12 Sampling Tech 4

Acceptance Sampling System

by Attributes

Topic Outcome:

At the end of this topic, students will be able to:

Use sampling system for lot-by-lot sampling (ANSI/ASQ Z1.4 1993 & Dodge-Romig sampling Tables)

Use sampling system for isolated lot sampling

(ANSI/ASQ Standard Q3 – 1998)

Topic Outline:

• Lot-by-lot Acceptance Sampling Plans:

1)AQL Sampling Plans (ANSI/ASQ Z1.4 1993)

- An Introduction.

- Inspection Level

- Types of Sampling Plans.

- Switching Procedures

- AOQL factors

- LQ

- ASN

2) Dodge-Romig Sampling Tables

Topic Outline:

• Isolated lot Acceptance Sampling Plan1) ANSI/ASQ Standard Q3 - 1998

3) Chain Sampling Inspection Plan (ChSP-1)

4)Sequential Sampling Plan

5) Skip-lot Sampling Plan

6) ANSI/ASQ SI - 1996

Refresh:

• Sampling plan, scheme, & system

Lot-by-lot Acceptance Sampling Plans

AQL Sampling Plans – ANSI/ASQ Z1.4 – 1993LQ Sampling PlansAOQL Sampling Plans

Dodge-Romig Sampling Tables

ANSI/ASQ Z1.4 – 1993

An Introduction ANSI/ASQ Z1.4 – 1993 ASQ = American Society for Quality A sampling plan for lot-by-lot inspection by attributes for

use by US government & industries1942 (@Bell Telephone Lab) – JAN-STD-105Since that time, there have been 5 revisions (MIL-

STD-105E).1973 – adopted by ISO (International Organization for

Standardization) and designated International Standard ISO/DIS-2859.

1993 – modifications were made by ASQ under designation ANSI/ASQ Z1.4 all tables and procedures remain unchanged, but with 3 basic changes:

ANSI/ASQ Z1.4 – 1993

1) Nonconformity (tak kesesuaian) and nonconforming (tak sesuai) unit are substituted for the words defect (cacat) and defective (kecacatan).

2) The switching rule that used a limit number for one of the reduced inspection criteria is an option.

3) Additional tables for AOQL,LQ,ASN,and OC curves are added. These tables reflect scheme performance, which is the combination of switching among normal, tightened, and reduced sampling plans.

The most widely used acceptance sampling plan in the world.

Quality Index is AQL.

In Malaysia Malaysian Standard, MS 567: Part 1 & 2 – 1978.

Inspection Level

Inspection level determines relationship between lot size (N) and sample size (n).

The levels of inspection offered are (Table 1: Sample size code letters (abjad)):

Level II – Normal Level, designated as N.

Level III – Tightened Level (tahap diperketat), designated as T.

Greater discrimination is required. Introduced when

there is deterioration of process capability.

Level I – Reduced Level (tahap dipermudah), designated as R.

Less discrimination is required. Introduced when there

is an improvement of process capability or no rejected

lots for a specified period.

4 additional special levels are given in the same table:S-1S-2S-3S-4

May be used where relatively small sample sizes are necessary and large sampling risks can be tolerated.

Types of Sampling Plans

This Standard offers:Single Sampling PlansDouble Sampling PlansMultiple Sampling Plans

Single Sampling Plans

Only a number of sample is taken per plan.

Double Sampling Plans

An initial sample is taken.

Based on the information, a decision is made to

A) Accept the lot,

B) Reject the lot, or

C) Take a second sample.

If the 2nd sample is taken, the information from 1st and 2nd is combined to reach a decision to accept or reject the lot.

Multiple Sampling PlansThis is an extension of the double sampling

concept.More than 2 samples are drawn to reach a

decision either to reject or accept the lot.

