11 Sampling Tech 3new

Lot-by-lot Acceptance Sampling

Techniques by

Attributes

Topic Outcome:At the end of this topic, student will be able to:

Describe the consumer-producer relationship

from a OC curve.

Determine and explain producer’s risk,

consumer’s risk, AQL, and LQ.

Determine and explain AOQ, ASN, and ATI.

Design and apply a sampling plan.

Topic Outline:Introduction

Consumer-Producer Relationship

Producer’s Risk () & AQL (Tahap Mutu

Kebolehterimaan)

Consumer’s Risk () & LQ (Mutu Terhad)

AOQ (Mutu Purata Pengeluaran)

Design of Sampling Plan.

(1) Introduction

Introduction An OC curve can be plotted for any combination of sample

size (n) and acceptance number (c).

Each combination results in a different curve.

Some of the most important things to remember about OC curves can be seen by comparing the curves.

DIFFERENT n: A larger sample size tends to result in a steeper curve such a plan is said to have greater “discriminating power” than plan with smaller sample size and shallower curve.

DIFFERENT c: a larger c tends to change the shape of the curve, creating a flat “shoulder” at the top while retaining a thin “tail” at the bottom.

OC curve has 3 parts

05.0 10.0

t of L

Percent Nonconforming (100p0)

Shoulder (or peak) at the topIt shows the quality of the product that

will be accepted by sampling plan without question.

Thin part of the tail at the bottom

It shows the quality of the product that is almost certain to be rejected by

the plan.

Middle portion between shoulder and tail

At exact centre of the curve, where the Pa is 50%, product of corresponding quality has 50-50 chance of being either rejected

or accepted.

In general, engineer should make sure that the “shoulder” of the curve corresponds to the product he/she is willing to be accepted and that the “tail” of the curve corresponds to the product he/she is willing to be rejected.

Maximum economy is likely to be obtained when the process is running at or near its capability level and when this level matches the shoulder of the OC curve to be used.

(2) Consumer–Producer Relationship

Producer’s RiskAQL

Producer’s Risk (Producer’s Risk ()) It is defined as the probability or risk of rejecting a lot

when the quality of the lot is acceptable.

What does this mean to an engineer?

The risk for a good product to be rejected a lost to an

organization.

Example:

A process capability is 99%. Customer is willing to accept 1%

nonconforming units. However, during inspection a certain

percentage of lots still rejected ever though the percent of

nonconforming is less than 1%.

How to estimate Producer’s Risk ()How to estimate Producer’s Risk ()

1) Plot OC curve for the sampling plan.

2) Find the percentage nonconforming (100p0) in the

process.

• Rough estimation can be done if no exact data is available.

• However for a more accurate check, a process process capability studycapability study is preferable.

3) Use the process capability percentage to determine the probability of acceptance (Pa) on the OC curve.

4) Producer’s Risk () = 1 - Pa

Q & A:

Process Capability = 99%

1.Sampling Plan, N=3000, n=80, c=0

Find the Producer’s Risk for the above sampling plans according to the stated process capability.

OC Curve

0 1 2 3 4 5 6

Process Capability = 99% 100p0 = 1%

c 100Pa 100

0 44.9% 55.1%

1 80.9% 19.1%

2 95.2% 4.8%

3 99.1% 0.9%

4 99.8% 0.2%

Percentage of Lots NOT Accepted (Pr)

It is associated with producer’s risk.

It is a numerical definition of an acceptable lot.

It is the maximum percent nonconforming that can be considered satisfactory for the purposes of acceptance sampling.

It is a reference point on the OC curve and is not meant to convey to the producer that any percent nonconforming is acceptable.

It is a statistical term and is not meant to be used by general public.

Producer only can guarantee an acceptable lot when 0% nonconforming or the number nonconforming in the lot less than or equal to acceptance number, c.

Acceptable Quality Level (AQL)

Producer’s quality goal is to meet or exceed the specifications so that no nonconforming units are present in the lot.

A sampling plan should have a low Producer’s risk for quality that is equal to or better than the AQL.

Example:

Sampling plan: N=4000, n=300, and c=4, AQL=0.7%. What is the non-acceptance probability?

