Ø If a batch with 5% leakers is submitted, there is a 99% chance of them being

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Setting an Inspection Level Ø Select the required AQL (as an average). Ø Decide what quality should have a high chance of rejection. Ø Choose the appropriate sampling plan by inspecting the OC curves.

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Setting an Inspection Level Example Ø An AQL of 1.5% defective items has been chosen. Ø It is desired to have at least an 80% chance of rejecting a 6%  defective batch under normal inspection. Ø Code letters A->J fail to meet the requirement. Ø Code letter K almost meets it, codes L-P more than meet it. Ø Decide the batch size. If batch size is 1000, check sample size code letters table - we can use general inspection level III. Ø The sampling plan would therefore be AQL=1.5%, General  Inspection Level=III. 

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Switching Rules Ø Normal to tightened: Shall be implemented as soon as two out  of five (or fewer than five) consecutive lots have been  non-acceptable on original inspection. Ø Tightened to normal: Shall be reinstated when five consecutive  lots have been considered acceptable on original inspection. Ø Normal to reduced: implemented when the switching score is at  least 30, production is at a steady state, and reduced inspection is  considered desirable by the responsible authority. Ø Discontinuation of inspection: is implemented when the  number of lots not accepted in a sequence of 33 consecutive lots on

Switching scores Single Sampling Plans Ø Used when deciding whether to switch to reduced inspection. Ø Set switching score to 0 at start of normal inspection. Ø Score updated following the inspection of each lot. Ø When the acceptance number is >= 2, add 3 to the switching  score if the lot would have been accepted if the AQL had been  one step tighter; otherwise reset score to 0. Ø When the acceptance number is 0 or 1, add 2 to the switching  score if the lot is accepted; otherwise reset score to 0. Ø As in the previous slide, when the switching score is at least 30  and production is at a steady state, switch to reduced 34 

Important points Ø Setting an AQL does not guarantee to the customer that batches of a worse quality will not be accepted. Ø If the average quality of batches being submitted are a little worse than the AQL, a number of batches will probably be accepted before a switch to tightened inspection is called for. Ø In general, the customer gets a quality which is, on average, better than the AQL, since the tables are arranged to provide an economic incentive -the manufacturer cannot afford to have more than a small proportion of batches rejected, and so will improve the quality if this proportion is exceeded. 35