Summary of Advantages & Disadvantages of Various Types of Sampling Plans

SINGLE DOUBLE MULTIPLE

Acceptability to Producer

One chance to pass

Better chance to pass

Indecisive

Number to be Inspected

Greatest10% to 50%

less than Single Sampling Plans

30% less than Double

Sampling Plans

Cost Lowest Fair Greatest

Information Most Less Least

Simplicity Easiest Hard Hardest

Estimation of Quality

Good Fair Fair

How to use this Standard for Single Sampling Plans?

The steps are:

1) Decide on AQL.

2) Decide on the inspection level.

3) Decide on the type sampling plan to be used SINGLE

4) Decide whether normal, tightened, or reduced inspection is to be used.

5) Determine Lot size (N).

6) Use Table 1 to find the sample size code letter.

7) Use the appropriate table to obtain the plan (Table II.A to II.C).

8) Look at OC curve.

Q & A

1) Determine a single sampling plan if AQL = 1.0%, the inspection level is II and the Lot size is 2500.

Table 1- Sample size code letter KTable II.A – Single sampling plans for normal inspection

(Master table) n=125, Ac=3, Re=4.

Q & A

2) Determine a single sampling plan if AQL = 0.4%, the inspection level is I and the Lot size is 230.

3) Determine a single sampling plan if AQL = 0.015%, the inspection level is III and the Lot size is 120.



Switching Procedures

This standard has provision for switching of sampling

plan to a tighter sampling plan when the supplier’s (or

producer’s) quality has deteriorated.

It is also provides for a switch to reduced inspection

when the product quality has improved.

NORMAL to TIGHTENEDTightened inspection is instituted where 2 out of 5

consecutive lots have been rejected on original

inspection.

TIGHTENED to NORMALWhen tightened inspection is in effect, normal

inspection shall be reinstated when 5 consecutive lots

have been accepted on original inspection.

NORMAL to REDUCEDTo qualify for reduced inspection, the following

conditions must be satisfied:

10 consecutive lots have been accepted on

normal and original inspection.

The total number of nonconforming in the 10

samples is equal or less than the limit number.

Production is at steady state.

Reduced inspection is considered desired by

inspecting authority.

REDUCED to NORMALWhen reduced inspection is in effect,normal inspection

is reinstated when:

A lot is rejected.

Sampling terminates without either acceptance or rejection criteria having been met. The lot is considered acceptable but normal inspection is reinstated.

Production becomes irregular or delayed.

Other conditions warrant that normal inspection shall be re-instated.

Summary of Switching Rules

• Preceding 10 lots Accepted with total nonconforming less than limit number,

• Steady production, and• Approval from

responsible authority.

• Lot not accepted, or• Lot accepted but

nonconformities found lie between Ac and Re of the plan, or

• Irregular production, or• Other conditions

warrant.

• 2 out of 5 consecutive lots not accepted

• 5 consecutive lots accepted

• 10 consecutive lots Remain on Tightened

• Discontinue inspection

NORMAL TIGHTENEDREDUCED

START

How to use this Standard for Double Sampling Plans?

Schematic Operation of a Double Sampling.

Inspect a 1st sample of n1 pieces

If the number of nonconforming found in the 1st sample

Exceeds c1, but does not exceed r1

Does not exceed c1 Exceed r1

Inspect a 2nd sample of n2 pieces

If the number of nonconforming found in the 1st and 2nd samples combined

Does not exceed c2 Equal or exceed r2

Accept the Lot Reject the Lot

The steps are:

1) Decide on AQL.


3) Decide on the type sampling plan to be used DOUBLE




7) Use the appropriate table to obtain the plan (Table III.A to III.C).


Q & A

1) Determine a double sampling plan if AQL = 1.0%, the inspection level is II and the Lot size is 2500.

Table 1- Sample size code letter KTable III.A – Double sampling plans for normal inspection (Master table) n1=80, cumulative sample size=80

n2=80, cumulative sample size=160

AQL=1.0%

Ac Re

First

Second

1 4

4 5

Q & A

2) Determine a double sampling plan if AQL = 0.4%, the inspection level is I and the Lot size is 230.