OC Curve

0 0.5 1 1.5 2 2.5 3

(AQL, 1-100Pa)

Comment:

• Producer’s risk = 7%

• Product that is 0.7% nonconforming will have a non-acceptance probability of 0.07 or 7%.

• 7 out of 100 lots that are 0.7% nonconforming will not be accepted by the sampling plan.

Consumer’s RiskLQ

Consumer’s Risk ()

It is defined as the probability or risk of accepting a “bad” lot based on a sampling plan.

It is expressed in terms of probability of acceptance.

To estimate consumer’s risk:

Plot OC curve for a sampling plan.

Find out the percent nonconforming that the consumer wants to reject.

Find this value on the OC curve and determine the probability of acceptance. This gives the risk of accepting unsatisfactory quality provided the product of such poor quality is actually submitted.

Q & A:

Consumer wants to reject product that is 4% nonconforming.

1. Sampling Plan, N=3000, n=80, c=0

Find the Consumer’s Risk () for the above sampling plans.

OC Curve

0 1 2 3 4 5 6

c 100Pa=100

0 4.1%

1 17.1%

2 38.0%

3 60.2%

4 78.1%

It is associated with consumer’s risk.

It is a numerical definition of a nonconforming lot.

It is the percent nonconforming in a lot for which (for acceptance purposes) the consumer wishes the probability of acceptance to be low.

For previous example, LQ = 4% for 100 = 5% (for c=0) lots that are 4% nonconforming will have a 5% chance of being accepted. 1 out of 20 lots that are 4% nonconforming will be accepted by this sampling plan.

Limiting Quality (LQ)

(3) Average Outgoing Quality (AOQ)

It is the average quality of outgoing product.

All accepted lots + All rejected lots after the rejected lots have been effectively 100% inspected (screened) and all nonconforming units replaced by conforming one.

Rectifying Inspection.

AOQ = (100p0)(Pa)

It is the quality that leaves the inspection operation.

When rectification/sorting does not occur, the AOQ is same as the incoming quality (or process quality).

Average Outgoing Quality (AOQ)

Rectifying Inspection Program

Fraction non-conforming

AOQ Curve

0 1 2 3 4 5 6Percent Nonconforming

Average Outgoing Quality Limit (AOQL)

N=3000, n=89, c=2

Incoming quality = 2% nonconforming AOQ

is ~1.45%

AOQ Curve

0 1 2 3 4 5 6Percent Nonconforming

Curve without rectification

Why AOQ is better than Incoming quality?

Suppose that over a period of time, 15 lots of 3000 each are shipped by producer to the consumer. The lots are 2% nonconforming and a sampling plan of n=89 and c=2 is used to determine acceptance.

OC Curve

0 1 2 3 4 5 6

• 100Pa for 2% nonconforming

lot (100p0) is 73.6%.

• How many lots are accepted

by the consumer?

• 0.736 x 15 = 11 lots

• 4 lots are NOT accepted by the

sampling plan and returned to

the producer for rectification.

• These 4 lots receive 100%

inspection and are returned to

the consumer with 0%

nonconforming.

ConsumerProducerN = 3000

n = 89c = 2

15 lots2% Nonconforming

Total Number Number Nonconforming11 lots (2% NC) 11(3000) =33,000 33,000(0.02) = 6604 lots (0% NC) 4(3000)(0.98)=11,760 0

------------ ------- 44,760 660

Percent Nonconforming (AOQ) = (660/44760)100 = 1.47%[consumer actually receives 1.47% nonconforming, whereas the producer’s quality is 2%]

OC Curve

0 1 2 3 4 5 6

• If the producer’s risk () is equal to 0.05, the AQL=1%

• Producer at 2% nonconforming is not achieving the desired quality level.

(4) Average Sample Number(ASN)

Average Sample Number(ASN) It is a comparison of the average amount inspected per lot

by the consumer for a certain sampling plan. The ASN depends on the type of sampling plan: Single Sampling Plan

ASN is constant Double Sampling Plan

ASN = n1 + n2(1+P1)Where n1 is the first sample size number

n2 is the second sample size number

P1 is the probability of a decision on first sample

Multiple Sampling PlanASN = n1P1 + (n1+n2)P2 + ….+ (n1+n2+…+nk)Pk

Where nk is the sample size of the last level, Pk is the probability of a decision at the last level.

ASN curves

ASN curves are a valuable tool for justifying double or multiple sampling.

When inspection costs are great due to inspection time, equipment cost, or equipment availability.

Q & AGiven the single sampling plan n=80 and c=2 and the equal effective double sampling plan n1=50, c1=0, r1=3, n2=50, c2=3, and r2=4, compare the ASN of the two by constructing their curves.

For single sampling, the ASN is a straight line at n=80.For double sampling, the solution is

PI =P0 + P3 or more

Assume that p0=0.01; then np0=50(0.01)=0.5. From Appendix C:P0=0.607P3 or more = 1-P2 or less = 1-0.986 = 0.014ASN = n1+n2[1-(P0+P3 or more)]ASN = 50 + 50[1-(0.607+0.014)]=69Repeating for different values of p0, the double sampling plan is plotted as shown in Fig. above.

The formula assumes that inspection continues even after the rejection number is reached.

It is frequently the practice to discontinue inspection after the rejection number is reached on either the first or second sample.

This practice is called curtailed inspection, and the formula is much more complicated. Thus, the ASN curve for double sampling is somewhat lower than what actually occurs.

Analysis of the ASN curve for double sampling. At p0 of 0.03 single and double sampling plans have

about the same amount of inspection. For p0<0.03 double sampling plan has less inspection

because a decision to accept on the first sample is more likely.

For p0>0.03 double sampling plan has less inspection because a decision not to accept on the first sample is more likely and a second sample is not required.

Note: in most ASN curves, the double sample curve does not get close to the single sample one.

Typical ASN curves

(5) Average Total Inspection (ATI)

Average Total Inspection (ATI) It is the amount inspected by both the consumer and the

producer. Provide information on the amount inspected and NOT the

effectiveness of the plan. Single Sampling Plan

ATI = n + (1 – Pa)(N – n)

Assumption:Rectified lots receive 100% inspection.

Lots with 0% nonconforming ATI = n Lots with 100% nonconforming ATI = N

Example Determine the ATI curve for the single sampling plan N =

3000, n = 89, and c = 2.

Assume that p0 = 0.02. From the OC curve, Pa = 0.731. ATI = n + (1 – Pa)(N – n) = 89 + (1 – 0.731)(3000 – 89)

ATI = 872 Repeat for other p0 values until a smooth curve is obtained.

(6) Design of Sampling Plan

Design of Sampling Plan

“Rule of Thumb”

A sampling plan should not be adopted without seeing

the OC curve!!!!!

Sampling Plans for Stipulated Producer’s Risk

When and its corresponding AQL are specified, a family of sampling plans can be determined.

05.0 10.0

t of L

Producer’s Risk

How to construct a family of sampling plan? [=0.05, AQL (or 100p0)=1.2%]

1) Arbitrarily select c values.

2) Find np0 values correspond to the c values (from

Table below) -- interpolation

3) Calculate n, from n=np0/p0

4) Construct OC curves with n and c values.

Construct a family of sampling plan [=0.05, AQL (or 100p0)=1.2%]

•Arbitrarily select c values.

• c=1, 2, and 6

• =0.05 Pa=0.95 np0 values correspond to the c values (Table 9-4). If np0 value is not available in

the Table, interpolation is needed to find the values using the pre-calculated values from Table above.

• From Table 9-4 obtain np0 values

• Calculate n, from n=np0/p0

• Pa = 0.95; p0.95 = 0.012

c np0 (from Table 9-4) n = np0/p0

1 0.355 29.6 (~30)

2 0.818 68.2 (~68)

6 3.286 273.9 (~274)

• Construct OC curves with n and c values.

OC Curve

0 1 2 3 4 5 6 7 8

Pa) c=1

c=2n=68

c=6n=274

AQL=1.2%

100=10

• All the plans provide the same protection for the producer, but the consumer’s risk () = 0.10 is quite different.