3) Determine a double sampling plan if AQL = 0.015%, the inspection level is III and the Lot size is 120.



How to use this Standard for Multiple Sampling Plans?

Schematic Operation of a Multiple Sampling.

Inspect a 1st sample of n1 pieces

If the number of nonconforming found in the 1st sample

Exceeds c1, but does not equal or exceed r1

Does not exceed c1 Exceed r1

Inspect a 2nd sample of n2 pieces

If the total number of nonconforming found in the 1st and 2nd samples

combined

Does not exceed c2

Equal or exceed r2

Accept the Lot

Do not Accept

the Lot*


Does not exceed c3

Equal or exceed r3

Accept the Lot


Inspect a 3rd sample of n3 pieces

If the total number of nonconforming found in the first three samples

combined

Do not Accept the Lot*Etc.

* Some of these plans continue to the ‘bitter end’, i.e., the taking of samples continues if necessary until the lot is fully inspected, unless the plan has meanwhile ‘made up its mind’.

The steps are:

1) Decide on AQL.


3) Decide on the type sampling plan to be used MULTIPLE




7) Use the appropriate table to obtain the plan (Table IV.A to IV.C).


Q & A

1) Determine a multiple sampling plan if AQL = 1.0%, the inspection level is II and the Lot size is 2500.

Table 1- Sample size code letter KTable IV.A – Multiple sampling plans for normal inspection (Master table)

sample Sample size Cumulative sample size

AQL=1.0%

Ac Re

First

Second

Third

Fourth

32

32

32

32

32

64

128

160

# 3

0 3

1 4

2 5

# acceptance not permitted at this sample size.

Q & A

2) Determine a multiple sampling plan if AQL = 0.4%, the inspection level is I and the Lot size is 230.

3) Determine a multiple sampling plan if AQL = 0.015%, the inspection level is III and the Lot size is 120.



Average outgoing quality limit factors Tables V-A and V-B.

Limiting Quality (LQ)

Tables VI-A and VI-B. (for consumer’s risk of 0.10) Tables VII-A and VII-B (for consumer’s risk of 0.05) A sampling plan for isolated lots can be obtained that will

come close to both the producers’ and consumers’ criteria. It is much easier to use ANSI/ASQ standard Q3 – 1988. Table VIII. (for reduced inspection)

Average Sample Number(ASN) Table IX – Average sample size curves for double and

multiple sampling. Assumption: no curtailment of inspection and are

approximate to Poisson distribution, and the sample size for double and multiple sampling are assumed to be 0.631n and 0.25n.

Arrow indicates the location of the AQL.

Dodge-Romig Tables

An Introduction 4 sets of attributes plans emphasizing either lot-by-lot

quality (LQ) or long-run quality (AOQL)LQ (or LTPD)

Single samplingDouble sampling

AOQLSingle samplingDouble sampling

1st Classification:LQ (Mutu terhad) Sampling

Plans [Lot Tolerance Percent

Defective (LTPD) (Peratus kecacatan toleran lot)

Sampling Plans]

An Introduction LQ (or LTPD) is the level of quality that is unsatisfactory

and therefore should be rejected by the sampling plan. A consumer’s risk of 0.1 is common and LQ is defined as

the lot quality for which the probability of acceptance is 0.1, that is, only 10% of these lots are accepted.

Consumer’s risk of 10% calculated for n=80 and c=0 does not mean that the consumer has 10% chance of receiving poor product.

The meaning of the consumer’s risk is that such product has

10% of being accepted if actually submitted to inspection.

These tables are based on LQ and AOQL concepts.

For each of these concepts there are tables for single and double sampling.

No provision is made for multiple sampling.

What is the main advantages of the Dodge-Romig tables?

Minimum amount of inspection for a given inspection procedure

desirable for in-house inspection.

Let us consider the following OC curves.A n=165, c=1B n=100, c=0

OC Curve

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8

Percent Nonconforming (100p0)

Pe

rce

nt

of

Lo

ts A

cc

ep

ted

(1

00

Pa)

A

B

Both of these OC curves have a low probability of acceptance for product 2% nonconforming.

These plans are fairly similar at the tail (the product would be rejected) but are not necessary similar at the shoulder (the product would be accepted).

Plans classified on this basis are called LQ (or LTPD) plans because their point of similarity is the quality level or percent of nonconforming which can just be tolerated in a small percentage of the product.

LQ is defined as “ an allowable percentage nonconforming” a number which may be considered as the borderline of distinction between a satisfactory lot and in unsatisfactory one.

To engineers, this means the percent nonconforming which will regularly be rejected by inspection, that is, the percentage nonconforming for which the probability of acceptance is very low.

When an engineer chooses a “2% LQ Sampling Plan”, he/she is choosing a plan which would regularly reject 2% nonconforming product. The fact that the plan is classified as “2% LQ” does not tell the characteristic of the remainder of the OC curve that is what quality of product will regularly be accepted.

Customers sometime specify a certain value of LQ for a particular product. In such case, the manufacturer of the product will try to select a sampling plan which has a low probability of acceptance at the specified LQ.

Sampling Plans on this basis have been published by Dodge and Romig (1920s). These tables provide a useful classification of sampling plans wherever we wish to make sure that product of a particular quality will have a low probability of acceptance.

How to use the table? Determine the usual lot size. Determine the process average (percent nonconforming at

which the products runs). Determine sample size, n Under the applicable process average, determine the

number c.

Q & AN=1500, process average=0.25%, required single sampling plan for LQ=1.0%

n=490, c=2, AQOL=0.21%

Analysis of the table. As the lot size increases, the relative sample size

decreases Inspection costs are more economical with large lot size.

The tables extend until the process average is one-half of the LQ. Additional process average is unnecessary, since 100% inspection becomes more economical than sampling inspection.

As the process average increases, a corresponding increase occurs in the amount inspected. Therefore, an improvement in the process average results in fewer inspections and a lower sampling inspection cost.

2nd Classification:AOQL (Mutu purata terhad

pengeluaran) Sampling Plans

AOQL is a value which can be calculated for any sampling plan.

It is a ‘limiting value of percent nonconforming’ which becomes associated with the sampling plan as soon as we make provision for doing 100% inspection on all lots rejected by the plan.

In any case, where 100% inspection cannot or will not be done on all rejected lots, do not attempt to select a sampling plan on the basis of its AOQL.

AOQL concept is applicable to inspection of lots in a convenient sub-division of a flow of product for material-handling purposes (non-homogeneous).

[when the lot quantity is specified (as is the case with customer lots (homogeneous)) LQ concept].

AOQL plans limit the amount of poor outgoing quality on an average basis but give no assurance on individual lots.

The meaning of the term AOQL can be illustrated as follows:

OC Curve

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8


Pe

rce

nt

of

Lo

ts A

cc

ep

ted

(1

00

Pa)

• Consider a sampling plan for n=18, c=0.

• When product is 2% nonconforming, this plan will accept approximately 70% of it.

• When product is 8% nonconforming, this plan will accept approx. 25%…etc.

Suppose we make the following rules: All rejected lots by this sampling plan must be 100%

inspected. All nonconforming unit found by this inspection must

be replaced with good units, and The rejected lots which have had all nonconforming

removed must then be considered together with the accepted lots in such a way as to make one total quality of product.

It is possible to calculate the percentage of nonconforming which will be left in the mass of product if this procedure is followed.

AOQ Curve

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15Percent Nonconforming

Ave

rag

e O

utg

oin

g Q

ual

ity-

%

AOQL~2%

This curve shows that when the incoming product is 10% nonconforming, the outgoing product will be only 1.6% nonconforming; provided the requirements of the sampling plan, including 100% inspection for rejected products, have been faithfully carried out.

Note that the AOQ curve rises until it reaches a certain maximum point, after which it falls off again as a result of more and more product being “100% inspected”.