• For plan c=6, n=274 product that is 3.8% nonconforming will be accepted 10% (= 0.10) of the time.

For consumer’s viewpoint, the plan with c=6, n=274 provides better protection,but the size is large.

Sampling Plans for Stipulated Consumer’s Risk

When and its corresponding Limiting Quality (LQ) are specified, a family of sampling plans can be determined.

05.0 10.0

t of L

Consumer’s Risk

Construct a family of sampling plans [=0.10, LQ (or 100p0)=6.0%]

• Arbitrarily select c values.

• c=1, 3, and 7

• Find np0 values correspond to the c values (from above Table).

=0.10 100Pa=10 np0 values correspond to the c values

in the Table.

• Calculate n, from n=np0/p0

• Pa = 0.10; p0.95 = 0.060

c np0 (from above Table) n

1 3.890 64.8 (~65)

3 6.681 111.4 (~111)

7 11.771 196.2 (~196)

• Construct OC curves with n and c values.

OC Curve

0 1 2 3 4 5 6 7 8

Pa) c=1

c=3n=111

c=7n=196

LQ=6.0%

100=10

• All the plans provide the

same protection for the

consumer, but the

producer’s risk () = 0.05 is

quite different.

• For plan c=1, n=65

product that is 0.5%

nonconforming will not be

accepted 5% (100= 5%) of

the time.

For producer’s viewpoint, the plan with c=1, n=65 provides better protection,but the size is large.

Sampling Plans for Stipulated Producer’s & Consumer’s Risk

It is difficult to obtain an OC curve that will satisfy both

conditions.

There will be 4 sampling plans that are close to meeting

both stipulations.

How to determine the plans?

Given: (decisions based on historical data, experimentation, or engineering judgment, purchasing contract)=0.05, AQL=0.9=0.10, LQ=7.8

667.8009.0

10.0 p

• From above Table, the ratio falls between the row for c=1 and c=2.

• Thus, plans that exactly meet the consumer’s stipulation of LQ = 7.8% for =0.10 are:

cnp0.10 (from Table 9-4)

n=(np0.10)/p0.10

1 3.890 49.9 (50)

2 5.322 68.2 (68)

Plans that exactly meet the producer’s stipulation of AQL =0.9% for =0.05 are:

cnp0.95 (from Table 9-4)

n=(np0.95)/p0.95

1 0.355 39.4 (39)

2 0.818 90.8 (91)

Construct OC curves.

OC Curve

0 1 2 3 4 5 6 7 8

100 = 5

AQL=0.9

100 = 10

LQ=7.8

c=1n=39

c=2n=91

c=1n=50

c=2n=68

Which of the 4 plans to select?Based on 4 criteria.

1st Criterion:The lowest sample size and the lowest acceptance numberbe selected.e.g. c=1, n=39

2nd Criterion:The greatest sample size and the largest acceptance numberbe selected.e.g. c=2, n=91

3rd Criterion:The plan exactly meets the consumer’s stipulation and comes as close as possible to the producer’s stipulation. e.g. c=1, n=50 and c=2, n=68.

Calculations to determine which plan is closest to the producer’s stipulation of AQL=0.9%, =0.05 are:

c n p0.95=np0.95/n

1 50 0.355/50=0.007

2 68 0.818/68=0.012

0.009This plan is selected

4th Criterion:The plan exactly meets the producer’s stipulation and comes as close as possible to the consumer’s stipulation. e.g. c=1, n=39 and c=2, n=91.

Calculations to determine which plan is closest to the producer’s stipulation of LQ=7.8%, =0.10 are:

c n p0.10=np0.10/n

1 39 0.3890/39=0.100

2 91 5.322/91=0.058 0.078This plan is selected

The task of designing a sampling plan system is a

tedious one.

Sampling plan systems are available.

E.g.: ANSI/ASQ Z1.4 – 1993 (equivalent to MS 567 –

universally used for acceptance

product.

This is an AQL or producer’s risk

system.

E.g.: Dodge-Romig

LQ or consumer’s risk system.

11 Sampling Tech 3new

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