The AOQL is the maximum point which is reached by the AOQ curve.

The AOQL of a sampling plan is therefore defined as follows: This limit is the worst average quality that can exist, in the long run, in the outgoing product, after the rejected lots have been 100% inspected and all the nonconformities have been replaced by good units.

How to calculate AOQL? Three ways:

1) From a series of AOQ values, a graph is plotted and AOQL is determined from the maximum point of the graph.

2) From Table V-A and V-B (MS 567)

3) From an equation.

N

y

n

yAOQL

where y is a factor depending on the acceptance number, c, of the sampling plan.

Table for values of “y” to be used in AOQL calculation

c

y

0

0.368

1

0.841

2

1.372

c

y

3

1.946

4

2.544

5

3.172

c

y

6

3.810

7

4.465

8

5.150

c

y

9

5.836

10

6.535

11

7.234

This table can be used wherever Poisson distribution is employed.

Q & A

N=2000, n=18, c=0. Calculate AOQL.

y=0.368

AOQL=(0.368/18)-(0.368/2000) = 0.204-0.0002 = 0.0202 or 2.02%

This agrees with the AOQ curve in the earlier figure.

Plan n c AOQL

A 165 1 0.47%

B 35 2 3.86%

C 18 0 2.02%

D 5 0 7.34%

E 220 7 1.81%

F 100 4 2.41%

G 100 0 0.35%

H 25 1 3.32%

• Suppose we wish to group together plans having similar AOQL’s.

• From the above list, we would group together Plans C and E, since both have AOQL’s of approx. 2%.

• These plans are not alike at

the shoulder. Neither are

they alike at the tail.

• The only similarity is that,

when they are used with

100% inspection of all the

rejected lots, the outgoing

product will not, on the

average, be worse than 2%.

OC Curve

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8


Pe

rce

nt

of

Lo

ts A

cc

ep

ted

(1

00

Pa)

Plan C

Plan E

When an engineer chooses a “2% AOQL sampling plan”, he/she is choosing a plan which will limit the outgoing product, in the long run, to a 2% average or less.

The outgoing quality may be, and frequently is, limited to some point much lower than the stated AOQL.

For example, the above curve is shown that if the submitted product were 20% nonconforming, this particular plan would force the shop to do sufficient screening to cut it down (theoretically) to 0.5%. The plan imposes, under these circumstances, a much tighter standard of quality than the stated 2%.

The fact that a sampling plan is classified as “2% AOQL” does not tell the engineer anything about any part of the OC curve.

He/she is not able to tell what quality of product will be regularly accepted, or rejected. (same as LQ and AQL)

When an AOQL is specified for a particular product, we try to select a sampling plan whose AOQL is equal to the specified value.

To make it easier to select plans on this basis, Dodge and Romig have developed tables of sampling plans classified according to their AOQLs. The values of AOQL range from 0.1% to 10% nonconforming.

The plans in these tables are arranged in such a way that, if the engineer selects a plan under the correct “process average”, he/she will minimize the total number of pieces which must be looked at, including both sampling and 100% inspection.

LQ values also included as supplementary information.

These tables are useful wherever we are interested in setting a fixed maximum limit on outgoing quality and are willing to achieve this by a combination of sampling and 100% inspection.

Table 10-9

Precautions to be taken in using AOQL Sampling Plan

Since this plans involve a combination of sampling and 100% inspection, they are often subjected to misinterpretation and misuse.

Engineers should guard against the following errors: (1) We sometime hear of “AOQL sampling plans”

being used in connection with destructive test or in Receiving Inspection where the rejected lots are junked, returned to the supplier or accepted on an outside limit basis, but where there is no intention whatever of doing 100% inspection on the lots which fail to meet the acceptance number of the plans. [IMPORTANT: Unless 100% inspection is done, we will not get the protection promised by an AOQL sampling plan]

(2) If the product contains nonconforming to begin with, an AOQL sampling plan will depend on the presence of the rejected lots, which have been made prefect by screening, to dilute the percentage of the nonconformities which may still be present in the lots accepted.

What the Operating organization should know about AOQL Sampling Plans?

Operating people often assume that a “2% AOQL sampling Plan” will accept product which is 2% nonconforming.

That is, if inspection is using a 2% AOQL plan, and if Operating submits product which is actually 2% nonconforming, they feel that all or most of the submitted product should pass the inspection plan.

Engineers also frequently make this assumption in discussing suitable quality levels for sampling or in agreeing to the use of some specific proposed plan.

However, unless the AOQL sampling plan is deliberately chosen with this in mind, a 2% plan may reject large portions of 2% nonconforming product.

The following example will show why it is necessary to restrict the choice of plan.

Suppose a sampling plan were to be chose solely for its AOQL. The following plans will have an AOQL of 2% when used for lots of approx. 1000.

(a) n=18, c=0(b) n=40,c=1(c) n=65,c=2(d) n=90,c=3

While all of these plans have a 2% AOQL, they will reject very different amounts of 2% nonconforming product.

In all cases, rejection is high. If the Operating department wishes to avoid these

rejections, it will have to maintain a quality level considerably better than 2%.

2% AOQL Sampling Plan

Approximate Percentage of Product Rejected

(if product is running at 2% nonconforming)

(a) n=18, c=0 30%

(b) n=40, c=1 19%

(c) n=65, c=2 14%

(d) n=90, c=3 11%

How good is the product to avoid rejection:

(1) Plot the OC curve of the sampling plan in question. (2) Find the point where the curve drops away from the

1.00 probability of acceptance (Pa).

(3) The corresponding percent nonconforming is the point at which Operating must aim if it wishes to have its product accepted regularly by Inspection.

The following are the points at which Operating should aim.

2% AOQL Sampling Plan

Necessary Level of Quality to assure regular acceptance

(“regular acceptance” means acceptance about 98% of the

time)

(a) n=18, c=0 0.1% nonconforming

(b) n=40, c=1 0.5% nonconforming

(c) n=65, c=2 0.9% nonconforming

(d) n=90, c=3 1.1% nonconforming

Summary Characteristics of a Good Acceptance Plan

1) The index (AQL, LQ, AOQL) used to define “quality” should reflect the needs of the consumer and producer and not chosen primarily for statistical convenience.

2) The sampling risks should be known in quantitative term (OC curve). The producer should have adequate protection against the rejection of good lots; the consumer should be protected against the acceptance of bad lots.

3) The plan should minimize the total cost of inspection or all products. This requires careful evaluation of the pros and cons of attributes and variables plans, and single, double, and multiple samplings.

4) The plan should have built-in flexibility to reflect changes in lot sizes, quality of product submitted, and any other pertinent factors.

5) The measurements required by the plan should provide information useful in estimating individual lot quality and long-run quality.

ANSI/ASQ Standard Q3 - 1988

ANSI/ASQ Standard Q3 - 1988 It is used for inspection of isolated lots by attributes.

It complements ANSI/ASQ Z1.4-1993 (for continuous stream of lots).

Indexes tables by Limiting Quality (LQ).Limiting Quality (LQ).

1st scheme:

It is designated to be used for lots that are isolated or mixed or that have an unknown history as far as both vendor and vendee know.

Lot size and LQ value must be known.

Table 10-6

Q & A:Q: Lot size = 295, LQ=3.15%, determine the sampling

plan.A: n=80, Ac=0

2nd scheme:

When a vendor is producing a continuous stream of lots

and sends one or a few to a customer who will consider

them as isolated lots (purchasing of small qualities of purchasing of small qualities of

a raw materiala raw material).

Example of Table for LQ=3.15%

Table 10-7

The last 5 columns can be used to plot the OC curves.

Q & A:Q: Lot size = 295, inspection level II, LQ=3.15%, where

the isolated lot is from a vendor with a continuous stream of product, determine the sampling plan.

A: n=125, Ac=1

12 Sampling Tech 4

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